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+Task-Guided IRL in POMDPs that Scales
+Franck Djeumou∗, Christian Ellis∗∗, Murat Cubuktepe∗, Craig Lennon, Ufuk Topcu∗
+Abstract
+In inverse reinforcement learning (IRL), a learning agent infers a reward function en-
+coding the underlying task using demonstrations from experts. However, many ex-
+isting IRL techniques make the often unrealistic assumption that the agent has access
+to full information about the environment. We remove this assumption by developing
+an algorithm for IRL in partially observable Markov decision processes (POMDPs).
+We address two limitations of existing IRL techniques. First, they require an exces-
+sive amount of data due to the information asymmetry between the expert and the
+learner. Second, most of these IRL techniques require solving the computationally in-
+tractable forward problem—computing an optimal policy given a reward function—in
+POMDPs. The developed algorithm reduces the information asymmetry while increas-
+ing the data efficiency by incorporating task specifications expressed in temporal logic
+into IRL. Such specifications may be interpreted as side information available to the
+learner a priori in addition to the demonstrations. Further, the algorithm avoids a com-
+mon source of algorithmic complexity by building on causal entropy as the measure of
+the likelihood of the demonstrations as opposed to entropy. Nevertheless, the resulting
+problem is nonconvex due to the so-called forward problem. We solve the intrinsic
+nonconvexity of the forward problem in a scalable manner through a sequential linear
+programming scheme that guarantees to converge to a locally optimal policy. In a series
+of examples, including experiments in a high-fidelity Unity simulator, we demonstrate
+that even with a limited amount of data and POMDPs with tens of thousands of states,
+our algorithm learns reward functions and policies that satisfy the task while inducing
+similar behavior to the expert by leveraging the provided side information.
+1. Introduction
+A robot can satisfy certain human-specified tasks by describing desired behavior
+through a reward function. However, the design of such a reward function is a non-
+trivial task. Inverse reinforcement learning (IRL) is an established technique that in-
+fers a reward function encoding the underlying task using expert demonstrations. IRL
+∗The University of Texas at Austin
+∗∗The University of Massachusetts: Dartmouth
+United States Army Research Laboratory
+Email addresses: fdjeumou@utexas.edu (Franck Djeumou), cellis3@umassd.edu
+(Christian Ellis), mcubuktepe@utexas.edu (Murat Cubuktepe),
+craig.t.lennon.civ@army.mil (Craig Lennon), utopcu@utexas.edu (Ufuk Topcu)
+Preprint submitted to Elsevier
+January 4, 2023
+arXiv:2301.01219v1  [cs.LG]  30 Dec 2022
+
+techniques have found a wide range of applications in various domains such as ac-
+robatic helicopter flight [1], inferring future actions of people [2], human-autonomy
+interaction [3, 4], robotic surgery [5, 6], and robotic manipulation tasks [7]. Most
+existing work [1, 8, 9, 10, 3, 7] has focused on Markov decision processes (MDPs),
+assuming that the learner can fully observe the state of the environment and expert
+demonstrations. However, the learner will not have access to full state observations in
+many applications. For example, a robot will never know everything about its envi-
+ronment [11, 12, 13] and may not observe the internal states of a human with whom it
+works [14, 15]. Such information limitations violate the intrinsic assumptions made in
+most existing IRL techniques.
+We investigate IRL in partially observable Markov decision processes (POMDPs),
+a widely used model for decision-making under imperfect information. Partial observ-
+ability brings two key challenges in IRL. The first challenge is due to the so-called
+information asymmetry between the expert and the learner. The expert typically has
+either full or partial information about the environment, while the learner has only a
+partial view of the state and the expert’s demonstrations. Even in the hypothetical
+case in which the underlying reward function is known to the learner, its optimal pol-
+icy under limited information may not yield the same behavior as an expert with full
+information due to such information asymmetry.
+The second challenge is due to the computational complexity of policy synthesis in
+POMDPs. Indeed, many standard IRL techniques rely on a subroutine that solves the
+so-called forward problem, i.e., computing an optimal policy for a given reward. Solv-
+ing the forward problem for POMDPs is significantly more challenging than MDPs,
+both theoretically and practically [16]. Optimal policies for POMDPs may require infi-
+nite memory of observations [17], whereas memoryless policies are enough for MDPs.
+An additional limitation in existing IRL techniques is due to the limited expressiv-
+ity and often impracticability of state-based reward functions in representing complex
+tasks [18]. For example, it will be tremendously difficult to define a merely state-based
+reward function to describe requirements such as “do not steer off the road while reach-
+ing the target location and coming back to home” or “monitor multiple locations with
+a certain order”. However, such requirements can be concisely and precisely speci-
+fied in temporal logic [19, 20]. Therefore, recent work has demonstrated the utility of
+incorporating temporal logic specifications into IRL in MDPs [21, 22].
+In this work, we address these challenges and limitations in state-of-the-art IRL
+techniques by investigating the following problem.
+Task-Guided IRL in POMDPs: Given a POMDP, a set of expert demonstrations,
+and, if available, a task specification expressed in temporal logic, learn a policy
+along with the underlying reward function that maximizes the causal entropy of
+the induced stochastic process, induces a behavior similar to the expert’s, and
+ensures the satisfaction of the specification.
+We highlight two parts of the problem statement. Using causal entropy as an opti-
+mization criterion instead of traditional entropy results in a least-committal policy that
+induces a behavior obtaining the same accumulated reward as the expert’s demonstra-
+tions while making no additional assumptions about the demonstrations. Task specifi-
+2
+
+cations given as task requirements guide the learning process by describing the feasible
+behaviors and allow the learner to learn performant policies with respect to the task re-
+quirements. Such specifications can be interpreted as side information available to
+the learner a priori in addition to the demonstrations aimed at partially alleviating the
+information asymmetry between the expert and the learner.
+Specifically, we tackle the IRL on POMDPs problem by a reformulation into a
+maximum causal entropy (MCE) problem. Then, we develop a new solver for the
+MCE problem that improves computational tractability over existing approaches. The
+developed solver can enforce prior task knowledge expressed as temporal logic specifi-
+cations, which guides the learning, improves the data efficiency, and partially alleviates
+the information asymmetry problem.
+Most existing work on IRL relies on entropy as a measure of the likelihood of the
+demonstrations, yet, when applied to stochastic MDPs, has to deal with nonconvex
+optimization problems [8, 10]. On the other hand, IRL techniques that adopt causal
+entropy as the measure of likelihood enjoy formulations based on convex optimiza-
+tion [9, 10, 23]. We show similar algorithmic benefits in maximum-causal-entropy
+IRL carry over from MDPs to POMDPs.
+A major difference between MDPs and POMDPs in maximum-causal-entropy IRL
+is, though, due to the intrinsic nonconvexity of policy synthesis in POMDPs, which
+yields a formulation of the task-guided IRL problem as a nonconvex optimization
+problem. It is known that this nonconvexity severely limits the scalability for syn-
+thesis in POMDPs [16]. We develop an iterative algorithm that solves the resulting
+nonconvex problem in a scalable manner by adapting sequential convex programming
+(SCP) [24, 25].
+In each iteration, it linearizes the underlying nonconvex problem
+around the solution from the previous iteration. The algorithm introduces several ex-
+tensions to alleviate the errors resulting from the linearization. One of these extensions
+is a verification step not present in existing SCP schemes. We show that the proposed
+algorithm computes a sound and locally optimal solution to the task-guided problem.
+Additionally, we empirically demonstrate that the algorithm scales to POMDPs
+with tens of thousands of states as opposed to tens of states in most existing work.
+In POMDPs, finite-memory policies that are functions of the history of the observa-
+tions outperform memoryless policies [26]. Besides, computing a finite-memory pol-
+icy for a POMDP is equivalent to computing a memoryless policy on a larger product
+POMDP [27]. Thus, we leverage the scalability of our algorithm to compute more per-
+formant policies that incorporate memory using finite-state controllers [28, 29]. On the
+other hand, the existing IRL techniques on POMDPs aforementioned cannot effectively
+utilize memory, as they do not scale to large POMDPs.
+We demonstrate the applicability of the approach through several examples, in-
+cluding a simulated wheeled ground robot operating in a high-fidelity, continuous, 3-
+D Unity simulation. We show that, without task specifications, the developed algo-
+rithm can compute more performant policies than a straight adaptation of the original
+GAIL [30] to POMDPs. Then, we demonstrate that by incorporating task specifications
+into the IRL procedure, the learned reward function and policy accurately describe
+the behavior of the expert while outperforming the policy obtained without the task
+specifications. We observe that with more limited data, the performance gap becomes
+more prominent between the learned policies with and without using task specifica-
+3
+
+tions. Most importantly, we empirically demonstrate the scalability of our approach
+for solving the forward problem through extensive comparisons with several state-of-
+the-art POMDP solvers and show that on larger POMDPs, the algorithm can compute
+more performant policies in significantly less time.
+2. Preliminaries
+The following section provides a review of prerequisite understanding for POMDPs,
+their accompanying policies and how a POMDP’s belief over states is updated using
+Bayesian techniques.
+Notation. We denote the set of nonnegative real numbers by R+, the set of all proba-
+bility distributions over a finite or countably infinite set X by Distr(X), the set of all
+(infinite or empty) sequences x0, x1, . . . , x∞ with xi ∈ X by (X)∗ for some set X,
+and the expectation of a function g of jointly distributed random variables X and Y by
+EX,Y [g(X, Y )].
+2.1. Partially Observable Markov Decision Process
+A partially observable Markov decision process (POMDP) is a framework for mod-
+eling sequential interaction between an agent and a partially observable environment,
+where the agent cannot perceive its underlying state but must infer it based on the given
+noisy observation.
+POMDPs. We define a POMDP by a tuple M = (S, A, P, Z, O, R, µ0, γ), where S,
+A, and Z are finite sets of states, actions, and observations, respectively. The function
+µ0 : S �→ R+ provides the initial distribution of the agent’s state and γ ∈ [0, 1) is
+a discount factor over a possibly infinite planning horizon. At each decision time, an
+agent selects an action α ∈ A and the transition function P : S × A �→ Distr(S)
+defines the probability P(s′|s, α) of reaching state s′ ∈ S given the current state s ∈ S
+and action α. After the state transition, the agent receives an observation z′ ∈ Z
+according to the function O : S �→ Distr(Z), which defines the probability O(z′|s′)
+of perceiving z′ at state s′. The agent also receives a reward function R(s, α) from the
+function R : S × A �→ R encoding the task specification. In the following, without
+loss of generality, we consider infinite-horizon problems.
+Policies. An observation-based policy σ : (Z × A)∗ × Z �→ Distr(A) for a POMDP
+M maps a sequence of observations and actions to a distribution over actions. A M-
+finite-state controller (M-FSC) is a tuple C = (Q, qI, η, δ), where Q = {q1, q2, . . . , qM}
+is a finite set of memory states, qI is the initial memory state, η : Q×Z �→ Distr(A) is
+a decision function, and δ : Q × Z × A �→ Distr(Q) is a memory transition function.
+The action mapping η(n, z) takes a FSC memory state n and an observation z ∈ Z,
+and returns a distribution over the POMDP actions. The memory update δ(n, z, α) re-
+turns a distribution over memory states and is a function of the action α selected by η.
+An FSC induces an observation-based policy by following a joint execution of these
+two functions upon a trace of the POMDP. An FSC is memoryless if there is a single
+4
+
+memory state. Memoryless FSCs, denoted by σ: Z → Distr(A), are observation-
+based policies, where σ(α|z) = σz,α is the probability of taking the action α given
+solely observation z.
+Remark 1 (REDUCTION TO MEMORYLESS POLICIES). In the remainder of the pa-
+per, for ease of notation, we synthesize optimal M-FSCs for POMDPs (so-called for-
+ward problem) by computing memoryless policies σ on theoretically-justified larger
+POMDPs obtained from the so-called product of the memory update δ and the original
+POMDPs. Indeed, the authors of [27] provide product POMDPs, whose sizes grow
+polynomially only with the size of the domain of δ.
+Belief Update. Given a history on the POMDP M as the perceived observation and
+executed action sequence τ = {(z0, α0), (z1, α1), . . . , (zT , αT )}, where zi ∈ Z, αi ∈
+A, i ∈ {0, . . . , T} and T is the length of the trajectory, the belief state specifies the
+probability of being in each state of the POMDP given an initial belief b0 = µ0. Such
+a belief state can be updated at each time step using the following Bayes rule
+bt+1(s′) =
+O(zt|s′) �
+s∈S P(s′|s, αt)bt(s)
+�
+s′′∈S O(zt|s′′) �
+s∈S P(s′′|s, αt)bt(s).
+(1)
+2.2. Causal Entropy in POMDPs.
+For a POMDP M, a policy σ induces the stochastic processes Sσ
+0:∞ := (Sσ
+0 , . . . , Sσ
+∞),
+Aσ
+0:∞ := (Aσ
+0, . . . , Aσ
+∞), and Zσ
+0:∞ := (Zσ
+0 , . . . , Zσ
+∞). At each time index t, the ran-
+dom variables Sσ
+t , Aσ
+t , and Zσ
+t take values st ∈ S, αt ∈ A, and zt ∈ Z, respec-
+tively. The probability P(A0:T ||S0:T ) of A0:T causally-conditioned on S0:T , given
+by [10, 31, 32] P(A0:T ||S0:T ) := �T
+t=0 P(At|S0:t, A0:t−1), defines a correlation be-
+tween the stochastic processes, where each variable At is conditionally influenced by
+only the earlier predicted variables S0:t, A0:t−1, and not the future variables St+1:T .
+Let H(A|S) ≜ EA,S[− log P(A|S)] be the conditional entropy of a random variable
+A given a random variable S. In the finite-horizon setting, the causal entropy Hσ in-
+duced by a given policy σ is defined as Hσ := EAσ
+0:T ,Sσ
+0:T [− log P(Aσ
+0:T ||Sσ
+0:T )] =
+�T
+t=0 H(Aσ
+t |Sσ
+0:t, Aσ
+0:t−1). Then, the causal entropy in the infinite-horizon setting,
+namely the discounted causal entropy [9, 33], is defined as
+Hγ
+σ :=
+�∞
+t=0 γtH(Aσ
+t |Sσ
+0:t, Aσ
+0:t−1) =
+�∞
+t=0 γtEAσ
+t ,Sσ
+t [− log P(Aσ
+t |Sσ
+t )],
+(2)
+where the second equality is due to the Markov property.
+Remark 2. The entropy of POMDPs (or MDPs) involves the future policy decisions [8],
+i.e., Sσ
+t+1:T , at a time index t, as opposed to the causal entropy in POMDPs (or MDPs).
+Thus, the authors of [8] show that the problem of computing a policy that maximizes
+the entropy is nonconvex, even in MDPs. Inverse reinforcement learning techniques
+that maximize the entropy of the policy rely on approximations or assume that the tran-
+sition function of the MDP is deterministic. On the other hand, computing a policy that
+maximizes the causal entropy can be formulated as a convex optimization problem in
+MDPs [10, 9].
+5
+
+2.3. LTL Specifications.
+We use general linear temporal logic (LTL) to express complex task specifications
+on the POMDP M. Given a set AP of atomic propositions, i.e., Boolean variables
+with truth values for a given state s or observation z, LTL formulae are constructed
+inductively as following:
+ϕ := true | a | ¬ϕ | ϕ1 ∧ ϕ2 | Xϕ | ϕ1Uϕ2,
+where a ∈ AP, ϕ, ϕ1, and ϕ2 are LTL formulae, ¬ and ∧ are the logic negation and
+conjunction, and X and U are the next and until temporal operators. Besides, temporal
+operators such as always (G) and eventually (F) are derived as Fϕ := trueUϕ and
+Gϕ := ¬F¬ϕ. We denote by Prσ
+M(ϕ) the probability of satisfying the LTL formula ϕ
+when following the policy σ on the POMDP M. A detailed description of the syntax
+and semantics of LTL is beyond the scope of this paper and can be found in [20, 19].
+3. Problem Formulation
+In this section, we formulate the problem of task-guided inverse reinforcement
+learning (IRL) in POMDPs. Given a POMDP M with an unknown reward function
+R, we seek to learn a reward function R along with an underlying policy σ that in-
+duces a behavior similar to the expert demonstrations.
+We define an expert trajectory on the POMDP M as the perceived observation and
+executed action sequence τ = {(z0, α0), (z1, α1), . . . , (zT , αT )}, where zi ∈ Z and
+αi ∈ A for all i ∈ {0, . . . , T}, and T denotes the length of the trajectory. Similarly to
+[34], we assume given or we can construct from τ (via Bayesian belief updates (1)) the
+belief trajectory bτ = {b0 := µ0, . . . , bT }, where bi(s) is the estimated probability of
+being at state s at time index i. In the following, we assume that we are given a set of
+belief trajectories D = {bτ1, . . . , bτN } from trajectories τ1, . . . , τN, where N denotes
+the total number of underlying trajectories.
+We parameterize the unknown reward function R by a differentiable function (with
+respect to the parameter) Rθ : S × A �→ Rd, where θ ∈ RF is a parameter that defines
+uniquely the reward function. Such an encoding includes traditional representations of
+the reward such as Rθ(s, α) = gθ(φ(s, α)), where φ : S × A �→ Rd is a known vector
+of basis functions with components referred to as feature functions, d is the number
+of features, and gθ can be any function approximator such as neural networks. For
+example, in the traditional linear encoding, we have gθ(z) = θTz.
+Specifically, we seek for a parameter θ defining Rθ and a policy σ such that its
+discounted return expectation Rθ
+σ matches an empirical discounted return expectation
+¯Rθ of the expert demonstration D. That is, we have that Rθ
+σ = ¯Rθ, where
+Rθ
+σ :=
+∞
+�
+t=0
+γtESσ
+t ,Aσ
+t [Rθ(Sσ
+t , Aσ
+t )|σ] and ¯Rθ = 1
+N
+�
+bτ ∈D
+�
+bi∈bτ
+γi �
+s∈S
+bi(s)Rθ(s, αi).
+In the case of linear encoding of the reward, the above condition is called feature match-
+ing expectation, and it can be simplified by replacing Rθ with the feature function φ.
+6
+
+Nevertheless, the problem is ill-posed and there may be infinitely many reward
+functions and policies that can satisfy the above matching condition. To resolve the
+ambiguities, we seek for a policy σ that also maximizes the discounted causal entropy
+Hγ
+σ. We now define the problem of interest.
+Problem 1. Given a reward-free POMDP M, a demonstration set D, and a feature φ,
+compute a policy σ and weight θ such that (a) The matching condition holds; (b) The
+causal entropy Hγ
+σ given by (2) is maximized by σ.
+Furthermore, we seek to incorporate, if available, a priori high-level side informa-
+tion on the task demonstrated by the expert in the design of the reward and policy.
+Problem 2. Given a linear temporal logic formula ϕ, compute a policy σ and weight
+θ such that the constraints (a) and (b) in Problem 1 are satisfied, and Prσ
+M(ϕ) ��� λ for
+a given parameter λ ≥ 0.
+Although the parameter λ that specifies the threshold for satisfaction of ϕ is as-
+sumed to be given, the approach can easily be adapted to compute the optimal λ.
+4. Nonconvex Formulation for IRL in POMDPs
+In this section, we formulate Problem 1 and Problem 2 as finding saddle points of
+a nonconvex functions. Then, we propose an algorithm based on solving a nonconvex
+optimization problem to compute such saddle points. We emphasize (see Remark 1)
+that we compute M-FSC for POMDPs by computing memoryless policies σ on larger
+product POMDPs. Indeed, in the remainder of the paper, we reason directly on the
+product POMDP, which is the product of a POMDP and an FSC, and it yields a POMDP
+with state memory pairs [27].
+Substituting Visitation Counts. We eliminate the (infinite) time dependency in Hγ
+σ
+and the matching condition by a substitution of variables involving the policy-induced
+discounted state visitation count µγ
+σ : S �→ R+ and state-action visitation count νγ
+σ :
+S×A �→ R+. For a policy σ, state s, and action α, the discounted state and state-action
+visitation counts are defined by
+µγ
+σ(s) := ESt[
+∞
+�
+t=1
+γt1{St=s}|σ] and νγ
+σ(s, α) := EAt,St[
+∞
+�
+t=1
+γt1{St=s,At=α}|σ],
+where 1{·} is the indicator function. From these definitions, it is straightforward to
+deduce that νγ
+σ(s, α) = πs,αµγ
+σ(s), where πs,α = P[At = a|St = s]. It is also
+straightforward to check that for all s ∈ S and α ∈ A, µγ
+σ(s) ≥ 0, νγ
+σ(s, α) ≥ 0, and
+µγ
+σ(s) = �
+α∈A νγ
+σ(s, α).
+We first provide a concave expression for the discounted causal entropy Hγ
+σ as a
+7
+
+function of the visitation counts µγ
+σ and νγ
+σ:
+Hγ
+σ :=
+�∞
+t=0 γtESσ
+t ,Aσ
+t [− log(πst,αt)]
+=
+�∞
+t=0
+�
+(s,α)∈S×A −(log πs,α)πs,αγtP[Sσ
+t = s]
+=
+�
+(s,α)∈S×A −(log πs,α)πs,αµγ
+σ(s)
+=
+�
+(s,α)∈S×A − log νγ
+σ(s, α)
+µγ
+σ(s) νγ
+σ(s, α),
+(3)
+where the first equality is due to the definition of the discounted causal entropy Hγ
+σ,
+the second equality is obtained by expanding the expectation. The third and fourth
+equalities follow by the definition of the state visitation count µγ
+σ, and the state-action
+visitation count νγ
+σ. We prove in the appendix that the above expression is indeed
+concave in the visitation counts. Next, we obtain a linear expression in νγ
+σ for the
+discounted return expectation Rθ
+σ as:
+Rθ
+σ =
+∞
+�
+t=0
+�
+(s,α)∈S×A
+Rθ(s, α)γtP[Sσ
+t = s, Aσ
+t = α]
+=
+�
+(s,α)∈S×A
+Rθ(s, α)νγ
+σ(s, α),
+(4)
+where the second equality is obtained by the definition of the visitation count νγ
+σ. The
+following nonconvex constraint in µγ
+σ(s) and σz,α ensures observation-based policies:
+νγ
+σ(s, α) = µγ
+σ(s)
+�
+z∈Z O(z|s)σz,α.
+(5)
+Finally, the variables for the discounted visitation counts must satisfy the so-called
+Bellman flow constraint [9] to ensure that the policy is well-defined. For each state
+s ∈ S,
+µγ
+σ(s) = µ0(s) + γ
+�
+s′∈S
+�
+α∈A
+P(s|s′, α)νγ
+σ(s′, α).
+(6)
+Saddle Point Formulation. Computing a policy σ that satisfies the return matching
+constraint Rθ
+σ = ¯Rθ might be infeasible due to ¯Rθ being an empirical estimate from
+the finite set of demonstrations D. Additionally, the feature matching constraint might
+also be infeasible due to the information asymmetry between the expert and the learner,
+e.g., the expert has full observation.
+We build on a saddle point computation problem to incorporate the return matching
+constraints into the objective of the forward problem, similar to other IRL algorithms
+in the literature. Specifically, the desired weight vector θ and policy σ of Problem 1
+and Problem 2 are the solutions of minθ f(θ) := maxσ Hγ
+σ +(Rθ
+σ − ¯Rθ). The function
+f corresponds to the inner optimization problem when the reward parameter is fixed.
+That is, f(θ) computes a policy σ that maximizes the sum Hγ
+σ + Rθ
+σ of the causal
+8
+
+Algorithm 1 Compute the weight vector θ and policy σ solution of the Lagrangian
+relaxation of the IRL problem.
+Input: Feature expectation ¯Rφ from D, initial weight θ0, step size η : N �→ R+, and
+(if available) a priori side information ϕ and λ ∈ [0, 1] imposing Prσ
+M(ϕ) ≥ λ .
+1: σ0 ← uniform policy
+▷ Initialize uniform policy
+2: for k = 0, 1, . . . , do
+▷ Compute θ via gradient descent
+3:
+σk+1 ← SCPForward(θk, σk, ϕ, λ)
+▷ Solve the forward problem (7)–(9)
+with optional ϕ and λ
+4:
+θk+1 ← θk − η(k)∇θf(θk; σk+1)
+▷ Gradient step
+5: end for
+6: return σk, θk
+entropy and the current estimate of the reward function. In other words, f(θ) returns
+the solution to the forward problem, i.e., finding optimal policy on the POMDP when
+the entropy term is removed.
+Algorithm 1 updates the reward weights by using gradient descent. Initially, the
+policy σ0 is a random uniform variable and the weight θ0 is a nonzero vector. At
+iteration k ≥ 0, the policy σk+1 = arg maxσ Hγ
+σ + (Rθk
+σ − ¯Rθk) is the optimal policy
+on the POMDP under the current reward estimate Rθk given by θk. That is, σk+1 is the
+solution to the forward problem. Then, to update the weight θ, Algorithm 1 computes
+the gradient ∇θf with respect to θ as follows:
+∇θf(θ; σ) =
+�
+s,α∈S×A
+νγ
+σ(s, α)∇θRθ(s, α) − 1
+N
+�
+bτ ∈D
+�
+bi∈bτ
+γi �
+s∈S
+bi(s)∇θRθ(s, αi).
+We develop the algorithm SCPForward, presented in next section, to solve the
+forward problem, i.e., computing σk+1 given θk, in an efficient and scalable manner
+while incorporating high-level task specifications to guide the learning.
+Nonconvex Formulation of the Forward Problem. Given a weight vector θk, we take
+advantage of the obtained substitution by the expected visitation counts to formulate
+the forward problem associated to Problem 1 as the nonconvex optimization problem:
+maximize
+µγ
+σ,νγ
+σ,σ
+�
+(s,α)∈S×A
+− log νγ
+σ(s, α)
+µγ
+σ(s) νγ
+σ(s, α) +
+�
+(s,α)∈S×A
+Rθk(s, α)νγ
+σ(s, α)
+(7)
+subject to
+(5) − (6),
+∀(s, α) ∈ S × A, µγ
+σ(s) ≥ 0, νγ
+σ(s, α) ≥ 0,
+(8)
+∀(s, α) ∈ S × A, µγ
+σ(s) =
+�
+α∈A νγ
+σ(s, α),
+(9)
+where the source of nonconvexity is from (5), and we remove the constant − ¯Rθk from
+the cost function of the above optimization problem.
+9
+
+5. Sequential Linear Programming Formulation
+We develop an algorithm, SCPForward, adapting a sequential convex program-
+ming (SCP) scheme to efficiently solve the nonconvex forward problem (7)–(9). In-
+deed, SCPForward involves a verification step to compute sound policies and visi-
+tation counts, which is not present in the existing SCP schemes. Additionally, we de-
+scribe in the next section how to take advantage of high-level task specification (Prob-
+lem 2) through slight modifications of the obtained optimization problem solved by
+SCPForward.
+5.1. Linearizing Nonconvex Optimization Problem
+SCPForward iteratively linearizes the nonconvex constraints in (5) around a pre-
+vious solution. However, the linearization may result in an infeasible or unbounded
+linear subproblem [25]. We first add slack variables to the linearized constraints to
+ensure feasibility. The linearized problem may not accurately approximate the non-
+convex problem if the solutions to this problem deviate significantly from the previous
+solution. Thus, we utilize trust region constraints [25] to ensure that the linearization is
+accurate to the nonconvex problem. At each iteration, we introduce a verification step
+to ensure that the computed policy and visitation counts are not just approximations but
+actually satisfy the nonconvex policy constraint (5), improves the realized cost function
+over past iterations, and satisfy the temporal logic specifications, if available.
+Linearizing Nonconvex Constraints and Adding Slack Variables. We linearize the
+nonconvex constraint (5), which is quadratic in µγ
+σ(s) and σz,α, around the previously
+computed solution denoted by ˆσ, µγ
+ˆσ, and νγ
+ˆσ. However, the linearized constraints may
+be infeasible. We alleviate this drawback by adding slack variables ks,α ∈ R for
+(s, α) ∈ S × A, which results in the affine constraint:
+νγ
+σ(s, α) + ks,α = µγ
+ˆσ(s)
+�
+z∈Z O(z|s)σz,α +
+(10)
+�
+µγ
+σ(s) − µγ
+ˆσ(s)
+� �
+z∈Z O(z|s)ˆσz,α.
+Trust Region Constraints. The linearization may be inaccurate if the solution deviates
+significantly from the previous solution. We add following trust region constraints to
+alleviate this drawback:
+∀(z, α) ∈ Z × A,
+ˆσz,α/ρ ≤ σz,α ≤ ˆσz,αρ,
+(11)
+where ρ is the size of the trust region to restrict the set of allowed policies in the lin-
+earized problem. We augment the cost function in (7) with the term −β �
+(s,α)∈S×A ks,α
+to ensure that we minimize the violation of the linearized constraints, where β is a large
+positive constant.
+10
+
+Linearized Problem. Finally, by differentiating x �→ x log x and y �→ x log(x/y),
+we obtain the coefficients required to linearize the convex causal entropy cost function
+in (7). Thus, we obtain the following linear program (LP):
+maximize
+µγ
+σ,νγ
+σ,σ
+�
+(s,α)∈S×A −
+�
+βks,α −
+�νγ
+ˆσ(s, α)
+µγ
+ˆσ(s)
+�
+µγ
+σ(s)
++
+�
+log νγ
+ˆσ(s, α)
+µγ
+ˆσ(s)
++ 1
+�
+νγ
+σ(s, α)
+�
++
+�
+(s,α)∈S×A
+Rθk(s, α)νγ
+σ(s, α) (12)
+subject to
+(6), (8) − (11).
+Verification Step. After each iteration, the linearization might be inaccurate, i.e, the
+resulting policy ˜σ and potentially inaccurate visitation counts ˜νγ
+˜σ, ˜µγ
+˜σ might not be fea-
+sible to the nonconvex policy constraint (5). As a consequence of the potential infea-
+sibility, the currently attained (linearized) optimal cost might significantly differ from
+the realized cost by the feasible visiation counts for the ˜σ. Additionally, existing SCP
+schemes linearizes the nonconvex problem around the previously inaccurate solutions
+for ˜νγ
+˜σ, and ˜µγ
+˜σ, further propagating the inaccuracy. The proposed verification step
+solves these issues. Given the computed policy ˜σ, SCPForward computes the unique
+and sound solution for the visitation count µγ
+˜σ by solving the corresponding Bellman
+flow constraints:
+µγ
+˜σ(s) =µ0(s) + γ
+�
+s′∈S
+�
+α∈A
+P(s|s′, α)µγ
+˜σ(s′)
+�
+z∈Z
+O(z|s)˜σz,α,
+(13)
+for all s ∈ S, and where µγ
+˜σ ≥ 0 is the only variable of the linear program. Then,
+SCPForward computes νγ
+˜σ(s, α) = µγ
+˜σ(s′) �
+z∈Z O(z|s)˜σz,α and the realized cost
+at the current iteration is defined by
+C(˜σ, θk) =
+�
+(s,α)∈S×A
+− log νγ
+˜σ(s, α)
+µγ
+˜σ
+νγ
+˜σ(s, α) +
+�
+(s,α)∈S×A
+Rθk(s, α)νγ
+˜σ(s, α),
+(14)
+where we assume 0 log 0 = 0. Finally, if the realized cost C(˜σ, θk) does not improve
+over the previous cost C(ˆσ, θk), the verification step rejects the obtained policy ˜σ, con-
+tracts the trust region, and SCPForward iterates with the previous solutions ˆσ, µγ
+ˆσ,
+and νγ
+ˆσ . Otherwise, the linearization is sufficiently accurate, the trust region is ex-
+panded, and SCPForward iterates with ˜σ, µγ
+˜σ and νγ
+˜σ. By incorporating this verifica-
+tion step, we ensure that SCPForward always linearizes the nonconvex optimization
+problem around a solution that satisfies the nonconvex constraint (5).
+5.2. Incorporating High-Level Task Specifications
+Given high-level side information on the agent tasks as the LTL formula ϕ, we first
+compute the product of the POMDP and the ω-automaton representing ϕ to find the
+set T ⊆ S of states, called target or reach states, satisfying ϕ with probability 1 by
+11
+
+using standard graph-based algorithms as a part of preprocessing step. We refer the
+reader to [19] for a detailed introduction on how LTL specifications can be reduced to
+reachability specifications given by T .
+As a consequence, the probability of satisfying ϕ is the sum of the probability of
+reaching the target states s ∈ T , which are given by the undiscounted state visitation
+count µsp
+σ . That is, Prσ
+M(ϕ) = �
+s∈T µsp
+σ (s). Unless γ = 1, µsp
+σ
+̸= µγ
+σ. Thus,
+we introduce new variables µsp
+σ , νsp
+σ , and the adequate constraints in the linearized
+problem (12).
+Incorporating Undiscounted Visitation Variables to Linearized Problem. We append
+new constraints, similar to (8), (9), and (10), into the linearized problem (12), where
+the variables µγ
+σ, νγ
+σ, ks,α, µγ
+ˆσ, νγ
+ˆσ are replaced by µsp
+σ , νsp
+σ , ksp
+s,α, µsp
+ˆσ , νsp
+ˆσ , respectively.
+Further, we add the constraint
+µsp
+σ (s) = µ0(s) +
+�
+s′∈S\T
+�
+α∈A
+P(s|s′, α)νsp
+σ (s′, α),
+(15)
+which is a modification of the Bellman flow constraints such that µsp
+σ (s) for all s ∈ T
+only counts transitions from non-target states. Finally, we penalize the introduced slack
+variables for feasibility of the linearization by augmenting the cost function with the
+term −β �
+(s,α)∈S×A ksp
+s,α.
+Relaxing Specification Constraints. To incorporate the probability of satisfying the
+specifications, We add the following constraint to the linearized problem:
+(spec) :=
+�
+s∈T
+µsp
+σ (s) + Γsp ≥ λ,
+(16)
+where we introduce Γsp ≥ 0 as a slack variable ensuring that the linearized problem
+is always feasible. Further, we augment the cost function with −βspΓsp to penalize
+violating ϕ, where βsp is a positive hyperparameter.
+Updating Verification Step. We modify the previously-introduced realized cost C(˜σ, θk)
+to penalize when the obtained policy does not satisfy the specification ϕ. This cost also
+accounts for the linearization inaccuracy of the new policy constraint due to σ, µsp
+σ ,
+and νsp
+σ . At each iteration, SCPForward computes the accurate µsp
+˜σ of current pol-
+icy ˜σ through solving a feasibility LP with constraints given by the modified Bellman
+flow constraints (15). Then, it augments Csp
+˜σ = min{0, (�
+s∈T µsp
+˜σ (s) − λ)βsp} to the
+realized cost to take the specification constraints into account.
+Convergence to Local Optimum Solution. The convergence guarantees of the pro-
+posed sequential convex scheme with trust regions follow straightforwardly from the
+general convergence of sequential convex programming (SCP) schemes as proved in
+Theorem 3.14 and Theorem 4.7 of [25]. Specifically, weak convergence is ensured as
+the SCP algorithm generates a set of convergent subsequences, all of which satisfy the
+first-order conditions [25]. This is not convergence in its strict sense due to potential
+oscillation between several limit points. Still, surprisingly most of the convergence
+12
+
+Algorithm 2 SCPForward: Linear programming-based algorithm to solve the for-
+ward problem (7)–(9), i.e., compute a policy σk+1 that maximizes the causal entropy,
+considers the matching constraint, and satisfies the specifications, if available.
+Input: Current weight estimate θk, current best policy ˆσ = σk, side information ϕ
+and λ, trust region ρ > 1, penalization coefficients β, βsp ≥ 0, constant ρ0 to
+expand or contract trust region, and a threshold ρlim for trust region contraction.
+1: Find µγ
+ˆσ via linear constraint (13) and νγ
+ˆσ = µγ
+ˆσ(s′) �
+z∈Z O(z|s)ˆσz,α, given ˆσ ▷
+Realized visitation counts
+2: Find µsp
+ˆσ via linear constraint (15) with νsp
+ˆσ = µsp
+ˆσ (s′) �
+z∈Z O(z|s)ˆσz,α, given ˆσ
+▷ If ϕ is available
+3: Compute the realized cost C(ˆσ, θk) ← (14) + Csp
+ˆσ , given ˆσ ▷ Add specifications’
+violation
+4: while ρ > ρlim do
+▷ Trust region threshold
+5:
+Find optimal ˜σ to the augmented LP (12) via ˆσ, µγ
+ˆσ, νγ
+ˆσ, µsp
+ˆσ , νsp
+ˆσ
+▷ We
+augment the LP with constraints (8), (9), (10), (15), and (16) induced by µsp
+σ , νsp
+σ ,
+and by adding −β �
+(s,α)∈S×A ksp
+s,α − βspΓsp to the cost (12).
+6:
+Compute the realized µγ
+˜σ, νγ
+˜σ,µsp
+˜σ , νsp
+˜σ , and C(˜σ, θk) via ˜σ as in lines 1–3
+7:
+{ˆσ ← ˜σ; ρ ← ρρ0} if C(˜σ, θk) ≥ C(ˆσ, θk) else {ρ ← ρ/ρ0}
+▷ Verification
+step
+8: end while
+9: return σk+1 := ˆσ
+claims of nonlinear optimization schemes fall into this category. Furthermore, under
+the right regularity assumptions on the cost function, the authors of [25] proved that
+SCP schemes with trust regions can converge to a local optimum solution with a super-
+linear convergence rate.
+6. Numerical Experiments
+We evaluate the proposed IRL algorithm on several POMDP instances from [35],
+and a simulated wheeled ground robot operating in a high-fidelity, continuous, and 3-D
+Unity simulation. We first compare our IRL algorithm with a straightforward variant
+of GAIL [30] adapted for POMDPs. Then, we provide results on the data-efficiency
+of the proposed approach when taking advantage of side information. Finally, we
+demonstrate the scalability of the routine SCPForward for solving the forward prob-
+lem through comparisons with state-of-the-art solvers such as SolvePOMDP [36],
+SARSOP [37], PRISM-POMDP [38]. We provide the code for reproducibility of the
+results in this paper at https://github.com/wuwushrek/MCE IRL POMDPS.
+6.1. Simulation on Hand-Crafted POMDP Instances
+We first evaluate the proposed IRL algorithm on several POMDP instances ex-
+tracted from the work [35].
+13
+
+1
+2
+3
+4
+5
+6
+9
+12
+7
+10
+13
+8
+11
+14
+Figure 1: Some examples from the benchmark set provided in [35]. From left to right, we have the Maze,
+Avoid, and Evade environments, respectively.
+Benchmark Set. The POMDP instances are as follows. Evade is a turn-based game
+where the agent must reach a destination without being intercepted by a faster player.
+In Avoid, the agent must avoid being detected by two other moving players following
+certain preset, yet unknown routes. In Intercept, the agent must intercept another player
+who is trying to exit a gridworld. In Rocks, the agents must sample at least one good
+rock over the several rocks without any failures. In Obstacle, an agent must find an exit
+in a gridworld without colliding with any static obstacles. In these instances, the agent
+only observes a fixed radius around its current position, see Figure 1. Finally, in Maze,
+the agent must exit a maze as fast as possible while observing only the walls around it
+and should not get stuck in any of the trap states.
+Variants of Learned Policies and Experts. We refer to four types of policies. The
+type of policy depends on whether it uses side information from a temporal specifi-
+cation ϕ or not, and whether it uses a memory size M = 1 or M = 10. We also
+consider two types of experts. The first expert has full information about the envi-
+ronment and computes an optimal policy in the underlying MDP. The second expert
+has partial observation and computes a locally optimal policy in the POMDP with a
+memory size of M = 15. Recall that the agent always has partial information. There-
+fore, the first type of expert corresponds to having information asymmetry between the
+learning agent and expert. Besides, we consider as a baseline a variant of GAIL where
+we learn the policy on the MDP without side information, and extend it to POMDPs
+via an offline computation of the belief in the states. Specifically, we find the optimal
+policy on the MDP by solving the convex optimization problem corresponding to the
+forward problem on MDPs. The resulting policy is a state-based policy that needs to
+be transformed in order to act on a POMDP. The transformation is done by exploiting
+the expert demonstrations to construct a belief state. That is, the trajectories τ of the
+expert are used in a Bayesian belief updates (1) to estimate the probability of being in
+each state of the POMDP. Thus, by combining the computed belief and the state-based
+policy, we obtain an observation-based policy for the POMDP. Doing so could provide
+a significant advantage to the GAIL variant since the state-based policy is the optimal
+policy on the MDP. However, despite the high performance in practice, the policy on
+the POMDP is generally suboptimal, even if the MDP policy were optimal.
+We discuss the effect of side information and memory in the corresponding policies.
+While we detail only on the Maze example, where the agent must exit a maze as fast as
+possible, we observe similar patterns for other examples. Detailed results for the other
+examples are provided in the appendix.
+14
+
+A low state-space Avoid instance
+0
+1
+2
+3
+4
+5
+x=0,y=0
+0
+X=2,y=2,d=E
+X=0,y=4,d=E
+1
+west
+east
+2
+north
+south
+3 
+adv
+placement
+5
+XA low state-space Evade instance
+0
+1
+2
+3
+4
+5
+x=1,y=0
+0
+x=2,y=3
+1
+scan
+adv
+2
+north
+east
+3
+placement
+west
+south
+4 
+5
+XNo information asymmetry
+Under information asymmetry
+GAIL
+0
+25
+50
+75
+100
+−20
+0
+20
+40
+60
+Finite-memory policy
+Without side
+information
+Rθ
+σ
+0
+25
+50
+75
+100
+Memoryless policy
+0
+25
+50
+75
+100
+−20
+0
+20
+40
+60
+Time Steps
+With side
+information
+Rθ
+σ
+0
+25
+50
+75
+100
+Time Steps
+Figure 2: Representative results on the Maze example; each sub-figure represents the average accumulated
+reward under the true reward function (Rθ
+σ) over 1000 runs as a function of time. Compare the two rows:
+The policies in the top row that do not utilize side information suffer a performance drop under information
+asymmetry. On the other hand, in the bottom row, the performance of policies incorporating side information
+into learning does not decrease under information asymmetry. Compare the two columns: The performance
+of the finite-memory policies in the left column is significantly better than memoryless policies. Except for
+the memoryless policies without side information, our algorithm outperforms GAIL. The expert reward on
+the MDP is in average 48.22, while we obtain the value 47.83 for an expert acting on the POMDP.
+6.1.1. Maze Example
+The POMDP M is specified by S = {s1, . . . , s14} corresponding to the cell labels
+in Figure 1. An agent in the maze only observes whether or not there is a wall (in blue)
+in a neighboring cell. That is, the set of observations is O = {o1, . . . , o6, o7}. For
+example, o1 corresponds to observing west and north walls (s1), o2 to north and south
+walls (s2, s4), and o5 to east and west walls (s6, s7, s8, s9, s10, s11). The observations
+o6 and o7 denote the target state (s13) and bad states(s12, s14). The transition model is
+stochastic with a probability of slipping p = 0.1. Further, the states s13 and s14 lead to
+the end of the simulation (trapping states).
+In the IRL experiments, we consider three feature functions. We penalize taking
+more steps with φtime(s, α) = −1 for all s, α. We provide a positive reward when
+reaching s13 with φtarget(s, α) = 1 if s = s13 and φtarget(s, α) = 0 otherwise. We
+penalize bad states s12 and s14 with φbad(s, α) = −1 if s = s12 or s = s14, and
+φbad(s, α) = 0 otherwise. Finally, we have the LTL formula ϕ = G ¬ bad as the
+task specification, where bad is an atomic proposition that is true if the current state
+s = s12 or s = s14. We constrain the learned policy to satisfy Prσ
+M(G ¬ bad) ≥ 0.9.
+Side Information Alleviates the Information Asymmetry. Figure 2 shows that if there
+is an information asymmetry between the learning agent and the expert, the policies
+that do not utilize side information suffer a significant performance drop. The policies
+15
+
+With side information
+Without side information
+GAIL
+0
+75
+150
+225
+300
+−20
+0
+20
+40
+Time Steps
+Total Reward
+Figure 3: Representative results on the Avoid example showing the reward of the policies under the true
+reward function (Rθ
+σ) versus the time steps.
+that do not incorporate side information into learning obtain a lower performance by
+57% under information asymmetry, as shown in the top row of Figure 2. On the other
+hand, as seen in the bottom row of Figure 2, the performance of the policies that use
+side information is almost unaffected by the information asymmetry.
+Memory Leads to More Performant Policies. The results in Figure 2 demonstrate that
+incorporating memory into the policies improves the performance, i.e., the attained
+reward, in all examples, both in solving the forward problem and learning policies
+from expert demonstrations. Incorporating memory partially alleviates the effects of
+information asymmetry, as the performance of the finite-memory policy decreases by
+18% under information asymmetry as opposed to 57% for the memoryless policy.
+We see that in Table 1, incorporating memory into policy on the Maze and Rocks
+benchmarks, allows SCPForward to compute policies that are almost optimal, evi-
+denced by obtaining almost the same reward as the solver SARSOP.
+Side Information Improves Data Efficiency. Figure 4 shows that even on a low data
+regime, learning with task specifications achieves significantly better performance than
+without the task specifications.
+5
+10
+15
+20
+30
+40
+Number of trajectories
+Total reward
+Without LTL
+With LTL
+Opt. Rew. POMDP
+5
+10
+15
+40
+42
+44
+46
+Number of trajectories
+Figure 4: We show the data efficiency of the proposed approach through the total reward obtained by the
+learned policies as a function of the number of expert demonstrations (No information asymmetry). The
+figure on the left shows the performance of learning memoryless policies, while the figure on the right shows
+the performance of a 5-FSC.
+16
+
+SCPForward
+SARSOP
+SolvePOMDP
+Problem
+|S|
+|S × O|
+|O|
+Rθ
+σ
+Time (s)
+Rθ
+σ
+Time (s)
+Rθ
+σ
+Time (s)
+Maze
+17
+162
+11
+39.24
+0.1
+47.83
+0.24
+47.83
+0.33
+Maze (3-FSC)
+49
+777
+31
+44.98
+0.6
+NA
+NA
+NA
+NA
+Maze (10-FSC)
+161
+2891
+101
+46.32
+2.04
+NA
+NA
+NA
+NA
+Obstacle[10]
+102
+1126
+5
+19.71
+8.79
+19.8
+0.02
+5.05
+3600
+Obstacle[10](5-FSC)
+679
+7545
+31
+19.77
+38
+NA
+NA
+NA
+NA
+Obstacle[25]
+627
+7306
+5
+19.59
+14.22
+19.8
+0.1
+5.05
+3600
+Rock
+550
+4643
+67
+19.68
+12.2
+19.83
+0.05
+−
+−
+Rock (3-FSC)
+1648
+23203
+199
+19.8
+15.25
+NA
+NA
+−
+−
+Rock (5-FSC)
+2746
+41759
+331
+19.82
+97.84
+NA
+NA
+−
+−
+Intercept[5, 2, 0]
+1321
+5021
+1025
+19.83
+10.28
+19.83
+13.71
+−
+−
+Intercept[5, 2, 0.1]
+1321
+7041
+1025
+19.81
+13.18
+19.81
+81.19
+−
+−
+Evade[5, 2, 0]
+2081
+13561
+1089
+97.3
+26.25
+97.3
+3600
+−
+−
+Evade[5, 2, 0.1]
+2081
+16761
+1089
+96.79
+26.25
+95.28
+3600
+−
+−
+Evade[10, 2, 0]
+36361
+341121
+18383
+94.97
+3600
+−
+−
+−
+−
+Avoid[4, 2, 0]
+2241
+5697
+1956
+9.86
+34.74
+9.86
+9.19
+−
+−
+Avoid[4, 2, 0.1]
+2241
+8833
+1956
+9.86
+14.63
+9.86
+210.47
+−
+−
+Avoid[7, 2, 0]
+19797
+62133
+3164
+9.72
+3503
+−
+−
+−
+−
+Table 1: Results for the benchmark sets for solving the forward problem. On larger benchmarks (e.g., Evade
+and Avoid), SCPForward can compute locally optimal policies, while the other solvers fail to provide
+solutions in the given time limit. In the environments Obstacle[n], Intercept[n, r, slip], Evade[n, r, slip],
+and Avoid[n, r, slip], the parameters n, r, and slip are the size of the gridworld, the view radius of the agent,
+and the probability of slippery, respectively. We set the time-out to 3600 seconds. An empty cell (denoted by
+−) represents the solver failed to compute any policy before the time-out, while NA refers to not applicable
+due to the approach being based on belief updates.
+Side Information Improves Performance. Besides, in a more complicated environ-
+ment such as Avoid, Figure 3 shows that task specifications are crucial to hope even
+to learn the task. Specifically, Avoid[n, r, slip] is a turn-based game, where the agent
+must reach an exit point while avoiding being detected by two other moving players
+following certain predefined yet unknown routes. The agent can only observe the play-
+ers if they are within a fixed radius from the agent’s current position when the action
+scan is performed. Besides, with the players’ speed being uncertain, their position in
+the routes can not be inferred by the agent. The parameters n, r, and slip specify the
+dimension of the grid, the view radius, and the slippery probability, respectively.
+We consider four feature functions to parameterize the unknown reward. The first
+feature provides a positive reward to the agent upon reaching the exit point. The second
+feature penalizes the agent if it collides with a player. The third feature penalizes the
+agent if it is detected by a player. The fourth feature imposes a penalty cost for each
+action taken. We encode the side information as the temporal logic task specification
+avoid being detected until reaching the exit point with probability greater than 0.98.
+Figure 3 shows that the algorithm is unable to learn without side information while
+side information induces a learned policy that is optimal. Specifically, the learned
+policy without side information seems to only focus on avoiding being detected and
+collision as the corresponding learned features were close to zero.
+17
+
+Figure 5: Left: A simulated Clearpath Warthog operating in a Unity simulation. Right: A demonstration
+provided by an expert.
+6.1.2. SCPForward Yields Better Scalability
+We highlight three observations regarding the scalability of SCPForward. First,
+the results in Table 1 show that only SARSOP is competitive with SCPForward on
+larger POMDPs. SolvePOMDP runs out of time in all but the smallest benchmarks,
+and PrismPOMDP runs out of memory in all benchmarks. Most of these approaches
+are based on updating a belief over the states, which for a large state space can become
+extremely computationally expensive.
+Second, in the benchmarks with smaller state spaces, e.g., Maze and Rock, SARSOP
+can compute policies that yield better performance in less time. This is due to the effi-
+ciency of belief-based approaches on small-size problems. On the other hand, SARSOP
+does not scale to larger POMDPs with a larger number of states and observations. For
+example, by increasing the number of transitions in Intercept benchmark from 5021 to
+7041, the computation time for SARSOP increases by 516%. On the other hand, the
+increase of the computation time of SCPForward is only 28%.
+Third, on the largest benchmarks, including tens of thousands of states and obser-
+vations, SARSOP fails to compute any policy before time-out, while SCPForward
+found a solution. Finally, we also note that SCPForward can also compute a policy
+that maximizes the causal entropy and satisfies an LTL specification, unlike SARSOP.
+6.2. Simulation on a Ground Robot
+We demonstrate an application of the proposed algorithm in a continuous 3-D Unity
+environment containing a ClearPath warthog operating in a semi-structured village. A
+screen shot of the robot operating in this environment and its corresponding trajectory
+can be seen in Figure 5. This environment contains a variety of obstacles including
+buildings, trees, and vehicles as well as three terrain types describing our features, φ,
+grass, gravel, and road. The simulated environment operates in a state space consisting
+of 3350 states, 33254 transitions and 944 total observations. This simulation is used to
+18
+
+0
+5
+10
+15
+20
+25
+30
+0
+10
+20
+30
+grass
+gravel
+road
+unknown
+Figure 6: Gridworld representation of the environment. The figure shows the area of the unity environment
+where we applied the developed algorithm.
+gather data for training, and test an agent’s ability to follow a policy from the learned
+reward function in two experimental scenarios. In this experiment, we demonstrate
+the agent’s ability to learn a reward function from demonstrations that are sub-optimal
+with respect to a known, true reward function. We also show how the learned policies
+perform compared to the optimal policies with full and partial observations obtained
+by solving the MDP or POMDP problem with the true reward function.
+The ground vehicle contains an autonomy stack consisting of three main subsys-
+tems—mapping, perception, and planning. The mapping subsystem based on Omni-
+Mapper[? ] performs simultaneous localization and mapping (SLAM) using LiDAR
+and IMU sensors, providing a map used during planning. The perception subsystem
+provides pixel level semantic segmentation for each image in a video stream from a
+RGB camera to an ontology of terrain and object classes. Each semantic image is
+passed to a terrain projection algorithm which builds N binary occupancy feature maps
+of the known environment used for reward learning where N is the number of features.
+The planning subsystem uses the maps produced from previous subsystems and the
+trajectory from a learned policy to autonomously navigate to a waypoint.
+Expert Demonstrations and Reward Feature Encoding. We collected 10 demonstra-
+tions of an expert teleoperating a robot to a predetermined waypoint (see Figure 6).
+The expert has an implicit preference to traverse the road followed by grass, and lastly
+gravel. Consequently, we encode the unknown reward function as a linear combination
+of known features: Rθ = θ1φroad + θ2φgravel + θ3φgrass + θ4φtime + θ5φgoal, where
+φi returns a value of 0 when the feature of the corresponding state is not feature i, or
+1 otherwise. In order to incentivize the shortest path, the feature time penalizes the
+number of actions taken in the environment before reaching the waypoint. Further-
+19
+
+(a) The trajectories resulting from executing each policy
+with and without task specifications. The learner exploit-
+ing task specifications (orange) is able to reach one of the
+target states, while avoiding the gravel along the path. In
+contrast, the learner without side information (purple) fails
+to avoid the gravel.
+0
+100
+200
+300
+−20
+0
+20
+Expert MDP
+Expert POMDP
+With LTL
+Without LTL
+(b) Evolution of the cumulative reward obtained by the
+learner as a function of the number of environment inter-
+actions.
+Expert MDP and Expert POMDP are the opti-
+mal policies on the MDP and POMDP, respectively for the
+ground truth reward function.
+Figure 7: Impact of incorporating task specifications into reward learning.
+more, goal provides a positive reward upon reaching the waypoint. For comparisons
+of the learned policy, we use the values θ = [0.2, −30, −2, −0.5, 50] as the ground
+truth reward weight vector. We emphasize that the demonstrations are sub-optimal
+with respect to the above ground truth reward as the vehicle often traverses gravel,
+corresponding to a high penalty reward.
+Modeling Robot Dynamics as POMDPs. From a ground truth map of the environment
+in the simulation, we obtain a high-level MDP abstraction of the learner’s behavior on
+the entire state space. Then, we impose a partial observability of the robot as follows:
+The robot does not see the entire map of the world but only see a fixed radius r = 4
+(in terms of the number of grid cells) around its current position. Furthermore, we also
+incorporate uncertainty on the sensor classification of terrain features such that with
+probability p = 0.9 the prediction is correct.
+Task Specifications. In addition to the expert demonstrations, we constrain the learned
+policy to satisfy Prσ
+M(¬ gravel U goal) ≥ 0.9, where gravel is an atomic proposition
+that is true for states having gravel as its feature, and goal is an atomic proposition that
+is true at each target state. Note that this side information does not necessarily enforce
+that the learner should reach the set of target states. Instead, if the learner reaches the
+target state, it should not drive on gravel with probability at least
+Results. Figure 7a shows how the learner with side information avoids the gravel com-
+pared to the learner without side information. Figure 7b further illustrates this result by
+empirically demonstrating that the proposed approach can efficiently take advantage
+of side information to compute policies that matches the expert’s desired behavior.
+Specifically, Figure 7b shows that the gain in the total reward of a learner without side
+20
+
+information increases by 294% with respect to a learner with side information. Ad-
+ditionally, it is important to note in Figure 6 how the initial state distribution of the
+demonstrator trajectories is different from the initial state distribution during the eval-
+uation of the learned policies (Figure 7a). Nevertheless, despite these distinctions, the
+learned policies can effectively navigate toward points present in the expert demonstra-
+tions and then maximally mimic these trajectories.
+7. Related work.
+The closest work to ours is by [34], where they extend classical maximum-margin-
+based IRL techniques for MDPs to POMDPs. However, even on MDPs, maximum-
+margin-based approaches cannot resolve the ambiguity caused by suboptimal demon-
+strations, and they work well when there is a single reward function that is clearly better
+than alternatives [39]. In contrast, we adopt causal entropy that has been shown [39, 10]
+to alleviate these limitations on MDPs. Besides, [34] rely on efficient off-the-shelf
+solvers to the forward problem. Instead, this paper also develops an algorithm that
+outperforms off-the-shelf solvers and can scale to POMDPs that are orders of magni-
+tude larger compared to the examples in [34]. Further, [34] do not incorporate task
+specifications in their formulations.
+One of the basic challenges in IRL, is that finding a reward function and a policy
+that induces a similar behavior to the expert is an ill-defined problem. Prior work has
+addressed this challenge using maximum margin formulations [40, 41, 42], as well as
+probabilistic models to compute a likelihood of the expert demonstrations [43, 8, 10].
+We build on the latter approach and build on the maximum-causal-entropy IRL [9,
+10, 23], which brings algorithmic benefits to IRL in POMDPs as mentioned in the
+introduction. We note that these maximum-causal-entropy IRL techniques assume that
+both the expert and the agent can fully observe the environment, and these approaches
+only apply for MDPs as opposed to POMDPs.
+IRL under partial information has been studied in prior work [2, 44, 45, 46, 47].
+Reference [44] considers the setting where the features of the reward function are par-
+tially specified as opposed to having partial information over the state of the environ-
+ment. The work in [2] considers a special case of POMDPs. It only infers a distribution
+over the future trajectories of the expert given demonstrations as opposed to computing
+a policy that induces a similar behavior to the expert. The works in [45, 46, 47] assume
+that the states of the environment are either fully observable, or fully hidden to the
+learning agent. Therefore, these approaches also consider a special case of POMDPs,
+like in [2]. We also note that none of these methods incorporate side information into
+IRL and do not provide guarantees on the performance of the policy with respect to a
+task specification.
+The idea of using side information expressed in temporal logic to guide and aug-
+ment IRL has been explored in some previous work. In [48, 22], the authors incor-
+porate side information as in temporal logic specification to learn policies that induce
+a behavior similar to the expert demonstrations and satisfies the specification. Refer-
+ence [21] iteratively infers an underlying task specification that is consistent with the
+expert demonstrations and learns a policy and a reward function that satisfies the task
+21
+
+specification. However, these methods also assume full information for both the expert
+and the agent.
+8. Conclusion
+We develop an algorithm for inverse reinforcement learning under partial obser-
+vation. We empirically demonstrate that by incorporating task specifications into the
+learning process, we can alleviate the information asymmetry between the expert and
+the learner while increasing the data efficiency of the learning scheme. Further, we
+empirically demonstrate that our main routine SCPForward, used inside the IRL al-
+gorithm, solves the forward problem in a scalable manner and outperforms state-of-
+the-art POMDP solvers on instances with a large number of states, observations, and
+transitions.
+Work Limitations. This work assumes that the transition and observation functions of
+the POMDP are known to the algorithm. Future work will investigate removing this
+assumption and developing model-free-based approaches. We will also integrate the
+framework with more expressive neural-network-based reward functions.
+Acknowledgements.. Research was sponsored by the Army Research Laboratory and
+Office of Naval Research accomplished under cooperative agreement number(s) ARL
+W911NF-20-2-0132, ARL W911NF-19-2-0285 and ONR N00014-22-1-2254. The
+views and conclusions contained in this document are those of the authors and should
+not be interpreted as representing the official policies; either expressed or implied,
+of the Army Research Laboratory, Office of Naval Research, or the U.S. Government.
+The U.S. Government is authorized to reproduce and distribute reprints for Government
+purposed notwithstanding any copyright notation herein.
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+Appendices
+In this appendix, we provide supplementary derivations for the results in the paper and
+more details on the numerical experiments.
+A. Concavity of Causal Entropy and Derivations of the Bellman Constraints
+In this section, we first recall the obtained expression of the causal entropy Hγ
+σ as a
+function of the visitation counts µγ
+σ and νγ
+σ. We then prove the concavity of the causal
+entropy, which enables convex-optimization-based formulation of the task-guided in-
+verse reinforcement learning (IRL) problem. Then, we provide additional details on
+the derivation of the affine constraint implied by the Bellman flow constraint.
+Concave Causal Entropy. We first recall the definitions of the state and state-action
+visitation counts. For a policy σ, state s, and action α, the discounted state visitation
+counts are defined by µγ
+σ(s) ≜ ESt[�∞
+t=1 γt1{St=s}] and the discounted state-action
+visitation counts are defined by νγ
+σ(s, α) ≜ EAt,St[�∞
+t=1 γt1{St=s,At=α}], where 1{·}
+is the indicator function and t is the time step. From the definitions of the state and
+state-action visitation counts µγ
+σ and νγ
+σ, it is straightforward to deduce that νγ
+σ(s, α) =
+σs,αµγ
+σ(s), where σs,α = P[At = a|St = s]. We use the visitation counts to prove in
+Section 4 that
+Hγ
+σ =
+�
+(s,α)∈S×A
+−(log πs,α)πs,αµγ
+σ(s) =
+�
+(s,α)∈S×A
+− log νγ
+σ(s, α)
+µγ
+σ(s) νγ
+σ(s, α),
+where the last inequality is obtained by using that πs,α = νγ
+σ(s, α)/µγ
+σ(s). We claim
+that Hγ
+σ is a concave fucntion of the visitation counts. Thus, we want to show that
+the function f(νγ
+σ, µγ
+σ) = �
+(s,α)∈S×A − log νγ
+σ(s,α)
+µγ
+σ(s) νγ
+σ(s, α) is concave. To this end,
+consider any λ ∈ (0, 1) and the two sets of variables νγ
+σ, µγ
+σ and ¯νγ
+σ, ¯µγ
+σ. Then, we have
+26
+
+the following result:
+f(λνγ
+σ + (1 − λ)¯νγ
+σ, λ¯µγ
+σ + (1 − λ)¯µγ
+σ)
+=
+�
+(s,α)∈S×A
+− log λνγ
+σ(s, α) + (1 − λ)¯νγ
+σ(s, α)
+λµγ
+σ(s) + (1 − λ)¯µγ
+σ(s, α) (λνγ
+σ(s, α) + (1 − λ)¯νγ
+σ(s, α))
+≥
+�
+(s,α)∈S×A
+−λνγ
+σ(s, α) log λνγ
+σ(s, α)
+λµγ
+σ(s, α) − (1 − λ)¯νγ
+σ(s, α) log (1 − λ)¯νγ
+σ(s, α)
+(1 − λ)¯µγ
+σ(s, α)
+=
+�
+(s,α)∈S×A
+−λνγ
+σ(s, α) log νγ
+σ(s, α)
+µγ
+σ(s, α) − (1 − λ)¯νγ
+σ(s, α) log ¯νγ
+σ(s, α)
+¯µγ
+σ(s, α)
+= λf(νγ
+σ, µγ
+σ) + (1 − λ)f(¯νγ
+σ, ¯µγ
+σ),
+where the first inequality is obtained by applying to the well-known log-sum inequality,
+i.e.,
+x1 log x1
+y1
++ x2 log x2
+y2
+≥ (x1 + x2) log x1 + x2
+y1 + y2
+,
+for nonnegative numbers x1, x2, y1, y2. Specifically, we apply the substitution x1 =
+λνγ
+σ, y1 = λµγ
+σ, x2 = (1 − λ)¯νγ
+σ, and y2 = (1 − λ)¯µγ
+σ. Note that the inequality
+f(λνγ
+σ + (1 − λ)¯νγ
+σ, λ¯µγ
+σ + (1 − λ)¯µγ
+σ) ≥ λf(νγ
+σ, µγ
+σ) + (1 − λ)f(¯νγ
+σ, ¯µγ
+σ)
+implies that f(νγ
+σ, µγ
+σ) is concave in νγ
+σ, and µγ
+σ.
+Bellman Flow Constraint. For the visitation count variables to correspond to a valid
+policy generating actions in the POMDP M , νγ
+σ and µγ
+σ must satisfy the bellman flow
+constraint given by
+µγ
+σ(s) = ESσ
+t
+� ∞
+�
+t=0
+γt1{Sσ
+t =s}
+�
+= µ0(s) + γESσ
+t
+� ∞
+�
+t=0
+γt1{Sσ
+t+1=s}
+�
+= µ0(s) + γ
+∞
+�
+t=0
+�
+s′∈S
+�
+α∈A
+γtP(s|s′, α)P[Sσ
+t = s′, Aσ
+t = α]
+= µ0(s) + γ
+�
+s′∈S
+�
+α∈A
+P(s|s′, α)νγ
+σ(s′, α).
+B. Experimental Tasks
+In this section, we first provide a detailed description of the POMDP models used
+in the benchmark. The simulations on the benchmark examples empirically demon-
+strate that side information alleviates the information asymmetry, and more memory
+leads to more performant policies. Then, we provide additional numerical simulations
+supporting the claim that SCPForward is sound and yields better scalability than
+off-the-shelf solvers for the forward problem, i.e., computing an optimal policy on a
+POMDP for a given reward function.
+27
+
+B.1. Computation Resources and External Assets
+All the experiments of this paper were performed on a computer with an Intel Core
+i9-9900 CPU 3.1GHz ×16 processors and 31.2 Gb of RAM. All the implementations
+are written and tested in Python 3.8, and we attach the code with the supplementary
+material.
+Required Tools. . Our implementation requires Stormpy of Storm [? ] and Gurobipy
+of Gurobi 9.1 [? ]. On one hand, we use Storm, a tool for model checking, to parse
+POMDP file specifications, to compute the product POMDP with the finite state con-
+troller in order to reduce the synthesis problem to the synthesis of memoryless policies,
+and to compute the set T of target states satisfying a specification ϕ via graph prepro-
+cessing. On the other hand, we use Gurobi to solve both the linearized problem in (7)
+and the feasible solution of the Bellman flow constraint needed for the verification step.
+Off-The-Shelf Solvers for Forward Problem.
+. In order to show the scalability of
+the developed algorithm SCPForward, we compare it to state-of-the-art POMDP
+solvers SolvePOMDP [36], SARSOP [37], and PRISM-POMDP [38].
+The solver
+SolvePOMDP implements both exact and approximate value iterations via incremen-
+tal pruning [? ] combined with state-of-the-art vector pruning methods [36]. Finally,
+PrismPOMDP discretizes the belief state and adopts a finite memory strategy to find
+an approximate solution of the forward problem. For all the solvers above, we use the
+default settings except from the timeout enforced to be 3600 seconds. These solvers
+are not provided with our implementation. However, we provide the POMDP models
+that each of the solvers can straightforwardly use. Further details are provided in the
+readme files of our implementation.
+B.2. Benchmark Set
+We evaluate the proposed learning algorithm on several POMDP instances adapted
+from [35]. We attached the modified instances in our code with the automatically
+generated models for each off-the-shelf solver that the reader can straightforwardly
+use to reproduce Table 1. The reader can refer to Table 1 for the number of states,
+observations, and transitions of each environment of the benchmark set. In all the
+examples, we gather 10 trajectories from an expert that can fully observe its current
+state in the environment and an expert having partial observation of the environment.
+Our algorithm learns reward functions from these trajectories under different memory
+policies and high-level side information.
+28
+
+Rocks Instance. In the environment Rocks, an agent navigates in a gridworld to sam-
+ple at least one valuable rock (if a valuable rock is in the grid) over the two possibly
+dangerous rocks, without any failures. When at least one valuable rock has been col-
+lected, or the agent realizes that all the rocks are dangerous, it needs to get to an exit
+point to terminate the mission. The partial observability is due to the agent can only
+observe if its current location is an exit point or a dangerous rock. Furthermore, the
+agent has noisy sensors enabling sampling neighbor cells.
+We consider three feature functions. The first feature provides a positive reward
+when reaching the exit point with at least one valuable rock or no rocks when all of
+them are dangerous. The second feature provides a negative reward when the agent
+is at the location of a dangerous rock. Finally, the third feature penalizes each action
+taken with a negative reward to promote reaching the exit point as soon as possible.
+No information asymmetry
+Under information asymmetry
+GAIL
+0
+75
+150
+225
+300
+0
+50
+100
+Finite-memory policy
+Without side
+information
+Rφ
+σ
+0
+75
+150
+225
+300
+0
+50
+100
+Memoryless policy
+Rφ
+σ
+0
+75
+150
+225
+300
+0
+50
+100
+Time Steps
+With side
+information
+Rφ
+σ
+0
+75
+150
+225
+300
+0
+50
+100
+Time Steps
+Rφ
+σ
+Figure 8: Representative results on the Rock example showing the reward of the policies under the true
+reward function (Rφ
+σ) versus the time steps.
+We compare scenarios with no side information and the a priori knowledge on the
+task such as the agent eventually reaches an exit point with a probability greater than
+0.995. Figure 8 supports our claim that side information indeed alleviates the informa-
+tion asymmetry between the expert and the agent. Additionally, we also observe that
+incorporating memory leads to more performant policies in terms of the mean accumu-
+lated reward.
+29
+
+Obstacle Instance. . In the environment Obstacle[n], an agent must find an exit in a
+gridworld without colliding with any of the five static obstacles in the grid. The agent
+only observes whether the current position is an obstacle or an exit state. The parameter
+n specifies the dimension of the grid.
+Similar to the Rocks example, the agent receives a positive reward if it successfully
+exits the gridworld and a negative reward for every taken action or colliding with an
+obstacle.
+As for the side information, we specify in temporal logic that while learning the
+reward, the agent should not collide any obstacles until it reaches an exit point with a
+probability greater than 0.9.
+No information asymmetry
+Under information asymmetry
+0
+25
+50
+75
+100
+−200
+0
+200
+400
+Finite-memory policy
+Without side
+information
+Rφ
+σ
+0
+25
+50
+75
+100
+−200
+0
+200
+400
+Memoryless policy
+Rφ
+σ
+0
+25
+50
+75
+100
+−200
+0
+200
+400
+Time Steps
+With side
+information
+Rφ
+σ
+0
+25
+50
+75
+100
+−200
+0
+200
+400
+Time Steps
+Rφ
+σ
+Figure 9: Representative results on the Obstacle example showing the reward of the policies under the true
+reward function (Rφ
+σ) versus the time steps.
+Similar to the Maze and Rock examples, Figure 9 supports our claim that side in-
+formation alleviates the information asymmetry and memory leads to more performant
+policies.
+30
+
+Evade Instance. Evade[n, r, slip] is a turn-based game where the agent must reach a
+destination without being intercepted by a faster player. The player cannot access the
+top row of the grid. Further, the agent can only observe the player if it is within a fixed
+radius from its current location and upon calling the action scan. The parameters n, r,
+and slip specify the dimension of the grid, the view radius, and the slippery probability,
+respectively.
+The feature functions are defined such that the agent receives a positive reward if
+at the destination, a high negative reward if it is intercepted by the player, and a small
+negative reward for each action taken, including the scan action.
+With side information
+Without side information
+GAIL
+0
+25
+50
+75
+100
+−10
+0
+10
+20
+Time Steps
+Mean accumulated reward
+Figure 10: Representative results on the Evade example showing the reward of the policies under the true
+reward function (Rφ
+σ) versus the time steps.
+As for the side information, we specify in temporal logic that while learning the
+reward, the agent must reach an exit point with probability greater than 0.98.
+Figure 10 shows that learning with side information provides higher reward than
+without side information. Besides, there is less randomness in the policy with side
+information compared to the policy without side information. Specifically, the standard
+deviation of the policy with side information is significantly smaller than the policy
+without side information.
+We did not discuss the impact of different memory size policies in this example
+since the performance of the memoryless policy is already near-optimal, as the policy
+obtains the same reward as SARSOP (see Table 1 for a reference. Specifically, we
+observe that the optimal policy on the underlying MDP yields comparable policies to
+the optimal memoryless policy on the POMDP. As a consequence, we observe that the
+information asymmetry between the expert and the agent does not hold here either, and
+the learned policies obtain a similar performance.
+31
+
+Intercept Instance. Intercept[n, r, slip] is a variant of Evade where the agent must
+intercept another player who is trying to exit the gridworld. The agent can move in 8
+directions and can only observe the player if it is within a fixed radius from the agent’s
+current position when the action scan is performed. Besides, the agent has a camera
+that enables it to observe all cells from west to east from the center of the gridworld. In
+contrast, the player can only move in 4 directions. The parameters n, r, and slip specify
+the dimension of the grid, the view radius, and the slippery probability, respectively.
+We consider three feature functions to parameterize the unknown reward. The first
+feature provides a positive reward to the agent upon intercepting the player. The second
+feature penalizes the agent if the player exits the gridworld. The third feature imposes
+a penalty cost for each action taken.
+With side information
+Without side information
+GAIL
+0
+25
+50
+75
+100
+−10
+0
+10
+20
+Time Steps
+Mean accumulated reward
+Figure 11: Representative results on the Intercept example showing the reward of the policies under the
+true reward function (Rφ
+σ) versus the time steps.
+We encode the high-level side information as the temporal logic task specification
+Eventually intercept the player with probability greater than 0.98, i.e., the agent should
+eventually reach an observation where its location coincides with the player’s location.
+Figure 11 demonstrates that side information does not improve the performance
+of the policy. This result is because memoryless policies are optimal in this example,
+and a combination of the given reward features can perfectly encode the temporal logic
+specifications, similar to the Evade example.
+B.3. Effects of Side Information
+In this section, we provide additional experiments on how the side information
+speeds up the learning process in terms of computation time and convergence to the
+optimal policy and reward parameters. Then, we quantify the effects of the side infor-
+mation by solving the POMDPs using only the task specifications and no demonstra-
+tions. All these experiments are performed on the Maze example, which is a relatively
+low-size POMDP example.
+32
+
+Convergence of the Learning With and Without Side Information. In the Maze ex-
+periments, we empirically observe that side information enables the learning algorithm
+to converge with eight times less number of iterations compared to learning without
+side information. The number of iterations here denotes both the number of lineariza-
+tions in the sequential convex scheme and the number of gradient steps during reward
+updates. However, the gain in computation time is not as prominent as the gain in
+the number of iterations. In fact, learning with side information is only approximately
+three times faster than learning without side information due to how side information
+almost doubles the number of variables in the convex optimization problem.
+Effects of Side Information Without Any Demonstrations.
+• Experiment 1.
+We consider the exact setting of the Maze example with no
+demonstrations and the LTL specification Prσ
+M(G ¬ bad) ≥ 0.9. Essentially,
+we seek policies that avoid the trapping states with high probability. Without any
+additional reward, the optimal policy is exactly what we expect: High probabil-
+ity for action enforcing no movements and low probability for the others. With
+respect to the true reward, this is clearly suboptimal as the reward will keep de-
+creasing due to no minimization of the amount of time spent in the environment.
+Now, we optimize the problem for a policy that satisfies the LTL specifications
+while penalizing spending time in the environment according to the ground truth
+reward for taking more steps in the environment. The obtained policy has the op-
+timal reward of 47.83, which corresponds to optimizing the ground truth reward
+in the POMDP. Indeed, the fastest way to clear the Maze is to exit through a goal
+state while not getting trapped.
+• Experiment 2.
+We consider the exact setting of the Maze example with no
+demonstrations and the LTL specification Prσ
+M(E target) ≥ 0.95. Essentially,
+we seek policies that eventually reach the target state (exit of the Maze) with
+high probability. Without any additional reward, the optimal policy is subopti-
+mal with respect to the optimal policy on the POMDP, given the ground truth
+reward. Indeed, the policy can reach the target with an optimal reward of 28.63
+due to the amount of time spent to reach the goal. By adding the reward on time
+spent in the environment to the LTL specifications, we can obtain the optimal
+reward on the POMDP again.
+B.4. Summary of the Results
+Side Information Alleviates the Information Asymmetry . As mentioned in the sub-
+mitted manuscript, side information can indeed alleviate the information asymmetry.
+Specifically, we observe that if there is an information asymmetry in the forward prob-
+lem, i.e., the obtained reward from an optimal policy on the underlying POMDP is
+lower than from an optimal policy on the underlying fully observable MDP, incor-
+porating side information in temporal logic specifications alleviates the information
+asymmetry between the expert and the agent. For example, we can see the effects of
+such information asymmetry in the Maze, Rocks, Obstacle, and Avoid examples. In
+33
+
+these examples, having partial observability reduces the obtained reward in the for-
+ward problem. The policies that do not incorporate side information into the learning
+procedure also obtain a lower reward under information asymmetry.
+Memory Leads to More Performance Policies. Similarly to the side information, we
+also observe that if incorporating memory improves the performance of the learned
+policies, if it also improves the obtained reward in the forward problem, as seen in the
+Maze, Rocks, and Obstacle instances. In Table 1, we can also see that incorporating
+memory helps to compute a better optimal policy in these examples, unlike computing
+a memoryless policy.
+34
+

3NAzT4oBgHgl3EQfR_tE/content/tmp_files/2301.01224v1.pdf.txt

@@ -0,0 +1,1827 @@
+An Empirical Investigation into the Use of Image
+Captioning for Automated Software Documentation
+Kevin Moran†, Ali Yachnes∗, George Purnell∗, Junayed Mahmud†,
+Michele Tufano‡, Carlos Bernal Cardenas‡, Denys Poshyvanyk∗, Zach H’Doubler∗
+†George Mason University, VA, USA, ∗William & Mary, VA, USA, ‡Microsoft, WA, USA
+kpmoran@gmu.edu, ayachnes@email.wm.edu, gwpurnell@email.wm.edu, jmahmud@gmu.edu
+michele.tufano@microsoft.com, carlosbe@microsoft.com, denys@cs.wm.edu, pzhdoubler@email.wm.edu
+Abstract—Existing automated techniques for software docu-
+mentation typically attempt to reason between two main sources
+of information: code and natural language. However, this reason-
+ing process is often complicated by the lexical gap between more
+abstract natural language and more structured programming
+languages. One potential bridge for this gap is the Graphical User
+Interface (GUI), as GUIs inherently encode salient information
+about underlying program functionality into rich, pixel-based
+data representations. This paper offers one of the first com-
+prehensive empirical investigations into the connection between
+GUIs and functional, natural language descriptions of software.
+First, we collect, analyze, and open source a large dataset of
+functional GUI descriptions consisting of 45,998 descriptions
+for 10,204 screenshots from popular Android applications. The
+descriptions were obtained from human labelers and underwent
+several quality control mechanisms. To gain insight into the
+representational potential of GUIs, we investigate the ability of
+four Neural Image Captioning models to predict natural language
+descriptions of varying granularity when provided a screenshot
+as input. We evaluate these models quantitatively, using common
+machine translation metrics, and qualitatively through a large-
+scale user study. Finally, we offer learned lessons and a discussion
+of the potential shown by multimodal models to enhance future
+techniques for automated software documentation.
+Index Terms—Software Documentation, Image Captioning,
+Deep Learning
+I. INTRODUCTION & MOTIVATION
+Proper documentation is generally considered to be an inte-
+gral component of building and distributing modern software
+systems. In fact, past studies have illustrated the general ben-
+efits of documentation during the development lifecycle [1],
+[2], [3], [4] and the importance of technical documentation
+to software maintenance and evolution [5]. However, despite
+the value of well-documented systems, modern development
+processes and constraints often lead to the disregard or aban-
+donment of a range of documentation tasks [6], [5], [2], [7],
+[8], [1]. These difficulties have given rise to a wealth of
+research on automated techniques that aim to ease the burden
+on stakeholders by generating various types of documentation
+for a given task. For example, existing approaches have been
+developed to automatically generate natural language sum-
+maries and documentation for code [9], [10], [11], [12], [13],
+[14], [15], APIs [16], [17], unit tests [18], bug reports [19],
+[20], release notes [21], [22], and commit messages [23], [24],
+among other artifacts [25], [26].
+Generally, existing techniques for automated software doc-
+umentation have been concerned with modeling relationships
+that exist between two primary information modalities: code
+and natural language (NL). Unfortunately, reasoning between
+these two information sources is difficult due to the lexical
+gap resulting from the often disparate conceptual associations
+that connect source code lexicon and the more abstract words
+and phrases used in NL descriptions
+[27], [28]. Recently,
+this lexical gap was acknowledged as an information inference
+problem in a report made by Robillard et al. [29], wherein
+key research challenges exist in (i) inferring undocumented
+program properties, and (ii) discovering latent abstractions
+and rationales. These challenges suggest that overcoming the
+semantic disconnect between code and NL may require new
+knowledge sources that encode distinct program properties
+typically absent from traditional software or NL lexicon.
+One source of information which has been left largely
+unexplored for the purposes of automated documentation is
+visual software data encoded into Graphical User Interfaces
+(GUIs). GUI-based applications predominate modern user-
+facing software, as can be readily seen in the popularity of
+desktop and mobile apps [30]. Furthermore, high quality ap-
+plications with well-designed GUIs allow technically-inclined
+users to instinctively understand underlying program features.
+Thus, intuitively, certain functional properties of applications
+are encoded into the visual, pixel-based representation of the
+GUI such that cognitive human processes can determine the
+computing tasks provided by the interface. This suggests that
+there are latent patterns that exist within visual GUI data
+which indicate the presence of natural use cases capturing core
+functionality [31].
+Given the inherent representational power of GUIs in con-
+veying program related information, we set forth the following
+hypothesis that serves as the basis for work in this paper:
+The representational power of graphical user interfaces to
+convey program-related information can be meaningfully
+leveraged to support automated techniques for software doc-
+umentation.
+While most existing work on automated documentation con-
+cerns itself with the dichotomy between code and NL, we posit
+that the latent information encoded within GUIs can aid in
+bridging the existing semantic documentation gap by providing
+arXiv:2301.01224v1  [cs.SE]  3 Jan 2023
+
+an additional source of knowledge that inherently reflects pro-
+gram functionality. In fact, GUI-based representations of soft-
+ware have the potential to address the two challenges set forth
+by Robillard et al. [29]. More specifically, GUIs can aid in
+inferring undocumented program properties that are inherently
+represented within the design of GUI controls or widgets (e.g.,
+capturing a feature which is otherwise poorly represented by
+low-quality code identifiers/comments). Further, GUIs could
+be used as source to mine abstractions or rationales that
+would otherwise remain obscure (e.g., providing a use case-
+based explanation of a block of code connected to a GUI
+screen). In overcoming these challenges, we see GUI-centric
+documentation having an impact on the following types of
+software documentation:
+Technical Documentation: Developers utilize technical docu-
+mentation, such as code comments or READMEs, in order
+to learn about the functionality and interfaces of software
+to support engineering tasks. Automatically generating such
+documentation accurately is a challenging inference problem.
+However, it has been shown that GUI-related code can com-
+prise as much as half of the code in user facing programs [32].
+This means that graphical software data is connected in some
+way to large portions of GUI-based software projects i.e.,
+through GUI-event handlers, or code stipulating GUI layouts
+such as html. Therefore, if automated techniques are able to
+effectively infer salient functionality from the GUIs, they could
+be combined with existing techniques and leveraged to provide
+automation to developers, such as comment generation or code
+summarization with greater feature-based context awareness.
+As we illustrate in this paper, GUI code/metadata appears to
+encode orthogonal information compared to visual GUI data
+(i.e., screenshots), which suggests that we may be able to infer
+documentation information from visual GUI data that likely
+can’t be inferred from GUI code alone.
+User Documentation: Developers typically provide users with
+documentation such as tutorials or walkthroughs to help
+clearly illustrate software features. While some experienced
+users can infer functionality directly from a GUI, end-users
+exhibit a range of technological expertise, and many rely upon
+various forms of end-user documentation [33]. Thus, building
+techniques capable of automatically generating such documen-
+tation would free up development effort for other critical tasks,
+such as bug fixing. Beyond typical user facing software aids,
+GUI-centric program documentation could also enable entirely
+new classes of automated accessibility features, which are
+sorely needed for mobile apps [34]. For example, rather than a
+typical text-to-speech engine, one could envision a screen-to-
+functionality engine that could aid a motor-impaired user with
+navigating the software, without extra development effort.
+To investigate the potential of automated GUI-centric soft-
+ware documentation, we offer one of the first comprehensive
+empirical investigations into this new research direction’s most
+fundamental task: generating a natural language descrip-
+tion given a screenshot (or screen-related information) of
+a software GUI. Given that this task underlies the various
+potential applications discussed above, we view this as a
+logical first step towards investigating the feasibility of fu-
+ture techniques. To accomplish this, we collect and analyze
+a dataset for Comprehending visuaL semAntics to pRedict
+applicatIon functionalTY (the CLARITY dataset) consisting
+of 45,998 functional descriptions of 10,204 screenshots of
+popular Android apps available on Google Play. We provide
+a descriptive analysis of this dataset that investigates the
+“naturalness” and semantic topics of the collected descriptions
+by measuring cross-entropy compared to other corpora and
+performing a topic modeling analysis. To learn functional
+descriptions of the screens from this dataset, we customize,
+train, and test four Deep Learning (DL) models for neural
+image captioning—three that learn from image data and one
+that learns from textual GUI metadata—to predict functional
+descriptions of software at different granularities. We evaluate
+the efficacy of these models both quantitatively, by measuring
+the widely used BLEU metric, and qualitatively through a
+large-scale user study. In summary, this paper’s contributions
+are as follows:
+• We collect the CLARITY dataset of GUIs annotated
+with 45,998 functional, NL descriptions from 10,204
+screenshots of popular Android apps. The NL captions
+were obtained from human labelers, underwent several
+quality control mechanisms, and contain both high- and
+low-level descriptions of screen functionality. While other
+GUI datasets exist [35], [36], the CLARITY dataset differs
+by providing an extensively labeled set of screens, akin
+to Flickr8K [37] or MSCOCO [38];
+• We illustrate the underlying, natural patterns that exist in
+the CLARITY dataset through topic modeling.
+• We provide an extensive quantitative and qualitative eval-
+uation of four tailored DL models for image captioning
+using standard metrics and a large scale user study;
+• We offer an online appendix with examples of model-
+generated descriptions and experimental data [39]. Our
+dataset, trained models, code, and evaluation scripts are
+open source and accessible via the appendix.
+II. BACKGROUND
+A. The Connection between Images and NL
+The task of image captioning is much more difficult than
+that of classification or labeling, as an effective model must
+be able to both learn salient features from images automati-
+cally and semantically equate these features with the proper
+NL words and grammar that describe them. This task of
+semantically aligning two completely different modalities of
+information has led to the development of multimodal DL
+architectures that jointly embed NL and pixel-based infor-
+mation in order to predict an appropriate description of a
+given input image. These techniques are typically trained
+on large-scale datasets that contain images annotated with
+multiple captions, such as MSCOCO [38], and have largely
+drawn inspiration from encoder-decoder neural language mod-
+els traditionally applied to machine translation tasks. In this
+2
+
+…
+…
+…
+Input Image
+CNN or RCNN
+BRNN 
+ or LSTM
+xt
+yt
+W
+st
+v
+Image “Encoder”
+NL “Decoder”
+Fig. 1: Generalized overview of multimodal DL architectures
+for image captioning (with RCNN)
+paper, we adapt three recent architectures for image caption-
+ing, neuraltalk2 [40], the im2txt [41], and the show,
+attend and tell (SAT) [42] frameworks to predict func-
+tional descriptions of software screenshots through the use of
+custom pre-training and fine-tuning procedures. Additionally,
+we explore the seq2seq neural language model.
+DL models for image captioning build upon the success
+of encoder-decoder neural language models. The im2txt
+framework treats image captioning as a machine translation
+problem, wherein the source “sentence” is an image, and
+the target “translation” is a NL description. The generalized
+architecture of such models is shown in Fig. 1. As illustrated,
+these architectures replace the encoder RNN with a Convolu-
+tional Neural Network (CNN), which have been shown to be
+highly capable of learning rich image features [43], [44], [45].
+Google’s implementation of im2txt uses a Long-Short Term
+Memory (LSTM) RNN [46] for the “decoder” module, which
+has also proven extremely effective when applied to machine
+translation tasks. The decoder module of the neuraltalk2
+architecture is composed of a Bidirectional RNN (BRNN) [47]
+as opposed to an LSTM. Finally, the show, attend, &
+tell (SAT) model [42] uses an LSTM decoder but with
+the addition of an attention mechanism that can “attend” to
+salient parts of the image representation by combining “hard”
+and “soft” attention mechanisms.
+III. OVERVIEW
+In this section, we provide an “at-a-glance” overview of the
+data-collection procedures and various analyses performed in
+this paper. Figure 2 illustrates the four major components of
+the paper. The first major task of our study is to derive a
+suitable dataset of screenshot-caption pairs. We describe this
+process in two parts: (i) the collection of screenshots (Sec.
+IV-A), and (ii) the collection of captions from human workers
+(Sec. IV-B). The result of this data-collection effort is the
+CLARITY dataset, which contains 45,998 captions of 10,204
+Android screenshots. Next, we aim to understand the lexical
+properties of our captions through an empirical analysis in
+order to better understand how easily they might be modeled
+(Sec. V). Thus, we perform both a comparison of the the
+cross-entropy of language models trained our caption corpus
+to other popular SE corpora, and perform an LDA-based
+topic analysis. Next, we discuss the process of configuring
+and training three neural image captioning models, and one
+1   Clarity Dataset Collection
+(Screenshots + GUI metadata + Captions)
+2  Naturalness & Topic Analysis
+Cross-Entropy
+Analysis
+LDA-based 
+Topic Analysis
+…
+…
+…
+Input Image
+CNN or RCNN
+BRNN 
+ or LSTM
+xt
+yt
+W
+st
+v
+Image “Encoder”
+NL “Decoder”
+3   Train Image-Captioning and 
+Metadata Captioning Models
+Image-Captioning
+Models
+Metadata-Captioning
+Models
+4  !antitative and !alitative
+Model Evaluations
+“Ground Truth”
+Captions
+“Predicted”
+Captions
++
+Screens
+Large-Scale
+Human Evaluation
+!antitative  
+Evaluation 
+with BLEU
+Fig. 2: Overview of Dataset Collection and Analysis
+sequence-based model to predict functional descriptions of
+software GUIs (Sec. VI). Finally, we conclude our analysis
+by measuring the accuracy of our trained models according to
+both automated reference-based metrics (i.e., BLEU@n) and
+via a large-scale human evaluation. (Sec. VII)
+IV. DATASET COLLECTION
+A. Screen & GUI Metadata Collection
+The first step in deriving the CLARITY dataset is the
+collection of a sizable and diverse dataset of screenshots and
+GUI-metadata. We chose to focus this dataset derivation on the
+Android platform for three main reasons: (i) Android is cur-
+rently the most popular OS in the world [30], (ii) Android apps
+are highly GUI-and gesture driven, making them a suitable
+target for our investigation, and (iii) the Android screencap
+and uiautomator tools facilitate the extraction of screenshots
+and GUI-metadata from running apps. Fortunately, large-scale
+datasets of Android screenshots and metadata are publicly
+available in related literature [48], [35]. For this work, we took
+advantage of the REDRAW [48], [36] dataset which contains
+nearly 17k unique screenshots from 8,655 of the top-rated
+apps from the Google Play Store. It should be noted that
+another large-scale Android GUI dataset that contains a larger
+number of screenshots, RICO, is also available [35]. However,
+we chose to utilize the REDRAW dataset as it aligned with
+one of our primary study objectives. That is, we aim to learn
+latent feature information from both screenshots and GUI-
+metadata. However, for the GUI-metadata to properly align
+with the displayed content on a screen, the app must make use
+of native Android components. Therefore, apps that primarily
+display their information using web technologies, so-called
+hybrid apps, would obscure the GUI-metadata and impact
+our study findings. The REDRAW dataset contains a set of
+screenshots that underwent several stages of filtering to remove
+instances of hybrid apps along with other noise. Furthermore,
+the REDRAW dataset contains a set of GUI-component images
+3
+
+2:04
+Q Search
+Stories
+ Play All
+Add
+Your Story
+George
+Amanda
+Colby
+[因
+What's on your mind?
+Photo
+Guillermo Moreno with Josephine
+Williams and 2 others.
+Yesterday at 10:14 PM · 
+Good friends, good food and a lot of laughs
+Colby Harris and 23 others
+2 CommentsHello
+4
+again!
+Password
+I forgot
+ENTER
+Don't have an account?Register
+wordsearch
+Animals-Countries-Cifies
+PLAY
+Uspresidents-Trademarks0000
+三12:34
+Kids A-Z
+Teacher Username
+No Username? Start Here!日#
+12:27
+The Dollar in Mexico
+=
+updated: Jun 19, 2017
+Order by:
+Sell
+V
+Select the bank of your choice
+BAsE
+Banco
+Buy: 17.4129
+Sell: 17.8129
+ BANCODE MEXICO
+Buy: 17.9895
+Sell: 17.9945
+*Interbank dollar to 48 hours
+OBANCO AZTECA
+Buy:
+16.95
+Sell:
+18.01
+HSBC 
+Buy:
+17.68
+Sell:
+18.17
+BANORTE
+Buy:
+16.85
+Sell:
+18.25
+Ixe
+Buy:
+16.85
+Sell:
+18.25
+monex
+Buy:
+17.67
+Sell:
+18.28
+Banamex
+Buy:
+17.50
+Sell:
+18.30
+INBURSA
+Buy:
+17.70
+Grupo Financierc
+Sell:
+18.30
+ Santander
+Buy:
+17.50
+Sell:
+18.30
+BBVA
+Bancomer
+Buy:
+17.16
+Sell:
+18.35
+B BANCO DEL BAJIO
+Buy:
+17.40
+Sell:
+18.40
+ Scotiabank
+Buy:
+16.80
+Sell:
+18.42
+Bx+
+Buy:
+17.50
+Sell:
+18.50
+BANREGIO
+Buy:
+17.40
+Sell:
+18.502:04
+Q Search
+Stories
+ Play All
+Add
+Your Story
+George
+Amanda
+Colby
+[因
+What's on your mind?
+Photo
+Guillermo Moreno with Josephine
+Williams and 2 others.
+Yesterday at 10:14 PM · 
+Good friends, good food and a lot of laughs
+Colby Harris and 23 others
+2 Comments000
+</>AHigh Level Caption
+The screen allows the user to 
+look at clothing categories
+Low Level Captions
+The top le! icon allows the user 
+to access the menu
+The top right icon allows the 
+user to access the shopping cart
+The center list of categories allows 
+the user to make a selection
+The heart icon to the le! of the 
+shopping cart allows the user to 
+view favorites
+Fig. 3: Example of captions from the CLARITY dataset.
+labeled with their corresponding types (e.g., Button) which
+we utilize later in our study (Sec. VI). The end result of this
+filtering process was a total set of 17,203 candidate screens for
+labeling. We refer readers to the REDRAW paper for complete
+details of the filtering process [48].
+B. Collection of Functional Descriptions
+Once we derived a suitable set of screens, we needed to
+manually label these screens with functional captions. This
+process occurred in two steps: (i) first, we conducted a pilot
+labeling study in order to develop and prove out a tagging
+methodology suitable for large scale caption collection; (ii)
+second, we performed a full scale data collection study using
+Amazon’s Mechanical Turk Crowd-worker platform to collect
+over 10k screens with functional descriptions.
+1) Caption Granularity: Intuitively, GUIs encode func-
+tional information at multiple levels of granularity. For exam-
+ple, if you were to ask a user or developer what the high-
+level purpose of a given screen is, they might say “This
+screen allows users to browse clothing categories”, as shown
+in Fig. 3. These types of descriptions constitute the “high-
+level” functionality of a given screen. However, a single screen
+rarely implements only one functionality, and there may be
+multiple functional properties that enable the screen’s high-
+level functional purpose. User descriptions of these types
+of functional properties are typically centered around the
+interactive components of a screen, since these represent the
+instances of actions (e.g., users “doing something”) that are
+easily attributed to implemented functions. For example, in
+the screen in Fig. 3, underlying functions include viewing
+favorites, accessing a shopping cart, or selecting an item from
+a list. These types of “low-level” screen properties centered
+around GUI-components describe key constituent functional-
+ity. Hence, in order to capture a holistic functional view of
+each screen, we tasked participants with labeling each screen
+with one “high-level” functional caption, and up to four “low-
+level” functional captions. Fig. 3 shows these two categories
+using actual captions collected as part of the CLARITY dataset.
+2) Pilot Data Collection Study: We developed an initial
+image caption collection platform using a Java-based web
+application. Using this system, the authors manually labeled
+743 screens with the caption granularities described earlier.
+During this study, we discovered some instances of screens
+with relatively little information displayed on them, making
+it difficult to label them with functional attributes, even after
+the filtering techniques discussed previously. Therefore, before
+moving onto the large-scale caption collection with Mechani-
+cal Turk (MTurk), at least one author manually inspected each
+of the 17,203 candidate screens, and discarded those with a
+severe lack of functionality. This resulted in a set of 16,311
+candidate screens for the next phase of the study.
+3) Mechanical Turk Data Collection Study: To set up our
+large-scale data-collection process, we adapted our web ap-
+plication caption collection mechanism to work with MTurk’s
+crowd worker platform. This involved configuring a Human
+Intelligence Task (HIT) that provided workers with a set of
+detailed instructions, displaying a screenshot from our dataset
+alongside text entry boxes for one high-level functional caption
+and up to four low-level functional captions (a limit was
+imposed to normalize the amount of time workers would spend
+on the HIT). This study was approved by the Institutional
+Review Board of the authors’ affiliated institution.
+Given that we aimed to collect high-quality functional
+descriptions of screens in natural English, we targeted MTurk
+users from primarily English speaking countries that had
+completed at least 1,000 HITs and had a HIT approval rate
+of at least 90%. We provided a detailed set of instructions
+for labeling images with captions that clearly explained the
+concept of high-level and low-level captions with examples,
+and provided users with explicit instructions as well as DOs
+and DONTs for the labeling task. The full set of instructions is
+available in our online appendix [39]. With regard to caption
+quality, we specifically had three major requirements: (i) that
+the caption describes the perceived functionality of a screen
+and not simply its appearance, (ii) that spatial references are
+given for low-level captions (e.g., “the button in the top-left
+corner of the screen”), and (iii) that captions be written in
+complete English sentences with reasonably proper grammar.
+We published batches of HIT tasks by sampling unique
+screens from our set of 16,311 candidate screens, ensuring
+that no user was assigned the same screen twice. The quality
+of work from crowd-sourced tasks is not always optimal,
+so as captions were submitted, they needed to be vetted for
+quality. Thus, the captions for each screen were examined by
+at least one author for the three quality attributes mentioned
+above. If an author was unsure about whether a screen met
+these quality attributes, it was reviewed by at least one other
+author to reach a consensus. In total, 2,419 screens were
+rejected and republished as new HITs due to quality issues.
+In summary, 2,150 MTurk workers collected 45,998 captions
+(across granularities) for 10,204 screens (≈5 screens per
+participant), and over $2,400 was paid out.
+V. EMPIRICAL DATASET ANALYSIS
+The CLARITY dataset provides a rich source of data for
+exploring the relationship between GUI-based and lexical
+4
+
+#？
+日9:47
+asos
+三
+HOME
+CATEGORIES
+NEWIN:CLOTHING
+ACTIVEWEAR
+TALL
+JEANS
+SHOES&SNEAKERS
+-SHIRTS
+SUNGIASSESTABLE I: LDA topics learned over high-level captions k = 15
+Assigned Label
+Top 7 Words
+”color options”
+screen show app option color book differ
+”login or create acccount”
+user screen allow account log creat app
+”select image from a list”
+user screen allow select view list imag
+”map search by location”
+screen locat search map user show find
+TABLE II: LDA Topics learned on low-level captions k = 25
+Assigned Label
+Top 7 Words
+”page button”
+page button top center bottom side left
+”select date”
+avail date select one option theme present
+”camera button”
+video imag photo pictur bottom camera
+”privacy policy banner”
+titl just term blue banner privaci polici
+software data. However, it is important to investigate the
+semantic makeup of the collected captions in order to better
+understand: (i) the latent topics they capture as well as (ii)
+their naturalness and, hence, predictability. In this section we
+carry out an empirical analysis of this phenomena guided by
+the following two Research Questions (RQs):
+• RQ1: What are the latent topics captured within the high-
+and low-level captions in the CLARITY dataset?
+• RQ2: How natural (i.e., predictable) are the high- and
+low-level captions in the CLARITY dataset?
+A. Analysis Methodology
+1) RQ1: Investigating Dataset Topics: To investigate the
+latent topics in the CLARITY dataset, we learned topic models
+over caption corpora representing different granularities of
+functional descriptions. More specifically, we applied Latent
+Dirichlet Allocation (LDA) [49] to both segmented high-
+and low- level captions from the CLARITY dataset. In our
+analysis, the set of captions for a specific screenshot in the
+CLARITY dataset represents a document, and the entire set
+of captions across screenshots for a given granularity (i.e.,
+high or low level) constitutes a corpus. LDA has several
+configurable hyper-parameters that impact the smoothing of
+generated topics. These include k, the number of topics,
+n which denotes the number of iterations of the sampling
+algorithm (Gibbs sampling [50], in our case), as well as α
+and β which impact topic distributions. We set α and β to
+standard values for NL corpora, set n to 500, which proved to
+be a sufficient for model convergence, and varied k between
+15, 25, 50, and 75 topics.
+2) RQ2: Analyzing the Naturalness of GUI Descriptions:
+Past work has pioneered the notion of the naturalness of
+software [51], which illustrated the fact that software, even
+more so than NL, exhibits repetitive patterns that make it
+predictable. This finding was recently further investigated and
+the existence of certain natural patterns was confirmed [52]. To
+illustrate naturalness, these past studies have learned statistical
+n-gram language models over software corpora, and measured
+the “perplexity” (or a log-transformed version, cross-entropy)
+of these models, which represents the degree to which a model
+is “surprised” by the patterns on a test corpus when trained on
+a corpus from the same domain. A model with lower measured
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+N-gram order
+Cross-Entropy
+High-Level Captions
+Low-Level Captions
+Java Raw Code
+Java w/o Syntax Tokens
+Stack Overflow
+Guntenberg
+Fig. 4: Cross-entropy of the CLARITY dataset’s high and low-
+Level captions compared to other corpora.
+cross-entropy represents a higher predictive power, and thus,
+a more natural underlying corpus.
+We follow the methodology of these past studies to explore
+the naturalness of the CLARITY dataset captions. Thus, similar
+to the methodology for the previous RQ, we split the collected
+captions into two corpora, one for the high-level descriptions,
+and one for the low-level descriptions. We then learned inter-
+polated n-gram models, using the mitlm [53] implementation
+of Kneser-Ney smoothing [54], which has been shown to be
+the most effective n-gram smoothing method [51], following
+a ten-fold cross-validation procedure. We report the average
+cross-entropy values across these experiments for both the high
+and low-level corpora, compared to prior results [51], [52] for
+other NL and software corpora.
+B. Analysis Results
+1) RQ1: Results of Dataset Topic Modeling: We present
+selected results of some of the most representative topics in
+Tables I & II, complete with descriptive labels that we provide
+for readability, and include all the results in our appendix [39].
+These topics help to provide a descriptive illustration of some
+of the latent patterns that exist in both the high and low level
+CLARITY captions. The high-level captions illustrate several
+screen-level topics, including searching on a map and adjusting
+app settings. The low-level captions conversely capture topics
+that describe component-level functionality, such as date selec-
+tors, camera buttons, and back buttons. These results indicate
+the existence of logical topics specific to the domain of GUIs
+in our collected captions.
+2) RQ2: The Naturalness of Clarity Descriptions: The
+results of our naturalness analysis are illustrated in Figure 4.
+This figure shows the average cross entropy of the high- and
+low- level captions from the CLARITY dataset compared to
+several other corpora as calculated by Rahman et al. [52].
+More specifically, the graph depicts the average ten-fold cross
+entropy for: (i) The Gutenberg corpus containing over 3k
+English books written by over a hundred different authors,
+(ii) Java code from over 134 open source projects on GitHub,
+(iii) Java without Syntax Tokens (i.e., separators, keywords,
+and operators), and (iv) a Stack Overflow corpus consisting of
+only the English descriptions from over 200k posts.
+5
+
+13
+high
+12
+low
+11
+javaraw
+java withoutsyntaxtokens
+10
+stack overflow
+9
+gutenberg
+entropy
+8
+cross
+6
+5
+4
+3
+2
+1
+1
+2
+m
+4
+5
+6
+7
+8
+9
+10
+n-gram lengthAs described earlier, the lower the cross-entropy is for a
+particular dataset, the more natural it is. That is, the corpora
+that exhibit lower cross entropy tend to exhibit stronger latent
+patterns that can be effectively modeled and predicted. As we
+see from Fig. 4, the CLARITY high and low level captions are
+more natural than every dataset excluding raw Java code. It
+should be noted that, comparatively, there are several factors
+that could account for the observed lower cross entropy of the
+CLARITY captions. For instance, such factors could include
+other corpora having a larger size or having a more diverse
+set of human authors and writing styles. However, we mainly
+provide entropy measures of other datasets to provide context
+for the predictability of the CLARITY dataset compared to
+other popular corpora. Regardless of dataset differences, the
+average ≈ 5 bits of entropy measured for the two datasets of
+CLARITY captions signals that our collected descriptions ex-
+hibit strong semantic patterns that can be effectively modeled
+for prediction. Additionally, we observe that the cross-entropy
+for the high and low-level captions are surprisingly similar.
+Intuitively, one might expect that the low-level CLARITY
+captions would exhibit more prevalent patterns due to the
+repetitive use cases of certain GUI-components such as menu
+buttons. This indicates the tendency of both datasets to exhibit
+patterns that can be appropriately modeled. However, as we
+illustrate in Sec. VII the ability for GUI-related information
+to predict captions differs according to granularity.
+VI. DEEP LEARNING FUNCTIONAL DESCRIPTIONS FROM
+SOFTWARE GUIS
+The results of the analysis from the previous section demon-
+strate the presence of the latent patterns in the CLARITY
+dataset of screenshots and captions. In this section, we detail
+our methodology for investigating the capability of different
+customized DL models to learn these patterns to predict
+functional descriptions from two GUI representations.
+A. Clarity Dataset Segmentation
+We collected two different granularities of captions from
+users to derive the CLARITY dataset (Sec. IV-B). For the
+experiments in this section, we want to explore the model’s
+ability to learn both high- and low-level functional descrip-
+tions. Thus, we split the CLARITY dataset into two groups,
+one containing only high level captions, and one containing
+only low level ones. We also created a third dataset combining
+the high and low level captions, in order to explore whether the
+predictive capabilities of the models improved by aggregating
+multiple granularities. It should be noted that each low-level
+caption was treated as a single caption (i.e. each low-level
+caption was treated as a separate data point) as is convention
+with datasets containing multiple captions [38]. Each screen
+in the dataset has both an associated screenshot and a GUI-
+metadata file. In order to make for a fair comparison of
+performance across various model configurations, we created
+consistent training, validation and test partitions (80%, 10%,
+10% according to the number of images/GUI metadata files) to
+be used across models. The NL text used as input to the models
+TABLE III: Image Captioning Model Configs. used in Study
+Model
+Identifier
+Caption Config.
+Model Config.
+im2txt-h-imgnet
+High
+im2txt-l-imgnet
+Low
+im2txt-c-imgnet
+Combined
+inception v3 trained on
+imagenet
+im2txt-h-comp
+High
+im2txt-l-comp
+Low
+im2txt-c-comp
+Combined
+Inception v3 fine-tuned on
+Component Dataset
+im2txt-h-fs
+High
+im2txt-l-fs
+Low
+im2txt
+im2txt-c-fs
+Combined
+Inception v3 fine-tuned on
+Full Screen Dataset
+ntk2-h-imgnet
+High
+ntk2-l-imgnet
+Low
+ntk2-c-imgnet
+Combined
+VGGNet pre-trained on
+ImageNet
+ntk2-h-ft
+High
+ntk2-l-ft
+Low
+NeuralTalk 2
+ntk2-c-ft
+Combined
+VGGNet pre-trained on
+ImageNet with Fine
+Tuning
+sat-h
+High
+sat-l
+Low
+SAT
+sat-c
+Combined
+VGGNet pre-trained on
+ImageNet
+TABLE IV: Subset of Model Hyper-paramters
+Hyperparameter
+im2txt
+NeuralTalk2
+SAT
+Seq2Seq
+Batch Size
+64
+16
+17
+64
+Embedding Size
+512
+512
+512
+128
+Decoder RNN Size/Units
+512
+512
+1024
+128
+Optimizer
+SGD
+SGD
+Adam
+Adam
+Initial Learning Rate
+2
+2
+0.001
+0.0001
+Dropout Probability
+0.7
+0.7
+0.3
+0.8
+was preprocessed according to the specific requirements for
+each model implementation [55], [56], [57].
+B. Image Captioning Model Configurations
+We customize, train, and test the three neural image
+captioning models, im2txt, neuraltalk2, and show,
+attend, & tell (SAT) (Sec. II-A), on the screenshots
+and captions of the CLARITY dataset. We choose to explore
+these three models due to their different underlying design
+decisions related to the type of utilized CNNs and RNNs
+(Sec. II-A), as these differences may affect their performance
+in our domain. It should be noted that in the course of our
+experiments, we make several customizations to these models
+through adaptions to pre-training and fine-tuning procedures.
+However, given the typical number of parameters that consti-
+tute these models, the training time can be quite prohibitive,
+even on modern hardware. Thus, to control our experimental
+complexity and investigate a number of model configurations
+that can be trained in a reasonable amount of time, we fix
+the values of the hyper-parameters for each model in our
+experiments. We derived our utilized hyper-parameter values
+by conducting random searches for optimal values of certain
+parameters, and chose optimal parameters reported in prior
+work for others. While we fix the hyper-parameters for these
+models, we instead customize the configurations of our image
+captioning models at the architectural level. Specifically, we
+investigate how training the “encoder” CNN using different
+datasets and training procedures effects the efficacy of the
+model predictions. This type of analysis allows us to more
+effectively flush out broader patterns related to the benefits and
+drawbacks of model design decisions. In the end, we trained
+more than 15 different configurations of the models (see Table
+III) over several machine months of computation.
+1) im2txt
+Model
+Configurations
+&
+Training:
+For
+im2txt, we adapted Google’s open source implementa-
+6
+
+tion of the model in TensorFlow [55]. Given the incredibly
+large number of parameters that need to be trained for the
+im2txt model, performing even relatively simple hyper-
+paramter searches proved to be computationally prohibitive
+for our experiments. Therefore, for this model we utilized the
+optimal set of parameters reported by Vinyals et al. [41] on
+similarly sized datasets. A subset of these hyper-parameter
+values are given in Table IV, whereas full configuration details
+can be found in our appendix. The publicly available imple-
+mentation of Google’s im2txt model utilizes the Inception
+v3 [58] image captioning architecture as its encoder CNN.
+In past work, the inception model weights were initialized
+by training on the large-scale image classification dataset
+ImageNet [59], which contains “commonplace” image cate-
+goires. However, given that we are applying these models to
+very particular domain (predicting descriptions of software)
+it is unclear if an Inception v3 model trained on the broader
+ImageNet dataset would capture subtle semantic patterns in
+the CLARITY dataset. Therefore, we explored three different
+model configurations to explore this phenomena: one with
+Inception v3 pre-trained on ImageNet, and two with Inception
+v3 fine-tuned on domain specific-datasets. The first domain
+specific image dataset we utilize is the ReDraw cropped
+image dataset outlined in Sec. IV-A, which contains over 190k
+images of native Android GUI-components labeled with their
+type (e.g., Button, TextView). The second domain specific
+image dataset we use consists of the full screenshots from the
+CLARITY dataset, labeled with their Google Play categories.
+2) NeuralTalk2 Model Configurations & Training: For
+neuraltalk2, we adapted Karpathy et al.’s implementation
+written in Torch and lua [56]. We performed a brief random-
+ized hyper-parameter search for this model, given its more
+efficient training time, using the optimal im2txt parameters as
+a starting point. The optimal values resulting from this search
+are provided in Table IV. For its CNN decoder, neuraltalk2
+makes use of a VGGNet [44] architecture pre-trained on
+the ImageNet [59] dataset. Unlike our im2txt configurations,
+we explore the effect of jointly fine-tuning neuraltalk2’s
+CNN and RNN. Thus, we explore two configurations of
+neuraltalk2, one that jointly fine tunes the pre-trained
+VGGNet on the CLARITY dataset, and one that does not
+perform fine-tuning. We followed a training procedure similar
+to that of our im2txt models, in that we trained our models
+on the high, low, and combined CLARITY caption training
+data for 500K iterations, saving model checkpoints every 2K
+iterations.
+3) Show, Attend and Tell Model Configurations & Train-
+ing: For the SAT model, we adapted the open-source imple-
+mentation of the model in Tensorflow [60]. The hyperparam-
+eters that we used to train our model are shown in Table IV.
+The implementation used VGG16 [44] as its encoder CNN.
+We trained the SAT model on the CLARITY dataset for the
+low, high and combined captions for 500K iterations and kept
+the checkpoints after every 1K iterations. Note that due to
+the prohibitive training cost of this model, we did not explore
+using a fine-tuned VGGNet as we did with neuraltalk2.
+TABLE V: Metadata Captioning Model Congfigurations
+Model
+Identifier
+Caption Config.
+Model Config.
+seq2seq-h-type
+High
+seq2seq-l-type
+Low
+seq2seq-c-type
+Combined
+Trained on GUI
+Component Types
+seq2seq-h-text
+High
+seq2seq-l-text
+Low
+seq2seq-c-text
+Combined
+Trained on
+GUI-Component Text
+seq2seq-h-tt
+High
+seq2seq-l-tt
+Low
+seq2seq-c-tt
+Combined
+Trained on
+GUI-Component Type +
+Text
+seq2seq-h-ttl
+High
+seq2seq-l-ttl
+Low
+Seq2Seq
+seq2seq-c-ttl
+Combined
+Trained on
+GUI-component Type +
+Text + Location
+C. Metadata Captioning Model Configurations
+To explore the ability to translate between the lexical
+representations of GUI-metadata and NL functional descrip-
+tions, we train and test an encoder-decoder neural language
+model using Google’s seq2seq [57] framework. Note that
+recent work has proposed new models that take advantage of
+structural text properties [61], however, implementations of
+such models are generally not available, hence we leave the
+study of more advanced models for future work. We chose
+to utilize the default general-purpose architecture and hyper-
+parameters for this model, as they have been shown to be
+effective across a wide-range of machine translation tasks [62].
+More specifically, our encoder network consists of a BRNN
+with Gated Recurrent Units (GRUs) and our decoder network
+consists of an RNN with LSTM units; hyperparameters are
+listed in Table IV.
+To investigate the representative power of different attributes
+included in Android GUI-metadata, we create four config-
+urations of GUI-metadata consisting of different attribute
+combinations (Table V). We chose to utilize these attribute
+combinations as they represent (i) the attributes that are most
+likely to have values, and (ii) represent a wide range of
+information types (e.g., displayed text, component types, and
+spatial information). Note that seq2seq did not consistently
+converge for the high level caption dataset, thus we do not
+report these results. Consistent with the training of the other
+models, our implementation of the seq2seq model was
+trained to 500k iterations, with checkpoints every 2k iterations.
+VII. DEEP LEARNING MODEL EVALUATION
+To explore our core hypothesis set forth at the beginning
+of this paper, and evaluate our DL models described in
+Sec. VI, we perform a comprehensive empirical evaluation
+with two main goals: (i) intrinsically evaluate the predictive
+power of the models according to a well accepted machine
+translation effectiveness metric, and (ii) extrinsically evaluate
+the models by examining and rating the quality of the pred-
+icated functional NL descriptions. The quality focus of this
+evaluation is our studied models’ ability to effectively predict
+accurate, concise, and complete functional descriptions. To aid
+in achieving our study goals, we define the following RQs:
+• RQ3: How accurate are our model’s predicted NL de-
+scriptions?
+• RQ4: How accurate, complete, & understandable are our
+model’s predicted NL descriptions from the viewpoint of
+evaluators?
+7
+
+TABLE VI: BLEU Score Evaluation Results for Models
+Model
+Capt.
+Model Type
+Bc
+B1
+B2
+B3
+B4
+High
+im2txt-h-fs
+12.4
+24.8
+12.6
+6.7
+5.3
+Low
+im2txt-l-comp
+27.0
+45.6
+31.8
+20.0
+10.1
+im2txt
+Comb.
+im2txt-c-comp
+30.3
+51.7
+35.9
+22.1
+11.6
+High
+ntk2-h-imgnet
+13.3
+27.4
+13.5
+7.3
+5.3
+Low
+ntk2-l-ft
+27.4
+47.5
+32.8
+19.5
+9.6
+NeuralTalk2
+Comb.
+ntk2-c-ft
+30.1
+52.1
+36.0
+21.8
+10.8
+Low
+seq2seq-l-type
+18.1
+44.6
+17.0
+7.9
+0.24
+seq2seq
+Comb.
+seq2seq-c-type
+16.9
+38.9
+14.7
+6.0
+0.08
+High
+sat-h
+17.7
+30.1
+18.3
+12.9
+9.8
+Low
+sat-l
+35.0
+52.5
+38.7
+28.1
+20.7
+SAT
+Comb.
+sat-c
+37.7
+56.8
+42.0
+30.5
+22.0
+NeuralTalk2
+Trained on Flickr8K
+34.0
+57.9
+38.3
+24.5
+16.0
+NeuralTalk2
+Trained on MSCOCO
+40.7
+62.5
+45.0
+32.1
+23.0
+im2txt
+42.6
+66.6
+46.1
+32.9
+24.6
+SAT
+45.7
+71.8
+50.4
+35.7
+25.0
+A. Evaluation Methodology
+1) RQ3: Empirically Evaluating Model Accuracy: To
+evaluate the accuracy of our trained model’s generated cap-
+tions, we follow past work [40], [41] and report BLEU
+scores [63] of the predicted captions on the shared CLARITY
+test set of images and GUI-metadata. The BLEU score is a
+standard metric used in machine translation research that mea-
+sures the textual similarity between a predicted caption (the
+output from a model) and a reference caption (the collected
+descriptions from humans in the CLARITY test set). The BLEU
+score can be measured according to the similarity of different
+subsequence lengths (i.e., BLEUn), and we report BLEU1
+through BLEU4, as well as a composite score calculated as the
+average of these, as is convention [40], [41]. For the image
+captioning models, we use the coco-caption implementation
+of the BLEU score adapted for the CLARITY test set. For
+each test image across all image captioning models, three
+captions were generated using a beam width of 3 for the
+beam search across candidate predictions. The seq2seq models
+were evaluated in the same manner. We chose to utilize a
+beam width of 3 as an initial qualitative examination of our
+models’ predictions showed this size to achieve a reasonable
+balance between prediction accuracy and model confidence.
+For the high-level captions, the three candidate captions were
+compared to the reference, and the overall average BLEUn
+scores were calculated for each model. For the low-level and
+combined captions, the predicted captions and reference cap-
+tions were compared in a pairwise manner and overall average
+BLEUn scores were calculated for each model configuration.
+2) RQ4: Human Perceptions of Predicted Captions: To
+qualitatively evaluate our studied model’s generated captions,
+we performed a large-scale study involving an additional 220
+participants recruited from MTurk. We randomly sampled
+220 screens from the CLARITY test set, and then predicted
+high, low, and combined captions for them using the opti-
+mal configurations of im2txt, NeuralTalk2, and seq2seq
+according to the composite BLEU score for each model
+and caption level combination. The SAT captions were not
+included in this study due to time constraints related to the
+model’s training. We created a HIT wherein each participant
+viewed 11 screenshots paired with captions. Two of the 11
+captions were reference high and low to serve as a control,
+while the other 9 captions came from the model predictions.
+Screens and caption pairs were arranged into HITs such that
+1) no single HIT had two of the same screenshot, 2) each
+of the 11 types of captions (2 reference, 9 model) were
+included only once per HIT. The order of these captions was
+randomized per HIT to prevent bias introduced by identical
+caption ordering between HITs. By this arrangement, each
+screen-caption pair was evaluated by 11 participants. After
+viewing these screenshot-caption pairs, participants were asked
+to answer six evaluation questions. Three of these questions
+(EQ1-EQ3) were adapted from prior work that assessed the
+quality of automatically generated code summaries [21], and
+inquired about accuracy, completeness, and understandability,
+respectively. The three remaining questions (EQ4-EQ6), were
+free response and asked participants to explain accuracies,
+inaccuracies, and improvements. The full set of questions and
+HIT are in our online appendix [39]. Similar to the CLARITY
+dataset collection, each participant’s response was thoroughly
+vetted by at least one author, and discarded if the answers
+were incomplete. Responses were collected until 220 HITs
+were completed by unique respondents.
+B. Evaluation Results
+1) RQ3 Results: Evaluating BLEU Scores: We illustrate
+the BLEU score results for the most effective model config-
+uration and checkpoint across all of our trained models in
+Table VI, whereas the results for other model configurations
+can be found in our online appendix [39] in addition to
+caption examples. The cells highlighted in blue illustrate the
+highest performing model configuration for each caption type.
+In general we observe that SAT exhibits the highest overall
+BLEU scores across all caption granularities. We speculate
+that this is attributable to the addition of the advanced attention
+mechanism in this model that is able to “focus” on varying
+image regions or features to effectively handle multiple caption
+granularities. In general, the seq2seq model performed quite
+poorly across the varying caption types, indicating a lower
+tendency for rich representation. Perhaps most interestingly,
+we see that the optimal model configurations for the im2txt
+framework were those where the CNN was conditioned on
+domain specific datasets. More specifically, the best high-level
+caption model was conditioned on full screenshots and the
+best low-level caption was conditioned on the cropped GUI-
+component screenshots.
+Another general trend that emerges is the low-level and
+combined caption models tend to exhibit higher overall BELU
+scores compared to the high-level captions. This is somewhat
+intuitive, as it indicates that there are more natural connections
+between visual GUI and lexical patterns in the low-level
+captions, compared to the high-level captions that reflect more
+abstract functional descriptions. When examining the captions
+generated by the optimal configurations of each model, it is
+clear that im2txt and SAT produces a more diverse set of out-
+put captions than neuraltalk2, which could be considered
+as more useful in many software documentation tasks.
+Finally, it is worth discussing how the BLEU scores of our
+models compare to those of the same models trained on the
+8
+
+None
+Some
+A Lot
+im2txt high
+im2txt low
+im2txt combined
+Easy to Read
+Somewhat
+Readable
+Hard to 
+Read
+im2txt high
+im2txt low
+im2txt combined
+EQ3: Understandability
+EQ2:  Unnecessary Information
+im2txt high
+im2txt low
+im2txt combined
+Strongly 
+Disagree
+ Disagree
+Neutral
+Agree
+Strongly
+Agree
+EQ1: Accuracy
+seq2seq high
+seq2seq low
+seq2seq combined
+Fig. 5: Responses across models for EQ1-EQ3
+more traditional Flickr8k [37] and MSCOCO [38] datasets
+given at the bottom of Table VI. Given the data-intensive
+nature of our DL models, and the much larger size of the
+MSCOCO dataset (≈123k images, each with 5 captions), we
+did not expect our models trained on the CLARITY dataset to
+outperform those trained on MSCOCO. Thus, unsurprisingly,
+we observe that on average, im2txt, neuraltalk2, and SAT
+models trained on the MSCOCO dataset outperform the same
+models trained on the CLARITY datasets by ≈ 10 BLEU score
+points for the combined and low level captions, and ≈ 27
+points on high-level captions. However, when we examine
+the performance of Neuraltalk2 on the more similarly sized
+Flickr8K dataset (≈ 8K images, each with 5 captions) we
+observe comparable performance to the CLARITY low-level
+and combined datasets, with the SAT model narrowly outper-
+forming the Flickr8K neuraltalk2 model, with a slightly
+bigger discrepancy for the high-level captions. Overall, these
+results indicate that when compared with datasets of similar
+size, DL models trained on the CLARITY dataset exhibit
+similar performance.
+2) RQ4 Results: Human Evaluations: The results of EQ1-
+EQ3 for the model configurations with the best performance
+during the human study, in addition to the seq2seq accuracy
+scores, are summarized in Fig. 5. Complete results across all
+model configurations can be found in our online appendix. The
+responses to EQ4-EQ6 varied by the type of caption, and are
+provided in our appendix in full. Generally, im2txt fared the
+best in terms of accuracy, and was followed by neuraltalk2
+and seq2seq respectively. For im2txt, despite mixed reac-
+tions from participants, in many cases respondents verified that
+the caption was accurate (e.g., ”The description accurately
+describes the screen, it is in fact a terms and conditions
+screen.”) and suggested minor improvements similarly to the
+reference captions (e.g., ”It could add specifics about what the
+settings pertain to (i.e. security)”). As illustrated in Fig. 5 the
+im2txt predictions were consistently rated as being readable
+and containing relevant information. It is also interesting to
+note that there appears to a mismatch between the performance
+as indicated by BLEU scores, and human perceptions, with the
+participants consistently rating the im2txt captions better
+than other models across EQ1-EQ3, despite neuraltalk2
+achieving a higher BLEU score for two model configurations.
+VIII. DISCUSSION & LEARNED LESSONS
+Lesson 1: Functional Descriptions of GUIs exhibit a
+high degree of naturalness and can be modeled using DL
+techniques. We observed that DL models trained on the low-
+level and combined datasets exhibit similar performance to
+models trained on general image captioning datasets of similar
+size (e.g., Flickr8K). This indicates that GUI screenshots could
+be used to augment approaches for automated documentation.
+Lesson 2: GUI-centric software documentation mod-
+els benefit from being pre-trained on domain specific
+GUI data, as opposed to general image datasets (e.g.,
+MSCOCO) The qualitative results of our model analysis
+illustrate that for im2txt, the most effective configurations
+were those trained on domain specific CNN datasets. This
+suggests a perceptible difference between the utility of image
+features learned from general datasets, compared to those
+learned on datasets more specific to software. This suggests
+that future work aiming to leverage DL models for GUI-
+centric program documentation should look to collecting and
+extracting features from large-scale GUI-related datasets.
+Lesson 3: Future automated approaches for GUI-centric
+program documentation would likely benefit from com-
+bining the orthogonal semantics of screenshots and GUI-
+metadata. Our evaluation in this paper illustrates that the rep-
+resentational power of screenshots appears to be superior when
+applied to a software documentation task. However, given stark
+differences between these two modalities of information, we
+also observed that they encode orthogonal semantic patterns
+that could be combined for more effective documentation
+generation. One property we observed of certain captions
+generated by the image-based models was the effect of their
+limited vocabulary. For example, certain predicted captions
+similar to the following: “The screen allows the user to select
+a <UNK>”, wherein the UNK token represents missing token,
+which should be mapped to some unobserved app property,
+such as a “album cover” or “store location”. However, such
+predictions could be combined with the vocabulary present in
+GUI metadata to help predict more complete, and accurate
+descriptions. Thus, a promising direction for future work is to
+jointly encode both screenshots and lexical GUI-metadata.
+Lesson 4: Training image captioning models to predict
+specific or diverse pieces of functionality is difficult.
+Practical models for GUI-centric documentation should able
+to predict both specific pieces of information (e.g. the func-
+tionality of a particular button for a given method handler),
+and diverse functionality (being able to generate descriptions
+of functionality anywhere on a given screen). However, one
+aspect we observed across our models is that the most
+common observed types of functionality (e.g., back buttons,
+menu buttons) corresponded to the functionalities that our
+9
+
+seq2seq high
+seq2seq low
+0
+0
+seq2seq combined
+0
+0models predicted most often and most confidently on unseen
+screenshots. This is somewhat expected, as the models saw the
+most examples of such functionalities during training. Thus,
+the diversity of predictions is an open problem for future
+research. This problem can be partially mitigated by larger,
+more diverse datasets with specifically curated descriptions
+(such as extensions to the CLARITY dataset). However, it is
+likely that domain-specific models, or ensembles of models,
+may be required to more effectively predict diverse features.
+Lesson 5: Future studies that evaluate automated GUI-
+centric documentation approaches should include human
+studies, as human perceptions of models may differ from
+automated reference-based metrics. One of the more surpris-
+ing results of our study is that there seems to be a mismatch
+between humans perceptions of the captions generated by our
+DL models and the BLEU score metrics typically used to asses
+the accuracy of model predictions. This signifies that there are
+aspects of human perception that are not effectively captured
+in the BLEU metric, and possibly other translation metrics.
+IX. LIMITATIONS & THREATS TO VALIDITY
+Internal Validity. Threats to internal validity correspond to
+unexpected factors in the experiments that may contribute to
+observed results. To derive our dataset we rely on MTurk and
+its workers to extract the high- and low-level descriptions per
+each screenshot. It should be noted that we did not ask MTurk
+workers to provide technical software documentation descrip-
+tions, but rather general descriptions of screen functionality at
+differing granularities. To minimize low quality captions we
+published the jobs for workers with more than 1k HITS, from
+English speaking countries, and HIT approval rate of more
+than 90%. Also, each successfully completed HIT was vetted
+by at least one of the authors to assure quality. If there was
+any question related to caption quality, at least one of the other
+authors stepped in to resolve the ambiguity. As a result 2,429
+HITs were rejected due to low quality descriptions.
+External Validity. Threats to external validity concern the
+generalization of the results. As with any collected dataset,
+there is a threat to external validity about the generalizability
+of the CLARITY dataset. However, we used a diverse set
+of popular apps from the Android domain, extracted popular
+screenshots from these apps, and the apps were captioned by
+a large and diverse set of MTurk workers. During our data
+collection process, we only collected 4 low-level captions per
+each screen in order to make the task feasible for MTurk
+workers as workers tend to abandon or perform poorly on long
+tasks. This means that, for certain screens with many GUI-
+components, some components may lack natural language
+descriptions. However, given the size of our dataset and the
+diversity of our screenshots and captions, we assert that our
+low-level captions are reasonably representative.
+X. RELATED WORK
+DL for Image Captioning and GUIs. Hossain et al. [64]
+recently performed a wide-ranging study on DL models for
+image captioning, surveying the many different architectures
+and datasets used to evaluate them. However, this survey
+did not examine the ability of any image captioning model
+to predict functional descriptions of software. There have
+been a limited number of papers in the SE community that
+have applied DL techniques to GUI related data. Chen et
+al. [65] designed an approach that uses an NMT to translate
+an Android screenshot into a GUI-skeleton. However, their
+technique is able to predict GUI structure given an image, not
+functional natural language descriptions. Recently, Zhang et.
+al. [66] created a dataset of iOS image captions to train a
+model for captioning accessibility data. However, the authors
+do not make their dataset publicly available and target a
+different goal of accessibility data compared our goal of
+generating functional captions. Chen et al. investigated the
+use of DL image captioning models for applying labels to
+GUI-components in mobile apps [67], however, this approach
+only aims to predict short labels for a limited subset of
+GUI-components, whereas our study focuses upon predicting
+functional descriptions consisting of complete sentences for
+both individual GUI-components and entire screenshots.
+GUI-based Analysis of Mobile Apps. GVT and GCat analyze
+the visual properties of GUIs to detect design violations and
+evolutionary changes [68], [69]. In contrast, we focus solely on
+image captioning techniques to provide functional program de-
+scriptions of screenshots. Approaches such as REMAUI [70],
+REDRAW [48], and pix2code [71] aim to automatically
+generate mobile app code given an app screenshot. Conversely,
+we leverage DL techniques to generate functional descriptions
+rather than source code using a pixel-based image as input.
+Chen et al. [72] introduced StoryDroid, for automatically
+generating visual storyboards of Android apps to help aid
+in the app design process. However, their approach is not
+capable of generating a functional description of an application
+from GUI data. Furthermore, Deka et al. showed how the
+Rico dataset could be navigated via semantic search using
+autoencoders
+[35]. UiRef [73] is an approach for resolving
+security and privacy concerns by considering semantics of
+GUI-components that request user’s inputs. Moreover, Liu et
+al. [74] presented an approach for automatically classifying
+mobile app icons according to semantic GUI patterns. Xiao et
+al. proposed IconIntent that combines program analysis and
+icon classification to detect privacy sensitive GUI-components
+[75]. Different from this body of work, we aim to predict
+functional descriptions of GUIs for software documentation.
+XI. CONCLUSION
+In this paper, we have conducted one of the first com-
+prehensive empirical investigations into the connection be-
+tween GUI-related information, and functional descriptions
+of programs. We have derived the CLARITY dataset of GUI
+screenshots/metadata and NL captions, trained DL models
+on this dataset, and demonstrated their ability to bridge the
+semantic gap between visual and lexical program information.
+ACKNOWLEDGMENT
+This work was supported by the NSF CCF-2007246 &
+CCF-1955853 grants. Any opinions, findings, and conclusions
+expressed herein are the authors’ and do not necessarily reflect
+those of the sponsors.
+10
+
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+arXiv:2301.08636v1  [math.NA]  20 Jan 2023
+CONVERGENT APPROACHES FOR THE DIRICHLET MONGE-AMPÈRE
+PROBLEM
+Hajri Imen 1 Fethi Ben Belgacem 2
+Abstract. In this article, we introduce and study three numerical methods for the Dirichlet
+Monge-Ampère equation in two dimensions. The approaches consist in considering new equivalent
+problems. The latter are discretized by a wide stencil finite difference discretization and monotone
+schemes are obtained. Hence, we apply the Barles-Souganidis theory to prove the convergence of
+the schemes and the Damped Newtons method is used to compute the solutions of the schemes.
+Finally, some numerical results are illustrated.
+Monge-Ampere, Monotone scheme, Newton method.
+1. Introduction
+We are interested in the numerical solution of the Monge-Ampère equation with Dirichlet bound-
+ary condition
+(1.1)
+(MAD)
+
+
+
+
+
+det
+�
+D2u (x)
+�
+= f (x) , for x in Ω,
+u (x) = ϕ (x) , for x on ∂Ω,
+u is convex.
+Where Ω is a convex bounded domain in R2, with boundary ∂Ω, (D2u) , is the Hessian of the
+function u, f and ϕ are given functions.
+We take the simplest boundary conditions. For more general operator of Monge-Ampère and
+other boundary conditions, we mention for instance [12]. The convexity constraint is crucial for
+the (MAD). It is required for the Monge-Ampère equation to be degenerate elliptic and for (MAD)
+to have a unique solution. It is also needed for numerical stability. The Monge-Ampère equation,
+has extensive applications, it is strictly related to the “prescribed Gauss curvature” problem, see
+for instance [12]. It appears also in affine geometry, precisely, in the affine sphere problem and the
+affine maximal surfaces problem, this was discussed in [5, 6, 25, 27, 28, 29]. Other applications
+appear in fluid mechanics, geometric optics, and meteorology : for example, in semigeostrophic
+equations, the Monge-Ampère equation is coupled with a transport equation, this is pointed out
+in [12]. The analysis of the regularity of the Monge-Ampere equation is essential in the study
+1Higher
+Institute
+of
+Applied
+Studies
+in
+Humanities
+of
+Mahdia,5121
+Mahdia,
+Tunisia,
+Email:hajri.imene2017@gmail.com.
+2Laboratory of partial differential equations (LR03ES04), ISIMM, University of Monastir, El Manar, TUNISIA.
+Email: fethi.benbelgacem@isimm.rnu.tn
+1
+
+2
+CONVERGENT APPROACHES FOR THE DIRICHLET MONGE-AMPÈRE PROBLEM
+of the regularity of the transp ort problem. This, latter, has been employed in many areas. We
+only briefly mention [8, 9, 4] for mesh geneartion,[15, 16, 17]for image registration, and [12] for
+reflector design. Developing an efficient numerical method has aroused a lot of interest, and large
+standard techniques have been proposed. A first method to do so was introduced in [24] by using
+a discretization of the geometric Alexandrov-Bakelman interpretation of solutions. Variational
+approaches have been presented in [10, 11], more precisely, the augmented Lagrangian approach
+and the least-squares approach. But these methods needed more regularity than can be predicted
+for solutions. A different approach was studied in [18], using the vanishing moment method. The
+periodic case was treated in [14].
+Although, the standard techniques, mentioned above, work well for smooth solutions, and they
+fail for singular solutions, for more details, see, for instance, the discussion in [2]. To overcome
+these difficulties, we have to use the notion of viscosity solution or Alexsandrov solution. In two
+dimension, a numerical method was introduced in [24], which is geometric in nature, and converges
+to the Alexsandrov solution. The method introduced in [22]„ in two dimension and improved in
+[19] for higher dimension, uses the wide stencil scheme that converges to the viscosity solution,
+which we briefly describe for this reason in the end of this section.
+The following variant of the AM-GM inequalities, is the keystone of our formulation introduced
+here.
+For A and B two symmetric matrices, such that, A, B ≥ 0. We have the following inequality
+2
+�
+det (AB) ≤ Tr (AB) .
+Where for symmetric matrices M ≥ 0 means xT Mx ≥ 0.
+Remark 1. We can deduce from the above inequality that for a smooth convex solution u of (1.1),
+one can deduce the following inequality
+∆u − 2
+�
+f ≥ 0.
+Let us define the function
+˜g := ∆u − 2
+�
+f.
+It is then straightforward to check that if u is a smooth solution of (1.1), then is indeed a solution
+of the linear Dirichlet Poisson problem
+(1.2)
+�
+P˜g� �
+∆u = 2√f + ˜g,
+u|Γ = ϕ,
+which can be easily descretized by any method of choice if the function ˜g is known.
+We finish this remark by mentioning that the convexity constraint is essential to ensure unique-
+ness (for example, u and −u are both solution of the Monge Ampère equation). For viscosity
+solution, this constraint can be required by the equation
+(1.3)
+λ1
+�
+D2u
+�
+≥ 0,
+
+CONVERGENT APPROACHES FOR THE DIRICHLET MONGE-AMPÈRE PROBLEM
+3
+in the viscosity sense, see for instance [21, 22], where λ1 (D2u) is the smallest eigenvalue of the
+Hessian of u. However, for a twice continuously differentiable function u, the convexity restriction
+is equivalent to requiring that the eigenvalues of the Hessian, D2u, are positives, which is approved
+by considering the linear Poisson Dirichlet problem
+�
+P˜g�
+.
+The approaches that we follow, in the present paper, are inspired by the idea developed in [3]
+and the wide stencil finite difference discretization introduced in [22] and [19] for viscosity solution
+of M-A equation in two and higher dimensions that relies on a framework developed in [1]. For
+clarity, we recall the full result in the next section.
+2. Viscosity solution and convergence theory of approximation schemes
+2.1. Degenerate elliptic equations. Let F (x, r, p, X) be a continuous real valued function de-
+fined on Ω × R × Rn × Sn, with Sn being the space of symmetric n × n matrices. Consider the
+nonlinear, partial differential equation with Dirichlet boundary conditions,
+�
+F (x, u (x) , Du (x) , D2u (x)) (x) = 0
+for x in Ω
+u (x) = g (x)
+for x in ∂Ω.
+Where Ω is a domain in Rn, Du and D2u denote the gradient and Hessian of u, respectively.
+Definition 2. [19]The equation F is degenerate elliptic if
+F (x, r, p, X) ≤ F (x, s, p, Y ) whenever r ≤ s and Y ≤ X.
+Where Y ≤ X means that Y − X is a nonnegative definite symmetric matrix.
+The viscosity solution for the Monge-Ampère equation is defined in [22].
+Definition 3. Let u ∈ C (Ω) be convex and f ≥ 0 be continuous. The function u is a viscosity
+subsolution (supersolution) of the Monge-Ampère equation in Ω if whenever convex ϕ ∈ C2 (Ω)
+and x0 ∈ Ω are such that (u − ϕ) (x) ≤ (≥) (u − ϕ) (x0) for all x in a neighborhood of x0, then we
+must have
+det
+�
+D2φ (x0)
+�
+≥ (≤) f (x0) .
+The function u is a viscosity solution if it is both a viscosity subsolution and supersolution.
+For the existence and uniqueness of viscosity solution for (1.1), we mention the next result in
+[7],
+Theorem 4. Let Ω ⊆ Rd be abounded and strictly convex, g ∈ C (∂Ω) , f ∈ C (Ω) , with f ≥ 0.
+Then there exists a unique convex viscosity solution u ∈ C
+� ¯Ω
+�
+of the problem (1.1).
+The advantage of considering viscosity solutions come from the following fundamental theorem,
+obtained in [1], which gives conditions for convergence of approximation schemes to viscosity
+solution.
+
+4
+CONVERGENT APPROACHES FOR THE DIRICHLET MONGE-AMPÈRE PROBLEM
+Theorem 5. (Convergence of Approximation Schemes). Consider a degenerate elliptic equation,
+for which there exist unique viscosity solutions. A consistent, stable approximation scheme con-
+verges uniformly on compact subsets to the viscosity solution, provided it is monotone.
+By the previous theorem, we need just a way to build a monotone finite difference schemes,
+which represents a new challenge. In the sequel, we recall here the basic framework introduced in
+[20], for building a monotone scheme.
+Firstly, a finite difference equation take the form
+F i [u] = F i (ui, ui − uj|i̸=j) .
+We say that a scheme is degenerate elliptic if the following holds [20]:
+Definition 6. The scheme F is degenerate elliptic if F i is non-decreasing in each variable.
+We are now ready to present the following theorem in [20]:
+Theorem 7. Under mild analytic conditions, degenerate elliptic schemes are monotone, and non-
+expansive in the uniform norm. The iteration
+(2.1)
+um+1 = um + dtF (um) ,
+is a contraction in L∞ provided dt ≤ K (F)−1 , where K (F) is the Lipschitz constant of the scheme,
+regarded as a function from RN −→ RN.
+We end this paragraphre by the next result, proven in [20]
+Theorem 8. A proper, locally Lipschitz continuous degenerate elliptic scheme has a unique solu-
+tion which is stable in the l∞ norm.
+2.2. Wide stencil schemes. We finish this section by noting that wide stencil schemes are re-
+quired to build consistent, monotone schemes of degenerate second order PDEs (see discussion in
+[22]). Wide stencil schemes were built for the two-dimensional Monge-Ampère equation in [22]
+and for the convex envelope in [21]. Each approach considered here is a function of eigenvalues of
+the Hessian. To fully discretize the equation (4.1) for the eigenvalues of the Hessian on a finite
+difference grid, we approximate the second derivatives by centered finite differences; this is the
+spatial discretization, with parameter h. We consider also a finite number of possible directions ν
+that lie on the grid; this is the directional discretization, with parameter dθ. The spatial resolution
+is improved by using more grid points, the directional resolution is improved by increasing the size
+of the stencil. So, a wide stencil is needed (see Fig 2.2)
+3. First Formulation of the (MAD) in two dimensions (method A)
+3.1. An equivalent problem. Let us begin with a simple approach to illustrate the ideas. We
+can rephrase, for instance, the (MAD) as the following:
+(3.1)
+�
+Find a positive function g, such that
+det (D2ug) = λ1 [D2ug] × λ2 [D2ug] = f,
+
+CONVERGENT APPROACHES FOR THE DIRICHLET MONGE-AMPÈRE PROBLEM
+5
+Figure 2.1. Grid for wide stencil 17 points, in two dimension
+where: ug is the solution of�
+∆ug = λ1 [D2ug] + λ2 [D2ug]
+= 2√f + g,
+ug
+|Γ
+= ϕ.
+We are now ready to state a first example of our approches.
+Lemma 9. Provided the solution, u, of (1.1) is in H2, there exists a unique positive function
+˜g ∈ L2, such that u = u˜g, where u˜g is the solution of (P˜g). Conversely, if u¯g is solution of (3.1)
+for some ¯g > 0, then u¯g = u.
+Proof. Let u be a solution of (1.1). From the above, one can see easily, that u = u˜g.
+Conversely, if u¯g is a solution of (3.1), we can clearly see that
+�
+det (D2u¯g) = f > 0
+∆u¯g ≥ 0.
+It follows that u¯g is convex and satisfies (1.1).
+□
+Remark 10. We notice that according to the result in [26], we have equivalence of viscosity and
+weak solutions for the Poisson problem. This motivates us to build a convergent scheme to the
+viscosity solution of Poisson problem
+�
+P˜g�
+through the discretization of the (MAD) problem. The
+viscosity solution u˜g of
+�
+P˜g�
+will be equivalent to the weak solution of (MAD) problem in the
+distributional sense.
+4. Discretization of the problem (3.1)
+Let us consider a regular and uniform cartesian grid, consider the stencil at the reference point
+x0 consist of the neighbors x1, ..., xN (as in Figure 1). We can define vi in polar coordinates by
+vi = xi − x0 = hivθi.
+
+6
+CONVERGENT APPROACHES FOR THE DIRICHLET MONGE-AMPÈRE PROBLEM
+We assume that the stencil is symetric and we define the local spatial resolution and the directional
+resolution respectively by
+¯h (x0) = max
+i
+hi
+and
+dθ = max
+θ∈[−π,π] min
+i
+|θ − θi|.
+First, the problem (3.1) can be written as function of the eigenvalues of the Hessian. We will
+then start by discretizing λ1 and λ2. Hence by a simple substitution we obtain the scheme for (3.1).
+We recall that the smallest and the largest eigenvalues of a symmetric matrix can be represented
+respectively by the Rayleigh-Ritz formula
+(4.1)
+λ1
+�
+D2u
+�
+(x) = min
+θ
+d2u
+dν2
+θ
+,
+λ2
+�
+D2u
+�
+(x) = max
+θ
+d2u
+dν2
+θ
+,
+where νθ = (cos θ, sin θ)is the unit vector in the direction of the angle θ.
+This formula was used in [22] to build a monotone scheme in two dimension for the (MAD).
+We begin by building monotone schemes for λ1 and λ2 on a wide stencil uniform grid. These
+operators are used to give schemes for all formulations in this paper.
+We discretize the eigenvalues of the Hessian by the following formula.
+(4.2)
+λh,dθ
+1
+�
+D2ug�
+(x) = min
+i
+ug (x + vi) − 2ug (x) + ug (x − vi)
+|vi|2
+and
+(4.3)
+λh,dθ
+2
+�
+D2ug�
+(x) = max
+i
+ug (x + vi) − 2ug (x) + ug (x − vi)
+|vi|2
+.
+Lemma 11. The schemes (4.2) and (4.3) are degenerate elliptic.
+Proof. We follow the same as in [22].
+Since each discrete second derivative in the direction vi is the average of the terms which have
+the form ug
+j − ug
+i , they are non-decreasing in ug
+j − ug
+i . Taking a minimum (or maximum) of non-
+decreasing functions furnishes a non-decreasing function.
+□
+We finally substitute (4.2) and (4.3) in (3.1) to obtain the wide stencil finite difference scheme
+of (3.1)
+(4.4)
+�
+Find a positive function gi, such that
+λh,dθ
+1
+�
+D2ugi�
+× λh,dθ
+2
+�
+D2ugi�
+= f i,
+
+CONVERGENT APPROACHES FOR THE DIRICHLET MONGE-AMPÈRE PROBLEM
+7
+with
+�
+λh,dθ
+1
+�
+D2ugi�
++ λh,dθ
+2
+�
+D2ugi�
+= 2
+�
+f i + gi.
+ug
+|Γ
+= ϕ.
+Where f i = f (xi) and gi = g (xi) .
+Lemma 12. The scheme (4.4) is degenerate elliptic.
+Proof. From the properties of nondecreasing functions, obtained in [22],
+that if G : R2 → R is a nondecreasing function, and if F1 and F2 are degenerate elliptic finite
+difference schemes, then so is F = G (F1, F2) . It is also clear that the discretization f i = f (xi)
+and gi = g (xi) does affect the ordering properties. We conclude that (4.4) is degenerate elliptic.
+□
+In the following, for simplicity, we omit the index i when there is no ambiguity.
+Definition 13. We say the scheme Hh,dθ is consistent with the equation (MAD) at x0 if for every
+twice continuously differentiable function ϕ (x) defined in a neighborhood of x0, Hh,dθ(ϕ) (x0) →
+H (ϕ) (x0) as h, dθ → 0. The global scheme defined on Ω is consistent if the limit above holds
+uniformly for all x ∈ Ω. (The domain is assumed to be closed and bounded).
+Lemma 14. The consistency holds for (4.2) and (4.3) and so for (4.4).
+Proof. Let x0 be a reference point with neighbors x1, ..., xN, and direction vectors vi = xi − x0, for
+i = 1, ..., N, arranged symmetrically, if vi is a direction vector, then so is −vi. By Taylor series one
+has
+ug (x0 + vi) − 2ug (x0) + ug (x0 − vi)
+|vi|2
+= d2ug
+dv2
+i
++ O
+�
+h2
+i
+�
+.
+Let M given symetric 2 × 2 matrix, that we can take it diagonal. Set vθ a unit vector. It follows
+from [22] (Lemma 3) that
+min
+θ∈{θ1,...,θN} vT
+θ Mvθ = λ1 + (λ2 − λ1) O
+�
+θ2�
+.
+Which implies that
+λ1 (ϕ) (x0) − λh,dθ
+1
+(ϕ) (x0) = O
+�¯h2 + (λ2 − λ1) dθ2�
+and thus consistency holds for (4.2).
+Similar argument gives consistency for (4.3) and so for
+(4.4).
+□
+Theorem 15. Suppose that unique viscosity solutions exist for the equation (3.1) Then the finite
+difference scheme given by (4.4) converges uniformly on compacts subsets of Ω to the unique
+viscosity solution of the equation.
+Proof. We need to verify consistency and monotonicity. Consistency follows from Lemma 14 and
+monotonicity follows from Lemma 12.
+□
+
+8
+CONVERGENT APPROACHES FOR THE DIRICHLET MONGE-AMPÈRE PROBLEM
+Finally, the scheme yields a fully nonlinear equation defined on grid functions. We perform the
+iteration (2.1) and by Theorem 7 will converge to a fixed point which is a solution of the equation.
+This approach is used in [22].
+5. TWO METHODS OF FIXED POINT
+5.1. The first method (Method B). Notice that from Lemma 9 if u is a solution of (1.1)
+u (x, y) = u˜g (x, y) it follows that det (D2u) = det
+�
+D2u˜g�
+, where u˜g is the solution of (1.2) for
+˜g ∈ L2.
+By writing
+△u˜g = 2
+�
+f + ˜g =
+�
+(∆u˜g)2 + 2 (f − det (D2u˜g))
+and expanding
+�
+∆u˜g�2 =
+�
+u˜g
+xx
+�2 +
+�
+u˜g
+yy
+�2 + 2u˜g
+xxu˜g
+yy we have
+△u˜g =
+��
+u˜g
+xx
+�2
++
+�
+u˜g
+yy
+�2
++ 2
+�
+u˜g
+xy
+�2
++ 2f = 2
+�
+f + ˜g
+Let us define the operator Q : L2 (Ω) → L2 (Ω) for Ω ⊂ R2 by
+Q (g) :=
+�
+(ug
+xx)2 + (ug
+yy)2 + 2 (ug
+xy)2 + 2f − 2
+�
+f,
+with ug solution of (Pg) . So, one has
+Lemma 16. ˜g is a fixed point of Q.
+Proof. It follows from above expansions.
+□
+5.1.1. The scheme. We consider the following scheme
+gn+1 = Q (gn) =
+�
+(ugn
+xx)2 + (ugn
+yy)2 + 2 (ugn
+xy)2 + 2f − 2
+�
+f.
+With initial value g0 > 0 close to zero and ug0 is the solution of (P g0) .
+Remark 17. The advantage of this method by comparing it to that in [19] and [2] is that it
+guarantees, at least, at each iteration that tr (D2ugn (x)) > 0, which is necessary to check the
+convexity.
+Although this method turns out to be simple to implement is well suited in the case where ug
+is in H2 (Ω) . If not, the method may not converge.
+5.1.2. Algorithm.
+• g0 ≥ 0 (close to 0), solve (P g0) , ((ug)0 = ug0being known),
+• For n ≥ 0, compute gn+1 and (ug)n+1 as follows
+gn+1 = Q (gn) ,
+(ug)n+1 solution of
+�
+P gn+1�
+.
+
+CONVERGENT APPROACHES FOR THE DIRICHLET MONGE-AMPÈRE PROBLEM
+9
+Where, the method involves simply discretising the second derivatives using standard central dif-
+ferences on a uniform Cartesian grid, as a result
+D2
+xxuij
+=
+1
+h2 (ui+1,,j + ui−1,j−, 2ui,,j) ,
+D2
+yyuij
+=
+1
+h2 (ui,,j+1 + ui,,j−1 − 2ui,,j) ,
+D2
+xxuij
+=
+1
+4h2 (ui+1,,j+1 + ui−1,,j−1 − ui−1,,j+1 − ui+1,,j−1) .
+5.2. The second method (Method C). In the same setting we define the next operator.
+Definition 18. Let Ω a bounded domain in R2. Define the operator F : L2 (Ω) → L2 (Ω) ,
+by
+(5.1)
+F (g) =
+�
+|det [D2ug] − f| + g,
+where ug is a solution of
+(5.2)
+(Pg)
+�
+∆u = 2√f + g,
+u|Γ = ϕ.
+For g ∈ L2 (Ω), the operator F is well defined and it is easy to verify that
+Lemma 19. �g is a fixed point of the operator F.
+Proof. Let u a smooth solution of (1.1). It follows from Lemma 9 that u = u�g. Which implies that
+det
+�
+D2u�g�
+= det [D2u] = f and therefore, F (�g) = �g.
+□
+5.3. The scheme. We define the following scheme
+gn+1 = F (gn) =
+�
+|det [D2ugn] − f| + gn.
+With initial value g0 > 0 close to zero and ug0 is the solution of (P g0) .
+Remark 20. The method is advantageous, it simply involves evaluating derivatives and solving the
+Poisson equation that preserves the convexity constraint.
+5.3.1. Algorithm.
+• g0 ≥ 0 (close to 0), solve (P g0) , ((ug)0 = ug0being known),
+• For n ≥ 0, compute gn+1 and (ug)n+1 as follows
+gn+1 = α
+�
+|det [D2ugn] − f| + gn,
+with 0 < α < 1.
+(ug)n+1 solution of
+�
+P gn+1�
+.
+As in the above method, second derivatives are descretized using standard central differences on a
+uniform Cartesian grid.
+
+10
+CONVERGENT APPROACHES FOR THE DIRICHLET MONGE-AMPÈRE PROBLEM
+N
+Results in [19]
+Method A
+Method B
+Method C
+31
+2.44 × 10−4
+2.965 × 10−4
+4.226 × 10−4
+18 × 10−4
+45
+1.52 × 10−4
+3.052 × 10−4
+2.202 × 10−4
+18 × 10−4
+63
+9.06 × 10−5
+2.801 × 10−4
+1.190 × 10−4
+17 × 10−4
+89
+5.32 × 10−5
+8.035 × 10−4
+6.494 × 10−5
+17 × 10−4
+127
+3.06 × 10−5
+2.015 × 10−4
+3.888 × 10−5
+17 × 10−4
+Table 1. Errors
+��u − uN��
+∞ for the exact solution of the first example on an N ×N
+grid. We include results from the wide stencil methods of [19] on seventeen point
+stencils.
+−1
+−0.5
+0
+0.5
+1
+−1
+−0.5
+0
+0.5
+1
+1
+1.5
+2
+2.5
+3
+30
+40
+50
+60
+70
+80
+90
+100
+110
+120
+130
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+N
+CPU Time
+ 
+ 
+Method A
+Method B
+Method C
+Figure 6.1. Results for example 1 on an N × N grid and total CPU time versus
+N for the methods A, B and C.
+6. Numerical experiments
+The three methods are tested on three different examples (smooth or singular solutions). The
+discretization is done in the wide stencil Finite Difference method with 17- points (see Figure 2.2).
+The number of noeuds meshing is equal to N ∗ N with N = 31, 45, 63, 89, 127, the step of meshing
+h = L/N, with L is the length of the side of the rectangular domain Ω. The results obtained are
+compared with those in [19].
+In the first example we study the regular solution given by :
+u (x, y) = exp
+�(x2 + y2)
+2
+�
+with f (x, y) =
+�
+x2 + y2 + 1
+�
+exp
+�
+x2 + y2�
+.
+The Table 1 summarizes the obtained results for different meshing.
+In Figure 6 we show the surface plot of the solution and the total CPU time versus N for the
+methods A, B and C.
+
+CONVERGENT APPROACHES FOR THE DIRICHLET MONGE-AMPÈRE PROBLEM
+11
+N
+Results in [19]
+Method A
+Method B
+Method C
+31
+1.22 × 10−3
+5.806 × 10−4
+6.853 × 10−4
+8.794 × 10−4
+45
+5.9 × 10−4
+4.92 × 10−4
+6.719 × 10−4
+8.727 × 10−4
+63
+4.2 × 10−4
+4.914 × 10−4
+2.733 × 10−4
+8.601 × 10−4
+89
+2.6 × 10−4
+4.085 × 10−4
+2.09 × 10−5
+8.173 × 10−4
+127
+2.0 × 10−4
+4.056 × 10−4
+1.08 × 10−5
+8.164 × 10−4
+Table 2. Errors
+��u − uN��
+∞ for the exact solution of the second example on an
+N × N grid. We include results from the wide stencil methods of [19] on seventeen
+point stencils.
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.5
+1
+0
+0.05
+0.1
+0.15
+0.2
+30
+40
+50
+60
+70
+80
+90
+100
+110
+120
+130
+0
+50
+100
+150
+200
+250
+300
+N
+CPU Time
+ 
+ 
+Method A
+Method B
+Method C
+Figure 6.2. Results for example 2 on an N × N grid and total CPU time versus
+N for the methods A, B and C.
+As a second example, which is C1 , we take the one considered in [19] which is given by
+u(x, y) = 1
+2((
+�
+(x − 0.5)2 + (y − 0.5)2 − 0.2)+)2 with f(x, y) = (1 −
+0.2
+�
+(x − 0.5)2 + (y − 0.5)2)+.
+The results are in Table 2.
+Finally, we consider a third example which is singular at the bord of the domain Ω = [0, 1] ×
+[0.1] ,defined by
+u(x, y) = −
+�
+(2 − x2 − y2) where f(x, y) =
+2
+(2 − x2 − y2)2.
+.
+The results are illustrated in Table 3 and
+
+12
+CONVERGENT APPROACHES FOR THE DIRICHLET MONGE-AMPÈRE PROBLEM
+N
+Results in [19]
+Method A
+Method B
+Method C
+31
+1.74 × 10−3
+1.7 × 10−3
+5.1 × 10−3
+5.7 × 10−3
+45
+9.8 × 10−4
+1.5 × 10−3
+4.8 × 10−3
+5.5 × 10−3
+63
+5.9 × 10−4
+8.9 × 10−4
+3.9 × 10−3
+5.5 × 10−3
+89
+3.5 × 10−4
+8.9 × 10−4
+3.1 × 10−3
+5.5 × 10−3
+127
+2.0 × 10−4
+8.2 × 10−4
+2.4 × 10−3
+5.5 × 10−3
+Table 3. Errors
+��u − uN��
+∞ for the exact solution of the third example on an N ×N
+grid. We include results from the wide stencil methods of [19] on seventeen point
+stencils.
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.5
+1
+−1.5
+−1
+−0.5
+0
+30
+40
+50
+60
+70
+80
+90
+100
+110
+120
+130
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+1800
+N
+CPU Time 
+ 
+ 
+Method A
+Method B
+Method C
+Figure 6.3. Results for example 2 on an N × N grid and total CPU time versus
+N for the methods A, B and C.
+Acknowledgments
+We are indebted to Pr. Pierre-Emmanuelle Jabin for his relevant remarks and his impressive
+comments which have greatly improved this work.
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+.
+
+40
+50
+60
+70
+80
+90
+100
+110
+N
+
+40
+50
+60
+70
+80
+90
+100
+110
+N
+
+40
+50
+60
+70
+80
+90
+100
+110
+N
+
+40
+50
+60
+70
+80
+90
+100
+110
+N
+
+10
+20
+30
+40
+50
+60
+70
+
+40
+50
+60
+70
+80
+90
+100
+110
+N
+
+40
+50
+60
+70
+80
+90
+100
+110
+N
+
+20
+40
+60
+80
+100
+
+20
+40
+60
+80
+100
+
+40
+50
+60
+70
+80
+90
+100
+110
+N
+

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+arXiv:2301.11595v1  [math-ph]  27 Jan 2023
+Exact solutions of Maxwell equations in homogeneous
+spaces with group of motions G3(IX)
+V. V. Obukhov
+Institute of Scietific Research and Development, Tomsk State Pedagogical University
+(TSPU). Tomsk State Pedagogical University, 60 Kievskaya St., Tomsk, 634041, Russia;
+Laboratory for Theoretical Cosmology, International Center of Gravity and Cosmos, Tomsk
+State University of Control Systems and Radio Electronics (TUSUR), 36, Lenin Avenue, Tomsk,
+634050, Russia
+Keywords: Maxwell equations, Klein-Gordon-Fock equation, algebra of symmetry operators,
+theory of symmetry, linear partial differential equations.
+Exact solutions of Maxwell equations in homogeneous spaces with group of motions G3(IX)
+1
+Introduction
+All known methods of integration of main differential equations of mathematical physics are
+based on complete reduction of these equations to a system of ordinary differential equations.
+Reduction is carried out using symmetry operators. For the equations of motion of classical
+or quantum sample particle in external electromagnetic and gravitational fields the symmetry
+operators are integrals of motion. It is known that a necessary condition for the existence of
+integrals of motion is the existence of spacetime symmetry given by the Killing fields.
+Thus the problem of exact integration is closely related to the study of space-time symmetry.
+At present two methods of exact integration of equations of motion are known. These are
+methods of commutative (CIM) and noncommutative (NCIM) integration. The first method is
+based on the theory of complete separation of variables, and it is applicable in stackel spaces.
+Stackel spaces admit complete sets consisting of mutually commuting Killing fields. Theory
+of Stackel spaces was developed in [1], [2], [3], [4], [5], [6],[7].
+A description of the theory
+and a detailed bibliography can be found in [8], [9] [10], [13] (see also [12]). Solutions of field
+equations, which are still used widely in the theory of gravitation, have been constructed on
+the basis of Stackel spaces. These solutions are often used in the study of various effects in
+gravitational fields (see, for example,[14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25],
+[26]).
+The second method (NCI method) is based on the use of noncommutative algebras of
+symmetry operators linear in moments and constructed using vector Killing fields. The method
+was proposed in [27]. The development of the method and its application to gravity theory can
+be found in [28], [29], [30], [31].
+1
+
+As in stackel spaces in the spaces with a noncommutative group of motions the equations
+of motion of a test particle admit the complete reduction to a system of ordinary differential
+equations.
+Therefore, we will call space-time manifolds admitting noncommutative groups
+Gr, r ≥ 3 as post-Stackel spaces (PSS).
+By analogy with stackel spaces we will call the PSS non-isotropic if a group Gr (or its
+subgroup of rank 3) acts transitive on a non-isotropic hypersurface of spacetime, or isotropic,
+if the hypersurface is isotropic. For non-isotropic post-stackel spaces we will also use the term
+”homogeneous post-stackel space (HPSS)”.
+The same classification problems can be considered for the PSS as for the stackel spaces.
+For example, in the papers [9] [10], [11] a complete classification is given for the case when
+the Hamilton-Jacobi equation for a charged test particle admits the complete separation of
+variables in the external electromagnetic field. A similar classification problem has been solved
+for PSS-spaces as well. In [32] PSS-spaces with transitive four-parameter groups of motions
+are considered; in [33] HPSS-spaces are considered (see also [34]); in [35] PSS-spaces with
+groups acting on isotropic hypersurfaces of transitivity are considered. PSS-spaces with four-
+parameter groups of motions are considered in [36], provided that these groups have transitive
+three-parameter subgroups. Thus, one has found the potentials of all admissible electromag-
+netic fields, for which the Hamilton-Jacobi and Klein-Gordon-Fock equations have algebras of
+symmetry operators given by groups of motions of post-stackel spaces. It was proved, that
+the Klein-Gordon-Fock equation admits the algebra of symmetry operators given by groups of
+motions of PSS if and only if the Hamilton-Jacobi equations admits the appropriate algebra of
+integrals of motion.
+Next classification problem is the classification of electrovacuum solutions of the Einstein-
+Maxwell equations for the case, when CIM and NCIM methods are applicable. During the
+century-long history of General relativity, many exact solutions of the vacuum and electrovac-
+uum Einstein equations have been found (see, for example,[42] ). Nevertheless, this problem has
+not lost its relevance up to now. The main purpose of the classification is not so much to find
+new exact solutions, as to list all gravitational and electromagnetic fields, in which equations
+of motion of test particles can be exactly integrated or at least reduced to systems of ordinary
+differential equations. This problem divided into two stages.
+At the first stage all non-equivalent classes of solutions of the vacuum Maxwell equations for
+the potentials of admissible electromagnetic fields are found. At the second stage the obtained
+classification is used to classify the corresponding electrovacuum spaces. Historically, for Stackel
+spaces this problem was solved before the problem of the first stage (see the bibliography given
+in [9], [10], [11]). The present article is devoted to solving the first stage of this classification
+problem. All non-equivalent solutions of empty Maxwell equations in homogeneous spaces of
+type IX according to Bianchi’s classification are found.
+2
+Admissible electromagnetic fields in homogeneous spaces
+There are two definitions of homogeneous spaces.
+According to the first a spacetime
+V4
+is homogeneous if its subspace
+V3,
+endowed with the Euclidean space signature, admits
+coordinate transformations (forming the group G3(N) of motions of spaces V4), that allow to
+connect any two points in
+V3
+(see [38]). This definition directly implies that metric of the
+2
+
+V4 in the semi-geodesic coordinate system
+[ui]
+can be represented as follows:
+ds2 = −du02 + ηabla
+αlb
+βduαduβ,
+gij = −δ0
+i δ0
+j + δa
+i δb
+jea
+αeb
+βηab(u0),
+det|ηab| > 0
+ea
+α,0 = 0.
+(1)
+The coordinate indices of the variables of the semi-geodesic coordinate system are denoted
+by lower case Latin letters:
+i, j, k = 0, 1 . . . 3.
+The coordinate indices of the variables of the
+local coordinate system on the hypersurface
+V3
+are denoted by lower case Greek letters:
+α, β, γ = 1, . . . 3.
+the time variable is denoted by a 0 index. Group indices and indices of
+nonholonomic frame are denoted by
+a, b, c = 1, . . . 3.
+Summation is performed over repeated
+upper and lower indices within the index range.
+The 1-form
+ea
+αduα
+is invariant under the acting of the group G3(N). The vectors of the
+frame ea
+α define a non-holonomic coordinate system in V3. The dual triplet of vectors
+eα
+a,
+eα
+aeb
+α = δb
+a,
+eα
+aea
+β = δα
+β
+defines the operators of the group algebra:
+ˆYa = eα
+a∂a,
+[ ˆYa, ˆYb] = Cc
+ab ˆYc.
+(2)
+According to another definition, space-time
+V4
+is homogeneous if it admits a three-parameter
+group of motions
+G3(N),
+whose hypersurface
+V3
+of transitivity has the Euclidean space
+signature. The Killing vector fields ξα
+a
+and their dual vector fields
+ξa
+α
+form another frame
+of the space V3 and another representation of the algebra of the group G3. The vector fields
+ξα
+a
+satisfy the Killing equations:
+gαβ
+,γ ξγ
+a = gαγξβ
+a,γ + gβγξα
+a,γ,
+(3)
+and sets the infinitesimal group operators of the algebra G3
+ˆXa = ξα
+a ∂α,
+[ ˆXa, ˆXb] = Cc
+ab ˆXc.
+(4)
+Let us consider electromagnetic field with potential Ai. For a charged test particle, moving in
+this external electromagnetic field, it has been proved, that the Hamilton-Jacobi equation:
+gijPiPj = m,
+Pi = pi + Ai.
+(5)
+and the Klein-Gordon-Fock equation:
+ˆHϕ = (gij ˆPi ˆPj)ϕ = m2ϕ,
+ˆPk = ˆpk + Ak.
+(6)
+admit the integrals of motion, which are given by Killing vectors:
+˜Xα = ξi
+αpi
+(or
+ˆ˜Xα = ξi
+αˆpi),
+if and only if the conditions:
+ξα
+a ( ˜A),α = Cc
+ab ˜A
+(7)
+are satisfied (see papers [33]). Here
+pi = ∂iϕ;
+ˆpk = −ı ˆ∇k;
+( ˆ∇k is the covariant deriva-
+tive operator corresponding to the partial derivative operator -
+ˆ∂i
+in the coordinate field
+ui),
+ϕ
+is a scalar function of particle with mass
+m;
+˜Aa = ξα
+a Aα.
+3
+
+The electromagnetic field whose potential satisfies condition (7) is called admissible. All
+admissible electromagnetic fields for groups of motion
+Gr(N)
+(r ≤ 4),
+acting transitively
+on hypersurfaces of the spacetime, have been found in [33], [35], [36].
+Solutions of the set of equations (7) for HPSS of type IX have the form:
+Aα = αa(u0)la
+α ⇒ Aa = lα
+aAα = αa(u0).
+(8)
+To prove this let’s find the frame vector. We will use the metric tensor of IX-type space by
+Bianchi, found in Petrov’s book [39]. As it is known, the Bianchi type IX metric contains as
+a special case the space of constant positive curvature and therefore is of special interest for
+cosmology.
+ds2 = du12[a11 − (a12 cos 2u3 + a22 sin 2u3)] + 2du1du3((a13 cos u3 − a23 sin u3)+
+(9)
++2du1du2[(a13 cos u3 − a23 sin u3) cos u1 + (a12 cos 2u3 − a22 sin 2u3) sin u1]
++du22[a33cos u12 + (a23 cos u3 + a13 sin u3) sin 2u1 + (a12 sin 2u3 + a22 cos 2u3 + a11)sin u12]
+2du2du3(a33 cos u1 + (a23 cos u3 + a13 sin u3) sin u1) + du32a33 + edu02.
+aab are arbitrary functions on u0.
+To obtain the functions
+lα
+a
+, it is sufficient to consider the components
+g13, g23
+from the system (3). The solution has the form:
+la
+α = δp
+αla
+p(u1, u3) + δ3
+αδa
+3
+.
+From the equations:
+g13 = a13 cos u3 − a23 sin u3 = η3ala
+1,
+g23 = a33 cos u1 + (a23 cos u3 + a13 sin u3) sin u1 = η3ala
+1
+it follows:
+la
+α =
+�
+�
+cos u3
+− sin u3
+0
+sin u1 sin u3
+sin u1 cos u3
+cos u1
+0
+0
+1
+�
+� , lα
+a =
+�
+�
+cos u3
+sin u3
+sin u1
+−cos u1 sin u3
+sin u1
+− sin u3
+cos u3
+sin u1
+−cos u1 cos u3
+sin u1
+0
+0
+1
+�
+� ,
+(10)
+la
+αlα
+b = δa
+b .
+The lower index numbers the lines. One can show that the vector fields (10) satisfy the
+equations (1), (2): We present the components of the vectors ξα
+a in the form of a matrix:
+||ξα
+a || =
+�
+�
+0
+1
+0
+cos u2
+−cos u1 sin u2
+sin u1
+sin u2
+sin u1
+− sin u2
+−cos u1 cos u2
+sin u1
+cos u2
+sin u1
+�
+�
+The components ˜Aα can be expressed through Aα as follows:
+˜Aa = Zb
+aAb,
+4
+
+where
+||Zb
+a = ξα
+a lb
+α|| =
+�
+�
+sin u1 sin u3
+sin u1 cos u3
+cos u1
+(cos u2 cos u3 − sin u2 sin u3 cos u1)
+−(cos u2 sin u3 + sin u2 cos u3 cos u1)
+sin u1 sin u2
+−(sin u2 cos u3 + cos u2 sin u3 cos u1)
+(sin u2 sin u3 − cos u2 cos u3 cos u1)
+cos u2 sin u1
+�
+� .
+It can be shown by direct calculation that the elements of the matrix Zb
+a satisfy the equation:
+Zb
+a|c = Ca1
+caZb
+a1,
+|a = lα
+a∂α.
+(11)
+Therefore, the equation (7) can be reduced to the form:
+ξα
+a Ab,α = 0 ⇒ Aa = αa(u0).
+(12)
+3
+Maxwell’s equations with zero electromagnetic field
+sources in a homogeneous spacetime
+All exact solutions of vacuum Maxwell equations for solvable groups have been found in the
+papers [40], [41]. In the present paper the problem is solved for the group G3(IX).
+We will use the first definition of homogeneous spaces. Note, that for the space-time with
+the groups of motions G3(I) − G3(V I), G3(IX) both definitions are equivalent
+Consider the Maxwell equations with zero electromagnetic field sources in homogeneous
+space in the presence of an electromagnetic field invariant with respect to the group Gr:
+1
+√−g(√−gF ij),j = 0,
+(13)
+The metric tensor is defined by relations (1), the electromagnetic potential by the relations (7).
+When i = 0, from the set of equations (13) it follows:
+1
+√−g(√−ggαβF0β)α = 1
+l (llα
+aηab ˙αb),α = ηabρa ˙αb = 0.
+(14)
+Here it is denoted g = − det ||gαβ|| = −(ηl)2,
+where
+η2 = det ||ηαβ||,
+l = det ||la
+α||,
+ρa =
+lα
+a,α + l|a/l,
+the dots means the time derivatives. Let
+i = α.
+Then from the equation (13)
+it follows:
+1
+η(ηgαβF0β),0 = 1
+l (lgνβgαγFβγ),ν ⇒ 1
+η(ηηablα
+a ˙αb),0 = 1
+l (llν
+alβ
+b ηablα
+˜alγ
+˜b η˜a˜bFβγ),ν ⇒
+(15)
+(ηηab ˙αb),0 = ηla
+α
+l (llβ
+b lα
+˜a1lγ
+˜b Fβγ)|a1ηa1bη˜a˜b.
+(16)
+Let us find components of Fαβ, using the relations (8).
+Fαβ = (la
+β,α − la
+β,α)αa = lc
+βlγ
+c ld
+αlν
+d(la
+γ,ν − la
+ν,γ)αa = lb
+βla
+αlc
+γ(lγ
+a|b − lγ
+b|a)αc = lb
+βla
+αCc
+baαc.
+(17)
+Then
+(lF αβ),β = ηabη˜a˜bCd
+˜bbαd((llα
+a)|˜a + llα
+alγ
+˜a,γ).
+(18)
+5
+
+Structural constants of a group
+G3
+can be present in the form:
+Cc
+ab = Cc
+12ε12
+˜a˜b + Cc
+13ε13
+˜a˜b + Cc
+23ε23
+˜a˜b,
+εAB
+ab = δA
+a δB
+b − δA
+b δB
+a .
+(19)
+Using the notations:
+σ1 = Ca
+23αa,
+σ2 = Ca
+31αa,
+σ3 = Ca
+12αa,
+γ1 = σ1η11 + σ2η12 + σ3η13,
+γ2 = σ1η12 + σ2η22 + σ3η23,
+γ3 = σ1η13 + σ2η23 + σ3η33,
+let us reduce Maxwell’s equations (13) to the form:
+η(ηab ˙αb),0 = δa
+1(γ1(C1
+32) − γ2(C1
+31 + ρ3) + γ3(C1
+21 + ρ2)) + δa
+2(γ1(C2
+32 + ρ3)+
+(20)
+γ2C2
+13 − γ3(C2
+12ρ1)) + δa
+3(−γ1(C3
+23 + ρ2) + γ2(C3
+13 + ρ1) + γ3C3
+21),
+The order of the equations (20) can be decreased by introducing a new independent functions:
+βa = βa = ηηab ˙αb
+⇒
+η ˙αa = ηabβb.
+(21)
+Let us consider the Maxwell equations for the group G3(IX). As in this case
+non zero
+structural constants are following:
+C3
+12 = C2
+31 = C1
+23 = 1,
+functions
+σa, γ1
+have the form:
+σ1 = α1,
+σ2 = α2,
+σ3 = α3.
+γ1 = α1η11 + α2η12 + α3η13,
+γ2 = α1η12 + α2η22 + α3η23,
+γ1 = α1η13 + α2η23 + α3η33.
+Using these relations, we obtain Maxwell’s equations (14), (20) as a system of linear algebraic
+equations on the unknown functions
+nab:
+nab = ηab
+η ⇒ η =
+1
+det nab
+.
+(22)
+ˆW ˆn = ˆω,
+(23)
+where
+ˆW =
+�
+�
+�
+�
+�
+�
+�
+�
+α1
+α2
+α3
+0
+0
+0
+β1
+β2
+β3
+0
+0
+0
+0
+α1
+0
+α2
+α3
+0
+0
+β1
+0
+β2
+β3
+0
+0
+0
+α1
+0
+α2
+α3
+0
+0
+β1
+0
+β2
+β3
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(24)
+ˆnT = (n11, n12, n13, n22, n23, n33);
+ˆωT = (− ˙β1, ˙α1, − ˙β2, ˙α2, − ˙β3, ˙α3),
+6
+
+index T means the transposition of a matrix. Let us find the algebraic complement of the
+matrix ˆW :
+ˆV =
+�
+�
+�
+�
+�
+�
+�
+�
+β1V 2
+1
+−α1V 2
+1
+β2V 2
+1
+−α2V 2
+1
+β3V 2
+1
+−α3V 2
+1
+β1V1V2
+−α1V1V2
+β2V1V2
+−α2V1V2
+β3V1V2
+−α3V1V2
+β1V1V3
+−α1V1V3
+β2V1V3
+−α2V1V3
+β3V1V3
+−α3V1V3
+β1V 2
+2
+−α1V 2
+2
+β2V 2
+2
+−α2V 2
+2
+β3V 2
+2
+−α3V 2
+2
+β1V2V3
+−α1V2V3
+β2V2V3
+−α2V2V3
+β3V2V3
+−α3V2V3
+β1V 2
+3
+−α1V 2
+3
+β2V 2
+3
+−α2V 2
+3
+β3V 2
+3
+−α3V 2
+3
+�
+�
+�
+�
+�
+�
+�
+�
+(25)
+As ˆW is singular matrix, ˆV is the annulling matrix for ˆW:
+ˆV ˆW = 0.
+(26)
+Therefore, one of the equations from the system (23) can be replaced by the equation:
+δab( ˙αa ˙αb + ˙βa ˙βb) ⇒ δab(αaαb + βaβb) = c2 = const.
+(27)
+Depending on the rank of the matrix ˆW, one or more functions nab are independent. The
+remaining functions nab can be expressed through them and through the functions αa, βa. For
+classification it is necessary to find non-equivalent solutions of the system (23). Obviously, this
+system are symmetric with respect to the transposition
+lα
+1 ↔ lα
+2 . Therefore the reference
+indices a = 1 and a = 2 can be interchanged. Taking this observation into account, let us
+consider all non-equivalent options.
+4
+Solutions of Maxwell equations
+1.
+a1V1 ̸= 0 ⇒ the minor ˆW12 and its inverse matrix ˆΩ = ˆW −1
+12 have the form:
+ˆW12 =
+�
+�
+�
+�
+�
+�
+α2
+α3
+0
+0
+0
+α1
+0
+α2
+α3
+0
+β1
+0
+β2
+β3
+0
+0
+α1
+0
+α2
+α3
+0
+β1
+0
+β2
+β3
+�
+�
+�
+�
+�
+�
+,
+(28)
+ˆΩ1 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− V2
+α1V1
+− α3β2
+α1V1
+α2α3
+α1V1
+− α3β3
+α1V1
+α2
+3
+α1V1
+− V3
+α1V1
+α2β2
+α1V1
+− α2
+2
+α1V1
+α2β3
+α1V1
+−α2α3
+α1V1
+− V 2
+2
+α1V 2
+1
+(α3β1V1−α2β3V3)
+α1V 2
+1
+α3(α2V2−α1V1)
+α1V 2
+1
+−α3β3V2
+α1V 2
+1
+α2
+2V2
+α1V 2
+1
+− V2V3
+α1V 2
+1
+α2β2V2
+α1V 2
+1
+− α2
+2V2
+α1V 2
+1
+−α3β3‘V3
+α1V 2
+1
+α2
+3V3
+α1V 2
+1
+− V 2
+3
+α1V 2
+1
+α2β2V3
+α1V 2
+1
+− α2
+2V3
+α1V 2
+1
+(α3β2V3−α2β1V1)
+α1V 2
+1
+α2(α1V1−α3V3)
+α1V 2
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(29)
+Then the solution of equation (23) can be represented as:
+ˆn1 = ˆΩ1ˆω1,
+(30)
+were
+ˆnT
+1 = (n12, n13, n22, n23, n33);
+ˆωT
+1 = (−( ˙β1 + α1n11), − ˙β2, ˙α2, −β3, ˙α3),
+7
+
+Function
+n11,
+as well as the functions
+αa,
+βa
+are arbitrary functions of
+u0,
+that
+obey the condition (27).
+2.
+α2V1 ̸= 0, ⇒ α1 = 0 ⇒ the minor ˆW −1
+14 and its inverse matrix ˆΩ2 = ˆW −1
+14 have the
+form:
+ˆW14 =
+�
+�
+�
+�
+�
+�
+α2
+α3
+0
+0
+0
+β2
+β3
+0
+0
+0
+0
+0
+α2
+α3
+0
+0
+0
+0
+α2
+α3
+0
+β1
+0
+β2
+β3
+�
+�
+�
+�
+�
+�
+,
+ˆΩ2 =
+�
+�
+�
+�
+�
+�
+�
+�
+β3
+V1
+−α3
+V1
+0
+0
+0
+− β2
+V1
+α2
+V1
+0
+0
+0
+a2
+3β1β2
+α2V 2
+1
+−α2
+3β1
+V 2
+1
+1
+α2
+− α3β3
+α2V1
+a2
+3
+α2V1
+−a3β1β2
+V 2
+1
+α2α3β1
+V 2
+1
+0
+β3
+V1
+− a3
+V1
+a2β1β2
+V1
+−α2
+2β1
+V1
+0
+− β2
+V1
+α2
+V1
+�
+�
+�
+�
+�
+�
+�
+�
+(31)
+Solution of the equation (23) can be represented as:
+ˆn2 = ˆΩˆω2,
+(32)
+were
+ˆnT
+2 = (n12, n13, n22, n23, n33);
+ˆω2 = (− ˙β1, −β1n11, − ˙β2, − ˙β3, ˙α3)
+Function
+n11,
+as well as the functions
+αa,
+βa
+are arbitrary functions of
+u0,
+that
+obey the condition (27).
+3.
+a3V1 ̸= 0, ⇒ a1 = a2 = 0 ⇒ the minor ˆW −1
+16 and its inverse matrix ˆΩ3 = ˆW −1
+16 have the
+form:
+ˆW16 =
+�
+�
+�
+�
+�
+�
+0
+a3
+0
+0
+0
+β2
+β3
+0
+0
+0
+0
+0
+0
+a3
+0
+β1
+0
+β2
+β3
+0
+0
+0
+0
+0
+a3
+�
+�
+�
+�
+�
+�
+,
+ˆΩ3 =
+�
+�
+�
+�
+�
+�
+�
+− β3
+a3β2
+1
+β3
+0
+0
+0
+1
+a3
+0
+0
+0
+0
+β1β3
+a3β2
+2
+− β1
+β2
+2
+− β3
+β2a3
+1
+β2
+0
+0
+0
+1
+a3
+0
+0
+0
+0
+0
+0
+1
+a3
+�
+�
+�
+�
+�
+�
+�
+(33)
+Then the solution of equation (23) can be represented as:
+ˆn3 = ˆΩ3ˆω3,
+(34)
+were
+ˆnT
+3 = (n12, n13, n22, n23, n33);
+ˆωT
+3 = (− ˙β1, −β1n11, − ˙β2, 0, − ˙β3)
+Function
+n11,
+as well as the functions
+α3,
+βa
+are arbitrary functions of
+u0,
+that
+obey the condition (27).
+4.
+a1V3 ̸= 0. ⇒ V1 = V2 = 0,
+otherwise, we get a solution equivalent to the previous
+ones. As
+V3 ̸= 0 ⇒
+α3 = β3 = 0.
+The minor ˆW62 and its inverse matrix ˆΩ4 = ˆW −1
+62 have
+8
+
+the form:
+ˆW26 =
+�
+�
+�
+�
+�
+�
+α1
+α2
+0
+0
+0
+0
+α1
+0
+a2
+0
+0
+β1
+0
+β2
+0
+0
+0
+α1
+0
+α2
+0
+0
+β1
+0
+β2
+�
+�
+�
+�
+�
+�
+,
+ˆΩ4 =
+�
+�
+�
+�
+�
+�
+�
+1
+α1
+− α2β2
+α1V3
+α2
+2
+α1V3
+0
+0
+0
+β2
+V3
+−α2
+V3
+0
+0
+0
+0
+0
+β2
+V3
+−α2
+V3
+0
+− β1
+V3
+α1
+V3
+0
+0
+0
+0
+0
+− β1
+V3
+α1
+V3
+�
+�
+�
+�
+�
+�
+�
+(35)
+Then the solution of equation (23) can be represented as:
+ˆn4 = ˆΩ4ˆω4.
+(36)
+were
+ˆnT
+4 = (n11, n12, n13, n22, n23);
+ˆωT
+4 = (− ˙β1, − ˙β2, ˙α2, 0, 0).
+Function
+n33,
+as well as the functions
+α1,
+α2
+βa
+are arbitrary functions of
+u0,
+that obey the condition (27).
+5.
+Va = 0.
+Let us represent the system of Maxwell equations in the form:
+ˆQIˆnI = ˆωI
+were
+ˆQ =
+�
+�
+�
+�
+�
+�
+�
+�
+α1
+α2
+α3
+0
+0
+0
+0
+α1
+0
+α2
+α3
+0
+0
+0
+α1
+0
+α2
+α3
+β1
+β2
+β3
+0
+0
+0
+0
+β1
+0
+β2
+β3
+0
+0
+0
+β1
+0
+β2
+β3
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(37)
+ˆωI = (ˆωβ, ˆωα);
+ˆωβ = −( ˙β1, ˙β2, ˙β3),
+ˆωα = ( ˙α1, ˙α2, ˙α3)
+ˆnI = (ˆnα, ˆnβ);
+ˆnα = (n11, n12, n13),
+ˆnβ = (n22, n23, n33).
+To provide the classification, it is sufficient to consider the options:
+1)
+a1 ̸= 0,
+2)
+a3 ̸=
+0,
+a1 = a2 = 0.
+a) a1 ̸= 0 ⇒ βa = αaβ1
+α1 .
+ˆWIˆnα = (ˆωβ − ˆQ1ˆnβ) ⇒ ˆnα = ˆW −1
+I (ˆωβ − ˆQ1ˆnβ),
+β1 ˆWIˆnα = α1ˆωα − β1 ˆQ1ˆnβ ⇒ β1ˆωβ − α1ˆωα = 0 ⇒
+�
+�
+�
+α1 ˙α2 + β1 ˙β2 = 0,
+α1 ˙α3 + β1 ˙β3 = 0,
+α1 ˙α1 + β1 ˙β1 = 0.
+⇒
+�
+�
+�
+α1 = e sin ϕ,
+β1 = e cos ϕ,
+e = const,
+α2 = ec2 sin ϕ,
+β1 = ec2 cos ϕ,
+e, c2 = const,
+α3 = ec3 sin ϕ,
+β1 = ec3 cos ϕ,
+e, c3 = const.
+(38)
+9
+
+Here:
+ˆWI =
+�
+�
+α1
+α2
+α3
+0
+α1
+0
+0
+0
+α1
+�
+� , ˆW −1
+I
+=
+�
+�
+1
+α1
+−α2
+α2
+1
+−α3
+α2
+1
+0
+1
+α1
+0
+0
+0
+1
+α1
+�
+� , ˆQI =
+�
+�
+0
+0
+0
+α2
+α3
+0
+0
+α2
+α3,
+�
+�
+α1 = e sin ϕ,
+β1 = e cos ϕ,
+e, ca = const,
+Then matrices ˆWI, ˆW −1
+I , ˆQI and lines ˆωT take the form:
+ˆWI = sin ϕ ˆP,
+ˆW −1
+I
+=
+1
+sin ϕ
+ˆP −1,
+ˆQI = sin ϕ ˆQ.
+ˆP =
+�
+�
+1
+c2
+c3
+0
+1
+0
+0
+0
+1
+�
+� ,
+ˆP −1 =
+�
+�
+1
+−c2
+−c3
+0
+1
+0
+0
+0
+1
+�
+� ,
+ˆQ =
+�
+�
+0
+0
+0
+c2
+c3
+0
+0
+c2
+c3,
+�
+�
+ˆωT
+α = ˆωT
+β = ˙ϕ sin ϕ ˆCT = ˙ϕ sin ϕ(1, c2, c3),
+ˆnα = ˆw−1( ˙ϕ ˆCT − ˆQˆnβ)
+Function
+n22, n23, n33,
+as well as the function
+ϕ
+are arbitrary functions of
+u0.
+b) Va = 0,
+α3 ̸= 0.
+⇒ β1 = β2 = 0.
+The system of Maxwell equations has the form:
+α3n13 = α3n23 = 0,
+α3n33 = − ˙β3,
+β3n33 = ˙α3.
+a3 ˙a3 + β3 ˙β3 = 0 ⇒ a3 = c sin ϕ,
+β3 = cos ϕ
+From here:
+n33 = ˙ϕ,
+n13 = n23 = α1 = α2 = β1 = β2 = 0,
+α3 = c sin ϕ,
+β3 = c cos ϕ.
+Functions
+ϕ,
+n11,
+n12,
+n22 - are arbitrary functions on
+u0,
+c = const.
+5
+Conclusion
+It is known that homogeneous spaces of IV and IX types according to Bianchi classification
+include as special cases the spaces of constant curvature.This causes a special interest to them
+in cosmology. In the Universe with the metric of homogeneous space all physical fields are
+invariant with respect to the group of motions of the space-time. Therefore, exactly such fields
+should be considered in the first place when solving the self-consistent Einstein equations,
+in particular the Einstein-Maxwell equations.
+The final goal of classification of PSS with
+admissible electromagnetic fields is to enumerate all electrovacuum solutions of the Einstein-
+Maxwell equations. In [40], [41] the complete classification of vacuum solutions of the Maxwell
+equations for homogeneous spaces with solvable groups of motions has been carried out. In the
+present paper the same problem is solved for HPSS of IX-type. For the final decision of the
+first stage of the classification problem it remains to consider HPSS V III-type, which will be
+10
+
+done in the next paper. The results obtained will be used in the second stage for integration
+of the corresponding Einstein-Maxwell equations.
+FUNDING: The work is supported by Russian Science Foundation, project number N 23-
+21-00275.
+INSTITUTIONAL REVIEW BOARD STATEMENT: Not applicable.
+INFORMED CONSENT STATEMENT: Not applicable.
+DATA AVAILABILITY STATEMENT: The data that support the findings of this study
+are available within the article.
+CONFLICTS OF INTEREST: The author declares no conflict of interest.
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+14
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+Draft version January 9, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Diverse Carbonates in Exoplanet Oceans Promote the Carbon Cycle
+Kaustubh Hakim
+,1 Meng Tian
+,1 Dan J. Bower
+,1 and Kevin Heng
+2, 3, 4
+1University of Bern, Center for Space and Habitability, Gesellschaftsstrasse 6, CH-3012 Bern, Switzerland
+2Ludwig Maximilian University, University Observatory Munich, Scheinerstrasse 1, Munich D-81679, Germany
+3University of Warwick, Department of Physics, Astronomy & Astrophysics Group, Coventry CV4 7AL, United Kingdom
+4University of Bern, ARTORG Center for Biomedical Engineering Research, Murtenstrasse 50, CH-3008, Bern, Switzerland
+ABSTRACT
+Carbonate precipitation in oceans is essential for the carbonate-silicate cycle (inorganic carbon cycle)
+to maintain temperate climates.
+By considering the thermodynamics of carbonate chemistry, we
+demonstrate that the ocean pH decreases by approximately 0.5 for a factor of 10 increase in the
+atmospheric carbon dioxide content. The upper and lower limits of ocean pH are within 1–4 of each
+other, where the upper limit is buffered by carbonate precipitation and defines the ocean pH when
+the carbon cycle operates. If the carbonate compensation depth (CCD) resides above the ocean floor,
+then carbonate precipitation and the carbon cycle cease to operate.
+The CCD is deep (>40 km)
+for high ocean temperature and high atmospheric carbon dioxide content. Key divalent carbonates of
+magnesium, calcium and iron produce an increasingly wider parameter space of deep CCDs, suggesting
+that chemical diversity promotes the carbon cycle. The search for life from exoplanets will benefit by
+including chemically more diverse targets than Earth twins.
+Keywords: Extrasolar rocky planets (511); Carbon dioxide (196); Habitable zone (696); Ocean-
+atmosphere interactions (1150); Geological processes (2288); (Unified Astronomy Thesaurus)
+1. INTRODUCTION
+The carbonate-silicate cycle, also known as the in-
+organic carbon cycle, is a negative climate feedback
+mechanism that stabilises the surface temperature via
+the greenhouse effect of carbon dioxide in response
+to changes in volcanism rates, stellar luminosity, at-
+mospheric composition and opacity, planetary orbital
+movements and spin axis tilt (Berner 2004; Catling
+& Kasting 2017).
+Continental silicate rocks and at-
+mospheric carbon dioxide react with water in a pro-
+cess known as silicate weathering to produce carbonate-
+forming ions that precipitate as carbonates onto the
+ocean floor (Walker et al. 1981).
+The carbon cycle
+is completed when carbonates are transferred into the
+mantle for deep storage or carbon is eventually re-
+leased back into the atmosphere by volcanism (Holland
+1978; Sleep & Zahnle 2001), although the degassing ef-
+Corresponding author: Kaustubh Hakim
+kaustubh.hakim@unibe.ch
+ficiency is debated (Kelemen & Manning 2015; Foley
+2015). Silicate weathering and carbonate precipitation
+are traditionally represented by the net chemical reac-
+tion (Walker et al. 1981),
+CaSiO3 + CO2 → CaCO3 + SiO2,
+(1)
+where wollastonite (CaSiO3), which serves as a proxy for
+silicate rocks, is converted into calcite (CaCO3). Cal-
+cium thus plays a crucial role in silicate weathering and
+carbonate precipitation and is present as Ca2+ cations
+in oceans (Sect. 2).
+The existence of habitable zones assumes that the
+carbon cycle operates on Earth analogues to stabilise
+their atmospheric carbon dioxide content (Kasting et al.
+1993).
+Implicitly, this assumes not only that silicate
+weathering operates, but that ocean floor precipitation
+and deep storage of carbonates also occur. There exists
+a critical ocean depth known as the carbonate compen-
+sation depth (CCD), below which carbonates are unable
+to exist in their solid form because carbonate solubility
+increases with pressure in the ocean (Zeebe & West-
+broek 2003, see also Sect. 2.3, Figure 1). In modern
+arXiv:2301.02652v1  [astro-ph.EP]  6 Jan 2023
+
+ID2
+CCD
+!"#$#
+%&'(
+Ocean	Depth
+%)*+,-
+%&'( ≤ %,/0
+1)*+,- = 13456 = 1
+!78'(,#$#
+Carbonates
+Silicates
+Dissolved	ions
+Figure 1. Model parameters, nDtot where D = Ca, Mg or
+Fe, nSiO2,tot, PCO2, Patm, Poc and T. See Sect. 2 and Table 1
+for a full list of output quantities and description.
+Earth oceans, the CCD is located between 4–5 km, be-
+low the average ocean depth of about 3.8 km (Zeebe
+2012). If the CCD resides at a depth above the ocean
+floor, then carbonates are unable to settle. This leads
+to the disruption of the carbon cycle—at least, as it is
+understood to operate on Earth. Moreover, there are
+currently no theoretical constraints on exoplanet ocean
+chemistry. We investigate the interplay between atmo-
+spheric carbon dioxide content, ocean acidity (pH) and
+carbonate precipitation.
+We then calculate the CCD
+over a broad range of physical conditions.
+2. METHODS
+2.1. Ocean chemistry model
+2.1.1. Ca system
+Ocean chemistry is modelled by considering thermo-
+chemical equilibrium for pure Ca, Mg, or Fe systems.
+The CO2 partial pressure PCO2, ocean–surface temper-
+ature T and local ocean pressure Poc are control pa-
+rameters (Figure 1, Table 1). In the Ca system, there
+are 13 unknowns, the number density n of H+, OH−,
+H2O, HCO−
+3 , CO2−
+3 , CO2(aq), Catot, Ca2+, SiO2,tot,
+SiO2(aq), quartz SiO2(s), wollastonite CaSiO3(s) and
+calcite CaCO3(s). Out of the 13 unknowns, 2 are conti-
+nental silicate weathering products, nCatot and nSiO2,tot,
+that depend on PCO2 and T (Sect.
+2.2).
+There are
+11 remaining unknowns. We solve for 3 mass conserva-
+tion equations (for H, Ca and SiO2), 1 charge balance
+equation, and 7 equations from 7 chemical reactions pro-
+viding relations between equilibrium constants (that de-
+pend on Poc and T), reactants and products.
+Table 1. Parameters and output quantities.
+Symbol
+Description
+Reference
+Parameters for ocean chemistry
+T
+Ocean–Surface temperature
+288 K
+PCO2
+CO2 partial pressure
+0.3 mbar
+Patm
+Atmospheric pressure
+1 bar
+Poc
+Ocean layer pressure
+1 bar
+Parameters for weathering
+nDtot,0
+Ca, Mg or Fe ref. number density
+1 m−3
+β
+Weathering power-law exponent
+0.3
+Te
+e-folding temperature
+13.7 K
+Output quantities
+nX
+Number density of X [m−3]
+pH
+–log10(nH+/n0); n0 = 103 m−3
+These 3 mass-conservation equations, 1 charge balance
+equation and 7 reactions (water dissociation, Henry’s
+law/physical CO2 dissolution, chemical CO2 dissolu-
+tion, bicarbonate ion dissociation, calcite precipitation,
+quartz precipitation and wollastonite precipitation) are
+specified below. Henry’s law gives the amount of CO2
+physically dissolved in ocean water in equilibrium with
+PCO2:
+CO2(g) ⇌ CO2(aq).
+(2)
+The chemical dissolution or dissociation of CO2 in ocean
+water leads to the production of HCO−
+3 and H+ ions and
+thereby increases the ocean acidity (and decreases ocean
+pH = − log10(nH+/n0), where the standard number den-
+sity n0 = 1 m−3) by the following reaction:
+CO2(g) + H2O ⇌ H+ + HCO−
+3 .
+(3)
+To maintain the charge balance in ocean water, the ad-
+dition of Ca2+ to oceans decreases the number density
+of H+ and hence increases the ocean pH. The charge
+balance equation is given by:
+2nCa2+ + nH+ = nHCO−
+3 + 2nCO2−
+3
++ nOH−,
+(4)
+where CO2−
+3
+is produced due to the bicarbonate disso-
+ciation reaction:
+HCO−
+3 ⇌ CO2−
+3
++ H+,
+(5)
+and where OH− is produced due to the water dissocia-
+tion reaction:
+H2O ⇌ H+ + OH−.
+(6)
+
+3
+The mass conservation of H is given by
+nHtot = 2nH2O + nH+ + nHCO−
+3 .
+(7)
+Catot partitions into Ca2+, calcite and wollastonite
+which is accounted for by mass conservation:
+nCatot = nCa2+ + nCal + nWo.
+(8)
+Calcite precipitation occurs when nCa2+ is saturated to
+a certain value determined by the equilibrium constant
+of the calcite precipitation reaction and the abundance
+of nCO2−
+3 :
+Ca2+ + CO2−
+3
+⇌ CaCO3(s).
+(9)
+SiO2,tot partitions into aqueous silica SiO2(aq), quartz
+SiO2(s) and wollastonite CaSiO3(s). The mass conser-
+vation for SiO2 is given by:
+nSiO2,tot = nSiO2(aq) + nQz + nWo.
+(10)
+The quartz precipitation reaction is:
+SiO2(aq) ⇌ SiO2(s).
+(11)
+The reaction of wollastonite precipitation is given by:
+Ca2+ + SiO2(aq) + H2O ⇌ 2H+ + CaSiO3(s).
+(12)
+These equilibrium chemistry calculations are per-
+formed using Reaktoro v2 (Leal 2015), a multi-phase
+(aqueous, gas and solid mineral phases) chemistry soft-
+ware.
+This software implements the extended law of
+mass action including the determination of stable and
+unstable species for a given set of species in the system
+(Leal et al. 2017).
+We use the SUPCRTBL database
+for thermodynamic data (Johnson et al. 1992; Zimmer
+et al. 2016), the Peng-Robinson activity model for gases
+(Peng & Robinson 1976), the HKF activity model for
+water (Helgeson et al. 1981) and the Drummond activ-
+ity model for CO2(aq) (Drummond 1981).
+2.1.2. Mg and Fe systems
+In the Mg system, Ca is replaced by Mg, calcite
+by magnesite MgCO3(s) and wollastonite by enstatite
+Mg2Si2O6(s). This includes replacing equilibrium con-
+stants of all reactions including Mg. Similarly, in the
+Fe system, Ca is replaced by Fe, calcite by siderite
+FeCO3(s) and wollastonite by fayalite Fe2SiO4(s). We
+limit our calculations to Fe2+ although its oxidation has
+inhibited the formation of siderite during Earth’s his-
+tory, particularly since the great oxidation event (Rye
+et al. 1995).
+2.2. Weathering model
+The introduction of carbonate-producing divalent
+cations in oceans is dictated by silicate weathering. Sil-
+icate weathering and therefore the total number density
+of divalent cations D2+ (D = Ca, Mg or Fe) must depend
+on the CO2 partial pressure PCO2 and surface tempera-
+ture T (Walker et al. 1981; Hakim et al. 2021),
+nDtot = fW (PCO2, T) = nDtot,0
+� PCO2
+PCO2,0
+�β
+exp
+�T − T0
+Te
+�
+,
+(13)
+where ‘0’ represents the Earth reference values (Table 1),
+Te = 13.7 K is the e-folding temperature and β = 0.3 is
+the weathering power-law exponent (Walker et al. 1981).
+However, not all added Ca (or Mg, Fe) in oceans re-
+mains in the form of divalent cations, a fraction of it
+precipitates as carbonates on the ocean floor and an-
+other fraction as silicates. For this reason, we perform
+partitioning calculations of Ca (or Mg, Fe) in different
+phases following the ocean chemistry model (Sect. 2.1).
+2.3. CCD model
+Carbonates are deposited onto the ocean floor as part
+of sediments. The transition from calcite-rich to calcite-
+free sediments is gradual. The carbonate compensation
+depth (CCD) for the Earth ocean is normally defined
+as the depth at which the dissolution flux of calcite bal-
+ances the precipitation flux (Zeebe 2012). The depth
+at which the rapid dissolution of calcite-rich sediments
+begins is known as the lysocline, which is a sediment
+property (Zeebe & Westbroek 2003). The lysocline and
+CCD serve as bounds on the transition zone (∼0.5 km)
+between calcite-rich and calcite-free sediments. Other
+definitions for the CCD exist (Berger et al. 1976; Ridg-
+well & Zeebe 2005; Zeebe 2012). The depth of ocean d
+[km] in terms of ocean pressure Poc [bar] at the equator
+is given by (Leroy & Parthiot 1998)
+d =
+1
+9.7803 × 103 + 0.011Poc (97.266Poc − 2.512 × 10−3P 2
+oc
++ 2.28 × 10−7P 3
+oc − 1.8 × 10−11P 4
+oc).
+(14)
+We consider the CCD to be the depth dCCD (equiv-
+alent to the ocean pressure where Poc = PCCD) at
+which 99.9% of near-surface (Poc = Psurf) Ca, Mg or
+Fe-carbonates dissolve,
+nCarb,CCD = 0.001 nCarb,surf.
+(15)
+Our calculations of CCD are performed up to dCCD =
+45 km because of the availability of thermodynamic data
+up to the pressure of 5000 bar (Zimmer et al. 2016). This
+limitation does not affect our conclusions.
+
+4
+2.4. Analytical solution of ocean pH
+Upper limit of ocean pH. For calcite precipitation, all
+reactions in Section 2 need to be satisfied. However, two
+of these reactions can be used to analytically constrain
+ocean pH: Equations 9 and 16 where Equation 16 is a
+combination of Equations 3 and 5,
+CO2(g) + H2O ⇌ 2H+ + CO2−
+3 .
+(16)
+The ocean pH can be written as a function of PCO2,
+nCa2+ and equilibrium constants of Equations 9 and 16
+(Appendix A):
+pH = −1
+2
+�
+log PCO2 + log K9K16 + log nCa2+
+n0
+�
+. (17)
+This equation demonstrates the reason for the slope of
+approximately −0.5 for the upper limit of ocean pH as a
+function of the logarithm (base 10) of PCO2. Because K9
+and K16 are constants at a fixed T and P, pH becomes
+a function of only PCO2 and nCa2+ in Equation 17. As a
+function of PCO2, nCa2+ at the limit of carbonate satura-
+tion varies between ∼0.1 m−3 (at PCO2 = 0.01 µbar) and
+∼6 m−3 (at PCO2 = 0.3 bar). This additional increase
+in nCa2+ of less than two orders of magnitude over seven
+orders of magnitude increase in PCO2, makes the slope
+of ocean pH slightly steeper than −0.5 (see Fig. A1).
+Using nCa2+ from the numerical solution in Equation 17
+results in a semi-analytical solution matching with the
+numerical solution until PCO2 = 0.1 bar, beyond which
+non-ideal effects accounted in the numerical solution ex-
+hibit a small deviation from the analytical equation.
+Lower limit of ocean pH. In the absence of divalent
+cations in ocean, the ocean pH is largely governed by the
+conversion of CO2 to protons (Equation 3). For PCO2 >
+1 µbar, the ocean is acidic, where the number density of
+H+ is larger than that of OH− and the number density
+of HCO−
+3 is larger than CO2−
+3
+(bicarbonate-carbonate-
+water equilibria, Wolf-Gladrow et al. 2007). Therefore,
+the charge balance equation can be approximated as
+nH+ = nHCO−
+3 .
+(18)
+In terms of the equilibrium constant of Equation 3, this
+leads to (Appendix A)
+pH = −1
+2 (log PCO2 + log K3) .
+(19)
+At a fixed T and P, K3 is constant and thus the ocean
+pH exhibits a slope of −0.5 for PCO2 > 1 µbar (Fig. A1).
+For PCO2 < 1 µbar, the analytical solution does not
+hold because the number density of OH− is significant
+enough to make the charge balance approximation in
+Equation 18 invalid.
+The lower limit of ocean pH is
+independent of the Ca, Mg or Fe systems considered.
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+4
+5
+6
+7
+8
+9
+10
+11
+Ocean pH
+(a)
+Carbon Cycle
+No Carbon Cycle
+Modern 
+Earth pH
+Forbidden
+Forbidden
+Ca
+nCa, tot = fW(PCO2)
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+4
+5
+6
+7
+8
+9
+10
+11
+Ocean pH
+(b)
+Carbon Cycle
+No Carbon Cycle
+Modern 
+Earth pH
+Forbidden
+Forbidden
+Mg
+nMg, tot = fW(PCO2)
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+4
+5
+6
+7
+8
+9
+10
+11
+Ocean pH
+(c)
+Carbon Cycle
+No Carbon Cycle
+Modern 
+Earth pH
+Forbidden
+Forbidden
+Fe
+nFe, tot = fW(PCO2)
+Figure 2.
+Sensitivity of ocean pH to PCO2 at T = 288 K for
+pure (a) Ca, (b) Mg, (c) Fe systems. Upper and lower bounds
+of ocean pH are represented by the blue shaded region. Pink
+shaded regions are forbidden.
+
+5
+3. RESULTS AND DISCUSSION
+We consider the ocean pH to be determined by the
+chemical dissolution of atmospheric carbon dioxide in
+a well-mixed ocean, which occurs at the atmosphere–
+ocean interface. The chemical dissolution of CO2 is gov-
+erned by the reaction between water and CO2 to produce
+H+, HCO−
+3 and CO2−
+3
+ions (Sect. 2). As PCO2 increases,
+the ocean becomes more acidic. We consider an atmo-
+spheric surface pressure of 1 bar, but allow the atmo-
+spheric carbon dioxide content to vary via PCO2. Atmo-
+spheric surface pressures up to 100 bar have a negligible
+effect on our results and those between 100–1000 bar
+exhibit a small effect (Fig. A2a).
+For a given value of PCO2, the ocean pH is bounded
+between two limits (Fig. 2a). The ocean pH is restricted
+to a narrow range between 7–11 at PCO2 = 0.01 µbar
+and 4–7 for PCO2 = 0.1 bar. These ocean pH ranges
+are consistent with the inferences for Earth’s history,
+transitioning from an acidic ocean during the Archean
+at high PCO2 to an alkaline ocean at present-day PCO2
+(Halevy & Bachan 2017; Krissansen-Totton et al. 2018).
+The lower limit corresponds to the complete absence of
+divalent cations and thus it is independent of the car-
+bonate system under investigation (Sect. 2). The upper
+limit corresponds to the saturation of calcium cations
+in ocean water such that more weathering does not pro-
+duce further changes in pH and simply produces more
+calcite. This upper limit is buffered by the precipita-
+tion of carbonates and hence it results in one solution
+of ocean pH when the carbon cycle is operational for
+a given carbonate system and PCO2. Both upper and
+lower limits of ocean pH follow a slope of approximately
+–0.5 as a function of PCO2 (see Sect.
+2.4).
+Between
+these two limits, the number density of calcium cations
+is below the threshold to precipitate carbonates onto the
+ocean floor; thus, the carbon cycle is not operational.
+Due to their high condensation temperatures, the
+relative abundances of refractory elements observed in
+the photosphere of stars are expected to be mirrored
+in the rocky exoplanets they host (Bond et al. 2010;
+Thiabaud et al. 2015).
+For example, the calcium-to-
+magnesium ratio of the solar photosphere and Earth are
+0.062 and 0.066, respectively (Lodders 2003; Elser et al.
+2012). The relative abundances of Ca, Mg and Fe, mea-
+sured from the spectra of stars, vary by up to an or-
+der of magnitude. For example, Ca/Mg=0.02–0.2 and
+Ca/Fe=0.04–0.2 in the Hypatia catalogue of more than
+7000 stars (Hinkel et al. 2014). Furthermore, carbonates
+involving Mg and Fe are known to have formed during
+Earth’s history: e.g., magnesite (MgCO3) and siderite
+(FeCO3); these carbonates have dissolution properties
+that differ from those of calcite.
+Siderite could have
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+280
+300
+320
+340
+360
+T [K]
+(a)
+nCa, tot = 100 m
+3
+nCa, tot = 1 m
+3
+Carbon Cycle
+No Carbon Cycle 
+(cations consumed 
+by silicates)
+No Carbon Cycle 
+(too little CO2)
+No Carbon Cycle 
+(too acidic)
+nCa, tot = fW(PCO2, T)
+Ca-CCD
+1
+2
+4
+10
+20
+40
+CCD [km]
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+280
+300
+320
+340
+360
+T [K]
+(b)
+nMg, tot = 100 m
+3
+nMg, tot = 1 m
+3
+Carbon Cycle
+No Carbon Cycle 
+(cations consumed 
+by slicates)
+No Carbon Cycle 
+(too little CO2)
+No Carbon Cycle 
+(too acidic)
+nMg, tot = fW(PCO2, T)
+Mg-CCD
+1
+2
+4
+10
+20
+40
+CCD [km]
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+280
+300
+320
+340
+360
+T [K]
+(c)
+nFe, tot = 100 m
+3
+nFe, tot = 1 m
+3
+Carbon Cycle
+No Carbon Cycle 
+(cations consumed 
+by slicates)
+nFe, tot = fW(PCO2, T)
+Fe-CCD
+1
+2
+4
+10
+20
+40
+CCD [km]
+Figure 3. Carbonate compensation depth (CCD) as a func-
+tion of PCO2 and T (Patm = 1 bar) for (a) Ca, (b) Mg and
+(c) Fe systems.
+Gray contours represent the weathering-
+dependent cation number density as a function of PCO2 and
+T (Eq. 13). Gray disc denotes modern Earth PCO2 and T.
+
+6
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+10
+1
+100
+101
+n [m
+3]
+(a)
+nCa, tot = fW(PCO2)
+Ca Partitioning
+Ca++
+Calcite
+Silicates
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+10
+1
+100
+101
+n [m
+3]
+(b)
+nMg, tot = fW(PCO2)
+Mg Partitioning
+Mg++
+Magnesite
+Silicates
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+10
+1
+100
+101
+n [m
+3]
+(c)
+nFe, tot = fW(PCO2)
+Fe Partitioning
+Fe++
+Siderite
+Silicates
+Figure 4.
+Partitioning of (a) Ca, (b) Mg and (c) Fe in
+aqueous, carbonate and silicate phases as a function of PCO2
+at T = 310 K (Patm = Poc = 1 bar) in pure Ca, Mg and Fe
+systems, respectively.
+played a key role in locking up CO2 in carbonates on
+Earth during the Archean (Rye et al. 1995; Sverjensky
+& Lee 2010). We calculate ocean pH for the pure Mg and
+Fe systems in addition to the Ca system (Fig. 2b,c). The
+upper limit of ocean pH for a given PCO2 varies when
+considering systems with purely Ca, Mg or Fe as the
+source of weathering cations. The upper limit of ocean
+pH for the Mg system is only 0.2 higher than for the Ca
+system, whereas it is more than unity lower for the Fe
+system.
+For PCO2 < 10 µbar, ocean chemistry and hence
+the CCD is sensitive to the addition of aqueous silica
+(SiO2) in the ocean (Fig. 3). Silica is another product
+of silicate weathering, which enables the locking up of
+cations in silicate minerals instead of carbonate minerals
+(Walker et al. 1981; Hakim et al. 2021). For instance,
+for T > 300 K and PCO2 < 0.1 µbar in the Ca sys-
+tem in the presence of aqueous silica, silicates impinge
+on the stability of calcite (Fig. 4a) and prevent carbon-
+ate precipitation at all depths (Fig. 3a).
+In contrast,
+when no silica is present in the ocean for T > 300 K
+and PCO2 < 0.1 µbar, calcite is stable (Fig. B2a) and
+deep CCDs are produced (Fig. B1a), thereby increasing
+the parameter-space where the carbon cycle is stable.
+Similarly, in the Mg and Fe systems, silicates are more
+stable than carbonates for PCO2 < 10 µbar (Fig. 4b,c).
+PCO2 > 10 µbar favours the thermodynamic stability of
+carbonates over silicates.
+Carbon cycle box models of exoplanets often omit self-
+consistent modelling of ocean chemistry and precipita-
+tion of carbonates. Carbonate precipitation is implicitly
+assumed to persist and is not expected to be a bottle-
+neck for carbon cycling.
+Our ocean chemistry model
+can be incorporated directly into carbon cycle box mod-
+els for exoplanets, which can couple via key parameters,
+PCO2, T, and the carbonate chemistry. Thermochemi-
+cal equilibrium calculations of our ocean model can be
+used to determine the carbon fluxes into or out of the
+near-surface reservoirs.
+The carbon cycle box models
+can also be informed of the effect of ocean chemistry
+and ocean depth on the efficiency of carbon degassing
+and recycling.
+Upcoming observations of terrestrial exoplanets from
+the James Webb Space Telescope, Atmospheric Remote-
+sensing Infrared Exoplanet Large-survey and Extremely
+Large Telescopes will put constraints on their atmo-
+spheric composition, for instance, the volume mixing
+ratio of atmospheric carbon dioxide (PCO2/P). Deter-
+mining the partial pressure of carbon dioxide (PCO2)
+requires the atmospheric surface pressure (P) which is
+not easily constrained. Nonetheless, our thermodynamic
+calculations provide strong constraints on ocean chem-
+
+7
+istry in the presence or absence of magnesium, calcium
+or iron carbonates; the relative abundances of these
+carbonate-forming elements in planetary systems can
+be deduced from observations of stellar photospheres.
+Our results suggest that the carbon cycle will oper-
+ate robustly on chemically-diverse terrestrial exoplanets
+exhibiting silicate weathering.
+This implies that the
+search for life from exoplanets with temperate climates
+or biospheres will benefit by broadening the target list
+to planets that are more chemically diverse than Earth.
+We acknowledge financial support from the European
+Research Council via Consolidator Grant (ERC-2017-
+CoG-771620-EXOKLEIN, awarded to K. Heng) and the
+Center for Space and Habitability, University of Bern.
+We thank Allan Leal for the support with Reaktoro.
+DATA AVAILABILITY
+All data generated or analysed during this study are
+included in the published article.
+CODE AVAILABILITY
+OCRA (Ocean Chemistry with Reaktoro And beyond):
+the open-source code developed in this work is hosted
+at https://github.com/kaustubhhakim/ocra. OCRA v1.0
+was used in this study and is also available on Zenodo
+(Hakim 2022).
+Software:
+numpy (Harris et al. 2020), scipy (Vir-
+tanen et al. 2020), pandas (The pandas development
+team 2020), astropy (Astropy Collaboration et al.
+2013, 2022), matplotlib (Hunter 2007), Reaktoro (Leal
+2015)
+APPENDIX
+A. ANALYTICAL SOLUTION OF OCEAN PH AND
+P–T SENSITIVITY
+The analytical solution for the upper limit of ocean
+pH is derived from the relations between the equilibrium
+constants and reactants and products (assuming water
+activity to be unity in diluted solutions) of reactions
+described by Equations 9 and 16,
+K9 =
+n2
+0
+nCa2+nCO2−
+3
+,
+(A1)
+K16 =
+n2
+H+nCO2−
+3
+PCO2n3
+0
+.
+(A2)
+By eliminating the carbonate ion number density from
+these two equations, proton number density is
+nH+
+n0
+=
+�
+PCO2K9K16
+nCa2+
+n0
+�1/2
+(A3)
+Because the pH is given by
+pH = − log(nH+/n0),
+(A4)
+the analytical upper limit of ocean pH is Equation 17.
+The analytical solution for the lower limit of ocean
+pH is derived from the the equilibrium constant of the
+reaction described by Equation 3,
+K3 =
+nH+nHCO−
+3
+PCO2n2
+0
+.
+(A5)
+Then the proton number density is
+nH+
+n0
+= K3PCO2n0
+nHCO−
+3
+(A6)
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+4
+5
+6
+7
+8
+9
+10
+11
+Ocean pH
+Carbon Cycle
+No Carbon Cycle
+Modern 
+Earth pH
+Forbidden
+Ca
+Up (numerical)
+Up (semi-analytical)
+Up (ana., nCa2 + = 1 m
+3)
+Low (numerical)
+Low (analytical)
+Figure A1. Numerical, analytical and semi-analytical so-
+lutions of the upper and lower limits of ocean pH in the Ca
+system.
+Thus, the lower limit of ocean pH is given by Equation
+19.
+The analytical solutions of upper and lower limits of
+ocean pH as a function of PCO2 result in a slope of –0.5
+(Fig. A1). Pressure and temperature have a negligible
+effect on ocean pH (Fig. A2).
+
+8
+100
+101
+102
+103
+P [bar]
+4
+5
+6
+7
+8
+9
+10
+11
+Ocean pH
+(a)
+Carbon Cycle
+No Carbon Cycle
+Modern 
+Earth pH
+Forbidden
+Forbidden
+Ca
+nCa, tot = fW(PCO2)
+280
+300
+320
+340
+360
+T [K]
+4
+5
+6
+7
+8
+9
+10
+11
+Ocean pH
+(b)
+Carbon Cycle
+No Carbon Cycle
+Modern 
+Earth pH
+Forbidden
+Forbidden
+Ca
+nCa, tot = fW(PCO2)
+Figure A2. The sensitivity of ocean pH to (a) P and (b) T
+in the Ca-system.
+B. CCD WITHOUT SILICATE PRECIPITATION
+When no silicates are allowed to precipitate, CCDs for
+the Ca, Mg and Fe systems become deeper for PCO2 <
+1 µbar (Fig. B1). This is reflected in the phase stability
+plots in Fig. B2.
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+280
+300
+320
+340
+360
+T [K]
+(a)
+nCa, tot = 100 m
+3
+nCa, tot = 1 m
+3
+Carbon Cycle
+No Carbon Cycle 
+(too little CO2)
+No Carbon Cycle 
+(too acidic)
+nCa, tot = fW(PCO2, T)
+Ca CCD
+1
+2
+4
+10
+20
+40
+CCD [km]
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+280
+300
+320
+340
+360
+T [K]
+(b)
+nMg, tot = 100 m
+3
+nMg, tot = 1 m
+3
+Carbon Cycle
+No Carbon Cycle 
+(too little CO2)
+No Carbon Cycle 
+(too acidic)
+nMg, tot = fW(PCO2, T)
+Mg CCD
+1
+2
+4
+10
+20
+40
+CCD [km]
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+280
+300
+320
+340
+360
+T [K]
+(c)
+nFe, tot = 100 m
+3
+nFe, tot = 1 m
+3
+Carbon Cycle
+nFe, tot = fW(PCO2, T)
+Fe CCD
+1
+2
+4
+10
+20
+40
+CCD [km]
+Figure B1. Same as Fig. 3 but with no silica nSiO2,tot = 0.
+
+9
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+10
+1
+100
+101
+n [m
+3]
+(a)
+nCa, tot = fW(PCO2)
+Ca Partitioning
+Ca++
+Calcite
+Silicates
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+10
+1
+100
+101
+n [m
+3]
+(b)
+nMg, tot = fW(PCO2)
+Mg Partitioning
+Mg++
+Magnesite
+Silicates
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+PCO2 [bar]
+10
+1
+100
+101
+n [m
+3]
+(c)
+nFe, tot = fW(PCO2)
+Fe Partitioning
+Fe++
+Siderite
+Silicates
+Figure B2. Same as Fig. 4 but with no silica nSiO2,tot = 0.
+
+10
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BdE3T4oBgHgl3EQftAtl/content/tmp_files/2301.04672v1.pdf.txt

@@ -0,0 +1,897 @@
+Astronomy & Astrophysics manuscript no. VMS
+©ESO 2023
+January 13, 2023
+Clues on the presence and segregation of very massive stars in
+the Sunburst Lyman-continuum cluster at z=2.37⋆
+U. Meštri´c1,⋆⋆, E. Vanzella1, A. Upadhyaya2, F. Martins3, R. Marques-Chaves2, D. Schaerer2, 4,
+J. Guibert2, A. Zanella5, C. Grillo6, 7, P. Rosati8, F. Calura1, G.B. Caminha9, 10, A. Bolamperti5, 11, 12,
+M. Meneghetti1, P. Bergamini1, 6, A. Mercurio13, 14, M. Nonino15, R. Pascale1
+1 INAF – OAS, Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, I-40129 Bologna, Italy
+2 Geneva Observatory, Department of Astronomy, University of Geneva, Chemin Pegasi 51, CH-1290 Versoix, Switzerland
+3 LUPM, Université de Montpellier, CNRS, Place Eugène Bataillon, F-34095 Montpellier, France
+4 CNRS, IRAP, 14 Avenue E. Belin, 31400 Toulouse, France
+5 INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122, Padova, Italy
+6 Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy
+7 INAF – IASF Milano, via A. Corti 12, I-20133 Milano, Italy
+8 Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, via Saragat 1, I-44122 Ferrara, Italy
+9 Technical University of Munich, TUM School of Natural Sciences, Department of Physics, James-Franck-Str 1, 85748 Garching,
+Germany
+10 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany
+11 Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Vicolo dell’Osservatorio 3, I-35122 Padova, Italy
+12 European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching bei München, Germany
+13 Dipartimento di Fisica “E.R. Caianiello”, Università Degli Studi di Salerno, Via Giovanni Paolo II, I–84084 Fisciano (SA), Italy
+14 INAF – INAF - Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy
+15 INAF – Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143, Trieste, Italy
+ABSTRACT
+We report on the identification of very massive stars (VMS, mass > 100 M⊙) possibly segregated in the center of the young massive
+star cluster at z=2.37 hosted in the Sunburst lensed galaxy. Such a result is based on two pieces of evidence: (1) the VLT/MUSE
+spectra of several multiple images of the same star cluster show key spectral signatures of VMS, like the Heiiλ1640 broad emission,
+Nivλ1486 emission and Nivλ1720 P-Cygni profile. In particular, Heiiλ1640 is broad (∼ 1610 ± 300 km s−1) with an equivalent width
+of 3Å and shows an asymmetric profile. Such features require an extremely young (∼ 2.5 Myr) stellar population component with
+masses of the stars exceeding 100 M⊙. Assuming a Salpeter IMF and BPASS models for normal massive stars, the observed spectral
+features require ∼400 VMS; (2) the same star cluster is detected at S/N ∼ 100 in the LyC domain (λ < 900Å). The LyC emission
+emerges from a region with a radius at least 2 times smaller than what is observed at 1700Å (independently from magnification)
+and is located in the center of the cluster. In absolute scales, after de-lensing, the effective radii are Reff[LyC] ∼ 4.7 ± 1.5 pc and
+Reff[1700] = 7.8 ± 1.4 pc. The LyC radiation is mainly produced by hot and massive stars, implying that their spatial distribution
+(including VMS) is preferentially more confined in the central parts of the cluster. Approximately 400 VMS hosted by a cluster of
+∼ 107 M⊙ are producing ∼15% of the escaping LyC photons, while the rest is produced from other massive early-type stars.
+Key words. galaxies: high-redshift – galaxies: star formation – galaxies: ISM – galaxies: star clusters: general – gravitational lensing:
+strong – galaxies: individual: Sunburst galaxy.
+1. Introduction
+For many years the existence and occurrence of very massive
+stars (VMS) was mostly associated with the early Universe and
+metal-free environments in the context of the so-called Popula-
+tion III stars (e.g. Abel et al. 2002). VMS are short-lived stars ∼
+2 – 3 Myr (e.g. Yusof et al. 2013) with mass M > 100 M⊙ (Vink
+et al. 2015) and predominantly populate the central regions of
+young massive star clusters (within the core radius rc ∼ 0.1−0.2
+pc, Portegies Zwart et al. 2010). Due to their narrow lifetime,
+⋆ Based on observations collected at the European Southern Observa-
+tory for Astronomical research in the Southern Hemisphere under ESO
+programmes DDT MUSE program ID 107.22SK.001 (PI E. Vanzella),
+X-Shooter program ID 0103.A-0688 (PI E. Vanzella) and DDT MUSE
+program ID 297.A-5012(A) (PI Aghanim).
+⋆⋆ E-mail: uros.mestric@inaf.it
+studies of VMS in Milky Way star clusters is limited only to
+few targets, for example, the Arches cluster (Martins et al. 2008)
+or NGC3603 (Crowther et al. 2010). Individual VMS have been
+investigated in the local Universe, with high spatial resolution,
+thanks to the Hubble Space Telescope (HST, Cignoni et al. 2015;
+Crowther et al. 2016; Calzetti et al. 2015; Smith et al. 2016,
+2020; Brands et al. 2022). Very massive stars with masses above
+100 M⊙ are recognized as objects with significant impact on the
+evolution of early galaxies, influencing their chemical enrich-
+ment and star formation through feedback (e.g. Goswami et al.
+2021). Therefore, extending upper masses beyond 100 M⊙ of the
+current population synthesis models is essential for investigating
+and understanding young massive star clusters and VMS at dif-
+ferent redshifts (Smith et al. 2016; Crowther et al. 2016).
+Article number, page 1 of 10
+arXiv:2301.04672v1  [astro-ph.GA]  11 Jan 2023
+
+A&A proofs: manuscript no. VMS
+Despite some progress, the maximum stellar mass attained
+and the conditions determining the presence of VMS remain
+largely unknown. Recent observations of local star clusters re-
+port initial stellar masses up to ∼ 270 M⊙ (Brands et al. 2022) in
+the star cluster R136, with a cluster age of ∼1.5 Myr (Crowther
+et al. 2016). Furthermore, observations of young stellar clus-
+ters have revealed the presence of peculiar spectroscopic fea-
+tures such as unusually strong broad Heiiλ1640 emission (with
+FWHM > 1000 km s−1), which suggests the presence of VMS
+in these objects (e.g., Wofford et al. 2014; Crowther et al. 2016;
+Senchyna et al. 2021).
+Alongside with the observations, different models are try-
+ing to predict and trace the evolution, through different ages
+and masses, of the various spectroscopic features characteristic
+to VMS (e.g., Köhler et al. 2015; Gräfener 2021). For exam-
+ple, Martins & Palacios (2022) have generated new evolution-
+ary models and synthetic spectra of stars with initial masses in
+the range 150 – 400 M⊙, taking into account the existence of
+stellar winds stronger than typical OB-type stars produce. The
+resulting models predict specific features in the UV and optical
+part of the spectra, which are characteristic signatures of VMS.
+The most robust ultraviolet spectral features associated to VMS
+are Nivλ1486, broad Heiiλ1640 emission, and the Nivλ1720 P-
+Cygni profile (Martins & Palacios 2022). Such lines are expected
+to have equivalent widths spanning the interval 0.1 − 7 Å rest-
+frame. High signal-to-noise spectra with well detected contin-
+uum are therefore required to identify them, as shown, e.g., by
+Crowther et al. (2016) in the R136 stellar cluster in the local
+Universe. At cosmological distance strong gravitational lensing
+is necessary to detect these faint spectral features, allowing us to
+further gain in spatial resolution at tens of parsec scale and depth
+(see also, Vanzella et al. 2016; Johnson et al. 2017; Rigby et al.
+2017, 2018b,a; Vanzella et al. 2017, 2021; Meštri´c et al. 2022;
+Vanzella et al. 2022b).
+In this paper, we present for the first time convincing spec-
+troscopic evidence for the presence of VMS in a stellar cluster
+at cosmological distance (z=2.37, Vanzella et al. 2020a, 2022a).
+The host galaxy is dubbed Sunburst (Rivera-Thorsen et al.
+2019, 2017), and Lyman continuum (LyC) radiation is detected
+from the same clumpy regions which are showing the presence
+of VMS. Those massive stars in the center of the stellar cluster
+are significant producers of LyC radiation and hence are the main
+culprits for creating porous interstellar medium (ISM) enabling
+LyC escape. For purpose of this work, we perform a compre-
+hensive analysis of deep VLT/MUSE, X-Shooter and synthetic
+spectra with aim to confirm presence of VMS. Additionally we
+investigate the existence of the segregation of VMS by modeling
+the morphology of the young massive star cluster (YMC) hosted
+in the Sunburst galaxy.
+The paper is organized as follows. In Section 2 we briefly de-
+scribe the Sunburst galaxy and the available observational data.
+In Section 3 we analyze the spectral signatures of very massive
+stars using MUSE/IFU and X-Shooter observations in combina-
+tion with the latest evolutionary models and synthetic spectra. In
+Section 4 we discuss the morphological properties of the YMC
+(dubbed 5.1) and the possible segregation of the (very) massive
+stars in its central parts. We present our conclusion in Section 5.
+We assume a flat cosmology with ΩM= 0.3, ΩΛ= 0.7 and
+H0 = 70 km s−1 Mpc−1. Within this model, one arcsec at z = 2.37
+corresponds to a projected physical scale of 8200 parsec. All
+magnitudes are given in the AB system.
+Fig. 1. Left: The HST F555W band image, showing the six aperture po-
+sitions where a MUSE 1D spectrum is extracted (red contours). White
+arrows point to the multiple images of the young stellar cluster. Right:
+The MUSE IFU image at ∼ 1800Å of the same region shown on the left
+with the same apertures in red.
+2. The Sunburst lensed galaxy
+The Sunburst is a galaxy at z=2.37, strongly lensed by the
+Planck cluster PSZ1 G311.65-18.48 at z=0.44, initially reported
+by Dahle et al. (2016). The strong gravitational lensing effect
+deflects the light from the background high-z Sunburst galaxy
+into four bright arcs. These bright arcs harbor at least 13 star-
+forming knots, which likely are stellar clusters. There are more
+than 50 multiple images of this system (Pignataro et al. 2021),
+whose physical properties are studied in detail in Vanzella et al.
+(2022a). Among the 13 young stellar clusters, one has been iden-
+tified 12 times (dubbed 5.1) and it is the subject of this work
+(see Figure 1). The source 5.1 shows a multi-peaked Lyα emis-
+sion consistent with an optically thin medium and Lyman con-
+tinuum (LyC) leakage along the line of sight (Rivera-Thorsen
+et al. 2017). Furthermore, the detection of LyC radiation emerg-
+ing from the 12 detected multiple images of the 5.1 young mas-
+sive star cluster is confirmed by HST multi-band observations
+(Rivera-Thorsen et al. 2019). Additional analyses of the 12 LyC
+multiple images of 5.1 have revealed that the star cluster has
+an age younger than 3 Myr and a stellar metallicity of 0.5Z⊙
+(Chisholm et al. 2019), with a physical size of ≃ 10 pc and a
+stellar (and dynamical) mass value of ≃ 107 M⊙ (Vanzella et al.
+2022a).
+The Sunburst was observed with HST, providing multi-
+band photometry in the F275W, F410M, F555W, F606W,
+F814W, F098M, F105W, F140W and F160W filters, under the
+programs 15101 (PI Dahle), 15949 (PI Gladders), and 15377
+(PI Bayliss). Sunburst has also been targeted with ground-
+based high resolution (R ∼ 5000 − 9000) VLT/X-Shooter spec-
+troscopy covering the spectral range 3000-22000Å in three main
+arms, UVB, VIS abd NIR. The observational strategy and the
+data reduction procedures applied to HST imaging and VLT/X-
+Shooter spectroscopy have been presented in Vanzella et al.
+(2020b, 2022a). VLT/MUSE integral field spectroscopy at res-
+olution R = 3000 and covering the spectral range 4800-9400Å
+was obtained during 2016 (1h integration, DDT, PI. Aghanim)
+and 2021 (1h integration, PI, Vanzella) in the wide field mode
+configuration. The final datacube which combines the two hours
+and the data reduction is described in Vanzella et al. (2022a).
+We also presented a first version of the lens model in Pignataro
+et al. (2021) based on the 62 spectroscopically confirmed mul-
+tiple images in the redshift range 1 < z < 3.5 (see also Sharon
+et al. 2022; Diego et al. 2022). A revised lens model will be com-
+puted once the new VLT/MUSE observations (7h integration)
+planned during 2023 will be performed (prog. 110.249D.001,
+PI. Vanzella).
+Article number, page 2 of 10
+
+PSF
+HST F555W
+PSF
+MUSE IFU
+AP1
+AP2
+AP3
+5.1a
+5.1b
+5.1c
+5.1d'
+ 5.1f
+5.1e
+AP4
+AP5
+AP5
+.5.1h
+AP6
+5.1iU. Mestric et al.: Very massive, spatially segregated stars at z=2.4
+Here we focus on the ≃ 3 Myr old, UV-bright and Ly-
+man continuum source with MUV = −18.6 (1700Å magnitude
+and ultraviolet slope β = −1.71 ± 0.01, Fλ ∼ λβ), massive
+(M ∼ 107 M⊙) star cluster 5.1, subjected to large magnification
+values (µ ∼ 10 − 70 over 12 multiple images, Pignataro et al.
+2021; Vanzella et al. 2022a). In the following we perform a new
+analysis focusing on the nature of the ionizing source (Sect. 3)
+and its morphology (Sect. 4).
+3. Spectral signatures of very massive stars in the
+Sunburst star cluster at z=2.37
+We aim to investigate the UV and optical spectroscopic prop-
+erties of the young stellar cluster 5.1. The VLT/MUSE one-
+dimensional spectra are extracted from six apertures enclosing
+nine multiple images of 5.1 (shown in Fig. 1) and subsequently
+combined to produce a continuum-detected high signal-to-noise
+ratio SNR (> 60) weighted-average spectrum (Figure 2). The
+stacked spectrum shown in Figure 2 is equivalent to an inte-
+gration time of (2 × 9) × 302 > 16, 000 hours without lensing
+amplification, adopting the minimum amplification among the 9
+multiple images (µ = 30).
+3.1. Observed VMS features with VLT MUSE and X-Shooter
+The very high SNR MUSE spectrum (Fig. 2) allows us to
+identify several emission and absorption lines. Among them
+we have the nebular emission lines associated to the interstel-
+lar medium of the galaxy, like Oiii]λ1661, 1666, Niii]λ1750,
+[Siiii]λ1883, 1892, and Ciii]λλ1907, 1909. The well detected
+continuum allows us to investigate faint line emissions (of a frac-
+tion of an Å rest-frame equivalent width), and to sample the de-
+tails of the line profiles, otherwise not accessible without lens-
+ing amplification. In particular, faint Nivλ1486 emission, the ev-
+ident P-Cygni profile of the Civλ1550, the prominent broad and
+asymmetric Heiiλ1640 line profile, and the P-Cygni signature of
+Nivλ1720 clearly stand out. All these lines are associated with
+young, hot and (very) massive stars.
+We report detection of Heiiλ1640 emission with measured
+rest frame EW=3.0±0.3Å and FWHM=8.8±1.7Å (∼ 1610±300
+km s−1). The broad shape of the Heiiλ1640 emission line ob-
+served in the Sunburst cluster is asymmetric and resembles a
+typical P-Cygni profile. The blue end of the emission line drops
+steeply, while the red end drops more gradually. The P-Cygni
+profile of Heiiλ1640 line is consistent with that predicted by
+models and synthetic spectra (see, Martins & Palacios 2022).
+Broad Heiiλ1640 emission observed in galaxies is usually re-
+lated to non-nebular origin, commonly associated with Wolf-
+Rayet (WR) stars, (e.g. Schaerer & Stasi´nska 1999; Brinchmann
+et al. 2008; Leitherer et al. 2018; Senchyna et al. 2021), though
+the failure of the synthesis models to reproduce some of the
+strong Heiiλ1640 lines might be related to missing ingredients in
+stellar evolution models (see, e.g., Leitherer et al. 2018). How-
+ever, far-UV spectroscopic investigation of ∼57 individual stars
+located within the R136 star cluster reveals that massive stars
+with M>100 M⊙ have a crucial role in producing the Heiiλ1640
+emission line (Gräfener & Vink 2015; Crowther et al. 2016).
+On the other hand, Martins & Palacios (2022) have shown that
+Heiiλ1640 can be produced in significant amount only when stel-
+lar winds are stronger than in normal O stars. VMS develop
+such strong winds because of their proximity to the Eddington
+limit (Vink et al. 2011; Bestenlehner 2020; Gräfener 2021). At
+the same time, these winds peel off the external layers of the
+stars and expose to the surface the products of hydrogen burn-
+ing through the CNO cycle. This results in a strong nitrogen
+(and helium) enrichment that boosts the strength of Nivλ1486
+and Nivλ1720. This typically happens after ∼1.5 Myr. Both
+mentioned emission lines are detected in the spectrum of the
+Sunburst cluster at SNR > 15 (Figure 2), with EW=0.2Å and
+FWHM=2.9Å for Nivλ1486 and EW=0.15Å and FWHM ∼ 2Å
+for Nivλ1720. The helium enrichment also contributes to the
+strength of Heiiλ1640. The same effects (strong winds combined
+with surface chemical enrichment) happen in normal evolved
+massive stars when they are seen as WR stars. The key difference
+compared to VMS is that helium enrichment takes place only af-
+ter the main sequence (>∼ 4 Myr), while the same process takes
+place at younger ages in VMS. Furthermore, VMS are more lu-
+minous than normal WR stars and hence their contribution to
+integrated light is larger.
+The nebular Hα equivalent width provides constraints on the
+cluster age and hence on whether the Heiiλ1640 line is primar-
+ily due to WR stars or VMS. According to the BPASS mod-
+els and results from Eldridge & Stanway (2012) (their Fig. 3)
+they predict that normal and WR stars produce EWHα < 1Å for
+ages <∼ 3Myr. The X-Shooter spectrum reveals a prominent Hα
+line and no continuum detection, which very conservatively im-
+plies an equivalent width larger than 200Å rest-frame at 1-sigma.
+However, if we assume for Hα the same continuum level ob-
+served at λ ∼ 5000Å rest-frame in the photometric spectral en-
+ergy distribution (SED) by Vanzella et al. (2022a) such a limit in-
+creases to ∼ 840Å. This value would be still a lower limit, even in
+the case of leakage of ionizing photons. After correcting the Hα
+flux for the fraction of escaping LyC photons (Hα/(1− f abs
+esc )), the
+resulting EW increases to EWHα ∼ 1231Å. We adopt f abs
+esc values
+from Rivera-Thorsen et al. (2019), where the corresponding ab-
+solute escape fraction of LyC photons along the line of sight is
+f abs
+esc = 32+2
+−4% . Such a large Hα equivalent width is consistent
+with a star-forming burst younger than ∼ 3 Myr (e.g., Leitherer
+et al. 2014). Furthermore as discussed in Chisholm et al. (2019)
+Nvλ1240 stellar wind profile predominantly depends on the stel-
+lar age while variations due to different metallicity are negligible
+and it is related to the young stellar populations (< 5 Myr). From
+the comparison of the observed Nvλ1240 with the models Figure
+3 and 4 we additionally demonstrate that the age of the cluster is
+< 3 Myr. From our age analysis, we can conclude that properties
+of both Nvλ1240 and Hα fit well with < 3 Myr age of the stel-
+lar cluster which requires other sources than WR stars to explain
+the observed strong Heiiλ1640 EW=3.0±0.3Å. Therefore these
+results strongly suggest that VMS are responsible for the pro-
+duction of the spectral ultraviolet features we observe in such a
+young massive star cluster. Moreover, Wofford et al. (2014) and
+Smith et al. (2016) have argued that the presence of Ovλ1371 in
+integrated light of the clusters was also a key feature of VMS.
+This line is not seen in the Sunburst cluster, see Figure 2. As
+demonstrated by Martins & Palacios (2022), this is not incom-
+patible with the presence of VMS, since Ovλ1371 disappears as
+VMS evolve to lower effective temperature. In their Fig. 4, we
+see that no sign of Ovλ1371 exists after ∼1 Myr. This, together
+with the presence of Nivλ1486, places a rather tight constraint
+on the cluster age.
+3.2. Comparing observations with models
+To investigate the rest-frame UV spectrum of the cluster, we have
+created an integrated VMS model following Martins & Palacios
+Article number, page 3 of 10
+
+A&A proofs: manuscript no. VMS
+Fig. 2. The MUSE IFU spectrum of the 5.1 young massive star cluster extracted from 6 apertures is shown in black (thin line) and the X-Shooter
+long slit spectrum of 5.1l knot is shown in blue (bold line). The key confirmed features indicating the presence of VMS in the stellar cluster
+are marked with shaded light red strips while the dark-orange line shows the best-fit Fλ ∼ λβ, with β = −1.71. The shaded grey strip indicates
+the (absence of) Ovλ1371 line, which usually is an indicator of VMS too. The prominent P-Cygni of Nvλ1240 and strong emission part of the
+Civλ1550 are present and indicate the young age of the stellar cluster (black bold markers). In the bottom, the 1-sigma errors of both spectra are
+shown. Other detected interstellar features and stellar features are marked with dashed red and green lines, respectively.
+(2022) that includes normal mass stars (0.1-100 M⊙) with differ-
+ent VMS (150 M⊙ and 200 M⊙).
+We have used the spectral energy distribution (SEDs) of
+BPASS (Eldridge et al. 2017; Stanway & Eldridge 2018) v2.2.1
+single-star population synthesis model. The model has an up-
+per mass limit of 100 M⊙ with the Salpeter IMF, metallicity of
+Z=0.006 (where 0.02 corresponds to solar metallicity), and in-
+stantaneous star formation history, with a burst of mass 106 M⊙.
+The adopted metallicity of the model is the closest to our mea-
+sured value based on N2 index (Marino et al. 2013), which is
+≃ 0.4Z⊙ and consistent with the estimate provided by Mainali
+et al. (2022) (see also Chisholm et al. 2019).
+We have extrapolated the Salpeter IMF to 225 M⊙ upper
+mass limit within a few mass bins given by Equation 1 from the
+BPASS manual1. Equation 1 gives the number of massive stars
+in the mass range [Ma; Mb].
+N(Ma; Mb) = C × Mα1
+1
+� Mb
+Ma
+Mα2 dM
+(1)
+Here, C is a constant and has a value of 1.23×105 for an arbitrary
+burst mass of 106 M⊙. Also, M1 = 0.5 M⊙ α1 = -1.3, α2 = -
+2.35. The mass bins are selected in a way to add the SEDs of
+appropriate numbers of single VMS stars, which are available
+for discrete sets of VMS with masses including 150 M⊙ and 200
+M⊙. In this manner we compute SEDs including VMS with IMFs
+extending up to 175 and 225 M⊙, respectively, following Martins
+& Palacios (2022).
+1 https://flexiblelearning.auckland.ac.nz/bpass/9.html
+From Figure 2, we can see that the cluster shows signifi-
+cant Nivλ1486 emission. From the VMS models and synthetic
+spectra, Nivλ1486 emission only appears after 1.5 Myr of VMS
+evolution (see, Martins & Palacios 2022) and VMS last approx-
+imately until 2.5 Myr. Based on this, we created the SEDs of in-
+tegrated VMS models at 1.5 Myr, 2 Myr, and 2.5 Myr. We have
+normalized the spectrum of the cluster and the models by fitting
+a UV power law by using the spectral windows provided by Rix
+et al. (2004).
+We have directly compared the cluster spectrum with the two
+VMS models at 3 different ages. The comparison shows that
+VMS are clearly needed to reproduce the observations (see, Fig-
+ures 3, 4, and A.1). However, the Heiiλ1640 and Nivλ1720 lines
+in the models appear stronger than observed even at the age of
+1.5 Myr and with a maximum mass of 175 M⊙. To match the
+observed Heiiλ1640 and Nivλ1720 profiles, we have therefore
+reduced the VMS contribution by decreasing their numbers. We
+find good agreement if we reduce the VMS contribution by a fac-
+tor of 6 in the VMS model, which includes only 150 M⊙ VMS
+at 2.5 Myr (Fig. 3). Alternatively, a similar match is also found
+by reducing the VMS contribution by a factor of 8 in models in-
+cluding also the 200 M⊙ VMS (Fig. 4). In short, the observations
+are compatible with an IMF extending up to ∼ 175 or 225 M⊙,
+but with an IMF slope steeper than Salpeter (α2 < −2.35) for
+M > 100 M⊙.
+Article number, page 4 of 10
+
+Aobs / A
+4000
+4500
+5000
+5500
+6000
+6500
+3.0
+1H13
+IIIS.+IO
+AS:
+{IIIN
+IIIS
+IIIS
+AID
+全13
+sv
+CIV1550
+2.5
+NV1240
+y1486
+ux
+fl
+Normalized
+1.0
+0.5
+0.0
+F
+1100
+1200
+1300
+1400
+1500
+1600
+1700
+1800
+1900
+2000
+Arest / AU. Mestric et al.: Very massive, spatially segregated stars at z=2.4
+3.3. VMS contribution to the LyC budget of the young stellar
+cluster
+After adopting the results from the previous section, we can now
+estimate the number of O-type stars hosted by the same stellar
+cluster and the percentage of LyC photons emitted by VMS only.
+Measurements are performed in the range of ∼730–900Å
+rest-frame (range covered by HST F275W filter in which LyC
+radiation is detected and fesc later on evaluated). First, we cal-
+culate the mean flux from the model which include both normal
+and VMS stars and, secondly, we calculate the mean flux from
+the model including only VMS. The resulting ratio of those two
+models gives us the fraction of the LyC photons produced by
+VMS, which is ∼15%. Since LyC photons are mainly produced
+by O-type and more massive stars, we can see that ∼15% of the
+LyC production is generated only from ∼1% of the stars capable
+of producing LyC photons. It is worth noting that the fraction
+of LyC ionizing radiation produced from VMS in the Sunburst
+5.1 stellar cluster is smaller than the predicted LyC fraction pro-
+duced by the VMS located in R136 stellar cluster, which is 25%
+(Doran et al. 2013). However, we also note that in the case of the
+Sunburst stellar cluster the light coming from the host galaxy
+could slightly decreases the inferred equivalent width of the key
+spectral features discussed above. While such dilution is difficult
+to address with the present ground-based spectroscopic data2, its
+effect implies a possible slightly higher contribution of VMS to
+the LyC radiation.
+4. Spatial segregation of the Lyman continuum
+radiation
+We now address the morphological properties of image 5.1l,
+which is the most magnified among the multiple images of the
+star cluster (µtot ≃ 76, Pignataro et al. 2021). 5.1l is the brightest
+image detected with a large SNR in the F275W (SNR ∼ 90) and
+F555W (SNR ≫100), allowing us to investigate and compare
+the morphology in these two spectral regions: the emitting LyC
+(λ < 900Å, in HST F275W band) and the non-ionizing radiation
+at 1700Å (HST F555W band). We follow two approaches: (1)
+we ran simulations injecting the sources in the F275W band and
+(2) we analyzed the curve of growth of the resulting images.
+Figure 5 shows the F555W image of 5.1l, in which the elon-
+gation is clearly visible in the direction of the tangential stretch
+produced by gravitational lensing. As discussed in Pignataro
+et al. (2021) (see also Vanzella et al. 2022a), the tangential am-
+plification largely dominates along the arc (µtang ≃ 57). We per-
+form here a relative comparison between images, to ensure that
+the the results do not depend on the magnification values.
+As a first step, we compute a realistic model of 5.1l on the
+HST F555W image using Galfit (Peng et al. 2010). The point
+spread function (PSF) has been extracted by combining non-
+saturated stars available in the field of view. While the fit with a
+single component does not produce acceptable residuals (larger
+than 20%), we reproduce quite well the light profile of the ob-
+ject by combining two components: a core with a Gaussian light
+profile and an effective radius (Reff) smaller than 0.5 pixels (in
+practice nearly unresolved) and an extended component with
+Reff = 6 pixels and Sersic index n=1 (similar results are ob-
+tained also with n=0.5). The combination of the two components
+produces an optimal shape which leaves normalized residuals
+smaller than 10% (see Figure 5). It is worth now investigating if
+2 JWST/NIRSpec-IFU and NIRCAM observations on the same YMC
+are planned during 2023, prog. 2555, PI. Rivera-Thorsen
+such a resolved shape (sampled at 1700Å) is recovered if placed
+in the F275W Lyman continuum image. For this check, we in-
+jected mock images of 5.1l into the F275W image on five dif-
+ferent positions around 5.1l, which are not contaminated by the
+flux coming from other sources. Such images are produced from
+the aforementioned two-component Galfit model constructed
+at 1700Å (F555W), but now accounting for the F275W PSF (in
+other words convolved by the F275W PSF) to allow for a proper
+comparison with the LyC 5.1l source (observed in HST F275W
+band). Such images have been added to F275W after rescaling
+each of them to the observed peak value of the LyC 5.1l ob-
+ject. This step has been performed with IRAF (Tody 1986) task
+IMARITH and IMCOPY. Figure 6 shows the results, in which all
+the injected images show a spatially-resolved morphology along
+the tangential magnification. Conversely, the observed LyC im-
+age (of 5.1l) appears nucleated, suggesting that the emitting LyC
+region is smaller than the one at 1700Å.
+To quantify this result, we calculate the curve of growth
+(CoG) of the images shown in Figure 6. The flux is then mea-
+sured in the F275W band in 34 circular apertures. The small-
+est aperture has a radius of 0.1 pixel. Intermediate apertures are
+drawn with increasing radii, with a step of 0.5 pix, up to largest
+one, which has a radius of 34 pixels. As a reference point-like
+source, we constructed the mean CoG from a selected sample of
+twenty non-saturated and non-contaminated stars. The resulting
+CoG is shown in Figure 7, where the y-axis reports the frac-
+tion of the flux enclosed at the corresponding radius in pixels
+(x-axis).
+The same procedure has been applied to the LyC emitting
+source 5.1l, while another CoG has been constructed by averag-
+ing the five CoG of the injected models resembling the morphol-
+ogy at 1700Å. Figure 7 compares all the CoG after normalizing
+them to the saturation value at the largest radius. The first re-
+sult which emerges from this test is the clear deviation of the
+CoG of the observed 5.1l LyC source from the behavior of a
+point-like source (stars). This was not explored before and sug-
+gests that in the most magnified image of the star cluster the
+LyC appears spatially resolved. This is the first evidence of a re-
+solved stellar LyC emission at cosmological distance. Second,
+such barely resolved LyC emission appears more nucleated than
+the one at 1700Å. Consequently, the sources of ionizing radi-
+ation appears located in the central part of the cluster. Indeed,
+from those curves, it emerges that 50% of the flux of the stars is
+enclosed within a radius of ∼1.9 pixel, while for 5.1l it lies within
+∼2.2 pixels. Additionally, we perform the Kolmogorov-Smirnov
+two-sample test (KS-test) to check if the CoGs derived from the
+stars and 5.1l source follow the same distribution (null hypothe-
+sis). For this purpose, we used the statistical function ks_2samp
+from scipy.stats. After comparing the average CoG of the
+stars with cyan and violet CoGs, the KS-test gives p << 0.05. It
+means that the null hypothesis is not satisfied with the LyC pro-
+file of 5.1l and it deviates from the CoG of a point-like source.
+Furthermore, we also find that the half-light size of 5.1l at 1700Å
+is larger than the ionizing region, ∼2.6 pixels compared to the
+∼2.2 pixels. If we correct such radii for the instrumental reso-
+lution (given by the stars) we obtain an effective radius for 5.1l
+at 1700Å ≃ 7.8 ± 1.4 pc after de-lensing3, in agreement with
+Vanzella et al. (2020a), while the LyC image (5.1l) has a smaller
+radius, Reff ≃ 4.7 ± 1.5 pc. We therefore find a LyC emission
+which is more compact than the non-ionizing UV continuum,
+3 Adopting the pixel scale of 0.03′′/pixel, 8200 pc per arcsecond at
+z=2.37 and µtang ≃ 57, Reff = 0.03∗8200∗((2.62−1.92)0.5)/57, adopting
+the same uncertainty on µtang reported by Vanzella et al. (2022a).
+Article number, page 5 of 10
+
+A&A proofs: manuscript no. VMS
+Fig. 3. The MUSE ultraviolet spectra of the young star cluster (blue) is shown in the bottom panel with the X-Shooter spectrum (green). The
+grey-shaded regions show specific UV features closely associated with the presence of the VMS (Nivλ1486, Heiiλ1640, and Nivλ1720) and some
+of them show a P-Cygni profile, characteristic of young and massive stars. Furthermore, the black line shows the single BPASS model including
+only normal stars at 2.5 Myr, while the red line shows the BPASS single-star model augmented by VMS with masses up to 150 M⊙. The upper
+panels show the zoom in VMS characteristic features compared with models and Nvλ1240 P-Cygni line, characteristic due to the presence of very
+young stellar populations.
+which we interpret as a spatial segregation of the most massive
+stars.
+5. Summary and Conclusions
+In this paper, we have presented a detailed spectroscopic and
+morphological analysis of the massive and young stellar cluster
+hosted in the Sunburst lensed galaxy at z=2.37, for which also
+LyC emission was confirmed in the literature. We used results
+from recent stellar evolutions and atmosphere models including
+VMS (Martins & Palacios 2022) to conduct extensive compar-
+isons with high spectral resolution observations performed with
+VLT/MUSE and X-Shooter. The main results of this work can
+be summarized as follows:
+– In the spectroscopic observations, the high signal-to-noise
+MUSE and X-Shooter spectra reveal features of broad (and
+asymmetric) Heiiλ1640 emission with EW ≃ 3Å rest-frame
+and line width of 1610 km s−1, and Nivλ1486 with EW ≃
+0.2Å emission. In addition, the P-Cygni profile of Nivλ1720
+(along with NV and CIV) is also observed. All these features
+suggest the presence of very massive (> 100M⊙) stars. The
+absence of Ovλ1371 provides a lower age limit of 1 Myr. On
+the other hand, the large Hα EW (> 1231Å after correcting
+for the escaping LyC radiation) indicates an age younger than
+∼ 3 Myr. These narrow age constraints strongly favor the
+existence of VMS over WR stars, implying that the strength
+of the Heiiλ1640 emission line is entirely due to VMS.
+– A comparison of the observations with the models reveals
+that the most plausible age of the star cluster is 2.5 Myr, and
+an estimated number of ∼ 370 − 400 VMS for a cluster mass
+of 107 M⊙. The observations are compatible with an IMF
+extending up to ∼ 175 − 225 M⊙, but with a slope which is
+steeper than the Salpeter IMF.
+– The fraction of LyC radiation emerging from the VMS com-
+ponent is not negligible. We estimate that in the 730Å – 900Å
+range (probed by the HST/F275W band) about 360 – 400
+VMS (or roughly 1% of the total population of O-type stars
+in the star cluster) account for 15% of the escaping LyC pho-
+tons, with the rest being produced mostly by the other less
+massive O-type stars.
+– Detailed morphological analysis of the most magnified im-
+age of the star cluster shows that the region emitting LyC
+is not point-like, with a light profile different from the av-
+erage profile of stars present in the same field of view. This
+is the first evidence of a resolved LyC emission at any red-
+shift. Remarkably, the physical scale of the LyC emitting re-
+gion appears also smaller (with a significant K-S probability
+p << 0.05) than the non-ionizing region (1700Å), suggest-
+ing that massive O-type stars responsible for the LyC radi-
+ation, and likely the VMS (significantly contributing to it),
+are segregated in the central part of the star cluster. After de-
+lensing the angular half-light radii, the LyC region appears
+barely resolved with Reff ≃ 4.7 ± 1.5pc, while at 1700Å it
+is Reff ≃ 7.8 ± 1.4 pc. The packaging of such a large num-
+ber of massive O-type stars per parsec cube in the central
+region, ≃ 70 pc−3, is likely a element which allowed to carve
+the ionizing channel and the development of a high-speed
+outflowing gas (Rivera-Thorsen et al. 2017; Vanzella et al.
+2022a; Mainali et al. 2022).
+Acknowledgements. We acknowledge financial support through grants PRIN-
+MIUR 2017WSCC32, 2020SKSTHZ and the INAF GO Grant 2022 “The rev-
+Article number, page 6 of 10
+
+Nv1240A
+NIV1486A
+NIV1720A
+HeI1640A
+1.6
+1.4
+ Sunburst Cluster Mus
+1.5
+1.3 E
+BPASS 100 M。+ 150 M。VMS (39.43) at 2.5 Myr
+1.4
+1.2
+ 0.7 E
+0.7 
+0.4E
+0.61705
+1220
+1492
+1640
+1650
+1240
+1480
+1484
+1488
+1660
+670
+1710
+1715
+1720
+1725
+1730
+Rest Frame Wavelength [A]
+Rest Frame Wavelength [A]
+ Rest Frame Wavelength [A]
+ Rest Frame Wavelength [A]
+2.0E
+ Sunburst Cluster MUSE
+Sunburst Cluster Xshooter
+BPASS 2.5 Myr
+1.8E
+Single BPASSup to 100 Mo + 39.43 number of 150 Mo VMS at 2.5 Myr 
+1.6E
+1.4
+1.2E
+1.0M
+W
+0.8E
+Nor
+0.6
+0.4
+0.2日
+1250
+1300
+1350
+1400
+1450
+1500
+1550
+1600
+1650
+1700
+1750
+Rest Frame Wavelength [A]U. Mestric et al.: Very massive, spatially segregated stars at z=2.4
+Fig. 4. All symbols as in Figure 3 except for the red lines, showing the BPASS single-star model augmented by VMS with masses up to 200 M⊙.
+Fig. 5. The results from Galfit modeling after using a two-component
+fit. From the left, the first panel shows the 5.1l source in the F555W
+band (UV1700Å). The second panel shows the model from Galfit. The
+third panel shows the residual and the fourth panel is the normalized
+residual produced after dividing the residual with the original image.
+The white contour encloses the region used to check the quality of the
+produced model.
+olution is around the corner: JWST will probe globular cluster precursors and
+Population III stellar clusters at cosmic dawn” (PI Vanzella). FC and RP ac-
+knowledge funding from PRIN INAF 1.05.01.85.01.
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+Gräfener, G. & Vink, J. S. 2015, A&A, 578, L2
+Fig. 6. The HST F275W image showing 5.1l LyC leaking source in
+its centre; other two multiple images, of the same source, are labeled
+as 5.1i and 5.1h. Around 5.1l, in the region not contaminated by other
+sources, five models are injected (see text for more details), enclosed
+in the white squares. In the bottom right corner, we shown the largest
+(34pix diameter) aperture size used to construct CoGs.
+Johnson, T. L., Rigby, J. R., Sharon, K., et al. 2017, ApJ, 843, L21
+Köhler, K., Langer, N., de Koter, A., et al. 2015, A&A, 573, A71
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+Marino, R. A., Rosales-Ortega, F. F., Sánchez, S. F., et al. 2013, A&A, 559, A114
+Article number, page 7 of 10
+
+Nv1240A
+NIV1486A
+HeII1640A
+NIV1720A
+- Sunburst Cluster Xshootel
+1.05
+1260
+1480
+1492
+1496
+1630
+1640
+1650
+1670
+1220
+1484
+1488
+1710
+1715
+1725
+1730
+1230
+Rest Frame Wavelength [A]
+Rest Frame Wavelength [A]
+Rest Frame Wavelength [A]
+Rest Frame Wavelength [A]
+2.0
+Sunburst Cluster MUSE
+1.8E
+Sunburst Cluster Xshooter
+BPASS 2.5 Myr
+1.6E
+1.4
+.2
+1.0
+MA
+0.8
+LLLLLLLLLLLLL
+LJON
+0.6
+0.4E
+0.2E
+一
+0.0
+1250
+1300
+1350
+1400
+1450
+1500
+1550
+1600
+1650
+1700
+1750
+Rest Frame Wavelength [A]5.11
+normalised
+model
+residual
+HST
+F555W
+residual5.1h
+1"
+Model5
+Model
+5.1i
+Model
+5.11
+Model
+ModelA&A proofs: manuscript no. VMS
+Fig. 7. Three curves of growth normalized to 1. The cyan CoG cor-
+responds to the 5.1l LyC leaking source located in the Sunburst arc
+(F275W band). The violet CoG is the PSF-convolved, best-fit model of
+the 5.1l source observed in F555W constructed averaging 5 CoGs (see
+text for more details). The orange growth curve is used for comparison
+and it is constructed averaging 20 single CoGs from randomly selected
+stars. In both cases (cyan and violet) error bars are 1σ. Vertical lines in
+the bottom left part of the figure mark the pixel radii at which 50% of
+the light is enclosed. The colors corresponds to the CoG colors.
+Martins, F., Hillier, D. J., Paumard, T., et al. 2008, A&A, 478, 219
+Martins, F. & Palacios, A. 2022, A&A, 659, A163
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+Rigby, J. R., Bayliss, M. B., Sharon, K., et al. 2018b, AJ, 155, 104
+Rigby, J. R., Johnson, T. L., Sharon, K., et al. 2017, ApJ, 843, 79
+Rivera-Thorsen, T. E., Dahle, H., Chisholm, J., et al. 2019, Science, 366, 738
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+arXiv:2209.03417
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+Vink, J. S., Heger, A., Krumholz, M. R., et al. 2015, Highlights of Astronomy,
+16, 51
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+Article number, page 8 of 10
+
+1.0
+0.8
+iction
+rd
+0.6
+ux
+0.4
+0.2
+observed CoG of 5.1l in F275W
+★
+avg CoG for 5 observed Stars
+0.0
+6AE
+CoG model n=0.5
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+Radius
+[pixels]U. Mestric et al.: Very massive, spatially segregated stars at z=2.4
+Appendix A: Initial models
+As described in Section 3.2, we compare the X-Shooter and
+MUSE spectroscopic observations with the BPASS models at
+different ages (1.5 Myr, 2 Myr, and 2.5 Myr). We narrow our
+models to the mentioned age range since the predicted age of the
+cluster is higher than 1.5 Myr (inferred from Nivλ1486 emission
+line) and the lifetime of the VMS is about 2.5 Myr (Martins &
+Palacios 2022). We started with the BPASS which has an upper
+mass limit of 100 M⊙ and added 236.56 stars in the 100 - 175
+M⊙ mass range and 60.30 stars in the mass range 175 – 225 M⊙.
+Resulting models (at different ages) are shown in A.1; all models
+produce significantly strong spectroscopic features characteristic
+of the presence of VMS. To better match models with observa-
+tions, we decreased the numbers of the VMS and the final results
+are presented in Section 3 (Figure. 3 and 4).
+Article number, page 9 of 10
+
+A&A proofs: manuscript no. VMS
+Fig. A.1. The six panels are showing the comparison of the observations (MUSE spectrum, red line) with BPASS models (black line). The left
+column shows three BPASS models with an IMF up to 100 M⊙, ages of 1.5 Myr, 2 Myr, 2.5 Myr and an added number of 236.56 150 M⊙ VMS.
+The right column shows BPASS model with an IMF up to 100 M⊙ at same ages (as shown in previous column) but with added nuber of 236.56
+and 60.30 VMS of 150 M⊙ and 200 M⊙, respectively. In all six panels we can see that the strength of feature characteristic to VMS is higher than
+in observed cluster, see the Section 3 for more details.
+Article number, page 10 of 10
+

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+1
+Privacy-Preserving Distributed Energy Resource
+Control with Decentralized Cloud Computing
+Xiang Huo, Graduate Student Member, IEEE, Mingxi Liu, Member, IEEE
+Abstract—The rapidly growing penetration of renewable en-
+ergy resources brings unprecedented challenges to power distri-
+bution networks – management of a large population of grid-
+tied controllable devices encounters control scalability crises and
+potential end-user privacy breaches. Despite the importance,
+research on privacy preservation of distributed energy resource
+(DER) control in a fully scalable manner is lacked. To fill the
+gap, this paper designs a novel decentralized privacy-preserving
+DER control framework that 1) achieves control scalability over
+DER population and heterogeneity; 2) eliminates peer-to-peer
+communications and secures the privacy of all participating
+DERs against various types of adversaries; and 3) enjoys higher
+computation efficiency and accuracy compared to state-of-the-
+art privacy-preserving methods. A strongly coupled optimization
+problem is formulated to control the power consumption and
+output of DERs, including solar photovoltaics and energy storage
+systems, then solved using the projected gradient method. Cloud
+computing and secret sharing are seamlessly integrated into the
+proposed decentralized computing to achieve privacy preserva-
+tion. Simulation results prove the capabilities of the proposed
+approach in DER control applications.
+Index Terms—Decentralized optimization, distributed energy
+resources, privacy preservation, secret sharing
+I. INTRODUCTION
+A. Related Works
+L
+ARGE-scale deployment of distributed energy resources
+(DERs) has proven efficacy in reducing carbon footprint
+and providing grid-edge services such as voltage control, load
+following, and backup power supply [1]. DERs, including
+energy storage systems (ESSs), solar photovoltaic (PV), and
+electric vehicles (EVs), along with other monitoring and
+controllable devices, can offer significant opportunities for
+advancing efficient, reliable, and cost-effective power grids [2],
+[3]. Though integrating DERs into power grids can provide
+multifarious benefits, such as enhanced energy efficiency and
+economic boost, the high penetration of DERs raises surging
+challenges on the scalability of existing control strategies [4].
+To address the aforementioned challenges in large-scale
+DER control problems, distributed and decentralized control
+strategies are drawing increased attention owing to their
+superior scalability. For instance, a distributed coordination
+method based on local droop control and consensus control
+was designed in [5] to deal with the voltage rise problem
+caused by the high penetration of solar PVs. Zhang et al. in [6]
+proposed an asynchronous distributed leader-follower control
+strategy that optimally schedules DERs to lower the voltage
+The authors are with the Department of Electrical and Computer Engineer-
+ing, University of Utah, Salt Lake City, UT 84112 USA (e-mail: xiang.huo,
+mingxi.liu@utah.edu).
+for peak load shaving and long-term energy saving. To reduce
+the communication burden, a distributed low-communication
+algorithm was proposed in [7] to control islanded PV-battery-
+hybrid systems. Though distributed methods can achieve
+scalability, they generically suffer from massive peer-to-peer
+communications. To overcome this issue, Navidi et al. in
+[8] developed a two-layer decentralized DER coordination
+architecture that can scale the solution to large networks, and
+no direct communication is required between local controllers.
+In [9], a decentralized stochastic control strategy was designed
+for radial distribution systems with controllable PVs and ESSs
+to minimize the demand balancing cost. Huo et al. in [10] pro-
+posed a decentralized shrunken primal-multi-dual subgradient
+algorithm with dimension reduction to achieve scalability w.r.t.
+both agent population size and network dimension.
+Despite the superior scalability and communication effi-
+ciency of decentralized methods, their implementation has
+been significantly hampered by the vulnerability to privacy
+breaches. Furthermore, both distributed and decentralized
+strategies rely heavily on mandatory communications which
+can disclose users’ sensitive information and expose system
+vulnerabilities to adversaries. Differential privacy (DP) has re-
+ceived substantial attention in addressing privacy concerns due
+to its rigorous mathematical formulation [11]. DP-based meth-
+ods add persistent randomized perturbations to the datasets,
+constraints, or objective functions for privacy preservation.
+In [12], a DP-based aggregation algorithm is proposed to
+compensate for solar power fluctuations and protect users’
+personal information. Han et al. in [13] developed a distributed
+optimization algorithm based on DP to preserve the privacy
+of the participating agents. Gough et al. in [14] designed an
+innovative DP-compliant algorithm to ensure that the data
+from consumers’ smart meters are protected. Despite the
+success in privacy preservation, DP-based methods inevitably
+suffer from accuracy loss due to the added perturbations.
+In contrast, encryption-based strategies achieve privacy
+preservation with high accuracy by encrypting the original
+data into cyphertexts, and only those holding private keys
+can decrypt the cyphertexts. Lu et al. in [15] proposed an
+efficient and privacy-preserving aggregation scheme for smart
+grid communications, in which the data is encrypted by Paillier
+cryptosystem. In [16], a privacy-preserving and fault-tolerant
+scheme was designed based on homomorphic cryptosystem
+to achieve secure aggregation of metering data. Similarly,
+Cheng et al. in [17] proposed a novel private collaborative
+distributed energy management system based on homomorphic
+encryption to solve the privacy issues in distribution systems
+and microgirds. Despite the high accuracy, the drawback
+arXiv:2301.02198v1  [math.OC]  5 Jan 2023
+
+2
+of encryption-based methods lies in the prevalent comput-
+ing overhead caused by encryption and decryption. Other
+hardware-integrated privacy-preserving methods, e.g., garbled
+circuit [18], [19], are deficient in flexibility and uneconomic
+due to the hardware cost.
+Secret sharing (SS) [20] is a lightweight cryptographic
+method that can securely distribute a secret among a group
+of participants. Each participant will be allocated a share
+of the secret, and only through the collaboration of certain
+participants where the number of participants is greater than a
+threshold can the secret be reconstructed from their shares.
+Adopting SS, Nabil et al. in [21] designed an SS-based
+detection scheme to identify malicious consumers who steal
+electricity, in which system operators only collect masked
+meter readings from the consumers to avoid privacy vio-
+lation. In [22], an SS-based EV charging control protocol
+was developed to achieve privacy-preserving EV charging
+control for overnight valley filling. Compared with encryption-
+based strategies, SS-based methods can preserve privacy while
+avoiding the heavy computational load. Despite the superiority,
+few research studied the integration of SS into DER control
+due to the highly complex distribution network structure, large
+DER population, and lack of theoretical support in privacy
+guarantees. To fill these gaps, this paper designs a novel SS-
+based privacy-preserving algorithm that merits high efficiency,
+security, and accuracy for large-scale DER control problems.
+B. Statement of Contributions
+The contribution of this paper is three-fold: 1) We propose
+a novel decentralized privacy-preserving algorithm that con-
+currently achieves scalability and privacy in large-scale DER
+control. To the best of our knowledge, this is the first paper
+that proposes a decentralized SS-based algorithm for DER
+privacy preservation, in which decentralized solutions, privacy
+guarantees, and rigorous security proofs are provided; 2) The
+proposed method eliminates the frequent peer-to-peer commu-
+nications and secures the privacy of the participating DERs
+against various types of adversaries. The designed framework
+serves as a benchmark for secure and scalable DER control. 3)
+Compared to state-of-the-art approaches, the proposed method
+can achieve lower computational overhead and identically
+accurate solutions as the non-privacy-concerned algorithms.
+The rest of this paper is organized as follows: In Sec-
+tion II, we construct the models of distribution networks,
+PVs, and ESSs, then formulate the DER control problem
+into a constrained optimization problem. Section III derives
+the decentralized solution via the projected gradient method
+and presents the corresponding DER aggregation and control
+strategies. The SS-based privacy-preserving DER control al-
+gorithm and privacy analyses are provided in Section IV. We
+give simulation results and analyses in Section V. Section VI
+concludes this paper.
+II. PROBLEM FORMULATION
+A. Branch Flow Model
+Consider an n-bus radial distribution network where B =
+{0, 1, . . . , n} denotes the set of buses. Let lij denote the line
+segment connecting buses i and j, L = {1, . . . , h} denote
+the set of lines, Cj denote the set of bus j’s child buses, Vj
+denote the voltage magnitude at bus j, Pij and Qij denote
+the active and reactive power flow from bus i to bus j,
+respectively, and rij and xij be the resistance and reactance of
+line lij, respectively. For bus j, let pc
+j and qc
+j denote the active
+and reactive power consumptions, respectively, and pg
+j and qg
+j
+denote its active and reactive power generations, respectively.
+To simplify the network model, a nonlinear DistFlow model
+[23] can be linearized to the LinDistFlow model by omitting
+the higher order terms with negligible error [24]. Therefore,
+this paper adopts the LinDistFlow model, represented as
+Pij −
+�
+u∈Cj
+Pju = pc
+j − pg
+j
+(1a)
+Qij −
+�
+u∈Cj
+Qju = qc
+j − qg
+j
+(1b)
+V 2
+i − V 2
+j = 2(rijPij + xijQij).
+(1c)
+A radial 13-bus distribution network connected with rooftop
+solar PVs and ESSs is shown in Fig. 1 and will be used as an
+example throughout this paper.
+10
+3
+2
+11
+6
+7
+5
+9
+8
+4
+1
+0
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+P3, Q3
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+TRAnHpIEjFvG2iwRhNCQNSUj7ZgTFLiMtNz+ba63HgXNArv5SAmdoC6IfUpRlJRjn5huRHzxCBQX2oFSPYwYmkty5yzEzhXq+eao5eNijEuOAvMX1Cu7g/r348Hw5qjf1lehJOAhBIzJETHNGJp4hLihnJSlYiSIxwH3VJR8EQBUTY6fi+DB4pxoN+xNULJRyzfydSFIjcpurMXYpLSfnaZ1E+ld2SsM4kSTEk0V+wqCMYB4W9CgnWLKBAghzqrxC3EMcYakiLakQzOmTZ0HztGKeV67rZrl6AyZVBHvgEBwDE1yCKrgDNdAGDyBF/AG3rVn7VX70D4nrQXtd2YX/Ct9AMnIaf+</latexit>
+12
+Fig. 1.
+A radial 13-bus distribution network connected with rooftop solar
+PVs and ESSs.
+In this paper, one control objective is to minimize the
+total power loss of the distribution network by controlling the
+dynamics of PVs and ESSs, which is approximated by
+f1(pg
+1, . . . , pg
+n) =
+�
+lij∈L
+rij
+�∥Pij∥2
+2 + ∥Qij∥2
+2
+V 2
+0
+�
+(2)
+where V0 denotes the nominal voltage magnitude, pg
+j, Pij, and
+Qij ∈ RT are augmented vectors of pg
+j, Pij, and Qij across T
+time intervals, respectively. Note that we only consider active
+power loss and assume reactive power flows Qij to be constant
+vectors. Though the reactive power loss is not included here for
+simplicity, it can be added without affecting algorithm design.
+The active power flows are constrained by
+0 ≤ Pij ≤ Pij
+(3)
+where Pij denotes the maximum active power flow limit.
+B. Solar Photovoltaic
+Let V denote the set of in total V solar PVs. During T time
+intervals of a day, the active power injection ˜pν ∈ RT from
+the νth PV inverter should satisfy
+0 ≤ ˜pν ≤ pv
+ν
+(4)
+
+田田3
+where pv
+ν denotes the maximum active power injection and is
+assumed to be known by the forecast. Herein, the curtailment
+cost can be calculated by [25]
+f2(˜pν) = ∥˜pν − pv
+ν∥2
+2.
+(5)
+C. Energy Storage System
+Let S denote the set of E ESSs. The charging/discharging
+power ˆpσ ∈ RT of the σth ESS is constrained by
+− ps
+σ ≤ ˆpσ ≤ ps
+σ
+(6)
+where ps
+σ and ps
+σ denote the maximum discharging and
+charging power, respectively. Let s0
+σ denote the initial state of
+charge (SoC) of the σth ESS and Hσ ≜ [s0
+σ, . . . , s0
+σ]T ∈ RT .
+Aggregate the charging/discharging power across T time in-
+tervals, then the capacity of the σth ESS is constrained by
+pa
+σ ≤ Hσ + Aˆpσ∆T ≤ pa
+σ
+(7)
+where pa
+σ and pa
+σ denote its lower and upper capacity
+bounds, respectively, ∆T denotes the sampling time, and
+the aggregation matrix A is lower triangular consisting of
+ones and zeros, i.e., element Aˆı,ˆȷ = 1 if ˆı ≥ ˆȷ, element
+Aˆı,ˆȷ = 0 if ˆı < ˆȷ, ∀ˆı, ˆȷ = 1, . . . , T. Therefore, the SoCs of
+ESS σ during T time slots are obtained by aggregating the
+charging/discharging power using A.
+Furthermore, the σth ESS’s degradation cost is calculated in
+terms of the smoothness of charging and discharging by [26]
+f3(ˆpσ) = ∥B ˆpσ∥2
+2.
+(8)
+where B calculates discharging/charging differences between
+adjacent times, i.e., Bˆı,ˆı = 1, ∀ˆı = 1, . . . , T, Bˆı,ˆı+1 =
+−1, ∀ˆı = 1, . . . , T − 1, and all other elements are zeros.
+D. Problem Formulation
+The optimization problem is then formulated to minimize
+the summation of total active power loss, PV curtailment cost,
+and ESS degradation cost within the distribution network as
+min
+˜p, ˆp
+δ1f1(pg) +
+V
+�
+ν=1
+δ2f2(˜pν) +
+E
+�
+σ=1
+δ3f3(ˆpσ)
+s.t.
+(1a), (3), (4), (6), (7)
+(P1)
+where
+˜p
+=
+[˜pT
+1 , . . . , ˜pT
+n]T,
+ˆp
+=
+[ˆpT
+1 , . . . , ˆpT
+n]T, pg
+=
+[pg
+1
+T, . . . , pg
+n
+T]T, and δα denotes the cost coefficient asso-
+ciated with the objective function fα(·). Note that the cost
+coefficients are constants that allow flexible adjustments on
+the weights of the global and local objective functions and
+regulate different units.
+III. DECENTRALIZED OPTIMIZATION
+A. Projected Gradient Method
+This paper achieves scalability in solving (P1) via projected
+gradient method (PGM). PGM decomposes a centralized opti-
+mization problem into local optimizations at agents, resulting
+in a paralleled computing structure. Let M = {1, . . . , m}
+denote the set of agents, e.g., buses or DERs, who work
+cooperatively in solving (P1). In this setting, the κth agent
+updates its decision variable xκ using PGM by
+x(ℓ+1)
+κ
+= PXκ[x(ℓ)
+κ − γ(ℓ)
+κ Φκ(x(ℓ))]
+(9)
+where
+ℓ
+denotes
+the
+iteration
+number,
+x(ℓ)
+=
+[x(ℓ)
+1
+T, . . . , x(ℓ)
+m
+T]T
+includes
+all
+decision
+variables,
+i.e.,
+˜pν and ˆpσ in problem (P1), γ(ℓ)
+k
+denotes the step size, Φκ(·)
+denotes the gradient of the Lagrangian w.r.t. x(ℓ)
+κ , and PXκ[·]
+denotes the projection operation onto set Xκ.
+In (P1), the local constraint of the νth PV in (4) and local
+constraints of the σth ESS in (6) and (7) can be represented
+by two feasible sets Pv
+ν and Pe
+σ as
+Pv
+ν ≜ {˜pν| 0 ≤ ˜pν ≤ pv
+ν}
+(10a)
+Pe
+σ ≜ {ˆpσ| − ps
+σ≤ˆpσ≤ps
+σ, pa
+σ≤H+Aˆpσ∆T ≤ pa
+σ}. (10b)
+In what follows, aiming at reducing the number of coupling
+terms, we rewrite the networked constraints in (1a) and (3) to
+a single inequality constraint based on the network topology.
+To this end, we first represent the active power flows in (1a)
+through active power generations of each bus using
+pi = ˜pi − ˆpi − pc
+i
+(11)
+where pi denotes the aggregated active power generation at
+bus i, ˜pi = �Vi
+ν=1 ˜pν and ˆpi = �Ei
+σ=1 ˆpσ denote the aggre-
+gated active power of all PVs and ESSs that are connected at
+bus i, respectively. Vi and Ei denote the total number of PVs
+and ESSs connected at bus i, respectively.
+For the ιth line flow Pι in the distribution network, the
+from-bus is defined by the bus where the flow begins, and the
+to-bus set is defined by the set of buses that the ιth line flow
+travels to till reaching the edge of the distribution network.
+Let Z ∈ Rn×n denote the adjacency matrix of the distribution
+network and Zι denote the ιth row of Z that represents the
+adjacency vector of the ιth line flow. Let Zι(i) denote the ith
+element of Zι, and Zι(i) = 1 if the ιth power flow has bus i
+as a to-bus, e.g., Z9 = [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0]. Then, the
+power flows in the distribution network can be represented by
+Z. Expand Z across T time slots, we have
+˜Z =
+�
+����
+Z1(1)I Z1(2)I · · · Z1(n)I
+...
+...
+...
+Zn(1)I Zn(2)I · · · Zn(n)I
+�
+����
+(12)
+where I∈RT ×T denotes the identity matrix and ˜Z ∈ RnT ×nT .
+In what follows, let ˜P ∈ RnT denote the aggregated active
+power generations defined in (11) from all buses, we have
+˜P =
+�
+��
+p1
+...
+pn
+�
+�� =
+�
+����
+�V1
+ν=1 ˜pν − �E1
+σ=1 ˆpσ − pc
+1
+...
+�Vn
+ν=Vn−1+1 ˜pν − �En
+σ=En−1+1 ˆpσ − pc
+n
+�
+���� .
+(13)
+Furthermore, ˜P can be rewritten compactly as
+˜P =
+n
+�
+i=1
+∆i (˜pi − ˆpi − pc
+i)
+(14)
+
+4
+where ∆i denotes the aggregation matrix whose ith block is
+represented by the identity matrix I, and all other blocks are
+zeros, e.g., ∆1 = [I, 0, . . . , 0]T ∈ RnT ×T . Then, the active
+power flow of the ιth line can be calculated by
+Pι = ˜Zι ˜P .
+(15)
+Consequently, the power flow limit constraint in (3) becomes
+0 ≤ ˜Zι ˜P ≤ Pι.
+(16)
+Therefore, problem (P1) can be written into
+min
+˜p, ˆp
+δ1f1(pg) +
+V
+�
+ν=1
+δ2f2(˜pν) +
+E
+�
+σ=1
+δ3f3(ˆpσ)
+s.t.
+pν ∈ Pv
+ν, ∀ν ∈ V
+pσ ∈ Pe
+σ, ∀σ ∈ S
+0 ≤ ˜Zι ˜P ≤ Pι, ∀ι ∈ L
+(P2)
+The optimization problem in (P2) seeks to find the optimal
+decision variables, i.e., charging and discharging power ˜pσ’s
+of the ESSs and the active power injection ˆpν’s of the PVs. In
+what follows, we focus on solving (P2) through a decentralized
+fashion based on PGM defined in (9). To solve (P2) via PGM,
+we firstly derive its relaxed Lagrangian as
+L(˜p, ˆp, µl, µu) = δ1f1(pg) +
+V
+�
+ν=1
+δ2f2(˜pν) +
+E
+�
+σ=1
+δ3f3(ˆpσ)
++
+L
+�
+ι=1
+µT
+uι( ˜Zι ˜P − Pι)−
+L
+�
+ι=1
+µT
+lι ˜Zι ˜P (17)
+where µl = [µT
+l1, . . . , µT
+lL]T and µu = [µT
+u1, . . . , µT
+uL]T, µlι
+and µuι denote the dual variables associated with lower and
+upper power flow limits of the line ι, respectively.
+Suppose ˜pν and ˆpσ are decision variables of the νth PV and
+σth ESS connected at bus i, respectively. Take the subgradients
+of (17) w.r.t. the primal variables ˜pν and ˆpσ, we have
+∇ ˜pνL(·) = 2δ2(˜pν − pv
+ν) + 2δ1
+V 2
+0
+L
+�
+ι=1
+rι( ˜Zι∆i)T( ˜Zι ˜P )
++
+L
+�
+ι=1
+( ˜Zι∆i)T(µuι − µlι)
+(18a)
+∇ ˆpσL(·) = 2δ3 ˆpσ − 2δ1
+V 2
+0
+L
+�
+ι=1
+rι( ˜Zι∆i)T( ˜Zι ˜P )
+−
+L
+�
+ι=1
+( ˜Zι∆i)T(µuι − µlι).
+(18b)
+Without affecting the efficacy of the algorithm design, we
+assume all power lines have the same resistance ¯r for the
+simplicity of presentation, herein (18) becomes
+∇ ˜pνL(·) = 2δ2(˜pν − pv
+ν) + ¯δ1πi ˜P + ψi(µu − µl)
+(19a)
+∇ ˆpσL(·) = 2δ3 ˆpσ − ¯δ1πi ˜P − ψi(µu − µl)
+(19b)
+where ¯δ1 = 2δ1
+V 2
+0 ¯r, πi = �L
+ι=1( ˜Zι∆i)T ˜Zι, and ψi denotes the
+ith column block of ˜Z.
+The detailed derivation of the Lagrangian subgradients in
+(19) can be found in APPENDIX A.
+Therefore, based on the calculated subgradients in (18), at
+the ℓth iteration, the νth PV and the σth ESS can update their
+decision variables using PGM by
+˜p(ℓ+1)
+ν
+= ΠPvν
+�
+˜p(ℓ)
+ν
+− αv
+ν,ℓ∇ ˜pνL(ℓ) (·)
+�
+(20a)
+ˆp(ℓ+1)
+σ
+= ΠPeσ
+�
+ˆp(ℓ)
+σ − αe
+σ,ℓ∇ ˆpσL(ℓ) (·)
+�
+(20b)
+where αv
+ν,ℓ and αe
+σ,ℓ denote the primal step sizes of the νth PV
+and the σth ESS, respectively, L(ℓ) (·) denotes the calculated
+Lagrangian in (17) at the ℓth iteration. The dual variables can
+be updated similarly using PGM.
+B. DER Aggregation and Control
+In PGM iterations, the ith agent needs to calculate Φi(xℓ)
+in (9) where the decision variables xi’s from all other agents
+are required. As indicated in (19), calculating subgradients
+∇ ˜pνL(·) and ∇ ˆpσL(·) indeed requires the decision variables
+˜P from all the agents. Specifically, the calculation of subgra-
+dients in (19a) and (19b) are coupled through
+C = Cp + Cd = ¯δ1πi ˜P + ψi(µu − µl)
+(21)
+where Cp and Cd denote the coupling terms associated with
+the primal and dual variables, respectively.
+To clearly demonstrate the information exchange needs in
+subgradient calculation, we exemplify the primal update of the
+ˆνth PV connected at bus 2. The ˆνth PV can update its decision
+variable ˜pˆν using the subgradient in (19a) which is
+∇ ˜pˆνL(·) = 2δ2(˜pˆν − pv
+ˆν) +
+2
+�
+ι=1
+�
+¯δ1πι ˜P + µuι + µlι
+�
+(22)
+where π1 ˜P = �n
+i=1 pi and π2 ˜P = p2 + p3. Therefore, the
+ˆνth PV requires the active power generations pi, ∀i = 1, . . . , n
+from all buses to conduct the update in (20a).
+Based on the above observations, two different aggregation
+and control strategies, i.e., Bus-level aggregation and control
+and DER-level aggregation and control, can be applied as
+shown in Fig. 2. In bus-level aggregation and control, the ith
+Each bus aggregates the decision 
+variables                   and 
+DERs exchange decision variables 
+with others to obtain 
+                 and 
+˜pi=
+XVi
+⌫=1 ˜p⌫
+<latexit sha1_base64="/FNjpGVvIa+B8d61OziIhaF3i9I=">ACR3icdVDLSgMxFM3Ud31VXboJFsFVmRFBXRENy4VbBU6dchkUg3mMSR3hBLm79y4decvuHGhiEsztQufF0IO59yT3HvSXHALYfgY1CYmp6ZnZufq8wu
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+x2gY3SCOoiO/SEXtBrcB8B2/B+2drLRh71tC3qgUf3V+2Gw=</latexit>
+ˆpi=
+XEi
+�=1 ˆp�
+<latexit sha1_base64="J1PqXdAVQ0WCWiXxcnVQymKUhI=">ACSXicbVDPSxtBFJ6N2qaprWl79DIYCp7CbhG0B0EshR4jGBWycX07mSD82OZeSuEYf+9Xnrf9DLx4s4snZuIf648EwH9/3vpn3vryQwmEc/4laK6tr163Ter97v9H98PHEmdIyPmRGnuWg+NSaD5EgZKfFZaDyiU/zS+/1frpFbdOGH2Mi4KPFcy0mAoGKise5HOAX2aGzlxCxUuX1RVJqjfr2jqSpX51ImZgv2kSrWRQgl05z5VgHMG0n8PvdWLTzS+Kuv24n68LPocJA3okaYGWfd
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+Each individual PV or ESS owns
+decision variable     or     to itself  
+˜p⌫
+<latexit sha1_base64="SM/x7D2mHQVQXsiJ9CAKr6xkWEY=">ACBXicbVC7TsMwFHXKq5RXgBEGiwqJqUpQJWCrYGEsEn1ITRQ5jtadezIdpCqKAsLv8LCAEKs/AMbf4PTZoCWI1k+
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+ˆp�
+<latexit sha1_base64="yRsJzl/IUYDZq3zl9fswdZn4Y=">ACBnicbVDLSsNAFJ3UV62vqEsRBovgqiRSUHdFNy4r2Ac0IUwm03boTCbMTIQSsnLjr7hxoYhbv8Gdf+OkzUJbDwxzOde
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+The ith bus performs the primal-
+dual updates for all the DERs
+Each DER performs the primal-
+dual update in Eqs. (22) and (23)
+End iteration if : DERs’ decision variables achieve convergence  
+Aim: Calculate subgradients               and               for the updates in PGM 
+r˜p⌫L(·)
+<latexit sha1_base64="NCdmlfZtlskHCRoWxGd7tDpT0mQ=">ACIHicbVBNS8NAEN34WetX1aOXYBH0UhIpqDfRiwcPFWwVmlIm61dutkNuxOhPwUL/4VLx
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+rˆp�L(·)
+<latexit sha1_base64="WTcG5eEwOkl6SUea4MtPTARM=">ACIXicbVBNS8NAEN34bf2qevQSLEK9lEQE9SZ68eBwX5AU8pks2XbnbD7kQoIX/Fi3/FiwdF
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+˜pi=
+XV
+⌫=1 ˜p⌫
+<latexit sha1_base64="sOXuDwMIMEucHRfVxl8qnlY3zU=">ACRHicdVDLSiQxFE05PtXO7N0E2wEV02VCOqiQZyNSwemW6GrLVKptAbzKJbQhPycW78AHfzBW5cKOJWJtX2wueFkM595CTk5eCW4jf9
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+Buses exchange decision variables 
+with others to obtain 
+8i=1 . . . , n
+<latexit sha1_base64="eYA2cw+pfmQ4bl+JGXOd6TBo2Y=">AB/nicbVDLSgMxFL3js9ZXVy5CRbBhZQZKagLoejGZQX7gM5QMpm0Dc0kQ5IRylDwV9y4UMSt3+HOvzFtZ6GtBwKHc+7lnpw4Uwb1/12lpZXVt
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+a65OQ7B/AHzucP2q2VcA=</latexit>
+ˆpi=
+XE
+�=1 ˆp�,
+<latexit sha1_base64="GuE1wgouyMWpJt3pSfGs7Vt7lQo=">ACSHicbVBNaxsxENW6zZeTpm57zEXEFHoZrcY2h4CoaXQYwJxEvA6y6ws2yL6WKTZgBH6eb302Ft/Qy89tJTeonX2kK8Bocd786SZV1ZSOEzTn0n
+ydO19Y3Nre72zrPd570XL0+dqS3jI2akseclOC6F5iMUKPl5ZTmoUvKz8vJzo59dceuE0Se4rPhEwVyLmWCAkSp6Rb4A9Hlp5NQtVbx8FUIhqD8INHe1KnzuxFzBQRZybaRQAt2FzxXgoH0X0J49IHWFd4WvX46SFdFH4KsBX3S1lHR+5FPDasV18gkODfO0gonHiwKJno5rXjFbBLmPNxhBoUdxO/CiLQ15GZ0pmx8WikK/a2w4NyzZSxs9nA3dca8jFtXOPsw8QLXdXINbv5aFZLioY2qdKpsJyhXEYAzIo4K2ULsMAwZt+NIWT3V34ITt8NsuHg4/
+Gwf/ipjWOT7JF98oZk5D05JF/JERkRr6RX+QP+Zt8T34n/5L/N62dpPW8Ineq07kGLPy2Kg=</latexit>
+ˆpi, ˜pi, 8i = 1 . . . , n
+<latexit sha1_base64="ChPanXivTdMBX2Dsthv7AEM2Hyg=">ACLnicbVDLSgMxFM3UV62vqks3wSK4KGVGCupCKIrgsoJ9QGcomUymDc1MhuSOUIZ+kRt/ReCirj1M0wfC217IeRwzr3JucdP
+BNdg2+9WbmV1bX0jv1nY2t7Z3SvuHzS1TBVlDSqFVG2faCZ4zBrAQbB2ohiJfMFa/uBmrLcemdJcxg8wTJgXkV7MQ04JGKpbvHX7BDLXlyLQw8hcWTIadXkZu8BFwJYqoVRECMyvHFcEnTZvFOyK/ak8CJwZqCEZlXvFl/dQNI0YjFQbTuOHYCXkYUcCrYqOCmiWEDkiPdQyMScS0l03WHeETwTYuDAnBjxh/05kJNJjy6YzItDX89qYXKZ1UgvIzHSQosptOPwlRgkHicHQ64YhTE0ABCFTdeMe0TRSi
+YhAsmBGd+5UXQPKs41crlfbVUu57FkUdH6BidIgedoxq6Q3XUQBQ9oRf0gT6tZ+vN+rK+p605azZziP6V9fMLNxmqbQ=</latexit>
+r˜p⌫L(·)
+<latexit sha1_base64="NCdmlfZtlskHCRoWxGd7tDpT0mQ=">ACIHicbVBNS8NAEN34WetX1aOXYBH0UhIpqDfRiwcPFWwVmlIm61dutkNuxOhPwUL/4VLx4U0Z
+v+GjdtDn4NLPt4b4Z58JEcIOe9+HMzM7NLyxWlqrLK6tr67WNzY5RqasTZVQ+iYEwSXrI0cBbtJNIM4FOw6HJ0V+vUd04YreYXjhPViuJV8wCmgpfq1w0BCKCfBchFxLIgVCIy49h+WZLnlpdpngcx4JCyC7yvYBGCvf7tbrX8Cbl/gV+CeqkrFa/9h5EiqYxk0gFGNP1vQR7GWjkVLC8GqSGJUBHcMu6FkqImelkwNzd9cykTtQ2j6J7oT9
+PpFBbArTtrNwan5rBfmf1k1xcNTLuExSZJOFw1S4aJyi7TciGtGUYwtAKq59erSIWigaDOt2hD83yf/BZ2Dht9sHF826yenZRwVsk12yB7xySE5IekRdqEknvySJ7Ji/PgPDmvztu0dcYpZ7bIj3I+vwD93qVW</latexit>
+rˆp�L(·)
+<latexit sha1_base64="WTcG5eEwOkl6SUea4MtPTARM=">ACIXicbVBNS8NAEN34bf2qevQSLEK9lEQE9SZ68eBwX5AU8pks2XbnbD7kQoIX/Fi3/FiwdFvIl/
+xk3bg7YOLPt4b4Z58JEcIOe9+UsLC4tr6yurZc2Nre2d8q7ew2jUk1ZnSqhdCsEwSXrI4cBWslmkEcCtYMh9eF3nxk2nAlH3CUsE4Mfcl7nAJaqls+DySEArpZMADMglCJyIxi+2VJnlvW8H4MeR7EgAMKIrvNqwGNFB53yxWv5o3LnQf+FTItO65c8gUjSNmUQqwJi27yXYyUAjp4LlpSA1LAE6hD5rWyghZqaTjS/M3SPLRG5PafskumP290QGsS
+l8287CqZnVCvI/rZ1i7yTcZmkyCSdLOqlwkXlFnG5EdeMohZAFRz69WlA9BA0YZasiH4syfPg8ZJzT+tXdyfVi6vpnGskQNySKrEJ2fktyQO1InlDyRF/JG3p1n59X5cD4nrQvOdGaf/Cn+wfAO6W5</latexit>
+Individual DER acts an agent to 
+calculate subgradients 
+and               
+r˜p⌫L(·)
+<latexit sha1_base64="NCdmlfZtlskHCRoWxGd7tDpT0mQ=">ACIHicbVBNS8NAEN34WetX1aOXYBH0UhIpqDfRiwcPFWwVmlIm61dutkNuxOhPwUL/4VLx4U0Z
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+rˆp�L(·)
+<latexit sha1_base64="WTcG5eEwOkl6SUea4MtPTARM=">ACIXicbVBNS8NAEN34bf2qevQSLEK9lEQE9SZ68eBwX5AU8pks2XbnbD7kQoIX/Fi3/FiwdFvIl/
+xk3bg7YOLPt4b4Z58JEcIOe9+UsLC4tr6yurZc2Nre2d8q7ew2jUk1ZnSqhdCsEwSXrI4cBWslmkEcCtYMh9eF3nxk2nAlH3CUsE4Mfcl7nAJaqls+DySEArpZMADMglCJyIxi+2VJnlvW8H4MeR7EgAMKIrvNqwGNFB53yxWv5o3LnQf+FTItO65c8gUjSNmUQqwJi27yXYyUAjp4LlpSA1LAE6hD5rWyghZqaTjS/M3SPLRG5PafskumP290QGsS
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+Individual bus acts an agent to 
+calculate subgradients 
+and               
+DER-level aggregation and control
+Bus-level aggregation and control
+Fig. 2.
+Aggregation and control of DERs via bus-level and DER-level
+architectures.
+bus (agent) aggregates the decision variables ˜pi = �Vi
+ν=1 ˜pν
+
+5
+and ˆpi = �Ei
+σ=1 ˆpσ where only aggregated decision variables
+are transmitted and used for the primal updates. In contrast,
+DER-level control strategies require each DER to act as an
+agent and receive all data of others that is demanded for
+updates in (20). However, due to the large number of DERs
+connected to the distribution network, DER-level control can
+suffer from massive data exchange and heavy local computa-
+tion. Therefore, we adopt the bus-level aggregation and control
+scheme which is more computing and communicating effi-
+cient. We will later show that the proposed privacy-preserving
+algorithm can be readily extended to the DER-level control
+(See Remark 1 for details).
+Apart from scalability and efficiency, the inevitable private
+information exposure in both bus-level and DER-level methods
+raises fundamental privacy concerns, e.g., the electrical load
+can reveal sensitive business activities and/or customer’s daily
+routines. To address the privacy concerns, we will develop
+a novel SS-based algorithm to achieve secure information
+exchange in executing (20).
+IV. SS-BASED PRIVACY-PRESERVING DER CONTROL
+A. Real Number to Integer Quantization
+Note that the SS scheme requires modular arithmetic instead
+of real arithmetic. However, decentralized optimization ge-
+netically requires real number calculations, e.g., real decision
+variables and parameters. Therefore, a real number to integer
+transformation is needed to integrate SS into decentralized
+optimization. We adopt the fixed-point number quantization
+[27] to map the real numbers onto the integer space and the
+fixed-point real-number set is defined by
+Qθ,γ,ζ≜
+�
+−θγ, −θγ + θ−ζ, . . . , θγ − 2θ−ζ, θγ − θ−ζ�
+(23)
+where θ ∈ N1+ denotes the basis, γ ∈ N denotes the
+magnitude, and ζ ∈ N denotes the resolution. Therefore, by
+defining a surjective mapping m(·) : R �→ Qθ,γ,ζ, a real
+number can be mapped to the closest point in Qθ,γ,ζ. To limit
+the quantization error, the mapping m(·) needs to satisfy
+|m(ϕ) − ϕ| ≤ θ−ζ, ∀ϕ ∈ [−θγ, θγ]
+(24)
+where the quantization error is restricted by the resolution
+within the range of Qθ,γ,ζ. To map the real-number set onto
+the integer set Z, we simply scale Qθ,γ,ζ by θζ as
+Zθ,γ,ζ = θζQθ,γ,ζ=
+�
+−θγ+ζ, −θγ+ζ+1, . . . , θγ+ζ−1
+�
+(25)
+where Zθ,γ,ζ ⊆ Z denotes the fixed-point set in the integer
+field. Moreover, the SS requires the inputs to be within the
+field E. Therefore, we further map each element in z ∈ Zθ,γ,ζ
+onto E with the modular operation as
+g(z) = z mod e.
+(26)
+Note that z ∈ Zθ,γ,ζ can be any negative integer, and the
+modular operation in (26) will change the sign of a negative
+input, i.e., g(ˆz) = ˆz + e for ˆz < 0. To address the negative
+integer operation, we introduce the partial inverse of g(·) as
+ψ(z) =
+� z − e
+if z ≥ e
+2,
+z
+otherwise.
+(27)
+Therefore, we can readily obtain z = ψ(g(z)), ∀z ∈ E.
+B. SS-based Privacy-Preserving Algorithm
+1) Shamir’s secret sharing scheme: Before introducing the
+privacy-preserving algorithm design, we first briefly intro-
+duce Shamir’s SS scheme [20] which merits an efficient and
+lightweight private information distribution structure. Suppose
+a manager (secret holder) seeks to distribute a secret ω to
+specific agents and mandates the cooperation of at least d
+agents to retrieve the secret. In such needs, Shamir’s SS is
+grounded on the following idea of Lagrange interpolation for
+secret distribution and recovery.
+Theorem 1 (Polynomial interpolation [28]). Let {(ς1, y1), . . . ,
+(ςd, yd)} ⊆ R2 be a set of points whose values of ςı are all
+distinct. Then there exists a unique polynomial Y of degree
+d − 1 that satisfies yı = Y(ςı), ∀ı = 1, . . . , d.
+■
+In SS-based schemes, the manager first constructs a random
+polynomial of degree d − 1 as
+y(z) = ω + a1z + · · · + ad−1zd−1
+(28)
+where ω denotes an integer secret, a1, . . . , ad−1 are random
+coefficients that are uniformly distributed in the field E ≜
+[0, e), and e denotes a prime number that is larger than ω.
+Secondly, the manager calculates the outputs of (28) with
+non-zero integer inputs, e.g., setting τ = 1, . . . , n to retrieve
+(τ, y(τ)) where yΠ
+τ
+= y(τ) mod e. Then, the share yΠ
+τ
+is
+distributed to agent τ. Lastly, at least d agents with shares
+are required to reconstruct the polynomial based on Theorem
+1 and hence recover the secret ω by
+ω =
+d
+�
+τ=1
+yΠ
+τ
+d
+�
+υ=0
+υ̸=τ
+υ
+υ − τ .
+(29)
+2) Proposed privacy-preserving algorithm: We next present
+the proposed two-layer decentralized privacy-preserving al-
+gorithm based on SS in a bus-level aggregation and control
+architecture, to achieve privacy preservation and scalability
+concurrently. In the distribution network layer, all DERs’ deci-
+sion variables are updated in parallel, and only masked data are
+sent from each bus to the servers. In the cloud computing layer,
+the servers calculate the aggregated messages and distribute
+them to the related buses. The computing structure of the
+proposed privacy-preserving algorithm is shown in Fig. 3.
+Cloud Computing
+Distribution 
+Network
+ESS
+Solar 
+PV
+Server
+Secure 
+Data Flow
+Secure 
+Data Flow
+Bus
+Fig. 3. Two-layer privacy-preserving computing structure for DER control in
+distribution networks.
+
+田田Compute20066
+Let C denote the set of clouds and c ≥ 2 denotes the total
+number of clouds. The ith bus generates a random polynomial
+of order d − 1 using (28) to obtain
+y(ℓ)
+i (z) = ω(ℓ)
+i
++ a(ℓ)
+i,1z + · · · + a(ℓ)
+i,d−1zd−1
+(30)
+where 2 ≤ d ≤ c, ω(ℓ)
+i
+denotes the secret of bus i at the ℓth
+iteration, ℓ denotes the iteration number, and a(ℓ)
+i,1, . . . , a(ℓ)
+i,d−1
+denote random coefficients that are uniformly distributed in the
+field E. Note that for a vector secret such as pi, we refer to an
+elementwise calculation of the vector using (30) by default.
+At the ℓth iteration, the uth cloud firstly generates a random
+integer α(ℓ)
+u , then it broadcasts α(ℓ)
+u
+to all the buses. Subse-
+quently, the ith bus can calculate y(ℓ)
+i (α(ℓ)
+u ), ∀u = 1, . . . , c
+using the received inputs based on (30). Finally, the ith bus
+sends y(ℓ)
+i (α(ℓ)
+u ) back to the uth cloud. Note that the coupling
+term πi ˜P in (21) is a linear combination of all pi’s that
+requires the private generation/consumption details from the
+buses. Therefore, a secure computation framework of πi ˜P is
+required to preserve the privacy of buses and DER owners.
+Suppose the clouds are aware of the network topology
+matrix Z which contains no private information of the buses
+or DERs. In order to calculate the aggregated information πi ˜P
+for bus i, the uth cloud firstly multiplies the received outputs
+y1(α(ℓ)
+u ), . . . , yn(α(ℓ)
+u ) utilizing the coefficients of πi to obtain
+{α(ℓ)
+u , πi(1)y(ℓ)
+1 (α(ℓ)
+u ), . . . , πi(n)y(ℓ)
+n (α(ℓ)
+u )}
+(31)
+Then, the uth cloud sums the outputs in (31) to obtain a new
+pair of input and output as
+¯
+Au,i = {α(ℓ)
+u ,
+n
+�
+ˆı=1
+πi(ˆı) y(ℓ)
+ˆı (α(ℓ)
+u )}.
+(32)
+Finally, the uth cloud calculates
+¯
+Au,i, ∀i = 1, . . . , n and
+broadcasts the new input-output share ¯
+Au,i to the ith bus.
+Therefore, after receiving new shares from in total c clouds
+servers, the ith bus now has access to
+˜
+Ai =
+�
+α(ℓ)
+ˆȷ ,
+n
+�
+ˆı=1
+πi(ˆı) y(ℓ)
+��ı (α(ℓ)
+ˆȷ ), ∀ˆȷ = 1, . . . , c
+�
+.
+(33)
+Note that ˜
+Ai contains in total c shares that can construct a
+new polynomial of the form
+˜y(ℓ)
+i (z) = πi ˜P + ˜a(ℓ)
+i,1z + · · · + ˜a(ℓ)
+i,d−1zd−1
+(34)
+whose constant term is exactly πi ˜P .
+During this information exchange process, each bus only
+sends a single share to each server so that a single cloud server
+is incapable of reconstructing the secret based on the received
+shares, and herein cannot infer agents’ true decision variables.
+The cloud servers only need to calculate aggregated messages
+using outputs of randomized polynomials. The details of the
+proposed method are presented via Algorithm 1.
+Algorithm 1 can achieve privacy preservation while main-
+taining exact solutions as non-privacy PGM-based methods.
+The decision variables will be continuously updated till the
+convergence errors ϵ(ℓ)
+ν
+≜ ∥˜p(ℓ)
+ν − ˜p(ℓ−1)
+ν
+∥2
+2 and ϵ(ℓ)
+σ
+≜ ∥ˆp(ℓ)
+σ −
+ˆp(ℓ−1)
+σ
+∥2
+2 are smaller than the threshold ϵ0. The correctness of
+Algorithm 1 is presented via Theorem 2.
+IEEE 13 bus network
+Decentralized 
+updates
+Cloud
+Aggregation
+y(`)
+3 (↵(`)
+1 )
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+Cloud1
+Cloud 2
+Cloud c
+DERs
+DERs
+DERs
+Bus 3
+Bus 2
+Bus 6
+Bus 1
+Bus 4
+Bus 5
+Bus 7
+Bus 9
+Bus 10
+Bus 12
+Bus 11
+Bus 8
+Bus 2
+Bus 1
+Bus 12
+¯
+A(`)
+1,1
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+>
+¯
+A(`)
+1,2
+<latexit sha1_base64="BX3MrG24Z53ODrxs+/8snufJ/7o=">AC
+CXicbVDLSsNAFJ3UV62vqEs3g0WoICUpBXVX68ZlBfuAJobJdNoOnUzCzEQoIVs3/obF4q49Q/c+TdO2iy09cCFwzn3cu89fsSoVJb1bRWVtf
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+DV/BmPBkvxrvxMW8tGPnMIfgD4/MHFgeZ9Q=</latexit>
+¯
+A(`)
+1,12
+<latexit sha1_base64="wbvr5FCeuxjCnzM3Uiy/T
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+Bus 0
+Fig. 4. Information exchange structure between the distribution network and
+cloud servers (only the messages sent from bus 3 and cloud 1 are labeled).
+Algorithm 1 Decentralized SS-based privacy-preserving DER
+control strategy
+1: Agents initialize decision variables, tolerance ϵ0, basis θ,
+magnitude γ, resolution ζ, iteration counter ℓ = 0, and
+maximum iteration ℓmax.
+2: while ϵ(ℓ)
+ν(σ) > ϵ0 and ℓ < ℓmax do
+3:
+Each bus performs real number to integer transforma-
+tion using (23)-(26), then obtains the integer secret ω(ℓ)
+i .
+4:
+The uth cloud generates a random integer α(ℓ)
+u , then
+broadcasts α(ℓ)
+u
+to all the buses.
+5:
+The ith bus generates a random polynomial y(ℓ)
+i (z)
+using (30) with ω(ℓ)
+i
+as the constant term, calculates the
+outputs using α(ℓ)
+1 , . . . , α(ℓ)
+c
+to obtain y(ℓ)
+i (α(ℓ)
+1 ), . . . ,
+y(ℓ)
+i (α(ℓ)
+c ), then sends y(ℓ)
+i (α(ℓ)
+u ) to the uth cloud.
+6:
+The uth cloud formulates ¯
+Au,i in (32), then broadcasts
+¯
+Au,i to the ith bus.
+7:
+The ith bus formulates
+˜
+Ai in (33), reconstructs the
+aggregated secrets using c shares to obtain πi ˜P , then
+calculates Cp in (21).
+8:
+The ith bus transforms Cp back to real numbers
+using (27), then decision variables ˜p(ℓ)
+ν
+or ˆp(ℓ)
+σ
+of DERs
+connected at bus i are updated by PGM using (9). The ith
+bus calculates the error ϵ(ℓ)
+ν
+or ϵ(ℓ)
+σ .
+9:
+ℓ = ℓ + 1.
+10: end while
+Theorem 2 (Correctness). Let E denote the domain of the
+input secrets ω1, . . . , ωn, and Cp denote the desired outputs.
+Then, Algorithm 1 satisfies:
+Pr
+�
+∀c ≥ d, Rec
+�
+A, E, Z, ¯δ1, θ, γ, ζ
+�
+= Cp
+�
+= 1
+(35)
+where A = { ˜
+A1, . . . , ˜
+Ac} denotes the set of shares from
+agents, Pr[·] denotes probability, and Rec(·) denotes the secret
+reconstruction operation.
+■
+Theorem 2 states that Algorithm 1 can correctly retrieve
+the aggregated information Cp which would be further used to
+achieve exact primal and dual updates.
+
+Compute田田7
+The detailed proof of Theorem 2 can be found in AP-
+PENDIX B.
+Remark 1 : Though Algorithm 1 is developed based on bus-
+level aggregation and control, it can also be extended to the
+DER-level aggregation and control. In DER-level aggregation
+and control, each DER is required to generate a polynomial in
+(30) and act as an independent agent in secret reconstruction
+using (33). Besides, depending on the practical applications,
+DERs can also be clustered and controlled by the household
+or district where the new clusters act as agents, following the
+similar design of Algorithm 1.
+□
+Remark 2: The multi-server architecture seamlessly integrates
+the SS scheme into DER aggregation and control. Shares
+generated from buses were aggregated and broadcasted to the
+buses by a group of servers for the purpose of secret retrieval.
+The aggregation task is distributed to multiple servers to ensure
+that a single server cannot retrieve any secrets.
+□
+C. Privacy Analysis
+The proposed approach aims at protecting the decision
+variables of the DERs whose disclosure can lead to the leakage
+of customers’ sensitive information. To resolve this issue, Al-
+gorithm 1 achieves privacy preservation against two types of
+adversaries, including honest-but-curious-agent who follows
+the algorithm but may utilize the possessed and received data
+to infer the private information of other agents, and external
+eavesdroppers who wiretap and intercept exchanged messages
+from communication channels.
+Proposition 1: (Secure cloud computing). In Algorithm 1, any
+cloud number less than d − 1 cannot infer any information of
+the aggregated decision variables Cp.
+■
+Proposition 1 presents the security of the proposed al-
+gorithm against corrupted clouds. Based on the polynomial
+interpolation in Theorem 1, at least d clouds are required to
+retrieve any secret through collusion.
+Proposition 1 is proved based on the correctness analysis.
+Please refer to APPENDIX C for the detailed proof.
+Assumption 1. At least one communication link of an indi-
+vidual agent is secure against external eavesdroppers.
+■
+Assumption 1 is essential and generically used in SS-
+based schemes. Given d pairs of shares sent via different
+communication links, i.e., {(ς1, y1), . . . , (ςd, yd)} ⊆ R2, if
+an external eavesdropper wiretap all communication links to
+gain access to the shares, then it can simply deduce the secret
+by Lagrangian interpolation using Theorem 1.
+Theorem 3 (Privacy preservation against adversaries). By
+using Algorithm 1, the following two statements stand:
+1) Algorithm 1 securely computes and updates the deci-
+sion variables between agents in the presence of honest-
+but-curious agents.
+2) External eavesdroppers learn no private information of
+the agents.
+■
+Theorem 3 gives privacy preservation guarantees in the
+presence of honest-but-curious agents and external eavesdrop-
+pers. The privacy preservation of Algorithm 1 can be proved
+from secure multi-party computation (SMC) perspective. Be-
+fore giving detailed privacy analyses and proofs, we first
+introduce some concepts of SMC.
+Definition 1 (Computational indistinguishability [29]). Let
+{Dκ}κ∈N and {Eκ}κ∈N be two distribution ensembles with
+security parameter κ; If for any non-uniform probabilistic
+polynomial-time algorithm G, δ(κ) is negligible, where
+δ(κ) =
+����
+Pr
+x1←Dκ[G(x1) = 1] −
+Pr
+x2←Eκ[G(x2) = 1]
+����
+(36)
+we say that {Dκ}κ∈N and {Eκ}κ∈N are computationally
+indistinguishable, denoted as Dκ
+c≡ Eκ.
+■
+Therefore, Definition 1 states that any polynomial-time
+algorithm cannot distinguish two computationally indistin-
+guishable ensembles because the outputs of those algorithms
+do not significantly differ. In what follows, Definition 2
+presents the standard privacy notion in SMC.
+Definition 2 ([30], [31]). Let Π be an m-party protocol
+for computing the outputs of function F(¯x) where ¯x =
+{x1, . . . , xm} and Fρ(¯x) denotes the ρth output of F(¯x). Let
+M = {M1, . . . , Mm} denote the set of parties. The view
+of the ρth party during the execution of Π is denoted by
+VIEWΠ
+ρ (¯x). We say that Π privately computes F(¯x) if there
+exists a polynomial-time algorithm S, such that for every party
+Mρ in M, we have
+S(ρ, xρ, Fρ(¯x))
+c≡ VIEWΠ
+ρ (¯x).
+(37)
+■
+Definition 2 states that the security of an m-party protocol
+can be evaluated based on computational indistinguishability,
+i.e., the view of the parties can be efficiently simulated based
+solely on their inputs and outputs. In other words, SMC allows
+a group of participants to learn the correct outputs of some
+agreed-upon function applied to their private inputs without
+revealing anything else. The theoretical underpinnings of Def-
+inition 1 and Definition 2 can help prove that Algorithm 1
+securely computes π1 ˜P , . . . , πn ˜P between the agents.
+The detailed proofs of Theorem 3 can be found in AP-
+PENDIX D.
+V. SIMULATION RESULTS
+A simplified single-phase IEEE 13-bus test feeder [32] is
+used to verify the proposed decentralized privacy-preserving
+DER control strategy. In specific, each bus, except the feeder
+head, is assumed to be connected with 2 houses and each house
+is equipped with an ESS and 5 solar panels that can generate
+maximum 2.5 kW solar output. The maximum capacity of all
+residential ESSs are 10 kWh, the initial SoCs of all ESSs are
+uniformly set to be 4 kWh, and the maximum charging and
+discharging rates are ±3 kW, respectively [33]. The forecasted
+solar PV generation is chosen from 01/01/2021 with ∆T = 15
+mins in California from CAISO [34].
+In total c = 4 clouds are responsible for message aggrega-
+tion and distribution. The degree of all polynomials is set to
+be d−1 = 3 and the integer field is chosen as E = [0, 231−1).
+For the fixed-point number quantization, the basis, magnitude,
+and resolution are uniformly set to be θ = 2, γ = 27, and
+ζ = 4, respectively. For the distribution network shown in
+Fig. 1, all 24 houses are assumed to be located in the same
+area with identical solar radiation. The baseline load profiles
+
+8
+00:00
+04:00
+08:00
+12:00
+16:00
+20:00
+24:00
+Time
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+Power (kW)
+(a) Heterogeneous baseline loads of 24
+houses
+00:00
+04:00
+08:00
+12:00
+16:00
+20:00
+24:00
+Time
+0.0
+0.5
+1.0
+1.5
+2.0
+Solar PV generations (kW)
+(b) Solar power injection of 24 houses
+00:00
+04:00
+08:00
+12:00
+16:00
+20:00
+24:00
+Time
+−1.00
+−0.75
+−0.50
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Charging/discharging (kW)
+(c) Charging and discharging power from
+24 ESSs
+00:00
+04:00
+08:00
+12:00
+16:00
+20:00
+24:00
+Time
+0
+5
+10
+15
+20
+25
+Power (kW)
+(d) Power flows of 12 lines in the
+distribution network
+Fig. 5. The optimal solutions of (P2) by controlling DERs in the distribution network.
+of all houses are shown in Fig. 5(a) [34]. The primal and dual
+step sizes are chosen based on experience to be αv
+ν,ℓ = 2.3,
+αe
+σ,ℓ = 1.8, and βµlι,ℓ = 5×10−4, respectively. Note that only
+the lower bound of power flow limits in (16) is active, herein,
+only the results related to µlι are presented.
+Fig. 5(b) and Fig. 5(c) show the active power generations
+and the charging/discharging power from the solar PVs and
+ESSs, respectively. At around 12:00, the solar PVs generate
+the maximum amount of energy, and the ESSs charge at
+peak rates. After 16:00, energy stored in ESSs is extracted to
+supply in-home use and compensate for the power loss in the
+distribution network. The power flows of 12 lines are shown
+in Fig. 5(d) where no inverse flows occur. Moreover, accurate
+primal and dual solutions are achieved without affecting the
+anticipated primal-dual convergence. The iterative solutions
+of the primal and dual variables are shown in Fig. 6.
+Fig.
+0
+20
+40
+60
+80
+100
+Iterations
+0.0
+0.5
+1.0
+1.5
+2.0
+˜pν
+(a) Convergence of solar PVs’ decision
+variables ˜pν
+0
+100
+200
+300
+400
+500
+600
+700
+Iterations
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+µlι
+(b) Convergence of the dual variable µlι
+Fig. 6. Convergence of the primal and dual variables
+Fig. 7. Random shares generated by Bus 6 at different iterations
+7 presents normalized shares generated by Bus 6 using the
+random polynomial y(ℓ)
+6 (z) = ω(ℓ)
+6
++ a(ℓ)
+1 z + a2z2 + a(ℓ)
+3 z3
+where the coefficients a(ℓ)
+i , i = 1, 2, 3 are randomized at each
+iteration and different time slots. The privacy preservation of
+Algorithm 1 against external eavesdroppers are guaranteed
+because external eavesdroppers have insufficient information
+in polynomial reconstruction by wiretapping the transmitted
+shares. Without loss of generality, suppose bus 6 is honest-
+but-curious. Fig. 8 shows the existence of a simulator that
+−1
+0
+1
+×1015
+True polynomial y6(z)
+−100
+−75
+−50
+−25
+0
+25
+50
+75
+100
+−1
+0
+1
+×1015
+Simulated polynomial ˜y′
+6(z)
+True constructed polynomial ˜y6(z)
+Fig. 8.
+Polynomials simulated by a simulator to achieve computational
+indistinguishability among agents
+can generate true polynomial y6(z) and simulated polyno-
+mials y′
+i(z) (dashed lines), ∀i = 1, . . . , n, i ̸= 6, such that
+(π6 ˜P )′ = π6 ˜P . Therefore, the computational indistinguisha-
+bility ˜y′
+6(αj)
+c≡ ˜y6(αj), ∀j = 1, . . . , c is satisfied at any
+iteration and any time slot, and herein π1 ˜P , . . . , πn ˜P can
+be securely computed among buses and the ith bus can only
+know the information contained in its own view VIEWi.
+VI. CONCLUSION
+This
+paper
+proposed
+a
+novel
+decentralized
+privacy-
+preserving algorithm with cloud computing architecture for
+DER control in distribution networks. The DER control prob-
+lem was formulated into a constrained optimization problem
+with the objectives of minimizing the line loss, PV curtailment
+cost, and ESS degradation cost. By integrating SS into the
+decentralized PGM, the proposed approach achieved privacy
+preservation for DER owners’ private data, including the
+DERs’ generation, consumption and daily electricity usage.
+The security of the proposed approach was proved rigor-
+ously with privacy guarantees and analyses against honest-but-
+curious agents and external eavesdroppers. Simulation results
+verified the applicability of the proposed approach on the
+modified IEEE 13-bus test feeder with controllable ESSs
+and solar PVs. Moreover, the designed methodology can be
+readily used in general large-scale decentralized optimization
+problems in the context of privacy preservation provisions.
+APPENDIX A
+DERIVATION OF THE PGM UPDATES
+We take the IEEE 13-bus test feeder in Fig. 1 for example
+to illustrate the derivation of subgradients in (18). To prove
+
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+00:00
+04:00
+08:00
+104
+12:00
+103
+16:00
+102
+Time
+20:00
+101
+24:009
+(18a), we firstly consider the subgradient of the power loss
+minimization objective, the active power loss is
+f1(pg
+1, . . . , pg
+n) = δ1
+�
+lij∈L
+rij
+�∥Pij∥2
+2
+V 2
+0
+�
+= δ1¯r
+V 2
+0
+�
+ι∈L
+∥Pι∥2
+2
+=
+¯δ1
+2
+�
+ι∈L
+∥Pι∥2
+2.
+(38)
+Take (15) into (38), we have
+f1(pg
+1, . . . , pg
+n) =
+¯δ1
+2
+�
+ι∈L
+∥ ˜Zι ˜P ∥2
+2.
+(39)
+Without loss of generality, assume the νth PV with decision
+variable ˜pν is connected at bus i, we have
+∇ ˜pνL(·) = δ1∇ ˜pνf1(pg
+1, . . . , pg
+n) + δ2∇ ˜pνf2(˜pν)
++
+L
+�
+ι=1
+∇ ˜pνµT
+uι( ˜Zι ˜P −Pι)−
+L
+�
+ι=1
+∇ ˜pνµT
+lι ˜Zι ˜P . (40)
+Substitute (14) and (38) into the first term of (40), we have
+δ1∇ ˜pνf1(·) =
+¯δ1
+2 ∇ ˜pν
+�
+ι∈L
+∥ ˜Zι ˜P ∥2
+2
+= ¯δ1
+�
+ι∈L
+�
+∇ ˜pν ˜Zι
+n
+�
+ˆı=1
+∆ˆı ˜pˆı
+� �
+˜Zι ˜P
+�
+= ¯δ1
+�
+ι∈L
+�
+˜Zι∆i
+�T �
+˜Zι ˜P
+�
+.
+(41)
+Take the subgradient of (5), the second term in (40) becomes
+δ2∇ ˜pνf2(˜pν) = δ2∇ ˜pν∥˜pν − pv
+ν∥2
+2 = 2δ2 (˜pν − pv
+ν) . (42)
+Then, substitute (14) into the third term of (40) on the right
+hand side, we have
+L
+�
+ι=1
+∇ ˜pνµT
+uι( ˜Zι ˜P − Pι) =
+L
+�
+ι=1
+∇ ˜pνµT
+uι ˜Zι(
+n
+�
+ˆı=1
+∆ˆı ˜pˆı)
+=
+L
+�
+ι=1
+( ˜Zι∆i)
+Tµuι.
+(43)
+Similarly, the last term of (40) can be readily obtained as
+−
+L
+�
+ι=1
+∇ ˜pνµT
+lι( ˜Zι ˜P ) = −
+L
+�
+ι=1
+( ˜Zι∆i)
+Tµlι.
+(44)
+Finally, by substituting (41), (42), (43), (44) into (40), (18a)
+is readily proved. Following similar lines, subgradients of the
+primal variable ˆpσ in (18b) can be readily proved.
+APPENDIX B
+PROOF OF THEOREM 2
+Proof: To prove the correctness of Algorithm 1, we show
+that the proposed method has the same primal and dual
+solutions as the non-privacy PGM. Recall that the uth cloud
+multiplies the received n outputs by the elements of πi
+according to (31), it yields
+�
+�
+�
+�
+�
+πi(1)y1(αu) = πi(1)
+�
+ω1 + a1,1αu + · · · + a1,d−1αd−1
+u
+�
+...
+πi(n)yn(αu) = πi(n)
+�
+ωn + an,1αu + · · · + an,d−1αd−1
+u
+�
+(45)
+Then, the aggregated outputs �n
+ˆı=1 πi(ˆı)yˆı(αu) in (31) can be
+obtained by summing the left hand side of (45). Therefore, in
+total c pairs of shares from all clouds as in (32) can be seen
+as the inputs and outputs of a polynomial
+˜y(z) =
+n
+�
+ˆı=1
+πi(ˆı)ωˆı + ˜a1z + · · · + ˜ad−1zd−1
+(46)
+where ˜aˆȷ = �n
+ˆı=1 πi(ˆı)aˆı,ˆȷ, ˆȷ = 1, . . . , d���1 and �n
+ˆı=1 πi(ˆı)ωˆı
+is exactly πi ˜P . Then, the aggregated secret πi ˜P can be
+readily retrieved by using c pairs of shares in (33) since d ≤ c,
+as stated by Theorem 1.
+APPENDIX C
+PROOF OF PROPOSITION 1
+Proof: Under the collusion of d − 1 clouds, they can
+construct the following set of equations
+�
+�
+�
+�
+�
+˜yi(α1) = ˜ω + ˜ai,1α1 + · · · + ˜ai,d−1αd−1
+1
+...
+˜yi(αd−1) = ˜ω + ˜ai,1αd−1 + · · · + ˜ai,d−1αd−1
+d−1
+(47)
+where ˜yi(z) is defined in (34) and ˜ω = πi ˜P . In (47), ˜ai,ı,
+∀ı = 1, . . . , d − 1 and ˜ω are unknown, therefore the d − 1
+clouds can yield in total d − 1 equations yet d unknowns that
+leads to underdetermined solutions.
+APPENDIX D
+PROOF OF THEOREM 3
+Proof: To prove the privacy preservation of Algorithm
+1 against honest-but-curious agents, we aim at verifying
+that whatever an honest-but-curious agent receives can be
+efficiently simulated. That being said, the honest-but-curious
+agent cannot retrieve useful information from others using the
+received data because it cannot distinguish the received data
+from its own. During the ℓth iteration of executing Algorithm
+1, the view of bus i can be described via
+VIEWi = {α1, . . . , αc, θ, γ, ζ, πi ˜P , yi(z), ωi, ¯
+Ai,
+˜yi(αj), ∀j = 1, . . . , c, Cp, Cd}.
+(48)
+Based on Definition 2, we need to prove the existence of a
+polynomial-time algorithm, denoted as simulator S, that can
+simulate VIEWi using the data of agent i, i.e.,
+S(Ξi)
+c≡ VIEWi
+(49)
+where Ξi ≜ {α1, . . . , αc, θ, γ, ζ, πi ˜P , yi(z), ωi, ¯
+Ai, ˜yi(αj),
+∀j = 1, . . . , c, Cp, Cd} denotes the set of data that agent
+i has access to. Manifesting (49) indicates that whatever
+agent i receives can be efficiently reconstructed based on its
+own knowledge Ξi. To this end, the simulator is required to
+generate ˜y′
+i(αj),∀j = 1, . . . , c that satisfy
+˜y′
+i(αj)
+c≡ ˜yi(αj), ∀j = 1, . . . , c.
+(50)
+To achieve this goal, the simulator firstly generates secrets
+w′
+j̸=i ∈ E of other agents such that
+πi ˜P = wi +
+�
+j̸=i
+w′
+j.
+(51)
+
+10
+Then it generates a set of random polynomials as in (30) to
+obtain y′
+j(z), ∀j ̸= i with w′
+j, ∀j ̸= i as the corresponding
+constant terms, i.e.,
+�
+yi(z) = wi + ai,1z + · · · + ai,d−1zd−1
+(52a)
+y′
+j(z) = w′
+j + a′
+i,1z + · · · + a′
+i,d−1zd−1, ∀j ̸= i.
+(52b)
+Consequently, the simulator can use {α1, . . . , αc} as inputs
+for (52) and obtain
+˜
+A′
+i =
+�
+�
+�αˆȷ, yi(αˆȷ) +
+�
+j̸=i
+y′
+j(αˆȷ), ∀, ˆȷ = 1, . . . , c
+�
+�
+� .
+(53)
+By Theorem 1 and Theorem 2, the shares in (53) can be
+used to construct a new polynomial in the form of
+˜y′
+i(x) = (πi ˜P )′ + ˜a′
+i,1z + · · · + ˜a′
+i,d−1zd−1
+(54)
+where (πi ˜P )′ = πi ˜P . Therefore, (50) and (49) hold, by
+Definition 2, Algorithm 1 securely computes π1 ˜P , . . . , πn ˜P
+between the agents.
+In what follows, we prove the privacy preservation of Al-
+gorithm 1 against external eavesdroppers. Under Assumption
+1, assume agent 1 is safe from external eavesdroppers, by
+wiretapping any other agents’ communication channels, an
+external eavesdropper can at most have access to
+Ξe=
+�
+α1,. . ., αc,yi(αu), ¯
+Au,i, ∀i=2,. . ., n,u=1, . . ., c
+�
+.
+(55)
+Since (55) is insufficient to formulate (33), the external eaves-
+dropper is incapable of inferring either yi(z)’s or ˜y′
+i(z)’s,
+i.e., unable to infer agents’ private information pi’s or the
+aggregated message πi ˜P ’s.
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+

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F9E3T4oBgHgl3EQfWAol/content/tmp_files/2301.04464v1.pdf.txt

@@ -0,0 +1,176 @@
+arXiv:2301.04464v1  [math.NT]  8 Jan 2023
+Runs of Consecutive Integers Having the
+Same Number of Divisors
+By Vlad-Titus Sp˘ataru
+Abstract
+Our principal objective is to provide an upper bound for the length ℓN of the
+longest run of consecutive integers smaller than N which have the same number of
+divisors. We prove that ℓN ⩽ exp �C√log N log log N� in an elementary manner.
+1. Introduction
+The equation d(n) = d(n + k) has been studied extensively.
+In 1981, Spiro [Spi81]
+showed that it has infinitely many solutions for k = 5040.
+Subsequently, Heath-Brown
+[HB84] extended Spiro’s work to deal with the case k = 1, and Pinner [Pin97] ultimately
+proved that, in fact, all values of k yield infinitely many solutions.
+As d(n) = d(n+1) infinitely often, one naturally wonders how many consecutive integers
+can there be, having the same number of divisors. Erd˝os and Mirsky [EM52] conjectured
+that there are arbitrarily long such runs of integers. They were not able to provide any
+estimates for the length of such sequences: “A related problem consists in the estimation of
+the longest run of consecutive integers ⩽ x all of which have the same number of divisors.
+This problem seems to be one of exceptional difficulty, and we [Erd˝os & Mirsky] have not
+been able to make any progress with it.”
+Our principal objective is to provide an upper bound for the length of the runs in
+question. We shall obtain the following result, in an elementary manner:
+Theorem 1. Let ℓN denote the length of the longest run of consecutive integers smaller
+than N, having the same number of divisors. Then,
+ℓN ⩽ exp
+Ä
+C
+�
+log N · log log N
+ä
+,
+for an absolute constant C.
+Keywords: divisor counting function, consecutive equidivisible integers.
+2020 Mathematics Subject Classification: Primary: 11A25, 11N37.
+1
+
+2
+VLAD-TITUS SP˘ATARU
+2. The main result
+In proving theorem 1, we will make use of the following lemmas, the first being proven
+in an elementary manner in [Far09] and the second being Mertens’ bound.
+Lemma 1. Let n be a positive integer. Then, lcm(1, 2, . . . , n + 1) ⩾ 2n.
+Lemma 2. There exists an absolute constant C1 such that for any positive integer n ⩾ 2,
+�
+p⩽n
+1
+p ⩽ C1 · log log n,
+the sum being over all prime numbers p not exceeding n.
+Note that it suffices to prove that theorem 1 holds for large enough N. Assume that
+there exist k > 2 consecutive numbers smaller than N, having the same number of divisors.
+Let them be n + 1, n + 2, . . . , n + k and write
+d(n + 1) = d(n + 2) = · · · = d(n + k) = D.
+We will firstly provide an estimate for D, in terms of k. For simplicity, let K = ⌊log2 k⌋.
+As k ⩾ 2K, all residues modulo 2K are among n + 1, n + 2, . . . , n + k. Therefore, for all
+1 ⩽ i ⩽ K − 1, there exists some 1 ⩽ ti ⩽ k such that n + ti ≡ 2i mod 2K. Consequently,
+ν2(n + ti) = i, so (i + 1) divides d(n + ti) = D.
+Hence, D is divisible by lcm(1, 2, . . . , K). Using lemma 1, we infer that
+D ⩾ lcm(1, 2, . . . , K) ⩾ 2K−1.
+Recall that K = ⌊log2 k⌋ ⩾ log2 k − 1, so D ⩾ k/4.
+Next, we will bound ω((n + 1) · · ·(n + k)). Choose 1 ⩽ l ⩽ k arbitrarily. As n + l ⩽ N,
+it follows that νp(n + l) ⩽ logp N ⩽ log2 N for all prime numbers p. Therefore,
+D = d(n + l) =
+�
+p
+(νp(n + l) + 1) ⩽
+�
+p|n+l
+(log2 N + 1) = (log2 N + 1)ω(n+l),
+where p always represents a prime number. Thus, ω(n+l) ⩾ log D/ log(log2 N +1). A prime
+number p can divide at most ⌈k/p⌉ ⩽ k/p + 1 numbers among n + 1, . . . , n + k, so
+ω((n + 1) · · ·(n + k)) ⩾
+k
+�
+i=1
+ω(n + i) −
+�
+p⩽k
+k
+p,
+
+RUNS OF CONSECUTIVE INTEGERS HAVING THE SAME NUMBER OF DIVISORS
+3
+the second sum being taken over all prime numbers p not exceeding k. Using lemma 2 and
+the inequality we have previously deduced for ω(n + l), we may finally infer that
+ω((n + 1) · · ·(n + k)) ⩾
+k · log D
+log(log2 N + 1) − C1k log log k.
+Further, we will write log(log2 N +1) ⩽ C2 log log N, for an absolute constant C2. Recall
+that D ⩾ k/4, so we have
+ω((n + 1) · · ·(n + k)) ⩾ k · log(k/4)
+C2 log log N − C1k log log k.
+(1)
+Write the right-hand side of equation 1 as k · fN(k). Clearly, if ω(a) ⩾ b then a ⩾ b!.
+Using this remark on equation 1, we get (n+1) · · · (n+k) ⩾ ⌈k ·fN(k)⌉!. Moreover, because
+Nk ⩾ (n + 1) · · ·(n + k), by taking logarithms and applying the well-known inequality
+log t! ⩾ t log t − t, we have
+k log N ⩾ log ((n + 1) · · · (n + k)) ⩾ log (⌈k · fN(k)⌉!)
+⩾ k · fN(k) · log(k · fN(k)) − k · fN(k).
+(2)
+Finally, dividing equation 2 by k we obtain
+log N ⩾ fN(k) · log(k · fN(k)) − fN(k).
+(3)
+Define the interval IN = [exp (C1 · C2 · log log N) , ∞). Using standard arguments, one
+may infer that fN is increasing on IN.
+Let us suppose, for the sake of contradiction, that k > exp �C√log N log log N�, where
+C > max(√C2, C1 · C2). Firstly, note that since log N > log log N and C > C1 · C2 then
+exp �C√log N log log N� and k are in IN. Therefore, we have
+fN(k) > fN
+Ä
+exp
+Ä
+C
+�
+log N · log log N
+ää
+= C
+C2
+ 
+log N
+log log N −
+log 4
+C2 log log N − C1 log
+Ä
+C
+�
+log N · log log N
+ä
+.
+(4)
+Viewing equation 4 as a function in N, it is evident that for large enough N (greater than
+some N1) we also have fN(k) > e. In what follows, we will assume that N > N1.
+As fN(k) > e, it follows from equation 3 that log N ⩾ fN(k) · log k. Further, applying
+equation 4 and the estimate for k and isolating the term log N, we get
+C log 4
+C2
+ 
+log N
+log log N + C1C
+�
+log N log log N log
+Ä
+C
+�
+log N log log N
+ä
+⩾
+ÅC2
+C2
+− 1
+ã
+log N.
+Recall that C > √C2, so the latter inequality is absurd for large enough N (greater than some
+N2), as the left-hand side is asymptotically much smaller than log N. Therefore, theorem 1
+holds for N > max(N1, N2) and C > max(√C2, C1 · C2).
+
+4
+VLAD-TITUS SP˘ATARU
+3. Acknowledgments
+The author thanks Alexandru Gica for his proofreading and valuable comments.
+References
+[EM52] P. Erd˝os and L. Mirsky, The distribution of values of the divisor function d(n), Pro-
+ceedings of the London Mathematical Society no. 1 (1952), 257–271.
+[Far09] B. Farhi, An identity involving the least common multiple of binomial coefficients and its
+application, The American Mathematical Monthly 116 no. 9 (2009), 836–839.
+[HB84] D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 no. 1
+(1984), 141–149.
+[Pin97] C. G. Pinner, Repeated values of the divisor function, The Quarterly Journal of Mathe-
+matics 48 no. 4 (1997), 499–502.
+[Spi81] C. A. Spiro, The Frequency with Which an Integral-Valued, Prime-Independent, Mul-
+tiplicative or Additive Function of n Divides a Polynomial Function of n, Ph.D. thesis,
+University of Illinois at Urbana-Champaign, 1981.
+V. T. Sp˘ataru, Bucharest, Romania
+E-mail : vtspataru@gmail.com
+

F9E3T4oBgHgl3EQfWAol/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf,len=106
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='04464v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='NT]  8 Jan 2023  Runs of Consecutive Integers Having the  Same Number of Divisors  By Vlad-Titus Sp˘ataru  Abstract  Our principal objective is to provide an upper bound for the length ℓN of the  longest run of consecutive integers smaller than N which have the same number of  divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' We prove that ℓN ⩽ exp �C√log N log log N� in an elementary manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Introduction  The equation d(n) = d(n + k) has been studied extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  In 1981, Spiro [Spi81]  showed that it has infinitely many solutions for k = 5040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  Subsequently, Heath-Brown  [HB84] extended Spiro’s work to deal with the case k = 1, and Pinner [Pin97] ultimately  proved that, in fact, all values of k yield infinitely many solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  As d(n) = d(n+1) infinitely often, one naturally wonders how many consecutive integers  can there be, having the same number of divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Erd˝os and Mirsky [EM52] conjectured  that there are arbitrarily long such runs of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' They were not able to provide any  estimates for the length of such sequences: “A related problem consists in the estimation of  the longest run of consecutive integers ⩽ x all of which have the same number of divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  This problem seems to be one of exceptional difficulty, and we [Erd˝os & Mirsky] have not  been able to make any progress with it.”  Our principal objective is to provide an upper bound for the length of the runs in  question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' We shall obtain the following result, in an elementary manner:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Let ℓN denote the length of the longest run of consecutive integers smaller  than N, having the same number of divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Then,  ℓN ⩽ exp  Ä  C  �  log N · log log N  ä  ,  for an absolute constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  Keywords: divisor counting function, consecutive equidivisible integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  2020 Mathematics Subject Classification: Primary: 11A25, 11N37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  1  2  VLAD-TITUS SP˘ATARU  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' The main result  In proving theorem 1, we will make use of the following lemmas, the first being proven  in an elementary manner in [Far09] and the second being Mertens’ bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Let n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Then, lcm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' , n + 1) ⩾ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' There exists an absolute constant C1 such that for any positive integer n ⩾ 2,  �  p⩽n  1  p ⩽ C1 · log log n,  the sum being over all prime numbers p not exceeding n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  Note that it suffices to prove that theorem 1 holds for large enough N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Assume that  there exist k > 2 consecutive numbers smaller than N, having the same number of divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  Let them be n + 1, n + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' , n + k and write  d(n + 1) = d(n + 2) = · · · = d(n + k) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  We will firstly provide an estimate for D, in terms of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' For simplicity, let K = ⌊log2 k⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  As k ⩾ 2K, all residues modulo 2K are among n + 1, n + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' , n + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Therefore, for all  1 ⩽ i ⩽ K − 1, there exists some 1 ⩽ ti ⩽ k such that n + ti ≡ 2i mod 2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Consequently,  ν2(n + ti) = i, so (i + 1) divides d(n + ti) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  Hence, D is divisible by lcm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' , K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Using lemma 1, we infer that  D ⩾ lcm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' , K) ⩾ 2K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  Recall that K = ⌊log2 k⌋ ⩾ log2 k − 1, so D ⩾ k/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  Next, we will bound ω((n + 1) · · ·(n + k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Choose 1 ⩽ l ⩽ k arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' As n + l ⩽ N,  it follows that νp(n + l) ⩽ logp N ⩽ log2 N for all prime numbers p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Therefore,  D = d(n + l) =  �  p  (νp(n + l) + 1) ⩽  �  p|n+l  (log2 N + 1) = (log2 N + 1)ω(n+l),  where p always represents a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Thus, ω(n+l) ⩾ log D/ log(log2 N +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' A prime  number p can divide at most ⌈k/p⌉ ⩽ k/p + 1 numbers among n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' , n + k, so  ω((n + 1) · · ·(n + k)) ⩾  k  �  i=1  ω(n + i) −  �  p⩽k  k  p,  RUNS OF CONSECUTIVE INTEGERS HAVING THE SAME NUMBER OF DIVISORS  3  the second sum being taken over all prime numbers p not exceeding k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Using lemma 2 and  the inequality we have previously deduced for ω(n + l), we may finally infer that  ω((n + 1) · · ·(n + k)) ⩾  k · log D  log(log2 N + 1) − C1k log log k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  Further, we will write log(log2 N +1) ⩽ C2 log log N, for an absolute constant C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Recall  that D ⩾ k/4, so we have  ω((n + 1) · · ·(n + k)) ⩾ k · log(k/4)  C2 log log N − C1k log log k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  (1)  Write the right-hand side of equation 1 as k · fN(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Clearly, if ω(a) ⩾ b then a ⩾ b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='.  Using this remark on equation 1, we get (n+1) · · · (n+k) ⩾ ⌈k ·fN(k)⌉!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='. Moreover, because  Nk ⩾ (n + 1) · · ·(n + k), by taking logarithms and applying the well-known inequality  log t!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' ⩾ t log t − t, we have  k log N ⩾ log ((n + 1) · · · (n + k)) ⩾ log (⌈k · fN(k)⌉!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=')  ⩾ k · fN(k) · log(k · fN(k)) − k · fN(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  (2)  Finally, dividing equation 2 by k we obtain  log N ⩾ fN(k) · log(k · fN(k)) − fN(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  (3)  Define the interval IN = [exp (C1 · C2 · log log N) , ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Using standard arguments, one  may infer that fN is increasing on IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  Let us suppose, for the sake of contradiction, that k > exp �C√log N log log N�, where  C > max(√C2, C1 · C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Firstly, note that since log N > log log N and C > C1 · C2 then  exp �C√log N log log N� and k are in IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Therefore, we have  fN(k) > fN  Ä  exp  Ä  C  �  log N · log log N  ää  = C  C2  log N  log log N −  log 4  C2 log log N − C1 log  Ä  C  �  log N · log log N  ä  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  (4)  Viewing equation 4 as a function in N, it is evident that for large enough N (greater than  some N1) we also have fN(k) > e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' In what follows, we will assume that N > N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  As fN(k) > e, it follows from equation 3 that log N ⩾ fN(k) · log k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Further, applying  equation 4 and the estimate for k and isolating the term log N, we get  C log 4  C2  log N  log log N + C1C  �  log N log log N log  Ä  C  �  log N log log N  ä  ⩾  ÅC2  C2  − 1  ã  log N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  Recall that C > √C2, so the latter inequality is absurd for large enough N (greater than some  N2), as the left-hand side is asymptotically much smaller than log N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Therefore, theorem 1  holds for N > max(N1, N2) and C > max(√C2, C1 · C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  4  VLAD-TITUS SP˘ATARU  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Acknowledgments  The author thanks Alexandru Gica for his proofreading and valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  References  [EM52] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Erd˝os and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Mirsky, The distribution of values of the divisor function d(n), Pro-  ceedings of the London Mathematical Society no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' 1 (1952), 257–271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  [Far09] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Farhi, An identity involving the least common multiple of binomial coefficients and its  application, The American Mathematical Monthly 116 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' 9 (2009), 836–839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  [HB84] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Heath-Brown, The divisor function at consecutive integers, Mathematika 31 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' 1  (1984), 141–149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  [Pin97] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Pinner, Repeated values of the divisor function, The Quarterly Journal of Mathe-  matics 48 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' 4 (1997), 499–502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  [Spi81] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Spiro, The Frequency with Which an Integral-Valued, Prime-Independent, Mul-  tiplicative or Additive Function of n Divides a Polynomial Function of n, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' thesis,  University of Illinois at Urbana-Champaign, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content=' Sp˘ataru, Bucharest, Romania  E-mail : vtspataru@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E3T4oBgHgl3EQfWAol/content/2301.04464v1.pdf'}

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+1 
+ 
+Catalytic action of two-dimensional layered materials (WS2, and MoS2) on hydrogen 
+sorption properties of MgH2 
+Satish Kumar Verma1, Mohammad Abu Shaz1, Thakur Prasad Yadav1,2* 
+1Hydrogen Energy Centre, Department of Physics, Banaras Hindu University, Varanasi-
+221005, India. 
+2Department of Physics, Faculty of Science, University of Allahabad, Prayagraj-211002, 
+India. 
+ 
+Abstract: 
+The present study reports the catalytic action of two-dimensional (2D) layered materials 
+(MoS2 and WS2) for improving the de/re-hydrogenation kinetics of MgH2. The MgH2 
+start desorbing at 277 ºC with a hydrogen storage capacity of 5.95 wt% in the presence of 
+WS2 catalyst whereas onset desorption temperature of MgH2 catalyzed by MoS2 is 330 
+ºC. The MgH2-WS2 absorbed hydrogen ~ 3.72 wt% within 1.3 minutes at 300 ºC under 13 
+atm hydrogen pressure and it desorbed ~5.57 wt% within 20 minutes at 300 ºC under 1 
+atm hydrogen pressure. We have performed 25 cycles of dehydrogenation (under 1 atm 
+hydrogen pressure at 300 ºC) and re-hydrogenation (under 13 atm hydrogen pressure at 
+300 °C) to ensure cyclic stability of catalyzed version of MgH2 where MgH2-WS2 shows 
+better cyclic stability than MgH2-MoS2. MgH2-WS2 also shows the lower reaction 
+activation energy ~117 kJ/mol as compare to other catalyzed and uncatalyzed samples. 
+On the other hand, these catalysts (WS2 and MoS2) do not have any impact on the 
+thermodynamical parameters that is change in enthalpy. 
+ 
+Key words: 2D layered materials, De/re-hydrogenation kinetics, Activation energy, 
+MgH2. 
+*Corresponding author Email: yadavtp@gmail.com 
+ 
+
+2 
+ 
+1. Introduction 
+A crucial and promising area of research for onboard hydrogen applications is the 
+development of safe and efficient hydrogen storage. The solid-state approach is one of 
+the most appropriate, secure, and effective ways to store hydrogen among the several 
+methods that can be used, including gaseous, liquid, and solid-state storage [1,2]. Due to 
+its high hydrogen storage capacity (110 g/L volumetric and 7.6 wt% gravimetric), low 
+cost, light weight, and large abundance (in the form of Mg) in earth crust (8th most) and 
+seawater (3rd most), MgH2 is a leading choice for hydrogen storage in the solid-state 
+mode [3–6]. According to the United States Department of Energy (US DOE) technical 
+targets for hydrogen storage systems [7], MgH2 has certain advantages that make it a 
+viable option. The high dehydrogenation temperature (above 400 ºC), slow kinetics 
+(hydrogen de/re-hydrogenation kinetics 0.4 kg-H2/min), and high thermodynamic 
+properties (high reaction enthalpy 74 kJ/mol) of MgH2 prevent it from being a suitable 
+material for onboard applications even with these advantages [8–10]. In recent years, the 
+creation of suitable catalyst(s), alloys, composite materials with complicated hydrides, 
+and scaffolding have all been used as feasible methods to improve the hydrogen storage 
+performance of MgH2 [11,12]. The use of various types of catalysts and additives to 
+enhance the performance of Mg/MgH2 has been the subject of several studies by various 
+research organizations [13–17]. 
+Another application for the 2D materials is as a catalyst for improving the hydrogen 
+characteristics of MgH2 [18–22]. Due to its enormous surface area, ballistic conduction, 
+thermal conductivity, mechanical stability, and light weight, graphene, which has a 2D 
+planer structure with sp2 carbon atoms arranged in a hexagonal framework, has attracted 
+a lot of attention as a catalyst and as a template material for hydrogen storage application 
+in MgH2 [23,24]. MgH2's de/rehydrogenation kinetics exhibit effective catalytic behavior 
+in the graphene layer, which also inhibits MgH2's agglomeration and grain growth 
+[3,5,25]. Liu et al., for instance, have created MgH2-5% Gr nanosheets [26]. They have 
+demonstrated that graphene nanosheets offer a significant hydrogen diffusion pathway 
+and prevent MgH2 from aggregating. According to Huang et al., [27] report's MgH2 
+nanoparticles supported by graphene exhibit remarkable hydrogen sorption kinetics and 
+
+3 
+ 
+cyclic stability. Due to the strong interaction between graphene and MgH2 nanoparticles 
+and the prevention of nanoparticle agglomeration, the MgH2 nanoparticles demonstrated 
+excellent hydrogen storage performance. Additionally, grapheme prevents the 
+aggregation of nanoparticles during the rehydrogenation of MgH2, according to a 
+theoretical study using molecular dynamics simulation [28]. Rough studies are still 
+required to determine the impact of graphene and other 2D layered materials on MgH2, 
+even though some prior studies have shown the remarkable catalytic/co-catalytic and 
+agglomeration blocking properties of Gr on MgH2. 
+We have examined a comparison between WS2 and MoS2 as a catalyst for enhancing 
+hydrogen sorption properties of MgH2. WS2 and MoS2 are suitable alternatives to 
+graphene for the catalytic action on MgH2 due to their high conductivity (metallic 
+nature), thermal stability, and strong catalytic behavior [29,30]. Tungsten (W) and 
+Molybdenum (Mo) are sandwiched between two Sulphur layers with weak Van der 
+Waals interactions in the family of layered transition-metal dichalcogenides (TMDs) 
+materials that include WS2 and MoS2. The re/de-hydrogenation kinetics, and catalytic 
+behavior of WS2 and MoS2 on MgH2 has been investigated in details. 
+2. Experimental section 
+2.1. Synthesis of a few layered WS2  
+The bulk tungsten sulfide (WS2) (99.80 %) powder was procured from the Alfa Aesar for 
+the present investigation. For the preparation of few layered WS2, WS2 powder was 
+dispersed in de-ionized water and sonicated it for 74 hours using ultrasonicator at 20 kHz 
+frequency. The sonicated sample was then dried at 50 °C under a dynamic vacuum of 
+order 10-2 torr to form the few layered WS2 powder. This preparation method can also be 
+understood by the schematic given in Fig. 1.  
+
+4 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig.1: Schematic diagram for the synthesis of a few layers WS2. 
+ 
+2.2. Synthesis of few-layer MoS2 
+The Otto Chemica bulk molybdenum disulfide (MoS2) (99 %) powder was used for the 
+present investigation. MoS2 powder was dispersed in de-ionized water and sonicate it for 
+74 hours using ultrasonicator at 20 kHz frequency to obtain the few-layered MoS2. The 
+sonicated sample was then dried at 50 °C under dynamic vacuum of order 10-2 torr to 
+form the few layered MoS2 powder. Fig. 2, shows the schematic diagram for preparation 
+of few-layered MoS2.    
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+BulkMoS2
+BulkMoS,
+FewlayeredMoS2
+DoublelayeredMoS2
+UltrasonicationofbulkMoS,BulkWS2
+BulkWs,
+FewlayeredwS2
+DoublelayeredWS2
+Ultrasonicationof bulkWS25 
+ 
+ 
+Fig.2: Schematic diagram for the synthesis of a few layers MoS2. 
+ 
+2.3. Synthesis of MgH2 catalyzed by WS2, and MoS2 
+The pure MgH2 was procured from Fujifilm (Japan) (99.9%) for the present investigation. 
+Mechanical ball-milling of MgH2 with graphene at 180 rpm for 24 hours with a ball-to-
+powder ratio of 50:1 (by weight) using a planetary ball-miller (Retsch PM 400) was used 
+to synthesize MgH2 catalyzed by WS2 (MgH2-WS2). To explore the optimum catalyst 
+concentration for hydrogen sorption kinetics of Mg/MgH2, we have synthesized a set of 
+different catalyst concentrations (5, 10, 12 wt%) to catalyze MgH2. For hydrogen 
+sorption in Mg/MgH2, 10 wt% catalysts were found to be optimal (in terms of desorption 
+temperature and hydrogen storage capacity). The ball-miller vials were filled with 5 atm 
+H2 pressure to compensate for the loss of hydrogen from MgH2 during milling. All the 
+loading and unloading of the samples was done inside the N2-filled glove box 
+(MBRAUM MB10 compact) with O2 and H2O levels < 1 ppm. The synthesis of MgH2 
+catalyzed by MoS2 (MgH2-MoS2) was done using the same synthesis route as MgH2-
+WS2. 
+2.4. Characterization techniques 
+The structural characterization of prepared samples was carried out by XRD technique 
+using Empyrean PANalytical X-ray diffractometer equipped with 2D detector with a Cu 
+Kα beam (λ = 1.5415 Å) operated at 40 kV and 40 mA. The microstructural and selected 
+area electron diffraction (SAED) analysis of as-prepared samples was carried out by 
+TEM (Technai-20G2) operating at the accelerating voltage of 200 kV. Perkin Elmer 
+(Spectrum 100) spectrometer in transmission mode with attenuated total reflectance 
+(ATR) sampling mode (wavenumber range 500–4000 cm-1) was used to carry out FTIR 
+spectroscopy. The Raman spectra have been acquired at -60 ºC using Horiba-Jobin-Yvon 
+LABRAM-HR800 spectrometer with diode LASER (532 nm). The desired thickness and 
+surface topography of the prepared samples were examined by using solver next AFM in 
+non-contact mode. The characterized samples then proceed for the hydrogen desorption 
+and absorption using automated two-channel volumetric sieverts type apparatus. The 
+temperature programmed desorption (TPD) was carried out with a heating rate of 5
+ oC-
+
+6 
+ 
+min-1. The activation energy (Ea) study of prepared catalyzed samples has been done by 
+using DSC (Perkin Elmer DSC 8000) with a heating rate of 15 
+oC/min, 18 
+oC/min, 21 
+oC/min, and 24 
+oC/min under nitrogen atmosphere (20 ml/min). 
+ 
+3. Results and discussion 
+3.1. Structural, microstructural, and spectroscopic characterization analysis 
+The structural characteristics of as-prepared samples have been examined using the XRD 
+characterization. Fig. 3(a) shows the XRD pattern of pristine MgH2, which matches well 
+with the tetragonal MgH2 with space group P42/mnm (136) and a=b= 4.516 Å, c = 3.020 
+Å (JCPDS no. 740934). Fig. 3(b) shows the XRD pattern of MoS2, which matches well 
+with the hexagonal structure of MoS2 with space group P63/mmc(194) and a=b= 3.1602 
+Å, c = 12.294 Å (Joint Committee on Powder Diffraction Standards (JCPDS) no. 
+651951). The XRD pattern of as-prepared WS2 is shown in Fig. 3(c), that matches well 
+with the hexagonal structure of WS2 with space group P63/mmc(194) and a=b= 3.1532 
+Å, c = 12.323 Å (JCPDS no. 841398). The usual diffraction pattern of MgH2-MoS2, and 
+MgH2-WS2 are shown in Fig. 3(d-e), respectively, where besides the tetragonal phase of 
+MgH2, some peaks of WS2 and MoS2 are either suppressed or masked by the peaks of 
+MgH2. The diffraction peaks of WS2 and MoS2 are identified and labeled in the Fig. 3(d-
+e), respectively.  
+The different bands position, shapes, and relative intensities of Raman spectra give us 
+essential information about the materials and stacking of layers, i.e., Raman spectroscopy 
+can determine the layer thickness at the atomic level. The Raman spectra of as-prepared 
+WS2, and MoS2 have shown in Fig. 4. In the case of MoS2, the two Raman modes are 
+appeared at ~ 345 cm-1 and ~ 370 cm-1 corresponds to E12g and A1g modes of vibrations 
+(labeled in Fig. 4(b)). The indicated modes of MoS2 have frequency difference of ~ 25 
+cm-1, that means the MoS2 as layered material with few layers of stacking (3-5 layers) 
+[31,32]. The FWHM of A1g mode is ~ 7 cm-1, which can also be referred to stacking a 
+few layers of MoS2 [33]. The Raman shifts at ~316 cm-1 and 384 cm-1 (shown in Fig. 
+4(a)) corresponds to the presence of E12g and A1g modes respectively in WS2 sample. The 
+
+7 
+ 
+intensity ratio of E12g and A1g modes was estimated E12g/A1g i.e. = 1.26, which is higher 
+than the intensity ratio of bulk WS2 (E12g/A1g = 0.47) and lower than the monolayer WS2 
+(E12g/A1g = 2.2) [34,35]. This calculated intensity ratio (E12g/A1g = 1.26) is compatible 
+with the range of 2-3 layers of WS2.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 3: XRD patterns of (a) Pristine MgH2, (b) MoS2, (c) WS2, (d) MgH2-MoS2, and (e) 
+MgH2-WS2.  
+ 
+
+o-Parafilm,
+*-MgH2,
+t-Ws2, u -Mos2
+(e)MgH2-Ws
+52
+T
+*
+*
+*
+*
+(d) mgh2-Mos2
+*
+Intensity (wt%)
+(c) WS 2
+(002)
+-(004)
+ (100)
+(101)
+(900)
+(105)
+(110)
+(112)
+-(114)
+(203)
+(116)
+8
+tt
+T
+T
+.1
+1
+T
+(b)Mos.
+(00L)2
+2
+(002)
+U
+C(105)
+(102)
+(103)
+C(110)
+(112)
+c(108)
+(203)
+U
+(a) Pristine
+MgH2
+*
+(200)
+(110)
+(220)
+*(002)
+(310)
+(112)
+(301)
+(202)
+(211)
+*
+8
+8
+*
+*
+￥
+10
+20
+30
+40
+50
+60
+70
+80
+2e(degree8 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 4: Raman spectra of (a) WS2 and (b) MoS2. 
+ 
+ 
+ 
+The information about stacking layers in 2D layered materials (like WS2 and 
+MoS2) can also be verified by AFM analysis. The surface topography and height profile 
+of prepared MoS2, and WS2 were examined along the blue dotted line as shown in Fig. 
+S1(a-b) (given in supporting information). The layered surface morphology along with 
+height profile (shown in Fig. S1(a-a1)) shown the average thickness of MoS2 is ~1.3 nm, 
+that indicates the presence of ~2 layers of stacking in the MoS2 sample [36,37]. The ~7-8 
+layers of stacking were present in the case of WS2 (shown in Fig. S1(b-b1)) with a 
+monolayer height of ~0.7 nm [38].       
+ 
+ 
+ 
+
+(b) Raman spectra of MoS2
+(370)
+A1g
+(a) Raman spectra of WS2
+(345)
+2g
+Intensity (a.u.)
+(384)
+2g
+A1
+(316)
+175200225250275300325
+350
+375400
+425
+450475500
+Raman shift (cm9 
+ 
+. 
+ 
+3.2 De/Re-hydrogenation kinetics of catalyzed MgH2 
+To identify the optimal percentage of catalyst in MgH2 with optimum temperature range 
+where material performed promptly, we have characterized as-prepared samples for the 
+temperature programmed desorption (TPD) analysis. The TPD curves of MgH2-MoS2 
+have seen in Fig. S2 (given in supporting information). The MgH2-5%MoS2, MgH2-
+10%MoS2, and MgH2-12%MoS2, starts releasing hydrogen at ~ 357 °C, ~ 330 °C, ~ 302 
+°C with ~ 6.41 wt%, ~ 6.00 wt%, ~ 4.88 wt% of hydrogen storage capacity respectively. 
+On the other hand, MgH2-5%WS2, MgH2-10%WS2, and MgH2-12%WS2, starts releasing 
+hydrogen at ~ 339 °C, ~ 277 °C, ~ 258 °C with ~ 6.54 wt%, ~ 5.95 wt%, ~ 5.14 wt% of 
+hydrogen storage capacity respectively (shown in Fig. S3 in supporting information). 
+Based on TPD analysis, the optimum catalyst concentration for catalyzing MgH2 is 10 
+wt% for all catalysts. 
+After getting information about the optimum catalyst for MgH2, we compared the TPD 
+analysis of all optimum catalyzed samples with pristine MgH2, as shown in Fig. 5.  The 
+TPD of pristine MgH2 (shown in Fig. 5(a)) was then carried out to compare hydrogen 
+storage properties with catalyzed samples. The pristine MgH2 has an onset desorption 
+temperature of 376 
+oC with a total release of ~7.45 wt% storage capacity. The onset 
+desorption temperature of MgH2-MoS2 (MgH2-10%MoS2) is ~ 330 
+oC, and it desorbs ~ 
+6.00 wt% hydrogen while the desorption gets completed at 396 
+oC (Fig. 5(b)). In the case 
+of MgH2-WS2 (MgH2-10%WS2), it starts desorbing hydrogen at ~ 277
+ oC with a storage 
+capacity of 5.95 wt% (Fig. 5(c)).  
+
+10 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 5: Comparative TPD analysis of (a) Pristine MgH2, (b) MgH2-MoS2 and (c) MgH2-
+WS2. 
+ 
+The desorbed samples then proceed for re/de-hydrogenation to check the cyclic stability 
+and reversibility of catalyzed and pristine MgH2. The re-hydrogenation kinetics was 
+carried out at 300 
+oC under 13 atm hydrogen pressures, as shown in Fig. 6. It can be seen, 
+the pristine MgH2 absorbed ~1.16 wt% hydrogen in 1.2 minutes whereas MgH2-MoS2, 
+MgH2-WS2 absorbed 4.60 wt%, 3.72 wt%, hydrogen, respectively, under similar 
+conditions of temperature and pressure.  
+ 
+ 
+ 
+
+0
+Hydrogen desorbed (wt%)
+(a)
+(b)
+(c)
+(a)PristineMgH
+5
+(b) MgH,-Mos,
+6
+(c) MgH,-WS,
+7
+8
+200
+225
+250
+275
+300
+325
+350
+375
+400
+425
+Temperature (C)11 
+ 
+ 
+Fig. 6: Rehydrogenation kinetics curves at 300 °C under 13 atm H2 pressure of (a) b) 
+MgH2-WS2, (c) MgH2-MoS2 and (e) Pristine MgH2. 
+ 
+The rehydrogenated samples were then dehydrogenated at 300 
+oC under 1 atm 
+hydrogen pressure. It can be seen clearly in Fig. 7, that the MgH2-WS2 sample releases 
+5.57 wt% hydrogen within 20 minutes while MgH2-MoS2 and pristine MgH2 releasees 
+2.25 wt%, and 0.23 wt% of hydrogen under similar temperature and pressure conditions, 
+which is 3.32 wt%, and 4.48 wt% more than pristine MgH2, MgH2-MoS2, respectively. 
+Based on the above re/de-hydrogenation kinetics study, it is clearly shown that WS2 
+works as a superior catalyst to MoS2 for catalyzing MgH2. Therefore, in present study 
+WS2 is a prominent catalyst to catalyze MgH2. 
+
+(b)
+5.
+Hydrogen absorbed (wt%)
+(a)
+(c)
+- (a) MgH,-WS
+ (b) MgH,-Mos
+ (c) Pristine MgH,
+0
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+Time (Min.)12 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 7: Dehydrogenation kinetics curves at 300 °C under 1 atm H2 pressure of (a) MgH2-
+WS2, (b) MgH2-MoS2, and (c) Pristine MgH2. 
+ 
+3.3. Study of kinetics: Estimation of activation energy 
+The DSC was carried out to determine the hydrogen desorption activation energy barrier 
+to convert MgH2 into Mg. The DSC profile of MgH2-MoS2, MgH2-WS2, are shown in 
+Figs. 8-9. In the case of MgH2-WS2, the peak desorption temperature found from DSC is 
+~ 380 
+oC, while the onset desorption temperature found from TPD is ~ 277 
+oC. There is a 
+difference in desorption temperature in TPD (Fig. 5(c)) and DSC (Fig. 9(a)) curves due to 
+the TPD being performed under vacuum with a temperature ramping rate of 5
+ oC/min 
+while DSC was performed under N2 atmosphere with a temperature ramping rate of 15
+ 
+oC/min. For calculating the desorption activation energy, we have performed DSC with a 
+set of the various rate of heating (15, 18, 21, 24 
+oC/min) and plotted the Kissinger curve 
+by using the Kissinger equation[39] as given: 
+
+(a) Pristine MgH,
+- (b) MgH,-MoS,
+Hydrogen desorbed (wt%)
+5
+ (c) MgH,-WS
+(a)
+(b)
+2
+(c)
+0
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+55
+60
+Time (min.)13 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(1) 
+Where β, Tp, and Ea are the heating rate, corresponding peak desorption temperature, and 
+activation energy, respectively. The slope of Kissinger plot (ln(β/Tp
+2) vs. 1000/Tp
+2 plot) 
+(Figs. 8-9) is used to calculate the desorption activation energy. The calculated activation 
+energy for MgH2-MoS2, and MgH2-WS2 is 117.09 kJ/mol (± 1.60 kJ/mol), and 104.00 
+kJ/mol (± 2.74 kJ/mol) respectively. This activation energy indicates that ~104 kJ/mol 
+energy is required to overcome the barrier to convert MgH2 into Mg in the presence of a 
+WS2 catalyst. These calculated activation energies are significantly lower than the 
+activation energy of pristine MgH2 [3,40]. 
+Table 1: Table for plateau pressures at corresponding temperatures, change in enthalpy, 
+and activation energy of MgH2-MoS2, and MgH2-WS2.  
+ 
+ 
+ 
+S.No.  
+Sample 
+name 
+Plateaus 
+pressure 
+          (atm) 
+Temperature 
+(
+ºC) 
+Change 
+in 
+enthalpy 
+ (kJ/mol) 
+Activation 
+energy 
+(kJ/mol) 
+1. 
+MgH2-
+MoS2 
+1.03 
+272.62 
+ 
+  -78.33 
+ 
+117.09 
+2.03 
+292.28 
+3.77 
+313.28 
+2. 
+MgH2-
+WS2 
+1.52 
+281.26 
+ 
+  -77.44 
+ 
+104.66 
+2.92 
+300.60 
+3.45 
+316.29 
+
+14 
+ 
+ 
+ 
+ 
+Fig. 8: (i) DSC profile for desorption of MgH2-MoS2 with the heating rate (a) 15 ºC/min, 
+(b) 18 ºC/min, (c) 21 ºC/min, (d) 24 ºC/min, and (ii) corresponding Kissinger plot for 
+evaluating the desorption activation energy of MgH2-MoS2. 
+ 
+
+DSCprofilefor MgH2-MoS2
+-9.8
+Kissinger plot for MgHb-MoS2
+Linear fit
+(d) 24°C/min
+-9.9
+(a.u.)
+-10.0
+(c)21cC/min
+Heatflow(
+P
+-10.1
+Endo up
+(b) 18 C/min
+Eguation
+y=a+b
+-10.2
+Adj. R-Squ
+0.99937
+Value 
+Standard Er
+-10.3
+In(beta/Tp2) Irtercept
+11.212
+0.30766
+(a) 15 C/min
+In(beta/Tp2) Slope 
+-14.083
+0.20379
+1.4881.4941.5001.5061.5121.5181.5241.530
+250
+275
+300
+325
+350
+375
+400
+425
+450
+1000/T,(K1)
+Temperature (cC)DSCprofileforMgH2-WS2
+Kissinger plot for MgH2-WS2
+-9.8
+(d) 24 C/min
+Linear fit
+-9.9
+Heat flow (a.u.)
+(c) 21 °C/min
+-10.0
+[β/T,
+(b) 18 °C/min
+-10.1
+Endo
+Equation
+=a+
+Adj. R-Sq
+0.9979
+-10.2
+Value
+ Standard
+(a) 15 °C/min
+In(beta/Tp Interce
+9.3313
+0.50735
+In(beta/Tp Slope
+-12.58
+0.33004
+-10.3
+1.520
+1.525
+1.530
+1.535
+1.540
+1.545
+1.550
+1.555
+1000/T(K
+320330340350360370380390400410420430440450
+Temperature (C)15 
+ 
+Fig. 9: (i) DSC profile for desorption of MgH2-WS2 with the heating rate (a) 15 ºC/min, 
+(b) 18 ºC/min, (c) 21 ºC/min, (d) 24 ºC/min, and (ii) corresponding Kissinger plot for 
+evaluating the desorption activation energy of MgH2-WS2. 
+ 
+3.4. Study of thermodynamics 
+After the kinetics and reversibility study, we have proceeded with the thermodynamic 
+analysis of catalyzed MgH2 for comparing the change in enthalpy and entropy of the 
+system using well known Van’t Hoff equation [41]. 
+ 
+ 
+ 
+lnP = (ΔH/RT) - (ΔS/R) 
+ 
+ 
+(2) 
+Where P, ∆H, R, T, and ∆S are the pressure, change in enthalpy, gas constant, absolute 
+temperature, and change in entropy, respectively. The PCI isotherms (Figs. 10(i)-11(i) 
+and Van’t Hoff plots (Figs. 10(ii)-11(ii)) were used for the calculation of change in 
+enthalpy of MgH2-MoS2 and MgH2-WS2, respectively. The calculated change in 
+desorption enthalpy was found to be 78.33 kJ/mol (± 1.40 kJ/mol), and 77.44 kJ/mol (± 
+1.13 kJ/mol), for MgH2-MoS2 and MgH2-WS2 respectively. It is clear from the above 
+estimation of change in enthalpy, that there is no significant enthalpy change in the 
+presence of a catalyst. Thus MoS2, WS2 have not positively impacted the thermodynamic 
+barrier of the MgH2. The plateau pressure at corresponding temperatures, change in 
+enthalpy, and activation energy has been tabulated in Table 1. 
+
+(i) PCl desorption for MgH2-MoS2
+1.4.
+(ii) Vant's Hoff plot for MgH2-MoS2
+6.
+- Linear fit
+1.2
+5.
+1.0
+(atm)
+320°C
+4
+0.8.
+Pressure (
+P
+三 0.6
+3.
+300°C
+0.4
+2
+Equation
+y=a+b*
+0.2-
+Adj. R-Squar0.99936
+280 °C
+Value
+Standard Err
+InP
+Intercept 
+17.3878
+0.298
+0.0
+InP
+Slope
+9.4207
+0.16804
+0
++
+1.70 1.72 1.74 1.76 1.78 1.80 1.82 1.84 1.86 1.88
+0
+1
+2
+3
+4
+5
+1000/T (K
+Hydropgen capacity (wt%)16 
+ 
+Fig. 10: (i) PCI desorption plots for MgH2-MoS2 at different temperatures and (ii) 
+corresponding Van't Hoff plot for calculating the change in enthalpy 
+ 
+Fig. 11: (i) PCI desorption plots for MgH2-WS2 at different temperatures and (ii) 
+corresponding Van't Hoff plot for calculating the change in enthalpy 
+3.5 Cyclic stability of catalyzed MgH2 
+The WS2 (optimum catalyst) plays a significant role in improving the kinetics of MgH2. 
+The cyclic stability is an essential characteristic of the hydride material (MgH2) besides 
+kinetic and thermodynamics, making it a worthy hydrogen storage material. Therefore, it 
+is crucial to look at the cyclic stability of the catalyzed MgH2 samples. We have 
+performed 25 cycles of dehydrogenation (under 1 atm hydrogen pressure at 300 °C) and 
+re-hydrogenation (under 13 atm hydrogen pressure at 300 °C) to ensure cyclic stability of 
+catalyzed MgH2. The cyclic stability curve of MgH2-MoS2 and MgH2-WS2 are shown in 
+Fig. 12. From Fig. 12(a) MgH2-MoS2 shows the ~ 0.42 wt% (from 5.77 wt% to 5.35 
+wt%) degradation in hydrogen storage capacity during rehydrogenation and ~ 0.38 wt% 
+(from 5.69 wt% to 5.31 wt%) in dehydrogenation. The MgH2-WS2 has the loss of 
+hydrogen storage capacity ~ 0.3 wt% (from 5.80 wt% to 5.50 wt%) during re-
+hydrogenation and ~ 0.36 wt% (from 5.76 wt% to 5.40 wt%) during dehydrogenation. 
+Thus, MgH2-WS2 has more substantial cyclic stability than MgH2-MoS2 under similar 
+
+8
+(i) PCI desorption for MgH2-WS2
+1.4
+(ii) Vant's Hoff plot for MgH2-WS2
+1.2
+Linear fit
+6
+(atm)
+1.0
+5
+Pressure
+320 °C
+InP
+0.8
+300°℃
+3
+0.6
+Equation
+y=a+
+Adj. R-Squ0.9995
+2
+280°C
+Value
+Standard E
+0.4 .
+InP
+Intercep
+17.038
+0.23617
+Inp
+Slope
+-9.314
+0.1362
+0.2
+1.68
+1.70
+1.72
+1.74
+1.76
+1.78
+0
+1.80
+1000/T (K-1)
+0
+1
+2
+3
+4
+5
+6
+Hydrogen capacity (wt%)17 
+ 
+temperature and pressure conditions. The comparative study for hydrogen storage 
+properties of different recently used 2D materials as the catalyst for MgH2 is explored in 
+Table 2.  
+ 
+Fig. 12: Cyclic stability of (a) MgH2-MoS2 and (b) MgH2-WS2. 
+ 
+Table 2: Table for different 2D materials as the catalyst for hydrogen storage application. 
+S. 
+No. 
+Material 
+2D- based 
+catalyst 
+Hydrogen 
+storage 
+capacity 
+(wt%) 
+Onset 
+dehydrogen
+ation 
+temperature 
+(ºC) 
+ 
+Activation 
+energy 
+(kJ/mol) 
+Change 
+in 
+enthalpy 
+(kJ/mol) 
+ 
+Ref. 
+1. 
+Mg6C2N 
+C2N 
+6.79 
+-- 
+-- 
+-- 
+[20] 
+2. 
+MgH2-LiAlH4-
+Ti3C2 
+Ti3C2 
+6.50 
+63.0 
+128.4 
+74.3 
+[22] 
+3. 
+MgH2-
+Nb4C3Tx 
+Nb4C3Tx 
+3.50 
+150.6 
+81.2 
+-- 
+[21] 
+4. 
+1T’-MoS2 
+ 
+3.90 
+-- 
+-- 
+-- 
+[42] 
+5. 
+MgH2-Gr 
+Graphene 
+5.80 
+300.0 
+-- 
+-- 
+[43] 
+
+Cyclic stability for MgH,-MoS
+capacity (wt%)
+6
+00300
+=0=0-0:
+5
+-I- Rehydrogenation
+- Dehydrogenation
+4
+3
+Degradation during rehydrogenation=0.42 wt%
+Degradationduring dehydrogenation=0.38 wt%
+Hydrogen :
+2
+1
+0
+6
+8
+10
+16
+18
+222426
+No.of cycleCyclic stability for MgH,-WS
+6
+5.
+ Rehydrogenation
+4.
++- Dehydrogenation
+3
+Degradation during rehydrogenation=0.30 wt%
+Degradation during dehydrogenation=0.36 wt%
+Hydrogen :
+2
+0
+10
+12
+16
+18.20
+222426
+No. of cycle18 
+ 
+6. 
+MgH2-
+TiH2@Gr 
+Graphene 
+6.77 
+204.0 
+88.89 
+74.54 
+[3] 
+MgH2-
+TiO2@Gr 
+Graphene 
+5.98 
+240.0 
+98.00 
+76.87 
+MgH2-Ti@Gr 
+Graphene 
+5.70 
+235.0 
+103.03 
+75.65 
+7. 
+MgH2-Gr 
+Graphene 
+6.14 
+300.0 
+134.95 
+77.90 
+[13] 
+8. 
+MgH2-VS2 
+VS2 
+6.51 
+242.0 
+98.10 
+76.83 
+9. 
+MgH2-WS2 
+WS2 
+5.95 
+277.0 
+104.66 
+77.44 
+Pres
+ent 
+stud
+y 
+10. 
+MgH2-MoS2 
+MoS2 
+6.00 
+330.0 
+117.09 
+78.33 
+Pres
+ent 
+stud
+y 
+ 
+4. Conclusions 
+ 
+ The catalytic effect of MoS2, and WS2 on MgH2 was evaluated and compared. 
+Based on the de/re-hydrogenation study, it is found that WS2 works as an optimum 
+catalyst over MoS2 for MgH2. The MgH2-WS2 has an onset de-hydrogenation ~277 oC 
+with a hydrogen storage capacity of 5.95 wt%. The MgH2-WS2 absorbed hydrogen ~ 3.72 
+wt% within 1.3 minutes at 300 oC under 13 atm hydrogen pressure and it desorbed ~5.57 
+wt% within 20 minutes at 300 oC under 1 atm hydrogen pressure. The MgH2-WS2 shows 
+a minimum degradation of hydrogen storage capacity ~ 0.3 wt% upto 25 cycles which 
+shows a better cyclic stability than cyclic stability of MgH2-MoS2 (~ 0.4 wt% loss in 
+hydrogen storage capacity). MgH2-WS2 also shows the lower reaction activation energy 
+~117 kJ/mol as compare to other catalyzed and uncatalyzed samples. On the other hand, 
+these catalysts (WS2 and MoS2) do not have any impact on the thermodynamical 
+parameters that is change in enthalpy. This study opens a new era to further applications 
+of 2D layered materials for various applications like template materials.  
+
+19 
+ 
+Acknowledgments 
+We gratefully accept funding assistance from the Department of Science and Technology 
+(DST), New Delhi, India. The Council of Scientific and Industrial Research (CSIR), New 
+Delhi, India, has awarded the author (S.K.V.) a CSIR-Senior Research Fellowship 
+(Award No. 09/013(0872)/2019-EMR-I), for which the author is grateful. 
+Conflict of Interest Declaration 
+There are no conflicts of interest among the authors. 
+ 
+ 
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+

GtE5T4oBgHgl3EQfWA9Z/content/tmp_files/2301.05555v1.pdf.txt

@@ -0,0 +1,1575 @@
+Magnetic phase diagram of the breathing-kagome antiferromagnet Nd3BWO9
+D. Flavi´an,1, ∗ J. Nagl,1 S. Hayashida,1, 2 M. Yan,1 O. Zaharko,3
+T. Fennell,3 D. Khalyavin,4 Z. Yan,1 S. Gvasaliya,1 and A. Zheludev1, †
+1Laboratory for Solid State Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland
+2Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany
+3Laboratory for Neutron Scattering and Imaging,
+Paul Scherrer Institut, 5232 Villigen, Switzerland
+4ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom
+(Dated: January 16, 2023)
+The highly-frustrated rare-earth based magnet Nd3BWO9 is a promising candidate in the search
+for proximate spin liquid physics. We present a thorough investigation on single crystals of this ma-
+terial using bulk and microscopic techniques. Magnetization data reveal a fractional magnetization
+plateau for three different investigated field directions. The magnetic phase diagram is mapped out
+from calorimetric data and exhibits several domes of magnetic order below 0.3 K. Propagation vec-
+tors for all ordered phases are presented. The results suggest complex ordering in this material, and
+unveil the existence of a commensuration transition of the propagation vector at zero magnetic field.
+A scenario where interplane exchange interactions are essential to a magnetic model of Nd3BWO9
+is discussed.
+I.
+INTRODUCTION
+Strongly frustrated quantum antiferromagnets (AFM)
+are known to realize a panoply of magnetic states due
+to the delicate equilibrium between the magnetic in-
+teractions.
+In the presence of magnetic fields, the
+large ground state degeneracy is lifted in subtle and
+diverse ways, which leads to extremely rich phase di-
+agrams. Realization of spin-density waves [1], magne-
+tization plateaus [2, 3] commensurate-incommensurate
+transitions [4], and even more exotic order like spin
+nematicity [5, 6] is not rare, particularly in quasi-low-
+dimensional systems.
+The archetypal model in 2D frustrated magnetism is
+the kagome lattice Heisenberg S = 1/2 AFM (KHAF).
+The impossibility of satisfying all magnetic interactions
+in this lattice results in a macroscopic degeneracy of the
+ground state already at a classical level [7]. Turning to
+S = 1/2 spins promotes the quantum fluctuations on the
+ground state giving rise to highly non-trivial phases [8].
+Arguably, the most intriguing state is the hypothesized
+Quantum Spin Liquid (QSL) [9] ground state. The pre-
+diction of fractionalization of quasiparticles in a 2D sys-
+tem triggered extensive effort from both theory and ex-
+perimental perspectives [10, 11]. Nevertheless, the QSL
+phase remains elusive [12] as it constitutes a very frag-
+ile state. One of the main causes of the instability of
+the QSL states is the presence of terms in the Hamilto-
+nian that lift the ground state degeneracy [13, 14]. The
+many different ways to lift this degeneracy have led to
+a flurry of new magnetic structures [15–18]. However,
+occasionally deviations from a putative KHAF tend to
+stabilize QSL phases. In particular, the so called breath-
+ing anisotropy has been predicted to favor a resonance
+∗ daniefla@ethz.ch
+† zhelud@ethz.ch; http://www.neutron.ethz.ch/
+valence bond solid ground state for a wide range of cou-
+pling parameters [19, 20].
+In
+this
+context,
+the
+recently
+discovered
+family
+R3BWO9 of rare-earth antiferromagnets is an optimal
+platform for the search of spin-liquid candidates [21].
+Here R is a trivalent rare-earth element and the large
+difference in size of the constituent atoms prevents anti-
+site chemical disorder. All of the members of the family
+realize a breathing kagome lattice in their basal plane
+and show no sign of magnetic ordering down to 2 K.
+The strong spin-orbit coupling in combination with crys-
+tal electric field effects opens the possibility of realizing
+effective Jeff = 1/2 magnetic moments.
+Among all compounds in the family, the most promis-
+ing is Nd3BWO9. A large Weiss temperature [21] has
+been reported and the total angular momentum of Nd3+
+(J = 9/2) makes it a Kramers-doublet system. No mag-
+netic long-range order has been found in previous studies
+down to 1.8 K. However, little is known so far about its
+magnetism. In this study we report on the low tempera-
+ture properties of single crystals of Nd3BWO9. We found
+static magnetic long-range order below 0.3 K. The ob-
+served magnetism suggests a three dimensional network
+of exchange interactions. Nonetheless, due to the highly
+frustrated interaction a complex phase diagram is real-
+ized.
+The paper is structured as follows. First, a summary
+of the various methods used is provided. Then, we out-
+line the main results of the experiments. Subsequently,
+a detailed discussion of the main outcome is provided,
+including a thorough description of the magnetic struc-
+ture and a detailed picture of the magnetic phase dia-
+gram under applied fields. Finally, the main conclusions
+are drawn and further steps in the search of QSL physics
+are examined.
+arXiv:2301.05555v1  [cond-mat.str-el]  13 Jan 2023
+
+(b)
+(e)
+2 mm
+a
+b
+c
+2.66 Å
+2.57 Å
+2.25 Å
+2.49 Å
+2.38 Å
+2.36 Å
+Nd
+2.55 Å
+2.42 Å
+c
+WO6
+B
+NdO8
+a
+b*a*
+b
+c
+(a)
+(d)
+Nd
+a
+b
+c
+l = 16.44 Å
+(c)
+3.95 Å
+4.92 Å
+4.25 Å
+FIG. 1.
+Crystal structure and superexchange topology in
+Nd3BWO9. (a) Schematic structure reflecting the purported
+kagome interaction in the crystallographic ab plane.
+Only
+atoms with 0 ≤ z ≤ 0.5 are shown here. There is an addi-
+tional kagome plane displaced by half lattice parameter along
+the c crystallographic direction. (b) The shortest superex-
+change Nd-O-Nd bond links neodymium atoms in different
+kagome planes, forming isolated spin tubes along the c axis
+arranged in a triangular lattice. The kagome bonds are shown
+for reference along with bond distances. (c) A typical single
+crystal sample of Nd3BWO9. (d) A single spin tube is unfrus-
+trated. However, further-neighbor interactions frustrate the
+system. An arrow indicates the size of the magnetic supercell
+at zero field. (e) The environment of neodymium has very
+low symmetry, resulting in a C1 point group for the magnetic
+ion. Nd-O distances are indicated.
+II.
+METHODS
+Nd3BWO9 crystallizes in a hexagonal structure, with
+space group P63 (No. 173), where the magnetism stems
+from the effective magnetic moment of the Nd3+ ions.
+Single crystal samples were grown by spontaneous crys-
+tallization using a flux method as described in [22]. Pur-
+ple transparent single crystals with well defined facets
+were obtained [Fig. 1(c)].
+Typical masses range from
+a few micrograms to 40 mg and different samples were
+used in this study, depending on the technique.
+The
+chemical structure of the different single-crystal samples
+used in this study was validated using single-crystal X-
+ray diffraction on a Bruker APEX-II instrument, and was
+found to be in agreement with previous reports [21]. The
+structure is schematically depicted in Fig. 1, where the
+kagome-lattice bonds can be readily identified. Powder
+samples of Nd3BWO9, as well as of the non-magnetic
+La3BWO9, were synthesized by a solid state reaction.
+The correct chemical structure and the quality of the
+powders was checked with powder X-ray diffraction in
+a Rigaku MiniFlex diffractometer.
+Boron-11 enriched
+samples (both powder and single crystals) were also pre-
+pared for their use in neutron scattering experiments.
+Measurements of heat capacity, magnetocaloric effect
+(MCE), magnetization and magnetic torque were carried
+out using a 3He-4He dilution refrigerator insert for the
+Quantum Design Physical Property Measurement Sys-
+tem (PPMS). A sample of mass 0.131 mg was used for
+both heat capacity and MCE measurements. Heat ca-
+pacity data were collected using a standard relaxation
+method from Quantum Design for temperatures 100 mK
+< T < 4 K in applied fields of 0 T < µ0H < 3 T. The
+magnetic field was applied along the crystallographic a∗,
+and c directions. In zero field, data were collected from
+100 mK to 300 K. Heat capacity data of La3BWO9 were
+measured down to 2 K and extrapolated to lower tem-
+peratures from an empirical fit to a T 3-power law. MCE
+data were measured using the same puck as for heat ca-
+pacity. The change of temperature of the sample was
+recorded as the magnetic field was swept up and down
+at a constant rate. In order to avoid self heating of the
+puck, the field change rate was optimized and a value of
+0.5 mT/s was selected. In the terminology of MCE mea-
+surements, our experiment was conducted under equilib-
+rium conditions.
+Magnetization was measured using an in house made
+Faraday-balance capacitive magnetometer [23] at 120
+mK and 2 K and magnetic fields applied along three ori-
+entations: a∗, and b, and c. Additional measurements
+of magnetization carried out in the MPMS system at 2
+K were used to calibrate the low temperature data and
+obtain absolute units (not shown here). Using the same
+setup, magnetic torque was measured up to 3 T and
+for temperature from 120 mK to 600 mK. The torque
+data correspond to the deflection of a small cantilever
+on which the sample is mounted.
+The magnetic field
+sweeping rate was also optimized to minimize heating
+due to eddy currents.
+Magnetic susceptibility was measured using the Quan-
+tum Design Magnetic Property Measurement System
+(MPMS) SQUID Magnetometer.
+The temperature
+range from 1.8 K to 300 K was probed using a small po-
+larizing field applied along three crystal directions: a∗,
+and b, and c. The probing field was µ0H = 0.1 T, where
+µ0 denotes the permeability of vacuum.
+Inelastic neutron scattering on powder samples of
+Nd3BWO9 was measured to investigate the crystal elec-
+tric field induced scheme of total angular momentum
+states. The instrument of choice was the thermal neu-
+tron triple-axis-spectrometer EIGER at PSI. 11.1 g of
+Nd3 11BWO9 was sealed in an aluminum can and in-
+stalled in a standard 4He orange cryostat. A final wave-
+length of kf= 2.66 ˚A−1 (λ = 2.36 ˚A) was chosen, us-
+ing a pyrolytic graphite filter to eliminate higher-order
+neutrons without further collimation. Data were mea-
+sured at constant scattering angle, 2θ. The background
+was investigated to select the optimal value for the scat-
+tering angle, sufficiently far from the direct beam and
+low enough to have good counting and small decay in
+the signals due to magnetic structure factors. A value
+of 2θ = 10◦ was chosen, and the incident energy was
+scanned at three different temperatures: 1.5 K, 100 K
+and 300 K.
+Neutron single crystal diffraction was used to investi-
+2
+
+J0
+100
+200
+300
+T (K)
+0
+50
+100
+150
+200
+-1 (mol T μB )
+H || a*
+θCW = -3.76 K
+μ0H = 0.1 T
+H || b
+H || c
+-1
+FIG. 2.
+Inverse magnetic susceptibility on single crystals.
+Data show measurements for three field orientations. A small
+probing field of 0.1 T was used for all measurements. The
+black solid line represents a Curie-Weiss model with the av-
+erage Weiss temperature and effective moment parameters,
+given in Table. I.
+gate the magnetic structures in the ordered phases. A
+single crystal sample of 18 mg in mass of Nd3 11BWO9
+and 5.5×1.4×0.8 mm3 was studied using two different
+instruments. Measurements with H ∥ a∗ were carried
+out at the Thermal Single Crystal Diffractometer ZE-
+BRA at the Swiss Spallation Neutron Source, SINQ,
+in the Paul Scherrer Institut (PSI, Switzerland).
+The
+diffractometer was used in conjunction with a 3He-4He
+dilution refrigerator and a 6-T magnet.
+The crystal
+was aligned with its a∗ axis vertical, the same direc-
+tion as the applied magnetic field. Neutron wavelengths
+of λ = 2.314 ˚A and 1.383 ˚A were selected, provided by
+the PG(200) and Ge(220) monochromators. Additional
+measurements with H ∥ c were carried out in the time-
+of-flight diffractometer WISH at the ISIS facility in the
+Rutherford Appleton Laboratory, in the United King-
+dom. The sample was mounted with its c axis vertical
+and parallel to the magnetic field. A 3He-4He dilution
+refrigerator and a 10-T magnet were used to access the
+ordered states in Nd3BWO9.
+III.
+EXPERIMENTAL RESULTS
+A.
+Magnetic susceptibility
+Figure 2 shows inverse susceptibility measurements for
+probing fields applied along the crystallographic direc-
+tions a∗b, and c.
+Down to the lowest accessible tem-
+perature of 1.8 K, these data show no sign of magnetic
+ordering.
+A fit of the experimental data to a Curie-Weiss model
+is shown overlaid on the experimental results. A good
+TABLE I. Fitting parameters from the Curie-Weiss model for
+data shown in Fig.
+2.
+200 K ≤ T ≤ 300 K 20 K ≤ T ≤ 60 K
+θW (K)
+µeff (µB)
+θW (K) µeff (µB)
+H ∥ a∗
+-54.3
+3.76
+-3.78
+2.94
+H ∥ b
+-54.7
+3.79
+-3.82
+2.90
+H ∥ c
+-59.2
+3.77
+-3.68
+2.91
+agreement is found for data above 130 K, with a large,
+negative Weiss temperature. The resulting Weiss tem-
+peratures, θW are given in Table. I, as well as the cor-
+responding effective magnetic moments extracted from
+the Curie constants as C = NAµ0µ2
+eff/(3kB). The ob-
+tained effective magnetic moments are close to the value
+expected for a free Nd3+ ion: µeff = gJ
+�
+J(J + 1)µB =
+3.6µB. Importantly, the susceptibility data show little
+dependence on the direction of the magnetic field, which
+suggests that, the resulting magnetic anisotropy remains
+quite small.
+Our results are consistent with those re-
+ported in Ref.[21] on polycrystal samples.
+Below 130 K a clear deviation from the high temper-
+ature fit is observed.
+This is roughly consistent with
+the existence of a crystal electric field (CEF) level at
+15.9 meV (see below), signaling the total depletion of
+the population of the first excited state. A Curie-Weiss
+analysis is heavily affected by the partial population of
+excited multiplets and lead to an overestimation of ex-
+change parameters and exchange couplings. Therefore,
+an additional fit to a Curie-Weiss law for a tempera-
+ture range far enough from the CEF resonance has been
+performed.
+The results are also summarized in Table
+I. Temperatures in the range between 20 K and 60 K
+were considered for this fit. The resulting Weiss temper-
+atures are much reduced compared to the high temper-
+ature fit. However, they still reflect a predominant an-
+tiferromagnetic interaction in Nd3BWO9. The effective
+magnetic moments are also reduced with respect to their
+high temperature value, yielding an average moment of
+µeff = 2.92µB.
+B.
+CEF level scheme
+The inelastic neutron scattering spectra are shown in
+Fig. 3.
+Large intensity at zero energy transfer corre-
+sponds to quasielastic scattering. Three resonances are
+identified at 15.9, 32.8, and 43.7 meV, which we ascribe
+to CEF induced levels due to their temperature depen-
+dence. Importantly, no resonance is found below 15.9
+meV. Since the total angular momentum J = 9/2 of the
+free Nd3+ is expected to be fully split into five Kramers
+doublets, this suggests that the low temperature physics
+of Nd3BWO9 can indeed be described in terms of the
+lowest laying doublet, giving rise to an effective two-level
+system well below ∆ = 15.9 meV ≈ 180 K.
+3
+
+0
+10
+20
+30
+40
+0
+0
+0
+ħω (meV)
+T = 1.5 K
+T = 100 K
+T = 300 K
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Ef = 14.7 meV
+2θ = 10°
+Intensity (arb. units)
+FIG. 3. Inelastic neutron scattering intensity at a constant
+scattering angle for three different temperatures. The final
+energy of Ef= 14.7 meV was fixed and incident energy var-
+ied, fixing a 10 degree scattering angle. CEF resonances are
+indicated by black arrows. An offset of 0.25 and 0.50 units
+was added for visibility, a dashed line indicates the reference
+zero for those data sets.
+C.
+Specific heat
+Specific heat as a function of temperature and mag-
+netic field is used to unveil the magnetic phase diagram
+of Nd3BWO9 at ultra-low temperatures. Data obtained
+at zero field are shown in Fig. 4. Nd3BWO9 shows an up-
+turn in specific heat below 4 K with two clearly distinct
+features [Fig. 4(a)]. Around 1 K, a hump in specific heat
+suggests the onset of short-range magnetic correlations
+[24]. At TN = 300 mK we found a sharp lambda anomaly
+representing the transition into magnetic long range or-
+der.
+Below TN the specific heat signal remains large
+down to the lowest accessible temperatures in our setup,
+likely due to nuclear specific heat from the rare-earth
+ions. In order to understand exactly the nature of the
+magnetic specific heat, we have examined the different
+contributions and subtracted them from the measured
+total specific heat.
+To estimate the phononic contribution, we synthesized
+the non-magnetic isostructural material La3BWO9 and
+measured its specific heat in the same range of temper-
+atures.
+This is shown in Fig. 4(a) and represents the
+lattice contribution, CL, in Fig. 4(b).
+An accurate estimation of the nuclear contribution to
+specific heat is usually much more complicated, as a
+1
+10
+100
+T (K)
+0
+10
+20
+30
+40
+50
+Cp (J mol-1 K-1)
+Nd3BWO9
+Nd3BWO9
+TN
+La3BWO9
+0.1
+0.4
+1
+4
+10
+T (K)
+Rln(2)
+0
+10
+20
+Cp/T (J mol-1 K-2)
+CL
+CN
+Ctot
+Cmag
+0
+1
+2
+3
+4
+T (K)
+0
+2
+4
+6
+8
+Smag (J mol-1
+NdK-1)
+(a) μ0H = 0 T
+(b)
+(c)
+TN
+FIG. 4.
+(a) Total specific heat at zero magnetic field for
+Nd3BWO9 and the nonmagnetic isostructural compound
+La3BWO9.
+Nd3BWO9 shows a substantial magnetic con-
+tribution to specific heat below 3 K. (b) Total specific heat
+(open circles) and magnetic specific heat (filled circles) after
+subtraction of lattice and nuclear degrees of freedom. Lat-
+tice (CL) and nuclear (CN) contribution are estimated as
+discussed in the text. A lambda anomaly can be found at
+TN = 0.30 K, signaling the onset of long-range magnetic or-
+der. (c) The magnetic entropy per Nd3+ ion saturates above
+3 K. A dashed line represents the expected value for a two-
+level system at infinite temperature.
+variety of effects has to be considered.
+These include
+dipole and quadrupolar splitting, or hyperfine coupling
+between nuclei and electrons (which can be quite signif-
+icant in magnetically ordered materials).
+Neodymium
+has two isotopes with nonzero dipolar and quadrupolar
+momenta, out of its 7 stable isotopes. Following the rea-
+soning in Ref. [25], the effect of quadrupolar splitting is
+assumed to be small compared to that of hyperfine cou-
+pling, and we neglect it here. In a magnetized phase,
+local fields are expected to be sizable and therefore hy-
+perfine coupling may significantly contribute to specific
+heat. The contribution from dipole field splitting from a
+single isotopic species is given by
+4
+
+Cp/T (J/molK-2)
+(a) H || a*
+0
+0.5
+1
+1.5
+T (K)
+0
+20
+40
+60
+80
+Cp/T (J/molK-2)
+0 T
+0.85 T
+0.55 T
+1.2 T
+0
+10
+20
+30
+40
+50
+60
+70
+0 T
+0.85 T
+0.975 T
+(b) H || c
+FIG. 5. Typical temperature scans of specific heat for differ-
+ent fixed values of magnetic field applied along (a) H ∥ c and
+(b) H ∥ a∗. An offset of 15 J/mol/K2 has been added for
+visibility. Solid filled triangles show features associated with
+the phase transitions discussed in the main text.
+CH,i =
+NAkB
+α2
+i
+4I2
+i
+�
+�
+1
+sinh2 �
+αi
+2Ii
+� −
+(2Ii + 1)2
+sinh2 �
+(2Ii+1)αi
+2Ii
+�
+�
+�
+(1)
+where αi = AH(µNd
+Hyp/gJ)Ii/kBT, and Ii is the nuclear
+spin, gJ = 8/11 (Land´e factor for Nd), NA is the Avo-
+gadro constant, and kB the Boltzmann constant. AHyp
+represents the strength of the hyperfine coupling and
+here we made a second approximation. We assume all
+the nuclei couple equally to the electron density and the
+value of AHyp is approximated as that of Nd metal [26].
+µNd
+Hyp denotes the static dipole moment of the Nd3+ ions.
+This is precisely the origin of the local field and for its
+value we chose the averaged effective magnetic moment
+from the magnetic susceptibility data at low tempera-
+tures µNd
+Hyp = 2.914µB. Finally, the different species are
+summed, weighted by their isotopical abundance to ob-
+tain the temperature dependence of nuclear specific heat.
+This model with no free parameters is in excellent
+agreement with the lowest temperature data, as shown in
+Fig. 4(b). Having modeled the nuclear specific heat, the
+magnetic specific heat can be extracted by subtraction.
+The magnetic specific heat was subsequently integrated
+to obtain the temperature dependence of magnetic en-
+Cp/T (J/mol K-2)
+150 mK
+250 mK
+350 mK
+500 mK
+800 mK
+0
+0.5
+1
+1.5
+2
+2.5
+3
+0H (T)
+0
+20
+40
+60
+Cp/T (J/mol K-2)
+(a) H || a*
+0
+20
+40
+60
+80
+150 mK
+250 mK
+350 mK
+500 mK
+800 mK
+(b) H || c
+*
+FIG. 6. Typical field scans of specific heat measured at con-
+stant temperature in Nd3BWO9 for (a) H ∥ c and (b) H ∥ a∗.
+An offset of 10 or 15 J/mol/K2 is added for visibility be-
+tween the scans for (a) and (b), respectively.
+Solid filled
+triangles show features associated with the phase transitions
+discussed in the main text. Black arrows signal the existence
+of broad double-hump features, described in the text.
+An
+asterisk shows a feature above the saturation transition.
+tropy, depicted in Fig. 4(c). The high temperature trend
+of this quantity approaches the value of R ln(2), the ex-
+pected value of a two-level system.
+In a magnetic field, a simple estimation of the contri-
+bution of the nuclear spin due to Zeeman splitting could
+not account for the effects observed here. Low tempera-
+ture data in Fig. 5 show that the effect of nuclear specific
+heat is of the same order of magnitude up to 1.2 T and it
+is not strongly field dependent. This suggests that also
+in a field the main contribution comes from hyperfine
+coupling. However, a quantitative determination of this
+effect under magnetic fields becomes paramount.
+The evolution of the specific heat of Nd3BWO9 under
+magnetic fields is shown in Fig. 6 for fields along two
+different crystallographic directions. The total heat ca-
+pacity is displayed here, without subtraction of lattice
+or nuclear degrees of freedom. Typical-field scans show
+a number of anomalies that are consistent with the exis-
+tence of three different phases with static magnetic order
+at low temperatures.
+Up to three distinct features can be observed for
+H ∥ a∗ at the lowest temperature, at 0.45, 0.62 and 1.05
+T and are marked with triangles in Fig. 6(a).These fea-
+5
+
+tures are rather spread in fields, specially at saturation.
+However, the existence of thermodynamic transitions has
+been confirmed by neutron diffraction (as discussed be-
+low). The two lower field anomalies move apart as the
+temperature is increased. The two higher field anoma-
+lies merge at 0.25 K, denoting the highest temperature
+of the ordered phase. Though the specific heat anoma-
+lies in Fig. 6(a) are too broad for a precise estimation
+of the upper critical field, this quantity can be deduced
+from magnetocaloric effect measurements (see below).
+For fields orthogonal to the hexagonal plane (H ∥ c)
+at the lowest temperature one finds two anomalies at 0.5
+T, 0.8 T and a sharper one at 0.95 T. [Fig. 6(b)] Notably,
+in this configuration the different anomalies appear nar-
+rower than for H ∥ a∗, especially at the saturation field.
+The first two anomalies move apart as the temperature
+is increased, while the higher field anomaly barely shifts
+in position up to 0.2 K. The low field anomaly shifts to-
+wards zero field and disappears as TN is reached. The
+two high-field anomalies merge at T = 0.2 K. From the
+high field anomaly we extract an estimate of the satura-
+tion field of µ0Hc = 0.975(3) T. Interestingly, an extra
+feature can be identified above saturation (asterisk in
+Fig. 6(a)).
+This feature shifts to higher fields as the
+temperature is increased and decreases rapidly in mag-
+nitude. Above 0.2 K it is hardly identifiable.
+Finally, double-hump features can be observed above
+0.3 K for both magnetic field configurations. These are
+significant up to the highest measured temperatures and
+particularly prominent around the saturation field (black
+arrows in Fig. 6).
+For H ∥ c the amplitude of these
+modulations is larger than in H ∥ a∗. Such features are
+often associated with a low-dimensional crossover from
+the zero field disordered phase to the fully polarized state
+without the occurrence of a phase transition [27–30].
+D.
+Magnetocaloric effect
+Magnetocaloric
+effect
+(MCE)
+measurements
+in
+Nd3BWO9 provide key information on the nature of the
+various phase transitions found with other techniques
+[31–33].
+Representative temperature profiles are sum-
+marized in Fig.
+7. Several crossings can be observed
+for both configurations.
+The observed anomalies are
+too broad to assign exactly a transition point.
+Due
+to the proximity of the thermodynamic transitions
+in the phase diagram, features corresponding to both
+transitions merge and overlap.
+In our measurements
+the field is swept slow enough as to ensure equilibrium
+conditions.
+Data measured with H ∥ a∗ show mostly symmetric
+features around the crossing points.
+Particularly, this
+suggests that the measured phase transitions are of sec-
+ond order. In contrast, the low temperature profiles for
+H ∥ c show two distinct behaviors. At 0.6 T one finds a
+roughly symmetric feature, suggesting again a second or-
+der phase transition. This is different at 0.975 T, where
+a very asymmetric feature appears, pointing to a first
+0
+μ
+μ
+1
+2
+0H (T)
+0.1
+0.2
+0.3
+0.5 mT/s
+0.5 mT/s
+0.4
+0.5
+0.6
+0.7
+T (K)
+0
+1
+2
+0H (T)
+(b) H || c
+(a) H || a*
+FIG. 7. Plots of the magnetocaloric effect in Nd3BWO9 for
+different base temperatures and fields applied along (a) H ∥
+a∗ and (b) H ∥ c. For all the scans, red (blue) color represents
+data measured while driving the magnetic field up (down).
+A ramping rate of 0.5 mT/s was used throughout all the
+measurements.
+Small prominent features (specially at low
+fields) are spurious and the result of an unstable platform.
+order or discontinuous transition.
+Finally, the absence of anomalies above the saturation
+field for H ∥ c must be noted. The features observed in
+the fully polarized phase in Fig. 6(b) leave no trace in
+the MCE data in the same configuration.
+The MCE technique is based on the change of entropy
+in a magnetic system as it is driven through a phase
+transition, crossover, level crossing, etc. Consequently,
+one can retrieve the change in entropy in a system from
+the change in temperature against magnetic field [31].
+Under equilibrium conditions, we obtain the entropy as
+∆S = S(H) − S0 = −
+�
+κT − Tbath
+T
+dt
+(2)
+where κ is the thermal conductivity of the thermal
+link in the calorimeter, T is the sample temperature, and
+Tbath is the thermal bath temperature. Integration of the
+data in Fig. 7 gives rise to the entropy maps displayed in
+Fig. 8. The data above 0.2 K are a good picture of the
+entropy stored in the magnetic subsystem. However, for
+temperatures below 0.15 K imperfect equilibrium condi-
+tions prevent a reliable estimation of entropy. A strong
+accumulation of entropy is observed above the saturation
+transitions for both field configurations. The position of
+the peaks in entropy match the estimated position of the
+critical fields from specific heat. For H ∥ a∗, the maxima
+in entropy at different temperatures were used to obtain
+an accurate estimate of the upper critical field.
+A fit
+to the data provides Hc,a∗ = 1.187(13) T. This value is
+consistent with the various probes used in this study.
+6
+
+0
+0.5
+1
+1.5
+2
+0
+0.2
+0.4
+0.6
+0.8
+T (K)
+μ0H (T)
+0
+0.2
+0.4
+0.6
+0.8
+T (K)
+(a) H || a*
+A
+B
+S (J molNd K-1)
+-1
+0
+1
+2
+3
+(b) H || c
+A
+C
+FIG. 8. Entropy maps in false color for two magnetic field
+orientations. In false color plots, the change in entropy ex-
+tracted from magnetocaloric data from Fig. 7. Filled circles
+(diamonds) denote phase anomalies associated with phase
+transitions from specific heat field (temperature) scans for
+(a) H ∥ a∗ and (b) H ∥ c.
+In (a) red squares show the
+maxima of entropy at the measured temperatures. A white
+dashed line is a power law fit to the data showing the best
+estimate for the upper critical field.
+E.
+Magnetization
+The evolution of magnetization under a magnetic field
+provides insight on the type of order in Nd3BWO9.
+Strikingly, a fractional magnetization plateau is observed
+for all measured configurations, as displayed in Fig. 9.
+The value of magnetization is consistent with a fractional
+m=1/3 plateau and spans a range of fields of 0.2-0.3 T.
+In addition, the zero field phase shows zero magnetiza-
+tion for all applied fields, which indicates the realization
+of a gapped phase at T = 0. Magnetization data for in-
+equivalent directions in the hexagonal plane show very
+similar behavior, but differ from the results perpendicu-
+lar to the plane.
+For H ∥ a∗ and H ∥ b the zero magnetization phase
+extends up to 0.4 T. Above 0.5 T the system transitions
+into the factional magnetization plateau state up to a
+0
+0.5
+1
+1.5
+2
+0H (T)
+0
+0.5
+1
+1.5
+2
+2.5
+M ( B/Nd3+)
+0
+2
+4
+6
+8
+10
+12
+Ms,c/3
+(a.u.)
+H || a*
+H || b
+H || c
+T = 120 mK
+Magnetometer
+H||c
+(0,2,0), 55 mK
+H||a*
+(0,0,2), 130 mK
+Ms,a*/3
+Ms,b/3
+FIG. 9. Magnetization per Nd3+ ion measured at 120 mK
+in Nd3BWO9 for magnetic fields along the crystallographic
+directions a∗, b, and c from bulk measurements (left axis).
+Magnetization extracted from neutron diffraction intensity
+of nuclear reflections is superimposed to the corresponding
+bulk data. Plotted is the rescaled square root of the static,
+magnetic structure factor S∞
+z,z(q) (right axis). The measured
+reflections (Q) are indicated in the figure. Two of the data
+sets have been offset vertically by 0.5 and 1.0 units to improve
+visibility (a dashed line indicates their respective zero). The
+magnetization value at 1/3 of saturation is indicated for each
+individual data set by an arrow next to the plateau state.
+marked limit at 1 T. The transition into the fully satu-
+rated phase is gradual between 1 T and 1.3 T.
+In contrast, much sharper features are found when
+fields H ∥ c are applied. A non-magnetizable phase ap-
+pears up to 0.5 T, above which the system jumps rapidly
+into the plateau state at 0.65 T. The plateau terminates
+in a first-order jump to saturation around 1 T. Notably,
+despite the presence of a first-order transition, our mea-
+surements did not show signatures of hysteresis across
+the saturation transition for H ∥ c.
+Saturation fields extracted from magnetization data
+are consistent with those found in the specific heat
+data. The values for saturation magnetization show little
+anisotropy, finding 1.34(6) µB, 1.31(3) µB, and 1.35(4)
+µB per magnetic ion for configurations a∗, b, and c re-
+spectively. It suggests a nearly isotropic g-tensor in the
+material.
+F.
+Magnetic torque
+Magnetic torque is arguably the most sensitive tech-
+nique to magnetic phase transitions.
+Raw data are
+presented as the change in the measured capacitance
+7
+
+-0.2
+-0.15
+-0.1
+-0.05
+0
+-0.5
+0
+0.5
+-0.4
+-0.2
+0
+0.2
+C (fF)
+-2
+-1
+0
+1
+dC/dH (fF/T)
+0
+1
+2
+0
+0.2
+0.4
+0.6
+0
+1
+2
+0H (T)
+-0.5
+0
+0.5
+1
+1.5
+2
+2.5
+175 mK
+200 mK
+225 mK
+250 mK
+275 mK
+300 mK
+350 mK
+400 mK
+500 mK
+600 mK
+(a) H || a*
+(b) H || a*
+(c) H || b
+(d) H || b
+(e) H || c
+(f) H || c
+FIG. 10. Magnetic torque (∆C) and its field derivative mea-
+sured at constant temperatures against magnetic field, for
+three field orientations: (a,b) a∗, (c,d) b, and (d,e) c. For all
+data sets, a reference value of capacitance at zero field has
+been chosen and subtracted. Black arrows indicate features
+that may be identified with phase transitions.
+∆C = C(H) − C(H = 0 T) as a function of magnetic
+field for each temperature (Fig. 10). The torque data
+show strong differences between the measurements in the
+basal plane and perpendicular to it, but the obtained re-
+sults are very similar for both measurements within the
+plane. The raw data show some structure, but not sharp
+features as is customary in such measurements. Phase
+transitions are best captured in the first derivative of the
+raw data 10.
+Field derivative data show features that correspond
+with transitions observed in the other techniques re-
+ported in this study. Direct comparison with the other
+data sets is necessary to pinpoint what anomalies repre-
+sent real phase transitions. These features are indicated
+with arrows in the field derivative data [Fig. 10(b), 10(d),
+and 10(f)]. For H ∥ a∗ two distinct anomalies can be ob-
+served in the scan at 175 mK, at 0.68 T and at 0.99 T.
+These correspond to the lower and upper boundaries of
+the plateau phase. Data for H ∥ b show three anomalies
+at 0.68 T, 1.02 T and 1.28 T. The lower fields correspond
+again to the boundaries of the plateau phase. Notably,
+these two anomalies come together as the temperature is
+increased and disappear above 300 mK. The higher field
+anomaly, which is broader and less sharp, corresponds
+to the crossover into the fully saturated state. Finally,
+fields applied along the c direction reveal a completely
+different structure. Three anomalies can be identified at
+0.48 T, 0.71 T, and 0.95 T. The associated transitions in
+this case are the boundaries of the plateau for the high
+field features and the transition from the low field phase
+to paramagnet for the low field anomaly. The low field
+features, though weak, fade away as the transition tem-
+perature is overcome. The high field anomaly remains up
+to the highest temperatures representing the crossover of
+the system into the fully polarized pseudospin.
+G.
+Neutron diffraction
+We resorted to single-crystal neutron diffraction to in-
+vestigate the magnetic structures realized in the low field
+and the plateau phases. Figure 11 summarizes the re-
+sults obtained from the different instruments. The field
+dependence of the order parameter is depicted for both
+field configurations, which is in perfect agreement with
+our thermodynamic measurement data.
+Zero field data from both experiments unveil a com-
+mensurate phase with propagation vector Q = (0, 0,1/3).
+Fig.
+11(a) and Fig.
+11(b) show that magnetic reflec-
+tion (1,1,-1/3) is present throughout phase A for both
+field orientations.
+The phase is consistent with fully
+commensurate order, which leads to the appearance of a
+magnetic supercell, as is shown in Fig. 1(d). Integrated
+intensity of reflection (1,1,-1/3) drops at the intermedi-
+ate transition field, above which a different type of or-
+der is found depending on the direction of the magnetic
+field. For phase B (H ∥ a∗) we found magnetic reflec-
+tions (0,1/2,1/2) and (1/2,0,1/2). These reflections van-
+ish at fields slightly below saturation. Finally, phase C
+(H ∥ c) has been found to realize order with propaga-
+tion vector (1/3,1/3,1/3). Magnetic reflection (1/3,1/3,-
+1/3), which is inequivalent to the former, has also been
+found. Fig. 11(b) shows an abrupt drop in the intensity
+of reflection (1/3,1/3,1/3), consistent with a first order
+transition to saturation.
+An external magnetic field induces a ferromagnetic
+component in every lattice site that gives rise to the
+bulk magnetization. This produces extra scattering pro-
+portional to the square of the induced magnetic mo-
+mentum at the position of each nuclear peak . Fig. 9
+shows the uniform magnetization density extracted from
+two nuclear reflections: (020) for H ∥ a∗ and (200) for
+H ∥ c. We selected reflections where nuclear contribu-
+tion is minimal while a measurable magnetic intensity
+can be observed. The zero-field integrated intensity is
+subtracted from the data in a field to obtain the cor-
+8
+
+0
+2
+4
+6
+0.1
+0
+0.5
+1
+1.5
+0H (T)
+0
+5
+10
+Integrated Intensity (arb. units)
+0.2
+0.3
+0.1
+0.3
+0.1
+0.4
+T (K)
+0
+1
+2
+3
+4
+5
+Int.(arb. units)
+0.2
+0.4
+T (K)
+0.36
+0.35
+0.34
+0.33
+0.32
+(1,1,-l) (r.l.u.)
+Q = (1, 1, -1/3)
+Q = (0, 1/2, 1/2)
+Q = (1, 1, -1/3)
+Q = (1/3, 1/3, 1/3)
+T = 120 mK
+A
+B
+C
+A
+(a)
+(b)
+μ0H || a*
+ZEBRA
+ZEBRA
+T = 55 mK
+μ0H || c
+WISH
+(d)
+µ0H = 0 T
+0.1
+1.0
+Int.
+(a. u.)
+(c)
+Q = (1, 1,-⅓-δ)
+µ0H = 0 T
+FIG. 11. Results from single crystal magnetic neutron diffrac-
+tion. (a,b) Field dependence of the integrated neutron inten-
+sity at the magnetic propagation vectors for: (a) H ∥ a∗
+(ZEBRA, PSI) and (b) H ∥ c (WISH, ISIS). Note that in (a)
+the intensity of the (0,1/2,1/2) reflection has been rescaled by
+×0.1. In (b) the limits of the ordered phases are highlighted
+and shown with arrows. (c) Evolution of the integrated in-
+tensity of the reflection (1,1,-1/3) with temperature at zero
+magnetic field. (d) Incommensuration of the propagation vec-
+tor at zero field against temperature, shown as a shift in the
+peak position of the (1,1,l) reflection.
+responding magnetic scattering. Longitudinal magneti-
+zation is then plotted as the square root of mangetic
+intensity. The agreement with bulk measurements is re-
+markable and further highlights the existence of magne-
+tization plateaus regardless of field orientation.
+Finally, zero field neutron diffraction reveals an incom-
+mensurate state between the low temperature ordered
+phase and the paramagnetic phase.
+The onset of in-
+commensurate magnetic order appears around 0.34 K at
+the wavevector Q = (0, 0, 1/3 + δ).
+Temperature de-
+pendence of the intensity around the (1,1,l) reflection
+in Fig. 11(d), where the peak position is superimposed,
+shows this incommensuration.
+Reduction of the tem-
+perature leads to a change in the incommensurate prop-
+agation vector roughly linearly with temperature.
+At
+0.26 K the propagation vector locks into the commensu-
+rate Q = (0, 0, 1/3), as observed for the low temperature
+structure. The robustness of this evolution to commen-
+suration has been verified for several additional magnetic
+reflections.
+IV.
+DISCUSSION
+The purported breathing kagome structure is shown in
+Fig. 1. Unequal Nd-Nd distances and Nd-O-Nd angles
+result in inequivalent exchange parameters for neighbor-
+ing corner-sharing triangles [21]. This is represented by
+the exchange constants J△ and J▽, respectively. How-
+ever, a crystallographic analysis cannot rule out the ex-
+istence of interaction between adjacent kagome planes.
+Due to the short distances between kagome planes, the
+topology of the exchange interaction in Nd3BWO9 is
+likely three dimensional. In fact, the shortest superex-
+change Nd-O-Nd pathway (nearest neighbors, J1) links
+rare-earth ions belonging to different kagome planes [Fig.
+1(b)]. These couplings are arranged into isolated twisted
+3-legged spin tubes, one-dimensional structures that ex-
+tend perpendicular to the kagome planes [see Fig. 1(c)].
+Noteworthy, the resulting structure considering only
+nearest neighbor coupling is bipartite.
+A single tube
+would show no frustration, highlighting the relevance
+of further neighbor interactions.
+A three-dimensional
+structure with several exchange parameters may have
+to be regarded, as opposed to the originally suggested
+kagome structure. Yet, the onset of static magnetic or-
+der is extremely suppressed by the strong magnetic frus-
+tration f = −θW /TN ≈ 12.6, confirmed from magnetic
+susceptibility.
+Six magnetic rare-earth Nd3+ ions occupy general
+Wyckoff positions in the unit cell. The reduced point
+symmetry around the Nd3+ ions [Fig.
+1(e)] fully lifts
+the degeneracy of the total angular momentum levels (J
+= 9/2) into five Kramers doublets. The strong CEF iso-
+lates a single Kramers doublet with a large gap to excited
+multiplets. The obtained zero-field entropy is consistent
+with a value of S = R ln(2).
+These two observations
+show that Nd3BWO9 can be described as an effective
+spin S = 1/2 system below 100 K. However, the low
+symmetry precludes attempts to identify unequivocally
+a CEF-Hamiltonian and to extract the eigenstates of the
+lowest energy multiplet.
+Both magnetization and susceptibility suggest very lit-
+tle magneto-crystalline anisotropy. Susceptibility mea-
+surements suggest no preferential direction in the high
+temperature paramagnetic state.
+In addition, low-
+temperature magnetization in the fully saturated pseu-
+dospin phase shows no increase up to the highest probed
+fields. The increase of magnetization may be a rough
+estimator of the eigenstate admixing due to anisotropies
+(via Van-Vleck terms). No appreciable change in mag-
+netization is observed up to 2 T, indicating the total
+magnetization in the restricted pseudospin subspace is
+likely to be an approximately good quantum number. It
+9
+
+is, thus, likely that the low energy physics in Nd3BWO9
+can be described in terms of a highly symmetric spin
+Hamiltonian. A small axial anisotropy may be needed
+to account for the sharp features found for H ∥ c.
+To map out the phase diagram in the low tempera-
+ture regime for Nd3BWO9 we use specific heat measure-
+ments. Using a combination of all outlined techniques,we
+identify several regions of magnetic order.
+As shown
+in Fig. 12, the system reveals complex behaviour, with
+two different domes of long-range order observed for each
+configuration.
+A low field phase (A) extends roughly up to 0.6 T
+for both studied orientations. This phase possesses com-
+mensurate order with propagation vector Q =(0,0,1/3).
+Magnetization measurements show that this phase is
+hardly magnetizable, suggesting a gapped state in this
+field range. Although further analysis is needed to un-
+derstand the magnetic structures of the different phases
+in detail, a series of general remarks can be deduced
+from the data. For phase A, the presence of reflections
+(0,0,±2/3) forbids the existence of a collinear structure
+with spins parallel to c. Thus, a coplanar structure in
+the ab plane is likely realized.
+By increasing the magnetic field the system transitions
+into a field-induced ordered phase. A field H ∥ a∗ leads
+to the fractional m = 1/3 plateau phase B, characterized
+by a propagation vector Q =(0,1/2,1/2). The additional
+presence of wavevectors (1/2,0,1/2) and equivalent sug-
+gests a multi-Q structure or the presence of domains in
+the B phase.
+Strikingly, the order realized in the plateau is com-
+pletely different when fields are applied in the basal ab
+plane or perpendicular to it. In a field H ∥ c, phase C
+is found with propagation vector Q =(1/3,1/3,1/3). In
+contrast, saturation H ∥ c occurs through a sharp first
+order phase transition. Magnetocaloric effect supports
+this claim. A tricritical termination point appears where
+first and second order transition lines converge as shown
+in Fig. 12(b), at 0.20 K and 0.975 T. The presence of
+magnetic reflections (1/3,1/3,-1/3) and equivalent also
+indicates a complex spin texture, with either a multi-Q
+structure or the presence of domains.While here domains
+may be consistent with the observed first order transition
+to saturation, it is not possible at this stage to exclude
+either possibility.
+The existence of a tricritical point only for one ori-
+entation may be related to the large spin-lattice inter-
+action stemming from strong spin-orbit coupling. The
+transition to saturation for H ∥ c can be prematurely
+precipitated via an ’order by distortion’ [34] mechanism.
+A gain in magnetic energy compensates a small loss in
+elastic energy, leading to a first order transition to sat-
+uration. Though our neutron diffraction data show no
+evident change in the space group or lattice parameters
+in the high field phase, a detailed study would be neces-
+sary to discard this possibility.
+Phases A and B appear to merge below 100 mK at
+0.56 T. A first order phase transition is speculated be-
+tween A and B, with a termination bicritical point where
+0
+0.2
+0.4
+0.6
+0.8
+T (K)
+0
+5
+10
+15
+Cp/T (J mol-1K-2)
+(b) µ0H || c
+(a) µ0H || a*
+A
+A
+C
+B
+0
+0.5 ?
+(0,0,⅓)
+(0,0,⅓)
+(0,½,½)
+(⅓,⅓,⅓)
+1
+μ0H (T)
+0
+0.2
+0.4
+0.6
+FPP
+20
+25
+30
+FPP
+1.5
+2
+FIG. 12. Magnetic phase diagram of Nd3BWO9 in a mag-
+netic field applied along the principal directions: (a) a∗ and
+(b) c. The background depicts false color maps of Cp(H, T),
+with a shared color scale. Symbols: white circles and dia-
+monds represent transitions obtained from field and tempera-
+ture scans of specific heat, respectively. Green squares repre-
+sent the phase boundaries extracted from neutron diffraction
+data in Fig. 11. Upward-facing blue triangles show transi-
+tions extracted from bulk magnetization, downward facing
+pink triangles transitions from magnetic torque. A red dia-
+mond denotes the estimated position of the tricritical point
+for H ∥ c. An orange star shows the upper critical field es-
+timated in Fig. 8. Solid and dashed lines are a guide to the
+eye, representing second and first order transitions, respec-
+tively. The different phases are labeled as: A, B, C and Fully
+Polarized Pseudospin (FPP). The ordered phases show their
+corresponding magnetic propagation vector, as discussed in
+the text.
+all phase boundaries meet. Neutron diffraction data in
+Fig. 11(a) indicate the phases will likely merge slightly
+below 120 mK. Interestingly, between A and C the phase
+boundaries seem to develop smoothly down to the lowest
+measured temperatures and converge at T = 0. Neutron
+data at 55 mK show the phases are still separated by
+paramagnetism at this temperature Fig. 11(b). A highly
+non-trivial order-to-order quantum phase transition may
+take place between A and C at zero temperature (indi-
+cated with a question mark). Precise measurements in
+10
+
+the vicinity of these phase transitions would provide im-
+portant insight on their nature. However, the strong sig-
+nal from nuclear degrees of freedom and the extremely
+low temperatures involved prevent further investigation.
+The double hump features in specific above the transi-
+tion temperature represent a crossover from the low field
+disordered phase to the high field polarized phase. Such
+features can be understood in terms of models of hard-
+core bosons and are usually associated with quantum
+critical behaviour in one dimensional magnets [35, 36].
+They can be observed in several quasi-1D antiferromag-
+nets [27, 28], and therefore suggest the relevance of one-
+dimensional correlations for the physics of Nd3BWO9.
+These modulations are accentuated when the field is
+applied along the direction of the spin tubes (H ∥ c).
+Notably, despite the first-order nature of the transition
+these modulations are still present and seem to be most
+prominent around the tricritical termination point.
+Plateaux in the magnetically ordered sector are a hall-
+mark of frustrated magnets. The existence of magneti-
+zation plateaus (and particularly at 1/3 of saturation)
+has been predicted for both kagome antiferromagnets
+[37, 38], as well as for a model of isolated spin tubes with
+a weak triangular rung interaction (see Fig. 1(c)) [39–41].
+The presence of magnetization plateaux independent of
+the orientation of the applied magnetic field suggests an
+stabilizing interplay between frustration mechanisms.
+Finally,
+we comment on the origin of the ob-
+served incommensurate-commensurate (IC-C) transi-
+tion.
+Dipolar interactions are not uncommon in the
+study of rare-earth based magnets due to their large
+magnetic moments (µ(Nd3+) = 3.6µB) [42]. Their sta-
+bilizing role on incommensurate structures at temper-
+atures above commensurate order has been argued in
+several systems with hexagonal structure [43–45]. The
+realization of a IC-C transition at zero field opens the
+question to the importance of dipolar coupling for the
+low temperature properties of Nd3BWO9.
+We conclude the discussion by comparing Nd3BWO9
+to its isostructural compounds. To this point, only two
+other systems in the R3BWO9 family have been studied
+at low temperatures. NMR spectra reveal an inconm-
+mensurate magnetic structure in Sm3BWO9[46], while a
+dynamical state has been proposed for Pr3BWO9 at tem-
+peratures as low as 90 mK [47]. These two systems have
+been analyzed in terms of 2D Hamiltonians based on
+the existence of the kagome planes. However, our work
+highlights the presence of three-dimensional couplings
+and the potential dominance of the one-dimensional spin
+tubes. The discussion outlined here is inevitably rele-
+vant for investigations on other members of the family
+of R3BWO9.
+V.
+CONCLUSION
+We have presented a comprehensive study of the low
+temperature physics of the highly frustrated quantum
+antiferromagnet Nd3BWO9. Calorimetric and neutron
+scattering data support the realization of strongly in-
+teracting effective spin-1/2 moments below 100 K. Our
+measurements reveal a complex magnetic phase diagram
+below 300 mK, featuring magnetization plateaux for all
+field orientations. The ordering brings about important
+insight about the relevant magnetic interactions. Differ-
+ent magnetic structures are realized in the plateau states,
+depending on the direction of the magnetic field. Even
+though the phase diagram is considerably anisotropic, it
+can be described in terms of an effective S = 1/2 pseu-
+dospin.
+The experimental framework provided here is key for
+future studies on Nd3BWO9 and in the remaining mem-
+bers of the R3BWO9.
+The presence of the spin-tube
+structures perpendicular to the kagome planes is indi-
+cates that the magnetic properties of these highly frus-
+trated systems cannot be understood in terms of kagome-
+lattice physics. Further work is needed to fathom the
+effective dimensionality of the magnetic lattice.
+VI.
+ACKNOWLEDGEMENTS
+This work is supported by a MINT grant of the Swiss
+National Science Foundation.
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+Disintegration of Long-Period Comet C/2021 A1 (Leonard)
+David Jewitt1, Yoonyoung Kim2, Michael Mattiazzo3, Max Mutchler4, Jing Li1
+and Jessica Agarwal2
+1Department of Earth, Planetary and Space Sciences, UCLA
+2Institute for Geophysics and Extraterrestrial Physics, TU Braunschweig, D-38106
+Braunschweig, Germany
+3Swan Hill Observatory, Australia
+4 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218
+jewitt@ucla.edu
+Received
+;
+accepted
+Revised 2022 January 16
+arXiv:2301.08673v1  [astro-ph.EP]  20 Jan 2023
+
+– 2 –
+ABSTRACT
+We present imaging observations of the disintegrating long-period comet
+C/2021 A1 (Leonard). High resolution observations with Hubble Space Tele-
+scope show no evidence for surviving fragments, and place a 3σ upper limit to
+their possible radius ∼60 m (albedo 0.1 assumed). In contrast, wide field ob-
+servations from the Swan Hill Observatory, Australia, show an extensive debris
+cloud, the cross-section and estimated mass of which are consistent with com-
+plete disintegration of the nucleus near mid- December 2021 (at about 0.8 au).
+Two methods give the pre-disruption nucleus radius, rn = 0.6 ± 0.2 km. Tidal,
+collisional, sublimation and pressure-confined explosion models provide implau-
+sible explanations of the disintegration. However, rotational instability driven
+by outgassing torques has a very short timescale (∼0.1 year) given the orbit and
+size of the C/2021 A1 nucleus, and offers the most plausible mechanism for the
+disruption. Initial rotational breakup is accelerated by the exposure and strong
+sublimation of previously buried volatiles, leading to catastrophic destruction of
+the nucleus.
+Subject headings: comets: general—comets: individual C/2021 A1
+
+– 3 –
+1.
+INTRODUCTION
+Comet C/2021 A1 (Leonard), hereafter “A1”, was discovered on UT 2021 January 3 as
+a diffuse V ∼ 19 magnitude object inbound to the Sun at heliocentric distance rH = 5 au
+(Leonard 2021). A1 is a long-period comet, with heliocentric osculating semimajor axis a
+= -6124 au, eccentricity e = 1.0001 and inclination i = 132.6◦, reaching perihelion (at rH
+= 0.615 au) on UT 2022 January 03.3, about a year after discovery. Although presently
+following a weakly hyperbolic orbit, the pre-entry orbital elements (corrected for planetary
+perturbations to 1900 January 1, when the heliocentric distance was 137 au) are those of a
+bound object, a = 2020 au, e = 0.999696 and i = 132.7◦. A1 is thus not a dynamically new
+comet, having passed through the planetary system ∼ 105 years ago.
+Comet A1 attained naked eye visibility in late 2021 and then displayed spectacular
+gas and dust tails. However, images and commentary recorded in public on-line archives1
+indicate that A1 became photometrically unstable in 2021 December and 2022 January.
+Measurements of the OH production rate from the Nancay radio telescope were steady near
+QOH = 2.6×1028 s−1 between UT 2021 December 9 and 12, but jumped by a factor of ∼8
+to QOH = 22×1028 s−1 on December 15, even as the heliocentric distance barely decreased
+from 0.80 au to 0.74 au (Crovisier et al. 2021). The morphology also changed, becoming
+more diffuse and with “the tail being more prominent than the head” on UT 2022 January
+222 at rH ∼ 0.74 au outbound. Based on these early observational reports we requested
+Director’s Discretionary Time on the Hubble Space Telescope (HST), with the science
+objective being to study the presumed breakup of this long-period comet at the highest
+angular resolution. Independently, coauthor Mattiazzo also obtained wide-field imaging
+data using a private telescope at the Swan Hill Observatory in Australia. The wide-field and
+1e.g. https://britastro.org/cometobs/2021a1/thumbnails.html
+2https://groups.io/g/comets-ml/message/30541
+
+– 4 –
+HST data are highly complementary, with the former providing sensitivity to low surface
+brightness debris over a wide angle and the latter providing ultra-high resolution and very
+deep imaging of the near-nucleus region.
+While the phenomenon of cometary breakup has been known for over a century, very
+few physical observations of disintegrating comets are to be found in the refereed literature.
+In this paper, we present the observations and consider possible causes of the breakup of
+comet A1.
+2.
+OBSERVATIONS
+2.1.
+Hubble Space Telescope
+The 2.4 m diameter Hubble Space Telescope was used to observe disintegrating A1
+under program GO 16929. We used the WFC3 camera, which houses two 2015×4096 pixel
+charge coupled devices separated by a 1.2′′ wide gap. The 0.04′′ pixel−1 image scale gives a
+full-frame 162′′×162′′ field of view. HST images were taken using the F350 LP filter in order
+to maximize throughput. This filter has an effective central wavelength λc = 6230˚A when
+observing a Sun-like (G2V) source and a FWHM ∆λ = 4758˚A. We secured four images
+each of 450 s duration in each of the first three orbits and five frames of 285 s, with a
+sub-frame readout, in the fourth. The first three orbits were obtained in 2022 April with
+spacings of one and four days, with the intention being to measure the sky-plane motions
+of fragments produced by the break-up of A1. The fourth orbit was scheduled on UT 2022
+June 7 to coincide with the passage of the Earth through the projected orbit plane of the
+comet. Observations from this vantage point provide a model-free measure of the thickness
+of the dust distribution perpendicular to the plane. Unfortunately, the images from the
+fourth orbit suffered from extreme field star contamination, as a result of the low (-6◦)
+
+– 5 –
+galactic latitude of the comet, and were not useful.
+2.2.
+Swan Hill Observatory
+Wide-field observations were taken by co-author Michael Mattiazzo using a 0.28 m
+diameter, f/2.2 wide-field telescope at the Swan Hill Observatory (observatory code Q38),
+located in Victoria, Australia. A 4655×3522 pixel CMOS imaging device (Panasonic model
+QHY163M) provided an image scale of 1.27′′ pixel−1, and a field of view approximately
+1.6◦×1.2◦. Each pixel of the 0.28 m telescope subtends a solid angle equal to 103 HST
+pixels. Ten images each of 30 s duration were obtained, during which time the comet moved
+relative to field stars by about 2.7′′, which is small compared to the 5.1′′ full width at half
+maximum of point source objects in the data. The wide field image shows evidence for loss
+of sensitivity due to vignetting, especially near the corners of the device. We removed this
+by fitting a cubic spline surface to the image, using the median signal within 50×50 pixel
+boxes (after checking that the procedure did not self-subtract the comet).
+No filter was employed in order to maximize the throughput of the system. The
+quantum efficiency of the detector peaks near a central wavelength 5500˚A, and has a
+FWHM estimated at ∼4000˚A. The central wavelength is close to that of Johnson V (see
+the discussion in Bessel 1990), but the response is so broad that it captures the same light
+as the Johnson B, V and R filters (or, equivalently, the Sloan g and r filters) combined.
+The large bandwidth and lack of a standard filter together limit the accuracy with which
+the measured magnitudes can be related to, for example, the V band magnitudes. We
+calibrated the data using measurements of field stars on the Sloan filter system, provided
+by the Skymapper southern survey (Wolf et al. 2018). For this purpose we extracted
+measurements using circular apertures of projected radius 12.7′′, with sky subtraction from
+the median signal within a concentric annulus having inner and outer radii 19.1′′ and 38.1′′,
+
+– 6 –
+respectively. In order to minimize the color term in our photometry, we selected stars with
+optical color g-r ∼0.4 to 0.5, so as to approximately match the color of the Sun (given as g-r
+= 0.45±0.02 by Holmberg et al. 2006). We further selected these stars to lie within ∼1′ of
+the comet in order to minimize spatial variations in the photometry caused by imperfect
+flatness of the data.
+The geometrical circumstances of observation are given in Table 1.
+3.
+RESULTS
+3.1.
+High Resolution Data
+We combined the four images from each orbit in order to reject cosmic rays, suppress
+trailed field objects, and reach a fainter limiting magnitude. The composite from UT 2022
+April 5 is shown in Figure 1; composites from April 6 and 10 look the same. The predicted
+location of the nucleus is indicated in the Figure. The JPL Horizons ephemeris for April
+5 gives 3σ positional uncertainties of ±1.3′′ in right ascension and ±1.0′′ in declination,
+both of which are negligible compared to the 160′′ field of view of WFC3. We searched
+for the principal nucleus and discrete fragments in the data by comparing image subsets
+to identify correlated motion, but found none. Instead, the images show evidence for
+diffuse light scattered from cometary dust, evident in Figure 1 as a region of slightly higher
+surface brightness in the south east quadrant of the image (marked by a dashed white
+line in the right-hand panel of the figure). Although it at first resembles a flat-field defect
+or a smudge of internally scattered light, two lines of evidence show that this region of
+diffuse brightness is neither. First, the enhanced region is fixed with respect to the daily
+predicted ephemeris position of A1. Second, the enhanced region moves on the detector as
+the telescope orientation angle changes. The enhancement appears at the same position in
+
+– 7 –
+image composites from all three dates in April, whereas scattered light from bright stars
+outside the WFC3 field of view would vary as the background stars are completely different
+from day to day. A flat-field defect would not rotate as the telescope orientation changes.
+We conclude that the diffuse light is sunlight scattered from cometary debris released from
+the now invisible nucleus of A1.
+The on-line WFC3 Exposure Time Calculator3 gives a 3σ limit for detection of point
+source objects at V = 26.7, in each of our orbits. This limiting magnitude is consistent
+with the measured sky noise in the data. Corrected to absolute magnitude using phase
+coefficient β = 0.04 magnitude degree−1, we find H ≥ 22.81. For a nominal albedo, pV =
+0.1, this corresponds to a 3σ limit to the fragment radius, r ≤ 60 m.
+3.2.
+Wide Field Data
+The composite wide field image is shown in Figure 2. A low surface brightness dust
+structure extends over at least 0.4◦ (2×106 km in the plane of the sky), with a position
+angle 120◦±2◦ and no indication of a brightness peak at the expected location of the
+nucleus. The latter was determined from the JPL Horizons ephemeris for the mid-time of
+the image, and is marked in the figure. Overall, the morphology is similar to that of C/2010
+X1 (Elenin), a long period comet which disintegrated when inbound near rH = 0.6 AU (Li
+and Jewitt 2015), and C/2019 J2 (Palomar), which disintegrated pre-perihelion near rH
+= 1.9 au (Jewitt and Luu 2019). Comparison with Figure 1 shows that the HST, which
+was pointed at the expected location of the nucleus, indeed recorded diffuse light from the
+western tip of this dust structure.
+We estimated the total light from the dust as follows. First, we rotated the image to
+3https://etc.stsci.edu/etc/input/wfc3uvis/imaging/
+
+– 8 –
+bring the long axis of the dust tail to the horizontal (upper panel in Figure 3). Next, we
+manually replaced field stars with the average of surrounding pixels. The median signal from
+the comet was then computed within a rectangular box, “A” in the lower panel of Figure
+3) 1105′′ long by 380′′ tall, and the background sky estimated from equal-sized photometry
+boxes contiguous with the comet box but displaced above and below it (“B” and “C” in
+Figure 3). Figure 3 shows that the tail extends beyond the left edge of the photometry box
+“A” but the increased uncertainty imposed by the sky rendered measurements of this very
+faint material impractical. The light from the tail was calculated from fT = fA−(fB+fC)/2,
+where fx is the flux in box “x”. Then, applying the calibration obtained from field stars,
+we find VT = 10.9±0.5, where the quoted error is our best estimate of the uncertainty
+resulting from non-flatness of the data, the transformation from the wide response of the
+camera and the effective V magnitude. With assumed phase function 0.02±0.02 magnitude
+degree−1 and the geometry given in Table 1, the corresponding absolute magnitude is H =
+7.6±0.6, where the larger uncertainty is introduced by the phase correction. The scattering
+cross-section needed to give this absolute magnitude is C = 1.4+1.0
+−0.8 × 1010 m2, assuming
+geometric albedo pV = 0.1 (appropriate for cometary dust; Zubko et al. 2017).
+Figure 4 shows the averaged surface brightness profile from the March 31 image,
+measured parallel to the long axis of region A in Figure 3. Most of the scatter in the
+surface brightness profile is statistical noise in the data, but larger oscillations (for example
+at ∼480′′ and 750′′) result from spatial background variations caused by the digital removal
+of field stars. In this plot, the peak of 1000 units corresponds to a surface brightness Σ =
+24.4 magnitudes arcsec−1, about 5% of the surface brightness of the night sky. The surface
+brightness shows a steep increase, reaching a maximum at about 100′′ from the ephemeris
+nucleus location, followed by a steady decline at larger projected angles. This profile shape
+is indicative of a suddenly terminated dust mass release, with the peak of the profile giving
+the distance traveled by the largest, slowest particles.
+
+– 9 –
+4.
+DISCUSSION
+4.1.
+Radius and Mass of the Nucleus
+We use the effective spherical nucleus radius of A1 ¯r = 0.6±0.2 km from Jewitt (2022).
+This estimate is based on independent measurements of QH2O(1), the gas production rate
+at 1 au, and of α1, the non-gravitational acceleration at 1 au. Comet A1 has QH2O(1) =
+1.9×1028 s−1 (only pre-perihelion observations are used because post-perihelion rates are
+clearly affected by the breakup) and α1 = 1.3×10−6 m s−2, provided by JPL Horizons. A
+substantially smaller nucleus would have a surface area insufficient to supply the QH2O(1),
+while a substantially larger nucleus would have too much mass to be accelerated at α1
+given the known gas production rate. Using ¯r and nominal nucleus density ρn = 500 kg
+m−3 (Groussin et al. 2019), we estimate the nucleus mass Mn = (4.5+6.5
+−3.2) × 1011 kg. The
+largest surviving fragments, with radii <60 m, individually contain < 10−3 of the mass of
+the primary.
+4.2.
+Time of Disruption
+Syndynes (the loci of particles having one size, released with zero initial relative
+velocity over a range of times; Finson & Probstein (1968)) are curved and do not match
+the linear shape of the debris cloud in A1. Instead, the morphology more resembles a
+set of synchrones as shown in Figure 5. Synchrones trace the loci of particles in the sky
+plane having a range of sizes (hence, radiation pressure accelerations) but released from the
+nucleus simultaneously. The position angle of the debris trail in A1 is most compatible with
+ejection 110±10 days before the image was taken, i.e. on UT 2021 December 11±10. This is
+about a month before reports of distinct morphological change appeared but coincides with
+a dramatic increase in the OH production rate from 4.4×1028 s−1 on UT 2021 December
+
+– 10 –
+19 to 14×1028 s−1 on UT 2021 December 21, in unpublished SOHO/SWAN data (personal
+communication M. Combi). It is also close to a reported OH outburst on UT 2021 December
+15 (Crovisier et al. 2021). While we lack continuous coverage of the gas production from
+A1, it is likely that the sublimation rate became highly unstable as a result of the breakup
+of the nucleus when close to perihelion.
+We assume that the disintegration began on UT 2021 December 11±10. To reach the
+far end of the measured debris cloud (an angular distance ∼1500′′, corresponding to linear
+distance L = 2.2 × 106 km) under the action of radiation pressure requires an average
+acceleration 2L/∆T 2, where ∆T = 111 days (9.6×106 s) is the interval between the time
+of disintegration and the Swan Hill image from UT 2022 March 31. In units of the solar
+gravitational acceleration at the average rH = 1.3 au heliocentric distance in this period,
+β =
+2Lr2
+H
+g⊙(1)∆T 2
+(1)
+where g⊙(1) = 0.006 m s−2 is the solar gravity at 1 au and rH is expressed in au.
+Substituting, we obtain β = 0.01. With β ∼ 1/aµm, where aµm is the particle radius
+expressed in microns (c.f. Bohren & Huffman (1983)), we infer that the particles at the far
+end of the tail in the March 31 image had aµm ∼ 75 µm. All particles in the visible debris
+cloud on UT 2022 March 31 must be larger, while smaller particles were presumably ejected
+but have been swept by radiation pressure beyond the visible extent of the tail. Particles
+near the peak of the surface brightness profile (angular distance ∼100′′, corresponding to
+L = 1.4 × 105 km) have β ∼ 10−3 by Equation 1 and, therefore, radii ∼1 mm.
+
+– 11 –
+4.3.
+Mass of the Optical Debris
+How does the mass of the debris compare with the mass of the nucleus prior to its
+disappearance? To answer this question, we treat the debris as consisting of a distribution
+of spherical particles with radii between a and a+da written as n(a)da. Then, the combined
+mass of the particles between minimum radius a1 and maximum radius a2 is
+Md =
+� a2
+a1
+4
+3πρa3n(a)da
+(2)
+while their combined cross-section is
+C =
+� a2
+a1
+πa2n(a)da
+(3)
+It is useful to represent the size distribution as a power law
+n(a)da = Γa−γda
+(4)
+where γ is the differential size distribution index and Γ is a normalizing constant.
+Substituting equation 4 into equations 2 and 3 and eliminating Γ, we obtain
+Md = 4
+3ρC
+� a2
+a1 a3−γda
+� a2
+a1 a2−γda
+(5)
+The minimum particle radius is selected as a1 = 75 µm, since all smaller particles
+would have been swept out of the image field in the time since ejection. The maximum
+radius, a2 = 60 m, is set by the non-detection of larger bodies in our deep HST imaging
+data. With these values for a1 and a2, we plot Equation 5 as a function of γ in the range 2.5
+≤ γ ≤ 4.0 (Figure 6). The particle mass required to account for the measured cross-section,
+
+– 12 –
+C, is seen to vary by orders of magnitude for modest changes in the index, γ, with smaller
+values (flatter distributions) hiding a larger fraction of the total mass in big bodies.
+Also plotted in the figure is the nucleus mass, Mn = (4.5+6.5
+−3.2) × 1011 kg, computed from
+the effective radius, rn = 0.6±0.2 km, (Section 4.1), and density, ρn = 500 kg m−3, with the
+mass uncertainty marked as a horizontal yellow band. The red point marks the intersection
+of the two curves where Md = Mn and shows that, for index γ = 3.5±0.1, the debris mass
+and nucleus mass are equal. The upper limit to the size distribution could be substantially
+smaller than the 0.6 km limit set by the Hubble data, in which case a smaller value of
+the index would be needed for the mass of the debris to equal the mass of the nucleus.
+A relevant comparison can be made with the size distribution of the Kreutz sungrazing
+comets, which are themselves produced by the fragmentation of a precursor body. The
+Kreutz objects have γ = 3.2 in the 5 m to 35 m radius range (Knight et al. 2010), plotted
+as a blue square in Figure 6. The uncertainty on γ for the Kreutz objects is not stated;
+we have plotted a nominal ±0.1 error bar for reference and note reasonable agreement
+with the index deduced for A1 within the uncertainties. Perhaps less relevant are radar
+measurements of the debris size distributions in six meteoroid streams, most associated
+with decaying comets. These are plotted for comparison using green triangular symbols
+(Blaauw et al. (2011)). The formal meteoroid stream index uncertainties are comparable to
+the size of the symbols in the figure. The measured indices span the range γ = 3.2 to 3.7,
+encompassing the values found for A1 and the Kreutz comets.
+We conclude that the optical cross-section presented by the debris in 2022 March is
+consistent with the complete disintegration of the original ∼0.6 km scale nucleus into a
+power law distribution (index γ = 3.5±0.1) of particle sizes. We emphasize that we possess
+no independent evidence that the debris mass and original nucleus mass are equal, although
+a consideration of the particle properties using more detailed considerations (section 4.4)
+
+– 13 –
+supports this result. It should also be noted that 60 m is an upper limit to the size of
+the largest post-disruption “particles” and our result would be changed if a2 ≪ 60 m, as
+it would if the size distribution of particles is not well represented by a single power law
+across the full range of sizes. It is also not obvious that the density of the particles should
+necessarily be the same as the bulk density of the nucleus, as we have assumed. These and
+other physically plausible possibilities lie beyond the observational constraints obtained
+from the data.
+4.4.
+Monte Carlo Simulation
+We next used a Monte Carlo simulation as developed by Ishiguro et al. (2007) (see
+also Kim et al. (2017)) to model the cometary debris in more detail. The model is
+under-constrained and cannot provide unique solutions for the particle properties. It
+is nevertheless valuable in allowing us to test the deductions made based on order of
+magnitude considerations, and also to more fully explore the range of plausible solutions.
+We particularly examined the effect of the particle size distribution index and the minimum
+and maximum particles sizes in the distribution.
+Figure 7 shows the data with results of simulations for γ = 3.3, 3.4 and 3.5 and size
+parameter in the range 7 × 10−4 ≤ β ≤ 0.07, with ejection on 2021 December 11. The
+upper limit to β (lower limit to particle radius) is set by the field of view, with smaller
+particles have already been pushed out of the field by radiation pressure. We obtain a ≥
+14 µm, different by a factor of five from the limit a ≥ 75 µm estimated by the order of
+magnitude procedure, above. The lower limit to β (upper limit to the particle size of ∼1.4
+mm) is determined from the location of the surface brightness peak in Figure 7. This is very
+small compared to the 60 m upper limit to the radius of the largest possible fragment, set
+by non-detection in the HST images. However, this difference is understandable since, for
+
+– 14 –
+commonly measured cometary size distributions, the scattering cross-section is dominated
+by the smallest particles; large particles contribute little to the cross-section and thus are
+poorly constrained by scattered light observations. In order to fit the data, we assumed
+that the particle ejection speed varies with size parameter as V = V0β1/2, with V0 = 550
+m s−1 being the gas thermal speed. Unlike the particle trails of weakly active comets and
+asteroids, a high ejection speed is required in order to fit the large width of the debris cloud
+in A1.
+As is evident in Figure 7, the plotted models do not perfectly reproduce the measured
+surface brightness profile, with larger γ models being 25% to 30% brighter than the data at
+large distances from the nucleus and smaller γ models being too sharply peaked compared
+to the measurements. If they are real, these differences could result from physical effects not
+included in the model. For example, we have ignored dust released before disintegration,
+reasoning that the dramatic outbursts and brightening starting in mid-December would
+swamp any signal from older material. As another example, large aggregate grains in the
+tail might break up into smaller particles which would be quickly swept from the field of
+view by radiation pressure, perhaps explaining the lower brightness of the tail ≳1000′′ from
+the nucleus. On the other hand, the differences between the models and the measured
+profile are certainly affected by systematic uncertainties intrinsic to the wide field data,
+particularly by imperfect flatness of the data and by the presence of scattered light from
+bright background sources. Rather than over-interpret the data, we conclude from the
+Monte Carlo simulation only that γ ∼ 3.4 ± 0.1 provides a broad match to the profile, while
+much steeper and much less steep distributions do not. The range of allowable indices
+deduced from Monte Carlo models is consistent with γ = 3.5 ± 0.1 as inferred from the
+debris mass in Section 4.3 (c.f. Figure 6).
+Lastly, we used the Monte Carlo model to test the possibility that the debris observed
+
+– 15 –
+in 2022 March could be long-lived material released before perihelion, in the form of a
+so-called “neck-line” structure (e.g. Pansecchi and Fulle 1990). We find that material
+ejected in the period 2021 November 15 to December 15 would produce a tail structure
+in March having position angle (113◦) distinctly different from that measured (120◦) or
+calculated from the impulsive ejection model (119◦). In addition, neck-line structures in
+other comets are most prominent when observed from near the projected orbital plane,
+whereas our observations were taken ∼20◦ from the orbital plane of C/2021 A1 (c.f. Table
+1). The combination of the unfavorable observing geometry, the failure to reproduce the
+measured position angle of the dust in 2022 March, and the obvious importance of the
+outbursts reported in 2021 December together show that pre-perihelion dust is a negligible
+contributor to the post-perihelion appearance.
+4.5.
+Disintegration Mechanism
+The preceding discussion shows that a ∼0.6 km scale nucleus disintegrated into
+fragments, the largest of which were no more than about 10% of the radius of the original
+body. What process could lead to such a dramatic outcome?
+Tidal Breakup: The 0.615 au perihelion distance of A1 far exceeds the Roche radius
+of the Sun (∼10−2 au), negating the possibility of a tidal breakup. Comet A1 did pass
+within a distance rV = 0.029 au from Venus on UT 2021 December 18 (Zhang et al. 2021)
+but this is still ∼300 times the Roche radius (∼10−4 au) of the planet. To within a numerical
+multiplier, the differential of the gravitational force on opposite sides of the nucleus is
+∆F ∼ (GMV ρnr3
+n/r2
+V )(rn/rV ) giving an order of magnitude tidal stress S ∼ ∆F/r2
+n or
+S ∼ GMV ρnr2
+n
+r3
+V
+,
+(6)
+
+– 16 –
+where G = 6.67 × 10−11 N kg−2 m2 is the gravitational constant, MV = 5 × 1024 kg is
+the mass of Venus and the other quantities are already defined. Substituting ρn = 500 kg
+m−3, rn = 600 m, and rV = 0.029 au, we estimate S ∼ 10−6 N m−2 at closest approach,
+which is orders of magnitude smaller even than the cohesive strengths of fine, unconfined
+powders (S ≳ 100 N m−2) measured in the laboratory (Garcia-Trinanes et al. 2019). The
+disintegration of A1 is very unlikely to be a consequence of tidally induced stresses.
+Equilibrium Sublimation: The rate of loss of surface material is drn/dt ∼ −fs/ρ,
+where fs ∼ 2 × 10−4 kg m−2 s−1, at 1 au. Substitution gives drn/dt ∼ -3 cm day−1. At this
+rate, the timescale for eroding the whole nucleus would be |rn/(drn/dt)| ∼ 40 years, which
+is very large compared to the ∼1 year spent by A1 in the vicinity of the Sun. In any case,
+sublimation would produce steady erosion of the comet not a catastrophic disintegration
+like that observed. Equilibrium sublimation cannot account for the sudden disintegration
+of A1.
+Collisional Disruption: Collisional disruption timescales for 0.6 km scale objects,
+even in the dense parts of the asteroid belt, are measured in hundreds of millions of years
+(Bottke et al. 2005). Comet A1 arrived from a high inclination orbit and disintegrated ∼0.5
+au from the ecliptic plane where there are no known objects with which to collide. We
+confidently dismiss the possibility that A1 was collisionally disrupted.
+Internal Pressure: Could internal pressure build-up from sublimated gases cause
+the nucleus to explode (Samarasinha 2001)? The core temperature of the nucleus of A1
+is comparable to the Oort cloud equilibrium temperature of just a few degrees above
+absolute zero. Heat transport from the surface to the interior by conduction is controlled
+by the thermal diffusivity, which is proportional to the conductivity and which, in turn,
+is strongly affected by the particulate nature and porosity of the cometary material.
+Laboratory measurements of porous, dielectric powders yield conductivities ∼ 102 to 103
+
+– 17 –
+times smaller than the solid material (Henke et al. 2012). The expected high porosities
+and low thermal diffusivities of cometary material lead to small thermal skin depths that
+make deep conduction impossible. Heat applied for a time τ will conduct over a distance
+d ∼ (κτ)1/2, where κ is the diffusivity. For example, with κ = 10−8 to 10−9 m2 s−1, even in
+the year between discovery at rH = 5 au and perihelion at 0.6 au, conducted heat travelled
+into the nucleus by a characteristic distance only d ∼ 0.2 m to 0.5 m. This distance is so
+small compared to the nucleus radius that it is difficult to see how subsurface gas produced
+by surface heating could have any relevance to the complete disintegration of the nucleus.
+Rotational Instability: The remaining possibility for nucleus break-up is also the
+most plausible. The timescale for changing the spin angular momentum of a spherical
+nucleus through outgassing torques is (Jewitt 2021)
+τs =
+�16π2
+15
+� � ρnr4
+n
+kTVthP
+� � 1
+˙M
+�
+,
+(7)
+where P is the instantaneous spin-period and kT is the dimensionless moment arm, equal
+to the fraction of the outflow momentum that exerts a torque on the nucleus. The median
+values in a sample of short-period comet nuclei with perihelia in the range 1 ≤ q ≤ 2 au
+are kT = 0.007 and P = 15 hours (5×104 s) (Jewitt 2021). We substitute
+˙M = 800 kg
+s−1, equal to the sublimation rate at 1 au as measured by Combi’s Lyman-α data, on the
+understanding that this sets a lower bound to the mass loss rate at smaller distances and
+therefore sets an upper limit to τs. With ρn = 500 kg m−3, rn = 600 m, and Vth = 500
+m s−1, substitution into Equation 7 gives τs < 5 × 106 s (0.16 year, or 2 months), which
+compares to the 6 weeks (0.12 year) spent by A1 with rH < 1 au. While this is not proof
+that A1 disintegrated through a rotational instability, given the nominal nucleus parameters
+and measured mass loss rate, rotational instability does offer a plausible mechanism for
+nucleus disintegration.
+
+– 18 –
+Rotational breakup is expected to launch fragments with a velocity dispersion
+comparable to the tangential speed of the nucleus due to its rotation. For a strengthless
+nucleus, this equals the gravitational escape speed from the primary, in this case ∼0.3
+m s−1. In contrast, the Monte Carlo models show that larger speeds are required to fit
+the head width of the debris trail. For example, with V = V0β1/2 and V0 = 550 m s−1,
+millimeter sized particles (β = 0.001) would have V ∼ 17 m s−1, about 60 times the escape
+speed. We conjecture that these higher speeds result from gas drag acceleration following
+the exposure and intense sublimation of previously buried ices caused by rotational breakup
+at rH ∼ 0.8 au.
+Very large particles and boulders would not be substantially accelerated by gas drag
+and should leave the disintegrating nucleus at about the escape velocity of the primary.
+In the ∼3 months elapsed between the first signs of breakup and the HST observations,
+such slow-moving fragments would travel ∼2000 km, a distance subtending 1′′ to 2′′ in the
+plane of the sky (c.f. Table 1). Large fragments should therefore be resolvable in the HST
+data (the resolution is ∼0.08′′) but, nevertheless, remain unseen. This might reflect the
+continued disintegration of the fragments, again aided by the new exposure to the heat of
+the Sun of previously buried volatiles. The breakup process would then be catastrophic.
+Smaller fragments produced by breakup of the primary nucleus would have progressively
+shorter and shorter spin-up times, owing to their smaller size (c.f. Equation 7) and to the
+sudden exposure of large areas of previously buried ice which could amplify the moment
+arm, kT, by orders of magnitude. The expected result is a runaway fragmentation cascade.
+4.6.
+Gas Production Resulting from Nucleus Disintegration
+Disintegration of the nucleus must suddenly expose previously buried ices to the heat
+of the Sun, leading to a burst in the gas production rate caused by sublimation. Indeed,
+
+– 19 –
+measurements of the gas production rate in the mid-December to January period are highly
+variable, peaking near QH2O = 2.4 × 1029 s−1 in radio (Crovisier et al. 2021), Lyman-α (M.
+Combi, (private communication)), and near-ultraviolet (Jehin et al. 2021, 2022a, 2022b)
+observations. At break up, A1 was about rH = 0.8 AU from the Sun and ∆ = 0.2 AU from
+the SWAN/SOHO observatory used to take the Lyman-α data. The latter has 1◦ wide
+pixels, corresponding to about w ∼ 6 × 105 km per pixel at the comet and 1.2×106 km
+for the nominal Nyquist (2 pixel) resolution of the data. With an isothermal blackbody
+temperature at 0.8 AU ∼310 K, the thermal velocity of hydrogen atoms is Vth ∼ 2.5 km
+s−1. This, however, is a strong lower limit to the outflow velocity because of photo-electric
+heating (e.g. Combi and Delsemme 1980, Combi et al. 2000). Based on published models,
+we adopt a hydrogen outflow speed Vth ∼ 10 km s−1 and estimate the residence time for
+hydrogen atoms within a Nyquist sampled resolution element as tr ∼ 2w/Vth ∼ 1.2 × 105
+s (about 1.4 days). This means that the peak rate inferred from SWAN/SOHO Lyman-α
+data should be understood as a measure of the production rate averaged over 1.4 days.
+We are interested to see how QH2O compares with estimates of the gas production
+expected from the break up of the nucleus. To this end, we consider an idealized model in
+which the nucleus consists of particles which are either refractory or ice, and in which the
+ratio of ice to refractory masses is fice. Both refractory and ice particles are assumed to
+occupy a differential power size distribution (Equation 4). To render the problem tractable,
+we make the simplifying assumption that the nucleus disintegrates instantaneously into
+power law distributions of ice and refractory particles, each having radii in the range
+a1 ≤ a ≤ a2. The icy component then sublimates at the rate fs [kg m−2 s−1], which we
+calculate from energy balance including terms for radiation and sublimation.
+In the residence time tr, an ice surface will sublimate over a layer thickness
+
+– 20 –
+as = fstr
+ρn
+,
+(8)
+where ρn is the density of the particle, assumed equal to the bulk density of the nucleus.
+All the ice particles with radii a ≤ as will sublimate away, releasing water molecules and,
+eventually, producing by photodissociation the hydrogen atoms detected using the SWAN
+instrument. Ice particles with a > as will also partially sublimate in time tr, but their
+contribution to the gas flux should be small because, for plausible power law distributions
+(in particular, for γ = 3.5 as determined in sections 4.3 and 4.4), large particles present a
+small fraction of the total particle cross-section.
+The fraction of the mass contained in ice particles having a ≤ as is given by
+F =
+� as
+a1 a3−γda
+� a2
+a1 a3−γda
+(9)
+which, for 3 < γ < 4 and as ≫ a1 and a2 ≫ a1, simplifies to
+F =
+�γ − 3
+4 − γ
+� �as
+a2
+�4−γ
+.
+(10)
+The total ice mass in the undisrupted nucleus, assumed to be spherical, is Mi =
+(4π/3)ρnr3
+nfice. The production rate averaged over time tr may be written QH2O =
+FMi/(trµmH), where µ = 18 is the molecular weight of the water molecule and
+mH = 1.67 × 10−27 kg is the mass of the hydrogen atom. Substitution of Equations 8 and
+10 into this expression gives
+QH2O = 4πρnr3
+nfice
+3trµmH
+�γ − 3
+4 − γ
+� � fstr
+ρna2
+�4−γ
+.
+(11)
+The equilibrium sublimation mass flux calculated for a blackbody water ice sphere at 0.8
+
+– 21 –
+AU is fs = 1.7 × 10−4 kg m−2 s−1. The flux could be smaller if the grain albedo is high, or
+larger if the grain is anisothermal (albeit then sublimating from a smaller fraction of the
+grain surface). We set a2 = 60 m, the largest “particle” allowed by the Hubble imaging,
+and a1 = 10−7 m (however, Equation 11 is insensitive to a1 and its value is unimportant
+provided a1 ≪ as). The nominal nucleus radius is rn = 600 m, and the size distribution
+index is γ = 3.5, as deduced above. Measured cometary ice/refractory ratios, fice, show a
+wide range of values, from fice ∼ 1 in 67P/Churyumov-Gerasimenko (Marschall et al. 2020),
+to fice < 0.2 in C/1995 O1 Hale-Bopp (Jewitt and Matthews 1999) and fice = 0.03 to
+0.1 in 2P/Encke (Reach et al. 2000). We adopt fice = 1/4, recognizing that this value is
+substantially uncertain.
+Substitution into Equation 11 gives QH2O = 8.8+12.0
+−6.2 × 1029 s−1, where the error bars
+reflect only the ±200 m uncertainty in the estimated radius of the nucleus. This is larger
+than the measured peak water production rate (2×1029 s−1) but shows acceptable agreement
+given the crude nature of the model calculation and the likelihood that the disintegration
+was in reality spread over a finite period not impulsive, as modeled. We conclude that
+complete disintegration of the nucleus into a power law particle size distribution is consistent
+both with the optical brightness of the debris cloud and with the surge in the water
+production rate measured using Lyman-α.
+Future improvements to this model could include a treatment of the initial, optically-
+thick phase of the expanding disintegration cloud, when self-shielding will suppress and
+delay the sublimation surge relative to the estimate given here. Also needed is a treatment
+of the gas drag interaction with cometary solids in a fully disintegrated body, responsible
+for the size-dependent acceleration of refractory particles into the coma and surviving
+debris field. Furthermore, several of the parameters needed to accurately model nucleus
+disintegration remain unmeasured, and most other disintegrating comets are observationally
+
+– 22 –
+even less-well characterized than A1. It is obvious, even from these simple considerations
+that many more detailed observations, across a wide range of wavelengths and with
+adequate temporal sampling, will be needed to better understand what is likely to be the
+dominant destructive cometary process.
+
+– 23 –
+5.
+SUMMARY
+We present both high resolution and wide field observations of disintegrating
+long-period comet C/2021 A1 (Leonard) taken to study the nature of its demise.
+• The pre-disintegration radius of the nucleus, estimated using two methods, was
+rn = 0.6 ± 0.2 km. After breakup, which began in mid-December 2021 and may have
+continued for weeks, no nucleus fragments larger than about rn = 0.06 km (i.e. < 10−3
+of the primary mass) survived.
+• The observed debris cloud consists of sub-millimeter and larger particles, with a
+differential power law size distribution having index γ = 3.4±0.1 and 3.5±0.1, as
+estimated by two different methods. The observational constraints are consistent with
+equality between the mass of the debris cloud and the mass of the primary nucleus,
+indicating a total disintegration.
+• Tidal disruption, sublimation, collisional disruption, and explosion following internal
+pressure build-up in the nucleus all offer implausible explanations of the disintegration
+of C/2021 A1.
+• The spin-up timescale due to outgassing torques for a 600 m nucleus in the orbit of
+C/2021 A1 is as short as ∼2 months, pointing to rotational instability as the likely
+cause of the disintegration.
+• A simple model of the exposure and rapid sublimation of previously buried ice
+indicates a peak gas production rate (QH2O = 9+12
+−6 × 1029 s−1) of the same order as
+the measured peak value (QH2O = 2.4 × 1029 s−1).
+We thank Michael Combi for a preview of his SWAN data on C/2021 A1 and the
+anonymous referee for prompt comments on the manuscript. Based on observations made
+
+– 24 –
+with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the
+Space Telescope Science Institute. STScI is operated by the Association of Universities for
+Research in Astronomy, Inc. under NASA contract NAS 5-26555. Support for this work
+was provided by NASA through grant number GO-16929 from the Space Telescope Science
+Institute, which is operated by auRA, Inc., under NASA contract NAS 5-26555.
+Facilities: HST.
+
+– 25 –
+REFERENCES
+Bessell, M. S. 1990, PASP, 102, 1181. doi:10.1086/132749
+Bohren, C. F. & Huffman, D. R. 1983, Absorption and scattering of light by small particles.
+New York: Wiley, 1983
+Bottke, W. F., Durda, D. D., Nesvorn´y, D., et al. 2005, Icarus, 179, 63.
+doi:10.1016/j.icarus.2005.05.017
+Blaauw, R. C., Campbell-Brown, M. D., & Weryk, R. J. 2011, MNRAS, 414, 3322.
+doi:10.1111/j.1365-2966.2011.18633.x
+Combi, M. R. & Delsemme, A. H. 1980, ApJ, 237, 633. doi:10.1086/157909
+Combi, M. R., Reinard, A. A., Bertaux, J.-L., et al. 2000, Icarus, 144, 191.
+doi:10.1006/icar.1999.6335
+Crovisier, J., Biver, N., and Bockelee-Morvan, D. 2021, Central Bureau Electronic Telegram
+5087 (2021 December 22)
+Finson, M. J. & Probstein, R. F. 1968, ApJ, 154, 327. doi:10.1086/149761
+Garcia-Trinanes, P., Luding, S. and Shi, H.
+2019. Advanced Powder Technology, 30,
+2868-2880
+Groussin, O., Attree, N., Brouet, Y., et al. 2019, Space Sci. Rev., 215, 29. doi:10.1007/s11214-
+019-0594-x
+Henke, S., Gail, H.-P., Trieloff, M., et al. 2012, A&A, 537, A45. doi:10.1051/0004-
+6361/201117177
+Holmberg, J., Flynn, C., & Portinari, L. 2006, MNRAS, 367, 449. doi:10.1111/j.1365-
+2966.2005.09832.x
+
+– 26 –
+Ishiguro, M., Sarugaku, Y., Ueno, M., et al. 2007, Icarus, 189, 169.
+doi:10.1016/j.icarus.2007.01.003
+Jehin, E., Moulane, Y., & Manfroid, J. 2021, The Astronomer’s Telegram, 15128
+Jehin, E., Moulane, Y., Manfroid, J., et al. 2022a, The Astronomer’s Telegram, 15186
+Jehin, E., Moulane, Y., Manfroid, J., et al. 2022b, The Astronomer’s Telegram, 15189
+Jewitt, D. & Matthews, H. 1999, AJ, 117, 1056. doi:10.1086/300743
+Jewitt, D. & Luu, J. 2019, ApJ, 883, L28. doi:10.3847/2041-8213/ab4135
+Jewitt, D., Kim, Y., Mutchler, M., et al. 2020, ApJ, 896, L39. doi:10.3847/2041-8213/ab99cb
+Jewitt, D. 2021, AJ, 161, 261. doi:10.3847/1538-3881/abf09c
+Jewitt, D. 2022, AJ, 164, 158. doi:10.3847/1538-3881/ac886d
+Kim, Y., Ishiguro, M., Michikami, T., et al. 2017, AJ, 153, 228. doi:10.3847/1538-
+3881/aa69bb
+Knight, M. M., A’Hearn, M. F., Biesecker, D. A., et al. 2010, AJ, 139, 926. doi:10.1088/0004-
+6256/139/3/926
+Leonard, G. J., Aschi, S., Pettarin, E., et al. 2021, Minor Planet Electronic Circulars,
+2021-A99
+Li, J. & Jewitt, D. 2015, AJ, 149, 133. doi:10.1088/0004-6256/149/4/133
+Marschall, R., Markkanen, J., Gerig, S.-B., et al. 2020, Frontiers in Physics, 8, 227.
+doi:10.3389/fphy.2020.00227
+Pansecchi, L. & Fulle, M. 1990, A&A, 239, 369
+
+– 27 –
+Reach, W. T., Sykes, M. V., Lien, D., et al. 2000, Icarus, 148, 80. doi:10.1006/icar.2000.6478
+Samarasinha, N. H. 2001, Icarus, 154, 540. doi:10.1006/icar.2001.6685
+Wolf, C., Onken, C. A., Luvaul, L. C., et al. 2018, PASA, 35, e010. doi:10.1017/pasa.2018.5
+Zhang, Q., Ye, Q., Vissapragada, S., et al. 2021, AJ, 162, 194. doi:10.3847/1538-3881/ac19ba
+Zubko, E., Videen, G. Shkuratov, Y., et al. 2017, JQSRT, 202, 104.
+doi:10.1016/j.jqsrt.2017.07.026
+This manuscript was prepared with the AAS LATEX macros v5.2.
+
+– 28 –
+Table 1.
+Observing Geometry
+UT Date & Time
+νa
+rH b
+∆c
+αd
+θ−⊙e
+θ−V f
+δ⊕g
+Telh
+Scalei
+Uncj
+2022 Mar 31 18:14-18:26
+107.4
+1.756
+1.942
+30.8
+243.6
+90.2
+-20.4
+Swan Hill
+1408
+±1.4
+2022 Apr 05 23:35-24:04
+109.2
+1.833
+1.910
+30.9
+246.4
+91.7
+-19.8
+HST
+1385
+±1.6
+2022 Apr 06 23:32-23:51
+109.5
+1.848
+1.902
+30.9
+246.9
+92.0
+-19.7
+HST
+1379
+±1.6
+2022 Apr 10 19:23-19:53
+110.7
+1.903
+1.875
+30.7
+249.0
+93.2
+-19.1
+HST
+1359
+±1.7
+2022 Jun 7 16:36-17:12
+123.0
+2.698
+1.715
+6.8
+319.3
+137.0
++0.2
+HST
+1243
+±3.7
+aTrue anomaly, in degrees
+bHeliocentric distance, in au
+cGeocentric distance, in au
+dPhase angle, in degrees
+ePosition angle of projected anti-solar direction, in degrees
+fPosition angle of negative heliocentric velocity vector, in degrees
+gAngle from orbital plane, in degrees
+hTelescope
+iImage scale, km arcsecond−1
+j3σ ephemeris uncertainty, arcsecond (from JPL Horizons)
+
+– 29 –
+Fig. 1.— A) Composite of four, 450 s HST images from UT 2022 April 5. Diffuse streaks
+are imperfectly removed field stars and galaxies.
+B) Same image, anotated to show the
+approximate boundary of the debris (white dashed line) and the expected location of the
+nucleus (yellow line segments). Two scale bars of 30′′ and 5×104 km in length are shown, as
+well as the projected anti-solar (−S) and negative heliocentric velocity (−V ) vectors. North
+is to the top, East to the Left.
+
+UT 2022 April 5
+30
+5x104 km
+B– 30 –
+Fig. 2.— Wide field image from Swan Hill Observatory showing C/2021 A1 on UT 2022
+March 31. 10′ and 106 km scale bars are shown, as well as the projected anti-solar (−S) and
+negative heliocentric velocity (−V ) vectors. Yellow lines mark the ephemeris location of the
+nucleus. The white square shows the size of the HST field of view. The image has North to
+the top, East to the left.
+
+10
+.UT 2022 March 31
+106.km– 31 –
+Fig. 3.— (Upper:) Same image as in Figure 2 but rotated to bring the axis of the dust tail
+to the horizontal and shown at a larger scale. Yellow lines mark the ephemeris location of
+the nucleus. (Lower:) Locations of the photometry regions A, B and C used to measure the
+scattering cross-section of particles in the tail.
+
+R
+380″
+1105"– 32 –
+-500
+0
+500
+1000
+1500
+0
+400
+800
+1200
+1600
+Surface Brightness
+Distance [arcsecond]
+Fig. 4.— Surface brightness profile parallel to the long axis of Box A (Figure 3) plotted
+against the distance from the nucleus ephemeris location (axis is reversed relative to Figure
+3). 1000 units correspond to a surface brightness Σ = 24.4 magnitudes arcsec−2. The linear
+distance scale is approximately 1500 km per arcsecond.
+
+– 33 –
+Fig. 5.— (Left:) Same image as Figure 2 with synchrones overplotted, for ejection dates 80,
+100, 120, 140 and 160 days prior to the date of the image. (Right:) Syndynes for particles
+with β = 0.0003, 0.001, 0.003, 0.01 and 0.03, as marked. The axis of the debris cloud is best
+matched by the 110±10 day synchrones, corresponding to ejection on UT 2021 December
+11±10.
+
+UT 2022 March 31
+0.0003
+0.001
+120
+0.003
+100-
+80
+10'0
+0.03– 34 –
+1011
+1012
+1013
+1014
+1015
+2.5
+3.0
+3.5
+4.0
+Debris Mass [kg]
+Differential Size index, γ
+rn = 0.6+/-0.2 km
+Blaauw et al. 2011
+Kreutz Sungrazers
+C/2021 A1
+Fig. 6.— Total mass of the debris cloud (assuming density ρn = 500 kg m−3) plotted as a
+function of the differential power law index, γ, is plotted as a solid black line. The equivalent
+spherical mass of the original 0.6±0.2 km radius nucleus is shown (assuming the same ρn),
+together with its uncertainty, as a yellow horizontal band. The debris and nucleus masses
+are equal at γ = 3.5 ± 0.1, shown by the red filled circle. For comparison we show, as a blue
+square, the size distribution of the Kreutz sungrazing comets (Knight et al. 2010) and, as
+green triangles, several radar-measured meteoroid streams (Blaauw et al. 2011). The vertical
+positions of the Kreutz and radar stream points have no meaning.
+
+– 35 –
+-200
+0
+200
+400
+600
+800
+1000
+1200
+0
+400
+800
+1200
+1600
+Surface Brightness
+Distance [arcsecond]
+3.5
+3.4
+3.3
+3.5
+3.3
+3.4
+Fig. 7.— Axial surface brightness profile on UT 2022 March 31 (yellow diamonds) compared
+with results from a Monte Carlo simulation. The models shown have size index γ = 3.3 (red
+curve), 3.4 (black curve) and 3.5 (blue curve), all with 7 × 10−4 ≤ β ≤ 0.07, corresponding
+to particle radii 14 µm to 1.4 mm.
+

M9AzT4oBgHgl3EQfWPwO/content/tmp_files/2301.01296v1.pdf.txt

@@ -0,0 +1,1618 @@
+TinyMIM: An Empirical Study of Distilling MIM Pre-trained Models
+Sucheng Ren
+Fangyun Wei*
+Zheng Zhang
+Han Hu
+Microsoft Research Asia
+Abstract
+Masked image modeling (MIM) performs strongly in pre-
+training large vision Transformers (ViTs). However, small
+models that are critical for real-world applications can-
+not or only marginally benefit from this pre-training ap-
+proach. In this paper, we explore distillation techniques to
+transfer the success of large MIM-based pre-trained mod-
+els to smaller ones. We systematically study different op-
+tions in the distillation framework, including distilling tar-
+gets, losses, input, network regularization, sequential dis-
+tillation, etc, revealing that: 1) Distilling token relations
+is more effective than CLS token- and feature-based distil-
+lation; 2) An intermediate layer of the teacher network as
+target perform better than that using the last layer when
+the depth of the student mismatches that of the teacher;
+3) Weak regularization is preferred; etc. With these find-
+ings, we achieve significant fine-tuning accuracy improve-
+ments over the scratch MIM pre-training on ImageNet-1K
+classification, using all the ViT-Tiny, ViT-Small, and ViT-
+base models, with +4.2%/+2.4%/+1.4% gains, respectively.
+Our TinyMIM model of base size achieves 52.2 mIoU in
+AE20K semantic segmentation, which is +4.1 higher than
+the MAE baseline. Our TinyMIM model of tiny size achieves
+79.6% top-1 accuracy on ImageNet-1K image classifica-
+tion, which sets a new record for small vision models of
+the same size and computation budget. This strong perfor-
+mance suggests an alternative way for developing small
+vision Transformer models, that is, by exploring better train-
+ing methods rather than introducing inductive biases into
+architectures as in most previous works. Code is available
+at https://github.com/OliverRensu/TinyMIM.
+1. Introduction
+Masked image modeling (MIM), which masks a large
+portion of the image area and trains a network to recover
+the original signals for the masked area, has proven to be a
+very effective self-supervised method for pre-training vision
+Transformers [2,12,18,53]. Thanks to its strong fine-tuning
+performance, MIM has now been a main-stream pre-training
+*Corresponding author: fawe@microsoft.com.
+ViT-T
+ViT-S
+ViT-B
+70
+74
+78
+82
+86
+Scratch
+MAE
+TinyMIM
+-0.6
++3.6
++0.7
++3.1
++2.4
++3.8
+Acc.
+72.2
+79.9
+81.2
+Figure 1. Comparison among TinyMIM (ours), MAE [18] and
+training from scratch by using ViT-T, -S and -B on ImageNet-1K.
+We report top-1 accuracy. We adopt DeiT [44] when training from
+scratch. For the first time, we successfully perform masked image
+modeling pre-training for smaller ViTs.
+Model
+Param.
+Flops
+Top-1
+mIoU
+(M)
+(G)
+(%)
+DeiT-T [44]
+5.5
+1.3
+72.2
+38.0
+PVT-T [46]
+13.0
+1.9
+75.1
+39.8
+CiT-T [39]
+5.5
+1.3
+75.3
+38.5
+Swin [32]
+8.8
+1.2
+76.9
+40.4
+EdgeViT-XS [35]
+6.4
+1.1
+77.5
+42.1
+MobileViTv1-S [34]
+4.9
+2.0
+78.4
+42.7
+MobileViTv3-S [45]
+4.8
+1.8
+79.3
+43.1
+TinyMIM⋆-T (Ours)
+5.8
+1.3
+79.6
+45.0
+Table 1. Comparison with state-of-the-art tiny Transformers with
+architecture variants. The parameters indicate the backbone pa-
+rameter excluding the parameters of the last classification layer
+in classification or the decoder in segmentation. We report top-1
+accuracy on ImageNet-1K classification and mIoU on ADE20K
+segmentation.
+method for vision Transformers, and numerous follow-ups
+have been carried out in this research line, such as study-
+ing how to set decoding architectures [25], reconstruction
+targets [11,36,48,60], etc., as well as revealing its proper-
+ties [49,52,54].
+1
+arXiv:2301.01296v1  [cs.CV]  3 Jan 2023
+
+Method
+ViT-T
+ViT-S
+ViT-B
+ViT-L
+Scratch
+72.2
+79.9
+81.2
+82.6
+MAE
+71.6
+80.6
+83.6
+85.9
+Gap
+-0.6
++0.7
++2.4
++3.3
+Table 2. Comparison between MAE pre-trained ViTs and ViTs
+trained from scratch by using ViT-T, -S, -B and -L on ImageNet-
+1K. We adopt DeiT when training from scratch. We report top-1
+accuracy. As model size shrinks, the superiority of MAE gradually
+vanishes. MAE even hurts the performance of ViT-T.
+However, as shown in Table 2, MIM pre-training [18]
+mainly effects for relatively large models. When the model
+size is as small as ViT-Tiny (5 million parameters), which
+is critical for real-world applications, MIM pre-training can
+even hurt the fine-tuning accuracy on ImageNet-1K classifi-
+cation. In fact, the accuracy drops by -0.6 compared to the
+counterpart trained from scratch. This raises a question: can
+small models also benefit from MIM pre-training, and how
+can this be achieved?
+In addition, the existing study on small vision Transform-
+ers mainly focus on introducing certain inductive bias into
+architecture design [6,26,34,35]. The additional architec-
+tural inductive biases facilitate optimization yet limit the
+expressive capacity. It’s natural to ask whether we can boost
+plain small vision Transformers to perform just as well.
+In this work, we present TinyMIM, which answers the
+above questions. Instead of directly training small ViT mod-
+els using a MIM pretext task, TinyMIM uses distillation
+technology [24] to transfer the knowledge of larger MIM
+pre-trained models to smaller ones. Distillation endows the
+nice properties of larger MIM pre-trained models to smaller
+ones while avoiding solving a “too” difficult MIM task. Not-
+ing that knowledge distillation has been well developed,
+especially for supervised models [16], our main work is to
+systematically study for the first time the effects of different
+design options in a distillation framework when using MIM
+pre-trained models as teachers. Specifically, we consider dis-
+tillation targets, data augmentation, network regularization,
+auxiliary losses, macro distillation strategy, etc., and draw
+several useful findings:
+• Distillation targets. There are two main findings re-
+lated to distillation targets: 1) Distilling token relations
+is more effective than distilling the CLS token and fea-
+ture maps. 2) Using intermediate layers as the target
+may perform better than using the last layer, and the
+optimal target layer for different down-stream tasks,
+e.g., classification and segmentation, can be different.
+• Data and network regularization. Weak augmentation
+and regularization is preferred: 1) The performance of
+using a masked image is worse than using the original
+image; 2) Relatively small drop path rate (0 for teacher
+and 0.1 for student) performs best.
+• auxiliary losses. We find that an auxiliary MIM loss
+does not improve fine-tuning accuracy.
+• Macro distillation strategy. We find that using a se-
+quential distillation strategy, i.e., “ViT-B → ViT-S →
+ViT-T”, performs better than that distilling directly from
+ViT-B to ViT-T.
+By selecting the best framework options, we achieve sig-
+nificant fine-tuning accuracy improvements over the direct
+MIM pre-training on ImageNet-1K classification, using ViT
+models of different sizes, as shown in Figure 1. Specifi-
+cally, the gains of TinyMIM on the ViT-Tiny, ViT-Small, and
+ViT-base models are +4.2%/+2.4%/+1.4%, respectively.
+In particular, our TinyMIM⋆-T model with knowledge
+distillation during finetune-tuning achieves a top-1 accuracy
+of 79.6% on ImageNet-1K classification (see Table 1), which
+performs better than all previous works that develop small
+vision Transformer models by introducing architectural in-
+ductive biases or smaller feature resolutions. It sets a new
+accuracy record using similar model size and computation
+budget. On ADE20K semantic segmentation, TinyMIM-T
+achieves 45.0 mIoU, which is +1.9 higher than the second
+best method, MobileViTv3-S [45]. The strong fine-tuning
+accuracy by TinyMIM⋆-T suggests an alternative way for
+developing small vision Transformer models, that is, by
+exploring better training methods rather than introducing
+inductive biases into architectures as most previous works
+have done.
+2. Related Works
+2.1. Masked Image Modeling
+Masked Language Modeling (MLM) [10] for self-
+supervised Transformer pre-training has achieved incredible
+success in natural language processing (NLP) field. Inspired
+by the same idea of masking and reconstruction, BEiT [2]
+is the pioneer to bring such success to computer vision filed
+by encoding masked images and predicting masked tokens
+generated by DALL-E [38]. SimMIM [53] and MAE [18]
+find that reconstructing RGB pixels results in favorable rep-
+resentations. MAE adopts an asymmetric encoder-decoder
+architecture. The encoder only encodes the visible tokens
+and drops a high portion of masked tokens to reduce the com-
+putation burden. A lightweight decoder then produces recon-
+structed patches. Different from tokens in natural language
+processing that have rich semantics, pixels in computer vi-
+sion are low-level information, therefore, a lot of recent
+works aim at looking for better supervisions. MaskFeat [48]
+takes local gradient features produced by the manually-
+crafted HOG descriptor [9] as supervisions. PeCo [11] trains
+2
+
+Masked Image
+Raw Image
+Factors
+Input
+Target 
+Feature
+Relation
+𝑄·𝑄𝑇
+𝐾·𝐾𝑇
+𝑉·𝑉𝑇
+𝑄·𝐾𝑇
+Head Number
+w/ or w/o Softmax
+Output Feature
+Block Feature
+QKV Features
+Attention Feature
+FFN Feature
+Res. Connection of FFN
+Block
+Last
+Intermediate
+…
+…
+Transformer Block-N
+Output Feature
+Multi-Head 
+Attention
+Add & Norm
+FFN
+Add & Norm
+Attention Feature
+FFN Feature
+Block Feature
+Raw Image
+Masked Image
+Feature of Last Block
+𝑄·𝑄𝑇
+𝐾·𝐾𝑇
+𝑉·𝑉𝑇
+𝑄·𝐾𝑇
+𝑄
+𝐾
+𝑉
+Softmax
+Transformer Block-n
+Transformer Block-1
+Teacher
+(Highlight 
+by Blue)
+Relations
+Figure 2. We comprehensively study a variety of factors (highlighted by Royal Blue) that may affect TinyMIM pre-training including input,
+distillation target (feature or relation) and target block.
+a new tokenizer by enforcing perceptual similarity. iBot [60]
+and data2vec [1] take exponential moving average (EMA)
+updated models as tokenizers. MILAN [25] adopts a pre-
+trained CLIP as the teacher. Similarly, BeiTv2 [36] also uses
+CLIP [37] for tokenizer training. Different from these works
+that use various tokenizers/teachers, we adopt a masked im-
+age modeling pre-trained model as our teacher.
+The MIM pre-training performs very well on relatively
+large models from base size to giant size [31,53]. However,
+it will hurt the fine-tuning when the model is as small as
+tiny size, probably because the limited capthe MIM task is
+“too” difficult for small model. This paper explores how to
+make small vision Transformer models also benefit from
+MIM training, through a systematic study of the distillation
+technology.
+2.2. Knowledge Distillation
+Knowledge distillation is a classical method to transfer
+the knowledge from cumbersome models to a small one, pi-
+oneered by [24]. The original knowledge distillation frame-
+work adopts the annealed classification logits of the teacher
+as the distilling target for the student. Since then, extensive
+variants have been carried out to improve the distilling ef-
+fectiveness [16], including changing the distilling targets as
+intermediate features [22,23,28,40] and relations [29,56],
+data augmentations of teacher and students [39, 50], regu-
+larization [50], distilling strategies [47, 55, 57, 58] and so
+on.
+While almost all studies are made for CNN architec-
+tures under supervised settings, recently, there have been
+a few works performing distilling technologies for vision
+Transformers [44,50] and contrastive learning based meth-
+ods [14, 50]. In DeiT [44], the teacher is set as a CNN
+architecture so as to transfer the inductive bias involved in
+CNNs to vision Transformers. It also propose to use hard
+distillation which uses hard pseudo class labels of the teacher
+network as the distilling targets, which performs better than
+the naive knowledge distillation [24]. In [14], a distillation
+method regarding the similarities between instances is ap-
+plied to transfer the power of contrastive pre-trained large
+CNN models to small CNNs. In [50], a method based on
+feature map distillation is proposed to generally improve
+vision transformers by different pre-training approaches in-
+cluding image classification, instance contrastive based self-
+sueprvised learning [3] and CLIP pre-training [37]. However,
+it shows no gains for MIM pre-trained models.
+This paper for the first time studies the distillation frame-
+work for MIM pre-trained vision Transformers. Through
+a systematic study, it draws several useful findings and the
+best options, under which, significant gains are achieved for
+vision Transformers of various sizes.
+2.3. Small Vision Transformers
+Designing efficient CNN models [27,42] has been widely
+studied in recent years.
+With the emergence of Vision
+Transformer (ViT), there have been several works study-
+ing how to develop efficient vision Transformer, with the
+majority focus on introduing inductive biases into the archi-
+tectures [17,26,30,34,35].
+Different from these works that develop small vision
+Transformers by introducing sophisticated components into
+architectures, we demonstrate that a plain vision Trans-
+former [12] at a small scale can perform just as well, or
+even better. Our main insight is that the MIM pre-training
+can implicitly incorporate necessary inductive biases, and
+thus avoids the need of explicit architecture bias. Our plain
+3
+
+vision Transformer of tiny size achieves the state-of-the-art
+accuracy for both ImageNet-1K image classification and
+ADE20K semantic segmentation using similar model size
+and computation budget.
+3. TinyMIM
+We adopt a larger, MIM pre-trained model as the teacher,
+and a smaller ViT as the student. The objective of TinyMIM
+is to train the randomly initialized student by mimicking the
+target produced by the teacher in a knowledge distillation
+manner. After pre-training, the TinyMIM pre-trained model
+can be transferred to various downstream tasks. In this work,
+we adopt MAE [18] as the MIM model due to its popularity
+and simplicity.
+In this section, we first describe the factors that may affect
+TinyMIM pre-training: distillation target in Section 3.1.1;
+input in Section 3.1.2; target block in Section 3.1.3. Then we
+present a series of distillation losses for different distillation
+target in Section 3.1.3. At last, a sequential distillation strat-
+egy is introduced to facilitate the performance in Section 3.3.
+3.1. Factors
+3.1.1
+Distillation Target
+Block Feature and Output Feature. Given an input image
+x, we first divide it into N non-overlapping patches and use
+a linear projection layer to map N patches into patch em-
+beddings F0 ∈ RN×D, where D is the dimension of hidden
+features. Suppose we have a ViT containing L Transformer
+blocks. Each Transformer block takes the output Fi−1 of the
+last Transformer block as the input and generates the feature
+Fi of the current block, which can be formulated as:
+Fi = Transformer(Fi−1), i ∈ [1, L].
+(1)
+We term Fi as the block feature of the i-th Transformer
+block. In particular, we name the feature FL from the last
+Transformer block as the output feature.
+Attention Feature and FFN Feature. Each Transformer
+block is composed of a self-attention layer and a feed for-
+ward layer, which can be defined as:
+Hi = Attention(LN(Fi−1)),
+�Hi = Hi + Fi−1,
+�Hi = FFN(LN( �Hi)),
+F i = �Hi + �Hi,
+(2)
+where Attention(·), FFN(·) and LN(·) denotes self-
+attention layer, feed forward layer and layer norm, respec-
+tively. We term �Hi and �Hi as attention feature and FFN
+feature of the i-th Transformer block.
+Query/Key/Value Features. Each self-attention layer con-
+sists of M head networks, each of which maps input feature
+Fi−1 to query (Q), key (K) and value (V):
+Qm
+i = LN(Fi−1)W Q
+i ,
+Km
+i
+= LN(Fi−1)W K
+i ,
+V m
+i
+= LN(Fi−1)W V
+i ,
+(3)
+where Qi, Ki, Vi ∈ RN× D
+M represent the query, key and
+value of the m-th head network. The query/key/value fea-
+tures (Qi, Ki, Vi ∈ RN×D) are the concatenation of M
+Qm
+i /Km
+i /V m
+i , respectively.
+Relations. For the m-th head network from the i-th Trans-
+former block, we could calculate its Q-Q, K-K, V-V and
+Q-K relations (RQQ
+i,m, RKK
+i,m , RV V
+i,m, RQK
+i,m ∈ RN×N), which
+are implemented as the scaled product relation:
+RQQ
+i,m = Softmax
+�
+Qm
+i Qm
+i
+T
+�
+D/M
+�
+,
+RKK
+i,m = Softmax
+�
+Km
+i Km
+i
+T
+�
+D/M
+�
+,
+RV V
+i,m = Softmax
+�
+V m
+i V m
+i
+T
+�
+D/M
+�
+,
+RQK
+i,m = Softmax
+�
+Qm
+i Km
+i
+T
+�
+D/M
+�
+.
+(4)
+The Q-Q/K-K/V-V/Q-K relations (RQQ
+i
+, RKK
+i
+, RV V
+i
+,
+RQK
+i
+∈ RM×N×N) of the i-th Transformer block is the
+stack of M RQQ
+i,m/RKK
+i,m /RV V
+i,m/RQK
+i,m , respectively.
+3.1.2
+Input
+MIM models randomly mask a high proportion of image
+patches on an input image x, yielding a masked image �x
+for pre-training. We also investigate the input of TinyMIM
+when performing knowledge distillation— the input could
+be either a raw image x or a masked image �x.
+3.1.3
+Target Block
+Consider a situation where we tend to use an MAE pre-
+trained ViT-L (teacher) containing 24 blocks to distill a ViT-
+B (student) containing 12 blocks. In this scenario, the block
+number of the student does not match that of the teacher. We
+investigate which block of the teacher can provide the most
+appropriate target. The selected block is referred to as the
+target block.
+3.2. Knowledge Distillation as MIM Pre-training
+In Section 3.1.1, we describe a variety of distillation target
+candidates. In this section, we introduce different knowledge
+distillation losses for various distillation targets. Let x de-
+note an input image, ft and fs represent a teacher model and
+4
+
+…
+Teacher
+𝑉·𝑉𝑇
+𝑄·𝐾𝑇
+Raw Image
+#Head
+𝑄·𝐾𝑇
+…
+…
+𝑉·𝑉𝑇
+#Head
+𝑉·𝑉𝑇
+𝑄·𝐾𝑇
+#Head
+𝑄·𝐾𝑇
+…
+…
+𝑉·𝑉𝑇
+#Head
+Block-1
+Block-n
+Block-N
+…
+…
+Student
+Block-1
+Block-L (Adaptive Block) 
+Loss
+…
+: Forward
+: Backward
+Figure 3. The default knowledge distillation strategy of TinyMIM. The student (e.g. ViT-B) is optimized to mimic the relations generated by
+the intermediate block of a MIM pre-trained teacher (e.g. ViT-L) with raw image as input. We replace the last block of the student with an
+adaptive block to match teacher’s head number (no extra computational cost). After pre-training (knowledge distillation), the student model
+can be transferred to various downstream tasks.
+a student model, respectively. The objective of knowledge
+distillation is to transfer the knowledge from ft to fs by
+optimizing fs while freezing ft. In general, the training is
+supervised by the KL divergence, which is defined as:
+LKL(p, t) = tlog t
+p,
+(5)
+where t denotes the target generated by ft(x), and p is the
+prediction produced by fs(x).
+Class Token Distillation. We use ct and cs to denote class
+token feature of ft and fs, respectively. The loss of class
+token distillation is formulated as:
+L = LKL(cs, ct).
+(6)
+Feature Distillation. In general, the feature dimension of
+the teacher network and the student network are mismatched.
+To tackle this problem, we adopt an extra linear layer on the
+output of the student network to match the feature dimension
+of the teacher’s target. Let F t and F s denote the target
+feature and the prediction yielded by the student followed by
+a linear projection layer, respectively. We could formulate
+the loss of feature distillation as follows:
+L = L1(F s, Norm(F t)),
+(7)
+where Norm(·) is the whitening operation implemented by
+layer norm without affiliation, and L1 is the smooth L1 loss
+defined as:
+L1(y, ˆy) =
+�
+1
+2(ˆy − y)2/β,
+|ˆy − y| ≤ β
+(|ˆy − y| − 1
+2β),
+otherwise
+,
+(8)
+where β is set to 2.0.
+Relation Distillation. This is our default knowledge distilla-
+tion strategy as illustrated in Figure 3. For the sake of clarity,
+we use RQK
+t→m to denote the m-th head generated Q-K rela-
+tion target (see Eq 4) from the teacher network, and RQK
+s→m to
+represent the corresponding Q-K relation prediction from the
+student network. We define RV V
+t→m and RV V
+s→m in a similar
+way. The loss of relation distillation is formulated as:
+LQK = 1
+M
+M
+�
+m=1
+LKL(RQK
+s→m, RQK
+t→m),
+LV V = 1
+M
+M
+�
+m=1
+LKL(RV V
+s→m, RV V,S
+t→m ),
+L = LQK + LV V .
+(9)
+Head Alignment for Relation Distillation. In general, the
+head number of the student network is lower than that of the
+teacher network. For instance, ViT-L (teacher) contains 16
+heads per block while ViT-B (student) only contains 12 heads
+per block. Recall that the relation distillation loss (Eq. 9)
+is calculated head by head, thus we have to solve the head
+misalignment issue before performing relation distillation.
+To this end, we replace the last block of the student with an
+adaptive block, which keeps the original hidden dimension
+but adjusts the head number to the teacher. Concretely, given
+a teacher network with Mt heads per block, and a student
+network with Ms heads per block, a hidden dimension of
+Ds, and a head dimension of Ds/Ms, the adaptive block is
+designed to be a Transformer block with Mt heads per block,
+a hidden dimension of Ds and a head dimension of Ds/Mt.
+3.3. Sequential Distillation
+When training a small model like ViT-S, the teacher has
+two options: a pre-trained ViT-B and a pre-trained ViT-
+L. Intuitively, the pre-trained ViT-L is a good teacher due
+to its higher representation capability. However, there is
+5
+
+a huge capacity gap between ViT-L and ViT-S, resulting
+in poor distillation results. Following [8, 15], we adopt a
+sequential distillation strategy to improve pre-training. For
+instance, when pre-training a ViT-S, the teacher is selected
+as a TinyMIM pre-trained ViT-B, which has been trained by
+TinyMIM with ViT-L as the teacher.
+4. Experiments
+4.1. Implementation Details
+Pre-training.
+All models are pre-trained under a 100-
+epoch schedule on ImageNet-1K [41] training set.
+We
+use a batch size of 4096 and a learning rate of lr=1.5e-
+4×batchsize/256. We adopt a cosine decay schedule with
+a warm-up for 5 epochs. We adopt AdamW [33] optimizer
+with a weight decay of 0.05. We use random resized crop-
+ping random horizontal flipping, color jitter for student only.
+The input size is set to 224 × 224.
+Fine-tuning. We transfer TinyMIM pre-trained models to
+ImageNet [41] image classification and ADE20K [59] se-
+mantic segmentation. For ImageNet, we use AdamW op-
+timizer with weight decay of 0.05. For data augmentation,
+we follow the settings in MAE [18]. We fine-tune ViT-B
+for 100 epochs with a batch size of 1024, a learning rate of
+2e-3, and a drop path rate of 0.1. We fine-tune ViT-S and
+ViT-T for 200 epochs with a batch size of 2048, a learning
+rate of 5e-3, and a drop path rate of 0.1. For ADE20K, we
+follow the same setting in MAE and adopt UperNet [51]
+as our framework with a TinyMIM pre-trained backbone.
+The input image resolution is 512 × 512 for training and
+evaluating. We use mIoU as the evaluation metric.
+Besides, we evaluate the robustness of TinyMIM on var-
+ious out-of-domain ImageNet datasets [19–21] which are
+generated by applying different perturbations on ImageNet,
+e.g. natural adversarial examples (ImageNet-A), semantic
+shift (ImageNet-R), common image corruptions (ImageNet-
+C). We report top-1 accuracy on ImageNet-A/R and mCE
+error on ImageNet-C (lower is better).
+Default Setting. By default, we adopt relation distillation
+formulated in Eq. 9, head alignment, raw image as input, se-
+quential distillation and the 18-th block of MAE pre-trained
+ViT-L as the target block for TinyMIM-ViT-B pre-training.
+4.2. Main Results
+As shown in Table 3, we compare our TinyMIM with
+previous methods on ImageNet image classification and
+ADE20K semantic segmentation using different ViTs. In
+particular, TinyMIM pre-trained ViT-T achieves 75.8% top-
+1 accuracy, outperforming MAE baseline by +4.2.
+An
+enhanced model named TinyMIM⋆-T, which retains the
+plain architecture and computation budget of ViT-T, fur-
+ther achieves 79.6% top-1 accuracy. See appendix for the
+details of TinyMIM⋆-T. Moreover, TinyMIM pre-trained
+ViT-S achieves 83.0% top-1 accuracy, outperforming MAE
+baseline and previous best method CIM [13] by +2.4, +1.4,
+respectively. By transferring the knowledge of an MAE pre-
+trained ViT-L, TinyMIM pre-trained ViT-B achieves 85.0%
+top-1 accuracy on ImageNet-1K.
+As for semantic segmentation, TinyMIM pre-trained ViT-
+B surpasses MAE baseline and state-of-the-art CAE [4] by
++4.1 and +2.0, respectively. An intermediate fine-tuning on
+ImageNet-1K classification before ADE20K segmentation
+fine-tuning further boosts the performance.
+We also evaluate our models on out-of-domain datasets
+in Table 4. Our TinyMIM pretrained models are more robust
+than MAE pre-trained ones. Specifically, TinyMIM-ViT-B
+outperforms MAE-ViT-B by +6.4 and +4.6 on ImageNet-A
+and ImageNet-R, respectively, and lower the mCE by -5.1.
+4.3. Ablation Study
+Unless otherwise specified, all ablation studies are con-
+ducted on TinyMIM-ViT-B, with a teacher of being an MAE
+pre-trained ViT-L, relation distillation strategy, raw image as
+input, the 18-th block of ViT-L as the target block, under a
+100-epoch pre-training schedule. We report top-1 accuracy
+on ImageNet-1K.
+Class Token Distillation. For this distillation strategy, we
+study two variants: 1) class token distillation as formulated
+in Eq.6; 2) class token distillation with an extra MAE re-
+construction loss. The results are shown in Table 5. Both
+variants perform worse than MAE baseline, indicting that
+the class token is improper to be served as the distillation
+target since there is no explicit supervision applied on class
+token during teacher’s pre-training.
+Feature Distillation. As described in Section 3.1.1, there
+are four types of features can be served as the targets for
+feature distillation formulated in Eq. 7: output feature, FFN
+feature, attention feature and Q/K/V features. Table 6 com-
+pares the results of using different features as distillation
+targets. We also report the results of FFN feature and atten-
+tion feature before the residual connection (see Eq. 2). An
+interesting finding is that distilling FFN feature and attention
+feature after the residual connection significantly degrades
+the performance.
+Relation Distillation. Eq. 9 formulates our default relation
+distillation, which jointly distills Q-K relation and V-V re-
+lation (see Eq. 4). Here we study a variant by changing the
+target relations from Q-K/V-V to Q-K/K-K/V-V. We also
+investigate that whether to apply a Softmax operator on each
+relation. The results are shown in Table 7.
+Comparison of Different Distillation Strategies. In this
+study, all models are pre-trained under a 300-epoch schedule.
+We compare three distillation strategies on ImageNet image
+classification (Table 8) and ADE20K semantic segmentation
+(Table 9). For each strategy, we use the target that yields
+the best result. We also highlight the improvements over the
+6
+
+Method
+Pretraining
+Tokenizer/
+Tokenizer/Teacher
+Classification
+Segmentation
+Epochs
+Teacher
+Data
+Top-1 Acc (%)
+mIoU
+Tiny-size models (ViT-T/16)
+Scratch [44]
+300
+Label
+IN1K
+72.2
+38.0
+MAE† [18]
+1600
+Pixel
+IN1K
+71.6
+37.6
+MoCo [5]
+1600
+EMA
+IN1K
+73.3
+39.3
+TinyMIM (Ours)
+300
+TinyMIM-ViT-S
+IN1K
+75.8
+44.0/44.6‡
+TinyMIM⋆ (Ours)
+300
+TinyMIM-ViT-S
+IN1K
+79.6
+45.0‡
+Small-size models (ViT-S/16)
+Scratch [44]
+300
+Label
+IN1K
+79.9
+43.1
+MAE† [18]
+1600
+Pixel
+IN1K
+80.6
+42.8
+MoCo [5]
+1600
+EMA
+IN1K
+81.4
+43.9
+DINO [3]
+1600
+EMA
+IN1K
+81.5
+45.3
+CIM [13]
+1600
+Pixel
+IN1K
+81.6
+-
+TinyMIM (Ours)
+300
+TinyMIM-ViT-B
+IN1K
+83.0
+48.4/48.9‡
+Base-size models (ViT-B/16)
+Scratch [44]
+300
+Label
+IN1K
+81.2
+47.2
+BeiT [2]
+800
+DALL-E
+DALLE250M+IN22K+IN1K
+83.2
+45.6
+MAE [18]
+1600
+Pixel
+IN1K
+83.6
+48.1
+SIM [43]
+1600
+EMA
+IN1K
+83.8
+-
+CAE [4]
+1600
+DALL-E
+DALLE250M+IN22K+IN1K
+83.9
+50.2
+MaskFeat [48]
+1600
+HOG
+IN1K
+84.0
+-
+SdAE [7]
+300
+EMA
+IN1K
+84.1
+48.6
+data2vec [1]
+800
+EMA
+IN1K
+84.2
+-
+PeCo [11]
+300
+VQGAN
+IN1K
+84.1
+46.7
+PeCo [11]
+800
+VQGAN
+IN1K
+84.5
+48.5
+TinyMIM (Ours)
+300
+MAE-ViT-L
+IN1K
+85.0
+52.2/52.6‡
+Table 3. Fine-tuning results on ImageNet-1K and ADE20K. All models are pre-trained on ImageNet-1K. “Tokenizer/Teacher Data”: training
+data of teacher and tokenizer. †: reproduced result using official code. ⋆: the model is fine-tuned for 1000 epochs with DeiT-style [44]
+knowledge distillation. ‡: the model adopts an intermediate fine-tuning on ImageNet-1K classification before ADE20K segmentation
+fine-tuning.
+Method
+Model Size
+ImageNet ↑
+IN-Adversarial↑
+IN-Rendition↑
+IN-Corruption ↓
+DeiT [44]
+ViT-T
+72.2
+8.0
+32.7
+54.0
+MAE [18]
+71.8
+7.0
+36.5
+55.2
+TinyMIM
+75.8
+11.0
+39.8
+50.1
+DeiT [44]
+ViT-S
+79.9
+18.3
+42.3
+41.4
+MAE [18]
+80.6
+20.1
+45.6
+40.6
+TinyMIM
+83.0
+27.5
+48.8
+35.8
+DeiT [44]
+ViT-B
+81.2
+25.8
+45.4
+36.8
+MAE [18]
+83.6
+33.6
+50.0
+37.8
+TinyMIM
+85.0
+43.0
+54.6
+32.7
+Table 4. Robustness evaluation on out-of-domain datasets.
+MAE baseline.
+Target Block. As described in Section 3.1.3, we consider
+a situation where the block number of the student does not
+match that of the teacher. Here we use an MAE pre-trained
+ViT-L containing 24 blocks to distill a ViT-B containing
+12 blocks. Here we examine the effects of using the 12th,
+15th, 18th, 21th and 24th (last) blocks of the ViT-L as the
+target blocks. The comparison is shown in Table 10. We
+7
+
+Method
+Reconstruction Loss
+Top-1 Acc.
+MAE
+✓
+83.6
+TinyMIM w/ Cls
+80.6
+TinyMIM w/ Cls
+✓
+82.1
+Table 5. Study of class token distillation formulated in Eq.6.
+Feature
+Res. Connection
+Top-1 Acc.
+MAE
+83.6
+Output Feature
+83.7
+FFN Feature
+84.2
+FFN Feature
+✓
+81.8
+Attention Feature
+84.1
+Attention Feature
+✓
+81.3
+Q/K/V Features
+84.3
+Table 6. Study of feature distillation formulated in Eq.7. See
+Section 3.1.1 and Eq. 2 for the definitions of different features.
+Relation
+Softmax
+Top-1 Acc.
+MAE
+83.6
+Q-Q, K-K, V-V
+84.4
+Q-Q, K-K, V-V
+✓
+84.5
+Q-K, V-V
+84.4
+Q-K, V-V
+✓
+84.6
+Table 7. Study of relation distillation formulated in Eq. 9. See
+Section 3.1.1 and Eq. 4 for the definitions of different relations.
+experimentally find that using 18th block yields the best
+result.
+Sequential Distillation. In Section 3.3, we advocate to
+adopt a sequential distillation strategy to enable distillation
+from a larger model (e.g. ViT-L) to a smaller model (e.g.
+ViT-S). Table 11 compares the result of adopting different
+teachers with or without the sequential distillation. We have
+two conclusions: 1) using a larger teacher (MAE-ViT-L) to
+distill a smaller student (ViT-S) degrades the performance; 2)
+sequential distillation significantly boosts the performance
+of ViT-T (MAE-ViT-B→TinyMIM-ViT-S as the teacher and
+ViT-T as the student).
+Integrating MAE into TinyMIM. MAE is a simple but ef-
+fective self-supervised pre-training paradigm that trains a
+model by requiring it to predict masked inputs. In contrast,
+TinyMIM pre-trains smaller ViTs in a knowledge distilla-
+tion manner. Here we integrate MAE into our TinyMIM,
+yielding an integrated model. This model is optimized under
+two losses: knowledge distillation loss from TinyMIM, and
+Method
+Model Size
+Top-1 Acc.
+Supervised (DeiT)
+ViT-T
+72.2
+MAE
+71.6
+Class Token Distillation
+70.6
+Feature Distillation
+73.4
+Relation Distillation
+75.8 (+4.2)
+Supervised (DeiT)
+ViT-S
+79.9
+MAE
+80.6
+Class Token Distillation
+79.6
+Feature Distillation
+80.8
+Relation Distillation
+83.0 (+3.1)
+Supervised (DeiT)
+ViT-B
+81.2
+MAE
+83.6
+Class Token Distillation
+82.6
+Feature Distillation
+83.8
+Relation Distillation
+85.0 (+1.6)
+Table 8. Comparison of three distillation strategies on ImageNet-1K
+image classification. The models are pre-trained under a 300-epoch
+schedule.
+Method
+Model Size
+mIoU
+Supervised (DeiT)
+ViT-B
+47.2
+MAE
+48.1
+Class Token Distillation
+46.2
+Feature Distillation
+47.7
+Relation Distillation
+52.2 (+4.1)
+Table 9. Comparison of three distillation strategies on ADE20K
+semantic segmentation. The models are pre-trained under a 300-
+epoch schedule.
+Task
+12th
+15th
+18th
+21th
+24th
+Classification
+83.6
+84.1
+84.6
+84.8
+84.4
+Segmentation
+48.7
+49.8
+52.2
+50.6
+50.0
+Table 10. Study of target block on ImageNet-1K and ADE20K.
+Student
+Teacher
+Acc.
+ViT-S
+MAE-ViT-B
+82.3
+MAE-ViT-L
+82.1
+MAE-ViT-L → TinyMIM-ViT-B
+82.6
+ViT-T
+MAE-ViT-S
+74.1
+MAE-ViT-B
+74.4
+MAE-ViT-B → TinyMIM-ViT-S
+75.0
+Table 11. Study of sequential distillation.
+8
+
+Masked Image
+Reconstruction Loss
+Top-1 Acc.
+84.6
+✓
+83.9
+✓
+✓
+84.0
+Table 12. Comparison between the TinyMIM-ViT-B (the first row)
+and the integrated model (the third row). We also study the input
+of TinyMIM-ViT-B, which could be raw image (the first row) or
+masked image (the second row).
+DPR (Teacher)
+DPR (Student)
+Top-1 Acc.
+0.0
+0.0
+84.3
+0.0
+0.1
+84.6
+0.0
+0.2
+84.3
+0.0
+0.3
+84.1
+0.1
+0.1
+83.9
+Table 13. Ablation study of drop path rate (DPR) used in teacher
+and student.
+reconstruction loss from MAE. To enable MAE pre-training,
+we randomly mask 75% image patches, and feed the visi-
+ble patches into the network to initiate the pre-training of
+the integrated model. Table 12 shows the comparison be-
+tween TinyMIM-ViT-B and the integrated model. From the
+Table, we could draw a conclusion—integrating MAE into
+our TinyMIM does not improve the performance. In addi-
+tion, we also investigate the input of TinyMIM-ViT-B, which
+could be either raw image or masked image, as shown in
+Table 12—taking raw image as input yields better result.
+Drop Path. Drop path is one of the most critical techniques
+in training Transformers [44]. Using an appropriate drop
+path rate could significantly alleviate the over-fitting issue.
+However, MAE disables this technique in its implementation.
+Here we verify the effects of applying drop path to our
+TinyMIM. The results are shown in Table 13. For the student
+model, the optimal drop path rate is 0.1. For the teacher
+model, disabling drop path yields best result.
+5. Conclusion
+In this paper, we present TinyMIM, which is the first to
+successfully perform masked image modeling (MIM) pre-
+training for smaller ViT models. In stead of adopting a
+mask-and-predict pretext task, we pre-train a small ViT by
+mimicking the relations of a large ViT in a knowledge dis-
+tillation manner. The success of TinyMIM can be attributed
+to a comprehensive study of various factors that may affect
+TinyMIM pretraining including distillation target, distillation
+input and target block. With extensive experiments, we draw
+a series of conclusions. For instance, relation distillation is
+superior than feature distillation and class token distillation;
+taking raw image as input is optimal; a sequential distillation
+is necessary for training smaller ViTs; etc. With its simplic-
+ity and strong performance, we hope our approach can serve
+as a solid baseline for future research.
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+[58] Shan You, Chang Xu, Chao Xu, and Dacheng Tao. Learning
+with single-teacher multi-student. In Proceedings of the AAAI
+Conference on Artificial Intelligence, volume 32, 2018. 3
+[59] Bolei Zhou, Hang Zhao, Xavier Puig, Tete Xiao, Sanja Fidler,
+Adela Barriuso, and Antonio Torralba. Semantic understand-
+ing of scenes through the ADE20K dataset. Int. J. Comput.
+Vis., 127(3):302–321, 2019. 6
+[60] Jinghao Zhou, Chen Wei, Huiyu Wang, Wei Shen, Cihang
+Xie, Alan Yuille, and Tao Kong. ibot: Image bert pre-training
+with online tokenizer. arXiv preprint arXiv:2111.07832, 2021.
+1, 3
+11
+
+A. Hyper-parameters
+Hyper-parameters of ImageNet-1K Pre-training. See Ta-
+ble 14.
+Hyper-parameters of ImageNet-1K Image Classification
+Fine-tuning. See Table 15. TinyMIM⋆-T retains the plain
+architecture and computation budget of ViT-T. We fine-tune
+TinyMIM⋆ for 1000 epochs with DeiT-style [44] knowledge
+distillation on ImageNet-1K. Following MobileNetV3 [26],
+an extra fully connected layer is placed before the classifi-
+cation layer to increase the feature dimension from 192 to
+1280. The head number is set to 12 instead of the default 3.
+Hyper-parameters for ADE20K Semantic Segmentation
+Fine-tuning. See Table 16.
+Hyperparameter
+ViT-T
+ViT-S
+ViT-B
+Layers
+12
+Hidden size
+192
+384
+768
+FFN inner hidden size
+768
+1536
+3072
+Attention heads
+3
+6
+12
+Patch size
+16 × 16
+Pre-training epochs
+100/300
+Batch size
+4096
+Adam ϵ
+1e-8
+Adam β
+(0.9, 0.999)
+Peak learning rate
+2.4e-3
+Minimal learning rate
+1e-5
+Learning rate schedule
+Cosine
+Warmup epochs
+5/15
+Stochastic depth
+0.1
+Dropout
+�
+Weight decay
+0.05
+Data augment
+RandomResizeAndCrop
+Input resolution
+224 × 224
+Color jitter (student only)
+0.4
+Table 14. Hyper-parameters of ImageNet-1K Pre-training.
+Hyperparameter
+ViT-T
+ViT-S
+ViT-B
+Peak learning rate
+5e-3
+5e-3
+2e-3
+Fine-tuning epochs
+200
+200
+100
+Warmup epochs
+5
+Layer-wise learning rate decay
+0.65
+0.65
+0.65/0.6∗
+Batch size
+2048
+2048
+1024
+Adam ϵ
+1e-8
+Adam β
+(0.9, 0.999)
+Minimal learning rate
+1e-6
+Learning rate schedule
+Cosine
+Stochastic depth
+0.1
+Weight decay
+0.05
+Label smoothing ε
+0.1
+Dropout
+�
+Gradient clipping
+�
+Erasing
+0.25
+Input resolution
+224 × 224
+Rand augment
+9/0.5
+Mixup
+0.8
+Cutmix
+1.0
+Table 15. Hyper-parameters of ImageNet-1K image classification
+fine-tuning. ∗ indicates that we use 0.65 and 0.6 for 100-epoch and
+300-epoch pre-trained models, respectively.
+Hyperparameter
+ViT-S
+ViT-B
+Input resolution
+512 × 512
+Peak learning rate
+1e-4
+Fine-tuning steps
+160K
+Batch size
+16
+Adam ϵ
+1e-8
+Adam β
+(0.9, 0.999)
+Layer-wise learning rate decay
+{0.65, 0.75, 0.8}
+Minimal learning rate
+0
+Learning rate schedule
+Linear
+Warmup steps
+1500
+Dropout
+�
+Stochastic depth
+0.1
+Weight decay
+0.05
+Table 16. Hyper-parameters of ADE20K semantic segmentation
+fine-tuning.
+12
+

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