Papers

Let $G$ be a locally finite multigraph that is bipartite and "unbalanced," meaning that it has a nontrivial bipartition $(A,B)$ with $\mathrm{deg}(x) > \mathrm{deg}(y)$ for all $x \in A$ and $y \in B$. We explore matchings in such graphs through the lens of descriptive set theory. In particular, we show that when $G$ is Borel and $\mu$ is a Borel probability measure on its vertex set, there is a Borel matching in $G$ that covers $\mu$-almost every vertex in $A$. This was previously known only under the assumption that $\mu$ is $G$-invariant, which we eliminate using a novel probabilistic approach. We also describe various extra conditions that imply the existence of a Borel matching covering every vertex in $A$. Along the way, we confirm a conjecture of the first and third named authors concerning the existence of Borel independent complete sections in Borel graphs of finite asymptotic separation index. In addition to their intrinsic interest, our results have applications to various other topics, such as edge-colorings, balanced orientations, and equidecomposition theory for group actions. For example, we show that the measurable edge-chromatic number of every Borel multigraph with finite maximum degree $\Delta$ is at most $\lfloor\frac{3\Delta}{2}\rfloor$, matching Shannon's optimal bound for finite multigraphs. Another example is that paradoxical Borel group actions with finite asymptotic separation index admit paradoxical decompositions with Borel pieces. This refines a result of Marks and Unger.

This paper investigates an inverse problem for the simultaneous reconstruction of two spatially varying reaction coefficients, the local proliferation rate and the competition (saturation) coefficient, together with the unknown initial condition, in a nonlinear, density-dependent reaction-diffusion model motivated by cell invasion and tumor growth dynamics. Using Carleman estimates, we establish a global uniqueness result together with a Lipschitz-type stability estimate for the reaction coefficients and a weaker, logarithmic stability estimate for the initial condition. For the numerical reconstructions, we develop a two-stage algorithm employing a time-shift strategy to decouple the coefficient and the initial condition. Numerical experiments are presented to illustrate the feasibility, accuracy, and robustness of the proposed inversion method.

We give exact Gelfand–Kirillov (GK) dimension formulas for simple highest weight modules of the basic classical Lie superalgebras considered in this paper. For the type II orthosymplectic algebras $D(m,n)$ and $B(m,n)$, we identify the PBW associated variety as a product of factor supports evaluated at highest weights for two compatible Borel subalgebras. This gives algorithms for arbitrary complex highest weights, without integrality or typicality hypotheses. In type $B$, the terminal factor is $\mathfrak{osp}(1|2n)$, for which we derive an insertion formula that distinguishes integer and half-integer shifted coordinates. For $G(3)$, $F(4)$ and $D(2,1;\alpha)$, finite odd-reflection procedures give exact formulas, their GK spectra, and finite-dimensionality criteria; we also describe the dependence on $\alpha$ in the last case. In type I, the GK dimension equals that of the simple even highest weight module for the distinguished Borel. The computations use the established ordinary Lie algebra formulas involving Lusztig's $\mathbf a$-function and Robinson–Schensted insertion.

Let $q$ be a prime power and write $\Omega$ for the bigraded algebra of regular differential forms over $\mathbb{F}_q^n$. The general linear group $GL_n(\mathbb{F}_q)$ acts on $\Omega$; write $SI \subseteq \Omega$ for the ideal generated by $GL_n(\mathbb{F}_q)$-invariants with vanishing constant term. The {\em $GL_n(\mathbb{F}_q)$-superspace coinvariant ring} is the quotient $SR := \Omega/SI$. We calculate the bigraded Hilbert series of $SR$ and give an operator-theoretic characterization of the inverse system $SI^\perp$. Our results extend to subgroups $G$ of $GL_n(\mathbb{F}_q)$ which contain $SL_n(\mathbb{F}_q)$.

Let $G$ be a graph with $p(G)$ pendant vertices and $q(G)$ quasi-pendant vertices. Denote by $m_{L(G)}(\lambda)$ the multiplicity of $\lambda$ as a Laplacian eigenvalue of $G$. A graph $G$ is called reduced, if $p(G)=q(G)$. It is known that deleting a pendant path $P_3$ from a graph $G$ cannot change $m_{L(G)}(1)$. By the reduction operation for a graph (defined by Tian and Wong, 2026), we could turn to the reduced graphs with each quasi-pendant vertex of degree 2 to investigate $m_{L(G)}(1)$. Then let $T$ be a reduced tree on $n(\geq 7)$ vertices with each quasi-pendant vertex of degree 2 and without pendant path $P_3$. We first prove that \begin{equation*} m_{L(T)}(1)\leq \frac{n-5}{6} \end{equation*} and the extremal trees attaining the upper bound are determined completely. In addition, let $G$ be an arbitrary connected reduced graph with order $n\geq 6$ and size $m$. Denote by $c=m-n+1$ the first Betti number of $G$, then we obtain \begin{equation*} m_{L(G)}(1)\leq c+\frac{n-2}{4}, \end{equation*} and the extremal graphs attaining the upper bound are characterized completely.

An $n$-partite graph is a graph such that every vertex has a color in $\{1,\ldots,n\}$ and every two vertices of the same color are not adjacent. We study the model comparisons of the theories of $n$-partite graph and $K_{\overline{m}}$-free $n$-partite graph, where $K_{\overline{m}}$ is a complete graph of a given size. The model companion of the theory of $n$-partite graph is simple and has IP. The model companion of the theory of $K_{\overline{m}}$-free $n$-partite graph has $\operatorname{TP}_2$, $\operatorname{SOP}_3$ and $\operatorname{NSOP}_4$ if $n > 2$. Forking independence coincides with dividing independence in this theory.

In this paper, we develop a generalized differentiation theory at infinity for convex sets and functions. In particular, we study several fundamental notions of convex analysis at infinity, including tangent cones, normal cones, and subdifferentials. Our work complements the recently developed theories of nonsmooth analysis at infinity by focusing on the convex setting, where these constructions preserve convexity and admit natural connections with recession analysis, polarity, and epigraphical geometry. We show that, in the convex case, many results can be established under weaker assumptions and admit simpler and more explicit representations than those available in the general nonsmooth framework. In addition, we develop calculus rules and geometric characterizations for these constructions and apply them to optimality conditions and attainment criteria for convex optimization problems over unbounded feasible sets. The results obtained in this paper provide new tools for the study of convex sets and functions at infinity and further strengthen the connections between convex analysis, variational analysis, and optimization.

We introduce the ladderpath index as a measure of language complexity grounded in algorithmic information theory. It counts the minimum steps needed to reconstruct a sequence through hierarchical reuse of repeated substructures, capturing an exactly computable but constrained form of algorithmic compressibility related to, but distinct from, Kolmogorov complexity. We apply the ladderpath approach to 21 parallel corpora from the Parallel Universal Dependencies dataset. The ladderpath index is approximately invariant across the languages, and varies much less than the corpus length. This is more pronounced when all corpora are mapped to a unified binary representation, providing evidence for the equi-complexity hypothesis from a representation-independent perspective. We also observe trade-offs between character inventory size and corpus length, and between vocabulary-level and corpus-level reconstruction complexity, supporting the trade-off hypothesis that total complexity is conserved and redistributed across linguistic levels. The reusable substructures identified by the ladderpath approach, without any linguistic input, overlap with words and morphological components attested in the natural vocabulary. The hierarchical reuse captured by the ladderpath approach parallels the chunking mechanisms proposed in cognitive science, where the human cognitive system compresses linguistic input into nested, reusable units under shared memory and processing constraints. This connection between cognitive chunking and the ladderpath approach provides a new interpretation for the equi-complexity and trade-off hypotheses, grounding both in the shared cognitive architecture that underlies language processing across human languages.

We give a dynamical proof of Benoist's non-arithmeticity theorem for Jordan spectra of Zariski dense subgroups of connected semisimple real algebraic groups. After passing to a Zariski dense Schottky subgroup, we use the coding of the limit set to realize Jordan projections as periods of a vector-valued Busemann return map for an expanding map on the Furstenberg boundary. The key step is to prove that a suitable two-branch asymptotic discrepancy is not locally constant on the limit set. We also show that the same criterion applies beyond Lie groups; in particular, it yields a direct density result for multiplier spectra of hyperbolic rational maps whose Julia set is not contained in a circle.

In this paper, we present a new approach for unmanned aerial vehicle (UAV) positioning and reconfigurable intelligent surface (RIS) partitioning to enhance connectivity of uplink RIS-assisted UAV networks. To achieve this, our approach optimizes RIS-aided link selection, RIS partitioning, and UAV positions to maximize network connectivity characterized by its Fiedler value. Meanwhile, it maintains a specific signal-to-interference plus noise ratio (SINR) constraint for user equipment (UE), which is influenced by RIS partitioning and UAV reliability. The network connectivity optimization problem is formulated using the Fiedler value subject to RIS elements allocation and SINR constraints. This problem is a computationally expensive combinatorial optimization, necessitating an efficient iterative approach. In particular, we propose a perturbation method for RIS-aided link selection, and derive a closed-form solution for RIS partitioning, with each partition tailored to optimize SINR for individual UAV. For the given RIS-aided links and RIS partitioning, we then show that the problem of UAV positioning can be formulated as a low complexity semi-definite programming (SDP) optimization problem, which can be solved using off-the-shelf CVX solvers. Our simulations show the potential gain of UAV positioning and RIS partitioning compared to the benchmark schemes from the literature.

The diamond transport is generated by the uniform law on a diamond-shaped copula support. Since a classical optimal transport (OT) objective is affine in the coupling, this transport cannot be the unique minimizer in the classical setting. We study a broader family of transports, called diamond-type transports, in non-classical settings such as quadratic-form optimal transport (QOT) and distorted optimal transport (DOT), which are generally nonconvex. Our main results are within the QOT framework: for symmetric one-dimensional marginals, the diamond transport is an optimizer for a large class of QOT problems whose costs depend on within-coordinate distances. Examples include product costs under positive-definiteness and convexity conditions and, in particular, mixed rectangular costs. For rectangular costs, we show that the diamond transport is the unique minimizer except for boundary cases. In the DOT framework, diamond-type transports are minimizers for a natural class of cost, and the diamond transport is the unique minimizer in specialized examples. We also identify the intersection between DOT and QOT, which corresponds precisely to quadratic distortion functions.

In this work, we introduce the idea of anisotropic Herz-Amalgam spaces. Then we find the atomic decomposition in these spaces. As an application, in these spaces, we demonstrate the boundedness of certain sublinear operators.

We show that the category of orthomodular posets is a full coreflective subcategory of the category of strong orthoposets, those orthoposets in which any two orthogonal elements have a join. This coreflection is obtained by building from a strong orthoposet $P$, an orthomodular poset with the same underlying set and same orthocomplementation as $P$, but with modified order. This coreflector restricts to a functor from the category of ortholattices to the category of orthomodular posets, and this functor is right adjoint.

This paper studies the propagation of fake news through a stochastic rumor spreading model based on Markov chains. Inspired by classical epidemiological SIR models, we consider a generalization of the Daley-Kendall framework for rumours that incorporates fact-checkers, following the Ignorant/Spreader/Checker/Stifler model introduced in Piqueira (2020). The model analyzes the influence of checkers on fake news dynamics. Numerical simulations are used to illustrate the behavior of the system and the impact of fact-checkers.

This article develops a mathematical framework to handle fuzzy partial differential equations (PDEs) using Sobolev spaces defined over A-linearly interactive complex fuzzy processes. We introduce the notion of weak derivative in the fuzzy sense to define fuzzy Sobolev spaces that preserve key analytical properties. Furthermore, we introduce a fuzzy Fourier transform adapted to this context and explore its main properties. Applications to fuzzy versions of the heat and Schrödinger equations are presented to demonstrate the effectiveness and generality of the proposed framework. This approach not only extends classical tools to the fuzzy setting, but also provides new insights into the treatment of uncertainty in dynamic systems.

In fuzzy-set theory, the notion of convexity has often been formulated through scalar reductions, typically via scores and aggregation functions. Although useful, such reductions may obscure relevant order-theoretic information in the codomain, especially in complex set-valued settings. This article develops a lattice-theoretic framework for convexity on lattice-valued mappings over point-convex segment-generated abstract convexity spaces. The framework separates the segment structure of the domain from the lattice structure of the codomain, distinguishing intrinsic convexity, defined through the lattice meet, from relational and observable convexities induced by preorders or scalar maps. The article characterizes when scalar observables preserve or reconstruct intrinsic convexity, and which codomain operators preserve it through their pointwise extensions. It then specializes the framework to hesitant fuzzy sets endowed with the symmetric lattice, recovering classical fuzzy and interval-valued convexities as natural restrictions. The structural result shows that the symmetric order cannot be represented by any finite family of scalar observables. This obstruction appears even on two-point typical hesitant fuzzy elements. Consequently, symmetric hesitant convexity cannot, in general, be reconstructed by any finite family of scalar observables. Moreover, every finite family of monotone scalar observables admits a three-point hesitant profile that is scalar-convex for all selected descriptions but not symmetrically convex.

We developed the third-order hydrodynamic equations using relativistic extended thermodynamics of gases with 14 independent fields. The resulting fluid equations are based on the relativity principle, the entropy principle, and the requirement of hyperbolic, and hence finite, propagation of disturbances, which is automatically incorporated. The expressions of entropy, four-current, shear-stress tensor, dynamic pressure, and heat flux are expanded up to third order (cubic). We explicitly present the newly calculated coefficients in the equilibrium properties of an ultra-relativistic gas regime and the non-degenerate relativistic gas. Contrary to the general cases, the non-degenerate regime eliminates fugacity from the coefficients, allowing for the easy normalization of these coefficients, and the ultra-relativistic regime provides us with the upper bounds of these coefficients. We found good agreement on some of the coefficients as compared to calculations from earlier models, specifically in kinetic theory, and other coefficients had slightly different values to those obtained in kinetic theory.

This paper investigates the utility of the movable antenna (MA) and reconfigurable intelligent surface (RIS) framework for downlink wireless communications. In the considered scenario, a base station (BS) is equipped with two sub-arrays of MAs transmits signals to the users via the RIS. By jointly exploiting the antenna-positioning flexibility of MAs and the RIS element selection capability, the proposed joint MA-RIS framework introduces additional design degrees of freedom to enhance desired signals and mitigate inter-user interference, thereby maximizing the network sum-rate. To this end, we formulate a joint optimization problem involving MA positioning, sub-array beamforming, and RIS element selection, subject to the minimum antenna separation and transmit power constraints. The resulting problem is highly non-convex and challenging to solve directly. To address this issue, an alternating optimization framework is developed that decomposes the problem into three tractable subproblems. Specifically, zero-forcing beamforming is employed for transmit beamformer design, a low-complexity one-dimensional search is derived for RIS element selection, and the MA positioning problem is solved using block coordinate descent (BCD) and convex optimization techniques implemented via CVX. Simulation results demonstrate that the proposed joint MA-RIS framework significantly improves the achievable sum-rate compared with conventional fixed MAs and benchmark schemes with random configurations.

We present a novel method for solving conditional value-at-risk (CVaR) optimization problems based on the dual representation of CVaR, which is defined as the worst-case expectation over a risk envelope. The method is based on the Bregman proximal point algorithm and alternates between stochastic primal and dual stages. Every (inner) primal stage involves a subproblem solved by sampling from a probability distribution updated at each dual stage (outer iteration). The likelihood ratio of the dual probability distributions relative to the distribution underlying the original problem converges to the risk identifier of the solution's CVaR. Thus, the dual distribution provides the algorithm with a built-in importance sampling mechanism that draws from the tail of the underlying distribution. Because only samples in the tail influence the CVaR, and samples outside the tail are drawn with decreasing probability, the algorithm delivers exceptional performance over other stochastic approximation methods. We prove the convergence of the algorithm for convex objective functions. Our numerical experiments target representative problems in financial mathematics and machine learning, focusing on portfolio optimization and support-vector machines, respectively.

We investigate the Schrödinger bridge problem on a finite state space with a strictly positive Markov reference kernel. Starting from the semi-dual convex formulation, we introduce a continuous-time evolution for the terminal Schrödinger potential and show that its equilibria coincide with the unique solution of the bridge problem. The proposed dynamics induces an evolution for the terminal marginal. This marginal equation is governed by a state-dependent nonlocal Onsager operator, identified with the Hessian of the semi-dual functional. We derive its associated Dirichlet form, establish coercivity estimates on the appropriate quotient space, and interpret the resulting equation as a nonlocal gradient-flow formulation of relative entropy. Under natural positivity assumptions, we prove global well-posedness of the SBOF, convergence to the Schrödinger bridge, convergence of the induced couplings and path measures, and exponential relaxation of the terminal marginal. The latter follows from a uniform Poincaré inequality on compact sublevel sets together with entropy–variance comparison estimates. We also discuss the connection with finite-state generative modeling through the Doob transform and illustrate the theory on finite-grid examples involving rare states.

We study a critical multitype branching particle system in $\mathbb{R}^N$ with finite type space $\mathcal{K}={1,\dots,K}$. Particles of type $i$ move according to a symmetric $\alpha_i$-stable process and reproduce according to a critical offspring law with irreducible stochastic mean matrix. All lifetime distributions are non-arithmetic. The type-$1$ lifetime distribution has infinite mean and a regularly varying tail $$ 1-F_1(t)\sim c_1t^{-\gamma}, \qquad 0<\gamma<1, $$ whereas the lifetime distributions of types $2,\dots,K$ satisfy the polynomial upper-tail bounds $$ 1-F_i(t)\le C t^{-\eta_i}, \qquad i=2,\dots,K, \qquad \eta_i>1, \qquad \eta:=\min_{2\le i\le K}\eta_i. $$ The offspring mechanism satisfies a $(1+\beta)$-stable-domain regular-variation condition, with $\beta\in(0,1]$. The initial population is a Poisson random measure with intensity $$ \Lambda=\sum_{i=1}^K a_i\,\lambda\otimes\delta_i, \qquad a_i\ge0, $$ and we set $A:=\sum_{i=1}^K a_i$. Under the space–lifetime condition $$ \rho:=\left(\eta-1\right)\wedge\frac{N}{\alpha_1} > \frac{\gamma}{\beta}, $$ we prove that the system converges to a Poisson random measure with intensity $A\lambda\otimes\delta_1$. Thus, although the initial population may assign positive spatial intensity to every type, the limiting population is supported entirely on the infinite-mean type while preserving the aggregate initial spatial intensity $A\lambda$.

We consider the classical $N$-body problem with the $\frac{1}{|x|^p}$-type potential, where $p>0$. We construct hyperbolic, parabolic and hyperbolic-parabolic solutions with prescribed asymptotic data as $t \to+\infty$.

We consider independent Bernoulli percolation on top of sequences of hierarchical graphs. Given a graph $G_{1}$ with two distinguished vertices $a_{1}$ and $b_{1}$, the hierarchical graph with seed $G_{1}$ is the sequence $\big( G_{k} \big)_{k \geq 1}$ resulting from the inductive procedure, where the graph $G_{k+1}$ is obtained from $G_{k}$ by replacing each of its edges with a copy of $G_{1}$, attached by the vertices $a_{1}$ and $b_{1}$. We prove that, under sharp hypotheses, percolation on these graphs presents a unique phase transition. Second, we establish the existence of several critical exponents in this context, such as the critical exponents for the correlation length $\nu$, the surface tension $\mu$, the one-arm exponent $\alpha_{1}$. Several results are also obtained for their infinite counterpart $G_\infty$, which is the Benjamini-Schramm limit of $G_k$: uniqueness of the infinite cluster, continuity of $\theta(p)$, existence of the percolation-probability exponent $\beta$ and scaling relations for the critical exponents $\alpha_1$, $\nu$ and $\beta$. Furthermore, we analyze noise sensitivity for crossing functions in $G_{k}$ and establish sharp noise sensitivity in this setting. Finally, we propose a setup where it is possible to verify the locality hypothesis, stating that the critical threshold for percolation is a local property, while critical exponents are determined by the global geometry of the graph. As a consequence of the techniques developed here, we also provide a necessary and sufficient condition for the existence of a unique fixed point for the map $p \mapsto \mathbb{E}_p[g]$ in $(0,1)$, where $g:\{0,1\}^n \to \{0,1\}$ is a nontrivial monotone Boolean function.

We investigate a free boundary PDE model for vascularized tumor growth with two time-periodic coefficients representing nutrient supply $\phi(t)$ and nutrient demand $\psi(t)$. Under radial symmetry, the model reduces to a nonautonomous ODE for the tumor radius. We prove a vanishing-persistence dichotomy governed by the averaged coefficients $\bar{\phi}$ and $\bar{\psi}$: if the average nutrient supply does not exceed the average nutrient demand, namely $\bar{\phi}\le \bar{\psi}$, the tumor radius tends to zero; if the average supply exceeds the average demand, namely $\bar{\phi}>\bar{\psi}$, the tumor persists and converges to a unique positive periodic solution. We also study the linear stability of this periodic solution when nonradial perturbations are imposed. Under a stronger condition on the nutrient supply and demand, we show that the radially symmetric periodic solution is linearly stable when the tumor aggressiveness parameter is sufficiently small. Numerical simulations are provided to illustrate the analytical results.

The success of large-scale deep learning models in neuroscience is fundamentally constrained by severe data heterogeneity. Native fMRI data aggregated from diverse sources exhibit substantial variation in both spatial and temporal resolutions. Consequently, most existing frameworks rely on lengthy, rigid preprocessing pipelines that enforce uniformity across datasets. This practice introduces two critical limitations: (1) potential degradation of subject-specific anatomical information; (2) significant computational overhead, often requiring hours of processing per subject. Here, we propose FlexiBrain, a resolution-agnostic voxel-level encoding framework for native fMRI based on Mamba-JEPA. FlexiBrain defines patch sizes in real-world physical units and employs a dynamic patch resizing, thereby bypassing destructive spatial standardization while enabling direct ingestion of data in native space. We instantiate the framework using an efficient Mamba-JEPA backbone to model high-dimensional 4D fMRI signals. Across five diverse downstream neuroscience tasks, FlexiBrain consistently outperforms recent state-of-the-art methods, achieving gains of up to 12 percentage points without external data augmentation. Importantly, FlexiBrain functions as a seamless plug-in module, substantially reducing preprocessing costs and accelerating the development of robust voxel-level fMRI foundation models. Code is available at https://github.com/OneMore1/FlexiBrain.

We continue to study ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances, together with the dc-embeddings, which we introduced in our earlier paper "Universal homogeneous two-sorted ultrametric spaces". The class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé whose limit we denote by $\mathbb{U}$. The main result of the article is that $\operatorname{Aut}(\mathbb{U})$ has a comeager conjugacy class. For that we show the cofinal amalgamation property of partial automorphisms and characterize amalgamation bases. In fact we develop a general strategy for showing cofinal amalgamation property for a broad class of categories. Furthermore, we show that there is no generic pair of automorphisms, we provide a detailed description of single orbits under dc-automorphisms, and we prove that any finite partial dc-automorphism, even in the presence of other orbits, can be extended to one that is closed or monotone.

We study the graded polynomial identities of the first Weyl algebra $W_1$ over an infinite field. The algebra $W_1$ satisfies no ordinary polynomial identities in characteristic 0. It admits a natural grading by the infinite cyclic group $\mathbb{Z}$. We construct a basis of the $\mathbb{Z}$-graded identities of $W_1$, which consists of a single identity. It expresses the fact that the degree 0 component in the grading is commutative. It is also well known that if the characteristic of the base field is $p>2$, then $W_1$ satisfies the same identities as the full matrix algebra of order $p$. In this situation, we describe the $\mathbb{Z}_p$-graded identities of $W_1$. Afterwards, using various combinatorial and algebraic tools we consider graded identities for various types of algebras generalizing the Weyl algebras. For example, we show that $\mathbb{Z}$-graded Galois rings in characteristic 0 satisfy the same graded identities as $W_1$ when they embed in a shift operator algebra $\mathcal{S}_1$, and as a consequence we obtain that these $\mathbb{Z}$-graded Galois rings are not PI. The same holds for the algebra of differential operators on $1$-dimensional torus. We obtain similar results for generalized Weyl algebras. We also deal with the graded identities for the quantum Weyl algebras and for the quantum plane. It turns out that in the latter case and when $q$ is the $\ell$-th primitive root of unity, one is led to study gradings by the group $\mathbb{Z}_\ell\times \mathbb{Z}_\ell$. In this case the quantum plane satisfies the same graded identities as the matrix algebra of order $\ell$. Finally we construct a natural $\mathbb{Z}$-grading on the universal enveloping algebra of $\mathfrak{sl}_2$, and prove that its $\mathbb{Z}$-graded identities are the same as those of $W_1$, in characteristic $0$.

We study the problem of fair division of indivisible goods with constraints imposed by a prescribed independence structure, that is, a family of subsets of goods closed under taking subsets. As a motivating example, imagine that the goods to be divided are the available connections in a logistic, financial, or social network. The admissible bundle of goods for each agent must correspond to an acyclic set of edges, corresponding to a basic feasible solution to a linear network problem to be solved. Suppose that all agents assign the same value to each good (in the example, the network connections are equally important for every agent) and evaluate each bundle by summing the values of its goods. Is there a fair partition of the goods into such acyclic bundles? Surprisingly, the answer is yes, provided that the number of agents is at least the arboricity of $G$, and the fairness requirement is envy-freeness up to one good (EF1). The situation becomes more mysterious when agents have arbitrary additive valuations. Our main result guarantees that, in this case, epistemic EF1 partitions always exist, which means that each agent receives an acyclic bundle for which there exists a feasible partition of the remaining goods into acyclic bundles that they do not envy up to one good. We derive this conclusion from a general result for abstract independence structures defined on the sets of goods. We also discuss connections with several conjectures concerning matroids. In particular, we prove that any Hamiltonian matroid partitionable into two independent sets admits an EF1 bipartition with respect to a common monotone valuation. We complement our results with a constructive perspective: we present explicitly two algorithms for computing the fair allocations described above. Finally, we provide illustrative examples to demonstrate these algorithms on specific instances.

We study a mean field game (MFG) of state and control with state dynamics described by stochastic differential equations driven by both idiosyncratic and common noise, and subject to the constraint that the terminal state variable belongs to a nonempty, convex, closed set. The mean-field interaction enters both the dynamics and the costs through state and control. We formulate an equivalent auxiliary MFG problem and derive the stochastic maximum principle for an auxiliary optimization problem under fixed flows. By means of a suitable forward-backward stochastic differential equation (FBSDE) of conditional McKean-Vlasov type, we establish a characterization of MFG solutions via such a system. Moreover, we prove the existence and uniqueness of solutions for the FBSDE system in the specific case where the running cost is zero and the state coefficients are linear. Additionally, we obtain uniqueness results in the sense of Lasry-Lions and we apply our findings to MFGs of optimal investment with a quadratic relative performance criterion.

We propose a new approach to unbiased estimation of the gradients of the stationary means associated with parametrized families of Markov chains. Our estimators are particularly efficient when the Markov chains have slow mixing rate. Our approach does not require a specific parametrization except for an oracle to evaluate the transition density and its gradient at a given data point without any additional knowledge about the density function itself. It makes our estimator suitable for parametrizations associated with neural networks. The estimator can potentially achieve large improvement in terms of efficiency. Numerical experiments confirm the good performance predicted by the theory.