Papers

This paper considers Sobolev-Morrey and Besov-Morrey spaces. For Morrey spaces, multipliers are studied, and a theorem on multipliers from M_p^lambda(T) to M_p^lambda(T) is obtained for 1 < p < infinity and 0 <= lambda < 1/p. Based on this result, embeddings between Besov-Morrey and Sobolev-type spaces are established, namely B_1^alpha(M_p^lambda) embedded into W^alpha(M_p^lambda) embedded into B_infinity^alpha(M_p^lambda) for 0 < p <= infinity, 0 <= lambda <= 1/p, and alpha in R. In addition, an interpolation theorem for Besov-Morrey spaces is proved.

This work establishes a bipartite generalization of the Harer-Zagier formula using non-Hermitian Random Matrix Theory. More specifically, we use a decomposition of powers of Ginibre eigenvalues as a superposition of independent point processes to identify all coefficients of the generating function of the genus of a surface obtained by a random bipartite pairing of the sides of one polygon with $kM$ sides and $k$ polygons with $M$ sides.

We introduce the complex equivariant spectral flow – a virtual $G$-representation assembling the eigenvalue winding numbers of the linearization at an isolated two-parameter critical point for $G = S^1 \times \Gamma$ bifurcation problems – and prove that, for maximal twisted orbit types, the coefficient of the local bifurcation invariant in $A_1^t(G)$ reduces to a closed-form dimension formula, bypassing the standard pipeline of basic degree factorization and Burnside ring multiplication entirely. When the eigenvalue dependence is holomorphic, topological cancellation among winding numbers is impossible, yielding unconditional local and global bifurcation guarantees. As applications, we establish the macroscopic escape of symmetric Hopf branches in $\Gamma$-equivariant systems and of patterned relative equilibria for the complex Ginzburg–Landau equation directly from spectral data.

Berwald-Finsler spacetimes are Finsler spacetimes that are closest to pseudo-Riemannian geometry, as their canonical nonlinear connection defines an affine connection on spacetime. In this paper, we find the first exact, non-Ricci flat, $SO(3)$-symmetric Berwald solutions to the Finsler gravity vacuum equation. To this aim, we first select from all spherically symmetric Berwald spacetimes, the only class that admits flat Finsler spacetime structures, making it particularly suitable for future investigations of asymptotic flatness. Then, for this class, we completely solve the Finsler gravity vacuum equation, and find three families of solutions. In particular, we show that in Finsler geometry there exist $SO(3)$-symmetric, vacuum solutions that are not Ricci-flat. These solutions are promising candidates to model the gravitational field around compact objects, beyond their Riemannian description.

The Ize Conjecture proposed that every absolutely irreducible representation of a compact Lie group admits a maximal isotropy subgroup with an odd-dimensional fixed-point space, which would provide a universal bifurcation guarantee via the equivariant degree. Its disproof by Lauterbach and Matthews necessitates a more targeted criterion. We introduce Ize pairs – pairs $(G, V)$ for which some maximal isotropy subgroup $H$ satisfies $\dim V^H - \dim V^G \equiv 1 \pmod{2}$ – and prove that this dimension-parity condition completely captures the algebraic obstruction to a non-trivial equivariant degree change at maximal orbit types. Integrating this criterion with a mod-2 equivariant spectral flow yields local and global bifurcation guarantees without recourse to Burnside ring arithmetic. As an application, we establish unbounded branches of non-stationary periodic solutions in a $\Gamma$-symmetric coupled oscillator network, where the bifurcation guarantees follow entirely from the crossing parity of the linearization at the boundary of a regular parameter window.

We study non-atomic Bayesian routing games whereby a transportation network is shared by two types of traffic: a coordinated fleet and a mass of selfish users. The links in the network are characterized by travel time functions that depend both on the aggregate flow on the link and on a random state of the world $W$ that, in general, is not directly observable. Rather, we assume that both the fleet coordinator and the selfish users know the prior distribution of $W$, observe (different) private messages that are correlated with $W$ and possibly among themselves, and make routing decisions based on such heterogeneous partial information. Under the assumption that both the state of the world and the private message sets are finite, we prove the existence and uniqueness of a Bayesian equilibrium for the ensuing Bayesian routing game.

We study universal deformations in incompressible isotropic Cauchy elastic solids with residual stress, without assuming any specific origin for the residual stress. Starting from the constitutive representation of the Cauchy stress as an isotropic tensor-valued function of strain and residual stress, we derive the universality constraints for residually-stressed incompressible isotropic Cauchy elastic solids. We show that for the six known families of universal deformations the set of universal deformations is identical to that of incompressible isotropic elasticity in the absence of residual stress. We also show that residual stress does not enlarge the space of universal deformations. We then determine the universal residual stress fields corresponding to the six known families of universal deformations. Assuming that the residual stress field has the same symmetry as the corresponding universal deformation, the universality constraints reduce to systems of ordinary differential equations that can be solved explicitly. The resulting universal residual stress fields are characterized and discussed for each family.

Quantum entanglement assistance is known to improve the Shannon capacity of classical communication networks but the largest gains noted thus far are rather modest (less than 6%), motivating the question: are large capacity gains ever possible? It is shown in this work that in the presence of causal channel state information at the transmitters (CSIT), quantum entanglement assistance provides a multiplicative capacity advantage that grows exponentially with the number of users K for certain classical K-user multiple access channels with fixed size (binary) alphabet for inputs, outputs and states. Similarly, in the presence of causal channel state information at the transmitters, quantum entanglement assistance is shown to provide a multiplicative capacity advantage that is unbounded as the size of the state alphabet grows, while the number of users (K=3) and the input and output alphabet (binary) are held fixed. Even with only a few users and small alphabet sizes, substantial multiplicative gains in capacity are found, e.g., with binary inputs, outputs and states, multiplicative gains by factors exceeding 21 and 88 are noted with K=5 and K=7 users, respectively. The gains are robust in the sense that they persist even with noisy quantum resources, e.g., an exponential (in K) capacity advantage from quantum entanglement assistance remains available even if each entangled qubit independently depolarizes completely with probability $\approx$ 30%. The gains are based on quantum entanglement assistance provided only to the transmitters.

We propose a unified framework for dimension reduction of partial differential equations posed on thin domains. Our approach combines three complementary ingredients: a careful boundary-condition analysis, an averaging-based splitting for general thin geometries, and a slow-manifold viewpoint for the resulting fast-slow system. Homogeneous Neumann conditions on the thin faces emerge as the most relevant and physical regime because they preserve the transverse zero mode and therefore lead to a genuine lower-dimensional reduced equation. For general thin domains we derive the averaged fast-slow system and isolate the geometry-induced correction term. We then formulate a splitting-based Lyapunov-Perron construction for an exact slow manifold when a suitable spectral decomposition of the slow variable is available, and we construct approximate slow manifolds and corrected reduced dynamics directly from the invariance equation by asymptotic expansion. Moreover, we propose the Schnakenberg reaction-diffusion system as a canonical test problem for comparing the full thin-domain dynamics, the averaged model, and the manifold-corrected reduced dynamics. Finally, we also extend the framework to thin tubular domains and derive the corresponding rescaled fast-slow system in curved geometry.

In this paper, we discuss a family of highly regular graphs, said to be distance-regular. We are particularly interested in the distance-regular graphs with valency three. It is known that there exist exactly 13 such graphs. Let $\Gamma$ denote a distance-regular graph with vertex set $X$. For any vertex $x \in X$, the corresponding Terwilliger algebra $T=T(x)$ is generated by the adjacency algebra $M$ of $\Gamma$ and the dual adjacency algebra $M^*=M^*(x)$ of $\Gamma$ with respect to $x$. It is known that the algebra $T$ is semisimple. By construction, the vector space $V=\mathbb{C}^X$ is a module for $T$, said to be standard. In this paper we have the following goal. For each of the 13 distance-regular graphs $\Gamma$ with valency three, we will decompose the standard module $V$ into a direct sum of irreducible $T$-modules. Using this information, we will work out the dimension of $T$.

We consider interacting vertex-reinforced random walks on a finite graph, each transitioning according to independent geometric holding times of parameter $p_i \in (0,1]$. Letting $x=X(n)$ be the vector of vertex-occupation proportions up to time $n$, the one-step transition probabilities of walk $i$ are governed by $Q^i(x,p_i)=p_i\Pi^i(x)+(1-p_i)I$, where $\Pi^i(x)$ has rows equal to a probability measure $\pi^i(x)$ on the vertex set and $I$ is the identity. Its unique invariant measure is thus $\pi^i(x)$, independent of $p_i$. Consequently, the limiting points of $X(n)$ coincide with those of the simultaneous-transition model ($p_i=1$): the solutions of $x=\pi(x)$. However, almost sure convergence is non-trivial: the standard stochastic-approximation approach requires the Clark-Kushner condition, which is not immediate since the stochastic input is biased by the walk current state. We overcome this via a decomposition of the input into a martingale and a geometrically decaying correction, establishing almost sure convergence.

We establish the Clark-Kushner condition for a large class of interacting vertex-reinforced random walks on finite graphs, where the transition matrix $Q^i(x)$ of each walk depends on the joint vector $x$ of vertex occupation proportions and may have distinct rows. This allows one to study the dynamics of the vertex occupation measure by using the tools of stochastic approximation theory. However, the standard approach fails because the noise inputs are in our case not a martingale difference: they retain memory of the previous state. Using the solution of the Poisson equation for Markov chains, we decompose the noise into a martingale difference minus the increment of a bounded process – a structure originating in Gordin's work on limit theorems for stationary processes. The key technical ingredient of our approach is a uniform geometric ergodicity bound derived from the Dobrushin contraction coefficient, which also controls the Lipschitz continuity of the solution of the Poisson equation. Our hypotheses require only that each $Q^i(x)$ be irreducible, aperiodic, and Lipschitz continuous in $x$; in particular, strictly positive entries are not assumed. Our results generalize and simplify previous arguments considered for single self-reinforced vertex-reinforced random walks.

Reduced-order modeling of high-dimensional dynamical systems is often hindered by closure effects arising from unresolved variables, which can introduce non-Markovian dependence into the resolved dynamics. Motivated by the history-dependent memory term arising in the Mori–Zwanzig formalism, we recast non-Markovian closure modeling as a sequence modeling problem and propose the Mamba-Assisted Closure (MAC) framework. MAC employs a Mamba-based sequence model to predict the closure from the resolved trajectory and couples the learned closure with the reduced-order governing equations through a numerical integrator to advance the resolved variables in time. During training, the selective scan mechanism in Mamba enables efficient parallel sequence processing with linear scaling in sequence length, while autoregressive inference proceeds through recurrent state updates at essentially constant per-step cost. We evaluate MAC on four benchmark systems with complementary characteristics: the viscous Burgers' equation, the chaotic two-scale Lorenz '96 system, the 3-bus DeMarco–Zheng power-grid system, and the dispersive Korteweg–de Vries equation. Across these benchmarks, MAC consistently improves predictive accuracy and long-time rollout stability relative to the comparison models, demonstrating an effective and computationally scalable approach to non-Markovian closure modeling.

On a platform with many sellers, should a pricing algorithm explicitly model competitors' prices when learning demand? Classical arguments suggest that ignoring competitors induces model misspecification and inefficiency, yet findings from algorithmic collusion suggest that ignoring competitor prices may, surprisingly, facilitate collusive outcomes and improve profits. We study this problem in a competitive market with unknown noisy demand, in which sellers repeatedly set prices, either incorporating competitor prices in learning their demand models (informed), or ignoring them (oblivious). We show that, relative to a monopolist, an oblivious seller in a competitive market must conduct more aggressive price exploration to compensate for the loss of dynamic competitor information. When all sellers are oblivious, prices converge to the competitive outcome under persistent exploration, while a continuum of pseudo-equilibria arises when exploration is "insufficient." In markets with a mix of oblivious and informed sellers, the informed strictly out-earn the oblivious. In game-theoretic terms, the unique Nash equilibrium is the all-informed market, in which prices converge to the competitive outcome efficiently, and oblivious modeling does not robustly lead to collusive patterns.

Real-world vehicle routing and scheduling problems involve complex operational rules and feasibility constraints typically formulated as mixed-integer linear programs (MILP). However, optimization tools are built around a fixed set of hard-coded constraints, while in practice this set evolves as new rules or preferences emerge, seasonally or permanently. Updating it requires modeling and operations research skills that planners rarely have, so generated plans are routinely adjusted by hand based on practical knowledge. Building on recent work that uses machine learning to recover such hidden constraints, we propose a data-driven constraint-learning approach that trains three complementary predictors, a Graph Neural Network (GNN), a decision tree, and a linear regression, on historical execution data from a log-truck routing and scheduling problem ($\mathcal{LTRSP}$), and embeds each inside a MILP through linearized constraints. We further introduce a stacking mechanism that combines all three within a single augmented optimization problem (AOP), letting the solver endogenously select the most reliable predictor for each decision. On real-world industrial data, each predictor already improves feasibility, but the stacked embedding consistently achieves the lowest objective degradation: it (i)~satisfies the operational rules on unseen instances with smaller degradation than any single-model variant, (ii)~picks the most appropriate predictor per decision without prior knowledge of the rule's nature, and (iii)~reduces daily manual adjustment effort while remaining tractable for daily use. Beyond this application, the framework enables optimization tools that adapt to evolving practice without recurrent manual remodeling.

Let $\mathbb{F}_q$ be a finite field of characteristic $p$ and $\pi\colon Y\to X$ be a finite $\mathbb{F}_q$-morphism of separated $\mathbb{F}_q$-schemes of finite type. Suppose $\pi$ is generically Galois with group $G$ of prime order $r\neq p$. We determine the mod-$r$ reduction of the zeta function of $Y$ in terms of the zeta function of $X$ and the branch locus $Z\subset X$ of $\pi$. We give applications to curves and to numerators of hyperelliptic/superelliptic curves.

In this paper, we study how the Frobenius–Perron dimension of finite-dimensional algebras behaves under tensor products and related constructions. We prove that Frobenius–Perron dimension is super-additive under tensor products and is additive whenever one tensor factor is local. In particular every non-negative integer occurs as a Frobenius–Perron dimension. We further show that the invariant equals $1$ for every representation-infinite cycle-finite algebra, such as a tame concealed or tubular algebra, and we determine it on the grids $\mathsf{k} A_m\otimes_{\mathsf{k}}\mathsf{k} A_n$, where it is $0$, $1$, or $\infty$ according to representation type. Finally we treat skew group algebras of local algebras, for which a McKay quiver computation gives a lower bound and shows that the dimension can jump from finite to infinite.

We conclude the classification work done in the two previous papers of the same name. Here we add flexibility to the construction, thereby viewing the graphs in full generality. Our goal, as ever, is to determine which graphs do or do not occur as the prime character degree graph of a solvable group.

We continue to investigate properties of the function $M_\nu(z)$ which is associated with the Macdonald function $K_\nu(z)$ in terms of the corresponding Fourier integral. In particular, recurrence relations for this function and its derivatives are obtained, involving properties of the associated Laguerre polynomials. Multiple orthogonal polynomials related to the scaled Macdonald-type weights $ \hat{\rho}_{\nu}(x)= 2 x^{\nu/2} M_\nu\left(2\sqrt x\right), x >0$ are investigated.

We study the dynamics of gradient descent in the Edge of Stability regime, where the learning rate is large enough to induce persistent oscillations in the trajectory, which has been linked to better generalization performance. We introduce the mean–fluctuation dynamics, a tractable continuous-time model coupling the window-averaged trajectory to its fluctuation covariance. Among our contributions, we rigorously derive this model from gradient descent in a sharp-valley framework, characterize its stationary states and their linear stability, and establish precise connections with other effective dynamics. Numerical experiments illustrate these predictions and their finite-time limitations. We also study our model in the overparametrized regime of wide two-layer networks at a fixed learning rate, where we rigorously derive a kinetic equation describing weights and their fluctuations as a Wasserstein-2 gradient flow, for which we prove well-posedness, a mean-field limit, and conditional convergence results.

Irreversible perturbations accelerate the convergence of Langevin dynamics, breaking detailed balance while preserving the invariant measure. The design of optimal irreversible perturbations has been studied in the continuous-time Gaussian setting, but extensions to non-Gaussian target distributions, and the impact of time discretization on the design of optimal perturbations, have not been well understood. Numerical discretizations of Langevin dynamics introduce bias, which is typically exacerbated by irreversible perturbations; handling this interaction demands a joint treatment of acceleration and accuracy. This paper develops a systematic framework for optimizing position-independent irreversible perturbations of the unadjusted Langevin algorithm (ULA). We formulate a constrained optimization problem that simultaneously accounts for mixing efficiency and discretization bias, where the former is characterized by a spectral gap analogue and the latter is quantified via a weighted expected squared jump distance. Within this framework, we derive an explicit characterization of the optimal position-independent irreversible perturbation. Extensive numerical experiments demonstrate that our design yields faster convergence with controlled bias, and improves mean squared estimation errors compared to other choices of irreversible perturbation.

A cataclysm deformation, that shears and twists a given Anosov representation according to data known as a twisted transverse cocycle, is an intuitive and powerful tool for studying Anosov representations. We show that if a sequence of twisted measured laminations converges weakly, the sequence of corresponding cataclysm deformations on the space of Anosov representations converges uniformly on compact sets. This result leads to two applications. First, we obtain an extension of the Goldman product formula. Second, we consider strongly dense representations, introduced by Breuillard–Green–Guralnick–Tao and Long–Reid. Using cataclysm deformations, we show that, for a split real form $\mathsf{G}$ whose Weyl group contains $-1$, the set of strongly dense $\mathsf{G}$-Hitchin representations is not open in the $\mathsf{G}$-Hitchin component.

We study a pair of exactly solvable, isospectral fermion chains, one strongly interacting and one quadratic, that nevertheless display remarkably different phase structures and operator dynamics. A nonlocal nonlinear unitary transformation maps one onto the other while preserving the entire many-body spectrum and converting local fermion operators into extended many-body strings. Thus, operators that evolve within a closed linear subspace in the quadratic model become interacting operators that generate increasingly higher-body terms and exhibit asymptotic Lanczos growth $b_n\propto\sqrt n$. Despite their identical spectra, the two models realize distinct phases and sharply different notions of operator complexity. Our results demonstrate that free many-body spectra and interacting operator dynamics are fundamentally compatible.

Isomorphic flops are topology-changing transitions connecting two diffeomorphic families of Calabi-Yau threefolds. They correspond to the generators of certain Coxeter groups acting on the moduli space. As a consequence of these symmetries, the prepotential of 4D $\mathcal{N} = 2$ Type IIA compactifications on such varieties must assemble into Coxeter-invariant functions. We construct a database of all Coxeter symmetries from isomorphic flops in Kähler-favorable CICYs. The action of the Coxeter group on the Kähler moduli space leaves a symmetric bilinear form invariant, which we interpret as a metric and construct its associated Laplace-Beltrami operator. We argue that the Coxeter-invariant functions featured in the prepotential solve the Helmholtz equation with this Laplacian, and that the prepotential can then be resummed into a decomposition in terms of eigenfunctions of the Laplace-Beltrami operator. The convergence rate of the raw orbit sums of worldsheet instanton contributions and the resummed expressions are complementary, with the latter sharply localizing around the first few terms in the interior of the moduli space.

A QFT is said to be relative if it lives at the boundary of a topological QFT in one higher dimension. We develop a general framework for working with noninvertible symmetries of relative theories in two spacetime dimensions, extending several well-known results for absolute QFTs. We emphasize various new features which arise in the relative setting, including the role of topological surfaces of the bulk, and the appearance of hypergroups and certain generalizations of tube algebras known as dome algebras. Our formalism is particularly well-suited for studying rational chiral algebras, where it predicts that finite symmetries are in explicit one-to-one correspondence with conformal embeddings of finite index. We describe several implications of our framework for absolute theories. First, we explain how to "glue" together symmetries of the left- and right-moving chiral algebras of a 2D CFT to produce topological line defects of the full theory. Second, we derive a precise correspondence between boundary conditions of a 2D CFT and symmetries of its chiral algebra. This correspondence has several structural corollaries: in diagonal rational CFTs, we demonstrate that the topological line defects of the theory act transitively on its boundary conditions, and further that the identity Cardy state has the smallest $g$-function amongst all boundary conditions, including those which only preserve Virasoro symmetry. We conclude by illustrating our results in a variety of examples. For instance, we show that, if there exists a rational chiral algebra with central charge $c=8$ whose modular tensor category is the Drinfeld center of the Haagerup fusion category, then it must arise as the fixed points of a rank-2 hypergroup acting on the $SU(3)_1\otimes (E_{6})_1$ chiral algebra.

We establish the first sharp thresholds for low-degree polynomial tests in planted-vs-planted settings, where the goal is to determine with vanishing error which of two structured planted mechanisms generated the observed data. We prove matching low-degree upper and lower bounds for counting communities in the planted submatrix and planted dense subgraph models. The resulting testing threshold coincides, down to the sharp constant, with the known low-degree recovery threshold. In contrast, the task of weak testing, where the goal is to outperform random guessing, does not have a sharp threshold but rather a smooth transition, which we identify. To prove our results, we develop a framework for planted-vs-planted testing that builds on a latent-variable expansion originating in low-degree recovery and employs new methods to identify and prune non-signal contributions.

Uncrewed aerial vehicles (UAVs) are increasingly considered as aerial platforms capable of providing both sensing and communication services, representing a promising paradigm for intelligent transportation systems. This paper investigates the optimal time allocation for a UAV-enabled integrated sensing and communication (ISaC) system operating in the X-band for vehicular networks. We analyze the trade-off between sensing accuracy and communication performance under practical UAV constraints and fading effects, considering both single-shadowing and double-shadowing channel models. An optimization framework is developed to allocate time between sensing and communication while guaranteeing minimum communication rates and sufficient sensing reliability. Simulation results demonstrate adaptive time allocation strategies, highlighting how UAV-to-ground channel conditions and target distances influence the balance between sensing and communication in smart mobility scenarios.

We exhibit a counterexample to a conjecture of Bhatt–Lurie on the cohomological dimension of the Hodge–Tate locus for regular local rings. The example arises from a non-excellent discrete valuation ring constructed by Datta–Smith, closely related to an earlier example of Bosch–Lütkebohmert–Raynaud. We also explain how the same mechanism yields broader families of counterexamples, while the expected bound is recovered under an excellence hypothesis.

We present a concise, elementary proof that the second smallest modulus of a covering system must be bounded, simplifying the known results through a maximality argument.

This paper investigates the joint resource block group (RBG) scheduling and beamforming optimization problem for weighted sum-rate (WSR) maximization in multi-cell multiuser multiple-input multiple-output (MU-MIMO) downlink networks. While the Fast Fractional Programming (FastFP) framework provides a reliable model-driven solution, it suffers from conservative continuous beamforming updates and prohibitive computational overhead during the discrete RBG matching phase. To address these bottlenecks, we propose a joint deep unfolding framework comprising two core modules: P-Net and K-Net. Specifically, P-Net learns an adaptive relaxation factor along the FastFP direction, strictly bounded within an ascent-preserving interval to accelerate convergence while retaining stationary-point guarantees. Meanwhile, K-Net learns a long-horizon priority policy to guide a low-complexity greedy assignment, maintaining high assignment quality while bypassing the computationally expensive Hungarian matching. Both networks leverage analytical algorithmic priors and utilize recurrent parameter sharing, enabling flexible inference beyond the training horizon. Extensive simulations demonstrate that the proposed joint framework achieves higher WSR and faster execution times than conventional model-driven baselines, while generalizing robustly across unseen network scales and channel conditions without retraining.