Papers

Krieger's classical embedding theorem gives necessary and sufficient conditions for embedding a subshift into a mixing shift of finite type (SFT) as a proper subshift. The same result does not hold if one replaces mixing SFT by a mixing sofic shift. In this paper, we generalize Krieger's conditions to give necessary and sufficient conditions for embedding a subshift into a mixing (in fact irreducible) near Markov sofic shift (a special conjugacy-invariant class of sofic shifts). We also show that if the subshift to be embedded is irreducible sofic, then the conditions are finitely decidable.

We study the dimension theory of expanding rational semigroups. An expanding rational semigroup is naturally described by a skew product on the product of the symbolic space and the Riemann sphere. For an invariant Borel probability measure of this skew product, we consider its disintegration over a symbolic factor and study the push-forwards of the conditional measures under the second coordinate projection. We prove a Ledrappier–Young type formula for the projected conditional measures. In particular, if the original invariant measure is ergodic, then these projected conditional measures are exact dimensional for almost every fiber. Applying this result to the trivial factor yields exact dimensionality of the projected invariant measure. This result can be viewed as a backward analogue of the result of [D.-J. Feng and H. Hu. Comm. Pure Appl. Math. 62 (2009), no. 11, 1435–1500].

Let $P=P_n^{(\alpha,\beta)}$ be the $n$-th degree Jacobi polynomial, which is orthogonal in $[-1,1]$ with respect to the weight function $(1-x)^{\alpha}(1+x)^{\beta}$, $\alpha,\beta>-1$. For parameters $(\alpha,\beta)$ satisfying either $\alpha\geq\beta\geq 1/2$ or $\alpha\geq 1/2$, $\beta=-1/2$, we prove the inequality $$ P(1)-P(x)\geq P^{\prime}(x)\,(1-x),\quad x\in [0,1], $$ which may be viewed as a complement to Finite Increment Theorem for Jacobi polynomials.

We study Riemann solitons and $\eta$-hyperbolic Ricci solitons on Bochner-flat Lorentzian Kähler spacetime manifolds. Under the Einstein field equations with cosmological constant and perfect fluid assumptions, explicit formulas for the soliton parameter are derived, yielding criteria for shrinking, steady, and expanding behaviors. Several physically relevant models, including dark fluid, stiff matter, dust, and radiation, are analyzed. We show that Bochner-flat Lorentzian Kähler spacetimes are Einstein and investigate the resulting geometric and dynamical consequences. In the context of generalized Robertson–Walker spacetimes, we obtain constraints on the warping function and classify soliton solutions. Global properties such as geodesic completeness, singularity formation, and stability are also examined.

This paper studies a singular dividend control problem for a firm with heterogeneous shareholders whose discount rates follow a given distribution. The central planner aggregates expected discounted payoffs using an ambiguity aggregation function $phi$, which captures shareholder heterogeneity and ambiguity attitudes but also leads to time inconsistency. To address this issue, we seek a time-homogeneous equilibrium dividend law characterized by a partition of the state space into waiting and dividend-paying regions. We provide a rigorous mathematical characterization by proving a verification theorem and deriving necessary conditions for the equilibrium law. We then analyze barrier-type equilibria, showing non-existence for a class of aggregation functions that includes power-type and logarithmic aggregation functions, and establishing existence and uniqueness under linear and exponential aggregation. In the linear case, the bounded-rate equilibrium is shown to converge to the singular barrier-type equilibrium as the dividend rate bound tends to infinity. Numerical examples illustrate the effects of discount-rate heterogeneity and ambiguity aversion on the equilibrium barrier.

The primary aim of this paper is to provide an explicit construction of Kontsevich graphs whose integrals give certain multiple zeta values. Furthermore, by using this construction, we explicitly determine the weights of $\mathbb{Z}$-linear combinations of graphs whose integrals yield normalized multiple zeta values of a given weight. The methods employed in this paper are based on those introduced by Ritland, and our results provide a refinement of his results.

This work introduces a class of abstract Orlicz-Morrey spaces endowed with a ball-basis on general measure spaces and defines the associated concept of \(\Psi\)-bounded oscillation operators. Within this framework, we establish pointwise estimates and norm inequalities for these operators via sparse domination techniques. As applications, we verify that this class of \(\Psi\)-bounded oscillation operators includes maximal operators and Carleson-type operators on general measure spaces, as well as \(\omega\)-Calderón-Zygmund operators and intrinsic square operators on \(\mathbb{R}^n\), thus providing a generalization of certain classical Orlicz-Morrey spaces and their associated operator theory.

We provide an upper-bound for the Lévy concentration function: $$ Q_{S}(\varepsilon):= \sup_{x \in\mathbb{R}}\mathbb{P} (x < S \leq x+\varepsilon) $$ where $S$ is a weighted sum of noncentral chi-square random variables: $$ S:= \sum_{k=1}^\infty \lambda_k (Z_k^2 - 1) + \mu_kZ_k $$ Here, $\{Z_k\}_{k=1}^\infty$ is a sequence of independent standard Gaussian random variables and $\{\lambda_k\}_{k=1}^\infty, \{\mu_k\}_{k=1}^\infty$ are real valued, square summable sequences. Random variables of this type often appear as limiting distributions of second order U-statistics. Our bound is adaptive, in that it recovers (up to constant factors) Gaussian type concentration function estimates if $\|\lambda\|_2$ is negligible compared to $\|\mu\|_2$ and chi-square estimates if $\|\mu\|_{2}$ is negligible compared to $\|\lambda\|_2$. Our bound generalizes existing bounds in various ways. In particular, we make no assumptions regarding the number of nonzero $|\lambda_k|$ or the size of the minimal $|\lambda_k|$, nor do we make any assumptions on the signs of $\lambda_k$. Finally, we apply our bound to some examples of interest, specifically quadratic forms that arise in limit theorems for second-order U-statistics.

A Littlewood polynomial is a polynomial whose coefficients lie in $\{- 1, +1\}$. While the majority of roots of a Littlewood polynomial of large degree are near the unit circle, numerical experiments suggest that when plotting the roots of all Littlewood polynomials of a given large degree, striking fractal structures appear away from the unit circle. These fractals resemble the attractor of a certain iterated function system and are known as dragon curves. In this note, we provide a rigorous explanation of this phenomenon, along with an analysis of a random variant, saying that such fractal behavior is typical.

We prove a three-dimensional positive-cone continuation criterion for the stress-diffusion-free Oldroyd-B system on the periodic torus. Writing the positive conformation tensor as A = exp(B), we show that finite-time breakdown of a strong H^s solution, s > 5/2, can occur only through loss of the logarithmic spectral envelope of A or divergence of the endpoint vorticity clock given by the time integral of the B^0_{infty,1} norm of curl u. The proof combines compact positive-cone envelopes, endpoint Biot-Savart estimates, and high-order logarithmic conformation estimates, without using stress diffusion. We also derive a positive-cone Reynolds admissibility criterion with an exact residual-work cost. The least L^2 conformation residual needed to pay positive pressure-free residual work is determined by the entropy-dual lever G = I - A^{-1}, and this cost degenerates quantitatively near the equilibrium A = I. Together, the two criteria identify the same positive-cone obstruction in the strong and relaxed regimes: before breakdown one must control the endpoint flow clock on a compact logarithmic cone, while after passage to a relaxed description positive residual work must be paid for by an exact entropy-dual conformation defect.

In this paper, we introduce Cesàro function spaces over $p$-adic fields and investigate their fundamental properties, such as the dilation operator and the Minkowski-type integral inequality. We establish boundedness result for $p$-adic Hardy–Hilbert-type integral operators acting on $p$-adic Cesàro function spaces, and as an application we derive $p$-adic analogue of the Hardy inequality, the Hilbert inequality, and the Hardy-Littlewood-Pólya inequality. Furthermore, we define the $p$-adic analogue of the Erdélyi–Kober fractional integral operators and prove their boundedness on $p$-adic Cesàro function spaces with the help of the obtained boundedness result.

A center-fed pinching antenna system (C-PASS)-enabled uplink environment sensing framework is proposed. Through the center-fed framework, doubled degrees of freedom is achieved compared to conventional end-fed PASS. Based on this, we consider an uplink sensing scenario, in which a linear inverse model is developed to reconstruct the environment through signals scattered by the environment object. In the proposed framework, the distance between the feed points for stable separation of the received signals is characterized in closed form. Furthermore, Ziv-Zakai bound (ZZB) expressions for the mean-squared reconstruction error are derived for C-PASS and end-fed PASS. Based on these theoretical results, it can be proved that C-PASS achieves a strictly lower reconstruction error bound than conventional PASS for uplink environment sensing. Finally, numerical results validate the accuracy of the derived ZZB expressions and 1) demonstrate that C-PASS provides more stable separation of the received signals, and 2) confirm the consistent performance advantages of C-PASS.

In this work, to address the challenge of joint channel estimation and data detection (JED) for orthogonal delay-Doppler (DD) division multiplexing (ODDM) in doubly selective channels, we propose an orthogonal approximate message passing (OAMP)-aided JED (OAMP-JED) receiver. We first formulate a bilinear cross-domain JED model, which can be linearized into separate channel estimation and data detection subproblems. The proposed OAMP-JED receiver alternately executes two OAMP modules for these subproblems, effectively coupled through a variational noise term to account for model uncertainty. Leveraging OAMP's error orthogonality, we derive closed-form scalar-variance updates to enable efficient and principled soft information exchange between the modules, thereby mitigating error propagation during JED. Simulation results show that, for both uncoded and coded ODDM, OAMP-JED achieves a lower bit error rate (BER) than benchmark schemes. Moreover, its BER performance closely approaches that of OAMP with perfect CSI.

We present amalgam-type characterisations of compactness in quasi-Banach function spaces. We further treat, in the same spirit, several related topics, namely convergence, (uniform) absolute continuity of the norm, and almost-compact embeddings. We also treat the case of rearrangement-invariant spaces and show that most of our conditions can be formulated in terms of the non-increasing rearrangement and the representation space.

This paper presents a formalization of line search methods in the Lean 4 theorem prover. Our goal is to advance machine verification of nonlinear optimization theory by translating standard textbook definitions and convergence arguments into rigorous Lean code. We formalize fundamental notions related to gradient descent and descent directions, adaptive step-size selection via backtracking line search, and several classical line search criteria, including the Armijo, Goldstein, and Wolfe conditions, as well as nonmonotone variants. We further formalize a key convergence result, namely the Zoutendijk theorem, which plays a central role in the global convergence analysis of gradient-based iterative methods. By providing machine-checkable definitions and proofs for line search theory, this work complements existing formalizations of first-order optimization methods and establishes a foundation for the verified development of more advanced algorithms in nonlinear programming.

We establish a family of linear relations for the coefficients of the Tutte polynomial of a semimatroid. We also introduce ranked central sets, a mild generalization of semimatroids, and prove that the same identities hold in this broader setting. The proof relies on the specialization $x=\frac{z}{z-1}$, $y=z$ along the hyperbola $(x-1)(y-1)=1$; under this specialization, the edge subgraph expansion over central sets reduces to the $f$-polynomial of the central set complex. As applications, we obtain explicit formulas for Tutte polynomials of affine hyperplane arrangements and for balanced Tutte polynomials of biased graphs, whose central sets are central subarrangements and balanced edge sets, respectively. Our results recover the generalized Brylawski's identities for Tutte polynomials of ranked sets, matroids and graphs.

As part of Lutwak's broadening of the Brunn-Minkowski theory, and extending the notion of affine quermassintegrals and dual curvature measure discussed by Milman, Yehudayoff and Huang, Lutwak, Yang and Zhang, centro-sectional measures with real parameter q have been recently introduced by Cai, Leng, Wu, Xi. In this paper, we introduce the Lp cross sectional Minkowski problem analogously to the Lp dual Minkowski problem formulated by Lutwak, Yang and Zhang. We solve the Lp centro-sectional Minkowski problem for p>1 and q>0, discuss the regularity and uniqueness of the solution, and prove Lp Brunn-Minkowski-type inequalities when $p$ is relatively large.

Using operator inequalities, we study a Wold-type decomposition of covariant representations. Building on this decomposition, we prove a Beurling-type theorem showing that every nonzero invariant subspace is uniquely determined by its wandering subspace. Our results extend classical theorems of Beurling and subsequent developments for left-invertible operators to the setting of covariant representations of $C^*$-correspondences, providing a unified framework for invariant subspace theory under operator inequalities.

We characterize the kernel of Iwahori-Whittaker averaging in microlocal terms. Applying this to automorphic sheaves, we generalize Faergeman and Raskin's theorem that anti-temperedness is equivalent to having irregular singular support. Moreover, using a Radon transform argument of Bezrukavnikov and Morton-Ferguson, we extend the tilting property of Whittaker averaged central sheaves to integer coefficients.

This paper studies polynomial extensions and localization of HMCM rings, where HMCM means Cohen–Macaulay in the sense of Hamilton–Marley, a notion for non-Noetherian rings. We show that the HMCM property is not preserved either under polynomial extensions or under localization in general. More precisely, we construct an HMCM ring $A$ such that $A[X]$ is not HMCM, and an HMCM ring $B$ with a prime ideal $\mathfrak q$ such that $B_{\mathfrak q}$ is not HMCM. We also prove a positive result: polynomial rings over stably coherent rings of finite weak global dimension are HMCM. In addition, we revisit polynomial grade, give a counterexample to the “Moreover” assertion in [HM07, Proposition 2.7], and study localization of torsion-free modules via regular saturation and Krull primes.

Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^N$, $W_0^{1,N} \left( \mathbb{B} \right)$ is a standard Sobolev space. Zhang proved the extremal function of the Moser-Trudinger inequality as follows, \begin{align*} \int_{ \mathbb{B}} h_\epsilon(x) e^{ \alpha_N \left( 1 + \epsilon \right) |u_{\epsilon}|^{ \frac{N}{N-1} } } dx, \quad u_{\epsilon} \in W_0^{1,N} ( \mathbb{B} ) \cap \mathcal{S}, \end{align*} where $\alpha_N = \omega_N^{ \frac{1}{N-1} }$, $\omega_N $ is the area of the unit sphere in $\mathbb{R}^N$(see \citep{26}) . In this paper, we consider the compactness of the sequence $\{ u_{\epsilon} \}_{\epsilon} $ and prove that it has a subsequence converging to a function in $C^1 \left(\overline{ \mathbb{B}} \right)$.

Mathematical knowledge is organized around statements and their dependencies, but this structure is exposed unevenly: informal papers cite mostly at the document level, while formal libraries record fine-grained dependencies over a much smaller body of mathematics. We introduce TheoremGraph, a unified statement-level dependency graph spanning both informal and formal mathematics. On the informal side, we parse 11.7M theorem-like environments from mathematics arXiv and recover 18.3M candidate directed dependencies, each labeled by the extractor that proposed it so downstream users can trade coverage for precision. On the formal side, we release LeanGraph, a Lean 4 elaborator-level extractor producing 388,105 declaration nodes and 11.3M typed edges across 25 Lean projects. We bridge the two graphs by embedding generated natural-language slogans into a shared semantic space, linking related statements across papers and across the informal/formal divide; an LLM judge affirms 47,952 such matches above a 0.8 cosine floor, with the judge-acceptance rate rising from 48% across the floor to 87% in the >=0.9 tier. On formal concept retrieval, our name-and-signature representation with graph expansion comes within 0.5pp of LeanSearch v2's reranked Recall@10 (0.775 vs. 0.780) without an LM reranker. We release the dataset, extractors, HTTP API, and MCP interface as infrastructure for mathematical search, attribution, and retrieval-augmented reasoning, available at theoremsearch.com and huggingface.co/datasets/uw-math-ai/theorem-matching.

Sequential contextual stochastic programs model real-time decision systems in which each time epoch commits to an action under uncertainty whose consequences propagate into future decisions. In many practical contexts, these programs require obtaining solutions rapidly as new information becomes available. These problems can be represented through scenario approximations to be solved by off-the-shelf optimization solvers, which achieve high decision quality offline but typically run in seconds to minutes per instance, falling short of the sub-second responses that peak periods of planning require. This paper develops a learning-based optimization proxy: a scenario-embedded neural network trained offline on solver-generated labels, paired online with a decoder that enforces feasibility, replacing the per-epoch solve with a single forward pass. The framework is specialized to omnichannel order fulfillment, where each arriving order requires a sub-second assignment of products to distribution centers and carrier services under stochastic delivery times and future demand. A two-stage contextual stochastic program is introduced to formulate this problem, and its contextual sample average approximation (C-SAA) supplies the offline labels, while a composite training loss combines label imitation, a constraint-violation penalty, and self-supervised cost alignment. In a calibrated simulator built from JD.com transactional records, a detailed computational study is provided. The proxy reduces decision latency by roughly 2800x relative to the online finite-sample C-SAA reference and improves over it by 3.3% in realized fulfillment cost. Relative to established fulfillment policies, the proxy lowers total realized cost by at least 10.7% and roughly halves the late-delivery rate.

State abstraction plays a key role in scaling reinforcement learning to complex but structured systems. In studying such systems, a wide range of behavioral structures have been studied in reinforcement learning, including value functions, invariants, bisimulation relations, and behavioral metrics. However, a general principle for determining what structures are provably preserved under state abstraction is still lacking. In this paper, we present a unified framework for defining and analyzing behavioral structures in reinforcement learning. Our framework provides a compositional way to specify behavioral semantics based on local, one-step descriptions of system dynamics. Using this framework, we establish results showing how behavioral structures can be safely transferred between abstract and concrete systems. We further show how to construct quantitative metrics from logical behavioral semantics with soundness guarantees. Together, these results provide a principled foundation for reasoning about behaviors under state abstraction in reinforcement learning and offer reusable definition and proof principles for a broad class of behavioral structures in reinforcement learning.

We tackle the challenging problem of stochastic weakly convex optimization subject to mixed (equality and inequality) expected-value constraints. While optimal $\mathcal{O}(\epsilon^{-3})$ sample complexity algorithms exist for unconstrained weakly convex problems, dealing with complex functional constraints typically requires cumbersome multi-loop penalty or augmented Lagrangian methods, which suffer from high inner-loop complexity and sensitive parameter tuning. To bridge this fundamental gap, we propose the primal-dual Halpern-PAGE (PD-HP) algorithm. As a purely single-loop method, PD-HP completely bypasses the computational burden of nested iterations. At each step, it merely requires solving a simple strongly convex surrogate subproblem alongside a straightforward dual projection, making it exceptionally efficient and convenient to implement. Crucially, we prove that this computationally lightweight algorithm achieves the optimal $\mathcal{O}(\epsilon^{-3})$ sample complexity for mixed-constrained stochastic weakly convex problems, successfully matching the theoretical lower bounds. Furthermore, when the primal domain is a compact polyhedral convex set, we establish the deterministic stability of the dual multipliers by exploiting the generalized Mangasarian-Fromovitz constraint qualification (MFCQ) alongside Hoffman's error bound. This ensures that our optimal complexity bound holds strictly under the standard, unbounded KKT residual metric without any theoretical gaps or artificial residual truncations.

This paper studies the secrecy capacity of a class of degraded broadcast channels in the presence of an external eavesdropper, where a transmitter aims to deliver two independent confidential messages to two legitimate receivers. The transmitter is assumed to have non-causal access to the channel state information (CSI), and each legitimate receiver possesses prior knowledge of the other receiver's message, referred to as receiver message side information (RMSI). We consider two distinct scenarios: complementary RMSI, where each receiver knows only the other's message, and non-complementary RMSI, where the side information does not perfectly align. For both scenarios, we derive novel inner bounds on the achievable secrecy rate region and present tight outer bounds, establishing the secrecy capacity region for the considered degraded channel settings. Unlike prior works, which primarily address general broadcast settings without secrecy constraints or omit key interactions between RMSI and CSI, our results provide a complete characterization of the secure communication limits under these conditions. Moreover, we extend our analysis to the Gaussian degraded broadcast channel, highlighting the pivotal role of CSI in enhancing secure transmission performance. Our findings demonstrate that the combination of RMSI and CSI can be strategically leveraged to expand the secrecy capacity region, thus offering new insights into secure multiuser communication system design.

We study sharp exponents in inequalities for pairs of finite geometric blocks. We characterize exactly when the endpoint $t=1$ determines the optimal exponent and compute the exponent that is uniform in the length of one block. For a two-term first block the answer is $p_0=\log 4/\log 6$. This yields a uniform two-slice max-convolution inequality and, for every $m,d\ge1$, the dimension-free mixed-alphabet sumset bound \[ |A+B|\ge (|A||B|)^{p_0}, \qquad A\subset\{0,1\}^d,\quad B\subset\{0,1,\ldots,m\}^d. \] For every $m\ge2$, the exponent $p_0$ is best possible; for $m=1$, a larger exponent is available.

We develop a general probabilistic framework for analyzing propagation of chaos in transport ensemble filters (TEFs), a broad class of interacting particle systems that are used to approximate the sequence of state distributions in hidden Markov models given a history of observations. This class of transport-based filtering algorithms includes the widely used ensemble Kalman filter (EnKF), based on affine updates at each filtering step, as well as the ensemble stochastic map filter (EnSMF), which employs nonlinear updates. For this class, we identify the limiting mean-field dynamics. We then establish non-asymptotic, high-probability, pathwise Wasserstein convergence of the interacting particle system to an i.i.d. ensemble drawn from this mean-field limit at the Monte Carlo rate. Convergence to the mean-field law itself follows with the usual dimension-dependent empirical Wasserstein rate. The proof combines a synchronous coupling construction with stability of moments and tails under conditioning, together with quantitative estimates for the propagation of the underlying dynamics through the interacting particle system. Applying our theory to both the EnKF and the EnSMF yields the first non-asymptotic, high-probability convergence guarantees for TEFs.

This paper introduces a novel and general framework for constructing arbitrary-order, energy-stable, and decoupled schemes for the anisotropic phase-field dendritic crystal growth model. The proposed framework combines a time-dependent auxiliary variable, an algebraically stable Runge-Kutta method, and a matrix diagonalization technique. This unique combination yields a scheme that is proven to be unconditionally energy-dissipative while requiring only the solution of independent, constant-coefficient elliptic equations at each time step, which can be solved in parallel. The auxiliary variable is recovered from a uniquely solvable $q\times q$ algebraic system. We provide a rigorous proof of $q$th-order convergence in time for a $q$-stage Runge-Kutta method within a fixed Fourier-Galerkin space. Numerical experiments in 2D and 3D not only confirm the theoretical convergence and energy stability but also demonstrate the efficiency of the decoupled approach and provide a detailed investigation of key parameters' effects on dendritic morphology.

We audit a finite-scale program for the local regularity problem of the three-dimensional incompressible Navier–Stokes equations. The program develops critical ledgers, coarse-grained defect decompositions, pressure–flux work identities, quotient cleanings, and bad-scale counting mechanisms. These results form an obstruction calculus: they locate how Caffarelli–Kohn–Nirenberg badness may be transported, hidden, or reproduced across scales, but they do not by themselves provide a coercive estimate excluding a surviving obstruction. We prove a resolution lemma separating full CKN badness into coarse badness and subfilter residual, and show that no unconditional single-scale domination by a signed combined-work detector is available. The audit therefore identifies the next necessary target: a filtered stretching–diffusion estimate, including subgrid forcing, leakage, pressure tails, and direction-incoherence defects, capable of converting the existing decomposition theory into a genuine regularity or obstruction-exclusion mechanism.