Papers

We study the spectrum of the $f$-Laplacian on complete gradient Ricci shrinkers. Upper and lower bounds for the $k$-th eigenvalue are established in terms of the volume growth rate. Both bounds are shown to be sharp in the exponent. The method extends to $f$-Laplace-type operators on vector bundles; as an application we obtain explicit upper bounds for the Betti numbers.

Since Krengel's work [Kre78] in 1978, it has been widely known that no effective rate of convergence exists in the ergodic theorem. For toral translations, however, by choosing appropriate weights one can accelerate the convergence of ergodic averages to an exponential rate, but this intuitively requires both highly nonresonant frequencies and very regular observables. In this paper, we uncover a new phenomenon: even for any given nonresonant frequency, there exists a non-trivial family of weights and observables of low regularity such that the weighted Birkhoff averages along quasi-periodic orbits converge at a quantitative, uniform, and exponential rate. This not only yields a finer understanding of the deep interaction between nonresonance and regularity in ergodic theory, but also stands as a weighted counterpart to a Yoccoz-type result [Yoc80,Yoc95].

We study destination inference for adversarial unmanned underwater navigation in a spatially varying current field. The red vehicle is modeled as approximately following a Hamilton-Jacobi (HJ) time-optimal path toward an unknown destination, while a blue vehicle observes a noisy realization of that trajectory. We derive a continuous-time likelihood model for this problem and obtain a closed-form local maximum likelihood estimator, a constant-memory multi-period estimator, and an asymptotic Cramér-Rao efficiency result. The Fisher information is governed by a combined score kernel with two additive components, a policy-mean sensitivity and a reference-path sensitivity, and a multiplicative drift sensitivity whose leading geometric contribution is a contraction of the current Hessian with the Jacobi field of the HJ characteristic flow. All three sensitivities are induced by that Jacobi field, yielding a computable link between current-field geometry and destination identifiability. We further extend the framework to a sweep-dependent observation model and formulate an active sweep-design problem for the blue team. Numerical experiments in vortex and channel-shear currents validate the Cramér-Rao prediction and illustrate how current geometry determines which destinations can be reliably inferred.

In this paper, we identify the dual spaces of variable Dunkl–Hardy spaces using a discrete Dunkl–Calderón reproducing formula and the Frazier and Jawerth's method adapted to the variable exponent setting. Furthermore, we prove a $T1$ theorem for Dunkl–Calderón–Zygmund operators acting on variable Dunkl–Hardy spaces and on their dual spaces via using almost orthogonal estimate, the duality result and the density argument.

The integrated sensing, communication, and computing (ISCC) system overcomes the limitations of conventional standalone architectures. Through resource sharing and collaborative design, it dynamically optimizes and jointly enhances communication, sensing, and computing performance, thereby significantly improving overall system efficiency. This work investigates the performance trade-off among secure communication rate, radar estimation rate, and computational energy efficiency in an uncrewed aerial vehicle (UAV)-assisted ISCC system. By jointly optimizing the UAV's three-dimensional (3D) trajectory, beamforming, user scheduling, and computational frequency, three optimization problems are formulated to maximize the average secrecy rate, sensing rate, and computational energy efficiency, respectively, thus establishing the system's performance boundaries under diverse scenarios. On this basis, the trade-off among security, sensing, and computation is further explored with the goal of maximizing the normalized weighted sum of the three performance metrics, which provides a theoretical basis for the performance-coordinated design of aerial ISCC systems.

We construct a dense set of homeomorphisms with infinite topological entropy whose associated pseudo-singular suspension flows have finite entropy – and indeed, arbitrarily small positive values can be achieved – showing that infinite entropy is not preserved under singular time changes. Complementing this, we prove that for a residual set of homeomorphisms, all pseudo-singular suspensions retain positive entropy. We also prove that for any $n \geq 2$, there exists a compact n-dimensional manifold admitting a minimal homeomorphism with infinite topological entropy; for such minimal homeomorphisms, a suitably chosen pseudo-singular suspension with a single singularity has zero entropy. Our results reveal that while positivity of entropy is generically stable, its infinitude is fragile under singular reparametrizations.

We establish a convex ordering result for count distributions arising from a parasite acquisition model with density-dependent deaths. If the ratio of two death-rate sequences is non-increasing, then the corresponding mean-parametrised distributions are ordered in the convex order. This yields, as a special case, the convex ordering of the mean-parametrised Conway-Maxwell-Poisson distribution in its dispersion parameter.

We investigate the initial-boundary value problem for the three-dimensional isentropic compressible Navier–Stokes equations in the upper half-space with the slip boundary conditions. We prove the global existence and uniqueness of strong solutions in the presence of vacuum and large oscillations. Although scaling frameworks for compressible flows in domains with boundaries have been developed in several settings, the global well-posedness result in the half-space remains far from complete. The system with far-field vacuum admits a natural scaling structure that preserves both the half-space geometry and the slip boundary conditions. Motivated by this scaling, we identify the following scaling-invariant initial quantity: $$ \left[ \|\rho_{0}\|_{L^{\infty}}^3 \left( \frac12\|\sqrt{\rho_0} u_0\|_{L^{2}}^{2} +\frac{1}{\gamma-1}\|P(\rho_{0})\|_{L^{1}} \right) +\|\rho_{0}\|_{L^{\infty}}^{\frac{3-\gamma}{2}} \right] \left( \|\nabla u_0\|_{L^2}^2 +\|P(\rho_{0})\|_{L^2}^2 \right). $$ Under the assumption that this quantity is sufficiently small, we establish the global well-posedness of strong solutions. This result provides a half-space counterpart of the scaling-invariant global theory for the Cauchy problem established by Wen (Adv. Math. 482 (2025), Paper No. 110628) and shows that the slip boundary condition is compatible with the system.

Let $K$ be an algebraically closed field and let $n, d \geq 2$. We show that the critical scheme of a general endomorphism of $\mathbb{P}^n_K$ of degree $d$ is an integral hypersurface. This extends a result of Ingram–Ramadas–Silverman to arbitrary characteristic. We use basic facts about polynomial rings to show that certain carefully chosen examples have absolutely irreducible Jacobian. To handle the wild case where the characteristic of $K$ divides $d$, we introduce a polynomial that imitates the homogeneous Jacobian determinant.

Many scientific and engineering systems are observed as time-indexed probability distributions whose governing dynamics are unknown and whose individual trajectories are unavailable. These settings challenge conventional system-identification approaches that rely on trajectory correspondence or prescribed evolution equations. This work presents a population-level inference framework that recovers latent stochastic dynamics directly from snapshot probability distributions by decomposing the observed evolution into an intrinsic latent stochastic process and a discrepancy transport map that captures geometric deformation between the latent and observed probability spaces. The latent dynamics are modeled using an Ornstein–Uhlenbeck process, providing a closed-form solution to the associated Fokker–Planck equation, while the discrepancy transport map is parameterized through the Knothe–Rosenblatt rearrangement with monotone neural networks. To mitigate the non-uniqueness inherent in the latent–transport decomposition, the transport map is regularized using a deformation energy motivated by hyperelasticity, promoting smooth, physically interpretable deformations while reducing unnecessary complexity. The latent stochastic model and discrepancy transport map are learned jointly through a unified optimization problem defined over probability distributions. Numerical examples involving nonlinear and multimodal distributional dynamics demonstrate that the proposed framework accurately reconstructs complex probability evolution while preserving a compact and analytically tractable latent representation. The proposed formulation provides a general framework for population-level dynamical inference and establishes a foundation for extending latent stochastic models and transport-based learning to more general and higher-dimensional systems.

Scientists increasingly express their ideas as dynamic models of complex processes. It is often much easier to simulate these models than to calculate the probability of their generating a particular outcome, making likelihood-based estimation infeasible. Existing likelihood-free approaches rely either on manually chosen summary statistics or on representations learned by neural networks. The former is error-prone and laborious, while the latter is computationally intensive, leaving many scientists in a difficult position. We show that, for a large class of dynamic models, parameters can be estimated by matching a small number of random features of the observed and simulated data. In particular, we show that models with a $p$-dimensional parameter can be identified from just $2p+1$ generic random Fourier features. We introduce two estimators for stationary and nonstationary processes, respectively, and we establish their consistency under mild regularity conditions. More broadly, our results serve as the foundation for a new class of methods for simulation-based inference.

Erdős and Gyárfás conjectured in 1995 that, in every red–blue edge-coloring of a complete graph $K_n$, the vertex set can be covered by at most $\sqrt n$ monochromatic paths, all of the same color. Pokrovskiy, Versteegen and Williams (JCT-B, 2026) proved the conjecture for all sufficiently large $n$. In this paper, by using minimal counterexample method, we confirm the conjecture completely.

We introduce regularized bivariate affine $q$-Krawtchouk polynomials and show that they give the first eigenmatrix of a translation association scheme on $\operatorname{Mat}_{d\times n}(\operatorname{GR}(p^2,r))$. The relations are defined by Smith type, and the corresponding eigenvalues are written as character sums over Smith type classes. The proof is based on recurrence relations obtained from the transition numbers describing how Smith types change when one row and one column are added to a matrix. These recurrences identify the character sums with the regularized bivariate affine $q$-Krawtchouk polynomials.

The notion of the factorial extends to abelian groups under the group operation and any permutation of the elements of the group. Notably, for some finite groups, it is possible to order elements in these groups such that no two “factorials” are the same. In this paper we count the number of ways to order elements to achieve this result for the integers modulo 20 and 22 under addition, as well as introduce an improved algorithm to count these permutations for the integers modulo $n$. We relate these numbers to the values in an already known sequence via a bijection which we prove in this paper. This proof and other lemmas allow us to calculate new terms for the sequence.

In this paper we study a Stackelberg game with one leader and multiple followers. Given the leader's control, the followers solve a Nash game with possibly multiple equilibria. We consider a robust leader who considers the worst scenario, namely the followers would select the equilibrium worst for the leader. By using the weak formulation, the problem induces a zero sum game problem with open loop controls, which is time inconsistent. We shall characterize the last problem through an HJB equation on the Wasserstein space of probability measures. The principal-agent problem with one principal and multiple agents can be viewed as a special case of our problem, but with certain constraints.

Coarsening-based training for graph neural networks (GNNs), i.e.\ training on coarsened graphs rather than the original large ones, has become a promising direction for scaling GNNs to massive graphs. However, prior work has been evaluated almost exclusively on homophilic graphs, leaving the more challenging heterophilic settings underexplored. We show, both empirically and theoretically, that existing coarsening-based training methods suffer significant performance degradation on heterophilic graphs due to inevitable loss of graph information during coarsening. To address this, we propose Adaptive Complementary Enhancement, a plug-and-play, model-agnostic strategy that reintegrates the information discarded in coarsening: ACE learns a projector for re-constructing original node features and applies anisotropic structural regularization to embed local heterophily. We further adopt homoscedastic uncertainty weighting to adaptively balance the combined training objective of primary coarsened-graph training loss and full-graph auxiliary loss with augmented node features re-constructed by the heterophily-aware projector. Extensive experiments show that ACE drives consistent gains on heterophilic benchmarks while preserving competitive results on homophilic graphs with minimal computational overhead. Code is available at the GitHub repository: https://github.com/vasile-paskardlgm/ACE.

Bacterial motility is a fundamental trait for territorial expansion and resource acquisition. While existing models of nutrient-dependent motility often do not explicitly account for the metabolic costs associated with motility, these costs become critical in nutrient-limited or closed systems. In this study, we developed a mathematical framework using partial differential equations (PDEs) that explicitly incorporates the energetic trade-offs of motility. By formulating an optimization problem focused on maximizing population yield—defined by the total cell count—we evaluated various motility strategies across different environmental contexts. Our results demonstrate that the optimal motility response is highly sensitive to resource distribution. Specifically, we show that in unpredictable environments, a non-monotonic motility response emerges as the optimal strategy, providing a robust theoretical explanation for dose-response curves observed in experimental microbiology. This framework serves as a powerful, interpretable tool for predicting bacterial behavior in resource-constrained ecosystems and offers new insights into how such adaptive strategies are shaped by environmental pressures.

Buchstab's function $\omega(u)$ describes the distribution of integers without small prime factors. We establish numerically explicit upper and lower bounds for $\omega(u)$ that are easy to evaluate, without the need to solve the delay differential equation numerically.

This work establishes a theoretical foundation for joint feature selection with multivariate outcomes, positioning the permutation-based variable importance measure (PVIM) of multivariate random forests (MRF) as a principled tool for high-dimensional feature selection. We establish the first consistency guaranty for MRF, showing that it retains all truly influential features with probability tending to one as the sample size grows to infinity under mild regularity conditions. Incomplete U-statistics is employed to incorporate three layers of randomness: subsampling of subjects for training each tree, subsampling of features at each split, and permutation of each feature for the out-of-bag (OOB) samples. Unlike independence-based screening that evaluates each feature in isolation, PVIM is a joint screening approach that accounts for multicollinearity, nonlinear, high-order interactions, and subject heterogeneity via ensemble aggregation. Moreover, we demonstrate the practical utility of MRF through a genome-wide association study (GWAS) of human facial morphology (with 2,342 subjects and 453,273 SNPs), where MRF identifies several novel loci and interaction hubs that extend prior findings. Extensive simulations show that MRF accurately identifies truly influential signals while producing parsimonious feature sets with well controlled false selection rates, outperforming canonical correlation analysis (CCA) and several other independence multivariate screening approaches. In addition, we also propose a novel simulation framework, including image outcomes, that more closely mimic the intricate nature of real-world data and provide rigorous testbeds for machine learning research.

We study semigroups of composition operators acting on the Besov spaces $B_p$, where they exhibit some new behaviors relative to many classical spaces. Often for a Banach space $X$ of analytic functions on the unit disk, the maximal closed space of strong continuity, $[ \varphi_t, X ]$, exists for every semigroup $\{ \varphi_t \}$ of analytic self-maps of the disk, and the question whether $[\varphi_t , X ]$ equals $X$ itself has an answer independent of $\{\varphi_t\}$. Such is the case for the Hardy and Bergman spaces, Bloch, BMOA, and $H^{\infty}$. For the disk algebra $A$, $[\varphi_t , A ] = A$ precisely when $\{\varphi_t\} \subset A$. For $B_p$ with $p \geq 2$, every $\{\varphi_t\} \subset B_p$ and always $[ \varphi_t, B_p ] = B_p$, but this fails when $1 < p < 2$. We give an example where $\{\varphi_t\} \subset B_p$ and yet the induced composition operators $\{C_t\}$ are not bounded on $B_p$ and we do not know if $[\varphi_t,B_p]$ exists. If it does exist, it cannot be equal to $B_p$. Under the hypothesis that there is a uniform bound for the operator norms of the $\{C_t\}$, $0 \leq t \leq 1$, we characterize the semigroups $\{ \varphi_t \}$ such that $[ \varphi_t, B_p ] = B_p$.

We prove that $$\max_{t \in {\Bbb R}}{T_n(t)} \geq \frac{1}{60}n^{1/3} \qquad \text {and} \qquad -\min_{t \in {\Bbb R}}{T_n(t)} \geq \frac{1}{60}n^{1/3}$$ for every trigonometric polynomial $T_n$ of the form $$T_n(t) = \sum_{j=1}^n{\cos(jt-\theta_j)}\,, \quad \theta_j \in {\Bbb R}\,, \quad t \in {\Bbb R}\,.$$

We study overdetermined boundary problems for the graphical translating-soliton equation \[ -\operatorname{div}\!\left(\frac{Du}{\sqrt{1+|Du|^2}}\right) =\frac{1}{\sqrt{1+|Du|^2}} \quad\text{in }\Omega, \qquad \partial_\nu u=\Gamma H+C \quad\text{on }\partial\Omega, \] where $\Gamma, C$ are constants, $H$ is the mean curvature of the boundary $\partial\Omega$ of the regular bounded domain in $R^n$ such that $H_{\partial B_R}=-1/R$. For $\Gamma\geq0$, we prove that constant Dirichlet data force a bounded domain to be a ball. We also prove a spherical-shell rigidity theorem for a doubly connected domain with two ordered boundary heights and $a<u<b$ in the interior. The argument combines linearization under reflection, Reichel's critical-plane and annular continuation principles, curvature comparison, Serrin's corner lemma, and a radial ODE that excludes the annular alternative in the one-height problem. Finally, we give explicit counterexamples showing the sharpness of the sign, ordering, connectedness, and nesting assumptions.

This paper studies product spectral sets and functional tiles in the $p$-adic spaces $\mathbb{Q}_p^d$ within the framework of the product spectral set conjecture. We establish a stability result for functional tiles under weak convergence of tiling complements, characterize product spectral pairs with product spectra, and fully resolve the conjecture for cylindric sets by proving that spectrality of a cylindric set $\Omega=B_\gamma(a)\times\Theta$ is equivalent to spectrality of $\Theta$.

Let $P_4^r$ denote the $r$-uniform expansion of the graph path with four edges. We prove that every $n$-vertex linear $r$-uniform $P_4^r$-free hypergraph has at most $(r+1)n/r$ edges, resolving a conjecture of Adak and Verma for every $r \geq 2$. Equality holds precisely for vertex-disjoint unions of Steiner systems $S(2,r,r^2)$. The main ingredient is a sharp incidence-rank inequality. If $N(H)$ is the edge-vertex incidence matrix of a linear $r$-uniform hypergraph whose line graph is a cograph, then $(r+1)\operatorname{rank}_{\mathbb{R}} N(H) \geq r|E(H)|$. Equality holds exactly when every edge-containing component is an $S(2,r,r^2)$. The proof follows the union-join decomposition of cographs. At a join node, the row-difference spaces of the co-components are mutually orthogonal, and the possible rank defect is determined by balanced co-components. Perron-Frobenius theory identifies the smallest balanced pieces as parallel classes of $r$ disjoint $r$-sets, while an orthogonality argument bounds their number by $r+1$. The equality case then reconstructs the Steiner system. The linear Turán bound follows from $\operatorname{rank}_{\mathbb{R}} N(H) \leq |V(H)|$.

Low Earth orbit (LEO) satellite constellations are poised to become a cornerstone of the sixth-generation (6G) Internet of Things (IoT), providing truly global coverage and ubiquitous connectivity. This article presents a holistic two-dimensional system architecture for 6G LEO satellite IoT that incorporates composition and functional perspectives to facilitate the seamless integration of LEO satellites and terrestrial networks. Building upon this architecture, we evaluate three pivotal enabling technologies targeting the uplink, downlink, and inter-satellite links (ISLs). Specifically, we analyze massive grant-free random access for efficient uplink connectivity, investigate deep learning-based multibeam precoding for robust downlink transmission, and examine distributed cooperative routing for resilient ISL data delivery. Furthermore, we present an on-orbit verification platform that validates the real-world feasibility and performance of the proposed solutions. Finally, we outline key open challenges and future research directions to guide the realization of future LEO satellite IoT.

In 1978, Yu. F. Borisov presented an axiom system using a few basic assumptions and four explicit axioms, the fourth being a formulation of the relativity principle; and he demonstrated that this axiom system had (up to choice of units) only two models: a relativistic one in which worldview transformations are Poincaré transformations and a classical one in which they are Galilean. In this paper, we reformulate Borisov's original four axioms within an intuitively simple, but strictly formal, first-order logic framework, and convert his basic background assumptions into explicit axioms. Instead of assuming that the structure of physical quantities is the field of real numbers, we assume only that they form an ordered field. This allows us to investigate how Borisov's theorem depends on the structure of quantities. We demonstrate (as our main contribution) how to construct Euclidean, Galilean, and Poincaré models of Borisov's axiom system over every non-Archimedean field. We also demonstrate the existence of an infinite descending chain of models and transformation groups in each of these three cases, something that is not possible over Archimedean fields. As an application, we note that there is a model of Borisov's axioms that satisfies the relativity principle, and in which the worldview transformations are Euclidean isometries. Over the field of reals it is easy to eliminate this model using natural axioms concerning time's arrow and the absence of instantaneous motion. In the case of non-Archimedean fields, however, the Euclidean isometries appear intrinsically as worldview transformations in models of Borisov's axioms and neither the assumption of time's arrow, nor the rejection of instantaneous motion, can eliminate them.

We develop a structural credit-risk model under incomplete information in which investors observe firm value only indirectly through noisy market signals and scheduled corporate disclosures. While disclosure dates are known in advance, their informational content is random, leading to stochastic discontinuities in the observation process. We derive the Kushner-Stratonovich equation for structural credit-risk models with endogenous default by applying the nonlinear filtering framework with predictable jumps. We then study the valuation and local risk-minimization hedging for default-sensitive securities under partial information. The interaction between predictable disclosure events and endogenous default produces discrete adjustments in the conditional default compensator, leading to announcement-driven distortions in credit spreads and hedge ratios that are absent from classical diffusion-based and inaccessible-jump models. Numerical experiments illustrate how scheduled disclosures affect filtered default probabilities, Credit Default Swaps (CDS) spreads, and hedging strategies, generating characteristic pre-announcement dynamics in credit spreads.

This study evaluates the potential of electric vehicle (EV) charging flexibility to enhance both energy efficiency and power grid stability. Using real-world data from public charging stations in Prague, we analyze individual and aggregated charging sessions to explore how optimizing charging times can reduce energy waste, minimize grid imbalances, and support the integration of renewable energy. By aligning EV charging with periods of lower grid demand and higher renewable generation, we demonstrate a significant improvement in energy efficiency, reduc ing the need for costly system support and ancillary services. Our findings suggest that cooperation between power distributors and transmission system operators can unlock new opportunities for maintaining grid stability while promoting sustainable energy use in an increasingly uncertain energy landscape.

We develop a rigorous mathematical framework for statistical mechanics models on hypergraphs, and give conditions for lifting the classical connection between Potts model partition functions and the Tutte polynomial from graphs to hypergraphs. We define hypergraphical models and their associated partition functions, and extend these to special classes of models induced by families of interaction functions. For boolean interaction families, whose interaction functions map to $\{0,1\}$, we show that the partition function is determined by a combinatorial rank function, and we establish sufficient conditions for a hypergraph deletion-contraction recurrence and for when the rank function defines a polymatroid. We illustrate the theory by applying it to three hypergraph interaction families: Parity Ising, Delta Potts, and And Ising. The induced hypergraphical models are not isomorphic to each other, but the first two reduce to the same graphical Ising models. We identify three polymatroids naturally associated with hypergraphs for these models: respectively, the binary matroid of the incidence matrix over $\mathbb{F}_2$, the hypergraphical polymatroid, and the boolean polymatroid. For graphs, the first two reduce to the classical graphical matroid, and their partition functions recover the multivariate Tutte polynomial. The Tutte polynomial thus admits at least two distinct generalizations for hypergraphs, both satisfying a deletion-contraction recurrence: the Tutte polynomial of the binary matroid of the hypergraph incidence matrix, and a multivariate version of the Poincaré polynomial of the hypergraphical polymatroid. The partition functions of And Ising models are likewise multivariate versions of the Poincaré polynomial, here of the boolean polymatroid. These examples illustrate the much greater range of hypergraphical models and underscore the need for the unifying theory.

This is the fourth work in a series devoted to proving conjectures recorded in the On-Line Encyclopedia of Integer Sequences (OEIS). The problems considered here concern elementary and multiplicative number theory, Fibonacci numbers, decimal concatenation, Diophantine and Pell equations, binary representations and bitwise operations, lattice paths, parity patterns, recurrences, and formal power series. Several of the results give complete characterizations of the relevant sequences; others establish exact identities, recurrences, generating functions, asymptotic estimates, integrality properties, or nonoccurrence results. The proofs use combinatorial bijections, congruences, Möbius inversion, valuations, Fibonacci identities, Pell-type arguments, Lucas' theorem, Riordan arrays, Lagrange inversion, and generating-function methods.