Papers

We present a few ideas on the Riemann Hypothesis based on properties of analytic functions in the complex plane. In particular, we focus on the real and imaginary parts of the Riemann xi ($\xi$) function whose zeros coincide with those of the zeta ($\zeta$) function within the critical strip. We discuss the forms of the zero contour lines of the two conjugate harmonic functions (the real and imaginary parts of xi) and consider where their intersections could conceivably occur. Those intersections would be the roots of both $\xi$ and $\zeta$ functions. The question of whether a zero could occur away from the critical line becomes equivalent to whether the solutions of a pair of Laplace's equations with well-defined boundary conditions in some semi-infinite strip can possess zero contour lines that intersect within that strip.

We prove an $L^2$ two-weight testing theorem for the Dunkl–Poisson semigroup. The difficulty is geometric. The Dunkl orbit distance has several reflected diagonals, so a single orbit-box test may mix different chamber components. We avoid this by working on one Weyl chamber and keeping the chamber indices. Under the wall-null assumption the full operator becomes a finite matrix of scalar positive Poisson-type operators. In each entry the orbit diagonal is just the ordinary diagonal in the chamber variables. The scalar proof is then a principal-cube stopping-time argument, with two Dunkl kernel comparisons as the only new estimates. The resulting forward and backward tests are necessary and sufficient for the original Dunkl–Poisson inequality.

We consider the closure of three classical combination properties, namely, stable infiniteness, gentleness and shininess (or, equivalently for decidable theories, strong politeness), under intersection and combinability. We compute every possible intersection, and then compute the maximal set of theories that can be combined with each resulting intersection. We iterate this process until no new sets are identified. How many properties will we end up with?

The classical Liouville theorem states that every bounded harmonic function on Euclidean space is constant. On complete Riemannian manifolds, analogous conclusions hold under geometric assumptions such as nonnegative Ricci curvature. The quadratic Wasserstein space $\mathcal{P}_2(M)$ has no canonical infinite-dimensional Riemannian volume and hence no canonical Laplace–Beltrami operator. We introduce a natural finite-particle notion of harmonicity: a continuous function $u:\mathcal{P}_2(M)\to\mathbb{R}$ is called empirically harmonic if, for every $N\geq1$, its pullback under the empirical map $$ \iota_N(x_1,\ldots,x_N)=\frac1N\sum_{i=1}^N\delta_{x_i} $$ is weakly harmonic on $M^N$. We prove that if $M$ has the finite-product Liouville property, then every bounded empirically harmonic function on $\mathcal{P}_2(M)$ is constant. In particular, the result applies to $M=\mathbb{R}^d$ and to every complete connected Riemannian manifold with nonnegative Ricci curvature. We also derive a finite-particle chain rule for sufficiently regular functionals on $\mathcal{P}_2(\mathbb{R}^d)$ and show that the empirical Laplacian is exactly the Hessian trace of a discrete $N$-particle lift. Finally, if $M$ admits a nonconstant bounded harmonic function, then $\mathcal{P}_2(M)$ admits a nonconstant bounded empirically harmonic linear statistic.

We consider collisions of multiple random walks on the trace of a simple random walk on the four-dimensional integer lattice. For two independent walks (in continuous time), we apply a result of Noda to derive a scaling limit for the collision time process. For three independent walks (in discrete time), we demonstrate infinitely many triple collisions occur.

Motivated by the recent work of Harper–Kohli–Song–Tahar, we formulate a positivity, hole-free, and log-concavity conjecture for the Links–Gould polynomial of alternating links and verify it for all 51.3 million alternating knots with at most 19 crossings. All but 544 of those knots satisfy a stronger type-B log-concavity condition characterized by the slopes of edges in the subdivision of the monomial support induced by the log coefficients.

This paper studies integral forms for affine vertex superalgebras and their modules. We first obtain integral forms for the universal enveloping superalgebra $U(\hat{\g})$ of an affine Lie superalgebra $\hat{\g}$. For $\g$ a basic classical Lie superalgebra, we give another construction of integral forms for $U(\hat{\g})$ using Chevalley basis, generalizing Kostant-Garland integral form theory. Then we apply the theory to construct integral forms for vertex (operator) superalgebras based on affine Lie superalgebras and their modules, we also investigate when an integral form contains the conformal vector, and integral forms in contragredient modules for vertex operator superalgebras.

In this work, we define the interaction graph for a continuous-time Markov chain to analyze its dynamic structure. Under certain assumptions, we demonstrate that verifying Kolmogorov's reversibility criterion reduces to checking it for a cycle basis of the incidence matrix kernel associated with the interaction graph. This result provides an efficient tool for verifying reversibility in Markov continuous chains.

In tensor dynamics for liquid crystals derived from molecular models, a common problem is closure approximation. For rod-like molecules, the Bingham closure has proved to outperform other methods because it inherits the gradient flow structure of the molecular model, but is difficult to achieve efficient computations maintaining the gradient flow structure. We propose a closure approximation by the quasi-entropy that has been successfully applied to the free energy, based on which we construct the tensor gradient flow. The quasi-entropy closure has the same symmetry properties as the Bingham closure. The resulting tensor gradient flow is able to constrain the eigenvalues of the tensor within the physical range, guaranteeing the positive definiteness of the dissipation operator given by the higher-order tensors. The quasi-entropy closure is easy to implement since it can be reduced to minimizing an elementary function of three variables. As a result, we construct a numerical scheme preserving the eigenvalue constraints and energy dissipation, with the closure approximation decoupled from solving the scheme. Numerical simulations are carried out for the interface between the isotropic and the uniaxial nematic phase, as well as the defect evolutions, where the higher-order tensors indeed make a difference.

We define a quantum graph to be vertex-transitive if the join of its automorphism group is the maximum quantum relation on its quantum vertex set, in direct analogy with the classical case. All simple quantum graphs in $M_2(\mathbb C)$ are vertex-transitive, but many simple quantum graphs in $M_3(\mathbb C)$ are not vertex-transitive. We provide a complete classification of vertex-transitive quantum graphs in $M_3(\mathbb C)$ up to isomorphism. To do this, we introduce a polynomial invariant for quantum graphs in $M_n(\mathbb C)$, which we call the panoramic polynomial.

A $\mathbf{GL}$-variety is a (typically) infinite dimensional variety $X$ equipped with an action of the infinite general linear group. In recent work of the first two authors with Draisma, a candidate definition for the singular locus of $X$ was put forth. The approach there made use of auxiliary finite dimensional varieties associated to $X$. In this paper, we give a number of characterizations of the singular locus that are intrinsic to $X$. This work shows that the candidate definition is clearly correct, and helps clarify the geometric meaning of singular points.

We consider the directed polymer model on $\mathbb{Z}^d$, in an i.i.d.\ random environment $\omega=(\omega_{n,x})_{n\geq 0,x\in\mathbb Z^d}$, focusing on the critical dimension $d=2$. Our main contribution is to give a sharp lower bound on the free energy in the high-temperature regime. Our proof uses a percolation argument inspired by Lacoin (2010), for which we introduce a key property of bounded “$\log$-energy”: this property quantifies the regularity of the polymer measures at diffusive scales and we show that it propagates along open paths. Writing $\mathfrak{f}(\beta)$ for the quenched free energy, and setting $\lambda(\beta):=\log \mathbb E[e^{\beta\omega_{1,0}}]$ and $\sigma(\beta)^2:=e^{\lambda(2\beta)-2\lambda(\beta)}-1$, our lower bound combined with Theorem 2.8 of Berger, Caravenna, and Turchi (2025) gives $$ -\mathfrak{f}(\beta) \asymp \exp{\Big(- \frac{\pi}{\sigma^2(\beta)}\Big)},\quad \text{ as $\beta\downarrow 0$.} $$

An impulse response/equilibrium model (IREM) structure combines a linear convolution model with a nonlinear function that sets the current operating point via an equilibrium variable with integrator dynamics. This model structure is well suited for mildly nonlinear systems and in particular has been applied to battery fast charging control. This paper provides observability conditions for the IREM model structure and bounds on the prediction error. These conditions can be evaluated directly on the system impulse response.

The stochastic dynamics of dislocations on a one-dimensional periodic lattice domain are considered. Two models are studied: one without a collision rule, and one which annihilates colliding dislocations if they have opposite orientation. The behaviour of both models is investigated by means of a series of numerical simulations exploring the asymptotic behaviour of these models as the number of dislocations increases. From these simulations, evidence is obtained that the discrete model with annihilation tends to a PDE for the dislocation density that accounts for annihilation. However, the discrete model without a collision rule does not appear to exhibit consistent convergence behaviour; instead, it appears that the expected PDE with conserved dislocation density appears in the limit for some parameters, but that for other parameters the density appears to follow to the evolution of the PDE with annihilation. These findings provide evidence that a careful treatment of dislocation collisions is important in discrete dislocation dynamics models.

In this paper, we study a structural acoustic model consisting of a semilinear wave equation defined on a three-dimensional bounded domain, coupled with a Kirchhoff-Love plate equation acting on a flat portion of the boundary. Primarily for mathematical interest, we impose nonlinear damping terms and logarithmic-type supercritical source terms on the system. We investigate local and global well-posedness, energy decay rates of potential well solutions, and blow-up of solutions under different conditions on the parameters and initial data. The main novelties include the analysis of the interaction between nonlinear damping and logarithmic energy-amplifying source terms, as well as the development of techniques for handling logarithmic source terms within potential well theory. The logarithmic nonlinearities are not homogeneous under scaling, which creates difficulties in studying potential well solutions. The wave-plate coupling through the acoustic pressure also causes technical difficulties in the analysis, especially in the proof of blow-up of weak solutions.

For $(M,g)$ a compact Riemannian surface with Laplace-Beltrami operator $\Delta$, and for $\lambda,\delta \geq 0$, let $P_{\lambda,\delta}$ be the spectral projector on the frequency interval $[\lambda-\delta,\lambda+\delta]$ associated to $\sqrt{-\Delta}$. For the Euclidean disk, away from its boundary, we improve the upper bound on the $L^2\to L^4$ norm of $P_{\lambda,\delta}$ in the regime where the bandwidth $\delta$ is polynomially small compared to the target frequency $\lambda$. Decomposing on the explicit joint eigenbasis of $\left(\sqrt{-\Delta}, \frac{1}{i}\frac{\partial}{\partial \theta}\right)$ given in terms of Bessel eigenfunctions, which are well-approximated by oscillatory functions outside of their caustic set, we reduce the analysis to a number of precise quantitative estimates of nonstationary phase oscillatory integrals. We strongly use convexity phenomenon both for these estimates, and then for the summation of the contribution of all eigenfunctions through a new arithmetic estimate. The method extends to other $S^1$-symmetric surfaces satisfying similar conditions on the induced completely integrable structure.

We introduce asymptotically Hölder nonexpansive mappings and mappings of asymptotically Hölder nonexpansive type, in which the Hölder exponents converge to one along the iterates. We prove that if \(K\) is a nonempty closed bounded convex subset of a Banach space \(X\) with characteristic of convexity \(\epsilon_0(X)<1\), then every self-mapping of \(K\) of asymptotically Hölder nonexpansive type admits a point \(x\in K\) such that \(T^n x\to x\). Consequently, a fixed point exists whenever some positive iterate of \(T\) is continuous; in particular, this applies to asymptotically Hölder nonexpansive mappings. We also obtain fixed-point-free constructions in spaces containing copies of \(c_0\), and a related construction on a bounded convex, not necessarily closed, subset of every Banach space containing an isomorphic copy of \(\ell_1\). Further examples show that the new classes properly extend their classical counterparts and may contain mappings with no continuous positive iterate. Finally, we examine Lin's renorming of \(\ell_1\), identify an obstruction related to shift-type constructions, and establish a shrinking-diagonal criterion that reduces the remaining closed-set problem to the construction of a suitably controlled uniformly Lipschitzian mapping.

This study introduces SEGOMOE, a Bayesian optimization tool for optimizing complex, computationally expensive systems, especially in aeronautics. It efficiently handles mixed design variables (continuous, discrete, categorical, hierarchical) using adaptive Gaussian process models. SEGOMOE combines expert models to address nonlinearities in objectives and constraints, leveraging the open-source Surrogate Modeling Toolbox (SMT). The tool supports multi-fidelity data and solves both single- and multi-objective problems, including hidden constraints and high-dimensional decomposition. Validated through benchmarks and real-world aeronautical applications, SEGOMOE proves to be robust and versatile for tackling multidisciplinary challenges.

In this article, we present the new definition of the fuzzy Caputo-Katugampola derivative and its related concepts for the class of fuzzy functions and apply them to establish both necessary and sufficient optimality conditions for fractional fuzzy variational problems. By adopting the horizontal membership functional representation of fuzzy numbers, the granular Caputo-Katugampola derivatives offer higher computational efficiency than the previous approaches in certain scenarios. The optimality conditions for the fuzzy fractional variational problems under granular fuzzy Caputo fractional derivatives are also examined and illustrated with detailed examples. Another novel aspect of our results, even in the non-fuzzy case, is the application of special functions to represent the exact solutions of the fuzzy fractional variational problems.

Efficient thermal management is critical for the reliability and performance of power electronics systems in automotive applications. This work presents a computationally efficient modeling approach for transient thermal simulation of power electronic systems, with a focus on inverter modules using multiple MOSFETs mounted on a printed circuit board assembly (PCBA). A case study of an inverter module comprising six MOSFETs arranged as high-side and low-side pairs for a three phases system mounted on a PCBA, attached to a heat sink is considered. Computational fluid dynamic (CFD) simulations in Ansys Icepak are performed considering different heat transfer mechanisms, including natural convection, forced convection at constant velocity, and forced convection with varying flow velocity. A transient thermal model is developed using the Lumped Parameter Linear Superposition (LPLSP) method, a hybrid approach that combines lumped parameter modeling with the principle of linear superposition to capture transient thermal behavior efficiently. Temperatures of the components from the simulations are compared with temperatures from the LPLSP model and temperatures from a Linear Time Invariant (LTI) based reduced order model (ROM) developed for this system. It is observed that the LPLSP model is able to model a wide range of use cases very accurately with error of less than 5 %. This method enables rapid thermal performance evaluation of power electronics systems that have very fast transients in component level power dissipation and variations in ambient conditions, making it particularly well-suited for early-stage design iterations and long-duration mission profile simulations. The approach offers a practical path to reducing development cycles for automotive power electronics design.

Robust decision-making requires compression. A system that forms a rich support state cannot usually preserve its full structure at the point of action. It must retain only those distinctions needed to act, verify, abstain, or defer under the current consequence geometry. This paper formalizes support sufficiency as action-sufficient compression. Let $H$ denote a full support state, $\mathcal{A}$ a finite action set, and $Z$ a consequence geometry specifying payoff structure. For fixed $Z$, the coarsest exactly action-sufficient compression is the quotient of support space by policy equivalence. Two support states may be merged exactly when they require the same optimal action. This clarifies why content-only and scalar-confidence-only arbitration fail whenever their induced partitions cross action boundaries. Approximate sufficiency is then defined by bounded expected policy regret. In the finite single-cycle setting, this yields a rate-regret problem with source $H$, reproduction alphabet $\mathcal{A}$, and distortion given by consequence-sensitive regret. The optimal stochastic action channel inherits the standard rate-distortion Gibbs form, applied here to support states with regret distortion. The contribution is interpretive: action adequacy is distinguished from reconstruction fidelity, information-bottleneck prediction, and rational inattention. Robust single-cycle arbitration does not require preserving all support, but it does require preserving the distinctions that consequence geometry makes action-relevant.

We exhibit an exact correspondence between sampling with score-based diffusion models and adiabatic transport of ground states for a family of Schrödinger operators we call Score Hamiltonians, built from the learned score's quantum potential. We obtain novel density reconstruction bounds and principled annealing schedules via adiabatic theorems for Fokker-Planck equations with time-varying potentials. We find the fundamental limit of sampling is set by the ratio of squared score-matching error to Score Hamiltonian spectral gap - the inverse Poincaré constant of the data density.

Sixth-generation networks are expected to extend connectivity beyond terrestrial infrastructure through non-terrestrial networks (NTNs) comprising satellites, high-altitude platform stations, and unmanned aerial vehicles. However, these platforms operate in a regime that bit-fidelity-centric design handles poorly: high free-space path loss, massive round-trip delays, Doppler shifts of hundreds of kilohertz, limited visibility windows, and limited on-board computing capability compared with ground hardware. Semantic communication (SemCom), which transmits task-relevant meaning rather than exact bits, provides a promising way to address these constraints. This survey examines SemCom for NTNs from the perspective of how semantic mechanisms support different parts of the communication system. We first map five structural NTN constraints onto the semantic mechanisms that can address them, and we show that each platform imposes a distinct constraint vector that selects among those mechanisms. We then propose a five-plane taxonomy covering semantic representation and on-board encoding, channel-adaptive transmission, semantic networking, resource management, and distributed learning with knowledge-base maintenance, together with a cross-cutting trust plane, and we review the literature within it. Finally, we summarize current standardization efforts and available research resources, and identify open problems and future research directions for SemCom in NTNs.

These notes provide a rigorous and accessible introduction to measure and integration theory, with emphasis on the conceptual transition from the Riemann integral to the Lebesgue integral and the role played by limiting processes in modern analysis. The manuscript develops the basic theory of measurable sets, measurable functions, measures on $\sigma$-algebras, Lebesgue integration, convergence theorems, and $L^p$ spaces. Particular attention is devoted to the interaction between integration and convergence, as well as to the limitations of the Riemann integral that motivate the development of measure theory. The exposition seeks to balance mathematical rigor with pedagogical clarity through detailed proofs, examples, exercises, and supplementary projects. These notes are intended primarily for undergraduate students in mathematics and related areas encountering measure theory for the first time, although they may also serve as a reference for introductory graduate courses in analysis.

Forman has defined a discrete version of the Ricci curvature on Riemannian manifolds, known as the Forman–Ricci curvature. The Forman–Ricci curvature has found significant applications in several pattern recognition problems occurring in natural sciences. Domokos and Langi, on the other hand, have defined a notion of irregularity for convex mosaics, which has also found remarkable applications to the geological problem of fractures in rocks. We define a modification of the classical Forman–Ricci curvature for irregular convex mosaics and demonstrate how they can be used to distinguish between various fractures or cracking patterns appearing in nature.

Let $T(n)=n/2$ if $n$ is even and $T(n)=(3n+1)/2$ if $n$ is odd. We prove that for each $m\ge1$, exactly $F(m+1)$ odd integers $n$ in $\{1,\ldots,2^m\}$ have the property that none of the iterates $T(n),T^2(n),\ldots,T^{m-1}(n)$ lies in the residue class $4\pmod6$, where $F(m+1)$ is the $(m+1)$-th Fibonacci number; the proportion decays at rate $(\varphi/2)^m$, $\varphi=(1+\sqrt{5})/2$. Equivalently, these are the odd $n\le2^m$ for which no two consecutive terms of $n,T(n),\ldots,T^{m-1}(n)$ are even. The proof uses the directed graph $G$ of Collatz transitions modulo $6$ and its unique absorbing strongly connected component $G'=G[\{1,2,4,5\}]$. Removing vertex $4$ from $G'$ yields a subgraph of spectral radius $\varphi$, against $\rho(G')=2$; the Fibonacci count follows from this spectral gap. We construct an explicit bijection $\Psi_m:\{1,\ldots,6\cdot2^m\}\to\mathcal{P}_m(G)$ onto the directed paths of length $m$ in $G$. We further show that no vertex of $G'$ is dispensable: removing any single vertex reduces the spectral radius strictly below $2$, with hierarchy $1<\sqrt{2}<\varphi<2$. In particular, every positive cycle of $T$ must visit residue class $2\pmod6$, and a flow conservation identity forces this class to account for more than $18\%$ of the steps in any such cycle.

In this note we explore rational approximations to irrational numbers whose denominators are restricted to a subset of the natural numbers. We introduce a specific notion of density for such subsets and use it to establish a restricted analog of a theorem of Dirichlet's.

In the setting of multi-head finite-state dimensions, trailing heads lag behind a leading head, accessing past data to aid a finite-state gambler placing bets on successive bits read by the leading head. Cruz, Glashausser, Li, and Lutz (2026) proved that, for any fixed number of trailing heads, adaptive (data-dependent) movement rules can strictly outperform oblivious (data-independent) movement schedules. In this paper we strengthen that separation by proving that a single trailing head with adaptive movements can outperform, by a large and uniform margin, arbitrarily many trailing heads with oblivious movements. Formally, our main theorem states that there is a binary sequence whose adaptive two-head finite-state strong dimension is less than its oblivious multi-head finite-state dimension, and that the gap is greater than 0.3.

We approach image denoising from a perception-driven perspective: how can we select the parameters that are best suited for human visual perception? We combine research methods in mathematics and psychology to develop a mathematical framework for measuring perceived similarity. We construct a sample set of differently denoised photographs by using the same base image as input data and by tuning the parameter value in a total variation denoising algorithm. A comparison test is conducted with human participants to survey perceived differences between the images. Analyzing the results with psychometric scaling provides us with a HaarPSI value to use as a threshold in discretizing parameter grids. As a result, we obtain psychometrically scaled, openly available image sets that are ready to use in further experiments in perception-driven imaging, as well as a framework for ensuing experiments involving comparison tests.

We study the repulsive Hubbard model at half filling in the canonical ensemble and compare its thermodynamics with the spin-$\frac12$ antiferromagnetic Heisenberg model at superexchange coupling $J_0(U)=4t^2/U$. For fixed hopping $t\ne0$, bounded staggered fields $|h|\le h_0$, and inverse temperatures $\beta J_0(U)\ge\ell_0>0$, we establish a volume-uniform pressure comparison with error $O(U^{-3}+|h|U^{-2})$. On the Heisenberg field scale $h=J_0(U)x$, the error improves to $O(U^{-3})$ for bounded $x$. Using the quantum Lee–Yang theorem of Suzuki and Fisher, we obtain a volume-uniform $O(U^{-1})$ comparison of staggered magnetizations on compact positive intervals of the scaled field $x$. By combining a recent fermionic Lee–Yang theorem with analytic interpolation, we improve this estimate to $O(U^{-2}\log U)$ in the thermodynamic limit. The latter analysis also establishes analyticity of the limiting Hubbard pressure at nonzero real fields. The improved comparison extends to vanishing positive fields satisfying $U^{-3}\ll h(U)\ll U^{-1}$, with error $O((1+\log(U^3h(U)))/(U^3h(U)))$. Consequently, sufficiently low-temperature Heisenberg Néel order yields a quantitative positive lower bound on the Hubbard staggered magnetization in this regime. Furthermore, we prove $O(U^{-2})$ bounds on the double-occupancy and charge-defect densities and the normalized charge-density-wave second moment. At zero staggered field, this decay rate is optimal for both densities, even in the thermodynamic limit. Our approach uses finite-order Schrieffer–Wolff transformations and does not establish spontaneous Néel order at fixed finite $U$.