Papers

We introduce various cohomological obstructions for smooth integral varieties over $p$-adic function fields. We show that the unramified obstruction is the finest one among obstructions arising from arithmetic dualities. We also construct an explicit example whose unramified obstruction gives a proper obstruction while Manin obstruction does not. Finally, we compare the unramified obstruction with the descent obstruction.

We study the asymptotic behavior of global minimizers of a Ginzburg–Landau-type functional with general compact vacuum manifold $\mathcal{N}$ on bounded domains in $\mathbb{R}^3$, in the regime where the energy grows at a logarithmic rate. We show that the normalized energy measures converge, up to a subsequence, to a measure supported on a finite union of closed line segments connecting prescribed singularities on the boundary. The limit map is a harmonic map valued locally by minimizing $\mathcal{N}$ away from this singular set. We also establish uniform $W^{1,q}$-estimates with $ q\in(1,2) $ and uniform potential estimates for minimizers, independent of the parameter $\varepsilon$. Finally, we prove that the singular set of the limiting measure solves the homotopical Plateau problem in codimension $2$.

We study the dissipative Boussinesq problem, which extends the "good" Boussinesq equation by incorporating viscosity effects. It is well-known that this model supports monotone decreasing traveling kink solutions. We show that these kinks are orbitally stable in $H^1(\rone)$. Moreover, they are asymptotically stable in $\dot{H}^1(\rone)$ and $L^p(\rone), 2<p\leq \infty$.

We study energy quantization for a class of Dirac systems on compact spin Einstein manifolds of dimension \(n\). For a sequence of solutions to a nonlinear Dirac system with uniformly bounded energy on a fixed spin Riemannian manifold, we first establish an energy identity theorem. We then investigate the more complicated case of underlying domain manifolds being a sequence of non-collapsed degenerating spin Einstein manifolds. At an orbifold singular point, three types of bubble spinors can possibly appear, living respectively on \(\mathbb{R}^n\), on a Ricci-flat ALE bubble space, and on the flat cone \(\mathbb{R}^n/\Gamma\). By developing asymptotic analysis for solutions over degenerating neck regions, we establish that energy identity holds.

Enami and Maezawa give a complete characterization of $(s_1, s_2, \ldots, s_k)$-linked planar graphs for any $k$-tuple of positive integers. In this paper, we investigate linkage problems for optimal 1-planar graphs. In particular, we show that every optimal 1-planar graph with connectivity $6$ is $(5, 5)$-linked. Moreover, for an optimal $1$-planar graph $G$ that is not $(2,2,1)$-linked, we characterize disjoint vertex subsets $S_1, S_2, S_3$ in $G$ with $|S_1|=|S_2|=2$ and $|S_3|=1$ such that $G$ is not $\{S_1,S_2,S_3\}$-linked.

Body-fitted finite-element methods deliver high-order accuracy but hinge on a clean, watertight, conforming mesh, a requirement that breaks down for the geometrically imperfect CAD assemblies, image-based volumetric data, and voxel-native designs that pervade biomedical engineering and additive manufacturing, where mesh generation has become the dominant cost of the analysis cycle. Immersed methods on regular background Cartesian grids sidestep body-fitted meshing, but classical implementations integrate over irregular cut subdomains, destroying the tensor-product structure that enables separable, reduced-order methods such as tensor decomposition. In this paper we propose the Immersed Tensor Decomposition (ITD) framework, which couples a mesh-free geometric representation via body-fitted function with the separable C-HiDeNN-TD reduced-order solver to enable large-scale simulation directly on regular background voxel meshes. The geometry is encoded in three steps: a signed-distance function represents the boundary, a body-fitted function $\Phi$ approximates it with controllable error, and a low-rank Tucker decomposition provides model-order reduction; for a fixed grid spacing $h$, accuracy is improved by raising the approximation order of C-HiDeNN interpolation up to degree $p$ with a linear background mesh. The central contribution is an exact Dirichlet formulation that enforces the boundary condition strongly by multiplying the trial function with $\Phi$, so that $u=g$ holds by construction without any variational penalty or interface quadrature. We establish an a priori error estimate for the formulation and assess it on canonical 2D/3D domains, demonstrating optimal convergence and robustness on non-Cartesian geometries discretized by regular voxel meshes.

We consider species, consisting of a possibly infinite set of rings, and bimodules between them. Simson realised the category of representations as a functor category, which we prove is hereditary when each of the rings is semisimple. We use purity to provide sufficient conditions, in order for a representation to decompose into indecomposables with local endomorphism rings. For any bifunctor valued in bimodules, we functorially construct species equipped with commutativity conditions. This generates examples coming from a range of topics, such as subobject lattices in abelian length categories, the field choice problem in persistent homology, and topological field theories with defects.

In the context of hyperbolic formulations of Einstein's field equations obtained via gauge fixing, constraint damping is a desirable feature that ensures that violations of the gauge condition and thus of the constraint equations are suppressed in evolution. Besides its utility in numerical relativity, it has played a key role in several (linear and nonlinear) stability proofs of spacetimes as solutions of the Einstein equation. In this paper, we show that an enhanced form of constraint damping can be implemented for the linearization of the Einstein equation around any subextremal Kerr black hole metric: the constraint propagation wave operator satisfies mode stability and has an arbitrarily large indicial gap at zero energy. The results proved here are a key ingredient in the author's proof of the nonlinear stability of the subextremal Kerr family.

We study shadowing, with respect to the left uniformity, for continuous endomorphisms of Lie groups and totally disconnected locally compact groups. For Lie groups, an endomorphism has shadowing if and only if its differential is hyperbolic, with zero eigenvalues allowed. This includes singular maps outside the classical theory of Anosov endomorphisms. As consequences, positively expansive Lie group endomorphisms are topologically expanding, while on connected semisimple Lie groups the shadowing endomorphisms are precisely the nilpotent ones. On compact connected Lie groups, the nonsingular case agrees with classical Anosov theory. In sharp contrast, every continuous endomorphism of an arbitrary totally disconnected locally compact group has shadowing, without compactness, metrizability or invertibility assumptions; the proof uses Willis' tidy-above decomposition. Consequently, in this category topological expansion and the topologically Anosov property reduce to positive expansiveness and expansiveness, respectively. We also discuss group shifts and revisit Aoki's dense-orbit compactness result without assuming metrizability.

We study the two-dimensional stochastic Navier-Stokes equations on the torus with horizontal dissipation and additive noise. First, we prove a uniform large deviation principle for the solution paths in the energy space $C([0,T];H).$ The proof combines a large deviation principle for a linear Ornstein-Uhlenbeck process, a new Lipschitz estimate for the nonlinear map, and a contraction principle that avoids any exponential tightness condition. Second, we analyse the long-time dynamics. By exhibiting a deterministic steady state immune to the horizontal dissipation, we show that exponential mixing cannot hold. Finally, we investigate the invariant measures of the system. In the deterministic case, the infinite-dimensional kernel of the horizontal Laplacian contains infinitely many Dirac invariant measures. Under a suitably degenerate noise, however, the kernel supports no invariant measure at all; moreover, any invariant measure of this stochastic system, should it exist, must be orthogonal to this kernel. This sharply distinguishes the anisotropic equations on the torus both from the fully dissipative Navier-Stokes equations and from their counterparts on domains with no-slip boundaries.

The well-posedness of multidimensional quadratic backward stochastic differential equations (qBSDEs) remains one of the central open problems in BSDE theory. Motivated by a mean-field utility maximization model with price impact, we introduce a new class of multidimensional qBSDEs that lies beyond the scope of existing well-posedness results (see (2.15) for the generator). In order to study the limit as the number of players tends to infinity, we establish existence and uniqueness for a large class of such qBSDEs under suitable smallness conditions imposed on each individual dynamics. A key feature of our approach is that the smallness condition is independent of the dimension of the BSDE system. In particular, the system itself is not confined to a small neighborhood, which allows us to analyze the mean-field limit of the underlying utility maximization problem. Such a condition is also natural in view of the well-known fact that general multidimensional qBSDEs may fail to admit solutions in the absence of suitable smallness assumptions. In addition, we derive a stability result for this class of equations based on the application of Picard iterations. Finally, using this stability result, we establish quantitative convergence rates toward the corresponding mean-field equilibria in two settings: Nash and Radner equilibria.

We design and implement a Microscale Topological Surgery (MTS) algorithm to detect and enforce topological transitions in two-dimensional tracked interfaces. The method combines classical Lagrangian tracking with an intermittent topological processor that: (i) constructs Eulerian snapshots from which an interface family with microscale-resolved topology is extracted, (ii) infers adjacency topology between dual Lagrangian and Eulerian interface families, and (iii) performs interface surgery to stitch the two families together across microscale defect regions. A novel long-time nonlinear alternating-shear flow is introduced, in which repeated stretching and folding generate rich multiscale interface dynamics with filamentation at microscales. Using the MTS algorithm and a posteriori geometric and material diagnostics, we compute and visualize microscale filament-breakup dynamics. Error analysis and scaling studies demonstrate second-order geometric convergence and optimal computational scaling of the MTS algorithm, with topology-processing costs comparable to those of the underlying Lagrangian evolution. Ensemble simulations generated by pseudo-random perturbations of the flow further reveal coherent droplet size distributions and statistically robust filament-breakup dynamics.

For $p\geq 1$ and $m\geq 2$, let $c_p(m)$ be the least constant such that \[ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p \] for all square complex matrices. Tang and Zhang recently conjectured a closed formula for $c_p(m)$, obtained from a common-left equiangular rank-one family. We disprove that formula for every $m\geq2$ and every $1<p<2$.

Vehicle-to-Everything (V2X) communications require timely and reliable message delivery under highly dynamic wireless conditions. Existing approaches that integrate forward error correction (FEC) into QUIC rely mainly on proactive redundancy and fall back to retransmission once losses exceed the correction capability of the configured code, leading to inefficient recovery under burst loss and unnecessary overhead when network conditions are favorable. This paper presents a Bandit-Controlled Deadline-Aware Incremental Redundancy (BC-DIR) framework for QUIC-based V2X transport. BC-DIR combines rateless coding with a soft decoding deadline and a redundancy margin, enabling repair to be triggered within the available delivery budget while injecting additional repair symbols beyond the immediate deficit to improve recovery under burst loss. A contextual bandit controller further adapts the redundancy configuration online according to end-to-end feedback. We also develop a deadline-constrained reliability analysis under burst loss, showing the advantage of the proposed repair mechanism over conventional retransmission and the existence of an optimal redundancy margin. Monte Carlo simulations validate the analytical results. BC-DIR is implemented in a QUIC-based transport stack and evaluated in Veins/OMNeT++ under both congested urban V2X scenarios and stable network conditions. Experimental results show that, across different traffic congestion levels, BC-DIR improves completion ratio by 10%–40% over benchmark schemes in congested V2X scenarios, while under favorable network conditions it can even reduce overhead by about 1% compared with native QUIC.

We prove that the total area of the power circles associated with an odd $p$-gon inscribed in and circumscribed about a pair of homothetic ellipses remains invariant throughout the Poncelet family. Our proof is based on a simple affine averaging principle, which also yields several related quadratic invariants, including explicit formulas for the sums of the squared distances from the center to the vertices, to the side midpoints, as well as for the sum of the squared side lengths. As an application, we show that a convex Poncelet pentagon and the corresponding star Poncelet pentagon, both circumscribed about the same inellipse, have equal total power-circle area. These results unify several metric invariants of odd Poncelet polygons within a common affine-geometric framework.

We study matrix pencils of the form $\lambda E - (J-R)Q$, where $Q^*E = E^*Q \geq 0$, $J^* = -J$, $R^* = R \geq 0$, and $\lambda E - Q$ is regular, in the framework of Pokrzywa's ordering of strict equivalence orbits. We characterise the maximal elements, analyse some properties of the orbit closures, and derive a restriction on possible degenerations when $Q = I$. We also consider the case where $\lambda E - Q$ may be singular, which naturally arises in the study of orbit closures, and obtain a partial result on the Kronecker canonical form. The theory is illustrated by numerous examples.

The upper and lower approximations of the second order are defined for a function differentiable in the directions. Exhausters of the second order are introduced. The relation between the second-order Frechet subdifferentials and the second-order exhausters is given. A rule for constructing exhausters of the first and second orders is given for Lipschitz functions.Necessary and sufficient extremum conditions through exhausters are given. It is discussed about the use of Exhausters in optimization.

Some infinite series involving the positive zeros of Bessel functions of the first kind are investigated. The motivation behind these series lies in quantum mechanical perturbation problems, in which energies and matrix elements of unperturbed states are expressible in terms of zeros of Bessel functions of the first kind. The existing approach by Pedersen and Urbanowicz for calculating these series involves the Thomas-Reiche-Kuhn sum rule, differential recurrences for powers of Bessel function ratios or application of Lommel polynomials. In this paper an alternative approach is provided: a recursive algorithm is proposed that theoretically can produce the infinite series values in question. Our method is relatively simple and rely on three main ingredients: the Mittag-Leffler expansion and Riccati differential equation for the quotient of Bessel functions of the first kind, as well as the Taylor series coefficients of these ratios. The technique employed in the paper could be useful to treat similar problems where infinite series of zeros of special functions is involved.

We establish new connections between real and complex contact geometry via embeddings of 3-manifolds into $\C^3$. We introduce a new contact wedge construction combining two transverse real contact structures to make a new complex contact structure, subject to obstructions measured by the Nijenhuis tensor and Dolbeault cohomology. Dually, we form a hedge construction which extracts real contact structures from complex ones. Applying these tools, we prove that $\C^3$ admits uncountably many complex contact structures.

The reverse isoperimetric conjecture states that, at fixed thermodynamic volume, Schwarzschild–AdS black holes maximize entropy. We show that its local fixed-volume form admits a boundary-completed canonical-energy interpretation. The key observation is that the bulk Hollands–Wald canonical energy vanishes on exact stationary black-hole families, while the fixed-volume entropy Hessian is encoded in an asymptotic boundary contribution. The same structure governs the global comparison. We choose an affine parameterization of a fixed-volume stationary branch connecting Schwarzschild–AdS to the solution of interest. If the stationary boundary Hessian is non-negative along this comparison, entropy concavity gives the reverse isoperimetric bound, while equality is determined by the directions along which the boundary Hessian vanishes. In Einstein gravity this reproduces the area–volume inequality on the stated comparison branches, and the formulation extends naturally to Wald entropy in higher-derivative theories.

{We study nonautonomous systems whose equilibrium sets are nonisolated and move with time. Their motion is measured on each ball by a localized Attouch–Wets distance, which yields an equilibrium speed that may depend on the radius. The distance from a trajectory to the current equilibrium set is estimated in two ways: through a composite Lyapunov function compared with a power of this distance, and, without any Lyapunov function, through a decay condition on the frozen vector fields. In the first setting, we prove practical stability, and Lyapunov stability when the dissipation has no residual, a case that forces the equilibrium sets to be nested. In the second, we obtain, in the region where the decay condition holds, an explicit tracking estimate driven by the equilibrium speed and the inputs, and we prove asymptotic tracking whenever their averages over unit time intervals vanish, a condition that cannot be weakened among conditions on the speed alone. We also derive input-to-state stability bounds and explicit convergence rates. When the equilibrium speeds are integrable, the equilibrium sets converge in the Attouch–Wets sense to a limiting set at an explicit rate. If, moreover, the velocity is bounded by a multiple of the distance to the current equilibrium set, every trajectory remaining in the region considered has finite length and converges to a point of the limiting set; for a fixed equilibrium set, this yields semistability. The results are illustrated on a scalar example and on resource-allocation models with time-varying demand and with rotating weights.

This work introduces and studies strong affine semigroups, extending the notion of strong numerical semigroups to the higher-dimensional setting. We show that non-numerical strong affine semigroups present structural differences with respect to strong numerical semigroups. Special attention is devoted to strong $\mathcal C$-semigroups. We prove that the family of strong $\mathcal C$-semigroups with a given set of multiplicities $E$ admits a maximal element and has a tree structure. We characterize when this family is finite and provide an algorithm to compute all such semigroups up to a fixed genus. We also introduce the notion of special strong affine semigroups and obtain refined versions of several previous results. Finally, we study toric ideals arising from strong affine semigroups, determining their indispensable monomials and Betti elements for several families.

We describe an artistic project involving the fabrication of each of the 1812 distinct footballs that can be obtained by rearranging the way the 32 panels of a classic Telstar ball are stitched together. Each ball is one of a kind and individually numbered, and the entire series is sorted by decreasing roundness.

In 1976, J. H. Conway introduced Nim arithmetic which establishes an algebraically closed field structure over the class of ordinals and proved that the first transcendental ordinal is $\omega^{\omega^\omega}$. The problem of finding the next transcendental ordinal is still open. Two years later, H. Lenstra proved that $\varepsilon_0$ is the next quadratically closed field ordinal. In this paper, we prove that $\{\varepsilon_\alpha \mid \alpha \leq \omega^{\omega^\omega} \}$ are the next quadratically closed field ordinals.

Let $R(z)=\sum_{n=0}^{\infty} r_n z^n$ be a power series with $|r_n|=1$ for every $n\ge 0$. We show that for each integer $m\ge 2$, the coefficient sequence of $R(z)^m$ is unbounded. The proof combines Parseval's identity with Jensen's inequality. As a consequence, Conjecture~3.9 of Gawron, Miska, and Ulas \cite{gmu} is confirmed.

We study finite-horizon Markov decision processes under distributional uncertainty in the transition kernels and develop a policy-gradient framework for Wasserstein distributionally robust control. Ambiguity is modeled by Wasserstein balls of common radius centered at state–action-dependent nominal transition kernels, leading to a max–min problem over randomized policies and admissible transition laws. Because the worst-case transition law depends implicitly on the policy parameters, the standard policy-gradient argument does not apply directly. We address this difficulty by combining the dynamic programming recursion with Wasserstein duality and a primal envelope argument. In general, the right and left directional derivatives of the one-step worst-case value are obtained by taking the minimum or maximum expected downstream value derivative over the set of worst-case transition laws. In finite state–action spaces, this set is characterized through the optimal face of a transport linear program, yielding an exact directional-derivative recursion. Under the required stability conditions and uniqueness of the active dual and transport optimizers, the derivative becomes linear in the policy perturbation and admits an explicit vector valued policy-gradient recursion. Building on this representation, we propose a robust actor–critic implementation and evaluate it on benchmark examples.

We study a one-dimensional active-line equation motivated by thin stress-sheet dynamics in the high-Weissenberg Oldroyd-B regime. A positive periodic line density $\rho=m+\eta$ satisfies $\rho_t+cP_0\{\rho\Lambda\rho-(\mathcal H\rho)\rho_s\}+\gamma(\rho-m)=0$, where $\Lambda=\mathcal H\partial_s$ and $P_0f=f-\langle f\rangle$. We prove that every strictly positive smooth initial density of arbitrary size generates a unique global smooth solution. The key is the pointwise cancellation obtained after one differentiation: for $w=\rho_s$, $w_t+c\rho\Lambda w-c(\mathcal H\rho)w_s+\gamma w=0$. Its maximum principle controls the slope globally, while critical-drift Hölder and Schauder estimates close all higher derivatives. We also prove quantitative small-oscillation stability. Independently, an exact fourth-difference sum-of-squares identity gives $\int_{\mathbb T}\rho^2\Lambda^3\rho\,ds\geq0$ for every smooth nonnegative density. The stronger derivative-energy sign leads, in its sharp phase-opposed form, to the cubic convolution inequality $\mathcal C(x)\leq2AE_3$, where $A=\sum_{n\geq1}x_n$, $E_3=\sum_{n\geq1}n^3x_n^2$, and $\mathcal C(x)=\sum_{a,b\geq1}(a+b)(a^2+ab+b^2)x_ax_bx_{a+b}$. The constant $2$ is sharp along critical $n^{-3/2}$ plateaux. We prove the inequality for several cutoff-uniform and infinite-support classes, including selected Schur, dyadic-layer, Mellin, and moment families. The unrestricted inequality remains open, but it is not needed for the global regularity theorem.

We study the stability of a classical family of metrics defined over functions' Gaussian scale-space representations, focusing on the comparison of images (functions of two variables). These metrics have precedents both in harmonic analysis, specifically the theory of Besov spaces, and in classical methods of image processing; special cases are also known to be metrically equivalent to certain Wasserstein distances. We quantify these metrics' robustness to geometric deformations, and introduce rotationally-invariant versions that are stable to changes in angle when comparing tomographic projections. We also describe computationally efficient algorithms for evaluating the metrics from finite samples, and prove their robustness to additive noise. The results are illustrated through numerical experiments.

Power system planning models provide important guidance on long-term investment strategies with significant socio-economic impact. To remain computationally manageable, however, such planning models compromise on the level of complexity with which power system operations and physics are captured. A common approach in most planning models is to collapse multi-stage power system operational processes into a single stage and, as a result, give up on the ability to account for uncertainty in each operational stage. In light of newly emerging load patterns and the continuing adoption of weather-dependent stochastic renewable generation, this uncertainty, however, becomes increasingly impactful on operations, and ignoring it has been shown to cause underinvestment in transmission capacity and flexible resources. In this work, we present a computational approach for power system expansion planning that explicitly considers two-stage day-ahead (DA) and real-time (RT) operational decisions under uncertainty while retaining time-coupling constraints to allow modeling generator ramping and energy storage. To solve the resulting optimization problem efficiently, we employ a projected stochastic gradient descent algorithm combined with a primal-dual optimization framework and an exponential moving average smoothing strategy to improve convergence stability. We evaluate the resulting investment decisions within a two-stage DA and RT simulation framework and compare them with a classic expansion planning model that assumes perfect knowledge of renewable generation. Our experiments show that the proposed framework achieves lower total system costs while ensuring that the implemented technology portfolio achieves set renewable integration targets.

In this article, we derive explicit formulae expressing multiple orthogonal polynomials in terms of standard orthogonal polynomials. We treat both the real-line and unit-circle settings: multiple orthogonal polynomials on the real line (MOPRL) are reduced to orthogonal polynomials on the real line (OPRL), while multiple orthogonal polynomials on the unit circle (MOPUC) are reduced to orthogonal polynomials on the unit circle (OPUC). These formulae also yield corresponding reductions of the Christoffel–Darboux kernels, from the MOPRL kernel to the OPRL kernel and from the MOPUC kernel to the OPUC kernel. In particular, they recover Zinn-Justin's kernel for the external-source random matrix model [arXiv:cond-mat/9703033] and Baik's kernel reduction formula in the spiked source model [arXiv:0809.3970]. We also apply our general results to concrete examples: in the real-line setting, we obtain an explicit expression for the multiple Hermite Christoffel–Darboux kernel in terms of classical Hermite polynomials, while in the unit-circle setting, we use arc-indicator weights to exhibit resonance-type zero escape phenomena for type II MOPUC.