Papers

We study asymptotic probabilities of attaining the maximum in heterogeneous Gaussian samples. In the two-group setting, the first sample has variance $1$ and size $n_1$, while the second has variance $\sigma^2>1$ and size $n_2$. We investigate the probability that the maximum of the standard-variance group exceeds that of the high-variance group. Using the classical extreme-value normalization for Gaussian maxima together with a second-order comparison of the centering terms, we show that this probability admits a non-degenerate limit if and only if $n_1\sim C n_2^{\sigma^2}(\log n_2)^{-(\sigma^2-1)/2}$ as $n_1,n_2\to\infty$ for some $C\in(0,\infty)$. In that regime, the limit admits an integral representation. Outside the critical regime, the comparison necessarily degenerates to $0$ or $1$. We then extend the analysis to finitely many independent Gaussian groups and obtain a generalized integral representation for the limiting winning probabilities. The results provide a complete asymptotic classification for this maximum-comparison problem

Let $\Gamma$ be a minimal connected negative-definite plumbing tree with all vertices of genus zero, and let $Y_\Gamma$ be the oriented link of the corresponding normal complex surface singularity, equipped with its canonical contact structure $\xi_{\rm can}$. We give an explicit Legendrian surgery description of $\xi_{\rm can}$, showing that it is the unique consistent diagram-realizable contact structure on $Y_\Gamma$, up to isomorphism. We then derive a closed-form formula for Gompf's $\theta$-invariant of $\xi_{\rm can}$ in the Seifert fibered case, expressed purely in terms of the Hirzebruch–Jung continued fraction expansions of the normalized Seifert invariants, and prove a recursive leaf-to-root formula for arbitrary plumbing trees. The Seifert formula recovers previously known formulas for lens spaces, dihedral manifolds, and small Seifert fibered spaces with complementary legs, and agrees with the Némethi–Nicolaescu expression via the classical Hirzebruch–Zagier identity. As a final application we show that $\xi_{\rm can}$ strictly minimizes $\theta$ among all diagram-realizable contact structures on $Y_\Gamma$, and we use this to rule out symplectic rational homology ball fillings for a large class of Stein fillable contact rational homology $3$-spheres.

We study a coincidence between two enumerations governed by the same product formula, reminiscent of the Robbins numbers: the unweighted enumeration of twenty-vertex configurations on quadrangular domains with fixed west boundary, and the weighted enumeration of Gelfand–Tsetlin patterns avoiding three equal entries in a row. This coincidence naturally raises the question of whether there is a combinatorial explanation relating these two enumerations. In this paper, we provide such an explanation by constructing a probabilistic bijection between twenty-vertex configurations on quadrangular domains and Gelfand–Tsetlin patterns avoiding three equal entries in a row. Under this probabilistic bijection, the west boundary of a twenty-vertex configuration corresponds to the bottom row of Gelfand–Tsetlin patterns; in particular, the fixed boundary case corresponds to Gelfand–Tsetlin patterns with bottom row~$(1, 2, \ldots, n)$. Combining this correspondence with an enumeration formula of Fischer and Schreier-Aigner for Gelfand–Tsetlin patterns avoiding three equal entries in a row with bounded entries, we obtain an enumeration formula for twenty-vertex configurations with a free west boundary.

A new Bernoulli phase-fitted finite difference method for the Helmholtz equation is introduced, obtained by applying a complexified Scharfetter–Gummel flux to the one-way factors of the operator. The rigorous analysis is developed for the one-dimensional Helmholtz problem with impedance boundary conditions. For the homogeneous problem, the scheme reproduces sampled plane-waves exactly, both in the interior and at the discrete impedance boundary closures. For the inhomogeneous problem, we prove wavenumber-explicit stability, consistency, and second-order convergence estimates for all nondegenerate mesh wavenumbers \(kh\notin\pi\mathbb Z\). Under the fixed-resolution condition \(kh\le s_0<\pi\) and \(kL\ge\pi\), the estimates yield a pollution-free convergence theory. Numerical experiments confirm the plane-wave exactness and the predicted convergence behavior, and show favorable fixed-resolution performance compared with standard and dispersion-corrected finite difference methods.

We give a general method to build categories of combinatorial manifolds, i.e. categories of combinatorial objects satisfying some local property at every "point", as coreflective subcategories of categories of relational presheaves. To do this, we crucially rely on unique factorization systems, and we can interpet our technique as a way of building a model category whose cofibrant objects are exactly the combinatorial manifolds. We then illustrate the usefulness of this point of view by two applications. First we build a category of euclidean precubical sets, i.e. precubical sets that locally look like a grid (of some fixed dimension), and show that it is coreflective in the category of relational precubical sets. This is the combinatorial analog of eulidean locally ordered spaces and the blowup construction from directed topology. Secondly, we show how to give an abstract proof of Kleene's theorem from automata theory by defining "manifold automata" that behave well with respect to concatenation.

We give new upper bounds for the number of nonconstant holomorphic maps depending only on the genus. Our estimates improve previously known bounds. The proof is based on the study of pullbacks of holomorphic differentials, together with techniques from the geometry of numbers and the theory of Jacobian varieties.

We establish a crossed product decomposition theorem for stabilized Cuntz–Pimsner algebras. This extends Cuntz's classical decomposition for Cuntz algebras and reveals an implicit symmetric structure within these algebras. Exploiting this structure, we characterize their simplicity and classify their ideals, tracial weights, and KMS weights for generalized quasi-free flows, revisiting and refining seminal results by Kitamura, Schweizer, and Laca–Neshveyev. We also give a short, elementary solution to the reduced Hao–Ng isomorphism problem for locally compact groups in full generality. Bypassing non-self-adjoint operator algebra techniques used in prior work, our proof relies solely on C*-algebra theory. In contrast, we disprove the full Hao–Ng conjecture by constructing counterexamples. Combining these results, we investigate quasi-free actions on Cuntz algebras. Notably, we affirmatively answer a recent question posed by Izumi on isometric shift-absorption for compact groups.

This paper is concerned with ground states of the defocusing nonlinear Schrödinger equation with a point interaction, \[ \mathrm{i} \partial_t \psi = -\Delta_\alpha \psi + \psi |\psi|^{p - 2} \quad \text{in} \quad \mathbb{R} \times \mathbb{R}^N, \] where $- \Delta_\alpha$ denotes the Laplacian of point interaction centered at the origin with inverse s-wave scattering length $- 2 (N - 1) \pi \alpha$ and we suppose that either (i) $N = 2$, $\alpha \in \mathbb{R}$ and $p > 2$ or (ii) $N = 3$, $\alpha < 0$ and $2 < p < 3$. At sufficiently small masses, (i) we prove that this equation admits ground states, (ii) we obtain some qualitative properties of ground states and (iii) we obtain some results relating ground states with critical points of the associated action functional.

Reset systems can overcome fundamental limitations of linear time-invariant control. The recently introduced notion of scaled (relative) graphs provides a promising framework for developing graphical analysis and design tools for reset systems, in line with widely adopted loopshaping methods for linear systems. The aim of this paper is to derive techniques for over-bounding the scaled graph of reset systems, and obtain insights in their accuracy. We exploit connections between quadratic dissipativity and scaled graphs to recast the over-bounding problem as the search for piecewise quadratic storage functions. Using specific sampling techniques, we reveal a fundamental limitation of general scaled graph approximation methods that are based on quadratic dissipativity.

In 2008, Dan Romik studied in this journal Primitive Pythagorean Triples, or PPTs. In order to do so, he introduced a modified slow (subtractive) Euclidean algorithm, and showed that the underlying dynamical system of this Euclidean algorithm (the “Romik system”), is ergodic and has a $\sigma$-finite, infinite measure, of which is explicitly given. In this paper, the Romik system is further studied. Various basic properties are determined, such as the expansion of rational numbers and quadratic irrationals. Also (a version of) the planar natural extension of the Romik system is obtained, and the $\sigma$-finite, invariant measure is explicitly given, and it is shown that it is ergodic. Furthermore, for Lebesgue almost every $x$ asymptotically half of the regular continued fraction (RCF) convergents of $x$ are among the Romik convergents. We also show that related to the Romik map a “strange” continued fraction can be given. “Strange,” as the set of possible partial quotients (i.e., digits) for any $x\in [0,1]$ in this expansion is $\{ 0, \pm 2\}$. Various properties of this “Romik expansion” are given.

We study two nonlocal versions of the kinetic $p$-Laplace equation: a Gagliardo-type model defined through differences and a Bessel-type model defined via Fourier multiplication. Using critical kinetic trajectories, we derive representation formulas adapted to the kinetic transport-diffusion geometry and establish homogeneous and scale-invariant kinetic Gagliardo-Nirenberg inequalities for nonlocal diffusion, which yield gain-of-integrability estimates for weak solutions to the kinetic $p$-Laplace equations with nonlocal diffusion.

In this paper, we study the complete gradient Ricci solitons $(M^n, g,f)$ with zero radial Weyl curvature, which means that the interior product of $\nabla f$ with the Weyl tensor $W$ is zero, i.e., $i_{\nabla f}W=0$. We classify completely the complete gradient Ricci solitons with zero radial Weyl curvature for the dimension $n\geq 4$.

Galluccio–Loebl and Tesler showed that the perfect-matching polynomial of a graph embedded in an orientable surface of genus $g$ can be written as a linear combination of at most $4^g$ Pfaffians. We show that, in general, exponentially many Pfaffians are necessary. More precisely, for every $g\ge1$, there exists a graph of orientable genus at most $g$ whose perfect-matching polynomial requires at least $(8/3)^g$ Pfaffians in any such linear representation. In particular, for every even integer $n\ge6$, there is a graph on $n$ vertices with Pfaffian number at least $(8/3)^{\lfloor n/6\rfloor}$. Moreover, the lower bound is witnessed even by cubic bipartite matching-covered graphs. We prove this by showing that expressing the permanent of an $n\times n$ matrix of distinct variables as a linear combination of determinants obtained by changing signs of its entries requires exponentially many determinants. As a consequence, we improve a recent linear lower bound on the Pfaffian number due to Junchaya, Miranda, and Lucchesi to an exponential lower bound.

The dominant Earth-Moon tide is accurately described by Newtonian theory, but interpreting a possible off-diagonal residual requires separating its signature from errors in the standard tidal model. In a local lunar principal frame, the Newtonian plus channel varies as cos(2 beta), whereas a symmetric off-diagonal residual enters the sin(2 beta) quadrature. We extend this separation to a globally defined three-dimensional symmetric trace-free tensor with two transverse coefficients projected onto each station vertical. This formulation establishes an exact first-order identifiability limit: lunar-frame errors generate responses indistinguishable from the cross-residual signal, making finite constraints intrinsically prior limited. A joint-prior GLS/Fisher forecast incorporates frame, ephemeris, projection, terrestrial-response, loading, environmental, instrumental, and correlated-network uncertainties. The dimensionless worst-principal-axis uncertainties are 9.513 x 10^-3, 4.490 x 10^-4, and 4.182 x 10^-5 for conservative, moderate, and optimistic synthetic networks. Monte Carlo injection-recovery experiments assess robustness using independent calibration and validation halves. Under the declared response-expanded Stress budget, optimistic formal all-draw coverage falls to 0.317, with a false-positive rate of 0.685; calibrated robust covariance restores held-out coverage to 0.949 and the false-positive rate to 0.051. These results remain conditional on the synthetic assumptions. The Halilsoy cross-polarized sector provides only a local morphology template; no detection or constraint on gamma_H is claimed. The physical contribution is to establish how a generic lunar cross-residual would enter gravimetric observations and what independent constraints are required to distinguish it from standard tidal-model errors, providing a quantitative foundation for future observational tests.

We study the complex property $\partial\partial = 0$ of the boundary operator $\partial$ on a weighted, infinite, and possibly non-locally finite simplicial complex. We give a characterization of this property in $\ell^2$ in terms of the recurrence of the links of simplices. The complex property is essential to ensure that Hodge Laplacians $\Delta^H $ indeed act as $\delta\partial + \partial\delta$ and to decompose $\Delta^H$ into a direct sum of operators acting on $k$-forms. Furthermore, it allows us to define relative cohomology classes, show a respective weak Hodge decomposition, and prove the existence of harmonic Dirichlet eigenforms. We also discuss a transience property for simplicial complexes, that was introduced by Parzanchevski and Rosenthal.

We utilize the structure of quasiautomorphic forms over a Hecke triangle group to define a mapping from a quasiautomorphic form to a vector-valued automorphic form (vvaf). This kind of vvaf we call a Hecke vector-form. First we supply a proof of the functional equations that hold for Hecke vector-forms modulo the group generators. Then, utilizing the multiplier system for these Hecke vector-forms, we prove the opposite direction and complete the bijection. Since the modular group is a special instance of the Hecke triangle groups, our results hold for quasimodular forms.

We present a new family of information-theoretic generalization bounds within the framework of conditional mutual information (CMI). Most of our results are established based on the leave-$m$-out (L$m$O) cross-validation error, with $m$ denoting the number of the hold-out supersamples. Under this setting, we propose a unified CMI-based bound, allowing to envelop and reproduce many known CMI-based bounds and also bridge the gap between the MI- and CMI-based bounds when $m$ tends to infinity. The proposed framework not only provides a unified description of the existing bounds but also develops new, sharper bounds. We show the benefits of the proposed bounds through several simple examples, where the existing results are either inapplicable or looser. Moreover, under the premise that the loss function is bounded, we tighten the CMI quantities involved in the proposed bounds by reducing the number of conditional terms, thereby enhancing the proposed framework. We show empirically that the resulting new bounds improve upon the previously known ones.

For dynamical systems satisfying the approximate $\mathbb{Z}^{d}$ or $\mathbb{Z}_+^{d}$-product property and asymptotically entropy expansiveness, we establish a precise description of the structure of their space of invariant measures. In particular, we prove that the set of ergodic measures with any given intermediate entropy is generic in certain natural subspaces. As a consequence, this result confirms Katok's conjecture on the existence of ergodic measures with arbitrary intermediate entropy for such systems.

Stochastic Galerkin methods offer unexplored potential for the numerical simulation of parabolic problems with random variables, in particular if they are combined with variational discretizations of the space and time variables. Due to the high dimensionality, the solution of the arising algebraic systems do not become feasible without efficient solvers, preconditioners, and software architectures. A stochastic Galerkin discretization with an embedded slabwise finite element approximation of the space and time variables is proposed and analyzed numerically. For solving the linear systems, GMRES iterations are block-preconditioned by a geometric multigrid (GMG) technique using a local Vanka smoother for the space-time subsystems. Monte-Carlo methods are also used for solving random parabolic problems and studied here for the purpose of comparison. The Monte-Carlo approach is built on the space-time finite element formulation together with the GMRES-GMG solver technology. All algorithms have been implemented in a unified matrix-free framework based on the deal.II software library. Comparative numerical evaluations illustrate the performance properties of both approaches, including convergence of the discretizations and statistics of the algebraic solver. Superiority of the stochastic Galerkin approach is observed.

Data-Driven Continuum Mechanics – the continuous counterpart of Data-Driven Computational Mechanics – is a modern paradigm that enhances classical continuum mechanics by incorporating finite sets of experimental material data directly, avoiding any form of constitutive modeling. Despite recent progress, its analytical foundations remain at an early stage. In this work, we establish a rigorous functional-analytic framework for the data-driven stress problem under purely homogeneous normal Neumann boundary conditions. The problem is formulated as finding a stress field (satisfying the balance of linear and angular momenta and the boundary conditions) that is closest, in an $L^p$-sense, to an auxiliary stress field that is simultaneously sought and locally resembles a finite discrete set of experimental stress states. Our analysis relies on two key ingredients. First, the divergence operator induces a topological isomorphism between the space of symmetric stress fields modulo its kernel and the space of loads balanced by rigid-body motions, ensuring the existence of an equilibrated response. Second, the finiteness of the material data set guarantees proximinality in the stress space, which in turn yields a complete existence and uniqueness theory for solution equivalence classes. Together, these two properties provide a rigorous mathematical foundation for the data-driven stress problem under purely homogeneous normal Neumann boundary conditions.

We determine cuspidal character sheaves explicitly for all (GIT) stably graded exceptional Lie algebras.

In this article, we establish integration by parts formulas for the solutions of McKean-Vlasov stochastic differential equations with jumps under elliptic coefficients. The derived formulas accommodate both derivatives with respect to real-valued variables and measure-valued variables, interpreted through the Lions' derivative. As applications, we obtain estimates for the derivatives of the density functions of the McKean-Vlasov SDEs, and relying on the integration by parts formulas, we subsequently prove the existence and uniqueness of classical solutions to the associated PDEs with irregular terminal conditions.

Let $G$ be a finite group. Let $\mathcal{N}(G)$ be the lattice of normal subgroups ordered by inclusion, regarded as an abstract lattice. Define $\operatorname{LatAut}(G) := \operatorname{Aut}(\mathcal{N}(G))$. The LatAut tower is the sequence defined by $G_0 = G$, $G_{n+1} = \operatorname{LatAut}(G_n)$. Let $G$ be a tower group if $G \cong \prod_{k \geq 3} S_k^{a_k}$ with finitely many $a_k \neq 0$. We establish the following for tower groups. Product Formula. $\operatorname{LatAut}\!\bigl(\prod_{k \geq 3} S_k^{a_k}\bigr) \cong S_{a_4} \times S_B$, where $B = \sum_{k \geq 3,\, k \neq 4} a_k$. Termination Theorem. For every tower group $G_0$, we prove that $G_3 = 1$, and that this bound is sharp. The proof applies Goursat's lemma to classify $\mathcal{N}(G)$ into three families parameterised by admissible triples $(J,\mathbf{P},H)$ as sub-products, sign-parity elements, and mixed elements, and uses the Krull–Schmidt theorem to identify the direct factors $S_k^{(k,i)}$ as precisely the nontrivial indecomposable complemented elements of $\mathcal{N}(G)$ (the complemented elements being exactly the full sub-products). These results do not extend to groups outside the tower-group family.

In this paper, we study the notion of smooth $\infty$-categories within the framework of a six-functor formalism. By leveraging the theory of condensed mathematics and analytic stacks, we apply these results to demonstrate that a rigid analytic variety is smooth if and only if its associated category of nuclear sheaves is smooth. Furthermore, we relate the compact generation of the category of nuclear sheaves to the algebraization of the rigid analytic variety; these results are then employed to obtain an example of a non atomically generated but internally smooth category.

We construct a hypersimplicial subdivision of the $r$-dilation of the $i$-th hypersimplex of dimension $d$ that provides a geometric proof of the Brenti–Welker identity.

Multiple-input multiple-output (MIMO) radar has waveform diversity and large spatial degrees of freedom (DoFs), making it attractive for high-resolution sensing. Scaling MIMO radar to massive arrays can further improve sensing performance, but it also increases hardware cost, power consumption, and digital processing complexity. The microwave linear analog computer (MiLAC) can tackle these challenges by moving linear operations from the digital domain to the analog domain. MiLAC has shown promising benefits for communications in recent studies and this paper identifies its potential for radar sensing. Specifically, we consider both MiLAC-aided transmit beamforming and receiver-side two-dimensional discrete Fourier transform (2D-DFT)-based direction-of-arrival (DoA) estimation. For transmit beamforming, we formulate a weighted Cramer Rao bound (CRB) minimization problem under lossless and reciprocal MiLAC constraints and propose a penalty dual decomposition (PDD)-based iterative algorithm to address the non-convex problem. We further prove that MiLAC-aided and fully-digital beamforming achieve the same CRB. For receiver processing, we show that the 2D DFT can be implemented by a lossless reciprocal MiLAC, which enables analog-domain DoA estimation without digital optimization. Numerical results confirm the theoretical finding and show that the MiLAC-aided approach achieves the same CRB and DoA estimation performance as the fully-digital benchmark. Meanwhile, hardware cost and power consumption are reduced because only low-resolution DACs are required at the transmitter, while RF chains and ADCs are eliminated at the receiver. Moreover, performing the 2D DFT in the analog domain eliminates all digital DFT operations for DoA estimation.

There is a surge of need for fresh information with the overwhelming proliferation of the Internet of Things (IoT) applications. To characterize the information freshness perceived by the destination, the age of information (AoI) has been proposed. In this paper, we consider an IoT system with multiple devices sending status update packets to a central controller through time-correlated Markov channels and assume that the instantaneous channel states are not available to the central controller before making scheduling decisions. To ensure information freshness, we investigate a timely scheduling problem that minimizes the total expected time-average AoI under a strict communications bandwidth constraint. We formulate this problem as a partially observable restless multi-armed bandit problem. Using Lagrangian relaxation, we decouple the relaxed problem into multiple sub-problems and prove the threshold structure of their optimal policies. Armed with this property, we establish the indexability for the decoupled problem and design an algorithm to compute the Whittle's index. To reduce implementation complexity, we further derive the Whittle-like index in closed-form for low-complexity scheduling. Simulation results show that the proposed index-based policies outperform the baselines, remain close to the optimal policy or relaxed lower bound, and are especially effective when scheduling resources are limited or the network size is large.

In this paper, we show that the treewidth of the $n \times n$ toroidal grid is $2n-1$ for all $n \ge 5$. This closes the gap between the previously known upper bound of $2n-1$ (Ellis and Warren, DAM 2008) and the lower bound of $2n-2$ (Kiyomi, Okamoto, and Otachi, DAM 2016). To establish the matching lower bound, we construct a bramble of maximum order by utilizing maximum components obtained after removing $2n-1$ vertices. Our construction relies on the vertex-isoperimetric properties of the infinite grid to establish tight lower bounds on neighborhood sizes, combined with a careful analysis of balls of radius $n/2-1$ and their boundaries to overcome structural obstructions when $n$ is even.

We provide a first systematic treatment of so-called rectangular multispectral perturbation theory. With their paper from 2003, Hochstenbach and Plestenjak ["Backward Error, Condition Numbers, and Pseudospectra for the Multiparameter Eigenvalue Problem" in Linear Algebra and its Applications] extended perturbation theory from one-parameter eigenvalue problems to multiple spectral parameters. After two decades, we take it one step further and consider a different manifestation of the multiparameter eigenvalue problem that consists of one matrix equation with rectangular coefficient matrices. We perform a norm-wise backward error analysis, define condition numbers for both eigenvalues and eigenvectors, and introduce the pseudospectrum while also considering the computational implications of working with multiple spectral parameters. The rectangular shape hampers a direct application of the existing definitions and properties. For example, the left null space at a given eigenvalue is non-trivial and the dimensions of the left and right eigenvectors are different. Through numerical examples, we illustrate and link the different concepts from the perturbation theory. A system identification application seem to suggest that, in optimization-driven problems for which multiparameter reformulations exist, the globally optimal solutions tend to coincide with the best-conditioned eigenvalues.

In many stochastic models, the observables of interest are naturally encoded in double transforms (e.g., Laplace transforms) that couple spatial and temporal variables. Notably, the double transform often provides the only analytically tractable starting point for the study of processes with correlated increments or path constraints. We extend the Tauberian approach for precise large deviations of stochastic processes belonging to the domain of attraction of spectrally positive stable laws, previously developed for single-variable Laplace–Stieltjes transforms [9], to the bivariate setting. This methodology provides a direct route to asymptotic behaviour that is otherwise difficult to characterize using single-transform techniques. As illustrative examples, we derive precise large deviations for random sums with increments correlated to the stopping time and for path-dependent observables of random walks constrained to remain positive.