Papers

We introduce Bernstein constraint complexes, a representation of finite element differential complexes in which every triangle of a triangulation keeps its own Bernstein–Bézier coefficients and global continuity is imposed only through smoothness functionals attached to edges and vertices. Continuity of scalar splines, tangential continuity of spline vector fields, normal continuity of spline vector fields, and higher smoothness differ only in the edge functionals. No global basis is constructed. The structural result is a coefficient-level commuting identity: on every edge the smoothness functionals of a Bernstein derivative are explicit combinations of the smoothness functionals of its argument, through univariate Bernstein difference matrices. This gives a local proof that the kernels of the smoothness matrices form a subcomplex, for the de Rham traces and for componentwise $C^r$ spline profiles, and a computable row-space test for candidate smooth profiles. We realize the lowest-order Powell–Sabin exact sequence in this form and confirm its exactness by ranks. The constrained Galerkin problem is solved in the broken coefficients with exact smoothness equations; we prove equivalence with the conforming method, treat semidefinite element operators through a pseudoinverse interface system, and give a singular pencil whose finite spectrum is the constrained Maxwell spectrum. Under the covariant and contravariant surface Piola maps the reference functionals of the de Rham sequence transfer unchanged to conforming curved triangulations. Experiments compare the null-space method, the augmented Lagrangian iteration of Awanou, Lai and Wenston, and direct solution of the saddle system on the same problems, and cover planar source and eigenvalue problems, a $C^1$ biharmonic solve, the exact sphere, and an embedded hyperboloid.

We study the numerical evaluation of contour integral representations for solutions of the linearized second and fourth Painlevé equations. The construction combines a nonlinear background solution, continuation of canonical Lax-pair columns, normalization matching, and quadrature over cycles with decaying branches in distinct sectors. Accuracy is assessed using a reference fundamental matrix, differential-equation residuals, and variations of monodromy data computed independently from the spectral problem. An exact identity describes the propagation of initial-basis errors. A stagewise a posteriori refinement criterion uses increments in monodromy variations to allocate numerical accuracy. The contour algorithm is compared with the Dormand–Prince method on a regular background, in a region of rapid solution growth, and through hard loss of stability followed by approximately regular oscillations. Accumulated offsets in monodromy variations and subsequent drift are measured separately. The experiments show how drift depends on spectral conditioning, quadrature accuracy, and error allocation between initial and subsequent parts of the computation. Five additional initial-data and parameter cases demonstrate the dependence of comparative performance on the nonlinear background.

Transient pressure diffusion in heterogeneous porous media becomes difficult to resolve efficiently when permeability is discontinuous and spans several orders of magnitude. We develop a domain-decomposed random-feature evolutional deep neural network (DD RF-EDNN) that separates spatial approximation from temporal evolution. Permeability-informed random features are compressed into an orthonormal pressure space, and a conservative finite-volume operator is projected onto this space so that only the reduced coordinates are advanced in time. This formulation retains the dissipative structure of the discrete flow problem without repeated neural-network optimization. The analysis quantifies dictionary and singular-value truncation errors, proves contractivity under an energy-consistent dissipativity correction, and derives a conditional grid-level estimate for homogeneous dynamics and time-independent data admitting a steady lifting. Experiments on low-permeability inclusions, high-conductivity channels and blocks, and three-dimensional grids demonstrate consistent accuracy across distinct coefficient structures. The method reaches a final-time relative $L^2$ error of $9.54\times10^{-4}$ in the principal benchmark and maintains errors on the order of $10^{-2}$ at a permeability contrast of $10^3$. Numerical diagnostics further show that the observed error balance and temporal convergence are consistent with the analysis. These results support DD RF-EDNN as a structure-preserving and interpretable reduced formulation for transient simulations on prescribed heterogeneous media.

Fractional operator algebras provide a natural setting for linking algebraic normal forms to causal inversion, but arbitrary smooth multipliers and Volterra kernels destroy finite fractional rewriting. We construct an algebra generated by polynomial multipliers, fractional powers indexed by an additive subgroup $\Gamma\subset\mathbb{R}$ containing $\mathbb{Z}$, and flat causal convolution operators. Its finite polynomially weighted diagonal-flat convolution operators form the smallest two-sided ideal containing the flat generators, polynomial multipliers are intrinsic, and distributional separation yields a unique finite normal form with an additive order filtration. For monic normal forms, the leading power selects a canonical causal right factor that extends from the flat core to the natural range $J^\alpha C[0,T]$. Completing the negative-order convolution sector in $L^1$ admits integrable Sonine and Prabhakar kernels, while an independently prescribed continuous bivariate Volterra term remains external. Under injectivity and power-type kernel bounds, variable lower-order terms and this bivariate perturbation reduce to one weakly singular second-kind equation, whose classical resolvent exists on every finite interval without smallness assumptions. A finite boundary-jet block incorporates prescribed initial data without changing the reduced kernel. Sonine, distributed-order, and regularized-Prabhakar realizations follow. In the noninteger Prabhakar regime, an exact range criterion characterizes the classical $AC^m$ subdomain, and an explicit Riemann–Liouville example proves that it is strictly smaller than the continuous natural range.

We study root-of-unity sections of quotients formed from dilated Euler products. At the boundary point \(q=1\), odd sections tend to nonzero cyclotomic constants, whereas even sections have infinitely many simple real poles. We determine the locations, spacing, and residues of all sufficiently late poles. In the cubic case we find the leading oscillatory error in an elementary approximation and prove that its exponential scale is sharp. At the origin we determine exact contact, including a gap-dependent parity law, and integral divisibility with the first nonzero coefficient determined for every modulus \(|k-1|\ge2\). For Euler powers \(k\ge2\), a moving formal pole controls the section error at small fixed nomes; a real zero-free interval gives a further convergence regime. For the twenty-fourth power we prove explicit CM approximation bounds and a signed asymptotic as the dilation increases.

Let $$ M_{r,p}^{(d)}(g)=\int_{\mathbb R^d}|x|^r|g(x)|^p\,dx, \qquad 1<p<\infty. $$ Write $p'=p/(p-1)$ and $\omega_d=|B(0,1)|$. A direct step in the Wigderson–Wigderson moment method asks for a single-function estimate $$ \lVert g\rVert_1\le K\lVert g\rVert_q^{\,1-p\eta} \bigl(M_{r,p}^{(d)}(g)\bigr)^\eta, \qquad \eta=\frac{d(1-1/q)}{r+d-dp/q}. $$ We give a self-contained proof that, in its scale-invariant regime, this estimate holds exactly when $r>d(p-1)$. We then quantify the transition: the best constant blows up like $[r-d(p-1)]^{-1/p'}$ at the critical line, while on the annulus $1\le |x|\le R$ the endpoint loss is exactly $(d\omega_d\log R)^{1/p'}$. The critical line nevertheless has two sharp Lorentz endpoints. The strong moment controls $L^{1,p}$, $$ \lVert g\rVert_{L^{1,p}}\le \omega_d^{1/p'}M_{d(p-1),p}^{(d)}(g)^{1/p}, $$ and strengthening the weighted norm from $L^p$ to $L^{p,1}$ restores the strong target, $$ \lVert g\rVert_1\le \omega_d^{1/p'} \lVert |x|^{d/p'}g\rVert_{L^{p,1}}. $$ More generally, every power moment has a sharp target Lorentz space, and its secondary index is optimal. These results identify the endpoint lost by the direct same-$q$ reduction. They do not obstruct auxiliary-norm versions of the Wigderson framework, which reach the Cowling–Price range by a different route.

We prove some residually reducible $p$-adic Galois representations of number fields arise from modular forms. We study the universal deformation ring arising from deformations satisfying the Fontaine-Laffaille condition at primes over $p$. Under certain conditions, we establish the reduced universal deformation ring is a discrete valuation ring. The method uses pseudocharacters and certain bounds on Selmer groups, where the ideal of reducibility as defined by Bellaïche and Chenevier is shown to be maximal and principal. A self-dual assumption is not needed in our argument. The main result on deformations applies to $n$-dimensional representations. In applications, a congruence between Hermitian modular forms due to Klosin is used with our results to prove modularity of some 4-dimensional $p$-adic representations of the imaginary quadratic field $\mathbf{Q}(i)$.

Let $q_m=2m/(m+1)$ and put \[ \beta_0=\frac{3}{2}+\frac{1}{\log 2}=2.9426950408\ldots, \] where $\log$ is the natural logarithm. We give a proof scheme showing that, for every $\varepsilon>0$, there is $C_\varepsilon<\infty$ such that every complex-valued function $f:\{-1,1\}^n\to\C$ of Fourier degree at most $m$ satisfies \[ \left(\sum_{A\subseteq[n]}\abs{\wh f(A)}^{q_m}\right)^{1/q_m} \le C_\varepsilon m^{\beta_0+\varepsilon}\norm{f}_\infty. \] The improvement over the $m^9$ estimate of the earlier draft has two ingredients. The first three Fourier levels are estimated at the scales $m$, $m^{3/2}$, and $m^{5/3}$. These losses are encoded in the weight $M^{\mu_r/r}r^B\min\{r,L_M\}^8$, $L_M\asymp_B \log (M+1)$, with $\mu_r=\lceil3r/2\rceil$ for $r\ge2$. A parity-compatible central window handles $r<L_M$, while a shrinking balanced window handles $r\ge L_M\asymp_B\log(M+1)$, no parity is used in this regime. The leading high-degree loss–gain factor is $\ee\,2^{-B}$; careful uniform bounds close the bootstrap for every $B>1/\log2$. The formula for $\mu_r$ is explained below.

We study $(\mathbf{O}^\top \mathbf{A} \mathbf{O} + \mathbf{B})^{-1}$, where $\mathbf{A}, \mathbf{B} \in \mathbb{R}^{N \times N}$ are symmetric positive definite, with limiting eigenvalue distributions ${\rm F}_{\mathbf{A}}, {\rm F}_{\mathbf{B}}$ and spectra bounded and bounded away from zero, and $\mathbf{O}$ is Haar orthogonal. We analyze the deviation of deterministic-equivalent-type approximation $\boldsymbol{\Delta}\doteq (\mathbf{O}^\top \mathbf{A} \mathbf{O} + \mathbf{B})^{-1}- \mathbf{S}^{-1}$ where $\mathbf{S} = \mathbf{B} - z^\star \mathbf{I}$. First, let $\mathbf{O}$ be independent of $(\mathbf{A}, \mathbf{B}, \mathbf{u})$, $\|\mathbf{u}\| = 1$. There is sequence of a standard Gaussian vectors $\mathbf{g}_N$ independent of $(\mathbf{A}, \mathbf{B}, \mathbf{u})$ with $\| \mathbf{S} \boldsymbol{\Delta} \mathbf{S} \mathbf{u} - \sqrt{\frac \tau N}\, \mathbf{g}_N \| \to 0$ a.s. Here $z^\star = -{\rm R}(-\chi)$ and $\tau = {\rm R}'(-\chi)(1 + \eta {\rm R}'(-\chi))$, where ${\rm R}$ is the R-transform of ${\rm F}_{\mathbf{A}}$ and $(\chi, \eta)$ are the first two inverse moments of the free additive convolution of ${\rm F}_{\mathbf{A}}, {\rm F}_{\mathbf{B}}$. Let $\mathbf{K} = \sum_{i \le \lfloor N^q \rfloor} \lambda_i \mathbf{u}_i \mathbf{v}_i^\top$, $q \in [0, C]$, with unit-norm, not necessarily orthogonal $\mathbf{u}_i, \mathbf{v}_i$ and $\limsup_N \max_i |\lambda_i| < \infty$ a.s. If $\mathbf{O}$ is independent of $(\mathbf{A}, \mathbf{B}, \{\mathbf{u}_i, \mathbf{v}_i\})$, then $N^{-q} {\rm tr}(\mathbf{K} \boldsymbol{\Delta}) \to 0$ a.s. The proofs use a dynamical mean-field approach: the generating functions of the free Appell polynomials give a linear dynamics approximating the quantity of interest in the iterated large-$N$, large-time limit, whose fluctuations we analyze.

Octopus crawling motivates soft robots that exploit redundancy, yet discovering and organizing diverse coordination modes for adaptation remains challenging. To address this, we introduce a Diffusion-based Uncertainty-aware Optimization (DUO) algorithm that learns demonstration-free crawling controllers for a simulated, muscle-actuated CyberOctopus. This work represents the first application of diffusion-based control to soft multi-arm robots in contact-rich simulations. By embedding a variety of locomotion behaviors within a shared control distribution, this approach enables the simulated octopus to navigate dynamic physical constraints, demonstrating that learned coordination diversity inherently facilitates robust adaptation. The main contributions include: (i) a symmetry-structured policy representation that folds radially equivalent controllers into a canonical directional sector, (ii) an online black-box optimization strategy, the DUO algorithm, that discovers and retains diverse coordination modes, and (iii) a control editing technique that adapts existing controllers to novel actuator constraints without retraining. These results show how learned coordination diversity makes motor abundance a practical resource for adaptation in soft multi-arm robots.

Generative AI changes the conditions under which graduate mathematics is learned, assessed, written, and defended. The central claim of this paper is that mathematics graduate programs should respond to the moment by clarifying what graduate mathematics education is trying to teach and assess. In most ways, the goals of mathematics education have not changed. Rather, with changing tools it has become more essential than ever to make clear the goals of mathematical training. We use the term mathematical judgment to refer to the capacity to evaluate mathematics (e.g., claims, definitions, examples, proofs, analogies, computations, uses of tools, research directions) as mathematically sound, useful, well-posed, and appropriately justified. The recommendation is to center training on mathematical judgment, and we examine possible policies for graduate programs in Mathematics to this end.

Chui's conjecture asks whether the average electrostatic field generated by $N$ unit point charges on the unit circle is minimized when the charges are equally spaced. Abakumov, Borichev, and Fedorovskiy proved an analogue of this conjecture in weighted Bergman spaces, showing that the corresponding norm is minimized uniquely by the $N$th roots of unity. In this paper, we give two alternative approaches to this weighted minimization problem. The first is a direct calculus-based proof. We reduce the norm to a pairwise interaction energy, identify the equally spaced configuration as a critical point, and analyze the resulting gradient system and Hessian. The second approach connects the problem with discrete energy minimization on spheres by Cohn and Kumar. By rewriting the Bergman interaction as a completely monotonic function of squared Euclidean distance, we recover the minimizing property of the regular $N$-gon. This second argument also extends the minimization result beyond the concave-weight setting of Abakumov, Borichev, and Fedorovskiy, including the power weights $g_\alpha(t)=t^\alpha$ for every $\alpha>0$.

Let $\mathcal{A}_0$ be the set of finitely generated free groups on some countable alphabet and and define $\mathcal{A}_r$ inductively as the union of $\mathcal{A}_{r-1}$ with the set of amalgamated free products $G_1\ast_{\ZZ}G_2$, where $G_1,G_2\in \mathcal{A}_{r-1}$. We show, for all $r$, that every element of $\mathcal{A}_r$ has linear conjugator length, quantifying a result of Larsen and extending a result of Kharlampovich and Myasnikov. This is achieved by a linear conjugator length bound on a family of iterated HNN extensions of free groups. Also, in an appendix, we adapt a proof of A. Wei$\ss$ to show that all Generalized Baumslag-Solitar Groups have linear conjugator length. As a consequence, we obtain that one-relator groups with non-trivial center have linear conjugator length, giving evidence for a conjecture of Bridson, Riley, and Sale.

For a suitable class of structures $\mathcal{K}$, let ${\rm Sh}(\mathcal{K}_\lambda^{\rm op},J_{\rm at})$ denote the atomic sheaf topos over the opposite of the category $\mathcal{K}_\lambda$ of structures in $\mathcal{K}$ of cardinality at most $\lambda$, with embeddings as morphisms. In this article, we analyze the relationship between model-theoretic properties of $\mathcal{K}$ and logical properties of ${\rm Sh}(\mathcal{K}_\lambda^{\rm op},J_{\rm at})$. For an infinite cardinal $\lambda$, we show that $\mathcal{K}$ is Galois $\lambda$-stable if and only if the atomic sheaf topos ${\rm Sh}(\mathcal{K}_\lambda^{\rm op},J_{\rm at})$ satisfies the axiom of choice for $\lambda^+$-indexed families. We further show that every structure in $\mathcal{K}_\lambda$ extends to a maximal structure if and only if ${\rm Sh}(\mathcal{K}_\lambda^{\rm op},J_{\rm at})$ satisfies the internal axiom of choice. If $\mathcal{K}_{\rm fin}$ is a class of finite structures, the cardinalities of the structures in each connected component of $\mathcal{K}_{\rm fin}$ have a finite upper bound if and only if ${\rm Sh}(\mathcal{K}^{\rm op}_{\rm fin},J_{\rm at})$ satisfies the internal axiom of (countable) choice. Furthermore, via the topological Galois representation, we establish a correspondence between the criteria for the axiom of choice for atomic sheaf toposes and those for permutation models in set theory.

GPU implementations of the Muon optimizer and of first-order semidefinite programming (SDP) solvers share one computational pattern: a matrix factorization is replaced by a fixed-depth polynomial filter applied after normalization. When a few dominant spectral components carry most of the input's scale, normalization pushes the remaining spectrum toward zero, where the filter is least accurate. We propose spectral deflation: estimate the dominant components, remove them, filter the normalized residual with the unchanged filter, and restore them. Deflation preserves the target matrix function, and we prove that it strictly reduces the finite-step error of the classical Newton–Schulz family. Implemented with batched randomized SVD for Muon and warm-started subspace tracking for ADMM, deflation consistently improves GPT-2 pretraining over the corresponding Muon baselines with both the Newton–Schulz and Polar Express mappings, also in wall-clock time, and lowers the KKT residuals of factorization-free ADMM on large-scale SDPs within a similar projection time.

We address a problem on large signed sums of unit vectors that arose in work of Brugger, Fiedler, González Merino and Kirschbaum and was later formulated in its present form by Ambrus and Nietert. Given $d+1$ unit vectors $u_1,\ldots,u_{d+1}$ in $\mathbb R^d$, with $d\ge2$, the problem asks for the smallest possible value of \[ \max_{\varepsilon_i=\pm1} \left\|\sum_{i=1}^{d+1}\varepsilon_i u_i\right\|. \] We prove that this value is $\sqrt{d+2}$. We also determine all equality cases: up to independent sign changes and orthogonal transformations, they consist of the vertices of a centered regular simplex of positive even dimension together with an orthonormal basis of its orthogonal complement.

We study linear delay differential equations whose delay terms are represented by time-dependent finite signed measures. This framework accommodates discrete, distributed, and singular delay contributions without requiring continuity of the coefficient paths in time. We establish well-posedness in the Carathéodory sense and sequential dependence of solutions on the measure-valued parameters under almost-everywhere weak-* convergence and a common total variation bound. A central part of the analysis associates a signed measure on the time–delay domain with each coefficient path and establishes an equivalent integral formulation. This representation yields three equivalent characterizations of a $\sigma$-algebra on the parameter space suited to measurability of the continuation operator. A measurable fixed-point argument then provides measurability of the solution maps, despite the possible failure of continuity in the product weak-* topology. For coefficients generated by a measure-preserving base flow, these results lead to measurable linear skew-product semiflows on continuous and absolutely continuous phase spaces. Under additional ergodic hypotheses and the assumption of a finite top Lyapunov exponent, eventual compactness and regularization yield Oseledets decompositions with the same Lyapunov exponents and multiplicities on both spaces.

We consider a double flag variety of the form $L/Q\times G/P$ where $G$ is a connected reductive group and $L$ is a Levi subgroup of $G$. The Levi subgroup $L$ acts diagonally on this double flag variety and a basic problem is to classify pairs of parabolic subgroups $P\subset G$ and $Q\subset L$ such that there are finitely many orbits for the considered action. We solve this problem in the case where $G$ is the general linear group by using representations of quivers.

A qKZ-type additive discrete flat connection in characteristic $p$ has $p$-curvature operators. They are commuting automorphisms of the connection. We present a Bethe-ansatz-type construction of eigensections and eigenvalues of the $p$-curvature operators. An eigensection of $p$-curvature operators is a discrete hypergeometric sum over the finite lattice $\mathbb{Z}^r/p\mathbb{Z}^r$. Thus, the $p$-curvature eigensections are constructed by a finite, discrete analogue of a hypergeometric integral.

We present tight lower bounds for the size of any linear matrix inequality representation of the Euclidean ball, and on the semidefinite extension complexity of the ball. Specifically, we show that any linear matrix inequality representing the Euclidean ball must involve matrices of size at least $n$, and any spectrahedral extension of the Euclidean ball must involve matrices of size at least $\lceil 2 \sqrt{n-1}\rceil$. These match the upper bounds given by existing explicit constructions exactly. Our proofs rely on elementary facts about the dimension of the set of faces of spectrahedra, and indeed extend to any convex set whose extreme points form a semialgebraic set.

Gaussian processes (GPs) provide principled probabilistic predictions while encoding prior knowledge, including equivariances. Yet, their use in large-scale scientific problems is limited by computational cost. Equivariant neural networks are common but typically lack the uncertainty quantification offered by GPs, which is valuable in applications such as molecular research. High-dimensional inputs and large symmetry groups further demand scalability. We establish results pertaining to the interplay of GP equivariance and conditioning and leverage them to obtain equivariant sparse GPs through suitable mean functions and covariance kernels. We instantiate this framework with a flexible class of integration-free equivariant kernels, yielding scalable and data-efficient GP inference. In particular, we introduce triply scalable equivariant Gaussian processes. We employ equivariant sparse variational Gaussian processes for $\mathrm{SO}(2)$-equivariant vector fields and molecular property prediction. Alongside the SVGP, we develop a matrix-free equivariant full-GP implementation that combines an exact Kronecker reduction with preconditioned conjugate-gradient solves, enabling fast and scalable evaluation of the full joint predictive density. We further compare different approaches for selecting inducing points in the equivariant sparse GP models. Our test cases include synthetic $\mathrm{SO}(2)$-equivariant fields as well as the prediction of electric dipole moments of N-methylformamide based on quantum chemistry simulations, achieving accurate, uncertainty-aware predictions at a fraction of the computational cost of classical GP inference.

The permanent and mixed discriminant of positive matrices are classic problems for which we do not expect an efficient algorithm for exact computation. Thus much work has been done to understand how well we can bound and approximately compute these quantities. One line of research in this area begins with the results of the first author, where van der Waerden lower bounds of $\frac{n!}{n^n}$ are proven for doubly stochastic inputs for both problems, using a simple proof via stable polynomials. Along with the bound itself, the same techniques are used to show that the permanent and mixed discriminant are uniquely minimized at a certain natural symmetric input. In this paper, we generalize those results in two ways. First, we extend the unique minimization results beyond doubly stochastic inputs to other marginals which are near doubly stochastic. This yields the first such unique minimization results for the mixed discriminant beyond the doubly stochastic case. We also discuss why one cannot hope similar results to hold in general for all marginals. Second, we extend the unique minimization result for real stable polynomials to strongly log-concave (aka Lorentzian) polynomials in the doubly stochastic case. This captures an analogous previous result on unique minimization for the mixed volume. Finally, we discuss various open problems related to these results.

We prove the fixed-potential conjecture of Fefferman, Shapiro, and Weinstein for inversion-symmetric magnetic double wells. We construct a number $L_0>0$ and one smooth, compactly supported, nonpositive, nonradial single-well potential $v$, independent of both the coupling parameter $\lambda$ and the well displacement, such that for every fixed $L\ge L_0$ the associated magnetic double-well Hamiltonian has infinitely many exact degeneracies of its two lowest eigenvalues as $\lambda\to\infty$. The magnetic hopping coefficient also vanishes along an infinite sequence tending to infinity. The potential consists of a radial core and two small, reflection-related perturbations supported on one-sided quadratic cusps. Their log-flat profiles isolate two dominant hopping contributions, which we evaluate by steepest descent after inserting the radial ground-state asymptotics. The resulting asymptotic cosine expression, together with error bounds for the even–odd splitting, yields infinitely many exact eigenvalue crossings and changes of ground-state parity.

We use the series representation of the Bessel functions of type $B_N$ in terms of Jack polynomials and show that $$\lim_{k_1\to\infty} J_{(k_1,k_2)}^B(x,k_1y)^{1/k_1}= 2^N\prod\limits_{i=1}^N \Bigg( e^{\sqrt{1+x_i^2 y_i^2}-1}\cdot \frac{1}{\sqrt{1+x_i^2 y_i^2}+1} \Bigg)$$ for $x,y\in\mathbb R^N$ and $k_2\ge0$. Moreover, the known Laplace-type integral representations for $N\ge1$ and $k_2=0,1/2,1,2$ and for $N=2$ and $k_2>0$ by Rösler and Demni respectively lead to related limits for $$\lim_{k_1 \to \infty} \partial_{x_j} J_{(k_1,k_2)}^B(x,k_1y) /(k_1 \cdot J_{(k_1,k_2)}^B(x,k_1 y)) \quad (j=1,\ldots,N).$$ These limits lead to weak limit results for the associated Bessel processes with drift. For $k_2=1/2,1,2$, these limit results have applications to radial parts of Brownian motions with drift on the $M\times N$-dimensional matrices over $\mathbb R,\mathbb C$, and the quaternions for $M\to\infty$. We also discuss these limits in the Dunkl case $N=1$.

We prove sharp $L^{4}$ Strichartz estimates for two-dimensional periodic Schrödinger flows with integral diagonal symbols $ak_{1}^2+bk_2^2$ where $a,b \in \mathbb Z \setminus \{0\}$. The estimates exhibit a sharp arithmetic dichotomy according to whether $-ab$ is a square: for frequencies bounded by $N$, the optimal loss is $(\log N)^{1/4}$ when $-ab$ is not a square and $N^{1/4}$ when $-ab$ is a square. As a consequence, we show that arbitrarily small perturbations of the rectangular aspect ratio can lead to a transition between global well-posedness and norm inflation for cubic hyperbolic NLS.

In this paper, we shall use the $L^2$ existence theorems for $\partial\bar{\partial}$ provide by Guy Laville to etablish for existence theorem for the $\partial\bar{\partial}$-Neumann operator on any bounded pseudoconvex starred domaine $\Omega$ in $\mathbb{C}^n$ based on the work of D. Spencer on the $\bar{\partial}$-Neumann. As a result, we obtain a new method for resolving the $\partial\bar{\partial}. $

We prove that every finite Helly complex is isomorphic, at the level of its $1$-skeleton, to the Hellyfication (equivalently, the discrete injective hull) of its combinatorial boundary equipped with the metric induced from the complex. In particular, the boundary-rigidity phenomenon for Helly complexes proved by Blufstein-Chalopin-Chepoi admits a canonical injective-hull interpretation, independent of reconstruction procedures based on dismantling.

Accurate channel modeling and estimation of active reconfigurable intelligent surface (RIS)-assisted links with densely integrated elements are essential to fully unleashing this technology's potential. This work adopts a physically consistent model incorporating mutual coupling (MC) effects, modeled via scattering parameters, in RIS-aided communication. We formulate the MC-aware channel estimation as a compressed sensing (CS) problem. The MC effect leads to an increase in the sensing matrix dimensions. This increased dimensionality substantially elevates the complexity of the formulated CS problem. To overcome this, we propose a low-complexity estimator that leverages the structure of the scattering matrix and MC mechanisms to obtain a reduced-size design sensing matrix. Numerical results demonstrate that our approach outperforms MC-unaware estimators by several dBs, achieving accuracy comparable to fully MC-aware solutions but with significantly lower complexity.

We determine the multiplicative structure on the Tor algebras associated to the coordinate rings of general points in $\mathbb{P}^3$ in terms of the number of points. We show that in codimension four, the Waring rank of a form determines the Tor algebra class of its apolar Artinian Gorenstein ring for a dense subset of the forms of fixed Waring rank and sufficiently large degree.

Signal processing techniques for wireless communications and sensing fundamentally differ between near-field and far-field propagation regimes. Accurately identifying the applicable propagation region is therefore essential for enabling efficient beamforming and channel estimation in ultra-massive MIMO (UM-MIMO) systems. This paper proposes a fully unsupervised learning framework to distinguish near-field from far-field propagation based solely on received signal measurements, before estimating the communication distance, and without relying on channel state information. The proposed approach exploits spatial signal power variations across subarrays of a UM array as a physics-inspired feature extraction stage, followed by the OPTICS clustering algorithm to infer the communication region. Simulation results under various system configurations and signal-to-noise ratio (SNR) levels demonstrate that the proposed method accurately identifies the near-field and far-field regions, showing agreement with theoretical boundaries.