Papers

The precise characterization of the spatial degree of coherence of a radiation field is important for assessing its suitability for specific applications in optical communications, advanced imaging, and quantum information processing. However, measuring the full coherence matrix traditionally requires complex, phase-sensitive interferometric setups that are highly susceptible to noise and difficult to scale on integrated platforms. To address this, we propose an architecture-agnostic approach for analyzing partially coherent light that is compatible with any universal programmable photonic unitary circuit, regardless of its internal topology. Leveraging the Schur-Horn theorem, our method diagonalizes the output coherence matrix, enabling direct extraction of its eigenvalues from output power measurements alone. We numerically validate this framework across various universal topologies and demonstrate its efficacy even in under-parameterized, non-universal architectures with only minor loss in precision. Finally, our black-box optimization approach proves inherently resilient to arbitrary optical losses and component deviations, paving the way for robust, lower-depth, and programmable spatial coherence analyzers.

We resolve an open problem of Kumar, Scholtz, and Welch (1985) by constructing generalized bent functions from $(\mathbb{Z}/q\mathbb{Z})^m$ to $\mathbb{Z}/q\mathbb{Z}$ in the exceptional case $q\equiv2\pmod4$ with $m$ odd, the case their paper left without a construction and which four decades of subsequent work had addressed only through nonexistence results. Concretely, for all integers $m\geq d\geq 2$ with $m=dr+2k$, $r\geq 1$, $k\geq 0$, we construct an explicit generalized bent function from $(\mathbb{Z}/q\mathbb{Z})^m$ to $\mathbb{Z}/q\mathbb{Z}$, where $q=2(2^d-1)$. We further show that, when $d\ge3$, these generalized bent functions have Fourier coefficients that are not roots of unity — all of them when $r$ is odd — which gives a negative answer to a recent question of Armario, Egan, Kharaghani, and Ó~Catháin about bent vectors for character tables.

We study a continuous-time rational addiction model where addiction capital follows a piecewise deterministic Markov process with state-dependent jumps capturing relapse and recovery. The instantaneous utility combines consumption and addiction capital via a power specification, leading to Hamilton-Jacobi-Bellman (HJB) equations with nonlocal jump terms. In the capped, bounded-control case we obtain a unique bounded viscosity solution and prove that optimal policies are bang-bang, switching between minimal and maximal consumption, while in the uncapped case we derive explicit linear feedback controls and closed-form value functions in several parameter regimes.

Gromov's delta-hyperbolicity, the classical measure of how tree-like a metric space is, works well on spaces with unbounded diameter but behaves poorly on finite graphs, where it depends primarily on diameter rather than geometry. We introduce a function relating Gromov's delta of a quadruple to its diameter and use it to define a normalized invariant that characterizes the hyperbolicity of any finite graph, taking values between zero (trees) and one (large lattice graphs). We derive a closed-form formula for this function on the hyperbolic plane. Using this formula, we give an alternate proof of the best constant of Gromov's delta-hyperbolicity for the hyperbolic plane. We also give the first theoretical proof of the optimal constant for the scaled Gromov four-point condition under diameter scaling, previously known only from numerical computations.

We develop heat-flow estimates in critical homogeneous Triebel–Lizorkin spaces and apply them to two-dimensional Keller–Segel type equations. For $q>2$, we show that the heat-flow from $\dot F^0_{1,q}(\mathbb R^2)$ to $L^2(0,T;\dot F^1_{1,q}(\mathbb R^2))$ is unbounded. This motivates the introduction of new spaces, time-refined Triebel–Lizorkin spaces, in which the time norm is taken before the dyadic summation. In this framework, we establish a family of heat smoothing estimates together with the corresponding maximal regularity. We further prove Banach-valued Peetre estimates, Banach-valued Jawerth-type estimate, homogeneous Poisson estimate, and an endpoint bilinear estimate for the Keller–Segel drift. As an application, we obtain local well-posedness for arbitrary initial data and global well-posedness for sufficiently small initial data in the critical space $\dot F^0_{1,q}(\mathbb R^2)$, $2\le q<\infty$, for the two-dimensional parabolic–elliptic Keller–Segel equation. The same analytic framework also yields analogous well-posedness results for a related two-component drift–diffusion system.

In many applications, a diffusion trajectory is observed without distortion only while it remains inside a time-varying region; outside this region, only its nearest boundary point and an indicator of visibility are recorded. We formalize this setting through a censoring scheme for multidimensional stochastic differential equations, in which the latent state is observed via its Euclidean projection onto a random, time-dependent closed convex set. Classical arguments handle the one-dimensional case. In higher dimensions, however, the projection generates additional finite-variation terms, and it is not immediate that the local information required for drift estimation is preserved under censoring. We show that it is. Under a suitable regularity condition on the projection map, allowing the censoring region to be nonsmooth, the observed process is a continuous semimartingale and admits a generalized Itô decomposition, whose finite-variation correction term assigns no mass to visible times. Consequently, stochastic integrals over visible times coincide exactly for the observed and latent processes. Building on this identity, we construct a Nadaraya-Watson-type drift estimator with a data-driven bandwidth selection rule and establish oracle inequalities. We also derive anisotropic minimax lower bounds for both the full-observation and censored experiments.

A structure-preserving fully implicit Scharfetter-Gummel finite-volume scheme for a three-species drift-diffusion model for semiconductors is proposed and analyzed. The equations describe the evolution of the electron, hole, and oxygen vacancy densities in a (bounded) memristor device, coupled to the Poisson equation for the electric potential, with mixed-type boundary conditions. Recasting the Scharfetter-Gummel fluxes in an upwind form, a hidden Fisher information component is revealed. Owing to the degeneracy of the Bernoulli function appearing in the fluxes, additional edgewise coercivity estimates are required, leading to refined local and global dissipation estimates. Using these ideas, the existence of a discrete finite-volume solution, a discrete free energy inequality, and the convergence of the numerical scheme are established. Numerical simulations in two space dimensions confirm the structure-preserving properties of the scheme and illustrate the filament formation in a memristor device.

The main results of this paper are threefold. First, we prove an observability inequality for the Airy equation on the real line from Hausdorff-thick sets in a time-trace sense for every observation time $T>0$. The observation functional is a block supremum of $L^2(0,T)$ time traces over the Hausdorff-thick set. Second, we prove observability inequalities for the Airy equation on the real line with observations on some periodic sets, which in particular yields observability on a class of spatial point sequences. Third, we give a necessary and sufficient condition for finite point observability for the Airy equation on the torus with a bounded real-valued potential. Indeed, for a finite observation set $F$, a Kalman rank condition on a finite-dimensional invariant subspace is found. As a corollary, we obtain sharp point observability results for the Airy and linear KdV equations on the torus. Moreover, for potentials with finite order regularity assumptions we prove that every observation set with an accumulation point, in particular any set of positive Hausdorff dimension, gives an observability inequality for each observation time $T>0$. Since the Airy equation has neither high frequency exponential decay nor pointwise smoothing effects, which are essential in recent works on Hausdorff type observation results on heat equations, we introduce several new ideas adapted to the Airy case.

We study the query complexity of recovering a planted assignment from a random constraint-satisfaction instance with one-sided noise. We consider the following 1-CNF recovery problem: an unknown binary string with $n/2$ ones and $n/2$ zeros is queried at individual variables. A query to a $1$-variable returns "$1$" with probability $p$ and "$0$" otherwise, while a $0$-variable always returns "$0$" (each query is a fresh noisy draw). The goal is to recover the binary string with probability at least $1 - \delta$. While the naive counting argument may suggest a query complexity of $\log_2 \binom{n}{n/2}=\Theta(n)$, we show that the query complexity is $(1+o(1))c(p) \frac{n}{2} \left( \log_2 n + \log_2(1/\delta)\right)$, where $c(p) = \tfrac{1}{-\log_2(1-p)}$. We then study planted $k$-CNF satisfaction with one-sided noise. Each $k$-set containing a $1$-variable is included as a clause independently with probability $p$, and an algorithm may ask whether any given $k$-set is a clause. Unlike the $1$-CNF case, a clause-existence query is one-shot: each $k$-set either is or is not a clause, so repeating yields no new information. The model is one-sided because an observed clause certifies that at least one queried variable is assigned 1, whereas its absence does not certify all are assigned 0. The goal is to recover the planted assignment with probability at least $1 - \delta$. The counting baseline is $\Theta(n)$, yet we prove a query complexity of $(1+o(1))\,c(p,k)\, \frac{n}{2}\left( \log_2 n + \log_2(1/\delta)\right)$, where $c(p,k) = \tfrac{1}{k(-\log_2(1-p))}$. These bounds are for adaptive algorithms. We also prove bounds for nonadaptive algorithms, showing that for fixed $p$, adaptivity gives a factor $\exp(\Theta(k))$ improvement. Our results also imply lower bounds for noisy sorting of $\{0,1\}$-valued strings, and we study a variant of the model with negations.

We study the semiclassical propagation of edge-states for a two-dimensional Dirac operator with a spatially varying mass. The zero set of the mass models the interface between topological phases and is allowed either to consist of two disjoint smooth curves or to possess an isolated singular point. For disjoint interfaces, we prove that a wave packet initially localized on one component remains confined near this interface over long semiclassical time scales and we quantify the influence of the distance between the two components on the accuracy of the approximation. For singular interfaces, we determine the regime in which the wave-packet approximation remains valid as the packet approaches the singularity and identify the scaling at which our estimates lose their uniformity.

The Unmapped Tent Pitching (UTP) algorithm is a space-time domain decomposition method for the parallel solution of wave-type problems. We have recently extended UTP to heterogeneous settings and compared, at the continuous level, the computational cost of different space-time decompositions. In this note, we report discrete-level observations showing that the optimal decomposition strategy may differ from the one predicted by the continuous analysis.

We show that a variety of known and new results concerning connected plane graphs whose vertex and face degrees satisfy prescribed uniformity conditions with at most two exceptions can be deduced from recent results on the Hurwitz existence problem regarding the realizability of branch patterns of rational functions. Our method also yields a description of the Belyi functions corresponding to such graphs.

We study the variance of a local Weyl law over a fixed smooth energy window, when averaged over large Weil–Petersson hyperbolic surfaces. Our results are consistent with the predictions of Berry's random wave model. Our approach allows to explicitly integrate certain test functions which depend on lengths of based geodesic loops, and relate them to the associated lengths of the closed geodesics in their free-homotopy class. We thus utilise the work of Mirzakhani, with exact stationary phase arguments, to identify correct main and error terms, making explicit the asymptotic behaviour of the variance of the local Weyl law. Furthermore, we introduce the geometric notion of length-minimising geodesic loops and sequences, based at a point. We prove a complete characterisation of the topology of these, namely that they are simple. This forms a key ingredient in our study, and yields a new streamlined argument to bound the contributions of remainder terms which depend on lengths of pairs of different short primitive geodesic loops. To illustrate the generality of our results, we further introduce a family of "exploring" loops based at a point, which might be of independent interest.

We introduce a second-order accurate vectorial lattice Boltzmann scheme for the incompressible Navier-Stokes system, inspired by a discrete-velocity kinetic approximation proposed by Carfora and Natalini [ESAIM: M2AN, 42(1), 93-112, 2008]. Advantages and drawbacks compared to relaxation schemes are investigated by providing spectral analyses in the linearized case, and numerical validations on the genuinely non-linear problem.

In this paper, we develop an explicit multi-physics structure-preserving Strang splitting scheme for a four-dimensional relativistic charged-particle dynamical system in the presence of a strong magnetic field under the maximal ordering scaling. The proposed scheme not only preserves volume, energy, and Lorentz invariance, but also yields second-order uniform error bounds for the position and the parallel velocity component. We present rigorous theoretical proofs for these geometric properties and the error convergence, which are subsequently validated by several numerical experiments.

This paper studies a free boundary problem for the Boltzmann equation that models the interaction of rarefied gas with a moving wall—a classical piston problem in kinetic theory. The piston motion is governed by Newton's law under the drag force exerted by the gas, with or without an additional Hookean restoring force. A central challenge for such a problem is the strong coupling between the kinetic equation and the free moving boundary, for which the classical Lagrangian formulation is unavailable due to low-regularity of solutions. Our approach relies on two new ingredients: a conformal transformation is introduced to reduce the moving domain to a fixed domain, and a coupled energy structure linking the kinetic distribution and free boundary variables is uncovered to reveal intrinsic dissipation and cancellation mechanisms at the interface. These structural observations lead to a global nonlinear theory for the fully coupled system. As a result, we establish the global existence, uniqueness, nonlinear stability, and exponential convergence to equilibrium of solutions near a global Maxwellian. This provides the first rigorous global well-posedness theory for a fully coupled Boltzmann free boundary problem of piston type.

Let $A \subseteq \mathcal{O}_{K}$ be a set of algebraic integers of height up to $H$ such that $|A \mod{\mathfrak{p}}|\leq \alpha |\mathcal{O}_{K}/\mathfrak{p}|$ for every prime ideal $\mathfrak{p}$ with $N\mathfrak{p}>c$ for some $\alpha \in (0,1)$. It follows from a larger sieve due to Ellenberg, Elsholtz, Hall and Kowalski that $|A| \ll_{K,c,\alpha}H^{2\alpha}$. In this paper, we improve on this larger sieve bound by showing that $|A|\ll_{K,c,\alpha}H^{\alpha}(\log H)^r$. We also obtain a two-dimensional larger sieve of Helfgott and Venkatesh type over $\mathcal{O}_{K} \times \mathcal{O}_{K}$ and apply it to produce a Bombieri-Pila type bound over $\mathcal{O}_{K}$.

High-order methods can improve worst-case evaluation complexity, but for orders $p\geq3$ their Taylor subproblems are nonconvex polynomial optimization problems and are generally difficult to solve. We develop a radius-controlled boundary approach based on homogeneous tensor representations. By augmenting the step with a constant coordinate, any $p$th-order Taylor polynomial can be represented exactly as an order-$p$ homogeneous tensor form; at a prescribed radius, the boundary model is a spherical polynomial optimization problem. The representation applies to arbitrary $p$, while the algorithmic development focuses on the cubic case $p=3$. For an inhomogeneous cubic on the sphere, we introduce a quadratic shift and prove, under an explicit shift bound, equivalence with a three-block multilinear formulation at global optimality. This motivates a proximal alternating minimization (PAM) method with closed-form block updates; its objective values decrease and every accumulation point is stationary. We embed the boundary-step mechanism in an Adaptive Homogeneous Tensor Method (Ada–HTM). Under explicit smoothness, safeguarded-decrease, weak-curvature nondegeneracy, and local-refinement conditions, Ada–HTM attains the adaptive-regularization-type (AR$p$-type) evaluation complexity $\mathcal{O}(\epsilon^{-(p+1)/p})$ for first-order stationarity. Numerically, PAM matches order-$2$ moment–sum-of-squares (SOS) certificates on the structured cubic instances for which certification is tractable, scales particularly well for low-rank tensors, and makes Ada–HTM competitive with trust-region and cubic-regularization methods, with its largest gains on ill-conditioned and badly-scaled problems.

Over-parameterized linear regression has been widely studied over the last decade. However, most existing works assume that the covariates are independent and that their covariance matrices are non-degenerate. In this paper, we relax both assumptions and derive deterministic equivalents for the prediction risk in a vanishing-ridge regime. We show that degeneracy of the covariance matrices and dependence can lead to multiple descent, and characterize where the corresponding peaks can occur. Our proofs use a novel graph representation of the variance profile. We show that maximum matchings and the Dulmage–Mendelsohn decomposition of the associated bipartite graph identify the configurations at which the variance becomes singular.

Let $H$ be a pointed Hopf algebra and let $A$ be a right $H$-simple left $H$-comodule algebra. We show that every relative $(H,A)$-Hopf module is free as an $A$-module and that this freeness characterizes the class of pointed Hopf algebras. We give a criterion for the category of relative $(H,A)$-Hopf modules to be semisimple. We also show that $A$ can be embedded into a left $H$-comodule algebra of a specific form when $H$ and $A$ are $\mathbb{N}_0$-graded. As a consequence, we prove that if $H$ is finite-dimensional and $A^{\mathrm{co} H}=\Bbbk$, then $A$ is finite-dimensional and $\dim A$ divides $\dim H$.

In this paper, the continuous solutions of the complex Kac–Bernstein functional equation \[ f_1 ( u + v ) f_2 ( u - v ) f_1 ( u' + v' ) f_2 ( u' - v' ) = f_1 ( u + v' ) f_2 ( u - v' ) f_1 ( u' + v ) f_2 ( u' - v ) \] are classified for $ \mathbb{ Z } $ and $ \mathbb{ R } $. By using this classification for $ \mathbb{ Z } $, we consider a generalization of the Kac–Bernstein theorem on the one-dimensional torus $ \mathbb{ T } $ by Baryshnikov–Eisenberg–Stadje from probability Borel measures to complex Borel measures. Similarly, the original Kac–Bernstein theorem on $ \mathbb{ R } $ is generalized from probability Borel measures to complex Borel measures.

We prove Graffiti.pc Conjecture 2, a 1996 conjecture listed as open on the Written on the Wall II page marked “Last update 7/23/26.” Let $G$ be a finite simple connected graph. For $v\in V(G)$, let $I(v)=\alpha(G[N_G(v)])$, and let $I_{\mathrm{avg}}(G)$ be the average of these local independence numbers. The conjecture states that the maximum number $L_s(G)$ of leaves in a spanning tree of $G$ satisfies $L_s(G)\ge 2\bigl(I_{\mathrm{avg}}(G)-1\bigr)$. We establish this inequality by extracting a triangle-free spanning subgraph that retains at least half of the total local-independence mass. A degree-square argument then produces a double star with sufficiently many leaves, and this tree extends to a spanning tree without losing leaves. Balanced complete bipartite graphs show that the bound is sharp.

In this paper, we slightly extend the framework of nearly Gorenstein graded rings introduced by Herzog-Hibi-Stamate, and investigate the nearly Gorenstein property of the blow-up algebras associated with an ideal, namely the Rees algebra, the extended Rees algebra, and the associated graded ring. We establish characterizations of the nearly Gorenstein property for these algebras and clarify the relations among them. In particular, for two-dimensional regular local rings, we obtain a complete classification of the integrally closed ideals for which these blow-up algebras are nearly Gorenstein.

The Plancherel-random infinite Young tableau arises from applying the infinite Robinson-Schensted-Knuth (RSK) correspondence to a sequence of i.i.d. random variables distributed uniformly on the unit interval [0,1]. Building on the isomorphism of dynamical systems established in prior work, we provide an explicit geometric characterization of the inverse map. Our approach makes use of a previously unexplored structure, the jeu de taquin forest on the infinite tableau, where edges connect boxes according to local comparison rules. We prove that each tree in this forest almost surely extends toward infinity with a well-defined asymptotic direction, establishing a canonical bijection between trees and values in the i.i.d. input sequence. The ordering is recovered through tree lifetimes under iterated jeu de taquin transformations.

In this paper, we investigate the following quasi-linear weighted $N$-Laplacian Liouville equation \begin{equation*}\label{0} -\Delta_N u=|x|^{N\alpha}e^{u}, \qquad x\in \R^N, \end{equation*} where $N \geq 2$. For $\al>0$, by carefully studying the linearized problem and applying the approximation method and bifurcation theory, we prove that, when the parameter $\alpha$ equals to the critical values $\alpha(k):=\frac{\sqrt{k(N-1)(k+N-2)}}{N-1}-1$ for $k \geq 2$, there exist non-radial solutions $u$ (bifurcating from $U_{\alpha(k)}$) to the above quasi-linear Hénon type Liouville equation such that $u\sim \ln|x|$, $|\nabla u|= O(|x|^{-1})$ at $\infty$ and $\int_{\R^N}|x|^{N\alpha}e^{u}\md x=N\left(\frac{N^2}{N-1}\right)^{N-1}(\alpha+1)^{N-1}\omega_N$. One should note that, $\alpha(k)=k-1$ for $k\geq2$ when $N=2$. Our results successfully extend the existence result of J. Prajapat and G. Tarantello in \cite{PT} concerning the $2$-dimension and Laplacian case (i.e., $N=2$) to the more general $N$-dimension and $N$-Laplacian cases ($N\geq 2$), and extend the results of F. Gladiali, M. Grossi, and S. L. N. Neves in \cite{GGN} and the authors in \cite{DDGL} from $1<p<N$ to the much more complicated limiting case $p=N$. We introduced some new ideas and overcame a series of crucial difficulties, including the nonlinearity nature of the $N$-Laplacian $\Delta_N$, the lack of Green integral representation formula and critical weighted Sobolev embedding inequality, the absence of Kelvin type transforms for linearized/difference equations, the invariance of the total mass under scalings of $u$, and the signs-changing and divergence (to $-\infty$) at $\infty$ of the solutions, which makes the suitable choices of the approximate problems, the (normalized) approximate function sequences and the working space to be quite difficult.

We revisit contextual optimization from the perspective of policy class design. A desirable policy class should be expressive enough to learn rich context-decision relationships, should enforce hard feasibility constraints rather than soft penalty terms, and should remain smooth enough for gradient-based training on downstream decision losses. Existing approaches usually emphasize only part of these requirements. We propose Legendre-regularized policies, which parameterize decisions as solutions of regularized optimization problems over the original feasible region. This construction yields policies that are feasible by construction and differentiable with respect to learned latent parameters. We prove that the associated optimizer map is single-valued, maps onto the relative interior of the feasible set, admits an explicit Jacobian, is Lipschitz continuous, and can be made arbitrarily smooth. We also establish a universal approximation result showing that the proposed class can approximate any continuous feasible policy on compact context sets. The framework unifies explicitly regularized optimizers and implicit perturbation-based smooth optimizers. Experiments on contextual newsvendor and resource allocation problems show that our approach improves prescriptive performance relative to the benchmark methods.

We prove that every bounded measurable subset of the real line that is both spectral and a union of three intervals tiles the line by translations, thereby completing Fuglede's conjecture for this class. The proof develops a new cofactor-rigidity method based on the secular determinant of an associated self-adjoint derivative.

This paper investigates a class of stochastic linear-quadratic (SLQ) control problems over an infinite horizon for Markov regime-switching jump-diffusion systems. Unlike classical diffusion models modulated by a Markov chain, we assume that the state process undergoes abrupt jumps that are synchronous with the regime switches of the Markov chain. In contrast to conventional Poisson jump-diffusion models, the jumps in the state process are entirely induced by the state transitions of the Markov chain, which can be interpreted as losses or gains of state process incurred during regime changes. Under this formulation, we thoroughly discuss the closed-loop solvability of the SLQ control problem and provide a feedback representation of the optimal control via the stabilizing solution of a system of coupled algebraic Riccati equations (CAREs). Finally, we further apply our results to a lifetime wealth tracking problem and derive the corresponding optimal investment strategy.

As a key technology for 6G, integrated sensing and communication (ISAC) is receiving considerable attention, and deploying a reconfigurable intelligent surface (RIS) can enhance both communication performance and sensing capability of ISAC by providing additional degrees of freedom. In this paper, we investigate a beam training framework for RIS-aided ISAC systems where beam alignment for a communication user equipment (UE) is conducted while simultaneously detecting a single target through its echo signal. Using codebooks constructed according to the principles of the 5G standard, we propose a partial search procedure that achieves low training overhead and mathematically show that this strategy is sufficient to identify a suitable codeword combination to serve the UE. By applying the auxiliary beam pair method, the target's angle information from the perspectives of the base station and RIS is obtained. Then, a high-accuracy closed-form localization is proposed based on the angle estimates, and we further extend the proposed technique to multi-target localization scenarios. Numerical results highlight the advantages of the proposed technique in the ISAC context, showing that the training procedure can effectively find a codeword combination and that the target localization technique outperforms the benchmarks.

Gil-Medrano asked whether real projective subspaces remain minimal and volume-minimising under Berger deformations in dimensions greater than three. We answer the minimality question and make substantial progress on the corresponding minimisation problem. The fundamental observation is that the full linear geometry of a real subspace $V\subset\mathbb{C}^N$ is controlled by the single skew-adjoint endomorphism $A_V=\operatorname{pr}_VJ|_V$. For every non-round Berger metric, this yields a complete classification: $\mathbb{RP}(V)$ is minimal if and only if $V$ is a linear CR subspace, equivalently if all of its multiple Kähler angles are $0$ or $\pi/2$. Thus the answer to Gil-Medrano's minimality question is negative in general, although every equatorial hypersurface remains minimal. The same structure determines the critical points and global extrema of the linear volume functional. Combining it with projective integral geometry, we also compute exact mod-two homological systoles in the pure stretched and squashed regimes. Our final contribution is a conjecture that all remaining cases consist of mixed CR equators.