Papers

In this paper, we consider a spiked Wigner type matrix with a heteroscedastic and unknown variance profile. It is well known that in the supercritical regime of the BBP transition, strong spikes can create outliers in the spectrum. Unfortunately, in the heteroscedastic case, in general it is not possible to estimate the spike strength from these observed outlier consistently, as the latter is a solution to a Dyson equation with unknown parameters from the variance profile. In this paper, inspired by the work on sparse matrix completion \citep{BordenaveCosteNadakuditi2023}, we introduce an asymmetrized model by randomly splitting the spiked matrix into two parts, which transforms the noisy Wigner type matrix into a non Hermitian random matrix, while preserving the Hermitian spikes at the cost of a dilution. We establish a BBP type transition for the asymmetrized model, from which we can estimate the strength of the spikes precisely, even without knowing the variance profile of the noise part. We then further apply our approach to study the correlation between two correlated spiked models, where the spike/signal parts of the two models are correlated, and the noise parts are independent but may both be heteroscedastic. By applying our asymmetrization approach to the two models separately and also jointly, we are able to obtain a precise estimate of the correlation between the signal parts of the two models.

We consider a class of sixth-order elliptic partial differential equations in two and three dimensions subject to simply supported and Cahn–Hilliard-type boundary conditions. Based on the Ciarlet–Raviart reformulation, we recast the sixth-order problem as an equivalent mixed system involving second- and fourth-order equations and establish its well-posedness under suitable assumptions. For the resulting mixed formulation, we propose and analyse two discretization frameworks: a fully non-conforming hybrid high-order (HHO) method on general polytopal meshes and a $C^0$-conforming HHO–finite element method on simplicial meshes. We prove stability and derive optimal-order a priori error estimates for the primary variables under appropriate regularity assumptions. For the fully non-conforming HHO method, we further derive a reliable residual-based a posteriori error estimator. Numerical experiments confirm the theoretical convergence rates and illustrate the robustness of the proposed methods for a range of mobility parameters.

We ask how much of the geometry a charged particle moves in is fixed by its motion, and how much a particle must bring. We ask of a symplectic structure only that it relate velocity to momentum as Hamilton's equations do, and we ask it of every energy at once. In particular, we show that the structures meeting that demand are the canonical one and its twists by a closed two-form of the base. A field provides the two-form, a particle the multiplier before it, which we identify constitutively with its charge. Thus, a single energy governs a family of structures, and each particle takes the one its charge fixes. We then ask what a particle must bring to be given a momentum, and we show that the degree of that map settles the degree at which a field enters Newton's Second Law. An antisymmetric bilinear form returns no Lorentz force, whilst a Randers metric returns one — a length whose difference from a Riemannian one is linear in the velocity. Moreover, we find that metric already within the twisted structure, as its primitive over a level of the free energy, and its law of transport to be nonlinear, no affine connection being known to serve. Under an indefinite signature the length parts from the dynamics, and the extremals turn from shortest to longest. On the round sphere a monopole flux leaves no such metric, whilst the transport remains and prequantisation, given a unit of action, restricts the charge to a lattice. In this manner, we conclude that each charge-to-mass ratio receives a geometry of its own, so that by a functionalist criterion none of them is the spacetime of a charged particle.

We study parametric estimation for second order linear parabolic stochastic partial differential equations in two space dimensions with a small volatility parameter driven by a $Q$-Wiener process with an unknown damping parameter using high frequency spatio-temporal data. We first provide an estimator for the damping parameter of the $Q$-Wiener process utilizing realized quadratic variations based on spatial and temporal increments. We next propose minimum contrast estimators of the diffusive and advective parameters in the SPDE using a contrast function with the proposed estimator of the damping parameter. We then construct a quasi-maximum likelihood estimator of the reaction parameter in the SPDE using the approximate coordinate process derived from the estimators of the diffusive and advective parameters. We also provide simulation results of the proposed estimators.

Let $C$ be a smooth projective complex curve of genus $g$ and gonality $k$, and $L$ be a very ample line bundle on $C$. When $L$ has sufficiently large degree, the vanishing and nonvanishing of the Koszul cohomology groups $K_{p,q}(C,L)$ have been determined previously, but the exact values of the graded Betti numbers $\kappa_{p,q}(C, L)$ remain largely unknown. In this paper, we give explicit closed formulas for all graded Betti numbers $\kappa_{p,q}(C, L)$ when the Brill–Noether locus $W_k^1(C)$ has the expected dimension and $H^1(C, L \otimes \omega_C^{-1})=0$. Consequently, we determine the complete Betti table for a general curve when $\deg L \geq 4g-3$ or when $\deg L \geq 3g-3$ and $L$ is general. We also explicitly compute the Boij–Söderberg coefficient of the section ring $R(C, L)$ governing asymptotic purity, and show eventual monotonicity of the remaining coefficients: they decrease for hyperelliptic curves and increase under a natural generic reducedness assumption on the relevant Brill–Noether loci.

For a twice differentiable function $f:\left( a,b\right) \rightarrow \mathbb{R}$ define $v\left( f\right) =f^{\prime}f^{\prime}-f^{\prime\prime }f.$ It is well known that the positivity of $v\left( f\right) $ implies that the function $\left\vert f\right\vert $ is strictly log-concave on each subinterval which does not contain zeros of $f.$ In this paper we provide criteria for the positivity of $v\left( F\right) $ for the composite Bessel function $F\left( x\right) =J_{\alpha,\beta,\gamma,\nu}\left( x\right) :=x^{\alpha}J_{\nu}\left( \beta x^{\gamma}\right) $ for positive numbers $\beta$ and $\gamma$ and real numbers $\alpha$ and $\nu.$

In this paper, we establish the saturation order and a local inverse result for the uniform approximation of non-negative, bounded, and uniformly continuous functions on $\mathbb{R}$ by max-product generalized sampling operators based on suitable kernel functions. In particular, assuming that the kernel is an even centered bell-shaped function, we first show that $1/w$, $w>0$, is the uniform saturation order, with the corresponding saturation class coinciding with the class of non-negative constant functions. This means that $1/w$ is the best possible rate of convergence that the max-product generalized sampling operators can achieve when approximating non-trivial (i.e., non-constant) non-negative, bounded, and uniformly continuous functions on $\mathbb{R}$. Moreover, it is known that, for Lipschitz continuous functions on $\mathbb{R}$, the approximation order is $1/w$ as $w \to +\infty$. Here, we show that this result can be locally reversed. Specifically, we prove that if $f$ can be approximated at the rate $1/w$ on a compact interval $[a,b]\subset\mathbb{R}$, then $f$ is Lipschitz continuous on $[a,c]$ for every $c \in [a,b)$ whenever $0<a<b$, and on $[c,b]$ for every $c \in (a,b]$ whenever $ a<b<0$. Finally, under the same assumptions on the kernel, we establish a strong localization result for sequences of truncated max-product generalized sampling operators in the case of strictly positive and bounded functions defined on $[0,1]$. All these results extend previous results of Coroianu and Gal, which were established only for specific sinc-type kernels, to a broader class of kernel functions.

The weak dominance of aggregation operators, particularly between triangular norms (t-norms) and triangular conorms (t-conorms), has attracted considerable attention in aggregation operator theory. While several characterizations have been obtained for Archimedean and continuous cases, a general criterion for continuous t-conorms over continuous t-norms remains to be fully clarified. In this paper, we provide a complete characterization of a continuous t-conorm weakly dominating a continuous t-norm. We first reduce the problem for ordinal sum operators to that for their single Archimedean components, and then express the weak dominance condition entirely in terms of the additive generators of these components. Our approach covers both strict and nilpotent cases uniformly, and recovers the known results for Archimedean operators as a special case.

We prove the black hole uniqueness conjecture in the axially symmetric, stationary, electrovacuum setting, subject to the refined asymptotic analysis of the associated singular harmonic maps, which includes an analyticity hypothesis at the axes. More precisely, it is shown that any asymptotically flat solution of the Einstein–Maxwell equations in this class, with more than one black hole horizon component is either: Majumdar–Papapetrou, up to a duality rotation, in which case all logarithmic angle defects vanish, or every finite axis rod logarithmic angle defect is strictly negative and hence every interaction force is strictly attractive. The proof extends the singular harmonic map method used for vacuum Kerr uniqueness in [18].

Characterizing all PID gains that guarantee uniform exponential regulation of uncertain nonlinear systems remains a fundamental problem. Most existing results provide only sufficient conditions derived from a particular Lyapunov construction. This paper gives exact gain regions, defined directly by uniform regulation over the entire uncertainty class, for two classes of nonaffine MIMO systems with $m\ge2$ controlled coordinates and input channels. For systems of second order under PID control, we independently characterize a common quadratic inner region and an outer region based on a linear subclass. The two regions coincide and therefore characterize the PID gain region exactly at the system level. For systems of first order under PI control, a lossless reduction to a linear family on the uncertainty boundary, combined with a common quadratic certificate, yields the exact PI region. Both regions strictly enlarge existing sufficient regions, and the PID result also provides a sequential tuning rule. A compact linear family with one uncertain parameter shows that common quadratic certification is not generally lossless. These results establish endpoint coincidence and boundary family equivalence as two routes to exact gain regions and motivate the broader question of which nonlinear uncertainty classes admit such characterizations.

Hyperbolic partial differential equations (PDEs) with time delay are essential mathematical tools used to model numerous physical systems where wave propagation or oscillations depend heavily on their historical states. As these applications scale in physical complexity, efficient parallel computing techniques become essential; however, developing highly scalable parallel solvers for delayed PDEs remains a significant computational challenge. To address this gap, this study extends the Dirichlet-Neumann Waveform Relaxation (DNWR) method to a multi-subdomain framework explicitly designed for solving time-delayed hyperbolic PDEs in parallel. We advance the underlying mathematical framework and provide a rigorous convergence analysis for both one-dimensional and two-dimensional spatial configurations. A central theoretical contribution of this work is the establishment of finite-step convergence for the multi-domain DNWR algorithm. These theoretical guarantees are firmly corroborated through comprehensive numerical experiments. Furthermore, a detailed comparative analysis against alternative domain decomposition techniques namely NNWR, OSWR, and Classical SWR, highlight the advantages of the proposed method. Finally, this approach provides a robust, highly parallelizable solver capable of efficiently handling complex hyperbolic PDEs with time delay.

Recently, Li obtained a mean Lindelöf estimate for the cubic moment of the central values of Maass form symmetric square $L$-functions over short intervals $(T-H, T+H)$ of length $H \ge T^{18/19+\epsilon}$. We improve this result by showing that the estimate holds for $H \gg T^{6/7+\epsilon}$. The key ingredient in our proof is a new asymptotic formula for the second twisted moment of Maass symmetric square $L$-functions. Based on this formula, we also improve the lower bound for the proportion of non-vanishing central $L$-values in short intervals. Previously, even under the assumption of the Lindelöf hypothesis for Dirichlet $L$-functions, the proportion of non-vanishing values in intervals of length $H = T^{\beta}$ was only known to be at least $\frac{3\beta-1}{4}$. We establish an unconditional lower bound of $\frac{7\beta-2}{8}$.

In this work, we study the null controllability by one-sided boundary controls of one-dimensional hyperbolic balance laws with constant coefficients. Our first result shows that, when the system has only one negative or positive speed, the minimal null control time of such systems depends on some orthogonality conditions for a particular sequence. This sequence is explicit in function of the coefficients of the system but it is defined by a nonlinear recurrence relation. Our second result then completes the previous one by giving explicit bounds on the number of orthogonality conditions that have to be checked in two nontrivial situations. The proofs rely on a careful analysis of the so-called kernel equations associated with the system, including a new well-posedness result. Our results are also valid for the finite-time stabilization property.

We introduce cooperative integer programming games (CIPGs), in which agents pool budget constraints to accomplish indivisible tasks jointly and the characteristic function maps every coalition to the optimal value of a pooled integer program. Our goal is to identify an optimal coalition structure (OCS) and a stable one (OSCS). We derive a stability inequality that keeps each formed coalition in the Core with respect to itself, and present two mixed-integer OCS formulations, aggregated and disaggregated, proving that the disaggregated formulation is integer-equivalent yet yields a tighter LP relaxation. Building on the stability inequality we develop lifted stability cuts, several separation strategies inside a cutting-plane algorithm, an SCS-feasible primal heuristic that constructs warm starts with guaranteed stability, and a payoff-refinement step computing the Shapley value and the nucleolus of every formed coalition. On benchmark cooperative knapsack games, the method certifies optimality with up to 16 players and reaches MIP gaps below 1% at 30 players while evaluating 766 of the roughly $10^9$ coalition values.

We introduce asymptotic multiplier ideal sheaves and log canonical thresholds associated with tuples of pseudoeffective divisors on a projective klt pair. We prove that the threshold of a coupled potentially klt tuple is computed by a quasi-monomial valuation. For varieties of potentially klt type, we prove that every big divisor admits a birational Zariski decomposition with semiample positive part. We also prove finite generation of multisection rings of big divisors and give a criterion for a variety of potentially klt type to be a Mori dream space.

We prove that every de Rham $p$-adic local system on a smooth projective curve over a $p$-adic field is potentially semistable; that is, it becomes semistable after pulling back along a finite cover of the curve. This establishes a relative version of the classical $p$-adic monodromy theorem of Berger and André–Kedlaya–Mebkhout. Along the way, we show that every $p$-adic differential equation near a type I\!V point on a curve becomes trivial after a finite étale extension.

We extend the algorithm and several results published in the celebrated 1959 paper of Cheney and Goldstein about the best approximation pair (BAP) problem in two separate directions. One is, for the first time, the ability to consider more than two sets, a task which has been in the mind of researchers for many years without much success in fulfilling it because of a certain theoretical obstacle related to cycles and minimizers of general functionals. The other direction is the ability to handle each set as an intersection of a finite family of sets. We call the resulting problem the "Best Approximation Tuple (BAT) problem". The fundamental observation that leads to these extensions is to recognize and handle one set (the "pivot set") as different from the remaining sets (the "satellite sets") instead of seeking cycles as the minimizers of a target functional. We prove the convergence of the algorithm to the unique solution of the problem in the Euclidean case with strictly convex and compact satellite sets. Because of the lack of Fejér monotonicity, our convergence analysis is not standard, and is based on almost unknown properties of orthogonal projections regarding equality and inequality in the definition of nonexpansiveness.

We study the two-dimensional Matkowski–Sutô equation, which asks for two quasi-arithmetic means whose sum is twice the arithmetic mean, in two settings. For holomorphic injective generators with convex images on a convex domain in the complex plane, the solutions are exactly the affine pairs and the exponential pairs with a nonzero complex exponent, up to affine changes of the generators. The admissible exponents depend on the shape of the domain and are described by a curvature criterion for its boundary. In the monotone-operator framework of Tóth, we construct an infinite-dimensional family of non-affine shear pairs on the whole plane. Their generators are strictly increasing in the sense of monotone operators and need not be differentiable. These pairs solve the weighted equation for any number of variables. The rigidity of the one-dimensional problem, due to Daróczy and Páles, persists under holomorphy but not under monotonicity.

Hyperbolic PDEs with time delay are essential for modeling a wide range of real-world applications. Since closed-form solutions are often not attainable, developing accurate and efficient numerical methods becomes crucial. In this work a novel substructuring waveform relaxation method namely Neumann-Neumann Waveform Relaxation (NNWR) has been implemented for numerically solving hyperbolic delay PDEs in the case of non-uniform subdomain partitioning. The convergence behavior is first analyzed using Fourier analysis to obtain an estimate, followed by a demonstration of finite-step convergence through Laplace transform techniques. Various test cases are presented as numerical illustrations.

We introduce a class of additive reduced basis preconditioners designed to accelerate the iterative solution of large-scale linear systems arising from discretized parametrized PDEs. The main idea is to regularize the inherently singular reduced-order approximation by adding simple correction terms: either a scaled identity correction or a projected correction based on a basic preconditioner. This yields nonsingular preconditioners under explicit and easily checked conditions. The construction and application of the preconditioners are integrated into an FGMRES framework through an offline strategy that dynamically builds the reduced-basis component by proper orthogonal decomposition at each FGMRES step. We establish sufficient conditions for the nonsingularity of the preconditioners and derive error bounds for the preconditioned Richardson iteration. Numerical results for convection-diffusion, anisotropic vortex, Stokes, and Helmholtz problems are provided to verify the efficiency and convergence of the proposed ARB preconditioners. The method consistently converges in a few iterations and substantially reduces online solve time, supporting its efficiency for multi-query engineering scenarios.

We construct modified wave operators for the nonlinear Schrödinger equation $i\partial_tu+\frac12\Delta u=|u|^pu$ in the full subcritical long-range case $0<p<2/d$, with small, nonvanishing, analytic final data $W(x)$ with bounded logarithmic gradients. Previous results established large-time asymptotics for selected classes of Cauchy data. Moreover, the exact asymptotic expansion for $p<1/d$ remained unknown. When $1/d<p<2/d$, our result gives the approximation $$\frac{1}{(it)^{\frac{d}{2}}} e^{\frac{i|x|^2}{2t}} W\left(\frac{x}{t}\right) \exp\left[ -i\frac{t^{1-\frac{dp}{2}}-1}{1-\frac{dp}{2}} \left|W\left(\frac{x}{t}\right)\right|^p \right]$$ The wave operator is constructed by an iteration in the analytic spaces with decreasing radius. When $p\le 1/d$, we construct the profile from a finite truncation of a Fuchsian equation coupled with a transport equation. This profile still leaves a long-range triangular coupling whose terminal integral does not preserve the required fast decay class. The construction yields quantitative $L^q$ asymptotics for $2\le q\le\infty$ and uniqueness in the prescribed analytic asymptotic classes. The central new ingredients are a nonlinear final-state normal form that removes this long-range coupling and a mixed iteration in particular analytic spaces.

The recursive-line Zarankiewicz number maximizes the number of squares in a structured irreducible sum-of-squares representation encoded by an augmentation of an extremal $C_4$-free bipartite graph. We determine its four-column behavior under the strengthened recursive definition in the manuscript of Löfberg and Qi dated 9 September 2026. Combining AI-assisted discovery with exact certificate verification and finite exclusion computations, we determine eighteen of the nineteen values for $2\le m\le20$ and isolate the only unresolved case to $37\le\zr(14,4)\le38$. More significantly, we prove the first eventual exact formula in the four-column setting: \[ \zr(m,4)=\floor{\frac{5m+6}{2}}\qquad(m\ge15). \] The upper bound follows from the classical identity $z(m,4)=m+6$ and a sharp cell count. For the matching lower bound, we construct a two-row extension chain from an explicit $20\times4$ seed and derive the odd orders by a fixed deletion. Analytic propagation, together with two independently audited symbolic certificate tables, proves the construction for arbitrary chain length rather than merely for a finite computational range. Thus every extremal configuration has no holes when $m$ is even and exactly one hole when $m$ is odd, and the same exact formula holds for the second-order number $z_2(m,4)$.

Bargmann invariants constructed from a bipartite density operator and its two lifted marginals are polynomial invariants of local-unitary conjugation. We determine the precise information encoded in these invariants. Whenever one subsystem is a qubit, the ordinary marginal-word family determines the full spectrum of the partial transpose and hence decides whether the state has the positive-partial-transpose property. In $2\otimes2$ and $2\otimes3$ systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decomposition

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

We exhibit an annular knot with wrapping number four whose Kauffman bracket has annular degree at most two. This disproves the wrapping number conjecture.

We completely characterize the denseness of strongly minimum-attaining Lipschitz functions, a minimum analogue for strongly norm-attaining Lipschitz functions, in terms of bi-Lipschitz embeddings. More precisely, our main result shows that the set of strongly minimum-attaining Lipschitz functions defined on a complete metric space $M$ fails the denseness if and only if $M$ is bi-Lipschitz equivalent to a subset of $\mathbb{R}$ with positive Lebesgue measure, or equivalently, if $M$ admits a bi-Lipschitz embedding into $\mathbb{R}$ and $M$ has positive 1-dimensional Hausdorff measure. As a consequence, we provide an isometric characterization of the pure 1-unrectifiability of $M$ in terms of strongly minimum-attaining Lipschitz maps defined on bi-Lipschitz copies of closed subsets of $M$. Several counterexamples showing that the main result cannot be naturally extended to the vector-valued setting are also presented.

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\geq2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=ABA^{-1}B^{-1}$ of the free group $\mathbb{F}_{2}=\langle A,B\rangle$ is freely complemented in the group factor: $L(\mathbb{F}_{2})=W^{*}(w)*W^{*}(v)$ for some Haar unitary $v\in L(\mathbb{F}_{2})$ that is freely independent from $w$. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup $\Gamma\subset PSL_{2}(\mathbb{R})$, $L(\Gamma)$ is a free group factor, settling a conjecture of de la Harpe and Voiculescu. This result was obtained using OpenAI's ChatGPT Pro 6.0.

Data-driven distributionally robust optimization (DRO) typically treats the conditional outcome law as fixed and uses ambiguity sets to capture estimation error. This paper studies latent distributional heterogeneity, where each instance has an unobserved law but contributes only one observation, so uncertainty persists even if the mixture law is known. We propose Conformal-DRO, which uses nested conformal regions to construct an ambiguity set for the future latent law. Under exchangeability, the set covers this law with probability at least $1-\alpha$ in finite samples, without estimating underlying latent laws or their mixing mechanism. The conformal path induces a data-driven transport geometry, while $\alpha$ determines the radius. The worst-case problem reduces to a finite linear program over conformal shells and admits sparse adversarial solutions. The resulting robust value provides a finite-sample certificate for the selected decision's expected cost.

In this paper, we will study a trivariate extension of the Cauchy numbers of both the first kind (also called Gregory coefficients) and the second kind (also called Nörlund numbers) via the Laurent expansion of the reciprocal of any positive integer power (which is called the order) of multiple polylogarithms. In the case of logarithm, we will show by the WZ method that for each order $\ell>1$ some Gregory coefficient of order $\ell$ must vanish, in contrast to the fact that all classical Gregory coefficients are nonzero. We also prove in this higher order logarithm case that the sequence is eventually alternating for each fixed order, a property enjoyed by the classical Gregory coefficients. In the most general setting, we conjecture that these new sequences are all eventually positive, which is supported by strong numerical evidence. Finally, we confirm this conjecture in the special case of polylogarithms and double polylogarithms. As a by product, for each zeta value and double zeta value, we find an infinite family of identities expressing its reciprocal as a sum of a rational number and an improper integral.