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arXiv:2606.00233v1 Announce Type: new Abstract: Density estimation is often presented as a choice among parametric summaries, finite mixtures, and nonparametric smoothers. This review argues for a complementary view: a data set can be studied through a path of densities indexed by smoothing scale, diffusion time, model complexity, density level, or noise level. We call this perspective density evolution. Under this lens, Gaussian kernel de....
arXiv:2606.00246v1 Announce Type: new Abstract: The Fourier Entropy-Influence (FEI) conjecture states that the Fourier entropy of Boolean functions is uniformly bounded by their total influence. It has been verified for canonical examples such as disjoint tribes and for some classes of Boolean functions such as symmetric functions and read-$k$ decision trees (with a constant that depends linearly on $k$). In this note we present new classe....
arXiv:2606.00247v1 Announce Type: new Abstract: We show that Grothendieck's real constant $K_G$ can be upper bounded by projecting vectors onto a random plane through the origin and thresholding a degree five Hermite polynomial. This resolves a conjecture of Braverman-Makarychev-Makarychev-Naor from 2011, who required an extra randomization step in their rounding scheme and proved $K_G<\frac{\pi}{2\log(1+\sqrt{2})}-10^{-500}$. As a corolla..
arXiv:2606.00273v1 Announce Type: new Abstract: The goal of this work is to study the combinatorics of the lattices of higher torsion classes of the higher Auslander and Nakayama algebras of type \textbf{A}. Combinatorial descriptions of these higher torsion classes were recently obtained by August \textit{et al.} (2025), and it was observed that the lattices that they form are not semidistributive. We study in some depth these lattices, p....
arXiv:2606.00274v1 Announce Type: new Abstract: In the design of computational methods for Bayesian inverse problems, costly forward model evaluations make it difficult to sample from or compute the posterior. This motivates the need for approximate forward models that are cheaper to evaluate. We consider reduced-order forward models which exploit the lower-dimensional structure in the Bayesian inverse problem by projecting to the "likelih....
arXiv:2606.00276v1 Announce Type: new Abstract: We derive dispersive-dissipative estimates for multipliers associated to a general class of wave-type equations, in particular for the viscous Boussinesq equation. The dispersion is related to the geometric hypotheses on the phase func tion and on the degeneracies that may happen at low and high frequencies. The dissipation interacts with the dispersion, influencing the decay rate of the solu..
arXiv:2606.00283v1 Announce Type: new Abstract: We present a constructive version of the Br{\o}ndsted-Rockafellar theorem in general geodesic metric spaces. Applications include a constructive form of the Caristi theorem and quantitative relations between metric slope error bounds and the global growth of functionals.
arXiv:2606.00290v1 Announce Type: new Abstract: The inducibility constant $\lambda_{F}$ of a graph $F$ is the asymptotically maximum induced density of $F$ in a growing sequence of graphs. This paper systematically investigates the case when $F$ has 6 vertices (and there are 78 cases to consider up to isomorphism and complementation). We show that flag algebras can compute the sharp upper bound on $\lambda_F$ in 36 cases of which, as far a..
arXiv:2606.00298v1 Announce Type: new Abstract: Data-driven reduced-order modeling is an essential component in the computer-aided design of control systems. In this work, we present a novel symmetric Hermite formulation of the quadrature-based balanced truncation algorithm that constructs linear reduced-order models from evaluations of the full-order system's transfer function and its derivative. Significantly, the Hermite formulation pre..
arXiv:2606.00312v1 Announce Type: new Abstract: Over the past half-dozen years, stochastic rounding (SR) has regained significant attention as a quantization scheme for low-precision floating-point arithmetic, with applications spanning numerical analysis and modern machine learning systems. Recent work has shown that SR acts as an implicit regularizer by increasing the smallest singular value of extremely tall-and-thin (or, symmetrically,....
arXiv:2606.00316v1 Announce Type: new Abstract: We study smooth functions on Euclidean space whose Hessian is positive definite outside a bounded set, with emphasis on products of squared distance functions. More precisely, we first prove a simple convexity principle: if the superlevel region $f^{-1}([c,\infty))$ is contained in the Hessian-positive region of $f$, then the sublevel set $\{f\le c\}$ is convex. We apply this to finite produc..
arXiv:2606.00325v1 Announce Type: new Abstract: This work deals with the ergodicity for the stochastic dissipative Hasegawa-Mima equations in a bounded domain and perturbed by additive noise. The uniqueness of the invariant measure is obtained by the asymptotic coupling technique.
arXiv:2606.00330v1 Announce Type: new Abstract: We study the fluctuations of logarithmic Green singularities in the spherical ensemble, viewed as a random discretization of the two-sphere. Smooth observables exhibit the usual Sobolev or Gaussian free field fluctuations, whereas logarithmic singularities live on a larger logarithmic scale and asymptotically decouple in high-dimension, producing an explicit white-noise limit. The result give..
arXiv:2606.00335v1 Announce Type: new Abstract: Perelman's proof of the Poincare conjecture shows that every simply connected closed 3-manifold is homeomorphic to the 3-sphere. The fundamental groups of 3-manifolds attract lots of interest from mathematicians of different fields. As it was stated in a famous survey of Allen Hatcher "The classification of 3-manifolds", one would want to know exactly which groups occur as fundamental groups ....
arXiv:2606.00343v1 Announce Type: new Abstract: Motivated by the analysis of the behaviour of extremes from multivariate heavy-tailed distributions, we introduce a novel notion of statistical depth, referred to as Polar Depth. The polar depth function is naturally expressed in polar coordinates, as is the limiting distribution of a regularly varying random variable, beyond asymptotically large thresholds, once its marginals have been appro....
arXiv:2606.00347v1 Announce Type: new Abstract: We study empirical $L_p$ moments of a random vector $\pmb\varphi$ based on its i.i.d.\ copies $\pmb\varphi^1,\ldots,\pmb\varphi^m$, that is, $\frac1m\sum_{j=1}^m |\langle \pmb\varphi^j,y\rangle|^p$. Our main result is a new estimate for the expected uniform deviation \[ \mathbb{E}\sup_{y\in D}\biggl| \frac1m\sum_{j=1}^m |\langle \pmb\varphi^j,y\rangle|^p -\mathbb{E}|\langle \pmb\varphi,y\rang....
arXiv:2606.00354v1 Announce Type: new Abstract: We demonstrate that the three prevailing notions of completeness for Cartan geometries are not actually equivalent and provide methods for constructing many examples in the affine case, modernizing and extending results due to Yeaton Clifton. In an effort to rectify the folklore on the topic, we also provide detailed explanations of how specious proofs in the literature that would claim that ..
arXiv:2606.00363v1 Announce Type: new Abstract: The task of inductive logic is to develop a formal framework to analyze inductive reasoning. Historically this was accomplished by assigning probabilities to sentences of a logical language. Two natural criteria for such a system are: (i) the underlying language should be rich enough to express scientific hypotheses, and (ii) the probabilities should be, in some sense, accessible. The first c....
arXiv:2606.00375v1 Announce Type: new Abstract: Complementing a theorem of \v{S}krekovski, we characterize the $(h-1)$-critical graphs embeddable in surfaces of Euler genus at least $5$, where $h$ denotes the Heawood number of the surface. Outside of a few small cases, the bulk of our proof is determining the genus of the join of a complete graph and the 5-cycle. As a byproduct of our proof, we also provide a simpler solution to the minimu..
arXiv:2606.00378v1 Announce Type: new Abstract: Many researchers have proposed their own potential-energy functions for the study of diatomic molecules, some of which have been proved to be equivalent. In this paper we suggest the application of a few simple rules to determine the equivalence between molecular potentials. The resulting systematic procedure is illustrated by means of several examples.
arXiv:2606.00381v1 Announce Type: new Abstract: The classic Besicovitch projection theorem asserts that if a set is purely $1$-unrectifiable with finite length in $\mathbb{R}^2$, its orthogonal projection has Lebesgue measure zero in almost every direction. In the opposite direction, the two-projection theorem states that if a Borel set has zero measure under orthogonal projections onto two distinct non-antipodal directions, it must be pur..
arXiv:2606.00385v1 Announce Type: new Abstract: Let $\Bbbk$ be a field, $H$ a colour Hopf algebra and $A$ a graded $H$-comodule colour algebra. We give a sufficient condition for a colour $(A,H)$-Hopf module to be injective as a graded $H$-comodule and we deduce relative projectivity in the category of colour $(A,H)$-Hopf modules. We generalize the Fundamental Theorem of $(A,H)$-Hopf modules to the context of colour $(A,H)$-Hopf modules. U..
arXiv:2606.00388v1 Announce Type: new Abstract: High-resolution energy system capacity expansion models (CEMs) for energy transition planning often result in large-scale mixed-integer linear programming (MILP) formulations. Benders decomposition (BD) offers a scalable solution approach by iteratively solving a master problem (MP) for investment decisions and multiple subproblems (SPs) for operational decisions. However, accumulated Benders....
arXiv:2606.00389v1 Announce Type: new Abstract: We consider the periodic quintic nonlinear Schr\"odinger and prove small-mass global well-posedness in $H^s(\mathbb{T})$ for $s>0$. The proof relies on a new derivative-loss-free $L^6_{t,x}$ Strichartz estimate which is established using the high-low method, an asymmetric superlevel set estimate and a new refined broad-narrow argument. Although our $L^6_{t,x}$ Strichartz estimate is not sharp..
arXiv:2606.00396v1 Announce Type: new Abstract: We establish basic properties of the deformed mass and phase functions on the space of stability conditions. We prove that these functions are continuous and deduce that the space of stability conditions admits a homeomorphic embedding into a product space of finite measures. Subsequently, we give a proof of the triangle inequality for deformed mass functions and provide estimates for the def..
arXiv:2606.00406v1 Announce Type: new Abstract: This paper introduces a new framework for quantum simultaneous information and power transfer (QSIPT), enabling the joint use of quantum states for classical information and energy transfer in quantum communication systems. We propose a novel model in which quantum states are simultaneously used to transmit classical information through a quantum channel and transfer energy to an energy harve....
arXiv:2606.00410v1 Announce Type: new Abstract: We prove that the set of positive integers having between $1$ and $n$ prime factors, counted with multiplicity, has degree of accessibility $2^n$. In particular, the case $n=1$ answers a question of Landman and Robertson asking whether the set of prime numbers is $2$-accessible.
arXiv:2606.00420v1 Announce Type: new Abstract: Ballantine, Beck, and Merca defined the elementary symmetric partition map pre$_j$ that sends a partition $\lambda$ to a larger partition whose parts are the summands appearing in the evaluation of the $j$-th elementary symmetric polynomial on $\lambda$. They conjectured that pre$_j$ is injective on the set of partitions of $n$ with length $\ell \geq j$. The $\ell = j$ case was disproved by D..
arXiv:2606.00421v1 Announce Type: new Abstract: We construct explicit isomorphisms between three models for the $B(\infty)$ crystal in type ${\tt A}_1^{(1)}$: affine Mirkovi\'c--Vilonen polytopes, Kleshchev multipartitions, and a new model we call upper ledge diagrams. We also present some clarifying results on these crystals, giving a direct method for completing an affine MV polytope from the data of one of its boundary root partitions, ..
arXiv:2606.00466v1 Announce Type: new Abstract: We use sieving arguments to estimate the frequency of $s$-tuples of rational points $$(P_1,\dots,P_s)\in E_1(\mathbb{Q})\times\cdots\times E_s(\mathbb{Q}),$$ where $E_1,\dots,E_s$ are (not necessarily distinct) elliptic curves over $\mathbb{Q}$, for which the product of their denominators is a perfect $\ell$th power for a fixed prime $\ell$. We consider two settings: one in which the points a..
arXiv:2606.00468v1 Announce Type: new Abstract: We present a lifting-based interface reduction framework for nonlinear transmission and eigenvalue problems. The method represents the solution as a sum of a bulk component and a lifting component that carries the interface jump, thereby reducing the original problem to a nonlinear system posed on the interface. A low-dimensional approximation is obtained by restricting the interface unknown....
arXiv:2606.00469v1 Announce Type: new Abstract: We study supervised regression with neural ODEs (NODEs) from a control-theoretic perspective to derive explicit population-risk bounds. We focus on a widely used class of non-autonomous models with constant parameters and explicit time dependence, which we call semi-autonomous NODEs (SA-NODEs). We constructively prove that SA-NODEs are capable of \emph{exact} interpolation of admissible finit....
arXiv:2606.00475v1 Announce Type: new Abstract: This paper introduces interval $B_{\pi}^{R^I}$-tensors as a natural extension of $B_{\pi}^{R}$-tensors to the interval setting. We provide two practical verifiable criteria for an interval tensor to be an interval $B_{\pi}^{R^I}$-tensor, one based on endpoint inequalities and another constructing an explicit vector $\pi$. Connections with interval $P$-tensors, positive definite interval tenso..
arXiv:2606.00478v1 Announce Type: new Abstract: Online high-dimensional regression has gained increasing attention in recent years, yet existing methods typically assume that all candidate features, including important ones, are observed from the outset of data collection. This assumption is often violated in real-world scenarios, where new variables become available gradually as data accumulate. To address this gap, we introduce a novel f....
arXiv:2606.00480v1 Announce Type: new Abstract: Continuous data assimilation seeks to estimate the state of a dynamical system from partial observations. In many applications, however, the state dynamics are unknown or prohibitively expensive to simulate at the required resolution, leading to model error. Motivated by this challenge and the increasing adoption of machine learning surrogates in data assimilation, this paper develops a unifi....
arXiv:2606.00482v1 Announce Type: new Abstract: Let $k$ be a real quadratic field. Let $k_n$ be the $n$-th layer of the $\mathbb{Z}_2$-extension of $k$, and $2^{a_n}$ be the $2$-part of the class number of $k_n$. In this paper, we derive many laws concerning the sequence $(a_0,a_1,\dotsc)$, by iteratively extending the group of cyclotomic units.
arXiv:2606.00484v1 Announce Type: new Abstract: We characterize dominating sets for weighted Fock spaces $F^p_{\alpha,w}$ induced by Muckenhoupt $A_{\infty}$ weights, which leads to a sufficient condition for reverse Carleson measures for the spaces $F^p_{\alpha,w}$ with restricted $A_{\infty}$ weights. Applications to invertibility of Toeplitz operators, closed range property of Volterra operators, and atomic decomposition of $F^p_{\alpha..
arXiv:2606.00505v1 Announce Type: new Abstract: Exceptional longest-run behaviour in continued fraction expansions is studied through the interaction between fixed-symbol runs and the overall longest run. For every prescribed partial quotient value and every admissible growth scale, a full Hausdorff dimensional set of irrational numbers is constructed on which the longest run of the prescribed value has exactly the prescribed asymptotic gr..
arXiv:2606.00517v1 Announce Type: new Abstract: This paper studies the regularization of ill-posed inverse problems by deep neural networks (DNNs). We extend architecture-based regularization from shallow networks to deep models by developing a deterministic framework in which the admissible network class is enlarged adaptively and the resulting architecture complexity acts as the regularization mechanism. We propose two discrepancy-princi....
arXiv:2606.00520v1 Announce Type: new Abstract: Many stochastic gradient methods are believed not to converge when the noise in stochastic gradients has only a finite $p$-th moment for $p\in\left(1,2\right)$, a setting known as the heavy-tailed noise assumption. However, some recent studies have found that Stochastic Gradient Descent ($\textsf{SGD}$), without any modification to its update rule, can surprisingly converge in expectation for....
arXiv:2606.00525v1 Announce Type: new Abstract: We introduce a family of constants \[ C_m := \lim_{n\to\infty} \left( \sum_{k=1}^n \operatorname{Li}_m\!\left(\frac1k\right) - \log n \right), \] which may be regarded as polylogarithmic analogues of Euler's constant. We study their basic properties and derive representations in terms of iterated logarithmic integral structures associated with the gamma function. We further introduce associat..
arXiv:2606.00526v1 Announce Type: new Abstract: We study perturbation and stability properties of solution mappings associated with convex regularized least-squares problems. We first establish an implicit function theorem for generalized equations governed by maximally monotone operators and smooth perturbations, yielding conditions for local Lipschitz continuity, directional differentiability, and semismoothness of solution mappings. We ....
arXiv:2606.00528v1 Announce Type: new Abstract: Let $E$ be a uniformly smooth and uniformly convex real Banach space. We study perturbation-resilient inertial Krasnosel'skii-type hybrid retraction schemes for a countable family of generalized nonexpansive mappings satisfying the NST-condition with a family $\Gamma$. The main result shows that strong convergence is preserved when the exact $\phi$-Fej\'er decrease condition is replaced by a ....
arXiv:2606.00533v1 Announce Type: new Abstract: A \textit{cut-edge} of a connected graph is an edge whose deletion increases the number of components. In this paper, we first state some conditions for a planar graph to have a few cut-edges. After that, we use the main results to study the conditions for a colored planar graph to have a bounded conflict-free connection number.
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