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arXiv:2606.02513v1 Announce Type: new Abstract: We provide a classification of neighborly Murai spheres, which implies that all of them are polytopal. Furthermore, we show that each neighborly $d$-sphere with no more than $d+4$ vertices is combinatorially equivalent to a Murai sphere for any $d\geq 1$.
arXiv:2606.02517v1 Announce Type: new Abstract: We study the flow of an isothermal compressible Newtonian fluid around a body that performs a (time-independent) rigid motion. We derive a weak-strong uniqueness principle, and show that in the low Mach number limit, the governing equation is well approximated by the Navier-Stokes equations for incompressible rotating flow. Both results are based on the derivation of a relative energy inequal..
arXiv:2606.02525v1 Announce Type: new Abstract: In the first part of our paper we give an abstract characterization of matrix convex sets, and compact matrix convex sets. We also extend the real case of this for classical sets (due to Marshall Stone and others) to the complex case. Our approach is in some part via a universal Banach space (resp.\ operator space $X_K$ of an abstract compact convex set (resp.\ matrix convex set) $K$. This tu....
arXiv:2606.02529v1 Announce Type: new Abstract: Incentive design studies how a central authority can influence strategic agents through payments, subsidies, or taxes, so that individual objectives align with collective welfare. This paper introduces a No-Regret Adaptive Incentive Design (RAID) framework for nonlinear games with continuous action spaces and private agent costs. In this framework, the authority (planner) designs incentives t....
arXiv:2606.02541v1 Announce Type: new Abstract: For $k$ sufficiently large, we show that there is a three-coloring of the first $2^{k (\log^* k)/4}$ positive integers without any monochromatic $k$-term arithmetic progressions. Thus, the three-color van der Waerden number $w(k;3)$ grows faster than any exponential in $k$. We further prove a new lower bound on multicolor van der Waerden numbers which resolves a problem of Erd\H{o}s and Graha..
arXiv:2606.02546v1 Announce Type: new Abstract: Given $k$-uniform hypergraphs $G$ and $H$, we say that $G$ is a unique subhypergraph of $H$ if $H$ contains exactly one subhypergraph isomorphic to $G$. For an $n$-vertex $k$-graph $H$, let $f_k(H)$ be the number of non-isomorphic unique subhypergraphs of $H$, normalized by $2^{\binom n k}/n!$, and let $f_k(n)$ be the maximum of $f_k(H)$ over all $n$-vertex $k$-graphs $H$. In the graph case $....
arXiv:2606.02558v1 Announce Type: new Abstract: We show solubility of the conjugacy problem for Dehn twist automorphisms of finitely generated free products of free abelian groups.
arXiv:2606.02561v1 Announce Type: new Abstract: We study pure unital completely positive maps on the finite Toeplitz operator system $ T_{d}$ of $d \times d$ Toeplitz matrices. Our first main result gives an explicit characterization of pure UCP maps from $T_{d}$ to $M_n$ in terms of positive $n\times n$ matrix-valued trigonometric polynomials of degree at most $d-1$. This characterization provides a checkable criterion for deciding when a..
arXiv:2606.02567v1 Announce Type: new Abstract: We show that for any set of $n$ unit vectors $v_1,\ldots,v_n$ in a real Hilbert space and positive numbers $p_1,\ldots,p_n$ satisfying $\sum_j p_j = 1$, there exists a unit vector $u$ such that \[ \sum_{j=1}^n \frac{p_j^2}{\langle v_j, u\rangle^2}\leq 1. \] This inequality is a weighted version of the strong polarization inequality. As immediate corollaries, it yields a polarization inequ..
arXiv:2606.02570v1 Announce Type: new Abstract: We study stochastic completeness for landmark spaces equipped with Riemannian metrics induced by right-invariant metrics on subgroups of the diffeomorphism group of the shape domain. We extend a previous stochastic completeness result, which only covers the case of exactly two landmarks, to landmark spaces with any number of landmarks. This succeeds the characterization of geodesic completene....
arXiv:2605.28052v1 Announce Type: cross Abstract: We analyze a stochastic 5-neighbor cellular automaton with several conserved quantities, including the particle density. By examining the eigenvalue problem of the associated transition matrix, we derive an explicit formula for the stationary distribution on each irreducible component, in which the weight of each configuration is expressed in terms of the numbers of occurrences of two speci..
arXiv:2605.28496v2 Announce Type: cross Abstract: For any positive integer $n$, the author previously constructed several minimal simplicial $n$-complexes which necessarily contain a non-splittable two-component link, consisting of an $(n-1)$-sphere and an $n$-sphere, in any embedding into $\mathbb{R}^{2n}$. In this paper, we present an additional simplicial $n$-complex with the same property.
arXiv:2605.28952v2 Announce Type: cross Abstract: E-values have attracted considerable interest in recent years as flexible tools for enabling anytime-valid and adaptive data analysis. Hypothesis testing is at the core of many of these applications, which can often involve private or sensitive data. In this work, we answer a simple but important question: given two distributions $\mathbb{P}$ and $\mathbb{Q}$, what is the maximum achievable....
arXiv:2605.30797v1 Announce Type: cross Abstract: We revisit the shifted-tree decomposition formula proposed in our previous work arXiv:2410.17192 for two-site cosmological wavefunction coefficients. For the two-site bubble-like family at arbitrary loop order, we show that the nontrivial central part of the decomposition reduces to an alternating subset sum over shifted diagonal divisors. This subset sum is naturally organized by the Boole....
arXiv:2605.31169v1 Announce Type: cross Abstract: This paper studies sum-of-squares (SoS) representations of nonnegative polynomials over the hypercube $[0,1]^n$. De Klerk and Laurent (SIAM J. Optim., 2010) conjectured that the smallest constant $C_n$ such that the polynomial $x_1\cdots x_n +C_n$ is contained in the degree-$n$ truncated quadratic module $M_{n,n}(x_1-x_1^2,\ldots,x_n-x_n^2)$ of the hypercube is $C_n=1/(n(n+2))$, for $n$ eve..
arXiv:2606.00055v1 Announce Type: cross Abstract: Tensor train (TT) methods have recently gained popularity for accelerating the solving of systems of PDEs. Here, we evaluate the performance of TT methods in the context of geophysical fluid dynamics (GFD) using the shallow water equations and a discretization scheme employed by the ocean component of the Energy Exascale Earth System Model (E3SM). Through a suite of four test cases of incre..
arXiv:2606.00063v1 Announce Type: cross Abstract: Systems moving in low Reynolds number fluid regimes are known to be governed by a ``motility map'' which linearly relates their shape change rates to they body frame velocity moving through the fluid. A consequence of this is ``Purcell's Scallop Theorem'' -- a locomotion system that undergoes shape changes that follow the same path forward and backward in time (reciprocal body deformations)....
arXiv:2606.00102v1 Announce Type: cross Abstract: Over the centuries, probability theory has grown from the calculus of games of chance into a central framework for reasoning under uncertainty. This article interprets that evolution not merely as a mathematical history, but as a transformation of rationality itself. From Pascal and Fermat's combinatorial symmetry to the inductive logic of Bayes and Laplace, from Poisson's statistics of eve....
arXiv:2606.00114v1 Announce Type: cross Abstract: Image semantic communication is a critical component in next-generation wireless communication systems. However, such systems typically suffer from large memory footprints and high computational complexity, making them difficult to deploy on resource-constrained devices. To address these challenges, we propose a vision transformer (ViT)-enabled image semantic communication system. In this s....
arXiv:2606.00122v1 Announce Type: cross Abstract: We approach image denoising from a perception-driven perspective: how can we select the parameters that are best suited for human visual perception? We combine research methods in mathematics and psychology to develop a mathematical framework for measuring perceived similarity. We construct a sample set of differently denoised photographs by using the same base image as input data and by tu..
arXiv:2606.00127v1 Announce Type: cross Abstract: In the setting of multi-head finite-state dimensions, trailing heads lag behind a leading head, accessing past data to aid a finite-state gambler placing bets on successive bits read by the leading head. Cruz, Glashausser, Li, and Lutz (2026) proved that, for any fixed number of trailing heads, adaptive (data-dependent) movement rules can strictly outperform oblivious (data-independent) mov..
arXiv:2606.00157v1 Announce Type: cross Abstract: We consider establishing the interpretability theory of deep learning through constructing a corresponding relationship between the renormalization group (RG) method in statistical physics and the training process of deep neural networks (DNNs). We have proved the constructed relationship using the one-dimensional Ising model as the input data. In this paper we generalize our results to the....
arXiv:2606.00175v1 Announce Type: cross Abstract: In this work we study the general relativistic massless Boltzmann equation on Friedmann-Lema\^itre-Robertson-Walker spacetimes with spatial topology $\mathbb{T}^3$ in the linear and decelerated expanding regimes, where the scale factor is $t^{\mathfrak{q}}$ with $\mathfrak{q}\in [0,1]$. The massless Boltzmann equation on these backgrounds admits non-stationary Maxwell-J\"uttner equilibria o....
arXiv:2606.00176v1 Announce Type: cross Abstract: Stochastic inflation is a powerful technique for calculating the probability distribution function (PDF) of large inflationary perturbations, which may collapse to form Primordial Black Holes. The PDF, $P({\cal N})$, of the stochastic number of e-folds, ${\cal N}$, satisfies an adjoint Fokker-Planck Equation. We develop a new self-contained eigenvalue technique which can be used to determin....
arXiv:2606.00179v1 Announce Type: cross Abstract: In this paper, we investigate contraction analysis for nonlinear time-delay systems described by functional differential equations. We first extend the concept of Lyapunov-Krasovskii functionals within the differential framework. We then show that its existence is equivalent to that of an incremental Lyapunov-Krasovskii functional and guarantees uniform incremental exponential stability. Ne..
arXiv:2606.00183v1 Announce Type: cross Abstract: Tree search is a central abstraction behind many language-agent reasoning and decision-making tasks: agents must explore actions, remember failures, and backtrack toward promising alternatives. Yet, we lack a theoretical understanding of how transformer-based policies acquire such search capabilities from the training dynamics of reinforcement learning (RL). We study this question in a stoc....
arXiv:2606.00245v1 Announce Type: cross Abstract: We develop a new algorithm for computing the second discrete homology group of a graph which is much faster when compared to existing algorithms. To do so, we identify five basic shapes, which are quotient graphs of the 3-cube with the property that the injective maps from them detect all possible 2-boundaries in the singular chain complex computing discrete homology.
arXiv:2606.00292v1 Announce Type: cross Abstract: In 2010, Steurer conjectured that any family of $n$ unit-norm vectors $v_1,\dots,v_n$ with polynomially small average correlation $\mathbb{E}_{i,j}|\langle v_i,v_j\rangle|\leq n^{-\epsilon}$ contains linear-sized constant-separated sets. We refute this conjecture in a strong sense using the machinery of sparse high-dimensional expanders: such vector families do not even have linear-sized $\..
arXiv:2606.00296v1 Announce Type: cross Abstract: Neural operators are often reported to exhibit zero-shot super-resolution, a phenomenon in which a model trained on coarse grids produces accurate predictions on finer testing grids without additional retraining. Despite strong empirical evidence, the theoretical foundations of this phenomenon remain unclear. In this work, we provide a systematic theoretical study of zero-shot super-resolut..
arXiv:2606.00311v1 Announce Type: cross Abstract: The causal propagator (or Pauli-Jordan function), which multiplied by $i$ is the spacetime commutator of the field $[\phi(x),\phi(x')]$, plays an essential role in scalar quantum field theory. We discuss the role of the causal propagator and its spectrum in recent developments in defining quantum field theory in a more explicitly covariant manner, as well as in causal set theory. We then pr..
arXiv:2606.00317v1 Announce Type: cross Abstract: Model Predictive Path Integral (MPPI) control is a powerful sampling-based method for solving stochastic optimal control problems and has enabled real-time control in complex robotic systems. Despite its empirical success, its theoretical understanding remains limited. In this work, we show that MPPI can be interpreted as a special case of the Expectation-Maximization (EM) algorithm applied....
arXiv:2606.00401v1 Announce Type: cross Abstract: Simulating large molecular systems comprising thousands of atoms requires highly scalable methodologies. While modern Density Functional Theory (DFT) codes exhibit linear scaling, solving the associated large, sparse generalized eigenproblems remains a critical computational bottleneck on exascale architectures. In the context of the LimitX project, we propose a data-driven framework to acc....
arXiv:2606.00412v1 Announce Type: cross Abstract: Global parameter identifiability is a property of a parametric ODE model to recover the parameter values uniquely from the input-output data. Not all parametric ODE models have this property, and checking for parameter identifiability is a prerequisite to perform numerical parameter estimation. There are many algorithms and software packages for global parameter identifiability, and frequen..
arXiv:2606.00436v1 Announce Type: cross Abstract: Clustering is a central tool for discovering latent structure in unlabeled data; yet modern clustering pipelines often end with a hard assignment of each observation to a cluster without rigorous measures of assignment uncertainty. We propose a novel weighted conformal approach for constructing valid confidence sets for cluster labels. The key difficulty is that the labels available for cal....
arXiv:2606.00442v1 Announce Type: cross Abstract: Many machine learning techniques rely on approximating a loss function's curvature, but this is notoriously hard to do at the scale of modern deep networks. Surprisingly, no previous work has exploited the curvature constraints that arise from well known weight-space symmetries in loss landscapes. By analytically averaging over group actions that leave the loss invariant, we construct struc....
arXiv:2606.00458v1 Announce Type: cross Abstract: The Black Scholes equation provides a fundamental model for the no arbitrage pricing of financial derivatives. After finite difference discretisation, the pricing problem can be formulated as a finite dimensional linear algebra problem involving the inverse of a non Hermitian time step matrix. Recent advances in quantum linear algebra algorithms, particularly the generalised quantum signal ....
arXiv:2606.00481v1 Announce Type: cross Abstract: This research presents a novel stochastic framework for proactive cybersecurity defense timing under a single attack scenario. The approach models the defense process as a continuous observation mechanism in which the defense instant and the subsequent observation slot follow independent exponential distributions. Laplace-Carson transforms combined with first-excess theory yield the joint d....
arXiv:2606.00500v1 Announce Type: cross Abstract: We present a simple and efficient algorithm for robust approximate message passing (AMP) in the spiked matrix setting. In particular, let $\varepsilon$ be a sufficiently small constant, and suppose that $X \in \mathbb R^{n \times n}$ is a Gaussian matrix with a planted rank-$1$ spike, and $E \in \mathbb R^{n \times n}$ is an adversarially chosen matrix supported on an $\varepsilon n \times ....
arXiv:2606.00512v1 Announce Type: cross Abstract: In many modern machine learning pipelines, abundant pretrained representations serve as noisy proxy covariates, while task-specific labels remain scarce. We study semi-supervised regression in this setting, and propose a simple two stage estimator that learns kernel eigenfeatures from all proxy covariates and fits a ridge predictor on labeled data. We derive finite sample bounds showing tha..
arXiv:2606.00539v1 Announce Type: cross Abstract: Training stability is a key bottleneck in low-precision language model training: efficient low-cost paths can still produce short-lived numerical risks at a small set of operators. We formulate this as runtime stability control and present Gradient Norm-to-Mean Ratio (GNMR), a lightweight controller that compares each recoverable unit's current gradient norm with its historical mean. Togeth..
arXiv:2606.00632v1 Announce Type: cross Abstract: Control science is a core representative of the third industrial revolution and is so important to modern civilization. Control systems are the main subject of control science and may involve many aspects of consideration, such as hardware consideration, software consideration, operation consideration, maintenance consideration, economy consideration, society consideration. However, besides....
arXiv:2606.00643v1 Announce Type: cross Abstract: Physics-Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill-conditioned loss landscapes inherited from the underlying differential operator. We study PINNs augmented with a pointwise data-fidelity term, added at a few points in the domain to the standard residual and boundary....
arXiv:2606.00718v1 Announce Type: cross Abstract: While Large Language Models (LLMs) have recently shown promise in Automated Heuristic Design (AHD), existing methods typically generate and evolve heuristics as a single operator or search strategy, limiting their ability to model strong coupling among multiple decision substructures in problems such as the Traveling Thief Problem (TTP) and the Traveling Purchaser Problem (TPP). In this wor....
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