arXiv preprints from January 1, 2026 through July 20, 2026 — 10:39:00 EST
We study Johnson amenability for unconditional direct sums of Banach algebras. Given a family $(A_i)_{i\in I}$ of Banach algebras and a Banach sequence lattice $E$ on~$I$, the $E$-sum $\bigl(\bigoplus_{i\in I} A_i\bigr)_{\!E}$ carries a natural Banach algebra structure via coordinatewise multiplication. Under the hypothesis that $C_E...
Given a coaction $δ$ of a locally compact group $G$ on a $\mathrm{C}^*$-algebra $A$, we study the relationship between two different forms of coaction invariance of ideals of $A$ and the ideals of the corresponding crossed product $\mathrm{C}^*$-algebra $A \rtimes_δ G$. In particular, we characterize when these two notions of...
We study the class structure of finite-alphabet Markov chains with arbitrary memory length. To capture the structural constraints induced by prohibited transitions, we introduce the skeleton of a higher-order transition kernel, defined as a reduced set of contexts encoding all essential zero-probability patterns. To each skeleton we...
A constructive modification of the moving frame method is developed in this paper for the construction of relative invariants of regular Lie group actions. Let a relative invariant $I$ of weight $ω$ transform according to the rule $$ I(g \cdot \boldsymbol x) = μ(g, \boldsymbol x)^ω I(\boldsymbol x), $$ where $μ: G \times \mathcal{M}...
We construct a new family of surgery exact triangles in Heegaard Floer theory over the field with two elements. This family generalizes both Ozsváth and Szabó's $n$- and $1/n$-surgery exact triangles for positive integers $n$ and the author's recent 2-surgery exact triangle to all positive rational slopes. The construction reduces...
This paper proposes an efficient algorithm for testing copositivity of homogeneous polynomials over the positive semidefinite cone. The algorithm is based on a novel matrix optimization reformulation and requires solving a hierarchy of semidefinite programs. Notably, it always terminates in finitely many iterations. If a homogeneous...
The asymptotic behavior of the Mellin transform of the associated $B$-splines $B_N^*(t) :=t^{-N}B_N(t)$ with special knots in terms of theta-like functions is found. The proof is based on polynomial interpolation of power functions and properties of certain theta-like functions. Pointwise asymptotics of $B_N^*$ and $B_N$ are...
Pure-breakage population balance equations (PBEs) give the standard deterministic description of fragmentation and comminution. They predict mean particle size distributions, but they do not determine fluctuations, size-size correlations, or universality under coarse-graining. We develop a field-theoretic framework anchored in the PBE...
A subgraph $H$ of an edge-colored graph $G$ is rainbow if all the edges of $H$ receive different colors. If $G$ does not contain a rainbow subgraph isomorphic to $H$, we say that $G$ is rainbow $H$-free. For connected graphs $H_1$ and $H_2$, if there exists an integer $t=t(H_1,H_2)$ such that every rainbow $H_1$-free edge-colored...
In [36, Section 8], the present author proposed the hypergraph obstruction for the existence of k-regular embeddings. In this paper, we develop the hypergraph obstruction concretely and give some homological obstructions for the k-regular embeddings of graphs by using the embedded homology of sub-hypergraphs of the (k-1)-skeleton of...
Given a stratified topological space, we answer the question whether the functor from the derived category of constructible sheaves to the derived category of sheaves with constructible cohomology is an equivalence. We also establish basic facts on the category of locally constant sheaves and on the category of constructible sheaves.
We relate hook-length products for adjacent staircase partitions to special values of Jacobi polynomials. This connection expresses the number of semistandard tableaux in terms of Jacobi polynomials defined via Gauss hypergeometric functions. From this identity, we derive the special values of stable Grothendieck polynomials and...
This paper mainly establishes the finite-horizon stochastic bounded real lemma, and then solves the $H_{\infty}$ control problem for discrete-time stochastic linear systems defined on the separable Hilbert spaces, thereby unifying the relevant theoretical results previously confined to the Euclidean space $\mathbb{R}^n$. To achieve...
We establish a new $T1$ theorem for the compactness of bi-parameter Calderón-Zygmund singular integral operators. Namely, we show that if a bi-parameter CZO $T$ satisfies the product weak compactness property, the mixed weak compactness/CMO property, and $T1, T^t1,$ $T_t1, T_t^t1 \in...
The online Ramsey number $\tilde r(G,H)$ is defined via a Builder--Painter game on an empty graph with countably many vertices. In each round, Builder reveals an edge, which Painter immediately colors either red or blue. Builder wins once a red copy of $G$ or a blue copy of $H$ appears, and $\tilde r(G,H)$ is the minimum number of...
This paper establishes a rigorous theoretical foundation for the function class implicitly learned by XGBoost, bridging the gap between its empirical success and our theoretical understanding. We introduce an infinite-dimensional function class $\mathcal{F}^{d, s}_{\infty-\text{ST}}$ that extends finite ensembles of bounded-depth...
We study a quantitative Ramsey-type problem on 3-term arithmetic progressions: how should the set of integers $[n] = \{1, 2, \dots, n\}$ be colored using 3 colors in order to maximize the number of rainbow 3-term arithmetic progressions? By "rainbow", we mean progressions whose elements are each assigned a distinct color. We determine...
We investigate the Assouad spectrum and dimension of graphs of functions lying in certain Banach spaces. We find the typical values in the sense of Baire category, proving that these values are often as large as possible, given the constraints of the particular function space. For example, we demonstrate that in the little $α$-Hölder...
In the paper, we mainly determine the structures, counting formulas, and density sets of representations for binary and ternary ADC quadratic lattices over arbitrary non-archimedean local fields. In the binary case, we show that under certain conditions, there are finitely many primitive positive definite ADC lattices and infinitely...
Cubic forms $C$ are constructed in the work of R. Aguilar, M. Green and P. Griffiths to establish the generic global Torelli theorem for Fano-K3 pairs $(X,Y)$, where $X: F=0$ is a cubic threefold in $\mathbb{P}^4$ and $Y\in|-K_X|$ is an anticanonical smooth section of $X$ defined by a quadratic form $Q$. In this article, we prove the...
There exists a complex structure $J$ on a connected open subset $S^3_δ\times S^3$ of $S^6$. The present paper proves that: (1) $J$ can be extended to a global almost complex structure $\widetilde{J}$ on $S^6$; (2) any extension to $S^6$ is necessarily non-integrable. Therefore, it is impossible to deform $\widetilde{J}$ to an...
A kind of two-variable elliptic genus for almost-complex manifolds was introduced by Ping Li and its various properties were established by him. In this paper, we define a two-variable elliptic genus for odd dimensional spin manifolds which is the index for some Toeplitz operator and a holomorphic $SL(2,Z)$-Jacobi form. We also define...
Neural networks (NNs) can be viewed as approximation tools. Traditionally, NNs are relying on gradient and stochastic gradient (SG) methods. There are a number of available computational packages for constructing least squares approximations, while uniform (minimax) approximations are hard due to their nonsmooth nature. It was...
We establish a stable form of the general Euclidean Brascamp-Lieb inequality in all cases in which the Lebesgue exponents are strictly between 1 and 2, asserting that all near-extremizers are nearly Gaussian.
In this paper, we investigate the existence and nonexistence of entire solutions to a general class of Cauchy problems in the positive half line. Our results provide a unified approach to proving sharp local and entire solvability of nonlinear partial differential equations in n-dimensional Euclidean space. As applications of the...