arXiv subjects

Ángel Chávez

Publications and source records attributed to Ángel Chávez.

7 recordsLinked to original sources

arXiv2026

Extremal Polynomial Norms of Graphs

Recent work shows that a new family of norms on Hermitian matrices arise by evaluating the even degree complete homogeneous symmetric (CHS) polynomials on the eigenvalues of a Hermitian matrix. The CHS norm of a graph is then defined by evaluating the even degree CHS polynomials on the eigenvalues of the adjacency matrix of a graph. The fact that these norms are defined in terms of eigenvalues (as opposed to singular values) ensures they can distinguish between graphs that other norms cannot. In addition, we prove that the CHS norms are minimized over all connected graphs by the path and maximized over all connected graphs by the complete graph. Finally, we prove that the CHS norms are minimized over all trees by the path and maximized over all trees by the star. Our paper is intended for a wide mathematical audience and we assume no prior knowledge about graphs or symmetric polynomials.

math.CO↗

arXiv2026

Stability of Khintchine-type inequalities via log-monotonicity

We investigate Khintchine-type inequalities for the weighted sums $S=\sum_ka_kX_k$ of independent copies of a symmetric random variable $X$. We show how log-monotonicity of the sequence $r_k(X)=k! \mathbb{E}[X^{2k}]/(2k)!$ implies sharp comparisons between the $L_p$ and $L_2$ norms of $S$ for every even integer $p\geq 2$, extending classic Khintchine-type inequalities and yielding new results in the log-convex setting. We also investigate the stability of our inequalities. Our first stability inequality sharpens the classic inequality by a deviation of the coefficient vector from the coordinate extremizers, while the second quantifies deviation from the Gaussian limit. Our results recover recent stability inequalities for random signs and apply to a broad class of distributions, including type-$\mathscr{L}$ random variables, ultra sub-Gaussian random variables and Gaussian mixtures.

math.PR↗

arXiv2026

A general framework for inequalities on simple graphs

A general framework is developed for deriving sharp inequalities on simple graphs from majorization and Schur-convexity. After establishing majorization relations between the spectrum of an arbitrary graph and the spectra of the complete, complete bipartite, and matching graphs, it is shown that every positive Schur-convex spectral functional yields several sharp inequalities relating $λ_1$, $|λ_n|$, and $\|G\|_\ast$. This reduces the problem of proving graph inequalities to the choice of a suitable Schur-convex function. This optimization problem is then studied within the family of random vector norms, whose moment and cumulant expansions connect the framework to the numbers of closed walks. This yields new sharp results, recovers classical inequalities from a unified viewpoint, and produces further bounds in settings such as triangle-free and square-free graphs.

math.CO↗

arXiv2024

Hunter's positivity theorem and random vector norms

A theorem of Hunter ensures that the complete homogeneous symmetric polynomials of even degree are positive definite functions. A probabilistic interpretation of Hunter's theorem suggests a broad generalization: the construction of so-called random vector norms on square complex matrices. This paper surveys these ideas, starting from the fundamental notions and developing the theory to its present state. We study numerous examples and present a host of open problems.

math.FA↗

arXiv2023

Norms on complex matrices induced by random vectors II: extension of weakly unitarily invariant norms

We improve and expand in two directions the theory of norms on complex matrices induced by random vectors. We first provide a simple proof of the classification of weakly unitarily invariant norms on the Hermitian matrices. We use this to extend the main theorem in [7] from exponent $d\geq 2$ to $d \geq 1$. Our proofs are much simpler than the originals: they do not require Lewis' framework for group invariance in convex matrix analysis. This clarification puts the entire theory on simpler foundations while extending its range of applicability.

math.FA↗

arXiv2022

Norms on Complex Matrices Induced by Random Vectors

We introduce a family of norms on the $n \times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter's positivity theorem for the complete homogeneous symmetric polynomials.

math.FA↗

arXiv2022

Norms on complex matrices induced by complete homogeneous symmetric polynomials

We introduce a remarkable new family of norms on the space of $n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. Our norms enjoy many desirable analytic and algebraic properties, such as an elegant determinantal interpretation and the ability to distinguish certain graphs that other matrix norms cannot. Furthermore, they give rise to new dimension-independent tracial inequalities. Their potential merits further investigation.

math.CO↗