Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.17902

Entrywise preservers of sign regularity

Projesh Nath Choudhury·Shivangi Yadav
View original source ↗Read source document ↗Export metadata ↓

Abstract

Entrywise functions preserving positivity and related notions have a rich history, beginning with the seminal works of Schur, P\'olya-Szeg\H{o}, Schoenberg, and Rudin. Following their classical results, it is well-known that entrywise functions preserving positive semidefiniteness for matrices of all dimensions must be real analytic with non-negative Taylor coefficients. These works were taken forward in the last decade by Belton, Guillot, Khare, Putinar, and Rajaratnam. Recently, Belton-Guillot-Khare-Putinar [J. d'Analyse Math. 2023] classified all functions that entrywise preserve totally positive (TP) and totally non-negative (TN) matrices. In this paper, we study entrywise preservers of strictly sign regular and sign regular matrices - a class that includes TP/TN matrices as special cases and was first studied by Schoenberg in 1930 to characterize variation diminution. Our main results provide complete characterizations of entrywise transforms of rectangular matrices which preserve: (i)~sign regularity and strict sign regularity, as well as (ii)~sign regularity and strict sign regularity with a given sign pattern.

Explore related subjects

math.FAmath.CAmath.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Projesh Nath Choudhury, Shivangi Yadav. 2025-09-22. Entrywise preservers of sign regularity. https://arxiv.org/abs/2509.17902

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Typical dynamical properties of operators on $\ell_p$

We investigate the typical dynamical properties of hypercyclic operators in $\mathcal{L}_M(X)$, the set of all bounded linear operators on $X$ whose norms are at most $M$, when $X=\ell_p$, $1< p<\infty$. We show that, with respect to SOT$^*$, a typical operator $T\in \mathcal{L}_M(X)$ is weakly mixing, is weakly disjoint from a given hypercyclic operator $S$, is not topologically ergodic, and satisfies $(T,T^2,\dotsc,T^k)$ is disjoint hypercyclic for any $k\geq 2$. We also study the typical dynamical properties for the concrete family $\mathcal{M}=\{I+B_w\in \mathcal{L}(X)\colon w\in c_0(\mathbb{Z})\}$, endowed with the norm topology, where $B_w$ is a bilateral weighted backward shift.

math.FA↗
arXiv2026

A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps

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.

math.FA↗
arXiv2026

On weak dominance of t-conorms over t-norms

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.

math.FA↗