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Samir Mondal

Publications and source records attributed to Samir Mondal.

8 recordsLinked to original sources
arXiv2026

Linear Preservers of Infinitely Divisible Matrices

An infinitely divisible nonnegative matrix is an entry-wise nonnegative matrix that admits an entry-wise nonnegative $m$th root, with respect to usual matrix multiplication, for every positive integer $m$; it is called strongly infinitely divisible when it is, in addition, invertible. The study of such matrices has its origins in the theory of infinitely divisible probability distributions and the embedding problem for Markov matrices and is closely connected with continuous one-parameter semigroups; in the invertible case, this connection admits a natural description in terms of the exponential map. In this paper, we characterize the bijective linear maps on $M_n(\mathbb{R})$ that preserve strongly infinitely divisible matrices and infinitely divisible nonnegative matrices. In the former case, we combine the Inverse Function Theorem with Zariski-density techniques, using the Zariski-density approach recently developed in linear preserver theory by Fallat and Mondal [\textit{Proc. Amer. Math. Soc.}, 2026]. In the latter case, we first show that preserving infinite divisibility forces preservation of the cone of entry-wise nonnegative matrices and then exploit the additional structure afforded by infinite divisibility to complete the characterization.

math.RA↗
arXiv2026

Powers, Roots, and Positivity

There is a remarkable variety of natural generalizations of the notion of scalar positivity to square matrices. The lists and tables indicate those that we consider here; they are practically exhaustive of those that arise in practice and theory. Since positive scalars are closed under taking arbitrary powers and roots, it is natural to ask which of these matrix generalizations enjoy analogous closure properties. Our purpose here is to examine and advance these generalizations with respect to their closure under integer powers and fractional matrix roots. In addition to surveying known results, we establish for several important classes of matrices that positivity is preserved under fractional powers, showing that key structural features are retained under matrix roots. The role of Pick functions in the analysis is emphasized throughout. Proofs, counterexamples, and related results are presented, and summary tables are included to provide a concise and useful reference.

math.FA↗
arXiv2026

Variations on Majorization of Vectors and Connections to Determinantal Inequalities

Majorization is a fundamental tool for comparing vectors, with connections to convexity, doubly stochastic matrices, eigenvalues, singular values, and zeros of polynomials. In matrix analysis, it plays a central role in the study of eigenvalue inequalities, particularly those arising from classical determinantal inequalities such as those attributed to Hadamard and Fischer in the context of positive semidefinite matrices. A result of Fischer and Holbrook shows that equality in the Hardy--Littlewood--P\'olya theorem for non-affine convex functions is closely linked to block structure in the associated doubly stochastic transformations. Motivated by this, we introduce $*$-majorization, a structured extension of majorization that respects prescribed block decompositions of vectors. This framework naturally corresponds to block diagonal doubly stochastic matrices and provides a refinement of the classical Hardy--Littlewood--P\'olya and Rado theorem. We show that such transformations are precisely the linear operators that preserve $*$-majorization, and we extend fundamental constructions such as $T$-transforms and convex combinations to this setting. In an application, we study the eigenvalue relations associated with the principal submatrices of positive definite matrices. Classical majorization does not, in general, capture determinantal inequalities such as those of Koteljanskii, whereas $*$-majorization provides a natural framework for structured comparisons of eigenvalue vectors. This leads to new insights into the interplay between majorization theory, determinantal inequalities, and spectral properties of matrices.

math.GM↗
arXiv2026

Determinant Bounds for $(n-1)$-Locally Positive Semidefinite Matrices

In this framework, the extremal case corresponds to the tightest nontrivial relaxation in this hierarchy, in which every proper principal submatrix is constrained to be positive semidefinite, while the global positive semidefiniteness condition is governed by the determinant. In this paper, we study the determinants of locally positive semidefinite matrices and derive sharp lower bounds on their determinants that quantify the gap between local and global positive semidefiniteness. We further obtain analogous extensions of classical determinant inequalities, including Fisher and Koteljanskii inequalities, providing tight lower bounds in each case. In a sense, these results quantify, via determinant bounds, how far the class of locally positive semidefinite matrices can be from being positive semidefinite.

math.OC↗
arXiv2025

Linear Preservers of Real Matrix Classes Admitting a Real Logarithm

In real Lie theory, matrices that admit a real logarithm reside in the identity component $\mathrm{GL}_n(\mathbb{R})_+$ of the general linear group $\mathrm{GL}_n(\mathbb{R})$, with logarithms in the Lie algebra $\mathfrak{gl}_n(\mathbb{R})$. The exponential map \[ \exp : \mnr \to \mathrm{GL}_n(\mathbb{R}) \] provides a fundamental link between the Lie algebra and the Lie group, with the logarithm as its local inverse. In this paper, we characterize all bijective linear maps $\varphi : \mnr \to \mnr$ that preserve the class of matrices admitting a real logarithm (principal logarithm). We show that such maps are exactly those of the form \[ \varphi(A) = c\, P A P^{-1} \quad \text{or} \quad \varphi(A) = c\, P A^{T} P^{-1}, \] for some $P \in \mathrm{GL}_n(\mathbb{R})$ and $c > 0$. The proof proceeds in two stages. First, we analyze preservers within the class of standard linear transformations. Second, using Zariski denseness, we prove that any bijective linear map preserving matrices with real logarithms (principal logarithm) must preserve $\mathrm{GL}_n(\mathbb{R})$, which then implies the map is of the standard form.

math.RT↗
arXiv2023

A short survey of $Z$-matrices and new results on the subclass of $F_0$-matrices

A real square matrix $A$ of order $n \times n~ (n \geq 3)$ is called an $F_0$-matrix, if it is a $Z$-matrix (off-diagonal entries nonpositive), all of whose principal submatrices of orders at most $n-2$ are $M$-matrices while there is at least one principal submatrix of order $n-1$, which is an $N_0$-matrix. An $M$-matrix is a $Z$-matrix with the property that the real parts of all its eigenvalues are nonnegative. An $N_0$-matrix, in turn, is characterized by the fact that it is an invertible $Z$-matrix whose inverse is (entrywise) nonpositive. The first aim of this article is to present a short survey of some subclasses of $Z$-matrices, pertinent to the second objective, where new results concerning $F_0$-matrices are presented.

math.RA↗
arXiv2021

A packaged whispering gallery resonator device based on an optical nanoantenna coupler

In this work, we present the design and fabrication of a packaged whispering gallery mode (WGM) device based on an optical nanoantenna as the coupler and a glass microsphere as the resonator. The microspheres were fabricated from SiO$_2$ fiber or Er$^{3+}$-doped fiber, the latter creating a WGM laser with a threshold of 93 $μ$W at 1531 nm. The coupler-resonator WGM device is packaged in a glass capillary. The performance of the packaged microlaser is characterized, with lasing emission both excited in and collected from the WGM cavity via the nanoantenna. The packaged system provides isolation from environmental contamination, a small size, and unidirectional coupling while maintaining a high quality (Q-) factor ($\sim$10$^8$). It opens up new possibilities for practical applications of WGM microdevices in a variety of fields such as low threshold lasers, filters, and sensors.

physics.optics↗