Related papers: Schur's Lemma for Coupled Reducibility and Coupled…

McNamara and Pylyavskyy conjectured precisely which connected skew shapes are maximal in the Schur-positivity order, which says that $B\leq _s A$ if $s_A-s_B$ is Schur-positive. Towards this, McNamara and van Willigenburg proved that it…

For an $m \times n$ complex matrix $X$ of rank $r$ with Schur multiplier $S_X$ we show that there exist an $ r \times m $ complex matrix $L$ and an $ r\times n $ complex matrix $R$ such that $X = L^*R$ and $\|S_X\|\, =\, \|\mathrm{diag}…

Lapid and M\'{i}nguez gave a criterion of the irreducibility of the parabolic induction $\sigma \times \pi$, where $\sigma$ is a ladder representation and $\pi$ is an arbitrary irreducible representation of the general linear group over a…

Let $C$ be a curve with two smooth components and a single node. Let $\mathcal{U}_C(r,w,\chi)$ be the moduli space of $w$-semistable classes of depth one sheaves on $C$ having rank $r$ on both components and Euler characteristic $\chi$. In…

We consider $n\times n$ real-valued matrices $A = (a_{ij})$ satisfying $a_{ii} \geq a_{i,i+1} \geq \dots \geq a_{in} \geq a_{i1} \geq \dots \geq a_{i,i-1}$ for $i = 1,\dots,n$. With such a matrix $A$ we associate a directed graph $G(A)$. We…

In this article, we give two different sufficient conditions for the irreducibility of a polynomial of more than one variable, over the field of complex numbers, that can be written as a sum of two polynomials which depend on mutually…

It is common in stability analysis to linearize a system and investigate the spectrum of the Jacobian matrix. This approach faces the challenge of determining the matrix spectrum when the coefficients depend on parameters or when the…

P. Flajolet and B. Salvy \cite{FS1998} prove the famous theorem that a nonlinear Euler sum $S_{i_1i_2\cdots i_r,q}$ reduces to a combination of sums of lower orders whenever the weight $i_1+i_2+\cdots+i_r+q$ and the order $r$ are of the…

Inspired by the classic apolarity theory of symmetric tensors, the aim of this paper is to introduce the Schur apolarity theory, i.e. an apolarity for any irreducible representation of the special linear group $SL(V)$. This allows to…

In this paper we propose a new scaling method to study the Schur complements of $SDD_{1}$ matrices. Its core is related to the non-negative property of the inverse $M$-matrix, while numerically improving the Quotient formula. Based on the…

Friedland (1981) showed that for a nonnegative square matrix A, the spectral radius r(e^D A) is a log-convex functional over the real diagonal matrices D. He showed that for fully indecomposable A, log r(e^D A) is strictly convex over D_1,…

In this paper, we prove that the variety $C_m(L)$ of commuting $m$-tuples of elements of simple Lie algebra $L$ is often reducible. Explicitely, we prove it is reducible for all simple Lie algebra $L$ not isomorphic to $\mathfrak{sl}_2$ and…

We call a matrix blocky if, up to row and column permutations, it can be obtained from an identity matrix by repeatedly applying one of the following operations: duplicating a row, duplicating a column, or adding a zero row or column.…

Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent…

For a positive integer $n$ and a real number $\alpha$, the generalized Laguerre polynomials are defined by \begin{align*} L^{(\alpha)}_n(x)=\sum^n_{j=0}\frac{(n+\alpha)(n-1+\alpha)\cdots (j+1+\alpha)(-x)^j}{j!(n-j)!}. \end{align*} These…

The Schur decomposition of a square matrix $A$ is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following…

We conjecture a formula for the Schur index of N=2 four-dimensional theories in the presence of boundary conditions and/or line defects, in terms of the low-energy effective Seiberg-Witten description of the system together with massive BPS…

A non-isospectral linear problem for an integrable 2+1 generalization of the non linear Schr\"odinger equation, which includes dispersive terms of third and fourth order, is presented. The classical symmetries of the Lax pair and the…

We look for solutions to the Schr\"odinger equation \[ -\Delta u + \lambda u = g(u) \quad \text{in } \mathbb{R}^N \] coupled with the mass constraint $\int_{\mathbb{R}^N}|u|^2\,dx = \rho^2$, with $N\ge2$. The behaviour of $g$ at the origin…

Schur-type methods in \cite{Chu2} and \cite{GCQX} solve the robust pole assignment problem by employing the departure from normality of the closed-loop system matrix as the measure of robustness. They work well generally when all poles to…