Related papers: On singular value inequalities for matrix means

Given $B_1(0)$ the unit ball of $\mathbb{R}^n$ ($n\geq 3$), we study smooth positive singular solutions $u\in C^2(B_1(0)\setminus \{0\})$ to $-\Delta u=\frac{u^{2^\star(s)-1}}{|x|^s}-\mu u^q$. Here $0< s<2$, $2^\star(s):=2(n-s)/(n-2)$ is…

Positive semidefinite Hermitian matrices that are not fully specified can be completed provided their underlying graph is chordal. If the matrix is positive definite the completion can be uniquely characterized as the matrix that maximizes…

We consider the polynomial equation $$X^n + a_{n-1}\cdot X^{n-1} + \dots + a_1 \cdot X + a_0 \cdot I = O,$$ over $(2 \times 2)$-matrices $X$ with the real entries, where $I$ is the identity matrix, $O$ is the null matrix, $a_i \in \mathbb…

In this paper, we obtain two new lower bounds for the smallest singular value of nonsingular matrices which is better than the bound presented by zou \cite{zou2012lower}, Lin, Minghua and Xie, Mengyan \cite{lin2021some} under certain…

We consider the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the $n$-sphere involving an operator of order $2s> n$. In this case the Sobolev exponent is negative. Our results…

The purpose of this article is two-fold. First, we investigate the inequality $$ -\Delta u+V(x) u\geq f\quad\mbox{ in } B_1\setminus\{0\}\subset \mathbb{R}^N , N \geq 2, $$ where $f\in L^1_{loc}(B_1)$. If $V\geq 0$ is radially symmetric, we…

A $n\times n$ matrix $A$, which has a certain sign-symmetric structure ($J$--sign-symmetric), is studied in this paper. It is shown that such a matrix is similar to a nonnegative matrix. The existence of the second in modulus positive…

In this paper, the authors establish a new type integral inequalities for differentiable s-convex functions in the second sense. By the well-known H\"older inequality and power mean inequality, they obtain some integral inequalities related…

For each irreducible finite dimensional representation of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ of $2\times 2$ traceless matrices, an explicit uniform upper bound is given for the multiplicities in the cocharacter sequence of the…

Let $f(x)=x^TAx+2a^Tx+c$ and $h(x)=x^TBx+2b^Tx+d$ be two quadratic functions having symmetric matrices $A$ and $B$. The S-lemma with equality asks when the unsolvability of the system $f(x)<0, h(x)=0$ implies the existence of a real number…

Let $F(x, y)$ be a binary form with integer coefficients, degree $n\geq 3$ and irreducible over the rationals. Suppose that only $s + 1$ of the $n + 1$ coefficients of $F$ are nonzero. We show that the Thue inequality $|F(x,y)|\leq m$ has…

Let $M$ be an $n\times n$ random i.i.d. matrix. This paper studies the deviation inequality of $s_{n-k+1}(M)$, the $k$-th smallest singular value of $M$. In particular, when the entries of $M$ are subgaussian, we show that for any…

For $k=1,\ldots,K$, let $A_k$ and $B_k$ be positive semidefinite matrices such that, for each $k$, $A_k$ commutes with $B_k$. We show that, for any unitarily invariant norm, \[ |||\sum_{k=1}^K A_kB_k||| \le ||| (\sum_{k=1}^K…

We prove a conjecture of Zagier about the inverse of a $(K-1)\times (K-1)$ matrix $A=A_{K}$ using elementary methods. This formula allows one to express the the product of single zeta values $\zeta(2r)\zeta(2K+1-2r)$, $1\leq r\leq K-1$, in…

We prove bilinear inequalities for differential operators in $\mathbb{R}^2$. Such type inequalities turned out to be useful for anisotropic embedding theorems for overdetermined systems and the limiting order summation exponent. However,…

Over the finite field with two elements, we present a method for obtaining explicit expressions for the number of rank i matrices of the form A above B, where A is persymmetric (A matrix [a(i,j)] is persymmetric if a(i,j) = a(r,s) for i+j =…

Let $\mathcal{S}$ be the set of all positive-definite, symmetrizable integer matrices with non-zero upper and lower diagonal and $\mathcal{T}$ to be the set of all positive-definite real symmetric matrices with nonzero upper diagonal such…

We study the rate of convergence to a normal random variable of the real and imaginary parts of Tr(AU), where U is an N x N random unitary matrix and A is a deterministic complex matrix. We show that the rate of convergence is O(N^{-2 +…

We characterize sequences of positive integers $(a_1,a_2,\ldots,a_n)$ for which the $2\times2$ matrix $\left( \begin{array}{cc} a_n&-1 1&0 \end{array} \right) \left( \begin{array}{cc} a_{n-1}&-1 1&0 \end{array} \right) \cdots \left(…

This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar…