Related papers: On some mean matrix inequalities of dynamical inte…

We prove that for any $n\times n$ matrix, $A$, and $z$ with $|z|\geq \|A\|$, we have that $\|(z-A)^{-1}\|\leq\cot (\frac{\pi}{4n}) \dist (z, \spec(A))^{-1}$. We apply this result to the study of random orthogonal polynomials on the unit…

Let $\rho=(\frac{p}{q})^{\frac{1}{r}}<1$ for some $p,q,r\in\mathbb{N}$ with $(p,q)=1$ and $\mathcal{D}_{n}=\{0,1,\cdot\cdot\cdot,N_{n}-1\}$, where $N_{n}$ is prime for all $n\in\mathbb{N}$, and denote…

In [Israel J. Math, 2014], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions $\mathcal F$ in a domain $D\subset \mathbb C,$ and for a…

We prove that, given $\epsilon>0$ and $k\geq 1$, there is an integer $n$ such that the following holds. Suppose $G$ is a finite group and $A\subseteq G$ is $k$-stable. Then there is a normal subgroup $H\leq G$ of index at most $n$, and a…

Let $A_n$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\det A_n|$ satisfies a central limit theorem. More…

For a nonnegative matrix A and real diagonal matrix D, two known inequalities on the spectral radius, r(A^2 D^2) >= r(AD)^2 and r(A) r(A D^2) >= r(AD)^2, leave open the question of what determines the order of r(A^2 D^2) with respect to…

We show that assuming the standard conjectures, for any smooth projective variety $X$ of dimension $n$ over an algebraically closed field, there is a constant $C>0$ such that for any positive rational number $r$ and for any polarized…

A simple closed curve in the Euclidean plane is said to have property C_n(R) if at each point we can inscribe a unique regular $n$-gon with edges length $R$. C_2(R) is equivalent to having constant diameter. We show that smooth curves…

We show that the fixed elements for the natural GL_m-action on the universal division algebra UD(m,n) of m generic n x n matrices form a division subalgebra of degree n, assuming n >= 3 and 2 <= m <= n^2 - 2. This allows us to describe the…

Recht and R\'{e} introduced the noncommutative arithmetic geometric mean inequality (NC-AGM) for matrices with a constant depending on the degree $d$ and the dimension $m$. In this paper we prove AGM inequalities with a dimension-free…

Representations of the symmetry group D_n of the n-sided regular polygon have generic multiplication rules if n is prime. Using D_n with n=5 or greater, a particular well-known form of the Majorana neutrino mass matrix is derived.

Let X be a finite set of complex numbers and let A be a normal operator with spectrum X that acts on a separable Hilbert space H. Relative to a fixed orthonormal basis e_1,e_2, ... for H, A gives rise to a matrix whose diagonal is a…

Let $R$ be a ring generated by $l$ elements with stable range $r$. Assume that the group $EL_d(R)$ has Kazhdan constant $\epsilon_0>0$ for some $d > r $. We prove that there exist $\epsilon(\epsilon_0,l) >0$ and $k \in N$, s.t. for every $n…

We show that the monotonicity conjecture of Merriss and Watkins is true in average when taking the set of matrices of given non negative spectra as probability space with respect to the Haar measure of the unitarian group .

For $d\geq 2$ and any norm on $\mathbb R^d$, we prove that there exists a set of $n$ points that spans at least $(\tfrac d2-o(1))n\log_2n$ unit distances under this norm for every $n$. This matches the upper bound recently proved by Alon,…

Let $A$ be a square random matrix of size $n$, with mean zero, independent but not identically distributed entries, with variance profile $S$. When entries are i.i.d. with unit variance, the spectral radius of $n^{-1/2}A$ converges to $1$…

The present paper deals with lower bounds for the norm of products of linear forms. It has been proved by J. Arias-de-Reyna \cite{ARIAS}, that %for ${\mathbb C}^n$, the so-called $n^{\rm th}$ linear polarization constant $c_n({\mathbb…

We determine precisely when the branching coefficients arising from the restriction of irreducible representations of the symmetric group $S_n$ to the dihedral subgroup $D_n$ are nonzero, and we establish uniform linear lower bounds outside…

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$ and let $(Q_t)$ be the Ornstein--Ulhenbeck semigroup. Eldan and Lee recently established that for every non--negative function $f$ of integral $1$ and any time $t$ the…

The exponential of an NxN matrix can always be expressed as a matrix polynomial of order N-1. In particular, a general group element for the fundamental representation of SU(N) can be expressed as a matrix polynomial of order N-1 in a…