Related papers: A Result for Orthogonal Plus Rank-1 Matrices
From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric…
We describe a simple machinery which translates results on algebraic sums of sets of reals into the corresponding results on their cartesian product. Some consequences are: 1. The product of a meager/null-additive set and a strong measure…
Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets $\mathcal A$ of finite rank multiplicative groups infields of characteristic zero. We…
In this paper, we propose an efficient and scalable low rank matrix completion algorithm. The key idea is to extend orthogonal matching pursuit method from the vector case to the matrix case. We further propose an economic version of our…
We study some sum-product problems over matrix rings. Firstly, for $A, B, C\subseteq M_n(\mathbb{F}_q)$, we have $$ |A+BC|\gtrsim q^{n^2}, $$ whenever $|A||B||C|\gtrsim q^{3n^2-\frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(\mathbb{F}_q)$…
For a square matrix $A$, the resolvent of $A$ at a point $z \in \mathbb{C}$ is defined as $(A-zI )^{-1}$. We consider the set of points $z \in \mathbb{C}$ where the relative difference in 2-norm between the resolvent and the nearest rank…
We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating…
An ensemble of random unistochastic (orthostochastic) matrices is defined by taking squared moduli of elements of random unitary (orthogonal) matrices distributed according to the Haar measure on U(N) (or O(N), respectively). An ensemble of…
In this work and its sequel, we study the expanding phenomenon of matrices over a finite chain ring of large residue field. A sum-product estimate is proved. It is showed that $x+yz$ is a moderate expander on $n\times n$ matrices with…
In this paper, we obtain some new matrix inequalities involving Hadamard product. Also some Hadamard product inequalities for accretive matrices involving the matrix means, positive unital linear maps and matrix concave functions are…
I give the explicit formula for the (set-theoretical) system of Resultants of m+1 homogeneous polynomials in n+1 variables.
For t a positive integer, the t-term rank of a (0,1)-matrix A is defined to be the largest number of 1s in A with at most one 1 in each column and at most t 1s in each row. Thus the 1-term rank is the ordinary term rank. We generalize some…
In this paper we show under weak assumptions that for $R\stackrel{d}{=}1+M_1+M_1M_2+\ldots$, where $P(M\in[0,1])=1$ and $M_i$ are independent copies of $M$, we have $\ln P(R>x)\sim C\, x\ln P(M>1-\frac1x)$ as $x\to\infty$. The constant $C$…
In this note we investigate the existence of flat orthogonal matrices, i.e. real orthogonal matrices with all entries having absolute value close to $\frac{1}{\sqrt{n}}$. Entries of $\pm \frac{1}{\sqrt{n}}$ correspond to Hadamard matrices,…
In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We…
We apply methods from algebraic geometry to study uniform matrix product states. Our main results concern the topology of the locus of tensors expressed as uMPS, their defining equations and identifiability. By an interplay of theorems from…
The main goal in this manuscript is to present a class of functions satisfying a certain orthogonality property for which there also exists a three term recurrence formula. This class of functions, which can be considered as an extension to…
We describe an error in the proof of a key proposition, which was necessary for the proof of the main result. Alternate proofs of the main result are given by Ozsvath-Stipsicz-Szabo and Dai-Hom-Stoffregen-Truong.
We announce numerous new results in the theory of orthogonal polynomials on the unit circle.
This thesis reviews recent progress on products of random matrices from the perspective of exactly solved Gaussian random matrix models. We derive exact formulae for the correlation functions for the eigen- and singular values at arbitrary…