Related papers: 11 can be reduced to 10

The present paper considers extremal combinatorics questions in the language of matrices. An $s$-matrix is a matrix with entries in $\{0,1,\ldots, s-1\}$. An $s$-matrix is simple if it has no repeated columns. A matrix $F$ is a…

We proved in previous work that all real nilpotent Lie algebras of dimension up to $10$ carrying an ad-invariant metric are nice. In this paper we show by constructing explicit examples that nonnice irreducible nilpotent Lie algebras…

This paper presents an algorithm for the integer multiplicative inverse (mod $2^w$) which completes in the fewest cycles known for modern microprocessors, when using the native bit width $w$ for the modulus $2^w$. The algorithm is a…

Let B be an n by n doubly substochastic matrix. We show that B can be written as a convex combination of no more than {\sigma}(B)+t subpermutation matrices, where {\sigma}(B) is the number of nonzero elements in B and t is the number of…

In this article we compute the number of invertible $2\times 2$ matrices with integer entries modulo $n$ whose permanents are congruent modulo $n$ to a given integer $x$.

We define variants of Pisier's similarity degree for unital C*-algebras and use direct integral theory to obtain new results. We prove that if every II$_{1}$ factor representation of a separable C*-algebra $\mathcal{A}$ has property…

Given an arithmetical function $f$, by $f(a, b)$ and $f[a, b]$ we denote the function $f$ evaluated at the greatest common divisor $(a, b)$ of positive integers $a$ and $b$ and evaluated at the least common multiple $[a, b]$ respectively. A…

A matrix is given in ``shredded'' form if we are presented with the multiset of rows and the multiset of columns, but not told which row is which or which column is which. The matrix is reconstructible if it is uniquely determined by this…

Let $A={bmatrix} A_{11} &A_{12} A_{21} & A_{22} {bmatrix}$ be an $n\times n$ accretive-dissipative matrix, $k$ and l be the orders of $A_{11}$ and $A_{22}$, respectively, and let $m=\min\{k,l\}$. Then $$|\det A|\le a|\det A_{11}|\cdot|\det…

Given positive integers $h, N$ satisfying $1 \leqslant h \leqslant 2N^2$, we define $T(h,N)$ to be the number of $2\times 2$ integer matrices with determinant equal to $h$ whose entries lie in $[-N,N]$. Our main result states that for any…

For a $m\times n$ matrix $B=(b_{ij})_{m\times n}$ with nonnegative entries $b_{ij}$ and any $k\times l-$submatrix $B_{ij}$ of $B$, let $a_{B_{ij}}$ and $g_{B_{ij}}$ denote the arithmetic mean and geometric mean of elements of $B_{ij}$…

We recall four open problems concerning constructing high-order matrix-exponential approximations for the infimum of a spectrally negative Levy process (with applications to first-passage/ruin probabilities, the waiting time distribution in…

Let $A=(a_{ij})$ be an $n\times n$ random matrix with i.i.d. entries such that $\mathbb{E} a_{11} = 0$ and $\mathbb{E} {a_{11}}^2 = 1$. We prove that for any $\delta>0$ there is $L>0$ depending only on $\delta$, and a subset $\mathcal{N}$…

We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer…

The alternate row and column scaling algorithm applied to a positive $n\times n$ matrix $A$ converges to a doubly stochastic matrix $S(A)$, sometimes called the \emph{Sinkhorn limit} of $A$. For every positive integer $n$, a two parameter…

We prove a conjecture posted in the Online Encyclopedia of Integer Sequences, namely that there are exactly five positive integers that can be written in more than one way as the sum of a nonnegative power of 2 and a nonnegative power of 3.…

Let $X=(x_{ij})\in\mathbb{R}^{N\times n}$ be a rectangular random matrix with i.i.d. entries (we assume $N/n\to\mathbf{a}>1$), and denote by $\sigma_{min}(X)$ its smallest singular value. When entries have mean zero and unit second moment,…

We show that, for any $r\geq 1$, if $g_1,\ldots,g_r$ are distinct coprime integers, sufficiently large depending only on $r$, then for any $\epsilon>0$ there are infinitely many integers $n$ such that all but $\epsilon \log n$ of the digits…

A new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix is presented; this is a contribution to the classical spectral inverse problem for symmetric doubly stochastic matrices that is…

We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a…