Related papers: Zero-sum squares in $\{-1, 1\}$-matrices with low …

We prove that for every $\epsilon>0$ and predicate $P:\{0,1\}^k\rightarrow \{0,1\}$ that supports a pairwise independent distribution, there exists an instance $\mathcal{I}$ of the $\mathsf{Max}P$ constraint satisfaction problem on $n$…

We consider words over a binary alphabet. A word $w$ is overlap-free if it does not have factors (blocks of consecutive letters) of the form $uvuvu$ for nonempty $u$. Let $M(w)$ denote the number of positions that are middle positions of…

Let a and x denote tuples of (jointly) freely noncommuting variables. A square matrix valued polynomial p in these variables is naturally evaluated at a tuple (A,X) of symmetric matrices with the result p(A,X) a square matrix. The…

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…

A set of integers is sum-free if it contains no solution to the equation $x+y=z$. We study sum-free subsets of the set of integers $[n]=\{1,\ldots,n\}$ for which the integer $2n+1$ cannot be represented as a sum of their elements. We prove…

We realize many sharp spectral bounds of the spectral radius of a nonnegative square matrix $C$ by using the largest real eigenvalues of suitable matrices of smaller sizes related to $C$ that are very easy to find. As applications, we give…

We propose a model that all quark and lepton mass matrices originally have the same zero texture. Namely their (1,1), (1,3) and (3,1) components are zeros. For the neutrino mass matrix, we further impose the symmetry between the second and…

In the setting of real square matrices, it is known that, if $A$ is a singular irreducible $M$-matrix, then the only nonnegative vector that belongs to the range space of $A$ is the zero vector. In this paper, we prove an analogue of this…

We consider the set $\mathcal M_n\left(\mathbb Z; H\right)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain and asymptotic formula on the number of matrices from $\mathcal M_n\left(\mathbb Z; H\right)$ with…

An $n$-by-$n$ ($n\ge 3$) weighted shift matrix $A$ is one of the form $$[{array}{cccc}0 & a_1 & & & 0 & \ddots & & & \ddots & a_{n-1} a_n & & & 0{array}],$$ where the $a_j$'s, called the weights of $A$, are complex numbers. Assume that all…

Let $\mathcal S$ be a multiset of integers. We say $\mathcal S$ is a $\textit{zero-sum sequence}$ if the sum of its elements is 0. We study zero-sum sequences whose elements lie in the interval $[-k,k]$ such that no subsequence of length…

A lonesum matrix is a $(0,1)$-matrix that is uniquely determined by its row and column sum vectors. In this paper, we introduce lonesum decomposable matrices and study their properties. We provide a necessary and sufficient condition for a…

We prove that, for all even $n\geq10$, there exists a latin square of order $n$ with at least one transversal, yet all transversals coincide on $ \big\lfloor n/6 \big\rfloor$ entries. These latin squares have at least $ 19 n^2/36 + O(n)$…

Let $m\neq0,\pm1$ and $n\geq 2$ be integers. The ring of algebraic integers of the pure fields of type $\mathbb{Q}(\sqrt[n]{m})$ is explicitly known for $n=2,3,4$. It is well known that for $n=2$, an integral basis of the pure quadratic…

We compute the singular values of an $m \times n$ sparse matrix $A$ in a distributed setting, without communication dependence on $m$, which is useful for very large $m$. In particular, we give a simple nonadaptive sampling scheme where the…

Let \svec = (s_1,...,s_m) and \tvec = (t_1,...,t_n) be vectors of nonnegative integer-valued functions of m,n with equal sum S = sum_{i=1}^m s_i = sum_{j=1}^n t_j. Let M(\svec,\tvec) be the number of m*n matrices with nonnegative integer…

We give an algorithm that generates a uniformly random contingency table with specified marginals, i.e. a matrix with non-negative integer values and specified row and column sums. Such algorithms are useful in statistics and combinatorics.…

We show that there is a positive constant $c_0$ such that \[\sum_{n\le x}\mu^2(n^2+1)c_0x+O_{\varepsilon}(x^{7/12+\varepsilon})\] for any fixed $\varepsilon>0$. This improves a result of Estermann [3] from 1931, in which the error term had…

Let $\alpha\in \mathbb{R}\setminus\mathbb{Q}$ and $\beta\in \mathbb{R}$ be given. Suppose that $a_1,\ldots,a_s$ are distinct positive integers that do not contain a reduced residue system modulo $p^2$ for any prime $p$. We prove that there…

A number $s$ is the sum of the entries of the inverse of an $n \times n, (n \geq 3)$ upper triangular matrix with entries from the set $\{0, 1\}$ if and only if $s$ is an integer lying between $2-F_{n-1}$ and $2+F_{n-1}$, where $F_n$ is the…