Related papers: On an Algorithm for Obtaining All Binary Matrices …
The paper considers implementations of some randomized algorithms in connection with obtaining a random $n^2 \times n^2$ Sudoku matrix with programming language C++. For this purpose we describes the set $\Pi_n$ of all $(2n) \times n$…
An analytical method for getting new complex Hadamard matrices by using mutually unbiased bases and a nonlinear doubling formula is provided. The method is illustrated with the n=4 case that leads to a rich family of eight-dimensional…
Inspired by [4] we present a new algorithm for uniformly random generation of ordered trees in which all occuring outdegrees can be specified by a given sequence of numbers. The method can be used for random generation of binary or n-ary…
In this paper, a class of combinatorial identities is proved. A method is used which is based on the following rule: counting elements of a given set in two ways and making equal the obtained results. This rule is known as "counting in two…
A binary triangle of size $n$ is a triangle of zeroes and ones, with $n$ rows, built with the same local rule as the standard Pascal triangle modulo $2$. A binary triangle is said to be balanced if the absolute difference between the…
The long-standing hypothesis formulated by Goreinov, Tyrtyshnikov and Zamarashkin \cite{GTZ1997} has recently been solved positively by Sengupta and Pautov \cite{SP2026} in the case of two-column matrices. In this paper, we complement their…
Unitary matrices which are zero below the secondary diagonal (Hessenberg unitary matrices) have many uses in analysis. Given a set of needed conditions on a unitary matrix, this algorithm will give the sparsest unitary matrix. We give an…
We describe algorithms for computing maximal determinants of binary circulant matrices of small orders. Here "binary matrix" means a matrix whose elements are drawn from $\{0,1\}$ or $\{-1,1\}$. We describe efficient parallel algorithms for…
It is widely known that the lower bound for the algorithmic complexity of square matrix multiplication resorts to at least $n^2$ arithmetic operations. The justification builds upon the following reasoning: given that there are $2 n^2$…
We continue developing the theory around the twin-width of totally ordered binary structures, initiated in the previous paper of the series. We first introduce the notion of parity and linear minors of a matrix, which consists of…
Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…
We describe a dynamic programming algorithm for exact counting and exact uniform sampling of matrices with specified row and column sums. The algorithm runs in polynomial time when the column sums are bounded. Binary or non-negative integer…
A formula for the number of toroidal m x n binary arrays, allowing rotation of the rows and/or the columns but not reflection, is known. Here we find a formula for the number of toroidal m x n binary arrays, allowing rotation and/or…
The McCarty Conjecture states that any McCarty Matrix (an $n\times n$ matrix $A$ with positive integer entries and each of the $2n$ row and column sums equal to $n$), can be additively decomposed into two other matrices, $B$ and $C$, such…
In this paper we provide an algorithm, similar to the simplex algorithm, which determines a rational cp-factorization of a given matrix, whenever the matrix allows such a factorization. This algorithm can be used to show that every integral…
Given a matrix with partitions of its rows and columns and entries from a field, we give the necessary and sufficient conditions that it has a non--singular submatrix with certain number of rows from each row partition and certain number of…
The problem of efficiently characterizing degree sequences of simple hypergraphs is a fundamental long-standing open problem in Graph Theory. Several results are known for restricted versions of this problem. This paper adds to the list of…
The current paper is dedicated to the problem of finding the number of mutually non isomorphic bipartite graphs of the type $g=\langle R_g ,C_g ,E_g \rangle$ at given $n=|R_g |$ and $m=|C_g |$, where $R_g$ and $C_g$ are the two disjoint…
In this paper, a matrix is said to be prime if the row and column of this matrix are both prime numbers. We establish various necessary and sufficient conditions for developing matrices into the sum of tensor products of prime matrices. For…
Matrix multiplication is a fundamental task in almost all computational fields, including machine learning and optimization, computer graphics, signal processing, and graph algorithms (static and dynamic). Twin-width is a natural complexity…