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The set of morphisms mapping any Sturmian sequence to a Sturmian sequence forms together with composition the so-called monoid of Sturm. For this monoid, we defne a faithful representation by $(3\times 3)$-matrices with integer entries. We…

A $(p,q,r)$-board that has $pq+pr+qr$ squares consists of a $(p,q)$-, a $(p,r)$-, and a $(q,r)$-rectangle. Let $S$ be the set of the squares. Consider a bijection $f : S \to [1,pq+pr+qr]$. Firstly, for $1 \le i \le p$, let $x_i$ be the sum…

In this paper, we prove that each matrix in $M_{m\times n}(\mathbb Z_{\geq0})$ is uniformly column sign-coherent with respect to any $n\times n$ skew-symmetrizable integer matrix. Using such matrices, we introduce the definition of…

For every $m,n \in \mathbb{N}$ and every field $K$, let $M(m \times n, K)$ be the vector space of the $(m \times n)$-matrices over $K$ and let $S(n,K)$ be the vector space of the symmetric $(n \times n)$-matrices over $K$. We say that an…

Let $n,k$ be fixed natural numbers with $1\le k\le n$ and let $A_{n+1,k,2k,\dots,sk}$ denote an $(n+1)\times (n+1)$ complex multidiagonal matrix having $s=[n/k]$ sub- and superdiagonals at distances $k,2k,\dots,sk$ from the main diagonal.…

Let $X$ be a real normed space with unit sphere S. We prove that $X$ is an inner product space if and only if there exists a real number $\rho=\sqrt{(1+\cos\frac{2k\pi}{2m+1})/2}$, $(k=1,2,\ldots , m ;\:m=1,2,\ldots)$, such that every chord…

Given a matrix $M = (a_{i,j})$ a square is a $2 \times 2$ submatrix with entries $a_{i,j}$, $a_{i, j+s}$, $a_{i+s, j}$, $a_{i+s, j +s}$ for some $s \geq 1$, and a zero-sum square is a square where the entries sum to $0$. Recently,…

Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $n\times n$ matrix ring, so $R\cong M_{n}(S)$ for some ring $S$, if and only if it contains a…

An integer Heffter array $H(m,n;s;k)$ is an $m\times n$ partially filled array whose entries are the elements of a subset $\Omega\subset \mathbb{Z}$ such that $\{\Omega,-\Omega\}$ is a partition of the set $\{1,2,\ldots,2nk\}$ and such that…

We study the properties of rectangular constant $ \mu(\mathbb{X}) $ in a normed linear space $\mathbb{X}$. We prove that $ \mu(\mathbb{X}) = 3$ iff the unit sphere contains a straight line segment of length 2. In fact, we prove that the…

Given a real symmetric $n\times n$ matrix, the sepr-sequence $t_1\cdots t_n$ records information about the existence of principal minors of each order that are positive, negative, or zero. This paper extends the notion of the sepr-sequence…

A $(v,k,\lambda)$ difference set in a group $G$ of order $v$ is a subset $\{d_1, d_2, \ldots,d_k\}$ of $G$ such that $D=\sum d_i$ in the group ring $\mathbb{Z}[G]$ satisfies $$D D^{-1} = n + \lambda G,$$ where $n=k-\lambda$. If $D=\sum s_i…

An alternating sign matrix, or ASM, is a $(0, \pm 1)$-matrix where the nonzero entries in each row and column alternate in sign. We generalize this notion to hypermatrices: an $n\times n\times n$ hypermatrix $A=[a_{ijk}]$ is an {\em…

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…

We prove a general result on completing objects similar to Latin rectangles in which the number of occurrences of each symbol is prescribed, each cell contains multiple symbols, and no cell contains repeated symbols. This generalizes…

Let $m\neq n$. An $m\times n\times p$ proper array is a three-dimensional array composed of directed cubes that obeys certain constraints. Because of these constraints, the $m\times n\times p$ proper arrays may be classified via a schema in…

A ring $R$ is strongly clean provided that every element in $R$ is the sum of an idempotent and a unit that commutate. Let $T_n(R,\sigma)$ be the skew triangular matrix ring over a local ring $R$ where $\sigma$ is an endomorphism of $R$. We…

A tight Heffter array H(m,n) is an m x n matrix with nonzero entries from Z_{2mn+1} such that i) the sum of the elements in each row and each column is 0, and ii) no element from {x,-x\ appears twice. We prove that H(m,n) exist if and only…

Let $m\neq n$. An $m\times n\times p$ {\it proper array} is a three-dimensional rectangular array composed of directed cubes that obeys certain constraints. Because of these constraints, the $m\times n\times p$ proper arrays may be…

A matrix is incomplete when some of its entries are missing. A Robinson incomplete symmetric matrix is an incomplete symmetric matrix whose non-missing entries do not decrease along rows and columns when moving toward the diagonal. A…