Related papers: Circular bidiagonal pairs

A square (0,1)-matrix X of order n > 0 is called fully indecomposable if there exists no integer k with 0 < k < n, such that X has a k by n-k zero submatrix. A stable set of a graph G is a subset of pairwise nonadjacent vertices. The…

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show…

A curious number is a palindromic number whose base ten representation has the form $a \ldots a b \ldots b a \ldots a$. In this paper, we determine all curious numbers that are perfect squares. Our proof involves reducing the search for…

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,…

Fix a nonnegative integer $d$, a field $\mathbb{F}$, and a vector space $V$ over $\mathbb{F}$ with dimension $d+1$. Let $T$ denote an invertible upper triangular matrix in ${\rm Mat}_{d+1}(\mathbb{F})$. Using $T$ we construct three flags on…

Bicircular lift matroids are a class of matroids defined on the edge set of a graph. For a given graph $G$, the circuits of its bicircular lift matroid are the edge sets of those subgraphs of $G$ that contain at least two cycles, and are…

We give a necessary and sufficient condition for two circles, each with finitely many points added inside, to be betweenness isomorphic. We fully characterize the betweenness isomorphism classes in the family consisting of all circles with…

A P-matrix is a square matrix $X$ such that all principal submatrices of $X$ have positive determinant. Such matrices appear naturally in instances of the linear complementarity problem, where these are precisely the matrices for which the…

A square is a factor $S = (S_1; S_2)$ where $S_1$ and $S_2$ have the same pattern, and a permutation is said to be square-free if it contains no non-trivial squares. The permutation is further said to be bicrucial if every extension to the…

There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is…

Working over a field of characteristic other than $2$, we examine a relationship between quadrilaterals and the pencil of conics passing through their vertices. Asymptotically, such a pencil of conics is what we call a bisector field, a set…

On the set of mappings of the given set, we define the product of mappings. If A is associative algebra, then we consider the set of matrices, whose elements are linear mappings of algebra A. In algebra of matrices of linear mappings we…

Let $\mathcal A:U\to V$ be a linear mapping between vector spaces $U$ and $V$ over a field or skew field $\mathbb F$ with symmetric, or skew-symmetric, or Hermitian forms $\mathcal B:U\times U\to\mathbb F$ and $\mathcal C:V\times…

We consider a Leonard pair $A, A^*$ of linear maps on a vector space $V$ that has finite positive dimension. This Leonard pair $A,A^*$ is said to have spin whenever there exist invertible linear maps $W : V \to V$ and $W^* : V \to V$ such…

A map $f: \ff^n \to \ff^n$ over a field $\ff$ is called affine if it is of the form $f(x)=Ax+b$, where the matrix $A \in \ff^{n\times n}$ is called the linear part of affine map and $b \in \ff^n$. The affine maps over $\ff=\rr$ or $\cc$ are…

In 1840 Jacob Steiner on Christian Rudolf's request proved that a triangle with two equal bisectors is isosceles. But what about changing the bisectors to cevians? Cevian is any line segment in a triangle with one endpoint on a vertex of…

Two (real or complex) $m\times n$ matrices $A$ and $B$ are said to be parallel (resp. triangle equality attaining, or TEA in short) with respect to the spectral norm $\|\cdot\|$ if $\|A+ \mu B\| = \|A\| + \|B\|$ for some scalar $\mu$ with…

For a positive linear map F and a normal matrix N, we show that |F(N)| is bounded by some simple linear combinations in the unitary orbit of F(|N|). Several elegant sharp inequalities are derived, especially for the Schur product.

Multiplicative matrix semigroups with constant spectral radius (c.s.r.) are studied and applied to several problems of algebra, combinatorics, functional equations, and dynamical systems. We show that all such semigroups are characterized…

We characterize pairs of orthogonal countable ordinals. Two ordinals $\alpha$ and $\beta$ are orthogonal if there are two linear orders $A$ and $B$ on the same set $V$ with order types $\alpha$ and $\beta$ respectively such that the only…