-matrices, discrepancy and preservers
Let and be positive integers, and let and be nonnegative integral vectors. Let be the set of all -matrices with row sum vector and column vector...
Let and be positive integers, and let and be nonnegative integral vectors. Let be the set of all -matrices with row sum vector and column vector...
For integers , Brietzke (2008) defined the -central coefficients of an infinite lower triangular matrix as , with , and the -central coefficient triangle of as It is known that the -central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array with and , we obtain the generating function of its -central coefficients and give an explicit representation for the -central Riordan array in terms of the Riordan array . Meanwhile, the...
An sign pattern is said to be potentially nilpotent if there exists a nilpotent real matrix with the same sign pattern as . Let be an sign pattern with such that the superdiagonal and the entries are positive, the
Let be a commutative ring with nonzero identity, let be the set of all ideals of and an expansion of ideals of defined by . We introduce the concept of -primary ideals in commutative rings. A proper ideal of is called a -primary ideal if whenever and , then or . Our purpose is to extend the concept of -ideals to -primary ideals of commutative rings. Then we investigate the basic properties of -primary ideals and also discuss the relations among -primary, -primary and...
As a general case of molecular graphs of benzenoid hydrocarbons, we study plane bipartite graphs with Kekulé structures (1-factors). A bipartite graph G is called elementary if G is connected and every edge belongs to a 1-factor of G. Some properties of the minimal and the maximal 1-factor of a plane elementary graph are given. A peripheral face f of a plane elementary graph is reducible, if the removal of the internal vertices and edges of the path that is the intersection of...
We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.
The simple incidence structure formed by points and unordered pairs of distinct parallel lines of a finite affine plane of order is a design. If , is the complementary design of . If , is isomorphic to the geometric design (see [2; Theorem 1.2]). In this paper we give necessary and sufficient conditions for a design to be of the form for some finite affine plane of order . As a consequence we obtain a characterization of small designs .