Consider a local one-parameter family of Noetherian schemes. More precisely, let f : X → S be a flat map of Noetherian schemes, where S stands for the spectrum of a discrete valuation ring. Let s and η denote the special and generic points of S; putXs := f (s) andXη := f (η). Assume thatXs is of pure dimension and has no embedded components. Let D be an effective Cartier divisor of X. View it a...

This paper presents an algorithm and its implementation for computing the approximate GCD (greatest common divisor) of multivariate polynomials whose coefficients may be inexact. The method and the companion software appears to be the first practical package with such capabilities. The most significant features of the algorithm are its robustness and accuracy as demonstrated in the results of c...

We give a new proof of a theorem of Mansour and Sun by using number theory and Rothe’s identity. It is well-known that the number of ways of choosing k points, no two consecutive, from a collection of n points arranged on a cycle is n n−k ( n−k k ) (see [11, Lemma 2.3.4]). A generalization of this result was obtained by Kaplansky [5], who proved that the number of k-subsets {x1, . . . , xk} of ...

The Cartesian product of two hamiltonian graphs is always hamiltonian. For directed graphs, the analogous statement is false. We show that the Cartesian product C,,, x C,, of directed cycles is hamiltonian if and only if the greatest common divisor (g.c.d.) d of n, and n, is a t least two and there exist positive integers d,, d, so that d, + d, = d and g.c.d. (n,, d,) = g.c.d. (n,, d,) = 1. We ...

The squarefree decomposition of a univariate polynomial, P (y) ∈ K[y], K being a characteristic zero field, is one of the most basic problems in computer algebra (after the greatest common divisor computation, probably the most basic one). The parametric case has not usually been considered in the literature and this paper is dedicated to showing how the case of a single parameter can easily be...

Similarity relations, i.e., compatibility relations on words induced by relations on letters were introduced as a generalization of partial words in [9](i). There and in [10, 11] the main focus was on the effect of these relations on coding properties and on the defect theorem of words. In the article [8] we started the study of the interaction properties of periods with respect to similarity r...

A bipartite graph G is called (α, β)-biregular if all vertices in one part of G have the degree α and all vertices in the other part have the degree β. An edge coloring of a graph G with colors 1, 2, 3, . . . , t is called an interval t-coloring if the colors received by the edges incident with each vertex of G are distinct and form an interval of integers and at least one edge of G is colored ...

INTRODUCTION ...................................................................................................................... 1 ARITHMETIC ........................................................................................................................... 2 Integer and Polynomial Addition, Multiplication, and Division with Remainder ............. 2 Integer and Polynomial Greatest C...

A novel method guaranteeing nondecreasing girth is presented for constructing longer low-density parity-check (LDPC) codes from shorter ones. The parity-check matrix of a shorter base code is decomposed into N (N ≥ 2) nonoverlapping components with the same size. Then, these components are combined together to form the parity-check matrix of a longer code, according to a given N×N Latin square....

For n ∈ N and D ⊆ N, the distance graph P n has vertex set {0, 1, . . . , n− 1} and edge set {ij | 0 ≤ i, j ≤ n − 1, |j − i| ∈ D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs. A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of ...