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We consider the periodicity of recursive sequences defined by linear homogeneous recurrence relations of arbitrary order, when they are reduced modulo a positive integer . We show that the period of such a sequence with characteristic polynomial can be expressed in terms of the order of as a unit in the quotient ring . When is prime, this order can be described in terms of the factorization of in the polynomial ring . We use this connection to develop efficient algorithms for determining the factorization types of monic polynomials of degree in .
We investigate the Gram determinant of the pairing arising from curves in a planar surface, with a focus on the disk with two holes. We prove that the determinant based on curves divides the determinant based on curves. Motivated by the work on Gram determinants based on curves in a disk and curves in an annulus (Temperley–Lieb algebra of type and , respectively), we calculate several examples of the Gram determinant based on curves in a disk with two holes, and advance conjectures on the complete factorization of Gram determinants.
This article begins with an introduction to a conjecture made around 1930 in the area of Diophantine approximation: the Littlewood Conjecture. The conjecture asks whether any two real numbers can be simultaneously well approximated by rational numbers with the same denominator. The introduction also focuses briefly on an analogue of this conjecture, regarding power series and polynomials with coefficients in an infinite field. Harold Davenport and Donald Lewis disproved this analogue of the Littlewood Conjecture in 1963. Following the introduction we focus on a claim relating to another analogue of this conjecture. In 1970, John Armitage believed that he had disproved an analogue of the Littlewood Conjecture, regarding power series and polynomials with coefficients in a finite field. The remainder of this article shows that Armitage’s claim was false.
The discrete logarithm is a problem that surfaces frequently in the field of cryptography as a result of using the transformation . Analysis of the security of many cryptographic algorithms depends on the assumption that it is statistically impossible to distinguish the use of this map from the use of a randomly chosen map with similar characteristics. This paper focuses on a prime modulus, , for which it is shown that the basic structure of the functional graph produced by this map is largely dependent on an interaction between and . We deal with two of the possible structures, permutations and binary functional graphs. Estimates exist for the shape of a random permutation, but similar estimates must be created for the binary functional graphs. Experimental data suggest that both the permutations and binary functional graphs correspond well to the theoretical predictions.
In the field of reliability, a lot has been written on the analysis of phenomena that are related. Estimation of the difference of two population means have been mostly formulated under the no-correlation assumption. However, in many situations, there is a correlation involved. This paper addresses this issue. A sequential estimation method for linearly related lifetime distributions is presented. Estimations for the scale parameters of the exponential distribution are given under square error loss using a sequential prediction method. Optimal stopping rules are discussed using concepts of mean criteria, and numerical results are presented.
Let denote the probability that two randomly chosen monic polynomials , of degrees and , respectively, are relatively prime. Let be a prime power. We establish an explicit formula for when , the ring of integers mod .
For a group with generating set , the -graph of , denoted by , is the graph whose vertices are distinct cosets of in . Two distinct vertices are joined by an edge when the set intersection of the cosets is nonempty. In this paper, we explore the planarity of .