This paper is the second part of a new proof Bel’tyukov–Lipshitz theorem, which states that existential theory structure $$\left\langle {\mathbb{Z};0,1, + , - \leqslant ,|{\kern 1pt} } \right\rangle $$ decidable. We construct quasi-quantifier elimination algorithm (the notion was introduced in first proof) to reduce decision problem for ,{\text{GCD}}} positive {{{\mathbb{Z}}_{{ > 0}}};1,{{{\{ a...

the unitary cayley graph xn has vertex set zn = {0, 1,…, n-1} and vertices u and v areadjacent, if gcd(uv, n) = 1. in [a. ilić, the energy of unitary cayley graphs, linear algebraappl. 431 (2009) 1881–1889], the energy of unitary cayley graphs is computed. in this paperthe wiener and hyperwiener index of xn is computed.

Let p, l be distinct odd primes, q be an odd prime power with gcd(q, p) = gcd(q, l) = 1, and m, n be positive integers. In this paper, we determine all self-dual, self-orthogonal and complementarydual negacyclic codes of length 2pl over the finite field Fq with q elements. We also illustrate our results with some examples.

In this paper, we consider the Diophantine equation b + (a+ b) + · · ·+ (a (x− 1) + b) = = d + (c+ d) + · · ·+ (c (y − 1) + d) , where a, b, c, d, k, l are given integers with gcd(a, b) = gcd(c, d) = 1, k 6= l. We prove that, under some reasonable assumptions, the above equation has only finitely many solutions.

PURPOSE We analyzed the preoperative conjunctival goblet cell density (GCD), MUC5AC, and HLA-DR in glaucomatous patients undergoing trabeculectomy, using laser scanning confocal microscopy (LSCM) and impression cytology (IC). METHODS We enrolled 57 patients undergoing trabeculectomy. At baseline LSCM and IC were performed at the site planned for surgery; LSCM was repeated after 12 months at t...

Let a, b and n be positive integers and let S = {x1, x2, . . . , xn} be a set of distinct positive integers. The n × n matrix (Sf ) = (f ((xi, xj))), having f evaluated at the greatest common divisor (xi, xj) of xi and xj as its ij−entry, is called the GCD matrix associated with f on the set S. Similarly, the n × n matrix [Sf ] = (f ([xi, xj ])) is called the LCM matrix associated with f on S. ...