In this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. We study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. For strongly z-ideals, we analyze prime ideals using the concept of zero sets. Moreover, it is proven that the intersection of all zero sets of a prime ideal of C(L),...

Let m > 0 and q > 1 be relatively prime integers. We find an explicit period ν m (q) such that for any integers n 0 and r we have n + ν m (q) r m (a) ≡ n r m (a) (mod q), provided that a = −1 and n = 0, or a is an integer with 1 − (−a) m relatively prime to q, where n r m (a) = k≡r (mod m) n k a k. This is a further extension of a congruence of Glaisher.

A graph G(V,E) with vertex set V is said to have a prime labeling if there exist a bijection f : V (G) → {1, 2, . . . , |V |} such that for each edge xy ∈ E(G), f(x) and f(y) are relatively prime. It becomes an interesting problem to investigate the maximum possible edges that can be constructed in a graph with n vertices having prime labeling property. We interpret this problem in number theor...

In ‎this ‎paper, ‎we ‎study ‎the ‎relation ‎between ‎the ‎soft ‎sets ‎and ‎soft ‎Lie ‎algebras ‎with ‎the ‎lattice theory. ‎We ‎introduce ‎the ‎concepts ‎of ‎the ‎lattice ‎of ‎soft ‎sets, ‎full ‎soft ‎sets ‎and ‎soft ‎Lie ‎algebras ‎and next, we ‎verify ‎some ‎properties ‎of ‎them. We ‎prove ‎that ‎the ‎lattice ‎of ‎the ‎soft ‎sets ‎on ‎a fixed parameter set is isomorphic to the power set of a ...

Let p be an odd prime number, k an imaginary abelian field containing a primitive p-th root of unity, and k∞/k the cyclotomic Zp-extension. Denote by L/k∞ the maximal unramified pro–p abelian extension, and by L′ the maximal intermediate field of L/k∞ in which all prime divisors of k∞ over p split completely. Let N/k∞ (resp. N ′/k∞) be the pro–p abelian extension generated by all p-power roots ...

The authors present a method for decomposition of Bayesian networks into their maximal prime subgraphs. The correctness of the method is proven and results relating the maximal prime subgraph decomposition (MPD) to the maximal complete subgraphs of the moral graph of the original Bayesian network are presented. The maximal prime subgraphs of a Bayesian network can be organized as a tree which c...

in this work, by employing the krasnosel'skii fixed point theorem, we study the existence of positive solutions of a three-point boundary value problem for the following fourth-order differential equation begin{eqnarray*} left { begin{array}{ll} u^{(4)}(t) -f(t,u(t),u^{prime prime }(t))=0 hspace{1cm} 0 leq t leq 1, & u(0) = u(1)=0, hspace{1cm} alpha u^{prime prime }(0) - beta u^{prime prime...