Volume 63, Issue 6

In this paper, some properties of prime elements, pseudoprime elements, irreducible elements and coatoms in posets are investigated. We show that the four kinds of elements are equivalent to each other in finite Boolean posets. Furthermore, we demonstrate that every element of a finite Boolean poset can be represented by one kind of them. The example presented in this paper indicates that this result may not hold in every finite poset, but all the irreducible elements are proved to be contained in each order generating set. Finally, the multiplicative auxiliary relation on posets and the notion of arithmetic poset are introduced, and some properties about them are generalized to posets.

Let n > 1 and k > 1 be positive integers. We show that if $$\left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right) \equiv 1 (\bmod k)$$ for each integer m with 0 ≤ m ≤ n − 1, then k is a prime and n is a power of this prime. In particular, this assertion under the hypothesis that n = k implies that n is a prime. This was proved by Babbage, and thus our result may be considered as a generalization of this criterion for primality.

The goal of this paper is to use algebraic techniques to compute sums of powers of roots of certain polynomials and derive congruences of Ankeny-Artin-Chowla types modulo p 3.

Let φ be a homomorphism from the partially ordered abelian group (S, v) to the partially ordered abelian group (G, u) with φ(v) = u, where v and u are order units of S and G respectively. Then φ induces an affine map φ* from the state space St(G, u) to the state space St(S, v). Firstly, in this paper, we give some suitable conditions under which φ* is injective, surjective or bijective. Let R be a semilocal ring with the Jacobson radical J(R) and let π: R → R/J(R) be a canonical map. We discuss the affine map (K 0 π)*. Secondly, for a semiprime right Goldie ring R with the maximal right quotient ring Q, we consider the relations between St(R) and St(Q). Some results from [ALFARO, R.: State spaces, finite algebras, and skew group rings, J. Algebra 139 (1991), 134–154] and [GOODEARL, K. R.-WARFIELD, R. B., Jr.: State spaces of K 0 of noetherian rings, J. Algebra 71 (1981), 322–378] are extended.

A finite group whose irreducible complex characters are rational valued is called a rational group. Thus, G is a rational group if and only if N G(〈x〉)/C G(〈x〉) ≌ Aut(〈x〉) for every x ∈ G. For example, all symmetric groups and their Sylow 2-subgroups are rational groups. Structure of rational groups have been studied extensively, but the general classification of rational groups has not been able to be done up to now. In this paper, we show that a full symmetric group of prime degree does not have any rational transitive proper subgroup and that a rational doubly transitive permutation group containing a full cycle is the full symmetric group. We also obtain several results related to the study of rational groups.

We show some interesting results concerning entire functions sharing two sets of small functions CM with their difference operators or shifts.

In this paper we study q-subharmonic and q-plurisubharmonic functions in ℂn. Next as an application, we give the notion of q-convex domains in ℂn which is an extension of weakly q-convex domains introduced and investigated in [10]. In the end of the paper we show that the q-convexity is the local property and give some examples about q-convex domains.

In this paper, we deal with the existence and multiplicity of periodic solutions for the p(t)-Laplacian Hamiltonian system. Some new existence theorems are obtained by using the least action principle and minmax methods in critical point theory, and our results generalize and improve some existence theorems.

In this paper, we investigate stability of the zero solution of differential equations with maximum by using Lyapunov functions and Razumikhin techniques. Sufficient conditions for stability, uniform stability and asymptotic stability of the zero solution of such equations are found.

Some new criteria for the oscillation of all solutions of certain fourth-order functional differential equations are established.

This paper deals with the Dirichlet problem for second order linear elliptic equations in unbounded domains of the plane in weighted Sobolev spaces. We prove an a priori bound and an existence and uniqueness result.

In many classical tests for convergence of number series monotonicity of terms of series is a basic assumption. It was shown by Liflyand, Tikhonov and Zeltser that many of these tests are applicable not only to monotone sequences but also to those from a wider class, called weak monotone. Being more general this class still does not allow zeros and too much oscillation. In this paper we extend the class of weak monotonicity to include the mentioned cases and verify that the convergence tests considered by the mentioned authors still hold on this weaker assumption.

The goal of the present work is to give an existence result for a nonlinear integral equation on time scales by considering the Banach space endowed with its weak topology. More precisely, we obtain the existence of weakly continuous solutions for an integral equation that has on the right hand side the sum of two operators, one of them continuous while the other one satisfies a partial continuity condition and some integrability (in a nonabsolute sense) assumptions.

In this paper, we obtain some sufficient conditions for a 3-dimensional compact trans-Sasakian manifold of type (α, β) to be homothetic to a Sasakian manifold. A characterization of a 3-dimensional cosymplectic manifold is also obtained.

We extend the well-known and important fact that “a topological space X is compact if and only if every ideal in C(X) is fixed”, to more general topological spaces. Some interesting consequences are also observed. In particular, the maximality of compact Hausdorff spaces with respect to the property of compactness is generalized and the topological spaces with this generalized property are characterized.

We present a short proof that if two gradient maps on the two-dimensional disc have the same degree, then they are gradient homotopic.

Each member of an n-person team has a secret, say a password. The k out of n gruppen secret sharing requires that any group of k members should be able to recover the secrets of the other n − k members, while any group of k − 1 or less members should have no information on the secret of other team member even if other secrets leak out. We prove that when all secrets are chosen independently and have size s, then each team member must have a share of size at least (n − k)s, and we present a scheme which achieves this bound when s is large enough. This result shows a significant saving over n independent applications of Shamir’s k out of n − 1 threshold schemes which assigns shares of size (n − 1)s to each team member independently of k. We also show how to set up such a scheme without any trusted dealer, and how the secrets can be recovered, possibly multiple times, without leaking information. We also discuss how our scheme fits to the much-investigated multiple secret sharing methods.

In this paper, we define and study the notions of ΛIs-sets, ΛIs-closed sets and I-generalized semi-closed (briefly I-gs-closed) sets by using semi-I-open sets in an ideal topological space. Moreover, we present and characterize two new low separation axioms using the above notions.