Volume 65, Issue 1

Let R be a commutative ring and M be an R-module, and let Z(M) be the set of all zero-divisors on M. In 2008, D. F. Anderson and A. Badawi introduced the regular graph of R. In this paper, we generalize the regular graph of R to the M-regular graph of R, denoted by M-Reg(Γ(R)). It is the undirected graph with all M-regular elements of R as vertices, and two distinct vertices x and y are adjacent if and only if x+y ∈ Z(M). The basic properties and possible structures of M-Reg(Γ(R)) are studied. We determine the girth of the M-regular graph of R. Also, we provide some lower bounds for the independence number and the clique number of M- Reg(Γ(R)). Among other results, we prove that for every Noetherian ring R and every finitely generated module M over R, if 2 ∉ Z(M) and the independence number of M-Reg(Γ(R)) is finite, then R is finite.

The notion of torsion class of abelian cyclically ordered groups has been introduced and fundamental properties of the collection T of all such classes, ordered by the class-theoretical inclusion, have been proved by the second author in 2011. The present paper can be considered as a continuation of the above mentioned one. We describe all atoms of T , show that T does not have any dual atom and prove complete distributivity of T .

A groupoid is called power-commutative if every mono-generated subgroupoid is commutative. The class P c of power-commutative groupoids is a variety. A description of free objects in this variety and their characterization by means of injective groupoids in P c are obtained.

The classical probability that a randomly chosen number from the set {n ∈ N : n ≤ n0} belongs to a set A ⊆ N can be approximated for large number n0 by the asymptotic density of the set A. We say that the events are independent if the probability of their intersection is equal to the product of their probabilities. By analogy we define the independence of sets. We say that the sets are independent if the asymptotic density of their intersection is equal to the product of their asymptotic densities. In the article is described a generalisation of one of the criteria of independence of sets and one interesting case in which sets are not independent

Let K/Q be a cyclic tamely ramified extension, then any ambiguous order of K has a normal basis if and only if any ambiguous ideal has a normal basis.

Let q be a prime number, k an algebraically closed field of characteristic 0, and H a non-trivial semisimple Hopf algebra of dimension 2q 3 . This paper proves that H can be constructed either from group algebras and their duals by means of extensions, or from Radford’s biproduct H ≅R#kG, where kG is the group algebra of G of order 2, R is a semisimple Yetter-Drinfeld Hopf algebra in kG kG YD of dimension q 3 .

We provide a characterization of two families of real functions, namely, of those functions f such that the series ∑f(xn) diverges whenever the series ∑xn diverges, or, respectively, whenever the series ∑xn non-absolutely converges. This solves two open problems of J. Borsík. We also reformulate known results on families of functions preserving or changing the type of convergence of series, and add some results about divergent series of terms converging to zero.

In this paper, we deal with multiple solutions of fractional differential equations with p-Laplacian operator and nonlinear boundary conditions. By applying the Amann theorem and the method of upper and lower solutions, we obtain some new results on the multiple solutions. An example is given to illustrate our results.

In this paper, we investigate the relations between two meromorphic functions that share some pairs of small functions. In particular, we show that two meromorphic functions that share six pairs of small functions IM∗ must be linked by quasi-Möbius transformation. We also give some properties of two meromorphic functions sharing five pairs of values or small functions.

In the paper, the authors review some inequalities and the (logarithmically) complete monotonicity concerning the gamma and polygamma functions and, more importantly, present a sharp double inequality for bounding the polygamma function by rational functions.

In this paper we establish the existence of C (n) -pseudo almost automorphic mild solutions to some n-order evolution equations involving not only time-dependent operator coefficients but also pseudo almost automorphic forcing terms. To illustrate our abstract results, the existence of C (2) -pseudo almost automorphic mild solutions to some damped second-order nonautonomous differential equations, which among other things model the nonautonomous damped beam equation, will be discussed.

In this paper we consider the following Hamiltonian elliptic terns in R N : [XXX] Where V(x) > 0 is a periodic continuous real Function, b̅(x) = (b 1 ....b N ) € C 1 (R N ,R N ) satisfies the gauge condition div b̅(x) =0, g(x,v), f(x,u) are super- linear at infinity. We establish the existence of ground state solutions without the classical Ambrosetti-Rabinowitz superlinear condition.

In this paper, we prove the stability of higher ring derivations associated with a general Cauchy-Jensen functional inequality in the class of mappings from non-Archimedean normed algebras to non-Archimedean Banach algebras.

In this note we obtain some characterizations of Banach spaces which are not isomorphic to any of their proper subspaces. In particular, we show that a Banach space X is not isomorphic to any of its proper subspaces if and only the equality σ RD (LT) = σ LD (T) holds for every bounded linear operator T on X if and only if int(σ(T)) ⊆ σ LD (T) holds for every bounded linear operator T on X, where L T denotes the left multiplication operator by T.

We study the operator Q-class functions, present some Hermite- Hadamard inequalities for operator Q-class functions and give some Kantorovich and Jensen type operator inequalities involving Q-class functions

In this paper, we consider locally dually flat Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We find some conditions under which such a Finsler metric is Berwaldian, Riemannian or locally Minkowskian.

We consider the number of eigenvalues which fall into an interval for the complex sample covariance matrices. The central limit theorem and a moderate deviation principle are established when the endpoint of the interval is close to the edge of the spectrum. The proofs depend on the Four Moment Theorem about the local statistics of eigenvalues up to edge, and the rigidity theorem of the eigenvalues for sample covariance matrices.

In 2009 Jakubec gave two determinantal formulas for the relative class number of the pth cyclotomic field, p an odd prime. We generalize one of the formulas to an arbitrary cyclotomic field and also determine the sign of the formula, which he had not given.