Volume 46, Issue 1

A tree language of a fixed type τ is any set of terms of type τ . We consider here a binary operation + n on the set W τ ( X n ) of all n -ary terms of type τ , which results in semigroup ( W τ ( X n ), + n ). We characterize languages which are idempotent with respect to this binary operation, and look at varieties of tree languages containing idempotent languages. We also compare properties of semigroup homomorphisms from (𝒫( W τ ( X n ));+ n ) to (𝒫( W τ ( X m ));+ m ) with properties of homomorphisms between the corresponding absolutely free algebras ℱ τ ( X n ) and ℱ τ ( X m ).

The current paper represents a suplement for papers [7] and [8]. Many of the new summation formulae connecting Lucas numbers with binomials are presented here. All these relations are obtained by using definition and simple properties of the so called δ -Lucas numbers.

The paper considers the anticommutative multiplication property for the matrix algebra M 2 ( G ) over the infinite Grassmann algebra G . We define as well some classes of *-symmetric and * skew-symmetric matrices in ( M 2 ( G ), * ) for * being the transpose or the symplectic involution. As a consequence we give some *-identities for M 2 ( G ) of degree < 8. Examples of the application of finite-dimensional Grassmann algebras in quantum field theory are mentioned and some properties of the algebras G 4 and M 2 ( G 4 ) are given as well.

In this paper, we prove a fuzzy Korovkin-type approximation theorem for fuzzy positive linear operators by using A -statistical convergence for four-dimensional summability matrices. Also, we obtain rates of A -statistical convergence of a double sequence of fuzzy positive linear operators for four-dimensional summability matrices.

We employ the notion of weighted sharing to investigate the uniqueness of meromorphic functions when two nonlinear differential polynomials share fixed points. The results of the paper improve and generalize the recent results due to Xu–Lu–Yi [10].

The purpose of the present paper is to investigate some interesting properties on generalized convolutions of functions for the classes HP ∗ ( α ), HS ( α ) and HC ( α ). Further, an application of the convolution on certain integral operator are mentioned.

Let ρ p ( f ) and σ p ( f ) denote respectively the iterated p -order and the iterated p -type of an entire function f. In this paper, we study the iterated order and the fixed points of some differential polynomials generated by solutions of the differential equation f″+A1(z)f′+A0(z)f=0,$$f'' + A_1 (z)f^\prime + A_0 (z)f = 0,$$ where A 1 ( z ), A 0 ( z ) are entire functions of finite iterated p -order such that ρ p ( A 1 ) = ρ p ( A 0 ) = ρ (0 < ρ < + ∞) and σ p ( A 1 ) < σ p ( A 0 ) = σ (0 < σ < + ∞).

In the paper, Guo–Krasnoselskii’s fixed point theorem is adapted to study the existence of positive solutions to a class of boundary value problems for higher order differential equations with delay. The sufficient conditions, which assure that the equation has one positive solution or two positive solutions, are derived. These conclusions generalize some existing ones.

In this paper, we establish some new oscillation criteria for the higher-order half-linear neutral differential equation [a(t)  ([x(t)+p(t)x(σ(t))](n−1))γ]′+q(t)xγ(τ(t))=0.$$\left[ {a(t)\,\,\left( {[x(t) + p(t)x(\sigma (t))]^{(n - 1)} } \right)^\gamma } \right]^\prime + q(t)x^\gamma (\tau (t)) = 0.$$ An example is given to illustrate the main results.

In this paper, we analyze the dynamical behaviors of Liu system using the complementary-cluster energy-barrier criterion (CCEBC). Moreover, the Hopf bifurcation of this system is investigated using the first Lyapunov coefficient. Also, it is proved that this system has two Hopf bifurcation points, at which these Hopf bifurcations are non degenerate and subcritical.

In this paper, some results on the equi-boundedness of solutions, the stability of the zero and the existence of positive periodic solutions of nonlinear difference equation with variable delay xn+1=λnxn+αnF(xn−mn),    n=0, 1,…$$x_{n + 1} = \lambda _n x_n + \alpha _n F(x_{n - m_n } ),\,\,\,\,n = 0,\,1, \ldots $$ are obtained.

In this paper we introduce and study the concepts of I -divergence and I ∗ -divergence of sequences as well as double sequences in an asymmetric metric spaces. We investigate the interrelationship between I -divergence and I ∗ -divergence and show that they are equivalent under some condition and prove some basic properties of these concepts.

A new general theorem concerning some kind of absolute summability is presented.

The object of this paper is to introduce the sequence space ces ( f, p ) using a modulus function f . Various algebraic and topological properties of this space, and certain inclusion relations have been discussed which generalize several known results of Shiue [10], Sanhan and Suantai [9], and Leibowitz [2].

We introduce the class of Banach lattices with the AM-compactness property and we use it to characterize Banach lattices on which each positive weak Dunford–Pettis operator is almost Dunford–Pettis and conversely.

A generalized metric space has been defined by Branciari as a metric space in which the triangle inequality is replaced by a more general inequality. Subsequently, some classical metric fixed point theorems have been transferred to such a space. In this paper, we continue in this direction and prove a version of Fisher’s fixed point theorem in generalized metric spaces.

The Demyanov metric in the family of convex, compact sets in finite dimensional spaces has been recently extended to the family of convex, bounded sets – not necessarily closed. In this note it is shown that these spaces are not complete and a model for the completion is proposed. A full answer is given in ℝ 2 and the situation in higher dimensions is discussed.

In this paper we shall study a density-type topology generated by the convergence everywhere except for a finite set similarly as the classical density topology is generated by the convergence in measure. Among others it is shown that the set of finite density points of a measurable set need not be measurable.

This paper investigates the notion of approach nearness spaces. Using clusters, completion of an approach nearness space is constructed, which is a unified study of completion in the context of metric spaces, uniform approach spaces, weakly symmetric approach spaces and nearness spaces. Another generalization of completeness, called ultrafilter completeness is introduced to prove the Niemytzki–Tychonoff theorem for approach nearness spaces. Both definitions of completions are shown to be equivalent in a limit-regular approach space. Various examples are given to support the present study.

A new class of functions called ‘ R cl -supercontinuous functions’ is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of R cl -supercontinuous functions properly contains the class of cl-supercontinuous (≡ clopen continuous) functions (Applied Gen. Topology 8(2) (2007), 293–300; Indian J. Pure Appl. Math. 14(6) (1983), 767–772) and is strictly contained in the class of R δ -supercontinuous functions which in its turn, is properly contained in the class of R -supercontinuous functions (Demonstratio Math. 43(3) (2010), 703–723).