Volume 61, Issue 2

Pseudo-BCK algebras were introduced by G. Georgescu and A. Iorgulescu as a generalization of BCK algebras in order to give a corresponding structure to pseudo-MV algebras, since the bounded commutative BCK algebras correspond to MV algebras. Properties of pseudo-BCK algebras and their connections with other fuzzy structures were established by A. Iorgulescu and J. Kühr. The aim of this paper is to define and study the local pseudo-BCK algebras with pseudo-product. We will also introduce the notion of perfect pseudo-BCK algebras with pseudo-product and we will study their properties. We define the radical of a bounded pseudo-BCK algebra with pseudo-product and we prove that it is a normal deductive system. Another result consists of proving that every strongly simple pseudo-hoop is a local bounded pseudo-BCK algebra with pseudo-product.

A generalized Hermitian (GH) algebra is a special Jordan algebra that is at the same time a spectral order-unit space. In this paper we characterize the von Neumann regular elements in a GH-algebra, relate maximal pairwise commuting subsets of the algebra to blocks in its projection lattice, and prove a Gelfand-Naimark type representation theorem for commutative GH-algebras.

The object of this paper is to derive some interesting properties of certain meromorphically multivalent functions.

The purpose of the paper is to study the uniqueness of meromorphic function when certain non-linear differential polynomials share the same 1-points. As a consequence of the main result we improve and supplement the following recent result: [LAHIRI, I.—PAL, R.: Nonlinear differential polynomials sharing 1-points, Bull. Korean Math. Soc. 43 (2006), 161–168].

We prove some uniqueness theorems concerning the derivatives of meromorphic functions when they share two or three sets which will improve some existing results.

This paper develops the approach to the evaluation of a class of infinite series that involve special products of binomial type, generalized harmonic numbers of order 1 and rational functions. We give new summation results for certain infinite series of non-hypergeometric type. New formulas for the number π are included.

In this paper we define the sequence space ℓ M(Δυm, p, q, s) on a seminormed complex linear space, by using a sequence of Orlicz functions. We study some algebraic and topological properties. We prove some inclusion relations involving ℓ M(Δυm, p, q, s). spaces

In this paper, we obtain a statistical Voronovskaya-type theorem for the Szász-Mirakjan-Kantorovich (SMK) operators by using the notion of A-statistical convergence, where A is a non-negative regular summability matrix.

This paper begins with new definitions for double sequence spaces. These new definitions are constructed, in general, by combining modulus function and nonnegative four-dimensional matrix. We use these definitions to establish inclusion theorems between various sequence spaces such as: If A = (a m,n,k,l) be a nonnegative four-dimensional matrix such that $$ \mathop {\sup }\limits_{m,n} \sum\limits_{k,l = 0,0}^{\infty ,\infty } {a_{m,n,k,l} < \infty } $$ and let f be a modulus, then ω″(A, f) ⊂ ω″∞(A, f) and ω″0(A, f) ⊂ ω″∞(A, f).

We use viscosity approximation methods to obtain strong convergence to common fixed points of monotone mappings and a countable family of nonexpansive mappings. Let C be a nonempty closed convex subset of a Hilbert space H and P C is a metric projection. We consider the iteration process {x n} of C defined by x 1 = x ∈ C is arbitrary and $$ x_{n + 1} = \alpha _n f(x_n ) + (1 - \alpha _n )S_n P_C (x_n + \lambda _n Ax_n ) $$ where f is a contraction on C, {S n} is a sequence of nonexpansive self-mappings of a closed convex subset C of H, and A is an inverse-strongly-monotone mapping of C into H. It is shown that {x n} converges strongly to a common element of the set of common fixed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping which solves some variational inequality. Finally, the ideas of our results are applied to find a common element of the set of equilibrium problems and the set of solutions of the variational inequality problem, a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space. The results of this paper extend and improve the results of Chen, Zhang and Fan.

Considering the positive d-dimensional lattice point Z +d (d ≥ 2) with partial ordering ≤, let {X k: k ∈ Z +d} be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $$ S_n = \sum\limits_{k \leqslant n} {X_k } $$, n ∈ Z +d. Let σ i2, i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ 2. Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $$ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) $$. We show that when l ≥ 2 and b > −l/2, E[‖X‖2(log ‖X‖)d−2(log log ‖X‖)b+4] < ∞ implies $$ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} $$, where Γ(·) is the Gamma function and $$ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } $$.

Bayes’ estimator of truncated Poisson distribution (TPD) has been obtained by using gamma prior. Furthermore, recurrence relations for the estimator of the parameter are obtained. R-software has been used for comparing the estimates with the corresponding maximum likelihood estimator (MLE).