Issue 3, Volume 39, September 2014

2014

1. Inclusion Relations among methods of four-dimensional summability compounded from given four-dimensional methods Richard F. Patterson PDF

   ▸ Abstract

   The goals of this paper include the introduction of a new four-dimensional summability method construction by compounding a single four-dimensional method. The examination of this method begins with the characterization of its RH-regularity properties. In addition, the following inclusion and consistent theorems will be presented.

   If $\{\alpha_m^\prime\}$, $\{\beta_n^\prime\}$, $\{\alpha_m\}$, and $\beta_n$ are sequences such that $\{\alpha_m\}$ and $\{\beta_n\}$ are monotone increasing with $\alpha_m^\prime\geq\alpha_m$ and $\beta_n^\prime\geq\beta_n$ for all sufficiently large $m$ and $n$ and if the transformations $B(\alpha_m^\prime, \beta_n^\prime)$ and $B(\alpha_m, \beta_n)$ are factorable and RH-regular then $B(\alpha_m^\prime, \beta_n^\prime)$ includes $B(\alpha_m, \beta_n)$.

   The RH-regular matrix transformations of the form $B(r_m ,s_n)$ for which $r_1\leq r_2\leq r_3\leq\dots$ and $s_1\leq s_2\leq s_3\leq\dots$ constitute a double sequence of consistent family. Other implications and variations will also be presented.

   277-285
2. Some inequalities for continuous functions of selfadjoint operators in hilbert spaces S. S. Dragomir PDF

   ▸ Abstract

   If $\left\{ E_{\lambda }\right\} _{\lambda \in \mathbb{R}}$ is the spectral family of the bounded selfadjoint operator $A$ on the Hilbert space $H$ and $ m=\min Sp\left( A\right) $ and $M=\max Sp\left( A\right)$, we show that for any continuous function $\varphi $: $\left[ m,M\right] \rightarrow \mathbb{C}$ we have the inequality \begin{eqnarray*} \left\vert \left\langle \varphi \left( A\right) x,y\right\rangle \right\vert ^{2} &\leq &\left( \int_{m-0}^{M}\left\vert \varphi \left( t\right) \right\vert d\left( \bigvee_{m-0}^{t}\left( \left\langle E_{\left( \cdot \right) }x,y\right\rangle \right) \right) \right) ^{2} \\ &\leq &\left\langle \left\vert \varphi \left( A\right) \right\vert x,x\right\rangle \left\langle \left\vert \varphi \left( A\right) \right\vert y,y\right\rangle \end{eqnarray*} for any vector $x$ and $y$ from $H.$ Some related results and applications are also given.

   287-303
3. Beurling spectrum of a function in a Banach space Dang Vu Giang PDF

   ▸ Abstract

   We investigate the Beurling spectra of $\mathbb X$-valued functions with application to functional delay differential equations.

   305-312
4. Trace fields of subgroups of $ { m SU}(n,1)$ Juliette Genzmer PDF

   ▸ Abstract

   In this note, we study the field generated by the traces of subgroups of ${\rm SU}(n,1)$. Under some hypotheses, the trace field of a group ${\rm \Gamma}\subset {\rm SU}(2,1)$ is equal to the field generated by the coefficients of the matrices in ${\rm \Gamma}$. If the group is the image of a representation of the fundamental group of a triangulated $3$-manifold, we can relate the trace field to a geometric invariant. For an arithmetic group of the first type in ${\rm SU}(n,1)$, up to conjugacy, the trace field and the field of the coefficients are the same.

   313-323
5. The stability of property ($gw$) under compact perturbation M.H.M. Rashid , T. Prasad PDF

   ▸ Abstract

   Let $\mathcal H$ be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is given for an operator $T$ on $\mathcal H$ to satisfy that $f(T)$ obeys property $(gw)$ for each function $f$ analytic on some neighborhood of $\sigma(T)$. Also we investigate the stability of property $(gw)$ under (small) compact perturbations.

   325-336
6. Harmonic starlikeness and convexity of integral operators generated by generalized Bessel functions Saurabh Porwal PDF

   ▸ Abstract

   The present paper establishes connections between various subclasses of harmonic univalent functions by applying certain integral operator involving the generalized Bessel functions of first kind. To be more precise, we investigate such connections with harmonic starlike and harmonic convex mappings in the plane.

   337-346
7. Stolarsky-type means related to generalizations of Steffensen’s and Gauss’ inequality Josip Pečarić , Ksenija Smoljak PDF

   ▸ Abstract

   In this paper, functionals defined as the difference between the left-hand and the right-hand sides of generalizations of Steffensen’s and Gauss’ inequality are studied. n-exponential and exponential as well as logarithmic convex functions related to defined functionals are produced. Furthermore, several families of exponentially convex functions are investigated, and new Stolarsky-type means related to defined functionals are generated.

   347-358
8. Factorization of quasi-variational relations systems Daniela Inoan PDF

   ▸ Abstract

   Variational relation problems allow a general approach for variational inequalities, equilibrium problems, optimization problems, and variational inclusions. In this paper, we consider a system of quasi-variational relations and determine some conditions in which the solvability of the independent problems implies the existence of a solution for the system. We particularize then the result for a system of variational inequalities and for a constrained Nash equilibrium problem.

   359-365
9. On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank Edoardo Ballico PDF

   ▸ Abstract

   Let $\sigma _b(X_{m,d}(\mathbb {C}))(\mathbb {R})$, $b(m+1) < {m+d \choose m}$, denote the set of all degree $d$ real homogeneous polynomials in $m+1$ variables (i.e., real symmetric tensors of format $(m+1)\times \cdots \times (m+1)$, $d$ times) which have border rank $b$ over $\mathbb {C}$. It has a partition into manifolds of real dimension $\le b(m+1)-1$ in which the real rank is constant. A typical rank of $\sigma _b(X_{m,d}(\mathbb {C}))(\mathbb {R})$ is a rank associated to an open part of dimension $b(m+1)-1$. Here we classify all typical ranks when $b\le 7$ and $d, m$ are not too small. For a larger set of $(m,d,b)$ we prove that $b$ and $b+d-2$ are the two first typical ranks. In the case $m=1$  (real bivariate polynomials) we prove that $d$ (the maximal possible a priori value of the real rank) is a typical rank for every $b$.

   367-378
10. A strongly convergent shrinking descent-like Halpern’s method for monotone variational inequality and fixed point problems Nguyen Thi Thu Thuy PDF

    ▸ Abstract

    In this paper, we introduce a new iteration method based on the hybrid method in mathematical programming, the descent-like method and the Halpern’s iteration method for finding a common element of the solution set for a variational inequality and the set of common fixed points of a countably infinite family of nonexpansive mappings in Hilbert spaces. The result in this paper modifies and improves some well-known results in the literature for a more general problem.

    379-391
11. On difference ideal convergence of double sequences in random 2-normed spaces Bipan Hazarika PDF

    ▸ Abstract

    An ideal $I$ is a family of subsets of positive integers $\mathbb{N}$ which is closed under taking finite unions and subsets of its elements. We define and study the notions of $\Delta^{n}$-ideal convergence and $\Delta^{n}$-ideal Cauchy double sequences in random $2$-normed spaces, and prove some interesting theorems.

    393-404
12. Hybrid proximal point and extragradient algorithms for solving equilibrium problems Pham N. Anh , Nguyen D. Hien PDF

    ▸ Abstract

    We propose new iteration methods for finding a common point of the solution set of a pseudomonotone equilibrium problem and the solution set of a monotone equilibrium problem. The methods are based on both the extragradient-type method and the viscosity approximation method. We obtain weak convergence theorems for the sequences generated by these methods in a real Hilbert space.

    405-423