Let $\mathbb D^n$ be the unit polydisk and $B$ be the unit ball in $\mathbb C^n$ respectively. In this paper, we extend the Cesàro operator to the unit polydisk and the unit ball. We prove that the generalized Cesàro operator $\mathcal{C}^{\vec b,\vec c}$ is bounded on the Hardy space $H^p(\mathbb D^n)$ and the mixed norm space $A^{p,q}_{\vec \mu}(\mathbb D^n)$, when $0 \lt q \lt \infty$, $p \in (0,1]$ and $Re\, (b_j+1) \gt Re\, c_j \gt 0$, $j = 1,\ldots,n$, or if $0 \lt q \lt \infty$, $p \gt 1$ and $Re\, (b_j+1) \gt Re\, c_j \geq 1$, $j = 1,\ldots,n$. Here $\vec\mu = (\mu_1,\ldots,\mu_n)$ and each $\mu_j$, $j \in \{1,\ldots,n\}$ is a positive Borel measure on the interval $[0,1)$. We also introduce a new class of averaging integral operators $\mathcal{C}^{b,c}_{\zeta_0}$ (the generalized Cesàro operators) on $B$ and prove the boundedness of the operator on the Hardy space $H^p(B)$, $p \in (0,\infty)$, the mixed-norm space $\mathcal{A}^{p,q}_\mu(B)$, $0 \lt p,q \lt \infty$ and the $\alpha$-Bloch space, when $\alpha \gt 1$. Finally, we study the boundedness and compactness of recently introduced Riemann-Stieltjes type operators $T_g$ and $L_g$, from $H^\infty$ and Bergman type spaces to $\alpha$-Bloch spaces and little $\alpha$-Bloch spaces on $B$.

In this paper we present some notions of parametric well-posedness for Stampacchia and Minty vector variational inequalities. We show that under suitable conditions, the parametric well-posedness of a Stampacchia vector variational inequality is equivalent to the parametric well-posedness of a vector optimization problem. Further, we introduce some concepts of well-posedness for a vector optimization problem with a Stampacchia vector variational inequality constraint. We prove that the well-posedness of this constrained vector optimization problem is closely related to the parametric well-posedness of its constrained vector variational inequality.

In this paper, we mainly investigate the radial distribution of the Julia set of a transcendental meromorphic function with finitely many deficient values.

In this paper, we first establish the Brunn-Minkowski type inequalities for the volume of the $L_p$ centroid body and its polar body with respect to the normalized $L_{p}$ radial addition. Furthermore, we prove some properties for operator $\Gamma_{-p}$ and obtain some inequalities for it.

Let $p := \{p_j\}_{j=0}^{\infty}$ and $q := \{q_k\}_{k=0}^{\infty}$ be complex sequences with $p,q \in$ SVA. Assume that $\{s_{mn}\}_{m,n=0}^\infty$ is a double sequence in $\mathbf{C}$ (or one of $\mathbf{R}$, a Banach space, and an ordered linear space), with $s_{mn} \stackrel{st}{\rightarrow} s \hspace{6 pt} (\bar{\mathrm{N}},p,q;\alpha,\beta)$, where $(\alpha,\beta) = (1,1)$, $(1,0)$ or $(0,1)$. We give sufficient and/or necessary conditions under which $s_{mn} \stackrel{st}{\rightarrow} s$. The theory developed here is the statistical version of the work of Chen and Hsu in [Anal. Math. 26 (2000), 243-262]. Our results generalize Móricz, [J. Math. Anal. Appl. 286 (2003), 340-350].

In this paper we derive a set of sufficient conditions for the existence of mild solutions of second order nonlinear integrodifferential evolution equations with nonlocal conditions in Banach spaces. The results are obtained by applying the Schaefer fixed point theorem. An application is provided to illustrate the technique.

We introduce weakly semicommutative rings which are a generalization of semicommutative rings, and give some examples which show that weakly semicommutative rings need not be semicommutative. Also we give some relations between semicommutative rings and weakly semicommutative rings.

We give a criterion of discrete stability for polynomials of degree $n \leq 4$ in terms of polynomial inequalities which contain linear inequalities and only one inequality of degree $n-1$. The result then applies to study stability regions for fixed points of the Arneodo-Coullet-Tresser maps.

In this paper we describe superderivations in certain superalgebras by their actions on elements satisfying some special conditions. One of the main results is applied to local superderivations on some certain superalgebras.

Let $R$ be a 2-torsion free semiprime ring. In this paper we will show that every Jordan triple $(\theta,\phi)$-derivation on $R$ is a $(\theta,\phi)$-derivation. Also every Jordan triple left centralizer on $R$ is a left centralizer. As a consequence, every generalized Jordan triple $(\theta,\phi)$-derivation on $R$ is a generalized $(\theta,\phi)$-derivation. This result gives an affirmative answer to the question posed by Wu and Lu in [14].

In this paper we prove that an oriented hypersurface $M$ of a Euclidean space $E^{n+1}$ has pointwise 1-type Gauss map of the first kind if and only if $M$ has constant mean curvature. Then we conclude that all oriented isoparametric hypersurfaces of $E^{n+1}$ has 1-type Gauss map. We also show that a rational hypersurface of revolution in a Euclidean space $E^{n+1}$ has pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone.

Let $A$ be a normed algebra and $X$ a normed $A$-bimodule. By a ternary derivation we mean a triple $(D_1, D_2, D_3)$ of linear mappings $D_1, D_2, D_3: A \rightarrow X$ such that $D_1(ab) = D_2(a)b + aD_3(b)$ for all $a, b \in A$. Our aim is to establish the stability of ternary derivations associated with the extended Jensen functional equation \[ qf (\frac{\sum_{k=1}^q x_k} {q}) = \sum_{k = 1}^{q} f(x_k) \] for all $x_1, \cdots, x_q \in A$, where $q \gt 1$ is a fixed positive integer.

We define generalized Lie derivations on rings and prove that every generalized Lie derivation on a prime ring $R$ is a sum of a generalized derivation from $R$ into its central closure $R_C$ and an additive map from $R$ into the extended centroid $C$ sending commutators to zero.

In this paper we prove the following result: Let $R$ be a prime ring of characteristic different from two and let $T: R \to R$ be an additive mapping satisfying the relation $T(x^3) = xT(x)x$ for all $x \in R$. In this case $T$ is a two-sided centralizer.

We consider the subspace of Segal-Bargmann space which is invariant under the action of the special orthogonal group. We establish a pointwise bound for a function in this space which is polynomially better than the pointwise bound for a function in the Segal-Bargmann space.

In this paper, we give a new and useful criterion for the positivity of generalized functions and study positive operators on test function space of entire functions on the dual space of a nuclear space with a certain exponential growth condition. This new criterion is used to prove that every positive operator has an integral representation given by positive Borel measure, which can be characterized by integrability conditions. Moreover, this new criterion of positivity can be easily applied to operators such as the identity, the translation, the multiplication, and the convolution operators. This enable us to obtain characterization and integral representation of the associated measure. We also apply the above results to study regularity property of the solution of some quantum stochastic differential equations.

In this paper we introduce and investigate $\mathcal{I}-$convergence in $2-$normed spaces, and also define and examine some new sequence spaces using $2-$norm.

We propose an algorithm for the state feedback pole assignment problem. The algorithm is the first of its kind, making use of the Schur form and minimizing the departure from normality of the closed-loop poles by Newton’s method. Eleven illustrative examples, comparing our algorithm with three other existing ones, are given.

We study the Hyers-Ulam stability of a functional equation of Pexider type associated with a functional equation $f(xy) = xf(y) + f(x)y$ which defines derivations in algebras.

Let $(M, \rho)$ be a metric space and $\tau$ a Hausdorff topology on $M$ such that $\{ M, \tau \}$ is sequentially compact. Let $T$ be a $\rho$-asymptotically nonexpansive type self-mapping of $M$ and $u = \{ x_n \}$ a $\rho$-bounded almost-orbit of $T$. We study the $\tau$-convergence of $u$ in $M$ when the triplet $\{ M, \rho, \tau \}$ satisfies various types of $\tau$-Opial conditions. Our results, which also hold for the continuous case of one-parameter semigroups, extend and unify many previously known results [1, 5, 9, 10, 12-19, 21, 22], and answers affirmatively an open question of S. Reich [20, p.550] in the very general context of a metric space.

Recently Lavoie, Grondin and Rathie obtained ten results closely related to the classical Kummer's theorem as special cases from generalized Whipple's theorem on the sum of a ${}_3F_2$ with unit argument. The aim of this paper is to provide general summation formulas contiguous to the Kummer's theorem by simply using a known integral representation of ${}_2F_1$. As by-product, two classes of summation formulas closely related to the Kummer's theorem were obtained. Some simplified special cases were also given for later easy use.

This note begins a study of the structure of quasigroup semisymmetrizations. For the class of quasigroups isotopic to abelian groups, a fairly complete description is available. The multiplication group is the split extension of the cube of the abelian group by a cyclic group whose order is identified as the semisymmetric index of the quasigroup. For a finite abelian group isotope, the dual of the semisymmetrization is isomorphic to the opposite of the semisymmetrization. The character table of the semisymmetrization is readily computed. The simplicity question for semisymmetrizations is raised. It is shown that a simple, non-abelian quasigroup need not have a simple semisymmetrization.

B. S. Mordukhovich Variational Analysis and Generalized Differentiation, I. Basic Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], A Series of Comprehensive Studies in Mathematics Vol. 330, Springer-Verlag, Berlin, 2006, xxii+579 pp. Variational Analysis and Generalized Differentiation, II. Applications, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], A Series of Comprehensive Studies in Mathematics Vol. 331, Springer-Verlag, Berlin, 2006, xxii+610 pp.