An operator T on a Hilbert space is called half-centered if the sequence T * T, (T * ) 2 T 2 , ... consists of mutually commuting operators. It is a subclass of the well-studied centered operators. In this paper we give a condition for when a half-centered operator is centered and prove a structure theorem for those half-centered operators that satisfies a criteria of a technical nature.

MV-algebras were introduced by Chang to prove the completeness of the infinite-valued Łukasiewicz propositional calculus. Recently, algebraic theory of MV-algebras has been intensively studied. Wajsberg algebras are just a reformulation of Chang MV-algebras where implication is used instead of disjunction. Using these equivalence, in this paper we provide conditions for the existence of an epimorphism between two finite MV-algebras $A$ and $B$. Specifically, we define the mv-functions with domain in the ordered set of prime elements of $B$ and with range in the ordered set of prime elements of $A$, and prove that every epimorphism from $A$ to $B$ can be uniquely constructed from an mv-function.

This paper studies the uniqueness of meromorphic functions f^n\prod_{i=1}^k(f^{(i)})^{n_i}\quadand\quad g^n\prod_{i=1}^k(g^{(i)})^{n_i} that share two values, where $n, n_k, k\in\mathbb{N}$, $n_i\in\mathbb{N}\cup\{0\}$, $i=1,2,\ldots,k-1$. The results significantly rectify, improve and generalize the results due to Cao and Zhang (2012).

In this paper, we consider Lipschitz conditions for tri-quadratic functional equations. We introduce a new notion similar to that of the left invariant mean and prove that a family of functions with this property can be approximated by tri-quadratic functions via a Lipschitz norm.

The spaces of entire functions represented by Dirichlet series have been studied by Hussein and Kamthan and others. In the present paper we consider the space $X$ of all entire functions defined by vector-valued Dirichlet series and study the properties of a sequence space which is defined using the type of an entire function represented by vector-valued Dirichlet series. The main result concerns with obtaining the nature of the dual space of this sequence space and coefficient multipliers for some classes of vector-valued Dirichlet series.

We introduce and study some new subclasses of starlike, convex and close-to-convex functions defined by the generalized Bessel operator. Inclusion relations are established and integral operator in these subclasses is discussed.

Given a groupoid $< G, \star >$, and $k \geq 3$, we say that $G$ is antiassociative iff for all $x_1, x_2, x_3 \in G$, $(x_1 \star x_2) \star x_3$ and $x_1 \star (x_2 \star x_3)$ are never equal. Generalizing this, $< G, \star >$ is $k$-antiassociative iff for all $x_1, x_2, ... x_k \in G$, any two distinct expressions made by putting parentheses in $x_1 \star x_2 \star x_3 \star ...x_k$ are never equal. We prove that for every $k \geq 3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.

Let $\sum \limits _{n=1}^\infty a_n$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $(a_n)$ is non-increasing, then $\lim \limits _{n \to \infty } n a_n = 0$. In the present paper: \endgraf (a) We formulate and prove a necessary and sufficient condition for having $\lim \limits _{n \to \infty } n a_n = 0$; Olivier's theorem is a consequence of our Theorem \ref {import}. \endgraf (b) We prove properties analogous to Olivier's property when the usual convergence is replaced by the $\mathcal I$-convergence, that is a convergence according to an ideal $\mathcal I$ of subsets of $\mathbb N$. Again, Olivier's theorem is a consequence of our Theorem \ref {Iol}, when one takes as $\mathcal I$ the ideal of all finite subsets of $\mathbb N$.

A vertex $k$-coloring of a graph $G$ is a \emph {multiset $k$-coloring} if $M(u)\neq M(v)$ for every edge $uv\in E(G)$, where $M(u)$ and $M(v)$ denote the multisets of colors of the neighbors of $u$ and $v$, respectively. The minimum $k$ for which $G$ has a multiset $k$-coloring is the \emph {multiset chromatic number} $\chi _{m}(G)$ of $G$. For an integer $\ell \geq 0$, the $\ell $-\emph {corona} of a graph $G$, ${\rm cor}^{\ell }(G)$, is the graph obtained from $G$ by adding, for each vertex $v$ in $G$, $\ell $ new neighbors which are end-vertices. In this paper, the multiset chromatic numbers are determined for \mbox {$\ell $-\emph {coronas}} of all complete graphs, the regular complete multipartite graphs and the Cartesian product $K_{r}\square K_2$ of $K_r$ and $K_2$. In addition, we show that the minimum $\ell $ such that $\chi _{m}({\rm cor}^{\ell }(G))=2$ never exceeds $\chi (G)-2$, where $G$ is a regular graph and $\chi (G)$ is the chromatic number of $G$.

A prominent Czech scientist, Jaroslav Kurzweil, an excellent mathematician, reaches ninety years of age on May 7, 2016.His previous anniversaries were mentioned among other in the papers [1]-[5]; their electronic versions are now available online at DML-CZ: The Czech Digital Mathematics Library 1 .In particular, [5] contains a detailed survey of Kurzweil's results achieved before 2006 as well as his complete bibliography up to 2006.Jaroslav Kurzweil significantly contributed e.g. to the metric theory of Diophantine approximations, geometry of Banach spaces, stability theory of differential equations, theory of differential inclusions, control theory, invariant manifolds flows or global solutions of functional differential equations.Particularly valuable is his impact on the theory of differential equations.The lecture notes [9] are still very actual and widely used for advanced courses of ordinary differential equations.However, in mathematical world he is now famous primarily as the creator of a new approach to the integration and qualitative theory of differential equations.His Riemann-type definition of an integral was first published in 1957 in the Czechoslovak Mathematical Journal (see [11]).Its main idea is similar to the classical Riemann's approach: The integral of a function over an interval [a, b] is approximated by the sum of the lengths of subintervals of a division of [a, b] multiplied by the value of the function in particular points called the tags.The novelty is that the tags are chosen first, while the division points are allowed to vary in a controlled neighborhood of the tag.This made it possible to control the singularities and integrate very general classes of functions.This integral, now generally called the Kurzweil (or more often Henstock-Kurzweil) integral has proved to be very strong and inspiring, not only for the integration theory itself but also for differential and integral equations.It includes the classical concepts of the Riemann, Newton, Lebesgue and