‎This is a study of derivations constructed from conditionally negative type functions on groupoids which illustrates Sauvageot's theory of non-commutative Dirichlet forms‎.

‎In this paper we deal with the notion of regulated functions with‎ ‎values in a $C^*$-algebra $ \mathcal A $ and present examples using‎ ‎a special bi-dimensional $C^*$-algebra of triangular matrices‎. ‎We‎ ‎consider the Dushnik integral for these functions and shows that a‎ ‎convenient choice of the integrator function produces an integral‎ ‎homomorphism on the $C^*$-algebra of all regulated functions $‎ ‎G([a,b],\mathcal A) $‎. ‎Finally we construct a family of linear‎ ‎integral functionals on this $C^*$-algebra‎.

‎Let $S$ be a semigroup and $X$ a Banach space‎. ‎The functional‎ ‎equation $\varphi (xyz)‎+ ‎\varphi (x)‎ + ‎\varphi (y)‎ + ‎\varphi (z) =‎ ‎\varphi (xy)‎ + ‎\varphi (yz)‎ + ‎\varphi (xz)$ is said to be stable for‎ ‎the pair $(X‎, ‎S)$ if and only if $f‎: ‎S\to X$ satisfying $\|‎ ‎f(xyz)+f(x)‎ + ‎f(y)‎ + ‎f(z)‎ - ‎f(xy)‎- ‎f(yz)-f(xz)\| \leq \delta $ for‎ ‎some positive real number $\delta$ and all $x‎, ‎y‎, ‎z \in S$‎, ‎there is‎ ‎a solution $\varphi‎ : ‎S \to X$ such that $f-\varphi$ is bounded‎. ‎In‎ ‎this paper‎, ‎among others‎, ‎we prove the following results‎: ‎1) this‎ ‎functional equation‎, ‎in general‎, ‎is not stable on an arbitrary‎ ‎semigroup; 2) this equation is stable on periodic semigroups; 3)‎ ‎this equation is stable on abelian semigroups; 4) any semigroup with‎ ‎left (or right) law of reduction can be embedded into a semigroup‎ ‎with left (or right) law of reduction where this equation is stable‎. ‎The main results of this paper generalize the works of Jung [J‎. ‎Math‎. ‎Anal‎. ‎Appl‎. ‎222 (1998)‎, ‎126--137]‎, ‎Kannappan [Results Math‎. ‎27‎ ‎(1995)‎, ‎368--372] and Fechner [J‎. ‎Math‎. ‎Anal‎. ‎Appl‎. ‎322 (2006)‎, ‎774--786]‎.

‎The classical moment problem is formulated for commutative hypergroups and the uniqueness is proved for polynomial hypergroups in a single variable and for Sturm--Liouville hypergroups‎.

‎I‎. ‎Kaplansky showed in 1947 that every submultiplicative norm $\Vert‎ . ‎\Vert$ on the algebra ${\mathcal C}(K)$ of complex--valued‎ ‎functions on an infinite compact space $K$ satisfies $\Vert f \Vert‎ ‎\ge \Vert f \Vert _K$ for every $f \in {\mathcal C}(K),$ where‎ ‎$\Vert f \Vert _K=max_{t \in K} \vert f(t)\vert $ denotes the‎ ‎standard norm on ${\mathcal C}(K).$ He asked whether all‎ ‎submultiplicative norms $\Vert‎ . ‎\Vert$ were in fact equivalent to‎ ‎the standard norm (which is obviously true for finite compact‎ ‎spaces)‎, ‎or equivalently‎, ‎whether all homomorphisms from ${\mathcal‎ ‎C}(K)$ into a Banach algebra were continuous‎. ‎This problem turned‎ ‎out to be undecidable in ZFC‎, ‎and we will discuss here some recent‎ ‎progress due to Pham and open questions concerning the structure of‎ ‎the set of nonmaximal prime ideals of ${\mathcal C}(K)$ which are‎ ‎closed with respect to a discontinuous submultiplicative norm on‎ ‎${\mathcal C}(K)$ when the continuum hypothesis is assumed‎. ‎We will‎ ‎also discuss the existence of discontinuous characters on Fr\'echet‎ ‎algebras (Michael's problem)‎, ‎a long standing problem which remains‎ ‎unsolved‎. ‎The Mittag--Leffler theorem on inverse limits of complete‎ ‎metric spaces plays an essential role in the literature concerning‎ ‎both problems‎.

‎This article presents the result on existence of mild solutions for random impulsive semilinear functional differential inclusions under sufficient conditions‎. ‎The results are obtained by using the Martelli fixed point theorem and the fixed point theorem due to Covitz and Nadler‎.

‎In this paper‎, ‎we give two unique common coupled fixed point‎ ‎theorems for mappings satisfying a generalized condition in‎ ‎$d$-complete topological spaces‎.

‎In this paper‎, ‎we consider the ranks of four real‎ ‎matrices $G_{i}(i=0,1,2,3)$ in $M^{\dagger},$ where $M=M_{0}+M_{1}%‎ ‎i+M_{2}j+M_{3}k$ is an arbitrary quaternion matrix‎, ‎and $M^{\dagger}%‎ ‎=G_{0}+G_{1}i+G_{2}j+G_{3}k$ is the Moore-Penrose inverse of $M$‎. ‎Similarly‎, ‎the ranks of four real matrices in Drazin inverse of a‎ ‎quaternion matrix are also presented‎. ‎As applications‎, ‎the necessary‎ ‎and sufficient conditions for $M^{\dagger}$ is pure real or pure‎ ‎imaginary Moore-Penrose inverse and $N^{D}$ is pure real or pure‎ ‎imaginary Drazin inverse are presented‎, ‎respectively‎.

‎Some Hardy--Hilbert type inequalities for Hilbert space operators are established‎. ‎Several particular cases of interest are given as well‎.

‎We study spectral properties of the Manhattan products of‎ ‎the path graphs and show the concentration of zero-eigenvelues‎.

‎In 1955‎, ‎H‎. ‎Dye defined certain projections of a‎ ‎$C^*$-matrix algebra by‎ $P_{i,j}(a)= (1+aa^*)^{-1}\otimes E_{i,i}‎ + ‎(1+aa^*)^{-1}a \otimes‎ ‎E_{i,j}+ a^*(1+aa^*)^{-1} \otimes E_{j,i}‎ + ‎a^*(1+aa^*)^{-1}a\otimes E_{j,j}‎$, ‎which was used to show that in the case of factors not of type‎ ‎$I_{2n}$‎, ‎the unitary group determines the algebraic type of that‎ ‎factor‎. ‎We study these projections and we show that in‎ ‎$\mathbb{M}_2(\mathbb{C})$‎, ‎the set of such projections includes all‎ ‎the projections‎. ‎For infinite $C^*$-algebra $A$‎, ‎having a system of‎ ‎matrix units‎, ‎we have $A\simeq \mathbb{M}_n(A)$‎. ‎M‎. ‎Leen proved that‎ ‎in a simple‎, ‎purely infinite $C^*$-algebra $A$‎, ‎the $*$-symmetries‎ ‎generate $\mathcal{U}_0(A)$‎. ‎Assuming $K_1(A)$ is trivial‎, ‎we revise‎ ‎Leen's proof and we use the same construction to show that any‎ ‎unitary close to the unity can be written as a product of eleven‎ ‎$*$-symmetries‎, ‎eight of such are of the form $1-2P_{i,j}(\omega ),\‎ ‎\omega \in \mathcal{U}(A)$‎. ‎In simple‎, ‎unital purely infinite‎ ‎$C^*$-algebras having trivial $K_1$-group‎, ‎we prove that all‎ ‎$P_{i,j}(\omega )$ have trivial $K_0$-class‎. ‎Consequently‎, ‎we prove‎ ‎that every unitary of $\mathcal{O}_n$ can be written as a finite‎ ‎product of $*$-symmetries‎, ‎of which a multiple of eight are‎ ‎conjugate as group elements‎.

‎In this paper‎, ‎we define a new cone ball-metric and get fixed points and common fixed points for the Meir-Keeler type functions in cone ball-metric spaces‎.

‎The aim of this work is to clarify a new viewpoint of connection between system of integral‎ ‎equations and matrix polynomials‎. ‎A procedure is described for transforming a‎ ‎linear system of integral equations to an independent‎ ‎system‎. ‎The latter is converted to equalities by considering its‎ ‎equivalent matrix polynomial equation which employs the integral operator as its variable‎, ‎and admits a normal form‎ ‎for simplifying the system‎. ‎We will show that under certain suitable‎ ‎conditions‎, ‎an independent reduced system is obtained‎, ‎which can be shown to have the same unknowns as the main system‎, ‎and‎ ‎has only one unknown in each equation‎. ‎In fact‎, ‎the basic idea enables us to develop a methodology to solve general systems of linear integral equations‎.

‎In this paper we consider the classes of starlike functions of‎ ‎order $\alpha $‎, ‎convex functions of order $\alpha $ and we study‎ ‎the convexity and $\alpha $-order convexity for some general‎ ‎integral operators‎. ‎Several corollaries of the main results are‎ ‎also considered‎.