If $f$ is $\mathcal{M}$-harmonic and integrable with respect to a weighted radial measure $\nu_{\h}$ over the unit ball $B_n$ of $\mathbb{C}^n$, then $\int_{B_n} (f\circ\psi)\ d\nu_{\h}=f(\psi(0))$ for every $\psi \in {\hbox{Aut}(B_n)}$. Equivalently $f$ is fixed by the weighted Berezin transform; $T_{\h}f=f$. In this paper, we show that if a function $f$ defined on $B_n$ satisfies $R(f \circ \phi) \in L^{\infty}(B_{n})$ for every $\phi \in {\hbox{Aut}(B_n)}$ and $Sf=rf$ for some $|r|=1$, where $S$ is any convex combination of the iterations of ${T_{\h}}'s$, then $f$ is $\mathcal{M}$-harmonic.

A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are balanced and all prime-valent regular Cayley maps on abelian groups are either balanced or anti-balanced. Furthermore, we prove that there is no prime-valent regular Cayley map on any dicyclic group.

A characterization of the holomorphic function spaces of me\-an Bloch type on the unit disc is deduced in terms of the induced distance.

Let $P$ be a Jacobian polynomial such as $\deg P=\deg_y P$. Suppose the Jacobian polynomial $P$ satisfies the intersection condition satisfying $\dim_{\CC} \CC[x,y]/\langle P, P_y\rangle=\deg P-1$, we can prove that the Keller map which has $P$ as one of coordinate polynomial always has its inverse.

In this paper, we prove the generalized Hyers-Ulam stability of bi-homomorphisms in $C^*$-ternary algebras and of bi-derivations on $C^*$-ternary algebras for the following bi-additive functional equation $$f(x+y,z-w)+f(x-y,z+w)=2f(x,z)-2f(y,w).$$ This is applied to investigate bi-isomorphisms between $C^*$-ternary algebras.

Let $M$ be a real hypersurface with almost contact metric structure $(\phi, g, \xi, \eta)$ in a complex space form $\mn$, $c \neq 0$. In this paper we prove that if $\rx\lx g=0$ holds on $M$, then $M$ is a Hopf hypersurface in $\mn$, where $\rx$ and $\lx$ denote the structure Jacobi operator and the operator of the Lie derivative with respect to the structure vector field $\xi$ respectively. We characterize such Hopf hypersurfaces of $\mn$.

This paper concerns the problem of global robust stability of a time-delay discontinuous system with a positive-defined connection matrix under polytopic-type uncertainty. In order to give the stability condition, we firstly address the existence of solution and equilibrium point based on the properties of $M$-matrix, Lyapunov-like approach and the theories of differential equations with discontinuous right-hand side as introduced by Filippov. Second, we give the delay-independent and delay-dependent stability condition in terms of linear matrix inequalities (LMIs), and based on Lyapunov function and the properties of the convex sets. One numerical example demonstrate the validity of the proposed criteria.

In this paper, we analyze a zero-knowledge identification scheme presented in~\cite{Cab05}, which is based on an average-case hard problem, called \emph{distributional matrix representability problem}. On the contrary to the soundness property claimed in~\cite{Cab05}, we show that a simple impersonation attack is feasible. \keywords{zero-knowledge, computational complexity, matrix representability problem}

In 1992, the author introduced the definition and the properties of Wiener measure over paths in abstract Wiener space and this measure was investigated extensively by some mathematicians. In 2002, the author and Dr. Im presented an article for analogue of Wiener measure and its applications which is the generalized theory of Wiener measure theory. In this note, we will derive the analogue of Wiener measure over paths in abstract Wiener space and establish two integration formulae, one is similar to the Wiener integration formula and another is similar to simple formula for conditional Wiener integral. Furthermore, we will give some examples for our formulae.

Using a degree theory for countably condensing multivalued maps, we show that under certain conditions an invariance of domain theorem holds for countably condensing or countably $k$-contractive multivalued vector fields.

In this note, we obtain range inclusion results for left Jordan derivations on Banach algebras: (i) Let $\d$ be a spectrally bounded left Jordan derivation on a Banach algebra $A$. Then $\d$ maps $A$ into its Jacobson radical. (ii) Let $\d$ be a left Jordan derivation on a unital Banach algebra $A$ with the condition sup$\{r(c^{-1}\d(c)): c \in A \text{ invertible} \}< \infty$. Then $\d$ maps $A$ into its Jacobson radical. Moreover, we give an exact answer to the conjecture raised by Ashraf and Ali in \cite[p.\,260]{Ash08}: every generalized left Jordan derivation on 2-torsion free semiprime rings is a generalized left derivation.

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP) \begin{gather*} -\mathbf{D}_{0+}^\alpha u(t)=\lambda [f(t,u(t))+q(t)],\quad 00$ is a parameter, $1<\alpha\leq 2$, $\mathbf{D}_{0+}^\alpha$ is the standard Riemann-Liouville differentiation, $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous, and $q(t):(0,1)\to [0, +\infty)$ is Lebesgue integrable. We obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of BVP when $\lambda$ in some interval. Our approach is based on Leray-Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity which was essential for the technique used in almost all existed literature on $f$.

We discuss the critical points of the functional $\mathcal {F}_{\lambda, \mu} (J, g) = \int_M (\lambda \tau + \mu \tau^* ) dv_g$ on the spaces of all almost Hermitian structures $\mathcal{AH}(M)$ with ${(\lambda, \mu)} \in \mathbb{R}^2 - (0,0)$, where $\tau$ and $\tau^*$ being the scalar curvature and the $*$-scalar curvature of $(J, g)$, respectively. We shall give several characterizations of K\"{a}hler structure for some special classes of almost Hermitian manifolds, in terms of the critical points of the functionals $\mathcal {F}_{\lambda, \mu} (J, g)$ on $\mathcal{AH}(M)$. Further, we provide the almost Hermitian analogy of the Hilbert's result.

We establish the existence of weak solutions in an infinite subsonic channel in the self-similar plane to the two-dimensional Burgers system. We consider a boundary value problem in a fixed domain such that a part of the domain is degenerate, and the system becomes a second order elliptic equation in the channel. The problem is motivated by the study of the weak shock reflection problem and 2-D Riemann problems. The two-dimensional Burgers system is obtained through an asymptotic reduction of the 2-D full Euler equations to study weak shock reflection by a ramp.

In this paper, the moments of the Riesz-N{\'a}gy-Tak{\'a}cs(RNT) distribution over a general interval $[a,b]\subset [0,1]$, are found through the moments of the RNT distribution over the unit interval, $[0,1]$. This is done using some special features of the distribution and the fact that $[0,1]$ is a self-similar set in a dynamical system generated by the RNT distribution. The results are important for the study of the orthogonal polynomials with respect to the RNT distribution over a general interval.

A pair of point vortices of the same strength but opposite sign is called a vortex dipole. We consider the limiting case where two vortices approach infinitely close while the ratio of the strength to the distance kept constant. The motion of such {\em point} vortex dipole on the ellipsoid of revolution is investigated geometrically to conclude that the trajectory draws a geodesic up to the leading order of perturbation, whose direction is determined by the initial orientation of the dipole. Related issues are also remarked.

This paper concerns the propagation phenomenon of weight\-ed shifts. We here establish the existence of positive real numbers $b$ and $c$ ($1

In this article, we discuss the reflective function which can be represented by three exponential matrixes and apply the results to studying the existence of periodic solutions of these systems. The obtained conclusions extend and improve the foregoing results.

In this paper we consider the uniform central limit theorem for a martingale-difference array of a function-indexed stochastic process under the uniformly integrable entropy condition. We prove a maximal inequality for martingale-difference arrays of process indexed by a class of measurable functions by a method as Ziegler~\cite{ReferZIEGLER} did for triangular arrays of row wise independent process. The main tools are the Freedman inequality for the martingale-difference and a sub-Gaussian inequality based on the restricted chaining. The results of present paper generalizes those of Ziegler~\cite{ReferZIEGLER} and other results of independent problems. The results also generalizes those of Bae and Choi~\cite{ReferBAECHOI} to martingale-difference array of a function-indexed stochastic process. Finally, an application to classes of functions changing with $n$ is given.

In the present paper, we give two new proofs for the necessary and sufficient condition $\alpha\le1$ such that the function $x^\alpha[\ln x-\psi(x)]$ is completely monotonic on $(0,\infty)$.