Let $S(\mathbb{D})$ be the collection of all the holomorphic self-maps of open unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$, and $C_{\varphi}$, the composition operator induced by $\varphi\in S(\mathbb{D})$. For $\alpha>0,\;\lambda\in \mathbb{C}$, we give some sufficient and necessary conditions for the hypercyclicity of multiples of composition operators $\lambda C_\varphi$ acting on the weighted Banach spaces of entire functions $H_{\alpha,0}^\infty$. Moreover, we obtain a partial characterization for the frequent hypercyclicity of $\lambda C_\varphi$ on $H_{\alpha,0}^\infty$.

We have recently shown that a nilpotent Lie algebra $L$ of dimension $n \ge1 $ satisfies the inequality $\mathrm{dim} \ H_2(L,\mathbb{Z}) \leq \frac{1}{2}(n+m-2)(n-m-1)+1$, where $\mathrm{dim} \ L^2=m \ge 1$ and $H_2(L,\mathbb{Z})$ is the 2-nd integral homology Lie algebra of $L$. Our first main result correlates this bound with the $i$-th Betti number $\mathrm{dim} \ H^i(L,\mathbb{C}^\times)$ of $L$, where $H^i(L,\mathbb{C}^\times)$ denotes the $i$-th complex cohomology Lie algebra of $L$. Our second main result describes a more general restriction, which follows an idea of Ellis in [G. Ellis, The Schur multiplier of a pair of groups, Appl. Categ. Structures 6 (1998), 355--371].

We are interested in the existence of travelling waves for the Benjamin-Bona-Mahony equation on a network. First we construct an explicit wave, defined in $\mathbb{R}$. Then, we use this wave to derive some conditions on the coefficients appearing in the equations and on the geometry of the network to ensure the existence of travelling waves on the network.

In this paper, relationships between the Ricci curvature and the squared mean curvature for integral submanifolds of an $f.p.k.$-space form by a basic inequality, are studied. We show that if an integral submanifold of maximum dimension of an $f.p.k.$-space form satisfies the equality case, then it must be minimal.

In this paper we determine some examples of the domains of $n$-valence for functions which are analytic with classical normalization and locally univalent. This problem has not been studied yet. It seems to be rather difficult. We formulate a few conjectures concerning some subclasses of typically real functions and odd typically real functions.

Following Andrei's approach, a modified scaled memoryless BFGS preconditioned conjugate gradient method is proposed based on the modified secant equation suggested by Li and Fukushima. It is shown that the method is globally convergent without convexity assumption on the objective function. Furthermore, for uniformly convex objective functions, sufficient descent property of the method is established based on an eigenvalue analysis. Numerical experiments are employed to demonstrate the efficiency of the method.

In this paper, it is proved that a non-locally compact paratopological group $G$ has a remainder which is a $p$-space if and only if $G$ is either a Lindelöf $p$-space or a $\sigma$-compact space. We show that if $G$ is a non-locally compact paratopological group with a compactification $bG$ such that the remainder $bG\setminus G$ is locally metrizable, then both $G$ and $bG$ are separable and metrizable. It is proved that if $G$ is a cosmic paratopological group with a paracompact remainder, then $G$ is separable and metrizable.

Rank one groups were introduced by F. G. Timmesfeld as the building blocks of Lie type groups. Division pairs are algebraic objects categorically equivalent to Moufang sets. We define a functor $\varDelta$ from rank one groups to division pairs and prove that $\varDelta$ has a left adjoint $\varSigma$, given by a Steinberg type construction. We also extend the theory of the quasi-inverse and the Bergmann operators, well known from Jordan pairs, to this setting. As an application, we show that identities proved by T. De Medts and Y. Segev for Moufang sets hold in arbitrary rank one groups.

Let $G$ be a locally compact amenable group. The goal of this paper is to investigate the problem of surjective zero products preserving maps from the Fourier algebra of $G$ into a completely contractive Banach algebra. We show that if $B$ is a completely contractive Banach algebra which is faithful and factors weakly, then every surjective completely bounded linear map from $A(G)$ into $B$ which preserves zero products is a weighted homomorphism. Moreover an equivalent condition is given for such a map to be a homomorphism. In particular, this result implies that if $B$ is a commutative C$^*$-algebra or a matrix space and $T:A(G)\rightarrow B$ is a continuous surjective linear map which preserves zero products, then $T$ is a weighted homomorphism and there is an equivalent condition for $T$ to be a homomorphism.

We investigate the initial value problem for a semilinear heat equation with exponential-growth nonlinearity in two space dimension. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space $H^1(\R^2)$. The uniqueness part is non trivial although it follows Brezis-Cazenave's proof in the case of monomial nonlinearity in dimension $d\geq3$. Next, we show that in the defocusing case our solution is bounded, and therefore exists for all time. In the focusing case, we prove that any solution with negative energy blows up in finite time.

Recently, Abardia and Bernig introduced the notion of mixed complex projection bodies and established a number of important geometric inequalities for them. In the present paper we prove several new isoperimetric type inequalities for volume differences of mixed complex projection bodies.

We characterize Banach lattices under which each AM-compact (resp. b-AM-compact) operator is Dunford-Pettis. Also, we study the AM-compactness of limited completely continuous operators.