We show that there exists a holomorphic function, continuous to the boundary in a bounded, balanced, strictly pseudoconvex domain $\Omega$ with $C^{2}$ boundary such that almost every slice function has a series of Taylor coefficients divergent with every power $p\in(0,2)$.

We study the dynamics of the multi-dimensional Ces\`aro integral operator on $L^p(I^n)$, for $I$ the unit interval, $1<p<\infty$, and $n\ge 2$, that is defined as \begin{multline*} \displaystyle \mathcal{C}(f)(x_1,\ldots,x_n)=\frac {1} {x_1x_2\cdots x_n} \int_0^{x_1}\ldots\int_{0}^{x_n} f(u_1,\ldots,u_n)du_1\ldots du_n\\ \quad \text{ for } f\in L^p(I^n). \end{multline*} This operator is already known to be bounded. As a consequence of the Eigenvalue Criterion, we show that it is hypercyclic as well. Moreover, we also prove that it is Devaney chaotic and frequently hypercyclic.

We consider a planar vector field $X$ near a saddle type $p:-q$ resonant singular point. Assuming that it has a normal form with a Gevrey-$d$ expansion (like $d=p+q$ which is in particular the case when starting from an analytic vector field) we show that $X$ can be linearized working with a change of coordinates that is of Gevrey order $d$ in certain $\log$-like variables, called compensators or also tags, multiplied by the first integral $u=x^qy^p$ of the linear part. Next we consider the unfolding of such a resonance, and provide (weaker) Gevrey-type linearization using compensators.

In this paper we study a model of population which is described by positive solutions to the nonlinear boundary value problem $$ \left\{\begin{array}{ll} -\Delta u = au-bu^{2}-c\frac{u^{2}}{1+u^{2}}-\epsilon, & x\in \Omega,\\ \mathbf{n}.\nabla u+ g(u)=0 , & x\in\partial \Omega.\\ \end{array}\right. $$ Here $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N},$ $\Delta$ is the Laplacian operator, $a,$ $b,$ $c,$ $\epsilon$ are positive parameters and $g \in C^{1}\Big([0,\infty),$ $[\theta,\infty)\Big)$ is decreasing for some $\theta > 0.$ This model describes the dynamics of the fish populations. Our existence results are established via the well-known sub-super solution method.

In this paper, we study a Korovkin type approximation theorem for a sequence of positive linear operators acting from $L_{p,q}\left( loc\right) $ into itself with the use of Abel method which is a sequence-to-function transformation. Using the modulus of continuity for $L_{p,q}\left( loc\right) $ we also give the rate of Abel convergence of these operators.

Let $G$ be a finite group. We say that an element $g$ in $G$ is a vanishing element if there exists some irreducible character $\chi$ of $G$ such that $\chi(g)=0$. In this paper, we prove that if the set of vanishing elements of $G$ is the union of at most three conjugacy classes, then $G$ is solvable.

In this work we consider the non local evolution problem \[ \begin{cases} \partial_t u(x,t)=-u(x,t)+g(\beta K(f\circ u)(x,t)+\beta h), ~x \in\Omega, ~t\in[0,\infty[;\\ u(x,t)=0, ~x\in\mathbb{R}^N\setminus\Omega, ~t\in[0,\infty[;\\ u(x,0)=u_0(x),~x\in\mathbb{R}^N, \end{cases} \] where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N; ~g,f: \mathbb{R}\to\mathbb{R}$ satisfying\linebreak certain growing condition and $K$ is an integral operator with symmetric kernel, $ Kv(x)=\int_{\mathbb{R}^{N}}J(x,y)v(y)dy.$ We prove that Cauchy problem above is well posed, the solutions are smooth with respect to initial conditions, and we show the existence of a global attractor. Furthermore, we exhibit a Lyapunov's functional, concluding that the flow generated by this equation has the gradient property.

In this paper, we introduce the concept of pointwise compactness for a locally compact group $G,$ and among other results, we show that pointwise compactness of $G$ is a necessary condition for pointwise contractibility of $L^1(G)$ in a commutative case. Also, pointwise contractibility of measure algebras in a general case is studied. Finally, applying the results, we study the pointwise contractibility of Fourier and Fourier-Stieltjes algebras in a commutative case.

By using Nevanlinna theory and linear algebra, we show that the number one is a lower bound of the hyper-order of any meromorphic solution of a non-linear delay differential equation under certain conditions.

We consider the semidirect product $G = K \ltimes V$ where $K$ is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space $V$ equipped with an inner product $\langle,\rangle$. We denote by $\widehat{G}$ the unitary dual of $G$ (note that we identify each representation $\pi\in\widehat{G}$ to its classes $[\pi]$) and by $\mathfrak{g}^\ddag/G$ the space of admissible coadjoint orbits, where $\mathfrak{g}$ is the Lie algebra of $G.$ It was pointed out by Lipsman that the correspondence between $\mathfrak{g}^\ddag/G$ and $\widehat{G}$ is bijective. Under some assumption on $G,$ we prove that the Lipsman mapping \begin{eqnarray*} \Theta:\mathfrak{g}^\ddag/G &\longrightarrow&\widehat{G}\\ \mathcal{O}&\longmapsto&\pi_\mathcal{O} \end{eqnarray*} is a homeomorphism.