In this paper we study the integral cohomology of pure mapping class groups of surfaces, and other related groups and spaces, as $\mathsf{FI}$-modules. We use recent results from Church, Miller, Nagpal and Reinhold to obtain explicit linear bounds for their presentation degree and to give an inductive description of these $\mathsf{FI}$-modules. Furthermore, we establish new results on representation stability, in the sense of Church and Farb, for the rational cohomology of pure mapping class groups of non-orientable surfaces.

We continue the study of asymmetry in $T_0$-quasi-metric spaces. In this context we introduce the property of symmetric connectedness for a $T_0$-quasi-metric space. We present some methods in order to find the symmetrically connected pairs of a $T_0$-quasi-metric space. We also show that the problem to determine the symmetry components of points turns out to be easier when formulated for the induced $T_0$-quasi-metric of an asymmetrically normed real vector space. In addition, as a kind of opposite to the notion of a metric space, we define antisymmetric $T_0$-quasi-metric spaces. Subsequently some useful results about antisymmetry can be emphasized by describing the property of antisymmetric connectedness for a $T_0$-quasi-metric space. Finally, we observe that there are natural relations between the theory of (anti)symmetrically connected $T_0$-quasi-metric spaces and the theory of connectedness in the sense of graph theory.

In this paper we introduce a generalization of Bernstein-Chlodovsky operators that preserves the exponential function $e^{-2x}$ $(x \geq 0)$. We study its approximation properties in several function spaces, and we evaluate the rate of convergence by means of suitable moduli of continuity. Throughout some estimates of the rate of convergence, we prove better error estimation than the original operators on certain intervals.

In this work we shall prove new results on the theory of convolution operators on spaces of entire functions. The focus is on hypercyclicity results for convolution operators on spaces of entire functions of a given type and order; and existence and approximation results for convolution equations on spaces of entire functions of a given type and order. In both cases we give a general method to prove new results that recover, as particular cases, several results of the literature. Applications of these more general results are given, including new hypercyclicity results for convolution operators on spaces on entire functions on $\mathbb{C}^n.$

We show that amenability, the Haagerup property, the Kazhdan's property (T) and exactness are preserved under taking second nilpotent product of groups. We also define the restricted second nilpotent wreath product of groups, this is a semi-direct product akin to the restricted wreath product but constructed from the second nilpotent product. We then show that if two discrete groups have the Haagerup property, the restricted second nilpotent wreath product of them also has the Haagerup property. We finally show that if a discrete group is abelian, then the restricted second nilpotent wreath product constructed from it is unitarizable if and only if the acting group is amenable.

In this paper, we study the limit distribution of the record values under nonlinear normalization of the form $${\cal T}_n(x)=\exp\{u_{n}(| \log |x||)^{v_{n}}\mbox{sign}(\log |x|)\}\mbox{sign}(x),$$ which is called exponential norming ($e-$norming). The corresponding limit laws of the upper extremes are called $e$-max stable laws (denoted by $U(.)$). In this paper, we show that the limit distributions of the record values under exponential norming are of the form $ {\cal N}(-\log (-\log U(x))),$ where ${\cal N}(.)$ is the standard normal distribution. Moreover, we study the domains of attraction for these types of limit laws. Finally, some illustrative examples are given.

In this paper, we prove a refined version of a compactness lemma and we use it to establish mass-concentration for the focusing nonlinear Schrödinger equation with an inverse-square potential.

In this paper, we consider a class of pseudo-parabolic equation, which was studied extensively in recent years, and the main results on blow-up depend on the initial energy $J(u_0)$ (see (1.5)) and the mountain-pass level $d$ (see (1.7)). By using eigenfunction, we give a new blow-up result which does not depend on $J(u_0)$ and $d$. Moreover, we obtain the energy functional $J(u)$ (see (1.5)) decays exponentially. These results extend the previous studies on this equation.

Let $\field{F}$ be a number field with ring of adeles $\field{A}$, let $\field{K}$ be a quadratic extension of $\field{F}$ with ring of adeles $\field{A}_\field{K}$. Let $\eta$ be an irreducible, automorphic, cuspidal, self-dual representation of $GL_{2n}(\field{A})$ and let $\phi \in Ind_{P(\field{A})}^{Sp_{2n}(\field{A})}\eta$, where $P$ is the standard Siegel parabolic. We present an identity between the $Sp_n(\field{A}_\field{K})$-period of the residual Eisenstein series on $Sp_{2n}(\field{A})$ associated to $\phi$ and the $GL_n(\field{A}_\field{K})$- period of $\phi$. This is a non-split version of a result of Ginzburg, Soudry and Rallis.