We study systematically when $C_p(X)$ has a topological property $\mathcal{P}$ if $C_p(X)$ is discretely $\mathcal{P}$, i.e., the set $\overline {D}$ has $\mathcal{P}$ for every discrete subspace $D\subset C_p(X)$. We prove that it is independent of ZFC whether discrete metrizability of $C_p(X)$ implies its metrizability for a compact space $X$. We show that it is consistent with ZFC that countable tightness and Lindelöf $\Sigma$-property are not discretely reflexive in spaces $C_p(X)$. It is also established that a space $X$ must be countable and discrete if $C_p(X)$ is discretely Čech-complete. If $C_p(X)$ is discretely $\sigma$-compact then $X$ has to be finite.

Let $M^{2n+1}$ be an almost Kenmotsu manifold with the characteristic vector field belonging to the $(k,\mu)'$-nullity distribution. We prove that the curvature tensor of $M^{2n+1}$ is harmonic if and only if $M^{2n+1}$ is locally isometric to either a product space $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^n$, or an Einstein warped product $C\times_{f}N^{2n}$ of an open interval and a Ricci-flat almost Kähler manifold.

This is a synthetic presentation of several barrelledness notions, in locally convex algebras. These are characterized, as in locally convex spaces, via (algebra) seminorms. This approach reveals a new notion of barrelledness. The latter shows to be what is needed to have meaningful statements in locally uniformly convex algebras.

We estimate from below the lower density of the set of prime numbers $p$ such that $p-1$ has a prime factor of size at least $p^c$, where $1/4 \le c \leq 1/2$. We also establish upper and lower bounds on the counting function of the set of positive integers $n\le x$ with exactly $k$ prime factors, counted with or without multiplicity, such that the largest prime factor of ${\text{\rm gcd}}(p-1: p\mid n)$ exceeds $n^{1/2k}$.

Symmetric cohomology of groups, defined by M. Staic in [2], is similar to the way one defines the cyclic cohomology for algebras. We show that there is a well-defined restriction, conjugation and transfer map in symmetric cohomology, which form a Mackey functor under a restriction. Some new properties for the symmetric cohomology group using normalized cochains are also given.

Let $\varphi$ be a holomorphic self-map of the unit disk $\mathbb{U}:=\{z\in \mathbb{C}: |z| < 1\}$, and the composition operator with symbol $\varphi$ is defined by $C_\varphi f=f \circ \varphi.$ In this paper we present formula for the adjoint of composition operators in some Hilbert spaces of analytic functions, in the case that $\varphi$ is a finite Blaschke product or a rational univalent holomorphic self-map of the unit disk $\mathbb{U}$.

In this short note we show that there exist uncountably generated algebras every non-zero element of which is a smooth function having uncountably many zeros. This result complements some recent ones by Enflo et al.

In this paper, we establish an exponential inequality for negatively associated random variables, which improves known results. These results concern in particular the previous work of Kim and Kim (2007), Sung (2009 and 2011) as well as Xing et al. (2009). We also try to put our results in a more general context by showing where they may touch the sphere of interest of other authors.

Embedding Calculus, as described by Weiss, is a calculus of functors, suitable for studying contravariant functors from the poset of open subsets of a smooth manifold $M$, denoted $\mathcal{O}(M)$, to a category of topological spaces (of which the functor $Emb(-,N)$ for some fixed manifold $N$ is a prime example). Polynomial functors of degree $k$ can be characterized by their restriction to $\mathcal{O}_k(M)$, the full subposet of $\mathcal{O}(M)$ consisting of open sets which are a disjoint union of at most $k$ components, each diffeomorphic to the open unit ball. In this work, we replace $\mathcal{O}_k(M)$ by more general subposets and see that we still recover the same notion of polynomial cofunctor.

The purpose of this work is to establish a new generalization of Darbo type fixed point theorem using the concept of the so-called $\alpha$-admissibility and the Schauder fixed point theorem. We also include an example which shows that our results are applicable where the previous ones are not. Moreover, we apply our main result to the problem of existence of solutions for a class of infinite systems of second order differential equations.

IWe extend the nonstandard hull construction of Luxemburg and we show that the proposed extension covers various other constructions such as the tracial and the weak nonstandard hull. We provide a novel proof that the tracial nonstandard hull of an internal von Neumann algebra is itself a von Neumann algebra. We prove that some extensively studied properties of $C^*$--algebras are preserved and reflected by the nonstandard hull construction. Eventually, we show how to obtain hull--like representations of the two $K$--groups associated to the nonstandard hull of an internal $C^*$--algebra.

In this article we consider a family of orientation preserving expanding maps on the circle. We associate to each member of this family a geodesic lamination on the disc, endowed with a transversal measure. The map on the circle induces an expanding dynamical systems on the lamination. We explore relations between the geometry of the lamination and the symbolic dynamics of the circle map. We present a list of open problems.

Let $\Omega$ be a measurable space and $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra which is also a second dual space. On the set of functions from $\Omega$ into $\mathcal{M}$, it is supposed to give a criterion to illustrate $\tau$-measurability where $\tau$ runs over some well-known locally convex topologies on $\mathcal{M}$ which is stronger than weak operator topology and weaker than the Arens-Mackey topology.

We discuss the curvature ellipse of minimal surfaces in the product space $N^3(c)\times R$, where $N^3(c)$ is the $3$-dimensional simply connected space form of constant curvature $c$.