For $0\leq\alpha\leq 1,$ let $H_{\alpha}(x,y)$ be the convex weighted harmonic mean of $x$ and $y.$ We establish differential subordination implications of the form $$H_{\alpha}(p(z),p(z)\Theta(z)+zp'(z)\Phi(z))\prec h(z)\Rightarrow p(z)\prec h(z),$$ where $\Phi$, $\Theta$ are analytic functions and $h$ is a univalent function satisfying some special properties. Further, we prove differential subordination implications involving a combination of three classical means. As an application, we generalize many existing results and obtain sufficient conditions for starlikeness and univalence.

We give a unified approach to handle the problem of extending functions with values in a locally convex Hausdorff space $E$ over a field $\mathbb{K}$, which have weak extensions in a space $\mathcal{F}(\Omega,\mathbb{K})$ of scalar-valued functions on a set $\Omega$, to functions in a vector-valued counterpart $\mathcal{F}(\Omega,E)$ of $\mathcal{F}(\Omega,\mathbb{K})$. The results obtained are based upon a representation of vector-valued functions as linear continuous operators and extend results of Bonet, Frerick, Gramsch and Jordá. In particular, we apply them to obtain a sequence space representation of $\mathcal{F}(\Omega,E)$ from a known representation of $\mathcal{F}(\Omega,\mathbb{K})$.

Our goal is firstly to derive a revision of the basic theory of $D_{-\omega}$-semi-classical orthogonal polynomials and secondly to study the discrete semiclassical character of two standard perturbations of a $D_{\omega}$-semiclassical form into detail. Some illustrative examples are given.

We construct an action of the Thompson group $F$ on a compact space built from pairs of infinite, binary rooted trees. The action arises as an $F$-equivariant compactification of the action of $F$ by translations on one of its homogeneous spaces, $F/H_2$, corresponding to a certain subgroup $H_2$ of $F$. The representation of $F$ on the Hilbert space $\ell^2(F/H_2)$ is faithful on the complex group algebra $\mathbb{C}[F]$.

The aim of this paper is to study global dynamics of $C^\infty$-smooth slow-fast systems on the $2$-torus of class $C^\infty$ using geometric singular perturbation theory and the notion of slow divergence integral. Given any $m\in\mathbb{N}$ and two relatively prime integers $k$ and $l$, we show that there exists a slow-fast system $Y_{\epsilon}$ on the $2$-torus that has a $2m$-link of type $(k,l)$, i.e. a (disjoint finite) union of $2m$ slow-fast limit cycles each of $(k,l)$-torus knot type, for all small $\epsilon>0$. The $(k,l)$-torus knot turns around the $2$-torus $k$ times meridionally and $l$ times longitudinally. There are exactly $m$ repelling limit cycles and $m$ attracting limit cycles. Our analysis: a) proves the case of normally hyperbolic singular knots, and b) provides sufficient evidence to conjecture a similar result in some cases where the singular knots have regular nilpotent contact with the fast foliation.

It is proved that, under GCH, for any Hausdorff space $X$, we have the inequality $|X| \leq sh(X)^{\pi\chi(X) \cdot \psi_c(X)}$ and therefore $|X\leq sh(X)^{\chi(X)}$; here $sh(X)=\min\{\kappa\geq \omega: \kappa^+$ is a caliber of $X\}$ is the Shanin number of $X$. If $X$ is regular, and GCH holds, then $d(X) \leq sh(X)^{t(X)\cdot \pi\chi(X)}$. We also establish, in ZFC, that $|X| \leq wL(X)^{\CL\Delta(X)\cdot 2^{\pi\chi(X)}}$ whenever $X$ is a Urysohn space.

This paper considers von Neumann-Schatten p-Bessel sequences, von Neumann-Schatten p-frames and von Neumann-Schatten q-Riesz bases. Sequences, consisting of operators from a separable Banach space $\mathcal{X}$ into the von Neumann-Schatten p-class $\mathcal{C}_p$, which are von Neumann-Schatten p-Bessel sequences are characterized. It is shown that the set of all von Neumann-Schatten p-Bessel sequences can be considered as a Banach algebra. Also, some necessary and sufficient conditions for a reflexive Banach space to possess von Neumann-Schatten p-frames or von Neumann-Schatten q-Riesz bases are presented. Moreover, the stability of von Neumann-Schatten p-frames and von Neumann-Schatten q-Riesz bases under different kinds of perturbations is studied, especially some sufficient conditions for the preservation of the frame property under perturbations are derived.