We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every countably subcompact. (ii) A $\mathbf{X}=(X,T)$ compact if only it relative to $T$. (iii) For $\mathbf{X}=(X,T)$, the following are equivalent: \noindent(a) $\mathbf{X}$ compact; \noindent(b) for open filter $\mathcal{F}$ of $\mathbf{X}$, $\bigcap \{\overline{F}\colon F\in \mathcal{F}\}\neq \emptyset $...

Motivated by a recent investigation of Costakis et al. on the notion recurrence in linear dynamics, we study various stronger forms for operators, particular that frequent recurrence. We study, among other things, relationship between type and corresponding hypercyclicity, influence power boundedness, interplay spectral properties. obtain, particular, Ansari- L\'eon-M\"uller-type theorems $\mat...

Abstract In this paper we prove a result on the effective generation of pluri-canonical linear systems foliated surfaces general type. Fix function {P:\mathbb{Z}_{\geq 0}\to\mathbb{Z}} , then there exists an integer {N>0} such that if {(X,{\mathcal{F}})} is canonical or nef model foliation type with Hilbert polynomial {\chi(X,{\mathcal{O}}_{X}(mK_{\mathcal{F}}))=P(m)} for all {m\in\mathbb{Z}...

We prove that a function in several variables is the local Zygmund class $\mathcal{Z}^{m,1}$ if and only its composite with every smooth curve of $\mathcal{Z}^{m,1}$. This complements well-known analogous result for Hölder–Lipschitz classes $\mathcal{C}^{m,\alpha}$, which we reprove along way. demonstrate these results generalize to mappings between Banach spaces use them study regularity super...

Let $\mathcal{F}$ be an antichain of finite subsets $\mathbb{N}$. How quickly can the quantities $|\mathcal{F}\cap 2^{[n]}|$ grow as $n\to\infty$? We show that for any sequence $(f_n)_{n\ge n_0}$ positive integers satisfying $\sum_{n=n_0}^\infty f_n/2^n \le 1/4$, $f_{n_0}=1$ and $f_n\le f_{n+1}\le 2f_n$, there exists infinite $\mathbb{N}$ such 2^{[n]}| \geq f_n$ all $n\ge n_0$. It follows $\var...

We say that a class $${\mathcal {G}}$$ of analytic functions f the form $$f(z)=\sum _{n=0}^{\infty } a_{n}z^{n}$$ in unit disk $${\mathbb {D}}:=\{z\in {\mathbb {C}}: |z|<1\}$$ satisfies Bohr phenomenon if for largest radius $$R_{f}<1$$ , following inequality $$\begin{aligned} \sum \limits _{n=1}^{\infty |a_{n}z^{n}| \le d(f(0),\partial f({\mathbb {D}}) ) \end{aligned}$$ holds $$|z|=r\le R_{f}$$...

Azéma associated with an honest time L the supermartingale $Z_{t}^{L}=\mathbb{P}[L>t|\mathcal{F}_{t}]$ and established some of its important properties. This supermartingale plays a central role in the general theory of stochastic processes and in particular in the theory of progressive enlargements of filtrations. In this paper, we shall give an additive characterization for these supermarting...

Abstract In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, show that if there exists relatively compact set K (in weak operator topology) such $0$ is an element its closure but it not norm-closed convex hull, then can guarantee does attain norm. This ...