Let $T_{\rho}$ be an irrational rotation on a unit circle $S^{1}\simeq [0,1)$. Consider the sequence $\{\mathcal{P}_{n}\}$ of increasing partitions $S^{1}$. Define hitting times $N_{n}(\mathcal{P}_n;x,y):= \inf\{j\geq 1\mid T^{j}_{\rho}(y)\in P_{n}(x)\}$, where $P_{n}(x)$ is element $\mathcal{P}_{n}$ containing $x$. D. Kim and B. Seo in [9] proved that rescaled $K_n(\mathcal{Q}_n;x,y):= \frac{\...

Abstract Given a bounded convex subset C of Banach space X and free ultrafilter $${\mathcal {U}}$$ U , we study which points $$(x_i)_{\mathcal ( x i ) are extreme the ultrapower $$C_{\mathcal <mml...

‎let ${x_{alpha}:alphainlambda}$ be a collection of topological spaces‎, ‎and $mathcal {g}_{alpha}$ be a grill on $x_{alpha}$ for each $alphainlambda$‎. ‎we consider tychonoffrq{}s type theorem for $x=prod_{alphainlambda}x_{alpha}$ via the above grills and a natural grill on $x$ ‎related to these grills, and present a simple proof to this theorem‎. ‎this immediately yields the classical theorem...

‎let ${x_{alpha}:alphainlambda}$ be a collection of topological spaces‎, ‎and $mathcal {g}_{alpha}$ be a grill on $x_{alpha}$ for each $alphainlambda$‎. ‎we consider tychonoffrq{}s type theorem for $x=prod_{alphainlambda}x_{alpha}$ via the above grills and a natural grill on $x$ ‎related to these grills, and present a simple proof to this theorem‎. ‎this immediately yields the classical theorem...

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 $...

In this article the notions of semi weak orthogonality and semi weak factorization structure in a category $mathcal X$ are introduced. Then the relationship between semi weak factorization structures and quasi right (left) and weak factorization structures is given. The main result is a characterization of semi weak orthogonality, factorization of morphisms, and semi weak factorization structur...

In this manuscript, new algebraic and analytic aspects of the orthogonal polynomials satisfying $R_{II}$ type recurrence relation given by \begin{align*} \mathcal{P}_{n+1}(x) = (x-c_n)\mathcal{P}_n(x)-\lambda_n (x-a_n)(x-b_n)\mathcal{P}_{n-1}(x), \quad n \geq 0, \end{align*} where $\lambda_n$ is a positive chain sequence $a_n$, $b_n$, $c_n$ are sequences real or complex numbers with $\mathcal{P...