In this paper, we define and study the notion of the real-valued functions on a frame $L$. We show that $F(L) $, consisting of all frame homomorphisms from the power set of $mathbb{R}$ to a frame $ L$, is an $f$-ring, as a generalization of all functions from a set $X$ into $mathbb R$. Also, we show that $F(L) $ is isomorphic to a sub-$f$-ring of $mathcal{R}(L)$, the ring of real-valued continu...

Descriptive set theory was originally developed on Polish spaces. It was later extended to $\omega$-continuous domains [Selivanov 2004] and recently to quasi-Polish spaces [de Brecht 2013]. All these spaces are countably-based. Extending descriptive set theory and its effective counterpart to general represented spaces, including non-countably-based spaces has been started in [Pauly, de Brecht ...

This note provides a conditional Berry-Esseen bound for the sum of martingale difference sequence $\{X_i\}_{i=1}^n$ in $\mathbb{R}^d$, $d\ge 1$, adapted to filtration $\{\mathcal{F}_i\}_{i=1}^n$. We approximate distribution $S=\sum_{i=1}^n X_i$ given some $\sigma$-field $\mathcal{F}_0\subset \mathcal{F}_1$ by that mean-zero normal random vector having same variance $\mathcal{F}_0 $ as $S$. Assu...

Abstract Let $f$ be a degree $d$ bicritical rational map with critical point set $\mathcal{C}_f$ and value $\mathcal{V}_f$. Using the group $\textrm{Deck}(f^k)$ of deck transformations $f^k$, we show that if $g$ is shares an iterate $f$, then $\mathcal{C}_f = \mathcal{C}_g$ $\mathcal{V}_f \mathcal{V}_g$. this, two maps even share iterate, they second both belong to symmetry locus maps.

Let $\mathcal{F}$ denote a singular holomorphic foliation on $\mathbb{P}^2$ having finite automorphism group $\mbox{aut}(\mathcal{F})$. Fixed the degree of $\mathcal{F}$, we determine maximal value that $|\mbox{aut}(\mathcal{F})|$ can take and explicitly exhibit all foliations attaining this value. Furthermore, classify with large but group.

Given a family of graphs $\mathcal{F}$, we define the $\mathcal{F}$-saturation game as follows. Two players alternate adding edges to an initially empty graph on $n$ vertices, with only constraint being that neither player can add edge creates subgraph in $\mathcal{F}$. The ends when no more be added graph. One wishes end quickly possible, while other prolong game. We let $\textrm{sat}_g(n,\mat...

Let $Omega_X$ be a bounded, circular and strictly convex domain of a Banach space $X$ and $mathcal{H}(Omega_X)$ denote the space of all holomorphic functions defined on $Omega_X$. The growth space $mathcal{A}^omega(Omega_X)$ is the space of all $finmathcal{H}(Omega_X)$ for which $$|f(x)|leqslant C omega(r_{Omega_X}(x)),quad xin Omega_X,$$ for some constant $C>0$, whenever $r_{Omega_X}$ is the M...