Let $(X,T)$ be a topological dynamical system, $n\geq 2$ and $\mathcal{F}$ Furstenberg family of subsets $\mathbb{Z}_+$. is called broken $\mathcal{F}$-$n$-sensitive if there exist $\delta>0$ $F\in\mathcal{F}$ such that for every opene (non-empty open) subset $U$ $X$ $l\in\mathbb{N}$, $x_1^l,x_2^l,\dotsc,x_n^l\in U$ $m_l\in \mathbb{Z}_+$ satisfying $d(T^k x_i^l, T^k x_j^l)> \delta,\ \forall 1\l...

Abstract For a braided fusion category $\mathcal{V}$, $\mathcal{V}$-fusion is $\mathcal{C}$ equipped with monoidal functor $\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$. Given fixed $(\mathcal{C}, \mathcal{F})$ and $G$-graded extension $\mathcal{C}\subseteq \mathcal{D}$ as an ordinary category, we characterize the enrichments $\widetilde{\mathcal{F}}:\mathcal{V} Z(\mathcal{D})$ of $\mathcal{D}$ ...

Let $\mathcal{A}= \{A_t \}_{t \in G}$ and $\mathcal{B}= \{B_t \}_{t\in be $C^*$-algebraic bundles over a finite group $G$. $C=\oplus_{t G}A_t$ $D=\oplus_{t\in G}B_t$. Also, let $A=A_e$ $B=B_e$, where $e$ is the unit element in We suppose that $C$ $D$ are unital $A$ $B$ have elements $D$, respectively. In this paper, we shall show if there an equivalence $\mathcal{A}-\mathcal{B}$-bundle $G$ with...

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...

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 ...