We study the topological centers of $nth$ dual of Banach $mathcal{A}$-modules and we extend some propositions from Lau and "{U}lger into $n-th$ dual of Banach $mathcal{A}-modules$ where $ngeq 0$ is even number. Let   $mathcal{B}$   be a Banach  $mathcal{A}-bimodule$. By using some new conditions, we show that $ Z^ell_{mathcal{A}^{(n)}}(mathcal{B}^{(n)})=mathcal{B}^{(n)}$ and $ Z^ell_{mathcal{B}...

For two metric spaces $\mathbb X$ and $\mathcal Y$, the chromatic number $\chi(\mathbb X;\mathcal Y)$ of with forbidden Y$ is smallest $k$ such that there a coloring points colors no monochromatic copy Y$. In this paper, we show for each finite space $\mathcal{M}$ contains at least value $\chi\left(\mathbb R^n_\infty; \mathcal M \right)$ grows exponentially $n$. We also provide explicit lower u...

We extend the notion of Rauzy induction interval exchange transformations to case toral $\mathbb{Z}^2$-rotation, i.e., $\mathbb{Z}^2$-action defined by rotations on a 2-torus. If $\mathcal{X}_{\mathcal{P},R}$ denotes symbolic dynamical system corresponding partition $\mathcal{P}$ and $R$ such that is Cartesian sub-domain $W$, we express 2-dimensional configurations in as image under $2$-dimensi...

We study the relationship between metric thickenings and simplicial complexes associated to coverings of spaces. Let $${\mathcal {U}}$$ be a cover separable space X by open sets with uniform diameter bound. The Vietoris complex {V}}({\mathcal {U}})$$ contains all simplices vertex set contained in some $$U \in {\mathcal , thickening {V}}^\textrm{m}({\mathcal is probability measures support equip...

Let $mathcal{H}$ and $mathcal{K}$ be infinite dimensional Hilbert spaces, while $mathcal{B(H)}$ and $mathcal{B(K)}$ denote the algebras of all linear bounded operators on $mathcal{H}$ and $mathcal{K}$, respectively. We characterize the forms of additive mappings from $mathcal{B(H)}$ into $mathcal{B(K)}$ that preserve the nonzero idempotency of either Jordan products of operators or usual produc...

assume that $mathbb{d}$ is the open unit disk. applying ozaki's conditions, we consider two classes of locally univalent, which denote by $mathcal{g}(alpha)$ and $mathcal{f}(mu)$ as follows begin{equation*}  mathcal{g}(alpha):=left{fin mathcal{a}:mathfrak{re}left( 1+frac{zf^{prime prime }(z)}{f^{prime }(z)}right)

For trace class operators A , B ? <mml:mi class="MJX-te...

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...

Let $d(n)$ and $d^{\ast}(n)$ be the numbers of divisors unitary integer $n\geq1$. In this paper, we prove that \[ \underset{n\in\mathcal{B}}{\underset{n\leq x}{\sum}}\frac{d(n)}{d^{\ast}% (n)}=\frac{16\pi% %TCIMACRO{\U{b2}}% %BeginExpansion {{}^2}% %EndExpansion }{123}\underset{p}{\prod}(1-\frac{1}{2p% }+\frac{1}{2p^{3}})x+\mathcal{O}\left( x^{\frac{\ln8}{\ln10}+\varepsilon }\right) ,~\left( x\...