arXiv subjects
Publications and source records attributed to Fatemah Ayatollah Zadeh Shirazi.
In infinite topological Fort space $X$, for nonempty subsets $C,D$ of $X$ in the following text we answer to this question "Is there any $λ$ and Top--design $C-(X,D,λ)$ of type $i$?" for $i=1,2,3,4$. We prove there exist $λ$ and $C-(X,D,λ)$, Top--design of type 2 (resp. type 4) if and only if $C$ can be embedded into $D$.
In the following text for cardinal number $τ>0$, and self--map $φ:τ\toτ$ we show the generalized shift operator $σ_φ(\ell^2(τ))\subseteq\ell^2(τ)$ (where $σ_φ((x_α)_{α<τ})=(x_{φ(α)})_{α<τ}$ for $(x_α)_{α<τ}\in{\mathbb C}^τ$) if and only if $φ:τ\toτ$ is bounded and in this case $σ_φ\restriction_{\ell^2(τ)}:\ell^2(τ)\to\ell^2(τ)$ is continuous, consequently $σ_φ\restriction_{\ell^2(τ)}:\ell^2(τ)\to\ell^2(τ)$ is a compact operator if and only if $τ$ is finite.
In the following text for arbitrary $X$ with at least two elements, nonempty set $Γ$ and self-map $φ:Γ\toΓ$ we prove the set-theoretical entropy of generalized shift $σ_φ:X^Γ\to X^Γ$ ($σ_φ((x_α)_{α\inΓ})=(x_{φ(α)})_{α\inΓ}$ (for $(x_α)_{α\inΓ}\in X^Γ$)) is either zero or infinity, moreover it is zero if and only if $φ$ is quasi-periodic. We continue our study on contravariant set-theoretical entropy of generalized shift and motivate the text using counterexamples dealing with algebraic, topological, set-theoretical and contravariant set-theoretical positive entropies of generalized shifts.
In the following text we prove that for finite discrete $X$ with at least two elements and infinite $Γ$, the generalized shift transformation semigroup $({\mathcal S},X^Γ)$ is equicontinuous (resp. has at least an equicontinuous point, is not sensitive) if and only if for all $w\inΓ$, $\{φ(w):σ_φ\in{\mathcal S}\}$ is finite. We continue our study regarding distality and expansivity of $({\mathcal S},X^Γ)$.
In the following text we prove that there exists a Fort transformation group with indicator sequence $(p_0,\ldots,p_n)$ if and only if $0=p_0\leq p_1\leq\cdots\leq p_n=1$, moreover we characterize all possible indicator topological spaces of Fort transformation groups.The text will study indicator sequences and indicator topologies of Fort transformation semigroups too.
For every finite-to-one map $λ:Γ\toΓ$ and for every abelian group $K$, the generalized shift $σ_λ$ of the direct sum $\bigoplus_ΓK$ is the endomorphism defined by $(x_i)_{i\inΓ}\mapsto(x_{λ(i)})_{i\inΓ}$. In this paper we analyze and compute the algebraic entropy of a generalized shift, which turns out to depend on the cardinality of $K$, but mainly on the function $λ$. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples.