arXiv subjects
Publications and source records attributed to Mohammad Reza Mardanbeigi.
In the following text for Khalimsky $n-$dimensional space $\mathcal{K}^n$ we show self--map $f:\mathcal{K}^n\to\mathcal{K}^n$ has closed graph if and only if there exist integers $λ_1,\ldots,λ_n$ such that $f$ is a constant map with value $(2λ_1,\cdots,2λ_n)$. We also show each self--map on Khalimsky circle and Khalimsky sphere which has closed graph is a constant map. The text is motivated by examples.
For a finite discrete topological space $X$ with at least two elements, a nonempty set $Γ$, and a map $φ:Γ\toΓ$, $σ_φ:X^Γ\to X^Γ$ with $σ_φ((x_α)_{α\inΓ})= (x_{φ(α)})_{α\inΓ}$ (for $(x_α)_{α\inΓ}\in X^Γ$) is a generalized shift. In this text for $\mathcal{S}=\{σ_ψ:ψ\inΓ^Γ\}$ and $\mathcal{H}=\{σ_ψ: Γ\mathop{\rightarrow}\limits^ψΓ$ is bijective$\}$ we study proximal relations of transformation semigroups $(\mathcal{S},X^Γ)$ and $(\mathcal{H},X^Γ)$. Regarding proximal relation we prove: \[P({\mathcal S},X^Γ)=\{((x_α)_{α\inΓ},(y_α)_{α\inΓ}) \in X^Γ\times X^Γ: \existsβ\inΓ\:(x_β=y_β)\}\] and $P({\mathcal H},X^Γ)\subseteq \{((x_α)_{α\inΓ},(y_α)_{α\inΓ}) \in X^Γ\times X^Γ: \{β\inΓ:x_β=y_β\}$ is infinite~$\}\cup\{ (x,x):x\in \mathcal{X}\}$. \\ Moreover, for infinite $Γ$, both transformation semigroups $({\mathcal S},X^Γ)$ and $({\mathcal H},X^Γ)$ are regionally proximal, i.e., $Q({\mathcal S},X^Γ)=Q({\mathcal H},X^Γ)=X^Γ\times X^Γ$, also for sydetically proximal relation we have $L({\mathcal H},X^Γ)=\{((x_α)_{α\inΓ},(y_α)_{α\inΓ}) \in X^Γ\times X^Γ: \{γ\inΓ:x_γ\neq y_γ\}$ is finite$\}$.
For every frame spectral measure $ μ$, there exists a discrete measure $ ν$ as a frame measure. Since if $ μ$ is not a frame spectral measure, then there is not any general statement about the existence of frame measures $ ν$ for $ μ$, we were motivated to examine Bessel and frame measures. We construct infinitely many measures $ μ$ which admit frame measures $ ν$, and we show that there exist infinitely many frame spectral measures $ μ$ such that besides having a discrete frame measure, they admit continuous frame measures too.
Considering a finite Borel measure $ μ$ on $ \mathbb{R}^d $, a pair of conjugate exponents $ p, q $, and a compatible semi-inner product on $ L^p(μ) $, we introduce $ (p,q) $-Bessel and $ (p,q) $-frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we define notions of $ q $-Bessel and $ q$-frame in the semi-inner product space $ L^p(μ) $. Every finite Borel measure $ν$ is a $(p,q)$-Bessel measure for a finite measure $ μ$. We construct a large number of examples of finite measures $ μ$ which admit infinite $ (p,q) $-Bessel measures $ ν$. We show that if $ ν$ is a $ (p,q) $-Bessel/frame measure for $ μ$, then $ ν$ is $ σ$-finite and it is not unique. In fact, by using convolutions of probability measures, one can obtain other $ (p,q) $-Bessel/frame measures for $ μ$. We present a general way of constructing a $ (p,q) $-Bessel/frame measure for a given measure.