We study the expected number of zeros $$P_n(z)=\sum_{k=0}^n\eta_kp_k(z),$$ where $\{\eta_k\}$ are complex-valued i.i.d standard Gaussian random variables, and $\{p_k(z)\}$ polynomials orthogonal on unit disk. When $p_k(z)=\sqrt{(k+1)/\pi} z^k$, $k\in \{0,1,\dots, n\}$, we give an explicit formula for $P_n(z)$ in a disk radius $r\in (0,1)$ centered at origin. From our establish limiting value ze...

In this paper, we propose to extend the hierarchical bivariateHermite Interpolant to the spherical case. Let $T$ be an arbitraryspherical triangle of the unit sphere $S$ and  let $u$ be a functiondefined over the triangle $T$. For $kin mathbb{N}$, we consider aHermite spherical Interpolant problem $H_k$ defined by some datascheme $mathcal{D}_k(u)$ and which admits a unique solution $p_k$in the ...

in this paper, we propose to extend the hierarchical bivariatehermite interpolant to the spherical case. let $t$ be an arbitraryspherical triangle of the unit sphere $s$ and  let $u$ be a functiondefined over the triangle $t$. for $kin mathbb{n}$, we consider ahermite spherical interpolant problem $h_k$ defined by some datascheme $mathcal{d}_k(u)$ and which admits a unique solution $p_k$in the ...

The definition of $L$-fuzzy Q-convergence spaces is presented by Pang and Fang in 2011. However, Cartesian-closedness of the category of $L$-fuzzy Q-convergence spaces is not investigated. This paper focuses on Cartesian-closedness of the category of $L$-fuzzy Q-convergence spaces, and it is shown that  the category $L$-$mathbf{QFCS}$ of $L$-fuzzy Q-convergence spaces is Cartesian-closed.

A category $mathbf{C}$ is called Cartesian closed  provided that it has finite products and for each$mathbf{C}$-object $A$ the functor $(Atimes -): Ara A$ has a right adjoint. It is well known that the category $mathbf{TopFuzz}$  of all topological fuzzes is both complete  and cocomplete, but it is not Cartesian closed. In this paper, we introduce some Cartesian closed subcategories of this cat...

In path factorization, H. Wang [1] gives the necessary and sufficient conditions for the existence of P_k-factorization of a complete bipartite graph for k, an even integer. Further, Beiling Du [2] extended the work of H. Wang, and studied the P_2k-factorization of complete bipartite multigraph. For odd value of k the work on factorization was done by a number of researchers. P_3-factorization ...

"Let $P : \CI(M; E) \to F)$ be an order $\mu$ differential operator with coefficients $a$ and $P_k := P H^{s_0 + k +\mu}(M; k}(M; F)$. We prove polynomial norm estimates for the solution $P_0^{-1}f$ of form $$\|P_0^{-1}f\|_{H^{s_0 \mu}(M; E)} \le C \sum_{q=0}^{k} \, \| P_0^{-1} \|^{q+1} \,\|a \|_{W^{|s_0|+k}}^{q} f \|_{H^{s_0 - q}},$$ (thus in higher Sobolev spaces, which amounts also to a para...

‎let $g$ be a graph and $chi^{prime}_{aa}(g)$ denotes the minimum number of colors required for an‎ ‎acyclic edge coloring of $g$ in which no two adjacent vertices are incident to edges colored with the same set of colors‎. ‎we prove a general bound for $chi^{prime}_{aa}(gsquare h)$ for any two graphs $g$ and $h$‎. ‎we also determine‎ ‎exact value of this parameter for the cartesian product of ...