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...

The purpose of this note is to characterize all the sequences orthogonal polynomials \((P_n)_{n\ge 0}\) such that $$\begin{aligned} \frac{\triangle }{\mathbf{\triangle } x(s-1/2)}P_{n+1}(x(s-1/2))=c_n(\triangle +2\,\mathrm {I})P_n(x(s-1/2)), \end{aligned}$$where \(\,\mathrm {I}\) identity operator, x defines a class lattices with, generally, nonuniform step-size, and \(\triangle f(s)=f(s+1)-f(s...

Let$mathbb{C}^n$ be the space of $n$ complex variables. Let$Omega_{n,p_2,ldots,p_n}$ be a complete Reinhardt on$mathbb{C}^n$. The Minkowski functional on complete Reinhardt$Omega_{n,p_2,ldots,p_n}$ is denoted by $rho(z)$. The concept ofspirallike mapping of type $beta$ and order $alpha$ is defined.So, the concept of the strong and almost spirallike mappings o...

Abstract We present a modification of the Depth first search algorithm, suited for finding long induced paths. use it to give simple proofs following results. show that size-Ramsey number paths satisfies $\hat{R}_{\mathrm{ind}}(P_n)\leq 5 \cdot 10^7n$ , thus giving an explicit constant in linear bound, improving previous bound with large from regularity lemma argument by Haxell, Kohayakawa and ...

Motivated by classical nontransitivity paradoxes, we call an $n$-tuple $(x_1,\dots,x_n) \in[0,1]^n$ \textit{cyclic} if there exist independent random variables $U_1,\dots, U_n$ with $P(U_i=U_j)=0$ for $i\not=j$ such that $P(U_{i+1}>U_i)=x_i$ $i=1,\dots,n-1$ and $P(U_1>U_n)=x_n$. We the tuple $(x_1,\dots,x_n)$ \textit{nontransitive} it is cyclic in addition satisfies $x_i>1/2$ all $i$. Let $p_n$...

Abstract Let $$ (P_n)_{n\ge 0}$$ ( P n ) ? 0 be the sequence of Perrin numbers defined by ternary relation P_0=3 = 3 , P_1=0 1 P...

We study the orbit space $$X_n = G_{n,2}/T^n$$ of standard action compact torus $$T^n$$ on complex Grassmann manifold $$G_{n,2}$$ . describe structure set critical points $$\operatorname{Crit}G_{n,2}$$ generalized moment map $$\mu_n \colon\, G_{n,2}\to \mathbb{R}^n$$ whose image is a hypersimplex $$\Delta_{n,2}$$ The canonical projection $$G_{n,2}\to X_n$$ maps $$\operatorname{Crit} G_{n,2}$$ t...