By applying the Euler–Rayleigh method to a specific representation of Jacobi polynomials as hypergeometric functions, we obtain new bounds for their largest zeros. In particular, derive upper and lower bound $$1-x_{nn}^2(\lambda )$$ , with $$x_{nn}(\lambda being zero n-th ultraspherical polynomial $$P_n^{(\lambda )}$$ . For every fixed $$\lambda >-1/2$$ limit ratio our does not exceed 1.6. This...

Ramsey proved that for every positive integer $n$, sufficiently large graph contains an induced $K_n$ or $\overline{K}_n$. Among the many extensions of Ramsey's Theorem there is analogue connected graphs: $K_n$, $K_{1,n}$, $P_n$. In this paper, we establish 2-connected graphs. particular, prove exceeding two, one following as subgraph: a subdivision $K_{2,n}$, $K_{2,n}$ with edge between two ve...

The crossing number of a graph G is the minimum number of pairwise intersections of edges in a drawing of G. In this paper, we study the crossing numbers of Km ×Pn and Km × Cn.

По выбранной ортогональной последовательности полиномов $\{p_n(s)\}_{n=0}^{\infty}$, которая имеет дискретный спектр, строится энергетический спектр $E_k=f(s_k)$, где $\{s_k\}$ - точки конечного или бесконечного дискретного спектра полиномов. С помощью подхода к квантовой механике, который основан не на функциях потенциала, а ортогональных полиномах, зависящих от энергии, построена локальная чи...

We study a class of stochastic processes the type $\frac{d^n x}{dt^n}= v_0\, \sigma(t)$ where $n>0$ is positive integer and $\sigma(t)=\pm 1$ represents an `active' telegraphic noise that flips from one state to other with constant rate $\gamma$. For $n=1$, it reduces standard run tumble process for active particles in dimension. This can be analytically continued any including non-integer valu...