Let $P=(p_1, p_2, \dots, p_n)$ be a polygonal chain in $\mathbb{R}^d$. The stretch factor of $P$ is the ratio between total length and distance its endpoints, $\sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|$. For parameter $c \geq 1$, we call $c$-chain if $|p_ip_j|+|p_jp_k| \leq c|p_ip_k|$, for every triple $(i,j,k)$, $1 i<j<k n$. global property: it measures how close to straight line, involves al...

The system of equations $$\begin{aligned}&u_1p_1^2 + \cdots u_sp_s^2 = 0,\\&v_1p_1^3 v_sp_s^3 0 \end{aligned}$$ has prime solutions $$(p_1, \ldots , p_s)$$ for $$s \ge 12$$ assuming that the modulo each p. This is proved via Hardy–Littlewood circle method, building on Wooley’s work corresponding over integers and recent results Vinogradov’s mean value theorem. Additionally, a set sufficient ...

We introduce a natural class of multicomponent local Poisson structures $$\mathcal P_k + \mathcal P_1$$ , where is bracket order one and P_k$$ homogeneous odd $$k$$ under the assumption that has Darboux coordinates (Darboux–Poisson bracket) nondegenerate. For such brackets, we obtain general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples), provide description...

Bresse-Timoshenko beam model with thermal, mass diffusion and theormoelastic effects is studied. We state prove the well-posedness of problem. The global existence uniqueness solution proved by using classical Faedo-Galerkin approximations along two a priori estimates. an exponential stability estimate for problem under unusual assumption, multiplier technique frictional damping in vertical dis...

The paper studies the interpolation properties of anisotropic net spaces $N_{\bar{p},\bar{q}}(M)$, where $\bar{p}=(p_1, p_2)$, $\bar{q}=(q_1, q_2)$. It is shown that following equality holds with respect to multidimensional method $$ (N_{\bar{p}_0,\bar{q}_0}(M), N_{\bar{p}_1,\bar{q}_1}(M))_{\bar{\theta},\bar{q}}=N_{\bar{p},\bar{q}}(M),\;\;\; \frac{1}{\bar{p}}=\frac{1-\bar{\theta}}{\bar{p}_0}+\f...

Abstract In this paper, we consider the Schrödinger equation involving fractional $$(p,p_1,\dots ,p_m)$$ ( p , 1 ⋯ m ) -Laplacian as follows $$\begin{aligned} (-\...