For any given integer $$r\geqslant 3$$ , let $$k=k(n)$$ be an with $$r\leqslant k\leqslant n$$ . A hypergraph is r-uniform if each edge a set of r vertices, and said to linear two edges intersect in at most one vertex. Let $$A_1,\ldots ,A_k$$ k-partition [n] $$|A_i|=n_i\geqslant 1$$ An H called k-partite e satisfies $$|e\cap A_i|\leqslant for $$1\leqslant i\leqslant k$$ In this paper, the numbe...

Let $$G_n$$ be a non-extensible, flexible closed curve of length n in the 3-space $${\mathbb {R}}^3$$ with particles $$A_1$$ ,..., $$A_n$$ evenly fixed (according to arc ) on curve. $$f:(0, \infty )\rightarrow {\mathbb {R}}$$ an increasing and continuous function. Define energy function $$\begin{aligned} E^f_n(G_n)= \sum _{p< q} f(|A_pA_q|), \end{aligned}$$ where $$|A_pA_q|$$ is distance betwee...

Abstract Given a finite set $A \subseteq \mathbb{R}^d$ , points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$ -hole in A if they are the vertices of convex polytope, which contains no its interior. We construct arbitrarily large point sets general position $\mathbb{R}^d$ having holes size $O(4^dd\log d)$ or more. This improves previously known upper bound order $d^{d+o(d)}$ due to Valtr. The b...

Recently there has been considerable interest in studying the length and depth of finite groups, algebraic groups Lie groups. In this paper we introduce study similar notions for algebras. Let $k$ be a field let $A$ an associative, not necessarily unital, algebra over $k$. An unrefinable chain is subalgebras $A=A_0>A_1>...>A_t=0$ some integer $t$ where each $A_i$ maximal subalgebra $A_{i-1}$. T...

For a polynomial F(t, A1, …, An) ∈ $$\mathbb{F}$$ p[t, An] (p being prime number) we study the factorization statistics of its specializations $$F\left({t,{a_1}, \ldots ,{a_n}} \right) \in {\mathbb{F}_p}\left[t \right]$$ with (a1, an) S, where $$S \subset \mathbb{F}_p^n$$ is subset, in limit p → ∞ and deg F fixed. We show that for sufficiently large regular subset , e.g., product n intervals le...

There is a well known result from the Fierz-Pauli (FP) theory in de Sitter background that existence of lower bound for mass $m$ spin 2 particle, Higuchi bound. It establishes $m^2\geq 2H^2$, where $H$ Hubble parameter, order to presents no ghost-like instabilities. In this sense, should be unacceptable high fulfill condition at time inflationary epoch Universe, posing difficulty conciliate FP ...

We classify the coefficients $$(a_1, \ldots , a_n) \in {\mathbb {F}}_q[t]^n$$ that appear in a linear relation $$\sum _{i=1}^n a_i \gamma _i =0$$ among Galois conjugates $$\gamma \overline{{\mathbb {F}}_q(t)}$$ . call such an n-tuple Smyth tuple. Our main theorem gives affirmative answer to function field analogue of 1986 conjecture (J Numb Theory 23, 243–254, 1986) over $${\mathbb {Q}}$$ showe...

Abstract The natural maximal and minimal functions commute pointwise with the logarithm on $$A_\infty $$ A ∞ . We use this observation to characterize spaces $$A_1$$ 1 $$RH_\infty R H metric measu...

We consider the Cauchy problem for focusing nonlinear Schrödinger equation with initial data approaching different plane waves $$A_j\mathrm {e}^{\mathrm {i}\phi _j}\mathrm {e}^{-2\mathrm {i}B_jx}$$ , $$j=1,2$$ as $$x\rightarrow \pm \infty $$ . The goal is to determine long-time asymptotics of solution, according value $$\xi =x/t$$ general situation analyzed in a recent paper where we develop Ri...

Abstract Consider the primitive equations on $$\mathbb {R}^2\times (z_0,z_1)$$ R2×(z0,z1</mml:...