In the present paper, we study normalized solutions with least energy to following system: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1u=\mu _1 |u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &{}\hbox {in}~{{\mathbb {R}}^N},\\ v+\lambda _2v=\mu _2 |v|^{q-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v\quad \int _{{{\mathbb {R}}^N}}u^2=a_1^2\quad \hbox {and}\quad {R}}^N}}v^2=a_2^2, \end{a...

For a graph G and integers $$a_i\ge 1$$ , the expression $$G \rightarrow (a_1,\ldots ,a_r)^v$$ means that for any r-coloring of vertices there exists monochromatic $$a_i$$ -clique in some color $$i \in \{1,\ldots ,r\}$$ . The vertex Folkman numbers are defined as $$F_v(a_1,\ldots ,a_r;H) = \min \{|V(G)| : G$$ is H-free ,a_r)^v\}$$ where H graph. Such have been extensively studied $$H=K_s$$ with...

Let $f(X_1,\dots, X_n)$ be a nonzero multilinear noncommutative polynomial. If $A$ is unital algebra with surjective inner derivation, then every element in can written as $f(a_1,\dots,a_n)$ for some $a_i\in A$.

Abstract There has been recent interest in a hybrid form of the celebrated conjectures Hardy–Littlewood and Chowla. We prove that for any $k,\ell \ge 1$ distinct integers $h_2,\ldots ,h_k,a_1,\ldots ,a_\ell $ , we have: $$ \begin{align*}\sum_{n\leq X}\mu(n+h_1)\cdots \mu(n+h_k)\Lambda(n+a_1)\cdots\Lambda(n+a_{\ell})=o(X)\end{align*} all except $o(H)$ values $h_1\leq H$ so long as $H\geq (\log X...

We prove bounds for the number of solutions to$$a_1 + \dots a_k = a_1' a_k'$$ over $N$-element sets reals, which are sufficiently convex or near-convex. A near-convex set will be image a with small additive doubling under function many strictly monotone derivatives. show, roughly, that every time terms in equation is doubled, an additional saving $1$ exponent trivial bound $N^{2k-1}$ made, star...

For a compact connected Riemannian n-manifold $$(\Omega ,g)$$ with smooth boundary, we explicitly calculate the first two coefficients $$a_0$$ and $$a_1$$ of asymptotic expansion $$\sum _{k=1}^\infty \mathrm{{e}}^{-t \tau _k^{\mp }}= a_0t^{-n/2} {\mp } a_1 t^{-(n-1)/2} +O(t^{1-n/2})$$ as $$t\rightarrow 0^+$$ , where $$\tau ^-_k$$ (respectively, ^+_k$$ ) is k-th Navier–Lamé eigenvalue on $$\Omeg...