A ($2k+1$)$-$dimensional contact Lie algebra is one which admits a one-form $\varphi$ such that $\varphi \wedge (d\varphi)^k\ne0$. Such algebras have index one, but this not generally sufficient condition. Here we show index-one type-A seaweed are necessarily contact. Examples, together with method for their explicit construction, provided.

‎in this paper, based on [a. razani, v. rako$check{c}$evi$acute{c}$ and z. goodarzi, nonself mappings in modular spaces and common fixed point theorems, cent. eur. j. math. 2 (2010) 357-366.] a fixed point theorem for non-self contraction mapping $t$ in the modular space $x_rho$ is presented. moreover, we study a new version of krasnoseleskii's fixed point theorem for $s+t$, where $t$ is a cont...

We establish curvature estimates and a convexity result for mean convex properly embedded $\[\varphi,\vec{e}{3}]$-minimal surfaces in $\mathbb{R}^3$, i.e., $\varphi$-minimal when $\varphi$ depends only on the third coordinate of $\mathbb{R}^3$. Led by works 3-manifolds, due to White minimal surfaces, Rosenberg, Souam Toubiana stable CMC Spruck Xiao translating solitons we use compactness argume...

Suppose that $f$ is a $K$-quasiconformal self-mapping of the unit disk $\mathbb{D}$, which satisfies following: $(1)$ biharmonic equation $\Delta(\Delta f)=g$ $(g\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) boundary condition $\Delta f=\varphi$ ($\varphi\in\mathcal{C}(\mathbb{T})$ and $\mathbb{T}$ denotes circle), $(3)$ $f(0)=0$. The purpose this paper to prove Lipschitz continuos, and, furthe...

Let E be a vector lattice, $$\lambda _1,\lambda _2,\ldots ,\lambda _k$$ scalars and $$\varphi _1,\ldots ,\varphi pairwise independent regular linear functionals on E. We show that if $$k<m$$ then $$\sum _{j=1}^k\lambda _j\varphi _j^m$$ is orthogonally additive only _j$$ or $$-\varphi lattice homomorphism for each j, $$1\le j\le k$$ . Moreover, $$m\ge 2$$ , we provide an example to this result d...

Nonreciprocal responses of noncentrosymmetric quantum materials attract recent intensive interests, which is essential for the rectification function in diodes. A breakthrough discovery superconducting diode effect. The principle to enlarge effect highly desired guide design diode. Here, we study theoretically Josephson junction S/FI/S (S: $d$-wave superconductor, FI: ferromagnetic insulator) o...

We consider a skew product $F_{A} = (\sigma_{\omega}, A)$ over irrational rotation $\sigma_{\omega}(x) x + \omega$ of circle $\mathbb{T}^{1}$. It is supposed that the transformation $A: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ being $C^{1}$-map has form $A(x) R(\varphi(x)) Z(\lambda(x))$, where $R(\varphi)$ in $\mathbb{R}^{2}$ angle $\varphi$ and $Z(\lambda)= diag\{\lambda, \lambda^{-1}\}$ diagona...

Let $\varphi=\{\varphi_k\}_{k=-\infty}^\infty$ denote the extended Takenaka--Malmquist system on unit circle $\mathbb T$ and let $\sigma_{n,\varphi}(f),$ $f\in L^1(\mathbb T)$, be Fej\'er-type operator based $\varphi$, introduced by V. N. Rusak. We give convergence criteria for $\sigma_{n,\varphi}(f)$ in Banach space $X(\mathbb T):=L^p(\mathbb T)\vee C(\mathbb $p\ge 1$. Also we prove Voronovska...

–a notion of amenability for topological semigroups is introduced. a topological semigroup s iscalled johnson amenable if for every banach s -bimodule e , every bounded crossed homomorphism froms to e* is principal. in this paper it is shown that a discrete semigroup s is johnson amenable if and only if1(s) is an amenable banach algebra. also, we show that if a topological semigroup s is johns...