The weak commutativity group $\chi(G)$ is generated by two isomorphic groups $G$ and $G^{\varphi }$ subject to the relations $[g,g^{\varphi}]=1$ for all $g \in G$. an extension of $D(G) = [G,G^{\varphi}]$ $G \times We prove that if a finite solvable derived length $d$, then $\exp(D(G))$ divides $\exp(G)^{d}$ $|G|$ odd $2^{d-1}\cdot \exp(G)^{d}$ even. Further, $p$ prime $p$-group class $p-1$, $\...

(3) JC'C' and (3) JHNHα couplings are related to the intervening backbone torsion angle ${\varphi }$ by standard Karplus equations. Although these couplings are known to be affected by parameters other than ${\varphi }$, including H-bonding, valence angles and residue type, experimental results and quantum calculations indicate that the impact of these latter parameters is typically very small....

Emulsions of water in as-recovered native crude oils of diverse geographical origin evidently possess some common morphological features. At low volume fractions varphi of water, the viscosity behavior of emulsions is governed by the presence of flocculated clusters of water droplets, whereas characteristic tight gels, composed of visually monodisperse small droplets, are responsible for the vi...

The scalar field theory with potential $V(\ensuremath{\varphi})=\frac{1}{2}{m}^{2}{\ensuremath{\varphi}}^{2}\ensuremath{-}\frac{1}{4}g{\ensuremath{\varphi}}^{4}$ ($g>0$) is ill defined as a Hermitian but in non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric framework it well defined, and has positive real energy spectrum for the case of spacetime dimension $D=1$. While methods used literature ...

We report an experimental investigation of the caging motion in a uniformly heated granular fluid for a wide range of filling fractions, varphi. At low varphi the classic diffusive behavior of a fluid is observed. However, as varphi is increased, temporary cages develop and particles become increasingly trapped by their neighbors. We statistically analyze particle trajectories and observe a num...

Let $K$ be an algebraically closed field of characteristic zero and let $G$ a finitely generated subgroup the multiplicative group $K$. We consider $K$-valued sequences form $a_n:=f(\varphi^n(x_0))$, where $\varphi\colon X\to X$ $f\colon X\to\mathbb{P}^1$ are rational maps defined over $x_0\in is point whose forward orbit avoids indeterminacy loci $\varphi$ $f$. Many classical from number theor...

Abstract In this paper, using the notion of w -distance in metric spaces, we introduce two types nonlinear contractions, $(\iota ,\psi ,\varphi ,\phi ,p)$ ( ι , ψ φ ϕ p ) and rational- , \var...

The purpose of the paper is to analyze g-frames form \(\{\varphi T^{i} \in B(\mathcal {H},\mathcal {K})\}_{i=0}^\infty \), where \(T\in {H})\) and \(\varphi {K})\), discuss properties operator T. We consider stability g-Riesz sequences \). Finally, a weighted representation g frame introduced some its are presented. provide sufficient condition for given g-frame \(\Lambda =\{\Lambda _{i}\in {B(...