Let $G$ be a subgroup of $S_{n}$, the symmetric group degree $n$. For any field $k$, acts naturally on rational function $k(x_{1},\cdots,x_{n})$ via $k$-automorphisms defined by $\sigma\cdot x_{i}:=x_{\sigma\cdot i}$ for $\sigma\in G$ and $1\leq i\leq n$. In this article, we will show that if is solvable transitive $S_{14}$ $\text{char}(k)=7$, then fixed subfield $k(x_{1},\cdots,x_{14})^{G}$ (i...

Adding self-loops at each vertex of a graph improves the performance quantum walks algorithms over loopless algorithms. Many works approach to search for single marked vertex. In this article, we experimentally address several problems related walk in hypercube with multiple vertices. We first investigate hypercube. saw that neighbor vertices are also amplified and approximately $1/2$ system en...

Abstract We consider the nonlocal double phase equation $$\begin{align*} \textrm{P.V.} &\int_{\mathbb{R}^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{sp}(x,y)\,\textrm{d}y\\ &+\textrm{P.V.} \int_{\mathbb{R}^n} a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{tq}(x,y)\,\textrm{d}y=0, \end{align*}$$where $1<p\leq q$ and modulating coefficient $a(\cdot ,\cdot )\geq 0$. Under some suitable hypotheses, we fi...

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 $p-1$, $\...

5d transition metal Pt is the canonical spin Hall material for efficient generation of spin-orbit torques (SOTs) in Pt/ferromagnetic layer (FM) heterostructures. However, a long while with tremendous engineering endeavors, damping-like SOT efficiencies (${\xi}_{DL}$) and alloys have still been limited to ${\xi}_{DL}$<0.5. Here we present that proper alloying elements, particularly 3d metals V C...

Let $mathcal {A} $ and $mathcal {B} $ be C$^*$-algebras. Assume that $mathcal {A}$ is of real rank zero and unital with unit $I$ and $k>0$ is a real number. It is shown that if $Phi:mathcal{A} tomathcal{B}$ is an additive map preserving $|cdot|^k$ for all normal elements; that is, $Phi(|A|^k)=|Phi(A)|^k $ for all normal elements $Ainmathcal A$, $Phi(I)$ is a projection, and there exists a posit...

Let C be an arithmetic circuit of size s, given as input that computes a polynomial \(f\in {\mathbb {F}}[x_1,x_2,\ldots ,x_n]\), where \({\mathbb {F}}\) is finite field or the rationals. Using Hadamard product polynomials, we obtain new algorithms for following two problems first studied by Koutis and Williams (Faster algebraic path packing problems, 2008, https://doi.org/10.1007/978-3-540-7057...