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Register a new userSome combinatorial aspects of differential operation compositions on space $R^n$
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Branko J. Malesevic
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In this paper we present a recurrent relation for counting meaningful compositions of the higher-order differential operations on the space $R^{n}$ (n=3,4,...) and extract the non-trivial compositions of order higher than two.
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In this paper we consider successive iterations of the first-order differential operations in space ${bf R}^3.$
Let $R^{n+1, n}$ be the vector space $R^{2n+1}$ equipped with the bilinear form $(X,Y)=X^t C_n Y$ of index $n$, where $C_n= sum_{i=1}^{2n+1} (-1)^{n+i-1} e_{i, 2n+2-i}$. A smooth $gamma: Rto R^{n+1,n}$ is {it isotropic} if $gamma, gamma_x, ldots, gamma_x^{(2n)}$ are linearly independent and the span of $gamma, ldots, gamma_x^{(n-1)}$ is isotropic. Given an isotropic curve, we show that there is a unique up to translation parameter such that $(gamma_x^{(n)}, gamma_x^{(n)})=1$ (we call such parameter the isotropic parameter) and there also exists a natural moving frame. In this paper, we consider two sequences of curve flows on the space of isotropic curves parametrized by isotropic parameter. We show that differential invariants of these isotropic curves satisfy Drinfeld-Sokolovs KdV type soliton hierarchies associated to the affine Kac-Moody algebra $hat B_n^{(1)}$ and $hat A_{2n}^{(2)}$ Then we use techniques from soliton theory to construct bi-Hamiltonian structure, conservation laws, Backlund transformations and permutability formulas for these curve flows.
Some of the observational aspects related to the evolutionary status and dust production in R Cor Bor stars are discussed. Recent work regarding the surface abundances, stellar winds and evidence for dust production in these high luminosty hydrogen deficient stars are also reviewed. Possibility of the stellar winds being maintained by surface magnetic fields is also considered.
Langer and Perline proved that if x is a solution of the geometric Airy curve flow on R^n then there exists a parallel normal frame along x(. ,t) for each t such that the corresponding principal curvatures satisfy the (n-1) component modified KdV (vmKdV_n). They also constructed higher order curve flows whose principal curvatures are solutions of the higher order flows in the vmKdV_n soliton hierarchy. In this paper, we write down a Poisson structure on the space of curves in R^n parametrized by the arc-length, show that the geometric Airy curve flow is Hamiltonian, write down a sequence of commuting Hamiltonians, and construct Backlund transformations and explicit soliton solutions.
Let $C^{[M]}$ be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence $M=(M_k)$ is log-convex and has moderate growth. We prove that the groups ${operatorname{Diff}}mathcal{B}^{[M]}(mathbb{R}^n)$, ${operatorname{Diff}}W^{[M],p}(mathbb{R}^n)$, ${operatorname{Diff}}{mathcal{S}}{}_{[L]}^{[M]}(mathbb{R}^n)$, and ${operatorname{Diff}}mathcal{D}^{[M]}(mathbb{R}^n)$ of $C^{[M]}$-diffeomorphisms on $mathbb{R}^n$ which differ from the identity by a mapping in $mathcal{B}^{[M]}$ (global Denjoy--Carleman), $W^{[M],p}$ (Sobolev-Denjoy-Carleman), ${mathcal{S}}{}_{[L]}^{[M]}$ (Gelfand--Shilov), or $mathcal{D}^{[M]}$ (Denjoy-Carleman with compact support) are $C^{[M]}$-regular Lie groups. As an application we use the $R$-transform to show that the Hunter-Saxton PDE on the real line is well-posed in any of the classes $W^{[M],1}$, ${mathcal{S}}{}_{[L]}^{[M]}$, and $mathcal{D}^{[M]}$. Here we find some surprising groups with continuous left translations and $C^{[M]}$ right translations (called half-Lie groups), which, however, also admit $R$-transforms.
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