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تسجيل مستخدم جديدGeneralized analytic functions, Moutard-type transforms and holomorphic maps
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We continue the studies of Moutard-type transform for generalized analytic functions started in our previous paper: arXiv:1510.08764. In particular, we suggest an interpretation of generalized analytic functions as spinor fields and show that in the framework of this approach Moutard-type transforms for the aforementioned functions commute with holomorphic changes of variables.
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271 -
P.G. Grinevich L.D. Landau Institute forn Theoretical Physics
, Lomonosov Moscow State University
2015
We continue studies of Moutard-type transforms for the generalized analytic functions started in arXiv:1510.08764, arXiv:1512.00343. In particular, we show that generalized analytic functions with the simplest contour poles can be Moutard transformed
to the regular ones, at least, locally. In addition, the later Moutard-type transforms are locally invertible.
78 -
P.G. Grinevich L.D. Landau Institute forn Theoretical Physics
, Lomonosov Moscow State University
2015
We construct a Moutard-type transform for the generalized analytic functions. The first theorems and the first explicit examples in this connection are given.
A Moutard type transformation for matrix generalized analytic functions is derived. Relations between Moutard type transforms and gauge transformations are demonstrated.
Via a unified geometric approach, a class of generalized trigonometric functions with two parameters are analytically extended to maximal domains on which they are univalent. Some consequences are deduced concerning commutation with rotation, continu
ation beyond the domain of univalence, and periodicity.
Let $mathcal{G}$ resp. $M$ be a positive dimensional Lie group resp. connected complex manifold without boundary and $V$ a finite dimensional $C^{infty}$ compact connected manifold, possibly with boundary. Fix a smoothness class $mathcal{F}=C^{infty}
$, Holder $C^{k, alpha}$ or Sobolev $W^{k, p}$. The space $mathcal{F}(V, mathcal{G})$ resp. $mathcal{F}(V, M)$ of all $mathcal{F}$ maps $V to mathcal{G}$ resp. $V to M$ is a Banach/Frechet Lie group resp. complex manifold. Let $mathcal{F}^0(V, mathcal{G})$ resp. $mathcal{F}^{0}(V, M)$ be the component of $mathcal{F}(V, mathcal{G})$ resp. $mathcal{F}(V, M)$ containing the identity resp. constants. A map $f$ from a domain $Omega subset mathcal{F}_1(V, M)$ to $mathcal{F}_2(W, M)$ is called range decreasing if $f(x)(W) subset x(V)$, $x in Omega$. We prove that if $dim_{mathbb{R}} mathcal{G} ge 2$, then any range decreasing group homomorphism $f: mathcal{F}_1^0(V, mathcal{G}) to mathcal{F}_2(W, mathcal{G})$ is the pullback by a map $phi: W to V$. We also provide several sufficient conditions for a range decreasing holomorphic map $Omega$ $to$ $mathcal{F}_2(W, M)$ to be a pullback operator. Then we apply these results to study certain decomposition of holomorphic maps $mathcal{F}_1(V, N) supset Omega to mathcal{F}_2(W, M)$. In particular, we identify some classes of holomorphic maps $mathcal{F}_1^{0}(V, mathbb{P}^n) to mathcal{F}_2(W, mathbb{P}^m)$, including all automorphisms of $mathcal{F}^{0}(V, mathbb{P}^n)$.
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