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ВИДЕОТЕКА |
Международная молодежная конференция «Геометрия и управление»
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Classification of Binary Forms with Control Parameter Pavel Bibikov Institute of Control Sciences RAS, Moscow, Russia |
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Аннотация: The aim of the talk is to classify binary forms, whose coefficients depend on control parameter, with respect to the action of some pseudogroup. We solve this problem in two steps. Firstly, we consider the action of our pseudogroup on the infinite prolongation of the differential Euler equation and find differential invariant algebra of this action. Secondly, using methods from geometric theory of differential equations, we prove that three dependencies between basic differential invariants and their invariant derivatives uniquely define the equivalent class of binary forms with control parameter. Let us consider the space $$ f(x,y;u)=\sum\limits_{i=0}^n a_i(u)x^iy^{n-i}, \quad \text{where $a_i$ are holomorphic functions.} $$ The pseudogroup $G:=\mathrm{SL}_2\leftthreetimes (\mathcal{F}(u)\times \mathrm{T}(u))$ acts on the space 1) “semisimple part” $$ \mathrm{SL}_2\ni A\colon \left( \begin{matrix} x\\y \end{matrix}\right)\mapsto A^{-1}\left( \begin{matrix} x\\y \end{matrix}\right); $$ 2) “functional part” 3) “torus” Consider space Binary forms with control parameter can be considered as solutions of the Euler differential equation $$ \mathcal{E}:=\{x h_x+y h_y=nh\}\subset J^1$$ (see also [2]). The action of the pseudogroup Definition 1. Differential invariant of the action of pseudogroup Remark. Function Definition 2. Invariant derivative is a combination of total derivatives, which commutes with the action of group Theorem 1. Differential invariant algebra of the action of pseudogroup $$ H:=\frac{h_{xx}h_{yy}-h^2_{xy}}{h^2}$$ of order 2 and by invariant derivatives $$\nabla_1:=\frac{h_y}{h}D_x-\frac{h_x}{h}D_y \quad \text{and} \quad \nabla_2:=\frac{h^2}{h_xh_{yu}-h_yh_{xu}}\cdot D_u $$ (where Definition 3. Binary form Consider the regular binary form $$ H_{11}=A(H,H_1,H_2), \;\;H_{12}=B(H,H_1,H_2), \;\;H_{22}=C(H,H_1,H_2). $$ The triple Theorem 2. Two regular binary forms $$ (A,B,C)=(\widetilde{A},\widetilde{B},\widetilde{C}). $$ The author is supported by RFBR, grand mol_a-14-01-31045. Язык доклада: английский Список литературы
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