Equivalent Activity Price Theory And Its Symbolic Calculation Algorithm And Application Research

In resent years, with the continuous development of nonlinear science, people put forward and created new theoretical systems and algorithms based on the classical theory and algorithms to look for new research fields. In this study, Peter Olver and Mark Fels have not only done much pioneering work but also put forward a new theory of equivalent moving frames, with the help of the powerful features of symbolic computation and new theory system, which extended moving frame from lie group action to infinite dimension-al Lie pseudo group action. Via construction of equivalent moving frames, we can obtain differential invariants, prolongation form of infinitesimal generator, recursive formula and so on, which also can be further applied in polynomial equivalence, classification of dif-ferential equations, invariant theory and other related application of study.Motivated by the equivalent moving frame method and the infinitesimal generators of group action and the universal recurrence formulae for differential invariants, we got high prolongation theory of infinitesimal generators. Furthermore, we used features of sym-bolic computation to design corresponding algorithm and found the fundamental sets of normalized differential invariants. Then we construct infinite recursion formula of high-er order normalized differential invariants and syzygies on the generating set of them. Furthermore, the explicit Monge-Taylor forms are constructed from the obtained higher order differential invariants for submanifolds under group actions. Secondly, we used the theory of moving frame and then improved related algorithm to get differential invariants and recursion formula on action of Lie pseudo-group. The method is mainly used induc-tive method and improved algorithm to construct the equivalent moving frame, which can not only reduce tedious math computation, but also get more simpler algorithm to con-struct moving frame on the Lie pseudo-group action and get its differential invariants. At last, based on the method and invariant process and infinitesimal theorem of Lie group, we mainly applied it to classify differential equations of Lie symmetry pseudo-group generat-ed differential invariants and also analyze the algebraic structure of differential invariants. Meanwhile, we used the higher prolongation of infinitesimal generator, differential in-variants and syzygies of chapter 3 to study the application in computer image processing in terms of basic moving frames and computational algorithm.The text is mainly studied by the equivalent moving frame and is applied in specific examples, then we obtained following results:minimum generated system of differen- tial invariants for submanifolds under group actions, (higher order)differential invariants and recursion formula, normal recursion formula, syzygies, higher order prolongation of infinitesimal generators on group action and the relation of them, and the applica-tion research of image processing of curve for submanifold under equiaffine group action. Meanwhile, we used improved algorithm of equivalent moving frame to construct moving frame on action of finite Lie group and infinite dimensional Lie pseudo-group as well and got related differential invariants, which will be better to classify differential invariants on action of symmetry group of differential equations and to get more research achievement, which provides more theory research for hydromechanics, elastic mechanics, geometry, computer vision, soliton theory and so on.