Moving Coframes: I. A Practical Algorithm

This is the first in a series of papers devoted to the development and applications of a new general theory of moving frames. In this paper, we formulate a practical and easy to implement explicit method to compute moving frames, invariant differential forms, differential invariants and invariant differential operators, and solve general equivalence problems for both finite-dimensional Lie group actions and infinite Lie pseudo-groups. A wide variety of applications, ranging from differential equations to differential geometry to computer vision are presented. The theoretical justifications for the moving coframe algorithm will appear in the next paper in this series.     

Maurer–Cartan forms and the structure of Lie pseudo-groups

This paper begins a series devoted to developing a general and practical theory of moving frames for infinite-dimensional Lie pseudo-groups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations for a pseudo-group. Our approach is completely explicit and avoids reliance on the theory of exterior differential systems and prolongation. The second paper [60] will apply these constructions in order to develop the moving frame algorithm for the action of the pseudo-group on submanifolds. The third paper [61] will apply Gr?bner basis methods to prove a fundamental theorem on the freeness of pseudo-group actions on jet bundles, and a constructive version of the finiteness theorem of Tresse and Kumpera for generating systems of differential invariants and also their syzygies. Applications of the moving frame method include practical algorithms for constructing complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, analyzing invariant variational principles, and solving equivalence and symmetry problems arising in geometry and physics.     

Poisson brackets associated to the conformal geometry of curves

In this paper we present an invariant moving frame, in the group theoretical sense, along curves in the Möbius sphere. This moving frame will describe the relationship between all conformal differential invariants for curves that appear in the literature. Using this frame we first show that the Kac-Moody Poisson bracket on can be Poisson reduced to the space of conformal differential invariants of curves. The resulting bracket will be the conformal analogue of the Adler-Gel'fand-Dikii bracket. Secondly, a conformally invariant flow of curves induces naturally an evolution on the differential invariants of the flow. We give the conditions on the invariant flow ensuring that the induced evolution is Hamiltonian with respect to the reduced Poisson bracket. Because of a certain parallelism with the Euclidean case we study what we call Frenet and natural cases. We comment on the implications for completely integrable systems, and describe conformal analogues of the Hasimoto transformation.

     

Algorithms for Differential Invariants of Symmetry Groups of Differential Equations

We develop new computational algorithms, based on the method of equivariant moving frames, for classifying the differential invariants of Lie symmetry pseudo-groups of differential equations and analyzing the structure of the induced differential invariant algebra. The Korteweg-deVries (KdV) and Kadomtsev-Petviashvili (KP) equations serve to illustrate examples. In particular, we deduce the first complete classification of the differential invariants and their syzygies of the KP symmetry pseudo-group.     

Classification of Differential Invariant Submodels

Calculation of differential invariants and invariant differentiation operators of a subalgebra of the Lie algebra admitted by a system of differential equations enables us to construct differential invariant submodels. We classify submodels for every subalgebra of an optimal system of subalgebras. Classification includes the invariant submodels and partially invariant submodels considered earlier. We give examples of classification for three-dimensional subalgebras admitted by the equations of gas dynamics.     

Lie group analysis of two-dimensional variable-coefficient Burgers equation

The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ans?tze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws.     

Lie group analysis of two-dimensional variable-coefficient Burgers equation

The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ansätze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws.     

Differential invariants of the transformation group of a homogeneous space

We demonstrate that the invariant operators on a homogeneous space generate differential invariants and invariant differentiation operators. The coordinate-free method of this article makes it possible to simply the computations essentially, namely to reduce them to operations of linear algebra. Some examples are exhibited.     

Perturbation Theory for Simultaneous Bases of Singular Subspaces

New perturbation theorems are proved for simultaneous bases of singular subspaces of real matrices. These results improve the absolute bounds previously obtained in [6] for general (complex) matrices. Unlike previous results, which are valid only for the Frobenius norm, the new bounds, as well as those in [6] for complex matrices, are extended to any unitarily invariant matrix norm. The bounds are complemented with numerical experiments which show their relevance for the algorithms computing the singular value decomposition. Additionally, the differential calculus approach employed allows to easily prove new sin perturbation theorems for singular subspaces which deal independently with left and right singular subspaces.     

A hierarchy of submodels of differential equations

We consider a system of differential equations admitting a group of transformations. The Lie algebra of the group generates a hierarchy of submodels. This hierarchy can be chosen so that the solutions to each of submodels are solutions to some other submodel in the same hierarchy. For this we must calculate an optimal system of subalgebras and construct a graph of embedded subalgebras and then calculate the differential invariants and invariant differentiation operators for each subalgebra. The invariants of a superalgebra are functions of the invariants of the algebra. The invariant differentiation operators of a superalgebra are linear combinations of invariant differentiation operators of a subalgebra over the field of invariants of the subalgebra. The comparison of the representations of group solutions gives a relation between the solutions to the models of the superalgebra and the subalgebra. Some examples are given of embedded submodels for the equations of gas dynamics.     

The work of Gian-Carlo rota on invariant theory

The purpose of this paper is twofold: first, to explain Gian-Carlo Rotas work on invariant theory; second, to place this work in a broad historical and mathematical context. Rotas work falls under three specific cases: vector invariants, the invariants of binary forms, and the invariants of skew-symmetric tensors. We discuss each of these cases and show how determinants and straightening play central roles. In fact, determinants constitute all invariants in the vector case; for binary forms and skew-symmetric tensors, they constitute all invariants when invariants are represented symbolically. Consequently, we explain the symbolic method both for binary forms and for skew-symmetric tensors, where Rota developed generalizations of the usual notion of a determinant. We also discuss the Grassmann algebra, with its two operations of meet and join, which was a theme which ran through Rotas work on invariant theory almost from the very beginning.To the memory of Gian-Carlo Rota     

Conformal invariants associated with quadratic differentials

We associate a functional of pairs of simply-connected regions D₂ ? D₁ to any quadratic differential on D₁ with specified singularities. This functional is conformally invariant, monotonic, and negative. Equality holds if and only if the inner domain is the outer domain minus trajectories of the quadratic differential. This generalizes the simply-connected case of results of Z. Nehari [20], who developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle for harmonic functions. Nehari’s method corresponds to the special case that the quadratic differential is of the form (?q)² for a singular harmonic function q on D₁.As an application we give a one-parameter family of monotonic, conformally invariant functionals which correspond to growth theorems for bounded univalent functions. These generalize and interpolate the Pick growth theorems, which appear in a conformally invariant form equivalent to a two-point distortion theorem of W. Ma and D. Minda [16].     

流形上的微分Harnack 估计

In the thesis, we study the differential Harnack estimate for the heat equation of the Hodge Laplacian deformation of (p, p)-forms on both fixed and evolving (by Kähler-Ricci flow) Kähler manifolds, which generalize the known differential Harnack estimates for (1, 1)-forms. On a Kähler manifold, we define a new curvature cone C_p and prove that the cone is invariant under Kähler-Ricci flow and that the cone ensures the preservation of the nonnegativity of the solutions to Hodge Laplacian heat equation. After identifying the curvature conditions, we prove the sharp differential Harnack estimates for the positive solution to the Hodge Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler-Ricci flow after obtaining some interpolating matrix differential Harnack type estimates for curvature operators between Hamilton’s and Cao’s matrix Harnack estimates. Similarly, we define another new curvature cone, which is invariant under Ricci flow, and prove another interpolating matrix differential Harnack estimates for curvature operators on Riemannian manifolds.     

Differential invariants of a class of Lagrangian systems with two degrees of freedom

We consider systems of Euler–Lagrange equations with two degrees of freedom and with Lagrangian being quadratic in velocities. For this class of equations the generic case of the equivalence problem is solved with respect to point transformations. Using Lie?s infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems. We describe certain types of Lagrangian systems in terms of their invariants. The results are illustrated by several examples.     

Joint Invariant Signatures

A new, algorithmic theory of moving frames is applied to classify joint invariants and joint differential invariants of transformation groups. Equivalence and symmetry properties of submanifolds are completely determined by their joint signatures, which are parametrized by a suitable collection of joint invariants and/or joint differential invariants. A variety of fundamental geometric examples are developed in detail. Applications to object recognition problems in computer vision and the design of invariant numerical approximations are indicated. August 25, 1999. Final version received: May 3, 2000. Online publication: xxxx.     

Pseudo-almost Periodic and Pseudo-almost Automorphic Solutions to Evolution Equations in Hilbert Spaces

In this work we prove some existence and uniqueness results for pseudo-almost periodic and pseudo-almost automorphic solutions to a class of semi-linear differential equations in Hilbert spaces using theoretical measure theory. The main technique is based upon some appropriate composition theorems combined with the Banach contraction mapping principle and the method of the invariant subspaces for unbounded linear operators. A few illustrative examples will be discussed at the end of the paper.     

Curvature invariants, differential operators and local homogeneity

We first prove that a Riemannian manifold with globally constant additive Weyl invariants is locally homogeneous. Then we use this result to show that a manifold whose Laplacian commutes with all invariant differential operators is a locally homogeneous space.