A元不变量及其复合

在一个变换群下有许多的变换不变量,同时也有任意元的不变量或称为A元不变量,本文提出基于A元不变量,使所有的A元不变量都可以则基本A元不变量复合而成,证明A元基本不变量是存在的;给同一个充分必要条件,用于判定不变量的基本性,还对欧氏空间中各种常见变换群下的基本不变量进行稳定。     

Approximation of 3-Edge-Coloring of Cubic Graphs

The problem to find a 4-edge-coloring of a 3-regular graph is solvable in polynomial time but an analogous problem for 3-edge-coloring is NP-hard. We study complexity of approximation algorithms for invariants measuring how far is a 3-regular graph from having a 3-edge-coloring. We prove that it is an NP-hard problem to approximate such invariants by a power function with exponent smaller than 1.     

Group classification of projective type second-order ordinary differential equations

Group classification with respect to admitted point transformation groups is carried out for second-order ordinary differential equations with cubic nonlinearity of the first-order derivative. The result is obtained with use of the invariants of the equivalence transformation group of the family of equations under consideration. The corresponding Riemannian metric is found for the equations that are the projection of the system of geodesics to a two-dimensional surface.     

Invariants of Conformai and Projective Structures

While in Euclidean, equiaffine or centroaffine differential geometry there exists a unique, distinguished normalization of a regular hypersurface immersion x: M ⁿ → Aⁿ⁺¹, in the geometry of the general affine transformation group, there only exists a distinguished class of such normalizations, the class of relative normalizations. Thus, the appropriate invariants for speaking about affine hypersurfaces are invariants of the induced classes, e.g. the conformai class of induced metrics and the projective class of induced conormal connections. The aim of this paper is to study such invariants. As an application, we reformulate the fundamental theorem of affine differential geometry.     

Gauged Gromov–Witten theory for small spheres

We relate the genus zero gauged Gromov–Witten invariants of a smooth projective variety for sufficiently small area with equivariant Gromov–Witten invariants. As an application we deduce a gauged version of abelianization for Gromov–Witten invariants in the small area chamber. In the symplectic setting, we prove that any sequence of genus zero symplectic vortices with vanishing area has a subsequence that converges after gauge transformation to a holomorphic map with zero average moment map.     

Smooth and Algebraic Invariants of a Group Action: Local and Global Constructions

We provide an algebraic formulation of the moving frame method for constructing local smooth invariants on a manifold under an action of a Lie group. This formulation gives rise to algorithms for constructing rational and replacement invariants. The latter are algebraic over the field of rational invariants and play a role analogous to Cartan's normalized invariants in the smooth theory. The algebraic algorithms can be used for computing fundamental sets of differential invariants.     

Parametric Bäcklund transformations I: Phenomenology

We begin an exploration of parametric Bäcklund transformations for hyperbolic Monge-Ampère systems. (The appearance of an arbitrary parameter in the transformation is a feature of several well-known completely integrable PDEs.) We compute invariants for such transformations and explore the behavior of four examples, two of which are new, in terms of their invariants, symmetries, and conservation laws. We prove some preliminary results and indicate directions for further research.

     

Moving Coframes: II. Regularization and Theoretical Foundations

The primary goal of this paper is to provide a rigorous theoretical justification of Cartans method of moving frames for arbitrary finite-dimensional Lie group actions on manifolds. The general theorems are based a new regularized version of the moving frame algorithm, which is of both theoretical and practical use. Applications include a new approach to the construction and classification of differential invariants and invariant differential operators on jet bundles, as well as equivalence, symmetry, and rigidity theorems for submanifolds under general transformation groups. The method also leads to complete classifications of generating systems of differential invariants, explicit commutation formulae for the associated invariant differential operators, and a general classification theorem for syzygies of the higher order differentiated differential invariants. A variety of illustrative examples demonstrate how the method can be directly applied to practical problems arising in geometry, invariant theory, and differential equations.     

Exact solutions to the equations of a dynamic asymmetric pseudoelasticity model

Using a group foliation of the equations of the dynamic asymmetric model of pseudoelasticity effectively used in studying the elastic materials made of polymers, we obtain a system that, after renaming the functions, becomes equivalent to these equations and contains fewer additional functions than the union of the resolving and automorphic systems of the accomplished group foliation. Among the first-order systems equivalent to these equations, it contains the least number of additional functions and is the only such a system up to a nondegenerate linear transformation of the additional functions. For this system, we find the main Lie group of transformation, an optimal system of its subgroups, and their universal invariants. Some invariant and partially invariant exact solutions are obtained, and their physical meaning is explained.     

Classification of Curves in 2D and 3D via Affine Integral Signatures

We propose new robust classification algorithms for planar and spatial curves subjected to affine transformations. Our motivation comes from the problems in computer image recognition. To each planar or spatial curve, we assign a planar signature curve. Curves, equivalent under an affine transformation, have the same signature. The signatures are based on integral invariants, which are significantly less sensitive to small perturbations of curves and noise than classically known differential invariants. Affine invariants are derived in terms of Euclidean invariants. We present two types of signatures: the global and the local signature. Both signatures are independent of curve parameterization. The global signature depends on a choice of the initial point and, therefore, cannot be used for local comparison. The local signature, albeit being slightly more sensitive to noise, is independent of the choice of the initial point and can be used to solve local equivalence problem. An experiment that illustrates robustness of the proposed signatures is presented.     

线性系统若当不变量与标准形

1.引言1980年 Bosgra 和 Weiden 在满秩线性变换意义下,首先给出了能控能观线性系统的所谓输入输出不变量及其相应的标准形。这一结果在满秩线性变换群意义下将能控能观系统参数化,也即从输入输出方面来看,将系统参数化,给定一组参数,即可唯一确定一个输入输出关系;不同的参数组所确定的输入输出关系不同。它一方面揭示了线性系统的内在本质,同时在实现理论、部分实现理论、编码理论和有限自动机方面有着重要应用。本文依据矩阵的若当形,除了给出能控系统的一组完备独立不变量及其相应的标准形外,还给出了能控能观系统一组完备独立不变量及其相应的标准形,在某些方面这一结果比     

一些特殊变换群的联合不变量和联合微分不变量

The theory of moving frames developed by Peter J Olver and M Fels has importaut applications to geometry,classical invariant theory.We will use this theory to classify joint invariants and joint differential invariants of some transformation groups.     

On the generalized Davenport constant and the Noether number

Known results on the generalized Davenport constant relating zero-sum sequences over a finite abelian group are extended for the generalized Noether number relating rings of polynomial invariants of an arbitrary finite group. An improved general upper degree bound for polynomial invariants of a non-cyclic finite group that cut out the zero vector is given.     

Change-of-bases abstractions for non-linear hybrid systems

We present abstraction techniques that transform a given non-linear dynamical system into a linear system, or more generally, an algebraic system described by polynomials of bounded degree, so that invariant properties of the resulting abstraction can be used to infer invariants for the original system. The abstraction techniques rely on a change-of-bases transformation that associates each state variable of the abstract system with a function involving the state variables of the original system. We present conditions under which a given change-of-bases transformation for a non-linear system can define an abstraction. Furthermore, the techniques developed here apply to continuous systems defined by Ordinary Differential Equations (ODEs), discrete systems defined by transition systems and hybrid systems that combine continuous as well as discrete subsystems.The techniques presented here allow us to discover, given a non-linear system, if a change-of-bases transformation involving degree-bounded polynomials yielding an algebraic abstraction exists. If so, our technique yields the resulting abstract system, as well. Our techniques enable the use of analysis techniques for linear systems to infer invariants for non-linear systems. We present preliminary evidence of the practical feasibility of our ideas using a prototype implementation.     

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.     

Invariants and approximate solutions for certain non-linear oscillators by means of the field method

Certain strongly non-linear conservative oscillators are approached with the field method, which is combined with the convolution integral method. A complete set of their adiabatic invariants are derived, on the basis of which approximate solutions for motion can be obtained.     

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 Calculating the Inverse of a Given R-Automorphism of R[x]

Associated to a toric variety X of dimension r over a field k is a fan Δ on R¹. The fan Δ is a finite set of cones which are in one-to-one correspondence with the orbits of the torus action on X. The fan Δ inherits the Zariski topology from X. In this article some cohomological invariants of X are studied in terms of whether or not they depend only on Δ and not k. Secondly some numerical invariants of X are studied in terms of whether or not they are topological invariants of the fan Δ. That is, whether or not they depend only on the finite topological space defined on Δ. The invariants with which we are mostly concerned are the class group of Weil divisors, the Picard group, the Brauer group and the dimensions of the torsion free part of the etale cohomology groups with coefficients in the sheaf of units. The notion of an open neighborhood of a fan is introduced and examples are given for which the above invariants are sufficiently fine to give nontrivial stratifications of an open neighborhood of a fan all of whose maximal cones are nonsimplicia.     

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.     

Universal Affine Classification of Boolean Functions

In this paper we advance a practical solution of the classification problem of Boolean functions by the affine group – the largest group of linear transformations of variables. We show that the affine types (equivalence classes) can be arranged in a unique infinite sequence which contains all previous lists of types. The types are specified by their minimal representatives, spectral invariants, and stabilizer orders. A brief survey of the fundamental transformation groups is included.