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.     

三维流形不变量θ(Ωp)(M3)

本文利用Jones-Kauffman模和Kirby技巧等给出了某些三维流形不变量的一般计算方法.     

Dickson-Mui Invariants

Let V be a finite-dimensional vector space over a finite field.The general and special linear groups, GL(V) and SL(V), acton the exterior algebras ^*V and ^*V^* of V and its dual V^*, andon the symmetric algebra S^*V. The subring of SL(V)-invariantsof ^*VS^*V was determined by Dickson and Mui. This paper describesthe equivalent, but simpler, calculation of the invariant subringof ^*VS^*V as a representation of GL(V)/SL(V). 2000 MathematicsSubject Classification 13A50.     

Homological Invariants and Quasi-Isometry

Building upon work of Y. Shalom we give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain that the cohomological dimension cd_R over a commutative ring R satisfies the inequality if Λ embeds uniformly into Γ and holds. Another consequence of our results is that the Hirsch ranks of quasi-isometric solvable groups coincide. Further, it is shown that the real cohomology rings of quasi-isometric nilpotent groups are isomorphic as graded rings. On the analytic side, we apply the induction technique to Novikov-Shubin invariants of amenable groups, which can be seen as homological invariants, and show their invariance under quasi-isometry. Received: November 2004 Revision: April 2004 Accepted: April 2004     

Invariants and Chebyshev polynomials

On different compact sets from ? ⁿ , new multidimensional analogs of algebraic polynomials least deviating from zero (Chebyshev polynomials) are constructed. A brief review of the analogs constructed earlier is given. Estimates of values of the best approximation obtained by using extremal signatures, lattices, and finite groups are presented.     

Near-Rings of Invariants

For a group S acting on a group G we let I(S,G) = {?: G → G ? σ(x) = ? (x), ∈ G, σ ∈ S}, the near-ring of invariants of (S,G). In this paper we investigate the transfer of information between the ideal structure of I(S,G) and the group action (S, G).     

Invariants and Chebyshev polynomials

On different compact sets from ℝ ⁿ , new multidimensional analogs of algebraic polynomials least deviating from zero (Chebyshev polynomials) are constructed. A brief review of the analogs constructed earlier is given. Estimates of values of the best approximation obtained by using extremal signatures, lattices, and finite groups are presented.     

Invariants of modular groups

For a faithful linear representation of a finite group G over a field of characteristic p, we study the ring of invariants. We especially study the polynomial and Cohen–Macaulay properties of the invariant ring. We first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the Hilbert ideal conjecture is true for a class of groups. In particular, we prove that the conjecture is true for vector invariant rings of Abelian reflection p-groups. Then we study the relationships between the invariant ring of G and that of a subgroup of G. Finally, we study the invariant rings of affine groups and show that, over a finite field, if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group.     

Invariants of three-dimensional manifolds

The work of the second author was partially supported by an NSF grant.     

Invariants of Algebraic Automorphisms

Let R be a prime ring with extended centroid C and let σ be a C-algebraic automorphism of R. We let $R^{(\sigma)}\mathop{=}\limits^{\rm def.}\{x\in R\mid \sigma(x)=x\}$ , the subring of invariants of σ in R, and let Out-deg(σ) and Inn-deg(σ) denote the outer and inner degrees of σ, respectively. In the paper we first prove the nilpotence of the prime radical of R ^(σ) with a bound and characterize the semiprimeness and primeness of R ^(σ). Moreover, we show that if R ^(σ) is a prime PI-ring, then PI-deg(R)?=?PI-deg(R ^(σ)) × Inn-deg(σ).     

Loewy Modules with Finite Loewy Invariants and Max Modules with Finite Radical Invariants

Over a commutative ring R, a module is artinian if and only if it is a Loewy module with finite Loewy invariants [5 Facchini , A. ( 1981 ). Loewy and artinian modules over commutative rings . Ann. Mat. Pura Appl. 128 : 359 – 374 .[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that this is not necesarily true for modules over noncommutative rings R, though every artinian module is always a Loewy module with finite Loewy invariants. We prove that every Loewy module with finite Loewy invariants has a semilocal endomorphism ring, thus generalizing a result proved by Camps and Dicks for artinian modules [3 Camps , R. , Dicks , W. ( 1993 ). On semilocal rings . Israel J. Math. 81 : 203 – 211 .[Crossref], [Web of Science ®] , [Google Scholar]]. Finally, we obtain similar results for the dual class of max modules.     

Conservation Principles and Integral Invariants

Conservation laws in the form of integral invariants of linear systems form a particularly simple concept. This is developed into a teaching tool connecting various other fields with this physical notion in such a way that it may be introduced as a seminal first year topic in applied mathematics.     

Projective Curvature and Integral Invariants

In this paper, an extension of all Lie group actions on R ²to coordinates defined by potentials is given. This provides a new solution to the equivalence problems of curves under the projective group and two of its subgroups. The potentials correspond to integrals of higher and higher-order producing an infinite number of independent integral invariants. Applications to computer vision are discussed.     

Homomorphisms and Polynomial Invariants of Graphs

This paper initiates a study of the connection between graph homomorphisms and the Tutte polynomial. This connection enables us to extend the study to other important polynomial invariants associated with graphs, and closely related to the Tutte polynomial. We then obtain applications of these relationships in several areas, including Abelian Groups and Statistical Physics. A new type of uniqueness of graphs, strongly related to chromatically-unique graphs and Tutte-unique graphs, is introduced in order to provide a new point of view of the conjectures about uniqueness of graphs stated by Bollobas, Peabody and Riordan.     

Invariants and constructions of mendelsohn designs

In this paper we examine Mendelsohn designs and some connections to topology which lead to an easily described algorithm for computing invariants of these designs. The results are applied to designs which have natural group actions. We also use the topology to describe when ordinary two-fold triple systems with a group action lead to Mendelsohn designs with the same group action. Procedures for constructing Mendelsohn designs are also given. In particular, we give necessary and sufficient conditions for constructing 2-(v, 4, 1) Mendelsohn designs.     

Feedback Differential Invariants

The problem of feedback equivalence for control systems is considered. An algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found.