Equivariant K-Homology and Restriction to Finite Cyclic Subgroups

For a discrete group G, we prove that a G-map between proper G–CW-complexes induces an isomorphism in G-equivariant K-homology if it induces an isomorphism in C-equivariant K-homology for every finite cyclic subgroup C of G. As an application, we show that the source of the Baum–Connes assembly map, namely K _* ^G (E(G, in)), is isomorphic to K _* ^G (E(G, )), where E(G, ) denotes the classifying space for the family of finite cyclic subgroups of G. Letting be the family of virtually cyclic subgroups of G, we also establish that and related results.     

Classical Integrable Systems and Billiards Related to Generalized Jacobians

We study some classical integrable systems of dynamics (the Euler top in space, the asymptotic geodesic motion on an ellipsoid) which are linearized on unramified coverings of generalized Jacobian varieties. We find explicit expressions for so called root functions living on such coverings which enable us to solve the problems in terms of generalized theta-functions. In addition, general and asymptotic solutions for ellipsoidal billiards and the billiard in an ellipsoidal layer are obtained.     

Strip Bundles and the Highest Weight Crystals Over Quantum Infinite Rank Affine Algebras

We give new realizations of the highest weight crystals B(λ) over the quantum infinite rank affine algebras U _q (A _∞), U _q (B _∞), U _q (C _∞), and U _q (D _∞) using strip bundles, which are related with Nakajima monomials.     

Abnormal,Pronormal, Contranormal,and Carter Subgroups in Some Generalized Minimax Groups

ABSTRACT

Some properties of abnormal and pronormal subgroups in generalized minimax groups are considered. For generalized minimax groups (not only periodic) whose locally nilpotent residual is nilpotent and satisfies Min-G the existence of Carter subgroups and their conjugations have been proven. Some generalizations of results of J. Rose on abnormal and contranormal subgroups have been also obtained.     

A counterexample to the theorem of Beppo Levi in three dimensions

Supported by the Society of Fellows and NSF Grant DMS-86-06160     

Induced Surfaces and Their Integrable Dynamics Ii. Generalized Weierstrass Representations in 4-d Spaces and Deformations via Ds Hierarchy

Extensions of the generalized Weierstrass representation to generic surfaces in 4-D Euclidean and pseudo-Euclidean spaces are given. Geometric characteristics of surfaces are calculated. It is shown that integrable deformations of such induced surfaces are generated by the Davey–Stewartson hierarchy. Geometrically, these deformations are characterized by the invariance of an infinite set of functionals over surface. The Willmore functional (the total squared mean curvature) is the simplest of them. Various particular classes of surfaces and their integrable deformations are considered.     

Arrangements of minors in the positive Grassmannian and a triangulation of the hypersimplex

The structure of zero and nonzero minors in the Grassmannian leads to rich combinatorics of matroids. In this paper, we investigate an even richer structure of possible equalities and inequalities between the minors in the positive Grassmannian. It was previously shown that arrangements of equal minors of largest value are in bijection with the simplices in a certain triangulation of the hypersimplex that was studied by Stanley, Sturmfels, Lam and Postnikov. Here, we investigate the entire set of arrangements and its relations with this triangulation. First, we show that second largest minors correspond to the facets of the simplices. We then introduce the notion of cubical distance on the dual graph of the triangulation and study its relations with the arrangement of t-th largest minors. Finally, we show that arrangements of largest minors induce a structure of a partially ordered set on the entire collection of minors. We use this triangulation of the hypersimplex to describe a two-dimensional grid structure on this poset.     

Projective functors and the restriction of a Verma module to a subalgebra of Levi type

We observe that the restriction of a Verma module over a semi-simple Lie algebra to a subalgebra of Levi type may be viewed as a projective functor. By simple arguments we prove that this restriction can be decomposed into a direct sum of standard indecomposables in the category O. For the restriction problem from sl(n+1) to gl(n) we describe the complete answer. We study the properties of the modules with Verma flag also and prove that any module with Verma flag is a submodule of some projective.     

Hopf Algebras and Integrable Flows Related to the Heisenberg–Weil Coalgebra

On the basis of the structure of Casimir elements associated with general Hopf algebras, we construct Liouville–Arnold integrable flows related to naturally induced Poisson structures on an arbitrary coalgebra and their deformations. Some interesting special cases, including coalgebra structures related to the oscillatory Heisenberg–Weil algebra and integrable Hamiltonian systems adjoint to them, are considered.     

Affine surfaces in R 4 with planar geodesics with respect to the affine metric

It is proved that non-degenerate surfaces in R ⁴ with planar geodesics are only the complex paraboloid and the product of two parabolas. An erratum to this article is available at .     

Restriction of the cotangent bundle to elliptic curves and Hilbert-Kunz functions

We describe the possible restrictions of the cotangent bundle to an elliptic curve . We apply this in positive characteristic to the computation of the Hilbert-Kunz function of a homogeneous R₊-primary ideal in the graded section ring .     

Approximation and Spanning in the Hardy Space,by Affine Systems

We show that every function in the Hardy space can be approximated by linear combinations of translates and dilates of a synthesizer , provided only that and satisfies a mild regularity condition. Explicitly, we prove scale averaged approximation for each ,

Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the Sato Grassmannian

We consider cohomological and Poisson structures associated with the special tautological subbundles $TB_{W_{1,2, \ldots n} }$ for the Birkhoff strata of the Sato Grassmannian. We show that the tangent bundles of $TB_{W_{1,2, \ldots n} }$ are isomorphic to the linear spaces of two-coboundaries with vanishing Harrison cohomology modules. A special class of two-coboundaries is provided by a system of integrable quasilinear partial differential equations. For the big cell, it is the hierarchy of dispersionless Kadomtsev-Petvishvili (dKP) equations. We also demonstrate that the families of ideals for algebraic varieties in $TB_{W_{1,2, \ldots n} }$ can be viewed as Poisson ideals. This observation establishes a relation between families of algebraic curves in $TB_{W_{\hat S} }$ and coisotropic deformations of such curves of zero and nonzero genus described by hierarchies of systems of hydrodynamic type; the dKP hierarchy is such a hierarchy. We note the interrelation between cohomological and Poisson structures.     

Commutative Sequences of Integrable Functions and Best Approximation With Respect to the Weighted Vector Measure Distance

Let λ be a countably additive vector measure with values in a separable real Hilbert space H. We define and study a pseudo metric on a Banach lattice of integrable functions related to λ that we call a λ-weighted distance. We compute the best approximation with respect to this distance to elements of the function space by the use of sequences with special geometric properties. The requirements on the sequence of functions are given in terms of a commutation relation between these functions that involves integration with respect to λ. We also compare the approximation that is obtained in this way with the corresponding projection on a particular Hilbert space.     

On Generalized Orders and Generalized Types of Dirichlet Series in the Right Half-Plane

In the paper, generalized orders and generalized types of Dirichlet series in the right half-plane are given. Some interesting relationships on maximum modulus, the maximum term and the coefficients of entire function defined by Dirichlet series of in the right half-plane are obtained.     

Strict Versions of Integrable Hierarchies in Pseudodifference Operators and the Related Cauchy Problems

Theoretical and Mathematical Physics - In the algebra PsΔ of pseudodifference operators, we consider two deformations of the Lie subalgebra spanned by positive powers of an invertible constant...     

四元数矩阵的广义Schmidt分解与广酉空间中向量组的广义标准正交化

推广了四元数矩阵的Schmidt分解及广酉空间中向量组的标准正交化问题，给出了实四元数矩阵分解为广酉矩阵与生对角元全正的上三角阵乘积的实用方法．