CONSTRUCTIONS OF THE INVARIANT SETS AND INVARIANT MEASURES

In this paper, we will discuss the constructiOn problems about the invariant sets and invariant measures of continues maps~ which map complexes into themselves, using simplical approximation and Markov cbeirs. In [7], the author defined a matrix by using r-normal subdivi of the w,dimensional unit cube, considered it a Markov matrix, and constructed the invariantset and invafiant measure, In fact, the matrix he defined is not Markov matrix generally. So wewill give [7] and amendment in the last pert of this paper. We also construct an invariant set thatis the chain-recurrent set of the map by means of a non-negative matrix which only depends on themap. At hst, we will prove the higher dimension?Banach variation formuls that can simplify thetransition matrix.     

A smooth invariant not a smooth function of the invariant polynomials

An explicit example is given of a smooth function invariant under a linear group action that is not a smooth function of the invariant polynomials.

     

Separable and vector groups whose projectively invariant subgroups are fully invariant

Siberian Mathematical Journal - We study the abelian groups all of whose projectively invariant subgroups are fully invariant. We describe the separable and vector groups with this property.     

Weakly dual automorphism invariant and superfluous cotightness

Abstract

The aim of the present paper is to introduce and study the dual concepts of weakly automorphism invariant modules and essential tightness. These notions are non-trivial generalizations of both weakly projectivity, dual automorphism invariant property and cotightness. We obtain certain relations between weakly projective modules, weakly dual automorphism invariant modules and superfluous cotight modules. It is proved that: (1) for right perfect rings, every module is a direct summand of a weakly dual automorphism invariant module and (2) weakly dual automorphism invariant modules are precisely superfluous cotight modules.     

Algebraic links and skein relations of the Links-Gould invariant

We give certain skein relations of the Links-Gould invariant. We also show that the relations lead us to recursive calculation of the invariant for algebraic links. As an application we give a formula for the Links-Gould invariant of 2-bridge links.

     

Brukner-Zeilinger invariant information

We present an alternative interpretation of the notion of invariant information by establishing that it is directly related to the total ordinary variance of a quantum state. Here, “total” means summing the variance over any complete orthogonal set of observables or, equivalently, averaging over a certain sufficiently general ensemble of the observables. This simple, intuitive substratum of the Brukner-Zeilinger invariant information sheds further light on the informational and statistical nature of quantum measurements. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 302–310, May, 2007.     

Continuation of invariant subspaces

In this work we consider implementation and testing of an algorithm for continuation of invariant subspaces. Copyright © 2001 John Wiley & Sons, Ltd.     

On the ratio integral formula for the likelihood ratio of a maximal invariant

The usual ratio of an integral formula for the likelihood ratio of a maximal invariant in a group model is shown to be correct under assumption that the denominator integral is finite almost everywhere. The limitation of this assumption is discussed, and an application to invariant suffciency is given.     

Hypergroups with invariant metric

The purpose of this note is to extend the following classical result from groups to hypergroups in the sense of C.F. Dunkl, R.I. Jewett, and R. Spector: If a hypergroup has a countable neighborhood base of its identity, then admits a left- or a right-invariant metric. Moreover, it admits an invariant metric if and only if there exists a countable conjugation-invariant neighborhood base of the identity.

     

An invariant for unbounded operators

For a class of unbounded operators, a deformation of a Bott projection is used to construct an integer-valued invariant measuring deviation of the non-commutative deformations from the commutative originals, and its interpretation in terms of -theory of -algebras is given. Calculation of this invariant for specific important classes of unbounded operators is also presented.

     

The structure of shift invariant spaces on a locally compact abelian group

We investigate shift invariant subspaces of L²(G), where G is a locally compact abelian group. We show, among other things, that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame.     

Diagonal invariant ideals of Toeplitz algebras on discrete groups

Diagonal invariant ideals of Toeplitz algebras defined on discrete groups are introduced and studied. In terms of isometric representations of Toeplitz algebras associated with quasi-ordered groups, a character of a discrete group to be amenable is clarified. It is proved that whenG is Abelian, a closed twosided non-trivial ideal of the Toeplitz algebra defined on a discrete Abelian ordered group is diagonal invariant if and only if it is invariant in the sense of Adji and Murphy, thus a new proof of their result is given.     

On principal invariant subspaces

Let F and G be closed subspaces of the complex Hilbert space H, and U and V be closed subspaces of F and G, respectively. In this paper, using the technique of operator block, we present the necessary and sufficient conditions under which (U, V) is a pair of (strictly, non-degenerate) principal invariant subspaces for (F, G).     

Eta invariant and holonomy: The even dimensional case

In previous work, we introduced eta invariants for even dimensional manifolds. It plays the same role as the eta invariant of Atiyah–Patodi–Singer, which is for odd dimensional manifolds. It is associated to K¹$K^1$ representatives on even dimensional manifolds and is closely related to the so called WZW theory in physics. In fact, it is an intrinsic interpretation of the Wess–Zumino term without passing to the bounding 3-manifold. Spectrally the eta invariant is defined on a finite cylinder, rather than on the manifold itself. Thus it is an interesting question to find an intrinsic spectral interpretation of this new invariant. We address this issue here using adiabatic limit technique. The general formulation relates the (mod Z$\mathbb{Z}$ reduction of) eta invariant for even dimensional manifolds with the holonomy of the determinant line bundle of a natural family of Dirac type operators. In this sense our result might be thought of as an even dimensional analogue of Witten's holonomy theorem proved by Bismut–Freed and Cheeger independently.     

Attractive invariant fiber bundles

This work on implicit nonautonomous difference equations (iterations) is devoted to an abstract technical result on the existence of attractive invariant manifolds. We investigate their existence, smoothness and their asymptotic phase property using invariant foliations. Such results lay the basic foundation of further investigations on discretization methods of evolutionary differential equations.     

Conservation and bifurcation of an invariant torus of a vector field

We consider small perturbations with respect to a small parameter ε≥0 of a smooth vector field in ℝ^n+m possessing an invariant torusT ^m. The flow on the torusT ^m is assumed to be quasiperiodic withm basic frequencies satisfying certain conditions of Diophantine type; the matrix Ω of the variational equation with respect to the invariant torus is assumed to be constant. We investigate the existence problem for invariant tori of different dimensions for the case in which Ω is a nonsingular matrix that can have purely imaginary eigenvalues. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 34–44, January, 1997. Translated by S. K. Lando     

Lipschitzian retracts and curves as invariant sets

By an invariant set in a metric space we mean a non-empty compact set K such that K = ⋃ _i=1 ⁿ T _i (K) for some contractions T ₁, …, T _n of the space. We prove that, under not too restrictive conditions, the union of finitely many invariant sets is an invariant set. Hence we establish collage theorems for non-affine invariant sets in terms of Lipschitzian retracts. We show that any rectifiable curve is an invariant set though there is a simple arc which is not an invariant set.      

The Rost invariant has trivial kernel for quasi-split groups of low rank

For G a simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map . This includes the Arason invariant for quadratic forms and Rost's mod 3 invariant for exceptional Jordan algebras as special cases. We show that R _G has trivial kernel if G is quasi-split of type E ₆ or E ₇. A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank. Received: November 1, 2000