Baire's Category Theoretic Classification of Compact Expansive Dynamical Systems

2002 ◽  
Vol 09 (04) ◽  
pp. 315-323 ◽  
Author(s):  
Kazutaka Sakai
Keyword(s):  
Dynamical Systems ◽  
Classification Method ◽  
Topological Classification ◽  
Finite Dimensional ◽  
Theoretic Classification

A revised version of the estimation inequality of Akashi [2] is given, and this result is applied to Baire's category theoretic classification of ∊-expansive dynamical systems. Moreover, this classification method is applied to topological classification of shift dynamical systems on finite-dimensional compact domains.

Classification of AF Flows

2003 ◽  
Vol 46 (2) ◽  
pp. 164-177 ◽  
Author(s):  
Andrew J. Dean
Keyword(s):  
Dynamical Systems ◽  
Automorphism Group ◽  
Bratteli Diagram ◽  
Finite Dimensional

AbstractAn AF flow is a one-parameter automorphism group of an AF C*-algebra A such that there exists an increasing sequence of invariant finite dimensional sub-C*-algebras whose union is dense in A. In this paper, a classification of C*-dynamical systems of this form up to equivariant isomorphism is presented. Two pictures of the actions are given, one in terms of a modified Bratteli diagram/pathspace construction, and one in terms of a modified K0 functor.

scholarly journals Dynamics of Bose-Einstein Condensates: Exact Representation and Topological Classification of Coherent Matter Waves

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Leilei Jia ◽  
Qihuai Liu ◽  
Shengqiang Tang
Keyword(s):  
Angular Momentum ◽  
Dynamical Systems ◽  
Bifurcation Theory ◽  
Topological Classification ◽  
Exact Representation ◽  
Matter Waves ◽  
Existence And Multiplicity ◽  
Bose Einstein ◽  
The Waves

By using the bifurcation theory of dynamical systems, we present the exact representation and topological classification of coherent matter waves in Bose-Einstein condensates (BECs), such as solitary waves and modulate amplitude waves (MAWs). The existence and multiplicity of such waves are determined by the parameter regions selected. The results show that the characteristic of coherent matter waves can be determined by the “angular momentum” in attractive BECs while for repulsive BECs; the waves of the coherent form are all MAWs. All exact explicit parametric representations of the above waves are exhibited and numerical simulations support the result.

scholarly journals A topological classification method of the superior mesenteric venous system affluent branches

2018 ◽  
Vol 10 (3) ◽  
pp. 36-44
Author(s):  
Semen A. Simbirtsev ◽  
Evgeniy M. Trunin ◽  
Aleksandr A. Smirnov ◽  
Rustem E. Topuzov ◽  
Azat I. Nazmiev ◽  
...  
Keyword(s):  
Software Package ◽  
Venous Return ◽  
Block Diagram ◽  
Anatomical Study ◽  
Classification Method ◽  
Topological Classification ◽  
Anatomical Variants ◽  
New Type ◽  
The Right

The authors propose a new type of classification of various anatomical variants of venous return from the right half of the colon based on the application of the principles of topology and combinatorics. The article presents data obtained from the topographic and anatomical study of the venous return from the right hemicolon collected from anatomical material (25 observations), and describes the coding algorithm in each case, allocating it to a particular class according to the proposed classification. A block diagram of a software package for semi-automatic retopology of venous return from the right half of the colon is also proposed.

MINIMAL DYNAMICS AND $\mathcal{Z}$-STABLE CLASSIFICATION

2011 ◽  
Vol 22 (01) ◽  
pp. 1-23 ◽  
Author(s):  
KAREN R. STRUNG ◽  
WILHELM WINTER
Keyword(s):  
Dynamical Systems ◽  
Compact Metric Space ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Minimal Dynamical Systems ◽  
Uniquely Ergodic ◽  
Stable Classification

Let X be an infinite compact metric space, α : X → X a minimal homeomorphism, u the unitary that implements α in the transformation group C*-algebra C(X) ⋊α ℤ, and [Formula: see text] a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in [Formula: see text]. Motivated by the success of tracial approximation by finite dimensional C*-algebras as an abstract characterization of classifiable C*-algebras and the idea that classification results for C*-algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra [Formula: see text], we prove that (C(X) ⋊α ℤ) ⊗ Mq∞ is tracially approximately [Formula: see text] if there exists a y ∈ X such that the C*-subalgebra (C*(C(X), uC0(X\{y}))) ⊗ Mq∞ is tracially approximately [Formula: see text]. If the class [Formula: see text] consists of finite dimensional C*-algebras, this can be used to deduce classification up to tensoring with [Formula: see text] for C*-algebras associated to minimal dynamical systems where projections separate tracial states. This is done without making any assumptions on the real rank or stable rank of either C(X) ⋊α ℤ or C*(C(X), uC0(X\{y})), nor on the dimension of X. The result is a key step in the classification of C*-algebras associated to uniquely ergodic minimal dynamical systems by their ordered K-groups. It also sets the stage to provide further classification results for those C*-algebras of minimal dynamical systems where projections do not necessarily separate traces.

K-Theoretic Classification for Inductive Limit Z2 Actions on af Algebras

1996 ◽  
Vol 48 (5) ◽  
pp. 946-958 ◽  
Author(s):  
George A. Elliott ◽  
Hongbing Su
Keyword(s):  
Dynamical Systems ◽  
Inductive Limit ◽  
Finite Dimensional ◽  
Theoretic Classification ◽  
Af Algebras

AbstractIn this paper a K-theoretic classification is given of the C*-dynamical systems where An is finite-dimensional. Corresponding to the trivial action is the K-theoretic classification for AF algebras obtained in [3] (also see [1]).

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals Classification of Degenerate Verma Modules for E(5, 10)

2021 ◽  
Author(s):  
Nicoletta Cantarini ◽  
Fabrizio Caselli ◽  
Victor Kac
Keyword(s):  
Infinite Number ◽  
Verma Module ◽  
Lie Superalgebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Finite Dimensional ◽  
De Rham ◽  
Via Classification

AbstractGiven a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

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