Topological classification of integrable nondegenerate Hamiltonians on the isoenergy three-dimensional sphere

1991 ◽  
pp. 267-296 ◽  
Author(s):  
A. Fomenko ◽  
T. Nguyen
Keyword(s):  
Three Dimensional ◽  
Topological Classification ◽  
Dimensional Sphere

Crystal chemistry of zirconosilicates and their analogs: topological classification of MT frameworks and suprapolyhedral invariants

2002 ◽  
Vol 58 (2) ◽  
pp. 198-218 ◽  
Author(s):  
G. D. Ilyushin ◽  
V. A. Blatov
Keyword(s):  
Crystal Structure ◽  
Crystal Structures ◽  
Crystal Chemistry ◽  
Three Dimensional ◽  
Geometrical Interpretation ◽  
Topological Classification ◽  
Hierarchical Analysis ◽  
Framework Structure ◽  
Coordination Spheres

The first attempt is undertaken to consider systematically topological structures of zirconosilicates and their analogs (60 minerals and 34 synthetic phases), where the simplest structure units are MO6 octahedra and TO4 tetrahedra united by vertices ([TO4]:[MO6] = 1:1–6:1). A method of analysis and classification of mixed three-dimensional MT frameworks by topological types with coordination sequences {N k } is developed, which is based on the representation of crystal structure as a finite `reduced' graph. The method is optimized for the frameworks of any composition and complexity and implemented within the TOPOS3.2 program package. A procedure of hierarchical analysis of MT-framework structure organization is proposed, which is based on the concept of polyhedral microensemble (PME) being a geometrical interpretation of coordination sequences of M and T nodes. All 12 theoretically possible PMEs of MT 6 polyhedral composition are considered where T is a separate and/or connected tetrahedron. Using this methodology the MT frameworks in crystal structures of zirconosilicates and their analogs were analyzed within the first 12 coordination spheres of M and T nodes and related to 41 topological types. The structural correlations were revealed between rosenbuschite, lavenite, hiortdahlite, woehlerite, siedozerite and the minerals of the eudialyte family.

scholarly journals On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow

2020 ◽  
Vol 66 (2) ◽  
pp. 160-181
Author(s):  
V. Z. Grines ◽  
E. Ya. Gurevich ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Topological Classification ◽  
Higher Dimension ◽  
Topological Information ◽  
Topology Of Manifolds ◽  
Necessary And Sufficient

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Paliss problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

Topological classification of real three-dimensional cubics

2009 ◽  
Vol 85 (5-6) ◽  
pp. 841-847 ◽  
Author(s):  
V. A. Krasnov
Keyword(s):  
Three Dimensional ◽  
Topological Classification

scholarly journals Acoustic Möbius insulators from projective symmetry

2021 ◽  
Author(s):  
Chunyin Qiu ◽  
Tianzi Li ◽  
Juan Du ◽  
Qicheng Zhang ◽  
Yitong Li ◽  
...  
Keyword(s):  
Critical Role ◽  
Three Dimensional ◽  
Real Space ◽  
Extensive Study ◽  
Topological Classification ◽  
Translation Symmetry ◽  
Topological Crystalline Insulators ◽  
First Time ◽  
Projective Translation

Abstract Symmetry plays a critical role in classifying phases of matter. This is exemplified by how crystalline symmetries enrich the topological classification of materials and enable unconventional phenomena in topologically nontrivial ones. After an extensive study over the past decade, the list of topological crystalline insulators and semimetals seems to be exhaustive and concluded. However, in the presence of gauge symmetry, common but not limited to artificial crystals, the algebraic structure of crystalline symmetries needs to be projectively represented, giving rise to unprecedented topological physics. Here we demonstrate this novel idea by exploiting a projective translation symmetry and constructing a variety of Möbius-twisted topological phases. Experimentally, we realize two Möbius insulators in acoustic crystals for the first time: a two-dimensional one of first-order band topology and a three-dimensional one of higher-order band topology. We observe unambiguously the peculiar Möbius edge and hinge states via real-space visualization of their localiztions, momentum-space spectroscopy of their 4π periodicity, and phase-space winding of their projective translation eigenvalues. Not only does our work open a new avenue for artificial systems under the interplay between gauge and crystalline symmetries, but it also initializes a new framework for topological physics from projective symmetry.

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