Parallelohedrons and Stereohedrons of Delaunay

2021 ◽  
pp. 182-197
Keyword(s):  
Three Dimensional ◽  
Higher Dimension ◽  
Direct Verification

The works of Delaunay and the followers of his ideas about the geometry of n-dimensional parallelohedrons and stereohedrons are considered. It is proved that these representations do not take into account the conditions for the existence of polytopes of higher dimension and the properties characteristic of figures of higher dimension. They are an attempt to extend the properties of three-dimensional figures to figures of higher dimension. A direct verification of the parallelohedrons from the Delaunay classification taking into account the Shtogrin parallelohedron showed that these figures do not satisfy the Euler-Poincaré equation and therefore the assertion that they are parallelohedrons with dimension 4 is erroneous.

Stability and instability of some nonlinear dispersive solitary waves in higher dimension

1996 ◽  
Vol 126 (1) ◽  
pp. 89-112 ◽  
Author(s):  
Anne de Bouard
Keyword(s):  
Solitary Waves ◽  
Acoustic Waves ◽  
Vries Equation ◽  
Three Dimensional ◽  
Higher Dimension ◽  
Ion Acoustic Waves ◽  
Stability And Instability ◽  
Ion Acoustic ◽  
De Vries ◽  
The Stability

We study the stability of positive radially symmetric solitary waves for a three dimensional generalisation of the Korteweg de Vries equation, which describes nonlinear ion-acoustic waves in a magnetised plasma, and for a generalisation in dimension two of the Benjamin–Bona–Mahony equation.

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.

The Structure of the Polytope of Hereditary Information

2019 ◽  
Vol 8 (2) ◽  
pp. 7-22
Author(s):  
Gennadiy Vladimirovich Zhizhin
Keyword(s):  
Nucleic Acid ◽  
Hydrogen Bonds ◽  
Phosphoric Acid ◽  
High Intensity ◽  
Information Exchange ◽  
Three Dimensional ◽  
Higher Dimension ◽  
Nitrogen Bases ◽  
Higher Dimensional ◽  
Low Dimensional

The representations of the sugar molecule and the residue of phosphoric acid in the form of polytopes of higher dimension are used. Based on these ideas and their simplified three-dimensional images, a three-dimensional image of nucleic acids is constructed. The geometry of the neighborhood of the compound of two nucleic acid helices with nitrogen bases has been investigated in detail. It is proved that this neighborhood is a cross-polytope of dimension 13 (polytope of hereditary information), in the coordinate planes of which there are complementary hydrogen bonds of nitrogenous bases. The structure of this polytope is defined, and its image is given. The total incident flows from the low-dimensional elements to the higher-dimensional elements and vice versa of the hereditary information polytope are calculated equal to each other. High values of these flows indicate a high intensity of information exchange in the polytope of hereditary information that ensures the transfer of this information.

Joint Normal Partitions and Hierarchical Filling of N-Dimensional Spaces

2021 ◽  
pp. 261-272
Keyword(s):  
Three Dimensional ◽  
Translational Symmetry ◽  
Higher Dimension

The structures arising in spaces of various dimensions with simultaneous normal partitioning of spaces and their hierarchical fillings are considered. The conditions for the appearance of translational symmetry in these structures are investigated. It is shown that simultaneous hierarchical filling and normal tiling in three-dimensional spaces do not lead to the formation of translational symmetry. Such consistent transformations lead to many elements of translational symmetry in spaces of higher dimension. The higher the dimension of space, the more complex the emerging structure and the more symmetry the elements.

Translating solitons foliated by spheres

2017 ◽  
Vol 28 (01) ◽  
pp. 1750006 ◽  
Author(s):  
Daehwan Kim ◽  
Juncheol Pyo
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Higher Dimension ◽  
Translating Soliton ◽  
Surface Of Revolution ◽  
Translating Solitons

In this paper, we consider translating solitons in [Formula: see text] which is foliated by spheres. In three-dimensional Euclidean space, we show that such a translating soliton is a surface of revolution and the axis of revolution is parallel to the translating direction of the translating soliton. We also show that the same result holds for a higher dimension case with a hypersurface foliated by spheres in parallel hyperplanes that are perpendicular to the translating direction.

scholarly journals Linked Weyl surfaces and Weyl arcs in photonic metamaterials

Science ◽  
2021 ◽  
Vol 373 (6554) ◽  
pp. 572-576
Author(s):  
Shaojie Ma ◽  
Yangang Bi ◽  
Qinghua Guo ◽  
Biao Yang ◽  
Oubo You ◽  
...  
Keyword(s):  
Electromagnetic Waves ◽  
Three Dimensional ◽  
Electromagnetic Properties ◽  
Higher Dimension ◽  
Surface Phenomena ◽  
Physical Systems ◽  
Higher Dimensional ◽  
Topological Singularities ◽  
Lower Dimensional

Generalization of the concept of band topology from lower-dimensional to higher-dimensional (n > 3) physical systems is expected to introduce new bulk and boundary topological effects. However, theoretically predicted topological singularities in five-dimensional systems—Weyl surfaces and Yang monopoles—have either not been demonstrated in realistic physical systems or are limited to purely synthetic dimensions. We constructed a system possessing Yang monopoles and Weyl surfaces based on metamaterials with engineered electromagnetic properties, leading to the observation of several intriguing bulk and surface phenomena, such as linking of Weyl surfaces and surface Weyl arcs, via selected three-dimensional subspaces. The demonstrated photonic Weyl surfaces and Weyl arcs leverage the concept of higher-dimension topology to control the propagation of electromagnetic waves in artificially engineered photonic media.

Higher Dimensions of Clusters of Intermetallic Compounds

2019 ◽  
Vol 4 (1) ◽  
pp. 8-25 ◽  
Author(s):  
Gennadiy V Zhizhin
Keyword(s):  
Diffraction Pattern ◽  
Intermetallic Compounds ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Translational Symmetry ◽  
Higher Dimension ◽  
Internal Geometry ◽  
Metallic Nanoclusters

The author has previously proven that diffraction pattern of intermetallic compounds (quasicrystals) have translational symmetry in the space of higher dimension. In this paper, it is proved that the metallic nanoclusters also have a higher dimension. The internal geometry of clusters was investigated. General expressions for calculating the dimension of clusters is obtained, from which it follows that the dimension of metallic nanoclusters increases linearly with increasing number of cluster shells. The dimensions of many experimentally known metallic nanoclusters are determined. It is shown that these clusters, which are usually considered to be three - dimensional, have a higher dimension. The Euler-Poincaré equation was used, the internal geometry of clusters was investigated.

The Geometry of the Structure of Nucleic Acids With Regard to the Higher Dimension of the Components

2019 ◽  
pp. 203-218
Keyword(s):  
Protein Synthesis ◽  
Nucleic Acid ◽  
Phosphoric Acid ◽  
Nucleic Acids ◽  
Three Dimensional ◽  
Acid Form ◽  
Higher Dimension

Using three-dimensional visualization of nucleic acid molecules, obtained in the previous chapter, an analysis of the geometry of nucleic acid molecules in the space of higher dimension is carried out. It is shown that phosphoric acid residues and five-carbon sugar molecules in a double-stranded nucleic acid form polytopes of higher dimension with anti-parallel edges. These polytopes are of type n-cross-polytope (n = 5 for phosphoric acid residues, n = 13 for sugar molecules). It was found that these n-cross-polytopes located in right- and left-twisted spirals are enantiomorphic. It has been found that in cross-polytopes constructed of two sugar molecules there are 12 coordinate planes, each of which may contain a bond of nitrogenous bases (one of the 12 known ones). The formation of codons (triplets) corresponds to the separation in space of the highest dimension of nucleic acids of three-dimensional regions. This also occurs in the ribosomes upon contact with transport and adapter RNA during protein synthesis.

Higher Dimensions of Clusters of Intermetallic Compounds

2021 ◽  
pp. 31-57
Keyword(s):  
Intermetallic Compounds ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Translational Symmetry ◽  
Higher Dimension ◽  
Internal Geometry ◽  
Metallic Nanoclusters ◽  
Diffraction Patterns

The author has previously proved that diffraction patterns of intermetallic compounds (quasicrystals) have translational symmetry in the space of higher dimension. In this chapter, it is proved that the metallic nanoclusters also have a higher dimension. The internal geometry of clusters was investigated. General expressions for calculating the dimension of clusters are obtained from which it follows that the dimension of metallic nanoclusters increases linearly with increasing number of cluster shells. The dimensions of many experimentally known metallic nanoclusters are determined. It is shown that these clusters, which are usually considered to be three-dimensional, have a higher dimension. The Euler-Poincaré equation was used, and the internal geometry of clusters was investigated.

BIANCHI IX GROUP-MANIFOLD REDUCTIONS OF GRAVITY

2005 ◽  
Vol 20 (40) ◽  
pp. 3115-3125
Author(s):  
ROMAN LINARES
Keyword(s):  
Lie Algebras ◽  
Three Dimensional ◽  
Einstein Gravity ◽  
Spin Connection ◽  
Higher Dimension ◽  
Dimensional Theory ◽  
Bianchi Ix ◽  
Group Manifold ◽  
Manifold Reduction ◽  
Kaluza Klein

We exhibit a new way to perform the group-manifold reduction of pure Einstein gravity in the vielbein formulation when the compactification group manifold is S3. The new Bianchi IX group-manifold reduction is obtained by exploiting the two three-dimensional Lie algebras that the S3 group manifold admits. As an application of the new reduction we show that there exists a domain wall solution to the lower-dimensional theory which upon uplifting to the higher-dimension turns out to be the self-dual (in the nonvanishing components of both curvature and spin connection) Kaluza–Klein monopole.

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