scholarly journals INNER STRUCTURE OF GAUSS–BONNET–CHERN THEOREM AND THE MORSE THEORY

2001 ◽  
Vol 16 (39) ◽  
pp. 2483-2493 ◽  
Author(s):  
YI-SHI DUAN ◽  
PENG-MING ZHANG
Keyword(s):  
Morse Theory ◽  
Topological Structure ◽  
Fundamental Tensor ◽  
Chern Form ◽  
Topological Current ◽  
Tensor Formula ◽  
The Relationship ◽  
Poincare Theorem ◽  
Mapping Theory ◽  
Theory Formula

We define a new one-form HA based on the second fundamental tensor [Formula: see text], the Gauss–Bonnet–Chern form can be novelly expressed with this one-form. Using the ϕ-mapping theory we find that the Gauss–Bonnet–Chern density can be expressed in terms of the δ-function δ(ϕ) and the relationship between the Gauss–Bonnet–Chern theorem and Hopf–Poincaré theorem is given straightforwardly. The topological current of the Gauss–Bonnet–Chern theorem and its topological structure are discussed in details. At last, the Morse theory formula of the Euler characteristic is generalized.

ϕ-MAPPING TOPOLOGICAL THEORY IN TWO-GAP SUPERCONDUCTOR

2004 ◽  
Vol 18 (09) ◽  
pp. 1309-1318 ◽  
Author(s):  
YISHI DUAN ◽  
XUGUANG SHI
Keyword(s):  
Magnetic Field ◽  
Topological Structure ◽  
Topological Theory ◽  
Topological Current ◽  
Composite Vortex ◽  
Mapping Theory ◽  
The Magnetic Field ◽  
Zero Points

In this paper, the topological structure of the two-gap superconductor is discussed in detail based on the ϕ-mapping theory. The expression of the vorticity ∇×V of the composite vortex is given and the relation between the vorticity and the magnetic field which is carried by the composite vortex is discussed. The curl of velocity ∇×V has important relation to δ2(ϕ) or we can say that ∇×V has an important relation to the zero points of ϕ. The inner structure of the topological current is characterized by the ϕ-mapping topological numbers Hopf-index and Brouwer degrees.

scholarly journals On the representation of metric spaces

1979 ◽  
Vol 20 (3) ◽  
pp. 367-375 ◽  
Author(s):  
G.J. Logan
Keyword(s):  
Algebraic Structure ◽  
Topological Structure ◽  
Dual Space ◽  
Metric Spaces ◽  
Closure Operator ◽  
The Family ◽  
Necessary And Sufficient ◽  
The Relationship

A closure algebra is a set X with a closure operator C defined on it. It is possible to construct a topology τ on MX, the family of maximal, proper, closed subsets of X, and then to examine the relationship between the algebraic structure of (X, C) and the topological structure of the dual space (MX τ) This paper describes the algebraic conditions which are necessary and sufficient for the dual space to be separable metric and metric respectively.

scholarly journals Exploring Spatiotemporal Variation in Hourly Metro Ridership at Station Level: The Influence of Built Environment and Topological Structure

2018 ◽  
Vol 10 (12) ◽  
pp. 4564 ◽  
Author(s):  
Zhuangbin Shi ◽  
Ning Zhang ◽  
Yang Liu ◽  
Wei Xu
Keyword(s):  
Built Environment ◽  
Topological Structure ◽  
Regression Models ◽  
Closeness Centrality ◽  
Spatiotemporal Variation ◽  
Spatial And Temporal Variation ◽  
Structure Factors ◽  
The Relationship ◽  
Best Fit

Reliable and accurate estimates of metro demand can provide metro authorities with insightful information for the planning of route alignment and station locations. Many existing studies focus on metro demand from daily or annual ridership profiles, but only a few concern the variation in hourly ridership. In this paper, a geographically and temporally weighted regression (GTWR) model was used to examine the spatial and temporal variation in the relationship between hourly ridership and factors related to the built environment and topological structure. Taking Nanjing, China as a case study, an empirical study was conducted with automatic fare collection (AFC) data in three weeks. With an analysis of variance (ANOVA), it was found that the GTWR model produced the best fit for hourly ridership data compared with traditional regression models. Four built-environment factors, namely residence, commerce, scenery, and parking, and two topological-structure factors, namely degree centrality and closeness centrality, were proven to be significantly related to station-level ridership. The spatial distribution pattern and temporal nonstationarity of these six variables were further analyzed. The result of this study confirmed that the GTWR model can provide more realistic and useful information by capturing spatiotemporal heterogeneity.

Skyrmion Excitations in Graphene

2014 ◽  
Vol 887-888 ◽  
pp. 960-965 ◽  
Author(s):  
Bu Da Zhao ◽  
Ming Xiang
Keyword(s):  
Potential Theory ◽  
Limit Point ◽  
Bifurcation Theory ◽  
Current Theory ◽  
Gauge Potential ◽  
Topological Current ◽  
Mapping Theory

By making use of theφ-mapping topological current theory and the decomposition of gauge potential theory, we investigate the skyrmion excitations of (2+1)-dimensional graphene. It is shown that the topological numbers are Hopf indices and Brower degrees. Based on the bifurcation theory of theφ-mapping theory, it is founded that the skyrmions can be generated or annihilated at the limit point (the generation and annihilation of skyrmion-antiskyrmion pairs).

GRAVITATIONAL POTENTIALS AND THE EXISTENCE OF GRAVITATIONAL GREEN'S TENSORS

1998 ◽  
Vol 13 (29) ◽  
pp. 2347-2353 ◽  
Author(s):  
P. DOLAN ◽  
B. MURATORI
Keyword(s):  
Wave Equation ◽  
Gravitational Field ◽  
Degrees Of Freedom ◽  
Tensor Potential ◽  
Non Local ◽  
Relationship Of ◽  
Tensor Formula ◽  
The Relationship ◽  
Insight Into ◽  
Gauge Conditions

The non-local part of the gravitational field Cabcd can be generated by the 16-component Lanczos tensor potential Labc. When six gauge conditions are imposed, Labe;e=0, its ten degrees of freedom match those of the Weyl tensor. The Penrose wave equation for Cabcd can be independently derived from that for Labc. The consistency between Labc and Cabcd is also shown by the compatibility of their algebraic classifications. An unexpected insight into the relationship of Labc and Cabcd is found in "Euclidean gravity" which in turn leads to the introduction of a gravitational Green's tensor [Formula: see text] corresponding to the potential Labc.

TOPOLOGICAL STRUCTURE OF ENTROPY OF (3+1)-DIMENSIONAL SPHERICALLY SYMMETRIC BLACK HOLES

2001 ◽  
Vol 16 (22) ◽  
pp. 1457-1464 ◽  
Author(s):  
GUO-HONG YANG
Keyword(s):  
Black Holes ◽  
Vector Field ◽  
Physical Meaning ◽  
Topological Structure ◽  
Killing Vector ◽  
Entropy Density ◽  
Topological Invariants ◽  
Spherically Symmetric ◽  
Killing Vector Field ◽  
The Relationship

Using the relationship between the entropy and the Euler characteristic, an entropy density is introduced to describe the inner topological structure of the entropy of (3+1)-dimensional spherically symmetric black holes. It is pointed out that the density of entropy is determined by the singularities of the timelike Killing vector field of space–time, and these singularities carry the topological numbers, Hopf indices and Brouwer degrees, naturally, which are topological invariants. Taking account of the physical meaning in statistics, the entropy of black holes is given by the Hopf indices merely, which will lead to the increasing principle of entropy of black holes.

scholarly journals Relational Topology-Based Heterogeneous Network Embedding for Predicting Drug-Target Interactions

2021 ◽  
Author(s):  
Fuyu Hu ◽  
Chunping Ouyang ◽  
Yongbin Liu ◽  
Zheng Gao ◽  
Yaping Wan
Keyword(s):  
Heterogeneous Network ◽  
Topological Structure ◽  
Drug Target ◽  
Computational Models ◽  
Representation Learning ◽  
Network Embedding ◽  
Relationship Type ◽  
Target Proteins ◽  
Analytical Approaches ◽  
The Relationship

Abstract Background: Predicting interactions between drugs and target proteins is a key task in drug discovery. Although the method of validation via wet-lab experiments has become available, experimental methods for drug-target interactions (DTIs) identification remain either time consuming or heavily dependent on domain expertise. Therefore, various computational models have been proposed to predict possible interactions between drugs and target proteins. Usually, we construct a heterogeneous network with drugs and target proteins to calculate the relationship between them. However, most calculation methods do not consider the topological structure of the relationship between drugs and target proteins. Fortunately, Network Embedding Learning provides new and powerful graph analytical approaches for predicting drug-target interaction, which is considering both content and topology of network.Results: In this article, we propose a relational topology-based heterogeneous network embedding method to predict DITs, abbreviated as RTHNE_DTI. We use the ideas of word embeddings to turn heterogeneous network with drugs and target proteins into dense, low-dimensional real-valued vectors. Furthermore, according to two different topological structure of the relationship between the nodes, we represent them separately by training two different models. Then the meaningful vectors represented for drugs and target proteins can be used to calculate the interaction of them easily. Results show that by considering topological structure and different relationship type of drugs and target proteins, RTHNE_DTI outperforms other state-of-the-art methods on both labeled network and unlabeled network.Conclusions: This work proposes heterogeneous network representation learning for DITs prediction. To the best of our knowledge, this study first introduces relation classification to heterogeneous network embedding to improve predicting DTIs efficiently.

scholarly journals Theory of Alamin (A Formation of Universal Communication Formula)

2018 ◽  
Vol 1 (2) ◽  
pp. 161-180
Author(s):  
Muhammad Aminullah
Keyword(s):  
Methodological Approach ◽  
Human Beings ◽  
Human Relationships ◽  
Good Communication ◽  
Scientific Approach ◽  
Universal Relationship ◽  
Universal Communication ◽  
The Relationship ◽  
The Universe ◽  
Theory Formula

Studies of this study need to be done to understand the basic nature of communication, so that the formation of a universal communication formula. The communication formula constructed in the theory of nature, explains that communication can be done in the form of a universal relationship ,not limited to the relationship between fellow human beings. It turns out that communication can be done in the form of relationships with all the elements that exist in the Universe of the universe is based on only one aspect of the relationship can be done on the need. Relationships that are built on necessities are an absolute right for humans to survive in their lives. The methodological approach in this research is the scientific approach that is built based on the discipline of ALAMTOLOGI. The communication formulas that are awakened in natural theory based on the discipline of ALAMTOLOGI communication and applied on all aspects of human relationships with others scientifically, systematically and universally in everyday life in a harmony way. The NAMORY THEORY formula is X + Z (Y) ⤇ g Hp and X - Z (0) ⤇ g Cp. Based on this concept the implementation of communication of this formula in the implementation of communication can ensure its attainment to the value of harmony or lameness value. Good communication is harmony communication. 

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