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Published in S.P. Bhattacharyya, L.H. Keel, of UNCERTAIN DYNAMIC SYSTEMS, 2020
Edmond A. Jonckheere, Jonathan R. Bar-on
This is the so-called 0-th order homology group. The dimension of that group, also referred to as Betti number bo, is the number of connected components [9,2.4]. One of the accomplishments of algebraic topology is the ability to compute the dimension of that group by algebraic manipulation without the need to “visualize” the situation. However, the number of connected components is not enough to completely describe the topology of the problem. Clearly, the 3-torus of Fig. 1 has “higher connectivity” properties, e.g. “holes”, that are less trivial to visualize and that are formalized in the notion of n-th order homology group Hn.
Topology
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Steve Huntsman, Jimmy Palladino, Michael Robinson
The chief advantage that topological techniques provide is a sort of invariance with respect to well-behaved maps on an underlying space. The more general manifestation of this idea is called functoriality: it is at the root of category theory, algebraic topology, modern algebraic geometry, and many other major disciplines within mathematics. This sort of invariance yields qualitative and/or robust measures of structures that can be particularly useful in the face of uncertain or missing data, a lack of canonical coordinates or parameters, etc.
Elements of topology and homology
Published in Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama, Computational Topology for Biomedical Image and Data Analysis, 2019
Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama
Like all branches of mathematics, algebraic topology and homology have a particular lexicon. This chapter intends to immerse the reader in the essential techniques and vocabulary of group theory, topology, and homology as well as providing illustrative examples for an accessible yet rigorous introduction to these topics. In this regard, the essential components of the theory required to present homology are introduced in an accessible way. This chapter does not aim to be a mathematical compendium.
Imperfect Defects in Smectics A
Published in Liquid Crystals Reviews, 2023
Yuriy A. Nastishin, Claire Meyer
Types of induced defects topologically permitted in a given phase are prescribed by the symmetry of the phase in its ground state. The latter means that being per se distortions of the ground state, the defects in the LC phase are markers identifying the ground state of the phase. As a consequence, LC phases can be identified via polarization microscopy textures of their samples. The discoveries of the smectic and nematic phases by George Friedel are impressive examples of the realization of such a possibility. A theoretical basis for the relation between the symmetry group of the phase and its defects is the theory of homotopy, a branch of algebraic topology. Homotopy classification of defects, one of the most prominent achievements of Maurice Kleman [52-54] allows one to predict the types and properties of defects in a given ordered phase knowing the symmetry group of the phase.
Les vertus des défauts: The scientific works of the late Mr Maurice Kleman analysed, discussed and placed in historical context, with particular stress on dislocation, disclination and other manner of local material disbehaviour
Published in Liquid Crystals Reviews, 2022
It was Maurice Kleman, who died on 29 January 2021, who converted the investigation of defects in various materials, and particularly in liquid crystals, from a set of accidental and unconnected studies – albeit extremely interesting studies – into a professional and coherent science of defects. His distinguished career lasted more than half a century, with at least one article in the press at the time of his death. In this article I celebrate his scientific contribution by giving some account of his personal background, briefly reviewing some of the highlights of his own work, and also, perhaps primarily, by trying to place his work in a larger scientific context. A particular feature of this context is the wide set of scientific fields in which the concept of defect either makes a significant contribution or is required in order to construct a conceptual framework. These fields range from the pure mathematical algebraic topology, through standard ideas of partial differential equations in applied mathematics, continuum mechanics in physics, material science and metallurgy, as well as physical chemistry and finally as we shall see, rather more surprisingly, into various aspects of biology.
Topological properties of solution sets for Sobolev-type fractional stochastic differential inclusions with Poisson jumps
Published in Applicable Analysis, 2020
Solutions to a differential inclusion usually can not be unique for a given initial point. Therefore, it is of importance to study the topological structure of solution sets. In the investigation of the topological properties of solution sets to differential inclusions, an important issue is the -property. The -property of a set shows that the set may not be a singleton, but in the point of view algebraic topology, it is equivalent to a point in the sense that it has the same homology groups as one-point space. Recently, this problem has been treated in several work, and we can refer to [15–21] and the references therein. In particular, Gabor et al. [18] studied the topological structure of solution sets to impulsive functional differential inclusions on the half-line; Wang et al. [22] studied the topological structure of solution sets of compactness and trajectories to fractional evolution inclusions in Caputo fractional derivative of order on the closed domain; Zhou et al. [23] studied the topological structure of solution sets to fractional stochastic evolution inclusions driven by Brownian motion in Caputo fractional derivative of order . However, as far as we know, there are few results available on the topological structure of solution sets for Sobolev-type fractional stochastic differential inclusions with Poisson jumps in Caputo and Riemann–Liouville fractional derivatives of order , respectively.