Some applications of homology theory

2001 ◽  
pp. 129-135
Author(s):  
V. Vassiliev
Keyword(s):  
Homology Theory

A Geometric Approach to Homology Theory

1976 ◽  
Author(s):  
S. Buonchristiano ◽  
C. P. Rourke ◽  
B. J. Sanderson
Keyword(s):  
Geometric Approach ◽  
Homology Theory

scholarly journals Blackbox computation of A ∞-algebras

2010 ◽  
Vol 17 (2) ◽  
pp. 391-404
Author(s):  
Mikael Vejdemo-Johansson
Keyword(s):  
Homology Theory ◽  
Algebra Structure ◽  
Finite Amount ◽  
Computational Work ◽  
Dg Algebra ◽  
Dg Algebras

Abstract Kadeishvili's proof of theminimality theorem [T. Kadeishvili, On the homology theory of fiber spaces, Russ. Math. Surv. 35:3 (1980), 231–238] induces an algorithm for the inductive computation of an A ∞-algebra structure on the homology of a dg-algebra. In this paper, we prove that for one class of dg-algebras, the resulting computation will generate a complete A ∞-algebra structure after a finite amount of computational work.

scholarly journals Path homology theory of edge-colored graphs

2021 ◽  
Vol 19 (1) ◽  
pp. 706-723
Author(s):  
Yuri V. Muranov ◽  
Anna Szczepkowska
Keyword(s):  
Spectral Sequence ◽  
Edge Coloring ◽  
Homology Theory ◽  
Homotopy Category ◽  
Homology Groups ◽  
Colored Graphs ◽  
Natural Filtration ◽  
Definition Of ◽  
Exact Sequences

Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

On the Homology Theory of Algebraic Varieties, I

1956 ◽  
Vol 63 (2) ◽  
pp. 248 ◽  
Author(s):  
Andrew H. Wallace
Keyword(s):  
Homology Theory ◽  
Algebraic Varieties

A New Homology Theory

1942 ◽  
Vol 43 (2) ◽  
pp. 370 ◽  
Author(s):  
W. Mayer
Keyword(s):  
Homology Theory

Persistent Homology for Land Cover Change Detection

2021 ◽  
Author(s):  
Djamel Bouchaffra ◽  
Faycal Ykhlef
Keyword(s):  
Land Cover ◽  
Change Detection ◽  
Land Cover Change ◽  
Vegetation Index ◽  
Homology Theory ◽  
Urban Morphology ◽  
Detection Methods ◽  
Landsat Images ◽  
Data Repositories ◽  
Land Cover Change Detection

The need for environmental protection, monitoring, and security is increasing, and land cover change detection (LCCD) can aid in the valuation of burned areas, the study of shifting cultivation, the monitoring of pollution, the assessment of deforestation, and the analysis of desertification, urban growth, and climate change. Because of the imminent need and the availability of data repositories, numerous mathematical models have been devised for change detection. Given a sample of remotely sensed images from the same region acquired at different dates, the models investigate if a region has undergone change. Even if there is no substantial advantage to using pixel-based classification over object-based classification, a pixel-based change detection approach is often adopted. A pixel can encompass a large region, and it is imperative to determine whether this pixel (input) has changed or not. A changed image is compared to the available ground truth image for pixel-based performance evaluation. Some existing change detection systems do not take into account reversible changes due to seasonal weather effects. In other words, when snow falls in a region, the land cover is not considered as a change because it is seasonal (reversible). Some approaches exploit time series of Landsat images, which are based on the Normalized Difference Vegetation Index technique. Others evaluate built-up expansion to assess urban morphology changes using an unsupervised approach that relies on labels clustering. Change detection methods have also been applied to the field of disaster management using object-oriented image classification. Some methodologies are based on spectral mixture analysis. Other techniques invoke a similarity measure based on the evolution of the local statistics of the image between two dates for vegetation LCCD. Probabilistic approaches based on maximum entropy have been applied to vegetation and forest areas, such as Hustai National Park in Mongolia. Researchers in this field have proposed an LCCD scheme based on a feed-forward neural network using backpropagation for training. This paper invokes the new concept of homology theory, a subfield of algebraic topology. Homology theory is incorporated within a Structural Hidden Markov Model.

Infinitesimals, Microsimplexes and Elementary Homology Theory

1986 ◽  
Vol 93 (7) ◽  
pp. 540
Author(s):  
Rade T. Zivaljevic
Keyword(s):  
Homology Theory
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