Uncovering Patterns Beneath the Surface:A Glimpse into the Applications of Algebraic Topology

This paper explores the significant impact of algebraic topology on diverse real-world applications.Starting with an introduction to the historical development and essence of algebraic topology,it delves into its applications in neuroscience,physics,biology,engineering,data analysis,and Geographic Information Systems(GIS).Remarkable applications incorporate the analysis of neural networks,quantum mechanics,materials science,and disaster management,showcasing its boundless significance.Despite computational challenges,this study outlines prospects,emphasizing the requirement for proficient algorithms,noise robustness,multi-scale analysis,machine learning integration,user-friendly tools,and interdisciplinary collaborations.In essence,algebraic topology provides a transformative lens for uncovering stowed-away topological structures in complex data,offering solutions to perplexing problems in science,engineering,and society,with vast potential for future exploration and innovation.

Toward a Neurophysiological Measure of Image Quality Perception Based on Algebraic Topology Analysis

The bandwidth of internet connections is still a bottleneck when transmitting large amounts of images,making the image quality assessment essential.Neurophysiological assessment of image quality has highlight advantages for it does not interfere with natural viewing behavior.However,in JPEG compression,the previous study is hard to tell the difference between the electroencephalogram(EEG)evoked by different quality images.In this paper,we propose an EEG analysis approach based on algebraic topology analysis,and the result shows that the difference between Euler characteristics of EEG evoked by different distortion images is striking both in the alpha and beta band.Moreover,we further discuss the relationship between the images and the EEG signals,and the results implied that the algebraic topological properties of images are consistent with that of brain perception,which is possible to give birth to braininspired image compression based on algebraic topological features.In general,an algebraic topologybased approach was proposed in this paper to analyze the perceptual characteristics of image quality,which will be beneficial to provide a reliable score for data compression in the network and improve the network transmission capacity.

TANGENT UNIT-VECTOR FIELDS:NONABELIAN HOMOTOPY INVARIANTS AND THE DIRICHLET ENERGY

Let O be a closed geodesic polygon in S2. Maps from O into S2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S2, we compute the infimum Dirichlet energy 6（H） for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H. The expression for C（H） involves a topological invariant - the spelling length - associated with the （non-abelian） fundamental group of the n-times punctured two-sphere, π1（S2 - {s1,..., sn}, ＊）. The lower bound for C（H） is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes （classes containing wholly conformal and anticonformal representatives respectively）, the expression for C（H） reduces to a previous result involving the degrees of a set of regular values sl,…… sn in the target 82 space. These degrees may be viewed as invariants associated with the abelianization of vr1（S2 - {s1,..., sn}, ＊）. For nonconformal classes, however, ε（H） may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees. This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy （one-constant Oseen-Frank energy） of reflection-symmetric tangent unitvector fields in a rectangular prism.

Application of topology-based structure features for machine learning in materials science

Structure features play an important role in machine learning models for the materials investigation.Here,two topology-based features for the representation of material structure,specifically structure graph and algebraic topology,are introduced.We present the fundamental mathematical concepts underlying these techniques and how they encode material properties.Furthermore,we discuss the practical applications and enhancements of these features made in specific material predicting tasks.This review may provide suggestions on the selection of suitable structural features and inspire creativity in developing robust descriptors for diverse applications.

Completely Positive Definite Maps Over Topological-algebras

In this paper, we discuss completely positive definite maps over topological algebras. A Schwarz type inequality for n-positive definite maps, and the Stinespring representation theorem for completely positive definite maps over topological algebras are given.

A CLASS OF INFINITE MATRIX TOPOLOGICAL ALGEBRAS

Let λ and μ be sequence spaces and have both the signed weak gliding hump property, (λ,μ) the algebra of the infinite matrix operators which transform λ into μ . In this paper, it is proved that if λ and μ are β spaces and λ β and μ β have also the signed weak gliding hump property, then for any polar topology τ, ((λ,μ),τ) is always sequentially complete locally convex topological algebra.

Computational Tools in Weighted Persistent Homology

In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology are introduced. For applications, the authors show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. They also prove a theorem to calculate the mod p^(2) weighted persistent homology provided with some information on the mod p weighted persistent homology.

Cohomology in 3D Magneto-Quasistatics Modeling

Electromagnetic modeling provides an interesting context to present a link between physical phenomena and homology and cohomology theories.Over the past twenty-five years,a considerable effort has been invested by the computational electromagnetics community to develop fast and general techniques for defining potentials.When magneto-quasi-static discrete formulations based on magnetic scalar potential are employed in problems which involve conductive regions with holes,cuts are needed to make the boundary value problem well defined.While an intimate connection with homology theory has been quickly recognized,heuristic definitions of cuts are surprisingly still dominant in the literature.The aim of this paper is first to survey several definitions of cuts together with their shortcomings.Then,cuts are defined as generators of the first cohomology group over integers of a finite CW-complex.This provably general definition has also the virtue of providing an automatic,general and efficient algorithm for the computation of cuts.Some counter-examples show that heuristic definitions of cuts should be abandoned.The use of cohomology theory is not an option but the invaluable tool expressly needed to solve this problem.

On the Supercyclicity and Hypercyclicity of the Operator Algebra

Let B（X） be the operator algebra for a separable infinite dimensional Hilbert space H, endowed with the strong operator topology or ＊-strong topology. We give sufficient conditions for a continuous linear mapping L ： B（X） →B（X） to be supercyclic or ,-supercyclic. In particular our condition implies the existence of an infinite dimensional subspace of supercyclic vectors for a mapping T on H. Hypercyclicity of the operator algebra with strong operator topology was studied＇ by Chan and here we obtain an analogous result in the case of ＊-strong operator topology.

Derivation algebra and automorphism group of generalized topological N = 2 superconformal algebra

We determine the derivation algebra and the automorphism group of the generalized topological N = 2 superconformal algebra.

Harish-Chandra Modules of the Intermediate Series over the Topological N = 2 Superconformal Algebra

The topological N=2 superconformal algebra was introduced by Dijkgraaf,Verlinde and Verlinde as the symmetry algebra of topological strings at d<1.We give a classification of irreducible Z×Z-graded modules of the intermediate series over this infinite-dimensional Lie superalgebra.

Homology Groups of Simplicial Complements

This paper deals with homology groups induced by the exterior algebra generated by the simplicial compliment of a simplicial complex K. By using ech homology and Alexander duality, the authors prove that there is a duality between these homology groups and the simplicial homology groups of K.