Abstract
This paper presents an abstract approach of analysing population growth in the field of algebraic topology using the tools of homology theory. For a topological space X and any point vn∈X, where vn is the n-dimensional surface, the group η=X,vn is called population of the space X. The increasing sequence from vin∈X to vjn∈X for i<j provides the bases for the population growth. A growth in population η=X,vn occurs if vin<vjn for all vin∈X and vjn∈X. This is described by the homological invariant Hηk=1. The aim of this paper is to construct the homological invariant Hηk and use Hηk=1 to analyse the growth of the population. This approach is based on topological properties such as connectivity and continuity. The paper made extensive use of homological invariant in presenting important information about the population growth. The most significant feature of this method is its simplicity in analysing population growth using only algebraic category and transformations.
Highlights
Algebraic topology which provides computable property for identifying topological properties has gained importance and popularity in many diverse fields of mathematics and other branches of science. ough the subject algebraic topology is a relatively new field of mathematics, its contribution to the study of the dynamics in both metric and topological spaces cannot be overruled
Does the information gained helps us to understudy both topological and geometric properties of the space, and provides a method of reducing topological problems and continuous maps into algebraic problems, thereby making it appropriate to determine some homological invariants. e essence of the homological invariant in this study is that it does not change under the deformations for the same property
The novelty is the introduction of a homological invariant outside statistical and differential equations. e homological invariants we have introduced in this paper will describe the continuous paths between the 2-dimensional surfaces or groups and will be referred to as the growth of the population
Summary
Algebraic topology which provides computable property for identifying topological properties has gained importance and popularity in many diverse fields of mathematics and other branches of science. ough the subject algebraic topology is a relatively new field of mathematics, its contribution to the study of the dynamics in both metric and topological spaces cannot be overruled. Does the information gained helps us to understudy both topological and geometric properties of the space, and provides a method of reducing topological problems and continuous maps into algebraic problems, thereby making it appropriate to determine some homological invariants. Homological invariant in algebraic topology shortens this process and provides convenient techniques to the study of continuous process such as population growth. E homological invariants we have introduced in this paper will describe the continuous paths between the 2-dimensional surfaces or groups and will be referred to as the growth of the population. In some previous works [6, 7], statistical and differential equation methods were used to study the growth of populations from a given data set. These statistical and differential equation models have proven to be successful methods in the study of both linear and nonlinear growth of population. ough there are other important methods which attempt to establish the growth of population, none of these methods adopted the concept of groups in the algebraic category. e algebraic topology used the concept of the homological invariant which preserves certain topological structures of the 2-dimensional surface (algebraic category). e objects of the group are limited to only the 2-dimensional surface. e paths (continuous maps) are helpful in producing the extension. e extension of the groups by the continuous maps in this study is referred to as the population growth
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