Original Research Article

DE-NOISING HIGH-DIMENSIONAL DATA VIA PERSISTENT HOMOLOGY: ALGORITHM & APPLICATION

H. N. ALAA, A. M. RASHAD, M. S. MOHAMMED

Asian Journal of Mathematics and Computer Research, Page 1-13

In this paper, we introduce the homological de-noising algorithm whose purpose is preparing point cloud data (PCD) sets that are noisy to analyze topological data. We essentially, base this method on the homological persistence whose using avoids choosing a parameter ε. Alternatively, the interval of ε must be defined for which the topological feather occurs. In computational, this algorithm is an efficient. We can run it easily. We improve and generalize the topological de-noising algorithm. We can apply this algorithm to noisy sampling, for example, the circle and sphere in the Euclidean space. Finally, we apply the new algorithm to the noisy sampling of the torusℝ3 in  and the Klein bottle in ℝ4 .

 

Original Research Article

APPLICATIONS OF PERSISTENT HOMOLOGY IN FINITE METRIC SPACES AND CLUSTERING

A. M. RASHAD, H. N. ALAA, M. S. MOHAMMED

Asian Journal of Mathematics and Computer Research, Page 14-27

Finite data sets are considered as metric spaces with defining the dissimilarity measure between the data points of them. In this work, we describe the persistent homology flowchart (PHF) in detail and study the persistence barcodes of Vietoris-Rips. PHF starts with a finite metric space and ends with a persistence diagram encoding features of the input data set. We compute  and -dimensional persistence diagrams and barcodes of the -point metric space, . Finally, as an application on PH flowchart, we interpret the single linkage clustering of    via -dimensional persistence diagram , we then deduce that persistent homology generalizes clustering.

Original Research Article

MATHEMATICAL MODELING OF THE DYNAMICS OF INFECTIOUS DISEASES WITH RELAPSE

HALSON OGETO NYABERI, VICTOR WANGILA WAKWABUBI

Asian Journal of Mathematics and Computer Research, Page 28-37

In this paper, we have formulated a mathematical model based on a series of ordinary dierential equations to study the transmission dynamics of infectious diseases that exhibit relapse. The basic reproduction number of the model was computed using the next generation matrix method. The existence of the equilibrium points of the model was investigated and stability analysis carried out. The disease free equilibrium point was found to be locally asymptotically stable when R0 < 1 and unstable when R0 > 1 and globally asymptotically stable when R0 < 1 and unstable when R0 > 1. The endemic equilibrium point was found to be locally and globally asymptotically stable when R0 > 1. The center manifold theory was used to investigate the type of bifurcation at R0 = 1.

Original Research Article

ON THE DUAL SPACE OF CONE NORMED SPACES

ANAS YUSUF, ABOR ISA GARBA, BELLO NAKONE

Asian Journal of Mathematics and Computer Research, Page 38-45

Cone normed spaces are the generalization of the normed spaces with many authors adjusting the theory to the classical one. Despite all the efforts of researchers in generalizing the theory, there is no specific research on the duality of cone normed space in the literature. In this paper, we investigate and study some properties of the space of all continuous linear mappings between cone normed spaces, this allows us to define the concept of dual in the setting of cone normed spaces, state some of its properties and used the properties to prove the Hahn-Banach Theorem in cone normed space.

Short Communications

NOTES ON THE DIVISIBILITY BY NINE OF THE SUMS OF EVEN AMICABLE PAIRS

PEDRO HENRIQUE COLOMBINI DELPINO

Asian Journal of Mathematics and Computer Research, Page 46-49

This article presents a perspective on the development of the theory of amicable numbers, focusing particularly on the contributions of Poulet, Gardner and Elvin Lee on the divisibility by nine of the sums of even amicable pairs.

In this way, this manuscript, after evaluating the contributions brought by these authors, retrieves Elvin Lee's indication that not all sums of even amicable pairs is divisible by nine, highlighting the eleven examples that refuted Gardner, and which will be called, in this article, exceptional even amicable pairs.

Finally, based on the eleven counterproofs mentioned, the article will propose two conjectures: 1) The final digits of the exceptional even amicable pairs follows a pattern; and 2) They will never end with the digits 2-2; 2-4; 2-6; 4-2; 4-4; 4-8; 6-2; 6-6; 8-4; 8-6; 8-8.