Applications of Cluster and Factorial Analysis |Chapter 11 | Research Updates in Mathematics and Computer Science Vol. 7

The primary objective of the cluster analysis is to form clusters such that each cluster is an homogeneous as possible with respect to items of interest and as different as possible between clusters. Cluster analysis is a mathematical technique in Multivariate Data Analysis which indicates the proper guidelines in grouping the data into clusters. Factor Analysis is a combination of intrinsic decomposition and recurrent factor decomposition [1]. It is surpassing Statistical approaches, factor decomposition has endured from incertitude review justification. We can understand the concept with illustrated notations of cluster Analysis and various Clustering Techniques in this Research paper. Similarity and Dissimilarity measures and Dendrogram Analysis will be computed as required measures for Analysis. The approach of factor analysis is helpful in figuring out the underlying, hidden factors that underlie the correlations between the variables. Sometimes it's crucial to identify and isolate these facts for use in a variety of statistical techniques across different domains. With the help of illustrated factor analysis methodologies, we may comprehend the significance of factor analysis and its key concepts. We can estimate the Basic Factor Modeling and Factor Loadings, and also Factor Rotation process. Provides the complete application process and approaches of Principal Factor M.L. Factor and PCA comparison of Factor Analysis in this Research paper Since all these methods use some sort of similarity measures. The same thing can be carried out with a similarity measure matrix also. In the behavioral and social sciences, researchers need to develop scales for the various un-observable factors such as attitudes, images, intelligence, personality and patriotism.

 

Author(s) Details

Ramya Nemania

Department of Mathematics, Gitam School of Science, GITAM (Deemed to be University), Visakhapatnam, India and  I/c - Research Department, OPJS University, Churu, Rajasthan, India.

Daruri Venugopalb

I/c - Research Department, OPJS University, Churu, Rajasthan, India.

 

 

Please see the book here:- https://doi.org/10.9734/bpi/rumcs/v7/299

vSimplification of Lie's Similarity Transformation Method of Solving Coupled Nonlinear Partial Differential Equations and Exact Dyon Solutions SU(2)Yang-Mills-Higg's Field Equations | Chapter 5 | Research Updates in Mathematics and Computer Science Vol. 9

 

For any \(\mathrm{n}\) number of coupled nonlinear partial differential equations for spherically symmetric field equations of the typer \({ }^2\left(\frac{\partial^2 \phi_j}{\partial r^2}-\frac{\partial^2 \phi_j}{\partial t^2}\right)=F \_i(\phi \mathrm{j})\), where \(\mathrm{j}=1,2, \ldots \mathrm{n}\), are the number of dependent variables and \(\mathrm{F} \mathrm{i}\left(\phi \_\mathrm{j}\right)\) are any functions of dependent variables \(\phi_j, \mathrm{j}=1,2, . . \mathrm{n}\). and free of independent variables \(\mathrm{r}\) and \(\mathrm{t}\) then a similarity variable is found as \(\mathrm{s}(\mathrm{r}, \mathrm{t})=\mathrm{r} /\left[\left(r^2-t^2\right)-\kappa t / \tau+\kappa^2 /\left(4 \tau^2\right)\right]\), where \(\kappa\) and \(\tau \neq 0\) are arbitrary integration constants. Using \(s(r, t)\) above coupled partial differential equations can be transformed into coupled ordinary differential equations. This result may reduce lengthy calculations for finding similarity transformations of coupled partial differential equations. Using this similarity variable two exact Dyon solutions of spherically symmetric Yang-Mills-Higg's field equations are found with 'circular functions.' For which known solutions are with hyperbolic functions.

 

Author(s) Details

B.V. Baby

3/88, Jadkal Post, Udupi District, Karnataka State -576 233, India.

 

Please see the link:- https://doi.org/10.9734/bpi/rumcs/v9/607

Simplification of Lie's Similarity Transformation Method of Solving Coupled Nonlinear Partial Differential Equations and Exact Dyon Solutions SU(2)Yang-Mills-Higg's Field Equations | Chapter 5 | Research Updates in Mathematics and Computer Science Vol. 9

For any n number of coupled nonlinear partial differential equations for spherically symmetric field equations of the typer^2 ( (ð^2 ϕ_j)/(∂r^2 )-(∂^2 ϕ_j)/(∂t^2 ))=F_i (ϕ_j ), where j=1,2,…n, are the number of dependent variables and F_i (ϕ_j ) are any functions of dependent variables ϕ_j,j=1,2,..n. and free of independent variables r and t then a similarity variable is found as s(r,t)=r/[(r^2−t^2 )−κt/τ+κ^2/(4τ^2 )], where κ and τ≠0 are arbitrary integration constants. Using s(r,t) above coupled partial differential equations can be transformed into coupled ordinary differential equations. This result may reduce lengthy calculations for finding similarity transformations of coupled partial differential equations. Using this similarity variable two exact Dyon solutions of spherically symmetric Yang-Mills-Higg's field equations are found with ‘circular functions.' For which known solutions are with hyperbolic functions.

 

Author (s) Details

 

B.V. Baby
3/88, Jadkal Post, Udupi District, Karnataka State -576 233, India.

 

Please see the book here:- https://doi.org/10.9734/bpi/rumcs/v9/607

Lie’s Infinitesimal Continuous Parameter Similarity Transformation Method of Solving Coupled Nonlinear Partial Differential Equations and Exact Solutions of SU (2) Yang-Mills Field Equations | Chapter 5 | Research Updates in Mathematics and Computer Science Vol. 6

 

Using Lie’s Infinitesimal continuous parameter similarity transformation method of solving nonlinear coupled partial differential equations is used to find two new exact solutions of Wu-Yang-t’ Hooft-Julia-Zee ansatz reduced SU(2) Yang-Mills-Higgs field equations at Prasad-Sommerfield limits. These new solutions are with trigonometric circular functions unlike known hyperbolic functions. A general method of solving coupled nonlinear partial differential equations by reducing the number of independent variables by one, this method is explained.

 

Author(s)details:-

 

B. V. Baby
3/88, Jadkal Post, Udupi District, Karnataka State,576 233, India.

 

Please See the book here :- https://doi.org/10.9734/bpi/rumcs/v6/315

Vascular Flora Analysis in the Southern Part of Chott El Hodna Wetland, Algeria: An Advanced Research | Chapter 4 | Challenging Issues on Environment and Earth Science Vol. 9

 The goal of the study was to offer vegetation data for a wetland of international significance, which necessitated knowledge of its flora, particularly endemic or uncommon species, as well as those that highlight the value of local or regional biodiversity. The wetland in the southern portion of Chott El Hodna (M'Sila province, Algeria) corresponds to the mediterranean desert bioclimatic stage, which has moderate winters. 79 floristic samples were taken along two transects from north to south and east to west throughout the period 2009-2013 as part of an inventory of the natural vegetation in this area. We discovered 116 species, which were divided into 85 genera and 29 botanical families. The Liliopsida has just 02 families and 17 genera, while the Magnoliopsida has 27 families and 68 genera. There are six plant families that account for 65.52 percent of the overall flora (Poaceae, Asteraceae, Fabaceae, Chenopodiaceae, Brassicaceae and Caryophyllaceae). There are 20 endemic taxa in this flora, including eight uncommon species. Annuals (therophytes) account for 60.34 percent of the biological spectrum, whereas perennials account for 36.66 percent (chamaephytes: 18.10 percent , hemicryptophytes: 11.20 percent , phanerophytes: 6.03 percent and geophytes: 4.31 percent ). This is a feature of arid environments. The numerical analysis of vegetation using Srensen's similarity index and Detrended Correspondence Analysis (DCA) resulted in the identification of three primary groups of samples that reflect diverse environmental conditions and are sufficiently specific to the research area. The plant appeared to be well adapted to its surroundings. The preservation of this environment and flora must be prioritised in the event of an emergency.

Author(S) Details

Abdelghani Zedam
Department of Agricultural Sciences, University of M’Sila, M’Sila, Algeria.

Mohamed Fenni
Department of Agricultural Sciences, University of Setif1, Setif, Algeria.

View Book:- https://stm.bookpi.org/CIEES-V9/article/view/4937