The binary quadratic equation () representing the hyperbola is studied for its non-zero distinct integer solutions. A few interesting properties among the solutions are presented. Employing the integer solutions of the equation under consideration, integer solutions for special straight lines, hyperbolas and parabolas are exhibited.
{"title":"On the Positive Pell Equation","authors":"N. Thiruniraiselvi, M. Gopalan","doi":"10.32861/ajams.67.85.92","DOIUrl":"https://doi.org/10.32861/ajams.67.85.92","url":null,"abstract":"The binary quadratic equation () representing the hyperbola is studied for its non-zero distinct integer solutions. A few interesting properties among the solutions are presented. Employing the integer solutions of the equation under consideration, integer solutions for special straight lines, hyperbolas and parabolas are exhibited.","PeriodicalId":375032,"journal":{"name":"Academic Journal of Applied Mathematical Sciences","volume":"208 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116515396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we employ variational iterative method (VIM) to develop a suitable Algorithm for the numerical solution of systems of Volterra integro-differential equations. The formulated algorithm is used to solve first and second order linear and nonlinear system of Volterra integrodifferential equations which demonstrated a good numerical approach to overcome lengthen computational and integral simplification involves. Moreover, the comparison of the exact solution with the approximated solutions are made and approximate solutions p(x) q(t) proved to converge to the exact solutions p(x) q(t) respectively. The results reveal that the formulated algorithm are simple, effective and faster than analytical approach of solving Volterra integro-differential equations.
{"title":"Computatıonal Algorıthm for the Numerıcal Solutıon of Systems of Volterra Integro-Dıfferentıal Equatıons","authors":"F. Iyanda, Tiamiyu Abdgafar Tunde","doi":"10.32861/ajams.66.66.76","DOIUrl":"https://doi.org/10.32861/ajams.66.66.76","url":null,"abstract":"In this paper, we employ variational iterative method (VIM) to develop a suitable Algorithm for the numerical solution of systems of Volterra integro-differential equations. The formulated algorithm is used to solve first and second order linear and nonlinear system of Volterra integrodifferential equations which demonstrated a good numerical approach to overcome lengthen computational and integral simplification involves. Moreover, the comparison of the exact solution with the approximated solutions are made and approximate solutions p(x) q(t) proved to converge to the exact solutions p(x) q(t) respectively. The results reveal that the formulated algorithm are simple, effective and faster than analytical approach of solving Volterra integro-differential equations.","PeriodicalId":375032,"journal":{"name":"Academic Journal of Applied Mathematical Sciences","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125431437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using the Lorentz model and Hamiltonian systems without dissipation as an example, spectral methods for analyzing the dynamics of systems with chaotic behavior are considered. The insufficiency of the traditional approach to the study of perturbation dynamics based on an analysis of the roots of the classical spectral equation is discussed. It is proposed to study nonlinear systems using the method of constructing spectral equations with different eigenvalues, which allows one to take into account the randomness and multiplicity of states. The spectral features of instability and chaos for systems without dissipation are shown by the example of short-wave perturbations of a flow of a weakly ionized plasma gas.
{"title":"Spectral Features of Systems With Chaotic Dynamics","authors":"Perevoznikov E. N.","doi":"10.32861/ajams.66.58.65","DOIUrl":"https://doi.org/10.32861/ajams.66.58.65","url":null,"abstract":"Using the Lorentz model and Hamiltonian systems without dissipation as an example, spectral methods for analyzing the dynamics of systems with chaotic behavior are considered. The insufficiency of the traditional approach to the study of perturbation dynamics based on an analysis of the roots of the classical spectral equation is discussed. It is proposed to study nonlinear systems using the method of constructing spectral equations with different eigenvalues, which allows one to take into account the randomness and multiplicity of states. The spectral features of instability and chaos for systems without dissipation are shown by the example of short-wave perturbations of a flow of a weakly ionized plasma gas.","PeriodicalId":375032,"journal":{"name":"Academic Journal of Applied Mathematical Sciences","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121885495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let δ: Cp→Cp be normal, then the linear map ( ) attains a local minimum at Cp if and only if z Cp such that ( )( ( )≥0. Also let Cp, and let ( ) have the polar decomposition ( ) ( ) then the map attains local minimum on Cp at T if and only if ( ) . Regarding orthogonality, let S Cp and let N(S) have the polar decomposition N(S) = U|N(S)|, then ( ) ( ) for X Cp if ( ) . Moreover, the map has a local minimum at if and only if ( )( ( )) for y . In this paper, we give some results on local minimum and orthogonality of normal derivations in Cp-Classes. We employ some techniques for normal derivations due to Mecheri, Hacene, Bounkhel and Anderson.
{"title":"On Properties of Derivations in Normed Spaces","authors":"Benard Okelo","doi":"10.32861/ajams.66.77.79","DOIUrl":"https://doi.org/10.32861/ajams.66.77.79","url":null,"abstract":"Let δ: Cp→Cp be normal, then the linear map ( ) attains a local minimum at Cp if and only if z Cp such that ( )( ( )≥0. Also let Cp, and let ( ) have the polar decomposition ( ) ( ) then the map attains local minimum on Cp at T if and only if ( ) . Regarding orthogonality, let S Cp and let N(S) have the polar decomposition N(S) = U|N(S)|, then ( ) ( ) for X Cp if ( ) . Moreover, the map has a local minimum at if and only if ( )( ( )) for y . In this paper, we give some results on local minimum and orthogonality of normal derivations in Cp-Classes. We employ some techniques for normal derivations due to Mecheri, Hacene, Bounkhel and Anderson.","PeriodicalId":375032,"journal":{"name":"Academic Journal of Applied Mathematical Sciences","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114018950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper a three-step two hybrid block method with two offgrid hybrid points chosen within interval [Xn,Xn+1] and [Xn+1,Xn+2] was developed to solve second Order Ordinary Differential Equations directly, using the power series as the basic function to approximate and generate some continuous schemes. The basic properties of the method was investigated and was found to converge. Numerical Solution of our method was tested on some stiff equations and was found to give better approximation than the existing method.
{"title":"Three-Step Two-Hybrid Block Method for the Direct Solution of Second-Order Ordinary Differential Equations","authors":"Raymond Dominic, T. K. Yusuf","doi":"10.32861/ajams.63.15.23","DOIUrl":"https://doi.org/10.32861/ajams.63.15.23","url":null,"abstract":"In this paper a three-step two hybrid block method with two offgrid hybrid points chosen within interval [Xn,Xn+1] and [Xn+1,Xn+2] was developed to solve second Order Ordinary Differential Equations directly, using the power series as the basic function to approximate and generate some continuous schemes. The basic properties of the method was investigated and was found to converge. Numerical Solution of our method was tested on some stiff equations and was found to give better approximation than the existing method.","PeriodicalId":375032,"journal":{"name":"Academic Journal of Applied Mathematical Sciences","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132110592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Principal component analysis (PCA) is one of the successful dimensionality reduction approaches for color face recognition. For various PCA methods, the experiments show that the contribution of eigenvectors is different and different weights of eigenvectors can cause different effects. Based on this, a modified and simplified color two-dimensional quaternion principal component analysis (M2D-QPCA) method is proposed along the framework of the color two-dimensional quaternion principal component analysis (2D-QPCA) method and the improved two-dimensional quaternion principal component analysis (2D-GQPCA) method. The shortcomings of 2D-QPCA are corrected and the CPU time of 2D-GQPCA is reduced. The experiments on two real face data sets show that the accuracy of M2D-QPCA is better than that of 2D-QPCA and other PCA-like methods and the CPU time of M2D-QPCA is less than that of 2D-GQPCA.
{"title":"M2D-QPCA: An Improved Quaternion Principal Component Analysis Method for Color Face Recognition","authors":"Song Song, Kaisong Sun, Minghui Wang","doi":"10.32861/ajams.62.5.14","DOIUrl":"https://doi.org/10.32861/ajams.62.5.14","url":null,"abstract":"Principal component analysis (PCA) is one of the successful dimensionality reduction approaches for color face recognition. For various PCA methods, the experiments show that the contribution of eigenvectors is different and different weights of eigenvectors can cause different effects. Based on this, a modified and simplified color two-dimensional quaternion principal component analysis (M2D-QPCA) method is proposed along the framework of the color two-dimensional quaternion principal component analysis (2D-QPCA) method and the improved two-dimensional quaternion principal component analysis (2D-GQPCA) method. The shortcomings of 2D-QPCA are corrected and the CPU time of 2D-GQPCA is reduced. The experiments on two real face data sets show that the accuracy of M2D-QPCA is better than that of 2D-QPCA and other PCA-like methods and the CPU time of M2D-QPCA is less than that of 2D-GQPCA.","PeriodicalId":375032,"journal":{"name":"Academic Journal of Applied Mathematical Sciences","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127504428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Quantitative and qualitative analysis of the Averaging methods for the parabolic partial differential equation appears as an exciting field of the investigation. In this paper, we generalize some known results due to Krol on the averaging methods and use them to solve the parabolic partial differential equation.
{"title":"Method of Averaging for Some Parabolic Partial Differential Equations","authors":"M. El-Borai, Hamed Kamal Awad, R. H. M. Ali","doi":"10.32861/ajams.61.1.4","DOIUrl":"https://doi.org/10.32861/ajams.61.1.4","url":null,"abstract":"Quantitative and qualitative analysis of the Averaging methods for the parabolic partial differential equation appears as an exciting field of the investigation. In this paper, we generalize some known results due to Krol on the averaging methods and use them to solve the parabolic partial differential equation.","PeriodicalId":375032,"journal":{"name":"Academic Journal of Applied Mathematical Sciences","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132752279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-02DOI: 10.32861/ajams.512.168.173
Eli Innocent Cleopas, G. C. Abanum
In this paper, we consider the existence and uniqueness for the controllability of a dynamical system. Here, measure of non-compactness of set was employed to examine the conditions for darbo’s fixed point theorem which is used to established the existence and uniqueness solution for nonlinear integro-differential equation with implicit derivatives.
{"title":"Existence and Uniqueness for the Controllability of a Dynamical System","authors":"Eli Innocent Cleopas, G. C. Abanum","doi":"10.32861/ajams.512.168.173","DOIUrl":"https://doi.org/10.32861/ajams.512.168.173","url":null,"abstract":"In this paper, we consider the existence and uniqueness for the controllability of a dynamical system. Here, measure of non-compactness of set was employed to examine the conditions for darbo’s fixed point theorem which is used to established the existence and uniqueness solution for nonlinear integro-differential equation with implicit derivatives.","PeriodicalId":375032,"journal":{"name":"Academic Journal of Applied Mathematical Sciences","volume":"277 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122953250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-05DOI: 10.32861/ajams.511.150.163
O. Maxwell, G. A. Osuji, Ibeakuzie Precious Onyedikachi, Chinelo Ijeoma Obi-Okpala, I. U. Chinedu, O. I. Frank
In regression analysis, it is relatively necessary to have a correlation between the response and explanatory variables, but having correlations amongst explanatory variables is something undesired. This paper focuses on five methodologies for handling critical multicollinearity, they include: Partial Least Square Regression (PLSR), Ridge Regression (RR), Ordinary Least Square Regression (OLS), Least Absolute Shrinkage and Selector Operator (LASSO) Regression, and the Principal Component Analysis (PCA). Monte Carlo Simulations comparing the methods was carried out with the sample size greater than or equal to the levels (n>p) considered in most cases, the Average Mean Square Error (AMSE) and Akaike Information Criterion (AIC) values were computed. The result shows that PCR is the most superior and more efficient in handling critical multicollinearity problems, having the lowest AMSE and AIC values for all the sample sizes and different levels considered.
在回归分析中，反应和解释变量之间的相关性是相对必要的，但解释变量之间的相关性是不希望的。本文重点介绍了处理关键多重共线性的五种方法，包括:偏最小二乘回归(PLSR)，岭回归(RR)，普通最小二乘回归(OLS)，最小绝对收缩和选择算子(LASSO)回归以及主成分分析(PCA)。在大多数情况下，在样本量大于或等于考虑的水平(n>p)的情况下，对方法进行蒙特卡罗模拟比较，计算平均均方误差(AMSE)和明池信息准则(Akaike Information Criterion)值。结果表明，PCR在处理关键多重共线性问题时最优、最有效，在所有样本量和不同水平下，其AMSE和AIC值都最低。
{"title":"Handling Critical Multicollinearity Using Parametric Approach","authors":"O. Maxwell, G. A. Osuji, Ibeakuzie Precious Onyedikachi, Chinelo Ijeoma Obi-Okpala, I. U. Chinedu, O. I. Frank","doi":"10.32861/ajams.511.150.163","DOIUrl":"https://doi.org/10.32861/ajams.511.150.163","url":null,"abstract":"In regression analysis, it is relatively necessary to have a correlation between the response and explanatory variables, but having correlations amongst explanatory variables is something undesired. This paper focuses on five methodologies for handling critical multicollinearity, they include: Partial Least Square Regression (PLSR), Ridge Regression (RR), Ordinary Least Square Regression (OLS), Least Absolute Shrinkage and Selector Operator (LASSO) Regression, and the Principal Component Analysis (PCA). Monte Carlo Simulations comparing the methods was carried out with the sample size greater than or equal to the levels (n>p) considered in most cases, the Average Mean Square Error (AMSE) and Akaike Information Criterion (AIC) values were computed. The result shows that PCR is the most superior and more efficient in handling critical multicollinearity problems, having the lowest AMSE and AIC values for all the sample sizes and different levels considered.","PeriodicalId":375032,"journal":{"name":"Academic Journal of Applied Mathematical Sciences","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128037966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-25DOI: 10.32861/ajams.510.140.149
O. Olawale
This paper proposes and describes the acumen on alternate two covariates linear Cosine and Sine regression functions that possessed a noisy-wave or tone frequencies via wave-trend of actualized observations of regressors and responsive variable needed in fitting a wavy equation of trigonometry regression. The method of maximum likelihood was used in estimating parameters associated to the Cosine and Sine alternate functions via vector coefficients as well as their distributional and residual properties. The estimations obtained via the method were enthralled to the noisy-wave mesokurtic observations of babies’ rate of heartbeats exactly an hour after birth (HR1), two hours after birth (HR2) and three hours after birth (HR3). The implementation and illustrative application was via R using the heartbeat dataset. It was gleaned that the trigonometry equation line .......
{"title":"On Two Covariates Cosine and Sine Noisy-Wave Trigonometry Regression of Heartbeats","authors":"O. Olawale","doi":"10.32861/ajams.510.140.149","DOIUrl":"https://doi.org/10.32861/ajams.510.140.149","url":null,"abstract":"This paper proposes and describes the acumen on alternate two covariates linear Cosine and Sine regression functions that possessed a noisy-wave or tone frequencies via wave-trend of actualized observations of regressors and responsive variable needed in fitting a wavy equation of trigonometry regression. The method of maximum likelihood was used in estimating parameters associated to the Cosine and Sine alternate functions via vector coefficients as well as their distributional and residual properties. The estimations obtained via the method were enthralled to the noisy-wave mesokurtic observations of babies’ rate of heartbeats exactly an hour after birth (HR1), two hours after birth (HR2) and three hours after birth (HR3). The implementation and illustrative application was via R using the heartbeat dataset. It was gleaned that the trigonometry equation line .......","PeriodicalId":375032,"journal":{"name":"Academic Journal of Applied Mathematical Sciences","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122124844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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