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New Versions of the Hermite Bieler Theorem in Stability Contexts
Issue:
Volume 7, Issue 1, February 2019
Pages:
1-4
Received:
3 January 2019
Accepted:
24 January 2019
Published:
25 February 2019
Abstract: The Hermite-Bieler theorem played key roles in several control theory problems including the proof of Kharitonov’s theorem and derivations of elementary proofs of the Routh’s algorithm for determining the Hurwitz stability of a real polynomial. In the present work, we explore the stability of complex continuous-time systems of differential equations. Using the theory of positive paraodd functions, we obtain Hermite-Bieler like conditions for the Routh-Hurwitz stability of such systems. We also look at the problem of stability of discrete-time systems of difference equations. By using suitable conformal mappings, we also establish Hermite-Bieler like conditions for the Schur-Cohn stability of these systems. In both cases, the conditions are necessary as well as sufficient.
Abstract: The Hermite-Bieler theorem played key roles in several control theory problems including the proof of Kharitonov’s theorem and derivations of elementary proofs of the Routh’s algorithm for determining the Hurwitz stability of a real polynomial. In the present work, we explore the stability of complex continuous-time systems of differential equation...
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Effects of Gross Domestic Product and Inflation Rate on Unemployment Rate in Ghana: Comparative Analysis of Multiple Regression and Covariance Matrix Models
Brew Lewis,
Crankson Monica Veronica,
Nyarko Francis,
Ampofi Isaac
Issue:
Volume 7, Issue 1, February 2019
Pages:
5-12
Received:
4 February 2019
Accepted:
11 March 2019
Published:
22 April 2019
Abstract: This paper analyses the effects of Gross Domestic Product growth (GDP) and Inflation rate (INF) on Unemployment rate (UMP) in Ghana’s economy using covariance matrix and multiple regression models. The two models were examined separately on the same data of three variables and the different outputs analysed to determine the effectiveness among the two models. The analyses of the outputs highlight the significance of both predictor variables on unemployment rate in Ghana. Scatterplot and normal probability distribution (pnorm) graphs were used to analyse the normality of the predictor variables. Data on inflation rate and GDP growth spanning from 1991 to 2017 was used. The data was transformed to n X m matrix form for covariance –variance matrix analysis. The rows in the n by m data matrix were the multivariate observations on n units. Multiple regression analysis was performed on the data. Both the two methods provided the long-run effects of the two predictor variables on the unemployment rate. However, while multiple regression model could quantify the effect of each predictor variable on the predicted variable, the covariance matrix model only quantifies the relation existing between predictor variables and the predicted variable.
Abstract: This paper analyses the effects of Gross Domestic Product growth (GDP) and Inflation rate (INF) on Unemployment rate (UMP) in Ghana’s economy using covariance matrix and multiple regression models. The two models were examined separately on the same data of three variables and the different outputs analysed to determine the effectiveness among the ...
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Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease
Birke Seyoum Desta,
Purnachandra Rao Koya
Issue:
Volume 7, Issue 1, February 2019
Pages:
13-20
Received:
18 June 2018
Accepted:
28 April 2019
Published:
29 May 2019
Abstract: In this study the infected population is classified into two categories viz., chronic and acute and thus developed a five compartmental SEICIAR model. Also, both vaccination and treatments are included and studied their impact on the spread of hepatitis B virus. The present model is biologically meaningful and mathematically well posed since the solutions are proved to be positive as well as bounded. The basic reproduction number RO of the model is derived using the next generation matrix method. Further, the equilibrium points of the model are identified and mathematical analysis pertaining to their stability is conducted using Routh – Hurtiz criteria. It is shown that the disease free equilibrium point is locally and globally stable If RO＜1. On the other hand, the endemic equilibrium point is proved to be stable if RO＞1. Also, the numerical simulation study of the model is carried out using ode45 of MATLAB: Rung – Kutta order four. It is observed that, if the vaccination and treatment rates are increased then the infective population size decreases and evenfall to zero over time. Hence, it is concluded that the use of vaccination and treatment at the highest possible rates is essential so as to control the spread hepatitis B virus.
Abstract: In this study the infected population is classified into two categories viz., chronic and acute and thus developed a five compartmental SEICIAR model. Also, both vaccination and treatments are included and studied their impact on the spread of hepatitis B virus. The present model is biologically meaningful and mathematically well posed since the so...
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Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation
Issue:
Volume 7, Issue 1, February 2019
Pages:
21-29
Received:
1 April 2019
Accepted:
10 May 2019
Published:
3 June 2019
Abstract: In this paper, we study a class of the long-time behavior of solutions to initial-boundary value problems for higher order equations with nonlinear source term and strong damping term. First of all, give some space and marks as well as the basic assumption of stress and nonlinear source term, take the inner product on both sides of the equation and obtain a priori estimate of the global smooth solution of the equation by using Holder inequality, Yong inequality, Poincare inequality and Gronwall inequality. Then prove the existence of the global solution of the equation by using the Galerkin finite element method. The uniqueness of the global solution of the equation is proved, and then the bounded absorption set of the solution semi-group is constructed by a priori estimate. It is proved that the solution semi-group is uniformly bounded and completely continuous in the interior, thus the global attractor family of the equation is obtained. Then the original equation is linearized, and the differentiability of the solution semi-group is proved, and the line is further proved. The decay of the volume element of the sexualization problem is studied, and the finite Hausdorff dimension and Fractal dimension of the global attractor family are obtained.
Abstract: In this paper, we study a class of the long-time behavior of solutions to initial-boundary value problems for higher order equations with nonlinear source term and strong damping term. First of all, give some space and marks as well as the basic assumption of stress and nonlinear source term, take the inner product on both sides of the equation and...
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A Hybrid Structure Solution of Quaternion Lyapunov Equation and Its Optimal Approximation
Wang Yun,
Huang Jingpin,
Lan Jiaxin
Issue:
Volume 7, Issue 1, February 2019
Pages:
30-36
Received:
22 April 2019
Published:
15 June 2019
Abstract: Recently, the establishment of a multi-structure control system has demonstrated vital significance in practice. Its stability analysis are mostly determined by Lyapunov matrix equation. Tridiagonal-arrow matrix (TA matrix for short) is a special matrix with hybrid structure. In this paper, the problem of TA constraint solution to continuous Lyapunov equation A*X+XA=C over quaternion field is discussed. By using the representation of vectors of a TA matrix and Kronecker product of matrices, a constrained problem will be transformed into an unconstrained equation. Then the necessary and sufficient conditions for the equation with TA and self-conjugate TA solutions as well as the expression of general solution are obtained. Meanwhile, when the solution set is nonempty, by using invariance of Frobenius norm of orthogonal matrix product, the optimal approximation solution with minimal Frobenius norm for a given TA matrix is derived. Finally, two numerical examples are provided to verify the algorithm.
Abstract: Recently, the establishment of a multi-structure control system has demonstrated vital significance in practice. Its stability analysis are mostly determined by Lyapunov matrix equation. Tridiagonal-arrow matrix (TA matrix for short) is a special matrix with hybrid structure. In this paper, the problem of TA constraint solution to continuous Lyapun...
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