Pub Date : 2020-06-17DOI: 10.5772/intechopen.89341
Ivan I. Kyrchei
Generalized inverse matrices are important objects in matrix theory. In particu-lar, they are useful tools in solving matrix equations. The most famous generalized inverses are the Moore-Penrose inverse and the Drazin inverse. Recently, it was introduced new generalized inverse matrix, namely the core inverse, which was late extended to the core-EP inverse, the BT, DMP, and CMP inverses. In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, even for basic generalized inverses, there exist different determinantal representations as a result of the search of their more applicable explicit expressions. In this chapter, we give new and exclusive determinantal representations of the core inverse and its generalizations by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author.
{"title":"Determinantal Representations of the Core Inverse and Its Generalizations","authors":"Ivan I. Kyrchei","doi":"10.5772/intechopen.89341","DOIUrl":"https://doi.org/10.5772/intechopen.89341","url":null,"abstract":"Generalized inverse matrices are important objects in matrix theory. In particu-lar, they are useful tools in solving matrix equations. The most famous generalized inverses are the Moore-Penrose inverse and the Drazin inverse. Recently, it was introduced new generalized inverse matrix, namely the core inverse, which was late extended to the core-EP inverse, the BT, DMP, and CMP inverses. In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, even for basic generalized inverses, there exist different determinantal representations as a result of the search of their more applicable explicit expressions. In this chapter, we give new and exclusive determinantal representations of the core inverse and its generalizations by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author.","PeriodicalId":322265,"journal":{"name":"Functional Calculus","volume":"222 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124413382","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 : 2020-06-17DOI: 10.5772/intechopen.90888
Elimhan N. Mahmudov
{"title":"Optimal Control of Evolution Differential Inclusions with Polynomial Linear Differential Operators","authors":"Elimhan N. Mahmudov","doi":"10.5772/intechopen.90888","DOIUrl":"https://doi.org/10.5772/intechopen.90888","url":null,"abstract":"","PeriodicalId":322265,"journal":{"name":"Functional Calculus","volume":"181 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114589018","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 : 2020-04-23DOI: 10.5772/intechopen.90322
F. Soltani
In this paper, we establish an uncertainty inequality for a Hilbert space H . The minimizer function associated with a bounded linear operator from H into a Hilbert space K is provided. We come up with some results regarding Hardy and Dirichlet spaces on the unit disk .
{"title":"Analytical Applications on Some Hilbert Spaces","authors":"F. Soltani","doi":"10.5772/intechopen.90322","DOIUrl":"https://doi.org/10.5772/intechopen.90322","url":null,"abstract":"In this paper, we establish an uncertainty inequality for a Hilbert space H . The minimizer function associated with a bounded linear operator from H into a Hilbert space K is provided. We come up with some results regarding Hardy and Dirichlet spaces on the unit disk .","PeriodicalId":322265,"journal":{"name":"Functional Calculus","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115534046","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 : 2020-02-28DOI: 10.5772/intechopen.91479
Baver Okutmustur
The main purpose of this chapter is to provide a brief review of Hilbert space with its fundamental features and introduce reproducing kernels of the corresponding spaces. We separate our analysis into two parts. In the first part, the basic facts on the inner product spaces including the notion of norms, pre-Hilbert spaces, and finally Hilbert spaces are presented. The second part is devoted to the reproducing kernels and the related Hilbert spaces which is called the reproducing kernel Hilbert spaces (RKHS) in the complex plane. The operations on reproducing kernels with some important theorems on the Bergman kernel for different domains are analyzed in this part.
{"title":"A Survey on Hilbert Spaces and Reproducing Kernels","authors":"Baver Okutmustur","doi":"10.5772/intechopen.91479","DOIUrl":"https://doi.org/10.5772/intechopen.91479","url":null,"abstract":"The main purpose of this chapter is to provide a brief review of Hilbert space with its fundamental features and introduce reproducing kernels of the corresponding spaces. We separate our analysis into two parts. In the first part, the basic facts on the inner product spaces including the notion of norms, pre-Hilbert spaces, and finally Hilbert spaces are presented. The second part is devoted to the reproducing kernels and the related Hilbert spaces which is called the reproducing kernel Hilbert spaces (RKHS) in the complex plane. The operations on reproducing kernels with some important theorems on the Bergman kernel for different domains are analyzed in this part.","PeriodicalId":322265,"journal":{"name":"Functional Calculus","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114955889","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 : 2020-02-12DOI: 10.5772/intechopen.91140
Z. Dahmani, Meriem Mansouria Belhamiti
In this chapter, fractional calculus is used to develop some results on integral inequalities and differential equations. We develop some results related to the Hermite-Hadamard inequality. Then, we establish other integral results related to the Minkowski inequality. We continue to present our results by establishing new classes of fractional integral inequalities using a family of positive functions; these classes of inequalities can be considered as generalizations of order n for some other classical/fractional integral results published recently. As applications on inequalities, we generate new lower bounds estimating the fractional expectations and variances for the beta random variable. Some classical covariance identities, which correspond to the classical case, are generalised for any α ≥ 1, β ≥ 1. For the part of differential equations, we present a contribution that allow us to develop a class of fractional chaotic electrical circuit. We prove recent results for the existence and uniqueness of solutions for a class of Langevin-type equation. Then, by establishing some sufficient conditions, another result for the existence of at least one solution is also discussed.
{"title":"Integral Inequalities and Differential Equations via Fractional Calculus","authors":"Z. Dahmani, Meriem Mansouria Belhamiti","doi":"10.5772/intechopen.91140","DOIUrl":"https://doi.org/10.5772/intechopen.91140","url":null,"abstract":"In this chapter, fractional calculus is used to develop some results on integral inequalities and differential equations. We develop some results related to the Hermite-Hadamard inequality. Then, we establish other integral results related to the Minkowski inequality. We continue to present our results by establishing new classes of fractional integral inequalities using a family of positive functions; these classes of inequalities can be considered as generalizations of order n for some other classical/fractional integral results published recently. As applications on inequalities, we generate new lower bounds estimating the fractional expectations and variances for the beta random variable. Some classical covariance identities, which correspond to the classical case, are generalised for any α ≥ 1, β ≥ 1. For the part of differential equations, we present a contribution that allow us to develop a class of fractional chaotic electrical circuit. We prove recent results for the existence and uniqueness of solutions for a class of Langevin-type equation. Then, by establishing some sufficient conditions, another result for the existence of at least one solution is also discussed.","PeriodicalId":322265,"journal":{"name":"Functional Calculus","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131005938","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 : 2020-01-27DOI: 10.5772/intechopen.90302
K. Shah, T. Abdeljawad, H. Khalil, R. Khan
In this chapter, we develop an efficient numerical scheme for the solution of boundary value problems of fractional order differential equations as well as their coupled systems by using Bernstein polynomials. On using the mentioned polynomial, we construct operational matrices for both fractional order derivatives and integrations. Also we construct a new matrix for the boundary condition. Based on the suggested method, we convert the considered problem to algebraic equation, which can be easily solved by using Matlab. In the last section, numerical examples are provided to illustrate our main results.
{"title":"Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices of Bernstein Polynomials","authors":"K. Shah, T. Abdeljawad, H. Khalil, R. Khan","doi":"10.5772/intechopen.90302","DOIUrl":"https://doi.org/10.5772/intechopen.90302","url":null,"abstract":"In this chapter, we develop an efficient numerical scheme for the solution of boundary value problems of fractional order differential equations as well as their coupled systems by using Bernstein polynomials. On using the mentioned polynomial, we construct operational matrices for both fractional order derivatives and integrations. Also we construct a new matrix for the boundary condition. Based on the suggested method, we convert the considered problem to algebraic equation, which can be easily solved by using Matlab. In the last section, numerical examples are provided to illustrate our main results.","PeriodicalId":322265,"journal":{"name":"Functional Calculus","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127488704","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 : 2019-11-19DOI: 10.5772/intechopen.88279
Thomas Plocoste, R. Calif
During the last 20 years, many megacities have experienced air pollution leading to negative impacts on human health. In the Caribbean region, air quality is widely affected by African dust which causes several diseases, particularly, respiratory diseases. This is why it is crucial to improve the understanding of PM10 fluctuations in order to elaborate strategies and construct tools to predict dust events. A first step consists to characterize the dynamical properties of PM10 fluctuations, for instance, to highlight possible scaling in PM10 density power spectrum. For that, the scale-invariant properties of PM10 daily time series during 6 years are investigated through the theoretical Hilbert frame. Thereafter, the Hilbert spectrum in time-frequency domain is considered. The choice of theoretical frame must be relevant. A comparative analysis is also provided between the results achieved in the Hilbert and Fourier spaces.
{"title":"Spectral Observations of PM10 Fluctuations in the Hilbert Space","authors":"Thomas Plocoste, R. Calif","doi":"10.5772/intechopen.88279","DOIUrl":"https://doi.org/10.5772/intechopen.88279","url":null,"abstract":"During the last 20 years, many megacities have experienced air pollution leading to negative impacts on human health. In the Caribbean region, air quality is widely affected by African dust which causes several diseases, particularly, respiratory diseases. This is why it is crucial to improve the understanding of PM10 fluctuations in order to elaborate strategies and construct tools to predict dust events. A first step consists to characterize the dynamical properties of PM10 fluctuations, for instance, to highlight possible scaling in PM10 density power spectrum. For that, the scale-invariant properties of PM10 daily time series during 6 years are investigated through the theoretical Hilbert frame. Thereafter, the Hilbert spectrum in time-frequency domain is considered. The choice of theoretical frame must be relevant. A comparative analysis is also provided between the results achieved in the Hilbert and Fourier spaces.","PeriodicalId":322265,"journal":{"name":"Functional Calculus","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132947926","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: 10.5772/intechopen.88553
M. Saleem
Our aim in the present chapter is to introduce a new type of operations on the chaotic graph, namely, chaotic connected edge graphs under the identification topology. The concept of chaotic foldings on the chaotic edge graph will be discussed from the viewpoint of algebra and geometry. The relation between the chaotic homeomorphisms and chaotic foldings on the chaotic connected edge graphs and their fundamental group is deduced. The fundamental group of the limit chaotic chain of foldings on chaotic. Many types of chaotic foldings are achieved. Theorems governing these relations are achieved. We also discuss some applications in chemistry and biology.
{"title":"Folding on the Chaotic Graph Operations and Their Fundamental Group","authors":"M. Saleem","doi":"10.5772/intechopen.88553","DOIUrl":"https://doi.org/10.5772/intechopen.88553","url":null,"abstract":"Our aim in the present chapter is to introduce a new type of operations on the chaotic graph, namely, chaotic connected edge graphs under the identification topology. The concept of chaotic foldings on the chaotic edge graph will be discussed from the viewpoint of algebra and geometry. The relation between the chaotic homeomorphisms and chaotic foldings on the chaotic connected edge graphs and their fundamental group is deduced. The fundamental group of the limit chaotic chain of foldings on chaotic. Many types of chaotic foldings are achieved. Theorems governing these relations are achieved. We also discuss some applications in chemistry and biology.","PeriodicalId":322265,"journal":{"name":"Functional Calculus","volume":"353 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":"123401163","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 : 2019-09-30DOI: 10.5772/INTECHOPEN.89342
I. Ciric
The solution of certain differential equations is expressed using a special type of matrix series and is directly related to the solution of general systems of algebraic equations. Efficient formulae for matrix exponentials are derived in terms of rapidly convergent series of the same type. They are essential for two new solution methods, especially beneficial for large linear systems, namely an iterative method and a method based on an exact matrix product formula. The computational complexity of these two methods is analysed, and for both of them, the number of matrix exponential-vector multiplications required for an imposed accuracy can be predetermined in terms of the system condition. The total number of arithmetic operations involved is roughly proportional to n 2 , where n is the matrix dimension. The common feature of all the series in the results presented is that starting with a first term that is already well-conditioned, each subsequent term is computed by multiplication with an even better conditioned matrix, tending quickly to the identity matrix. This contributes substantially to the stability of the numerical computation. A very efficient method based on the numerical integration of a special kind of differential equations, applicable to even ill-conditioned systems, is also presented.
{"title":"New Matrix Series Formulae for Matrix Exponentials and for the Solution of Linear Systems of Algebraic Equations","authors":"I. Ciric","doi":"10.5772/INTECHOPEN.89342","DOIUrl":"https://doi.org/10.5772/INTECHOPEN.89342","url":null,"abstract":"The solution of certain differential equations is expressed using a special type of matrix series and is directly related to the solution of general systems of algebraic equations. Efficient formulae for matrix exponentials are derived in terms of rapidly convergent series of the same type. They are essential for two new solution methods, especially beneficial for large linear systems, namely an iterative method and a method based on an exact matrix product formula. The computational complexity of these two methods is analysed, and for both of them, the number of matrix exponential-vector multiplications required for an imposed accuracy can be predetermined in terms of the system condition. The total number of arithmetic operations involved is roughly proportional to n 2 , where n is the matrix dimension. The common feature of all the series in the results presented is that starting with a first term that is already well-conditioned, each subsequent term is computed by multiplication with an even better conditioned matrix, tending quickly to the identity matrix. This contributes substantially to the stability of the numerical computation. A very efficient method based on the numerical integration of a special kind of differential equations, applicable to even ill-conditioned systems, is also presented.","PeriodicalId":322265,"journal":{"name":"Functional Calculus","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114457336","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 : 2019-06-20DOI: 10.5772/INTECHOPEN.86940
M. Shubov, Laszlo P. Kindrat
Analytic and numerical results of the Euler-Bernoulli beam model with a two-parameter family of boundary conditions have been presented. The co-diagonal matrix depending on two control parameters ( k 1 and k 2 ) relates a two-dimensional input vector (the shear and the moment at the right end) and the observation vector (the time derivatives of displacement and the slope at the right end). The following results are contained in the paper. First, high accuracy numerical approximations for the eigenvalues of the discretized differential operator (the dynamics generator of the model) have been obtained. Second, the formula for the number of the deadbeat modes has been derived for the case when one control parameter, k 1 , is positive and another one, k 2 , is zero. It has been shown that the number of the deadbeat modes tends to infinity, as k 1 ! 1 þ and k 2 ¼ 0. Third, the existence of double deadbeat modes and the asymptotic formula for such modes have been proven. Fourth, numerical results corroborating all analytic findings have been produced by using Chebyshev polynomial approximations for the continuous problem.
{"title":"Spectral Analysis and Numerical Investigation of a Flexible Structure with Nonconservative Boundary Data","authors":"M. Shubov, Laszlo P. Kindrat","doi":"10.5772/INTECHOPEN.86940","DOIUrl":"https://doi.org/10.5772/INTECHOPEN.86940","url":null,"abstract":"Analytic and numerical results of the Euler-Bernoulli beam model with a two-parameter family of boundary conditions have been presented. The co-diagonal matrix depending on two control parameters ( k 1 and k 2 ) relates a two-dimensional input vector (the shear and the moment at the right end) and the observation vector (the time derivatives of displacement and the slope at the right end). The following results are contained in the paper. First, high accuracy numerical approximations for the eigenvalues of the discretized differential operator (the dynamics generator of the model) have been obtained. Second, the formula for the number of the deadbeat modes has been derived for the case when one control parameter, k 1 , is positive and another one, k 2 , is zero. It has been shown that the number of the deadbeat modes tends to infinity, as k 1 ! 1 þ and k 2 ¼ 0. Third, the existence of double deadbeat modes and the asymptotic formula for such modes have been proven. Fourth, numerical results corroborating all analytic findings have been produced by using Chebyshev polynomial approximations for the continuous problem.","PeriodicalId":322265,"journal":{"name":"Functional Calculus","volume":"43 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115699066","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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