In this paper, we establish sixteen interesting generalized fractional integral and derivative fo... more In this paper, we establish sixteen interesting generalized fractional integral and derivative formulas including their composition formulas by using certain integral transforms involving generalized (p, q)-Mathieu-type series.
The Gamma function) (x Γ and the associated Gamma functions) (± Γ x are defined as distributions ... more The Gamma function) (x Γ and the associated Gamma functions) (± Γ x are defined as distributions and neutrix product
International Journal of Applied and Computational Mathematics, Jun 3, 2015
In this paper we investigate the solution of a generalized nonlinear fractional reaction diffusio... more In this paper we investigate the solution of a generalized nonlinear fractional reaction diffusion equation by the application of Sumudu Transform.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
The aim of this work is to introduce and study some new types of generalizations of pairwise para... more The aim of this work is to introduce and study some new types of generalizations of pairwise paralindeloff spaces, pairwise nearly paralindeloff and almost paralindeloff spaces. Some of their characterizations, properties and subsets are investigated and studied. Furthermore, the relation of among them with some pairwise separation axioms are considered.
In this paper, a family of geometric means for positive matrices is studied; we discuss possible ... more In this paper, a family of geometric means for positive matrices is studied; we discuss possible definitions of the geometric means of positive matrices, and some counter examples are given. It is still an open problem to find a completely satisfactory definition. Other problems are how to define the geometric, arithmetic, harmonic, $\alpha-$power and operator means of finitely many positive matrices. We generalize these means of two positive matrices to arrive the definitions of the weighted means of $k$ positive matrices. We recover and develop the relationship between the Ando's geometric mean and the Kronecker product to the Tracy-Singh product and other means. Some new attractive inequalities for the Tracy-Singh product, Khatri-Rao product and geometric means of several positive matrices are established. The results lead to the case of Kronecker and Hadamard products of any finite numbers of matrices. Keywords: Tracy-Singh product; Khatri-Rao product; Kronecker product; Had...
The aim of this work is to introduce and study some new types of generalizations of pairwise para... more The aim of this work is to introduce and study some new types of generalizations of pairwise paralindeloff spaces, pairwise nearly paralindeloff and almost paralindeloff spaces. Some of their characterizations, properties and subsets are investigated and studied. Furthermore, the relation of among them with some pairwise separation axioms are considered.
Fourier series is a method that can solve for many problems especially for solving various differ... more Fourier series is a method that can solve for many problems especially for solving various differential equations of interest in science and engineering. However, Gibbs oscillations will be occurs when using a truncated Fourier series. Thus, we solve the problems by using the corrected Fourier series. Here, we want to compare the results between the solutions that we get from Fourier series method and Corrected Fourier series method. The comparison between these two methods will be our finding.
In credibility models, the so-called structural parameters must be estimated before the credibili... more In credibility models, the so-called structural parameters must be estimated before the credibility estimators can be calculated. Several existing methods provide estimators, but these estimators are not necessarily unbiased or simple to use. In this paper, we introduce alternative unbiased estimators for structural parameters in credibility models with dependence induced by common effects. The main advantage of our estimators is their ease of application.
Computers & Mathematics with Applications, 2007
This paper is concerned with a long list of matrix products for partitioned and non-partitioned m... more This paper is concerned with a long list of matrix products for partitioned and non-partitioned matrices, including Kronecker, Khatri-Rao of first kind, Hadamard, Tracy-Singh, Khatri-Rao, block Kronecker, block Hadamard,and box products (sums). Our contribution is to gather the most useful connections of the matrix products from various sources and present them in one place. Several new attractive connections are also added to this collection. We give, as an example, two applications in order to show that these connections play a central role in many applications. These applications are: A block diagonal least-squares problem and a generalization of matrix inequalities involving Khatri-Rao products of positive definite matrices.
In this paper, the result are established in the following four ways: First, we present a general... more In this paper, the result are established in the following four ways: First, we present a general representation for the weighted Drazin inverse A d,W of an arbitrary rectangular matrix A ∈ M m,n involving Moore-Penrose inverse, which reduces to the well-known result if the matrix A is a square and W = I n. Second, we find represenations for the weighted Drazin inverse of the Tracy-Singh product A B of the two matrices A ∈ M m,n and B ∈ M p,q by using our approach. Third, the results are extended to the case of Tracy-Singh product of any finite number of matrices.The result lead to equalities involving Kronecker product, Drazin inverse and group inverse, as a special case. Finally, We apply our result to present the solution of restricted singular matrix equations.
The Khatri-Rao and Tracy-Singh products for partitioned matrices are viewed as generalized Hadama... more The Khatri-Rao and Tracy-Singh products for partitioned matrices are viewed as generalized Hadamard and generalized Kronecker products, respectively. We define the Khatri-Rao and Tracy-Singh sums for partitioned matrices as generalized Hadamard and generalized Kronecker sums and derive some results including matrix equalities and inequalities involving the two sums. Based on the connection between the Khatri-Rao and Tracy-Singh products (sums) and use mainly Liu's, Mond and Pečarić's methods to establish new inequalities involving the Khatri-Rao product (sum). The results lead to inequalities involving Hadamard and Kronecker products (sums), as a special case.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
In this paper, we introduce some new $I_\lambda$-lacunary statistically convergent sequence space... more In this paper, we introduce some new $I_\lambda$-lacunary statistically convergent sequence spaces of order $\alpha$ defined by a Musielak-Orlicz function. We study some relations between $I_\lambda$-lacunary statistically convergence with lacunary $I_\lambda$-summable sequences. Moreover we also study about the $I_\lambda$-lacunary statistically convergence of sequences in topological groups and give some important inclusion theorems.
This paper demonstrates a miraculousness of word Allah SWT in Quran based on the ideas from Risal... more This paper demonstrates a miraculousness of word Allah SWT in Quran based on the ideas from Risalei Nur, pioneer by Bediuzaman Saeed Nursi, a famous religion scholar from Turkey. Few examples will be shown to support the ideas on how word Allah SWT show wonder in Quran such, how every surah and its total verses can form an illustration pattern in graph of word Allah SWT, how word Allah SWT portray a numerical link with the total number of verse in Quran, how unique position of word Allah are patterned in Quran and also tawafuq or coincidence that happen within first few surah's in Quran that relate with word Allah. This study hopefully can boost the confidence of the readers of the majesty of Allah SWT contained in the Quran and increase the faith of the readers about the greatness of Quran itself.
In this article, we defined the generalized fractional differential Tremblay operator in the open... more In this article, we defined the generalized fractional differential Tremblay operator in the open unit disk that by usage the definition of the generalized Srivastava-Owa operator. In particular, we established a new operator denoted by H b;s;c z based on the normalized generalized fractional differential operator and represented by convolution product. Moreover, we studied the coefficient criteria of univalence, starlikeness and convexity for the last operator mentioned. Keywords Univalent function Á Subclasses of univalent function Á Generalized fractional differential operator Á Convolution Mathematics Subject Classification 30C45 Á 30C55 w à hðzÞ ¼z þ X 1 j¼mþ1 a j b j z j : ð4Þ
International Journal of Applied and Computational Mathematics, Aug 11, 2016
In this paper, we deal with the existence outcomes for a fractional iterative differential equati... more In this paper, we deal with the existence outcomes for a fractional iterative differential equation involving first order derivative in a Banach space. The appliance utilized in this study, is the non-expansive operator method and BGK (Browder-Ghode-Kirk) fixed point theorem. The fractional differential operator is taken in the sense of Riemann-Liouville.
The Journal of Nonlinear Sciences and Applications, Feb 19, 2017
The authors present some new inequalities of generalized Hermite-Hadamard's type for the class of... more The authors present some new inequalities of generalized Hermite-Hadamard's type for the class of functions whose second local fractional derivatives of order α in absolute value at certain powers are generalized s-convex functions in the second sense. Moreover, some applications are given.
This paper investigates the companion of Ostrowski's inequality in the framework of fractal s... more This paper investigates the companion of Ostrowski's inequality in the framework of fractal sets. First, a new identity related to local fractional integrals is introduced, serving as the foundation for establishing a set of inequalities applicable to functions with generalized $s$-convex and $s$-concave derivatives. An illustrative example is presented to validate the obtained results, demonstrating their accuracy. Additionally, the paper discusses several practical applications, highlighting the significance of the established inequalities. The research presented in this paper contributes to the growing field of studying functions on fractal sets, which has attracted considerable interest from scientists and engineers.
In the present work, we introduce two new local fractional integral operators involving Mittag–Le... more In the present work, we introduce two new local fractional integral operators involving Mittag–Leffler kernel on Yang’s fractal sets. Then, we study the related generalized Hermite–Hadamard-type inequality using generalized (E,h)-convexity and obtain two identities pertaining to these operators, and the respective first- and second-order derivatives are given. In terms of applications, we provide some new generalized trapezoid-type inequalities for generalized (E,h)-convex functions. Finally, some special cases are deduced for different values of δ, E, and h.
In this paper, we establish sixteen interesting generalized fractional integral and derivative fo... more In this paper, we establish sixteen interesting generalized fractional integral and derivative formulas including their composition formulas by using certain integral transforms involving generalized (p, q)-Mathieu-type series.
The Gamma function) (x Γ and the associated Gamma functions) (± Γ x are defined as distributions ... more The Gamma function) (x Γ and the associated Gamma functions) (± Γ x are defined as distributions and neutrix product
International Journal of Applied and Computational Mathematics, Jun 3, 2015
In this paper we investigate the solution of a generalized nonlinear fractional reaction diffusio... more In this paper we investigate the solution of a generalized nonlinear fractional reaction diffusion equation by the application of Sumudu Transform.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
The aim of this work is to introduce and study some new types of generalizations of pairwise para... more The aim of this work is to introduce and study some new types of generalizations of pairwise paralindeloff spaces, pairwise nearly paralindeloff and almost paralindeloff spaces. Some of their characterizations, properties and subsets are investigated and studied. Furthermore, the relation of among them with some pairwise separation axioms are considered.
In this paper, a family of geometric means for positive matrices is studied; we discuss possible ... more In this paper, a family of geometric means for positive matrices is studied; we discuss possible definitions of the geometric means of positive matrices, and some counter examples are given. It is still an open problem to find a completely satisfactory definition. Other problems are how to define the geometric, arithmetic, harmonic, $\alpha-$power and operator means of finitely many positive matrices. We generalize these means of two positive matrices to arrive the definitions of the weighted means of $k$ positive matrices. We recover and develop the relationship between the Ando's geometric mean and the Kronecker product to the Tracy-Singh product and other means. Some new attractive inequalities for the Tracy-Singh product, Khatri-Rao product and geometric means of several positive matrices are established. The results lead to the case of Kronecker and Hadamard products of any finite numbers of matrices. Keywords: Tracy-Singh product; Khatri-Rao product; Kronecker product; Had...
The aim of this work is to introduce and study some new types of generalizations of pairwise para... more The aim of this work is to introduce and study some new types of generalizations of pairwise paralindeloff spaces, pairwise nearly paralindeloff and almost paralindeloff spaces. Some of their characterizations, properties and subsets are investigated and studied. Furthermore, the relation of among them with some pairwise separation axioms are considered.
Fourier series is a method that can solve for many problems especially for solving various differ... more Fourier series is a method that can solve for many problems especially for solving various differential equations of interest in science and engineering. However, Gibbs oscillations will be occurs when using a truncated Fourier series. Thus, we solve the problems by using the corrected Fourier series. Here, we want to compare the results between the solutions that we get from Fourier series method and Corrected Fourier series method. The comparison between these two methods will be our finding.
In credibility models, the so-called structural parameters must be estimated before the credibili... more In credibility models, the so-called structural parameters must be estimated before the credibility estimators can be calculated. Several existing methods provide estimators, but these estimators are not necessarily unbiased or simple to use. In this paper, we introduce alternative unbiased estimators for structural parameters in credibility models with dependence induced by common effects. The main advantage of our estimators is their ease of application.
Computers & Mathematics with Applications, 2007
This paper is concerned with a long list of matrix products for partitioned and non-partitioned m... more This paper is concerned with a long list of matrix products for partitioned and non-partitioned matrices, including Kronecker, Khatri-Rao of first kind, Hadamard, Tracy-Singh, Khatri-Rao, block Kronecker, block Hadamard,and box products (sums). Our contribution is to gather the most useful connections of the matrix products from various sources and present them in one place. Several new attractive connections are also added to this collection. We give, as an example, two applications in order to show that these connections play a central role in many applications. These applications are: A block diagonal least-squares problem and a generalization of matrix inequalities involving Khatri-Rao products of positive definite matrices.
In this paper, the result are established in the following four ways: First, we present a general... more In this paper, the result are established in the following four ways: First, we present a general representation for the weighted Drazin inverse A d,W of an arbitrary rectangular matrix A ∈ M m,n involving Moore-Penrose inverse, which reduces to the well-known result if the matrix A is a square and W = I n. Second, we find represenations for the weighted Drazin inverse of the Tracy-Singh product A B of the two matrices A ∈ M m,n and B ∈ M p,q by using our approach. Third, the results are extended to the case of Tracy-Singh product of any finite number of matrices.The result lead to equalities involving Kronecker product, Drazin inverse and group inverse, as a special case. Finally, We apply our result to present the solution of restricted singular matrix equations.
The Khatri-Rao and Tracy-Singh products for partitioned matrices are viewed as generalized Hadama... more The Khatri-Rao and Tracy-Singh products for partitioned matrices are viewed as generalized Hadamard and generalized Kronecker products, respectively. We define the Khatri-Rao and Tracy-Singh sums for partitioned matrices as generalized Hadamard and generalized Kronecker sums and derive some results including matrix equalities and inequalities involving the two sums. Based on the connection between the Khatri-Rao and Tracy-Singh products (sums) and use mainly Liu's, Mond and Pečarić's methods to establish new inequalities involving the Khatri-Rao product (sum). The results lead to inequalities involving Hadamard and Kronecker products (sums), as a special case.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
In this paper, we introduce some new $I_\lambda$-lacunary statistically convergent sequence space... more In this paper, we introduce some new $I_\lambda$-lacunary statistically convergent sequence spaces of order $\alpha$ defined by a Musielak-Orlicz function. We study some relations between $I_\lambda$-lacunary statistically convergence with lacunary $I_\lambda$-summable sequences. Moreover we also study about the $I_\lambda$-lacunary statistically convergence of sequences in topological groups and give some important inclusion theorems.
This paper demonstrates a miraculousness of word Allah SWT in Quran based on the ideas from Risal... more This paper demonstrates a miraculousness of word Allah SWT in Quran based on the ideas from Risalei Nur, pioneer by Bediuzaman Saeed Nursi, a famous religion scholar from Turkey. Few examples will be shown to support the ideas on how word Allah SWT show wonder in Quran such, how every surah and its total verses can form an illustration pattern in graph of word Allah SWT, how word Allah SWT portray a numerical link with the total number of verse in Quran, how unique position of word Allah are patterned in Quran and also tawafuq or coincidence that happen within first few surah's in Quran that relate with word Allah. This study hopefully can boost the confidence of the readers of the majesty of Allah SWT contained in the Quran and increase the faith of the readers about the greatness of Quran itself.
In this article, we defined the generalized fractional differential Tremblay operator in the open... more In this article, we defined the generalized fractional differential Tremblay operator in the open unit disk that by usage the definition of the generalized Srivastava-Owa operator. In particular, we established a new operator denoted by H b;s;c z based on the normalized generalized fractional differential operator and represented by convolution product. Moreover, we studied the coefficient criteria of univalence, starlikeness and convexity for the last operator mentioned. Keywords Univalent function Á Subclasses of univalent function Á Generalized fractional differential operator Á Convolution Mathematics Subject Classification 30C45 Á 30C55 w à hðzÞ ¼z þ X 1 j¼mþ1 a j b j z j : ð4Þ
International Journal of Applied and Computational Mathematics, Aug 11, 2016
In this paper, we deal with the existence outcomes for a fractional iterative differential equati... more In this paper, we deal with the existence outcomes for a fractional iterative differential equation involving first order derivative in a Banach space. The appliance utilized in this study, is the non-expansive operator method and BGK (Browder-Ghode-Kirk) fixed point theorem. The fractional differential operator is taken in the sense of Riemann-Liouville.
The Journal of Nonlinear Sciences and Applications, Feb 19, 2017
The authors present some new inequalities of generalized Hermite-Hadamard's type for the class of... more The authors present some new inequalities of generalized Hermite-Hadamard's type for the class of functions whose second local fractional derivatives of order α in absolute value at certain powers are generalized s-convex functions in the second sense. Moreover, some applications are given.
This paper investigates the companion of Ostrowski's inequality in the framework of fractal s... more This paper investigates the companion of Ostrowski's inequality in the framework of fractal sets. First, a new identity related to local fractional integrals is introduced, serving as the foundation for establishing a set of inequalities applicable to functions with generalized $s$-convex and $s$-concave derivatives. An illustrative example is presented to validate the obtained results, demonstrating their accuracy. Additionally, the paper discusses several practical applications, highlighting the significance of the established inequalities. The research presented in this paper contributes to the growing field of studying functions on fractal sets, which has attracted considerable interest from scientists and engineers.
In the present work, we introduce two new local fractional integral operators involving Mittag–Le... more In the present work, we introduce two new local fractional integral operators involving Mittag–Leffler kernel on Yang’s fractal sets. Then, we study the related generalized Hermite–Hadamard-type inequality using generalized (E,h)-convexity and obtain two identities pertaining to these operators, and the respective first- and second-order derivatives are given. In terms of applications, we provide some new generalized trapezoid-type inequalities for generalized (E,h)-convex functions. Finally, some special cases are deduced for different values of δ, E, and h.
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Papers by Adem Kilicman