MATHEMATICAL RESEARCHES
Year:2020 | Volume:5 | Issue:2|Papers Count:13

Introduction: In the last few years there is a growing interest in the theory of quasi-metric spaces and other related structures such as quasi-normed cones and asymmetric normed linear spaces, because such a theory provides an important tool in the study of several problems in theoretical computer science, approximation theory, applied physics, convex analysis and optimization. Many works on general topology and functional analysis have recently been obtained in order to extend the well-known results of the classical theory of normed linear spaces to the framework of asymmetric normed linear spaces and quasi-normed cones. An abstract cone is analogous to a real vector space, except that we take R+ as the set of scalars. In 2004, O. Valero introduced the normed cones and proved some closed graph and open mapping results for normed cones. Also Valero defined and studied some properties of quotient normed cones. P. Selinger studied the norm properties of a cone with its order properties and proved Hahn-Banach theorems in these cones under the appropriate conditions. Valero and his colleagues discussed the metrizability of the unit ball of the dual of a normed cone and the isometries of normed cones. Other properties are investigated in a series of papers by Romaguera, Sanchez Perez and Valero. The Bishop-Phelps theorem is a fundamental theorem in functional analysis which has many applications in the geometry of Banach spaces and optimization theory. The classical Bishop-Phelps theorem states that “ the set of support functionals for a closed bounded convex subset B of a real Banach space X, is norm dense in X* and the set of support points of B is dense in the boundary of B". Indeed, E. Bishop and R. R. Phelps answer a question posed by Victor Klee in 1958. We give an analogue to the normed cones, in fact we show that in a continuous normed cone the set of support points of a closed convex set is a dense subset of the boundary under the appropriate hypothesis. Conclusion: In this paper the notion of support points of convex sets in normed cones is introduced and it is shown that in a continuous normed cone, under the appropriate conditions, the set of support points of a bounded Scott-closed convex set is nonempty. We also present a Bishop-Phelps type Theorem for normed cones.

Introduction: A system of integral equations can describe different kind of problems in sciences and engineering. There are many different methods for numerical solution of linear and nonlinear system of integral equations. Material and methods: This paper proposed a numerical method based on modification of Hat functions for solving system of Fredholm-Hammerstein integral equations. The proposed method reduced a system of integral equation to a system of algebraic equations that can be solved easily by known methods. Results and discussion: For showing the accuracy and capability of the proposed method, some numerical examples are proposed that their results compared by results of other methods, and shows the capability and the superiority of this method to other existed methods. Also this paper derived the computational cost and the error analysis of the proposed method. Conclusion: The following conclusions were drawn from this research. This paper proposed a numerical method based on modification of Hat functions for solving system of Fredholm-Hammerstein integral equations. The proposed method reduced a system of integral equation to a system of algebraic equations that can be solved easily by known methods. The presented error analysis and solved problems show capability and the superiority of this method to other existed methods.

In this paper, we develop some stationary iterative schemes in block forms for solving double saddle point problem. To this end, we first generalize the Jacobi iterative method and study its convergence under certain condition. Moreover, using a relaxation parameter, the weighted version of the Jacobi method together with its convergence analysis are considered. Furthermore, we extend a method from the class of Gauss-Seidel iterative method and establish its convergence properties under a certain condition. In addition, the block successive overrelaxation (SOR) method is used to construct an iterative scheme to solve the mentioned double saddle point problem and its convergence properties are analyzed. In order to illustrate the efficiency of the proposed methods, we report some numerical experiments for a class of saddle point problems arising from the modeling of liquid crystal directors using finite elements...

Introduction: In this paper and by employing some certain techniques and results arisen in the theory of Tanaka, we provide a short proof for the maximum conjecture on the rigidity of Beloshapka's models of the specific CR dimension one and of length. As a consequence, we realize that the Lie group of biholomorphisms (CR automorphisms) associated with each of these models only consists of linear maps. Material and methods: Two major approaches employed so far to study Beloshapka's maximum conjecture have been 1) the approach of envelope of holomorphy, applied by Ilya Kossovskii to confirm the mentioned conjecture in the specific lengths three and four and 2) Cartan's approach of solving (biholomorphic) equivalence problems, employed by the present author in CR dimension one. Both of these approaches are geometric. In this paper, we introduce and employ the algebraic approach of applying some techniques in Tanaka's theory of transitive prolongations to study the already mentioned conjecture in the specific CR dimension one. The proofs are based upon some results concerning Tanaka prolongation of rank two fundamental algebras of lengths, achieved by Medori and Nacinovich (1997). Here, we prove first an algebraic parallel version of Beloshapka's conjecture and employ the results to solve this geometric open problem in CR dimension one. Results and discussion: As is the main goal of this paper, we confirm the maximum conjecture on the rigidity of Beloshapka's CR models in CR dimension one. As a consequence, we realize that each biholomorphic deformation of these models is linear. It may be worth to notice that the maximum conjecture in CR dimension one has been confirmed before by the present author, using the Cartan approach of solving equivalence problems. But, here we provide a much shorter proof for this result. Conclusion: The following conclusions were drawn from this research. We confirm Beloshapka's maximum conjecture in CR dimension one. It is introduced a new approach to consider this conjecture. Comparing the performance of the Tanaka approach, employed in this paper, with two approaches of "envelope of holomorphy" and "Cartan's theory", this algebraic approach seems the best weapon, introduced so far, to attack the maximum conjecture.

Throughout this paper, (R, m) is a commutative Noetherian local ring with the maximal ideal m. The following conjecture proposed by Bass [1], has been proved by Peskin and Szpiro [2] for almost all rings: (B) If R admits a finitely generated R-module of finite injective dimension, then R is Cohen-Macaulay. The problems treated in this paper are closely related to the following generalization of Bass conjecture which is still wide open: (GB) If R admits a finitely generated R-module of finite Gorenstein-injective dimension, then R is Cohen-Macaulay. Our idea goes back to the first steps of the solution of Bass conjecture given by Levin and Vasconcelos in 1968 [3] when R admits a finitely generated R-module of injective dimension ≤ 1. Levin and Vasconcelos indicate that if x m\m2 is a non-zerodivisor, then for every finitely generated R/xR-module M, there is id R M= id R/xr M+1. Using this fact, they construct a finitely generated R-module of finite injective dimension in the case where R is Cohen-Macaulay (the converse of Conjecture B). In this paper we study the Gorenstein injective dimension of local cohomology. We also show that if R is Cohen-Macaulay with minimal multiplicity, then every finitely generated module of finite Gorenstein injective dimension has finite injective dimension. We prove that a Cohen-Macaulay local ring has a finitely generated module of finite Gorenstein injective dimension.

In this paper we study the concept of Arveson spectrum on locally compact hypergroups and for an important class of compact countable hypergroups. In thiis paper we study the concept of Arveson spectrum on locally compact hypergroups and develop its basic properties for an important class of compact countable hypergroups...

In this paper, we consider the problem of means in several multivariate log-normal distributions and propose a useful method called as generalized variable method. Simulation studies show that suggested method has a appropriate size and power regardless sample size. To evaluation this method, we compare this method with traditional MANOVA such that the actual sizes of the two methods are close but the power of test and coverage probability of proposed methods are better than MANOVA in most cases specially when the sample sizes are small. Therefore, we can use this method when the variance-covariance matrices are not equal and there is not a suitable method...

Determining the order and the structure of the automorphism group of a finite p-group is an important problem in group theory. There have been a number of studies of the automorphism group of-p groups. Most of them deal with the order of Aut (G) the automorphism group of G, . Moreover various attempts have been made to find a structure for the automorphism group of a finite p-group. Following [2], [29], two groups are isoclinic if their commutator subgroups and central quotients are isomorphic and their commutator operations are essentially the same. We use the classification of p-groups by James [5], which based on isoclinism. By using [5] we have ten non-isoclinic families of non-abelian groups of order p^4. Let G be a finite non-abelian group of order p^4, Aut p (G)be the Sylow p--subgroup of Aut (G) and  = (G) be the Frattini subgroup of G. It is well-known [7, Satz III. 3. 17] that the Aut^  (G) the group of all automorphisms of Gcentralizing G/ (G)', is normal p-subgroup of Aut (G). In this paper we give a structure theorem for Aut p (G). Also we prove that if G is p-group of maximal class then Aut p (G)=Aut  (G) or Aut p (G) is a split extension of Aut  (G) by a cyclic group of order p.

Introduction: The modeling of many real-life physical systems leads to a set of fractional differential equations. Also fractional differential equations appear in various physical processes such as viscoelasticity and viscoplasticity, modeling of polymers and proteins, transmission of ultrasound waves, signal processing, control theory, etc. Most of fractional differential equations especially their nonlinear types do not have exact analytic solution, so numerical methods must be used. Therefore many authors have worked on the numerical solutions of this kind of equations. In recent years, many numerical methods have emerged, such as, the Adomian decomposition method, the Homotopy method, the multistep method, the extrapolation method, the spline collocation method, the product integration method and the predictor-corrector method. But most of the aforementioned methods consider the linear type of equations without a reliable theoretical justification. Then providing an efficient numerical scheme to approximate the solutions of nonlinear fractional differential equations is worthwhile and new in the literature. The main object of this paper is to develop and analyze a high order numerical method based on the collocation method when applies the orthogonal Jacobi polynomials as bases functions for the single term nonlinear fractional differential equations. Material and methods: Due to the well-known existence and uniqueness theorems the solutions of the fractional differential equations typically suffer from singularity at the origin. Consequently direct application of the Jacobi collocation method may lead to very weak numerical results. To fix this difficulty, we introduce a smoothing transformation that removes the singularity of the exact solution and enables us to approximate the solution with a satisfactory accurate result. Convergence analysis of the proposed scheme is also presented which demonstrates that the regularization process improves the smoothness of the input data and thereby increases the order of convergence. Results and discussion: We illustrate some test problems to show the effectiveness of the proposed scheme and to confirm the obtained theoretical predictions. In overall, the reported results justify that the proposed regularization strategy works well and the obtained approximate solutions have a good accuracy. To show the applicability of our approach we solve a practical example which is developed for a micro-electrical system (MEMS) instrument that has been designed primary to measure the viscosity of fluids that are encounter during oil well exploration using the proposed scheme. Moreover, we make a comparison between our scheme and the operational Tau method to show the efficiency of our technique. The reported results approve the superiority of the proposed approach. Finally, we consider a problem that we do not have access to its exact solution. In this case, we use the “ Variational Iteration Method (VIM)” as a qualitatively correct picture of the exact solution (the source solution) to evaluate the precision of the proposed technique. The obtained results approve that our approach produces the approximate solution which is in a good agreement with source ones. Conclusion: The following conclusions were drawn from this research. A reliable numerical method based on the Jacobi collocation method to approximate the solutions of a class of nonlinear fractional differential equations was developed. To achieve an efficient approximation a regularization strategy was proposed that improves the smoothness of the input data and enables us to obtain an approximate solution with a satisfactory accuracy. Convergence analysis of the proposed method was investigated which confirmed the high order of convergence of the proposed method.

Let G be a locally compact group with a fixed left Haar measure λ and ω be a weight function on G; that is a Borel measurable function ω : G→ (0,  ) with ω (x, y)≤ ω (x) ω (y) for all x, y  G...

Introduction: The stochastic calculus plays an important role in the study of stochastic integral equations and stochastic differential equations. The fractional Brownian motion has many applications in different branches of sciences such as economics, physics and biology. In many situations, the exact solution of these equations are not available or finding their exact solution is a very difficult process. Thus, finding an accurate and efficient numerical method for solving stochastic differential equations, and stochastic integral equations is important. Researchers have applied various numerical methods such as Dirichlet forms, Euler approximation, Skorohod integral, etc. In this paper, we used Haar wavelet functions for solving fractional Wiener integrals. Moreover, the error analysis of the proposed method is investigated. Material and methods: In this scheme, first we present the properties of the Haar wavelet functions then an efficient method based on these functions is proposed to estimate the solution of fractional Wiener integral with Hurst parameter H (1/2, 1). Results and discussion: We solve two numerical examples by using present method to demonstrate the efficiency and simplicity of the present method. For different values of, mean of error and standard deviation of error are shown in the tables. The obtained results confirm that proposed method enables us to find reasonable approximate solutions. Conclusion: The Haar wavelet is the simplest possible wavelet, so proposed method is easy to implement and it is a powerful mathematical tool to obtain the numerical solution of various kind of problems.

Introduction: Frames for Hilbert spaces were first introduced by Duffin and Schaeffer in 1952 to study some problems in nonharmonic Fourier series, reintroduced in 1986 by Daubechies, Grossmann and Meyer. Various generalizations of frames have been introduced and many applications of them in different branches have been presented. Bessel multipliers in Hilbert spaces were introduced by Peter Balazs. As we know in frame theory, the composition of the synthesis and analysis operators of a frame is called the frame operator. A multiplier for two Bessel sequences is an operator that combines the analysis operator, a multiplication pattern with a fixed sequence, called the symbol, and the synthesis operator. Bessel multipliers have useful applications, for example they are used for solving approximation problems and they have applications as time-variant filters in acoustical signal processing . We mention that many generalizations of Bessel multipliers have been introduced, also multipliers have been studied for non-Bessel sequences. Approximate duals in frame theory have important applications, especially are used for the reconstruction of signals when it is difficult to find alternate duals. Approximate duals are useful for wavelets, Gabor systems and in sensor modeling. Approximate duality of frames in Hilbert spaces was recently investigated by Christensen and Laugesen and some interesting applications of approximate duals were obtained. For example, it was shown that how approximate duals can be obtained via perturbation theory and some applications of approximate duals to Gabor frames especially Gabor frames generated by the Gaussian were presented. Afterwards, many authors studied approximate duals of Bessel sequences and many properties and generalizations of them were presented. In this note, we consider approximate duals for arbitrary sequences. Results and discussion: In this paper, we introduce some new kinds of duals and approximate duals in Hilbert spaces using multipliers, invertible operators and symbols. Many papers about approximate duals and their applications have been written so far which in these papers approximate duals have been considered for Bessel sequences. Here, we introduce approximate duals for arbitrary sequences in a Hilbert space, compare them with Bessel approximate duals and we show that they can be useful for the reconstruction of signals though they do not have all of the properties of Bessel approximate duals. Moreover, we obtain some new results for Bessel approximate duals. Conclusion: The following conclusions were drawn from this research. New kinds of duals and approximate duals for arbitrary sequences are introduced using multipliers, invertible operators and symbols. Duals and approximate duals of non-Bessel sequences are compared with the Bessel ones and some differences between them are shown by presenting various examples. Some properties and applications of duals and approximate duals of non-Bessel sequences are stated. Some new results about duals and approximate duals of Bessel sequences are obtained especially some important concepts such as closeness of Bessel sequences, nearly Parseval frames and multipliers with constant symbols are related to approximate duals of frames.