# International Journal of Pure Mathematics

**E-**ISSN: 2313-0571

Volume 1, 2014

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of NAUN Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

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**Volume 1, 2014**

Title of the Paper: **Local and Nonlocal Symmetries and Inverse Problems for Ordinary Differential Equations**

Authors: **Valentin F. Zaitsev, Lidiya V. Linchuk, Alexander V. Flegontov**

Pages: **72-76**

Abstract: Some recent results and perspectives for development of contemporary group analysis for ordinary differential equations are considered. The article deals with regular algorithms of searching for the higher symmetries (tangential, Lie-Backlund and nonlocal – exponential and non-exponential). The solution for an inverse problem in class of equations admitting some non-exponential nonlocal operator is given.

Title of the Paper: **An Extended Newton’s Method with Free Second-Order Derivatives**

Authors: **Young Hee Geum, Young Ik Kim**

Pages: **68-71**

Abstract: We propose the cubic-order numerical method free of second derivatives and derive the asymptotic error constant in terms of control parameters. Applying this proposed scheme to various test functions, numerical results show a good agreement with the theory analyzed in this paper and are proven using Mathematica with its high-precision computability.

Title of the Paper: **Spectral Equivalence of As-Scalar Operators. As-Decomposable and As-Spectral Operators**

Authors: **Cristina Şerbănescu, Ioan Bacalu**

Pages: **56-67**

Abstract: This paper is dedicated to the study of some properties of the operators which admit residually non-analytic functional calculus initiated in [16]. We shall also define and study the spectral s-capacities, and give several s-decomposability criteria. We shall further study the restrictions and the S-decomposable operators’ quotients. The concepts of As-spectral function, respectively As-decomposable and As-spectral operators are introduced and characterized here and several elementary properties concerning them are studied. These operators are natural generalizations of the notions of A-scalar, A-decomposable and A-spectral operators studied in [8] and appear, in generally, as restrictions or quotients of the last one.

Title of the Paper: **Analytic and Numeric Solution of Nonlinear Partial Differential Equations of Fractional Order**

Authors: **M. A. Abdou, M. M. El–kojok, S. A. Raad**

Pages: **47-55**

Abstract: The existence and uniqueness solution of the Cauchy problem are discussed and proved in a Banach space E due to Bielecki method and Picard method depending on the properties we expect a solution to possess. Moreover, some properties concerning the stability of solutions are obtained. The product Nyström method is used as a numerical method to obtain a nonlinear system of algebraic equations. Also, many important theorems related to the existence and uniqueness solution of the algebraic system are derived. Finally, an application is given and numerical results are obtained.

Title of the Paper: **Integral Fourier Transforms with Discontinuous Coefficients**

Authors: **Oleg Yaremko, Natalia Yaremko**

Pages: **43-46**

Abstract: This paper presents new methods for direct and inverse Fourier integral transform in piecewise- homogeneous axis. New formulas are obtained in the form of Hermite- type polynomial series. Method proposed by authors has features that distinguish it from well- known Fourier integral transform method. In particular, obtained formulas for direct and inverse Fourier integral transform in the form of Hermite- type polynomial series have symmetry and can be the basis for regularizing algorithms. In the article it is proved that the analogues of Hermite polynomials and Hermite functions form biorthogonal system.

Title of the Paper: **Approximation to Hypergeometric Distribution with Modified Binomial Distributions**

Authors: **Juthaphorn Sinsomboonthong**

Pages: **35-42**

Abstract: Two modified binomial approximations to a hypergeometric distribution—modified binomial distributions 2 and 3—are proposed and their accuracy is investigated in terms of the total variation distance. In addition, an efficiency comparison with a binomial approximation was conducted using a simulation study for 162 situations. It is found that the total variation distances of the two modified binomial approximations are less than that of a binomial approximation for almost all situations and tend to zero for a small sampling fraction whatever the levels of population size. Even for the large population size of 20,000, there seems to be no difference in the efficiencies of the two modified binomial approximations and the binomial approximation at all levels of the sampling fraction and the proportion of the population that has the specified attribute.

Title of the Paper: **Infinite Series Forms of the Derivatives of Two Types of Functions**

Authors: **Chii-Huei Yu**

Pages: **30-34**

Abstract: This article takes the mathematical software Maple as the auxiliary tool to study the differential problem of two types of functions. We can obtain the infinite series forms of any order derivatives of these two types of functions by using binomial series and differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order derivative values. On the other hand, we provide two examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying our answers by using Maple.

Title of the Paper: **A Method of Estimating the p-Adic Sizes Polynomials**

Authors: **S. H. Sapar, S. S. Aminudin, K. A. Mohd Atan**

Pages: **22-29**

Abstract: The exponential sum associated with f is defined as S (f; q) = ∑x mod q , where the sum is taken over a complete set of residues modulo q. The value of S (f; q) depends on the estimate of cardinality in the set V = {x mod q|fx 0 mod q} where fx is the partial derivatives of f with respect to x. In order to determine the cardinality, the p-adic sizes of common zeros of the partial derivative polynomials need to be obtained. This paper will give an estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree eight in p by using Newton polyhedron technique.

Title of the Paper: **The Stability of Equilibriums of a Fifth Order Ordinary Differential Equation**

Authors: **Chung-Hsien Tsai, Shy-Jen Guo, Jeng-Chi Yan**

Pages: **14-21**

Abstract: The objective of this paper is to study the stability of equilibrium points of a model equation, which governs two- dimensional steady capillary-gravity waves of an ideal fluid flow with Bond number near 1/3 and Froude number close to one.

Title of the Paper: **Generalized Bivariate Fibonacci-Like Polynomials**

Authors: **Yashwant K. Panwar, Mamta Singh**

Pages: **8-13**

Abstract: In this paper, we introduce a generalized bivariate Fibonacci-Like polynomials sequence, from which specifying initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Also we define some properties of generalized bivariate Fibonacci-Like polynomials.

Title of the Paper: **Monomiality Principle and Related Operational Techniques for Orthogonal Polynomials and Special Functions**

Authors: **Clemente Cesarano**

Pages: **1-7**

Abstract: The concepts and the related aspects of the monomiality principle are presented in this paper to explore different approaches for some classes of orthogonal polynomials. The associated operational calculus introduced by the monomiality principle allows us to reformulate the theory of Hermite, Laguerre and Legendre polynomials from a unified point of view. They are indeed shown to be particular cases of more general polynomials, whose usefulness in purely mathematical and applied context is discussed. The powerful tool represented by the Hermite and Laguerre polynomials allows us to derive classes of isospectral problems in applied mathematics and economics.