• Bulletin of the Korean Mathematical Society
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• v.35 no.2
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• pp.235-258
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• 1998

We define a crossing of a link without referring to a specific projection of the link and describe a construction of a non-normalized Alexander polynomial associated to collections of such crossings of oriented links under an equivalence relation, called homology relation. The polynomial is computed from a special Seifert surface of the link. We prove that the polynomial is well-defined for the homology equivalence classes, investigate its relationship with the combinatorially defined Alexander polynomials and study some of its properties.

• Nonlinear Functional Analysis and Applications
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• v.28 no.3
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• pp.647-657
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• 2023

The aim of this paper is to study certain subclass ${\tilde{S^q_{\Sigma}}}({\lambda},\,{\alpha},\,t,\,s,\,p,\,b)$ of analytic and bi-univalent functions which are defined by using symmetric q-derivative operator. We estimate the second and third coefficients of the Taylor-Maclaurin series expansions belonging to the subclass and upper bounds for Feketo-Szegö inequality. Furthermore, some relevant connections of certain special cases of the main results with those in several earlier works are also pointed out.

• Honam Mathematical Journal
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• v.43 no.2
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• pp.183-197
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• 2021

Recently several authors have extended the Beta function by using its integral representation. However, in many cases no expression of these extended functions in terms of classic special functions is known. In the present paper, we introduce a further extension by defining a family of functions G_r,s : ℝ^*₊ → ℂ, with r, s ∈ ℂ and ℜ(r) > 0. For given r, s, we prove that this function satisfies a second-order linear differential equation with rational coefficients. Solving this ODE, we express G_r,s as a combination of confluent hypergeometric functions. From this we deduce a new integral relation satisfied by Tricomi's function. We then investigate additional specific properties of G_r,1 which take the form of new non trivial integral relations involving exponential and error functions. We discuss the connection between G_r,1 and Stokes' first problem (or Rayleigh problem) in fluid mechanics which consists in determining the flow created by the movement of an infinitely long plate. For $r{\in}{\frac{1}{2}}{\mathbb{N}}^*$, we find additional relations between G_r,1 and Hermite polynomials. In view of these results, we believe the family of extended beta functions G_r,s will find further applications in two directions: (i) for improving our knowledge of confluent hypergeometric functions and Tricomi's function, (ii) and for engineering and physics problems.

• Structural Engineering and Mechanics
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• v.14 no.3
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• pp.263-285
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• 2002

This paper presents a numerical procedure for solving initial-value problems using the special functions which belong to a class of Rvachev's basis functions $R_{bf}$ based on algebraic and trigonometric polynomials. Because of infinite derivability of these functions, derivatives of all orders, required by differential equation of the problem and initial conditions, are used directly in the numerical procedure. The accuracy and stability of the proposed numerical procedure are proved on an example of a single degree of freedom system. Critical time step was also determined. An algorithm for solving multiple degree of freedom systems by the collocation method was developed. Numerical results obtained by $R_{bf}$ functions are compared with exact solutions and results obtained by the most commonly used numerical procedures for solving initial-value problems.

• Journal for History of Mathematics
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• v.15 no.3
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• pp.17-24
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• 2002

It will be described the discovery of fundamental algebras such as complex numbers and the quaternions. Cardano(1539) was the first to introduce special types of complex numbers such as 5$\pm$$\sqrt{-15}$. Girald called the number a$\pm$$\sqrt{-b}$ solutions impossible. The term imaginary numbers was introduced by Descartes(1629) in “Discours la methode, La geometrie.” Euler knew the geometrical representation of complex numbers by points in a plane. Geometrical definitions of the addition and multiplication of complex numbers conceiving as directed line segments in a plane were given by Gauss in 1831. The expression “complex numbers” seems to be Gauss. Hamilton(1843) defined the complex numbers as paire of real numbers subject to conventional rules of addition and multiplication. Cauchy(1874) interpreted the complex numbers as residue classes of polynomials in R[x] modulo $x^2$＋1. Sophus Lie(1880) introduced commutators [a, b] by the way expressing infinitesimal transformation as differential operations. In this paper, it will be studied general quaternion algebras to finding of algebraic structure in Algebras.

• Journal of the Korean Mathematical Society
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• v.55 no.1
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• pp.185-210
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• 2018

The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: $${\zeta}_E(s,x)={\sum_{n=0}^{\infty}}{\frac{(-1)^n}{(n+x)^s}}$$. In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions ${\zeta}_E(s,x)$. Furthermore, the relations between the values of a class of the Hurwitz-type (or Lerch-type) Euler zeta functions at rational arguments have also been given.

• Kyungpook Mathematical Journal
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• v.58 no.1
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• pp.19-35
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• 2018

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [30] and the second Appell function [6], we introduce here the incomplete Lauricella functions ${\gamma}^{(n)}_A$ and ${\Gamma}^{(n)}_A$ of n variables. We then systematically investigate several properties of each of these incomplete Lauricella functions including, for example, their various integral representations, finite summation formulas, transformation and derivative formulas, and so on. We provide relevant connections of some of the special cases of the main results presented here with known identities. Several potential areas of application of the incomplete hypergeometric functions in one and more variables are also pointed out.

• Bulletin of the Korean Mathematical Society
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• v.47 no.1
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• pp.187-193
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• 2010

In this paper, the moments of the Riesz-N$\acute{a}$gy-Tak$\acute{a}$cs(RNT) distribution over a general interval [a, b] $\subset$ [0, 1], are found through the moments of the RNT distribution over the unit interval, [0, 1]. This is done using some special features of the distribution and the fact that [0, 1] is a self-similar set in a dynamical system generated by the RNT distribution. The results are important for the study of the orthogonal polynomials with respect to the RNT distribution over a general interval.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1996.11a
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• pp.459-463
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• 1996

The lateral-shearing interferometer specially devised for production line inspection lenses is presented. The interferometer is composed with immersion oil and four prisms whose relative sliding motion provide lateral-shearing and phase-shifting. A special phase-measuring algorithm of a-bucket is adopted to compensate the phase-shifting error caused by the thickness reduction in the immersion oil Three different algorithm for determinating quantitative aberration of aspherical lenses are presented and compared with one another.

• Journal of the Korean Society for Aviation and Aeronautics
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• v.16 no.2
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• pp.21-27
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• 2008

An inverse method is introduced to construct the problem for the error analysis of the numerical solution of initial value problem. These problems constructed through this method have a known exact solution, even though analytical solutions are generally not obtainable. The process leading to the exact solution makes use of an initially available approximate numerical solution. A smooth interpolation of the approximate solution is forced to exactly satisfy the differential equation by analytically deriving a small forcing function to absorb all of the errors in the interpolated approximate solution. Using this special case exact solution, it is possible to investigate the relationship between global errors of a candidate numerical solution process and the associated tuning parameters for a given problem. Under the assumption that the original differential equation is well-posed with respect to the small perturbations, we thereby obtain valuable information about the optimal choice of the tuning parameters and the achievable accuracy of the numerical solution.