• 대한수학회논문집
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• 제12권3호
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• pp.645-653
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• 1997

We study centrally symmetric orthogonal polynomials satisfying an admissible partial differential equation of the form $$ Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y = \lambda_n y, $$ where $A, B, \cdots, E$ are polynomials independent of n and $\lambda_n$ is the eignevalue parameter depending on n. We show that they are either the product of Hermite polymials or the circle polynomials up to a complex linear change of variables. Also we give some properties of them.

• Journal of applied mathematics & informatics
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• 제17권1_2_3호
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• pp.787-796
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• 2005

We generalize a characterization of classical orthogonal polynomials and obtain a two-variable analogue. Also an example is given arising from a second order partial differential equation.

• 대한수학회지
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• 제41권6호
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• pp.1049-1070
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• 2004

We classify all partial differential equations with polynomial coefficients in $\chi$ and y of the form A($\chi$) $u_{{\chi}{\chi}}$ ＋ 2B($\chi$, y) $u_{{\chi}y}$ ＋ C(y) $u_{yy}$ ＋ D($\chi$) $u_{{\chi}}$ ＋ E(y) $u_{y}$ = λu, which has weak orthogonal polynomials as solutions and show that partial derivatives of all orders are orthogonal. Also, we construct orthogonal polynomials in d-variables satisfying second order partial differential equations in d-variables.s.

• Journal of applied mathematics & informatics
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• 제17권1_2_3호
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• pp.797-802
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• 2005

We generate simplex polynomials by using a method, which produces an OPS in (d + 1) variables from an OPS in d variables and the Jacobi polynomials. Also we obtain a partial differential equation of the form $${\Sigma}_{i,j=1}^{d+1}\;A_ij{\frac{{\partial}^2u}{{\partial}x_i{\partial}x_j}}+{\Sigma}_{i=1}^{d+1}\;B_iu\;=\;{\lambda}u$$, which has simplex polynomials as solutions, where ${\lambda}$ is the eigenvalue parameter.

• 대한수학회논문집
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• 제31권4호
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• pp.827-844
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• 2016

Here, in this paper, we aim at establishing some new unified integral and differential formulas associated with the $\bar{H}$-function. Each of these formula involves a product of the $\bar{H}$-function and Srivastava polynomials with essentially arbitrary coefficients and the results are obtained in terms of two variables $\bar{H}$-function. By assigning suitably special values to these coefficients, the main results can be reduced to the corresponding integral formulas involving the classical orthogonal polynomials including, for example, Hermite, Jacobi, Legendre and Laguerre polynomials. Furthermore, the $\bar{H}$-function occurring in each of main results can be reduced, under various special cases.

• Journal of Mechanical Science and Technology
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• 제20권3호
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• pp.366-375
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• 2006

In this paper, it is intended to introduce a method to solve multi-objective optimization problems and to evaluate its performance. In order to verify the performance of this method it is applied for a vertical roller mill for Portland cement. A design process is defined with the compromise decision support problem concept and a design process consists of two steps: the design of experiments and mathematical programming. In this process, a designer decides an object that the objective function is going to pursuit and a non-linear optimization is performed composing objective constraints with practical constraints. In this method, response surfaces are used to model objectives (stress, deflection and weight) and the optimization is performed for each of the objectives while handling the remaining ones as constraints. The response surfaces are constructed using orthogonal polynomials, and orthogonal array as design of experiment, with analysis of variance for variable selection. In addition, it establishes the relative influence of the design variables in the objectives variability. The constrained optimization problems are solved using sequential quadratic programming. From the results, it is found that the method in this paper is a very effective and powerful for the multi-objective optimization of various practical design problems. It provides, moreover, a reference of design to judge the amount of excess or shortage from the final object.

• 한국수학사학회지
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• 제15권1호
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• pp.83-92
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• 2002

Gator Szego was one of the most brilliant Mathematicians. Mathematical science owes him several fundamental contributions in such fields as theory of functions of a complex variables, conformal mapping, Fourier series, theory of orthogonal polynomials, and many others. He wrote the famous Polya-Szego Problems and Theorem in Analysis which is the two volume of concentrated mathematical beauty. In this paper, we mention Szego's life, Szego's work, and Szego reproducing kernel.