• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1781-1797
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• 2013

Let x : $M{\rightarrow}\mathbb{R}^n$ be an n - 1-dimensional hypersurface in $\mathbb{R}^n$, L be the Laguerre Blaschke tensor, B be the Laguerre second fundamental form and $D=L+{\lambda}B$ be the Laguerre para-Blaschke tensor of the immersion x, where ${\lambda}$ is a constant. The aim of this article is to study Laguerre Blaschke isoparametric hypersurfaces and Laguerre para-Blaschke isoparametric hypersurfaces in $\mathbb{R}^n$ with three distinct Laguerre principal curvatures one of which is simple. We obtain some classification results of such isoparametric hypersurfaces.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.875-884
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• 2016

Let x : $^{Mn-1}{\rightarrow}{\mathbb{R}}^n$ ($n{\geq}4$) be an umbilical free hyper-surface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. We denote the Laguerre scalar curvature by R and the trace-free Laguerre tensor by ${\tilde{L}}:=L-{\frac{1}{n-1}}tr(L)g$. In this paper, we prove a local classification result under the assumption of parallel Laguerre form and an inequality of the type $${\parallel}{\tilde{L}}{\parallel}{\leq}cR$$ where $c={\frac{1}{(n-3){\sqrt{(n-2)(n-1)}}}$ is appropriate real constant, depending on the dimension.

• The Transactions of the Korean Institute of Electrical Engineers A
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• v.48 no.6
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• pp.728-734
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• 1999

An on-line system identification scheme of unknown system is proposed based on a Laguerre models representation. The unknown parameters are detemined using recursive least-square identification. The proposed method have the advantage that an unknown system can be modelled without structural knowledge and assumption about the true model order and time delay. Therefore, the proposed method can make the design procedure very when compared to widely-used conventional method.

• Communications of the Korean Mathematical Society
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• v.30 no.3
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• pp.191-200
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• 2015

Generating functions play an important role in the investigation of various useful properties of the sequences which they generate. Exton [13] presented a very general double generating relation involving products of two Laguerre polynomials. Motivated essentially by Exton's derivation [13], the authors aim to show how one can obtain nineteen new generating relations associated with products of two Laguerre polynomials in the form of a single result. We also consider some interesting and potentially useful special cases of our main findings.

• Proceedings of the Acoustical Society of Korea Conference
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• 1994.06a
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• pp.692-697
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• 1994

In the actual sound environmental systems, it seems to be essentially difficult to exactly evaluate a whole probability distribution form of its response fluctuation, owing to various types of natural, social and human factors. Up to now, we very often reported two kinds of unified probability density expressions in the standard expansion from of Hermite and Laguerre type orthonormal series to generally evaluate non-Gaussian, non-linear correlation and/or non-stationary properties of the fluctuation phenomenon. However, in the real sound environment, there still remain many actual problems on the necessity of improving the above two standard type probability expressions for practical use. In this paper, first, a central point is focused on how to find a new probabilistic theory of practically evaluating the variety and complexity of the actual random fluctuations, especially through introducing some equivalence transformation toward two standard probability density expressions mentioned above in the expansion from of Hermite and Laguerre type orthonormal series. Then, the effectiveness of the proposed theory has been confirmed experimentally too by applying it to the actual problems on the response probability evaluation of various sound insulation systems in an acoustic room.

• The Journal of Korean Institute of Electromagnetic Engineering and Science
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• v.22 no.10
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• pp.1005-1011
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• 2011

In this paper, we present a marching-on-in-degree(MOD) finite difference method(FDM) based on the Helmholtz wave equation for analyzing transient electromagnetic responses in a general dispersive media. The two issues related to the finite difference approximation of the time derivatives and the time consuming convolution operations are handled analytically using the properties of the Laguerre functions. The basic idea here is that we fit the transient nature of the fields, the flux densities, the permittivity with a finite sum of orthogonal Laguerre functions. Through this novel approach, not only the time variable can be decoupled analytically from the temporal variations but also the final computational form of the equations is transformed from finite difference time-domain(FDTD) to a finite difference formulation through a Galerkin testing. Representative numerical examples are presented for transient wave propagation in general Debye, Drude, and Lorentz dispersive medium.

• Korean Journal of Mathematics
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• v.16 no.4
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• pp.481-494
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• 2008

Iteration functions $K_m(z)$ and $U_m(z)$, $m{\geq}2$are defined recursively using the determinant of a matrix. We show that the fixed-iterations of $K_m(z)$ and $U_m(z)$ converge to a simple zero with order of convergence m and give closed form expansions of $K_m(z)$ and $U_m(z)$: To show the convergence, we derive a recursion formula for $L_m$ and then apply the idea of Ford or Pomentale. We also find a Toeplitz matrix whose determinant is $L_m(z)/(f^{\prime})^m$, and then we adapt the well-known results of Gerlach and Kalantari et.al. to give closed form expansions.

• Proceedings of the Acoustical Society of Korea Conference
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• 1994.06a
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• pp.686-691
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• 1994

For evaluating the response fluctuation of the actual environmental acoustic system excited by arbitrary random inputs, it is important to predict a whole probability distribution form closely connected with evaluation indexes Lx, Leq and so on. In this paper, a new type evaluation method is proposed by introducing three functional models matched to the prediction of the response probability distribution from a problem-oriented viewpoint. Because of the positive variable of the sound intensity, the response probability density function can be reasonably expressed theoretically by a statistical Laguerre expansion series form. The relationship between input and output is described by the regression relationship between the distribution parameters(containing expansion coefficients of this expression) and the stochastic input. These regression functions are expressed in terms of the orthogonal series expansion and their parameters are determined based on the least-squares error criterion and the measure of statistical independency.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.973-1008
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• 1997

We reconsider the problem of calssifying all classical orthogonal polynomial sequences which are solutions to a second-order differential equation of the form $$ \ell_2(x)y"(x) + \ell_1(x)y'(x) = \lambda_n y(x). $$ We first obtain new (algebraic) necessary and sufficient conditions on the coefficients $\ell_1(x)$ and $\ell_2(x)$ for the above differential equation to have orthogonal polynomial solutions. Using this result, we then obtain a complete classification of all classical orthogonal polynomials : up to a real linear change of variable, there are the six distinct orthogonal polynomial sets of Jacobi, Bessel, Laguerre, Hermite, twisted Hermite, and twisted Jacobi.cobi.

• Kyungpook Mathematical Journal
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• v.63 no.3
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• pp.485-506
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• 2023

This paper investigates an extended form Hurwitz-Lerch zeta function, as well as related integral images, ordinary and fractional derivatives, and series expansions, using the term extended beta function. We establish a connection between the extended Hurwitz-Lerch zeta function and the Laguerre polynomials. Furthermore, we present a probability distribution application of the extended Hurwitz-Lerch zeta function ζ^𝛿,𝜇_𝜈,λ. Several results, both known and new, are shown to follow as special cases of our findings.