• Title/Summary/Keyword: Gauss transformation

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THE PARITIES OF CONTINUED FRACTION

  • Ahn, Young-Ho
    • Honam Mathematical Journal
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    • v.30 no.4
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    • pp.733-741
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    • 2008
  • Let T be Gauss transformation on the unit interval defined by T (x) = ${\frac{1}{x}}$ where {x} is the fractional part of x. Gauss transformation is closely related to the continued fraction expansions of real numbers. We show that almost every x is mod M normal number of Gauss transformation with respect to intervals whose endpoints are rational or quadratic irrational. Its connection to Central Limit Theorem is also shown.

UNIQUE CONTINUATION FOR SCHRӦDINGER EQUATIONS

  • Shin, Se Chul;Lee, Kyung Bok
    • Journal of the Chungcheong Mathematical Society
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    • v.15 no.2
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    • pp.25-34
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    • 2003
  • We prove a local unique continuation for Schr$\ddot{o}$dinger equations with time independent coefficients. The method of proof combines a technique of Fourier-Gauss transformation and a Carleman inequality for parabolic operator.

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NUMERICAL EVALUATION OF CAUCHY PRINCIPAL VALUE INTEGRALS USING A PARAMETRIC RATIONAL TRANSFORMATION

  • Beong In Yun
    • The Pure and Applied Mathematics
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    • v.30 no.4
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    • pp.347-355
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    • 2023
  • For numerical evaluation of Cauchy principal value integrals, we present a simple rational function with a parameter satisfying some reasonable conditions. The proposed rational function is employed in coordinate transformation for accelerating the accuracy of the Gauss quadrature rule. The efficiency of the proposed rational transformation method is demonstrated by the numerical result of a selected test example.

NEW RESULTS FOR THE SERIES 2F2(x) WITH AN APPLICATION

  • Choi, Junesang;Rathie, Arjun Kumar
    • Communications of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.65-74
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    • 2014
  • The well known quadratic transformation formula due to Gauss: $$(1-x)^{-2a}{_2F_1}\[{{a,b;}\\\hfill{21}{2b;}}\;-\frac{4x}{(1-x)^2}\]={_2F_1}\[{{a,a-b+\frac{1}{2};}\\\hfill{65}{b+\frac{1}{2};}}\;x^2\]$$ plays an important role in the theory of (generalized) hypergeometric series. In 2001, Rathie and Kim have obtained two results closely related to the above quadratic transformation for $_2F_1$. Our main objective of this paper is to deduce some interesting known or new results for the series $_2F_1(x)$ by using the above Gauss's quadratic transformation and its contiguous relations and then apply our results to provide a list of a large number of integrals involving confluent hypergeometric functions, some of which are (presumably) new. The results established here are (potentially) useful in mathematics, physics, statistics, engineering, and so on.

ON DISTRIBUTIONS IN GENERALIZED CONTINUED FRACTIONS

  • AHN, YOUNG-HO
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.6 no.2
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    • pp.1-8
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    • 2002
  • Let $T_{\phi}$ be a generalized Gauss transformation and $[a_1,\;a_2,\;{\cdots}]_{T_{\phi}}$ be a symbolic representation of $x{\in}[0,\;1)$ induced by $T_{\phi}$, i.e., generalized continued fraction expansion induced by $T_{\phi}$. It is shown that the distribution of relative frequency of [$k_1,\;{\cdots},\;k_n$] in $[a_1,\;a_2,\;{\cdots}]_{T_p}$ satisfies Central Limit Theorem where $k_i{\in}{\mathbb{N}}$ for $1{\leq}i{\leq}n$.

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Analysis of Herringbone Grooved Journal Bearing Using Generalized Coordinate Transformation (일반좌표계 변환을 이용한 헤링본 그루브 베어링의 해석)

  • 박상신;한동철
    • Proceedings of the Korean Society of Tribologists and Lubrication Engineers Conference
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    • 1999.06a
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    • pp.317-324
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    • 1999
  • The present work is an attempt to calculate the steady state pressure and perturbed pressure of herringbone grooved journal bearings. A generalized coordinate system is introduced to handle the complex bearing geometry. The coordinates are fitted to the groove boundary and the Reynold's equation is transformed to be fitted to this coordinates system using the Gauss divergence theorem. This method makes it possible to deal with an arbitrary configuration of a lubricated surface. The characteristics of finite herringbone grooved journal are well calculated using this method.

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Analysis of Herringbone Grooved Journal Bearing Using Generalized Coordinate Transformation (일반좌표계 변환을 이용한 헤링본 그루브 베어링의 해석)

  • 박상신;김영진;유송민
    • Tribology and Lubricants
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    • v.16 no.6
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    • pp.432-439
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    • 2000
  • The present work is an attempt to calculate the steady state pressure and perturbed pressure of herringbone grooved journal bearings. A generalized coordinate system is introduced to handle the complex bearing geometry. The coordinates are fitted to the groove boundary and the Reynold's equation is transformed to be fitted to this coordinate system using the Gauss divergence theorem. This method makes it possible to deal with an arbitrary configuration of a lubricated surface. The caharacteristics of finite herringbone groove journal bearing are well calculated using this method.

TWO RESULTS FOR THE TERMINATING 3F2(2) WITH APPLICATIONS

  • Kim, Yong-Sup;Choi, June-Sang;Rathie, Arjun K.
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.3
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    • pp.621-633
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    • 2012
  • By establishing a new summation formula for the series $_3F_2(\frac{1}{2})$, recently Rathie and Pogany have obtained an interesting result known as Kummer type II transformation for the generalized hypergeometric function $_2F_2$. Here we aim at deriving their result by using a very elementary method and presenting two elegant results for certain terminating series $_3F_2(2)$. Furthermore two interesting applications of our new results are demonstrated.