• Title, Summary, Keyword: quadratic forms

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UNIVERSAL QUADRATIC FORMS OVER POLYNOMIAL RINGS

  • Kim, Myung-Hwan;Wang, Yuanhua;Xu, Fei
    • Journal of the Korean Mathematical Society
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    • v.45 no.5
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    • pp.1311-1322
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    • 2008
  • The Fifteen Theorem proved by Conway and Schneeberger is a criterion for positive definite quadratic forms over the rational integer ring to be universal. In this paper, we give a proof of an analogy of the Fifteen Theorem for definite quadratic forms over polynomial rings, which is known as the Four Conjecture proposed by Gerstein.

Saddlepoint approximation to the distribution function of quadratic forms based on multivariate skew-normal distribution (다변량 왜정규분포 기반 이차형식의 분포함수에 대한 안장점근사)

  • Na, Jonghwa
    • The Korean Journal of Applied Statistics
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    • v.29 no.4
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    • pp.571-579
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    • 2016
  • Most of studies related to the distributions of quadratic forms are conducted under the assumption of multivariate normal distribution. In this paper, we suggested an approximation to the distribution of quadratic forms based on multivariate skew-normal distribution as alternatives for multivariate normal distribution. Saddlepoint approximations are considered and the accuracy of the approximations are verified through simulation studies.

Saddlepoint Approximations to the Distribution Function of Non-homogeneous Quadratic Forms (비동차 이차형식의 분포함수에 대한 안장점근사)

  • Na Jong-Hwa;Kim Jeong-Soak
    • The Korean Journal of Applied Statistics
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    • v.18 no.1
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    • pp.183-196
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    • 2005
  • In this paper we studied the saddlepoint approximations to the distribution of non-homogeneous quadratic forms in normal variables. The results are the extension of Kuonen's which provide the same approximations to homogeneous quadratic forms. The CGF of interested statistics and related properties are derived for applications of saddlepoint techniques. Simulation results are also provided to show the accuracy of saddlepoint approximations.

A Note on Independence of Quadratic Forms

  • Han, Sung-Shin
    • Journal of the Korean Statistical Society
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    • v.7 no.2
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    • pp.99-100
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    • 1978
  • This note is to demonstrate that the extention of Theorem 4.15 of Graybill on the independence of quadratic forms is possible. For the self-contained exposition, Theorem 4.15 is rewritten.

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CUSP FORMS IN S40 (79)) AND THE NUMBER OF REPRESENTATIONS OF POSITIVE INTEGERS BY SOME DIRECT SUM OF BINARY QUADRATIC FORMS WITH DISCRIMINANT -79

  • Kendirli, Baris
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.3
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    • pp.529-572
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    • 2012
  • A basis of a subspace of $S_4({\Gamma}_0(79))$ is given and the formulas for the number of representations of positive integers by some direct sums of the quadratic forms $x^2_1+x_1x_2+20x^2_2$, $4x^2_1{\pm}x_1x_2+5x^2_2$, $2x^2_1{\pm}x_1x_2+10x^2_2$ are determined.

A LOWER BOUND FOR THE NUMBER OF SQUARES WHOSE SUM REPRESENTS INTEGRAL QUADRATIC FORMS

  • Kim, Myung-Hwan;Oh, Byeong-Kweon
    • Journal of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.651-655
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    • 1996
  • Lagrange's famous Four Square Theorem [L] says that every positive integer can be represented by the sum of four squares. This marvelous theorem was generalized by Mordell [M1] and Ko [K1] as follows : every positive definite integral quadratic form of two, three, four, and five variables is represented by the sum of five, six, seven, and eight squares, respectively. And they tried to extend this to positive definite integral quadratic forms of six or more variables.

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ON THE PUBLIC KEY CRYPTOSYSTEMS OVER CLASS SEMIGROUPS OF IMAGINARY QUADRATIC NON-MAXIMAL ORDERS

  • Kim, Young-Tae;Kim, Chang-Han
    • Communications of the Korean Mathematical Society
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    • v.21 no.3
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    • pp.577-586
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    • 2006
  • In this paper we will propose the methods for finding the non-invertible ideals corresponding to non-primitive quadratic forms and clarify the structures of class SEMIGROUPS of imaginary quadratic orders which were given by Zanardo and Zannier [8], and we will give a general algorithm for calculating power of ideals/classes via the Dirichlet composition of quadratic forms which is applicable to cryptography in the class semigroup of imaginary quadratic non-maximal order and revisit the cryptosystem of Kim and Moon [5] using a Zanardo and Zannier [8]'s quantity as their secret key, in order to analyze Jacobson [7]'s revised cryptosystem based on the class semigroup which is an alternative of Kim and Moon [5]'s.