• Title/Summary/Keyword: Quadratic Form

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MULTIPLICATIVE FUNCTIONS COMMUTABLE WITH BINARY QUADRATIC FORMS x2 ± xy + y2

  • Poo-Sung, Park
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.1
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    • pp.75-81
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    • 2023
  • If a multiplicative function f is commutable with a quadratic form x2 + xy + y2, i.e., f(x2 + xy + y2) = f(x)2 + f(x) f(y) + f(y)2, then f is the identity function. In other hand, if f is commutable with a quadratic form x2 - xy + y2, then f is one of three kinds of functions: the identity function, the constant function, and an indicator function for ℕ \ pℕ with a prime p.

EVEN 2-UNIVERSAL QUADRATIC FORMS OF RANK 5

  • Ji, Yun-Seong;Kim, Myeong Jae;Oh, Byeong-Kweon
    • Journal of the Korean Mathematical Society
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    • v.58 no.4
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    • pp.849-871
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    • 2021
  • A (positive definite integral) quadratic form is called even 2-universal if it represents all even quadratic forms of rank 2. In this article, we prove that there are at most 55 even 2-universal even quadratic forms of rank 5. The proofs of even 2-universalities of some candidates will be given so that exactly 20 candidates remain unproven.

ESTIMATION OF NON-INTEGRAL AND INTEGRAL QUADRATIC FUNCTIONS IN LINEAR STOCHASTIC DIFFERENTIAL SYSTEMS

  • Song, IL Young;Shin, Vladimir;Choi, Won
    • Korean Journal of Mathematics
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    • v.25 no.1
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    • pp.45-60
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    • 2017
  • This paper focuses on estimation of an non-integral quadratic function (NIQF) and integral quadratic function (IQF) of a random signal in dynamic system described by a linear stochastic differential equation. The quadratic form of an unobservable signal indicates useful information of a signal for control. The optimal (in mean square sense) and suboptimal estimates of NIQF and IQF represent a function of the Kalman estimate and its error covariance. The proposed estimation algorithms have a closed-form estimation procedure. The obtained estimates are studied in detail, including derivation of the exact formulas and differential equations for mean square errors. The results we demonstrate on practical example of a power of signal, and comparison analysis between optimal and suboptimal estimators is presented.

Robust Stable Conditions Based on the Quadratic Form Lyapunov Function (2차 형식 Lyapunov 함수에 기초한 강인한 안정조건)

  • Lee, Dong-Cheol;Bae, Jong-Il;Jo, Bong-Kwan;Bae, Chul-Min
    • Proceedings of the KIEE Conference
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    • 2004.07d
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    • pp.2212-2214
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    • 2004
  • Robust stable analysis with the system bounded parameteric variation is very important among the various control theory. This study is to investigate the robust stable conditions using the quadratic form Lyapunov function in which the coefficient matrix is affined linear system. The quadratic stability using the quadratic form Lyapunov function is not investigated yet. The Lyapunov unction is robust stable not to be dependent by the variable parameters, which means that the Lyapunov function is conservative. We suggest the robust stable conditions in the Lyapunov function in which the variable parameters are dependent in order to reduce the conservativeness of quadratic stability.

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A FUNCTIONAL EQUATION RELATED TO QUADRATIC FORMS WITHOUT THE CROSS PRODUCT TERMS

  • Park, Won-Gil;Bae, Jae-Hyeong
    • Honam Mathematical Journal
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    • v.30 no.2
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    • pp.219-225
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    • 2008
  • In this paper, we obtain the general solution and the stability of the 2-dimensional vector variable quadratic functional equation f( x + y, z - w) + f(x - y, z + w) = 2f(x, z ) + 2f(y, ${\omega}$). The quadratic form f( x, y) = $ax^2$ + $by^2$ without cross product terms is a solution of the above functional equation.

Saddlepoint Approximation to Quadratic Form and Application to Intraclass Correlation Coefficient

  • Na, Jong-Hwa
    • Journal of the Korean Data and Information Science Society
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    • v.19 no.2
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    • pp.497-504
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    • 2008
  • In this paper we studied the saddlepoint approximations to the distribution of quadratic forms in normal variables. We derived the approximations as a special case of Na & Kim (2005). Also applications to a statistic which concerns intraclass correlation coefficient are presented. Simulations show the accuracy and availability of the suggested approximations.

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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.

SOME RESULTS OF MOMENTS IN MULTIVARIATE STATISTICAL DISTRIBUTION

  • Chul Kang;Park, Sang-Don
    • Journal of applied mathematics & informatics
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    • v.12 no.1_2
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    • pp.323-334
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    • 2003
  • We review the developmental history of the moment matrix of matrix quadratic form. This paper also investigates, the moment matrix of (non-central) Wishart distribution, which is multi-version of X$^2$ distribution.

Quadratic GARCH Models: Introduction and Applications (이차형식 변동성 Q-GARCH 모형의 비교연구)

  • Park, Jin-A;Choi, Moon-Sun;Hwan, Sun-Young
    • The Korean Journal of Applied Statistics
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    • v.24 no.1
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    • pp.61-69
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    • 2011
  • In GARCH context, the conditional variance (or volatility) is of a quadratic function of the observation process. Examine standard ARCH/GARCH and their variant models in terms of quadratic formulations and it is interesting to note that most models in GARCH context have contained neither the first order term nor the interaction term. In this paper, we consider three models possessing the first order and/or interaction terms in the formulation of conditional variances, viz., quadratic GARCH, absolute value GARCH and bilinear GARCH processes. These models are investigated with a view to model comparisons and applications to financial time series in Korea