• Journal of the Chungcheong Mathematical Society
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• 제24권4호
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• pp.919-920
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• 2011

For two even unimodular positive definite integral quadratic forms A[X], B[X] in n-variables, J. K. Koo [1, Theorem 1] showed that ${\theta}_A(\tau)/{\theta}_B(\tau)$ is a rational function of J, satisfying a certain condition. Where ${\theta}_A(\tau)$ and ${\theta}_B(\tau)$ are theta series related to A[X] and B[X], respectively, and J is the classical modular invariant. In this paper we give a simple proof of Theorem 1 of [1].

• The Journal of the Korea Contents Association
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• 제3권3호
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• pp.103-109
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• 2003

Functional equations are useful in the expermental science because they play very important to formulate mathematical moods in general terms, through some not very restrictive equations, without postulating the forms of such functions. In this paper n solve one of a generalized quadratic functional equation (equation omitted) and prove the stability of this equation.

• The Journal of the Korea Contents Association
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• 제2권4호
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• pp.93-98
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• 2002

Functional equations are useful in the experimental science because they play very important role for researchers to formulate mathematical models in general terms, through some not very restrictive equations that only stipulate basic properties of functions showing in these equations, without postulating the exact forms of such functions. Of lots of such functional equations, in this paper we adopt and solve some generalized quadratic functional equation a$^2$f((x+y/a))+b$^2$f((x-y/b)) = 2f(x)+2f(y)

• Journal of the Korean Mathematical Society
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• 제48권6호
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• pp.1249-1268
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• 2011

For imaginary quadratic number fields F = $\mathbb{Q}(\sqrt{{\varepsilon}p_1{\ldots}p_{t-1}})$, where ${\varepsilon}{\in}${-1,-2} and distinct primes $p_i{\equiv}1$ mod 4, we give condition of 8-ranks of class groups C(F) of F equal to 1 or 2 provided that 4-ranks of C(F) are at most equal to 2. Especially for F = $\mathbb{Q}(\sqrt{{\varepsilon}p_1p_2)$, we compute densities of 8-ranks of C(F) equal to 1 or 2 in all such imaginary quadratic fields F. The results are stated in terms of congruence relation of $p_i$ modulo $2^n$, the quartic residue symbol $(\frac{p_1}{p_2})4$ and binary quadratic forms such as $p_2^{h+(2_{p_1})/4}=x^2-2p_1y^2$, where $h+(2p_1)$ is the narrow class number of $\mathbb{Q}(\sqrt{2p_1})$. The results are also very useful for numerical computations.

• Bulletin of the Korean Mathematical Society
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• 제20권2호
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• pp.81-85
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• 1983

It is the purpose of this paper to give some remarks on finite fields. We shall show that the little theorem of Fermat, Euler's criterion for quadratic residue mod p, and other few theorems in the number theory can be derived from the theorems in theory of finite field K=GF(p), where p is a prime. The forms of some irreducible ploynomials over K-GF(p) will be given explicitly.

• Korean Journal of Mathematics
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• 제30권2호
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• pp.335-339
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• 2022

In this paper, we find the forms of polynomials which generate consecutive prime values. Then, we consider about general form of polynomials and apply on the quadratics and tertiaries.

• International Journal of Ocean System Engineering
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• 제3권1호
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• pp.1-9
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• 2013

A design procedure for a ship with minimum resistance had been developed using a numerical optimization method called SQP (Sequential Quadratic Programming) combined with computational fluid dynamics (CFD) technique. The frictional resistance coefficient was estimated by the ITTC 1957 model-ship correlation line formula and the wave-making resistance coefficient was evaluated by the potential-flow panel method with the nonlinear free surface boundary conditions. The geometry of the hull surface was represented and modified by B-spline surface modeling technique during the optimization process. The Series 60 ($C_B$=0.60) hull was selected as a parent hull to obtain an optimized hull that produces minimum resistance. The models of the parent and optimized hull forms were tested at calm water condition in order to demonstrate the validity of the proposed methodolgy.

• The Korean Journal of Applied Statistics
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• 제22권1호
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• pp.171-179
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• 2009

In this paper we studied the effective approximations to the distribution of the sum of products of normal variables. Based on the saddlepoint approximations to the quadratic forms, the suggested approximations are very accurate and easy to use. Applications to the FSK (Frequency Shift Keying) communication are also considered.

• Journal of the Korean Mathematical Society
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• 제54권2호
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• pp.479-491
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• 2017

We investigate the zeros of Epstein zeta functions associated with positive definite quadratic forms with rational coefficients in the vertical strip ${\sigma}_1$ < ${\Re}s$ < ${\sigma}_2$, where 1/2 < ${\sigma}_1$ < ${\sigma}_2$ < 1. When the class number h of the quadratic form is bigger than 1, Voronin gave a lower bound and Lee gave an asymptotic formula for the number of zeros. Recently Gonek and Lee improved their results by providing a new upper bound for the error term when h > 3. In this paper, we consider the cases h = 2, 3 and provide an upper bound for the error term, smaller than the one for the case h > 3.

• Journal of Ocean Engineering and Technology
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• 제18권3호
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• pp.32-39
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• 2004

A design procedure for a ship with minimum total resistance has been developed using a numerical optimization method called SQP(sequential quadratic programming) to search for different optimal hull forms. The frictional resistance has been estimated using the ITTC 1957 model-ship correlation line formula, and the wave resistance has been evaluated using a potential-flow panel method that is based on Rankine sources with nonlinear free surface boundary conditions. The geometry of a hull surface has been modified using B-spline surface patches, during the whole optimization process. The numerical analyses have been carried out for the modified Wilgey hull at three different speeds (Fn=0.25, 0.316, 0.408), and the calculation results were compared.