• Journal of the Korean Mathematical Society
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• v.48 no.6
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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.

• Structural Engineering and Mechanics
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• v.66 no.3
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• pp.387-397
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• 2018

In this paper, a general closed-form solution for evaluating the dynamic behavior of a Timoshenko beam on elastic foundation under a moving harmonic line load is formulated in the frequency-wavenumber domain and in a moving coordinate system. It is found that the characteristic equation is quartic with real coefficients only, and its poles can be presented explicitly. This enables the substitution of these poles into Cauchy's residue theorem, leading to the general closed-form solution. The solution can be reduced to seven existing closed-form solutions to different sub-problems and a new closed-form solution to the subproblem of a Timoshenko beam on an elastic foundation subjected to a moving quasi-static line load. Two examples are included to verify the solution.

• Honam Mathematical Journal
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• v.34 no.1
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• pp.93-101
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• 2012

Let $E_A^B$ denote the elliptic curve $E_A^B:y^2=x^3+Ax+B$. In this paper, we calculate the number of points on elliptic curves $E_A^0:y^2=x^3+Ax$ over $\mathbb{F}_p$ mod 24. For example, if $p{\equiv}1$ (mod 24) is a prime, $3t^2{\equiv}1$ (mod p) and A(-1 + 2t) is a quartic residue modulo p, then the number of points in $E_A^0:y^2=x^3+Ax$ is congruent to 0 modulo 24.