We obtained the best constants of Sobolev-type inequalities corresponding to higher-order partial differential operators $L=(\partial_t-\Delta+a_0)\cdots(\partial_t-\Delta+a_{M-1})$ and $L_0=(-\Delta+a_0)\cdots(-\Delta+a_{M-1})$ with positive distinct characteristic roots $a_0,\dots,a_{M-1}$, under the suitable assumption on $M$ and $n$. The best constants are given by $L^2$-norm of Green's functions of the boundary value problem $Lu=f(x,t)$ and $L_0 u=f(x)$. The Green's functions are expressed by the elliptic theta function.

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We propose a new structure-preserving integrator for the Korteweg-de Vries (KdV) equation. In this integrator, two independent structure-preserving techniques are newly combined; the “discrete variational derivative method” for constructing invariants-preserving integrator, and the “compact finite difference method” which is widely used in the area of numerical fluid dynamics for resolving wave propagation phenomena. Numerical experiments show that the new integrator is in fact advantageous than the existing integrators.

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We propose a numerical algorithm to compute a Jordan basis as well as the Jordan canonical form for matrices of block Schur form (BSF). Combining it with the standard preprocessing which reduces a square matrix to BSF, we establish an efficient numerical algorithm only with unitary processes to reveal a full detail of the Jordan structure for an arbitrarily given square matrix.

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This paper presents a new methodology to compute VaR in the portfolio credit loss model. The Wavelet Approximation can be useful to compute non-smooth distributions, often arising in small or concentrated portfolios. We contribute to this technique by extending the Wavelet Approximation for Vasicek one-factor model to multi-factor model. Key features of our new algorithm are: (i) a finite series expansion of the wavelet scaling coefficients, (ii) Wynn's epsilon-algorithm to accelerate convergence of those series, and (iii) an efficient spline interpolation to calculate the Laplace transforms. We illustrate the effectiveness of our algorithm through numerical examples.

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Principal component analysis (PCA) is a general method to analyse the factors of the term structure of interest rates. There are usually two or three factors. However, it is shown by Liu that when we apply PCA to forward rates, not spot rates, we need more factors to explain $95\%$ of variability. In order to verify the robustness of this result, we introduce another method based on Fourier series, which is proposed by Malliavin and Mancino. The results reconfirm the observation of Liu with different data sets. In particular, the Fourier series method gives us similar results to PCA.

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In 1948, Ward defined elliptic divisibility sequences satisfying a certain recurrence relation. An elliptic divisibility sequence arises from any choice of elliptic curve and initial point on that curve. In this paper, we propose a factorization algorithm based on elliptic divisibility sequences. We then discuss our implementations of the algorithm and its optimization, and estimate the computational complexity.

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In this paper, the difference between the residual and the true residual caused by the computation errors that arise in matrix multiplications for solutions generated by the Block IDR($s$) method is analyzed. Moreover, in order to reduce the difference between the residual and the true residual, a modified Block IDR($s$) method is proposed. Numerical experiments demonstrate that the difference under the proposed method is smaller than that of the conventional Block IDR($s$) method.

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A new procedure to design numerical schemes for coupled partial differential equations is proposed. The resulting schemes have discrete counterparts of conservative or dissipative quantity in original system. They also enjoy another welcome feature that they are constructed on staggered time meshes, by which each variables can be computed alternately with less computational costs than usual schemes. The procedure is demonstrated in the case of the coupled KdV equations.

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Topology optimization is to determine a shape or topology, having minimum cost. We are devoted entirely to minimum compliance (maximum stiffness) as minimum cost. An optimal shape $\Omega$ is realized as a distribution of material on a reference domain $D$, strictly larger than $\Omega$ in general. The optimal shape $\Omega$ and an equilibrium $u(\Omega)$ on $\Omega$ are approximated by material distributions on the domain $D$ and equilibriums also on $D$, respectively. This note gives a sufficient setting to the existence of an optimal material distribution.

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We present an exhaustive search method to find all small solutions of a multivariate modular linear equation over the integers on the basis of lattice enumeration technique. Previous methods become ineffective when the bound in the definition of small solutions becomes large. Our algorithm can find all the solutions in a given bound; therefore, it can cope with problems with large bounds. We demonstrate the superiority of our algorithm by applying it to the attack on the RSA-CRT with small secret exponent.

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The Jacobi preconditioner is well known as a preconditioner with high parallel efficiency to solve very large linear systems. However, the Jacobi preconditioner does not always show the great improvement of the convergence rate, because of the poor convergence property of the Jacobi method. In this paper, in order to improve the quality of the Jacobi preconditioner without loss its parallel efficiency, we introduce a weighted Jacobi-type preconditioner, and propose an optimization technique for the weight parameter. The numerical experiments indicate that the proposed preconditioner has higher quality and is more efficient than the traditional Jacobi preconditioner.

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