The new Lanczos-Phillips type algorithm for computing the LDU decompositions of non-Hermitian Toeplitz matrices is presented by using Laurent biorthogonal polynomials. It can be applied to computation of the Thron continued fractions which are related to two point Padé approximations. We see that the new algorithm can compute the Thron continued fractions faster and more stably than the FG algorithm does.

View full abstract

Recently, Ogita and Aishima proposed an efficient eigendecomposition refinement algorithm for the symmetric eigenproblem. Their basic algorithm involves division by the difference of two approximate eigenvalues, and can become unstable when there are multiple eigenvalues. To resolve this problem, they proposed to replace those equations that cause instability with different equations and gave a convergence proof of the resulting algorithm. However, it is not straightforward to understand intuitively why the modified algorithm works, because it removes some of the necessary and sufficient conditions for obtaining the eigendecomposition. We give an answer to this question using Banach's fixed-point theorem.

View full abstract

We propose an almost periodic frequency arrangement (APFA) constructed by an irrational number group created using the power root of a prime number. Based on findings, it is possible to connect more than one million channels at a base station using the super-multicarrier APFA systems with the same communication quality characteristics as the current system. In this paper, we propose a new frame Lyapunov exponent (FLE) which estimates the phase difference sensitivity depending on the frame number. We reveal the universal feature of FLE and elucidate several characteristics such as short-term radiations of phases.

View full abstract

The double-exponential formula is known as a very efficient quadrature formula. An important point in eliciting its high performance is properly selecting mesh size and truncation numbers depending on a given positive integer. However, the standard selection formulas are not optimal, and there is room for improvement. In this paper, we propose improved selection formulas that reduce the error of the double-exponential formula. We also present a computable error bound of the modified double-exponential formula.

View full abstract

We discuss the size of the determinants, which appear in the determinant formulae of the relative class numbers of cyclotomic function fields. These are the determinants of integer symmetric matrices, whose entries are 0 or 1. We show that, for a smaller characteristic, the determinants are significantly large (in the absolute value) compared to the determinants of randomly generated such matrices, while for a larger characteristic, it is not the case. We explain why this happens by comparing some upper bounds.

View full abstract

We give a mathematical analysis of potential-theoretical numerical integration formulas mainly proposed by Tanaka et al. over weighted Hardy spaces, which are spaces of analytic functions with a certain decay. We investigate the case of choosing sampling points by discrete energy minimization. In order to bound the convergence rate of the numerical integration error, we make use of the recent result of Hayakawa and Tanaka on the function approximation formula.

View full abstract

In insurance mathematics, specifically in risk theory, mainly functional analytic techniques are used. In this paper, we give an alternative approach to deriving some well-known, basic, but important results on the classical collective risk model. Applying techniques based on Itô's calculus, we derive an integro-differential equation for the Gerber-Shiu function, under the Cramér-Lundberg model.

View full abstract

Hyper-dual numbers (HDN) are numbers defined by using nilpotent elements that differ from each other. The introduction of an operator to extend the domain of functions to HDN space based on Taylor expansion allows higher-order derivatives to be obtained from the coefficients. This study inductively defines matrix representations of HDN and proposes a numerical method for higher-order derivatives, called HDN-M differentiation, based on the matrix representations of HDN. The proposed method is characterized so that higher-order derivatives can be computed with matrix operation rules without implementations of the operation rules of HDN.

View full abstract

Dynamic mode decomposition (DMD) is a popular technique for extracting important information of nonlinear dynamical systems. In this paper, we focus on the DMD based on the total least squares (TLS), which is experimentally efficient for noisy datasets for a dynamical system, while the asymptotic analysis is not given. We propose a statistical model of random noise, adapting to the Koopman operator associated with the DMD. Moreover, under reasonable assumptions, we prove strong convergence of random variables, corresponding to the eigenpairs computed by the DMD based on the TLS.

View full abstract

Numerical verification methods are proposed in order to construct local Lyapunov functions around non-hyperbolic equilibria of dynamical systems described by ODEs in two dimensional space. The normal form theory in dynamical systems gives basic ideas of these methods. To prove negative definiteness of polynomials of higher degree than two, a new theorem on interval arithmetic is also proposed.

View full abstract

We study the arithmetic of bitangents of smooth quartics over global fields. With the aid of computer algebra systems and using Elsenhans--Jahnel's results on the inverse Galois problem for bitangents, we show that, over any global field of characteristic different from $2$, there exist smooth quartics which have bitangents over every local field, but do not have bitangents over the global field. We give an algorithm to find such quartics explicitly, and give an example over $\Q$. We also discuss a similar problem concerning symmetric determinantal representations.

View full abstract

The convergence of the BiConjugate Gradient (BiCG) method depends on its input matrices. We tried to predict the convergence of BiCG method by applying a Convolutional Neural Network to matrices that had been converted to grayscale images. Using 875 real non-symmetric matrices in the SuiteSparse Matrix Collection, we applied the 5-fold cross-validation method and were able to predict convergence with an average accuracy that exceeded 80\% for all cases in the test collection.

View full abstract

The best constants of discrete Sobolev inequalities corresponding to 1812 isomers of C60 fullerene are found. Classical mechanical models of these isomers with a linear spring on each edge are investigated. The best constants stand for rigidities of these models. We show the best constants of 1812 isomers are distinct rational numbers and among these, Buckyball (or equivalently truncated icosahedron) takes the least. In other words, one can say that the Buckyball is the most rigid among 1812 C60 fullerene isomers.

View full abstract

The Kuramoto--Sivashinsky equation for Jordan curves is used to model the smoldering combustion of a sheet of paper. Here, the behavior of a rotating wave bifurcating from an expanding circle solution to this equation is mathematically analyzed. We also present some numerical examples in which the rotating waves are visualized.

View full abstract

In the present paper, we introduce a generalization of Ho-Lee's binary tree interest rate model, which can be regarded as a Bayesian type extension, by combining the technique used by Akahori-Aoki-Nagata with a randomization by a Polya urn type reinforcement. Moreover, a multinomial extension of the generalization of the Ho-Lee model is also discussed.

View full abstract

In this paper, we address the problem of numerical integration, which can be solved by kernel quadrature. Existing methods have limitations. In particular, the nodes are not well-balanced when their number is small. We propose two new methods for generating nodes for quadrature in reproducing kernel Hilbert spaces. By using the explicit formula for the error of the quadrature, we improve a set of a fixed number of sampling points with a tractable optimization algorithm. We provide a theoretical analysis of the convergence rate of the error of our first method. Numerical experiments show that our methods are effective.

View full abstract

In this paper, we consider the asymptotic behavior of traveling wave solutions of a certain degenerate nonlinear parabolic equation for $\xi \equiv x - ct \to - \infty$ with $c>0$. We give a refined one of them, which was not obtained in the preceding work [Ichida-Sakamoto, J. Elliptic and Parabolic Equations, to appear], by an appropriate asymptotic study and properties of the Lambert $W$ function.

View full abstract

A new measure for disproportionality, especially for the case of parliament seat allocation, is proposed. After showing some good properties of the measure, we introduce three kinds of variation which work together for detailed investigation on the subject. Its performance is demonstrated by numerical studies using historical data of Japan.

View full abstract

In this study, texture shape optimization is performed to minimize the friction coefficient based on the adjoint variable method under constant load. The shape update equation based on the gradient descent method is regarded as a differential equation, and in order to improve the accuracy of the solution of the differential equation, a new shape update equation is proposed that adapts the second-order Taylor expansion. Shape optimization is performed for four different textures, and the results are compared with those based on the gradient descent method and the proposed shape update equation. FreeFEM++ is used for the calculation.

View full abstract