arXiv subjects
Publications and source records attributed to Kosuke Suzuki.
We establish formulas for the $b$-adic Walsh coefficients of functions in $C^α[0,1]$ for an integer $α\geq 1$ and give upper bounds on the Walsh coefficients of these functions. We also study the Walsh coefficients of periodic and non-periodic functions in reproducing kernel Hilbert spaces.
In this paper we study quasi-Monte Carlo integration of smooth functions using digital nets. We fold digital nets over $\mathbb{Z}_{b}$ by means of the $b$-adic tent transformation, which has recently been introduced by the authors, and employ such \emph{folded digital nets} as quadrature points. We first analyze the worst-case error of quasi-Monte Carlo rules using folded digital nets in reproducing kernel Hilbert spaces. Here we need to permit digital nets with "infinite digit expansions," which are beyond the scope of the classical definition of digital nets. We overcome this issue by considering the infinite product of cyclic groups and the characters on it. We then give an explicit means of constructing good folded digital nets as follows: we use higher order polynomial lattice point sets for digital nets and show that the component-by-component construction can find good \emph{folded higher order polynomial lattice rules} that achieve the optimal convergence rate of the worst-case error in certain Sobolev spaces of smoothness of arbitrarily high order.
In this paper, we study randomized quasi-Monte Carlo (QMC) integration using digitally shifted digital nets. We express the mean square QMC error of the $n$-th discrete approximation $f_n$ of a function $f\colon[0,1)^s\to \mathbb{R}$ for digitally shifted digital nets in terms of the Walsh coefficients of $f$. We then apply a bound on the Walsh coefficients for sufficiently smooth integrands to obtain a quality measure called Walsh figure of merit for root mean square error, which satisfies a Koksma-Hlawka type inequality on the root mean square error. Through two types of experiments, we confirm that our quality measure is of use for finding digital nets which show good convergence behaviors of the root mean square error for smooth integrands.
In this paper we investigate quasi-Monte Carlo (QMC) integration using digital nets over $\mathbb{Z}_b$ in reproducing kernel Hilbert spaces. The tent transformation, or the baker's transformation, was originally used for lattice rules by Hickernell (2002) to achieve higher order convergence of the integration error for smooth non-periodic integrands, and later, has been successfully applied to digital nets over $\mathbb{Z}_2$ by Cristea et al. (2007) and Goda (2014). The aim of this paper is to generalize the latter two results to digital nets over $\mathbb{Z}_b$ for an arbitrary prime $b$. For this purpose, we introduce the {\em $b$-adic tent transformation} for an arbitrary positive integer $b$ greater than 1, which is a generalization of the original (dyadic) tent transformation. Further, again for an arbitrary positive integer $b$ greater than 1, we analyze the mean square worst-case error of QMC rules using digital nets over $\mathbb{Z}_b$ which are randomly digitally shifted and then folded using the $b$-adic tent transformation in reproducing kernel Hilbert spaces. Using this result, for a prime $b$, we prove the existence of good higher order polynomial lattice rules over $\mathbb{Z}_b$ among the smaller number of candidates as compared to the result by Dick and Pillichshammer (2007), which achieve almost the optimal convergence rate of the mean square worst-case error in unanchored Sobolev spaces of smoothness of arbitrary high order.
We report the spontaneous generation of an Archimedean spiral pattern of fullerene via the evaporation of solvent. The self-organized spiral pattern exhibited equi-spacing on the order of micrometer between neighboring stripes. The characteristics of the spirals, such as the spacing between stripes, the number of stripes and the band width of stripes, could be controlled by tuning the thickness of the liquid bridge and the concentration of solution. The mechanism of pattern formation is interpreted in terms of a specific traveling wave on the liquid-solid interface accompanied by a stick-slip process of the contact line.
We report on a periodic precipitation pattern emerged from a drying meniscus via evaporation of a polystyrene solution in a Petri dish. It appeared a quasi-logarithmic spacing relation in the pattern as a result of stick-slip motion of the contact line towards the wall. A model based on the dynamics of the evaporating meniscus is proposed.