Super-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions

We investigate multivariate integration for a space of infinitely times differentiable functions $mathcal{F}_{s, boldsymbol{u}} := {f in C^infty [0,1]^s mid | f |_{mathcal{F}_{s, boldsymbol{u}}} < infty }$, where $| f |_{mathcal{F}_{s, boldsymbol{u}}} := sup_{boldsymbol{alpha} = (alpha_1, dots, alpha_s) in mathbb{N}_0^s} |f^{(boldsymbol{alpha})}|_{L^1}/prod_{j=1}^s u_j^{alpha_j}$, $f^{(boldsymbol{alpha})} := frac{partial^{|boldsymbol{alpha}|}}{partial x_1^{alpha_1} cdots partial x_s^{alpha_s}}f$ and $boldsymbol{u} = {u_j}_{j geq 1}$ is a sequence of positive decreasing weights. Let $e(n,s)$ be the minimal worst-case error of all algorithms that use $n$ function values in the $s$-variate case. We prove that for any $boldsymbol{u}$ and $s$ considered $e(n,s) leq C(s) exp(-c(s)(log{n})^2)$ holds for all $n$, where $C(s)$ and $c(s)$ are constants which may depend on $s$. Further we show that if the weights $boldsymbol{u}$ decay sufficiently fast then there exist some $1 < p < 2$ and absolute constants $C$ and $c$ such that $e(n,s) leq C exp(-c(log{n})^p)$ holds for all $s$ and $n$. These bounds are attained by quasi-Monte Carlo integration using digital nets. These convergence and tractability results come from those for the Walsh space into which $mathcal{F}_{s, boldsymbol{u}}$ is embedded.

Formulas for the Walsh coefficients of smooth functions and their application to bounds on the Walsh coefficients

We establish formulas for the $b$-adic Walsh coefficients of functions in $C^alpha[0,1]$ for an integer $alpha geq 1$ 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.

Self-organized Archimedean Spiral Pattern: Regular Bundling of Fullerene through Solvent Evaporation

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.

Quasi-logarithmic spacing law in dewetting patterns from the drying meniscus of a polymer solution

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.