Multiple-cell elements and regular multifractals

Based on fractal super fibers and binary fractal fibers, the following objectives are approached in this paper： First, the concept of multiple-cell elements is induced and abstracted. Second, through multiple-cell elements, the constructability of regular multifractals with strict self-similarities is confirmed, and the universality of the con- struction mode for regular multifractals is proved. Third, through the construction mode and multiple-cell elements, regular multifractals are demonstrated to be equivalent to generalized regular single fractals with multilayer fine structures. On the basis of such equivalence, the dimension formula of the regular single fractal is extended to that of the regular multifractal, and the geometry of regular single fractals is extended to that of regular multifractals. Fourth, through regular multifractals, a few golden fractals are constructed.     

分形插值曲面理论及其应用*

本文叙述了分形曲面的生成原理,给出了分形插值曲面的计算公式,证明了分形插值曲面迭代函数系唯一性定理,导出了分形插值曲面的维数定理,并应用实际数据进行了分形插值曲面的实例研究。     

Models for the Dynamics of Hyperbranched Macromolecules

Summary: We focus on the motion of hyperbranched macromolecules in solution, paying particular attention to the relation between underlying topological structure and dynamics; we consider especially the mechanical moduli. Under the prominent representatives of hyperbranched polymers are both regular structures (such as the dendrimers) as well as disordered structures (such as irregular Cayley-trees). Evidently, batch-prepared hyperbranched macromolecules are closer to the latter. In order to theoretically determine their mechanical moduli we employ the method of generalized Gaussian structures (GGS), which allows us to study the situation including or excluding the hydrodynamic interactions (HI). Disordered hyperbranched structures display a complex dynamics; here we recall several analytical and numerical schemes for determining it and compare our theoretical results to the experimental data.     

Photoinduced Reversible Topographical Changes on Diarylethene Microcrystalline Surfaces with Biomimetic Wetting Properties

Reversible topographical changes were observed on a photochromic diarylethene microcrystalline film surface by alternate irradiation with UV and visible light. Two types of surfaces were prepared from this film: 1) Storage of the film at 30 °C for 24 hours in the dark after UV irradiation afforded a surface that was covered with needle‐shaped crystals, whose diameter and length were approximately 1 μm and 10 μm, respectively, and showed a superhydrophobic lotus effect. 2) Storage of the film at 70 °C for 3 hours in the dark caused the needle‐shaped crystals to be converted into larger rod‐like crystals (5∼8 μm wide and 20∼30 μm long) by Ostwald ripening and a disappearance of the lotus effect. The obtained activation energy of the formation of the needle‐ and rod‐shaped crystals was 143 and 162 kJ mol⁻¹, respectively. Subsequent UV irradiation to the surface, which was followed by storage at 50 °C for 1 hour in the dark, gave a doubly rough structure; small needle‐shaped crystals were formed between the larger rod‐shaped crystals. The surface showed both superhydrophobic properties and the pinned effect of the water droplet: the petal effect. Fractal analysis of both surfaces were carried out using a box‐counting method, and the lotus effect was observed in the presence of smaller‐sized crystals, whilst the petal effect was observed with larger sized crystals (ca. 100 μm). We demonstrated that the hydrophobic property was controlled by the distribution in crystal size of the closed‐ring isomer of the diarylethene. Visible‐light irradiation of both rough surfaces afforded surfaces with cubic‐shaped micro‐crystals of the open‐ring isomer.     

Construction and Properties of Sierpiński Triangular Fractals on Surfaces

Fractal structures are of fundamental importance in science, engineering, mathematics, and aesthetics. Construction of molecular fractals on surfaces can help to understand the formation mechanism of fractals and a series of achievements have been acquired in the preparation of molecular fractals. This review focuses on Sierpiński triangles (STs), representatives of various prototypical fractals, on surfaces. They are investigated by Monte Carlo simulations and ultra-high vacuum scanning tunneling microscopy. STs are bonded through halogen bonds, hydrogen bonds, metal-organic coordination bonds and covalent bonds. The coexistence of and competition between fractals and crystals are realized for a hydrogen-bonded system. Electronic properties of two types of STs are summarized.     

A Grand Canonical Monte-Carlo Simulation Study of Xenon Adsorption in a Vycor-like Porous Matrix

We have performed atomistic Grand Canonical Monte-Carlo (GCMC) simulations of adsorption of xenon in a Vycor-like matrix at 195 K. The disordered mesoporous network is obtained by applying a numerical 3D off-lattice reconstruction procedure to a simulation box originally containing silicon and oxygen atoms of a non-porous silica solid. In order to reduce the computational cost, we have applied a homothetic decrease of the simulation box dimensions which preserves the morphology and the topology of the pore network (the average pore dimension is then around 30 Å). The surface chemistry is obtained in a realistic fashion by saturating all dangling bonds with hydrogen atoms. Small angle scattering spectra calculated on different numerical samples have evidenced a departure from Porod's law due to surface roughness. The simulated isotherms calculated on such disordered connected porous networks, show the capillary condensation phenomenon. The shape of the adsorption curves differs from that obtained for simple pore geometries. The analysis of the adsorbed quantity distribution indicates partial molecular-film formation depending on the local surface curvature and roughness.     

A plethora of strange nonchaotic attractors

We show that it is possible to devise a large class of skew-product dynamical systems which have strange nonchaotic attractors (SNAs): the dynamics is asymptotically on fractal attractors and the largest Lyapunov exponent is non-positive. Furthermore, we show that quasiperiodic forcing, which has been a hallmark of essentially all hitherto known examples of such dynamics is not necessary for the creation of SNAs.     

Discrete characterisations of Lipschitz spaces on fractals

A. Kamont has discretely characterised Besov spaces on intervals. In this paper, we give a discrete characterisation of Lipschitz spaces on fractals admitting a type of regular sequence of triangulations, and for a class of post critically finite self‐similar sets. This shows that on some fractals, certain discretely defined Besov spaces, introduced by R. Strichartz, coincide with Lipschitz spaces introduced by A. Jonsson and H. Wallin for low order of smoothness. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Potential spaces and traces of Lévy processes on <Emphasis Type="Italic">h</Emphasis>-sets

Potential spaces and Dirichlet forms associated with Lévy processes subordinate to Brownian motion in ℝ ⁿ with generator f(−Δ) are investigated. Estimates for the related Rieszand Bessel-type kernels of order s are derived which include the classical case f(r) = r ^α/2 with 0 < α < 2 corresponding to α-stable Lévy processes. For general (tame) Bernstein functions f potential representations of the trace spaces, the trace Dirichlet forms, and the trace processes on fractal h-sets are derived. Here we suppose the trace condition ∫₀¹ r ⁻⁽ⁿ⁺¹⁾ f(r ⁻²)⁻¹ h(r) dr < ∞ on f and the gauge function h. Dedicated to the 80th birthday of Klaus Krickeberg