Hierarchical Fractal Structure Research Articles

Fractal Design of Hierarchical PtPd with Enhanced Exposed Surface Atoms for Highly Catalytic Activity and Stability.

Hierarchical assembly of arc-like fractal nanostructures not only has its unique self-similarity feature for stability enhancement but also possesses the structural advantages of highly exposed surface-active sites for activity enhancement, remaining a great challenge for high-performance metallic nanocatalyst design. Herein, we report a facile strategy to synthesize a novel arc-like hierarchical fractal structure of PtPd bimetallic nanoparticles (h-PtPd) by using pyridinium-type ionic liquids as the structure-directing agent. Growth mechanisms of the arc-like nanostructured PtPd nanoparticles have been fully studied, and precise control of the particle sizes and pore sizes has been achieved. Due to the structural features, such as size control by self-similarity growth of subunits, structural stability by nanofusion of subunits, and increased numbers of exposed active atoms by the curved homoepitaxial growth, h-PtPd displays outstanding electrocatalytic activity toward oxygen reduction reaction and excellent stability during hydrothermal treatment and catalytic process.

Structure of zirconium dioxide based porous glasses

This study is devoted to investigation of the micro- and mesostructure (including fractal properties) of porous zirconia glasses synthesized by precipitation from zirconium n-propoxide solutions in the presence of different hydrolyzing-agent (H2O) quantities at different temperatures. Analysis of small-angle neutron, ultra-small-angle neutron and X-ray scattering, scanning electron microscopy allows concluding that the synthesized glasses are complex systems with a three-level hierarchical fractal structure. It is revealed that both the temperature of synthesis and the H2O concentration in the initial solution significantly affect the structural characteristics of the glasses.

Hierarchical fractal structure of perfect single-layer grapheme

The atomic lattice structure of perfect singlelayer graphene that can actually be regarded as a kind of hierarchical fractal structure from the perspective of fractal geometry was studied for the first time. Three novel and special discoveries on hierarchical fractal structure and sets were unveiled upon examination of the regular crystal lattices of the single-layer graphene. The interior fractaltype structure was discovered to be the fifth space-filling curve from physical realm. Two efficient methods for calculating the fractal dimension of this fresh member was also provided. The outer boundary curve had a fractal dimension equal to one, and a multi-fractal structure from a naturally existing material was found for the first time. A series of strict self-similar hexagons comprised a rotating fractal set. These hexagons slewed at a constant counterclockwise angle α of 19.1° when observed from one level to the next higher level. From the perspective of fractal geometry, these pioneering discoveries added three new members to the existing regular fractal structures and sets. A fundamental example of a multi-fractal structure was also presented.

A new proof of Honeycomb Conjecture by fractal geometry methods

Based on fractal geometry, we put forward a concise and straightforward method to prove Honeycomb Conjecture—a classical mathematic problem. Hexagon wins the most efficient covering unit in the two-dimensional space, compared with the other two covering units—triangle and square. From this point of view, honeycomb is treated as a hierarchical fractal structure that fully fills the plane. Therefore, the total side length and area are easily calculated and from the results, the covering efficiency of each possible unit is provided quantitatively.

Grazing incidence X-ray scattering study of sol–gel derived indium tin oxide thin films

Indium tin oxide (ITO) thin film as one of promising transparent conducting oxide (TCO) films has attracted ever increasing attention owing to its special optical, photocatalytic and optoelectronic properties. In this research, ITO films were prepared by sol–gel dip-coating method and annealed at different temperatures subsequently. The lateral and surface structures of ITO films as well as the structural evolution have been assessed by grazing incidence small angle X-ray scattering (GISAXS) technique. The films show pore fractal structure when annealed at low temperature (≤ 800 °C) which transforms to a hierarchical fractal structure at high temperature (1000 °C). As the temperature rises, films are densified due to the elimination of small pores on the surface at low temperature and the shrinkage of big pores buried inside at high temperature. However, the surface roughness and porosity near the surface are improved at high annealing temperature.

Analysing the Hierarchical Organization of Text by Using Biologically-Inspired Statistical Methods

We have binarily encoded written monologues in a similar manner to the encoding of a continuous electrical signal of neuronal networks. The encoding keys have been carefully designed to include phrases that are co-referents. Only such keys produce binary series, which have an hierarchical fractal structure. This fractal structure can be revealed by using a statistical tool named “Fano factor”. By outlining the analogy between the hierarchical structure of neuronal electrical activity and the hierarchical structure of text, we try to link the organization of text and neuronal activity.

Chaotic mixing and fractals in a geophysical jet current

We model Lagrangian lateral mixing and transport of passive scalars in meandering oceanic jet currents by two-dimensional advection equations with a kinematic streamfunction with a time-dependent amplitude of a meander imposed. The advection in such a model is known to be chaotic in a wide range of the meander’s characteristics. We study chaotic transport in a stochastic layer and show that it is anomalous. The geometry of mixing is examined and shown to be fractal-like. The scattering characteristics (trapping time of advected particles and the number of their rotations around elliptical points) are found to have a hierarchical fractal structure as functions of initial particle’s positions. A correspondence between the evolution of material lines in the flow and elements of the fractal is established.

Lagrangian coherent structures, transport and chaotic mixing in simple kinematic ocean models

Methods of dynamical system’s theory are used for numerical study of transport and mixing of passive particles (water masses, temperature, salinity, pollutants, etc.) in simple kinematic ocean models composed with the main Eulerian coherent structures in a randomly fluctuating ocean—a jet-like current and an eddy. Advection of passive tracers in a periodically-driven flow consisting of a background stream and an eddy (the model inspired by the phenomenon of topographic eddies over mountains in the ocean and atmosphere) is analyzed as an example of chaotic particle’s scattering and transport. A numerical analysis reveals a non-attracting chaotic invariant set Λ that determines scattering and trapping of particles from the incoming flow. It is shown that both the trapping time for particles in the mixing region and the number of times their trajectories wind around the vortex have hierarchical fractal structure as functions of the initial particle’s coordinates. Scattering functions are singular on a Cantor set of initial conditions, and this property should manifest itself by strong fluctuations of quantities measured in experiments. The Lagrangian structures in our numerical experiments are shown to be similar to those found in a recent laboratory dye experiment at Woods Hole. Transport and mixing of passive particles is studied in the kinematic model inspired by the interaction of a current (like the Gulf Stream or the Kuroshio) with an eddy in a noisy environment. We demonstrate a non-trivial phenomenon of noise-induced clustering of passive particles and propose a method to find such clusters in numerical experiments. These clusters are patches of advected particles which can move together in a random velocity field for comparatively long time. The clusters appear due to existence of regions of stability in the phase space which is the physical space in the advection problem.

Complex Wurtzite ZnSe Microspheres with High Hierarchy and Their Optical Properties

Complex wurtzite ZnSe microspheres with hierarchical fractal structure and showing strong quantum-size effects can be prepared easily by a mild solvothermal reaction in a diethylenetriamine (DETA)-deionized water (DIW) binary solution. The ZnSe microspheres are made up of nanosheets, each formed by the face-to-face pairing of two individual nanosheets. In addition, high-order, flowerlike hierarchical structures are formed by the attachment of flexible, uniform nanofibers to the exposed faces of each nanosheet pair. The surface morphology of the nanosheets changed as reaction time increased and the detailed phase-transformation and shape-evolution processes were studied. This approach could provide an effective strategy for tuning the electronic and optical properties of semiconductors, with special advantages over the traditional high-temperature approach, and could be extended to access other semiconductor materials with unusual morphologies and structures.

Chaotic scattering, transport, and fractals in a simple hydrodynamic flow

Advection of passive tracers in an unsteady hydrodynamic flow consisting of a background stream and a vortex is analyzed as an example of chaotic particle scattering and transport. A numerical analysis reveals a nonattracting chaotic invariant set Λ that determines the scattering and trapping of particles from the incoming flow. The set has a hyperbolic component consisting of unstable periodic and aperiodic orbits and a nonhyperbolic component represented by marginally unstable orbits in the particle-trapping regions in the neighborhoods of the boundaries of outer invariant tori. The geometry and topology of chaotic scattering are examined. It is shown that both the trapping time for particles in the mixing region and the number of times their trajectories wind around the vortex have hierarchical fractal structure as functions of the initial particle coordinates. The hierarchy is found to have certain properties due to an infinite number of intersections of the stable manifold in Λ with a material line consisting of particles from the incoming flow. Scattering functions are singular on a Cantor set of initial conditions, and this property must manifest itself by strong fluctuations of quantities measured in experiments.

連続および複合打球動作が打球成績に及ぼす影響 : ダイナミカルシステムアプローチからの検討

Previous experimental studies (Yamamoto and Gohara, 1999; 2000) have shown that continuous hitting movements have a hierarchical fractal structure, as predicted by dynamical systems approaches with external temporal input. The switching input condition in the previous experiments showed higher performance than when periodic input was applied. To extend this finding to motor learning, we therefore examined the influence of a similar complex hitting action on performance using novices as subjects. Hitting performances of novices were recorded under two conditions: one where two different hitting actions (forehand and backhand tennis strokes) were switched repeatedly-as was the case for the switching condition used in the previous experiment- and one where only one of the two strokes was repeated, corresponding to the periodic input condition. In the complex hitting condition, the end phase and the next preparatory phase were shown to be "fused" in the intermediate phase, as a consequence of exploiting the inertia of the trunk rotation movement, and subjects showed better performance under the complex hitting condition than under the continuous hitting condition. These results suggest that when more than two different classes of novel movements need to be acquired, the movement combination that exploits the physical inertia would be more effective than repeating each movement individually.

Fractal encoding in a chaotic neural network.

We analyze a model of a chaotic neural network consisting of three neurons, namely a chaotically forcing neuron and two neurons comprizing a stable response system with a contraction mapping property, for digital encoding with chaotic dynamics. We show that dynamics of the chaotically forcing neuron is embedded in the form of a code sequence on a fractal attractor of the two-neuron response system. We consider the relation between the state transition of the chaotically forcing neuron and the hierarchical fractal structure on the attractor in the state space of the contracting system. We also report hardware implementation of the presented model with an analog electronic circuit to investigate the fractal attractor of the chaotic neural network as a realistic system.

Continuous hitting movements modeled from the perspective of dynamical systems with temporal input

The kinematics of continuously repeated forehand and backhand tennis strokes were measured and analyzed in terms of a dynamical system with temporal input. The tennis strokes were performed under two conditions: when the same input pattern was repeated and when two different input patterns were switched stochastically. The condition with the two periodic inputs revealed that there were two different trajectories in cylindrical space, while the condition with switching input induced transitions between these two trajectories, which were considered as excited attractors. The transitions between the two excited attractors were characterized by a fractal-like feature that could be described by a simple Cantor set with rotation. This result suggests that continuous hitting movements have a hierarchical fractal structure, as was expected from the theory of dynamical systems with external temporal input.

SCAN: A Scalable Model of Attentional Selection

This paper describes the SCAN (Signal Channelling Attentional Network) model, a scalable neural network model for attentional scanning. The building block of SCAN is a gating lattice, a sparsely-connected neural network defined as a special case of the Ising lattice from statistical mechanics. The process of spatial selection through covert attention is interpreted as a biological solution to the problem of translation-invariant pattern processing. In SCAN, a sequence of pattern translations combines active selection with translation-invariant processing. Selected patterns are channelled through a gating network, formed by a hierarchical fractal structure of gating lattices, and mapped onto an output window. We show how the incorporation of an expectation-generating classifier network (e.g. Carpenter and Grossberg's ART network) into SCAN allows attentional selection to be driven by expectation. Simulation studies show the SCAN model to be capable of attending and identifying object patterns that are part of a realistically sized natural image. © 1997 Elsevier Science Ltd.

Extremal Hamiltonian for Directed Polymers

We study an extremal Hamiltonian for directed polymers. The ground states are degenerate and they form a cluster of directed percolation. This degeneracy has a hierarchical fractal structure and can be lifted by a progressive optimization.