This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The is based on the persistence diagram (PD), mathematical tool for capturing shapes multiscale data. input to PDs given by an atomic configuration and output expressed as 2D histograms. Then, specific distributions such curves islands in identify meaningful shape characteristics configuration. Although can be applied wide variety disordered systems, it here silica glass,...

To evaluate performance in a complex survival task, we studied the morphology of Physarum plasmodium transportation network when presented with multiple separate food sources. The comprises tubular elements through which chemical nutrient, intracellular signals and viscous body are transported circulated. When three sources were presented, located at vertices triangle, connected them via short pathway, was often analogous to mathematically shortest route known as Steiner's minimum tree...

When two food sources are presented to the slime mold Physarum in dark, a thick tube for absorbing nutrients is formed that connects through shortest route. light-avoiding organism partially illuminated, however, connecting follows different Defining risk as experimentally measurable rate of movement, minimum-risk path exhibited by organism, determined integrating along path. A model an adaptive-tube network good agreement with experimental observations.

We study the global structure of set bifurcating solutions a class coupled nonlinear reaction-diffusion systems. Our main result is that when one diffusion coefficients sufficiently large, branch emanating from uniform state continues to exist until it connected singularly perturbed which contain interior transition layers (Theorem 6.1). also present branching theorem for shows in general situation does not fall entirely on trivial 2.2).

The characterization of the medium-range (MRO) order in amorphous materials and its relation to short-range is discussed. A new topological approach extract a hierarchical structure presented, which robust against small perturbations allows us distinguish it from periodic or random configurations. This method called persistence diagram (PD) introduces scales many-body atomic structures facilitate size shape characterization. We first illustrate representation perfect crystalline PDs. Then,...

Stability theorem is presented for large amplitude singularly perturbed solutions (SPS) of reactiondiffusion systems on a finite interval. Spectral analysis shows that there exists unique real critical eigenvalue $\lambda _c (\varepsilon )$ which behaves like ) \simeq \tau \varepsilon $ as $\varepsilon \downarrow 0$, where small parameter contained in the system. All other noncritical eigenvalues have strictly negative parts independent $. The singular limit problem §2 plays key role to...

We report that various geometric patterns can be formed upon mechanical deformation of hexagonal micro polymer mesh. The patterning micromesh applied to the fabrication micropatterned soft-materials for cell culturing. A microporous film was prepared from a viscoelastic polymer, poly(ε-caprolactone). mesh 4 μm diameter. Plastic caused by loading tensile force in one direction. Geometrical such as elongated hexagons, rectangles, squares, and triangles were found stretched film. These four...

This paper considers a two-component system of reaction-diffusion equations involving two parameters $\varepsilon $ and $\tau $: \[ \varepsilon \tau u_1 = ^2 u_{xx} + f( u,v ),\qquad v_1 v_{xx} g( ). \]. The stability stationary internal layer-solutions when is sufficiently small investigated. It shown that becomes smaller than some critical value, such solutions are destabilized there appear layer-oscillating behave as does “breathing motion.” due to the instability via Hopf-bifurcation.

Scattering of particlelike patterns in dissipative systems is studied, especially we focus on the issue how input-output relation controlled at a head-on collision one-dimensional(1D) space where traveling pulses interact strongly. It remains an open problem due to large deformation colliding point. We found that special type steady or time-periodic solutions called separators and their stable unstable manifolds direct traffic flow orbits. Such are, general, highly even 1D case which causes...

The stability properties of the traveling front solutions to bistable reaction-diffusion systems in which there are big differences both diffusion rates and reaction between two species studied. In contrast scalar case, this system has multiple existence waves appropriate region parameters. Each wave can be constructed by using a singular perturbation method, its or instability is determined simple algebraic quantity appearing construction: namely, sign Jacobian inner outer matching...

One of the fundamental questions for self-organization in pattern formation is how spatial periodic structure spontaneously formed starting from a localized fluctuation. It known dissipative systems that splitting dynamics one driving forces to create many particle-like patterns single seed. On way final state there occur collisions among them and its scattering manner crucial predict whether realized or not. We focus on colliding traveling spots arising three-component system study...

One of the fundamental issues pulse dynamics in dissipative systems is clarifying how heterogeneity media influences propagating manner. Heterogeneity most important and ubiquitous type external perturbation. We focus on a class one-dimensional traveling pulses, associated parameters which are close to drift and/or saddle-node bifurcations. The advantage studying such twofold: First, it gives us perfect microcosm for variety outputs general setting when pulses encounter heterogeneities....

We numerically study a set of coupled Cahn–Hilliard equations as means to find morphologies diblock copolymers in three-dimensional spherical confinement. This approach allows us variety energy minimizers including rings, tennis balls, Janus balls and multipods among several others. Phase diagrams confined are presented. modify the size interface between microphases control number holes multipod morphologies. Comparison experimental observation by transmission electron microtomography...

Annealing of block copolymers has become a tool great importance for the reconfiguration nanoparticles. Here, we present experimental results annealing copolymer nanoparticles and theoretical model to describe morphological transformation ellipsoids with striped lamellae into onionlike spheres. A good correspondence between findings predictions was observed. The based on finding steepest direction descent an appropriate free energy leads set Cahn–Hilliard equations that correctly dynamical...

We are primarily concerned with the variational problem long-range interaction. This functional represents Gibbs free energy of microphase separation diblock copolymer melts. The critical points this can be regarded as thermodynamic equilibrium state phase phenomenon. Experimentally it is well-known in that final prefers periodic structures such lamellar, column, spherical, double-diamond geometries and so on. interested characterization structure global minimizer (corresponding to strong...