Cellular Automata

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Reazione-diffusione. Da Wikipedia, l'enciclopedia libera.

Reazione-diffusione

In analisi matematica il modello reazione-diffusione è l'equazione parabolica la cui omogenea associata è l'equazione della diffusione: il termine di sorgente viene chiamato termine di reazione, in quanto nell'applicazione più frequente dove la funzione incognita è la concentrazione di un composto è dovuto ad una reazione chimica in cui il composto è coinvolto. Modello generale di Reazione-Diffusione[modifica | modifica sorgente] Se nell'equazione di diffusione per la funzione u(x, t) è presente un termine non omogeneo di reazione si ha che l'equazione diviene: Leggi di Fick. Da Wikipedia, l'enciclopedia libera.

Leggi di Fick

In analisi matematica, l'equazione del calore, anche detta equazione di diffusione, è un'equazione differenziale alle derivate parziali che trova nelle scienze svariate applicazioni: per esempio in fisica modellizza l'andamento della temperatura in una regione dello spazio-tempo sotto opportune condizioni, e in chimica l'andamento della concentrazione chimica di una specie. La buona posizione dei problemi associati all'equazione del calore segue inoltre dall'analisi di buona posizione di un problema parabolico, di cui l'equazione è un classico esempio.

Anisotropic diffusion. In image processing and computer vision, anisotropic diffusion, also called Perona–Malik diffusion, is a technique aiming at reducing image noise without removing significant parts of the image content, typically edges, lines or other details that are important for the interpretation of the image.[1][2][3] Anisotropic diffusion resembles the process that creates a scale space, where an image generates a parameterized family of successively more and more blurred images based on a diffusion process.

Anisotropic diffusion

Each of the resulting images in this family are given as a convolution between the image and a 2D isotropic Gaussian filter, where the width of the filter increases with the parameter. This diffusion process is a linear and space-invariant transformation of the original image. Reaction-Diffusion Textures. Next: Diffusion Tensor Interpolation Up: Methods Previous: Hue-balls and Deflection Mapping Subsections Introduction Our goal in this section is to use reaction-diffusion textures as a means of visualizing three-dimensional diffusion tensor data.

Reaction-Diffusion Textures

We start by describing a simple model of reaction-diffusion texture that works in two and three dimensions, and then discuss how to modify its calculation to make the texture reflect measured diffusion tensor data. Gray-Scott reaction-diffusion java applet. How can patterns be formed by chemical reactions?

Gray-Scott reaction-diffusion java applet

A first answer to this question was provided by Alan Turing, who specified mathematical conditions necessary for it to be possible to form spatial patterns in two-component reaction-diffusion systems. The java applet on this page simulates diffusion and reaction between two chemicals U and V. Reaction: U + 2 V -> 3 V The chemical U diffuses faster than V, and is used as fuel to produce chemical V, while chemical V catalyzes its own production. Java demo: Gray-Scott Reaction-Diffusion. Most of these images are linked to an applet with the same parameters so you can watch and interact with the pattern evolution.

Java demo: Gray-Scott Reaction-Diffusion

Varying F, k, and diffusion parameters Parameters: F=0.035&k=0.065&diffuseU=0.16&diffuseV=0.08& simwidth=400&simheight=400& wrap=false&varyPhysics=true& diffuseU2=0.06&diffuseV2=0.03&F2=0.06&k2=0.062 This image shows how physical parameters can vary within a single environment. Here the diffusion constants vary on the vertical axis, and the reaction constants vary along the horizontal axis.

Reaction-Diffusion by the Gray-Scott Model: Pearson's Parameterization at MROB. Introduction Instructions: A click anywhere in the crescent-shaped complex region will take you to a page with images, a movie and a specific description.

Reaction-Diffusion by the Gray-Scott Model: Pearson's Parameterization at MROB

Each grid square leads to a different page. I have special pages for the uskate-world and certain other exotic patterns. This web page serves several purposes: Microemulsion structure. Nervous System – explorations in generative design and natural phenomena » reaction diffusion. Nervous System has released Reaction, their first collection of housewares.

Nervous System – explorations in generative design and natural phenomena » reaction diffusion

The collection includes porcelain cups and plates and matching 3D printed lamps. The pieces are intricately embossed with intertwining patterns […] Read Article → As part of a new experimental project we are working on we had to create a reaction-diffusion system that can run on a constantly changing surface. Cellular Automata – How the Leopard gets its spots. « Jonathan Pace. I’ve been reading up on how you can take a set of individual ‘things’, give each thing a rule to iterate, then sit back and watch them exhibit some interesting behaviour.

Cellular Automata – How the Leopard gets its spots. « Jonathan Pace

These systems are classified as ‘Cellular Automata’ ( and can be simulated in a number of different ways. One such simulation is an array of pixels in an image. By initially randomly assigning each pixel a value of 1 or 0 (corresponding to white or black) we end up with a ‘noisy’ image like this. Now come the rules. SIMULATION PICS -  Virtual Laboratory for Simulation and Analysis of Propagating Interfaces. Virtual Laboratory for Simulation and Analysis of Propagating Interfaces more...

SIMULATION PICS -  Virtual Laboratory for Simulation and Analysis of Propagating Interfaces

Sample Numerical Simulation Snapshots Produced with VLSAπ. Alan Turing’s Patterns in Nature, and Beyond | Wired Science. Turing Goes GalacticOnce one starts to look, there seems to be no end to Turing patterns: their forms can be seen in weather systems, the distribution of vegetation across landscapes and even the constellations of galaxies.Image: Galaxy N51, the Whirlpool Galaxy.

(European Space Agency)Turing Patterns in CellsTuring patterns can involve not just chemicals, but large, complex systems in which each unit — for example, a cell — is distributed like molecules of pigment.Pictured is a Turing pattern of cells in Dictyostelium, or a slime mold.Image: National Institutes of HealthTuring Patterns in 3-DThat markings on animals are produced by Turing systems of pigments is now generally accepted, but the origin of what appear to be Turing patterns in more complex settings — such as limb and tooth and lung development — is still debated.A basic step towards proving the existence of these three-dimensional Turing patterns is demonstrating a three-dimensional pattern in the lab.

Gierer-Meinhardt model. The Gierer-Meinhardt model Figure 1: Short-range activator and long-range inhibitor in Gierer-Meinhardt model \frac{\partial a}{\partial t} = \rho\frac{a^2}{h} - \mu_a a + D_a \frac{\partial^2 a}{\partial x^2} + \rho_a. Blending of animal colour patterns by hybridization : Nature Communications. Computer simulations Equations used for RD systems14 can be generally described as where u and v are the concentrations of hypothetical factors, f and g are the reaction kinetics and Du and Dv are the hypothetical diffusion coefficients (or their mathematical equivalents) for u and v, respectively15, 37.

The reaction rate R was introduced for convenience of parameter adjustment. The reaction kinetics and parameters used for each simulation are as follows: The linear model:14. Kondo labo. Simulation Programs (to download, just click the labyrinth pattern) reaction-diffusion simulator for windows Vista (much faster): This program calculates the original Turing's equation reaction-diffusion simulator for XP (or older OS): This program calculates the original Turing's equation reaction-diffusion simulator for XP (or older OS) :This program calculates four famous types of RD models; GM model, Schnakenberg model, GS model and oregonator. PapersPigment pattern formation by contact-dependent depolarization.Inaba M, Yamanaka H, Kondo S.