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For most infinite lattice graphs, p c cannot be calculated exactly, though in some cases p c there is an exact value. For example: For example: for the square lattice ℤ 2 in two dimensions, p c = 1 / 2 for bond percolation, a fact which was an open question for more than 20 years and was finally resolved by Harry Kesten in the early ...
Diffusion process is stochastic in nature and hence is used to model many real-life stochastic systems. Brownian motion , reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diffusion processes.
Diffusion curves are vector graphic primitives for creating smooth-shaded images. Each diffusion curve partitions the 2D graphics space through which it is drawn, defining different colors on either side. When rendered, these colors then spread into the regions on either side of the curve in a way analogous to diffusion. The colors may also be ...
An exponentially stable LTI system is one that will not "blow up" (i.e., give an unbounded output) when given a finite input or non-zero initial condition. Moreover, if the system is given a fixed, finite input (i.e., a step ), then any resulting oscillations in the output will decay at an exponential rate , and the output will tend ...
In recent times, reaction–diffusion systems have attracted much interest as a prototype model for pattern formation. [20] The above-mentioned patterns (fronts, spirals, targets, hexagons, stripes and dissipative solitons) can be found in various types of reaction–diffusion systems in spite of large discrepancies e.g. in the local reaction ...
Since diffusion goes as the Laplacian of velocity, the dissipation rate may be written in terms of the energy spectrum as: = (), with ν the kinematic viscosity of the fluid. From this equation, it may again be observed that dissipation is mainly associated with high wavenumbers (small eddies) even though kinetic energy is associated mainly ...
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion , resulting from the random movements and collisions of the particles (see Fick's laws of diffusion ).
Diffusion maps exploit the relationship between heat diffusion and random walk Markov chain.The basic observation is that if we take a random walk on the data, walking to a nearby data-point is more likely than walking to another that is far away.