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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 ...
It is used heavily in statistical physics, statistical analysis, information theory, data science, neural networks, finance and marketing. A sample path of a diffusion process models the trajectory of a particle embedded in a flowing fluid and subjected to random displacements due to collisions with other particles, which is called Brownian motion.
The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small oscillations as in the case of a pendulum. In a system with damping, a stable equilibrium state is moreover asymptotically stable. On the other hand ...
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 ...
Different from linear dimensionality reduction methods such as principal component analysis (PCA), diffusion maps are part of the family of nonlinear dimensionality reduction methods which focus on discovering the underlying manifold that the data has been sampled from. By integrating local similarities at different scales, diffusion maps give ...
The graph of the logistic map + = is the plane curve that plots the relationship between and +, with (or x) on the horizontal axis and + ( or f ( x ) ) on the vertical axis. The graph of the logistic map looks like this, except for the case r = 0 :
The green line is the cubic nullcline and the red line is the linear nullcline. The black lines are integral curves. Graph of v with parameters I=0.5, a=0.7, b=0.8, and τ=12.5. The FitzHugh–Nagumo model (FHN) describes a prototype of an excitable system (e.g., a neuron).
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 ...