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When an orbital diagram is drawn for the logistic map, it is possible to see how the branch representing the stable periodic orbit splits, which represents a cascade of period-doubling bifurcations. Bifurcation diagram of the period-doubling bifurcation cascade occurring between parameters r 1 = 3 {\displaystyle r_{1}=3} and a ∞ = 3.56994 ...
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.
Stable Diffusion, for example, imposes conditioning in the form of cross-attention mechanism, where the query is an intermediate representation of the image in the U-Net, and both key and value are the conditioning vectors. The conditioning can be selectively applied to only parts of an image, and new kinds of conditionings can be finetuned ...
A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).
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 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 1980s, [6] see ...
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 ...
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 ...
The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. [ 1 ] [ 2 ] Of the four parameters defining the family, most attention has been focused on the stability parameter, α {\displaystyle \alpha } (see panel).