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The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. The problem is to find the equilibrium position of an elastic membrane whose boundary is held fixed, and which is constrained to lie above a given obstacle.
Marginal stability, like instability, is a feature that control theory seeks to avoid; we wish that, when perturbed by some external force, a system will return to a desired state. This necessitates the use of appropriately designed control algorithms.
The Radial Unit Hypothesis (RUH) is a conceptual theory of cerebral cortex development, first described by Pasko Rakic. It states that the cerebral cortex develops during embryogenesis as an array of interacting cortical columns , or 'radial units', each of which originates from a transient stem cell layer called the ventricular zone , which ...
The physics of the system will be described by a certain formula, say the Hamiltonian H(T, J). Now proceed to divide the solid into blocks of 2×2 squares; we attempt to describe the system in terms of block variables , i.e., variables which describe the average behavior of the block.
In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random models by studying a simpler model that approximates the original by averaging over degrees of freedom (the number of values in the final calculation of a statistic that are free to vary).
Ergodic theory is often concerned with ergodic transformations.The intuition behind such transformations, which act on a given set, is that they do a thorough job "stirring" the elements of that set. E.g. if the set is a quantity of hot oatmeal in a bowl, and if a spoonful of syrup is dropped into the bowl, then iterations of the inverse of an ergodic transformation of the oatmeal will not ...
The Cauchy problem (sometimes called the initial value problem) is the attempt at finding a solution to a differential equation given initial conditions. In the context of general relativity , it means the problem of finding solutions to Einstein's field equations — a system of hyperbolic partial differential equations — given some initial ...
In the classical limit, ħ → 0, the Moyal bracket reduces to the Poisson bracket, while this evolution equation reduces to the Liouville equation of classical statistical mechanics. Formally, the classical Liouville equation can be solved in terms of the phase-space particle trajectories which are solutions of the classical Hamilton equations.