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This theorem has since been extended to the time-dependent domain to develop time-dependent density functional theory (TDDFT), which can be used to describe excited states. The second HK theorem defines an energy functional for the system and proves that the ground-state electron density minimizes this energy functional.
The formal foundation of TDDFT is the Runge–Gross (RG) theorem (1984) [1] – the time-dependent analogue of the Hohenberg–Kohn (HK) theorem (1964). [2] The RG theorem shows that, for a given initial wavefunction, there is a unique mapping between the time-dependent external potential of a system and its time-dependent density.
The definition given initially by Quillen was that of a closed model category, the assumptions of which seemed strong at the time, motivating others to weaken some of the assumptions to define a model category. In practice the distinction has not proven significant and most recent authors (e.g., Mark Hovey and Philip Hirschhorn) work with ...
Artin reciprocity law is a general theorem in number theory that forms a central part of global class field theory. Named after Emil Artin. Ashby's law of requisite variety, that the number of states in a control mechanism must be greater than or equal to the number of states in the system it controls.
The Moore–Aronszajn theorem goes in the other direction; it states that every symmetric, positive definite kernel defines a unique reproducing kernel Hilbert space. The theorem first appeared in Aronszajn's Theory of Reproducing Kernels, although he attributes it to E. H. Moore. Theorem. Suppose K is a symmetric, positive definite kernel on a ...
Let T be a complete L-theory.An L-formula φ(x,y) is said to have the independence property (with respect to x, y) if in every model M of T there is, for each n = {0,1,...,n − 1} < ω, a family of tuples b 0,...,b n−1 such that for each of the 2 n subsets X of n there is a tuple a in M for which
If H is normal, then H \ G is a group, and the right action of K on this group factors through the right action of H \ HK. It follows that H \ G / K = G / HK. Similarly, if K is normal, then H \ G / K = HK \ G. If H is a normal subgroup of G, then the H-double cosets are in one-to-one correspondence with the left (and right) H-cosets.
A series of images that represent research (left) and practice (right) in the field of academic kinesiology. Kinesiology (from Ancient Greek κίνησις (kínēsis) 'movement' and -λογία-logía 'study of') is the scientific study of human body movement.