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The second HK theorem defines an energy functional for the system and proves that the ground-state electron density minimizes this energy functional. In work that later won them the Nobel prize in chemistry , the HK theorem was further developed by Walter Kohn and Lu Jeu Sham to produce Kohn–Sham DFT (KS DFT).
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.
Actually, Hahn-Komorogorov Theorem is a stronger theorem than Caratheodory extension theorem, right? (I am using the Wikipedia's naming scheme here.) It looks to me that Caratheodory's Theorem is just HK Theorem with "ring" replaced with "algebra". And by definition, all rings are algebras. So HK implies Caratheodory. Kelvinator0
In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup H in a group G is a subgroup K of G such that = = {:,} = {}. Equivalently, every element of G has a unique expression as a product hk where h ∈ H and k ∈ K.
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 function L is slowly varying if and only if there exists B > 0 such that for all x ≥ B the function can be written in the form = (() + ())where η(x) is a bounded measurable function of a real variable converging to a finite number as x goes to infinity
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 ...
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 ...