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The Hertz–Knudsen equation describes the non-dissociative adsorption of a gas molecule on a surface by expressing the variation of the number of molecules impacting on the surfaces per unit of time as a function of the pressure of the gas and other parameters which characterise both the gas phase molecule and the surface: [1] [2]
The first HK theorem demonstrates that the ground-state properties of a many-electron system are uniquely determined by an electron density that depends on only three spatial coordinates. It set down the groundwork for reducing the many-body problem of N electrons with 3 N spatial coordinates to three spatial coordinates, through the use of ...
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.
Central limit theorem; Characterization of probability distributions; Cochran's theorem; Complete class theorem; Continuous mapping theorem; Cox's theorem; Cramér's decomposition theorem; Craps principle
Perelman's Geometrization theorem (3-manifolds) Perfect graph theorem (graph theory) Perlis theorem (graph theory) Perpendicular axis theorem ; Perron–Frobenius theorem (matrix theory) Peter–Weyl theorem (representation theory) Phragmén–Lindelöf theorem (complex analysis) Picard theorem (complex analysis)
Lagrange's Theorem implies the intersection of H and K is equal to 1. By definition,P 1 P 2 ···P t = HK, hence HK is isomorphic to H×K which is equal to P 1 ×P 2 ×···×P t. This completes the induction. Now take t = s to obtain (d). (d)→(e) Note that a p-group of order p k has a normal subgroup of order p m for all 1≤m≤k.
In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the Fundamental Theorem of Statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, describes the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows. [1]
While this example is simple, compare to the Nash embedding theorem, specifically the Nash–Kuiper theorem, which says that any short smooth embedding or immersion of in + or larger can be arbitrarily well approximated by an isometric -embedding (respectively, immersion). This is also a dense h-principle, and can be proven by an essentially ...