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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.
Henstock–Kurzweil integrals are linear: given integrable functions and and real numbers and , the expression + is integrable (Bartle 2001, 3.1); for example, (() + ()) = + (). If f is Riemann or Lebesgue integrable, then it is also Henstock–Kurzweil integrable, and calculating that integral gives the same result by all three formulations.
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
A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. A good article about a theorem should include the following elements: Description of the theorem; History of the theorem; Proof of the theorem; An example of the use of the theorem; Applications of the theorem in other fields (physics, economics, etc.)
Littlewood's three principles are quoted in several real analysis texts, for example Royden, [2] Bressoud, [3] and Stein & Shakarchi. [4] Royden [5] gives the bounded convergence theorem as an application of the third principle. The theorem states that if a uniformly bounded sequence of functions converges pointwise, then their integrals on a ...
Pólya’s theorem can be used to construct an example of two random variables whose characteristic functions coincide over a finite interval but are different elsewhere. Pólya’s theorem . If φ {\displaystyle \varphi } is a real-valued, even, continuous function which satisfies the conditions
In probability theory, an empirical process is a stochastic process that characterizes the deviation of the empirical distribution function from its expectation. In mean field theory, limit theorems (as the number of objects becomes large) are considered and generalise the central limit theorem for empirical measures.