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In many cases, a corollary corresponds to a special case of a larger theorem, [4] which makes the theorem easier to use and apply, [5] even though its importance is generally considered to be secondary to that of the theorem. In particular, B is unlikely to be termed a corollary if its mathematical consequences are as significant as those of A.
Corollary 1: the ground-state density uniquely determines the potential and thus all properties of the system, including the many-body wavefunction. In particular, the HK functional, defined as F [ n ] = T [ n ] + U [ n ] {\displaystyle F[n]=T[n]+U[n]} , is a universal functional of the density (not depending explicitly on the external potential).
From the Kolmogorov axioms, one can deduce other useful rules for studying probabilities. The proofs [6] [7] [8] of these rules are a very insightful procedure that illustrates the power of the third axiom, and its interaction with the prior two axioms. Four of the immediate corollaries and their proofs are shown below:
Kawasaki's theorem (mathematics of paper folding) Kelvin's circulation theorem ; Kempf–Ness theorem (algebraic geometry) Kepler conjecture (discrete geometry) Kharitonov's theorem (control theory) Khinchin's theorem (probability) Killing–Hopf theorem (Riemannian geometry) Kinoshita–Lee–Nauenberg theorem (quantum field theory)
The theorem states that if you have an infinite matrix of non-negative real numbers , such that the rows are weakly increasing and each is bounded , where the bounds are summable < then, for each column, the non decreasing column sums , are bounded hence convergent, and the limit of the column sums is equal to the sum of the "limit column ...
[7] [8] [9] However, he could not prove the latter theorem, which is shown in 1915 to be equivalent to the axiom of choice by Friedrich Moritz Hartogs. [2] [10] 1896 Schröder announces a proof (as a corollary of a theorem by Jevons). [11] 1897 Bernstein, a 19-year-old student in Cantor's Seminar, presents his proof. [12] [13]
Cl – conjugacy class. cl – topological closure. CLT – central limit theorem. cod, codom – codomain. cok, coker – cokernel. colsp – column space of a matrix. conv – convex hull of a set. Cor – corollary. corr – correlation. cos – cosine function. cosec – cosecant function. (Also written as csc.) cosech – hyperbolic ...
In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".