Search results
Results From The WOW.Com Content Network
The free will theorem states: Given the axioms, if the choice about what measurement to take is not a function of the information accessible to the experimenters (free will assumption), then the results of the measurements cannot be determined by anything previous to the experiments. That is an "outcome open" theorem:
A linear group is not amenable if and only if it contains a non-abelian free group (thus the von Neumann conjecture, while not true in general, holds for linear groups). The Tits alternative is an important ingredient [2] in the proof of Gromov's theorem on groups of polynomial growth. In fact the alternative essentially establishes the result ...
Nielsen–Schreier theorem (free groups) Orbit-stabilizer theorem (group theory) Schreier refinement theorem (group theory) Schur's lemma (representation theory) Schur–Zassenhaus theorem (group theory) Sela's theorem (hyperbolic groups) Stallings theorem about ends of groups (group theory) Superrigidity theorem (algebraic groups)
The Hawking singularity theorem is based on the Penrose theorem and it is interpreted as a gravitational singularity in the Big Bang situation. Penrose shared half of the Nobel Prize in Physics in 2020 "for the discovery that black hole formation is a robust prediction of the general theory of relativity".
Many mathematicians then attempted to construct elementary proofs of the theorem, without success. G. H. Hardy expressed strong reservations; he considered that the essential "depth" of the result ruled out elementary proofs: No elementary proof of the prime number theorem is known, and one may ask whether it is reasonable to expect one.
The notions of completely and absolutely monotone function/sequence play an important role in several areas of mathematics. For example, in classical analysis they occur in the proof of the positivity of integrals involving Bessel functions or the positivity of Cesàro means of certain Jacobi series. [ 6 ]
Given a discrete-time stationary ergodic stochastic process on the probability space (,,), the asymptotic equipartition property is an assertion that, almost surely, (,, …,) where () or simply denotes the entropy rate of , which must exist for all discrete-time stationary processes including the ergodic ones.
Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. [1] [2]Mathematical induction is a method for proving that a statement () is true for every natural number, that is, that the infinitely many cases (), (), (), (), … all hold.