According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

According to (), the first historical mention of the statement of this theorem appears in ().

According to Birkhoff's theorem, it is the only vacuum solution that is spherically symmetric.

According to the theorem, it is possible to expand the power ( x + y )< sup > n </ sup > into a sum involving terms of the form ax < sup > b </ sup > y < sup > c </ sup >, where the exponents b and c are nonnegative integers with, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

According to the theorem, it is possible to expand any power of x + y into a sum of the form

According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.

According to the theorem, within every sufficiently powerful logical system ( such as Principia ), there exists a statement G that essentially reads, " The statement G cannot be proved.

* According to a theorem of Nikolay Nikolov and Dan Segal, in any topologically finitely-generated profinite group ( that is, a profinite group that has a dense finitely-generated subgroup ) the subgroups of finite index are open.

According to the spin-statistics theorem, particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states ; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics.

According to Rice's theorem, if there is at least one computable function in a particular class C of computable functions and another computable function not in C then the problem of deciding whether a particular program computes a function in C is undecidable.

According to the Rouché – Capelli theorem, the system is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix.

According to the optical Raychaudhuri's theorem, this requires a violation of the averaged null energy condition.

According to the theorem, the reconstructed signal will match the original signal provided that the original signal contains no frequencies at or above this limit.

According to the Nyquist theorem, error-free detection of the line code requires a channel bandwidth of at least the Nyquist rate, which is half the line code pulse rate.

According to Shannon's source coding theorem, the optimal code length for a symbol is − log < sub > b </ sub > P, where b is the number of symbols used to make output codes and P is the probability of the input symbol.

According to the CPT theorem of particle physics, antihydrogen atoms should have many of the characteristics regular hydrogen atoms have ; i. e., they should have the same mass, magnetic moment, and transition frequencies ( see atomic spectroscopy ) between their atomic quantum states.

According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant.

According to Allen Weiss, in Mirrors of Infinity, this optical effect is a result of the use of the tenth theorem of Euclid ’ s Optics which asserts that “ the most distant parts of planes situated below the eye appear to be the most elevated .” In Fouquet ’ s time, interested parties could cross the canal in a boat, but walking around the canal provides a view of the woods that mark what is no longer the garden and shows the distortion of the grottos previously seen as sculptural.

According to the virial theorem, the gravitational binding energy of a star is about two times its internal thermal energy.

According to the correspondence principle and Ehrenfest's theorem as a system becomes larger or more massive ( action >> Planck's constant ) the classical dynamics tends to emerge, with some exceptions, such as superfluidity.

According to the Kutta – Joukowski theorem, lift is the product of circulation, airspeed, and air density.

According to the Shannon – Hartley theorem, the channel capacity of a properly-encoded signal is proportional to the bandwidth of the channel and the logarithm of the signal-to-noise ratio ( SNR ) ( assuming the noise is additive white Gaussian noise ).

According to Ptolemy's second theorem,

According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function is the Jacobian matrix of the inverse function.

According to one leading historian of statistics, he may have been the earliest discoverer of Bayes theorem.

According to Porphyrion, the hymn to Hermes was imitated by Horace in one of his own ' sapphic ' odes ( C. 1. 10: Mercuri, facunde nepos Atlantis ).

According to Steven J. Brams and Peter C. Fishburn, the IEEE board in 2002 rescinded its decision to use approval voting.

According to D. C. Congressional Delegate Eleanor Holmes Norton, Hill's treatment by the panel also contributed to the large number of women elected to Congress in 1992, " women clearly went to the polls with the notion in mind that you had to have more women in Congress ", she said.

According to his website, three extended periods of his fasting have been observed under control of scientific and medical teams, the first lasting 211 days in 1995 – 96, in Calicut, India, under the direction of Dr C. K. Ramachandran.

According to T. C.

According with the Yajnavalkya-smrti, as quoted in Srimad-Bhagavatam ( 6. 13-14 ) ( A. C. Bhaktivedanta's authorized commentary ), bramacarya means celibacy:

According to Cordwainer Smith scholar Alan C. Elms, this speculation first reached print in Brian Aldiss's 1973 history of science fiction, Billion Year Spree ; Aldiss, in turn, claimed to have gotten the information from Leon Stover.

According to C. Michael Hogan, the Daugava River began experiencing environmental deterioration in the era of Soviet collective agriculture ( producing considerable adverse water pollution runoff ) and a wave of hydroelectric power projects.

According to Biblia Sacra iuxta vulgatum versionem, there are Latin Vulgate manuscripts containing this epistle dating between the 6th and 12th century, including Latin manuscripts F ( Codex Fuldensis ), M, Q, B, D ( Ardmachanus ), C, and Lambda.

According to Wray, Jean Harlow had been RKO's original choice, but because MGM put Harlow under exclusive contract during the pre-production phase of the film, she became unavailable and Wray was approached by director Merian C. Cooper to play the role of Ann Darrow, the blonde captive of King Kong.

According to Arendt, the concept of freedom became associated with the Christian notion of freedom of the will, or inner freedom, around the 5th century C. E.

According to A. C. Hobbs:

According to the classicist J. C. Stobart, Tiberius's Greek education had caused him to overestimate the reliability of the people as a powerbase, causing him to overplay his hand.

According to Hegemonius ( 4th C .) Mani ( 3rd C .) vilified the creator God of the pagan philosophers ( Plato's demiurge ) and the creator God of Judeo-Christianity ( creator ).

According to A. C. Hobbs:

According to David Burner, C Wright Mills claimed that the proletariat were no longer the revolutionary force ; the new agent of revolutionary change were young intellectuals around the world.

According to Charles C. Mann's, " 1491 ", only the lower classes of Aztec society were polytheistic.

According to C. Stace, " slaves in Plautus account for almost twice as much monologue as any other character ... this is a significant statistic ; most of the monologues being, as they are, for purposes of humor, moralizing, or exposition of some kind, we can now begin to see the true nature of the slave's importance.

According to Herbert C. Brichtothe, writing in Reform Judaism's Hebrew Union College Annual, the family tomb is the central concept in understanding biblical views of the afterlife.

According to Gould the most influential political books he read were C. Wright Mills ' The Power Elite and the political writings of Noam Chomsky.

According to T. C. Skeat they suggest Caesarea as a place in which the manuscript was made.