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 one report, however, Mr. Hammarskjold was considered `` too controversial '' a figure to warrant bestowal of the coveted honor last spring.

According to Friends, the unit was organized by John Snook, a former World War 2, commando who is vice president and general manager of the telephone company.

According to the new theories, the nineteenth century corporate sovereign was `` sovereign '' in a quite new and different sense from his historical predecessors.

According to Stacy, she told him she was planning to remarry and she wanted him to ask Forbes for the lump sum.

According to the medical examiner, she was shot between eleven p.m. and one a.m..

According to the medical examiner she was shot between eleven p.m. and one a.m..

According to this doctrine, the universe was ruled by Heaven, T'ien -- as a natural force, or in the personification of a Supreme Sky-god -- governing all things by means of a process called the Tao, which can be roughly interpreted as `` the Order of the Universe '' or `` the Universal Way ''.

According to the Greek tradition the Dipylon master was named Daedalus, and in his statues the limbs were freed from the body, giving the impression that the statues could move.

According to Igor M. Diakonoff ( 1988: 33n ), Proto-Afroasiatic was spoken c. 10, 000 BC.

According to Christopher Ehret ( 2002: 35 – 36 ), Proto-Afroasiatic was spoken c. 11, 000 BC at the latest and possibly as early as c. 16, 000 BC.

According to the 2010 revison of the World Population Prospects the total population was 19 082 000 in 2010, compared to only 4 148 000 in 1950.

According to Burgess, the novel was a jeu d ' esprit written in just three weeks.

According to Plutarch's natural order of attribution of the vowels to the planets, alpha was connected with the Moon.

According to Lovecraft's " History of the Necronomicon " ( written 1927, first published 1938 ), Alhazred was:

According to a research report by the Centers for Disease Control and Prevention, Oran was decimated by the plague in 1556 and 1678, but outbreaks after European colonization, in 1921 ( 185 cases ), 1931 ( 76 cases ), and 1944 ( 95 cases ), were very far from the scale of the epidemic described in the novel.

( Despite common belief, he did not take a day from February ; see the debunked theory on month lengths ) According to a Senatus consultum quoted by Macrobius, he chose this month because it was the time of several of his great triumphs, including the conquest of Egypt.

According to the National Statistical Service, during the January – August 2007 period, Armenia's industrial sector was the single largest contributor to the country's GDP, but remained largely stagnant with industrial output increasing only by 1. 7 percent per year.

According to the National Statistical Service, Armenia's trade deficit in 2006 was $ 1. 2 billion with growth in exports being largely flat.

According to the state-owned Armenpress news agency, the average monthly wage in Armenia for the first half of 2007 was 70, 700 drams ( about $ 210 USD ).

According to research commissioned by the Yerevan office of the Organization for Security and Co-operation in Europe ( OSCE ), at least one in three working-age Armenians was unemployed as of February 2005 despite several consecutive years of double-digit economic growth.

According to estimates by the National Statistical Survey, the rate of labor emigration was twice as higher in 2001 and 2002.

According to Hesiod's Theogony, she was born when Cronus cut off Uranus ' genitals and threw them into the sea, and from the sea foam ( aphros ) arose Aphrodite.

According to the archaeologists, the door was reused in a structure in Roman Egypt.

According to Maimonides, an afterlife continues for the soul of every human being, a soul now separated from the body in which it was " housed " during its earthly existence.

According to Hartshorne people do not experience subjective ( or personal ) immortality in the afterlife, but they do have objective immortality because their experiences live on forever in God, who contains all that was.