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 ancient sources, ( Plutarch Theseus, Pausanias ), Amazon tombs could be found frequently throughout what was once known as the ancient Greek world.

According to a 2008 television programme, presented by Griff Rhys Jones, “ the flame has only been extinguished once, by a drunken Mexican football supporter on the night that France beat Brazil here in Paris ,” most likely referring to the 1998 FIFA World Cup Final.

According to the chroniclers, the Abbasid caliph al-Mansur once asked his courtiers who deserved the exalted title of " Falcon of the Quraysh " ( Saqr Quraish, foremost of the Prophet's tribe ).

According to Hyginus Artemis once loved Orion ( in spite of the late source, this version appears to be a rare remnant of her as the pre-Olympian goddess, who took consorts, as Eos did ), but was tricked into killing him by her brother Apollo, who was " protective " of his sister's maidenhood.

According to Strabo, this site was first called Sigeia ; around 306 BC Antigonus refounded the city as the much-expanded Antigonia Troas by settling the people of five other towns in Sigeia, including the once influential city of Neandria.

According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly.

According to this theory, the cycle of philosophical upheavals that in part drove the diversification of Buddhism into its many schools and sects only began once Buddhists began attempting to make explicit the implicit philosophy of the Buddha and the early suttas.

According to tradition, a claim was once made that the shattering of a pumpkin frozen in liquid nitrogen and dropped from a sufficient height would produce a triboluminescent spark.

According to Isaac Asimov, Gauss was once interrupted in the middle of a problem and told that his wife was dying.

According to Parton, he has only ever seen her perform once.

According to historical accounts from the first century AD, seven branches of the Nile once ran through the delta.

According to the Fulham Football Club, he once resided in the original Craven Cottage, today the site of their stadium.

According to the Buddhist point of view, once we have developed a genuine understanding of the causes of suffering, such as craving ( tanha ) and ignorance ( avijja ), then we can completely eradicate these causes and thus be free from suffering.

According to Samuel Angus ( 1920 ) gnosis in these early sects was distinct from the secret teachings revealed to initiates once they had reached a certain level of progression akin to arcanum.

According to Ludovico Ariosto's Orlando Furioso it once belonged to Hector of Troy, and was given to Roland by Malagigi ( Maugris ).

( According to chess rules, once a player has released a piece, he cannot make a different move.

According to his son ( Johnny Jr ), Johnny ( senior ) was named Peter by his parents ; but, once he began to be successful as a swimmer, he formally used his brother's name, Johnny, because his brother John was, by birth, an American citizen ( and had official records that verified this fact ), and Peter was not ( this was done so that non-citizen Peter could represent USA in the Olympics ).

According to Protector, the original colonists had planned to call their world " Flatland " as a sort of joke, but once settled on Home they had changed their minds — " a belated attack of patriotism ", Elroy Truesdale of Protector muses.

According to Mad writer Frank Jacobs, a letter was once successfully delivered to the magazine through the U. S. mail bearing only Neuman's face, without any address or other identifying information.

According to Christopher, Rhine did take precautions against cheating in response to criticisms of his methods, but once he did, he was unable to find the same high scores reported in earlier trials.

According to the myth, Pandora opened a jar ( pithos ), in modern accounts sometimes mistranslated as " Pandora's box " ( see below ), releasing all the evils of mankind — although the particular evils, aside from plagues and diseases, are not specified in detail by Hesiod — leaving only Hope inside once she had closed it again.

According to official Pente instructions, there are three formations that are customary ( although not mandatory ) for a player to inform his opponent of, once created:

According to the Supreme Court Historical Society, 20 goose-quill pens, neatly crossed, are placed at the four counsel tables each day the U. S. Supreme Court is in session ; " most lawyers appear before the Court only once, and gladly take the quills home as souvenirs.

According to the section, once, late in the evening, Bragi was driving through " a certain forest " when a troll woman aggressively asked him who he was, in the process describing herself:

According to his friend, the theologian Desiderius Erasmus of Rotterdam, More once seriously contemplated abandoning his legal career to become a monk.