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 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 the poet Saadi Shirazi: Arslan possessed a fort, which raised at the height of Alwand, from all were those within its walls, for its roads were a labyrinth, like the curls of a bride.

According to the American Institute of Architects, titles and job descriptions within American architectural offices might be as follows:

According to Asser, because of Pope Marinus ’ friendship with King Alfred, the pope granted an exemption to any Anglo-Saxons residing within Rome from tax or tribute.

According to the author of his biography in the Eleventh edition of the Encyclopædia Britannica: " Ambrose is interesting as typical of the new humanism which was growing up within the church.

According to the Book of Exodus, God instructed Moses on Mount Sinai during his 40-day stay upon the mountain within the thick cloud and darkness where God was ( Ex.

According to Critias, 9, 000 years before his lifetime a war took place between those outside the Pillars of Hercules at the Strait of Gibraltar and those who dwelt within them.

According to many traditions within Mahāyāna Buddhism, on the way to becoming a Buddha, a bodhisattva proceeds through ten, or sometimes fourteen, grounds or bhūmis.

According to a 2003 study undertaken by the International Diabetes Federation ( IDF ) on the access to and availability of insulin in its member countries, synthetic ' human ' insulin is considerably more expensive in most countries where both synthetic ' human ' and animal insulin are commercially available: e. g. within European countries the average price of synthetic ' human ' insulin was twice as high as the price of pork insulin.

According to the New Testament Jesus was crucified, died a physical death, was buried within a tomb, and rose from the dead three days later.

" According to West and Sanders, when talking about conspiracies in the Vietnam era, Pipes includes within the fringe element anyone who entertains the thought that conspiracies played a role in the major political scandals and assassinations that rocked American politics in the Vietnam era.

According to political scientist Arthur Stinchcombe, citizenship is based on the extent that a person can control one's own destiny within the group in the sense of being able to influence the government of the group.

According to Stephen Dorril, at about the same time, Special Branch officers recruited an informant within CND, Stanley Bonnett, on the instructions of MI5.

According to Rabbinic Judaism the Torah was revealed by God to Moses ; within it, Jews find 613 Mitzvot ( commandments ).

According to Bryan Wilson, functionally the definition of a " fellowship " within Christadelphian history has been mutual or unilateral exclusion of groupings of ecclesias from the breaking of bread.

According to Fitzgerald ( 1977 ) " Reinforcements arrived, and within fifteen minutes these ' vicious brutes ', who for some two hours had terrorized well-armed prison staff, and effectively controlled the prison, had surrendered and been locked up again ".

According to a March 2004 BBC profile, politics within the country are dominated by tensions between Obiang's son, Teodorin, and other close relatives with powerful positions in the security forces.

According to the Board of Governors, the Federal Reserve is independent within government in that " its monetary policy decisions do not have to be approved by the President or anyone else in the executive or legislative branches of government.

According to the WHO, it is practiced in 28 countries in western, eastern, and north-eastern Africa, in parts of the Middle East, and within some immigrant communities in Europe, North America, Australasia.

According to French law, the Foreign Legion was not to be used within Metropolitan France except in the case of a national invasion, and was consequently not a part of Napoleon III ’ s Imperial Army that capitulated at Sedan.

According to the regulation in force in 2010 this obligation has to be met within three months after the start of its activities in Spain.

According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure.

According to the constitution, the President of the National Assembly was to assume the Presidency of the Republic in the event of a vacancy, and a new presidential election was to be held within 60 days.

According to the findings of the programme, no one within of the blast could have survived, and all of the stained glass windows in Westminster Abbey would have been shattered, as would all of the windows in the vicinity of the Palace.

According to Levy's account, sharing was the norm and expected within the non-corporate hacker culture.

According to Kallistos Ware, some Western theologians, both Roman Catholic and Anglican, see the theology of Palamas as introducing an inadmissible division within God ; however, others have incorporated his theology into their own thinking, maintaining, as Jeffrey D. Finch reports, that there is no conflict between his teaching and Roman Catholic thought.