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 The Chicago Tribune News Service, State Atty. Gen. Stanley Mosk of California has devised a series of questions which the joiner might well ask about any organization seeking his money and his name: 1.

According to Richard Dawkins, a distinction between agnosticism and atheism is unwieldy and depends on how close to zero we are willing to rate the probability of existence for any given god-like entity.

According to Tylor, the more scientifically advanced the society, the less that society believed in Animism ; however, any remnant ideologies of souls or spirits, to Tylor, represented “ survivals ” of the original animism of early humanity.

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 Richard I. Pervo, " Townsend's methodologically adventurous but ultimately cautious essay is another valuable lesson in the danger of establishing the date of Acts – or any work – by arguing for the earliest possible time of origin.

According to encyclopaedia Iranica, AFGHAN ( afḡān ) in current political usage means any citizen of Afghanistan, whatever their ethnic, tribal, or religious affiliation is.

According to the Kun-bzang bla-ma ' i zhal-lung, a bodhisattva can choose any of three paths to help sentient beings in the process of achieving buddhahood.

According to its Memorandum & Articles of Association, its objectives are :- “ To act as Nominee or agent or attorney either solely or jointly with others, for any person or persons, partnership, company, corporation, government, state, organisation, sovereign, province, authority, or public body, or any group or association of them ....” Bank of England Nominees Limited was granted an exemption by Edmund Dell, Secretary of State for Trade, from the disclosure requirements under Section 27 ( 9 ) of the Companies Act 1976, because, “ it was considered undesirable that the disclosure requirements should apply to certain categories of shareholders .” The Bank of England is also protected by its Royal Charter status, and the Official Secrets Act.

According to Tacitus, he amassed a force including his own Legio XIV Gemina, some vexillationes ( detachments ) of the XX Valeria Victrix, and any available auxiliaries.

According to Unilever, " in blind taste tests, 10 % didn't notice any difference in taste, 40 % preferred the original and 50 % preferred the new product.

According to Moyra Grant, in political philosophy " collectivism " refers to any philosophy or system that puts any kind of group ( such as a class, nation, race, society, state, etc.

According to Charles Darwin's theory of natural selection, characteristics such as camouflage that help an animal to survive will tend to evolve in any population.

According to this view, any form of specific coercion is then unethical in itself as an injury to freedom, quite apart from its damaging effects on social progress.

According to Aquinas, the universe cannot, at any particular moment, be causing itself.

According to the United States Department of Energy, the wide range of its current and projected uses, together with the lack of any immediately suitable replacement, makes dysprosium the single most critical element for emerging clean energy technologies-even their most conservative projections predict a shortfall of dysprosium before 2015.

According to the natives of Peru, who by this time had observed the Spanish lust for gold, the territories of Chile had abundances of gold which would justify any effort.

According to a 2008 article in The Wall Street Journal, Dartmouth graduates also earn higher median salaries at least 10 years after graduation than alumni of any other American university surveyed.

According to Tacitus, Mucianus was not keen on this prospect but since he considered Domitian a liability in any capacity that was entrusted to him, he preferred to keep him close at hand rather than in Rome.

According to David Vine, " Today, at any given time, 3, 000 to 5, 000 U. S. troops and civilian support staff live on the island.

According to the relational interpretation of quantum mechanics, first proposed by Carlo Rovelli, observations such as those in the double-slit experiment result specifically from the interaction between the observer ( measuring device ) and the object being observed ( physically interacted with ), not any absolute property possessed by the object.

According to a survey by The Washington Post, The Thin Blue Line made dozens of critics ' top ten lists for 1988, more than any other film that year.

According to the 4th century Latin writer Servius, a fourth set of Etruscan books existed, dealing with animal gods, but it is unlikely that any scholar living in the 4th century could have read Etruscan.