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, then, to what I take to be the prevailing view, these rioters were merely a handful of irresponsible, Stalinist-corrupted provocateurs.

According to these needles, they're stone dead ''.

According to Tillich, the last of these three types of existential anxiety, i. e. spiritual anxiety, is predominant in modern times while the others were predominant in earlier periods.

According to Lucas, Bayley, and Valli, each sign has a handshape ( see handshape section in American Sign Language grammar ) and these handshapes can resemble signs of numbers or letters.

* According to a note of Isaac de Beausobre ’ s, Jean Hardouin accepted the first three of these, taking the four others for the initials of the Greek anthrōpoussōzōn hagiōi xylōi, “ saving mankind by the holy cross .”

According to the study, more than nine out of 10 consumers are familiar with and have tried these foods, and about half report eating them frequently.

According to these statements, claims that one or more denominations might be the " true Church " are nothing more than propaganda which has evolved over centuries to support authoritarian claims --- based on tradition or based on scripture --- of merely human institutions.

According to these models, a necessary ( but not sufficient ) condition for the occurrence of amorphous phases is that T < sub > h </ sub > has to be smaller than 0. 3, that is the deposition temperature must be below 30 % of the melting temperature.

According to the doctrine of some Tibetan schools ( like Theravāda but for different reasons ), only the first of these is recognized.

According to these, the SI prefixes would only be used in the decimal sense, even when referring to data storage capacities: kilobyte and megabyte would denote one thousand bytes and one million bytes respectively ( consistent with SI ), while new terms such as kibibyte, mebibyte and gibibyte,

According to this theory, the anonymous author attributed these events to Daniel, as prophecies that were witnessed by this writer in the 2nd century BCE.

According to these scholars, there was something they variously call " earliest Buddhism ", " original Buddhism " or " pre-canonical Buddhism ".

According to these critics who use data from the 1997 poll, the poverty rate rises to 29 %.

According to certain authorities, twelve teeth of the saint were taken from the tomb in the fifteenth century and kept in the treasury, but these have now disappeared ( Stokes, Apennines, p. 183 ).

According to biographer Ray Spangenburg, these early years as Sagan tried to understand the mysteries of the planets, became a " driving force in his life, a continual spark to his intellect, and a quest that would never be forgotten.

According to Josephus, these actions led to several failed conspiracies against Caligula.

According to medical intuitive and author, Caroline Myss, who described chakras in her work Anatomy of the Spirit ( 1996 ), " Every thought and experience you've ever had in your life gets filtered through these chakra databases.

According to Zacks, Kidd was unaware that Culliford had only about 20 crew with him, and felt ill manned and ill equipped to take the Mocha Frigate until his two prize ships and crews arrived, so he decided not to molest Culliford until these reinforcements came.

According to Bernard McGinn, these parts of the oracles were probably written by a Christian and incorporated " mythological language " in describing Nero's return.

According to Christadelphians, these terms are used in reference to specific political systems or individuals in opposition or conflict.

According to an article in the Massachusetts Institute of Technology's Technology Review, these Knuth reward checks are " among computerdom's most prized trophies ".

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 the committee, these Lectures were included in the compilation " in consequence of their embracing the important doctrine of salvation.

According to Lenz's law, the field of these electrons will oppose the magnetic field changes provided by the applied field.

According to researchers of the Ben Gurion University of the Negev, treatment with mineral-rich mud compresses can be used to augment conventional medical therapy in these patients.