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 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 him, while the first two visions ( the alter-globalism and the anti-globalism ) represent the reconstructed forms of old and new left ideologies, respectively, in the context of current globalization, only the third one has shown the capacity to respond more effectively to the intellectual requirements of today ’ s global complexities.

According to a reconstructed inscription found on the site, " the emperor Vespasian ordered this new amphitheatre to be erected from his general's share of the booty.

According to this theory, fairies are personified aspects of nature and deified abstract concepts such as ‘ love ’ and ‘ victory ’ in the pantheon of the particular form of animistic nature worship reconstructed as the religion of Ancient Western Europe .< ref >

According to Breuer, the slow and laborious progress of her “ remembering work ” in which she recalled individual symptoms after they had occurred, thus “ dissolving ” them, came to a conclusion on 7 June 1882 after she had reconstructed the first night of hallucinations in Ischl.

According to records, Shuri Castle burned several times and was reconstructed each time.

" According to Islamic tradition, the Kaaba was built by Adam as a place of worship, and then later reconstructed by Abraham and Ishmael.

* Compressible: According to Miller, " Uncompressed digital data is very large, and in its raw form would actually produce a larger signal ( therefore be more difficult to transfer ) than analog data.

According to one definition of digital signal, the modulated signal is a digital signal, and according to another definition, the modulation is a form of digital-to-analog conversion.

According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century.

According to the most common definition of digital signal, both baseband and passband signals representing bit-streams are considered as digital transmission, while an alternative definition only considers the baseband signal as digital, and passband transmission of digital data as a form of digital-to-analog conversion.

According to signaling theory, education does not lead to increased human capital, but rather acts as a mechanism by which workers with superior innate abilities can signal those abilities to prospective employers and so gain above average wages.

According to Dick Hebdige, members of a subculture often signal their membership through a distinctive and symbolic use of style, which includes fashions, mannerisms, and argot.

According to Julian Symons, " Ellery ... occasionally lost his father, as his exploits took place more frequently in the small town of Wrightsville ... where his arrival as a house guest was likely to be the signal for the commission of one or more murders.

According to a critical analysis in The Independent, the genre rose to prominence in the 1990s not as a continuance of the celebration of " sex and shopping reflected the materialism of the 1980s ", but as a signal of " disillusionment with those values ".

According to Thackery ( in a 2011 exchange on Facebook ), " A smart tech guy fixed us up with signal boosting cords to prevent a loss of signal due to the extraordinarily long guitar cables.

According to Stevens ' definition, a loudness of 1 sone is equivalent to the loudness of a signal at 40 phons, the loudness level of a 1 kHz tone at 40 dB SPL.

According to him, our ancestors, like many other social animals, used contact calls to maintain constant contact with the members of the group, and a signal of danger was communicated through becoming silent and freezing.

According to the jurisprudence of the Boards of Appeal of the EPO, a technical effect provided by a computer program can be, for example, a reduced memory access time, a better control of a robotic arm or an improved reception and / or decoding of a radio signal.

According to another telling of this story, Étaín, deciding that she still loved Eochaid, gave him a signal so he would choose correctly.

According to Article 5, George explains that Domon is not allowed to fight more than one fighter, however Domon claims he received the signal to fight before Neo Cuba's Gundam Fighter had arrived.

According to the sampling theorem, any signal with bandwidth B can be perfectly described by more than 2B samples per second.

According to local legend, this is the site where Robert the Bruce mistook farmers ' fires on the mainland as the signal to launch his campaign.

According to nepahdtv. com, this move was met with some dismay from viewers in areas where reception of signals from Penobscot Knob is difficult if not impossible, leaving many people in rural areas without their signal.