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 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 Edmund Gosse, his father's career was destroyed by his " strange act of wilfulness " in publishing Omphalos ; Edmund claimed his father had " closed the doors upon himself forever.

According to statute, Qing society was divided into relatively closed estates, of which in most general terms there were five.

According to Press-Enterprise Co. v. Superior Court,, trials can be closed at the behest of the government on account of there being " an overriding interest based on findings that closure is essential to preserve higher values and is narrowly tailored to serve that interest.

According to the Natural Law Party official web site, the national headquarters of the Natural Law Party closed effective on April 30, 2004 and the US Peace Government is now carrying forward the programs, policies, and ideals of the Natural Law Party.

According to the paper by Thiers, in certain climates and certain seasons, it may be an advantage to remain closed because moisture can be conserved in that way.

According to a current classification system, Hebeloma now belongs to family Hymenogastraceae, and is considered more narrowly related to the closed Hymenogaster fungi than, for instance, to the ordinary mushrooms in genus Cortinarius.

According to Kendall Bennett, developer of the VBE / AF standard, the VESA Software Standards Committee was closed down due to a lack of interest resulting from charging high prices for specifications.

According to the rendering Daphne prays for help either to the river god Peneus or to Gaia, and is transformed into a laurel ( Laurus nobilis ): " a heavy numbness seized her limbs, thin bark closed over her breasts, her hair turned into leaves, her arms into branches, her feet so swift a moment ago stuck fast in slow-growing roots, her face was lost in the canopy.

According to the 1967 constitution, Presidents were at that time elected by both Houses of Congress in joint session, with representatives of all State Legislatures ; this was widely recognised as a mere sham as, since the March 1964 coup d ' état, each " elected " President had been a retired general chosen in a closed military caucus.

According to the Dubuque Telegraph Herald, " the school was closed in 2005 because of budget problems and declining enrollment ," and on April 5, 2006 the 85-year-old Welton School building was demolished.

According to the District web site, Clarksville school will remain closed for the indefinite future due to declining enrollment and economic issues.

According to the FAO, about 7 % of the natural closed forest being lost in the tropics is land being converted to plantations.

According to the historian of science Norwood Russell Hanson: There is no bilaterally-symmetrical, nor excentrically-periodic curve used in any branch of astrophysics or observational astronomy which could not be smoothly plotted as the resultant motion of a point turning within a constellation of epicycles, finite in number, revolving around a fixed deferent. Any path — periodic or not, closed or open — can be represented with an infinite number of epicycles.

According to U. S. Air Force Captain Edward J. Ruppelt, Copies of Lt. Col. Halt's letter regarding the sighting at RAF Woodbridge ( see above ) to the U. K. Ministry of Defense were routinely released ( without addition comment ) by the American base public affairs staff throughout the 1980s until the base closed.

* According to a report in The New York Times, March 10 and 11, 1929, Isidore Fink, of 4 East 132nd Street, New York City, was in his Fifth Avenue Laundry on the night of March 9, 1929 with the windows closed and door of the room bolted.

According to Jigsaw Puzzles: An Illustrated History and Price Guide, by Anne D. Williams, Parker Bros. closed the Pastime line in the 1950's and their die-cut puzzles were phased out in the late 1970's.

According to Guaman Poma, quipucamayocs could " read " the quipus with their eyes closed.

According to a March 2004, Mellman Group poll, Gregoire would beat both Sims and Talmadge 36 % to 11 % and four percent in an open primary, and would beat Sims 55 % to 17 % in a closed primary.

According to bathory. se ( Site now closed ) ( link no longer available ), Quorthon ceased this spectacle due to overblown media attention to his image rather than music.

According to the New York Post, all 26 members of the musical cast brought charges that Quaid " physically and verbally abused his fellow performers " and that the show closed rather than continuing to Broadway because of Quaid's " oddball behavior ".

According to the nuclear shell model, radioactive nuclei that would decay into isotopes of these elements have closed neutron shells near the neutron drip line, where more neutrons cannot be added.

According to German records, of the total 17, 210 brought in with 58 transports, 6, 408 managed to survive long enough to be evacuated to other camps when the Germans closed the factory in 1944.

According to staff working at the company, The UK offices and the Canadian offices of Roadrunner Records are to be closed completely, with severe levels of cutbacks occurring across the rest of the label worldwide.

According to the rules of most air transportation authorities, such as the U. S. Federal Aviation Administration and European Union's Joint Aviation Authorities, should passengers flying internationally with checked baggage fail to arrive at the departure gate before the flight is closed, that person's baggage must be retrieved from the aircraft hold before the flight is permitted to take off.