According to Vizing's theorem, the chromatic index of any graph ( the minimum number of colors needed to color its edges so that no two adjacent edges have the same color ) is either the maximum degree of any vertex of the graph or one plus the maximum degree.

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 Maimonides, an afterlife continues for the soul of every human being, a soul now separated from the body in which it was " housed " during its earthly existence.

According to Suetonius, Domitius was a wealthy man with a despicable and dishonest character, who, according to Suetonius, was “ A man who was in every aspect of his life detestable ", and served as consul in 32.

* Drinking songs: According to the grammarian Athenaeus, Alcaeus made every occasion an excuse for drinking and he has provided posterity several quotes in proof of it.

According to Alcidamas, the highest aim of the orator was the power of speaking extempore on every conceivable subject.

According to the Buddha every Buddha in the past and to come will have two chief disciples and one attendant during his ministry.

According to the current Nobel Committee head Lars Gillensten, his nominee was discussed every year from 1946 to 1950, then again in 1957 ( it was finally awarded in 1958 ).

" According to Dhanis ' medical officer, Captain Hinde, their town of Ngandu had " at least 2, 000 polished human skulls " as a " solid white pavement in front " of its gates, with human skulls crowning every post of the stockade.

According to The General History of the Pirates, published more than 25 years after the event by an author whose very identity remains in dispute, Kidd made peaceful overtures to Culliford: he " drank their Captain's health ," swearing that " he was in every respect their Brother ," and gave Culliford " a Present of an Anchor and some Guns.

According to Strabo, by the time of emperor Augustus, up to 120 Roman ships were setting sail every year from Myos Hormos in Roman Egypt to India.

According to the Eclogae Propheticae, every thousand years every member of each order moves up a degree, and thus men can become angels.

According to Christian theologian Alister McGrath, the Jewish Christians affirmed every aspect of then contemporary Second Temple Judaism with the addition of the belief that Jesus was the messiah, with Isaiah 49: 6, " an explicit parallel to 42: 6 " quoted by Paul in Acts 13: 47 and reinterpreted by Justin the Martyr.

According to a study, " in most areas studied, the various ventures that prompted deforestation rarely generated more than US $ 5 for every ton of carbon they released and frequently returned far less than US $ 1 ".

According to the National Safety Council in the United States, 300, 000 injuries are caused by doors every year.

According to the UN, an area of fertile soil the size of Ukraine is lost every year because of drought, deforestation and climate change.

* According to the CSIRO, in the next decade, astronomers expect to be processing 10 petabytes of data every hour from the Square Kilometre Array ( SKA ) telescope.

According to IBM, the new SKA telescope initiative will generate over an exabyte of data every day.

According to Josephus, the Essenes had settled " not in one city " but " in large numbers in every town ".

According to the U. S. National Transportation Safety Board, in 2005 general aviation in the United States ( excluding charter ) suffered 1. 31 fatal accidents for every 100, 000 hours of flying in that country, compared to 0. 016 for scheduled airline flights.

According to Ruickbie ( 2004, p. 19 ) the Greeks observed two days sacred to Hecate, one on the 13th of August and one on the 30th of November, whilst the Romans observed the 29th of every month as her sacred day.

According to the Italian Ministry of Foreign Affairs, every year there are more than 200, 000 foreign students that are learning Italian language ; they are distributed in the 90 Institutes of Italian Culture in the world, in the 179 Italian schools abroad and in the 111 Italian sections that are open into foreign schools.

According to Technorati, the number of blogs doubles about every 6 months with a total of 35. 3 million blogs. This is an example of the early stages of logistic growth, where growth is approximately exponential, since blogs are a recent innovation.

According to the constitution, direct elections must be held for the parliament every 4 years.

According to her Foundation, " she oversaw every aspect of the making of her films, from hiring talent and crew to overseeing the script, the shooting, the editing, to the final release and promotion of each project.

According to Maya Lin, art should be an act of every individual willing to say something new and that which is not quite familiar.