According to Vizing's theorem every cubic graph needs either three or four colors for an edge coloring.

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 writer Philip Norman, when Groucho jokingly pointed his index fingers as if holding a pair of six-shooters, Elton John put up his hands and said, " Don't shoot me, I'm only the piano player ," thereby naming the album he had just completed.

According to the same index series, even " dedicated short bias " funds had a return of − 6. 08 % during September 2008.

According to the CIA the gini index which measures income inequality ( the higher the less equal the income distribution ) was clocked at 45. 0 in 2005, compared to 32. 0 in the European Union and 28. 3 in Germany.

According to the UN and the World Bank, it is a high income country with a very high human development index.

According to Ziemba and Ziemba ( 2007 ), Keynes risk-taking reached ' cowboy ' proportions, i. e. 80 % of the maximum rationally justifiable levels ( of the so called Kelly criterion ), with overall return volatility approximately three times higher than the stock market index benchmark.

According to the Big Mac index, the yen was 12 % undervalued against the dollar, whereas it was 13 % overvalued according to the tall latte index.

According to a May 2004 Time Magazine article on obesity in the United States, Jefferson County had the highest percentage of obese adult residents ( 26. 1 %) of any county in the U. S. In that context, the percentage measured were those with a body mass index between 30 and 40 ( any measure greater than 40 corresponds to morbid obesity ).

According to a study conducted by John Bogle over a sixteen-year period, investors get to keep only 47 % of the cumulative return of the average actively managed mutual fund, but they keep 87 % in a market index fund.

According to The Vanguard Group, a well run S & P 500 index fund should have a tracking error of 5 basis points or less, but a Morningstar survey found an average of 38 basis points across all index funds.

According to U. S. News & World Report's America's Best Colleges index, " In guidebooks and surveys by U. S. News & World Report, The Princeton Review and others, SU consistently ranks in the top 10 percent of public and private institutions nationwide.

According to this convention, when an index variable appears twice in a single term it implies summation of that term over all the values of the index.

According to the Financial Accounting Standards Board's FASB Statement of Financial Accounting Standards No. 133 ( FAS 133 )-Accounting for Derivative Instruments and Hedging Activities, an underlying is a specified interest rate, security price, commodity price, foreign exchange rate, index of prices or rates, or other variable ( including the occurrence or nonoccurrence of a specified event such as a scheduled payment under a contract ).

According to Reedy, he was told by Special Agent Frank Super to leave the sites in his index for later investigation.

According to folk medicine, dietary nopales may affect the glycemic index and be useful in diabetes management.

According to the ACORN index, 49. 6 % of residents describe Bewsey and Whitecross as a " hard-pressed " area.

According to Nielsen Media Research, Boston Legal drew the richest viewing audience on television, based on the concentration of high-income viewers in its young adult audience ( Adult 18 – 49 index w /$ 100k + annual income ).

According to the “ Medialogy ”, VTsIOM holds the top score in media citation index ( among other Russian polling organizations ).

According to Fihrist's brief preface, it is meant to be an index of all books written in Arabic, whether by Arabs or others.

According to a state report, for the 2011-2012 school year budgets, 247 school districts adopted a resolution certifying that tax rates would not be increased above their index ; 250 school districts adopted a preliminary budget.

According to writer Philip Norman in his early 1990s authorized biography, " Elton ," during a party in Los Angeles, John's friend Groucho Marx jokingly pointed his index fingers at the singer, as if holding a pair of six-shooters.