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 definition above, two relations with the same graph may be different, if they differ in the sets and.

According to this argument, the research around Dunbar's Number implies a limit on the number of inbound and outbound connections a human in a group-forming network can manage, so that the actual maximum-value structure is much sparser than the set-of-subsets measured by Reed's law or the complete graph measured by Metcalfe's Law.

According to DeVos, Nesetril, and Raspaud, " A cycle of a graph G is a set C E ( G ) so that every vertex of the graph ( V ( G ), C ) has even degree.

According to the strong perfect graph theorem, the perfect graphs have a forbidden graph characterization resembling that of bipartite graphs: a graph is bipartite if and only if it has no odd cycle as a subgraph, and a graph is perfect if and only if it has no odd cycle or its complement as an induced subgraph.

According to all major temperature reconstructions published in peer-reviewed journals ( see graph ), the increase in temperature in the 20th century and the temperature in the late 20th century is the highest in the record.

According to the Robertson – Seymour theorem, there exists a finite set H of minimal elements in S. These minimal elements form a forbidden graph characterization of F: the graphs in F are exactly the graphs that do not have any graph in H as a minor.

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 a result of, a graph with 3n vertices can have at most 3 < sup > n </ sup > maximal cliques.

According to Robbins theorem, an undirected graph may be oriented in such a way that it becomes strongly connected, if and only if it is 2-edge-connected.

According to former Omni Group president Wil Shipley, Graffle " sort of stands for ' graph layout '... but in general it was a nonsense word invented just to counter ' Visio '.

According to Brooks ' theorem every cubic graph other than the complete graph K < sub > 4 </ sub > can be colored with at most three colors.

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

According to the Foster census, the Heawood graph, referenced as F014A, is the only cubic symmetric graph on 14 vertices.

According to the graph the CPI stabilized during the 1990s at a fairly low rate never going above 5 percent during the Clinton presidency.

According to the Erdős – Pósa theorem, the size of a minimum feedback vertex set is within a logarithmic factor of the maximum number of vertex-disjoint cycles in the given graph.

According to the isothermal theorem, a graph has the same fixation probability as the corresponding Moran process if and only if it is isothermal, thus the sum of all weights that lead into a vertex is the same for all vertices.