According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant.

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 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 Alexander Technique instructor Michael J. Gelb, people tend to study the Alexander Technique either to rid themselves of pain, to increase their performance abilities, or for reasons of personal development and transformation.

According to Husserl, this view of logic and mathematics accounted for the objectivity of a series of mathematical developments of his time, such as n-dimensional manifolds ( both Euclidean and non-Euclidean ), Hermann Grassmann's theory of extensions, William Rowan Hamilton's Hamiltonians, Sophus Lie's theory of transformation groups, and Cantor's set theory.

According to special relativity, the Galilean transformation is a good approximation only at relative speeds much smaller than the speed of light.

According to Ahmadiyya thought, Messiahship is a phenomenon through which a special emphasis is given on the transformation of a people by way of offering suffering for the sake of God instead of giving suffering ( i. e. refraining from revenge ).

According to Crosland, it was now possible to achieve greater equality in society without the need for " fundamental " economic transformation.

According to James Tong, the regime aimed at both coercive dissolution of the Falun Gong denomination and " transformation " of the practitioners.

According to director Grace Phan, it's an " intimate insight into the personal transformation " of the man who helped shape and liberate East Timor.

" According to Debord, the Situationist International is so named because the construction of situations is the central idea of their theory, a process he describes as " the concrete construction of momentary ambiances of life and their transformation into a superior passional quality.

According to the United Nations Statistics Division, " wholesale " is the resale ( sale without transformation ) of new and used goods to retailers, to industrial, commercial, institutional or professional users, or to other wholesalers, or involves acting as an agent or broker in buying merchandise for, or selling merchandise to, such persons or companies.

According to present doctrine, that of " treaty monism ", treaties are in the Dutch legal system in principle self-executing ; no special transformation is needed by implementing special law, as in countries with a " dualistic " system ( such as the United Kingdom ).

According to Virgil, Cycnus lamented Phaethon's death till he grew old, so his gray hair became gray feathers upon his transformation.

According to Chairman Steve Van Andel and President Doug DeVos, " We are going through a global transformation of our business ; this includes rethinking our global approach to products, training, brands, and how we operate in all the countries in which Alticor operates.

According to Peter J. Bowler, Cuvier " ridiculed Lamarck's theory of transformation and defended the fixity of species.

According to Hugh Honour, " so far from being, as is sometimes supposed, the culmination of the Neo-classical movement, the Empire marks its rapid decline and transformation back once more into a mere antique revival, drained of all the high-minded ideas and force of conviction that had inspired its masterpieces ".

According to film critic Robin Wood Magnani's " persona as a great actress is built, not on transformation, but on emotional authenticity ... doesn't portray characters but expresses ' genuine ' emotions.

According to the website of The American Prospect, " at the dawn of a new progressive era and a time of economic transformation for the United States and the world, the magazine's founding purpose was to demonstrate that progressive ideas could animate a majority politics ; to restore to intellectual and political respectability the case for social investment ; to energize civic democracy and give voice to the disenfranchised ; and to counteract the growing influence of conservative media.

According to Archbishop Rowan Williams this transformation, accompanied by the proliferation of the tender image of Jesus in Madonna and Child paintings made an important impact within the Christian Ministry by allowing Christians to feel the living presence of Jesus as a loving figure " who is always there to harbor and nurture those who turn to him for help.

According to its constitution, " The YCL is devoted to the interests of all young people and is dedicated to the revolutionary cause of the working class of our country, the transformation of the United States through mass democratic struggle into a socialist society.

According to Coxeter, the transformation by inversion in circle was invented by L. I. Magnus in 1831.

According to his translations of the Vedas, a body functions because it contains three dosha or principles, namely movement ( vata ), transformation ( pitta ) and lubrication and stability ( kapha ).

According to Artzy, " The linear part of each affinity the real affine plane can be brought into one of the following standard forms by a coordinate transformation followed by a dilation from the origin:

According to Klaus, legislation and institutions cannot be created before economic transformation, they have to go hand in hand.

According to Elixir, Warren's transformation is permanent, implying that he is fully capable of transforming back into Archangel again at any time.