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 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 Katherine Litz, `` the becoming, the process of realization, is the dance ''.

According to the official interpretation of the Charter, a member cannot be penalized by not having the right to vote in the General Assembly for nonpayment of financial obligations to the `` special '' United Nations' budgets, and of course cannot be expelled from the Organization ( which you suggested in your editorial ), due to the fact that there is no provision in the Charter for expulsion.

According to Friends, the unit was organized by John Snook, a former World War 2, commando who is vice president and general manager of the telephone company.

According to Philippoff, the recoverable shear S is given by Af where **yc is the angle of extinction.

According to Whipple ( 1955 ), cometary debris is sufficient to replenish the material spiraling into the sun, maintaining a fairly steady state.

According to the theory proposed, this is a consequence of the severe condition of perceived threat that persists unabated for the anxious child in an ambiguous sort of school environment.

According to the theory underlying odd-lot indices, the trader who trades odd lots is most likely a small trader, one who can't afford to trade round lots.

According to Dr. Ellis, the average 20-year-old American woman is capable of far greater sexual arousal than her partner.

According to a newspaper report of the 1961 statistics of the Church of England, the `` total of confirmed members is 9,748,000, but only 2,887,671 are registered on the parochial church rolls '', and `` over 27 million people in England are baptized into the Church of England, but roughly only a tenth of them continue ''.

According to the fossil record, Lissamphibia, which includes all modern amphibians and is the only surviving lineage, may have branched off from the extinct groups Temnospondyli and Lepospondyli at some period between the Late Carboniferous and the Early Triassic.

According to the definition by IUPAC, the former two are alkanes, whereas the third group is called cycloalkanes.

According to adventurer André Malraux, in his La Condition Humaine ( 1933 ), " If a man is not ready to risk his life, where is his dignity ?".

According to the Catholic Encyclopedia, an Allocution is a solemn form of address or speech from the throne employed by the Pope on certain occasions.

According to Richard Dawkins, a distinction between agnosticism and atheism is unwieldy and depends on how close to zero we are willing to rate the probability of existence for any given god-like entity.

According to Tillich, the last of these three types of existential anxiety, i. e. spiritual anxiety, is predominant in modern times while the others were predominant in earlier periods.

According to Viktor Frankl, the author of Man's Search for Meaning, when a person is faced with extreme mortal dangers, the most basic of all human wishes is to find a meaning of life to combat the " trauma of nonbeing " as death is near.

According to one analyst, Armenia's economic system is anticompetitive due to the structure of the economy being a type of " monopoly or oligopoly ".

According to the estimate of a former prime minister, Hrant Bagratian, 55 percent of Armenia's GDP is controlled by 44 families.

According to the Central Bank of Armenia, in 2005, cash remittances from Armenians working abroad reached a record-high level of $ 1 billion, which is worth more than one fifth of the country ’ s 2005 Gross Domestic Product.

According to the ROA National Statistical Service, the average monthly salary during January-June 2008 is 86, 850 drams ( about $ 287 at the time ).