In mathematics, a Fuchsian model is a construction of a hyperbolic Riemann surface R as a quotient of the upper half-plane H. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic.

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

By setting K = ker ( f ) we immediately get the first isomorphism theorem.

* By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.

By " satisfactory " one would mean at least the equivalent of Plancherel theorem.

By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization.

By the Glivenko – Cantelli theorem, if the sample comes from distribution F ( x ), then D < sub > n </ sub > converges to 0 almost surely.

By this theorem, once a star's chemical composition and its position on the main sequence is known, so too is the star's mass and radius.

By Gödel's incompleteness theorem, Peano arithmetic is incomplete and its consistency is not internally provable.

By Rice's theorem, the 1-halting problem is undecidable.

By the corollary to the recursion theorem, there is an index such that returns.

By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space.

By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation.

By the first isomorphism theorem, the image of A under ƒ is a substructure of B isomorphic to the quotient of A by this congruence.

By the central limit theorem, this distribution approaches the normal distribution as n increases.

By analogy to the prior and posterior probability terms in Bayes ' theorem, Bayes ' rule can be seen as Bayes ' theorem in odds form.

By the extreme value theorem, a continuous function on a closed interval must attain its minimum and maximum values at least once.

By the rank-nullity theorem, a system of n vectors in k dimensions ( where all dimensions are necessary ) satisfies a ( p = n − k )- dimensional space of relations.

By Tychonoff's theorem we have that is compact since is, so the closure of in is a compactification of.

By Stone's representation theorem every Boolean ring is isomorphic to a field of sets ( treated as a ring with these operations ).

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.

By Euler's rotation theorem, we may replace the vector with where is a 3x3 rotation matrix and is the position of the particle at some fixed point in time, say t = 0.

By the divergence theorem Gauss's law can alternatively be written in the differential form:

By `` image '' is meant not only a visual presentation, but also remembered sensations of any of the five senses plus the feelings which are immediately conjoined therewith.

By limiting American strength too much to nuclear strength, this country limited its ability to fight any kind of war besides a nuclear war.

By this standard, it is determined that where two stations operating on the same frequency are involved, objectionable interference from station A exists at any point within the service area of station B where station A's signal is of an intensity one-twentieth or more of the strength of station B's signal at that point.

By using the appropriate version any one of these factors can be determined for any combination of the other three.

By varying the formula, this curve may be moved forward or backward along the coordinates to produce any desired compression strength / density ratio.

By means of charts showing wave-travel times and depths in the ocean at various locations, it is possible to estimate the rate of approach and probable time of arrival at Hawaii of a tsunami getting under way at any spot in the Pacific.

By the time Felix turned up it was early afternoon, which, one would think, would be late enough so that by then, except for small children and a few hardy souls who had not yet sobered up, it could have been expected that people would no longer be having any sort of active interest in the previous night's noisemakers and paper hats.

By March 1861, no leaders of the insurrection had proposed rejoining the Union on any terms.

By the Brønsted-Lowry definition, any compound which can easily be deprotonated can be considered an acid.

By contrast, the Rijndael specification per se is specified with block and key sizes that may be any multiple of 32 bits, both with a minimum of 128 and a maximum of 256 bits.

Whenever any result is sought by its aid, the question will then arise — By what course of calculation can these results be arrived at by the machine in the shortest time?

By the late 19th century, improved instruments were designed which could be mounted in any position and allowed accurate measurements in electric power systems.

By virtue of being a Java application, it is available on any platform supported by Java.

By symmetry, any algebra which satisfies any two of:

By a further act of 1541 — which was not repealed until 1845 — artificers, labourers, apprentices, servants and the like were forbidden to play bowls at any time except Christmas, and then only in their master's house and presence.

By 2011, 165 countries had signed the treaty and none are known-though some are still suspected-to possess any biological weapons.

By managing all four populations as one-through strategic transfers, gene-loss is reduced from 8 % per decade to 2 % per decade, without any increase in bongo numbers in Kenya.

By the Victorian era, ballad had come to mean any sentimental popular song, especially so-called " royalty ballads ", which publishers would pay popular singers to perform in Britain and the United States in " ballad concerts.

By extension, the term is also used to refer to any system administrator who displays ( or wishes he could get away with ) the qualities of the original.

By the end of World War 2, Canadians communicated by telephone, more than any other country.

By the early 2000s, the CD largely replaced the audio cassette player as standard equipment in new automobiles, with 2010 being the final model year for any car in the US to have a factory-equipped cassette player.

By contrast, the Fascist and Nazi regimes of Italy and Germany tended to confine their coercive activities to purely political aims, without any serious attempt to force the ideological conversion of their opponents.

By extension, it has come to include an extreme and unreasoning partisanship on behalf of any group to which one belongs, especially when the partisanship includes malice and hatred towards rival groups.