proved that the limit of a < sub > n </ sub >/ n exists, and has absolute value less than 1 unless f is a Koebe function.

( Milin later showed that 14 can be replaced by 1. 14., and Hayman showed that the numbers b < sub > k </ sub > have a limit less than 1 if φ is not a Koebe function, so Littlewood and Paley's conjecture is true for all but a finite number of coefficients of any function.

* A Kleinian group is called a Koebe group if all its factors are elementary or Fuchsian.

is achieved by the Koebe function z /( 1 − z )< sup > 2 </ sup >.

Related work was done by Emil Artin in 1924 ( for the system now called Artin billiard ), P. J. Myrberg, Paul Koebe, Jakob Nielsen, G. A. Hedlund.

Paul Koebe later gave several more proofs and generalizations.

No general installation should be made until a model installation has been proved and its maximum capability determined.

The 1976 definition of the astronomical unit was incomplete, in particular because it does not specify the frame of reference in which time is to be measured, but proved practical for the calculation of ephemerides: a fuller definition that is consistent with general relativity was proposed, and " vigorous debate " ensued until in August 2012 the International Astronomical Union adopted the current definition of 1 astronomical unit = 149597870700 meters.

Supported by the prestige of being the only general who had proved capable of defeating the French, he promptly initiated a far-reaching scheme of reform, which replaced the obsolete methods of the 18th century.

" Given its general release in January 1931, City Lights proved to be a popular and financial success — eventually grossing over $ 5 million.

In a sense it is more of a research program proposal: part of the research is to find a proper formal statement that is physically reasonable and that can be proved to be true or false ( and that is sufficiently general to be interesting ).

Promoted to general of cavalry he arrived in Turin with his friend the Prince of Commercy ; but it proved an inauspicious start.

His reign proved uneventful, and he took little public part in politics, devoting himself to military affairs, in which his advice and experience were of the greatest value, not only to the Saxon corps but to the German army in general.

The following more general statement will be proved.

It is named for Hans Hahn and Stefan Banach who proved this theorem independently in the late 1920s, although a special case was proved earlier ( in 1912 ) by Eduard Helly, and a general extension theorem from which the Hahn – Banach theorem can be derived was proved in 1923 by Marcel Riesz.

However, his discoveries were not appreciated by his contemporaries and came into general use only with discoveries of British surgeon Joseph Lister, who in 1865 proved the principles of antisepsis in the treatment of wounds ; However, medical conservatism on new breakthroughs in pre-existing science prevented them from being generally well received during the 19th century.

Alexander proved to be enthusiastic about the Canadian wilderness, as well as a popular governor general with the Canadian people, and he would be the last non-Canadian-born governor general before the appointment of Adrienne Clarkson in 1999.

In 1972, Intel launched the 8008, the first 8-bit microprocessor .< ref > using enhancement load PMOS logic ( demanding 14V, achieving TTL-compatibility by having V < sub > CC </ sub > at + 5V and V < sub > DD </ sub > at-9V )</ ref > It implemented an instruction set designed by Datapoint corporation with programmable CRT terminals in mind, that also proved to be fairly general purpose.

This was rigorously proved and extended by Vladimir Arnold ( in 1963 for analytic Hamiltonian systems ) and Jürgen Moser ( in 1962 for smooth twist maps ), and the general result is known as the KAM theorem.

Philby found that the award proved helpful in obtaining access to fascist circles: " Before then ," he later wrote, " there had been a lot of criticism of British journalists from Franco officers who seemed to think that the British in general must be a lot of Communists because so many were fighting with the International Brigades.

Mies had hoped his architecture would serve as a universal model that could be easily imitated, but the aesthetic power of his best buildings proved impossible to match, instead resulting mostly in drab and uninspired structures rejected by the general public.

In 1824, Niels Henrik Abel proved the striking result that there can be no general ( finite ) formula, involving only arithmetic operations and radicals, that expresses the roots of a polynomial of degree 5 or greater in terms of its coefficients ( see Abel-Ruffini theorem ).

:" We were justified intuitionistically in using the classical 2-valued logic, when we were using the connectives in building primitive and general recursive predicates, since there is a decision procedure for each general recursive predicate ; i. e. the law of the excluded middle is proved intuitionistically to apply to general recursive predicates.

He proved to be an able Pope, with an unlimited capacity for work and a lawyer's eye for detail, and a wise statesman, the general object of whose policy was to free the Papacy from its dependence upon Spain.

Some of these have proved more durable during periods of standardization and some have entered general public awareness.

These special cases were known long before the general theorem was proved.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

The Cook – Levin theorem states that the Boolean satisfiability problem is NP-complete, and in fact, this was the first decision problem proved to be NP-complete.

SAT was the first known NP-complete problem, as proved by Stephen Cook in 1971 ( see Cook's theorem for the proof ).

The term Bayesian refers to Thomas Bayes ( 1702 – 1761 ), who proved a special case of what is now called Bayes ' theorem in a paper titled " An Essay towards solving a Problem in the Doctrine of Chances ".

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Gauss proved the method under the assumption of normally distributed errors ( see Gauss – Markov theorem ; see also Gaussian ).

He also studied and proved some theorems on perfect powers, such as the Goldbach – Euler theorem, and made several notable contributions to analysis.

He also proved a result concerning Fermat numbers that is called Goldbach's theorem.

Groups of rank at least 3 are further subdivided into 3 classes by the trichotomy theorem, proved by Aschbacher for rank 3 and by Gorenstein and Lyons for rank at least 4.

J. Chernick proved a theorem in 1939 which can be used to construct a subset of Carmichael numbers.

In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 ( by Kenneth Appel and Wolfgang Haken, using substantial computer assistance ).

Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself.

Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute, often metaphysical, sense.

The Pythagorean theorem is proved.

Cantor supposed that Thales proved his theorem by means of Euclid book I, prop 32 after the manner of Euclid book III, prop 31.

It improves Dirichlet's theorem on prime numbers in arithmetic progressions, by showing that by averaging over the modulus over a range, the mean error is much less than can be proved in a given case.

It was the first major theorem to be proved using a computer.

In 1890, in addition to exposing the flaw in Kempe's proof, Heawood proved the five color theorem and generalized the four color conjecture to surfaces of arbitrary genus — see below.

The existence of this functor can be proved by realizing that it is the right-adjoint to the diagonal functor and invoking the Freyd adjoint functor theorem.