He had introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, and given a proof of the Riemann – Roch theorem with them ( a version appeared in his Basic Number Theory in 1967 ).

The self-adjoint requirement means finite-dimensional C *- algebras are semisimple, from which fact one can deduce the following theorem of Artin – Wedderburn type:

The oblate spheroid shape reflects, following Clairaut's theorem, the balance between containment by gravitational attraction and dispersal by centrifugal force.

The following theorem connects Euclidean relations and equivalence relations:

In the following, we state two equivalent forms of the theorem, and show their equivalence.

The following lemma, which Gödel adapted from Skolem's proof of the Löwenheim-Skolem theorem, lets us sharply reduce the complexity of the generic formula for which we need to prove the theorem:

The intermediate value theorem states the following: If f is a real-valued continuous function on the interval b, and u is a number between f ( a ) and f ( b ), then there is a c ∈ b such that f ( c ) = u.

For example, Rice's theorem shows that each of the following sets of computable functions is undecidable:

Applying the rational root theorem thus yields the following possible roots for t:

The following theorem represents positive linear functionals on C < sub > c </ sub >( X ), the space of continuous compactly supported complex-valued functions on a locally compact Hausdorff space X.

Stokes ' theorem is a vast generalization of this theorem in the following sense.

To simplify this greatly, we can use Green's theorem to simplify the left side to get the following:

The following theorem is named after Richard Kadison.

No closed tours exist for smaller square boards besides the trivial case ( this is a corollary of the following theorem ).

( following from it, and corresponding to the binomial theorem ), are included in the observations which matured to the system of the umbral calculus.

Indeed, the following theorem holds true ( provided that the deduction apparatus of the considered fuzzy logic satisfies some obvious effectiveness property ).

In practice, we might not have the distribution function but the Fisher – Tippett – Gnedenko theorem provides the following asymptotic result

Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy as the following: p divides a < sup > p − 1 </ sup > − 1 whenever p is prime and a is coprime to p.

The following argument proves Minkowski's theorem for the special case of L = Z < sup > 2 </ sup >.

For all of the above reasons, the following theorem, referred to as the Lasker – Noether theorem, may be seen as a certain generalization of the fundamental theorem of arithmetic:

That fall he submitted to Professor Baker the first acts and outlines of the following acts of several plays, six of them, according to some of his associates, and he also worked on a play that he first called Niggertown, the material for which he had collected during the summer at home.

Autosuggestibility, the reaction of the subject in such a way as to conform to his own expectations of the outcome ( i.e., that the arm-rise is a reaction to the pressure exerted in the voluntary contraction, because of his knowledge that `` to every reaction there is an equal and opposite reaction '' ) also seems inadequate as an explanation for the following reasons: ( 1 ) the subjects' apparently genuine experience of surprise when their arms rose, and ( 2 ) manifestations of the phenomenon despite anticipations of something else happening ( e.g., of becoming dizzy and maybe falling, an expectation spontaneously volunteered by one of the subjects ).

She also was the original GOP national committeewoman from New Jersey in the early 1920s following adoption of the women's suffrage amendment.

The Canadian Aboriginal syllabics are also an abugida rather than a syllabary as their name would imply, since each glyph stands for a consonant which is modified by rotation to represent the following vowel.

Note that any model of ZF ¬ C is also a model of ZF, so for each of the following statements, there exists a model of ZF in which that statement is true.

Since the administrative reforms in Bavaria in the 1970s, the town also encompasses the following Ortsteile:

" Through Methodism, Wesley's teachings also inspire a large scholarly following, with vocal proponents including J. Kenneth Grider, Stanley Hauerwas, Thomas Oden, Thomas Jay Oord, and William Willimon.

This word is usually conceded to be derived from the Hebrew ( Aramaic ), meaning " Thou art our father " ( אב לן את ), and also occurs in connection with Abrasax ; the following inscription is found upon a metal plate in the Carlsruhe Museum:

The war also marked a watershed in the geo-political history of the region, as together with the Lombard migration the following year, it signalled the end of six centuries of Germanic dominance in the Pannonian Basin.

Although not mentioned by Asser or by the Anglo-Saxon Chronicle, Alfred probably also paid the Vikings cash to leave, much as the Mercians were to do in the following year.

He also conquered an important part of the land south of the Tagus River, although this was lost again to the Moors in the following years.

Compare also the following lines of Elvis Costello's song " Oliver's Army ": Oliver's Army are on their way / Oliver's Army is here to stay.

Along with the Bill of Rights 1689, it remains today one of the main constitutional laws governing the succession to not only the throne of the United Kingdom, but, following British colonialism, the resultant doctrine of reception, and independence, also to those of the other Commonwealth realms, whether by willing deference to the act as a British statute or as a patriated part of the particular realm's constitution.

Tolstoy defined art ( and by no coincidence also characterized its value ) as the following: " Art is a human activity consisting in this, that one man consciously, by means of certain external signs, hands on to others feelings he has lived through, and that other people are infected by these feelings and also experience them.

In his book The Lost Ark of the Covenant ( 2008 ), Parfitt also suggests that the Ark was taken to Arabia following the events depicted in the Second Book of Maccabees, and cites Arabic sources which maintain it was brought in distant times to Yemen.

Dummett's writings on anti-realism also draw heavily on the later writings of Wittgenstein concerning meaning and rule following.

The couple moved to Paris the following year, where Guiler pursued his banking career and Nin began to pursue her interest in writing ; in her diaries she also mentions having trained as a flamenco dancer in Paris in the mid-to-late 1920s.

Scholars generally agree that many of the personalities of Beowulf also appear in Scandinavian sources ( specific works designated in the following section ).

The infection may also be introduced through a skull fracture following a head trauma or surgical procedures.

Quinte also houses a large number of industries in the plastics & packaging sector, transportation sector, logistics sector and advanced manufacturing sector, including the following ( just a few of over 350 industries located in the Bay of Quinte Region ):

In addition to winning the 1966 MVP, Robinson also won the Triple Crown ( leading the American League in batting average, home runs, and runs batted in ), a feat also achieved the following season by Boston's Carl Yastrzemski but never since.

His voice became a fixture of both Memorial Stadium and Camden Yards, and his expression " Give that fan a contract ", uttered whenever a fan caught a foul ball, was one of his trademarks – the other being his distinct " Thank Yooooou ..." following every announcement ( He was also known on occasion to say " Give that fan an error " after a dropped foul ball ).

Zahir-ud-din Muhammad Babur ( 14 February 148326 December 1530 ; sometimes also spelt Baber or Babar ) was a conqueror from Central Asia who, following a series of setbacks, finally succeeded in laying the basis for the Mughal dynasty in the Indian Subcontinent and became the first Mughal emperor.