By this time a procedure for determining whether the ring of all algebraic integers of a given quadratic field is a PID was well known to the quadratic form theorists.

By Sylvester's law of inertia, a basis of tangent vectors X < sub > i </ sub > can be chosen locally so that the quadratic form diagonalizes in the following manner

By the usual solution of quadratic equations, any extension of degree 2 of K has this form.

By the use of the Bohr-Sommerfeld (" old ") quantum theory Paul Epstein and Karl Schwarzschild were independently able to derive equations for the linear and quadratic Stark effect in hydrogen.

By the Fundamental theorem of projective geometry a reciprocity is the composition of an automorphic function of K and a homography.

By Frobenius reciprocity for compact groups, this is equivalent to finding the multiplicity of π in the unitary representation induced from σ. Branching rules for the classical groups were determined by

By Frobenius reciprocity, on K they decompose as a direct sum of the irreducible representations of K with dimensions | k | + 2m + 1 with m a non-negative integer.

By the same token, however, the surface terms do not in general vanish if one integrates over all space for this reciprocity variant, so a Rayleigh-Carson form does not hold without additional assumptions.

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.

By the end of the year Mobutu had created a rival government with its own prime minister.

By the 18th century, it had transformed itself into the Maratha Empire under the rule of the peshwas ( prime ministers ).

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

By the late 20th century the majority of the world's countries had a prime minister or equivalent minister, holding office under either a constitutional monarchy or a ceremonial president.

By the 1920s however the Senate had lost much of its original influence, and hence no sitting senator had been known to have serious aspirations of becoming prime minister whilst remaining in the Senate.

By Article 75 of the constitution of India, remuneration of the prime minister as well as other ministers are to be decided by the Parliament and is renewed from time to time.

By 2004 Neighbours was regularly attracting just under a million viewers per episode — low for Australian prime time television.

By April 1948, the coup group brought back Field Marshal Luang Phibulsongkram, the wartime Prime Minister, to prime ministership once again.

By the time of his presidency, Wilson hoped that Presidents could be party leaders in the same way British prime ministers were.

By 1971, the FCC introduced the Prime Time Access Rule, which freed local network affiliates in the top 50 markets ( in practice, the entire network ) to take a half hour of prime time from the networks on Mondays through Saturdays and one full hour on Sundays.

By 1976, 60 Minutes became the top-rated show on Sunday nights in the U. S. By 1979, it had achieved the number-one Nielsen rating for all television programs, unheard of before for a news broadcast in prime time.

By definition of prime element, if is a Gaussian prime, then it divides ( in Z ) some.

The ones whose numerators are relatively prime to 20 By definition this is φ ( 20 ) fractions.

By this time, his approval ratings had dipped into the teens, and were at 11 % in a 1992 Gallup poll, making him one of the most unpopular prime ministers since opinion polling began in Canada in the 1940s.

By English law and custom they may only elect the person who has been nominated by the monarch on the advice of the prime minister.

" By the time he gained widespread acceptance in the 1890s, Whistler was past his prime as a portrait painter.

By definition, every integer greater than one is either a prime number or a composite number.

By the fundamental theorem of arithmetic, every positive integer greater than 1 has a unique prime factorization.

By 1867, many more families had established homesteads, and two-thirds of the prime farming land had been claimed.

By the mid-1900s Houghton Lake had become known as a prime resort area.

By tradition and convention, only the Cabinet, a standing committee in the larger council, advises the governor general and this advice is usually expressed exclusively through a consultation with the prime minister.

By the Chinese Remainder Theorem, each can further be decomposed into a direct sum of submodules of the form, where is a power of a prime ideal.