Therefore both and are inverses of By Theorem 1. 5,

By the Chinese Remainder Theorem, this ring factors into the direct product of rings of integers mod p. Now each of these factors is a field, so it is clear that the only idempotents will be 0 and 1.

By Fermat's Little Theorem, 2 < sup >( q − 1 )</ sup > ≡ 1 ( mod q ).

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.

By the Hilbert Basis Theorem and some elementary properties of Noetherian rings, every affine or projective coordinate ring is Noetherian.

By applying the Coase Theorem two possible solutions arise for internalizing this externality.

By the First Fundamental Theorem, 0 ≤ δ ( a, f ) ≤ 1, if T ( r, f ) tends to infinity ( which is always the case for non-constant functions meromorphic in the plane ).

By Theorem 1. 3. 1 of Bekka et al., G is compactly generated.

By comparing the form of several yupanas, researchers found that calculations were based using the Fibonacci sequence 1, 1, 2, 3, 5 and powers of 10, 20 and 40 as place values for the different fields in the instrument.

By the 2004 season the Diamondbacks had dropped to a dismal 51 – 111 record, the worst in Major League Baseball and one of the 10 worst records in the modern era, despite Johnson pitching a perfect game on May 18 of that season.

By the time of the war, they fielded the SdKfz 10 / 4 and 6 / 2, cargo halftracks mounting single 20 mm or 37 mm AA guns ( respectively ).

By 2007, national branches of the Anthroposophical Society had been established in fifty countries, and about 10, 000 institutions around the world were working on the basis of anthroposophy.

By lowering the activation energy, the enzyme speeds up that reaction by a rate of 10 < sup > 11 </ sup > or more: a reaction that would normally take over 3, 000 years to complete spontaneously might take less than a second with an enzyme.

By the time the federal environmental clearances were delivered in 1994, A $ 10 million contract, signed off as a cost overrun, was used to repair these leaks.

By 1866 the collection consisted of some 10, 000 objects.

By the 19th Century, this body of legislation included some 10, 000 norms.

By then, the team had appeared in 10 straight championship games and won seven.

By quickly setting up for the next play ( often within 5 – 10 seconds after the last play despite being afforded 45 seconds ) this hindered the other team's defense from substituting situational players, regrouping for tactics, and, some suggest, increased the defense's rate of fatigue ( This is attributed to the belief that the offense dictates when a play starts so they tend to be more mentally relaxed and prepared for the start of a play where the defense must remain on a higher level of alert before the play starts ).

By 1992 the cost of typical recorders was down to $ 10 – 12, 000, and in September 1995 Hewlett-Packard introduced its model 4020i manufactured by Philips, which at $ 995 was the first recorder to cost less than $ 1000.

By the 1870s, British physicist William Crookes and others were able to evacuate tubes to a lower pressure, below 10 < sup >− 6 </ sup > atm.

By the 1860s, there were around 5, 000 residents, and the population grew to around 10, 000 by the late 1880s.

" Two Women Chatting By The Sea ," St. Thomas, ( 1856 ) Camille Pissarro was born on July 10, 1830 on the island of St. Thomas to Frederick and Rachel Pissarro.

By 1990, an estimated 50 % of islanders had some degree of proficiency in Corsican, and a small minority, perhaps 10 %, used Corsican as a first language.

By the end of the decade, similar calculators were priced less than $ 10 (£ 5 ).

By September 22 there were no disco songs in the US Top 10 chart

By 1964, Bluebird was being built at 10, 000 cars a month.

By the time he was about 10, Defoe's mother Annie had died.

Plaque explaining the history of Deception Pass " Deception Pass-Named By Captain George Vancouver 10 June 1792.

By using directional antennas and sacrificing some traffic capacity, cell coverage could extend to over 10 km.

By the mid-1980s, her record sales were still relatively strong, with " Save the Last Dance for Me ", " Downtown ", " Tennessee Homesick Blues " ( all 1984 ); " Real Love " ( another duet with Kenny Rogers ), " Don't Call It Love " ( both 1985 ); and " Think About Love " ( 1986 ) all reaching the country-singles Top 10.

By the 1530s, the writings of Erasmus accounted for 10 to 20 percent of all book sales.

* 1502 10 July – By a Royal Warrant passed in Toledo by Isabella I of Castile, Gibraltar was granted its coat of arms: " An escutcheon on which the upper two thirds shall be a white field and on the said field set a red castle, and below the said castle, on the other third of the escutcheon, which must be a red field in which there must be a white line between the castle and the said red field, there shall be a golden key which hangs by a chain from the said castle, as are here figured ".

By 1900 the demand for Beaux's work brought clients from Washington, D. C., to Boston, prompting the artist to move to New York City ; it was there she spent the winters, while summering at Green Alley, the home and studio she had built in Gloucester, Massachusetts.

By the 12th to 13th centuries A. D., monks were preparing illuminated manuscripts on vellum and parchment in monasteries throughout Europe and were using lead styli to draw lines for their writings and for the outlines for their illuminations.

By applying voltage to G, one can control I < sub > D </ sub >.

By this time Thomas Holt had left teaching and moved into theatrical and artist management in partnership with the noted entrepreneur Hugh D. McIntosh, owner of the Tivoli theatre circuit.

Eric D. Weitz wrote, " By 1948, according to Nicolas Werth, the mortality rate of the 600, 000 people deported from the Caucasus between 1943 and 1944 had reached 25 %.

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

By using the streamwise vector i parallel to the freestream in place of k in the integral, we obtain an expression for the pressure drag D < sub > p </ sub > ( which includes induced drag in a 3D wing ).

By the end of the war, MIT became the nation's largest wartime R & D contractor ( attracting some criticism of Bush ), employing nearly 4000 in the Radiation Laboratory alone and receiving in excess of $ 100 million ($ billion in 2012 dollars ) before 1946.

By early 1978, the D. C. jazz-fusion group Mind Power had transformed into Bad Brains, one of the first bands to be identified with hardcore punk.

By 1983, St. Paul's Hüsker Dü, Willful Neglect, Chicago's Naked Raygun and D. C .' s The Faith were taking the hardcore sound in experimental and ultimately more melodic directions.

By the Imperial Session held in Washington, D. C. in 1900, there were 55, 000 members and 82 Temples.

By 1972 the participating cities included Atlanta, Buffalo, Detroit, Washington D. C., Miami, and Philadelphia, as well as San Francisco.

By 1846, the Morse telegraph service was operational between Washington, D. C., and New York.

By the time of the Lancaster House Agreement in 1979 Rhodesia only had representative offices in London, Bonn, Pretoria, Washington, D. C. and Tokyo.

* Dr. J. Frick, Physical Technics ; Or Practical Instructions for Making Experiments in Physics Translated By John D. Easter, Ph. D .-J.

* By functions: production, selling, distribution, administration, R & D, development,

By the end of the American Civil War, the Observatory's clocks were linked via telegraph to ring the alarm bells in all of the Washington, D. C. firehouses three times a day, and by the early 1870s the Observatory's daily noon time signal was being distributed nationwide via the Western Union Telegraph Company.

Detaching one slug from its host, they seal it in a film canister and bring it back to headquarters in Washington, D. C. By the time they get there though, the remains of the slug are a stinking mess, and they are unable to convince the President that there is an invasion.

By Himself, published by D. Appleton & Co., in two volumes, began with the year 1846 ( when the Mexican War began ) and ended with a chapter about the " military lessons of the war " ( 1875 edition: Volume I ;

By mid 1883, financial difficulties due to aggressive growth and expenditures led to a shake up among the D & RG board of directors, and General Palmer resigned as president of the D & RG in August 1883, while retaining that position with the Western.

By 1956, the D & RGW's standard-gauge steam locomotives had been retired and scrapped.

* Handbook of Mathematical Cognition By Jamie I. D. Campbell