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Statement of the inequality. Suppose is a natural number and are positive numbers and: Then the Shapiro inequality states that. where and . For greater values of the inequality does not hold, and the strict lower bound is with . The initial proofs of the inequality in the pivotal cases [2] and [3] rely on numerical computations.
The Shapiro reaction or tosylhydrazone decomposition is an organic reaction in which a ketone or aldehyde is converted to an alkene through an intermediate hydrazone in the presence of 2 equivalents of organolithium reagent. [1] [2] [3] The reaction was discovered by Robert H. Shapiro in 1967. [4] The Shapiro reaction was used in the Nicolaou ...
As of 2010, 90.2% (10,710,239) of Pennsylvania residents age five and older spoke English at home as a primary language while 4.1% (486,058) spoke Spanish, 0.9% (103,502) spoke German including Pennsylvania Dutch, and 0.5% (56,052) spoke Chinese, which includes Mandarin of the population over the age of five.
Shapiro, Bernstein & Co., Inc. is an American music publishing company established in 1900. History [ edit ] The company was established 1900 in New York 's Tin Pan Alley by Maurice Shapiro (1872–1911), who had worked at Adelphi Music publishing company, and his brother-in-law, real-estate dealer Louis Bernstein (1873–1962) [1] (not to be ...
"Rat Pack" is the 54th episode of the HBO original series The Sopranos and is the second of the show's fifth season. Written by Matthew Weiner and directed by Alan Taylor , it originally aired on March 14, 2004.
The 1981 NCAA Division I lacrosse tournament was the 11th annual tournament hosted by the National Collegiate Athletic Association to determine the team champion of men's college lacrosse among its Division I programs at the end of the 1981 NCAA Division I lacrosse season. North Carolina capped off a 12-0 season with its first-ever NCAA ...
Under terms of the proposed agreement, the NCAA will pay $2.77 billion over 10 years to former and current college athletes who were denied by now-defunct rules the ability to earn money from ...
Gary Shapiro is an American journalist. A graduate of Harvard University and Columbia Law School he has written for numerous publications including The New York Sun (where he penned the "Knickerbocker" column) [1] , The Forward and The Wall Street Journal .
David Shapiro (born November 25, 1946) is an American economist at the Pennsylvania State University. He joined the Penn State faculty [1] in 1980. He is a leading academic in the field of Economic Demography , specializing in fertility transition in sub-Saharan Africa and in the study of children's schooling in Africa.
Ultima Online ( UO) is a fantasy massively multiplayer online role-playing game (MMORPG) released on September 24, 1997 [1] by Origin Systems . Set in the Ultima universe, it is known for its extensive player versus player combat system. Since its release, it has added eight expansion packs, a booster pack, and dozens of free content updates.