- published: 25 Oct 2009
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In mathematics, the phrase "almost all" has a number of specialised uses.
"Almost all" is sometimes used synonymously with "all but [except] finitely many" (formally, a cofinite set) or "all but a countable set" (formally, a cocountable set); see almost.
A simple example is that almost all prime numbers are odd. (Two is a prime number.)
When speaking about the reals, sometimes it means "all reals but a set of Lebesgue measure zero" (formally, almost everywhere). In this sense almost all reals are not a member of the Cantor set even though the Cantor set is uncountable.
In number theory, if P(n) is a property of positive integers, and if p(N) denotes the number of positive integers n less than N for which P(n) holds, and if
(see limit), then we say that "P(n) holds for almost all positive integers n" (formally, asymptotically almost surely) and write
For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N/ln N. Therefore the proportion of prime integers is roughly 1/ln N, which tends to 0. Thus, almost all positive integers are composite (not prime), however there are still an infinite number of primes.
Whitney Elizabeth Houston (August 9, 1963 – February 11, 2012) was an American recording artist, actress, producer, and model. In 2009, the Guinness World Records cited her as the most-awarded female act of all-time. Houston was one of the world's best-selling music artists, having sold over 170 million albums, singles and videos worldwide. She released seven studio albums and three movie soundtrack albums, all of which have diamond, multi-platinum, platinum or gold certification. Houston's crossover appeal on the popular music charts, as well as her prominence on MTV, starting with her video for "How Will I Know", influenced several African American female artists to follow in her footsteps.
Houston is the only artist to chart seven consecutive No. 1 Billboard Hot 100 hits. She is the second artist behind Elton John and the only female artist to have two number-one Billboard 200 Album awards (formerly "Top Pop Album") on the Billboard magazine year-end charts. Houston's 1985 debut album Whitney Houston became the best-selling debut album by a female act at the time of its release. The album was named Rolling Stone's best album of 1986, and was ranked at number 254 on Rolling Stone's list of the 500 Greatest Albums of All Time. Her second studio album Whitney (1987) became the first album by a female artist to debut at number one on the Billboard 200 albums chart.
Walter Elias "Walt" Disney (December 5, 1901 – December 15, 1966) was an American film producer, director, screenwriter, voice actor, animator, entrepreneur, entertainer, international icon, and philanthropist, well known for his influence in the field of entertainment during the 20th century. Along with his brother Roy O. Disney, he was co-founder of Walt Disney Productions, which later became one of the best-known motion picture producers in the world. The corporation is now known as The Walt Disney Company and had an annual revenue of approximately US$36 billion in the 2010 financial year.
Disney is particularly noted as a film producer and a popular showman, as well as an innovator in animation and theme park design. He and his staff created some of the world's most well-known fictional characters including Mickey Mouse, for whom Disney himself provided the original voice. During his lifetime he received four honorary Academy Awards and won 22 Academy Awards from a total of 59 nominations, including a record four in one year, giving him more awards and nominations than any other individual in history. Disney also won seven Emmy Awards and gave his name to the Disneyland and Walt Disney World Resort theme parks in the U.S., as well as the international resorts Tokyo Disney Resort, Disneyland Paris, and Hong Kong Disneyland.