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* Smith – Volterra – Cantor set, a set that is nowhere dense ( in particular it contains no intervals ), yet has positive measure
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Smith and Volterra
In mathematics, the Smith – Volterra – Cantor set ( SVC ), fat Cantor set, or ε-Cantor set is an example of a set of points on the real line R that is nowhere dense ( in particular it contains no intervals ), yet has positive measure.
The Smith – Volterra – Cantor set is named after the mathematicians Henry Smith, Vito Volterra and Georg Cantor.
Similar to the construction of the Cantor set, the Smith – Volterra – Cantor set is constructed by removing certain intervals from the unit interval.
Continuing indefinitely with this removal, the Smith – Volterra – Cantor set is then the set of points that are never removed.
By construction, the Smith – Volterra – Cantor set contains no intervals and therefore has empty interior.
The function is defined by making use of the Smith – Volterra – Cantor set and " copies " of the function defined by f ( x )
We then translate this function to the interval 5 / 8 so that the resulting function, which we call f < sub > 1 </ sub >, is nonzero only on the middle interval of the complement of the Smith – Volterra – Cantor set.
Volterra's function then results by repeating this procedure for every interval removed in the construction of the Smith – Volterra – Cantor set ; in other words, the function V is the limit of the sequence of functions f < sub > 1 </ sub >, f < sub > 2 </ sub >, ...
Thus there are points where V ′ takes values 1 and − 1 in every neighborhood of each of the endpoints of intervals removed in the construction of the Smith – Volterra – Cantor set S. In fact, V ′ is discontinuous at every point of S, even though V itself is differentiable at every point of S, with derivative 0.
Since the Smith – Volterra – Cantor set S has positive Lebesgue measure, this means that V ′ is discontinuous on a set of positive measure.
Smith and Cantor
It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
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Eddie Cantor, Fred Allen, Ethel Merman and Smith and Dale were among the top acts seen in Paramount shorts.
Other performers appearing at the Palace included Sarah Bernhardt, Al Jolson, Enrico Caruso, Helen Kane, Eddie Cantor, Frank Fay, Bob Hope, Sophie Tucker, George Jessel, Mae West, Vernon and Irene Castle, Gus Edwards, Eddie Leonard, Burns and Allen, Fred Astaire, Benny Fields, Kate Smith, Bill Robinson, Ethel Merman, Bing Crosby, Wheeler and Woolsey, Rudolph Valentino, and Jack Benny.
She has co-written with many named producers / songwriters including: Poet Name Life, Mick Mars, DJ Ammo, Tim Fagan, The Heavyweights, Xandy Berry, Trey Bruce, Mitch Allan, Brian Sperber, Brandon Barnes, Jimmy Robertson Landry, Randy Cantor, Jodi Marr, Jerome Brailey, Malcolm Springer, Andrew Lane, Eddie Galan, Jan Smith, Travis Wetherington, Melissa Mattey and Tony High.
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