In mathematical logic and computer science, the Kleene star is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics it is more known as the free monoid construction; the application of the Kleene star to a set V is written as V*. It is used for regular expressions, the context in which it was introduced by Stephen Kleene to characterize certain automata, where it means "zero or more". If V is a set of strings V* is defined as the smallest superset of V that contains the empty string ε and is closed under the string concatenation operation. If V is a set of symbols or characters V* is the set of all strings over symbols in V, including the empty string ε; the set V* can be described as the set of finite-length strings that can be generated by concatenating arbitrary elements of V, allowing the use of the same element multiple times. If V is either the empty set ∅ or the singleton set V* =; the operators are used in rewrite rules for generative grammars.
Given a set V define V0 =, V1 = Vand define recursively the set Vi+1 = for each i > 0. If V is a formal language Vi, the i-th power of the set V, is a shorthand for the concatenation of set V with itself i times; that is, Vi can be understood to be the set of all strings that can be represented as the concatenation of i strings in V. The definition of Kleene star on V is V ∗ = ⋃ i ≥ 0 V i = V 0 ∪ V 1 ∪ V 2 ∪ V 3 ∪ V 4 ∪ ⋯. Notice that the Kleene star operator is an idempotent unary operator: * = V* for any set V of strings or characters. In some formal language studies, a variation on the Kleene star operation called the Kleene plus is used; the Kleene plus omits the V0 term in the above union. In other words, the Kleene plus on V is V + = ⋃ i ≥ 1 V i = V 1 ∪ V 2 ∪ V 3 ∪ ⋯. For every set L, the Kleene plus of L equals the concatenation of L with L*. Example of Kleene star applied to set of strings: * =. Example of Kleene plus applied to set of characters: + =. Kleene star applied to the same character set: * =.
Example of Kleene star applied to the empty set: ∅* =. Example of Kleene plus applied to the empty set: ∅+ = ∅ ∅* = = ∅,where concatenation is an associative and noncommutative product, sharing these properties with the Cartesian product of sets. Example of Kleene plus and Kleene star applied to the singleton set containing the empty string: If V = also Vi = for each i, hence V* = V+ =. Strings ε the identity element; the Kleene star is defined for any monoid, not just strings. More let be a monoid, S ⊆ M. S* is the smallest submonoid of M containing S. Furthermore, the Kleene star is generalized by including the *-operation in the algebraic structure itself by the notion of complete star semiring. Wildcard character Glob Hopcroft, John E.. Introduction to Automata Theory and Computation. Addison-Wesley
Edward Jakobowski was an English composer of musical theatre, best known for writing the hit comic opera Erminie. Jakobowski was born in Islington, the only son of Israel Jakobowski, a salesman dealing in stationery and cigars, his wife Fanny, who were both Viennese of Polish extraction, he had Helena. At age six, he moved to Vienna, where he lived for some 15 years and was given a musical education. In the late 1870s he lived in Paris for three years. In 1881, he returned to London. Jakobowski's most successful work by far, opened in 1885 in London, it was revived extensively and toured internationally, playing with extraordinary success on Broadway from 1886. None of his other works had more than two, although many of them toured profitably. For two Victorian burlesques, The Three Beggars and Little Carmen, Jakobowski used the pen name Edward Belville, his principal shows were Dick, The Palace of Pearl, Mynheer Jan, Paola, La Rosiére, The Queen of Brilliants, The Devil's Deputy, Milord Sir Smith and Winsome Winnie.
He was one of eight composers who contributed to Pat in 1892. Two short operettas in 1893 with libretti by B. C. Stephenson, The Improvisatore and A Venetian Singer, made little impact. Jakobowski was married twice, the second time in New York in 1895 to Clara Brown, which ended in a London divorce in 1901. In 1902, he was declared bankrupt with debts of £1,090, he died at the Infirmary, Friern Barnet, north London, in 1929. His estate was valued at 8 shillings. Wearing, J. P. "Jakobowski, Edward " in The London Stage, 1890–1899: A Calendar of Plays and Players, The Scarecrow Press ISBN 0-8108-0910-9 Photo of Lillian Russell in The Queen of Brilliants
Stollings is a census-designated place in central Logan County, West Virginia, United States. Stollings is located on the Guyandotte River at the junction of West Virginia Route 10 and the southern terminus of West Virginia Route 17, 1.5 miles east-southeast of Logan. In addition to the two state highways that serve the community, the entire length of West Virginia Route 17 Truck is located within Stollings; the CDP is situated southeast of the city of Logan and north of the CDP of McConnell. Stollings has a post office with ZIP code 25646; as of the 2010 census, its population was 316. List of census-designated places in West Virginia Media related to Stollings, West Virginia at Wikimedia Commons