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Lusternik–Schnirelmann category

In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space is the homotopical invariant defined to be the smallest integer number such that there is an open covering of with the property that each inclusion map is nullhomotopic. For example, if is the circle, this takes the value two.

Sometimes a different normalization of the invariant is adopted, which is one less than the definition above Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below).

In general it is not easy to compute this invariant, which was initially introduced by Lazar Lyusternik and Lev Schnirelmann in connection with variational problems. It has a close connection with algebraic topology, in particular cup-length. In the modern normalization, the cup-length is a lower bound for LS category.

It was, as originally defined for the case of X a manifold, the lower bound for the number of critical points that a real-valued function on X could possess (this should be compared with the result in Morse theory that shows that the sum of the Betti numbers is a lower bound for the number of critical points of a Morse function).

Higher category theory

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.

Strict higher categories

An ordinary category has objects and morphisms. A 2-category generalizes this by also including 2-morphisms between the 1-morphisms. Continuing this up to n-morphisms between (n-1)-morphisms gives an n-category.

Just as the category known as Cat, which is the category of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms, the category n-Cat of (small) n-categories is actually an n+1-category.

An n-category is defined by induction on n by:

So a 1-category is just a (locally small) category.

The monoidal structure of Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite products can be given a monoidal structure. The recursive construction of n-Cat works fine because if a category C has finite products, the category of C-enriched categories has finite products too.

Plácido Domingo discography

Plácido Domingo has made hundreds of opera performances, music albums, and concert recordings throughout his career as an operatic tenor. From his first operatic leading role as Alfredo in La traviata in 1961, his major debuts continued in swift succession: Tosca at the Hamburg State Opera and Don Carlos at the Vienna State Opera in 1967; Adriana Lecouvreur at the Metropolitan Opera, Turandot in Verona Arena and La bohème in San Francisco in 1969; La Gioconda in 1970; Tosca in Royal Opera House, Covent Garden in 1971; La bohème at the Bavarian State Opera in 1972; Il trovatore at the Paris Opéra in 1973 and Don Carlo at the Salzburg Festival in 1975,Parsifal in 1992 at the Bayreuth Festival; and the list continues until today; the same role is often recorded more than once.

Other than full-length opera performance recordings, Domingo has also made many music albums, recording opera arias, live opera performances and concerts, and crossover songs in solo and duet. His albums have simultaneously appeared on Billboard charts of best-selling classical and crossover recordings; contributing to many gold and platinum records and nine Grammy awards.

Songs (Deptford Goth album)

Songs is an album by Deptford Goth.

Track listing

References

Kate Micucci

Kate Micucci (pronounced /mᵻˈkuːtʃi/; born March 31, 1980) is an American actress, comedian, singer-songwriter, and artist. She is one half of the musical comedy duo Garfunkel and Oates. Her first major television exposure was her role as Stephanie Gooch in Scrubs. Later she portrayed Shelley in Raising Hope and Raj's girlfriend, Lucy in The Big Bang Theory. She also provides the voice of Velma Dinkley in Be Cool, Scooby-Doo!, and Sadie in Steven Universe.

Early life

Born in Jersey City, New Jersey, of Italian ancestry, Micucci was raised in Nazareth, Pennsylvania, in the Lehigh Valley region of the state, where she first learned to play classical piano, taught by her mother. She graduated alongside rock musician Jordan White in 1998 from Nazareth Area High School. Micucci then received an A.A. in Fine Arts from Keystone College in La Plume, Pennsylvania, and a B.A. in Studio Art in 2003 from Loyola Marymount University, Los Angeles.

Career

Micucci's TV credits include numerous television commercials, as well as Toni the barista on NBC's Four Kings, guest roles on Malcolm in the Middle, 'Til Death, How I Met Your Mother, Cory in the House, and Campus Ladies, and recurring roles on Scrubs and Raising Hope. Her film credits include The Last Hurrah, Bart Got a Room and When in Rome. She plays "Lily the IT girl" on Elevator produced by HBO's Runawaybox. In early 2009, she released a five-track EP entitled Songs.

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