Group action

In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A group action formalizes the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.

In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices. It is the same as a group action of G on an ordered basis of a vector space.

A group action is an extension to the notion of a symmetry group in which every element of the group "acts" like a bijective transformation (or "symmetry") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.

Orbit (horse)

Orbit (foaled 1885) was a Thoroughbred racehorse. He was trained at Kingsclere by John Porter for the 1st Duke of Westminster. As a three-year-old he won the Eclipse Stakes.

Breeding

Orbit was the son of Epsom Derby and Champion Stakes winner Bend Or. His dam was Fair Alice, a daughter of July Stakes winner Cambuscan.

Racing career

Orbit won three races as a two-year-old; the Criterion Nursery Handicap at Newmarket, the Kempton Park Champion Nursery Handicap and the Daveridge Stakes. Orbit started his three-year-old career by winning the Craven Stakes at Newmarket by ¾ length from Cotillon. His next race came in the 2000 Guineas at Newmarket. Friar's Balsam started as the 1/3 favourite for the race, with Ayrshire at 100/12 and Orbit at 100/8. Orbit ran on well in the closing stages to finish in third place. Ayrshire won the race by two lengths from Johnny Morgan, who was a head in front of Orbit. After winning the 2000 Guineas win Ayrshire started as the 5/6 favourite for the Epsom Derby and Orbit was second favourite at 11/2. Orbit could only finish in fifth place, over seven lengths behind winner Ayrshire. He then finished second in the Triennial Stakes at Ascot. Orbit started as the 9/4 favourite for the Eclipse Stakes and in the final 100 yards of the race Orbit gradually edged away from stablemate Ossory and beat him by a length.

Orbit (band)

Orbit is a Boston, Massachusetts-based power trio. Formed in 1994, the band went on hiatus in late 2001. Their initial releases were on drummer Buckley's own Lunch Records label before the band moved to major label A&M Records. They completed recording their second major label album, "Guide To Better Living", but it was never released by A&M. The band then moved back to Lunch Records for the rest of their releases.

Perhaps the high point of the band's career was the hit, "Medicine", and their presence on the 1997 Lollapalooza tour. They also had the song, "XLR8R", included on the soundtrack of the PlayStation 2 game, FreQuency.

Orbit played two reunion shows on December 28 and 29, 2007, at the Paradise Rock Club in Boston, MA. They performed with also defunct Boston indie rock group The Sheila Divine.

Orbit performed a show on January 14, 2011, at the Paradise Rock Club in Boston, MA with The Sheila Divine.

Members

Factor

Factor, a Latin word meaning "who/which acts", may refer to:

Commerce

Science and technology

Biology

Computer science and information technology

Factoring (finance)

Factoring is a financial transaction and a type of debtor finance in which a business sells its accounts receivable (i.e., invoices) to a third party (called a factor) at a discount. A business will sometimes factor its receivable assets to meet its present and immediate cash needs.Forfaiting is a factoring arrangement used in international trade finance by exporters who wish to sell their receivables to a forfaiter. Factoring is commonly referred to as accounts receivable factoring, invoice factoring, and sometimes accounts receivable financing. Accounts receivable financing is a term more accurately used to describe a form of asset based lending against accounts receivable.

Factoring is not the same as invoice discounting (which is called an "Assignment of Accounts Receivable" in American accounting – as propagated by FASB within GAAP). Factoring is the sale of receivables, whereas invoice discounting ("assignment of accounts receivable" in American accounting) is a borrowing that involves the use of the accounts receivable assets as collateral for the loan. However, in some other markets, such as the UK, invoice discounting is considered to be a form of factoring, involving the "assignment of receivables", that is included in official factoring statistics. It is therefore also not considered to be borrowing in the UK. In the UK the arrangement is usually confidential in that the debtor is not notified of the assignment of the receivable and the seller of the receivable collects the debt on behalf of the factor. In the UK, the main difference between factoring and invoice discounting is confidentiality.

Von Neumann algebra

In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.

Two basic examples of von Neumann algebras are as follows. The ring L^∞(R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, which acts by pointwise multiplication on the Hilbert space L²(R) of square integrable functions. The algebra B(H) of all bounded operators on a Hilbert space H is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least 2.

Von Neumann algebras were first studied by von Neumann (1930) in 1929; he and Francis Murray developed the basic theory, under the original name of rings of operators, in a series of papers written in the 1930s and 1940s (F.J. Murray & J. von Neumann 1936, 1937, 1943; J. von Neumann 1938, 1940, 1943, 1949), reprinted in the collected works of von Neumann (1961).