Simple (abstract algebra)

In mathematics, the term simple is used to describe an algebraic structure which in some sense cannot be divided by a smaller structure of the same type. Put another way, an algebraic structure is simple if the kernel of every homomorphism is either the whole structure or a single element. Some examples are:

The general pattern is that the structure admits no non-trivial congruence relations.

The term is used differently in semigroup theory. A semigroup is said to be simple if it has no nontrivial ideals, or equivalently, if Green's relation J is the universal relation. Not every congruence on a semigroup is associated with an ideal, so a simple semigroup may have nontrivial congruences. A semigroup with no nontrivial congruences is called congruence simple.

Leaf

A leaf is an organ of a vascular plant and is the principal lateral appendage of the stem. The leaves and stem together form the shoot.Foliage is a mass noun that refers to leaves collectively.

Typically a leaf is a thin, dorsiventrally flattened organ, borne above ground and specialized for photosynthesis. Most leaves have distinctive upper (adaxial) and lower (abaxial) surfaces that differ in colour, hairiness, the number of stomata (pores that intake and output gases) and other features. In most plant species, leaves are broad and flat. Such species are referred to as broad-leaved plants. Many gymnosperm species have thin needle-like leaves that can be advantageous in cold climates frequented by snow and frost. Leaves can also have other shapes and forms such as the scales in certain species of conifers. Some leaves are not above ground (such as bulb scales). Succulent plants often have thick juicy leaves, but some leaves are without major photosynthetic function and may be dead at maturity, as in some cataphylls, and spines). Furthermore, several kinds of leaf-like structures found in vascular plants are not totally homologous with them. Examples include flattened plant stems (called phylloclades and cladodes), and phyllodes (flattened leaf stems), both of which differ from leaves in their structure and origin. Many structures of non-vascular plants, and even of some lichens, which are not plants at all (in the sense of being members of the kingdom Plantae), look and function much like leaves. The primary site of photosynthesis in most leaves (palisade mesophyll) almost always occurs on the upper side of the blade or lamina of the leaf but in some species, including the mature foliage of Eucalyptus palisade occurs on both sides and the leaves are said to be isobilateral.

Special classes of semigroups

In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ab = ba for all elements a and b in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large variety of special classes of semigroups have been defined though not all of them have been studied equally intensively.

In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semigroups and occasionally on the cardinality and similar properties of subsets of the underlying set. The underlying sets are not assumed to carry any other mathematical structures like order or topology.

Circus (Britney Spears album)

Circus is the sixth studio album by American recording artist Britney Spears. It was released on November 28, 2008, by JIVE Records. Looking to transition from her "darker and more urban" fifth studio album Blackout (2007), Spears wanted to make her next project "a little bit lighter", incorporating electropop and dance-pop styles. Spears recorded the record during the summer of 2008, after her much-publicized personal struggles saw her placement under a temporary conservatorship earlier that year. As executive producers, Larry Rudolph and Teresa LaBarbera Whites enlisted collaborators including Spears' longtime colleague Max Martin and Nate "Danja" Hills.

Upon its release, Circus received generally favorable reviews from music critics, who complimented its production but were ambivalent towards its lyrical content. It debuted at number one on the US Billboard 200 with first-week sales of over 505,000 copies, making it her fifth album to reach the top of the chart. The album topped charts in nine additional countries. The record eventually exceeded sales of 3.5 million copies and 15 million digital tracks, in doing so it became Spears' best-selling album since her fourth studio album In the Zone (2003). The project was promoted through a series of television performances and Spears' fifth concert tour The Circus Starring Britney Spears. The latter generated controversy during the Australian leg after accusations of lip-syncing surfaced.

Circus (1936 film)

Circus (Russian: Цирк; translit. Tsirk) is a 1936 Soviet melodramatic comedy musical film. It was directed by Grigori Aleksandrov and Isidor Simkov at the Mosfilm studios. In his own words, it was conceived as "an eccentric comedy...a real side splitter."

Starring the glamorous and immensely popular Lyubov Orlova (Aleksandrov's wife), the first recognized star of Soviet cinema and a gifted singer, the film contains several songs which instantly became Soviet classics. The most famous is the "Song of the Motherland" (Широка страна моя родная).

The film was based on a comedy play by Ilf and Petrov and Valentin Kataev, Under the Circus Dome (Под куполом цирка), which was seen and liked by Aleksandrov. They made the play into the plot, but during the initial film shooting they went to America. Upon return, they disliked the director's interpretation, and after a conflict they abandoned the work, forbade the mention of their names in the credits, and further work on the plot was continued by Isaac Babel.

Pilot (House)

"Pilot", also known as "Everybody Lies", is the first episode of the U.S. television series House. The episode premiered on the Fox network on November 16, 2004. It introduces the character of Dr. Gregory House (played by Hugh Laurie)—a maverick antisocial doctor—and his team of diagnosticians at the fictional Princeton-Plainsboro Teaching Hospital in New Jersey. The episode features House's attempts to diagnose a kindergarten teacher after she collapses in class.

House was created by David Shore, who got the idea for the curmudgeonly title character from a doctor's visit. Initially, producer Bryan Singer wanted an American to play House, but British actor Hugh Laurie's audition convinced him that a foreign actor could play the role. Shore wrote House as a character with parallels to Sherlock Holmes—both are drug users, aloof, and largely friendless. The show's producers wanted House handicapped in some way and gave the character a damaged leg arising from an improper diagnosis.

House (1995 film)

House is a Canadian drama film, released in 1995. Written and directed by Laurie Lynd as an adaptation of Daniel MacIvor's one-man play House, the film stars MacIvor as Victor, an antisocial drifter with some hints of paranoid schizophrenia, who arrives in the town of Hope Springs and invites ten strangers into the local church to watch him perform a monologue about his struggles and disappointments in life.

The original play was performed solely by MacIvor. For the film, Lynd added several other actors, giving the audience members some moments of direct interaction and intercutting Victor's monologue with scenes which directly depict the stories he describes. The extended cast includes Anne Anglin, Ben Cardinal, Patricia Collins, Jerry Franken, Caroline Gillis, Kathryn Greenwood, Nicky Guadagni, Joan Heney, Rachel Luttrell, Stephen Ouimette, Simon Richards, Christofer Williamson and Jonathan Wilson.

The film premiered at the 1995 Toronto International Film Festival in the Perspectives Canada series, before going into general release in 1996.