Connection (principal bundle)

In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal G-connection on a principal G-bundle P over a smooth manifold M is a particular type of connection which is compatible with the action of the group G.

A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any fiber bundle associated to P via the associated bundle construction. In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the frame bundle of a smooth manifold.

Formal definition

Let π:P→M be a smooth principal G-bundle over a smooth manifold M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P.

Connection (affine bundle)

Let be an affine bundle modelled over a vector bundle . A connection on is called the affine connection if it as a section of the jet bundle of is an affine bundle morphism over . In particular, this is the case of an affine connection on the tangent bundle of a smooth manifold .

With respect to affine bundle coordinates on , an affine connection on is given by the tangent-valued connection form

An affine bundle is a fiber bundle with a general affine structure group of affine transformations of its typical fiber of dimension . Therefore, an affine connection is associated to a principal connection. It always exists.

For any affine connection , the corresponding linear derivative of an affine morphism defines a unique linear connection on a vector bundle . With respect to linear bundle coordinates on , this connection reads

Since every vector bundle is an affine bundle, any linear connection on a vector bundle also is an affine connection.

If is a vector bundle, both an affine connection and an associated linear connection are connections on the same vector bundle , and their difference is a basic soldering form on . Thus, every affine connection on a vector bundle is a sum of a linear connection and a basic soldering form on .

Connection (Elastica song)

"Connection" is a song released by the Britpop group Elastica. It was originally released in 1994 as a single and the album version was not released until 1995 on their self-titled debut.

The song was the subject of controversy, due to its overt similarity to another band's work. The intro synthesizer part (later repeated as a guitar figure) is lifted from the guitar riff in Wire's "Three Girl Rhumba" and transposed down a semitone. A judgment resulted in an out-of-court settlement and the credits were rewritten.

The song is the theme to the UK television programme Trigger Happy TV. A live version of the song was featured on the MuchMusic live compilation album, Much at Edgefest '99.

The song was covered by Elastica's label-mates Collapsed Lung and released as a seven-inch vinyl single in 1995.

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External links

Metro-Goldwyn-Mayer

Metro-Goldwyn-Mayer Studios Inc. (abbreviated as MGM or M-G-M, also known as Metro-Goldwyn-Mayer Pictures, Metro-Goldwyn-Mayer, or simply Metro) is an American media company, involved primarily in the production and distribution of feature films and television programs.

Once the largest, most glamorous, and most revered film studio, MGM was founded in 1924 when the entertainment entrepreneur Marcus Loew gained control of Metro Pictures, Goldwyn Pictures and Louis B. Mayer Pictures. Its headquarters are in Beverly Hills, California. It is one of the oldest mini-major film studios and a former major film studio.

In 1971, it was announced that MGM would merge with 20th Century Fox, a plan which never came into fruition. Over the next thirty-nine years, the studio was bought and sold at various points in its history until, on November 3, 2010, MGM filed for Chapter 11 bankruptcy. MGM emerged from bankruptcy on December 20, 2010, at which time the executives of Spyglass Entertainment, Gary Barber and Roger Birnbaum, became co-Chairmen and co-CEOs of the holding company of Metro-Goldwyn-Mayer.

MGM-18 Lacrosse

The MGM-18 Lacrosse was a short-ranged tactical ballistic weapon intended for close support of ground troops. Its first flight test was in 1954 and was deployed by the United States Army beginning in 1959, despite being still in the development stage. The program's many technical hurdles proved too difficult to overcome and the missile was withdrawn from field service by 1964.

History

Development

The Lacrosse project began with a United States Marine Corps requirement for a short-range guided missile to supplement conventional field artillery. The navy's Bureau of Ordnance issued contracts to both the Johns Hopkins University Applied Physics Laboratory and the Cornell Aeronautical Laboratory in September 1947, for the study of design aspects pertaining to this mission.

The missile system was named the Lacrosse because it employed a forward observation station which had a direct view of the target. The forward observation station was mounted on a jeep and after the missile was launched control was passed to the forward station for final guidance to the target. Hence the name Lacrosse which is how the game of lacrosse is played with the ball being passed to players closer to the goal.

MGM-134 Midgetman

The MGM-134A Midgetman, also known as the Small Intercontinental Ballistic Missile (SICBM), was an intercontinental ballistic missile developed by the United States Air Force.

Overview

The Midgetman grew out of a requirement expressed in the mid-1980s by the U.S. Air Force for a small ICBM which could be deployed on road vehicles. Fixed silos are inherently vulnerable to attack, and with the increasing accuracy of submarine-launched ballistic missiles there was a growing threat that the Soviet Union could launch large numbers of missiles from off the coast, destroying most of the U.S. ICBM force before it could be used (first strike). By producing a mobile missile which could not easily be targeted by enemy forces, and thus survive a first strike attempt, the Air Force hoped to reduce this possibility and maintain the ability to deter (second strike). It was also a response to the Soviet development of SS-24 (rail mobile) and the SS-25 (road mobile) ICBMs.

System definition studies for the SICBM (Small Intercontinental Ballistic Missile) commenced in 1984 under an Air Force Program Office, located at Norton AFB, CA, with TRW providing System Engineering and Technical Assistance (SETA) support. Contracts were awarded by the end of 1986 to Martin Marietta, Thiokol, Hercules, Aerojet, Boeing, General Electric, Rockwell and Logicon and authorization to proceed with full scale development of the MGM-134A Midgetman was granted. The first prototype missile was launched in 1989, but tumbled off course and was destroyed over the Pacific Ocean after about 70 seconds. The first successful test flight took place on April 18, 1991.