Connection

Connection may refer to:

Technology

Mathematics

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 .

Purple drank

Purple drank is a slang term for a concoction which includes a prescription-strength cough syrup used in a manner inconsistent with its labeling as a recreational drug. The mixture became popular in the hip hop community in the southern United States, originating in Houston.

The prescription-strength cough syrup used in purple drank contains codeine and promethazine (not to be confused with dextromethorphan; DXM). The cough syrup, used in doses much higher than medically recommended, is typically mixed with ingredients such as the soft drinks Sprite or Mountain Dew and optionally "a Jolly Rancher hard fruit candy thrown in for extra sweetness." The purplish hue of purple drank comes from dyes in the cough syrup. The amount of cough syrup used "can exceed up to 25 times the recommended dose." The concoction is "Typically consumed out of Styrofoam cups".

There are numerous slang terms for purple drank, including sizzurp,lean,syrup,drank,barre,purple jelly,Texas tea,dirty Sprite, and Tsikuni.

Syrup (disambiguation)

Syrup is a thick, viscous liquid, containing a large amount of dissolved sugars, but showing little tendency to deposit crystals.

Syrup may also refer to:

See also

Syrup (novel)

Syrup is a novel written by Max Barry under the pseudonym Maxx Barry, which satirizes consumerism and marketing techniques. It is Barry's debut novel, published in 1999, and adapted into a movie in 2013.

Main Characters

Plot

Set in the present day, a young marketing graduate named Scat comes up with an idea for a new product for Coca-Cola called 'Fukk'. This causes him to go to Coca-Cola to sell his idea for $3 million, but he finds that Sneaky Pete has already claimed the trademark in a backstabbing move. This then leads him to leave his apartment with Sneaky Pete and move in with Cindy. Cindy eventually throws him out and he goes to live with 6 and Tina while managing the summer marketing campaign for Coca-Cola. He eventually succeeds with the campaign. After that Scat tries to undermine Sneaky Pete's effort to run a new secret project for Coca-Cola, the first feature length advertising movie.