Affine bundle

In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.

Formal definition

Let be a vector bundle with a typical fiber a vector space . An affine bundle modelled on a vector bundle is a fiber bundle whose typical fiber is an affine space modelled on so that the following conditions hold:

(i) All the fiber of are affine spaces modelled over the corresponding fibers of a vector bundle .

(ii) There is an affine bundle atlas of whose local trivializations morphisms and transition functions are affine isomorphisms.

Dealing with affine bundles, one uses only affine bundle coordinates possessing affine transition functions

There are the bundle morphisms

where are linear bundle coordinates on a vector bundle , possessing linear transition functions .

Properties

An affine bundle has a global section, but in contrast with vector bundles, there is no canonical global section of an affine bundle. Let be an affine bundle modelled on a vector bundle . Every global section of an affine bundle yields the bundle morphisms

Bundle

Bundle or Bundling may refer to:

In marketing:

In economics:

In mathematics:

In medicine:

In computing:

Bundle (OS X)

In NeXTSTEP, OPENSTEP, GNUstep, and their lineal descendants OS X and iOS, a bundle is a file directory with a defined structure and file extension, allowing related files to be grouped together as a conceptually single item.

Examples of bundles that contain executable code include applications, frameworks, and plugins. This kind of bundle usually contains one file representing executable code, and files that represent resources such as nibs, templates, images, sounds, and other media. On some other systems, such as Microsoft Windows, these resources are usually included directly in the executable file itself at compile time. On older Macintoshes, a similar technique is used, where additional metadata can be added to a file's resource fork. Similar in concept are the application directories used in RISC OS and on the ROX Desktop, and the RUNZ bundles in Super OS.

Examples of bundles that do not contain executable code include document packages (iWork documents) and media libraries (iPhoto Library).

Bundled software

In the software industry, bundled software is software distributed with another product such as a piece of computer hardware or other electronic device, or a group of software packages which are sold together. Software which is pre-installed on a new computer is an example of bundled software.

See also