Bundle (mathematics)

In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology. Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle π: E→ B with E and B sets. It is no longer true that the preimages π ^{− 1}(x) must all look alike, unlike fiber bundles where the fibers must all be isomorphic (in the case of vector bundles) and homeomorphic.

Definition

A bundle is a triple (E, p, B) where E, B are sets and p:E→B a map.

This definition of a bundle is quite unrestrictive. For instance, the empty function defines a bundle. Nonetheless it serves well to introduce the basic terminology, and every type of bundle has the basic ingredients of above with restrictions on E, p, B and usually there is additional structure.

For each b ∈ B, p⁻¹(b) is the fibre or fiber of the bundle over b.

Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),structure,space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.

Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

! (disambiguation)

! is a punctuation mark called an exclamation mark (33 in ASCII), exclamation point, ecphoneme, or bang.

! may also refer to:

Mathematics and computers

Music

Other

See also

Junior Certificate

The Junior Certificate (Irish: Teastas Sóisearach) is an educational qualification awarded in Ireland by the Department of Education and Skills to students who have successfully completed the junior cycle of secondary education, and achieved a minimum standard in their Junior Certification examinations. These exams, like those for the Leaving Certificate, are supervised by the State Examinations Commission. A "recognised pupil"<ref name"">Definitions, Rules and Programme for Secondary Education, Department of Education, Ireland, 2004</ref> who commences the Junior Cycle must reach at least 12 years of age on 1 January of the school year of admission and must have completed primary education; the examination is normally taken after three years' study in a secondary school. Typically a student takes 9 to 13 subjects – including English, Irish and Mathematics – as part of the Junior Cycle. The examination does not reach the standards for college or university entrance; instead a school leaver in Ireland will typically take the Leaving Certificate Examination two or three years after completion of the Junior Certificate to reach that standard.

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