Base (group theory)

Let be a finite permutation group acting on a set . A sequence

of k distinct elements of is a base for G if the only element of which fixes every pointwise is the identity element of .

Bases and strong generating sets are concepts of importance in computational group theory. A base and a strong generating set (together often called a BSGS) for a group can be obtained using the Schreier–Sims algorithm.

It is often beneficial to deal with bases and strong generating sets as these may be easier to work with than the entire group. A group may have a small base compared to the set it acts on. In the "worst case", the symmetric groups and alternating groups have large bases (the symmetric group S_n has base size n − 1), and there are often specialized algorithms that deal with these cases.

References

Base

Base or BASE may refer to:

Mathematics

Science and technology

Computing

Base (politics)

In politics, the term base refers to a group of voters who almost always support a single party's candidates for elected office. Base voters are very unlikely to vote for the candidate of an opposing party, regardless of the specific views each candidate holds. In the United States, this is typically because high-level candidates must hold the same stances on key issues as a party's base in order to gain the party's nomination and thus be guaranteed ballot access. In the case of legislative elections, base voters often prefer to support their party's candidate against an otherwise appealing opponent in order to strengthen their party's chances of gaining a simple majority, typically the gateway to overarching power, in a legislature.

Base (geometry)

In geometry, a base is a side of a polygon or a face of a polyhedron, particularly one oriented perpendicular to the direction in which height is measured, or on what is considered to be the "bottom" of the figure. This term is commonly applied to triangles, parallelograms, trapezoids, cylinders, cones, pyramids, parallelepipeds and frustums.

Role in area and volume calculation

Bases are commonly used (together with heights) to calculate the areas and volumes of figures. In speaking about these processes, the measure (length or area) of a figure's base is often referred to as its "base."

By this usage, the area of a parallelogram or the volume of a prism or cylinder can be calculated by multiplying its "base" by its height; likewise, the areas of triangles and the volumes of cones and pyramids are fractions of the products of their bases and heights. Some figures have two parallel bases (such as trapezoids and frustums), both of which are used to calculate the extent of the figures.

Group (auto racing)

An FIA Group is a category of car allowed to compete in auto racing. The FIA Appendix J to the international motor sports code defines the various Groups.

While a given racing car may fit into an FIA Group, local rules still may either prohibit the car, or allow additional modifications that aren't part of the FIA Group specifications.

See also

157 Group

The 157 Group is a group of 30 colleges of further education in the United Kingdom. The group was established in 2006 with an aim of promoting and maintaining high standards of education and management for this type of college. The group whose creation was announced at the 2006 Association of Colleges (AoC) annual conference chaired by the then Secretary of State for Education, Alan Johnson, aims to raise the profile and reputation of Further Education through policy influence. The group takes its name from paragraph No.157 of a British government paper on education, The Foster Report (formally the Review of the future role of FE colleges), published a year previously.

Together, the group's member colleges employ more than 37,000 staff and the total enrolment is around 670,000 students. The group develops projects and enterprises for its members and the wider FE sector.

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Theory

Theory is a contemplative and rational type of abstract or generalizing thinking, or the results of such thinking. Depending on the context, the results might for example include generalized explanations of how nature works. The word has its roots in ancient Greek, but in modern use it has taken on several different related meanings. A theory is not the same as a hypothesis. A theory provides an explanatory framework for some observation, and from the assumptions of the explanation follows a number of possible hypotheses that can be tested in order to provide support for, or challenge, the theory.

A theory can be normative (or prescriptive), meaning a postulation about what ought to be. It provides "goals, norms, and standards". A theory can be a body of knowledge, which may or may not be associated with particular explanatory models. To theorize is to develop this body of knowledge.

As already in Aristotle's definitions, theory is very often contrasted to "practice" (from Greek praxis, πρᾶξις) a Greek term for "doing", which is opposed to theory because pure theory involves no doing apart from itself. A classical example of the distinction between "theoretical" and "practical" uses the discipline of medicine: medical theory involves trying to understand the causes and nature of health and sickness, while the practical side of medicine is trying to make people healthy. These two things are related but can be independent, because it is possible to research health and sickness without curing specific patients, and it is possible to cure a patient without knowing how the cure worked.