Transition function
Encyclopedia
In mathematics
, a transition function has several different meanings:
Category:Homeomorphisms
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a transition function has several different meanings:
- In topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a transition function is a homeomorphismHomeomorphismIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
from one coordinate chartAtlas (topology)In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...
to another. Given two charts (Ui, φi) and (Uj, φj) a transition functionFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
normally takes the form
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- for some set F being covered by the topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. See fiber bundleFiber bundleIn mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
, in particular also vector bundle, and atlas (topology)Atlas (topology)In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...
for additional details.
- In computingComputingComputing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...
, a transition function is the function that defines the state transitions of a Turing machineTuring machineA Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...
, finite-state machine, or cellular automatonCellular automatonA cellular automaton is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off"...
.
- In statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
and probability theoryProbability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, a transition function is a stochastic kernelStochastic kernelIn statistics, a stochastic kernel estimate is an estimate of the transition function of a stochastic process. Often, this is an estimate of the conditional density function obtained using kernel density estimation...
, the conditional probability distribution function controlling the transitions of a stochastic processStochastic processIn probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
.
Category:Homeomorphisms