In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
- φ : R → G
from the real line R (as an additive group) to some other topological group G. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the That means that it is not in fact a group, strictly speaking; if φ is injective then φ(R), the image, will be a subgroup of G that is isomorphic to R as additive group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element That is, we start knowing only that
- φ (s + t) = φ(s)φ(t)
where s, t are the 'parameters' of group elements in G. We may have
- φ(s) = e, the identity element in G,
for some s ≠ 0. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that This happens for example if G is the unit circle and
- φ(s) = eis. In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex
In that case the kernel of φ consists of the integer multiples of 2π. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism
The action of a one-parameter group on a set is known as a flow. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a flow formalizes in mathematical terms the general idea of "a variable that depends on time" that occurs very frequently in Engineering
A technical complication is that φ(R) as subspace of G may carry a topology that is coarser than that on R; this may happen in cases where φ is injective. Subspace may refer to;Mathematics Euclidean subspace, in linear algebra a set of vectors in n -dimensional Euclidean space that is closed under addition In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation Think for example of the case where G is a torus T, and φ is constructed by winding a straight line round T at an irrational slope. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar
Therefore a one-parameter group or one-parameter subgroup has to be distinguished from a group or subgroup itself, for the three reasons
- it has a definite parametrization,
- the group homomorphism may not be injective, and
- the induced topology may not be the standard one of the real line. Parameterization (or parametrization parameterisation in British English) is the process of defining or deciding the Parameters - usually of some model - that are
Such one-parameter groups are of basic importance in the theory of Lie groups, for which every element of the associated Lie algebra defines such a homomorphism, the exponential map. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In differential geometry the exponential map is a generalization of the ordinary Exponential function of mathematical analysis to all differentiable manifolds with an Affine In the case of matrix groups it is given by the matrix exponential. In Mathematics, the matrix exponential is a Matrix function on square matrices analogous to the ordinary Exponential function.
Another important case is seen in functional analysis, with G being the group of unitary operators on a Hilbert space. For functional analysis as used in psychology see the Functional analysis (psychology article In Functional analysis, a branch of Mathematics, a unitary operator is a Bounded linear operator U H → This article assumes some familiarity with Analytic geometry and the concept of a limit. See Stone's theorem on one-parameter unitary groups. In Mathematics, Stone's theorem on one-parameter Unitary groups is a basic theorem of Functional analysis which establishes a One-to-one
Physics
In physics, one-parameter groups describe dynamical systems. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Dynamical systems theory is an area of Applied mathematics used to describe the behavior of complex Dynamical systems usually by employing Differential [1] Furthermore, whenever a system of physical laws admits a one-parameter group of differentiable symmetries, then there is a conserved quantity, by Noether's theorem. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has
See also
References
- ^ Zeidler, E. In Mathematics, an integral curve for a Vector field defined on a Manifold is a curve in the manifold whose tangent vector (i In Mathematics, a C 0-semigroup, also known as a (strongly continuous one-parameter semigroup, is a continuous morphism from ( R ++ Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has Applied Functional Analysis: Main Principles and Their Applications. Springer-Verlag, 1995.
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