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One-parameter group

Index One-parameter group

In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism from the real line \mathbb (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if \varphi is injective then \varphi(\mathbb), the image, will be a subgroup of G that is isomorphic to \mathbb as additive group. [1]

50 relations: Abelian group, Alexander Macfarlane, Alfred Robb, C0-semigroup, Circle group, Comparison of topologies, Conservation law, Continuous function, Derivative, Dynamical system, E. T. Whittaker, Exponential map (Lie theory), Flow (mathematics), Functional analysis, Group (mathematics), Group action, Group homomorphism, Hermann Minkowski, Hilbert space, Hyperbolic angle, Identity element, Infinitesimal transformation, Injective function, Integral curve, James Cockle, Kernel (algebra), Lie algebra, Lie group, Mathematics, Matrix exponential, Noether's theorem, Parametrization, Paul Cohn, Physics, Principle of relativity, Rapidity, Real line, Sophus Lie, Spacetime, Special relativity, Stack Exchange, Stone's theorem on one-parameter unitary groups, Subspace topology, Symmetry group, Topological group, Torus, Unit hyperbola, Unitary operator, William Kingdon Clifford, World line.

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

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Alexander Macfarlane

Prof Alexander Macfarlane FRSE LLD (21 April 1851 – 28 August 1913) was a Scottish logician, physicist, and mathematician.

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Alfred Robb

Alfred Arthur Robb FRS (18 January 1873 in Belfast – 14 December 1936 in Castlereagh) was a Northern Irish physicist.

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C0-semigroup

In mathematics, a C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function.

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Circle group

In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers.

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Comparison of topologies

In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set.

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Conservation law

In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time.

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Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

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Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.

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E. T. Whittaker

Edmund Taylor Whittaker FRS FRSE (24 October 1873 – 24 March 1956) was an English mathematician who contributed widely to applied mathematics, mathematical physics, and the theory of special functions.

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Exponential map (Lie theory)

In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra.

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Flow (mathematics)

In mathematics, a flow formalizes the idea of the motion of particles in a fluid.

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Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

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Group homomorphism

In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".

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Hermann Minkowski

Hermann Minkowski (22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Hyperbolic angle

In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle. The hyperbolic angle is first defined for a "standard position", and subsequently as a measure of an interval on a branch of a hyperbola. A hyperbolic angle in standard position is the angle at (0, 0) between the ray to (1, 1) and the ray to (x, 1/x) where x > 1. The magnitude of the hyperbolic angle is the area of the corresponding hyperbolic sector which is ln x. Note that unlike circular angle, hyperbolic angle is unbounded, as is the function ln x, a fact related to the unbounded nature of the harmonic series. The hyperbolic angle in standard position is considered to be negative when 0 a > 1 so that (a, b) and (c, d) determine an interval on the hyperbola xy.

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Identity element

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 set, which leaves other elements unchanged when combined with them.

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Infinitesimal transformation

In mathematics, an infinitesimal transformation is a limiting form of small transformation.

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Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

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Integral curve

In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.

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James Cockle

Sir James Cockle FRS FRAS FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician.

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Kernel (algebra)

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 fails to be injective.

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Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

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Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Matrix exponential

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function.

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Noether's theorem

Noether's (first) theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.

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Parametrization

Parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation.

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Paul Cohn

Paul Moritz Cohn FRS (8 January 1924 – 20 April 2006) was Astor Professor of Mathematics at University College London, 1986-9, and author of many textbooks on algebra.

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Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

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Principle of relativity

In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference.

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Rapidity

In relativity, rapidity is commonly used as a measure for relativistic velocity.

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Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers.

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Sophus Lie

Marius Sophus Lie (17 December 1842 – 18 February 1899) was a Norwegian mathematician.

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Spacetime

In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.

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Special relativity

In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time.

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Stack Exchange

Stack Exchange is a network of question-and-answer (Q&A) websites on topics in varied fields, each site covering a specific topic, where questions, answers, and users are subject to a reputation award process.

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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 that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space \mathcal and one-parameter families of unitary operators that are strongly continuous, i.e., and are homomorphisms, i.e., Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.

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Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).

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Symmetry group

In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.

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Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.

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Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

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Unit hyperbola

In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation x^2 - y^2.

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Unitary operator

In functional analysis, a branch of mathematics, a unitary operator is a surjective bounded operator on a Hilbert space preserving the inner product.

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William Kingdon Clifford

William Kingdon Clifford FRS (4 May 1845 – 3 March 1879) was an English mathematician and philosopher.

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World line

The world line (or worldline) of an object is the path that object traces in -dimensional spacetime.

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Redirects here:

1-parameter group, 1-parameter subgroup, One-parameter subgroup.

References

[1] https://en.wikipedia.org/wiki/One-parameter_group

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