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Operation measuring the failure of two entities to commute
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions
Commutator
Device for changing direction of current
A commutator is a rotary electrical switch in certain types of electric motors and electrical generators that periodically reverses the current direction
Commutator_(electric)
Smallest normal subgroup by which the quotient is commutative
algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is
Commutator_subgroup
Machine that converts electrical energy into mechanical energy
material like carbon press against the commutator. The brushes make sliding contact with successive commutator segments as the rotator turns, supplying
Electric_motor
Electrical generator that produces direct current with the use of a commutator
dynamo is an electrical generator that creates direct current using a commutator, and uses self-powering electromagnets for the stator field rather than
Dynamo
mathematics, the commutator collecting process is a method for writing an element of a group as a product of generators and their higher commutators arranged
Commutator_collecting_process
In mathematical physics, the ternary commutator is an additional ternary operation on a triple system defined by [ a , b , c ] = a b c − a c b − b a c
Ternary_commutator
Synchronous electric motor powered by an electronic controller
speed and torque of the motor. It is an improvement on the mechanical commutator (brushes) used in many conventional electric motors. The construction
Brushless_DC_electric_motor
Type of electric motor
field coils are connected in series with the rotor windings through a commutator. It is often referred to as an AC series motor. The universal motor is
Universal_motor
the commutator subspace of a two-sided ideal of bounded linear operators on a separable Hilbert space is the linear subspace spanned by commutators of
Commutator_subspace
A mercury swivel commutator is a connector mainly used in animal experimentation to prevent the twisting of electrode wires attached to moving animals
Mercury_swivel_commutator
Control panel using electrical patch cords
A plugboard or control panel (the term used depends on the application area) is an array of jacks or sockets (often called hubs) into which patch cords
Plugboard
Type of derivative in differential geometry
interior product defined above and it is clear whether [·,·] denotes the commutator or the Lie bracket of vector fields. Various generalizations of the Lie
Lie_derivative
Operator in differential topology
bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields
Lie_bracket_of_vector_fields
Electricity-producing machine
of alternating current. Unlike a dynamo, a magneto does not contain a commutator to produce direct current. It is categorized as a form of alternator,
Magneto
1925 physics article by Werner Heisenberg
In the history of physics, "On the quantum-theoretical reinterpretation of kinematical and mechanical relationships" (German: Über quantentheoretische
Umdeutung_paper
Quantum mechanical operator related to rotational symmetry
L_{x},\;\;\left[L_{z},L_{x}\right]=i\hbar L_{y},} where [ , ] denotes the commutator [ X , Y ] ≡ X Y − Y X . {\displaystyle [X,Y]\equiv XY-YX.} This can be
Angular_momentum_operator
Relation satisfied by conjugate variables in quantum mechanics
point particle in one dimension, where [x , px] = x px − px x is the commutator of x and px , i is the imaginary unit, and ℏ is the reduced Planck constant
Canonical commutation relation
Canonical_commutation_relation
Formulation of quantum mechanics
Schrödinger picture respectively, H is the Hamiltonian and [·,·] denotes the commutator of two operators (in this case H and A). Taking expectation values automatically
Heisenberg_picture
Brackets as used in mathematical notation
used to denote the commutator. In group theory, the commutator [g,h] is commonly defined as g−1h−1gh. In ring theory, the commutator [a,b] is defined as
Bracket_(mathematics)
Property of some binary operations
Poisson brackets. In quantum mechanics, it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of
Jacobi_identity
Device that converts other energy to electrical energy
AC was converted into DC with a commutator, a set of rotating switch contacts on the armature shaft. The commutator reversed the connection of the armature
Electric_generator
Mathematical group with trivial abelianization
specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients
Perfect_group
Mathematical operation on vectors in 3D space
corresponds exactly to the commutator product in geometric algebra and both use the same symbol × {\displaystyle \times } . The commutator product is defined
Cross_product
Algebraic structure used in analysis
gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, [ x , y ] = x y − y x {\displaystyle [x,y]=xy-yx} . Lie algebras
Lie_algebra
Historical society dedicated to the ship
history. The Society publishes a quarterly online magazine, The Titanic Commutator, and operates a museum in Indian Orchard, Massachusetts, featuring artifacts
Titanic_Historical_Society
Power-producing component of an electric machine
current. The armature windings conduct AC even on DC machines, due to the commutator action (which periodically reverses current direction) or due to electronic
Armature_(electrical)
Electrical machine
stationary field windings. This alternating current is rectified by means of a commutator, which allows direct current to be extracted from the rotor. This principle
Rotary_converter
English computer scientist (1912–1954)
pointing out that we would not really use them in that way. Their test (of commutators) can hardly be considered conclusive as they were not testing for the
Alan_Turing
Punctuation mark
brackets denote the commutator. In group theory, the commutator [g, h] is commonly defined as g −1 h −1 g h . In ring theory, the commutator [a, b] is defined
Bracket
Internally commutated electric motor
utilizing an electric brush for contact. Existing DC generators use a commutator to achieve DC output instead of AC output, without commutation of voltage
Brushed_DC_electric_motor
Nonabelian cryptographic protocol
Anshel–Anshel–Goldfeld protocol, also known as a commutator key exchange, is a key-exchange protocol using nonabelian groups. It was invented by Drs. Michael
Anshel–Anshel–Goldfeld key exchange
Anshel–Anshel–Goldfeld_key_exchange
Algebraic generalization of the derivative
algebra. If the algebra A {\displaystyle A} is noncommutative, then the commutator with respect to an element of the algebra A {\displaystyle A} defines
Derivation (differential algebra)
Derivation_(differential_algebra)
Motor which works on direct current
current paths. The ends of the wire winding are connected to a commutator. The commutator allows each armature coil to be energized in turn and connects
DC_motor
Electrical contact that conducts current
copper or brass commutator or 'slip ring' to the shaft, with springs pressing braided copper wire 'brushes' onto the slip rings or commutator which conduct
Brush_(electric)
Sum of elements on the main diagonal
similar to the commutator of any pair of matrices. Conversely, any square matrix with zero trace is a linear combination of the commutators of pairs of matrices
Trace_(linear_algebra)
Theorem in quantum mechanics
expectation of any quantum mechanical operator and the expectation of the commutator of that operator with the Hamiltonian of the system d d t ⟨ A ⟩ = 1 i
Ehrenfest_theorem
Devices in automobiles to charge the battery and power the electrical system
running. Until the 1960s, automobiles used DC dynamo generators with commutators. As silicon-diode rectifiers became widely available and affordable,
Alternator_(automotive)
Operators useful in quantum mechanics
the commutator of the creation and annihilation operators that are associated with the same boson state equals one, while all other commutators vanish
Creation and annihilation operators
Creation_and_annihilation_operators
Type of AC electric motor
brushes. Most commutator motors are limited to about 1,500 volts because higher voltages give rise to a risk of arcing across the commutator. Repulsion motors
Repulsion_motor
The Portable Commutator was a telecommunications device that was used by the military during World War I. It was manufactured by Northern Electric Company
Portable_Commutator
Operator shifting particles and fields by a certain amount in a certain direction
{x} +\mathbf {r} )|\mathbf {x} +\mathbf {r} \rangle } Therefore, the commutator between a translation operator and the position operator is: [ r ^ , T
Translation operator (quantum mechanics)
Translation_operator_(quantum_mechanics)
Stationary windings in a motor/generator
power switch known as the commutator is needed to keep the field correctly aligned across the spinning rotor. The commutator must become larger and more
Stator_(electric_machines)
{\displaystyle \textstyle \prod _{i=1}^{n}h_{i}} in H/H′, where H′ is the commutator subgroup of H. The order of the factors is irrelevant since H/H′ is abelian
Transfer_(group_theory)
Device for converting electrical power to another form
motor coils are driven from a commutator on one end of the shaft, while the generator coils provide output to another commutator on the other end of the shaft
Motor–generator
Externally controlled electric motor that runs by reluctance torque
to be delivered to the moving rotor, which eliminates the need for a commutator. However it complicates the electrical design, because a switching system
Switched_reluctance_motor
Serbian-American engineer and inventor (1856–1943)
patented in May 1888, was a simple self-starting design that did not need a commutator, thus avoiding sparking and the high maintenance of constantly servicing
Nikola_Tesla
Direct current electric motor
force being continuous in one direction, the homopolar motor needs no commutator but still requires slip rings. The word homopolar indicates that the electrical
Homopolar_motor
Construction providing a total order on a free monoid
also be used to give a total order to the elements of a group, via the commutator collecting process, which is a special case of the general construction
Hall_word
Formula in Lie theory
convergent) in X {\displaystyle X} and Y {\displaystyle Y} and iterated commutators thereof. The first few terms of this series are: Z = X + Y + 1 2 [ X
Baker–Campbell–Hausdorff formula
Baker–Campbell–Hausdorff_formula
Basis used to express spherical tensors
For higher ranks, one may use either the commutator, or rotation definition of a spherical tensor. The commutator definition is given below, any operator
Spherical_basis
Concept in mathematics
bracket operation in g {\displaystyle {\mathfrak {g}}} corresponds to the commutator x y − y x {\displaystyle xy-yx} in A {\displaystyle {\mathcal {A}}} and
Universal_enveloping_algebra
Mathematical group whose commutator subgroup is abelian
In mathematics, a metabelian group is a group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there is an abelian
Metabelian_group
Exponential representation for differential equations
an infinite series, whose terms involve multiple integrals and nested commutators. Given the n × n coefficient matrix A(t), one wishes to solve the initial-value
Magnus_expansion
a ring A {\displaystyle A} is the universal central extension of the commutator subgroup of the stable general linear group of A {\displaystyle A} . It
Steinberg_group_(K-theory)
connected to commutators that were 90 degrees offset from the commutators connected to the other armature. When one set of commutators is aligned with
Earth_inductor_compass
form a Lie algebra with the usual commutator bracket [ x , y ] = x y − y x {\displaystyle [x,y]=xy-yx} (graded commutator if C is graded). If A is a connected
Primitive element (co-algebra)
Primitive_element_(co-algebra)
Electrical switch operated by an applied magnetic field
transistors to act as a commutator, but without the contact problems, wear and electrical noise of a traditional DC commutator. The motor design could
Reed_switch
Theoretical physics phenomenon
_{x}K_{x}+\zeta _{y}K_{y}+\zeta _{x}\zeta _{y}K_{y}K_{x}+\cdots } and the group commutator is e − ζ y K y e − ζ x K x e ζ y K y e ζ x K x = I + ζ x ζ y [ K y , K
Wigner_rotation
Hypercomplex number system
group. They do, however, form a loop, specifically a Moufang loop. The commutator of two octonions x and y is given by [ x , y ] = x y − y x {\displaystyle
Octonion
Tensor field in Riemannian geometry
fields and [ ∇ X , ∇ Y ] {\displaystyle [\nabla _{X},\nabla _{Y}]} is a commutator of differential operators. It turns out that the right-hand side actually
Riemann_curvature_tensor
Physics property associated with symmetries
Q} , and so the invariance of the charge corresponds to the vanishing commutator [ Q , H ] = 0 {\displaystyle [Q,H]=0} , where H {\displaystyle H} is
Charge_(physics)
Frequency used on standard electricity grid in a given area
applications of commercial electric power were incandescent lighting and commutator-type electric motors. Both of them operate well on DC, but DC could not
Utility_frequency
Complex vector of electromagnetic fields
\cdot \mathbf {p} )^{2}} We can now readily calculate the commutator while calculating commutators of α i {\displaystyle \alpha _{i}} matrixes and scaled
Riemann–Silberstein_vector
Formulations of quantum mechanics
A(t)}{\partial t}},} where H is the Hamiltonian and [•,•] denotes the commutator of two operators (in this case H and A). Taking expectation values yields
Dynamical_pictures
Technique in electrical devices
voltage drop. Historically, vibrator-driven switches or motor-driven commutators have also been used for mechanical rectifiers and synchronous rectification
Active_rectification
Device converting mechanical into electrical energy
it was normally converted into direct current via the addition of a commutator in the generator. The early machines were developed by pioneers such as
Alternator
Electrical generator that produces direct current
The winding requires twice the number of turns and twice the number of commutator bars as an equivalent drum-wound armature. The Gramme machine used a ring
Gramme_machine
Electromechanical device
the term commutator; however, commutators are somewhat different and are specialized for use on DC motors and generators. While commutators are segmented
Slip_ring
Type of electronic circuit
motor coils are driven from a commutator on one end of the shaft, when the generator coils output to another commutator on the other end of the shaft
DC-to-DC_converter
Complex-valued function
{\displaystyle A} and B {\displaystyle B} are bounded operators with trace-class commutator A B − B A {\displaystyle AB-BA} , then det e A e B e − A e − B = exp
Fredholm_determinant
Electric motor for discrete partial rotations
motors with a reduced pole count. They generally employ closed-loop commutators. Brushed DC motors rotate continuously when DC voltage is applied to
Stepper_motor
Topics referred to by the same term
matrices whose products do not depend on the order of multiplication Commutator, a measure of the failure of two elements to be commutative in a group
Commute
Tensor product of algebras over a field; itself another algebra
\forall a\in A,b\in B:[f(a),g(b)]=0\rbrace ,} where [-, -] denotes the commutator. The natural isomorphism is given by identifying a morphism ϕ : A ⊗ B
Tensor_product_of_algebras
Miniature remote-controlled cars
A tiny 5-segment commutator less than 2 mm in diameter, on a direct-current motor in a toy radio control ZipZaps car.
ZipZaps
Group of unitary complex matrices with determinant of 1
of traceless anti‑Hermitian n × n complex matrices, with the regular commutator as a Lie bracket. Particle physicists often use a different, equivalent
Special_unitary_group
Concept in mathematics
bracket [ X , Y ] := X Y − Y X {\displaystyle [X,Y]:=XY-YX} given by the commutator. This algebra is well studied and understood, and is often used as a model
Special_linear_Lie_algebra
Class of American electric locomotives
equipment. Twelve 385-horsepower (287 kW) GEA-627-A1 traction motors (AC commutator motors, not AC induction motors) drove the GG1's 57-inch (1,448 mm) diameter
Pennsylvania Railroad class GG1
Pennsylvania_Railroad_class_GG1
Device that shreds food waste for disposal via plumbing
due to the higher speeds and partially because the commutator brushes rub on the slotted commutator. Inside the grinding chamber there is a rotating metal
Garbage_disposal_unit
Mathematical concept in algebra
commute if A B = B A {\displaystyle AB=BA} , or equivalently if their commutator [ A , B ] = A B − B A {\displaystyle [A,B]=AB-BA} is zero. Matrices A
Commuting_matrices
Theorem in quantum information theory
The main idea in the original argument of Solovay and Kitaev is that commutators of elements close to the identity can be approximated "better-than-expected"
Solovay–Kitaev_theorem
Noncommutative geometric structure
trace-class operators, Kalton developed a spectral condition for the commutator subspace of trace class operators following on from results of Gary Weiss
Singular_trace
Model railway control system
Conversion kit for drum commutator motor (DCM) AC locomotives, including motor upgrade components. 60922 Decoder for drum commutator motor (DCM) AC locomotives
Märklin_Digital
structure inside the braided category ("braided algebra") one has a braided commutator (e.g. for a superspace the anticommutator): [ x , y ] τ := μ ( ( x ⊗ y
Braided_vector_space
Friedrich Engel, if it satisfies the n-Engel condition that the repeated commutator [...[[x,y],y], ..., y] with n copies of y is trivial (where [x, y] means
Engel_group
specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity
Three_subgroups_lemma
Construct in theoretical physics
[Y3, Y1] = Y2. The contraction limit ε → 0 trivializes the first commutator and thus yields the non-isomorphic algebra of the plane Euclidean group
Group_contraction
certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index. Originally, such mappings
Artin_transfer_(group_theory)
Description of physical properties at the atomic and subatomic scale
self-adjoint operators A {\displaystyle A} and B {\displaystyle B} . The commutator of these two operators is [ A , B ] = A B − B A , {\displaystyle [A,B]=AB-BA
Quantum_mechanics
Type of Kac–Moody algebras
functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra g ^ {\displaystyle {\hat {\mathfrak {g}}}} is
Affine_Lie_algebra
British railway signalling scheme
signal and moves the commutator to "Train On Line". His lower indicator on the block indicator to A repeats the position of the commutator. B immediately offers
Absolute_block_signalling
Infinite dimensional Lie algebra occurring in quantum field theory
local symmetry – could still be encoded in an algebra of currents. The commutators involved in current algebra amount to an infinite-dimensional extension
Current_algebra
Form of a matrix
space is given by the commutator: [ A , B ] = A B − B A . {\displaystyle [A,B]=AB-BA.\,} It is easy to check that the commutator of two skew-symmetric
Skew-symmetric_matrix
Commutative ring with no zero divisors other than zero
Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric
Integral_domain
Algebra over a field where binary multiplication is not necessarily associative
identity. Every associative algebra gives rise to a Lie algebra by using the commutator as Lie bracket. In fact every Lie algebra can either be constructed this
Non-associative_algebra
Room in Teylers Museum, Haarlem, Netherlands
by Martin van Marum as a student Gerhard Kuyper Cabinet IV: Telegraphy commutator for Telegraph 1865 contact breaker 1865 philips lightbulbs Wireless telegraphy
Teylers_Instrument_Room
Electric motor with only two poles to its stationary field
stationary field. They are an example of the simple brushed DC motor, with a commutator. This field may be generated by either a permanent magnet or a field coil
Bipolar_electric_motor
Mathematical representation
_{2}^{2}\sigma _{1}\sigma _{4}^{5}.} Then an element of the kernel is given by the commutator [ ψ 1 − 1 σ 4 ψ 1 , ψ 2 − 1 σ 4 σ 3 σ 2 σ 1 2 σ 2 σ 3 σ 4 ψ 2 ] . {\displaystyle
Burau_representation
Group of symmetries of an n-dimensional hypercube
{\displaystyle n\geq 2} , the commutator subgroup of S n ± {\displaystyle S_{n}^{\pm }} has index 4; it is equal to the commutator subgroup of the even-signed
Hyperoctahedral_group
Module over a sheaf of differential operators
separately commute with each other, and xi and ∂j commute for i ≠ j, but the commutator satisfies the relation [∂i, xi] = ∂ixi − xi∂i = 1. For any polynomial
D-module
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