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Commutator (including recent related patents.)
CommutatorFor a electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g-1 h-1 gh, often denoted by [g,h]. It is equal to 1 if and only if g and h commute, i.e., if and only if gh = hg. The subgroup generated by all commutators is called the derived group or the commutator subgroup of G: we consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation. Commutators are used to define nilpotent groups. Commutators are also defined for rings and associative algebras. Here, the commutator [a,b] of two elements a and b is also called the Lie bracket and is defined by [a,b] = ab - ba. It is zero if and only if a and b commute. By using the Lie bracket, every associative algebra can be turned into a Lie algebra. The commutator of two operators defined on a Hilbert space is an important concept in quantum mechanics since it measures how well the two observables described by the operators can be measured simultaneously. The uncertainty principle is ultimately a theorem about these commutators. Likewise, the anticommutator is defined as ab + ba, often written {a,b}. See also Poisson algebra.This article is adapted from from Wikipedia All Wikipedia article text is available under the terms of the GNU Free Documentation License Co-Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians (Progress in Mathematics (Birkhauser Boston), Vol 135 by Werner O. Amrein Nonlinear Commutators in Interpolation Theory (Memoirs of the American Mathematical Society, 385) by Nigel G. Kalton On the Connection between Weighted Norm Inequalities, Commutators and by Jesus Bastero Commutator Calculus and Groups of Homotopy Classes by Hans Joachim Baues Commutator Theory for Congruence Modular Varieties by Ralph Freese Cb0S-Groups, Commutator Methods, and Spectral Theory of N-Body Hamiltonians (Progress in Mathematics (Boston, Mass.), Vol. 135.) by Werner O. Amrein Recent Commutator related patents From USPTO: 6717375: Discharge lamp lighting device and system comprising it 6717322: Motor housing having simplified cover plate and brush base 6717321: Electric motor comprising improved means of fixing an end flange to a stator shell 6717306: Coreless motor 6717301: Filtering and interference suppressing device for an electric motor 6717297: Electrical machine 6715986: Fuel pump 6715471: Electric fuel pump RE38486: Electric motor control circuit 6714069: Self-configurable amplifier circuit 6714007: Optically powered resonant integrated microstructure magnetic field gradient sensor 6713933: Electromagnetic motor 6713932: External connecting structure of electric motor 6713931: Cylindrical commutator securely fixed to mold resin 6713929: Series-wound motor 6713916: Electric motor assembly 6713915: Motor 6713911: Disc-shaped eccentric rotor and flat type vibrator motor having the rotor 6713902: Closed-path linear motor 6711965: Electric steering column apparatus 6711421: Structural reinforced superconducting ceramic tape and method of making 6710581: Constant-power brushless DC motor 6710574: Reversible DC motor drive including a DC/DC converter and four quadrant DC/DC controller 6710500: Commutation device, especially a commutator, and method for producing such a device 6710493: Dynamo-electric machine having tapered magnets secured to yoke 6710484: Motor having electronic control unit and method for manufacturing the same 6710482: Generator 6710480: Rotation angle measuring device with magnetized commutator 6710477: Motor 6708388: Method of making a permanent magnet field-type compact DC motor 6707268: Alternative drive power arrangement having an electric motor with multiple armatures 6707220: Electric motor brush holder for suppressing audible and electromagnetic noise 6707219: Small-sized motor and manufacturing method thereof 6707218: Rotor for small motors and manufacturing method therefor 6707213: Hybrid magnet type DC motor 6707208: Flux diode motor 6707198: Armature shaft retainer 6707193: Miniature motor with vibrator secured to an irregular portion of the motor shaft 6707188: Motor having rotational sensor 6707183: Connector with flux concentrator for electric motor and corresponding geared motor 6707177: Lubrication recirculation and wear protective member for electric motor 6705586: Control valve 6704349: Method and apparatus for canceling a transmit signal spectrum in a receiver bandwidth 6704307: Compact high-capacity switch 6703805: System for measuring the position of an electric motor 6703755: Brush-equipped motor and method of exchanging brushes thereof 6703754: Electric motor and brush retaining assembly 6703751: Dynamo-electric machine 6703742: Electric motor with rotor being a drive wheel |