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Commutative property
About: Commutative property is a research topic. Over the lifetime, 11851 publications have been published within this topic receiving 164424 citations. The topic is also known as: commutative operation & commutativity.
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19 Oct 2011
TL;DR: A detailed introduction to the theory of groups: finite and infinite; commutative and non-commutative is given in this article, where the reader is provided with only a basic knowledge of modern algebra.
Abstract: This is a detailed introduction to the theory of groups: finite and infinite; commutative and non-commutative. Presupposing only a basic knowledge of modern algebra, it introduces the reader to the different branches of group theory and its principal accomplishments.
3,406 citations
TL;DR: In this article, it was shown that it is possible to build up a fairly satisfactory theory of the emission of radiation and of the reaction of the radiation field on the emitting system on the basis of a kinematics and dynamics which are not strictly relativistic.
Abstract: The new quantum theory, based on the assumption that the dynamical variables do not obey the commutative law of multiplication, has by now been developed sufficiently to form a fairly complete theory of dynamics. One can treat mathematically the problem of any dynamical system composed of a number of particles with instantaneous forces acting between them, provided it is describable by a Hamiltonian function, and one can interpret the mathematics physically by a quite definite general method. On the other hand, hardly anything has been done up to the present on quantum electrodynamics. The questions of the correct treatment of a system in which the forces are propagated with the velocity of light instead of instantaneously, of the production of an electromagnetic field by a moving electron, and of the reaction of this field on the electron have not yet been touched. In addition, there is a serious difficulty in making the theory satisfy all the requirements of the restricted principle of relativity, since a Hamiltonian function can no longer be used. This relativity question is, of course, connected with the previous ones, and it will be impossible to answer any one question completely without at the same time answering them all. However, it appears to be possible to build up a fairly satisfactory theory of the emission of radiation and of the reaction of the radiation field on the emitting system on the basis of a kinematics and dynamics which are not strictly relativistic.
1,774 citations
TL;DR: In this article, the deformation theory for algebras is studied in terms of the set of structure constants as a parameter space, and an example justifying the choice of parameter space is given.
Abstract: CHAPTER I. The deformation theory for algebras 1. Infinitesimal deformations of an algebra 2. Obstructions 3. Trivial deformations 4. Obstructions to derivations and the squaring operation 5. Obstructions are cocycles 6. Additivity and integrability of the square 7. Restricted deformation theories and their cohomology theories 8. Rigidity of fields in the commutative theory CHAPTER II. The parameter space 1. The set of structure constants as parameter space for the deformation theory 2. Central algebras and an example justifying the choice of parameter space 3. The automorphism group as a parameter space, and examples of obstructions to derivations 4. A fiber space over the parameter space, and the upper semicontinuity theorem 5. An example of a restricted theory and the corresponding modular group CHAPTER III. The deformation theory for graded and filtered rings 1. Graded, filtered, and developable rings 2. The Hochschild theory for developable rings 3. Developable rings as deformations of their associated graded rings 4. Trivial deformations and a criterion for rigidity 5. Restriction to the commutative theory 6. Deformations of power series rings
1,565 citations
TL;DR: In this paper, the authors present a legal opinion on the applicability of commercial or impression systématiques in the context of the agreement of publication mathématique de l'I.H.É.S.
Abstract: © Publications mathématiques de l’I.H.É.S., 1985, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
1,149 citations
01 Jan 1988
TL;DR: For purely mathematical reasons, it is necessary to consider spaces which cannot be represented as point set sand where the coordinates describing the space do not commute as mentioned in this paper, i.e., spaces which are described by algebras of coordinates which are not commutative.
Abstract: For purely mathematical reasons it is necessary to consider spaces which cannot be represented as point set sand where the coordinates describing the space do not commute.In other words,spaces which are described by algebras of coordinates which arenot commutative.If you conside rsuch spaces,then it is necessary to rethink most of the notions of classical geometry and redefine them. Motivated from pure mathematics it turns out that there are very striking parallels to what is done in quantum physics In the following lectures, I hope to discuss some of these parallels.
1,025 citations