Set Of Commutators Research Articles

Policy-Sensitive Model for Mode Split Planning

Urban transportation mode choice is critically dependent upon residential walk times, transit frequency, and out-of-pocket costs. A policy-sensitive model of these attributes account for a significant portion of mode choice variation among a set of commuters. A discrimination-classification model was generalized and used to simulate the probability that an individual would choose an abstract mode, k, of policy-sensitive variables. The variables were used to define linear discriminant functions and mode choice probability was tested based on individual discriminant functions by a model. Data for calibration were provided by a hand out/mail back travel survey of commuters to the central business district of Vancouver, B.C. Presimulation tests showed that transportation system attributes were more important than socioeconomic characteristics in the mode choice decision.

Symmetry ascent eigenvectors in finite point groups and an expanded matrix element selection rule

A formalism is proposed in which wave-functions quantized in a finite point group may be unambiguously labelled by their relative behaviour under the mapping of individual components from the generic point group R 3. Such a mapping is only possible into highly symmetric finite groups including O h and D 6h . Mapping to lower point groups by conventional symmetry descent creates ambiguities which can be removed by retaining the effect of discriminating virtual operators as parity labels for components. With such labelled wave functions, the formation of unambiguous direct products is possible with the introduction of Symmetry Ascent V Coefficients. By quantizing the wave functions about the desired n-fold axis in complex space, a commutative set of components is obtained. This allows component combination rules similar to those for 3-j symbols to be stated, modified to accommodate the possible mappings in finite groups and to retain the effect of parity. Hamiltonian operators in complex tensor form are tre...

Commutators modulo the center in a properly infinite von Neumann algebra

1. Introduction. An element in a von Neumann algebra stf is said to be a commutator in stf if there are elements A and B in stf such that C=AB — BA. For finite homogeneous discrete algebras and for properly infinite factor algebras the set of commutators has been completely described [l]-[5], [10]. In each of these special cases any element is a commutator modulo a central element depending on C. In this paper we show that given any element in a properly infinite von Neumann algebra stf there is an element C0 in the center of stf depending on such that C— C0 is a commutator in stf. The element C0 is an arbitrary element in the intersection criTc of the center with the uniform closure of the convex hull of {U*CU | U unitary in ?tf} [6, III, §5]. We then present a few facts about those elements such that 0 e Xc or what is the same as far as determining commutators is concerned about those elements such that OeJf^-ics for some invertible S in stf. 2. Commutators. Let stf be a C*-algebra with identity and let / be a closed two-sided ideal in stf. The image of the element A e stf in the factor algebra stf(T) = stf/1 under the canonical homomorphism of stf onto stf/I will be denoted by A(T). If £ is a maximal ideal of the center of stf, the smallest closed two-sided ideal in stf containing £ is denoted by [{]. For simplicity we write A([Q) as A(t). The set of maximal (respectively, primitive) ideals of stf with the hull-kernel topology is called the strong structure space (respectively, structure space) of stf. If stf is a von Neumann algebra, then the strong structure space M(stf) of stf is homeomorphic with the spectrum of the center S of stf under the map M —> M C J? [13]. This means M(stf) is extremely disconnected. Proposition 1. Let stf be a properly infinite von Neumann algebra and let A be a fixed element of stf. The function M -> ||^(M)|| of the strong structure space M (stf) of stf into the real numbers is continuous. Proof. For every «?Owe know that the set X={M e M (stf) \\A(M) ^«} is closed. If /=p| X, then \\A(I)\\^a [8, Lemma 1.9] and so M(M)|| ^a for every MeM(stf) containing /. Thus X={M e M (stf) | I<=M}.

Existence of common fixed points for commutative sets of nonlinear operators

Existence of common fixed points for commutative sets of nonlinear operators

Commutators of Certain Finitely Generated Soluble Groups

Groups in which the commutator subgroup coincides with the set of commutators have been studied to a certain extent by several authors. It was shown in (2; 4; 6; 7) that various types of known simple groups have this property. In (3), Macdonald has considered certain soluble groups with this property, and Hall has shown that any group can be embedded as a subgroup of a simple group of this type. Here we shall be concerned with the class C of groups defined as follows.For any positive integer n, denote by Cn the class of all groups in which every element of the commutator subgroup can be expressed as a product of at most n commutators. It is not difficult to show that Cn is a proper subclass of Cn+1 for all n. Let so that a group G ∊ C if and only if G ∊ Cn for some n.

Currents as Dynamical Variables

A dynamical theory of currents is presented. For each current there are a complete set of field equations and a complete set of equal-time commutators. These are related by expressing the energy-momentum tensor solely in terms of the currents and not of derivatives of the currents. The requirements of Lorentz covariance place severe restrictions on the commutators. Several models with and without internal symmetry are discussed.

Equal-Time Commutators Involving Divergences of Currents

A study is made on the equal-time commutators between the time and space components of vector and axial-vector currents and also between the time components and the divergences of these currents. The only assumption made is that these commutators consist of terms proportional to the $\ensuremath{\delta}$ function and at most the first-order space derivatives of the $\ensuremath{\delta}$ function. Lorentz covariance and consistency among the commutators lead to partial determination of the above commutators, including the result that the Schwinger terms are symmetric in the $\mathrm{SU}(3)$ indices. The resulting set of commutators is sufficient to demonstrate that the off-mass-shell strong matrix element, in which two of the bosons are made off the mass shell by means of the divergences of the axial-vector currents, maintains manifest symmetry with respect to the boson variables even after partial integrations are done. Conservation of some of the currents leads to further determination of the commutators. Thus, for example, when $\mathrm{SU}(2)$ symmetry is exact, all the components and also the divergences of these currents have the same $\mathrm{SU}(2)$ transformation properties as the corresponding time components.

On the connexion between the poincaré group and internal symmetry groups

Three new theorems are proved from which can be deduced that the conditions [H l ,P μ ]=[H l ,M ij ]=0 are maximal in the following sense: the vanishing of any supplementary set of commutators implies that the group obtained in combining the Poincare group with an internal symmetry group reduces to a direct product.

Triangular representations of linear algebras

It is well known that the elements of any given commutative algebra (and hence of any commutative set) of n × n matrices, over an algebraically closed field K, have a common eigenvector over K; indeed, the elements of such an algebra can be simultaneously reduced to triangular form (by a suitable similarity transformation). McCoy (5) has shown that a triangular reduction is always possible even for matrix algebras satisfying a condition substantially weaker than commutativity. Our aim in this note is to extend these results to more general systems (our arguments being, incidentally, simpler than some used for the matrix case even by writers subsequent to McCoy).