Invariant theory and differential equations

Olver, Peter J.

doi:10.1007/bfb0078807

Cited by 19 publications

(23 citation statements)

References 21 publications

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“…A generalization of the Poincaré lemma states that the differential Bianchi-like identity (12) together with the previous algebraic irreducibility conditions on K imply that the Weinberg tensor is the sth derivative of a symmetric tensor field of rank s [7,8]. The same theorem states that the most general pure-gauge tensor field for which the curvature vanishes identically is a symmetrized derivative of a symmetric tensor field of rank s − 1.…”

Section: Curvature Tensors Of De Wit Freedman and Weinbergmentioning

confidence: 94%

Tensor Gauge Fields in Arbitrary Representations of GL(D, $${\mathbb{R})}$$ : II. Quadratic Actions

Bekaert

Boulanger

2007

Commun. Math. Phys.

100

155

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Quadratic, second-order, non-local actions for tensor gauge fields transforming in arbitrary irreducible representations of the general linear group in D-dimensional Minkowski space are explicitly written in a compact form by making use of Levi-Civita tensors. The field equations derived from these actions ensure the propagation of the correct massless physical degrees of freedom and are shown to be equivalent to non-Lagrangian local field equations proposed previously. Moreover, these actions allow a frame-like reformulationà la MacDowell-Mansouri, without any trace constraint in the tangent indices.

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Section: Curvature Tensors Of De Wit Freedman and Weinbergmentioning

confidence: 94%

Tensor Gauge Fields in Arbitrary Representations of GL(D, $${\mathbb{R})}$$ : II. Quadratic Actions

Bekaert

Boulanger

2007

Commun. Math. Phys.

100

155

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“…, S} containing the q elements (#I = q) corresponding to the difference between Y p+q and Y p . We "generalize" the definition (3.10) by introducing the differential operators d I as follows (see also [9,10])…”

Section: Generalized Nilpotencymentioning

confidence: 99%

“…Section 3 "N-complexes" gathers together the mathematical background needed for the following sections. Based on the works [9,10,11,12], it includes definitions and propositions together with a review of linearized gravity gauge structure in the language of N-complexes. Section 4 discusses linearized gravity field equations and their duality properties in the introduced mathematical framework.…”

Section: Introductionmentioning

confidence: 99%

Tensor Gauge Fields in Arbitrary Representations of GL(D, ?). Duality and Poincar� Lemma

Bekaert

Boulanger

2004

Communications in Mathematical Physics

186

318

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Using a mathematical framework which provides a generalization of the de Rham complex (well-designed for p-form gauge fields), we have studied the gauge structure and duality properties of theories for free gauge fields transforming in arbitrary irreducible representations of GL(D, R). We have proven a generalization of the Poincaré lemma which enables us to solve the above-mentioned problems in a systematic and unified way.

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“…The following discussion, taken from [34], shows the existence of a flow-based coordinate system for ideal flow fields which, by definition, have zero divergence and zero curl. The following theorems (stated without proof from [34]) are used to introduce a curvilinear coordinate system using complex analytic functions, and are reported here for the sake of completeness.…”

Section: Appendix a Existence Of A Flow-based Coordinate Systemmentioning

confidence: 98%

“…The following theorems (stated without proof from [34]) are used to introduce a curvilinear coordinate system using complex analytic functions, and are reported here for the sake of completeness.…”

Section: Appendix a Existence Of A Flow-based Coordinate Systemmentioning

confidence: 99%

Topology control in large-scale wireless sensor networks: Between information source and sink

2013

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With non-uniform traffic patterns in wireless sensor networks due to the manyto-one nature of communications, the traditional definition of connectivity in graph theory does not seem to be sufficient to satisfy the requirements of sensor networks. In this work, a new notion of connectivity (called pathimplementability) is defined which represents the ability of sensor nodes to relay traffic along a given direction field (referred to as information flow vector field) D. The magnitude of information flow is proportional to the traffic flux (per unit length) passing through any point in the network, and its direction is toward the flow of traffic. The flow field may be obtained from engineering knowledge or as a solution to an optimization problem. In either case, information flow flux lines are abstract paths that are assumed to represent desired paths for flow of traffic. In a sensor network with a given flow field D(x, y), we show that a density of n(x, y) = O(| D(x, y)| 2 ) sensor nodes is not sufficient to implement the flow field. On the other hand, by increasing the density of wireless nodes to n(x, y) = O(| D(x, y)| 2 log | D(x, y)|), the flow field becomes path-implementable. Path-implementability requires more nodes than simple connectivity. However, it guarantees existence of enough paths connecting the information source to the sink so that all the traffic can be transmitted to the sink. We also propose a joint MAC and routing protocol to forward traffic along the flow field; the proposed tier-based scheme can be further exploited to build lightweight protocol stacks which meet the specific requirements of dense sensor networks.

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Invariant theory and differential equations

Cited by 19 publications

References 21 publications

Tensor Gauge Fields in Arbitrary Representations of GL(D, $${\mathbb{R})}$$ : II. Quadratic Actions

Tensor Gauge Fields in Arbitrary Representations of GL(D, $${\mathbb{R})}$$ : II. Quadratic Actions

Tensor Gauge Fields in Arbitrary Representations of GL(D, ?). Duality and Poincar� Lemma

Topology control in large-scale wireless sensor networks: Between information source and sink

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