Graph complexes in deformation quantization

Kontsevich's formality theorem and the consequent star-product formula rely on the construction of an $L_\infty$-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential graded Lie algebras between the Kontsevich DGLA of admissible graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between polyvector fields and polydifferential operators. Kontsevich's proof of the formality morphism is reexamined in this light and an algebraic framework for discussing the tree-level reduction of Kontsevich's star-product is described.Comment: 39 pages; 3 eps figures; uses Xy-pic. Final version. Details added, mainly concerning the tree-level approximation. Typos corrected. An abridged version will appear in Lett. Math. Phy

Geometrical construction of quantum groups representations

We describe geometrically the classical and quantum inhomogeneous groups G_0=(SL(2, \BbbC)\triangleright \BbbC^2) and G_1=(SL(2, \BbbC)\triangleright \BbbC^2)\triangleright \BbbC by studying explicitly their shape algebras as a spaces of polynomial functions with a quadratic relations.Comment: 16 pages, 1 figur

Experimental analysis of the boundary layer transition with zero and positive pressure gradient

The influence of a positive pressure gradient on the boundary layer transition is studied. The mean velocity and turbulence profiles of four cases are examined. As the intensity of the pressure gradient is increased, the Reynolds number of the transition onset and the length of the transition region are reduced. The Tollmein-Schlichting waves disturb the laminar regime; the amplification of these waves is in good agreement with the stability theory. The three dimensional deformation of the waves leads finally to the appearance of turbulence. In the case of zero pressure gradient, the properties of the turbulent spots are studied by conditional sampling of the hot-wire signal; in the case of positive pressure gradient, the turbulence appears in a progressive manner and the turbulent spots are much more difficult to characterize

Experimental analysis and computation of the onset and development of the boundary layer transition

The transition of an incompressible boundary layer, with zero pressure gradient and low free-stream turbulence is studied. Mean velocity, turbulence and Reynolds shear stress profiles are presented. The development of the Tollmien-Schlichting waves is clearly shown until the turbulent spots appear. The intermittency phenomenon is studied by conditional sampling of the hotwire signal. The comparison with calculation results obtained by resolution of a set of transport equations shows a good agreement for the mean characteristics of the flow; discrepancies observed for the turbulent quantities evolution are due to the intermittency phenomenon

La convexité de l'application moment d'un groupe de Lie

AbstractLet π be a unitary representation of a Lie group G. The moment mapping Ψπ of π assigns to every C∞ vector ξ in the Hilbert space H of π the linear functional Ψπ(ξ) of the Lie algebra g of G by the rule ψπ(ξ)(X)=1i〈dπ(X)ξ, ξ〉H, X ϵ g In this paper, we study the moment set Iπ of π, i.e., the closure of the image of Ψπ. It is shown that for solvable G, Iπ is always convex and that if furthermore π is irreducible, then Iπ is the closure (in g∗) of the convex hull of the Kirillov-Pukanszky orbit of π. If G is compact and if π is irreducible, then we show that Iπ is the convex hull of the orbit of the highest weight Λ of π, if and only if the number Πi = 1n 〈2Λ − αi, αi〉 is different from 0. Here α1, …, αn denote the simple roots of g

Three-dimensional compressible stability-transition calculations using the spatial theory

The e(exp n)-method is employed with the spatial amplification theory to compute the onset of transition on a swept wing tested in transonic cryogenic flow conditions. Two separate eigenvalue formulations are used. One uses the saddle-point method and the other assumes that the amplification vector is normal to the leading edge. Comparisons of calculated results with experimental data show that both formulations give similar results and indicate that the wall temperature has a rather strong effect on the value of the n factor

Ultrarobust calibration of an optical lattice depth based on a phase shift

We report on a new method to calibrate the depth of an optical lattice. It consists in triggering the intrasite dipole mode of the cloud by a sudden phase shift. The corresponding oscillatory motion is directly related to the intraband frequencies on a large range of lattice depths. Remarkably, for a moderate displacement, a single frequency dominates this oscillation for the zeroth and first order interference pattern observed after a sufficiently long time-of-flight. The method is robust against atom-atom interactions and the exact value of the extra external confinement of the initial trapping potential.Comment: 7 pages, 6 figure