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Supplemental Material for Supervisor Cultural Humility and Supervisee Nondisclosure: The Supervisory Working Alliance Matters by Melissa M. Ertl, Michael V. Ellis and Lawrence P. Peterson in The Counseling Psychologist

We define a conformally invariant action S on gauge connections on a closed pseudo-Riemannian manifold M of dimension 6. At leading order this is quadratic in the gauge connection. The Euler-Lagrange equations of S, with respect to variation of the gauge connection, provide a higher-order conformally invariant analogue of the (source-free) Yang-Mills equations. For any gauge connection A on M, we define S(A) by first defining a Lagrangian density associated to A. This is not conformally invariant but has a conformal transformation analogous to a Q-curvature. Integrating this density provides the conformally invariant action. In the special case that we apply S to the conformal Cartan-tractor connection, the functional gradient recovers the natural conformal curvature invariant called the Fefferman-Graham obstruction tensor. So in this case the Euler-Lagrange equations are exactly the "obstruction-flat" condition for 6-manifolds. This extends known results for 4-dimensional pseudo-Riemannian manifolds where the Bach tensor is recovered in the Yang-Mills equations of the Cartan-tractor connection., Comment: 24 pages. Uses tensor.sty

The (Fefferman-Graham) ambient obstruction tensor is a conformally invariant symmetric trace-free 2-tensor on even-dimensional Riemannian and pseudo-Riemannian manifolds. The conformal deformation complex is a differential complex related to infinitesimal deformations of conformal structure. We construct a conformally invariant differential operator on algebraic Weyl tensors that gives special curved analogues of certain operators related to the deformation complex and that, upon application to the Weyl curvature, yields the obstruction tensor. This new definition of the obstruction tensor leads to simple direct proofs that the obstruction tensor is divergence free and vanishes for a class of metrics which (strictly) includes the class of conformally Einstein metrics. Our main constructions are based on the ambient metric of Fefferman-Graham and its relation to the conformal tractor connection. We prove that the obstruction tensor is an obstruction to the formal problem of obtaining a Ricci-flat ambient metric and that it may equivalently be viewed as the obstruction to finding an ambient metric with curvature harmonic for a certain (ambient) form Laplacian. This leads to a new ambient formula for the obstruction in terms of a power of this form Laplacian acting on the ambient curvature. This result leads us to construct Laplacian type operators that generalise the conformal Laplacians of Graham-Jenne-Mason-Sparling. As background to these issues, we give an explicit construction of the deformation complex in dimensions n>3, construct two related (detour) complexes, and establish essential properties of the operators in these., 40 pages. No figures. Record of changes: V1, 18 August 2004; V2, 9 November 2004, adding a reference and correcting some typographical errors; V3, 19 May 2005, shortening article, updating references. Version V3 accepted by Pacific Journal of Mathematics

The application of aerodynamics to the automobile in the last several years has grown considerably. Ford Motor Company, a forerunner in the use of this technology in both their production and concept vehicles, recently unveiled its latest advanced concept vehicle--Probe IV. This paper discusses the various engineering developments which made this vehicle a reality. Specifically addressed are those engineering developments which were necessary to accommodate the unique styling and aerodynamic functions.