On bifurcation of solutions of the Yamabe problem in product manifolds

We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non-homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds. (C) 2011 Elsevier Masson SAS. All rights reserved.

A NOTE ON THE UNIQUENESS OF SOLUTIONS FOR THE YAMABE PROBLEM

Using recent results on the compactness of the space of solutions of the Yamabe problem, we show that in conformal classes of metrics near the class of a nondegenerate solution which is unique (up to scaling) the Yamabe problem has a unique solution as well. This provides examples of a local extension, in the space of conformal classes, of a well-known uniqueness criterion due to Obata.

Positive solutions of a Neumann problem with competing critical nonlinearities

We present existence results for a Neumann problem involving critical Sobolev nonlinearities both on the right hand side of the equation and at the boundary condition.. Positive solutions are obtained through constrained minimization on the Nehari manifold. Our approach is based on the concentration 'compactness principle of P. L. Lions and M. Struwe.

An Efficient and Verifiable Solution to the Millionaire Problem

A new solution to the millionaire problem is designed on the base of two new techniques: zero test and batch equation. Zero test is a technique used to test whether one or more ciphertext contains a zero without revealing other information. Batch equation is a technique used to test equality of multiple integers. Combination of these two techniques produces the only known solution to the millionaire problem that is correct, private, publicly verifiable and efficient at the same time.

Finding order in chaos : a problem based collaborative assignment for 1300 students across 19 disciplines

The use of collaborative assignments for assessment is a risky undertaking for students and course designers. Yet the benefits, in terms of core learning outcomes, competencies, collaborative sense making and student involvement, suggest that the effort is worthwhile. Formal descriptions and rules do little to ameliorate the perception of risk and increased anxiety by students. (Ryan, 2007). BEB100 Introducing Professional Learning is a faculty-wide foundation unit with over 1300 students from 19 disciplines across the Faculty of the Built Environment and Engineering (“BEE”) at the Queensland University of Technology (“QUT”), Brisbane, Australia. Finding order in chaos outlines the approach and justification, assessment criteria, learning resources, teamwork tools, tutorial management, communication strategies, 2007-09 Student Learning Experience Survey results, annual improvements, findings and outcomes.