Representing measures on the Royden boundary for solutions of Δu=Pu on a Riemannian manifold

Consider the Royden compactification R* of a Riemannian n-manifold R, Γ = R*\R its Royden boundary, Δ its harmonic boundary and the elliptic differential equation Δu = Pu, P ≥ 0 on R. A regular Borel measure m^P can be constructed on Γ with support equal to the closure of Δ^P = {q ϵ Δ : q has a neighborhood U in R* with _U^ʃ_ᴖR^{P ˂ ∞} }. Every enegy-finite solution to u (i.e. E(u) = D(u) + ^ʃ_Ru²P ˂ ∞, where D(u) is the Dirichlet integral of u) can be represented by u(z) = ^ʃ_Γu(q)K(z,q)dm^P(q) where K(z,q) is a continuous function on ^Rx Γ . A _P^~_E-function is a nonnegative solution which is the infimum of a downward directed family of energy-finite solutions. A nonzero _P^~_E-function is called _P^~_E-minimal if it is a constant multiple of every nonzero _P^~_E-function dominated by it. THEOREM. There exists a _P^~_E-minimal function if and only if there exists a point in q ϵ Γ such that m^P(q) > 0. THEOREM. For q ϵ Δ^P , m^P(q) > 0 if and only if m⁰(q) > 0 .

A Novel System for Reducing Turbo-Lag by Injection of Compressed Gas into the Exhaust Manifold

A key challenge in achieving good transient performance of highly boosted engines is the difficulty of accelerating the turbocharger from low air flow conditions (“turbo lag”). Multi-stage turbocharging, electric turbocharger assistance, electric compressors and hybrid powertrains are helpful in the mitigation of this deficit, but these technologies add significant cost and integration effort. Air-assist systems have the potential to be more cost-effective. Injecting compressed air into the intake manifold has received considerable attention, but the performance improvement offered by this concept is severely constrained by the compressor surge limit. The literature describes many schemes for generating the compressed gas, often involving significant mechanical complexity and/or cost. In this paper we demonstrate a novel exhaust assist system in which a reservoir is charged during braking. Experiments have been conducted using a 2.0 litre light-duty Diesel engine equipped with exhaust gas recirculation (EGR) and variable geometry turbine (VGT) coupled to an AC transient dynamometer, which was controlled to mimic engine load during in-gear braking and acceleration. The experimental results confirm that the proposed system reduces the time to torque during the 3rd gear tip-in by around 60%. Such a significant improvement was possible due to the increased acceleration of turbocharger immediately after the tip-in. Injecting the compressed gas into the exhaust manifold circumvents the problem of compressor surge and is the key enabler of the superior performance of the proposed concept.

Consensus on homogeneous manifolds

The present paper considers distributed consensus algorithms for agents evolving on a connected compact homogeneous (CCH) manifold. The agents track no external reference and communicate their relative state according to an interconnection graph. The paper first formalizes the consensus problem for synchronization (i.e. maximizing the consensus) and balancing (i.e. minimizing the consensus); it thereby introduces the induced arithmetic mean, an easily computable mean position on CCH manifolds. Then it proposes and analyzes various consensus algorithms on manifolds: natural gradient algorithms which reach local consensus equilibria; an adaptation using auxiliary variables for almost-global synchronization or balancing; and a stochastic gossip setting for global synchronization. It closes by investigating the dependence of synchronization properties on the attraction function between interacting agents on the circle. The theory is also illustrated on SO(n) and on the Grassmann manifolds. ©2009 IEEE.

A Grassmann-Rayleigh quotient iteration for computing invariant subspaces

The classical Rayleigh quotient iteration (RQI) allows one to compute a one-dimensional invariant subspace of a symmetric matrix A. Here we propose a generalization of the RQI which computes a p-dimensional invariant subspace of A. Cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature.