Trace Operator Inequality for Superquadratic Functions

In this work, some operator trace inequalities are proved. An extension of Klein's inequality for all Hermitian matrices is proved. A non-commutative version (or Hansen-Pedersen version) of the Jensen trace inequality is provided as well. A generalization of the result for any positive Hilbert space operators acts on a positive unital linear map is established.

Relative pairing in cyclic cohomology and divisor flows

We construct invariants of relative K-theory classes of multiparameter dependent pseudodifferential operators, which recover and generalize Melrose's divisor flow and its higher odd-dimensional versions of Lesch and Pflaum. These higher divisor flows are obtained by means of pairing the relative K-theory modulo the symbols with the cyclic cohomological characters of relative cycles constructed out of the regularized operator trace together with its symbolic boundary. Besides giving a clear and conceptual explanation to the essential features of the divisor flows, namely homotopy invariance, additivity and integrality, this construction allows to uncover the previously unknown even-dimensional counterparts. Furthermore, it confers to the totality of these invariants a purely topological interpretation, that of implementing the classical Bott periodicity isomorphisms in a manner compatible with the suspension isomorphisms in both K-theory and in cyclic cohomology. We also give a precise formulation, in terms of a natural Clifford algebraic suspension, for the relationship between the higher divisor flows and the spectral flow.

Smoothing spline in a convex closed set of Hilbert space

A characterisation of a smoothing spline is sought in a convex closed set C of Hilbert space: , T and A are linear operators. A representation of the solution is obtained in the terms of the kernels of the above operators, of the dual operators T*, A* and of the dual cone C0. A particular case is considered when T is the differential operator and A is the operator-trace of a function.

Bituminous Pavement Smoothness: Statistically Based Approach to Acceptance

Since 1987 the Federal Lands Highway (FLH) Branch of FHWA has been evaluating acceptance of newly constructed bituminous pavements using California-type profilograph measurements. California Test Method 526 and FLH T504, as well as various acceptance plans, have been used in this evaluation. The purpose of this study was threefold: (a) to determine whether operator trace reduction variability was too large for the method to be suitable for acceptance testing; (b) to decide the type of acceptance plan to incorporate into the Standard Specifications for Construction of Roads and Bridges on Federal Highway Projects (FP 92); and (c) to evaluate two commercially available computerized trace reduction systems. The study concludes that when used in conjunction with statistical evaluation procedures, the test method is suitable for acceptance purposes and that computerized trace reduction is superior to manual reduction. Also presented are some fundamentals of statistically based acceptance that are not widely known or understood by highway engineers.