We shall rst show that the classi cation of factors, as seen in the unusual light of Andr e Weil's Basic Number Theory, is a natural substitute for the Brauer theory of central simple algebras in local class eld theory at Archimedian places. Passing to the global case provides a natural geometric framework in which the Frobenius, its eigenvalues and the Lefschetz formula interpretation of the e...

We study the notion of class refinement in a concurrent object-oriented setting. Our model is based on a combination of action systems and classes. An action system describes the behavior of a concurrent, distributed, or interactive system in terms of the atomic actions that can take place during the execution of the system. Classes serve as templates for creating objects. To express concurrenc...

and |ξA,B(λ)| ≤ n if A −B is rank n (2) are well known; see, for example, [5] or [6]. The Krein spectral shift function can also be defined for unbounded self-adjoint operators A,B and enjoys the same properties as long as their difference is trace class. The results of this paper extend to general unbounded operators A and B (as long as their difference is trace class) but for simplicity, we w...

This note represents an approach to the abstract Fredholm theory of trace class (and more generally ~ = {A [ Tr(] A [ ~ ) < oo}) operators on a separable Hi lber t space, ~v{,. There are few new results here but there are a set of new proofs which we feel sheds considerable light on the theory discussed. In part i cular, we would emphasize our proof of Lidskii 's theorem (see Sect. 4): I t was ...

In his 1953 paper [Matem. Sbornik 33 (1953), 597 – 626] Mark Krein presented an example of a symmetric rank one perturbation of a selfadjoint operator such that for all values of the spectral parameter in the interior of the spectrum, the difference of the corresponding spectral projections is not trace class. In the present note it is shown that in the case in question this difference has simp...

Interpolation inequalities of Gagliardo-Nirenberg type and compactness results for self-adjoint trace-class operators with finite kinetic energy are established. Applying these results to the minimization of various free energy functionals, we determine for instance stationary states of the Hartree problem with temperature corresponding to various statistics. Key-words. Compact self-adjoint ope...

It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix V is concave (convex) with respect to V . Using the theory of the spectral shift function we generalize this property to self-adjoint operators on a separable Hilbert space with an arbitrary spectrum. More precisely, we prove that the spectral shift function integrated with respect to the spectral p...

Abstract. Let A and B be almost commuting (i.e., the commutator AB−BA belongs to trace class) self-adjoint operators. We construct a functional calculus φ 7→ φ(A,B) for functions φ in the Besov class B ∞,1(R ). This functional calculus is linear, the operators φ(A,B) and ψ(A,B) almost commute for φ, ψ ∈ B ∞,1(R ), and φ(A,B) = u(A)v(B) whenever φ(s, t) = u(s)v(t). We extend the Helton–Howe trac...

In a recent paper we have presented a method to evaluate certain Hankel determinants as almost products; i.e. as a sum of a small number of products. The technique to find the explicit form of the almost product relies on differential-convolution equations and trace calculations. In the trace calculations a number of intermediate nonlinear terms involving determinants occur, but only to cancel ...

An important veriication problem for concurrent systems in general is that of establishing whether one system is a correct implementation, or reenement, of another system. For untimed systems, trace inclusion (or variants such as failure inclusion) is often used as a criterion of reenement. The natural extension of this criterion to timed systems is timed trace{inclusion. Unfortunately, this pr...