Among the research topics are the hypergroupoid of boundary conditions for local quantum observables, the asymptotic stability of connective groups, the K-theory of the flip automorphisms, the modular

cocycle from commensuration and its Mackey range, and the classification of gapped Hamiltonians in quantum spin chains.

The notion of a

cocycle over a (semi)flow appears naturally when taking into account the linearization along an invariant manifold of a dynamical system generated by an autonomous differential equation in an infinite dimensional space (see for instance [19] Chapter 4).

The antisymmetry of this relation in [alpha] and [beta] implies that d([[LAMBDA].sub.[alpha][beta]] + [[LAMBDA].sub.[beta][gamma]] + [[LAMBDA].sub.[gamma][alpha]]) = 0, making [[LAMBDA].sub.[alpha][beta]] + [[LAMBDA].sub.[beta][gamma]] + [[LAMBDA].sub.[gamma][alpha]] a constant in [U.sub.[alpha]] [intersection] [U.sub.[beta]] [intersection] [U.sub.[gamma]] that is an integer (since (4) is nothing but the

cocycle condition for a U(1) fibre bundle):

Schmalfuss, "Non-autonomous systems,

cocycle attractors and variable time-step discretization," Numerical Algorithms, vol.

It is induced on a Banach space by a skew-evolution

cocycle (see [13]) defined over a semiflow associated with a generalized dynamical system.

Moreover, let A be a linear

cocycle over this system which takes values in a family of compact and injective operators on some Banach space.

The minimum number of Fox colors and quandle

cocycle invariants.

if and only if The dual notion of a cycle is that of a cut or

cocycle. If is a partition of the vertex set, and the set , consisting of those edges with one end in and one end in , is not empty, then is called a cut.

In Section 2 we collect some notions and facts from the theory of dynamical systems (semigroup dynamical system,

cocycle, full trajectory, non-autonomous dynamical system, compact global attractor) used in our paper.

of Colorado) construct a retracted relative

cocycle representing the Connes-Chern character in relative cyclic cohomology and derive the ensuing pairing formulae with the K-theory, establishing a connection with the Atiyah-Patodi-Singer index theorem.

The mapping [PHI]: [R.sub.+] X [OMEGA] [right arrow] B(X) is said to be a stochastic

cocycle (over the semiflow [phi]) if it satisfies ([c.sub.1]) [PHI](0, [omega]) = I (the identity on X), for all [omega] [member of] [OMEGA]; ([c.sub.2]) [PHI](t + s, [omega]) = [PHI](t, [phi](s, [omega]))[PHI](s, [omega]), for all t, s [greater than or equal to] 0, and [omega] [member of] [OMEGA].