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  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 ) 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
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].