We study several duality isomorphisms between equivariant bivariant K-theory groups, generalising Kasparov’s first and second Poincaré duality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised self-maps. The second duality is related to the description of bivariant Kasparov theory for commutative C∗-algebras by families of elli...

Let $G$ be a linear Lie group acting properly on $G$-$\mathrm{spin}^c$ manifold $M$ with compact quotient. We give short proof that Poincaré duality holds between $G$-equivariant $K$-theory of $M$, defined using finite-dimensional $G$-vector bundles, and $K$-homology through the geometric model Baum Douglas.

Based on the Cayley, Grothendieck, Knudsen Mumford theory of determinants we extend the CM polarisation to the Hilbert scheme. The Baum Fulton Macpherson GRR Theorem enables us to show that on any flat, proper, local complete intersection family the restriction of the extension and the original CM sheaf are isomorphic (under a mild hypothesis on the base). As a consequence the CM stability impl...

Let {Xni , i≥ 1,n≥ 1} be an array of rowwise ρ̃-mixing random variables. Some sufficient conditions for complete convergence for weighted sums of arrays of rowwise ρ̃-mixing random variables are presented without assumptions of identical distribution. As applications, the Baum and Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of ρ̃-mixing random vari...

The complete convergence for pairwise negative quadrant dependent PNQD random variables is studied. So far there has not been the generalmoment inequality for PNQD sequence, and therefore the study of the limit theory for PNQD sequence is very difficult and challenging. We establish a collection that contains relationship to overcome the difficulties that there is no general moment inequality. ...