Thom form in equivariant Cech-de Rham theory

In the present paper, we provide the foundation of a $G$-equivariant Cech-de Rham theory for a compact Lie group $G$ by using the Cartan model of equivariant differential forms. Our approach is quite elementary without referring to the Mathai-Quillen framework. In particular, by a direct computation, we give an explicit formula of the $U(l)$-equivariant Thom form of C^l, which deforms the classical Bochnor-Martinelli kernel. Also we discuss a version of equivariant Riemann-Roch formula.Comment: 26pages, no figure

Inelastic tunneling in a double quantum dot coupled to a bosonic environment

Coupling a quantum system to a bosonic environment always give rise to inelastic processes, which reduce the coherency of the system. We measure energy dependent rates for inelastic tunneling processes in a fully controllable two-level system of a double quantum dot. The emission and absorption rates are well repro-duced by Einstein's coefficients, which relate to the spontaneous emission rate. The inelastic tunneling rate can be comparable to the elastic tunneling rate if the boson occupation number becomes large. In the specific semiconductor double dot, the energy dependence of the inelastic rate suggests that acoustic phonons are coupled to the double dot piezoelectrically.Comment: 6 pages, 4 figure

On sums of generalized Ramanujan sums

Ramanujan sums have been studied and generalized by several authors. For example, Nowak studied these sums over quadratic number fields, and Grytczuk defined that on semigroups. In this note, we deduce some properties on sums of generalized Ramanujan sums and give examples on number fields. In particular, we have a relational expression between Ramanujan sums and residues of Dedekind zeta functions.Comment: 10 page