Methods for measuring an integral of a classical field via local interaction of classical bits or local interaction of qubits passing through the field one at a time are analyzed. A quantum method, which has an exponentially better precision than any classical method we could see, is described.

It is demonstrated that Chandler–Silbey–Ladanyi integral equation theory for the site–site molecular fluids is the limiting case of complete association of more general two-density integral equation theory for associating fluids developed by Wertheim. The analysis is presented for a site–site molecular system with any number and geometrical arrangement of the sites in the molecule and arbitrary...

in this paper, we give a generalization of the integral dependence from rings to modules. we study the stability of the integral closure with respect to various module theoretic constructions. moreover, we introduce the notion of integral extension of a module and prove the lying over, going up and going down theorems for modules.