1-Density Operators and Algebraic Version of The Hohenberg-Kohn Theorem

Interrelation of the Coleman's representabilty theory for 1-density operators and abstract algebraic form of the Hohenberg-Kohn theorem is studied in detail. Convenient realization of the Hohenberg-Kohn set of classes of 1-electron operators and the Coleman's set of ensemble representable 1-density operators is presented. Dependence of the Hohenberg-Kohn class structure on the boundary properties of the ground state 1-density operator is established and is illustrated on concrete simple examples. Algorithm of restoration of many electron determinant ensembles from a given 1-density diagonal is described. Complete description of the combinatorial structure of Coleman's polyhedrons is obtained.Comment: AMSLaTex, 37 pages, 4 figures, 1 tabl

Prediction Markets to Forecast Electricity Demand

A preference is invariant with respect to a transformation tau if its ranking of acts is unaffected by a reshuffling of the states under tau. We show that any invariant preference must be parametric: there is a unique sufficient set of parameters such that the preference ranks acts according to their expected utility given the parameters. This property holds for all non-trivial preferences, provided only that they are reflexive, transitive, monotone, continuous and mixture linear.