44,252 research outputs found
Differential Chow Form for Projective Differential Variety
In this paper, a generic intersection theorem in projective differential
algebraic geometry is presented. Precisely, the intersection of an irreducible
projective differential variety of dimension d>0 and order h with a generic
projective differential hyperplane is shown to be an irreducible projective
differential variety of dimension d-1 and order h. Based on the generic
intersection theorem, the Chow form for an irreducible projective differential
variety is defined and most of the properties of the differential Chow form in
affine differential case are established for its projective differential
counterpart. Finally, we apply the differential Chow form to a result of linear
dependence over projective varieties given by Kolchin.Comment: 17 page
Decay process of quantum open system at finite-temperature
Starting from the formal solution to the Heisenberg equation, we revisit an
universal model for a quantum open system with a harmonic oscillator linearly
coupled to a boson bath. The analysis of the decay process for a Fock state and
a coherent state demonstrate that this method is very useful in dealing with
the problems in decay process of the open system. For finite temperature, the
calculations of the reduced density matrix and the mean excitation number for
the open system show that an initial coherent state will evolve into a
temperature-dependant coherent state after tracing over the bath variables.
Also in short-time limit, a temperature-dependant effective Hamiltonian for the
open system characterizes the decay process of the open system
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