2,173 research outputs found
Extending Whitney's extension theorem: nonlinear function spaces
We consider a global, nonlinear version of the Whitney extension problem for
manifold-valued smooth functions on closed domains , with non-smooth
boundary, in possibly non-compact manifolds. Assuming is a submanifold with
corners, or is compact and locally convex with rough boundary, we prove that
the restriction map from everywhere-defined functions is a submersion of
locally convex manifolds and so admits local linear splittings on charts. This
is achieved by considering the corresponding restriction map for locally convex
spaces of compactly-supported sections of vector bundles, allowing the even
more general case where only has mild restrictions on inward and outward
cusps, and proving the existence of an extension operator.Comment: 37 pages, 1 colour figure. v2 small edits, correction to Definition
A.3, which makes no impact on proofs or results. Version submitted for
publication. v3 small changes in response to referee comments, title
extended. v4 crucial gap filled, results not affected. v5 final version to
appear in Annales de l'Institut Fourie
The Langevin Equation for a Quantum Heat Bath
We compute the quantum Langevin equation (or quantum stochastic differential
equation) representing the action of a quantum heat bath at thermal equilibrium
on a simple quantum system. These equations are obtained by taking the
continuous limit of the Hamiltonian description for repeated quantum
interactions with a sequence of photons at a given density matrix state. In
particular we specialise these equations to the case of thermal equilibrium
states. In the process, new quantum noises are appearing: thermal quantum
noises. We discuss the mathematical properties of these thermal quantum noises.
We compute the Lindblad generator associated with the action of the heat bath
on the small system. We exhibit the typical Lindblad generator that provides
thermalization of a given quantum system.Comment: To appear in J.F.
From n+1-level atom chains to n-dimensional noises
In quantum physics, the state space of a countable chain of (n+1)-level atoms
becomes, in the continuous field limit, a Fock space with multiplicity n. In a
more functional analytic language, the continuous tensor product space over R
of copies of the space C^{n+1} is the symmetric Fock space Gamma_s(L^2(R;C^n)).
In this article we focus on the probabilistic interpretations of these facts.
We show that they correspond to the approximation of the n-dimensional normal
martingales by means of obtuse random walks, that is, extremal random walks in
R^n whose jumps take exactly n+1 different values. We show that these
probabilistic approximations are carried by the convergence of the basic matrix
basis a^i_j(p) of \otimes_N \CC^{n+1} to the usual creation, annihilation and
gauge processes on the Fock space.Comment: 22 page
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