30 research outputs found
The category of local algebras and points proches
Categorial methods for generating new local algebras from old ones are
presented. A direct proof of the differential structure of the prolongations of
a manifold is proposed
Polarizations and differential calculus in affine spaces
Within the framework of mappings between affine spaces, the notion of -th
polarization of a function will lead to an intrinsic characterization of
polynomial functions. We prove that the characteristic features of derivations,
such as linearity, iterability, Leibniz and chain rules, are shared -- at the
finite level -- by the polarization operators. We give these results by means
of explicit general formulae, which are valid at any order , and are based
on combinatorial identities. The infinitesimal limits of the -th
polarizations of a function will yield its -th derivatives (without
resorting to the usual recursive definition), and the above mentioned
properties will be recovered directly in the limit. Polynomial functions will
allow us to produce a coordinate free version of Taylor's formula
Convergence of a quantum nornal form and a generalization of Cherry's theorem
We consider on L-2(T-2) the Schrodinger operator family H-epsilon : epsilon is an element of R with the domain and action defined as follows:
D(H-epsilon) = H-1 (T-2); H(epsilon)u = -i (h) over bar omega . del u + Vu,
where epsilon is an element of R, omega = (omega(1), omega(2)) is a vector of complex frequencies, and V is a pseudodifferential operator of order zero. H-epsilon represents the Weyl quantization of the Hamiltonian family H-epsilon defined on the phase space R-2 x T-2: (xi, chi) (sic) R-2 x T-2 bar right arrow H-epsilon(xi, chi) = omega . xi + epsilon V(xi, chi), where V(xi, chi). C-2(R-2 x T-2; R). We prove the uniform convergence with respect to (h) over bar is an element of [0, 1] of the quantum normal form, which reduces to the classical one for (h) over bar = 0. As a consequence, we simultaneously obtain an exact quantization formula for the quantum spectrum as well as a convergence criterion for the classical Birkhoff normal form generalizing a well- known theorem of Cherry
Singular densities, test particles and generalized Hamiltonian systems in general relativity
We derive the motion equations and the structure equations of neutral and charged test particles, starting from the gravitational field equations. The method consists in the ap- plication of conservation laws to singular tensor densities, which represent regions of strong matter concentration. Moreover, a Hamiltonian formulation of the particle equa- tions is given, in the form of implicit differential equations generated by Hamiltonian Morse families