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 $n$-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 $n$, and are based on combinatorial identities. The infinitesimal limits of the $n$-th polarizations of a function will yield its $n$-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

A remark on eigenvalue perturbation theory at vanishing isolation distance

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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