17 research outputs found
Nonlinear Generalized Functions: their origin, some developments and recent advances
We expose some simple facts at the interplay between mathematics and the real
world, putting in evidence mathematical objects " nonlinear generalized
functions" that are needed to model the real world, which appear to have been
generally neglected up to now by mathematicians. Then we describe how a
"nonlinear theory of generalized functions" was obtained inside the Leopoldo
Nachbin group of infinite dimensional holomorphy between 1980 and 1985. This
new theory permits to multiply arbitrary distributions and contains the above
mathematical objects, which shows that the features of this theory are natural
and unavoidable for a mathematical description of the real world. Finally we
present direct applications of the theory such as existence-uniqueness for
systems of PDEs without classical solutions and calculations of shock waves for
systems in non-divergence form done between 1985 and 1995, for which we give
three examples of different nature: elasticity, cosmology, multifluid flows.Comment: 42 pages, 4 figure
The partition bundle of type A_{N-1} (2, 0) theory
Six-dimensional (2, 0) theory can be defined on a large class of
six-manifolds endowed with some additional topological and geometric data (i.e.
an orientation, a spin structure, a conformal structure, and an R-symmetry
bundle with connection). We discuss the nature of the object that generalizes
the partition function of a more conventional quantum theory. This object takes
its values in a certain complex vector space, which fits together into the
total space of a complex vector bundle (the `partition bundle') as the data on
the six-manifold is varied in its infinite-dimensional parameter space. In this
context, an important role is played by the middle-dimensional intermediate
Jacobian of the six-manifold endowed with some additional data (i.e. a
symplectic structure, a quadratic form, and a complex structure). We define a
certain hermitian vector bundle over this finite-dimensional parameter space.
The partition bundle is then given by the pullback of the latter bundle by the
map from the parameter space related to the six-manifold to the parameter space
related to the intermediate Jacobian.Comment: 15 pages. Minor changes, added reference
Multiplication of Distributions and Nonperturbative Calculations of Transition Probabilities
In a mathematical context in which one can multiply distributions the
"`formal"' nonperturbative canonical Hamiltonian formalism in Quantum Field
Theory makes sense mathematically, which can be understood a priori from the
fact the so called "`infinite quantities"' make sense unambiguously (but are
not classical real numbers). The perturbation series does not make sense. A
novelty appears when one starts to compute the transition probabilities. The
transition probabilities have to be computed in a nonperturbative way which, at
least in simplified mathematical examples (even those looking like
nonrenormalizable series), gives real values between 0 and 1 capable to
represent probabilities. However these calculations should be done numerically
and we have only been able to compute simplified mathematical examples due to
the fact these calculations appear very demanding in the physically significant
situation with an infinite dimensional Fock space and the QFT operators
Multiplication of distributions: a tool in mathematics, numerical engineering and theoretical physics
This book presents recent and very elementary developments of a theory of multiplication of distributions in the field of explicit and numerical solutions of systems of PDEs of physics (nonlinear elasticity, elastoplasticity, hydrodynamics, multifluid flows, acoustics). The prerequisites are kept to introductory calculus level so that the book remains accessible at the same time to pure mathematicians (as a smoothand somewhat heuristic introdcution to this theory) and to applied mathematicians, numerical engineers and theoretical physicists (as a tool to treat problems involving products of distributions)
Un survol des applications actuelles du calcul différentiel dans les espaces bornologiques
Differential calculus and holomorphy : real and complex analysis in locally convex spaces /
Bibliography: p. 431-452.Includes index