292 research outputs found
A topological approach to non-Archimedean Mathematics
Non-Archimedean mathematics (in particular, nonstandard analysis) allows to
construct some useful models to study certain phenomena arising in PDE's; for
example, it allows to construct generalized solutions of differential equations
and variational problems that have no classical solution. In this paper we
introduce certain notions of non-Archimedean mathematics (in particular, of
nonstandard analysis) by means of an elementary topological approach; in
particular, we construct non-Archimedean extensions of the reals as appropriate
topological completions of . Our approach is based on the notion of
-limit for real functions, and it is called -theory. It can
be seen as a topological generalization of the -theory presented in
\cite{BDN2003}, and as an alternative topological presentation of the
ultrapower construction of nonstandard extensions (in the sense of
\cite{keisler}). To motivate the use of -theory for applications we
show how to use it to solve a minimization problem of calculus of variations
(that does not have classical solutions) by means of a particular family of
generalized functions, called ultrafunctions.Comment: 22 page
Infinitesimals without Logic
We introduce the ring of Fermat reals, an extension of the real field
containing nilpotent infinitesimals. The construction takes inspiration from
Smooth Infinitesimal Analysis (SIA), but provides a powerful theory of actual
infinitesimals without any need of a background in mathematical logic. In
particular, on the contrary with respect to SIA, which admits models only in
intuitionistic logic, the theory of Fermat reals is consistent with classical
logic. We face the problem to decide if the product of powers of nilpotent
infinitesimals is zero or not, the identity principle for polynomials, the
definition and properties of the total order relation. The construction is
highly constructive, and every Fermat real admits a clear and order preserving
geometrical representation. Using nilpotent infinitesimals, every smooth
functions becomes a polynomial because in Taylor's formulas the rest is now
zero. Finally, we present several applications to informal classical
calculations used in Physics: now all these calculations become rigorous and,
at the same time, formally equal to the informal ones. In particular, an
interesting rigorous deduction of the wave equation is given, that clarifies
how to formalize the approximations tied with Hook's law using this language of
nilpotent infinitesimals.Comment: The first part of the preprint is taken directly form arXiv:0907.1872
The second part is new and contains a list of example
Non-radial sign-changing solutions for the Schroedinger-Poisson problem in the semiclassical limit
We study the existence of nonradial sign-changing solutions to the
Schroedinger-Poisson system in dimension N>=3. We construct nonradial
sign-changing multi-peak solutions whose peaks are displaced in suitable
symmetric configurations and collapse to the same point. The proof is based on
the Lyapunov-Schmidt reduction
Convex domains of Finsler and Riemannian manifolds
A detailed study of the notions of convexity for a hypersurface in a Finsler
manifold is carried out. In particular, the infinitesimal and local notions of
convexity are shown to be equivalent. Our approach differs from Bishop's one in
his classical result (Bishop, Indiana Univ Math J 24:169-172, 1974) for the
Riemannian case. Ours not only can be extended to the Finsler setting but it
also reduces the typical requirements of differentiability for the metric and
it yields consequences on the multiplicity of connecting geodesics in the
convex domain defined by the hypersurface.Comment: 22 pages, AMSLaTex. Typos corrected, references update
Nonlinear Klein-Gordon-Maxwell systems with Neumann boundary conditions on a Riemannian manifold with boundary
Let (M,g) be a smooth compact, n dimensional Riemannian manifold, n=3,4 with
smooth n-1 dimensional boundary. We search the positive solutions of the
singularly perturbed Klein Gordon Maxwell Proca system with homogeneous Neumann
boundary conditions or for the singularly perturbed Klein Gordon Maxwell system
with mixed Dirichlet Neumann homogeneous boundary conditions. We prove that
stable critical points of the mean curvature of the boundary generates
solutions when the perturbation parameter is sufficiently small.Comment: arXiv admin note: text overlap with arXiv:1410.884
Displacement energy of unit disk cotangent bundles
We give an upper bound of a Hamiltonian displacement energy of a unit disk
cotangent bundle in a cotangent bundle , when the base manifold
is an open Riemannian manifold. Our main result is that the displacement
energy is not greater than , where is the inner radius of ,
and is a dimensional constant. As an immediate application, we study
symplectic embedding problems of unit disk cotangent bundles. Moreover,
combined with results in symplectic geometry, our main result shows the
existence of short periodic billiard trajectories and short geodesic loops.Comment: Title slightly changed. Close to the version published online in Math
Zei
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