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Smooth bundles of generalized half-densities

Daniel Canarutto (2000)

Archivum Mathematicum

Smooth bundles, whose fibres are distribution spaces, are introduced according to the notion of smoothness due to Frölicher. Some fundamental notions of differential geometry, such as tangent and jet spaces, Frölicher-Nijenhuis bracket, connections and curvature, are suitably generalized. It is also shown that a classical connection on a finite-dimensional bundle naturally determines a connection on an associated distributional bundle.

Snakes and articulated arms in an Hilbert space

Fernand Pelletier, Rebhia Saffidine (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

The purpose of this paper is to give an illustration of results on integrability of distributions and orbits of vector fields on Banach manifolds obtained in [5] and [4]. Using arguments and results of these papers, in the context of a separable Hilbert space, we give a generalization of a Theorem of accessibility contained in [3] and [6] for articulated arms and snakes in a finite dimensional Hilbert space.

Some quantitative results in singularity theory

Y. Yomdin (2005)

Annales Polonici Mathematici

The classical singularity theory deals with singularities of various mathematical objects: curves and surfaces, mappings, solutions of differential equations, etc. In particular, singularity theory treats the tasks of recognition, description and classification of singularities in each of these cases. In many applications of singularity theory it is important to sharpen its basic results, making them "quantitative", i.e. providing explicit and effectively computable estimates for all the important...

Some remarks about the p -Dirichlet integral

Mariano Giaquinta, Giuseppe Modica, Jiří Souček (1994)

Commentationes Mathematicae Universitatis Carolinae

We discuss variational problems for the p -Dirichlet integral, p non integer, for maps between manifolds, illustrating the role played by the geometry of the target manifold in their weak formulation.

Spaces of upper semicontinuous multi-valued functions on complete metric spaces

Katsuro Sakai, Shigenori Uehara (1999)

Fundamenta Mathematicae

Let X = (X,d) be a metric space and let the product space X × ℝ be endowed with the metric ϱ ((x,t),(x’,t’)) = maxd(x,x’), |t - t’|. We denote by U S C C B ( X ) the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify φ U S C C B ( X ) with its graph which is a closed subset of X × ℝ. The space U S C C B ( X ) admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then U S C C B ( X ) is homeomorphic to a...

Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval

Kamil Kaleta (2012)

Studia Mathematica

We prove a uniform lower bound for the difference λ₂ - λ₁ between the first two eigenvalues of the fractional Schrödinger operator ( - Δ ) α / 2 + V , α ∈ (1,2), with a symmetric single-well potential V in a bounded interval (a,b), which is related to the Feynman-Kac semigroup of the symmetric α-stable process killed upon leaving (a,b). “Uniform” means that the positive constant C α appearing in our estimate λ - λ C α ( b - a ) - α is independent of the potential V. In the general case of α ∈ (0,2), we also find a uniform lower bound for...

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