39,417 research outputs found
Deformation Quantization of Almost Kahler Models and Lagrange-Finsler Spaces
Finsler and Lagrange spaces can be equivalently represented as almost Kahler
manifolds enabled with a metric compatible canonical distinguished connection
structure generalizing the Levi Civita connection. The goal of this paper is to
perform a natural Fedosov-type deformation quantization of such geometries. All
constructions are canonically derived for regular Lagrangians and/or
fundamental Finsler functions on tangent bundles.Comment: the latex 2e variant of the manuscript accepted for JMP, 11pt, 23
page
Range descriptions for the spherical mean Radon transform
The transform considered in the paper averages a function supported in a ball
in \RR^n over all spheres centered at the boundary of the ball. This Radon
type transform arises in several contemporary applications, e.g. in
thermoacoustic tomography and sonar and radar imaging. Range descriptions for
such transforms are important in all these areas, for instance when dealing
with incomplete data, error correction, and other issues. Four different types
of complete range descriptions are provided, some of which also suggest
inversion procedures. Necessity of three of these (appropriately formulated)
conditions holds also in general domains, while the complete discussion of the
case of general domains would require another publication.Comment: LATEX file, 55 pages, two EPS figure
Computing Teichm\"{u}ller Maps between Polygons
By the Riemann-mapping theorem, one can bijectively map the interior of an
-gon to that of another -gon conformally. However, (the boundary
extension of) this mapping need not necessarily map the vertices of to
those . In this case, one wants to find the ``best" mapping between these
polygons, i.e., one that minimizes the maximum angle distortion (the
dilatation) over \textit{all} points in . From complex analysis such maps
are known to exist and are unique. They are called extremal quasiconformal
maps, or Teichm\"{u}ller maps.
Although there are many efficient ways to compute or approximate conformal
maps, there is currently no such algorithm for extremal quasiconformal maps.
This paper studies the problem of computing extremal quasiconformal maps both
in the continuous and discrete settings.
We provide the first constructive method to obtain the extremal
quasiconformal map in the continuous setting. Our construction is via an
iterative procedure that is proven to converge quickly to the unique extremal
map. To get to within of the dilatation of the extremal map, our
method uses iterations. Every step of the iteration
involves convex optimization and solving differential equations, and guarantees
a decrease in the dilatation. Our method uses a reduction of the polygon
mapping problem to that of the punctured sphere problem, thus solving a more
general problem.
We also discretize our procedure. We provide evidence for the fact that the
discrete procedure closely follows the continuous construction and is therefore
expected to converge quickly to a good approximation of the extremal
quasiconformal map.Comment: 28 pages, 6 figure
Supersymmetric quantum theory and (non-commutative) differential geometry
We reconsider differential geometry from the point of view of the quantum
theory of non-relativistic spinning particles, which provides examples of
supersymmetric quantum mechanics. This enables us to encode geometrical
structure in algebraic data consisting of an algebra of functions on a manifold
and a family of supersymmetry generators represented on a Hilbert space. We
show that known types of differential geometry can be classified in terms of
the supersymmetries they exhibit. Replacing commutative algebras of functions
by non-commutative *-algebras of operators, while retaining supersymmetry, we
arrive at a formulation of non-commutative geometry encompassing and extending
Connes' original approach. We explore different types of non-commutative
geometry and introduce notions of non-commutative manifolds and non-commutative
phase spaces. One of the main motivations underlying our work is to construct
mathematical tools for novel formulations of quantum gravity, in particular for
the investigation of superstring vacua.Comment: 125 pages, Plain TeX fil
Difference Problems and Differential Problems
We state some elementary problems concerning the relation between difference
calculus and differential calculus, and we try to convince the reader that, in
spite of the simplicity of the statements, a solution of these problems would
be a significant contribution to the understanding of the foundations of
differential and integral calculus
On the classical geometry of embedded surfaces in terms of Poisson brackets
We consider surfaces embedded in a Riemannian manifold of arbitrary dimension
and prove that many aspects of their differential geometry can be expressed in
terms of a Poisson algebraic structure on the space of smooth functions of the
surface. In particular, we find algebraic formulas for Weingarten's equations,
the complex structure and the Gaussian curvature
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