571 research outputs found
Dirac operators and spectral triples for some fractal sets built on curves
We construct spectral triples and, in particular, Dirac operators, for the
algebra of continuous functions on certain compact metric spaces. The triples
are countable sums of triples where each summand is based on a curve in the
space. Several fractals, like a finitely summable infinite tree and the
Sierpinski gasket, fit naturally within our framework. In these cases, we show
that our spectral triples do describe the geodesic distance and the Minkowski
dimension as well as, more generally, the complex fractal dimensions of the
space. Furthermore, in the case of the Sierpinski gasket, the associated
Dixmier-type trace coincides with the normalized Hausdorff measure of dimension
.Comment: 48 pages, 4 figures. Elementary proofs omitted. To appear in Adv.
Mat
Improved Convergence Proof of the Delta Expansion and Order Dependent Mappings
We improve and generalize in several accounts the recent rigorous proof of
convergence of delta expansion - order dependent mappings (variational
perturbation expansion) for the energy eigenvalues of anharmonic oscillator.
For the single-well anharmonic oscillator the uniformity of convergence in
is proven. The convergence proof is extended also to complex
values of lying on a wide domain of the Riemann surface of . Via the
scaling relation \`a la Symanzik, this proves the convergence of delta
expansion for the double well in the strong coupling regime (where the standard
perturbation series is non Borel summable), as well as for the complex ``energy
eigenvalues'' in certain metastable potentials. Sufficient conditions for the
convergence of delta expansion are summarized in the form of three theorems,
which should apply to a wide class of quantum mechanical and higher dimensional
field theoretic systems.Comment: some bugs of uuencoded postscript figures are fixe
A walk in the noncommutative garden
This text is written for the volume of the school/conference "Noncommutative
Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in
noncommutative geometry, based on a discussion of significant examples of
noncommutative spaces in geometry, number theory, and physics. The paper also
contains an outline (the ``Tehran program'') of ongoing joint work with Consani
on the noncommutative geometry of the adeles class space and its relation to
number theoretic questions.Comment: 106 pages, LaTeX, 23 figure
Constrained -approximation by polynomials on subsets of the circle
We study best approximation to a given function, in the least square sense on
a subset of the unit circle, by polynomials of given degree which are pointwise
bounded on the complementary subset. We show that the solution to this problem,
as the degree goes large, converges to the solution of a bounded extremal
problem for analytic functions which is instrumental in system identification.
We provide a numerical example on real data from a hyperfrequency filter
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