3,932 research outputs found
Derivations and Dirichlet forms on fractals
We study derivations and Fredholm modules on metric spaces with a local
regular conservative Dirichlet form. In particular, on finitely ramified
fractals, we show that there is a non-trivial Fredholm module if and only if
the fractal is not a tree (i.e. not simply connected). This result relates
Fredholm modules and topology, and refines and improves known results on p.c.f.
fractals. We also discuss weakly summable Fredholm modules and the Dixmier
trace in the cases of some finitely and infinitely ramified fractals (including
non-self-similar fractals) if the so-called spectral dimension is less than 2.
In the finitely ramified self-similar case we relate the p-summability question
with estimates of the Lyapunov exponents for harmonic functions and the
behavior of the pressure function.Comment: to appear in the Journal of Functional Analysis 201
Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals
We provide two methods for constructing smooth bump functions and for
smoothly cutting off smooth functions on fractals, one using a probabilistic
approach and sub-Gaussian estimates for the heat operator, and the other using
the analytic theory for p.c.f. fractals and a fixed point argument. The heat
semigroup (probabilistic) method is applicable to a more general class of
metric measure spaces with Laplacian, including certain infinitely ramified
fractals, however the cut off technique involves some loss in smoothness. From
the analytic approach we establish a Borel theorem for p.c.f. fractals, showing
that to any prescribed jet at a junction point there is a smooth function with
that jet. As a consequence we prove that on p.c.f. fractals smooth functions
may be cut off with no loss of smoothness, and thus can be smoothly decomposed
subordinate to an open cover. The latter result provides a replacement for
classical partition of unity arguments in the p.c.f. fractal setting.Comment: 26 pages. May differ slightly from published (refereed) versio
Residual acceleration data on IML-1: Development of a data reduction and dissemination plan
A residual acceleration data analysis plan is developed that will allow principal investigators of low-gravity experiments to efficiently process their experimental results in conjunction with accelerometer data. The basic approach consisted of the following program of research: (1) identification of sensitive experiments and sensitivity ranges by order of magnitude estimates, numerical modelling, and investigator input; (2) research and development towards reduction, supplementation, and dissemination of residual acceleration data; and (3) implementation of the plan on existing acceleration data bases
Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates
With a view toward fractal spaces, by using a Korevaar-Schoen space approach,
we introduce the class of bounded variation (BV) functions in a general
framework of strongly local Dirichlet spaces with a heat kernel satisfying
sub-Gaussian estimates. Under a weak Bakry-\'Emery curvature type condition,
which is new in this setting, this BV class is identified with a heat semigroup
based Besov class. As a consequence of this identification, properties of BV
functions and associated BV measures are studied in detail. In particular, we
prove co-area formulas, global Sobolev embeddings and isoperimetric
inequalities. It is shown that for nested fractals or their direct products the
BV class we define is dense in . The examples of the unbounded Vicsek set,
unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.Comment: The notes arXiv:1806.03428 will be divided in a series of papers.
This is the third paper. v2: Final versio
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