92 research outputs found
Restoration of isotropy on fractals
We report a new type of restoration of macroscopic isotropy (homogenization)
in fractals with microscopic anisotropy. The phenomenon is observed in various
physical setups, including diffusions, random walks, resistor networks, and
Gaussian field theories. The mechanism is unique in that it is absent in spaces
with translational invariance, while universal in that it is observed in a wide
class of fractals.Comment: 11 pages, REVTEX, 3 postscript figures. (Compressed and encoded
figures archived by "figure" command). To appear in Physical Review Letter
Some results on embeddings of algebras, after de Bruijn and McKenzie
In 1957, N. G. de Bruijn showed that the symmetric group Sym(\Omega) on an
infinite set \Omega contains a free subgroup on 2^{card(\Omega)} generators,
and proved a more general statement, a sample consequence of which is that for
any group A of cardinality \leq card(\Omega), Sym(\Omega) contains a coproduct
of 2^{card(\Omega)} copies of A, not only in the variety of all groups, but in
any variety of groups to which A belongs. His key lemma is here generalized to
an arbitrary variety of algebras \bf{V}, and formulated as a statement about
functors Set --> \bf{V}. From this one easily obtains analogs of the results
stated above with "group" and Sym(\Omega) replaced by "monoid" and the monoid
Self(\Omega) of endomaps of \Omega, by "associative K-algebra" and the
K-algebra End_K(V) of endomorphisms of a K-vector-space V with basis \Omega,
and by "lattice" and the lattice Equiv(\Omega) of equivalence relations on
\Omega. It is also shown, extending another result from de Bruijn's 1957 paper,
that each of Sym(\Omega), Self(\Omega) and End_K (V) contains a coproduct of
2^{card(\Omega)} copies of itself.
That paper also gave an example of a group of cardinality 2^{card(\Omega)}
that was {\em not} embeddable in Sym(\Omega), and R. McKenzie subsequently
established a large class of such examples. Those results are shown to be
instances of a general property of the lattice of solution sets in Sym(\Omega)
of sets of equations with constants in Sym(\Omega). Again, similar results --
this time of varying strengths -- are obtained for Self(\Omega), End_K (V), and
Equiv(\Omega), and also for the monoid \Rel of binary relations on \Omega.
Many open questions and areas for further investigation are noted.Comment: 37 pages. Copy at http://math.berkeley.edu/~gbergman/papers is likely
to be updated more often than arXiv copy Revised version includes answers to
some questions left open in first version, references to results of Wehrung
answering some other questions, and some additional new result
Rigidity and Non-recurrence along Sequences
Two properties of a dynamical system, rigidity and non-recurrence, are
examined in detail. The ultimate aim is to characterize the sequences along
which these properties do or do not occur for different classes of
transformations. The main focus in this article is to characterize explicitly
the structural properties of sequences which can be rigidity sequences or
non-recurrent sequences for some weakly mixing dynamical system. For ergodic
transformations generally and for weakly mixing transformations in particular
there are both parallels and distinctions between the class of rigid sequences
and the class of non-recurrent sequences. A variety of classes of sequences
with various properties are considered showing the complicated and rich
structure of rigid and non-recurrent sequences
Electronic shot noise in fractal conductors
By solving a master equation in the Sierpinski lattice and in a planar
random-resistor network, we determine the scaling with size L of the shot noise
power P due to elastic scattering in a fractal conductor. We find a power-law
scaling P ~ L^(d_f-2-alpha), with an exponent depending on the fractal
dimension d_f and the anomalous diffusion exponent alpha. This is the same
scaling as the time-averaged current I, which implies that the Fano factor
F=P/2eI is scale independent. We obtain a value F=1/3 for anomalous diffusion
that is the same as for normal diffusion, even if there is no smallest length
scale below which the normal diffusion equation holds. The fact that F remains
fixed at 1/3 as one crosses the percolation threshold in a random-resistor
network may explain recent measurements of a doping-independent Fano factor in
a graphene flake.Comment: 6 pages, 3 figure
The universal Glivenko-Cantelli property
Let F be a separable uniformly bounded family of measurable functions on a
standard measurable space, and let N_{[]}(F,\epsilon,\mu) be the smallest
number of \epsilon-brackets in L^1(\mu) needed to cover F. The following are
equivalent:
1. F is a universal Glivenko-Cantelli class.
2. N_{[]}(F,\epsilon,\mu)0 and every probability
measure \mu.
3. F is totally bounded in L^1(\mu) for every probability measure \mu.
4. F does not contain a Boolean \sigma-independent sequence.
It follows that universal Glivenko-Cantelli classes are uniformity classes
for general sequences of almost surely convergent random measures.Comment: 26 page
Remarks on the Cauchy functional equation and variations of it
This paper examines various aspects related to the Cauchy functional equation
, a fundamental equation in the theory of functional
equations. In particular, it considers its solvability and its stability
relative to subsets of multi-dimensional Euclidean spaces and tori. Several new
types of regularity conditions are introduced, such as a one in which a complex
exponent of the unknown function is locally measurable. An initial value
approach to analyzing this equation is considered too and it yields a few
by-products, such as the existence of a non-constant real function having an
uncountable set of periods which are linearly independent over the rationals.
The analysis is extended to related equations such as the Jensen equation, the
multiplicative Cauchy equation, and the Pexider equation. The paper also
includes a rather comprehensive survey of the history of the Cauchy equation.Comment: To appear in Aequationes Mathematicae (important remark: the
acknowledgments section in the official paper exists, but it appears before
the appendix and not before the references as in the arXiv version);
correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of
Theorem 2.1; a few small improvements in various sections; added thank
Self-Organization and Complex Networks
In this chapter we discuss how the results developed within the theory of
fractals and Self-Organized Criticality (SOC) can be fruitfully exploited as
ingredients of adaptive network models. In order to maintain the presentation
self-contained, we first review the basic ideas behind fractal theory and SOC.
We then briefly review some results in the field of complex networks, and some
of the models that have been proposed. Finally, we present a self-organized
model recently proposed by Garlaschelli et al. [Nat. Phys. 3, 813 (2007)] that
couples the fitness network model defined by Caldarelli et al. [Phys. Rev.
Lett. 89, 258702 (2002)] with the evolution model proposed by Bak and Sneppen
[Phys. Rev. Lett. 71, 4083 (1993)] as a prototype of SOC. Remarkably, we show
that the results obtained for the two models separately change dramatically
when they are coupled together. This indicates that self-organized networks may
represent an entirely novel class of complex systems, whose properties cannot
be straightforwardly understood in terms of what we have learnt so far.Comment: Book chapter in "Adaptive Networks: Theory, Models and Applications",
Editors: Thilo Gross and Hiroki Sayama (Springer/NECSI Studies on Complexity
Series
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