146 research outputs found
Infinite partition monoids
Let and be the partition monoid and symmetric
group on an infinite set . We show that may be generated by
together with two (but no fewer) additional partitions, and we
classify the pairs for which is
generated by . We also show that may be generated by the set of all idempotent partitions
together with two (but no fewer) additional partitions. In fact,
is generated by if and only if it is
generated by . We also
classify the pairs for which is
generated by . Among other results, we show
that any countable subset of is contained in a -generated
subsemigroup of , and that the length function on
is bounded with respect to any generating set
Overlap properties of geometric expanders
The {\em overlap number} of a finite -uniform hypergraph is
defined as the largest constant such that no matter how we map
the vertices of into , there is a point covered by at least a
-fraction of the simplices induced by the images of its hyperedges.
In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph
expansion for higher dimensional simplicial complexes, it was asked whether or
not there exists a sequence of arbitrarily large
-uniform hypergraphs with bounded degree, for which . Using both random methods and explicit constructions, we answer this
question positively by constructing infinite families of -uniform
hypergraphs with bounded degree such that their overlap numbers are bounded
from below by a positive constant . We also show that, for every ,
the best value of the constant that can be achieved by such a
construction is asymptotically equal to the limit of the overlap numbers of the
complete -uniform hypergraphs with vertices, as
. For the proof of the latter statement, we establish the
following geometric partitioning result of independent interest. For any
and any , there exists satisfying the
following condition. For any , for any point and
for any finite Borel measure on with respect to which
every hyperplane has measure , there is a partition into measurable parts of equal measure such that all but
at most an -fraction of the -tuples
have the property that either all simplices with
one vertex in each contain or none of these simplices contain
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
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
A Metric Discrepancy Result With Given Speed
It is known that the discrepancy DN{ kx} of the sequence { kx} satisfies NDN{ kx} = O((log N) (log log N) 1 + ε) a.e. for all ε> 0 , but not for ε= 0. For nk= θk, θ> 1 we have NDN{ nkx} ≦ (Σ θ+ ε) (2 Nlog log N) 1 / 2 a.e. for some 0 0 , but not for ε 0 , there exists a sequence { nk} of positive integers such that NDN{ nkx} ≦ (Σ + ε) Ψ (N) eventually holds a.e. for ε> 0 , but not for ε< 0. We also consider a similar problem on the growth of trigonometric sums. © 2016, Akadémiai Kiadó, Budapest, Hungary
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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