2,612 research outputs found
Laplace Operators on Fractals and Related Functional Equations
We give an overview over the application of functional equations, namely the
classical Poincar\'e and renewal equations, to the study of the spectrum of
Laplace operators on self-similar fractals. We compare the techniques used to
those used in the euclidean situation. Furthermore, we use the obtained
information on the spectral zeta function to define the Casimir energy of
fractals. We give numerical values for this energy for the Sierpi\'nski gasket
Reversible boolean networks II: Phase transition, oscillation, and local structures
We continue our consideration of a class of models describing the reversible
dynamics of Boolean variables, each with inputs. We investigate in
detail the behavior of the Hamming distance as well as of the distribution of
orbit lengths as and are varied. We present numerical evidence for a
phase transition in the behavior of the Hamming distance at a critical value
and also an analytic theory that yields the exact bounds on
We also discuss the large oscillations that we observe in the Hamming
distance for as a function of time as well as in the distribution of
cycle lengths as a function of cycle length for moderate both greater than
and less than . We propose that local structures, or subsets of spins
whose dynamics are not fully coupled to the other spins in the system, play a
crucial role in generating these oscillations. The simplest of these structures
are linear chains, called linkages, and rings, called circuits. We discuss the
properties of the linkages in some detail, and sketch the properties of
circuits. We argue that the observed oscillation phenomena can be largely
understood in terms of these local structures.Comment: 31 pages, 15 figures, 2 table
Hidden structure in the randomness of the prime number sequence?
We report a rigorous theory to show the origin of the unexpected periodic
behavior seen in the consecutive differences between prime numbers. We also
check numerically our findings to ensure that they hold for finite sequences of
primes, that would eventually appear in applications. Finally, our theory
allows us to link with three different but important topics: the
Hardy-Littlewood conjecture, the statistical mechanics of spin systems, and the
celebrated Sierpinski fractal.Comment: 13 pages, 5 figures. New section establishing connection with the
Hardy-Littlewood theory. Published in the journal where the solved problem
was first describe
On the selection of materials for cryogenic seals and the testing of their performance
Three questions are addressed: what mission must a cryogenic seal perform; what are the contrasts between desirable and available seal materials; and how realistic must test conditions be. The question of how to quantify the response of a material subject to large strains and which is susceptible to memory effects leads to a discussion of theoretical issues. Accordingly, the report summarizes some ideas from the rational mechanics of materials. The report ends with a list of recommendations and a conclusion
Optimal Vertex Cover for the Small-World Hanoi Networks
The vertex-cover problem on the Hanoi networks HN3 and HN5 is analyzed with
an exact renormalization group and parallel-tempering Monte Carlo simulations.
The grand canonical partition function of the equivalent hard-core repulsive
lattice-gas problem is recast first as an Ising-like canonical partition
function, which allows for a closed set of renormalization group equations. The
flow of these equations is analyzed for the limit of infinite chemical
potential, at which the vertex-cover problem is attained. The relevant fixed
point and its neighborhood are analyzed, and non-trivial results are obtained
both, for the coverage as well as for the ground state entropy density, which
indicates the complex structure of the solution space. Using special
hierarchy-dependent operators in the renormalization group and Monte-Carlo
simulations, structural details of optimal configurations are revealed. These
studies indicate that the optimal coverages (or packings) are not related by a
simple symmetry. Using a clustering analysis of the solutions obtained in the
Monte Carlo simulations, a complex solution space structure is revealed for
each system size. Nevertheless, in the thermodynamic limit, the solution
landscape is dominated by one huge set of very similar solutions.Comment: RevTex, 24 pages; many corrections in text and figures; final
version; for related information, see
http://www.physics.emory.edu/faculty/boettcher
Fractal Tube Formulas for Compact Sets and Relative Fractal Drums: Oscillations, Complex Dimensions and Fractality
We establish pointwise and distributional fractal tube formulas for a large
class of relative fractal drums in Euclidean spaces of arbitrary dimensions. A
relative fractal drum (or RFD, in short) is an ordered pair of
subsets of the Euclidean space (under some mild assumptions) which generalizes
the notion of a (compact) subset and that of a fractal string. By a fractal
tube formula for an RFD , we mean an explicit expression for the
volume of the -neighborhood of intersected by as a sum of
residues of a suitable meromorphic function (here, a fractal zeta function)
over the complex dimensions of the RFD . The complex dimensions of
an RFD are defined as the poles of its meromorphically continued fractal zeta
function (namely, the distance or the tube zeta function), which generalizes
the well-known geometric zeta function for fractal strings. These fractal tube
formulas generalize in a significant way to higher dimensions the corresponding
ones previously obtained for fractal strings by the first author and van
Frankenhuijsen and later on, by the first author, Pearse and Winter in the case
of fractal sprays. They are illustrated by several interesting examples. These
examples include fractal strings, the Sierpi\'nski gasket and the 3-dimensional
carpet, fractal nests and geometric chirps, as well as self-similar fractal
sprays. We also propose a new definition of fractality according to which a
bounded set (or RFD) is considered to be fractal if it possesses at least one
nonreal complex dimension or if its fractal zeta function possesses a natural
boundary. This definition, which extends to RFDs and arbitrary bounded subsets
of the previous one introduced in the context of fractal
strings, is illustrated by the Cantor graph (or devil's staircase) RFD, which
is shown to be `subcritically fractal'.Comment: 90 pages (because of different style file), 5 figures, corrected
typos, updated reference
Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator
A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] by
the second author and H. Maier in terms of an inverse spectral problem for
fractal strings. This problem is related to the question "Can one hear the
shape of a fractal drum?" and was shown in [LaMa2] to have a positive answer
for fractal strings whose dimension is c\in(0,1)-\{1/2} if and only if the
Riemann hypothesis is true. Later on, the spectral operator was introduced
heuristically by M. L. Lapidus and M. van Frankenhuijsen in their theory of
complex fractal dimensions [La-vF2, La-vF3] as a map that sends the geometry of
a fractal string onto its spectrum. We focus here on presenting the rigorous
results obtained by the authors in [HerLa1] about the invertibility of the
spectral operator. We show that given any , the spectral operator
, now precisely defined as an unbounded normal
operator acting in a Hilbert space , is `quasi-invertible'
(i.e., its truncations are invertible) if and only if the Riemann zeta function
does not have any zeroes on the line . It follows
that the associated inverse spectral problem has a positive answer for all
possible dimensions , other than the mid-fractal case when ,
if and only if the Riemann hypothesis is true.Comment: To appear in: "Fractal Geometry and Dynamical Systems in Pure and
Applied Mathematics", Part 1 (D. Carfi, M. L. Lapidus, E. P. J. Pearse and M.
van Frankenhuijsen, eds.), Contemporary Mathematics, Amer. Math. Soc.,
Providence, RI, 2013. arXiv admin note: substantial text overlap with
arXiv:1203.482
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