8 research outputs found
Averaged shelling for quasicrystals
The shelling of crystals is concerned with counting the number of atoms on
spherical shells of a given radius and a fixed centre. Its straight-forward
generalization to quasicrystals, the so-called central shelling, leads to
non-universal answers. As one way to cope with this situation, we consider
shelling averages over all quasicrystal points. We express the averaged
shelling numbers in terms of the autocorrelation coefficients and give explicit
results for the usual suspects, both perfect and random.Comment: 4 pages, several figures, 2 tables; updated version with minor
corrections and improvements; to appear in the proceedings of ICQ
Modeling of waiting times and price changes in currency exchange data
A theory which describes the share price evolution at financial markets as a
continuous-time random walk has been generalized in order to take into account
the dependence of waiting times t on price returns x. A joint probability
density function (pdf) which uses the concept of a L\'{e}vy stable distribution
is worked out. The theory is fitted to high-frequency US$/Japanese Yen exchange
rate and low-frequency 19th century Irish stock data. The theory has been
fitted both to price return and to waiting time data and the adherence to data,
in terms of the chi-squared test statistic, has been improved when compared to
the old theory.Comment: 22 pages, 5 postscript figures, LaTeX2e using elsart.cl
Dynamics of Money and Income Distributions
We study the model of interacting agents proposed by Chatterjee et al that
allows agents to both save and exchange wealth. Closed equations for the wealth
distribution are developed using a mean field approximation. We show that when
all agents have the same fixed savings propensity, subject to certain well
defined approximations defined in the text, these equations yield the
conjecture proposed by Chatterjee for the form of the stationary agent wealth
distribution. If the savings propensity for the equations is chosen according
to some random distribution we show further that the wealth distribution for
large values of wealth displays a Pareto like power law tail, ie P(w)\sim
w^{1+a}. However the value of for the model is exactly 1. Exact numerical
simulations for the model illustrate how, as the savings distribution function
narrows to zero, the wealth distribution changes from a Pareto form to to an
exponential function. Intermediate regions of wealth may be approximately
described by a power law with . However the value never reaches values of
\~ 1.6-1.7 that characterise empirical wealth data. This conclusion is not
changed if three body agent exchange processes are allowed. We conclude that
other mechanisms are required if the model is to agree with empirical wealth
data.Comment: Sixteen pages, Seven figures, Elsevier style file. Submitted to
Physica
Exact Eigenstates of Tight-Binding Hamiltonians on the Penrose Tiling
We investigate exact eigenstates of tight-binding models on the planar
rhombic Penrose tiling. We consider a vertex model with hopping along the edges
and the diagonals of the rhombi. For the wave functions, we employ an ansatz,
first introduced by Sutherland, which is based on the arrow decoration that
encodes the matching rules of the tiling. Exact eigenstates are constructed for
particular values of the hopping parameters and the eigenenergy. By a
generalized ansatz that exploits the inflation symmetry of the tiling, we show
that the corresponding eigenenergies are infinitely degenerate. Generalizations
and applications to other systems are outlined.Comment: 24 pages, REVTeX, 13 PostScript figures include
Option pricing with log-stable L\'{e}vy processes
We model the logarithm of the price (log-price) of a financial asset as a random variable obtained by projecting an operator stable random vector with a scaling index matrix onto a non-random vector. The scaling index models prices of the individual financial assets (stocks, mutual funds, etc.). We find the functional form of the characteristic function of real powers of the price returns and we compute the expectation value of these real powers and we speculate on the utility of these results for statistical inference. Finally we consider a portfolio composed of an asset and an option on that asset. We derive the characteristic function of the deviation of the portfolio, \mbox{}, defined as a temporal change of the portfolio diminished by the the compound interest earned. We derive pseudo-differential equations for the option as a function of the log-stock-price and time and we find exact closed-form solutions to that equation. These results were not known before. Finally we discuss how our solutions correspond to other approximate results known from literature,in particular to the well known Black & Scholes equation.
Planar quasiperiodic Ising models
We investigate zero-field Ising models on periodic approximants of planar quasiperiodic tilings by means of partition function zeros and high-temperature expansions. These are obtained by employing a determinant expression for the partition function. The partition function zeros in the complex temperature plane yield precise estimates of the critical temperature of the quasiperiodic model. Concerning the critical behaviour, our results are compatible with Onsager universality, in agreement with the Harris–Luck criterion based on scaling arguments