1,165 research outputs found
Complexity for extended dynamical systems
We consider dynamical systems for which the spatial extension plays an
important role. For these systems, the notions of attractor, epsilon-entropy
and topological entropy per unit time and volume have been introduced
previously. In this paper we use the notion of Kolmogorov complexity to
introduce, for extended dynamical systems, a notion of complexity per unit time
and volume which plays the same role as the metric entropy for classical
dynamical systems. We introduce this notion as an almost sure limit on orbits
of the system. Moreover we prove a kind of variational principle for this
complexity.Comment: 29 page
A Numerical Study of the Hierarchical Ising Model: High Temperature Versus Epsilon Expansion
We study numerically the magnetic susceptibility of the hierarchical model
with Ising spins () above the critical temperature and for two
values of the epsilon parameter. The integrations are performed exactly, using
recursive methods which exploit the symmetries of the model. Lattices with up
to sites have been used. Surprisingly, the numerical data can be fitted
very well with a simple power law of the form for the {\it whole} temperature range. The numerical values for
agree within a few percent with the values calculated with a high-temperature
expansion but show significant discrepancies with the epsilon-expansion. We
would appreciate comments about these results.Comment: 15 Pages, 12 Figures not included (hard copies available on request),
uses phyzzx.te
Evidence for Complex Subleading Exponents from the High-Temperature Expansion of the Hierarchical Ising Model
Using a renormalization group method, we calculate 800 high-temperature
coefficients of the magnetic susceptibility of the hierarchical Ising model.
The conventional quantities obtained from differences of ratios of coefficients
show unexpected smooth oscillations with a period growing logarithmically and
can be fitted assuming corrections to the scaling laws with complex exponents.Comment: 10 pages, Latex , uses revtex. 2 figures not included (hard copies
available on request
A max-type recursive model: some properties and open questions
We consider a simple max-type recursive model which was introduced in the
study of depinning transition in presence of strong disorder, by Derrida and
Retaux. Our interest is focused on the critical regime, for which we study the
extinction probability, the first moment and the moment generating function.
Several stronger assertions are stated as conjectures.Comment: A version accepted to Charles Newman Festschrift (to appear by
Springer
A Guide to Precision Calculations in Dyson's Hierarchical Scalar Field Theory
The goal of this article is to provide a practical method to calculate, in a
scalar theory, accurate numerical values of the renormalized quantities which
could be used to test any kind of approximate calculation. We use finite
truncations of the Fourier transform of the recursion formula for Dyson's
hierarchical model in the symmetric phase to perform high-precision
calculations of the unsubtracted Green's functions at zero momentum in
dimension 3, 4, and 5. We use the well-known correspondence between statistical
mechanics and field theory in which the large cut-off limit is obtained by
letting beta reach a critical value beta_c (with up to 16 significant digits in
our actual calculations). We show that the round-off errors on the magnetic
susceptibility grow like (beta_c -beta)^{-1} near criticality. We show that the
systematic errors (finite truncations and volume) can be controlled with an
exponential precision and reduced to a level lower than the numerical errors.
We justify the use of the truncation for calculations of the high-temperature
expansion. We calculate the dimensionless renormalized coupling constant
corresponding to the 4-point function and show that when beta -> beta_c, this
quantity tends to a fixed value which can be determined accurately when D=3
(hyperscaling holds), and goes to zero like (Ln(beta_c -beta))^{-1} when D=4.Comment: Uses revtex with psfig, 31 pages including 15 figure
Thermodynamic Limit Of The Ginzburg-Landau Equations
We investigate the existence of a global semiflow for the complex
Ginzburg-Landau equation on the space of bounded functions in unbounded domain.
This semiflow is proven to exist in dimension 1 and 2 for any parameter values
of the standard cubic Ginzburg-Landau equation. In dimension 3 we need some
restrictions on the parameters but cover nevertheless some part of the
Benjamin-Feijer unstable domain.Comment: uuencoded dvi file (email: [email protected]
The Oscillatory Behavior of the High-Temperature Expansion of Dyson's Hierarchical Model: A Renormalization Group Analysis
We calculate 800 coefficients of the high-temperature expansion of the
magnetic susceptibility of Dyson's hierarchical model with a Landau-Ginzburg
measure. Log-periodic corrections to the scaling laws appear as in the case of
a Ising measure. The period of oscillation appears to be a universal quantity
given in good approximation by the logarithm of the largest eigenvalue of the
linearized RG transformation, in agreement with a possibility suggested by K.
Wilson and developed by Niemeijer and van Leeuwen. We estimate to be
1.300 (with a systematic error of the order of 0.002) in good agreement with
the results obtained with other methods such as the -expansion. We
briefly discuss the relationship between the oscillations and the zeros of the
partition function near the critical point in the complex temperature plane.Comment: 21 pages, 10 Postcript figures, latex file, uses revte
SYNTAX II and SYNTAX III trials: What is the take home message for surgeons?
Percutaneous coronary intervention (PCI) has evolved greatly in the last 40 years since its introduction by Andreas Grüntzig in 1977. Since then, we've observed an evolution in balloons, the development of stents, changes in stent structure, development of drug eluting stents, improvements in strut design, thickness and even their polymeric coating. Most recently we saw the rise and "fall" of bioabsorbable scaffolds for PCI. Trials with the most diverse devices for PCI and diagnostic techniques have been conducted. Two of the most recent trials were reported in the last year and deserve special attention-SYNTAX II and SYNTAX III. These trials are completely different in design but present valuable information for doctors managing coronary artery disease (CAD). Both trials take into account contemporary technology for assessing and treating CAD. The first uses so-called "state-of-the-art" PCI and compares the outcomes of that approach with the outcomes of the PCI arm of the pivotal SYNTAX trial. SYNTAX III Revolution on the other hand does not focus on clinical endpoints: it is a blinded trial that does not randomize patients but randomizes doctors ("the heart team") to make a decision on the best treatment for complex CAD. This decision was based either on multi-slice CT with physiological assessment using FFRCT or on conventional angiography. In this review we bring the most important aspects of those trials and the key messages for surgeons together; also, what the surgeon may expect in the future after the publication of these interesting concepts
High-Accuracy Calculations of the Critical Exponents of Dyson's Hierarchical Model
We calculate the critical exponent gamma of Dyson's hierarchical model by
direct fits of the zero momentum two-point function, calculated with an Ising
and a Landau-Ginzburg measure, and by linearization about the Koch-Wittwer
fixed point. We find gamma= 1.299140730159 plus or minus 10^(-12). We extract
three types of subleading corrections (in other words, a parametrization of the
way the two-point function depends on the cutoff) from the fits and check the
value of the first subleading exponent from the linearized procedure. We
suggest that all the non-universal quantities entering the subleading
corrections can be calculated systematically from the non-linear contributions
about the fixed point and that this procedure would provide an alternative way
to introduce the bare parameters in a field theory model.Comment: 15 pages, 9 figures, uses revte
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