Improved Smoothing Algorithms for Lattice Gauge Theory

The relative smoothing rates of various gauge field smoothing algorithms are investigated on ${\cal O}(a^2)$-improved \suthree Yang--Mills gauge field configurations. In particular, an ${\cal O}(a^2)$-improved version of APE smearing is motivated by considerations of smeared link projection and cooling. The extent to which the established benefits of improved cooling carry over to improved smearing is critically examined. We consider representative gauge field configurations generated with an ${\cal O}(a^2)$-improved gauge field action on \1 lattices at $\beta=4.38$ and \2 lattices at $\beta=5.00$ having lattice spacings of 0.165(2) fm and 0.077(1) fm respectively. While the merits of improved algorithms are clearly displayed for the coarse lattice spacing, the fine lattice results put the various algorithms on a more equal footing and allow a quantitative calibration of the smoothing rates for the various algorithms. We find the relative rate of variation in the action may be succinctly described in terms of simple calibration formulae which accurately describe the relative smoothness of the gauge field configurations at a microscopic level

Gluons, quarks, and the transition from nonperturbative to perturbative QCD

Lattice-based investigations of two fundamental QCD quantities are described, namely the gluon and quark propagators in Landau gauge. We have studied the Landau gauge gluon propagator using a variety of lattices with spacings from a = 0.17 to 0.41 fm. We demonstrate that it is possible to obtain scaling behavior over a very wide range of momenta and lattice spacings and to explore the infinite volume and continuum limits. These results confirm that the Landau gauge gluon propagator is infrared finite. We study the Landau gauge quark propagator in quenched QCD using two forms of the O(a)-improved propagator and we find good agreement between these. The extracted value of the infrared quark mass in the chiral limit is found to be 300 +/- 30 MeV. We conclude that the momentum regime where the transition from nonperturbative to perturbative QCD occurs is Q^2 approx 4GeV^2.Comment: 8 pages, 6 figures, 1 table. Talk presented by AGW at the Workshop on Lepton Scattering, Hadrons and QCD, March 26-April 5, 2001, CSSM, Adelaide, Australia. To appear in the proceeding

World Health Organization cardiovascular disease risk charts: revised models to estimate risk in 21 global regions

BACKGROUND: To help adapt cardiovascular disease risk prediction approaches to low-income and middle-income countries, WHO has convened an effort to develop, evaluate, and illustrate revised risk models. Here, we report the derivation, validation, and illustration of the revised WHO cardiovascular disease risk prediction charts that have been adapted to the circumstances of 21 global regions. METHODS: In this model revision initiative, we derived 10-year risk prediction models for fatal and non-fatal cardiovascular disease (ie, myocardial infarction and stroke) using individual participant data from the Emerging Risk Factors Collaboration. Models included information on age, smoking status, systolic blood pressure, history of diabetes, and total cholesterol. For derivation, we included participants aged 40-80 years without a known baseline history of cardiovascular disease, who were followed up until the first myocardial infarction, fatal coronary heart disease, or stroke event. We recalibrated models using age-specific and sex-specific incidences and risk factor values available from 21 global regions. For external validation, we analysed individual participant data from studies distinct from those used in model derivation. We illustrated models by analysing data on a further 123 743 individuals from surveys in 79 countries collected with the WHO STEPwise Approach to Surveillance. FINDINGS: Our risk model derivation involved 376 177 individuals from 85 cohorts, and 19 333 incident cardiovascular events recorded during 10 years of follow-up. The derived risk prediction models discriminated well in external validation cohorts (19 cohorts, 1 096 061 individuals, 25 950 cardiovascular disease events), with Harrell's C indices ranging from 0·685 (95% CI 0·629-0·741) to 0·833 (0·783-0·882). For a given risk factor profile, we found substantial variation across global regions in the estimated 10-year predicted risk. For example, estimated cardiovascular disease risk for a 60-year-old male smoker without diabetes and with systolic blood pressure of 140 mm Hg and total cholesterol of 5 mmol/L ranged from 11% in Andean Latin America to 30% in central Asia. When applied to data from 79 countries (mostly low-income and middle-income countries), the proportion of individuals aged 40-64 years estimated to be at greater than 20% risk ranged from less than 1% in Uganda to more than 16% in Egypt. INTERPRETATION: We have derived, calibrated, and validated new WHO risk prediction models to estimate cardiovascular disease risk in 21 Global Burden of Disease regions. The widespread use of these models could enhance the accuracy, practicability, and sustainability of efforts to reduce the burden of cardiovascular disease worldwide. FUNDING: World Health Organization, British Heart Foundation (BHF), BHF Cambridge Centre for Research Excellence, UK Medical Research Council, and National Institute for Health Research

Option pricing using path integrals.

It is well established that stock market volatility has a memory of the past, moreover it is found that volatility correlations are long ranged. As a consequence, volatility cannot be characterized by a single correlation time in general. Recent empirical work suggests that the volatility correlation functions of various assets actually decay as a power law. Moreover it is well established that the distribution functions for the returns do not obey a Gaussian distribution, but follow more the type of distributions that incorporate what are commonly known as fat–tailed distributions. As a result, if one is to model the evolution of the stock price, stock market or any financial derivative, then standard Brownian motion models are inaccurate. One must take into account the results obtained from empirical studies and work with models that include realistic features observed on the market. In this thesis we show that it is possible to derive the path integral for a non-Gaussian option pricing model that can capture fat–tails. However we find that the path integral technique can only be used on a very small set of problems, as a number of situations of interest are shown to be intractable.Thesis (Ph.D.) -- University of Adelaide, School of Electrical and Electronic Engineering, 201

Improved actions in lattice QCD.

In this thesis I explore the physical effects of improved actions combined with improved operators in the framework of lattice QCD. All calculations are done in the quenched approximation, that is, when all of the dynamical fermion interactions have been suppressed by setting the determinant of the fermion matrix to a constant. The thesis first briefly introduces lattice QCD to familiarize the reader with the basic concepts. It then describes the common numerical procedures used. It is made up of three major sections. The first is the exploration of gauge field configurations and the study of the role of instantons in lattice QCD. In this work the Wilson gauge action and a standard 1 loop topological charge operator are used to determine the relative rates of standard cooling and smearing algorithms in pure SUc(3)-color gauge theory. I consider representative gauge field configurations on 16³ × 32 lattices at β = 5.70 and 24³ × 36 lattices at β = 6.00. I find the relative rate of variation in the action and topological charge under various algorithms may be succinctly described in terms of simple formulae ¹. The results are in accord with recent suggestions from fat-link perturbation theory. This work is then extended to O(a²)-improved gauge action and O(a²)-improved operators ². In particular, an O(a²)-improved version of APE smearing is motivated by considerations of smeared link projection and cooling. The extent to which the established benefits of improved cooling carry over to improved smearing is critically examined. I consider representative gauge field configurations generated with an O(a²)-improved gauge field action on 16³ × 32 lattices at β = 4.38 and 24³ × 36 lattices at β = 5.00 having lattice spacings of 0.165(2) fm and 0.077(1) fm respectively. While the merits of improved algorithms are clearly displayed for the coarse lattice spacing, the fine lattice results put the various algorithms on a more equal footing and allow a quantitative calibration of the smoothing rates for the various algorithms. I find that the relative rate of variation in the action may also be described in terms of simple calibration formulae for O(a²)-improvement which accurately describes the relative smoothness of the gauge field configurations at a microscopic level. In the second section the first calculation of the gluon propagator using an O(a²)- improved action with the corresponding O(a²)-improved Landau gauge fixing ³ condition is presented ⁴. The gluon propagator obtained from the improved action and improved Landau gauge condition is compared with earlier unimproved results on similar physical lattice volumes of 3.2³ × 6.4⁴ fm. It is found that there is good agreement between the improved propagator calculated on a coarse lattice with lattice spacing a = 0.35 fm and the unimproved propagator calculated on a fine lattice with spacing a = 0.10 fm. This motivated us to calculate the gluon propagator on a coarse very large-volume lattice of 5.6³ × 11.2⁴fm. The infrared behavior observed in previous studies is confirmed. The gluon propagator is enhanced at intermediate momenta and suppressed at infrared momenta. The observed infrared suppression of the Landau gauge gluon propagator is not a finite volume effect. This work is then extended to a variety of lattices with spacing ranging from a = 0.17 to a = 0.4 fm ⁵ to further explore finite volume and discretization effects. In this work a technique previously used for minimizing lattice artifacts, known as “tree-level correction”, has also been extended. It is demonstrated that by using tree-level correction, determined by the tree-level behavior of the action being considered, it is possible to obtain scaling behavior over a very wide range of momenta and lattice spacings. This makes it possible to explore the infinite volume and continuum limits of the Landau-gauge gluon propagator. As a final part of this thesis I present the first results for the quark propagator using an Overlap fermionic quark action ⁶. I compare the results with those obtained from the standard Wilson fermion. The overlap quark action is O(a)-improved compared with the Wilson fermion. This action realizes exact chiral symmetry on the lattice unlike the Wilson fermion and it demonstrates that the fastest way forward in this field is with improved lattice operators. The idea of studying improved actions in lattice gauge theory was suggested to me by A/Prof. Anthony G. Williams during the “Nonperturbative Methods in Quantum Field Theory” workshop in early February 1998. Initially it was suggested to me that a calculation of the gluon propagator using improved action on large volumes, following a study just done with standard gauge action in Ref. [62]. The point of interest was to study the effect an improved gauge field action would have on the gluon propagator. This study would then be extended to quark actions. In the meantime when generating gauge field configurations using a computer code written in Fortran 77 (provided by Dr. Derek B. Leinweber), it occurred to me that it would be good to explore the content of these gauge field configurations. In order to do realistic calculations on large lattices we needed a gauge field configuration generator that would run on our CM5 computer and so Connection Machine Fortran (CMF) became the adopted language. I started writing the computer code to generate the gauge field configuration in the SUc(2) with the help of Dr. Derek B. Leinweber, who introduced me to the basic concepts in lattice QCD. I then extended this code to the SUc(3) gauge group. This is commonly known as the standard Wilson gauge action. After investigating with some of the optimization possibilities, I moved on to code an O(a²)-improved gauge action. The code uses a masking procedure for the link update. I have generalized the masking procedure for any planar gauge field action in SUc(N), Ref. [18]. From there it was very obvious that by applying a continuous repetition of some sections of code that I written, that some bigger Wilson loops could easily be included in the action and hence highly improved actions could be easily constructed. The only difficulty was to calculate the improvement coefficients. I then moved on to study smearing algorithms. I adapted the gauge field configuration code to a cooling and a 1 × 2 and 2 × 1 improved cooling code in which we inserted higher order loop operators. This was the tool used to explore gauge field configurations and their topological structures. Once the short range quantum fluctuations are removed it is possible to see instantons. Instantons are believed to play a crucial role in the spontaneous chiral symmetry breaking mechanism. We improved the topological charge operator from the clover term to an (1 × 2 and 2 × 1) O(a²)–improved topological charge operator (see Appendices, Sections E.16 and E.17). This code was subsequently adapted by Sundance Bilson-Thompson so that he could insert higher order loops. I have also inserted my O(a²)–improved operator to construct an O(a²)–improved smearing algorithm. Using these tools I have calibrated the relative rates of cooling and smearing. Another piece of work on gauge fixing, reviewed in Chapter 8, was led by Dr. Patrick O. Bowman, Ref. [63]. There I supplied the gauge field configurations and checked some of the analytical work. For the gluon propagator work I supplied all of the lattice configurations with the exception of the 32³ × 64 used in Ref. [62]. The analysis was primarily carried out by Dr. Patrick O. Bowman and partly inspired by the one carried out in hep-lat/0106023. While this gluon propagator work is not being presented here as my own Ph. D. qualifying work, I am a co author on the subsequent papers and so I have therefore decided to include a review of this work in Chapter 9. I have also made some contribution in the construction of the Fat–link quark action (with and without the clover term) developed by James M. Zanotti. These contributions involve the code for the Reunitarization of the smeared links, Appendix E.21. Because of the code developed for the improved lattice definition of the Fµν(x) term I have also made some contribution to the Fat–link clover quark action although I will not discuss about this work in the following thesis. My main contribution for the overlap quark propagator study was in the analysis of the propagator data. The overlap propagators were generated by Dr. Jianbo Zhang and the research was also carried out in collaboration with A/Prof. Anthony G. Williams and Dr. Derek B. Leinweber. The quark propagators for the Wilson fermion were generated by a computer code parallelized by James M. Zanotti and originally written by Prof. Frank X. Lee. The anisotropic lattice code has not been used in any calculations yet although it has been tested and verified. The code was extended from the isotropic improved generator code in SUc(3). After a literature search, we decided to implement the action described in Ref. [31] for the anisotropic Wilson action and in Ref. [11, 32] for the improved anisotropic case. Apart from the work on the gauge fixing and the gluon propagator, done in collaboration with Dr. Patrick O. Bowman, and which for completeness is briefly reviewed in Chapters 8 and 9 respectively, this thesis contains no material which has been accepted for the award of any other degree or diploma in any university or other institution and to the best of knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being available for loan and photocopying. Fr´ed´eric D. R. Bonnet Date: 20th of September 2001. ____________ [Footnotes]: ¹F. D. R. Bonnet, P. Fitzhenry, D. B. Leinweber, M. R. Stanford & A. G. Williams, Phys. Rev. D 62, 094509 (2000) [hep lat/0001018]. ²F. D. R. Bonnet, D. B. Leinweber, A. G. Williams & J. M. Zanotti, Submitted to Phys. Rev. D. [hep-lat/0106023]. ³F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, D. G. Richards & A. G. Williams, Aust. J. Phys. 52, 939 (1999). ⁴F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber & A. G. Williams, Infrared behavior of the gluon propagator on a large volume lattice, Phys. Rev. D 62, 051501, (2000). ⁵F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, A. G. Williams & J. M. Zanotti, Infinite volume and continuum limits of the landau gauge gluon propagator, Phys. Rev. D 64, 034501 (2001) [hep-lat/0101013]. ⁶F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, A. G. Williams & J. Zhang, Overlap Propagator in Landau Gauge, to be Submitted to Phys. Rev. D.Thesis (Ph.D.) -- University of Adelaide, School of Chemistry and Physics, 200

Discretisation Errors in Landau Gauge on the Lattice

Lattice discretisation errors in the Landau gauge condition are examined. An improved gauge fixing algorithm in which O(a 2) errors are removed is presented. It is shown that O(a 2) errors dominate standard gauge fixing procedures and an absolute estimate of the deviation from continuum Landau gauge is given. These results emphasise the importance of implementing an improved gauge fixing condition

Can a minority game follow real market dynamics?

It is commonly known in economics that markets follow both positive and/or negative trends, crashes and bubble effects. In general a strong positive trend is followed by a crash. Famous examples of these effects were seen in the recent crash on the NASDAQ (April 2000) and prior to the crash on the Hong Kong market, which was associated with the Asian crisis in the early 1994. In this paper we use real market data input into a minority game with a variable payoff function and a nonlinear super exponential model for bubbles, to explore financial bubbles. By changing the payoff function in the minority game we study how one can get the price function to follow the dynamics of a real market.Frederic D. R. Bonnet and Derek Abbot