Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise

The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces, such as that expounded by Meyn-Tweedie. Application of these Markov chain results leads to straightforward proofs of geometric ergodicity for a variety of SDEs, including problems with degenerate noise and for problems with locally Lipschitz vector fields. Applications where this theory can be usefully applied include damped-driven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. The two primary ingredients for ergodicity are a minorization condition and a Lyapunov condition. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov condition fails for explicit methods such as Euler-Maruyama; for pathwise approximations it is, in general, only inherited by specially constructed implicit discretizations. Examples of such discretization based on backward Euler methods are given, and approximation of the Langevin equation studied in some detail

Laguatan

The Laguatan (plural : Ilaguas) comprised a confederation of Berber tribes in late antiquity and though the history of the confederation cannot be reconstructed in detail, the importance of this tribal grouping must not be underestimated. From its first appearance in the late third century AD, the confederation played a significant role in the politics of late Roman,Vandal, Byzantine and early Arab Africa. It is to the Laguatan that we can look for a vital thread of continuity across this lon..

The origins and development of Zuwīla, Libyan Sahara: an archaeological and historical overview of an ancient oasis town and caravan centre

Zuwīla in southwestern Libya (Fazzān) was one of the most important early Islamic centres in the Central Sahara, but the archaeological correlates of the written sources for it have been little explored. This paper brings together for the first time a detailed consideration of the relevant historical and archaeological data, together with new AMS radiocarbon dates from several key monuments. The origins of the settlement at Zuwīla were pre-Islamic, but the town gained greater prominence in the early centuries of Arab rule of the Maghrib, culminating with the establishment of an Ibāḍī state ruled by the dynasty of the Banū Khaṭṭāb, with Zuwīla its capital. The historical sources and the accounts of early European travellers are discussed and archaeological work at Zuwīla is described (including the new radiocarbon dates). A short gazetteer of archaeological monuments is provided as an appendix. Comparisons and contrasts are also drawn between Zuwīla and other oases of the ash-Sharqiyāt region of Fazzān. The final section of the paper presents a series of models based on the available evidence, tracing the evolution and decline of this remarkable site

Convergence of the stochastic Euler scheme for locally Lipschitz coefficients

Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. The important case of superlinearly growing coefficients, however, has remained an open question. The main difficulty is that numerically weak convergence fails to hold in many cases of superlinearly growing coefficients. In this paper we overcome this difficulty and establish convergence of the Monte Carlo Euler method for a large class of one-dimensional stochastic differential equations whose drift functions have at most polynomial growth.Comment: Published at http://www.springerlink.com/content/g076w80730811vv3 in the Foundations of Computational Mathematics 201

TeV Astrophysics Constraints on Planck Scale Lorentz Violation

We analyze observational constraints from TeV astrophysics on Lorentz violating nonlinear dispersion for photons and electrons without assuming any a priori equality between the photon and electron parameters. The constraints arise from thresholds for vacuum Cerenkov radiation, photon decay and photo-production of electron-positron pairs. We show that the parameter plane for cubic momentum terms in the dispersion relations is constrained to an order unity region in Planck units. We find that the threshold configuration can occur with an asymmetric distribution of momentum for pair creation, and with a hard photon for vacuum Cerenkov radiation.Comment: 4 pages, RevTeX4, 1 figure. Some references and a footnote added, improved discussion on the photon annihilation and GZK cutoff. Minor changes of wording. Main results unchanged. Version to appear as a Rapid Communication in PR

Archaeology and Desertification in the Wadi Faynan: the Fourth (1999) Season of the Wadi Faynan Landscape Survey

Reproduced with permission of the publisher. © 2000 Council for British Research in the Levant. Details of the publication are available at: http://www.cbrl.org.uk/Publications/publications_default.shtmThis report describes the fourth season of fieldwork by an interdisciplinary team of archaeologists and geographers working together to reconstruct the landscape history of the Wadi Faynan in southern Jordan. The particular focus of the project is the long-term history of inter-relationships between landscape and people, as a contribution to the study of processes of desertification and environmental degradation. The 1999 fieldwork contributed significantly towards the five Objectives defined for the final two field seasons of the project in 1999 and 2000: to map the archaeology outside the ancient field systems flooring the wadi that have formed the principal focus of the archaeological survey in the previous seasons; to use ethnoarchaeological studies both to reconstruct modern and recent land use and also to yield archaeological signatures of land use to inform the analysis of the survey data; to complete the survey of ancient field systems and refine understanding of when and how they functioned; to complete the programme of geomorphological and palaeoecological fieldwork, and in particular to refine the chronology of climatic change and human impacts; and to complete the recording and classification of finds

The Fezzan Project 2001: Preliminary report on the fifth season of work

AbstractThe Fezzan Project completed its five-year fieldwork cycle in 2001. The geographical research team located numerous additional palaeolake sites within the Edeyen Ubari, using a combination of Remote Sensing technology and field visits. Additional samples were taken for analysis and dating from many lake edge locations, relating to both the large Pleistocene lake and to the numerous smaller Holocene lakes that have been identified by the team. The excavations at Old Germa were taken down through Garamantian occupation levels to the natural subsoil below the earliest cultural horizon. The earliest activity, represented by a few mudbrick walls and hearths built directly on the natural soil, is believed to date toc. 400-300 BC. Traces of several phases of Garamantian buildings were uncovered, along with numerous rubbish pits, which yielded a rich assemblage of finds, including, for the first time, examples of Garamantian figurines, small 3-D sculptures of humans and animals. Work on the various classes of finds (pottery, small finds, lithics and other stone artefacts, metallurgical evidence, etc.) complemented the excavation work. In addition, a small amount of further survey work was carried out on sites in the Wadi al-Ajal, along with a contour survey of Old Germa and standing building survey at a number of other sites.</jats:p

On the Behavior of the Effective QCD Coupling alpha_tau(s) at Low Scales

The hadronic decays of the tau lepton can be used to determine the effective charge alpha_tau(m^2_tau') for a hypothetical tau-lepton with mass in the range 0 < m_tau' < m_tau. This definition provides a fundamental definition of the QCD coupling at low mass scales. We study the behavior of alpha_tau at low mass scales directly from first principles and without any renormalization-scheme dependence by looking at the experimental data from the OPAL Collaboration. The results are consistent with the freezing of the physical coupling at mass scales s = m^2_tau' of order 1 GeV^2 with a magnitude alpha_tau ~ 0.9 +/- 0.1.Comment: 15 pages, 4 figures, submitted to Physical Review D, added references, some text added, no results nor figures change

Spiral Multi-component Structure in Pade - Approximant QCD

We present a graphical method of analyzing the infra-red fixed point structure of Pade approximant QCD. The analysis shows a spiral multi-component couplant structure as well as an infra-red attractor behavior of PQCD couplant for all flavors $0 \le N_{f} \le 16$.Comment: 78 pages, 4 tables, 44 graph

The renormalization group inspired approaches and estimates of the tenth-order corrections to the muon anomaly in QED

We present the estimates of the five-loop QED corrections to the muon anomaly using the scheme-invariant approaches and demonstrate that they are in good agreement with the results of exact calculations of the corresponding tenth-order diagrams supplemented by the additional guess about the values of the non-calculated contributions.Comment: LATEX 15 pages, figures available upon request; preprint CERN-TH.7518/9