Critical Exponents and Stability at the Black Hole Threshold for a Complex Scalar Field

This paper continues a study on Choptuik scaling in gravitational collapse of a complex scalar field at the threshold for black hole formation. We perform a linear perturbation analysis of the previously derived complex critical solution, and calculate the critical exponent for black hole mass, $\gamma \approx 0.387106$. We also show that this critical solution is unstable via a growing oscillatory mode.Comment: 15 pages of latex/revtex; added details of numerics, in press in Phys Rev D; 1 figure included, or available by anonymous ftp to ftp://ftp.itp.ucsb.edu/figures/nsf-itp-95-58.ep

Application of a simple multiplicative spatio-temporal stream water quality model to the river Conwy, North Wales

We use a simple multiplicative spatio-temporal model to describe variability in a sequence of water quality monitoring data from headwater streams in the Conwy catchment, North Wales. The spatial component of the model treats concentrations as due to simple mixing of a small number of distinct source types, each associated with particular upstream catchment characteristics. The temporal component allows concentration variability due to seasonal or hydrological change. We apply the model using three candidate catchment characteristic classifications to generate mixing concentrations, and a seasonal component to describe temporal variability, and test a range of sub-models. We identify a cross-classification of soil and land cover as providing the best spatial indicator of water quality of the classifications considered. The spatial model based on a selected grouped cross-classification was shown to account for between 35% and 90% of the spatial variability and the seasonal model accounted for between 45% and 100% of the temporal variability in the data. Analysis of residuals showed an inverse relationship between DOC and sulphate and between hydrogen ion concentration and calcium and magnesium. We also found residual correlations between sites which are strongly related to landscape class. These are attributed to landscape class by time interactions which are not accounted for in the simple multiplicative model

Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations

Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation

Criticality and Bifurcation in the Gravitational Collapse of a Self-Coupled Scalar Field

We examine the gravitational collapse of a non-linear sigma model in spherical symmetry. There exists a family of continuously self-similar solutions parameterized by the coupling constant of the theory. These solutions are calculated together with the critical exponents for black hole formation of these collapse models. We also find that the sequence of solutions exhibits a Hopf-type bifurcation as the continuously self-similar solutions become unstable to perturbations away from self-similarity.Comment: 18 pages; one figure, uuencoded postscript; figure is also available at http://www.physics.ucsb.edu/people/eric_hirschman

The liquid-vapor interface of an ionic fluid

We investigate the liquid-vapor interface of the restricted primitive model (RPM) for an ionic fluid using a density-functional approximation based on correlation functions of the homogeneous fluid as obtained from the mean-spherical approximation (MSA). In the limit of a homogeneous fluid our approach yields the well-known MSA (energy) equation of state. The ionic interfacial density profiles, which for the RPM are identical for both species, have a shape similar to those of simple atomic fluids in that the decay towards the bulk values is more rapid on the vapor side than on the liquid side. This is the opposite asymmetry of the decay to that found in earlier calculations for the RPM based on a square-gradient theory. The width of the interface is, for a wide range of temperatures, approximately four times the second moment correlation length of the liquid phase. We discuss the magnitude and temperature dependence of the surface tension, and argue that for temperatures near the triple point the ratio of the dimensionless surface tension and critical temperature is much smaller for the RPM than for simple atomic fluids.Comment: 6 postscript figures, submitted to Phys. Rev.

Finite covers of random 3-manifolds

A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface S on the set of quotients pi_1(S) -> Q. If Q is a simple group, we show that if the genus of S is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman's theorem that the action of the mapping class group on the SU(2) character variety is ergodic.Comment: 60 pages; v2: minor changes. v3: minor changes; final versio

Heat release by controlled continuous-time Markov jump processes

We derive the equations governing the protocols minimizing the heat released by a continuous-time Markov jump process on a one-dimensional countable state space during a transition between assigned initial and final probability distributions in a finite time horizon. In particular, we identify the hypotheses on the transition rates under which the optimal control strategy and the probability distribution of the Markov jump problem obey a system of differential equations of Hamilton-Bellman-Jacobi-type. As the state-space mesh tends to zero, these equations converge to those satisfied by the diffusion process minimizing the heat released in the Langevin formulation of the same problem. We also show that in full analogy with the continuum case, heat minimization is equivalent to entropy production minimization. Thus, our results may be interpreted as a refined version of the second law of thermodynamics.Comment: final version, section 2.1 revised, 26 pages, 3 figure

Universality of the thermodynamic Casimir effect

Recently a nonuniversal character of the leading spatial behavior of the thermodynamic Casimir force has been reported [X. S. Chen and V. Dohm, Phys. Rev. E {\bf 66}, 016102 (2002)]. We reconsider the arguments leading to this observation and show that there is no such leading nonuniversal term in systems with short-ranged interactions if one treats properly the effects generated by a sharp momentum cutoff in the Fourier transform of the interaction potential. We also conclude that lattice and continuum models then produce results in mutual agreement independent of the cutoff scheme, contrary to the aforementioned report. All results are consistent with the {\em universal} character of the Casimir force in systems with short-ranged interactions. The effects due to dispersion forces are discussed for systems with periodic or realistic boundary conditions. In contrast to systems with short-ranged interactions, for $L/\xi \gg 1$ one observes leading finite-size contributions governed by power laws in $L$ due to the subleading long-ranged character of the interaction, where $L$ is the finite system size and $\xi$ is the correlation length.Comment: 11 pages, revtex, to appear in Phys. Rev. E 68 (2003

Development of a polygenic risk score to improve screening for fracture risk: A genetic risk prediction study

Background Since screening programs identify only a small proportion of the population as eligible for an intervention, genomic prediction of heritable risk factors could decrease the number needing to be screened by removing individuals at low genetic risk. We therefore tested whether a polygenic risk score for heel quantitative ultrasound speed of sound (SOS)—a heritable risk factor for osteoporotic fracture—can identify low-risk individuals who can safely be excluded from a fracture risk screening program. Methods and findings A polygenic risk score for SOS was trained and selected in 2 separate subsets of UK Biobank (comprising 341,449 and 5,335 individuals). The top-performing prediction model was termed “gSOS”, and its utility in fracture risk screening was tested in 5 validation cohorts using the National Osteoporosis Guideline Group clinical guidelines (N = 10,522 eligible participants). All individuals were genome-wide genotyped and had measured fracture risk factors. Across the 5 cohorts, the average age ranged from 57 to 75 years, and 54% of studied individuals were women. The main outcomes were the sensitivity and specificity to correctly identify individuals requiring treatment with and without genetic prescreening. The reference standard was a bone mineral density (BMD)–based Fracture Risk Assessment Tool (FRAX) score. The secondary outcomes were the proportions of the screened population requiring clinical-risk-factor-based FRAX (CRF-FRAX) screening and BMD-based FRAX (BMD-FRAX) screening. gSOS was strongly correlated with measured SOS (r2 = 23.2%, 95% CI 22.7% to 23.7%). Without genetic prescreening, guideline recommendations achieved a sensitivity and specificity for correct treatment assignment of 99.6% and 97.1%, respectively, in the validation cohorts. However, 81% of the population required CRF-FRAX tests, and 37% required BMD-FRAX tests to achieve this accuracy. Using gSOS in prescreening and limiting further assessment to those with a low gSOS resulted in small changes to the sensitivity and specificity (93.4% and 98.5%, respectively), but the proportions of individuals requiring CRF-FRAX tests and BMD-FRAX tests were reduced by 37% and 41%, respectively. Study limitations include a reliance on cohorts of predominantly European ethnicity and use of a proxy of fracture risk. Conclusions Our results suggest that the use of a polygenic risk score in fracture risk screening could decrease the number of individuals requiring screening tests, including BMD measurement, while maintaining a high sensitivity and specificity to identify individuals who should be recommended an intervention

White-etching matter in bearing steel. Part II: Distinguishing cause and effect in bearing steel failure

The premature failure of large bearings of the type used in wind turbines, possibly through a mechanism called “white-structure flaking”, has triggered many studies of microstructural damage associated with “white-etching areas” created during rolling contact fatigue, although whether they are symptoms or causes of failure is less clear. Therefore, some special experiments have been conducted to prove that white-etching areas are the consequence, and not the cause, of damage. By artificially introducing a fine dispersion of microcracks in the steel through heat treatment and then subjecting the sample to rolling contact fatigue, manifestations of hard white-etching matter have been created to a much greater extent than samples similarly tested without initial cracks. A wide variety of characterization tools has been used to corroborate that the white areas thus created have the same properties as reported observations on real bearings. Evidence suggests that the formation mechanism of the white-etching regions involves the rubbing and beating of the free surfaces of cracks, debonded inclusions, and voids under repeated rolling contact. It follows that the focus in avoiding early failure should be in enhancing the toughness of the bearing steel in order to avoid the initial microscopic feature event.Funding by CONACyT, the Cambridge Overseas Trust, and the Roberto Rocca Education Programme is highly appreciated and acknowledged.This is the accepted manuscript version. The final published version is available from Springer at http://link.springer.com/article/10.1007%2Fs11661-014-2431-x