Spectral asymmetry of the massless Dirac operator on a 3-torus

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.

Selection-mutation balance models with epistatic selection

We present an application of birth-and-death processes on configuration spaces to a generalized mutation4 selection balance model. The model describes the aging of population as a process of accumulation of mu5 tations in a genotype. A rigorous treatment demands that mutations correspond to points in abstract spaces. 6 Our model describes an infinite-population, infinite-sites model in continuum. The dynamical equation which 7 describes the system, is of Kimura-Maruyama type. The problem can be posed in terms of evolution of states 8 (differential equation) or, equivalently, represented in terms of Feynman-Kac formula. The questions of interest 9 are the existence of a solution, its asymptotic behavior, and properties of the limiting state. In the non-epistatic 10 case the problem was posed and solved in [Steinsaltz D., Evans S.N., Wachter K.W., Adv. Appl. Math., 2005, 11 35(1)]. In our model we consider a topological space X as the space of positions of mutations and the influence of epistatic potentials

A simple variance formula for population size estimators by conditioning

This note considers the variance estimation for population size estimators based on capture–recapture experiments. Whereas a diversity of estimators of the population size has been suggested, the question of estimating the associated variances is less frequently addressed. This note points out that the technique of conditioning can be applied here successfully which also allows us to identify sources of variation: the variance due to estimation of the model parameters and the binomial variance due to sampling n units from a population of size N. It is applied to estimators typically used in capture–recapture experiments in continuous time including the estimators of Zelterman and Chao and improves upon previously used variance estimators. In addition, knowledge of the variances associated with the estimators by Zelterman and Chao allows the suggestion of a new estimator as the weighted sum of the two. The decomposition of the variance into the two sources allows also a new understanding of how resampling techniques like the Bootstrap could be used appropriately. Finally, the sample size question for capture–recapture experiments is addressed. Since the variance of population size estimators increases with the sample size, it is suggested to use relative measures such as the observed-to-hidden ratio or the completeness of identification proportion for approaching the question of sample size choice.

Asymptotic normality in mixtures of power series distributions

The problem of estimating the individual probabilities of a discrete distribution is considered. The true distribution of the independent observations is a mixture of a family of power series distributions. First, we ensure identifiability of the mixing distribution assuming mild conditions. Next, the mixing distribution is estimated by non-parametric maximum likelihood and an estimator for individual probabilities is obtained from the corresponding marginal mixture density. We establish asymptotic normality for the estimator of individual probabilities by showing that, under certain conditions, the difference between this estimator and the empirical proportions is asymptotically negligible. Our framework includes Poisson, negative binomial and logarithmic series as well as binomial mixture models. Simulations highlight the benefit in achieving normality when using the proposed marginal mixture density approach instead of the empirical one, especially for small sample sizes and/or when interest is in the tail areas. A real data example is given to illustrate the use of the methodology.

A simple variance formula for population size estimators by conditioning

This note considers the variance estimation for population size estimators based on capture–recapture experiments. Whereas a diversity of estimators of the population size has been suggested, the question of estimating the associated variances is less frequently addressed. This note points out that the technique of conditioning can be applied here successfully which also allows us to identify sources of variation: the variance due to estimation of the model parameters and the binomial variance due to sampling n units from a population of size N. It is applied to estimators typically used in capture–recapture experiments in continuous time including the estimators of Zelterman and Chao and improves upon previously used variance estimators. In addition, knowledge of the variances associated with the estimators by Zelterman and Chao allows the suggestion of a new estimator as the weighted sum of the two. The decomposition of the variance into the two sources allows also a new understanding of how resampling techniques like the Bootstrap could be used appropriately. Finally, the sample size question for capture–recapture experiments is addressed. Since the variance of population size estimators increases with the sample size, it is suggested to use relative measures such as the observed-to-hidden ratio or the completeness of identification proportion for approaching the question of sample size choice.

Architectural, construction and environmental matters of Bahrain's international formula 1 circuit

The Bahrain International Circuit (BIC) is considered its one of the best international racing car track in terms of technical aspects and architectural quality. Two Formula 1 races have been hosted in the Kingdom of Bahrain, in 2004 and 2005, at BIC. The BIC had recently won the award of the best international racing car circuit. This paper highlights on the elements that contributed to the success of such project starting from the architectural aspects, construction, challenges, tendering process, risk management, the workforce, speed of the construction method, and future prospects for harnessing solar and wind energy for sustainable electrification and production of water for the circuit, i.e. making BIC green and environment-friendly international circuit.

Bahrain's Formula-1 racing circuit: energy and environmental considerations

The Bahrain International Circuit (BIC) and complex, at latitude 26.00N and longitude 51.54E, was built in 483 days and cost 150 million US$. The circuit consists of six different individual tracks with a 3.66 km outer track (involving 10 turns) and a 2.55 km inner track (having six turns). The complex has been designed to host a variety of other sporting activities. Fifty thousand spectators, including 10,500 in the main grandstand, can be accommodated simultaneously. State-of-the art on-site media and broadcast facilities are available. The noise level emitted from vehicles on the circuit during the Formula-1 event, on April 4th 2004, was acceptable and caused no physical disturbance to the fans in the VIP lounges or to scholars studying at the University of Bahrain's Shakeir Campus, which is only 1.5 km away from the circuit. The sound-intensity level (SIL) recorded on the balcony of the VIP lounge was 128 dB(A) and was 80 dB(A) inside the lounge. The calculated SIL immediately outside the lecture halls of the University of Bahrain was 70 dB(A) and 65 dB(A) within them. Thus racing at BIC can proceed without significantly disturbing the academic-learning process. The purchased electricity demand by the BIC complex peaked (at 4.5 MW) during the first Formula-1 event on April 4th 2004. The reverse-osmosis (RO) plant at the BIC provides 1000 m(3) of desalinated water per day for landscape irrigation. Renewable-energy inputs, (i.e., via solar and wind power), at the BIC could be harnessed to generate electricity for water desalination, air conditioning, lighting as well as for irrigation. If the covering of the BIC complex was covered by adhesively fixed modern photovoltaic cells, then similar to 1.2 MW of solar electricity could be generated. If two horizontal-axis, at 150 m height above the ground, three 75m bladed, wind turbines were to be installed at the BIC, then the output could reach 4 MW. Furthermore, if 10,000 Jojoba trees (a species renowned for having a low demand for water, needing only five irrigations per year in Bahrain and which remain green throughout the year) are planted near the circuit, then the local micro-climate would be improved with respect to human comfort as well as the local environment becoming cleaner.

Global asymptotic stability in a class of Putnam-type equations

In this paper, we initiate the study of a class of Putnam-type equation of the form x(n-1) = A(1)x(n) + A(2)x(n-1) + A(3)x(n-2)x(n-3) + A(4)/B(1)x(n)x(n-1) + B(2)x(n-2) + B(3)x(n-3) + B-4 n = 0, 1, 2,..., where A(1), A(2), A(3), A(4), B-1, B-2, B-3, B-4 are positive constants with A(1) + A(2) + A(3) + A(4) = B-1 + B-2 + B-3 + B-4, x(-3), x(-2), x(-1), x(0) are positive numbers. A sufficient condition is given for the global asymptotic stability of the equilibrium point c = 1 of such equations. (c) 2005 Elsevier Ltd. All rights reserved.

Global asymptotic stability in a rational recursive sequence

In this paper, we study the global stability of the difference equation x(n) = a + bx(n-1) + cx(n-1)(2)/d - x(n-2), n = 1,2,....., where a, b greater than or equal to 0 and c, d > 0. We show that one nonnegative equilibrium point of the equation is a global attractor with a basin that is determined by the parameters, and every positive Solution of the equation in the basin exponentially converges to the attractor. (C) 2003 Elsevier Inc. All rights reserved.

Examination of faecal Bifidobacterium populations in breast- and formula-fed infants during the first 18 months of life.

Bifidobacteria in the infant faecal microbiota have been the focus of much interest, especially during the exclusive milk-feeding period and in relation to the fortification of infant formulae to better mimic breast milk. However, longitudinal studies examining the diversity and dynamics of the Bifidobacterium population of infants are lacking, particularly in relation to the effects of weaning. Using a polyphasic strategy, the Bifidobacterium populations of breast- and formula-fed infants were examined during the first 18 months of life. Bifidobacterium-specific denaturing gradient gel electrophoresis demonstrated that breast-fed infants harboured greater diversity than formula-fed infants and the diversity of the infants' Bifidobacterium populations increased with weaning. Twenty-seven distinctive banding profiles were observed from ∼1100 infant isolates using ribosomal intergenic spacer analysis, 14 biotypes of which were confirmed to be members of the genus Bifidobacterium. Two profiles (H, Bifidobacterium longum subsp. infantis; and I, Bifidobacterium bifidum) were common culturable biotypes, seen in 9/10 infants, while profile E (Bifidobacterium breve) was common among breast-fed infants. Overall, inter- and intra-individual differences were observed in the Bifidobacterium populations of infants between 1 and 18 months of age, although weaning was associated with increased diversity of the infant Bifidobacterium populations. Breast-fed infants generally harboured a more complex Bifidobacterium microbiota than formula-fed infants.

A unified titration formula

In a number of standard titrations, a volume of a monoprotic base MOH at a specific concentration is added to a volume of a monoprotic acid HA at a specific concentrations. Four different types of titration are possible, depending on whether the acid and base are strong or weak. A single unifying formula covering all four cases has been determined.