Secondary-school teachers’ noticing of aspects of mathematics teaching talk in the context of one-day workshops

In a research project with one-day teacher education workshops for secondary-school mathematics teachers, our study explores the potential of tool-supported discussions in helping them to notice important and critical aspects of mathematics teaching talk. Mathematical practices of naming and explaining in teaching talk, students’ content learning challenges, and noticing processes of identifying, interpreting and deciding are the components of our framework and the tools that guided the design and implementation of three workshops on linear equations, fractions and plane isometries. The data was collected during the discussions with the seven teachers and the teacher educator throughout these workshops. The coding of the discussions allowed us to see discourse moves that reveal the teachers’ noticing of: (i) challenges in the identification of mathematical naming, (ii) mathematical explaining that voices the students’ learning, (iii) classroom practice in relation to mathematical naming and explaining.     

The Weierstrass–Mandelbrot Process Revisited

We derive a functional central limit theorem for quasi-Gaussian processes. In particular, we prove that the limit of the Mandelbrot–Weierstrass process is a complex fractional Brownian motion.     

Beyond Black-Scholes: semimartingales and Lévy processes for option pricing

We consider the problem of option pricing when the underlying asset follows a general semimartingale process. After reviewing the foundations of arbitrage pricing theory for semimartingales and the link with Lévy processes, we introduce a general method to price options in this framework based on Fourier and Wavelet analysis. Received 4 September 2000     

Frequency dependence of the photonic noise spectrum in an absorbing or amplifying diffusive medium

A theory is presented for the frequency dependence of the power spectrum of photon current fluctuations originating from a disordered medium. Both the cases of an absorbing medium (“grey body”) and of an amplifying medium (“random laser”) are considered in a waveguide geometry. The semiclassical approach (based on a Boltzmann-Langevin equation) is shown to be in complete agreement with a fully quantum mechanical theory, provided that the effects of wave localization can be neglected. The width of the peak in the power spectrum around zero frequency is much smaller than the inverse coherence time, characteristic for black-body radiation. Simple expressions for the shape of this peak are obtained, in the absorbing case, for waveguide lengths large compared to the absorption length, and, in the amplifying case, close to the laser threshold. Received 8 August 2000     

The Smoluchowski limit for a simple mechanical model

We consider a vertical stick constantly accelerated along thex-axis by a forceF and which elastically collides with point particles of the same mass (atoms). The atoms are initially Poisson distributed and are allowed to have four velocities only. It is shown that under suitable scaling of the system the displacementQ(t) of the stick satisfies a nontrivial CLT:Q(t)=vFt+D ^1/2 W(t) (Smoluchowski equation), where the values ofv andD depend on the fact that one atom may collide several times.     

Three-dimensional integrable models and associated tangle invariants

In this paper we show that the Boltzmann weights of the three-dimensional Baxter-Bazhanov model give representations of the braid group if some suitable spectral limits are taken. In the trigonometric case we classify all possible spectral limits which produce braid group representations. Furthermore, we prove that for some of them we get cyclotomic invariants of links and for others we obtain tangle invariants generalizing the cyclotomic ones.     

Equivalent formulations of ＂the equation of life＂

Motivated by progress in theoretical biology a recent proposal on a general and quantitative dynamical framework for nonequilibrium processes and dynamics of complex systems is briefly reviewed. It is nothing but the evolutionary process discovered by Charles Darwin and Alfred Wallace. Such general and structured dynamics may be tentatively named "the equation of life". Three equivalent formulations are discussed, and it is also pointed out that such a quantitative dynamical framework leads naturally to the powerful Boltzmann–Gibbs distribution and the second law in physics. In this way, the equation of life provides a logically consistent foundation for thermodynamics. This view clarifies a particular outstanding problem and further suggests a unifying principle for physics and biology.     

Self-averaging in complex brain neuron signals

Nonlinear statistical properties of Ventral Tegmental Area (VTA) of limbic brain are studied in vivo. VTA plays key role in generation of pleasure and in development of psychological drug addiction. It is shown that spiking time-series of the VTA dopaminergic neurons exhibit long-range correlations with self-averaging behavior. This specific VTA phenomenon has no relation to VTA rewarding function. Last result reveals complex role of VTA in limbic brain. Received 17 April 2002 / Received in final form 30 September 2002 Published online 31 December 2002     

Pulsed-field separation of particles in a microfluidic device

We demonstrate the proof-of-principle of a new separation concept for micrometer-sized particles in a structured microfluidic device. Under the action of externally applied, periodic voltage-pulses two different species of like-charged polystyrene beads are observed to simultaneously migrate into opposite directions. Based on a theoretical model of the particle motion in the microdevice that shows good agreement with the experimental measurements, the underlying separation mechanism is identified and explained. Potential biophysical applications, such as cell sorting, are briefly addressed.     

Leading and higher twists in proton,neutron and deuteron unpolarized structure functions F<Subscript>2</Subscript>

We summarize the results of a recent global analysis of proton and deuteron F₂ structure function world data performed over a large range of kinematics, including recent measurements done at JLab with the CLAS detector. From these data the lowest moments (n≤10) of the unpolarized structure functions are determined with good statistics and systematics. The Q² evolution of the extracted moments is analyzed in terms of an OPE-based twist expansion, taking into account soft-gluon effects at large x. A clean separation among the leading- and higher-twist terms is achieved. By combining proton and deuteron measurements the lowest moments of the neutron F₂ structure function are determined and its leading-twist term is extracted. Particular attention is paid to nuclear effects in the deuteron, which become increasingly important for the higher moments. Our results for the non-singlet, isovector (p-n) combination of the leading-twist moments are used to test recent lattice simulations. We also determine the lowest few moments of the higher-twist contributions, and find these to be approximately isospin independent, suggesting the possible dominance of ud correlations over uu and dd in the nucleon.