Asymptotic Level Spacing of the Laguerre Ensemble: A Coulomb Fluid Approach

We determine the asymptotic level spacing distribution for the Laguerre Ensemble in a single scaled interval, $(0,s)$, containing no levels, E_{\bt}(0,s), via Dyson's Coulomb Fluid approach. For the $\alpha=0$ Unitary-Laguerre Ensemble, we recover the exact spacing distribution found by both Edelman and Forrester, while for $\alpha\neq 0$, the leading terms of $E_{2}(0,s)$, found by Tracy and Widom, are reproduced without the use of the Bessel kernel and the associated Painlev\'e transcendent. In the same approximation, the next leading term, due to a ``finite temperature'' perturbation (\bt\neq 2), is found.Comment: 10pp, LaTe

Action research in physical education: focusing beyond myself through cooperative learning

This paper reports on the pedagogical changes that I experienced as a teacher engaged in an action research project in which I designed and implemented an indirect, developmentally appropriate and child‐centred approach to my teaching. There have been repeated calls to expunge – or at least rationalise – the use of traditional, teacher‐led practice in physical education. Yet despite the advocacy of many leading academics there is little evidence that such a change of approach is occurring. In my role as teacher‐as‐researcher I sought to implement a new pedagogical approach, in the form of cooperative learning, and bring about a positive change in the form of enhanced pupil learning. Data collection included a reflective journal, post‐teaching reflective analysis, pupil questionnaires, student interviews, document analysis, and non‐participant observations. The research team analysed the data using inductive analysis and constant comparison. Six themes emerged from the data: teaching and learning, reflections on cooperation, performance, time, teacher change, and social interaction. The paper argues that cooperative learning allowed me to place social and academic learning goals on an even footing, which in turn placed a focus on pupils’ understanding and improvement of skills in athletics alongside their interpersonal development

Eigenvalue correlations on Hyperelliptic Riemann surfaces

In this note we compute the functional derivative of the induced charge density, on a thin conductor, consisting of the union of g+1 disjoint intervals, $J:=\cup_{j=1}^{g+1}(a_j,b_j),$ with respect to an external potential. In the context of random matrix theory this object gives the eigenvalue fluctuations of Hermitian random matrix ensembles where the eigenvalue density is supported on J.Comment: latex 2e, seven pages, one figure. To appear in Journal of Physics

Wigner-Dyson Statistics from the Replica Method

We compute the correlation functions of the eigenvalues in the Gaussian unitary ensemble using the fermionic replica method. We show that non--trivial saddle points, which break replica symmetry, must be included in the calculation in order to reproduce correctly the exact results for the correlation functions at large distance.Comment: 13 pages, added reference

Dyson's Brownian Motion and Universal Dynamics of Quantum Systems

We establish a correspondence between the evolution of the distribution of eigenvalues of a $N\times N$ matrix subject to a random Gaussian perturbing matrix, and a Fokker-Planck equation postulated by Dyson. Within this model, we prove the equivalence conjectured by Altshuler et al between the space-time correlations of the Sutherland-Calogero-Moser system in the thermodynamic limit and a set of two-variable correlations for disordered quantum systems calculated by them. Multiple variable correlation functions are, however, shown to be inequivalent for the two cases.Comment: 10 pages, revte

The Probability of an Eigenvalue Number Fluctuation in an Interval of a Random Matrix Spectrum

We calculate the probability to find exactly $n$ eigenvalues in a spectral interval of a large random $N \times N$ matrix when this interval contains $s \ll N$ eigenvalues on average. The calculations exploit an analogy to the problem of finding a two-dimensional charge distribution on the interface of a semiconductor heterostructure under the influence of a split gate.Comment: 4 pages, postscrip

Kepler and the Kuiper Belt

The proposed field-of-view of the Kepler mission is at an ecliptic latitude of ~55 degrees, where the surface density of scattered Kuiper Belt Objects (KBOs) is a few percent that in the ecliptic plane. The rate of occultations of Kepler target stars by scattered KBOs with radii r>10km is ~10^-6 to 10^-4 per star per year, where the uncertainty reflects the current ignorance of the thickness of the scattered KBO disk and the faint-end slope of their magnitude distribution. These occultation events will last only ~0.1% of the planned t_exp=15 minute integration time, and thus will appear as single data points that deviate by tiny amounts. However, given the target photometric accuracy of Kepler, these deviations will nevertheless be highly significant, with typical signal-to-noise ratios of ~10. I estimate that 1-20 of the 10^5 main-sequence stars in Kepler's field-of-view will exhibit detectable occultations during its four-year mission. For unresolved events, the signal-to-noise of individual occultations scales as t_exp^{-1/2}, and the minimum detectable radius could be decreased by an order of magnitude to ~1 km by searching the individual 3-second readouts for occultations. I propose a number of methods by which occultation events may be differentiated from systematic effects. Kepler should measure or significantly constrain the frequency of highly-inclined, ~10 km-sized KBOs.Comment: 5 pages, 1 figure. No changes. Accepted to ApJ, to appear in the August 1, 2004 issue (v610

Fluctuation properties of strength functions associated with giant resonances

We performed fluctuation analysis by means of the local scaling dimension for the strength function of the isoscalar (IS) and the isovector (IV) giant quadrupole resonances (GQR) in $^{40}$Ca, where the strength functions are obtained by the shell model calculation within up to the 2p2h configurations. It is found that at small energy scale, fluctuation of the strength function almost obeys the Gaussian orthogonal ensemble (GOE) random matrix theory limit. On the other hand, we found a deviation from the GOE limit at the intermediate energy scale about 1.7MeV for the IS and at 0.9MeV for the IV. The results imply that different types of fluctuations coexist at different energy scales. Detailed analysis strongly suggests that GOE fluctuation at small energy scale is due to the complicated nature of 2p2h states and that fluctuation at the intermediate energy scale is associated with the spreading width of the Tamm-Dancoff 1p1h states.Comment: 14 pages including 13figure

Exact calculation of the ground-state dynamical spin correlation function of a S=1/2 antiferromagnetic Heisenberg chain with free spinons

We calculate the exact dynamical magnetic structure factor S(Q,E) in the ground state of a one-dimensional S=1/2 antiferromagnet with gapless free S=1/2 spinon excitations, the Haldane-Shastry model with inverse-square exchange, which is in the same low-energy universality class as Bethe's nearest-neighbor exchange model. Only two-spinon excited states contribute, and S(Q,E) is found to be a very simple integral over these states.Comment: 11 pages, LaTeX, RevTeX 3.0, cond-mat/930903

Final state interaction phase in B decays

From an estimate of the meson-meson inelastic scatterin at 5 GeV it is concluded that a typical strong phase in B decays to two mesons is of order of 20 degrees. For a particular final state an estimate of the phase depends on whether that state is more or less probable as a final state compared to those states to which it is connected by the strong interaction S matrix.Comment: 10 pages in RevTex with 1 eps figur