Winding number correlation for a Brownian loop in a plane

A Brownian loop is a random walk circuit of infinitely many, suitably infinitesimal, steps. In a plane such a loop may or may not enclose a marked point, the origin, say. If it does so it may wind arbitrarily many times, positive or negative, around that point. Indeed from the (long known) probability distribution, the mean square winding number is infinite, so all statistical moments - averages of powers of the winding number - are infinity (even powers) or zero (odd powers, by symmetry). If an additional marked point is introduced at some distance from the origin, there are now two winding numbers, which are correlated. That correlation, the average of the product of the two winding numbers, is finite and is calculated here. The result takes the form of a single well-convergent integral that depends on a single parameter - the suitably scaled separation of the marked points. The integrals of the correlation weighted by powers of the separation are simple factorial expressions. Explicit limits of the correlation for small and large separation of the marked points are found.Comment: The right hand sides of various equations were missing factors of 1/2 or 1/4, now correcte

Lessons from Lesson Study: Exploring School Climate, Teacher Learning and Teacher Self-Efficacy in an Inner London Primary School

England has a school teacher recruitment and retention crisis. Fewer people are turning to teaching as a career and of those that do, nearly half of them leave the profession within a few years in the classroom. Common reasons for this include micromanagement, excessive workload, and low professional morale. School leaders must balance the weight of high-stakes external accountability through standardised assessment and inspection with a positive school climate where teachers deeply believe in their capacity to improve and impact upon pupils’ achievement. It is therefore important that school leaders are able to draw upon theories in action that positively impact on teachers’ perceptions of the school climate and self-efficacy that simultaneously support deep teacher learning and pupil outcomes. Professional capital theory posits that through the systematic development and integration of three kinds of capital – human, social and decisional – learning and achievement can improve everywhere (A. Hargreaves & Fullan, 2012). Lesson Study (LS) is a model of teacher development that has been widely researched for its impact on teacher learning and pupil outcomes, but with little evidence about its association with teachers’ perceptions of school climate and teacher self-efficacy. While a small number of recent studies have considered the impact of LS on school culture and teacher self-efficacy, they have focused primarily on quantitative measures and have been conducted by external researchers, without considering the voice of the teachers in an emerging picture of LS in shaping the school climate or teacher self-efficacy. LS is positioned within the study as an approach aligned with social capital while, crucially, the research is being conducted at a school situated within a system that is not conducive to professional capital in action. This is of importance to school leaders as teachers’ perspectives on school climate and self-efficacy as a result of improvement approaches are fundamental in teacher satisfaction, development, improvement and job performance. Teachers’ perspectives about school improvement are fundamental to its sustainability and long term impact. The aims of this study were: 1) To positively change school culture/climate through the introduction of Lesson Study as professional learning and development; 2) To improve teacher self-efficacy in teaching mixed-ability classes in mathematics, ultimately phasing out “ability grouping”; and, 3) To interrogate current teaching strategies being used with struggling and advanced learners in primary mathematics with regard to pupil progress. The following research questions were formulated to explore the aims: 1) Will initiating a programme of Lesson Study be associated with a positive impact on the climate of a primary school? 2) Will initiating a programme of Lesson Study be associated with a positive impact on teacher self-efficacy in implementing inclusive practice? 3) What conclusions did the teachers draw about improving the teaching following the Lesson Study cycle? 4) What changes to practice will teachers sustain after engaging in a wave of Lesson Study? 5) What changes in pupil maths attainment will follow a programme of Lesson Study? This research presents the case study of a primary school in inner London conducting LS for the first time in 2015/16, with a prologue discussing the events leading up to the study itself from 2012, concluding with an epilogue exploring the outcomes over time in 2020/21. Using professional capital theory, I collected data from semi-structured individual interviews, group interviews, pre- and post- LS questionnaires and a review of group research posters and pupil mathematics assessment data. I then critically examined this data to identify qualitative themes in teacher perspectives. Finally, these analyses were combined to consider what associations teachers perceived LS to have. Quantitative analysis showed both high initial ratings from teachers and overall mean score improvements to both climate and self-efficacy scales. These results were expanded upon through interview and teachers identified new potential domains for the analysis of the school climate and teacher self-efficacy. Teachers' responses to questions about their learning and sustained changes to practice were in line with relevant LS literature and pupil outcomes reflected a significant difference when comparing Wave 1 to Wave 2 and a difference between prior low-attaining pupils and prior high attainers. There is also evidence to support a change in teacher practice as it related to “ability grouping” due to the construct of LS itself. The research undertaken in this project is significant as it supports and furthers the work in the field of LS. It contends that LS is both a vehicle for teacher development and pupil achievement, but adds to the field that LS is a mechanism that can be used to positively influence the climate in a primary school and improve the self-efficacy of teachers in implementing inclusive practices in the context of professional capital theory over time. In addition to this, this study adds content to the body of knowledge about school climate and teacher self-efficacy beyond the realm of LS, which could be used in designing quantitative tools to measure climate and efficacy in other settings. It also provides a longitudinal look at the place of LS and professional capital theory in action at an English primary school over time, with work analysed in both the initial stages and five years on. Future research could be pursued about those elements that allow effective LS to be sustained in English primary schools and the factors that support or dissuade leaders from adopting Lesson Study in system-based cultures less conducive to LS. An analysis of current school climate and self-efficacy scales could be undertaken to further develop the coverage of school climate and teacher self-efficacy measures

An experiment on the shifts of reflected C-lines

An experiment is described that tests theoretical predictions on how C-lines incident obliquely on a surface behave on reflection. C-lines in a polarised wave are the analogues of the optical vortices carried by a complex scalar wave, which is the usual model for describing light and other electromagnetic waves. The centre of a laser beam that carries a (degenerate) C-line is shifted on reflection by the well-known Goos-H\"anchen and Imbert-Fedorov effects, but the C-line itself splits into two, both of which are shifted longitudinally and laterally; their shifts are different from that of the beam centre. To maximise the effect to be measured, internal reflection in a glass prism close to the critical angle was used. In a simple situation like this two recently published independent theories of C-line reflection overlap and it is shown that their predictions are identical. The measured differences in the lateral shifts of the two reflected C-lines are compared with theoretical expectations over a range of incidence angles.Comment: 9 pages, 2 figure

Negative moments of characteristic polynomials of random GOE matrices and singularity-dominated strong fluctuations

We calculate the negative integer moments of the (regularized) characteristic polynomials of N x N random matrices taken from the Gaussian Orthogonal Ensemble (GOE) in the limit as $N \to \infty$. The results agree nontrivially with a recent conjecture of Berry & Keating motivated by techniques developed in the theory of singularity-dominated strong fluctuations. This is the first example where nontrivial predictions obtained using these techniques have been proved.Comment: 13 page

Two-point correlations of the Gaussian symplectic ensemble from periodic orbits

We determine the asymptotics of the two-point correlation function for quantum systems with half-integer spin which show chaotic behaviour in the classical limit using a method introduced by Bogomolny and Keating [Phys. Rev. Lett. 77 (1996) 1472-1475]. For time-reversal invariant systems we obtain the leading terms of the two-point correlation function of the Gaussian symplectic ensemble. Special attention has to be paid to the role of Kramers' degeneracy.Comment: 7 pages, no figure

Semiclassical Approach to Parametric Spectral Correlation with Spin 1/2

The spectral correlation of a chaotic system with spin 1/2 is universally described by the GSE (Gaussian Symplectic Ensemble) of random matrices in the semiclassical limit. In semiclassical theory, the spectral form factor is expressed in terms of the periodic orbits and the spin state is simulated by the uniform distribution on a sphere. In this paper, instead of the uniform distribution, we introduce Brownian motion on a sphere to yield the parametric motion of the energy levels. As a result, the small time expansion of the form factor is obtained and found to be in agreement with the prediction of parametric random matrices in the transition within the GSE universality class. Moreover, by starting the Brownian motion from a point distribution on the sphere, we gradually increase the effect of the spin and calculate the form factor describing the transition from the GOE (Gaussian Orthogonal Ensemble) class to the GSE class.Comment: 25 pages, 2 figure

Barnett-Pegg formalism of angle operators, revivals, and flux lines

We use the Barnett-Pegg formalism of angle operators to study a rotating particle with and without a flux line. Requiring a finite dimensional version of the Wigner function to be well defined we find a natural time quantization that leads to classical maps from which the arithmetical basis of quantum revivals is seen. The flux line, that fundamentally alters the quantum statistics, forces this time quantum to be increased by a factor of a winding number and determines the homotopy class of the path. The value of the flux is restricted to the rational numbers, a feature that persists in the infinite dimensional limit.Comment: 5 pages, 0 figures, Revte

Real roots of Random Polynomials: Universality close to accumulation points

We identify the scaling region of a width O(n^{-1}) in the vicinity of the accumulation points $t=\pm 1$ of the real roots of a random Kac-like polynomial of large degree n. We argue that the density of the real roots in this region tends to a universal form shared by all polynomials with independent, identically distributed coefficients c_i, as long as the second moment \sigma=E(c_i^2) is finite. In particular, we reveal a gradual (in contrast to the previously reported abrupt) and quite nontrivial suppression of the number of real roots for coefficients with a nonzero mean value \mu_n = E(c_i) scaled as \mu_n\sim n^{-1/2}.Comment: Some minor mistakes that crept through into publication have been removed. 10 pages, 12 eps figures. This version contains all updates, clearer pictures and some more thorough explanation

Notes on Conformal Invisibility Devices

As a consequence of the wave nature of light, invisibility devices based on isotropic media cannot be perfect. The principal distortions of invisibility are due to reflections and time delays. Reflections can be made exponentially small for devices that are large in comparison with the wavelength of light. Time delays are unavoidable and will result in wave-front dislocations. This paper considers invisibility devices based on optical conformal mapping. The paper shows that the time delays do not depend on the directions and impact parameters of incident light rays, although the refractive-index profile of any conformal invisibility device is necessarily asymmetric. The distortions of images are thus uniform, which reduces the risk of detection. The paper also shows how the ideas of invisibility devices are connected to the transmutation of force, the stereographic projection and Escheresque tilings of the plane