Truncated Stochastic Approximation with Moving Bounds: Convergence

In this paper we propose a wide class of truncated stochastic approximation procedures with moving random bounds. While we believe that the proposed class of procedures will find its way to a wider range of applications, the main motivation is to accommodate applications to parametric statistical estimation theory. Our class of stochastic approximation procedures has three main characteristics: truncations with random moving bounds, a matrix valued random step-size sequence, and dynamically changing random regression function. We establish convergence and consider several examples to illustrate the results

A no-go theorem for accelerating cosmologies from M-theory compactifications

It is known that four-dimensional cosmologies exhibiting transient phases of acceleration can be obtained by compactifications of low-energy effective string or M-theory on time-varying manifolds. In the four-dimensional theory, the acceleration can be attributed to a quintessential scalar field with a positive effective potential. Recently, Townsend has conjectured that the potentials obtained by such compactifications cannot give rise to late-time accelerating universes which possess future event horizons. Such a `no-go' result would be desirable, since current string or M-theory seems unable to provide an adequate description of space-times with future event horizons. In this letter, we provide a proof of this conjecture for a class of warped compactifications with a single scalar modulus parametrising the volume of the compactification manifold.Comment: 11 pages, LaTeX; minor changes and clarifications mad

The Effectiveness in Measuring Character Development Outcomes in Singapore Schools Through the Character Development Award

In 2006, Singapore’s Ministry of Education started recognising schools for their effort in using effective character development programmes in producing holistic students who are both competent in their academic studies and possess good moral character. Although the history of character education in schools started in 1959, it is only in the recent years that formal awards with a set of evaluation criteria are used to recognise schools for their high quality character development programmes. Literature review shows that measuring the effectiveness of character development outcomes is a constant issue among critics of character education to validate the claim. The latest empirical research findings have given a strong indication that the outcomes of character development can be measured as long as the constructs of character are properly defined. The objective of the research is to examine if the current awarding process and criteria used for schools are accurately measuring the character development outcomes of schools. The quantitative research instrument used is the Collective Responsibility for Excellence and Ethics version 2.7 Short designed by Khemelkov and Davidson (2008a). The instrument is designed to capture the inputs and outputs of the key stakeholders of students, teachers and parents in the school community towards character development. A total of 1266 students, 210 staff and 396 parents were involved in the research across five schools with different awards. The findings highlighted the need for a constant review of the evaluation criteria used by awarding body to evaluate schools that are involved in character education. It is also imperative for schools that are involved in character education to have well defined constructs for character. These constructs will determine the outcomes to be measured, and hence, the character development programmes’ effectiveness. Through the findings, the study also made recommendations for policy makers and educators on areas that require careful consideration when implementing character development programmes

Universal Index Theorem on Mob(S^1)\Diff_+(S^1)

By conformal welding, there is a pair of univalent functions $(f,g)$ associated to every point of the complex K\"ahler manifold \Mob(S^1)\bk\Diff_+(S^1). For every integer $n\geq 1$, we generalize the definition of Faber polynomials to define some canonical bases of holomorphic $1-n$ and $n$ differentials associated to the pair $(f,g)$. Using these bases, we generalize the definition of Grunsky matrices to define matrices whose columns are the coefficients of the differentials with respect to standard bases of differentials on the unit disc and the exterior unit disc. We derive some identities among these matrices which are reminiscent of the Grunsky equality. By using these identities, we showed that we can define the Fredholm determinants of the period matrices of holomorphic $n$ differentials $N_n$, which are the Gram matrices of the canonical bases of holomorphic $n$-differentials with respect to the inner product given by the hyperbolic metric. Finally we proved that $\det N_n =(\det N_1)^{6n^2-6n+1}$ and \pa\bar{\pa}\log\det N_n is $-(6n^2-6n+1)/(6\pi i)$ of the Weil-Petersson symplectic form.Comment: 46 page

Recursive Parameter Estimation: Convergence

We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. We propose a wide class of recursive estimation procedures for the general statistical model and study convergence.Comment: 25 pages with 1 postscript figur