Monte Carlo Integration with Subtraction

This paper investigates a class of algorithms for numerical integration of a function in d dimensions over a compact domain by Monte Carlo methods. We construct a histogram approximation to the function using a partition of the integration domain into a set of bins specified by some parameters. We then consider two adaptations; the first is to subtract the histogram approximation, whose integral we may easily evaluate explicitly, from the function and integrate the difference using Monte Carlo; the second is to modify the bin parameters in order to make the variance of the Monte Carlo estimate of the integral the same for all bins. This allows us to use Student's t-test as a trigger for rebinning, which we claim is more stable than the \chi-squared test that is commonly used for this purpose. We provide a program that we have used to study the algorithm for the case where the histogram is represented as a product of one-dimensional histograms. We discuss the assumptions and approximations made, as well as giving a pedagogical discussion of the myriad ways in which the results of any such Monte Carlo integration program can be misleading.Comment: Code PANIC included as a set of ancillary file

Asymptotics of Fixed Point Distributions for Inexact Monte Carlo Algorithms

We introduce a simple general method for finding the equilibrium distribution for a class of widely used inexact Markov Chain Monte Carlo algorithms. The explicit error due to the non-commutivity of the updating operators when numerically integrating Hamilton's equations can be derived using the Baker-Campbell-Hausdorff formula. This error is manifest in the conservation of a ``shadow'' Hamiltonian that lies close to the desired Hamiltonian. The fixed point distribution of inexact Hybrid algorithms may then be derived taking into account that the fixed point of the momentum heatbath and that of the molecular dynamics do not coincide exactly. We perform this derivation for various inexact algorithms used for lattice QCD calculations.Comment: 24 pages, accepted for publication in Physics Review

Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

We consider the problem of minimising the $k$th eigenvalue, $k \geq 2$, of the ($p$-)Laplacian with Robin boundary conditions with respect to all domains in $\mathbb{R}^N$ of given volume $M$. When $k=2$, we prove that the second eigenvalue of the $p$-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For $p=2$ and $k \geq 3$, we prove that in many cases a minimiser cannot be independent of the value of the constant $\alpha$ in the boundary condition, or equivalently of the volume $M$. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions $\Delta u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$.Comment: 16 page

A study of The Rock by T.S. Eliot : a thesis presented in partial fulfilment of the requirements for the degree of Master of Arts in English at Massey University

Chapter I of this thesis is just a brief account of the genesis of The Rock: brief for fear of reproducing what has already been ably said before by Mr E. Martin Browne in his book The Making of T.S. Eliot's Plays. In this section the pageant and music-hall revue, which were the vehicles for this vaguely propagandist work, are treated, and what is so strangely important about the music-hall form of entertainment is that Eliot was very much attracted to it. This chapter, although it just sets the scene, shows the author working at a much more superficial level than ever before. Chapter II deals with the importance of Eliot's socio-religious thinking relative to The Rock in the 1930s. The authoritarian nature and very rigour of his orthodoxy may have been partly the reason why The Rock and After Strange Gods were never republished, but the important point is made clear, through these views on Christian orthodoxy and tradition, that Eliot was to be admired as perhaps the only poet and intellectual of great standing in England in the 1930s who gave his allegiance to something wholly outside himself. In addition, what is made explicit in this chapter, and held implicit throughout, is that Eliot was no turncoat who now gladly and facilely embraced the succour of Mother Church as some critics would have us believe. What is made plain is that this relatively new convert was finding a via media between facile hope and pointless despair - hence the very discipline and rigour of his Anglo-Catholic affirmations. Chapter III is about the requirements of the medium in what was in fact the first time Eliot had moved beyond a coterie audience. The demands and limitations are listed as criteria against which The Rock can only be measured; and although the choruses may be the first bad poetry Eliot had written, it is made clear that he was conscious of the seeming hollowness of ecclesiastical utterances. It may seem paradoxical that Eliot, in this propaganda setting, was actually trying to wring the neck of rhetoric, and the moral is even enforced by his inclusion of a verse-sketch which clearly shows an adulteration through rhetoric. Chapter IV reverberates on the two preceding chapters in its delineation of a return to a purified yet traditional language as well. Although the nobility of language from biblical books is still there, Eliot was for the first time using a democratic, and non-hieratic, language of ordinary man. There is a new distrust of the cunning and rhetorical, as contained in the 'objective correlative' of before, and the author is attempting a personal atone through what seems to be an authenticity and sincerity of tone. The Rock could conceivably exist without the choruses at all but they are important, unlike the prose episodes, because they were written without the various collaborators. Chapter V attempts, very briefly, to establish that The Rock was not a propagandist's hackwork but that the author was consciously groping for new forms of prosody and dramatic techniques preparatory to his later plays and poetry and, as such, the work is seen in the perspective of an important stage midway in Eliot's career as an artist and thinker

An investigation into unsteady base bleed for drag reduction in bluff two-box SUVs

This paper discusses a preliminary investigation into the use of base bleed on a production SUV using CFD analysis. The paper shows the methods used in creating the computational model and conducting the analysis, and present the findings to date. The paper shows that the reduction in drag increases as the mass flow rate of air is increased when the flow is deflected at the outlet. By controlling the turbulent wake to the rear of the vehicle, it is shown in the paper that mass flow rates of under 2kg/s can reduce drag coefficient by 8.2% with an outlet on the side of the vehicle, and that a mass flow rate of under 1.5kg/s can reduce the drag coefficient by 10.7% for an outlet on the upper section of the rear of the vehicle. The paper also discusses the feasibility of base bleed being applied to a production vehicle