Decline of the “Little Parliament”: Juries and Jury Reform in England and Wales

Lloyd-Bostock and Thomas take a historical look at the English jury and place the jury and jury reform in the context of the English legal and political system

Computing the Gamma function using contour integrals and rational approximations

Some of the best methods for computing the gamma function are based on numerical evaluation of Hankel's contour integral. For example, Temme evaluates this integral based on steepest-decent contours by the trapezoid rule. Here we investigate a different approach to the integral: the application of the trapezoid rule on Talbot-type contours using optimal parameters recently derived by Weideman for computing inverse Laplace transforms. Relatedly, we also investigate quadrature formulas derived from best approximations to exp(z) on the negative real axis, following Cody, Meinardus and Varga. The two methods are closely related and both converge geometrically. We find that the new methods are competitive with existing ones, even though they are based on generic tools rather than on specific analysis of the gamma function

Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals

Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of $f(A)$, where $A$ is a negative definite matrix and $f$ is the exponential function or one of the related ``$\varphi$ functions'' such as $\varphi_1(z) = (e^z-1)/z$. Building on previous work by Trefethen and Gutknecht, Gonchar and Rakhmanov, and Lu, we propose two methods for the fast evaluation of $f(A)$ that are especially useful when shifted systems $(A+zI)x=b$ can be solved efficiently, e.g. by a sparse direct solver. The first method method is based on best rational approximations to $f$ on the negative real axis computed via the Carathéodory-Fejér procedure, and we conjecture that the accuracy scales as $(9.28903\dots)^{-2n}$, where $n$ is the number of complex matrix solves. In particular, three matrix solves suffice to evaluate $f(A)$ to approximately six digits of accuracy. The second method is an application of the trapezoid rule on a Talbot-type contour

Large-scale computation of pseudospectra using ARPACK and eigs

ARPACK and its MATLAB counterpart, eigs, are software packages that calculate some eigenvalues of a large non-symmetric matrix by Arnoldi iteration with implicit restarts. We show that at a small additional cost, which diminishes relatively as the matrix dimension increases, good estimates of pseudospectra in addition to eigenvalues can be obtained as a by-product. Thus in large-scale eigenvalue calculations it is feasible to obtain routinely not just eigenvalue approximations, but also information as to whether or not the eigenvalues are likely to be physically significant. Examples are presented for matrices with dimension up to 200,000

Large eddy simulations of a circular cylinder at Reynolds numbers surrounding the drag crisis

Large eddy simulations of the flow around a circular cylinder at high Reynolds numbers are reported. Five Reynolds numbers were chosen, such that the drag crisis was captured. A total of 18 cases were computed to investigate the effect of gridding strategy, domain width, turbulence modelling and numerical schemes on the results. It was found that unstructured grids provide better resolution of key flow features, when a ‘reasonable’ grid size is to be maintained.When using coarse grids for large eddy simuation, the effect of the turbulence models and numerical schemes becomes more pronounced. The dynamic mixed Smagorinsky model was found to be superior to the Smagorinsky model, since the model coefficient is allowed to dynamically adjust based on the local flow and grid size. A blended upwind-central convection scheme was also found to provide the best accuracy, since a fully central scheme exhibits artificial wiggles which pollute the entire solution.Mean drag, fluctuating lift and Strouhal number are compared to experiments and empirical estimates for Reynolds numbers ranging from 6.31 × 104 ? 5.06 × 105. In terms of the drag coefficient, the drag crisis is well captured by the present simulations, although the other integral quantities (rms lift and Strouhal number) less so. For the lowest Reynolds number, the drag is seen to be most sensitive to the domain width, while at the higher Reynolds numbers the grid resolution plays a more important role

An investigation into the effects of gender, prior academic achievement, place of residence, age and attendance on first year undergraduate attainment

The number of people engaging in higher education (HE) has increased considerably over the past decade. However, there is a need to achieve a balance between increasing access and bearing down on rates of non-completion. It has been argued that poor attainment and failure within the first year are significant contributors to the overall statistics for non-progression and that, although research has concentrated on factors causative of student withdrawal, less attention has focused on students who fail academically. This study investigated the effects of a number of a number of factors on the academic attainment of first-year undergraduates within the Faculty of Humanities and Social Sciences at the University of Glamorgan. Results showed that gender and age had only minor impacts upon educational achievement, while place of residence, prior educational attainment and attendance emerged as significant predictors of attainment. Further analysis showed these three factors to be interrelated , with attendance correlating strongly with both entry points and place or residence. In turn, prior attainment was strongly linked to place of residence. Findings may be used to identify and proactively target students at risk of poor academic performance and dropout in order in order to improve rates of performance and progression

ACSB: A minimum performance assessment

Amplitude companded sideband (ACSB) is a new modulation technique which uses a much smaller channel width than does conventional frequency modulation (FM). Among the requirements of a mobile communications system is adequate speech intelligibility. This paper explores this aspect of minimum required performance. First, the basic principles of ACSB are described, with emphasis on those features that affect speech quality. Second, the appropriate performance measures for ACSB are reviewed. Third, a subjective voice quality scoring method is used to determine the values of the performance measures that equate to the minimum level of intelligibility. It is assumed that the intelligibility of an FM system operating at 12 dB SINAD represents that minimum. It was determined that ACSB operating at 12 dB SINAD with an audio-to-pilot ratio of 10 dB provides approximately the same intelligibility as FM operating at 12 dB SINAD

Quantum Hopfield neural network

Quantum computing allows for the potential of significant advancements in both the speed and the capacity of widely used machine learning techniques. Here we employ quantum algorithms for the Hopfield network, which can be used for pattern recognition, reconstruction, and optimization as a realization of a content-addressable memory system. We show that an exponentially large network can be stored in a polynomial number of quantum bits by encoding the network into the amplitudes of quantum states. By introducing a classical technique for operating the Hopfield network, we can leverage quantum algorithms to obtain a quantum computational complexity that is logarithmic in the dimension of the data. We also present an application of our method as a genetic sequence recognizer.Comment: 13 pages, 3 figures, final versio

Talbot quadratures and rational approximations

Many computational problems can be solved with the aid of contour integrals containing $e^z$ in the the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and operators, parabolic PDEs, and reaction-diffusion equations. One approach to the numerical quadrature of such integrals is to apply the trapezoid rule on a Hankel contour defined by a suitable change of variables. Optimal parameters for three classes of such contours have recently been derived: (a) parabolas, (b) hyperbolas, and (c) cotangent contours, following Talbot in 1979. The convergence rates for these optimized quadrature formulas are very fast: roughly $O(3^{-N})$, where $N$ is the number of sample points or function evaluations. On the other hand, convergence at a rate apparently about twice as fast, $O(9.28903^{-N})$, can be achieved by using a different approach: best supremum-norm rational approximants to $e^z$ for $z\in (-\infty,0]$, following Cody, Meinardus and Varga in 1969. (All these rates are doubled in the case of self-adjoint operators or real integrands.) It is shown that the quadrature formulas can be interpreted as rational approximations and the rational approximations as quadrature formulas, and the strengths and weaknesses of the different approaches are discussed in the light of these connections. A MATLAB function is provided for computing Cody--Meinardus--Varga approximants by the method of Carathèodory-Fejèr approximation

Tax Rate Variability and Public Spending as Sources of Indeterminacy

We consider a constant returns to scale, one sector economy with segmented asset markets of the Woodford (1986) type. We analyze the role of public spending, financed by labor income and consumption taxation, on the emergence of indeterminacy. We find that what is relevant for indeterminacy is the variability of the distortion introduced by government intervention. We further discuss the results in terms of the level of the tax rate, its variability with respect to the tax base and the degree of externalities in preferences due to the existence of a public good. We show that the degree of public spending externalities affects the combinations between the tax rate and its variability under which indeterminacy occurs. Moreover, in contrast to previous results, we find that consumption taxes can lead to local indeterminacy when asset markets are segmented.Indeterminacy ; public spending ; taxation ; segmented asset markets