A planar reconfigurable linear rigid-body motion linkage with two operation modes

A planar reconfigurable linear (also rectilinear) rigid-body motion linkage (RLRBML) with two operation modes, that is, linear rigid-body motion mode and lockup mode, is presented using only R (revolute) joints. The RLRBML does not require disassembly and external intervention to implement multi-task requirements. It is created via combining a Robert’s linkage and a double parallelogram linkage (with equal lengths of rocker links) arranged in parallel, which can convert a limited circular motion to a linear rigid-body motion without any reference guide way. This linear rigid-body motion is achieved since the double parallelogram linkage can guarantee the translation of the motion stage, and Robert’s linkage ensures the approximate straight line motion of its pivot joint connecting to the double parallelogram linkage. This novel RLRBML is under the linear rigid-body motion mode if the four rocker links in the double parallelogram linkage are not parallel. The motion stage is in the lockup mode if all of the four rocker links in the double parallelogram linkage are kept parallel in a tilted position (but the inner/outer two rocker links are still parallel). In the lockup mode, the motion stage of the RLRBML is prohibited from moving even under power off, but the double parallelogram linkage is still moveable for its own rotation application. It is noted that further RLRBMLs can be obtained from the above RLRBML by replacing Robert’s linkage with any other straight line motion linkage (such as Watt’s linkage). Additionally, a compact RLRBML and two single-mode linear rigid-body motion linkages are presented.

On classical and other methods of discriminant analysis and estimation of log-odds

For two multinormal populations with equal covariance matrices the likelihood ratio discriminant function, an alternative allocation rule to the sample linear discriminant function when n1 ≠ n2 ,is studied analytically. With the assumption of a known covariance matrix its distribution is derived and the expectation of its actual and apparent error rates evaluated and compared with those of the sample linear discriminant function. This comparison indicates that the likelihood ratio allocation rule is robust to unequal sample sizes. The quadratic discriminant function is studied, its distribution reviewed and evaluation of its probabilities of misclassification discussed. For known covariance matrices the distribution of the sample quadratic discriminant function is derived. When the known covariance matrices are proportional exact expressions for the expectation of its actual and apparent error rates are obtained and evaluated. The effectiveness of the sample linear discriminant function for this case is also considered. Estimation of true log-odds for two multinormal populations with equal or unequal covariance matrices is studied. The estimative, Bayesian predictive and a kernel method are compared by evaluating their biases and mean square errors. Some algebraic expressions for these quantities are derived. With equal covariance matrices the predictive method is preferable. Where it derives this superiority is investigated by considering its performance for various levels of fixed true log-odds. It is also shown that the predictive method is sensitive to n1 ≠ n2. For unequal but proportional covariance matrices the unbiased estimative method is preferred. Product Normal kernel density estimates are used to give a kernel estimator of true log-odds. The effect of correlation in the variables with product kernels is considered. With equal covariance matrices the kernel and parametric estimators are compared by simulation. For moderately correlated variables and large dimension sizes the product kernel method is a good estimator of true log-odds.

Portfolio of compositions with commentaries 2012-2015

This writing forms part of a portfolio of works which gathers together notations and recordings made between 2012 and 2015. The writing divides into four chapters. The first chapter undertakes an examination of abstraction. From the proposition obtained therein, that it is possible to extend within abstraction beyond a correspondence between its terms and concrete manifestations, I ask in what way abstraction may be considered to function therefrom. The second chapter performs an analysis of radical music notations from the nineteenth, twentieth, and twenty-first centuries. The central analysis identifies what is essentially an evolution of what I call ''figuration'' throughout three non-chronological phases of Christian Wolff's notations. The chapter closes by questioning what form of understanding might yet be pursued in respect of this work given (a) its nature, and (b) the nature of our inquiry. The third chapter reflects on some of the philosophical problematics raised in the preceding chapters, at the same time as it prepares for those that follow in the final chapter. The final chapter is a record of the development of my thought appertaining to my compositional work. It sets out two case studies (prepared piano and postcard notations) before proceeding to a selected set of commentaries. Finally, the notion of the functioning of written and acoustical ''marks'' developed in the preceding commentaries is generalised as I develop an aesthetics of scriptural experience.

Financial modelling with 2-EPT probability density functions

The class of all Exponential-Polynomial-Trigonometric (EPT) functions is classical and equal to the Euler-d’Alembert class of solutions of linear differential equations with constant coefficients. The class of non-negative EPT functions defined on [0;1) was discussed in Hanzon and Holland (2010) of which EPT probability density functions are an important subclass. EPT functions can be represented as ceAxb, where A is a square matrix, b a column vector and c a row vector where the triple (A; b; c) is the minimal realization of the EPT function. The minimal triple is only unique up to a basis transformation. Here the class of 2-EPT probability density functions on R is defined and shown to be closed under a variety of operations. The class is also generalised to include mixtures with the pointmass at zero. This class coincides with the class of probability density functions with rational characteristic functions. It is illustrated that the Variance Gamma density is a 2-EPT density under a parameter restriction. A discrete 2-EPT process is a process which has stochastically independent 2-EPT random variables as increments. It is shown that the distribution of the minimum and maximum of such a process is an EPT density mixed with a pointmass at zero. The Laplace Transform of these distributions correspond to the discrete time Wiener-Hopf factors of the discrete time 2-EPT process. A distribution of daily log-returns, observed over the period 1931-2011 from a prominent US index, is approximated with a 2-EPT density function. Without the non-negativity condition, it is illustrated how this problem is transformed into a discrete time rational approximation problem. The rational approximation software RARL2 is used to carry out this approximation. The non-negativity constraint is then imposed via a convex optimisation procedure after the unconstrained approximation. Sufficient and necessary conditions are derived to characterise infinitely divisible EPT and 2-EPT functions. Infinitely divisible 2-EPT density functions generate 2-EPT Lévy processes. An assets log returns can be modelled as a 2-EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2-Period Bermudan Options are also provided. Certain Greeks, including Delta and Gamma, of these options are also computed analytically. MATLAB scripts are provided for calculations involving 2-EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2-EPT approach to financial modelling.

Composition portfolio and commentaries

In an attempt to provide an analytical entry point into my compositional practice, I have identified eight themes which are significantly recurrent: reduction – the selection of a small number of elements; imperfection – a damaged or warped characteristic of sound; hierarchy – a concern with the roles of instruments with regard to their relative prominence; motion – apparently static sound masses consist of fine internal movement; listener perception – expectations for change influence the experience of affect; translation – the transitioning of electronic sounds to the acoustic realm, and vice versa; immersion – the creation of an accommodating soundscape; blurring – smearing and overlapping sounds or genres. Each of these eight factors is associated with relevant precedents in the history and theory of music that have been influential on my work. These include the minimalist compositions of Steve Reich and Arvo Pärt; the lo-fi aesthetic of Boards of Canada and My Bloody Valentine; concerns with political hierarchy in the work of Louis Andriessen; the variations of dynamics and microtonal shifts of Giacinto Scelsi; Leonard B. Meyer's account of expectation in music; cross-fertilisation of the acoustic and electronic in pieces by Gérard Grisey and Gyorgy Ligeti; the immersive technique of Brian Eno's ambient music; and the overlapping sounds of Aphex Twin. These eight factors are variously applicable to the eleven submitted pieces, which are individually analysed with reference to the most significant of the categories. Together they form a musical language that sustains the interaction of a variety of techniques, concepts and genres.