Finding the point on Bezier Curves with the normal vector passing an external point

Two algorithms for finding the point on non-rational/rational Bezier curves of which the normal vector passes through a given external point are presented. The algorithms are based on Bezier curves generation algorithms of de Casteljau's algorithm for non-rational Bezier curve or Farin's recursion for rational Bezier curve, respectively. Orthogonal projections from the external point are used to guide the directional search used in the proposed iterative algorithms. Using Lyapunov's method, it is shown that each algorithm is able to converge to a local minimum for each case of non-rational/rational Bezier curves. It is also shown that on convergence the distance between the point on curves to the external point reaches a local minimum for both approaches. Illustrative examples are included to demonstrate the effectiveness of the proposed approaches.

Comparison of the computational cost of a Monte Carlo and deterministic algorithm for computing bilinear forms of matrix powers

In this paper we consider bilinear forms of matrix polynomials and show that these polynomials can be used to construct solutions for the problems of solving systems of linear algebraic equations, matrix inversion and finding extremal eigenvalues. An almost Optimal Monte Carlo (MAO) algorithm for computing bilinear forms of matrix polynomials is presented. Results for the computational costs of a balanced algorithm for computing the bilinear form of a matrix power is presented, i.e., an algorithm for which probability and systematic errors are of the same order, and this is compared with the computational cost for a corresponding deterministic method.

Visual motion interferes with lexical decision on motion words

Embodied theories of cognition propose that neural substrates used in experiencing the referent of a word, for example perceiving upward motion, should be engaged in weaker form when that word, for example ‘rise’, is comprehended. Motivated by the finding that the perception of irrelevant background motion at near-threshold, but not supra-threshold, levels interferes with task execution, we assessed whether interference from near-threshold background motion was modulated by its congruence with the meaning of words (semantic content) when participants completed a lexical decision task (deciding if a string of letters is a real word or not). Reaction times for motion words, such as ‘rise’ or ‘fall’, were slower when the direction of visual motion and the ‘motion’ of the word were incongruent — but only when the visual motion was at nearthreshold levels. When motion was supra-threshold, the distribution of error rates, not reaction times, implicated low-level motion processing in the semantic processing of motion words. As the perception of near-threshold signals is not likely to be influenced by strategies, our results support a close contact between semantic information and perceptual systems.

Adaptive designs for Phase I dose-finding studies

Improving methodology for Phase I dose-finding studies is currently of great interest in pharmaceutical and medical research. This article discusses the current atmosphere and attitude towards adaptive designs and focuses on the influence of Bayesian approaches.

Closed form approximations for spread options

This article expresses the price of a spread option as the sum of the prices of two compound options. One compound option is to exchange vanilla call options on the two underlying assets and the other is to exchange the corresponding put options. This way we derive a new closed form approximation for the price of a European spread option and a corresponding approximation for each of its price, volatility and correlation hedge ratios. Our approach has many advantages over existing analytical approximations, which have limited validity and an indeterminacy that renders them of little practical use. The compound exchange option approximation for European spread options is then extended to American spread options on assets that pay dividends or incur costs. Simulations quantify the accuracy of our approach; we also present an empirical application to the American crack spread options that are traded on NYMEX. For illustration, we compare our results with those obtained using the approximation attributed to Kirk (1996, Correlation in energy markets. In: V. Kaminski (Ed.), Managing Energy Price Risk, pp. 71–78 (London: Risk Publications)), which is commonly used by traders.

Reducing contamination by exposure plus safety behaviour

Background and objectives: It has been proposed that the judicious use of safety behaviour can facilitate improvements in the acceptability of cognitive behaviour therapy (CBT). It was decided to explore the possibility of facilitating CBT by introducing a form of safety behaviour.We sought to assess the degree to which Exposure plus Safety Behaviour (E þ SB) is an effective intervention for contamination fears. Methods: A comparison was made between the effects of a control condition (Exposure and Response Prevention; ERP) and an experimental condition (Exposure plus Safety Behaviour; E þ SB) in which each exposure to a contaminant was followed by the use of a hygienic wipe in a sample of (n ¼ 80) undergraduate students. In session one, each participant touched a confirmed contaminant 20 times. After each exposure participants were asked to report their feelings of contamination, fear, disgust, and danger. In the second session, two weeks later, the same procedure was carried out for a further 16 trials. Results: The ERP and the E þ SB conditions both produced large, significant and stable reductions in contamination. Significant reductions in fear, danger and disgust were also reported in both conditions. Limitations: The treatment was provided to an analogue sample and over two sessions. Conclusions: The use of hygienic wipes, the safety behaviour used in this experiment, did not preclude significant reductions in contamination, disgust, fear and danger. If it is replicated and extended over a longer time-frame, this finding may enable practitioners to enhance the acceptability of cognitive behavioural treatments and boost their effectiveness.