Probability, propensity and probabilities of propensities (and of probabilities)

The process of doing Science in condition of uncertainty is illustrated with a toy experiment in which the inferential and the forecasting aspects are both present. The fundamental aspects of probabilistic reasoning, also relevant in real life applications, arise quite naturally and the resulting discussion among non-ideologized, free-minded people offers an opportunity for clarifications.Comment: Invited contribution to the proceedings MaxEnt 2016 based on the talk given at the workshop (Ghent, Belgium, 10-15 July 2016), supplemented by work done within the program Probability and Statistics in Forensic Science at the Isaac Newton Institute for Mathematical Sciences, Cambridg

Recent progress in exact geometric computation

AbstractComputational geometry has produced an impressive wealth of efficient algorithms. The robust implementation of these algorithms remains a major issue. Among the many proposed approaches for solving numerical non-robustness, Exact Geometric Computation (EGC) has emerged as one of the most successful. This survey describes recent progress in EGC research in three key areas: constructive zero bounds, approximate expression evaluation and numerical filters

Finding All Solutions of Equations in Free Groups and Monoids with Involution

The aim of this paper is to present a PSPACE algorithm which yields a finite graph of exponential size and which describes the set of all solutions of equations in free groups as well as the set of all solutions of equations in free monoids with involution in the presence of rational constraints. This became possible due to the recently invented emph{recompression} technique of the second author. He successfully applied the recompression technique for pure word equations without involution or rational constraints. In particular, his method could not be used as a black box for free groups (even without rational constraints). Actually, the presence of an involution (inverse elements) and rational constraints complicates the situation and some additional analysis is necessary. Still, the recompression technique is general enough to accommodate both extensions. In the end, it simplifies proofs that solving word equations is in PSPACE (Plandowski 1999) and the corresponding result for equations in free groups with rational constraints (Diekert, Hagenah and Gutierrez 2001). As a byproduct we obtain a direct proof that it is decidable in PSPACE whether or not the solution set is finite.Comment: A preliminary version of this paper was presented as an invited talk at CSR 2014 in Moscow, June 7 - 11, 201

Generally speaking : exploring expressions of generality in secondary mathematics classrooms

It is widely recognised that generality is at the heart of the learning and teaching of mathematics. Motivated by a desire to understand what it is about generality which presents such an obstacle for so many students, this study examines the variety and complexity of ways in which generality is expressed in mathematics classrooms. Systematic reflection on my own experience of teaching over a year revealed a wide range of types of generalisation taking place in mathematics classrooms. The main study then analyses transcripts of fifty-two lessons taught by six teachers teaching at least four hundred students, sampled over a period of two months. The focus is on 'ordinary' lessons where expression of generality is not the main objective. Infonned by the literature, observation notes and student work, a framework is developed with five categories used to distinguish between types of generalisations, which emerge from the transcribed data . These categories are: the object of generalisation, its presumed longevity of relevance, its justification, its origin and the awareness being promoted. Having established the Ubiquitous richness and complexity of expression of generality in mathematics classrooms, the study looks in closer detail at the expression of generality pertinent to mathematical procedures and to mathematical concepts. The study uses the framework, and draws on second language education literature, to re-examine the fifty-two main study lessons. This analysis highlights the complexity of expressing generality through natural language, and suggests that natural language exhibits many of the pitfalls and ambiguities of algebraic expression. Further, it suggests that algebraic notation might offer a clearer means of expressing generality in many cases. The framework developed for considering characteristics of expressions of generality is then applied to the researcher's own classroom, demonstrating how awareness of ways in which generality is expressed can inform pedagogic choices as well as provide a structure for reflection on practice

Fast and Accurate Computation of Orbital Collision Probability for Short-Term Encounters

International audienceThis article provides a new method for computing the probability of collision between two spherical space objects involved in a short-term encounter under Gaussian-distributed uncertainty. In this model of conjunction, classical assumptions reduce the probability of collision to the integral of a two-dimensional Gaussian probability density function over a disk. The computational method presented here is based on an analytic expression for the integral, derived by use of Laplace transform and D-finite functions properties. The formula has the form of a product between an exponential term and a convergent power series with positive coefficients. Analytic bounds on the truncation error are also derived and are used to obtain a very accurate algorithm. Another contribution is the derivation of analytic bounds on the probability of collision itself, allowing for a very fast and - in most cases - very precise evaluation of the risk. The only other analytical method of the literature - based on an approximation - is shown to be a special case of the new formula. A numerical study illustrates the efficiency of the proposed algorithms on a broad variety of examples and favorably compares the approach to the other methods of the literature

Intra-procedural Optimization of the Numerical Accuracy of Programs

Numerical programs performing oating-point computations are very sensitive to the way formulas are written. These last years, several techniques have been proposed concerning the transformation of arithmetic expressions in order to improve their accuracy and, in this article, we go one step further by automatically transforming larger pieces of code containing assignments and control structures. We define a set of transformation rules allowing the generation, under certain conditions and in polynomial time, of larger expressions by performing limited formal computations, possibly among several iterations of a loop. These larger expressions are better suited to improve the numerical accuracy of the target variable. We use abstract interpretation-based static analysis techniques to over-approximate the roundoff errors in programs and during the transformation of expressions. A prototype has been implemented and experimental results are presented concerning classical numerical algorithm analysis and algorithm for embedded systems

Teacher's use of exemplification and explanations in mediating the object of learning

A research report submitted to the Faculty of Science, University of the Witwatersrand, in partial fulfilment of the requirements for the degree of Master of Science. November 2017.This study examined teachers' use of exemplification and explanations in the teaching of algebraic expressions. In particular the focus was on the selection and sequencing of examples as well as what a teacher does with these examples in terms of their explanations to maintain the focus or object of learning. Adler and Ronda's (2015) Mathematical Discourse in Instruction (MDI) framework was the foundation of this study's conceptual and analytical framework and was complemented with the work of Stein (2000) and Moschovich (1999, 2015). Data was collected from two Grade 8 teachers through the use of video-recordings and transcripts. The data was then analysed based on the themes that emerged from the conceptual framework. The findings revealed that the examples themselves had the potential to restrict the object of learning and together with the teacher's corresponding explanatory talk could reduce or shift the object of learning from translating algebraic expressions to focusing on procedures. The findings show how each component of MDI worked separately and then together to mediate the of the object of learning, but this study has additionally highlighted how the components themselves, namely exemplification and explanatory talk, have a direct effect on each other.LG201

Spartan Daily, November 24, 1980

Volume 75, Issue 60https://scholarworks.sjsu.edu/spartandaily/6695/thumbnail.jp

Transformation of a PID Controller for Numerical Accuracy

Numerical programs performing floating-point computations are very sensitive to the way formulas are written. Several techniques have been proposed concerning the transformation of expressions in order to improve their accuracy and now we aim at going a step further by automatically transforming larger pieces of code containing several assignments and control structures. This article presents a case study in this direction. We consider a PID controller and we transform its code in order to improve its accuracy. The experimental data obtained when we compare the different versions of the code (which are mathematically equivalent) show that those transformations have a significant impact on the accuracy of the computation