Empirical positivism,an epistemological obstacle in the learning of calculus

Using Chevallard’s anthropological approach to the didactics of mathematics considered as an evolution of Brousseau’s theory of didactic situations, we envision the development of calculus as an epistemological transition between two types of praxeologies, pragmatic and deductive, a praxeology being an anthropological and epistemological model of knowledge. This allows us to depart from a form of dichotomy between formal and intuitive aspects of limits where a mathematical activity should finally become rigorous on some formal definition: we give credit to limits being a pragmatic model of magnitudes relying on mental objects. This understanding of limits is used to argue the relevance of empirical positivism, an epistemology held by Belgian students, as well as pupils, as an obstacle to learning calculus, and show how it is reinforced by learning institutions as a consequence of their inability to give credit to a pragmatic level of rationality.     

Rough functions: p-Variation, calculus, and index estimation

In this paper, we give an overview of several dissipated results on the p-variation property of a function presented in a suitable way. More specifically, we attempt to show: (1) usefulness of this property in a calculus of rough functions; (2) a relatively thorough knowledge of the p-variation property of a sample function of basic stochastic processes; and (3) an almost unexplored area of statistical analysis seeking to estimate the p-variation index. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 1, pp. 123–134, January–March, 2006.     

Minimax and minimax adaptive estimation in multiplicative regression: locally bayesian approach

This paper deals with the non-parametric estimation in the regression with the multiplicative noise. Using the local polynomial fitting and the bayesian approach, we construct the minimax on isotropic H?lder class estimator. Next, applying Lepski’s method we propose an estimator which is optimally adaptive over the collection of isotropic H?lder classes. To prove the optimality of the proposed procedure, we establish in particular the exponential inequality for the deviation of locally bayesian estimator since the parameter estimated. These theoretical results are illustrated by simulation study.     

Finite product representation via multiplicative calculus and its applications to exponential signal processing

In this paper, the multiplicative least square method is introduced and is applied to integrals for the finite product representation of the positive functions. Hence, many nonlinear functions can be represented by well-behaved exponential functions. Product representation produces an accurate representation of signals, especially where exponentials occur. Some real applications of nonlinear exponential signals will be selected to demonstrate the applicability and efficiency of proposed representation.     

Integrating quantitative and qualitative forecasting approaches: organizational learning in an action research case

This article examines the integration of quantitative and judgmental forecasting, focusing on the implementation process and its impacts on the organization. To this end, the study is based on an action research case study in the cement industry. Empirical evidence highlights the critical change management issues that need to be dealt with to implement an integrated forecasting system. The implementation phase needs to be carried out carefully to gain acceptance within the organization and to provide the best results. In addition, the forecasting process and organization need to be aligned to allow a two-way flow of information from the periphery to the centre and vice versa to allow the integration of the two approaches. In this way, not only can forecasting accuracy be improved, but better knowledge and consensus within the organization can also be achieved.     

Optimal estimation and control of discrete multiplicative systems with unknown second-order statistics

The problem of estimation and control for discrete-time systems with multiplicative noise is examined. Such systems occur naturally in the modeling of stochastic systems with random or unknown coefficients and appear to be robust in contrast to LQG regulators which are sensitive to errors in the coefficients.The statistics of the white sequences of the system are unknown. The problem of stochastic estimation and control of such a system is difficult not only because of the unknown statistics but also because the state is not Gaussian.The approach of this work is to convert the stochastic problem to a deterministic game-theoretic one. We find the estimator and controller so as to minimize a suitable performance measure assuming the worst behavior of nature.A set of necessary and sufficient conditions is developed for the existence of a saddle-point estimator. When both estimation and control are considered, two difficulties appear: the optimality conditions are only necessary and the separation principle collapses. As a result, the saddle-point conditions are only necessary. If the covariances belong to sets with maximal points, then the necessary conditions are satisfied at these points. If, on the other hand, they belong to convex and compact sets and the system has a steady state, then the estimation problem alone has always a saddle-point solution.     

A nonparametric estimation of the infection curve

Predicting the future course of an epidemic depends on being able to estimate the current numbers of infected individuals.However,while back-projection techniques allow reliable estimation of the numbers of infected individuals in the more distant past,they are less reliable in the recent past.We propose two new nonparametric methods to estimate the unobserved numbers of infected individuals in the recent past in an epidemic.The proposed methods are noniterative,easily computed and asymptotically normal wit...     

Derivation of linear estimation algorithms from measurements affected by multiplicative and additive noises

This paper addresses the problem of estimating signals from observation models with multiplicative and additive noises. Assuming that the state-space model is unknown, the multiplicative noise is non-white and the signal and additive noise are correlated, recursive algorithms are derived for the least-squares linear filter and fixed-point smoother. The proposed algorithms are obtained using an innovation approach and taking into account the information provided by the covariance functions of the process involved.     

A connectionist model of the organizational learning curve

Organizational learning can be understood as a spontaneous development of routines. Mathematically, this process can be described as a search for better paths on a graph whose nodes are humans and machines. Since the rules for connecting nodes depend on their ability to process goods, the slope of the learning curve may be connected to physical and psychological properties. Two suggestive examples are discussed. Guido Fioretti, born 1964, graduated in Electronic Engineering and obtained a PhD in Economics from the University of Rome “La Sapienza”. He is currently an assistant professor at the University of Bologna, Italy.His research interests span from decision theory to economics and organization science. In particular, he is interested in linking structural development to cognitive processes. The present article has been conceived as a theoretical underpinning of agent-based simulations of organizations. In particular, future applications of the Java Enterprise Simulator () may test the usefulness of the results derived herein.     

On the statistical origins of the learning curve

This paper presents a statistical rationale for the existence of the learning curve phenomenon. We consider a firm which allocates a fixed amount of input into several activities under uncertainty concerning the values of the parameters of the production function. It is shown, under fairly reasonable assumptions, that if the firm learns about the parameters of the production function from previous observations of allocations and outputs, then a learning curve phenomenon will emerge. This result occurs since the estimates of the parameters become more precise over time, and thus the allocation of the production factor into the various activities becomes more efficient (i.e. closer to the optimum allocation that would have been determined if the parameters were known with certainty). Output, therefore, increases and inputs per unit of output decrease as a function of time (and cumulative output), and a learning curve emerges.‘Plateauing’ of the learning curve is discussed, as are the conditions sufficient for the existence of this phenomenon, for which the model presented herein is offered as a possible explanation.     

Model-based estimation of the stress-strain curve of metal strips

The identification of the stress-strain curve of metal strips is a common task in the metals industry. As an alternative to commonly used tensile test machines, an inexpensive, model-based optical measurement method is presented. Particular importance was placed on the cost and usability of the method. The indirect approach computes the stress-strain curve based on a measured strip bending line. For the measurement, a metal strip is bent over a solid roll. A defined weight can be mounted at the end of the strip to control the local bending moment in the strip. The bending line of the strip is optically measured by a camera. The identification is carried out based on an optimization problem, where the quadratic error between the measured and the modelled strip bending line is minimized. Experimental results and measurements from a tensile test machine show a good agreement and thus verify the proposed identification method.     

Integrating transportation and production: an international study case

The problem we study is inspired by the real case of Mesdan S.p.A., an Italian company worldwide leader in the textile machinery sector, which has two production units with two warehouses, one located in Italy (Brescia) and the other in China (Foshan). The critical point in this logistic system is the integration between production and transportation management, given the long distance between Brescia and Foshan. Shipments are performed by the means of different types of vehicles with different unit costs and significantly different lead times. Variable production costs, variable and fixed transportation costs and, possibly, inventory costs are charged in the objective function. Different production policies are compared. Our aim is to determine integrated policies that minimize the total cost of the system. We formulate integer linear programming models for the solution of these problems, and we solve the real instance and carry out a sensitivity analysis of the optimal solution.     

Comparison of non-linear optimization algorithms for yield curve estimation

The yield curve is a very important financial tool used in investment and policy decisions. Its estimation from market data is essentially a non-linear optimization problem. In this paper, we compare a diversity of non-linear optimization algorithms for estimating yield curves based on actual bond market data and conclude that certain classes of algorithms are more effective due to the nature of the problem.     

Analytical solution for a class of learning by doing models with multiplicative uncertainty

We find the closed form optimal solution for a class of learning by doing models, where multiplicative uncertainty is introduced in a piecewise linear cost reduction function. Previous literature does not find the closed form optimal solution for these models. We consider a monopolist, facing a linear demand function. The optimal policy for the resulting problem is shown to be piecewise linear and continuous. The optimal output increases with unit cost for certain values of the latter. Numerical examples are provided.     

Early error detection: an action-research experience teaching vector calculus

This paper describes an action-research experience carried out with second year students at the School of Engineering of the National University of Entre Ríos, Argentina. Vector calculus students played an active role in their own learning process. They were required to present weekly reports, in both oral and written forms, on the topics studied, instead of merely sitting and watching as the teacher solved problems on the blackboard. The students were also asked to perform computer assignments, and their learning process was continuously monitored. Among many benefits, this methodology has allowed students and teachers to identify errors and misconceptions that might have gone unnoticed under a more passive approach.     

Integrating learning theories and application-based modules in teaching linear algebra

The research team of The Linear Algebra Project developed and implemented a curriculum and a pedagogy for parallel courses in (a) linear algebra and (b) learning theory as applied to the study of mathematics with an emphasis on linear algebra. The purpose of the ongoing research, partially funded by the National Science Foundation, is to investigate how the parallel study of learning theories and advanced mathematics influences the development of thinking of individuals in both domains. The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted, in some students, a rich understanding of both domains and that had a mutually reinforcing effect. Furthermore, there is evidence that the deeper insights will contribute to more effective instruction by those who become high school math teachers and, consequently, better learning by their students. The courses developed were appropriate for mathematics majors, pre-service secondary mathematics teachers, and practicing mathematics teachers. The learning seminar focused most heavily on constructivist theories, although it also examined socio-cultural and historical perspectives. A particular theory, Action-Process-Object-Schema (APOS) [10], was emphasized and examined through the lens of studying linear algebra. APOS has been used in a variety of studies focusing on student understanding of undergraduate mathematics. The linear algebra courses include the standard set of undergraduate topics. This paper reports the results of the learning theory seminar and its effects on students who were simultaneously enrolled in linear algebra and students who had previously completed linear algebra and outlines how prior research has influenced the future direction of the project.