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This paper investigates the role of tools in the formation of mathematical practices and the construction of mathematical meanings in the setting of a telecommunication organization through the actions undertaken by a group of technicians in their working activity. The theoretical and analytical framework is guided by the first-generation activity theory model and Leont’ev’s work on the three-tiered explanation of activity. Having conducted a 1-year ethnographic research study, we identified, classified, and correlated the tools that mediated the technicians’ activity, and we studied the mathematical meanings that emerged. A systemic network was generated, presenting the categories of tools such as mathematical (communicative, processes, and concepts) and non-mathematical (physical and written texts). This classification was grounded on data from three central actions of the technicians’ activity, while the constant interrelation and association of these tools during the working process addressed the mathematical practices and supported the construction of mathematical meanings that this group developed from the researchers’ perspective. Technicians’ emerging mathematical meanings referred to place value, spatial, and algebraic relations and were expressed through personal algorithms and metaphorical and metonymic reasoning. Finally, the educational implications of the findings are discussed. [ABSTRACT FROM AUTHOR]

Abstract: The relation between the discrete time Lainiotis filter on the one side and the golden section and the Fibonacci sequence on the other is established. As far as the random walk system is concerned, the relation between the Lainiotis filter and the golden section is derived through the Riccati equation since the steady state estimation error covariance is related to the golden section. The relation between the closed form of the Lainiotis filter and the Fibonacci sequence is also derived. It is shown that the steady state Lainiotis filter computes the state estimate using a linear combination of the previous estimate and of the current measurement with coefficients related to the golden section. A Finite Impulse Response (FIR) implementation of the steady state Lainiotis filter is also proposed, where the filter computes the state estimate as a linear combination of a well-defined set of the last measurements with coefficients which are powers of the golden section. Finally, the scalar generic stochastic dynamic system is considered and the relation between its parameters and the golden section is investigated. [Copyright &y& Elsevier]