Search

This paper is devoted to the conception of mathematical objects and methods according to d'Alembert. We first recall his vision of the place of mathematics in the knowledge of nature, then the internal hierarchy of the various fields of this science, based on their degree of abstraction from sensations (§1 and 2). Then we come to the ideas of definitions, primitive ideas , simple ideas , and their generation as well as their generalization (§3 and 4). Then, having looked at what he means by quantities, numbers, quantities, as well as his conception of the objects and rules of algebra as abstract ideas by generalization (§5), we approach the question of the reality of mathematical objects with the example of the irrational (§6). The following paragraphs of the text are devoted to the difficulties encountered in various fields and the way d'Alembert tries to solve them: algebra and negative quantities (§7); principles of geometry (§8); the notion of limit as the basis of infinitesimal calculus (§9). His reflections, even if unfinished, were not without posterity (§10). [ABSTRACT FROM AUTHOR]

Abstract: One of the particularities of information encoded as DNA strands is that a string u contains basically the same information as its Watson–Crick complement, denoted here as . Thus, any expression consisting of repetitions of u and can be considered in some sense periodic. In this paper, we give a generalization of Lyndon and Schützenberger’s classical result about equations of the form , to cases where both sides involve repetitions of words as well as their complements. Our main results show that, for such extended equations, if , then all three words involved can be expressed in terms of a common word t and its complement . Moreover, if , then is an optimal bound. These results are established based on a complete characterization of all possible overlaps between two expressions that involve only some word u and its complement , which is also obtained in this paper. [Copyright &y& Elsevier]

Energy efficiency in irrigation distribution networks can be estimated by following the method proposed by Abadia et al (2008) [Biosystems Engineering, 101, 21–27], in which energy efficiency is quantified in terms of global energy efficiency (GEE), a parameter that is a compound of the pumping energy efficiency (PEE) and the supply energy efficiency (SEE). PEE is related to pump functioning, while SEE is related to the design and management of the distribution system. Calculation of SEE by this method is limited to irrigation networks that have a single water intake at an elevation below that of the irrigated area and a single supplied sector. Results from the method are confusing for networks with more than one water supply point or irrigation sectors supplied by intakes located at elevations higher than that of the irrigated area. This paper proposes a modification of the earlier method that is appropriate to any type of layout and configuration of the irrigation network, with one or several water intake points and supplied sectors. The new model is applied to four different types of collective pressurised irrigation network, and the results are compared with those of the previous model. The influence of different measures taken to improve the SEE and determine the variables that affect this efficiency was analysed. The results show that adequate SEE values can be obtained with the proposed model for all the situations analysed, and that it is possible to quantify the improvements in SEE with the model. Highlights • Knowing the energy efficiency of a network can improve design and energy management. • Previous Supply Energy Efficiency (SEE) definition unfit for some network designs. • New method proposed and applied to four distribution networks. • New method is coherent and appropriate for all networks designs. • New SEE definition allows for identification of badly designed or managed networks. [ABSTRACT FROM AUTHOR]

Abstract: In the paper we generalize the main results presented in Bentkus and Paulauskas (2004) by giving rates of approximation of some semigroups of operators of the order , . Also two classes of operators, generalizing sectorial and quasi-sectorial operators, are introduced and their properties are studied. [Copyright &y& Elsevier]

Abstract: In this paper we attempt to generalize the notion of “unique factorization domain” in the spirit of “half-factorial domain”. It is shown that this new generalization of UFD implies the now well-known notion of half-factorial domain. As a consequence, we discover that one of the standard axioms for unique factorization domains is slightly redundant. [Copyright &y& Elsevier]

In this paper we consider a learning rule whose underlying space, possibly infinite dimensional, is equipped with an inner product. The rule proposed is a generalization of Oja''s maximum eigenfilter algorithm. We study its convergence properties and iterative behavior. We observe a whole variety of dynamical behaviors. We establish conditions on parameter values generating chaoticity as well as asymptotic convergence. [Copyright &y& Elsevier]

In this paper, using the notion of τ-distance, we prove a generalization of Downing–Kirk''s theorem. [Copyright &y& Elsevier]