The Experts below are selected from a list of 2460 Experts worldwide ranked by ideXlab platform
Wim Fias - One of the best experts on this subject based on the ideXlab platform.
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Interacting neighbors: A connectionist model of retrieval in single-digit multiplication
Memory & Cognition, 2005Co-Authors: Tom Verguts, Wim FiasAbstract:For most adults, retrieval is the most common way to solve a single-digit multiplication problem (Campbell & Xue, 2001). Many theories have been proposed to describe the underlying mechanism of Arithmetical fact retrieval. Testing their validity hinges on evaluating how well they account for the basic findings in mental arithmetic. The most important findings are the problem size effect (small multiplication problems are easier than larger ones; cf. 3 × 2 and 7 × 8), the five effect (problems with 5 are easier than can be accounted for by their size), and the tie effect (problems with identical operands are easier than other problems; cf. 8 × 8 and 8 × 7). We show that all existing theories have difficulties in accounting for one or more of these phenomena. A new theory is presented that avoids these difficulties. The basic assumption is that candidate answers to a particular problem are in cooperative/competitive interactions and these interactions favor small, five, and tie problems. The theory is implemented as a connectionist model, and simulation data are described that are in good accord with empirical data.
Liesbeth C. De Wreede - One of the best experts on this subject based on the ideXlab platform.
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A dialogue on the use of arithmetic in geometry: Van Ceulen’s and Snellius’s Fundamenta Arithmetica et Geometrica
Historia Mathematica, 2010Co-Authors: Liesbeth C. De WreedeAbstract:Snellius’s Fundamenta Arithmetica et Geometrica (1615) is much more than a Latin translation of Ludolph van Ceulen’s Arithmetische en Geometrische Fondamenten. Willebrord Snellius both adapted and commented on the Dutch original in his Fundamenta, and thus his Latin version can be read as a dialogue between representatives of two different approaches to mathematics in the early modern period: Snellius’s humanist approach and Van Ceulen’s practitioner’s approach. This article considers the relationship between the Dutch and Latin versions of the text and, in particular, puts some of their statements on the use of numbers in geometry under the microscope. In addition, Snellius’s use of the Fundamenta as an instrument to further his career is explained.
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a dialogue on the use of arithmetic in geometry van ceulen s and snellius s fundamenta Arithmetica et geometrica
Historia Mathematica, 2010Co-Authors: Liesbeth C. De WreedeAbstract:Snellius’s Fundamenta Arithmetica et Geometrica (1615) is much more than a Latin translation of Ludolph van Ceulen’s Arithmetische en Geometrische Fondamenten. Willebrord Snellius both adapted and commented on the Dutch original in his Fundamenta, and thus his Latin version can be read as a dialogue between representatives of two different approaches to mathematics in the early modern period: Snellius’s humanist approach and Van Ceulen’s practitioner’s approach. This article considers the relationship between the Dutch and Latin versions of the text and, in particular, puts some of their statements on the use of numbers in geometry under the microscope. In addition, Snellius’s use of the Fundamenta as an instrument to further his career is explained.
Peter Borwein - One of the best experts on this subject based on the ideXlab platform.
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wallis Arithmetica infinitorum
2004Co-Authors: Lennart Berggren, Jonathan M Borwein, Peter BorweinAbstract:HInc Fequitur, quod Si ex Tabellae prop. 184. locis vacuis unus quilibet numero noto fuppleatur, erunt & reliqui omnes cogniti.
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wallis Arithmetica infinitorum 1655
2000Co-Authors: Lennart Berggren, Jonathan M Borwein, Peter BorweinAbstract:H Inc fequitur, quod. Si ex Tabellae prop. 184. locis vacuis unus quilibet numero noto fuppleatur, enurt & reliqui omnes cogniti.
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Arithmetica infinitorum 1655
1997Co-Authors: Lennart Berggren, Jonathan M Borwein, Peter BorweinAbstract:Hinc fequitur, quod Si ex Tabellae prop. 184. locis vacuis unus quilibet numero noto fuppleatur, erunt & reliqui omnes cogniti.
Jean Christianidis - One of the best experts on this subject based on the ideXlab platform.
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Tracing the early history of algebra: Testimonies on Diophantus in the Greek-speaking world (4th–7th century CE)
Historia Mathematica, 2019Co-Authors: Jean Christianidis, Athanasia MegremiAbstract:Abstract The transmission and reception of the mathesis carried by Diophantus' Arithmetica has not attracted much attention among historians of Greek mathematics, who have devoted their scholarly activity almost exclusively to questions about the proper understanding of the character of the mathematical undertaking of the Alexandrian mathematician. As a result, the common belief is that Diophantus' Arithmetica is presented as an isolated, and thus uncontextualized phenomenon in the history of ancient Greek mathematics. The aim of this paper is to investigate testimonies and other piece of evidence suggesting that Diophantus' heritage was present in intellectual milieus of the Greek-speaking world during the late antique and early medieval times. Special emphasis is given to a number of scholia to the Arithmetical epigrams of the Palatine Anthology which witness the persistence of the method of problem solving taught by Diophantus in the late antique world.
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theory of ratios in nicomachus Arithmetica and series of Arithmetical problems in pachymeres quadrivium reflections about a possible relationship
SHS Web of Conferences, 2015Co-Authors: Athanasia Megremi, Jean ChristianidisAbstract:The voluminous Treatise of the four mathematical sciences of Georgios Pachymeres is the most renowned quadrivium produced in Byzantium. Among its specific features, historians of mathematics have pointed out, is the inclusion of Diophantus, besides Nicomachus and Euclid, in the sources for the Arithmetical section and, accordingly, the incorporation of series of problems and problem-solving in its contents. The present paper investigates the “Diophantine portion” of Pachymeres' treatise and it shows that it is structured according to two criteria intrinsically characterized by seriality : on one hand, the arrangement in which the problems are presented in book I of Diophantus' Arithmetica ; on the other hand, for those problems of which the enunciation involves ratio, the order in which Nicomachus discusses the kinds of ratios in his Arithmetical introduction . Furthermore, it analyses the solutions that Pachymeres offers and argues that Nicomachus' Arithmetical introduction provides the necessary tools for pursuing them.
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the meaning of hypostasis in diophantus Arithmetica
2015Co-Authors: Jean ChristianidisAbstract:Historians of ancient philosophy and theological writers often come up against the puzzling issue of understanding the meaning of the term hypostasis used by different ancient authors. One could hardly expect that the same issue would be of interest for historians of ancient mathematics. Indeed, altogether absent from the works of Euclid, Archimedes, and Apollonius, and scarcely appearing in a nonmathematical context in the works of Heron and Nicomachus, the term hypostasis and its cognates appear 127 times in the six books of Diophantus’ Arithmetica preserved in Greek. This chapter examines Diophantus’ use of the term hypostasis and argues in favour of interpreting it as a term for numbers qua specific, individual entities. It is composed of three parts. The first part discusses the different statuses of numbers in a worked-out problem according to Diophantus’ general method, and the relevant issue of the Diophantine conception of an Arithmetical problem; the second part investigates all instances of the term within Diophantus’ text; and the third part surveys briefly the testimonies of the Byzantine commentators of the Arithmetica, which provide further evidence supporting the interpretation proposed in this paper.
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The way of Diophantus: Some clarifications on Diophantus' method of solution
Historia Mathematica, 2007Co-Authors: Jean ChristianidisAbstract:Abstract In the introduction of the Arithmetica Diophantus says that in order to solve Arithmetical problems one has to “follow the way he (Diophantus) will show.” The present paper has a threefold objective. Firstly, the meaning of this sentence is discussed, the conclusion being that Diophantus had elaborated a program for handling various Arithmetical problems. Secondly, it is claimed that what is analyzed in the introduction is definitions of several terms, the exhibition of their symbolism, the way one may operate with them, but, most significantly, the main stages of the program itself. And thirdly, it is argued that Diophantus' intention in the Arithmetica is to show the way the stages of his program should be practically applied in various Arithmetical problems.
Andreas J. Fallgatter - One of the best experts on this subject based on the ideXlab platform.
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Changes in cortical blood oxygenation during Arithmetical tasks measured by near-infrared spectroscopy
Journal of Neural Transmission, 2009Co-Authors: Melany M. Richter, Kathrin C. Zierhut, Thomas Dresler, Michael M. Plichta, Ann-christine Ehlis, Kristina Reiss, Reinhard Pekrun, Andreas J. FallgatterAbstract:Solving Arithmetical problems is a core skill which is learned starting early in childhood and has been shown to involve a temporo-parietal network. In this study, we investigated hemodynamic concentration changes in oxygenated (O_2Hb) and deoxygenated hemoglobin (HHb) within cortical brain regions by means of near-infrared spectroscopy (NIRS). Ten healthy subjects had to calculate or just read two-digit addition tasks that were either presented as numeric formulas or embedded in text. We found higher increases for O_2Hb in parietal brain regions of both hemispheres for the calculation compared to the reading-only condition. Furthermore, these increases were more pronounced during text-embedded tasks than during numeric tasks. Corresponding decreases of HHb could also be detected. These first NIRS findings on that topic confirm that parietal regions are involved in the processing of arithmetic tasks while the amount of activation seems to depend on task modalities like difficulty or complexity.
