Symbolical Algebra as a Foundation for Calculus: D. F. Gregory's Contribution

Duncan Farquharson Gregory (1813–1844) was a mathematician and founder of the Cambridge Mathematical Journal. His 1841 Examples of the Processes of the Differential and Integral Calculus was an extensive revision of Peacock's 1820 textbook of a similar title. Among the new material in Gregory's version is an exposition of symbolical algebra, prominently featuring the method of “separation of symbols.” We examine Gregory's career; the mathematicians he influenced; and Servois and Murphy, who influenced him. We consider Gregory's use of separation of symbols in the Examples and consider whether he believed these techniques to be capable of providing an adequate foundation for calculus. © 2002 Elsevier Science (USA).Le mathématicien Duncan Farquharson Gregory (1813–1844) fut également fondateur du Cambridge Mathematical Journal. Son ouvrage publié en 1841, Examples of the Processes of the Differential and Integral Calculus, constitue une révision substantielle du livre de Peacock au titre similaire. L'un des apports du livre de Gregory est un exposé de l'algèbre symbolique, et en particulier de la méthode de séparation des symboles. Dans cet article nous passons en revue la carrière de Gregory et nous parlons des mathématiciens qu'il a influencés, ainsi que de l'influence que Servois et Murphy ont eu sur ses travaux. Nous considérons son usage de la séparation des symboles dans ses Examples, et nous examinons si Gregory croyait que ses techniques pouvaient constituer un fondement pour le calcul différentiel. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 01A55, 01A70, 01A50, 26-03, 26A03, 34-03, 34A30.     

The Influence of Positivism on the Teaching of Mathematics in Brazil: 1870–1930

This article describes the strong influence of positivism on the teaching of mathematics in Brazil. The dissemination of positivism occurred in a very intensive way from 1870 to 1930, due mainly to the strong leadership of teachers at the military and engineering academies. From its firmly entrenched position in these institutions, the positivistic ideology affected the social, political, pedagogical, and ideological life in Brazil. Here, I identify the main representatives of positivism, who focused their research on Auguste Comte's concept of mathematics. They oriented curricula and programs according to Comte's principles as well as produced mathematics with a distinct positivist bent. Although a marked decline occurred after 1930, the positivistic phenomenon was not exhausted as a research topic, and, indeed, it still has not been entirely extinguished in Brazilian life. Copyright 1999 Academic Press.Este trabalho descreve a forte influência do positivismo no ensino da Matemática no Brasil. A difusão do positivismo aconteceu de forma muita intensa entre 1870 e 1930, devido principalmente a atuação dos docentes-militares, que mantinham uma liderança forte nas academias militares e de engenharia. Nestas instituições a ideologia positivista encontrou uma forte sustentação e pode, então, ter efeitos na vida social, polı́tica, pedagógica e ideológica brasileira. Identificamos os principais representantes do positivismo no cı́rculo acadêmico. Detectamos as primeiras manifestações da concepção de Matemática de Auguste Comte em livros-texto. Identificamos a orientação de currı́culos e programas segundo os preceitos de Comte e analisamos principalmente as obras de Matemática de autores positivistas. O declı́nio do positivismos depois de 1930 também é registrado. O fenômeno positivismo não foi esgotado como tema de pesquisa e tudo indica que ainda não se extinguiu completamente da vida brasileira. Copyright 1999 Academic Press.MSC 1991 subject classifications: 01A55, 01A70.     

Mathematics for Independence: From Spanish Liberal Exile to the Young American Republics

This paper studies one period in the history of mathematical relationships between Spain and Latin America, when right after the first liberating movements at the beginning of the 19th century, the Spanish-exiled liberals in London produced a peculiar kind of mathematical textbook for the new republics, the catechisms. Copyright 1999 Academic Press.Este trabajo estudia un periodo de la historia de los intercambios matemáticos entre España y Latinoamérica inmediatamente posterior a los movimientos liberadores de principios del siglo XIX, cuando los liberales españoles exiliados en Londres produjeron un tipo muy peculiar de textos matemáticos para las nuevas repúblicas americanas, los catecismos. Copyright 1999 Academic Press.AMS 1991 subject classifications: 00A05, 01A55, 01A80.     

Problèmes linéaires dans le Compendy de la praticque des nombres de Barthélemy de Romans et Mathieu Préhoude (1471): Une approche nouvelle basée sur des sources proches du Liber abbaci de Léonard de Pise

The Compendy de la praticque des nombres (1471) is one of a number of commercial arithmetics produced in southern France during the medieval period. Its interest and originality rest in its treatment of problem-solving. The author of the text limited his treatment to an in-depth analysis of only a few types of problems, not treating particular cases but rather emphasizing general methods. The sources from which he drew were very close to the Liber abbaci of Leonardo Fibonacci and many of them were new to the southern French arithmetic tradition. Thus, the Compendy sheds new light on the transmission of arithmetic thought into Europe. Copyright 2000 Academic Press.Le Compendy de la praticque des nombres (1471) est un traité qui appartient au groupe des arithmétiques commerciales du Sud de la France. L'intérêt et l'originalité du texte résident dans la partie consacrée à la résolution de problèmes. L'auteur sélectionne quelques types de problèmes seulement, auxquels il consacre une longue étude, délaissant les cas particuliers pour privilégier les méthodes. Ce faisant, il utilise de nouvelles sources, proches du “Liber abbaci” de Léonard de Pise, étrangères aux autres arithmétiques françaises de la même famille. Le Compendy nous apporte ainsi un éclairage nouveau sur la transmission de l'algorisme. Copyright 2000 Academic Press.MSC subject classifications: 01A35; 01A40.     

Blue Sky and Hot Piles: The Evolution of Radiative Transfer Theory from Atmospheres to Nuclear Reactors

Radiative transfer theory has mirrored many of the trends in the development of applied mathematics during the past century. It began with the development of the phenomenological equation of transfer at the start of the century, designed to treat the passage of light through foggy opaque atmospheres. By the mid-1940s, the theory provided detailed modeling of stellar and planetary atmospheres and subsequently transformed into the remote sensing problem of remote diagnoses of physical conditions and abundances in such environments. During the late 1940s, the need to treat neutron transfer in complex geometries led to an interest in applying these methods to nuclear reactors, using methods of invariant imbedding. Monte Carlo simulations, and integral equations, all areas of continuing study. The formative period, mainly analytic in thrust, ended by the mid-1960s with the comprehensive treatises by Chandrasekhar, Ambartsumian, Kourganoff, Busbridge, Sobolev, and Davison and the 11th AMS Applied Mathematics Symposium on nuclear reactors. Invariant imbedding and doubling methods have been widely developed for scattering problems, mainly in planetary atmospheres and nebular transfer, while improvements in computational methods over the past 30 years have introduced advances in multidimensional, multigroup transfer codes for neutron physics (Boltzmann equation solvers), neutrinos, and significant advances in radiative transfer and the growth of the field of radiation hydrodynamics. © 2002 Elsevier Science (USA).Die Theorie des Strahlungstransport reflektiert viele der Strömungen der angewandten Mathematik des letzten Jahrhunderts wieder. Es begann am Anfang des 20sten Jahrhunderts mit der Entwicklung der phänomenologischen Transportgleichung, die darauf abzielte den Durchgang von Licht duch eine nebelig undurchlässige Atmosphäre zu beschreiben. Mitte der 1940er Jahre erlaubte die Theorie detailliertes modellieren sowohl von Sternatmosphären als auch von planetaren Atmosphären, und entwickelte sich in der Folge in ein diagnostisches Werkzeug das es ermöglichte physikalische Bedingungen und Elementeverteilungen in solchen Situationen zu analysieren. Gegen Ende der 1940er erweckte die Notwendigkeit Neutronentransport in komplizierten Geometrien zu behandeln Interesse an der Anwendung dieser Methoden in Kernreaktoren, insbesondere mittels der Techniken der invarianten Einbettung. Monte Carlo Simulationen und Integralgleichungen, allesamt Gebiete denen fortwährende Aufmerksamkeit gewidmet wurde. Die formgebende Periode, die hauptsächlich analytischer Natur war, ging Mitte der 1960er zu Ende, einerseits mit den umfassenden Monographien von Chandrasekar, Ambartsumian, Kourganoff, Busbridge, Sobolev, und Davison, und andererseits mit dem 11ten AMS Symposium über Angewandte Mathematik im Kontext von Kernreaktoren. Sowohl Methoden der invarianten Einbettung als auch Verdopplungsmethoden wurden weiter entwickelt, insbesondere für Streuprobleme hauptsächlich in Planetenatmosphären und Transport in Nebeln, während Verbesserungen in numerischen Methoden in den letzten 30 Jahren nicht nur zu Fortschritten in höherdimensionalen, multigruppen Transportprogrammen für Neutronenphysik (Boltzmann Gleichung) und Neutrinos geführt haben, sondern auch zu entscheidendem Fortschritt beim Strahlungstransportproblem und zur Entwicklung des Feldes der Strahlungshydrodynamik. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 01A60; 80A20; 85-03; 85A25.     

A Contextualized Historical Analysis of the Kuhn–Tucker Theorem in Nonlinear Programming: The Impact of World War II

When Kuhn and Tucker proved the Kuhn–Tucker theorem in 1950 they launched the theory of nonlinear programming. However, in a sense this theorem had been proven already: In 1939 by W. Karush in a master's thesis, which was unpublished; in 1948 by F. John in a paper that was at first rejected by the Duke Mathematical Journal; and possibly earlier by Ostrogradsky and Farkas. The questions of whether the Kuhn-Tucker theorem can be seen as a multiple discovery and why the different occurences of the theorem were so differently received by the mathematical communities are discussed on the basis of a contextualized historical analysis of these works. The significance of the contexts both mathematically and socially for these questions is discussed, including the role played by the military in the shape of Office of Naval Research (ONR) and operations research (OR). Copyright 2000 Academic Press.En démontrant, en 1950, le théorème qui porte aujourd'hui leur nom, Kuhn et Tucker ont donné naissance à la théorie de la programmation non-linéaire. Cependant, en un sens, ce théorème avait été démontré auparavant, d'abord par W. Karush en 1939 dans un mémoire de maîtrise inédit, par la suite par F. John en 1948 dans un article qui avait d'abord été rejeté par le Duke Mathematical Journal, et peut-être même plus tôt par Ostrogradsky et aussi par Farkas. Le présent article cherche à élucider deux questions: Peut-on considérer le théorème Kuhn–Tucker comme un exemple de découverte multiple? Et pourquoi le théorème a-t-il été reçu si différemment dans les diverses communautés mathématiques? Notre discussion se base sur une analyse historique contextuelle des différents ouvrages. Nous examinons ici l'importance du contexte, tant du point de vue des mathématiques que du point de vue social, y compris le rôle joué par le secteur militaire dans le cadre de l'Office of Naval Research et de la recherche opérationnelle. Copyright 2000 Academic Press.MSC 1991 subject classification: 01A60; 49-03; 52-03; 90-03; 90C30.     

Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead

A. N. Whitehead (1861–1947) contributed notably to the foundations of pure and applied mathematics, especially from the late 1890s to the mid 1920s. An algebraist by mathematical tendency, he surveyed several algebras in his book Universal Algebra (1898). Then in the 1900s he joined Bertrand Russell in an attempt to ground many parts of mathematics in the newly developing mathematical logic. In this connection he published in 1906 a long paper on geometry, space and time, and matter. The main outcome of the collaboration was a three-volume work, Principia Mathematica (1910–1913): he was supposed to write a fourth volume on parts of geometries, but he abandoned it after much of it was done. By then his interests had switched to educational issues, and especially to space and time and relativity theory, where his earlier dependence upon logic was extended to an ontology of events and to a general notion of “process,” especially in human experience. These innovations led to somewhat revised conceptions of logic and of the philosophy of mathematics. © 2002 Elsevier Science (USA).A. N. Whitehead (1861–1947) contribuiu de forma marcante para os Fundamentos da Matemática Pura e Aplicada, especialmente entre o fim da década de 1890 e meados da década de 1920. Sendo um algebrista na sua vertente matemática, fez um levantamento de diversas álgebras no seu livro Universal Algebra (1898). Pouco depois de 1900 juntou-se a Bertrand Russell numa tentativa para basear várias partes da matemática sobre a lógica matemática, que se começava então a desenvolver. Nesse âmbito publicou em 1906 um longo artigo sobre geometria, espaço e tempo, e matéria. O principal resultado da colaboração foi um trabalho em três volumes, Principia Mathematica (1910–1913): estava previsto que Whitehead escrevesse um quarto volume sobre aspectos das geometrias, mas abandonou-o depois de uma boa parte já estar escrita. Por essa altura os seus interesses tinham-se voltado para questões educacionais; especialmente para o espaço e o tempo e para a teoria da relatividade, onde a sua anterior dependência da lógica se estendeu a uma ontologia de acontecimentos e a uma noção geral de “processo” especialmente na experiência humana. Estas inovações levaram a concepções um pouco revistas da lógica e da filosofia da matemática. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 00A30; 01A60; 03-03; 03A05.     

A New Hypothesis on Dürer's Enigmatic Polyhedron in His Copper Engraving “Melencolia I”

A hitherto unnoticed property of Dürer's truncated rhombohedron, a way he could have constructed it, and a possible motivation for this construction are discussed. Copyright 1999 Academic Press.Eine bisher nicht beachtete Eigenschaft von Dürers gestutztem Rhomboeder, ein Weg, wie er ihn konstruieren konnte, und eine mögliche Motivation für diese Konstruktion werden diskutiert. Copyright 1999 Academic Press.1991 AMS subject classification: 01A40.     

Les débuts de l'histoire des mathématiques sur les scènes internationales et le cas de l'entreprise encyclopédique de Felix Klein et Jules Molk

At the beginning of the 20th century, the history of mathematics first appeared on different international scenes as a result of different initiatives of mathematicians, historians, and philosophers. By studying the way in which historians of mathematics took advantage of these opportunities, we have sought the contours, the lines of force, and the equilibria of this process of the internationalization of the history of mathematics on an institutional as well as intellectual plane.This search was focused through the study of the German- and French-language editions of the Encyklopädie der mathematischen Wissenschaften, which is one of the most remarkable examples dating from this period of an international collaboration for mathematics and its history. Copyright 1999 Academic Press.Au début du vingtième siècle l'histoire des mathématiques fait ses débuts sur différentes scènes internationales à l'occasion de différentes initiatives de mathématiciens, d'historiens, de philosophes. En étudiant la façon dont les historiens des mathématiques se saisirent de ces opportunités, nous avons cherché à comprendre les contours, les lignes de force, les équilibres de ce processus d'internationalisation de l'histoire des mathématiques, tant sur le plan institutionnel qu'intellectuel.Cette enquête a été précisée par l'étude des éditions en langues allemande et française de l'Encyklopä-die der mathematischen Wissenschaften qui est un des exemples les plus marquant de cette période d'une collaboration internationale, tant pour les mathématiques que leur histoire. Copyright 1999 Academic Press.MSC subject classifications: 01A60, 01A74.     

The “Error” in the Indian “Taylor Series Approximation” to the Sine

It has been repeatedly noted, but not discussed in detail, that certain so-called “third-order Taylor series approximations” found in the school of the medieval Keralese mathematician M dhava are inaccurate. That is, these formulas, unlike the other series expansions brilliantly developed by M dhava and his followers, do not correspond exactly to the terms of the power series subsequently discovered in Europe, by whose name they are generally known. We discuss a Sanskrit commentary on these rules that suggests a possible derivation explaining this discrepancy, and in the process re-emphasize that the Keralese work on such series was rooted in geometric approximation rather than in analysis per se. © 2001 Elsevier Science (USA).Es ist mehrfach festgestellt bisher aber nicht ausführlich diskutiert worden, daß einige sogenannte Taylor-reihennäherungswerte dritter Ordnung, die in der mittelalterlichen Schule keralesischen M dhava gefunden werden, ungenau sind. Das heißt, diesc Formeln sind den Termen der Potenzreihe, die später in Europa entwickelt wurde und unter dem Namen Taylorreihe bekannt ist, nicht äquivalent, im Gegensatz zu den anderen Entwicklungen von Reihen, die glänzend von M dhava und seinen Nachfolgern entwickelt werden. Wir behandeln einen Sanskritkommentar zu den Regeln, der eine mögliche Herleitung suggeriert, die diese Diskrepanz erklärt. Dabei betonen wir nochmals, daß die keralesische Arbeit über solche Reihen eher in geometrischen Näherungen als in der Analysis an sich ihre Wurzeln hat. © 2001 Elsevier Science (USA).MSC subject classification: 01A32.     

On Birget''s regular embedding

This paper gives another construction of (S)_reg, and a different proof of the uniqueness of coded normal forms.     

On non-Archimedean hilbertian spaces

We consider a special class of non-Archimedean Banach spaces, called Hilbertian, for which every one-dimensional linear subspace has an orthogonal complement. We prove that all immediate extensions of c_o, contained in l^∞, are Hilbertian. In this way we construct examples of Hilbertian spaces over a non-spherically complete valued field without an orthogonal base.     

On the irrationality of factorial series III

In this paper we derive some irrationality and linear independence results for series of the form where is either a non-negative integer sequence with υ_n = o(log n/log log n) or a non-decreasing integer sequence with .     

On the number of groups in clustering

Clustering is the problem of partitioning data into a finite number k of homogeneous and separate groups, called clusters. A good choice of k is essential for building meaningful clusters. In this paper, this task is addressed from the point of view of model selection via penalization. We design an appropriate penalty shape and derive an associated oracle-type inequality. The method is illustrated on both simulated and real-life data sets.     

On open real polynomial maps

We study open polynomial maps from ⁿ to ^p. For n = p we give a complete characterization, and for p = 2, n ≥ 3 we obtain some partial information.     

Empirical Bayes predictive densities for high-dimensional normal models

This paper addresses the problem of estimating the density of a future outcome from a multivariate normal model. We propose a class of empirical Bayes predictive densities and evaluate their performances under the Kullback–Leibler (KL) divergence. We show that these empirical Bayes predictive densities dominate the Bayesian predictive density under the uniform prior and thus are minimax under some general conditions. We also establish the asymptotic optimality of these empirical Bayes predictive densities in infinite-dimensional parameter spaces through an oracle inequality.     

On coefficients of polynomials over finite fields

In this paper we study the relation between coefficients of a polynomial over finite field Fq and the moved elements by the mapping that induces the polynomial. The relation is established by a special system of linear equations. Using this relation we give the lower bound on the number of nonzero coefficients of polynomial that depends on the number m of moved elements. Moreover we show that there exist permutation polynomials of special form that achieve this bound when m|q−1. In the other direction, we show that if the number of moved elements is small then there is an recurrence relation among these coefficients. Using these recurrence relations, we improve the lower bound of nonzero coefficients when m?q−1 and . As a byproduct, we show that the moved elements must satisfy certain polynomial equations if the mapping induces a polynomial such that there are only two nonzero coefficients out of 2m consecutive coefficients. Finally we provide an algorithm to compute the coefficients of the polynomial induced by a given mapping with O(q3/2) operations.