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ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
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Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
Harmonic Analysis and Applications: Appendix B: Functional Analysis
, 1998
"... last property is the triangle inequality , and the function # is a metric. An excellent reference for metric spaces is [Gle91]. b. Let {x n : n # Z d } # M be a sequence. For N = (N 1 , , N d ) # N d , let RN = {n = (n 1 , , n d ) # Z d : N j # n j # N j for j = 1, ..."
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last property is the triangle inequality , and the function # is a metric. An excellent reference for metric spaces is [Gle91]. b. Let {x n : n # Z d } # M be a sequence. For N = (N 1 , , N d ) # N d , let RN = {n = (n 1 , , n d ) # Z d : N j # n j # N j for j = 1, , d}. 1 Also, if n = (n 1 , , n d ), let n 2<F11.97
On the origin and early history of functional analysis
"... In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations in the 18 th and 19 th century to see why there was a need to develop the concepts of functions and limits. We will see how a general theory of ..."
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In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations in the 18 th and 19 th century to see why there was a need to develop the concepts of functions and limits. We will see how a general theory of infinite systems of equations and determinants by Helge von Koch were used in Ivar Fredholm’s 1900 paper on the integral equation ϕ(s) = f(s) + λ
“But you have to remember P. J. Daniell of Sheffield”
"... Abstract: P. J. Daniell is a mysterious figure who appears at several turns in the history of mathematics in the 20th century, in the fields of integration, stochastic processes, statistics, control engineering and even in the history of English mathematical education. The main focus of this paper i ..."
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Abstract: P. J. Daniell is a mysterious figure who appears at several turns in the history of mathematics in the 20th century, in the fields of integration, stochastic processes, statistics, control engineering and even in the history of English mathematical education. The main focus of this paper is on Daniell’s work in relation to the development of probability in the twentieth century. But as it seems that no survey of his work and life has been attempted for 60 years I try to consider all his contributions and place them in an appropriate historical context. Résumé: P. J. Daniell est un personnage mystérieux qui apparaît à plusieurs moments clefs de l’histoire des mathématiques du 20ème siècle, dans le domaine de l’intégration, des processus stochastiques, des statistiques, de la commande optimale et même dans l’histoire de l’éducation mathématique en Angleterre. Ce papier se concentre sur le travail de Daniell en relation avec le développement des probabilités au vingtième siècle. Comme aucune description de sa vie et de son œuvre n’a sembletil été réalisée depuis 60 ans, nous essayons de dresser un tableau de l’ensemble de ses contributions et de les placer dans un contexte historique approprié.
PARTNERS: FUNCTIONAL ANALYSIS AND TOPOLOGY
, 2005
"... Functional analysis and topology were born in the first two decades of the twentieth century and each has greatly influenced the other. Identifying the dual space—the space of continuous linear functionals—of a normed space played an especially important role in the formative years of functional ..."
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Functional analysis and topology were born in the first two decades of the twentieth century and each has greatly influenced the other. Identifying the dual space—the space of continuous linear functionals—of a normed space played an especially important role in the formative years of functional
Press, 2008. “THERE IS NO ONTOLOGY HERE”: VISUAL AND STRUCTURAL GEOMETRY IN ARITHMETIC
"... In Diophantine geometry one looks for solutions of polynomial equations which lie either in the integers, or in the rationals, or in their analogues for number fields. Such polynomial equations {Fi(T1,... Tn)} define a subscheme of affine space An over the integers which can have points in an arbitr ..."
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In Diophantine geometry one looks for solutions of polynomial equations which lie either in the integers, or in the rationals, or in their analogues for number fields. Such polynomial equations {Fi(T1,... Tn)} define a subscheme of affine space An over the integers which can have points in an arbitrary commutative ring R. (Faltings, 2001, 449) Structuralists in philosophy of mathematics can learn from the current heritage of the ancient arithmetician Diophantus. A list of polynomial equations defines a kind of geometric space called a scheme. By one definition these schemes are countable sets built from integers in very much the way that Leopold Kronecker approached pure arithmetic. In another version every scheme is a functor as big as the universe of sets. The two versions are often mixed together because they give precisely the same structural relations between schemes. The practice was vividly put by André Joyal in conversation: “There is no ontology here. ” Mathematicians work rigorously with relations among schemes without choosing between the definitions. The tools
URL: www.emis.de/journals/AFA/ SOME REMARKS ON THE HISTORY OF FUNCTIONAL ANALYSIS
"... Abstract. Several information on the beginning of functional analysis as an important and powerful chapter of mathematics, on the results and people, are given. 1. introduction Annals of Functional Analysis is a new established journal devoted to publication of high quality papers mainly in function ..."
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Abstract. Several information on the beginning of functional analysis as an important and powerful chapter of mathematics, on the results and people, are given. 1. introduction Annals of Functional Analysis is a new established journal devoted to publication of high quality papers mainly in functional analysis and operator theory. As the first paper of the first issue of the journal, it is natural to recall some information on the beginning of functional analysis. Nobody can summarize the whole History of Functional Analysis in a few pages, so that we try only to present some remarks on the history of functional analysis by focusing on Banach spaces and works of Stefan Banach. 2. “Studia mathematica ” and “Théorie des opérations linéaires” It is almost always difficult and controversial to decide what period can be regarded as the beginning of a mathematical theory. Of course, to make a theory, several preparations must take place, sometimes for several centuries. Not always several mathematical events lead to the birth of the whole theory. One of possible definitions of the beginning of a theory may be that the moment of the birth of a theory is when the first fundamental monograph about the theory was published and, moreover, it is the basis to further important research. This
Polycephalic Euclid? Collective aspects in the history of mathematics in the Bourbaki era.
, 2013
"... In 1961, the French historian of mathematics Jean Itard ventured the idea that Euclid might have been no more than a nom de plume for a collective mathematical enterprise. This was anything but innocent, at the time when Bourbaki was so successful and wellknown, and, more generally, collective aspe ..."
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In 1961, the French historian of mathematics Jean Itard ventured the idea that Euclid might have been no more than a nom de plume for a collective mathematical enterprise. This was anything but innocent, at the time when Bourbaki was so successful and wellknown, and, more generally, collective aspects determined more and more the mathematical life of the period. In this paper, we look both at the place of collective practices in the historical writing of mathematicians around Bourbaki and at the role played by concepts representing the collective in their historiography. At first sight, although written collectively, Bourbaki’s Elements of the History of Mathematics, which has been seen as an “internalist history of concepts”, is an unlikely candidate for exhibiting collective aspects. But, as we shall show tension between individuals and collective notion, such as, most famously, “Zeitgeist ” which is presented as orchestrating the development of infinitesimal calculus, are constant. It is interesting to unpack the way in which changes in mathematical practices impacted conceptions of the history of mathematics.