Arithmetical Investigations

Arithmetical Investigations

Author: Shai M. J. Haran

Publisher: Springer Science & Business Media

Published: 2008-04-25

Total Pages: 224

ISBN-13: 3540783784

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In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Zp which are the inverse limit of the finite rings Z/pn. This gives rise to a tree, and probability measures w on Zp correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L2(Zp,w). The real analogue of the p-adic integers is the interval [-1,1], and a probability measure w on it gives rise to a special basis for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For special (gamma and beta) measures there is a "quantum" or "q-analogue" Markov chain, and a special basis, that within certain limits yield the real and the p-adic theories. This idea can be generalized variously. In representation theory, it is the quantum general linear group GLn(q)that interpolates between the p-adic group GLn(Zp), and between its real (and complex) analogue -the orthogonal On (and unitary Un )groups. There is a similar quantum interpolation between the real and p-adic Fourier transform and between the real and p-adic (local unramified part of) Tate thesis, and Weil explicit sums.


Disquisitiones Arithmeticae

Disquisitiones Arithmeticae

Author: Carl Friedrich Gauss

Publisher: Springer

Published: 2018-02-07

Total Pages: 491

ISBN-13: 1493975609

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Carl Friedrich Gauss’s textbook, Disquisitiones arithmeticae, published in 1801 (Latin), remains to this day a true masterpiece of mathematical examination. .


Arithmetical Investigations

Arithmetical Investigations

Author: Shai M. J. Haran

Publisher: Springer

Published: 2008-04-25

Total Pages: 224

ISBN-13: 3540783792

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In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Zp which are the inverse limit of the finite rings Z/pn. This gives rise to a tree, and probability measures w on Zp correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L2(Zp,w). The real analogue of the p-adic integers is the interval [-1,1], and a probability measure w on it gives rise to a special basis for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For special (gamma and beta) measures there is a "quantum" or "q-analogue" Markov chain, and a special basis, that within certain limits yield the real and the p-adic theories. This idea can be generalized variously. In representation theory, it is the quantum general linear group GLn(q)that interpolates between the p-adic group GLn(Zp), and between its real (and complex) analogue -the orthogonal On (and unitary Un )groups. There is a similar quantum interpolation between the real and p-adic Fourier transform and between the real and p-adic (local unramified part of) Tate thesis, and Weil explicit sums.


Arithmetical Properties of Commutative Rings and Monoids

Arithmetical Properties of Commutative Rings and Monoids

Author: Scott T. Chapman

Publisher: CRC Press

Published: 2005-03-01

Total Pages: 410

ISBN-13: 1420028243

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The study of nonunique factorizations of elements into irreducible elements in commutative rings and monoids has emerged as an independent area of research only over the last 30 years and has enjoyed a recent flurry of activity and advancement. This book presents the proceedings of two recent meetings that gathered key researchers from around the w


Philosophy of Arithmetic

Philosophy of Arithmetic

Author: Edmund Husserl

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 558

ISBN-13: 9401000603

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This volume is a window on a period of rich and illuminating philosophical activity that has been rendered generally inaccessible by the supposed "revolution" attributed to "Analytic Philosophy" so-called. Careful exposition and critique is given to every serious alternative account of number and number relations available at the time.


Grounding Concepts

Grounding Concepts

Author: C. S. Jenkins

Publisher: OUP Oxford

Published: 2008-08-14

Total Pages: 306

ISBN-13: 0191552402

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Grounding Concepts tackles the issue of arithmetical knowledge, developing a new position which respects three intuitions which have appeared impossible to satisfy simultaneously: a priorism, mind-independence realism, and empiricism. Drawing on a wide range of philosophical influences, but avoiding unnecessary technicality, a view is developed whereby arithmetic can be known through the examination of empirically grounded concepts. These are concepts which, owing to their relationship to sensory input, are non-accidentally accurate representations of the mind-independent world. Examination of such concepts is an armchair activity, but enables us to recover information which has been encoded in the way our concepts represent. Emphasis on the key role of the senses in securing this coding relationship means that the view respects empiricism, but without undermining the mind-independence of arithmetic or the fact that it is knowable by means of a special armchair method called conceptual examination. A wealth of related issues are covered during the course of the book, including definitions of realism, conditions on knowledge, the problems with extant empiricist approaches to the a priori, mathematical explanation, mathematical indispensability, pragmatism, conventionalism, empiricist criteria for meaningfulness, epistemic externalism and foundationalism. The discussion encompasses themes from the work of Locke, Kant, Ayer, Wittgenstein, Quine, McDowell, Field, Peacocke, Boghossian, and many others.


The Way It Was

The Way It Was

Author: Donald Saari

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 336

ISBN-13: 9780821826720

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The formative years of the American Mathematical Society coincided with a time of remarkable development in mathematics. During this period, the Bulletin of the American Mathematical Society and its predecessor, The Bulletin of the New York Mathematical Society, served as a primary vehicle for reporting mathematics to American mathematicians. As a result, some of the most important and fundamental work of early twentieth-century mathematics found its way into the Bulletin. Milestone articles include Hilbert's problems presented at the 1900 Paris International Congress of Mathematicians (ICM), Poincare's 1904 lecture on the future of mathematical physics (with commentary suggesting that he was tantalizingly close to capturing the notion of relativity), and Klein's Erlangen program; all of these articles received added publicity when the first English translation was published in the Bulletin. This book reproduces these and other well-written articles from the early Bulletin, offering readers the best way to experience a slice of that time. Other articles in the book include, in particular, a report to American mathematicians about what happened at that important 1900 ICM meeting and three articles from the scientific portion of the 1904 centennial celebration of the Louisiana Purchase: Darboux describing the development of geometry, Pierpont focusing on nineteenth-century mathematics, and Poincare emphasizing the significance of mathematical physics. Accompanying the transition from the nineteenth to twentieth century was that new important thing called ``mathematical rigor''. Included is an article by Klein reflecting the beliefs of the time with his promotion of rigor. These are just some of the many topics characterizing the early days of the developing American mathematical community. The book offers a captivating review of mathematics through the early years of the Bulletin.