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Arithmetic Certainty

José Ferreirós

in Mathematical Knowledge and the Interplay of Practices

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691167510
eISBN:
9781400874002
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691167510.003.0007
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter considers the idea that we have certainty in our basic arithmetic knowledge. The claim that arithmetical knowledge enjoys certainty cannot be extended to a similar claim about number ... More


On non-parametric statistical methods

Peter Hall

in Celebrating Statistics: Papers in honour of Sir David Cox on his 80th birthday

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198566540
eISBN:
9780191718038
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566540.003.0007
Subject:
Mathematics, Probability / Statistics

This chapter describes some contemporary topics in non-parametric statistics and outlines their relation to computation, particularly some topics on pure mathematics. It opens with a brief comparison ... More


The Number of Infinity

Svetla Slaveva-Griffin

in Plotinus on Number

Published in print:
2009
Published Online:
May 2009
ISBN:
9780195377194
eISBN:
9780199869572
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195377194.003.0004
Subject:
Classical Studies, Ancient Greek, Roman, and Early Christian Philosophy

This chapter analyzes Plotinus’ refutation of the Aristotle’s criticism of Plato’s view of number in the Parmenides. By rejecting any quantitative value of number in the intelligible realm, Plotinus ... More


Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory (AM-175)

Ben Brubaker, Daniel Bump, and Solomon Friedberg

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691150659
eISBN:
9781400838998
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691150659.001.0001
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, ... More


Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Nicholas M. Katz

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.001.0001
Subject:
Mathematics, Number Theory

This book explores an important aspect of number theory—the theory of exponential sums over finite fields and their Mellin transforms—from a new, categorical point of view. The book presents ... More


‘There is no Ontology Here’: Visual and Structural Geometry in Arithmetic

Colin McLarty

in The Philosophy of Mathematical Practice

Published in print:
2008
Published Online:
February 2010
ISBN:
9780199296453
eISBN:
9780191711961
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199296453.003.0015
Subject:
Philosophy, Logic/Philosophy of Mathematics

Today's number theory solves classical problems by structural tools that violate standard philosophical expectations in ontology. The far-reaching practical demands of this mathematics require on one ... More


What Structuralism Achieves

Colin McLarty

in The Philosophy of Mathematical Practice

Published in print:
2008
Published Online:
February 2010
ISBN:
9780199296453
eISBN:
9780191711961
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199296453.003.0014
Subject:
Philosophy, Logic/Philosophy of Mathematics

Mathematicians in practice use a quite different idea of structure than is discussed by many philosophers of mathematics. Great work in geometry and number theory made category theory the standard ... More


Convolution of Perverse Sheaves

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0003
Subject:
Mathematics, Number Theory

Let k be a finite field, q its cardinality, p its characteristic, 𝓁 ≠ p a prime number, and 𝓖/k a smooth commutative group scheme which over k¯ becomes isomorphic to 𝔾ₘ/k¯. This chapter is concerned ... More


Results of Goursat-Kolchin-Ribet Type

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0014
Subject:
Mathematics, Number Theory

This chapter takes up the proofs of Theorems 13.1–13.5. Suppose we are given some number r ≤ 2 of objects N₁; N₂, Nᵣ in Garith of some common “dimension” d ≤ 1. Suppose they are all ι‎-pure of weight ... More


Quasi-Periodicity and Chaos

Robert C. Hilborn

in Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers

Published in print:
2000
Published Online:
January 2010
ISBN:
9780198507239
eISBN:
9780191709340
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507239.003.0006
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter explores the quasi-periodicity route to chaos in which the behaviour is characterized by two (or more) incommensurate frequencies. Poincare sections, frequency-locking, winding numbers, ... More


Epilogue

A. W. Johnson

in Ben Jonson: Poetry and Architecture

Published in print:
1995
Published Online:
March 2012
ISBN:
9780198117599
eISBN:
9780191671005
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198117599.003.0010
Subject:
Literature, 16th-century and Renaissance Literature

This chapter summarizes the preceding discussions and presents some concluding thoughts from the author. The foregoing chapters have revealed close links between Jonson and the Vitruvian aesthetic. ... More


Frobenius Conjugacy Classes

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0006
Subject:
Mathematics, Number Theory

This chapter analyzes Frobenius conjugacy classes. It shows that in either the split or nonsplit case, when χ‎ is good for N, the conjugacy class FrobE,X has unitary eigenvalues in every ... More


A First Construction of Autodual Objects

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0011
Subject:
Mathematics, Number Theory

This chapter presents the construction of autodual objects. It begins with a geometrically irreducible middle extension sheaf Ƒ on G/k which is ι‎-pure of weight zero, and which is not geometrically ... More


An <i>O</i>(2<i>n</i>) Example

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0025
Subject:
Mathematics, Number Theory

This chapter works over a finite field k of odd characteristic. Fix an even integer 2n ≤ 4 and a monic polynomial f(x) ɛ k[x] of degree 2n, f(x)=∑i=02nAixi, A2n=1. The following three assumptions are ... More


Frobenius Tori and Weights; Getting Elements of <i>G<sub>arith</sub></i>

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0017
Subject:
Mathematics, Number Theory

This chapter works on Gₘ/k. It considers an arithmetically semisimple object N ɛ Garith which is pure of weight zero. It assumes it is of the form G[1], with G a middle extension sheaf. Thus, for ... More


Overview

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0002
Subject:
Mathematics, Number Theory

This chapter provides an overview of the present analysis, which grew out of two questions, raised by Ron Evans and Zeev Rudnick respectively, in May and June of 2003. Evans and Rudnick performed ... More


Orthogonal Examples, Especially <i>SO</i>(<i>n</i>) Examples

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0020
Subject:
Mathematics, Number Theory

The orthogonal case is more diffcult than the symplectic one because of the need to distinguish between SO(n) and O(n), which we do not in general know how to do. This chapter works on either the ... More


Introduction

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0001
Subject:
Mathematics, Number Theory

This introductory chapter sets out the book's focus, namely equidistribution results over larger and larger finite extensions of a given finite field. Emanuel Kowalski drew attention to the interest ... More


Fibre Functors

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0004
Subject:
Mathematics, Number Theory

This chapter introduces the fiber functor suggested by Pierre Deligne. The proof that it is in fact a fiber functor is given in the Appendix.


Isogenies, Connectedness, and Lie-Irreducibility

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0009
Subject:
Mathematics, Number Theory

This chapter takes up the proofs of Theorems 8.1 and 8.2. For each prime to p integer n, we have the n'th power homomorphism [n] : G → G. Formation of the direct image is an exact functor from Perv ... More


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