## 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

## 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

## 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

## 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

##
Frobenius Tori and Weights; Getting Elements of *G*_{arith}

_{arith}

*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

##
An *O*(2*n*) 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

## 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 *SO*(*n*) 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