## THE VIRTUES OF CRACKED REASONING

*Mark Wilson*

### in Wandering Significance: An Essay on Conceptual Behaviour

- Published in print:
- 2006
- Published Online:
- January 2007
- ISBN:
- 9780199269259
- eISBN:
- 9780191710155
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199269259.003.0006
- Subject:
- Philosophy, Philosophy of Language

Drawing upon the rich experience gathered within applied mathematics, various ‘facade patterns’ are examined that frequently develop when an originating usage enlarges its descriptive scope through ... 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

## Computations, Transition Formulae, and the Local Index Formula

*Simon Scott*

### in Traces and Determinants of Pseudodifferential Operators

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780198568360
- eISBN:
- 9780191594748
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568360.003.0004
- Subject:
- Mathematics, Analysis

In this chapter the constructions of chapters 1 and 2 for are illustrated with explicit computations. The first part of the chapter presents various methods of computation commonly used in the ... More

## Holomorphic harmonic morphisms

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0008
- Subject:
- Mathematics, Pure Mathematics

The first section of this chapter investigates the general question of when holomorphic maps between almost Hermitian manifolds are harmonic maps or harmonic morphisms. In particular, a holomorphic ... More

## Multivalued harmonic morphisms

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0009
- Subject:
- Mathematics, Pure Mathematics

Multivalued analytic functions are considered to be functions on a Riemann surface, which can be defined either by cutting and pasting or by a graph construction. Multivalued maps in general are ... More

## Nineteenth‐ and Twentieth‐Century Theories of Metrical Conflict

*Harald Krebs*

### in Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann

- Published in print:
- 1999
- Published Online:
- May 2008
- ISBN:
- 9780195116236
- eISBN:
- 9780199871308
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195116236.003.0001
- Subject:
- Music, Theory, Analysis, Composition

This chapter begins by describing the 19th-century origins of some of the basic ideas that underlie the book's approach: the first metaphorical application of the terms “consonance” and “dissonance” ... More

## Riemann Surfaces

*Simon Donaldson*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.001.0001
- Subject:
- Mathematics, Geometry / Topology, Analysis

The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an ... More

## Riemann Surfaces, Coverings, and Hypergeometric Functions

*Paula Tretkoff*

### in Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0003
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and ... More

## Advanced Math: The Hypothetical Conception

*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/9780691167510.003.0006
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter proposes the idea that advanced mathematics is based on hypotheses—that far from being a priori, it is based on hypothetical assumptions. The concept of quasi-empiricism is often linked ... More

## Lorentz Geometry

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0001
- Subject:
- Mathematics, Applied Mathematics

This chapter presents a survey of the basic definitions of Riemannian and Lorentzian differential geometry used in this book. The first nine sections use the simplest formulations, in local ... More

## Meromorphic functions and the Main Theorem for compact Riemann surfaces

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0008
- Subject:
- Mathematics, Geometry / Topology, Analysis

This chapter explains a strategy for proving the fundamental structural results about Riemann surfaces. It first considers the consequence of the Main Theorem for compact Riemann surfaces. It then ... More

## RELATIVISTIC HYDRODYNAMICS

*Miguel Alcubierre*

### in Introduction to 3+1 Numerical Relativity

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199205677
- eISBN:
- 9780191709371
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199205677.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter presents the derivation of dynamical equations governing the motion of a perfect fluid, known as the Euler equations, both in the special and general relativistic cases. It considers ... More

## Integration

*James Davidson*

### in Stochastic Limit Theory: An Introduction for Econometricians

- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.003.0004
- Subject:
- Economics and Finance, Econometrics

The concept of an abstract integral is developed from first principles. Riemann‐Stieltjes and Lebesgue‐Stieltjes integrals are defined. The monotone convergence theorem, fundamental properties of ... More

## TOPOLOGICAL STRINGS

*Marcos Mariño*

### in Chern-Simons Theory, Matrix Models, and Topological Strings

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568490
- eISBN:
- 9780191717604
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568490.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Type-A and type-B topological sigma models are two topological field theories in two dimensions. Although they contain a lot of information in genus 0, they turn out to be trivial for g > 1. This is ... More

## TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION

*Kazuo Fujikawa and Hiroshi Suzuki*

### in Path Integrals and Quantum Anomalies

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198529132
- eISBN:
- 9780191712821
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198529132.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses the quantum anomalies in two-dimensional field theory. Two-dimensional field theory is important in connection with conformal field theory and its applications to string theory ... More

## Holomorphic functions

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0001
- Subject:
- Mathematics, Geometry / Topology, Analysis

This chapter reviews some examples of holomorphic functions in complex analysis. It emphasizes the idea of ‘analytic continuation’, which is a fundamental motivation for Riemann surface theory.

## Facades

*Christopher Pincock*

### in Mathematics and Scientific Representation

- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780199757107
- eISBN:
- 9780199932313
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199757107.003.0013
- Subject:
- Philosophy, Philosophy of Science

Mark Wilson’s book Wandering Significance: An Essay on Conceptual Behavior is considered in this chapter. Wilson argues that common views on mathematical and scientific concepts distort our ... More

## Mathematical Concepts: Fruitfulness and Naturalness

*Jamie Tappenden*

### 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/9780199296453.003.0011
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

This chapter addresses the question of when one concept or definition is properly regarded as more ‘natural’ than another with reference to the principle that a mark of good definitions is their ... More

## Hexatonic Cycles

*Richard Cohn*

### in Audacious Euphony: Chromatic Harmony and the Triad's Second Nature

- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780199772698
- eISBN:
- 9780199932238
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199772698.003.0002
- Subject:
- Music, Theory, Analysis, Composition

Chapter 2 presents the first of two preliminary models of the triadic universe, based on voice-leading proximity, On this conception, the most proximate triads are those whose roots are related by ... More

## Tensors in general relativity

*Ta-Pei Cheng*

### in Relativity, Gravitation and Cosmology: A Basic Introduction

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199573639
- eISBN:
- 9780191722448
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573639.003.0013
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Differentiation of tensor components in a curved space must be handled with extra care. By adding another term (related to Christoffel symbols) to the ordinary derivative operator, we can form a ... More