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## Riemannian manifolds and conformality

*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/9780198503620.003.0002
- Subject:
- Mathematics, Pure Mathematics

This chapter introduces fundamental concepts from Riemannian geometry, including the Laplacian and harmonic functions on a Riemannian manifold. The next two sections discuss weak conformality and the ... More

## HARMONIC FUNCTIONS AND RADON TRANSFORM ON ALGEBRAIC VARIETIES

*Leon Ehrenpreis*

### in The Universality of the Radon Transform

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198509783
- eISBN:
- 9780191709166
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509783.003.0004
- Subject:
- Mathematics, Mathematical Physics

This chapter looks at a variation of the “harmonic function” introduced by Chevalley. It deals with a noncompact version of the theory described in Chapter 3 (which deals with a compact version). One ... More

## Prolongationl Functions

*Fred Lerdahl*

### in Tonal Pitch Space

- Published in print:
- 2005
- Published Online:
- May 2008
- ISBN:
- 9780195178296
- eISBN:
- 9780199870370
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195178296.003.0005
- Subject:
- Music, Theory, Analysis, Composition

This chapter elaborates a new approach to tonal and harmonic function. To do this, it establishes a frame of reference; so it first develops a tonic-finding method, based on the principle of the ... More

## Harmonic Syntax

*Drew Nobile*

### in Form as Harmony in Rock Music

- Published in print:
- 2020
- Published Online:
- August 2020
- ISBN:
- 9780190948351
- eISBN:
- 9780190948399
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780190948351.003.0001
- Subject:
- Music, Popular, Theory, Analysis, Composition

This chapter adapts traditional Schenkerian analytical methodology to form a theory of rock harmony rooted in the concept of prolongation. The chapter begins with the premise that focusing on ... More

## Harmonic Functions and the Mean-Value Property

*Ali Taheri*

### in Function Spaces and Partial Differential Equations: Volume 1 - Classical Analysis

- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198733133
- eISBN:
- 9780191797712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198733133.003.0001
- Subject:
- Mathematics, Analysis

This chapter introduces harmonic functions and discussing some of their basic properties, including the mean-value property, strong and weak maximum principles, uniqueness of solutions to the ... More

## Perron’s Method of Sub-Harmonic Functions

*Ali Taheri*

### in Function Spaces and Partial Differential Equations: Volume 1 - Classical Analysis

- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198733133
- eISBN:
- 9780191797712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198733133.003.0010
- Subject:
- Mathematics, Analysis

The solvability of the classical Dirichlet problem via the method of sub-harmonic functions (or Perron’s method) is presented in this chapter. Boundary regularity via existence of barriers and ... More

## Navigating the Triadic Universe: Three Compositional Scripts

*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/9780199772698.003.0006
- Subject:
- Music, Theory, Analysis, Composition

Chapter 6 presents three analytic scripts for navigating the triadic universe, and shows how those scripts and their compositional instantiations are represented using the concepts and geometries ... More

## Function Spaces and Partial Differential Equations: Volume 1 - Classical Analysis

*Ali Taheri*

- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198733133
- eISBN:
- 9780191797712
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198733133.001.0001
- Subject:
- Mathematics, Analysis

This book, which is the first volume of two, presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of ‘partial differential equations’ and the associated ... More

## Reflected Dirichlet Spaces

*Zhen-Qing Chen and Masatoshi Fukushima*

### in Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691136059
- eISBN:
- 9781400840564
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691136059.003.0006
- Subject:
- Mathematics, Probability / Statistics

This chapter turns to reflected Dirichlet spaces. It first introduces the notion of terminal random variables and harmonic functions of finite energy for a Hunt process associated with a transient ... More

## Function Spaces and Partial Differential Equations: Volume 2: Contemporary Analysis

*Ali Taheri*

- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198733157
- eISBN:
- 9780191797729
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198733157.001.0001
- Subject:
- Mathematics, Analysis

This book presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of ‘partial differential equations’ and the associated ‘function spaces’. It begins with a ... More

## Harmony

*Daniel Harrison*

### in Pieces of Tradition: An Analysis of Contemporary Tonal Music

- Published in print:
- 2016
- Published Online:
- June 2016
- ISBN:
- 9780190244460
- eISBN:
- 9780190244484
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780190244460.003.0004
- Subject:
- Music, Theory, Analysis, Composition

Common-Practice composition relies on a rule-based distinction between consonance and dissonance. The first move into contemporary tonality was relaxation of the rule that tonic chords had to be ... More

## Formal Structure and Harmonic Function

*Donna Louise Gunn*

### in Discoveries from the Fortepiano: A Manual for Beginners and Seasoned Performers

- Published in print:
- 2015
- Published Online:
- December 2015
- ISBN:
- 9780199396634
- eISBN:
- 9780199396672
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199396634.003.0003
- Subject:
- Music, Performing Practice/Studies

The chapter presents the critical role that formal structure and harmonic function play in understanding Classical Era repertoire, particularly in the galant style. Focus is on interesting atypical ... More

## Cadences

*Ken Stephenson*

### in What to Listen For in Rock: A Stylistic Analysis

- Published in print:
- 2002
- Published Online:
- October 2013
- ISBN:
- 9780300092394
- eISBN:
- 9780300128239
- Item type:
- chapter

- Publisher:
- Yale University Press
- DOI:
- 10.12987/yale/9780300092394.003.0004
- Subject:
- Music, Theory, Analysis, Composition

This chapter examines the cadence of rock music. It shows that when cadences occur they frequently do not follow the formulas of earlier music and that the traditional perfect authentic cadence ... More

## Calculus on 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/9780198526391.003.0005
- Subject:
- Mathematics, Geometry / Topology, Analysis

This chapter develops the theory of differential forms on smooth surfaces and Riemann surfaces. The analysis begins with smooth surfaces, covering cotangent spaces and 1-forms; 2-forms and ... More

## Toward an Analysis of the Music of Prokofiev’s Middle Period

*Konrad Harley*

### in Rethinking Prokofiev

- Published in print:
- 2020
- Published Online:
- March 2020
- ISBN:
- 9780190670764
- eISBN:
- 9780190670801
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780190670764.003.0020
- Subject:
- Music, History, Western

Prokofiev’s music is so individual that it is not uncommon for analysts to come up with new terms and concepts to describe it. Many of his devices have their origin in eighteenth- and ... More

## Moving, Chanting, and Singing for All Students in Instrumental Music

*Susan A. Harvey and David A. Stringham*

### in Winding It Back: Teaching to Individual Differences in Music Classroom and Ensemble Settings

- Published in print:
- 2016
- Published Online:
- April 2016
- ISBN:
- 9780190201616
- eISBN:
- 9780190201654
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780190201616.003.0010
- Subject:
- Music, History, American

Moving, chanting, and singing offer all students at varied developmental levels opportunities to develop their musicianship in instrumental music classrooms. These activities can be wound forward and ... More

## The electrostatics of conductors

*J. Pierrus*

### in Solved Problems in Classical Electromagnetism: Analytical and Numerical Solutions with Comments

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198821915
- eISBN:
- 9780191861055
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198821915.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter begins by proving some important properties of (i) conductors in electrostatic equilibrium, and (ii) harmonic functions. These results underpin most of the remaining questions of Chapter ... More

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