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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 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


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


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


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


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


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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