Stravinsky's Chords (II)

Tempo ◽  
1966 ◽  
pp. 2-9
Author(s):  
G. W. Hopkins
Keyword(s):  
Harmonic Analysis ◽  
Harmonic Functions ◽  
Extended Study ◽  
Middle Period ◽  
Soldier's Tale

In The Soldier's Tale there were instances of musical parody which could not possibly be elucidated by simple harmonic analysis within the work's own stylistic terms of reference; for example, it was necessary to allude to Bach's style in order to discuss the harmonies of Stravinsky's chorales. While it is not always correct to view neo-classical works as parodies, generically they present a ‘gloss’ on earlier styles—so that it becomes impossible to analyse such works without reference back, sometimes even to a particular work of a particular forerunner. Clearly an extended study of the use of classical harmonic functions in the works of Stravinsky's middle period would be an undertaking worthy of some months of research and several chapters of exegesis. Here I shall content myself with some instances of his use of cadential formulae.

Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle

2005 ◽  
Vol 11 (4) ◽  
pp. 517-525
Author(s):  
Juris Steprāns
Keyword(s):  
Harmonic Analysis ◽  
Set Theory ◽  
Harmonic Functions ◽  
Maximal Functions ◽  
Pigeonhole Principle ◽  
Cardinal Invariants ◽  
Geometric Objects

AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

scholarly journals Harmonic Functions on Multiplicative Graphs and Interpolation Polynomials

10.37236/1506 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Alexei Borodin ◽  
Grigori Olshanski
Keyword(s):  
Symmetric Group ◽  
Harmonic Analysis ◽  
Harmonic Functions ◽  
Symmetric Functions ◽  
Multivariate Interpolation ◽  
Infinite Symmetric Group ◽  
Interpolation Polynomials

We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur's S and P functions and with Jack symmetric functions. As a by–product, we compute certain Selberg–type integrals.

scholarly journals A harmonic analysis of exponential decay

2020 ◽  
Author(s):  
David Lawunmi ◽  
Soodamani Ramalingam
Keyword(s):  
Harmonic Analysis ◽  
Exponential Function ◽  
Exponential Decay ◽  
Harmonic Functions ◽  
Exponential Functions ◽  
Single Exponential Function ◽  
Single Exponential

We analyse the decay of a single exponential function and develop an algorithm to determine the exponent and the constant, C, (C exp(-kt)) associated with this function . In essence this approach involves `transforming' exponential functions into harmonic functions. This manoeuvre allows techniques that are used to analyse harmonic functions to be used to characterise decaying exponential functions.

scholarly journals V. On the integrals of the squares of ellipsoidal surface harmonic functions

1904 ◽  
Vol 203 (359-371) ◽  
pp. 111-137
Keyword(s):  
Harmonic Analysis ◽  
Harmonic Functions ◽  
Essential Step ◽  
Ellipsoidal Harmonic ◽  
Surface Harmonic ◽  
In Series ◽  
Ellipsoidal Surface ◽  
The Stability

This paper forms a sequel to three others published in Series A of the. ‘ Philosophical Transactions,’ namely, “ On Ellipsoidal Harmonic Analysis,” vol. 197, pp. 461-557, “ On the Pear-shaped Figure of Equilibrium of a Rotating Mass of Liquid,” vol. 198, pp. 301-381, and “ On the Stability of the Pear-shaped Figure of Equilibrium, &c.,” vol. 200, pp. 251-314. 1 shall refer to these three papers as “ Harmonics,” “ The Pear-shaped Figure,” and “ Stability.” In “ Harmonics,” the functions being expressed approximately, approximate formulae are found for the integrals over the surface of the ellipsoid of the squares of all the surface harmonics. These integrals are of course required whenever it is proposed to make practical use of this method of analysis, and the evaluation of them is therefore an absolutely essential step towards any applications.

scholarly journals On the integrals of the squares of ellipsoidal surface harmonic functions

1904 ◽  
Vol 72 (477-486) ◽  
pp. 492-492
Keyword(s):  
Harmonic Analysis ◽  
Harmonic Functions ◽  
Practical Applications ◽  
Ellipsoidal Harmonic ◽  
Surface Harmonic ◽  
In Series ◽  
Ellipsoidal Surface

This paper is a sequel to three others on ellipsoidal harmonic analysis and its applications, published in Series A of the ‘Philosophical Transactions,’ vol. 197, pp. 461—557; vol. 198, pp. 301—331 and vol. 200, pp. 251—314. The integrals referred to in the title are absolutely essential for practical applications of this method of analysis. A table of all such integrals is given in the first of the above-named papers, but the results are only approximate. In the present paper the rigorous forms of the integrals, numbering 1 + 3 + 5 + 7, are given for the surface harmonics of orders 0, 1, 2, 3.

Representation Theory and Harmonic Analysis of Wreath Products of Finite Groups

2009 ◽  
Author(s):  
Tullio Ceccherini-Silberstein ◽  
Fabio Scarabotti ◽  
Filippo Tolli
Keyword(s):  
Representation Theory ◽  
Harmonic Analysis ◽  
Finite Groups ◽  
Wreath Products

Classical and Multilinear Harmonic Analysis

2009 ◽  
Author(s):  
Camil Muscalu ◽  
Wilhelm Schlag
Keyword(s):  
Harmonic Analysis
Sign in / Sign up

Export Citation Format

Share Document