Techniques of Multivariate Calculation
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520 Roger H. Farrell
Techniques of Multivariate Calculation
Springer-Verlag Berlin. Heidelberg. New York 19?6
Author Roger H. Farrell Department of Mathematics Cornell University Ithaca, New York 14850 USA
Library of Congress Cataloging in Publication Data
~arrell, Roger U 1929Techniques of multivariate calculation. (Lecture notes in mathematics ; 520) Bibliography: p. Includes index. i. Multivariate analysis. 2. Distribution (Probability theory) 3- Measure theory. I. Title. II. Se~ ties: Lecture notes in mathematics (Berlin) ; 520. QA3.L28 no. 520 [Q&278] 510'.8s [519.5'3176_14~39
AMS Subject Classifications(19?0): 62A05,62E15,62H10,62J10 ISBN 3-540-07695-6 Springer-Verlag Berlin Heidelberg 9 New 9 York ISBN 0-387-076/95-6 Springer-Verlag New York" Heidelberg Berlin 9 /
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin. Heidelberg 1976 Printed in Germany. Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
CONTENTS
Chapter i.
Introduction
Section i.I. 1.2. 1.3. 1.4. 1. 5 . 1.6. Chapter 2. Section 2.0. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. Chapter 3. Section 3.0. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. Chapter 4. Section 4.0. 4.1. 4.2. Chapter 5. Section 5.0. 5.1. 5.2. 5.3. Chapter 6. Section 6.0. 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7.
and brief survey.
The aspects of multivariate analysis The literature History Inference On the organization of these notes Notations
i.i
1.4 1.5 1.5
1.8
1.9
Trazsforms. Introduction Definitions and uniqueness The multivariate normal density functions Noncentral Chi-square, F-, and t-density functions Inversion of transforms and Hermite polynomials Inversion of the Laplace and Mellin transforms Examples in the literature
2.1 2.2 2.6 2.10
2.13 2. i8 2.19
Locally compact groups and Haar measure. Introduction Basic point set topology Quotient spaces Haar measure Factorization of measures Modular functions A remark on matrix group~ Cross-sections Solvability Wishart's
3. i 3.2
3.4 3.5 3.13 3.i7 3.i9
3.2o 3.21
paper.
Introduction Wishart's argument Related problems The noncentral Wishart Introduction James' method James on series, Problems Manifolds
4.1
4. i 4.4 density function. 5.1 5.1 5-5 5.6
rank 3
and exterior differential
forms.
Introduction Basic structural definitions and assumptions Multilinear forms, algebraic theory Differential forms and the operator d. Theory of integration Transformation of manifolds A matrix lemma Problems
6.1 6.2
6.4
6.10
6.16 6.18 6.22 6.24
IV Chapter 7. Section
Invarismt measures on manifolds.
7.1.
~nh
Introduction
7.i
7.2. 7.3. 7.4. 7.5. 7.6.
Lower triangular matr
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