Author | Farrell, Roger H. author |
---|---|
Title | Multivariate Calculation [electronic resource] : Use of the Continuous Groups / by Roger H. Farrell |
Imprint | New York, NY : Springer New York, 1985 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-8528-8 |
Descript | 376 p. online resource |
1 Introduction and Brief Survey -- 1.1. Aspects of Multivariate Analysis -- 1.2. On the Organization of the Book -- 1.3. Sources and the Literature -- 1.4. Notations -- 2 Transforms -- 2.0. Introduction -- 2.1. Definitions and Uniqueness -- 2.2. The Multivariate Normal Density Functions -- 2.3. Noncentral Chi-Square, F-, and t-Density Functions -- 2.4. Inversion of Transforms and Hermite Polynomials -- 2.5. Inversion of the Laplace and Mellin Transforms -- 2.6. Examples in the Literature -- 3 Locally Compact Groups and Haar Measure -- 3.0. Introduction -- 3.1. Basic Point Set Topology -- 3.2. Quotient Spaces -- 3.3. Haar Measure -- 3.4. Factorization of Measures -- 3.5. Modular Functions -- 3.6. Differential Forms of Invariant Measures on Matrix Groups -- 3.7. Cross-Sections -- 3.8. Solvability, Amenability -- 4 Wishartโs Paper -- 4.0. Introduction -- 4.1. Wishartโs Argument -- 4.2. The Noncentral Wishart Density Function -- 4.3. James on Series, Rank 3 -- 4.4. Related Problems -- 5 The Fubini-Type Theorems of Karlin -- 5.0. Introduction -- 5.1. The Noncentral t-Density -- 5.2. The Wishart Density Function -- 5.3. The Eigenvalues of the Covariance Matrix -- 5.4. The Generalized T -- 5.5. Remarks on Noncentral Problems -- 5.6. The Conditional Covariance Matrix -- 5.7. The Invariant S22-1/2S21S11-1S12S22-1/2 -- 5.8. Some Problems -- 6 Manifolds and Exterior Differential Forms -- 6.0. Introduction -- 6.1. Basic Structural Definitions and Assumptions -- 6.2. Multilinear Forms, Algebraic Theory -- 6.3. Differential Forms and the Operator d -- 6.4. Theory of Integration -- 6.5. Transformation of Manifolds -- 6.6. Lemmas on Multiplicative Functionals -- 6.7. Problems -- 7 Invariant Measures on Manifolds -- 7.0. Introduction -- 7.1. ?nh -- 7.2. Lower Triangular Matrices, Left and Right Multiplication -- 7.3. S(h) -- 7.4. The Orthogonal Group O(n) -- 7.5. Grassman Manifolds Gk,n-k -- 7.6. Stiefel Manifolds Vk,n -- 7.7. Total Mass on the Stiefel Manifold, k = 1 -- 7.8. Mass on the Stiefel Manifold, General Case -- 7.9. Total Mass on the Grassman Manifold Gk,n-k -- 7.10. Problems -- 8 Matrices, Operators, Null Sets -- 8.0. Introduction -- 8.1. Matrix Decompositions -- 8.2. Canonical Correlations -- 8.3. Operators and Gaussian Processes -- 8.4. Sets of Zero Measure -- 8.5. Problems -- 9 Examples Using Differential Forms -- 9.0. Introduction -- 9.1. Density Function of the Critical Angles -- 9.2. Hotelling T2 -- 9.3. Eigenvalues of the Sample Covariance Matrix XtX -- 9.4. Problems -- 10 Cross-Sections and Maximal Invariants -- 10.0. Introduction -- 10.1. Basic Theory -- 10.2. Examples -- 10.3. Examples: The Noncentral Multivariate Beta Density Function -- 10.4. Modifications of the Basic Theory -- 10.5. Problems -- 11 Random Variable Techniques -- 11.0. Introduction -- 11.1. Random Orthogonal Matrices -- 11.2. Decomposition of the Sample Covariance Matrix Using Random Variable Techniques. The Bartlett Decomposition -- 11.3. The Generalized Variance, Zero Means -- 11.4. Noneentral Wishart, Rank One Means -- 11.5. Hotelling T2 Statistic, Noneentral Case -- 11.6. Generalized Variance, Nonzero Means -- 11.7. Distribution of the Sample Correlation Coefficient -- 11.8. Multiple Correlation, Algebraic Manipulations -- 11.9. Distribution of the Multiple Correlation Coefficient -- 11.10. BLUE: Best Linear Unbiased Estimation, an Algebraic Theory -- 11.11. The GaussโMarkov Equations and Their Solution -- 11.12. Normal Theory. Idempotents and Chi-Squares -- 11.13. Problems -- 12 The Construction of Zonal Polynomials -- 12.0. Introduction -- 12.1. Kronecker Products and Homogeneous Polynomials -- 12.2. Symmetric Polynomials in n Variables -- 12.3. The Symmetric Group Algebra -- 12.4. Youngโs Symmetrizers -- 12.5. Realization of the Group Algebra as Linear Transformations -- 12.6. The Center of the Bi-Symmetric Matrices, as an Algebra -- 12.7. Homogeneous Polynomials II. Two-Sided Unitary Invariance -- 12.8. Diagonal Matrices -- 12.9. Polynomials of Diagonal Matrices X -- 12.10. Zonal Polynomials of Real Matrices -- 12.11. Alternative Definitions of Zonal Polynomials. Group Characters -- 12.12. Third Construction of Zonal Polynomials. The Converse Theorem -- 12.13. Zonal Polynomials as Eigenfunctions. Takemuraโs Idea -- 12.14. The Integral Formula of Kates -- 13 Problems for Users of Zonal Polynomials -- 13.0. Introduction -- 13.1. Theory -- 13.2. Numerical Identities -- 13.3. Coefficients of Series -- 13.4. On Group Representations -- 13.5. First Construction of Zonal Polynomials -- 13.6. A Teaching Version -- 14 Multivariate Inequalities -- 14.0. Introduction -- 14.1. Lattice Ordering of the Positive Definite Matrices -- 14.2. Majorization -- 14.3. Eigenvalues and Singular Values -- 14.4. Results Related to Optimality Considerations -- 14.5. Loewner Ordering -- 14.6. Concave and Convex Measures -- 14.7. The FKG-Inequality -- 14.8. Problems