《Applied Group-Theoretic and Matrix Methods》

PART ⅠFINITE GROUPS1

Ⅰ．ABSTRACT FINITE GROUPS1

1．2．The cyclic group of order2

1．3．The dihedral groups3

1．4．S-groups5

1．5．Permutation groups5

1．6．Basic definitions and theorems7

Ⅱ．MATRIX ALGEBRA10

2．2．Linear operators10

2．3．Combinations of operators12

2．4．Matrix algebra and group theory13

2．5．Non-square matrices14

2．6．The theory of vector spaces14

2．7．Eigenvectors and eigenvalues16

2．8．Functions of a diagonal operator18

Ⅲ．COMPLEX AND HYPERCOMPLEX NUMBERS20

3．2．Scalar products and vector duals20

3．3．Related matrices:(i)the adjoint and reciprocal22

3．4．Related matrices:(ii)the transpose and associate23

3．5．Related matrices:(iii)orthogonal and unitary matrices23

3．6．Eigenvalues of special matrices24

3．7．Reciprocal vectors25

3．8．Dyads and dyadics26

3．9．Linear algebras30

3．10．Nomenclature32

Ⅳ．CONJUGATION AND EQUIVALENCE33

4．2．Conjugation34

4．3．Factor groups34

4．4．The implications of equivalence36

4．5．The algebra of classes37

4．6．Oblique axis theory37

4．7．Reduction of a matrix to diagonal form41

4．8．Functions of an arbitrary operator42

Ⅴ．REPRESENTATIONS--THE HEART OF THE MATTER45

5．2．Reducibility45

5．3．The fundamental theorems48

5．4．Some simple corollaries50

5．5．Group characters51

5．6．Induced and Kronecker product representations54

Ⅵ．REVIEW OP GROUPS TO ORDER 2458

6．2．Cyclic groups58

6．3．S-groups58

6．4．Groups up to order 660

6．5．Groups of orders 7 and 862

6．6．Direct product and generalized dihedral groups64

6．7．Groups of orders 9 to 24;symmetric groups66

Ⅶ．MISCELLANEOUS ADDENDA AND NUMERICAL METHODS72

7．2．Matrices not reducible to diagonal form72

7．3．Numerical evaluation of determinants74

7．4．The reciprocal of a matrix76

7．5．Computation of eigenvalues and eigenvectors78

7．6．The orthogonality relations83

7．7．Operator space86

7．8．Some miscellaneous proofs89

PART ⅡAPPLICATIONS OF FINITE GROUPS90

Ⅷ．THE EXTERNAL FORMS OF CRYSTALS90

8．2．Forms without multiple axes93

8．3．Forms with one multiple axis95

8．4．Forms with more than one multiple axis96

8．5．Graphical representation of symmetry types99

Ⅸ．THE INTERNAL STRUCTURE OF CRYSTALS102

9．2．The lattice hypothesis104

9．3．The three-dimensional point lattices:(i)from lattice to symmetry108

9．4．The complete lattice group110

9．5．The three-dimensional point lattices:(ii)from symmetry to lattice118

9．6．The conventional axes and matrices122

9．7．The space groups125

9．8．The reciprocal lattice133

Ⅹ．THE VIBRATIONS OF MOLECULES136

10．2．Procedure for transformation of coordinates137

10．3．The normal modes of symmetrical molecules139

10．4．The structure of the ozone molecule(i)140

10．5．Numerical determination of natural frequencies147

Ⅺ．FACTOR ANALYSIS149

11．2．The basic problem of factor analysis149

11．3．A simple solution151

11．4．Correlation and rank152

11．5．Correlation and error154

11．6．Transformations of the f-space157

11．7．Rotation of axes and oblique factors161

11．8．Application to the problem of aromatic activity163

11．9．Spearman＇s approach170

PAST ⅢCONTINUOUS GROUPS AND APPLICATIONS174

Ⅻ．CONTINUOUS GROUPS:INTRODUCTION174

12．2．Representations and characters in continuous groups174

12．3．The groups u2 and r3177

12．4．Representations of u2 and r3182

12．5．The numerical groups185

12．6．Infinitesimal operators and the group manifold187

12．7．Differential operators and Hilbert space189

12．8．Eigenvector theory in Hilbert space196

12．9．Functions of two or more variables202

ⅩⅢ．THE SYMMETRIC AND FULL LINEAR GROUPS205

13．2．The simple symmetric functions205

13．3．Relations between the symmetric functions207

13．4．The Kronecker mth power209

13．5．The simple characters of fn212

13．6．The characters of the symmetric groups215

13．7．Schur functions219

13．8．Further development of Schur functions222

13．9．Subgroups of the symmetric and full linear groups226

13．10．The spinor group231

ⅩⅣ．TENSORS237

14．2．The metric tensor237

14．3．General notions239

14．4．The volume element and the Laplace operator244

14．5．Tensor properties of matter:(i)symmetrical tensors249

14．6．Tensor properties of matter:(ii)in crystals255

14．7．The identification of tensor types259

ⅩⅤ．RELATIVITY THEORY266

15．2．The Galilean transformation267

15．3．The Lorentz group268

15．4．The representations of the Lorentz group270

15．5．The four-dimensional principle273

15．6．Curvature of space277

15．7．The basic principles of general relativity283

15．8．Relativistic and quantum-relativistic units284

ⅩⅥ．QUANTUM THEORY288

16．2．The basic postulates289

16．3．Vector observables292

16．4．The commutation rules and wave functions296

16．5．The Schrodinger equations301

16．6．The free particle and the simple harmonic oscillator306

16．7．Central force fields and spherical harmonics308

16．8．Perturbation theory315

16．9．Symmetry considerations320

16．10．n-Electron systems335

ⅩⅦ．MOLECULAR STRUCTURE AND SPECTRA344

17．2．Diatomic molecules345

17．3．Methods of molecular analysis352

17．4．Tetrahedral molecules358

17．5．Covalency maxima and stereochemistry366

17．6．Unsaturated compounds374

17．7．Spectra and selection rules385

17．8．The structure of ozone(ii)393

17．9．Related problems in crystals399

ⅩⅧ．EDDINGTON＇S QUANTUM RELATIVITY402

18．2．The nature of measurement403

18．3．Measures and measurables404

18．4．The E-frame and sedenion algebra406

18．5．Rotations and reality conditions409

18．6．Anchoring the E-frame411

18．7．The primary analysis417

18．8．k-Factor theory419

18．9．Strain vectors and quantum theory425

18．10．The double frame and wave tensors430

18．11．The cosmic number440

18．12．Conclusions441

BIBLIOGRAPHY444

INDEX449