Applied Group-Theoretic and Matrix MethodsPDF电子书下载

PART Ⅰ FINITE GROUPS 1

Ⅰ．ABSTRACT FINITE GROUPS 1

1．2．The cyclic group of order 2

1．3．The dihedral groups 3

1．4．S-groups 5

1．5．Permutation groups 5

1．6．Basic definitions and theorems 7

Ⅱ．MATRIX ALGEBRA 10

2．2．Linear operators 10

2．3．Combinations of operators 12

2．4．Matrix algebra and group theory 13

2．5．Non-square matrices 14

2．6．The theory of vector spaces 14

2．7．Eigenvectors and eigenvalues 16

2．8．Functions of a diagonal operator 18

Ⅲ．COMPLEX AND HYPERCOMPLEX NUMBERS 20

3．2．Scalar products and vector duals 20

3．3．Related matrices:(i)the adjoint and reciprocal 22

3．4．Related matrices:(ii)the transpose and associate 23

3．5．Related matrices:(iii)orthogonal and unitary matrices 23

3．6．Eigenvalues of special matrices 24

3．7．Reciprocal vectors 25

3．8．Dyads and dyadics 26

3．9．Linear algebras 30

3．10．Nomenclature 32

Ⅳ．CONJUGATION AND EQUIVALENCE 33

4．2．Conjugation 34

4．3．Factor groups 34

4．4．The implications of equivalence 36

4．5．The algebra of classes 37

4．6．Oblique axis theory 37

4．7．Reduction of a matrix to diagonal form 41

4．8．Functions of an arbitrary operator 42

Ⅴ．REPRESENTATIONS--THE HEART OF THE MATTER 45

5．2．Reducibility 45

5．3．The fundamental theorems 48

5．4．Some simple corollaries 50

5．5．Group characters 51

5．6．Induced and Kronecker product representations 54

Ⅵ．REVIEW OP GROUPS TO ORDER 24 58

6．2．Cyclic groups 58

6．3．S-groups 58

6．4．Groups up to order 6 60

6．5．Groups of orders 7 and 8 62

6．6．Direct product and generalized dihedral groups 64

6．7．Groups of orders 9 to 24;symmetric groups 66

Ⅶ．MISCELLANEOUS ADDENDA AND NUMERICAL METHODS 72

7．2．Matrices not reducible to diagonal form 72

7．3．Numerical evaluation of determinants 74

7．4．The reciprocal of a matrix 76

7．5．Computation of eigenvalues and eigenvectors 78

7．6．The orthogonality relations 83

7．7．Operator space 86

7．8．Some miscellaneous proofs 89

PART Ⅱ APPLICATIONS OF FINITE GROUPS 90

Ⅷ．THE EXTERNAL FORMS OF CRYSTALS 90

8．2．Forms without multiple axes 93

8．3．Forms with one multiple axis 95

8．4．Forms with more than one multiple axis 96

8．5．Graphical representation of symmetry types 99

Ⅸ．THE INTERNAL STRUCTURE OF CRYSTALS 102

9．2．The lattice hypothesis 104

9．3．The three-dimensional point lattices:(i)from lattice to symmetry 108

9．4．The complete lattice group 110

9．5．The three-dimensional point lattices:(ii)from symmetry to lattice 118

9．6．The conventional axes and matrices 122

9．7．The space groups 125

9．8．The reciprocal lattice 133

Ⅹ．THE VIBRATIONS OF MOLECULES 136

10．2．Procedure for transformation of coordinates 137

10．3．The normal modes of symmetrical molecules 139

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

10．5．Numerical determination of natural frequencies 147

Ⅺ．FACTOR ANALYSIS 149

11．2．The basic problem of factor analysis 149

11．3．A simple solution 151

11．4．Correlation and rank 152

11．5．Correlation and error 154

11．6．Transformations of the f-space 157

11．7．Rotation of axes and oblique factors 161

11．8．Application to the problem of aromatic activity 163

11．9．Spearman＇s approach 170

PAST Ⅲ CONTINUOUS GROUPS AND APPLICATIONS 174

Ⅻ．CONTINUOUS GROUPS:INTRODUCTION 174

12．2．Representations and characters in continuous groups 174

12．3．The groups u2 and r3 177

12．4．Representations of u2 and r3 182

12．5．The numerical groups 185

12．6．Infinitesimal operators and the group manifold 187

12．7．Differential operators and Hilbert space 189

12．8．Eigenvector theory in Hilbert space 196

12．9．Functions of two or more variables 202

ⅩⅢ．THE SYMMETRIC AND FULL LINEAR GROUPS 205

13．2．The simple symmetric functions 205

13．3．Relations between the symmetric functions 207

13．4．The Kronecker mth power 209

13．5．The simple characters of fn 212

13．6．The characters of the symmetric groups 215

13．7．Schur functions 219

13．8．Further development of Schur functions 222

13．9．Subgroups of the symmetric and full linear groups 226

13．10．The spinor group 231

ⅩⅣ．TENSORS 237

14．2．The metric tensor 237

14．3．General notions 239

14．4．The volume element and the Laplace operator 244

14．5．Tensor properties of matter:(i)symmetrical tensors 249

14．6．Tensor properties of matter:(ii)in crystals 255

14．7．The identification of tensor types 259

ⅩⅤ．RELATIVITY THEORY 266

15．2．The Galilean transformation 267

15．3．The Lorentz group 268

15．4．The representations of the Lorentz group 270

15．5．The four-dimensional principle 273

15．6．Curvature of space 277

15．7．The basic principles of general relativity 283

15．8．Relativistic and quantum-relativistic units 284

ⅩⅥ．QUANTUM THEORY 288

16．2．The basic postulates 289

16．3．Vector observables 292

16．4．The commutation rules and wave functions 296

16．5．The Schrodinger equations 301

16．6．The free particle and the simple harmonic oscillator 306

16．7．Central force fields and spherical harmonics 308

16．8．Perturbation theory 315

16．9．Symmetry considerations 320

16．10．n-Electron systems 335

ⅩⅦ．MOLECULAR STRUCTURE AND SPECTRA 344

17．2．Diatomic molecules 345

17．3．Methods of molecular analysis 352

17．4．Tetrahedral molecules 358

17．5．Covalency maxima and stereochemistry 366

17．6．Unsaturated compounds 374

17．7．Spectra and selection rules 385

17．8．The structure of ozone(ii) 393

17．9．Related problems in crystals 399

ⅩⅧ．EDDINGTON＇S QUANTUM RELATIVITY 402

18．2．The nature of measurement 403

18．3．Measures and measurables 404

18．4．The E-frame and sedenion algebra 406

18．5．Rotations and reality conditions 409

18．6．Anchoring the E-frame 411

18．7．The primary analysis 417

18．8．k-Factor theory 419

18．9．Strain vectors and quantum theory 425

18．10．The double frame and wave tensors 430

18．11．The cosmic number 440

18．12．Conclusions 441

BIBLIOGRAPHY 444

INDEX 449