|
|
 |
| Авторизация |
|
|
|
| Поиск по указателям |
|
|
|
|
|
|
|
 |
|
|
|
| Campbell J.E. — Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups |
|
|
|
| Предметный указатель |
Abelian group, definition of 17
Abelian group, simplest form of, when all its operators are unconnected 85
Abelian operations of a group, condition that a group may have 71
Abelian operations of a group, definition of 16
Abelian operations of a group, if a group has none, it has the structure of the linear group (the adjoint group) given 73
Abelian sub-system of functions, definition of 218
Admit, a contact transformation by a function or equation 278
Admit, an infinitesimal transformation, by a function 82
Admit, by a complete system of operators or of differential equations 93
Admit, when an operation is said to be admitted by a group 16
Alternant, of two functions 196
Alternant, of two linear operators, definition of 8
Ampere's equation, the group then admitted 307
Ampere's equation, when it can be transformed to s=0 243
Bilinear equations, defining a contact transformation 257
Bilinear equations, simplified by projective transformation 257 268
Burnside, quoted 2 165 406
Canonical equations of a group 45
Canonical equations of a group, canonical form varies with choice of fundamental operators 162
Canonical equations of a group, relation between canonical parameters of an operation and its inverse 46
Characteristic function of an infinitesimal contact transformation 277
Characteristic function of the alternant of  and  285
Characteristic function of the contact operator of the plane x, z regarded as an operator in space x, y, z 371
Characteristic manifold of an equation or function, definition of 279
Characteristic manifold of an equation or function, one passes through every element of space 279
Characteristic manifold of an equation or function, properties of 279 280
Co-gredient transformation schemes, definition of 15
Complete system of homogeneous functions, can be transformed by a homogeneous contact transformation to any other system of the same structure 235
Complete system of homogeneous functions, definition of 213 215
Complete system of homogeneous functions, if of degree zero, in involution 215
Complete system of homogeneous functions, is a subsystem within a system not containing Abelian functions 224
Complete system of homogeneous functions, reduced to simplest form 222 223
Complete system of linear partial differential equations, condition that they should admit an infinitesimal transformation 93
Complete system of operators, definition of 82
Complete system of operators, in normal form 83
Complete system of operators, when permutable 84
Complex numbers, connexion of, with a class of linear groups 406—410
Complexes, linear, of lines, elementary properties of 255—257
Complexes, tetrahedral 269
Conformal group 32
Conformal group, isomorphic with the projective group of a linear complex 305 396
Conjugate elements, definition of 260
Conjugate operations, definition of 16
Conjugate sub-group, definition of 17
Conjugate sub-group, method of finding all 183—185
Contact groups, connexion with Pfaff's problem 293
Contact groups, fundamental theorems on 287—290
Contact groups, in the plane regarded as point groups in space 302
Contact groups, when reducible 292
Contact groups, when similar 290
Contact transformation which transforms straight lines into sphere 262
Contact transformation with symmetrical generating equations 268
Contact transformation with symmetrical generating equations, examples on this method of transformation 274
Contact transformation with symmetrical generating equations, planes, to twisted cubies 269
Contact transformation with symmetrical generating equations, straight lines, to quadrics 271
Contact transformation with symmetrical generating equations, transforms points to lines of tetrahedral complex 269
Contact transformations, generate a group 241
Contact transformations, geometrical interpretation 280
Contact transformations, given when  given 229
Contact transformations, homogeneous, definition of 228
Contact transformations, how the infinitesimal operator is transformed 286
Contact transformations, infinitesimal 276. See also under Extended
Contact transformations, nonhomogeneous, definition of 240
Contact transformations, points into minimum lines 261
Contact transformations, positive and negative correspondents to a sphere 263
Contact transformations, spheres in contact 264
Contact transformations, when one set of functions can be transformed to another by aid of 236
Continuous group, definition of 3
Contracted operators of a group with respect to equations admitting the group 128
Contracted operators of a group with respect to equations admitting the group, generate a group 129
Contracted operators of a group with respect to equations admitting the group, number of unconnected operators in this group 130
Coordinates of a surface, definition of 135
Correspondence between manifolds in two spaces 262 268 304
Correspondence established between the points of two spaces 151 152
Correspondence of isomorphic groups 162 163
Correspondents, positive and negative, of a sphere, definition of 263
Dependent, when an operator is said to be, on others 7
Differential equation  transformations admitted by 28
Differential equation, of the conic given by tbe general Cartesian equation 324
Differential equation, of the cuspidal cubic 326
Differential equations, method of finding the complete integral of non-linear 204
Differential equations, partial of first order, theory of the solution of linear, admitting known infinitesimal transformations 90—112
Differential invariants of a group defined 320
Differential invariants of a group defined, how obtained 320
Differential invariants of the group  321
Differential invariants of the group of movements in non-Euclidean space 330
Differential invariants of the projective group of the plane 324
Differential invariants, absolute 324
Distinct, when infinitesimal transformations are said to be 95
Dupin's cyclide, transformed into a quadric 265
Effective parameters, definition of 7
Element, linear element, definition of 280
Element, of space, and united elements, definitions of 194
Elliott, quoted 55
Engel's theorem 36
Equations admitting a given group, examples on method 132
Equations admitting a given group, how to obtain 130
Equivalent, when two function or equation systems are said to be 197
Euler's transformation formulae 20
Extended contact transformations in explicit form for the plane 296
Extended contact transformations, circles into circles 300
Extended contact transformations, explicit form of operators in space 305
Extended contact transformations, operators of 295
Extended contact transformations, transformation of this group 302—304
Extended contact transformations, transforming straight line to straight lines 297
Extended operators of projective group of the plane 322 323
Extended operators of, the group 321 322
Extended operators of, the group of movements in non-Euclidean space 327
Extended point transformations, circles to circles 298
Extended point transformations, explained 24
Extended point transformations, extended point group 288
Extended point transformations, formulae for 24
Extended point transformations, illustrative example 25
Extended point transformations, structure of 290
Extended point transformations, transforming straight lines to straight lines 297
Finite continuous transformation groups, contact groups 287
Finite continuous transformation groups, definition of 5
Finite continuous transformation groups, origin of theory of 100
Finite operations of a group generated from infinitesimal ones 45
Finite operations of a group generated from infinitesimal ones, example on method 48
Finite operations of a group generated from infinitesimal ones, method of obtaining 47
Forsyth, quoted 36 77 88 211 217
Fundamental functions used in invariant theory of groups 119
Fundamental functions used in invariant theory of groups, how found 121
Fundamental theorems on groups, first 38
Fundamental theorems on groups, first, and its converse 66
Fundamental theorems on groups, resume 80
Fundamental theorems on groups, second 51
Fundamental theorems on groups, second, and converse 57—59
Fundamental theorems on groups, similar theorems hold for contact groups 287—290
Fundamental theorems on groups, third 68
Fundamental theorems on groups, third, converse 75
Generating equations of a contact transformation, definition of 245
Generating equations of a Pfaffian system, definition of 196
Generating equations, applications of 252 259 268
Generating equations, interpretation of limitation 247
Generating equations, limitations on 246
Generating equations, property of 246
Generators of a quadric are divided in a constant anharmonic ratio bv any inscribed tetrahedron 272
Goursat, quoted 244
Group of a point, definition of 140
Group of a point, group locus, definition of 141
Group of a point, stationary and non-stationary groups 141
Group of a point, when the point is the origin 332
Group of movements in non-Euclidean space 327 395
Group of movements of a net on a surface 317
Group of movements of a rigid body in a plane 18
Group of transformations, continuous 3 example 4
Group of transformations, discontinuous 3
Group of transformations, discontinuous, example 4
| Group of transformations, finite and continuous 5
Group of transformations, finite and continuous, example 6
Group of transformations, general definition of 2
Group of transformations, infinite 3
Group of transformations, infinite, example 4
Group of transformations, mixed group 3
Groups of the linear complex 304 388
Groups, in cogredient sets of variables 115
Groups, possible types of, in a single variable 335
Hamiltonian Quaternion system 410
Homogeneous function systems, condition that a system should be homogeneous 214. See also under Complete
Homogeneous function systems, defined 198
Homogeneous function systems, equation systems 198
Identical transformation, definition of 3
Identical transformation, parameters defining 34
Imprimitive groups, admitted by a complete system 139
Imprimitive groups, all types of these groups found 354—364
Imprimitive groups, arranged into mutually exclusive types 368
Imprimitive groups, definition of 137
Imprimitive groups, of the plane, divided into four classes 353
Independent, functions 81
Independent, infinitesimal transformations 7
Independent, linear operators 7
Index of sub-group, definition of 183
Infinitesimal transformation, are unconnected 45
Infinitesimal transformation, definition of 6
Infinitesimal transformation, operator, definition of 6
Infinitesimal transformation, operators of first parameter group 41
Integral cones, associated differential equation 282
Integral cones, elementary, definition of 281
Integral of a differential equation, Lie's extension of definition 202 231 232
Integration operations, definition of 88
Invariant See under Differential
Invariant curve systems of the imprimitive groups of the plane 366 367
Invariant, decomposition of space 137
Invariant, equations with respect to a group 128
Invariant, how obtained 130
Invariant, theory of binary quantics 118
Invariants, geometrical interpretation 114
Invariants, of a complete system of operators 87
Invariants, of an intransitive group 114
Invariants, transformed to other invariants by any transformation which the system admits 94
Inverse transformation scheme 1
Involution, contact transformation admitted by equation system in 278
Involution, equations in 197
Involution, functions in, definition of 197
Involution, if any equation system is in involution, so is any equivalent system 197
Irreducible contact groups of the plane obtained 371—378
Irreducible contact groups of the plane obtained, types of, enumerated 378 380
Isomorphic, two groups are simply isomorphic when they have the same parameter group 162
Isomorphism of two groups, multiple, definition of 163
Isomorphism of two groups, simple, definition of 10
Isomorphism of two groups, simple, example of 10
Isomorphism of two groups, when a group is multiply isomorphic with another, a self-conjugate sub-group in the first corresponds to the identical transformation in the second 164
Jacobian identity, definition of 67
Jacobian identity, identity deduced from 216
Linear complex, complexes in involution 257
Linear complex, definition of 255
Linear complex, form to which it can be reduced 256
Linear complex, lines conjugate with respect to 256
Linear complex, projective group of 304
Linear groups connexion with the theory of higher complex numbers 406—410
Linear groups enumeration of such groups in three variables 405 406
Linear groups must contain an Abelian operator 401
Linear groups whose finite equations involve the parameters linearly 398—401
Linear groups, standard form of such a group 399
Linear homogeneous group, general 14
Linear homogeneous group, possible types of, in two variables 339 341
Linear homogeneous group, simplification of the form of an operator of 336—338
Linear homogeneous group, special 17
Linear operators, any one is of type  84
Linear operators, formal laws of combination of 54—57
Linear operators, transformation formula for any operator 91
Lines of curvature transformed to lines of inflection 266
Manifolds of united elements, definition of 201
Manifolds of united elements, different classes of 201
Manifolds of united elements, in ordinary 3-way space 250
Manifolds of united elements, the symbol  201
Maximum sub-group, definition of 101
Measure of curvature unaltered by transformations which do not alter length of arc 310
Measure of curvature, constant along lines of motion of points of a net 312
Measure of curvature, expression for 315
Minimum curves, definition of 28
Mongian equations, associated with an equation of first order 28 282
Mongian equations, defined 29
Mongian equations, of tetrahedral complex 282
Non-homogeneous contact transformation 240
Non-stationary group, defined 141
Normal form of complete system of operators 83
Normal operators are permutable 84
Normal structure constants, defined 72
Null plane, definition of 256
Operators of a group, arranged in classes according to their degrees in the variables 332
Operators of a group, condition that one may be self-conjugate, Abelian 93
Operators of a group, definition of 37
Operators of a group, examples on finding 40 41
Operators of a group, fundamental theorem on 38
Operators of a group, number of independent 38
Operators of a group, of a Pfaffian system of equations 196
Operators of a group, of an integration operation 88
Order of a group, definition of 18
Parameter group, any operation of the first permutable with any operation of the second 13
Parameter group, first and second, definitions of 13
Parameter group, of two simply isomorphic groups identical 162
Parameter group, operators of 160 161
Parameter group, parameter groups of general linear homogeneous group 15
Parameter group, structure constants of 65 159
Permutable operations, condition that two linear transformations may be 400
Permutable operations, definition of 2
Pfaff's equation, definition of 194
Pfaff's equation, in nonhomogeneous form 238
Pfaff's equation, solution 195
Pfaff's problem, in relation to contact transformation 293
Pfaffian system, condition that given system of equations should form 201
Pfaffian system, definition of 196
Pfaffian system, transformation of 231
Poincare, quoted 36
Polar system of functions to a given complete system 217
Polar system of functions, if given system is homogeneous, polar is also 217
Primitive groups, definition of 137
Primitive groups, in space 397
Primitive groups, possible types of, in the plane 352
Projective groups and sub-groups 18 20
Projective groups and sub-groups, examples of non-projective groups 19 22
Projective groups and sub-groups, of the linear complex 304 388
Projective groups and sub-groups, of the plane, property of sub-group of 385
Reciprocal groups, definition of 62
Reciprocal groups, structure constants of 158
Reciprocation, a case of contact transformation 252
Reduced, operators, definition of 97
Reducible contact groups 292
Reducible contact groups, of the plane, condition for 370
Salmon, quoted 265 266 315
Seheffera, quoted 272 398
Self-conjugate operator, condition for 93
Self-conjugate sub-group, condition that a given sub-group may be 92
Self-conjugate sub-group, definition of 17
Similar groups, are simply isomorphic 16
Similar groups, definition of 16
Similar groups, necessary and sufficient conditions that two groups may be similar 149—154
Similar groups, that two contact groups may be 290 291
Similar operations, definition of 2
Simple group, definition of 165
Special elements, definition of 249
Special elements, equations satisfied by 249 254
Special envelope, definition of 249
Special equations, definition of 247
Special linear homogeneous group, definition of 17
Special position, points of, with respect to a complete system of operators 110
Special position, transformed to points of the same special order by transformations admitted by system 127
|
|
|
| Реклама |
|
|
|
|
|
|
|
|
О проекте