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Campbell J.E. — Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups
Campbell J.E. — Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups

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Название: Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups

Автор: Campbell J.E.

Язык: en

Рубрика: Математика/Симметрия и группы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1903

Количество страниц: 416

Добавлена в каталог: 13.12.2004

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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 $\overline{W}_1$ and $\overline{W}_2$      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 $X_1, \ldots, \ X_n$ 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 $\left(\frac{\partial u}{\partial x}\right)^2+\ldots=0$ 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 $x'=x, \ y'=\frac{ay+b}{cy+d}$      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 $x'=x, \ y'=\frac{ay+b}{cy+d}$      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 $\frac{\partial}{\partial x}$      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 $M_{n-1}$      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
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