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 Главная    Ex Libris    Книги    Журналы    Статьи    Серии    Каталог    Wanted    Загрузка    ХудЛит    Справка    Поиск по индексам    Поиск    Форум Авторизация Поиск по указателям     Campbell J.E. — Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups Обсудите книгу на научном форуме Нашли опечатку?Выделите ее мышкой и нажмите Ctrl+Enter Название: Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups Автор: Campbell J.E. Язык: 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 Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID 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
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