.

0 (hodnocen0 x )

(4) Půjčeno:4x 

BK

Brno : Masaryk University, 1998

x,296 s.

ISBN 80-210-1819-4 (brož.)

Folia Facultatis scientiarium naturalium Universitatis Masarykianae Brunensis. Biologia ; [vol.] 6

Obsahuje rejstřík.

Bibliografie: s. 293-296.

Rovnice diferenciální - učebnice vysokošk.

000034410

Contents // I. Fundamental concepts // 1. Linear algebra...1 // 2. Underlying spaces...3 // 3. Localization and genericity...4 // 4. The blanket principle...5 // 5. Remarks to modules...6 // 6. Differential forms...7 // 7. Two generalizations...8 // 8. Vector fields...9 // 9. The Lie bracket...10 // 10. The exterior differential...10 // 11. The effect of a mapping...11 // 12. The Lie derivative intuitively...13 // 13. Infinitesimal transformations...14 // 14. The Lie derivative again...15 // 15. A digest of vector fields...16 // Comments...18 // II. Around the Frobenius theorem // 1. Flatness and complete integrability...19 // 2. Classical Frobenius theorem...20 // 3. Adjoint variables...20 // 4. The Pfaff-Darboux normal shapes...24 // 5. Derived modules...25 // 6. Groups of transformations...25 // 7. On infinitesimal transformations...27 // 8. Groups of substitutions...27 // 9. Theorem...28 // 10. Approximation property for groups...30 // 11. Generalized Frobenius theorem...30 // 12. Other generalizations and perspectives...31 // Comments...32 // III. Ordinary differential equations // 1. From Monge systems to diffieties...33 // 2. From diffieties to Monge systems... 35 // 3. The Cartan filtrations...36 // 4. Standard filtrations...38 // 5. Simple remarks...39 // 6. Morphisms of diffieties... 41 // 7. Elementary approach to subdiflfieties...42 viii // 8. Elementary approach to factordiffieties...42 // 9. The contact forms...43 // 10. Automorphisms of contact forms...46 // 11.

Continuation: explicit calculations...48 // 12. Contact forms and parameters...51 // 13. The Sturm-Liouville equation...51 // 14. Continuation: solution with quadratures...54 // 15. Infinitesimal symmetries...55 // 16. Miscellanery of symmetries...57 // Comments...59 // IV. The Monge problem // 1. An outlook and the strategy of solution...61 // 2. Factordiffieties through Cartan filtrations...62 // 3. The comparison lemma...63 // 4. The cross-sections method...66 // 5. Application: a reduction principle...67 // 6. Solvability with one function...68 // 7. Continuation: another approach...69 // 8. Solvability with one function and constants...71 // 9. Solvability with two functions...72 // 10. Continuation: the semi-structural lemma...73 // 11. Termination: the structural lemma...76 // 12. Solvability with one function and one quadrature...77 // 13. Continuation: a particular example...82 // 14. Continuation: yet more particular examples...84 // 15. Identification of diffieties...85 // 16. Identification of contact diffieties...86 // 17. Identification of quadratures...87 // Comments...89 // V. Formal calculus of variations // 1. Stationary points...91 // 2. The Lagrange problem...92 // 3. The moving boundary problem...94 // 4. Generalized Poincaré-Cartan forms...95 // 5. Definitions, simple results, future tasks...97 // 6. A general strategy for the inverse problem...100 // 7. First order non-degenerate integrals...102 // 8. Two remarks on the resolving system...105 // 9. An alternative

approach...106 // 10. Continuation: a few particular subcases...107 // 11. A degenerate variational problem...109 // 12. The underdetermined Euler-Lagrange system...113 // 13. Notes to higher order variational problems...114 // 14. Two examples of constrained variational integrals...116 ix // Comments...118 // VI. On commutative algebra // 1. Notation and terminology...119 // 2. The Koszul complex...121 // 3. The Hilbert resolution...123 // 4. Associated ideals and regularity of sequences...126 // 5. The Chevalley-Krull-Samuel theorem...129 // 6. Remarks on the multiplicity...129 // 7. The technique of generic sequences...130 // 8. The homologies once more...132 // 9. The regularity of sequences and involutiveness...133 // 10. Introduction of characteristics...138 // 11. Continuation: four remarks...140 // 12. On alternative homologies...141 // Comments...144 // VII. Partial differential equations // 1. Diffieties, morphisms, infinitesimal symmetries...145 // 2. Links between diffieties and differential equations ...146 // 3. Evolution diffieties...148 // 4. Links to commutative algebra...149 // 5. The multiplicity result...153 // 6. The absolute Cauchy characteristics...155 // 7. Lemma ...156 // 8. Toward the integrability theorem...157 // 9. Pseudodifferential operators and fractions...159 // 10. Proof of the main theorem...162 // 11. Integrability of characteristics...163 // 12. The Cartan filtration ...164 // 13. The second acyclicity existence theorem...167 // 14. The involutiveness

existence theorem ...170 // 15. Subdiffieties of contact diffiety ...170 // 16. Standard filtrations ... 176 // 17. On symmetries and infinitesimal symmetries...177 // 18. Remarks to linearization ...182 // 19. Remarks to underdetermined diffieties ...183 // Comments ...183 // VIII. The Lie-Cartan pseudogroups // 1. The classical equivalence problem ...185 // 2. Illustrative examples ...190 // 3. Slightly generalized diffieties ...195 // 4. Groupieties ...196 // 5. Some properties of Maurer-Cartan forms...197 // 6. Axioms for Maurer-Cartan forms ... 200 x // 7. The uniqueness of Maurer-Cartan forms ...200 // 8. Dual axioms ...201 // 9. Essential invariants and adapted coordinates ...202 // 10. The existence of adapted coordinates ...204 // 11. Simple examples ... 207 // 12. Subgroupieties ...212 // 13. Normal subgroupieties ...213 // 14. Intrinsical filtrations and composition series ...215 // 15. Pseudogroups of diffeomorphisms ...216 // 16. Examples of subgroupieties ...220 // 17. Homogeneous spaces and related concepts...225 // 18. The left and the right cosets ... 227 // 19. Missed topics...229 // Comments... 229 // IX. The multiple variational integrals // 1. A general class of variational problems ...231 // 2. Few notes to the classical approach...232 // 3. The case of contact diffieties again...235 // 4. To the Poincaré lemma...242 // 5. The variational bicomplex...243 // 6. Towards the spectral sequences...246 // 7. One-dimensional variational bicomplex ...247 // 8.

Continuation: yet the inverse problem ...249 // 9. Two-dimensional variational bicomplex ...250 // 10. Continuation: few examples ... 255 // 11. Links to commutative algebra... 258 // 12. Few notes to general bicomplex ...260 // 13. Comeback of calculus of variations ...262 // Comments...264 // X. Appendices // 1. The Lie’s main theorems ...265 // 2. A comparison of Lie groups and pseudogroups...273 // 3. The spectral sequences ...274 // 4. A report on diffieties with symmetries ...277 // 5. To lower order differential equations ...281 // 6. Concluding examples...286 // Comments ...290 // Index // References