Table of contents

CONTENTS Introduction §1. Billiard trajectories in a plane domain §2. Fagnano's problem. Mechanical interpretations of periodic trajectories in triangles §3. An extremal property of billiard trajectories. Birkhoff's theorem. The non-existence of a unified construction of periodic trajectories in obtuse triangles §4. 'Perpendicular' trajectories in obtuse triangles of special shape §5. 'Perpendicular' trajectories in rational polygons and polyhedra §6. Stable trajectories §7. Stable perpendicular trajectories §8. Isolated trajectories §9. Isolated trajectories in acute and obtuse triangles. The bifurcation diagram of isolated trajectories (a 'hang-glider' configuration) §10. The density of F-triangles in a neighbourhood of (0, 0) §11. Generalization of the construction of isolated trajectories in obtuse triangles §12. Stable and unstable billiard trajectories in plane Weyl chambers §13. A criterion for the stability of periodic trajectories in a regular hexagon Conclusion References

CONTENTS Introduction Chapter I. Modules over soluble groups §1. Finitely generated modules over groups of finite rank §2. Quasifinite and co-quasifinite modules §3. Direct complements in modules Chapter II. Extensions, splitting, and conjugacy of complements Chapter III. Factorizations of infinite groups §1. Existence of factorizations §2. The inverse factorization problem §3. Abelian factorizations §4. General results on factorizable infinite groups §5. Soluble factorizations with finiteness conditions Chapter IV. The weak minimum and maximum conditions §1. General properties §2. Groups of finite rank §3. Minimax rank §4. Min-∞ and Max-∞ on various types of subgroups §5. Min-∞ and Max-∞ on normal subgroups References

CONTENTS Introduction Chapter I. Motivation for the reader §1.1. Examples of factorization for partial differential equations Chapter II. Factorization of evolution equations §2.1. Symmetries of evolution equations §2.2. The Lie algebra of classical symmetries and its invariants §2.3. Invariant derivations §2.4. How to choose invariants for the factorization of evolution equations §2.5. Canonical forms for the principal part of a factor-equation §2.6. Decomposition of a factorization Chapter III. What can be derived from the simple heat equation? §3.1. Classification of linear equations via Lie groups §3.2. Factorization in a class of linear equations §3.3. Construction of an optimal system of subalgebras in the algebra of essential symmetries of the heat equation §3.4. Classification of "tail" subalgebras §3.5. Optimal systems of subalgebras in the algebra of symmetries for the heat equation with potential Appendix. Differential substitutions arising as a result of a factorization of the heat equation References