Hyperbolic Differential Polynomials: and their Singular Perturbations

Approach your problems from It isn't that they can't see the the right end and begin with the solution. It is that they can't see answers. Then, one day, perhaps the problem. you will find the final question. 'The Hermit Clad in Crane Feathers' G. K. Chesterton, The scandal of in R. Van Gulik's The Chinese Maze Father Brown 'The point of a Murders. pin" Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be com­ pletely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homo­ topy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

Table of Contents:
I. Generalities.- I.1. Emission Cones.- I.2. The Topological Algebra D?(?).- I.3. The Set U(?) of Polynomial Distributions with Inverse in D?(?).- I.4. Bounded Subsets of U(?) with Bounded Inverse.- I.5. First Consequences of U being Invertible in D?(?)with Bounded Inverse.- 1.6. Remarks.- II. The Semi-algebraic Case. Criterion forUto be Invertible with Bounded Inverse.- II.1. Semi-algebraic Subsets of ?n.- II.2. Polynomial Mappings of ?n into ?m. Theorem of Seidenberg.- II.3. Asymptotic Behavior of Semi-algebraic Subsets of ?2.- II.4. If U is Invertible with Bounded Inverse, then the Union of the V(a) can be Localized.- II.5. Hyperbolicity of A or of $$ \bigcup\limits_{a \in A} {V\left( a \right)} $$ the V(a). Criterion for U to be Invertible with Bounded Inverse.- II.6. Differential Polynomials that are a Polynomial Function of a Parameter (? ? ?p).- III. A Sufficient Condition thatUis Invertible with Bounded Inverse. The Cauchy Problem in Hsloc.- III.1. Upper Bounds for |a(?)|?1.- III.2. Laplace Transforms and Supports of Distributions.- III.3. A Sufficient Condition that U is Invertible with Bounded Inverse.- III.4. The Cauchy Problem with Data in Hsloc.- IV. Hyperbolic Hypersurfaces and Polynomials.- IV.0. Preliminary Notations and Definitions.- IV.1. First Properties of 0?-Hyperbolic V(a).- IV.2. First Properties of 0?-Hyperbolic Cones V(am).- IV.3. 0?-Hyperbolicity and ?-Hyperbolicity.- IV.4. Polars with respect to ? ? 0? of 0?-Hyperbolic V(a).- IV.5. Successive Multiplicities of a Series in ?[[X]] with respect to a Polynomial with Roots in ?[[X]]..- IV.6. Relations between V(am?k) and V(am) that follow from Hyperbolicity of V(a).- IV.7. Relations between V(am?k) and the Polars of V(am) Implied by Hyperbolicity of V(a).- IV.8. Functions Rm?k, ? on V(am(k), ?) ? ?n. A Sufficient Condition that V(a) is Hyperbolic.- IV.9. Local Properties of the Functions Rm?k, ?.- IV.10. Real Ordered Sheets of Hyperbolic Cones.- IV.11. Locally Constant Multiplicity on V(am)* ? ?n. A Hyperbolicity Criterion for V(a).- IV.12. n = 3. A Criterion for Hyperbolicity.- IV.13. Hyperbolicity and Strength of Polynomials.- IV.14. The Cauchy Problem.- V. Examples.- V.1. Sets of Homogeneous Polynomials of the Same Degree.- V.2. Sets of Polynomials of the Same Degree.- V.3. Lowering the Degree by One.- V.4. Lowering the Degree by Two.- V.5. An Example with Arbitrary Lowering of Degree.- V.6. Conclusion.- Appendix 1. On a Conjecture of Lars Gårding and Lars Hörmander.- Appendix 2. A Necessary and Sufficient Condition For Hyperbolicity.- Name Index.