Author | Chaillou, Jacques. author |
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Title | Hyperbolic Differential Polynomials [electronic resource] : and their Singular Perturbations / by Jacques Chaillou |
Imprint | Dordrecht : Springer Netherlands, 1979 |
Connect to | http://dx.doi.org/10.1007/978-94-009-9506-2 |
Descript | XV, 168 p. online resource |
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 (?)|?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 Gรฅrding and Lars Hรถrmander -- Appendix 2. A Necessary and Sufficient Condition For Hyperbolicity -- Name Index