Ebook: Ramified Integrals, Singularities and Lacunas

This volume contains an introduction to the Picard--Lefschetz theory, which controls the ramification and qualitative behaviour of many important functions of PDEs and integral geometry, and its foundations in singularity theory.
Solutions to many problems of these theories are treated. Subjects include the proof of multidimensional analogues of Newton's theorem on the nonintegrability of ovals; extension of the proofs for the theorems of Newton, Ivory, Arnold and Givental on potentials of algebraic surfaces. Also, it is discovered for which d and n the potentials of degree d hyperbolic surfaces in Rⁿ are algebraic outside the surfaces; the equivalence of local regularity (the so-called sharpness), of fundamental solutions of hyperbolic PDEs and the topological Petrovskii--Atiyah--Bott--Gårding condition is proved, and the geometrical characterization of domains of sharpness close to simple singularities of wave fronts is considered; a `stratified' version of the Picard--Lefschetz formula is proved, and an algorithm enumerating topologically distinct Morsifications of real function singularities is given.
This book will be valuable to those who are interested in integral transforms, operational calculus, algebraic geometry, PDEs, manifolds and cell complexes and potential theory.

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This volume contains an introduction to the Picard--Lefschetz theory, which controls the ramification and qualitative behaviour of many important functions of PDEs and integral geometry, and its foundations in singularity theory. Solutions to many problems of these theories are treated. Subjects include the proof of multidimensional analogues of Newton's theorem on the nonintegrability of ovals; extension of the proofs for the theorems of Newton, Ivory, Arnold and Givental on potentials of algebraic surfaces. Also, it is discovered for which d and n the potentials of degree d hyperbolic surfaces in R n are algebraic outside the surfaces; the equivalence of local regularity (the so-called sharpness), of fundamental solutions of hyperbolic PDEs and the topological Petrovskii--Atiyah--Bott--Gårding condition is proved, and the geometrical characterization of domains of sharpness close to simple singularities of wave fronts is considered; a `stratified' version of the Picard--Lefschetz formula is proved, and an algorithm enumerating topologically distinct Morsifications of real function singularities is given. This book will be valuable to those who are interested in integral transforms, operational calculus, algebraic geometry, PDEs, manifolds and cell complexes and potential theory.

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This volume contains an introduction to the Picard--Lefschetz theory, which controls the ramification and qualitative behaviour of many important functions of PDEs and integral geometry, and its foundations in singularity theory. Solutions to many problems of these theories are treated. Subjects include the proof of multidimensional analogues of Newton's theorem on the nonintegrability of ovals; extension of the proofs for the theorems of Newton, Ivory, Arnold and Givental on potentials of algebraic surfaces. Also, it is discovered for which d and n the potentials of degree d hyperbolic surfaces in R n are algebraic outside the surfaces; the equivalence of local regularity (the so-called sharpness), of fundamental solutions of hyperbolic PDEs and the topological Petrovskii--Atiyah--Bott--Gårding condition is proved, and the geometrical characterization of domains of sharpness close to simple singularities of wave fronts is considered; a `stratified' version of the Picard--Lefschetz formula is proved, and an algorithm enumerating topologically distinct Morsifications of real function singularities is given. This book will be valuable to those who are interested in integral transforms, operational calculus, algebraic geometry, PDEs, manifolds and cell complexes and potential theory.
Content:
Front Matter....Pages i-xvii
Picard—Lefschetz—Pham Theory and Singularity Theory....Pages 1-85
Newton’s Theorem on the Nonintegrability of Ovals....Pages 87-114
Newton’s Potential of Algebraic Layers....Pages 115-147
Lacunas and the Local Petrovski? Condition for Hyperbolic Differential Operators with Constant Coefficients....Pages 149-179
Calculation of Local Petrovski? Cycles and Enumeration of Local Lacunas Close to Real Function Singularities....Pages 181-247
Back Matter....Pages 249-294

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Content:
Front Matter....Pages i-xvii
Picard—Lefschetz—Pham Theory and Singularity Theory....Pages 1-85
Newton’s Theorem on the Nonintegrability of Ovals....Pages 87-114
Newton’s Potential of Algebraic Layers....Pages 115-147
Lacunas and the Local Petrovski? Condition for Hyperbolic Differential Operators with Constant Coefficients....Pages 149-179
Calculation of Local Petrovski? Cycles and Enumeration of Local Lacunas Close to Real Function Singularities....Pages 181-247
Back Matter....Pages 249-294

---

Content:
Front Matter....Pages i-xvii
Picard—Lefschetz—Pham Theory and Singularity Theory....Pages 1-85
Newton’s Theorem on the Nonintegrability of Ovals....Pages 87-114
Newton’s Potential of Algebraic Layers....Pages 115-147
Lacunas and the Local Petrovski? Condition for Hyperbolic Differential Operators with Constant Coefficients....Pages 149-179
Calculation of Local Petrovski? Cycles and Enumeration of Local Lacunas Close to Real Function Singularities....Pages 181-247
Back Matter....Pages 249-294
....