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E-raamat: Introduction To The Geometrical Analysis Of Vector Fields, An: With Applications To Maximum Principles And Lie Groups

(Marche Polytechnic Univ, Italy), (Univ Of Bologna, Italy)
  • Formaat: 452 pages
  • Ilmumisaeg: 05-Dec-2018
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789813276635
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Introduction To The Geometrical Analysis Of Vector Fields, An: With Applications To Maximum Principles And Lie GroupsSuurem pilt
  • Formaat: 452 pages
  • Ilmumisaeg: 05-Dec-2018
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789813276635
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This book provides the reader with a gentle path through the multifaceted theory of vector fields, starting from the definitions and the basic properties of vector fields and flows, and ending with some of their countless applications, in the framework of what is nowadays called Geometrical Analysis. Once the background material is established, the applications mainly deal with the following meaningful settings: 1. ODE theory; 2. Maximum Principles (weak, strong and propagation principles); 3. Lie groups (with an emphasis on the construction of Lie groups). This book also provides an introduction to the basic theory of Geometrical Analysis, with a new foundational presentation based on Ordinary Differential Equation techniques, in a unitary and self-contained way. The book also contains: •58 figures; •182 exercises; •3 Appendices; •a Further Reading section.

Preface vii
1 Flows of Vector Fields in Space 1(36)
1.1 Notations for vector fields in space
2(4)
1.2 The flow of a vector field
6(12)
1.2.1 The semigroup property
11(2)
1.2.2 Global vector fields
13(1)
1.2.3 Regular and singular points
14(4)
1.3 Differentiation along a flow
18(1)
1.4 The equation of variation for the flow
19(4)
1.4.1 A Liouville Theorem for ODES
20(1)
1.4.2 Further regularity of the flow
21(2)
1.5 Flowing through X, Y, -X, -Y: commutators
23(2)
1.6 The product of exponentials: motivations
25(6)
1.7 Exercises
31(6)
2 The Exponential Theorem 37(34)
2.1 Main algebraic setting
38(5)
2.2 The Exponential Theorem for K(x, y) [ t]
]42
2.2.1 Two crucial lemmas of non-commutative algebra
43(3)
2.2.2 Poiricare's ODE in the formal power series setting
46(3)
2.3 The Exponential Theorem for K((x, y))
49(1)
2.4 Dynkin's Formula
50(5)
2.4.1 A Dynkin-type formula
50(2)
2.4.2 Dynkin's original formula
52(3)
2.5 Identities from the Exponential Theorem
55(3)
2.6 The Exponential Theorem for K(x, y) [ s, t]
58(4)
2.6.1 The algebra K(x, y)[ s,t]
58(1)
2.6.2 The Exponential Theorem for K(x, y)[ s, t]
59(2)
2.6.3 Poincare's PDEs on K(x,y) [ s,t]
61(1)
2.7 More identities
62(2)
2.8 Appendix: manipulations of formal series
64(1)
2.9 Exercises
65(6)
3 The Composition of Flows of Vector Fields 71(18)
3.1 Again on commutators
72(3)
3.2 Composition of flows of vector fields
75(3)
3.3 Approximation for higher order commutators
78(4)
3.4 Appendix: another identity between formal power series
82(2)
3.5 Exercises
84(5)
4 Hadamard's Theorem for Flows 89(24)
4.1 Preliminaries on derivations and differentials
90(3)
4.1.1 Time-dependent vector fields
93(1)
4.2 Relatedness of vector fields and flows
93(5)
4.2.1 Invariance of a vector field under a map
96(2)
4.3 Commutators and Lie-derivatives
98(5)
4.4 Hadamard's Theorem for flows
103(3)
4.5 Commuting vector fields
106(1)
4.6 Hadamard's Theorem for flows in space
107(3)
4.6.1 Series expansibility
107(2)
4.6.2 Conjugation of flows
109(1)
4.7 Exercises
110(3)
5 The CBHD Operation on Finite Dimensional Lie Algebras 113(20)
5.1 Local convergence of the CBHD series
114(2)
5.2 Recursive identities for Dynkin's polynomials
116(2)
5.3 Poincare's ODE on Lie algebras
118(3)
5.3.1 More Poincare-type ODEs
120(1)
5.4 The local associativity of the CBHD series
121(3)
5.5 Appendix: multiple series in Banach spaces
124(4)
5.6 Exercises
128(5)
6 The Connectivity Theorem 133(14)
6.1 Hormander systems of vector fields
134(2)
6.2 A useful Linear Algebra lemma
136(1)
6.3 The Connectivity Theorem
137(6)
6.3.1 X-subunit curves and X-connectedness
137(3)
6.3.2 Connectivity for Hormander vector fields
140(3)
6.4 Exercises
143(4)
7 The Carnot-Caratheodory distance 147(22)
7.1 The X-control distance
148(5)
7.2 Some equivalent definitions of dx
153(3)
7.3 Basic topological properties of the CC-distance
156(7)
7.3.1 Euclidean boundedness of the dx balls
159(2)
7.3.2 Length space property
161(2)
7.4 Exercises
163(6)
8 The Weak Maximum Principle 169(28)
8.1 Main definitions
170(2)
8.2 Picone's Weak Maximum Principle
172(10)
8.3 Existence of L-barriers
182(6)
8.4 The parabolic Weak Maximum Principle
188(2)
8.5 Appendix: semiellipticity and the WMP
190(2)
8.6 Exercises
192(5)
9 Corollaries of the Weak Maximum Principle 197(14)
9.1 Comparison principles
198(2)
9.2 Maximum-modulus and Maximum Principle
200(4)
9.2.1 The parabolic case
202(2)
9.3 An a priori estimate
204(2)
9.4 Application: Green and Poisson operators
206(2)
9.5 Appendix: Another Maximum Principle
208(1)
9.6 Exercises
209(2)
10 The Maximum Propagation Principle 211(26)
10.1 Assumptions on the operators
212(1)
10.2 Principal vector fields
213(2)
10.3 Propagation and Strong Maximum Principle
215(3)
10.4 Invariant sets and the Nagumo-Bony Theorem
218(7)
10.5 The Hopf Lemma
225(5)
10.6 The proof of the Propagation Principle
230(3)
10.6.1 Conclusions and a resume
232(1)
10.7 Exercises
233(4)
11 The Maximum Propagation along the Drift 237(22)
11.1 Propagation along the drift
238(8)
11.2 A resume of drift propagation
246(2)
11.3 The point of view of reachable sets
248(8)
11.3.1 Examples of propagation sets for a PDO
251(5)
11.4 Exercises
256(3)
12 The Differential of the Flow wrt its Parameters 259(8)
12.1 The non-autonomous equation of variation
260(4)
12.1.1 The autonomous equation of variation
264(1)
12.2 More on flow differentiation
264(1)
12.3 Appendix: A review of linear ODEs
265(1)
12.4 Exercises
266(1)
13 The Exponential Theorem for ODEs 267(10)
13.1 Finite-dimensional algebras of vector fields
268(1)
13.2 The differential of the flow wrt the vector field
269(4)
13.3 The Exponential Theorem for ODEs
273(3)
13.4 Exercises
276(1)
14 The Exponential Theorem for Lie Groups 277(12)
14.1 The differential of the Exponential Map
278(3)
14.2 The Exponential Theorem for Lie groups
281(1)
14.3 An alternative approach with analytic functions
282(3)
14.4 Exercises
285(4)
15 The Local Third Theorem of Lie 289(10)
15.1 Local Lie's Third Theorem
290(4)
15.2 Global Lie's Third Theorem in the nilpotent case
294(3)
15.2.1 The Exponential Map of G
296(1)
15.3 Exercises
297(2)
16 Construction of Carnot Groups 299(6)
16.1 Finite-dimensional stratified Lie algebras
300(1)
16.2 Construction of Carnot groups
301(3)
16.3 Exercises
304(1)
17 Exponentiation of Vector Field Algebras into Lie Groups 305(26)
17.1 The assumptions for the exponentiation
306(3)
17.2 Construction of the local Lie group
309(6)
17.2.1 The local Lie-group multiplication
309(5)
17.2.2 The local left invariance of g
314(1)
17.3 Local to global
315(11)
17.3.1 Schur's ODE on g and prolongation of solutions
316(10)
17.4 Exercises
326(5)
18 On the Convergence of the CBHD Series 331(10)
18.1 A domain of convergence for the CBHD series
332(5)
18.2 Exercises
337(4)
Appendix A Some prerequisites of Linear Algebra 341(16)
A.1 Algebras and Lie algebras
341(7)
A.1.1 Stratified Lie algebras
346(2)
A.2 Positive semidefinite matrices
348(1)
A.3 The Moore-Penrose pseudo-inverse
349(4)
A.4 Exercises
353(4)
Appendix B Dependence Theory for ODEs 357(30)
B.1 Review of basic ODE Theory
357(10)
B.1.1 Preliminaries
357(5)
B.1.2 Maximal solutions
362(3)
B.1.3 ODEs depending on parameters
365(2)
B.2 Continuous dependence
367(7)
B.2.1 The Arzela-Ascoli Theorem
367(2)
B.2.2 Dependence on the equation
369(3)
B.2.3 Dependence on the datum
372(1)
B.2.4 Dependence on the parameters
373(1)
B.3 Ck dependence
374(6)
B.3.1 The equation of variation
378(2)
B.4 Cw dependence
380(4)
B.5 Exercises
384(3)
Appendix C A brief review of Lie Group Theory 387(22)
C.1 A short review of Lie groups
387(8)
C.1.1 The Lie algebra of G
388(2)
C.1.2 The exponential map of G
390(2)
C.1.3 Right invariant vector fields
392(1)
C.1.4 Lie's First Theorem
393(2)
C.2 Homomorphisms
395(5)
C.3 A few examples
400(5)
C.4 Exercises
405(4)
Further Readings 409(5)
List of abbreviations 414(1)
Bibliography 415(6)
Index 421
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