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Calculus Without Derivatives 2013 ed. [Kõva köide]

  • Formaat: Hardback, 524 pages, kõrgus x laius: 235x155 mm, kaal: 9339 g, XX, 524 p., 1 Hardback
  • Sari: Graduate Texts in Mathematics 266
  • Ilmumisaeg: 09-Nov-2012
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 146144537X
  • ISBN-13: 9781461445371
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Calculus Without Derivatives 2013 ed.Suurem pilt
  • Formaat: Hardback, 524 pages, kõrgus x laius: 235x155 mm, kaal: 9339 g, XX, 524 p., 1 Hardback
  • Sari: Graduate Texts in Mathematics 266
  • Ilmumisaeg: 09-Nov-2012
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 146144537X
  • ISBN-13: 9781461445371
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781461445371.html
Märksõnad:

Calculus Without Derivatives expounds the foundations and recent advances in nonsmooth analysis, a powerful compound of mathematical tools that obviates the usual smoothness assumptions. This textbook also provides significant tools and methods towards applications, in particular optimization problems. Whereas most books on this subject focus on a particular theory, this text takes a general approach including all main theories.

In order to be self-contained, the book includes three chapters of preliminary material, each of which can be used as an independent course if needed. The first chapter deals with metric properties, variational principles, decrease principles, methods of error bounds, calmness and metric regularity. The second one presents the classical tools of differential calculus and includes a section about the calculus of variations. The third contains a clear exposition of convex analysis.



Here is a wide-ranging introduction to the foundations of nonsmooth analysis, a powerful compound of mathematical tools that obviates the usual smoothness assumptions. Covers recent progress and methods of implementation, especially in optimization problems.

Arvustused

The book collects three different branches of analysis: differential calculus, convex analysis, and nonsmooth analysis. What makes Penots work stand out is his path through the material and the clean and scholarly presentation. It is well suited for individual study or a classroom . As preparation for the rough road ahead of us in the coming decades, it might be worth the investment. (Russell Luke, SIAM Review, Vol. 57 (2), June, 2015)

This very good book is an treatise on approximate calculus and justifies the authors claim that the rules of this calculus are as important and useful as those for exact calculus. The book is notable not only for its exposition but also for the notes at the end of each chapter explaining the historical and other relevant backgrounds of the material. There are many exercises throughout the book. (Peter S. Bullen, Zentralblatt MATH, Vol. 1264, 2013)

By collecting together a lot of results in nonsmooth analysis and presenting them in a coherent and accessible way, the author rendered a great service to the mathematical community. The book can be considered as an incentive for newcomers to enter this area of research . The specialists will find also a lot of systematized information, and the first three chapters can be used for independent graduate courses. (S. Cobza¸ Studia Universitatis Babes-Bolyai, Mathematica, Vol. 58 (1), 2013)

1 Metric and Topological Tools
1(116)
1.1 Convergences and Topologies
2(22)
1.1.1 Sets and Orders
2(1)
1.1.2 A Short Refresher About Topologies and Convergences
3(5)
1.1.3 Weak Topologies
8(7)
1.1.4 Semicontinuity of Functions and Existence Results
15(7)
1.1.5 Baire Spaces and the Uniform Boundedness Theorem
22(2)
1.2 Set-Valued Mappings
24(10)
1.2.1 Generalities About Sets and Correspondences
25(4)
1.2.2 Continuity Properties of Multimaps
29(5)
1.3 Limits of Sets and Functions
34(6)
1.3.1 Convergence of Sets
34(4)
1.3.2 Supplement: Variational Convergences
38(2)
1.4 Convexity and Separation Properties
40(20)
1.4.1 Convex Sets and Convex Functions
40(11)
1.4.2 Separation and Extension Theorems
51(9)
1.5 Variational Principles
60(15)
1.5.1 The Ekeland Variational Principle
61(4)
1.5.2 Supplement: Some Consequences of the Ekeland Principle
65(1)
1.5.3 Supplement: Fixed-Point Theorems via Variational Principles
66(2)
1.5.4 Supplement: Metric Convexity
68(2)
1.5.5 Supplement: Geometric Principles
70(2)
1.5.6 Supplement: The Banach-Schauder Open Mapping Theorem
72(3)
1.6 Decrease Principle, Error Bounds, and Metric Estimates
75(29)
1.6.1 Decrease Principle and Error Bounds
76(7)
1.6.2 Supplement: A Palais-Smale Condition
83(1)
1.6.3 Penalization Methods
84(3)
1.6.4 Robust and Stabilized Infima
87(5)
1.6.5 Links Between Penalization and Robust Infima
92(3)
1.6.6 Metric Regularity, Lipschitz Behavior, and Openness
95(4)
1.6.7 Characterizations of the Pseudo-Lipschitz Property
99(1)
1.6.8 Supplement: Convex-Valued Pseudo-Lipschitzian Multimaps
100(1)
1.6.9 Calmness and Metric Regularity Criteria
101(3)
1.7 Well-Posedness and Variational Principles
104(8)
1.7.1 Supplement: Stegall's Principle
110(2)
1.8 Notes and Remarks
112(5)
2 Elements of Differential Calculus
117(70)
2.1 Derivatives of One-Variable Functions
118(4)
2.1.1 Differentiation of One-Variable Functions
118(2)
2.1.2 The Mean Value Theorem
120(2)
2.2 Primitives and Integrals
122(4)
2.3 Directional Differential Calculus
126(7)
2.4 Frechet Differential Calculus
133(11)
2.5 Inversion of Differentiable Maps
144(25)
2.5.1 Newton's Method
145(3)
2.5.2 The Inverse Mapping Theorem
148(6)
2.5.3 The Implicit Function Theorem
154(4)
2.5.4 The Legendre Transform
158(1)
2.5.5 Geometric Applications
159(7)
2.5.6 The Method of Characteristics
166(3)
2.6 Applications to Optimization
169(11)
2.6.1 Normal Cones, Tangent Cones, and Constraints
170(5)
2.6.2 Calculus of Tangent and Normal Cones
175(2)
2.6.3 Lagrange Multiplier Rule
177(3)
2.7 Introduction to the Calculus of Variations
180(6)
2.8 Notes and Remarks
186(1)
3 Elements of Convex Analysis
187(76)
3.1 Continuity Properties of Convex Functions
188(6)
3.1.1 Supplement: Another Proof of the Robinson-Ursescu Theorem
192(2)
3.2 Differentiability Properties of Convex Functions
194(10)
3.2.1 Derivatives and Subdifferentials of Convex Functions
194(7)
3.2.2 Differentiability of Convex Functions
201(3)
3.3 Calculus Rules for Subdifferentials
204(8)
3.3.1 Supplement: Subdifferentials of Marginal Convex Functions
208(4)
3.4 The Legendre-Fenchel Transform and Its Uses
212(14)
3.4.1 The Legendre-Fenchel Transform
213(3)
3.4.2 The Interplay Between a Function and Its Conjugate
216(3)
3.4.3 A Short Account of Convex Duality Theory
219(5)
3.4.4 Duality and Subdifferentiability Results
224(2)
3.5 General Convex Calculus Rules
226(22)
3.5.1 Fuzzy Calculus Rules in Convex Analysis
227(8)
3.5.2 Exact Rules in Convex Analysis
235(4)
3.5.3 Mean Value Theorems
239(3)
3.5.4 Application to Optimality Conditions
242(6)
3.6 Smoothness of Norms
248(5)
3.7 Favorable Classes of Spaces
253(7)
3.8 Notes and Remarks
260(3)
4 Elementary and Viscosity Subdifferentials
263(94)
4.1 Elementary Subderivatives and Subdifferentials
264(24)
4.1.1 Definitions and Characterizations
264(6)
4.1.2 Some Elementary Properties
270(2)
4.1.3 Relationships with Geometrical Notions
272(7)
4.1.4 Coderivatives
279(3)
4.1.5 Supplement: Incident and Proximal Notions
282(3)
4.1.6 Supplement: Bornological Subdifferentials
285(3)
4.2 Elementary Calculus Rules
288(8)
4.2.1 Elementary Sum Rules
288(1)
4.2.2 Elementary Composition Rules
289(3)
4.2.3 Rules Involving Order
292(2)
4.2.4 Elementary Rules for Marginal and Performance Functions
294(2)
4.3 Viscosity Subdifferentials
296(6)
4.4 Approximate Calculus Rules
302(24)
4.4.1 Approximate Minimization Rules
302(5)
4.4.2 Approximate Calculus in Smooth Banach Spaces
307(4)
4.4.3 Metric Estimates and Calculus Rules
311(6)
4.4.4 Supplement: Weak Fuzzy Rules
317(3)
4.4.5 Mean Value Theorems and Superdifferentials
320(6)
4.5 Soft Functions
326(3)
4.6 Calculus Rules in Asplund Spaces
329(6)
4.6.1 A Characterization of Frechet Subdifferentiability
330(1)
4.6.2 Separable Reduction
331(3)
4.6.3 Application to Fuzzy Calculus
334(1)
4.7 Applications
335(20)
4.7.1 Subdifferentials of Value Functions
335(8)
4.7.2 Application to Regularization
343(3)
4.7.3 Mathematical Programming Problems and Sensitivity
346(4)
4.7.4 Openness and Metric Regularity Criteria
350(2)
4.7.5 Stability of Dynamical Systems and Lyapunov Functions
352(3)
4.8 Notes and Remarks
355(2)
5 Circa-Subdifferentials, Clarke Subdifferentials
357(50)
5.1 The Locally Lipschitzian Case
358(11)
5.1.1 Definitions and First Properties
358(3)
5.1.2 Calculus Rules in the Locally Lipschitzian Case
361(4)
5.1.3 The Clarke Jacobian and the Clarke Subdifferential in Finite Dimensions
365(4)
5.2 Circa-Normal and Circa-Tangent Cones
369(6)
5.3 Subdifferentials of Arbitrary Functions
375(15)
5.3.1 Definitions and First Properties
375(7)
5.3.2 Regularity
382(1)
5.3.3 Calculus Rules
383(7)
5.4 Limits of Tangent and Normal Cones
390(4)
5.5 Moderate Subdifferentials
394(10)
5.5.1 Moderate Tangent Cones
394(4)
5.5.2 Moderate Subdifferentials
398(3)
5.5.3 Calculus Rules for Moderate Subdifferentials
401(3)
5.6 Notes and Remarks
404(3)
6 Limiting Subdifferentials
407(56)
6.1 Limiting Constructions with Firm Subdifferentials
408(12)
6.1.1 Limiting Subdifferentials and Limiting Normals
408(5)
6.1.2 Limiting Coderivatives
413(3)
6.1.3 Some Elementary Properties
416(3)
6.1.4 Calculus Rules Under Lipschitz Assumptions
419(1)
6.2 Some Compactness Properties
420(6)
6.3 Calculus Rules for Coderivatives and Normal Cones
426(21)
6.3.1 Normal Cone to an Intersection
427(3)
6.3.2 Coderivative to an Intersection of Multimaps
430(5)
6.3.3 Normal Cone to a Direct Image
435(2)
6.3.4 Normal Cone to an Inverse Image
437(2)
6.3.5 Coderivatives of Compositions
439(5)
6.3.6 Coderivatives of Sums
444(3)
6.4 General Subdifferential Calculus
447(2)
6.5 Error Bounds and Metric Estimates
449(5)
6.5.1 Upper Limiting Subdifferentials and Conditioning
449(4)
6.5.2 Application to Regularity and Openness
453(1)
6.6 Limiting Directional Subdifferentials
454(6)
6.6.1 Some Elementary Properties
457(2)
6.6.2 Calculus Rules Under Lipschitz Assumptions
459(1)
6.7 Notes and Remarks
460(3)
7 Graded Subdifferentials, Ioffe Subdifferentials
463(16)
7.1 The Lipschitzian Case
463(12)
7.1.1 Some Uses of Separable Subspaces
464(1)
7.1.2 The Graded Subdifferential and the Graded Normal Cone
465(3)
7.1.3 Relationships with Other Subdifferentials
468(2)
7.1.4 Elementary Properties in the Lipschitzian Case
470(5)
7.2 Subdifferentials of Lower Semicontinuous Functions
475(3)
7.3 Notes and Remarks
478(1)
References 479(40)
Index 519
Jean-Paul Penot is an Emeritus Professor at Université Paris 6.  He has taught in Paris, Pau and Canada.