4 Idempotent Analysis......Page 5
5.2 The Cauchy Problem for the Hamilton–Jacobi Equations......Page 9
7 Idempotent Functional Analysis......Page 11
7.2 Basic Results......Page 15
1 Introduction......Page 1
3 Semirings and Semifields: The Idempotent CorrespondencePrinciple......Page 4
2.1.3 Discrete Fokker–Planck Equation......Page 7
2.3 A Fundamental Identity......Page 14
7.3 Idempotent b-semialgebras......Page 16
3 Examples of Convergence Results......Page 19
5.3 Finite Speed of Propagation......Page 24
8.1 Dequantization Transform: Algebraic Properties......Page 26
8.2 Generalized Polynomials and Simple Functions......Page 27
10 Dequantization of Geometry......Page 31
6.1 Description of the Planning Problem......Page 35
6.2 The Finite Difference Scheme and an Optimal Control Formulation......Page 36
10.1 Introduction......Page 44
16 The Correspondence Principle for Software Design......Page 45
2 The Maslov Dequantization......Page 3
5.1 Heuristics......Page 6
3.3 Back to the Running Example (I): The Value Function U is a Viscosity Solution of (7)......Page 8
6 Convolution and the Fourier–Legendre Transform......Page 10
7.1 Idempotent Semimodules and Idempotent Linear Spaces......Page 12
7.8 The b-approximation Property and b-nuclear Semimodules and Spaces......Page 22
8.3 Subdifferentials of Sublinear Functions......Page 28
8.4 Newton Sets for Simple Functions......Page 29
11.3 Matrices Over Semirings......Page 33
11.5 Weighted Directed Graphs and Matrices Over Semirings......Page 34
13 Universal Algorithms of Linear Algebra Over Semirings......Page 38
9 Convex Hamiltonians, Barron–Jensen Solutions......Page 41
17 Interval Analysis in Idempotent Mathematics......Page 46
10.4 Asymptotic Behavior of u(x,t)-ct......Page 49
10.5 The Namah–Roquejoffre Framework......Page 50
10.6 The ``Strictly Convex'' Framework......Page 52
10.7 Concluding Remarks......Page 58
References......Page 59
7.5 Functional Semimodules......Page 18
3.2 Additive Eigenvalue Problems......Page 42
4 Stationary Problem: Weak KAM Aspects......Page 55
4.1 Aubry Sets and Representation of Solutions......Page 56
4.2 Proof of Theorem 4.2......Page 64
5.1 Skorokhod Problem......Page 75
5.2 Value Function I......Page 81
5.3 Basic Lemmas......Page 84
5.4 Value Function II......Page 92
5.5 Distance-Like Function d......Page 98
6 Large-Time Asymptotic Solutions......Page 101
6.1 Preliminaries to Asymptotic Solutions......Page 104
6.2 Proof of Convergence......Page 109
6.3 Representation of the Asymptotic Solution u∞......Page 112
6.4 Localization of Conditions (A9)......Page 116
A.1 Local maxima to global maxima......Page 119
A.2 A Quick Review of Convex Analysis......Page 120
A.3 Global Lipschitz Regularity......Page 125
A.4 Localized Versions of Lemma 4.2......Page 128
A.5 A Proof of Lemma 5.4......Page 132
A.6 Rademacher's Theorem......Page 135
References......Page 137
7.4 Linear Operator, b-semimodulesand Subsemimodules......Page 17
7.6 Integral Representations of Linear Operators in Functional Semimodules......Page 20
7.10 Integral Representations of Operators in Abstract Idempotent Semimodules......Page 23
8 The Dequantization Transform, Convex Geometry and the Newton Polytopes......Page 25
9 Dequantization of Set Functions and Measures on Metric Spaces......Page 30
11.2 Closure Operations......Page 32
12 Universal Algorithms......Page 37
15 The Correspondence Principle for Hardware Design......Page 43
References......Page 47