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| Foreword to the Classics Edition | |
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| Preface to the First Edition | |
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| Preface to the Second Edition | |
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| Errata | |
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| Preliminaries | |
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| Preliminaries | |
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| Basic theorems | |
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| Smooth approximations | |
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| Change of integration variables | |
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| Notes | |
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| Existence | |
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| The Picard-Lindelof theorem | |
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| Peano's existence theorem | |
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| Extension theorem | |
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| H. Kneser's theorem | |
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| Example of nonuniqueness | |
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| Notes | |
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| Differential inequalities and uniqueness | |
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| Gronwall's inequality | |
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| Maximal and minimal solutions | |
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| Right derivatives | |
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| Differential inequalities | |
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| A theorem of Wintner | |
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| Uniqueness theorems | |
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| van Kampen's uniqueness theorem | |
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| Egress points and Lyapunov functions | |
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| Successive approximations | |
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| Notes | |
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| Linear differential equations | |
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| Linear systems | |
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| Variation of constants | |
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| Reductions to smaller systems | |
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| Basic inequalities | |
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| Constant coefficients | |
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| Floquet theory | |
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| Adjoint systems | |
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| Higher order linear equations | |
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| Remarks on changes of variables | |
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| Analytic Linear Equations | |
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| Fundamental matrices | |
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| Simple singularities | |
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| Higher order equations | |
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| A nonsimple singularity | |
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| Notes | |
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| Dependence on initial conditions and parameters | |
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| Preliminaries | |
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| Continuity | |
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| Differentiability | |
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| Higher order differentiability | |
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| Exterior derivatives | |
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| Another differentiability theorem | |
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| S- and L-Lipschitz continuity | |
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| Uniqueness theorem | |
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| A lemma | |
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| Proof of Theorem 8.1 | |
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| Proof of Theorem 6.1 | |
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| First integrals | |
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| Notes | |
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| Total and partial differential equations | |
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| A theorem of Frobenius | |
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| Total differential equations | |
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| Algebra of exterior forms | |
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| A theorem of Frobenius | |
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| Proof of Theorem 3.1 | |
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| Proof of Lemma 3.1 | |
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| The system (1.1) | |
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| Cauchy's method of characteristics | |
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| A nonlinear partial differential equation | |
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| Characteristics | |
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| Existence and uniqueness theorem | |
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| Haar's lemma and uniqueness | |
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| Notes | |
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| The Poincare-Bendixson theory | |
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| Autonomous systems | |
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| Umlaufsatz | |
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| Index of a stationary point | |
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| The Poincare-Bendixson theorem | |
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| Stability of periodic solutions | |
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| Rotation points | |
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| Foci, nodes, and saddle points | |
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| Sectors | |
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| The general stationary point | |
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| A second order equation | |
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| Poincare-Bendixson theory on 2-manifolds | |
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| Preliminaries | |
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| Analogue of the Poincare-Bendixson theorem | |
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| Flow on a closed curve | |
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| Flow on a torus | |
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| Notes | |
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| Plane stationary points | |
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| Existence theorems | |
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| Characteristic directions | |
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| Perturbed linear systems | |
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| More general stationary point | |
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| Notes | |
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| Invariant manifolds and linearlizations | |
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| Invariant manifolds | |
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| The maps T[superscript t] | |
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| Modification of F([xi]) | |
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| Normalizations | |
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| Invariant manifolds of a map | |
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| Existence of invariant manifolds | |
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| Linearizations | |
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| Linearization of a map | |
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| Proof of Theorem 7.1 | |
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| Periodic solution | |
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| Limit cycles | |
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| Smooth equivalence maps | |
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| Smooth linearizations | |
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| Proof of Lemma 12.1 | |
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| Proof of Theorem 12.2 | |
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| Smoothness of stable manifolds | |
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| Notes | |
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| Perturbed linear systems | |
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| The case E = 0 | |
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| A topological principle | |
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| A theorem of Wazewski | |
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| Preliminary lemmas | |
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| Proof of Lemma 4.1 | |
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| Proof of Lemma 4.2 | |
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| Proof of Lemma 4.3 | |
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| Asymptotic integrations. Logarithmic scale | |
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| Proof of Theorem 8.2 | |
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| Proof of Theorem 8.3 | |
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| Logarithmic scale (continued) | |
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| Proof of Theorem 11.2 | |
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| Asymptotic integration | |
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| Proof of Theorem 13.1 | |
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| Proof of Theorem 13.2 | |
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| Corollaries and refinements | |
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| Linear higher order equations | |
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| Notes | |
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| Linear second order equations | |
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| Preliminaries | |
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| Basic facts | |
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| Theorems of Sturm | |
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| Sturm-Liouville boundary value problems | |
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| Number of zeros | |
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| Nonoscillatory equations and principal solutions | |
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| Nonoscillation theorems | |
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| Asymptotic integrations. Elliptic cases | |
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| Asymptotic integrations. Nonelliptic cases | |
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| Disconjugate systems | |
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| Disconjugate systems | |
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| Generalizations | |
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| Notes | |
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| Use of implicit function and fixed point theorems | |
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| Periodic solutions | |
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| Linear equations | |
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| Nonlinear problems | |
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| Second order boundary value problems | |
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| Linear problems | |
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| Nonlinear problems | |
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| A priori bounds | |
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| General theory | |
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| Basic facts | |
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| Green's functions | |
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| Nonlinear equations | |
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| Asymptotic integration | |
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| Notes | |
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| Dichotomies for solutions of linear equations | |
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| General theory | |
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| Notations and definitions | |
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| Preliminary lemmas | |
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| The operator T | |
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| Slices of [double vertical line]Py(t)[double vertical line] | |
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| Estimates for [double vertical line]y(t)[double vertical line] | |
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| Applications to first order systems | |
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| Applications to higher order systems | |
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| P(B, D)-manifolds | |
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| Adjoint equations | |
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| Associate spaces | |
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| The operator T' | |
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| Individual dichotomies | |
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| P'-admissible spaces for T' | |
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| Applications to differential equations | |
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| Existence of P D-solutions | |
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| Notes | |
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| Miscellany on monotony | |
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| Monotone solutions | |
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| Small and large solutions | |
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| Monotone solutions | |
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| Second order linear equations | |
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| Second order linear equations (continuation) | |
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| A problem in boundary layer theory | |
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| The problem | |
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| The case [lambda] > 0 | |
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| The case [lambda] < 0 | |
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| The case [lambda] = 0 | |
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| Asymptotic behavior | |
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| Global asymptotic stability | |
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| Global asymptotic stability | |
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| Lyapunov functions | |
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| Nonconstant G | |
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| On Corollary 11.2 | |
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| On "J(y)x . x [less than or equal] 0 if x . f(y) = 0" | |
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| Proof of Theorem 14.2 | |
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| Proof of Theorem 14.1 | |
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| Notes | |
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| Hints for exercises | |
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| References | |
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| Index | |