Table of contents

Contents §1. Introduction §2. Linear equation for average times: the case Associated branching process The simplest queue §3. Fundamental non-linear equation: the case Minimal solution of the equation §4. The main criterion. The case §5. Classification of Markov chains by means of a minimal solution §6. Appendix

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Contents §1. Introduction §2. The case A §3. The case B §4. The case C

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Contents §1. Introduction §2. The principle of the largest term 2.1. The general setting 2.2. The principle of the largest term 2.3. Upper and lower deviation functions 2.4. Concentration of measures on compact spaces §3. Vague large deviation principles and Ruelle-Lanford functions 3.1. Vague large deviation principles 3.2. Ruelle-Lanford functions §4. Examples §5. Narrow large deviation principles and exponential tightness 5.1. Narrow large deviation principles 5.2. Exponential tightness 5.3. Concentration of exponentially tight measures §6. Large deviation principles and Varadhan's theorems 6.1. Large deviation principles 6.2. Varadhan's theorems §7. Convexity 7.1. The scaled generating function 7.2. Weak law of large numbers and the differentiability of the pressure

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Contents §1. Introduction §2. Statements of the problems §3. The space and §4. The space for and some open problems §5. and

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Contents Introduction §1. The abstract KAM theorem §2. Remarks §3. Hamiltonian systems §4. Divergence-free systems §5. Reversible systems

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Contents Introduction §1. Harmonics and splines 1.1. Spaces of generalized harmonics 1.2. Generalized splines on groups 1.3. Interpolation of generalized harmonics 1.4. Interpolation properties of spaces of generalized splines 1.5. Harmonics as limits of splines 1.6. Average dimension of harmonics and splines §2. Harmonics as optimal tools for the approximation and recovery of functions 2.1. Widths of generalized Sobolev classes in §3. Splines as optimal tools for approximation 3.1. Introduction 3.2. Stationary points of the Rayleigh relation 3.3. Example 3.4. Main theorems §4. Smoothness and approximation. Asymptotic results 4.1. What is smoothness? 4.2. Orthoprojectional and non-linear widths 4.3. Statement of the problem of asymptotically best approximation methods 4.4. Upper bounds 4.5. Some comments Appendix by G.G. Magaril-Il'yaev §1. A criterion for an element of best approximation 1.1. Statement of the problem 1.2. Statement of the result 1.3. Proof of the theorem 1.4. Consequences §2. Asymptotics of the average width of a Sobolev class of functions on with a mixed norm

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Contents §1. Introduction and results §2. Laces and the idea of the proof §3. Induction hypothesis and initial step §4. Induction step: estimation of and §5. Induction step: convergence of polymer expansions for and §6. Induction step: estimate of §7. Proof of Theorem 1.1 Appendix. The polymer expansion theorem

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