Linked e-resources
Details
Table of Contents
Preface; Backbone of the Book; Acknowledgements; Contents; Introduction and History; Historical Comments; Contents of the Book; Part I The General Theory; 1 Probabilistic Background; 1.1 Markov Chains; 1.2 Random Walks in a Quarter Plane; 1.3 Functional Equations for the Invariant Measure; 2 Foundations of the Analytic Approach ; 2.1 Fundamental Notions and Definitions; 2.1.1 Covering Manifolds; 2.1.2 Algebraic Functions; 2.1.3 Elements of Galois Theory; 2.1.4 Universal Covering and Uniformization; 2.1.5 Abelian Differentials and Divisors; 2.2 Restricting the Equation to an Algebraic Curve
2.2.1 First Insight (Algebraic Functions)2.2.2 Second Insight (Algebraic Curve); 2.2.3 Third Insight (Factorization); 2.2.4 Fourth Insight (Riemann Surfaces); 2.3 The Algebraic Curve Q(x,y) = 0; 2.3.1 Branches of the Algebraic Functions on the Unit Circle; 2.3.2 Branch Points; 2.4 Galois Automorphisms and the Group of the Random Walk; 2.4.1 Construction of the Automorphisms and on S; 2.5 Reduction of the Main Equation to the Riemann Torus; 3 Analytic Continuation of the Unknown Functions in the Genus 1 Case ; 3.1 Lifting the Fundamental Equation onto the Universal Covering
4.5 Various Comments4.6 Rational Solutions; 4.6.1 The Case N(f) neq1; 4.6.2 The Case N(f) = 1; 4.7 Algebraic Solutions; 4.7.1 The Case N(f)=1; 4.7.2 The Case N(f) neq1 ; 4.8 Final Form of the General Solution; 4.9 The Problem of the Poles and Examples ; 4.9.1 Rational Solutions; 4.10 An Example of an Algebraic Solution by Flatto and Hahn; 4.11 Two Queues in Tandem; 5 Solution in the Case of an Arbitrary Group; 5.1 Informal Reduction to a Riemann
Hilbert
Carleman BVP; 5.2 Introduction to BVPs in the Complex Plane; 5.2.1 A Bit of History; 5.2.2 The Sokhotski
Plemelj Formulae
5.2.3 The Riemann Boundary Value Problem for a Closed Contour5.2.4 The Riemann BVP for an Open Contour; 5.2.5 The Riemann
Carleman Problem with a Shift; 5.3 Further Properties of the Branches Defined by Q(x,y) = 0; 5.4 Index and Solution of the BVP (5.1.5); 5.5 Complements; 5.5.1 Analytic Continuation; 5.5.2 Computation of w; 6 The Genus 0 Case; 6.1 Properties of the Branches; 6.2 Case 1: p01 = p-1,0 = p-1,1 = 0; 6.3 Case 3: p11 = p10 = p01 = 0; 6.4 Case 4: p-1,0 = p0,-1 = p-1,-1 = 0; 6.4.1 Integral Equation; 6.4.2 Series Representation; 6.4.3 Uniformization
2.2.1 First Insight (Algebraic Functions)2.2.2 Second Insight (Algebraic Curve); 2.2.3 Third Insight (Factorization); 2.2.4 Fourth Insight (Riemann Surfaces); 2.3 The Algebraic Curve Q(x,y) = 0; 2.3.1 Branches of the Algebraic Functions on the Unit Circle; 2.3.2 Branch Points; 2.4 Galois Automorphisms and the Group of the Random Walk; 2.4.1 Construction of the Automorphisms and on S; 2.5 Reduction of the Main Equation to the Riemann Torus; 3 Analytic Continuation of the Unknown Functions in the Genus 1 Case ; 3.1 Lifting the Fundamental Equation onto the Universal Covering
4.5 Various Comments4.6 Rational Solutions; 4.6.1 The Case N(f) neq1; 4.6.2 The Case N(f) = 1; 4.7 Algebraic Solutions; 4.7.1 The Case N(f)=1; 4.7.2 The Case N(f) neq1 ; 4.8 Final Form of the General Solution; 4.9 The Problem of the Poles and Examples ; 4.9.1 Rational Solutions; 4.10 An Example of an Algebraic Solution by Flatto and Hahn; 4.11 Two Queues in Tandem; 5 Solution in the Case of an Arbitrary Group; 5.1 Informal Reduction to a Riemann
Hilbert
Carleman BVP; 5.2 Introduction to BVPs in the Complex Plane; 5.2.1 A Bit of History; 5.2.2 The Sokhotski
Plemelj Formulae
5.2.3 The Riemann Boundary Value Problem for a Closed Contour5.2.4 The Riemann BVP for an Open Contour; 5.2.5 The Riemann
Carleman Problem with a Shift; 5.3 Further Properties of the Branches Defined by Q(x,y) = 0; 5.4 Index and Solution of the BVP (5.1.5); 5.5 Complements; 5.5.1 Analytic Continuation; 5.5.2 Computation of w; 6 The Genus 0 Case; 6.1 Properties of the Branches; 6.2 Case 1: p01 = p-1,0 = p-1,1 = 0; 6.3 Case 3: p11 = p10 = p01 = 0; 6.4 Case 4: p-1,0 = p0,-1 = p-1,-1 = 0; 6.4.1 Integral Equation; 6.4.2 Series Representation; 6.4.3 Uniformization