Contents Preface vi i 1. Mathematica l model 1 1.1. Filtration flo w of an incompressible fluid 1 1.2. The moving boundary proble m 2 2. First integrals of boundary motion 5 5 2.2. Reconstruction o f a domain fro m th e values of its moments, and th e inverse problem of two-dimensional potentia l theory 6 2.3. Results on the uniqueness of a domain with given moments (potential) 7 2.4. The resul t o f injectio n doe s not depen d o n th e orde r o f wor k o f th e sources and sink s 9 Problems 1 0 3. Algebraic solutions 1 1 3.1. Algebraic and abelian domains 1 1 3.2. Algebraic solutions 1 1 3.3. The Cauchy transform an d its properties 1 2 3.4. Singularity correspondence theore m 1 2 3.5. Proof o f the theorem o n algebraic solutions 1 4 3.6. Construction o f algebraic solutions 1 4 iii

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