These topics are of theoretical and applied interest in mathematics and statistics. There are applications to mathematical finance, cryptology, and applied statistics. This research-level monograph surveys the theoretical and applied aspects.

From Probabilistic Diophantine Approximation to Quadratic Fields.- 1 Part I: Super Irregularity.- 2 Part II: Probabilistic Diophantine Approximation.- 3 Part III: Quadratic Fields and Continued Fractions.- 4 Part IV: Class Number One Problems.- 5 Part V: Cesaro Mean of % MathType!MTEF!2!1!+-
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olimits_n {\left( {\left\{ {n\alpha } \right\} - 1/2} \right)} $$.- 6 References.- On the Assessment of Random and Quasi-Random Point Sets.- 1 Introduction.- 2 Chapter for the Practitioner.- 3 Mathematical Preliminaries.- 4 Uniform Distribution Modulo One.- 5 The Spectral Test.- 6 The Weighted Spectral Test.- 7 Discrepancy.- 8 Summary.- 9 Acknowledgements.- 10 References.- Lattice Rules: How Well Do They Measure Up?.- 1 Introduction.- 2 Some Basic Properties of Lattice Rules.- 3 A General Approach to Worst-Case and Average-Case Error Analysis.- 4 Examples of Other Discrepancies.- 5 Shift-Invariant Kernels and Discrepancies.- 6 Discrepancy Bounds.- 7 Discrepancies of Integration Lattices and Nets.- 8 Tractability of High Dimensional Quadrature.- 9 Discussion and Conclusion.- 10 References.- Digital Point Sets: Analysis and Application.- 1 Introduction.- 2 The Concept and Basic Properties of Digital Point Sets.- 3 Discrepancy Bounds for Digital Point Sets.- 4 Special Classes of Digital Point Sets and Quality Bounds.- 5 Digital Sequences Based on Formal Laurent Series and Non-Archimedean Diophantine Approximation.- 6 Analysis of Pseudo-Random-Number Generators by Digital Nets.- 7 The Digital Lattice Rule.- 8 Outlook and Open Research Topics.- 9 References.- Random Number Generators: Selection Criteria and Testing.- 1 Introduction.- 2 Design Principles and Figures of Merit.- 3 Empirical Statistical Tests.- 4 Examples of Empirical Tests.- 5 Collections of Small RNGs.- 6 Systematic Testing for Small RNGs.- 7 How Do Real-Life Generators Fare in These Tests?.- 8 Acknowledgements.- 9 References.- Nets, (ts)-Sequences, and Algebraic Geometry.- 1 Introduction.- 2 Basic Concepts.- 3 The Digital Method.- 4 Background on Algebraic Curves over Finite Fields.- 5 Construction of (ts)-Sequences.- 6 New Constructions of (tms)-Nets.- 7 New Algebraic Curves with Many Rational Points.- 8 References.- Financial Applications of Monte Carlo and Quasi-Monte Carlo Methods.- 1 Introduction.- 2 Monte Carlo Methods for Finance Applications.- 3 Speeding Up by Quasi-Monte Carlo Methods.- 4 Future Topics.- 5 References.