This document describes the software library TestU01, implemented in the ANSI C language, and offering a collection of utilities for the (empirical) statistical testing of uniform random number generators (RNG). The library implements several types of generators in generic form, as well as many specific generators proposed in the literature or found in widely-used software. It provides general implementations of the classical statistical tests for random number generators, as well as several others proposed in the literature, and some original ones. These tests can be applied to the generators predefined in the library and to user-defined generators. Specific tests suites for either sequences of uniform random numbers in [0, 1] or bit sequences are also available. Basic tools for plotting vectors of points produced by generators are provided as well. Additional software permits one to perform systematic studies of the interaction between a specific test and the structure of the point sets produced by a given family of RNGs. That is, for a given kind of test and a given class of RNGs, to determine how large should be the sample size of the test, as a function of the generator’s period length, before the generator starts to fail the test systematically.

Abstract. Fast and reliable pseudo-random number generators are required for simulation and other applications in Scientific Computing. We outline the requirements for good uniform random number generators, and describe a class of generators having very fast vector/parallel implementations with excellent statistical properties. We also discuss the problem of initialising random number generators, and consider how to combine two or more generators to give a better (though usually slower) generator. 1

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We evaluate the virial coefficients Bk for k ≤ 10 for hard spheres in dimensions D = 2, · · ·,8. Virial coefficients with k even are found to be negative when D ≥ 5. This provides strong evidence that the leading singularity for the virial series lies away from the positive real axis when D ≥ 5. Further analysis provides evidence that negative virial coefficients will be seen for some k> 10 for D = 4, and there is a distinct possibility that negative virial coefficients will also eventually occur for D = 3.

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The first part of this introductory lecture on Monte Carlo methods gives an overview of theessential ideas of Monte Carlo, discusses the relation between the basic sampling concept and statistical mechanics and the probabilistic interpretation of quantum mechanics, and provides aclassification of existing Monte Carlo methods. This part is followed by a summary of essential concepts of probability and statistics, the construction of random walks, and the application ofrandom sampling in the estimation of integrals. 1 Introduction The original title of this lecture was supposed to be "Classical Monte Carlo", which issomewhat surprising for a winter school about quantum simulations. Since this is the first scientific lecture of this week I decided to interpret the title in a more general sense of(i) providing a sort of classification of the existing Monte Carlo methods, (ii) discussing certain algorithmic elements which were originally invented for classical Monte Carlo sim-ulations but turn out to be useful also for quantum simulations, and last but not least (iii) giving some guidelines of how to handle the stochastic nature of the results. Many of thesesubjects will be revisited by other speakers in specialized lectures during this week.

as simulation input, and they strongly influence the results. Thus, their usage and the usage of their generator need to be taken care of very well. Qualified generators are available on the web as source code or libraries. However, they require an additional middleware to adapt them to the running environments, and this can lead to misuses. This paper proposes Pseudo Random Number Generators embedded inside a Web Service for the use in simulations. This service, while offering a unique interface accessible from any platform, eases the important task of retrieving correct pseudo random numbers. The general architecture of the service is presented, as well as a reference implementation. The performance of the Web Service generator is compared to the performance of local generators.