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Glück, Logik und Bluff Welche Gewinnaussichten bietet ein Spiel? Und wie sollte man am besten spielen? Die beiden Fragen führen je nach Typ eines Spiels zu ganz unterschiedlichen mathematischen Mechanismen: Die Wahrscheinlichkeitsrechnung erlaubt es, zufällige Einflüsse in Glücksspielen zu kalkulieren, um so die Gewinnchancen der Spieler abzuschätzen. Wie ein Schachcomputer funktioniert und welchen Grenzen die zugrundeliegenden Algorithmen unterworfen sind, davon handelt die Theorie der kombinatorischen Spiele. Ganz andere Optimierungsansätze, nämlich solche aus der mathematischen Spieltheorie, sind gefragt, wenn Kartenspieler ihre Entscheidungen in Unkenntnis der Karten ihrer Mitspieler treffen müssen. Die drei genannten Theorien werden anhand konkreter (Bei-)Spiele erörtert, darunter Roulette, Lotto, Monopoly, Risiko, Black Jack, das Leiterspiel, Schach, Mühle, Go-Moku, Nim, Backgammon, Go, Mastermind, Memory, Pokern und Baccarat. Trotz der populären Darstellung, die mathematisches Interesse aber kaum Vorkenntnisse voraussetzt, sind die Methoden so konkret beschrieben, dass eine entsprechende Programmierung oder eine Übertragung auf andere Fälle möglich ist. In der 6. Auflage wurden Erläuterungen zu Zwischenschritten bei der Berechnung von Black Jack sowie neuere spieltheoretische Resultate über Mastermind ergänzt. Der Inhalt Glücksspiele - Kombinatorische Spiele - Strategische Spiele Die Zielgruppe - Mathematisch vorgebildete Leser, die Interesse an Spielen haben - Mathematiklehrer - Studierende und Dozenten der Mathematik Der Autor Dr. Jörg Bewersdorff, Dipl. Mathematiker, ist seit mehreren Jahren Geschäftsführer der Firma MEGA-Spielgeräte in Limburg.

Probabilities. --- Game theory. --- Probability Theory and Stochastic Processes. --- Game Theory, Economics, Social and Behav. Sciences.

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Problems in Probability  comprises one of the most comprehensive, nearly encyclopedic, collections of problems and exercises in probability theory. Albert Shiryaev has skillfully created, collected, and compiled  the exercises in this text over the course of many years while working on topics which interested him the most.   A substantial  number of the exercises resulted from diverse sources such as textbooks, lecture notes, exercise manuals, monographs, and discussions that took place during special seminars for graduate and undergraduate students. Many problems contain helpful hints and other relevant comments and a portion of the material covers some important applications from optimal control and mathematical finance. Readers of diverse backgrounds—from students to researchers—will find a great deal of value in this book and can treat the work as an exercise manual, a handbook, or as a supplementary text to a course in probability theory, control, and mathematical finance. The problems and exercises in this book vary in nature and degree of difficulty. Some problems are meant to test the reader’s basic understanding, others are of medium-to-high degrees of difficulty and require more creative thinking. Other problems are meant to develop additional theoretical concepts and tools or to familiarize the reader with various facts that are not necessarily covered in mainstream texts. Additional problems are related to the passage from random walk to Brownian motions and Brownian bridges. The appendix contains a summary of the main results, notation and terminology that are used throughout the book.  It also contains additional material from combinatorics, potential theory and Markov chains—subjects that are not covered in the book, but are nevertheless needed for many of the exercises included here.

Probabilities --- Mathematics --- Mathematics. --- Probabilities. --- Combinatorics. --- Probability Theory and Stochastic Processes. --- Distribution (Probability theory. --- Combinatorics --- Algebra --- Mathematical analysis --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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Probabilistic approaches have played a prominent role in the study of complex physical systems for more than thirty years. This volume collects twenty articles on various topics in this field, including self-interacting random walks and polymer models in random and non-random environments, branching processes, Parisi formulas and metastability in spin glasses, and hydrodynamic limits for gradient Gibbs models. The majority of these articles contain original results at the forefront of contemporary research; some of them include review aspects and summarize the state-of-the-art on topical issues – one focal point is the parabolic Anderson model, which is considered with various novel aspects including moving catalysts, acceleration and deceleration and fron propagation, for both time-dependent and time-independent potentials. The authors are among the world’s leading experts. This Festschrift honours two eminent researchers, Erwin Bolthausen and Jürgen Gärtner, whose scientific work has profoundly influenced the field and all of the present contributions.

Stochastic processes. --- Mathematics --- Physics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Atomic Physics --- Statistical physics. --- Mathematical statistics. --- Statistical inference --- Statistics, Mathematical --- Statistical methods --- Mathematics. --- Probabilities. --- Statistics. --- Probability Theory and Stochastic Processes. --- Statistics, general. --- Statistics --- Probabilities --- Sampling (Statistics) --- Mathematical statistics --- Distribution (Probability theory. --- Statistical analysis --- Statistical data --- Statistical science --- Econometrics --- Distribution functions --- Frequency distribution --- Characteristic functions --- Statistics . --- Probability --- Combinations --- Chance --- Least squares --- Risk

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Many notions and results presented in the previous editions of this volume have since become quite popular in applications, and many of them have been “rediscovered” in applied papers.   In the present 3rd edition small changes were made to the chapters in which long-time behavior of the perturbed system is determined by large deviations. Most of these changes concern terminology. In particular, it is explained that the notion of sub-limiting distribution for a given initial point and a time scale is identical to the idea of metastability, that the stochastic resonance is a manifestation of metastability, and that the theory of this effect is a part of the large deviation theory. The reader will also find new comments on the notion of quasi-potential that the authors introduced more than forty years ago, and new references to recent papers in which the proofs of some conjectures included in previous editions have been obtained.   Apart from the above mentioned changes the main innovations in the 3rd edition concern the averaging principle. A new Section on deterministic perturbations of one-degree-of-freedom systems was added in Chapter 8. It is shown there that pure deterministic perturbations of an oscillator may lead to a stochastic, in a certain sense, long-time behavior of the system, if the corresponding Hamiltonian has saddle points. The usefulness of a joint consideration of classical theory of deterministic perturbations together with stochastic perturbations is illustrated in this section. Also a new Chapter 9 has been inserted in which deterministic and stochastic perturbations of systems with many degrees of freedom are considered. Because of the resonances, stochastic regularization in this case is even more important.

Perturbation (Mathematics). --- Stochastic processes. --- Stochastic processes --- Perturbation (Mathematics) --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Perturbation equations --- Perturbation theory --- Random processes --- Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Approximation theory --- Dynamics --- Functional analysis --- Mathematical physics --- Probabilities --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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This monograph provides an extensive treatment of the theory and applications of the celebrated Borel-Cantelli Lemma. Starting from some of the basic facts of the axiomatic probability theory, it embodies the classical versions of these lemma, together with the well known as well as the most recent extensions of them due to Barndorff-Nielsen, Balakrishnan and Stepanov, Erdos and Renyi, Kochen and Stone, Petrov and the present author. The versions of the second Borel-Cantelli Lemma for pair wise negative quadrant dependent sequences, weakly *-mixing sequences, mixing sequences (due to Renyi) and for many other dependent sequences are all included. The special feature of the book is a detailed discussion of a strengthened form of the second Borel-Cantelli Lemma and the conditional form of the Borel-Cantelli Lemmas due to Levy, Chen and Serfling. All these results are well illustrated by means of many interesting examples. All the proofs are rigorous, complete and lucid. An extensive list of research papers, some of which are forthcoming, is provided. The book can be used for a self study and as an invaluable research reference on the present topic.

Number theory. --- Stochastic approximation. --- Probabilities --- Measure theory --- Mathematical statistics --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Probabilities. --- Measure theory. --- Mathematical statistics. --- Statistical inference --- Statistics, Mathematical --- Lebesgue measure --- Measurable sets --- Measure of a set --- Probability --- Statistical methods --- Statistics. --- Statistical Theory and Methods. --- Probability Theory and Stochastic Processes. --- Combinations --- Chance --- Least squares --- Risk --- Statistics --- Sampling (Statistics) --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Statistics . --- Statistical analysis --- Statistical data --- Statistical science --- Econometrics --- Borell-Cantelli Lemma.