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Optimization problems involving uncertain data arise in many areas of industrial and economic applications. Stochastic programming provides a useful framework for modeling and solving optimization problems for which a probability distribution of the unknown parameters is available. Motivated by practical optimization problems occurring in energy systems with regenerative energy supply, Debora Mahlke formulates and analyzes multistage stochastic mixed-integer models. For their solution, the author proposes a novel decomposition approach which relies on the concept of splitting the underlying scenario tree into subtrees. Based on the formulated models from energy production, the algorithm is computationally investigated and the numerical results are discussed.

Mathematical optimization. --- Stochastic approximation. --- Stochastic processes. --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Applied Mathematics --- Mathematical Statistics --- Stochastic programming. --- Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Mathematics, general. --- Linear programming --- Distribution (Probability theory. --- Math --- Science --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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The purpose of these lecture notes is to provide an introduction to the general theory of empirical risk minimization with an emphasis on excess risk bounds and oracle inequalities in penalized problems. In recent years, there have been new developments in this area motivated by the study of new classes of methods in machine learning such as large margin classification methods (boosting, kernel machines). The main probabilistic tools involved in the analysis of these problems are concentration and deviation inequalities by Talagrand along with other methods of empirical processes theory (symmetrization inequalities, contraction inequality for Rademacher sums, entropy and generic chaining bounds). Sparse recovery based on l_1-type penalization and low rank matrix recovery based on the nuclear norm penalization are other active areas of research, where the main problems can be stated in the framework of penalized empirical risk minimization, and concentration inequalities and empirical processes tools have proved to be very useful.

Regression analysis --- Estimation theory --- Nonparametric statistics --- Probabilities --- Inequalities (Mathematics) --- Sparse matrices --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Machine learning --- Risk management --- Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions

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This book explores the remarkable connections between two domains that, a priori, seem unrelated: Random matrices (together with associated random processes) and integrable systems. The relations between random matrix models and the theory of classical integrable systems have long been studied. These appear mainly in the deformation theory, when parameters characterizing the measures or the domain of localization of the eigenvalues are varied. The resulting differential equations determining the partition function and correlation functions are, remarkably, of the same type as certain equations appearing in the theory of integrable systems. They may be analyzed effectively through methods based upon the Riemann-Hilbert problem of analytic function theory and by related approaches to the study of nonlinear asymptotics in the large N limit. Associated with studies of matrix models are certain stochastic processes, the "Dyson processes", and their continuum diffusion limits, which govern the spectrum in random matrix ensembles, and may also be studied by related methods. Random Matrices, Random Processes and Integrable Systems provides an in-depth examination of random matrices with applications over a vast variety of domains, including multivariate statistics, random growth models, and many others. Leaders in the field apply the theory of integrable systems to the solution of fundamental problems in random systems and processes using an interdisciplinary approach that sheds new light on a dynamic topic of current research.

Random matrices. --- Stochastic processes. --- Random matrices --- Stochastic processes --- Hamiltonian systems --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Applied Physics --- Random processes --- Matrices, Random --- Physics. --- Probabilities. --- Theoretical, Mathematical and Computational Physics. --- Probability Theory and Stochastic Processes. --- Probabilities --- Matrices --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Mathematical physics. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Physical mathematics --- Physics

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Since the publication of the first edition of this seminar book in 1994, the theory and applications of extremes and rare events have enjoyed an enormous and still increasing interest. The intention of the book is to give a mathematically oriented development of the theory of rare events underlying various applications. This characteristic of the book was strengthened in the second edition by incorporating various new results. In this third edition, the dramatic change of focus of extreme value theory has been taken into account: from concentrating on maxima of observations it has shifted to large observations, defined as exceedances over high thresholds. One emphasis of the present third edition lies on multivariate generalized Pareto distributions, their representations, properties such as their peaks-over-threshold stability, simulation, testing and estimation. Reviews of the 2nd edition: "In brief, it is clear that this will surely be a valuable resource for anyone involved in, or seeking to master, the more mathematical features of this field." David Stirzaker, Bulletin of the London Mathematical Society "Laws of Small Numbers can be highly recommended to everyone who is looking for a smooth introduction to Poisson approximations in EVT and other fields of probability theory and statistics. In particular, it offers an interesting view on multivariate EVT and on EVT for non-iid observations, which is not presented in a similar way in any other textbook." Holger Drees, Metrika.

Extreme value theory. --- Poisson processes. --- Distribution (Probability theory) --- Random variables --- Processes, Poisson --- Point processes --- Distribution (Probability theory. --- Mathematical statistics. --- Probability Theory and Stochastic Processes. --- Statistical Theory and Methods. --- Mathematics --- Statistical inference --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Distribution functions --- Frequency distribution --- Characteristic functions --- Statistical methods --- Probabilities. --- Statistics . --- Statistical analysis --- Statistical data --- Statistical science --- Econometrics --- Probability --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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Inequality has become an essential tool in many areas of mathematical research, for example in probability and statistics where it is frequently used in the proofs. "Probability Inequalities" covers inequalities related with events, distribution functions, characteristic functions, moments and random variables (elements) and their sum. The book shall serve as a useful tool and reference for scientists in the areas of probability and statistics, and applied mathematics. Prof. Zhengyan Lin is a fellow of the Institute of Mathematical Statistics and currently a professor at Zhejiang University, Hangzhou, China. He is the prize winner of National Natural Science Award of China in 1997. Prof. Zhidong Bai is a fellow of TWAS and the Institute of Mathematical Statistics; he is a professor at the National University of Singapore and Northeast Normal University, Changchun, China.

Inequalities (Mathematics) --- Probabilities. --- Distribution (Probability theory) --- Mathematics. --- Statistics. --- Probability Theory and Stochastic Processes. --- Statistical Theory and Methods. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Processes, Infinite --- Distribution (Probability theory. --- Mathematical statistics. --- Statistics, Mathematical --- Statistics --- Sampling (Statistics) --- Statistical methods --- Statistics . --- Statistical analysis --- Statistical data --- Statistical science --- Econometrics