Wolfgang Trutschnig – författare

Ancient times witnessed the origins of the theory of continued fractions. Throughout time, mathematical geniuses such as Euclid, Aryabhata, Fibonacci, Bombelli, Wallis, Huygens, or Euler have made significant contributions to the development of this famous theory, and it continues to evolve today, especially as a means of linking different areas of mathematics.

This book, whose primary audience is graduate students and senior researchers, is motivated by the fascinating interrelations between ergodic theory and number theory (as established since the 1950s). It examines several generalizations and extensions of classical continued fractions, including generalized Lehner, simple, and Hirzebruch-Jung continued fractions. After deriving invariant ergodic measures for each of the underlying transformations on [0,1] it is shown that any of the famous formulas, going back to Khintchine and Levy, carry over to more general settings. Complementing these results, the entropy of the transformations is calculated and the natural extensions of the dynamical systems to [0,1]2 are analyzed.

Features

Ancient times witnessed the origins of the theory of continued fractions. Throughout time, mathematical geniuses such as Euclid, Aryabhata, Fibonacci, Bombelli, Wallis, Huygens, or Euler have made significant contributions to the development of this famous theory, and it continues to evolve today, especially as a means of linking different areas of mathematics.

This book, whose primary audience is graduate students and senior researchers, is motivated by the fascinating interrelations between ergodic theory and number theory (as established since the 1950s). It examines several generalizations and extensions of classical continued fractions, including generalized Lehner, simple, and Hirzebruch-Jung continued fractions. After deriving invariant ergodic measures for each of the underlying transformations on [0,1] it is shown that any of the famous formulas, going back to Khintchine and Levy, carry over to more general settings. Complementing these results, the entropy of the transformations is calculated and the natural extensions of the dynamical systems to [0,1]2 are analyzed.

Features

This volume contains more than 65 peer-reviewed papers corresponding to presentations at the 11th Conference on Soft Methods in Probability and Statistics (SMPS) held in Salzburg, Austria, in September 2024. It covers recent advances in the field of probability, statistics, and data science, with a particular focus on dealing with dependence, imprecision and incomplete information. Reflecting the fact that data science continues to evolve, this book serves as a bridge between different groups of experts, including statisticians, mathematicians, computer scientists, and engineers, and encourages interdisciplinary research. The selected contributions cover a wide range of topics such as imprecise probabilities, random sets, belief functions, possibility theory, and dependence modeling. Readers will find discussions on clustering, depth concepts, dimensionality reduction, and robustness, reflecting the conference''s commitment to addressing real-world challenges through innovative methods.