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This book includes a self-contained approach of the general theory of quadratic forms and integral Euclidean lattices, as well as a presentation of the theory of automorphic forms and Langlands' conjectures, ranging from the first definitions to the recent and deep classification results due to James Arthur. Its connecting thread is a question about lattices of rank 24: the problem of p-neighborhoods between Niemeier lattices. This question, whose expression is quite elementary, is in fact very natural from the automorphic point of view, and turns out to be surprisingly intriguing. We explain how the new advances in the Langlands program mentioned above pave the way for a solution. This study proves to be very rich, leading us to classical themes such as theta series, Siegel modular forms, the triality principle, L-functions and congruences between Galois representations. This monograph is intended for any mathematician with an interest in Euclidean lattices, automorphic forms or number theory. A large part of it is meant to be accessible to non-specialists.

Automorphic forms. --- Automorphic functions --- Forms (Mathematics) --- Number theory. --- Algebra. --- Number Theory. --- Discrete Mathematics. --- Mathematics --- Mathematical analysis --- Number study --- Numbers, Theory of --- Algebra --- Discrete mathematics. --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis

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This proceedings volume presents a diverse collection of high-quality, state-of-the-art research and survey articles, written by leading experts, on low-dimensional topology and its applications. The content addresses a wide range of historical and contemporary invariants of knots and links, as well as related topics including: three- and four-dimensional manifolds, braids, virtual knot theory, quantum invariants, braids, skein modules and knot algebras, link homology, quandles and their homology, hyperbolic knots and geometric structures of three-dimensional manifolds, the mechanism of topological surgery in physical processes, knots in nature in the sense of physical knots with applications to polymers, DNA enzyme mechanisms, and protein structure and function. The chapters are based on contributions presented at the International Conference on Knots, Low-Dimensional Topology and Applications – Knots in Hellas 2016, which was held at the International Olympic Academy in Greece in July 2016. The goal of the conference was to promote the exchange of methods and ideas across disciplines and generations, from graduate students to senior researchers, and to explore fundamental research problems in the broad fields of knot theory and low-dimensional topology. This book will benefit all researchers who wish to take their research in new directions, to learn about new tools and methods, and to discover relevant and recent literature for future study.

Algebra. --- Geometry. --- Topology. --- Discrete mathematics. --- Biomathematics. --- Statistical physics. --- Algebra. --- Geometry. --- Topology. --- Discrete Mathematics. --- Mathematical and Computational Biology. --- Statistical Physics and Dynamical Systems. --- Physics --- Mathematical statistics --- Physics --- Biology --- Biology --- Mathematics --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Mathematics --- Euclid's Elements --- Mathematics --- Mathematical analysis --- Statistical methods --- Mathematics

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This volume collects together research and survey papers written by invited speakers of a conference celebrating the 70th birthday of László Lovász. The topics covered include classical subjects such as extremal graph theory, coding theory, design theory, applications of linear algebra and combinatorial optimization, as well as recent trends such as extensions of graph limits, online or statistical versions of classical combinatorial problems, and new methods of derandomization. László Lovász is one of the pioneers in the interplay between discrete and continuous mathematics, and is a master at establishing unexpected connections, “building bridges” between seemingly distant fields. His invariably elegant and powerful ideas have produced new subfields in many areas, and his outstanding scientific work has defined and shaped many research directions in the last 50 years. The 14 contributions presented in this volume, all of which are connected to László Lovász's areas of research, offer an excellent overview of the state of the art of combinatorics and related topics and will be of interest to experienced specialists as well as young researchers.

Combinatorics. --- Computer science—Mathematics. --- Computer mathematics. --- Mathematical Applications in Computer Science. --- Mathematics --- Math --- Science --- Computer mathematics --- Electronic data processing --- Combinatorics --- Algebra --- Mathematical analysis --- Discrete mathematics. --- Computer science --- Discrete Mathematics. --- Mathematics. --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis

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This book outlines the scientific career of Arto Salomaa, a pioneer in theoretical computer science and mathematics. The author first interviewed the subject and his family and collaborators, and he then researched this fascinating biography of an intellectual who was key in the development of these fields. Early chapters progress chronologically from Academician Salomaa's origins, childhood, and education to his professional successes in science, teaching, and publishing. His most impactful direct research efforts have been in the areas of automata and formal languages. Beyond that he has influenced many more scientists and professionals through collaborations, teaching, and books on topics such as biocomputing and cryptography. The author offers insights into Finnish history, culture, and academia, while historians of computer science will appreciate the vignettes describing some of the people who have shaped the field from the 1950s to today. The author and his subject return throughout to underlying themes such as the importance of family and the value of longstanding collegial relationships, while the work and achievements are leavened with humor and references to interests such as music, sport, and the sauna.

Computer science. --- Information theory. --- History of Computing. --- Theory of Computation. --- History of Mathematical Sciences. --- Discrete Mathematics. --- Mathematics of Computing. --- Communication theory --- Communication --- Cybernetics --- Informatics --- Science --- Computers. --- Mathematics. --- History. --- Discrete mathematics. --- Computer science—Mathematics. --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis --- Annals --- Auxiliary sciences of history --- Math --- Automatic computers --- Automatic data processors --- Computer hardware --- Computing machines (Computers) --- Electronic brains --- Electronic calculating-machines --- Electronic computers --- Hardware, Computer --- Computer systems --- Machine theory --- Calculators --- Cyberspace --- Mathematicians --- Salomaa, Arto. --- Scientists --- Salomaa, A.

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The book presents the proceedings of the 23rd International Conference on Difference Equations and Applications, ICDEA 2017, held at the West University of Timișoara, Romania, under the auspices of the International Society of Difference Equations (ISDE), July 24 - 28, 2017. It includes new and significant contributions in the field of difference equations, discrete dynamical systems and their applications in various sciences. Disseminating recent studies and related results and promoting advances, the book appeals to PhD students, researchers, educators and practitioners in the field.

Difference equations. --- Functional equations. --- Dynamics. --- Ergodic theory. --- Biomathematics. --- Applied mathematics. --- Engineering mathematics. --- Discrete mathematics. --- Difference and Functional Equations. --- Dynamical Systems and Ergodic Theory. --- Mathematical and Computational Biology. --- Applications of Mathematics. --- Discrete Mathematics. --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis --- Engineering --- Engineering analysis --- Mathematical analysis --- Biology --- Biology --- Mathematics --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Equations, Functional --- Functional analysis --- Calculus of differences --- Differences, Calculus of --- Equations, Difference --- Mathematics --- Mathematics