Hopf algebras have important connections to quantum theory, Lie algebras, knot and braid theory, operator algebras, and other areas of physics and mathematics. They have been intensely studied in the last decade; in particular, the solution of a number of conjectures of Kaplansky from the 1970s has led to progress on the classification of semisimple Hopf algebras and on the structure of pointed Hopf algebras. There has been much progress also on actions and coactions of Hopf algebras and on Hopf Galois extensions. Many new methods have been used for these results: modular and braided categories, representation theory, algebraic geometry, and Lie methods such as Cartan matrices.

The contributors to this volume of expository papers were participants in the Hopf Algebras Workshop held at MSRI as part of the 1999—2000 Year on Noncommutative Algebra. Together the papers give a clear picture of the current trends in this active field, with a focus on what is likely to be important in future research.

Among the topics covered are results toward the classification of finite-dimensional Hopf algebras (semisimple and non-semisimple), as well as what is known about the extension theory of Hopf algebras. Some papers consider Hopf versions of classical topics, such as the Brauer group, while others are closer to recent work in quantum groups. The book also explores the connections and applications of Hopf algebras to other fields.

Σημειώσεις για το μάθημα Μαθηματικά II στο Χημικό Τμήμα Πατρών.

Περιέχει τα κεφάλαια:

• Διανυσματική Ανάλυση
• Διανυσματικές Συναρτήσεις
• Πίνακες, Ορίζουσες, Γραμμικά συστήματα
• Σειρές Φουριέ

Οι σημειώσεις αυτές διδάσκονται στο μάθημα Πραγματική Ανάλυση IV του Μαθηματικού Τμήματος Πατρών.

Είναι ασκήσεις που συμπληρώνουν το βιβλίο “Μαθηματική Ανάλυση” του L. Brand.

Les statistiques inférentielles utilisent la théorie des probabilités pour restreindre ce nombre en faisant des sondages sur des échantillons.
D’autre part, la théorie des tests d’hypothèses permet de prendre des décisions dans des situations faisant intervenir une part de hasard.

Non-cooperative game theory is an abstract framework for analyzing strategic situations that involve multi-person interdependent decision making. Conflict, cooperation, coordination, bargaining, auctions, and (tacit) communication are all topics that can be usefully analyzed within this framework.

This graduate course will teach the fundamentals of game theory. It will be a rigorous introduction that does not shy away from technical detail but that emphasizes modeling issues and solution concepts. Game theory emerged as a branch of applied mathematics and is still quite mathematical. Although we shall rarely use more than algebra, the course will be analytically demanding. The hard part of game theory is not the math but the logic, and mastering this takes time and effort. There are no formal prerequisites for this course, but mathematical thinking will be indispensable.

The formal theory of bargaining originated with John Nash’s work in the early 1950s. In this book we discuss two recent developments in this theory.

The first uses the tool of extensive games to construct theories of bargain- ing in which time is modeled explicitly.
The second applies the theory of bargaining to the study of decentralized markets. We do not attempt to survey the field.

Rather, we select a small number of models, each of which illustrates a key point. We take the approach that a thorough analysis of a few models is more rewarding than short discussions of many models. Some of our selections are arbitrary and could be replaced by other models that illustrate similar points.

Course notes for analytic number theory by Graham Everest

This series of lectures introduces some important conjectures in Arithmetic Algebraic Geometry. Our central objects of study will be elliptic curves and abelian varietes, along with modular curves, Shimura curves, modular forms, automorphic forms, L-functions, and l-adic representations.

This is an introductory course in Elementary Number Theory.

The first two thirds will include such topics as rational and irrational numbers, the division algorithm, the Euclidean Algorithm, continued fractions, modulo arithmetic, Fermat’s Little Theorem, public-key encryptions, the Chinese Remainder Theorem, primitive roots, and quadratic reciprocity.

These topics will be introduced and various applications will be given to show how they relate to the subject of Number Theory as a whole.

This book gives a treatment of exterior differential systems. It will include both the general theory and various applications.

An exterior differential system is a system of equations on a manifold defined by equating to zero a number of exterior differential forms. When all the forms are linear, it is called a pfaffian system. Our object is to study its integral manifolds, i.e., submanifolds satisfying all the equations of the system. A fundamental fact is that every equation implies the one obtained by exterior differentiation, so that the complete set of equations associated to an exterior differential system constitutes a differential ideal in the algebra of all smooth forms. Thus the theory is coordinate-free and computations typically have an algebraic character; however, even when coordinates are used in intermediate steps, the use of exterior algebra helps to efficiently guide the computations, and as a consequence the treatment adapts well to geometrical and physical problems.