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This Invitation to Mathematics consists of 14 contributions, many from the world's leading mathematicians, that introduce the readers to exciting aspects of current mathematical research. The contributions are as varied as the personalities of active mathematicians, but together they show mathematics as a rich and lively field of research.   The contributions are written for interested students at the age of transition between high school and university who know high school mathematics and perhaps competition mathematics and who want to find out what current research mathematics is about. We hope that it will also be of interest to teachers or more advanced mathematicians who would like to learn about exciting aspects of mathematics outside of their own work or specialization.   Together with a team of young ``test readers'', editors and authors have taken great care, through a substantial ``active editing'' process, to make the contributions understandable by the intended readership.

Mathematics -- Ability testing. --- Mathematics -- Study and learning. --- Mathematics. --- Mathematics --- Physical Sciences & Mathematics --- Mathematics - General --- Mathematical Theory --- Competitions. --- Research. --- Mathematical research --- Math --- Mathematics, general. --- Science

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The IMO Compendium is the ultimate collection of challenging high school level mathematics problems. It is an invaluable resource, not only for students preparing for competitions, but for anyone who loves and appreciates math. Training for mathematical olympiads is enjoyed by talented students throughout the world. Olympiads have become one of the primary ways to recognize and develop talented youth with a potential to excel in areas that require abstract thinking. Although the problems appearing at IMO do not involve advanced mathematics, they must be difficult and their solutions must arise from creative and clever insights rather than tedious calculations. In preparation for the distinguished International Mathematical Olympiad (IMO) competition, each participating country selects the top six high school students every year through a series of national olympiads. These students are invited to participate in the IMO, usually held in July. The IMO is a two-day contest where each day competitors are given three problems which they work on independently. The IMO host country appoints a special committee to which each country submits up to six problems. From this composite “longlist” of problems, a “shortlist” of 25-30 problems is created. The jury, consisting of one professor from each country, makes the final selection from the shortlist a few days before the IMO begins. The IMO has sparked a burst of creativity among enthusiasts to create new and interesting mathematics problems. It can be safely said that the IMO and shortlisted problems are among the well-crafted problems created in a given year. This book attempts to gather all of these problems with their solutions. In addition, the book contains all the available longlist problems, for a total of more than 2000 problems.   From the reviews of the first edition: "The International Mathematical Olympiad, or IMO is the premier international competition for talented high school mathematics students. … This book collects statements and solutions of all of the problems ever set in the IMO, together with many problems proposed for the contest. … serves as a vast repository of problems at the Olympiad level, useful both to students … and to faculty looking for hard elementary problems. No library will want to be without a copy, nor will anyone involved in mathematics competitions …" — (Fernando Q. Gouvêa, MathDL, March, 2006).

International Mathematical Olympiad. --- Mathematics -- Competitions. --- Mathematics -- Problems, exercises, etc. --- Mathematics --- Physical Sciences & Mathematics --- Mathematics - General --- Mathematical Theory --- Competitions --- Competitions. --- IMO (Conference) --- International Mathematics Olympiad --- Mathematics. --- Mathematics, general. --- Math --- Science

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"Visions in Mathematics - Towards 2000" was one of the most remarkable mathematical meetings in recent years. It was held in Tel Aviv from August 25th to September 3rd, 1999, and united some of the leading mathematicians worldwide. The goals of the conference were to discuss the importance, the methods, the past and the future of mathematics as we enter the 21st century and to consider the connection between mathematics and related areas. The aims of the conference are reflected in the present set of survey articles, documenting the state of art and future prospects in many branches of mathematics of current interest. This is the first part of a two-volume set that will serve any research mathematician or advanced student as an overview and guideline through the multifaceted body of mathematical research in the present and near future.

Optoelectronics. --- Organic electronics. --- Mathematics. --- Mathematics, general. --- Mathematics --- Math --- Science --- Organic solid state chemistry --- Solid state electronics --- Electronics --- Photonics

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In July 2009 Germany hosted the 50th International Mathematical Olympiad (IMO). For the very first time the number of participating countries exceeded 100, with 104 countries from all continents. Celebrating the 50th anniversary of the IMO provides an ideal opportunity to look back over the past five decades and to review its development to become a worldwide event. This book is a report about the 50th IMO as well as the IMO history. A lot of data about all the 50 IMOs are included. We list the most successful contestants, the results of the 50 Olympiads and the 112 countries that have ever taken part. It is impressive to see that many of the world’s leading research mathematicians were among the most successful IMO participants in their youth. Six of them gave presentations at a special celebration: Bollobás, Gowers, Lovász, Smirnov, Tao and Yoccoz. This book is aimed at students in the IMO age group and all those who have interest in this worldwide leading competition for high school students.

International Mathematical Olympiad. --- Mathematics -- Competitions. --- Number theory. --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Competitions. --- IMO (Conference) --- International Mathematics Olympiad --- Mathematics. --- Mathematics, general. --- Math --- Science

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This book presents new accurate and efficient exponentially convergent methods for abstract differential equations with unbounded operator coefficients in Banach space. These methods are highly relevant for practical scientific computing since the equations under consideration can be seen as the meta-models of systems of ordinary differential equations (ODE) as well as of partial differential equations (PDEs) describing various applied problems. The framework of functional analysis allows one to obtain very general but at the same time transparent algorithms and mathematical results which then can be applied to mathematical models of the real world. The problem class includes initial value problems (IVP) for first order differential equations with constant and variable unbounded operator coefficients in a Banach space (the heat equation is a simple example), boundary value problems for the second order elliptic differential equation with an operator coefficient (e.g. the Laplace equation), IVPs for the second order strongly damped differential equation as well as exponentially convergent methods to IVPs for the first order nonlinear differential equation with unbounded operator coefficients.  For researchers and students of numerical functional analysis, engineering and other sciences this book provides highly efficient algorithms for the numerical solution of differential equations and applied problems.

Algebra. --- Boundary value problems -- Textbooks. --- Differential equations, Partial. --- Differential equations. --- Differential equations --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Calculus --- Algorithms. --- 517.91 Differential equations --- Algorism --- Mathematics. --- Mathematics, general. --- Math --- Science --- Algebra --- Arithmetic --- Foundations