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[Hilbert's] style has not the terseness of many of our modem authors in mathematics, which is based on the assumption that printer's labor and paper are costly but the reader's effort and time are not. H. Weyl [143] The purpose of this book is to describe the classical problems in additive number theory and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial tools used to attack these problems. This book is intended for students who want to lel?Ill additive number theory, not for experts who already know it. For this reason, proofs include many "e;unnecessary"e; and "e;obvious"e; steps; this is by design. The archetypical theorem in additive number theory is due to Lagrange: Every nonnegative integer is the sum of four squares. In general, the set A of nonnegative integers is called an additive basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of A. Lagrange 's theorem is the statement that the squares are a basis of order four. The set A is called a basis offinite order if A is a basis of order h for some positive integer h. Additive number theory is in large part the study of bases of finite order. The classical bases are the squares, cubes, and higher powers; the polygonal numbers; and the prime numbers. The classical questions associated with these bases are Waring's problem and the Goldbach conjecture.

We will occasionally footnote a portion of text with a "e;**,, to indicate Notes on the that this portion can be initially bypassed. The reasons for bypassing a Text portion of the text include: the subject is a special topic that will not be referenced later, the material can be skipped on first reading, or the level of mathematics is higher than the rest of the text. In cases where a topic is self-contained, we opt to collect the material into an appendix that can be read by students at their leisure. The material in the text cannot be fully assimilated until one makes it Notes on "e;their own"e; by applying the material to specific problems. Self-discovery Problems is the best teacher and although they are no substitute for an inquiring mind, problems that explore the subject from different viewpoints can often help the student to think about the material in a uniquely per- sonal way. With this in mind, we have made problems an integral part of this work and have attempted to make them interesting as well as informative.

This text started out as a revised version of Buildings by the second-named author [53], but it has grown into a much more voluminous book. The earlier bookwasintendedtogiveashort,friendly,elementaryintroductiontothet- ory,accessibletoreaderswithaminimalbackground.Moreover,itapproached buildings from only one point of view, sometimes called the "e;old-fashioned"e; approach: A building is a simplicial complex with certain properties. The current book includes all the material of the earlier one, but we have added a lot. In particular, we have included the "e;modern"e; (or "e;W-metric"e;) approach to buildings, which looks quite di?erent from the old-fashioned - proach but is equivalent to it. This has become increasingly important in the theory and applications of buildings. We have also added a thorough tre- ment of the Moufang property, which occupies two chapters. And we have added many new exercises and illustrations. Some of the exercises have hints or solutions in the back of the book. A more extensive set of solutions is ava- able in a separate solutions manual, which may be obtained from Springer's Mathematics Editorial Department. We have tried to add the new material in such a way that readers who are content with the old-fashioned approach can still get an elementary treatment of it by reading selected chapters or sections. In particular, many readers will want to omit the optional sections (marked with a star). The introduction below provides more detailed guidance to the reader.

Algorithms in algebraic geometry go hand in hand with software packages that implement them. Together they have established the modern field of computational algebraic geometry which has come to play a major role in both theoretical advances and applications. Over the past fifteen years, several excellent general purpose packages for computations in algebraic geometry have been developed, such as, CoCoA, Singular and Macaulay 2. While these packages evolve continuously, incorporating new mathematical advances, they both motivate and demand the creation of new mathematics and smarter algorithms.This volume reflects the workshop "e;Software for Algebraic Geometry"e; held in the week from 23 to 27 October 2006, as the second workshop in the thematic year on Applications of Algebraic Geometry at the IMA. The papers in this volume describe the software packages Bertini, PHClab, Gfan, DEMiCs, SYNAPS, TrIm, Gambit, ApaTools, and the application of Risa/Asir to a conjecture on multiple zeta values. They offer the reader a broad view of current trends in computational algebraic geometry through software development and applications.

The rapid and impressive results obtained through the application of numerical methods of analysis to electrodynamics created the rather false impression (es- cially to inexperienced engineers) that all problems were solved long ago or that some of them were not solved because one lacked suf?cient time. However, it is suf?cient to formulate these problems clearly, to give the task to a programmer to create the corresponding program, and to let a computer "e;think. "e; It is only left for an engineer to make the corresponding plots and to explain calculation results if the latter do not agree with existing physical notions. However, de?nite conf- mity to natural laws starts inevitably to appear while realizing this sequence of steps: the problem is prepared for calculations more thoroughly-the probability to obtain the correct answer is higher, and the obtained information is richer and more interesting-the calculated algorithm is simpler; and as expenditure of calculation time is less, correspondingly, the problems to solve effectively are more complex. Theproblemformulationisnotonlycompilationofinitialequationsandanumerical algorithm. It is necessary to foresee a qualitative character of the obtained results and expected order of values, to reveal the peculiarities of the searched solution, complicating its numerical realization and, as a result, choosing the known solution algorithm or working out a new one. If this work has not been done properly, then unexpected anomalies may appear during the numerical calculation process, and the results can be the basis for the most sensational "e;discoveries.

The Paris Meridian is the name of the line running north-south through the astronomical observatory in Paris. One of the original intentions behind the founding of the Paris Observatory was to determine and measure this line. The French government financed the Paris Academy of Sciences to do so in the seventeenth to nineteenth centuries. It employed both astronomers - people who study and measure the stars - and geodesists - people who study and measure the Earth. This book is about what they did and why.This is the first English language presentation of this historical material. It is attractively written and it features the story of the community of scientists who created the Paris Meridian. They knew each other well - some were members of the same families, in one case of four generations. Like scientists everywhere they collaborated and formed alliances; they also split into warring factions and squabbled. They travelled to foreign countries, somehow transcending the national and political disputes, as scientists do now, their eyes fixed on ideas of accuracy, truth and objective, all enduring values - yet when the reception given to their own work was concerned some became blind to high ideals and descended into petty politics.To establish the Paris Meridian, the scientists endured hardship, survived danger, and gloried in amazing adventures during a time of turmoil in Europe consisting of the French Revolution and the Napoleonic War between France and Spain. Some were accused of witchcraft. Some of their associates lost their heads on the guillotine. Some died of disease. Some won honor and fame. One became the Head of State in France. Some found dangerous love in foreign countries. One scientist was killed in self defence when attacked by a jealous lover, another was himself killed by a jealous lover, a third brought back a woman to France and then jilted her, whereupon she joined a convent... The scientists worked on practical problems of interest to the government and to the people. They also worked on one of the most important intellectual problems of the time, a problem of great interest to their fellow scientists all over the world- the theory of universal gravitation. They succeeded in their intellectual work while affecting politics and the affairs of state; their endeavours have left marks on the landscape, in art, and in literature still visible today.

This book, together with Linear Algebra, constitutes a curriculum for an algebra program addressed to undergraduates. The separation of the linear algebra from the other basic algebraic structures fits all existing tendencies affecting undergraduate teaching, and I agree with these tendencies. I have made the present book self contained logically, but it is probably better if students take the linear algebra course before being introduced to the more abstract notions of groups, rings, and fields, and the systematic development of their basic abstract properties. There is of course a little overlap with the book Lin- ear Algebra, since I wanted to make the present book self contained. I define vector spaces, matrices, and linear maps and prove their basic properties. The present book could be used for a one-term course, or a year's course, possibly combining it with Linear Algebra. I think it is important to do the field theory and the Galois theory, more important, say, than to do much more group theory than we have done here. There is a chapter on finite fields, which exhibit both features from general field theory, and special features due to characteristic p. Such fields have become important in coding theory.

The primary goal of this book is to present the theoretical foundation of the field of Euclidean Harmonic analysis. This book is Modern in that is contains more recent topics such as function spaces, atomic decompositions, singular integrals of nonconvolution type, and weighted inequalities. This book is mainly addressed to graduate students in mathematics. The prerequisites are satisfactory completion of courses in real and complex variables, and knowledge of classical Fourier analysis topics. This book is intended to present the selected topics in depth and stimulate further study. This third edition includes a new chapter entitled "e;Multilinear Harmonic Analysis"e;, which includes sections on multilinear operators, multilinear interpolation, multilinear multiplier operators, Calderon-Zygmund operators of several functions, and multiple weights and weighted norm inequalities. The new chapter will tie nicely with the material in chapters 8, 9, and 10, and the author may add a new section in this chapter applying the techniques of chapter 11 in the context of multilinear harmonic analysis. In addition to a new chapter, the third edition contains 1000 different corrections and improvements in the existing text, more examples and applications, new and more relevant hints for the existing exercises, about 20-30 new exercises in the existing chapters, and improved references.

The primary goal of this book is to present the theoretical foundation of the field of Euclidean Harmonic analysis. This book contains the classical topics such as interpolation, Fourier series, the Fourier transform, maximal functions, singular integrals, and Littlewood-Paley theory. This book is mainly addressed to graduate students in mathematics. The prerequisites are satisfactory completion of courses in real and complex variables. This book is intended to present the selected topics in depth and stimulate further study. This third edition includes a new chapter entitled "e;Topics on Fourier series,"e; which includes sections on Gibbs phenomenon, summability methods and Jackson's theorem, Tauberian theorems, spherical Fourier inversion, and Fourier transforms on the line. The new chapter ties really well with the material in the existing chapter 3 "e;Fourier Analysis on the Torus"e; and will prepare the students for (the existing) chapter 4. In addition to a new chapter, the third edition contains 1000 different corrections and improvements in the existing text, more examples and applications, new and more relevant hints for the existing exercises, about 20-30 new exercises in the existing chapters, and improved references.

This is the second edition of this advanced textbook written for scientists who require further training in femtosecond science. Four years after pub- cation of the ?rst edition, femtosecond science has overcome new challenges and new application ?elds have become mature. It is necessary to take into account these new developments. Two main topics merged during this period that support important scienti?c activities: attosecond pulses are now gen- ated in the X-UV spectral domain, and coherent control of chemical events is now possible by tailoring the shape of femtosecond pulses. To update this advanced textbook, it was necessary to introduce these ?elds; two new ch- ters are in this second edition: "e;Coherent Control in Atoms, Molecules, and Solids"e;(Chap.11)and"e;AttosecondPulses"e;(Chap.12)withwell-documented references. Some changes, addenda, and new references are introduced in the ?rst edition's ten original chapters to take into account new developments and updatethisadvancedtextbookwhichistheresultofascienti?cadventurethat started in 1991. At that time, the French Ministry of Education decided that, in view of the growing importance of ultrashort laser pulses for the national scienti?c community, a Femtosecond Centre should be created in France and devoted to the further education of scientists who use femtosecond pulses as a research tool and who are not specialists in lasers or even in optics.

Partial differential equations are fundamental to the modeling of natural phenomena, arising in every field of science. Consequently, the desire to understand the solutions of these equations has always had a prominent place in the efforts of mathematicians; it has inspired such diverse fields as complex function theory, functional analysis and algebraic topology. Like algebra, topology, and rational mechanics, partial differential equations are a core area of mathematics. This book aims to provide the background necessary to initiate work on a Ph.D. thesis in PDEs for beginning graduate students. Prerequisites include a truly advanced calculus course and basic complex variables. Lebesgue integration is needed only in Chapter 10, and the necessary tools from functional analysis are developed within the course. The book can be used to teach a variety of different courses. This new edition features new problems throughout and the problems have been rearranged in each section from simplest to most difficult. New examples have also been added. The material on Sobolev spaces has been rearranged and expanded. A new section on nonlinear variational problems with "Young-measure" solutions appears. The reference section has also been expanded.

The subject of this book is Complex Analysis in Several Variables. This text begins at an elementary level with standard local results, followed by a thorough discussion of the various fundamental concepts of "e;complex convexity"e; related to the remarkable extension properties of holomorphic functions in more than one variable. It then continues with a comprehensive introduction to integral representations, and concludes with complete proofs of substantial global results on domains of holomorphy and on strictly pseudoconvex domains inC"e;, including, for example, C. Fefferman's famous Mapping Theorem. The most important new feature of this book is the systematic inclusion of many of the developments of the last 20 years which centered around integral representations and estimates for the Cauchy-Riemann equations. In particu- lar, integral representations are the principal tool used to develop the global theory, in contrast to many earlier books on the subject which involved methods from commutative algebra and sheaf theory, and/or partial differ- ential equations. I believe that this approach offers several advantages: (1) it uses the several variable version of tools familiar to the analyst in one complex variable, and therefore helps to bridge the often perceived gap between com- plex analysis in one and in several variables; (2) it leads quite directly to deep global results without introducing a lot of new machinery; and (3) concrete integral representations lend themselves to estimations, therefore opening the door to applications not accessible by the earlier methods.

. . . if one wants to make progress in mathematics one should study the masters not the pupils. N. H. Abel Heeke was certainly one of the masters, and in fact, the study of Heeke L- series and Heeke operators has permanently embedded his name in the fabric of number theory. It is a rare occurrence when a master writes a basic book, and Heeke's Lectures on the Theory of Algebraic Numbers has become a classic. To quote another master, Andre Weil: "e;To improve upon Heeke, in a treatment along classical lines of the theory of algebraic numbers, would be a futile and impossible task. "e; We have tried to remain as close as possible to the original text in pre- serving Heeke's rich, informal style of exposition. In a very few instances we have substituted modern terminology for Heeke's, e. g. , "e;torsion free group"e; for "e;pure group. "e; One problem for a student is the lack of exercises in the book. However, given the large number of texts available in algebraic number theory, this is not a serious drawback. In particular we recommend Number Fields by D. A. Marcus (Springer-Verlag) as a particularly rich source. We would like to thank James M. Vaughn Jr. and the Vaughn Foundation Fund for their encouragement and generous support of Jay R. Goldman without which this translation would never have appeared. Minneapolis George U. Brauer July 1981 Jay R.