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Adam 've trade of an other vice point. The Nyugat download information technology for the practicing physician health informatics in turn-of-thecentury Budapest boosted out of this interesting immigration concerning side, access, the events, intuition, border and trend. Introduction to combinatorial algorithms and Boolean functions. Generating all tuples and permutations.

Generating all combinations and partitions. History of combinatorial generation. Upper Saddle River, NJ: Fascicle 0, xii, p. Volume 2 on Seminumerical Algorithms and Volume 3 on Searching and Sorting followed in and , respectively. On the other hand, most of the algorithms that we have to apply to real applications, e. This is the subject of our review, a task similar to writing a review for the Bible.

After introducing a rich set of exemplary combinatorial structures, Knuth focuses on the combinatorial structure generation problem in Section 7. To let readers better understand his intention, he makes a distinction among several related tasks at the very beginning of the subject [see p. Cambridge Studies in Advanced Mathematics. Cambridge University Press ; Zbl On the other hand, although this generating problem has been discussed in some of the applied combinatoric studies, e. Wiley ], it gets an encyclopedic treatment in this set of fascicles, where Knuth studies the generation of all the tuples, permutations, combinations, integer partitions, set partitions and trees, with the focus being placed on methods, i.

We do need such a generating mechanism often. For example, when looking for the number of all the shortest paths between two nodes in an arrangement graph [K. It is clear that Step 1 is indirectly involved with an integer partition generation problem [Algorithm H, p.

In both cases, it is not enough to merely enumerate such partitions, but there is no need to explicitly list all such partitions, either. This subsection ends with a historic review of combinatorial generation with an international scope. Although we have to wait for many other interesting subjects to come out later, e.

By the way, if you have a good deal of time to spare, you can certainly try some of the exercises, of them for just Section 7. In fact, Knuth has provided an answer, either a hint, a reference, or a complete solution, to almost all of the exercises as contained in these fascicles, with Exercise 92 of Section 7. As a result, e. This is a remarkable book. The analytic and the algebraic theory of abelian varieties are brought together, which up to now did not happen before in one volume. The last 30 years it seems to be a sophisticated custom in algebraic geometry to hide geometric ideas and analytic motivations by wrapping them up into certainly useful, but often rather dry and complicated algebraic or functorial terminology.

This sounds rather contradictory, but the result is a book which carries the reader immediately to the heart of the matter. An abelian variety is a complete irreducible group variety; the group structure is automatically commutative under these conditions; these varieties turned up naturally in the study of abelian integrals and their periods, which seems to explain the terminology.

An abelian variety over the complex numbers is a complex torus, i. Abelian varieties not only turn up in analytic theories but also in algebraic number theory and in various aspects of algebraic geometry. A connected group variety is an extension of an abelian variety by a linear group; thus the study of group varieties naturally splits into two. In contrast with the theory of linear algebraic groups, for abelian varieties the group theory is easy, but the geometry is complicated. Anyone who has struggled through the book of F. Weil in the algebraic case, obvious in the analytic case, is proved here in a short way.

One of the new ideas exposed in the book is a direct construction of the dual abelian variety, i. The third chapter, algebraic theory via schemes, centers around this technical, and certainly important point. The proof of the strengthened version of the theorem of the cube pp. The methods of descent are applied to give a short proof of the duality theorem: The methods of sheaf cohomology are used to prove the vanishing theorem and the Riemann Roch theorem. The last chapter deals with the endomorphism ring: The last two sections cover the theory of theta groups and the Riemann form of a line bundle corollary: Thus the circle closes: The author is somewhat brief an the point of references; for two reasons it seems desirable to have more: We admire the style of exposition.

In this way the book favourably contrasts with some modern volumes in which the real issues get lost in too many details and expanded notations. Experience has shown that a student with basic knowledge in algebraic geometry and some guidance is able to enjoy the treasures hidden in these pages. There are several new results. Ramanujam when writing up the notes after the course given at the Tata Institute of Fundamental Research.

We strongly recommed this book, if not to every mathematician, certainly to analysts, geometers and algebraists. Frans Oort Zbl With appendices by C. Ramanujam and Yuri Manin. Steele Prize for Mathematical Exposition is one of the highest distinctions in Mathematics. Mumford in recognition of his pioneering and beautiful accounts of a host of aspects of algebraic geometry, published over a period of more than two decades. The prize citation emphasizes that, particularly in D.

Springer ; Zbl The book under review is the reprint of one of these distinguished expository writings of D. The second edition, with two appendices added by C. Ramanujam and by Yu. Manin, was brought out in Zbl In the current new reprint, the welltried text has been left entirely intact. However, some appropriate corrections have been supplied by B. Conrad and by Ching-Li Chai, and the book appears now in modern LaTeX typesetting, with the original page numbers indicated in the margins of the present edition. Cartier, thereby clarifying the picture of the situation also in positive characteristics.

Mumford himself pointed out in the preface, this book covers roughly half of the material that would be appropriate for a reasonably complete treatment of the theory of abelian varieties. Therefore, apart from the various recent excellent textbooks on complex tori and complex abelian varieties Birkenhake-Lange, Kempf, Debarre, Polishchuk, and others and from J. As the current new edition of this book is an unaltered reprint of the second edition from , we may refer to the review of the latter D. However, after so many decades, it might be worthwile to recall that the text comprises four chapters and two appendices discussing the following topics: Analytic Theory complex tori, line bundles, and algebraizabilty of tori ; Chapter II: All in all, it is more than rewarding that D.

In its modern typesetting and in its corrected version, this jewel in the mathematical literature has become even more attractive and valuable. Werner Kleinert Berlin Zbl Invariant theory, old and new. This book is reprinted from an article of the authors [Adv. Because it has now appeared in bock form making it more readily available, because of the renewed interest in the subject and because it is a well written, exciting introduction to invariant theary, it is worth another review. The preface gives a brief history of invariant theory which serves also as an outline of the book itself.

Chapter 1 introduces the terminology and ends with the important restitution process and the reduction from the multilinear to the linear case. Often the space E is a vector space and the action of G is required to be linear. The question then is whether the space of orbits is again a k-scheme. In Chapter 2, entitled: A complete proof is given of the Hilbert-Nagata theorem: Suppose GL n, k actc in the usual manner on the ring of polynomials k[X1 ,.

Let G be a subgroup of GL n, k. Is the ring k[X1 ,. Nagata [Notes by M. Tata Institute of Fundamental Research. The chapter ends with several illustrative examples. This material is used only in Chapter 2. This book is an excellent introduction to the subject giving not only the important contributions of the last century, but also the most important contributions of the last decades, as well as an introduction to the recent developments due to Mumford.

Recent successes of invariant theory have been made by Hochster, Eagon and Roberts. Hochster has shown that the ring of invariants of a torus is Cohen-Macaulay [Ann. Fossum Champaign Zbl Hp spaces of several variables. This paper is a major contribution to the study of Hp spaces, singular integrals, and harmonic analysis an Rn.

Classically the theory of Hp spaces arose from analytic function theory. The authors present several intrinsic descriptions of Hp , of a real variable nature, not involving conjugate functions. These results greatly clarify the meaning of Hp , as well as throwing new light on the behaviour of convolution operators an Lp. The following is a summary of some of the main results. If T is a convolution operator i. The main result is as follows: Finally the authors consider boundary values.

Denote the set of such f also by Hp. The last result above characterises Hp in terms of Poisson integrals as u is the Poisson integral of f. Then for 0 Zbl If the isomorphism classes of objects of M form a set, the geometric realization of the nerve of Q M is called the classifying space, BQ M ; it is determined up to homotopy equivalence. For a ring resp. Among the important results are: For a scheme resp. In this monograph the author presents a concise and beautiful account of the Large Sieve and some of its important applications to analytic number theory, in particular, the density theorems, the distribution of primes in arithmetical progressions and the connection with the small sieve of BrunSelberg.

The method of the Large Sieve, which was invented by Yu. Gallagher and others, is, as is widely recognized, one of the most powerful tools in multiplicative number theory.

## 80 Years of Zentralblatt MATH by Olaf Teschke, Bernd Wegner & Dirk Werner on Apple Books

The booklet consists of Introduction, Notes in twelve paragraphs, Bibliography and Summary in English. The twelve paragraphs are: Le crible de Linnik et Renyi. La forme analytique additive du grand crible. La forme multiplicative du grand crible. La forme analytique multiplicative du grand crible. Le theoreme de Linnik. Le crible de Selberg II. Application du crible de Selberg. The results demonstrated are not always the strongest ones known at present, but the exposition is an the whole very clear and readable.

Period three implies chaos. Let F be a continuous function of an interval J into itself. Lanee, Tensor products of operator algebras, to appear in Advances Math. He also proved further equivalence of these properties to those of the property P by J. The equivalences of those properties are further shown to be valid for any factor in a separable Hilbert spare.

The paper also contains characterizations of an automorphism which lies in the closure of the inner automorphism J. Tomiyama group, Int N, for a factor of type II1. Numerical analysis of spectral methods: A 25, The proof depends on the theorem of K. Carsten Thomassen Lyngby Zbl Grundlehren der mathematischen Wissenschaften. Berlin, Heidelberg, New York: This approach has been since widely used to construct and handle various types of Markov processes. Other aspects of the same approach, as well as a complementary bibliography, can be found in [J.

Lecture Notes in Mathematics. The following questions are studied by the authors: Then, the martingale problem is studied, starting with existence. The Feller property of the process is proved under similar conditions. Also, the Cameron-Martin-Girsanov formula is established.

## 80 Years of Zentralblatt MATH

Basically, the results are the same provided the process does not explode. Conditions for explosion and non explosion are given. The last-but-one chapter is concerned with limit theorems: Here, the martingale formulation proves to be very powerful since martingale methods are very useful in proving limit theorems. Some of these are part of the preliminary material in probability, which forms Chapter 1. An appendix at the end of the book establishes some estimates from the theory of singular integrals.

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A good part of the theory presented in the book does need these results from analysis. The text itself is complemented with a great number of exercises, for which the main ideas of proof are often indicated. The book is very carefully written. Many introductions and comments explain the ideas, which might unless be hidden behind the often very intricate technicalities. The mathematical rigour is kept at a very high level. It should prove to be very useful to both theoretical and applied probabilists.

Though all the results needed from the theory of Markov processes, martingales, weak convergence of probability measures are proved in Chapter 1, the reader should have a good background in probability theory as well as in analysis, with some elementary functional analysis. Etienne Pardoux Marseille Zbl The book is organized as follows.

Chapter 1 provides an introduction to those parts of measure and probability theories which are considered most important for an understanding of the book. Basic tools are developed which are necessary for the construction of measures on functional spaces and introduces some notions and results which will play an important role in what follows. The relationship between the martingale problem and the strong Markov property, as well as the formula of Cameron, Martin and Girsanov are shown.

Chapter 7 contains a proof of the general theorem about uniqueness for the solution of martingale problem. Standard conditions that can be used to test for explosion are given. In Chapter 11 the stability results for the Markov processes are studied. These results can be naturally divided into two categories: Chapter 12 takes up the question of what can be done in those circumstances when existence of solutions to a martingale problem can be proved but uniqueness cannot.

It is also shown that every solution to a given martingale problem can, in some sense, be built out of those solutions which are part of a Markov family. In the Appendix some results from outside probability theory and concerning to the theory of singular integrals that are relied in Chapters 7 and 9 are proved in order to make the book self-contained.

Pavel Gapeev Berlin Zbl Seminaire de Mathematiques Superieures, Embedding theorems for various classes of spaces are also given. A space X is a Lefschetz space for a given class of maps if to each map f: In Chapter 7, Lefschetz spaces for various classes of maps mentioned above are studied and general theorems of Lefschetz type are proved. Chapter 9 contains results due to Bowszyc, Hajek and Halpern on the existence of periodic points of a map f: A version of the HopfLefschetz theorem for maps of compact attraction is also given.

Representation and Alexander duality theorems for this theory are stated and applications to questions of non-linear analysis are indicated. This part of the monograph is closed with historical comments and a bibliography list divided into several topics according to the organization of the book. An appendix, written by K. It is shown that GLc E has exactly two components. Next, a proof of a theorem due to Itze on the multiplicity of the characteristic values of a compact operator is given.

The set U on p. There is also a confusing misprint an p. Jan Jaworowski Zbl Annals of Mathematics Studies, This work investigates the a. There is also a converse to this result. This material is developed in Chapter 3. To conclude, let me quote the authors: Translated from the Chinese by Peter Shiu. Wright [An introduction to the theory of numbers. Oxford at the Clarendon Press ; Zbl Three dimensional manifolds, Kleinian groups and hyperbolic geometry.

In the last few years the face of three dimensional topology has changed completely, due mainly to deep and fascinating discoveries of the author who had revealed for topologists an entirely new world of beautiful geometrical phenomena. Now many deep connections of three dimensional topology with hyperbolic geometry, theory of Kleinian groups and theory of Riemannian surfaces come to light.

The interior of every compact 3-manifold has a canonical decomposition into pieces which have geometric structures. The author proved this conjecture in many important cases, the most notable one is the case of Haken manifolds for example, exteriors of all knots in S3 are Haken manifolds.

There are eight types of geometric structures in question. The most interesting, important and complicated structure is the hyperbolic one a hyperbolic manifold is a Riemannian manifold of constant negative curvature. The paper is intended for a wide audience but extremely interesting for specialists in three dimensional topology or Kleinian groups, too. The paper is beautifully illustrated with excellent pictures by G. Firstly, the Geometrization Conjecture is precisely formulated and discussed. After this many cases in which the Conjecture is proved are discussed.

A new exciting result is formulated in a footnote added in proof: The next section is devoted to applications. Ivanov Zbl Der Autor hat in dieser Arbeit seinen vielbeachteten und gefeierten Beweis der Mordellschen Vermutung niedergelegt: Diese Aussagen wurden von J. Zarkhin [insbesondere in Math. Mit denselben Methoden Zarkhin, loc. Silverman, Arithmetic Geometry, Proc. Storrs Springer Verlag ; Zbl Die weiteren Schaltstellen im Beweisgang sind wie folgt: Das erledigt Satz 5.

Norbert Schappacher Zbl Maximal subgroups and ordinary characters for simple groups. With computer assistance from J. The following informations about a simple group G are provided: The alternating groups occur up to degree Thus with the exception of small rank Chevalley groups all simple groups of order less than are listed. In the second part of the introduction the authors introduce some terminology of their own to give the desired economic presentation.

The critical point in understanding the tables in the main part of the ATLAS lies in the readability of this terminology. In particular the notation for character tables of groups Y, where Y has both nontrivial Schur multiplier and outer automorphism group, becomes quite intricate. To understand these tables fully will not be always immediate! Secondly methods of constructing G or a covering group are given.

This is done in a very brief way. Then a presentation with generators and relations is given this presentation is one which is easy to compute with but normally not one with a minimum number of generators or relations. The maximal subgroups normally all are given not only in a structural description but also in the role they play in various constructions. These tables contain even subtle details: Harmonic analysis on the Heisenberg nilpotent Lie group, with applications to signal theory.

Pitman Research Notes in Mathematics Series, Furthermore the author establishes a connection to signal theory: Among the results, obtained in this way, is the solution of the synthesis problem. In chapter II various inducing procedures for representations are given, in particular the Mackey machinery is introduced. Chapter IV is devoted to nilpotent topological groups.

Here every irreducible representation can be constructed by inducing 1-dimensional representations of suitable subgroups. In chapter V various presentations of this group are given and its unitary dual is explicitly determined. The irreducible unitary representations occur as actions on L2 R. In the last chapter the author exploits the connections between representation theory and signal theory. The functions on the diagonal, i. The main result of the present paper is another proof of the LBT and a proof of this conjecture about the case of equality.

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The author remarks that the basic relation between the LBT and rigidity has been observed independently by M. Related results about manifolds with boundary, pseudomanifolds and polyhedral manifolds are also included. At the end various conjectures are given. This is where the review goes. Introduction to coding theory and algebraic geometry. Coding theory and algebraic geometry, Lect. The principal achievement so far of this line of research is the construction of families of linear block codes that meet or even go beyond the classical Gilbert-Varshamov bound. In the Deutsche MathematikerVereinigung organized a seminar on this topic which featured two series of lectures, one on the coding theory background and the other on the algebraic geometry background.

It contains a masterful exposition of the theory of linear block codes which requires only minimal algebraic prerequisites. Besides basic concepts, the article covers BCH codes, classical Goppa codes, bounds on codes such as the Gilbert-Varshamov bound and the Plotkin bound, self-dual codes, and some examples of codes obtained from algebraic curves. The elegant, succinct style allows the presentation of a lot of material in a relatively short space.

This paper reviews the rich evolution of such models in physics with special regard to their interaction with mathematics. This paper reviews many interesting results relating the geometry of a space e. Polynomial invariants for smooth four-manifolds. These restrictions are not detected by classical invariants. The Yang-Mills equations depend on the Riemannian geometry of the underlying 4-manifold; however, certain homological properties of the moduli space of solutions are invariant under continuous change of metric. These invariants were shown to distinguish distinct smooth structures on homeomorphic 4-manifolds.

Van de Ven [Invent. The paper under review extends these ideas. The most striking results are: Let S be a simply-connected, smooth, complex projective surface. This follows immediately from two other theorems: Then all the Donaldson invariants for X vanish. Let S be a simply-connected complex projective surface and H a hyperplane class in H2 S.

Then for a large enough k, qk,S H,.

## 80 Years of Zentralblatt Math: 80 Footprints of Distinguished Mathematicians in Zentralblatt

These results are impressive, as are their proofs. Kronheimer and the author [The geometry of four-manifolds, Oxford: Clarendon Press ; Zbl Besicovitch type maximal operators and applications to Fourier analysis. Wavelet theory found many applications during the past decade. This book is one of the few recent books on wavelets. The authoress has made important contributions to the theory of wavelets. The book consists of an introduction, 10 chapters, a bibliography, a subject index and an author index.

In Chapter 1 the authoress explains why the wavelets are of interest. In Chapter 2 the notion and properties of the continuous wavelet transform are studied. In Chapter 3 the discrete wavelet transforms are studied. In Chapter 4 the role of time-frequency density in wavelet transforms is discussed. Orthonormal wavelet bases are constructed. These bases have good localization properties in both time and frequency domains. In Chapter 6 orthonormal bases of compactly supported wavelets are studied. Examples of compactly supported wavelets, generating an orthonormal basis, are given. In Chapter 7 the regularity of the compactly supported wavelets is discussed.

In Chapter 8 symmetry and lack of it for compactly supported wavelet bases is discussed. Except the Haar basis, no symmetric or antisymmetric compactly supported real valued wavelet bases exist. Symmetric biorthogonal wavelet bases do exist. ICM, Warsawa , Vol. I, ; Zbl Thurston, Word processing in groups ; Zbl The author has again opened up new vistas that geometers and geometric group theorists may explore in the years to come.

The major section titles are: This list of headings hardly indicates the breadth of discussion. Apanasov Norman Zbl Motivated by his joint work with N. These new invariants have since revolutionized 4-manifold topology as they have led to simpler approaches to results previously proved using Donaldson invariants as well as providing a tool to prove many new results. One of the major breakthroughs in Donaldson theory due to P. Independent of this conjecture, these new in- variants have provided an extremely powerful tool for studying 4-manifolds that has proved easier to use than the Donaldson invariant.

Let X be an oriented, closed 4-manifold with a Riemannian metric g. As usual, the proper analytic formulation requires the use of appropriate Sobolev spaces. In particular, all Seiberg-Witten invariants must vanish if X has a metric of positive scalar curcature.

More general wall crossing formulas have since been given by T. Liu [General wall crossing formula, Math. There has been a lot of important work that has been based on the Seiberg-Witten invariants introduced in this paper, and we want to inform the reader of a few of the more important papers that have appeared.

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They had earlier proved forms of the generalized Thom conjecture and given a generalized adjunction inequality for other manifolds using Donaldson theory, but these proofs did not work for CP2. Their earlier results can be reproven using Seiberg-Witten theory. Taubes has published a number of papers [Math. From the Seiberg-Witten equations to pseudo-holomorphic curves, J. There are also now available expository treatments of the Seiberg-Witten invariants, including a paper by S.

Morgan [The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Princeton Math. Notes 44 ; Zbl Lawson New Orleans Zbl The main result is the proof of the Taniyama-Weil conjecture for a large class of elliptic curves over Q. These include semistable curves, and thus the result implies the famous Fermat conjecture. To achieve this one shows that in many cases the Hecke algebra of a modular curve is the base of a universal deformation of the associated p-adic Galois representation.

In the meantime there has been more progress, extending the result to elliptic curves with semistable reduction at 3 and 5. The restriction stems from the argument above, and limitations of our present crystalline techniques. The contents in more detail. Chapter I introduces the universal deformation ring, various local conditions on representations, and the corresponding tangent spaces.

Chapter II treats Hecke algebras. This is easy for primes of good reduction, but involved for the others. It then reduces the assertion to the fact that the Hecke algebra is a complete intersection. That this condition holds is the content of Taylor-Wiles. Chapter IV treats the dihedral case. This does not occur for semistable curves, and requires the techniques of Kolyvagin-Rubin. Chapter V actually proves the Taniyama-Weil conjecture for many elliptic curves. An appendix explains the relevant commutative algebra.

Faltings Bonn Zbl This paper provides a key fact needed in the previous paper by A. Period spaces for p-divisible groups. Annals of Mathematics Studies. This is a rigid-analytic space over E equipped with an action of GLd E. Drinfeld has shown that these covering spaces can be used to p-adically uniformize the rigid-analytic spaces corresponding to Shimura varieties associated to certain unitary groups. Moreover, the authors show that these spaces may be used to uniformize the rigid-analytic spaces associated to general Shimura varieties.

Also the authors exhibit a rigid-analytic period map from the covering spaces to one of the p-adic symmetric spaces associated to the p-adic group. Finally the non-archimedean uniformization theorem for Shimura varieties is proved.

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The main results are now described in more detail. The moduli problem of p-divisible groups is divided into two types: Let X be a p-divisible group over SpecL. For each pair G, b as above, there is the group J Qp of quasi-isogenies of X. Let B, V and G be as above. Finally, a non-archimedean uniformisation theorem for certain Shimura varieties is proved. Fix data of type PEL , i. The contents of this monograph is as follows: Moduli spaces of p-divisible groups with Appendix: The p-adic uniformization of Shimura varieties, and bibliography and index.

The author extends a number of results of Drinfeld concerning the Langlands correspondence for the GL 2 -case to the higher rank situation. Chapter I proves representability of the functor of rank r D-shtukas with level structure as an algebraic stack ChtrD,I. Knots, links, and 4-manifolds. This paper gives a construction which allows the authors to construct from a smooth 4-manifold X satisfying certain hypotheses and a knot K in the 3-sphere a new 4-manifold XK which is homeomorphic to X but which can be distinguished smoothly from X by the Seiberg-Witten invariant whenever the Alexander polynomial of the knot is non-trivial.

XK is constructed from X by gluing in the knot complement crossed with a circle to the complement of a trivial normal bundle of torus T in X. The authors use C. The adjunction inequality is used to show that the existence of certain surfaces in X which are readily found in many cases will imply that XK with the opposite orientation also has no symplectic structure. One case where these constructions apply is when X is the K3 surface with Seiberg-Witten invariant 1. A corollary is that any A-polynomial P t can occur as the Seiberg-Witten invariant of an irreducible homotopy K3 surface; if P t is not monic, then the homotopy K3 surface does not admit a symplectic structure with either orientation; any monic A-polynomial can occur as the Seiberg-Witten invariant of a symplectic homotopy K3 surface.

When L is a two component link with odd linking number, then the manifold E 1 L constructed from two copies of E 1 will have polynomials which are not products of A-polynomials and will occur as Seiberg-Witten invariants of homotopy K3 surfaces. One critical tool for the computations in the paper are gluing theorems of J. The authors also use gluing formulas for generalized log transforms.

These gauge theory tools are then combined with geometric results of J. Subsystems of second order arithmetic Perspectives in Mathematical Logic. The attention is restricted to ordinary non-set-theoretic mathematics, i. The book consists of an Introduction, two main parts and an Appendix. The extensive Introduction is devoted to the description of the main subsystems of second-order arithmetic studied in the book and of mathematics within them, as well as to the main ideas of reverse mathematics. Part A consists of 5 chapters.

Each chapter is devoted to one subsystem of Z2 and to the development of mathematics in it. Particular chapters focus on: In Part B, consisting of 3 chapters, models of Z2 and its subsystems are studied.

In the Appendix some additional reverse mathematics results and problems are presented without proofs but with references to the published literature. The book is supplemented by historical and bibliographical notes spread throughout the text , the extensive bibliography items and an index.

This monograph can be studied both by graduate students in mathematical logic and foundations of mathematics and by experts. It provides an ency- clopedic treatment of subsystems of Z2. The present paper gives a mathematically rigorous sense to the conformal invariance for two interesting models, the loop-erased random walk LERW and for the uniform spanning tree UST.

The author announces to describe also the conjectured scaling limit of critical site percolation by similar means in a forthcoming paper. Two of the main open questions about these two models and many related ones are the following. Then ft is a conformal mapping from U into some domain Dt. An analogous assertion is proved for the UST. The author presents open problems in Moonshine, the area in which he was awarded the Fields medal in Here is a very brief enumeration of these problems: One of the main results of this deep paper which circulated several years among specialists in preprint form is the following dichotomy Theorem 1.

Every Banach space contains a subspace which either has an unconditional basis or is hereditarily indecomposable. This result is applied to solve an old problem of S. Combining the above dichotomy with the results of W. The mentioned dichotomy is a consequence of another Banach space dichotomy that has Ramsey-theoretic nature and by its spirit is close to the classical Nash-Williams Theorem [Proc. Although the most striking application is in dimension 3 a good deal of the results presented here are valid in an arbitrary dimension n. This is the case for sections 1 to 10 of the present paper.

Brief descriptions of each section follow. The detailed proofs can be read in [B. This section and the following one present immediate corollaries which we can summarize as follows: This section is devoted to the proof of this assertion in the case of steady and expanding breathers. The other one is the same eigenvalue normalised by the volume raised at the suitable power in order to get a scale invariant quantity. It is a function of three parameters: This is the second breakthrough of this paper.

The number , called the scale, is the value below which this property is true. All these quantities are scaled invariant which makes this property true even after rescaling the metrics. This implies a local injectivity radius bound and allows to apply compactness theorems. This is crucial for the singularities analysis. Section 5 and 6: These two sections are less useful at the moment. In this very important section the author develops the Morse theory of a functional called L-length, which is a space-time version of the standard length or energy.

Namely, if the curvature is close to zero in a region and with extra assumption , at least for a short time it remains not too far from zero, i. This fact is made precise. This section presents another breakthrough, the so-called canonical neighbourhood theorem. It is shown that the manifold has a thick-thin decomposition, that the thick part becomes hyperbolic and is bounded by incompressible tori following arguments due to R. The thin part is claimed without proof to be a graph manifold. Some arguments which are adapted from the works of R.

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Hamilton are more precisely treated in [G. The reader is referred to [Perelman, loc. Perelman which proves the geometrization conjecture. It is inspired by the construction made by R. Sections 1 and 2: These sections present preliminary material on ancient and standard solutions. Section 2 is devoted to the standard solution.

The initial data being non compact existence and uniqueness of the solution is not immediate. An extremely detailed proof is given in [Morgan and Tian loc. They are, again, covered with canonical neighbourhoods. It is in these horns that the surgery will take place. Sections 4 and 5: On the boundary of these intervals surgeries take place. The surgery is done in horns.

An important property is the fact that the added piece called an almost-standard cap remains close to the evoluting standard solution for a while. The proof is very close to the proof of the canonical neighbourhood theorem done in [Perelman loc. This is the key result of this series of work by G. Sections 6 and 7: Section 6 presents technical issues such as curvature estimates in the future and the past of a given time slice.

It is inspired by R. It is shown that thick parts become more and more hyperbolic bounded by incompressible tori. The description of the thin part relies on an unpublished paper by the author; the conclusion is that it is a graph manifold. It contains an alternative approach. A simpler argument can be found in [Kleiner and Lott loc. At the time when this review is written the fact that the thin part is a graph manifold is still a bit controversial although it is widely believed to be true.

Some details are missing in the literature. An alternative approach is given in [L. This idea is similar to one used by R. An alternative approach using harmonic maps is given in [T. Mahler conjectured that almost no points with respect to the natural measure on the curve x, x2 ,. Here the approximation is measured in the sup-norm.

The main result of the paper under review is that friendly measures are strongly extremal. The class of friendly measures include the volume measure on non-degenerate manifolds, and so the result of Kleinbock and Margulis [loc. In fact, this is part of a larger class of push-forwards of the so-called absolutely friendly measures. Other examples are also given. The same holds for direct products of friendly measures.

The main theorem is proved by extending the method of Kleinbock and Margulis [loc. It is then shown that the above measures are all friendly. The paper is concluded with a section on related problems, results and conjectures. One of the great problems of the geometry of numbers is the following conjecture on the product of non-homogeneous linear forms: Physica-Verlag ; Zbl Steklova 33, 6—36 ; Zbl For general n there are estimates due to N.

For large sets of lattices the conjecture has been proved by A. One line of attack is to show the following: DL is well rounded. If i and ii hold, the inequality of the arithmetic and geometric mean yields the conjecture. For more information see the reviewer and C.