From the Publisher's description: "Since the publication of the first edition of the book Topological Methods in Hydrodynamics (over 20 years ago) and its Russian edition (over 10 years ago) there has appeared an enormous body of literature on topological fluid mechanics. Many problems and open questions posed or discussed in the book have been solved or substantially advanced. It would be natural to revise the text to update the reader on recent developments in this vast area for the second edition of the book.
   "Instead, the second edition, in addition to editorial corrections, contains a specially prepared survey of recent developments in topological, geometric, and grouptheoretic hydrodynamics with an independent bibliography.''

From the preface: "This volume of the Collected Works includes papers written by V. I. Arnold mostly during the period from 1980 to 1985. Arnold's work of this period is so multifaceted that it is almost impossible to give a single unifying theme for it. It ranges from properties of integral convex polygons to the large-scale structure of the Universe. Also during this period Arnold wrote eight papers related to magnetic dynamo problems, which were included in Volume II, mostly devoted to hydrodynamics. We have chosen the topic of singularities in symplectic and contact geometry only as a `marker' for this Volume IV.
   "There are many articles specifically translated for this volume. They include problems for the Moscow State University alumni conference, papers on magnetic analogues of Newton's and Ivory's theorems, on attraction of dust-like particles, on singularities in variational calculus, on Poisson structures, and others. We would like to draw the reader's attention to the translations of Arnold's comments to Selected works of H. Weyl and those of A. N. Kolmogorov. The latter were included along with short prefaces by A. N. Kolmogorov himself.''
   {For Vol. III see [V. I. Arnolʹd, Vladimir I. Arnold—collected works. Vol. III. Singularity theory 1972–1979, Springer, Berlin, 2016; MR3618837].}

From the preface: "This volume of the Collected Works contains papers written by V. I. Arnold mostly during the period from 1972 to 1979. While the papers collected here cover a variety of research directions, the main theme emerging in Arnold's work of this period is the development of singularity theory of smooth functions and mappings.
   "Among the articles specifically translated for this volume, the reader will find papers by V. I. Arnold on catastrophe theory and on A. N. Kolmogorov's school, his prefaces to Russian editions of several books related to singularity theory, a report on the first All-Union mathematical student competition in the USSR, as well as a hard-to-get preprint presenting notes by A. P. Shapiro of V. I. Arnold's lectures on bifurcations of discrete dynamical systems. The volume also contains a translation of the review by V. I. Arnold and Ya. B. Zeldovich of V. V. Beletsky's book on celestial mechanics.''
   {For Volume II see [V. I. Arnolʹd, Vladimir I. Arnold—collected works. Vol. II. Hydrodynamics, bifurcation theory, and algebraic geometry 1965–1972, Springer, Berlin, 2014; MR3185029].}

Papers in this collection include the following:
  

Dorin Andrica and Oana Liliana Chender, "A new way to compute the Rodrigues coefficients of functions of the Lie groups of matrices”, 1–24. MR3526912

M. Baldo and F. Raciti, "Quasimodes in integrable systems and semi-classical limit”, 25–47. MR3526913

Giovanni Bazzoni and Vicente Muñoz, "Manifolds which are complex and symplectic but not Kähler”, 49–69. MR3526914

O. Chau, D. Goeleven and R. Oujja, "Solvability of a nonclamped frictional contact problem with adhesion”, 71–87. MR3526915

Alessandro Fortunati and Stephen Wiggins, "The Kolmogorov-Arnold-Moser (KAM) and Nekhoroshev theorems with arbitrary time dependence”, 89–99. MR3526916

B. Jadamba, A. A. Khan, F. Raciti, C. Tammer and B. Winkler, "Iterative methods for the elastography inverse problem of locating tumors”, 101–131. MR3526917

Dumitru Motreanu and Viorica Venera Motreanu, "Transversality theory with applications to differential equations”, 133–157. MR3526918

A. B. Németh and S. Z. Németh, "Lattice-like subsets of Euclidean Jordan algebras”, 159–179. MR3526919

Werner Georg Nowak, "Simultaneous Diophantine approximation: searching for analogues of Hurwitz's theorem”, 181–197. MR3526920

Kaoru Ono and Andrei Pajitnov, "On the fixed points of a Hamiltonian diffeomorphism in presence of fundamental group”, 199–228. MR3526921

Nihal Yılmaz Özgür and Nihal Taş, "Some generalizations of fixed-point theorems on S-metric spaces”, 229–261. MR3526922

Choonkil Park, Jung Rye Lee and Themistocles M. Rassias, "Functional inequalities in Banach spaces and fuzzy Banach spaces”, 263–310. MR3526923

Agostino Prástaro, "The Maslov index in PDEs geometry”, 311–359. MR3526924

Biagio Ricceri, "On the infimum of certain functionals”, 361–367. MR3526925

Teerapong Suksumran, "The algebra of gyrogroups: Cayley's theorem, Lagrange's theorem, and isomorphism theorems”, 369–437. MR3526926

Árpád Száz and Amr Zakaria, "Mild continuity properties of relations and relators in relator spaces”, 439–511. MR3526927

Mihai Turinici, "Contraction maps in pseudometric structures”, 513–562. MR3526928

Abraham Albert Ungar, "Novel tools to determine hyperbolic triangle centers”, 563–663. MR3526929

It is always an exultant privilege to review a book that deals with the splendor of mathematics, be it a work created by an author who is a "problem-solver'', or one who is a "theory-creator'', as delightfully categorized by P. R. Halmos [Am. Sci. 56 (1968), no. 4, 375–389]. Thus, when an oeuvre comes along by a writer, such as Arnold, who combines both of these two characteristics, it is fair to assume that we are in for a marvelous manifestation of the preeminence of mathematics in the world of science.
   Vladimir Igorevich Arnold (1937–2010), born in Odessa, Ukraine (then in USSR), was one of the most prolific and talented mathematicians of the twentieth century. In fact, while only 19 years old and still a student of Andrei Kolmogorov at Moscow State University, he solved Hilbert's thirteenth problem. In that particular problem, Hilbert had considered the seventh-degree equation and had asked whether its solution, x, considered as a function of the three variables a, b, and c, could be expressed as the composition of a finite number of two-variable functions.
   A natural generalization of the problem is the following question: Can every continuous function of three variables be expressed as a composition of finitely many continuous functions of two variables? In 1956, Kolmogorov had shown that any function of several variables could be constructed with a finite number of three-variable functions. In 1957, Arnold expanded on this work to show that only bi-variate functions were in fact required, thus answering Hilbert's question affirmatively.
   After graduating from Moscow State University in 1959, Arnold taught there until 1986. From then on, up until his death, he worked at Steklov Mathematical Institute in Moscow and at Paris Dauphine University, two premier research institutes. He was made a Foreign Honorary Member of the American Academy of Arts and Sciences in 1987, a Foreign Member of the Royal Society in 1988, a member of the Academy of Sciences of the Soviet Union (renamed Russian Academy of Sciences in 1991) in 1990, and was awarded numerous prestigious prizes. For more information on Arnold see [V. V. Goryunov and V. M. Zakalyukin, Mosc. Math. J. 11 (2011), no. 3, 409–411; MR2894422], [Notices Amer. Math. Soc. 59 (2012), no. 3, 378–399; MR2931629], [Arnold: swimming against the tide, Amer. Math. Soc., Providence, RI, 2014; MR3223035], or M. White's entry in the Encyclopaedia Britannica on Arnold.
   Arnold was also very interested in the history of mathematics and liked to study the classics, most notably the works of Huygens, Newton, and Poincaré, and was a fan of Felix Klein's Development of mathematics in the 19th century [translated from the German by M. Ackerman, Lie Groups: History, Frontiers and Applications, IX, Math Sci Press, Brookline, MA, 1979; MR0549187]. He was also a socially conscientious individual, as can be seen from his open and public opposition to the persecution of dissidents in the USSR at the risk of creating a rancorous discord between himself and some prominent Soviet officials, and making him a target of the very same persecution; indeed, he was not allowed to leave the Soviet Union during most of the 1970s and 1980s (again, see the entry in the Encyclopaedia Britannica on Arnold). Thus, it is no surprise that among his numerous works that reflect his interest in various esoteric and diverse topics in mathematics, namely, differential equations, ergodic problems of classical mechanics, catastrophe theory, continued fractions, singularity theory, algebraic geometry, partial differential equations, experimental mathematics, and topology, there are also some gems that deal with the philosophical and pedagogical aspects of mathematics. Within the latter group, one should mention [Yesterday and long ago, translated from the 2006 Russian original by Leonora P. Kotova and Owen L. deLange, Springer, Berlin, 2007; MR2269569], [Uspekhi Mat. Nauk 53 (1998), no. 1(319), 229–234; MR1618209], [Arnold's problems, translated and revised edition of the 2000 Russian original, Springer, Berlin, 2004; MR2078115], [Mathematical understanding of nature, translated from the 2011 Russian original by Alexei Sossinsky and Olga Sipacheva, Amer. Math. Soc., Providence, RI, 2014; MR3237395] and of course, the subject of this review, Lectures and problems: a gift to young mathematicians.
   Lectures and problems was expertly translated into English by two well-known mathematicians, Dmitry Fuchs of UC Davis and Mark Saul of the Courant Institute, who managed to preserve Arnold's well-known lucid writing style. It also contains a preface by Saul and an extensive bibliography.
   Although the book is divided into four parts, it is more expedient to envision it in two parts: lectures and problems. The lectures involve sections on continued fractions; geometry of complex numbers, quaternions, and spins; and Euler groups and arithmetic of geometric progressions. The problems section, intended for children five to fifteen years old, contains 79 very interesting problems of varying degrees of difficulty. There is also a section on solutions to selected problems, composed by Fuchs.
   The chapter on continued fractions focuses mostly on the geometric approach to the topic, thus reviving Hermann Minkowski's "geometry of numbers'' and leading to interesting lemmas such as the one that states that if a parallelogram with vertices at lattice points has no other lattice points either inside or on the sides, then its area must be 1. This approach eventually culminates in Kuzmin's Theorem. This theorem asserts that in the continued fraction the probability pk that the integer k will appear as a term is with the interesting consequence that p1≅0.41, that is, about half of the time the terms in a continued fraction are 1's.
   For some reason, although Arnold credits Gauss with "the fundamental discovery needed for the proof of the theorem'' (p. 13), he calls it Kuzmin's Theorem and not the Gauss-Kuzmin Theorem as it is usually referred to in the literature. The chapter continues with a discussion of the golden section, and Lagrange's Theorem—a continued fraction is periodic if and only if the value represented by the fraction is a number of the form a+bc√, where a, b, c are rational numbers—and concludes with a generalization of Lagrange's Theorem.
   The chapter on the geometry of complex numbers, quaternions, and spins is equally interesting. Note the following two minor typos: on page 53, in Hamilton's multiplication table, we should have and and on page 63 we should have
   The chapter on Euler groups and arithmetic of geometric progressions is analogously thought-provoking and exciting. Starting with the simple definition of the Euler group Γ(n) as the multiplicative group on the set Zn=Z/nZ, Arnold introduces the order of this group, φ(n), where φ is the Euler function. As is well known, if then Arnold then uses this result to prove the Euler product formula, namely, the formula which Leonhard Euler proved in his thesis Variae observationes circa series infinitas, published by St. Petersburg Academy in 1737.
   The chapter then continues with Fermat-Euler dynamic systems, statistics of geometric progressions, measurement of the degree of randomness of a subset, and a discussion of quadratic residues.
   Caveat emptor: This is not a recreational, "fireside'' reading type of a book. Throughout the lectures, we are confronted by Arnold's demanding and challenging yet carefully constructed, cogent, and logically complete style of exposition. Thus, although the topics are within the realm of an undergraduate student in mathematics, the coverage makes the book exceedingly enriching, for it requires a careful analysis of each paragraph. Too many details are omitted, even for undergraduate students—in my opinion deliberately so, to limit the readership to those who truly wish to understand and learn mathematics.
   Arnold also uses the lectures to expose us to the historic development of the topics covered, and to introduce us to his views on mathematics:
   "In my view, mathematics and physics are parts of the same experimental science. When the experiments cost billions of dollars we call this science physics. When they are cheap, we call it mathematics.'' (p. 4)
   The problems section opens with a note from Arnold that ends with a mild rebuke of the (French) educational system:
   "I have even noticed that five-year-olds can solve problems like this better than can school children, who have been ruined by coaching, but who, in turn, find them easier than college students who are busy cramming at their universities. (And Nobel prize [sic] or Fields Medal winners are the worst at all in solving problems.)'' (p. 125)
   Problems range from ones that require nothing more than solving simple systems of equations to summing of infinite series and to performing numerical integrations of the type
   Overall, the book is a masterful combination of mathematical rigor and geometrical and physical intuition. Only a master mathematician could take such ordinary topics and extend them in such sophisticated directions while keeping the proofs as intuitive as possible. It is a veritable tour de force that will expand the horizons of all those who are genuinely interested in mathematics.

Experimental mathematics by V. I. Arnold would make for an unusual and delightful addition to any mathematician's library. Arnold is a masterful expositor, and in this book we get a glimpse of how Arnold thinks about what kinds of questions in mathematics are important and worth exploring, and how those first important explorations might be carried out, in this case with the aid of a computer. This book is a translation by Dmitry Fuchs and Mark Saul of four lectures given by Arnold to advanced high school and undergraduate students, and it contains many conjectures in a variety of fields that would be of interest to undergraduate and graduate students as well as researchers in mathematics. Below I will describe just a few of these open questions. But even if these kinds of questions are not of immediate interest to the reader, this book is worth exploring, for as Saul writes about Arnold, "Few mathematicians—indeed few scientists in any field—open their minds so completely as he has to their students.'' This read offers a rare opportunity to see how a master mathematician does mathematics.
   Lecture 1: The Statistics of Topology and Algebra.
   Let f(x,y) be an algebraic polynomial of degree n. How many closed curves ("ovals'') can its parabolic curve f(x,y)=0 on the real projective plane be made of? An upper bound is known to be (n−2)(n−1)2+1 connected components. And if one has the maximal number of ovals, in how many topologically distinct ways can this happen? The answer is currently not known if n=8, for example. It is known that the number of ovals is bounded above by an2 and below by bn2, where a>b. Arnold hopes that the reader will find the exact rate of growth of the number of ovals as n increases by moving the constants a and b closer together. Arnold then moves on to the statistics of smooth functions, the statistics and topology of periodic functions and trigonometric polynomials, and the algebraic geometry of trigonometric polynomials. Here for example computer experiments can be used to analyze graphs of Morse functions.
   Lecture 2: Combinatorial Complexity and Randomness.
   If we have a finite sequence of 0's and 1's, how can we assign a degree of randomness to the sequence? The operator A:Zn2→Zn2 of taking differences (for example (1,1,0,1,0,0,1) is sent to (0,1,1,1,0,1,0)) sends one binary sequence of length n to another. We can then form a directed graph of 2n vertices where each vertex is one of the 2n possible binary sequences, and there is an edge from x to y if and only if y=Ax. Every connected component of the graph has exactly one cycle Om consisting of m edges where attached to each vertex of Om is a copy of some tree T, and so there are m copies of T in the connected component, one attached to each vertex of Om. Furthermore, these trees are binary root trees, where if we were to traverse the tree "backwards'' from a vertex of Om, each time we reach a vertex of the tree there are two edges leading away from it. We will then say that the further a sequence (vertex) is from a cycle, and the larger the size of the cycle, the more random the corresponding sequence. Arnold tabulated what happens for n≤12, but the topology of the larger graphs remains unknown; however, there are several conjectures that could be initially investigated with the aid of a computer. Arnold then goes on to talk about logarithmic functions and their complexity, as well as the randomness and complexity of tables of Galois fields.
   Lecture 3: Random Permutations and Young Diagrams of Their Cycles.
   If we randomly permute n objects, which permutations among the n! many permutations occur the most often? They could be said to be the most random of the permutations. Each permutation can be broken into cycles, and the collection of these cycles can be represented by a Young diagram with a certain height, width, and fullness. Computer experiments suggest, for example, that for large n the average height is yˆ∼c2lnn and the average fullness is λˆ∼c3lnn. Arnold goes on to discuss experimentation with random permutations of larger numbers of elements, random permutations of p2 elements generated by Galois fields, and statistics of cycles of Fibonacci automorphisms.
   Lecture 4: The Geometry of Frobenius Numbers for Additive Semigroups.
   This lecture begins with a discussion of additive semigroups, Sylvester's Theorem, Frobenius numbers, and trees blocked by others in a forest. Topics considered include the geometry of numbers, an upper bound estimate of the Frobenius number, average values of the Frobenius numbers, a proof of Sylvester's Theorem, the geometry of continued fractions of Frobenius numbers, and the distribution of points of an additive semigroup on the segment preceding the Frobenius number.
   {For the Russian original see [V. I. Arnold, Experimental observation of mathematical facts (Russian), Izdat. MTsNMO, Moscow, 2006].}

This paper is the English translation of an earlier article by V. I. Arnolʹd (contained in [Selecta-60 (Russian), FAZIS, Moscow, 1997 (727–740); MR1647728]) containing his recollections about two research problems that he tackled dealing with superpositions of continuous functions and quasi-periodic motions in dynamical systems. The vivid presentation of Russian studies about the KAM theory, as well as other topics considered by Kolmogorov and collaborators (including Arnolʹd himself and Gelʹfand), is particularly interesting also just from a historical point of view. Quite intriguing is Arnolʹd's predilection for a "KAM theory'' rather than just a "KAM theorem''.

This little collection of essays belongs to the class of books many mathematicians would have been happy to write. It conveys a vision of mathematics, elementary and advanced, pure and applied, that any reader would enjoy. Back in the 1970s, when translations of Arnolʹd's text on ordinary differential equations became available, a breath of fresh air wiped out the rigid presentation based on the then canonical text by Coddington and Levinson: ODEs were simple, understandable and real-world based. The same spirit pervades the book under review. On browsing through Mathematical understanding..., the reader is tempted to consider it as (part of) a mathematical diary with sketches and ideas on a number of topics that might be classified into two groups: (a) Very short and schematic "chapters'' where the reader feels invited to try some back-of-the-envelope mathematics and enjoy self-exploration with simple-though-deep ideas and tools of a mathematical and physico-mathematical nature. In this group, Optics is the favourite leading thread. (b) Longer "chapters'' developing special topics to some extent in a self-contained manner. Mechanics and Fluid Mechanics appear in this second group. Along this line, Chapter 34, "The Mathematical Notion of Potential'', has been the reviewer's favourite. All through the book are comments about people—mostly Russian or Soviet—and glimpses of everyday life, as well as remarks on the senselessness of straightforward axiomatisation and the role of aesthetic considerations in mathematical development, reminiscent of the caustic comment attributed to Felix Klein: "when a mathematician does not have any more ideas, then he enters Axiomatics''.
   Just a last observation on the translation: It is of course very difficult to render into English the whole spirit of the Russian original, but sometimes the text is rather clumsy and hard to understand.

From the preface: "Vladimir Igorevich Arnold is one of the most influential mathematicians of our era. Arnold launched several mathematical domains (such as modern geometric mechanics, symplectic topology, and topological fluid dynamics) and contributed, in a fundamental way, to the foundations and methods in many subjects, from ordinary differential equations and celestial mechanics to singularity theory and real algebraic geometry.''
   The preface also contains a brief biography of Arnold as well as an (admittedly) incomplete list of results and concepts that bear Arnold's name. Part 1, "By Arnold'', contains seven articles, of which five are by Arnold. Included in this section is a translation into English of the complete interview given by Arnold to "Kvant'' in 1990. Two of his three lectures at the Fields Institute in 1997 in honor of his sixtieth birthday are included here (Chapters 2 and 4). This collection also contains a reprint of Arnold's "A Mathematical Trivium''—a collection of one hundred mathematical problems that, in the author's belief, delineate standards of undergraduate mathematical education.
   Part 1, "By Arnold'', includes the following: 1. Arnold in His Own Words; 2. From Hilbert's Superposition Problem to Dynamical Systems; 3. Recollections (J. Moser); 4. Polymathematics: Is Mathematics a Single Science or a Set of Arts?; 5. A Mathematical Trivium; 6. Comments on "A Mathematical Trivium'' (B. Khesin, S. Tabachnikov); 7. About Vladimir Abramovich Rokhlin. A generous selection of photographs of Arnold is included at the end of this section.
   Part 2, "About Arnold'', contains fifteen articles written by Arnold's colleagues, students, and friends: 8. To Whom It May Concern (A. Givental); 9. Remembering Vladimir Arnold: Early Years (Y. Sinai); 10. Vladimir I. Arnold (S. Smale); 11. Memories of Vladimir Arnold (M. Berry); 12. Dima Arnold in My Life (D. Fuchs); 13. V. I. Arnold, As I Have Seen Him (Y. Ilyashenko); 14. My Encounters with Vladimir Igorevich Arnold (Y. Eliashberg); 15. On V. I. Arnold and Hydrodynamics (B. Khesin); 16. Arnold's Seminar, First Years (A. Khovanskii, A. Varchenko); 17. Topology in Arnold's Work (V. Vassiliev); 18. Arnold and Symplectic Geometry (H. Hofer); 19. Some Recollections of Vladimir Igorevich (M. Sevryuk); 20. Remembering V. I. Arnold (L. Polterovich); 21. Several Thoughts about Arnold (A. Vershik); 22. Vladimir Igorevich Arnold: A View from the Rear Bench (S. Yakovenko).

From the preface: "This volume of the Collected Works appears in print after Vladimir Arnold's untimely death in June 2010. His passing was a terrible loss for mathematics and science in general.
   "We hope that this project of Collected Works, which is needed now more than ever, will contribute to establishing the tremendous legacy of V. I. Arnold, a remarkable mathematician and human being. Some memories of V. I. Arnold can be found in the recent March and April 2012 issues of the Notices of the AMS.
   "Our Editorial team has also suffered an unprecedented blow since Volume I was published in 2009 [V. I. Arnolʹd, Vladimir I. Arnold—collected works. Vol. I, Springer, Berlin, 2009; MR2640495]. Jerry Marsden passed away in September 2010, and Vladimir Zakalyukin passed away in December 2011. We dedicate this volume to their memory.''

This book is a translation of lecture notes from Russian with additional comments and notes by the editors. The book is aimed at advanced high school students, but works its way to the forefront of current research and unsolved problems. It introduces very advanced topics in a very relaxed and informal style. The book includes some exposition and definitions, some theorems and proofs and a lot of problems with hints or solutions. There are many illustrations to lead the reader to an intuitive understanding of the concepts being developed. The general theme of the book is, given an algebraic curve, what can be said of its topological structure in the plane? This is a special case of the first part of Hilbert's 16th problem, a topic taken up in detail in Chapter 4.
   After a brief introduction, Chapter 2 takes up conic sections. It begins with standard results on foci and eccentricity, but continues to more advanced ideas including work in the complex plane and in 3 dimensions.
   Chapter 3 does physics applications of conic sections and ellipsoids. It begins with an interesting (and historical) application to jet engines, then moves on to gravitational and magnetic fields.
   Chapter 4 moves into much deeper mathematics. Projective geometry is introduced via perspective and its use in art. The chapter quickly moves to defining the real projective plane, conic sections therein, the Möbius strip and genus of the Riemann surface of a curve. Perhaps because Arnold has himself done considerable research in this area, the chapter is quite nontrivial. It has a discussion of an incorrect statement by Hilbert in his 16th problem and proceeds to discuss the correction and generalizations which have been proved, thanks to a very seminal paper by Arnold. This concerns the possible arrangements of ovals of an algebraic sixth degree curve in the real projective plane. There is then a detailed discussion of the number of topologically different polynomials of degree n+1 with n critical points. The discussion goes on to polynomials in more variables, primarily in the form of open problems which lend themselves to enumerative computational results.
   Chapter 5 pushes further, considering algebraic curves in complex projective space. Remarkably, there is even a proof of the Riemann-Hurwitz Theorem on the genus of the Riemann surface of a smooth algebraic plane curve in the complex projective plane. The chapter ends with a quick discussion of elliptic functions and abelian integrals.
   Chapter 6, which claims to be accessible to preschool children, leads the reader to compute the possible number of regions that can be obtained when the plane is cut by n lines. In the Appendix following this, the author's paper on this topic is reproduced.

The Russian original has been reviewed [V. I. Arnolʹd, A. N. Varchenko and S. M. Gusein-Zade, Singularities of differentiable mappings. II (Russian), "Nauka'', Moscow, 1984; MR0755329].
   {For Volume 1 see [V. I. Arnolʹd, S. M. Gusein-Zade and A. N. Varchenko, Singularities of differentiable maps. Volume 1, reprint of the 1985 edition, translated from the Russian by Ian Porteous based on a previous translation by Mark Reynolds, Mod. Birkhäuser Class., Birkhäuser/Springer, New York, 2012; MR2896292].}

This is a reprint of [V. I. Arnolʹd, S. M. Gusein-Zade and A. N. Varchenko, Singularities of differentiable maps. Vol. I, translated from the Russian by Ian Porteous and Mark Reynolds, Monogr. Math., 82, Birkhäuser Boston, Boston, MA, 1985; MR0777682].
   REVISED (January, 2014)

Current version of review. Go to earlier version.

Courant proved that the zeros of the n-th eigenfunction of the Laplace operator on a compact manifold divide the manifold at most into n parts. Courant conjectured that the linear span of the first n eigenfunctions possesses the same property. Later, it was proven that this Courant hypothesis is not always true. The article contains a discussion and some results devoted to this Courant problem. Moreover, one can also find some historical facts here.

The author determines the structure of the group of invertible residue classes mod 3n in the ring Z[i]. This is a particular case of a result of K. Hensel [J. Reine Angew. Math. 146 (1916), 216–228; JFM 46.0251.02].

In this short and engaging book, Arnold presents a collection of observations, conjectures, and computations on the structure of finite fields (or Galois fields, as they are called here). Arnold's vast scientific body of work lies mostly far outside the area of algebra, and the viewpoint presented here is clearly one of an enthusiastic outsider. Indeed, the book derives from a 2-hour presentation to high-school students, and builds up its examination carefully in a way that one imagines a strong high school student easily following. This does not require sacrificing depth, however, and the book discusses and gives data on many questions that remain open. These questions come from dynamical systems, statistics, and projective geometry applied to the seemingly mundane notion of finite fields.
   Arnold's inquiry starts from a down-to-earth construction of finite fields with a cyclic multiplicative group. It illustrates well the flavor of this book that Arnold adds to his construction the axiom that the multiplicative group is cyclic, for his interest is in science-minded exploration rather than the development of theory required to prove that every finite field has this property.
   Arnold represents elements of a given finite field by the corresponding power of the primitive element, thereby turning field multiplication into addition of exponents. Field addition then becomes a tropical, or lower-level, operation, whose workings are less clear than multiplication. Arnold treats at length the case where the field has p2 elements for some prime p, and shows that in this case every element is a linear combination of 1 and a fixed primitive element. This leads to the presentation of an addition table for the field, where entries are the distinct powers of the primitive element, i.e., 1,…,p2−1, plus ∞ to represent the zero element. Naturally one obtains similar higher-dimensional "tables'' for fields with pn elements, where n≥3. Arnold asks the general question of whether these tables are arranged randomly as one increases p and n. In Chapter 3 he makes several specific conjectures that say the tables have certain qualities in common with randomly arranged sets. In Chapter 4 he gives detailed computations of some examples that support these randomness conjectures. (He also includes a fascinating aside on the complexity of the log function on finite fields, giving some intricate and visually appealing calculations.)
   The above is but one example of the kind of question discussed in the book. Another principal line of inquiry comes from consideration of the projective line over the finite field with p elements, and the dynamical structure of certain power maps acting on it.
   Throughout, Arnold's characteristic style of writing and thinking are evident. Ideas, intuitions, and well-presented examples abound, joined in only a few places by formal proofs. This is not a book in which to look for abstract theorems or complete studies. But both students and working mathematicians will find it accessible, provocative, and maybe even inspiring.

The Gibbs phenomenon for a periodic, piecewise differentiable function describes behavior of partial sums of the Fourier series at the discontinuity points of the function. In particular, the jump of the partial sums overshoots the jump of the function; as the number of terms tends to infinity, the overshoot approaches approximately 9 percent.
   The author generalizes this idea to a function F from Zp (p an odd prime) to the complex plane. The function is first represented as F=∑p−1k=0ckfk, where fk(z)=zk. The terms in which k are farthest away from 0, namely k=(p−1)/2 and k=(p+1)/2, are considered to be the "highest harmonics'', and the remaining p−2 terms are used in the partial sum. With this convention, the jump of the partial sum is approximately 1+1/p times the jump of the original function.

Let us denote by T(a,n) the minimal period of the sequence of numbers at(modn), t=1,2,…, where a and n are natural numbers. Obviously, all T(a,n) divide the value of Euler's function ϕ(n) and T(a+n,n)=T(n). The author studies the distribution of T(a,n) for a∈Z/nZ among all the divisors of ϕ(n). Let Then the equality n=∑tm(t) can be represented by a Young diagram. The author provides an empirical study of the periods T(a,n) and related quantities. For example, numerical values and suggestions are given concerning the averaged period The distribution of the periods T(a,n) for the invertible elements a(modn) is also studied.

At present, there is no known general topological classification of polynomials of low degree in two real variables, despite the obvious connections to Hilbert's 16th problem. The classification problem for polynomials in two variables is closely related to the problem of topological classification of smooth functions f:S2→R on the sphere. In the paper under review the author discusses the topological classification of polynomials of degree 4 in two variables having nondegenerate critical points and distinct critical values. The author proves that out of 17746 topological types of smooth Morse functions f:S2→R with the same number of critical points, at most 426 types can be realised by polynomials of degree 4.

This paper consists of the talk V. I. Arnold gave at the prize ceremony for the Shaw Prize. The paper reviews some results by the author and others on statistics in the continued fraction expansion and related topics. The paper also contains a biography of Arnold.

Let k=k(n) be the number of all distinct quadratic residues modulo n. These k points divide the finite circle of length n into k arcs with lengths aj, i.e. Let #(m) be the number of arcs with lengths m, which are bounded by all distinct distributions of k points lying on the finite circle Zn, and #′(m) the number of "arcs'' of length m, which are bounded by all distinct distributions of k points (not necessarily distinct) lying on the finite circle Zn. The author studies the following identities: and

From the preface: "Vladimir Igorevich Arnold is one of the most influential mathematicians of our time. Arnold launched several mathematical domains (such as modern geometric mechanics, symplectic topology, and topological fluid dynamics) and contributed, in a fundamental way, to the foundations and methods in many subjects, from ordinary differential equations and celestial mechanics to singularity theory and real algebraic geometry.
   "Arnold wrote some 700 papers, and many books, including 10 university textbooks. He is known for his lucid writing style, which combines mathematical rigour with physical and geometric intuition.
   "With this volume Springer starts an ongoing project of putting together Arnold's work since his very first papers (not including Arnold's books).''

In this paper random permutations of a large number of elements are considered and their statistics are investigated. It is shown that statistically there is a big difference between decompositions into cycles of such permutations from the same decompositions of algebraic permutations defined by linear or projective transformations of finite sets. Tables giving both these and other statistics as well as a comparison of them with the statistics of involutions or permutations with all their cycles of even length are given in the paper. The inclusions of a point in cycles of various lengths turn out to be equiprobable events for random permutations. The number of permutations on n elements with all cycles of even length turns out to be the square of an integer (namely, of (2n−1)!!). The number of cycles of projective permutations (over a field with an odd prime number of elements) is always even. These and other empirically discovered theorems are proved in the paper.

This is a survey article, in which the author provides some historical background and questions on the ordinary continued fractions and a version of multidimensional continued fractions.
   Summary (from the translation journal): "We study finite length sequences of numbers which, at first glance, look like realizations of a random variable (for example, sequences of fractional parts of arithmetic and geometric progressions, last digits of sequences of prime numbers, and incomplete periodic continuous fractions). The degree of randomness of a finite length sequence is measured by the parameter introduced by Kolmogorov in his 1933 Italian article published in an actuarial journal. Unexpectedly, fractional parts of terms of a geometric progression behave much more randomly than terms of an arithmetic progression, and the statistics of periods of continuous fractions for eigenvalues of unimodular matrices turns out to be different from the classical Gauss-Kuzmin statistics of partial continuous fractions of random real numbers. Empirically, the lengths of the period of continuous fractions for the roots of quadratic equations with leading coefficient 1 and increasing other (integer) coefficients grow, on the average, as the square root of the discriminant of the equation.''

In this paper Arnold comments on the following open problems: (1) Is it possible to obtain any false R4 (manifold homeomorphic to but not diffeomorphic to R4) as the orbit space of a polynomial vector field on R5? (2) Is it true that any pseudo-periodic curve has a unique noncompact component? (3) Is it true that almost always the number of periodic points of a smooth map f on a compact manifold M grows at most as an exponential function of the period? (4) One considers two submanifolds X,Y⊂M. If Z(n)=fn(X)∩Y, is it true that almost always the topological complexity of Z(n) grows at most as an exponential function of n? (5) One considers two germs of holomorphic curves (X,0),(Y,0) and f a germ of holomorphic map at 0∈C2. Is it true that in general the Milnor number μ(n) of fn(X)∩Y grows at most as an exponential function of n? (6) The infinitesimal version of the Hilbert sixteenth problem. (7) The materialization of resonances in holomorphic dynamics. (8) Analytic and geometric unsolvability in chaos theory (a problem is geometrically unsolvable if, among all the problems that are obtained by diffeomorphic changes in coordinates and parameters, there is no problem that is analytically solvable).

The paper under review is focused on an investigation of the length T(Q) of the period of the continued fraction of the square root of an integer Q, initiated by empirical study.
   Induced by special features suggested by data on T(Q), the author was led to some reasonable conclusions on T(Q), which are then proved by rigorous arguments. For instance,
   1. T(Q) is odd only when the number Q can be represented as Q=a2+b2 (integers a and b being relatively prime).
   2. T(Q)≤cQ−−√logQ, for some uniformly constant c.
   3. T(Q)=2 if and only if Q=x2y2+x (x>1, y≥1), or Q=x2y2+2x (x≥1, y≥1).
   4. The statistics of the elements of T(Q) differ substantially from those of the continued fractions of random real numbers.

A. N. Kolmogorov [Giorn. Inst. Italiano Attuari 4 (1933) 83–91; Zbl 0006.17402] established that the empirical statistics of several independent samplings of a random variable differ from its distribution function in a universal way, in the sense that the random distribution of the distance of one of these statistics from the other is asymptotically described by a universal law, now called "Kolmogorov's distribution''. Here an empirical study is made of the distribution of residues modulo a fixed modulus for a sequence of powers, and compared with the Kolmogorov distribution. This is a setting in which a highly deterministic sequence may be expected to exhibit some random behaviour, and is part of a broad study by the author of chaotic phenomena for discrete deterministic systems, in which empirically observed statistical phenomena may be extremely difficult to prove.

Summary: "The article describes the interrelations between the minimal integer number N(a,b,c), which belongs to the additive semigroup of integers generated by a, b, c together with all greater integers, on the one hand, and the geometrical theory of continued fractions describing the convex hulls of sets of integer points in simplicial cones, on the other hand. It also provides some hints on the extension of N to non-integral arguments.''

Summary (translated from the Russian): "A vector in an integer space is said to be divisible if it is the product of another vector in this space and an integer greater than one.
   "The uniform distribution of a set of integer vectors means that the number of points of this set in an N-fold homothetically expanded domain in an n-dimensional space is asymptotically proportional to the product of Nn and the volume of the domain as N→∞.
   "The coefficient of this proportionality (density) is equal to 1/ζ(n) for the set of indivisible vectors in an n-dimensional integer space (where n>1). For example, the density of the set of indivisible vectors on the plane is equal to 1/ζ(2)=6/π2≈2/3. This was the discovery that led Euler to the definition of the zeta function.
   "Here we present a proof of the uniform distribution of the set of indivisible integer vectors because there are arbitrarily large domains that have no indivisible vectors.
   "We show that such domains are located only far from the origin and are infrequent even there. Their distribution is also uniform and has a peculiar fractal character, which has not been studied even at the empirical computer-aided level or even for n=2.''

Let X1(ω),X2(ω),…,Xn(ω) be a sample of values of independent random variables, having the same distribution function F(x). Let Fn(ω,x) be an empirical distribution function, i.e., and A. N. Kolmogorov proved in 1933 that the distribution functions of the random variables λn converge to some limit distribution, not depending on F(x) as n→∞. The mean value of this asymptotic distribution is π/2−−−√.
   The author asks whether the interpretation of the sequence of fractional parts of members of arithmetical progression as a sample of a uniformly distributed random variable is in accordance with Kolmogorov's criterion. The positive answer would follow if the values of λn were in some sense close to the mean value of the limit distribution.
   The author proves that the behaviour of λn computed for the sequence {kx}(x=1,…,n) with the F(x) of uniform distribution is different. If k is rational, then λn→0 as n→∞. It is proved, too, that for any K an irrational value of k can be chosen such that λn>K holds for an infinite number of values of n. The value of such k is given explicitly.

Summary: "We present geometrical arguments suggesting that the part of the segment {0,1,…,N−1} covered by the additive semigroup generated by (a,b,c) between 0 and the Frobenius number N(a,b,c) should exceed λV for some constant λ (which might be 1/3 or even more).''

Let be an integer matrix, where a2+b2+c2+d2≤M. The eigenvalues of A satisfy the quadratic equation x2+px+q=0, where −p=trA=a+d and q=detA. Moreover, assume that p2≥4q, in which case the eigenvalues are real.
   The purpose of the paper is to study asymptotic properties of the periods of the continued fractions for the eigenvalues, in particular for the length T(p,q). For irrational eigenvalues the continued fractions are of the form where b0 is an integer and bn, n≥1, are positive integers. For rational eigenvalues the continued fraction terminates, and in that case T(p,q) is defined to be 0.
   Several questions are raised, and interesting observations are made, in some cases leading to a theorem, in other cases to a conjecture.
   Example of a theorem:
   Let, for M>2, Q(M) be the number of matrices A∈SL(2,Z) with trA=2 and with the sum of the squared elements ≤M. Then Q(M)≤1+30M−2−−−−−√(1+lnM).
   (Properties of the periods are contained in the condition A∈SL(2,Z).)
   The conjectures include a growth statement on sets of matrices satisfying conditions like a2+b2+c2+d2≤M or A∈SL(2,Z). Other conjectures deal with growth conditions for certain mean values of period length.

Summary: "This paper shows how the measurement of the stochasticity degree of a finite sequence of real numbers, published by A. N. Kolmogorov in Italian in a journal of insurance statistics [G. Ist. Ital. Attuari 4 (1933), 83–91; Zbl 0006.17402], can be usefully applied to measure the objective stochasticity degree of sequences, originating from dynamical systems theory and from number theory.
   "Namely, whenever the value of Kolmogorov's stochasticity parameter of a given sequence of numbers is too small (or too big), one may conclude that the conjecture describing this sequence as a sample of independent values of a random variables is highly improbable.
   "Kolmogorov used this strategy fighting [C. R. (Doklady) Acad. Sci. URSS (N.S.) 27 (1940), 37–41; MR0003556] against Lysenko, who had tried to disprove the classical genetics' law of Mendel experimentally.
   "Calculating his stochasticity parameter value for the numbers from Lysenko's experiment reports, Kolmogorov deduced, that, while these numbers were different from the exact fulfilment of Mendel's 3 : 1 law, any smaller deviation would be a manifestation of the report's number falsification.
   "The calculation of the values of the stochasticity parameter would be useful for many other generators of pseudorandom numbers and for many other chaotically looking statistics, including even the distribution of prime numbers (discussed in this paper as an example).''