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A Japanese mathematician
Kunihiko Kodaira is one of the greatest japanese mathematicians.

He is one of the last students of Teiji Takagi who is famous
for class field theory.

Kunihiko Kodaira died at Kohu, Yamanashi prefecture, Japan in 1997.

He also wrote essays.

Superb
Of importance to applications such as superstring theories in high-energy physics, the theory of complex manifolds and the deformation of complex structures are explained in great detail in this book by one of the major contributors to the subject. One of the valuable features of the book that is actually rare in more recent books on mathematics is that the author tries (and succeeds) to give motivation for the subject. This feature is actually quite common in older books on mathematics, for with few exceptions writers at that time believed that a proper understanding of mathematics can only come with explanations that are given outside the deductive structures that are created in the process of doing mathematics. These explanations frequently involve the use of diagrams, pictures, intuitive arguments, and historical analogies, and so are not held to be rigorous from a mathematical standpoint. They are however extremely valuable to students of mathematics and those who are interested in applying it, like physicists and engineers. There seems to be an inverse relationship between rigor and understanding of mathematics, and given the emphasis on the former in modern works of mathematics, one can expect students to have more trouble learning a particular branch of mathematics than those students of a few decades ago.

Luckily though the author of this book has given the reader valuable insights into the nature of complex manifolds and what is means to deform a complex structure. Complex manifolds are different from real manifolds due to the notion of holomorphicity, but are similar in the sense that they are constructed from domains that are "glued together". In complex manifolds, the "glue" is provided by biholomorphic maps between the domains, the latter of which are open sets called `polydisks'. A `deformation' of the complex manifold is then considered to be a glueing of the same polydisks but via a different identification. For an n-dimensional complex manifold, the maps could thus be dependent on say m parameters, which are labeled as "t" by the author. This dependence on t would result in a differentiable family of complex manifolds. One thus expects the complex manifold to be dependent on t, but the author discusses a counterexample that indicates that one must not be cavalier about this approach.

The definition that is arrived at involves letting t be an element of a domain B in m-dimensional Euclidean space, and considering a collection of compact complex n-dimensional manifolds that depends on t. This collection will be a `differentiable family' if: 1. There exists a differentiable manifold M and a C-infinity map W from M onto B such that the rank of the Jacobian matrix of W is equal to m at every point of M. 2. M(t), the inverse image of t under W is a compact connected subset of M, and in fact is equal to a member of the collection. 3. M has a locally finite open covering along with smooth coordinate functions on the covering that have non-empty intersection with each member of the covering. Beginning with an initial element of B, each member of the inverse image of t under W is viewed as a deformation of the initial member. The crucial point made by the author is that the restricting the domain of the parameter t to a sufficiently small interval allows the representation of the member M(t) as a union of polydisks that are independent of t. Therefore only the coordinate transformations depend on t, and thus only the way of glueing the polydisks depends on t.

To show that these constructions are meaningful, namely that the complex structure of M(t) actually depends on t, the author studies the case of m = 1. In the process he constructs the infinitesimal deformation of M(t), and interprets it as the derivative of the complex structure of M(t) with respect to t. He also shows that the infinitesimal deformation does not depend on the choice of systems of local coordinates, and that the infinitesimal deformation vanishes when M(t) does not vary with t. The author then defines, using a notion of equivalence between two differentiable families, a differentiable family (M, B, W) to be `trivial' if it is equivalent to a product (M x B, B, P). Restricting this triviality to a subdomain gives a notion of `local triviality', which implies immediately that each M(t) will be biholomorphically equivalent to a fixed M. He then shows that if the infinitesimal deformation vanishes then the differentiable family is locally trivial. A more substantial statement of this result is encapsulated in the Frolicher-Nijenhuis theorem, which follows from the results that the author proves in the book. These results involve the theory of strongly elliptic differential operators and considerations of the first cohomology group of M(t) with coefficients in the sheaf of germs of holomorphic vector fields over M(t).

The case of a complex analytic family of compact complex manifolds entails that B will be domain in complex n-space. The author shows that a complex analytic family will be trivial if it is trivial as a differentiable family. As expected, because of the nature of analyticity learned from the theory of complex variables, the proof of these results involves the theory of harmonic differential forms. The author gives these proofs in detail in the book. He also considers the question whether if given an element b of the first cohomology group of a compact complex manifold Mwith coefficients in the sheaf of germs of holomorphic vector fields over M, one can find a complex analytic family that takes M as its initial element and the derivative equal to b. This question, as expected, involves the use of obstruction theory, which the author develops in great detail. In these considerations, the reader will see the and origin and role of the moduli of complex structures. These are essentially the number of parameters m, as long as the complex analytic family is `complete.' ... Read more

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This textbook is an introduction to the classical theory of functions of a complex variable. The author's aim is to explain the basic theory in an easy to understand and careful way. He emphasizes geometrical considerations, and, to avoid topological difficulties associated with complex analysis, begins by deriving Cauchy's integral formula in a topologically simple case and then deduces the basic properties of continuous and differentiable functions. The remainder of the book deals with conformal mappings, analytic continuation, Riemann's mapping theorem, Riemann surfaces and analytic functions on a Riemann surface. The book is profusely illustrated and includes many examples. Problems are collected together at the end of the book. It should be an ideal text for either a first course in complex analysis or more advanced study. ... Read more

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This is the translation of the Japanese textbook for the grade 11 course, "Algebra and Geometry", which is one of three electives offered at this level in Japanese high schools. The book gives an extensive treatment of plane and solid coordinate geometry, vectors, and matrices. ... Read more

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An algebraic approach to geometry.
This text, which Japanese students study simultaneously with Basic Analysis:Japanese Grade 11, demonstrates the power and utility of algebraic techniques in the study of geometry.Its exercises, which require sophisticated algebra skills that you should acquire by first working through its prerequisite text Mathematics 1:Japanese Grade 10, require ingenuity to solve.Working through this text will give you a solid foundation for courses in calculus, multivariable calculus, and linear algebra.

The exposition is clear.After definitions and examples are introduced, demonstration problems are given.These demonstration problems are followed by problems embedded within the text in order to test your understanding of the material.There are also exercises at the end of each section and of each chapter.Answers are provided only for the chapter exercises.Therefore, you will have to check your own answers by substituting them back into the problem, which is good practice.The exercises are quite challenging.Working through them will enhance your understanding of the material and help you develop problem-solving skills that will prove useful in future mathematics courses and contests.

The text begins with a discussion of conic sections.Analytic geometry is used to derive equations for parabolas, ellipses, and hyperbolas.With the exception of a discussion of how to create an ellipse from a mixed dilation of a circle, much of the material is covered in American pre-calculus texts.However, the problems in this text are more difficult.

The succeeding chapters deal with linear algebra.It is in these chapters on vectors and matrices that you come to appreciate just how useful algebra can be in the study of geometry and how powerful the concepts of a vector and a matrix are.The problems in these chapters will test your ingenuity.

Vectors are initially introduced in the plane.Vectors in space are covered later in the text.The text covers the definition of a vector, operations on vectors, including the inner (dot) product, orthogonal vectors, the parametric form of a vector, the use of vectors in describing geometric figures including lines, circles, planes, and spheres, how to determine the distance of a point from a line or plane, and applications of vectors in physics.

The coverage of matrices includes their definition, operations on matrices, inverse matrices, and determinants.Matrices are applied to the solution of simultaneous linear equations.While all of the above material on matrices is taught in pre-calculus courses, this text also discusses linear transformations, the matrix of a linear transformation, composite and inverse transformations, and mappings of the plane.

I highly recommend this text to prospective mathematics majors.You would benefit from working through Mathematics 1: Japanese Grade 10 (Mathematical World, V. 8), this text, and Basic Analysis: Japanese Grade 11 (Mathematical World, V. 11). If you find the problems in this text intractable even after working through Mathematics 1:Japanese Grade 10, put this text aside, work through Mathematics 2: Japanese Grade 11 (Mathematical World), and then return to this text.It is worth the effort. ... Read more

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This volume serves as an introduction to the Kodaira-Spencer theory of deformations of complex structures. Based on notes taken by James Morrow from lectures given by Kunihiko Kodaira at Stanford University in 1965-1966, the book gives the original proof of the Kodaira embedding theorem, showing that the restricted class of Kähler manifolds called Hodge manifolds is algebraic. Included are the semicontinuity theorems and the local completeness theorem of Kuranishi. Readers are assumed to know some algebraic topology. Complete references are given for the results that are used from elliptic partial differential equations. The book is suitable for graduate students and researchers interested in abstract complex manifolds. ... Read more

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This is the translation of the Japanese textbook for the grade 11 course, "Basic Analysis", which is one of three elective courses offered at this level in Japanese high schools. The book includes a thorough treatment of exponential, logarithmic, and trigonometric functions, progressions, and induction method, as well as an extensive introduction to differential and integral calculus. ... Read more

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Provides preparation for a rigorous course in calculus.
This text, which Japanese students study concurrently with Algebra and Geometry:Japanese Grade 11, prepares you well for a rigorous course in calculus.The exposition is clear, although you are expected to perform some algebra to fill in the missing steps in the demonstration problems.The exercises are quite challenging.They require sophisticated algebra skills, which you should develop by first working through the text Mathematics 1:Japanese Grade 10, and some ingenuity.

The text begins by covering exponential, logarithmic, and trigonometric functions.The material covered in these chapters, with the exception of the discussion of the relationship between base 10 logarithms and computation in the decimal numeration system, will be familiar to American students who have studied pre-calculus.However, the problems are more difficult.The text does not discuss natural logarithms, presumably because the natural logarithm of a positive number x is defined in calculus as the area under the curve 1/x between 1 and x.Some proofs and derivations are left as exercises to the reader.

The following chapter on sequences and series covers arithmetic and geometric sequences and series, recursive definition, how to obtain explicit formulas for recursively defined sequences, how to find formulas for a sequence from the progression of differences of its terms, and proofs by mathematical induction.The problems in this chapter are particularly difficult.

The remaining chapters introduce differential and integral calculus, with attention restricted to polynomial functions.While this limits the scope of the coverage, you will get a sense of the power of calculus techniques and the utility of calculus from working through this material.The topics discussed include limits, derivatives, equations of tangent lines, increasing and decreasing functions, maxima and minima, velocity, equations and inequalities, antiderivatives, definite integrals, area, volume, and distance.

I should warn you that Demonstration 3 in the section on limits contains a step that looks like magic.As the text states, since the limit of the denominator x - 3 is 0 when x approaches 3 and the limit of the quotient exists, the limit of the numerator as x approaches 3 must also be zero.Therefore, the authors assume that you realize that x - 3 is a factor of the numerator.Consequently, you are expected to divide the numerator by x - 3, and realize that the remainder, an expression involving the constants a and b, equals zero.This process yields their substitution of -3a - 9 for b.

Fortunately, the exposition elsewhere in the text is much clearer than in the example I cited above.While other demonstration problems require you to do some algebra to fill in the omitted steps, it is clear what you have to do.The demonstration problems are followed by exercises embedded within the text so that you can check your understanding of the material.There are also exercises at the end of each section and the end of each chapter.Answers are provided only for the exercises at the end of each chapter, so you have to check your own answers, which is good practice.The exercises require you to think hard and should serve as good preparation both for a rigorous course in calculus such as that offered by Apostol's text and for mathematics contests.

I highly recommend this text to prospective mathematics majors.You will benefit from working through Mathematics 1: Japanese Grade 10 (Mathematical World, V. 8), this text, and Algebra and Geometry: Japanese Grade 11 (Mathematical World, V. 10).If you find the problems in this text intractable even after working through Mathematics 1:Japanese Grade 10, put this text aside,work through the text Mathematics 2: Japanese Grade 11 (Mathematical World) in order to familiarize yourself with the material in this text, and then return to this text.It is worth the effort.

GOOD FOR STUDY
If you want to study basic mathematics or know how Japanese student study, you should buy this book. ... Read more

Product Description
This is the translation from the Japanese textbook for the grade 10 course, "Basic Mathematics". The book covers the material which is a compulsory for Japanese high school students. The course comprises algebra (including quadratic functions, equations, and inequalities), trigonometric functions, and plane coordinate geometry. ... Read more

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A good way for algebraically inclined students to prepare for calculus.
This clearly written concise text provides the reader with a good foundation for calculus. The problems in the text are challenging and require sophisticated algebra skills. The text covers real and complex numbers, set theory, logic, algebraic proofs, polynomial and rational expressions, quadratic equations, linear and nonlinear systems of equations, inequalities, coordinate systems, equations of lines and circles, planar regions represented by inequalities, quadratic functions, rational functions, square root functions, trigonometric ratios, and applications of trigonometry.

Concepts are illustrated with examples and demonstration problems. The demonstration problems are followed by problems embedded within the text so that the reader can test her/his understanding of the material. In addition, there are exercises at the end of each section and chapter that require the reader to explore each topic further. The exercises are thought-provoking. Doing them will enhance the reader's algebra skills and provide the reader with a thorough understanding of the material.

The text was written for Japanese students, so there are a few differences in terminology and the prerequisites are not necessarily those covered in American algebra classes. The prerequisite material for this volume from grades 7, 8, and 9 of this series, also edited by Kodaira, is available through the University of Chicago School Mathematics Project.

I do have a few caveats. The authors rarely state explicitly that division by zero is not permitted. In addition, the reader may find it necessary to fill in omitted steps in the demonstration problems. While answers are provided to the chapter exercises, no answers are provided for the other problems. Consequently, the reader will have to check her/his work by making sure that the answer satisfies the given conditions, which is good practice.

That said, I think this is a valuable text. Concepts are covered in depth. For instance, rather than just giving the midpoint formula, a formula is derived for the coordinates of a point c which splits an interval [a, b] in the ratio m:n. The midpoint formula follows by setting m:n = 1:1. The demonstration problems illustrate different techniques for obtaining a solution. The exercises force the reader to decide how best to use the given information to solve the problem. In short, the book provides excellent mathematical training.

A good way to supplement this text would be to work through the texts in the Gelfand School Outreach Program. They include The Method of Coordinates, Algebra, Functions and Graphs (Dover Books on Mathematics), and Trigonometry. This text is the prerequisite for Mathematics 2: Japanese Grade 11 (Mathematical World), Algebra and Geometry: Japanese Grade 11 (Mathematical World, V. 10), and Basic Analysis: Japanese Grade 11 (Mathematical World, V. 11). ... Read more

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This is the translation from the Japanese textbook for the grade 11 course, "General Mathematics". It is part of the easier of the three elective courses in mathematics offered at this level and is taken by about 40% of students. The book covers basic notions of probability and statistics, vectors, exponential, logarithmic, and trigonometric functions, and an introduction to differentiation and integration. ... Read more

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Provides a challenging course in general mathematics.
This Japanese textbook provides a challenging general mathematics course.It is designed for an elective course taken by eleventh grade Japanese students who do not wish to prepare for rigorous courses in calculus and probability and statistics.It covers a variety of topics, including exponential, logarithmic, and trigonometric functions, vectors, sequences and series, differential and integral calculus, probability and statistics, and computer flowcharts.The coverage of each topic is necessarily narrow, but it is deep enough to provide the reader with numerous challenging problems.

The treatment of exponential, logarithmic, and trigonometric functions is cursory.These functions are explored in considerably greater depth in American pre-calculus texts.Still, there is some interesting material relating base 10 logarithms to the decimal representation of numbers.

Vectors are covered in greater depth.In addition to discussing operations on vectors, the text delves into conditions for parallelism and perpendicularity, parametric equations of lines, the distance between a point and a line, and applications to figures.

The following chapter on progressions addresses arithmetic and geometric sequences and series, summation notation, and recursive definition.The latter topic is a source for some challenging problems.

Topics from differential and integral calculus are introduced.Attention is restricted to polynomial functions.While this limits the scope of what can be covered, the text does address velocity, linear motion, the tangent line problem, computing areas and volumes, increasing and decreasing functions, and local extrema.

Probability and statistics are introduced through a discussion of basic combinatorics, including permutations and combinations.This leads to a discussion of basic probabilistic principles, including sample spaces, complementary events, mutually exclusive events, conditional probability, and independent events.The succeeding discussion of random distributions, expected value, variance, standard deviation, and the normal distribution leads to a discussion of statistics.There are some notational problems at the end of the chapter of probability that are clarified in the chapter on statistics, in which sampling is discussed.

The final chapter, which is somewhat dated, discusses how simple calculators and computer programs work.The problems in this chapter entail writing computer programs or flowcharts to solve mathematical problems.

There are problems embedded within the text to test your understanding of the material.There are also exercise sets at the end of each section and the end of each chapter.Answers are only provided to the chapter exercises.While there are routine problems, many of the problems are challenging. Some of the problems require some ingenuity to solve.

Some of the topics in this book are covered in more depth in the texts Algebra and Geometry:Japanese Grade 11 and Basic Analysis:Japanese Grade 11.However, other topics including probability, statistics, and computer flowcharts are not addressed in those texts since students who take the simultaneous courses based on those texts have the option of taking both a rigorous course on calculus and a rigorous course on probability and statistics in twelfth grade.

If you are a student planning to study mathematics or a related field in college, this text is somewhat superfluous.After studying pre-calculus, you could work through the through the text Mathematics 1: Japanese Grade 10 (Mathematical World, V. 8), and then study the texts Algebra and Geometry: Japanese Grade 11 (Mathematical World, V. 10) and Basic Analysis: Japanese Grade 11 (Mathematical World, V. 11).That will provide you with a strong foundation for studying mathematics in college.Teachers of high school mathematics would benefit from using this text as a source of enrichment problems for their honors pre-calculus students. ... Read more

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The articles in this volume cover some developments in complex analysis and algebraic geometry. The book is divided into three parts. Part I includes topics in the theory of algebraic surfaces and analytical surfaces. Part II covers topics in moduli and classification problems, as well as structure theory of certain complex manifolds. Part III is devoted to various topics in algebraic geometry analysis and arithmetic. A survey article by Ueno serves as an introduction to the general background of the subject matter of the volume. The volume was written for Kunihiko Kodaira, on the occasion of his sixtieth birthday, by his friends and students. Professor Kodaira was one of the world's leading mathematicians in algebraic geometry and complex manifold theory: and the contributions reflect those concerns. ... Read more

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Chapters: Heisuke Hironaka, Goro Shimura, Teiji Takagi, Seki Kōwa, Toshikazu Sunada, Yozo Matsushima, Kunihiko Kodaira, Kenjiro Shoda, Kiyoshi Itō, Yutaka Taniyama, Shokichi Iyanaga, Toshiyuki Kobayashi, Tadatoshi Akiba, Hōjō Tokiyuki, Daihachiro Sato, Shigefumi Mori, Kenkichi Iwasawa, Shizuo Kakutani, Tosio Kato, Shoshichi Kobayashi, Mikio Sato, Tetsuya Miyamoto, Michio Suzuki, Toru Kumon, Kazuya Kato, Masaki Kashiwara, Taro Morishima, Masahiko Fujiwara, Yoshio Mikami, Masayoshi Nagata, Hiroshi Haruki, Mitsuhiro Shishikura, Kiyoshi Oka, Wada Nei, Yasumasa Kanada, Joichi Suetsuna, Tadashi Nakayama, Takebe Kenko, Hiraku Nakajima, Hidehiko Yamabe, Kiiti Morita, Kikuo Takano, Gisiro Maruyama, Arima Yoriyuki, Aida Yasuaki, Kōsaku Yosida, Tomio Kubota, Michio Kuga, Tadao Tannaka, Michio Jimbo, Shikao Ikehara, Nobuo Yoneda, Hiroshi Okamura, Ajima Naonobu, Katsuya Eda, Sōichi Kakeya, Kambei Mori, Yoshida Koyu, Tsuruichi Hayashi, Kazuoki Azuma, Yoichi Miyaoka, Goro Azumaya, Hiroshi Toda. Source: Wikipedia. Pages: 167. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Seki Takakazu , 1642 December 5, 1708), also known as Seki Kwa ), was a Japanese mathematician in the Edo period. Seki laid foundations for the subsequent development of Japanese mathematics known as wasan; and he has been described as Japan's "Newton." He created a new algebraic notation system, and also, motivated by astronomical computations, did work on infinitesimal calculus and Diophantine equations. A contemporary of Gottfried Leibniz and Isaac Newton, Seki's work was independent. His successors later developed a school dominant in Japanese mathematics until the end of the Edo era. While it is not clear how much of the achievements of wasan are actually Seki's, since many of them appear only in writings with his pupils, some of t...More: http://booksllc.net/?id=532653 ... Read more

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In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions P?, where ? is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach of differential calculus to solving a problem with constraints. One can think of a structure that is not completely rigid, and that deforms slightly to accommodate forces applied from outside; this explains the name.Some characteristic phenomena are: the derivation of first-order equations by treating the ? quantities as having negligible squares; the possibility of isolated solutions, in that varying a solution may not be possible, or does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. ... Read more

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This digital document, covering the life and work of Kunihiko Kodaira, is an entry from Contemporary Authors, a reference volume published by Thompson Gale. The length of the entry is 1753 words. The page length listed above is based on a typical 300-word page. Although the exact content of each entry from this volume can vary, typical entries include the following information:

• Place and date of birth and death (if deceased)
• Family members
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• Professional associations and honors
• Employment
• Writings, including books and periodicals
• A description of the author's work
• References to further readings about the author

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Les achats comprennent une adhésion à l'essai gratuite au club de livres de l'éditeur, dans lequel vous pouvez choisir parmi plus d'un million d'ouvrages, sans frais. Le livre consiste d'articles Wikipedia sur : Kunihiko Kodaira, Michio Morishima, Masahiko Fujiwara, Kenkichi Iwasawa, Kiyoshi Itō, Yutaka Taniyama, Mikio Satō, Teiji Takagi, Heisuke Hironaka, Kowa Seki, Masayoshi Nagata, Kiyoshi Oka, Gorō Shimura, Yasumasa Kanada. Non illustré. Mises à jour gratuites en ligne. Extrait : Kunihiko Kodaira (小平 邦彦, 16 mars 1915 - 26 juillet 1997) était un mathématicien japonais connu pour son travail en géométrie algébrique et en théorie des variétés complexes et aussi en tant que fondateur de l'école japonaise de géomètrie algébrique. Il a reçu une médaille Fields en 1954, le premier Japonais à avoir cet honneur. Il est né à Tōkyō. Ses premiers travaux étaient pour la plupart en analyse fonctionnelle. Pendant les années de guerre, il a travaillé seul, mais a pu maîtriser la théorie de Hodge. Il a écrit son doctorat sur ce sujet, soutenu en 1949. Il avait été impliqué dans le travail cryptographique en 1944, tout en ayant un poste académique à Tōkyō. En 1949, il a fait un séjour à l'IAS à Princeton, à l'invitation de Hermann Weyl. À ce moment-là, les bases de la théorie de Hodge étaient mises en conformité avec la technique contemporaine de la théorie des opérateurs. Kodaira s'est rapidement mis à exploiter les nouveaux outils ainsi forgés en géométrie algébrique, ajoutant la théorie des faisceaux lorsqu'elle est devenue disponible. Ce travail a été particulièrement influent, par exemple sur Hirzebruch. Dans une deuxième série de recherches, Kodaira a écrit une longue suite d'articles en collaboration avec D. C. Spencer, fondant la théorie des déformations des structures complexes sur les variétés. Ceci a donné la possibilité de construire les espaces de modules, puisqu'en général de telles structures dépendent des paramètres continu...http://booksllc.net/?l=fr ... Read more

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This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 725 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

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Teiji Takagi one of the leading number theorists of this century, is most renowned as the founder of class field theory. This volume reflects the stages of his development of this theory. Inspired by a genial idea related to analytic number theory, he developed a beautiful general theory of abelian extensions of algebraic number fields which he addressed at the ICM 1920 at Strasbourg. This report ends with a problem to generalize the results to the case of normal, not necessarily abelian extensions. Up to now this problem has stimulated research. This second edition incorporates the whole contents of "The Collected Papers of Teiji Takagi" edited by S. Kuroda, published by Iwanami Shoten in 1974. Following additions have been made: Note on Eulerian squares (1946).- Concept of numbers.- K. Iwasawa: On arithmetical papers of Takagi.- K. Yosida: On analytical papers of Takagi.- S. Iyanaga: On life and works of Takagi. ... Read more