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Saunders Mac Lane, one of the founders of category theory, wrote this exposition to bring categories to the masses. Mac Lane brings to the fore the important concepts that make category theory useful, such as adjoint functors and universal properties. This purpose of this book is twofold: to provide a general introduction to higher category theory using the formalism of "quasicategories" or "weak Kan complexes" , and to apply this theory to the study of higher versions of Grothendieck topoi.

A few applications to classical topology are included. Contains the first proof that the set of all real numbers is uncountable; also contains a proof that the set of algebraic numbers is countable. See Georg Cantor's first set theory article. First published in , this was the first comprehensive introduction to set theory. Besides the systematic treatment of known results in set theory, the book also contains chapters on measure theory and topology, which were then still considered parts of set theory. Here Hausdorff presents and develops highly original material which was later to become the basis for those areas.

Also, in the process, introduces the class L of constructible sets , a major influence in the development of axiomatic set theory. Cohen's breakthrough work proved the independence of the continuum hypothesis and axiom of choice with respect to Zermelo—Fraenkel set theory. In proving this Cohen introduced the concept of forcing which led to many other major results in axiomatic set theory. Published in , The Laws of Thought was the first book to provide a mathematical foundation for logic. Its aim was a complete re-expression and extension of Aristotle's logic in the language of mathematics.

Boole's work founded the discipline of algebraic logic and would later be central for Claude Shannon in the development of digital logic. Published in , the title Begriffsschrift is usually translated as concept writing or concept notation ; the full title of the book identifies it as " a formula language , modelled on that of arithmetic , of pure thought ". Frege's motivation for developing his formal logical system was similar to Leibniz 's desire for a calculus ratiocinator.

Frege defines a logical calculus to support his research in the foundations of mathematics. Begriffsschrift is both the name of the book and the calculus defined therein. It was arguably the most significant publication in logic since Aristotle. First published in , the Formulario mathematico was the first mathematical book written entirely in a formalized language. It contained a description of mathematical logic and many important theorems in other branches of mathematics.

Many of the notations introduced in the book are now in common use. The Principia Mathematica is a three-volume work on the foundations of mathematics , written by Bertrand Russell and Alfred North Whitehead and published in — It is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic. The questions remained whether a contradiction could be derived from the Principia's axioms, and whether there exists a mathematical statement which could neither be proven nor disproven in the system. The first incompleteness theorem states:.

Provides a detailed discussion of sparse random graphs , including distribution of components, occurrence of small subgraphs, and phase transitions. Presents the Ford-Fulkerson algorithm for solving the maximum flow problem , along with many ideas on flow-based models. See List of important publications in theoretical computer science. See list of important publications in statistics. This book led to the investigation of modern game theory as a prominent branch of mathematics. This work contained the method for finding optimal solutions for two-person zero-sum games.

The zeroth part is about numbers, the first part about games — both the values of games and also some real games that can be played such as Nim , Hackenbush , Col and Snort amongst the many described.

From Geometry to Topology (Dover Books on Mathematics)

A compendium of information on mathematical games. It was first published in in two volumes, one focusing on Combinatorial game theory and surreal numbers , and the other concentrating on a number of specific games. A discussion of self-similar curves that have fractional dimensions between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until Shows Mandelbrot's early thinking on fractals, and is an example of the linking of mathematical objects with natural forms that was a theme of much of his later work.

Method of Fluxions was a book written by Isaac Newton. The book was completed in , and published in Within this book, Newton describes a method the Newton—Raphson method for finding the real zeroes of a function. Major early work on the calculus of variations , building upon some of Lagrange's prior investigations as well as those of Euler. Contains investigations of minimal surface determination as well as the initial appearance of Lagrange multipliers. Kantorovich wrote the first paper on production planning, which used Linear Programs as the model.

He received the Nobel prize for this work in Dantzig's is considered the father of linear programming in the western world. He independently invented the simplex algorithm. Dantzig and Wolfe worked on decomposition algorithms for large-scale linear programs in factory and production planning.

Klee and Minty gave an example showing that the simplex algorithm can take exponentially many steps to solve a linear program. Khachiyan's work on the ellipsoid method. This was the first polynomial time algorithm for linear programming. These are publications that are not necessarily relevant to a mathematician nowadays, but are nonetheless important publications in the history of mathematics. One of the oldest mathematical texts, dating to the Second Intermediate Period of ancient Egypt. It was copied by the scribe Ahmes properly Ahmose from an older Middle Kingdom papyrus.

Lectures on Classical Differential Geometry Second Edition Dover Books on Mathematics

It laid the foundations of Egyptian mathematics and in turn, later influenced Greek and Hellenistic mathematics. Even though it would be a strong overstatement to suggest that the papyrus represents even rudimentary attempts at analytical geometry, Ahmes did make use of a kind of an analogue of the cotangent.

Although the only mathematical tools at its author's disposal were what we might now consider secondary-school geometry , he used those methods with rare brilliance, explicitly using infinitesimals to solve problems that would now be treated by integral calculus. Among those problems were that of the center of gravity of a solid hemisphere, that of the center of gravity of a frustum of a circular paraboloid, and that of the area of a region bounded by a parabola and one of its secant lines.

For explicit details of the method used, see Archimedes' use of infinitesimals. The first known European system of number-naming that can be expanded beyond the needs of everyday life. Contains over theorems of mathematics, assembled by George Shoobridge Carr for the purpose of training his students for the Cambridge Mathematical Tripos exams.

Studied extensively by Ramanujan. One of the most influential books in French mathematical literature. Characterized by an extreme level of rigour, formalism and generality up to the point of being highly criticized for that , its publication started in and is still unfinished today. Written in , it was the first really popular arithmetic book written in the English Language. Textbook of arithmetic published in by John Hawkins, who claimed to have edited manuscripts left by Edward Cocker, who had died in This influential mathematics textbook used to teach arithmetic in schools in the United Kingdom for over years.

An early and popular English arithmetic textbook published in America in the 18th century. The book reached from the introductory topics to the advanced in five sections. The most widely used and influential textbook in Russian mathematics. See Kiselyov page and MAA review. A classic textbook in introductory mathematical analysis , written by G.

It was first published in , and went through many editions.

Mathematics

It was intended to help reform mathematics teaching in the UK, and more specifically in the University of Cambridge , and in schools preparing pupils to study mathematics at Cambridge. The book contains a large number of difficult problems. The content covers introductory calculus and the theory of infinite series. The first introductory textbook graduate level expounding the abstract approach to algebra developed by Emil Artin and Emmy Noether. First published in German in by Springer Verlag. A definitive introductory text for abstract algebra using a category theoretic approach.

Both a rigorous introduction from first principles, and a reasonably comprehensive survey of the field. The first comprehensive introductory graduate level text in algebraic geometry that used the language of schemes and cohomology. Published in , it lacks aspects of the scheme language which are nowadays considered central, like the functor of points.

An undergraduate introduction to not-very-naive set theory which has lasted for decades.

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It is still considered by many to be the best introduction to set theory for beginners. While the title states that it is naive, which is usually taken to mean without axioms, the book does introduce all the axioms of Zermelo—Fraenkel set theory and gives correct and rigorous definitions for basic objects. Where it differs from a "true" axiomatic set theory book is its character: There are no long-winded discussions of axiomatic minutiae, and there is next to nothing about topics like large cardinals. Instead it aims, and succeeds, in being intelligible to someone who has never thought about set theory before.

The nec plus ultra reference for basic facts about cardinal and ordinal numbers. If you have a question about the cardinality of sets occurring in everyday mathematics, the first place to look is this book, first published in the early s but based on the author's lectures on the subject over the preceding 40 years. This book is not really for beginners, but graduate students with some minimal experience in set theory and formal logic will find it a valuable self-teaching tool, particularly in regard to forcing.

It is far easier to read than a true reference work such as Jech, Set Theory. It may be the best textbook from which to learn forcing, though it has the disadvantage that the exposition of forcing relies somewhat on the earlier presentation of Martin's axiom. First published round , this text was a pioneering "reference" text book in topology, already incorporating many modern concepts from set-theoretic topology, homological algebra and homotopy theory. First published in , for many years the only introductory graduate level textbook in the US, teaching the basics of point set, as opposed to algebraic, topology.

Prior to this the material, essential for advanced study in many fields, was only available in bits and pieces from texts on other topics or journal articles. This short book introduces the main concepts of differential topology in Milnor's lucid and concise style. While the book does not cover very much, its topics are explained beautifully in a way that illuminates all their details. An historical study of number theory, written by one of the 20th century's greatest researchers in the field. The book covers some thirty six centuries of arithmetical work but the bulk of it is devoted to a detailed study and exposition of the work of Fermat, Euler, Lagrange, and Legendre.

The author wishes to take the reader into the workshop of his subjects to share their successes and failures. A rare opportunity to see the historical development of a subject through the mind of one of its greatest practitioners. An Introduction to the Theory of Numbers was first published in , and is still in print, with the latest edition being the 6th It is likely that almost every serious student and researcher into number theory has consulted this book, and probably has it on their bookshelf. It was not intended to be a textbook, and is rather an introduction to a wide range of differing areas of number theory which would now almost certainly be covered in separate volumes.

The writing style has long been regarded as exemplary, and the approach gives insight into a variety of areas without requiring much more than a good grounding in algebra, calculus and complex numbers. Escher and composer Johann Sebastian Bach interweave. I tried to reconstruct the central object, and came up with this book.

The World of Mathematics was specially designed to make mathematics more accessible to the inexperienced. It comprises nontechnical essays on every aspect of the vast subject, including articles by and about scores of eminent mathematicians, as well as literary figures, economists, biologists, and many other eminent thinkers. In addition, an informative commentary by distinguished scholar James R. Newman precedes each essay or group of essays, explaining their relevance and context in the history and development of mathematics.

Originally published in , it does not include many of the exciting discoveries of the later years of the 20th century but it has no equal as a general historical survey of important topics and applications. From Wikipedia, the free encyclopedia. The lowest-priced item in unused and unworn condition with absolutely no signs of wear. The item may be missing the original packaging such as the original box or bag or tags or in the original packaging but not sealed. The item may be a factory second or a new, unused item with defects or irregularities. See details for description of any imperfections.

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