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  • Algebraic Geometry
    Algebraic Geometry

    Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P.Serre and A. Grothendieck in Paris. After receiving his Ph.D. from Princeton in 1963, Hartshorne became a Junior Fellow at Harvard, then taught there for several years.In 1972 he moved to California where he is now Professor at the University of California at Berkeley.He is the author of "Residues and Duality" (1966), "Foundations of Projective Geometry (1968), "Ample Subvarieties of Algebraic Varieties" (1970), and numerous research titles.His current research interest is the geometry of projective varieties and vector bundles.He has been a visiting professor at the College de France and at Kyoto University, where he gave lectures in French and in Japanese, respectively.Professor Hartshorne is married to Edie Churchill, educator and psychotherapist, and has two sons.He has travelled widely, speaks several foreign languages, and is an experienced mountain climber.He is also an accomplished amateur musician: he has played the flute for many years, and during his last visit to Kyoto he began studying the shakuhachi.

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  • Algebraic Topology
    Algebraic Topology

    In most mathematics departments at major universities one of the three or four basic first-year graduate courses is in the subject of algebraic topology.This introductory textbook in algebraic topology is suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises. The four main chapters present the basic material of the subject: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally.The author emphasizes the geometric aspects of the subject, which helps students gain intuition.A unique feature of the book is the inclusion of many optional topics which are not usually part of a first course due to time constraints, and for which elementary expositions are sometimes hard to find.Among these are: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and a full exposition of Steenrod squares and powers.Researchers will also welcome this aspect of the book.

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  • Algebraic Number Theory
    Algebraic Number Theory


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  • Complex Algebraic Threefolds
    Complex Algebraic Threefolds

    The first book on the explicit birational geometry of complex algebraic threefolds arising from the minimal model program, this text is sure to become an essential reference in the field of birational geometry.Threefolds remain the interface between low and high-dimensional settings and a good understanding of them is necessary in this actively evolving area.Intended for advanced graduate students as well as researchers working in birational geometry, the book is as self-contained as possible.Detailed proofs are given throughout and more than 100 examples help to deepen understanding of birational geometry.The first part of the book deals with threefold singularities, divisorial contractions and flips.After a thorough explanation of the Sarkisov program, the second part is devoted to the analysis of outputs, specifically minimal models and Mori fibre spaces.The latter are divided into conical fibrations, del Pezzo fibrations and Fano threefolds according to the relative dimension.

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  • What are algebraic fractions?

    Algebraic fractions are fractions where the numerator and/or denominator are algebraic expressions, such as polynomials. They involve variables and can be simplified, added, subtracted, multiplied, and divided just like numerical fractions. Algebraic fractions are commonly used in algebra to solve equations and simplify expressions.

  • What are algebraic equations?

    Algebraic equations are mathematical expressions that contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. These equations are used to represent relationships between different quantities and are solved to find the values of the variables that satisfy the equation. Algebraic equations can be simple, like 2x + 5 = 11, or more complex, involving multiple variables and operations. They are fundamental in algebra and are used in various fields of mathematics and science to model real-world situations and solve problems.

  • What is an algebraic structure?

    An algebraic structure is a set equipped with one or more operations that satisfy certain properties. These properties define how the elements of the set interact with each other under the given operations. Common examples of algebraic structures include groups, rings, and fields, each with their own specific set of rules and properties. Algebraic structures are fundamental in abstract algebra and provide a framework for studying mathematical objects and their relationships.

  • What are algebraic word equations?

    Algebraic word equations are mathematical expressions that use words to describe a problem or situation, and then represent that problem using algebraic symbols and operations. These equations can be used to solve real-world problems by translating the given information into mathematical expressions. By using algebraic word equations, we can represent relationships between different quantities and solve for unknown variables. This allows us to analyze and solve a wide range of problems in various fields such as physics, engineering, and economics.

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  • First Course In Algebraic Geometry And Algebraic Varieties, A
    First Course In Algebraic Geometry And Algebraic Varieties, A

    This book provides a gentle introduction to the foundations of Algebraic Geometry, starting from computational topics (ideals and homogeneous ideals, zero loci of ideals) up to increasingly intrinsic and abstract arguments, such as 'Algebraic Varieties', whose natural continuation is a more advanced course on the theory of schemes, vector bundles, and sheaf-cohomology.Valuable to students studying Algebraic Geometry and Geometry, this title contains around 60 exercises (with solutions) to help students thoroughly understand the theories introduced in the book.Proofs of the results are carried out in full detail.Many examples are discussed in order to reinforce the understanding of both the theoretical elements and their consequences, as well as the possible applications of the material.

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  • Algebraic Combinatorics on Words
    Algebraic Combinatorics on Words

    Combinatorics on words has arisen independently within several branches of mathematics, for instance number theory, group theory and probability, and appears frequently in problems related to theoretical computer science.The first unified treatment of the area was given in Lothaire's book Combinatorics on Words. Originally published in 2002, this book presents several more topics and provides deeper insights into subjects discussed in the previous volume.An introductory chapter provides the reader with all the necessary background material.There are numerous examples, full proofs whenever possible and a notes section discussing further developments in the area.This book is both a comprehensive introduction to the subject and a valuable reference source for researchers.

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  • Introduction to Algebraic Geometry
    Introduction to Algebraic Geometry

    This book presents a readable and accessible introductory course in algebraic geometry, with most of the fundamental classical results presented with complete proofs.An emphasis is placed on developing connections between geometric and algebraic aspects of the theory.Differences between the theory in characteristic $0$ and positive characteristic are emphasized.The basic tools of classical and modern algebraic geometry are introduced, including varieties, schemes, singularities, sheaves, sheaf cohomology, and intersection theory.Basic classical results on curves and surfaces are proved.More advanced topics such as ramification theory, Zariski's main theorem, and Bertini's theorems for general linear systems are presented, with proofs, in the final chapters. With more than 200 exercises, the book is an excellent resource for teaching and learning introductory algebraic geometry.

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  • Elements Of Algebraic Topology
    Elements Of Algebraic Topology

    Elements of Algebraic Topology provides the most concrete approach to the subject.With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorems of point-set topology, this book is perfect for comunicating complex topics and the fun nature of algebraic topology for beginners.

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  • What are algebraic expressions involving fractions?

    Algebraic expressions involving fractions are mathematical expressions that contain variables, constants, and fractions. These expressions can include operations such as addition, subtraction, multiplication, and division of fractions. For example, (3/4)x + (1/2)y - (1/3)z is an algebraic expression involving fractions, where x, y, and z are variables. These expressions are used to represent relationships and solve equations in algebra.

  • How can one simplify algebraic fractions?

    To simplify algebraic fractions, you can start by factoring the numerator and denominator to look for common factors. Then, you can cancel out any common factors from the numerator and denominator. After canceling out the common factors, you can multiply out any remaining factors to simplify the fraction further. Finally, you can check if the fraction can be simplified further by repeating the process until no common factors remain.

  • What is the rule for algebraic transformations?

    The rule for algebraic transformations is that you can perform the same operation on both sides of an equation without changing the equality. This means you can add, subtract, multiply, or divide both sides by the same number or expression. By following this rule, you can simplify equations, solve for unknown variables, and manipulate expressions to make them easier to work with. Just remember to apply the same operation to both sides to maintain the balance of the equation.

  • What is the algebraic sum in mathematics?

    The algebraic sum in mathematics refers to the sum of all the numbers in a given set, taking into account their signs (positive or negative). It involves adding and subtracting numbers based on their sign to determine the overall result. The algebraic sum helps in understanding the net effect of combining different quantities, whether they are positive or negative.

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