Products related to Algebraic:
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Algebraic Topology
Algebraic Topology is an introductory textbook based on a class for advanced high-school students at the Stanford University Mathematics Camp (SUMaC) that the authors have taught for many years.Each chapter, or lecture, corresponds to one day of class at SUMaC.The book begins with the preliminaries needed for the formal definition of a surface.Other topics covered in the book include the classification of surfaces, group theory, the fundamental group, and homology. This book assumes no background in abstract algebra or real analysis, and the material from those subjects is presented as needed in the text.This makes the book readable to undergraduates or high-school students who do not have the background typically assumed in an algebraic topology book or class.The book contains many examples and exercises, allowing it to be used for both self-study and for an introductory undergraduate topology course.
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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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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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Algebraic Geometry II
Several generations of students of algebraic geometry have learned the subject from David Mumford's fabled "Red Book" containing notes of his lectures at Harvard University.Their genesis and evolution are described in the preface as:Initially notes to the course were mimeographed and bound and sold by the Harvard math department with a red cover.These old notes were picked up by Springer and are now sold as the "Red book of Varieties and Schemes".However, every time I taught the course, the content changed and grew.I had aimed to eventually publish more polished notes in three volumes... This book contains what Mumford had then intended to be Volume II.It covers the material in the "Red Book" in more depth with several more topics added.The notes have been brought to the present form in collaboration with T.Oda.
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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.
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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.
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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.
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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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Introduction to Algebraic Geometry
The goal of this book is to provide an introduction to algebraic geometry accessible to students.Starting from solutions of polynomial equations, modern tools of the subject soon appear, motivated by how they improve our understanding of geometrical concepts.In many places, analogies and differences with related mathematical areas are explained.The text approaches foundations of algebraic geometry in a complete and self-contained way, also covering the underlying algebra.The last two chapters include a comprehensive treatment of cohomology and discuss some of its applications in algebraic geometry.
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Lectures On Algebraic Topology
Algebraic Topology and basic homotopy theory form a fundamental building block for much of modern mathematics.These lecture notes represent a culmination of many years of leading a two-semester course in this subject at MIT.The style is engaging and student-friendly, but precise.Every lecture is accompanied by exercises. It begins slowly in order to gather up students with a variety of backgrounds, but gains pace as the course progresses, and by the end the student has a command of all the basic techniques of classical homotopy theory.
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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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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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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.
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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.
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What is the rule for algebraic manipulations?
The rule for algebraic manipulations is to perform the same operation on both sides of the equation in order to maintain equality. This means that when simplifying or solving an equation, you can add, subtract, multiply, or divide both sides by the same number or expression. Additionally, you can also use the distributive property to expand or factor expressions. It's important to remember that whatever operation is performed on one side of the equation, it must also be performed on the other side to maintain the equality.
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How does algebraic square root extraction work?
Algebraic square root extraction involves finding the square root of a number by breaking it down into its prime factors. The process involves identifying perfect squares within the number and simplifying the square root expression by taking out the square roots of those perfect squares. By simplifying the expression in this way, we can find the square root of the original number. This method is particularly useful when dealing with complex numbers or expressions involving variables.
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