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1. |
Record Nr. |
UNICAMPANIAVAN00268617 |
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Autore |
Larson, Loren C. |
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Titolo |
Problem-solving through problems / Loren C. Larson |
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Pubbl/distr/stampa |
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New York, : Springer, 1983 |
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Titolo uniforme |
Problem-solving through problems |
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Descrizione fisica |
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xi, 332 p. : ill. ; 24 cm |
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Soggetti |
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00A07 - Problem books [MSC 2020] |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Nota di contenuto |
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The purpose of this book is to isolate and draw attention to the most important problem-solving techniques typically encountered in undergradu ate mathematics and to illustrate their use by interesting examples and problems not easily found in other sources. Each section features a single idea, the power and versatility of which is demonstrated in the examples and reinforced in the problems. The book serves as an introduction and guide to the problems literature (e.g., as found in the problems sections of undergraduate mathematics journals) and as an easily accessed reference of essential knowledge for students and teachers of mathematics. The book is both an anthology of problems and a manual of instruction. It contains over 700 problems, over one-third of which are worked in detail. Each problem is chosen for its natural appeal and beauty, but primarily to provide the context for illustrating a given problem-solving method. The aim throughout is to show how a basic set of simple techniques can be applied in diverse ways to solve an enormous variety of problems. Whenever possible, problems within sections are chosen to cut across expected course boundaries and to thereby strengthen the evidence that a single intuition is capable of broad application. Each section concludes with "Additional Examples" that point to other contexts where the technique is appropriate. |
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2. |
Record Nr. |
UNINA9910136809303321 |
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Autore |
Eisenbud David |
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Titolo |
Minimal Free Resolutions over Complete Intersections / / by David Eisenbud, Irena Peeva |
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Pubbl/distr/stampa |
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Cham : , : Springer International Publishing : , : Imprint : Springer, , 2016 |
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ISBN |
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Edizione |
[1st ed. 2016.] |
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Descrizione fisica |
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1 online resource (X, 107 p.) |
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Collana |
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Lecture Notes in Mathematics, , 1617-9692 ; ; 2152 |
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Disciplina |
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Soggetti |
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Commutative algebra |
Commutative rings |
Geometry, Algebraic |
Algebra, Homological |
Mathematical physics |
Commutative Rings and Algebras |
Algebraic Geometry |
Category Theory, Homological Algebra |
Theoretical, Mathematical and Computational Physics |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Note generali |
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Bibliographic Level Mode of Issuance: Monograph |
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Nota di bibliografia |
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Includes bibliographical references and index. |
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Nota di contenuto |
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Intro -- Preface -- Acknowledgments -- Contents -- 1 Introduction and Survey -- 1.1 How We Got Here -- Revealing the Pattern -- 1.2 What is a Higher Matrix Factorization? -- Matrix Factorizations of a Sequence of Elements -- 1.3 What's in This Book? -- High Syzygies are Higher Matrix Factorization Modules -- Minimal R-Free and S-Free Resolutions -- Syzygies over Intermediate Quotient Rings -- 1.4 Notation and Conventions -- 2 Matrix Factorizations of One Element -- 2.1 Matrix Factorizations and Resolutions over a Hypersurface -- 3 Finite Resolutions of HMF Modules -- 3.1 The Minimal S-Free Resolution of a Higher Matrix Factorization Module -- 3.2 Consequences -- 3.3 Building a Koszul Extension -- 3.4 Higher Homotopies -- 4 CI Operators -- 4.1 CI Operators -- 4.2 The Action of the CI Operators on Ext -- 4.3 Resolutions with a Surjective CI Operator -- 5 Infinite Resolutions of HMF Modules -- 5.1 The Minimal R-Free |
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Resolution of a Higher Matrix Factorization Module -- 5.2 Betti Numbers -- 5.3 Strong Matrix Factorizations -- 5.4 Resolutions over Intermediate Rings -- 6 Far-Out Syzygies -- 6.1 Pre-stable Syzygies and Generic CI Operators -- 6.2 The Graded Case -- 6.3 The Box Complex -- 6.4 From Syzygies to Higher Matrix Factorizations -- 6.5 Betti Numbers of Pre-stable Matrix Factorizations -- 7 The Gorenstein Case -- 7.1 Syzygies and Maximal Cohen-Macaulay Modules -- 7.2 Stable Syzygies in the Gorenstein Case -- 7.3 Maximal Cohen-Macaulay Approximations -- 7.4 Stable Matrix Factorizations over a Gorenstein Ring -- 8 Functoriality -- 8.1 HMF Morphisms -- References -- Index. |
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Sommario/riassunto |
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This book introduces a theory of higher matrix factorizations for regular sequences and uses it to describe the minimal free resolutions of high syzygy modules over complete intersections. Such resolutions have attracted attention ever since the elegant construction of the minimal free resolution of the residue field by Tate in 1957. The theory extends the theory of matrix factorizations of a non-zero divisor, initiated by Eisenbud in 1980, which yields a description of the eventual structure of minimal free resolutions over a hypersurface ring. Matrix factorizations have had many other uses in a wide range of mathematical fields, from singularity theory to mathematical physics. |
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