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Record Nr. |
UNINA9911107078803321 |
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Autore |
Göbel R (Rüdiger), <1940-> |
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Titolo |
Approximations and endomorphism algebras of modules / / by Rudiger Gobel and Jan Trlifaj |
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Pubbl/distr/stampa |
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Berlin ; ; New York, : Walter de Gruyter, 2006 |
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ISBN |
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9786612194832 |
9781282194830 |
1282194836 |
9783110199727 |
3110199726 |
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Edizione |
[1st ed.] |
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Descrizione fisica |
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1 online resource (664 p.) |
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Collana |
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De Gruyter expositions in mathematics ; ; 41 |
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Classificazione |
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Altri autori (Persone) |
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Disciplina |
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Soggetti |
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Modules (Algebra) |
Moduli theory |
Approximation theory |
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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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Description based upon print version of record. |
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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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Front matter -- Contents -- Chapter 1. Some useful classes of modules -- Chapter 2. Approximations of modules -- Chapter 3. Complete cotorsion pairs -- Chapter 4. Deconstruction of cotorsion pairs -- Chapter 5. Tilting approximations -- Chapter 6. 1-tilting modules and their applications -- Chapter 7. Tilting approximations and the finitistic dimension conjectures -- Chapter 8. Cotilting modules -- Chapter 9. The Black Box and its relatives -- Chapter 10. Independence results for cotorsion pairs -- Chapter 11. The lattice of cotorsion pairs -- Chapter 12. Realizing algebras - by algebraically independent elements and by prediction principles -- Chapter 13. E(R)-algebras -- Chapter 14. Modules with distinguished submodules -- Chapter 15. Some useful classes of algebras -- Backmatter |
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Sommario/riassunto |
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The category of all modules over a general associative ring is too |
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complex to admit any reasonable classification. Thus, unless the ring is of finite representation type, one must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions and these are generally viewed as obstacles to the classification. Realization theorems have thus become important indicators of the non-classification theory of modules. In order to overcome this problem, approximation theory of modules has been developed over the past few decades. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by ones from C. These approximations are neither unique nor functorial in general, but there is always a rich supply available appropriate to the requirements of various particular applications. Thus approximation theory has developed into an important part of the classification theory of modules. In this monograph the two methods are brought together. First the approximation theory of modules is developed and some of its recent applications, notably to infinite dimensional tilting theory, are presented. Then some prediction principles from set theory are introduced and these become the principal tools in the establishment of appropriate realization theorems. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory. |
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2. |
Record Nr. |
UNINA9911143995303321 |
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Autore |
Fox William P. <1949-> |
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Titolo |
Advanced problem solving with Maple : a first course / / William P. Fox and William C. Bauldry |
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Pubbl/distr/stampa |
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Boca Raton : , : Taylor & Francis, CRC Press, , 2020 |
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ISBN |
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0-429-89134-2 |
0-429-89135-0 |
0-429-46963-2 |
9780429469633 |
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Edizione |
[1st ed.] |
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Descrizione fisica |
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1 online resource (347 pages) |
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Disciplina |
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Soggetti |
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Problem solving - Data processing |
Quantitative research - Data processing |
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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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Nota di contenuto |
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Cover -- Half Title -- Title Page -- Copyright Page -- Table of Contents -- Preface -- 1: Introduction to Problem Solving and Maple -- 1.1 Problem Solving -- 1.2 Introduction to Maple -- 1.3 The Structure of Maple -- 1.4 General Introduction to Maple -- 1.5 Maple Training -- 1.6 Maple Applications Center -- 2: Introduction, Basic Concepts, and Techniques in Problem Solving with First-Order, Ordinary Differential Equations -- 2.1 Introduction -- 2.2 Applied First-Order Differential Equations and Solution Methods -- 2.3 Slope Fields and Qualitative Assessments -- 2.4 Analytical Solution of First-Order Ordinary Differential Equations -- 2.5 First-Order Ordinary Differential Equations and Maple -- 2.6 Numerical Methods for First-Order Ordinary Differential Equations -- 3: Introduction, Basic Concepts, and Techniques in Problem Solving with Systems of Ordinary Differential Equations -- 3.1 Systems of Differential Equations -- 3.2 Applied Systems of Differential Equations -- 3.3 Phase Portraits and Qualitative Assessment -- 3.4 Solving Homogeneous and Nonhomogeneous Systems of ODEs -- 3.5 Numerical Solutions to Systems of Ordinary Differential Equations -- 4: Problem Solving with Linear, Integer, and Mixed Integer Programming -- 4.1 Formulating Linear Programming |
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Problems -- 4.2 Understanding Two-Variable Linear Programming: A Graphical Simplex -- 4.3 Solving the Linear Program: The Simplex Method and Maple -- 4.4 Linear Programming with Maple's Commands -- 4.5 Sensitivity Analysis with Maple -- 4.6 Integer and Mixed Integer Problems with Maple -- 5: Model Fitting and Linear Regression -- 5.1 Introduction -- 5.2 The Different Curve Fitting Criterion -- 5.3 Plotting the Residuals for a Least-Squares Fit -- 5.4 Case Studies -- 6: Statistical and Probabilistic Problem Solving with Maple -- 6.1 Introduction -- 6.2 Basic Statistics: Univariate Data. |
6.3 Introduction to Classical Probability -- 6.4 Reliability in Engineering and Business -- 6.5 Case Study: Airlines Overbooking Model -- 6.6 Continuous Probability Models -- 6.7 The Normal Distribution -- 6.8 Confidence Intervals and Hypothesis Testing -- 7: Problem Solving with Simulation -- 7.1 Introduction -- 7.2 Monte Carlo Simulation -- 7.3 Probability and Monte Carlo Simulation Using Deterministic Behavior -- 7.4 Probability and Monte Carlo Simulation Using Probabilistic Behavior -- 7.5 Case Studies: Applied Simulation Models -- Index. |
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Sommario/riassunto |
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Problem Solving is essential to solve real-world problems. Advanced Problem Solving with Maple: A First Course applies the mathematical modeling process by formulating, building, solving, analyzing, and criticizing mathematical models. It is intended for a course introducing students to mathematical topics they will revisit within their further studies. The authors present mathematical modeling and problem-solving topics using Maple as the computer algebra system for mathematical explorations, as well as obtaining plots that help readers perform analyses. The book presents cogent applications that demonstrate an effective use of Maple, provide discussions of the results obtained using Maple, and stimulate thought and analysis of additional applications. Highlights: The book's real-world case studies prepare the student for modeling applications Bridges the study of topics and applications to various fields of mathematics, science, and engineering Features a flexible format and tiered approach offers courses for students at various levels The book can be used for students with only algebra or calculus behind them About the authors: Dr. William P. Fox is an emeritus professor in the Department of Defense Analysis at the Naval Postgraduate School. Currently, he is an adjunct professor, Department of Mathematics, the College of William and Mary. He received his Ph.D. at Clemson University and has many publications and scholarly activities including twenty books and over one hundred and fifty journal articles. William C. Bauldry, Prof. Emeritus and Adjunct Research Prof. of Mathematics at Appalachian State University, received his PhD in Approximation Theory from Ohio State. He has published many papers on pedagogy and technology, often using Maple, and has been the PI of several NSF-funded projects incorporating technology and modeling into math courses. He currently serves as Associate Director of COMAP's Math Contest in Modeling (MCM). |
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