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4434Student Solutions Manual for Elementary Differential Equations and Elementary Differential Equations with Boundary Value Problems
https://www.merlot.org/merlot/viewMaterial.htm?id=1209146
This Student Solutions Manual provides worked solutions to the even-numbered problems in Trench's Elementary Differential Equations with Boundary Value Problems.Thu, 28 Jul 2016 19:47:01 -0700Discrete Mathematics: An Open Introduction
https://www.merlot.org/merlot/viewMaterial.htm?id=1209141
This text was written to be used as the primary text for the class Discrete Mathematics (Math 228) at the University of Northern Colorado. The course serves as the role of a transitions course (introduction to proof), as well as an introduction to topics in discrete mathematics. While we have a few students each semester who will go on to study computer science, pure mathematics or applied mathematics, the majority of students are studying to be elementary or secondary math teachers. For this reason, most of the standard discrete textbooks are not appropriate for us. For many years we used Discrete and Combinatorial Mathematics by Richard Grassl and Tabitha Mingus. This is a very nice book in many ways (Grassl taught at UNC) but the print-on-demand publishing was expensive for students and some sections needed updating and (as I saw it) rearranging.While the book began as a set of lecture notes, it now contains a number of features that should support its use as a primary textbook:299 exercises, all with hints or solutions, as well as 45 more involved problems suitable for homework.Investigate! activities throughout the text to support active, inquiry-based learning.A full index and list of symbols.Consistent and (hopefully) helpful page layout and formatting (i.e., examples are easy to identify, important definitions and theorems in boxes, etc.)Thu, 28 Jul 2016 19:28:47 -0700A Course In Commutative Algebra
https://www.merlot.org/merlot/viewMaterial.htm?id=1209133
This is a text for a basic course in commutative algebra, written in accordance with the following objectives. The course should be accessible to those who have studied algebra at the beginning graduate level. For general algebraic background, see my online text “Abstract Algebra: The Basic Graduate Year”, which can be downloaded from my web site www.math.uiuc.edu/∼ r-ash This text will be referred to as TBGY.The idea is to help the student reach an advanced level as quickly and efficiently as possible. In Chapter 1, the theory of primary decomposition is developed so as to apply to modules as well as ideals. In Chapter 2, integral extensions are treated in detail, including the lying over, going up and going down theorems. The proof of the going down theorem does not require advanced field theory. Valuation rings are studied in Chapter 3, and the characterization theorem for discrete valuation rings is proved. Chapter 4 discusses completion, and covers the Artin-Rees lemma and the Krull intersection theorem. Chapter 5 begins with a brief digression into the calculus of finite differences, which clarifies some of the manipulations involving Hilbert and Hilbert-Samuel polynomials. The main result is the dimension theorem for finitely generated modules over Noetherian local rings. A corollary is Krull’s principal ideal theorem. Some connections with algebraic geometry are established via the study of affine algebras. Chapter 6 introduces the fundamental notions of depth, systems of parameters, and M-sequences. Chapter 7 develops enough homological algebra to prove, under appropriate hypotheses, that all maximal M-sequences have the same length. The brief Chapter 8 develops enough theory to prove that a regular local ring is an integral domain as well as a Cohen-Macaulay ring. After completing the course, the student should be equipped to meet the Koszul complex, the AuslanderBuchsbaum theorems, and further properties of Cohen-Macaulay rings in a more advanced course.Thu, 28 Jul 2016 19:11:31 -0700A Course In Algebraic Number Theory
https://www.merlot.org/merlot/viewMaterial.htm?id=1209130
This is a text for a basic course in algebraic number theory, written in accordance with the following objectives. 1. Provide reasonable coverage for a one-semester course. 2. Assume as1. Provide reasonable coverage for a one-semester course. 2. Assume as2. Assume as prerequisite a standard graduate course in algebra, but cover integral extensions and localization before beginning algebraic number theory. For general algebraic background, see my online text “Abstract Algebra: The Basic Graduate Year”, which can be downloaded from my web site www.math.uiuc.edu/∼ r-ash/ The abstract algebra material is referred to in this text as TBGY.3. Cover the general theory of factorization of ideals in Dedekind domains, as well as the number field case.4. Do some detailed calculations illustrating the use of Kummer’s theorem on lifting of prime ideals in extension fields.5. Give enough details so that the reader can navigate through the intricate proofs of the Dirichlet unit theorem and the Minkowski bounds on element and ideal norms.6. Cover the factorization of prime ideals in Galois extensions. 7. Cover local as well as global fields, including the Artin-Whaples approximation theorem and Hensel’s lemmaThu, 28 Jul 2016 18:59:41 -0700Complex Variables
https://www.merlot.org/merlot/viewMaterial.htm?id=1209122
This book represents a substantial revision of the first edition which was published in 1971. Most of the topics of the original edition have been retained, but in a number of instances, the material has been reworked so as to incorporate alternative approaches to these topics that have appeared in the mathematical literature in recent years. The book is intended as a text, appropriate for use by advanced undergraduates or graduate students who have taken a course in introductory real analysis, or as it is often called advanced calculus. No background in complex variables is assumed, thus making the text suitable for those encountering the subject for the first time. It should be possible to cover the entire book in two semesters.Thu, 28 Jul 2016 18:41:39 -0700Calculus with Applications
https://www.merlot.org/merlot/viewMaterial.htm?id=1209119
The course 18.013A as it appears here is intended as a one and a half term course in calculus for students who have studied calculus in high school. It is intended to be self contained, so that it is possible to follow it without any background in calculus, for the adventurous. It makes use of some tools that are relatively new, such as applets, which are intended to make the subject easier to learn and more fun.However it was not our intention to make this course merely an easy calculus course, covering all the same material as the traditional course, but easier to assimilate because of the applets and use of spreadsheets.Modern labor saving devices in practice do not make life easier and simpler. Instead they save us time that allows us to do more in our lives and to make life more complicated and busier than ever.In the same sense it is our hope that these new tools allow the student to learn more and learn more thoroughly with the same amount of, and even perhaps more, effort than before.Thus, the material covered is substantially greater in depth and variety than is normally attempted in a calculus course.In fact, we have attempted to inject new material into almost every chapter of the text as it appears on this site.Why have we done so?In part it is mere human frailty: to keep from going crazy while creating this material.Also, though, it is in order to maintain interest in the subject among those who have already been exposed to it. Finally, it is to show how easy it now is to absorb and use material that not long ago was utterly inaccessible to students, rarely taught, and when taught rarely mastered.Thu, 28 Jul 2016 18:33:44 -0700CK-12 Basic Probability and Statistics
https://www.merlot.org/merlot/viewMaterial.htm?id=1208965
This FlexBook® textbook is an introduction to theoretical probability and data organization. Students learn about events, conditions, random variables, and graphs and tables that allow them to manage data.Thu, 28 Jul 2016 09:01:39 -0700Basic Analysis: Introduction to Real Analysis
https://www.merlot.org/merlot/viewMaterial.htm?id=1208960
This book started its life as my lecture notes for Math 444 at the (UIUC) in the fall semester of 2009, and was later enhanced to teach Math 521 at University of Wisconsin-Madison. (UW-Madison). A prerequisite for the course is a basic proof course. It should be possible to use the book for both a basic course for students who do not necessarily wish to go to graduate school, but also as a first semester of a more advanced course that also covers topics such as metric spaces.Thu, 28 Jul 2016 08:50:59 -0700Applied Probability
https://www.merlot.org/merlot/viewMaterial.htm?id=1208952
Although this is classified as a course, it can also be used as a textbook.This is a "first course" in the sense that it presumes no previous course in probability. The units are modules taken from the unpublished text: Paul E. Pfeiffer, ELEMENTS OF APPLIED PROBABILITY, USING MATLAB. The units are numbered as they appear in the text, although of course they may be used in any desired order. For those who wish to use the order of the text, an outline is provided, with indication of which modules contain the material.The mathematical prerequisites are ordinary calculus and the elements of matrix algebra. A few standard series and integrals are used, and double integrals are evaluated as iterated integrals. The reader who can evaluate simple integrals can learn quickly from the examples how to deal with the iterated integrals used in the theory of expectation and conditional expectation. Appendix Bprovides a convenient compendium of mathematical facts used frequently in this work. And the symbolic toolbox, implementing MAPLE, may be used to evaluate integrals, if desired.In addition to an introduction to the essential features of basic probability in terms of a precise mathematical model, the work describes and employs user defined MATLAB procedures and functions (which we refer to as m-programs, or simply programs) to solve many important problems in basic probability. This should make the work useful as a stand alone exposition as well as a supplement to any of several current textbooks.Most of the programs developed here were written in earlier versions of MATLAB, but have been revised slightly to make them quite compatible with MATLAB 7. In a few cases, alternate implementations are available in the Statistics Toolbox, but are implemented here directly from the basic MATLAB program, so that students need only that program (and the symbolic mathematics toolbox, if they desire its aid in evaluating integrals).Since machine methods require precise formulation of problems in appropriate mathematical form, it is necessary to provide some supplementary analytical material, principally the so-called minterm analysis. This material is not only important for computational purposes, but is also useful in displaying some of the structure of the relationships among events.Thu, 28 Jul 2016 08:44:04 -0700Advanced Calculus
https://www.merlot.org/merlot/viewMaterial.htm?id=1208948
Some of the topics that this book addresses are: Vector spaces; finite-dimensional vector spaces; differential calculus; compactness and completeness; scalar product space; differential equations; multilenear functionals; integration; differentiable manifolds; integral calculus on manifolds; exterior calculus.Thu, 28 Jul 2016 08:37:53 -0700