An Introduction to Mathematical Modeling (Dover Books on Computer Science) 1st Edition by Edward A. Bender (PDF)

Description

Employing a practical, “learn by doing” approach, this first-rate text fosters the development of the skills beyond the pure mathematics needed to set up and manipulate mathematical models. The author draws on a diversity of fields — including science, engineering, and operations research — to provide over 100 reality-based examples. Students learn from the examples by applying mathematical methods to formulate, analyze, and criticize models. Extensive documentation, consisting of over 150 references, supplements the models, encouraging further research on models of particular interest. The lively and accessible text requires only minimal scientific background. Designed for senior college or beginning graduate-level students, it assumes only elementary calculus and basic probability theory for the first part, and ordinary differential equations and continuous probability for the second section. All problems require students to study and create models, encouraging their active participation rather than a mechanical approach. Beyond the classroom, this volume will prove interesting and rewarding to anyone concerned with the development of mathematical models or the application of modeling to problem solving in a wide array of applications.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Purchased this as an upper-division student’s foray into mathematical modeling. I rarely leave reviews for literature, especially within the realm of mathematics, as I truly believe that they are all triumphant; however, this one did not suit my needs. My experience with leisurely reading in mathematics is that the actual discussion of the given topic is exceedingly lofty and requires rigorous and intensive interpretation. This was a unique experience insomuch that while I viewed the writing as entirely accessible, the examples and analogies provided were far removed from what I had been led to believe this book would help me to achieve. In short sum, long winded with respect to examplesanalogies with very little insight in actually constructing a modelStill mülling a return for refund or credit.

⭐Material is good overview of modeling methods as they evolved from pure, mathematical methods. The material does not cover current, computationally intensive techniques…but if you master this material, it makes current methods more effective and much more appreciated.

⭐excellent!

⭐It is a simple book, all the the content is very clear, you might understand the basic of mathematical modeling, a good book for undergraduate.

⭐I needed some kind of support for a modeling class that I took, so I decided to buy this book. However, I didn’t have too much information different than the book I was using.

⭐Thanks!

⭐This book is a description (with worked examples and discussions!) of various mathematical modeling methods. I can’t stress enough that this is about truly designing intelligent models for problems; you won’t find the term “machine learning” anywhere in the text. After a brief intro to “what is modeling”, it dives right into various techniques with real worked problems. Early sections build a few basic tools, but most chapters are self-contained. It’s easy to jump into a chapter or specific problem and come out with a full understanding. In terms of difficulty, most of the math is actually straightforward. Any technical undergrad or sharp teen can understand. And if you don’t follow, an appendix on probability fills most gaps. By the end, you’ll have explored a swath of applied models.I want to say again that this text is not about data analysis. Models are fit and backed up with data as appropriate, but the emphasis is on understanding how to construct intelligent models. Your latest neural net might reach the same conclusions, but it will require mountains of data, and you still won’t know why the model is what it is. This really reflects the era of the book. In 1978, computing power was limited, so you invested proportionally more effort into building a great model to minimize required processing. A skill that’s still useful today.Without giving away spoilers, I want to highlight a few of the worked examples. “The Nuclear Missile Arms Race” in Graphical Methods is one of my favorites. Without needing any numerical data, Bender derives a qualitative effect of missile technology and disarmament. His simple figures make crystal clear his argument, much better than many modern distracting graphics. Also relevant these days is “Are Fair Elections Possible?” in Potpurri. While this is one of the more theoretical math heavy sections, I still found it very accessible. It features a logical proof about trying to avoid a dictator in an election with more than two candidates. And it even gives a hypothetical real world example of deciding between contract choices. This is another situation where we can learn something even in absence of data. Finally, close to my heart is “Dynamics of Car Following” in Local Stability Theory. This is a “typical” problem in the book, with a three part description. After laying out the basic problem, it shows a short table of data that we’ll use to build or verify our model. Then it derives a reasonable model from first principles in (physics, biology, economics, etc). Finally, it applies the data, discusses the results, and acknowledges the limitations. No pretentions, just analysis. If you’re interested in self-driving cars, you also can check out my model (with fun graphics!) on github. This book contains dozens of worked problems; you’ll probably identify with several of them.While this is a wonderful insightful book, it doesn’t cover everything under the sun. I mentioned it’s from 1978; modeling has advanced since then. Cheap computing opens up new possibilities, and traps. One great thing about easy computing is exploratory data analysis. If you can visualize the data the right way, the model may become obvious. Also, Bender admits that no single model is deeply developed. He even mentions that he had to cut an “in-depth” chapter, I think it’s still worthwhile. My final tally, 97/100 (5 point scales are an archaic blight on the world). A wide variety of math and science topics, all accessible and modular make this a must read!

⭐Aimed at senior level undergraduates, the first chapter briefly discusses at a high level what mathematical models are, how they¡¯re formulated and rules of thumb as to how to evaluate them. The rest of the book surveys simple to moderately complex models applied to problems taken from a wide variety of disciplines in business, science, and engineering. As a survey course, brevity and breadth take precedent over depth and the examples are watered-down versions of problems taken from a plethora of sources cited throughout the text. However, the problems and models are not too superficial that they don¡¯t retain the essential issues modelers are likely to encounter.Although the book is intended primarily for college seniors and first year graduate students, ¡°Part I: Elementary Methods¡± requires only first year calculus and basic probability whereas ¡°Part II: More Advanced Methods¡± also requires differential equations. Therefore, the book will appeal to various levels.The book is rather dated as is evident by its lack of emphasis on numerical methods and no one should expect to be ready for any serious real world modeling as a result of reading this text alone. However, the book does not pretend to be anything more than what it is and the author cautions that it should merely supplement and not substitute mathematics and science coursework. (I would also add that a few courses in numerical methods and computer science would also be the order of the day.)Although the first chapter outlines a quick four-step process for formulating mathematical models, the author stresses the role of discussion and research behind each high level step. Any attempt to provide detailed cookbook heuristics would be a sham. Professor Bender also makes a good point about addressing the ambiguity associated with complex problems raised by clients. Indeed, two themes that resonate throughout the examples are redefining the problem by clarifying objectives through discussion as well as iteratively refining a model by adding (useful) detail to an initially crude one. If nothing else, iteratively modeling elucidates the subtleties of the problem under discussion.Success as an applied mathematician for industry thus requires excellent interpersonal skills and the author clearly reflects this sentiment by requiring group discussions for the book¡¯s exercises containing vaguely stated and open-ended problems having multiple answers. He also notes the crucial role the applied mathematician must play in helping a client clarify his/her objectives.A must read for any aspiring industrial mathematician.

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