Mathematical Association of America
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enOrdinary Differential Equations
http://www.maa.org/publications/ebooks/ordinary-differential-equations
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from Calculus to Dynamical Systems</h1>
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Virginia W. Noonburg</h2>
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TEXTBOOK*</h5>
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<p><strong>Ordinary Differential Equations</strong> is, first and foremost, a text for the introductory course in ordinary differential equations, usually taken by sophomore engineering and science majors after a two or three term calculus sequence. The driving idea behind this particular text is that if all science majors can be convinced to take the differential equations course along with the engineers, they will be in a much better position if they go on to graduate school or even if they just want to read the modern literature in their own field. An understanding of dynamical systems is gradually becoming a necessity for everyone in the sciences.</p>
<p>One way to encourage science majors of all kinds to take a course in differential equations is to show students that problems involving differential equations are often a lot more interesting than the problems seen in calculus. This is especially true in the area of nonlinear equations, and is one reason why this text contains much more than the usual amount of material on the geometry of nonlinear systems.</p>
<p>Recognizing that not all students are fortunate enough to take a differential equations course in their undergraduate career, another aim has been to make this book as readable as possible so it can be used for self-study. Each section of the book also has many examples followed by their worked out solution. Those two things (readability and full solutions to the examples) also make this text a likely candidate for a professor who wants to teach a “flipped” course in differential equations.</p>
<p>Each section of the book has its own set of exercises. Answers to odd-numbered exercises are included in the back of the book.</p>
<p>* As a textbook, <em>Ordinary Differential Equations</em> does have DRM. Our DRM protected PDFs can be downloaded to <strong>three</strong> computers.</p>
<p>Electronic ISBN: 9781614446149</p>
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</div><div class="field field-name-field-ber-category field-type-taxonomy-term-reference field-label-above"><div class="field-label">Publication Category: </div><div class="field-items"><div class="field-item even">Featured</div></div></div><div class="field field-name-field-book-series field-type-taxonomy-term-reference field-label-above"><div class="field-label">Book Series: </div><div class="field-items"><div class="field-item even"><a href="/book-series/maa-textbooks">MAA Textbooks</a></div></div></div>Mon, 17 Mar 2014 21:02:00 +0000bruedi363737 at http://www.maa.orghttp://www.maa.org/publications/ebooks/ordinary-differential-equations#commentsTextbooks, Testing, Training: How We Discourage Thinking
http://www.maa.org/publications/ebooks/textbooks-testing-training-how-we-discourage-thinking
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Stephen S. Willoughby</h2>
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<p><em>Willoughby's essay is a gem. It should be in the hands of every young teacher. I wish that I had read it many years ago. I have no doubt that many of his observations and the information he imparts will remain with me for a while. I certainly hope so. A collection of reminiscences from other teachers with their valuable insights and experiences (who could write with such expertise as he does) would make a fine addition to the education literature.</em> — James Tattersall, Providence College</p>
<p>Stephen S. Willoughby has taught mathematics for 59 years and he has seen everything. Some of it has annoyed him, some has inspired him. This little book is something of a valedictory and he shares some parting thoughts as he contemplates the end of his teaching career. Willoughby has strong, cogent and mostly negative opinions about textbooks, standardized testing, and teacher training. These opinions have been forged in the cauldron of the classroom of a deeply caring teacher. They might not please you, but they ought to make you think. They should spark needed debate in our community. Ultimately this is a human tale with rough parallels to Hardy's Apology; replace "Mathematician's" with "Teacher's" perhaps. Every teacher will sympathize with Willoughby's frustrations and empathize with the humanity and compassion that animated his life's work and that beat at the center of this book.</p>
<p>Print on demand (POD) books are not returnable because they are printed at your request. The POD version of this book is saddle stitched. Damaged books will, of course, be replaced (customer support information will be on your receipt).</p>
<p>Electronic edition ISBN: 9781614448037</p>
<p>Paperbound ISBN: 9780883859148</p>
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</div><div class="field field-name-field-ber-category field-type-taxonomy-term-reference field-label-above"><div class="field-label">Publication Category: </div><div class="field-items"><div class="field-item even">Featured</div></div></div><div class="field field-name-field-book-series field-type-taxonomy-term-reference field-label-above"><div class="field-label">Book Series: </div><div class="field-items"><div class="field-item even"><a href="/book-series/spectrum">Spectrum</a></div></div></div><div class="field field-name-field-ebook-category field-type-taxonomy-term-reference field-label-above"><div class="field-label">Category: </div><div class="field-items"><div class="field-item even"><a href="/ebook-category/new">New</a></div><div class="field-item odd"><a href="/ebook-category/popular-exposition">Popular Exposition</a></div><div class="field-item even"><a href="/ebook-category/resources-for-teachers">Resources for Teachers</a></div><div class="field-item odd"><a href="/ebook-category/teacher-education">Teacher Education</a></div></div></div>Mon, 10 Mar 2014 21:50:58 +0000bruedi352504 at http://www.maa.orghttp://www.maa.org/publications/ebooks/textbooks-testing-training-how-we-discourage-thinking#comments101 Careers in Mathematics
http://www.maa.org/publications/books/101-careers-in-mathematics
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<h3>
Andrew Sterrett, Editor</h3>
<p>3rd Edition<br />
Catalog Code: OCM-3<br />
Print ISBN: 978-0-88385-786-1<br />
Electronic ISBN: 978-1-61444-116-8<br />
334 pp., Paperbound, 2014<br />
List Price: $35.00<br />
MAA Member: $28.00<br />
Series: Classroom Resource Materials</p>
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<p>This third edition of the immensely popular,<em> 101 Careers in Mathematics</em>, contains updates on the career paths of individuals profiled in the first and second editions, along with many new profiles. No career counselor should be without this valuable resource.</p>
<p>The authors of the essays in this volume describe a wide variety of careers for which a background in the mathematical sciences is useful. Each of the jobs presented shows real people in real jobs. Their individual histories demonstrate how the study of mathematics was useful in landing well-paying jobs in predictable places such as IBM, AT&T, and American Airlines, and in surprising places such as FedEx Corporation, L.L. Bean, and Perdue Farms, Inc. You will also learn about job opportunities in the Federal Government as well as exciting careers in the arts, sculpture, music, and television. There are really no limits to what you can do if you are well prepared in mathematics.</p>
<p>The degrees earned by the authors profiled here range from bachelor’s to master’s to PhD in approximately equal numbers. Most of the writers use the mathematical sciences on a daily basis in their work. Others rely on the general problem-solving skills acquired in mathematics as they deal with complex issues.</p>
<!---<h3>
Table of Contents</h3>
<p style="margin-left:20px">
Preface<br />
</p>--->
<h3>
Excerpt: Amanda Quiring, Project Manager (p.206)</h3>
<p>‘Oh, so you’re both good with numbers!’ I’m an accountant and my twin sister is a math teacher and I can’t tell you how many times I’ve heard that statement. Although the comment considerably understates the nature, diversity and complexity of both accounting and mathematics, there is some truth behind it. The skill set that I developed in my math degree has been extremely useful as I have pursued a career as an accountant.</p>
<p>In my role as a project manager at the International Accounting Standards Board, I spend the majority of my time researching complex accounting issues and writing papers with my related analyses and proposed solutions. Without even realizing it, I find myself applying the same approach to my work as I would to a mathematical proof¬—laying out all of the known relevant information, fitting it together in a logical order, and ending up with an understandable solution to a complex problem.</p>
<p>My mathematical background also has been useful in studying academic research related to accounting. While I have seen some people’s eyes glaze over when they flip through the pages of statistical analyses, I am able to approach the research with confidence and with a passion for understanding the mathematical tests performed as well as the accounting implications. My mathematical background has given me a great foundation should I decide to pursue a PhD in accounting.</p>
<p>I would highly recommend a math degree (or a double-major in math and another field) to anyone who is considering a career that requires problem-solving and logic skills. I came really close to not majoring in math. In fact, my first week of college I was so convinced that I wasn't going to pursue a math degree that I dropped out of Calculus III. However, thanks to the persistence of one of my math professors, I signed up for the course again the next year and the rest is history. Even though it was extremely challenging and I had to take summer classes in order to graduate in four years, it is a decision that I never have regretted.</p>
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</div></div></div><div class="field field-name-field-book-series field-type-taxonomy-term-reference field-label-above"><div class="field-label">Book Series: </div><div class="field-items"><div class="field-item even"><a href="/book-series/classroom-resource-materials">Classroom Resource Materials</a></div></div></div>Wed, 05 Mar 2014 15:28:51 +0000swebb349084 at http://www.maa.orghttp://www.maa.org/publications/books/101-careers-in-mathematics#commentsCollege Mathematics Journal Contents—March 2014
http://www.maa.org/publications/periodicals/college-mathematics-journal/college-mathematics-journal-contents-march-2014
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><img alt="" src="/sites/default/files/images/pubs/cmj_march2014.jpg" style="width: 175px; height: 250px; float: left; margin-left: 5px; margin-right: 5px;" /></p>
<p>The March issue of <em>The College Mathematics Journal</em> features an abundance of student authors and several articles on algebraic topics. In that intersection, Arthur Benjamin and Ethan Brown (who was in middle school when they wrote the article) help prepare us for the upcoming Mathematics Awareness Month on "Mathematics, Magic, and Mystery." The cover art resonates with two articles: the quartet Field, Ivison, Reyher, and Warner help you determine the best direction to run from an oncoming truck, while trio Bolt, Meyer, and Visser consider how to run fast without ever breaking a four minute mile (six of those seven authors are students). Keep reading for topics ranging from permutations and matroids to golden triangles and cookies. <span style="line-height: 1.25em;">—</span><em style="line-height: 1.25em;">Brian Hopkins</em><span style="line-height: 1.25em;"> </span></p>
<p>Vol. 45, No. 2, pp. 82-159.</p>
<h5 style="font-size: 18px;">
JOURNAL SUBSCRIBERS AND MAA MEMBERS:</h5>
<p>To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.</p>
<h2 style="font-size: 28px;">
FROM THE EDITOR</h2>
<h3 style="font-size: 24px;">
Anonymity and Youth</h3>
<p>Brian Hopkins</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.082">http://dx.doi.org/10.4169/college.math.j.45.2.082</a></p>
<h2 style="font-size: 28px;">
ARTICLES</h2>
<h3 style="font-size: 24px;">
Power Series for Up-Down Min-Max Permutations</h3>
<p>Fiacha Heneghan and T. Kyle Petersen</p>
<p>Calculus and combinatorics overlap, in that power series can be used to study combinatorially defined sequences. In this paper, we use exponential generating functions to study a curious refinement of the Euler numbers, which count the number of “up-down” permutations of length <em>n</em>.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.083">http://dx.doi.org/10.4169/college.math.j.45.2.083</a></p>
<h3 style="font-size: 24px;">
Challenging Magic Squares for Magicians</h3>
<p>Arthur T. Benjamin and Ethan J. Brown</p>
<p>We present several effective ways for a magician to create a 4-by-4 magic square where the total and some of the entries are prescribed by the audience.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.092">http://dx.doi.org/10.4169/college.math.j.45.2.092</a></p>
<h3 style="font-size: 24px;">
The Fastest Way Not to Run a Four-Minute Mile</h3>
<p>Michael Bolt, Anthony Meyer, and Nicholas Visser</p>
<p>In this manuscript we present the mathematics that is needed to answer three counterintuitive problems related to the averaging of functions. The problems are manifestations of the question, “Is the average rate of change on a given interval determined by the average rate of change on subintervals of a fixed length?” We also ask questions in higher dimensions that may have interesting geometric significance.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.101">http://dx.doi.org/10.4169/college.math.j.45.2.101</a></p>
<h3 style="font-size: 24px;">
Classifying Nilpotent Maps via Partition Diagrams</h3>
<p>Nicholas Loehr</p>
<p>This note uses a visual analysis of partition diagrams to give an elementary, pictorial proof of the classification theorem for nilpotent linear maps. We show that any nilpotent map is represented by a matrix with ones in certain positions on the first super-diagonal and zeroes elsewhere.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.108">http://dx.doi.org/10.4169/college.math.j.45.2.108</a></p>
<h3 style="font-size: 24px;">
PROOF WITHOUT WORDS: Componendo et Dividendo, a Theorem on Proportions</h3>
<p>Yukio Kobayashi</p>
<p>We provide a geometric proof of a classical result on proportions.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.120">http://dx.doi.org/10.4169/college.math.j.45.2.120</a></p>
<h3>
Truck Versus Human: Mathematics Under Pressure</h3>
<p>Elizabeth Field, Rachael Ivison, Amanda Reyher, and Steven Warner</p>
<p>If you are ever faced with an oncoming truck, this paper could save your life. We investigate the optimal path that you should take from the middle of the road to the curb in order to avoid being hit by an oncoming truck. Although your instincts may tell you to run directly toward the curb, it turns out that this path, although the shortest, is not generally the safest.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.116">http://dx.doi.org/10.4169/college.math.j.45.2.116</a></p>
<h3 style="font-size: 24px;">
Proof Without Words: An Infinite Series Using Golden Triangles</h3>
<p>Steven Edwards</p>
<p>We give a visual proof of an infinite series involving the golden ratio.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.120">http://dx.doi.org/10.4169/college.math.j.45.2.120</a> </p>
<h3 style="font-size: 24px;">
Matroids on Groups?</h3>
<p>Jeremy S. LeCrone and Nancy Ann Neudauer</p>
<p>By carefully defining independence, we create two structures on a finite group that satisfy the matroid axioms. Both of these matroids are transversal and graphic, they are duals of each other, and are fundamental transversal matroids. The matroids capture some of the group structure, but two isomorphic matroids may have come from non-isomorphic groups, so we may not be able to recapture the group from the matroid. Our definitions of independent sets cannot be extended in what seems the natural way based on independence of vectors. Finding a definition of independence that satisfies the matroid axioms may not always be possible, though there are always more possibilities on the horizon.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.121">http://dx.doi.org/10.4169/college.math.j.45.2.121</a></p>
<h3>
Cookie Monster Devours Naccis</h3>
<p>Leigh Marie Braswell and Tanya Khovanova</p>
<p>The Cookie Monster wants to empty a set of jars filled with various numbers of cookies. On each of his moves, he may choose any subset of jars and take the same number of cookies from each of those jars. The minimal number of moves to accomplish this depends on the initial distribution of cookies in the jars. We discuss bounds of these Cookie Monster numbers and explicitly find them for jars containing numbers of cookies in the Fibonacci, Tribonacci, and other nacci sequences.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.129">http://dx.doi.org/10.4169/college.math.j.45.2.129</a></p>
<h3 style="font-size: 24px;">
Proof Without Words: The Difference of Consecutive Integer Cubes Is Congruent to 1 Modulo 6</h3>
<p>Claudi Alsina, Roger Nelsen, and Hasan Unal</p>
<p>We prove wordlessly that the difference of consecutive integer cubes is congruent to 1 modulo 6.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.135">http://dx.doi.org/10.4169/college.math.j.45.2.135</a></p>
<h3 style="font-size: 24px;">
A Single Family of Semigroups with Every Positive Rational Commuting Probability</h3>
<p>Michelle Soule</p>
<p>A semigroup is a set with an associative binary operation (which may not contain an identity element). The <em>commuting probability </em>of a semigroup is the probability that two elements chosen at random commute with each other. In this paper, we construct a single family of semigroups which achieves each positive rational in the interval (0, 1] as a commuting probability.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.136">http://dx.doi.org/10.4169/college.math.j.45.2.136</a></p>
<h3 style="font-size: 24px;">
<span style="color: rgb(196, 18, 48); font-size: 28px; line-height: 0.95em;">CLASSROOM CAPSULES</span></h3>
<h3>
On the Differentiation Formulae for Sine, Tangent, and Inverse Tangent</h3>
<p>Daniel McQuillan and Rob Poodiack</p>
<p>We prove the derivative formula for sine via a geometric argument and the symmetric derivative, and then use similar techniques for tangent and inverse tangent.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.140">http://dx.doi.org/10.4169/college.math.j.45.2.140</a></p>
<h3 style="font-size: 24px;">
A Proof for a Quadratic Function without Using Calculus</h3>
<p>Connie Xu</p>
<p>We strengthen a result on tangent lines of parabolas, originally proved with Taylor series, without making use of calculus.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.143">http://dx.doi.org/10.4169/college.math.j.45.2.143</a></p>
<h2 style="font-size: 28px;">
PROBLEMS AND SOLUTIONS</h2>
<p>Problems 1021-1025<br />
Solutions 996-1000</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.145">http://dx.doi.org/10.4169/college.math.j.45.2.145</a></p>
<h2 style="font-size: 28px;">
MEDIA HIGHLIGHTS</h2>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/college.math.j.45.2.152">http://dx.doi.org/10.4169/college.math.j.45.2.152</a></p>
</div></div></div>Tue, 18 Feb 2014 17:24:10 +0000kmerow336550 at http://www.maa.orghttp://www.maa.org/publications/periodicals/college-mathematics-journal/college-mathematics-journal-contents-march-2014#commentsAmerican Mathematical Monthly Contents—March 2014
http://www.maa.org/publications/periodicals/american-mathematical-monthly/american-mathematical-monthly-contents-march-2014
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p class="rtecenter" style="font-size: 14px;"><img alt="" src="/sites/default/files/images/pubs/amm_logo2014.jpg" style="width: 400px; height: 70px;" /></p>
<p>March clearly comes in like a lion in this month’s <em>Monthly</em>. We honor Pi Day with a special article by <em>Monthly</em> Associate Editor Jon Borwein and David Bailey: "Pi Day Is Upon Us Again and We Still Do Not Know if Pi Is Normal." This paper is based on an annual lecture that Professor Borwein gives at his home institution in Australia which has gained a “cult” following. We also honor Joan Leitzel, the winner of the 2014 Gung and Hu Award for distinguished service to mathematics.</p>
<p>In our Notes section, don’t miss Bill Johnston’s student-friendly proof that the weighted Hermite polynomials form a basis for $$L^2(\mathbb{R})$$. Mark Kozek reviews <em>Mathematics in Popular Culture: Essays on Appearances in Film, Fiction, 274 Games, Television and Other Media</em> by Jessica Sklar and Elizabeth Sklar. Need something to do over Spring Break? Take our Problem Section along for the ride. Stay tuned for the April edition when Andrew Simoson shows us when to expect the next transit of Venus. <span style="line-height: 1.25em;">—</span><em style="line-height: 1.25em;">Scott Chapman</em></p>
<div>
<p style="font-size: 14px;">Vol. 121, No. 3, pp.187-278.</p>
<h5 style="font-size: 18px;">
JOURNAL SUBSCRIBERS AND MAA MEMBERS:</h5>
<p style="font-size: 14px;">To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.</p>
<h3>
Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2014 to Joan Leitzel for Distinguished Service to Mathematics</h3>
<p>Kenneth A. Ross and Ann E. Watkins</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.187">http://dx.doi.org/10.4169/amer.math.monthly.121.03.187</a></p>
<h2 style="font-size: 28px;">
ARTICLES</h2>
<h3 style="font-size: 24px;">
Pi Day Is Upon Us Again and We Still Do Not Know if Pi Is Normal</h3>
<p>David H. Bailey and Jonathan Borwein</p>
<div>
<p>The digits of $$\pi$$ have intrigued both the public and research mathematicians from the beginning of time. This article briefly reviews the history of this venerable constant, and then describes some recent research on the question of whether $$\pi$$ is normal, or, in other words, whether its digits are statistically random in a specific sense.</p>
</div>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.191">http://dx.doi.org/10.4169/amer.math.monthly.121.03.191</a> <span style="font-size: 16px; line-height: 18px;">(</span><a href="/sites/default/files/pdf/pubs/BaileyBorweinPiDay.pdf" style="line-height: 18px;"><span class="style5">pdf</span></a><span style="font-size: 16px; line-height: 18px;">)</span></p>
<h3 style="font-size: 24px;">
The Length of Spirographic Curves</h3>
<p>Stephan Berendonk</p>
<p>In this note we will show, by elementary means, that we can express the length of curves drawn with a spirograph in terms of the perimeter of certain ellipses. We will present the proof in the language of Euclidean geometry.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.207">http://dx.doi.org/10.4169/amer.math.monthly.121.03.207</a></p>
<h3>
Martin Gardner’s Minimum No-3-in-a-Line Problem</h3>
<p>Alec S. Cooper, Oleg Pikhurko, John R. Schmitt, and Gregory S.Warrington</p>
<p>In Martin Gardner’s October 1976 Mathematical Games column in <em>Scientific American</em>, he posed the following problem: “What is the smallest number of [queens] you can put on an [$$n\times n$$ chessboard] such that no [queen] can be added without creating three in a row, a column, or, except in the case when $$n$$ is congruent to 3 modulo 4, in which case one less may suffice.” We use the Combinatorial Nullstellensatz to prove that this number is at least $$n$$. A second, more elementary proof is also offered in the case that $$n$$ is even. </p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.213">http://dx.doi.org/10.4169/amer.math.monthly.121.03.213</a></p>
<h3 style="font-size: 24px;">
Level Sets on Disks</h3>
<p>Aleksander Maliszewski and Marcin Szyszkowski</p>
<p>We prove that every continuous function from a disk to the real line has a level set containing a connected component of diameter at least $$\sqrt{3}$$<img src="file:///C:/Users/kmerow/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" />. We also show that if the disk is split into two sets—one open and the other closed—then one of them contains a component of diameter at least $$\sqrt{3}$$<img src="file:///C:/Users/kmerow/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" />.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.222">http://dx.doi.org/10.4169/amer.math.monthly.121.03.222</a></p>
<h3 style="font-size: 24px;">
Series Involving the Zeta Functions and a Family of Generalized Goldbach–Euler Series</h3>
<p>Junesang Choi and Hari M. Srivastava</p>
<p>We first present the corrected expression for a certain widely-recorded generalized Goldbach–Euler series. The corrected forms are then shown to be connected with the problem of closed-form evaluation of series involving the Zeta functions, which happens to be an extensively-investigated subject since the time of Euler as (for example) in the classical three-century old-Goldbach theorem.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.229">http://dx.doi.org/10.4169/amer.math.monthly.121.03.229</a></p>
<h3 style="font-size: 24px;">
Mathbit: A Polynomial Parent to a Fibonacci–Lucas Relation</h3>
<p>B. Sury</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.236">http://dx.doi.org/10.4169/amer.math.monthly.121.03.236</a></p>
<h3 style="font-size: 24px;">
A Wallis Product on Clovers</h3>
<p>Trevor Hyde</p>
<p>The $$m$$-clover is the plane curve defined by the polar equation $$r^{m/2}=\cos{\left(\frac{m}{2}\theta\right)}$$. In this article we extend a well-known derivation of the Wallis product to derive a generalized Wallis product for arc lengths of $$m$$-clovers.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.237">http://dx.doi.org/10.4169/amer.math.monthly.121.03.237</a></p>
<h2 style="font-size: 28px;">
NOTES</h2>
<h3 style="font-size: 24px;">
On Some Power Sums of Sine or Cosine</h3>
<p>Mircea Merca</p>
<p>In this note, using the multisection series method, we establish the formulas for various power sums of sine or cosine functions.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.244">http://dx.doi.org/10.4169/amer.math.monthly.121.03.244</a></p>
<h3>
The Weighted Hermite Polynomials Form a Basis for $$L^{2}(\mathbb{R})$$</h3>
<p>William Johnston</p>
<p>We present a student-friendly proof that the weighted Hermite polynomials form a complete orthonormal system (a basis) for the collection <img src="file:///C:/Users/kmerow/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" />of $$L^{2}(\mathbb{R})$$ real-valued functions.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.249">http://dx.doi.org/10.4169/amer.math.monthly.121.03.249</a></p>
<h3 style="font-size: 24px;">
A Proof of Lie’s Product Formula</h3>
<p>Gerd Herzog</p>
<p>For $$d\times d$$ matrices $$A$$, $$B$$ and entire functions $$f$$, $$g$$ with $$f(0) = g(0) = 1$$, we give an elementary proof of the formula</p>
<p>$$\lim_{k\to\infty}(f(A/k)g)(B/k))^{k}=\exp (f'(0)A+g'(0)B)$$</p>
<p><img src="file:///C:/Users/kmerow/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" /><img src="file:///C:/Users/kmerow/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" /><span style="line-height: 1.25em;">For the case $$f = g = \exp$$, this is Lie’s famous product formula for matrices.</span></p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.254">http://dx.doi.org/10.4169/amer.math.monthly.121.03.254</a></p>
<h3>
Hadamard’s Determinant Inequality</h3>
<p>Kenneth Lange</p>
<p>This note is devoted to a short, but elementary, proof of Hadamard’s determinant inequality.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.258">http://dx.doi.org/10.4169/amer.math.monthly.121.03.258</a></p>
<h3>
Mathbit: Mean of the Mean of the Mean . . .</h3>
<p>S. Kocak and M. Limoncu</p>
<p><span style="line-height: 17.016998291015625px;">To purchase the article from JST</span>OR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.259">http://dx.doi.org/10.4169/amer.math.monthly.121.03.259</a></p>
<h3 style="font-size: 24px;">
A Zorn’s Lemma Proof of the Dimension Theorem for Vector Spaces</h3>
<p>Justin Tatch Moore</p>
<p>This note gives a “Zorn’s Lemma” style proof that any two bases in a vector space have the same cardinality.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.260">http://dx.doi.org/10.4169/amer.math.monthly.121.03.260</a></p>
<h3 style="font-size: 24px;">
A Short Proof of McDougall’s Circle Theorem</h3>
<p>Marc Chamberland and Doron Zeilberger</p>
<p>This note offers a short, elementary proof of a result similar to Ptolemy’s theorem. Specifically, let $$d_{i, j}$$ denote the distance between $$P_{i}$$ and $$P_{j}$$ . Let $$n$$ be a positive integer and $$P_{i}$$ , for $$1\leq i \leq 2n$$, be cyclically ordered points on a circle. If</p>
<p>$$\sum_{i=1}^{n}\frac{1}{R_{2i}}=\sum_{i=1}^{n}\frac{1}{R_{2i-1}}$$ <img src="file:///C:/Users/kmerow/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" /></p>
<p><img src="file:///C:/Users/kmerow/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" /><span style="line-height: 1.25em;">Then</span></p>
<p>$$\sum_{i=1}^{n}\frac{1}{R_{2i}}=\sum_{i=1}^{n}\frac{1}{R_{2i-1}}$$</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.263">http://dx.doi.org/10.4169/amer.math.monthly.121.03.263</a></p>
<h2 style="font-size: 28px;">
PROBLEMS AND SOLUTIONS</h2>
<p>Problems 11761-11767<br />
Solutions 11613, 11620, 11623, 11624, 11625</p>
<p>To purchase from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.266">http://dx.doi.org/10.4169/amer.math.monthly.121.03.266</a></p>
<h2 style="font-size: 28px;">
REVIEWS</h2>
<p><em>Mathematics in Popular Culture: Essays on Appearances in Film, Fiction, Games, Television and Other Media</em>. Edited by Jessica K. Sklar and Elizabeth S. Sklar; Foreword by Keith Devlin.</p>
<p>Reviewed by Mark Kozek</p>
<p>To purchase from <span style="line-height: 1.25em;">JS</span>TOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.03.274">http://dx.doi.org/10.4169/amer.math.monthly.121.03.274</a></p>
</div>
<div>
</div>
</div></div></div>Tue, 18 Feb 2014 16:37:03 +0000kmerow336493 at http://www.maa.orghttp://www.maa.org/publications/periodicals/american-mathematical-monthly/american-mathematical-monthly-contents-march-2014#commentsMathematics Magazine Contents—February 2014
http://www.maa.org/publications/periodicals/mathematics-magazine/mathematics-magazine-contents-february-2014
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><img alt="" src="/sites/default/files/images/pubs/mm-feb14.jpg" style="width: 175px; margin: 5px; float: left; height: 250px;" /></p>
<p>In this issue we have a window into the eighteenth century, as Antonella Cupillari shows us how Maria Gaetana Agnesi graphed rational and algebraic functions in her influential textbook of 1748. We also have a rare hat trick in the Notes section: three new theorems. There is a new way to characterize parabolas that Archimedes would have appreciated, a new way to fill in a rectangle with digits so that each L-shaped triple occurs exactly once, and—illustrated on the cover—a new construction involving a star, a circle, and five concurrent lines. Or, if you prefer, you can start at the back of the issue with the 2013 Putnam solutions. <span style="line-height: 1.25em;">—</span><em style="line-height: 1.25em;">Walter Stromquist</em></p>
<p>Vol. 87, No. 1, pp.2-78.</p>
<h5 style="font-size: 18px;">
</h5>
<h5 style="font-size: 18px;">
JOURNAL SUBSCRIBERS AND MAA MEMBERS:</h5>
<p>To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.</p>
<h2 style="font-size: 28px;">
ARTICLES</h2>
<h3 style="font-size: 24px;">
Maria Gaetana Agnesi’s Other Curves (More Than Just the Witch)</h3>
<p>Antonella Cupillari</p>
<p>Maria Gaetana Agnesi’s name is commonly associated with the curve known as “the Witch of Agnesi.” However, the versiera (or versoria), as Agnesi called it, is only one of many curves she introduced in her mathematical compendium, the <em>Instituzioni Analitiche</em> (1748). Some of the other curves are much more interesting and complex than the versiera, and they are grouped in the lengthy section (pp. 351–415) titled “On the construction of Loci of degree higher than second degree.” This article showcases Agnesi’s presentation of some of them, made without using calculus. Instead, her tools of choice were geometry, algebra, and the method of using easier and already-known curves to build more challenging ones.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.3">http://dx.doi.org/10.4169/math.mag.87.1.3</a></p>
<h3 style="font-size: 24px;">
Maclaurin’s Inequality and a Generalized Bernoulli Inequality</h3>
<p>Iddo Ben-Ari and Keith Conrad</p>
<p>Maclaurin’s inequality is a natural, but nontrivial, generalization of the arithmetic-geometric mean inequality. We present a new proof that is based on an analogous generalization of Bernoulli’s inequality. Applications of Maclaurin’s inequality to iterative sequences and probability are discussed, along with graph-theoretic versions of the Maclaurin and Bernoulli inequalities.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.14.">http://dx.doi.org/10.4169/math.mag.87.1.14</a></p>
<h3 style="font-size: 24px;">
When Can You Factor a Quadratic Form?</h3>
<p>Brian G. Kronenthal and Felix Lazebnik</p>
<p>Consider the problem of determining, without using a computer or calculator, whether a given quadratic form factors into the product of two linear forms. A solution derived by inspection is often highly nontrivial; however, we can take advantage of equivalent conditions. In this article, we prove the equivalence of five such conditions. Furthermore, we discuss vocabulary such as “reducible,” “degenerate,” and “singular” that are used in the literature to describe these conditions, highlighting the inconsistency with which this vocabulary is applied.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.25">http://dx.doi.org/10.4169/math.mag.87.1.25</a></p>
<h2 style="font-size: 28px;">
NOTES</h2>
<h3 style="font-size: 24px;">
Defining Exponential and Trigonometric Functions Using Differential Equations</h3>
<p>Harvey Diamond</p>
<p>This note addresses the question of how to rigorously define the functions exp(<em>x</em>), sin(<em>x</em>), and cos(<em>x</em>), and develop their properties directly from that definition. We take a differential equations approach, defining each function as the solution of an initial value problem. Assuming only the basic existence/uniqueness theorem for solutions of linear differential equations, we derive the standard properties and identities associated with these functions. Our target audience is undergraduates with a calculus background.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.37">http://dx.doi.org/10.4169/math.mag.87.1.37</a></p>
<h3 style="font-size: 24px;">
Some Logarithmic Approximations for $$\pi$$ and $$e$$</h3>
<p>Poo-Sung Park</p>
<p>We offer the approximations $$\pi\approx\log_{5}157$$ and $$e\approx\log_{8}285$$, which may be useful where other logarithms are involved.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.43">http://dx.doi.org/10.4169/math.mag.87.1.43</a></p>
<h3 style="font-size: 24px;">
A 5-Circle Incidence Theorem</h3>
<p>J. Chris Fisher, Larry Hoehn, and Eberhard M. Schröder</p>
<p>We state and prove a surprising incidence theorem that was discovered with the help of a computer graphics program. The theorem involves sixteen points on ten lines and five circles; our proof relies on theorems of Euclid, Menelaus, and Ceva. The result bears a striking resemblance to Miquel’s 5-circle theorem, but as far as we can determine, the relationship of our result to known incidence theorems is superficial.</p>
<p>To purchase from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.44">http://dx.doi.org/10.4169/math.mag.87.1.44</a></p>
<h3 style="font-size: 24px;">
A Simple Proof that <em>e<sup>p</sup></em><sup>/<em>q</em></sup> Is Irrational</h3>
<p><span style="line-height: 1.25em;">Thomas Beatty and Timothy W. Jones</span></p>
<p>Using a simple application of the mean value theorem, we show that rational powers of <em>e</em> are irrational.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.50">http://dx.doi.org/10.4169/math.mag.87.1.50</a></p>
<h3 style="font-size: 24px;">
A Unique Area Property of the Quadratic Function</h3>
<p>Connie Xu</p>
<p>Suppose that a function $$f$$ defined on the real line is convex or concave with $$f''(x)$$ continuous and nonzero for all $$x$$. Let $$(x_{1}$$, $$f(x_{1}))$$ and $$(x_{2}, f(x_{2}))$$ be two arbitrary points on the graph of $$f$$ with $$x_{1} < x_{2}$$. For $$i = 1, 2$$, let $$L_{i}$$ denote the tangent line to $$f$$ at the point $$(x_{i}, f(x_{i}))$$ and let $$A_{i}$$ be the area of the region $$R_{i}$$ bounded by the graph of $$f$$, the tangent line $$L_{i}$$, and the line $$x =\hat{x}$$, the $$x$$-coordinate of the intersection of $$L_{1}$$ and $$L_{2}$$. It is proved that $$f$$ is a quadratic function if and only if $$A_{1} = A_{2}$$ for every choice of $$x_{1}$$ and $$x_{2}$$.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.52">http://dx.doi.org/10.4169/math.mag.87.1.52</a></p>
<h3 style="font-size: 24px;">
de Bruijn Arrays for L-Fillings</h3>
<p><span style="line-height: 1.25em;">Lara Pudwell and Rachel Rockey</span></p>
<p>We use modular arithmetic to construct a de Bruijn L-array, which is a $$k\times k^{2}$$ array consisting of exactly one copy of each L-shaped pattern (a $$2\times2$$ array with the upper right corner removed) with digits chosen from $$\{0,\dots,k-1\}$$.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.57">http://dx.doi.org/10.4169/math.mag.87.1.57</a></p>
<h2 style="font-size: 28px;">
PROBLEMS</h2>
<p>Proposals 1936-1940<br />
Quickies 1037 & 1038<br />
Solutions 1911-1915</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.61">http://dx.doi.org/10.4169/math.mag.87.1.61</a></p>
<h3>
Escape the Square</h3>
<p>Mark Dalthorp</p>
<p>To purchase from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.68">http://dx.doi.org/10.4169/math.mag.87.1.68</a></p>
<h2 style="font-size: 28px;">
REVIEWS</h2>
<p>Algebra: imperative? STEM: crisis? Alive: in 10 years?</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.69">http://dx.doi.org/10.4169/math.mag.87.1.69</a></p>
<h2 style="font-size: 28px;">
NEWS AND LETTERS</h2>
<h3>
74th Annual William Lowell Putnam Mathematical Competition</h3>
<p>To purchase from JSTOR: <a href="http://dx.doi.org/10.4169/math.mag.87.1.71">http://dx.doi.org/10.4169/math.mag.87.1.71</a></p>
</div></div></div>Mon, 10 Feb 2014 13:09:12 +0000kmerow327204 at http://www.maa.orghttp://www.maa.org/publications/periodicals/mathematics-magazine/mathematics-magazine-contents-february-2014#commentsMath Horizons Contents—February 2014
http://www.maa.org/publications/periodicals/math-horizons/math-horizons-contents-february-2014
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p style="font-size: 14px;"><img alt="" src="/sites/default/files/images/pubs/mh-feb2014-cover.JPG" style="margin-right: 5px; margin-left: 5px; float: left; width: 195px; height: 249px;" /></p>
<div>
<p style="font-size: 14px;">I am very pleased and honored to be the new editor of <a href="/publications/periodicals/math-horizons" title="Math Horizons"><em>Math Horizons</em></a>. In the February issue, Steven Strogatz talks to Patrick Honner about teaching, writing, and mathematics; Heidi Hulsizer uses linear algebra and modular arithmetic to beat a popular shoot-'em-up video game; James Hamblin and Doug McInvale use data to analyze a controversial decision by the New England Patriots in Super Bowl XLVI; and Erik Tou writes about the mathematics of juggling. I hope you enjoy reading this issue as much as I enjoyed putting it together. —<em>David Richeson, editor</em></p>
</div>
<p style="font-size: 14px;">Volume 21, Issue 3</p>
<h5 style="font-size: 18px;">
JOURNAL SUBSCRIBERS AND MAA MEMBERS:</h5>
<p style="font-size: 14px;">To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.</p>
<h3 style="font-size: 24px;">
Do the Math! Juggling with Numbers</h3>
<p>Erik R. Tou</p>
<p>Generate interesting juggling patterns using mathematics.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.21.3.5">http://dx.doi.org/10.4169/mathhorizons.21.3.5</a></p>
<h3 style="font-size: 24px;">
A Conversation with Steven Strogatz</h3>
<p>Mathematician and author Steven Strogatz talks about writing, teaching, Twitter, and what makes mathematics great in this interview by Patrick Honner. (<a href="/sites/default/files/pdf/horizons/strogatz_feb14.pdf"><span class="style5">pdf</span></a>)</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.21.3.8">http://dx.doi.org/10.4169/mathhorizons.21.3.8</a></p>
<h3 style="font-size: 24px;">
A ‘Mod’ern Mathematical Adventure in <em>Call of Duty: Black Ops</em></h3>
<p>Heidi Hulsizer</p>
<p>A shoot-’em-up video game contains an Easter egg that can be solved using linear algebra and modular arithmetic.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.21.3.12">http://dx.doi.org/10.4169/mathhorizons.21.3.12</a></p>
<h3 style="font-size: 24px;">
The View from Here: Who's That Teaching My Class?</h3>
<p>Josh Pabian</p>
<p>This student initially struggles with, but is eventually inspired by, his inquiry-based learning class.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.21.3.16">http://dx.doi.org/10.4169/mathhorizons.21.3.16</a></p>
<h3 style="font-size: 24px;">
Math Experts Split the Check</h3>
<p>Ben Orlin</p>
<p>A mathematician, a physicist, an engineer, a computer scientist, and an economist tackle the tricky mathematical problem of determining how much to tip the waiter.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.21.3.18">http://dx.doi.org/10.4169/mathhorizons.21.3.18</a></p>
<h3 style="font-size: 24px;">
Polynomial Long Division and Root Power Sums</h3>
<p>Dan Kalman and Stacy Langton</p>
<p>A clever algorithm involving a derivative and polynomial long division generates the sums of the powers of the roots of a polynomial.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.21.3.20">http://dx.doi.org/10.4169/mathhorizons.21.3.20</a></p>
<h3>
The Bookshelf:<em>The Simpsons and Their Mathematical Secrets</em></h3>
<p>review by Jim Wiseman</p>
<p>Simon Singh writes about the hidden mathematical nuggets in <em>The Simpsons</em>.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.21.3.24">http://dx.doi.org/10.4169/mathhorizons.21.3.24</a></p>
<h3 style="font-size: 24px;">
The Bookshelf: <i>Math on Trial: How Numbers Get Used and Abused in the Courtroom</i></h3>
<p>review by Darren Glass</p>
<p>Leila Schneps and Coralie Colmez write about the dangers of misusing mathematics and statistics in the courtroom.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.21.3.25">http://dx.doi.org/10.4169/mathhorizons.21.3.25</a></p>
<h3 style="font-size: 24px;">
Monday-Morning Math Modeling: Could the Patriots Have Won Super Bowl XLVI?</h3>
<p>James Hamblin and Doug McInvale</p>
<p>The New England Patriots let the New York Giants score a touchdown. Was this the right call?</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.21.3.26">http://dx.doi.org/10.4169/mathhorizons.21.3.26</a></p>
<h2 style="font-size: 28px;">
THE PLAYGROUND!</h2>
<p style="font-size: 14px;">The <em>Math Horizons</em> problem section, edited by Gary Gordon</p>
<p style="font-size: 14px;">JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.21.3.30">http://dx.doi.org/10.4169/mathhorizons.21.3.30</a></p>
<h3 style="font-size: 24px;">
AFTERMATH: Steven Strogatz on Math Education</h3>
<p>Steven Strogatz discusses the current state of mathematics <span style="line-height: 1.25em;">education in this interview with Patrick Honner.</span></p>
<p style="font-size: 14px;">To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.21.3.34">http://dx.doi.org/10.4169/mathhorizons.21.3.34</a></p>
</div></div></div>Tue, 04 Feb 2014 20:15:40 +0000kmerow322868 at http://www.maa.orghttp://www.maa.org/publications/periodicals/math-horizons/math-horizons-contents-february-2014#commentsIllustrated Special Relativity Through Its Paradoxes
http://www.maa.org/publications/ebooks/illustrated-special-relativity-through-its-paradoxes
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><h5>
A Fusion of Linear Algebra, Graphics, and Reality</h5>
<p> </p>
<h2>
John dePillis and José Wudka</h2>
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<p><strong>Illustrated Special Relativity</strong> shows that linear algebra is a natural language for special relativity. It illustrates and resolves several apparent paradoxes of special relativity including the twin paradox and train-and-tunnel paradox. Assuming a minimum of technical prerequisites the authors introduce inertial frames and use them to explain a variety of phenomena: the nature of simultaneity, the proper way to add velocities, and why faster-than-light travel is impossible. Most of these explanations are contained in the resolution of apparent paradoxes, including some lesser-known ones: the pea-shooter paradox, the bug-and-rivet paradox, and the accommodating universe paradox. The explanation of time and length contraction is especially clear and illuminating.</p>
<p>At the outset of his seminal paper on special relativity, Einstein acknowledges the work of James Clerk Maxwell whose four equations unified the theories of electricity, optics, and magnetism. For this reason, the authors develop Maxwell’s equations which lead to a simple calculation for the frame-independent speed of electromagnetic waves in a vacuum. (Maxwell did not realize that light was a special case of electromagnetic waves.) Several chapters are devoted to experiments of Roemer, Fizeau, and de Sitter to measure the speed of light and the Michelson-Morley experiment abolishing the aether.</p>
<p>Throughout the exposition is thorough, but not overly technical, and often illustrated by cartoons. The volume might be suitable for a one-semester general-education introduction to special relativity. It is especially well-suited to self-study by interested laypersons or use as a supplement to a more traditional text.</p>
<p>Electronic ISBN: 9781614445173</p>
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</div><div class="field field-name-field-ber-category field-type-taxonomy-term-reference field-label-above"><div class="field-label">Publication Category: </div><div class="field-items"><div class="field-item even">Featured</div></div></div><div class="field field-name-field-book-series field-type-taxonomy-term-reference field-label-above"><div class="field-label">Book Series: </div><div class="field-items"><div class="field-item even"><a href="/book-series/spectrum">Spectrum</a></div></div></div><div class="field field-name-field-ebook-category field-type-taxonomy-term-reference field-label-above"><div class="field-label">Category: </div><div class="field-items"><div class="field-item even"><a href="/ebook-category/algebraabstract-algebra">Algebra/Abstract Algebra</a></div><div class="field-item odd"><a href="/ebook-category/new">New</a></div><div class="field-item even"><a href="/ebook-category/resources-for-teachers">Resources for Teachers</a></div></div></div>Tue, 26 Nov 2013 17:09:44 +0000bruedi244516 at http://www.maa.orghttp://www.maa.org/publications/ebooks/illustrated-special-relativity-through-its-paradoxes#commentsMore Fallacies, Flaws and Flimflam
http://www.maa.org/publications/books/more-fallacies-flaws-and-flimflam
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<h3>
By Edward J. Barbeau</h3>
<p>Catalog Code: MFFL<br />
Print ISBN: 978-0-88385-580-5<br />
Electronic ISBN: 978-1-61444-515-9<br />
188 pp., Paperbound, 2013<br />
List Price: $35.00<br />
MAA Member: $28.00<br />
Series: Spectrum</p>
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<p><em>More Fallacies, Flaws, and Flimflam</em> is the second volume of selections drawn mostly from <a href="/publications/periodicals/college-mathematics-journal/the-college-mathematics-journal" title="The College Mathematics Journal "><em>The College Mathematics Journal</em> </a>column “Fallacies, Flaws, and Flimflam” from 2000 through 2008. The MAA published the first collection, <a href="/publications/ebooks/mathematical-fallacies-flaws-and-flimflam" title="Mathematical Fallacies, Flaws, and Flimflam"><em>Mathematical Flaws, Fallacies, and Flimflam</em></a>, in 2000.</p>
<p>As in the first volume, <em>More Fallacies, Flaws, and Flimflam</em> contains items ranging from howlers (outlandish procedures that nonetheless lead to a correct answer) to deep or subtle errors often made by strong students. Although some are provided for entertainment, others challenge the reader to determine exactly where things go wrong.</p>
<p>Items are sorted by subject matter. Elementary teachers will find chapter 1 of most use, while middle and high schoolteachers will find chapters 1, 2, 3, 7, and 8 applicable to their levels. College instructors can delve for material in every part of the book.</p>
<p>There are frequent references to <em>The College Mathematics Journal</em>; these are denoted by <em>CMJ</em>.</p>
<h3>
Table of Contents</h3>
<p style="margin-left:20px">Preface<br />
1. Arithmetic<br />
2. School Algebra<br />
3. Geometry<br />
4. Limits, Sequences and Series<br />
5. Differential Calculus<br />
6. Integral Calculus<br />
7. Combinatorics<br />
8. Probability and Statistics<br />
9. Complex Analysis<br />
10. Linear and Modern Algebra<br />
11. Miscellaneous<br />
Index<br />
About the Author</p>
<h3>
Excerpt: 5.8 A standard box problem (p.89)</h3>
<p>Dale R. Buske put on a recent calculus examination the following standard problem:</p>
<p><strong>Problem</strong> A crate open at the top has vertical sides, a square bottom, and a volume of 4 cubic meters. If the crate is to be constructed so as to have minimal surface area, find its dimensions.</p>
<p>One student started with this formula for the surface area of the crate: <em>SA</em> = 4<em>x</em> + 4<em>y</em>, where <em>x</em> was the length of one side of the base and <em>y</em> was the height of the crate. (After all, there are four line segments of length <em>x</em> on the bottom of the crate, four line segments of length <em>y</em> on the sides, and the four line segments on top do not count since the crate has an open top.) The student then correctly went on to use the volume constraint 4 = <em>x</em><sup>2</sup><em>y</em> to find <em>y</em> = 4/<em>x</em><sup>2</sup> and arrive at the formula <em>SA</em> = 4<em>x</em> + 16/<em>x</em><sup>2</sup>. Taking the derivative and applying the condition for a maximum leads to the correct answer: the crate should have a base with dimensions 2 meters by 2 meters and a height of 1 meter.</p>
<h3>
About the Author</h3>
<p><strong>Ed Barbeau</strong> graduated from the University of Toronto (BA, MA) and received his doctorate from the University of Newcastle-upon-Tyne, England in 1964. He was a Postdoctoral Fellow at Yale University in 1966-67, and taught at the University of Toronto from 1967 until 2003. He is currently retired.</p>
<p>Barbeau was named Fellow of the Ontario Institute for Studies in Education (1989), and has received the David Hilbert Award (1991) from the World Federation of National Mathematics Competitions and the Adrien Pouliot Award (1995) from the Canadian Mathematical Society. He has published a number of books with the MAA as well as two books, <em>Polynomials</em> and <em>Pell’s Equation</em> with Springer. He has been invited to give talks frequently to groups of teachers, students and the general public, and has presented three radio broadcasts, “Proof and truth in mathematics,” in the Canadian Broadcasting Corporation series, <em>Ideas</em>.</p>
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http://www.maa.org/publications/ebooks/calculus-modeling-and-application-2nd-edition
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><h2>
David A. Smith and Lawrence C. Moore</h2>
<hr />
<p><img src="/sites/default/files/images/ebooks/textbooks/boat_online_calc.png" style="border-width: 1px; border-style: solid; margin-left: 15px; margin-right: 15px; float: right; width: 134px; height: 187px;" /></p>
<p><strong>Calculus: Modeling and Application, 2nd Edition,</strong> responds to advances in technology that permit the integration of text and student activities into a unified whole. In this approach, students can use mathematics to structure their understanding of and investigate questions in the world around them, to formulate problems and find solutions, then to communicate their results to others.</p>
<p><span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: 13px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;"><strong>Calculus: Modeling and Application</strong> covers two semesters of single-variable calculus. Its features include </span></p>
<ul>
<li>
<span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: 13px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;">use of real-world contexts for motivation, </span></li>
<li>
<span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: 13px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;">guided discovery learning, </span></li>
<li>
<span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: 13px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;">hands-on activities (including built-in applets), </span></li>
<li>
<span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: 13px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;">a problem-solving orientation, </span></li>
<li>
<span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: 13px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;">encouragement of teamwork, </span></li>
<li>
<span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: 13px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;">written responses to questions, </span></li>
<li>
<span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: 13px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;">tools for self-checking of results, </span></li>
<li>
<span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: 13px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;">intelligent use of technology, and </span></li>
<li>
<span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: 13px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;">high expectations of students.</span></li>
</ul>
<p>It is important to note that <strong>this textbook is a website</strong>. When purchasing this textbook you will have access to two versions:</p>
<p><strong>Computer/CAS Version.</strong> This version requires the Firefox browser for proper display of mathematics and one of the commercial computer algebra systems (Maple, Mathematica, or Mathcad) for most of the interactivity in the text. Some activities require access to a printer.</p>
<p><strong>Tablet/Sage Version.</strong> This version was written for the iPad but will run on other tablets or computers in either Safari or Firefox (but not Internet Explorer or Chrome). It uses the free computer algebra system Sage to provide interactivity through remotely-processed interacts. Sage interacts require only calculator-form entry of functions and numbers. You need not have Sage installed on your machine, and no printer is required.</p>
<p>At the end of each section there are pages labeled Exercises and Problems. The Exercise pages each have a WebWork button that links to MAA's WebWork courses page. Adopters who set up WebWork courses (at MAA or on their local servers) get access to the <strong>Calculus: Modeling and Application</strong> Library (in addition to the national library), and it has problems that are more or less like the Exercises in <strong>Calculus: Modeling and Application</strong>. Instructors can make up homework sets from both the <strong>Calculus: Modeling and Application</strong> and national libraries. The authors plan to continue to add to the <strong>Calculus: Modeling and Application</strong> collection, so it will get stronger over time. All of the Exercises are potentially machine-gradable (suitable for WebWork), whereas the Problems need human responses.</p>
<p>Three sample chapters (chapters 2, 5, and 8) are available at <a href="http://calculuscourse.maa.org/sample">calculuscourse.maa.org/sample</a>.</p>
<p>Access to <strong>Calculus: Modeling and Application </strong>can be purchased at <a href="http://maa.pinnaclecart.com">maa.pinnaclecart.com</a>. Instructors wishing to adopt the text for their course should contact the MAA Service Center (1-800-331-1MAA) about obtaining desk access.</p>
<p>To see how one professor uses this text and other technologies to create a paperless classroom watch this YouTube video: <a href="http://www.youtube.com/watch?v=4m4WKefT1Mg" style="color: rgb(17, 85, 204); font-family: arial, sans-serif; font-size: 13px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);" target="_blank">http://www.youtube.com/watch?<wbr />v=4m4WKefT1Mg</a></p>
<p><strong>Note to professors:</strong> <span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: 13px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;">Some problems do not work in both versions (due to differing technologies), so if your students are using both versions, be sure to check problem numbers in both places when making homework assignments.</span></p>
<p><strong>Note to purchasers of the text: </strong>To gain easy access to the book website in the future</p>
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Login at<span class="Apple-converted-space"> </span><a href="http://maa.pinnaclecart.com">maa.pinnaclecart.com</a> using the username and password you created when ordering the book.</li>
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Click on My Orders.</li>
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Click on View Order (right side of page).</li>
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The page that comes up will look like a receipt page and it will have a link to the text (Access Calculus ebook). BOOKMARK this page so that in the future all you have to do is click on the bookmark, the login window will pop up, you log in, and then the page with the link pops up.</li>
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</div><div class="field field-name-field-ber-category field-type-taxonomy-term-reference field-label-above"><div class="field-label">Publication Category: </div><div class="field-items"><div class="field-item even">Featured</div></div></div><div class="field field-name-field-book-series field-type-taxonomy-term-reference field-label-above"><div class="field-label">Book Series: </div><div class="field-items"><div class="field-item even"><a href="/book-series/maa-textbooks">MAA Textbooks</a></div></div></div><div class="field field-name-field-ebook-category field-type-taxonomy-term-reference field-label-above"><div class="field-label">Category: </div><div class="field-items"><div class="field-item even"><a href="/ebook-category/calculus">Calculus</a></div></div></div>Thu, 29 Aug 2013 14:11:14 +0000bruedi166212 at http://www.maa.orghttp://www.maa.org/publications/ebooks/calculus-modeling-and-application-2nd-edition#commentsDistilling Ideas: An Introduction to Mathematical Thinking
http://www.maa.org/publications/books/distilling-ideas-an-introduction-to-mathematical-thinking
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<h3>
By Brian P. Katz and Michael Starbird</h3>
<p>Catalog Code: DIMT<br />
Print ISBN: 978-1-93951-203-1<br />
Electronic ISBN: 978-1-61444-613-2<br />
171 pp., Paperbound, 2013<br />
List Price: $54.00<br />
MAA Member: $45.00<br />
Series: MAA Textbooks</p>
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<p>Mathematics is not a spectator sport: successful students of mathematics grapple with ideas for themselves. <em>Distilling Ideas</em> presents a carefully designed sequence of exercises and theorem statements that challenge students to create proofs and concepts. As students meet these challenges, they discover strategies of proofs and strategies of thinking beyond mathematics. In order words, <em>Distilling Ideas</em> helps its users to develop the skills, attitudes, and habits of mind of a mathematician and to enjoy the process of distilling and exploring ideas.</p>
<p><em>Distilling Ideas</em> is an ideal textbook for a first proof-based course. The text engages the range of students' preferences and aesthetics through a corresponding variety of interesting mathematical content from graphs, groups, and epsilon-delta calculus. Each topic is accessible to users without a background in abstract mathematics because the concepts arise from asking questions about everyday experience. All the common proof structures emerge as natural solutions to authentic needs. <em>Distilling Ideas</em> or any subset of its chapters is an ideal resource either for an organized Inquiry Based Learning course or for individual study.</p>
<p>A student response to <em>Distilling Ideas</em>: "I feel that I have grown more as a mathematician in this class than in all the other classes I've ever taken throughout my academic life."</p>
<h3>
Table of Contents</h3>
<p style="margin-left:20px">1. Introduction<br />
2. Graphs<br />
3. Groups<br />
4. Calculus<br />
5. Conclusion<br />
Annotated Index<br />
List of Symbols<br />
Abouth the Authors</p>
<h3>
Excerpt: Ch. 3.12 The Man Behind the Curtain (p. 85)</h3>
<p>Many people mistakenly believe that mathematics is arbitrary and magical, or at least that there is some secret knowledge that math teachers have but won't share with their students. Mathematics is no more magical that the Great and Powerful Wizard of Oz, who was just behind a curtain. The development of mathematics is directed by a few simple principles and a strong sense of aesthetics. To develop the ideas of group theory we followed a path of guided discovery. Let's look back on the journey and let the guiding strategies emerge from behind the curtain.</p>
<p>Group theory has many applications, ranging from internet credit card security and cracking the Enigma Code to solving the Rubik's Cube and generalizing the quadratic formula. Our introduction to group theory has only exposed us to a tiny portion of the theorems living in the wild.</p>
<p>We started with familiar activities: adding, multiplying, telling time, and putting blocks in a matching hole. The definition of a group emerged by distilling the essential features and commonalities from these specific examples in the process that we call abstraction. These examples generalized to produce cyclic groups, the dihedral groups, and symmetric groups in the process we call exploration. We defined generators, subgroups, products, and homomorphisms to help us discover and describe the wide variety of groups that we caught in our definition-net. We saw patterns that led us to conjectures about the size of subgroups; we justified these insights, producing theorems. We applied those insights to extend our exploration of the concept of groups.</p>
<p>This investigation represents a complete cycle of inquiry, and it especially emphasizes and deepens the skill of definition exploration. We explored a mathematical idea in its own right, hoping to describe the rich mosaic of possibilities that an abstract diefinitiion could capture. In particular, trying to <em>classify</em> all groups was one of the motivations that guided our explorations. Let us reflect on these new tools in our inquiry tool belt...</p>
<p><br />
<a href="/sites/default/files/pdf/pubs/books/DIMT_p117.pdf">Preview: p. 117</a></p>
<h3>
About the Authors</h3>
<p><strong>Brian Katz</strong> is an Assistant Professor of Mathematics at Augustana College in Rock Island, Illinois. He received his BA from Williams College in 2003 with majors in mathematics, music and chemistry and his PhD from The University of Texas at Austin in 2011, concentrating in algebraic geometry. While at UT Austin, Brian received the Frank Gerth III Graduate Excellence Award and the<span style="line-height: 1.25em;"> Frank Gerth III Graduate Teaching Excellence Award from the Department of Mathematics. Brian is a Project NExT Fellow, supported by </span>Harry Lucas, Jr. and <span style="line-height: 1.25em;">the Educational Advancement Foundation.</span></p>
<p>Brian is a member of the Inquiry Based Learning community, which focuses on helping students to ask and explore mathematical questions for themselves. Brian has given many talks about mathematics, teaching, and technology at national conferences and workshops, including as a plenary speaker for the 2012 MAA IBL PREP workshop. In particular, Brian has written and spoken about ways to use student-written wikis to extend the power of IBL beyond the classroom. Brian has helped to organize the R.L. Moore Legacy Conference hosted by the Academy for Inquiry Based Learning, and serves as a mentor for new practitioners within the organization.</p>
<p>Brian is liberally educated and is passionate about engaging in the Liberal Arts. He sees mathematics at the core of this educational perspective: connecting to the deductive reasoning of philosophy, the structure of communication of linguistics, the abstract beauty of the arts, and all forms of critical writing and speaking. Outside of mathematics, Brian has offered an interdisciplinary first-year course called Mind and Meaning, and one of his mathematics courses has been designated as part of the core curriculum at Augustana College as a "Perspective on Human Values and Existence."</p>
<p>As an educator, Brian enjoys helping all students become clearer thinkers and communicators and empowering people to move from consuming to producing knowledge.</p>
<p><strong>Michael Starbird</strong> is Professor of Mathematics and a University Distinguished Teaching Professor at The University of Texas at Austin. He received his BA degree from Ponoma College and his PhD in mathematics from the University of Wisconsin, Madison. He has been on the faculty of the Department of Mathematics of The University of Texas at Austin except for leaves including as a Visiting Member of the Institute for Advanced Study in Princeton, New Jersey, and as a member of the technical staff at the Jet Propulsion Laboratory in Pasadena, California. He served as Associate Dean in the College of Natural Sciences at UT from 1989 to 1997.</p>
<p>Starbird is a member of the Academy of Distinguished Teachers at The University of Texas of Austin and is an inaugural member of The University of Texas System Academy of Distinguished Teachers. He has won many teaching awards, including the 2007 Mathematical Association of America Deborah and Franklin Tepper Haimo National Award for Distinguished College or University Teaching of Mathematics; a Minnie Stevens Piper Professorship, which is awarded each year to 10 professors from any subject at any college or university in the state of Texas; the inaugural award of the Dad's Association Centennial Teaching Fellowship; the Excellence Award from the Eyes of Texas, twice; the President's Associates Teaching Excellence Award; the Jean Holloway Award for Teaching Excellence, which is the oldest teaching award at UT and is presented to one professor each year; the Chad Oliver Plan II Teaching Award, which is student-selected and awarded each year to one professor in the Plan II liberal arts honors program; and the Friar Society Centennial Teaching Fellowship, which is awarded to one professor at UT annually. He is an inaugural year Fellow of the AMS. In 1989, Professor Starbird was the UT Recreational Sports Super Racquets Champion.</p>
<p>Starbird's mathematical research is in the field of topology. He has served as a member-at-large of the Council of the American Mathematical Society and on the national education committees of both the American Mathematical Society and the Mathematical Association of America. He currently serves on the MAA's CUPM Committee and on its Steering Committee for the next CUPM Curriculum Guide. He directs UT's Inquiry Based Learning Project. He has given more than 200 invited lectures at the colleges and universities throughout the country and more than 20 minicourses and workshops to mathematics teachers.</p>
<p>Starbird strives to present higher-level mathematics authentically to students and the general public and to teach thinking strategies that go beyond mathematics as well. With those goals in mind, he wrote, with co-author Edward B. Burger, <em>The Heart of Mathematics: An invitation to effective thinking</em> (now in its 4th edition), which won a 2001 Robert W. Hamilton Book Award. Burger and Starbird have also written a book that brings intriguing mathematical ideas to the public, entitled <em>Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas</em>, published by W.W. Norton, 2005, and translated into eight foreign languages. In 2012, Burger and Starbird published <em>The 5 Elements of Effective Thinking</em>, which describes practical strategies for creating innovation and insight.</p>
<p>Starbird has produced five courses for The Teaching Company in their Great Courses Series: <em>Change and Motion: Calculus Made Clear </em>(1st edition, 2001, and 2nd edition, 2007)<em>; Meaning from Data: Statistics Made Clear, </em>2005<em>; What are the Chances? Probability Made Clear, </em>2007<em>; Mathematics from the Visual World, </em>2009; and, with collaborator Edward Burger, <em>The Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas, </em>2003<em>.</em>These courses bring an authentic understanding of significant ideas in mathematics to many people who are not necessarily mathematically oriented. Starbird loves to see real people find the intrigue and fascination that mathematical thinking can bring.</p>
<p>David Marshall, Edward Odell, and Michael Starbird wrote <em>Number Theory Through Inquiry</em>, which appeared in 2007 in the MAA's Textbook Series. <em>Number Theory Through Inquiry</em> and the new Katz-Starbird book <em>Distilling Ideas: An Introduction to Mathematical Thinking</em> are books in the newly created MAA Textbook Subseries called "Mathematics Through Inquiry." This subseries contains materials that foster an Inquiry Based Learning strategy of instruction that encourages students to discover and develop mathematical ideas on their own.</p>
<!---<h3>
MAA Review</h3>
<p>Continued...</p>---></div>
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</div></div></div><div class="field field-name-field-book-series field-type-taxonomy-term-reference field-label-above"><div class="field-label">Book Series: </div><div class="field-items"><div class="field-item even"><a href="/book-series/maa-textbooks">MAA Textbooks</a></div></div></div>Wed, 07 Aug 2013 16:28:09 +0000swebb151866 at http://www.maa.orghttp://www.maa.org/publications/books/distilling-ideas-an-introduction-to-mathematical-thinking#commentsApplications of Mathematics in Economics
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Warren Page, Editor</h2>
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<p><strong>Applications of Mathematics in Economics</strong> presents an overview of the (qualitative and graphical) methods and perspectives of economists. Its objectives are not intended to teach economics, but rather to give mathematicians a sense of what mathematics is used at the undergraduate level in various parts of economics, and to provide students with the opportunities to apply their mathematics in relevant economics contexts.</p>
<p>The volume’s applications span a broad range of mathematical topics and levels of sophistication. Each article consists of self-contained, stand-alone, expository sections whose problems illustrate what mathematics is used, and how, in that subdiscipline of economics. The problems are intended to be richer and more informative about economics than the economics exercises in most mathematics texts. Since each section is self-contained, instructors can readily use the economics background and worked-out solutions to tailor (simplify or embellish) a section’s problems to their students’ needs. Overall, the volume’s 47 sections contain more than 100 multipart problems. Thus, instructors have ample material to select for classroom uses, homework assignments, and enrichment activities.</p>
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<p>Electronic ISBN: 9781614443179</p>
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<a href="/sites/default/files/pdf/ebooks/NTE82_preface.pdf">Preface</a><br />
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Models, Processes, and Directions</h1>
<h2>
Glenn Ledder, Jenna P. Carpenter, Timothy D. Comar, Editors</h2>
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<p>There is a gap between the extensive mathematics background that is beneficial to biologists and the minimal mathematics background biology students acquire in their courses. The result is an undergraduate education in biology with very little quantitative content. New mathematics courses must be devised with the needs of biology students in mind.</p>
<p>In this volume, authors from a variety of institutions address some of the problems involved in reforming mathematics curricula for biology students. The problems are sorted into three themes: Models, Processes, and Directions. It is difficult for mathematicians to generate curriculum ideas for the training of biologists so a number of the curriculum models that have been introduced at various institutions comprise the Models section. Processes deals with taking that great course and making sure it is institutionalized in both the biology department (as a requirement) and in the mathematics department (as a course that will live on even if the creator of the course is no longer on the faculty). Directions looks to the future, with each paper laying out a case for pedagogical developments that the authors would like to see.</p>
<p>The authors represent a wide variety of academic institutions, from universities to community colleges, and all of the articles begin with information about the institutional context. Many of the articles also include links to resources that can be found on the internet, and some have associated books in print as well. All emphasize features that could be applied to similar projects at other institutions and offer useful advice for the newcomer to mathematics curriculum development for life science students.</p>
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<a href="/sites/default/files/pdf/ebooks/NTE81_Preface.pdf">Preface</a><br />
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http://www.maa.org/publications/books/learning-modern-algebra
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<h3>
By Al Cuoco and Joseph J. Rotman</h3>
<p>Catalog Code: LMA<br />
Print ISBN: 978-1-93951-201-7<br />
Electronic ISBN: 978-1-61444-612-5<br />
456 pp., Hardbound, 2013<br />
List Price: $60.00<br />
MAA Member: 48.00<br />
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<p><em><strong>Learning Modern Algebra</strong> aligns with the CBMS Mathematical Education of Teachers–II recommendations, in both content and practice. It emphasizes rings and fields over groups, and it makes explicit connections between the ideas of abstract algebra and the mathematics used by high school teachers. It provides opportunities for prospective and practicing teachers to experience mathematics for themselves, before the formalities are developed, and it is explicit about the mathematical habits of mind that lie beneath the definitions and theorems.</em></p>
<p>This book is designed for prospective and practicing high school mathematics teachers, but it can serve as a text for standard abstract algebra courses as well. The presentation is organized historically: the Babylonians introduced Pythagorean triples to teach the Pythagorean theorem; these were classified by Diophantus, and eventually this led Fermat to conjecture his Last Theorem. The text shows how much of modern algebra arose in attempts to prove this; it also shows how other important themes in algebra arose from questions related to teaching. Indeed, modern algebra is a very useful tool for teachers, with deep connections to the actual content of high school mathematics, as well as to the mathematics teachers use in their profession that doesn't necessarily "end up on the blackboard."</p>
<p>The focus is on number theory, polynomials, and commutative rings. Group theory is introduced near the end of the text to explain why generalizations of the quadratic formula do not exist for polynomials of high degree, allowing the reader to appreciate the more general work of Galois and Abel on roots of polynomials. Results and proofs are motivated with specific examples whenever possible, so that abstractions emerge from concrete experience. Applications range from the theory of repeating decimals to the use of imaginary quadratic fields to construct problems with rational solutions. While such applications are integrated throughout, each chapter also contains a section giving explicit connections between the content of the chapter and high school teaching.</p>
<h3>
Table of Contents</h3>
<p style="margin-left:20px">Preface<br />
Some features of this book<br />
A Note to Students<br />
A Note to Instructors<br />
Notation<br />
1. Early Number Theory<br />
2. Induction<br />
3. Renaissance<br />
4. Modular Arithmetic<br />
5. Abstract Algebra<br />
6. Arithmetic of Polynomials<br />
7. Quotients, Fields, and Classical Problems<br />
8. Cyclotomic Integers<br />
9. Epilog References<br />
Index</p>
<!---<h3>
Excerpt: Ch. 8 Dauntless Courage (p. 109)</h3>--->
<h3>
About the Authors</h3>
<p><strong>Al Cuoco</strong> is Distinguished Scholar and Director of the Center for Mathematics Education at Education Development Center. He is lead author for The CME Project, a four-year NSF-funded high school curriculum, published by Pearson. He also co-directs Focus on Mathematics, a mathematics-science partnership that has established a mathematical community of mathematicians, teachers, and mathematics educators. The partnership evolved from his 25-year collaboration with Glenn Stevens (BU) on Boston University’s PROMYS for Teachers, a professional development program for teachers based on the Ross program (an immersion experience in mathematics). Al taught high school mathematics to a wide range of students in the Woburn, Massachusetts public schools from 1969 until 1993. A student of Ralph Greenberg, Cuoco holds a Ph.D. from Brandeis, with a thesis and research in Iwasawa theory. In addition to this book, MAA published his <em>Mathematical Connections: a Companion for Teachers and Others</em>. But his favorite publication is a 1991 paper in the <em>American Mathematical Monthly</em>, described by his wife as an attempt to explain a number system that no one understands with a picture that no one can see.</p>
<p><strong>Joseph Rotman</strong> was born in Chicago on May 26, 1934. He studied at the University of Chicago, receiving the degrees BA, MA, and Ph.D. there in 1954, 1956, and 1959, respectively; his thesis director was Irving Kaplansky.</p>
<p>Rotman has been on the faculty of the mathematics department of the University of Illinois at Urbana-Champaign since 1959, with the following ranks: Research Associate 1959-1961; Assistant Professor 1961-1963; Associate Professor 1963-1968; Professor 1968-2004; Professor Emeritus 2004-present. He has held the following visiting appointments: Queen Mary College, London, England 1965, 1985; Aarhus University, Denmark, Summer 1970; Hebrew University, Jerusalem, Israel 1070; University of Padua, Italy, 1972; Technion, Israel Institute of Technology and Hebrew University, Jerusalem (Lady Davis Professor), 1977-78; Tel Aviv University, Israel, 1982; Bar Ilan University, Israel, Summer 1982; Annual visiting lecture, South African Mathematical Society, 1985; University of Oxford, England, 1990. Professor Rotman was an editor of <em>Proceedings of American Mathematical Society</em>, 1970, 1971; managing editor, 1972, 1973.</p>
<p>Aside from writing research articles, mostly in algebra, he has written the following textbooks: <em>Group Theory</em> 1965, 1973, 1984, 1995; <em>Homological Algebra</em> 1970, 1979, 2009; <em>Algebraic Topology</em> 1988; <em>Galois Theory</em> 1990, 1998; <em>Journey into Mathematics</em> 1998, 2007; <em>First Course in Abstract Algebr</em>a 1996, 2000, 2006; <em>Advanced Modern Algebra</em> 2002.</p>
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<p> </p>
</div></div></div><div class="field field-name-field-book-series field-type-taxonomy-term-reference field-label-above"><div class="field-label">Book Series: </div><div class="field-items"><div class="field-item even"><a href="/book-series/maa-textbooks">MAA Textbooks</a></div></div></div>Wed, 26 Jun 2013 19:22:21 +0000swebb135679 at http://www.maa.orghttp://www.maa.org/publications/books/learning-modern-algebra#commentsBeyond the Quadratic Formula
http://www.maa.org/publications/books/beyond-the-quadratic-formula
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<div id="rightcolumn">
<h3>
By Ron Irving</h3>
<p>Catalog Code: BQF<br />
Print ISBN: 978-0-88385-783-0<br />
Electronic ISBN: 978-1-61444-112-0<br />
244 pp., Hardbound, 2013<br />
List Price: $55.00<br />
MAA Member: $44.00<br />
Series: Classroom Resource Materials</p>
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<p>The quadratic formula for the solution of quadratic equations was discovered independently by scholars in many ancient cultures and is familiar to everyone. Less well known are formulas for solutions of cubic and quartic equations whose discovery was the high point of 16th century mathematics. Their study forms the heart of this book, as part of the broader theme that a polynomial’s coefficients can be used to obtain detailed information on its roots. A closing chapter offers glimpses into the theory of higher-degree polynomials, concluding with a proof of the fundamental theorem of algebra. The book also includes historical sections designed to reveal key discoveries in the study of polynomial equations as milestones in intellectual history across cultures.</p>
<p>The book is designed for self-study, with many results presented as exercises and some supplemented by outlines for solution. The intended audience includes in-service and prospective secondary mathematics teachers, high school students eager to go beyond the standard curriculum, undergraduates who desire an in-depth look at a topic they may have unwittingly skipped over, and the mathematically curious who wish to do some work to unlock the mysteries of this beautiful subject.</p>
<h3>
Table of Contents</h3>
<p style="margin-left:20px">Preface<br />
1. Polynomials<br />
2. Quadratic Polynomials<br />
3. Cubic Polynomials<br />
4. Complex Numbers<br />
5. Cubic Polynomials, II<br />
6. Quartic Polynomials<br />
7. Higher-Degree Polynomials<br />
References<br />
Index<br />
About the Author</p>
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<h3>
About the Author</h3>
<p><strong>Ron Irving</strong> is a mathematics professor at the University of Washington in Seattle. He was born in suburban New York City, studied mathematics and philosophy at Harvard, and received his Ph.D. in mathematics at MIT. Following a postdoctoral position at Brandeis and a National Science Foundation postdoctoral fellowship year at the University of Chicago and UC San Diego, Irving came to Seattle. He has been a visiting faculty member at UCSD and Aarhus and a member of the Institute for Advanced Study in Princeton. His research interests have ranged over several areas of algebra, including ring theory and the representation theory of Lie groups and Lie algebras.</p>
<p>When Irving began teaching the department’s senior algebra course for majors planning on secondary teaching careers, he developed an interest in the preparation of pre-service and in-service teachers. His work with this audience led to receipt of the university’s Distinguished Teaching Award and to his book <em>Integers, Polynomials, and Rings</em>.</p>
<p>Irving spent seven years in academic administration, serving as department chair for a year, divisional dean of natural sciences for four, and interim dean of the College of Arts and Sciences for two. During this time, he established the Summer Institute for Mathematics at UW, a six-week residential program that brings talented high school students in the Pacific Northwest to the university to share in the excitement of doing mathematics. He continues to serve as the program’s executive director. He also began the planning for a new undergraduate degree in integrated sciences designed to meet the needs of future secondary science teachers.</p>
<p>Since returning to the department, Irving has taught a variety of algebra courses and written this book. He has been the director of the Integrated Sciences program, completing its planning, approval, and implementation. He serves as the secretary-treasurer of the Astrophysical Research Consortium, which oversees Apache Point Observatory in the Sacramento Mountains of New Mexico, and is president of the board of the Burke Museum of Natural History and Culture.</p>
<p>Irving is a member of the Mathematical Association of America and the American Mathematical Society. As this book goes to press, he is beginning a second stint as department chair.</p>
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</div></div></div><div class="field field-name-field-book-series field-type-taxonomy-term-reference field-label-above"><div class="field-label">Book Series: </div><div class="field-items"><div class="field-item even"><a href="/book-series/classroom-resource-materials">Classroom Resource Materials</a></div></div></div>Tue, 25 Jun 2013 13:39:01 +0000swebb134942 at http://www.maa.orghttp://www.maa.org/publications/books/beyond-the-quadratic-formula#commentsExploring Advanced Euclidean Geometry with GeoGebra
http://www.maa.org/publications/books/exploring-advanced-euclidean-geometry-with-geogebra
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<h3>
By Gerard Venema</h3>
<p>Catalog Code: EAEG<br />
Print ISBN: 978-0-88385-784-7<br />
Electronic ISBN: 978-1-61444-111-3<br />
129 pp., Hardbound, 2013<br />
List Price: $50.00<br />
Member Price: $40.00<br />
Series: Classroom Resource Materials</p>
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<p>This book provides an inquiry-based introduction to advanced Euclidean geometry. It utilizes dynamic geometry software, specifically GeoGebra, to explore the statements and proofs of many of the most interesting theorems in the subject. Topics covered include triangle centers, inscribed, circumscribed, and escribed circles, medial and orthic triangles, the nine-point circle, duality, and the theorems of Ceva and Menelaus, as well as numerous applications of those theorems. The final chapter explores constructions in the Poincaré disk model for hyperbolic geometry.</p>
<p>The book can be used either as a computer laboratory manual to supplement an undergraduate course in geometry or as a stand-alone introduction to advanced topics in Euclidean geometry. The text consists almost entirely of exercises (with hints) that guide students as they discover the geometric relationships for themselves. First the ideas are explored at the computer and then those ideas are assembled into a proof of the result under investigation. The goals are for the reader to experience the joy of discovering geometric relationships, to develop a deeper understanding of geometry, and to encourage an appreciation for the beauty of Euclidean geometry.</p>
<h3>
Table of Contents</h3>
<p style="margin-left:20px">Quick Review of Elementary Euclidean Geometry<br />
1. The Elements of GeoGebra<br />
2. The Classical Triangle Centers<br />
3. Advanced Techniques in GeoGebra<br />
4. Circumscribed, Inscribed, and Escribed Circles<br />
5. The Medial and Orthic<br />
6. Quadrilaterals<br />
7. The Nine-Point Circle<br />
8. Ceva's Theorem<br />
9. The Theorem of Menelaus<br />
10. Circles and Lines<br />
11. Applications of the Theorem of Menelaus<br />
12. Additional Topics in Triangle Geometry<br />
12.1. Napoleon's theorem and the Napoleon point<br />
13. Inversions in Circles<br />
14. The Poincare Disk</p>
<!---<h3>
Excerpt: Ch. 8 Dauntless Courage (p. 109)</h3>--->
<h3>
About the Author</h3>
<p><strong>Gerard Venema</strong> earned an A.B. in mathematics from Calvin College and a Ph.D. from the University of Utah. After completing his education he spent two years in a postdoctoral position at the University of Texas at Austin and another two years as a Member of the Institute for Advanced Study in Princeton, NJ. He then returned to his alma mater, Calvin College, and has been a faculty member there ever since. While on the Calvin College faculty he has also held visiting faculty positions at the University of Tennessee, the University of Michigan, and Michigan State University. He spent two years as a Program Director in the Division of Mathematical Sciences at the National Science Foundation.</p>
<p>Venema is a member of the American Mathematical Society and the Mathematical Association of America. He served for ten years as an Associate Editor of<em> <a href="/publications/periodicals/american-mathematical-monthly" title="The American Mathematical Monthly">T</a></em><a href="/publications/periodicals/american-mathematical-monthly" title="The American Mathematical Monthly"><em>he</em> <em>American Mathematical Monthly</em></a> and currently sits on the editorial board of <a href="/publications/periodicals/maa-focus" title="MAA FOCUS"><em>MAA FOCUS</em></a>. Venema has served the Michigan Section of the MAA as chair and is the 2013 recipient of the section’s distinguished service award. He currently holds the position of MAA Associate Secretary and is a member of the Association’s Board of Governors.</p>
<p>Venema is the author of two other books. One is an undergraduate textbook, <em>Foundations of Geometry</em>, published by Pearson Education, Inc., which is now in its second edition. The other is a research monograph coauthored by Robert J. Daverman. It is titled <em>Embeddings in Manifolds</em> and was published by the American Mathematical Society as volume 106 in its Graduate Studies in Mathematics series. In addition to these books, Venema is author of over 30 research articles in geometric topology.</p>
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MAA Review</h3>
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</div></div></div><div class="field field-name-field-book-series field-type-taxonomy-term-reference field-label-above"><div class="field-label">Book Series: </div><div class="field-items"><div class="field-item even"><a href="/book-series/classroom-resource-materials">Classroom Resource Materials</a></div></div></div>Fri, 31 May 2013 18:39:14 +0000swebb129789 at http://www.maa.orghttp://www.maa.org/publications/books/exploring-advanced-euclidean-geometry-with-geogebra#commentsAmerican Mathematical Monthly Contents—June/July 2013
http://www.maa.org/publications/periodicals/american-mathematical-monthly/american-mathematical-monthly-contents-junejuly-2013
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<p>Need some reading material for the beach? Look no further than the friendly green cover appearing soon in your mailbox (or on your computer screen)! In the June/July issue of the <em>Monthly</em>, check out Andrew Hwang's paper on paper geometry construction models, Papakonstantinou and Tapia's discussion of the evolution of the secant method, and Tadashi Tokieda's perpetual motion machine. Don't forget that the <em>Monthly</em> is not only about articles and problems: David Bressoud reviews not one, but FIVE calculus texts.</p>
<p>Stay tuned for the August/September issue: David Aldous explains how to use market data to illustrate undergraduate probability. Enjoy the beach! —<em>Scott Chapman</em></p>
<div>
<p>Vol. 120, No. 6, pp.487-580.</p>
<p><strong>Journal subscribers and MAA members:</strong> Please login into the member portal by clicking on 'Login' in the upper right corner.</p>
<h2>
ARTICLES</h2>
<h3>
Paper Surface Geometry: Surveying a Locally Euclidean Universe</h3>
<p>Andrew D. Hwang</p>
<p>The concepts of parallel transport and intrinsic (Gaussian) curvature arising in the differential geometry of surfaces may be pleasantly and concretely investigated using paper models and familiar notions of length and angle. This paper introduces parallel transport and curvature in the context of "locally Euclidean" surfaces: polyhedra, for which curvature is concentrated at isolated points, and "polycones," for which curvature is concentrated along circular arcs. The geometry of polycones is used to recover a strikingly simple intrinsic formula for the Gaussian curvature of a surface of rotation. We give instructions for building paper models of the catenoid and surfaces of constant Gaussian curvature.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.487">http://dx.doi.org/10.4169/amer.math.monthly.120.06.487</a></p>
<h3>
Origin and Evolution of the Secant Method in One Dimension</h3>
<p>Joanna M. Papakonstantinou and Richard A. Tapia</p>
<p>Many in the mathematical community believe that the secant method arose from Newton's method using a finite difference approximation to the derivative, most likely because that is the way that it is taught in contemporary texts. However, we were able to trace the origin of the secant method all the way back to the Rule of Double False Position described in the 18th-century B.C. Egyptian Rhind Papyrus, by showing that the Rule of Double False Position coincides with the secant method applied to a linear equation. As such, it predates Newton's method by more than 3,000 years. In this paper, we recount the evolution of the Rule of Double False Position as it spanned many civilizations over the centuries leading to what we view today as the contemporary secant method. Unfortunately, throughout history naming confusion has surrounded the Rule of Double False Position. This naming confusion was primarily a product of the last 500 years or so and became particularly troublesome in the past 50 years, creating confusion in the use of the terms Double False Position method, Regula Falsi method, and secant method. We elaborate on this confusion and clarify the names used.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.500">http://dx.doi.org/10.4169/amer.math.monthly.120.06.500</a></p>
<h3>
Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials</h3>
<p>Nathan Bliss, Ben Fulan, Stephen Lovett, and Jeff Sommars</p>
<p>Almost all algebra texts define cyclotomic polynomials using primitive $$n$$th roots of unity. However, the elementary formula $$\gcd(x^{m}-1,x^{n}-1)=x^{\gcd(m,n)}-1$$ in $$\mathbb{Z}[x]$$ can be used to define the cyclotomic polynomials without reference to roots of unity. In this article, partly motivated by cyclotomic polynomials, we prove a factorization property about strong divisibility sequences in a unique factorization domain. After illustrating this property with cyclotomic polynomials and sequences of the form $$A^{n}-B^{n}$$, we use the main theorem to prove that the dynamical analogues of cyclotomic polynomials over any unique factorization domain $$R$$ are indeed polynomials in $$R[x]$$. Furthermore, a converse to this article's main theorem provides a simple necessary and sufficient condition for a divisibility sequence to be a rigid divisibility sequence.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.519">http://dx.doi.org/10.4169/amer.math.monthly.120.06.519</a></p>
<h3>
Several Proofs of the Irreducibility of the Cyclotomic Polynomials</h3>
<p>Steven H. Weintraub</p>
<p>We present a number of classical proofs of the irreducibility of the $$n$$th cyclotomic polynomial $$\Phi_{n}(x)$$. For n prime we present proofs due to Gauss (1801), in both the original and a simplified form, Kronecker (1845), and Schönemann/Eisenstein (1846/1850). For general $$n$$, we present proofs due to Dedekind (1857), Landau (1929), and Schur (1929).</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.537">http://dx.doi.org/10.4169/amer.math.monthly.120.06.537</a></p>
<h3>
The Equation $$x^{2}+\overline{m}y^{2}=\overline{k}$$ in $$\mathbb{Z}/p\mathbb{Z}$$</h3>
<p>Kurt Girstmair</p>
<p>We present a simple and character-free approach to the determination of the number of nontrivial solutions of the equation $$x^{2}+\overline{m}y^{2}=\overline{k}$$ in $$\mathbb{Z}/p\mathbb{Z}$$, where $$m$$ and $$k$$ are integers not divisible by $$p$$.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.546">http://dx.doi.org/10.4169/amer.math.monthly.120.06.546</a></p>
<h2>
NOTES</h2>
<h3>
Coxeter Friezes and Triangulations of Polygons</h3>
<p>Claire-Soizic Henry</p>
<p>We study relations between Coxeter's frieze patterns and discrete Sturm-Liouville equations. We present a short proof of the Coxeter-Conway classification theorem.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.553">http://dx.doi.org/10.4169/amer.math.monthly.120.06.553</a></p>
<h3>
A Probabilistic Proof of a Binomial Identity</h3>
<p>Jonathon Peterson</p>
<p>We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.558">http://dx.doi.org/10.4169/amer.math.monthly.120.06.558</a></p>
<h3>
The Euclidean Algorithm and the Linear Diophantine Equation $$ax+by=\gcd(a,b)$$</h3>
<p>S. A. Rankin</p>
<p>In this note, we prove that for any positive integers $$a$$ and $$b$$, with $$d=\gcd(a,b)$$, among all integral solutions to the equation $$ax+by=d$$, the solution $$(x_{0},y_{0})$$ that is provided by the Euclidean algorithm lies nearest to the origin. In fact, we prove that $$(x_{0},y_{0})$$ lies in the interior of the circle centered at the origin with radius $$\frac{1}{2d}\sqrt{a^{2}+b^{2}}$$.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.562">http://dx.doi.org/10.4169/amer.math.monthly.120.06.562</a></p>
<h3>
A Buoyancy-Driven Perpetual Motion Machine</h3>
<p>Tadashi Tokieda</p>
<p>A perpetual motion machine driven by buoyancy is presented. We see that it does not quite work, and we also discuss how it would move in reality. Debunking the trick behind most such machines is a nice exercise in exact and non-exact differentials.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.564">http://dx.doi.org/10.4169/amer.math.monthly.120.06.564</a></p>
<h3>
Example of a Monotonic, Everywhere Differentiable Function on $$\mathbb{R}$$ Whose Derivative Is Not Continuous</h3>
<p>Manas R. Sahoo</p>
<p>We construct an example of a monotonic function which is differentiable everywhere, but the derivative is not continuous. This is done using a nonnegative discontinuous integrable function whose every point is a Lebesgue point.</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.566">http://dx.doi.org/10.4169/amer.math.monthly.120.06.566</a></p>
<h2>
PROBLEMS AND SOLUTIONS</h2>
<p>Problems 11712-11718<br />
Solutions 11595, 11605, 11607, 11608, 11612, 11614</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.569">http://dx.doi.org/10.4169/amer.math.monthly.120.06.569</a></p>
<h2>
REVIEWS</h2>
<p><em>Calculus </em>by Michael Spivak; <em>Calculus Deconstructed: A Second Course in First-Year Calculus</em> by Zbigniew Nitecki; <em>Approximately Calculus</em> by Shahriar Sharhriari; <em>A Guide to Cauchy's Calculus: A Translation and Analysis of Calcul Infinitesimal</em> by Dennis M. Cates; and <em>The Calculus Integral</em> by Brian S. Thomson</p>
<p>Reviewed by David Bressoud</p>
<p>To purchase the article from JSTOR: <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.577">http://dx.doi.org/10.4169/amer.math.monthly.120.06.577</a></p>
</div>
</div></div></div>Thu, 23 May 2013 15:03:21 +0000kmerow129270 at http://www.maa.orghttp://www.maa.org/publications/periodicals/american-mathematical-monthly/american-mathematical-monthly-contents-junejuly-2013#commentsMath Horizons Contents—April 2013
http://www.maa.org/publications/periodicals/math-horizons/math-horizons-contents-april-2013
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><img alt="" src="/sites/default/files/images/pubs/graphics/MH1304.jpg" style="width: 196px; height: 250px; margin-left: 10px; margin-right: 10px; float: left;" /></p>
<div>
<p>Highlights from the <em>Math Horizons</em> April issue include a mathematical exposé of recent psychology research, a homemade recipe for solving Rubik's cube, a mind-bending foray into high-dimensional space, and a nod to Paul Erdős on the occasion of his 100th birthday. —<em>Stephen Abbott</em> and <em>Bruce Torrence</em></p>
</div>
<p>Volume 20, Issue 4</p>
<h3>
The Mathematics of Measuring Self-Delusion</h3>
<p>Kristopher Tapp</p>
<p>Confusion around conditional probabilities calls into question several decades of psychology research. <a href="/sites/default/files/pdf/tapp_apr13.pdf"><span class="style5">(pdf)</span></a></p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.20.4.5">http://dx.doi.org/10.4169/mathhorizons.20.4.5</a></p>
<h3>
The 100th Birthday of Paul Erdős/Remembering Erdős</h3>
<p>Bruce Torrence and Ron Graham</p>
<p><em>Math Horizons</em> marks the centennial year of the prolific mathematician from Budapest with some reminiscing from an Erdős number one fan.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.20.4.10">http://dx.doi.org/10.4169/mathhorizons.20.4.10</a></p>
<h3>
THE VIEW FROM HERE: Manga Guides Can Make You Wise—Or at Least Smile</h3>
<p>Urchin Colley</p>
<p>Does a Japanese graphic narrative book series successfully animate the undergraduate mathematics curriculum? Our student reviewer decides.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.20.4.13">http://dx.doi.org/10.4169/mathhorizons.20.4.13</a></p>
<h3>
Tails in High Dimensions</h3>
<p>Avner Halevy</p>
<p>Fitting a ball into a box and mowing the lawn take on a whole new dimension.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.20.4.14">http://dx.doi.org/10.4169/mathhorizons.20.4.14</a></p>
<h3>
THE DISGRUNTLED MATH MAJOR: Keep the Math Flame Burning</h3>
<p>Nick Boredaki</p>
<p>Whether it’s qualifying exams or happy hour, surviving in graduate school is all about bringing the passion.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.20.4.18">http://dx.doi.org/10.4169/mathhorizons.20.4.18</a></p>
<h3>
And the Winners Are…</h3>
<p>Stephen Morris, Richard Stong, and Stan Wagon</p>
<p>A notorious interview question for Facebook job candidates sends the authors off to the races.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.20.4.20">http://dx.doi.org/10.4169/mathhorizons.20.4.20</a></p>
<h3>
Write Your Own Recipe for Rubik’s Cube</h3>
<p>Burkard Polster</p>
<p>Maybe you learned how to “solve” the cube—now learn how to <em>solve</em> the cube.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.20.4.23">http://dx.doi.org/10.4169/mathhorizons.20.4.23</a></p>
<h3>
The Fundamental Theorem of Algebra for Artists</h3>
<p>Bahman Kalantari and Bruce Torrence</p>
<p>Modulus plots reveal subtle truths about complex polynomials, roots, and even level curve intersections.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.20.4.26">http://dx.doi.org/10.4169/mathhorizons.20.4.26</a></p>
<h2>
THE PLAYGROUND!</h2>
<p>The <em>Math Horizons</em> problem section, edited by <em>Derek Smith</em> and <em>Gary Gordon</em></p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.20.4.30">http://dx.doi.org/10.4169/mathhorizons.20.4.30</a></p>
<h3>
Aftermath: Mathematical Habits of Mind</h3>
<p>Karen King</p>
<p>How does the way we learn mathematics at an early age set us up for success—or failure—down the road? <a class="style2 style5" href="http://horizonsaftermath.blogspot.com/">(Blogger)</a></p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.20.4.34">http://dx.doi.org/10.4169/mathhorizons.20.4.34</a></p>
<h3>
Math’s Life Lessons</h3>
<p>Tim Chartier</p>
<p>Words of wisdom for graduates wondering whether their mathematical studies have prepared them for life beyond the classroom.</p>
<p>JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.20.4.35">http://dx.doi.org/10.4169/mathhorizons.20.4.35</a></p>
<p> </p>
</div></div></div>Tue, 21 May 2013 15:36:43 +0000kmerow129195 at http://www.maa.orghttp://www.maa.org/publications/periodicals/math-horizons/math-horizons-contents-april-2013#commentsParadoxes and Sophisms in Calculus
http://www.maa.org/publications/books/paradoxes-and-sophisms-in-calculus
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<h3>
By Sergiy Klymchuk and Susan Staples</h3>
<p>Catalog Code: PAS<br />
Print ISBN: 978-0-88385-781-6<br />
Electronic ISBN: 978-1-61444-110-6<br />
108 pp., Paperbound, 2013<br />
List Price: $40.00<br />
Member Price: $32.00<br />
Series: Classroom Resource Materials</p>
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<p><em>Paradoxes and Sophisms in Calculus</em> offers a delightful supplementary resource to enhance the study of single variable calculus. By the word <em>paradox</em> the authors mean a surprising, unexpected, counter-intuitive statement that <em>looks</em> invalid, but in fact is <em>true</em>. The word <em>sophism</em> describes intentionally invalid reasoning that <em>looks</em> formally correct, but in fact contains a subtle mistake or flaw. In other words, a sophism is a <em>false</em> proof of an incorrect statement. A collection of over 50 paradoxes and sophisms showcases the subtleties of this subject and leads students to contemplate the underlying concepts. A number of the examples treat historically significant issues that arose in the development of calculus, while others more naturally challenge readers to understand common misconceptions. Sophisms and paradoxes from the areas of functions, limits, derivatives, integrals, sequences, and series are explored. The book could be useful for high school teachers and university faculty as a teaching resource; high school and college students as a learning resource; and a professional development resource for calculus instructors.</p>
<h3>
Table of Contents</h3>
<p style="margin-left:20px">Introduction<br />
Acknowledgments<br />
I. Paradoxes<br />
II. Sophisms<br />
III. Solutions to Paradoxes<br />
IV. Solutions to Sophisms<br />
Bibliography<br />
About the Authors</p>
<!---<h3>Excerpt: </h3>--->
<h3>
About the Authors</h3>
<p><b>Sergiy Klymchuk</b> is an Associate Professor of the School of Computing and Mathematical Sciences at the Auckland University of Technology, New Zealand. He has 32 years broad experience teaching university mathematics in different countries. His PhD (1988) was in differential equations. At present his main research interests are in mathematics education. He is a Fellow of the Institute of Mathematics and its Applications (IMA) based in the UK and an Associate Editor of the international journal Teaching Mathematics and its Applications. He is a member of the Royal Society of New Zealand (RSNZ). He is also a member of three affiliated groups of the International Commission on Mathematical Instruction (ICMI) of the International Mathematical Union (IMU): the International Community of Teachers of Mathematical Modeling and Applications (ICTMA), the International Group for the Psychology of Mathematics Education (PME) and the International Commission for the Study and Improvement of Mathematics Education (CIEAEM). He has more than 180 publications including several books on popular mathematics and science that have been, or are being, published in 11 countries. His book <em><a href="/publications/books/counterexamples-in-calculus" title="Counterexamples in Calculus">Counterexamples in Calculus</a></em>, also published by the MAA, received an Outstanding Academic Title award from Choice Magazine in 2010.</p>
<p><b>Susan Staples</b>, Associate Professor in Mathematics at Texas Christian University, traces her connections to the MAA back to her high school days in Canton, Massachusetts, where she was an active participant in mathematics competitions. She subsequently received a B.S. in Mathematics from Case Western Reserve University and a Ph. D. (1988) in mathematics from The University of Michigan in the area of complex analysis. Susan served ten years as graduate director of the TCU MAT program and continues to enjoy working with local teachers. She is the proud recipient of teaching awards from The University of Michigan, The University of Texas, and TCU. Her professional memberships include the MAA, AMS, AWM, NCTM, and the AAUW. Her graduate work was supported by an AAUW fellowship. Currently Susan holds positions on two editorial boards for the MAA — <a href="/publications/periodicals/american-mathematical-monthly" title="The American Mathematical Monthly"><em>The American Mathematical Monthly</em></a> and the Classroom Resources Material book series; she previously served as assistant editor for the problem section of the <a href="/publications/periodicals/mathematics-magazine" title="Mathematics Magazine"><em>Mathematics Magazine</em></a>. Susan has also directed the Actuarial Program at TCU since its inception. She is pleased to have been selected or the national Committee on the Undergraduate Program in Mathematics (CUPM) Actuarial Sciences group.</p>
<!---<h3>
MAA Review</h3>
<p>Review Pending</p>---></div>
<p> </p>
</div></div></div><div class="field field-name-field-book-series field-type-taxonomy-term-reference field-label-above"><div class="field-label">Book Series: </div><div class="field-items"><div class="field-item even"><a href="/book-series/classroom-resource-materials">Classroom Resource Materials</a></div></div></div>Fri, 10 May 2013 15:45:35 +0000swebb129092 at http://www.maa.orghttp://www.maa.org/publications/books/paradoxes-and-sophisms-in-calculus#commentsA Guide to Functional Analysis
http://www.maa.org/publications/books/a-guide-to-functional-analysis
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<h3>
By Steven G. Krantz</h3>
<p>Catalog Code: DOL-49<br />
Print ISBN: 978-0-88385-357-3<br />
Electronic ISBN: 978-1-61444-213-4<br />
149 pp., Hardbound, 2013<br />
List Price: $49.95<br />
Member Price: $39.95<br />
Series: Dolciani Mathematical Expositions</p>
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<p>This book is a quick but precise and careful introduction to the subject of functional analysis. It covers the topics that can be found in a basic graduate analysis text and it also covers more sophisticated topics such as spectral theory, convexity, and fixed-point theorems. A special feature of the book is that it contains a great many examples and even some applications. It concludes with a statement and proof of Lomonosov’s dramatic result about invariant subspaces.</p>
<h3>
Table of Contents</h3>
<p style="margin-left:20px">Preface<br />
1. Fundamentals<br />
2. Ode to the Dual<br />
3. Hilbert Space<br />
4. The Algebra of Operators<br />
5. Banach Algebra Basics<br />
6. Topological Vector Spaces<br />
7. Distributions<br />
8. Spectral<br />
9. Convexity<br />
10. Fixed-Point Theorems<br />
Table of Notation<br />
Glossary<br />
Bibliography<br />
Index<br />
About the Author</p>
<h3>
Excerpt: 1.1 What is Functional Analysis? (p. 1)</h3>
<p>The mathematical analysts of the nineteenth century (Cauchy, Riemann, Weierstrass, and others) contented themselves with studying one function at a time. As a sterling instance, the Weierstrass nowhere differentiable function is a world-changing example of the real function theory of “one function at a time.” Some of Riemann’s examples in Fourier analysis give other instances. This was the world view 150 years ago. To be sure, Cauchy and others considered sequences and series of functions, but the end goal was to consider the single limit function.</p>
<p>A major paradigm shift took place, however, in the early twentieth century. For then people began to consider <em>spaces</em> of functions. By this we mean a linear space, equipped with a norm. The process began slowly. At first people considered very specific spaces, such as the square-integrable real functions on the unit circle. Much later, people branched out to more general classes of spaces. An important feature of the spaces under study was that they must be complete. For we want to pass to limits, and completeness guarantees that this process is reliable.</p>
<p>Thus was born the concepts of Hilbert space and Banach space.</p>
<h3>
About the Author</h3>
<p><strong>Steven G. Krantz</strong> was born in San Francisco, California in 1951. He received a B.A. degree from the University of California at Santa Cruz in 1971 and the Ph.D. from Princeton University in 1974. Krantz has taught at UCLA, Penn State, Princeton University, and Washington University in St. Louis. He served as Chair of the latter department for five years. Krantz has published more than 60 books and more than 160 scholarly papers. He is the recipient of the Chauvenet Prize and the Beckenbach Book Award of the MAA. He has received the UCLA Alumni Foundation Distinguished Teaching Award and the Kemper Award. He has directed 18 Ph.D. theses and 9 Masters theses.</p>
<!--<h3>
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</div></div></div><div class="field field-name-field-book-series field-type-taxonomy-term-reference field-label-above"><div class="field-label">Book Series: </div><div class="field-items"><div class="field-item even"><a href="/book-series/dolciani">Dolciani</a></div><div class="field-item odd"><a href="/book-series/maa-guides">MAA Guides</a></div></div></div>Tue, 30 Apr 2013 16:07:33 +0000swebb129024 at http://www.maa.orghttp://www.maa.org/publications/books/a-guide-to-functional-analysis#comments