Amherst College - mathematics
https://www.amherst.edu/taxonomy/term/1701
enCantor and a Perspective on the Infinite
https://www.amherst.edu/users/L/u5klefebvre/the_dude_imbibes/node/611771
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Georg Cantor made many invaluable contributions to mathematics, it is to him that we owe the earliest technical explanation of <a href="https://en.wikipedia.org/wiki/Transcendental_number">transcendental numbers</a>, the formulation of the <a href="http://en.wikipedia.org/wiki/Heine-Cantor_theorem">Heine–Cantor theorem</a>, and most famously his development of <a href="http://en.wikipedia.org/wiki/set_theory">set theory</a>. The latter is a mainstay in grade school mathematics today, but it remains difficult to explain at its most basic level absent some analogous comparison.<br><br><img src="http://i.imgur.com/Q9KwAJj.gif" alt="" width="223" height="223" style="line-height:19.2000007629395px;max-height:100%;"></p>
<p><!--break--></p>
<p>In middle school al<span style="line-height:19.2000007629395px;">gebra, most of us learned that-</span></p>
<p>n {0, 1e-10, 2e-10...1e-9... 1...∞} > n {1, 2, 3, 4, 5...∞}<br>or rather that is to say there are an infinitely greater number of real numbers than there are integers, even though there are already an infinite number of integers. While we can correlate different sets of integers such as odd numbers and even numbers by offsetting their values, as in--<br><br>1 ----> 2<br>3 ----> 4<br>5 ----> 6<br>∞ ----> ∞+1 <sup>(yes I realize how ridiculous this notation is)<br><br>ratio: 1:1<br><br></sup>the number of values we can equate any whole integer with in an set of infinite decimals between integers, is, in itself infinite (ratio 1:<span>∞)</span>. So there are far more possible values in a set of real numbers approaching infinity than there are <a href="http://en.wikipedia.org/wiki/rational_numbers">rational numbers</a>, even as a <a href="http://en.wikipedia.org/wiki/dense_set">dense set</a>. However, at the same time, in a set approaching infinity there are as many rational numbers as there are integers, because they make use of defined units that have one-to-one correspondence with all integers. For more reading on how infinitely divisible real numbers are an exception to this, you can refer to the Wikipedia article on <a href="http://en.wikipedia.org/wiki/Cantor's_diagonal_argument">Cantor's diagonal argument</a>.<br><br>So if I have even made any remote sense in explaining the concept of infinite sets and "more infinite" (uncountable) sets, how can we translate this into something that high school students can understand, let alone most adults? I submit one of the best ways this can be explained- sine functions and radio frequencies.</p>
<p><br><img src="http://i.imgur.com/AYFzZjE.png" alt="" width="480" height="138" style="max-height:593px;max-width:636px;"></p>
<p>With the above sine function we have a simple graphical representation of correlating numbers in infinite sets; each corresponding hue represents a single whole digit of a set of 10. The gradient transitions between these represent infinitessimally smaller and smaller decimal values between each whole digit. Now, let's say this axis goes on infinitely in either direction to represent the entire spectrum of potential frequencies, not just the visible light spectrum. There will always be units by which we can express frequency, and in a sense we can always tune into these units.<br><br>For example, 1000 hertz go into 1 kilohertz.<br><img src="http://i.imgur.com/0ZWOJ6q.png" alt="" style="max-height:100%;"><br>When we realize that hertz are an arbitrary unit that can be broken down further, approaching an infinite number of divisions, we can see how<br>-1, graphically represented on an infinite set of real numbers, the spectrum in hZ appears as densely lines instead of a perfect gradient<br>-2, how non-matching sets of infinity ("more infinity"), are like tuning into a specific frequency lacking units in a broader spectrum of possible numbers.<br><br>Is this a proof? Nope. Not in the least.<br>Does it show some semblance of how you can have infinite and uncountably infinite sets? I hope so. I also hope that this conflation of the electromagnetic spectrum with mathematical numbers does a good job of illustrating this concept for teaching. I'll try to reword this in the future to make it more concise, but until then, I welcome any and all criticism on this lesson design.<br><br></p>
<hr><p><span class="fine-print">Image credits: 1. "Infinite Tangent Circles", Florida Center for Instruction Technology <em>Clipart ETC</em> (Tampa: U of S. Florida, 2007)</span></p>
<p><span class="fine-print">2. Self made, using Grapher on OS X</span></p>
<p><span class="fine-print">3. Derivative of work from AmateurSpectroscopy.com</span></p></div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1701">mathematics</a></div><div class="field-item odd"><a href="/taxonomy/term/22747">Cantor, Georg</a></div><div class="field-item even"><a href="/taxonomy/term/22748">illustrations</a></div><div class="field-item odd"><a href="/taxonomy/term/22749">infographics</a></div><div class="field-item even"><a href="/taxonomy/term/22750">teaching methods</a></div></div></div>Mon, 03 Aug 2015 05:43:13 +0000u5klefebvre611771 at https://www.amherst.eduhttps://www.amherst.edu/users/L/u5klefebvre/the_dude_imbibes/node/611771#commentsRenowned New York Times Math Blogger Steven Strogatz to Speak at Amherst College Oct. 4
https://www.amherst.edu/news/news_releases/2012/10/node/433582
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span class="fine-print">September 21, 2012</span> </p>
<p>AMHERST, Mass. – Steven Strogatz, Cornell University mathematics professor and <em>New York Times</em> math blogger, will give a talk titled “Bringing Math to the Masses” in the Cole Assembly Room of Amherst College’s Converse Hall on Thursday, Oct. 4, from 7 to 8 p.m.<!--break--> The lecture, which is free and open to the public, will be followed by a reception.</p>
<p>Strogatz—whose new book <em><a href="http://www.hmhbooks.com/hmh/site/hmhbooks/bookdetails?isbn=9780547517667">The Joy of x</a> </em>will be released Oct. 2—works in the areas of nonlinear dynamics and complex systems, often on topics inspired by the curiosities of everyday life. He is perhaps best known for the series of <em>New York Times</em> weekly <a href="http://topics.nytimes.com/top/opinion/series/steven_strogatz_on_the_elements_of_math/index.html" target="_blank" title="">blog entries about mathematics</a> that the <em>Harvard Business Review</em> <a href="http://blogs.hbr.org/schrage/2010/04/sorry-paul-too-bad-tom.html" target="_blank" title="">described</a> as “must reads for entrepreneurs and executives” and “a model for how mathematics needs to be popularized.” (On Sept. 10, 2012, he began blogging again for the<em> Times</em> in a series titled <a href="http://opinionator.blogs.nytimes.com/category/me-myself-and-math/">“Me, Myself and Math.”</a>) </p>
<p>His 1998 paper in the journal <em>Nature</em> on “small-world” networks ignited a flurry of research into the topic. He has received numerous awards for his research, teaching and public service including a Presidential Young Investigator Award from the National Science Foundation (1990); MIT’s highest teaching prize, the Everett Moore Baker Award for Excellence in Undergraduate Teaching (1991); the J.P. and Mary Barger ’50 Teaching Award (1997), the Robert ’55 and Vanne ’57 Cowie Teaching Award (2001), the Tau Beta Pi Teaching Award (2006) and the Swanson Teaching Award (2009), all from Cornell’s College of Engineering; and the Communications Award from the Joint Policy Board for Mathematics (2007), a lifetime achievement award for the communication of mathematics to the general public. In 2009 he was elected a Fellow of the Society for Industrial and Applied Mathematics for his “investigations of small-world networks and coupled oscillators and for outstanding science communication.” In 2012 he was elected a Fellow of the American Academy of Arts and Sciences. <br><br> In addition to contributing to the <em>New York Times</em>, Strogatz frequently serves as a <a href="http://www.radiolab.org/people/steve-strogatz/" target="_blank" title="">guest on National Public Radio’s <em>RadioLab</em></a>. He also filmed a series of 24 lectures on <a href="http://www.teach12.com/ttcx/coursedesclong2.aspx?cid=1333" target="_blank" title="">chaos</a> for the Teaching Company’s <em>Great Courses</em> series, available on DVD. In addition to <em>The Joy of x, </em>he is the author of <a href="http://www.westviewpress.com/book.php?isbn=9780738204536" target="_blank" title=""><em>Nonlinear Dynamics and Chaos</em> </a>(1994), <a href="http://www.hyperionbooks.com/book/sync-the-emerging-science-of-spontaneous-order-2/" target="_blank" title=""><em>Sync</em> </a>(2003) and <em><a href="http://press.princeton.edu/titles/8859.html" target="_blank" title="">The Calculus of Friendship </a></em>(2009).</p>
<p>After graduating summa cum laude in mathematics from Princeton in 1980, Strogatz studied at Trinity College in the United Kingdom, where he was a Marshall Scholar. He did his doctoral work in applied mathematics at Harvard, and then spent several years at Harvard and Boston University on a National Science Foundation postdoctoral fellowship. From 1989 to 1994, Strogatz taught in the Department of Mathematics at MIT. He joined the Cornell faculty in 1994 and now serves there as the Jacob Gould Schurman Professor of Applied Mathematics. For more on Strogatz, visit his website, <a href="http://www.stevenstrogatz.com">www.stevenstrogatz.com</a>.</p>
<p align="center">###</p></div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1701">mathematics</a></div><div class="field-item odd"><a href="/taxonomy/term/2457">Cole Assembly Room</a></div><div class="field-item even"><a href="/taxonomy/term/2907">converse hall</a></div><div class="field-item odd"><a href="/taxonomy/term/18023">Steven Strogatz</a></div><div class="field-item even"><a href="/taxonomy/term/18024">Bringing Math to the Masses</a></div></div></div><ul class="links inline"><li class="sharethis first last"><a href="/sharethis-ajax/433582" class="mm-sharethis">Share</a></li>
</ul>Fri, 21 Sep 2012 16:00:33 +0000channa433582 at https://www.amherst.eduDavid A. Cox Wins Ford Award from Mathematical Association of America
https://www.amherst.edu/academiclife/faculty_achievements/node/429599
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span class="drop-cap2">T</span>he Mathematical Association of America (MAA) has named <a href="https://www.amherst.edu/people/facstaff/dacox">David A. Cox</a>, the William J. Walker of Mathematics at Amherst, one of this year’s winners of its Lester R. Ford Award honoring the author of an outstanding paper published in the previous year. Cox was recognized for his article “Why Eisenstein Proved the Eisenstein Criterion and Why Schönemann Discovered It First,” which appeared in the January 2011 issue of the MAA’s <em>The American Mathematical Monthly</em>, and accepted the award during the <a href="http://www.maa.org/mathfest/othermath.html">MAA Prize Session</a> on Aug. 3 at the <a href="http://www.maa.org/mathfest/">2012 MAA MathFest </a>in Madison, Wisc.</p>
<div class="mediainline"><span class="inline"><img src="/media/view/429601/original/Cox-MAA.jpg" alt="Cox-MAA" title="Cox-MAA" style="display:block;margin-left:auto;margin-right:auto;" class="image original" height="382" width="400"></span></div>
<p><br><br> According to Cox, “Why Eisenstein” discusses the historical context of what is called “the Eisenstein Criterion” in the theory of polynomials. Cox explains in the piece that Gotthold Eisenstein, for whom the criterion was named, was not actually the first to discover it; that honor goes to Theodor Schönemann. “The amazing thing is that Eisenstein and Schönemann were led to their discoveries by completely independent paths,” he said. “They really should both get credit.” He also explored some of the related developments in algebra and number theory that were occurring in the 19<sup>th</sup> century.</p>
<p>In <a href="http://maa.org/news/MathFest2012awards/Ford.html">the citation</a> for Cox’s award, the MAA described the Amherst professor’s paper as “an engrossing tale” and “an amazingly rich story, beautifully told, not of a priority dispute but of a grand sweeping flow of ideas beginning with [Carl Friedrich] Gauss (who partially scooped both Schönemann and Eisenstein) and extending into the beating heart of modern-day mathematics. It is a tour de force of mathematical and historical scholarship.”<br><br> For Cox, such praise and the honor itself are huge compliments. “I put a high value on quality expository writing in mathematics, so it is very satisfying when my peers recognize my contribution.” What is also gratifying, said Cox, is that the same mathematics that led to his paper also resulted in a senior thesis. “[My paper] mentions Niels Henrik Abel’s wonderful theorem about geometric constructions on a curve called the lemniscate,” he noted. “Eisenstein proved his criterion in the course of trying to understand Abel’s proof. I liked Eisenstein’s argument so much that I included a proof of Abel’s theorem in a book on Galois theory that I wrote in 2004. But my treatment had one loose end –there was one Galois group I couldn't compute. Last September, I gave this problem to Trevor Hyde ’12 for his senior thesis in mathematics. Trevor solved the problem in spectacular fashion—his thesis was awarded summa cum laude, and he received an Amherst College Post-Baccalaureate Research Fellowship.” What’s more, said Cox, he and Hyde will write up his thesis for publication in a mathematical journal. <br><br> Cox, a member of the Amherst faculty since 1979, received his bachelor of arts degree from Rice University and Ph.D. from Princeton University. His research interests include algebraic geometry, commutative algebra, geometric modeling, number theory and the history of mathematics.</p>
<p>According to the MAA’s website, the Lester R. Ford Awards were established in 1964 to recognize authors of articles of “expository excellence published in <em>The American Mathematical Monthly</em> or <em>Mathematics Magazine</em>.” Named for mathematician Lester R. Ford, Sr., editor of the <em>American Mathematical Monthly </em>from 1942 to 1946 and president of the Mathematical Association of America from 1947 to 1948, the prize and $500 are given to up to five mathematicians annually at the summer meeting of the MAA. <br><br> Cox is not the first member of Amherst’s math department to win a Ford Prize. Dan Velleman<em>, </em><em>Julian H. Gibbs ’46 Professor of Mathematics, </em>and Tanya Leise, assistant professor of mathematics, received the award in 1994 and 2008, respectively.</p>
<p> </p></div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1701">mathematics</a></div><div class="field-item odd"><a href="/taxonomy/term/1956">math</a></div><div class="field-item even"><a href="/taxonomy/term/4821">David Cox</a></div><div class="field-item odd"><a href="/taxonomy/term/17901">Cox</a></div><div class="field-item even"><a href="/taxonomy/term/17902">Ford Award</a></div><div class="field-item odd"><a href="/taxonomy/term/17903">MAA</a></div></div></div>Tue, 28 Aug 2012 16:35:35 +0000kdduke429599 at https://www.amherst.eduTally Me Banana
https://www.amherst.edu/amherst-story/magazine/issues/2011winter/collegerow/bananas/node/293609
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span class="drop-cap2">H</span>ere’s a weighty question: How many pounds of bananas does Valentine Dining Hall go through in a single year? <br> After the college posted that very question on its Facebook page, more than 40 people contributed guesses. Responses ranged from 1,500 pounds (not even close) to 47,000 pounds (very warm) to 100,000 pounds (way too many).</p><p>One poster said he multiplied the weight of a banana (source: the International Banana Association) by the number of students, faculty and staff, and then, assuming three bananas per person per week that Valentine is open, settled on exactly 61,486.7486 pounds of bananas per year.</p><p>Sadly (not only for him and but also for the International Banana Association), he overestimated the popularity of bananas on campus.</p><p>Not to fear, though: another poster, Rhea Ghosh ’10, guessed the exact amount—49,000 pounds. Here was her reasoning: “Just in case you’re curious, I Googled the average weight of a banana—7 ounces, or 7/16 pounds—settled on 40 weeks (or 280 days) as the amount of time Valentine would be in use during the year including school and camps, and decided that approximately one quarter of the student body (hoping that the faculty and staff banana-eaters would even out some other off-base assumption), or 400 students, ate bananas every day. 7/16 x 400 x 280 = 49,000. Thankfully for me it seems that banana eaters consume in round numbers.”</p></div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1701">mathematics</a></div><div class="field-item odd"><a href="/taxonomy/term/1775">bananas</a></div><div class="field-item even"><a href="/taxonomy/term/1956">math</a></div><div class="field-item odd"><a href="/taxonomy/term/2442">valentine dining hall</a></div><div class="field-item even"><a href="/taxonomy/term/14789">weight</a></div></div></div><ul class="links inline"><li class="sharethis first last"><a href="/sharethis-ajax/293609" class="mm-sharethis">Share</a></li>
</ul>Mon, 14 Feb 2011 05:00:00 +0000kdduke293609 at https://www.amherst.eduhttps://www.amherst.edu/amherst-story/magazine/issues/2011winter/collegerow/bananas/node/293609#commentsNorton Starr Retires
https://www.amherst.edu/amherst-story/magazine/issues/2009summer/collegerow/starr/node/120274
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p></p><table class="table-align-right-gradient" border="0" cellpadding="10" width="200"><tbody><tr><td><div class="mediainline"><img class="image original" src="/media/view/120903/original/139340070.jpg" border="0" width="238" height="300" alt="image"></div><div class="fine-print" align="left">Norton Starr retired after 43 years at Amherst.</div></td></tr></tbody></table><span class="fine-print">By Emily Gold Boutilier</span><br><br><span class="drop-cap2">N</span>orton Starr, the Brian E. Boyle ’69 Professor of Mathematics and Computer Science, has retired after 43 years at Amherst.<br><br> Starr came to Amherst from MIT, where he earned a Ph.D. in mathematics, served as an instructor and received the Goodwin Medal for “conspicuously effective teaching.” In his early years at Amherst, he remembers, there were pitchers of milk and unleashed dogs in Valentine, nearly every professor lived within blocks of campus, and he once felt out of place for wearing a sweater rather than a coat and tie to a faculty meeting.<br><br> Named a full professor at Amherst in 1978, Starr has taught courses in advanced calculus, complex variables, probability, statistics and data analysis, among other subjects. He also taught a first-year seminar in computers and society. His most frequently cited publication is an article about the 1970 Vietnam draft lottery, published in the <i>Journal of Statistics Education</i> in 1997. In recent years, he has published on mathematical puzzles. <br><br> During a 1972-73 sabbatical at Canada’s University of Waterloo, he chanced into the young field of computer graphics and created a wide variety of computer drawings based on mathematical themes. A number of them have appeared in books and magazines. In 1989, he displayed his work <i>Experiment in Shading</i>, drawn by a ball-point pen under computer control, and <i>Tecumseh</i>, a lithograph of a graph-theoretic drawing made by computer-controlled fountain pen, as part of an invited exhibit at The Print Club in Philadelphia. <i>Tecumseh </i>had previously been displayed in the juried National Prints and Drawings Exhibition at Mount Holyoke in 1976. Most recently, <i>Tecumseh </i>appeared in a 2007 exhibit at Kunsthalle Bremen, where it is in the permanent collection. <br><br> When the math department moved to the Seeley Mudd Building in 1984, Starr could have picked an office with a view of the Holyoke Range. Instead, he chose a room facing the Quad, where students who walked by in the evening could see his light on and know he was available for questions. <br><br> This summer, Starr spent weeks clearing out that office, including 50 years of files and nine bookcases full of books. (The files include articles about nonacademic careers for philosophers and a list of at least 22 alumni who believe they were taught by him when, in fact, they were in none of his courses.)<p>In retirement, Starr will continue to write for a column in the <i>College Mathematical Journal</i>. “I look forward to simply being able to read a book, walk into town, see a movie,” he says. First, though, he must organize the papers and files that now overwhelm his house. Once he accomplishes that task, he says, he’ll feel retired.</p><p><span class="fine-print">Photo by Frank Ward</span></p></div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/931">retirement</a></div><div class="field-item odd"><a href="/taxonomy/term/1701">mathematics</a></div><div class="field-item even"><a href="/taxonomy/term/1956">math</a></div><div class="field-item odd"><a href="/taxonomy/term/4822">Norton Starr</a></div><div class="field-item even"><a href="/taxonomy/term/10751">Starr</a></div></div></div><ul class="links inline"><li class="sharethis first last"><a href="/sharethis-ajax/120274" class="mm-sharethis">Share</a></li>
</ul>Mon, 10 Aug 2009 04:00:00 +0000kdduke120274 at https://www.amherst.eduhttps://www.amherst.edu/amherst-story/magazine/issues/2009summer/collegerow/starr/node/120274#commentsFall 2007 Math 19 Website
https://www.amherst.edu/academiclife/departments/courses/0708F/MATH/MATH-19-0708F/node/20397
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even">Prof. Leise's webpage for <a href="http://www.amherst.edu/~tleise/Math19Fall2007/Fall2007Math19.html"><span class="Apple-style-span">Fall 2007 Math 19</span></a>.</div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1701">mathematics</a></div><div class="field-item odd"><a href="/taxonomy/term/4923">Fourier analysis</a></div><div class="field-item even"><a href="/taxonomy/term/4924">wavelets</a></div><div class="field-item odd"><a href="/taxonomy/term/4925">Math 19</a></div></div></div>Wed, 29 Aug 2007 16:50:49 +0000tleise20397 at https://www.amherst.eduhttps://www.amherst.edu/academiclife/departments/courses/0708F/MATH/MATH-19-0708F/node/20397#commentsMath 5 Fall 2007 Website
https://www.amherst.edu/academiclife/departments/courses/0708F/MATH/MATH-05-0708F/node/20396
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even">Prof. Leise's page for <a href="/www.amherst.edu/~tleise/Math5Fall2007/Fall2007Math5.html"> Fall 2007 Math 5</a>.</div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1701">mathematics</a></div><div class="field-item odd"><a href="/taxonomy/term/4926">calculus</a></div><div class="field-item even"><a href="/taxonomy/term/4927">algebra</a></div><div class="field-item odd"><a href="/taxonomy/term/4928">Math 5</a></div></div></div>Wed, 29 Aug 2007 16:46:16 +0000tleise20396 at https://www.amherst.eduhttps://www.amherst.edu/academiclife/departments/courses/0708F/MATH/MATH-05-0708F/node/20396#commentsFall 2007/Spring 2008 Course Catalog
https://www.amherst.edu/academiclife/departments/mathematics-statistics/courses/course_catalog/node/9690
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even">The information below is taken from the printed catalog the college produces each year. For more up to date information, including links to course websites, faculty homepages, reserve readings, and more, use the 'courses' or semester specific link to your left. <p class="MsoBodyText"><strong>05.</strong><span> <strong>Calculus with Algebra.</strong> Mathematics 05 and 06 are designed for students whose background and algebraic skills are inadequate for the fast pace of Mathematics 11. In addition to covering the usual material of beginning calculus, these courses will have an extensive review of algebra and trigonometry. There will be a special emphasis on solving word problems.</span></p> <p>Mathematics 05 starts with a quick review of algebraic manipulations, inequalities, absolute values and straight lines. Then the basic ideas of calculus--limits, derivatives, and integrals--are introduced, but only in the context of polynomial and rational functions. As various applications are studied, the algebraic techniques involved will be reviewed in more detail. When covering related rates and maximum-minimum problems, time will be spent learning how to approach, analyze and solve word problems. Four class hours per week. Note: While Mathematics 05 and 06 are sufficient for any course with a Mathematics 11 requisite, Mathematics 05 alone is not. However, students who plan to take Mathematics 12 should consider taking Mathematics 05 and then Mathematics 11, rather than Mathematics 06.</p> <p>First semester. Visiting Professor Leise.</p> <p class="MsoBodyText"><strong>06.</strong><span> <strong>Calculus with Elementary Functions.</strong> Mathematics 06 is a continuation of Mathematics 05. Trigonometric, logarithmic and exponential functions will be studied from the point of view of both algebra and calculus. The applications encountered in Mathematics 05 will reappear in problems involving these new functions. The basic ideas and theorems of calculus will be reviewed in detail, with more attention being paid to rigor. Four class hours per week.</span></p> <p>Second semester. Visiting Professor Leise.</p> <p class="MsoBodyText"><strong>09.</strong><span> <strong>Lies, Damned Lies, and Statistics. </strong>In 1895 H.G. Wells wrote that "Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write." Today, statistics are cited to sway our opinion on everything from which toothbrush dentists prefer to how crime rates have changed from one political administration to the next. This seminar focuses not on statistical calculations, but on the critical evaluation of statistics that are presented every day in mass media. Topics to be discussed include proper survey and study methodologies, accurate visual displays of information, fundamentals of probability, the basics of hypothesis testing and confidence intervals, as well as the true meaning of correlation and the limitations of regression models. Three class meetings per week.</span></p> <p>Limited to 20 students. Omitted 2007-08.</p> <p class="MsoBodyText"><strong>11.</strong><span> <strong>Introduction to the Calculus.</strong> Basic concepts of limits, derivatives, anti-derivatives; applications, including max/min problems and related rates; the definite integral, simple applications; trigonometric functions; logarithms and exponential functions. Four class hours per week.</span></p> <p>First and second semesters. Professor Benedetto.</p> <p class="MsoBodyText"><strong>12.</strong><span> <strong>Intermediate Calculus.</strong> A continuation of Mathematics 11. Inverse trigonometric and hyperbolic functions; methods of integration, both exact and approximate; applications of integration to volume and arc length; improper integrals; l’Hôpital's rule; infinite series, power series and the Taylor development; and polar coordinates. Four class hours per week.</span></p> <p>Requisite: A grade of C or better in Mathematics 11 or consent of the Department. First and second semesters. The Department.</p> <p class="MsoBodyText"><strong>13.</strong><span> <strong>Multivariable Calculus.</strong> Elementary vector calculus; introduction to partial derivatives; multiple integrals in two and three dimensions; line integrals in the plane; Green’s theorem; the Taylor development and extrema of functions of several variables; implicit function theorems; Jacobians. Four class hours per week.</span></p> <p>Requisite: A grade of C or better in Mathematics 12 or the consent of the instructor. First and second semesters. The Department.</p> <p class="MsoBodyText"><strong>15.</strong><span> <strong>Discrete Mathematics.</strong> This course is an introduction to some topics in mathematics that do not require the calculus. Emphasis is placed on topics that have applications in computer science, including elementary set theory, logic, mathematical induction; basic counting principles; relations and equivalence relations; graph theory; and rates of growth. Additional topics may vary from year to year. This course not only serves as an introduction to mathematical thought but it is also recommended background for advanced courses in computer science. Four class hours per week.</span></p> <p>Second semester. Professor Cox.</p> <p class="MsoBodyText"><strong>16.</strong><span> <strong>Chaotic Dynamical Systems.</strong> Given a system such as the weather, the stock market or the population of a large city, there are many questions that can be asked about its long-term behavior. A Dynamical System is a mathematical model of such a system, and in this course, we will study dynamical systems from a mathematical point of view. In particular, we will describe the various ways in which a dynamical system can behave, and we will discover that some very simple systems can have surprisingly complex behavior. This will lead to the notion of a chaotic dynamical system. We will also discuss Newton's method, fractals, and iterations of complex functions. Three class hours per week plus a weekly one-hour computer laboratory. Offered in alternate years.</span></p> <p>Requisite: Mathematics 13 or consent of the instructor. Omitted 2007-08.</p> <p class="MsoBodyText"><strong>17. Introduction to Statistics.</strong><span> This course is an introduction to applied statistical methods useful for the analysis of data from all fields. Brief coverage of data summary and graphical techniques will be followed by elementary probability, sampling distributions, the central limit theorem and statistical inference. Inference procedures include confidence intervals and hypothesis testing for both means and proportions, non-parametric alternatives to standard hypothesis tests of the mean, the chi-square test, simple linear regression, and a brief introduction to analysis of variance (ANOVA). Three class hours plus one hour of laboratory per week.</span></p> <p>Limited to 20 students. First and second semesters. Professor Tranbarger.</p> <p class="MsoBodyText"><strong>18. Regression Modeling and Design of Experiments.</strong><span> This continuation of Mathematics 17 includes more detailed regression modeling using both linear and multiple regression techniques. Also covered are categorial data analysis techniques such as chi-square tests, regression modeling with indicator variables and logistic regression, followed by one and two factor analysis of variance (ANOVA). Two class hours plus two hours of laboratory per week.</span></p> <p>Not open to students who took Math 36 in 2005-06. Requisite: Mathematics 17. Second semester. Professor Tranbarger. </p> <p class="MsoBodyText"><strong>19.</strong><span> <strong>Wavelet and Fourier Analysis.</strong> The first half of the course covers continuous and discrete Fourier transforms (including convolution and Plancherel’s formula), Fourier series (including convergence and the fast Fourier transform algorithm), and applications like heat conduction along a rod and signal processing. The second half of the course is devoted to wavelets: Haar bases, the discrete Haar transform in 1 and 2 dimensions with application to image analysis, multiresolution analysis, filters, and wavelet-based image compression like JPEG2000. Three class hours per week plus a weekly one-hour computer laboratory.</span></p> <p>Requisite: Mathematics 13 and one of 21, 22, or 25. First semester. Visiting Professor Leise.</p> <p class="MsoBodyText"><strong>20. Topics in Differential Equations.</strong><span> The study of differential equations is an important part of mathematics that involves many topics, both theoretical and practical. The precise subject matter of this course will vary from year to year. For spring 2007, the topics will be nonlinear dynamics and chaos. We will study the dynamics of one- and two-dimensional flows. The focus of the course will be on bifurcation theory: how do solutions of nonlinear differential equations change qualitatively as a control parameter is varied, and how does chaos arise? To illustrate the analysis, we will consider examples from physics, biology, chemistry, and engineering. The course will also cover basic theorems concerning existence and uniqueness of solutions and continuous dependence on parameters. Three class hours per week plus a weekly one-hour computer laboratory.</span></p> <p>Requisite: Mathematics 13 or consent of the instructor. Limited to 20 students. Omitted 2007-08. </p> <p class="MsoBodyText"><strong>21.</strong><span> <strong>Linear Algebra. </strong>The study of vector spaces over the real and complex numbers, introducing the concepts of subspace, linear independence, basis, and dimension; systems of linear equations and their solution by Gaussian elimination; matrix operations; linear transformations and their representations by matrices; eigenvalues and eigenvectors; and inner product spaces. Special attention will be paid to the theoretical development of the subject. Four class meetings per week.</span></p> <p>Requisite: Mathematics 12 or consent of the instructor. This course and Mathematics 22 or 25 may not both be taken for credit. First semester. Professor Velleman.</p> <p class="MsoBodyText"><strong>22.</strong><span> <strong>Linear Algebra with Applications. </strong>The study of vector spaces over the real and complex numbers, introducing the concepts of subspace, linear independence, basis, and dimension; systems of linear equations and their solution by Gaussian elimination; matrix operations; linear transformations and their representations by matrices; eigenvalues and eigenvectors; and inner product spaces. Additional topics include ill-conditioned systems of equations, the LU decomposition, covariance matrices, least squares, and the singular value decomposition. Recommended for Economics majors who wish to learn linear algebra. Four class hours per week, with occasional in-class computer labs.</span></p> <p>Requisite: Mathematics 12 or consent of the instructor. This course and Mathematics 21 or 25 may not both be taken for credit. Second semester. Visiting Professor Leise.</p> <p class="MsoBodyText"><strong>24.</strong><span> <strong>Theory of Numbers.</strong> An introduction to the theory of rational integers; divisibility, the unique factorization theorem; congruences, quadratic residues. Selections from the following topics: cryptology; Diophantine equations; asymptotic prime number estimates; continued fractions; algebraic integers. Four class hours per week. Offered in alternate years.</span></p> <p>Requisite: Mathematics 12 or consent of the instructor. Omitted 2007-08.</p> <p class="BodyIndent1stLine"><strong>26.</strong><span> <strong>Groups, Rings and Fields.</strong> A brief consideration of properties of sets, mappings, and the system of integers, followed by an introduction to the theory of groups and rings including the principal theorems on homomorphisms and the related quotient structures; integral domains, fields, polynomial rings. Four class hours per week.</span></p> <p>Requisite: Mathematics 21 or 25 or both 15 and 22 or consent of the instructor. Second semester. Professor Benedetto.</p> <p class="MsoBodyText"><strong>27.</strong><span> <strong>Set Theory. </strong>Most mathematicians consider set theory to be the foundation of mathematics, because everything that is studied in mathematics can be defined in terms of the concepts of set theory, and all the theorems of mathematics can be proven from the axioms of set theory. This course will begin with the axiomatization of set theory that was developed by Ernst Zermelo and Abraham Fraenkel in the early part of the twentieth century. We will then see how all of the number systems used in mathematics are defined in set theory, and how the fundamental properties of these number systems can be proven from the Zermelo-Fraenkel axioms. Other topics will include the axiom of choice, infinite cardinal and ordinal numbers, and models of set theory. Four class hours per week.</span></p> <p>Requisite: Mathematics 15, 21 or 25 or 28, or consent of the instructor. Second semester. Professor Velleman.</p> <p class="MsoBodyText"><strong>28.</strong><span> <strong>Introduction to Analysis.</strong> Completeness of the real numbers; topology of n-space including the Bolzano-Weierstrass and Heine-Borel theorems; sequences, properties of functions continuous on sets; infinite series, uniform convergence. The course may also study the Gamma function, Stirling’s formula, or Fourier series. Four class hours per week.</span></p> <p>Requisite: Mathematics 13. Second semester. Professor TBA.</p> <p class="MsoBodyText"><strong>29. Probability.</strong><span> This course explores the nature of probability and its use in modeling real world phenomena. The course begins with the development of an intuitive feel for probabilistic thinking, based on the simple yet subtle idea of counting. It then evolves toward the rigorous study of discrete and continuous probability spaces, independence, conditional probability, expectation, and variance. Distributions covered include the Bernoulli and Binomial, Hypergeometric, Poisson, Normal, Gamma, Beta, Multinomial, and bivariate Normal. Four class hours per week.</span></p> <p>Not open to students who have previously taken Mathematics 14. Requisite: Mathematics 12 or consent of the instructor. Omitted 2007-08.</p> <p class="MsoBodyText"><strong>30. Mathematical Statistics.</strong><span> This course examines the theory behind common statistical inference procedures including estimation and hypothesis testing. Beginning with exposure to Bayesian inference, the course will cover Maximum Likelihood Estimators, sufficient statistics, sampling distributions, joint distributions, confidence intervals, hypothesis testing and test selection, non-parametric procedures, and linear models. Four class hours per week.</span></p> <p>Requisite: Probability (Mathematics 14 or 29) or consent of the instructor. Omitted 2007-08. </p> <p class="MsoBodyText"><strong>31.</strong><span> <strong>Functions of a Complex Variable.</strong> An introduction to analytic functions; complex numbers, derivatives, conformal mappings, integrals. Cauchy’s theorem; power series, singularities, Laurent series, analytic continuation; Riemann surfaces; special functions. Four class hours per week.</span></p> <p>Requisite: Mathematics 13. First semester. Professor Starr.</p> <p class="MsoBodyText"><strong>34.</strong><span> <strong>Mathematical Logic.</strong> Mathematicians confirm their answers to mathematical questions by writing proofs. But what, exactly, is a proof? This course begins with a precise definition specifying what counts as a mathematical proof. This definition makes it possible to carry out a mathematical study of what can be accomplished by means of deductive reasoning and, perhaps more interestingly, what cannot be accomplished. Topics will include the propositional and predicate calculi, completeness, compactness, and decidability. At the end of the course we will study Gödel’s famous Incompleteness Theorem, which shows that there are statements about the positive integers that are true but impossible to prove. Four class hours per week. Offered in alternate years.</span></p> <p>Requisite: Mathematics 15, 21 or 25 or 28, or consent of the instructor. Omitted 2007-08. </p> <p class="MsoBodyText"><strong>36.</strong><span> <strong>Advanced Applied Statistics.</strong> This continuation of Mathematics 17 includes more detailed regression modeling using both linear and multiple regression techniques. Also covered are categorical data analysis techniques such as chi-square tests, regression modeling with indicator variables and logistic regression, followed by one and two factor analysis of variance (ANOVA). Four class meetings per week.</span></p> <p>Requisite: Mathematics 17. Omitted 2007-08.</p> <p class="MsoBodyText"><strong>37.</strong><span> <strong>Topics in Mathematics.</strong> The topics may vary from year to year. The topic for fall 2007 is Galois Theory, which is the systematic study of the roots of polynomials. The key idea, first glimpsed by Lagrange and later brought to fruition by Galois, is that there is a deep relation between group theory and the structure of the set of roots of a given polynomial. One of the most famous results of the theory is that there is no analogue of the quadratic formula for polynomials of degree five and higher; another is the impossibility of trisecting an angle with straightedge and compass. This course will develop Galois Theory from the basics of groups and fields. Four class hours per week. Offered in alternate years. </span></p> <p>Requisite: Mathematics 26. First semester. Professor Benedetto.</p> <p class="MsoBodyText"><strong>42.</strong><span> <strong>Functions of a Real Variable.</strong> An introduction to Lebesgue measure and integration; topology of the real numbers, inner and outer measures and measurable sets; the approximation of continuous and measurable functions; the Lebesgue integral and associated convergence theorems; the Fundamental Theorem of Calculus. Four class hours per week. Offered in alternate years.</span></p> <p>Requisite: Mathematics 28. Second semester. Professor Cox.</p> <p class="MsoBodyText"><strong>44.</strong><span> <strong>Topology.</strong> An introduction to general topology; the topology of Euclidean, metric and abstract spaces, with emphasis on such notions as continuous mappings, compactness, connectedness, completeness, separable spaces, separation axioms, and metrizable spaces. Additional topics may be selected to illustrate applications of topology in analysis or to introduce the student briefly to algebraic topology. Four class hours per week. Offered in alternate years.</span></p> <p>Requisite: Mathematics 28. Omitted 2007-08. </p> <p class="MsoBodyText"><strong>77, 78.</strong><span> <strong>Senior Departmental Honors.</strong></span></p> <p>Open to seniors with the consent of the Department. First and second semesters. The Department.</p> <p class="MsoBodyText"><strong>97, 98.</strong><span> <strong>Special Topics.</strong> Independent Reading Course.</span></p> <p>First and second semesters. The Department.</p> <h4 class="rule-above">Related Course</h4> <p class="MsoBodyText"><strong>Philosophy of Mathematics. </strong><span>See Philosophy 50.</span></p> <p>Omitted 2007-08. Professors A. George and Velleman.</p> </div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1701">mathematics</a></div><div class="field-item odd"><a href="/taxonomy/term/1825">courses</a></div></div></div>Wed, 20 Jun 2007 15:25:14 +0000blong9690 at https://www.amherst.eduAmherst College Math Professor Daniel Velleman To Edit American Mathematical Monthly
https://www.amherst.edu/news/news_releases/2006/05_2006/node/8767
<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="fine-print">May 12, 2006 <br> Director of Media Relations<br> 413/542-8417</p> <p class="text">AMHERST, Mass.—Daniel J. Velleman, professor of mathematics at Amherst College, has been selected to become the next editor of the <em>American Mathematical Monthly</em>. His term will begin with the January 2007 issue. The <em>American Mathematical Monthly</em> publishes articles, as well as notes and other features, about mathematics and the profession for readers with a broad range of mathematical interests, from professional mathematicians to undergraduate students of mathematics. The <em>American Mathematical Monthly</em> is written to be read, enjoyed and discussed.<br><br> Velleman came to Amherst in 1983, having earned a B.A. at Dartmouth College, and M.A. and Ph.D. degrees in mathematics at the University of Wisconsin-Madison. He is author of <em>How to Prove It: A Structured Approach</em> (1994), and co-author of <em>Which Way Did the Bicycle Go? And Other Intriguing Mathematical Mysteries</em> (with Joseph Konhauser and Stan Wagon, 1996) and <em>Philosophies of Mathematics</em> (with Alexander George, 2002).</p><p class="text" align="center">###</p></div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/55">editor</a></div><div class="field-item odd"><a href="/taxonomy/term/552">news releases</a></div><div class="field-item even"><a href="/taxonomy/term/782">faculty</a></div><div class="field-item odd"><a href="/taxonomy/term/1701">mathematics</a></div><div class="field-item even"><a href="/taxonomy/term/1956">math</a></div><div class="field-item odd"><a href="/taxonomy/term/2857">Daniel Velleman</a></div><div class="field-item even"><a href="/taxonomy/term/2858">American Mathematical Monthly</a></div></div></div>Fri, 15 Jun 2007 13:50:31 +0000daustin098767 at https://www.amherst.edu