UNDERGRADUATE MATHEMATICS SEMINAR

Spring ’10 Preregistration Process Begins this Weekend

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Department of Mathematics February 12, 2010

The next seminar of the winter term will be: DATE: MONDAY, February 15th Time & 4:15pm – Refreshments in the Math Common Room, Bailey 204 Location: 4:30pm – Seminar in Bailey 207 In this seminar, Union College’s Professor Kim Plofker will deliver the following talk:

ABSTRACT: The development of modern calculus wasn't the smooth transition pictured in today's textbooks, but rather a long struggle between practicality and precision. Handling the tricky quantities known as infinitesimals, which might be zero or not as the situation required, led early modern mathematicians into some awkward contradictions and some vehement disputes. This talk surveys some of the peculiar innovations in calculus that you won't find in your calculus book, and the controversies, shock, and outrage that they provoked in their day.

TITLE: Embarrassing Moments in the History of Calculus

The Timeline

• Petition course signup: Sat. Feb. 13 - Tue. Feb. 16. Log into webadvising.union.edu to request a slot in a petition course. This spring term, NO MATH COURSES ARE PETITION COURSES.

• Acceptance period: Tue. Feb. 23 – Wed Feb. 24. Log into webadvising.union.edu and change the ones marked “Faculty Approved” to “Student Accepted” if you wish to register for the course.

• Registration Period on Web at Hale House: Fri. Feb. 26 – Wed. Mar. 3.

The Courses: This spring, the Math Department is offering several interesting courses beyond the calculus sequence that are suitable for math majors and minors.

Math 130 is a course in Differential Equations. Math 115 is a prerequisite. A more theoretical version of this course, Math 234, is also being offered. Be aware that students may not take both 130 and 234.

Math 177 is a new course entitled “Historical Development of Math”, taught by an expert in this field, Prof. Kim Plofker, this week’s seminar speaker.

Math 199 is the department’s “bridge course,” intended to help students make the transition from computationally oriented courses to more theoretical proof-writing courses. It is a required course for all math majors and minors that is usually taken after a student has taken Math 115.

Beyond Math 199: There are four courses being offered this spring that have a Math 199 prerequisite: Math 221 (Cryptology), Math 234 (Differential Equations), Math 332 (Abstract Algebra), and Math 480 (Foundations of Mathematics).

Math 221 and Math 234, as 200-level courses, are particularly appropriate for students coming from Math 199. [Be aware that taking Math 221, students are generally ineligible to enroll in Math 235 (Number Theory) next spring, and also that Math 234 is not open to students who have passed Math 130.] Math 332 is a beautiful course that generalizes what you know about algebra in the integers and real numbers to a more abstract setting and is required for the major; one should have had at least one 200-level course before enrolling in Math 332. Math 480 is a course that studies the logical underpinnings of mathematics. Students considering graduate school in math or students who seek honors in the major should consider this offering.

Pieces of Theses: A View from Frank Cassano, ‘10 Senior Thesis: Friend or Foe? Most people would relate a senior thesis to an enemy rather than a friend. Almost every senior I see on campus right now is so busy stressing over his or her thesis; I have not felt that way at all. I don’t know how many other times you will hear this, but my thesis experience was enjoyable.

As one of the few applied math majors on campus, my thesis was different than most. Instead of working on a problem or proof, I studied a technique Professor Jue Wang is using in her research on Backscatter-Contour-Attenuation Joint Estimation Model for Attenuation Compensation in Ultrasound Imagery. My thesis was not limited to learning and writing about a certain problem; Professor Wang encouraged me to understand the material and then present my work to her in a professional manner. Without her help and her knowledge in the subject, my thesis would have not have gone as smoothly as it did.

Here is a brief description of level sets: a level set method is a numerical technique for tracking interfaces and shapes. We use level set methods to perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects. We solve these systems to create computer models for propagating interfaces. The propagating interfaces include, but are not limited to, waves, burning flames and material boundaries. So in the case of Professor Wang’s research, this method will be used to help find the boundary of ultrasound images.

The main idea of level set methods is to use partial differential equations to design computational techniques to an associated initial value problem and compute interface motion. In other words, level set methods are designed to track these moving interfaces. Therefore, we can use level set methods for problems where the speed function is either positive or negative; thus the interface can move either forwards or backwards.

The level set method was developed in the 1980s by American mathematicians Stanley Osher and James Sethian. Their main idea came from the analysis of curve and surface evolution. After noticing the link between evolving fronts and hyperbolic conservation laws was established, Osher and Sethian went on to use numerical technology of shock schemes to move the interface.

Their creation of level set methods led to many applications. Some of these applications included noise removal, medical scans, semiconductors, optimal design, seismic analysis tumor modeling and wave propagation.

If I could give any advice to future math thesis writers, it would be to try to work on something you really enjoy and have interest in. I started out with a different topic at the start of fall term. With the help and encouragement of Professor Wang, I changed my focus and learned a tremendous amount about a unique and interesting topic. As long as you enjoy your topic and are willing to do the work required to understand the new material, your senior thesis will be nowhere as difficult as some people might think!

Last week’s problem was a tough one to which no one, unfortunately, submitted a correct solution. A solution has been posted on the bulletin boards around Bailey Hall. Here is this week’s problem: Look at the picture to the right. Notice that in this stack of magazines, the top magazine lies completely beyond the edge of the desk. (The ruler is at the end of the magazine and the purple folder is at the edge of the desk.) The question(s): What is the least number of magazines that can be used so that a stack of magazines will balance AND the top magazine lies completely beyond the edge of the desk? How should the magazines be stacked so that the top magazine is maximally extended? What if the top two magazines are to lie beyond the desk’s edge? Can it be done? How so? Play with this/these problem/s and try to derive a super-extendo strategy.

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Problem of the Newsletter: February 12, 2010

Professor Friedman will accept solutions to this problem until 12:00 noon Thursday, February 18th. Email your solution to him (friedmap@union.edu) or put it in his mailbox in the Math office in Bailey Hall.