This has come just at the right time for Prof. Tibor Beke, who is teaching a section of Explorations in Math to students in the Humanities, Fine Arts and Social Sciences who have accepted the challenge to explore some mathematics as a somewhat higher level than is normally offered to students on our South Campus.

We all know the formula \[ 1+ 2 +3+ \dots + n = \frac{n(n+1)}{2} \]

that can be justified in a number of ways --- by induction, but adding the left-hand sum to itself in the reverse order, or by decomposing an \(n\) by \(n+1\) rectangle into two congruent pieces, each of which contains \(1+2+3+\dots+n\) unit squares. But what about \(1^2 + 2^2 + 3^3 + \dots + n^2 \)? A nice way to visualize such a sum is as the number of cubes in a skewed "Mayan pyramid." Here is are six Mayan pyramids printed by Rida that are a visual representation of \(1^2 + 2^2 + 3^2 + 4^2 \).

Three such pyramids can be combined to form a cuboid with a set of steps next to one of the faces. The steps in two such formations can, if you orient the pieces correctly, be fit together.

When this is done, you get a single cuboid. In this case, it's a \(4 \times 5 \times 9\) cuboid, demonstrating that \(6(1^2 + 2^2 + 3^2 + 4^2) = 4 \times 5 \times 9 \). The 5 in this equality is one more the 4 and 9 is one more than \(2 \times 4 \).

This configuration works for the sum of the first \(n\) squares for all positive values of \(n\), which demonstrates a general identity, after dividing by 6: \[ \sum _{k=1}^{n } {k^2} = \frac{n(n+1)(2n+1)}{6} \]

The nice thing about having a tactile representation of this fact is that students can actually put the pieces together and see how it is really not dependent on the number of squares. "Proofs with no words" such as this one have traditionally been accepted as valid proofs. They are limited to our three dimensions, but the printing of complex objects opens up possibilities that we haven't had until now.

Early in January of this year we were authorized to do faculty searches for two tenure-track positions, a statistician and an applied mathematician. Although it was a late start, we are happy to announce that both searches were successful.

Dr. Jong Soo Lee, a statistician who was most recently at the University of Delaware, has joined us this fall. His general research areas are functional data analysis, nonparametric statistics and the application of statistics. His Ph. D. was earned at Rice University with a thesis titled Aspects of Functional Data Inference and Its Applications (Advisor: Dennis Cox).

Dr. Hung Phan, an applied mathematician who was most recently at the University of British Columbia, Okanagan, will join us in the spring. His general research areas are Optimization, Numerical Methods, and Variational Analysis. His Ph. D. was earned at Wayne State University, with a thesis titled New Variational Principles with Applications in Optimization Theory and Algorithms (Advisor: Boris Mordukhovich).

This infusion of new personnel comes just in time to offset losses due to retirement. Charlie Byrne retired after being with us since 1986. He served as department chair for a term, and graduate coordinator for many years. His expertise in areas such as optimization and image processing is hard to replace.

Congratulations to Charlie on his recently published book, An Introduction to Optimization.

In addition, we've lost the services of Alan Kaplan, who retired after forty years of service. His many contributions to the department will be missed.

More good news is that we've been authorized to do two more searches for tenure-track faculty. We've decided to search for another statistician and a mathematician. The research area for the mathematician is a bit less focused that in our earlier search. To see details of our postings, go to the UML Jobs site.

There is plenty of other news to report, but I intend to make these postings more frequent. So I'll close here.

Naturally, the plot labeled MRKT BSKT is our spending at Market Basket. The SHAW graph is actually a combination of our spending at Shaw's and Hannaford's. I think I stopped at Shaw's around February and bought a few items, but other than that, spending there started at the same time as the boycott.

Instead of making this a one-time exercise, I decided to develop a function that would create other graphs like this one. My main summer project has been to rebuild the stairs at my son's house in Chelmsford, adding a small deck. Out of convenience, my main spending on the project, which started in mid-July, has been at three locations. They are my local NH Home Depot (no sales tax!), where I've bought most of the smaller items I could fit into my Subaru, the Lowell Lowe's, which is five minute from the project and where I've bought several last minute items I didn't expect to need, and finally, Friend Lumber from whom I've had lumber delivered. Here is the spending plot at those locations for 2014, with another ~$100 spending to go at this point:

One more example, here is my coffee spending - this is flawed in that it is only the debit card spending, so a lot of my purchases at Einstein's during the year are not included. A&E is my local coffee roaster, and I definitely spend more there that anywhere else, even if I took into account cash sales. Also, I use the Starbucks app on my phone, which is why larger, less frequent purchases are recorded there.

If you'd like to try this with your spending, you can download a copy of the Mathematica Package that creates these graphs. The package assumes you have a Bank of America debit card. You would download activity from your account and use the file path to that .csv file on your computer to specify the data file. With a bit of Mathematica expertise, it shouldn't be hard to adapt the package to other banking systems.

Erinn McLaughlin, together with Michelle Scribner-MacLean (Graduate College of Education) and William Morton (Lowell National Historical Park), described how she and her classmates teamed with the National Park to create science and math projects for high school students. The work was part of the required UTeach course, Project Based Instruction.

At a poster session that included dozens of student submissions, John Romano won the *best student poster for a research project* for the work he did as an intern with M2D2 and Lowell High School. Erinn also displayed a poster on the mathematics of water wheels.

- The
**CUPM Curriculum Guide**is produced by the MAA Committee on the Undergraduate Program in Mathematics to guide mathematics departments in designing curricula for their undergraduate students. The 2004 version was the last to come out. The new version will be out in 2015. We were told that a draft will appear at maa.org/cupm in the near future. - In 2012, the Conference Board of the Mathematical Sciences issued the latest recommendations for teacher preparation in mathematics:
**The Mathematical Education of Teachers II (MET2)**. A few highlights of the new recommendations:

- Elementary teachers should take four mathematics courses on elementary school mathematics. This doesn't mean that the mathematics they are taught are elementary. The objective is to give teachers a deeper understanding of the mathematics that is taught in elementary grades. For example, while an elementary school teacher may teach division, coursework might include continued fractions or a study of the periodic nature of decimal fractions.
- Recommendations for middle school teachers include at least 24 credits of mathematics, including at least 15 credits designed specifically for future middle grades teachers that address essential ideas in the middle school curriculum.
- It is still recommended that prospective High school teachers complete coursework equivalent to that of a mathematics major. One change is that at least nine credits involve advanced study of secondary mathematics.
- The American Statistical Association (ASA) will be releasing
**The Statistical Education of Teachers (SET)**in 2014. It is expected to put a greater emphasis on data analysis.

I think that a few developments at UMass Lowell have put us in a good position with respect to these recommendations. A few years ago, the College of Education and Mathematical Sciences Department collaborated with other UMass campuses on the development of mathematics courses for prospective elementary school teachers. This gives us a good start toward being in line with recommendations at that level.

There will be more for us to do to address these recommendations, but I think we are on the right track!

]]>I hadn’t expected to be writing this column up until last May, but here I am serving as chair of Mathematical Sciences again after a 14 year break. I had been chair for nine years, roughly coinciding with the Clinton administration. There are several news items to report, but first I’d like to reflect on what’s changed in 14 years.

- The last time I was chair we were housed in Olsen, now we’re in Olney. With new buildings starting to pop up on campus, several departments, including us, could be moving again in the coming years.
- Since my first “retirement” as chair, we’ve seen the retirements of Professors Yin, Makovoz, Weinberg, Spiegel, Mueller, Winslow, Berkovits, Ruskai, and Samarov. We currently have seven tenured faculty and seven Instructors who have joined us since then.
- When I left the chair’s position in the 20th century, we offered a two semester version of Calculus I, Calculus IA and Calculus IB. After trying “Preparation for Calculus” for a few years, we now we have a new two semester version of Calculus I. Same name, but different catalog numbers. Everything is cyclic.
- Scholarships: Toward the end of my first run as chair, Russ and Mary Bedell’s generosity had just brought us our first endowed scholarship. As you can see below, we’ve been fortunate to have two more endowed scholarships started since then and hope to be able to announce more in the future.
- Finally, Tangents didn’t exist in 1999 - the first issue was in 2002. So now I will move to the present.

One of the main reasons why I agreed to take on the chair’s position is that Kiwi has agreed to help me out as associate chair. In addition, I’m getting lots of help from other members of the faculty and, of course, Cori Lee, our Administrative Assistant. I thank all of them for helping make my return reasonably smooth.

It’s helped that I’ve been able to divest some of the things I was doing. Steve Pennell has started his “chair’s retirement” by taking over as coordinator of the Industrial Math PSM program. Also, I had been editing Tangents since 2002 and I’m glad to announce that Jim Propp has agreed to take over as editor.

Tibor Beke has returned to us after a year on sabbatical traveling through Europe and the States in 2012-13 and Ravi Montenegro has left us for a 2013-14 sabbatical to collaborate with researchers in Japan and New Zealand.

Math majors Olivia Demers and Mary Mersereau had been working with Shelley Rasmussen in doing research on the mathematics of weaving. They recently shared their enthusiasm for the subject with some children in a local summer camp.

In the past year, Industrial Mathematics PSM Students Isaac Duodu (Putnam Investments) and Lauren Edwards (Genscape) completed internships that gave them valuable experience to further their careers. Undergraduates Gifty Bado and Tyler Gilzinger spent their summers as Co-Op students at Putnam Investments. After completing a co-op position at Mercury Computer, recent B. S. graduate Owen Welch accepted a full-time position at Mercury. Krithika Manohar, who graduated last May, is in her first semester of the Applied Math doctoral program at the University of Washington.

Last May, we held our annual Awards Ceremony and Alumni Reception where we presented the following awards:

- Outstanding Graduate Student: Nour Almansour
- Shapiro Scholarship: Chris Leger
- Bedell Scholarship: Tyler Gilzinger
- Zamanakos Scholarships: David Campbell, Mary Mersereau and Kevin Southwick
- Hall Prize: Kevin Cerritelli.

The 2014 awards ceremony will be held at the UML Inn and Conference Center on April 25 from 5:30 to 7:30. Alumni are always welcome to attend. It’s a great setting.

]]>Here is a visualization of the Fall 2013 Math Schedule showing the times at which each course is offered.

]]>

One of the neat things about Sage, the open-source computer algebra system is that you can easily embed it into any web page. Here is an example of some code that can be evaluated to plot a function and it's derivative. For more information about Sage: http://sagemath.org. To learn how to embed Sage into your web page: http://aleph.sagemath.org/static/about.html.

- The book is now listed as part of the American Math Institute's Open Textbook Initiative.
- It has its first outside adoptions: classes at the University of the Puget Sound, Grinnell College, Casper College and Luzurne Community College are all using the book in the spring of 2013.

Resources, such as this dynamic demonstration of cyclic subgroups of the group of integers mod n are continuing to be developed. Move the sliders to see the different subgroups with varying moduli:

Applied Discrete Structures by Alan Doerr & Kenneth Levasseur is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License.]]>

Wildness wasn't always seen as beautiful. Prior to the Romantic era, the spectacle of untamed nature was a source of fear, not pleasure. Likewise, randomness wasn't always seen as mathematical. If math is the ultimate form of certain knowledge, what could be more un-mathematical than the things we don't know, like the face that will be shown by a die that we haven't rolled yet? If math is about perfection of form, what could be more un-mathematical than the shaggy, unkempt curves we see in turbulent water or in swirling smoke?

We live in a Romantic era of mathematics, when the wildness of random processes isn't just a respectable branch of mathematics, but part of the main trunk, close to the heart of the subject, where all the different sub-disciplines of mathematics converge and converse.

There are many reasons for this change: one of them is the electronic computer, which is responsible for all the images you'll see tonight. Like Galileo's telescope, the computer lets us explore worlds that were hitherto invisible. But unlike the moons of Jupiter, which were there before Galileo thought to look at them, the worlds of modern mathematics are to a great extent the creation of human minds, and their variety is limited only by what our minds are capable of imagining. In choosing which imaginary worlds to explore, we're guided by a desire to understand our own world, of course; and like physicists, we often create simplified worlds that capture, and teach us about, some aspect of the world we live in. But pure mathematicians are artists as much as they are scientists, so we often choose worlds based on esthetic criteria. Mathematicians vary in their ideas of mathematical beauty, but we tend to value mathematical systems that combine the classical and the baroque esthetics in a particular way: a classical simplicity in the underlying rules, and a baroque richness of the phenomena that the rules give rise to. We value surprise, and we find symmetry beautiful. So when symmetry catches us by surprise, we are especially gratified, as is the case for some of the worlds I'll show you tonight.

Usually the "worlds" of mathematics are purely conceptual worlds: through familiarity, they become vivid to a mathematician who has studied them, and can come to seem solid, even habitable. But these intuitions are very personal and usually hard to convey. This semester, MSRI has focussed on an area of mathematics that is unusually visual in nature: the study of random spatial processes. The worlds researchers like me study actually look like something, and what they look like, sometimes, are worlds. I'm here to share with you some images that illustrate what we've been up to. I call these images postcards for several reasons.

First, I don't make claims for them as art. There is such a thing as mathematical art, and it's a booming enterprise, but most of the images you'll see are more like nature photography: attempts to render something natural as accurately as possible. The fact that the natural objects being rendered are imaginary doesn't change the underlying representational intention.

Second, a postcard is something you receive from someone far away, and it's a given that getting a postcard from a place is not the same as visiting it. We mathematicians spend years learning the tools of mathematics and our reward is the opportunity to spend more years toiling in the application of those tools to problems we have come to care about; so if effort is an index of ascent, then we speak to you from underpopulated mountaintops. I am anxious to disclaim any implication that our elevation is a mark of superiority; it's just a fact about where we've chosen to go. And there is something unbridgeable, or very hard to bridge, about the distance between the mindstate of an intelligent person with no specialized knowledge and another person, possibly not as intelligent, who has spent years exploring some particular domain and acquainting himself or herself with its quirks. Indeed, for many of us researchers, the visual appeal of pictures like this is not the beauty that drives us; instead, we are driven to find beautiful trajectories of thought that let us prove that what these pictures tell us is really true.

So, I am not going to be able to tell you everything you may want to know about the images; partly because of the lack of time, and partly because of the intricacy of the backstories of some of the pictures, and partly because of limitations in my own knowledge or my limitations as an explainer. But if I succeed in making you want to know more than I can tell you, I'll judge the evening a success.

It's all well and good to invent new worlds, but without powerful ideas that tie our invented worlds together, mathematics would be in danger of splintering into a hundred hobbies, mutually well-disposed but not having a lot to say to each other. Fortunately there are unifying ideas. One that I've already talked about is the way chance at the microscale gives rise to determinism, or fate, at the macroscale. A central example of this is the Wiener process, which is what mathematicians call Norbert Wiener's idealization of Brownian motion. In the real world, the dance of pollen grains on a microscope slide has much the same sort of shagginess across many scales of magnitude, from the cellular down to the molecular. What if this shagginess looked the same at even smaller scales? In the real world this is false --- quantum weirdness is different from Brownian shagginess --- but as mathematicians we are free to imagine a world in which there is no preferred length-scale, atomic or otherwise, and Brownian motions have Brownian shagginess no matter how closely we zoom in. Turning this into mathematics was Norbert Wiener's contribution. Just as the bell-shaped or Gaussian curve gives a kind of universal law for many sorts of random quantities, the Wiener process is a universal law for many sorts of processes that evolve in time, and even when a process doesn't look like a Wiener path, it's often a modified Wiener path in disguise. For instance, the erratically changing prices seen in the stock market are well-modeled using a one-dimensional Wiener process, like a pollen-grain moving to and fro in a one-dimensional world.

Here's a final thought for you: These images aren't just postcards from other mathematical worlds. I like to think that they're also postcards from our world's future, twenty or fifty years from now, when a new generation of mathematicians, building on our work, will be able to prove things that our generation can only conjecture. Thank you for coming, and enjoy the rest of your evening here at MSRI!

A link to download a .zip archive containing the book and instructions on how to install it is

]]>In this second example, the number of faces on a die varies, and the generating function to the third power is expanded:

]]>The Spring 2012 issue of |

\[\int_0^x f(t) k(x,t) \, dt\] |

A Brown-Bag Applied Math Seminar has been launched in the UMass Lowell Department of Mathematical Sciences. Meetings are scheduled every Friday from noon to 1 PM in Olney 218. The first meeting is scheduled for Friday, February 17, 2012.

Interested students (both undergraduate and graduate) as well as interested faculty are welcome to attend.

This is a lunch-time working Seminar, so please feel free to bring your lunch or your favorite beverage, or just bring yourself!!

The seminar is being organized by Dimitris Christodoulou, Dimitris_Christodoulou@uml.edu]]>
http://blog.uml.edu/cotangents/2012/02/brown-bag_applied_math_seminar.html