College Math Teaching

May 20, 2016

Student integral tricks…

Ok, classes ended last week and my brain is way out of math shape. Right now I am contemplating how to show that the complements of this object

bingsling

and of the complement of the object depicted in figure 3, are NOT homeomorphic.

brinknot

I can do this in this very specific case; I am interested in seeing what happens if the “tangle pattern” is changed. Are the complements of these two related objects *always* topologically different? I am reasonably sure yes, but my brain is rebelling at doing the hard work to nail it down.

Anyhow, finals are graded and I am usually treated to one unusual student trick. Here is one for the semester:

\int x^2 \sqrt{x+1} dx =

Now I was hoping that they would say u = x +1 \rightarrow u-1 = x \rightarrow x^2 = u^2-2u+1 at which case the integral is translated to: \int u^{\frac{5}{2}} - 2u^{\frac{3}{2}} + u^{\frac{1}{2}} du which is easy to do.

Now those wanting to do it a more difficult (but still sort of standard) way could do two repetitions of integration by parts with the first set up being x^2 = u, \sqrt{x+1}dx =dv \rightarrow du = 2xdx, v = \frac{2}{3} (x+1)^{\frac{3}{2}} and that works just fine.

But I did see this: x =tan^2(u), dx = 2tan(u)sec^2(u)du, x+1 = tan^2(x)+1 = sec^2(u) (ok, there are some domain issues here but never mind that) and we end up with the transformed integral: 2\int tan^5(u)sec^3(u) du which can be transformed to 2\int (sec^6(u) - 2 sec^4(u) + sec^2(u)) tan(u)sec(u) du by elementary trig identities.

And yes, that leads to an answer of \frac{2}{7}sec^7(u) +\frac{4}{5}sec^5(u) + \frac{2}{3}sec^3(u) + C which, upon using the triangle

integraltrick

Gives you an answer that is exactly in the same form as the desired “rationalization substitution” answer. Yeah, I gave full credit despite the “domain issues” (in the original integral, it is possible for x \in (-1,0] ).

What can I say?

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January 26, 2016

The walk of shame…but

Filed under: academia, research — Tags: , — collegemathteaching @ 9:07 pm

Well, I walked to our university library with a whole stack of books that I had checked out to do a project…one which didn’t work out.

But I did check out a new book to get some new ideas…and in the book I found a little bit of my work in it (properly attributed). That was uplifting.

Now to get to work…

November 19, 2014

Tension between practitioners and theoretical mathematicians…

Filed under: academia, applied mathematics, mathematician, research — Tags: — collegemathteaching @ 2:01 am

I follow Schneier’s Security Blog. Today, he alerted his readers to this post about an NSA member’s take on the cryptography session of a mathematics conference. The whole post is worth reading, but these comments really drive home some of the tension between those of us in academia :

Alfredo DeSantis … spoke on “Graph decompositions and secret-sharing schemes,” a silly topic which brings joy to combinatorists and yawns to everyone else. […]

Perhaps it is beneficial to be attacked, for you can easily augment your publication list by offering a modification.

[…]

This result has no cryptanalytic application, but it serves to answer a question which someone with nothing else to think about might have asked.

[…]

I think I have hammered home my point often enough that I shall regard it as proved (by emphatic enunciation): the tendency at IACR meetings is for academic scientists (mathematicians, computer scientists, engineers, and philosophers masquerading as theoretical computer scientists) to present commendable research papers (in their own areas) which might affect cryptology at some future time or (more likely) in some other world. Naturally this is not anathema to us.

I freely admit this: when I do research, I attack problems that…interests me. I don’t worry if someone else finds them interesting or not; when I solve such a problem I submit it and see if someone else finds it interesting. If I solved the problem correctly and someone else finds it interesting: it gets published. If my solution is wrong, I attempt to fix the error. If no one else finds it interesting, I work on something else. 🙂

November 1, 2014

Ok Graduate Student, do you want a pure math Ph. D.???

Filed under: academia, calculus, editorial, research — collegemathteaching @ 2:19 am

appliedvspuremath

This slide made me chuckle (click to see a larger version). But here is the point of it: it is very, very difficult to earn your living by researching in pure mathematics.

Is it a reasonable expectation for you?

Ask yourself this: look at your advisor. Is your advisor considerably smarter than you are, or even moderately smarter than you are? If so, then forget about earning your living as a research professor in pure math. It. Is. NOT. Going. To. Happen.

Yeah, you might get a post-doc. You might even manage to get one of those “tenure track with little hope for tenure” jobs at a D-I research university…maybe (perhaps unlikely?).

I’ve been on search committees. I’ve seen the letters for those who didn’t get tenure; often these folks had decent publication records but didn’t get large enough external grants.

It is brutal out there.

If you get a pure math Ph. D. and you aren’t your advisor’s intellectual equal, about your only hope for a tenured academic job is at the “teaching intensive” universities; basically you’ll spend the vast majority of your time attempting to teach calculus to students of very average ability; after all, most of the teaching load in mathematics is teaching service courses rather than majors courses.

It does have its charm at times, but after 20+ years, it gets very, very old. I’ll discuss how to alleviate the boredom in a responsible way in another post. (e. g., it is probably a bad idea to, say, spice it up by teaching integration via hyperbolic trig functions or to try to teach residue integrals).

So, ask yourself: is your passion research and discovery? Or, is it teaching average students? If it is the latter: well, go ahead and get that theoretical math Ph. D.; after all, there ARE jobs out there, and we’ve hired a couple of people last year and might hire some more in the next couple of years.

IF your passion is research and mathematical discovery and you aren’t your advisor’s intellectual equal, either switch to applied mathematics (more demand for such research) OR enhance your education with sellable skills such as computer programming/modeling, software engineering or perhaps picking up a masters in statistics. Make yourself more marketable to industry.

July 17, 2014

I am going to celebrate this…

Filed under: mathematical ability, mathematician, research — Tags: — collegemathteaching @ 8:10 pm

This marks the second summer in a row I got news that a paper of mine has been accepted for publication. Last year, it was the College Mathematics Journal; this year it is the Journal of Knot Theory and its Ramifications.

Sure, that is a big “yawn”, “so what”, or “is that all?” for faculty at Division I research universities. But I teach at a 11-12 hour load institution which also has committee requirements.

And, to be blunt: I got my Ph. D. in 1991 and has a somewhat long slump in publication; I was beginning to wonder if my intellect had atrophied with time.

Ok, it has, in the sense that I don’t pick up material as quickly as I once did. But to counter that, the years of teaching across the curriculum (from business calculus to operations research to numerical analysis to differential equations) and the years of attending talks and attempting to learn new things has given me a bit more perspective. I make fewer “hidden assumptions” now.

So, I am going to celebrate this one…and then get back to work on spin-off ideas.

January 17, 2014

The New Semester: Spring 2014

Filed under: academia, advanced mathematics, algebraic curves, analysis, knot theory, research — Tags: — collegemathteaching @ 11:34 pm

The new semester is almost upon us here; our classes start up next Wednesday. I am ashamed to report that I am delinquent with a referee’s report; I’ll work some weekends to catch up.

Of course, we come in with “new ideas” which include evaluating things like this:

“Most people like to talk about how in college we need to develop critical thinking skills”, said Mike Starbird near the beginning of this talk yesterday, “but really, who wants to hear “Oh, yeah, Soandso, he’s really critical”?”. This, Starbird says, is what led him and coauthor Ed Burger to coin the phrase “effective thinking”. Because that is something one would like to be called.

The talk was affected by some technical difficulties, which meant that the slides Starbird had prepared with mathematical examples were unavailable to us. But, following his own advice, Starbird rose to the challenge and gave a talk, without slides, and using the overhead projector for the examples he needed to draw. As usual, his delivery and demeanor were both charming and informative (I am lucky enough to have both taken a class from him and taught a class with him), and the message on what strategies to follow for effective thinking, and to get our own students to be involved in effective thinking, was received loud and clear.

The 5 elements of effective thinking, as Starbird and Burger describe in their eponymous book, are the following: understand simple things deeply, fail to succeed, raise questions, follow the flow of ideas, and everything changes. The first couple he described by using examples of mathematics in which each strategy led to deep insights about a problem. For “understanding simple things deeply”, Starbird showed us a new, purely geometric, way of proving that the derivative of sin(x) is cos(x).

Note: Professor Starbird was one of my professors at the University of Texas. I took a summer class from him which involved the class going over his technical paper called A diagram oriented proof of Dehn’s Lemma

(Roughly speaking: Dehn’s Lemma says that if a polygonal closed curve bounds an immersed polygonal disk whose self intersections lie in the interior of the disk, then that given curve also bounds an embedded polygonal disk (e. g. one without self intersections). Dehn’s Lemma is especially interesting because the first widely accepted “proof” proved to be false; it wasn’t rigorously proved true into years later.)

Ed Burger was a Ph. D. classmate of mine; I consider him a friend. He has won all sorts of awards and is now President of Southwestern University.

I have to chuckle at the goals; at my institution we mostly teach calculus, which is mostly for engineers and scientists. The engineering faculty would blow a gasket if we spent the necessary time for finding deeper proofs that the derivative of sine is cosine.

And yes, we are terribly busy with this or that: on the plate, right off of the bat, is a meeting on “reforming” (read: watering down) our general education program, a visit day, among other things (such as search).

It has gotten to the point to where things like a “department lunch” went from being something fun to do to being “yet another frigging obligation”.

I’ll have to find a way to keep my creative energy up.

So, what I’d like to “think about”:

1. I have a couple of papers out about limits of functions of two variables. Roughly speaking: I gave new proofs of the following:

1. A real valued function of two variables can be continuous when evaluated over all real analytic curves going through the origin and yet still fail to be continuous. (see here)
2. If a real valued function of two variables is continuous when evaluated over all convex C^1 functions running through a point, then that function is continuous at that point. This result does NOT extend to C^2 .
(see here)

So, what is so special about C^1 ? Is this really a theorem about curves through a planar set of points with a limit point? Or is more going on….can this result extend to results about differentiablity?

Then there is something that sparked my interest.

There is this very interesting result about Bezier curves and their control polygons in 3-space: it is known that a Bezzier simple closed curve can be unknotted but have a knotted control polygon. What else is there to explore here? Can only certain differences appear (say, in terms of crossing number or other invariants?) Here is another reference.

I’d like to sink my teeth into this. It doesn’t hurt that I am teaching a numerical methods course. 🙂

October 22, 2013

The worst kind of paper to referee

Filed under: academia, advanced mathematics, research — Tags: — collegemathteaching @ 9:59 pm

Of course, refereeing journal articles is an expected duty; I’ve published a few and therefore benefited from the service of referees.

And it is very important that referees do their jobs responsibly.

I’ve refereed a few articles and some were very easy to reject: they either contained gross errors OR contained proofs of items that were already well known…and the existing “known” proofs were simpler (e. g. appeared in widely read textbooks).

But the most difficult articles to referee are those that are both

1. Poorly written and
2. contain some content that might have mathematical value.

These sorts of articles are time-sinks; one has to read them carefully because those ideas might well be worth seeing in print…but my goodness they are painful to read.

September 22, 2013

Mathematics journal articles: terse but is it the author?

Filed under: advanced mathematics, point set topology, research — Tags: , — collegemathteaching @ 11:54 pm

Via Recursivity:

It’s a sad truth, but the mathematics research literature is very tough going for beginners. By “beginners” I mean bright high-school students, or university students, or beginning graduate students, or even professional mathematicians who are trained in an area different from the article he/she is trying to read. […]

Things like this permeate the mathematical literature. Take compactness, for example. Compactness is a marvelous tool that lets you deduce — usually in a non-constructive fashion — the existence of objects (particularly infinite ones) from the existence of finite “approximations”. Formally, compactness is the property that a collection of closed sets has a nonempty intersection if every finite subcollection has a nonempty intersection; alternatively, if every open cover has a finite subcover.

Now compactness is a topological property, so to use it, you really should say explicitly what the topological space is, and what the open and closed sets are. But mathematicians rarely, if ever, do that. In fact, they usually don’t specify anything at all about the setting; they just say “by the usual compactness argument” and move on. That’s great for experts, but not so great for beginners.

I really wonder who was the very first to take this particular lazy approach to mathematical exposition.

Hmmm, often it is the reviewer, referee or editor. They accept your paper, but make you take out some details (and, to be fair, add others)

A colleague and I are thinking of starting a journal called “The Journal of Omitted Details”.

But yes, this practice makes some mathematics very difficult for the non-expert to read.

Note: the usual definition of a compact set (given some topology) is: X is compact if, given any collection of open sets U_{\alpha} where X \subset \cup_{i \in \alpha} U_{\alpha},there exists a finite number of the U_{\alpha} where X \subset \cup^{k}_{i=1} U_{\alpha i}. That is, any open cover has a finite subcover. This is equivalent to saying that any infinite set of points in X has a limit point, and in a metric space this means that X is both closed and bounded.

June 12, 2013

Just a 3 page paper…

Filed under: academia, advanced mathematics, editorial, research, topology — Tags: — collegemathteaching @ 9:17 pm

I just sent off the final (I think) revisions of a 3 page paper that has been accepted for publication.
Now, as I get ready to start writing another paper (different area entirely), I picked up an old notebook: 65+ pages of notes of work are related to this paper!
I submitted it; had a rejection (due to writing form), re did it, resubmitted it, had to do more revisions, etc. I talked about this in a seminar, had a colleague show me that a similar result had appeared (but mine was different enough to warrant publication)

Of course, I did things a different way and I created a “new to me” technique for “smoothing” a piecewise linear half-line in such a way that the resulting curve is C^1 (has a continuous first derivative) and remains convex.

And the result: 3 pages in a journal. In terms of time, that is one page per year!

So what was in all of these notes?

Some of these notes were about the idea itself (some elementary point set topology of the plane was involved); the idea: if one has a collection of points in the plane that has a limit point, can one run a locally piecewise linear arc though a selected convergent subset of points to the limit point, while keeping the resulting arc convex? (Yes, you can)

Now is there a way to “smooth” this arc (round off the corners) so as to pass though an infinite subset of this convergent sub-sequence of points so as to produce a C^1 curve? (Yes, there is; that is where the spline construction came in).

Can this curve be made into a C^2 curve? NO!!! The counter example is rather indirect.

Anyway, the details of the above is what fills up 65+ pages of my little notebook.

My point: expect research to take a while; there are starts, false starts, dead ends, revisions and revisions to the revisions, BEFORE you send off the first draft to the journal for consideration!

What comes next
I have one result ready to be proofed and another that I am writing up; hopefully these will be sent to a journal in a month’s time. But I won’t be surprised if these papers also take quite a bit of time.

So, for my academic year: what do I do as far as research?

1. Work on a specific problem?
2. Learn something “new to me” but related to my research?
3. Explore new topics (new to me) at a shallow level?
4. Work on a lower division book?

We shall see.

May 29, 2013

Thoughts about Formal Laurent series and non-standard equivalence classes

I admit that I haven’t looked this up in the literature; I don’t know how much of this has been studied.

The objects of my concern: Laurent Series, which can be written like this: \sum^{\infty}_{j = -\infty} a_j t^j ; examples might be:
...-2t^{-2} + -1t^{-1} + 0 + t + 2t^2 ... = \sum^{\infty}_{j = -\infty} j t^j . I’ll denote these series by p(t) .

Note: in this note, I am not at all concerned about convergence; I am thinking formally.

The following terminology is non-standard: we’ll call a Laurent series p(t) of “bounded power” if there exists some integer M such that a_m = 0 for all m \ge M ; that is, p(t) = \sum^{k}_{j = -\infty} j t^j for some k \le M .

Equivalence classes: two Laurent series p(t), q(t) will be called equivalent if there exists an integer (possibly negative or zero) k such that t^k p(t) = q(t) . The multiplication here is understood to be formal “term by term” multiplication.

Addition and subtraction of the Laurent series is the usual term by term operation.

Let p_1(t), p_2(t), p_3(t)....p_k(t).... be a sequence of equivalent Laurent series. We say that the sequence p_n(t) converges to a Laurent series p(t) if for every positive integer M we can find an integer n such that for all k \ge n, p(t) - p_k = t^M \sum^{\infty}_{j=1} a_j t^j ; that is, the difference is a non-Laurent series whose smallest power becomes arbitrarily large as the sequence of Laurent series gets large.

Example: p_k(t) = \sum^{k}_{j = -\infty} t^j converges to p(t) = \sum^{\infty}_{j = -\infty} t^j .

The question: given a Laurent series to be used as a limit, is there a sequence of equivalent “bounded power” Laurent series that converges to it?
If I can answer this question “yes”, I can prove a theorem in topology. 🙂

But I don’t know if this is even plausible or not.

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