ChapterZero

Apparently, the Canadian government was considering putting a strange hiring policy in place in one of their departments: only females and minorities would be hired for a period of time no shorter than 6 months. Of course, they had to back down on this proposal, due to public outcry. There’s a discussion going on about it at k5.

Predictably, some folk are drawing parallels between this policy and affirmative action policies. I don’t think it’s fair to AA to compare it to such a discriminatory policy. Under AA, a minority or female applicant is given preference over an equally competent white male. Under the proposed hiring policy, totally incompentent minority applicants would have been given infinitely more consideration than superlatively qualified white males. How could one possibly miss the important difference here? AA guarantees the best applicant gets the job, with race/sex only being referred to in the case of a tie, while the other policy appeals to race/sex first, and then capability.

As I see it, there are four reasons for disliking AA: a belief that AA encourages sloth on the part of minorities, a belief that AA doesn’t select the best candidate for the job, a belief that AA introduces an unfair bias against whites, or an aversion to ‘invasive’ social policies. The latter, I can’t argue with– I am a little against AA on that account myself; however, not enough so that I think the invasiveness of the policy outweighs its social benefits– but the first three are groundless worries. First, as I stated above, AA selects minorities over whites only when they are equally qualified, so minorities have to work just as hard as whites; this eliminates the first two concerns. As for introducing a bias against whites, I think that’s seeing the glass half empty instead of half full: the point of AA is to introduce a counterweight to society’s inclination to favor the majority, which can only be done by introducing a bias towards the minority in some way. I think AA, with its ceteris paribus clause, is the best way of introducing such a bias.

This year, let us give thanks for linear algebra, one of the greatest tools available to mathematicians. In particular, let us be thankful for matrix theory, which helps us to painlessly solve linear recursion equations such as:

Note that therefore, this method can be used to painlessly derive the mysterious formula for the Fibonacci series. How, you ask? By what serendipitous conspiracy of fate can matrices help us with these sorts of problems? Well, read on.

Given , let , then we have

so clearly .

In general, I guess this is as far as you can go easily, then you are faced with the problem of finding a general form for . However, if is diagonalizable, as it is in this case, we have , which is easily calculated. Doing so,

so .

Now you may enjoy your mathematically enhanced turkey feast. (Actually, now I see this method is just a slight retasking of the technique I learned oh-so-long-ago for solving a system of ODEs)

Lately, I’ve been thinking a lot about inpainting. Well, ‘a lot’ is a relative term, so maybe not.

I first ran into the concept of inpainting while looking frantically for papers on directional representations to write up my NSF fellowship application. Classically, inpainting is a technique for using information from the surrounding region to fill in missing regions of a painting— in some way, painters would transfer/propagate/diffuse ‘information’ into the corrupted region. Mathematically, inpainting refers to techniques that do the same thing, but automatically, without human intervention. The most common kind of inpainting is done using PDEs, as I tried to suggest by my choice of words.

Here’s a somewhat mathematical formulation of the inpainting problem that lends itself to a PDE approach. Let indicate the -th image generated by an iteration through the inpainting algorithm, with being the original image. Then define

where , the corrupted region, controls the rate of improvement, and is the -th update term. The hope is that as , , some reasonable image.

Of course, the most important term in this equation is , because it incorporates information from outside the region, and determines what can be reached. Here’s where the PDE formulations come in: you can define according to a PDE with boundary conditions that correspond to the values in the surrounding regions. That’s where the actual math comes in, and I start getting fuzzy.

Inpainting clearly has many applications besides image restoration: it can be used to coherently remove portions of an image in a hopefully more intelligent manner than Photoshop’s clone tool, to resize images smoothly, to compensate for the artifacts of JPEG encodings, etc… But one significant drawback of the naieve PDE-based inpainting algorithms is that they can’t be used on areas with significant texture or large areas, because of their diffusive, smoothing nature. There are hybrid algorithms designed to compensate for this shortcoming: ones that adaptively switch between variational and purely (stochastic?) texture-based inpainting, and ones that somehow merge the two. I don’t know anything about them, other than that they exist.

Today a professor visited UH, who had been one of Papadakis’ students, and mentioned an inpainting method he and someone else devised, based on the diffusion of framelet coefficients. I didn’t understand too much of what he talked about, but I got the impression that his is the only inpainting method with a proven convergence. Viz., other inpainting algorithms have great empirical performance, but nothing is known about their convergence. Something like that.

I’ve actually been giving a little thought to the design of an inpainting algorithm myself, based on wavelet coefficients. In my concocted scheme, you would decompose the image into several subbands, and mark the coefficients which correspond to areas overlapping with the corrupted region. Then you would perform a standard inpainting method on the subbands, and recombine them. This seems like such a simple method that someone must have thought of it before me.

P.M. Dirac on the relationship between mathematics and physics. I’ve noticed this, in my abortive attempts to read mathematical physics books— namely, they seem more abstract than most abstract math books—, but wouldn’t know how to put it quite so elegantly.

The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. This is only natural and to be expected. What however was not expected by the scientific workers of the last century was the particular form that the line of advancement of mathematics would take, namely it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation.
Paper on Magnetic Monopoles (1931)

Makes me want to be a physicist… wait, no, it makes me want to be a mathematician.

Another quote from P.M.:

I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it.
Quoted in F Wilczek, B Devine, Longing for the Harmonies

So obviously, I have a long way to go.

Did you know that it was Norbert Wiener who proved that linear time-invariant systems are characterized by convolution with a kernel? It surprised me to hear that: the proof of this principle is so simple — even though I can’t recall it off the top of my head — that I thought it was discovered a long time ago. It’s strange that the classical sampling theorem was known in 1841 to Cauchy, yet we had to wait for Wiener to come along to discover impulse responses, probably sometime in the 1920s. Which of the two is more intuitive?

I’ve been hearing a lot about Norbert lately— just now, when Papadakis told me that little fact, and in my stochastic processes class where we just talked about Wiener filters— so I looked him up on the MacTutor History of Mathematics site. Interesting guy. Some highlights: he graduated high school at 11, got his undergrad degree at 14, and originally studied zoology! Then he switched to philosophy, and wrote his graduate thesis on mathematical logic. From there, he found his way into mathematics.

He also invented the field of cybernetics, which despite having half-heartedly looked up, I have no real conception of. All I know is that it has something to do with control and systems theory, and nothing to do with the modern sci-fi inspired use of the term. From MacTutor, here’s one guy’s speculation on the origin of Wiener’s cybernetics:

While studying anti-aircraft fire control, Wiener may have conceived the idea of considering the operator as part of the steering mechanism and of applying to him such notions as feedback and stability, which had been devised for mechanical systems and electrical circuits. … As time passed, such flashes of insight were more consciously put to use in a sort of biological research … [Cybernetics] has contributed to popularising a way of thinking in communication theory terms, such as feedback, information, control, input, output, stability, homeostasis, prediction, and filtering.

Seems like what we would call control systems at UH. I wonder if he could be considered the ‘originator’ of that field, in asmuch as any one person could. At any rate, although he was a weird one— read some of the anecdotes—, he seems to have been the kind of mathematician I aspire to be. Certainly, the results he’s known for are all ones that I’d like to be known for

I generally don’t have the level of interest required to sit down and read a huge treatise on optimization. That eliminates just about all the optimization books I’ve seen. Someday, in graduate school, I hope to take a series of classes that will open up that world to me, but until then, I’ll have to resort to ad hoc solutions.

In my latest telecomm homework assignment, we had to design an indirect Armstrong FM modulator— a device used to create wideband FM signals. The problem is that it involves a variable number of frequency multiplication steps, with a frequency mixing somewhere in the middle of the process. I must determine, among other things, the number of frequency doublings and triplings to do before mixing, what frequency to mix at , and how many frequency doublings and triplings to do after mixing ( and ). If you consider it as a system to be optimized, there are 6 parameters to be optimized, five of which are related by the equation The other is a frequency deviation parameter which I’d like to satisfy the equation .

The interesting thing about this problem is the fact that have to be positive integers, and have to be within certain ranges. I ended up coding this up in Python: the program accepts the two continuous variables as input, and steps through the discrete ones to find optimal values. The engineer in me was satisfied with the results, but I wonder what kind of theories out there address problems like that. The mathematician in me would like to be able to find a provably optimal, acceptable solution.

Since I’m sure someone will ask me about this question tomorrow, I’ve attached it
(more…)

Who knew a multiple choice math test could be so hard? I skipped 19 out of 66 questions. Yet somehow, I feel I did well: maybe it is my skepticism about the capabilities of most math students, combined with the fact the test is ‘rescaled’. Of course, this may be completely unfounded skepticism, since only 2500 students or so take the test, so I may be in competition with the cream of the crop. That makes me all the more eager to see how I fared!

Here are two of the harder problems, that I had no idea how to approach:

• How many invertible 2-by-2 matrices exist over a finite field with q elements?
• If is an -by- matrix such that , which of the following are true: , , or ?

Papadakis showed me that a brute-force counting technique may be applicable to resolving the first question– a fact I don’t like, but I can appreciate the approach– and constructed an elegant argument to show that, in the second problem, we can make the statements about the trace and determinant, but despite this, is not necessarily the identity!

You know how sometimes you write something, and it develops from an utter piece of crap into something that you’re actually proud of, an almost perfect expression of what you intended to say? I just had one of those moments, writing my personal statement for the NSF Graduate Research Fellowship. Here it:

The major impetuses for my decision to pursue an advanced degree in mathematics are my pleasure in working with mathematics, and my desire to elicit the same sense of wonder in others. Consequently, my career objective is to become a math professor. An NSF Graduate Research Fellowship would assist me in surmounting the financial obstacles I will encounter in pursuing this objective.

I have always felt an affinity for math, but I was initially more fascinated by the physical sciences and their applications. It was in high school that I realized there was deeper content to mathematics than arithmetic and symbol manipulation— the determinant raised puzzling questions: why is it defined in such a strange way, and why is it connected to the solvability of systems of equations? In the same vein, why does Descartes’ rule of signs work? These were my first memorable encounters with non-intuitive mathematical results (pi excluded).

In my junior year of high school, I became fascinated with neural networks; I enrolled in college, majoring in electrical engineering, to follow that interest. In my first two semesters, I took calculus courses taught by Dr. Johnson, who invited a couple of students to take a Special Problems course on Fractional Derivatives. We never got around to discussing them, but that course served as my initiation to higher mathematics: I learned why proofs are useful and necessary, and how to construct them, as well as the basics of analysis. At the same time, I got a feel for the internal motivations that drive a lot of mathematical research; before this it seemed to me that all the interesting mathematical questions had been addressed already. But even within the microcosm of our classroom, I saw that every significant discovery or definition opens up new panoramas of challenging questions.

The course was taught using the Moore style, in which the professor provides definitions, propositions, and occasional motivating comments, while the students provide proofs or refutations of the propositions, and critique each other’s proofs. For some time, I thought this was the optimal method of learning mathematics—so much so that I took 2 other courses taught in the same style, also by Dr. Johnson. Inevitably, I found that the Moore style is definitely not suitable for teaching every mathematics course– for one, it is virtually impossible to cover a sizable amount of material– but it was the fun I had puzzling over and defending proofs in these classes that motivated me to continue taking math courses past the requirements for my engineering degree.

In the summer of my sophomore year, I obtained an internship with the Adaptive Optics group at Lawrence Livermore National Laboratories. I took full advantage of my access to the LLNL library, and in the process of reading background material for my project, I discovered the world of image processing. I was thrilled to learn of this part of the electrical engineering spectrum, and I decided to focus on signal processing. On returning from California, I signed up for an EE analysis course dealing with the Fourier and Laplace transforms. The material was exciting, but I desired a more comprehensive, theoretical development of the mathematical tools we used. The next semester, I took the new “Mathematics of Signal Representation” course being offered in the mathematics department. In that course, I learned the mathematical principles— inner product spaces, Hilbert spaces, orthonormal bases with infinite elements, generalized Fourier series, and so forth– that underlie Fourier analysis. The most titillating classes were those in which the professor detailed the historical developments that culminated in modern harmonic analysis, or placed his research projects in the context of the material we were studying in class. For me, that was motivational material! Dr. Papadakis, the professor teaching the course, recognized my interest, and that summer, recommended I attend the Wavelets and Matrix Theory REU being conducted at Texas A&M University by Dr. David Larson.

The Wavelets REU consisted of two parts: in the first, we attended two three-hour seminars weekly, where we learned the basics of operator theory, sampling and frame theory, and Hilbert spaces. In the second, we continued to attend the seminars while working on our research projects. The material I learned that summer was fascinating, the problems were challenging, and I had fun working with people who felt the same degree of enthusiasm about the mathematics we were doing as I did. While at A&M I read two engaging books of essays by Gian-Carlo Rota: Discrete Thoughts and Indiscrete Thoughts; his discussion of the New Math and New New Math systems and the declining efficacy of the modern educational system in the former motivated me to seriously consider becoming a mathematics professor.

It was over the course of this summer that I made the decision to pursue a graduate degree in mathematics, with the goal of becoming a mathematics professor. This decision has only been reinforced by the research I’ve conducted and the mathematics I’ve learned since returning to my home university. Observing Johnson, Papadakis, Larson, and my other professors, I have seen how rewarding it is to work in the field you love while sharing your delight with students. At the same time, as a professor, I will have a unique opportunity to enhance the scientific, and in particular, mathematical, understanding of younger generations. I believe this is vital not only to ensure continued public understanding of the importance of mathematical research, but also to inspire future generations of mathematicians, engineers, and scientists. It was my professors who nurtured a lifelong interest in mathematics within me, and I’d like to do the same for other students.

I wanted to put a section in there about how I want to write really badly (please interpret that in the way I intended!), but I couldn’t think of any non-cheesy way to develop that within the two page limit. But I do… want to write really badly, that is. Just about every time I pick up or put down a well-written expository book, whether on science or math, I get bit by that bug.

The latest book I’ve been reading is “Algebras of Linear Transformations”, by Douglas Farenick. The first two chapters are supposed to be a review of linear algebra, but I’ve already picked up a lot of interesting ideas from the first quarter of chapter one. Not necessarily new ideas, but ones I haven’t seen in a while, and certainly not in this interesting a light.

Here’s an example: to prove the rank-nullity theorem, the author takes a much more scenic and abstract route than Hoffman and Kunze— the authors of the infamous “Linear Algebra” textbook we’re using in my advanced linear algebra class. First he shows that if is a subspace of , then , which is pretty intuitive, then he shows the first isomorphism theorem: if is a linear operator, then , which is again intuitive. From there it’s a short step to take to get the rank-nullity theorem; for me this exposition has the advantage of introducing two results that are interesting in their own right, and then showing how they are not only interesting, but also useful. Wonderful stuff.

One more example:

Let be a linear functional on .

1. There is a unique matrix such that for all
2. If for all , then there is an such that for all

This is a great result, because it gives one use for the trace operator— determining functionals. It’s also interesting in that it provides another way of determining functionals, as opposed to inner products via the Reisz Representation Theorem.

Next are inner product spaces, which I look forward to with glee…