Routh Arrays

So, this week, we learned about Routh arrays. It's evidently a theorem about the sign changes of the real part of the roots of a polynomial based on the coefficients. I knew a few of these from my undergrad and graduate math days, but didn't think they were very important. I kind-of knew there would be some interest in the roots of polynomials, but didn't know exactly how it would come into play.

There's a MATLAB function someone evidently wrote to compute Routh arrays https://www.mathworks.com/matlabcentral/fileexchange/58-routh-m. It's super-useful. I got my homework done this week in ~30-45 minutes using it, which I think that might have take 4 hours or more computing all the Routh arrays by hand.

I always knew there was more to Brandon Routh than met the eye, and now I found out what it is. I was always very sorry he never got to reprise his role as The Man of Steel, and was very happy to see he was cast in the Arroverse "Crisis on Infinite Earths" story arc. More "Arne Saknussemm" notes on my way to the center of the control engineering universe.

Control Engineering Degree Ahoy!

So, this Fall 2019 semester, I matriculated at UW-Platteville as a distance education MS student in their Control Engineering program. I am taking Engineering 7310 Control Engineering I. It's simply fascinating. Chapter 2 of the book should be entitled, "Everything You Wanted to Know About Transfer Functions (but Were Afraid to Ask)" - it's simply fascinating.

It appears that in according-to-Hoyle control engineering, one has ODEs on manifolds, but simply takes a coordinate patch about a point of interest, pulls back the vector field along the coordinate patch, takes the Jacobian matrix of the vector field at the point of interest and evaluates it, then finds the transfer function for that linear system of ODEs with constant coefficients via a Laplace transform. [I peeked ahead in Bullo and Lewis, and this appears to be exactly what they do (at least in certain places). This simultaneously took the wind out of my sails and gave me hope as to what can be done in the real world.] It appears there are even ways to approximate a higher-order than second-order system by the transfer function of a second-order system, so mass-spring-dashpot systems/RLC circuits are completely general by some points-of-view.

I really think control engineering is my calling. This is all just fascinating. Next semester, we go over state-space representations of control systems, which is where manifold topology evidently comes into the picture. I really wish I had know about this in my undergrad. Of course, then I likely wouldn't have a Ph.D. in manifold topology; there's always a trade-off.

But, I think I'm on the right path now. We'll see if we can get it to pay actual dividend in terms of an increased salary in the future.

Week Unknown: "Open" Problems

The four problems on which I want to work when I have time:

On the circle, S¹, with its standard Riemannian metric it inherits as a subspace of ℝ², find a second-order ODE that is

1. gradient but not Hamiltonian
2. Hamiltonian but not gradient
3. both gradient and Hamiltonian ("harmonic") (I think I have a heuristic proof there are no non-trivial ones)
4. neither gradient nor Hamiltonian

Week 62: Self-Realization

[This post doesn't really belong here, but I've been thinking about it for a while, and I want to preserve the post, not let it just get lost amongst my Twitter postings.]

So, I've done some thinking about my teaching/teaching style, and I guess what I really like and what "got me into teaching" is that I like teaching new concepts to students. This is probably a mistake to say, especially given the School at which I'm currently teaching, but I appear to dislike teaching skills: I like teaching new concepts and the new worlds of ideas to which they lead for students, but I dislike working with students to develop skills and real proficiencies with the new topics. I had professors in the past (well, at least one professor :/ ) who would give exams that emphasized knowledge over skill - he would design his exams so if you had a good theoretical knowledge of the course, you would do well, but didn't need a good skill set to be able to perform well on the exam in the time allotted - and I kind-of thought that was math. But developing skills - both for myself and in my students - is a very important part of the game, I'm beginning to realize, even if you're not teaching them any new ideas.

[On an unrelated note, I (finally!) began Tony Robbins's Ultimate Edge program today; I intended to start it 10/09/2016, evidently, but never "felt I had the resources", I correct myself, until today. I'm very excited about it and can't wait to finish it; I hope it will bring a lasting improvement in me.]

Week 39: DeRham (Homology and) Cohmology with Coefficients in the Canonical Line Bundle

So, I studied differential topology from Kosinski's book and cellular homology from Geoghegan's book and, for a long time, I thought handlebody homology was the be-all and end-all for homology theories for smooth manifolds. Then that changed, basically with the Asian professor from video series embedded in my Week 01 posts in the 'blog, and I began to think that DeRham cohomology and "DeRham homology" were the be-all and the end-all homology and cohomology theories, at least for orientable, closed, connected, (smooth) Riemannian manifolds.

So, I've been toying around with a construction for a while, based on an idea from a paper by my dissertation advisor, Craig Guilbault. In a paper he co-authors with Fred Tinsley, they discuss homology (and cohomology, implicitly) with "twisted integer" coefficients.

This gave me the idea to try DeRham cohomology (and homology, in a meaningful sense of the phrase, using Poincare duals of differential forms) with "twisted real" coefficients, that is H^*_DR(M; ℝ [ℤ₂]) with "group field" coefficients; one essentially works in the double-cover of the manifold -- which is 2 disjoint copies of the manifold, if the initial manifold is orientable, or the orientation double-cover, if the initial manifold is non-orientable -- and does all of one's calculations there, then one "divides by 2".

I talked with Craig, and in Hatcher's book, there is a notion of homology with "twisted integer" coefficients that is different from both the ordinary, untwisted homology and from the twisted "group field" coefficients homology of the double cover; evidently, all 3 homologies fit into a long exact sequence. Davis and Kirk's book gives a nice treatment of the topic of homology with twisted coefficients in general. This coincides with the notion of De Rham cohomology with coefficients in the canonical line bundle over a manifold (which gives the cohomology with twisted coefficients from Davis and Kirk in the case M is non-orientable and the usual cohomology of M in the case M is orientable - just what the doctor ordered!) from Bott and Tu's book.

So, I discovered a way to rescue an orientation form in the case that M is non-orientable with De Rham cohomology. Now, I just need a way to recover torsion from De Rham cohomology, and I'll be in business with an all-purpose version of De Rham cohomology.

Week 27: A New Job

I got hired on May 25, 2016 at the Milwaukee School of Engineering as a full-time adjunct faculty member ("lecturer" is how they technically classify me, I guess; I'm hoping to move up to "adjunct assistant professor" quickly, if my teaching goes well and then hopefully to "assistant professor" or "associate professor" if I can get a research program off the ground). I'm going to be teaching 2 sections of Calc I, 2 sections of a calculus-based probability and statistics course, and, as a recent addition, 1 section of a Calc IV course (they run their Calculus courses differently there; this course covers multiple integral in the plane and 3-space - but not line or surface integrals - and infinite series).

So, I got the Calc I book the day I signed the contract and the statistics book a few days later, and I've been LaTeX'ing up my lecture notes for the Calc I and statistics course since then, so I haven't had any time to devote to my "old" research (I have one paper on which I just gave a presentation at WGT earlier this month that's almost finished and another one on 1-sided h-cobordisms with non-split total group of the side with the more complicated fundamental group that I haven't really started working on) or my "new" research (the topic of this blog).

I have a week's worth of lecture notes done in each of the Calc I and statistics courses, and hope to get a jump start on the Calc IV course this weekend. With those in the books, I'm hoping I can settle into a schedule of working on lecture notes, typing up old papers and working on old research, and working on new research, all the while tutoring at my part-time tutoring gig. But, until then, I'm going to be a little light on research.

But, it's great news for me that I got this full-time gig. The job market is really dismal for math Ph.D.'s right now, as nearly as I can figure. The job postings I was looking at on Vitae right before I landed the MSOE gig were requiring out-of-state hires to pay for their own travel expenses to apply for the job; who can afford that? I mean, even if you're desperate and willing to do that, you can really only afford doing that 2 or 3 times before you're out of money. I can only guess that they're going only to get local candidates to apply after a few months or a year of that as an industry-wide policy; if they were able to find viable candidates with only a local job search, they really shouldn't have been advertising in Vitae with which to begin. I just don't see that as an equilibrium industry-wide policy, but time will tell.

Anyways, I'm very excited about my new job, and I'm pouring all my energies into getting ready for that in the Fall. It's kinda slow going LaTeX'ing up my lecture notes, so I may need to punt on that and start doing chalk-talk lecture notes in a few weeks if I don't think I will be able to finish the semester in Beamer slides in time for the end of the semester. They gave me a spiffy laptop on which they said I can install Ubuntu Linux if I want, and they use this spiffy cloud storage system Box that plays well with Linux (Marquette only used OneDrive for their cloud storage, so I had to "roll my own" cloud storage, especially after my hard drive on my desktop-server crashed in the Fall of last year), so I'm just really loving this new job - and, it hasn't even started yet!

Week 19: The Symplectic Form and Symplectic Manifolds

So, in my undergraduate biology text, there was a chapter entitled, "Biology, Having Found Its Holy Grail, Drinks Deeply From It" (at least, that's how I remember the chapter title), and that sums up my last week. I found a book at the library, Foundations of Mechanics by Abraham and Marsden (the same Abraham as in Transversal Mappings and Flows by Abraham and Robbin), and it has a wonderful exposition of symplectic manifolds as it pertains to Lagrangian and Hamiltonian mechanics. It turns out, associated to the cotangent bundle, T^*(Q), of a configuration space (or smooth manifold, as we differential topologists like to call them) Q, there is a canonical 1-form -θ, and its exterior derivative, ω = -dθ, which is the canonical symplectic form associated to T^*(Q). All the hullabaloo associated to symplectic and contact geometry stems from this fact.

So, it took me all of Monday (05/02/2016) after class to wrap my brain around the definition of the canonical 1-form and symplectic form: evidently, if (q, u^*, v, w) denotes the general point of TT^*(Q), then θ = u^*(v) (using the canonical flip on TT(M), around which took me 5 months to wrap my brain - nice to see some of that theory pay off), and ω is then -dθ. A technically precise, but difficult to understand, definition of the canonical 1-form is that it is (u^*)o(Dτ^*_Q)(q,u^*), where τ^*_Q is the cotangent bundle projection map. To say the same thing slightly differently, as it is tradition to use q to represent an arbitrary point of Q ("generalized coordinates") and α to represent an arbitrary (co)tangent vector at q and using a c-patch φ with φ(q, α) = (q₁(q, α), ..., q₁(q, α), p₁(q, α), ...., p_n(q, α)), we have θ₀ = p₁dq₁ + ... + p_ndq_n so -dθ₀ =-(dp₁∧dq₁ + ... + dp_n∧dq_n) = dq₁∧dp₁ + ... + dq_n∧dp_n = ω - neat.

So, I also discovered this great video series on YouTube on symplectic and contact geometry:



The first video has a nice overview of symplectic geometry, most of which I already knew, but the second video takes a dog-leg into things more "theory of symplectic manifold"-oriented and less "building up to geometric control theory"-oriented, so I'm not sure how much more of the series I'm going to watch.

So, that's it for this week. I hope to start on Geometric Control of Mechanical Systems by Bullo and Lewis soon, and then onto trying to try to publish something in the field.