By roundabout means, I came across a post on faith and rationality by a SF author. One of the commenters, another sf author, made a suggestion for what I think could be a very interesting novel:
Here’s a story idea for you. Maybe it’s already been written, of course. Scientists discover a “religion” gene. People who have it experience God in their lives. People who don’t have it end up stinkin’ secular humanists or whatever.
The scientist who discovers the gene (or maybe he’s just a science fiction writer) has been comfortable with his religious beliefs. Now he discovers that there could be a simple biochemical explanation for it. What does he do? What happens to his belief?
You could call it “God of the Gaps.” If you don’t want it, maybe I’ll write it myself, one of these years.
We’ve already heard about the ‘religious’ spot in our brain, and other genetic dispositions toward religiousness; I think the idea of a book that dramatizes what impact these realizations could have is excellent.
In a way, it reminds me of Robert Sawyer’s Calculating God. In that book, when several races of technologically advanced humanoids make first contact with Earth, they are shocked to discover that the question of the existence of God is still open. For those races, not only does God obviously exist, the goal of science is to contact God– or something like that. This turns the spiritual and intellectual world of the museum curator who is their primary human contact, and who’s also a terminal cancer patient, upside down. Of course, the aliens’ ideas of God differ from most human conceptions.
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A bit of vocabulary I learned today: a plurale tantum is a noun that only appears in the plural. Some pluralia tantum (that’s the plural of plurale tantum) include pants (an appreviation for pantaloons), and scissors. Likewise singulare tantum are nouns which occur only in the singular, like dust or wealth. All of this is from Wikipedia: follow the link on uncountable nouns for loads of fun (which I think is a singulare tantum).
How did I come across this? I finally started reading Financial Markets and Martingales: Observations on Science and Speculation, a book which I have mixed feelings about– on the one hand, it proportedly explains the ideas of financial markets without lapsing into technicalities like the stochastic calculus, and has won awards– on the other hand, what’ve I read so far suggests that either the author writes like he’s stoned, or the translator was overly cautious: the flow of the words and sentences is very nonstandard. The introduction resembles a stream of consciousness novel. Anyhow– after a strange digression into the gambling addiction of Dostoyevsky– the author mentions that the term martingale is derived from the term martingale chausses, which are a medieval form of culotte. Then Wikipedia for culottes -> pantaloons -> plural tantum.
Another linguistic comment: apparently in Russia, instead of addressing people with a formal title ‘Mr. ‘ or ‘Mrs. ‘, they’re addressed as ‘ .’ So Fiodor Dostoievski is formally Fiodor Mikhailovitch. Interesting, no?
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Again, I’ve tossed out a couple of library books and started reading new ones. I guess this is my particular implementation of the advice I keep getting of not attempting to read any math book from cover to cover 
Today’s replacements: Elements of Nonlinear Analysis by Chipot, and Fundamentals of Applied Functional Analysis by Mitrovic and Zubrinic. The former I picked up almost at random, and happened to open it to the proof of the Lax-Milgram theorem, which I had just been thinking about an hour earlier– surely a sign. The latter I picked up because I keep passing it by, and saying eventually I’ll pick it up and learn distributional theory from– I gave it another look see, and I think I’ll like their approach to some PDE theory also.
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I’m suddenly interested in the connections between PDEs and functionals. Specifically, if I have a PDE and associated BCs/ICs on a given domain, how can I find a functional (and on what space) such that the minimizer of this functional is a solution to the PDE, and in what sense is it a solution? Conversely, what PDE (if any), does the minimizer of a given functional satisfy?
I’m not sure under what conditions these questions even make sense: e.g. why would a given functional have a unique minimizer? But, at least the latter is an important question to me, because it would allow me to convert optimization problems– in particular, I’m thinking of variational image processing– into PDEs, which, thanks to this term, I have confidence in my ability to attack numerically.
Here’s an example of the flavor of what I’d like to do. In the appendix of Variational Image Segmentation using Boundary Functionals, the authors show (sort of– the argument is neat, but not very rigorous) that the minimizer of the functional
satisfies the PDE
In the image processing context, the minimizer of 
/ the solution of the PDE represents a balance between fidelity to the original image 
and being slowly changing.
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I began writing up an exposition of the stochastic version of Lyapunov functions a while ago, when we were assigned a problem relating to them on one of my stochastic controls problem sets. It seemed rather random at the time, but now– maybe a month later– we’ve started talking about stochastic stability, and I see that we were just proving the same theorems that’re in the notes, but ahead of time and out of context. Quite an ego booster, really.
The set up: let 
be a compact ’state space’, and 
be a continuous function which, in conjunction with 
a sequence of real-valued i.i.d r.v.s, determines the dynamics of the system: 
. Note, at least in passing, that this system is a markov chain. Done noting? Introduce the concept of an equilibrium point 
as one satisfying 
for any value of 
. We’d like to talk about various types of stability of this point: is
- stable if an orbit which starts close to it remains close with high probability — for any
and 
there exists a 
such that we have 
whenever 
.
- asymptotically stable if it is highly probable that an orbit which starts close to it converges to it — if it is stable and for every there exists a
such that 
whenever 
.
- globally stable if any orbit converges to it a.s. — if it is stable and
a.s. for any 
.
Now, for the cool result:
Suppose that there is a continuous function 
with 
and 
for 
such that 
for all 
. Then is stable.
The big two tricks in the proof of this theorem are the fact that 
is a supermartingale, and then applying the supermartingale inequality.
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We’ve started to cover FEM in my numerical PDE course. Of course I’m enjoying it because we get to talk about functionals on Hilbert spaces, and how to minimize discretized versions of said functionals. There are clearly many applications for the technique besides solving PDEs. Oddly enough, the instructor (a postdoc), who usually seems a virtual font of information (after all, numerical methods are his specialty), doesn’t seem to know the basic functional analysis behind FEM. Like, he specified the Hilbert space we’re minimizing the variational form of the PDE over as a Hilbert space of functions where the 
-th derivative is square integrable. He didn’t specify in what sense he meant derivative, or even what the norm is, and admitted he’s not sure when I asked. I suspect we’re using Sobolev spaces and weak derivatives, but I’m not sure what either of those are, and I don’t really have the inclination to check, since it doesn’t matter for numerical methods (well, at least I suspect it doesn’t for what we’re doing). But still… instructors should know everything, right? And I thought functional analysis was a basic tool for numerical analyists…
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on 
, 
on 
.