Ganesh Swami

Had a chance to hike at the Dennett-Munroe Lake trail last week. It was described to me as “moderately-hard,” but it was one of the hardest I’ve done. All of the 1300m evelation is in the first two hours, after which it levels off. Our exact coordinates were here.

The areas surrounding the lakes had mostly dried up. It has slippery and I fell into a swamp twice. You can see my dirty pants in the one of the photographs.

Thanks to Calin for the pictures.

If we want to compute the fourier series expansion of , then our coefficients are readily computed on paper as:

and

Now we are going to call the functions fourier, foursimp and fourexpand, which obviously calculates the coefficients, simplifies it and expands them into a series. Alternatively, you could call totalfourier which is just the composition of the three functions.

foursimp(fourier(x^2,x,%pi));

This will spit out the coefficients , and . We can now use fourexpand to expand this into a finite series (four terms here).

fourexpand(%o7,x,%pi,4);

Now plot the series approximation along with the original to compare:

plot ([x^2,%o8],[x,-3,3]);

That is all.

I had a chance to attend three amazing talks since the last time I posted anything here.

Imaging Network

A new networking group was recently established for the purpose of better collaboration between practicing radiologists, mathematicians, computer scientists and device manufacturers. The inaugural meet was at the the BC Cancer Research Center. I got to meet a lot of interesting people - from chemists (who prepare gels for imaging), people who write monte carlo and motion planning software for needle insertion to mathematicians. From a business perspective, think of this meet as a company going out and finding out what the pain points are in the market.

One of the scientists I was talking to was doing some very interesting work. She was trying to find correlations between thoughts in the mind and deformations/activity in the brain with the use of f-MRI. This could be used for example to find if a person is undergoing depression. That’d be very cool.

Geometric PDEs

Another excellent talk by Dr. Guillermo Sapiro on the use of Geometric Partial Differential Equations in Imaging Science. He used classes of parabolic differential equations for image inpainting. Sapiro has been making waves in the image processing community recently. His work being inter-disciplinary was well attended (lots of people were standing.) I was expecting his talk to be overly technical, but fortunately he only had two equations in his entire presentation.

Something else of interest is that the movie industry spends about 36 man hours per frame for special effects. That’s about 30 frames per second and many thousands of seconds per movie. No wonder guys like George Lucas want to move away from Hollywood:

“We don’t want to make movies. We’re about to get into television. As far as Lucasfilm is concerned, we’ve moved away from the feature film thing–it’s too expensive, and it’s too risky,” Lucas told Daily Variety.

The most interesting part of his talk was his method to Djikstra’s algorithm for finding the shortest path. This was extremely useful for finding distances in the solution to Hamilton-Jacobi equations. Naïve implementations is of order complexity (with the simplifying assumption that the number of nodes and edges are equal) which can be a significant deal when N=4000.

Novel Nanotube structures

Another well attended talk by Dr. Meyya Meyyappan, who is the director and senior scientist at Ames Center for Nanotechnology, NASA. His talk was on growing novel one-dimensional structures with carbon nanotubes (CNT). Two main applications I can recall:

• The use of CNTs as a tip in nano-lithography and Scanning Probe Microscopy. Tips are usually made of Silicon Nitride, and this can wear down (reduced resolution) or break. CNTs being stronger than steel is a natural application.
• Growth of vertical transistors instead of horizontal to increase the packing density.

Quite interesting to see where nanotechnology is taking us in the future.

It’s only the second week of classes, and I’m already falling behind my reading schedule.

The week started out with some heavy mathematical optimization tricks in my optics class. Based on Fermat’s principle, we did some trajectory tracing problems. The path of a light ray can be traced through any arbitrary medium of varying refractive index.

L is the optical path length, n is the refractive index of the medium which is a function of the position. I like how transferable the solution is to other problem domains.

The other physics course has quickly dived into the depths of heavy theoretical mathematics. This section of the course deals with functions with complex arguments. The prof. kept stressing the point that Riemann sheets are very elegant - I wonder if he was being sarcastic. This becomes very ugly for branch cuts of multi-valued complex functions. Anyone who has used a symbolic program like Maple knows that these programs miss obvious (to a human) simplifications. Read W. Kahan’s “Branch Cuts for Complex Elementary Functions” (in Iserles and Powell (eds.), State of the Art of Numerical Analysis, 1987) to see how zero’s sign bit changes the way we think about math.

My other course Financial Accounting is the most interesting this semester. Just keeping up with the lingo is a full-time job. Operating margin, profit margin, marginal cost, minority interest, etc. The assignments for this course are to be submitted through an online system. I think this is very good - instant feedback. Below is a picture of a balance sheet I completed. Ofcourse, I couldn’t do any Enron-style accounting….(signified by all green.)

Some time back, a colleague introduced me to del.icio.us. Ever since, I’ve been hooked. If you’ve been to the portal section of this website, you’ll see a list of links that I frequently use. This includes links to course home pages, libraries and commonly read websites (that can’t be aggregated.) This list is dynamically generated from specially tagged links on del.icio.us.

The biggest advantage for me is that I don’t have to mess around with the hypertext behind that page. I made the page static as these links don’t change that often. The page is generated by a 10 line script that I wrote in PHP using the MagpieRSS library.

There’s also an Emacs client for del.icio.us. I haven’t used it myself, but the list of features look very tempting.

I’m always looking for ideas on further integration between my website and del.icio.us, so if you think of something, drop me a line.