Ganesh Swami

The guest speaker for this week was Dr. Dirk Bouwmeester, a scientist from the University of California, Santa Barbara. Most of his expertise is in experimental quantum optics. His website also mentions that he was “involved in the first experimental demonstrations of quantum teleportation, quantum cloning, 3-particle entanglement and stimulated emission of entangled photons.” His personal opinion of these experiments is that they don’t excite him so much - it’s mostly observing what you’d predict theoretically. He was looking for something more challenging.

Disclaimer: I didn’t understand a major portion of his talk. I’m only writing about things that I was able to understand. (If that isn’t obvious already.)

The first topic that made the audience uncomfortable was Penrose’s theories of state reduction in quantum mechanics. People familiar with quantum mechanics know that “objects” on the microscopic level exist in multiple quantum states simultaneously. These states somehow magically “collapse” when an observation is made. Penrose’s explanation for this collapse is due to gravity. When the gravitational force between these super-positioned states become significant, they collapse into one that can be physically measured. Bouwmeester’s team is conducting experiments to test this theory and their basis is that our measurement time resolution far outweighs the energy difference between the two superpositions. Mathematically, this is the Heisenberg uncertainty principle.

The second topic that was new to me was about entanglement of two or more photons. Young’s double slit experiment definitely showed that a single photon can interfere with itself. Most introductory textbooks on optics only treat interference with coherent sources (sources with no phase difference.) In fact, Paul Dirac in his book “The principles of Quantum Mechanics” says

Each photon interferes only with itself. Interference between two different photons never occurs.

Dr. Bouwmeester wrote a Nature article about research that has shown entanglement between two or more photons. You can read more about this at the ScienceWeek archive.

That is all.

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.