For some reason, I’m having a hard time getting going today. Well, let me just put one word in front of another and see how it goes.

Let me start with compressed sensing – because last Monday’s post has been very popular. I have, of course, ordered two books. More interestingly, I think, there are two more problems I want to work out… a sound signal composed of sinusoids and spikes… and a 2-D image.

I think I’ve almost sorted out in my head what I have to do for such a sound signal… but that means I haven’t even started this problem.

I often think that I am the kind of person who plunges directly into computation before he figures out what he really needs to do. Well, that’s not quite true – in the case of this sound signal, I’m still trying to figure out just what I need to know about it. On the other hand, once I have an idea of something I can compute, I’ll try it out… and learn as I go.

I have long known that people learn in different ways. When I was an undergraduate, a friend and I were studying together for a test. It was macroeconomics, i.e., not something as precise as the mathematics and physics we were majoring or minoring in, and we decided that we might do better at making sense of it together. (For me, at least, it was unusual to study with someone else.)

We took advantage of an opportunity to do our studying in the professor’s library. At one point, my friend was browsing through several books… and then we continued our discussion. She quoted a paragraph from the book… and I responded with, “What about the story he told in class about a small winery in Northern California?” I proceeded to tell her the story.

She looked at me and asked, “How can you remember that story from two months ago?” I, in turn, asked her, “How can you read several paragraphs you do not understand – and quote them back to me?”

Easy enough – I had a much better verbal memory, and she had a much better visual memory. We could each retain raw data – i.e. material we did not understand – better in one preferred form.

Fortunately, I can learn from books, too. Boy, is that a good thing! After all, I’ve got 1500 math and science books in the next room.

I was chasing down some chemical engineering links during the week, and I ended up on the homepage of an author I recognized. He has a number of interesting articles about the different ways in which people learn… and, in fact, he is the co-inventor of an index of learning. Let me quote from the abstract of one of his papers; he’s speaking about teaching to an audience whose members learn best in different ways.

“The paper concludes that the choice of a model is almost irrelevant: teaching designed to address all dimensions on any of the models is likely to be effective, and all of the models lead to more or less the same instructional approach.”

On the other hand, I wasn’t chasing different ways of learning – instead, I was chasing an article entitled “Imposters Everywhere”… which points out that an awful lot of people have the feeling that if people knew how hard they worked at something, they would not be respected. I found it rather interesting that when he brings the subject up in class, almost all of the students react the same way: “Oh, no! He’s talking about me! Oh, wait, everyone else thinks he’s talking about them!” I like the way he put it: “the audience resonates like a plucked guitar string”.

I can tell you it’s not a recent phenomenon.

If people knew how hard I have had to work to gain my mastery, it wouldn’t seem wonderful at all. – Michelangelo.

As for my own mathematics, my alter ego the grad student is still computing internal forces and moments in beams. (OK, he’s been temporarily demoted to a sophomore. Well, perhaps he’s the TA for a sophomore course he never took.) The latest two problems have been particularly frustrating. One of them worked on Thursday night… but didn’t work Friday night. And I mean it doesn’t work: Mathematica® just sits there running and delivering nothing. What the heck did I change? Error messages I can handle… strange answers I can handle (they usually mean I mistyped the name of a built-in function… but taking over my entire computer? Just what did I ask it to do? (Some of you will quibble that in fact it is working, and that’s the problem – it’s clearly using my entire machine to work on something other than what I intended.)

And the other one? Yikes! My intermediate equation, and therefore my solution, for the reaction forces, are both wrong – but my final solution for the internal forces matches the book. But then, in order to get his graph for the internal moments, I have to change the sign on an applied moment. In one respect, that’s good – I disagreed with his sign in the first place! But, overall, these calculations are a mess… and yet they should be perfectly straightforward. Grrr.

To put that in other words: I can get the right answers but I don’t believe them!

They really should be a piece of cake… I’m enjoying using “singularity functions”, i..e Dirac delta functions and their derivatives (for applied moments) and unit steps (DiracDelta and HeavisideTheta in Mathematica.

OK, maybe some lunch and then I’ll see what I can do mathematically.

April 2, 2011 at 12:32 pm

Hopefully one of the books you ordered is Mallat’s third edition of A wavelet tour of signal processing – the Sparse way. I remember you said you own an edition of the book, but this one has about 1/3 new material on compressed sensing – it’s an excellent book.

Enjoy the signal separation problem you intend to work on!

April 3, 2011 at 12:30 pm

Hi Sper,

Thanks for the recommendation. I saw the book on Amazon and wondered if I should buy it. I’ll wait, now, but it’s good to know that it’s a plausible purchase.