Happenings – 2012 Oct 28

I got so caught up in doing control theory yesterday that I completely forgot about a happenings post. Here it is a day late… and perhaps even a dollar short.

Here’s a link to the use of the golden ratio in logo designs.

I’ve finished the course in R and I’ve begin a new one, on computational investing…. I just got distracted for a couple of hours looking at the detailed requirements for homework 1, and for the grading specification, by which we are each to grade 5 of our peers. I’ve posted to the discussion fora asking a few questions. (I forgot to ask how we choose who to grade, and why only 1/5 of us do it all.) I’m also not happy that the info was posted 2 and 4 days ago – to an auxiliary web site without even an announcement on the course web site!

We’ll see. I had the temerity to practice the grading rules on the professor’s example. I did not give him a passing grade…. We’ll see if I’m allowed to stay in the class much longer.

I only just learned that Bill Thurston died back in August. He was a Field’s Medalist, and he tried to make his work accessible to non-specialists. I own his prize-winning book attempting this. I’ve mentioned him and that book a few times: in books added, in a happenings post, and in a review of the Poincare’ conjecture. Peter Woit has some info, as does Wikipedia.

I also learned that the verdict came back in the Italian seismologists’ trial: 6 years each. I don’t know what to say. I keep seeing conflicting statements out there. Sometimes I think the only reliable source would be to translate and read the actual charges for myself. Nevertheless, here’s a follow-up by Nature, and another one by CBCNews.

I don’t know if I’ve got enough for a post on Monday… yes, I’ve been doing control theory… but I have not been working on the second-order system. As usual, we’ll see how it plays out.

Elementary Number Theory: the algebra of arithmetic functions

intro & Euler’s \varphi and…

I am going to look at several arithmetic functions, and we will see how Mathematica knows them. We will see the convolution product of arithmetic functions for dealing with particularly appropriate summations, the Möbius Inversion Formula, and we will see a conceptual basis for the Möbius \mu\ function, which basis is part of a useful more conceptual way to deal with all these summations.

If you get bogged down along the way, skip to the final section entitled “Way too Easy!”

In fact, I have filed this post under “abstract algebra” because we’ll be doing just a little algebra to get a good grip on these functions. And yet that little will go a long, long way.

A function \alpha\ defined on the positive integers – which we would usually call a sequence! – is said to be multiplicative if

\alpha(m\ n) = \alpha(m)\ \alpha(n)\

whenever m and n are relatively prime. (That is, whenever the greatest common divisor (GCD) of m and n is 1: we often use (m,n) for the GCD and write that condition as (m,n) = 1.) Be careful: we require that the function of a product be the product of the functions only when the integers are relatively prime, not for all pairs of integers m and n.
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Happenings – 2012 Oct 20

Well, when I left you last Saturday morning, I was on my way to join some friends for a look inside the Lawrence Livermore National Laboratory.

“Credit: Lawrence Livermore National Laboratory.”

It went well… very well, in fact.
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Control Theory – Simple Bode Plots

Minor edits 10/28/12: replaced “poles -1/tau” by “poles s = -1/tau” and at least one -1/tau by the absolute value of -1/tau.

We’re going to be looking at multiple Bode plots in one image. Despite everything I’ve shown you, it seems impossible to set both the color and the thickness. I believe this is a bug.

Here’s what happens if I try. (This happens to be a second-order system.)

So, we lost the color spec in the phase plots. I’ve tried a few other possibilities, but they don’t work either.
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Happenings – 2012 Oct 13

I’m going to keep this real short this morning.

Today is “family day” at the Lawrence Livermore National Laboratory, and I’m going to visit the lab with a group of friends. I won’t get much, if any, math done today.

Nevertheless, I have gotten some math done this week, so I should have a post for this coming Monday. In fact, I’ve done some work on both control theory and on number theory functions… but I think the control theory is closer to publication.

Let’s see. An asteroid missed us yesterday, as it was supposed to. (Did they have the orbit right? I don’t know.)

The defense’s closing arguments in the Italian seismology trial did not encourage me. Apparently they said, “No one actually believed the assurance that there would be no earthquake.” I think that statement flies in the face of evidence. It also seems that there is no record of what was actually said by the man who made the statements. The verdict should be back by Oct 23rd, they say.

And with that, I’m off to meet some friends.

Mathematica Notes – Coloring functions

Sometimes I wish life were always perfectly straightforward. As I said recently, I have figured out how to change the default sequence of colors for multiple graphs. I thought this would be a very short post… but there are a few additional things I need to cover in order to show that to you.

Since I usually use Mathematica® version 7, let me note that I am using version 8 for this work.

Let me start by showing you where we’re headed. If we make a list of the first five Legendre polynomials and ask Mathematica to plot them all, it chooses a set of five colors, one for each function:

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Happenings – 2012 Oct 6

It’s been a mildly interesting week.

I have figured out how to change the default sequence of colors for multiple graphs. I’ll explain this in more detail in a Mathematica note, but let me show it to you now. Suppose we take the first 5 Legendre polynomials… and plot them.

We actually got 5 distinct colors, but the fifth and first are rather close. I can select a better sequence – more importantly, I can select a better sequence without explicitly enumerating the colors. One of several possibilities is:
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Control Theory – Closing the feedback loop

open loop

This example comes from Carstens, pp. 177 and 182. He did not, however, ask us to close the loop – and that’s where it gets extremely interesting. That’s why I’m posting this. And yes, this is the example described in this diary post.

I am, as usual for control theory, using Mathematica version 8… instead of version 7 with the Control System Professional add-on.

We have a transfer function with a linear term (in the denominator) and a double pole at 0. As usual, I write the bare transfer function (called “GK”), because sometimes I need it bare… and then I wrap it into a TransferFunctionModel “tf”, because sometimes I need the Model structure… and, finally, I ask for the poles. The third pole is at -5.

(Why “GK”? In principle GK is the product of a controller K and a system G.)

OK, get the Bode Plot… and extract gain margin and phase margin… and convert phase margin to degrees:
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