Blog Archives

We got the definition wrong

When I was in grad school, I had a course in Banach spaces with Haskell Rosenthal. One day he said “We got the definition wrong.” It took a while to understand what he meant. There’s nothing logically inconsistent about the

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Sensitive dependence on initial conditions

The following problem illustrates how the smallest changes to a problem can have large consequences. As explained at the end of the post, this problem is a little artificial, but it illustrates difficulties that come up in realistic problems. Suppose

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Relating Airy and Bessel functions

The Airy functions Ai(x) and Bi(x) are independent solutions to the differential equation For negative x they act something like sin(x) and cos(x). For positive x they act something like exp(x) and exp(-x). This isn’t surprising if you look at

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Hilbert space methods for PDE

When I was in grad school, my advisor asked me to study his out-of-print book, Hilbert Space Methods in Partial Differential Equations. I believe I had a photocopy of a photocopy; I don’t recall ever seeing the original book. I

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Mean residual time

If something has survived this far, how much longer is it expected to survive? That’s the question answered by mean residual time. For a positive random variable X, the mean residual time for X is a function eX(t) given by

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Negative damping

An earlier post looked at the effect of damping on free vibrations. We looked at the equation m u” + γ u’ + k u = 0 where the coefficients m, γ, and k were all positive. But what if

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Pulp science fiction and vibrations

This morning I ran across Pulp-o-mizer and decided my series of posts on mechanical vibrations could use a little sensational promotion. The posts are Part I: Introduction and free, undamped vibrations. Part II: Free, damped vibrations (under-damping, critical damping, over-damping)

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Damped, driven oscillations

This is the final post in a four-part series on vibrating systems. The first three parts were free, undamped vibrations free, damped vibrations driven, undamped vibrations and now we consider driven, damped vibrations. We are looking at the equation m

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Undamped forced vibrations

This is the third in a series of four blog posts on mechanical vibrations. The first two posts were Part I: Introduction and free undamped vibrations Part II: Free undamped vibrations

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Free damped vibrations

This is the second post in a series on vibrations determine by the equation m u” + γ u’ + k u = F cos ωt The first post in the series looked at the simplest case, γ = 0

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Fourier series before Fourier

I always thought that Fourier was the first to come up with the idea of expressing general functions as infinite sums of sines and cosines. Apparently this isn’t true. The idea that various functions can be described in terms of

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Mechanical vibrations

My favorite topic in an introductory differential equations course is mechanical and electrical vibrations. I enjoyed learning about it as a student and I enjoyed teaching it later. (Or more accurately, I enjoyed being exposed to it as a student

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Differential Equations and the City

This afternoon I got a review copy of X and the City: Modeling Aspects of Urban Life by John A. Adam. It’s a book about mathematical model, taking all its examples from urban life: public transportation, growth, pollution, etc. I’ve only

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Castles and quantum mechanics

How are castles and quantum mechanics related? One connection is rook polynomials. The rook is the chess piece that looks like a castle, and used to be called a castle. It can move vertically or horizontally, any number of spaces.

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Easiest and hardest classes to teach

I’ve taught a variety of math classes, and statistics has been the hardest to teach. The thing I find most challenging is coming up with homework problems. Most exercises are either blatantly artificial or extremely tedious. It’s hard to find

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