Happenings Jun 27

A friend of mine sent me a link (to a link) last week. The second link is an essay on the teaching of mathematics. It breaks my heart, both for its truth and its beauty.

I think the author is dead on about what’s wrong in the teaching of mathematics — and not just in K-12. He is also absolutely on the mark when he describes what mathematics means to me.

It is very well written, and I think it is very worthwhile reading.

There was an interesting question and answer out on the sci.math newsgroup (and probably sci.physics). The question was posed in John Baez’ June 20 “This Week’s Finds in Mathematical Physics (Week 276)” and answered in the very next post in the thread, by Phillip Helbig.
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Wavelet properties: orthogonality & counterexample

orthogonality

Edit 3 Aug. It is embarrassing to be so wrong in my guesses. Well, I understand more today than I did yesterday.

The two forms of orthogonality are not the same. We saw in the subsequent post about semi-orthogonality that we could have an orthogonal direct sum V_0 \oplus W_0 while having non-orthonormal bases in V_0 and W_0\ . Conversely, if we had a direct sum which was not orthogonal, we could still choose orthonormal bases in the two spaces. Okay, let me rephrase that: if we have bases in V_0 and W_0\ at all, I know we can make them orthonormal.

I am no longer sure about the following guess about integer translates of elements of W_0\ . And I am completely wrong about the conjecture in red.
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Wavelet Properties: one more from the dilation equation

Introduction

I want to add one more property to the previous post. This is long enough that I do not want to insert it as an edit. It’s amazing that I ever considered even for a moment to insert it as an edit!

Just as we deduce what is called a partition of unity \sum_k { \varphi(k)} = 1 from the dyadic sum

\sum_k { \varphi(\frac{k}{2^j})} = 2^j

(by setting j = 0)

we can get a more general equation, if \varphi is continuous. We can show that

\sum_k { \varphi(t-k)} = 1\ ,

where the sum, as before, is over all integer k. (The proof uses the previous dyadic sum and continuity to say something about non-integer values.) This may be called a generalized partition of unity.
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Wavelet properties: consequences of the dilation equation

discussion
edited 8 Jun to cross out the last line and correct it. see edit.
edited 13 Jun to correct the last line again. see edit. Sorry about that, but I shot from the hip and hit myself in the foot.

We have seen in the previous post that the idea of a set of scaling functions \{\varphi(t-k)\} spanning a space V_0\ , and such that the set \{\varphi(2t-k)\} span a space V_1\ , gives rise to a dilation equation, which we have been writing as

\varphi(t) = \sum_n {h(n)\ \sqrt{2}\ \ \varphi(2\ t - n)}.

(We also get more spaces V_j\ for both positive and negative integers j.)

We have also seen that if we impose an inner product, then wavelets live in the orthogonal complements W_j\ :

V_{j+1} =: V_j \oplus W_j\ .

I should probably remark, first, that a solution \varphi(t)\ of the dilation equation is determined only to within a constant factor. We describe that, however, by describing its integral (effectively, its average) rather than, for example, its maximum value. We write

\int\ \varphi(t)\ dt = E\ ,

where E is a nonzero constant which we often choose to be 1: E = 1.

To put that another way: if there is a solution, it is not unique.

Two examples we’ve seen repeatendly are Haar and Daubechies. The Haar scaling fuction is a unit step function on the unit interval, so its integral is 1. Similarly, the integrals of the Daubechies scaling functions are 1, although I have not shown that.

There are 3 consequences of the dilation equation (mostly with other conditions added).
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Happenings Jun 6

I’m pretty much working on only two things: wavelets and finite fields. In fact, right now I’m looking at field extensions, things like Q(\sqrt{2}) = \{a + b\ \sqrt{2} | \text{a,b rational}\}\ . (Just can’t seem to shake off those \sqrt{2}\text{'s}\ ).

I even asked a careless question last week out on sci.math. I won’t call it a stupid question — I insist that there are none — but I did forget one of the hypotheses. The topic was “Visualizing Finite Fields” in sci.math. My post was on May 29, and a reply pointing out my carelessness appeared several hours later, in the evening.
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