Transpose & Adjoint: review & example


Let me pick up an old problem. The mathematics comes from the two posts “transpose matrix and adjoint operator” part 1 & part 2. The problem itself comes from the schur’s lemma post.

Upon further reflection, I am going to change the problem a little bit. Do not expect to see the same answers as before.

I am also going to work it twice, assuming that we are given different information as our starting point, but I’ll do it for the very same problem.

As I have said, the appropriate question for an introduction to ABO blood groups is: Can your mother donate blood to you? Until you can answer that question, you’re missing something about blood groups.

This example is not that good, but it does tie together the following concepts and computations: Read the rest of this entry »

Correction to Happenings Aug 29

For any of you who get an RSS feed, the first publication of the “happenings” post did not actually include two links, which have just been added.

Happenings Aug 29

It is way past time to put out a diary entry. No, they don’t seem to be useful or popular — but that is your business, not mine. My business is that I want this blog to reflect my doing mathematics, although its main focus is the presentation of mathematics.

The two latest posts — color, and surfaces — should suggest, correctly, that I have put down wavelets for a while. Well, I have put them down, but I am still working on them indirectly. It is way past time to move into the frequency domain for wavelets, and I have begin to do so.
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Color: HSB (HSV, HSI) again

I need to learn to write better summaries.

Here are the summaries I should have written for the HSB post, and for the tint-tone-shade post.

First, I am going to describe what “saturation” and “brightness- intensity- value” mean in the HSB (HSI, HSV) color space.

I cannot over emphasize that I am discussing what these words mean in a particular color space. As far as I know, these are not universal truths, but particular approaches to them. (I could be wrong: these definitions may be more general than I know, but that’s the point – I don’t know that they are universal.)

I still don’t like the non-technical definitions which I see all over the place, but I will admit that there is a place for them. Color is a physiological response, so many questions of color boil down to describing our response to it. And that is, ultimately, personal and subjective; we don’t all have the same eyes.

Someday I may find other quantitative definitions of “saturation” and “intensity”. This is what I have, so let me summarize it. (Of the three equivalent terms — brightness, intensity, value — I am going to use “intensity” in this post.)

Let me say that again with different emphasis. Brightness, intensity, and value are equivalent terms in these color spaces.
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12 pentagons!

I’ve been reading Sternberg’s “Group Theory and Physics”, Cambridge University, reprinted 1999. On pages 43 to 44 he says, “… every fullerene has exactly 12 pentagons. This is not an accident.”

The stable structure of carbon which has 60 carbon atoms arranged like the vertices of a soccer ball is called a buckyball. It turns out that, in similar structures, we can have any even number, greater than 18, of carbon atoms except for 22. This is equivalent to polyhedra having 12 pentagons and any number of hexagons except 1.

This family of structures consists of polyhedra whose faces are either pentagons or hexagons. In chemistry they are labeled by the number of carbon atoms, so they talk about C_{20},  C_{22}, ...  C_{60}, ...  C_{72}, ....\

I find it unforgettable and marvelous that the number of pentagons is always exactly 12. And I can prove it.
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Wavelets: Consequences of Orthogonality and Review II

digression: eigenvalue 1

Before I proceed with consequences of orthogonality, I need to mention an omission. For one thing, I have gotten so caught up in the properties I’ve been looking at, that I have forgotten one of the crucial ones we used earlier. For another thing, the consequence which I have forgotten is still a little strange to me.

The consequence (or consequences)?

  • That the dilation equation could be written as an eigenvalue equation,
  • that the existence of a scaling function seems to be equivalent to an eigenvalue = 1,
  • and that the values of the scaling function at the integers are given by the corresponding eigenvector.

This has been crucial to some of our work: the recursion which I use for computing the scaling function is initialized by setting the values of the scaling function at the integers — that is, by finding the eigenvector with eigenvalue 1. Recursion — especially when combined with a lookup table — is very easy and very powerful; but we absolutely had to have initial values, and that eigenvector provided them. (I did this for the Daubechies D4, and for four wavelet systems with 6 nonzero h’s.)

The strangeness? Read the rest of this entry »

Wavelets: Review I and Going Forward a Little

Let us recall what we have.

We have a collection of nested spaces…

\dotsm\ V_{-3} \subset\  V_{-2} \subset\  V_{-1} \subset\  V_{0} \subset\  V_{1} \subset\  V_{2}\ \dotsm\ \

… whose intersection is the trivial space and whose union is all square integrable functions on the real line:

\cap\ V_i = \{0\}\ and \cup\ V_i = L^2(R)\ .

We assume that the spaceV_0\ is translation invariant and has the scaling property:

f(x) \in V_0\ \text{ if and only if } f(x-k) \in V_0\ for all integers k;

f(x) \in V_j \text{ if and only if } f(2^{-j} x) \in V_0\ .

Finally, the only real theorem I have shown you says that if we also have an orthonormal basis for V_0\ , then we can get an orthonormal basis for L^2(R)\ :

\psi_{j,k} (x) := 2^{j/2)} \psi(2^j\ x - k)\
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