Monthly Archives: August 2007

Good Math/Bad Math goes quiet for a week

Bright and early tomorrow morning, I’m leaving on vacation. I’ll be spending the week in
Yellowstone National Park. I’m not sure what the network situation is there, but I’m not expecting much. Unfortunately, this year I didn’t have time to prepare some reruns of old posts before I left.
If I manage to find a data connection, I’ll try to schedule some interesting reruns. But don’t hold your breath. See you in september!

Directed Graphs


The next interesting variant on graphs is directed graphs, or digraphs for
short. A digraph is a graph where each edge distinguished between its source and its target – so an edge is from one node, and to another node. Unlike a simple graph, where if A is adjacent to B, then you can follow the edge either from A to B, or from B to A, in a directed graph, A to B and B to A are different edges.

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The Julia Set Fractals


Aside from the Mandelbrot set, the most famous fractals are the Julia sets. You’ve almost definitely seen images of the Julias (like the ones scattered through this post), but what you might not have realized is just how closely related the Julia sets are to the Mandelbrot set.

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Yet Another Idiotic "Proof of God"

A bunch of readers, and one commenter in another thread, have all hit me with a pathetic
monstrosity of a purported proof of God. Several have even been misled by the URL where the
dreadful thing is posted, thinking that ScienceBlogs have picked up a creationist. Rest assured, this bozo and his blog have nothing to do with our beloved ScienceBlogs (note the “S”); it’s just some jerk who wants to try to capitalize on our reputation.

If you want to find the original page, you can go to “” yourself and find it. I’m not going to link to this slime – his blog name is an attempt to use SBs reputation to pump up his credibility, so I’m not going to send hits his way.

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Meeting My Sciblings

This weekend, Seed Media, our benevolent and beloved corporate overlords, sponsored a Scibling gathering: ScienceBloggers from all over the country (and outside) all gathered in New York, ate, drank, and partied.

It made for quite an interesting weekend. I didn’t end up being able to hang around nearly as much
as I would have liked (I missed the drunken Karaoke! As someone who never gets drunk, watching
my drunken sciblings singing badly would have been a kick!) Alas, as the father of two small kids,
I’m subject to the schedule of family/babysitters, so I couldn’t hang aronud. (Plus, to make matters worse, my wife became sick friday night, and I started feeling sick saturday afternoon. I’m writing this from bed.)

But I did manage to meet quite a lot of folks, even in my limited time there. It’s quite an odd experience in its way; between our blogs, and our back-channel forums, we’ve become a tight-knit community, and the people there were my friends, even though I’d never seen them before. It
was a whole lot of fun. My impressions are below the fold. They’re just off the top of my head; I’ll probably edit this
later as I remember more.

By the way, that ScienceBlogs mug that Seed is offering to give away in the subscription ads? They gave us each one as a gift, and theyre great. It’s a very nice, heavy glass mug that looks like a cross between a mug and a beaker.

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Friday Random Ten

  1. Sonic Youth, “Or”: Very smooth for SY. But great. They’re an amazing band.
  2. The Flower Kings, “Blue Planet”: A typical track from one of my favorite neo-progressive
    rock bands. For the Flower Kings, this is a short one at only 10 minutes.
  3. The Clogs, “Lantern”: once again, one of my favorite classical-leaning post-rock bands. Slow,
    beautiful track, featuring steel drums and oboe, and even some light vocals.
  4. Mariliion, “Fantastic Place”: one of my least favorite tracks from an amazing Marillion
    album. Overall, the “Marbles” album was fantastic, but this track just leaves me cold.
  5. Darol Anger’s Republic of Strings, “Higher Ground”: quite a change from the last
    couple. Very up-tempo jazzy string-ensemble cover of a Stevie Wonder tune.
  6. The National, “Baby We’ll Be Fine”: The National are the alter ego of the Clogs; they’ve got
    nearly the same personnel. But the National is a slightly country-ish alt-rock band. This track
    has a great psuedo-minimalist melodic pulse under it. Very cool.
  7. Naftule’s Dream, “Free Klez 1 & 2”: what happens when a group of really talented klezmer
    musicians try to an Ornette Coleman style free improv? Weird stuff.
  8. Navan, “Thig an T-Eathor”: very traditional a-capella Irish songs. Not what you’d probably
    expect if you’re used to instrumental Irish; songs are a totally different form
  9. Mark Knopfler, “Boom, Like That”: The guitar wizard of Dire Straights has been doing mostly
    solo work lately. This is a track from his latest solo album, about (of all people) Roy Kroc, founder
    of McDonalds. Good song, with a nice guitar hook.
  10. Marillion, “Ocean Cloud”: another track from Marillion’s “Marbles” album – this one is an
    example of just why I like this album so much: it’s an 18 minute long opus, just terrific stuff.

Bad Homeopathic Differential Equations. Yech.

My friend and blog-father Orac sent me a truly delectable piece of bad math today. It’s just
astonishing: a supposed mathematical model for why homeopathic dilution works, and for why the
standard dilutions are correct. It’s called “The octave potencies convention: a mathematical model of dilution and succussion”, and I got a copy of it via the Bad Science blog. The only part of it that’s depressing is the location of the authors: this piece of dreck was published by someone from the Harvard medical school.

To give you an idea of what you’re in for, here’s the abstract:

Several hypothesized explanations for homeopathy posit that remedies contain a concentration of discrete information-carrying units, such as water clusters, nano-bubbles, or silicates. For any such explanation to be sustainable, dilution must reduce and succussion must restore the concentration of these units. Succussion can be modeled by a logistic equation, which leads to mathematical relationships involving the maximum concentration, the average growth of information-carrying units rate per succussion stroke, the number of succussion strokes, and the dilution factor (x, c, or LM). When multiple species of information-carrying units are present, the fastest-growing species will eventually come to dominate, as the potency is increased.

An analogy is explored between iterated cycles dilution and succussion, in making homeopathic remedies, and iterated cycles of reseeding and growth, in bacterial cultures. Drawing on this analogy, the active ingredients in low and medium potency remedies may be present at early dilutions but only gradually come to ‘dominate’, while high potencies may develop from the occurrence of low-probability but faster-growing ‘mutations.’ Conclusions from this model include: ‘x’ and ‘c’ potencies are best compared by the amount of dilution, not the amount of succussion; the minimum number of succussion strokes needed per cycle is proportional to the logarithm of the dilution factor; and a plausible interpretation of why potencies at approximately regular ratios are traditionally used (the octave potencies convention).

What you find in this paper is both an astonishingly bad example of mathematical modeling, and
a dreadful abuse of differential equations. It’s pathetic to realize that anyone thought
that this piece of dreck was not too embarrasingly bad to publish.

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A Bid to Shut Up the Religious Bigots

This post is quite thoroughly off-topic for this blog. But as someone who is openly religious and who
has written a number of posts that criticize Christian institutions, I get a fair bit of mail from cretins
who make demands that I speak up to defend their pathetic insistence that all religious people
must support discrimination. In the hopes that I can get these jackasses to leave me alone by
demonstrating that I’m so far beyond the pale that pestering me is a waste of time, I present this

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Representing Graphs

One of the reasons that I like about graph theory so much is because of
how it ties into computer science. Graphs are fundamental to many problems in
computer science, and a lot of the work in graph theory has direct implications for
algorithm design. It also has a lot of resonance for me, because the work I do involves
tons and tons of graphs: I don’t think I’ve gotten through a week of work in the last decade without
implementing some kind of graph code.

Since I’ve described a lot of graph algorithms, and I’m going to
describe even more, today I’m going to talk a bit about how to represent graphs
in programs, and some of the tradeoffs between different representations.

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Traversing Graphs

One amusing thing in graph theory is graph traversal. Many of the interesting algorithms on graph
are ultimately based on the idea of iterating through the nodes of the graph in some order that is
related to the structure of the graph.

There are two fundamental orders of graph traversal, known as breadth-first and depth-first.

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