Friday Random 10

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  1. Naftule’s Dream, “Something is There”: What do you get when you mix up a traditional Klezmer band with Ornette Coleman, plus just a bit of thrash? Naftule’s Dream.
  2. Genesis, “Counting out Time”: a catchy little tune from Peter Gabriel’s opus with
    Genesis, “The Lamb Lies Down on Broadway”: It’s astonishing just how undated this sounds.
  3. Mogwai, “Glasgow Mega-Snake”: one of my favorite tracks by one of my favorite post-rock bands. This one should be listened to loud to really get the full effect.
  4. Pink Floyd, “Another Brick in the Wall (Part 3)”: Wow, now here’s one I haven’t listened
    to in a very long time. It’s as good as I remembered.
  5. Porcupine Tree, “Prepare Yourself”: Brilliant stuff from Porcupine Tree.
  6. New Grange, “Going to Boston”: This is what happens when Darol Anger gets goofy: a sort
    of old-timey rap song based on a traditional New England fiddle tune. This is great fun to see
    live.
  7. Hugh Blumenfield, “Longhaired Radical Socialist Jew”: the greatest gospel song of all time. “Jesus was a homeless lad with an unwed mother and an absent dad; And I don’t really think he’d
    have gotten that far, if Newt, Pat, and Jesse had followed that star. So let’s all sing out praised to that Longhaired Radical Socialist Jew”.
  8. Rush, “Bravest Face”: It’s really good to hear Rush back in full form. This is a great song, and so obviously Rush – the Peart drums, the Lifeson guitar playing, and above it all Geddy Lee’s voice and amazing bass playing.
  9. Spock’s Beard, “All That’s Left”: SB is a great neo-progressive band. They went through
    a bit of a rough patch a few years ago, when the former leader of the band found Jesus and quit the
    band. This album is the first where they really feel like they’re comfortable with the new lineup. It’s definitely got a different sound from a lot of their older stuff, but it’s still recognizable as the same band. This is a really nice track – it’s almost a ballad, but with a really solid structure, some odd rythyms, and a lot of transitions.
  10. Jonathan Coulton, “Todd the T1000”: A very silly song from a very geeky pop-singer.

Fractal Mountains

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When you mention fractals, one of the things that immediately comes to mind for most people
is fractal landscapes. We’ve all seen amazing images of mountain ranges, planets, lakes, and things
of that sort that were generated by fractals.

mount2.gif

Seeing a fractal image of a mountain, like the one in this image (which I found
here via a google image search for “fractal mountain”), I expected to find that
it was based on an extremely complicated fractal. But the amazing thing about fractals is how
complexity emerges from simplicity. The basic process for generating a fractal mountain – and many other elements of fractal landscapes – is astonishingly simple.

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Graph Searches and Disjoint Sets: the Union-Find Problem

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Suppose you’ve got a huge graph – millions of nodes. And you know that it’s not connected – so the graph actually consists of some number of pieces (called the connected components of the graph). And there are constantly new vertices and edges being added to the graph, but nothing is ever removed. Some questions you might want to ask about this graph at a particular point in time are:

  • How many components are there in the graph?
  • Which component is vertex X in?
  • Are vertices X and Y in the same component?
  • How many components are there?

All of these questions are variants of a classic computer science problem, called
union-find, which also comes up in an astonishing number of different contexts. The reason for the name is that in the representation of the solution, there
are two basic operations: union, and find. Basically, the division of the graph into
components is also a partition of the vertices of the graph into disjoint sets: union find
is a problem which focuses on a particular kind of disjoint set problem, where you can modify
the sets over time.

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Amortized Complexity – a Tool for Graph Algorithms (among others)

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There are a lot of very cool problems in computer science that can be solved by using
an appropriate data structure; and the data structures are often easiest to describe in terms
of graphs. And of those data structures, one thing that often comes up is amortized algorithmic complexity. Amortized complexity is something which has been occupying my thoughts lately,
because it’s come up in several real problems, so I’m in the mood to write about it, and it’ll be
useful later.

The idea of amortized complexity is that for some structures, the worst case complexity
cost of a series of operations is different from the worst-case complexity of a single operation. In amortized complexity, you consider cases where some operation is inexpensive most of the time – but to keep it inexpensive most of the time, you need to periodically do something expensive.

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More Pictures

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I’ve now uploaded nearly all of the Yellowstone pictures. There are two additional
photo albums on Picasa: one for the Mammoth Hot Springs area, and one for the Yellowstone canyon. Here are the links:

Yellowstone Vacation, Mammoth Hot Springs area
Yellowstone Vacation, Canyon Area

Yellowstone Vacation Photos

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I’m back from vacation! There was no network access in Yellowstone, and virtually no cellphone service.
Anyway, a bunch of people asked me to post photos. I’ve got three sets of photographs, for the three main areas of the park that I visited: the Geyser area, the Terrace area, and the Canyon area.
The first (and largest) batch – the Geyser area – is uploaded into an album on Picasa.
The album still needs a lot of work – many things are out of order, mislabeled, etc; but the photos are there, and some of them are amazing! (No credit to me – I’m a mediocre photographer at best, using a crappy camera, but when the subject of the photos is sufficiently magnificent, even a crappy photographer can take a beautiful photo.)
Just click on the album cover below, and enjoy. Feel free to comment.

Yellowstone Vacation, Geyser Area

Good Math/Bad Math goes quiet for a week

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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

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weighted-digraph.png

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

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julia2.jpeg

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"

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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 “scienceblog.com” 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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