Learning Computation

Counter-intuitive science

2008-12-05T10:19:00.005-05:00

What do you do when you come across a bit of science and engineering that seems totally counter-intuitive? Do you try to construct arguments for why it can't work and must be a hoax of some kind? Or do you accept that maybe you just don't (yet) understand the underlying concepts and need to work on improving your knowledge? Maybe you could even try experimentally verifying the concepts yourself, if no special equipment is required? Personally I have always thought that the concept of angular momentum is totally mind-bending, but children verify that this is a real force every time they spin a top, or ride a bicycle without falling over.

I imagine that your response will depend a lot on the context involved. If you're sitting in a science class and you generally trust your instructor, you're probably going to accept whatever you're told. But in fact, at least for college-level courses, you're not supposed to just take the instructor's word for things--that's why they have labs, so you can excruciatingly recreate classic experiments to verify the concepts for yourself.

On the other hand, if you distrust your source, you're liable to have the other kind of response. For example, adherents of intelligent design construct all sorts of ad hoc and easily debunked reasons why evolution cannot work as described, and seem to think that mainstream biologists are all either self-deluded or participants in a grand conspiracy. Many of the rest of us might consider the Internet, taken as a whole, to be a not-completely-trustworthy source, and if we see something peculiar on the Web we might be tempted to assume that it's just a joke or a hoax of some kind.

I mention this as a lead-up to the following hilarious little story that's current bouncing around some science and tech blogs: A couple of years ago, Jack Goodman posted a YouTube video showing a wind-powered cart he built to demonstrate the concept of "direct downwind faster than the wind" (or DDFTTW for short) travel. It is certainly counter-intuitive at first glance: if you are traveling directly downwind faster than the ambient wind speed, then you are going to be feeling the wind in your face! How could that wind possibly be providing the energy source to make you move forward? And yet (not to give away the story), this is a real effect that does not somehow violate the laws of physics.

Recently, debate over this was reignited when a new video was posted on YouTube to demonstrate the effect beyond any doubt and prove that it is not a hoax:

Perversely, it seems to have only increased the debate about what's going on. The story was picked up by the tech blog BoingBoing, which took a somewhat skeptical view of things. The pseudonymous poster of the video, spork33, has patiently responded to questions and counter-claims in the comments over there, which seems to have only made the skeptics more vocal in their complaints.

Science blogger Mark C. Chu-Carroll posted a blog entry at Good Math, Bad Math in which he attempted to intellectually eviscerate spork33 and his colleagues:

Via BoingBoing comes a bunch of bozos who believe that they can create a "wind-powered" vehicle that moves faster the wind that powers it. This is, obviously, stupid.

Chu-Carroll goes on to argue that the treadmill demonstration in the video has no relationship to conditions "in the field", using the same kind of dismissive language. He finishes up with this gem:

Everyone should be able to understand the physics involved here. My third grade daughter can understand this. This isn't difficult. There's nothing tricky or subtle about it. If you have a vehicle moving at the same velocity as the wind, the wind cannot possibly exert any force on the vehicle. No force, no acceleration. Period. How can supposedly intelligent, educated people not know this?

Unfortunately for MarkCC, since this phenomenon is completely on the level, he is the one who comes off looking like, well, a bozo.

[UPDATE: Mark has printed a retraction on his blog and apologized for his intemperate language. Curiously, he says he needed to work through the math to convince himself that the device could actually work. Personally, the "math" is the last thing I'd trust in a situation like this--the various comment threads are full of mathematical calculations by skeptics who claimed that their results show that the device can't work. Their calculations might have even been correct, but the way they formulated their models was wrong, so the results were meaningless. I'd much rather trust my eyes, but evidently Mark and many other skeptics just couldn't bring themselves to believe that the demonstration in the video wasn't using some sleight-of-hand.]

For a proper explanation of the physics involved, you could browse through the lengthy comment threads at BoingBoing or Good Math, Bad Math. It's a fascinating read if you have the time. Fortunately, Dave Munger has a post at Word Munger where he gives a succinct explanation of what's going on, and in which he addresses the major questions that have been raised on the other threads. (Dave is perhaps better known for his blogging at Cognitive Daily, and as a founder of ResearchBlogging.org.) Of course, there are still many skeptics who continue to argue in the comments thread on Dave's post.

Even though DWFTTW travel is totally cool just by itself, the thing I find most intriguing about this whole episode is the mindset of skeptics. I know that I've been guilty of this sort of thing too, where I'm totally convinced that such-and-such is a fact, and I discover later that just the opposite is true. What psychological factors cause us to believe what we believe? And what techniques can be used to help change the mind of a skeptic, when it is clear that simply presenting a factual exposition is not going to do the trick? Dave, if you see this, perhaps this question would make a good topic for your Cognitive Daily blog.

The Original of Laura

2008-05-02T11:09:00.007-04:00

This item is a bit far afield for this blog, but I bring it up because I mentioned the author Vladimir Nabokov a few posts back (scroll down to resolution #3). Nabokov was working on a new novel, to be titled The Original of Laura, at the time of his death in 1977. Nabokov had indicated to his family that he wanted the unfinished novel destroyed if he died before he completed it, and it has sat unpublished ever since.

Now Nabokov's son Dmitri has decided to go ahead with publication. In an interview on NPR, Dmitri Nabokov talks about the process leading up to the decision. This little bit of trivia caught my ear: when asked how long the novel fragment was, Dmitri said that his father had completed 138 index cards, which would come out to be about 100 pages including some supplemental material. A date for publication has not been set yet, but it should happen fairly soon.

Circular Puzzle

2008-04-26T01:13:00.005-04:00

Puzzles of various sorts have been popular on the math blogosphere lately, so I thought I'd make my own modest contribution. As puzzles go, there is not very much to this one -- no calculations are necessary, you can just look at it and visualize the answer. But the backstory that goes with it is mildly amusing. This appeared as a question on the SAT exam, back many years ago when I was an undergrad.

Imagine a circle of radius 1 unit, sitting on top of a larger circle of radius 3 units, like so: Now suppose that the smaller circle rolls around the perimeter of the larger circle until it returns to its starting position. In the course of doing this, how many rotations about its axis will the smaller circle make?

Before going on, you might want to take a moment to think about what the answer should be... Okay, ready? This particular problem was featured in a story in the news because the answer key used to score the SAT had the wrong value for this question. Let's just call it x for the time being. A high school student who sat for the exam realized that the correct answer should be y, and the math scores had to be retabulated for everyone who took the SAT that session.

The story appeared in our local paper over the weekend. The following Monday, the professor for my differential equations class, who also happened to be serving as the department chairperson that semester, mentioned that he had received a phone call from a newspaper reporter asking about the question. The professor sketched out the problem for us, and said, "I told him that, of course the answer was x." Naturally as soon as class was over, several of us students who had read about the problem over the weekend ran up to the professor and explained to him why he was wrong.

For the next week, the problem continued to stir up debate in our math lab. About half the students who looked at it initially believed the answer was x. Most people quickly changed their minds once it was explained to them, but we did have one hold-out who refused to budge. If I remember correctly, he was a Ph.D. candidate in differential geometry, who insisted that since he had advanced training in a geometrical field, we should defer to his judgment on the problem.

(Now, as to what x and y are... You have almost certainly figured out the correct values for those by now. If you are still unsure, though, I'll put a hint in the comments for you.)

A Short Note on the Halting Problem

2008-04-22T11:24:00.003-04:00

Over at Shtetl-Optimized, Scott Aaronson recently had an open thread where he invited readers to pose questions for him (this apparently is what Scott considers relaxation after a hard week). There was an interesting exchange in the comments that caught my eye, but since Scott has already closed the comments on that post, I thought I'd write about it here.

A commenter named Abel posed the following question:

Are there any interesting results concerning subsets of Turing Machines for which the Halting Problem can be solved?

For example, it is trivial to see that machines for which you cannot go back to a previously visited state do not halt. But it would be interesting if there were any non-trivial result for a subset in which there are both machines that halt and that doesn’t halt.

A little further down in the comments, Scott replies that a reader has emailed a response:

Matthew P. Johnson writes in to answer Abel’s question, on whether there exists a “nontrivial” but structurally-defined subset of Turing machines for which the halting problem is decidable:
Supposedly, the subset is extremely close to trivial, i.e., almost all of them (!):
“The halting problem for Turing machines is decidable on a set of asymptotic probability one. Specifically, there is a set B of Turing machine programs such that (i) B has asymptotic probability one, so that as the number of states n increases, the proportion of all n-state programs that are in B goes to one; (ii) B is polynomial time decidable; and (iii) the halting problem H intersect B is polynomial time decidable. The proof is sensitive to the particular computational model.”
“The halting problem is decidable on a set of asymptotic probability one”
http://arxiv.org/abs/math/0504351

Since that's a seemingly remarkable result, I'd like to look at it a little closer, and then offer a different response to Abel's original question.

The halting problem is decidable on a set of asymptotic probability one is written by Joel David Hamkins and Alexei Miasnikov, both of the City University of New York. The paper appears to be based on a talk by Hamkins at the Fall 2004 CUNY Logic Workshop.

Since they mention that their proof is sensitive to the particular computational model they use (actually, it would be more precise to say result instead of proof), let's start by taking a look at that. They use a fairly standard, if bare-bones, Turing Machine model. Their TM uses an alphabet consisting of {0,1} and a single tape which extends infinitely to the right.If Q is the set of (non-halting) states of the TM, then the transition function is given by:

δ: Q×{0,1} → (Q ∪ {halt})×{0,1}×{L,R}.

This allows the easy calculation that for a given size n (of number of states), there are (4(n+1))²ⁿ distinct TMs.

Now we come to the one little (but critical) feature of their model: if the head is located on the left-most cell, and the TM attempts to move the head to the left, then the computation fails in a non-halting condition. (We can suppose that this TM is implemented in Windows NT, and instead of failing gracefully, we get a "blue screen of death" when the TM attempts an illegal move.) We're ready to look at their main result.

Define B to be the set of TMs that, on an input tape initialized to all zeros, either halt or fall off the left edge of the tape before repeating a state.

Now, I imagine you can already see where this is going. We can obviously tell if an arbitrary TM is an element of B and if so whether it halts or not, by simply simulating it for n steps. (I wouldn't call that a structural property of the TM, but in fairness the authors do not describe it that way, either.) The only interesting question is whether B has asymptotic probability one. For this, the authors invoke a result on random walks due to Polya:

In the random walk with equal likelihood of moving left or right on a one-way infinite tape, beginning on the left-most cell, the probability of eventually falling off the left edge is 1.

This can be seen intuitively by imagining the behavior of the TM's tape head. At each step it can move either left or right. Assuming the TM doesn't halt first, as n gets larger it becomes increasingly likely that the head will return to the starting point; and each time it does there is a 50% chance it will fall off the left edge on the next step. There is some work involved in making this into a rigorous argument, but basically that's the entire paper right there. (By the way, for some more interesting background on random walks, see Brian Hayes' recent post at bit-player.)

I wouldn't blame you if you find that result less than satisfying.

I think I can offer a somewhat more satisfying answer for Abel. The canonical example of a computing model for which the halting problem is decidable is the linear-bounded automata, or LBA. In an LBA, as well as other space-bounded models, it is possible to tell if a TM is going to loop by simply letting it run long enough to exhaust all possible configurations. It is possible to enforce this syntactically for arbitrary TMs by adopting some conventions for the presence of an outer "control module" that enforces tape boundaries, etc. This same approach also works for syntactically checking for membership in time-bounded classes like P and NP, although of course the halting question is moot for those classes.

This agrees with our intuition about real-life computing. Physical computers are, as a practical matter, space-bounded machines, and they may as well be treated as time-bounded also. For most algorithms we have a pretty good idea of how long they should run, and if a computer program is taking too long to finish we don't hesitate to pull the plug on it and start examining our code for where we screwed up.

The Physiology of Religious Experience

2008-03-13T09:27:00.007-04:00

I posted a couple of months ago about TED - Technology, Entertainment, Design, and the wonderful presentations that can be viewed on their website. Great new material is continually being posted there, and a case in point is a recent talk by brain researcher Jill Bolte Taylor. From the description:

Neuroanatomist Jill Bolte Taylor had an opportunity few brain scientists would wish for: One morning, she realized she was having a massive stroke. As it happened -- as she felt her brain functions slip away one by one, speech, movement, understanding -- she studied and remembered every moment. This is a powerful story about how our brains define us and connect us to the world and to one another.

The language that Dr. Taylor uses to relate her experience is very much like what one might expect to hear from someone describing a mystical or religious experience, and it would be hard not to draw connections between the two. This is simply a fascinating talk. (It also deserves mention for Dr. Taylor's use of a certain visual aid.)

To view this talk at the TED website instead of using the embedded video above, go here. You can learn more about Dr. Taylor at her website, and she also has a book about her experience available from Lulu. (Hat tip to Mo at Neurophilosophy.)

New Year's To-Do List

2008-01-16T11:27:00.000-05:00

Well, it's once again the time of year when I review the state of my life, and resolve to do something about all the shortcomings that inevitably present themselves. Unfortunately, although I have no trouble coming up with worthwhile resolutions, I've never had much success in keeping them, to paraphrase Jerry Seinfeld. Therefore I don't think I'm going to waste time with making any resolutions in the traditional sense. Resolve is such a weighty word. I'm going to try taking a less serious approach here. I see that Jane has given herself a to-don't list, but I'm a little skeptical of this approach. The standard truism applicable here is that it is better to prescribe the desired behavior than to proscribe the unwanted behavior, and I have to agree with this advice. However, I do like the idea of a to-do list -- it's much more matter-of-fact than a resolution. A to-do item is like doing the laundry or paying the bills; it's something you do every day. Most importantly, it's something you know that you can do. So, here is my to-do list for 2008.

Where to start? According to Dave Munger at Cognitive Daily, having too many resolutions reduces your chances of success, so I'm going to limit myself to five items. His list of most-commonly made resolutions looks depressingly predictable, but as I am not one to break with tradition, I will start out with a health-related to-do.

One

I'm currently about 20 pounds over my "ideal" weight. I figure that if I set my goal at the modest target of losing about one pound per week, and allowing for a little bit of back-sliding, I should reach my ideal weight by mid-year. I debated about whether to include this item in this post (since, after all, who out there cares about how much I weigh?), but it gives me an opportunity to demonstrate the neat little chart API from Google. This API lets you generate graphs on-the-fly just by specifying a URL with the necessary parameters. So for example, here's a line graph I can use to chart my weight progress:

I'll have to post an updated graph every month or so, so that you can shame me into staying on target. (Hat tip to Brian Hayes at bit-player for this.)

If you need an extra incentive to get going on your own health to-do, this news item about a recent health study might help. Researchers in England found that middle-aged men and women who followed four good-health practices (don't smoke, consume alcohol in moderation, get some exercise, and eat lots of fruits and vegetables) were 4 times more likely to have survived through the duration of the study than those who followed none. Participants who only followed 1, 2, or 3 of the practices still showed (correspondingly smaller) benefits. Not surprising, but I think that one of the points of the study is that a few simple, relatively easily implemented practices can make a big difference in your health. You can read the whole article at PLoS Medicine.

Two

Also high on Dave Munger's list of typical resolutions is getting organized. If you've ever seen the HGTV show Mission: Organization, you have an idea of what my personal space looks like. I can't think of an aspect of my life that couldn't benefit from some organization, but just to keep things simple I'm going to focus on one item: my filing cabinet. I need to redo my filing system, and I'm going to set a target date of March 1st, which will force me to get my tax records together in plenty of time for tax season. I don't have any fancy links to help with setting up the physical files themselves (although I do like the looks of these MAGNIfiles). However, once the files are set up the tricky part is keeping them in order. I've been told that the adage to keep in mind is "handle each piece of paper only once", and if you Google that expression you'll get hundreds of relevant links. This one (somewhat randomly chosen) seems pretty sensible. Of course, since most of this is just common-sense stuff, the trick is not knowing what to do, but actually doing it.

One final link on filing, just to keep this post from being too un-mathy: Jeff Erickson had a neat post a couple of years ago on the time-complexity of various filing strategies. It turns out that the system I've been using isn't so bad after all!

Three

Next up, I want to try to improve my productivity. To keep this concrete, I'm going to set as my goal to write for 2 hours a day. I'm using "write" very loosely here, to mean pretty much any form of output in which I'm creating something instead of just passively reading or viewing. Now, this kind of goal is notoriously hard to keep, and I'm deliberately choosing it in order to try out some motivational techniques. On his blog, Jim Gibbon describes the Seinfeld method, which consists of getting a big, full-year wall calendar and making a large mark on each day you meet your goal. The aim is to try to keep the resulting string of marks from being broken. I love the simplicity of this technique, and I can use it to track my progress on my other goals as well. Jim Gibbon also describes a technique called contingency management, in which you give yourself a reward each time you complete your daily goal (or a punishment when you fail). I suspect that I would be too tempted to cheat using this method, but I'll give it a try as well.

One general problem I've had with writing is that ideas invariably occur to me at the most inopportune times, and I end up forgetting them before I have a chance to write them down. I can't be bothered with carrying a journal around with me all the time, and I tried using a voice recorder but was too self-conscious to make good use of it. But I recently read that Vladimir Nabokov did his writing using index cards, and this seems like such a great solution that I can't wait to give it a try.

Four

One of the things I plan on doing with that writing time is to produce some actual mathematical content for this blog instead of the typical link-fests (like this entry). But to present the math properly, I think I need to bite the bullet and convert this blog to WordPress. After all, if it's good enough for Terry Tao's What's New, Scott Aaronson's Shtetl-Optimized, Tyler DiPietro's PowerUp, and Foxy's FoxMaths! (among others), then it ought to be good enough for me.

Now, just setting up a new blog takes almost no time at all. But it's been ages since I've updated anything else on this site, so I think I need to give the whole place an overhaul. Since I've committed myself to getting my files in order first, I think I'll set a leisurely goal of May 1st for my new web digs.

Five

So if all goes according to plan, by mid-year I'll be fit, organized, and churning out math posts on my remodeled blog. That leaves the second half of the year open for something perhaps a little more ambitious.

Bill Gasarch recently posted his predictions for 2008, and I couldn't help noticing a hint of a contradiction between his 2nd and 3rd items:

There will be a big breakthrough in theory. Very hard to predict what it will be- note that this years big breakthrough, faster algorithm for integer multiplication, would have been hard to predict.

P vs NP, P vs BPP, will not be solved.

Now, to me that seems almost like a dare. So, this gives me the perfect project for the second half of this year: write a proof that P≠NP.

To help guide me with my proof-writing, Scott Aaronson recently posted a list of Ten Signs a Claimed Mathematical Breakthrough is Wrong. If I take care to avoid the pitfalls on this list, I'll be sure to get everyone to read my results. Actually, the one item that I think I would deliberately indulge in is this:

8. The paper wastes lots of space on standard material. If you’d really proved P≠NP, then you wouldn’t start your paper by laboriously defining 3SAT, in a manner suggesting your readers might not have heard of it.

Since a proof of something like P≠NP would be of interest to a wider audience than your typical math result, I would try to make it as self-contained as possible.

So to recap, here's my to-do list for 2008:

Lose 20 pounds.
Organize my files.
Commit to writing for 2 hours a day.
Update this web site and switch to WordPress.
Write a proof that P≠NP.

As I mentioned near the beginning of this post, since these are to-do items and not resolutions, I can be confident that I can matter-of-factly complete each task. We'll check back in December and see how I did!

What have you changed your mind about?

2008-01-06T01:19:00.000-05:00

A couple of months ago I wrote about Edge Foundation's What is your formula? question. Well, the results of the Edge annual question for 2008 are now online. The question posed to Edge's contributing scientists and thinkers this time around was, What have you changed your mind about? Why? (You'll need to scroll down past a bunch of blurbs on that page before you come to the list of responses.)

It's a challenging question to answer, and I was expecting some intriguing responses. There are some interesting answers, to be sure, but not as many as I initially expected. In retrospect, I guess that's not too surprising. (Ha! Look, I changed my mind about that!) Since a change of thinking is required, it kind of implies that the holder's original point of view was wrong. So, if someone has an important idea, but it is their original point of view on a subject, then it just isn't going to show up in this year's mix. In fact, in browsing through the responses I frequently found myself thinking, "Of course you changed your mind--your original ideas were clearly wrong!" And there were also a couple of entries were I'm pretty sure the authors changed their minds in the wrong direction.

When I try to answer this question for myself, I have a hard time coming up with a clear-cut example. Of course, there are lots of little mundane day-to-day opinions that change, like what kind of toppings I like on my hot dogs. But for the "big ideas", it's more a matter of gradual refinement over time as I learn more about a subject, than a complete change in direction. (I am of course limiting myself to my opinions as a mature adult. If my opinions during my teen years were included, I'm sure my parents could produce many examples where I've since done a complete about-face, but thankfully I have pretty much total amnesia about that time of my life.)

Here are just a few selections that struck me as I read through them...

Susan Blackmore talks about how she stopped believing in paranormal phenomenon. I've never been a real believer in this kind of stuff, but I did spend a lot of time researching it in college because it's the kind of thing that would be so cool if only it were true. I quickly realized, however, that nearly all the research published in parapsychology journals was just plain crap. It made me wonder how any self-respecting scientist could continue to work in the field.

I had to re-read Daniel Hillis's entry more than once to convince myself that I understood him correctly. He talks about how, as a child, he was told that hot water froze faster than cold water and he refused to believe it because it defied common sense. He challenges the reader to try the experiment for themselves in order to convince themselves otherwise. I'm pretty sure that he's just trying to taunt us into adopting an experimentalist mindset, but I'll be damned if I'm not going to have to try this for myself now.

Many respondents talked about changes in religious belief. Conflicts between fundamentalist religious beliefs and science have been in the news a lot lately (for example, in the context of the primary election debates), but moderate religious beliefs are generally portrayed as compatible with modern science. Clay Shirky has come to disagree with this perspective and argues that even moderate beliefs are not reconcilable with science, and we are entering a long period of societal restructuring on this issue.

Finally, there were a number of mathematically oriented entries. These tended to deal with philosophical shifts, such as Keith Devlin's move away from platonism and toward socially or evolutionarily constructed mathematics. But Bart Kosko had a very specific change of mind: he argues that the median should be the preferred measure of central tendency, instead of the mean.

Some inspirational videos to close out the year

2007-12-31T22:55:00.000-05:00

2007 has been a depressing year, and I don't think that it's just me. I've pretty much given up on watching TV news or reading newspapers. Doing so just adds to the malaise. And because of the writers' strike, the one remaining palatable news source has been unavailable. (Now if I could only wean myself from the Web, I might start feeling better...but that's a topic for another post.) So to close out the year, here are a couple of items that have managed to cheer me up and inspire me, in the form of Internet videos.

First up is TED. TED is an acronym for Technology, Entertainment, Design, and at its core it is an annual conference where the movers and shakers of the intellectual world gather to make short (15-20 minute) presentations on what is challenging and inspiring them. The best of these presentations have been made available as streaming video on the TED website. There is a truly eclectic assortment of topics, so there's bound to be something of interest for everyone. The TED staff maintains a blog where you can find out about the latest talks to go online. A while back they posted a list of all the talks that are available, which makes a handy reference point even though it's a couple of months out of date now.

Here's a quick sampling of some TED talks: Neuroscientist V.S. Ramachandran describes how we can learn about how the brain works by looking at how it fails to work normally in patients with neurological disorders. Ramachandran is a great lecturer and I was blown away by this talk. Biology student Eva Vertes talks about her ideas for finding a cure for cancer. Watch this not so much for the specific ideas presented but just to be reminded of the promise of youth. (Hat tips to Mo and Bora, respectively, for these two talks.) A couple of my own picks: Computer scientist and entrepreneur Jeff Hawkins talks about modeling the human brain on a computer. This talk is basically an executive summary of his book, On Intelligence. In the time constraints of the talk, Hawkins comes across as less than convincing, although in his book he develops a much stronger case for his ideas. Finally, for the math-ed crowd, be sure to check out "mathemagician" Arthur Benjamin, who puts on an amazing display of rapid-fire mental arithmetic. It's interesting to look at the comments for that presentation--some of the viewers seemed to think that Benjamin must have had a secret radio receiver in his ear with an accomplice sending him the answers! Michael Shermer joins in the comments to point out that he and Benjamin co-authored a book on how he performs his mental feats.

The second item for your consideration is the Last Lecture by Carnegie-Mellon computer scientist Randy Pausch. Pausch is suffering from incurable pancreatic cancer, and in his recorded farewell lecture at CMU, he recapped his career and his perspective on life so that his now-young children would one day be able to know a little more about him. This talk has become something of an Internet sensation after being reported about on network news and in other media. The "last lecture" is also going to be developed into a book, with the assistance of Wall Street Journal writer Jeff Zaslow. (Hat tip to my mom for this one--as I mentioned, I don't follow the main-stream media much any more.)

Pausch described his lecture as being for his children, and I think he's absolutely right about that. The lecture will be extremely meaningful for his kids as well as for other family and friends. But for outsiders, I'm not sure I understand the appeal. I mean, Randy Pausch is an extremely positive person, and there is no doubt that his attitude played an important part in helping him achieve his goals in life. But this is a lesson that pretty much everyone already understands. The problem is that for someone who feels beaten down by life, telling him or her to be more positive simply isn't going to help. It's just not inspirational to someone who's not already on the program.

So why am I including this video here? Well, one of the accomplishments that Pausch mentions in his lecture is a programming language named Alice. Pausch's research areas included human-computer interfaces (HCI) and virtual reality, and he was also interested in attracting kids, especially girls, to computer science. These combined interests gave birth to Alice, which is a programming platform designed to be easy to learn, and which uses story-telling as its central paradigm, instead of a more abstract treatment of algorithms. This is supposed to work as a hook to draw girls into programming, and it seems to have been very successful at Carnegie-Mellon. My daughter has taken an interest in learning Alice, and I'll be keeping an eye on how things progress. You can find a demo video of Alice on this page. I would recommend skipping the promo video at the top of that page, and scrolling down to the demo at the bottom.

**********************************

So that's it for another year. I hope that you survived 2007, and here's looking forward to a better year in 2008.

String Theory

2007-12-28T23:40:00.000-05:00

Physicists in the lab have produced hard data for string theory! Well, for a theory about strings. Actually, knot theory might be technically a more accurate label. Okay, the truth is that the only reason I'm writing this post at all is so I could make a pun about string theory in the title.

This research evidently made the news when it was published a couple of months ago, but I missed it until Keith Devlin mentioned it in passing last week on NPR. Physicists Doug Smith and Dorian Raymer at the University of California at San Diego have developed a mathematical model for how loose string in a box becomes tangled when the box is jostled. Their paper, Spontaneous knotting of an agitated string, was published this past October in the Proceedings of the National Academy of Sciences (PNAS). (The full article can be accessed without a subscription here.) There have been numerous write-ups about it in science and math publications; a couple of particularly nice articles were in Science News Online and in The Mathematical Tourist column in MAA Online.

The experimental apparatus consisted of clear acrylic boxes of various sizes which were rotated by a computer-controlled motor. Strings of various lengths and stiffnesses were placed in the boxes, and after being spun, the strings were analyzed to see what knots had formed. This was done in large part by taking digital photos of the ending configuration of the strings and using a computer analysis to compute the Jones polynomials of the knots. The PNAS website contains Quicktime movies of the apparatus in action, which are definitely worth a look.

As for any practical advice from the study, the bottom line seems to be that you should pack your loose cords tightly to prevent them from moving around. I don't think that's going to be of much help when I put away the Christmas lights next week. Each year they inevitably come out of the box a tangled mess, in spite of being packed more snuggly than sardines. Perhaps Smith and Raymer can do a follow-up study using Christmas lights instead of string, that are shaken side-to-side instead of being spun. (My Christmas lights, despite being tangled, are usually not knotted at all in a strict mathematical sense, and I don't think that Smith and Raymer examined this type of phenomenon in their study.)

A Search Meme

2007-10-26T00:41:00.000-04:00

David Ng of The World's Fair has been playing with viral marketing and Google bombing, and the experience has given him an idea for a nifty new meme. The way it works is this: Try to come up with 5 different search phrases for which, when entered into Google, your blog is the number one hit. The search phrase can be enclosed in quotes if necessary, but obviously it is even better if it works without the quotes. A commenter at David Ng's post (and by the way, read those comments at your own risk--some of them aren't pretty) suggests scoring the search phrases based on the number of hits. So, the ideal search phrase would have your blog being number one out of something like a million hits returned.

As it turns out, this is much easier to do than it might sound. Looking back at my previous post, Steven Pinker's formula gives an estimate of the number of distinct sentences one can utter, which comes out to be the enormous value of 10²⁰. Given that, it is perhaps not too surprising that we might have some catch phrases that appear more or less uniquely in our writing. (The key to getting the hit counts high at the same time is to have those catch phrases made up of a few words that individually are common.) So without further ado, here are some search phrases that (as of this writing) give this blog as the number one hit:

intractable yet easy

theory dilettante

ultimate collector's ipod

homage to procrastination

decay unto death

Each of those searches returns more than 100,000 hits. I'll add one more search string for good measure, even though it only returned a few hundred hits, because it links to one of my favorite posts (and because at heart I'm a nasty person): Bringsjord parody.

So that's it. Consider yourself tagged if you're so inclined, and have fun playing with search strings!

What is your formula?

2007-10-23T10:58:00.000-04:00

[Note: I goofed up in describing What is your formula? as an Edge annual question. It's not, and I've updated the post below to correct this.]

Edge Foundation does a yearly thing where they pose a question to an array of scientists and other intellectuals, and collect together all their answers. They have also had occasion to pose a question outside of the normal cycle, when after the 9/11 terrorist attacks they asked, What Now? Earlier this month, as part of a collaboration with the Serpentine Gallery in London, Edge posed another question to their collection of thinkers. The question was What is your formula? Your equation? Your algorithm? and browsing through the answers is a great way to waste an afternoon. Because the replies consisted mainly of diagrams and equations (many of which were hand-drawn), they are displayed as scanned images, which adds a personal touch to the results.

Unfortunately, the format of the question also limited the form the answers could take, most of which express some kind of heuristic relationship rather than a true formula. I did like this entry by psychology professor Danny Kahneman, which seems to explain a lot of popular culture.

As might be expected, the entries by mathematicians tended to have actual formulas in them. I can relate to the sentiment expressed by Nassim Taleb in his entry. Over a typeset page of equations on how to price stock derivatives, he hand-wrote, "I spent a large part of my life working on this equation. I am glad it is over."

Many of the entries were just plain silly. I'll single out this one by cognitive scientist Donald Hoffman, because of his abuse of mathematics. He has a nice diagram illustrating composition of functions that would make any algebra teacher proud, but follows it with a fallacious argument that shows he has not mastered basic logic.

Let me mention one more entry before I stop. Popular math writer Keith Devlin uses an analogy to speculate on how math will be taught in the future, and it looks like he would like to see virtual manipulatives taken to a whole new level.

Be sure to also check out the archive of annual answers to questions. Because they are not constrained by the "formula" format imposed on the current question, the answers are better developed and far more insightful.

SocialRank (and MathBloggers) is now live

2007-10-01T11:09:00.000-04:00

A few weeks ago I wrote a post about SocialRank, which seems to be a new blog search engine that identifies which blogs entries are currently generating a lot of interest. These entries are categorized by subject matter and daily 'top 15' lists are created for the top blogs and top blog entries. The result for each category appears under its own domain name, and for math blogs the end product is MathBloggers.

At the time I last wrote about this, the SocialRank sites were all password protected while under development, but an anonymous commenter has pointed out that as of today SocialRank is now live. The first 30 blog categories are now open, and this includes MathBloggers.

So how good of a job did MathBloggers do on its first outing? Well, in addition to several blog entries that I had already read and might expect to see on a 'top 15' list, there were several entries (and blogs) listed that I was not familiar with. So MathBloggers succeeded in broadening my reading. The MathBloggers list also included a nice mix of pure math, math ed, mathematical physics, and theoretical computer science blogs. The blog Good Math, Bad Math, while being a very good blog, seemed to be over-represented on the list, so SocialRank may need to fine-tune their ranking system to keep very popular blogs from overwhelming the little ones. There was also one noticeable bug: blog URLs only contained the actual domain name and not any path info. As a result, Good Math, Bad Math shows up as ScienceBlogs, for example, and Jacques Distler's Musings appears simply as the host server name Golem. (This problem did not affect the URLs for individual blog entries, only for the blog home URLs.)

Overall, it seems like an interesting effort, and I'm going to keep checking MathBloggers on a regular basis to see how it develops.

Update

Hmmm, maybe I spoke a little too soon on this. There is definitely something a little "off" about the Top Blog Posts list. A couple of entries currently listed there are from Skrentablog, which is a tech blog with practically no math content. The other blogs represented on the list seemed to fit the category better, but some of the particular entries seemed like peculiar choices. Vlorbik wonders how his own blog got so highly rated. His particular entry that made the list was about the Excel number formatting error that was a big news item last week; maybe the overall blogosphere interest in the story somehow rubbed off indirectly on his post. In any event, the SocialRank ranking algorithm seems to be doing some weird things. It will be interesting to see how well it tracks blog interest later this week when the next Carnival of Mathematics comes out.

Carnival of Mathematics XVI

2007-09-08T20:00:00.000-04:00

Welcome to the 16th Carnival of Mathematics, the comments and opinions edition. The host of the previous edition of the carnival set up some pretty high expectations, what with providing both interesting background information on the fortnight's posts, and pretty pictures to accompany them. We'll be having none of that here. No, this is the blogosphere we're talking about, and even if the subject matter is the noble quest for mathematical knowledge, our stock and trade are comments and opinions. That high bar actually has me thinking "limbo", and I want to see just how low I can go. So without further ado, here are this edition's entries.

The Good

I have nothing but admiration for the good souls who struggle day after day with teaching our children, so we begin our list with the mathematical educators...

Starting us off is Denise at Let's play math!, who gets us in the mood with quotations on the joy of mathematics. This batch of quotations ends with the line from G.H.Hardy, "Beauty is the first test: there is no permanent place in this world for ugly mathematics." We'll see about that; I'm pretty sure that some ugly mathematics (or is that ugly mathematicians?) lie in wait for us a bit later. But I digress... Denise then gets to the meat of pre-algebra problem solving with posts on the tools and 2nd grade. In the comments to that first post, commenter Bill Howdle describes a (long ago) homework math problem that he was unable to help his daughter with, and after 25 years it still nags at him!

Probably everyone here has heard at some point the story of how, as a young child, Gauss discovered a formula for the sum of the first n positive integers. In Pairing up with Gauss, Math Mom describes how she teaches this to 10- to 12-year-old kids, using different arithmetic progressions to reinforce the pairing technique. While you're visiting, be sure to also check out the link she provides to an American Scientist article which dissects the Gauss myth.

For our next entry, Dave Marain of MathNotations shows how to find the Golden Ratio in the geometry of a certain isosceles triangle, in Searching for Gold in Geometry. Be sure to scroll down through the comments (this will be a recurring theme in this edition) to read Eric Jablow's nifty introduction to algebraic number theory.

If someone poses a problem about a random number between 1 and 10, I think it's safe to say that everyone would assume that the number is going to be uniformly distributed. But what if we're talking about some random mathematical object where there is not an obvious, intuitive choice for the distribution? Jonathan discusses What is a random triangle? over at jd2718. The post is short, but (you guessed it) much discussion is generated in the comments.

To end this category, Jonathan also recommends reading Pissed Off's discussion of whether partial credit should be given on math tests. Now, having spent some time grading student exams, I'm a big fan of multiple choice questions--preferably using one of those machine-readable bubble answer sheets so that one doesn't even have to look at the students' work. Pissed Off, to the contrary, seems to be basing her opinion on what might be best for the student and not the instructor!

The Bad

Next up for your viewing pleasure are the mathematicians, either as the subjects or the authors of blog entries...

John Kemeny of A Mispelt Bog has a nice post about an obscure result relating to patterns of digits in repeating decimals, that has recently been getting renewed attention. In The Secret Theorem of M. E. Midy = Casting In Nines, he tells how Midy's theorem has recently been rediscovered and extended. John is also offering up a short post looking at some statistics on Ph.D. graduation rates and SAT scores by state.

Next, John Armstrong of The Unapologetic Mathematician takes a break from his ongoing series on category theory to respond to a question that reader posed in a comment. In A little aside on linear algebra, John discusses the relationship between inner products, norms, and the choice of a basis in vector spaces.

Hmmm, only two math submitters this edition? Well, that just won't do, and fortunately there has been some other juicy math blogging recently that we can highlight.

Let's start with the low-brow... I've often thought that the way to attract more public interest in mathematics is, like just about everything else that is "marketed" these days, to sex it up a bit. The New York Times helped out in this regard with an article describing an argument by UC Berkeley mathematician David Gale, that the commonly reported statistics on male and female sex partners cannot possibly be correct. This was all over the blogosphere last week, but since no one else has submitted a relevant entry, I'll suggest taking a look at Men and Women Cannot Can Have Different Average Numbers of Sexual Partners by Jake Young at Pure Pedantry. Jake gives a nice recap of the arguments back and forth, and has so many updates based on reader comments that I lost count. The very first commenter gives a nice little example of how, while the means must be equal, the medians definitely might not.

For something a bit more high-brow, I'd recommend Ben Webster's post at the Secret Blogging Seminar on Zeta function relations and linearly equivalent group actions. Ben mentions that the subject of this post comes from some research he did as an undergrad, and it contains the prettiest looking math blogging I've seen recently. But what caught my attention was that the comments were devoted entirely to a discussion of using mathematicians' names as nouns versus adjectives when referring to eponymous mathematical structures.

To shift back to the low-brow for a moment, I noticed that Ben is also making a call to keep the sarong theorem archive growing. I have to admit that this is a nice picture of Ben, and math ed content is welcome, too.

I recently bought Danica McKellar's new book, Math Doesn't Suck, for my daughter. This book has already been the subject of much discussion on the web and I won't try to add anything here. But I've been looking without much luck for a comprehensible explanation of the research paper she co-authored, "Percolation and Gibbs states multiplicity for ferromagnetic Ashkin-Teller models on Z²". So, I was delighted to see that Terry Tao has done just that in his blog entry “Math Doesn’t Suck”, and the Chayes-McKellar-Winn theorem. Tao gives both a high-level and a fairly detailed explanation of what the theorem means, and also provides some recollections about Danica's time as a student.

Finally, we close the mathematicians' portion of our program with Greg Muller at The Everything Seminar railing against the bad math behind a classic gambling scheme that's "guaranteed" to pay off, in Infinite Series vs. Reason. Specifically, if you are, say, flipping a coin to determine whether you win a bet, and you double your wager after each round, then you're bound to come out ahead at some point, right? What could possibly go wrong?

The Ugly

Finally, we come to the computer science portion of our carnival...

...and there is nothing better to make the segue from bad math to ugly computer science than the following little brouhaha. Mathematician Neal Koblitz has an article in the September Notices of the AMS in which he criticizes the cryptographic community, and it evidently has many computer scientists up in arms. None of them felt compelled to submit their entries to the carnival, unfortunately; but being the lover of soap opera that I am, I will do the honors. Luca Trevisan at in theory writes about The Swift-Boating of Modern Cryptography, and it goes without saying that you must read the comments on this one. Lots of ugly trash-talking there, but no doubt Koblitz is a bad, bad, naughty mathematician who deserves it all :)

On a more positive note, Julie Rehmeyer at the Science News blog MathTrek has a couple of entries timed just perfectly for this edition. (And Julie is being promoted from math to computer science as well!)

In Squashing Worms, Julie describes research by Microsoft theoretician Jennifer Chayes into how computer worms spread through the internet, in order to calculate which hosts should get patched first to best slow the spread. There may also be implications of this research for biological organisms, and Chayes plans to work with epidemiologists to gain a better understanding of this.

In Julie's second entry for this edition, Kidney Matchmaking, we learn about new algorithms being designed to match kidney donors with patients needing transplants. An estimated 1,000–2,000 additional patients per year might receive transplants that otherwise wouldn't, once the techniques are implemented nationwide.

In the comments to Julie's second post, Suresh from The Geomblog mentions some additional research being done on the kidney transplant matching problem. In fact, he recently did a post on the subject himself, Saving lives with exact algorithms, which deserves its own mention here in the Carnival. The amazing take-home point from Julie's and Suresh's posts is that results from the study of algorithms are actually saving patients lives--concrete benefits that one would probably not expect from such an abstract topic.

Most of us are familiar with the history of Euclid's fifth (parallel) postulate, and how relaxing this requirement allows us to come up with different geometries in which "lines" differ from our usual intuition--for example, lines might correspond to great circles on a sphere. Now, what if we were to really relax the definition of what constitutes a line, to create something that "behaves like" a line but isn't required to be straight? According to David Eppstein at 0xDE, computational geometers do just this sort of thing, and the result is a pseudoline. However, different researchers have used different definitions for pseudolines, and in his post Was sind und was sollen die Pseudogeraden?, David attempts to clarify some of the resulting confusion about what the proper definition should be.

For our next entry, Aaron Sterling recommends an item from Scott Aaronson's blog Shtetl-Optimized. In Shor, I'll do it, Scott gives a plain-language explanation for Peter Shor's quantum factoring algorithm, which was one of the first algorithmic results exploiting the power of quantum computing. This is an older post, but one of Scott's most popular ones. Now, the comment section is always lively at Shtetl-Optimized, so it goes without saying that you want to check that out. Aaron mentions that you should particularly look for Robin Blume-Kohout's comment explaining the Fourier transform.

Finally, Aw, shucks is a very short post by Jeff Erickson at Ernie's 3D Pancakes. It seems that Jeff has a difference of opinion with Doron Zeilberger over the value and philosophical implications of computer-generated proofs, and these two gentlemen are not shy about, ahem, expressing their opinions. I can't quite decide if they love each other or hate each other's guts, but either way you've got to follow those links and read what they have to say...it seriously just doesn't get any better than this.

This concludes the 16th edition of the Carnival of Mathematics. The 17th edition will be hosted on 9/21 over at MathNotations by Dave Marain, who will undoubtedly handle the proceedings in a more timely fashion than moi.

Let me leave you with one final parting link, some words of encouragement from Professor Mark Sapir at Vanderbilt University. Be sure to turn up your speaker and click on the sound file at the bottom of the page!

Tick, tock...

2007-09-07T00:37:00.000-04:00

I'm feeling a little bit under the weather, so rather than trying to force myself to finish gestating the next Carnival of Mathematics, I'm going to get some rest. But don't worry, the delivery shouldn't be more than a few hours overdue.

Let me take this moment to remind everyone that the Blog Carnival submission form will continue to forward submissions to me until midnight tonight, so please do send in those last-minute entries. Or you can email them to me at kurt (at) learningcomputation (dot) com.

And now, back to bed. Zzzzzzzzz.

Math Carnivalia: Present, Future and Past

2007-08-26T00:21:00.000-04:00

The 15th Carnival of Mathematics is now up at John Kemeny's A Mispelt Bog, and it looks like a good one. I especially like how he has taken the time to provide some background on the posted subjects above and beyond what was in the post entries themselves. It definitely sets a high bar for whatever poor schlemiel is going to be hosting the next one.

Which reminds me, the 16th Carnival of Mathematics will be hosted by yours truly right here in two weeks, on September 7th. The theme for the next fortnight's edition will be:

How has the mathematical mindset shaped your self-actualization in a post-postmodern world?

I don't know about you, but I could write a few thousand words on that without even thinking... Hey, wait, come back! I'm only joking! The Carnival of Mathematics is not a "themed" carnival beyond the mission statement that applies to all editions. The short version, from the Blog Carnival entry for the CoM is as follows:

Everything math-related goes in here: proofs, explanations of basic concepts, puzzles, writings about math education, mathematical anecdotes, refutations of bad math, applications of math, reviews of popular math... Note that sufficiently mathematized portions of other disciplines, especially physics and computer science, are acceptable.

For a slightly longer version, check out Alon Levy's description at the CoM homepage. And for anyone who's stumbled upon this page without having encountered the notion of a blog carnival before, more information can be found at Wikipedia.

So, start sending those submissions my way. You can use the Blog Carnival submission form, or you can email your links to me at kurt (at) learningcomputation (dot) com. Please include "CoM" or "Math Carnival" or the like in your subject line so I don't mistake your message for spam.

Now, I'd like to make a couple of extra requests of you for the upcoming carnival, beyond what may have been done in the past:

First, note that Alon has added a bit of extra instruction for this edition on the submission form:

It's perfectly acceptable to nominate entries from other blogs. The limit of three posts per individual means that a single blogger may not have more than three posts in an edition; it does not mean you may not nominate many more posts from many blogs or bloggers.

Several people in the past have made the observation that there is a lot of interesting math blogging going on that for one reason or another never gets submitted. I think it was this little exchange with Noah Snyder at the Secret Blogging Seminar that finally convinced Alon to codify this into the carnival description. So if you come across an example of someone else's math blogging that you think deserves a wider audience, by all means submit it to the carnival. Let me know whether or not you've already mentioned it to the author, because I'd like to give them advance notice before linking to them.

Second, when you submit a link to me, please include a paragraph or so describing what the linked post is about. I'll certainly read through all of the submitted posts, but I know from reading past carnivals that there are some posts for which there is just no way I could do them justice in my own words. Heck, a couple of them might as well have been Greek to me...

Finally, there has been some discussion around the past few editions of the carnival concerning the mix of "math ed" posts and "math research" (for lack of a better term) posts, for example here, here and here. So as a little experiment, I'll be grouping the submissions by these categories. However, I know from past carnivals that some posts don't fit cleanly into one category or another, so let me know how you think your submissions should be grouped. If I receive enough suitable entries, I'll also create a "comp sci" category. The first few carnivals had several submissions dealing with algorithms and computational geometry, but comp sci has been AWOL lately. I'd really like to see that change.

Now if you'll excuse me, I'm going to return to perusing the 15th edition ...

Link this, SocialRank

2007-08-23T23:31:00.000-04:00

[Update: I thought, since I mentioned Snap Shots in this entry, that I should just turn this feature on in this blog instead of sending you elsewhere. Done. Unfortunately, it turns out not to be a magic bullet, as you'll see if you scroll over the mathbloggers.com link. However, the cached Snap Shots image at The Unapologetic Mathematician that I refer to below still worked the last time I checked.]

I've noticed in the past couple of days, trackbacks appearing on many math blogs for a site named mathbloggers.com. Well neat, I thought, a new math blog. Unfortunately, that site is password protected, so clicking on that link is not going to get you very far. However, the site is evidently not shielded from search engines and the like, and if you roll over the trackback link on a blog that has the Snap Shots feature enabled, you can see that it appears to be just a link aggregator site. (For example, as of this writing you can use this post at The Unapologetic Mathematician and scroll to near the bottom of the comments.) There is one other thing, though, and that is the phrase "powered by SocialRank" on the trackback link.

What is SocialRank? On the Snap Shots image I can make out the blurb, "SocialRank brings you the most popular stories that people are paying attention to right now." If I try Googling "socialrank", I find a great number of these aggregator sites on all variety of topics, and all apparently password protected. Well now, SocialRank isn't going to be bringing me much of anything if I can't get to any sites powered by it... Could this be some bizarre new kind of comment spam? Did someone figure out a new way to hack search engine rankings or something? I tried doing a whois lookup on some of the sites and found that they were registered to someone named Michael Reining at a company called MindValley.

Ah, now we're getting somewhere. MindValley appears to be a "Web 2.0" type of company dealing in e-commerce and web marketing and the like, based in Kuala Lumpur. And, they have a blog. On it, they explain a little bit about SocialRank:

We develop breakthrough applications that have the power to be very disruptive. We are currently working on a project called SocialRank that will instantly be able to surface the top blog posts for 1,000 blogging communities online. To help us pull this off, we have hired one of the smartest mathematicians from the leading technical school in India and assembled an all start team of developers. The algorithm is nothing short of breathtaking because for the first time you will instantly be able to see 1) the top stories coming out of every blog community and 2) see which blogs are the hottest right now to see where the conversations are happening online.

So there you have it. I'm guessing that when they feel their product is ready for public consumption, the password protection will come off those sites.

Oh, and that mathematician, a fellow named Talat, has a blog entry with a math anecdote from his youth. Maybe we can get him to submit something to the Carnival of Mathematics :)

A simple plea to all computer scientists

2007-07-25T23:01:00.000-04:00

I was trying to do a little background reading on a problem in complexity theory (this problem, to be precise), and I was dismayed to find myself thwarted by the unavailability of papers. Not being an academic type, I don't have ready access to a research library. (I can, in theory, get papers from my local public library via inter-library loan, but it takes a couple of weeks. I can also drive to the nearest university that has a CS graduate program, but I'm not going to count that as "ready access".) So I depend upon papers being available online, and a lot of researchers are very obliging in this regard. However, there are still a lot of exceptions.

For example, Richard Karp's 1972 paper "Reducibility among combinatorial problems" [2] does not seem to be online. (By the way, it is of course possible that this or any the subsequent papers I mention are in fact online but I wasn't thorough enough in my searches. Please let me know if this is the case.) This paper is mainly of historical interest at this point, but it was such an important paper that I'm really surprised by its absence.

Moving forward in time 16 years to 1988, we have another highly cited paper, "How easy is local search?" by Johnson, Papadimitriou and Yannakakis [1]. Again, given how many citations this paper has, I'm surprised no one has been moved to scan it and put it online. (Well, actually, someone has, and I'll get to that in a moment.)

Going forward another 4 years to 1992 takes us to Woeginger and Yu's paper "On the equal-subset-sum problem" [4]. Here we're starting to get recent enough that the authors might still have the source versions of the paper in electronic form. Not to pick on anyone (especially since I'm a fan of his P vs. NP page), but why doesn't Woeginger include this paper on his publications page?

Finally, for a recent example, we jump forward 11 years to 2003 for a paper by Kellerer, Mansini, Pferschy and Speranza [3]. Given the recent date of this paper I would have thought that it would be a matter of routine for the authors to upload a preprint of it to their web pages.

Out of a total of 9 papers I tried to locate online, I was unable to access 4 of them. The good news is that more than half of the papers were in fact freely available on the web. The bad news is the following: with the exception of the Karp paper, the others are in fact online and they're for sale at Elsevier's ScienceDirect. Now, I have nothing against capitalism, and I actually think it's great that these articles are available for sale on an individual basis. But Elsevier charges $30 per article, and that is a bit steep for something that I might skim through once and then discard. Unfortunately, even after reading the abstracts it can be hard to tell which papers will turn out to be worth the expense and which ones won't.

So, my simple request is that researchers take the time to upload papers to their web pages. I'm not sure what kinds of legal issues might be involved, but some of the articles that I did find online were also published in Elsevier journals, so I don't think that copyright concerns are the main impediment. It may be more just a matter of developing the right mindset. So get those papers out there, and students and hobbyists everywhere will be grateful. (If copyright issues are the main problem, I'd like to hear more about that, too.)

1. D.S. Johnson, C.H. Papadimitriou and M. Yannakakis, How easy is local search?, J. Comput. System Sci. 37 (1988) 79-100.

2. R.M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations, R.E. Miller and J.W. Thatcher (eds.), Plenum Press, New York (1972), 85-104.

3. H. Kellerer, R. Mansini, U. Pferschy and M.G. Speranza, An efficient fully polynomial approximation scheme for the Subset-Sum Problem, J. Comput. System Sci. 66 (2003) 349-370.

4. G.J. Woeginger and Z.L. Yu, On the equal-subset-sum problem, Inform. Proc. Letters 42 (1992) 299-302.

Updates

Evidently Shiva Kintali would also like to see more papers uploaded to their authors' web pages, and he would like to see it done really, really quickly!

Commenter BarrosH pointed out to me that Karp's paper can in fact be found online, at Papadimitriou's Reading the Classics course.

Carnival of Mathematics

2007-07-02T00:07:00.000-04:00

I was just thinking to myself that it's been a long time since I've seen a new Carnival of Mathematics, and sure enough when I checked the calendar I found that the 11th edition is now up at Grey Matters. Hmmm, none of the math/cs blogs I read regularly posted any announcements about this; how am I supposed to remember these dates on my own? Maybe everyone is on vacation this week.

Perhaps it's just the summer doldrums, but this edition of the carnival seems a bit thin. There was a little bit of discussion back in the 10th edition about whether it might make sense to split into two separate carnivals: one for math ed and one for college and research level math. However, the current edition is almost all math ed, so perhaps the carnival will just evolve in that direction on its own. (There were also a couple of entries that really left me wondering what the authors were smoking, but that's another matter.)

There was one entry that struck me as being worth mentioning, though. John Armstrong at The Unapologetic Mathematician writes about categorification: the process of recasting a mathematical abstraction into the language of category theory, as a means of solidifying one's understanding of the topic. He gives some simple examples expressing addition and multiplication in terms of set operations, and reinterpreting the results in terms of category theory. As someone who doesn't know anything about category theory, I find this both intriguing and mystifying. He concludes his post with the adage, "If you want to understand something, try to categorify it!" I think that I first need to understand categories, and Armstrong has a series of posts on the basics of category theory that might help me in that regard.

While we're on the topic (sort of), I wonder if it would be worthwhile to try to split off the TCS-related posts from the Carnival of Mathematics into their own carnival? That might sound a little strange seeing as how there were exactly zero computer science posts in this edition, but I'm thinking that having our own carnival would encourage more submissions. Or is the TCS blogosphere so small (and we already read each other's blogs anyway) that having a carnival would be superfluous?

Open Problems

2007-07-01T03:06:00.000-04:00

One topic that comes up on a regular basis is the question of how to come up with open problems that might be suitable for tackling. Of course everyone knows about P vs. NP, and like Fermat's Last Theorem in number theory, this problem can serve as motivation for studying the foundations of computational complexity. However, as was the case with FLT, it is not a problem that a student would hope to make any real progress against. One difficulty with identifying problems that are hard enough to be interesting, but not so hard as to be hopeless, is that the people who are familiar with such problems might be reluctant to advertise them--they understandably might want to tackle the problems themselves first, or in the case of faculty, to give the problems to their students to work on.

So it was especially nice to learn (via David Eppstein) about the Open Problem Garden created by Matt DeVos and Robert Šámal. The Garden is a wiki-style repository of open problems from math and computer science, so anyone can register and contribute their own entries. Problems are categorized by subject (graph theory, combinatorics, etc.), and rated according to importance. Problems are not rated for difficulty per se, but there is a yes/no indicator for whether the problem might be suitable for an undergraduate student. It looks like the database was preloaded with the problems from Bojan Mohar's Problem of the month page, so it is currently heavily weighted toward graph theory. I'm really hoping that as time goes on, people in the TCS community will use this repository to store their own open problems.

There is as of this writing a single entry in the TCS subject category: Subset-sums equality. This search variant on the NP-complete decision problem SUBSET-SUM is one which I haven't seen before. Unfortunately, the entry does not currently contain much background or any references, so perhaps if someone reading this is in the know, they could flesh out the entry a bit. (A quick search turns up this article on finding approximate solutions to the problem.)

While I'm searching around, perhaps I should look for other sources of open problems... Unfortunately, a lot of the hits returned by Google are pages describing P vs. NP and not much more (including, surprisingly, this Wikipedia article). Open problems have been discussed on at least a couple of occasions on the Computational Complexity blog, here and here. In fact, in the comments to that second post, Matt DeVos puts in a plug for the Open Problem Garden (which was in beta at the time). That was back in March, and evidently no one in the meantime has been sufficiently motivated to add any additional TCS open problems. Pity, that.

Finally, for something of a slightly different flavor, here is David Johnson's Challenges for Theoretical Computer Science. Instead of a list of conjectures to be proven, this is a list of broad challenges for computer science, perhaps reminiscent of David Hilbert's list of problems for mathematics.

Uncertainty

2007-05-18T01:35:00.000-04:00

The title of this post is not a reference to quantum uncertainty, but rather to the notion of Platonic idealism suggested by today's xkcd comic titled Certainty (click on the image to see a bigger version so you can actually read the words):

It is, unfortunately, untrue.

Now, in practical terms, math as a school subject has a lot less uncertainty associated with it than, say, literary criticism. If you're asked on an exam to solve a given problem, chances are there is a specific answer which the instructor has in mind. However, even at this basic level, issues of interpretation come into play, as illustrated by what happened to Heath Raftery on a probability test. (It's particularly funny given how many people commented on that post to try to explain to Heath why he was wrong, when in fact he was absolutely correct.)

Even on a foundational level, mathematical truth is not as absolute as it is often assumed to be. Ever since Kurt Gödel's famous incompleteness theorems, it's been known that any sufficiently powerful axiomatic system cannot be both consistent and complete. But it is not necessary to invoke such a deep result to understand the potential limitations of mathematics. The rules of inference used to generate theorems are themselves crafted from empirical observation of the world. We learn from experience that if, say, A implies B and B implies C, then A implies C. These rules seem so natural that it can be hard to imagine that they are not of necessity true. The truth is that the rules of inference must be assumed to be true, along with any other axioms we are using. And not all mathematicians agree on which rules of inference should be included; for example the field of constructive mathematics rejects certain types of inferences used in classical mathematics.

The nicest explication of this idea that I've come across is an essay written in 1895 by Lewis Carroll, titled What the Tortoise said to Achilles. This dialog between the fictional Tortoise and Achilles was also included in Douglas Hofstadter's book Godel, Escher, Bach: An Eternal Golden Braid and served as the jumping-off point for Hofstadter's own set of dialogs. (If you haven't seen it before, I strongly urge you to read Carroll's essay; it's short, it's funny, and as they say, it contains no mathematics!)

Why I Blog

2007-04-25T12:30:00.000-04:00

There's a new meme making its way through the blogosphere that I think I will indulge myself with. Although I have not been tagged (as usual) for the "Why do I blog" meme, I have come across it in a few different spots such as Aetiology and A Blog Around The Clock. This meme seems to me to be spreading more slowly than other recent memes, perhaps because the reason why people blog is often self-evident from their blogs themselves. Having said that, I can see at the Blog Meme Tracker that it has been spreading. (One problem with that page is that it doesn't seem to have any facility for adding 'dotted line' links for people like me who weren't officially tagged.)

I actually had a rather specific purpose in mind when I started this blog, so maybe this post will help focus my attention back on that goal. It is said that the best way to test, or reinforce, your understanding of a subject is to explain it to someone else. My hope was that by explaining computability and complexity theory in this blog as I worked through the material, I would force myself to understand the material at a deeper level.

It would also serve a secondary interest I have in pedagogy. One thing I've found while trying to self-instruct from reading textbooks is that often the most frustrating stumbling blocks are the most superficial. I probably spend more time trying to decipher unfamiliar notation than I do with trying to grasp concepts. In fact, although there certainly are some very deep and difficult concepts in TCS, a lot of the material would be quite intuitive if only it was taught in a more concrete fashion. A big part of the problem is that textbooks are not generally meant to be used in isolation; something that might take an hour to parse while reading a text can often be explained away in a minute or two by a knowledgeable teacher in the classroom. So another goal I have with this blog is to explain the things I'm learning in such a concrete fashion that, say, a motivated high-school student could understand it.

As an aside, I've noticed some other bloggers touching on pedagogical issues recently... Bill Gasarch talks about teaching binary search to 8-year-olds. More ambitiously, Andy Drucker ponders teaching topology to kids. Much more ambitiously, Andy has also been working on simplifying the exposition of the PCP Theorem. This latter item is definitely not for kids. I don't know the PCP Theorem, and by that I mean that I do not even understand the statement of the theorem. (I know there are relatively simple "high level" explanations of the theorem, but I am not convinced that they really get to the meat of the theorem.) This might actually be a good end goal for this blog: If I can learn the PCP theorem, and explain the proof here in terms that a high-school student could follow, then I'll know it's time to quit! (I am assuming this will come after I prove that P ≠ NP.)

I've drifted away from my goals with this blog. The main reason for this, I think, is that my usual mode of discourse is to respond to other people instead of initiating the conversation myself. As a result, I've been spending my on-line time mostly by reading and leaving comments on other blogs. (I don't often provoke responses from others, so I sometimes wonder if anyone ever reads those comments... I have to add a thanks here to Joshua at The Adventures of Tobasco da Gama for the positive acknowledgment!) But blogging is all about initiating the conversation, so this is something I am going to work on. I'm also going to try to stay a little more focused on the goal, instead of getting distracted by all those other shiny objects on the Web.

Now for our little exercise in distributed peer pressure. I'm not sure if I know 5 different people who read this blog, but let me at least tag a couple of people. Let's see... I'm too late for Tyler, but I can still tag Foxy. And Antztein, you're being worse than me with your posting--consider this an excuse to make another entry. I know she won't see this, but since I'm sure she's been thinking about it, I'll tag Kathy Sierra. Okay, Kathy, now you have to post again since you've been tagged. And to anyone else out in the aether reading this, consider yourself tagged too.

Boys will be boys will be sociopaths

2007-03-29T10:14:00.000-04:00

A recurring theme on computer science blogs is what might be done to encourage more women to enter (or remain in) the field, and to promote more diversity in general. For example, a recent post at On being a scientist and a woman talks about why women leave the academic career path (this is for science generally, but I'm sure it applies equally well to comp sci). As difficult as the academic world may be, however, it appears that things can be much worse for women working on the 'nuts and bolts' side of the aisle. Blogger and author Kathy Sierra writes about how email and blog sexual harassment and death threats have caused her to cancel a public appearance and step back from blogging. (Warning, that linked post contains material that is not appropriate for kids.) There are by now hundreds of posts by others providing commentary on this; some background information on the people involved is given by Doc Searls.

One weird thing about the whole situation is that the harassment grew out of a couple of websites set up by peers of Kathy Sierra, specifically for the purpose of being critical of, and generally nasty about, other technology authors. (The actual death threats appear to be from an interloper who hacked the identity of one of the participants.) Death threats and misogyny aside, why does the internet seem to spawn such general nastiness? Okay, that's a naive question. More to the point, does the anything-goes atmosphere encourage misogyny and racism and other bad stuff, or is that something totally apart from general snarkiness found on the internet? For example, on a site like ScienceBlogs, more than a bit of gratuitous obscenity and sarcasm can be found mixed in with the science and analysis, but it is a very progressive site where any hint of sexism or racism is liable to get smacked down hard.

Still, sexism does seem to be a particular problem within the tech community. Robert Scoble writes in response to Kathy Sierra's problems,

It’s this culture of attacking women that has especially got to stop. I really don’t care if you attack me. I take those attacks in stride. But, whenever I post a video of a female technologist there invariably are snide remarks about body parts and other things that simply wouldn’t happen if the interviewee were a man.

It makes me realize just how ascerbic this industry and culture are toward women. This just makes me ill.

Update

Well, the response on the blogosphere to Kathy Sierra's post has been, as you might imagine, enormous. I get the impression that Sierra would, with the benefit of hindsight, have responded to the threats a bit differently. However, some good might come out of all the brouhaha. Publisher Tim O'Reilly is calling for a blogger code of conduct that would help maintain a certain level of civility in web discussions. I think that could be a tough sell, but just raising people's consciousness about the problem may help.

It looks like Kathy Sierra and one of the other involved bloggers, Chris Locke, will be on CNN Monday morning. Should be ~~an interesting segment~~ a couple of minutes of soundbites with little content.

The End of Blogging?

2007-03-26T01:05:00.000-04:00

Late last year the IT consulting firm Gartner issued its 'top ten predictions' for 2007, and among them was the claim that blogging would level off as the number of people quitting their blogs equaled the number of new bloggers.

I didn't think much of it at the time, but now 3 different 'mathy' blogs have abruptly ceased operation this month. Foremost was Lance Fortnow's Computational Complexity, the premier blog for theoretical computer science. The other two blogs, more political but with math as a sideline, were Alon Levy's Abstract Nonsense, and Tyler DiPietro's Growth Rate O(n lg n).

If you click on that link for Tyler's blog, you'll find that it does not even exist anymore; evidently he felt the temptation to blog would be too strong if he didn't just chuck the whole thing. While I can certainly sympathize with that sentiment, I feel obligated to mention for the benefit of anyone else who might be considering this, Bora Zivkovic's plea for the Proper Procedure For Shutting Down A Blog.

Although I wouldn't attempt to compare these blogs, one thing that they shared was a high output. For me, writing is a slow, tedious, and sometimes downright painful process, so I am always amazed by people who are able to write with such apparent ease. Unfortunately, it would also appear that they have simply burned themselves out in the process (at least as far as blogging is concerned). Of course, there is no need to worry about that happening here. At the rate of an entry per month, it would be almost superfluous for me to ever quit.

Finally, by way of balancing the karmic blogging equation, last month the mathematician Terence Tao started his own blog, What's new. Tao is a recipient of both a Fields medal and a MacArthur fellowship, and his blog should be of great interest to mathematically-minded readers.

Frances Allen receives 2006 Turing Award

2007-02-21T14:11:00.000-05:00

The 2006 Turing Award has been given to Frances Allen. This is the first time that this honor has been awarded to a woman. See the ACM press release for details. I have to admit that I am not familiar with Frances Allen. Her background is in compiler design and program optimization, amongst other things. More information about Allen can be found on the IBM Women in technology site. (Hat tip to Suresh.)

I hadn't thought about it before, but I'm surprised that this is only the first time a woman has received the Turing Award. I would have expected that Grace Hopper, for one, would have been a recipient, but perhaps she passed away before the opportunity came up. Past Turing Award winners can be found here.

The Turing Award will be presented to Allen at a ceremony in June. I'm sure that her Turing lecture will be much more edifying than the previous recipient's.

Update

I just noticed that this entry was included in a blog round-up over on ComputerWorld, so if you found your way here from there, welcome. If you haven't already seen it, Mark Chu-Carroll has a personal recollection about Frances Allen over on Good Math, Bad Math. Another item worth checking out is an interview with Allen from 1996 by Ann Devlin. They talk about how the Web will streamline commerce and empower users, and it's kind of weird listening to this from a 10-year-old perspective.

Up-up-date

Also check out Lance Fortnow's blog entry. The post itself is minimal; but read the comments, and you can either weep or get angry as seems fit.

Learning Computation

Counter-intuitive science

The Original of Laura

Circular Puzzle

A Short Note on the Halting Problem

The Physiology of Religious Experience

New Year's To-Do List

One

Two

Three

Four

Five

What have you changed your mind about?

Some inspirational videos to close out the year

String Theory

A Search Meme

What is your formula?

SocialRank (and MathBloggers) is now live

Update

Carnival of Mathematics XVI

The Good

The Bad

The Ugly

Tick, tock...

Math Carnivalia: Present, Future and Past

Link this, SocialRank

A simple plea to all computer scientists

Updates

More Open Problems

Update:

Carnival of Mathematics

Open Problems

Uncertainty

Why I Blog

Boys will be boys will be sociopaths

Update

The End of Blogging?

Frances Allen receives 2006 Turing Award

Update

Up-up-date