On the day Apollo 11 left for the moon, Wernher von Braun said “You give me 10 billion dollars and a 10 years and I’ll have a man on Mars.” Perhaps he could have solved the rocketry problem in time and under budget, but the biggest obstacles to visiting Mars are not rocket science. The biggest obstacle may be psychology. How do you keep a crew from sane all the way to Mars and back? Or politics: How do you stir up sufficient public interest in the project without a cold war? Or biology: How would astronauts handle years of exposure to cosmic radiation?
Next week I’m starting @CompSciFact. This Twitter account will post one fact from computer science per day, Monday through Friday. I’ll also have a few unscheduled posts from time to time. (I announced @StatFact earlier today. There are no more announcements coming! I don’t plan to start any more Twitter accounts any time soon.)
The account may change over time in response to your feedback, but what I have in mind for now is keeping the scheduled facts theoretical: analysis of algorithms, grammars, computability, etc. The unscheduled posts may be less theoretical but at least somewhat related to computer science.
I’m using a “big-Oh” symbol as the image for CompSciFact since a big part of computer science is determining the asymptotic order of the runtime of an algorithm.
I have several other daily tip accounts. @RegexTip and @TeXtip are the ones most closely related to computer science. I also have several mathematical accounts: @AlgebraFact, @AnalysisFact, @ProbFact, @TopologyFact, and also starting next week, @StatFact. For more details about these accounts, please see the FAQ.
I’m starting a new daily tip account on Twitter. @StatFact will post one statement from statistics per day, drawing from Bayesian and frequentist statistics. Like my other daily tip accounts, StatFact will post Monday through Friday on a regular schedule with a few unscheduled tweets sprinkled in occasionally.
I’m using a product sign as the symbol for StatFact.
I thought the product sign might suggest a likelihood function. The most obvious symbol for a statistics account would be a bell curve, but that’s been overused.
If you’re interested in StatFact, here are some things you could do.
- Follow StatFact on Twitter.
- Tell friends about StatFact.
- Suggest topics, or even better, specific tweets.
- Propose a better icon.
- Let me know if I say anything ambiguous or wrong.
To find out about my other daily tip accounts, please see the FAQ post.
Imagine a bank of three elevators along a wall. The elevators are in a straight line but they are not evenly spaced. Where do you stand in order to minimize the expected distance you’ll need to walk to catch the first elevator that arrives?
This scenario comes from the opening paragraph of a paper by James Handley et al.
If asked where they would stand and wait for the next of three elevators, unequally spaced along a wall, many students would choose to stand at the mean position. They think that by doing so they are minimizing the average distance to the elevator. They do not recognize that standing at the mean minimizes the average squared distance and that the minimal average distance to the elevator is in fact achieved by standing at the median.
Suppose you start out standing in front of the second elevator. If you move one foot to the left, you decrease the distance to first elevator by a foot, but you increase the distance to the other two elevators by one foot as each, so your average distance increases. Similarly, if you move one foot to the right, you decrease your distance to the third elevator but increase your distance to the other two so that you increase the average distance. Since you can’t move without increasing the average distance, you must have started at the best spot.
So standing in front of the second elevator minimizes the expected distance to the next elevator, assuming all three elevators are equally likely to arrive next.
What if you want to minimize the worst case instead of the average case? Stand half way between the first and third elevators. As before, you can see that if you were to move from that position, you’d increase your distance to at least one elevator and thus increase the maximum distance.
This problem illustrates three optimization theorems:
- The sample mean minimizes the total squared distance to a set of points.
- The sample median minimizes the mean absolute distance to a set of points.
- The mid-range minimizes the maximum distance to a set of points.
These theorems have numerous applications. For example, they are foundational in the study of robust estimators.
The mission that first landed men on the moon could easily have been aborted. From Rocket Men:
Though Apollo 11 is commonly believed to have been a perfect mission, so many things in fact went wrong Kennedy Space Center directory Jay Honeycutt later admitted, “I tell you, it would have been damn easy to abort that mission. Damn easy.”
David Jacobs has written a long blog post Ruby is beautiful (but I’m moving to Python). [Update: link no longer available.] Here’s my summary.
Ruby is much better than Java, but the Ruby community is too focused on web development and the language has no scientific library. Python has a lot of the same advantages as Ruby, is used for more than web programming, and has SciPy.
Update: There is now a fledgling SciRuby project.
When I was in college, a friend of mine gave me a math book that I found hard to get through. When I complained about it, he told me “You’re going to finish a PhD someday. When you do, do you think there’s going to be fairy dust on the diploma that’s going to enable you to do anything you can’t do now?”
That conversation stuck with me. I realized that I just needed to work hard rather than wait for my intelligence to mysteriously rise at graduation.
Doug Hoyte’s book Let Over Lambda is refreshingly opinionated. I don’t share the author’s opinions, but I appreciate his conviction. Hoyte is a zealous advocate for Lisp, and yet he admires Perl as a sort of anti-Lisp. He even calls Perl “beautiful” as far as non-Lisp languages go.
Hoyte argues that Lisp is the greatest programming language because its minimal (i.e. practically non-existent) syntax makes Lisp macro programming powerful. But if you’re going to have language syntax that prevents this style of programming, you might as well go for broke and have lots and lots of syntax.
If we have to have [non-Lisp] syntax — eliminating the possibility of macros — we may as well extend it as far as possible. Let’s throw in all the possible conveniences and power-user tricks we can think of. If Lisp is the result of taking syntax away, Perl is the result of taking syntax all the way.
I understand that Lisp’s lack of syntax opens interesting possibilities. I also understand the advantages of a rich syntax — provided you (and everyone you work with) have mastered the language and use it frequently enough to keep it loaded in your head. However, I prefer a moderate amount of syntax, somewhere between Lisp and Perl, though I admit this may simply be because that’s what I am accustomed to.
Related post: Periodic table of Perl operators
Seth Roberts and Daniel Lemire have both written blog posts on prestige. Roberts argues that high-prestige work almost necessarily has low practical value. Lemire takes this idea further and explains why high-prestige work can be unsatisfying.
From George Pólya:
There are two kinds of generalizations. One is cheap and the other is valuable. It is easy to generalize by diluting a little idea with a big terminology. It is much more difficult to prepare a refined and condensed extract from several good ingredients.
Related post: Jenga mathematics
There are incentives to use the latest technology, just because it’s the latest, even if it’s no better than its predecessor. Being up-to-date makes it easier to
- Find a job
- Work on new projects
- Demonstrate enthusiasm for your profession.
In addition, there are advantages to staying with the mainstream. If most people think something new is better but you disagree, you might do well to acquiesce. When you’re in the mainstream, it’s easier to find parts, documentation, people to answer questions, etc.
That said, here are some bad reasons to adopt the latest thing:
- Believing marketing hype
- Not considering your particular circumstances
- Under-estimating learning time
- Fearing a technology will die
Not every new release of every product is an improvement. If a new product truly is an improvement for most people, that doesn’t mean it’s necessarily better for your particular needs. And if your are sure the new thing will make you more productive, you have to also ask whether you will use it long enough to repay the time you invest learning it.
Many programmers live in inordinate fear that a technology will die. But technologies seldom disappear. They may become less fashionable, less visible, less common, or less lucrative, but hardly anything ever goes away. Programmers may suffer more pain from technology that won’t die than from technology that does.
Technologies don’t drop out of use nearly as quickly as they drop out of fashion or out of sight.
Update: As an example, this podcast claims that 72% of financial transactions are still processed in COBOL.
Daniel Lemire has an insightful article on good uses and bad uses of XML:
Richard Weaver argues in Visions of Order that our privacy and dignity depend on our being rooted in space. He predicted that as people become less attached to a geographical place, privacy and dignity erode.
There is something protective about “place”; it means isolation, privacy, and finally identity. … we must again become sensitive enough to realize that “place” means privacy and dignity …
When Weaver wrote those words in 1964, he was concerned about physical mobility. Imagine what he would have thought of online life.
I find it interesting that Weaver links privacy and dignity. There is a great deal of talk about loss of privacy online, but not much about loss of dignity. The loss of dignity is just as real, and more under our control. We may lose privacy through a third party mishandling data, but our loss of dignity we often bring on ourselves.
Related post: Emily Dickinson versus Paris Hilton
Here are a couple details of UNIX history I ran across this week.
Why AT&T first licensed UNIX to universities:
At this time , AT&T held a government-sanctioned monopoly on the US telephone system. The terms of AT&T’s agreement with the US government prevented it from selling software, which meant that it could not sell UNIX as a product. Instead … AT&T licensed UNIX for use in universities for a nominal distribution fee.
And why later they turned it into a commercial product:
… US antitrust legislation forced the breakup of AT&T (… the break-up became effective in 1982) with the consequence that, since it no longer held a monopoly on the telephone system, the company was permitted to market UNIX.
Source: The Linux Programming Interface
For daily tips on using Unix, follow @UnixToolTip on Twitter.
Donald Knuth recommends using the symbol ⊥ between two numbers to indicate that they are relatively prime. For example:
The symbol is denoted
perp in TeX because it is used in geometry to denote perpendicular lines. It corresponds to Unicode character U+27C2.
I mentioned this on TeXtip yesterday and someone asked for the reason for the symbol. I’ll quote Knuth’s original announcement and explain why I believe he chose that symbol.
When gcd(m, n) = 1, the integers m and n have no prime factors in common and we way that they’re relatively prime.
This concept is so important in practice, we ought to have a special notation for it; but alas, number theorists haven’t agreed on a very good one yet. Therefore we cry: Hear us, O Mathematicians of the World! Let us not wait any longer! We can make many formulas clearer by adopting a new notation now! Let us agree to write ‘m ⊥ n ’, and to say “m is prime to n,” if m and n are relatively prime.
This comes from Concrete Mathematics. In the second edition, the text is on page 115. The book explains why some notation is needed, but it does not explain why this particular symbol.
[Correction: The book does explain the motivation for the symbol. The justification is in a marginal note and I simply overlooked it. The note says “Like perpendicular lines don’t have a common direction, perpendicular numbers don’t have common factors.”]
Here’s what I believe is the reason for the symbol.
Suppose m and n are two positive integers with no factors in common. Now pick numbers at random between 1 and mn. The probability of being divisible by m and n is 1/mn, the product of the probabilities of being divisible by m and n. This says that being divisible by m and being divisible by n are independent events. Also, independent events are analogous to perpendicular lines. The analogy is made precise in this post where I show the connection between correlation and the law of cosines.
So in summary, the ideas of being relatively prime, independent, and perpendicular are all related, and so it makes sense to use a common symbol to denote each.