New Scientist has an interesting article, Where do science supermachines go when they die? that talks about what happens to the pieces of particle accelerators after they are decommissioned (or in the case of the Superconducting Supercollider, never turned on).
Afterlife of Particle Accelerators
November 13th, 2008Elsevier’s Chaos, Solitons, and Fractals
November 11th, 2008In the comments at n-category cafe, Zoran Skoda presents evidence that the journal Chaos, Solitons, and Fractals, a peer-reviewed journal published by Elsevier, is publishing pseudo-science. John Baez collects more evidence here. The journal is included in some of Elsevier’s journal bundles, so if you are at a school with a big library, you probably have access to the journal in electronic form, and can check it out yourself.
What is a Statistic?
November 1st, 2008From the request thread, I was hoping for a nice easy softball, maybe from an undergraduate or mathematical amateur. Apparently, though, I have finally scared off anyone other than procrastinating professional mathematicians, who want me to actually write the posts I promised.
In the comments here I promised a post explaining why most statistics satisfy the Central Limit Theorem. I thought I’d start slowly with an explanation of what a statistic is.
A statistic is just something you compute from the data. This definition is so uninteresting that statistics books are a little apologetic about how contentless the definition sounds. (This usage of the term “statistic” was coined by Fisher. There is a cutting quote by Pearson on the terminology that is impossible to Google for, since all I remember is that it’s about the word statistic, and it involves Fisher and Pearson, who are probably the two most famous statisticians.)
Probability distributions are mathematical abstractions, while statistics are numbers we compute from actual data. If we believe that we can model that data as if it is generated by a random variable, then we have to relate the statistic to some property of the probability distribution. Usually, we are interested in some property of the underlying distribution, and using statistics to estimate it. For example, we may be interested in the mean of the underlying random variable, which we can approximate by using the mean of data.
Approximating the mean of the random variable in this way is a special case of a general technique to compute a property of a random variable. A random sample drawn from a probability distribution can be thought of as a (discrete) probability distribution in its own right. The property for the sample distribution can be used as an estimate of the property for the true distribution — this is known as the plug-in estimate for the property. An analog of the law of large numbers shows that this estimate converges to the true value.
Next time: the analogue of the central limit theorem.
Quantum Hyperion
October 24th, 2008Sean at Cosmic Variance has an interesting article on decoherence called Quantum Hyperion. It describes a paper by Zurek and Paz that calculates that if Saturn’s moon Hyperion were an isolated system, then within twenty years it would evolve into a non-localized quantum state. It is only the interaction with the outside world that keeps Hyperion looking like a moon.
Lévy processes revisited
October 22nd, 2008I’ve been thinking about Lévy processes, a topic that I mentioned once before. A Lévy process is a generalization of both Brownian motion and a Poisson process. Brownian motion and Poisson processes are both continuous-time stochastic processes but have very different behavior. A Brownian motion follows a very jagged path that is almost always continuous. A Poisson process stays at one place for a long time, and then suddenly jumps to a new place. What they have in common is that changes over two disjoint time intervals are independent of each other, and if the two time intervals are the same the changes have the exact same distribution.
Lévy processes include generalizations such as various combinations of Brownian motions and a Poisson process, but they also include more exotic possibilities. The sample path of a combination of a Brownian motion and a Poisson process will almost always have only finite number of discontinuities. In general, Lévy processes can generate sample paths with infinitely many jump discontinuities in almost every interval. Over a finite time horizon, this can give rise to fait-tailed distributions such as the Cauchy distribution.
Requests Thread
October 11th, 2008In lieu of actually finishing any of the half-finished posts, I thought I’d see if anyone has any requests for posts on particular topics. Anything on your mind?
Sword of Damocles, Banking Edition
October 9th, 2008Sorry for the light posting. I have fifty million half-finished posts that I haven’t been able to find the energy to finish, mainly because I spend all my time hitting “refresh” on financial news sites. I normally use mathematics to take my mind off the news, but this time it isn’t working. The last time I was this distracted from math was in the immediate aftermath of September 11th, and before that my own birth.
Dyson’s Lectures on Quantum Field Theory
September 29th, 2008For the historically-minded physicist: Dyson’s 1951 lectures on Advanced Quantum Mechanics (quantum field theory) are available on arXiv. I looked them over, and they seem eminently understandable to someone who’s had a course on quantum mechanics.
Unreasonable Effectiveness of Mathematics
September 23rd, 2008Peter Woit observes that mathematicians and physicists had a prominent role to play in the current financial crises that threaten to take down the world’s banks. Stochastic calculus has become the main tool of evaluating financial derivatives, which are financial instruments whose payoffs are (usually nonlinear) functions of underlying assets. Overly-optimistic assumptions in pricing derivatives have led to large losses throughout the financial sector. The worst case scenario is the kind of widespread economic dislocation not seen since the Great Depression.
Hey, at least it means we have something to be more embarrassed about than Theodore Kaczynski.
Standard Borel Spaces
September 19th, 2008I just spotted this article on arxiv: Some Notes on Standard Borel and Related Spaces. A standard Borel space is a set with a σ-algebra which can be realized as the set of Borel sets of a complete metric space. The paper is an attempt to describe the theory of standard Borel spaces with the minimum of reliance on metric or topological ideas.