Digital computing and catastrophic failures

I just read a wonderful article on Discover magazine. The article centers around Kwabena Boahen (and other members of the school of Carver Mead) in creating electronic circuitry modeled more after the human brain. The main claim is that these types of neurocircuits have the potential in significantly lowering the power consumption for computing. If the claim were correct, though, it will imply there are certain nontrivial relationship between the voltage applied to a transistor and the noise experienced.

The idea, I think, if I understood correctly just from the lay explanation, is a trade-off between error rates versus power. Let us consider the completely simplified and idealized model given by the following. A signal is sent in at voltage . The line introduces thermal noise in the form of a Gaussian distribution. So the signal that comes out at the other end has a distribution , where the Gaussian family is defined as

Definition 1 (Noisy signal)

(Note: our definition is not the standard definition, in particular our Gaussian is centered at ! This definition makes calculations later simpler, as we shall see.)

Aharonov-Bohm effect’s analogue in water waves

Rather indirectly through Claude (a not-so-short story there) I learned of a paper by Michael Berry and collaborators titled “Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue” (a copy can be found on Berry’s website at number 96). It is among the most pleasurable papers I have read in physics to date.

To understand why this Chandrasekhar-esque paper tickles me so, I need to explain a little bit of the physics.

The Aharonov-Bohm effect
The Aharonov-Bohm effect is one of the classic illustrations of the difference between classical and quantum mechanics. More importantly though, it also demonstrated that the use of the vector magnetic potential is not a mere computational convenience, but that the potential also manifests itself in physical effects.

“No Hair” theorems

Hum, now I am a bit confused. I doubt any “prominent mathematician” will read this blog and comment, so I guess I’ll ask them in person next time I go to a conference.

The question is about the term “no hair theorem”.

Healthy skepticism

Besicovitch once said that “A mathematician’s reputation rests on the number of bad proofs he has given.” Of course, originating from someone educated in the Russian school, the word “bad” in the quote should probably be taken to mean “inelegant”. However, lack of beauty is certainly not the only possible deficiency in the quality of a published result. Cases abound where a proof is bad not in the aesthetics, but in something more fundamental.

Straightedge and compass constructions

Classical Euclidean geometry is based on the familiar five postulates. The first two are equivalent to the assumption of the existence of a straightedge; the third gives the existence of a compass. The fourth essentially states that space is locally Euclidean, while the fifth, the infamous parallel postulate, assumes that space is globally Euclidean.

A quick digression: by now, most people are aware of the concept of non-Euclidean geometry as described by Bolyai and Lobachevsky, and independently by Gauss. These types of geometries make also the first four postulates. By simply abandoning the fifth postulate, we can arrive a geometries in which every pair of “lines” intersect (in other words, geometries in which “parallel lines” do not exist), or in which parallel lines are non-unique. (As a side note, one of the primary preoccupations of mathematicians is the existence-and-uniqueness of objects. With our obsession of classifying things, we ask often the question, “Does an object exist with properties X and Y?” and follow it up with, “Is there only one object that possesses properties X and Y?” One may have been subjected to this analysis as early as high school algebra when one is asked to classify whether a system of linear, algebraic equations have solutions, and whether the solution is a point or a line.) With even this relaxation, the space is still locally Euclidean (by the fourth postulate): a small portion of the surface will look “flat” when you blow it up (the same way how from our point of view, the Earth often looks flat). It is, however, possible to also relax the fourth postulate. The easiest example to imagine is by taking a piece of paper and rolling it up into a cone. At the vertex of the cone, going around one full turn is no longer going around 360 degrees. The “total angle” at the vertex will be smaller. So if we define a “right angle” as an angle that is exactly one quarter of the “total angle” at the point, the right angle at the vertex will be smaller! You can experiment with it yourself by drawing four rays from the vertex and then unfurling the piece of paper. Geometries with this kind of structures are often studied under the name of orbifolds.

In any case, let us return to the topic at hand.

Classical Euclidean geometry is filled with constructive proofs (the modern methods of mathematics–proof by contradiction, principle of strong induction, and existence proofs without construction–are only really popular after the 18th century).