Randall's Blog

Novel way of explaining binary counting to absolute beginners

2011-12-11T22:56:00.001-08:00

Binary is explained very well on thousands of excellent web pages out there, so this is just a snippet of an idea that I think is novel. I just thought of a cool way to teach someone to count in binary, by just swapping the digit "1" for "9" and back, as follows:

A) Replace all the "1"s with "9"s;
B) Add 1 as you would in familiar decimal counting
C) Then change all "9"s back to "1"s.

For example, starting with say 1011
After step A) we have 9099
After step B) we have 9100
After step C) we have 1100

The goal is to very quickly just skip the step of actually writing any "9"s at all, and just know to treat the "1" as the "9", i.e. in binary, the "1" is the last digit. Works for any base.

Perhaps nature does not use the square root of negative one

2011-12-01T11:02:00.001-08:00

I've recently finished reading "Introduction to Smooth Manifolds" by John M. Lee, which I highly recommend if you want a solid and readable introduction to some aspects of differential geometry, Lie groups and Lie algebras, and a whole lot more. (I must be honest and admit that I think I would probably learn just as much as I did the first time if I read it a second time.)

Thoughts on quantum physics: The use of complex algebra in quantum physics is puzzling because sqrt(-1) does not make sense in our "classical" "real world". However, it may be as simple as accepting that complex algebra is just isomorphic to whatever the algebra is of the quantities that quantum physics is modelling. In other words, nature doesn't take the square root of negative one, but whatever is going on works in the same way as if it did.

A proposal for improving the Wikipedia's utility

2011-03-01T13:00:00.001-08:00

Wikipedia is wonderful. But when trying to use it for mathematical or other highly technical subjects, there is a problem. The topics tend to be correct, but contain too much material aimed at experts in the specific topic. Too much of the terminology in a given article requires reference to defining articles, that themselves have the same problem, and so it is very time consuming, and often leaves the user unsatisfied.

A proposal:

Allow the user to rank their own "satisfaction" with any article, and store this with their profile.

Also, for each article or article section, specify the prerequisite articles.

Develop a suggestion system that suggests the best path from the body of articles that the user is already satisfied with to the article on the topic that they require to understand.

In other words, instead of the user having to work backwards from the article that they require to understand, allow them to work in a more natural forward progression, similar to the way the topic would be covered in a text book or a teaching course.

Theoretical Minimum - Physics

2009-11-03T17:14:00.000-08:00

iTunes-U has lectures from Stanford called "The Theoretical Minimum".

I've watched 3 of the topics so far (about 10 lectures per topic, about 90 minutes per lecture). I am enjoying them. Topics so far:

Classical mechanics
Quantum theory
Quantum entanglment

My highlights from Classical Mechanics:
Principle of Least Action. Deriving the Euler-Lagrange equations from the principle of Least Action.

Noether's Theorem: differentiable symmetries (i.e. smoothly parametrised transformations that leave the Action invariant) imply conserved quantities, e.g. x->x+ε for linear momentum, θ->θ+ε for angular momentum, t->t+ε for energy.

Derivation of Hamiltonian formulation (used Legendre transformation L+H=pv). Liouville's theorem: flows in pq phase space are divergence free (like incompressible fluid flow).

Electromagnetic example: particle in magnetic field: guage field needed for Least Action formulation; guage invariance and guage transformations.

From Quantum Entanglement:

Definite highlight from quantum entanglement is Bell's theorem, and the violation of locality: Making an "observation" of one component of an entangled system instantaneously determines the outcome of the other component, irrespective of their spatial separation. I.e. the one component affects the other component instantaneously, and notably: faster than the speed of light!

Note: I also highly recommend the Quantum Mechanics lectures from Oxford by James Binney.

How cells represent their environment

2008-05-13T14:29:00.000-07:00

This book was recommended to me: "An Introduction to Systems Biology" by Uri Alon. I have just started on it.

I like this idea from it: think of each transcription factor (TF) as a variable. The cell represents its environment (both external and internal) by means of these variables.

Background:

A transcription factor is a protein that is either active or inactive. Some aspect of the environment will cause a molecule to be either active or inactive, e.g. a small molecule may bind to the TF, and cause the TF to undergo a conformational change. When active, it binds to a specific site on the DNA and causes transcription, which results in the production of a specific protein, which performs a function appropriate for the level of the environment variable.

The value of the variable that represents the environment is the proportion of TF in the active state.

Thus, the millions of real environment variables are projected onto a smaller parameter space: the number of distinct TFs.

E.Coli has 300 TFs, according to Alon. That's quite a small parameter space compared to the physical environment.

One comment: the kinetics of the TFs and the chemical species with which they interact is modulated by the physical environment, so the above view is simplified.

Real-time ray tracing

2006-12-16T03:31:00.000-08:00

My last adventure the interesting places in the universe of maths and software engineering involved real-time ray tracing. I took an existing demo from the JOGL (Java OpenGL) code, and converted it from the ARB shading language to the OpenGl shading language (GLSL). The demo shows a glass rabbit, which you can interactively rotate. Minimalistic ray tracing is done to create the effects of fresnel reflection variance, and chromatically variant refraction. The ray tracing is minimalistic, since only 4 rays are traced per fragment, and they are only traced to a cube-map texture.

First, I extended the the original demo by making the indices of refraction variable, as well as making the fresnel factors variable. Thus in real-time you can make the object become invisible (index = 1 ), or totally reflective (i.e. appears silver coated).

Then, once I got it going in GLSL, I extended it by adding colour filtering to both the reflected and refracted components. Now you can change the "glass" colour. Setting reflective to red, and transmissive to blue looked really cool too.

Aesthetically, taking such a minimalistic approach to ray tracing works fine, since in real-time ray tracing, each frame is transient, so there is "no time" to spend finding the flaws due to the gross simplification.

Genomics and bioinformatics

2006-06-15T15:12:00.000-07:00

I spent the last few months learning Bioinformatics, genomics, computational biology and similar stuff from mp3's that I downloaded from Massachusetts Institute of Technology.

What was quite cool was that none of the lectures had mentioned Array CGH (Comparitive Genomic Hybridization) yet, and one night, just thinking about how to detect cancer mutations genomically, I independently "invented" it. Although I must confess I was a teensy bit dissappointed when I discovered that it was already invented, I was chuffed to see my idea was valid. Turns out that sequencing the cancer genome is now a big project, on a similar scale to the Human genome project. Some critics (e.g. Garth Anderson, a cancer geneticist at Roswell Park Cancer Institute in Buffalo) point out that only the initial mutations are significant, since they allow a higher mutation rate in the cells, and by a kind of "miniaturised" version of evolution/selection, eventually become the dangerous cancer cells.

Some other recent learning:

Recently I found some good stats resources on the web, and I am using AT&T natural voices to convert the text to mp3's, so now I am updating my stats knowledge in my spare time.

A few days ago I read up on and played around a bit with Principle Component Analysis, which looks cool.

Today I am also reading up about Support Vector Machines, which a colleague of mine mentioned to me.

All this in my free time mostly, but it is just so interesting. Would really be cool to be doing this full time, I reckon I would have so much more time to learn this stuff.

Knotted Spheres

2006-01-15T15:23:00.000-08:00

I think I have figured out from first principles, how to tie a sphere into a simple knot. I spent about 20 minutes this morning just thinking about it. As far as I can tell, it requires 5 spatial dimensions. First stretch the sphere into a long sausage shape. Then bend the sausage into a C shape. Now twist the one end of the sausage into the 4th dimension, and the other end into the 5th dimension. Now all that remains in the original 3space of the ends are two sections of a loop. Interlink these (i.e. remove a point from one of the loops (i.e. cut it), pass the other loop through the gap, and replace the point (i.e. re-connect it). Viola.

Actually, later in the day, I read up a bit, and tried to explain it to a work colleague, Ben. While explaining it, I realised that that was not a knot, but would become untangled, so here is how to knot it.

Start with a normal sphere. Rotate by 90 degrees around a plane through its centre. You now have a series of circles, one in each 3-space. Create a knot with each circle simultaneously.

Now you have a knotted sphere in 4-space.