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If two matrices and are related by for an invertible matrix , (all matrices are n-by-n), then and are called similar. What this means is that the two matrices represent the same linear transformation but under different basis, is the change of basis matrix.

Some properties:
* If is an eigenvector for then is an eigenvector for , but both matrices will have the same eigenvalues.

if is an eigenvector and corresponding eigenvalue of then hence hence is an eigenvector to ….

* They share the same determinant, same trace (same characteristic polynomial).

….

Well, this is something which I was thinking about when learning about curvature of curves and surfaces. The concept of creating a straight edge without a reference.

Let us start of by having three objects in 2D, for example we can have the following


Say we have three stones A,B,C. And let us say that we choose to rub A,B together, due to friction they will eventually have a constant curvature all over the contact surface, the result is to the right. So what does constant curvature mean? Well this means that in 2D the contact surface is part of a circle (infinite diameter included). Also note that A and B must have opposite concavity, say A has positive and B has negative.

Let us now do the following scheme, rub A and B, then B and C then C and A. Then start over and repeat indefinitely =)

Since the curvature is constant we conclude that

the only possible solution is that all edges have both positive and negative concavity and hence they must be parts of lines.

Of course one cannot do this in finite time, although one can get very close, note that this is used still today to create straight edges.

Based on a recommendation, I saw a documentary about “dangerous knowledge” made by BBC. David Malone looks at four brilliant researchers - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing. In the documentary David talks about what questions they were asking, and how understanding these questions tragically drove them insane and ultimately led to their early demise.

The journey starts with Georg Cantor, who started looking at the concept of infinity, a concept that scared his predecessors and his likes. However Cantor understood that since most analysis is build upon infinity, he needed to understand this. This led Cantor to develop the set theory that we all use, and he also coined the different cardinalities of infinite sets. One example being the the power set of the natural numbers and the natural numbers does not have the same cardinality, the previous having the same cardinality as the real numbers.All this led Cantor to study the Continuum hypotheses, i.e. there is no set with a cardinality strictly between the cardinality of and the cardinality of .

A question which drove him mad.This documentary is worth seeing.

Link

Can you guess what this is?

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Just published a Wikipedia article about Frege systems, as a long overdue hand in assignment for a course I took in SAT. Lets hope it stays…

Recently I have started looking at something we call an expanding Cantor repeller. Now how do we define such a thing?Well basically it’s a dynamical system where the repeller of the system will become a fractal set of cantor type. Notice that a Julia set can be constructed by iteration.Let us describe the process more clearly, Let us start with a topological disc (think of a standard disc), inside this disc we place topological discs that are disjoint contained inside . Assume now that we have a defining function , i.e. a function

but we require more on the function , that is we require it to be a conformal isomorphism, that is univalent on the closure of each .Now we are ready to define the repeller,.So, the points of is all points that cannot escape under the mapping , and all other points never enter , that is why it’s a repeller.Let us now take a simple example of the Cantor set in two dimensions. First let us start with a unit square, let that be , secondly let be squares of sidelength and place them inside in each corner of , like such

Let the function be the function that takes , hence a linear function in each of the squares. The repeller will hence become the Cantor dust 2D set.

Just like the title says.

I have now corrected the last exam, and now I have a Christmas break. This essentially means that I can focus on research, yehaw.

halting my computer, I should get an UPS.