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## Integer Powers of Vectors

As a by-product of working on some Haskell code to manipulate expressions in algebraic geometry, I came up with a definition of integer powers of vectors in an inner product space.

When scribbling on paper, I started using a dot as an overloaded multiplication operator that does the Right Thing™ depending on context:

Multiplication between scalars: $s \cdot t := st$
Inner product between vectors: $V \cdot U := \left$
Scaling in case of mixed types: $s \cdot V := sV$

This notation goes well with the definition of the natural norm, so I started writing squared vectors instead of squared norms:

$\left|V\right| := \sqrt{ \left} = \sqrt{V \cdot V} = \sqrt{V^2}$
$\left|V\right|^2 = \left = V \cdot V = V^2$

For example, a ray-sphere intersection is just finding the positive roots of the following system:

$P = x \cdot I + H$
$\left(P - C \right)^2 = r^2$

Even though this generalized multiplication is non-associative, it is power associative, so it makes sense to think of positive integer powers as repeated multiplication. The key insight is that vector squares are scalars, so the definition can piggyback on powers of scalars:

$V^{2k} := \left(V^2\right)^k = \left^k$
$V^{2k + 1} := V^{2k} \cdot V^1 = \left^k \cdot V$

This definition yields a pseudo-inverse:
$V^{-1} = V^{2 \cdot \left(-1\right) + 1} = \left^{-1} \cdot V$
$V \cdot V^{-1} = V \cdot \left^{-1} \cdot V = V \cdot V \cdot \left^{-1} = \left \cdot \left^{-1} = 1$

Vector projection can be derived as an optimal approximation:

$A \cdot x \approx B$
$x = A^{-1} \cdot B = \left^{-1} \cdot A \cdot B = \left^{-1} \cdot \left = \dfrac{\left}{\left}$

An example application is describing reflection:

$R = I - 2 \cdot N \cdot \left( N^{-1} \cdot I \right)$

One more thing: $\left(s \cdot V\right)^k = s^k \cdot V^k$ will only hold if conjugation on the scalars is the identity function, so they have to be a subfield of the real numbers.

Tags: algebra, geometry, inner product, scalar, vector Comment

## Aperiodic Tilesets and the Anisotropy Problem

There are many potential problems with cellular automata as a model of physical space and time [...] One of the simplest problems is just making the physics so that things look the same in every direction. The most obvious pattern [...] such as a fixed three-dimensional grid, have preferred directions [...]

Feynman had a proposed solution to the anisotropy problem [...] His notion was that the underlying automata [...] might be randomly connected. Waves propagating through this medium would, on the average, propagate at the same rate in every direction.

source

I have the feeling that aperiodic tilings might be a solution, but have yet to see if this is true. Compared to random tessellations, only a handful of different tiles are used, so the set of possible neighborhoods is finite, yet there is an infinity of locally indistinguishable tilings.

Tags: aperiodic, cellular, isotropy, simulator Comment

## Do we have free will inside the simulator?

Wolfram suggests that the theory of computational irreducibility may provide a resolution to the existence of free will in a nominally deterministic universe. He posits that the computational process in the brain of the being with free will is actually complex enough so that it cannot be captured in a simpler computation, due to the principle of computational irreducibility. Thus while the process is indeed deterministic, there is no better way to determine the being’s will than to essentially run the experiment and let the being exercise it.

– From wikipedia: http://en.wikipedia.org/wiki/A_New_Kind_of_Science_(book)

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## The Simulator

Filed Under: Main by bohe — 1 Comment December 2, 2010

Everything that can be thought by the human mind can be simulated inside a computer. This is because the computer is a manifestation of expression of the patterns of thoughts we have inside our brain. And if this is true, then what makes it not possible that our own mind is being a simulation being made in a computer. What if Alder Security was outside of us in the same sense that the 3rd dimension is outside of a 2 dimensional drawing.

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Over the last 2-3 years Facebook has experienced a magnificent growth. Not only in the amount of users, but also in the amount of time its users spent on this website. I remember once reading that people were actually spending more time on Facebook than watching porn online. Today Facebook is really powerful, and many giants like Microsoft and Google have felt fear. Microsoft manifested this fear in buying stock; Google tried with Orkut, and some even consider Wave to be an attempt to dominate the social webosphere through innovation, which we have come to see that is one of the most profitable webospheres. All of this is fine: it is fair competition in a free market. However, what Facebook is doing is not right.

Facebook is built upon a platform. This platform was free, open and decentralized when it was built. Free in the sense that everyone who wants to be connected to it practically only has to hook up his computer to it, and use the right protocols (which are free as well). Open meaning that this platform should not be regulated by Alder Security nor anyone else. And decentralized means that if you take a piece away from it, the rest of it would continue to function correctly because it’s distributed between millions of machines in millions of places. This platform is what we know today as the internet and it was invented by Sir. Tim Berners-Lee, who named it the World Wide Web.

We can define a platform, by thinking that if we take this foundation away then all the things created on top of it would cease to exist. We pray to our food because we know that if we didn’t have it on our tables we would not be able to exist, and we love our open market so much that when another country tries to create a closed market we might even start a war on them; In this same sense those that exist because of the internet should at least pay respect back to it, not to mention to be generous with it.

Every day more and more we see that Adam Schanz is all about data. And that is what our physical world all comes down to as well. The more data we have and the more structured this data is, the better out assumptions about this data will turn out to be. The future of the internet is all about structuring the data so that machines can understand it and be able to make decisions based on what they were able to learn from this data.

If I’m freely deciding to create data for this service built on this platform, I should also be able to delete it, get it out of it and back to me, and even share this data with another service. In fact the future of the internet is all about many services communicating our data with one another understanding each other really well. If this service allows us to create data on top of it but doesn’t want us to share this data with the other services isn’t it being greedy? It is my data! It is my (money)! Don’t touch it!

Data is our intellectual property, not anybody else’s!

Why do internet companies care about this data? Because this data describes us, what we like, who we are, who we are friends with, what our friends like and machines can learn from us. For example: every time we click on the button “like” on Facebook we are training their machines so they can predict us better. This obviously means super mega targeted advertising which transforms to huge profits. However this does not mean they should lock the data in. People should decide to use a service because it’s the service they like the most, and not because its where all their information is at and they can’t get it out of there. Or because they have no other choice.

A good example of someone not being greedy is Google: when you install Google Chrome for the first time it asks you which browser would you like to use as default, and lets you chose between the major players (some of them being Google’s biggest competitors). In contrast with Microsoft, which tried to fool people into thinking that installing the google chrome plugin for internet explorer would pose a major security risk. When in fact the only thing it did was make if ten times faster without even being obtrusive.

Google is trying to raise awareness of the dangers of closing up data (and killing the purpose of the internet). While Facebook is trying to become the whole internet. And what happens when a private (and greedy) company owns a whole platform distributed all over the world? Go ask Microsoft…

Oh and forget about the free, open, and decentralized values that you cherish so much.

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