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Wednesday, May 11, 2011

3D without glasses for more than one viewer, with eye tracker


I think the title and the image says it all. However I'll have spill some details out for myself because I might forget what I was thinking. The thing is, having multiple viewers, is just a matter of solving a system of equations. Such that every viewer's eye will see certain pixels. With eye tracking we can control the blinds accordingly, up to a fixed number of viewers. The engineering factor of how to implement those blinds may be regarded as layering several blinds, those same ones used for one viewer 3D glassless stereoscopy.

Sunday, May 1, 2011

Generalization of quantum computing ?

This is not particularly related to quantum computing itself but rather to the notion of extending the manipulating a single value to manipulating a linear combination of such values, having the combinations denote the amplitude of measurement of a single value. One extension approach would be to consider non-linear combinations, but that's not what I mean here. I am asking what if this was generalized somehow to the linear combinations of such linear combinations ? What is the possible operations we could do in a manner similar to that of quantum computing and what useful outcomes would that mean to us ? Quantum computing allowed things not thought to be possible like sub-linear search complexity (Grover's algorithm) and others like Shor's algorithm. What would further generalizations like the proposed one allow us to do that we didn't think was possible with quantum computing ? Another research direction I may want to pursue after PhD en shaa Allah =)

Algebra view point on computation ??

Today I was trying to make a theoretical model for my peer to peer network simulator. Basically its a network of peers which is modified each loop according to certain rules (either random peer sampling or according to similarity-based clustering). The network is modeled as an adjacency matrix and to modify this matrix I may multiply it by a certain matrix; since the network is closed then it is possible to utilize a permutation matrix for that (peers exchange neighbors with each others). I also use a randomization matrix between each two operations to account for randomness in choice. So if we denote the network (adjacency matrix of the network graph) at cycle $n$ as $G(n)$, then the operation that happens in each cycle of the simulation is $$G(n+1) \leftarrow E \times R \times G(n) .$$ Where $R$ is the random permutation matrix and $E$ is the exchange matrix.

Okay, here comes an observation. Matrix multiplication is not commutative, and hence it allows us to express a sequence of operations (operators) and to express an adaptive computation. So it was fair enough to ask my self a question, non-commutativity actually added more power in this case, unlike the view common in algebras where it gets less interesting as you lose properties. So I though, what if I use Cayley-Dickson construction to make a higher algebra that is not even associative ? What would we get ? Well, I am not if this will be clear to you but I can clearly see that this allows us easily to express a tree, not just a sequence, of operations. The nature of operations might be different than a permutation or rotation, I am not sure of that, but the open question now is since a tree resembles recursion (in some sense), then could this allow us to construct a sort of universal recursive function (which would be turing-complete) ? Even more interestingly, can we construct a certain algebra in such a way to express all and only all such computations that can recognize a certain language, i.e. given a class of languages or a turing machine that recognizes them, can we construct an algebra equivalent to this machine ?

Why would this be interesting ? well, one reason that might occur to me is that if we were able to explicitly construct an algebra for $P$ and another for $NP$ then we might have another way to attack the famous $P=NP$ problem, which would then be equivalent to whether the two algebras are equivalent. This might be an interesting research direction for me, and in fact, I wrote this article as a sort of a TODO to remind me of this idea later when I finish my PhD and be able to pursue a research direction of my own. Thanks for reading and any comments or references are welcome!