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Advanced Mathematics for Computer Science


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#21 daviangel   Members   -  Reputation: 600

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Posted 21 March 2012 - 12:38 AM

For a lot of the topics mentioned (topology, differential geometry, nonlinear dynamics, etc), basically anything where there is a continuum instead of just finite structures, it will be difficult to make much progress without a solid grounding in real analysis. There's a great set of video lectures by Francis Su from Harvey Mudd where I did my undergrad,
http://beta.learnstream.org/course/6/
(or http:/ /www.youtube.com/watch?v=sqEyWLGvvdw and click through to the other videos)

Cool beans Posted Image
Been looking for an easy i.e. video intro to real analysis. I tried to read Mandelbrot's Fractal book and Kip Thorne's gravitation books a while back and both of them quickly lost me since right off the bat they both go into metric spaces Posted Image
I can also brush up on monoids too now Posted Image
Don't talk about writing games, don't write design docs, don't spend your time on web boards. Sit in your house write 20 games when you complete them you will either want to do it the rest of your life or not * Andre Lamothe

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#22 taby   Members   -  Reputation: 286

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Posted 21 March 2012 - 06:08 PM

I have used a few things from Abstract Algebra and Number Theory for hashing and pseudo-random number generators. I've used statistics quite a bit (Are those numbers really random? Is this version of my chess program stronger than the previous one?). Probability is really important for almost anything in Artificial Intelligence. If you do graphics, Euclidean and Projective Geometry are pretty important.

I only know a little bit about Chaos Theory, but my understanding is that it's pretty useless. It's just one of these things that have a sexy name and produce pretty pictures, but I don't think you can really do a whole lot with it.


Alvaro, surely you know that chaos theory applies to any system with more than two parts. A very interesting article is "On the nature of turbulence". Another interesting book is "Galactic Dynamics". Surely Poincare would be offended if he had heard you disparaging his theory of chaos as such. Posted Image

Anyway, otherwise inexplicable behaviour in physics, chemistry, and biology that was once labeled as noise and swept under the rug is now labeled as chaos. It's similar to how lightning was once labeled as the anger of the gods, but is now labeled as electrons and photons. I personally have no interest in knowing what makes up a lightning bolt, but science isn't about subjective opinion.

#23 taby   Members   -  Reputation: 286

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Posted 21 March 2012 - 06:12 PM

Category theory is basically object oriented programming on steroids.

#24 Álvaro   Crossbones+   -  Reputation: 10803

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Posted 21 March 2012 - 08:23 PM

Alvaro, surely you know that chaos theory applies to any system with more than two parts. A very interesting article is "On the nature of turbulence". Another interesting book is "Galactic Dynamics". Surely Poincare would be offended if he had heard you disparaging his theory of chaos as such.


I am not saying that you won't encounter chaotic systems. If you do pretty much anything that involves iterating a function, you'll get there pretty soon. However, there isn't all that much that is useful that you can do with "Chaos Theory".

Anyway, otherwise inexplicable behaviour in physics, chemistry, and biology that was once labeled as noise and swept under the rug is now labeled as chaos. It's similar to how lightning was once labeled as the anger of the gods, but is now labeled as electrons and photons

We understand lightning and we have the lightning rod to show for it. What has Chaos Theory given us? And if you do find some application, is it relevant to a CS professional?

Category theory is basically object oriented programming on steroids.

They are both forms of abstraction, and the people that invented OOP borrowed lots of terms from category theory, although I don't think you can push the parallels beyond superficial similarities.

#25 taby   Members   -  Reputation: 286

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Posted 22 March 2012 - 10:45 AM

While googling for topological mixing and bifurcation diagrams, I ran across this PhD thesis "CHAOTIC COMPUTATION By ABRAHAM MILIOTIS" from 2009:
http://etd.fcla.edu/UF/UFE0024234/miliotis_a.pdf

References 25-28, and the entire thesis actually, seem to be about the marriage of chaos theory and computation.

Then again, I really don't put any faith whatsoever in people with university educations, so it might be as much bunk as anything else. I'm sure there are many university educated people in this thread who have a subjective opinion on the matter. Hopefully they'll grace us with their knowledge.

#26 taby   Members   -  Reputation: 286

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Posted 02 April 2012 - 11:51 AM

This came across the Twitter today via @CompSciFact ...

"First of all, one of the most important features of category theory is that it is a guide to computation."

http://www.cs.man.ac.uk/~david/categories/book/book.pdf

Just an FYI for those who are curious.

#27 Álvaro   Crossbones+   -  Reputation: 10803

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Posted 02 April 2012 - 12:41 PM

I had to learn category theory in college and putting that stuff in my brain felt like hammering a square peg into a round hole. I had a similar feeling when I tried to learn functional programming. So I am not too surprised that the two are somehow connected. Posted Image

#28 taby   Members   -  Reputation: 286

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Posted 02 April 2012 - 01:22 PM

:)

#29 Nick Alger   Members   -  Reputation: 486

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Posted 02 April 2012 - 07:35 PM

I had to learn category theory in college and putting that stuff in my brain felt like hammering a square peg into a round hole. I had a similar feeling when I tried to learn functional programming. So I am not too surprised that the two are somehow connected. Posted Image


Do you have any recommendations for category theory learning materials (books, videos, websites, online courses, etc..)? I've been trying to learn the basics for a while now without much success..

#30 Álvaro   Crossbones+   -  Reputation: 10803

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Posted 03 April 2012 - 07:06 AM

Do you have any recommendations for category theory learning materials (books, videos, websites, online courses, etc..)? I've been trying to learn the basics for a while now without much success..

No, sorry. I ended up learning enough of our text book to pass the exam, but it's about the only time in my life when I memorized a bunch of material without understanding most of it. At the exam they asked me about the Yoneda lemma, of which by now I only remember the name.

#31 taby   Members   -  Reputation: 286

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Posted 03 April 2012 - 03:10 PM

You could try this one... I also saw it come across the Twitter the other day. It's very recent, and seems to be written by someone who has worked with category theory for a while now.

http://jscategory.wordpress.com/

The truth is, I don't know if an easy introduction to category theory exists.

#32 daviangel   Members   -  Reputation: 600

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Posted 05 April 2012 - 07:33 PM


Do you have any recommendations for category theory learning materials (books, videos, websites, online courses, etc..)? I've been trying to learn the basics for a while now without much success..

No, sorry. I ended up learning enough of our text book to pass the exam, but it's about the only time in my life when I memorized a bunch of material without understanding most of it. At the exam they asked me about the Yoneda lemma, of which by now I only remember the name.

Wow how'd we get on this tangent Posted Image
Don't talk about writing games, don't write design docs, don't spend your time on web boards. Sit in your house write 20 games when you complete them you will either want to do it the rest of your life or not * Andre Lamothe

#33 Nick Alger   Members   -  Reputation: 486

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Posted 12 April 2012 - 06:09 PM

Wow how'd we get on this tangent Posted Image


Haha, this thread got out of hand rather fast didn't it! I actually just got this book for the basics of category theory and it seems pretty readable so far.

#34 taby   Members   -  Reputation: 286

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Posted 14 April 2012 - 09:55 AM

Hehe, you stay away from those discrete dynamical systems. ;)




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