Commutative Hypercomplex Numbers 36
A reader writes: "The Generalized Number System (N+) implements commutative hypercomplex arithmetic to provide an alternative to vectors for processing multivariate data in three or more dimensions. Because of similarities between N+ and the complex number system, software for processing multivariate signals is readily derived from that for processing complex (or real) numbers. The derivation involves replacing operators on complex (or real) numbers with corresponding operators on hypercomplex numbers similar to the way in which steel replaced bronze as the ingredient for making swords during the Renaissance. In both cases, improved performance and capabilities of the product are attributed to properties of the new ingredient while many aspects of making and using the product remain the same. N+ and its application to signal and image processing is described on the website at www.hypercomplex.us ".
Too Deep For /. ? (Score:2)
Maybe it was that, by the time they got to the end of the text, they were too zonked out to post.
Congrats to Hemos for finding & posting this one - that's one hell of a subject/concept to get yer head around over a midnight snack
Re:Too Deep For /. ? (Score:2)
I love how they'll sell you the necessary software to incorporate "this wonderful new way to compute".
It's bullshit (Score:3, Informative)
There's a bunch of handwavy stuff about higher dimension number systems and getting communtative multiplication (needed for a bunch of signal processing algorithms and most other "real" math applications), but not proof.
Number systems with more than 2 coordinates are treated like matrices, because no one is able to find rules of muliplication and addition for say a 5-coordinate system that makes the numbers associative, commutative and distributive. Abstract algebra is basically only concerned with figuring out which of those properties hold for different sets. But in their stuff, they don't once show how their "N+" system will allow a 10-coordinate number (for instance) to be commutative with another one. From their site:
That sure sounds like they've found a system form making quaternions commutative. But quarternions aren't. I read everything in the left two columns of links. And there wasn't more than vague promises of N+ solving signal processing problems and basic descriptions of real number field theory. But nothing saying how their N+ numbers are associative, commutative and distributive. So this is just bullshit.
Rachael
It's renaming (Score:3, Informative)
Number systems with more than 2 coordinates are treated like matrices, because no one is able to find rules of muliplication and addition for say a 5-coordinate system that makes the numbers associative, commutative and distributive.
I read everything in the left two columns of links. And there wasn't more than vague promises of N+ solving signal processing problems and basic descriptions of real number field theory. But nothing saying how their N+ numbers are associative, commutative and distributive. So this is just bullshit.
I had the same reaction, but after digging a little deeper on their site [hypercomplex.us] (hurah Google) I did turn up a explanation, and guess what? They're just a shorthand for a sub-ring of the matricies.
-- MarkusQ
Re:It's renaming (Score:3, Interesting)
In fact, even just looking at the web site, you can see how from a computational point of view they aren't just matrices. A dimension N hypercomplex number is represeted by an NxN matrix. An O(N) algorithm isn't the same as an O(N^2) one. Etc.
Re:It's renaming (Score:2)
MarkusQ: They're just a shorthand for a sub-ring of the matricies
exp(pi*sqrt(163)) (aka Mr. 262537412640768744-epsilon): Now that's bullshit if I ever read it. Any algebra that has a faithful matrix representation can be considered a sub-ring of a matrix algebra.
Agreed. But that isn't anything new. I wasn't disputing their claims, just their hype. What is it that's so wonderful here?
- That, for any integer N>=1 you can find N orthagonal matricies to form the basis of a sub-ring?
- That the sub-ring so formed will have the properties you'd expect of a ring?
- That, knowing (choosing) the form of your basis, you can find computational shortcuts?
I guess it feels to me as if Bob Shmoe anounced the discovery of a new group of order 163, which he dubed "Shmo", and looking at it I saw it was {f^163=I}. I'd say "hey, that's just C-163. That wouldn't mean I was dismissing all modulo arithmetic.-- MarkusQ
P.S. If you think I'm missing something here, I'd be glad to know what. I think I read through more of their web pages than most posters, but I did it with a 13-month old on my lap, so I may well have bounced past some key point.
Re:It's renaming (Score:2)
However, I, like you, suspect that the whole thing is hype anyway.
Re:It's renaming (Score:1)
It's still bullshit. The subrings that are commutative are very small. I don't see how they would be generally applicable to as large a problem area as they seem to be claiming. Plus, they could at least show how it's important (e.g. they mention gaussian elimination 2-3 times -- maybe they could show how their "N+" system helps in that. But they don't. I wonder why?)
Rachael
No sir, I don't like it (Score:3, Insightful)
corresponding operators on hypercomplex numbers similar to the way in which steel replaced bronze as the ingredient for making swords during the Renaissance...
How the hell did Hemos let this one by?
Let's rewrite the article to be useful instead of stupid:
"The Generalized Number System (N+) implements commutative hypercomplex arithmetic to provide an alternative to vectors for processing multivariate data in three or more dimensions. Because of similarities between N+ and the complex number system, software for processing multivariate signals is readily derived from that for processing complex (or real) numbers. The derivation involves replacing operators on complex (or real) numbers with corresponding operators on hypercomplex numbers similar to the way in which steel replaced bronze as the ingredient for making swords during the Renaissance. In both cases, improved performance and capabilities of the product are attributed to properties of the new ingredient while many aspects of making and using the product remain the same. N+ and its application to signal and image processing is described on the website at www.hypercomplex.us".
becomes:
The Generalized Number System [hypercomplex.us] (N+) is numerical processing software that uses commutative hypercomplex arithmetic to solve multivariate data problems in more than two dimensions. This outperforms the more traditional vector-based approach.
I mean, all the info thrown out here can be comfortably mentioned on the website, and it doesn't look like a blatant attempt to get "wow, that's complicated" comments.
Slashdot should keep away from mathematics (Score:1, Funny)
Next they are going to post an article about a new incredible 10000:1 lossless compression system developed by some guy living in a garage in Seattle that REALLY WORKS and which all the information theorists in the world just never ever came to think about.
Commutativity important? (Score:2)
Re:Commutativity important? (Score:1)
Re:Commutativity important? (Score:2)
Commutativity is (one of the things) that distinguishes R^(N^2) from GL(N). It's a fundamentally important property. Commutative rings are by their very nature smaller than non-commutative rings, given the same set of generators.
I could go on, but...
Re:Commutativity important? (Score:2)
Please do. I was hoping to learn something. It seems like most extensions from scalars into multiple dimensions (matrices, quaternions) shed commutativity like last year's skin.
Can somebody please give an example of an important and comprehensible theorem that depends on commutativity? That is, what is a consequence that would be meaningful to a non-mathematician?
Re:Commutativity important? (Score:2)
Angular Momentum: Yup, the stuff that keeps you upright while on a bike or motorcycle, anti-commutative. The Coriolis force, a corollary of angular momentum, this is the same effect that causes tornados and hurricanes (but not toilettes) to always rotate in the same direction in the northern hemisphere. To understand or prove any of these effects you must minimally know about the cross product, which is (as you might guess) non-commuting.
Maxwell's equations have an inherent twist in them, consider Ampere's right-hand and left-hand rule. All motors and generators rely on this effect, and without it there could be no propagation of light.
Polarized sunglasses - that's right, photons have polarity and polarizing filters are caused by non-commutating interactions with materials.
The list of everyday phenomenon that require non-commutation to prove is much longer than this, but I hope this gives you a taste of how boring the world would be without this quirk of mathematics.
Re:Commutativity important? (Score:2)
I asked about examples of results for which commutativity is an essential quality. I don't doubt there are some, or plenty, but what are a few of them?
Re:Commutativity important? (Score:2)
but you have well defined inverses. And you can
even invert Matrixes of Octerions. P.S. I've
miss spelled Quaterions and Octerions, its late.
Re:Breathing important? (Score:3, Interesting)
Nor are these "trivial" uses, either; if you couldn't use commutivity as part of the equation re-writing process, many very common transforms become impossible... even the simple act of dividing "3x+2" out of an equation becomes difficult to set up if you can't re-order anything. (Remember that if you have x * 3 and you don't know multiplication is communitive, you can't rewrite that as 3*x, and thus you couldn't use that as part of 3x+2.)
In fact one would be hard pressed to find a non-trivial proof where commutivity isn't used implicitly, and you may find it very challenging (possibly even beyond your skill or downright impossible) to correctly re-write the proof without using commutivity.
(I speak in this post of "traditional" math, such as a normal person sees in school, somewhere up through low-level Calc. As others have pointed out, as you get higher into math, you encounter number and symbol systems where communitivity does not always hold. You typically meet one, "Matrix Math", in high school.)
Re:Commutativity important? (Score:2, Informative)
And if you don't know what what I just said means, then you shouldn't be commenting on the existence of 'important' theorems, should you?
Re:Commutativity important? (Score:2)
If you don't know the answer, just say so, and let somebody else speak up.
I'm just a dumbass (Score:2)
Dude develops a "new" system of mathematics. Dude isn't looking for a Nobel prize, or (excuse me!) (is he under 40?) a Fields Medal. He doesn't (prominently enough to notice) post any link to citations in any journal, let alone a peer-reviewed journal.
But he is willing to sell you a MATLab add-on for $250.00?
Does he sell tinfoil hats and perpetual motion machines too?
Re:I'm just a dumbass (Score:1)
Re:I'm just a dumbass (Score:1)
Re:I'm just a dumbass (Score:1)
Re:I'm just a dumbass (Score:2)
Sigh. Yeah, that's why I said, "(excuse me!) Fields Medal".
Steel replaced what? (Score:1)
Like airplanes replaced horses after the Viet Nam war, no doubt.
Excuse me? (Score:3)
looks like bullshit to me.
For starters, the change from bronze to steel was so huge, in terms of technological consequences, that any analogy in mathematics has to be something as novel, powerful, and versatile as "calculus". Furthermore, any bronze sword in the Renaissance would have been centuries out of date; iron and steel had been available in those parts for a long time.
Did they invent anything nearly as important as calculus? Not likely. Did they understand history well enough to properly state their analogy even if their invention is that important? No.
I guess nothing livens up a press release like a heaping helping of hubris. Credibility, meet toilet.
Hmmm (Score:2)
For the computational I'd expect to see the one and for the mathematical I'd expect to see the other.
Worse yet, there's mention of a patent. On the MATLAB toolbox, I'd expect. Certainly not on the mathematics, since that is not allowed in US Patents (I'm carefully hiding any laughter at the thought of something being patented that should not be).
On the plus side the russian side (see above) does seem to have some math and that doesn't look (at first glimpse) to be the kind of junk thats often generated like this.