+1! I'd love to attend. Lets see if we can reach quorum. Dean may not be on the list (he may be on wedtech, discuss, or sfx_graphics instead), or he may have multiple email addresses and sent from one not associated with his friam membership. That's the typical problem getting bounces from the list.
-- Owen On Tue, Jan 24, 2012 at 8:21 PM, Frank Wimberly <[email protected]>wrote: > This is a message from Dean Gerber. For some reason it didn’t reach the > List when he sent it. I forward it at his request. I will certainly > attend the lecture he offers.**** > > ** ** > > ** ** > > Algebras**** > > ** ** > > Owen-- > I think what you are looking for is the theory of algebras, generally > known as non-associative algebras. These structures are vector spaces V(F) > defined over a field of scalars F satisfying the usual axioms of a vector > space with respect to the operations of vector addition and scalar > multiplication, and an additional binary operation of vector multiplication > (called a product) that is distributive with respect the vector space > operations. To be specific, let x,y,z be vectors in V(F), let a,b be > scalars in F, and denote the product of x and y by x!y. Then V(F) and > the product (!) define an algebra if and only if > > i) x!y is in V. (x!y is a vector - the very meaning of "binary > composition") > ii) a(x!y) = (ax)!y = x!(ay). ( Scalar multiplication distributes with > vector multiplication) > iii) x!(y + z) = x!y + x!z and (y + z)!x = y!x + z!x. (Vector > multiplication distributes with vector addition) > > Since a vector space is always equivalent to a set of tuples, this > provides the multiplication of tuples you are looking for. For an > n-dimensional vector space the generic (general) product is completely > defined by n-cubed parameters, known as the structure constants. Specific > choices of these parameters from the field F define specific algebras and > the properties of these algebras vary greatly over the possible choices. > For example, for n = 2 there are 8 free parameters and the complex numbers > represent a single point in this 8 dimensional space of structure > constants. That particular choice implies that the algebra of complex > numbers is itself field. Generally, algebras have no properties other than > i) to iii) above, i.e. they are generally not commutative or associative.* > *** > > ** ** > > The Caley-Dickson procedure is a process by which a "normed" algebra can > be extended to a normed algebra of twice the dimension. The only real one > dimensional normed algebra is the real number field itself. The > Caley-Dickson extension is just the complex numbers as a 2 dimensional > algebra, and it is also a field; in fact the only 2 dimensional field..*** > * > > ** ** > > The Caley-Dickson extension of the complex number algebra is the 4 > dimensional quaternion algebra. But, the quaternions are NOT a field: they > are not commutative even though they are associative and a division > algebra. They are often known as a "skew" field, **** > > ** ** > > The Caley-Dickson extension of the quaternions is the 8 dimensional > octonion algebra, and these are neither commutative or associative, but > they are a division algebra.**** > > ** ** > > The next step gives the nonions of dimension 16 at which point we lose > the last semblance of a field because they are not commutative, not > associative, and not a division algebra. Thus, if we want fields, the > complex numbers are indeed the end point. All real division algebras are of > dimensions 1,2,4, or 8! There are many division algebras in the dimensions > 2,4,8, but only in n = 2 are all of them classified up to isomorphism.**** > > > I could go on, if you could gather up an audience of at least ten for a > (free) three hour blackboard lecture with two breaks. For an audience of > fewer than ten I would have to collect ten hours of Santa Fe minimum wages > for prep and lecture time. Its a beautiful subject with a very colorful > history, and includes the quaternions, octonians, Lie algebras, Jordan > algebras, associative algebras, everything mentioned by the FRIAM > commentariat. **** > > ** ** > > Regards- Dean Gerber**** > > ** ** > > > > **** > > *From:* [email protected] [mailto:[email protected]] *On > Behalf Of *Owen Densmore > *Sent:* Tuesday, January 24, 2012 9:34 AM > *To:* Complexity Coffee Group > > *Subject:* Re: [FRIAM] Complex Numbers .. the end of the line?**** > > ** ** > > Arlo:**** > > ...Would it not be better to say, "are there number(data?)-structures that > provide for interesting algebras not yet considered?"**** > > ** ** > > Yes indeed. I was fumbling for a way to say that but ran out of steam!*** > * > > ** ** > > Roger Critchlow:**** > > http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf **** > > Now that is interesting, and nice to know this is a broader conversation > than I had known. GA's .. gotta look into them and their unification of > complex numbers and vectors.**** > > ** ** > > Roger/Carl: **** > > Suspect you are about to pop out of algebra and end up someplace else as > interesting.**** > > As you say, I think this is the more fruitful approach.**** > > ** ** > > All: The Cayley Dickson generalizations discussed in wikipedia: R C H O > did present an "answer" in that there are successful numeric extensions, > that complex numbers "are not alone". As much as I wish computer graphics > had used them for their transformations rather than 4-tuples (homogeneous > coordinates) and 4-matrices, I'm not sure just how quaternions differ in > theory from linear algebra, which simply started in on generalized n-tuples. > **** > > ** ** > > In other words, simple n-tuple algebras might have put all these > generalizations from R into a single framework. Why *aren't* complex > numbers simply our first use of 2-tuples, unified with the rest of linear > algebra. Possibly the answer is that, yes linear algebras uses n-tuples, > but they focus on very different matters such as linear independence, > spanning sets, projections, subspaces, null spaces and so on. **** > > ** ** > > Fun! So now I hope I can find some interesting problems that ONLY can be > handled with some of these non linear algebraic higher number systems. > Interestingly enough, I believe all of the extensions mentioned, as well > as all of linear algebra, have the same cardinality .. the continuum, right? > **** > > ** ** > > Thanks for the insights,**** > > ** ** > > -- Owen**** > > ** ** > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org >
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
