tag:blogger.com,1999:blog-11295132.post115886175429092174..comments2014-11-13T21:22:20.572-08:00Comments on A Neighborhood of Infinity: Practical Synthetic Differential GeometryDan Piponihttps://plus.google.com/107913314994758123748noreply@blogger.comBlogger15125tag:blogger.com,1999:blog-11295132.post-12889966610820062882014-06-25T07:05:59.204-07:002014-06-25T07:05:59.204-07:00Actually you get higher derivative for free, you d...Actually you get higher derivative for free, you don't need explicitly to do the trick! Just define <br /><br />> im (D _ a) = a<br />> derivative f x = im $ f (D x 1)<br /><br />Then <br />derivative (\x -> (x-1)^3 ) 0 == 3<br />and you can immediately calculate higher derivatives!<br />(derivative . derivative) (\x -> (x-1)^3 ) 0 == -6<br />(derivative . derivative . derivative) (^5) 1 == 60Alexei Kopylovhttp://www.blogger.com/profile/15073177772478487515noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-21314542633487244592013-06-13T15:59:59.214-07:002013-06-13T15:59:59.214-07:00Did i succeed in leaving a comment or not? I gues...Did i succeed in leaving a comment or not? I guess i'll have to wait and see.<br /><br />I hope (?) i might get notified by email if i did succeed in leaving a comment ... ?John Roodhttp://www.blogger.com/profile/16126451592989412577noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-69118508877979590302013-06-13T15:58:19.736-07:002013-06-13T15:58:19.736-07:00Hello. Several comments.
Koch's book (nice l...Hello. Several comments.<br /><br />Koch's book (nice link, btw--i have a first edition, but did not know there existed a second) defines higher order nilpotent (?) differentials (or these little dual number guys, whatever you want to call them).<br /><br />And I WILL ask the question: What does this stuff have to do with "Geometric Algebra", Clifford Algebras, Grassmann Algebras, Supersymmetry, anything you can handle ... ?John Roodhttp://www.blogger.com/profile/16126451592989412577noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-50676693391850643612012-09-22T03:55:50.460-07:002012-09-22T03:55:50.460-07:00...because Pythogoras's theorem gave only one ......because Pythogoras's theorem gave only one solution...<br />Is this a spelling mistake, is *Pythagoras* spelled like this or is this just another guy?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1161741947129078302006-10-24T19:05:00.000-07:002006-10-24T19:05:00.000-07:00If your having trouble displaying math you might w...If your having trouble displaying math you might want to look at ASCIMathML at<BR/><BR/>http://www1.chapman.edu/%7Ejipsen/mathml/asciimath.html<BR/><BR/>It's a javascript that automatically parses a web page and converts standard LaTeX notation (or it's own simplifed notation) between `` or $$ delimiters into MathML on the fly.<BR/><BR/>You should be able to use it with Bloggger but I don't know how.Chris Wittehttp://www.blogger.com/profile/17977972230615509660noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1159288783977394562006-09-26T09:39:00.000-07:002006-09-26T09:39:00.000-07:00I'd ask how SDG compares to geometric calculus...G...<EM><BR/>I'd ask how SDG compares to geometric calculus...<BR/></EM><BR/>Geometric calculus is what you get when you combine calculus with geometric algebra. SDG provides an alternative approach to calculus. So the two should happily coexist. In fact, the code <A HREF="http://sigfpe.blogspot.com/2006/08/geometric-algebra-for-free_30.html" REL="nofollow">here</A> can be combined with this latest code without problem and I guess that some of the language of geometric calculus turns directly into usable code in a nice way.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1159288345956485862006-09-26T09:32:00.000-07:002006-09-26T09:32:00.000-07:00Michi,d(x)1 and 1(x)d commute so their commutator ...Michi,<BR/><BR/>d(x)1 and 1(x)d commute so their commutator is zero. In fact, both of the algebraic structures I define are commutative. The Lie bracket comes from the non-commutativity of the vector field functions, not the underlying algebras.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1159283508805860762006-09-26T08:11:00.000-07:002006-09-26T08:11:00.000-07:00Soooo... your Lie bracket is "just" the commutator...Soooo... your Lie bracket is "just" the commutator of d(x)1 and 1(x)d in the tensor product, right?<BR/><BR/>Is this just me being WAY to used to the algbraic point of view thinking that this is almost tautological?Michihttp://www.blogger.com/profile/04492458231737217248noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1159247076375367592006-09-25T22:04:00.000-07:002006-09-25T22:04:00.000-07:00Michael Shulman has an introductory lecture on syn...Michael Shulman has an <A HREF="http://www.math.uchicago.edu/~shulman/exposition/sdg/pizza-seminar.pdf" REL="nofollow">introductory lecture on synthetic differential geometry</A> that starts out with dual numbers and eventually describes a Lie bracket. I don't know enough about Lie groups to know how his formulation compares to yours.<BR/><BR/>I'd ask how SDG compares to geometric calculus, but I don't think I'd understand the answerDave Menendezhttp://www.blogger.com/profile/10628628100970152906noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1158942098275684212006-09-22T09:21:00.000-07:002006-09-22T09:21:00.000-07:00augustss,Thanks for the link. That's the only pape...augustss,<BR/><BR/>Thanks for the link. That's the only paper I've seen that formulates AD the way I do.<BR/><BR/>Judging by the date of that paper, I think that my approach to Lie algebras must be novel so maybe I should write a paper on it. (It's slightly different to what I wrote here because Kock's 'functional' definition of a vector field is pretty, but less useful for calculation.)sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1158938774023312682006-09-22T08:26:00.000-07:002006-09-22T08:26:00.000-07:00A reference to AD:http://www.bcl.hamilton.ie/~bap/...A reference to AD:<BR/>http://www.bcl.hamilton.ie/~bap/papers/popl2007-multi-forward-AD.pdfaugustsshttp://www.blogger.com/profile/05153404423721072935noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1158934579424705832006-09-22T07:16:00.000-07:002006-09-22T07:16:00.000-07:00Michi, the code's correct, the 'comments' weren't....Michi, the code's correct, the 'comments' weren't. Now fixed.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1158934224630590982006-09-22T07:10:00.000-07:002006-09-22T07:10:00.000-07:00For the second derivtive you need an element such ...For the second derivtive you need an element such that d^3=0 but d^2=0. Such an element can be found in R⊗R, as you've discovered. But, as you point out, you get the 1st derivative twice and it gets worse with higher derivatives. You can implement an appropriate algebra directly. One paper that does this is <A HREF="http://portal.acm.org/citation.cfm?id=74895" REL="nofollow">here</A>. (That implements exactly the right thing, but doesn't give an algebraic description.) For arbitrary higher derivatives you're probably need power series code and there are many implementations of that in Haskell, at least for the single variable case.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1158933767738243982006-09-22T07:02:00.000-07:002006-09-22T07:02:00.000-07:00Ummmm, I seem to be running into problems quite ea...Ummmm, I seem to be running into problems quite early on. I defined the duals following your exposition closely (I added a signum and an abs because ghci was complaining about those), and then got <BR/>*DiffGeo> let f x = x^3+2*x^2-3*x+1<BR/>*DiffGeo> f (1+e)<BR/>D 1 4<BR/>wgucg seems to contradict your calculation of that very example.Michihttp://www.blogger.com/profile/04492458231737217248noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1158904740391566982006-09-21T22:59:00.000-07:002006-09-21T22:59:00.000-07:00What's the preferred way to do higher derivatives?...What's the preferred way to do higher derivatives? I found that e2 = D e 1 works, but it also calculates the first derivative twice.Dave Menendezhttp://www.blogger.com/profile/10628628100970152906noreply@blogger.com