January 2005 - mathfun@mailman.xmission.com

[math-fun] Re: harmonic sum
by James Propp 19 Jan '05

19 Jan '05

>I've always used Harmonic Sum for xy/(x+y). >I can't say what "standard" is. >It does violate the heuristic that any kind of sum operation >should exceed its inputs. (x+y)/(1+xy) might be called the "relativistic sum" of x and y (since this is how velocities add in special relativity). Does this operation arise in other places? Note that (x+y)/(1+xy) = tanh (arctanh x + arctanh y). Jim Propp

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[math-fun] Harmonic Sum
by Richard Schroeppel 19 Jan '05

19 Jan '05

I've always used Harmonic Sum for xy/(x+y). I can't say what "standard" is. It does violate the heuristic that any kind of sum operation should exceed its inputs. Rich rcs(a)cs.arizona.edu

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Re: [math-fun] ~s harmonic sum?
by Mike Stay 19 Jan '05

19 Jan '05

RWG wrote: > What is the conventional name for 1/(1/x+1/y)? Don't know. Perhaps capacitive sum (since capacitors in series follow that addition law)? Parallel sum (since resistors in parallel follow that addition law)? The paper "Associativity of the Secant Method" by Sam Northshield refers to this sum. Given a function f, define x oplus y = ( x f(y) - y f(x) )/( f(y)-f(x) ) If f has a pole at e, then define p(x) = f(x)(x-e); otherwise p(x)=f(x). If f has a pole at e, then define z=e; otherwise z=+inf. The homomorphism F(x oplus y) = F(x)+F(y) holds if F'=1/p and F(z)=0. oplus is the capacitive sum for f(x)=x^2, and F(x)=-1/x. oplus is the nirmal sum when f(x)=1/x. oplus is multiplication when f(x) = x/(1-x), and F(x) = ln x. atan is a homomorphism for f(x) = (x^2+1)/x. etc. -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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[math-fun] ~s harmonic sum?
by R. William Gosper 19 Jan '05

19 Jan '05

What is the conventional name for 1/(1/x+1/y)? (Twice the harmonic mean.) A few dubious websites call this "harmonic sum", but most waste that term on harmonic numbers. --rwg disharmony <-> hydramnios

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Re: [math-fun] nucleotide tetrahedron
by Mark D. Niemiec 18 Jan '05

18 Jan '05

Michael Kleber <michael.kleber(a)gmail.com> wrote: > David Kephart already mentioned the basic idea that what we > really care about is strings over the alphabet {A,C,G,T}, and > that the fundamental operation on this set is the self-inverse > "reverse complement" action, the product of the double > transposition (AT)(CG) with reversal of the entire string. > This is the operation that really embodies Watson-Crick > base pairing, because (again as David already said), DNA > strands are oriented and the two strands in the double-helix > are opposite to one another, so "reverse complement" takes > one strand of your chromosome to the other. > > It's worth noting that the Watson-Crick pairing is inherently > meaningful in the DNA or RNA polymer context -- that is, as > part of a string, not a single base -- since the fact that these > guys pair with one another is a result of the precise alignment > and separation enforced by the helical backbone of the polymer. > In other words, the base pairing rules follow from the reverse- > complement operation, not the other way around! Back in the 70s, I did a lot of work with PDP-11s. They had this device called Dectape that was a magnetic tape storage medium, but designed for random access. The drives were very good at moving the tape in either direction, and reading it the direction of motion. Whenever the tape was being read in the 'backwards' direction, the tape driver did a reverse-and-complement operation on the data coming from the drive, because the bits were naturally in the wrong order, but their sense was also inverted. -- Mark D. Niemiec <mniemiec(a)interserv.com>

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Re: [math-fun] nucleotide tetrahedron
by James Propp 18 Jan '05

18 Jan '05

Cool! Thanks. Now if we could just find out who made that cardboard model, and why... Jim

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[math-fun] nucleotide tetrahedron
by James Propp 18 Jan '05

18 Jan '05

While at the Joint Winter Meetings in Atlanta, I saw a rather curious object sitting on a table: a regular tetrahedron made of white cardboard, whose sides were labelled "A", "C", "G", and "T". (Does anyone know who created it, and why?) This sighting prompted a question which is perhaps more suited to Michael Kleber than to any other person alive, since it concerns both polyhedral models (an old interest of his) and nucleotides (a newer interest of his), but since the answer may interest lots of people besides the two of us, I thought I'd ask it in this forum: What are the biologically relevant group-actions on the set {A,C,G,T}? Leaving aside the symmetric group action ("they're all necleotides, aren't they?") and the trivial group action ("yeah, but they're DIFFERENT nucleotides"), there's the 4-element group action that stabilizes the set {A,G} and the set {C,T} ("sure, but the two purines are more like each other than they are like the two pyrimidines, and vice versa"), and there's the 8-element group action that stabilizes the partition {{A,T},{C,G}} ("okay, but what really matters is that A pairs with T and C pairs with G") and there's the 4-element group action that stabilizes the sets {A,T} and {C,G} ("fine, but don't forget that the two pairs behave differently vis-a-vis transcription of DNA into RNA (the whole uracil thing)"), and ... I'd be surprised if there were some biologically relevant action of the 2-element group that leaves T and G alone but acts like Sym({A,C}) on the other two elements (to give just one example of a group-action that probably isn't biologically meaningful). But, like most of you, I like to be surprised, which is why I'm asking. Also, do any linear representations (as opposed to permutation representations) of Sym({A,C,G,T}) play a role in genomics? Jim Propp

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[math-fun] Re: nucleotide tetrahedron
by James Propp 17 Jan '05

17 Jan '05

David E. Kephart replied to my query as follows: >From dkephart(a)csee.usf.edu Mon Jan 17 11:06:29 2005 >To: propp(a)math.wisc.edu >CC: Natasha Jonoska <jonoska(a)math.usf.edu> > >James, > >I was forwarded your musing about biologically relevant (or otherwise) >group actions on DNA. I know you had polyhedral groups in mind, >originally. Now, everybody loves groups, but semi-groups have their good >points, too, you know. > >I say this because single-stranded DNA molecules can be quite naturally >modelled as elements of the free semigroup generated by {A,C,T,G} with >the antimorphic involution $\theta$, which maps A to T, T to A, C to G, >G to C. And why a semigroup? Because the nucleotide chain itself is >oriented, 5'-3', just as a word over {A,C,T,G}* has a first symbol, >second symbol, etc. > >As you might guess, I'm not just thinking this up on the spur of the >moment, but echoing some research I have been involved in. If you want >to check out more about this, and it is worth it, particularly since, >yes, the uracil thing has to be accounted for, there should be a way of >modeling the melting temperature problem, etc., some of which you refer >to, I would suggest checking out http://math.usf.edu/~jonoska, the Web >site of Natasha Jonoska here at USF and hunting through the links (some >of them are on her "Classes" page, some of them elsewhere) to find >papers by Lila Kari, et al, Natasha, Kalpana Mahalingam, and myself. Can >I put in a good word for the CodeGen DNA word-generating software >available there which I wrote, based on these ideas. > >Now if you only will be happy with group actions, then there is one >little problem when it comes to biology...what is a meaningful inverse >operation? Taking apart a DNA strand (as in $S_4$)? Dehybridizing it? >The closest thing I can think of is the various splicing operations, >with their obvious inverses, that are specific to the enzymes that act >on double-stranded DNA. With respect to that, rather than group actions, >I believe Natasha has done some work on the four-dimensional topological >implications...I can give you a reference on that, too, if you like. > >Anyhow, I would be interested to find out what other responses you may >get regarding this, and I hope that what I mentioned is of interest to you! > >Yrs, >David E. Kephart >graduate student at University of South Florida Jim

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[math-fun] What's this operation called?
by M. Stay 17 Jan '05

17 Jan '05

Given an n-dimensional vector v over a field K, and a basis such that all the components are nonzero. (Or over a ring R, and all the components are units.) Take the componentwise inverse to get v'; next, multiply componentwise by a nonzero row of the n-dimensional FFT to get v''. Has anyone seen this before? Does it have a name? -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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[math-fun] Re: nucleotide tetrahedron
by James Propp 17 Jan '05

17 Jan '05

Dave Halitsky replied to my query as follows: >From dhalitsky(a)www.cumulativeinquiry.com Mon Jan 17 11:02:21 2005 >To: <propp(a)math.wisc.edu> >Subject: In regard the symmetries of the bases, please note > >that (t,g) vs (a,c) is determined by the keto vs amino distinction >which is sufficiently well-known to be enshrined in IUPAC notation >as > >E = keto = t,g >M = amino = a,c > >You may be interested in references discussion at > >http://www.cumulativeinquiry.com/Problems >http://www.cumulativeinquiry.com/Forums > >Best regards >Dave Halitsky >256-426-6243 I responded: >Thanks! May I forward your reply to others? > >Also: I looked at the links you gave me, but couldn't find the relevant >thread. He replied: >From dhalitsky(a)www.cumulativeinquiry.com Mon Jan 17 11:40:27 2005 >To: "'James Propp'" <propp(a)math.wisc.edu> >Subject: 1) by all means; 2) relevant thread > >Jim - > >Please feel free to do whatever you like with the information. >It is good to see the question arising independently of the >CumulativeInquiry research team's investigations. > >Also, at the link: > >http://www.cumulativeinquiry.com/Problems > >under the heading: > >The "oDAG/Bioinformatic Sequence" Problem > >the following set of links appears: > >Original CI Posting to sci.math Re Ordered Bioinformatic Sequences >G. Sterten's Questions Concerning Above sci.math Posting >CI's Responses to G. Sterten's Questions >Note from C. Brown re building of oDAGs from sequences over {s,p,d,t} (or >{++,-+,--,+-}) >Bill Mann's PERL program "problist" for expected probabilities of sequences >building half-turn invariant oDAGs >Notes from Bill Mann on problist program >1994 Halitsky MathBioSci paper; relevant only for a characterization of the >four D(mR)NA bases via 2 binary biochemical features) > >The last link is to a 1994 paper which builds specifically >on the pyrimidine:purine and keto:amino distinction; the other >links can be read in order as a discussion of the more general >issue in which you are interested. > >You may also wish to see the somewhat obscure development of some >related ideas at the URL: > >http://www.cumulativeinquiry.com/services_products/index.html > >under the link: > >Technical documentation (PDF format) of the PABP protocol. > >Finally, please note that the (t,a) vs (c,g) opposition >is often referred to as: > >W(eak) = t,a >S(trong) = c,g > >because c,g pair canonically with three hydrogen bonds and >a,t pair canonically with two hydrogen bonds, respectively >"strongly" and "weakly". My co-researcher Jacques Fresco at >Princeton will be more than happy to discuss the details of this >classification. > >Dave Jim

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