Is _Brave New World_ really science fiction?
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From: Christopher Henrich on 3 Apr 2010 14:59 In article <967bdc68-5d20-4a0b-854b-5964d8897e33(a)l25g2000yqd.googlegroups.com>, Butch Malahide <fred.galvin(a)gmail.com> wrote: > On Apr 2, 10:36�am, Gene Wirchenko <ge...(a)ocis.net> wrote: > > > >"discrete mathematics" means (among other things such as graph theory, > > >finite geometry, etc.) above all *counting*, i.e., providing exact or > > >asymptotic answers to such puzzles as "how many ways can you rank- > > >order n things, if ties are permitted (e.g. 13 ways if n = 3 but we > > >are interested in general n); and for that it is really useful to have > > > > � � �Yes, of course. > > > > >a working knowledge of derivatives and integrals, infinite series, > > >elementary differential equations, and elementary complex variables, > > >especially contour integrals and the method of residues. > > > > � � �What on earth for? > > Because the answer to the question "how many ways can you rank-order n > things if ties are permitted" (a fairly typical counting problem) is > equal to the value at x = 0 of the nth derivative of the function > f(x) = 1/(2 - e^x). This can be handled with "formal derivatives," "formal power series," and so on, without any reference to epsilon, delta, tangents to curves, areas under curves, or the other stuff of "calculus." Until something strange happens and you discover a need for estimating a contour integral or something irremediably analytic like that. I am cross-posting this to sci.math, which needs a shot in the arm. -- Christopher J. Henrich chenrich(a)monmouth.com http://www.mathinteract.com "A bad analogy is like a leaky screwdriver." -- Boon
From: Butch Malahide on 3 Apr 2010 16:14 On Apr 3, 1:59 pm, Christopher Henrich <chenr...(a)monmouth.com> wrote: > In article > <967bdc68-5d20-4a0b-854b-5964d8897...(a)l25g2000yqd.googlegroups.com>, > Butch Malahide wrote: > > On Apr 2, 10:36 am, Gene Wirchenko <ge...(a)ocis.net> wrote: > > > > >"discrete mathematics" means (among other things such as graph theory, > > > >finite geometry, etc.) above all *counting*, i.e., providing exact or > > > >asymptotic answers to such puzzles as "how many ways can you rank- > > > >order n things, if ties are permitted (e.g. 13 ways if n = 3 but we > > > >are interested in general n); and for that it is really useful to have > > > > Yes, of course. > > > > >a working knowledge of derivatives and integrals, infinite series, > > > >elementary differential equations, and elementary complex variables, > > > >especially contour integrals and the method of residues. > > > > What on earth for? > > > Because the answer to the question "how many ways can you rank-order n > > things if ties are permitted" (a fairly typical counting problem) is > > equal to the value at x = 0 of the nth derivative of the function > > f(x) = 1/(2 - e^x). > > This can be handled with "formal derivatives," "formal power series," > and so on, without any reference to epsilon, delta, tangents to curves, > areas under curves, or the other stuff of "calculus." Yes (and you must be a mathematician), but: 1. That "formal" stuff *is* calculus to most undergraduates, especially the ones who complain about having to take calculus; the epsilons and deltas mostly just bounce off harmlessly. 2. While generatingfunctionology *can* be handled with "formal" calculus, it seems to me that e.g. the differentiation and integration rules are more easily motivated and absorbed in the setting of slopes, velocities, and areas. Maybe that's just me. > Until something strange happens and you discover a need for estimating a > contour integral or something irremediably analytic like that. Let a_n be the answer to the counting problem I mentioned above. I already posted the exact answer in terms of power series coefficients. Now, please find a simple asymptotic expression for a_n; let me know if anything strange happens! (Reference: Lovasz, _Combinatorial Problems and Exercises_, Chapter 1.)
From: Butch Malahide on 3 Apr 2010 16:19 On Apr 3, 3:14 pm, Butch Malahide <fred.gal...(a)gmail.com> wrote: > On Apr 3, 1:59 pm, Christopher Henrich <chenr...(a)monmouth.com> wrote: > > > In article > > <967bdc68-5d20-4a0b-854b-5964d8897...(a)l25g2000yqd.googlegroups.com>, > > Butch Malahide wrote: > > > On Apr 2, 10:36 am, Gene Wirchenko <ge...(a)ocis.net> wrote: [NOT] > > > > > >"discrete mathematics" means (among other things such as graph theory, > > > > >finite geometry, etc.) above all *counting*, i.e., providing exact or > > > > >asymptotic answers to such puzzles as "how many ways can you rank- > > > > >order n things, if ties are permitted (e.g. 13 ways if n = 3 but we > > > > >are interested in general n); and for that it is really useful to have Gene Wirchenko didn't write that, I did. Many apologies for my slipshod editing.
From: Butch Malahide on 3 Apr 2010 16:31 On Apr 3, 1:59 pm, Christopher Henrich <chenr...(a)monmouth.com> wrote: > > I am cross-posting this to sci.math, which needs a shot in the arm. But you should have changed the subject to something like "calculus for discrete mathematicians"; as it is, the sci.math people are probably going to ignore this, another cross-posted thread with an off- topic title.
From: Sea Wasp (Ryk E. Spoor) on 3 Apr 2010 16:55
Butch Malahide wrote: > On Apr 3, 1:59 pm, Christopher Henrich <chenr...(a)monmouth.com> wrote: >> I am cross-posting this to sci.math, which needs a shot in the arm. > > But you should have changed the subject to something like "calculus > for discrete mathematicians"; as it is, the sci.math people are > probably going to ignore this, another cross-posted thread with an off- > topic title. Or maybe for indiscrete mathematicians? -- Sea Wasp /^\ ;;; Live Journal: http://seawasp.livejournal.com |