March 2011 - mathfun@mailman.xmission.com

Re: [math-fun] Puzzle: Skew planes in 4-space?
by Tom Karzes 21 Mar '11

21 Mar '11

I'd say yes. In general, you can have two skew (n-2)-dimensional "planes" in n-space. In 3-space you can have two skew lines, in 4-space you can have two skew planes, etc. For lines in 3-space, coordinates (x1, x2, x3): Line 1: x1 = 0, x2 = 0, x3 = free Line 2: x1 = 1, x2 = free, x3 = 0 For planes in 4-space, coordinates (x1, x2, x3, x4): Plane 1: x1 = 0, x2 = 0, x3 = free, x4 = free Plane 2: x1 = 1, x2 = free, x3 = 0, x4 = free For higher dimensions, just keep adding more xi = free instances as needed. The x1 restriction prevents them from intersecting, and the x2/x3 restrictions prevent them from being parallel. Tom > Fred's query reminds me of the time in 1994 that I gave a lecture -- about > 4-dimensional space -- to the Hampshire College summer math program for > mathematically talented high school students. > > Some brilliant youngster asked me whether there could be skew (2D) planes > in 4-space. (Skew meaning neither parallel nor intersecting.) I thought > about it for a few seconds . . . and then a few more seconds . . . and had > to admit I didn't know. > > But I went home and soon figured it out. Can you? > > --Dan > > ________________________________________________________________________________________ > "Outside of a dog, a book is man's best friend. Inside of a dog, it's too > dark to read." --Groucho Marx > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > >

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Re: [math-fun] perpendicular planes
by Dan Asimov 19 Mar '11

19 Mar '11

Let's see: We assume that L, M do not intersect. Then there are unique points P on L and Q on M such that the distance ||P-Q|| is the least distance between any point of L and any point of M. Then the segment PQ must be perpendicular to both L and M (and conversely, and segment from L to M that's perpendicular to both lines has the least distance between them). Such a segment us unique. Then the planes mentioned below are the planes U through L and PQ, V through M and PQ, S through L and perpendicular to L, and T through M perpendicular to M. These can be found analytically by letting L: {y0 + sy} and M: {z0 + tz} for fixed vectors y0, y, z0, z and employing standard vector operations. (E.g., if w is an arbitrary segment from L to M, and r is the unique (up to +-) unit vector perpendicular to L and M, given by r = y x z, then R := <r,w> r is a vector whose length is the least distance from L to M in the direction of segment PQ, etc. --Dan Fred asks: << Given non-parallel lines L,M in Euclidean 3-space, there exist unique planes S,U meeting in L, and T,V in M, with both S,T perpendicular to both U,V. An algebraic proof of this turned out to be surprisingly nontrivial --- is there a more intuitive synthetic demonstration? >> ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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[math-fun] perpendicular planes
by Fred lunnon 19 Mar '11

19 Mar '11

Given non-parallel lines L,M in Euclidean 3-space, there exist unique planes S,U meeting in L, and T,V in M, with both S,T perpendicular to both U,V. An algebraic proof of this turned out to be surprisingly nontrivial --- is there a more intuitive synthetic demonstration? [The appropriate generalisation of this this apparently tedious little lemma provides one crucial link in the classification of isometries for a quadratic inner-product space. It may well be a (special case of) some well-known theorem concerning bases of vector subspaces, though I didn't recognise it as such.] Fred Lunnon

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[math-fun] Simplifying IF
by rcs＠xmission.com 18 Mar '11

18 Mar '11

Here's a Logic & Programming challenge: Simplify IF expressions. The general problem is NP-complete, since nested IF expressions can easily encompass Satisfiability problems. It might be even harder, since simplifying an IF expression down to True or False (when applicable) would require a short (or at least feasible) proof of equivalence, while SAT requires only an example. Nevertheless, it's reasonable to ask for partial solutions. The "simpler-than" ordering includes a lot of beholder input, and also depends on the application at hand. I'll offer a few "rules" to start the bidding. If(A,B,C) with A a predicate, B and C arbitrary types (usually the same), has the value B when A is true, and C when A is false. Other notations include A?B:C, Choose(A,B,C), Boole(A,C,B), with Choose and Boole sometimes interpreted per individual bit. If(True,B,C) -> B If(False,B,C) -> C If(A,B,B) -> B If(A,B,~B) -> A xor B (B a predicate) If(A,B,C) <-> If(~A,C,B) If(A,B,C) <-> If(A, A&B, (~A)&C ) If(A,B,C) <-> B + If(A,0,C-B) (B and C numerical) If(A,B&C,BvC) <-> Majority(A,B,C) (B and C predicates) If(A<B,A,B) <-> Min(A,B) If(A&B,C,D) <-> If(A,If(B,C,D),D) A similar challenge can be made for Loop expressions. We've adopted a convention for some binary operations, to represent a variable-length version as a kind of summation, using a larger or boldface version of the operator, with upper & lower limits specifying a range, and an expression for argument_k. Would it be useful to change to a generic Loop expression? Most of the simple manipulations for sums and products are applicable to Loops too. Rich

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[math-fun] prodigal and quiverative
by Bill Gosper 18 Mar '11

18 Mar '11

I was juat talking to the kids about this on Tuesday (apropos "What comes after Plus and Tmes?"). Mathematica's math palette includes extinct Greek characters like curly rho, koppa, stigma, digamma, and sampi. There's also Hebrew, e.g. Qoph. An oversize lowercase rho makes a decent prodigal. --rwg On Fri, Mar 18, 2011 at 3:05 PM, Marc LeBrun <mlb(a)well.com> wrote: > [just some of us] > > >="quad" <quadricode(a)gmail.com> > > I love mathematical typography, with a passion. > > > Yay! Double geek score! > > Me too, but mainly as a spectator sport 'cause the tools are so painful. > > I've long wanted to be able to typeset some silly notation of mine for the > "product integral" operator P and its inverse q: > > P f(x) ^ qx := exp integral ln f(x) dx > > (ie superscript qx) analogous to the everyday "Leibniz" notation for > integrals and derivatives: > > P would look like an integral sign with the top continuing to loop around > and joining back to the stem--a stretchy "P" like how the integral sign is > a > stretchy "S". > > q might just be a regular or boldface q, analogous to d, but there also > should be a "curly q" (ha ha), analogous to the partial derivative symbol. > > BTW I also like to call P the "prodigal", but I've never found a good name > for the q operator--"quiverative"? I dunno. > > It might also make sense to adopt the Greek rho in some form instead of q, > since the operator is to ratios like what delta is to differences... > > Anyway, could you actually make a typeface with such symbols in it? > Perhaps > even teach TeX how to use it? Or maybe even just MS Word? I love the > look, > but I gave up trying to actually use Metafont long ago... > > >

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Re: [math-fun] all the results
by N. J. A. Sloane 18 Mar '11

18 Mar '11

Simon Plouffe said: > Hello Mr Gosper, > Why you do not publish all of your results ? Me: I support this! The math-fun archive would have all of Bill's posted messages. Perhaps someone could help Bill edit them. Best regards Neil

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Re: [math-fun] symbolic algebra handling of (+/-) signs
by Henry Baker 16 Mar '11

16 Mar '11

Hmmm... In Maxima 5.19.1 (8/17/2009 -- I haven't updated to the latest version in a while), I find that trigsign is 'true' by default, but it doesn't work. The documentation sez trigsign allows asinh(-x) to become -asinh(x), which works, but the Maxima simplifier apparently doesn't recognize signum() as a sign function suitable for movement. I guess Maxima needs some work. I'll download the latest Maxima & see if this has been fixed. Thanks! [Of course, this doesn't provide the complete answer to my question, but it will help.] At 04:06 PM 3/15/2011, Bill Gosper wrote: >Over the years, I've had occasion to use Macsyma/Maxima to produce formulae that >were then utilized in my computer code (Lisp/C/...). While these symbolic algebra >tools are extremely useful, there is one area where symbolic algebra systems could >be improved quite a lot -- the handling of signs (e.g., +/-). > >While Maxima does a pretty good job of understanding the absolute value function >('abs'), it doesn't seem to know very much about the intimately related "sign" >('signmum') function. For example, although 'asinh' is a perfectly good odd >function, Maxima doesn't know how to move 'signum' in and out of the 'asinh' >function. > >Dealing with 'signum' in a high quality way should be considerably easier than >attempting to handle 'if-then-else' expressions (aka "()?:" expressions in C/C++, >etc.). But there are occasions where symbolic algebra systems would be extremely >helpful if they could handle such if-then-else expressions. The alternative is >pretty ugly: it may be necessary to create a software "harness" and >then evaluate >the alternative expressions with every conceivable boolean combination >-- a strategy >that quickly blows up exponentially, but it is still better if this strategy is >implemented mechanically, to make sure that all combinations are checked. > >I haven't had a lot of experience with other symbolic algebra systems. Do Maple & >Mathematica (or any others) do a better job of handling these signum and/or >if-then-else expressions? > >Alternatively, has anyone here had any experience in teaching the Macsyma/Maxima >simplifier about such expressions, so that I could expand its capabilities ? > >In Macsyma 2.4, > >(c16) TRIGSIGN; > >(d16) true > >(c17) ASINH(SIGNUM(A)*B); > >(d17) signum(a) asinh(b) > >(c18) BLOCK([TRIGSIGN : FALSE],ASINH(SIGNUM(A)*B)); > >(d18) asinh(signum(a) b) > >(c19) ASINH(SIGNUM(A)*B*(-1)^INTEGER); > > integer >(d19) signum(a) asinh(b) (- 1) > >(c20) MAKELIST(THETA[K](-X,Q),K,1,4); > >(d20) [- theta (x, q), theta (x, q), theta (x, q), theta (x, q)] > 1 2 3 4 > >with similar control by trigsign. I actually saw a proposal to remove trigsign from Maxima. >I wonder if those people know what they're doing. (They should maybe collect a bunch of > Macsyma code and notebooks, and just string search to see if anybody uses such-and-such > symbol. I found 14 *conversational* uses of trigsign in my .mfes.) >--rwg

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[math-fun] A000041, the partitions : the classical example of p(200), macmahon, hardy and ramanujan
by Bill Gosper 16 Mar '11

16 Mar '11

Simon Plouffe>Hello, I have been musing around the expansion of certain numbers in the exp(Pi) base. Some results are I think interesting. If I use F(x) = product(1/(1-x^n),n=1..infinity), then when x -> exp(-Pi*x) the value for F(10) is easily found to be 1/2 1/2 GAMMA(3/4) (5 + 5 5 ) -------------------------- = F(10). 1/4 5 Pi Pi exp(----) 12 and lprinted : F(10) = GAMMA(3/4)*(5+5*5^(1/2))^(1/2)/(Pi^(1/4)*exp(5/12*Pi)) If I expand this number in base exp(Pi*10) then the first 205 coefficients can be obtained. In other words, I can compute p(n) for up to n=205 using 1 mathematical constant and enough precision. (rwg)>Clever! SP> note : all the coefficients from n=1 to n=205 are computed at once. The <enough precision> is 2800 digits, unfortunately. I can only think at the Major Percy MacMahon that took 6 months to compute p(200) by hand and later Hardy and Ramanujan, using a very clever method to get that number! The method I use is not very clever, I only use a numerical trick, but it does work. With the base exp(Pi*16) I could obtain the algebraic value also and get : F(16) = 2*2^(3/4)*(560+396*2^(1/2)+3*(69708+49291*2^(1/2))^(1/2))^(1/8)*GAMMA(3/4)/Pi^(1/4)/exp(2/3*Pi) (rwg)>F(16) = GAMMA(3/4)*2^(57/32)*%E^-(Pi*2/3)/(Pi^(1/4)*(-3*2^(1/2)-4*2^(1/4)+9)^(1/8)) Also, F(24) = 2^(25/48)*3^(3/8)*((3^(1/4)+SQRT(2))*SQRT(3)+3^(1/4)+2)*Pi^(3/4)/ (exp(Pi)*(SQRT(2)-1)^(3/4)*(SQRT(3)-1)^(1/6)*GAMMA(1/4)) isn't too bad. SP>well, by using 11000 digits, I can get the first 500 values, not bad. NOTE 2) the computation of F(x) when x is small is more suitable when I use the infinite product, it converges quite fast, As fast as F(x) = 1/SUM((-1)^n*(1-exp(-(2*n+1)*Pi*x))*exp(-n*(3*n+1)*Pi*x/2),n,0,INF) ? it is nice to have the exact value of F(x) but faster to use the classical formula. Best regards, Simon Plouffe (rwg)>If you use a series or product formula, you can use F(arbitrarily large).

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[math-fun] symbolic algebra handling of (+/-) signs
by Bill Gosper 15 Mar '11

15 Mar '11

Over the years, I've had occasion to use Macsyma/Maxima to produce formulae that were then utilized in my computer code (Lisp/C/...). While these symbolic algebra tools are extremely useful, there is one area where symbolic algebra systems could be improved quite a lot -- the handling of signs (e.g., +/-). While Maxima does a pretty good job of understanding the absolute value function ('abs'), it doesn't seem to know very much about the intimately related "sign" ('signmum') function. For example, although 'asinh' is a perfectly good odd function, Maxima doesn't know how to move 'signum' in and out of the 'asinh' function. Dealing with 'signum' in a high quality way should be considerably easier than attempting to handle 'if-then-else' expressions (aka "()?:" expressions in C/C++, etc.). But there are occasions where symbolic algebra systems would be extremely helpful if they could handle such if-then-else expressions. The alternative is pretty ugly: it may be necessary to create a software "harness" and then evaluate the alternative expressions with every conceivable boolean combination -- a strategy that quickly blows up exponentially, but it is still better if this strategy is implemented mechanically, to make sure that all combinations are checked. I haven't had a lot of experience with other symbolic algebra systems. Do Maple & Mathematica (or any others) do a better job of handling these signum and/or if-then-else expressions? Alternatively, has anyone here had any experience in teaching the Macsyma/Maxima simplifier about such expressions, so that I could expand its capabilities ? In Macsyma 2.4, (c16) TRIGSIGN; (d16) true (c17) ASINH(SIGNUM(A)*B); (d17) signum(a) asinh(b) (c18) BLOCK([TRIGSIGN : FALSE],ASINH(SIGNUM(A)*B)); (d18) asinh(signum(a) b) (c19) ASINH(SIGNUM(A)*B*(-1)^INTEGER); integer (d19) signum(a) asinh(b) (- 1) (c20) MAKELIST(THETA[K](-X,Q),K,1,4); (d20) [- theta (x, q), theta (x, q), theta (x, q), theta (x, q)] 1 2 3 4 with similar control by trigsign. I actually saw a proposal to remove trigsign from Maxima. I wonder if those people know what they're doing. (They should maybe collect a bunch of Macsyma code and notebooks, and just string search to see if anybody uses such-and-such symbol. I found 14 *conversational* uses of trigsign in my .mfes.) --rwg

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Re: [math-fun] Factorials in non-hypergeometric sums
by Dan Asimov 15 Mar '11

15 Mar '11

Jim wrote: << I remember the number-theorist Michael Larsen as a teen-ager (back when I was a teen-ager too), sharing with me his excitement over his graduating from rediscovering nineteenth-century theorems to rediscovering twentieth- century theorems >> Ha! --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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