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[math-fun] non-adjacent :: final + insight
by Meeussen Wouter (bkarnd) 10 Mar '03

10 Mar '03

a chessboard shows it all in an instant : (just found it, thanks COS) http://www.theory.csc.uvic.ca/~cos/inf/setp/SetPartitions.html sorry to have bothered you all, Wouter (Blush...) =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

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[math-fun] non-adjacent, again ...
by Meeussen Wouter (bkarnd) 10 Mar '03

10 Mar '03

Set Partitions excluding singletons (sets of size one) are counted by N. Sloane and Christian G. Bower (bowerc(a)usa.net) in A000296 (Formerly M3423 and N1387) Sequence : 1, 0, 1, 1, 4, 11, 41, 162, 715, 3425, 17722, 98253 (with Bell[n]=a[n-1]+a[n-2] ) But, Set Partitions excluding neighbours (k excludes k+/-1 in partition) where 1 is taken to neighbour n (making it cyclic) are also counted by A000296. Anyone for a translation from one to the other? no_singletons & cyclic_neighbours_excluded {{{1, 2, 3, 4, 5}} & {{1}, {2, 4}, {3, 5}}}, {{{1, 2}, {3, 4, 5}} & {{1, 4}, {2}, {3, 5}}}, {{{1, 2, 3}, {4, 5}} & {{1, 3}, {2, 4}, {5}}}, {{{1, 4, 5}, {2, 3}} & {{1, 3}, {2, 5}, {4}}}, {{{1, 3}, {2, 4, 5}} & {{1, 4}, {2, 5}, {3}}}, {{{1, 5}, {2, 3, 4}} & {{1}, {2}, {3, 5}, {4}}}, {{{1, 2, 5}, {3, 4}} & {{1}, {2, 4}, {3}, {5}}}, {{{1, 3, 4}, {2, 5}} & {{1}, {2, 5}, {3}, {4}}}, {{{1, 4}, {2, 3, 5}} & {{1, 3}, {2}, {4}, {5}}}, {{{1, 2, 4}, {3, 5}} & {{1, 4}, {2}, {3}, {5}}}, {{{1, 3, 5}, {2, 4}} & {{1}, {2}, {3}, {4}, {5}}} Playfully yours, Wouter =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

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[math-fun] A near vector space - example
by Guy Haworth 10 Mar '03

10 Mar '03

To answer my own question, Noam Elkies suggested: K = C(omplex Numbers); one-dimensional V = K Define k * [v] = [Re(k).v], the . being multiplication in K Then all axioms are satisfied except (k1.k2) * v = k1 * (k2 * v) Are there sources on the web of neat examples like this where the independence of the axioms is demonstrated? I know of Gelbaum/Olmstead books I (Analysis) & II. - guy

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Re: [math-fun] Why is e the "best" base? + A symmetrical way to treat digits 0,1,2 base e
by Henry Baker 09 Mar '03

09 Mar '03

Here's the original paper, with corrections: http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html At 09:23 AM 3/7/03 -0800, Marc LeBrun wrote: >I'll try to dig out my copy of Shannon's original paper and see what he said.

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[math-fun] Why is e the "best" base? + A symmetrical way to treat digits 0,1,2 base e
by asimovd＠aol.com 07 Mar '03

07 Mar '03

I forget who mentioned that e was found to be the "best" number base. I would like to know in what sense it is best -- or is this just some kind of offhand, non-rigorous comment? I chose to look at e simply because it seems to be the math constant (other than 2) whose powers (not necessary integer) arise most commonly. There is the old chestnut that (modulo details) if we start with a real number K >> 0 and wish to break K up into the sum of positive pieces with maximum product, this is attained by making each piece = e. (This is exactly true if K = ne, but needs some modification otherwise, of course.) Is this relevant here? ------------------------------------------------------------------------------ ---------------------------------------- ALSO: There's something unappealing about the asymmetry inherent in using the greedy algorithm to represent a number to (say) base e, since 0 and 1 play a role different from the other digit, 2. One way to avoid this would be as follows. Suppose we want to represent K > 0 to the base e, and suppose that e^r is the largest integer power of e that does not exceed K. Hence K = c (e^r) where 0 <= c < e. Then we use as highest-order digit d = 0, 1, 2 according as 0 <= c < e/3; e/3 <= c < 2e/3; 2e/3 <= c < e, or in brief, d = [3c/e]. (Of course, the calculation of digits continues by applying the above to the quantity K - d(e^r), now using e^(r-1) in place of e^r, ad infinitum.) I wonder if anything nifty comes from treating the digits in this symmetrical fashion. --Dan

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RE: [math-fun] Why is e the "best" base? + A symmetrical way to treat digits 0,1,2 base e
by Marc LeBrun 07 Mar '03

07 Mar '03

>=Andy Latto > You get the quantity -ln(p) of information from a message that > occurs with probability p, but you only get that information > with probability p, so the expected amount of information > received is the sum of -p(ln(p)) over all possible messages > you might receive. Andy, thanks for the clarification of the provenance of this formula. So this sum measures the average information delivered by the channel, versus that of a single message. But the maximum of such a sum isn't necessarily the maximum of the summand. Applying the above by viewing a number as a stream of digits: for a given base b we sum over the b possible "messages", each of which (presumably) occurs with equal probability p=1/b, giving b (-1/b ln(1/b)) = -ln(1/b) = ln(b) which says the larger the base the higher the information, not that the maximum occurs when b=e. What's wrong with this picture?

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[math-fun] New topic: a near vector space please
by Guy Haworth 07 Mar '03

07 Mar '03

Is there a simple example of a structure that satisfies every vector space property except: k1 * (k2 @ V) = (k1 * k2) @ V k1, k2 in the underlying field K; @ == Field/Vector operation TIA - Guy

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Re: [math-fun] Why is e the "best" base?
by asimovd＠aol.com 07 Mar '03

07 Mar '03

I don't know much about information entropy E , but I do know that it's usually defined as a function of a *partition* of the unit interval into disjoint subintervals, aka the probaiblities p_i of a discrete probability measure. So E is a function of the collection (multiset) of values P = {p_1,p_2,p_3,...} and is defined via E(P) = sum over i of (-p_i log(p_i)) Among partitions of size n, the maximum-entropy partition P_M(n) is into equal intervals, giving E(P_M(n)) = log(n). --Dan

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[math-fun] Question
by David Wilson 07 Mar '03

07 Mar '03

Let p be a finite sequence of positive integers, WLOG nondecreasing. Call p "good" (for lack of a better term), if for each p(i) there exists 0 < q(i) < p(i) such that for every integer n, n == q(i) (mod p(i)) for some i. For example, let p = (3, 3, 6, 6), and let q = (0, 1, 2, 5). Since every integer n is of the form 0 (mod 3), 1 (mod 3), 2 (mod 6) or 5 (mod 6), p is good. Clearly, for p to be good, it is necessary that SUM 1/p(i) >= 1, but this is not sufficient. Is there an efficient method to tell if a given p is good?

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RE: [math-fun] Why is e the "best" base? + A symmetrical way to treat digits 0,1,2 base e
by Andy Latto 07 Mar '03

07 Mar '03

> -----Original Message----- > From: Marc LeBrun [mailto:mlb@fxpt.com] > Sent: Friday, March 07, 2003 12:24 PM > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] Why is e the "best" base? + A > symmetrical way to > treat digits 0,1,2 base e > > > >=mcintosh(a)servidor.unam.mx > > Entropy is p ln(p) summed over alternatives. It vanishes for > > p = 0 and p = 1, with a maximum somewhere inbetween, say > at p = 1/e. > > An interesting thing about this formula is that the maximum > is independent > of the base used for the logarithm. > > However, are you sure about that p factor? I thought the information > content of a message was, roughly, how "surprising" it was, > hence simply > -ln(p) (which is why information gets called "negative > entropy"). Getting > an impossible message would be a miraculous epiphany, > containing infinite > information (alas of the form "everything you know is wrong!"<;-) You get the quantity -ln(p) of information from a message that occurs with probability p, but you only get that information with probability p, so the expected amount of information received is the sum of -p(ln(p)) over all possible messages you might receive. Andy Latto andy.latto(a)pobox.com

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