(!+/~)@i.11 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 1 3 6 10 15 21 28 36 45 55 66 1 4 10 20 35 56 84 120 165 220 286 1 5 15 35 70 126 210 330 495 715 1001 1 6 21 56 126 252 462 792 1287 2002 3003 1 7 28 84 210 462 924 1716 3003 5005 8008 1 8 36 120 330 792 1716 3432 6435 11440 19448 1 9 45 165 495 1287 3003 6435 12870 24310 43758 1 10 55 220 715 2002 5005 11440 24310 48620 92378 1 11 66 286 1001 3003 8008 19448 43758 92378 184756
Very neat! According with the first link that I provided " he [Wallis] was unacquainted with the binomial theorem." At least he knew "Pascal's triangle" as a rectangle ;) On Fri, Mar 28, 2014 at 5:10 AM, Marc Simpson <m...@0branch.com> wrote: > Simplified: > > (!+/~)i.11 > > > On Fri, Mar 28, 2014 at 9:06 AM, Pascal Jasmin <godspiral2...@yahoo.ca> > wrote: > > well done, > > > > the tacit version: > > > > (] !"1 +/~) i.11 > > > > > > > > ----- Original Message ----- > > From: Bo Jacoby <bojac...@yahoo.dk> > > To: "programm...@jsoftware.com" <programm...@jsoftware.com> > > Cc: > > Sent: Friday, March 28, 2014 3:23:06 AM > > Subject: Re: [Jprogramming] Applied APL - How to think like an APL > programmer? > > > > "his table on page 105 looks interesting. I wonder what is the shortest > J expression that can reproduce it" > > > > This one may not be the shortest, but it works: > > > > n!"1 n+/n=.i.11 > > > > > > > > > > > > > > Den 0:07 fredag den 28. marts 2014 skrev Jose Mario Quintana < > jose.mario.quint...@gmail.com>: > > > > Was Wallis himself the first to assume x^0 =1 even for x=0? See, > >> > >> > >> > >> > http://www.maths.tcd.ie/pub/HistMath/People/Wallis/RouseBall/RB_Wallis.html > >> > >> > >> > >>Perhaps a Latin fluent member of the forum could shed some light into the > >>dark: > >> > >> > >> > >>https://archive.org/details/ArithmeticaInfinitorum ? > >> > >> > >> > >>At any rate, his table on page 105 looks interesting. I wonder what is > the > >>shortest J expression that can reproduce it... :) > >> > >> > >> > >> > >> > >> > >>On Fri, Jan 17, 2014 at 4:22 PM, Roger Hui <rogerhui.can...@gmail.com > >wrote: > >> > >>> Come to think of it, the insight that 1=0^0 is required for the > standard > >>> statement of polynomials may have come from Ken Iverson. Knuth doesn't > >>> mention this point and only mentions the binomial theorem. (Same with > "Ask > >>> a Mathematician".) But the polynomial argument is more convincing > because > >>> polynomials are ubiquitous. > >>> > >>> > >>> > >>> On Fri, Jan 17, 2014 at 12:56 PM, Roger Hui <rogerhui.can...@gmail.com > >>> >wrote: > >>> > >>> > Found it. It is in the very same paper. > >>> > http://arxiv.org/PS_cache/math/pdf/9205/9205211v1.pdf . > >>> > > >>> > On page 6, Knuth wrote: > >>> > > >>> > ... The debate stopped there, apparently with the conclusion that 0^0 > >>> > should be undefined. > >>> > > >>> > But no, no, ten thousand times no! Anybody who wants the binomial > >>> theorem > >>> > ... to hold for at least one non-negative integer n _must_ before > that > >>> 0^0 > >>> > = 1, ... > >>> > > >>> > > >>> > "Ask a Mathematican" > >>> > > >>> > > >>> > http://www.askamathematician.com/2010/12/q-what-does-00-zero-raised-to-the-zeroth-power-equal-why-do-mathematicians-and-high-school-teachers-disagree/ > >>> > has an interesting and useful discussion on this issue. In it the > >>> > "Mathematician" wrote: > >>> > > >>> > Zero raised to the zero power is one. Why? Because mathematicians > said > >>> > so. No really, it's true. > >>> > > >>> > And then goes on to explain why 1=0^0 is a good idea. > >>> > > >>> > > >>> > > >>> > > >>> > > >>> > > >>> > On Fri, Jan 17, 2014 at 12:30 PM, Roger Hui < > rogerhui.can...@gmail.com > >>> >wrote: > >>> > > >>> >> BTW, Knuth did something else which typifies APL thinking. In a > note or > >>> >> paper (I can not find it now), he argued strongly that 1=0^0, not > >>> >> undefined, not 0, not anything else. The common conventional > statement > >>> of > >>> >> a polynomial, p(x)=sigma(k=0;k<=n) a[k]*x^k, requires that x^0 be 1. > >>> Some > >>> >> writers are aware of this dependency and, being careful, write > instead > >>> the > >>> >> ugly p(x)=a[0]+sigma(k=1;k<=n)a[k]*x^k. > >>> >> > >>> >> Attention to edge cases is typical of APL thinking. It's another > way to > >>> >> stay in the world of expressions and away from the world of > statements. > >>> >> You know: > >>> >> > >>> >> if k=0 then > >>> >> a[0] > >>> >> else > >>> >> a[k]*x^k > >>> >> endif > >>> >> > >>> >> > >>> >> > >>> >> > >>> >> On Wed, Jan 15, 2014 at 6:20 PM, Roger Hui < > rogerhui.can...@gmail.com > >>> >wrote: > >>> >> > >>> >>> One aspect: J/APL programmers tend to stay in the nice world of > >>> >>> expressions and avoid the nastier world of statements. This > tendency > >>> >>> pushes you towards array thinking and away from scalar thinking. > >>> >>> > >>> >>> For example, if b is a boolean array, and you want 4 where b is 0 > and > >>> 17 > >>> >>> where b is 1, write: > >>> >>> > >>> >>> (4*0=b)+(17*1=b) > >>> >>> > >>> >>> And of course the signs of real numbers x are: > >>> >>> > >>> >>> (x>0)-(x<0) > >>> >>> > >>> >>> Even Knuth, an eminent mathematician and computer scientist but > not an > >>> >>> APL programmer, knows to <strike>steal</strike> adopt this idea. > See: > >>> Knuth, > >>> >>> *Two Notes on Notation*< > >>> http://arxiv.org/PS_cache/math/pdf/9205/9205211v1.pdf>, > >>> >>> 1992-05-01. In the first half of the paper he describes how > "Iverson's > >>> >>> convention" can be used to simplify the statement and manipulation > of > >>> sums. > >>> >>> > >>> >>> See also: > >>> >>> > >>> >>> http://www.jsoftware.com/papers/perlis77.htm > >>> >>> http://www.jsoftware.com/papers/perlis78.htm > >>> >>> http://www.jsoftware.com/papers/APLQA.htm#Perlis-foreword > >>> >>> > >>> >>> > >>> >>> > >>> >>> > >>> >>> > >>> >>> On Wed, Jan 15, 2014 at 5:32 PM, Joe Bogner <joebog...@gmail.com> > >>> wrote: > >>> >>> > >>> >>>> I went googling for some deeper material on how to think like an > APL > >>> >>>> programmer. I have read/skimmed through a good set of the > material on > >>> >>>> http://jsoftware.com/papers/ and have skimmed through many of the > >>> >>>> books listed on http://www.jsoftware.com/jwiki/Books. > >>> >>>> > >>> >>>> Are there any specific recommendations, free or for purchase? Or, > >>> >>>> perhaps I should spend more time with the list above. > >>> >>>> > >>> >>>> I found this, The APL Idiom List by Perlis and Rugaber, which > looks > >>> >>>> similar to what I'm looking for: > >>> >>>> http://archive.vector.org.uk/resource/yaleidioms.pdf. > >>> >>>> > >>> >>>> The review of this book looks like what I'm after, > >>> >>>> > >>> >>>> > >>> > http://www.amazon.com/Handbook-APL-programming-Clark-Wiedmann/dp/0884050262 > >>> >>>> , > >>> >>>> constructing useful programs and going into more depth. > >>> >>>> > >>> >>>> Or something of the style of The Little Schemer, > >>> >>>> http://scottn.us/downloads/The_Little_Schemer.pdf > >>> >>>> > >>> >>>> I searched the forum and had trouble finding a relevant post > >>> >>>> > ---------------------------------------------------------------------- > >>> >>>> For information about J forums see > >>> http://www.jsoftware.com/forums.htm > > > >> > >>> >>>> > >>> >>> > >>> >>> > >>> >> > >>> > > >>> ---------------------------------------------------------------------- > >>> For information about J forums see http://www.jsoftware.com/forums.htm > >>> > >>---------------------------------------------------------------------- > >>For information about J forums see http://www.jsoftware.com/forums.htm > >> > >> > >> > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
