"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
