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From: SourceForge.net <noreply@so...>  20030822 05:48:56

Bugs item #792514, was opened at 20030821 10:06 Message generated for change (Tracker Item Submitted) made by Item Submitter You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=792514&group_id=4933 Category: None Group: None Status: Open Resolution: None Priority: 5 Submitted By: Stavros Macrakis (macrakis) Assigned to: Nobody/Anonymous (nobody) Summary: Subscripted literal array doesn't display Initial Comment: matrix([a,b],[c,d])[2,1] is a perfectly legitimate Maxima expression which evaluates to 'c'. And it displays fine with display2d:false: display2d:false$ '( matrix([a,b],[c,d])[2,1] ); => matrix([a,b],[c,d])[2,1] But 2d display causes an error: display2d:true$ '( matrix([a,b],[c,d])[2,1] ) => Error: (DMATRIX RIGHT 2 ...) is not of type CHARACTER. Error signalled by DIMENSIONARRAY. Maxima 5.9.0 gcl 2.5.0 mingw32 Windows2000 Athlon  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=792514&group_id=4933 
From: SourceForge.net <noreply@so...>  20030822 04:16:07

Bugs item #781657, was opened at 20030801 20:30 Message generated for change (Comment added) made by wjenkner You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=781657&group_id=4933 Category: None Group: None Status: Open Resolution: None Priority: 5 Submitted By: Barton Willis (willisb) Assigned to: Nobody/Anonymous (nobody) Summary: binomial(x,x) => 1, but binomial(1,1) => 0 Initial Comment: binomial(x,x) simplifies to 1 yet binomial(1,1) simplifies to 0. (C1) binomial(x,x); (D1) 1 (C2) binomial(1,1); (D2) 0 I agree with (d2) because: 1/(1  x) = 1 + x + x^2 + ... = sum(binomial(1,k) (x) ^k,k,0,inf) implies binomial(1,k) = (1)^k, for integers k >= 0. In the recursion relation [Knuth Vol 1, 1.2.6 Eq. (20)] binomial(r,k) = binomial(r1,k) + binomial(r1,k1) set r > 0, k > 0, and use binomial(0,0) = 1 and binomial (1,0) = 1. From this we get binomial(1,1) = 0. Also see, Knuth Vol. 1 (third edition), Section 1.2.6 Exercise 9. There may be other approaches, but I think using the recursion relations and other identities to extend the domain of the binomial function is the best method. In short, I think the simplification binomial(x,x) ==> 1 should happen only for real x with x >= 0. Barton  >Comment By: Wolfgang Jenkner (wjenkner) Date: 20030822 06:16 Message: Logged In: YES user_id=581700 Unless I am mistaken, the (finite) limit of BINOMIAL(x,y) at some latticepoint (a,b) in Z^2 exists if and only if a >= 0. The "if" part is easy: Just write the binomial as GAMMA(x+1)/(GAMMA(y+x+1)*GAMMA(y+1)) and observe that Gamma is certainly continuous in the open right halfplane while 1/Gamma is continuous everywhere. Now assume a <= 1. We have the following identity (for the proof see the Maxima code below) %PI*BINOMIAL(x1,y)*BINOMIAL(x,y)*y = SIN(%PI*y)*SIN(%PI*(xy))/SIN(%PI*x) We have a1 >= 0, so we already know that lim BINOMIAL(x1,y) for (x,y)>(a,b) exists. If lim BINOMIAL(x,y) also existed, lim of the whole left hand side expression of this identity would exist. Now, looking at the right hand side expression we observe that it is Zperiodic with respect to x and y (except for a possible sign change). So lim of it at (a,b) exists if and only if lim of it at (0,0) exists, which is clearly not the case. Note: Strictly speaking, the identity is valid on, say, the complement of the union of the parallels through latticepoints to the first and second axis and to the median of the first quadrant, but this leaves us with enough space :) Nevertheless, of course, it's quite customary to define BINOMIAL(a,b)=a*(a1)* ... *(ab+1)/b! for all a in R and b in Z with b >= 0 (for b=0 the numerator is an empty product), in accordance with the binomial series. This definition happens to coincide with limit(limit(BINOMIAL(x,y),y,b),x,a). matchdeclare([%%u,%%v],true)$ sum_is_1(u,v):=is(u+v = 1)$ let(gamma(%%u)*gamma(%%v),%pi/sin(%pi*%%u),sum_is_1,%%u,%%v); let(%%v/gamma(%%u),1/gamma(%%v),sum_is_1,%%u,%%v); letrat:true; lhs:%pi*y*binomial(x,y)*binomial(x1,y); makegamma(%); letsimp(%); letsimp(%); num(%)/trigexpand(expand(denom(%))); lhs=%;  Comment By: Stavros Macrakis (macrakis) Date: 20030814 18:16 Message: Logged In: YES user_id=588346 I am not sure what you mean by "there may be other approaches". binomial(a,a) should simplify to Q if and only if the double limit exists: limit binomial(x,y) = Q x>a y>a A necessary (but in general not sufficient) condition for this to exist is that the two single limits exist and are equal: limit binomial(a,y) = limit binomial(x,a) = Q y>a x>a If the limit is not welldefined, then binomial(x,x) is not well defined, and it will cause incorrect results in some case or another to arbitrarily set it to some value. I don't know if this limit is or isn't welldefined. I do know that depending on identities with unspecified domains of validity is dangerous....  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=781657&group_id=4933 
From: SourceForge.net <noreply@so...>  20030822 00:17:04

Bugs item #792862, was opened at 20030821 20:17 Message generated for change (Tracker Item Submitted) made by Item Submitter You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=792862&group_id=4933 Category: None Group: None Status: Open Resolution: None Priority: 5 Submitted By: Stavros Macrakis (macrakis) Assigned to: Nobody/Anonymous (nobody) Summary: factor(r^23*r+3,a^3+1) fatal error Initial Comment: factor(r^23*r+3,a^3+1) Error: Caught fatal error [memory may be damaged] Error signalled by PFACTORALG1. (should be irreducible) This happens regardless of the setting of GCD. Note that various closeby cases work just fine: ..., a^2+3 (factors) ..., a^2+2 (irreducible) ..., a^2+1 (irreducible) ..., a^3+3 (irreducible) ..., a^4+1 (irreducible) ..., a^4+3 (factors) ..., a^6+3 (factors) Maxima 5.9.0 gcl 2.5.0 mingw32 Windows 2000 Athlon  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=792862&group_id=4933 