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From: Pedro Giffuni <pfg@FreeBSD.org>  20150505 04:29:21

Hello; I had a moment of inspiration and spare time this weekend and updated the FreeBSD port:  We are using the 20141130 snapshot now.  The PDF documentation is now included.  There is an option to build the (amd64) PSL version instead of the CSL version. Not sure there is interest, but if there is something I can do to help get more architectures supported on the PSL version, feel free to let me know. Enjoy! Pedro. 
From: Kostas Oikonomou <ko@re...>  20150429 22:27:30

Thanks for investigating this. The removal of this optimization seems necessary, since it is not correct in general. Kostas On 04/29/2015 13:32, Rainer Schöpf wrote: > On Sun, 26 Apr 2015 at 09:23 0400, Kostas Oikonomou wrote: > > > Rainer, > > > > Is this related to the problems with 'limit'? Also, when you say > > "sub(x=0,expression) is a valid substitution", > > does this mean that Reduce will raise an error if it is not valid? > > Yes to both. With the current code, the substitution x=0 has this special > optimization so that > > x^n*f(x) > > is replaced by 0 without looking at f(x). Try > > y := (x1)^2*e^(1/(x1)^2); > > sub(x=1,y); > > for comparison. I propose to remove the optimization. > > Rainer 
From: Rainer Schöpf <rainer.schoepf@gm...>  20150429 17:32:55

On Sun, 26 Apr 2015 at 09:23 0400, Kostas Oikonomou wrote: > Rainer, > > Is this related to the problems with 'limit'? Also, when you say > "sub(x=0,expression) is a valid substitution", > does this mean that Reduce will raise an error if it is not valid? Yes to both. With the current code, the substitution x=0 has this special optimization so that x^n*f(x) is replaced by 0 without looking at f(x). Try y := (x1)^2*e^(1/(x1)^2); sub(x=1,y); for comparison. I propose to remove the optimization. Rainer 
From: Kostas Oikonomou <ko@re...>  20150426 13:23:54

Rainer, Is this related to the problems with 'limit'? Also, when you say "sub(x=0,expression) is a valid substitution", does this mean that Reduce will raise an error if it is not valid? In any case, it would be interesting to know how much of a slowdown (if any) would be incurred by the fix. Kostas On 04/5/2015 05:08, Rainer Schöpf wrote: > I came across the following behaviour of the substitution code: > >  > 1: y := x^2*e^(1/x^2); > > 2 > 1/x 2 > y := e *x > > 2: sub(x=0,y); > > 0 >  > > The reason is an optimization in the code: when x=0, any term of the form > > expression*x^n > > is considered to be 0  which is correct only if > > sub(x=0,expression) > > is a valid substitution. If x is not 0, the substitution is carried out. > > I believe it would be better to go through the substitution even for x=0. I > cannot see a problem with backward compatibility. Increased computation time > may be an issue, but that can be tested. > > What do others think? > > Rainer > >  > One dashboard for servers and applications across PhysicalVirtualCloud > Widest outofthebox monitoring support with 50+ applications > Performance metrics, stats and reports that give you Actionable Insights > Deep dive visibility with transaction tracing using APM Insight. > http://ad.doubleclick.net/ddm/clk/290420510;117567292;y > _______________________________________________ > Reducealgebradevelopers mailing list > Reducealgebradevelopers@... > https://lists.sourceforge.net/lists/listinfo/reducealgebradevelopers 
From: Raymond Rogers <raymond.rogers72@gm...>  20150425 20:28:15

On 04/25/2015 10:00 AM, Rainer Schöpf wrote: > On Sat, 25 Apr 2015 at 09:32 0400, Raymond Rogers wrote: > > > > > > > On 04/25/2015 05:08 AM, Rainer Schöpf wrote: > > > I came across the following behaviour of the substitution code: > > > > > >  > > > 1: y := x^2*e^(1/x^2); > > > > > > 2 > > > 1/x 2 > > > y := e *x > > > > > > 2: sub(x=0,y); > > > > > > 0 > > >  > > > > > > The reason is an optimization in the code: when x=0, any term of the form > > > > > > expression*x^n > > > > > > is considered to be 0  which is correct only if > > > > > > sub(x=0,expression) > > > > > > is a valid substitution. If x is not 0, the substitution is carried out. > > > > > > I believe it would be better to go through the substitution even for x=0. > > > I > > > cannot see a problem with backward compatibility. Increased computation > > > time > > > may be an issue, but that can be tested. > > > > > > What do others think? > > > > > > Rainer > > I disagree; > > Disagree with what? > > It seems that I wasn't clear enough. > > At the moment the substitution sub(x=0,x^2*exp(1/x^2)) returns zero since > substition of the first factor yields 0. By going through the full substitution, > the first factor will be substituted by 0, but trying the substitution of the > second factor will cause an error. > > > 1) Consider the expression S=sum(x^n,n=0..) i.e. 1/(1x) and let x go to 0. > > The correct result for subsequent evaluation would be S=1 for x = 0; whereas > > substituting x=0 without thought would yield terms 0^n which is indeterminate > > at n=0. > > and therefore signal the error "0**0 formed". That behaviour won't change. > > Rainer Okay; but I am just "user" and would appreciate a handle on how (and where) to fix it, if necessary, if it resulted inside of a more complicated form. A clue if you will. Although there is probably a trace ability I haven't used yet. For example: sin(F({a};{b};x^2*e^(1/x^2)) ; a composition with a hypergeometric term. Ray x^2*e^(1/x^2) 
From: Rainer Schöpf <rainer.schoepf@gm...>  20150425 14:00:31

On Sat, 25 Apr 2015 at 09:32 0400, Raymond Rogers wrote: > > > On 04/25/2015 05:08 AM, Rainer Schöpf wrote: > > I came across the following behaviour of the substitution code: > > > >  > > 1: y := x^2*e^(1/x^2); > > > > 2 > > 1/x 2 > > y := e *x > > > > 2: sub(x=0,y); > > > > 0 > >  > > > > The reason is an optimization in the code: when x=0, any term of the form > > > > expression*x^n > > > > is considered to be 0  which is correct only if > > > > sub(x=0,expression) > > > > is a valid substitution. If x is not 0, the substitution is carried out. > > > > I believe it would be better to go through the substitution even for x=0. > > I > > cannot see a problem with backward compatibility. Increased computation > > time > > may be an issue, but that can be tested. > > > > What do others think? > > > > Rainer > I disagree; Disagree with what? It seems that I wasn't clear enough. At the moment the substitution sub(x=0,x^2*exp(1/x^2)) returns zero since substition of the first factor yields 0. By going through the full substitution, the first factor will be substituted by 0, but trying the substitution of the second factor will cause an error. > 1) Consider the expression S=sum(x^n,n=0..) i.e. 1/(1x) and let x go to 0. > The correct result for subsequent evaluation would be S=1 for x = 0; whereas > substituting x=0 without thought would yield terms 0^n which is indeterminate > at n=0. and therefore signal the error "0**0 formed". That behaviour won't change. Rainer 
From: Raymond Rogers <raymond.rogers72@gm...>  20150425 13:33:03

On 04/25/2015 05:08 AM, Rainer Schöpf wrote: > I came across the following behaviour of the substitution code: > >  > 1: y := x^2*e^(1/x^2); > > 2 > 1/x 2 > y := e *x > > 2: sub(x=0,y); > > 0 >  > > The reason is an optimization in the code: when x=0, any term of the form > > expression*x^n > > is considered to be 0  which is correct only if > > sub(x=0,expression) > > is a valid substitution. If x is not 0, the substitution is carried out. > > I believe it would be better to go through the substitution even for x=0. I > cannot see a problem with backward compatibility. Increased computation time > may be an issue, but that can be tested. > > What do others think? > > Rainer I disagree; I think some context is necessary to have a consistent mathematical system. 1) Consider the expression S=sum(x^n,n=0..) i.e. 1/(1x) and let x go to 0. The correct result for subsequent evaluation would be S=1 for x = 0; whereas substituting x=0 without thought would yield terms 0^n which is indeterminate at n=0. Of course the value of S=1 for x=0 is arbitrary and could S = anything. Providing the need for context. 2) Actually along the same lines I can construct continuous expressions f(x) such that the only way to reach a solution in the continuous domain at x=0 is to apply L'Hopital's rule. We surely don't want to add extra special cases the fall outside of standard mathematics, like putting holes in continuous functions, that has been validated by long usage (and hopefully by an axiomatic framework). Speed is nice unless your falling off of _something_ then it matters when you hit. Ray / / 
From: Rainer Schöpf <rainer.schoepf@gm...>  20150425 09:08:38

I came across the following behaviour of the substitution code:  1: y := x^2*e^(1/x^2); 2 1/x 2 y := e *x 2: sub(x=0,y); 0  The reason is an optimization in the code: when x=0, any term of the form expression*x^n is considered to be 0  which is correct only if sub(x=0,expression) is a valid substitution. If x is not 0, the substitution is carried out. I believe it would be better to go through the substitution even for x=0. I cannot see a problem with backward compatibility. Increased computation time may be an issue, but that can be tested. What do others think? Rainer 
From: Arthur Norman <acn1@ca...>  20150423 08:49:56

> From: scarpaflorent [mailto:scarpaflorent@...] > Sent: Tuesday, April 07, 2015 10:40 AM > To: reduce@... > Subject: A question about factorization in Reduce > > To whom it may concern, > > My name is Florent Scarpa, and I am using Reduce to realize calculations in physics. It seems very efficient, but I have a problem that really bothers me. It is about factorizing my expressions ; indeed, all functions doing factorizations, substitutions, limits, Taylor developments... only accept kernels as argument to be used as the common factor to find in expressions. My problem is that I would need to be able to factorize with a factor which is a product, or quotient, of kernels. Unfortunately, I did not find any solution in the user's manual, and I can not find a way to replace my product of kernels by a simple one; the consequence is that I am always confronted to the error message : "... invalid as kernel". Would you have any solution to propose for this problem ? Thank you in advance. Yours faithfully, > > Florent Scarpa > It is indeed the case that you can not do limits or series expansions etc with respect to something that is not a kernel. You can sometimes have rules that let you may a product of kernels into a single variable, as in 1: (1 + x*y)^3; 3 3 2 2 x *y + 3*x *y + 3*x*y + 1 2: ws where x*y => xy; 3 2 xy + 3*xy + 3*xy + 1 (I hope your email reader displays stuff using a fixedpitch font so that the exponents appear where they are wanted!). A different strategy that ends up with a similar result is to set y to xy/x. These two transformations would lead to different results if the original formula had terms using different powers of x and y. 1: (1 + x*y)^3; 3 3 2 2 x *y + 3*x *y + 3*x*y + 1 2: ws where y => xy/x; 3 2 xy + 3*xy + 3*xy + 1 But still the idea is to map or rename or reexpress things so that the thing you then need to bring out as a factor or expand in terms of (etc etc) is a single symbol. Without knowing more about exactly what you are trying to do I can just hope this helps you move forward! Arthur 
From: Rainer Schöpf <rainer.schoepf@gm...>  20150422 18:58:13

On Wed, 22 Apr 2015 at 13:53 0400, Kostas Oikonomou wrote: > Thanks. I don't pretend to understand what is going on, but I also > discovered this: > > 1: z := exp(2*x)/x^2; > > 2*x > e > z :=  > 2 > x > > 2: limit(z,x,infinity); > > 0 Oh dear, the limit code needs a thorough code review. Also this particular problem can be traced to this one: 1: y := x^2*e^(1/x^2); 2 1/x 2 y := e *x 2: sub(x=0,y); 0 Rainer 
From: Kostas Oikonomou <ko@re...>  20150422 17:53:52

Thanks. I don't pretend to understand what is going on, but I also discovered this: 1: z := exp(2*x)/x^2; 2*x e z :=  2 x 2: limit(z,x,infinity); 0 On 04/22/2015 12:45, Rainer Schöpf wrote: > On Tue, 21 Apr 2015 at 15:26 0400, Kostas Oikonomou wrote: > > > I just found the following problem with integrals: > > > > Reduce (Free CSL version), 06Apr15 ... > > > > 1: int(e^(x^2),x,infinity,infinity); > > > > sqrt(pi) > > > > > > 2: int(e^((xmu)^2/(2*sigma^2)),x,infinity,infinity); > > > > 2 > > ***** sigma invalid as number > > > It's more a problem with the limits package. Try this: > > y := e^((xmu)^2/(2*sigma^2)); > > z := df(y,x)/x; > > limit(z,x,infinity); > > on complex; > > limit(z,x,infinity); > > Rainer 
From: Hearn, Anthony <hearn@ra...>  20150422 17:17:39

Original Message From: scarpaflorent [mailto:scarpaflorent@...] Sent: Tuesday, April 07, 2015 10:40 AM To: reduce@... Subject: A question about factorization in Reduce To whom it may concern, My name is Florent Scarpa, and I am using Reduce to realize calculations in physics. It seems very efficient, but I have a problem that really bothers me. It is about factorizing my expressions ; indeed, all functions doing factorizations, substitutions, limits, Taylor developments... only accept kernels as argument to be used as the common factor to find in expressions. My problem is that I would need to be able to factorize with a factor which is a product, or quotient, of kernels. Unfortunately, I did not find any solution in the user's manual, and I can not find a way to replace my product of kernels by a simple one; the consequence is that I am always confronted to the error message : "... invalid as kernel". Would you have any solution to propose for this problem ? Thank you in advance. Yours faithfully, Florent Scarpa __________________________________________________________________________ This email message is for the sole use of the intended recipient(s) and may contain confidential information. Any unauthorized review, use, disclosure or distribution is prohibited. If you are not the intended recipient, please contact the sender by reply email and destroy all copies of the original message. 
From: Rainer Schöpf <rainer.schoepf@gm...>  20150422 16:46:09

On Tue, 21 Apr 2015 at 15:26 0400, Kostas Oikonomou wrote: > I just found the following problem with integrals: > > Reduce (Free CSL version), 06Apr15 ... > > 1: int(e^(x^2),x,infinity,infinity); > > sqrt(pi) > > > 2: int(e^((xmu)^2/(2*sigma^2)),x,infinity,infinity); > > 2 > ***** sigma invalid as number It's more a problem with the limits package. Try this: y := e^((xmu)^2/(2*sigma^2)); z := df(y,x)/x; limit(z,x,infinity); on complex; limit(z,x,infinity); Rainer 
From: Kostas Oikonomou <ko@re...>  20150421 19:27:34

I just found the following problem with integrals: Reduce (Free CSL version), 06Apr15 ... 1: int(e^(x^2),x,infinity,infinity); sqrt(pi) 2: int(e^((xmu)^2/(2*sigma^2)),x,infinity,infinity); 2 ***** sigma invalid as number 2 ***** sigma invalid as number 2 ***** sigma invalid as number Kostas 
From: Rainer Schöpf <rainer.schoepf@gm...>  20150417 10:39:26

On Thu, 9 Apr 2015 at 08:31 0000, Andrea Magnoni wrote: > Somebody are able to compile correctly this package written for a old version of Reduce ? > package: http://www.kfki.hu/~kadlec/sw/riccir/riccir.tar.gz > instructions: http://wwwold.kfki.hu/cnc/szhkpub/riccir/riccir.html The package build process has changed in the years since the riccir package was written. You might try this: 1. In subsmsq.red, move the powlis1!* variable from the global to the fluid declaration. 2. In ini, comment out the first in statement: %in "riccir.red"$ Regards, Rainer 
From: Andrea Magnoni <andrea_magnoni@ya...>  20150409 12:15:28

He doesn't know if the package is still operating with the newer version of Reduce "That's a very old package for Reduce and I have no idea whether it still works or not. You can download it from http://www.kfki.hu/~kadlec/sw/riccir/riccir.tar.gz Best regards, Jozsef  Email : kadlecsik.jozsef@... PGP key: http://www.kfki.hu/~kadlec/pgp_public_key.txt Address: Wigner Research Centre for Physics, Hungarian Academy of Sciences H1525 Budapest 114, POB. 49, Hungary" Il Giovedì 9 Aprile 2015 10:31, Andrea Magnoni <andrea_magnoni@...> ha scritto: Somebody are able to compile correctly this package written for a old version of Reduce ? package: http://www.kfki.hu/~kadlec/sw/riccir/riccir.tar.gz instructions: http://wwwold.kfki.hu/cnc/szhkpub/riccir/riccir.html Regards Andrea Magnoni 
From: Eberhard Schruefer <eschruefer@ca...>  20150409 10:18:01

It is probably the best to contact the author of RICCIR. To my knowledge Jozsef can be reached at the addresses above. Eberhard On 04/09/2015 10:31 AM, Andrea Magnoni wrote: > Somebody are able to compile correctly this package written for a old > version of Reduce ? > > package: http://www.kfki.hu/~kadlec/sw/riccir/riccir.tar.gz > <http://www.kfki.hu/%7Ekadlec/sw/riccir/riccir.tar.gz>; > > instructions: http://wwwold.kfki.hu/cnc/szhkpub/riccir/riccir.html > > > Regards > > Andrea Magnoni > > >  > BPM Camp  Free Virtual Workshop May 6th at 10am PDT/1PM EDT > Develop your own process in accordance with the BPMN 2 standard > Learn Process modeling best practices with Bonita BPM through live exercises > http://www.bonitasoft.com/bepartofit/events/bpmcampvirtual event?utm_ > source=Sourceforge_BPM_Camp_5_6_15&utm_medium=email&utm_campaign=VA_SF > > > _______________________________________________ > Reducealgebradevelopers mailing list > Reducealgebradevelopers@... > https://lists.sourceforge.net/lists/listinfo/reducealgebradevelopers 
From: Andrea Magnoni <andrea_magnoni@ya...>  20150409 08:32:24

Somebody are able to compile correctly this package written for a old version of Reduce ? package: http://www.kfki.hu/~kadlec/sw/riccir/riccir.tar.gz instructions: http://wwwold.kfki.hu/cnc/szhkpub/riccir/riccir.html Regards Andrea Magnoni 
From: Raymond Rogers <raymond.rogers72@gm...>  20150407 14:38:39

I asked on the TeXmacs site but they thought that I should ask here. First I would like to say that the the texmacs/reduce version works very well. There is one small thing with the TeXmacs display: with "on zb_trace" some of the long outputs terminate prematurely. Copying the same input into the reduce monitor app continues the output to an answer. In fact the premature termination in texmacs is erratic and usually after three or four tries sometimes displays the whole output. Attached is my particular program; if you like I will try to shrink it but I don't think the problem is with the input; but some random thing in the display module. You can switch the problem on and off by switching "on zb_tracs" and "off zb_trace" at the first line. The problem is with the output of the last line 719 of input. Thanks  Two views on life: life is an art not to be learned by observation. George Santayana:Interpretations of Poetry and Religion It's kinda nice to participate in your life Raymond Rogers 
From: Raffaele Vitolo <raffaele.vitolo@un...>  20150407 13:55:53

On 07/04/15 13:44, Alan Barnes wrote: > Recently I have experienced a few problems with the handling of large > amounts of output in the GUI version of redcsl. > > If output is directed to a file there is no problem. However, problems > occur with direct output. When scrolling back the first part of the > output is corrupt: wrong values and/or raw TeX displayed. This often > leads to a subsequent crash. > > The problems occur on Ubuntu with a fairly recent version of redcsl > (midMarch)  I haven't tried redcsl on Windows to see if the same > problem occurs. No problems with rfcsl or with PSLbased versions as > far as I can see. > > One way to reproduce this problem fairly predictably (but not to crash > the system immediately) is to load the file 20150407invfn3.tst from > packages/regressions and then scroll back to the beginning of the output. > > Has anyone experienced similar problems? I tried your file but I do not have the same problem (at least for the moment). However, the CSL GUI has some known bugs, eg: if you try to display a 'big' matrix it will refuse with an error; cutandpaste menu items are broken; etc. I am trying to help Arthur Norman to spot the bugs, I suggest you to ask him directly. Best regards, Raffaele Vitolo. > Alan Barnes > > >  > BPM Camp  Free Virtual Workshop May 6th at 10am PDT/1PM EDT > Develop your own process in accordance with the BPMN 2 standard > Learn Process modeling best practices with Bonita BPM through live exercises > http://www.bonitasoft.com/bepartofit/events/bpmcampvirtual event?utm_ > source=Sourceforge_BPM_Camp_5_6_15&utm_medium=email&utm_campaign=VA_SF > _______________________________________________ > Reducealgebradevelopers mailing list > Reducealgebradevelopers@... > https://lists.sourceforge.net/lists/listinfo/reducealgebradevelopers > 
From: Alan Barnes <alan.barnes45678@gm...>  20150407 11:44:52

Recently I have experienced a few problems with the handling of large amounts of output in the GUI version of redcsl. If output is directed to a file there is no problem. However, problems occur with direct output. When scrolling back the first part of the output is corrupt: wrong values and/or raw TeX displayed. This often leads to a subsequent crash. The problems occur on Ubuntu with a fairly recent version of redcsl (midMarch)  I haven't tried redcsl on Windows to see if the same problem occurs. No problems with rfcsl or with PSLbased versions as far as I can see. One way to reproduce this problem fairly predictably (but not to crash the system immediately) is to load the file 20150407invfn3.tst from packages/regressions and then scroll back to the beginning of the output. Has anyone experienced similar problems? Alan Barnes 
From: Kostas Oikonomou <ko@re...>  20150331 18:44:28

Great, thanks very much. Kostas On 03/31/2015 03:27, Rainer Schöpf wrote: > > > > 6: rank matrix_augment(A,B); > > > > +++ Error undefined function (2 args): matrix_augment > > I believe I corrected this problem as well. Changes are committed. > > Rainer 
From: Rainer Schöpf <rainer.schoepf@gm...>  20150331 07:27:29

Hi Kostas, > Rainer, thank you. This is what happens: > > Reduce (Free CSL version), 23Mar15 ... > > 1: load_package linalg; > > 2: put('matrix_augment,'rtypefn,'quotematrix); > > quotematrix > > 3: A := mat((a),(b))$ B := mat((c),(d))$ > > 5: tp matrix_augment(A,B); > > [a b] > [c d] > > 6: rank matrix_augment(A,B); > > +++ Error undefined function (2 args): matrix_augment I believe I corrected this problem as well. Changes are committed. Rainer 
From: Kostas Oikonomou <ko@re...>  20150330 22:57:58

Rainer, thank you. This is what happens: Reduce (Free CSL version), 23Mar15 ... 1: load_package linalg; 2: put('matrix_augment,'rtypefn,'quotematrix); quotematrix 3: A := mat((a),(b))$ B := mat((c),(d))$ 5: tp matrix_augment(A,B); [a b] [c d] 6: rank matrix_augment(A,B); +++ Error undefined function (2 args): matrix_augment 7: Kostas On 03/30/2015 04:59, Rainer Schöpf wrote: > On Sun, 29 Mar 2015 at 19:08 0400, Kostas Oikonomou wrote: > > > I would expect that I could write > > > > 39: x := mat((a),(b))$ > > > > 40: y := mat((c),(d))$ > > > > and then ask for the transpose: > > > > 41: tp matrix_augment(x,y); > > > > ***** tp(matrix_augment(x,y)) invalid as scalar > > Ah, well, that's one of the reasons why linalg is listed as contributed > package... > > Seriously, please try this > > put('matrix_augment,'rtypefn,'quotematrix); > > (and likewise for matrix_stack). > > If this works to your satisfaction, I'll commit the change. > > Rainer 
From: Tony Roberts <anthony.roberts@ad...>  20150330 19:16:22

BEGIN PGP SIGNED MESSAGE Hash: SHA1 Hi Rainer, many thanks, it appears to be working now for my programs, Tony On 30/03/2015 5:54 pm, Rainer Schöpf wrote: > Hi Tony, > > > have been using a 2012 version of Reduce, but just upgraded and now > > all my let rules invoking expressions stored in arrays break. The > > simplest example is the following code > > > array u(9); > > operator v; > > let v(~p)=>u(p); > > end; > > > which generates the output > > > 1: in "temp.red"; > > array u(9); > > operator v; > > let v(~p)=>u(p); > > > ***** ~p invalid as array index > > Cont? (Y or N) > > > Is this a bug to be corrected? or do I have to change all my code? > > I think it is a consequence of a change made in August 2014, to properly handle lists as array elements. > > I'm just committed a correction to allow free rule variables in array indices on the right side of rules. Your example does no longer produce the error; please test. > > Rainer > > > > > > I cannot find anything in the manual saying that the construct is > > invalidthe manual says that arrays on the lefthand side are > > evaluated, but my arrays are on the righthand side. > > > Tony > > > Rainer Schöpf     Professor A.J. Roberts School of Mathematical Sciences phone: +61 8 8313 3035 University of Adelaide fax: +61 8 8313 3696 South Australia 5005. mailto:anthony.roberts@... http://www.maths.adelaide.edu.au/anthony.roberts/ ==.0000001000000100000110001000011010001111110010111011101000010000== BEGIN PGP SIGNATURE Version: GnuPG v1 Comment: Using GnuPG with Thunderbird  http://www.enigmail.net/ iEYEARECAAYFAlUZoPcACgkQ7TX8dTbro1vs6QCgjXYOba4qisFzlpFjfK7T1bSK ruAAnj6njoc7o1n8v4gV6sgZFC/9knKw =Ab91 END PGP SIGNATURE 