I'd been running some computations on sagenb.org which involved
(implicitly) asking Singular to make some groebner basis
calculations. I was never able to complete it because it would
eventually appear to finish, but at that point it acted as though the
worksheet was restarted (and silently --
I'm running sage-4.7 64 bit on my macbook pro. I have a copy of magma
2.15 installed also. In the past I've been able to use the magma
interface from Sage, but today when I try something like
sage: Im = magma(I)
I get the error unable to start magma
I have the magma script in /usr/local/bin
computed up
to that point was the same.
Victor
On Aug 3, 8:38 am, VictorMiller victorsmil...@gmail.com wrote:
Robert, I'll see what I can do. As you suspected, the files are not
disclosable :-(.
Victor
On Aug 3, 3:03 am, Robert Bradshaw rober...@math.washington.edu
wrote
My vote is for an Overflow exception. Certainly doing nothing is not
good.
Victor
On Aug 4, 3:05 pm, Robert Bradshaw rober...@math.washington.edu
wrote:
On Thu, Aug 4, 2011 at 10:49 AM, Victor Miller victorsmil...@gmail.com
wrote:
There's a real bug in Cython. It looks like it's some sort
could come up
with a whittled-down example.
On Tue, Aug 2, 2011 at 7:22 PM, VictorMiller victorsmil...@gmail.com wrote:
Robert, The .py and .pyx files are identical. I copied one to the
other, and just in case I checked with diff. It's very puzzling.
Victor
On Aug 2, 8:19 pm
I'm doing a lot of finite field computations in finite fields of the
form GF(2^n). Since I'd like to speed it up I want to use cython.
When I copy over the .py file to a .pyx and compile (after excising a
particular bug) it works fine. Since I want more speed, I decided to
use ntl_GF2E. So I
Yesterday I started up sage 4.6.2 on one of our compute servers,
started the notebook interface, and connected to from firefox on my
workstation. I started up a notebook session, where I constructed a
particular elliptic curve over GF(2^351), and a point on it. I
(unwisely) tried
P1.order()
Is there any documentation for the decorators that are available in
Sage? I couldn't find anything about them in the reference manual?
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I've written a bunch of functions (some organized in classes) to do
some large computations in a particular finite field (always GF(2^n)
for some odd n). This seems to work fine. I'd like the computation
to be as fast as possible, so the first thing I did was to copy
the .py file to a .pyx file.
I'm unable to get some magma output converted back into sage. This
has to do with group invariants. Here's a script that produces the
error
sage: G =magma(SymmetricGroup(5))
sage: x = G.InvariantsOfDegree(2)
sage: print x
[
x1^2 + x2^2 + x3^2 + x4^2 + x5^2,
x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4
Whoops, I was copying the lines manually. The line should have been
sage: x = G.InvariantsOfDegree(GF(2),2)
Victor
On Aug 2, 5:32 pm, William Stein wst...@gmail.com wrote:
On Tue, Aug 2, 2011 at 1:29 PM, VictorMiller victorsmil...@gmail.com wrote:
I'm unable to get some magma output
2, 2011 at 11:09 AM, VictorMiller victorsmil...@gmail.com
wrote:
I've written a bunch of functions (some organized in classes) to do
some large computations in a particular finite field (always GF(2^n)
for some odd n). This seems to work fine. I'd like the computation
to be as fast
More bugs:
sage: a = [ [[1,1],[0,1]], [[0,-1],[1,0]] ]
sage: G = MatrixGroup([Matrix(_) for _ in a])
sage: Gm = magma(G)
TypeError: Error evaluating Magma code.
IN:_sage_[2]:=Matrix group over Integer Ring with 2 generators:
[[[1, 1], [0, 1]], [[0, -1], [1, 0]]];
OUT:
In file
I want to do a lot of finite field computations, and want to use
Cython to speed things up. It's not clear to me what the details are
that I need to adhere to. I noticed from the comments in
element_givaro.pyx that the givaro library is fastest from fields of
size 2^16. However, some of the
Suppose that I have field (it's actually a finite field, but it would
be nice to know if this works in more generality) F, and an extension
field K. If a[0], ..., a[k] are elements of K, I'd like to generate
the field F obtained by adjoining these elements to F as a sage
object. Is there a
Is sagenb.org having problems? I tried to log on yesterday (and
today, just now), and it said that my username was unknown!
Victor
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For
There are some defects in dual abelian group:
1) The base ring is currently CC. It would be nice if it were
CyclotomicField of the exponent of the of group. There's an old
(2006) from David Joyner about this on sage-support, and Kiran asked
for it. I'd like to pop that request to the top of
If you give the wrong number of arguments to a cached function you get
a rather mystifying error message:
sage: @cached_function
sage: def A(i,j):
sage:return i+j
sage: def B(i,j):
sage:return i+j
sage: A(3)
Traceback (click to the left of this block for traceback)
...
KeyError: 'j'
Thanks! Then perhaps Sage should raise an exception when it gets back
genus -1 from singular.
Victor
On Nov 22, 3:12 am, luisfe lftab...@yahoo.es wrote:
On Nov 21, 6:22 am, VictorMiller victorsmil...@gmail.com wrote:
sage: T.t1,t2,u1,u2 = QQ[]
sage: TJ = Ideal([t1^2 + u1^2 - 1,t2^2 + u2^2
I have the following:
sage: R.a,b = QQ[]
sage: K = FractionField(R)
sage: S.x,y = K[]
I then create an ideal, J, in S. I'd like to take various
specializations of the base. That is I have a homomorphism which maps
a and b to specific values in Q, and I'd like to form the ideal in a
bivariate
sage: T.t1,t2,u1,u2 = QQ[]
sage: TJ = Ideal([t1^2 + u1^2 - 1,t2^2 + u2^2 - 1, (t1-t2)^2 + (u1-
u2)^2 -1])
sage: TJ.genus()
4294967295
sage: TJ.dimension()
1
I'm very skeptical about that answer (I realize that Singular is doing
the calculation) especially since (for example). Even if it's
Are there any Sage functions and/or classes for computing Newton
Polygons?
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Observe (using sagenb.org)
sage: M = matrix(ZZ,4,4,sparse=True)
sage: M.norm()
Traceback (click to the left of this block for traceback)
...
AttributeError:
'sage.matrix.matrix_generic_sparse.Matrix_generic_sparse' object has
no
attribute 'SVD'
sage: M.norm(1)
Traceback (click to the left of
Observe (using sage 4.5.2 on my mac):
sage: import numpy
sage: a = numpy.array([1,2,3])
sage: v = vector(a)
Traceback (click to the left of this block for traceback)
...
TypeError: unsupported operand type(s) for ** or pow(): 'NoneType' and
'int'
The error message is particularly weird.
Victor
Last night I logged into sagenb.org and created a new worksheet. At
the end I pressed Save and Quit. However, when I logged in this
morning there was no trace of the worksheet. Can it be recovered?
Victor
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I'm a bit embarrassed (but only a small bit). It turns out that I
have two ids at sagenb.org: victorsmiller and VictorSMiller. I never
realized that they are different!
Victor
On Jul 29, 7:47 am, VictorMiller victorsmil...@gmail.com wrote:
Last night I logged into sagenb.org and created a new
, it reverts back to VictorSMiller, so I can't log
into the other account. Any suggestions as to how to deal with this?
Victor
On Jul 29, 8:54 am, VictorMiller victorsmil...@gmail.com wrote:
I'm a bit embarrassed (but only a small bit). It turns out that I
have two ids at sagenb.org: victorsmiller
There's a bug in assigning 1 x 1 submatrices. assigning any
submatrices with dimensions bigger than 1 seems to work as expected:
sage: A = matrix(GF(2),100,100)
sage: C1 = matrix(GF(2),[[1]])
sage: C2 = matrix(GF(2),[[0,1],[1,0]])
sage: A[88:90,88:90] = C2 # this is ok however
sage:
I'd like to construct a sparse vector (say a unit vector in 100
dimensional space) by saying
a = vector(GF(2),100,dict([(22,1)]),sparse=True)
However, this gives the error message below.
This seems like a bug to me.
Victor
ValueError: incompatible degrees in vector constructor
Traceback
I had thought that repr(A) (where A is some object) was supposed to be
a string with property that eval(repr(A)) == A.
However, if A is a dense matrix (whose size is above some threshold)
what I get instead is
something like:
'212 x 212 dense matrix over Finite Field of size 2'
I could
Alex, Thanks. I had discovered is_FreeModule after I posted my query,
but didn't know how to get rid of the deprecation warnings.
Victor
On May 29, 7:37 am, Alex Ghitza aghi...@gmail.com wrote:
On Sat, 29 May 2010 04:26:31 -0700 (PDT), John Cremona
john.crem...@gmail.com wrote:
Does the
John, Thanks for the explanation. It would have never occurred to me
that polynomial rings would have gotten involved here.
Victor
On May 29, 7:24 am, John Cremona john.crem...@gmail.com wrote:
This is now being tracked athttp://trac.sagemath.org/sage_trac/ticket/9085
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I have magma and sage 4.4.1 installed on my macbook pro. I've made
sure that the magma command is on my PATH. Yet when I run sage, and I
try and call to magma.eval, it says that it can't find magma. This
has worked without problem on my linux workstation at work. So what
needs to be done to
Here's something a bit odd:
sage: F = GF(2)
sage: A = CartesianProduct(F,F)
sage: print A.random_element()
This gets a trace back and the message
TypeError: You must specify the names of the variables
Victor
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To
wst...@gmail.com wrote:
On Fri, May 28, 2010 at 11:03 AM, VictorMiller victorsmil...@gmail.com
wrote:
I have magma and sage 4.4.1 installed on my macbook pro. I've made
sure that the magma command is on my PATH. Yet when I run sage, and I
try and call to magma.eval, it says that it can't
, May 28, 2010 at 3:33 PM, VictorMiller victorsmil...@gmail.com wrote:
More data: If I start sage and then type
!magma
it starts up Magma. I then exit from it (by typing exit;) and start
the notebook.
However, if I try to run magma from the notebook, I get the message
Type Error
in command line mode.
Victor
On May 28, 7:27 pm, William Stein wst...@gmail.com wrote:
On Fri, May 28, 2010 at 4:15 PM, VictorMiller victorsmil...@gmail.com wrote:
I'm in the notebook and I typed
os.system('magma')
It's now over 5 minutes, and it still hasn't come back! While I
Ah, that's it. I had put the .magmapass file in my home directory,
but never set up MAGMAPASSFILE as an environment variable. For some
reason when I started magma from the command line, or with !magma this
didn't bother it, but from the notebook it did. I added something to
set MAGMAPASSFILE
On May 28, 9:24 pm, VictorMiller victorsmil...@gmail.com wrote:
Ah, that's it. I had put the .magmapass file in my home directory,
but never set up MAGMAPASSFILE as an environment variable. For some
reason when I started magma from the command line, or with !magma this
didn't bother it, but from
eval to a 1 long
loop of
a,c,d := IsConsistent(A,v)
I found that Magma was faster than sage by a factor of 44. I haven't
looked at the sage code yet, but I hope that most of the difference
comes from the fact that it's not written in Cython.
Victor
On May 25, 2:48 pm, VictorMiller victorsmil
fast is it to do
b in A.column_space()
?
John
On May 25, 7:48 pm, VictorMiller victorsmil...@gmail.com wrote:
I have a sage program which involves a lot of calculations of the
form:
solve for x, Ax = b, where A is an integral matrix and x and b are
column vectors. A is an integer
] do a,b,c :=
IsConsistent(%s,%s); end for; Cputime()-t;'%(Am.name(),bm.name()))
Victor
On May 26, 3:58 pm, William Stein wst...@gmail.com wrote:
On Wed, May 26, 2010 at 12:55 PM, VictorMiller victorsmil...@gmail.com
wrote:
It was even slower than A.solve_right(check=False). The results from
...@nuigalway.ie wrote:
On 26 Mai, 22:09, VictorMiller victorsmil...@gmail.com wrote:
A = Matrix([[1,0,1,0],[1,1,0,1],[0,1,0,1],[1,1,0,0],[1,0,0,0]])
b=vector([13,1,7,5,14])
timeit('A.solve_right(b,check=False)')
B = A.change_ring(QQ)
timeit('B.solve_right(b,check=False)')
bm = magma(v)
Am
I have a sage program which involves a lot of calculations of the
form:
solve for x, Ax = b, where A is an integral matrix and x and b are
column vectors. A is an integer matrix, but the solution (if it
exists) for x might be rational. Actually I want more than this:
I want to also determine
Does the Matrix class have methods for vertical and horizontal joins
of matrices (as in Magma)? That is
if A is an m by n matrix and B is an r by n matrix then
VerticalJoin(A,B) would by the (m+r) by n matrix with A on top and B
on the bottom. Similarly, if A is m by n and B is m by r then
Thanks!
On May 20, 6:19 pm, Robert Bradshaw rober...@math.washington.edu
wrote:
On May 20, 2010, at 3:15 PM, VictorMiller wrote:
Does the Matrix class have methods for vertical and horizontal joins
of matrices (as in Magma)? That is
if A is an m by n matrix and B is an r by n matrix
If I write a function in a cell of a notebook like:
@interact
def foo(a = input_box(default=0, type=Integer)):
# do something here
pass
And the user enters something that cannot be coerced to Integer, then
I get a verbose (and rather unhelpful) exception, which, as far as I
can see,
I just downloaded, what I thought, was SAGE 4.4 (I went to
sagemath.org, download, and clicked on the server from Boston) for Mac
OS X 64 bit intel. After installing it, and running it the banner
says version 4.3.5. Has the new version not made it out to all the
servers?
Victor
--
To post to
Thanks for the suggestion. It turned out (for a rather convoluted
reason) that the program that I thought I'd be running didn't exist.
My bad :-).
Victor
On Apr 22, 2:58 pm, Jason Grout jason-s...@creativetrax.com wrote:
On 04/22/2010 12:18 PM, VictorMiller wrote:
I tried using sage
Is it possible to do the following: have SAGE be a server so that when
someone goes to a particular URL that they'll attach to SAGE only
running a particular worksheet (which would probably have @interactive
stuff)? If so, how would I do it?
Victor
PS. What I'd like to do is to have a server
I have some old Python programs that I've been using, and would like
to use them as part of SAGE. One particular function I have uses the
standard python module subprocess to call an external program, and
then process the output from that. This has worked fine in python for
a number of years
:
On 04/22/2010 11:38 AM, VictorMiller wrote:
I have some old Python programs that I've been using, and would like
to use them as part of SAGE. One particular function I have uses the
standard python module subprocess to call an external program, and
then process the output from
I should say that my call is
subprocess.Popen(myargs,stdin=subprocess.PIPE,stdout=subprocess.PIPE,stderr=subprocess.STDOUT)
where myargs is the usual list of tokens in the command
On Apr 22, 12:58 pm, Jason Grout jason-s...@creativetrax.com wrote:
On 04/22/2010 11:38 AM, VictorMiller wrote
Also, I'm running on Redhat Linux
On Apr 22, 1:18 pm, VictorMiller victorsmil...@gmail.com wrote:
I tried using sage -python
It still bombs out, but slightly differently: it now does a
raise child_exception in the _execute_child method of subprocess, and
gets the error string
OSError
I've been running Sage 4.3.1 on my macbook pro. Today, the following
happened twice:
The notebook server stopped responding. When I looked at the console
I had the following error on it:
2010-02-10 23:02:21-0500 [HTTPChannel,86,127.0.0.1] /Applications/sage/
:36 am, William Stein wst...@gmail.com wrote:
On Wed, Feb 10, 2010 at 9:34 PM, VictorMiller victorsmil...@gmail.com wrote:
I've been running Sage 4.3.1 on my macbook pro. Today, the following
happened twice:
The notebook server stopped responding. When I looked at the console
I had
I can't find a method for a vector space (or perhaps a subspace of a
vector space) for orthogonal subspace. Since vector space seem to
be equipped with an inner product this seems like a natural function
to have. Am I missing something? If not, I vote that it should be
added.
Victor
--
To
Is there any easy way of building up what I'd call a ring of
exponentials (maybe there's a better word)? For example I'd like to
work in the ring QQ[[2^j for j in Integers()]]: the ring with
coefficients in Q and elements 2^j where j is an integer (or possibly
just a non-negative integer). Such
, 31 Dec 2009 08:37:55 -0800 (PST)
VictorMiller victorsmil...@gmail.com wrote:
Is there any easy way of building up what I'd call a ring of
exponentials (maybe there's a better word)? For example I'd like to
work in the ring QQ[[2^j for j in Integers()]]: the ring with
coefficients in Q
I'm in the middle of implementing SAGE classes for binary decision
diagrams using the library called CUDD (which is included in
Polybori). Each BDD is actually a labeled DiGraph. It's standard in
the BDD literature to draw pictures of the digraph as follows:
the vertex label of a non-leaf is a
I just ran the following on sagenb.org (so the latest release):
PP.x,y,z,w = ProjectiveSpace(3,QQ)
f = x^3 + y^3 + z^3 + w^3
R = f.parent()
I = [f] + [f.derivative(zz) for zz in PP.gens()]
V = PP.subscheme(I)
V.irreducible_components()
The output is:
[
Closed subscheme of Projective Space of
I was looking at a hypersurface in projective 3 space, so I did
sage: PP.x,y,z,w = ProjectiveSpace(3,QQ)
and then defined a homogeneous polynomial f in x,y,z,w.
I wanted to find the singularities, so I did
sage: I = [f] + [f.derivative(zz) for zz in PP.gens()]
sage: V = PP.subscheme(I)
and
divisor_of_function appears to be badly broken:
E = EllipticCurve(j=1)
xx,yy,zz = E.coordinate_ring().gens()
E.divisor_of_function(yy)
gives the output
Type Error: A positive bound (=0) must be specified.
If I then do
E0 = E.change_ring(GF(144169))
xxx,yyy,zzz = E0.coordinate_ring().gens()
In calculating the Weil Pairing of two points I get a pari division by
zero error. Needless to say this shouldn't happen. I think that the
solution is that if either of the miller_functions that are calculated
yield a 0 then the value of the Weil Pairing must be 1.
Here's some code that
:
break
P.weil_pairing(2*P,3)
On Aug 6, 1:05 pm, VictorMiller victorsmil...@gmail.com wrote:
In calculating the Weil Pairing of two points I get a pari division by
zero error. Needless to say this shouldn't happen. I think that the
solution is that if either of the miller_functions
)/K(0)
L(1)/L(0)
Note that the errors given by K and L and different. And, if we work
over small characteristic, say p=7, we get a different set of errors.
Victor
On Aug 6, 1:13 pm, VictorMiller victorsmil...@gmail.com wrote:
Slight change. The code I gave works, but the following fails. I
I was asking SAGE to do a calculation that I knew was probably
laborious -- I had a plane curve (over Q) and I wanted its genus. I
defined it with C=Curve(equation_in_two_variables) and then typed
C.genus()
after a while (I was in the notebook) I just got the mysterious error
message:
)*[S(0)] + [b1,b2])
return Curve(phi(J.gens()[0]))
C = Curve1(9,3)
print C.genus()
On Aug 5, 4:12 pm, William Stein wst...@gmail.com wrote:
On Wed, Aug 5, 2009 at 1:09 PM, VictorMiller victorsmil...@gmail.comwrote:
I was asking SAGE to do a calculation that I knew was probably
I had a small typo in the last code. The line that said
F = FractionField(K)
should have read
F = FractionField(R)
also after the line that starts with vars = should be the line
R = PolynomialRing(Rationals(),k,vars)
On Aug 5, 4:59 pm, VictorMiller victorsmil...@gmail.com wrote:
Okay
Another small oops. The offending example should be
C = Curve1(9,4)
On Aug 5, 5:02 pm, VictorMiller victorsmil...@gmail.com wrote:
I had a small typo in the last code. The line that said
F = FractionField(K)
should have read
F = FractionField(R)
also after the line that starts
the calculation completed.
Victor
On Aug 5, 5:02 pm, VictorMiller victorsmil...@gmail.com wrote:
I had a small typo in the last code. The line that said
F = FractionField(K)
should have read
F = FractionField(R)
also after the line that starts with vars = should be the line
R = PolynomialRing
I was trying to find out how fast a calculation was (applying an
isogeny of degree on an elliptic curve over
a finite field). At first I noticed that when I repeated a timeit
call with the same expression I was getting monotonically increasing
numbers, so I decided to try something more
As far as I know Maxima isn't involved -- I don't think that isogenies
uses Maxima.
Victor
On Aug 3, 6:58 pm, Simon King simon.k...@nuigalway.ie wrote:
On 4 Aug., 00:29, VictorMiller victorsmil...@gmail.com wrote:
...
phi = E.isogeny([E(0),P,-P])
for i in xrange(20): timeit('phi(Q
Here are the commands I used:
qq = [z for z in primes(10,10+100) if (z%12) == 11]
E = EllipticCurve(j=GF(qq[0])(1728))
# E has qq[0]+1 points over GF(qq[0])
factor(qq[0]+1)
P = ((qq[0]+1)//3)*E.random_element()
K = [E(0),P,-P]
phi = E.isogeny(K)
for i in xrange(20): timeit('phi(Q)')
On
Sorry, here's the definition of Q:
Q = E.random_element()
Victor
On Aug 3, 8:45 pm, Simon King simon.k...@nuigalway.ie wrote:
Hi!
On 4 Aug., 02:31, VictorMiller victorsmil...@gmail.com wrote:
Here are the commands I used:
qq = [z for z in primes(10,10+100) if (z%12) == 11]
E
Suppose that q is a prime power, and I have an elliptic curve E over GF
(q)
(say created by E = EllipticCurve(coefficient_list))
and P,Q = E.gens()
How can I find E just given P (say if I pass just P to a function)?
If I say
print P.parent()
I get something like
Abelian group of point on
I just did a test of SAGE versus Magma on the same computer.
I had a finite field GF(19991^2), and timed generating a random
element in SAGE and in Magma.
I found, much to my surprise, that Magma was a factor of 7 times
faster. Does anyone know what
method they use?
Victor
, VictorMiller wrote:
I just did a test of SAGE versus Magma on the same computer.
I had a finite field GF(19991^2), and timed generating a random
element in SAGE and in Magma.
I found, much to my surprise, that Magma was a factor of 7 times
faster. Does anyone know what
method
I have a sage program in a file in one of my directories called
calc.sage. It uses a class that I wrote called Table, which I've put
in a file called Table.py in the same directory. In the sage notebook
I load calc.sage (by explicitly giving the path to the directory), and
calc.sage has a line
lies somewhere in the wealth of online python
documentation!
Of course someone else might give a more helpful answer...
John Cremona
On Jul 23, 5:16 pm, VictorMiller victorsmil...@gmail.com wrote:
I have a sage program in a file in one of my directories called
calc.sage. It uses a class
I have a finite dimensional vector space V/k, and various
automorphisms (i.e. members of GL(V)). I would like to construct the
field k(X), where the variables in X correspond to a basis of V, along
with the operation of GL(V) on k(X). Is there some existing way to do
this in SAGE, or do I need
Ok, I've absorbed all of that. This morning it occured to me that it
would be nice if one could define a class method called _html_. If
present, it would try to render an htmlized (is that a word?) version
of the class (just as __str__ renders a printable version or _latex_
renders a latex
Ok, I've absorbed all of that. This morning it occured to me that it
would be nice if one could define a class method called _html_. If
present, it would try to render an htmlized (is that a word?) version
of the class (just as __str__ renders a printable version or _latex_
renders a latex
Ok, I've absorbed all of that. This morning it occured to me that it
would be nice if one could define a class method called _html_. If
present, it would try to render an htmlized (is that a word?) version
of the class (just as __str__ renders a printable version or _latex_
renders a latex
William, Thanks. That works ok -- except, for example if I do
latex.eval('$N_0$',{})
I get what I expect plus a line with two single quotes before what I
wanted. This seems to happen with any latex string. Do you know
what's happening?
Victor
On Jul 20, 6:58 pm, William Stein
Jason, thanks for the suggestion. I didn't know about that function.
Victor
On Jul 21, 4:02 am, Jason Grout jason-s...@creativetrax.com wrote:
VictorMiller wrote:
I have a program which calculates a table of values, and I'd like to
display it nicely formatted. I've written a function
Is there a way to get tex formulas as cells in the table?
Victor
On Jul 21, 4:02 am, Jason Grout jason-s...@creativetrax.com wrote:
VictorMiller wrote:
I have a program which calculates a table of values, and I'd like to
display it nicely formatted. I've written a function to produce
I have a program which calculates a table of values, and I'd like to
display it nicely formatted. I've written a function to produce latex
for it (using tabular), but I can't figure out how to get SAGE to
display this in a notebook. I've tried the html command but that
doesn't work. Here's a
In converting some of my old python programs to run on SAGE I
expressions like:
0.5**numpy.arange(10,1,-1)
which works fine in python, but gives a type error in SAGE. I
eventually figured out that I could
get this to work by doing
float(0.5)**numpy.arange(10,1,-1)
but that's a pain. Any
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