Re: [sage-support] checking if a function is odd, even, or periodic
Dear Vince,
Thank you for your helpful reply.
I'm trying to do something similar to check whether a function is periodic,
I tried:
> bool(sin(x) == sin(x + 2*pi))
which gave 'True' as 2*pi is the period of sine.
However, I wanted to figure out what is the period of the function, so I
tried:
> var('p')
> solve(sin(x) == sin(x + p), p)
I though this would give me 2*pi (and I guess it should also give 4*pi,
6*pi, etc.) but it gave me:
[p == -x + arcsin(sin(x))]
Am I doing something wrong?
Regards, and thanks,
Avril
On Saturday, October 12, 2013 11:08:18 AM UTC+1, Vincent Knight wrote:
>
> Hi Avril,
>
> One approach to check parity for example could be:
>
> sage: f(x) = 1/(1-x^2)
> sage: bool(f(x) == f(-x))
> True
>
> whereas f is not odd as:
>
> sage: bool(f(x) == -f(-x))
> False
>
> There might well be an attribute, method on functions that does this also
> but I'm not aware of it.
> Vince
>
>
> On 12 October 2013 11:03, Avril Coghlan
> > wrote:
>
>> Dear all,
>>
>> I'm wondering whether there is a way to use SAGE to test whether a
>> function such as:
>> f(x) = 1/(1 - x^2)
>> or
>> f(x) = sin(x)
>> is an odd function, even function, or periodic?
>>
>> I'm a Maths student, and am not sure if this is possible, or is a silly
>> question (it seems to me quite a hard thing to check, but I could be
>> totally wrong),
>>
>> Regards,
>> Avril
>> Avril Coghlan
>>
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>
>
>
> --
> Dr Vincent Knight
> Cardiff School of Mathematics
> Senghennydd Road,
> Cardiff
> CF24 4AG
> (+44) 29 2087 5548
> www.vincent-knight.com
> +Vincent Knight
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> Skype: drvinceknight
>
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[sage-support] checking if a function is odd, even, or periodic
Dear all, I'm wondering whether there is a way to use SAGE to test whether a function such as: f(x) = 1/(1 - x^2) or f(x) = sin(x) is an odd function, even function, or periodic? I'm a Maths student, and am not sure if this is possible, or is a silly question (it seems to me quite a hard thing to check, but I could be totally wrong), Regards, Avril Avril Coghlan -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
[sage-support] Re: a multiplicative cipher in SAGE
Dear Keshav,
Thanks very much, that helps a lot and makes lots of sense now.
Using what you said, I was able to encode 'HAPPYNEWYEAR' using a 1-based
system (A=1, B=2, C=3,Z=26) by applying (i) a multiplicative cipher
with a=1,b=1 to change to a 1-based system (ii) then a multiplicative
cipher with a=9,b=0, (iii) then a multiplicative cipher with a=1,b=25 to
change back to the 0-based system to get the answer. I did this by typing:
A = AffineCryptosystem(AlphabeticStrings())
P = A.encoding("HAPPYNEWYEAR"); P
a, b = (1, 1)
C = A.enciphering(a, b, P); C
a, b = (9, 0)
C2 = A.enciphering(a, b, C); C2
a, b = (1, 25)
C3 = A.enciphering(a, b, C2); C3
This gives 'TINNQVSYQSIF', which is what I expected.
Thanks again for your help, I appreciate it a lot.
Regards,
Avril
On Tuesday, April 17, 2012 11:50:22 AM UTC+1, Avril Coghlan wrote:
>
> Dear all,
>
> I'm trying to figure out how to make a multiplicative cipher in SAGE.
> For example, to make a multiplicative cipher:
> M_9{m} = 9 x_{26} m
> and to use it to encipher the text 'HAPPYNEWYEAR', I tried typing:
>
>A = AffineCryptosystem(AlphabeticStrings())
>P = A.encoding("HAPPYNEWYEAR"); P
>a, b = (9, 0)
>C = A.enciphering(a, b, P); C
>
> I get back:
> LAFFINKQIKAX
>
> However, I think I should get back:
> TINNQVSYQSIF
> assuming A=1, B=2, C=3, D=4 Y=25, Z=26
>
> In order to get 'TINNQVSYQSIF', I have to type:
>a, b = (9, 8)
> instead of
>a, b = (9, 0)
> I am wondering why this is?
>
> I had thought:
>a, b = (9, 0)
> should give the multiplicative cipher:
> M_9{m} = 9 x_{26} m
> Is this wrong?
>
> I would be grateful for any pointers.
>
> Kind Regards,
> Avril
>
> Avril Coghlan
>
>
>
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[sage-support] a multiplicative cipher in SAGE
Dear all,
I'm trying to figure out how to make a multiplicative cipher in SAGE.
For example, to make a multiplicative cipher:
M_9{m} = 9 x_{26} m
and to use it to encipher the text 'HAPPYNEWYEAR', I tried typing:
A = AffineCryptosystem(AlphabeticStrings())
P = A.encoding("HAPPYNEWYEAR"); P
a, b = (9, 0)
C = A.enciphering(a, b, P); C
I get back:
LAFFINKQIKAX
However, I think I should get back:
TINNQVSYQSIF
assuming A=1, B=2, C=3, D=4 Y=25, Z=26
In order to get 'TINNQVSYQSIF', I have to type:
a, b = (9, 8)
instead of
a, b = (9, 0)
I am wondering why this is?
I had thought:
a, b = (9, 0)
should give the multiplicative cipher:
M_9{m} = 9 x_{26} m
Is this wrong?
I would be grateful for any pointers.
Kind Regards,
Avril
Avril Coghlan
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[sage-support] colouring documents about SAGE, written using sphinx
Dear all, I'd like to write a little document about using SAGE, using sphinx to write it, and host my document at readthedocs.org I'm wondering how can I get the beautiful colouring of sphinx documentation on SAGE, which I see at http://www.sagemath.org/doc/ ? As far as I can see, there is no specific 'sage' formatting available in sphinx (like there is for R, python, and many other languages), is that true? By the way, I wasn't sure if this query should go to the SAGE 'edu' mailing list, I hope I haven't sent it to the wrong one. Kind Regards, Avril Avril Coghlan Cork, Ireland -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
[sage-support] Re: finding a p-cycle of a real function
Thank you very much to both of you for your help with this, that is great, I appreciate it a lot. On Feb 11, 9:43 pm, John Cremona wrote: > Numerical precision issues would make it preferable (surely?) to use > exact arithmetic by replacing 1.76 by 176/100. > > Do you know the standard trick (as used for instance in the Pollard > rho method of factorization) which avoids storing lots of terms of the > sequence and instead runs two variables through the sequence > simultaneously, starting at the same point, but with one sequence > applying the iteration twice? then you just have to check when the > two values coincide. > > Details left to the reader, or > seehttp://en.wikipedia.org/wiki/Floyd%27s_cycle-finding_algorithm#Tortoi... > > John > > On 11 February 2012 20:34, D. S. McNeil wrote: > > > > > > > > > Warning: I haven't thought this through, but the ideas might be useful. > > > It looks like you'd call something a 3-cycle if for some x_0 we had > > f(f(f(x_0))) == x_0, right? Then we should be able to do this > > numerically, with some caveats: > > > # can't remember the slick way, so brute force > > def iter_apply(f0, n): > > fs = [f0()] > > for i in xrange(n-1): > > last = fs[-1] > > fs.append(f0(last)) > > return fs > > > def find_pcycles(f0, n): > > fs = iter_apply(f0, n) > > req = fs[-1] == x # defining equation of the cycle > > roots = req.roots(ring=RR) > > for root, mult in roots: > > yield [fi(x=root) for fi in fs] > > > which gives > > > sage: f(x) = x^2-1.76 > > sage: > > sage: p1 = list(find_pcycles(f, 1)) > > sage: p1 > > [[-0.917744687875782], [1.91774468787578]] > > sage: f(p1[0][0]) > > -0.917744687875783 > > sage: f(p1[1][0]) > > 1.91774468787578 > > sage: list(find_pcycles(f, 2)) > > [[0.504987562112089, -1.50498756211209], [-0.917744687875783, > > -0.917744687875782], [-1.50498756211209, 0.504987562112089], > > [1.91774468787578, 1.91774468787578]] > > sage: list(find_pcycles(f, 3)) > > [[1.33560128916886, 0.0238308036295167, -1.75943209279837], > > [1.27545967679486, -0.133202612870350, -1.74225706392451], > > [-0.917744687875783, -0.917744687875782, -0.917744687875783], > > [-1.74225706392451, 1.27545967679486, -0.133202612870348], > > [-1.75943209279837, 1.33560128916886, 0.0238308036295143], > > [-0.133202612870349, -1.74225706392451, 1.27545967679486], > > [0.0238308036295150, -1.75943209279837, 1.33560128916886], > > [1.91774468787578, 1.91774468787578, 1.91774468787578]] > > > Note that this says [-0.917744687875783, -0.917744687875782, > > -0.917744687875783] is a 3-cycle, which it kind of is.. > > > Anyway, maybe playing off of something like that would help? > > > Doug > > > -- > > To post to this group, send email to sage-support@googlegroups.com > > To unsubscribe from this group, send email to > > sage-support+unsubscr...@googlegroups.com > > For more options, visit this group > > athttp://groups.google.com/group/sage-support > > URL:http://www.sagemath.org -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
[sage-support] finding a p-cycle of a real function
Dear SAGE users and developers,
Is it possible for me to find a p-cycle of a real function using SAGE?
For example, the sequence x_{n+1} = x_{n}^2 - 1.76, x_{0} =0 (n = 0,1,2...)
has a three-cycle, so it alternates indefinitely between three numbers
(approximately 1.3, 0.0, and -1.8).
How can I find this three-cycle, or any p-cycle, using SAGE?
I would be very grateful for any advice.
Kind Regards,
Avril
Avril Coghlan
Cork, Ireland
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[sage-support] finding the closed form for a linear second order recurrence sequence
Dear all,
I am new to SAGE (since today) but am excited about learning how to use it!
I'm wondering is it possible to use SAGE to find the close form for a linear
second order recurrence sequence?
For example, can SAGE find a closed form for an equation such as:
u_{n+2} = 2 * u_{n+1} + 8 * u_{n}, where we know u_{0} = 2 and u_{1} = 7 ?
I will be very grateful for any help, or if you can point me towards the
right documentation page to look at.
Kind Regards,
Avril
Avril Coghlan
Cork, Ireland
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