Trac #10448: Extensions of ZZ are not unique parents

With an unpatched sage-4.6, we have:

{{{
sage: R1 = ZZ.extension(x**2+1,'i')
sage: R1 is ZZ.extension(x**2+1,'i')
False
sage: R1 is ZZ.extension(x**2+1,'i')
False
}}}

I think that should be changed.

URL: http://trac.sagemath.org/10448
Reported by: SimonKing
Ticket author(s): Julian Rüth
Reviewer(s): Peter Bruin
Trac #14543: Implement compositional inverses of cycle index series

In #14348, I submitted code which computes the cycle index of the
virtual species Ω, which is the compositional inverse of the species E+
of nonempty sets. It is possible to compute the compositional inverse of
(most) arbitrary cycle indices (subject to some conditions on the low-
order terms). The attached patch implements this calculation by adding a
compositional_inverse() method to the class CycleIndexSeries. This code
is doctested and builds HTML docs without errors.

It should be noted that this algorithm is very slow; however, it is the
best one I know for the general case. The algorithm for computing the
cycle index of Ω in #14348 is *much* faster, and that specific species
has real computational utility, so I believe it is worthwhile to have
both in Sage.

Apply [attachment:trac_14543_cycle_index_compositional_inverse.patch].

URL: http://trac.sagemath.org/14543
Reported by: agd
Ticket author(s): Andrew Gainer-Dewar
Reviewer(s): Frédéric Chapoton
Trac #11894: problems with infinite sum

A recent post on the number theory list asked to compute the value of
the infinite sum of `1/(m^4 + 2m^3 + 3m^2 + 2m)^2` for `m` between 1 and
infinity.

[https://listserv.nodak.edu/cgi-
bin/wa.exe?A2=ind1109&L=nmbrthry&T=0&P=1149]

Trying it to sage :
{{{
sage: var('m')
sage: s = sum(1/(m^4 + 2*m^3 + 3*m^2 + 2*m)^2, m, 1, infinity)
sage: s
1/12*pi^2 + 9/196*I*sqrt(7)*psi(1/14*(3*sqrt(7) - 7*I)*sqrt(7)) -
9/196*I*sqrt(7)*psi(1/14*(3*sqrt(7) + 7*I)*sqrt(7)) - 1/28*psi(1,
-1/2*I*sqrt(7) + 3/2) - 1/28*psi(1, 1/2*I*sqrt(7) + 3/2) - 1
}}}

The formula is less elegant than the formulas given by people who
answered using two proprietary sfotwares, but does not seem false. Sage
is not able to regognize it:
{{{
sage: bool(s == (-(19/16) + 1/84 * pi^2 * (7 - 3 * sech((sqrt(7) *
pi)/2)^2) + ( 9 * pi * tanh((sqrt(7) * pi)/2))/(28 * sqrt(7))))
False
sage: bool(s == -19/16 + 1/28*pi^2*tanh(1/2*pi*7^(1/2))^2 +
9/196*7^(1/2)*pi*tanh(1/2*pi*7^(1/2)) + 1/21*pi^2)
False
}}}

It is also not able to take the real part of a real number:
{{{
sage: CC(s)
0.0161011600422853
sage: RR(s)
[...]
TypeError: cannot convert -7*I to real number
}}}

Moreover, if we let `m` start to zero, sage does not provide an error
but a value:
{{{
sage: var('m')
sage: s = sum(1/(m^4 + 2*m^3 + 3*m^2 + 2*m)^2, m, 0, infinity)
sage: s
1/12*pi^2 + 9/196*I*sqrt(7)*psi(1/14*(sqrt(7) - 7*I)*sqrt(7)) -
9/196*I*sqrt(7)*psi(1/14*(sqrt(7) + 7*I)*sqrt(7)) - 1/28*psi(1,
-1/2*I*sqrt(7) + 1/2) - 1/28*psi(1, 1/2*I*sqrt(7) + 1/2)
sage: CC(s)
1.20360116004229
}}}

URL: http://trac.sagemath.org/11894
Reported by: tmonteil
Ticket author(s): Peter Bruin
Reviewer(s): Karl-Dieter Crisman
Trac #15386: Maxima is confused about limits at infinity

From https://groups.google.com/forum/#!topic/sage-support/dwR4kuBmiQo :
{{{
n = var('n')
assume(n>0)
series = -(3*n^2 + 1)*(-1)^n/sqrt(n^5 + 8*n^3 + 8)
limit(series, n=infinity)
38/25*pi^2*und
}}}
...when the limit is clearly zero.

The problem seems related to the `n^2` in the numerator:

{{{
sage: series = -(1)*(-1)^n/sqrt(n^5 + 8*n^3 + 8)
sage: limit(series, n=infinity)
0
sage: series = (n^2)*(-1)^n/sqrt(n^5 + 8*n^3 + 8)
sage: limit(series, n=infinity)
-38/75*pi^2*und
}}}

This seems to be an upstream problem with Maxima (see the sage-support
thread).

URL: http://trac.sagemath.org/15386
Reported by: ddrake
Ticket author(s): Karl-Dieter Crisman
Reviewer(s): Travis Scrimshaw
Trac #14209: Extra assumption kept by Maxima

From: https://groups.google.com/forum/?fromgroups=#!topic/sage-
devel/jIjYSSCPrrY

{{{
Initial symptom:
sage: integral(e^(-abs(x))/cosh(x),x,-infinity,infinity)
2*log(2)
sage: integral(e^(-abs(x))/cosh(x),x,-infinity,infinity)
------------------------------------------------------------------------
---
ValueError                                Traceback (most recent call
last)
}}}

A closer look:
{{{
sage: from sage.interfaces.maxima_lib import sr_to_max
sage: sr_to_max(integral(e^(-abs(x))/cosh(x),x,-infinity,infinity))
<ECL: ((MTIMES) ((%LOG) 2) 2)>
sage: sr_to_max(integral(e^(-abs(x))/cosh(x),x,-infinity,infinity))
------------------------------------------------------------------------
---
ValueError                                Traceback (most recent call
last)
}}}

URL: http://trac.sagemath.org/14209
Reported by: pipedream
Ticket author(s): Karl-Dieter Crisman
Reviewer(s): Travis Scrimshaw, Peter Bruin