https://github.com/uberparagon/mgn
Tip revision: bebc795e6778db315f0b7144c07024f8a666722f authored by Drew Johnson on 12 September 2017, 18:57:37 UTC
maybe some changes
maybe some changes
Tip revision: bebc795
index.doctree
cdocutils.nodes
document
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q2)q3}q4(U rawsourceq5XÒ StrataAlgegra documentation master file, created by
sphinx-quickstart on Thu Aug 31 18:54:11 2017.
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contain the root `toctree` directive.Uparentq6hUsourceq7X. /Users/drewjohnson/mgn/strataalgebra/index.rstq8Utagnameq9Ucommentq:U
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qGXÒ StrataAlgegra documentation master file, created by
sphinx-quickstart on Thu Aug 31 18:54:11 2017.
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contain the root `toctree` directive.qH
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ql)qm}qn(h5X strataalgebra Documentationqoh6heh7h8h9Utitleqph;}qq(hA]hB]h@]h?]hC]uhDK
hEhh0]qrhGX strataalgebra Documentationqs
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qv)qw}qx(h5X% Welcome! The ``strataalgebra`` module is designed to compute products in the strata algebra using sage. The strata algebra is of interest
because it has a natural surjective morphism to the tautological ring of the moduli space of curves. The kernel is described by the Faber-Zagier relations.h6heh7h8h9U paragraphqyh;}qz(hA]hB]h@]h?]hC]uhDK hEhh0]q{(hGX
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q)q}q(h5X ``strataalgebra``h;}q(hA]hB]h@]h?]hC]uh6hwh0]qhGX
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}q(h5U h6hubah9UliteralqubhGX module is designed to compute products in the strata algebra using sage. The strata algebra is of interest
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q}q(h5X module is designed to compute products in the strata algebra using sage. The strata algebra is of interest
because it has a natural surjective morphism to the tautological ring of the moduli space of curves. The kernel is described by the Faber-Zagier relations.h6hwubeubhv)q}q(h5X The product structure was implemented by `Drew Johnson`_, based on `S. Yang's note`_ describing the algorithm of Graber and
Pandharipande.
The program also computes the FZ relations using code copied from A. Pixton's `tautrel.sage`_ program.
The code was integrated into this package (with some modifications and optimizations) by Drew Johnson
(who takes responsibility for any bugs introduced!).h6heh7h8h9hyh;}q(hA]hB]h@]h?]hC]uhDKhEhh0]q(hGX) The product structure was implemented by q
q}q(h5X) The product structure was implemented by h6hubcdocutils.nodes
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q)q}q(h5X `Drew Johnson`_UresolvedqKh6hh9U referenceqh;}q(UnameX Drew JohnsonUrefuriqX http://pages.uoregon.edu/drewj/qh?]h@]hA]hB]hC]uh0]qhGX Drew Johnsonq
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q³}q´(h5U h6h¬ubaubhGX¤ program.
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q¸)q¹}qº(h5X1 .. _Drew Johnson: http://pages.uoregon.edu/drewj/U
referencedq»Kh6heh7h8h9Utargetq¼h;}q½(hhh?]q¾h(ah@]hA]hB]hC]q¿hauhDKhEhh0]ubh¸)qÀ}qÁ(h5XD .. _tautrel.sage: http://math.mit.edu/~apixton/programs/tautrel.sageh»Kh6heh7h8h9h¼h;}qÂ(hh°h?]qÃh-ah@]hA]hB]hC]qÄhauhDKhEhh0]ubh¸)qÅ}qÆ(h5X3 .. _S. Yang's note: https://arxiv.org/abs/0808.1974h»Kh6heh7h8h9h¼h;}qÇ(hh¤h?]qÈh*ah@]hA]hB]hC]qÉh
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qÖ}q×(h5hÒh6hÐubaubhv)qØ}qÙ(h5Xh Installation should be (hopefully) easy. ``strataalgebra`` is distributed as part of the `mgn package`_.qÚh6hÊh7h8h9hyh;}qÛ(hA]hB]h@]h?]hC]uhDKhEhh0]qÜ(hGX) Installation should be (hopefully) easy. qÝ
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qð}qñ(h5U h6hêubaubhGX .
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r^ }r_ (h5X First import the module:h6jX ubaubj )r` }ra (h5X! sage: from strataalgebra import *h6j} h7j
h9j h;}rb (h=h>h?]h@]hA]hB]hC]uhDKhEhh0]rc hGX! sage: from strataalgebra import *rd
re }rf (h5U h6j` ubaubhv)rg }rh (h5X& Construct a :class:`StrataAlgebra`: ::ri h6j} h7j
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StrataAlgebrar
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r }r
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Strata algebra with genus 1 and markings (1, 2) over Rational Fieldh6j} h7j
h9j h;}r (h=h>h?]h@]hA]hB]hC]uhDKhEhh0]r hGXl sage: SA = StrataAlgebra(QQ,1,(1,2)); SA
Strata algebra with genus 1 and markings (1, 2) over Rational Fieldr
r }r (h5U h6j ubaubhv)r }r (h5X+ Print the basis in a certain codimension ::r h6j} h7j
h9hyh;}r (hA]hB]h@]h?]hC]uhDKhEhh0]r hGX( Print the basis in a certain codimensionr
r }r (h5X( Print the basis in a certain codimensionh6j ubaubj )r }r (h5Xu sage: SA.print_strata(2)
**** i: 0
[0 1 2 0 0]
[0 0 0 2 1]
[0 1 1 0 1]
<BLANKLINE>
**** i: 1
[0 1 2 0 0]
[0 0 1 1 1]
[0 1 0 1 1]
<BLANKLINE>
**** i: 2
[ 0 1 2 0]
[ka1 1 1 2]
<BLANKLINE>
**** i: 3
[ 0 1 2 0]
[ 0 1 1 ps + 2]
<BLANKLINE>
**** i: 4
[ 0 1 2 0]
[ 0 ps + 1 1 2]
<BLANKLINE>
**** i: 5
[ 0 1 2 0]
[ 0 1 ps + 1 2]
<BLANKLINE>
**** i: 6
[ 0 1 2 0]
[ 0 1 1 1]
[ka1 + 1 0 0 1]
<BLANKLINE>
**** i: 7
[ 0 1 2 0]
[ 0 1 1 1]
[ 1 0 0 ps + 1]
<BLANKLINE>
**** i: 8
ka2
<BLANKLINE>
**** i: 9
ka1^2
<BLANKLINE>
**** i: 10
ka1*ps1
<BLANKLINE>
**** i: 11
ka1*ps2
<BLANKLINE>
**** i: 12
ps1^2
<BLANKLINE>
**** i: 13
ps1*ps2
<BLANKLINE>
**** i: 14
ps2^2
<BLANKLINE>h6j} h7j
h9j h;}r (h=h>h?]h@]hA]hB]hC]uhDKhEhh0]r hGXu sage: SA.print_strata(2)
**** i: 0
[0 1 2 0 0]
[0 0 0 2 1]
[0 1 1 0 1]
<BLANKLINE>
**** i: 1
[0 1 2 0 0]
[0 0 1 1 1]
[0 1 0 1 1]
<BLANKLINE>
**** i: 2
[ 0 1 2 0]
[ka1 1 1 2]
<BLANKLINE>
**** i: 3
[ 0 1 2 0]
[ 0 1 1 ps + 2]
<BLANKLINE>
**** i: 4
[ 0 1 2 0]
[ 0 ps + 1 1 2]
<BLANKLINE>
**** i: 5
[ 0 1 2 0]
[ 0 1 ps + 1 2]
<BLANKLINE>
**** i: 6
[ 0 1 2 0]
[ 0 1 1 1]
[ka1 + 1 0 0 1]
<BLANKLINE>
**** i: 7
[ 0 1 2 0]
[ 0 1 1 1]
[ 1 0 0 ps + 1]
<BLANKLINE>
**** i: 8
ka2
<BLANKLINE>
**** i: 9
ka1^2
<BLANKLINE>
**** i: 10
ka1*ps1
<BLANKLINE>
**** i: 11
ka1*ps2
<BLANKLINE>
**** i: 12
ps1^2
<BLANKLINE>
**** i: 13
ps1*ps2
<BLANKLINE>
**** i: 14
ps2^2
<BLANKLINE>r
r }r (h5U h6j ubaubhv)r }r (h5XZ Classes that are monomials in :math:`\psi` and :math:`\kappa` have self-explanatory names.r h6j} h7j
h9hyh;}r (hA]hB]h@]h?]hC]uhDKOhEhh0]r (hGX Classes that are monomials in r¡
r¢ }r£ (h5X Classes that are monomials in h6j ubcsphinx.ext.mathbase
math
r¤ )r¥ }r¦ (h5U h;}r§ (UlatexX \psih?]h@]hA]hB]hC]uh6j h0]h9Umathr¨ ubhGX and r©
rª }r« (h5X and h6j ubj¤ )r¬ }r (h5U h;}r® (UlatexX \kappah?]h@]hA]hB]hC]uh6j h0]h9j¨ ubhGX have self-explanatory names.r¯
r° }r± (h5X have self-explanatory names.h6j ubeubhv)r² }r³ (h5X> More complicated classes are displayed in a matrix as follows:r´ h6j} h7j
h9hyh;}rµ (hA]hB]h@]h?]hC]uhDKQhEhh0]r¶ hGX> More complicated classes are displayed in a matrix as follows:r·
r¸ }r¹ (h5j´ h6j² ubaubhv)rº }r» (h5X~ Each row after the first corresponds to a vertex of the graph.
The constant term in the entry in the first column is the genus.
The kappa classes also appear in the first column.
Each column beyond the first corresponds to an edge or a half edge.
The entry in the first row gives the label of the half edge, or 0 for a full edge.
The constant term of the entry in location (v,e) gives the number of times (0, 1, or 2) that edge e touches vertex v.
A `ps` in entry (v,e) means a :math:`\psi`-class associated to the half edge coming out of v.
A `ps_` may occur when there is a loop at the vertex.
The entry in the top left is just padding.h6j} h7j
h9hyh;}r¼ (hA]hB]h@]h?]hC]uhDKShEhh0]r½ (hGXÂ Each row after the first corresponds to a vertex of the graph.
The constant term in the entry in the first column is the genus.
The kappa classes also appear in the first column.
Each column beyond the first corresponds to an edge or a half edge.
The entry in the first row gives the label of the half edge, or 0 for a full edge.
The constant term of the entry in location (v,e) gives the number of times (0, 1, or 2) that edge e touches vertex v.
A r¾
r¿ }rÀ (h5XÂ Each row after the first corresponds to a vertex of the graph.
The constant term in the entry in the first column is the genus.
The kappa classes also appear in the first column.
Each column beyond the first corresponds to an edge or a half edge.
The entry in the first row gives the label of the half edge, or 0 for a full edge.
The constant term of the entry in location (v,e) gives the number of times (0, 1, or 2) that edge e touches vertex v.
A h6jº ubjØ )rÁ }r (h5X `ps`h;}rà (hA]hB]h@]h?]hC]uh6jº h0]rÄ hGX psrÅ
rÆ }rÇ (h5U h6jÁ ubah9jà ubhGX in entry (v,e) means a rÈ
rÉ }rÊ (h5X in entry (v,e) means a h6jº ubj¤ )rË }rÌ (h5U h;}rÍ (UlatexX \psih?]h@]hA]hB]hC]uh6jº h0]h9j¨ ubhGX6 -class associated to the half edge coming out of v.
A rÎ
rÏ }rÐ (h5X6 -class associated to the half edge coming out of v.
A h6jº ubjØ )rÑ }rÒ (h5X `ps_`h;}rÓ (hA]hB]h@]h?]hC]uh6jº h0]rÔ hGX ps_rÕ
rÖ }r× (h5U h6jÑ ubah9jà ubhGXY may occur when there is a loop at the vertex.
The entry in the top left is just padding.rØ
rÙ }rÚ (h5XY may occur when there is a loop at the vertex.
The entry in the top left is just padding.h6jº ubeubhv)rÛ }rÜ (h5XÂ To create classes, you can use their codimension and index.
Boundary strata and psi-kappa monomials are represented by their special names,
and the rest are represented by `s_{codim},{index}` ::h6j} h7j
h9hyh;}rÝ (hA]hB]h@]h?]hC]uhDK]hEhh0]rÞ (hGX¬ To create classes, you can use their codimension and index.
Boundary strata and psi-kappa monomials are represented by their special names,
and the rest are represented by rß
rà }rá (h5X¬ To create classes, you can use their codimension and index.
Boundary strata and psi-kappa monomials are represented by their special names,
and the rest are represented by h6jÛ ubjØ )râ }rã (h5X `s_{codim},{index}`h;}rä (hA]hB]h@]h?]hC]uh6jÛ h0]rå hGX s_{codim},{index}ræ
rç }rè (h5U h6jâ ubah9jà ubeubj )ré }rê (h5Xo sage: a = SA(2, 1); a
s_2,1
sage: b = SA(2, 7); b
s_2,7
sage: c = SA(2, 11); c
ka1*ps2
sage: d = SA(1,1); d
Dg1h6j} h7j
h9j h;}rë (h=h>h?]h@]hA]hB]hC]uhDKehEhh0]rì hGXo sage: a = SA(2, 1); a
s_2,1
sage: b = SA(2, 7); b
s_2,7
sage: c = SA(2, 11); c
ka1*ps2
sage: d = SA(1,1); d
Dg1rí
rî }rï (h5U h6jé ubaubhv)rð }rñ (h5X( Vector space arithmetic is supported. ::rò h6j} h7j
h9hyh;}ró (hA]hB]h@]h?]hC]uhDKjhEhh0]rô hGX% Vector space arithmetic is supported.rõ
rö }r÷ (h5X% Vector space arithmetic is supported.h6jð ubaubj )rø }rù (h5X: sage: 3*b
3*s_2,7
sage: a*72 - b/17
72*s_2,1 + -1/17*s_2,7h6j} h7j
h9j h;}rú (h=h>h?]h@]hA]hB]hC]uhDKphEhh0]rû hGX: sage: 3*b
3*s_2,7
sage: a*72 - b/17
72*s_2,1 + -1/17*s_2,7rü
rý }rþ (h5U h6jø ubaubhv)rÿ }r (h5Xo Use :meth:`~strataalgebra.StrataAlgebra.get_stratum` if you need to know what an unamed basis element means. ::r h6j} h7j
h9hyh;}r (hA]hB]h@]h?]hC]uhDKqhEhh0]r (hGX Use r
r }r (h5X Use h6jÿ ubj¿ )r }r (h5X0 :meth:`~strataalgebra.StrataAlgebra.get_stratum`r h6jÿ h7Nh9jÏ h;}r
(UreftypeX methjs jt X' strataalgebra.StrataAlgebra.get_stratumU refdomainX pyr h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]r h)r
}r (h5j h;}r (hA]hB]r (j~ j X py-methr eh@]h?]hC]uh6j h0]r hGX
get_stratum()r
r }r (h5U h6j
ubah9hubaubhGX8 if you need to know what an unamed basis element means.r
r }r (h5X8 if you need to know what an unamed basis element means.h6jÿ ubeubj )r }r (h5Xs sage: SA.get_stratum(2,7)
[ 0 1 2 0]
[ 0 1 1 1]
[ 1 0 0 ps + 1]h6j} h7j
h9j h;}r (h=h>h?]h@]hA]hB]hC]uhDKwhEhh0]r hGXs sage: SA.get_stratum(2,7)
[ 0 1 2 0]
[ 0 1 1 1]
[ 1 0 0 ps + 1]r
r }r (h5U h6j ubaubhv)r }r! (h5X You can construct :math:`\psi,\;\kappa` monomials and boundary divisors with the methods :meth:`~strataalgebra.StrataAlgebra.kappa`, :meth:`~strataalgebra.StrataAlgebra.psi`,
:meth:`~strataalgebra.StrataAlgebra.boundary`, and :meth:`~strataalgebra.StrataAlgebra.irr`.h6j} h7j
h9hyh;}r" (hA]hB]h@]h?]hC]uhDKxhEhh0]r# (hGX You can construct r$
r% }r& (h5X You can construct h6j ubj¤ )r' }r( (h5U h;}r) (UlatexX
\psi,\;\kappah?]h@]hA]hB]hC]uh6j h0]h9j¨ ubhGX2 monomials and boundary divisors with the methods r*
r+ }r, (h5X2 monomials and boundary divisors with the methods h6j ubj¿ )r- }r. (h5X* :meth:`~strataalgebra.StrataAlgebra.kappa`r/ h6j h7Nh9jÏ h;}r0 (UreftypeX methjs jt X! strataalgebra.StrataAlgebra.kappaU refdomainX pyr1 h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]r2 h)r3 }r4 (h5j/ h;}r5 (hA]hB]r6 (j~ j1 X py-methr7 eh@]h?]hC]uh6j- h0]r8 hGX kappa()r9
r: }r; (h5U h6j3 ubah9hubaubhGX , r<
r= }r> (h5X , h6j ubj¿ )r? }r@ (h5X( :meth:`~strataalgebra.StrataAlgebra.psi`rA h6j h7Nh9jÏ h;}rB (UreftypeX methjs jt X strataalgebra.StrataAlgebra.psiU refdomainX pyrC h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]rD h)rE }rF (h5jA h;}rG (hA]hB]rH (j~ jC X py-methrI eh@]h?]hC]uh6j? h0]rJ hGX psi()rK
rL }rM (h5U h6jE ubah9hubaubhGX ,
rN
rO }rP (h5X ,
h6j ubj¿ )rQ }rR (h5X- :meth:`~strataalgebra.StrataAlgebra.boundary`rS h6j h7Nh9jÏ h;}rT (UreftypeX methjs jt X$ strataalgebra.StrataAlgebra.boundaryU refdomainX pyrU h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]rV h)rW }rX (h5jS h;}rY (hA]hB]rZ (j~ jU X py-methr[ eh@]h?]hC]uh6jQ h0]r\ hGX
boundary()r]
r^ }r_ (h5U h6jW ubah9hubaubhGX , and r`
ra }rb (h5X , and h6j ubj¿ )rc }rd (h5X( :meth:`~strataalgebra.StrataAlgebra.irr`re h6j h7Nh9jÏ h;}rf (UreftypeX methjs jt X strataalgebra.StrataAlgebra.irrU refdomainX pyrg h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]rh h)ri }rj (h5je h;}rk (hA]hB]rl (j~ jg X py-methrm eh@]h?]hC]uh6jc h0]rn hGX irr()ro
rp }rq (h5U h6ji ubah9hubaubhGX .
rr }rs (h5X .h6j ubeubhv)rt }ru (h5Xv You can construct an element using the matrix notation. Just pass a list of lists into your
:class:`StrataAlgebra`. ::h6j} h7j
h9hyh;}rv (hA]hB]h@]h?]hC]uhDK{hEhh0]rw (hGX\ You can construct an element using the matrix notation. Just pass a list of lists into your
rx
ry }rz (h5X\ You can construct an element using the matrix notation. Just pass a list of lists into your
h6jt ubj¿ )r{ }r| (h5X :class:`StrataAlgebra`r} h6jt h7Nh9jÏ h;}r~ (UreftypeX classjs jt X
StrataAlgebraU refdomainX pyr h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]r h)r }r (h5j} h;}r (hA]hB]r (j~ j X py-classr
eh@]h?]hC]uh6j{ h0]r hGX
StrataAlgebrar
r }r (h5U h6j ubah9hubaubhGX .
r }r (h5X .h6jt ubeubj )r }r (h5X¤ sage: var('ps') #This is usually be done automatically, but we have to do it manually for the dotests to work.
ps
sage: SA([[0,1,2,0],[1,0,0,ps+1],[0,1,1,1]])
s_2,7h6j} h7j
h9j h;}r (h=h>h?]h@]hA]hB]hC]uhDKhEhh0]r hGX¤ sage: var('ps') #This is usually be done automatically, but we have to do it manually for the dotests to work.
ps
sage: SA([[0,1,2,0],[1,0,0,ps+1],[0,1,1,1]])
s_2,7r
r }r (h5U h6j ubaubhv)r }r (h5X# Here is an example of the `ps_`: ::r h6j} h7j
h9hyh;}r (hA]hB]h@]h?]hC]uhDKhEhh0]r (hGX Here is an example of the r
r }r (h5X Here is an example of the h6j ubjØ )r }r (h5X `ps_`h;}r (hA]hB]h@]h?]hC]uh6j h0]r hGX ps_r
r }r¡ (h5U h6j ubah9jà ubhGX :
r¢ }r£ (h5X :h6j ubeubj )r¤ }r¥ (h5Xå sage: s = StrataAlgebra(QQ,2,())
sage: s.get_stratum(3,9)
[ 0 0]
[ 1 ps^2 + 2]
sage: s.get_stratum(3,10)
[ 0 0]
[ 1 ps*ps_ + 2]
sage: var('ps_')
ps_
sage: s([[0,0],[1,ps_*ps+2]])
s_3,10h6j} h7j
h9j h;}r¦ (h=h>h?]h@]hA]hB]hC]uhDKhEhh0]r§ hGXå sage: s = StrataAlgebra(QQ,2,())
sage: s.get_stratum(3,9)
[ 0 0]
[ 1 ps^2 + 2]
sage: s.get_stratum(3,10)
[ 0 0]
[ 1 ps*ps_ + 2]
sage: var('ps_')
ps_
sage: s([[0,0],[1,ps_*ps+2]])
s_3,10r¨
r© }rª (h5U h6j¤ ubaubhv)r« }r¬ (h5X> One of the main features is the computation of the product. ::r h6j} h7j
h9hyh;}r® (hA]hB]h@]h?]hC]uhDKhEhh0]r¯ hGX; One of the main features is the computation of the product.r°
r± }r² (h5X; One of the main features is the computation of the product.h6j« ubaubj )r³ }r´ (h5X sage: a*b
0h6j} h7j
h9j h;}rµ (h=h>h?]h@]hA]hB]hC]uhDKhEhh0]r¶ hGX sage: a*b
0r·
r¸ }r¹ (h5U h6j³ ubaubhv)rº }r» (h5XJ Of course, the codimension was too big. Lets do some less trivial ones. ::r¼ h6j} h7j
h9hyh;}r½ (hA]hB]h@]h?]hC]uhDKhEhh0]r¾ hGXG Of course, the codimension was too big. Lets do some less trivial ones.r¿
rÀ }rÁ (h5XG Of course, the codimension was too big. Lets do some less trivial ones.h6jº ubaubj )r }rà (h5Xt sage: SA = StrataAlgebra(QQ,1,(1,2,3)); SA
Strata algebra with genus 1 and markings (1, 2, 3) over Rational Field
sage: SA.psi(1) * SA.psi(2)
ps1*ps2
sage: SA.get_stratum(2,7) #just so you can see what it is
[0 1 2 3 0 0]
[0 0 1 1 1 0]
[0 1 0 0 1 1]
[1 0 0 0 0 1]
sage: SA(2,7)*SA.psi(1)
0
sage: SA(2,7)*SA.kappa(1)
s_3,36
sage: SA.get_stratum(3,36)
[ 0 1 2 3 0 0]
[ 0 0 1 1 1 0]
[ 0 1 0 0 1 1]
[ka1 + 1 0 0 0 0 1]
sage: SA.irr()^3
6*s_3,4 + -3*s_3,15 + -3*s_3,23 + -3*s_3,35 + s_3,48 + s_3,49h6j} h7j
h9j h;}rÄ (h=h>h?]h@]hA]hB]hC]uhDKhEhh0]rÅ hGXt sage: SA = StrataAlgebra(QQ,1,(1,2,3)); SA
Strata algebra with genus 1 and markings (1, 2, 3) over Rational Field
sage: SA.psi(1) * SA.psi(2)
ps1*ps2
sage: SA.get_stratum(2,7) #just so you can see what it is
[0 1 2 3 0 0]
[0 0 1 1 1 0]
[0 1 0 0 1 1]
[1 0 0 0 0 1]
sage: SA(2,7)*SA.psi(1)
0
sage: SA(2,7)*SA.kappa(1)
s_3,36
sage: SA.get_stratum(3,36)
[ 0 1 2 3 0 0]
[ 0 0 1 1 1 0]
[ 0 1 0 0 1 1]
[ka1 + 1 0 0 0 0 1]
sage: SA.irr()^3
6*s_3,4 + -3*s_3,15 + -3*s_3,23 + -3*s_3,35 + s_3,48 + s_3,49rÆ
rÇ }rÈ (h5U h6j ubaubhv)rÉ }rÊ (h5X6 Everything should work with distributive laws, etc. ::rË h6j} h7j
h9hyh;}rÌ (hA]hB]h@]h?]hC]uhDKhEhh0]rÍ hGX3 Everything should work with distributive laws, etc.rÎ
rÏ }rÐ (h5X3 Everything should work with distributive laws, etc.h6jÉ ubaubj )rÑ }rÒ (h5X_ sage: SA.kappa(2)*(3*SA.psi(3)-SA(1,3)) == -SA(1,3) *SA.kappa(2) + SA.psi(3)*SA.kappa(2)*3
Trueh6j} h7j
h9j h;}rÓ (h=h>h?]h@]hA]hB]hC]uhDK³hEhh0]rÔ hGX_ sage: SA.kappa(2)*(3*SA.psi(3)-SA(1,3)) == -SA(1,3) *SA.kappa(2) + SA.psi(3)*SA.kappa(2)*3
TruerÕ
rÖ }r× (h5U h6jÑ ubaubhv)rØ }rÙ (h5X It should work over any ring. ::rÚ h6j} h7j
h9hyh;}rÛ (hA]hB]h@]h?]hC]uhDK²hEhh0]rÜ hGX It should work over any ring.rÝ
rÞ }rß (h5X It should work over any ring.h6jØ ubaubj )rà }rá (h5X. sage: R.<t> = PolynomialRing(ZZ)
sage: SA = StrataAlgebra(R, 2); SA
Strata algebra with genus 2 and markings () over Univariate Polynomial Ring in t over Integer Ring
sage: (3+t)*SA.kappa(1) + t^2 + t*SA.kappa(1)
(t^2)one + (2*t + 3)ka1
sage: _^2
(t^4)one + (4*t^3 + 6*t^2)ka1 + (4*t^2 + 12*t + 9)ka1^2h6j} h7j
h9j h;}râ (h=h>h?]h@]hA]hB]hC]uhDK¸hEhh0]rã hGX. sage: R.<t> = PolynomialRing(ZZ)
sage: SA = StrataAlgebra(R, 2); SA
Strata algebra with genus 2 and markings () over Univariate Polynomial Ring in t over Integer Ring
sage: (3+t)*SA.kappa(1) + t^2 + t*SA.kappa(1)
(t^2)one + (2*t + 3)ka1
sage: _^2
(t^4)one + (4*t^3 + 6*t^2)ka1 + (4*t^2 + 12*t + 9)ka1^2rä
rå }ræ (h5U h6jà ubaubhv)rç }rè (h5X3 There may be problems over a non-divisible ring. ::ré h6j} h7j
h9hyh;}rê (hA]hB]h@]h?]hC]uhDK½hEhh0]rë hGX0 There may be problems over a non-divisible ring.rì
rí }rî (h5X0 There may be problems over a non-divisible ring.h6jç ubaubj )rï }rð (h5Xß sage: SA.irr()
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *: 'Rational Field' and 'Strata algebra with genus 2 and markings () over Univariate Polynomial Ring in t over Integer Ring'h6j} h7j
h9j h;}rñ (h=h>h?]h@]hA]hB]hC]uhDKÃhEhh0]rò hGXß sage: SA.irr()
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *: 'Rational Field' and 'Strata algebra with genus 2 and markings () over Univariate Polynomial Ring in t over Integer Ring'ró
rô }rõ (h5U h6jï ubaubhv)rö }r÷ (h5X³ Also, repeated names of the markings are allowed. The following corresponds to :math:`\overline{\mathcal M}_{1,2}/S_2`.
Compare the codimension 2 strata to our earlier example. ::h6j} h7j
h9hyh;}rø (hA]hB]h@]h?]hC]uhDKÄhEhh0]rù (hGXO Also, repeated names of the markings are allowed. The following corresponds to rú
rû }rü (h5XO Also, repeated names of the markings are allowed. The following corresponds to h6jö ubj¤ )rý }rþ (h5U h;}rÿ (UlatexX \overline{\mathcal M}_{1,2}/S_2h?]h@]hA]hB]hC]uh6jö h0]h9j¨ ubhGX: .
Compare the codimension 2 strata to our earlier example.r
r }r (h5X: .
Compare the codimension 2 strata to our earlier example.h6jö ubeubj )r }r (h5XS sage: SA = StrataAlgebra(QQ,1,(1,1)); SA
Strata algebra with genus 1 and markings (1, 1) over Rational Field
sage: SA.print_strata(2)
**** i: 0
[0 1 1 0 0]
[0 0 0 2 1]
[0 1 1 0 1]
<BLANKLINE>
**** i: 1
[0 1 1 0 0]
[0 0 1 1 1]
[0 1 0 1 1]
<BLANKLINE>
**** i: 2
[ 0 1 1 0]
[ka1 1 1 2]
<BLANKLINE>
**** i: 3
[ 0 1 1 0]
[ 0 1 1 ps + 2]
<BLANKLINE>
**** i: 4
[ 0 1 1 0]
[ 0 ps + 1 1 2]
<BLANKLINE>
**** i: 5
[ 0 1 1 0]
[ 0 1 1 1]
[ka1 + 1 0 0 1]
<BLANKLINE>
**** i: 6
[ 0 1 1 0]
[ 0 1 1 1]
[ 1 0 0 ps + 1]
<BLANKLINE>
**** i: 7
ka2
<BLANKLINE>
**** i: 8
ka1^2
<BLANKLINE>
**** i: 9
ka1*ps1
<BLANKLINE>
**** i: 10
ps1^2
<BLANKLINE>
**** i: 11
ps1*ps1
<BLANKLINE>h6j} h7j
h9j h;}r (h=h>h?]h@]hA]hB]hC]uhDKËhEhh0]r hGXS sage: SA = StrataAlgebra(QQ,1,(1,1)); SA
Strata algebra with genus 1 and markings (1, 1) over Rational Field
sage: SA.print_strata(2)
**** i: 0
[0 1 1 0 0]
[0 0 0 2 1]
[0 1 1 0 1]
<BLANKLINE>
**** i: 1
[0 1 1 0 0]
[0 0 1 1 1]
[0 1 0 1 1]
<BLANKLINE>
**** i: 2
[ 0 1 1 0]
[ka1 1 1 2]
<BLANKLINE>
**** i: 3
[ 0 1 1 0]
[ 0 1 1 ps + 2]
<BLANKLINE>
**** i: 4
[ 0 1 1 0]
[ 0 ps + 1 1 2]
<BLANKLINE>
**** i: 5
[ 0 1 1 0]
[ 0 1 1 1]
[ka1 + 1 0 0 1]
<BLANKLINE>
**** i: 6
[ 0 1 1 0]
[ 0 1 1 1]
[ 1 0 0 ps + 1]
<BLANKLINE>
**** i: 7
ka2
<BLANKLINE>
**** i: 8
ka1^2
<BLANKLINE>
**** i: 9
ka1*ps1
<BLANKLINE>
**** i: 10
ps1^2
<BLANKLINE>
**** i: 11
ps1*ps1
<BLANKLINE>r
r }r (h5U h6j ubaubcsphinx.addnodes
seealso
r
)r }r (h5X :meth:`strataalgebra.StrataAlgebraElement.integrate`
:meth:`~strataalgebra.StrataAlgebra.betti`
:meth:`~strataalgebra.StrataAlgebra.FZ_betti`h6j} h7j
h9Useealsor
h;}r (hA]hB]h@]h?]hC]uhDNhEhh0]r (hv)r }r (h5X4 :meth:`strataalgebra.StrataAlgebraElement.integrate`r h6j h7j
h9hyh;}r (hA]hB]h@]h?]hC]uhDKþh0]r j¿ )r }r (h5j h6j h7Nh9jÏ h;}r (UreftypeX methjs jt X, strataalgebra.StrataAlgebraElement.integrateU refdomainX pyr h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]r h)r }r (h5j h;}r (hA]hB]r (j~ j X py-methr eh@]h?]hC]uh6j h0]r hGX. strataalgebra.StrataAlgebraElement.integrate()r
r! }r" (h5U h6j ubah9hubaubaubhv)r# }r$ (h5X* :meth:`~strataalgebra.StrataAlgebra.betti`r% h6j h7j
h9hyh;}r& (hA]hB]h@]h?]hC]uhDM h0]r' j¿ )r( }r) (h5j% h6j# h7Nh9jÏ h;}r* (UreftypeX methjs jt X! strataalgebra.StrataAlgebra.bettiU refdomainX pyr+ h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]r, h)r- }r. (h5j% h;}r/ (hA]hB]r0 (j~ j+ X py-methr1 eh@]h?]hC]uh6j( h0]r2 hGX betti()r3
r4 }r5 (h5U h6j- ubah9hubaubaubhv)r6 }r7 (h5X- :meth:`~strataalgebra.StrataAlgebra.FZ_betti`r8 h6j h7j
h9hyh;}r9 (hA]hB]h@]h?]hC]uhDMh0]r: j¿ )r; }r< (h5j8 h6j6 h7Nh9jÏ h;}r= (UreftypeX methjs jt X$ strataalgebra.StrataAlgebra.FZ_bettiU refdomainX pyr> h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]r? h)r@ }rA (h5j8 h;}rB (hA]hB]rC (j~ j> X py-methrD eh@]h?]hC]uh6j; h0]rE hGX
FZ_betti()rF
rG }rH (h5U h6j@ ubah9hubaubaubeubj )rI }rJ (h5U h6j} h7Nh9j h;}rK (h?]h@]hA]hB]hC]Uentries]rL (j X/ FZ_betti() (strataalgebra.StrataAlgebra method)hU NtrM auhDNhEhh0]ubj )rN }rO (h5U h6j} h7Nh9j# h;}rP (j% j& X pyh?]h@]hA]hB]hC]j( X methodrQ j* jQ uhDNhEhh0]rR (j, )rS }rT (h5X" StrataAlgebra.FZ_betti(codim=None)h6jN h7Xg /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_bettirU h9j0 h;}rV (h?]rW haj3 j4 X
strataalgebrarX
rY }rZ bh@]hA]hB]hC]r[ haj9 X StrataAlgebra.FZ_bettij; j: j< uhDNhEhh0]r\ (jP )r] }r^ (h5X FZ_bettih6jS h7jU h9jS h;}r_ (hA]hB]h@]h?]hC]uhDNhEhh0]r` hGX FZ_bettira
rb }rc (h5U h6j] ubaubjY )rd }re (h5X
codim=Noneh6jS h7jU h9j\ h;}rf (hA]hB]h@]h?]hC]uhDNhEhh0]rg j_ )rh }ri (h5X
codim=Noneh;}rj (hA]hB]h@]h?]hC]uh6jd h0]rk hGX
codim=Nonerl
rm }rn (h5U h6jh ubah9jg ubaubeubj| )ro }rp (h5U h6jN h7jU h9j h;}rq (hA]hB]h@]h?]hC]uhDNhEhh0]rr (hv)rs }rt (h5X® Give the dimension of the cohomology of moduli space of curves, as predicted by the Faber-Zagier-Pixton relations.
If the codimension is omited, return a list of all of them.ru h6jo h7Xg /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_bettirv h9hyh;}rw (hA]hB]h@]h?]hC]uhDKhEhh0]rx hGX® Give the dimension of the cohomology of moduli space of curves, as predicted by the Faber-Zagier-Pixton relations.
If the codimension is omited, return a list of all of them.ry
rz }r{ (h5ju h6js ubaubj )r| }r} (h5U h6jo h7Nh9j h;}r~ (hA]hB]h@]h?]hC]uhDNhEhh0]r j )r }r (h5U h;}r (hA]hB]h@]h?]hC]uh6j| h0]r (j )r }r
(h5X
Parametersh6j h7jU h9j h;}r (hA]hB]h@]h?]hC]uhDK h0]r hGX
Parametersr
r }r (h5U h6j ubaubj )r }r (h5U h;}r (hA]hB]h@]h?]hC]uh6j h0]r hv)r }r (h5X, codim -- Optional. The codimension you want.h;}r (hA]hB]h@]h?]hC]uh6j h0]r (j² )r }r (h5X codimh;}r (h?]h@]j¶ hA]hB]hC]uh6j h0]r hGX codimr
r }r (h5U h6j ubah9j» ubhGX -- r
r }r (h5U h6j ubhGX# Optional. The codimension you want.r
r }r (h5X# Optional. The codimension you want.h6j ubeh9hyubah9jV ubeh9jW ubaubj )r }r¡ (h5X sage: from strataalgebra import *
sage: StrataAlgebra(QQ, 2, (1,)).FZ_betti()
[1, 3, 5, 3, 1]
sage: StrataAlgebra(QQ, 2, (1,)).FZ_betti(2)
5h6jo h7jv h9j h;}r¢ (h=h>h?]h@]hA]hB]hC]uhDMhEhh0]r£ hGX sage: from strataalgebra import *
sage: StrataAlgebra(QQ, 2, (1,)).FZ_betti()
[1, 3, 5, 3, 1]
sage: StrataAlgebra(QQ, 2, (1,)).FZ_betti(2)
5r¤
r¥ }r¦ (h5U h6j ubaubj
)r§ }r¨ (h5X* :meth:`~strataalgebra.StrataAlgebra.betti`r© h6jo h7jv h9j
h;}rª (hA]hB]h@]h?]hC]uhDNhEhh0]r« hv)r¬ }r (h5j© h6j§ h7jv h9hyh;}r® (hA]hB]h@]h?]hC]uhDKh0]r¯ j¿ )r° }r± (h5j© h6j¬ h7Nh9jÏ h;}r² (UreftypeX methjs jt X! strataalgebra.StrataAlgebra.bettiU refdomainX pyr³ h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]r´ h)rµ }r¶ (h5j© h;}r· (hA]hB]r¸ (j~ j³ X py-methr¹ eh@]h?]hC]uh6j° h0]rº hGX betti()r»
r¼ }r½ (h5U h6jµ ubah9hubaubaubaubeubeubj )r¾ }r¿ (h5U h6j} h7Nh9j h;}rÀ (h?]h@]hA]hB]hC]Uentries]rÁ (j X0 FZ_matrix() (strataalgebra.StrataAlgebra method)hU Ntr auhDNhEhh0]ubj )rà }rÄ (h5U h6j} h7Nh9j# h;}rÅ (j% j& X pyh?]h@]hA]hB]hC]j( X methodrÆ j* jÆ uhDNhEhh0]rÇ (j, )rÈ }rÉ (h5X StrataAlgebra.FZ_matrix(r)h6jà h7Xh /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_matrixrÊ h9j0 h;}rË (h?]rÌ haj3 j4 X
strataalgebrarÍ
rÎ }rÏ bh@]hA]hB]hC]rÐ haj9 X StrataAlgebra.FZ_matrixj; j: j< uhDNhEhh0]rÑ (jP )rÒ }rÓ (h5X FZ_matrixh6jÈ h7jÊ h9jS h;}rÔ (hA]hB]h@]h?]hC]uhDNhEhh0]rÕ hGX FZ_matrixrÖ
r× }rØ (h5U h6jÒ ubaubjY )rÙ }rÚ (h5X rh6jÈ h7jÊ h9j\ h;}rÛ (hA]hB]h@]h?]hC]uhDNhEhh0]rÜ j_ )rÝ }rÞ (h5X rh;}rß (hA]hB]h@]h?]hC]uh6jÙ h0]rà hGX r
rá }râ (h5U h6jÝ ubah9jg ubaubeubj| )rã }rä (h5U h6jà h7jÊ h9j h;}rå (hA]hB]h@]h?]hC]uhDNhEhh0]ræ (hv)rç }rè (h5X3 Return the matrix of Faber-Zagier-Pixton relations.ré h6jã h7Xh /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_matrixrê h9hyh;}rë (hA]hB]h@]h?]hC]uhDKhEhh0]rì hGX3 Return the matrix of Faber-Zagier-Pixton relations.rí
rî }rï (h5jé h6jç ubaubj )rð }rñ (h5U h6jã h7Nh9j h;}rò (hA]hB]h@]h?]hC]uhDNhEhh0]ró (j )rô }rõ (h5U h;}rö (hA]hB]h@]h?]hC]uh6jð h0]r÷ (j )rø }rù (h5X
Parametersh6jô h7jÊ h9j h;}rú (hA]hB]h@]h?]hC]uhDK h0]rû hGX
Parametersrü
rý }rþ (h5U h6jø ubaubj )rÿ }r (h5U h;}r (hA]hB]h@]h?]hC]uh6jô h0]r hv)r }r (h5X r -- h;}r (hA]hB]h@]h?]hC]uh6jÿ h0]r (j² )r }r (h5X rh;}r (h?]h@]j¶ hA]hB]hC]uh6j h0]r
hGX r
r }r (h5U h6j ubah9j» ubhGX -- r
r }r (h5U h6j ubeh9hyubah9jV ubeh9jW ubj )r }r (h5U h;}r (hA]hB]h@]h?]hC]uh6jð h0]r (j )r }r (h5X Return typeh6j h7jÊ h9j h;}r (hA]hB]h@]h?]hC]uhDK h0]r hGX Return typer
r }r (h5U h6j ubaubj )r }r (h5U h;}r (hA]hB]h@]h?]hC]uh6j h0]r hv)r }r (h5X Matrixh;}r! (hA]hB]h@]h?]hC]uh6j h0]r" j¿ )r# }r$ (h5X :class:`Matrix`r% h6j h7Nh9jÏ h;}r& (UreftypeX classjs jt X MatrixU refdomainX pyr' h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]r( h)r) }r* (h5j% h;}r+ (hA]hB]r, (j~ j' X py-classr- eh@]h?]hC]uh6j# h0]r. hGX Matrixr/
r0 }r1 (h5U h6j) ubah9hubaubah9hyubah9jV ubeh9jW ubeubhv)r2 }r3 (h5X_ The columns correspond to the basis elements of the Strata algebra, and each row is a relation.r4 h6jã h7jê h9hyh;}r5 (hA]hB]h@]h?]hC]uhDKhEhh0]r6 hGX_ The columns correspond to the basis elements of the Strata algebra, and each row is a relation.r7
r8 }r9 (h5j4 h6j2 ubaubhv)r: }r; (h5X Notice that this matrix considers the kappa classes to be in the monomial basis. Thus, is different than the
output of Pixton's original `taurel.sage` program.h6jã h7jê h9hyh;}r< (hA]hB]h@]h?]hC]uhDKhEhh0]r= (hGX Notice that this matrix considers the kappa classes to be in the monomial basis. Thus, is different than the
output of Pixton's original r>
r? }r@ (h5X Notice that this matrix considers the kappa classes to be in the monomial basis. Thus, is different than the
output of Pixton's original h6j: ubjØ )rA }rB (h5X
`taurel.sage`h;}rC (hA]hB]h@]h?]hC]uh6j: h0]rD hGX taurel.sagerE
rF }rG (h5U h6jA ubah9jà ubhGX program.rH
rI }rJ (h5X program.h6j: ubeubj
)rK }rL (h5X@ :meth:`~strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basis`rM h6jã h7jê h9j
h;}rN (hA]hB]h@]h?]hC]uhDNhEhh0]rO hv)rP }rQ (h5jM h6jK h7jê h9hyh;}rR (hA]hB]h@]h?]hC]uhDK
h0]rS j¿ )rT }rU (h5jM h6jP h7Nh9jÏ h;}rV (UreftypeX methjs jt X7 strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basisU refdomainX pyrW h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]rX h)rY }rZ (h5jM h;}r[ (hA]hB]r\ (j~ jW X py-methr] eh@]h?]hC]uh6jT h0]r^ hGX FZ_matrix_pushforward_basis()r_
r` }ra (h5U h6jY ubah9hubaubaubaubeubeubj )rb }rc (h5U h6j} h7Xz /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basisrd h9j h;}re (h?]h@]hA]hB]hC]Uentries]rf (j XB FZ_matrix_pushforward_basis() (strataalgebra.StrataAlgebra method)h U Ntrg auhDNhEhh0]ubj )rh }ri (h5U h6j} h7jd h9j# h;}rj (j% j& X pyh?]h@]hA]hB]hC]j( X methodrk j* jk uhDNhEhh0]rl (j, )rm }rn (h5X, StrataAlgebra.FZ_matrix_pushforward_basis(r)h6jh h7Xz /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basisro h9j0 h;}rp (h?]rq h aj3 j4 X
strataalgebrarr
rs }rt bh@]hA]hB]hC]ru h aj9 X) StrataAlgebra.FZ_matrix_pushforward_basisj; j: j< uhDNhEhh0]rv (jP )rw }rx (h5X FZ_matrix_pushforward_basish6jm h7jo h9jS h;}ry (hA]hB]h@]h?]hC]uhDNhEhh0]rz hGX FZ_matrix_pushforward_basisr{
r| }r} (h5U h6jw ubaubjY )r~ }r (h5X rh6jm h7jo h9j\ h;}r (hA]hB]h@]h?]hC]uhDNhEhh0]r j_ )r }r (h5X rh;}r (hA]hB]h@]h?]hC]uh6j~ h0]r
hGX r
r }r (h5U h6j ubah9jg ubaubeubj| )r }r (h5U h6jh h7jo h9j h;}r (hA]hB]h@]h?]hC]uhDNhEhh0]r (hv)r }r (h5X Return the matrix of Faber-Zagier-Pixton relations, using the "pushforward" basis, NOT the kappa monomial basis
that the rest of the code uses.r h6j h7jd h9hyh;}r (hA]hB]h@]h?]hC]uhDKhEhh0]r hGX Return the matrix of Faber-Zagier-Pixton relations, using the "pushforward" basis, NOT the kappa monomial basis
that the rest of the code uses.r
r }r (h5j h6j ubaubj )r }r (h5U h6j h7jd h9j h;}r (hA]hB]h@]h?]hC]uhDNhEhh0]r (j )r }r (h5U h;}r (hA]hB]h@]h?]hC]uh6j h0]r (j )r }r (h5X
Parametersh6j h7jd h9j h;}r (hA]hB]h@]h?]hC]uhDK h0]r hGX
Parametersr
r¡ }r¢ (h5U h6j ubaubj )r£ }r¤ (h5U h;}r¥ (hA]hB]h@]h?]hC]uh6j h0]r¦ hv)r§ }r¨ (h5X r -- h;}r© (hA]hB]h@]h?]hC]uh6j£ h0]rª (j² )r« }r¬ (h5X rh;}r (h?]h@]j¶ hA]hB]hC]uh6j§ h0]r® hGX r
r¯ }r° (h5U h6j« ubah9j» ubhGX -- r±
r² }r³ (h5U h6j§ ubeh9hyubah9jV ubeh9jW ubj )r´ }rµ (h5U h;}r¶ (hA]hB]h@]h?]hC]uh6j h0]r· (j )r¸ }r¹ (h5X Return typeh6j´ h7jd h9j h;}rº (hA]hB]h@]h?]hC]uhDK h0]r» hGX Return typer¼
r½ }r¾ (h5U h6j¸ ubaubj )r¿ }rÀ (h5U h;}rÁ (hA]hB]h@]h?]hC]uh6j´ h0]r hv)rà }rÄ (h5X Matrixh;}rÅ (hA]hB]h@]h?]hC]uh6j¿ h0]rÆ j¿ )rÇ }rÈ (h5X :class:`Matrix`rÉ h6jà h7Nh9jÏ h;}rÊ (UreftypeX classjs jt X MatrixU refdomainX pyrË h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]rÌ h)rÍ }rÎ (h5jÉ h;}rÏ (hA]hB]rÐ (j~ jË X py-classrÑ eh@]h?]hC]uh6jÇ h0]rÒ hGX MatrixrÓ
rÔ }rÕ (h5U h6jÍ ubah9hubaubah9hyubah9jV ubeh9jW ubeubhv)rÖ }r× (h5XÃ The columns correspond to the basis elements of the Strata algebra, and each row is a relation.
This matrix should be the same as Pixton's original `tautrel.sage` program after permuting columns.h6j h7jd h9hyh;}rØ (hA]hB]h@]h?]hC]uhDKhEhh0]rÙ (hGX The columns correspond to the basis elements of the Strata algebra, and each row is a relation.
This matrix should be the same as Pixton's original rÚ
rÛ }rÜ (h5X The columns correspond to the basis elements of the Strata algebra, and each row is a relation.
This matrix should be the same as Pixton's original h6jÖ ubjØ )rÝ }rÞ (h5X `tautrel.sage`h;}rß (hA]hB]h@]h?]hC]uh6jÖ h0]rà hGX tautrel.sagerá
râ }rã (h5U h6jÝ ubah9jà ubhGX! program after permuting columns.rä
rå }ræ (h5X! program after permuting columns.h6jÖ ubeubeubeubj )rç }rè (h5U h6j} h7Xj /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.MgnLb_classré h9j h;}rê (h?]h@]hA]hB]hC]Uentries]rë (j X2 MgnLb_class() (strataalgebra.StrataAlgebra method)hU Ntrì auhDNhEhh0]ubj )rí }rî (h5U h6j} h7jé h9j# h;}rï (j% j& X pyh?]h@]hA]hB]hC]j( X methodrð j* jð uhDNhEhh0]rñ (j, )rò }ró (h5X StrataAlgebra.MgnLb_class(index)h6jí h7Xj /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.MgnLb_classrô h9j0 h;}rõ (h?]rö haj3 j4 X
strataalgebrar÷
rø }rù bh@]hA]hB]hC]rú haj9 X StrataAlgebra.MgnLb_classj; j: j< uhDNhEhh0]rû (jP )rü }rý (h5X MgnLb_classh6jò h7jô h9jS h;}rþ (hA]hB]h@]h?]hC]uhDNhEhh0]rÿ hGX MgnLb_classr
r }r (h5U h6jü ubaubjY )r }r (h5X indexh6jò h7jô h9j\ h;}r (hA]hB]h@]h?]hC]uhDNhEhh0]r j_ )r }r (h5X indexh;}r (hA]hB]h@]h?]hC]uh6j h0]r
hGX indexr
r }r
(h5U h6j ubah9jg ubaubeubj| )r }r (h5U h6jí h7jô h9j h;}r (hA]hB]h@]h?]hC]uhDNhEhh0]r (hv)r }r (h5X| Returns the class corresponding to the index from Carl Faber's Mgn_Lb Maple program.
This is useful for testing purposes. ::h6j h7jé h9hyh;}r (hA]hB]h@]h?]hC]uhDKhEhh0]r hGXy Returns the class corresponding to the index from Carl Faber's Mgn_Lb Maple program.
This is useful for testing purposes.r
r }r (h5Xy Returns the class corresponding to the index from Carl Faber's Mgn_Lb Maple program.
This is useful for testing purposes.h6j ubaubj )r }r (h5X· sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,2))
sage: s.MgnLb_class(1)
ps1
sage: s.MgnLb_class(4)
ka2
sage: s.MgnLb_class(6)
1/2*D_irr
sage: s.MgnLb_class(2)
ps2h6j h7jé h9j h;}r (h=h>h?]h@]hA]hB]hC]uhDMFhEhh0]r hGX· sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,2))
sage: s.MgnLb_class(1)
ps1
sage: s.MgnLb_class(4)
ka2
sage: s.MgnLb_class(6)
1/2*D_irr
sage: s.MgnLb_class(2)
ps2r
r }r (h5U h6j ubaubeubeubj )r }r! (h5U h6j} h7Nh9j h;}r" (h?]h@]hA]hB]hC]Uentries]r# (j X0 MgnLb_int() (strataalgebra.StrataAlgebra method)hU Ntr$ auhDNhEhh0]ubj )r% }r& (h5U h6j} h7Nh9j# h;}r' (j% j& X pyh?]h@]hA]hB]hC]j( X methodr( j* j( uhDNhEhh0]r) (j, )r* }r+ (h5X# StrataAlgebra.MgnLb_int(index_list)h6j% h7Xh /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.MgnLb_intr, h9j0 h;}r- (h?]r. haj3 j4 X
strataalgebrar/
r0 }r1 bh@]hA]hB]hC]r2 haj9 X StrataAlgebra.MgnLb_intj; j: j< uhDNhEhh0]r3 (jP )r4 }r5 (h5X MgnLb_inth6j* h7j, h9jS h;}r6 (hA]hB]h@]h?]hC]uhDNhEhh0]r7 hGX MgnLb_intr8
r9 }r: (h5U h6j4 ubaubjY )r; }r< (h5X
index_listh6j* h7j, h9j\ h;}r= (hA]hB]h@]h?]hC]uhDNhEhh0]r> j_ )r? }r@ (h5X
index_listh;}rA (hA]hB]h@]h?]hC]uh6j; h0]rB hGX
index_listrC
rD }rE (h5U h6j? ubah9jg ubaubeubj| )rF }rG (h5U h6j% h7j, h9j h;}rH (hA]hB]h@]h?]hC]uhDNhEhh0]rI (hv)rJ }rK (h5XJ Computes an integral of boundary divisors, kappa classes, and psi classes.rL h6jF h7Xh /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.MgnLb_intrM h9hyh;}rN (hA]hB]h@]h?]hC]uhDKhEhh0]rO hGXJ Computes an integral of boundary divisors, kappa classes, and psi classes.rP
rQ }rR (h5jL h6jJ ubaubj )rS }rT (h5U h6jF h7Nh9j h;}rU (hA]hB]h@]h?]hC]uhDNhEhh0]rV (j )rW }rX (h5U h;}rY (hA]hB]h@]h?]hC]uh6jS h0]rZ (j )r[ }r\ (h5X
Parametersh6jW h7j, h9j h;}r] (hA]hB]h@]h?]hC]uhDK h0]r^ hGX
Parametersr_
r` }ra (h5U h6j[ ubaubj )rb }rc (h5U h;}rd (hA]hB]h@]h?]hC]uh6jW h0]re hv)rf }rg (h5XÑ index_list -- A list of indices of classes, according the the scheme of Carl Faber's MgnLb Maple program. This function is useful because so you can test our implementation of the product and the FZ_relations.h;}rh (hA]hB]h@]h?]hC]uh6jb h0]ri (j² )rj }rk (h5X
index_listh;}rl (h?]h@]j¶ hA]hB]hC]uh6jf h0]rm hGX
index_listrn
ro }rp (h5U h6jj ubah9j» ubhGX -- rq
rr }rs (h5U h6jf ubhGXÃ A list of indices of classes, according the the scheme of Carl Faber's MgnLb Maple program. This function is useful because so you can test our implementation of the product and the FZ_relations.rt
ru }rv (h5XÃ A list of indices of classes, according the the scheme of Carl Faber's MgnLb Maple program. This function is useful because so you can test our implementation of the product and the FZ_relations.h6jf ubeh9hyubah9jV ubeh9jW ubj )rw }rx (h5U h;}ry (hA]hB]h@]h?]hC]uh6jS h0]rz (j )r{ }r| (h5X Return typeh6jw h7j, h9j h;}r} (hA]hB]h@]h?]hC]uhDK h0]r~ hGX Return typer
r }r (h5U h6j{ ubaubj )r }r (h5U h;}r (hA]hB]h@]h?]hC]uh6jw h0]r
hv)r }r (h5X Rationalh;}r (hA]hB]h@]h?]hC]uh6j h0]r jØ )r }r (h5X
`Rational`h;}r (hA]hB]h@]h?]hC]uh6j h0]r hGX Rationalr
r }r (h5U h6j ubah9jà ubah9hyubah9jV ubeh9jW ubeubhv)r }r (h5X Examples: ::h6jF h7jM h9hyh;}r (hA]hB]h@]h?]hC]uhDKhEhh0]r hGX Examples:r
r }r (h5X Examples:h6j ubaubj )r }r (h5X sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,2))
sage: s.MgnLb_int([1,6])
1/2
sage: s.MgnLb_int([1,2])
1/24
sage: s.MgnLb_int([4])
1/24h6jF h7jM h9j h;}r (h=h>h?]h@]hA]hB]hC]uhDM]hEhh0]r hGX sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,2))
sage: s.MgnLb_int([1,6])
1/2
sage: s.MgnLb_int([1,2])
1/24
sage: s.MgnLb_int([4])
1/24r
r }r (h5U h6j ubaubj
)r }r (h5X0 :meth:`~strataalgebra.StrataAlgebra.MgnLb_class`r¡ h6jF h7jM h9j
h;}r¢ (hA]hB]h@]h?]hC]uhDNhEhh0]r£ hv)r¤ }r¥ (h5j¡ h6j h7jM h9hyh;}r¦ (hA]hB]h@]h?]hC]uhDKh0]r§ j¿ )r¨ }r© (h5j¡ h6j¤ h7Nh9jÏ h;}rª (UreftypeX methjs jt X' strataalgebra.StrataAlgebra.MgnLb_classU refdomainX pyr« h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]r¬ h)r }r® (h5j¡ h;}r¯ (hA]hB]r° (j~ j« X py-methr± eh@]h?]hC]uh6j¨ h0]r² hGX
MgnLb_class()r³
r´ }rµ (h5U h6j ubah9hubaubaubaubeubeubj )r¶ }r· (h5U h6j} h7Xn /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.basis_integralsr¸ h9j h;}r¹ (h?]h@]hA]hB]hC]Uentries]rº (j X7 basis_integrals (strataalgebra.StrataAlgebra attribute)h U Ntr» auhDNhEhh0]ubj )r¼ }r½ (h5U h6j} h7j¸ h9j# h;}r¾ (j% j& X pyh?]h@]hA]hB]hC]j( X attributer¿ j* j¿ uhDNhEhh0]rÀ (j, )rÁ }r (h5X StrataAlgebra.basis_integralsh6j¼ h7Xn /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.basis_integralsrà h9j0 h;}rÄ (h?]rÅ h aj3 j4 X
strataalgebrarÆ
rÇ }rÈ bh@]hA]hB]hC]rÉ h aj9 X StrataAlgebra.basis_integralsj; j: j< uhDNhEhh0]rÊ jP )rË }rÌ (h5X basis_integralsh6jÁ h7jà h9jS h;}rÍ (hA]hB]h@]h?]hC]uhDNhEhh0]rÎ hGX basis_integralsrÏ
rÐ }rÑ (h5U h6jË ubaubaubj| )rÒ }rÓ (h5U h6j¼ h7jà h9j h;}rÔ (hA]hB]h@]h?]hC]uhDNhEhh0]rÕ (hv)rÖ }r× (h5XR File: /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py (starting at line 417)rØ h6jÒ h7j¸ h9hyh;}rÙ (hA]hB]h@]h?]hC]uhDKhEhh0]rÚ hGXR File: /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py (starting at line 417)rÛ
rÜ }rÝ (h5jØ h6jÖ ubaubhv)rÞ }rß (h5Xe Return a list of numbers corresponding to the integrals of the basis elements in the top codimension.rà h6jÒ h7j¸ h9hyh;}rá (hA]hB]h@]h?]hC]uhDKhEhh0]râ hGXe Return a list of numbers corresponding to the integrals of the basis elements in the top codimension.rã
rä }rå (h5jà h6jÞ ubaubhv)ræ }rç (h5X_ This is computed via the FZ relations, so it is probably not fast. However, it is a nice check.rè h6jÒ h7j¸ h9hyh;}ré (hA]hB]h@]h?]hC]uhDKhEhh0]rê hGX_ This is computed via the FZ relations, so it is probably not fast. However, it is a nice check.rë
rì }rí (h5jè h6jæ ubaubhv)rî }rï (h5XE The value is cached, so you only have to compute it once per session.rð h6jÒ h7j¸ h9hyh;}rñ (hA]hB]h@]h?]hC]uhDKhEhh0]rò hGXE The value is cached, so you only have to compute it once per session.ró
rô }rõ (h5jð h6jî ubaubhv)rö }r÷ (h5Xo This is used by :meth:`StrataAlgebraElement.integrate` which is the more likely way you will want to use it.
::h6jÒ h7j¸ h9hyh;}rø (hA]hB]h@]h?]hC]uhDK hEhh0]rù (hGX This is used by rú
rû }rü (h5X This is used by h6jö ubj¿ )rý }rþ (h5X& :meth:`StrataAlgebraElement.integrate`rÿ h6jö h7Nh9jÏ h;}r (UreftypeX methjs jt X StrataAlgebraElement.integrateU refdomainX pyr h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]r h)r }r (h5jÿ h;}r (hA]hB]r (j~ j X py-methr eh@]h?]hC]uh6jý h0]r hGX StrataAlgebraElement.integrate()r
r
}r (h5U h6j ubah9hubaubhGX6 which is the more likely way you will want to use it.r
r
}r (h5X6 which is the more likely way you will want to use it.h6jö ubeubj )r }r (h5XÖ sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,))
sage: s.print_strata(1)
**** i: 0
D_irr
<BLANKLINE>
**** i: 1
ka1
<BLANKLINE>
**** i: 2
ps1
<BLANKLINE>
sage: s.basis_integrals()
[1, 1/24, 1/24]h6jÒ h7j¸ h9j h;}r (h=h>h?]h@]hA]hB]hC]uhDMzhEhh0]r hGXÖ sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,))
sage: s.print_strata(1)
**** i: 0
D_irr
<BLANKLINE>
**** i: 1
ka1
<BLANKLINE>
**** i: 2
ps1
<BLANKLINE>
sage: s.basis_integrals()
[1, 1/24, 1/24]r
r }r (h5U h6j ubaubeubeubj )r }r (h5U h6j} h7Xd /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.bettir h9j h;}r (h?]h@]hA]hB]hC]Uentries]r (j X, betti() (strataalgebra.StrataAlgebra method)hU Ntr auhDNhEhh0]ubj )r }r (h5U h6j} h7j h9j# h;}r (j% j& X pyr h?]h@]hA]hB]hC]j( X methodr j* j uhDNhEhh0]r! (j, )r" }r# (h5X StrataAlgebra.betti(codim=None)h6j h7Xd /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.bettir$ h9j0 h;}r% (h?]r& haj3 j4 X
strataalgebrar'
r( }r) bh@]hA]hB]hC]r* haj9 X StrataAlgebra.bettij; j: j< uhDNhEhh0]r+ (jP )r, }r- (h5X bettih6j" h7j$ h9jS h;}r. (hA]hB]h@]h?]hC]uhDNhEhh0]r/ hGX bettir0
r1 }r2 (h5U h6j, ubaubjY )r3 }r4 (h5X
codim=Noneh6j" h7j$ h9j\ h;}r5 (hA]hB]h@]h?]hC]uhDNhEhh0]r6 j_ )r7 }r8 (h5X
codim=Noneh;}r9 (hA]hB]h@]h?]hC]uh6j3 h0]r: hGX
codim=Noner;
r< }r= (h5U h6j7 ubah9jg ubaubeubj| )r> }r? (h5U h6j h7j$ h9j h;}r@ (hA]hB]h@]h?]hC]uhDNhEhh0]rA (hv)rB }rC (h5X Give the number of basis elements in the Strata algebra for the given codimension.
If the codimension is omitted, return a list of all of them.rD h6j> h7j h9hyh;}rE (hA]hB]h@]h?]hC]uhDKhEhh0]rF hGX Give the number of basis elements in the Strata algebra for the given codimension.
If the codimension is omitted, return a list of all of them.rG
rH }rI (h5jD h6jB ubaubhv)rJ }rK (h5Xy This is NOT the Betti numbers for the moduli space of curves! For that see :meth:`~strataalgebra.StrataAlgebra.FZ_betti`.h6j> h7j h9hyh;}rL (hA]hB]h@]h?]hC]uhDKhEhh0]rM (hGXK This is NOT the Betti numbers for the moduli space of curves! For that see rN
rO }rP (h5XK This is NOT the Betti numbers for the moduli space of curves! For that see h6jJ ubj¿ )rQ }rR (h5X- :meth:`~strataalgebra.StrataAlgebra.FZ_betti`rS h6jJ h7Nh9jÏ h;}rT (UreftypeX methjs jt X$ strataalgebra.StrataAlgebra.FZ_bettiU refdomainX pyrU h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]rV h)rW }rX (h5jS h;}rY (hA]hB]rZ (j~ jU X py-methr[ eh@]h?]hC]uh6jQ h0]r\ hGX
FZ_betti()r]
r^ }r_ (h5U h6jW ubah9hubaubhGX .
r` }ra (h5X .h6jJ ubeubj )rb }rc (h5U h6j> h7j h9j h;}rd (hA]hB]h@]h?]hC]uhDNhEhh0]re j )rf }rg (h5U h;}rh (hA]hB]h@]h?]hC]uh6jb h0]ri (j )rj }rk (h5X
Parametersh6jf h7j h9j h;}rl (hA]hB]h@]h?]hC]uhDK h0]rm hGX
Parametersrn
ro }rp (h5U h6jj ubaubj )rq }rr (h5U h;}rs (hA]hB]h@]h?]hC]uh6jf h0]rt hv)ru }rv (h5X2 codim (int) -- Optional. The codimension you want.h;}rw (hA]hB]h@]h?]hC]uh6jq h0]rx (j² )ry }rz (h5X codimh;}r{ (h?]h@]j¶ hA]hB]hC]uh6ju h0]r| hGX codimr}
r~ }r (h5U h6jy ubah9j» ubhGX (r
r }r (h5U h6ju ubj¿ )r }r (h5U h;}r
(Ureftypejà j¶ U reftargetX intr U refdomainj h?]h@]U refexplicithA]hB]hC]uh6ju h0]r jÆ )r }r (h5j h;}r (hA]hB]h@]h?]hC]uh6j h0]r hGX intr
r }r (h5U h6j ubah9jÎ ubah9jÏ ubhGX )
r }r (h5U h6ju ubhGX -- r
r }r (h5U h6ju ubhGX# Optional. The codimension you want.r
r }r (h5X# Optional. The codimension you want.h6ju ubeh9hyubah9jV ubeh9jW ubaubj )r }r (h5X sage: from strataalgebra import *
sage: StrataAlgebra(QQ, 2, (1,)).betti()
[1, 4, 17, 49, 92]
sage: StrataAlgebra(QQ, 2, (1,)).betti(2)
17h6j> h7j h9j h;}r (h=h>h?]h@]hA]hB]hC]uhDMhEhh0]r hGX sage: from strataalgebra import *
sage: StrataAlgebra(QQ, 2, (1,)).betti()
[1, 4, 17, 49, 92]
sage: StrataAlgebra(QQ, 2, (1,)).betti(2)
17r
r }r (h5U h6j ubaubeubeubj )r }r (h5U h6j} h7Xg /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.boundaryr h9j h;}r¡ (h?]h@]hA]hB]hC]Uentries]r¢ (j X/ boundary() (strataalgebra.StrataAlgebra method)h U Ntr£ auhDNhEhh0]ubj )r¤ }r¥ (h5U h6j} h7j h9j# h;}r¦ (j% j& X pyr§ h?]h@]hA]hB]hC]j( X methodr¨ j* j¨ uhDNhEhh0]r© (j, )rª }r« (h5X( StrataAlgebra.boundary(g1, markings1=())h6j¤ h7Xg /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.boundaryr¬ h9j0 h;}r (h?]r® h aj3 j4 X
strataalgebrar¯
r° }r± bh@]hA]hB]hC]r² h aj9 X StrataAlgebra.boundaryj; j: j< uhDNhEhh0]r³ (jP )r´ }rµ (h5X boundaryh6jª h7j¬ h9jS h;}r¶ (hA]hB]h@]h?]hC]uhDNhEhh0]r· hGX boundaryr¸
r¹ }rº (h5U h6j´ ubaubjY )r» }r¼ (h5X g1, markings1=()h6jª h7j¬ h9j\ h;}r½ (hA]hB]h@]h?]hC]uhDNhEhh0]r¾ (j_ )r¿ }rÀ (h5X g1h;}rÁ (hA]hB]h@]h?]hC]uh6j» h0]r hGX g1rÃ
rÄ }rÅ (h5U h6j¿ ubah9jg ubj_ )rÆ }rÇ (h5X markings1=()h;}rÈ (hA]hB]h@]h?]hC]uh6j» h0]rÉ hGX markings1=()rÊ
rË }rÌ (h5U h6jÆ ubah9jg ubeubeubj| )rÍ }rÎ (h5U h6j¤ h7j¬ h9j h;}rÏ (hA]hB]h@]h?]hC]uhDNhEhh0]rÐ (hv)rÑ }rÒ (h5XQ Return a boundary divisor with genus `g1` and `markings1` points on one componenth6jÍ h7j h9hyh;}rÓ (hA]hB]h@]h?]hC]uhDKhEhh0]rÔ (hGX% Return a boundary divisor with genus rÕ
rÖ }r× (h5X% Return a boundary divisor with genus h6jÑ ubjØ )rØ }rÙ (h5X `g1`h;}rÚ (hA]hB]h@]h?]hC]uh6jÑ h0]rÛ hGX g1rÜ
rÝ }rÞ (h5U h6jØ ubah9jà ubhGX and rß
rà }rá (h5X and h6jÑ ubjØ )râ }rã (h5X `markings1`h;}rä (hA]hB]h@]h?]hC]uh6jÑ h0]rå hGX markings1ræ
rç }rè (h5U h6jâ ubah9jà ubhGX points on one componentré
rê }rë (h5X points on one componenth6jÑ ubeubj )rì }rí (h5U h6jÍ h7j h9j h;}rî (hA]hB]h@]h?]hC]uhDNhEhh0]rï (j )rð }rñ (h5U h;}rò (hA]hB]h@]h?]hC]uh6jì h0]ró (j )rô }rõ (h5X
Parametersh6jð h7j h9j h;}rö (hA]hB]h@]h?]hC]uhDK h0]r÷ hGX
Parametersrø
rù }rú (h5U h6jô ubaubj )rû }rü (h5U h;}rý (hA]hB]h@]h?]hC]uh6jð h0]rþ j¤ )rÿ }r (h5U h;}r (hA]hB]h@]h?]hC]uh6jû h0]r (j© )r }r (h5U h;}r (hA]hB]h@]h?]hC]uh6jÿ h0]r hv)r }r (h5X& g1 (int) -- The genus on one componenth;}r (hA]hB]h@]h?]hC]uh6j h0]r
(j² )r }r (h5X g1h;}r
(h?]h@]j¶ hA]hB]hC]uh6j h0]r hGX g1r
r }r (h5U h6j ubah9j» ubhGX (r
r }r (h5U h6j ubj¿ )r }r (h5U h;}r (Ureftypejà j¶ U reftargetX intr U refdomainj§ h?]h@]U refexplicithA]hB]hC]uh6j h0]r jÆ )r }r (h5j h;}r (hA]hB]h@]h?]hC]uh6j h0]r hGX intr
r }r (h5U h6j ubah9jÎ ubah9jÏ ubhGX )
r! }r" (h5U h6j ubhGX -- r#
r$ }r% (h5U h6j ubhGX The genus on one componentr&
r' }r( (h5X The genus on one componenth6j ubeh9hyubah9jã ubj© )r) }r* (h5U h;}r+ (hA]hB]h@]h?]hC]uh6jÿ h0]r, hv)r- }r. (h5Xa markings1 (list) -- A list or tuple of markings on the one component. Defaults to the empty list.h;}r/ (hA]hB]h@]h?]hC]uh6j) h0]r0 (j² )r1 }r2 (h5X markings1h;}r3 (h?]h@]j¶ hA]hB]hC]uh6j- h0]r4 hGX markings1r5
r6 }r7 (h5U h6j1 ubah9j» ubhGX (r8
r9 }r: (h5U h6j- ubj¿ )r; }r< (h5U h;}r= (Ureftypejà j¶ U reftargetX listr> U refdomainj§ h?]h@]U refexplicithA]hB]hC]uh6j- h0]r? jÆ )r@ }rA (h5j> h;}rB (hA]hB]h@]h?]hC]uh6j; h0]rC hGX listrD
rE }rF (h5U h6j@ ubah9jÎ ubah9jÏ ubhGX )
rG }rH (h5U h6j- ubhGX -- rI
rJ }rK (h5U h6j- ubhGXM A list or tuple of markings on the one component. Defaults to the empty list.rL
rM }rN (h5XM A list or tuple of markings on the one component. Defaults to the empty list.h6j- ubeh9hyubah9jã ubeh9jU ubah9jV ubeh9jW ubj )rO }rP (h5U h;}rQ (hA]hB]h@]h?]hC]uh6jì h0]rR (j )rS }rT (h5X Return typeh6jO h7j h9j h;}rU (hA]hB]h@]h?]hC]uhDK h0]rV hGX Return typerW
rX }rY (h5U h6jS ubaubj )rZ }r[ (h5U h;}r\ (hA]hB]h@]h?]hC]uh6jO h0]r] hv)r^ }r_ (h5X StrataAlgebraElementh;}r` (hA]hB]h@]h?]hC]uh6jZ h0]ra j¿ )rb }rc (h5X :class:`StrataAlgebraElement`rd h6j^ h7Nh9jÏ h;}re (UreftypeX classjs jt X StrataAlgebraElementU refdomainX pyrf h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]rg h)rh }ri (h5jd h;}rj (hA]hB]rk (j~ jf X py-classrl eh@]h?]hC]uh6jb h0]rm hGX StrataAlgebraElementrn
ro }rp (h5U h6jh ubah9hubaubah9hyubah9jV ubeh9jW ubeubhv)rq }rr (h5Xf Note that if the two components are the same, it will return the stratum with a coefficient of 1/2. ::h6jÍ h7j h9hyh;}rs (hA]hB]h@]h?]hC]uhDKhEhh0]rt hGXc Note that if the two components are the same, it will return the stratum with a coefficient of 1/2.ru
rv }rw (h5Xc Note that if the two components are the same, it will return the stratum with a coefficient of 1/2.h6jq ubaubj )rx }ry (h5XÇ sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ, 1, (1,2,3))
sage: s.boundary(0, (1,2))
Dg0m1_2
sage: s.boundary(1,[3])
Dg0m1_2
sage: s = StrataAlgebra(QQ, 2)
sage: s.boundary(1)
1/2*Dg1h6jÍ h7j h9j h;}rz (h=h>h?]h@]hA]hB]hC]uhDM¬hEhh0]r{ hGXÇ sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ, 1, (1,2,3))
sage: s.boundary(0, (1,2))
Dg0m1_2
sage: s.boundary(1,[3])
Dg0m1_2
sage: s = StrataAlgebra(QQ, 2)
sage: s.boundary(1)
1/2*Dg1r|
r} }r~ (h5U h6jx ubaubhv)r }r (h5X' Each component must still be stable. ::h6jÍ h7j h9hyh;}r (hA]hB]h@]h?]hC]uhDKhEhh0]r hGX$ Each component must still be stable.r
r }r
(h5X$ Each component must still be stable.h6j ubaubj )r }r (h5XH sage: s.boundary(2)
Traceback (most recent call last):
...
KeyError: Dg0h6jÍ h7j h9j h;}r (h=h>h?]h@]hA]hB]hC]uhDM¸hEhh0]r hGXH sage: s.boundary(2)
Traceback (most recent call last):
...
KeyError: Dg0r
r }r (h5U h6j ubaubeubeubj )r }r (h5U h6j} h7Xn /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.do_all_productsr h9j h;}r (h?]h@]hA]hB]hC]Uentries]r (j X6 do_all_products() (strataalgebra.StrataAlgebra method)hU Ntr auhDNhEhh0]ubj )r }r (h5U h6j} h7j h9j# h;}r (j% j& X pyh?]h@]hA]hB]hC]j( X methodr j* j uhDNhEhh0]r (j, )r }r (h5X StrataAlgebra.do_all_products()h6j h7Xn /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.do_all_productsr h9j0 h;}r (h?]r haj3 j4 X
strataalgebrar
r }r bh@]hA]hB]hC]r haj9 X StrataAlgebra.do_all_productsj; j: j< uhDNhEhh0]r¡ (jP )r¢ }r£ (h5X do_all_productsh6j h7j h9jS h;}r¤ (hA]hB]h@]h?]hC]uhDNhEhh0]r¥ hGX do_all_productsr¦
r§ }r¨ (h5U h6j¢ ubaubjY )r© }rª (h5U h6j h7j h9j\ h;}r« (hA]hB]h@]h?]hC]uhDNhEhh0]ubeubj| )r¬ }r (h5U h6j h7j h9j h;}r® (hA]hB]h@]h?]hC]uhDNhEhh0]r¯ (hv)r° }r± (h5X Compute all the products of basis elements of the ring (thus caching their values).
This could take a long time for some rings. ::h6j¬ h7j h9hyh;}r² (hA]hB]h@]h?]hC]uhDKhEhh0]r³ hGX Compute all the products of basis elements of the ring (thus caching their values).
This could take a long time for some rings.r´
rµ }r¶ (h5X Compute all the products of basis elements of the ring (thus caching their values).
This could take a long time for some rings.h6j° ubaubj )r· }r¸ (h5X^ sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,))
sage: s.do_all_products()h6j¬ h7j h9j h;}r¹ (h=h>h?]h@]hA]hB]hC]uhDMÅhEhh0]rº hGX^ sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,))
sage: s.do_all_products()r»
r¼ }r½ (h5U h6j· ubaubeubeubj )r¾ }r¿ (h5U h6j} h7Xj /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.get_stratumrÀ h9j h;}rÁ (h?]h@]hA]hB]hC]Uentries]r (j X2 get_stratum() (strataalgebra.StrataAlgebra method)hU Ntrà auhDNhEhh0]ubj )rÄ }rÅ (h5U h6j} h7jÀ h9j# h;}rÆ (j% j& X pyh?]h@]hA]hB]hC]j( X methodrÇ j* jÇ uhDNhEhh0]rÈ (j, )rÉ }rÊ (h5X StrataAlgebra.get_stratum(r, j)h6jÄ h7Xj /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.get_stratumrË h9j0 h;}rÌ (h?]rÍ haj3 j4 X
strataalgebrarÎ
rÏ }rÐ bh@]hA]hB]hC]rÑ haj9 X StrataAlgebra.get_stratumj; j: j< uhDNhEhh0]rÒ (jP )rÓ }rÔ (h5X get_stratumh6jÉ h7jË h9jS h;}rÕ (hA]hB]h@]h?]hC]uhDNhEhh0]rÖ hGX get_stratumr×
rØ }rÙ (h5U h6jÓ ubaubjY )rÚ }rÛ (h5X r, jh6jÉ h7jË h9j\ h;}rÜ (hA]hB]h@]h?]hC]uhDNhEhh0]rÝ (j_ )rÞ }rß (h5X rh;}rà (hA]hB]h@]h?]hC]uh6jÚ h0]rá hGX r
râ }rã (h5U h6jÞ ubah9jg ubj_ )rä }rå (h5X jh;}ræ (hA]hB]h@]h?]hC]uh6jÚ h0]rç hGX j
rè }ré (h5U h6jä ubah9jg ubeubeubj| )rê }rë (h5U h6jÄ h7jË h9j h;}rì (hA]hB]h@]h?]hC]uhDNhEhh0]rí (hv)rî }rï (h5XE Get the :class:`StrataGraph` associated a a codimension and an index.h6jê h7jÀ h9hyh;}rð (hA]hB]h@]h?]hC]uhDKhEhh0]rñ (hGX Get the rò
ró }rô (h5X Get the h6jî ubj¿ )rõ }rö (h5X :class:`StrataGraph`r÷ h6jî h7Nh9jÏ h;}rø (UreftypeX classjs jt X StrataGraphU refdomainX pyrù h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]rú h)rû }rü (h5j÷ h;}rý (hA]hB]rþ (j~ jù X py-classrÿ eh@]h?]hC]uh6jõ h0]r hGX StrataGraphr
r }r (h5U h6jû ubah9hubaubhGX) associated a a codimension and an index.r
r }r (h5X) associated a a codimension and an index.h6jî ubeubj )r }r (h5U h6jê h7jÀ h9j h;}r (hA]hB]h@]h?]hC]uhDNhEhh0]r
(j )r }r (h5U h;}r
(hA]hB]h@]h?]hC]uh6j h0]r (j )r }r (h5X
Parametersh6j h7jÀ h9j h;}r (hA]hB]h@]h?]hC]uhDK h0]r hGX
Parametersr
r }r (h5U h6j ubaubj )r }r (h5U h;}r (hA]hB]h@]h?]hC]uh6j h0]r j¤ )r }r (h5U h;}r (hA]hB]h@]h?]hC]uh6j h0]r (j© )r }r (h5U h;}r (hA]hB]h@]h?]hC]uh6j h0]r! hv)r" }r# (h5X r -- The codimensionh;}r$ (hA]hB]h@]h?]hC]uh6j h0]r% (j² )r& }r' (h5X rh;}r( (h?]h@]j¶ hA]hB]hC]uh6j" h0]r) hGX r
r* }r+ (h5U h6j& ubah9j» ubhGX -- r,
r- }r. (h5U h6j" ubhGX The codimensionr/
r0 }r1 (h5X The codimensionh6j" ubeh9hyubah9jã ubj© )r2 }r3 (h5U h;}r4 (hA]hB]h@]h?]hC]uh6j h0]r5 hv)r6 }r7 (h5X j -- The indexh;}r8 (hA]hB]h@]h?]hC]uh6j2 h0]r9 (j² )r: }r; (h5X jh;}r< (h?]h@]j¶ hA]hB]hC]uh6j6 h0]r= hGX j
r> }r? (h5U h6j: ubah9j» ubhGX -- r@
rA }rB (h5U h6j6 ubhGX The indexrC
rD }rE (h5X The indexh6j6 ubeh9hyubah9jã ubeh9jU ubah9jV ubeh9jW ubj )rF }rG (h5U h;}rH (hA]hB]h@]h?]hC]uh6j h0]rI (j )rJ }rK (h5X Return typeh6jF h7jÀ h9j h;}rL (hA]hB]h@]h?]hC]uhDK h0]rM hGX Return typerN
rO }rP (h5U h6jJ ubaubj )rQ }rR (h5U h;}rS (hA]hB]h@]h?]hC]uh6jF h0]rT hv)rU }rV (h5X StrataGraphh;}rW (hA]hB]h@]h?]hC]uh6jQ h0]rX j¿ )rY }rZ (h5X :class:`StrataGraph`r[ h6jU h7Nh9jÏ h;}r\ (UreftypeX classjs jt X StrataGraphU refdomainX pyr] h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]r^ h)r_ }r` (h5j[ h;}ra (hA]hB]rb (j~ j] X py-classrc eh@]h?]hC]uh6jY h0]rd hGX StrataGraphre
rf }rg (h5U h6j_ ubah9hubaubah9hyubah9jV ubeh9jW ubeubhv)rh }ri (h5X6 See :class:`StrataAlgebra` documentation for examples.h6jê h7jÀ h9hyh;}rj (hA]hB]h@]h?]hC]uhDKhEhh0]rk (hGX See rl
rm }rn (h5X See h6jh ubj¿ )ro }rp (h5X :class:`StrataAlgebra`rq h6jh h7Nh9jÏ h;}rr (UreftypeX classjs jt X
StrataAlgebraU refdomainX pyrs h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]rt h)ru }rv (h5jq h;}rw (hA]hB]rx (j~ js X py-classry eh@]h?]hC]uh6jo h0]rz hGX
StrataAlgebrar{
r| }r} (h5U h6ju ubah9hubaubhGX documentation for examples.r~
r }r (h5X documentation for examples.h6jh ubeubeubeubj )r }r (h5U h6j} h7Xb /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.irrr h9j h;}r (h?]h@]hA]hB]hC]Uentries]r
(j X* irr() (strataalgebra.StrataAlgebra method)hU Ntr auhDNhEhh0]ubj )r }r (h5U h6j} h7j h9j# h;}r (j% j& X pyh?]h@]hA]hB]hC]j( X methodr j* j uhDNhEhh0]r (j, )r }r (h5X StrataAlgebra.irr()h6j h7Xb /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.irrr h9j0 h;}r (h?]r haj3 j4 X
strataalgebrar
r }r bh@]hA]hB]hC]r haj9 X StrataAlgebra.irrj; j: j< uhDNhEhh0]r (jP )r }r (h5X irrh6j h7j h9jS h;}r (hA]hB]h@]h?]hC]uhDNhEhh0]r hGX irrr
r }r (h5U h6j ubaubjY )r }r (h5U h6j h7j h9j\ h;}r (hA]hB]h@]h?]hC]uhDNhEhh0]ubeubj| )r }r¡ (h5U h6j h7j h9j h;}r¢ (hA]hB]h@]h?]hC]uhDNhEhh0]r£ (hv)r¤ }r¥ (h5XP Make the irreducible boundary. It will be returned with a coefficient of 1/2. ::h6j h7j h9hyh;}r¦ (hA]hB]h@]h?]hC]uhDKhEhh0]r§ hGXM Make the irreducible boundary. It will be returned with a coefficient of 1/2.r¨
r© }rª (h5XM Make the irreducible boundary. It will be returned with a coefficient of 1/2.h6j¤ ubaubj )r« }r¬ (h5X_ sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ, 1, (1,2))
sage: s.irr()
1/2*D_irrh6j h7j h9j h;}r (h=h>h?]h@]hA]hB]hC]uhDMÜhEhh0]r® hGX_ sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ, 1, (1,2))
sage: s.irr()
1/2*D_irrr¯
r° }r± (h5U h6j« ubaubeubeubj )r² }r³ (h5U h6j} h7Xd /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.kappar´ h9j h;}rµ (h?]h@]hA]hB]hC]Uentries]r¶ (j X, kappa() (strataalgebra.StrataAlgebra method)hU Ntr· auhDNhEhh0]ubj )r¸ }r¹ (h5U h6j} h7j´ h9j# h;}rº (j% j& X pyr» h?]h@]hA]hB]hC]j( X methodr¼ j* j¼ uhDNhEhh0]r½ (j, )r¾ }r¿ (h5X StrataAlgebra.kappa(a)rÀ h6j¸ h7Xd /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.kapparÁ h9j0 h;}r (h?]rà haj3 j4 X
strataalgebrarÄ
rÅ }rÆ bh@]hA]hB]hC]rÇ haj9 X StrataAlgebra.kappaj; j: j< uhDNhEhh0]rÈ (jP )rÉ }rÊ (h5X kappah6j¾ h7jÁ h9jS h;}rË (hA]hB]h@]h?]hC]uhDNhEhh0]rÌ hGX kapparÍ
rÎ }rÏ (h5U h6jÉ ubaubjY )rÐ }rÑ (h5X ah6j¾ h7jÁ h9j\ h;}rÒ (hA]hB]h@]h?]hC]uhDNhEhh0]rÓ j_ )rÔ }rÕ (h5X ah;}rÖ (hA]hB]h@]h?]hC]uh6jÐ h0]r× hGX a
rØ }rÙ (h5U h6jÔ ubah9jg ubaubeubj| )rÚ }rÛ (h5U h6j¸ h7jÁ h9j h;}rÜ (hA]hB]h@]h?]hC]uhDNhEhh0]rÝ (hv)rÞ }rß (h5X Return a kappa class.rà h6jÚ h7j´ h9hyh;}rá (hA]hB]h@]h?]hC]uhDKhEhh0]râ hGX Return a kappa class.rã
rä }rå (h5jà h6jÞ ubaubj )ræ }rç (h5U h6jÚ h7j´ h9j h;}rè (hA]hB]h@]h?]hC]uhDNhEhh0]ré (j )rê }rë (h5U h;}rì (hA]hB]h@]h?]hC]uh6jæ h0]rí (j )rî }rï (h5X
Parametersh6jê h7j´ h9j h;}rð (hA]hB]h@]h?]hC]uhDK h0]rñ hGX
Parametersrò
ró }rô (h5U h6jî ubaubj )rõ }rö (h5U h;}r÷ (hA]hB]h@]h?]hC]uh6jê h0]rø hv)rù }rú (h5XC a (int) -- The subscript (codimension) of the kappa class you want.h;}rû (hA]hB]h@]h?]hC]uh6jõ h0]rü (j² )rý }rþ (h5X ah;}rÿ (h?]h@]j¶ hA]hB]hC]uh6jù h0]r
hGX a
r
}r
(h5U h6jý ubah9j» ubhGX (r
r
}r
(h5U h6jù ubj¿ )r
}r
(h5U h;}r
(Ureftypejà j¶ U reftargetX intr
U refdomainj» h?]h@]U refexplicithA]hB]hC]uh6jù h0]r
jÆ )r
}r
(h5j
h;}r
(hA]hB]h@]h?]hC]uh6j
h0]r
hGX intr
r
}r
(h5U h6j
ubah9jÎ ubah9jÏ ubhGX )
r
}r
(h5U h6jù ubhGX -- r
r
}r
(h5U h6jù ubhGX8 The subscript (codimension) of the kappa class you want.r
r
}r
(h5X8 The subscript (codimension) of the kappa class you want.r
h6jù ubeh9hyubah9jV ubeh9jW ubj )r
}r
(h5U h;}r
(hA]hB]h@]h?]hC]uh6jæ h0]r
(j )r
}r
(h5X Return typeh6j
h7j´ h9j h;}r!
(hA]hB]h@]h?]hC]uhDK h0]r"
hGX Return typer#
r$
}r%
(h5U h6j
ubaubj )r&
}r'
(h5U h;}r(
(hA]hB]h@]h?]hC]uh6j
h0]r)
hv)r*
}r+
(h5X StrataAlgebraElementh;}r,
(hA]hB]h@]h?]hC]uh6j&
h0]r-
j¿ )r.
}r/
(h5X :class:`StrataAlgebraElement`r0
h6j*
h7Nh9jÏ h;}r1
(UreftypeX classjs jt X StrataAlgebraElementU refdomainX pyr2
h?]h@]U refexplicithA]hB]hC]jv hZjw j: jx NuhDNh0]r3
h)r4
}r5
(h5j0
h;}r6
(hA]hB]r7
(j~ j2
X py-classr8
eh@]h?]hC]uh6j.
h0]r9
hGX StrataAlgebraElementr:
r;
}r<
(h5U h6j4
ubah9hubaubah9hyubah9jV ubeh9jW ubeubj )r=
}r>
(h5X% sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,2,())
sage: s.kappa(1)
ka1
sage: s.kappa(3)
ka3
sage: s.kappa(1)*s.kappa(2)
ka1*ka2
sage: s.kappa(2)*s.kappa(2)
0
sage: s.boundary(1,())*s.kappa(1)
s_2,4
sage: s.get_stratum(2,4)
[ 0 0]
[ka1 + 1 1]
[ 1 1]h6jÚ h7j´ h9j h;}r?
(h=h>h?]h@]hA]hB]hC]uhDMìhEhh0]r@
hGX% sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,2,())
sage: s.kappa(1)
ka1
sage: s.kappa(3)
ka3
sage: s.kappa(1)*s.kappa(2)
ka1*ka2
sage: s.kappa(2)*s.kappa(2)
0
sage: s.boundary(1,())*s.kappa(1)
s_2,4
sage: s.get_stratum(2,4)
[ 0 0]
[ka1 + 1 1]
[ 1 1]rA
rB
}rC
(h5U h6j=
ubaubhv)rD
}rE
(h5XQ The subscript must be less than or equal to the dimension of the moduli space. ::rF
h6jÚ h7j´ h9hyh;}rG
(hA]hB]h@]h?]hC]uhDKhEhh0]rH
hGXN The subscript must be less than or equal to the dimension of the moduli space.rI
rJ
}rK
(h5XN The subscript must be less than or equal to the dimension of the moduli space.h6jD
ubaubj )rL
}rM
(h5XC sage: s.kappa(4)
Traceback (most recent call last):
...
KeyError: 4h6jÚ h7j´ h9j h;}rN
(h=h>h?]h@]hA]hB]hC]uhDM hEhh0]rO
hGXC sage: s.kappa(4)
Traceback (most recent call last):
...
KeyError: 4rP
rQ
}rR
(h5U h6jL
ubaubeubeubj )rS
}rT
(h5U h6j} h7Xk /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.print_stratarU
h9j h;}rV
(h?]h@]hA]hB]hC]Uentries]rW
(j X3 print_strata() (strataalgebra.StrataAlgebra method)hU NtrX
auhDNhEhh0]ubj )rY
}rZ
(h5U h6j} h7jU
h9j# h;}r[
(j% j& X pyh?]h@]hA]hB]hC]j( X methodr\
j* j\
uhDNhEhh0]r]
(j, )r^
}r_
(h5X StrataAlgebra.print_strata(r)h6jY
h7Xk /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.print_stratar`
h9j0 h;}ra
(h?]rb
haj3 j4 X
strataalgebrarc
rd
}re
bh@]hA]hB]hC]rf
haj9 X StrataAlgebra.print_strataj; j: j< uhDNhEhh0]rg
(jP )rh
}ri
(h5X print_stratah6j^
h7j`
h9jS h;}rj
(hA]hB]h@]h?]hC]uhDNhEhh0]rk
hGX print_stratarl
rm
}rn
(h5U h6jh
ubaubjY )ro
}rp
(h5X rh6j^
h7j`
h9j\ h;}rq
(hA]hB]h@]h?]hC]uhDNhEhh0]rr
j_ )rs
}rt
(h5X rh;}ru
(hA]hB]h@]h?]hC]uh6jo
h0]rv
hGX r
rw
}rx
(h5U h6js
ubah9jg ubaubeubj| )ry
}rz
(h5U h6jY
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sage: s = StrataAlgebra(QQ,0,(1,2,3,4,5))
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sage: (psi1*psi2).integrate()
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sage: s = StrataAlgebra(QQ,0,(1,2,3,4,5))
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...
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...
ValueError: tuple.index(x): x not in tuple
sage: s = StrataAlgebra(QQ,1,(1,1,1))
sage: s.psi(1)^2
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sage: s.psi(1)^3
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sage: var('ps') #This should be done automatically.
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sage: s = StrataAlgebra(QQ,0,(1,1,1,2,2))
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sage: s.psi(1)^2
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sage: s.psi(2)
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...
ValueError: tuple.index(x): x not in tuple
sage: s = StrataAlgebra(QQ,1,(1,1,1))
sage: s.psi(1)^2
2*ps1^2 + 4*ps1*ps1
sage: s.psi(1)^3
4*ps1^3 + 24*ps1^2*ps1 + 8*ps1*ps1*ps1
sage: var('ps') #This should be done automatically.
ps
sage: s([[0,1,1,1],[1,ps^3+1,1,1]])
ps1^3
sage: s = StrataAlgebra(QQ,0,(1,1,1,2,2))
sage: s.psi(1)*s.psi(2)
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sage: s.psi(1)*s.psi(1)
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