Content - 4afc308cefbfb0894ec78747545163a3eb80d3f4 - 30c6726/index.doctree

index.doctree
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q:)q;}q<(U	rawsourceq=XÒStrataAlgegra documentation master file, created by
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qOXÒStrataAlgegra documentation master file, created by
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q~)q}q(h=X{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 (conjecturally) described by the Faber-Zagier relations. For more detail, see for example `R. Pandharipande's exposition`_.h>hmh?h@hAU	paragraphqhC}q(hI]hJ]hH]hG]hK]uhLKhMhh8]q(hOX
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because it has a natural surjective morphism to the tautological ring of the moduli space of curves. The kernel is (conjecturally) described by the Faber-Zagier relations. For more detail, see for example h>hubh)q¢}q£(h=X `R. Pandharipande's exposition`_hKh>hhAhhC}q¤(UnameXR. Pandharipande's expositionhX$https://arxiv.org/pdf/1603.05151.pdfq¥hG]hH]hI]hJ]hK]uh8]q¦hOXR. Pandharipande's expositionq§q¨}q©(h=Uh>h¢ubaubhOX.qª}q«(h=X.h>hubeubcdocutils.nodes
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Pandharipande.
The program also computes the FZ relations using code copied from A. Pixton's `tautrel.sage`_ program.
The code was integrated into this package (with some modifications and optimizations) by Drew Johnson
(who takes responsibility for any bugs introduced!).h>hmh?h@hAhhC}q»(hI]hJ]hH]hG]hK]uhLKhMhh8]q¼(hOX)The product structure was implemented by q½q¾}q¿(h=X)The product structure was implemented by h>h¹ubh)qÀ}qÁ(h=X`Drew Johnson`_hKh>h¹hAhhC}qÂ(UnameXDrew JohnsonhXhttp://pages.uoregon.edu/drewj/qÃhG]hH]hI]hJ]hK]uh8]qÄhOXDrew JohnsonqÅqÆ}qÇ(h=Uh>hÀubaubhOX, based on qÈqÉ}qÊ(h=X, based on h>h¹ubh)qË}qÌ(h=X`S. Yang's note`_hKh>h¹hAhhC}qÍ(UnameXS. Yang's notehXhttps://arxiv.org/abs/0808.1974qÎhG]hH]hI]hJ]hK]uh8]qÏhOXS. Yang's noteqÐqÑ}qÒ(h=Uh>hËubaubhOX describing the algorithm of Graber and
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The program also computes the FZ relations using code copied from A. Pixton's h>h¹ubh)qÖ}q×(h=X`tautrel.sage`_hKh>h¹hAhhC}qØ(UnameXtautrel.sageqÙhX2http://math.mit.edu/~apixton/programs/tautrel.sageqÚhG]hH]hI]hJ]hK]uh8]qÛhOXtautrel.sageqÜqÝ}qÞ(h=Uh>hÖubaubhOX¤ program.
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The code was integrated into this package (with some modifications and optimizations) by Drew Johnson
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The genus.h>j)ubehAhubahAj$ubjì)rJ}rK(h=UhC}rL(hI]hJ]hH]hG]hK]uh>jèh8]rMh~)rN}rO(h=Xkmarkings (tuple) -- The markings should be positive integers. Repeats are allowed. Defaults to no markings.hC}rP(hI]hJ]hH]hG]hK]uh>jJh8]rQ(jõ)rR}rS(h=XmarkingshC}rT(hG]hH]jùhI]hJ]hK]uh>jNh8]rUhOXmarkingsrVrW}rX(h=Uh>jRubahAjþubhOX (rYrZ}r[(h=Uh>jNubj)r\}r](h=UhC}r^(UreftypejjùU	reftargetXtupler_U	refdomainjjhG]hH]UrefexplicithI]hJ]hK]uh>jNh8]r`j	)ra}rb(h=j_hC}rc(hI]hJ]hH]hG]hK]uh>j\h8]rdhOXtuplererf}rg(h=Uh>jaubahAjubahAjubhOX)rh}ri(h=Uh>jNubhOX -- rjrk}rl(h=Uh>jNubhOXWThe markings should be positive integers. Repeats are allowed. Defaults to no markings.rmrn}ro(h=XWThe markings should be positive integers. Repeats are allowed. Defaults to no markings.h>jNubehAhubahAj$ubjì)rp}rq(h=UhC}rr(hI]hJ]hH]hG]hK]uh>jèh8]rsh~)rt}ru(h=X´make_vars (bool) -- Defaults to True. If True, creates variables ps, ps_, ka1, ... , ka{d} (where d is the dimension of the moduli space) that can be used to create basis elements.hC}rv(hI]hJ]hH]hG]hK]uh>jph8]rw(jõ)rx}ry(h=X	make_varshC}rz(hG]hH]jùhI]hJ]hK]uh>jth8]r{hOX	make_varsr|r}}r~(h=Uh>jxubahAjþubhOX (rr}r(h=Uh>jtubj)r}r(h=UhC}r(UreftypejjùU	reftargetXboolrU	refdomainjjhG]hH]UrefexplicithI]hJ]hK]uh>jth8]rj	)r}r(h=jhC}r(hI]hJ]hH]hG]hK]uh>jh8]rhOXboolrr}r(h=Uh>jubahAjubahAjubhOX)r}r(h=Uh>jtubhOX -- rr}r(h=Uh>jtubhOX-Defaults to True. If True, creates variables rr}r(h=X-Defaults to True. If True, creates variables h>jtubh)r}r(h=X``ps``hC}r(hI]hJ]hH]hG]hK]uh>jth8]rhOXpsrr}r(h=Uh>jubahAhubhOX, rr}r(h=X, h>jtubh)r }r¡(h=X``ps_``hC}r¢(hI]hJ]hH]hG]hK]uh>jth8]r£hOXps_r¤r¥}r¦(h=Uh>j ubahAhubhOX, r§r¨}r©(h=X, h>jtubh)rª}r«(h=X``ka1``hC}r¬(hI]hJ]hH]hG]hK]uh>jth8]rhOXka1r®r¯}r°(h=Uh>jªubahAhubhOX, ... , r±r²}r³(h=X, ... , h>jtubh)r´}rµ(h=X	``ka{d}``hC}r¶(hI]hJ]hH]hG]hK]uh>jth8]r·hOXka{d}r¸r¹}rº(h=Uh>j´ubahAhubhOXZ (where d is the dimension of the moduli space) that can be used to create basis elements.r»r¼}r½(h=XZ (where d is the dimension of the moduli space) that can be used to create basis elements.h>jtubehAhubahAj$ubehAUbullet_listr¾ubahAU
field_bodyr¿ubehAUfieldrÀubaubh~)rÁ}rÂ(h=XFirst import the module: ::rÃh>jÀh?jÈhAhhC}rÄ(hI]hJ]hH]hG]hK]uhLKhMhh8]rÅhOXFirst import the module:rÆrÇ}rÈ(h=XFirst import the module:h>jÁubaubjD)rÉ}rÊ(h=X!sage: from strataalgebra import *h>jÀh?jÈhAjGhC}rË(hEhFhG]hH]hI]hJ]hK]uhLKhMhh8]rÌhOX!sage: from strataalgebra import *rÍrÎ}rÏ(h=Uh>jÉubaubh~)rÐ}rÑ(h=X&Construct a :class:`StrataAlgebra`: ::rÒh>jÀh?jÈhAhhC}rÓ(hI]hJ]hH]hG]hK]uhLKhMhh8]rÔ(hOXConstruct a rÕrÖ}r×(h=XConstruct a h>jÐubj)rØ}rÙ(h=X:class:`StrataAlgebra`rÚh>jÐh?h@hAjhC}rÛ(UreftypeXclassUrefwarnrÜU	reftargetrÝX
StrataAlgebraU	refdomainXpyrÞhG]hH]UrefexplicithI]hJ]hK]UrefdocrßhbUpy:classràj}U	py:moduleráNuhLKh8]râh)rã}rä(h=jÚhC}rå(hI]hJ]ræ(UxrefrçjÞXpy-classrèehH]hG]hK]uh>jØh8]réhOX
StrataAlgebrarêrë}rì(h=Uh>jãubahAhubaubhOX:rí}rî(h=X:h>jÐubeubjD)rï}rð(h=Xlsage: SA = StrataAlgebra(QQ,1,(1,2)); SA
Strata algebra with genus 1 and markings (1, 2) over Rational Fieldh>jÀh?jÈhAjGhC}rñ(hEhFhG]hH]hI]hJ]hK]uhLKhMhh8]ròhOXlsage: SA = StrataAlgebra(QQ,1,(1,2)); SA
Strata algebra with genus 1 and markings (1, 2) over Rational Fieldrórô}rõ(h=Uh>jïubaubh~)rö}r÷(h=XSPrint the basis elements in a certain codimension, with their (arbitrary) index: ::røh>jÀh?jÈhAhhC}rù(hI]hJ]hH]hG]hK]uhLKhMhh8]rúhOXPPrint the basis elements in a certain codimension, with their (arbitrary) index:rûrü}rý(h=XPPrint the basis elements in a certain codimension, with their (arbitrary) index:h>jöubaubjD)rþ}rÿ(h=Xusage: 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>h>jÀh?jÈhAjGhC}r(hEhFhG]hH]hI]hJ]hK]uhLKhMhh8]rhOXusage: 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>rr}r(h=Uh>jþubaubh~)r}r(h=XZClasses that are monomials in :math:`\psi` and :math:`\kappa` have self-explanatory names.rh>jÀh?jÈhAhhC}r(hI]hJ]hH]hG]hK]uhLKOhMhh8]r	(hOXClasses that are monomials in r
r}r(h=XClasses that are monomials in h>jubcsphinx.ext.mathbase
math
r
)r}r(h=UhC}r(UlatexX\psihG]hH]hI]hJ]hK]uh>jh8]hAUmathrubhOX and rr}r(h=X and h>jubj
)r}r(h=UhC}r(UlatexX\kappahG]hH]hI]hJ]hK]uh>jh8]hAjubhOX have self-explanatory names.rr}r(h=X have self-explanatory names.h>jubeubh~)r}r(h=X>More complicated classes are displayed in a matrix as follows:rh>jÀh?jÈhAhhC}r(hI]hJ]hH]hG]hK]uhLKQhMhh8]rhOX>More complicated classes are displayed in a matrix as follows:r r!}r"(h=jh>jubaubh~)r#}r$(h=XEach 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.h>jÀh?jÈhAhhC}r%(hI]hJ]hH]hG]hK]uhLKShMhh8]r&(hOX7Each 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 r'r(}r)(h=X7Each 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 h>j#ubh)r*}r+(h=X``0``hC}r,(hI]hJ]hH]hG]hK]uh>j#h8]r-hOX0r.}r/(h=Uh>j*ubahAhubhOX 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 r0r1}r2(h=X 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 h>j#ubh)r3}r4(h=X``ps``hC}r5(hI]hJ]hH]hG]hK]uh>j#h8]r6hOXpsr7r8}r9(h=Uh>j3ubahAhubhOX in entry (v,e) means a r:r;}r<(h=X in entry (v,e) means a h>j#ubj
)r=}r>(h=UhC}r?(UlatexX\psihG]hH]hI]hJ]hK]uh>j#h8]hAjubhOX6-class associated to the half edge coming out of v.
A r@rA}rB(h=X6-class associated to the half edge coming out of v.
A h>j#ubh)rC}rD(h=X``ps_``hC}rE(hI]hJ]hH]hG]hK]uh>j#h8]rFhOXps_rGrH}rI(h=Uh>jCubahAhubhOXY may occur when there is a loop at the vertex.
The entry in the top left is just padding.rJrK}rL(h=XY may occur when there is a loop at the vertex.
The entry in the top left is just padding.h>j#ubeubh~)rM}rN(h=XÄTo 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}`` ::h>jÀh?jÈhAhhC}rO(hI]hJ]hH]hG]hK]uhLK]hMhh8]rP(hOX¬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 rQrR}rS(h=X¬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 h>jMubh)rT}rU(h=X``s_{codim},{index}``hC}rV(hI]hJ]hH]hG]hK]uh>jMh8]rWhOXs_{codim},{index}rXrY}rZ(h=Uh>jTubahAhubeubjD)r[}r\(h=Xosage: 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
Dg1h>jÀh?jÈhAjGhC}r](hEhFhG]hH]hI]hJ]hK]uhLKehMhh8]r^hOXosage: 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
Dg1r_r`}ra(h=Uh>j[ubaubh~)rb}rc(h=X(Vector space arithmetic is supported. ::rdh>jÀh?jÈhAhhC}re(hI]hJ]hH]hG]hK]uhLKjhMhh8]rfhOX%Vector space arithmetic is supported.rgrh}ri(h=X%Vector space arithmetic is supported.h>jbubaubjD)rj}rk(h=X9sage: 3*b
3*s_2,7
sage: a*72 - b/17
72*s_2,1 - 1/17*s_2,7h>jÀh?jÈhAjGhC}rl(hEhFhG]hH]hI]hJ]hK]uhLKphMhh8]rmhOX9sage: 3*b
3*s_2,7
sage: a*72 - b/17
72*s_2,1 - 1/17*s_2,7rnro}rp(h=Uh>jjubaubh~)rq}rr(h=XoUse :meth:`~strataalgebra.StrataAlgebra.get_stratum` if you need to know what an unamed basis element means. ::rsh>jÀh?jÈhAhhC}rt(hI]hJ]hH]hG]hK]uhLKqhMhh8]ru(hOXUse rvrw}rx(h=XUse h>jqubj)ry}rz(h=X0:meth:`~strataalgebra.StrataAlgebra.get_stratum`r{h>jqh?NhAjhC}r|(UreftypeXmethjÜjÝX'strataalgebra.StrataAlgebra.get_stratumU	refdomainXpyr}hG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]r~h)r}r(h=j{hC}r(hI]hJ]r(jçj}Xpy-methrehH]hG]hK]uh>jyh8]rhOX
get_stratum()rr}r(h=Uh>jubahAhubaubhOX8 if you need to know what an unamed basis element means.rr}r(h=X8 if you need to know what an unamed basis element means.h>jqubeubjD)r}r(h=Xssage: SA.get_stratum(2,7)
[     0      1      2      0]
[     0      1      1      1]
[     1      0      0 ps + 1]h>jÀh?jÈhAjGhC}r(hEhFhG]hH]hI]hJ]hK]uhLKwhMhh8]rhOXssage: SA.get_stratum(2,7)
[     0      1      2      0]
[     0      1      1      1]
[     1      0      0 ps + 1]rr}r(h=Uh>jubaubh~)r}r(h=XYou 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`.h>jÀh?jÈhAhhC}r(hI]hJ]hH]hG]hK]uhLKxhMhh8]r(hOXYou can construct rr}r(h=XYou can construct h>jubj
)r}r(h=UhC}r(UlatexX
\psi,\;\kappahG]hH]hI]hJ]hK]uh>jh8]hAjubhOX2 monomials and boundary divisors with the methods rr}r(h=X2 monomials and boundary divisors with the methods h>jubj)r}r (h=X*:meth:`~strataalgebra.StrataAlgebra.kappa`r¡h>jh?NhAjhC}r¢(UreftypeXmethjÜjÝX!strataalgebra.StrataAlgebra.kappaU	refdomainXpyr£hG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]r¤h)r¥}r¦(h=j¡hC}r§(hI]hJ]r¨(jçj£Xpy-methr©ehH]hG]hK]uh>jh8]rªhOXkappa()r«r¬}r(h=Uh>j¥ubahAhubaubhOX, r®r¯}r°(h=X, h>jubj)r±}r²(h=X(:meth:`~strataalgebra.StrataAlgebra.psi`r³h>jh?NhAjhC}r´(UreftypeXmethjÜjÝXstrataalgebra.StrataAlgebra.psiU	refdomainXpyrµhG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]r¶h)r·}r¸(h=j³hC}r¹(hI]hJ]rº(jçjµXpy-methr»ehH]hG]hK]uh>j±h8]r¼hOXpsi()r½r¾}r¿(h=Uh>j·ubahAhubaubhOX,
rÀrÁ}rÂ(h=X,
h>jubj)rÃ}rÄ(h=X-:meth:`~strataalgebra.StrataAlgebra.boundary`rÅh>jh?NhAjhC}rÆ(UreftypeXmethjÜjÝX$strataalgebra.StrataAlgebra.boundaryU	refdomainXpyrÇhG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rÈh)rÉ}rÊ(h=jÅhC}rË(hI]hJ]rÌ(jçjÇXpy-methrÍehH]hG]hK]uh>jÃh8]rÎhOX
boundary()rÏrÐ}rÑ(h=Uh>jÉubahAhubaubhOX, and rÒrÓ}rÔ(h=X, and h>jubj)rÕ}rÖ(h=X(:meth:`~strataalgebra.StrataAlgebra.irr`r×h>jh?NhAjhC}rØ(UreftypeXmethjÜjÝXstrataalgebra.StrataAlgebra.irrU	refdomainXpyrÙhG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rÚh)rÛ}rÜ(h=j×hC}rÝ(hI]hJ]rÞ(jçjÙXpy-methrßehH]hG]hK]uh>jÕh8]ràhOXirr()rárâ}rã(h=Uh>jÛubahAhubaubhOX.rä}rå(h=X.h>jubeubh~)ræ}rç(h=XvYou can construct an element using the matrix notation. Just pass a list of lists into your :class:`StrataAlgebra`. ::rèh>jÀh?jÈhAhhC}ré(hI]hJ]hH]hG]hK]uhLK{hMhh8]rê(hOX\You can construct an element using the matrix notation. Just pass a list of lists into your rërì}rí(h=X\You can construct an element using the matrix notation. Just pass a list of lists into your h>jæubj)rî}rï(h=X:class:`StrataAlgebra`rðh>jæh?NhAjhC}rñ(UreftypeXclassjÜjÝX
StrataAlgebraU	refdomainXpyròhG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]róh)rô}rõ(h=jðhC}rö(hI]hJ]r÷(jçjòXpy-classrøehH]hG]hK]uh>jîh8]rùhOX
StrataAlgebrarúrû}rü(h=Uh>jôubahAhubaubhOX.rý}rþ(h=X.h>jæubeubjD)rÿ}r(h=X¦sage: var('ps') #This is usually done automatically, but we have to do it manually here for the dotests to work.
ps
sage: SA([[0,1,2,0],[1,0,0,ps+1],[0,1,1,1]])
s_2,7h>jÀh?jÈhAjGhC}r(hEhFhG]hH]hI]hJ]hK]uhLKhMhh8]rhOX¦sage: var('ps') #This is usually done automatically, but we have to do it manually here for the dotests to work.
ps
sage: SA([[0,1,2,0],[1,0,0,ps+1],[0,1,1,1]])
s_2,7rr}r(h=Uh>jÿubaubh~)r}r(h=X%Here is an example of the ``ps_``: ::rh>jÀh?jÈhAhhC}r	(hI]hJ]hH]hG]hK]uhLKhMhh8]r
(hOXHere is an example of the rr}r
(h=XHere is an example of the h>jubh)r}r(h=X``ps_``hC}r(hI]hJ]hH]hG]hK]uh>jh8]rhOXps_rr}r(h=Uh>jubahAhubhOX:r}r(h=X:h>jubeubjD)r}r(h=Xåsage: 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_3,10h>jÀh?jÈhAjGhC}r(hEhFhG]hH]hI]hJ]hK]uhLKhMhh8]rhOXåsage: 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_3,10rr}r(h=Uh>jubaubh~)r}r(h=X>One of the main features is the computation of the product. ::r h>jÀh?jÈhAhhC}r!(hI]hJ]hH]hG]hK]uhLKhMhh8]r"hOX;One of the main features is the computation of the product.r#r$}r%(h=X;One of the main features is the computation of the product.h>jubaubjD)r&}r'(h=Xsage: a*b
0h>jÀh?jÈhAjGhC}r((hEhFhG]hH]hI]hJ]hK]uhLKhMhh8]r)hOXsage: a*b
0r*r+}r,(h=Uh>j&ubaubh~)r-}r.(h=XJOf course, the codimension was too big. Lets do some less trivial ones. ::r/h>jÀh?jÈhAhhC}r0(hI]hJ]hH]hG]hK]uhLKhMhh8]r1hOXGOf course, the codimension was too big. Lets do some less trivial ones.r2r3}r4(h=XGOf course, the codimension was too big. Lets do some less trivial ones.h>j-ubaubjD)r5}r6(h=Xqsage: 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_3,49h>jÀh?jÈhAjGhC}r7(hEhFhG]hH]hI]hJ]hK]uhLKhMhh8]r8hOXqsage: 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_3,49r9r:}r;(h=Uh>j5ubaubh~)r<}r=(h=X6Everything should work with distributive laws, etc. ::r>h>jÀh?jÈhAhhC}r?(hI]hJ]hH]hG]hK]uhLK¬hMhh8]r@hOX3Everything should work with distributive laws, etc.rArB}rC(h=X3Everything should work with distributive laws, etc.h>j<ubaubjD)rD}rE(h=X_sage: SA.kappa(2)*(3*SA.psi(3)-SA(1,3)) == -SA(1,3) *SA.kappa(2) + SA.psi(3)*SA.kappa(2)*3
Trueh>jÀh?jÈhAjGhC}rF(hEhFhG]hH]hI]hJ]hK]uhLK²hMhh8]rGhOX_sage: SA.kappa(2)*(3*SA.psi(3)-SA(1,3)) == -SA(1,3) *SA.kappa(2) + SA.psi(3)*SA.kappa(2)*3
TruerHrI}rJ(h=Uh>jDubaubh~)rK}rL(h=X It should work over any ring. ::rMh>jÀh?jÈhAhhC}rN(hI]hJ]hH]hG]hK]uhLK±hMhh8]rOhOXIt should work over any ring.rPrQ}rR(h=XIt should work over any ring.h>jKubaubjD)rS}rT(h=X'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)*ka1^2h>jÀh?jÈhAjGhC}rU(hEhFhG]hH]hI]hJ]hK]uhLK·hMhh8]rVhOX'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)*ka1^2rWrX}rY(h=Uh>jSubaubh~)rZ}r[(h=X3There may be problems over a non-divisible ring. ::r\h>jÀh?jÈhAhhC}r](hI]hJ]hH]hG]hK]uhLK¼hMhh8]r^hOX0There may be problems over a non-divisible ring.r_r`}ra(h=X0There may be problems over a non-divisible ring.h>jZubaubjD)rb}rc(h=Xßsage: 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'h>jÀh?jÈhAjGhC}rd(hEhFhG]hH]hI]hJ]hK]uhLKÂhMhh8]rehOXßsage: 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'rfrg}rh(h=Uh>jbubaubh~)ri}rj(h=X³Also, 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. ::h>jÀh?jÈhAhhC}rk(hI]hJ]hH]hG]hK]uhLKÃhMhh8]rl(hOXOAlso, repeated names of the markings are allowed. The following corresponds to rmrn}ro(h=XOAlso, repeated names of the markings are allowed. The following corresponds to h>jiubj
)rp}rq(h=UhC}rr(UlatexX\overline{\mathcal M}_{1,2}/S_2hG]hH]hI]hJ]hK]uh>jih8]hAjubhOX:.
Compare the codimension 2 strata to our earlier example.rsrt}ru(h=X:.
Compare the codimension 2 strata to our earlier example.h>jiubeubjD)rv}rw(h=XSsage: 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>h>jÀh?Xg/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_bettirxhAjGhC}ry(hEhFhG]hH]hI]hJ]hK]uhLKÊhMhh8]rzhOXSsage: 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}(h=Uh>jvubaubj[)r~}r(h=Uh>jÀh?NhAj^hC}r(hG]hH]hI]hJ]hK]Uentries]r(jaX/FZ_betti() (strataalgebra.StrataAlgebra method)hUNtrauhLNhMhh8]ubjc)r}r(h=Uh>jÀh?NhAjfhC}r(jhjiXpyhG]hH]hI]hJ]hK]jkXmethodrjmjuhLNhMhh8]r(jo)r}r(h=X"StrataAlgebra.FZ_betti(codim=None)h>jh?Xg/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_bettirhAjshC}r(hG]rhajvjwX
strataalgebrarr}rbhH]hI]hJ]hK]rhaj|XStrataAlgebra.FZ_bettij~j}juhLNhMhh8]r(j)r}r(h=XFZ_bettih>jh?jhAjhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]rhOXFZ_bettirr}r(h=Uh>jubaubj)r}r(h=X
codim=Noneh>jh?jhAjhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]rj¢)r}r(h=X
codim=NonehC}r(hI]hJ]hH]hG]hK]uh>jh8]r hOX
codim=Noner¡r¢}r£(h=Uh>jubahAjªubaubeubj¿)r¤}r¥(h=Uh>jh?jhAjÂhC}r¦(hI]hJ]hH]hG]hK]uhLNhMhh8]r§(h~)r¨}r©(h=X§Give the dimension of the cohomology of moduli space of curves, as predicted by the Faber-Zagier relations.
If the codimension is omited, return a list of all of them.rªh>j¤h?Xg/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_bettir«hAhhC}r¬(hI]hJ]hH]hG]hK]uhLKhMhh8]rhOX§Give the dimension of the cohomology of moduli space of curves, as predicted by the Faber-Zagier relations.
If the codimension is omited, return a list of all of them.r®r¯}r°(h=jªh>j¨ubaubjÎ)r±}r²(h=Uh>j¤h?NhAjÑhC}r³(hI]hJ]hH]hG]hK]uhLNhMhh8]r´jÔ)rµ}r¶(h=UhC}r·(hI]hJ]hH]hG]hK]uh>j±h8]r¸(jÙ)r¹}rº(h=X
Parametersh>jµh?jhAjÜhC}r»(hI]hJ]hH]hG]hK]uhLKh8]r¼hOX
Parametersr½r¾}r¿(h=Uh>j¹ubaubjâ)rÀ}rÁ(h=UhC}rÂ(hI]hJ]hH]hG]hK]uh>jµh8]rÃh~)rÄ}rÅ(h=X,codim -- Optional. The codimension you want.hC}rÆ(hI]hJ]hH]hG]hK]uh>jÀh8]rÇ(jõ)rÈ}rÉ(h=XcodimhC}rÊ(hG]hH]jùhI]hJ]hK]uh>jÄh8]rËhOXcodimrÌrÍ}rÎ(h=Uh>jÈubahAjþubhOX -- rÏrÐ}rÑ(h=Uh>jÄubhOX#Optional. The codimension you want.rÒrÓ}rÔ(h=X#Optional. The codimension you want.h>jÄubehAhubahAj¿ubehAjÀubaubjD)rÕ}rÖ(h=Xsage: from strataalgebra import *
sage: StrataAlgebra(QQ, 2, (1,)).FZ_betti()
[1, 3, 5, 3, 1]
sage: StrataAlgebra(QQ, 2, (1,)).FZ_betti(2)
5h>j¤h?j«hAjGhC}r×(hEhFhG]hH]hI]hJ]hK]uhLMhMhh8]rØhOXsage: from strataalgebra import *
sage: StrataAlgebra(QQ, 2, (1,)).FZ_betti()
[1, 3, 5, 3, 1]
sage: StrataAlgebra(QQ, 2, (1,)).FZ_betti(2)
5rÙrÚ}rÛ(h=Uh>jÕubaubcsphinx.addnodes
seealso
rÜ)rÝ}rÞ(h=X,:meth:`~strataalgebra.StrataAlgebra.hilbert`rßh>j¤h?j«hAUseealsoràhC}rá(hI]hJ]hH]hG]hK]uhLNhMhh8]râh~)rã}rä(h=jßh>jÝh?j«hAhhC}rå(hI]hJ]hH]hG]hK]uhLKh8]ræj)rç}rè(h=jßh>jãh?NhAjhC}ré(UreftypeXmethjÜjÝX#strataalgebra.StrataAlgebra.hilbertU	refdomainXpyrêhG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rëh)rì}rí(h=jßhC}rî(hI]hJ]rï(jçjêXpy-methrðehH]hG]hK]uh>jçh8]rñhOX	hilbert()ròró}rô(h=Uh>jìubahAhubaubaubaubeubeubj[)rõ}rö(h=Uh>jÀh?NhAj^hC}r÷(hG]hH]hI]hJ]hK]Uentries]rø(jaX0FZ_matrix() (strataalgebra.StrataAlgebra method)hUNtrùauhLNhMhh8]ubjc)rú}rû(h=Uh>jÀh?NhAjfhC}rü(jhjiXpyhG]hH]hI]hJ]hK]jkXmethodrýjmjýuhLNhMhh8]rþ(jo)rÿ}r(h=XStrataAlgebra.FZ_matrix(r)h>júh?Xh/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_matrixrhAjshC}r(hG]rhajvjwX
strataalgebrarr}rbhH]hI]hJ]hK]rhaj|XStrataAlgebra.FZ_matrixj~j}juhLNhMhh8]r(j)r	}r
(h=X	FZ_matrixh>jÿh?jhAjhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]rhOX	FZ_matrixr
r}r(h=Uh>j	ubaubj)r}r(h=Xrh>jÿh?jhAjhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]rj¢)r}r(h=XrhC}r(hI]hJ]hH]hG]hK]uh>jh8]rhOXrr}r(h=Uh>jubahAjªubaubeubj¿)r}r(h=Uh>júh?jhAjÂhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]r(h~)r}r(h=X,Return the matrix of Faber-Zagier relations.r h>jh?Xh/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_matrixr!hAhhC}r"(hI]hJ]hH]hG]hK]uhLKhMhh8]r#hOX,Return the matrix of Faber-Zagier relations.r$r%}r&(h=j h>jubaubjÎ)r'}r((h=Uh>jh?NhAjÑhC}r)(hI]hJ]hH]hG]hK]uhLNhMhh8]r*(jÔ)r+}r,(h=UhC}r-(hI]hJ]hH]hG]hK]uh>j'h8]r.(jÙ)r/}r0(h=X
Parametersh>j+h?jhAjÜhC}r1(hI]hJ]hH]hG]hK]uhLKh8]r2hOX
Parametersr3r4}r5(h=Uh>j/ubaubjâ)r6}r7(h=UhC}r8(hI]hJ]hH]hG]hK]uh>j+h8]r9h~)r:}r;(h=Xr -- The codimension.hC}r<(hI]hJ]hH]hG]hK]uh>j6h8]r=(jõ)r>}r?(h=XrhC}r@(hG]hH]jùhI]hJ]hK]uh>j:h8]rAhOXrrB}rC(h=Uh>j>ubahAjþubhOX -- rDrE}rF(h=Uh>j:ubhOXThe codimension.rGrH}rI(h=XThe codimension.h>j:ubehAhubahAj¿ubehAjÀubjÔ)rJ}rK(h=UhC}rL(hI]hJ]hH]hG]hK]uh>j'h8]rM(jÙ)rN}rO(h=XReturn typeh>jJh?jhAjÜhC}rP(hI]hJ]hH]hG]hK]uhLKh8]rQhOXReturn typerRrS}rT(h=Uh>jNubaubjâ)rU}rV(h=UhC}rW(hI]hJ]hH]hG]hK]uh>jJh8]rXh~)rY}rZ(h=XMatrixhC}r[(hI]hJ]hH]hG]hK]uh>jUh8]r\j)r]}r^(h=X:class:`Matrix`r_h>jYh?NhAjhC}r`(UreftypeXclassjÜjÝXMatrixU	refdomainXpyrahG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rbh)rc}rd(h=j_hC}re(hI]hJ]rf(jçjaXpy-classrgehH]hG]hK]uh>j]h8]rhhOXMatrixrirj}rk(h=Uh>jcubahAhubaubahAhubahAj¿ubehAjÀubeubh~)rl}rm(h=X_The columns correspond to the basis elements of the Strata algebra, and each row is a relation.rnh>jh?j!hAhhC}ro(hI]hJ]hH]hG]hK]uhLKhMhh8]rphOX_The columns correspond to the basis elements of the Strata algebra, and each row is a relation.rqrr}rs(h=jnh>jlubaubh~)rt}ru(h=X¤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. ::h>jh?j!hAhhC}rv(hI]hJ]hH]hG]hK]uhLKhMhh8]rw(hOXNotice that this matrix considers the kappa classes to be in the monomial basis. Thus, is different than the
output of Pixton's original rxry}rz(h=XNotice that this matrix considers the kappa classes to be in the monomial basis. Thus, is different than the
output of Pixton's original h>jtubh)r{}r|(h=X``taurel.sage``hC}r}(hI]hJ]hH]hG]hK]uh>jth8]r~hOXtaurel.sagerr}r(h=Uh>j{ubahAhubhOX	 program.rr}r(h=X	 program.h>jtubeubjD)r}r(h=Xsage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,0,(1,2,3,4))
sage: s.FZ_matrix(1)
[  -9/4   -9/4   -9/4 -153/4   45/4   45/4   45/4   45/4]
[   3/2    3/2    3/2  -33/2  -21/2   15/2   15/2   15/2]
[   3/2    3/2    3/2  -33/2   15/2  -21/2   15/2   15/2]
[   3/2    3/2    3/2  -33/2   15/2   15/2  -21/2   15/2]
[   3/2    3/2    3/2  -33/2   15/2   15/2   15/2  -21/2]
[ -15/4   21/4   21/4  -15/4  -21/4  -21/4   15/4   15/4]
[  21/4  -15/4   21/4  -15/4  -21/4   15/4  -21/4   15/4]
[  21/4   21/4  -15/4  -15/4  -21/4   15/4   15/4  -21/4]
[  21/4   21/4  -15/4  -15/4   15/4  -21/4  -21/4   15/4]
[  21/4  -15/4   21/4  -15/4   15/4  -21/4   15/4  -21/4]
[ -15/4   21/4   21/4  -15/4   15/4   15/4  -21/4  -21/4]
[ -15/4  -15/4  -15/4  -15/4   15/4   15/4   15/4   15/4]h>jh?j!hAjGhC}r(hEhFhG]hH]hI]hJ]hK]uhLM hMhh8]rhOXsage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,0,(1,2,3,4))
sage: s.FZ_matrix(1)
[  -9/4   -9/4   -9/4 -153/4   45/4   45/4   45/4   45/4]
[   3/2    3/2    3/2  -33/2  -21/2   15/2   15/2   15/2]
[   3/2    3/2    3/2  -33/2   15/2  -21/2   15/2   15/2]
[   3/2    3/2    3/2  -33/2   15/2   15/2  -21/2   15/2]
[   3/2    3/2    3/2  -33/2   15/2   15/2   15/2  -21/2]
[ -15/4   21/4   21/4  -15/4  -21/4  -21/4   15/4   15/4]
[  21/4  -15/4   21/4  -15/4  -21/4   15/4  -21/4   15/4]
[  21/4   21/4  -15/4  -15/4  -21/4   15/4   15/4  -21/4]
[  21/4   21/4  -15/4  -15/4   15/4  -21/4  -21/4   15/4]
[  21/4  -15/4   21/4  -15/4   15/4  -21/4   15/4  -21/4]
[ -15/4   21/4   21/4  -15/4   15/4   15/4  -21/4  -21/4]
[ -15/4  -15/4  -15/4  -15/4   15/4   15/4   15/4   15/4]rr}r(h=Uh>jubaubjÜ)r}r(h=Xw:meth:`~strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basis`, :meth:`~strataalgebra.StrataAlgebraElement.in_kernel`rh>jh?j!hAjàhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]rh~)r}r(h=jh>jh?j!hAhhC}r(hI]hJ]hH]hG]hK]uhLKh8]r(j)r}r(h=X@:meth:`~strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basis`rh>jh?NhAjhC}r(UreftypeXmethjÜjÝX7strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basisU	refdomainXpyrhG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rh)r}r(h=jhC}r(hI]hJ]r(jçjXpy-methrehH]hG]hK]uh>jh8]r hOXFZ_matrix_pushforward_basis()r¡r¢}r£(h=Uh>jubahAhubaubhOX, r¤r¥}r¦(h=X, h>jubj)r§}r¨(h=X5:meth:`~strataalgebra.StrataAlgebraElement.in_kernel`r©h>jh?NhAjhC}rª(UreftypeXmethjÜjÝX,strataalgebra.StrataAlgebraElement.in_kernelU	refdomainXpyr«hG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]r¬h)r}r®(h=j©hC}r¯(hI]hJ]r°(jçj«Xpy-methr±ehH]hG]hK]uh>j§h8]r²hOXin_kernel()r³r´}rµ(h=Uh>jubahAhubaubeubaubeubeubj[)r¶}r·(h=Uh>jÀh?NhAj^hC}r¸(hG]hH]hI]hJ]hK]Uentries]r¹(jaXBFZ_matrix_pushforward_basis() (strataalgebra.StrataAlgebra method)h"UNtrºauhLNhMhh8]ubjc)r»}r¼(h=Uh>jÀh?NhAjfhC}r½(jhjiXpyhG]hH]hI]hJ]hK]jkXmethodr¾jmj¾uhLNhMhh8]r¿(jo)rÀ}rÁ(h=X,StrataAlgebra.FZ_matrix_pushforward_basis(r)h>j»h?Xz/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basisrÂhAjshC}rÃ(hG]rÄh"ajvjwX
strataalgebrarÅrÆ}rÇbhH]hI]hJ]hK]rÈh"aj|X)StrataAlgebra.FZ_matrix_pushforward_basisj~j}juhLNhMhh8]rÉ(j)rÊ}rË(h=XFZ_matrix_pushforward_basish>jÀh?jÂhAjhC}rÌ(hI]hJ]hH]hG]hK]uhLNhMhh8]rÍhOXFZ_matrix_pushforward_basisrÎrÏ}rÐ(h=Uh>jÊubaubj)rÑ}rÒ(h=Xrh>jÀh?jÂhAjhC}rÓ(hI]hJ]hH]hG]hK]uhLNhMhh8]rÔj¢)rÕ}rÖ(h=XrhC}r×(hI]hJ]hH]hG]hK]uh>jÑh8]rØhOXrrÙ}rÚ(h=Uh>jÕubahAjªubaubeubj¿)rÛ}rÜ(h=Uh>j»h?jÂhAjÂhC}rÝ(hI]hJ]hH]hG]hK]uhLNhMhh8]rÞ(h~)rß}rà(h=XReturn the matrix of Faber-Zagier relations, using the "pushforward" basis, NOT the kappa monomial basis that the rest of the code uses.ráh>jÛh?Xz/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basisrâhAhhC}rã(hI]hJ]hH]hG]hK]uhLKhMhh8]rähOXReturn the matrix of Faber-Zagier relations, using the "pushforward" basis, NOT the kappa monomial basis that the rest of the code uses.råræ}rç(h=jáh>jßubaubjÎ)rè}ré(h=Uh>jÛh?NhAjÑhC}rê(hI]hJ]hH]hG]hK]uhLNhMhh8]rë(jÔ)rì}rí(h=UhC}rî(hI]hJ]hH]hG]hK]uh>jèh8]rï(jÙ)rð}rñ(h=X
Parametersh>jìh?jÂhAjÜhC}rò(hI]hJ]hH]hG]hK]uhLKh8]róhOX
Parametersrôrõ}rö(h=Uh>jðubaubjâ)r÷}rø(h=UhC}rù(hI]hJ]hH]hG]hK]uh>jìh8]rúh~)rû}rü(h=Xr -- The codimension.hC}rý(hI]hJ]hH]hG]hK]uh>j÷h8]rþ(jõ)rÿ}r(h=XrhC}r(hG]hH]jùhI]hJ]hK]uh>jûh8]rhOXrr}r(h=Uh>jÿubahAjþubhOX -- rr}r(h=Uh>jûubhOXThe codimension.rr	}r
(h=XThe codimension.h>jûubehAhubahAj¿ubehAjÀubjÔ)r}r(h=UhC}r
(hI]hJ]hH]hG]hK]uh>jèh8]r(jÙ)r}r(h=XReturn typeh>jh?jÂhAjÜhC}r(hI]hJ]hH]hG]hK]uhLKh8]rhOXReturn typerr}r(h=Uh>jubaubjâ)r}r(h=UhC}r(hI]hJ]hH]hG]hK]uh>jh8]rh~)r}r(h=XMatrixhC}r(hI]hJ]hH]hG]hK]uh>jh8]rj)r}r(h=X:class:`Matrix`r h>jh?NhAjhC}r!(UreftypeXclassjÜjÝXMatrixU	refdomainXpyr"hG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]r#h)r$}r%(h=j hC}r&(hI]hJ]r'(jçj"Xpy-classr(ehH]hG]hK]uh>jh8]r)hOXMatrixr*r+}r,(h=Uh>j$ubahAhubaubahAhubahAj¿ubehAjÀubeubh~)r-}r.(h=XÈThe 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. ::h>jÛh?jâhAhhC}r/(hI]hJ]hH]hG]hK]uhLKhMhh8]r0(hOXThe 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 r1r2}r3(h=XThe 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 h>j-ubh)r4}r5(h=X``tautrel.sage``hC}r6(hI]hJ]hH]hG]hK]uh>j-h8]r7hOXtautrel.sager8r9}r:(h=Uh>j4ubahAhubhOX! program after permuting columns.r;r<}r=(h=X! program after permuting columns.h>j-ubeubjD)r>}r?(h=X(sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,0,(1,2,3,4))
sage: s.FZ_matrix_pushforward_basis(1)
[  -9/4   -9/4   -9/4 -153/4   45/4   45/4   45/4   45/4]
[   3/2    3/2    3/2  -33/2  -21/2   15/2   15/2   15/2]
[   3/2    3/2    3/2  -33/2   15/2  -21/2   15/2   15/2]
[   3/2    3/2    3/2  -33/2   15/2   15/2  -21/2   15/2]
[   3/2    3/2    3/2  -33/2   15/2   15/2   15/2  -21/2]
[ -15/4   21/4   21/4  -15/4  -21/4  -21/4   15/4   15/4]
[  21/4  -15/4   21/4  -15/4  -21/4   15/4  -21/4   15/4]
[  21/4   21/4  -15/4  -15/4  -21/4   15/4   15/4  -21/4]
[  21/4   21/4  -15/4  -15/4   15/4  -21/4  -21/4   15/4]
[  21/4  -15/4   21/4  -15/4   15/4  -21/4   15/4  -21/4]
[ -15/4   21/4   21/4  -15/4   15/4   15/4  -21/4  -21/4]
[ -15/4  -15/4  -15/4  -15/4   15/4   15/4   15/4   15/4]h>jÛh?jâhAjGhC}r@(hEhFhG]hH]hI]hJ]hK]uhLMBhMhh8]rAhOX(sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,0,(1,2,3,4))
sage: s.FZ_matrix_pushforward_basis(1)
[  -9/4   -9/4   -9/4 -153/4   45/4   45/4   45/4   45/4]
[   3/2    3/2    3/2  -33/2  -21/2   15/2   15/2   15/2]
[   3/2    3/2    3/2  -33/2   15/2  -21/2   15/2   15/2]
[   3/2    3/2    3/2  -33/2   15/2   15/2  -21/2   15/2]
[   3/2    3/2    3/2  -33/2   15/2   15/2   15/2  -21/2]
[ -15/4   21/4   21/4  -15/4  -21/4  -21/4   15/4   15/4]
[  21/4  -15/4   21/4  -15/4  -21/4   15/4  -21/4   15/4]
[  21/4   21/4  -15/4  -15/4  -21/4   15/4   15/4  -21/4]
[  21/4   21/4  -15/4  -15/4   15/4  -21/4  -21/4   15/4]
[  21/4  -15/4   21/4  -15/4   15/4  -21/4   15/4  -21/4]
[ -15/4   21/4   21/4  -15/4   15/4   15/4  -21/4  -21/4]
[ -15/4  -15/4  -15/4  -15/4   15/4   15/4   15/4   15/4]rBrC}rD(h=Uh>j>ubaubjÜ)rE}rF(h=X.:meth:`~strataalgebra.StrataAlgebra.FZ_matrix`rGh>jÛh?jâhAjàhC}rH(hI]hJ]hH]hG]hK]uhLNhMhh8]rIh~)rJ}rK(h=jGh>jEh?jâhAhhC}rL(hI]hJ]hH]hG]hK]uhLKh8]rMj)rN}rO(h=jGh>jJh?NhAjhC}rP(UreftypeXmethjÜjÝX%strataalgebra.StrataAlgebra.FZ_matrixU	refdomainXpyrQhG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rRh)rS}rT(h=jGhC}rU(hI]hJ]rV(jçjQXpy-methrWehH]hG]hK]uh>jNh8]rXhOXFZ_matrix()rYrZ}r[(h=Uh>jSubahAhubaubaubaubeubeubj[)r\}r](h=Uh>jÀh?Xj/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.MgnLb_classr^hAj^hC}r_(hG]hH]hI]hJ]hK]Uentries]r`(jaX2MgnLb_class() (strataalgebra.StrataAlgebra method)h!UNtraauhLNhMhh8]ubjc)rb}rc(h=Uh>jÀh?j^hAjfhC}rd(jhjiXpyhG]hH]hI]hJ]hK]jkXmethodrejmjeuhLNhMhh8]rf(jo)rg}rh(h=X StrataAlgebra.MgnLb_class(index)h>jbh?Xj/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.MgnLb_classrihAjshC}rj(hG]rkh!ajvjwX
strataalgebrarlrm}rnbhH]hI]hJ]hK]roh!aj|XStrataAlgebra.MgnLb_classj~j}juhLNhMhh8]rp(j)rq}rr(h=XMgnLb_classh>jgh?jihAjhC}rs(hI]hJ]hH]hG]hK]uhLNhMhh8]rthOXMgnLb_classrurv}rw(h=Uh>jqubaubj)rx}ry(h=Xindexh>jgh?jihAjhC}rz(hI]hJ]hH]hG]hK]uhLNhMhh8]r{j¢)r|}r}(h=XindexhC}r~(hI]hJ]hH]hG]hK]uh>jxh8]rhOXindexrr}r(h=Uh>j|ubahAjªubaubeubj¿)r}r(h=Uh>jbh?jihAjÂhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]r(h~)r}r(h=XReturns the class corresponding to the index from Carl Faber's ``MgnLb`` Maple program.
This is useful for testing purposes. ::h>jh?j^hAhhC}r(hI]hJ]hH]hG]hK]uhLKhMhh8]r(hOX?Returns the class corresponding to the index from Carl Faber's rr}r(h=X?Returns the class corresponding to the index from Carl Faber's h>jubh)r}r(h=X	``MgnLb``hC}r(hI]hJ]hH]hG]hK]uh>jh8]rhOXMgnLbrr}r(h=Uh>jubahAhubhOX4 Maple program.
This is useful for testing purposes.rr}r(h=X4 Maple program.
This is useful for testing purposes.h>jubeubjD)r}r(h=X·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)
ps2h>jh?j^hAjGhC}r(hEhFhG]hH]hI]hJ]hK]uhLM]hMhh8]rhOX·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)
ps2rr}r(h=Uh>jubaubeubeubj[)r}r (h=Uh>jÀh?NhAj^hC}r¡(hG]hH]hI]hJ]hK]Uentries]r¢(jaX0MgnLb_int() (strataalgebra.StrataAlgebra method)hUNtr£auhLNhMhh8]ubjc)r¤}r¥(h=Uh>jÀh?NhAjfhC}r¦(jhjiXpyhG]hH]hI]hJ]hK]jkXmethodr§jmj§uhLNhMhh8]r¨(jo)r©}rª(h=X#StrataAlgebra.MgnLb_int(index_list)h>j¤h?Xh/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.MgnLb_intr«hAjshC}r¬(hG]rhajvjwX
strataalgebrar®r¯}r°bhH]hI]hJ]hK]r±haj|XStrataAlgebra.MgnLb_intj~j}juhLNhMhh8]r²(j)r³}r´(h=X	MgnLb_inth>j©h?j«hAjhC}rµ(hI]hJ]hH]hG]hK]uhLNhMhh8]r¶hOX	MgnLb_intr·r¸}r¹(h=Uh>j³ubaubj)rº}r»(h=X
index_listh>j©h?j«hAjhC}r¼(hI]hJ]hH]hG]hK]uhLNhMhh8]r½j¢)r¾}r¿(h=X
index_listhC}rÀ(hI]hJ]hH]hG]hK]uh>jºh8]rÁhOX
index_listrÂrÃ}rÄ(h=Uh>j¾ubahAjªubaubeubj¿)rÅ}rÆ(h=Uh>j¤h?j«hAjÂhC}rÇ(hI]hJ]hH]hG]hK]uhLNhMhh8]rÈ(h~)rÉ}rÊ(h=XJComputes an integral of boundary divisors, kappa classes, and psi classes.rËh>jÅh?Xh/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.MgnLb_intrÌhAhhC}rÍ(hI]hJ]hH]hG]hK]uhLKhMhh8]rÎhOXJComputes an integral of boundary divisors, kappa classes, and psi classes.rÏrÐ}rÑ(h=jËh>jÉubaubjÎ)rÒ}rÓ(h=Uh>jÅh?NhAjÑhC}rÔ(hI]hJ]hH]hG]hK]uhLNhMhh8]rÕ(jÔ)rÖ}r×(h=UhC}rØ(hI]hJ]hH]hG]hK]uh>jÒh8]rÙ(jÙ)rÚ}rÛ(h=X
Parametersh>jÖh?j«hAjÜhC}rÜ(hI]hJ]hH]hG]hK]uhLKh8]rÝhOX
ParametersrÞrß}rà(h=Uh>jÚubaubjâ)rá}râ(h=UhC}rã(hI]hJ]hH]hG]hK]uh>jÖh8]räh~)rå}ræ(h=XÑindex_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.hC}rç(hI]hJ]hH]hG]hK]uh>jáh8]rè(jõ)ré}rê(h=X
index_listhC}rë(hG]hH]jùhI]hJ]hK]uh>jåh8]rìhOX
index_listrírî}rï(h=Uh>jéubahAjþubhOX -- rðrñ}rò(h=Uh>jåubhOXGA list of indices of classes, according the the scheme of Carl Faber's rórô}rõ(h=XGA list of indices of classes, according the the scheme of Carl Faber's h>jåubh)rö}r÷(h=X	``MgnLb``hC}rø(hI]hJ]hH]hG]hK]uh>jåh8]rùhOXMgnLbrúrû}rü(h=Uh>jöubahAhubhOXw Maple program. This function is useful because so you can test our implementation of the product and the FZ_relations.rýrþ}rÿ(h=Xw Maple program. This function is useful because so you can test our implementation of the product and the FZ_relations.h>jåubehAhubahAj¿ubehAjÀubjÔ)r}r(h=UhC}r(hI]hJ]hH]hG]hK]uh>jÒh8]r(jÙ)r}r(h=XReturn typeh>jh?j«hAjÜhC}r(hI]hJ]hH]hG]hK]uhLKh8]rhOXReturn typerr	}r
(h=Uh>jubaubjâ)r}r(h=UhC}r
(hI]hJ]hH]hG]hK]uh>jh8]rh~)r}r(h=XRationalhC}r(hI]hJ]hH]hG]hK]uh>jh8]rcdocutils.nodes
title_reference
r)r}r(h=X
`Rational`hC}r(hI]hJ]hH]hG]hK]uh>jh8]rhOXRationalrr}r(h=Uh>jubahAUtitle_referencerubahAhubahAj¿ubehAjÀubeubh~)r}r(h=XExamples: ::h>jÅh?jÌhAhhC}r(hI]hJ]hH]hG]hK]uhLKhMhh8]rhOX	Examples:r r!}r"(h=X	Examples:h>jubaubjD)r#}r$(h=Xsage: 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])
1/24h>jÅh?jÌhAjGhC}r%(hEhFhG]hH]hI]hJ]hK]uhLMthMhh8]r&hOXsage: 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)(h=Uh>j#ubaubjÜ)r*}r+(h=X0:meth:`~strataalgebra.StrataAlgebra.MgnLb_class`r,h>jÅh?jÌhAjàhC}r-(hI]hJ]hH]hG]hK]uhLNhMhh8]r.h~)r/}r0(h=j,h>j*h?jÌhAhhC}r1(hI]hJ]hH]hG]hK]uhLKh8]r2j)r3}r4(h=j,h>j/h?NhAjhC}r5(UreftypeXmethjÜjÝX'strataalgebra.StrataAlgebra.MgnLb_classU	refdomainXpyr6hG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]r7h)r8}r9(h=j,hC}r:(hI]hJ]r;(jçj6Xpy-methr<ehH]hG]hK]uh>j3h8]r=hOX
MgnLb_class()r>r?}r@(h=Uh>j8ubahAhubaubaubaubeubeubj[)rA}rB(h=Uh>jÀh?Xn/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.basis_integralsrChAj^hC}rD(hG]hH]hI]hJ]hK]Uentries]rE(jaX7basis_integrals (strataalgebra.StrataAlgebra attribute)h#UNtrFauhLNhMhh8]ubjc)rG}rH(h=Uh>jÀh?jChAjfhC}rI(jhjiXpyhG]hH]hI]hJ]hK]jkX	attributerJjmjJuhLNhMhh8]rK(jo)rL}rM(h=XStrataAlgebra.basis_integralsh>jGh?Xn/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.basis_integralsrNhAjshC}rO(hG]rPh#ajvjwX
strataalgebrarQrR}rSbhH]hI]hJ]hK]rTh#aj|XStrataAlgebra.basis_integralsj~j}juhLNhMhh8]rUj)rV}rW(h=Xbasis_integralsh>jLh?jNhAjhC}rX(hI]hJ]hH]hG]hK]uhLNhMhh8]rYhOXbasis_integralsrZr[}r\(h=Uh>jVubaubaubj¿)r]}r^(h=Uh>jGh?jNhAjÂhC}r_(hI]hJ]hH]hG]hK]uhLNhMhh8]r`(h~)ra}rb(h=XRFile: /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py (starting at line 491)rch>j]h?jChAhhC}rd(hI]hJ]hH]hG]hK]uhLKhMhh8]rehOXRFile: /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py (starting at line 491)rfrg}rh(h=jch>jaubaubh~)ri}rj(h=XeReturn a list of numbers corresponding to the integrals of the basis elements in the top codimension.rkh>j]h?jChAhhC}rl(hI]hJ]hH]hG]hK]uhLKhMhh8]rmhOXeReturn a list of numbers corresponding to the integrals of the basis elements in the top codimension.rnro}rp(h=jkh>jiubaubh~)rq}rr(h=X_This is computed via the FZ relations, so it is probably not fast. However, it is a nice check.rsh>j]h?jChAhhC}rt(hI]hJ]hH]hG]hK]uhLKhMhh8]ruhOX_This is computed via the FZ relations, so it is probably not fast. However, it is a nice check.rvrw}rx(h=jsh>jqubaubh~)ry}rz(h=XEThe value is cached, so you only have to compute it once per session.r{h>j]h?jChAhhC}r|(hI]hJ]hH]hG]hK]uhLKhMhh8]r}hOXEThe value is cached, so you only have to compute it once per session.r~r}r(h=j{h>jyubaubh~)r}r(h=X~This is used by :meth:`~strataalgebra.StrataAlgebraElement.integrate` which is the more likely way you will want to use it.
::h>j]h?jChAhhC}r(hI]hJ]hH]hG]hK]uhLK	hMhh8]r(hOXThis is used by rr}r(h=XThis is used by h>jubj)r}r(h=X5:meth:`~strataalgebra.StrataAlgebraElement.integrate`rh>jh?NhAjhC}r(UreftypeXmethjÜjÝX,strataalgebra.StrataAlgebraElement.integrateU	refdomainXpyrhG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rh)r}r(h=jhC}r(hI]hJ]r(jçjXpy-methrehH]hG]hK]uh>jh8]rhOXintegrate()rr}r(h=Uh>jubahAhubaubhOX6 which is the more likely way you will want to use it.rr}r(h=X6 which is the more likely way you will want to use it.h>jubeubjD)r}r(h=XÖsage: 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]h>j]h?jChAjGhC}r(hEhFhG]hH]hI]hJ]hK]uhLMhMhh8]rhOXÖsage: 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]rr}r (h=Uh>jubaubeubeubj[)r¡}r¢(h=Uh>jÀh?Xg/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.boundaryr£hAj^hC}r¤(hG]hH]hI]hJ]hK]Uentries]r¥(jaX/boundary() (strataalgebra.StrataAlgebra method)hUNtr¦auhLNhMhh8]ubjc)r§}r¨(h=Uh>jÀh?j£hAjfhC}r©(jhjiXpyrªhG]hH]hI]hJ]hK]jkXmethodr«jmj«uhLNhMhh8]r¬(jo)r}r®(h=X(StrataAlgebra.boundary(g1, markings1=())h>j§h?Xg/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.boundaryr¯hAjshC}r°(hG]r±hajvjwX
strataalgebrar²r³}r´bhH]hI]hJ]hK]rµhaj|XStrataAlgebra.boundaryj~j}juhLNhMhh8]r¶(j)r·}r¸(h=Xboundaryh>jh?j¯hAjhC}r¹(hI]hJ]hH]hG]hK]uhLNhMhh8]rºhOXboundaryr»r¼}r½(h=Uh>j·ubaubj)r¾}r¿(h=Xg1, markings1=()h>jh?j¯hAjhC}rÀ(hI]hJ]hH]hG]hK]uhLNhMhh8]rÁ(j¢)rÂ}rÃ(h=Xg1hC}rÄ(hI]hJ]hH]hG]hK]uh>j¾h8]rÅhOXg1rÆrÇ}rÈ(h=Uh>jÂubahAjªubj¢)rÉ}rÊ(h=Xmarkings1=()hC}rË(hI]hJ]hH]hG]hK]uh>j¾h8]rÌhOXmarkings1=()rÍrÎ}rÏ(h=Uh>jÉubahAjªubeubeubj¿)rÐ}rÑ(h=Uh>j§h?j¯hAjÂhC}rÒ(hI]hJ]hH]hG]hK]uhLNhMhh8]rÓ(h~)rÔ}rÕ(h=XQReturn a boundary divisor with genus `g1` and `markings1` points on one componenth>jÐh?j£hAhhC}rÖ(hI]hJ]hH]hG]hK]uhLKhMhh8]r×(hOX%Return a boundary divisor with genus rØrÙ}rÚ(h=X%Return a boundary divisor with genus h>jÔubj)rÛ}rÜ(h=X`g1`hC}rÝ(hI]hJ]hH]hG]hK]uh>jÔh8]rÞhOXg1rßrà}rá(h=Uh>jÛubahAjubhOX and rârã}rä(h=X and h>jÔubj)rå}ræ(h=X`markings1`hC}rç(hI]hJ]hH]hG]hK]uh>jÔh8]rèhOX	markings1rérê}rë(h=Uh>jåubahAjubhOX points on one componentrìrí}rî(h=X points on one componenth>jÔubeubjÎ)rï}rð(h=Uh>jÐh?j£hAjÑhC}rñ(hI]hJ]hH]hG]hK]uhLNhMhh8]rò(jÔ)ró}rô(h=UhC}rõ(hI]hJ]hH]hG]hK]uh>jïh8]rö(jÙ)r÷}rø(h=X
Parametersh>jóh?j£hAjÜhC}rù(hI]hJ]hH]hG]hK]uhLKh8]rúhOX
Parametersrûrü}rý(h=Uh>j÷ubaubjâ)rþ}rÿ(h=UhC}r(hI]hJ]hH]hG]hK]uh>jóh8]rjç)r}r(h=UhC}r(hI]hJ]hH]hG]hK]uh>jþh8]r(jì)r}r(h=UhC}r(hI]hJ]hH]hG]hK]uh>jh8]r	h~)r
}r(h=X&g1 (int) -- The genus on one componenthC}r(hI]hJ]hH]hG]hK]uh>jh8]r
(jõ)r}r(h=Xg1hC}r(hG]hH]jùhI]hJ]hK]uh>j
h8]rhOXg1rr}r(h=Uh>jubahAjþubhOX (rr}r(h=Uh>j
ubj)r}r(h=UhC}r(UreftypejjùU	reftargetXintrU	refdomainjªhG]hH]UrefexplicithI]hJ]hK]uh>j
h8]rj	)r}r(h=jhC}r(hI]hJ]hH]hG]hK]uh>jh8]r hOXintr!r"}r#(h=Uh>jubahAjubahAjubhOX)r$}r%(h=Uh>j
ubhOX -- r&r'}r((h=Uh>j
ubhOXThe genus on one componentr)r*}r+(h=XThe genus on one componenth>j
ubehAhubahAj$ubjì)r,}r-(h=UhC}r.(hI]hJ]hH]hG]hK]uh>jh8]r/h~)r0}r1(h=Xamarkings1 (list) -- A list or tuple of markings on the one component. Defaults to the empty list.hC}r2(hI]hJ]hH]hG]hK]uh>j,h8]r3(jõ)r4}r5(h=X	markings1hC}r6(hG]hH]jùhI]hJ]hK]uh>j0h8]r7hOX	markings1r8r9}r:(h=Uh>j4ubahAjþubhOX (r;r<}r=(h=Uh>j0ubj)r>}r?(h=UhC}r@(UreftypejjùU	reftargetXlistrAU	refdomainjªhG]hH]UrefexplicithI]hJ]hK]uh>j0h8]rBj	)rC}rD(h=jAhC}rE(hI]hJ]hH]hG]hK]uh>j>h8]rFhOXlistrGrH}rI(h=Uh>jCubahAjubahAjubhOX)rJ}rK(h=Uh>j0ubhOX -- rLrM}rN(h=Uh>j0ubhOXMA list or tuple of markings on the one component. Defaults to the empty list.rOrP}rQ(h=XMA list or tuple of markings on the one component. Defaults to the empty list.h>j0ubehAhubahAj$ubehAj¾ubahAj¿ubehAjÀubjÔ)rR}rS(h=UhC}rT(hI]hJ]hH]hG]hK]uh>jïh8]rU(jÙ)rV}rW(h=XReturn typeh>jRh?j£hAjÜhC}rX(hI]hJ]hH]hG]hK]uhLKh8]rYhOXReturn typerZr[}r\(h=Uh>jVubaubjâ)r]}r^(h=UhC}r_(hI]hJ]hH]hG]hK]uh>jRh8]r`h~)ra}rb(h=XStrataAlgebraElementhC}rc(hI]hJ]hH]hG]hK]uh>j]h8]rdj)re}rf(h=X:class:`StrataAlgebraElement`rgh>jah?NhAjhC}rh(UreftypeXclassjÜjÝXStrataAlgebraElementU	refdomainXpyrihG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rjh)rk}rl(h=jghC}rm(hI]hJ]rn(jçjiXpy-classroehH]hG]hK]uh>jeh8]rphOXStrataAlgebraElementrqrr}rs(h=Uh>jkubahAhubaubahAhubahAj¿ubehAjÀubeubh~)rt}ru(h=XfNote that if the two components are the same, it will return the stratum with a coefficient of 1/2. ::h>jÐh?j£hAhhC}rv(hI]hJ]hH]hG]hK]uhLKhMhh8]rwhOXcNote that if the two components are the same, it will return the stratum with a coefficient of 1/2.rxry}rz(h=XcNote that if the two components are the same, it will return the stratum with a coefficient of 1/2.h>jtubaubjD)r{}r|(h=XÇsage: 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*Dg1h>jÐh?j£hAjGhC}r}(hEhFhG]hH]hI]hJ]hK]uhLM®hMhh8]r~hOXÇsage: 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*Dg1rr}r(h=Uh>j{ubaubh~)r}r(h=X'Each component must still be stable. ::h>jÐh?j£hAhhC}r(hI]hJ]hH]hG]hK]uhLKhMhh8]rhOX$Each component must still be stable.rr}r(h=X$Each component must still be stable.h>jubaubjD)r}r(h=XHsage: s.boundary(2)
Traceback (most recent call last):
...
KeyError: Dg0h>jÐh?j£hAjGhC}r(hEhFhG]hH]hI]hJ]hK]uhLMºhMhh8]rhOXHsage: s.boundary(2)
Traceback (most recent call last):
...
KeyError: Dg0rr}r(h=Uh>jubaubeubeubj[)r}r(h=Uh>jÀh?Xn/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.do_all_productsrhAj^hC}r(hG]hH]hI]hJ]hK]Uentries]r(jaX6do_all_products() (strataalgebra.StrataAlgebra method)hUNtrauhLNhMhh8]ubjc)r}r(h=Uh>jÀh?jhAjfhC}r(jhjiXpyhG]hH]hI]hJ]hK]jkXmethodrjmjuhLNhMhh8]r(jo)r}r(h=XStrataAlgebra.do_all_products()h>jh?Xn/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.do_all_productsrhAjshC}r(hG]rhajvjwX
strataalgebrar r¡}r¢bhH]hI]hJ]hK]r£haj|XStrataAlgebra.do_all_productsj~j}juhLNhMhh8]r¤(j)r¥}r¦(h=Xdo_all_productsh>jh?jhAjhC}r§(hI]hJ]hH]hG]hK]uhLNhMhh8]r¨hOXdo_all_productsr©rª}r«(h=Uh>j¥ubaubj)r¬}r(h=Uh>jh?jhAjhC}r®(hI]hJ]hH]hG]hK]uhLNhMhh8]ubeubj¿)r¯}r°(h=Uh>jh?jhAjÂhC}r±(hI]hJ]hH]hG]hK]uhLNhMhh8]r²(h~)r³}r´(h=XCompute all the products of basis elements of the ring (thus caching their values).
This could take a long time for some rings. ::h>j¯h?jhAhhC}rµ(hI]hJ]hH]hG]hK]uhLKhMhh8]r¶hOXCompute all the products of basis elements of the ring (thus caching their values).
This could take a long time for some rings.r·r¸}r¹(h=XCompute all the products of basis elements of the ring (thus caching their values).
This could take a long time for some rings.h>j³ubaubjD)rº}r»(h=X^sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,))
sage: s.do_all_products()h>j¯h?jhAjGhC}r¼(hEhFhG]hH]hI]hJ]hK]uhLMÇhMhh8]r½hOX^sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,))
sage: s.do_all_products()r¾r¿}rÀ(h=Uh>jºubaubeubeubj[)rÁ}rÂ(h=Uh>jÀh?Xj/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.get_stratumrÃhAj^hC}rÄ(hG]hH]hI]hJ]hK]Uentries]rÅ(jaX2get_stratum() (strataalgebra.StrataAlgebra method)hUNtrÆauhLNhMhh8]ubjc)rÇ}rÈ(h=Uh>jÀh?jÃhAjfhC}rÉ(jhjiXpyhG]hH]hI]hJ]hK]jkXmethodrÊjmjÊuhLNhMhh8]rË(jo)rÌ}rÍ(h=XStrataAlgebra.get_stratum(r, j)h>jÇh?Xj/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.get_stratumrÎhAjshC}rÏ(hG]rÐhajvjwX
strataalgebrarÑrÒ}rÓbhH]hI]hJ]hK]rÔhaj|XStrataAlgebra.get_stratumj~j}juhLNhMhh8]rÕ(j)rÖ}r×(h=Xget_stratumh>jÌh?jÎhAjhC}rØ(hI]hJ]hH]hG]hK]uhLNhMhh8]rÙhOXget_stratumrÚrÛ}rÜ(h=Uh>jÖubaubj)rÝ}rÞ(h=Xr, jh>jÌh?jÎhAjhC}rß(hI]hJ]hH]hG]hK]uhLNhMhh8]rà(j¢)rá}râ(h=XrhC}rã(hI]hJ]hH]hG]hK]uh>jÝh8]rähOXrrå}ræ(h=Uh>jáubahAjªubj¢)rç}rè(h=XjhC}ré(hI]hJ]hH]hG]hK]uh>jÝh8]rêhOXjrë}rì(h=Uh>jçubahAjªubeubeubj¿)rí}rî(h=Uh>jÇh?jÎhAjÂhC}rï(hI]hJ]hH]hG]hK]uhLNhMhh8]rð(h~)rñ}rò(h=XEGet the :class:`StrataGraph` associated a a codimension and an index.h>jíh?jÃhAhhC}ró(hI]hJ]hH]hG]hK]uhLKhMhh8]rô(hOXGet the rõrö}r÷(h=XGet the h>jñubj)rø}rù(h=X:class:`StrataGraph`rúh>jñh?NhAjhC}rû(UreftypeXclassjÜjÝXStrataGraphU	refdomainXpyrühG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rýh)rþ}rÿ(h=júhC}r	(hI]hJ]r	(jçjüXpy-classr	ehH]hG]hK]uh>jøh8]r	hOXStrataGraphr	r	}r	(h=Uh>jþubahAhubaubhOX) associated a a codimension and an index.r	r	}r		(h=X) associated a a codimension and an index.h>jñubeubjÎ)r
	}r	(h=Uh>jíh?jÃhAjÑhC}r	(hI]hJ]hH]hG]hK]uhLNhMhh8]r
	(jÔ)r	}r	(h=UhC}r	(hI]hJ]hH]hG]hK]uh>j
	h8]r	(jÙ)r	}r	(h=X
Parametersh>j	h?jÃhAjÜhC}r	(hI]hJ]hH]hG]hK]uhLKh8]r	hOX
Parametersr	r	}r	(h=Uh>j	ubaubjâ)r	}r	(h=UhC}r	(hI]hJ]hH]hG]hK]uh>j	h8]r	jç)r	}r	(h=UhC}r	(hI]hJ]hH]hG]hK]uh>j	h8]r 	(jì)r!	}r"	(h=UhC}r#	(hI]hJ]hH]hG]hK]uh>j	h8]r$	h~)r%	}r&	(h=Xr -- The codimensionhC}r'	(hI]hJ]hH]hG]hK]uh>j!	h8]r(	(jõ)r)	}r*	(h=XrhC}r+	(hG]hH]jùhI]hJ]hK]uh>j%	h8]r,	hOXrr-	}r.	(h=Uh>j)	ubahAjþubhOX -- r/	r0	}r1	(h=Uh>j%	ubhOXThe codimensionr2	r3	}r4	(h=XThe codimensionh>j%	ubehAhubahAj$ubjì)r5	}r6	(h=UhC}r7	(hI]hJ]hH]hG]hK]uh>j	h8]r8	h~)r9	}r:	(h=Xj -- The indexhC}r;	(hI]hJ]hH]hG]hK]uh>j5	h8]r<	(jõ)r=	}r>	(h=XjhC}r?	(hG]hH]jùhI]hJ]hK]uh>j9	h8]r@	hOXjrA	}rB	(h=Uh>j=	ubahAjþubhOX -- rC	rD	}rE	(h=Uh>j9	ubhOX	The indexrF	rG	}rH	(h=X	The indexh>j9	ubehAhubahAj$ubehAj¾ubahAj¿ubehAjÀubjÔ)rI	}rJ	(h=UhC}rK	(hI]hJ]hH]hG]hK]uh>j
	h8]rL	(jÙ)rM	}rN	(h=XReturn typeh>jI	h?jÃhAjÜhC}rO	(hI]hJ]hH]hG]hK]uhLKh8]rP	hOXReturn typerQ	rR	}rS	(h=Uh>jM	ubaubjâ)rT	}rU	(h=UhC}rV	(hI]hJ]hH]hG]hK]uh>jI	h8]rW	h~)rX	}rY	(h=XStrataGraphhC}rZ	(hI]hJ]hH]hG]hK]uh>jT	h8]r[	j)r\	}r]	(h=X:class:`StrataGraph`r^	h>jX	h?NhAjhC}r_	(UreftypeXclassjÜjÝXStrataGraphU	refdomainXpyr`	hG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]ra	h)rb	}rc	(h=j^	hC}rd	(hI]hJ]re	(jçj`	Xpy-classrf	ehH]hG]hK]uh>j\	h8]rg	hOXStrataGraphrh	ri	}rj	(h=Uh>jb	ubahAhubaubahAhubahAj¿ubehAjÀubeubh~)rk	}rl	(h=XESee :class:`~strataalgebra.StrataAlgebra` documentation for examples.h>jíh?jÃhAhhC}rm	(hI]hJ]hH]hG]hK]uhLKhMhh8]rn	(hOXSee ro	rp	}rq	(h=XSee h>jk	ubj)rr	}rs	(h=X%:class:`~strataalgebra.StrataAlgebra`rt	h>jk	h?NhAjhC}ru	(UreftypeXclassjÜjÝXstrataalgebra.StrataAlgebraU	refdomainXpyrv	hG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rw	h)rx	}ry	(h=jt	hC}rz	(hI]hJ]r{	(jçjv	Xpy-classr|	ehH]hG]hK]uh>jr	h8]r}	hOX
StrataAlgebrar~	r	}r	(h=Uh>jx	ubahAhubaubhOX documentation for examples.r	r	}r	(h=X documentation for examples.h>jk	ubeubeubeubj[)r	}r	(h=Uh>jÀh?Xf/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.hilbertr	hAj^hC}r	(hG]hH]hI]hJ]hK]Uentries]r	(jaX.hilbert() (strataalgebra.StrataAlgebra method)hUNtr	auhLNhMhh8]ubjc)r	}r	(h=Uh>jÀh?j	hAjfhC}r	(jhjiXpyr	hG]hH]hI]hJ]hK]jkXmethodr	jmj	uhLNhMhh8]r	(jo)r	}r	(h=X!StrataAlgebra.hilbert(codim=None)h>j	h?Xf/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.hilbertr	hAjshC}r	(hG]r	hajvjwX
strataalgebrar	r	}r	bhH]hI]hJ]hK]r	haj|XStrataAlgebra.hilbertj~j}juhLNhMhh8]r	(j)r	}r	(h=Xhilberth>j	h?j	hAjhC}r	(hI]hJ]hH]hG]hK]uhLNhMhh8]r	hOXhilbertr	r	}r 	(h=Uh>j	ubaubj)r¡	}r¢	(h=X
codim=Noneh>j	h?j	hAjhC}r£	(hI]hJ]hH]hG]hK]uhLNhMhh8]r¤	j¢)r¥	}r¦	(h=X
codim=NonehC}r§	(hI]hJ]hH]hG]hK]uh>j¡	h8]r¨	hOX
codim=Noner©	rª	}r«	(h=Uh>j¥	ubahAjªubaubeubj¿)r¬	}r	(h=Uh>j	h?j	hAjÂhC}r®	(hI]hJ]hH]hG]hK]uhLNhMhh8]r¯	(h~)r°	}r±	(h=X¦Give the number of basis elements in the Strata algebra for the given codimension (the Hilbert function).
If the codimension is omitted, return a list of all of them.r²	h>j¬	h?j	hAhhC}r³	(hI]hJ]hH]hG]hK]uhLKhMhh8]r´	hOX¦Give the number of basis elements in the Strata algebra for the given codimension (the Hilbert function).
If the codimension is omitted, return a list of all of them.rµ	r¶	}r·	(h=j²	h>j°	ubaubh~)r¸	}r¹	(h=XyThis is NOT the Betti numbers for the moduli space of curves! For that see :meth:`~strataalgebra.StrataAlgebra.FZ_betti`.h>j¬	h?j	hAhhC}rº	(hI]hJ]hH]hG]hK]uhLKhMhh8]r»	(hOXKThis is NOT the Betti numbers for the moduli space of curves! For that see r¼	r½	}r¾	(h=XKThis is NOT the Betti numbers for the moduli space of curves! For that see h>j¸	ubj)r¿	}rÀ	(h=X-:meth:`~strataalgebra.StrataAlgebra.FZ_betti`rÁ	h>j¸	h?NhAjhC}rÂ	(UreftypeXmethjÜjÝX$strataalgebra.StrataAlgebra.FZ_bettiU	refdomainXpyrÃ	hG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rÄ	h)rÅ	}rÆ	(h=jÁ	hC}rÇ	(hI]hJ]rÈ	(jçjÃ	Xpy-methrÉ	ehH]hG]hK]uh>j¿	h8]rÊ	hOX
FZ_betti()rË	rÌ	}rÍ	(h=Uh>jÅ	ubahAhubaubhOX.rÎ	}rÏ	(h=X.h>j¸	ubeubjÎ)rÐ	}rÑ	(h=Uh>j¬	h?j	hAjÑhC}rÒ	(hI]hJ]hH]hG]hK]uhLNhMhh8]rÓ	jÔ)rÔ	}rÕ	(h=UhC}rÖ	(hI]hJ]hH]hG]hK]uh>jÐ	h8]r×	(jÙ)rØ	}rÙ	(h=X
Parametersh>jÔ	h?j	hAjÜhC}rÚ	(hI]hJ]hH]hG]hK]uhLKh8]rÛ	hOX
ParametersrÜ	rÝ	}rÞ	(h=Uh>jØ	ubaubjâ)rß	}rà	(h=UhC}rá	(hI]hJ]hH]hG]hK]uh>jÔ	h8]râ	h~)rã	}rä	(h=X2codim (int) -- Optional. The codimension you want.hC}rå	(hI]hJ]hH]hG]hK]uh>jß	h8]ræ	(jõ)rç	}rè	(h=XcodimhC}ré	(hG]hH]jùhI]hJ]hK]uh>jã	h8]rê	hOXcodimrë	rì	}rí	(h=Uh>jç	ubahAjþubhOX (rî	rï	}rð	(h=Uh>jã	ubj)rñ	}rò	(h=UhC}ró	(UreftypejjùU	reftargetXintrô	U	refdomainj	hG]hH]UrefexplicithI]hJ]hK]uh>jã	h8]rõ	j	)rö	}r÷	(h=jô	hC}rø	(hI]hJ]hH]hG]hK]uh>jñ	h8]rù	hOXintrú	rû	}rü	(h=Uh>jö	ubahAjubahAjubhOX)rý	}rþ	(h=Uh>jã	ubhOX -- rÿ	r
}r
(h=Uh>jã	ubhOX#Optional. The codimension you want.r
r
}r
(h=X#Optional. The codimension you want.h>jã	ubehAhubahAj¿ubehAjÀubaubjD)r
}r
(h=Xsage: from strataalgebra import *
sage: StrataAlgebra(QQ, 2, (1,)).hilbert()
[1, 4, 17, 49, 92]
sage: StrataAlgebra(QQ, 2, (1,)).hilbert(2)
17h>j¬	h?j	hAjGhC}r
(hEhFhG]hH]hI]hJ]hK]uhLMåhMhh8]r
hOXsage: from strataalgebra import *
sage: StrataAlgebra(QQ, 2, (1,)).hilbert()
[1, 4, 17, 49, 92]
sage: StrataAlgebra(QQ, 2, (1,)).hilbert(2)
17r	
r

}r
(h=Uh>j
ubaubeubeubj[)r
}r
(h=Uh>jÀh?Xb/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.irrr
hAj^hC}r
(hG]hH]hI]hJ]hK]Uentries]r
(jaX*irr() (strataalgebra.StrataAlgebra method)hUNtr
auhLNhMhh8]ubjc)r
}r
(h=Uh>jÀh?j
hAjfhC}r
(jhjiXpyhG]hH]hI]hJ]hK]jkXmethodr
jmj
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(h=XStrataAlgebra.irr()h>j
h?Xb/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.irrr
hAjshC}r
(hG]r
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bhH]hI]hJ]hK]r
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}r"
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(hI]hJ]hH]hG]hK]uhLNhMhh8]ubeubj¿)r+
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(hI]hJ]hH]hG]hK]uhLNhMhh8]r.
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(h=XPMake the irreducible boundary. It will be returned with a coefficient of 1/2. ::h>j+
h?j
hAhhC}r1
(hI]hJ]hH]hG]hK]uhLKhMhh8]r2
hOXMMake the irreducible boundary. It will be returned with a coefficient of 1/2.r3
r4
}r5
(h=XMMake the irreducible boundary. It will be returned with a coefficient of 1/2.h>j/
ubaubjD)r6
}r7
(h=X_sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ, 1, (1,2))
sage: s.irr()
1/2*D_irrh>j+
h?j
hAjGhC}r8
(hEhFhG]hH]hI]hJ]hK]uhLMóhMhh8]r9
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sage: s = StrataAlgebra(QQ, 1, (1,2))
sage: s.irr()
1/2*D_irrr:
r;
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(h=Uh>j6
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(h=Uh>jÀh?Xd/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.kappar?
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(h=XStrataAlgebra.kappa(a)h>jC
h?Xd/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.kapparK
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(h=XReturn a kappa class.rj
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hAhhC}rk
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hOXReturn a kappa class.rm
rn
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Parametersh>jt
h?j?
hAjÜhC}rz
(hI]hJ]hH]hG]hK]uhLKh8]r{
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Parametersr|
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(h=UhC}r
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(h=XCa (int) -- The subscript (codimension) of the kappa class you want.hC}r
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(h=XahC}r
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(h=Uh>j
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(h=Uh>j
ubhOX8The subscript (codimension) of the kappa class you want.r¡
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(h=X8The subscript (codimension) of the kappa class you want.h>j
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(h=XStrataAlgebraElementhC}rµ
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j)r·
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}rÅ
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}rÇ
(h=X%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]h>jd
h?j?
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(hEhFhG]hH]hI]hJ]hK]uhLMhMhh8]rÉ
hOX%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]rÊ
rË
}rÌ
(h=Uh>jÆ
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(h=XQThe subscript must be less than or equal to the dimension of the moduli space. ::h>jd
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(hI]hJ]hH]hG]hK]uhLKhMhh8]rÐ
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rÒ
}rÓ
(h=XNThe subscript must be less than or equal to the dimension of the moduli space.h>jÍ
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}rÕ
(h=XCsage: s.kappa(4)
Traceback (most recent call last):
...
KeyError: 4h>jd
h?j?
hAjGhC}rÖ
(hEhFhG]hH]hI]hJ]hK]uhLMhMhh8]r×
hOXCsage: s.kappa(4)
Traceback (most recent call last):
...
KeyError: 4rØ
rÙ
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(h=Uh>jÔ
ubaubeubeubj[)rÛ
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(h=Uh>jÀh?Xk/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.print_stratarÝ
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(h=XStrataAlgebra.print_strata(r)h>já
h?Xk/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.print_stratarè
hAjshC}ré
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bhH]hI]hJ]hK]rî
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(h=Xprint_stratah>jæ
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hAjÂhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]r(h~)r}r(h=X=Print all the strata, with their indexes, in codimension `r`.h>jh?jÝ
hAhhC}r(hI]hJ]hH]hG]hK]uhLKhMhh8]r(hOX9Print all the strata, with their indexes, in codimension r	r
}r(h=X9Print all the strata, with their indexes, in codimension h>jubj)r}r
(h=X`r`hC}r(hI]hJ]hH]hG]hK]uh>jh8]rhOXrr}r(h=Uh>jubahAjubhOX.r}r(h=X.h>jubeubjÎ)r}r(h=Uh>jh?jÝ
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Parametersh>jh?jÝ
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Parametersr r!}r"(h=Uh>jubaubjâ)r#}r$(h=UhC}r%(hI]hJ]hH]hG]hK]uh>jh8]r&h~)r'}r((h=Xr -- The codimensionhC}r)(hI]hJ]hH]hG]hK]uh>j#h8]r*(jõ)r+}r,(h=XrhC}r-(hG]hH]jùhI]hJ]hK]uh>j'h8]r.hOXrr/}r0(h=Uh>j+ubahAjþubhOX -- r1r2}r3(h=Uh>j'ubhOXThe codimensionr4r5}r6(h=XThe codimensionh>j'ubehAhubahAj¿ubehAjÀubjÔ)r7}r8(h=UhC}r9(hI]hJ]hH]hG]hK]uh>jh8]r:(jÙ)r;}r<(h=XReturnsh>j7h?jÝ
hAjÜhC}r=(hI]hJ]hH]hG]hK]uhLKh8]r>hOXReturnsr?r@}rA(h=Uh>j;ubaubjâ)rB}rC(h=UhC}rD(hI]hJ]hH]hG]hK]uh>j7h8]rEh~)rF}rG(h=XNonehC}rH(hI]hJ]hH]hG]hK]uh>jBh8]rIhOXNonerJrK}rL(h=XNoneh>jFubahAhubahAj¿ubehAjÀubeubh~)rM}rN(h=XESee :class:`~strataalgebra.StrataAlgebra` documentation for examples.h>jh?jÝ
hAhhC}rO(hI]hJ]hH]hG]hK]uhLKhMhh8]rP(hOXSee rQrR}rS(h=XSee h>jMubj)rT}rU(h=X%:class:`~strataalgebra.StrataAlgebra`rVh>jMh?NhAjhC}rW(UreftypeXclassjÜjÝXstrataalgebra.StrataAlgebraU	refdomainXpyrXhG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rYh)rZ}r[(h=jVhC}r\(hI]hJ]r](jçjXXpy-classr^ehH]hG]hK]uh>jTh8]r_hOX
StrataAlgebrar`ra}rb(h=Uh>jZubahAhubaubhOX documentation for examples.rcrd}re(h=X documentation for examples.h>jMubeubeubeubj[)rf}rg(h=Uh>jÀh?Xb/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.psirhhAj^hC}ri(hG]hH]hI]hJ]hK]Uentries]rj(jaX*psi() (strataalgebra.StrataAlgebra method)hUNtrkauhLNhMhh8]ubjc)rl}rm(h=Uh>jÀh?jhhAjfhC}rn(jhjiXpyrohG]hH]hI]hJ]hK]jkXmethodrpjmjpuhLNhMhh8]rq(jo)rr}rs(h=XStrataAlgebra.psi(mark)rth>jlh?Xb/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebra.psiruhAjshC}rv(hG]rwhajvjwX
strataalgebrarxry}rzbhH]hI]hJ]hK]r{haj|XStrataAlgebra.psij~j}juhLNhMhh8]r|(j)r}}r~(h=Xpsih>jrh?juhAjhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]rhOXpsirr}r(h=Uh>j}ubaubj)r}r(h=Xmarkh>jrh?juhAjhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]rj¢)r}r(h=XmarkhC}r(hI]hJ]hH]hG]hK]uh>jh8]rhOXmarkrr}r(h=Uh>jubahAjªubaubeubj¿)r}r(h=Uh>jlh?juhAjÂhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]r(h~)r}r(h=XReturn a psi class.rh>jh?jhhAhhC}r(hI]hJ]hH]hG]hK]uhLKhMhh8]rhOXReturn a psi class.rr}r(h=jh>jubaubjÎ)r}r(h=Uh>jh?jhhAjÑhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]r(jÔ)r}r (h=UhC}r¡(hI]hJ]hH]hG]hK]uh>jh8]r¢(jÙ)r£}r¤(h=X
Parametersh>jh?jhhAjÜhC}r¥(hI]hJ]hH]hG]hK]uhLKh8]r¦hOX
Parametersr§r¨}r©(h=Uh>j£ubaubjâ)rª}r«(h=UhC}r¬(hI]hJ]hH]hG]hK]uh>jh8]rh~)r®}r¯(h=X8mark (int) -- The mark that the -class is associated to.hC}r°(hI]hJ]hH]hG]hK]uh>jªh8]r±(jõ)r²}r³(h=XmarkhC}r´(hG]hH]jùhI]hJ]hK]uh>j®h8]rµhOXmarkr¶r·}r¸(h=Uh>j²ubahAjþubhOX (r¹rº}r»(h=Uh>j®ubj)r¼}r½(h=UhC}r¾(UreftypejjùU	reftargetXintr¿U	refdomainjohG]hH]UrefexplicithI]hJ]hK]uh>j®h8]rÀj	)rÁ}rÂ(h=j¿hC}rÃ(hI]hJ]hH]hG]hK]uh>j¼h8]rÄhOXintrÅrÆ}rÇ(h=Uh>jÁubahAjubahAjubhOX)rÈ}rÉ(h=Uh>j®ubhOX -- rÊrË}rÌ(h=Uh>j®ubhOXThe mark that the rÍrÎ}rÏ(h=XThe mark that the h>j®ubj
)rÐ}rÑ(h=UhC}rÒ(UlatexX\psihG]hH]hI]hJ]hK]uh>j®h8]hAjubhOX-class is associated to.rÓrÔ}rÕ(h=X-class is associated to.h>j®ubehAhubahAj¿ubehAjÀubjÔ)rÖ}r×(h=UhC}rØ(hI]hJ]hH]hG]hK]uh>jh8]rÙ(jÙ)rÚ}rÛ(h=XReturn typeh>jÖh?jhhAjÜhC}rÜ(hI]hJ]hH]hG]hK]uhLKh8]rÝhOXReturn typerÞrß}rà(h=Uh>jÚubaubjâ)rá}râ(h=UhC}rã(hI]hJ]hH]hG]hK]uh>jÖh8]räh~)rå}ræ(h=XStrataAlgebraElementhC}rç(hI]hJ]hH]hG]hK]uh>jáh8]rèj)ré}rê(h=X:class:`StrataAlgebraElement`rëh>jåh?NhAjhC}rì(UreftypeXclassjÜjÝXStrataAlgebraElementU	refdomainXpyríhG]hH]UrefexplicithI]hJ]hK]jßhbjàj}jáNuhLNh8]rîh)rï}rð(h=jëhC}rñ(hI]hJ]rò(jçjíXpy-classróehH]hG]hK]uh>jéh8]rôhOXStrataAlgebraElementrõrö}r÷(h=Uh>jïubahAhubaubahAhubahAj¿ubehAjÀubeubh~)rø}rù(h=XExamples ::rúh>jh?jhhAhhC}rû(hI]hJ]hH]hG]hK]uhLKhMhh8]rühOXExamplesrýrþ}rÿ(h=XExamplesh>jøubaubjD)r}r(h=XÃsage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,0,(1,2,3,4,5))
sage: psi1 = s.psi(1); psi1
ps1
sage: psi2 = s.psi(2); psi2
ps2
sage: psi1*psi2
ps1*ps2
sage: (psi1*psi2).integrate()
2h>jh?jhhAjGhC}r(hEhFhG]hH]hI]hJ]hK]uhLM2hMhh8]rhOXÃsage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,0,(1,2,3,4,5))
sage: psi1 = s.psi(1); psi1
ps1
sage: psi2 = s.psi(2); psi2
ps2
sage: psi1*psi2
ps1*ps2
sage: (psi1*psi2).integrate()
2rr}r(h=Uh>jubaubh~)r}r(h=XPNote that the psi variables are not automatically injected into the namespace ::r	h>jh?jhhAhhC}r
(hI]hJ]hH]hG]hK]uhLKhMhh8]rhOXMNote that the psi variables are not automatically injected into the namespacerr
}r(h=XMNote that the psi variables are not automatically injected into the namespaceh>jubaubjD)r}r(h=XUsage: ps1
Traceback (most recent call last):
...
NameError: name 'ps1' is not definedh>jh?jhhAjGhC}r(hEhFhG]hH]hI]hJ]hK]uhLM?hMhh8]rhOXUsage: ps1
Traceback (most recent call last):
...
NameError: name 'ps1' is not definedrr}r(h=Uh>jubaubh~)r}r(h=XtIn case of repeated marks, notice that the :math:`\psi`-class is the pushforward under the quotient map. Observe: ::rh>jh?jhhAhhC}r(hI]hJ]hH]hG]hK]uhLKhMhh8]r(hOX+In case of repeated marks, notice that the rr}r(h=X+In case of repeated marks, notice that the h>jubj
)r}r(h=UhC}r (UlatexX\psihG]hH]hI]hJ]hK]uh>jh8]hAjubhOX:-class is the pushforward under the quotient map. Observe:r!r"}r#(h=X:-class is the pushforward under the quotient map. Observe:h>jubeubjD)r$}r%(h=Xsage: s = StrataAlgebra(QQ,1,(1,1))
sage: s.psi(1)^2
ps1^2 + ps1*ps1
sage: s.psi(2)
Traceback (most recent call last):
...
ValueError: tuple.index(x): x not in tuple
sage: s = StrataAlgebra(QQ,1,(1,1,1))
sage: s.psi(1)^2
2*ps1^2 + 4*ps1*ps1
sage: s.psi(1)^3
4*ps1^3 + 24*ps1^2*ps1 + 8*ps1*ps1*ps1
sage: var('ps') #This should be done automatically.
ps
sage: s([[0,1,1,1],[1,ps^3+1,1,1]])
ps1^3
sage: s = StrataAlgebra(QQ,0,(1,1,1,2,2))
sage: s.psi(1)*s.psi(2)
12*ps1*ps2
sage: s.psi(1)*s.psi(1)
4*ps1^2 + 8*ps1*ps1h>jh?jhhAjGhC}r&(hEhFhG]hH]hI]hJ]hK]uhLMFhMhh8]r'hOXsage: s = StrataAlgebra(QQ,1,(1,1))
sage: s.psi(1)^2
ps1^2 + ps1*ps1
sage: s.psi(2)
Traceback (most recent call last):
...
ValueError: tuple.index(x): x not in tuple
sage: s = StrataAlgebra(QQ,1,(1,1,1))
sage: s.psi(1)^2
2*ps1^2 + 4*ps1*ps1
sage: s.psi(1)^3
4*ps1^3 + 24*ps1^2*ps1 + 8*ps1*ps1*ps1
sage: var('ps') #This should be done automatically.
ps
sage: s([[0,1,1,1],[1,ps^3+1,1,1]])
ps1^3
sage: s = StrataAlgebra(QQ,0,(1,1,1,2,2))
sage: s.psi(1)*s.psi(2)
12*ps1*ps2
sage: s.psi(1)*s.psi(1)
4*ps1^2 + 8*ps1*ps1r(r)}r*(h=Uh>j$ubaubh~)r+}r,(h=X@This maybe looks surprising, but it makes sense with the formular-h>jh?jhhAhhC}r.(hI]hJ]hH]hG]hK]uhLK2hMhh8]r/hOX@This maybe looks surprising, but it makes sense with the formular0r1}r2(h=j-h>j+ubaubcsphinx.ext.mathbase
displaymath
r3)r4}r5(h=Uh>jh?jhhAUdisplaymathr6hC}r7(Ulatexr8Xx\pi_*(\alpha) \pi_*(\beta) = \frac{1}{|G|} \pi_* \left( \sum_{\sigma, \tau \in G} \sigma_*(\alpha) \tau_*(\beta) \right)Unowrapr9Unumberr:NhG]hH]hI]hJ]hK]Udocnamer;hbUlabelr<NuhLK4hMhh8]ubh~)r=}r>(h=Xfor two class :math:`\alpha,\;\beta \in H^*(X)` where :math:`\pi: X \rightarrow X/G` is the quotient map and :math:`G` is a finite group.r?h>jh?jhhAhhC}r@(hI]hJ]hH]hG]hK]uhLK8hMhh8]rA(hOXfor two class rBrC}rD(h=Xfor two class h>j=ubj
)rE}rF(h=UhC}rG(UlatexX\alpha,\;\beta \in H^*(X)hG]hH]hI]hJ]hK]uh>j=h8]hAjubhOX where rHrI}rJ(h=X where h>j=ubj
)rK}rL(h=UhC}rM(UlatexX\pi: X \rightarrow X/GhG]hH]hI]hJ]hK]uh>j=h8]hAjubhOX is the quotient map and rNrO}rP(h=X is the quotient map and h>j=ubj
)rQ}rR(h=UhC}rS(UlatexXGhG]hH]hI]hJ]hK]uh>j=h8]hAjubhOX is a finite group.rTrU}rV(h=X is a finite group.h>j=ubeubeubeubeubeubj[)rW}rX(h=Uh>jMh?Xo/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebraElement.integraterYhAj^hC}rZ(hG]hH]hI]hJ]hK]Uentries]r[(jaX7integrate() (strataalgebra.StrataAlgebraElement method)hUNtr\auhLNhMhh8]ubjc)r]}r^(h=Uh>jMh?jYhAjfhC}r_(jhjiXpyhG]hH]hI]hJ]hK]jkXmethodr`jmj`uhLNhMhh8]ra(jo)rb}rc(h=X StrataAlgebraElement.integrate()h>j]h?Xo/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebraElement.integraterdhAjshC}re(hG]rfhajvjwX
strataalgebrargrh}ribhH]hI]hJ]hK]rjhaj|XStrataAlgebraElement.integratej~XStrataAlgebraElementjuhLNhMhh8]rk(j)rl}rm(h=XStrataAlgebraElement.h>jbh?jdhAjhC}rn(hI]hJ]hH]hG]hK]uhLNhMhh8]rohOXStrataAlgebraElement.rprq}rr(h=Uh>jlubaubj)rs}rt(h=X	integrateh>jbh?jdhAjhC}ru(hI]hJ]hH]hG]hK]uhLNhMhh8]rvhOX	integraterwrx}ry(h=Uh>jsubaubj)rz}r{(h=Uh>jbh?jdhAjhC}r|(hI]hJ]hH]hG]hK]uhLNhMhh8]ubeubj¿)r}}r~(h=Uh>j]h?jdhAjÂhC}r(hI]hJ]hH]hG]hK]uhLNhMhh8]r(h~)r}r(h=XJReturn the integral of this class, i.e. its degree in the top codimension.rh>j}h?jYhAhhC}r(hI]hJ]hH]hG]hK]uhLKhMhh8]rhOXJReturn the integral of this class, i.e. its degree in the top codimension.rr}r(h=jh>jubaubh~)r}r(h=XWClasses of codimension less that the dimension of the moduli space will integrate to 0.rh>j}h?jYhAhhC}r(hI]hJ]hH]hG]hK]uhLKhMhh8]rhOXWClasses of codimension less that the dimension of the moduli space will integrate to 0.rr}r(h=jh>jubaubh~)r}r(h=XéThis uses the FZ relations to perform the integration. It is probably not very efficient. But it provides a nice check of the implementation. Consider using the ``topintersections`` module if you need to compute something quickly. ::h>j}h?jYhAhhC}r(hI]hJ]hH]hG]hK]uhLKhMhh8]r(hOX¡This uses the FZ relations to perform the integration. It is probably not very efficient. But it provides a nice check of the implementation. Consider using the rr}r(h=X¡This uses the FZ relations to perform the integration. It is probably not very efficient. But it provides a nice check of the implementation. Consider using the h>jubh)r}r(h=X``topintersections``hC}r(hI]hJ]hH]hG]hK]uh>jh8]rhOXtopintersectionsrr}r(h=Uh>jubahAhubhOX1 module if you need to compute something quickly.rr }r¡(h=X1 module if you need to compute something quickly.h>jubeubjD)r¢}r£(h=X\sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,2))
sage: (s.psi(1)*s.psi(2)).integrate()
1/24
sage: s.kappa(2).integrate()
1/24
sage: s.get_stratum(2,0) #just so you can see it.
[0 1 2 0 0]
[0 0 0 2 1]
[0 1 1 0 1]
sage: s(2,0).integrate()
1
sage: s(1,2).integrate()
0
sage: (42*s(2,0) + s(1,1) + 48*s.kappa(2)).integrate()
44    h>j}h?jYhAjGhC}r¤(hEhFhG]hH]hI]hJ]hK]uhLKhMhh8]r¥hOX\sage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,2))
sage: (s.psi(1)*s.psi(2)).integrate()
1/24
sage: s.kappa(2).integrate()
1/24
sage: s.get_stratum(2,0) #just so you can see it.
[0 1 2 0 0]
[0 0 0 2 1]
[0 1 1 0 1]
sage: s(2,0).integrate()
1
sage: s(1,2).integrate()
0
sage: (42*s(2,0) + s(1,1) + 48*s.kappa(2)).integrate()
44    r¦r§}r¨(h=Uh>j¢ubaubeubeubj[)r©}rª(h=Uh>jMh?Xj/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebraElement.dictr«hAj^hC}r¬(hG]hH]hI]hJ]hK]Uentries]r(jaX2dict() (strataalgebra.StrataAlgebraElement method)h
UNtr®auhLNhMhh8]ubjc)r¯}r°(h=Uh>jMh?j«hAjfhC}r±(jhjiXpyhG]hH]hI]hJ]hK]jkXmethodr²jmj²uhLNhMhh8]r³(jo)r´}rµ(h=XStrataAlgebraElement.dict()h>j¯h?Xj/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebraElement.dictr¶hAjshC}r·(hG]r¸h
ajvjwX
strataalgebrar¹rº}r»bhH]hI]hJ]hK]r¼h
aj|XStrataAlgebraElement.dictj~XStrataAlgebraElementjuhLNhMhh8]r½(j)r¾}r¿(h=XStrataAlgebraElement.h>j´h?j¶hAjhC}rÀ(hI]hJ]hH]hG]hK]uhLNhMhh8]rÁhOXStrataAlgebraElement.rÂrÃ}rÄ(h=Uh>j¾ubaubj)rÅ}rÆ(h=Xdicth>j´h?j¶hAjhC}rÇ(hI]hJ]hH]hG]hK]uhLNhMhh8]rÈhOXdictrÉrÊ}rË(h=Uh>jÅubaubj)rÌ}rÍ(h=Uh>j´h?j¶hAjhC}rÎ(hI]hJ]hH]hG]hK]uhLNhMhh8]ubeubj¿)rÏ}rÐ(h=Uh>j¯h?j¶hAjÂhC}rÑ(hI]hJ]hH]hG]hK]uhLNhMhh8]rÒ(h~)rÓ}rÔ(h=XReturn a dictionary with keys as :class:`StrataGraph` objects and values as the coefficient of that stratum in this element. ::h>jÏh?j«hAhhC}rÕ(hI]hJ]hH]hG]hK]uhLKhMhh8]rÖ(hOX!Return a dictionary with keys as r×rØ}rÙ(h=X!Return a dictionary with keys as h>jÓubj)rÚ}rÛ(h=X:class:`StrataGraph`rÜh>jÓh?h@hAjhC}rÝ(UreftypeXclassjÜjÝXStrataGraphU	refdomainXpyrÞhG]hH]UrefexplicithI]hJ]hK]jßhbjàXStrataAlgebraElementjáNuhLKh8]rßh)rà}rá(h=jÜhC}râ(hI]hJ]rã(jçjÞXpy-classräehH]hG]hK]uh>jÚh8]råhOXStrataGraphrærç}rè(h=Uh>jàubahAhubaubhOXG objects and values as the coefficient of that stratum in this element.rérê}rë(h=XG objects and values as the coefficient of that stratum in this element.h>jÓubeubjD)rì}rí(h=Xõsage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,2,3)); s
Strata algebra with genus 1 and markings (1, 2, 3) over Rational Field
sage: a = s.psi(1)*s.psi(2) - 7* s.kappa(3); a
ps1*ps2 - 7*ka3
sage: a.dict()
{ps1*ps2: 1, ka3: -7}h>jÏh?j«hAjGhC}rî(hEhFhG]hH]hI]hJ]hK]uhLKhMhh8]rïhOXõsage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,1,(1,2,3)); s
Strata algebra with genus 1 and markings (1, 2, 3) over Rational Field
sage: a = s.psi(1)*s.psi(2) - 7* s.kappa(3); a
ps1*ps2 - 7*ka3
sage: a.dict()
{ps1*ps2: 1, ka3: -7}rðrñ}rò(h=Uh>jìubaubeubeubj[)ró}rô(h=Uh>jMh?Xk/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebraElement.codimrõhAj^hC}rö(hG]hH]hI]hJ]hK]Uentries]r÷(jaX3codim() (strataalgebra.StrataAlgebraElement method)hUNtrøauhLNhMhh8]ubjc)rù}rú(h=Uh>jMh?jõhAjfhC}rû(jhjiXpyhG]hH]hI]hJ]hK]jkXmethodrüjmjüuhLNhMhh8]rý(jo)rþ}rÿ(h=XStrataAlgebraElement.codim()h>jùh?Xk/Users/drewjohnson/mgn/strataalgebra/strataalgebra.py:docstring of strataalgebra.StrataAlgebraElement.codimr
hAjshC}r
(hG]r
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bhH]hI]hJ]hK]r
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(h=Xcodimh>jþh?j
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(h=Uh>jùh?j
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(hI]hJ]hH]hG]hK]uhLNhMhh8]r
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(h=X?Returns the codimensions of this :class:`StrataAlgebraElement`.h>j
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(hI]hJ]hH]hG]hK]uhLKhMhh8]r 
(hOX!Returns the codimensions of this r!
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(h=X!Returns the codimensions of this h>j
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}r%
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(UreftypeXclassjÜjÝXStrataAlgebraElementU	refdomainXpyr(
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h)r*
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(h=Uh>j*
ubahAhubaubhOX.r3
}r4
(h=X.h>j
ubeubh~)r5
}r6
(h=XoIf it is not homogeneous, it retunrs the maximum codimension of a basis element with a non-zero coefficient. ::h>j
h?jõhAhhC}r7
(hI]hJ]hH]hG]hK]uhLKhMhh8]r8
hOXlIf it is not homogeneous, it retunrs the maximum codimension of a basis element with a non-zero coefficient.r9
r:
}r;
(h=XlIf it is not homogeneous, it retunrs the maximum codimension of a basis element with a non-zero coefficient.h>j5
ubaubjD)r<
}r=
(h=XÛsage: from strataalgebra import *
sage: s = StrataAlgebra(QQ,2)
sage: s(1,0).codim()
1
sage: s(2,1).codim()
2
sage: s(0,0).codim()
0
sage: (35*s(2,3) - 7*s(1,1) + s.kappa(2) - s(0,0)).codim()
2
sage: s.zero().codim()
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