€cdocutils.nodes
document
q
)q}q(U	nametypesq}q(Xbasic operationqNXsome more examplesqNXinstalling and startingqNXacknowledgmentsq	NX9topintersections: computing top intersections on mbar_g,nq
NXsaving and loadingqNXoptionsqNuUsubstitution_defsq
}qUparse_messagesq]qcdocutils.nodes
system_message
q)q}q(U	rawsourceqUU
attributesq}q(Udupnamesq]UlevelKUidsq]Ubackrefsq]UsourceX2/Users/drewjohnson/mgn/topintersections/readme.rstqUclassesq]Unamesq]UlineK1UtypeUWARNINGquUparentqcdocutils.nodes
block_quote
q)q }q!(hUhcdocutils.nodes
section
q")q#}q$(hUhhUsourceq%hUtagnameq&Usectionq'h}q((h]h]h]h]q)Ubasic-operationq*ah]q+hauUlineq,K&Udocumentq-hUchildrenq.]q/(cdocutils.nodes
title
q0)q1}q2(hXBasic operationq3hh#h%hh&Utitleq4h}q5(h]h]h]h]h]uh,K&h-hh.]q6cdocutils.nodes
Text
q7XBasic operationq8…q9}q:(hh3hh1ubaubcdocutils.nodes
paragraph
q;)q<}q=(hXµYou can use the space command to set the space you are working in. The first argument is the genus, and the last is the number of marked points. The space command does three things:q>hh#h%hh&U	paragraphq?h}q@(h]h]h]h]h]uh,K(h-hh.]qAh7XµYou can use the space command to set the space you are working in. The first argument is the genus, and the last is the number of marked points. The space command does three things:qB…qC}qD(hh>hh<ubaubh h;)qE}qF(hX8
The class Dg1m1 is the class corresponding to the reducible boundary divisor where one component has genus 1 and the marked point 1. The class irr corresponds to the class of the irreducible boundary divisor. The classes psi, ka, ch, and la represent psi, kappa, chern character, and lambda classes respectively.qGhh#h%hh&h?h}qH(h]h]h]h]h]uh,K?h-hh.]qIh7X8
The class Dg1m1 is the class corresponding to the reducible boundary divisor where one component has genus 1 and the marked point 1. The class irr corresponds to the class of the irreducible boundary divisor. The classes psi, ka, ch, and la represent psi, kappa, chern character, and lambda classes respectively.qJ…qK}qL(hhGhhEubaubh;)qM}qN(hXsNow we are ready to compute some things. You can type in a polynomial in the classes given by the space command: ::qOhh#h%hh&h?h}qP(h]h]h]h]h]uh,KAh-hh.]qQh7XpNow we are ready to compute some things. You can type in a polynomial in the classes given by the space command:qR…qS}qT(hXpNow we are ready to compute some things. You can type in a polynomial in the classes given by the space command:hhMubaubcdocutils.nodes
literal_block
qU)qV}qW(hX
sage: intnum(irr^3*psi1)
Computing the intersection of irr^3*psi1 over Mbar_2_1...
-11/6
sage: intnum(ka2^2)
Computing the intersection of ka2^2 over Mbar_2_1...
53/5760
sage: intnum(3*irr^3*psi1 + 6*ka2^2)
Computing the intersection of 3*irr^3*psi1 + 6*ka2^2 over Mbar_2_1...
-5227/960hh#h%hh&U
literal_blockqXh}qY(U	xml:spaceqZUpreserveq[h]h]h]h]h]uh,KCh-hh.]q\h7X
sage: intnum(irr^3*psi1)
Computing the intersection of irr^3*psi1 over Mbar_2_1...
-11/6
sage: intnum(ka2^2)
Computing the intersection of ka2^2 over Mbar_2_1...
53/5760
sage: intnum(3*irr^3*psi1 + 6*ka2^2)
Computing the intersection of 3*irr^3*psi1 + 6*ka2^2 over Mbar_2_1...
-5227/960q]…q^}q_(hUhhVubaubh;)q`}qa(hX´If you are just computing a monomial with no coefficient, you can pass in the indexes as a list. Thus, the following command computes the same number as the first example above. ::qbhh#h%hh&h?h}qc(h]h]h]h]h]uh,KMh-hh.]qdh7X±If you are just computing a monomial with no coefficient, you can pass in the indexes as a list. Thus, the following command computes the same number as the first example above.qe…qf}qg(hX±If you are just computing a monomial with no coefficient, you can pass in the indexes as a list. Thus, the following command computes the same number as the first example above.hh`ubaubhU)qh}qi(hXWsage: intnum([8,8,8,1])
Computing the intersection of irr^3*psi1 over Mbar_2_1...
-11/6hh#h%hh&hXh}qj(hZh[h]h]h]h]h]uh,KOh-hh.]qkh7XWsage: intnum([8,8,8,1])
Computing the intersection of irr^3*psi1 over Mbar_2_1...
-11/6ql…qm}qn(hUhhhubaubh;)qo}qp(hX¹You can also specify the space you wish to work over in the function call. Any classes that are in the namespace will be interpreted as being in the space you specified. For example: ::qqhh#h%hh&h?h}qr(h]h]h]h]h]uh,KSh-hh.]qsh7X¶You can also specify the space you wish to work over in the function call. Any classes that are in the namespace will be interpreted as being in the space you specified. For example:qt…qu}qv(hX¶You can also specify the space you wish to work over in the function call. Any classes that are in the namespace will be interpreted as being in the space you specified. For example:hhoubaubhU)qw}qx(hXVsage: intnum(2,2, psi1^5)
Computing the intersection of psi1^5 over Mbar_2_2...
1/1152hh#h%hh&hXh}qy(hZh[h]h]h]h]h]uh,KUh-hh.]qzh7XVsage: intnum(2,2, psi1^5)
Computing the intersection of psi1^5 over Mbar_2_2...
1/1152q{…q|}q}(hUhhwubaubh;)q~}q(hXXThe following syntax should be very similar to Carel Faber's MgnLb.txt Maple program: ::q€hh#h%hh&h?h}q(h]h]h]h]h]uh,KYh-hh.]q‚h7XUThe following syntax should be very similar to Carel Faber's MgnLb.txt Maple program:qƒ…q„}q…(hXUThe following syntax should be very similar to Carel Faber's MgnLb.txt Maple program:hh~ubaubhU)q†}q‡(hXZsage: intnum(2,2,[1,1,1,1,1])
Computing the intersection of psi1^5 over Mbar_2_2...
1/1152hh#h%hh&hXh}qˆ(hZh[h]h]h]h]h]uh,K[h-hh.]q‰h7XZsage: intnum(2,2,[1,1,1,1,1])
Computing the intersection of psi1^5 over Mbar_2_2...
1/1152qŠ…q‹}qŒ(hUhh†ubaubh;)q}qŽ(hXêHowever, in order to type in a polynomial in classes, the names must have been created by a previous space command. Thus, if you have not called space with marked points at least 2 in this session, the following will give an error: ::qhh#h%hh&h?h}q(h]h]h]h]h]uh,K_h-hh.]q‘h7XçHowever, in order to type in a polynomial in classes, the names must have been created by a previous space command. Thus, if you have not called space with marked points at least 2 in this session, the following will give an error:q’…q“}q”(hXçHowever, in order to type in a polynomial in classes, the names must have been created by a previous space command. Thus, if you have not called space with marked points at least 2 in this session, the following will give an error:hhubaubhU)q•}q–(hXksage: intnum(2,2, psi1*psi2^4)
Traceback (most recent call last):
...
NameError: name 'psi2' is not definedhh#h%hh&hXh}q—(hZh[h]h]h]h]h]uh,Kah-hh.]q˜h7Xksage: intnum(2,2, psi1*psi2^4)
Traceback (most recent call last):
...
NameError: name 'psi2' is not definedq™…qš}q›(hUhh•ubaubh;)qœ}q(hXInstead, do something like: ::qžhh#h%hh&h?h}qŸ(h]h]h]h]h]uh,Kfh-hh.]q h7XInstead, do something like:q¡…q¢}q£(hXInstead, do something like:hhœubaubhU)q¤}q¥(hX

sage: space(2,2)
[1]  psi1
[2]  psi2
[3]  ka1
[4]  ka2
[5]  ka3
[6]  ka4
[7]  ka5
[8]  ch1
[9]  ch3
[10]  irr
[11]  Dg0m1_2
[12]  Dg1m1
[13]  Dg1m1_2
[14]  la1
[15]  la2
Mbar_2_2
sage: intnum(psi1*psi2^4)
Computing the intersection of psi1*psi2^4 over Mbar_2_2...
1/384hh#h%hh&hXh}q¦(hZh[h]h]h]h]h]uh,Khh-hh.]q§h7X

sage: space(2,2)
[1]  psi1
[2]  psi2
[3]  ka1
[4]  ka2
[5]  ka3
[6]  ka4
[7]  ka5
[8]  ch1
[9]  ch3
[10]  irr
[11]  Dg0m1_2
[12]  Dg1m1
[13]  Dg1m1_2
[14]  la1
[15]  la2
Mbar_2_2
sage: intnum(psi1*psi2^4)
Computing the intersection of psi1*psi2^4 over Mbar_2_2...
1/384q¨…q©}qª(hUhh¤ubaubh;)q«}q¬(hX[If the degree is not correct, you will know. (The code only computes top intersections.) ::q­hh#h%hh&h?h}q®(h]h]h]h]h]uh,K}h-hh.]q¯h7XXIf the degree is not correct, you will know. (The code only computes top intersections.)q°…q±}q²(hXXIf the degree is not correct, you will know. (The code only computes top intersections.)hh«ubaubhU)q³}q´(hX›sage: intnum(2,2, psi1^2)
Computing the intersection of psi1^2 over Mbar_2_2...
The monomial psi1^2 has degree 2, while the space Mbar_2_2 has dimension 5.hh#h%hh&hXh}qµ(hZh[h]h]h]h]h]uh,Kh-hh.]q¶h7X›sage: intnum(2,2, psi1^2)
Computing the intersection of psi1^2 over Mbar_2_2...
The monomial psi1^2 has degree 2, while the space Mbar_2_2 has dimension 5.q·…q¸}q¹(hUhh³ubaubeubh%hh&Ublock_quoteqºh}q»(h]h]h]h]h]uh,Nh-hh.]q¼(cdocutils.nodes
enumerated_list
q½)q¾}q¿(hUh}qÀ(UsuffixqÁU
.h]h]h]UprefixqÂUh]h]UenumtypeqÃUarabicqÄuhh h.]qÅ(cdocutils.nodes
list_item
qÆ)qÇ}qÈ(hX;It prints a list of classes and indexes for easy reference.qÉh}qÊ(h]h]h]h]h]uhh¾h.]qËh;)qÌ}qÍ(hhÉhhÇh%hh&h?h}qÎ(h]h]h]h]h]uh,K*h.]qÏh7X;It prints a list of classes and indexes for easy reference.qЅqÑ}qÒ(hhÉhhÌubaubah&U	list_itemqÓubhÆ)qÔ}qÕ(hX{It sets the defaut space. This will be the space that the code will work in if you don't specify one in the function calls.qÖh}q×(h]h]h]h]h]uhh¾h.]qØh;)qÙ}qÚ(hhÖhhÔh%hh&h?h}qÛ(h]h]h]h]h]uh,K+h.]qÜh7X{It sets the defaut space. This will be the space that the code will work in if you don't specify one in the function calls.q݅qÞ}qß(hhÖhhÙubaubah&hÓubhÆ)qà}qá(hX\It injects the variables names into your namespace so you can use them to make polynomials.
h}qâ(h]h]h]h]h]uhh¾h.]qãh;)qä}qå(hX[It injects the variables names into your namespace so you can use them to make polynomials.qæhhàh%hh&h?h}qç(h]h]h]h]h]uh,K,h.]qèh7X[It injects the variables names into your namespace so you can use them to make polynomials.q酁qê}që(hhæhhäubaubah&hÓubeh&Uenumerated_listqìubh;)qí}qî(hXsage: space(2,1)
[1]  psi1
[2]  ka1
[3]  ka2
[4]  ka3
[5]  ka4
[6]  ch1
[7]  ch3
[8]  irr
[9]  Dg1m1
[10]  la1
[11]  la2
Mbar_2_1qïhh h%hh&h?h}qð(h]h]h]h]h]uh,K1h.]qñh7Xsage: space(2,1)
[1]  psi1
[2]  ka1
[3]  ka2
[4]  ka3
[5]  ka4
[6]  ch1
[7]  ch3
[8]  irr
[9]  Dg1m1
[10]  la1
[11]  la2
Mbar_2_1qò…qó}qô(hhïhhíubaubeubh.]qõh;)qö}q÷(hX#Literal block expected; none found.h}qø(h]h]h]h]h]uhhh.]qùh7X#Literal block expected; none found.qú…qû}qü(hUhhöubah&h?ubah&Usystem_messageqýubaUcurrent_sourceqþNU
decorationqÿNUautofootnote_startr
K
Unameidsr

}r
(hh*hUsome-more-examplesr
hUinstalling-and-startingr
h	Uacknowledgmentsr
h
U8topintersections-computing-top-intersections-on-mbar-g-nr
hUsaving-and-loadingr
hUoptionsr
uh.]r	
(h")r

}r
(hUhhh%hh&h'h}r
(h]h]h]h]r

j
ah]r
h
auh,Kh-hh.]r
(h0)r
}r
(hX9topintersections: Computing top intersections on Mbar_g,nr
hj

h%hh&h4h}r
(h]h]h]h]h]uh,Kh-hh.]r
h7X9topintersections: Computing top intersections on Mbar_g,nr
…r
}r
(hj
hj
ubaubh;)r
}r
(hXr
This worksheet demonstrates how to compute top intersections on Deligne-Mumford compactification of the moduli space of curves using Sage code written by Drew Johnson. A pdf description of the algorithms can be found in the BitBucket repository (see link below). If you have any questions or problems, or if you find the code useful, please contact the author by email .r
hj

h%hh&h?h}r
(h]h]h]h]h]uh,Kh-hh.]r
h7Xr
This worksheet demonstrates how to compute top intersections on Deligne-Mumford compactification of the moduli space of curves using Sage code written by Drew Johnson. A pdf description of the algorithms can be found in the BitBucket repository (see link below). If you have any questions or problems, or if you find the code useful, please contact the author by email .r
…r
}r
(hj
hj
ubaubeubh")r 
}r!
(hUhhh%hh&h'h}r"
(h]h]h]h]r#
j
ah]r$
h	auh,Kh-hh.]r%
(h0)r&
}r'
(hXAcknowledgmentsr(
hj 
h%hh&h4h}r)
(h]h]h]h]h]uh,Kh-hh.]r*
h7XAcknowledgmentsr+
…r,
}r-
(hj(
hj&
ubaubh;)r.
}r/
(hXò
I worked on this project as a graduate student at Brigham Young University while being funded by the NSA grant of my adviser, Tyler Jarvis. Dr. Jarvis also answered many questions and pointed me to helpful resources during this project. The mathematical basis of the algorithms implemented here come from the paper accompanying Carel Faber's Maple code, and also from Stephanie Yang's write up about her Macaulay 2 code.  The code computes the Witten tau function using the recursion of Liu and Xu.r0
hj 
h%hh&h?h}r1
(h]h]h]h]h]uh,K	h-hh.]r2
h7Xò
I worked on this project as a graduate student at Brigham Young University while being funded by the NSA grant of my adviser, Tyler Jarvis. Dr. Jarvis also answered many questions and pointed me to helpful resources during this project. The mathematical basis of the algorithms implemented here come from the paper accompanying Carel Faber's Maple code, and also from Stephanie Yang's write up about her Macaulay 2 code.  The code computes the Witten tau function using the recursion of Liu and Xu.r3
…r4
}r5
(hj0
hj.
ubaubeubh")r6
}r7
(hUhhh%hh&h'h}r8
(h]h]h]h]r9
j
ah]r:
hauh,Kh-hh.]r;
(h0)r<
}r=
(hXInstalling and startingr>
hj6
h%hh&h4h}r?
(h]h]h]h]h]uh,Kh-hh.]r@
h7XInstalling and startingrA
…rB
}rC
(hj>
hj<
ubaubh;)rD
}rE
(hXt``topintersections`` is now distributed as part of the ``mgn`` package. `Click here`_ for installation instructions.rF
hj6
h%hh&h?h}rG
(h]h]h]h]h]uh,Kh-hh.]rH
(cdocutils.nodes
literal
rI
)rJ
}rK
(hX``topintersections``h}rL
(h]h]h]h]h]uhjD
h.]rM
h7XtopintersectionsrN
…rO
}rP
(hUhjJ
ubah&UliteralrQ
ubh7X# is now distributed as part of the rR
…rS
}rT
(hX# is now distributed as part of the hjD
ubjI
)rU
}rV
(hX``mgn``h}rW
(h]h]h]h]h]uhjD
h.]rX
h7XmgnrY
…rZ
}r[
(hUhjU
ubah&jQ
ubh7X
 package. r\
…r]
}r^
(hX
 package. hjD
ubcdocutils.nodes
problematic
r_
)r`
}ra
(hX
`Click here`_rb
hjD
h%Nh&Uproblematicrc
h}rd
(h]re
Uid2rf
ah]h]h]h]UrefidUid1rg
uh,Nh-hh.]rh
h7X
`Click here`_ri
…rj
}rk
(hUhj`
ubaubh7X for installation instructions.rl
…rm
}rn
(hX for installation instructions.hjD
ubeubh;)ro
}rp
(hX If you accessed this worksheet online, you are probably viewing a read-only version. If you would like to use the code yourself, you have the following options.rq
hj6
h%hh&h?h}rr
(h]h]h]h]h]uh,Kh-hh.]rs
h7X If you accessed this worksheet online, you are probably viewing a read-only version. If you would like to use the code yourself, you have the following options.rt
…ru
}rv
(hjq
hjo
ubaubh;)rw
}rx
(hX5After installing, load the code in a sage session: ::ry
hj6
h%hh&h?h}rz
(h]h]h]h]h]uh,Kh-hh.]r{
h7X2After installing, load the code in a sage session:r|
…r}
}r~
(hX2After installing, load the code in a sage session:hjw
ubaubhU)r
}r€
(hXÇsage: from topintersections import *
***************************************************************************
Welcome to the intersection number computer for the moduli space of curves!

Written by Drew Johnson, based on Carel Faber's exposition and Maple code.

The main commands are "intnum" to compute the intersection number, and
"space" to select the space.  Type help(intnum) or help(space) for more
information.

Type save_data("filename") to save the results computed this session and
load_data("filname") to retrieve previously saved data.  The default filename is
"mgn_top_intersect.dat" if not specified.

Ctrl-D to quit.
***************************************************************************hj6
h%hh&hXh}r
(hZh[h]h]h]h]h]uh,Kh-hh.]r‚
h7XÇsage: from topintersections import *
***************************************************************************
Welcome to the intersection number computer for the moduli space of curves!

Written by Drew Johnson, based on Carel Faber's exposition and Maple code.

The main commands are "intnum" to compute the intersection number, and
"space" to select the space.  Type help(intnum) or help(space) for more
information.

Type save_data("filename") to save the results computed this session and
load_data("filname") to retrieve previously saved data.  The default filename is
"mgn_top_intersect.dat" if not specified.

Ctrl-D to quit.
***************************************************************************rƒ
…r„
}r…
(hUhj
ubaubeubh#h")r†
}r‡
(hUhhh%hh&h'h}rˆ
(h]h]h]h]r‰
j
ah]rŠ
hauh,K„h-hh.]r‹
(h0)rŒ
}r
(hXSome more examplesrŽ
hj†
h%hh&h4h}r
(h]h]h]h]h]uh,K„h-hh.]r
h7XSome more examplesr‘
…r’
}r“
(hjŽ
hjŒ
ubaubh;)r”
}r•
(hX@Here are some of the intersection numbers from Faber's paper: ::r–
hj†
h%hh&h?h}r—
(h]h]h]h]h]uh,K†h-hh.]r˜
h7X=Here are some of the intersection numbers from Faber's paper:r™
…rš
}r›
(hX=Here are some of the intersection numbers from Faber's paper:hj”
ubaubhU)rœ
}r
(hX´sage: intnum(4,0, irr^9)
Computing the intersection of irr^9 over Mbar_4_0...
-251987683/4320
sage: intnum(4,0, la1^9)
Computing the intersection of la1^9 over Mbar_4_0...
1/113400hj†
h%hh&hXh}rž
(hZh[h]h]h]h]h]uh,Kˆh-hh.]rŸ
h7X´sage: intnum(4,0, irr^9)
Computing the intersection of irr^9 over Mbar_4_0...
-251987683/4320
sage: intnum(4,0, la1^9)
Computing the intersection of la1^9 over Mbar_4_0...
1/113400r 
…r¡
}r¢
(hUhjœ
ubaubeubh")r£
}r¤
(hUhhh%hh&h'h}r¥
(h]h]h]h]r¦
j
ah]r§
hauh,Kh-hh.]r¨
(h0)r©
}rª
(hXOptionsr«
hj£
h%hh&h4h}r¬
(h]h]h]h]h]uh,Kh-hh.]r­
h7XOptionsr®
…r¯
}r°
(hj«
hj©
ubaubh;)r±
}r²
(hXqYou can suppress the helpful message that tells you what you are computing using the confirm keyword argument. ::r³
hj£
h%hh&h?h}r´
(h]h]h]h]h]uh,K’h-hh.]rµ
h7XnYou can suppress the helpful message that tells you what you are computing using the confirm keyword argument.r¶
…r·
}r¸
(hXnYou can suppress the helpful message that tells you what you are computing using the confirm keyword argument.hj±
ubaubhU)r¹
}rº
(hX0sage: intnum(2,2,psi1^5, confirm = False)
1/1152hj£
h%hh&hXh}r»
(hZh[h]h]h]h]h]uh,K”h-hh.]r¼
h7X0sage: intnum(2,2,psi1^5, confirm = False)
1/1152r½
…r¾
}r¿
(hUhj¹
ubaubh;)rÀ
}rÁ
(hXŠYou can have the program return zero if the degree is wrong instead of raising an exception by using the check_degree keyword argument. ::rÂ
hj£
h%hh&h?h}rÃ
(h]h]h]h]h]uh,K—h-hh.]rÄ
h7X‡You can have the program return zero if the degree is wrong instead of raising an exception by using the check_degree keyword argument.rÅ
…rÆ
}rÇ
(hX‡You can have the program return zero if the degree is wrong instead of raising an exception by using the check_degree keyword argument.hjÀ
ubaubhU)rÈ
}rÉ
(hXesage: intnum(2,2, psi1^2, check_degree = False)
Computing the intersection of psi1^2 over Mbar_2_2...hj£
h%hh&hXh}rÊ
(hZh[h]h]h]h]h]uh,K™h-hh.]rË
h7Xesage: intnum(2,2, psi1^2, check_degree = False)
Computing the intersection of psi1^2 over Mbar_2_2...rÌ
…rÍ
}rÎ
(hUhjÈ
ubaubeubh")rÏ
}rÐ
(hUhhh%hh&h'h}rÑ
(h]h]h]h]rÒ
j
ah]rÓ
hauh,Kh-hh.]rÔ
(h0)rÕ
}rÖ
(hXSaving and loadingr×
hjÏ
h%hh&h4h}rØ
(h]h]h]h]h]uh,Kh-hh.]rÙ
h7XSaving and loadingrÚ
…rÛ
}rÜ
(hj×
hjÕ
ubaubh;)rÝ
}rÞ
(hX}
So far we have computed numbers from scratch. The program automatically saves any answers that it has computed in this session, including numbers computed in recursion steps. For example, if you computed the example la1^9 above, you probably noticed that it took a few seconds. If we compute it again, it will be really fast because the program just looks it up in a dictionary. ::rß
hjÏ
h%hh&h?h}rà
(h]h]h]h]h]uh,Kžh-hh.]rá
h7Xz
So far we have computed numbers from scratch. The program automatically saves any answers that it has computed in this session, including numbers computed in recursion steps. For example, if you computed the example la1^9 above, you probably noticed that it took a few seconds. If we compute it again, it will be really fast because the program just looks it up in a dictionary.râ
…rã
}rä
(hXz
So far we have computed numbers from scratch. The program automatically saves any answers that it has computed in this session, including numbers computed in recursion steps. For example, if you computed the example la1^9 above, you probably noticed that it took a few seconds. If we compute it again, it will be really fast because the program just looks it up in a dictionary.hjÝ
ubaubhU)rå
}ræ
(hXfsage: timeit("intnum(4,0, la1^9)", number =1, repeat = 1) #random
1 loops, best of 1: 7.06 ms per loophjÏ
h%hh&hXh}rç
(hZh[h]h]h]h]h]uh,K h-hh.]rè
h7Xfsage: timeit("intnum(4,0, la1^9)", number =1, repeat = 1) #random
1 loops, best of 1: 7.06 ms per loopré
…rê
}rë
(hUhjå
ubaubh;)rì
}rí
(hXXWe can save this data to a file to avoid computing it over again in our next session. ::rî
hjÏ
h%hh&h?h}rï
(h]h]h]h]h]uh,K£h-hh.]rð
h7XUWe can save this data to a file to avoid computing it over again in our next session.rñ
…rò
}ró
(hXUWe can save this data to a file to avoid computing it over again in our next session.hjì
ubaubhU)rô
}rõ
(hX.sage: save_data("testsave.dat")
Save suceeded.hjÏ
h%hh&hXh}rö
(hZh[h]h]h]h]h]uh,K¥h-hh.]r÷
h7X.sage: save_data("testsave.dat")
Save suceeded.rø
…rù
}rú
(hUhjô
ubaubh;)rû
}rü
(hXZIf you don't specify a file name, the data is saved to the file ``mgn_top_intersect.dat``.rý
hjÏ
h%hh&h?h}rþ
(h]h]h]h]h]uh,K¨h-hh.]rÿ
(h7X@If you don't specify a file name, the data is saved to the file r…r
}r(hX@If you don't specify a file name, the data is saved to the file hjû
ubjI
)r}r(hX``mgn_top_intersect.dat``h}r(h]h]h]h]h]uhjû
h.]rh7Xmgn_top_intersect.datr…r}r	(hUhjubah&jQ
ubh7X
.…r
}r(hX
.hjû
ubeubh;)r}r
(hXCTo load a previously saved data file, use the following command: ::rhjÏ
h%hh&h?h}r(h]h]h]h]h]uh,Kªh-hh.]rh7X@To load a previously saved data file, use the following command:r…r}r(hX@To load a previously saved data file, use the following command:hjubaubhU)r}r(hX,sage: load_data("testsave.dat")
Data loaded.hjÏ
h%hh&hXh}r(hZh[h]h]h]h]h]uh,K¬h-hh.]rh7X,sage: load_data("testsave.dat")
Data loaded.r…r}r(hUhjubaubeubehUUtransformerrNU
footnote_refsr}rUrefnamesr}rX
click herer ]r!cdocutils.nodes
reference
r")r#}r$(hjb
h}r%(UnameX
Click hereh]h]h]Urefnamer&j h]h]uhjD
h.]r'h7X
Click herer(…r)}r*(hUhj#ubah&U	referencer+ubasUsymbol_footnotesr,]r-Uautofootnote_refsr.]r/Usymbol_footnote_refsr0]r1U	citationsr2]r3h-hUcurrent_liner4NUtransform_messagesr5]r6h)r7}r8(hUh}r9(h]UlevelKh]r:jg
ah]r;jf
aUsourcehh]h]UlineKUtypeUERRORr<uh.]r=h;)r>}r?(hUh}r@(h]h]h]h]h]uhj7h.]rAh7X"Unknown target name: "click here".rB…rC}rD(hUhj>ubah&h?ubah&hýubaUreporterrENUid_startrFKU
autofootnotesrG]rHU
citation_refsrI}rJUindirect_targetsrK]rLUsettingsrM(cdocutils.frontend
Values
rNorO}rP(Ufootnote_backlinksrQK
Urecord_dependenciesrRNUrfc_base_urlrSUhttps://tools.ietf.org/html/rTU	tracebackrUˆUpep_referencesrVNUstrip_commentsrWNU
toc_backlinksrXUentryrYU
language_coderZUenr[U	datestampr\NUreport_levelr]KU_destinationr^NU
halt_levelr_KU
strip_classesr`Nh4NUerror_encoding_error_handlerraUbackslashreplacerbUdebugrcNUembed_stylesheetrd‰Uoutput_encoding_error_handlerreUstrictrfU
sectnum_xformrgK
Udump_transformsrhNU
docinfo_xformriK
Uwarning_streamrjNUpep_file_url_templaterkUpep-%04drlUexit_status_levelrmKUconfigrnNUstrict_visitorroNUcloak_email_addressesrpˆUtrim_footnote_reference_spacerq‰UenvrrNUdump_pseudo_xmlrsNUexpose_internalsrtNUsectsubtitle_xformru‰Usource_linkrvNUrfc_referencesrwNUoutput_encodingrxUutf-8ryU
source_urlrzNUinput_encodingr{U	utf-8-sigr|U_disable_configr}NU	id_prefixr~UUcharacter_level_inline_markupr‰U	tab_widthr€KUerror_encodingrUUTF-8r‚U_sourcerƒhUgettext_compactr„ˆU	generatorr…NUdump_internalsr†NUsmart_quotesr‡‰Upep_base_urlrˆU https://www.python.org/dev/peps/r‰Usyntax_highlightrŠUlongr‹Uinput_encoding_error_handlerrŒjfUauto_id_prefixrUidrŽUdoctitle_xformr‰Ustrip_elements_with_classesrNU
_config_filesr‘]Ufile_insertion_enabledr’ˆUraw_enabledr“K
U
dump_settingsr”NubUsymbol_footnote_startr•Kh}r–(j
j6
j
j

jf
j`
j
j 
j
j†
h*h#jg
j7j
j£
j
jÏ
uUsubstitution_namesr—}r˜h&h-h}r™(h]h]h]Usourcehh]h]uU	footnotesrš]r›Urefidsrœ}rub.