version 1.2, 2021/01/20 08:17:54 |
version 1.4, 2021/03/29 05:08:01 |
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%% $OpenXM: OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v 1.1 2021/01/20 03:05:29 takayama Exp $ |
%% $OpenXM: OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v 1.3 2021/03/02 10:57:17 takayama Exp $ |
%% xetex mt_gkz-en.texi (.texi までつける. ) |
%% xetex mt_gkz-en.texi (.texi までつける. ) |
%% @math{tex形式の数式} |
%% @math{tex形式の数式} |
%% 参考: http://www.fan.gr.jp/~ring/doc/texinfo/texinfo-ja_14.html#SEC183 |
%% 参考: http://www.fan.gr.jp/~ring/doc/texinfo/texinfo-ja_14.html#SEC183 |
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@comment --- おまじない終り --- |
@comment --- おまじない終り --- |
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@comment --- GNU info ファイルの名前 --- |
@comment --- GNU info ファイルの名前 --- |
@setfilename mt_gkz_man |
@setfilename asir-contrib-mt_gkz_man |
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@comment --- タイトル --- |
@comment --- タイトル --- |
@settitle GKZ hypergeometric system |
@settitle GKZ hypergeometric system |
Line 1194 Here, @math{n_{2_{1_{3_1}}}} and @math{d_{6_1}} are ar |
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Line 1194 Here, @math{n_{2_{1_{3_1}}}} and @math{d_{6_1}} are ar |
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@comment --- 引数の簡単な説明 --- |
@comment --- 引数の簡単な説明 --- |
@table @var |
@table @var |
@item return |
@item return |
a rational function which is the cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[{dx\over x}],[{dx\over x}]\rangle_{ch}} in terms of the regular triangulation T. Here, @math{n} is the number of integration variables and @math{dx\over x} is the volume form @math{{dx_1\over x_1}\wedge\cdots\wedge{dx_n\over x_n}} of the complex @math{n}-torus. |
a rational function which is the cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[{dt\over t}],[{dt\over t}]\rangle_{ch}} in terms of the regular triangulation T. Here, @math{n} is the number of integration variables and @math{dt\over t} is the volume form @math{{dt_1\over t_1}\wedge\cdots\wedge{dt_n\over t_n}} of the complex @math{n}-torus. |
@item A,Beta |
@item A,Beta |
see @code{pfaff_eq}. |
see @code{pfaff_eq}. |
@item T |
@item T |
Line 1208 The number of polynomial factors in the integrand. see |
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Line 1208 The number of polynomial factors in the integrand. see |
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@comment --- @bullet は黒点付き --- |
@comment --- @bullet は黒点付き --- |
@itemize @bullet |
@itemize @bullet |
@item |
@item |
This function is useful when the basis of the cohomology group @math{\{\omega_i\}_{i=1}^r} is given so that @math{\omega_1=[{dx\over x}]}. |
This function is useful when the basis of the cohomology group @math{\{\omega_i\}_{i=1}^r} is given so that @math{\omega_1=[{dt\over t}]}. |
@item |
@item |
@code{leading_term} can be used more generally. |
One can find a regular triangulation by using a function @code{mt_gkz.regular_triangulation}. |
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@item |
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@code{mt_gkz.leading_terms} can be used more generally. |
@end itemize |
@end itemize |
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@comment --- @example〜@end example は実行例の表示 --- |
@comment --- @example〜@end example は実行例の表示 --- |
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@comment --- 参照(リンク)を書く --- |
@comment --- 参照(リンク)を書く --- |
@table @t |
@table @t |
@item Refer to |
@item Refer to |
@ref{mt_gkz.leading_term}. |
@ref{mt_gkz.leading_terms}. |
@end table |
@end table |
@comment おわり. |
@comment おわり. |
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@node mt_gkz.leading_term,,, Normalizing constant |
@node mt_gkz.leading_terms,,, Normalizing constant |
@subsection @code{mt_gkz.leading_term} |
@subsection @code{mt_gkz.leading_terms} |
@comment --- 索引用キーワード |
@comment --- 索引用キーワード |
@findex mt_gkz.leading_terms |
@findex mt_gkz.leading_terms |
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@table @t |
@table @t |
@item mt_gkz.leading_term(@var{A},@var{Beta},@var{W},@var{Q1},@var{Q2},@var{K},@var{N}) |
@item mt_gkz.leading_terms(@var{A},@var{Beta},@var{W},@var{Q1},@var{Q2},@var{K},@var{N}) |
:: It returns the W-leading terms of a cohomology intersection number specified by Q1 and Q2 up to W-degree=(minimum W-degree)+N. |
:: It returns the W-leading terms of a cohomology intersection number specified by Q1 and Q2 up to W-degree=(minimum W-degree)+N. |
@end table |
@end table |
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@comment --- 引数の簡単な説明 --- |
@comment --- 引数の簡単な説明 --- |
@table @var |
@table @var |
@item return |
@item return |
a list [[C1,DEG1],[C2,DEG2],...]. Each CI is a rational function depending on Beta times a monomial @math{x^m} in x-variables. DEGI is the W-degree of @math{x^m}. The cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[h^{-q_1^\prime}x^{q_1^{\prime\prime}}{dx\over x}],[h^{-q_2^\prime}x^{q_2^{\prime\prime}}{dx\over x}]\rangle_{ch}} has a Laurent expansion of the form C1+C2+.... |
a list [[C1,DEG1],[C2,DEG2],...]. Each CI is a rational function depending on Beta times a monomial @math{x^m} in x-variables. DEGI is the W-degree of @math{x^m}. The cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[h^{-q_1^\prime}t^{q_1^{\prime\prime}}{dt\over t}],[h^{-q_2^\prime}t^{q_2^{\prime\prime}}{dt\over t}]\rangle_{ch}} has a Laurent expansion of the form C1+C2+.... |
@item A,Beta |
@item A,Beta |
see @code{pfaff_eq}. |
see @code{pfaff_eq}. |
@item W |
@item W |
Line 1271 A positive integer. |
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Line 1273 A positive integer. |
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@item |
@item |
For a monomial @math{x^m=x_1^{m_1}\cdots x_n^{m_n}} and a weight vector @math{W=(w_1,\dots,w_n)}, the W-degree of @math{x^m} is given by the dot product @math{m\cdot W=m_1w_1+\cdots +m_nw_n}. |
For a monomial @math{x^m=x_1^{m_1}\cdots x_n^{m_n}} and a weight vector @math{W=(w_1,\dots,w_n)}, the W-degree of @math{x^m} is given by the dot product @math{m\cdot W=m_1w_1+\cdots +m_nw_n}. |
@item |
@item |
The W-leading terms of the cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[h^{-q_1^\prime}x^{q_1^{\prime\prime}}{dx\over x}],[h^{-q_2^\prime}x^{q_2^{\prime\prime}}{dx\over x}]\rangle_{ch}} can be computed by means of Theorem 2.6 of [GM2020]. See also Theorem 3.4.2 of [SST2000]. |
The W-leading terms of the cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[h^{-q_1^\prime}t^{q_1^{\prime\prime}}{dt\over t}],[h^{-q_2^\prime}t^{q_2^{\prime\prime}}{dt\over t}]\rangle_{ch}} can be computed by means of Theorem 2.6 of [GM2020]. See also Theorem 3.4.2 of [SST2000]. |
@item |
@item |
If the weight vector is not generic, you will receive an error message such as "WARNING(initial_mon): The weight may not be generic". In this case, the output may be wrong and you should retake a suitable W. To be more precise, W should be chosen from an open cone of the Groebner fan. |
If the weight vector is not generic, you will receive an error message such as "WARNING(initial_mon): The weight may not be generic". In this case, the output may be wrong and you should retake a suitable W. To be more precise, W should be chosen from an open cone of the Groebner fan. |
@item Option xrule. |
@item Option xrule. |
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2 |
2 |
[2927] N=2; |
[2927] N=2; |
2 |
2 |
[2928] NC=leading_term(A,Beta,W,Q[0],Q[1],K,N|xrule=[[x2,1],[x3,1],[x4,1]])$ |
[2928] NC=mt_gkz.leading_terms(A,Beta,W,Q[0],Q[1],K,N|xrule=[[x2,1],[x3,1],[x4,1]])$ |
--snip-- |
--snip-- |
[2929] NC; |
[2929] NC; |
[[(-3)/(x1),-5],[0,-4],[0,-3]] |
[[(-3)/(x1),-5],[0,-4],[0,-3]] |
Line 1316 intersection matrix is given by |
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Line 1318 intersection matrix is given by |
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@comment --- 参照(リンク)を書く --- |
@comment --- 参照(リンク)を書く --- |
@table @t |
@table @t |
@item Refer to |
@item Refer to |
@ref{mt_gkz.leading_term}, @ref{mt_gkz.generate_maple_file_IC}, @ref{mt_gkz.generate_maple_file_MR}. |
@ref{mt_gkz.leading_terms}, @ref{mt_gkz.generate_maple_file_IC}, @ref{mt_gkz.generate_maple_file_MR}. |
@end table |
@end table |
@comment おわり. |
@comment おわり. |
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Line 1351 a list of variables of P. |
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Line 1353 a list of variables of P. |
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@comment --- @bullet は黒点付き --- |
@comment --- @bullet は黒点付き --- |
@itemize @bullet |
@itemize @bullet |
@item |
@item |
This function is supposed to be combined with @code{leading_term} to compute the leading term of a cohomology intersection number. |
This function is supposed to be combined with @code{leading_terms} to compute the leading term of a cohomology intersection number. |
@item |
@item |
If W is chose so that there are several initial terms, you will receive an error message "WARNING(leading_term_rat):The weight vector may not be generic." |
If W is chose so that there are several initial terms, you will receive an error message "WARNING(leading_term_rat):The weight vector may not be generic." |
@end itemize |
@end itemize |
Line 1360 If W is chose so that there are several initial terms, |
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Line 1362 If W is chose so that there are several initial terms, |
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@comment --- 参照(リンク)を書く --- |
@comment --- 参照(リンク)を書く --- |
@table @t |
@table @t |
@item Refer to |
@item Refer to |
@ref{mt_gkz.leading_term}. |
@ref{mt_gkz.leading_terms}. |
@end table |
@end table |
@comment おわり. |
@comment おわり. |
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Line 1413 As for the definition of the standard pair, see Chapte |
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Line 1415 As for the definition of the standard pair, see Chapte |
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@item |
@item |
We set n=length(A) and set BS1:=@math{\{ 1,2,...,n\}\setminus S1}. Then, each L1[I] is an exponent @math{\bf k} of a top-dimensional standard pair @math{(\partial^{\bf k}_{BS1},S1)}. Here, @math{\bf k} is a list of length n-length(S1) and @math{\partial_{BS1}=(\partial_J)_{J\in BS1}}. |
We set n=length(A) and set BS1:=@math{\{ 1,2,...,n\}\setminus S1}. Then, each L1[I] is an exponent @math{\bf k} of a top-dimensional standard pair @math{(\partial^{\bf k}_{BS1},S1)}. Here, @math{\bf k} is a list of length n-length(S1) and @math{\partial_{BS1}=(\partial_J)_{J\in BS1}}. |
@item |
@item |
If the weight vector is not generic, you will receive an error message such as "WARNING(initial_mon): The weight may not be generic". See also @code{leading_term}. |
If the weight vector is not generic, you will receive an error message such as "WARNING(initial_mon): The weight may not be generic". See also @code{leading_terms}. |
@item |
@item |
These functions are utilized in @code{leading_term}. |
These functions are utilized in @code{leading_terms}. |
@end itemize |
@end itemize |
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@comment --- @example〜@end example は実行例の表示 --- |
@comment --- @example〜@end example は実行例の表示 --- |
Line 1425 Example: An example of a non-unimodular triangulation |
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Line 1427 Example: An example of a non-unimodular triangulation |
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[[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]] |
[[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]] |
[3257] W=[2,0,1,2,2]; |
[3257] W=[2,0,1,2,2]; |
[2,0,1,2,2] |
[2,0,1,2,2] |
[3258] toric_gen_initial(A,W); |
[3258] mt_gkz.toric_gen_initial(A,W); |
--snip-- |
--snip-- |
[[x1*x5,x1*x4,x3^2*x4],[x1,x2,x3,x4,x5]] |
[[x1*x5,x1*x4,x3^2*x4],[x1,x2,x3,x4,x5]] |
[3259] regular_triangulation(A,W); |
[3259] mt_gkz.regular_triangulation(A,W); |
--snip-- |
--snip-- |
[[2,4,5],[2,3,5],[1,2,3]] |
[[2,4,5],[2,3,5],[1,2,3]] |
[3260] top_standard_pairs(A,W); |
[3260] mt_gkz.top_standard_pairs(A,W); |
--snip-- |
--snip-- |
[[[[0,0],[0,1]],[2,4,5]],[[[0,0]],[2,3,5]],[[[0,0]],[1,2,3]]] |
[[[[0,0],[0,1]],[2,4,5]],[[[0,0]],[2,3,5]],[[[0,0]],[1,2,3]]] |
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Line 1448 are 2,1 and 1. Moreover, the top-dimensional standard |
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Line 1450 are 2,1 and 1. Moreover, the top-dimensional standard |
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@comment --- 参照(リンク)を書く --- |
@comment --- 参照(リンク)を書く --- |
@table @t |
@table @t |
@item Refer to |
@item Refer to |
@ref{mt_gkz.leading_term}. |
@ref{mt_gkz.leading_terms}. |
@end table |
@end table |
@comment おわり. |
@comment おわり. |
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