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version 1.3, 2000/01/23 05:28:33 version 1.4, 2000/01/24 02:48:24
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 %% $OpenXM: OpenXM/doc/OpenXM-specs/cmo-basic1.tex,v 1.1.1.1 2000/01/20 08:52:46 noro Exp $  %% $OpenXM: OpenXM/doc/OpenXM-specs/cmo-basic1.tex,v 1.3 2000/01/23 05:28:33 noro Exp $
 //&jp \section{ 数, 多項式 の  CMO 表現 }  //&jp \section{ 数, 多項式 の  CMO 表現 }
 //&eg \section{ CMOexpressions for numbers and polynomials }  //&eg \section{ CMOexpressions for numbers and polynomials }
   \label{sec:basic1}
 /*&C  /*&C
 @../SSkan/plugin/cmotag.h  @../SSkan/plugin/cmotag.h
 \begin{verbatim}  \begin{verbatim}
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 */  */
   
 /*&jp  /*&jp
 以下, グループ CMObject/Basic1, CMObject/Tree  以下, グループ CMObject/Basic, CMObject/Tree
 および CMObject/DistributedPolynomial  および CMObject/DistributedPolynomial
 に属する CMObject の形式を説明する.  に属する CMObject の形式を説明する.
 \noroa{ tagged list を導入すべきか?  cf. SSkan/plugin/cmo.txt }  \noroa{ tagged list を導入すべきか?  cf. SSkan/plugin/cmo.txt }
Line 32 
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 */  */
 /*&eg  /*&eg
 In the sequel, we will explain on the groups  In the sequel, we will explain on the groups
 CMObject/Basic1, CMObject/Tree  CMObject/Basic, CMObject/Tree
 and CMObject/DistributedPolynomial.  and CMObject/DistributedPolynomial.
 */  */
   
Line 40  and CMObject/DistributedPolynomial.
Line 40  and CMObject/DistributedPolynomial.
   
 \bigbreak  \bigbreak
 \noindent  \noindent
 Group CMObject/Basic1 requires CMObject/Basic0. \\  Group CMObject/Basic requires CMObject/Primitive. \\
 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic1. \\  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic. \\
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\  \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\
 & & \mbox{ --- ユニバーサルな ゼロを表す. } \\  & & \mbox{ --- ユニバーサルな ゼロを表す. } \\
Line 60  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
Line 60  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
   
 \bigbreak  \bigbreak
 \noindent  \noindent
 Group CMObject/Basic1 requires CMObject/Basic0. \\  Group CMObject/Basic requires CMObject/Primitive. \\
 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic1. \\  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic. \\
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\  \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\
 & & \mbox{ --- Universal zero } \\  & & \mbox{ --- Universal zero } \\
Line 102  in one to one correspondence.
Line 102  in one to one correspondence.
 /*&jp  /*&jp
   
 \noindent  \noindent
 Group CMObject/Tree requires CMObject/Basic1. \\  Group CMObject/Tree requires CMObject/Basic. \\
 Tree, Lambda $\in$ CMObject/Basic1. \\  Tree, Lambda $\in$ CMObject/Basic. \\
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{Tree}        &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},  \mbox{Tree}        &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},
  {\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\   {\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\
Line 119  Tree, Lambda $\in$ CMObject/Basic1. \\
Line 119  Tree, Lambda $\in$ CMObject/Basic1. \\
 /*&eg  /*&eg
   
 \noindent  \noindent
 Group CMObject/Tree requires CMObject/Basic1. \\  Group CMObject/Tree requires CMObject/Basic. \\
 Tree, Lambda $\in$ CMObject/Basic1. \\  Tree, Lambda $\in$ CMObject/Basic. \\
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{Tree}        &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},  \mbox{Tree}        &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},
  {\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\   {\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\
Line 160  For example,
Line 160  For example,
 $\sin(x+e)$ is expressed as  $\sin(x+e)$ is expressed as
 {\tt (sin, (plus, x, e))}  {\tt (sin, (plus, x, e))}
 as a tree.  as a tree.
 Tree may be expressed by putting itself between  Tree may be expressed by putting the expression between
 {\tt SM\_beginBlock} and {\tt SM\_endBlock}, which are  {\tt SM\_beginBlock} and {\tt SM\_endBlock}, which are
 stack machine commands for delayed evaluation.  stack machine commands for delayed evaluation.
 (cf. {\tt \{ }, {\tt \} } in PostScript).  (cf. {\tt \{ }, {\tt \} } in PostScript).
Line 213  Class.tree [    $plus$ , $Basic$ , [    123 , 345 ]  ]
Line 213  Class.tree [    $plus$ , $Basic$ , [    123 , 345 ]  ]
   
 \bigbreak  \bigbreak
 //&jp 次に, 分散表現多項式に関係するグループを定義しよう.  //&jp 次に, 分散表現多項式に関係するグループを定義しよう.
 //&eg Let us define a group for distributed polynomials.  /*&eg
   Let us define a group for distributed polynomials. In the following
   DMS stands for Distributed Monomial System.
   */
   
 \medbreak  \medbreak
 \noindent  \noindent
 Group CMObject/DistributedPolynomials requires CMObject/Basic0,  Group CMObject/DistributedPolynomials requires CMObject/Primitive,
 CMObject/Basic1. \\  CMObject/Basic. \\
 Monomial, Monomial32, Coefficient, Dpolynomial, DringDefinition,  Monomial, Monomial32, Coefficient, Dpolynomial, DringDefinition,
 Generic DMS ring, RingByName, DMS of N variables $\in$  Generic DMS ring, RingByName, DMS of N variables $\in$
 CMObject/DistributedPolynomials. \\  CMObject/DistributedPolynomials. \\
Line 289  $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$
Line 292  $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$
 \mbox{Generic DMS ring}  \mbox{Generic DMS ring}
                  &:& ({\tt CMO\_DMS\_GENERIC}) \\                   &:& ({\tt CMO\_DMS\_GENERIC}) \\
 \mbox{RingByName}&:& ({\tt CMO\_RING\_BY\_NAME}, {\sl Cstring} s) \\  \mbox{RingByName}&:& ({\tt CMO\_RING\_BY\_NAME}, {\sl Cstring} s) \\
                  & & \mbox{ --- The ring definition refered by the name ``s''.} \\                   & & \mbox{ --- The ring definition referred by the name ``s''.} \\
 \mbox{DMS of N variables}  \mbox{DMS of N variables}
                  &:& ({\tt CMO\_DMS\_OF\_N\_VARIABLES}, \\                   &:& ({\tt CMO\_DMS\_OF\_N\_VARIABLES}, \\
                  & & \ ({\tt CMO\_LIST}, {\sl int32}\, \mbox{m},                   & & \ ({\tt CMO\_LIST}, {\sl int32}\, \mbox{m},
Line 394  a CMO\_ZZ $14$ is expressed by
Line 397  a CMO\_ZZ $14$ is expressed by
 \mbox{(CMO\_ZZ, 1, 0, 0, 0, e)},  \mbox{(CMO\_ZZ, 1, 0, 0, 0, e)},
 \]  \]
 //&jp と表わす. これはバイト列では  //&jp と表わす. これはバイト列では
 //&egThe corresponding byte sequence is  //&eg The corresponding byte sequence is
 \[  \[
 \mbox{\tt 00 00 00 14 00 00 00 01 00 00 00 0e}  \mbox{\tt 00 00 00 14 00 00 00 01 00 00 00 0e}
 \]  \]
Line 442  Ring by Name を用いた場合, 現在の名前空間で変数 yyy に
Line 445  Ring by Name を用いた場合, 現在の名前空間で変数 yyy に
 /*&eg  /*&eg
 We treat polynomial rings and their elements as follows.  We treat polynomial rings and their elements as follows.
   
 An element of a generic DMS ring is an element of  Generic DMS ring is an $n$-variate polynomial ring $K[x_1, \ldots, x_n]$,
 an $n$-variate polynomial ring $K[x_1, \ldots, x_n]$,  
 where $K$ is some coefficient set. $K$ is unknown in advance  where $K$ is some coefficient set. $K$ is unknown in advance
 and it is determined when coefficients are received.  and it is determined when coefficients of an element are received.
 When a server has received an element in a generic DMS ring,  When a server has received an element in Generic DMS ring,
 the server has to translate it into the corresponding local object  the server has to translate it into the corresponding local object
 on the server. Each server has its own translation scheme.  on the server. Each server has its own translation scheme.
   
 In Asir such an element are translated into a distributed polynomial.  In Asir such an element are translated into a distributed polynomial.
   
 In {\tt kan/sm1} things are complicated.  In {\tt kan/sm1} things are complicated.
 {\tt kan/sm1} does not have any class corresponding to a generic DMS ring.  {\tt kan/sm1} does not have any class corresponding to Generic DMS ring.
 {\tt kan/sm1} translates a DMS of N variables into an element of  {\tt kan/sm1} translates a DMS of N variables into an element of
 the CurrentRing.  the CurrentRing.
 If the CurrentRing is $n'$-variate and $n' < n$, then  If the CurrentRing is $n'$-variate and $n' < n$, then
 a polynomial ring is newly created. Optional informations such as  an $n$-variate polynomial ring is newly created. Optional informations such as
 the term order are all ignored.  the term order are all ignored.
   
 If RingbyName ({\tt CMO\_RING\_BY\_NAME}, yyy)  If RingByName ({\tt CMO\_RING\_BY\_NAME}, yyy)
 is specified as the second field of DMS,  is specified as the second field of DMS,
 it requests a sever to use a ring object whose name is yyy  it requests a sever to use a ring object whose name is yyy
 as the destination ring for the translation.  as the destination ring for the translation.
Line 504  $3 x^2 y$ is regarded as an element of a six-variate p
Line 504  $3 x^2 y$ is regarded as an element of a six-variate p
 #define CMO_POLYNOMIAL_IN_ONE_VARIABLE  33  #define CMO_POLYNOMIAL_IN_ONE_VARIABLE  33
 \end{verbatim}  \end{verbatim}
   
 Group CMObject/RecursivePolynomial requires CMObject/Basic0, CMObject/Basic1.\\  Group CMObject/RecursivePolynomial requires CMObject/Primitive, CMObject/Basic.\\
 Polynomial in 1 variable, Coefficient, Name of the main variable,  Polynomial in 1 variable, Coefficient, Name of the main variable,
 Recursive Polynomial, Ring definition for recursive polynomials  Recursive Polynomial, Ring definition for recursive polynomials
 $\in$ CMObject/RecursivePolynomial \\  $\in$ CMObject/RecursivePolynomial \\
Line 543  $\in$ CMObject/RecursivePolynomial \\
Line 543  $\in$ CMObject/RecursivePolynomial \\
 \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\  \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\
 & & \quad \mbox{ Name of the main variable }, \\  & & \quad \mbox{ Name of the main variable }, \\
 & & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} \\  & & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} \\
 & & \mbox{ --- m is the number of monimials. } \\  & & \mbox{ --- m is the number of monomials. } \\
 & & \mbox{ --- A pair of e and Coefficieint represents a monomial. } \\  & & \mbox{ --- A pair of e and Coefficient represents a monomial. } \\
 & & \mbox{ --- The pairs of e and Coefficient are sorted in the } \\  & & \mbox{ --- The pairs of e and Coefficient are sorted in the } \\
 & & \mbox{ \quad decreasing order, usually with respect to e.} \\  & & \mbox{ \quad decreasing order, usually with respect to e.} \\
 & & \mbox{ ---  e denotes an exponent of a monomial with respect to } \\  & & \mbox{ ---  e denotes an exponent of a monomial with respect to } \\
Line 591  $$   x^3 (1234 y^5 + 17 ) +  x^1 (y^{10} + 31 y^5)  $$
Line 591  $$   x^3 (1234 y^5 + 17 ) +  x^1 (y^{10} + 31 y^5)  $$
 /*&eg  /*&eg
 We intend to represent non-commutative polynomials with the  We intend to represent non-commutative polynomials with the
 same form. In such a case, the order of products are defined  same form. In such a case, the order of products are defined
 as above, that is a power of the mail variable $\times$ a coeffcient.  as above, that is a power of the main variable $\times$ a coeffcient.
 */  */
   
 \noindent  \noindent
Line 613  Class.recursivePolynomial h * ((-1)) + (x^2  * (1))
Line 613  Class.recursivePolynomial h * ((-1)) + (x^2  * (1))
 \end{verbatim}  \end{verbatim}
   
 \noindent  \noindent
 Group CMObject/MachineDouble requires CMObject/Basic0.\\  Group CMObject/MachineDouble requires CMObject/Primitive.\\
 64bit machine double, Array of 64bit machine double  64bit machine double, Array of 64bit machine double
 128bit machine double, Array of 128bit machine double  128bit machine double, Array of 128bit machine double
 $\in$ CMObject/MachineDouble \\  $\in$ CMObject/MachineDouble \\
Line 693  conformant float.
Line 693  conformant float.
 */  */
   
 \noindent  \noindent
 Group CMObject/Bigfloat requires CMObject/Basic0, CMObject/Basic1.\\  Group CMObject/Bigfloat requires CMObject/Primitive, CMObject/Basic.\\
 Bigfloat  Bigfloat
 $\in$ CMObject/Bigfloat \\  $\in$ CMObject/Bigfloat \\
   
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