===================================================================
RCS file: /home/cvs/OpenXM/src/kan96xx/Kan/usage.sm1,v
retrieving revision 1.1.1.1
retrieving revision 1.8
diff -u -p -r1.1.1.1 -r1.8
--- OpenXM/src/kan96xx/Kan/usage.sm1	1999/10/08 02:12:02	1.1.1.1
+++ OpenXM/src/kan96xx/Kan/usage.sm1	2004/09/10 22:21:27	1.8
@@ -1,3 +1,4 @@
+%% $OpenXM: OpenXM/src/kan96xx/Kan/usage.sm1,v 1.7 2003/06/26 13:00:11 takayama Exp $
 %% usage.sm1
 %%  [(key word)
 %%    [(   )
@@ -102,6 +103,16 @@
    (The user cannot use the variable names h, e, H and E.)
    (cf. show_ring, system_variable --- CurrentRing)
    (    print_switch_status, switch_function)
+   (Example: [(x,y) ring_of_differential_operators [[(Dx) 1 (Dy) 1]])
+   (           weight_vector 0] define_ring)
+   (   Define the ring of differential operators Q<x,y,Dx,Dy> with the order)
+   (   obtained by refining the partial order defined by the weight vector)
+   (   [x,y,Dx,Dy] = [0,0,1,1] ) 
+   (Example: [(x,y) ring_of_differential_operators 0 [(weightedHomogenization) 1]])
+   (          define_ring )
+   (Example: [(x,y) ring_of_differential_operators [[(x) -1 (Dx) 1]] weight_vector)
+   (          0 [(degreeShift) [[1 0 1]]] ])
+   (          define_ring )
    (Example: [(x,y,z) ring_of_differential_operators (Dx,Dy) elimination_order 0] define_ring)
    (         Dx,Dy, and Dz are corresponding differential operators to the )
    (         space variables x, y and z.)
@@ -110,6 +121,7 @@
 
 [(ring_of_difference_operators)
  [ (string ring_of_difference_operators)
+   (!!! OBSOLETE !!!)
    (  )
    (This command has not been maintained since 1996. So, there may be)
    (troubles if it is used with new features of kan/sm1.)
@@ -145,6 +157,22 @@
  ]
 ] putUsages
 
+[(ring_of_differential_difference_operators)
+ [ ([string1 string2]  ring_of_differential_difference operators)
+   (This command is used to define a ring of differential-difference operators )
+   (with the macros define_ring, weight_vector and elimination_order.)
+   (string1 is a set of variables for differential operators.)
+   (string2 is a set of variables for difference operators.)
+   (The user cannot use the variable names h, e, H and E.)
+   (cf. show_ring, system_variable --- CurrentRing)
+   (    print_switch_status, switch_function)
+   (Example: [[(x,y) (s)] ring_of_differential_difference_operators )
+   (          [[(Es) 1] [(s) 1]] weight_vector 0] define_ring)
+   (         Dx and Dy are corresponding differential operators to the )
+   (         space variables x and y.  Try show_ring to see the commutation)
+   (         relations.)
+ ]
+] putUsages
 
 [(define_ring)
   [( [varlist ring_of_??? order characteristic options] define_ring)
@@ -157,7 +185,7 @@
    (    <<ring_of_???>> ring_of_polynomials, ring_of_differential_operators, )
    (    ring_of_difference_operators)
    (    <<order>> elimination_order, weight_vector)
-   (    ring_def,ring , << ,, >>)
+   (    ring_def,ring , << __ >>)
   ]
 ] putUsages
 
@@ -277,6 +305,7 @@ def
  [(obj toVectors vec)
   (Convert the internal expression of vector into the array <<vec>>.)
   (obj can be  [n [g1 ... gm]] where n is the length of the vector.)
+  $cf. [(toe_) array_of_poly] gbext $
   $Example 1: [(x) ring_of_polynomials 0] define_ring  $
   $           (e_ + 2).  toVectors :: $
   (        --->  [2  , 1] )