1.1 A list of functions

1.1.1 !ReservedNames

!ReservedNames

:: reserved symbols.

• The names k00*, K00*, sm1* , arg1,arg2,arg3,arg4,...., Helplist, Arglist, FunctionValue, @*, db.*, k.*, tmp002*, tmp00* are used for system functions. Percent, LeftBracket, RightBracket, Dollar, Newline are constants

1.1.2 AddString

AddString(list):: concatenate strings.

• AddString(list) returns the concatnated string (array list).

1.1.3 Annfs

Annfs (?)

::

• Annfs(f,v) computes the annihilating ideal of f^r and the Bernstein-Sato polynomial b(s) of f Return value: [Ann(f^r), r, b(s)] where r is the minimal integral root of b(s) = 0.

 
Example:  Annfs(x^2+y^2,"x,y"): 

See also

Not yet written.

1.1.4 Append

Append(f1,fn,g):: append two lists or arrays.

• Append([f1,...,fn],g) returns the list [f1,...,fn,g]

See also

Join, NewArray, Rest

1.1.5 AsciiToString

AsciiToString(ascii_code):: translate from ascii code to a string.

• AsciiToString(ascii_code) returns the string of which ascii code is ascii_code (integer ascii_code).

See also

!ReservedNames, AddString, StringToAsciiArray

1.1.6 BfRoots1

BfRoots1 (?)

::

• BfRoots1(g,v) returns the integral roots of g with respect to the weight vector (1,1,...,1) and the b-function itself

 
Example:  BfRoots1([x*Dx-2, y*Dy-3],[x,y]);

See also

Not yet written.

1.1.7 Boundp

Boundp(s):: check if a symbol is assigned a value or not.

• Boundp(s) checks if the symbol s is bounded to a value or not (string s).

1.1.8 Cancel

Cancel (?)

::

• Cancel the greatest common multiple of the denominators and the numerator.

See also

Not yet written.

1.1.9 CancelNumber

CancelNumber(rn):: factor out the greatest common divisor.

• CancelNumber(rn) reduces the rational number rn (rational rn).

 
 CancelNumber( 2/6 ) : 

See also

Cancel

1.1.10 Characteristic

Characteristic(ring):: get the characteristic of a given ring.

• Characteristic(ring) returns the characteristic of the ring .

See also

GetRing, RingD, RingPoly, SetRing

1.1.11 Coefficients

Coefficients(f,v):: get the exponents and coefficients.

• Coefficients(f,v) returns [exponents, coefficients] of f with respect to the variable v (polynomial f,v).

 
  Coefficients(Poly("(x+1)^2"),Poly("x")): 

See also

Exponent, Init, Init_w

1.1.12 CopyArray

CopyArray (?)

::

• It duplicates the argument array recursively.

 
Example: m=[1,[2,3]];
         a=CopyArray(m); a[1] = "Hello";
         Println(m); Println(a);

See also

Not yet written.

1.1.13 DC

DC(obj,key):: translate data types.

• DC(obj,key) converts obj to a new object in the primitive class key (object obj, string key)

 
  DC(" (x+1)^10 ", "polynomial"): 

See also

ToString

1.1.14 DeRham4

DeRham4(f):: de Rham cohomology groups of the complement of the zero set of a polynomial

• DeRham4(f) gives the dimensions of de Rham cohomology groups of C^4-V(f). f (string) is a polynomial in Z[x,y,z,www]. DeRham2 is for Z[x,y], DeRham3 is for Z[x,y,z].

 
Example: c=DeRham4("x^3+y^2+z^2+www^2-1"); c:
         It will take more than 10 minutes. Answer will be [2,0,0,1,1]
                           standing for dim of [H^4,H^3,H^2,H^1,H^0].

See also

ToString, Sintegration

1.1.15 Denominator

Denominator(f):: denominator

• Denominator(f) returns the denominator of the rational f.

See also

Cancel, Numerator

1.1.16 Dvar

Dvar(v):: differential operator variables standing for space variables.

• Dvar(v) gives dvars standing for v. vv is a list of variables

 
Example: RingD("x,y"); Dvar([x,y]):
         Dvar("z"): Dvar(["z","s"]):

See also

UseSmallD, @.Dsymbol

1.1.17 Eliminatev

Eliminatev(list,var)::

• Eliminatev(list,var) prunes polynomials in list (array of polynomials) which contains the variables in var ( array of strings )

 
 Eliminatev([Poly(" x+h "),Poly(" x ")],[ "h" ]): 

See also

eliminatev (sm1)

1.1.18 Error

Error(s):: cause an error.

• Error(s) causes an error and outputs a message s.

1.1.19 Exponent

Exponent(f,vars):: exponents of a given polynomial.

• Expoent(f,vars) returns the vector of exponents of the polynomial f

 
 Exponent( x^2*y-1,[x,y])

See also

Coefficients

1.1.20 Factor

Factor (?)

::

• Factor a given polynomial.

See also

Not yet written.

1.1.21 GKZ.GKZ

GKZ.GKZ (?)

::

• GKZ(a,b) returns the GKZ systems associated to the matrix a and the vector b The answer is given by strings.

 
Example: GKZ([[1,1,1,1],[0,1,3,4]],[0,2]);

See also

Not yet written.

1.1.22 Gb

Gb (?)

::

• Gb(f) returns the Groebner basis of f. cf. Kernel, Weyl.

See also

Not yet written.

1.1.23 GetEnv

GetEnv(s):: value of an environmental variable.

• GetEnv(s) returns the value of the environmental variable string s.

1.1.24 GetPathName

GetPathName(s):: find a file in the search path.

• GetPathName(s) checks if the file s exists in the current directory or in LOAD_K_PATH. If there exists, it returns the path name (string s).

See also

GetEnv

1.1.25 Gmp.

Gmp.

:: class Gmp

• Gmp is a class which supports the following methods: BitAnd, BitOr, ModuloPower, ProbabilisticPrimeP, Sqrt, Gcd, Div, Mod. These methods call functions of GNU MP package. Note that there is no method to create an instance.

 
  r = Gmp.Gcd(5,8); 

1.1.26 Groebner

Groebner(input):: compute the Grobner basis.

• Groebner(input) returns Groebner basis of the left ideal or the submodule defined by input (array of polynomials) The order is that of the ring to which each element of input belongs. The input is automatically homogenized.

 
 RingD("x,y",[["x", 10, "y", 1]]);
          Groebner([Poly(" x^2+y^2-4"),Poly(" x*y-1 ")]):
cf. RingD, Homogenize

See also

Gb, Init_w, RingD, groebner (sm1)

1.1.27 Help

Help(key):: display a help message.

• Help(key) or help(key) shows an explanation on the key (string key).

See also

HelpAdd, help, man

1.1.28 HelpAdd

HelpAdd(key,explanation):: Add a help message in the system.

• HelpAdd([key,explanation]) (string key, string explanation) or (string key, array explanation).

  HelpAdd([key,explanation,category]) is used to specify the category of the topics.

See also

help, man

1.1.29 Init

Init(f):: return the initial term.

• Init(f) returns the initial term of the polynomial f (polynomial) Init(list) returns the array of initial terms of the array of polynomials list (array)

See also

Init_w

1.1.30 Init_w

Init_w(f,vars,w):: return the initial terms.

• Init_w(f,vars,w) returns the initial terms with respect to the weight vector w (array of integer) of the polynomial f (polynomial). Here, f is regarded as a polynomial with respect to the variables vars (array of polynomials).

 
 RingD("x,y"); Init_w(x^2+y^2+x,[x,y],[1,1]):

See also

Gb, Init

1.1.31 Init_w_m

Init_w_m (?)

::

• Init_w_m(f,w,m) returns the initial of f with respect to w with the shift m. Note that the order of the ring and the weight w must be the same.

 
Example:  Sweyl("x,y",[["x",-1,"Dx",1]]); 
          Init_w_m([x*Dx+1,Dx^2+x^5],["x",-1,"Dx",1],[2,0]):

See also

Not yet written.

1.1.32 IntegerToSm1Integer

IntegerToSm1Integer(i):: translates integer to 32 bit integer (sm1 integer)

• IntegerToSm1Integer(i) translates integer i to sm1 integer (integer i).

1.1.33 IsArray

IsArray(f):: check if an given object is an array.

• If f is an array (or list) object, then IsArray(f) returns true, else IsArray(f) returns false.

See also

Is*, Tag

1.1.34 IsConstant

IsConstant(f):: check if a given object is a constant.

• IsConstant(f) returns true if the polynomial f is a constant.

See also

DC, Is*, Tag

1.1.35 IsDouble

IsDouble(a):: check if a given object is double.

• IsDouble(a) returns true if a is a double (object a). It returns false if a is not.

See also

DC, Is*, Tag

1.1.36 IsExact_h

IsExact_h (?)

::

• IsExact_h(complex,var): bool It checks the given complex is exact or not in D<h> (homogenized Weyl algebra) cf. ReParse

See also

Not yet written.

1.1.37 IsInteger

IsInteger(a):: check if a given object is an integer.

• IsInteger(a) returns true if a is an integer (object a). It returns false if a is not. cf. IsSm1Integer

See also

DC, Is*, Tag

1.1.38 IsObject

IsObject(a):: check if a given object is primitive or not.

• IsObject(a) return true if a is an Object.

See also

Tag

1.1.39 IsPolynomial

IsPolynomial(obj):: check if a given object is a polynomial.

• IsPolynomial(f) returns true if f (object) is a polynomial.

See also

DC, Is*, Tag

1.1.40 IsRational

IsRational(a):: check if a given object is a rational.

• IsRational(a) returns true if a is a rational (object a). It returns false if a is not.

See also

DC, Is*, Tag

1.1.41 IsRing

IsRing(obj):: check if a given object is a ring.

• IsRing(obj) returns true if obj is a ring (object obj).

See also

DC, Is*, Tag

1.1.42 IsSameIdeal_h

IsSameIdeal_h (?)

::

• IsSameIdeal_h(ii,jj,var): bool It checks the given ideals are the same or not in D<h> (homogenized Weyl algebra) cf. ReParse

See also

Not yet written.

1.1.43 IsSm1Integer

IsSm1Integer(obj):: check if a given object is a 32 bit integer.

• IsSm1Integer(obj) returns true if obj is an integer of sm1(object obj).

See also

DC, Is*, Tag

1.1.44 IsString

IsString(obj):: check if a given object is a string.

• IsString(obj) returns true if obj is a string (object obj).

 
  if (IsString("abc")) Println("Hello"); ;

See also

DC, Is*, Tag

1.1.45 Join

Join(f1,fn,g1,gm):: join two lists or arrays

• Join([f1,...,fn],[g1,...,gm]) returns the list [f1,...,fn,g1,...,gm]

See also

Append, NewArray, Rest

1.1.46 Kernel

Kernel (?)

::

• Kernel(f) returns the syzygy of f. Return value [b, c]: b is a set of generators of the syzygies of f : c=[gb, backward transformation, syzygy without dehomogenization

 
Example:  Weyl("x,y",[["x",-1,"Dx",1]]); 
          s=Kernel([x*Dx+1,Dx^2+x^5]); s[0]:

See also

Not yet written.

1.1.47 Length

Length(vec):: length of a given array or a list.

• Length(vec) returns the length of the array vec (array vec)

1.1.48 Ln

Ln():: print newline.

• Print newline.

See also

Print, Println

1.1.49 Load_sm1

Load_sm1(s,flag):: load a sm1 program

• Load_sm1(s,flag) loads a sm1 program from s[0], s[1], .... If loading is succeeded, the already-loaded flag is set to true. (list s, string flag).

See also

load

1.1.50 Map

Map(karg,func):: apply a function to each element of a list.

• Map(karg,func) applies the function func to the karg (string func).

 
  Map([82,83,85],"AsciiToString"):

1.1.51 Mapto

Mapto(obj,ring):: parse a polynomial in a given ring.

• Mapto(obj,ring) parses obj as elements of the ring . (ring ring , polynomial obj or array of polynomial obj ).

 
 R = RingD("x,y"); SetRingVariables();
    f = (x+y)^2; R2 = RingD("x,y,z",[["y",1]]); 
    f2 = Mapto(f,R2); f2: 

See also

ReParse

1.1.52 Max

Max (?)

::

• Max(v) returns the maximal element in v.

See also

Not yet written.

1.1.53 Mod

Mod(f,p):: modulo

• Mod(f,p) returns f modulo n where f (polynomial) and p (integer).

See also

gbext (sm1)

1.1.54 NewArray

NewArray(n):: it returns an array of a given size.

• NewArray(n) returns an array of size integer n.

See also

NewMatrix

1.1.55 NewMatrix

NewMatrix(m,n):: generate a matrix of a given size.

• NewMatrix(m,n) returns the (m,n)-matrix (array) with the entries 0.

See also

NewArray

1.1.56 Numerator

Numerator(f):: numerator

• Numerator(f) returns the numerator of the rational f.

See also

Cancel, Denominator

1.1.57 Ord_w_m

Ord_w_m (?)

::

• Ord_w_m(f,w,m) returns the order of f with respect to w with the shift m. Note that the order of the ring and the weight w must be the same. When f is zero, it returns -intInfinity = -999999999.

 
Example:  Sweyl("x,y",[["x",-1,"Dx",1]]); 
          Ord_w_m([x*Dx+1,Dx^2+x^5],["x",-1,"Dx",1],[2,0]):

See also

Not yet written.

1.1.58 OutputPrompt

OutputPrompt

:: output the prompt.

• Output the prompt. Files should end with this command.

1.1.59 Pmat

Pmat(m):: print an given array in a pretty way.

• Pmat(m): Print the array m in a pretty way.

See also

Println

1.1.60 Poly

Poly(name):: translate a string to a polynomial.

• Poly(name) parses a given string name in the current ring and returns a polynomial.

See also

DC, PolyR, ReParse, RingD

1.1.61 PolyR

PolyR(name,r):: translate a string to a polynomial in a given ring.

• PolyR(name,r) parses a string name in the ring r

 
 r = RingD("x,y"); y = PolyR("x+2*y",r); 

See also

Poly

1.1.62 Position

Position(list,elem):: find a position of an element in a list.

• Position(list,elem) returns the position p of the element elem in the array list. If elem is not in list, it return -1 (array list).

 
 Position([1,34,2],34): 

1.1.63 Print

Print(f):: display a given object.

• Print(f) prints f without the newline.

See also

Println

1.1.64 Println

Println(f):: display a given object with the newline.

• Println(f) prints f and goes to the new line.

See also

Ln, Print, Stderr.

1.1.65 Protect

Protect(name,level):: add read-only property for a given variable.

• Protect(name) protects the symbol name (string) Protect(name,level) protects the symbol name (string) with level

See also

extension (sm1)

1.1.66 QuoteMode

QuoteMode(number):: Change into the quote mode.

• QuoteMode(1) sets the parser in the quotemode; if unknown function symbol comes, it automatically translates the expression into a tree.

 
Example 1: class polymake extends PrimitiveObject {local ; def hogera() { return(1);} } 
         QuoteMode(1); polymake.afo(1,2): 
Example 2: polymake="polymake"; 
         QuoteMode(1); polymake.afo(1,2): 
QuoteMode(0) turns off the quotemode.

See also

Tag

1.1.67 ReParse

ReParse(obj):: parses a given object in the current ring.

• Reparse(obj): It parses the given object obj in the current ring.

See also

Mapto, RingD

1.1.68 ReducedBase

ReducedBase(base):: remove unnecessary elements.

• ReducedBase(base) prunes redundant elements in the Grobner basis base (array).

See also

Gb, Groebner

1.1.69 Reduction

Reduction(f,g):: get the remainder and the quotients.

• Reduction(f,G) returns the remainder and sygygies when f is devided by G (polynomial f, array G).

See also

Gb

1.1.70 Replace

Replace(f,rule):: substitute variables by given values

• Replace(f,rule) rewrites f by the rule (polynomial f, array rule).

 
 RingD("x,y"); Replace( (x+y)^3, [[x,Poly("1")]])

See also

replace (sm1)

1.1.71 Rest

Rest(a):: it returns the rest of a given list.

• Rest(a) returns the rest (cdr) of a (list a).

See also

Append, Join

1.1.72 RingD

RingD(names,weight_vector):: define a new ring of differential operators.

• RingD(names) defines a new ring of differential operators (string names). RingD(names,weight_vector) defines a new ring with the weight vector (string names, array weight_vector). RingD(names,weight_vector,characteristic) Dx is the associated variable to x where Dx*x - x*Dx = 1 holds.

 
  RingD("x,y",[["x",2,"y",1]]) 

See also

GetRing, PolyR, ReParse, SetRing

1.1.73 RingPoly

RingPoly(names,weight_vector):: define a ring of polynomials.

• RingPoly(names) defines a ring of polyomials (string names). The names of variables of that ring are names and the homogenization variable h. cf. SetRingVariables, RingD

 
 R=RingPoly("x,y");
  
RingPoly(names,weight_vector) defines a ring of polynomials
with the order defined by the weight_vector
(string names, array of array weight_vector).
RingPoly(names,weight_vector,characteristic)
Example: R=RingPoly("x,y",[["x",10,"y",1]]);
         (x+y)^10: 

See also

GetRing, RingD, SetRing

1.1.74 Sannfs2

Sannfs2 (?)

::

• Sannfs2(f) constructs the V-minimal free resolution for the weight (-1,1) of the Laplace transform of the annihilating ideal of the polynomial f in x,y. See also Sminimal, Sannfs3.

 
Example: a=Sannfs2("x^3-y^2");
         b=a[0]; sm1_pmat(b);
         b[1]*b[0]:
Example: a=Sannfs2("x*y*(x-y)*(x+y)");
         b=a[0]; sm1_pmat(b);
         b[1]*b[0]:

See also

Not yet written.

1.1.75 Sannfs3

Sannfs3 (?)

::

• Sannfs3(f) constructs the V-minimal free resolution for the weight (-1,1) of the Laplace transform of the annihilating ideal of the polynomial f in x,y,z. See also Sminimal, Sannfs2.

 
Example: a=Sannfs3("x^3-y^2*z^2");
         b=a[0]; sm1_pmat(b);
         b[1]*b[0]: b[2]*b[1]:

See also

Not yet written.

1.1.76 Save

Save(obj):: write a given object to a file.

• Save(obj) appends obj to the file sm1out.txt (object obj).

1.1.77 SetRingVariables

SetRingVariables()::

• Set the generators of the current ring as global variables. You do not need explicitly call this function which is called from RingD. cf. RingD(), Poly(), PolyR()

See also

RingD

1.1.78 SgetShift

SgetShift (?)

::

• SgetShift(mat,w,m) returns the shift vector of mat with respect to w with the shift m. Note that the order of the ring and the weight w must be the same.

 
Example:  Sweyl("x,y",[["x",-1,"Dx",1]]); 
          SgetShift([[x*Dx+1,Dx^2+x^5],[Poly("0"),x],[x,x]],["x",-1,"Dx",1],[2,0]):

See also

Not yet written.

1.1.79 SgetShifts

SgetShifts (?)

::

• SgetShifts(resmat,w) returns the shift vectors of the resolution resmat with respect to w with the shift m. Note that the order of the ring and the weight w must be the same. Zero row is not allowed.

 
Example:   a=Sannfs2("x^3-y^2");
           b=a[0]; w = ["x",-1,"y",-1,"Dx",1,"Dy",1];
           Sweyl("x,y",[w]); b = Reparse(b);
           SgetShifts(b,w):

See also

Not yet written.

1.1.80 Sinit_w

Sinit_w (?)

::

• Sinit_w(resmat,w) returns the initial of the complex resmat with respect to the weight w.

 
Example:   a=Sannfs2("x^3-y^2");
           b=a[0]; w = ["x",-1,"y",-1,"Dx",1,"Dy",1];
           Sweyl("x,y",[w]); b = Reparse(b);
           c=Sinit_w(b,w); c:

See also

Not yet written.

1.1.81 Sintegration

Sintegration (?)

::

• Ans=Sintegration(gg,v,intv) gives the cohomology group of the integration (complex) of gg along the list of variables intv. Here, v is a list of variables. Interation variables intv must be in the first part of v. Ans[1] gives the cohomology groups. When v=intv, Ans[0] gives the dimension of the cohomology groups. Sintegration uses the function Srestall. cf. Srestriction

 
Example: RingD("x,t"); II=Sintegration([Dt-(3*t^2-x),Dx+t],[t,x],[t]); II:

See also

Not yet written.

1.1.82 Sminimal

Sminimal (?)

::

• It constructs the V-minimal free resolution by LaScala’s algorithm option: "homogenized" (no automatic homogenization) : "Sordinary" (no (u,v)-minimal resolution) Options should be given as an array.

 
Example:  Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
          v=[[2*x*Dx + 3*y*Dy+6, 0],
             [3*x^2*Dy + 2*y*Dx, 0],
             [0,  x^2+y^2],
             [0,  x*y]];
         a=Sminimal(v);
         Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
         b = ReParse(a[0]); sm1_pmat(b); 
         IsExact_h(b,[x,y]):
Note:  a[0] is the V-minimal resolution. a[3] is the Schreyer resolution.
 ---> D^{m_3} --b[2]--> D^{m_2} --b[1]--> D^{m_1} --b[0]--> D^{m_0} 
  Here D^{m_i} are the set of row vectors. 

See also

Not yet written.

1.1.83 Sord_w

Sord_w (?)

::

• Sord_w(f,w) returns the w-order of f

 
Example: Sord_w(x^2*Dx*Dy,[x,-1,Dx,1]):

See also

Not yet written.

1.1.84 SresolutionFrameWithTower

SresolutionFrameWithTower (?)

::

• It returs [resolution of the initial, gbTower, skelton, gbasis] option: "homogenized" (no automatic homogenization)

 
Example: Sweyl("x,y");
         a=SresolutionFrameWithTower([x^3,x*y,y^3-1]);

See also

Not yet written.

1.1.85 Srestall

Srestall (?)

::

• Srestall(gg,v,rv,k1) evaluates the dimensions of all restictions of gg along the list of variables rv. Here, v is a list of variables and k1 is the maximal integral root of the b-function of gg. Srestall uses the function Sminimal to get a (-w,w)-minimal free resolution. cf. Bfroots1(ii,vv)

See also

Not yet written.

1.1.86 Srestriction

Srestriction (?)

::

• Ans=Srestriction(gg,v,restv) gives the cohomology group of the restriction (complex) of gg along the list of variables restv. Here, v is a list of variables. Restriction variables restv must be in the first part of v. Ans[1] gives the cohomology groups. Eliminate module base 1,e_,e_^2, ... in Ans[1,0,1] by the order 1>e_>e_^2>..., then the restriction ideal is obtained. When v=restv, Ans[0] gives the dimension of the cohomology groups. Srestriction uses the function Sintegration.

 
Example: RingD("x,y"); R=Srestriction([x*Dx-1,y*Dy-1],[x,y],[x]); R:

See also

Not yet written.

1.1.87 Stderr.

Stderr.

:: class Stderr

• Stderr is a class which supports printing to stderr. Methods are Print, Println, Ln, Flush. Stderr_fd is a global variable to save a file descriptor.

1.1.88 StringToAsciiArray

StringToAsciiArray(s):: translate a string to an array of ascii codes.

• StringToAsciiArray(s) decomposes the string s into an array of ascii codes of s. cf. AsciiToString.

See also

AsciiToString

1.1.89 Substitute

Substitute(f,xx,g)::

• Substitute(f,xx,g) replaces xx in f by g . This function takes coeffients of f with respect to xx and returns the inner product of the vector of coefficients and the vector of which elements are g^(corresponding exponent). Note that it may cause an unexpected result in non-commutative rings.

See also

Replace

1.1.90 System

System(comm):: call the unix shell.

• System(comm) executes the unix system command comm (string comm)

 
 System("ls");

1.1.91 Tag

Tag(f):: return the tag of a given object.

• Tag(f) returns the datatype tag of f where 0: null, 5: string, 6: array, 9: polynomial, 15: integer(big-num), 16: rational, 18:double, 257: Error

 
 Tag([Poly("0"), 0]):

See also

Is*

1.1.92 ToDouble

ToDouble(f):: translate a given object to double.

• ToDouble(f) translates f into double when it is possible object f .

 
 ToDouble([1,1/2,[5]]): 

See also

DC, Is*, Tag

1.1.93 ToString

ToString(obj):: translate a given object to a string.

• ToString(obj) transforms the obj to a string.

See also

DC

1.1.94 ToricIdeal

ToricIdeal (?)

::

• ToricIdeal(a) returns the affine toric ideal associated to the matrix a The answer is given by a list of strings.

 
Example: ToricIdeal([[1,1,1,1],[0,1,3,4]]);

See also

Not yet written.

1.1.95 Translate.to_int0

Translate.to_int0 (?)

::

• to_int0(a) : as same as sm1_push_int0.

See also

Not yet written.

1.1.96 Transpose

Transpose(m):: transposition

• Transpose(m.) return the transpose of the matrix m (array of array m).

See also

NewMatrix

1.1.97 UseSmallD

UseSmallD (?)

::

• UseSmallD() changes the Dsymbol to d and E,H symbols to ee0,hh @.Dsymbol can be refered by aaaDsymbol.

See also

Not yet written.

1.1.98 Weyl

Weyl (?)

::

• Weyl(v,w) defines the Weyl algebra (the ring of differential operators) with the weight vector w.

 
Example:  Weyl("x,y",[["x",-1,"Dx",1]]); 

See also

Not yet written.

1.1.99 false

false

::

• false returns sm1 integer 0.

1.1.100 generic_bfct

generic_bfct(ii,vv,dd,ww):: b-function for a weight vector

• generic_bfct(ii,vv,dd,ww) gives the b-function of the ideal ii for the weight vector ww. vv is a list of variables and dd is a list of D-variables. It calls ox_asir and returns a string

 
Example: RingD("x,y"); f=generic_bfct([Dx^2+Dy^2-1,Dx*Dy-4],[x,y],[Dx,Dy],[1,1]); Println(f);
      RingD("s"); ff=ReParse(f); Factor(ff);

See also

Factor, Srestriction, Sintegration

1.1.101 load

load(fname):: load a given file.

• load(fname) loads the file fname (string fname). load fname loads the file fname . load[fname] loads the file fname with the preprocessing by /lib/cpp.

See also

Load_sm1

1.1.102 sm1

sm1(arg1,arg2):: execute sm1 commands

• sm1(arg1,arg2,...) is used to embed sm1 native code in the kxx program.

 
 sm1( 2, 2, " add print "); 
 def myadd(a,b) { sm1(" a b add /FunctionValue set "); }

See also

usage (sm1)

1.1.103 sm1_deRham

sm1_deRham (?)

::

• sm1_deRham(f,v) computes the dimension of the deRham cohomology groups of C^n - V(f) This function does not use (-w,w)-minimal free resolution.

 
Example:  sm1_deRham("x^3-y^2","x,y");

See also

Not yet written.

1.1.104 true

true

::

• true returns sm1 integer 1.

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