#
# Chapter 17.  Overview of Other Packages
#
#
#
# MAPLE SESSION 1
#
>    ?index[packages]
#
# 17.1   Finite fields
#
#
#
# MAPLE SESSION 2
#
>    13 +  15 mod 17;
>    13 -  15 mod 17;
>    13/15 mod 17;
>    13^(-1) mod 17;
#
#
# MAPLE SESSION 3
#
>    P := x^4+3*x+1;
>    factor(P);
>    Factor(P) mod 43;
#
#
# MAPLE SESSION 4
#
>    alias(alpha=RootOf(y^2+1)): 
>    Normal(1/(1+alpha)) mod 5;
#
#
# MAPLE SESSION 5
#
>    Factor(x^3+x+1) mod 2;
#
#
# MAPLE SESSION 6
#
>    G8 := GF(2,3,alpha^3+alpha+1); 
#
#
# MAPLE SESSION 7
#
>    seq(G8[input](k),k=1..8);
#
#
# MAPLE SESSION 8
#
>    a := G8[ConvertIn](1+alpha);
>    b := G8[ConvertIn](alpha^2);
#
#
# MAPLE SESSION 9
#
>    G8[`^`](a,4);
>    G8[`/`](a,b);
>    G8[`*`](a,b);
#
# 17.2   Polynomials
#
#
#
# MAPLE SESSION 10
#
>    p:= x^4-4*x^2*y^2+4*y^4-6*x^2-12*y^2+9;
>    factor(p);
>    factor(p,sqrt(2));
>    factor(p,sqrt(2),sqrt(3));
#
#
# MAPLE SESSION 11
#
>    beta := RootOf(Z^3+Z+1);
>    Q := x^6-2*x^4-x^3+2*x^2-1;
>    factor(Q,beta);
#
#
# MAPLE SESSION 12
#
>    factor(Q,complex);
#
#
# MAPLE SESSION 13
#
>    c := 0.3178372451957822447:
>    p := polytools[minpoly](c,4);
>    r := solve(p);
>    map(evalf,[r]);
#
#
# MAPLE SESSION 14
#
>    P := 1 + x^2 + x^4; 
>    polytools[split](P,x);
#
# 17.3   Group theory
#
#
#
# MAPLE SESSION 15
#
>    sigma := [3,4,6,2,5,1];
#
#
# MAPLE SESSION 16
#
>    dcp := convert(sigma,'disjcyc'); 
#
#
# MAPLE SESSION 17
#
>    convert(dcp,'permlist',6);
#
#
# MAPLE SESSION 18
#
>    with(group): 
>    G:=permgroup(4,{sigma=[[1,2,3,4]],tau=[[1,3],[2,4]]});
#
#
# MAPLE SESSION 19
#
>    grouporder(G);
#
#
# MAPLE SESSION 20
#
>    elements(G);
#
#
# MAPLE SESSION 21
#
>    with(group): 
>    G := grelgroup(a,b,[a,a,a,a,a],[b,b],[a,b,a,1/b]);
>    grouporder(G);
#
#
# MAPLE SESSION 22
#
>    p:= 2*(x^2+6*x-3)^3-27*(x+1)^3*(x^2-1);
>    galois(p);
#
# 17.4   Combinatorics
# 17.4.1   The combinat package
#
#
#
# MAPLE SESSION 23
#
>    with(combinat): 
>    T := cartprod([{1,2,3},{a,b},{A,B}]);
#
#
# MAPLE SESSION 24
#
>    with(combinat): 
>    C := {}: 
>    T := cartprod([1,2,3,a,b,A,B]): 
>    while not T[finished] do 
>   C := C union T[nextvalue]():
> xend do: 
>    C;
#
#
# MAPLE SESSION 25
#
>    with(combinat): 
>    permute(4,2);
#
#
# MAPLE SESSION 26
#
>    permute(4); 
>    permute([a,b,c,d],3); 
>    permute([a,a,c,d],3);
#
#
# MAPLE SESSION 27
#
>    randperm([a,b,c,d]);
#
#
# MAPLE SESSION 28
#
>    numbperm(12,5); 
>    12!/7!;
#
#
# MAPLE SESSION 29
#
>    with(combinat): 
>    choose(4,2);
#
#
# MAPLE SESSION 30
#
>    choose([a,b,c,d,e,f],3); 
>    choose([a,a,c,d,d,d],3);
#
#
# MAPLE SESSION 31
#
>    with(combinat): 
>    numbcomb(12,5); 
>    12!/5!/7!; 
>    randcomb(12,5); 
>    randcomb([a,b,c,d,e],3);
#
#
# MAPLE SESSION 32
#
>    with(combinat): 
>    P:=partition(6);
>    nops(P);
#
#
# MAPLE SESSION 33
#
>    P := sum(numbpart(n)*q^ n,n=0..20);
>    series(1/P,q,21);
#
#
# MAPLE SESSION 34
#
>   p:= [1,1,1,3,5,5,12,21,21]:
>   conjpart(p);
#
#
# MAPLE SESSION 35
#
>    with(combinat): 
>    composition(6,3);
>    nops(%);
#
#
# MAPLE SESSION 36
#
>    numbcomp(6,3);
#
#
# MAPLE SESSION 37
#
>    with(combinat): 
>    powerset(4);
>    nops(%);
#
#
# MAPLE SESSION 38
#
>    PS := subsets({a,b,c,d}); 
>    while not PS[finished] do 
>   PS[nextvalue](); 
> xend do;
#
#
# MAPLE SESSION 39
#
>    with(combinat): 
>    for j from 1 to 6 do 
>   inttovec(j,3); 
> xend do;
#
#
# MAPLE SESSION 40
#
>    with(combinat): 
>    lambda := [1,1,3]: 
>    rho := [1,1,1,1,1]: 
>    Chi(lambda,rho);
#
#
# MAPLE SESSION 41
#
>    with(combinat): 
>    character(5);
#
#
# MAPLE SESSION 42
#
>    with(combinat): 
>    g := graycode(4);
>    printf(cat(` %.4d`$16,`\n`), op(map(convert, g, binary))); 
#
#
# MAPLE SESSION 43
#
>    with(combinat): 
>    multinomial(12,3,4,5); 
>    12!/3!/4!/5!;
#
# 17.4.2   The networks package
#
#
#
# MAPLE SESSION 44
#
>    with(networks): 
>    G:=new(): 
>    addvertex({A,B,C,D,E,F},G);
#
#
# MAPLE SESSION 45
#
>    addedge([{A,B},{B,C},{D,F},{E,F},{A,D},{C,E},{A,C}],G);
#
#
# MAPLE SESSION 46
#
>    draw(G);
#
#
# MAPLE SESSION 47
#
>    show(G);
#
#
# MAPLE SESSION 48
#
>    ends(e1,G);
#
#
# MAPLE SESSION 49
#
>    vertices(G); 
>    ends(G);
#
#
# MAPLE SESSION 50
#
>    H := duplicate(G):
#
#
# MAPLE SESSION 51
#
>    delete(A,G): 
>    draw(G); 
>    delete(e1,H): 
>    draw(H);
#
#
# MAPLE SESSION 52
#
>    with(networks): 
>    G:=new(): 
>    addvertex({A,B,C,D},G): 
>    addedge([[A,B],[B,A],[B,C],[C,D]],G);
#
#
# MAPLE SESSION 53
#
>    head(e4,G); 
>    tail(e4,G);
#
#
# MAPLE SESSION 54
#
>    with(networks): 
>    G := new(): 
>    addvertex({A,B,C,D,E},G): 
>    addedge([[A,B],[A,C],[A,D],[B,C],[B,E],[C,E],[D,E]], 
>  names=[AB,AC,AD,BC,BE,CE,DE], 
>  weights=[7,3,10,5,6,4,12],G);
#
#
# MAPLE SESSION 55
#
>    eweight(G);
#
#
# MAPLE SESSION 56
#
>    flow(G,A,E);
#
#
# MAPLE SESSION 57
#
>    flow(G,A,E,sateds): 
>    sateds;
#
#
# MAPLE SESSION 58
#
>    with(networks): 
>    G := new(): 
>    addvertex({A,B,C},weights=[0,1,2],G); 
>    addedge([[A,B],[B,C],[A,C]],G); 
>    vweight(G);
#
# 17.4.3   The combstruct package
# 17.5   Number theory
# 17.5.1   The numtheory  package
#
#
#
# MAPLE SESSION 59
#
>    n := 144;
>    ifactor(n);
#
#
# MAPLE SESSION 60
#
>    with(numtheory): 
>    n := 144;
>    divs := divisors(n);
>    factorset(n);
>    tau(n);
>    nops(divs);
#
#
# MAPLE SESSION 61
#
>    with(numtheory): 
>    factorEQ(30,-7);
#
#
# MAPLE SESSION 62
#
>    with(numtheory): 
>    nthpow(23152500);
#
#
# MAPLE SESSION 63
#
>    with(numtheory): 
>    p := mersenne([10]);
>    ifactor(p+1);
#
#
# MAPLE SESSION 64
#
>    with(numtheory): 
>    L := [seq([n,pi(n)/(n/log(n))],n=2..1000)]: 
>    plot(L, style=point,labels=[n," "]);
#
#
# MAPLE SESSION 65
#
>    with(numtheory): 
>    safeprime(100);
#
#
# MAPLE SESSION 66
#
>    with(numtheory): 
>    g := primroot(17);
>    [seq(modp(g^ k,17),k=1..16)];
>    sort(%);
#
#
# MAPLE SESSION 67
#
>    with(numtheory): 
>    order(2,17);
>    modp(2^8,17);
#
#
# MAPLE SESSION 68
#
>    with(numtheory): 
>    g := primroot(47);
>    d := index(6,5,47);
>    modp(g^ d,47);
#
#
# MAPLE SESSION 69
#
>    with(numtheory): 
>    mroot(2,5,13);           
>    modp(6^5,13);
#
#
# MAPLE SESSION 70
#
>    with(numtheory): 
>    rootsunity(7,43);
>    map(`^`,[%],7);  
>    map(modp,%,43);
#
#
# MAPLE SESSION 71
#
>    with(numtheory): 
>    legendre(15,23);
>    msqrt(15,23);
#
#
# MAPLE SESSION 72
#
>    with(numtheory): 
>    legendre(-1,41);
>    imagunit(41);
>    modp(%^2,41);
#
#
# MAPLE SESSION 73
#
>    with(numtheory): 
>    mcombine(4,3,5,4);
#
#
# MAPLE SESSION 74
#
>    with(numtheory): 
>    cfrac(exp(1),6);
#
#
# MAPLE SESSION 75
#
>    with(numtheory): 
>    cfrac(exp(1),30,'quotients');
#
#
# MAPLE SESSION 76
#
>    with(numtheory): 
>    y := product( 1-q^(5*n-1))*(1-q^(5*n-4))/(1-q^(5*n-2))
>  /(1-q^(5*n-3)),n=1..10): 
>    cfrac(y,q,6);
#
# 17.5.2   The GaussInt  package
# 17.5.3   $p$-adic numbers
#
#
#
# MAPLE SESSION 77
#
>    with(padic): 
>    r := 1705725;
>    s := 5561328861;
>    ifactor(r);
>    ifactor(s);
>    ordp(r,3);
>    ordp(s,3);
>    x:=r/s;
>    ordp(x,3);
#
#
# MAPLE SESSION 78
#
>    valuep(x,3);
#
#
# MAPLE SESSION 79
#
>    x;
>    evalp(x,3);
#
#
# MAPLE SESSION 80
#
>    evalp(x,3,20);
#
#
# MAPLE SESSION 81
#
>    with(padic): 
>    px := p_adic(3,-4,[1,2,2,1,2,1,2,2,1,0]);
>    evalp(px,3);
#
#
# MAPLE SESSION 82
#
>    with(padic): 
>    z := sqrtp(10,3);
>    evalp(z^2,3);
>    sqrtp(2,3);
#
#
# MAPLE SESSION 83
#
>    ?padic,function
#
# 17.6   Numerical approximation
#
#
#
# MAPLE SESSION 84
#
>    with(numapprox): 
>    f := exp(x): 
>    padf := pade(f,x,[3,3]);
>    taylor(f-padf,x,10);