# You must provide correct assumptions
restart;
with(inttrans):
#with(gfun);
#assume(alpha + beta = 1):
f := a*t^(-alpha) + b*t^(-beta):
inttrans[laplace](f, t, s) assuming alpha<1,beta<1;

This cannot be simplified, even if alpha+beta=1

P:=proc(pmax::posint)
   local p,q,u,v,n,sn, U, R, nR:=0;
   for q to pmax-1 do for p from q+1 to pmax do
     n:=p!/q!; sn:=isqrt(n);
     U:=NumberTheory:-Divisors(n);
     for u in U while u<sn do
       v:=n/u;
       if issqr((v-u)/2) then  nR:=nR+1; R[nR]:=[p, q, (u+v)/2, isqrt((v-u)/2)] fi;
     od:
   od od;
  print('num'=nR);
  entries(R, nolist)
  end proc:

P(13);
                            num = 56
[4, 1, 5, 1], [5, 1, 11, 1], [7, 1, 71, 1], [8, 1, 524, 22],  [8, 1, 204, 6], [8, 1, 201, 3], [9, 1, 684, 18], 
[11, 1, 7911, 69], [10, 1, 2304, 36], [3, 2, 2, 1], [12, 1, 22716, 78], [6, 2, 19, 1], [6, 2, 21, 3], 
[7, 2, 51, 3], [7, 2, 131, 11], [5, 3, 6, 2], [6, 3, 11, 1], [10, 2, 1620, 30], [11, 2, 4905, 45], [10, 3, 916, 22], 
[11, 3, 2956, 38], [7, 3, 29, 1], [8, 3, 104, 8], [8, 4, 76, 8], [13, 3, 32444, 62], [12, 3, 8935, 5], 
[12, 3, 12060, 90], [9, 4, 209, 13], [9, 4, 284, 16], [8, 4, 41, 1], [8, 4, 44, 4], [11, 5, 601, 13], 
[12, 5, 2511, 39], [13, 5, 10449, 87], [10, 6, 71, 1], [9, 4, 124, 4], [12, 4, 4905, 45], [8, 5, 31, 5], 
[9, 5, 55, 1], [11, 6, 236, 4], [11, 6, 244, 8], [11, 7, 336, 18], [8, 7, 3, 1], [11, 6, 431, 19], 
[11, 6, 2316, 48], [11, 6, 249, 9], [11, 6, 276, 12], [13, 8, 659, 23], [12, 9, 61, 7], [9, 8, 5, 2], 
[12, 8, 109, 1], [11, 7, 89, 1], [13, 7, 1231, 23],  [11, 7, 191, 13], [11, 7, 96, 6], [13, 9, 131, 1]
 

Download sys1-vv.mw

H0:=x->piecewise(x<=0,0,x<1/2,1,x<1,-1,0):
H_:=(k,m,x)-> H0(2^k*x-m):
H:=(n,x)->`if`(n=1,1, H_(ilog2(n), n-2^ilog2(n),x)):
seq(plot(H(n,x),x=0..1, color=red),n=1..7);

f:=(x^2)-6*x+p+3: g:=4*x^2-(p-8)*x+7:
minimize(f, x) = minimize(g, x)  assuming p::real;

           -6 + p = 4*(p/8 - 1)^2 - (p - 8)*(p/8 - 1) + 7

solve(%, p);
                            12, -12

It is a surprising bug. Maplesoft should fix it as soon as possible. (It probably comes due to the changes in add in Maple 2021).

A workaround is to replace add(p[], p=P)  with `+`(seq(p[], p=P));
Note that just adding compile=false (tom's suggestion)  is not enough, because removing the seq,  another error appears.

P := CartesianProduct([1,2,3,4],[1,2,3,4], compile=false);
#  seq(p[], p=P);
   add(p[], p=P);  # ==> error

The library must exist, or create a new one.

restart;
MyPackage := module() export f1, f2; local loc1; option package; 
  f1 := proc() loc1 end proc; 
  f2 := proc( v ) loc1 := v end proc; 
  loc1 := 2; 
end module:
mylib:="C:/tmp/test.mla";
LibraryTools:-Create(mylib);

#                   mylib := "C:/tmp/test.mla"

LibraryTools:-Save(MyPackage, mylib);
LibraryTools:-ShowContents(mylib); # optional
#    [["MyPackage.m", [2021, 9, 15, 12, 20, 8], 41984, 126]]

restart;
libname:=libname, "C:/tmp/test.mla":
with(MyPackage);
#                            [f1, f2]

f1();
#                               2

f2(123);
#                              123

f1();
#                              123

restart;
nat_param:=proc(ff, xx ::(name=anything), yy::(name=anything), t::name)
  local K,xt,yt, ode;
  xt:= diff(ff, op(1,xx)); 
  yt:= diff(ff, op(1,yy));
  K := 1/sqrt(xt^2 + yt^2);
  ode := {'diff'( op(1,xx),t ) = -K*yt, 'diff'( op(1,yy),t ) = K*xt};
  ode := subs( [op(1,xx)=op(1,xx)(t), op(1,yy)=op(1,yy)(t) ], ode);
  dsolve(ode union {op(1,xx)(0)=op(2,xx), op(1,yy)(0)=op(2,yy)},numeric, output=listprocedure);
  eval([lhs(xx(t)),lhs(yy(t))],%)[]
end:

f:=x1^4 + x2^4 + 0.4*sin(7*x1) + 0.3*sin(4*Pi*x2) - 1: 
xx,yy:=nat_param(f, x1=0,x2=1, t):
L:=fsolve(xx(t)=xx(0),t=8..10): N:=6:
plots:-display(seq(plot([xx,yy,k*L/N..(k+1)*L/N],color=COLOR(HUE,k/(N+2))), k=0..N-1), thickness=6);

op(1, xLp_tmp)

It's a bug (but actually the generators are correct).

restart;
with(GroupTheory):
H:=PermutationGroup({[[2, 3], [4, 5]], [[2, 5], [3, 4]]}, degree = 5, supergroup = PermutationGroup({[[2, 3, 4]], [[1, 2], [4, 5]]}, degree = 5));
#                    H := ((23)(45)(25)(34))

MinPermRepDegree(H);
#"GENS" = [[Perm([[1, 2]])], [Perm([[3, 4]])]]

#Error, (in GroupTheory:-Subgroup) not all the provided generators belong to the group

Workaround:

GroupOrder(H);
 #                              4

H1:=DirectProduct(CyclicGroup(2)$2):
AreIsomorphic(H, H1);
#                              true

MinimumPermutationRepresentationDegree(H1);
#"GENS" = [[Perm([[1, 2]])], [Perm([[3, 4]])]]

#                               4

The command GroupTheory:-IdentifySmallGroup identifies a group returning (n,d) where n is its order and d an integer (1 <= d <= NumGroups(n)). Then inspect https://en.wikipedia.org/wiki/List_of_small_groups

The computation is slow on my (old) computer. It needs several hours.
To  interrupt and then continue later, just save the partial results. 

restart;
read "d:/temp/Jfile.mpl";
max(indices(J,nolist));
smax := 20;  R := 1680;  k := 1/4;  gam := 1/2;    vmax:=R/2;
n:=0;
Digits:=15:
f:=proc(v,s) 
  evalf(Re(  sinh(s)*coth(1/2*s)^(2*I/k)/(gam-I*k*cosh(s))^5*exp(-(k^2*sinh(s)^2/(gam-I*
  k*cosh(s))+I*k*(1-cosh(s)))*v)*Hypergeom([I/k],[1],I*k*v)*v^n  ))
end:
V0:=0;  dV:=20;  # <-----
for V from V0 to vmax-dV by dV do
  J[V]:=evalf(1/R * Int(f, [V..V+dV, 0..smax], method=_CubaCuhre, epsilon=1e-5));
od;
save J, "d:/temp/Jfile.mpl";
add(entries(J,nolist));
max(indices(J,nolist));

Similar for the imaginary part if needed.

Download SimpJP-vv.mw