a = b mod n;

is interpreted as

   a = (b mod n)
or (by default) as
    a = modp(b, n)

see, the help page ?operators,precedence

So, 

3 = 0 mod 3;
                             3 = 0
0 = 3 mod 3;
                             0 = 0

eqs:=t^2*x^2 + t*x + 2*x - 1=0, t^2*y^3 + t*y^2 + 2*y - 1=0, 2*t^2*z^3 + t*z + 3*z - t^2=0:
sol:=solve({eqs,x>0},{x,y,z}, explicit,allsolutions):
plots:-spacecurve(eval([x,y,z],sol), t=0..4,color=red);

Note that x>0 eliminates the "x" solution not defined at t=0.

A:=op(1,A11):
B:=op(2,A11)*10^(-8470):C:=op([3,1],A11)*10^(-8470):
B:=convert(B,rational):
C:=convert(C,rational):

fB:=factor(B,complex):
fC:=factor(C,complex):
A*fB/fC;   # Here it is (it's a rational function, not a polynomial)

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Of course.
factor(A11);  # real factors
factor(A11,complex);  # complex factors

It is simpler without geom3d:

line1:=[1 + t, t, -2 + 2*t]:
line2:=[k*s+4, 2*s, 3*s]:
solve(line1=~line2);       # or solve(line1 - line2)  or  solve(line1 - line2, [k,s,t]);

        {k = 1/2, s = 2, t = 4}

But your identity is trivial. M(p,q) does not depend on p, so actually M(p,q) = M(q) = frac(q*(q-1)/4).
Now, both M(q) and D(q) have period 4, so it is enough to check M(q) = D(q)  for q in {0,1,2,3}.

It seems that HeatMap cannot be used with display,insequence.
[Also, the PLOT structure generated by HeatMap  contains some undocumented stuff].

But Explore works; however sometimes it may happen that the gridlines are missing (I don't know why).

restart;
with(Statistics):# with(LinearAlgebra): randomize(): with(plots):
f:= rand( 0..1 ):
ee:=proc()
  local RM := Matrix(10, 10, (i, j) ->2*f()-1 );
  Threads:-Sleep(0.5);
  HeatMap(RM, axis = [gridlines = [10, color = blue], thickness = 2]) ;
end:
Explore(ee(n), parameters=[[n=1..10, animate]], numframes=10);

1. You have many parameters which must be given numeric values in order to obtain a numeric solution.

indets([sys],name(Not(constant))) minus {t};

2. Please check if you mean pi or Pi.

3. Use
sol := dsolve(...)
because p is already a parameter.

4. You also have a syntax error in odeplot.

plots[odeplot](sol, [[t, S(t)], [t, V(t)], [t, C(t)], [t, I(t)], [t, R(t)]], t = 0 .. 200, numpoints = 25)

(do you really want numpoints=25?).

For such problems you should use one of the packages combinat or Iterator.

P.S. O simple method to test e.g. that i,j,k,l,m,n,o,p are distinct is to use nops({i,j,k,l,m,n,o,p}) = 8.

pdsolve looks for classical solutions, C^1 in this case. Your pde has not such solutions: the solution without bc, ic is u(x,t)=_F(t-x).

The solution given by Mathematica is not differentiable.
 

f:=int(sqrt(sin(x)),x):
a:=limit(f,x=Pi/2,left):
answer:=piecewise(x<Pi/2,f-a, x=Pi/2,0, -eval(f-a,x=Pi-x));

n:=4:
Z:=Matrix(n):  # zero matrix
LinearAlgebra:-MatrixExponential(Z);

This should be mathematically obvious because exp(Z) = I + Z^1/1! + Z^2/2! + ... = I

You should be aware that an expression has many equivalent forms.

SUBS:=proc(f,x)
local F,N:=0, g;
g:=subs(x=F(),f);
F:=proc() N:=N+1; x[N] end;
eval(g)
end:

SUBS(exp(-I*Pi*(n+2*n*(m-1))/m), n);