@james1482 

if would like to try to iterate just first top 10 of group

and display them as matrix group and then permutation group

how to do?

would like to try following groups, or even more in Group Theory Package

HardaNortonGroup
Monster Group
LyonsGroup
ThompsonGroup
ONanGroup
HeldGroup
RudvalisGroup
JankoGroup
ExceptionGroup
FischerGroup
GeneralLinearGroup
GeneralOrthogonalGroup
HigmanSimsGroup

@Carl Love 

COMPUTING GENUS 2 CURVES FROM INVARIANTS ON THE HILBERT MODULI SPACE

this paper use four points in example in page 3

but the example in

http://www.math.univ-montp2.fr/~rs/genus2unifdoc.html

use 5 or 6 points, 

any one know which is correct?

i believe 5 points correct because i tried example give genus 2 result

@Carl Love 

i find that the tool in

http://www.math.univ-montp2.fr/~rs/genus2unifdoc.html

it use solution to find invariant

but i find many papers mention to use invariant to find back the solution

if we know solution before using invariant, there is no need to use this invariant

@Carl Love 

sorry, i type wrong, i can see that the result is genus 2 now.

thank you very much, 

is there any skills to design a isomorphoric one, as this example is not isomorphism.

any tips about how to search with igusa invariant ?

@roman_pearce 

i use an example which has result from igusa invariant which is a genus 2

but i run the following , no result return

with(algcurves):
f := expand(y^2-x*(x-1)*(x-2)*(x-3/2)*(x-3));
genus(f, x, y):

@Markiyan Hirnyk 

i find a free free tool can compute this invariant but only a few fraction return

actually what kind of result it search for?

is it a beautiful fraction?

it is not easy to guess.

is it useful for search y^2 = f(x)

@John Fredsted 

it seems nullspace is not real eigenvector.

why secondary school book teaching nullspace as eigenvector?

what is the actual procedure of calculating eigenvector in maple?

@Markiyan Hirnyk 

mmm is variable

@Preben Alsholm 

Here it is the full command,

phaseportrait([a(t)*(diff(c(t), t))+b(t) = x^2+3*x+2, a(t)*(diff(b(t), t))+c(t)*(diff(b(t), t)) = exp(t), a(t)*(diff(c(t), t))+a(t)*(diff(b(t), t))+b(t) = x^2+2*x+1], [a(t), b(t), c(t)], t = -2 .. 2, [[a(0) = 1, b(0) = 0, c(0) = 2]], stepsize = 0.5e-1, scene = [c(t), a(t)], linecolour = sin((1/2)*t*Pi), method = classical[foreuler]);

Error, (in DEtools/phaseportrait) the ODE system does not contain derivatives of the unknown function a

@Carl Love 

how to write one differential system with H(x)+H(y) = H(x+y)?

eq2a := a(x)*(diff(c(x), x))+b(x);
eq3a := a(x)*(diff(b(x), x))+c(x)*(diff(b(x), x));
eq4a := a(x)*(diff(c(x), x))+a(x)*(diff(b(x), x))+b(x);
inis := diff(a(x),x)=H(a(x)), diff(b(x),x)=H(b(x)), diff(c(x),x)=H(c(x)), a(1) = b(1), b(1) = c(1), a(0) = b(0), b(0) = c(0);
inis := diff(a(x),x)=H(a(x)), diff(b(x),x)=H(b(x)), diff(c(x),x)=H(c(x));
inio := H(x) + H(y) = H(x);
sol := dsolve([eq2a=1, eq3a=exp(a), eq4a=0, inis, inio], [a(x), b(x), c(x), H(x), H(y)])

Error, (in dsolve) found differentiated functions with same name but depending on different arguments in the given DE system: {H(x), H(y)}

and

sol := dsolve([eq2a=1, eq3a=exp(a), inis, inio], [a(x), b(x), c(x), H(a(x)), H(b(x))])

Error, (in dsolve) ambiguous input: the variables {a} and the functions {H(a(x)), a(x)} cannot both appear in the system

i change to use pdsolve, but got this error

Error, (in pdsolve/sys/info) ambiguous input: the variables {a, b} and the functions {H(a(x)+b(x)), H(a(x)), H(a(y)), H(b(x)), H(b(y)), a(x), a(y), b(x), b(y)} cannot both appear in the system

then try again,

eq2a := a(x)*(diff(c(x), x))+b(x);

eq3a := a(x)*(diff(b(x), x))+c(x)*(diff(b(x), x));
eq4a := a(x)*(diff(c(x), x))+a(x)*(diff(b(x), x))+b(x);
inis := diff(a(x),x)=H(a(x)), diff(b(x),x)=H(b(x)), diff(c(x),x)=H(c(x)), a(1) = b(1), b(1) = c(1), a(0) = b(0), b(0) = c(0);
inis := diff(a(x),x)=H(a(x)), diff(b(x),x)=H(b(x)), diff(c(x),x)=H(c(x));
inio := H(a(x)) + H(b(x)) = H(b(x)+a(x)), H(b(x)) + H(c(x)) = H(b(x)+c(x)), H(a(x)) + H(c(x)) = H(a(x)+c(x));
sol := dsolve([eq2a=1, eq3a=exp(a), eq4a=0 inis, inio], [a(x), b(x), c(x), H(x)]);

Error, (in dsolve) ambiguous input: the variables {a} and the functions {H(a(x)+c(x)), H(b(x)+a(x)), H(a(x)), a(x)} cannot both appear in the system

@tomleslie 

moreover, how to plot it on the sphere?

it has error invalid function argument

plots:-display(
DEplot:-intersectplot(ODE1, [b(t), c(t)], t = 0 .. 30, b = -10 .. 10, c = -10 .. 10, arrows = comet, color = blue, numframes = 100);
plot3d(1, theta= 0..2*Pi, phi= 0..Pi, coords= spherical),
scaling= constrained, axes= boxed
);

@tomleslie 

if i use only two equations below to plot , it plot a river like up and down arrow, how do it embed a geometric object into it such as circle and put it in the middle of graph to show how flow move?

with(DEtools):
eq2 :=...;
eq3 :=...;
eq4 := ...;
eq2a := subs(a(t) = 1, eq2);
eq3a := subs(a(t) = 1, eq3);
eq4a := subs(a(t) = 1, eq4);

ODE1 := [eq2a, eq3a];
dsolve({eq2a = exp(t), eq3a = exp(t)});
DEplot(ODE1, [b(t), c(t)], t = 0 .. 30, b = -10 .. 10, c = -10 .. 10, arrows = comet, color = blue, numframes = 100);
ODE1 := [eq3a, eq4a];
DEplot(ODE1, [b(t), c(t)], t = 0 .. 30, b = -10 .. 10, c = -10 .. 10, arrows = comet, color = blue, numframes = 100);
ODE1 := [eq2a, eq4a];
DEplot(ODE1, [b(t), c(t)], t = 0 .. 30, b = -10 .. 10, c = -10 .. 10, arrows = comet, color = blue, numframes = 100);

@tomleslie 

dsolve([eq2=exp(t),eq3=exp(t),eq4=exp(t)],[a(t),b(t),c(t)]);

[{c(t) = 1}, {b(t) = exp(t)}, {a(t) = 0}]

then i set

dsolve([eq2=1,eq3=exp(t),eq4=0],[a(t),b(t),c(t)]);

it got a good unique solutionfor a(t),b(t) and c(t)

but it has so many unknown constant _C1, _C2 etc

in fact, it is possible to solve

how to further find these _C1, _C2 etc,

how to find this initial condition ?

update1 is rewrite rules for finding initial condition, 

after appending this result to above, it do not have output.

with(diffalg):
syst := [...];
R := differential_ring(derivations = [t], notation = diff, ranking = [[a,b,c]]);
ideal := Rosenfeld_Groebner(syst, R);
rewrite_rules(ideal[1]);

dsolve([eq2=1,eq3=exp(t),eq4=0, diff(b(t), t) = 0, diff(c(t), t) = -b(t)/a(t)],[a(t),b(t),c(t)]);

@Kitonum 

if use your method, how to plot for this case

with(DEtools):
eq2 := a(t)*(diff(c(t), t))+b(t);
eq3 := a(t)*(diff(b(t), t))+c(t)*(diff(b(t), t));
eq4 := a(t)*(diff(c(t), t))+a(t)*(diff(b(t), t))+b(t);
eq2a := subs(a(t) = 1, eq2);
eq3a := subs(a(t) = 1, eq3);
eq4a := subs(a(t) = 1, eq4);
ODE1 := [eq2a, eq3a];
dsolve({eq2a = exp(t), eq3a = exp(t)});

{diff(c(t), t, t) = c(t)*exp(t)/(c(t)+1)}, {b(t) = -(diff(c(t), t))+exp(t)};

A := seq(seq(plots[arrow]([diff(c(t), t, t) = c(t)*exp(t)/(c(t)+1), b(t) = -(diff(c(t), t))+exp(t)], color = red), t = 0 .. evalf(2*Pi), evalf(Pi/12)), t = 0 .. evalf(Pi), evalf(Pi/12));

Error, invalid input: diff received 0, which is not valid for its 2nd argument

@tomleslie 

i can give you input and output now,

i notice if b(t) and c(t) are constant, diff(b(t),t) and diff(c(t),t) are zero

then i choose a(t) be constant

originally i hope to solve and result two functions in terms of t

but solution have differential terms again, how to find vector field and

plot it, if this result can also be plot, how to plot?

with(DEtools):
eq2a := subs(a(t) = 1, eq2);
eq3a := subs(a(t) = 1, eq3);
eq4a := subs(a(t) = 1, eq4);
ODE1 := [eq2a, eq3a];
dsolve({eq2a = exp(t), eq3a = exp(t)});

{diff(c(t), t, t) = c(t)*exp(t)/(c(t)+1)}, {b(t) = -(diff(c(t), t))+exp(t)};

i have already given the initial condition, there is only c(t), why it ask for same of number of dependant variables DE, 

what do it require?

dfieldplot([diff(c(t), t, t) = c(t)*exp(t)/(c(t)+1), c(0) = 1], c(t), t = -3 .. 3, color = (1/2)*c);
Error, (in DEtools/dfieldplot) system must have same number of dependent variables as DE's.