# Define the slit map, algebraically again slitmap = build_slitmap(omega) # Test the slit map if we want -- IS THIS ALLOWED FOR ALGEBRAICALLY DEFINED # OMEGA? if slitmap_tests: test_slitmap(slitmap) # Build the slitmap piece by piece for diagnostic purposes. -- IS THIS # ALLOWED FOR ALGEBRAICALLY DEFINED OMEGA? if slitmap_full: build_slitmap_detailed(omega, delta, q) # Map the pre_branch_pts to the branch points under the slit map. The result # is an algebraic expression which we compare to the true branch_pts. There # are 2g alg_branch_pts alg_branch_pts = map(lambda x: slitmap(z=x),pre_branch_pts) # Now we want to solve this system. There are 2g unknown alg_branch_pts and # 2g known branch_pts. Therefore we have 2g equations in 2g unknowns. Solve # it up!!! # I don't know if this will just work like this. I imagine it will be # slow... # We also want to be careful to organize these well. We want the first # equation to be delta_1 - q_1 = e_1, delta_1 + q_1 = e_2 etc. The numbering # doesn't matter but we do want the two adjacent branch points to be labeled # by delta - q and delta + q respectively. eqns = [ alg_branch_pts[k] == branch_pts[k] for k in xrange(len(branch_pts)) ] sol = solve(eqns, delta+q, solution_dict=True) # Solve and get a dictionary # The above thing might give multiple answers, i.e. len(sol)>1. If so, we # need to look through all possible answers for now since I do not now know # how to determine which is which.
# Build the slitmap piece by piece for diagnostic purposes.
if slitmap_full:
build_slitmap_detailed(omega, delta, q, circle_colors=circle_colors)
# this thing can output some stuff for use, but for now it just plots.
# Define the points of intersection of the Cj with the real axis. The image
# of these points under the slitmap (5.19) are the branch points of the
# curve.
# For the hyperelliptic case this is easy, we know what the circles look
# like so this is where they intersect. For the nonhyperelliptic case we
# need to do something else maybe. For example, evaluate Cj(t=0)?
pre_branch_pts = [delta[j] - q[j] for j in xrange(genus)]
pre_branch_pts += [delta[j] + q[j] for j in xrange(genus)]
# Branch points are the image of pre_branch_pts
branch_pts = [slitmap(z=bp) for bp in pre_branch_pts]
# Simple plot of branch points if interested. Shows location.
if plot_branch_pts:
branch_plot = branch_point_plot(branch_pts)
branch_plot.show(axes=True, title='Branch Points')
return branch_pts
def forward_problem_on_Points_and_Lines(delta,
q,
Points,
Lines,
plot_circles=True,
plot_F=False,
# Define the slit map, algebraically again
slitmap = build_slitmap(omega)
# Test the slit map if we want -- IS THIS ALLOWED FOR ALGEBRAICALLY DEFINED
# OMEGA?
if slitmap_tests: test_slitmap(slitmap)
# Build the slitmap piece by piece for diagnostic purposes. -- IS THIS
# ALLOWED FOR ALGEBRAICALLY DEFINED OMEGA?
if slitmap_full: build_slitmap_detailed(omega, delta, q)
# Map the pre_branch_pts to the branch points under the slit map. The result
# is an algebraic expression which we compare to the true branch_pts. There
# are 2g alg_branch_pts
alg_branch_pts = map(lambda x: slitmap(z=x), pre_branch_pts)
# Now we want to solve this system. There are 2g unknown alg_branch_pts and
# 2g known branch_pts. Therefore we have 2g equations in 2g unknowns. Solve
# it up!!!
# I don't know if this will just work like this. I imagine it will be
# slow...
# We also want to be careful to organize these well. We want the first
# equation to be delta_1 - q_1 = e_1, delta_1 + q_1 = e_2 etc. The numbering
# doesn't matter but we do want the two adjacent branch points to be labeled
# by delta - q and delta + q respectively.
eqns = [
alg_branch_pts[k] == branch_pts[k] for k in xrange(len(branch_pts))
]
sol = solve(eqns, delta + q,
solution_dict=True) # Solve and get a dictionary
if slitmap_full: build_slitmap_detailed(omega, delta, q,
circle_colors=circle_colors)
# this thing can output some stuff for use, but for now it just plots.
# Define the points of intersection of the Cj with the real axis. The image
# of these points under the slitmap (5.19) are the branch points of the
# curve.
# For the hyperelliptic case this is easy, we know what the circles look
# like so this is where they intersect. For the nonhyperelliptic case we
# need to do something else maybe. For example, evaluate Cj(t=0)?
pre_branch_pts = [ delta[j]-q[j] for j in xrange(genus) ]
pre_branch_pts += [ delta[j]+q[j] for j in xrange(genus) ]
# Branch points are the image of pre_branch_pts
branch_pts = [slitmap(z = bp) for bp in pre_branch_pts]
# Simple plot of branch points if interested. Shows location.
if plot_branch_pts:
branch_plot=branch_point_plot(branch_pts)
branch_plot.show(axes = True, title='Branch Points')
return branch_pts
def forward_problem_on_Points_and_Lines(
delta, q, Points, Lines, plot_circles=True, plot_F=False,
slitmap_full=True, slitmap_direct=False, prec='double',
product_threshold=5, max_time=200, prime_function_tests=False,
slitmap_tests=False, point_size=80, save_plots=False
):
#