def test_canonical_structure_generation():
"""
Check the general signature function of 2D, 3D,
4D, and 5D canonical objects
"""
d = Seq([1, 1])
b = Block.power(d)
b_f = signature_function(b)
print(b_f)
print(b_f.i())
print()
c = Cube.canonical(d)
c_f = signature_function(c)
print(c_f)
print(c_f.i())
print()
h = canonical(d)
h_f = signature_function(h)
print(h_f)
print(h_f.i())
print(Sig(d).iter_add(2))
print()
t = canonical(d, dim=5, l=8)
t_f = signature_function(t)
print(t_f)
print(t_f.i())
print(Sig(d).iter_add(3))
print()
def tesseract_trapezoid_test():
d = Seq([1, 1])
g = Seq([1, 1])
print(d, ":::", g)
s = g_prism(d, g, dim=5)
s_f = signature_function(s)
print(s_f)
s_f = signature_function(s)
print(s_f)
print(s_f.i())
print(general_trapezoid_identity(d, g, 5))
def block_multiplication_hypercube():
l = 10
d = Seq([1, 1])
bd = Block.power(d, l)
g0 = Seq([2, -1])
bg0 = Block.power(g0, l)
g1 = Seq([1, -1])
bg1 = Block.power(g1, l)
s = [Cube.blank(l) for n in range(l)]
print("structure built")
temp0 = Block.power(x, l)
for n in range(l):
temp1 = bg1
s[n][0] = bd * temp0
for k in range(1, l):
s[n][k] = bd * (temp0 * temp1)
temp1 *= bg1
print(f"completed {n}")
temp0 *= bg0
s_f = signature_function(s)
print(s_f)
print(s_f.i())
print()
def power_hypercube():
"""
A demonstration of the power hypercube
"""
l = 20
d0 = Seq([1, 1])
d1 = Seq([2, -1])
d2 = Seq([1, -1])
s = [Cube.blank(l) for n in range(l)]
temp0 = Seq([1])
for n in range(l):
temp1 = Seq([1])
for k in range(l):
temp2 = Seq([1])
for t in range(l):
s[n][k][t] = temp0 * temp1 * temp2
temp2 *= d2
temp1 *= d1
print(f"completed {n}")
temp0 *= d0
s_f = signature_function(s)
print(s_f)
print(s_f.i())
print(Sig(d0) + Sig(d1) + Sig(d2))
def binomial_simplex_6D(d: Seq):
l = 10
s = [[[Cube.blank(l) for n in range(l)] for k in range(l)]
for t in range(l)]
temp_n = Seq(1)
for n in range(l):
temp_k = Seq(1)
for k in range(n + 1):
temp_t = Seq(1)
for t in range(k + 1):
temp_p = Seq(1)
for p in range(t + 1):
temp_y = Seq(1)
for y in range(p + 1):
temp_p_y = d**(p - y)
for r in range(p - y + 1):
s[n][k][t][p][y][r] = temp_n[k] * temp_k[
t] * temp_t[p] * temp_p[y] * temp_p_y[r]
temp_y *= d
temp_p *= d
temp_t *= d
temp_k *= d
print(f"completed {n} {k}")
temp_n *= d
s_f = signature_function(s)
# print(s_f)
# print(s_f.i())
return s_f.i()
def binomial_simplex_tesseract(d: Seq):
l = 10
s = [[Cube.blank(l) for n in range(l)] for k in range(l)]
temp_n = Seq(1)
for n in range(l):
temp_k = Seq(1)
for k in range(n + 1):
temp_t = Seq(1)
for t in range(k + 1):
temp_p = Seq(1)
for p in range(t + 1):
temp_t_p = d**(t - p)
for y in range(t - p + 1):
s[n][k][t][p][y] = temp_n[k] * temp_k[t] * temp_t[
p] * temp_t_p[y]
temp_p *= d
temp_t *= d
temp_k *= d
print(f"completed {n} {k}")
temp_n *= d
s_f = signature_function(s)
print(s_f)
print(s_f.i())
out = d[0]
out += x * d[1] * d[0]
out += x**2 * d[1]**2 * d[0]**2
out += x**3 * (d[0] + 1) * d[1]**3
return out
def binomial_simplex_hypercube():
l = 20
d = Seq([2, -1])
s = [Cube.blank(l) for n in range(l)]
temp_n = Seq(1)
for n in range(l):
temp_k = Seq(1)
for k in range(n + 1):
temp_t = Seq(1)
for t in range(k + 1):
temp_k_t = d**(k - t)
for p in range(k - t + 1):
s[n][k][t][p] = temp_n[k] * temp_k[t] * temp_k_t[p]
temp_t *= d
temp_k *= d
print(f"completed {n} {k}")
temp_n *= d
s_f = signature_function(s)
print(s_f)
print(s_f.i())
print(d[0] + (x * d[0]**2 * d[1]) + (x**2 * ((d[0] + 1) * d[1]**2)))
def seq_block_multi_signature_5D():
l = 8
d = Seq([1, 1])
b = Block.power(d, l)
g1 = Seq([1, 1])
g2 = Seq([2, 1])
g3 = Seq([1, 1, 1])
s = [[Cube.blank(l) for n in range(l)] for k in range(l)]
for n in range(l):
for k in range(l):
for t in range(l):
for p in range(l):
s[n][k][t][p] = ((g1**n)*(g2**k)*(g3**t)*b).f()
print(f"completed {n} {k}")
s_f = signature_function(s)
print(s_f)
print(s_f.i())
# The identity
print(Sig(d) + Sig(g1) + Sig(g2) + Sig(g3) + Sig(1))
def hypercube_trapezoid_test():
d = Seq([1, 1])
g = Seq([1, 1])
s = g_prism(d, g, dim=4)
s_f = signature_function(s)
print(s_f)
print(s_f.i())
print(general_trapezoid_identity(d, g, dim=4))
def six_dimensional_test():
d = Seq([1, 1])
g = Seq([1, 1])
print(d, ":::", g)
s = g_prism(d, g, dim=6, l=5)
s_f = signature_function(s)
print(s_f)
print(s_f.i())
print(general_trapezoid_identity(d, g, dim=6))
def catalan_triangle_4D_prism():
l = 15
b = catalan_triangle()
s = [Cube.prism(b, l) for n in range(l)]
s_f = signature_function(s)
print(s_f)
print(s_f.i())
print()
def test_prism_structure_generation():
"""
Generate and test an N-prism
"""
l = 10
d = Seq([1, 1, 1])
b = Block.power(d, l)
dim = 6
s = prism(b, dim=dim)
s_f = signature_function(s)
print(s_f)
print(s_f.i())
print(Sig(d) + Sig(1).iter_add(dim - 2))
def g_prism_sen_test_5D():
l = 10
d = Seq([1, -1])
g0 = Seq([1])
g1 = Seq([1])
g2 = Seq([1])
s = g_prism(d, [g0, g1, g2], l=l, block=Block.sen(d, l).truncate(l))
s_f = signature_function(s)
print(s_f[::2])
print(s_f[::2].i())
print(Sig(g0) + Sig(g1) + Sig(g2) + Sig(1).iter_add(5))
def sen_hypercube():
l = 20
d = Seq([1, -1])
g0 = Seq([1, -1])
g1 = Seq([1, 1])
b = Block.sen(d, l)
s = [Cube.blank(l) for n in range(l)]
for n in range(l):
for k in range(l):
s[n][k] = g0**n * g1**k * b
print(f"completed {n}")
s_f = signature_function(s)
print(s_f)
print(s_f.i())
def sen_tesseract():
l = 15
d = Seq([1, -1])
g0 = Seq([-1])
g1 = Seq([1])
g2 = Seq([-1])
b = Block.sen(d, l)
s = [[Cube.blank(l) for n in range(l)] for k in range(l)]
for n in range(l):
for k in range(l):
for t in range(l):
s[n][k][t] = g0**n * g1**k * g2**t * b
print(f"completed {n}")
s_f = signature_function(s)
print(s_f)
print(s_f.i())
def seq_block_4D():
l = 10
d = Seq([1, 1])
g = Seq([1, 1])
b = Block.power(d, l)
out = canonical(d, l=l)
for n in range(l):
for k in range(l):
for t in range(l):
out[n][k][t] = (g**(n + k + t) * b).f()
print(f"completed {n}")
out_f = signature_function(out)
print(out_f)
print(out_f.i())
# The identity
print(Sig(d) + Sig(g).iter_add(3))
def special_hypercube(d, g0, g1):
l = 15
b = Block.power(d, l).truncate(l)
s = [Cube.blank(l) for n in range(l)]
temp0 = Seq(1)
for n in range(l):
temp1 = Seq(1)
for k in range(l):
s[n][k] = temp0 * temp1 * b
temp1 *= g1
temp0 *= g0
s_f = signature_function(s)
print(s_f)
print(s_f.i())
return s
def alt_power_trapezoid(d, g0, g1):
l = 15
s = [Cube.blank(l) for n in range(l)]
g0_f = g0.f(l)
g1_f = g1.f(l)
b = Block.power(d, l).truncate(l)
for n in range(l):
for k in range(l):
s[n][k] = b * g0_f[:n + 1] * g1_f[:k + 1]
print(f"completed {n}")
s_f = signature_function(s)
print(s_f)
print(s_f.i())
return s_f
def g_prism_5D(d, g0, g1, g2, l=10):
s = [[Cube.blank(l) for n in range(l)] for k in range(l)]
g0_f = g0.f(l)
g1_f = g1.f(l)
g2_f = g2.f(l)
b = Block.power(d, l).truncate(l)
for n in range(l):
for k in range(l):
for t in range(l):
s[n][k][t] = b * g0_f[:n + 1] * g1_f[:k + 1] * g2_f[:t + 1]
print(f"completed {n} {k}")
s_f = signature_function(s)
print(s_f)
print(s_f.i())
return s_f
def sen_hypercube_g_prism():
"""
Again d must be fixed at {1, -1} for this identity
"""
l = 15
d = Seq([1, -1])
g0 = Seq([1, 1])
g1 = Seq([1, -1])
b = Block.sen(d, l).truncate(l)
s = [Cube.blank(l) for n in range(l)]
g0_f = g0.f(l)
g1_f = g1.f(l)
for n in range(l):
for k in range(l):
s[n][k] = g0_f[:n+1] * g1_f[:k+1] * b
print(f"completed {n}")
s_f = signature_function(s)
print(s_f)
print(s_f.i())
print()
s_f0 = s_f[::2]
s_f1 = s_f[1::2]
print(s_f0)
print(s_f1)
print()
# The first half identity
print(s_f0.i())
print(Sig(g0) + Sig(g1) + Sig(1).iter_add(2))
print()
# The second half identity
print(s_f1)
print((Seq([2, -1]) * (Sig(g0) + Sig(g1) + Sig(1).iter_add(2)).f())[:l//2])
def seq_block_multi_signature_4D():
l = 10
d = Seq([1, 1])
b = Block.power(d, l)
g1 = Seq([1, 1])
g2 = Seq([2, 1])
g3 = Seq([1, 1])
s = [Cube.blank(l) for n in range(l)]
for n in range(l):
for k in range(l):
for t in range(l):
s[n][k][t] = ((g1**n)*(g2**k)*b).f()
s_f = signature_function(s)
print(s_f)
print(s_f.i())
# The identity
print(Sig(d) + Sig(g1) + Sig(g2) + Sig(1))
def g_prism_sen_test_4D():
l = 25
d = Seq([1, -1])
g0 = Seq([2, -1])
g1 = Seq([1, 1])
s = g_prism(d, [g0, g1], l=l, block=Block.sen(d, l).truncate(l))
s_f = signature_function(s)
print(s_f)
print(s_f[::2])
print(s_f[::2].i())
print(Sig(g0) + Sig(g1) + Sig(1).iter_add(2))
s_f1 = s_f[1::2]
print(s_f1)
print((s_f1 / Seq([2, -1])).i())
print(Seq([2, -1]) * (Sig(g0) + Sig(g1) + Sig(1).iter_add(2)).f())