def circle_image(A, B):
G = Graphics()
G += circle((0, 0), 1, color='grey')
from collections import defaultdict
tmp = defaultdict(int)
for a in A:
for j in range(a):
if gcd(j, a) == 1:
rational = Rational(j) / Rational(a)
tmp[(rational.numerator(), rational.denominator())] += 1
for b in B:
for j in range(b):
if gcd(j, b) == 1:
rational = Rational(j) / Rational(b)
tmp[(rational.numerator(), rational.denominator())] -= 1
C = ComplexField()
for val in tmp:
if tmp[val] > 0:
G += text(str(tmp[val]),
exp(C(-.2 + 2 * 3.14159 * I * val[0] / val[1])),
fontsize=30,
axes=False,
color="green")
if tmp[val] < 0:
G += text(str(abs(tmp[val])),
exp(C(.2 + 2 * 3.14159 * I * val[0] / val[1])),
fontsize=30,
axes=False,
color="blue")
return G
def dc_calc_gauss(modulus, number):
arg = request.args.get("val", [])
if not arg:
return flask.abort(404)
try:
if modulus == 1:
# there is a bug in sage for modulus = 1
return r"""\(\displaystyle \tau_{%s}(\chi_{1}(1,·)) =
\sum_{r\in \mathbb{Z}/\mathbb{Z}} \chi_{1}(1,r) 1^{%s}= 1. \)""" % (int(arg), int(arg))
from dirichlet_conrey import DirichletGroup_conrey
chi = DirichletGroup_conrey(modulus)[number]
chi = chi.sage_character()
g = chi.gauss_sum_numerical(100, int(arg))
real = round(g.real(), 10)
imag = round(g.imag(), 10)
if imag == 0.:
g = str(real)
elif real == 0.:
g = str(imag) + "i"
else:
g = latex(g)
from sage.rings.rational import Rational
x = Rational('%s/%s' % (int(arg), modulus))
n = x.numerator()
n = str(n) + "r" if not n == 1 else "r"
d = x.denominator()
return r"\(\displaystyle \tau_{%s}(\chi_{%s}(%s,·)) = \sum_{r\in \mathbb{Z}/%s\mathbb{Z}} \chi_{%s}(%s,r) e\left(\frac{%s}{%s}\right) = %s. \)" % (int(arg), modulus, number, modulus, modulus, number, n, d, g)
except Exception, e:
return "<span style='color:red;'>ERROR: %s</span>" % e
def dc_calc_gauss(modulus, number):
arg = request.args.get("val", [])
if not arg:
return flask.abort(404)
try:
if modulus == 1:
# there is a bug in sage for modulus = 1
return r"""\(\displaystyle \tau_{%s}(\chi_{1}(1,·)) =
\sum_{r\in \mathbb{Z}/\mathbb{Z}} \chi_{1}(1,r) 1^{%s}= 1. \)""" % (
int(arg), int(arg))
from dirichlet_conrey import DirichletGroup_conrey
chi = DirichletGroup_conrey(modulus)[number]
chi = chi.sage_character()
g = chi.gauss_sum_numerical(100, int(arg))
real = round(g.real(), 10)
imag = round(g.imag(), 10)
if imag == 0.:
g = str(real)
elif real == 0.:
g = str(imag) + "i"
else:
g = latex(g)
from sage.rings.rational import Rational
x = Rational('%s/%s' % (int(arg), modulus))
n = x.numerator()
n = str(n) + "r" if not n == 1 else "r"
d = x.denominator()
return r"\(\displaystyle \tau_{%s}(\chi_{%s}(%s,·)) = \sum_{r\in \mathbb{Z}/%s\mathbb{Z}} \chi_{%s}(%s,r) e\left(\frac{%s}{%s}\right) = %s. \)" % (
int(arg), modulus, number, modulus, modulus, number, n, d, g)
except Exception, e:
return "<span style='color:red;'>ERROR: %s</span>" % e
def circle_image(A,B):
G = Graphics()
G += circle((0,0), 1 , color = 'grey')
from collections import defaultdict
tmp = defaultdict(int)
for a in A:
for j in range(a):
if gcd(j,a) == 1:
rational = Rational(j)/Rational(a)
tmp[(rational.numerator(),rational.denominator())] += 1
for b in B:
for j in range(b):
if gcd(j,b) == 1:
rational = Rational(j)/Rational(b)
tmp[(rational.numerator(),rational.denominator())] -= 1
C = ComplexField()
for val in tmp:
if tmp[val] > 0:
G += text(str(tmp[val]),exp(C(2*3.14159*I*val[0]/val[1])), fontsize = 30, axes = False, color = "green")
if tmp[val] < 0:
G += text(str(abs(tmp[val])),exp(C(2*3.14159*I*val[0]/val[1])), fontsize = 30, axes = False, color = "red")
return G
def circle_image(A, B):
G = Graphics()
G += circle((0, 0), 1, color='black', thickness=3)
G += circle(
(0, 0), 1.4, color='black', alpha=0
) # This adds an invisible framing circle to the plot, which protects the aspect ratio from being skewed.
from collections import defaultdict
tmp = defaultdict(int)
for a in A:
for j in range(a):
if gcd(j, a) == 1:
rational = Rational(j) / Rational(a)
tmp[(rational.numerator(), rational.denominator())] += 1
for b in B:
for j in range(b):
if gcd(j, b) == 1:
rational = Rational(j) / Rational(b)
tmp[(rational.numerator(), rational.denominator())] -= 1
C = ComplexField()
color1 = (41 / 255, 95 / 255, 45 / 255)
color2 = (0 / 255, 0 / 255, 150 / 255)
for val in tmp:
if tmp[val] > 0:
G += text(str(tmp[val]),
exp(C(-.2 + 2 * 3.14159 * I * val[0] / val[1])),
fontsize=30,
axes=False,
color=color1)
if tmp[val] < 0:
G += text(str(abs(tmp[val])),
exp(C(.2 + 2 * 3.14159 * I * val[0] / val[1])),
fontsize=30,
axes=False,
color=color2)
return G
def gauss_sum(self, val):
val = int(val)
mod, num = self.modulus, self.number
chi = self.chi.sage_character()
g = chi.gauss_sum_numerical(100, val)
g = complex2str(g)
from sage.rings.rational import Rational
x = Rational('%s/%s' % (val, mod))
n = x.numerator()
n = str(n) + "r" if not n == 1 else "r"
d = x.denominator()
Gtex = '\Z/%s\Z' % mod
chitex = self.char2tex(mod, num, tag=False)
chitexr = self.char2tex(mod, num, 'r', tag=False)
deftex = r'\sum_{r\in %s} %s e\left(\frac{%s}{%s}\right)'%(Gtex,chitexr,n,d)
return r"\(\displaystyle \tau_{%s}(%s) = %s = %s. \)" % (val, chitex, deftex, g)
def gauss_sum(self, val):
val = int(val)
mod, num = self.modulus, self.number
chi = self.chi.sage_character()
g = chi.gauss_sum_numerical(100, val)
real = round(g.real(), 10)
imag = round(g.imag(), 10)
if imag == 0.:
g = str(real)
elif real == 0.:
g = str(imag) + "i"
else:
g = latex(g)
from sage.rings.rational import Rational
x = Rational('%s/%s' % (val, mod))
n = x.numerator()
n = str(n) + "r" if not n == 1 else "r"
d = x.denominator()
Gtex = '\mathbb{Z}/%s\mathbb{Z}' % mod
chitex = self.char2tex(mod, num, tag=False)
chitexr = self.char2tex(mod, num, 'r', tag=False)
deftex = r'\sum_{r\in %s} %s e\left(\frac{%s}{%s}\right)'%(Gtex,chitexr,n,d)
return r"\(\displaystyle \tau_{%s}(%s) = %s = %s. \)" % (val, chitex, deftex, g)
