def test():
"""
We are considering electron muon scattering, following the feinmann diagram below.
Assuming all particles have helicity one, and the CM-fram, motion in z-dir, meaning
line 1 and 4 have spin up, line 3 and 4 have spin down. This gives the spinnors below.
p2\ /p4
\ ^
\ / μ
γ §
§
/ \e
^ \.
p1/ \p3
"""
# This is to make sure sympy knows these are real numbers
me, Ee, Em, mm = sp.symbols("me, Ee, mm, Em", positive=True)
ap = sp.symbols("ap", positive=True) # sqrt(Ee + me)
am = sp.symbols("am", positive=True) # sqrt(Ee - me)
bp = sp.symbols("bp", positive=True) # sqrt(Em + mm)
bm = sp.symbols("bm", positive=True) # sqrt(Em - mm)
u1 = np.array([ap, 0, am, 0], dtype=type(I))
u2 = np.array([0, bp, 0, bm], dtype=type(I))
u3 = np.array([0, ap, 0, am], dtype=type(I))
u4 = np.array([bp, 0, bm, 0], dtype=type(I))
M = 0
for i in range(dim):
M += g[i, i] * ((adj(u3) @ γ[i] @ u1) * (adj(u4) @ γ[i] @ u2))
M *= g ** 2 / (4 * am * bp * bm * ap)
pprint(M)
def deriveMetricFromConnection(self, nameOfTargetCoordinateSystem):
# check if the patch stores a coordsystem with the right name
matches = [
cs for cs in self.coord_systems
if cs.name == nameOfTargetCoordinateSystem
]
if len(matches) < 1:
raise "no coordinate system of the specified name found"
elif len(matches) > 1:
raise "more than one coord system with the specified name found"
else:
targetCoordinateSystem = matches[0]
connections = targetCoordinateSystem.transforms
#connections is a dictionary whose keys are CoordSystems
# check if there is at least one metric representation in a connected coord system
connectionsWithMetricRepresentation = {
key: val
for key, val in connections.items()
if key in self.metricRepresentations
}
# in most cases there will be only one match (e.g. the cartesian cs) but it is possible to have more
# so we chose the first that comes our way
firstCoordSystem = list(connectionsWithMetricRepresentation.keys())[0]
# now the question is to which of the metric representations of the patch it refers
symbols, equations = connectionsWithMetricRepresentation[
firstCoordSystem]
#compute the jacobian
J = targetCoordinateSystem.jacobian(firstCoordSystem, symbols)
pprint(J)
# the first column of the jacobian contains the components of the
# new base vectors in the old base
# therefor the components of the new metric tensor
# can be derived from the components of the old
# by J.transposeLU()*g_old*J
g_old = self.metricRepresentations[firstCoordSystem]
oldBases = g_old.bases
oB0 = oldBases[0] # first base
print("oldBases")
print(oldBases)
if isinstance(oB0, VectorFieldBase):
# the second base must be of the same type
# in this case also
newBase = VectorFieldBase("newBase", targetCoordinateSystem)
newBase.connect(oB0, J)
i, j = map(Idx, ['i', 'j'])
g_new = TensorIndexSet("g_new")
g_new[i, j] = g_old[i, j].rebase2([newBase, newBase])
return (g_new)
def _interactive_traversal(expr, stage):
if stage > 0:
print
cprint("Current expression (stage ", BYELLOW, stage, END, "):")
print BCYAN
pprint(expr)
print END
if isinstance(expr, Basic):
if expr.is_Add:
args = expr.as_ordered_terms()
elif expr.is_Mul:
args = expr.as_ordered_factors()
else:
args = expr.args
elif hasattr(expr, "__iter__"):
args = list(expr)
else:
return expr
n_args = len(args)
if not n_args:
return expr
for i, arg in enumerate(args):
cprint(GREEN, "[", BGREEN, i, GREEN, "] ", BLUE, type(arg), END)
pprint(arg)
print
if n_args == 1:
choices = '0'
else:
choices = '0-%d' % (n_args - 1)
try:
choice = raw_input("Your choice [%s,f,l,r,d,?]: " % choices)
except EOFError:
result = expr
print
else:
if choice == '?':
cprint(
RED,
"%s - select subexpression with the given index" % choices)
cprint(RED, "f - select the first subexpression")
cprint(RED, "l - select the last subexpression")
cprint(RED, "r - select a random subexpression")
cprint(RED, "d - done\n")
result = _interactive_traversal(expr, stage)
elif choice in ['d', '']:
result = expr
elif choice == 'f':
result = _interactive_traversal(args[0], stage + 1)
elif choice == 'l':
result = _interactive_traversal(args[-1], stage + 1)
elif choice == 'r':
result = _interactive_traversal(random.choice(args), stage + 1)
else:
try:
choice = int(choice)
except ValueError:
cprint(BRED,
"Choice must be a number in %s range\n" % choices)
result = _interactive_traversal(expr, stage)
else:
if choice < 0 or choice >= n_args:
cprint(BRED, "Choice must be in %s range\n" % choices)
result = _interactive_traversal(expr, stage)
else:
result = _interactive_traversal(
args[choice], stage + 1)
return result
def _interactive_traversal(expr, stage):
if stage > 0:
print
cprint("Current expression (stage ", BYELLOW, stage, END, "):")
print BCYAN
pprint(expr)
print END
if isinstance(expr, Basic):
if expr.is_Add:
args = expr.as_ordered_terms()
elif expr.is_Mul:
args = expr.as_ordered_factors()
else:
args = expr.args
elif hasattr(expr, "__iter__"):
args = list(expr)
else:
return expr
n_args = len(args)
if not n_args:
return expr
for i, arg in enumerate(args):
cprint(GREEN, "[", BGREEN, i, GREEN, "] ", BLUE, type(arg), END)
pprint(arg)
print
if n_args == 1:
choices = '0'
else:
choices = '0-%d' % (n_args-1)
try:
choice = raw_input("Your choice [%s,f,l,r,d,?]: " % choices)
except EOFError:
result = expr
print
else:
if choice == '?':
cprint(RED, "%s - select subexpression with the given index" % choices)
cprint(RED, "f - select the first subexpression")
cprint(RED, "l - select the last subexpression")
cprint(RED, "r - select a random subexpression")
cprint(RED, "d - done\n")
result = _interactive_traversal(expr, stage)
elif choice in ['d', '']:
result = expr
elif choice == 'f':
result = _interactive_traversal(args[0], stage+1)
elif choice == 'l':
result = _interactive_traversal(args[-1], stage+1)
elif choice == 'r':
result = _interactive_traversal(random.choice(args), stage+1)
else:
try:
choice = int(choice)
except ValueError:
cprint(BRED, "Choice must be a number in %s range\n" % choices)
result = _interactive_traversal(expr, stage)
else:
if choice < 0 or choice >= n_args:
cprint(BRED, "Choice must be in %s range\n" % choices)
result = _interactive_traversal(expr, stage)
else:
result = _interactive_traversal(args[choice], stage+1)
return result
The cnot performs the following transformation:
|00> -> |00>
|01> -> |01>
|10> -> |11>
|11> -> |10>
"""
theta = Symbol('theta')
X = np.array([[0, 1j], [-1j, 0]])
CNOT = Matrix(
np.array([[1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0], [0, 1, 0, 0]]))
I = Matrix(np.eye(2))
sigmax = Matrix(1j * X * theta)
sigmax_inv = Matrix(-1j * X * theta)
R = simplify(exp(sigmax))
R_ = simplify(exp(sigmax_inv))
IX = simplify(TensorProduct(I, R))
IX_ = simplify(TensorProduct(I, R_))
U = IX_ * CNOT * IX
M_ij = Matrix([[U[0, 0], U[0, 2]], [U[2, 0], U[2, 2]]])
pprint(simplify(M_ij))
Algumas funções são conhecidas:
- exponenciação: use ** em vez de ^
- multiplicação: sempre escreva explicitamente 2*a, nunca 2a
- algumas funções suportadas: cos(x), sin(x), ln(x), log(x), etc.
exemplo: 2*(ρ - ρl)*g / (9*a)
""")
ok = False
while not ok:
print("escreva a fórmula:\n")
try:
form = parse(input())
print("\nSua expressão:")
pprint(form)
if input("\nParece OK? (s/n)\n") == 's':
ok = True
except SyntaxError:
print("Há um problema com a fórmula... tente de novo")
print()
print(f"""Como você quer a formatação da incerteza?
(r): relativa (padrão) - a resposta sai na forma '{simb} * raiz de alguma coisa'
(a): absoluta - a resposta sai na forma 'raiz de alguma coisa'""")
if input() == 'a':
inc = incerteza_absoluta(form)
else:
inc = incerteza_relativa(form, simb)
print()
#from LAPM.linear_autonomous_pool_model import LinearAutonomousPoolModel
from sympy.printing import pprint
from sympy import var
var("lambda_1 lambda_2 t")
M=Matrix([[lambda_1,0,0],[1,lambda_1,0],[0,0,lambda_2]])
pprint(M.eigenvects())
var("t")
pprint(M.exp())
M.jordan_form()
pprint((M*t).exp())
T=Matrix([[1,2,0],[2,1,0],[1,1,1]])
T**-1
A=T*M*T**-1
from sympy import sqrt,exp,log,Symbol,oo,Rational,sin,cos,limit,I,pi,Mul
from sympy.core import basic
from sympy.printing import print_pygame, pprint
x=Symbol("x")
y=Symbol("y")
expressions = (
x**x,
x+y+x,
sin(x)**x,
sin(x)**cos(x),
sin(x)/(cos(x)**2 * x**x +(2*y)),
sin(x**2+exp(x)),
sqrt(exp(x)),
#print (1/cos(x)).series(x,10)
)
print "sympy print:"
for expr in expressions:
print expr
print ''
print "pygame print:"
for expr in expressions:
pprint(expr)
print_pygame(expr)
# Parameters
params.update(eq_params())
params.update({**vy(), **vq()})
# Forward operator
forward = forward_op(config.misc['symbolic'])
# Mapped backward operator
factor_x, factor_y, factor = factors(config.misc['symbolic'], config.num['λ'])
backward = eq.map_operator(forward, f, factor)
forward, backward = evaluate([forward, backward])
factor_x, factor_y, factor = evaluate([factor_x, factor_y, factor])
if config.misc['verbose']:
pprint(forward)
pprint(backward)
for key in params:
params[key].value = params[key].eval()
degree = config.num['degree']
n_points_num = config.num['n_points_num']
# }}}
# DEFINE QUADRATURES {{{
vprint("Defining quadratures")
def compute_quads():
μx = params['μx'].value
import numpy
import sympy
from sympy.printing import pprint
from sympy import init_printing
from matplotlib import pyplot
init_printing(use_latex=True)
x, nu, t = sympy.symbols(
'x nu t') # Setting symbols for x, nu, and t from symbol library.
# An example equation to show the look of output.
phi = (sympy.exp(-(x - 4 * t)**2 / (4 * nu * (t + 1))) +
sympy.exp(-(x - 4 * t - 2 * numpy.pi)**2 / (4 * nu * (t + 1))))
pprint("Phi") # Labeling the following equation.
pprint(
phi) # Prints Phi using pretty print, outputs the result in equation form.
pprint("Phi prime") # Labeling the following equation.
phiprime = phi.diff(x) # Solves for the derivative of phi.
pprint(
phiprime
) # Prints Phi' using pretty print, outputs the result in equation form.
from sympy.utilities.lambdify import lambdify
u = -2 * nu * (phiprime / phi) + 4
pprint(u)
ufunc = lambdify((t, x, nu), u)
pprint(ufunc(1, 4, 3))
# Burger's Equation
nx = 101 # Set number of grid points in x dir