You mean I should make independent variables acceptable as args to
minimizer,?
Zohreh Karimzadeh
Contact me on
+989102116325
and at
z.karimza...@gmail.com
🌧️🌍🌱
On Sat, 20 Aug 2022, 09:35 Peter Stahlecker, <peter.stahlec...@gmail.com>
wrote:
> But would then this not be a simpler way, without any lambdify ?
>
> def func_to_be_minimized(alpha, beta, gamma, eta, L, K, VA):
> ……
> ……
> return np.sum(…….)
>
> X0 = (…..)
> args = (L. K, VA)
> resultat = minimize(func_to_be_minimized, X0, args = args, bounds =…)
>
> args contains the variables of your function, which will not be minimized.
>
> You may have to play around with args = (L. K, VA) a bit, because minimize
> ‚hands over‘ whatever is in args to your function in a certain way, and
> your function must accept them this way. (I never know exactly how to do
> it, as I do not use it often at all, so I just play around until it works)
>
>
>
> On Sat 20. Aug 2022 at 11:48 Zohreh Karimzadeh <z.karimza...@gmail.com>
> wrote:
>
>> Exactly
>>
>> Zohreh Karimzadeh
>>
>> Contact me on
>> +989102116325
>> and at
>> z.karimza...@gmail.com
>> 🌧️🌍🌱
>>
>>
>> On Sat, 20 Aug 2022, 06:15 Peter Stahlecker, <peter.stahlec...@gmail.com>
>> wrote:
>>
>>> Maybe a dumb question from my part:
>>>
>>> do I understand you correctly:
>>>
>>> For *given* L, K, VA you try to find the alpha, beta, gamma, eta which
>>> minimize
>>> the function np.sum(….) ?
>>>
>>> Is my understanding correct?
>>>
>>> On Fri 19. Aug 2022 at 22:11 Zohreh Karimzadeh <z.karimza...@gmail.com>
>>> wrote:
>>>
>>>> I am new at python and using your comment seems hard to me could
>>>> possibly let me know know it by example or any key words if is there.
>>>>
>>>> Zohreh Karimzadeh
>>>>
>>>> Contact me on
>>>> +989102116325
>>>> and at
>>>> z.karimza...@gmail.com
>>>> 🌧️🌍🌱
>>>>
>>>>
>>>> On Fri, 19 Aug 2022, 00:03 Aaron Meurer, <asmeu...@gmail.com> wrote:
>>>>
>>>>> Instead of generating a separate lambdified function for every input,
>>>>> you may find it simpler to lambdify a single function with your params as
>>>>> extra symbolic parameters, then pass those in using the args() argument to
>>>>> minimize().
>>>>>
>>>>> Aaron Meurer
>>>>>
>>>>> On Thu, Aug 18, 2022 at 4:35 AM Zohreh Karimzadeh <
>>>>> z.karimza...@gmail.com> wrote:
>>>>>
>>>>>> the following code is ok when expression is passed as :
>>>>>>
>>>>>> import numpy as np
>>>>>> from scipy.optimize import minimize, curve_fit
>>>>>> from lmfit import Model, Parameters
>>>>>>
>>>>>> L = np.array([0.299, 0.295, 0.290, 0.284, 0.279, 0.273, 0.268, 0.262,
>>>>>> 0.256, 0.250])
>>>>>> K = np.array([2.954, 3.056, 3.119, 3.163, 3.215, 3.274, 3.351, 3.410,
>>>>>> 3.446, 3.416])
>>>>>> VA = np.array([0.919, 0.727, 0.928, 0.629, 0.656, 0.854, 0.955, 0.981,
>>>>>> 0.908, 0.794])
>>>>>>
>>>>>>
>>>>>> def f(param):
>>>>>> gamma = param[0]
>>>>>> alpha = param[1]
>>>>>> beta = param[2]
>>>>>> eta = param[3]
>>>>>> VA_est = gamma - (1 / eta) * np.log(alpha * L ** -eta + beta * K **
>>>>>> -eta)
>>>>>>
>>>>>> return np.sum((np.log(VA) - VA_est) ** 2)
>>>>>>
>>>>>>
>>>>>> bnds = [(1, np.inf), (0, 1), (0, 1), (-1, np.inf)]
>>>>>> x0 = (1, 0.01, 0.98, 1)
>>>>>> result = minimize(f, x0, bounds=bnds)
>>>>>> print(result.message)
>>>>>> print(result.x[0], result.x[1], result.x[2], result.x[3])
>>>>>>
>>>>>> but when the expression is passed as the following way:
>>>>>>
>>>>>> import numpy as np
>>>>>> import sympy as sp
>>>>>> from scipy.optimize import minimize, curve_fit
>>>>>> from lmfit import Model, Parameters
>>>>>>
>>>>>> L = np.array([0.299, 0.295, 0.290, 0.284, 0.279, 0.273, 0.268, 0.262,
>>>>>> 0.256, 0.250])
>>>>>> K = np.array([2.954, 3.056, 3.119, 3.163, 3.215, 3.274, 3.351, 3.410,
>>>>>> 3.446, 3.416])
>>>>>> VA = np.array([0.919, 0.727, 0.928, 0.629, 0.656, 0.854, 0.955, 0.981,
>>>>>> 0.908, 0.794])
>>>>>>
>>>>>>
>>>>>> def f(param):
>>>>>> gamma, alpha, beta, eta = sp.symbols('gamma, alpha, beta, eta')
>>>>>> gamma = param[0]
>>>>>> alpha = param[1]
>>>>>> beta = param[2]
>>>>>> eta = param[3]
>>>>>> Vi_est = gamma - (1 / eta) * sp.log(alpha * L ** -eta + beta * K **
>>>>>> -eta)
>>>>>> Vlam_est = sp.lambdify((gamma, alpha, beta, eta), Vi_est)
>>>>>>
>>>>>> return np.sum((np.log(VA) - Vlam_est) ** 2)
>>>>>>
>>>>>>
>>>>>> bnds = [(1, np.inf), (0, 1), (0, 1), (-1, np.inf)]
>>>>>> x0 = (1, 0.01, 0.98, 1)
>>>>>>
>>>>>> result = minimize(f, x0, bounds=bnds)
>>>>>>
>>>>>> print(result.message)
>>>>>> print(result.x[0], result.x[1], result.x[2], result.x[3])
>>>>>>
>>>>>>
>>>>>> I face difficulty:
>>>>>> *********************************************
>>>>>> Traceback (most recent call last):
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\core\cache.py",
>>>>>> line 70, in wrapper
>>>>>> retval = cfunc(*args, **kwargs)
>>>>>> TypeError: unhashable type: 'numpy.ndarray'
>>>>>>
>>>>>> During handling of the above exception, another exception occurred:
>>>>>>
>>>>>> Traceback (most recent call last):
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\core\cache.py",
>>>>>> line 70, in wrapper
>>>>>> retval = cfunc(*args, **kwargs)
>>>>>> TypeError: unhashable type: 'numpy.ndarray'
>>>>>>
>>>>>> During handling of the above exception, another exception occurred:
>>>>>>
>>>>>> Traceback (most recent call last):
>>>>>> File
>>>>>> "F:\Zohreh\MainZohreh\postdoc-field\CSU\pythonProject\fit_test_2.py",
>>>>>> line
>>>>>> 26, in <module>
>>>>>> result = minimize(f, x0, bounds=bnds)
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_minimize.py",
>>>>>> line 692, in minimize
>>>>>> res = _minimize_lbfgsb(fun, x0, args, jac, bounds,
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_lbfgsb_py.py",
>>>>>> line 308, in _minimize_lbfgsb
>>>>>> sf = _prepare_scalar_function(fun, x0, jac=jac, args=args,
>>>>>> epsilon=eps,
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_optimize.py",
>>>>>> line 263, in _prepare_scalar_function
>>>>>> sf = ScalarFunction(fun, x0, args, grad, hess,
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_differentiable_functions.py",
>>>>>> line 158, in __init__
>>>>>> self._update_fun()
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_differentiable_functions.py",
>>>>>> line 251, in _update_fun
>>>>>> self._update_fun_impl()
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_differentiable_functions.py",
>>>>>> line 155, in update_fun
>>>>>> self.f = fun_wrapped(self.x)
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_differentiable_functions.py",
>>>>>> line 137, in fun_wrapped
>>>>>> fx = fun(np.copy(x), *args)
>>>>>> File
>>>>>> "F:\Zohreh\MainZohreh\postdoc-field\CSU\pythonProject\fit_test_2.py",
>>>>>> line
>>>>>> 17, in f
>>>>>> Vi_est = gamma - (1 / eta) * sp.log(alpha * L ** -eta + beta * K
>>>>>> ** -eta)
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\core\cache.py",
>>>>>> line 74, in wrapper
>>>>>> retval = func(*args, **kwargs)
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\core\function.py",
>>>>>> line 476, in __new__
>>>>>> result = super().__new__(cls, *args, **options)
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\core\cache.py",
>>>>>> line 74, in wrapper
>>>>>> retval = func(*args, **kwargs)
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\core\function.py",
>>>>>> line 288, in __new__
>>>>>> evaluated = cls.eval(*args)
>>>>>> File
>>>>>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\functions\elementary\exponential.py",
>>>>>> line 718, in eval
>>>>>> coeff = arg.as_coefficient(I)
>>>>>> AttributeError: 'ImmutableDenseNDimArray' object has no attribute
>>>>>> 'as_coefficient'
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> Zohreh Karimzadeh
>>>>>> https://www.researchgate.net/profile/Zohreh-Karimzadeh
>>>>>> Skype Name 49a52224a8b6b38b
>>>>>> Twitter Account @zohrehkarimzad1
>>>>>> z.karimza...@gmail.com
>>>>>> +989102116325
>>>>>> ((((((((((((((((Value Water)))))))))))))))
>>>>>>
>>>>>> Zohreh Karimzadeh
>>>>>> *https://www.researchgate.net/profile/Zohreh-Karimzadeh*
>>>>>> <https://www.researchgate.net/profile/Zohreh-Karimzadeh>
>>>>>> Skype Name 49a52224a8b6b38b
>>>>>> Twitter Account @zohrehkarimzad1
>>>>>> z.karimza...@gmail.com
>>>>>> +989102116325
>>>>>>
>>>>>> ((((((((((((((((Value Water)))))))))))))))
>>>>>>
>>>>>>
>>>>>> On Thu, Aug 18, 2022 at 10:42 AM Peter Stahlecker <
>>>>>> peter.stahlec...@gmail.com> wrote:
>>>>>>
>>>>>>> I use lambdify quite a bit, on rather large expressions.
>>>>>>> Basically, it always works like this for me:
>>>>>>>
>>>>>>> import sympy as sm
>>>>>>> x1, x2, …, xn = sm.symbols(‚x1, x2, ….., xn‘)
>>>>>>> ….
>>>>>>> …
>>>>>>> expr = some expression of generally with me: sm.sin, sm.cos, sm.exp,
>>>>>>> sm.sqrt,
>>>>>>> sm.Heaviside, etc..
>>>>>>> This expression may have 50,000 terms, may be an (axb) matrix,
>>>>>>> whatever.
>>>>>>>
>>>>>>> expr_lam = sm.lambdify([x1, x2, …,xn], expr)
>>>>>>>
>>>>>>> Now I can evaluate expr_lam(…) like I would evaluate any numpy
>>>>>>> function.
>>>>>>>
>>>>>>> I have no idea, what expr_lam looks like, I would not know how to
>>>>>>> look at it.
>>>>>>> I assume, it converts sm.sin(..) to np.sin(…), etc
>>>>>>>
>>>>>>> This is how it works for me.
>>>>>>> As I do not really understand your points, like ‚dynamically
>>>>>>> created‘, ‚parse and subs‘, this may be of not help at all for you.
>>>>>>>
>>>>>>> Peter
>>>>>>>
>>>>>>>
>>>>>>> On Thu 18. Aug 2022 at 09:21 Zohreh Karimzadeh <
>>>>>>> z.karimza...@gmail.com> wrote:
>>>>>>>
>>>>>>>> Before run I import sp.sqrt or sp.exp but after run they get
>>>>>>>> disappeared. My expression is big and dynamically created and not
>>>>>>>> possible to parse and subs np.exp or sp.exp.
>>>>>>>>
>>>>>>>> Zohreh Karimzadeh
>>>>>>>>
>>>>>>>> Contact me on
>>>>>>>> +989102116325
>>>>>>>> and at
>>>>>>>> z.karimza...@gmail.com
>>>>>>>> 🌧️🌍🌱
>>>>>>>>
>>>>>>>>
>>>>>>>> On Thu, 18 Aug 2022, 01:17 Aaron Meurer, <asmeu...@gmail.com>
>>>>>>>> wrote:
>>>>>>>>
>>>>>>>>> Your expression uses "sqrt" but you haven't imported it from
>>>>>>>>> anywhere, since you only did "import sympy as sp". You need to use
>>>>>>>>> sp.sqrt.
>>>>>>>>>
>>>>>>>>> Aaron Meurer
>>>>>>>>>
>>>>>>>>> On Wed, Aug 17, 2022 at 11:02 AM Zohreh Karimzadeh <
>>>>>>>>> z.karimza...@gmail.com> wrote:
>>>>>>>>>
>>>>>>>>>> Here is my code:
>>>>>>>>>>
>>>>>>>>>> import matplotlib.pyplot as plt
>>>>>>>>>> import numpy as np
>>>>>>>>>> import sympy as sp
>>>>>>>>>> import pandas as pd
>>>>>>>>>> #exp_NaCl path: F:\Zohreh\MainZohreh\postdoc-field\CSU\Duplicat_Pure
>>>>>>>>>> df =
>>>>>>>>>> pd.read_excel(r'F:\Zohreh\MainZohreh\postdoc-field\CSU\Duplicat_Pure\data.xlsx',
>>>>>>>>>> sheet_name='NaCl_exp')
>>>>>>>>>> XNa = df['XNa']
>>>>>>>>>> XCl = df['XCl']
>>>>>>>>>> Xwater = df['Xwater']
>>>>>>>>>> Y = df['gama_x']
>>>>>>>>>> L=['WwaterNaCl', 'UwaterNaCl', 'VwaterNaCl', 'XCl', 'XNa', 'Xwater',
>>>>>>>>>> 'BNaCl']
>>>>>>>>>> for j in range(len(L)):
>>>>>>>>>> locals()[L[j]] = sp.symbols(L[j])
>>>>>>>>>> expr =
>>>>>>>>>> -0.0118343195266272*BNaCl*XCl*XNa*(-2*(9.19238815542512*sqrt(XNa) +
>>>>>>>>>> 9.19238815542512*sqrt(XCl + XNa) +
>>>>>>>>>> 1)*exp(-9.19238815542512*sqrt(XNa) - 9.19238815542512*sqrt(XCl +
>>>>>>>>>> XNa)) + 2)/((XCl + XNa)*(sqrt(XNa) + sqrt(XCl + XNa))**2) +
>>>>>>>>>> 0.00591715976331361*BNaCl*XCl*(-2*(9.19238815542512*sqrt(XNa) +
>>>>>>>>>> 9.19238815542512*sqrt(XCl + XNa) +
>>>>>>>>>> 1)*exp(-9.19238815542512*sqrt(XNa) - 9.19238815542512*sqrt(XCl +
>>>>>>>>>> XNa)) + 2)/(sqrt(XNa) + sqrt(XCl + XNa))**2 +
>>>>>>>>>> 0.00591715976331361*BNaCl*XNa*(-2*(9.19238815542512*sqrt(XNa) +
>>>>>>>>>> 9.19238815542512*sqrt(XCl + XNa) +
>>>>>>>>>> 1)*exp(-9.19238815542512*sqrt(XNa) - 9.19238815542512*sqrt(XCl +
>>>>>>>>>> XNa)) + 2)/(sqrt(XNa) + sqrt(XCl + XNa))**2 -
>>>>>>>>>> 1.0*Cl*WwaterNaCl*Xwater*(0.5*XCl + 0.5*XNa + 0.5)/XCl -
>>>>>>>>>> 0.5*Cl*WwaterNaCl/XCl - 4.0*UwaterNaCl*XCl*XNa*Xwater +
>>>>>>>>>> 2.0*UwaterNaCl*XCl*Xwater + 2.0*UwaterNaCl*XNa*Xwater -
>>>>>>>>>> 4.0*UwaterNaCl*XNa - 6.0*VwaterNaCl*XCl*XNa*Xwater**2 -
>>>>>>>>>> 4.0*VwaterNaCl*XCl*Xwater**2 + 2.0*VwaterNaCl*XNa*Xwater**2 -
>>>>>>>>>> 1.0*WwaterNaCl*Xwater*(0.5*XCl + 0.5*XNa + 0.5) +
>>>>>>>>>> 2.0*WwaterNaCl*Xwater - 0.5*WwaterNaCl -
>>>>>>>>>> 1.45739430799067*(0.707106781186548*sqrt(XNa) +
>>>>>>>>>> 0.707106781186548*sqrt(XCl + XNa))*(-XCl - XNa +
>>>>>>>>>> 1)/(9.19238815542512*sqrt(XNa) + 9.19238815542512*sqrt(XCl + XNa) +
>>>>>>>>>> 1) - 1.45739430799067*(0.707106781186548*sqrt(XNa) +
>>>>>>>>>> 0.707106781186548*sqrt(XCl + XNa))*(-1.4142135623731*sqrt(XNa) -
>>>>>>>>>> 1.4142135623731*sqrt(XCl + XNa) + 1)/(9.19238815542512*sqrt(XNa) +
>>>>>>>>>> 9.19238815542512*sqrt(XCl + XNa) + 1) -
>>>>>>>>>> 0.448429017843282*log(9.19238815542512*sqrt(XNa) +
>>>>>>>>>> 9.19238815542512*sqrt(XCl + XNa) + 1)
>>>>>>>>>> model_func = sp.lambdify(L, expr )
>>>>>>>>>>
>>>>>>>>>> def f(param):
>>>>>>>>>> BNaCl = param[0]
>>>>>>>>>> UwaterNaCl = param[1]
>>>>>>>>>> VwaterNaCl = param[2]
>>>>>>>>>> WwaterNaCl = param[3]
>>>>>>>>>> Y_est = model_func
>>>>>>>>>> return np.sum((np.log(Y) - Y_est)**2)
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> bnds = [(1, np.inf), (0, 1), (0, 1), (-1, np.inf)]
>>>>>>>>>> x0 = (1, 0.01, 0.98, 1)
>>>>>>>>>> con = {"type": "eq", "fun": c}
>>>>>>>>>>
>>>>>>>>>> result = minimize(f, x0, bounds=bnds)
>>>>>>>>>>
>>>>>>>>>> print(result.fun)
>>>>>>>>>> print(result.message)
>>>>>>>>>> print(result.x[0], result.x[1], result.x[2], result.x[3])
>>>>>>>>>>
>>>>>>>>>> while I got :
>>>>>>>>>> NameError: name 'sqrt' is not defined
>>>>>>>>>>
>>>>>>>>>> Zohreh Karimzadeh
>>>>>>>>>> *https://www.researchgate.net/profile/Zohreh-Karimzadeh*
>>>>>>>>>> <https://www.researchgate.net/profile/Zohreh-Karimzadeh>
>>>>>>>>>> Skype Name 49a52224a8b6b38b
>>>>>>>>>> Twitter Account @zohrehkarimzad1
>>>>>>>>>> z.karimza...@gmail.com
>>>>>>>>>> +989102116325
>>>>>>>>>>
>>>>>>>>>> ((((((((((((((((Value Water)))))))))))))))
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> On Wed, Aug 17, 2022 at 7:46 PM Peter Stahlecker <
>>>>>>>>>> peter.stahlec...@gmail.com> wrote:
>>>>>>>>>>
>>>>>>>>>>> I use lambdify(....) a lot, but always like this:
>>>>>>>>>>>
>>>>>>>>>>> x = sympy.symbols('x')
>>>>>>>>>>> expr = symy.S(10.) * sympy.sqrt(x)
>>>>>>>>>>> expr_lam = sympy.lambdify([x], expr)
>>>>>>>>>>>
>>>>>>>>>>> a = expr_lam(10.)
>>>>>>>>>>>
>>>>>>>>>>> This seems to work for me.
>>>>>>>>>>>
>>>>>>>>>>> On Wed 17. Aug 2022 at 20:38, Zohreh Karimzadeh <
>>>>>>>>>>> z.karimza...@gmail.com> wrote:
>>>>>>>>>>>
>>>>>>>>>>>> Dear sympy group
>>>>>>>>>>>> Thanks for your sympy.
>>>>>>>>>>>>
>>>>>>>>>>>> I am working on a code, after creating my big expression using
>>>>>>>>>>>> sympy it includes sqrt.
>>>>>>>>>>>>
>>>>>>>>>>>> I need to lambdify my expression to make it consistent with
>>>>>>>>>>>> numpy and other suffs.
>>>>>>>>>>>>
>>>>>>>>>>>> expr =10 * sp.sqrt(sp.symbols('x'))
>>>>>>>>>>>>
>>>>>>>>>>>> model_func = sp.lambdify('x', expr)
>>>>>>>>>>>>
>>>>>>>>>>>> But I found my expression after lambdifying becomes somethings
>>>>>>>>>>>> like this:
>>>>>>>>>>>>
>>>>>>>>>>>> 10*sqrt(x)
>>>>>>>>>>>>
>>>>>>>>>>>> while I need :
>>>>>>>>>>>>
>>>>>>>>>>>> 10*numpy.sqrt(x)
>>>>>>>>>>>>
>>>>>>>>>>>> Could possibly let me know how get sqrt to work with numpy?
>>>>>>>>>>>>
>>>>>>>>>>>> Regards,
>>>>>>>>>>>>
>>>>>>>>>>>> Zohreh
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> --
>>>>>>>>>>>> You received this message because you are subscribed to the
>>>>>>>>>>>> Google Groups "sympy" group.
>>>>>>>>>>>> To unsubscribe from this group and stop receiving emails from
>>>>>>>>>>>> it, send an email to sympy+unsubscr...@googlegroups.com.
>>>>>>>>>>>> To view this discussion on the web visit
>>>>>>>>>>>> https://groups.google.com/d/msgid/sympy/1f0b313f-31c5-402e-991e-142a556016f4n%40googlegroups.com
>>>>>>>>>>>> <https://groups.google.com/d/msgid/sympy/1f0b313f-31c5-402e-991e-142a556016f4n%40googlegroups.com?utm_medium=email&utm_source=footer>
>>>>>>>>>>>> .
>>>>>>>>>>>>
>>>>>>>>>>> --
>>>>>>>>>>> Best regards,
>>>>>>>>>>>
>>>>>>>>>>> Peter Stahlecker
>>>>>>>>>>>
>>>>>>>>>>> --
>>>>>>>>>>> You received this message because you are subscribed to the
>>>>>>>>>>> Google Groups "sympy" group.
>>>>>>>>>>> To unsubscribe from this group and stop receiving emails from
>>>>>>>>>>> it, send an email to sympy+unsubscr...@googlegroups.com.
>>>>>>>>>>> To view this discussion on the web visit
>>>>>>>>>>> https://groups.google.com/d/msgid/sympy/CABKqA0ZoGwsadsk4SWCbJVMbCDwXcO_gNGumJH00GAeEFod7Cw%40mail.gmail.com
>>>>>>>>>>> <https://groups.google.com/d/msgid/sympy/CABKqA0ZoGwsadsk4SWCbJVMbCDwXcO_gNGumJH00GAeEFod7Cw%40mail.gmail.com?utm_medium=email&utm_source=footer>
>>>>>>>>>>> .
>>>>>>>>>>>
>>>>>>>>>> --
>>>>>>>>>> You received this message because you are subscribed to the
>>>>>>>>>> Google Groups "sympy" group.
>>>>>>>>>> To unsubscribe from this group and stop receiving emails from it,
>>>>>>>>>> send an email to sympy+unsubscr...@googlegroups.com.
>>>>>>>>>> To view this discussion on the web visit
>>>>>>>>>> https://groups.google.com/d/msgid/sympy/CA%2B1XYLPRvXZ6jiJbUS_xpWNKqMuUH7Kt5evue%2BwKEwDMvGekBQ%40mail.gmail.com
>>>>>>>>>> <https://groups.google.com/d/msgid/sympy/CA%2B1XYLPRvXZ6jiJbUS_xpWNKqMuUH7Kt5evue%2BwKEwDMvGekBQ%40mail.gmail.com?utm_medium=email&utm_source=footer>
>>>>>>>>>> .
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>> --
>>>>>>>>> You received this message because you are subscribed to the Google
>>>>>>>>> Groups "sympy" group.
>>>>>>>>> To unsubscribe from this group and stop receiving emails from it,
>>>>>>>>> send an email to sympy+unsubscr...@googlegroups.com.
>>>>>>>>> To view this discussion on the web visit
>>>>>>>>> https://groups.google.com/d/msgid/sympy/CAKgW%3D6JfUmU7Uu%2BSrcA1STxVvWWm7bGWE%3Dit8CTchksTC0Qk7g%40mail.gmail.com
>>>>>>>>> <https://groups.google.com/d/msgid/sympy/CAKgW%3D6JfUmU7Uu%2BSrcA1STxVvWWm7bGWE%3Dit8CTchksTC0Qk7g%40mail.gmail.com?utm_medium=email&utm_source=footer>
>>>>>>>>> .
>>>>>>>>>
>>>>>>>> --
>>>>>>>> You received this message because you are subscribed to the Google
>>>>>>>> Groups "sympy" group.
>>>>>>>> To unsubscribe from this group and stop receiving emails from it,
>>>>>>>> send an email to sympy+unsubscr...@googlegroups.com.
>>>>>>>> To view this discussion on the web visit
>>>>>>>> https://groups.google.com/d/msgid/sympy/CA%2B1XYLPiCR%3DS2Fac3FZtjMpspqB7BRKtYEi45BVWPjkizVbNvw%40mail.gmail.com
>>>>>>>> <https://groups.google.com/d/msgid/sympy/CA%2B1XYLPiCR%3DS2Fac3FZtjMpspqB7BRKtYEi45BVWPjkizVbNvw%40mail.gmail.com?utm_medium=email&utm_source=footer>
>>>>>>>> .
>>>>>>>>
>>>>>>> --
>>>>>>> Best regards,
>>>>>>>
>>>>>>> Peter Stahlecker
>>>>>>>
>>>>>>> --
>>>>>>> You received this message because you are subscribed to the Google
>>>>>>> Groups "sympy" group.
>>>>>>> To unsubscribe from this group and stop receiving emails from it,
>>>>>>> send an email to sympy+unsubscr...@googlegroups.com.
>>>>>>> To view this discussion on the web visit
>>>>>>> https://groups.google.com/d/msgid/sympy/CABKqA0b%3DF0akMH4oyg5%2By9dGvgrf_vvVJTnVhVduMP1f%2Bp1pFw%40mail.gmail.com
>>>>>>> <https://groups.google.com/d/msgid/sympy/CABKqA0b%3DF0akMH4oyg5%2By9dGvgrf_vvVJTnVhVduMP1f%2Bp1pFw%40mail.gmail.com?utm_medium=email&utm_source=footer>
>>>>>>> .
>>>>>>>
>>>>>> --
>>>>>> You received this message because you are subscribed to the Google
>>>>>> Groups "sympy" group.
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>>>>>> <https://groups.google.com/d/msgid/sympy/CA%2B1XYLMK-fgpxc71GYzue5gJvd%3Dfj2sV6Dvhj8zrmVpPhiVk%2Bw%40mail.gmail.com?utm_medium=email&utm_source=footer>
>>>>>> .
>>>>>
>>>>>
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>>>>> <https://groups.google.com/d/msgid/sympy/CAKgW%3D6%2BBQo_nWKJtbxPmi40V0Y6OgAaT78jSNSWKnwW8L3qmZQ%40mail.gmail.com?utm_medium=email&utm_source=footer>
>>>>> .
>>>>>
>>>> --
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>>>> <https://groups.google.com/d/msgid/sympy/CA%2B1XYLNg4ScH5bK%3DeW7%3DVL_2%2BWdWT4kf5ud41LR5hh%2BJkCJR2g%40mail.gmail.com?utm_medium=email&utm_source=footer>
>>>> .
>>>>
>>> --
>>> Best regards,
>>>
>>> Peter Stahlecker
>>>
>> --
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>>> <https://groups.google.com/d/msgid/sympy/CABKqA0aP_Rx9rDQ-%2BumFq_9W8C4LDTRMGrFif_5GZ_dRTpA2VA%40mail.gmail.com?utm_medium=email&utm_source=footer>
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>> <https://groups.google.com/d/msgid/sympy/CA%2B1XYLOmpYg0iHjow4D5hvp_%3DFSfmKEgaWBiDFoweEX4Vo4jxw%40mail.gmail.com?utm_medium=email&utm_source=footer>
>> .
>>
> --
> Best regards,
>
> Peter Stahlecker
>
> --
> You received this message because you are subscribed to the Google Groups
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> .
>
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