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
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>>>>>>>>> <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
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>>>>>>>> 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
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>>>>>>>> <https://groups.google.com/d/msgid/sympy/CA%2B1XYLPRvXZ6jiJbUS_xpWNKqMuUH7Kt5evue%2BwKEwDMvGekBQ%40mail.gmail.com?utm_medium=email&utm_source=footer>
>>>>>>>> .
>>>>>>>
>>>>>>>
>>>>>>>> --
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>>>>>>> Groups "sympy" group.
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>>>>>>> send an email to sympy+unsubscr...@googlegroups.com.
>>>>>>> To view this discussion on the web visit
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>>>>>>> <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
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>>>>> <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
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>>>> 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%2B1XYLMK-fgpxc71GYzue5gJvd%3Dfj2sV6Dvhj8zrmVpPhiVk%2Bw%40mail.gmail.com
>>>> <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
>
> --
> You received this message because you are subscribed to the Google Groups
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