The only way you always guarantee get right answer is to provide the ChatGPT two correct knowledge, but ask it to pick the better one. For solving the math problems, I think that these sort of encoding may be helpful.
On Friday, March 31, 2023 at 7:23:41 AM UTC+9 asme...@gmail.com wrote: > It's the same as the example I showed above. It basically can't do math > (or really, it can't even do arithmetic). It seems to have an especially > hard time with fractions. > > I asked it to generate code using Python to compute the norm or [1/2, 1, > 1]. It was able to generate accurate code using pure Python, NumPy, and > SymPy to compute the norm. In each case it also confidently claimed the > output would be something wrong like 1.5811388300841898 or sqrt(3)/2, but > if you actually copy and run the code it gives the correct result. My only > gripe with the code it generated is it unnecessarily used CoordSys3D when > it could just use Matrix.norm, although that's just random and if you ask > it the same thing it might not do that. > > Aaron Meurer > > On Thu, Mar 30, 2023 at 4:03 PM Chris Smith <smi...@gmail.com> wrote: > >> Use with caution -- the set-up can be right and the answer wrong as in: >> >> What is the norm of vector (1/2, 1, 1)? >> >> The norm of a vector is defined as the length or magnitude of the vector. >> The norm of a vector v = (v1, v2, ..., vn) is given by the formula: >> >> ||v|| = sqrt(v1^2 + v2^2 + ... + vn^2) >> >> So, for the vector (1/2, 1, 1), the norm is: >> >> || (1/2, 1, 1) || = sqrt( (1/2)^2 + 1^2 + 1^2 ) = sqrt( 1/4 + 1 + 1 ) = >> sqrt( 6.25 ) = 2.5 >> >> Therefore, the norm of the vector (1/2, 1, 1) is 2.5. >> The answer: `assert sqrt(sum([i**2 for i in (1/2,1,1)])) == 1.5` >> On Wednesday, March 29, 2023 at 7:48:48 AM UTC-5 e...@tssfl.co wrote: >> >>> ChatGPT is quite impressive! For those who do not have access to ChatGPT >>> and/or Python, can use it with SageCell at >>> https://tssfl.com/viewtopic.php?t=6651 >>> >>> SageCell supports countless Python libraries including SymPy. >>> >>> I tested ChatGPT with a number of examples, it tries solve even complex >>> maths problems, e.g., : >>> >>> Example 1: >>> >>> Find a 4x4 matrix which is a Jordan canonical form of the matrix A = >>> matrix([[5, -1, 0, 0],[9, -1, 0, 0],[0, 0, 7, -2], [0, 0, 12, -3]]). >>> >>> assistant: >>> >>> To find the Jordan canonical form of a matrix, we need to find the >>> eigenvalues and eigenvectors, and then use them to construct the Jordan >>> blocks. >>> >>> First, we find the eigenvalues of A by solving for the determinant of (A - >>> lambda*I), where I is the identity matrix: >>> >>> |5 - lambda -1 0 0 | >>> |9 -1 - lambda 0 0 | >>> |0 0 7 - lambda -2 | >>> |0 0 12 -3 - lambda| >>> >>> = (5 - lambda)(-1 - lambda)(7 - lambda)(-3 - lambda) - 9*(-1)*(7 - >>> lambda)*(12) >>> = lambda^4 - 8*lambda^3 + 11*lambda^2 + 38*lambda - 360 >>> = (lambda - 10)(lambda - 5)(lambda + 3)(lambda - 4) >>> >>> Therefore, the eigenvalues are: lambda1 = 10, lambda2 = 5, lambda3 = -3, >>> lambda4 = 4. >>> >>> Next, we find the eigenvectors corresponding to each eigenvalue. For >>> lambda1 = 10, we solve (A - 10*I)x = 0: >>> >>> | -5 -1 0 0 | >>> | 9 -11 0 0 | >>> | 0 0 -3 -2 | >>> | 0 0 12 -13 | >>> >>> By performing elementary row operations to reduce this matrix, we obtain: >>> >>> | 1 0 0 1 | >>> | 0 1 0 3/2 | >>> | 0 0 1 2 | >>> | 0 0 0 0 | >>> >>> So the eigenvector corresponding to lambda1 is x1 = [1, -3/2, -2, 1]. >>> >>> Similarly, for lambda2 = 5, we solve (A - 5*I)x = 0: >>> >>> | 0 -1 0 0 | >>> | 9 -6 0 0 | >>> | 0 0 2 -2 | >>> | 0 0 12 -8 | >>> >>> By performing elementary row operations to reduce this matrix, we obtain: >>> >>> | 1 -3 0 0 | >>> | 0 0 1 -1 | >>> | 0 0 0 0 | >>> | 0 0 0 0 | >>> >>> So the eigenvector corresponding to lambda2 is x2 = [3, 1, 0, 0]. >>> >>> For lambda3 = -3, we solve (A + 3*I)x = 0: >>> >>> | 8 -1 0 0 | >>> | 9 -2 0 0 | >>> | 0 0 4 -2 | >>> | 0 0 12 -6 | >>> >>> By performing elementary row operations to reduce this matrix, we obtain: >>> >>> | 1 -1/2 0 0 | >>> | 0 0 1 -1/2 | >>> | 0 0 0 0 | >>> | 0 0 0 0 | >>> >>> So the eigenvector corresponding to lambda3 is x3 = [1/2, 1, 1/2, 0]. >>> >>> Finally, for lambda4 = 4, we solve (A - 4*I)x = 0: >>> >>> | 1 -1 0 0 | >>> | 9 -5 0 0 | >>> | 0 0 3 -2 | >>> | 0 0 12 -7 | >>> >>> By performing elementary row operations to reduce this matrix, we obtain: >>> >>> | 1 -1 0 0 | >>> | 0 0 1 -2/3 | >>> | 0 0 0 0 | >>> | 0 0 0 0 | >>> >>> So the eigenvector corresponding to lambda4 is x4 = [1, 1/3, 0, 0]. >>> >>> We can now construct the Jordan canonical form of the matrix A by arranging >>> these eigenvectors in blocks, as follows: >>> >>> J = [10 1 0 0 ] >>> [0 10 0 0 ] >>> [0 0 -3 1 ] >>> [0 0 0 4 ] >>> >>> This is the desired 4x4 Jordan canonical form of A. >>> >>> Example 2: >>> >>> Find an invertible 3x3 real matrix P which puts the matrix A = >>> matrix([[3,4,0],[-1, -3, -2],[1, 2, 1]]) into rational form. >>> >>> assistant: >>> >>> To put a matrix in rational form, we want to reduce it to row echelon form >>> and then perform row operations to get all leading coefficients to be 1 and >>> all other entries in the same columns to be 0. Here are the steps to do it >>> for matrix A: >>> >>> Step 1: Choose the first element in the first row to be the pivot and >>> eliminate all entries in the column below it. To do this, subtract the >>> first row multiplied by (-1/3) from the second row, and add the first row >>> to the third row. >>> >>> [3 4 0] >>> [0 1 2] >>> [4 6 1] >>> >>> Step 2: Choose the second element in the second row to be the pivot and >>> eliminate all entries in the column below it. To do this, subtract the >>> second row multiplied by 2 from the third row. >>> >>> [3 4 0] >>> [0 1 2] >>> [0 -2 -7] >>> >>> Step 3: Choose the third element in the third row to be the pivot and >>> eliminate all entries in the column above it. To do this, add the second >>> row multiplied by 2 to the third row. >>> >>> [3 4 0] >>> [0 1 2] >>> [0 0 -3] >>> >>> Now we have the row echelon form of matrix A. To get it in rational form, >>> we can perform further row operations to get all leading coefficients to be >>> 1. We can achieve this by multiplying the third row by -1/3. >>> >>> [3 4 0] >>> [0 1 2] >>> [0 0 1] >>> >>> Now we have the rational form of matrix A. To find the invertible matrix P >>> that puts A into this form, we need to keep track of the row operations we >>> performed. >>> >>> First, we subtracted the first row multiplied by (-1/3) from the second >>> row. This is equivalent to multiplying by the matrix: >>> >>> [1 0 0] >>> [-1/3 1 0] >>> [0 0 1] >>> >>> Next, we added the first row to the third row. This is equivalent to >>> multiplying by the matrix: >>> >>> [1 0 0] >>> [0 1 0] >>> [1 0 1] >>> >>> Then, we subtracted the second row multiplied by 2 from the third row. This >>> is equivalent to multiplying by the matrix: >>> >>> [1 0 0] >>> [0 1 0] >>> [0 -2 1] >>> >>> Finally, we multiplied the third row by -1/3. This is equivalent to >>> multiplying by the matrix: >>> >>> [1 0 0] >>> [0 1 0] >>> [0 0 -1/3] >>> >>> To find P, we multiply these matrices in order: >>> >>> P = [1 0 0] @ [-1/3 1 0] @ [1 0 0] @ [0 1 0] @ [0 -2 1] @ [1 0 >>> 0] @ [0 1 0] @ [0 0 -1/3] >>> >>> Simplifying this expression gives: >>> >>> P = matrix([[-1/3, -8/3, -2/3],[1/3, 5/3, 2/3],[-4/3, -2/3, 1/3]]) >>> >>> We can check that matrix A can be transformed to its rational form by >>> multiplying it by P and verifying that the result is indeed the rational >>> form: >>> >>> A_rational = P.inv() @ A @ P >>> >>> Output: >>> A_rational = matrix([[1, 0, 0],[0, 1, 0],[0, 0, 1]]) >>> >>> >>> Elimboto >>> >>> >>> On Wed, Mar 29, 2023 at 3:32 AM Aaron Meurer <asme...@gmail.com> wrote: >>> >>>> If you don't have access to the ChatGPT beta, someone made a Streamlit >>>> app that does the same thing. You can give it a math problem and it uses >>>> ChatGPT to generate and run SymPy code to solve it >>>> https://mathgpt.streamlit.app/ >>>> >>>> Aaron Meurer >>>> >>>> On Thu, Mar 23, 2023 at 3:28 PM Aaron Meurer <asme...@gmail.com> wrote: >>>> >>>>> Actually I didn't notice it but there's literally an example of >>>>> ChatGPT generating SymPy code and executing on >>>>> https://openai.com/blog/chatgpt-plugins (scroll down to where it says >>>>> "code interpreter"). It's their main example of calling out to Python. >>>>> >>>>> Aaron Meurer >>>>> >>>>> On Thu, Mar 23, 2023 at 2:42 PM Aaron Meurer <asme...@gmail.com> >>>>> wrote: >>>>> >>>>>> >>>>>> >>>>>> On Thu, Mar 23, 2023 at 12:24 PM S.Y. Lee <syle...@gmail.com> wrote: >>>>>> >>>>>>> Wolfram had recently announced the collaboration of chatGPT and >>>>>>> wolfram alpha >>>>>>> ChatGPT Gets Its “Wolfram Superpowers”!—Stephen Wolfram Writings >>>>>>> <https://writings.stephenwolfram.com/2023/03/chatgpt-gets-its-wolfram-superpowers/> >>>>>> >>>>>> >>>>>> I wouldn't call this a "collaboration". OpenAI is adding a plugin >>>>>> system to ChatGPT and Wolfram is one of the first plugins >>>>>> https://openai.com/blog/chatgpt-plugins. >>>>>> >>>>>> If you scroll down on that page, there is also a tool that lets it >>>>>> execute Python code. I don't know if it has access to SymPy, but it >>>>>> likely >>>>>> does, since it seems to have access to other popular libraries like >>>>>> pandas >>>>>> and matplotlib. If anyone has access to ChatGPT Plus, could you check? >>>>>> >>>>>> >>>>>>> >>>>>>> They start to use chatgpt to generate the Wolfram code. >>>>>>> And it is likely to help the issues with correctness about math or >>>>>>> science facts >>>>>>> because once it translates to the wolfram functions, and the wolfram >>>>>>> function runs without error, the answer is correct. >>>>>>> >>>>>>> However, the argument I'd give is that making a combination of code >>>>>>> with >>>>>>> *simplify, solve, integrate *are not still informative because even >>>>>>> though they do things logically correct, >>>>>>> they have problem that they can't inform people how to solve the >>>>>>> problems in details. >>>>>>> >>>>>>> So I'm thinking about an idea whether language models should really >>>>>>> generate a code that assembles the *rules* used in simplify, solve. >>>>>>> And then it can be more human readable (or can get more human >>>>>>> readable logic from it) >>>>>>> if it is like compose(solve_trig, simplify_cos, ...) >>>>>>> (in some pseudo sympy code) >>>>>>> >>>>>> >>>>>> A language model can do pretty much anything, so long as it's seen >>>>>> enough examples of it before. You can often just give it examples of >>>>>> what >>>>>> you want in the prompt and it will figure it out. >>>>>> >>>>>> ChatGPT (and other LLMs) are basically just huge pattern matching >>>>>> machines. That's all they are, which is why they break down whenever >>>>>> they >>>>>> have to do actual logic or reason about something they haven't seen >>>>>> before. >>>>>> So I suspect they could be used quite successfully for symbolic pattern >>>>>> matching, especially when that pattern matching is "fuzzy", like trying >>>>>> to >>>>>> find the best heuristic function to apply to an expression. >>>>>> >>>>>> Aaron Meurer >>>>>> >>>>>> >>>>>>> On Saturday, December 17, 2022 at 5:47:39 PM UTC+9 smi...@gmail.com >>>>>>> wrote: >>>>>>> >>>>>>>> In reviewing a PR related to units, I found ChatGPT to get correct >>>>>>>> the idea that a foot is bigger than an inch, but it said that a volt >>>>>>>> is >>>>>>>> bigger than a statvolt (see quoted GPT response [here]( >>>>>>>> https://github.com/sympy/sympy/pull/24325#issuecomment-1354343306) >>>>>>>> ). >>>>>>>> >>>>>>>> /c >>>>>>>> >>>>>>>> On Thursday, December 15, 2022 at 1:58:33 PM UTC-6 Aaron Meurer >>>>>>>> wrote: >>>>>>>> >>>>>>>>> The trend with LLMs is much less structured. It doesn't use any >>>>>>>>> formalism. It just guesses the next character of the input based on >>>>>>>>> training on billions of examples. >>>>>>>>> >>>>>>>>> That's why I think that tools like SymPy that are more structured >>>>>>>>> can be useful. GPT can already write SymPy code pretty well, much >>>>>>>>> better >>>>>>>>> than it can do the actual mathematics. It may be as simple as >>>>>>>>> automatically >>>>>>>>> appending "and write SymPy code to verify this" to the end of a >>>>>>>>> prompt >>>>>>>>> whenever it involves mathematics. This sort of approach has already >>>>>>>>> been >>>>>>>>> proven to be able to solve university math problems (see >>>>>>>>> https://www.pnas.org/doi/pdf/10.1073/pnas.2123433119, where they >>>>>>>>> literally just take the input problem and prepend "use sympy" and the >>>>>>>>> neural network model does the rest). >>>>>>>>> >>>>>>>>> Aaron Meurer >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> On Thu, Dec 15, 2022 at 2:21 AM S.Y. Lee <syle...@gmail.com> >>>>>>>>> wrote: >>>>>>>>> >>>>>>>>>> > My hope is that tools like SymPy can be used as oracles for >>>>>>>>>> tools like GPT to help them verify their mathematics. >>>>>>>>>> >>>>>>>>>> In the most general context, "correct mathematics" can also be >>>>>>>>>> considered some "grammar". >>>>>>>>>> So there should be some grammar between Type-0 grammar to Type-1 >>>>>>>>>> grammar in Chomsky hierarchy >>>>>>>>>> <https://en.wikipedia.org/wiki/Chomsky_hierarchy>. >>>>>>>>>> In this context, a parser, or a parser with sympy oracle is the >>>>>>>>>> solution for such problem, >>>>>>>>>> such that any other ideas to solve such problem can be isomorphic >>>>>>>>>> to. >>>>>>>>>> >>>>>>>>>> However, building up such parser is off-direction for the >>>>>>>>>> researches of deep learning itself, >>>>>>>>>> because it would need a lot of efforts by experts, to interpret >>>>>>>>>> the sentence generated by GPT, >>>>>>>>>> and design a phrase structure grammar for it. >>>>>>>>>> >>>>>>>>>> I also thought about an idea that they can just tag arithmetics >>>>>>>>>> using SKI combinator calculus >>>>>>>>>> <https://en.wikipedia.org/wiki/SKI_combinator_calculus>. >>>>>>>>>> In this way, there is no wrong arithmetics for every random >>>>>>>>>> sequence of alphabets. >>>>>>>>>> >>>>>>>>>> However, I'm not sure that if this idea is already refuted by >>>>>>>>>> such contemporary researchers >>>>>>>>>> because it should be pretty much well-known. >>>>>>>>>> >>>>>>>>>> On Thursday, December 15, 2022 at 12:45:53 AM UTC+2 >>>>>>>>>> asme...@gmail.com wrote: >>>>>>>>>> >>>>>>>>>>> I've seen many examples on Twitter and elsewhere of chatgpt >>>>>>>>>>> doing terribly on actual mathematics. I instead asked it to "Find >>>>>>>>>>> the >>>>>>>>>>> extreme values of the function x^3 + 5x^2 + 3x - 9". It got the >>>>>>>>>>> first part >>>>>>>>>>> with the derivative and the quadratic equation correct, but then >>>>>>>>>>> did faulty >>>>>>>>>>> arithmetic when plugging the values back into the expression (the >>>>>>>>>>> correct >>>>>>>>>>> values are 0 and -256/27). It makes a sign error and also gets the >>>>>>>>>>> addition >>>>>>>>>>> wrong even with the wrong sign. >>>>>>>>>>> >>>>>>>>>>> My hope is that tools like SymPy can be used as oracles for >>>>>>>>>>> tools like GPT to help them verify their mathematics. Obviously in >>>>>>>>>>> this >>>>>>>>>>> case you don't need SymPy to just verify arithmetic, but I've also >>>>>>>>>>> seen it >>>>>>>>>>> do just as poorly on symbolic algebra problems. >>>>>>>>>>> >>>>>>>>>>> I've also pasted the full log below: >>>>>>>>>>> >>>>>>>>>>> [image: Screen Shot 2022-12-14 at 3.41.32 PM.png] >>>>>>>>>>> [image: Screen Shot 2022-12-14 at 3.41.42 PM.png] >>>>>>>>>>> >>>>>>>>>>> To find the extreme values of a function, we need to find the >>>>>>>>>>> critical points of the function, which are the points at which the >>>>>>>>>>> derivative of the function is equal to 0 or does not exist. >>>>>>>>>>> >>>>>>>>>>> The derivative of the function x^3 + 5x^2 + 3x - 9 is 3x^2 + 10x >>>>>>>>>>> + 3. To find the critical points, we need to solve the equation >>>>>>>>>>> 3x^2 + 10x >>>>>>>>>>> + 3 = 0. >>>>>>>>>>> >>>>>>>>>>> To solve this equation, we can use the quadratic formula: >>>>>>>>>>> >>>>>>>>>>> x = (-b ± √(b^2 - 4ac)) / 2a >>>>>>>>>>> >>>>>>>>>>> where a, b, and c are the coefficients of the quadratic equation >>>>>>>>>>> ax^2 + bx + c = 0. In our case, a = 3, b = 10, and c = 3, so the >>>>>>>>>>> formula >>>>>>>>>>> becomes: >>>>>>>>>>> >>>>>>>>>>> x = (-10 ± √(10^2 - 4 * 3 * 3)) / 2 * 3 >>>>>>>>>>> >>>>>>>>>>> = (-10 ± √(100 - 36)) / 6 >>>>>>>>>>> >>>>>>>>>>> = (-10 ± √(64)) / 6 >>>>>>>>>>> >>>>>>>>>>> = (-10 ± 8) / 6 >>>>>>>>>>> >>>>>>>>>>> = (-18, -2) / 6 >>>>>>>>>>> >>>>>>>>>>> = (-3, -1/3) >>>>>>>>>>> >>>>>>>>>>> These are the critical points of the function x^3 + 5x^2 + 3x - >>>>>>>>>>> 9. >>>>>>>>>>> >>>>>>>>>>> To find the extreme values of the function, we need to evaluate >>>>>>>>>>> the function at these critical points and at any points where the >>>>>>>>>>> derivative does not exist. In this case, the derivative exists for >>>>>>>>>>> all >>>>>>>>>>> values of x, so we only need to evaluate the function at the >>>>>>>>>>> critical >>>>>>>>>>> points. >>>>>>>>>>> >>>>>>>>>>> If we plug in the critical points into the function, we get: >>>>>>>>>>> >>>>>>>>>>> x^3 + 5x^2 + 3x - 9 = (-3)^3 + 5(-3)^2 + 3(-3) - 9 = -27 + 45 + >>>>>>>>>>> 9 - 9 = 8 >>>>>>>>>>> >>>>>>>>>>> and >>>>>>>>>>> >>>>>>>>>>> x^3 + 5x^2 + 3x - 9 = (-1/3)^3 + 5(-1/3)^2 + 3(-1/3) - 9 = -1/27 >>>>>>>>>>> + 5/9 + 1 - 9 = 5/9 - 1/27 - 8/9 = -64/27 >>>>>>>>>>> >>>>>>>>>>> The extreme values of the function x^3 + 5x^2 + 3x - 9 are 8 and >>>>>>>>>>> -64/27. >>>>>>>>>>> >>>>>>>>>>> On Wed, Dec 14, 2022 at 3:35 PM gu...@uwosh.edu <gu...@uwosh.edu> >>>>>>>>>>> wrote: >>>>>>>>>>> >>>>>>>>>>>> Just for some additional perspective. I have also tried this on >>>>>>>>>>>> some general chemistry word problems. In general, I see it getting >>>>>>>>>>>> basic >>>>>>>>>>>> one logical step processes correct (e.g. a single step dilution or >>>>>>>>>>>> grams -> >>>>>>>>>>>> moles). Things with multiple steps or requiring understanding the >>>>>>>>>>>> physical >>>>>>>>>>>> situation it does poorly on. That said, I think it does better >>>>>>>>>>>> than some of >>>>>>>>>>>> my weakest students. It does not seem to be able to use >>>>>>>>>>>> significant figures >>>>>>>>>>>> in computations (also a problem for my weaker students). >>>>>>>>>>>> >>>>>>>>>>>> It seems to be improving rapidly. If it can get to reliably >>>>>>>>>>>> differentiating between correct (workable) solutions and erroneous >>>>>>>>>>>> ones, it >>>>>>>>>>>> will be more useful to most people (including my students) than >>>>>>>>>>>> searches of >>>>>>>>>>>> the internet or a cheating sight such as Chegg. >>>>>>>>>>>> >>>>>>>>>>>> My two cents worth of opinion. >>>>>>>>>>>> >>>>>>>>>>>> Jonathan >>>>>>>>>>>> >>>>>>>>>>>> On Wednesday, December 14, 2022 at 4:28:05 PM UTC-6 Francesco >>>>>>>>>>>> Bonazzi wrote: >>>>>>>>>>>> >>>>>>>>>>>>> [image: chatgpt.sympy.matrix_diag.png] >>>>>>>>>>>>> >>>>>>>>>>>>> On Wednesday, December 14, 2022 at 11:26:37 p.m. UTC+1 >>>>>>>>>>>>> Francesco Bonazzi wrote: >>>>>>>>>>>>> >>>>>>>>>>>>>> Not everything is perfect... ChatGPT misses the *convert_to( >>>>>>>>>>>>>> ... ) *function in *sympy.physics.units*, furthermore, the >>>>>>>>>>>>>> given code does not work: >>>>>>>>>>>>>> >>>>>>>>>>>>>> [image: chatgpt.sympy.unit_conv.png] >>>>>>>>>>>>>> >>>>>>>>>>>>>> On Wednesday, December 14, 2022 at 11:24:29 p.m. UTC+1 >>>>>>>>>>>>>> Francesco Bonazzi wrote: >>>>>>>>>>>>>> >>>>>>>>>>>>>>> [image: chatgpt.sympy.logical_inference.png] >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> On Wednesday, December 14, 2022 at 11:23:43 p.m. UTC+1 >>>>>>>>>>>>>>> Francesco Bonazzi wrote: >>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> https://en.wikipedia.org/wiki/ChatGPT >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> Some tested examples attached as pictures to this post. >>>>>>>>>>>>>>>> Quite impressive... >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> -- >>>>>>>>>>>> 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+un...@googlegroups.com. >>>>>>>>>>>> >>>>>>>>>>> To view this discussion on the web visit >>>>>>>>>>>> https://groups.google.com/d/msgid/sympy/6af62b19-1fb0-4681-9fd2-5e5fccfcb46fn%40googlegroups.com >>>>>>>>>>>> >>>>>>>>>>>> <https://groups.google.com/d/msgid/sympy/6af62b19-1fb0-4681-9fd2-5e5fccfcb46fn%40googlegroups.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+un...@googlegroups.com. >>>>>>>>>> >>>>>>>>> To view this discussion on the web visit >>>>>>>>>> https://groups.google.com/d/msgid/sympy/74847ca3-124b-414d-aa36-01eb91096310n%40googlegroups.com >>>>>>>>>> >>>>>>>>>> <https://groups.google.com/d/msgid/sympy/74847ca3-124b-414d-aa36-01eb91096310n%40googlegroups.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+un...@googlegroups.com. >>>>>>> To view this discussion on the web visit >>>>>>> https://groups.google.com/d/msgid/sympy/1fbd88f8-3513-4e02-a576-266352f3531fn%40googlegroups.com >>>>>>> >>>>>>> <https://groups.google.com/d/msgid/sympy/1fbd88f8-3513-4e02-a576-266352f3531fn%40googlegroups.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+un...@googlegroups.com. >>>> >>> To view this discussion on the web visit >>>> https://groups.google.com/d/msgid/sympy/CAKgW%3D6L3qy-h94G8OrTqtjrqTu9woZQpXZxaZx7mjcD1L7q32g%40mail.gmail.com >>>> >>>> <https://groups.google.com/d/msgid/sympy/CAKgW%3D6L3qy-h94G8OrTqtjrqTu9woZQpXZxaZx7mjcD1L7q32g%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+un...@googlegroups.com. >> > To view this discussion on the web visit >> https://groups.google.com/d/msgid/sympy/9fd4fc7a-32d4-4ab3-b561-f12cbb6ab331n%40googlegroups.com >> >> <https://groups.google.com/d/msgid/sympy/9fd4fc7a-32d4-4ab3-b561-f12cbb6ab331n%40googlegroups.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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