If there are no issues with the proposal or the timeline mentioned(which I will update soon due to the GSoC program timeline changes), then I am planning on submitting the proposal in 2 days. Feedback would be appreciated if possible.
On Monday, March 23, 2020 at 10:00:22 PM UTC+5:30, Milan Jolly wrote: > > Thank you for your feedback. I have added another paragraph in the > Motivation section where the I have added how these new solvers are > advantageous to the end users. > > On Monday, March 23, 2020 at 1:25:31 AM UTC+5:30, Oscar wrote: >> >> I took a quick look. It's long so I didn't read it fully but it looks >> good. There is a lot of detail about what you would do but perhaps the >> motivation section can be strengthened. What does all of this mean for >> end users etc? If you completed the work described then sympy's >> capabilities for systems of ODEs would be expanded enormously. >> >> On Sun, 22 Mar 2020 at 19:26, Milan Jolly <milan....@gmail.com> wrote: >> > >> > Here is the link to my proposal: >> https://docs.google.com/document/d/12QN19LSjwEvYoSukyq-BWd76ZrI24FQuU0CGIOIx6Ww/edit?usp=sharing >> >> > >> > On Saturday, March 21, 2020 at 3:22:00 AM UTC+5:30, Oscar wrote: >> >> >> >> Stating clearly what the different parts do in high-level terms should >> >> be sufficient. >> >> >> >> On Fri, 20 Mar 2020 at 16:57, Milan Jolly <milan....@gmail.com> >> wrote: >> >> > >> >> > Thanks for clearing my doubt. >> >> > >> >> > Now, I have started preparing my GSOC proposal and it will be ready >> soon. But, I wanted to know that will it be ok that I don't give details >> about the implementations of the helper functions and solvers and simply >> state what they do, which parameters they take, what they return and how >> they fit in the solving process while I give more details about how they >> fit together more generally. I would like to elucidate more on how the main >> function ode_sol handles the system of equations using the helper functions >> and various solvers as it is the only thing that is not clearly mentioned >> in the roadmap. >> >> > >> >> > On Friday, March 20, 2020 at 7:33:17 PM UTC+5:30, Oscar wrote: >> >> >> >> >> >> It's not always the case that symmetric matrices commute so >> actually >> >> >> checking if it is symmetric is not sufficient e.g.: >> >> >> >> >> >> In [83]: M = Matrix([[2*x**2, x], [x, x**2]]) >> >> >> >> >> >> In [84]: M.is_symmetric() >> >> >> Out[84]: True >> >> >> >> >> >> In [85]: M*M.diff(x) == M.diff(x)*M >> >> >> Out[85]: False >> >> >> >> >> >> Maybe there is something that can be said more generally about >> >> >> `exp(M(t)).diff(t)` when `M` is symmetric but does not necessarily >> >> >> commute with `M.diff(t)`... >> >> >> >> >> >> >> >> >> On Thu, 19 Mar 2020 at 18:34, Milan Jolly <milan....@gmail.com> >> wrote: >> >> >> > >> >> >> > In ODE systems roadmap, you have mentioned that for system of >> ODEs where the coefficient matrix is non-constant, if the coefficient >> matrix A(t) is symmetric, then A(t) and its anti derivative B(t) commute >> and thus we get the solution based on this fact. But it is also mentioned >> that if A and B commuting is more general than when A is symmetric, that >> is, it is possible that A is not symmetric but A and B commute. So, for >> that solver, should we first compute its anti derivative and test it that >> commutes with A or just check if A is symmetric and use the solution? >> >> >> > >> >> >> > On Wednesday, March 18, 2020 at 3:18:31 AM UTC+5:30, Oscar wrote: >> >> >> >> >> >> >> >> That sounds reasonable. >> >> >> >> >> >> >> >> Note that we can't start raising NotImplementedError yet. You >> will >> >> >> >> need to think about how to introduce the new code gradually >> while >> >> >> >> still ensuring that dsolve falls back on the old code for cases >> not >> >> >> >> yet handled by the new code. >> >> >> >> >> >> >> >> On Tue, 17 Mar 2020 at 17:51, Milan Jolly <milan....@gmail.com> >> wrote: >> >> >> >> > >> >> >> >> > So, I have made a rough layout of the main function that will >> be used to solve ODEs with the methods like >> neq_nth_order_linear_constant_coeff_homogeneous/nonhomogeneous, >> neq_nth_linear_symmetric_coeff_homogeneous/nonhomogeneous, special case >> non-linear solvers, etc. >> >> >> >> > >> >> >> >> > Some notations used: >> >> >> >> > eqs: Equations, funcs: dependent variables, t: independent >> variable, wcc: weakly connected component, scc: strongly connected >> component >> >> >> >> > >> >> >> >> > Introduction to helper functions that will be used(these are >> temporary names, parameters and return elements and may be changed if >> required): >> >> >> >> > >> >> >> >> > 1. match_ode:- >> >> >> >> > Parameters: eqs, funcs, t >> >> >> >> > Returns: dictionary which has important keys like: >> order(a dict that has func as a key and maximum order found as value), >> is_linear, is_constant, is_homogeneous, eqs, funcs. >> >> >> >> > >> >> >> >> > 2. component_division:- >> >> >> >> > Paramters: eqs, funcs >> >> >> >> > Returns: A 3D list where the eqs are first divided >> into its wccs and then into its sccs. >> >> >> >> > This function is suggested to be implemented later. >> So, until all the other solvers are not ready(tested and working), this >> function will just take eqs and return [[eqs]]. >> >> >> >> > >> >> >> >> > 3. get_coeff_matrix:- >> >> >> >> > Parameters: eqs, funcs >> >> >> >> > Returns: coefficient matrix A(t) and f(t) >> >> >> >> > This function takes in a first order linear ODE and >> returns matrix A(t) and f(t) from X' = A(t) * X + f(t). >> >> >> >> > >> >> >> >> > 4. nth_order_to_first_order:- >> >> >> >> > Parameters: eqs, order >> >> >> >> > Returns: first order ODE with new introduced dependent >> variables. >> >> >> >> > >> >> >> >> > And all the first order linear solvers mentioned above. >> >> >> >> > >> >> >> >> > Now, besides the main function, there are two separate >> functions depending on whether the system of ODEs is linear or not, namely >> _linear_ode_sol and _non_linear_ode_sol. >> >> >> >> > >> >> >> >> > 1. _first_order_linear_ode_sol:- >> >> >> >> > Parameters: match dict(obtained earlier and maybe >> modified in ode_sol) >> >> >> >> > Returns: Dict with keys as func and value as its >> solution that solves the ODE. >> >> >> >> > Working: First, extracts A(t) and f(t) using >> get_coeff_matrix, then using match dict, identify which solver is required >> and solve the ODE if it is possible to do so. For example: Case where A(t) >> is not symmetric isn't solved. >> >> >> >> > >> >> >> >> > 2. _non_linear_ode_sol has similar Parameters and Returns but >> the function operates differently that's why it is essential to use a >> different function. But I don't have a clear understanding >> >> >> >> > of how to design _non_linear_ode_sol yet but here is what I >> have came up with: First match the condition where it is possible seperate >> out the independent variable to get a relationship >> >> >> >> > between the dependent variables and then finally, just use >> the special solver to solve the ODE. >> >> >> >> > >> >> >> >> > Now, coming to the main function ode_sol(for now, I haven't >> considered initial values):- >> >> >> >> > Parameters: eqs, funcs, t >> >> >> >> > Returns: Solution in a dict form where func is the key and >> value is the solution for that corresponding func. >> >> >> >> > >> >> >> >> > Working: >> >> >> >> > The steps of its working- >> >> >> >> > 1. Preprocess the equations. >> >> >> >> > 2. Get the match dict using match_ode function. >> >> >> >> > 3. Convert nth order equations to first order equations >> using nth_order_to_first_order while storing the funcs seperately so that >> we can later filter out the dependent variables that were introduced in >> this step. >> >> >> >> > 4. Get the 3D list of equations using component_division >> function. >> >> >> >> > 5. Iterate through the wccs and solve and store >> solutions seperately but for sccs, first solve the first set of equations >> in a scc, then substitute the solutions found in the first set of the >> current scc to the second >> >> >> >> > set of current scc. Keep doing this until the all the >> sets for a particular scc is solved. >> >> >> >> > 6. For solving a component, choose either >> _linear_ode_sol or _non_linear_ode_sol depending upon the set of equations >> to be solved. >> >> >> >> > 7. Return a dict by taking out values from the solution >> obtained using all the dependent variables in funcs as there may be more >> variables introduced when we made the system into first order. >> >> >> >> > >> >> >> >> > For now, this is what I have came up with. Obviously the order >> in which we will proceed is, build the basic layout of the main function >> and component_division will just increase the number of dimensions to 3 >> rudimentarily as we >> >> >> >> > will have to first ensure that the general solvers work well >> since working on both of them simultaneously will make it tough to pinpoint >> the errors. Along with that, non-linear solvers can be implemented later, >> we can just raise a >> >> >> >> > NotImplementedError for now till we have completed both the >> general linear solvers and the component_division and then add the special >> case solvers. >> >> >> >> > >> >> >> >> > On Tuesday, March 17, 2020 at 3:02:29 AM UTC+5:30, Oscar >> wrote: >> >> >> >> >> >> >> >> >> >> There are possibilities to go from nonlinear to linear e.g.: >> >> >> >> >> >> >> >> >> >> In [6]: x, y = symbols('x, y', cls=Function) >> >> >> >> >> >> >> >> >> >> In [7]: eqs = [x(t).diff(t)**2 - y(t)**2, y(t).diff(t)**2 - >> x(t)**2] >> >> >> >> >> >> >> >> >> >> In [8]: eqs >> >> >> >> >> Out[8]: >> >> >> >> >> ⎡ 2 2⎤ >> >> >> >> >> ⎢ 2 ⎛d ⎞ 2 ⎛d ⎞ ⎥ >> >> >> >> >> ⎢- y (t) + ⎜──(x(t))⎟ , - x (t) + ⎜──(y(t))⎟ ⎥ >> >> >> >> >> ⎣ ⎝dt ⎠ ⎝dt ⎠ ⎦ >> >> >> >> >> >> >> >> >> >> In [9]: solve(eqs, [x(t).diff(t), y(t).diff(t)], dict=True) >> >> >> >> >> Out[9]: >> >> >> >> >> ⎡⎧d d ⎫ ⎧d d >> ⎫ >> >> >> >> >> ⎧d d ⎫ ⎧d d >> ⎫⎤ >> >> >> >> >> ⎢⎨──(x(t)): -y(t), ──(y(t)): -x(t)⎬, ⎨──(x(t)): -y(t), >> ──(y(t)): >> >> >> >> >> x(t)⎬, ⎨──(x(t)): y(t), ──(y(t)): -x(t)⎬, ⎨──(x(t)): y(t), >> ──(y(t)): >> >> >> >> >> x(t)⎬⎥ >> >> >> >> >> ⎣⎩dt dt ⎭ ⎩dt dt >> ⎭ >> >> >> >> >> ⎩dt dt ⎭ ⎩dt dt >> ⎭⎦ >> >> >> >> >> >> >> >> >> >> On Mon, 16 Mar 2020 at 15:48, Milan Jolly < >> milan....@gmail.com> wrote: >> >> >> >> >> > >> >> >> >> >> > Thanks for the suggestion, I have started with the design >> for these solvers. But I have one doubt, namely since now we are using >> linear_eq_to_matrix function to check if the system of ODEs is linear or >> not, would we require the canonical rearrangements part? Or rather are >> there other cases when we can reduce non-linear ODEs into linear ODEs. >> >> >> >> >> > >> >> >> >> >> > On Monday, March 16, 2020 at 2:53:57 AM UTC+5:30, Oscar >> wrote: >> >> >> >> >> >> >> >> >> >> >> >> That seems reasonable to me. Since the plan is a total >> rewrite I think >> >> >> >> >> >> that it would be good to put some time in at the beginning >> for >> >> >> >> >> >> designing how all of these pieces would fit together. For >> example even >> >> >> >> >> >> if the connected components part comes at the end it would >> be good to >> >> >> >> >> >> think about how that code would fit in from the beginning >> and to >> >> >> >> >> >> clearly document it both in issues and in the code. >> >> >> >> >> >> >> >> >> >> >> >> Getting a good design is actually more important than >> implementing all >> >> >> >> >> >> of the pieces. If the groundwork is done then other >> contributors in >> >> >> >> >> >> future can easily implement the remaining features one by >> one. Right >> >> >> >> >> >> now it is not easy to improve the code for systems because >> of the way >> >> >> >> >> >> that it is structured. >> >> >> >> >> >> >> >> >> >> >> >> On Sun, 15 Mar 2020 at 19:27, Milan Jolly < >> milan....@gmail.com> wrote: >> >> >> >> >> >> > >> >> >> >> >> >> > Thanks for your reply. I have planned a rough layout for >> the phases. I took a lot of time this past month to understand all the >> mathematics that will be involved and have grasped some part of it. >> >> >> >> >> >> > >> >> >> >> >> >> > If I am lucky and get selected for GSOC'20 for this >> organisation, then the below is the rough plan. Please comment on >> suggestions if necessary. >> >> >> >> >> >> > >> >> >> >> >> >> > Community Bonding phase: >> >> >> >> >> >> > 1. Using matrix exponential to solve first order linear >> constant coefficient homogeneous systems(n equations). >> >> >> >> >> >> > 2. Adding new test cases and/or updating old ones. >> >> >> >> >> >> > 3. Removing and closing related issues if they are >> solved by the addition of this general solver. Identifying and removing the >> special cases solvers which are covered by this general solver. >> >> >> >> >> >> > >> >> >> >> >> >> > Phase I: >> >> >> >> >> >> > 1. Adding technique to solve first order constant >> coefficient non-homogeneous systems(n equations). >> >> >> >> >> >> > 2. Adding the functionality that reduces higher order >> linear ODEs to first order linear ODEs(if not done already, and if done, >> then incorporating it to solve higher order ODEs). >> >> >> >> >> >> > 3. Adding a special case solver when non-constant linear >> first order ODE has symmetric coefficient matrix. >> >> >> >> >> >> > >> >> >> >> >> >> > Phase II: >> >> >> >> >> >> > 1. Adding technique to solve non-constant >> non-homogeneous linear ODE based off the solver added by the end of Phase >> I. >> >> >> >> >> >> > 2. Evaluating and eliminating unnecessary solvers. >> >> >> >> >> >> > 3. Closing related issues solved by the general solvers >> and identifying and removing unwanted solvers. >> >> >> >> >> >> > 4. Adding basic rearrangements to simplify the system of >> ODEs. >> >> >> >> >> >> > >> >> >> >> >> >> > Phase III: >> >> >> >> >> >> > 1. Dividing the ODEs by evaluating which sub-systems are >> weakly and strongly connected and handling both of these cases accordingly. >> >> >> >> >> >> > 2. Adding a special case solver where the independent >> variable can be eliminated and thus solving the system becomes easier. >> >> >> >> >> >> > 3. Wrapping things up: adding test cases, eliminating >> unwanted solvers and updating documentation. >> >> >> >> >> >> > >> >> >> >> >> >> > This is the rough layout and my plan for summer if I get >> selected. If this plan seems ok then I would include this plan in my >> proposal. >> >> >> >> >> >> > >> >> >> >> >> >> > On Saturday, March 14, 2020 at 9:37:31 PM UTC+5:30, >> Oscar wrote: >> >> >> >> >> >> >> >> >> >> >> >> >> >> It's hard to say how much time each of these would >> take. The roadmap >> >> >> >> >> >> >> aims to completely replace all of the existing code for >> systems of >> >> >> >> >> >> >> ODEs. How much of that you think you would be able to >> do is up to you >> >> >> >> >> >> >> if making a proposal. >> >> >> >> >> >> >> >> >> >> >> >> >> >> None of the other things described in the roadmap is >> implemented >> >> >> >> >> >> >> anywhere as far as I know. Following the roadmap it >> should be possible >> >> >> >> >> >> >> to close all of these issues I think: >> >> >> >> >> >> >> >> https://github.com/sympy/sympy/issues?q=is%3Aopen+is%3Aissue+label%3Asolvers.dsolve.system >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> On Fri, 13 Mar 2020 at 22:30, Milan Jolly < >> milan....@gmail.com> wrote: >> >> >> >> >> >> >> > >> >> >> >> >> >> >> > I have mostly read and understood matrix >> exponentials and Jordan forms along with the ODE systems roadmap. But I am >> unclear as to what has already been done when it comes to implementing the >> general solvers. For example: The matrix exponentials part has already been >> implemented and now I have a PR that has revived the matrix exponential >> code. >> >> >> >> >> >> >> > >> >> >> >> >> >> >> > I want to make a proposal and contribute to make >> these general solvers during this summer if my proposal gets accepted. But >> I am unclear what should be the parts I need to work during community >> bonding period, phase 1, phase 2 and phase 3 as I am unaware how much time >> each part of the general solvers would take. >> >> >> >> >> >> >> > >> >> >> >> >> >> >> > If someone can help me in this regard(helping me with >> these 2 questions) then it would be great. >> >> >> >> >> >> >> > >> >> >> >> >> >> >> > >> >> >> >> >> >> >> > On Tue, Feb 25, 2020, 5:09 AM Milan Jolly < >> milan....@gmail.com> wrote: >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> I will go through the roadmap. Also, I will work on >> reviving and finishing the stalled PRs namely the matrix exponential one >> for now as I am interested in working towards this. Thanks. >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> On Mon, Feb 24, 2020, 9:56 PM Oscar Benjamin < >> oscar.j...@gmail.com> wrote: >> >> >> >> >> >> >> >>> >> >> >> >> >> >> >> >>> This section in the roadmap refers to existing >> stalled PRs trying to >> >> >> >> >> >> >> >>> fix the n-equations solver for constant coefficient >> homogeneous ODEs >> >> >> >> >> >> >> >>> which is the first step: >> >> >> >> >> >> >> >>> >> https://github.com/sympy/sympy/wiki/ODE-Systems-roadmap#constant-coefficients---current-status >> >> >> >> >> >> >> >> >>> >> >> >> >> >> >> >> >>> A first step would be to attempt to revive one or >> both of those PRs >> >> >> >> >> >> >> >>> and finish them off. >> >> >> >> >> >> >> >>> >> >> >> >> >> >> >> >>> On Mon, 24 Feb 2020 at 05:59, Milan Jolly < >> milan....@gmail.com> wrote: >> >> >> >> >> >> >> >>> > >> >> >> >> >> >> >> >>> > So, I am interested in rewriting parts of the >> current ODE as discussed in the roadmap. Is there any work started in that >> direction and if not then can I create a PR for the same? >> >> >> >> >> >> >> >>> > >> >> >> >> >> >> >> >>> > On Mon, Feb 24, 2020, 2:52 AM Oscar Benjamin < >> oscar.j...@gmail.com> wrote: >> >> >> >> >> >> >> >>> >> >> >> >> >> >> >> >> >>> >> The current refactoring effort applies only to >> the case of solving >> >> >> >> >> >> >> >>> >> *single* ODEs. The ODE systems code also needs >> to be refactored but >> >> >> >> >> >> >> >>> >> (in my opinion) needs a complete rewrite. That >> is what the roadmap is >> >> >> >> >> >> >> >>> >> about (it describes how to rewrite everything). >> The code for systems >> >> >> >> >> >> >> >>> >> of ODEs should also get refactored in the >> process but there is no need >> >> >> >> >> >> >> >>> >> to "refactor" it in its current form if it is in >> fact being >> >> >> >> >> >> >> >>> >> *completely* rewritten: we can just make sure >> that the new code is >> >> >> >> >> >> >> >>> >> written the way we want it to be. >> >> >> >> >> >> >> >>> >> >> >> >> >> >> >> >> >>> >> On Sun, 23 Feb 2020 at 19:52, Milan Jolly < >> milan....@gmail.com> wrote: >> >> >> >> >> >> >> >>> >> > >> >> >> >> >> >> >> >>> >> > Ok so I have gone through the links suggested >> and I have realised that as far as ODE module is concerned, refactoring is >> the most important task. But, as far as that is concerned, I think Mohit >> Balwani is working on this for a while and I want to limit any collisions >> with my co-contributors. So, I have couple of ideas to work on: >> >> >> >> >> >> >> >>> >> > 1. Helping to extend the solvers, >> i.e.implementing a fully working n-equations solver for constant >> coefficient homogeneous systems. This is from the ODE systems map. I am >> interested in working on this but I understand that it might be hard to >> work upon it while refactoring takes place. Still, if its possible to work >> on this and if no one else has started to work in this direction yet then I >> am willing to work for this. >> >> >> >> >> >> >> >>> >> > 2. Using connected components function >> implemented by Oscar Benjamin in >> https://github.com/sympy/sympy/pull/16225 to enhance ODE solvers and >> computing eigen values faster as mentioned here >> https://github.com/sympy/sympy/issues/16207 . >> >> >> >> >> >> >> >>> >> > 3. This idea is not mentioned in the ideas >> page and is something of my own. If there is anything possible, then I can >> also work on extending functions like maximum, minimum, argmax, argmin, etc >> in calculus module. I have been working on the issue >> https://github.com/sympy/sympy/pull/18550 and I think there is some >> scope to extend these functionalities. >> >> >> >> >> >> >> >>> >> > >> >> >> >> >> >> >> >>> >> > On Sunday, February 23, 2020 at 1:32:20 AM >> UTC+5:30, Milan Jolly wrote: >> >> >> >> >> >> >> >>> >> >> >> >> >> >> >> >> >> >>> >> >> Hello everyone, >> >> >> >> >> >> >> >>> >> >> >> >> >> >> >> >> >> >>> >> >> My name is Milan Jolly and I am an >> undergraduate student at Indian Institute of Technology, Patna. For the >> past 2 month, I have been learning and exploring sympy through either >> contributions, reading documentation or trying examples out. This last >> month I have learned a lot of new things thanks to the well designed >> code-base, the structured way this community works and most importantly the >> maintainers who make it work. It has been a pleasure to be a part of the >> community. >> >> >> >> >> >> >> >>> >> >> >> >> >> >> >> >> >> >>> >> >> I am interested in participating for GSoC >> this year and I would like to work for this org during the summers if I am >> lucky. I particularly want to work on improving the current ODE module as >> it is given in the idea list. There is a lot of work that needs to be taken >> care of like: >> >> >> >> >> >> >> >>> >> >> 1. Implementing solvers for solving constant >> coefficient non-homogeneous systems >> >> >> >> >> >> >> >>> >> >> 2. Solving mixed order ODEs >> >> >> >> >> >> >> >>> >> >> 3. Adding rearrangements to solve the system >> >> >> >> >> >> >> >>> >> >> >> >> >> >> >> >> >> >>> >> >> These are not my ideas but I have taken >> inspiration from the ideas page but I am up for working on these. If >> someone can guide me regarding this then it would be really helpful. >> >> >> >> >> >> >> >>> >> > >> >> >> >> >> >> >> >>> >> > -- >> >> >> >> >> >> >> >>> >> > 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 sy...@googlegroups.com. >> >> >> >> >> >> >> >>> >> > To view this discussion on the web visit >> https://groups.google.com/d/msgid/sympy/1033f581-abbb-4be5-a5b2-1988f4261535%40googlegroups.com. >> >> >> >> >> >> >> >> >> >>> >> >> >> >> >> >> >> >> >>> >> -- >> >> >> >> >> >> >> >>> >> 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 sy...@googlegroups.com. >> >> >> >> >> >> >> >>> >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/sympy/CAHVvXxTeWturK6WmtHKxakLqbV1yhp5_KoTPs8vPtmbu8%3D2VxQ%40mail.gmail.com. >> >> >> >> >> >> >> >> >> >>> > >> >> >> >> >> >> >> >>> > -- >> >> >> >> >> >> >> >>> > 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 sy...@googlegroups.com. >> >> >> >> >> >> >> >>> > To view this discussion on the web visit >> https://groups.google.com/d/msgid/sympy/CAMrWc1BZKcwjFvVQwZZ80P6s-CP62Tn_VN30zBEdjmqhrN44DA%40mail.gmail.com. >> >> >> >> >> >> >> >> >> >>> >> >> >> >> >> >> >> >>> -- >> >> >> >> >> >> >> >>> 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 sy...@googlegroups.com. >> >> >> >> >> >> >> >>> To view this discussion on the web visit >> https://groups.google.com/d/msgid/sympy/CAHVvXxRGFJC%2BGeowaNJcVFQsQsGq%2Bh0SW-jmYezsSzU5%3DvipcA%40mail.gmail.com. >> >> >> >> >> >> >> >> >> > >> >> >> >> >> >> >> > -- >> >> >> >> >> >> >> > 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 sy...@googlegroups.com. >> >> >> >> >> >> >> > To view this discussion on the web visit >> https://groups.google.com/d/msgid/sympy/CAMrWc1CYmTAwU4mFX%3DkOdZnmkzosN14VUAQBdteUETXas58n1w%40mail.gmail.com. >> >> >> >> >> >> >> >> > >> >> >> >> >> >> > -- >> >> >> >> >> >> > 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 sy...@googlegr > > -- You received this message because you are subscribed to the Google Groups "sympy" group. 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