de ? On approche de la fin mais il faut encore répondre aux mails/messages des étudiants (qui travaillent eux aussi pendant les "vacances") et on on est pas à l'abri d'avoir une idée intéressante à rédiger pour des cours de l'an prochain !

simulation d'une solution de l'équation de Burger avec propagation d'un choc

another simulation for this year : a equation with harmonic potential and starting with a superposition of 2 orthogonal states

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resolution of a partial differentiel equation with Crank Nicolson method : every time I have to do this , new exercice for my students, it is an opportunity to play with to create a nice simulation (here a non-homogeneous advection/transport equation )

learning the simplex method to help my daughter for her courses

in numerical Analysis Crank-Nicolson Method is a finite difference scheme, stable, consistent of order 2 in space and time for linear partial differential equations created by Phyllis Nicolson to solve numerically the heat equation :

photos.app.goo.gl/T6442pXqngaL

en.wikipedia.org/wiki/Phyllis_

I made a on the simplify_sum bug I discovered 3 weeks ago. I don't know if the problem comes from the Zeilberger/Gosper algorithm or a lack of knowledge on some special functions ....

sourceforge.net/p/maxima/bugs/

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I've just learned today that finding a closed form for the limit of some series relies on Gosper's algorithm which was first implemented in (the kernel). The algorithm reduce the problem to finding some polynomial solving a system of linear equations ...❤️

en.wikipedia.org/wiki/Gosper%2

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the way to prove the formulas I gave yesterday is to study the Fourier series of the 2π periodic function equals to cos(x) on [0,π[ and 0 on [-π,0[ and apply the Dirichlet theorem to suitable x values

photos.app.goo.gl/AZXHViLWirhj

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so I've a simple closed form for the series
∑ k*sin(k*π/n)/(4*k^2-1) ( k=1 to ∞)
for all n≥2 can you guess it ?

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there was a last way to check where does the problem comes from : compute the numerical value of closed form given by and using ... so there is a bug in simplify_sum used by and !

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I also tried webapp , the closed form for the series is even more complicated , but the numerical value looks compatible with my result !

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for those who think that the problem comes from only you can check that do the same computations . But the question is why ? is the simplification false or is the numerical value of erf function at complex value i wrong?

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when I want to check such identities I try some computer algebra system like . the software often give a closed form more complicated than mine, so I compare numerical values and strange things can happend ...

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I've found these two identities while working on an examination about Fourier Series. Do you think the same formula is true when replacing the number in red by 4? If not what should be the result ?

implementing and solving numerically epidemiologic models, that's a good training for students and an example of systems of differential equations useful in real life ...

simples proof of irrationnality for logarithms

for example x=log_2(3)∉Q if not x=p/q with p,q non-zero naturals then
2^x=3=> 2^p=3^q => p=q=0 impossible!

the scheme above extends to a lot of case log_a(b) and is very similar to the irrationnality of nth-roots of rationals

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