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Trying to learn math :)


Oct
28
comment Help defining an open cover
@HagenvonEitzen the maps $\Phi_i$ will be solutions of an ODE. Such ODE is well defined on $\mathbb R^2$ but I have some an interest on making a distinction for initial conditions in the three of the $A_i$'s.
Oct
28
comment Help defining an open cover
Say, I want $U_0$ be an open set a bit bigger than $A_0$, and so on for the other ones. That has the "same cone shape".
Sep
24
comment How many solutions does the equation $2i+j+3k=l$ have in nonnegative integers?
thanks a lot for this nice answer!
Sep
18
comment critical points, differential equation
To linearise you take derivative, or the so called Jacobian, and then evaluate the equilibrium point $(x,y)=(a,b/a)$. You can try it yourself and then ask if you have problems.
Sep
18
comment critical points, differential equation
there you go then.
Sep
18
comment Proving that maximal interval of existence exists and that solution is unque
Is Gronwall's inequality useful? encyclopediaofmath.org/index.php/Gronwall_lemma
Sep
12
comment Stability analysis for a system of two differential equations
You need to check the eigenvalues, and eigenvectors (at each equilibrium point), they will give you a local picture near each equilibrium point. Then you might be able to draw global conclusions. Of course everything will depend on the choice of the parameters, so you may have several distinct cases to consider, e.g. all parameters are positive, or all parameters are negative, and so on.
Aug
26
comment Every solution of the system is attracted to the center manifold
**tex text is the extension.
Aug
24
comment Every solution of the system is attracted to the center manifold
I made it on Inkscape together with an extension called text ext, which allows you to "write Latex".
Aug
8
comment Central manifold theorem => Stable/unstable manifold?
I don't think so, if the centre manifold is of dimension zero, then you have the common results of stable/unstable manifold theory. I must remark though that centre manifolds are very important as the dynamics there are given by the nonlinear terms of your system. Furthermore, centre manifolds are very useful, for example: 1) in singular perturbation problems, or the so called slow-fast systems; 2) In normal form theory; and related to it, 3) In reducing the dimension of the problem you are studying.
Aug
4
comment Stable manifold for bidimensional nonlinear dynamic system with complex eigenvalues
You are correct in your idea. Imagine a node. Every orbit is an invariant manifold. In fact every orbit is an invariant stable manifold. Their union form a 2 dimensional invariant stable manifold. The advantage of having real eigenvectors is that it is enough to study the generated spaces, in the sense that if you know what happens there, then you know what happens everywhere (in the linear case of course). For the complex case, recall that you can see a complex number in the real plane. So a 2d real system with complex eigs. can be seen in a 1d complex system (like passing to polar coords.)
Jul
29
comment Doubt with smooth extensions
I guess $\tilde\pi^{−1}(x)=(x,\pm\sqrt{-x})=(-y^2,\pm y)$ is the correct way of taking $\tilde\pi^{-1}$, but since I know which $y$ we are taking since the beginning, we could set $\tilde\pi^{-1}(y)=(-y^2,y)$.
Jul
27
comment Help with operator $f(x^q)=\frac{1}{q+1}x^q$.
Thanks, I think this helps a lot. I'll accept the answer once I've checked everything carries on well in my project.
Jul
27
comment Help with operator $f(x^q)=\frac{1}{q+1}x^q$.
@ellya indeed, as you say, that is how I am thinking of the operator $f$.
Jul
27
comment Help with operator $f(x^q)=\frac{1}{q+1}x^q$.
This is indeed good for $g\in\mathbb R[x]$ as you say. What about $g\in\mathbb R[x_1,\ldots,x_n]$? ... is it possible to define $f_k(g)=\frac{1}{x_k}\int_0^{x_k}g(x_1,\ldots,x_{k-1},t,x_{k+1}\ldots,x_n)dt$ ?
Jul
27
comment Help with operator $f(x^q)=\frac{1}{q+1}x^q$.
I've edited. I didn't know about formal integration, but I just googled it and I will check it, thank you.
Jul
27
comment Help with operator $f(x^q)=\frac{1}{q+1}x^q$.
I need to think on this as I do not understand very well your argument. I thought I was sure I was working on "maps from the space of polynomials onto itself". But it seems I am doing something wrong. Thanks.
Jul
27
comment Help with operator $f(x^q)=\frac{1}{q+1}x^q$.
that is correct, I am assuming/thinking of $f$ as an operator that receives polynomials, not numbers.
Jul
27
comment Help with operator $f(x^q)=\frac{1}{q+1}x^q$.
Thanks. Do you think there is some way to require that the function "acts on $x^q$ and not in factorisations"? Or something like that? Does it makes a difference that I am not defining $f(P^q)=P/(q+1)$ but requiring that in the particular case that the argument is something "exactly like" $x^q$ I then obtain $x^q/(q+1)$ ? I set $f(P)=P^2/\phi(P)$, this is well defined right?
Jul
25
comment Help with function $f_r(x^q)=q^rx^{q-1}$
I'll use your first suggestion. Thanks a lot.