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Aug
12
revised Conditions to preserve Laplacian matrix
added 118 characters in body
Aug
12
comment Conditions to preserve Laplacian matrix
No, those are the principal properties that a Laplacian satisfies, in fact $\ker L=1_n$ is a consequence of $L1_n=0$ plus multiplicity assumptions. The definition of $L$ that I use is $L=D-A$ where $D$ is a diagonal "degree" matrix and $A$ is an adjacency matrix. Sorry, my syntax was misleading.
Aug
12
comment Conditions to preserve Laplacian matrix
Indeed, even more, I may assume (generically) that $\ker L=\left\{ 1_n\right\}$. However, concluding things from there is where I am stuck.
Aug
12
comment Conditions to preserve Laplacian matrix
Checking again your answer, I think it is not true since $W$ is or size $r\times n$. Then, the equation $Wx=0$ has non zero solutions.
Aug
11
comment Diagonalization of A Dynamical System with Multiple Zero Eigenvalues
exactly which part of Perko's book are you referring to? Under no extra assumption, what you claim may not be true (for example, depending on the multiplicity of the eigenvalues). It may interest you a result regarding the topological conjugacy (and decoupling) of systems having a center manifold, see e.g., staff.science.uu.nl/~kouzn101/NLDV/Lect12.pdf
Aug
5
revised Finding inhomogeneous solution given homogeneous solution.
Correct title
Aug
5
suggested approved edit on Finding inhomogeneous solution given homogeneous solution.
Aug
4
revised Conditions to preserve Laplacian matrix
added 2 characters in body
Aug
4
asked Conditions to preserve Laplacian matrix
May
13
accepted Poincaré lemma and conservative vector fields
May
13
comment Poincaré lemma and conservative vector fields
Of course! So silly of me, thank you. So should I thought as if $X$ satisfies $\frac{\partial X_i}{\partial x_j}=\frac{\partial X_j}{\partial x_i}$, then it is conservative?
May
13
asked Poincaré lemma and conservative vector fields
Mar
24
comment When does two curves do not intersect in the phase space
Recall the existence and uniqueness of solutions of differential equations. If $F$ is at least $C^1$ in $x$ (one time differentiable with continuous derivative) then the equation $x'=F$ has a unique solution for a given initial condition $x(t_0)$. If two solutions were to intersect, that would mean that to a given initial condition (the point of intersection) correspond two orbits or solution curves. This contradicts unicity, and thus, it cannot happen.
Feb
4
comment Why are equilibria so important?
I guess Poincaré would be the best person to answer. I think he was the person who started to not care about explicit solution, but to look at the behaviour. Indeed, the main problem is that nonlinear systems are rarely solvable. Still, we do not quit in trying to understand them. So the next best thing to do is to study a simplified and local version of the nonlinear problem. For this, we may use the flow-box theorem away from equilibria, and so, it remains to study what happens near equilibria.
Jan
16
awarded  Yearling
Sep
24
awarded  Autobiographer
Sep
24
accepted How many solutions does the equation $2i+j+3k=l$ have in nonnegative integers?
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
21
accepted self-adjoint operator without eigenvalues?
Sep
21
asked How many solutions does the equation $2i+j+3k=l$ have in nonnegative integers?