Recent content by JohnSimpson

Consider the system of linear equations (A+D)x=b, where D is a positive semidefinite diagonal matrix. Assume for simplicity that (A+D) is full rank for any D that we care about. In my particular case of interest, D has the form D = blkdiag(0,M) for some positive diagonal matrix M. So, a subset...

Consider the map f : \mathbb{R}^2 \rightarrow \mathbb{R}^2 defined by (x,y) \mapsto (xy-x^2, xy-y^2) I'm interested in figuring out the range of this function, but I keep thinking myself in circles. What would be a systematic method for approaching something like this?

Consider a generalized Eigenvalue problem Av = \lambda Bv where $A$ and $B$ are square matrices of the same dimension. It is known that $A$ is positive semidefinite, and that $B$ is diagonal with positive entries. It is clear that the generalized eigenvalues will be nonnegative. What else can...

So is a G only defined if you have an R, or is an R only defined when you have a finite G? I guess I would like a clear and unambiguous mathematical definition of how and when R,X,G, and B are defined. (I have read statements before such as one cannot always define an impedance and admittance...

I have somehow worked myself into a mental loop that I need a push to break out of. Consider an inductor in series with a resistor. In sinusoidal steady state, the combination has an impedance Z = R + jωL. The admittance is given by (1/Z) = (R-jωL)/(R2+(ωL)2), and if R is zero, it is simply...

Thanks!

In the context of power systems, what does it mean for a load to be unbalanced? Or for that matter, balanced?

Please Ignore. I found my problem!

I'm having a problem with NDSolve. See attached picture. I have a package generating a set of ODE's, which I display, and then the next line is the NDSolve integration. I get an "Encountered non-numerical value for a derivative at t==0" error, and I can't spot the mistake. The one thing that...

Thanks for the replies. Let me try to clear up where my confusion now lies. For a path \gamma(t) on M, and a vector field \xi(t) along gamma, one can show using the properties of a Levi-Civita connection that \frac{d}{d t} \Big \langle \Big \langle \xi(t),\xi(t) \Big \rangle \Big...

The length is nothing but the inner product of v with itself under g. Sorry, I tried to write something simplified when I probably should have just written out the whole thing. Let X : \mathbb{R} \times M \rightarrow TM be a smooth time dependent vector field on a smooth Riemannian manifold...

Hi, I'm trying to attack a problem where the Riemannian metric depends explicitly on time, and is therefore a time-dependent assignment of an inner product to the tangent space of each point on the manifold. Specifically, in coordinates I encounter a term which looks like...

Many thanks, will grab it tomorrow from the library!

Can anyone provide a nice intuitive explanation for the main properties of homothetic vector fields? Alternatively, could anyone point me in the direct of a thorough reference?

Right, sorry. What I want to do is say that epsilon is small compared to some other number, in this case 1, but to keep epsilon finite. 0 < \varepsilon << 1 Therefore, -1 + 2epsilon is ROUGHLY -1. So the first matrix above simplifies under this approximation to the second one...