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...
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...
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...