Suppose, for a suitable class of real-valued test functions T(\mathbb{R}^n), that \{G_x\} is a one-parameter family of distributions. That is, \forall x \in \mathbb{R}^n, G_x: T(\mathbb{R}^n) \to \mathbb{R}.
Now, suppose L is a linear differential operator. That is, \forall g \in...
I have a PDE of the following form:
f_t(t,x,y) = k f + g(x,y) f_x(t,x,y) + h(x,y) f_y(t,x,y) + c f_{yy}(t,x,y) \\
\lim_{t\to s^+} f(t,x,y) = \delta (x-y)
Here k and c are real numbers and g, h are (infinitely) smooth real-valued functions. I have been trying to learn how to do this...
Hi everyone. I have a copy of Ordinary Differential Equations by Vladimir Arnold. I'm hoping to learn more about differential equations, building up to differential equations on manifolds.
I've heard that this is a great book, but I've also heard Arnold sometimes leaves out important details...
Suppose I have a smooth curve \gamma:[0,1] \to M, where M is a smooth m-dimensional manifold such that \gamma(0) = \gamma(1), and \hat{\gamma}:=\gamma|_{[0,1)} is an injection. Suppose further that \gamma is an immersion; i.e., the pushforward \gamma_* is injective at every t\in [0,1].
Claim...
I'm trying to analyze the following Ito stochastic differential equation:
$$dX_t = \|X_t\|dW_t$$
where X_t, dX_t, W_t, dW_t \in \mathbb{R}^n. Here, dW_t is the standard Wiener process and \|\bullet\| is the L^2 norm. I'm not sure if this has an analytical solution, but I am hoping to at...
Homework Statement
Let M and N be two metric spaces. Let f:M \to N. Prove that a function that is locally Lipschitz on a compact subset W of a metric space M is Lipschitz on W.
A similar question was asked here...
I was wondering if there is a generalization of the following (roughly stated) theorem to n-dimensional systems:
Given some restrictions on the functions f and g \in \Re, let y_s(t) and x_s(t) \in \Re be solutions to the initial value problems:
\dot{x}(t) = f(x,t), x(t_0) = x_0\\...
When is the following equivalence valid?
$$\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x))$$
I was told that continuity of f is key here, but I'm not positive.
This question comes up, for instance in one proof showing the equivalence of the limit definition of the number e to the...
Hi all,
I'm having trouble finding a certain generalization of the mean value theorem for integrals. I think my conjecture is true, but I haven't been able to prove it - so maybe it isn't.
Is the following true?
If F: U \subset \mathbb{R}^{n+1} \rightarrow W \subset \mathbb{R}^{n}...
Homework Statement
Given:
|x-y| < K
x+y > K - 2
0 < K < 1
Prove:
\frac{|1-K+x|}{|1+y|} < 1
The Attempt at a Solution
I have tried using the fact that |x-y| < K \Rightarrow -K < x-y < K \Rightarrow y-K < x < y+K to write \frac{1-K+x}{1+y} < \frac{1+y}{1+y} = 1
But I can't figure...
I've been trying to search for some sort of list of top schools for graduate study in control theory. While there are rankings available for electrical engineering in general, I'm having a hard time finding rankings for specific specializations.
So far I've heard Carnegie Mellon, Stanford...
Here's the deal: I took the GRE a few weeks ago. Didn't do as well as I'd hoped in the quantitative section. I received
164 (90th percentile) Quantitative (93rd Percentile) 164 Verbal (92nd percentile) 5 Writing
I was planning on applying to top grad schools in electrical engineering...
Between control theory, photonics/optoelectronics/semiconductor devices, electromagnetics (antennas or other application), which of these fields has the potential to expose me to the most mathematics in graduate school?
Hey everyone. I'm hoping to get some advice from you guys. I am a senior studying Electrical Engineering. I am currently applying for graduate schools and for graduate fellowships (everything is mostly due mid-December). I plan to pursue a PhD. The only problem is, I'm not entirely sure which...
Example: A common problem in undergraduate electromagnetic classes is calculating the electric field inside a solid spherical charge distribution with known charge density. The common method of solution in my experience is as follows:
1) Place the origin of a coordinate system at the center...
Why exactly is there zero current flowing through a diode when it is short circuited, given the presence of the barrier potential? My current understanding is that the drift current precisely balances the diffusion current. But if you take a diode and short it with a wire, wouldn't you then also...
Hey everyone, I'm new to these forums. Being an electrical engineering major, most of my teachers aren't very concerned with the "physics" side of things. I'm hoping I can gain some insight on Maxwell's equations.
When first stating Gauss's Law for Magnetism, the only reason my electromagnetics...