Geometrically, think of f as a vector field. Imagine it as a collection of arrows describing the velocity of a fluid; that is, for each x, f(x) is the velocity of the fluid at the point x.
Now, imagine a small pebble moving in this fluid. At each point x, the trajectory of the pebble must be...
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...
Oh okay, that makes sense. So that seems to solve that problem. Now I just need to find some software to do this in since apparently with the Matlab toolboxes I have I can only solve 1-dimensional PDEs
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...
Thanks for the suggestion, but this looks like a book almost entirely devoted to linear differential equations. I don't think this is what I'm looking for, unfortunately.
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...
Thanks everyone!
I'm not very comfortable with Lie groups yet, but I think I understand what you're saying. Basically, I just need to require that \dot{\gamma}(1) = \dot{\gamma}(0), right? It sounds like requiring the curve to be a periodic function ensures this, and eliminates the issues of...
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...
Thanks a lot for the answer. Can you explain why
\theta(t) = \theta(0) + \int_0^t b_\theta(s)\,ds, \\
\phi(t) = \phi(0) + \int_0^t \frac{b_\phi(s)}{\sin \theta(s)}\,ds, becomes invalid if
ever \theta(t) < 0 or \theta(t) > \pi?
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...
Interesting. I'd never heard of sequential compactness before, but I just found some notes online and read about it. Did you mean to say "assume f is not Lipschitz"? We are given in the problem that f is locally Lipschitz, so I don't know why you would assume otherwise.
If I'm right in...
I was thinking along these lines earlier. Then d_N(f(x),f(y)) \leq d_N(f(x),f(z)) + d_N(f(z),f(y)) \leq L_1*d_M(x,z) + L_2*d_M(z,y)\\ \leq \max(L_1,L_2)*(d_M(x,z)+d_M(z,y))
But I'm stuck here. How would I use x_i and x_j? Also, how can I show that there is always a "chain" of overlapping balls...
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...
Sorry -- I was a little hasty in writing this and meant to say the functions f and g map \Re^2 to \Re, or, in the second possible theorem, map \Re^{n+1} to \Re^n.
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\\...
Ahh... Thank you for the detailed proof! I was able to use the ideas from your proof to prove the result in my original post (basically the exact same proof, but mine involved a function ##g(x)##, whereas yours involved the sequence ##\{x_n\}##)
Anyway, thanks again! And thanks to everyone else :)
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}...
It tells me that ##1 - K + x > -(y+1)##
which I actually had written on my paper before, but thanks to you I think I've realized the connection...
Since I already had from the first assumption ##1-K+x<1+y##, I now have
##-(y+1)<(1-K+x)<(y+1)##, which seems to imply ##|1-K+x| < (y+1)## as...
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...