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  1. O

    Geometrical interpretation of this property

    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...
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    Distributional derivative of one-parameter family of distributions

    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...
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    How to numerically solve a PDE with delta function boundary condition?

    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
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    How to numerically solve a PDE with delta function boundary condition?

    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...
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    ODE textbook recommentation (Arnold or other?)

    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.
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    ODE textbook recommentation (Arnold or other?)

    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...
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    Is every smooth simple closed curve a smooth embedding of the circle?

    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...
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    Is every smooth simple closed curve a smooth embedding of the circle?

    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...
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    What is the solution to this ODE (and SDE)?

    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?
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    What is the solution to this ODE (and SDE)?

    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...
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    Prove that Locally Lipschitz on a Compact Set implies Lipschitz

    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...
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    Prove that Locally Lipschitz on a Compact Set implies Lipschitz

    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...
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    Prove that Locally Lipschitz on a Compact Set implies Lipschitz

    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...
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    How can I prove this theorem on differential inclusions?

    Consider the following differential equations with initial conditions at time $t_0$ specified: \dot{x}_1 = f_1(x_1,t) ; \,\,\,x_1: [t_0,T]\to\mathbb{R}^n, f_1:\mathbb{R}^n\times[t_0,T]\to\mathbb{R}^n \\ \vdots \\ \dot{x}_k = f_k(x_k,t); \,\,\,x_1: [t_0,T]\to\mathbb{R}^n...
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    Generalization of Comparison Theorem

    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.
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    Generalization of Comparison Theorem

    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\\...
  17. O

    Moving limits in and out of functions

    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 :)
  18. O

    Moving limits in and out of functions

    Interesting. But how would you show that this definition is equivalent to, for example, the epsilon-delta definition?
  19. O

    Moving limits in and out of functions

    What exactly do you mean by the sequence x_n here?
  20. O

    Moving limits in and out of functions

    Thanks for the specific example. Can you prove (or point me to the proof) of the general case?
  21. O

    Moving limits in and out of functions

    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...
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    Generalization of Mean Value Theorem for Integrals Needed

    That makes sense. Thanks a lot for the help.
  23. O

    Generalization of Mean Value Theorem for Integrals Needed

    I'm actually fairly certain now that my conjecture is false...
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    Generalization of Mean Value Theorem for Integrals Needed

    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}...
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    Inequality problem

    That makes sense to me. Thanks again for the help.
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    Inequality problem

    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...
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    Inequality problem

    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...
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    Schools GPA and Graduate School Admittance

    I'm interested in this as well. Only, I am on the other end of the spectrum (higher GPA, probably less stellar letters of recommendation)
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    Schools Best grad schools for control theory/control systems engineering?

    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...
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