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    Uniform Convergence of a Sequence of Functions

    I don't remember, but in looking at your argument, why don't you include [\itex]|x|\geq 0[\itex]. Secondly, if you're going to do it for x<0, use the definition of |\cdot|, do the same thing, multiply both top and bottom by the conjugate and I think you have what you got. I hope this helps.
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    Michelson Interferometer IR Lab Question

    And the sample is hit with radiation and the radiation goes through the sample to which this radiation goes through the same Michelson Interferometer path?
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    Michelson Interferometer IR Lab Question

    I wish I knew how to draw a diagram. Let me see if I understand what you're saying. So the radiation goes through the interferometer first (as a reference), then the radiation goes through the sample. Do you then compare intensity values of the beam no sample to the intensity values of the beam...
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    Michelson Interferometer IR Lab Question

    I've been reading up on Fourier Transform Infrared Spectroscopy and the Michelson Interferometer. My main sources are "Principles of Instrumental Analysis" by Skoog etc and Fourier - Transform Infrared Spectrometry by Griffiths and Haseth. I believe I understand the theoretical principles...
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    Uniform Convergence question

    I see, very cool, I mean I always felt the "speed of convergence thing" but that does not really stick to my head. It definitely will now. I definitely like the numerical analysis as opposed to analysis proper mention. That definitely cleared the air a bit. I don't know if I felt the water...
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    Uniform Convergence question

    I really like this example, but I was thinking in the sense of a more physics like interpretation if that makes any sense.
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    Uniform Convergence question

    Yes, but I think Erland hit that point, which makes more sense. I totally feel you on your "not quite proof". Do you have a physical meaning for uniform convergence vs. pointwise convergence by any chance?
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    Uniform Convergence question

    That makes so much more sense Erland. I didn't look at it that way.
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    Uniform Convergence question

    So I'm reading "An Introduction to Wavelet Analysis" by David F. Walnut and it's saying that the following sequence " (x^n)_{n\in \mathbb{N}} converges uniformly to zero on [-\alpha, \alpha] for all 0 < \alpha < 1 but does not converge uniformly to zero on (-1, 1) " My problem is that isn't...
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    Equilibria in Nonlinear Systems

    Thanks so much! I know we want to do that in order to make the linearization work correct?
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    Equilibria in Nonlinear Systems

    So I'm reading the Example on page 161 of Differential Equations, Dynamical Systems and an Introduction to Chaos by Hirsh, Smale, and Devaney. I'm not understanding everything. So given the system x' = x + y^2 y' = -y we see this is non-linear. I get it that near the origin, y^2 tends to...
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    Solving z'(t) = az(t)

    Yeah that looks better!
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    Solving z'(t) = az(t)

    Thanks Mark, shortly after I saw that I did the work and Now I'm convinced x' = ax and y' = ay is now justified. Now suppose we chose the principal value log, will that then justify the separation of variables? I feel like what I'm about to do is a little naive, but let's give it a shot...
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    Solving z'(t) = az(t)

    By directly do you mean separation of variables? \dfrac{dz}{z} = a dt . First we need to define complex integration which would require line integration. I think that's what you mean right? Thing is, antiderivatives are a little more trickier in \mathbb{C} than they are with straight up real...
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    Solving z'(t) = az(t)

    So z'(t) = x'(t) + i y'(t), I just want to make sure that what I'm going to do is OK. I'm trying to solve z' = az where a = \alpha + i \beta z'(t) = az(t) \Rightarrow x'(t) + iy'(t) = a(x(t) + iy(t)) \Rightarrow x'(t) = a x(t), y'(t) = a y(t) If you could give me a justification that'll...
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    Non-autonomous diff eq and periodicity

    Hmm, ##(\Rightarrow)## We know that ##x(t+T) = x(t)##, and in particular we have that ##x(T) = x(0)##. So ##x(T) = x(0) e^{\int_0^T p(s) ds} = x(0) = x(0) e^0##. We see that ##e^{\int_0^T p(s) ds} = e^0##, which implies ##\int_0^T p(s) ds = 0##. We already have the fact that ##p## is ##T##...
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    Non-autonomous diff eq and periodicity

    ##(\Leftarrow)## Well ##x(T) = x(0) e^{\int_0^T p(s) ds} \Rightarrow x(T) = x(0) \Rightarrow x(0 + T) = x(0)## so when t = 0 we have that situation covered. Now we want to show periodicity for all t, ##x(t+T) = x(0) e^{\int_T^{t+T} p(s) ds} = x(0) e^{\int_T^{t+T}p(s)ds} = x(0) e^{\int_0^t...
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    Non-autonomous diff eq and periodicity

    Homework Statement Consider the first-order non-autonomous equation ##x' = p(t) x##, where ##p(t) ## is differentiable and periodic with period ##T##. Prove that all solutions of this equation are period with period ##T## if and only if ##\int_0^T p(s) ds = 0##. Homework Equations The...
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    Monotone Convergence Theorem

    I think I'm covered, because I'll have to consider the case when f is finite and f is infinite correct?
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    Monotone Convergence Theorem

    Sorry I just made some edits, I saw my mistake.
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    Monotone Convergence Theorem

    yes f \wedge n = \min(f, n) so define f_1(x) = \min(f(x),1) f_2(x) = \min(f(x),2) \vdots f_n(x) = \min(f(x),n) In short we still have that f_1(x) \leq f_2(x) \leq \cdots \leq f_n(x) for all x. Well I want \lim_{n \rightarrow \infty}f_n \rightarrow f for all x because if I have this...
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    Monotone Convergence Theorem

    Homework Statement Let f be a non-negative measurable function. Prove that \lim _{n \rightarrow \infty} \int (f \wedge n) \rightarrow \int f. The Attempt at a Solution I feel like I'm supposed to use the monotone convergence theorem. I don't know if I'm on the right track but I created...
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    Uniform Convergence of a Sequence of Functions

    mmmmmmm OK I'll do that! but yes that's what I meant
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    Uniform Convergence of a Sequence of Functions

    Sorry to bring up old news but I haven't really thought about this problem since, but anyway, pasmith all I'm getting from you is that g_n is one-to-one. g_n(x) = f_n(x) - f(x) = \left( x^2 - \dfrac{1}{n} \right)^{\frac{1}{2}} - |x| = \left(x^2 - \dfrac{1}{n} \right)^{\frac{1}{2}} - \sqrt{x}...
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    Uniform Convergence of a Sequence of Functions

    Thanks, I think I'm gonna have to think about this a little while
  26. B

    Uniform Convergence of a Sequence of Functions

    Just to be clear, I need to find the \lim_{n \rightarrow \infty} \text{sup}_{x \in \mathbb{R}} | f_n(x) - f(x) | and show that as n \rightarrow \infty then |f_n - f(x) | \rightarrow 0
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    Uniform Convergence of a Sequence of Functions

    Well I suppose we should then go to the definition right. Let \epsilon > 0 , I need to find the right N(\epsilon) such that whenever n\geq N, we have that \left| \sqrt{x^2 + \dfrac{1}{n}} - |x| \right| < \epsilon . So I have a hard time messing with that \left| \sqrt{x^2 + \dfrac{1}{n}}...
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    Uniform Convergence of a Sequence of Functions

    Homework Statement Define f_n : \mathbb{R} \rightarrow \mathbb{R} by f_n(x) = \left( x^2 + \dfrac{1}{n} \right)^{\frac{1}{2}} Show that f_n(x) \rightarrow |x| converges uniformly on compact subsets of \mathbb{R} Show that the convergence is uniform in all of \mathbb{R}...
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    Question on Definition of Cover of a Set

    So when we have an open cover of a set X means we have a collection of sets \{ E_\alpha\}_{\alpha \in I} such that X \subset \bigcup_{\alpha \in I} E_\alpha . My question comes from measure theory, on the question of finite \sigma -measures, The definition I'm readying says \mu is \sigma...
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    Question on Sigma Algebras

    Sorry I meant \bigcap B_i is not in \mathcal{A}_1 \cup \mathcal{A}_2
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