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  1. E

    Show that x and y are independent in this joint distribution

    Ah, I see. So we have: \begin{align} Pr(x,w) &= \iint Pr(w,x,y,z)\,dydz \\ &= \iint Pr(w)Pr(z|y)Pr(y|x,w)Pr(x)\,dydz \\ &= Pr(w)Pr(x) \iint Pr(z|y)Pr(y|x,w)\,dydz \\ &= Pr(w)Pr(x) \end{align}
  2. E

    Show that x and y are independent in this joint distribution

    Given that the joint probability Pr(w,x,y,z) over four variables factorizes as Pr(w,x,y,z) = Pr(w) Pr(z|y) Pr(y|x,w)Pr(x) show that x is independent of w by showing that Pr(x,w) = Pr(x)Pr(w). Attempt: if we simply assume Pr(x,w) = Pr(x)Pr(w), then: \begin{align} Pr(w,x,y,z) &= Pr(w)...
  3. E

    Use SVD to show rank(XGY) = rank (G)

    1. Use the Singular Value Decomposition (SVD) of G to prove: rank(XGY^T) = rank (G) Given that X and Y are two full column-rank matrices, but may not have the same rank. 2. The attempt at a solution \begin{eqnarray*} XGY^T & = & X(U\Sigma V^T)Y^T \\ & = & XU \left(...
  4. E

    E to the pi * i

    Sorry, I misread your first question. So, no, the complex exponential is not an invertible function. Where does my initial post break down then?
  5. E

    E to the pi * i

    The function must be 1-1, right?
  6. E

    E to the pi * i

    something is wrong with LaTeX... it isn't displaying my tex right...
  7. E

    E to the pi * i

    1. Compute all the values of e^ {\pi i} , indicating clearly whether there is just one or many of them. Trivially, exp(pi * i) = -1. However, we can also consider e to be the complex number z, and pi * i to be the complex number alpha. Then we get: e^{\pi i} = z^{\alpha} = e^{\alpha...
  8. E

    Pigeonhole problem?

    Homework Statement Let f be a one-to-one function from X = {1,2,...n} onto X. Let f k = f(f(f(...f(x))) be the k-fold composition of f with itself. Show that there are distinct positive integers i and j such that f i (x) = f j (x) for all x in X. Homework Equations pigeonhole...
  9. E

    Potential difference Capacitance problem

    But if Q1 + Q2 = Q3, and Q_tot = Q1 + Q2 + Q3, then Q_tot = Q3 + Q3 = 2*Q3 ... right?
  10. E

    Potential difference Capacitance problem

    Homework Statement http://img509.imageshack.us/img509/8805/problem5fw8.th.png [Broken] The potential difference V = 100 V is applied to the capacitor arrangement shown in the figure. Here C1 = 10 microF, C2 = 5 microF, and C3 = 4microF. If capacitor C3 undergoes electrical breakdown (i.e...
  11. E

    Speeds of particles of a wave on a cord

    I guess 0 m/s would be the minimum. But if you count -40.32 m/s, couldn't you say it is also the minimum (it's the smallest number). However, speed is technically always positive...but then why does it say "speeds"?
  12. E

    Speeds of particles of a wave on a cord

    Homework Statement A transverse traveling wave on a cord is represented by D = 0.48 sin (5.6x + 84t) where D and x are in meters and t in seconds. Determine the maximum and minimum speeds of particles of the cord. The Attempt at a Solution I'm guessing these speeds are found by the...
  13. E

    Fractional energy in a damped harmonic oscillator

    Using the Taylor series of e^x, e^{-(2\pi b)/(m\omega_0)} = 1 + -\frac{2\pi b}{m\omega_0} Can you explain why I would drop the rest of the terms? So \frac{\Delta E}{E} = -\frac{2\pi b}{m\omega_0} first of all, is this right? second, how do i account for the negative sign?
  14. E

    Fractional energy in a damped harmonic oscillator

    Homework Statement Show that the fractional energy lost per period is \frac{\Delta E}{E} = \frac{2\pi b}{m\omega_0} = \frac{2\pi}{Q} where \omega_0 = \srqt{k/m} and Q = m\omega_0 / b Homework Equations E = 1/2 k A^2 e^{-(b/m)t} = E_0 e^{-(b/m)t} The Attempt at a Solution \Delta E = 1/2 k A^2...
  15. E

    Simple harmonic motion and diatomic molecules

    Then F = - C4/D3 * (r - r0). k = C4/D3 T = 2 pi * square root of (m/k), but what's m?
  16. E

    Simple harmonic motion and diatomic molecules

    Yeah, I think so. So do I just say that for small displacements, F is approximately equal to that first term, which is proportional to r - r0?
  17. E

    Simple harmonic motion and diatomic molecules

    use open bracket, then "tex", then close bracket
  18. E

    Simple harmonic motion and diatomic molecules

    What does a Taylor series expansion mean? Not too long ago, I learned about Taylor series, and they're still a little fuzzy in my memory. I feel pretty dumb, because everything you say I seem to have questions about how to do it :frown: ... F = -C/r2 + D/r3 = -k * (r - r0) solving for k...
  19. E

    Simple harmonic motion and diatomic molecules

    I'm not sure how to do that... How would I plug in delta r? Do I need to use the equation F = -GMm/r2?
  20. E

    Simple harmonic motion and diatomic molecules

    Homework Statement In some diatomic molecules, the force each atom exerts on the other can be approximated by F = -C/r2 + D/r3, where r is the atomic separation and C and D are positive constants. Let delta r = r - r0 be a small displacement from equilibrium, where delta r << r0. Show that...
  21. E

    SHO energy problem

    A 0.0125 kg bullet strikes a 0.300 kg block attached to a fixed horizontal spring whose spring constant is 2.25 * 10^3 N/m and sets it into vibration with an amplitude of 12.4 cm. What was the speed of the bullet if the two objects move together after impact? E = .5 k A2 = .5 m v2 Do I use m =...
  22. E

    Conservation of energy related question

    Homework Statement A skier of mass m starts from test at the top of a solid sphere of radius r and slides down its frictionless surface. (a) At what angle \theta will the skier leave the sphere? (b) If friction were present, would the skier fly off at a greater or lesser angle...
  23. E

    Work done by air resistance

    Someone told me that the work done by the air resistance would be the kinetic energy of the man if he did not reach a terminal velocity minus the kinetic energy of the man when he's at terminal velocity. So that would be: W_a = \frac{1}{2}m(2gh) - \frac{1}{2}mv_t^2 with h = 370m. does this...
  24. E

    Work done by air resistance

    Homework Statement An airplane pilot fell 370 m after jumping without his parachute opening. He landed in a snowbank, creating a crater 1.1 m deep, but survived with only minor injuries. Assuming the pilot's mass was 80 kg and his terminal velocity was 50 m/s, estimate: (a) the work done by...
  25. E

    Compressed spring with work

    The force required to compress an imperfect horizontal spring an amount x is given by F = 150x + 12x3, where x is in meters and F in newtons. If the spring is compressed 2.0m, what speed will it give to a 3.0 kg ball held against it and then released? I know how to integrate F(x) to get the...
  26. E

    Determine a formula for the acceleration of each block

    Oh I see... So under the conditions of part (a), a1 = a2. And under the conditions of part (b), there is no tension. So now there's only two unknowns in two equations.
  27. E

    Determine a formula for the acceleration of each block

    Homework Statement For two blocks, connected by a cord and sliding down the incline shown in the figure, describe the motion (a) if \mu_1 < \mu_2, and (b) if \mu_1 > \mu_2. (c) Determine a formula for the acceleration of each block and the tension FT in teh cord in terms of m1, m2, and \theta...
  28. E

    Quantum mechanics - probability of finding an electron

    I don't know how to integrate it, but my teacher said although the correct way is to integrate, we won't need to integrate....
  29. E

    Quantum mechanics - probability of finding an electron

    Homework Statement The wave function of an electron in the lowest (that is, ground) state of the hydrogen atom is \psi(r) = (\frac{1}{\pi a_0^3})^{1/2} exp(-\frac{r}{a_0}) a_0 = 0.529 \times 10^{-10} m (a) What is the probability of finding the electron inside a sphere of volume 1.0 pm3...
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