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  1. 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)...
  2. 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(...
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. 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...
  8. 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...
  9. 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 =...
  10. 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...
  11. 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...
  12. 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...
  13. 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...
  14. 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...
  15. E

    Radial wave function

    "(a) Use the radial wave function for the 3p orbital of a hydrogen atom (see Table 15.2) to calculate the value of r for which a node exists. (b) Find the values of r for which nodes exist for the 3s wave function of the hydrogen atom." For part a, I looked at Table 15.2 and found the equation...
  16. E

    Limits w/ l'hospital

    Homework Statement Compute the limit \lim_{x \rightarrow 0} (1 + 3x)^{3/x}. Homework Equations The Attempt at a Solution \lim_{x \rightarrow 0} (1 + 3x)^{3/x} = \lim_{x \rightarrow 0} e^{\frac{3}{x} \ln (1 + 3x)} \lim_{x \rightarrow 0} \frac{3}{x} \ln (1 + 3x) = 3 \lim_{x \rightarrow 0}...
  17. E

    L'Hospital's Rule

    Homework Statement Find the limit. \lim_{x\rightarrow -\infty} x^2e^x Homework Equations L'Hospital's Rule. The Attempt at a Solution I rewrite the limit: \lim_{x\rightarrow -\infty} \frac{e^x}{\frac{1}{x^2}} applying L'Hospital's Rule: \lim_{x\rightarrow -\infty}...
  18. E

    Find the speed (2d)

    Homework Statement A stunt driver wants to make his car jump over 8 cars parked side by side below a horizontal ramp (Fig. 3-46). (a) With what minimum speed must he drive off the horizontal ramp? The vertical height of the ramp is 1.5 m above the cars and the horizontal distance he must clear...
  19. E

    Police car and speeder when do they meet?

    Homework Statement An unmarked police car traveling a constant 95km/h is passed by a speeder traveling 140 km/h. Precisely 1.00s after the speeder passes, the policeman steps on the accelerator; if the police car's acceleration is 2.00m/s2, how much time passes before the police car overtakes...
  20. E

    Does wind affect how far sound can travel?

    I feel that this may be a silly question, but, Does wind affect how far sound can travel? I know that the air temperature affects how far a sound can travel (does wind affect air temperature?), but what about wind? I tried to find an answer via google, but I haven't found anything...
  21. E

    Is this right? (limits w/ trig)

    Determine whether f'(0) exists. f(x) = { x2sin (1/x) if x does not equal 0, 0 if x = 0} f'(0) = \lim_{x\rightarrow 0} \frac{x^2 \sin \frac{1}{x}}{x - 0} f'(0) = \lim_{x\rightarrow 0} \frac{\sin \frac{1}{x}}{\frac{1}{x}} Let u = 1/x. As x approaches 0, u approaches infinity. So I can't use lim...
  22. E

    Trigonometry in this calculus problem

    Find all the points on the graph of the function f(x) = 2 \sin x + \sin^2 x at which the tangent is horizontal. f'(x) = 2 \cos x + 2 \sin x \cos x = 0 2 \cos x (1 + \sin x) = 0 \cos x = 0 or \sin x = -1 x = \frac{\pi}{2} + n\pi or x = \frac{3\pi}{2} + 2n\pi I then plug in to find the...
  23. E

    Electromagnetic induction-help with a problem

    "A metal rod 20 cm long moves in a straight line at a speed of 4.0 m/s with its length parallel to a uniform magnetic field of 1.2 T. Find the resulting potential difference between the ends of the rod." I don't know how to solve this...
  24. E

    Alpha particle magnetism

    "An alpha particle is accelerated in the +x direction through a voltage of 1000V. The particle then moves in an undeflected path between two oppositely charged parallel plates in a uniform magnetic field of 50 mT in the +y direction. (a) If the plates are parallel to the xy plane, what is the...
  25. E

    Magnetic forces on two current carrying wires

    "Find the force per unit length on each of two long, straight, parallel wires that are 24 cm apart when one carries a current of 2.0 A and the other a current of 4.0 A in the same direction." Tell me if I'm doing this right: I use d = 0.24m, and the equations B = (mag. perm. * I)/(2*pi*d)...
  26. E

    Find the current in the coil

    "The magnetic field at the center of a 50-turn coil of radius 15 cm is 0.60mT. find the current in the coil." N=50, r = .15m, B = 6.0 * 10^-4 T, magnetic permeability = 4 * pi * 10^-7. B = (mag.per. * N * I)/ L However, I don't have L, which is the length. I do have r, the radius, but it's...
  27. E

    Using implicit differentiation to differentiate log_a (x)

    Could someone please make sure I'm doing this right. I want to find the derivative of the logarithm to the base a of x, using implicit differentiation. Let y = \log_{a} x a^y = x \frac{d}{dx} (a^y) = 1 (implicit differentiation) \frac{d}{dx} (e^{\ln a})^y = 1 \frac{d}{dx} (e^{(\ln a)y}) = 1...
  28. E

    Find the currents in the circuit branches

    http://img335.imageshack.us/img335/8383/physics4a8ua.th.png [Broken]http://img335.imageshack.us/img335/8000/physics4b6vo.th.png [Broken] "Find the currents in the circuit branches in [the first figure on the left above]." V1 = 20V, V2 = 10V, R1 = 5.0 ohms, R2 = 4.0 ohms, R3 = 6.0 ohms...
  29. E

    Intervals of increase/decrease with a exponential function

    find the intervals of increase and decrease of f(x) = e3x + e-2x. f'(x) = 3e3x - 2e-2x I set f'(x) = 0 to find the critical numbers: 3e3x = 2e-2x 3 ln e3x = 2 ln e-2x 9x = -4x x = 0, which is obviously wrong, (3e^0 - 2e^0 = 1). I found out that I had to combine the two terms using a...
  30. E

    Light bulb Circuit problem

    http://img224.imageshack.us/img224/5535/physics31id.png [Broken] "Light bulbs with the wattage ratings shown in the figure above are connected in a circuit. (a) What current does the voltage source deliver to the circuit? (b) Find the power dissipated by each bulb. (Asusume normal operating...
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