Recent content by endeavor

1. 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. 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. 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 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 to the pi * i

The function must be 1-1, right?
6. E to the pi * i

something is wrong with LaTeX... it isn't displaying my tex right...
7. 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. 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. Potential difference Capacitance problem

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