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)...
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(...
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...
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...
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...
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"?
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...
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?
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...