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...
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...
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...
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 =...
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...
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...
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...
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...
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.
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...
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...