Shifted 1's Problem

This will be a blog post on the shifted 1's problem. I reference it in my thesis but haven't finished writing it yet. I am publishing it now because I need a URL to reference.

Meaning of Saucony

Today, I ran across a posting on wiki.answers.com entitled "What is the native American meaning of Saucony?" The answer they gave is "running water." The responder was probably trying to be funny due to the Saucony running shoe company and the nearby Saucony Creek.

The real meaning is "a place of outlet, the place where a smaller stream enters into a larger one."

You can find this on page 1 of "The Centennial History of Kutztown, Pennsylvania: Celebrating the Centennial of the Incorporation of the Borough", one of those public domain works Google has preserved,

Excerpt from "The Centennial History of Kutztown" defining Saucony

As I was writing this (November 2013), I noticed that Google Maps misspells Saucony Creek as "Sacony Creek".

Why MathJax isn't working on Blogger

There are countless pages saying MathJax works under Blogger. But I couldn't get it to work, despite following the instructions carefully. My problem: I was hitting preview. I followed these instructions and then hit Publish, and everything worked great.

Everything breaks eventually. So, below I have the Laplace Equation in Cartesian coordinates so readers will know whether my method still works.

\[ \frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} + \frac{\partial^2 U}{\partial z^2}= 0 \]

Posterior Mean and Mode

Two common Bayesian point estimators are the posterior mean and the posterior mode.

The posterior mode is the most likely value of the parameter given the data. It is the highest point in the posterior pdf. (The value, y, that maximizes P(y | data)). With uninformative priors, the posterior mode is frequently the same as the MLE.

The posterior mean is the expected value of the parameter given the data. E(y | data)

The posterior mode is useful when the parameter itself is your interest. For example, when measuring the fraction of voters who support candidate x, the posterior mode is the most likely fraction after you did your survey.

The posterior mean is useful when you want to predict the future based on the parameter. For example, the probability that an individual voter will support candidate x is the posterior mean.

Visualizing why the determinant changes sign on column swap

If a matrix is broken into columns, the determinant of a matrix can be visualized as the volume enclosed by those vectors. Ask A Mathematician/Ask a Physicist had a very nice post explaining this. However, it was not clear to me why the volume changes signs when you interchange the vectors. It seems to me that:

Parallelepiped with one set of labels (image from the Ask A Mathematician article)

Is the same parallelepiped as:

Changing the labels doesn't change the parallelepiped

So, they have the same volume. But they do not have the same determinant. So, something is lacking in my intuition about determinants and volumes.

As a first step to understanding this aspect of determinants, I worked out the proof that swapping two vectors negates the determinant for the specific example of the determinant of a two-dimensional matrix, keeping track of the geometric intuition at each step.

For a two dimensional matrix, the determinant is the signed area of the enclosed parallelogram.

Rooting a Nexus One under Linux

For anyone else rooting a Nexus One under Linux

Peter Williams gives the best, most readable instructions I've found. I was able to use them successfully with three modifications

1. I had to replace the su binary/zip he mentions with another one. I unzipped it to quickly look over the contents. I used Superuser-3.1.3-arm-signed.zip which can be found at http://androidsu.com/superuser/ (The one mentioned in Peter's instructions is no longer available - but there are virus-laden packages with that name)
2. I used recovery-clockwork-5.0.2.0-passion.img from http://www.clockworkmod.com/rommanager because I am not comfortable giving some program I found in a forum link root access. clockworkmod seems to have actual products for which people pay real money and thus has a reputation to uphold. Note that you use the trackball to select in clockworkmod, not the power button.
3. I booted the recovery image using different commands. They did two things. The first command saved me from holding down the power button to go into fastboot mode. The second command boots the recovery image only in memory. The normal commands flash it in and then the OS overwrites it on full boot.

• adb reboot-bootloader
• fastboot boot recovery-clockwork-5.0.2.0-passion.img