### Cuneiform to Computers and the MAA Lattice Points Problem

Imagine living thousands of years ago in ancient Sumeria as a mathematician.  Your medium for storing infomration is cuneiform on clay tablets.  As you work, you stamp each equation into wet clay by making wedge shaped marks using the blunt end of a reed to make a finished document that looks like this (picture 1)[1]

When your instructor tells you to investigate the properties of a table of let's say, a hundred numbers or so, you might sigh in resignation, and plan on having results by sometime next week.

With the advent of paper and pencil, things become much easier.  There's still lots of work to be done, but the recording of the information so that it can be viewed and worked with is, comparatively speaking, a piece of cake.

Finally, though, the computer comes along and getting a table of 100 numbers is more like playing.  With +The SageMathCloud  the 100 number task suggested in the +Mathematical Association of America video below can be done easily by anyone with a web browser.  Watch the video[2] and then read on for an example of just that.

The problem is to figure out the maximum slope, a, for a line described by y = 2 + m*x where 1/2 < m < a.  The game is that the line cannot intersect with any lattice points.  These are points where the x and y coordinates both have integer values.  The first minute or so of the video has a great illustration of this.  So, we don't want to hit any of the blue lattice points shown in picture 2 as we crank up the slope of the line to the value m < a.

Dr. Tanton suggested that we take a look at the value of the line with slope m = 1/2 and how far it was from each of the lattice points directly above it out to the x coordinate of 100.  Here's the graph of the m = 1/2 line as well as the lattice points that we need to avoid as we crank the slope up from 1/2. (picture 2)

and, here's the sage code if you'd like to play with it

Now, instead of figuring out the slope that would hit each successive 'bad' lattice point, we can have Sage do it for us, and dump it out as a list of slopes at each point.  Cick the blue button below for the table of slopes necessary to just hit each lattice point.  In other words, by staying less than each successive slope, we can just miss that lattice point.

We see that the slope which will hit the 99th lattice point we're trying to miss is 50/99.  So, in the problem as asked, the maximum the slope, a can be is a < 50/99.

Another snazzy thing we can look at is how the maximum slope progressed as we went from lattice point to lattice point out towards 99.  Here's the graph: (picture 3)

Here's the Sage code used to generate it.  Remember you can modify the sage code in any of these cells and hit the blue button again to find out what effect your modifications had.

References:

2.  Video
http://youtu.be/j2J6NEGTMsY

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### Lost Phone

We were incredibly lucky to have both been in university settings when our kids were born.  When No. 1 arrived, we were both still grad students.  Not long after No. 2 arrived, (about 10 days to be exact), mom-person defended her dissertation and gained the appellation prependage Dr.

While there are lots of perks attendant to grad school, not the least of them phenomenal health insurance, that’s not the one that’s come to mind for me just now.  The one I’m most grateful for at the moment with respect to our kids was the opportunities for sheer independence.  Most days, we’d meet for lunch on the quad of whatever university we were hanging out at at the time, (physics research requires a bit of travel), to eat lunch.  During those lunches, the kids could crawl, toddle, or jog off into the distance.  There were no roads, and therefore no cars.  And, I realize now with a certain wistful bliss I had no knowledge of at the time, there were also very few people at hand that new what a baby…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…