Whatever your development platform of choice, if you work with large amounts of data, you should be running a database management system. In the modern day, high quality DBMSs are available for whatever platform you’re probably using, and some are even available for free. For an important project on which my team is currently working, we have implemented MySQL, in part because it runs on all the same platforms that Mathematica does. Mathematica can interface with it easily, and it comes with decent administration tools. This means that we can program on Windows or Mac and run on either one, transparently.

All of our Macs also have Windows partitions, and we have Mathematica licenses for both. While it is possible to install MySQL under both operating systems, to do so would almost inevitably lead to inconsistencies between the datasets on the two sides. In development, this can complicate debugging, and it’s completely unacceptable in production.

A nice solution is to run the MySQL server under OSX, and to set up the Windows virtual machine as a client to it. This is not hard to do with a virtualization tool like VMWare Fusion, though it is not expressly documented.

To allow the windows account to see the mySQL server on the Mac side, you need to do three things

1. Create permissions on the MySQL server under OSX. For example, create a new account for root, limiting connectivity to hosts matching whatever the windows partition identifies itself as in its network control panel (with wildcards, this often comes to something like win-qae% or win-OFT%).

2. Create a server connection from the client (windows) side. When establishing the connection from windows, you can refer to the Mac by IP address or by its local network name, as appears at the top of the Sharing control panel (Mac-Pro.local in the example below).

If you use DHCP, it may be necessary to use a wildcarded version of the laptop’s IP address (e.g. 192.168.%), though this obviously reduces security.

3. In Workbench under Windows, create a connection to a remote MySQL server. This can be pure TCP/IP (which requires you open port 3306 or any other you have set for your MySQL server) or TCP/IP over SSH (which only requires port 22 and encrypts the communication).

This is set up in the Connection Manager (left column, last entry on the WB home screen). Alternatively you can use the New Connection Wizard (click the “New Connection” link in that column).

If you want to be able to administer the database from Windows, including making database backups, you’ll need a couple more steps.

4. Go to the Mac Sharing control panel and enable Remote Login, which turns on SSH.

5. In Workbench under Windows, create a remote “command connection.” This is set up in the Server Instance Manager, which can be reached via the last link in the right column on the workbench home screen. Alternatively you can also use the New Server Instance Wizard.

When you open the server instance manager for an existing server instance, you will see a block of settings titled “Remote Adminstration” or “Remote Management” (select your instance in the left-hand list first). There are the 3 possible type settings (none, win-based, ssh-based) and you need ssh-based for this technique. The SSH hostname is your computer’s name:SSH port number (Mac-Pro.local:22 for example). SSH username is your account name on in OSX.

For the sake of your sanity, do not use the same username for your computer account and your database account. When MySQL asks for the password for user ‘fred’, you need to know which one it’s asking about.

—

If you have done all these things correctly, it will be possible to query the OSX database from within the Windows partition on the same machine, and to administer the database from either operating system.

I did on a previous occasion write about the seemingly powerful relationship between the number of radios in the U.K, and the number of registered mental defectives in that country. In Google books, I recently found a 1958 book titled Principles of Statistical Techniques, which contains examples based on that remarkable dataset, the number of mental defectives in England and Wales.

The passage in question uses the “mental defectives” data from 1935 through 1946 to illustrate the ability to mislead through the use of graphs and charts. The graphs below are courtesy of Mathematica but closely replicate those in the original text.

In[]:= mentalDefectives = {{1935, 86086}, {1936, 88060}, {1938,
92299}, {1939, 99144}, {1940, 101364}, {1941, 100876}, {1942,
98125}, {1943, 98434}, {1944, 99608}, {1945, 102225}, {1946,
102390}};

In[]:= ListPlot[mentalDefectives, PlotRange -> {86000, 102500},
AxesOrigin -> {1934, 86000},
AxesLabel -> {"Year", "Number of\nmental defectives"},
AspectRatio -> 2, PlotMarkers -> {Automatic, Small}]

In[]:= ListPlot[mentalDefectives, PlotRange -> {0, 120000},
AxesOrigin -> {1935, 0},
AxesLabel -> {"Year", "Number of\nmental defectives"},
PlotMarkers -> {Automatic, Small}]

As explained in the original text, “The impression obtained from the first graph is that there was a staggering rise in the number of mental defectives in little more than a decade, whereas the second graph gives the impression of a much slower and more gradual increase. These rather different impressions are obtained, of course, by tampering with the horizontal and vertical scales used. . . . it is essential to recognize the importance of supplying graphs, as well as tables, with full and clear labelling and if possible, the source of the information.”

Frankly, their efforts seem pretty basic. We could further steepen the first graph through population adjustment, adjustment for number of doctors, and other techniques.

In googling for more on this remarkable dataset, I found letters to the editor of The British Medical Journal in which one R.A. Gibbons of London, S.W., argues that forced sterilization of mental defectives is not sufficient to protect “the race,” and that it is necessary to sterilize also those people who “from pronounced family history” are certain to “produce idiots.” Dr. Gibbons apparent delivered a paper on this topic to the Obstetrical Section of the Royal Society of Medicine and had in published in the Journal on March 18, 1922. In this letter, he refers to “over 12,000 registered mental defectives for whom [British citizens] are taxed.”

The reference to taxation feels terribly modern (“certainly their plight tugs at the heartstrings, but you just can’t care for everybody and it wouldn’t be fiscally prudent to allow them to run amok.”)

Germany is most famous for eugenics programs, but similar sterilizations of mental defectives were taking place in the United States in the early and mid 20th century. I’ll leave that for another post.

I posted a request to mathgroup asking for guidance on why Solve[] and its brethren failed on the puzzle of the opposing clock hands. Two people have already responded with working versions from which we can draw the following conclusions,

1. We have to unwind the linedupedness[], unitTimeToHourAngle[], and unitTimeToMinuteAngle[] functions, and

2. We need to restrict the domain of the solution to the range from 0 to 1.

When we do this, we go from

In[]:= Reduce[unstraightness[unitTime] == 0 && 0 < unitTime < 1]
to
In[]:= Solve[Abs[Abs[4320 Mod[unitTime, 1/12] - unitTime*360] - 180] == 0 &&
0 < unitTime < 1, unitTime]

which does indeed produce the right answer. There are other ways to get there too; unwinding and restricting the solution to Reals or Rationals will work, though we get some redundant solutions that way.

Wolfram tech support had an even nicer solution, pointing out that my wrapper function (unstraightness[]) could take any kind of input, while the internal functions (unitTimeToHourAngle[] and unitTimeToMinuteAngle[]) are restricted to real numbers. This means that when these functions are introduced into the Reduce function, unitTime is not defined as a number, so the function is never evaluated. The simplest solution is just to change the definition of the internal functions to

In[]:= minuteAngle[unitTime_] := 4320 Mod[unitTime, 1/12]
and
In[]:= hourAngle[unitTime_] := unitTime*360 // N

whereupon

In[]:=Reduce[Abs[minuteAngle[unitTime] - hourAngle[unitTime]] == 0 &&
0 < unitTime < 1]

works like a charm.

I was recently sitting in a poorly produced play called That Hopey Changey Thing when, looking wistfully at my watch, I noticed that the hour hand and the minute hand were perfectly lined up, pointing in opposite directions. I stayed awake for the rest of the show in part thanks to the thought puzzle of how many times a day this happens, and how best to determine exactly when.

Here’s an answer using Mathematica.

First, I found it useful to think of time in terms of a unit clock cycle — the hands on our theoretical clock run from 0 to 1, rather than to 12. The following Manipulate[] command draws a clock, allowing the user to set the unit time, and translating it into both conventional time and the angle of the minute and hour hands.

In[]:= Manipulate[ Graphics[{{Opacity[0], Disk[{0, 0}, .9]}, {Opacity[.8], Thickness[.02], Line[{{0, 0}, .65 {Sin[2 Pi m], Cos[2 Pi m]}}]}, Line[{{0, 0}, .85 {Sin[12*2 Pi m], Cos[12*2 Pi m]}}]}, PlotLabel -> ToString[Floor[Mod[11 + m*12, 12] + 1]] <> " hours, " <> ToString[360.*m] <> "\[Degree]\n" <> ToString[Floor[Mod[m, 1/12]*720]] <> " minutes, " <> ToString[4320 Mod[m, 1/12]] <> "\[Degree]"], {{m, 0, "time of day"}, 0, 1}]

The code when running looks like

This was useful when debugging the formulae for the conversions; any error was immediately apparent.

Breaking out the conversions explicitly, we have

In[]:= unitTimeToListTime[unitTime_Real] := Module[{hour, minute, seconds}, hour = Floor[Mod[11 + unitTime*12, 12] + 1]; minute = Floor[Mod[unitTime, 1/12]*720]; seconds = Mod[unitTime, 1/720]*60*720; {hour, minute, seconds} ]
In[]:= unitTimeToMinuteAngle[unitTime_Real] := 4320 Mod[unitTime, 1/12]

and

In[]:= unitTimeToHourAngle[unitTime_Real] := unitTime*360 // N

These in hand, I naively thought the problem was as good as solved. I defined my problem in functional form,

In[]:= unstraightness[unitTime_] := Abs[Abs[unitTimeToMinuteAngle[unitTime] - unitTimeToHourAngle[unitTime]] - 180]

checked it with

Plot[unstraightness[unitTime], {unitTime, 0, .999999}]

and we see as expected a periodic function. There are 11 solutions when “unstraightness” == 0 per day, which makes perfectly good sense when you think about it.

I thought I could determine these with NSolve[unstraightness[unitTime] == 0, unitTime]

but sadly, this yields nothing but error messages.
NSolve::nsmet: This system cannot be solved with the methods available to NSolve. >>

What to do? I decided on brute force. Slice the day into 1,000,000 pieces and identify the 11 times which produce the most opposed clock hands.

In[]:= handangle = Table[{unitTime, Abs[unitTimeToMinuteAngle[unitTime] - unitTimeToHourAngle[unitTime]]}, {unitTime, 0, .999999, .000001}];

In[]:= closeFits = Select[handangle, Abs[#[[2]] - 180] < .002 &];

In[]:= TableForm[
Map[{#, ToString[unitTimeToListTime[#]], unitTimeToHourAngle[#],
unitTimeToMinuteAngle[#]} &, Transpose[closeFits][[1]]],
TableHeadings -> {None, {"unit time", "clock time", "hour angle",
"minute angle"}}]

Ah-hah!

As looked to be the case from the graph, these solutions are clearly spaced exactly 1/11 of the clock apart from each other. This means that we can get a more perfect solution more quickly with

In[]:= Table[ToString[unitTimeToListTime[i // N][[1]]] <> ":" <> ToString[unitTimeToListTime[i // N][[2]]] <> ":" <> ToString[unitTimeToListTime[i // N][[3]]], {i, 1/22, .9999, 1/11}] // TableForm

which returns

12:32:43.6364
1:38:10.9091
2:43:38.1818
3:49:5.45455
4:54:32.7273
6:00:00.
7:5:27.2727
8:10:54.5455
9:16:21.8182
10:21:49.0909
11:27:16.3636

Clearly, my bored glance at my watch had been at 8:10:54 pm.

One of the nice things about this solution method is that it easily generalizes to solve other “clock face” problems. For example, in a second we can determine that the hands of a clock are perfectly lined up, facing the same direction, at the following times:
12:00:00.
1:5:27.2727
2:10:54.5455
3:16:21.8182
4:21:49.0909
5:27:16.3636
6:32:43.6364
7:38:10.9091
8:43:38.1818
9:49:5.45455
10:54:32.7273

I have been playing with creating pseudotexts via markov chains recently, and may post some code soon. Creating random text via markov chains has two steps — first you analyze the source material and make a tree that describes the likelihood of different words following each other, and second you create a random walk through that tree, respecting the relative probability of the different paths.

The first step is easily done via many different methods, but if you want a tree of arbitrary depth, it becomes most elegant to use recursion. This is no harder in Mathematica than any other language, and for the sake of any novices googling around for examples of using Mathematica recursively, note the following example —

If you want to compute factorials, in practice you would probably use Mathematica’s postfix operator, !

In[1]:= 4!
Out[1]= 24


You could program your own version about a thousand different ways. I find this to be natural:

In[2]:= functionalFactorial[x_] := Apply[Times, Range[x]]
In[3]:= functionalFactorial[4]
Out[3]= 24


And if you wanted a recursive version, you could do

In[4]:= recursiveFactorial[x_] := If[x == 1, 1, x*recursiveFactorial[x - 1]]
In[5]:= recursiveFactorial[4]
Out[5]= 24


As in any recursive code, you need a test in the inner loop that defines at least one stopping point (in this case when we're multiplying by 1). Without that branch, all recursions would continue indefinitely.

Either of the user-defined functions above could be compiled for additional speed.