Mathematica

The previous post showed how to list all limerick primes. This post shows how to list all sonnet primes. These are primes of the form ababcdcdefefgg, the rhyme scheme of an English (Shakespearean) sonnet, where the letters a through g represent digits and a is not zero.

Here’s the Mathematica code.

number[s_] := 10100000000000 s[[1]] + 1010000000000 s[[2]] +
    1010000000 s[[3]] + 101000000 s[[4]] +
    101000 s[[5]] + 10100 s[[6]] + 11 s[[7]]

test[n_] := n > 10^13 && PrimeQ[n]

candidates = Permutations[Table[i, {i, 0, 9}], {7}];

list = Select[Map[number, candidates], test];

The function Permutations[list, {n}] creates a list of all permutations of list of length n. In this case we create all permutations the digits 0 through 9 that have length 7. These are the digits a through g.

The function number turns the permutation into a number. This function is applied to each candidate. We then select all 14-digit prime numbers from the list of candidates using test.

If we ask for Length[list] we see there are 16,942 sonnet primes.

As mentioned before, the smallest of these primes is 10102323454577.
The largest is 98987676505033.

Related post:

Limerick primes

Rosenbrock’s banana function is a famous test case for optimization software. It’s called the banana function because of its curved contours.

The definition of the function is

The function has a global minimum at (1, 1). If an optimization method starts at the point (-1.2, 1), it has to find its way to the other side of a flat, curved valley to find the optimal point.

Rosenbrock’s banana function is just one of the canonical test functions in the paper “Testing Unconstrained Optimization Software” by Moré, Garbow, and Hillstrom in ACM Transactions on Mathematical Software Vol. 7, No. 1, March 1981, pp 17-41. The starting point (-1.2, 1) mentioned above comes from the starting point required in that paper.

I mentioned a while back that I was trying out Python alternatives for tasks I have done in Mathematica. I tried making contour plots with Python using matplotlib. This was challenging to figure out. The plots did not look very good, though I imagine I could have made satisfactory plots if I had explored the options. Instead, I fired up Mathematica and produced the plot above easily using the following code.

Banana[x_, y_] := (1 - x)^2 + 100 (y - x*x)^2
ContourPlot[Banana[x, y], {x, -1.5, 1.5}, {y, 0.7, 2.0}]

Everything I do regularly in Mathematica can be done in Python. Even though Mathematica has a mind-boggling amount of functionality, I only use a tiny proportion of it. I skimmed through some of my Mathematica files to see what functions I use and then looked for Python counterparts. I found I use less of Mathematica than I imagined.

The core mathematical functions I need are in SciPy. The plotting features are in matplotlib. The SymPy library appears to have the symbolic functionality I need, though I’m as not sure about this one.

I don’t have much experience with the Python libraries listed above. I haven’t used SymPy at all; I’ve only browsed its web site. Maybe I’ll find I’d rather work in Mathematica, particularly when I’m just trying out ideas. But I want to experiment with using Python for more tasks.

As I’ve blogged about before, I’d like to consolidate my tools. I started using Emacs again because I was frustrated with using a different editor for every kind of file. One of the things I find promising about Python is that I may be able to do more in Python and reduce the number of programming languages I use regularly.

Related posts:

Languages that are easy to pick back up
Where the Unix philosophy breaks down

Regular expressions are fairly portable. There are two main flavors of regular expressions — POSIX and Perl — and more languages these days use the Perl flavor. There are some minor differences in what it means to be “like Perl” but for the most part languages that say they follow Perl’s lead specify regular expressions the same way. The differences lie in how you use regular expressions: how you form matches, how you replace strings, etc.

Mathematica uses Perl’s regular expression flavor. But how do you use regular expressions in Mathematica? I’ll give a few tips here and give more details in the notes Regular expressions in Mathematica.

First of all, unlike Perl, Mathematica specifies regular expressions with ordinary strings. This means that metacharacters have to be doubly escaped. For example, to represent the regular expression \d{4} you must use the string "\\d{4}".

The function StringCases returns a list of all matches of a regular expression in a string. If you simply want to know whether there was a match, you can use the function StringFreeQ. However, note the you probably want the opposite of the return value from StringFreeQ because it returns whether a string does not contain a match.

By default, the function StringReplace replaces all matches of a regular expression with a given replacement pattern. You can limit the number of replacements it makes by specifying an addition argument.

Related links:

Regular expressions in Mathematica
Tips for getting started with regular expressions
Languages that are easy to pick back up
Regular expressions in C++, Python, R, PowerShell

I posted some notes this evening on working with probability distributions in Mathematica and R/S-PLUS.

I much prefer Mathematica’s syntax. The first time I had to read some R code I ran across a statement something like runif(1, 3, 4). I thought it was some sort of conditional executation statement: run something if some condition holds. No, the code generates a random value uniformly from the interval (3, 4). The corresponding Mathematica syntax is Random[ UniformDistribution[3,4] ].

Another example. The statement pnorm(x, m, s) in R corresponds to PDF[ NormalDistribution[m, s], x ] in Mathematica. Both evaluate the PDF of a normal random variable with mean m and standard deviation s at the point x.

It’s a matter of taste. Some people prefer terse notation, especially for things they use frequently. I’d rather type more and remember less.

Some programming languages are much easier to come back to than others. In my previous post I mentioned that Mathematica is easy to come back to, put Perl is not.

I found it easy to come back LaTeX after not using it for a while. It has a few quirks, but it’s basically consistent. The LaTeX commands for Greek letters are their names, lower case names for lower case letters, upper case names for upper case letters. The command for a mathematical symbol is usually the name a mathematician would give the symbol. Modes always begin with \begin and end with \end.

Python also has a consistent syntax that make it easier to come back to the language after a break. Someone has said that Python is similar to Perl, except that the word “except” does not appear nearly so often in the Python documentation.

It’s more important that a language be internally consistent than conventional. Each of the languages I mentioned have their peculiarities. Mathematica uses square brackets for function argument arguments. LaTeX uses percent signs for comments. Python uses indention to denote blocks. Each of these take a little getting used to, but each makes sense in its own context.

A special case of consistency is using full names for keywords. Mathematica always spells out words in full. For example, the gamma distribution object is named GammaDistribution. I don’t mind a little extra typing. I’d rather optimize for recall and readability than minimize keystrokes since I spend more time recalling and reading than typing. (One flaw in LaTeX is that it occasionally uses unnecessary abbreviations. For example, \infty for infinity. The corresponding Mathematica keyword is Infinity.)

Mathematica was first released June 23, 1988. I started using Mathematica not long after it came out and used it for a few years. Then for several years after that I didn’t touch it. When I began using Mathematica again several years after that, I was afraid that like Rip Van Winkle, I’d find many things had changed while I was gone. Instead, I was pleasantly surprised how easy it was to start using it again.

Mathematica syntax is simple, consistent, and predictable. They got this right twenty years ago and stuck to it. They’ve managed to grow over the years without alienating users, even those of us who take a long hiatus from using the product. I’ve used Mathematica more or less regularly over the last few years, but I’ll still go for weeks at a time without using it. It’s easy to pick up every time I return to it. (The opposite of my experience with Perl.)