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## Tag: Mathematica

New Asymptotic function in Mathematica 12.1
One of the new features in Mathematica 12.1 is the function Asymptotic. Here’s a quick example of using it. Here’s an asymptotic series for the log of the gamma function I wrote about here. If we ask Mathematica Asymptotic[LogGamma[z], z -> Infinity] we get simply the first term: But we …
Chebyshev approximation
In the previous post I mentioned that Remez algorithm computes the best polynomial approximation to a given function f as measured by the maximum norm. That is, for a given n, it finds the polynomial p of order n that minimizes the absolute error || f – p ||∞. The …
Remez algorithm conspicuously missing
The best polynomial approximation, in the sense of minimizing the maximum error, can be found by the Remez algorithm. I expected Mathematica to have a function implementing this algorithm, but apparently it does not have one. It has a function named MiniMaxApproximation which sounds like Remez algorithm, and it’s close, …
Analogies between Weierstrass functions and trig functions
If you look at the Wikipedia article on Weierstrass functions, you’ll find a line that says “the relation between the sigma, zeta, and ℘ functions is analogous to that between the sine, cotangent, and squared cosecant functions.” This post unpacks that sentence. Weierstrass p function First of all, what is …
Total curvature of a knot
Tie a knot in a rope and join the ends together. At each point in the rope, compute the curvature, i.e. how much the rope bends, and integrate this over the length of the rope. The Fary-Milnor theorem says the result must be greater than 4π. This post will illustrate …
What I’m going to present here is not exactly a paradox, but I couldn’t think of a better way to describe it in the space of a title. I’ll discuss two theorems about uniform convergence that seem to contradict each other, then show by an example why there’s no contradiction. …
Software to factor integers
In my previous post, I showed how changing one bit of a semiprime (i.e. the product of two primes) creates an integer that can be factored much faster. I started writing that post using Python with SymPy, but moved to Mathematica because factoring took too long. SymPy vs Mathematica When …
Sine of a googol
How do you evaluate the sine of a large number in floating point arithmetic? What does the result even mean? Sine of a trillion Let’s start by finding the sine of a trillion (1012) using floating point arithmetic. There are a couple ways to think about this. The floating point …
Sine of a googol
How do you evaluate the sine of a large number in floating point arithmetic? What does the result even mean? Sine of a trillion Let’s start by finding the sine of a trillion (1012) using floating point arithmetic. There are a couple ways to think about this. The floating point …
Spherical trig, Research Triangle, and Mathematica
This post will look at the triangle behind North Carolina’s Research Triangle using Mathematica’s geographic functions. Spherical triangles A spherical triangle is a triangle drawn on the surface of a sphere. It has three vertices, given by points on the sphere, and three sides. The sides of the triangle are …
Spherical trig, Research Triangle, and Mathematica
This post will look at the triangle behind North Carolina’s Research Triangle using Mathematica’s geographic functions. Spherical triangles A spherical triangle is a triangle drawn on the surface of a sphere. It has three vertices, given by points on the sphere, and three sides. The sides of the triangle are …
Continued fraction cryptography
Every rational number can be expanded into a continued fraction with positive integer coefficients. And the process can be reversed: given a sequence of positive integers, you can make them the coefficients in a continued fraction and reduce it to a simple fraction. In 1954, Arthur Porges published a one-page …
Group statistics
I just ran across FiniteGroupData and related functions in Mathematica. That would have made some of my earlier posts easier to write had I used this instead of writing my own code. Here’s something I find interesting. For each n, look at the groups of order at most n and …
Projecting the globe onto regular solids
I was playing around with some geographic features of Mathematica this morning and ran across an interesting example in the documentation for the PolyhedronProjection function given here. Here’s what you get when you project a map of the earth onto each of the five regular (Platonic) solids. How the images …
Sine of five degrees
Today’s the first day of a new month, which means the exponential sum of the day will be simpler than usual. The exponential sum of the day plots the partial sums of where m, d, and y are the month, day, and (two-digit) year. The n/d term is simply n, …