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Tag: Math

Solar Day vs Sidereal Day
How long does it take the earth to complete one rotation on its axis? The answer depends on your frame of reference. A solar day is the time it takes for the sun to appear at the same position in the sky. A sidereal day is the time it takes …
Coding theory posts
Here are some posts I’ve written that fall under the general heading of coding theory. Although coding theory can overlap with secret codes, it’s more concerned with things like Morse code, Reed-Solomon codes, and Unicode. Radio related Frequency Shift Keying Morse code numbers and abbreviations How efficient is Morse code? …
Polynomial approximations to sine
Taylor polynomials are terrific local approximations but poor global approximations. Taylor polynomials are optimal in some sense near their center, but are seldom the best choice over a large interval. This post will look at approximating sin(πx) over [-1, 1] with fifth degree polynomials. First, this plot compares the approximation …
Lagrange’s quintic and Descartes’ rule
Do fifth degree polynomial equations come up in applications? Yes, and this post will give an example. In general the three-body problem, describing the motion of three objects interacting under gravity, does not have a closed-form solution. However, Euler and Lagrange discovered a few special cases that do have closed-form …
Poisson distribution tail bounds
Yesterday Terence Tao published a blog post on bounds for the Poisson probability distribution. Specifically, he wrote about Bennett’s inequalities and a refinement that he developed or at least made explicit. Tao writes This observation is not difficult and is implicitly in the literature … I was not able to …
Developing Technology for Studying Quantum Phase Transitions
Researchers worldwide are developing new technologies for studying, documenting and triggering quantum phase transitions. Learning more about them could open up new possibilities in materials science, electronics and quantum computing. […] The post Developing Technology for Studying Quantum Phase Transitions appeared first on Datafloq.
Mentally calculating the day of the week in 2023
Mentally calculating the day of the week will be especially easy in 2023. The five-step process discussed here reduces to three steps in 2023. One of the steps involves leap years, and 2023 is not a leap year. Another step involves calculating and adding in the “year share,” and the …
Jacobi functions with complex parameter
Jacobi functions are complex-valued functions of a complex variable z and a parameter m. Often this parameter is real, and 0 ≤ m < 1. Mathematical software libraries, like Python’s SciPy, often have this restriction. However, m could be any complex number. The previous couple of posts spoke of the …
Conformal map from rectangles to half plane
As discussed in the previous post, the Jacobi elliptic function sn(z, m) is doubly periodic in the complex plane, with period 4K(m) in the horizontal direction and period 2K(1-m) in the vertical direction. Here K is the complete elliptic integral of the first kind. The function sn(z, m) maps the …
Solve for Jacobi sn parameter to have given period(s)
Here’s a calculation that I’ve had to do a few times. I’m writing it up here for my future reference and for the benefit of anybody else who needs to do the same calculation. The Jacobi sn function is doubly periodic: it has one period as you move along the …
Why determinants with columns of ones?
Geometric equations often involve a determinant with a column of 1s. For example, the equation of a line through two points or a circle through three points Or a general conic section through five points Why all the determinants and why all the 1s? When you see a determinant equal …
Test whether three complex numbers lie on an equilateral triangle
Let a, b, and c be three complex numbers. These numbers form the vertices of an equilateral triangle in the complex plane if and only if This theorem can be found in [1]. If we rotate the matrix above, we multiply its sign by -1. If we then swap two …
Arbitrary precision math in gawk
The idea of using awk for any math beyond basic arithmetic is kinda strange, and yet it has some nice features. Awk was designed for file munging, a task it does well with compact syntax. GNU awk (gawk) supports the original minimalist version of awk and adds more features. It …
Unicode arrows: math versus emoji
I used the character ↔︎︎ (U+2194) in a blog post recently and once again got bit by the giant pawn problem. That’s my name for when a character intended to be rendered as text is surprisingly rendered as an emoji. I saw when what I intended was I ran into …
What does rotating a matrix do to its determinant?
This post will look at rotating a matrix 90° and what that does to the determinant. This post was motivated by the previous post. There I quoted a paper that had a determinant with 1s in the right column. I debated rotating the matrix so that the 1s would be …
Area of a triangle in the complex plane
I recently ran across an elegant equation for the area of a triangle in the complex plane with vertices at z1, z2, and z3. [1]. This formula gives the signed area: the area is positive if the points are given in countclockwise order and negative otherwise. I’ll illustrate the formula …
Conservative vector fields
The previous post discussed the fact that the curl of a divergence is zero. The converse is true: a vector field F is the gradient of some potential φ function if ∇×F = 0. In that case we say F is a conservative vector field. There is one bit of …
{div, grad, curl} of a {div, grad, curl}
The various combinations of divergence, gradient, and curl are confusing to someone seeing them for the first time, and even for someone having seen them many times. Is the divergence of a curl zero or is it the divergence of a gradient that’s zero? And there’s another one, Is it …