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Tag: Math

How it started, how it’s going
This morning I created several mathematical versions of the popular “How it started, how it’s going” meme and posted them on Twitter. First, the Koch snowflake on @AlgebraFact: Then the logistic bifurcation on @AnalysisFact: Then cellular automaton “Rule 90” on @CompSciFact: And finally, the Lorenz system on @Diff_eq: The post …
Newton’s method spirals
In [1] the authors look at applying Newton’s root-finding method to the function f(z) = zp where p = a + bi. They show that if you start Newton’s method at z = 1, the kth iterate will be (1 – 1/p)k. This converges when a > 1/2, runs around …
Sums of consecutive reciprocals
The sum of the reciprocals of consecutive integers is never an integer. That is, for all positive integers m and n with n ≥ m, the sum is never an integer. This was proved by József Kürschák in 1908. This means that the harmonic numbers defined by are never integers. …
Peaks of Sinc
Yesterday I wrote two posts about finding the peaks of the sinc function. Both focused on numerical methods, the first using a contraction mapping and the second using Newton’s method. This post will focus on the locations of the peaks rather than ways of computing them. The first post included …
Rate of convergence for Newton’s method
In the previous post I looked at the problem of finding the peaks of the sinc function. In this post we use this problem to illustrate how two formulations of the same problem can behave very differently with Newton’s method. The previous post mentioned finding the peaks by solving either …
Reverse iteration root-finding
The sinc function is defined by sinc(x) = sin(x)/x. This function comes up constantly in signal processing. Here’s a plot. We would like to find the location of the function’s peaks. Let’s focus first on the first positive peak, the one that’s somewhere between 5 and 10. Once we can …
Zeros of trigonometric polynomials
A periodic function has at least as many real zeros as its lowest frequency Fourier component. In more detail, the Sturm-Hurwitz theorem says that has at least 2n zeros in the interval [0, 2π) if an and bn are not both zero. You could take N to be infinity if …
Quaternion square roots
If y is a quaternion, how many solutions does x² = y have? That is, does every quaternion have a square root, and if so, how many square roots does it have? A quaternion can have more than two roots. There is a example right in the definition of quaternions: …
Bootstrapping a minimal math library
Sometimes you don’t have all the math functions available that you would like. For example, maybe you have a way to calculate natural logs but you would like to calculate a log base 10. The Unix utility bc is a prime example of this. It only includes six common math …
Average sum of digits
The smaller the base you write numbers in, the smaller their digits sums will be on average. This need not be true of any given number, only for numbers on average. For example, let n = 2021. In base 10, the sum of the digits is 5, but in base …
Random polynomials revisited
A few days ago I wrote about the expected number of roots in a random polynomial where each coefficient is drawn from a standard normal, i.e. a Gaussian distribution with mean 0 and variance 1. Another class of random polynomials, one that comes up in applications to physics, draws each …
One infinity or two?
If you want to add ∞ to the real numbers, should you add one infinity or two? The answer depends on context. This post gives several examples each of when its appropriate to add one or two infinities. Two infinities: relativistic addition A couple days ago I wrote about relativistic …
Radio Frequency Bands
The radio spectrum is conventionally [1] divided into frequency bands that seem arbitrary at first glance. For example, VHF runs from 30 to 300 MHz. All the frequency band boundaries are 3 times a power of 10. Why all the 3’s? Two reasons: 3 is roughly the square root of …
Relativistic addition
Let c be a positive constant and define a new addition operation on numbers in the interval (-c, c) by This addition has several interesting properties. If x and y are small relative to c, then x ⊕ y is approximately x + y. But the closer x or y …
Galileo’s polygon theorem
William J. Milne [1] attributes the following theorem to Galileo: The area of a circle is a mean proportional between the areas of any two similar polygons, one of which is circumscribed about the circle and the other is isoparametric with the circle. So imagine a polygon P and let …
Visualizing real roots of a high degree polynomial
The previous post looked at finding the expected number of real zeros of high degree polynomials. If you wanted to see how many real roots a particular high degree polynomial has, you run into several difficulties. If you use something like Descartes’ rule of signs, you’re likely to greatly over-estimate …
Expected number of roots
Suppose you create a 100th degree polynomial by picking coefficients at random from a standard normal. How many real roots would you expect? There are 100 complex roots by the fundamental theorem of algebra, but how many would you expect to be real? A lot fewer than I would have …
Large matrices rarely have saddlepoints
A matrix is said to have a saddlepoint if an element is the smallest element in its row and the largest element in its column. For example, 0.546 is a saddlepoint in the matrix below because it is the smallest element in the third row and the largest element in …