What exactly is modified about a modified Bessel function?

Posted on by John

Special functions often have arcane names that not very helpful without some context. The previous post goes into some reasons for this. This post will expand on a point at the end of the post about “modified” functions.

Things are given their names for a reason. Discovering that reason helps you understand their motivation and use.

Pure math perspective

For each integer n, the modified Bessel function In is essentially the Bessel function Jn evaluated along the imaginary axis. Specifically,

I_n(x) = i^{-n} J_n(ix)

From a certain shallow perspective, that’s the end of the story: modified Bessel functions are modified in the sense that the argument is multiplied by i. And there’s a fiddly constant term up front for no apparent reason.

But of course that’s not the end of the story or else this wouldn’t be worth an entire post.

The equation above is analogous to the relationships between circular and hyperbolic functions

\begin{align*} \sin(ix) &= i \sinh(x) \\ \cos(ix) &= \phantom{i} \cosh(x) \\ \tan(ix) &= \phantom{i} \tanh(x) \end{align*}

These relationships are interesting because the circular and hyperbolic functions are independently meaningful. If you view these equations merely as definitions you lose their significance. Circular and hyperbolic functions were widely used before Euler discovered the connection between them.

Similarly, there’s a reason the modified Bessel functions were given a name their own. If you were led to Bessel functions and modified Bessel functions separately by different applications, you would regard the equation

I_n(x) = i^{-n} J_n(ix)

as a discovery rather than just a definition. The following section explains why someone would be interested in modified Bessel functions.

Before we move on, I’d like to explain the reason for the term i−n term. In general

I_\nu(x) = \exp(\nu\pi i/2) J_n(ix)

for all real ν. The reason for the exp(νπi/2) term is that it makes Iν(x) real for all real x.

Applied math perspective

Bessel functions often arise from solving problems with radial symmetry. Solving the wave equation in cylindrical coordinates using separation of variables leads to Bessel’s differential equation

x^2 y'' + x y' + (x^2 - \nu^2) y = 0

and its solutions Jn and Yn, Bessel functions of the first and second kind.

Solving the heat equation in cylindrical coordinates with separation of variables leads to the modified Bessel equation

x^2 y^{\prime \prime} + x y^{\prime} - (x^2 + \nu^2) y = 0

and its solutions In and Kn, the modified Bessel functions of the first and second kind.

This is the reason behind the complex analysis perspective above: the change of variables sending x to ix changes the sign of the x² term in Bessel’s equation.

Bessel functions describe radially symmetric oscillations, such as the vibrations of a drum head. Modified Bessel functions describe radially symmetric exponential growth or decay, such as in the heat in a cylinder.

Other modified functions

Struve functions are closely related to Bessel functions. The (modified) Struve functions also satisfy Bessel’s (modified) differential equation, but with a non-zero right hand side. The modified Struve functions are proportional to the unmodified Struve functions evaluated along the imaginary axis, with a proportionality constant that makes the modified Struve functions real for real arguments.

There’s a similar relationship between the Mathieu functions and modified Mathieu functions. The general pattern is that “modified” in the context of special functions means “evaluated at ix and multiplied by a constant to make the function real for real arguments.”

Why special function terminology is arcane

Posted on by John

Special functions are special because they’re useful. They can also be shrouded in arcane terminology. These two facts are related.

The more widely useful a function is, the more likely it is that the function will be discovered independently multiple times. Independent discoveries lead to varying definitions and notations. For example, there are two widely used definitions of Hermite polynomials, one used in probability and another used in physics, that only differ by a scaling factor. This also explains why there are so many variations on the definitions of the Fourier transform and spherical coordinates.

Special functions were discovered and applied before they were studied systematically. As with most mathematics, practice preceded theory. In hindsight, some names and conventions were less than ideal, at least from the perspective of someone seeking to organize a theory.

Functions can have arcane names for several reasons, one being that their usefulness became apparent long ago. If you’re instinct is that things with strange names are no longer important, you’re instinct might be backward. The strange name may be an indication that something is so important that its usefulness became apparent long ago.

Sometimes special functions have bland, uninformative names because the names stuck before anybody could think of something better. Bob looks into an interesting family of functions, then later he finds another interesting family of functions. These become known as “Bob’s functions of the first kind” and “Bob’s functions of the second kind.” These names are quite understandable at the time, though in the future people will want to know what distinguishes the functions, other than the fact that Bob discovered them, and what the groupings have in common other than the order in which Bob found them.

I started this post intending to discuss modified Bessel functions and explain what exactly is modified about them, but my preface became its own post. “Modified” is an example of the bland terminology mentioned above. There are Bessel functions and modified Bessel functions. Without more context, the “modified” term isn’t very informative. But it does provide a clue that there’s some kind of close relationship between the modified and unmodified functions. That’ll be the topic of my next post.

The difference orbit inclination makes

Posted on by John

Suppose you wanted to find the distance between Earth and Mars over time. To first approximation, both planets orbit the sun in elliptic orbits in the same plane.

If you wanted to be more accurate, you’d need to take into account the fact that the orbit of Mars is tilted about 1.85° relative to the Earth’s orbit. How much difference does that make?

To simplify things, let’s assume the Earth orbits the sun in a circle of radius 1 and Mars orbits the sun in a circle of radius 1.5. The distance between Earth and Mars over time would be basically sinusoidal.

How much does inclination contribute to this distance? In other words, what is the difference between the distance accounting for the inclination of Mars’ orbit and the distance if we assume the two orbits are in the same plane?

This plot gives the answer.

The effect is not large, about three orders of magnitude smaller than the main effect, but it’s interesting how erratic it is.

The plots were made with the following code.

from numpy import *

R = 1.5
T = R**1.5 # Kepler's third law

def f(t, theta):
    return sqrt(
        (cos(t) - R*cos(t/T)*cos(theta))**2 +
        (sin(t) - R*sin(t/T))**2 +
        (R*sin(theta)*cos(t/T))**2
    )

The first plot graphs f(t, θ) and the second graphs f(t, θ) − f(t, 0).

Coming soon

Posted on by John

There’s a pizza shop near my home with a sign out front that says “Coming Soon.” When I drove by it this morning I thought about how you would model the time until an event happens that is “coming soon.”

Suppose I look at the sign one day and guess how many days until the pizza shop will open. When I drive by a week later and guess again, should my guess be smaller? You might argue that the shop will open some day, fixed in time but unknown to me, and so every day I’m one day closer to the eventual opening.

You might model the pizza shop opening like radioactive decay and say that the estimated number of days until it opens is always the same until the day it actually opens.

Now I think this shop has been “coming soon” for over a year. So instead of decreasing, every day I increase my estimate of the time until the shop opens. Something has gone wrong that the owners didn’t expect when they put up the sign.

Maybe the reasonable thing would be for estimated days until opening to decrease over time, but only up to a point. After some point, the longer a business has been “coming soon” the less like that it is coming soon, or coming at all.

This brings up an interesting point about modeling. There are two probability distributions at work: the probability that the shop will eventually open, and the time until opening assuming it eventually opens.

When the sign first goes up saying the business is coming soon, there’s some change that it is in fact not coming. Maybe you’re optimistic and think this probability is small, but it would seem unreasonable to think the probability is zero. That means the expected number of days until opening is always infinite. If there’s a probability ε that the shop never opens, the expected time to opening is

ε × ∞ + (1 − ε) × something = ∞.

How would you know whether an ancient culture had zero?

Posted on by John

A few weeks ago I wrote about the number system used in labeling spreadsheet columns. Labels run from A through Z, then AA through AZ, etc. This looks a lot like base 26, but it’s not quite the same. It has no analog of zero. If Z were like zero, Y would be followed by AZ. The Excel labeling system is not base 26, but what’s called bijective base 26.

If you found fragments of writing from an ancient culture and inferred that five symbols were used as digits, how could you distinguish base 5 from bijective base 5? Suppose you believe these five symbols were digits

★ ☂︎ ☘︎ ☗ ☢︎

but you don’t know in what order. You just see sequences like ☂︎☘︎☢︎ and ★★☂︎ and believe they’re numbers.

If you noticed that numbers often contain ☘︎, but ☘︎ never appears at the beginning of a number, you might infer that ☘︎ is a zero. But this would take a fairly large sample. If you found only 20 numbers, for example, you could hardly conclude ☘︎ never appears at the beginning of a number just because it doesn’t come at the beginning of any number you’ve seen.

Now suppose you’ve found writing with more number symbols. Say you’ve found 17 numeric symbols. You might infer that the writing used a base 20 system, because it would be hard to imagine a human culture using base 17. Now imagine you find more fragments and confirmed that indeed there are 20 numeric symbols. Approached as a purely statistical problem, you’d need a very large sample to infer what the digits correspond to and whether they use a base 20 or bijective base 20 system (or some other system).

You’re best hope is to find numbers in some context where you know what number is being represented. If you knew somehow that some symbol corresponds to 20, then you’d know they didn’t use base 20 because base b doesn’t have a single symbol for b.

If you had a huge collection of numbers but no context, which is highly unlikely, you could use Benford’s law to infer the meaning of the number symbols: the most common leading digit is probably 1, the next most common is probably 2, etc. This is interesting to think about, but it seems much more realistic that a number system would be decoded by finding context, such as a list of consecutive numbers or numbers with known meaning.

 

AI-generated ASCII diagrams

Posted on by John

I like AI-generated ASCII diagrams. Because nobody would ask AI to generate ASCII diagrams, and so, it’s congruous. I like incongruity [1].

Aside from the incongruity of using a gazillion-parameter neural network to make 1970’s style ASCII art, ASCII diagrams have some uses. They’re absolutely tiny compared to image files. But more importantly they can be inserted into plain text files, such as source code or markdown. A diagram embedded directly into a source file cannot become separated from the code.

ASCII diagrams are tedious to create, though there are tools to mitigate the tedium. But if an AI can generate the diagram, the tedium goes away.

I was curious how well Claude could create ASCII diagrams, so I tried a few examples. I hope these render well in whatever format you’re reading this post. They look fine for me previewing the post in a browser. I expect they might not turn out so well in an RSS reader.

I asked it to reproduce the graphs from my recently post on the graph imbalance theorem and the first diagram turned out nicely.

                 +-------+                                +-------+
                 |   A   |--------------------------------|   B   |
                 +-------+                                +-------+
                     |                                        |
                     |                                        |
   ------------------|------------------             ---------|---------
   |        |        |        |        |             |        |        |
   |        |        |        |        |             |        |        |
+-----+  +-----+  +-----+  +-----+  +-----+       +-----+  +-----+  +-----+
| a0  |  | a1  |  | a2  |  | a3  |  | a4  |       | b0  |  | b1  |  | b2  |
+-----+  +-----+  +-----+  +-----+  +-----+       +-----+  +-----+  +-----+

The second network is more complicated and so the corresponding ASCII diagram is hard to read.

   +----------------------------------------------------------------+
   |                                                                |
   |+-----------------------------------------------+               |
   ||                                               |               |
   ||+-------------------------------+              |               |
  +-------+       +-------+       +-------+       +-------+       +-------+
  |  R1   |-------|  R2   |-------|  R3   |-------|  R4   |-------|  R5   |
  +-------+       +-------+       +-------+       +-------+       +-------+
      |             | | |          |   |           |   |           |  |  |
     ++             | | |          |   |           |   |           |  |  |
     | +---------------------------+   |           |   |           |  |  |
     | |            ++| |              |           |   |           |  |  |
     | |             || +--------------+           |   |           |  |  |
     | |             || |  +---------------------------------------+  |  |
     | |             |+-|--|-------------+         |   |              |  |
     | |             |  |  |             |  +------+   |              |  |
     | |             |  |  |             |  |  +----------------------+  |
     | |             |  +--|-------------|--|--|-------------+           |
     | |             |  |  |             |  |  |       +-----|--+        |
     | |             |  |  |             |  |  |             |  |  +-----+
     | |             |  |  |             |  |  |             |  |  |
  +-------+         +-------+           +-------+           +-------+
  |  G1   |         |  B1   |           |  B2   |           |  B3   |
  +-------+         +-------+           +-------+           +-------+

For a third example, here is a fairly complicated diagram that nevertheless lends itself to a readable ASCII diagram. It’s a Feistel network diagram for DES encryption.

   +-------------+                    +-------------+
   |   L(i-1)    |                    |   R(i-1)    |--------
   +-------------+                    +-------------+       |
          |                                  |              |
          |                                  |              |
          |                      +-----------------------+  |
          |                      |   E (expand 32->48)   |  |
          |                      +-----------------------+  |
          |                                  |              |
          |                      +-----------------------+  |
          |                      |     XOR with K(i)     |  |
          |                      +-----------------------+  |
          |                                  |              |
          |                      +-----------------------+  |
          |                      |    S-boxes S1..S8     |  |
          |                      +-----------------------+  |
          |                                  |              |
          |                      +-----------------------+  |
          |                      |    P (permutation)    |  |
          |                      +-----------------------+  |
          |                                  |              |
          |                                  |              |
          |            +-------+             |              |
          +------------|  XOR  |-------------+              |
                       +-------+                            |
                           |                                |
          +----------------|--------------------------------+
          |                +------------------+
          |                                   |
   +-------------+                    +-------------+
   |    L(i)     |                    |    R(i)     |
   +-------------+                    +-------------+

Related posts

[1] See Christian Wolff’s discussion of dogs playing poker in The Accountant (2016).

Big little hexagon

Posted on by John

A new paper just came out, The Maximum-Area Small Polygon Problem. The paper solves the problem of finding, for each n, the n-gon with diameter 1 and maximum area.

For odd n, the solution is what you might expect: a regular n-gon. I would expect this to be the solution for even n as well, but it’s not.

In 1974 [1] Ron Graham found a solution for n = 6, a hexagon with unit diameter and area larger than a regular hexagon with unit diameter. Polygons with diameter ≤ 1 are called “small”, and he found the “largest” (i.e. maximum area) small hexagon.

The vertices of Graham’s hexagon are given below.

  A = (0.0000000000,  0.0000000000)
  C = (0.4023506913, -0.5000000000)
  F = (0.9390533483, -0.3437714489)
  B = (1.0000000000,  0.0000000000)
  E = (0.9390533483,  0.3437714489)
  D = (0.4023506913,  0.5000000000)

You can verify that the distance between any pair of vertices is no more than 1 and that the area of Graham’s hexagon is 0.674981.

The area of a regular hexagon of diameter 1 is (3/8)√3 = 0.649519, and the area of Graham’s hexagon is about 3.9% larger.

[1] R. L. Graham. The Largest Small Hexagon. Journal of Combinatorial Theory (A) 18, 165–170 (1975). The paper was submitted February 22, 1974 and published in 1975.

The imbalance theorem

Posted on by John

The imbalance conjecture is now a theorem. James Alexander Schreib and Yousof Yavari posted a proof last week.

What does the conjecture theorem say? Start with a graph G with no edge between two nodes of the same degree. Then for every edge, calculate the absolute value of the difference of the degree of each end. The imbalance theorem says there exists another graph H whose vertices have degrees corresponding to the differences of degrees in G.

For example, let G be the graph below.

The edges from the top red vertex A to each of the blue vertices around it all have degree difference 5 because A has degree 6 and the vertices a0 to a4 have degree 1. The edge between the two red vertices, A and B, has degree difference 2. The remaining vertices have degree difference 3.

So the multiset of degree differences is

{5, 5, 5, 5, 5, 2, 3, 3, 3}

The imbalance theorem says there exists a graph H whose nodes have these degrees. Here is an example of such an H.

Note that in H, the 5 red nodes have degree 5, the single green node has degree 2, and the three blue nodes have degree 3.

More graph posts

Mean distance to the sun

Posted on by John

Suppose you have a planet in an elliptical orbit around a star. The math is identical for any light object orbiting a heavy object, such as a moon or satellite orbiting a planet, but we’ll call the heavy object a star and the light object a planet.

The center of the star is not quite the center of the orbit. The planet moves along an ellipse with the star at one focus of that ellipse.

Let a be the semi-major axis of planet’s orbit, the maximum distance from the center of the ellipse to a point on the ellipse. Then the distance of a focus to the center of the ellipse is ae where e is the eccentricity of the ellipse. This defines eccentricity. The center of earth’s orbit is between three and four solar radii away from the center of the sun [1].

The planet is farthest from the star when it is along the major axis of the ellipse on the opposite side as the star. The distance is then a + ae, the distance to the center plus the distance from the center to the star. On the opposite side of its orbit, the planet is closest to the star. There the distance is a − ae. In summary the maximum distance to the star is

a(1 + e)

and the minimum distance is

a(1 − e).

If you had to guess the average distance between the planet and its star, a would be a good guess since it’s the average of the maximum and minimum distance. And that’s a good approximation, provided e is small. The mean distance over time is

a(1 + ½e²).

See derivation. The average distance is greater than a because the planet moves faster when nearest the star and slower when further from the star.

The relative error in approximating the mean distance by a is then ½e². When e is small, ½e² is very small. For the earth’s orbit, e = 0.01671, and so the approximation is off by around 0.014%.

The eccentricity of Pluto’s orbit is 0.2488, and so in that case the approximation is off by about 3.1%. The eccentricity of a Molniya orbit, used by some Russian satellites, is 0.74 [2]. For such satellites the error in approximating the mean distance to earth as the semimajor axis is around 27%.

Related posts

[1] For earth’s orbit, e = 0.01671, a = 1.496×1011 m, and the sun’s radius is r = 6.957×108 m. And so ea / r = 3.59.

[2] An object in such a highly elliptical orbit will spend a long time at the far side of its orbit, i.e. over Russia. Sort of a poor man’s geostationary orbit.

Proportion of 1s in a Hadamard matrix

Posted on by John

The first post in the recent series of posts on Hadamard matrices describes a way of constructing new Hadamard matrices from two other Hadamard matrices by taking their Kronecker product.

Starting with a Hadamard matrix H0 and a Hadamard matrix G, you can construct a sequence of Hadamard matrices by

Hn+1 = G ⊗ Hn

for  positive integers n. This is known as the generalized Sylvester method.

Let pn be the proportion of 1s in Hn and let q be the proportion of 1s in G. Then you can show that the recurrence holds

pn+1 = q pn + (1 − q)(1 − pn).

You can solve the recurrence to show that

limn → ∞ pn = ½

and so as the iterations proceed, the ratio of number of 1s to the number of −1s approaches 1.

This doesn’t say anything Hadamard matrices in general, but it does apply to all Hadamard matrices created by repeatedly applying the generalized Sylvester method.

If you set G and H equal to the matrix

\begin{bmatrix} 1 & 1\\ 1 & -1 \end{bmatrix}

then p0 = q = ¾. Then for n = 1, 2, 3, …, 8 the values of pn are

0.625
0.5625
0.53125
0.515625
0.5078125
0.50390625
0.501953125
0.5009765625.