Mathematics

For my publications and preprints, see my academic website.

Below I have described some of my first discoveries in mathematics. These are the things that inspired me to become a mathematician.

Symmetric Functions and the Binomial Theorem

The Binomial Theorem

The binomial theorem gives the following expansion:

(1+t)^r = 1+rt+1/2!*r(r-1)t^2+...

The Binomial Product Formula

There is a generalization of the binomial theorem that gives the expansion of a binomial product. The formula involves two operators:

dot F = Sum(x_i*F(i),i=1,r) (dia F)(n) = x_n*F(n)

The first operator takes a function and returns a number, whereas the second operator takes a function and returns another function (like, for example, the differential operator). Using these operators, we can provide the following generalization of the binomial theorem:

Product(1+x_i*t,i=1,r) = 1+dot*t+1/2!*dot(dot-dia)t^2+...

This is a paper I wrote describing this and related discoveries:

Factorizations of Symmetric Function Relations - 2004

Ambiguity

Division

In examining a table of fractions, we readily observe that many of its elements are repeated. These are the fractions whose numerator and denominator have a common factor, allowing them to be reduced. In the figure below, all repeated fractions have been highlighted:

$fraction table$

This is a wider view:

$fraction table$

It turns out that the reciprocal of the portion of fractions that is repeated in this way (represented by the ratio of white to black above) is

pi^2/6

Multiplication

A similar problem exists for multiplication. In looking at a multiplication table, we notice that it has many repeated elements. For example, 1*12 = 2*6 = 3*4 = 4*3 = 6*2 = 12*1. Therefore, any multiplication table that goes up to 12 will include it in six different places. If we compare the number of elements with duplicates removed to the total number of elements in the table, we can calculate a ratio between them. This is represented as the white portion of the following figure.

ambiguity of multiplication

As it turns out, this ratio is equal to 0. That is, as the table becomes very large, the portion of entries that are unique approaches 0.

Tetration

Introduction

Three basic operations in mathematics are:

addition:
multiplication:
exponentiation:

But what comes next? Well, multiplication is nothing more then repeated addition, and exponentiation is nothing more than repeated multiplication. So what is repeated exponentiation? Tetration. We write "a tetrated to b" as:

a^^b = a^a^a^...[b times]

So for example:

5^^3 = 5^5^5 = 5^3125 = ~1.9*10^2184

Notice that this is different from (5^5)^5 = 5^25 = ~3.0*10^17

Fractional Tetration

We can certainly calculate a^^b if b is a positive integer, but what if it is not an integer? In order to calculate fractional (and real) tetrations, we have to expand our definition of tetration.

Consider the sequence of functions

T^1(x)=x, T^2(x)=e^x, T^3(x)=e^e^x, ...

which are related to tetration by the equation T^n(e)=e^^n . Notice that all of these functions satisfy the relation T(e^x)=e^T(x) . By taking derivatives of both sides of this relation and evaluating at a fixed point (a [complex] number satisfying e^x=x ) we can construct a Taylor series with a parameter a satisfying T^a(x)=e^^x . This function provides a possible definition for fractional tetration.

Another possible expansion of the definition of tetration is obtained by letting (a^^b)^^(1/b) = a . This definition allows us to calculate tetrations for all positive rational tetrates.

This is a plot of y=2^^x where x ranges from 0 to 1 (exclusive) using the above definition:

plot

This is a plot from 0 to 3 (exclusive):

plot

Notice that this function is not at all continuous but it is monotonic (always increasing).

Continued Fraction Algorithms

The process of calculating a general fractional tetration (with this definition) involes a decomposition of the tetrate into a continued fraction. The number of steps in this decomposition varies in an interesting way. This is a plot of the number of steps in a continued fraction decomposition of the rational number dictated by x/y:

plot

Using a similar process with 3-ratios (fractions are 2-ratios) gives the following plot, where the third element of the ratio is fixed (at a large number with few divisors):

plot

Infinite Tetration

This is a graph of y=x^^[Infinity]

plot

When x is very small, the value of y oscillates between the two values shown.

The positive domain of the above (multivalued) function is from 0 to e^(1/e) = ~1.444667 . This function is analogous to the implicit equation: y=x^x^y which is quite interesting because it contains four separate intervals wherein there are 3, 1, 2, and 0 solutions respectively. It is graphed below (the x-axis is vertical).

plot

Complex Tetration

If we let z=(x+iy)^^t then we get the following plot for t = 5:

plot

This is a different part of the same function:

plot

This is a wider view of t = 6:

plot

The 3x+1 Problem

Introduction

The 3x+1 Problem is also known as the Collatz Problem. Take any positive integer, n, and:

If n is even, divide it by two.
If n is odd, multiply it by three and add one.

It is conjectured that, if you do this over and over, you will always end up with 1. For example, if we start with 14, we have: 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

This conjecture has never been proven.

Fulldivision

We will define a new operation called fulldivison, such that "n fulldivided by p," or n//p is the only integer n/p^k not divisible by p.

So the sequence of numbers in the Collatz Problem is generated by a repeated composition of the function:

3*n//2+1

Now, to simplify the notation of multiple fulldivisions, we will say:

(n//a)//b = n//(a,b)

I have found several functions similar to the Collatz function which also appear to converge on 1. I have tested this one for n<2.8*10^7 .

5*n//(2,3)+1

I have tested this one for n<1.0*10^5 .

3*n//(2,5,7)+7

Here are a few more that I think are convergent. It seems likely that all such functions can be made convergent by selecting appropriate fulldenominators.

4*n//(3,5,7)+1

7*n//(2,3,5)+1

11*n//(2,3,17)+1

Harmonics

Since Pythagoras, many methods have been developed to describe the consonance of musical tones (chords). Such methods are useful in algorithmic composition.

This is an early paper I wrote detailing the development of one such method.

A Mathematical Theory of Sensory Harmonics - Dec. 2000

plot plot

Fractals

The following animations were created by drawing a curve whose curliness is dictated by certain functions.

The following images were created by joining regular polygons together. Each polygon connects to a previous polygon at one edge. Which edge it connects to depends on a summation of the following sequence (or a related sequence). This sequence is obtained by counting the number of times the number indicated by the x-axis is divisible by two.

plot