Four Color Theorem and Pixels

In 1976 the Four Color Theorem was proved by Appel and Haken, with the assistance of computers to go through the large quantity of special cases. Most experts consider it proved. There had been some controversy regarding whether this constitutes a true proof if no human expert or team of experts in the field can possibly work through the proof themselves. According to the Wikipedia article on this theorem there have been independent proofs, and all require a computer to aid the human expert. It is intriguing to imagine that there may be simpler proofs of any theorem.  Here is my proposal for a way forward to a simpler proof of the Four Color Theorem. Read more…

Binding Energy in Atoms

The concept of binding energy was something that I never intuitively grasped, so I decided to finally understand it to my satisfaction.  Why is it that for lighter elements, there is a net creation of energy during fusion, and for heavy elements above iron, there is a net creation of energy during fission, and for iron there is no net creation of energy.  This explanation below I believe is correct.  However, in my search of the internet I only find the potential energy curve for the light elements that I show in the first figure.  For some reason I don’t see the curve for elements heavier than iron.  Since I guessed right on the first diagram, I feel confident of my explanation of the heavier elements.  I will have to do more study of this to confirm it.  Read more…

Least Squares in Euler Math Toolbox

For an over-determined linear system expressed in matrix form, the matrix A has more rows than columns.  Therefore the inverse of A cannot be found. The trick with least squares is to multiply both sides of the equation by transpose of A and then solve.  This makes the matrix square and invertible.  In Matlab the back slash operator will do matrix division, and if the matrix is over determined, it will attempt to do least squares.

So for the equation Ax = b, Matlab will give the least squares solution:

x = A\b

In Euler Math Toolbox (EMT) A\b will only work for invertible matrices.  So for EMT to do least squares I need to do the extra steps:

x = (A’.A)\(A’.b)

Rightmost points of the Mandelbrot Set

Using Euler Math Toolbox with the plotting function given at the end of this article, I did a search for the rightmost points of the Mandelbrot sets.  My candidate for the rightmost points are in the neighborhood of 0.4708247 + i*0.3478550 (and its conjugate).  Not sure what the significance of those values is.

right-most points of mandelbrot set zoomed in

function Mandelbrot(a,b,c,d,n=500,m=100)
z=linspace(a,b,n)+1i*linspace(c,d,n)’;
h=zeros(size(z));
loop 1 to m
h = z + (h)^2;
h=(abs(h)>10)*10+(abs(h)<=10)*h;
end
density(log(max(abs(h),1))>0.001);
setplot(a,b,c,d);
xplot();
return h;
endfunction

Making a connection between zeta(3) and zeta(6)

There is no known ‘closed form’ expressions for the Riemann zeta function at odd positive integers similar to the even integers.  It is not known whether the pattern with the even powers of the constant π extends to the odd integers.  After more than 300 hundred years of the best minds in mathematics trying to solve this riddle, not much headway has been made. I recall reading in posting in Stack Exchange that there is not likely any obvious relationship between the odd and even integer zeta values.  The following article explores how to find the values of zeta for the even integers using Fourier series.  It also makes a connection between the even and odd zeta values using a manipulation of Fourier series. Read more…Fourier Series and zeta(n)

Statistics of Car Design

I have read articles on the mathematically averaged features of human faces.  The thesis of these studies is that the more faces that are added to the averaging calculation, the more attractive the face.  There have also been counter arguments by others that individual attractive faces are more appealing than mathematically averaged faces.  It got me thinking, what about applying that same technique to averaging car shapes?  So, for example, a car designer could ask the computer to combine a Porsche, a Ferrari, and a McLaren F1, and Dodge Viper, let’s say 25% Porsche 911 Coupe, 25% Ferrari 348 TS, 25% McLaren F1, and 25% Dodge Viper. This analysis will be worked out to a small extent below. Other questions that one can be ask.  What is the average looking American sedan by model year? How does that average shape evolve over the years?  Certainly we know that cars from each era have a characteristic look.  A mathematical approach could make that design evolution very precise.  What is the one standard deviation from the average sedan look like by model year, the two standard deviations car by model year?  Is there a correlation between popularity of a car design (as measured by sales volume or popularity among collectors) and standard deviation from the average?

An example of combining two car shapes is shown in Figure 1. It is a three quarter view of a combination of 40% 1997 Ferrari F355 Berlinetta and 60% 1967 Alfa Romeo Giulia Sprint GT Veloce.  I created this image with morphing software years ago when I first thought of this idea. The car color is purple because the original images were red and blue respectively. Ultimately one needs to scan 3 dimensional coordinates of the entire surface of each car to do full justice to this idea.  Read more…Mathematically combining car shapes

Discrete Convolution in Euler Math Toolbox

I did a search for discrete convolution functions in EMT, and there is no function with the letters ‘conv’ in them that I could find.  Matlab has the function conv(a,b) to perform convolution on two vectors a and b. The function deconv(a,b) in Matlab performs deconvolution. My college textbook Introduction to Applied Mathematics by Gilbert Strang indicates that discrete convolution is essentially polynomial multiplication.

Fortunately EMT has the function polymult(a,b) which does the same thing as conv(a,b) in Matlab, and polydiv(a,b) does the same thing as deconv(a,b) in Matlab.

Desert Island Formulas

In anticipation of ever being stuck on a deserted island, and wanting to have a minimum of math formulas to remember (from which other formulas can be derived), what would they be? Let’s say that you expect to spend a few years on the island before getting rescued, and your job when you get home depends on math, and you don’t want your math knowledge to get rusty. Here are a couple candidates for useful formulas.

Trigonometric Formulas

The following equation can be used to find many trig identities, and also to derive what many mathematicians consider to be the most beautiful formula in math. Read more…Desert_Island_Formulas

Vectorized calculations in Euler Math Toolbox (EMT)

I am impressed with how computers and software opens up the exploration of math to someone like me. I certainly don’t have the mental calculating ability of an Euler or Ramanujan. But at least modern technology levels the playing field a little in the sense that I can gain insights from numerical results in a method similar to what they probably used. Granted, computers can not help one to acquire the brilliant math talent that they had in abundance. But I can do some calculations with software and an average laptop PC that Euler or Ramanujan could only have dreamed of.

One thing I like about Matlab-type software such as EMT is that vectorized calculations speeds things up considerably. Vectorized means that, for example, the function cos(kx), where x is a vector and k is an integer, cos(kx) is treated as a vector and is calculated automatically and efficiently by EMT, so I don’t have to loop through the vector x components to calculate cos(kx). I ran a calculation of a fourier series for x from 0 to for 10 000 terms and 300 points along the interval [0, 2π], comparing the vectorized time with the non-vectorized time.

Here is the sample code:
// vectorized calculation
x = linspace(0,2*pi,299);
S = zeros(1,300);
Start = time();
for k = 1 to 10000
S = S + cos(k*x)/k^3;
end;

Time1 = time()-Start
// non-vectorized calculation
Start = time();
x = linspace(0,2*pi,299);
S = zeros(1,300);
for k = 1 to 10000
for n = 1 to 300
S[n] = S[n] + cos(k*x[n])/k^3;
end;
end;
Time2 = time()-Start

Comparing Time1 and Time2 for these two cases, Time1 was about 67 times quicker than Time2. So wherever possible take advantage of the vectorized methods that EMT provides. You might start running in to trouble with memory if you calculate with a matrix or vector of a million components or so, so there is a trade off in speed versus memory at some point. According to the EMT website there are languages that one can use to make EMT loops even faster, but what comes natively with EMT is fast enough for me so far.

Methods of Integration:  Mobius Transformations

There are three general types of definite integrals for which the complex contour integration methods are useful.  Integrals of functions of cosines and sines over the interval [0, 2π] are transformed to the complex unit circle.  The second and third type of integrals are over the intervals [-∞,∞] and [0, ∞], and are solved by including the interval on the real axis as part of the contour of integration in the complex plane.

This article works out an example of a complex contour integration method that I have not seen in my textbooks or in online articles I have read. I don’t know if this is a novel method or not. Maybe someone can point me to the relevant literature. The method transforms an infinite integral on the real line from –∞ to ∞ to a circle in the complex plane using Mobius transformations. Read more…Methods of Integration Using Mobius Transformations