Mixture of Lebesque and Riemann integral

Here are 3 types of integration, the first is the usual Riemann integration where the domain is split evenly into rectangles and the second is Lebesque where the range is split into even intervals. The bottom one in this picture is the one I thought of where you use whichever results in fewer rectangles for a given delta size over an interval. I was thinking for functions that are very expensive on a computer to evaluate it would speed up the evaluation of the integral.

possible simplification of 2d heat equation

I've been looking around at numerical solutions to the 2 dimensional heat equation, and I thought I'd make an effort at an explicit solution.
I start here with an isolated plane and a radially symmetric heat map f. The height indicates higher temperature. I then calculate a final state function g that is the integral of f over the plane divided by the total area which is 4 in my plot.

Because the plane is isolated I figure the total thermal energy is invariant. Then I consider the homeomorphism of the heat map f transitioning to the final state map g. To simulate the fact that the closer a heat map f(t) gets to g the slower it transitions, I use a logarithmic proportionality factor A.
The final command for my simulation:

Here is the animation:

It at least looks similar to simulations of the numerical heat equation, it seems to go along with the fact that the heat flow is in the opposite direction of the temperature gradient. And it goes along well with Newton's law of cooling that each small area cools slower the closer its temperature is to it's surroundings. 

Thurston Labs

I've been running several experiments in my laboratory. Here's a picture of the gently restrained chaos.

 My first experiment was using the plastic from yogurt containers and melting it down in acetone. This makes a gooey substance that can be formed into shapes and then allowed to dry and it goes back to a flimsy plastic like you started off with. But more interesting is I found that if you grind graphite from pencil leads down very fine and knead it into the goo when it dries it hardens to much harder than the plastic started off. Here's a picture of unadulterated and the graphite hardened plastics.

The one on the left is so hard and strong that I haven't been able to bend or break it using all my force. The plain on the right bends easily.
    The next experiment was using the same kind of plastic as above but I mixed in aluminum hydroxide. I made the aluminum hydroxide by disolving aluminum foil in sodium hydroxide that I bought online. The interesting result of this experiment was that the resulting plastic conducts electricity! But on the downside it made the plastic weak and brittle. So I'm going to try and combine these two experiments and see if I can make a strong plastic that conducts.

I thought of a lot of uses for conductive plastic. One is considering that usually a flashlight has wires and metal tabs inside it touching the battery, with a conductive plastic it could be built so the "wire" is a stripe in the all-plastic body. Or a circuit board could be made with a plastic foundation and something like a hot glue gun to put down a layer of plastic instead of solder. Searching online I did find there are some examples of conductive plastic already being produced but they are I think being produced with different base materials and methods. The graphite hardening step is something similar to how they currently make carbon reinforced products but I'm using plastics instead of epoxies and ground graphite instead of fiber woven into mats.

Flag Generator

I had this idea as a way to generate some AC electricity. Basically as a flag ripples in the wind it is connected to a framework of springs that will tend to oscillate at their resonant frequency. This oscillation is converted to electricity by moving a magnet through a coil. The picture might help:

If you've ever held a flag in the wind you'll know that even a moderate breeze can get it jerking your arms pretty hard. Ideally you'd want to construct the flag so it ripples at the same frequency to a given wind speed that the spring's vibrate when you press and release one. In the picture above the magnet is connected to the bar that the flag is on I wasn't sure how to make that clearer. 

Update:

This is the first prototype. I'm ordering the wire for the electrical part and it's a little tough to see in the video because it was nighttime, but basically I've got a plastic streamer inducing a vibration in the springs that I'll try and capture the electricity from. The second part of the video I'm just showing how the springs thread onto the screws. 

Cylindrical Chess

Prime number wave

I never knew if this was possible or not but apparently so. I started with considering an infinite square wave only 0 every say, 3rd number and then multiplying all the waves of that type gives this nice wave that is only 1 where t doesn't divide any number but 1 and itself. In other words it is prime. In practice there are easier ways to find the prime numbers but I still think this function is interesting. 

Tell if a series is generated by a Polynomial with integer coefficents

Theorem: A series of integers is generated by a polynomial P(x) with integer coefficients if for every choice of x:
m mod P(x) = m mod P(x+m)
Proof: Consider the polynomials:
P(x) = a*x^n + b*x^(n-1) + ... + k
P(x+m) = a*(x+m)^n + b*(x+m)^(n-1)+...+k
When P(x+m) is expanded, there are two types of terms, those that do not contain m and those that do. The collection of terms that do not contain m are exactly those terms found in P(x). The rest we said all divide m. Therefore:
P(x+m) = P(x) + m*(B(x)) for some polynomial B(x) with integer coefficients.
So whatever the remainder of P(x) when divided by m is, because m*(B(x)) divides m evenly and B(x) is an integer at any x,  P(x+m) will have the same remainder when divided by m.

In practice what that means is if you look at a series of integers, and then the remainders when divided by for example 3, if it produces a pattern that repeats over and over again every 3 remainders it is produced by a polynomial with integer coefficients.
  For example:
 3*x^3 + 2*x^2 + x + 4 for x from 1 to 10 has remainders when divided by 3 of:


1,2,1,1,2,1,1,2,1

This pattern repeats every 3 as it should.

Try a random series:
5,12,21,27,35,45
The remainders when divided by 3:
2,0,0,0,2,0
Since this pattern does not repeat every 3 it can't be formed by any polynomial with integer coefficents. Of course there is a polynomial going through these 6 points for x from 0 to 6 but it is:
 5-(7/6)*x+(5/3)*x^4+(43/3)*x^2-(185/24)*x^3-(1/8)*x^5
It doesn't have all integer coefficients.

Now I'm not sure if this result has ever been discovered before, thanks to Dr. Rose for the idea for the proof that I've tried to generalize here.