Imagine that you need to find the arithmetic mean of many thousands or even millions of data values, let’s call that number of values N. Also imagine that these values are arriving as a stream and not as a completed list. The first way one could do it is to store all N data values in a table and do a one-time calculation of all the values using the formula …(read article)
January 29, 2018
This posting is Part 1 of an exploration of a way to diagram special relativity that I found in a book. Future postings will delve into it more deeply. Part 1 introduces the idea.
I recall reading a book a while back called Relativity Visualized by Lewis Carroll Epstein. In it he has his explanation for why nothing can exceed the speed of light. The reason is that everything travels at the speed of light, whether it be purely through space (photons) or purely through time (for a stationary frame of reference, and we multiply the time axis by c to get units of space), or some combination of the two. He has an interesting diagram which shows what he means by this concept. Recently on YouTube on the FermiLab channel, the host, physicist Don Lincoln, used the same argument to explain the speed of light, but cautioned that it was just a rough intuitive explanation. The real explanation involves Minkowski diagrams. This paper will explore this concept and diagram, and see how far I can take the diagram as an explanation of the results of special relativity. I will eventually tie it back to the Minkowski diagram, and show how some improvements to the diagram can be made. Read more…
December 3, 2017
Considerations for (far) future engineers… In the Stars Wars movies vehicles are shown to have some kind of anti-gravity mechanism to ‘float’ above the surface of the planet. I realize that there is no known physics that allows for that. But my question is, if such a mechanism existed, what properties must this force have to allow for vehicles to float above the planet? To my mind there are two possibilities. One is a long range anti-gravity, where the vehicle acts against the mass of the entire planet. The second is a short range force analogous to magnets that maglev vehicles use to float above a track.
The long range force would not work well near the planet’s surface because the vehicle would bounce around like a balloon in the wind, unless the vehicle uses ailerons to deflect air, which the vehicles in the movies don’t appear to have. And a vehicle could not react to the local changes in terrain because a hill has negligable gravity. Another issue would be the problem that hovercraft have, namely no lateral constraints against sliding left or right or down a slope.
The second option, a local force, has a better chance of working near a planet’s surface. The vehicle would need to have an ability to adjust the upward force at every moment to smooth the motion. The control systems must ignore a small rock, but react to a overall change in elevation. And it would need the ability to create lateral forces to prevent the hovercraft problem. One thing to consider with short range force is flying over water, the water below the vehicle would be displaced by the downward reaction force that balances the upward force that holds the vehicle up, so the vehicle must be less dense than water, or if not, it must travel at high speed over water.
The problem with a short range force is that it can’t give the high flying spacecraft their anti-gravity. No spacecraft in Star Wars entering a planet’s atmosphere has a problem with braking to slow it down. So maybe the spacecraft would need both a short range and a long range force.
This post is about a connection between the Riemann zeta function and dot products of infinite dimensional vectors. Zeta and Infinite Dimensional Dot Products
May 29, 2017
We know that the integration rules for exponents breaks down for the case x^-1, and that this case is the log function.
We can recover the exponent rule in the following way: How To Use The Exponent Rules Of Integration To Approximate Log Function
Mathematica has the ability to do arbitrary precision arithmetic, which is an essential thing when attempting to find closed form expressions of decimal numbers. The problem with double precision (16 significant digit) calculations is that it is not near enough precision in most cases to see if there is a repeating pattern in the digits, which is an indication of a rational number. If a decimal number has a repeating pattern of let’s say 100 or 200 digits or more, one cannot see that with 16 digits. This article is going to explore numerical searches for closed form expressions for decimal numbers using the tools provided by Mathematica. Part 1 will be introducing the basic approach and some easy examples. Part 2 will get to harder examples and figuring out how to automate the search in these complicated cases. Using Mathematica to do numerical searches (part 1)
A closed form expression for Zeta(2) and other even integer inputs of the Riemann Zeta function was found by Euler in the 1730’s. This article is going to use Mathematica to re-create the solution that Euler derived. The derivation below is based on the article in Wikipedia on the Basel Problem. Mathematica helped me see the pattern to solve Zeta(4) and in principle any positive even value. I don’t know if this is the approach Euler used to solve for all even integers. The Wikipedia article did not get into how Euler solved for the larger even integers.
The function I found in Mathematica which expands the infinite product (for a finite subset of the infinite product) is the Expand function. Unfortunately it also collects the terms for like powers of x. So one cannot see the first few terms of the zeta series evident in the product transformed into a series. So I had to get a little creative on how to show those terms. Read more…
From time to time I read articles on the true cost of producing a product, or of consuming a product. For example, I have read that the true pollution costs of all the components required to make an electric car far outweighs any advantage in pollution reduction of using the car. Off hand I can’t remember the exact reasons, but I recall that there are some exotic materials required for electric cars that gasoline powered cars don’t require. Also, there are the pollution costs of creating the electricity to make an electric car run. An argument can be made that at least the pollution is not being produced in the middle of the city where the electric car is being used.
There are similar kinds of arguments for why wind power is not truly a ‘free’ energy source. There are also practical cost calculations that can be made as to how much energy is saved by having large shade trees surrounding a house.
My idea is to have a Wiki site purely dedicated to analyzing the true costs of anything. One problem with this type of site is that it would get very political as each side of the debate tries to score points for their side. The site needs to be dedicated to be an ideologically neutral as possible. It should be a resource that anyone from any side of the argument could use to support their case.
Here is how one could try to keep the site neutral and objective. One author, one article. That is, each person who wants to write an article would write their own article. Others are free to write comments and suggest improvements to that article, but they cannot directly change the content of the article, as can be done in Wikipedia. In the comments section other authors can link to articles they believe better state the true cost.
The following is a discussion of mapping of the points both inside and outside the Mandelbrot Set to its corresponding “root”. Read more…