MathPages

Kevin Brown's mathpages [0] are really something else. I keep falling into them again and again when trying to solve very different technical problems. Each time, he surprises me with several crucial insights that were already there 20 years ago. The wonky math equations are a lovely plus. [0] https://www.mathpages.com/

Great argument. I found this generalization to higher dimension: https://www.mathpages.com/home/kmath327/kmath327.htm

I believe the following has the symmetry based solution you mention (and more!) https://www.mathpages.com/home/kmath668/kmath668.htm

Looks really impressive. For those interested in this sort of thing, another huge and varied collection of mathematical/physics articles is located at https://www.mathpages.com/

If you took out the bits of relativity that were developed by others, you'd be left with not very much. Relativity without the Lorentz transform... I'm not even sure how one would resolve that. As I said, the vast majority of the insight was just taking what the math was already saying seriously. Einstein put it into a final form, and this is important and valid work, but it's not like he invented all the mathematical tools from scratch. Either you've misinterpreted Adam Frank's work, or it isn't excellent. I'll counter-cite Reflections on Relativity, which contains a lot of deep analysis of the mathematical history of relativity both before and after Einstein: https://mathpages.com/rr/rrtoc.htm which makes it quite clear both what he did, and did not do.

Here's a well-explained article about fractional derivatives and integrals (1). One freaky outcome is that the factorials of fractional numbers can be found. (Example: negative one half factorial = square root of pi.) It can be applied to all things quantum mechanical (2) so there's a fractional Schrödinger equation etc. I had to look it up, but the links are worth the bother. (1) https://www.mathpages.com/home/kmath616/kmath616.htm (2) https://arxiv.org/abs/1009.5533 https://www.mathpages.com/home/kmath616/kmath616.htm

Or a letter to a fellow scientist: In July of 1610 Galileo was still making discoveries faster than he could publish descriptions of them. On the 25th he discovered that Saturn was apparently situated between two smaller companions that always moved together. Wanting to establish his priority of discovery, but not yet ready to reveal what he had found, he sent to Kepler (and others) the following jumble of letters, which he informed them was a coded description of his latest discovery (...) http://www.mathpages.com/home/kmath151/kmath151.htm

Awesome. Some concepts in math are so much easier to grasp with an interactive visual. I once made an entire game around matrix multiplication -- such that players didn't even know they learned how to do this. And sometimes, visualizations can help cause discoveries. A lot of progress in theoretical physics came about because physicists 'guessed' what nature's equations should look like...and then got shocked when they discovered they were right. Take James Clerk Maxwell, for instance: "He had achieved his greatest success by relying on just one empirical fact known to every school boy. Yet now he tried to do without any empirical facts, by pure thinking. He believed in the power of reason to guess the laws according to which God has built the world." http://www.mathpages.com/home/kmath103/kmath103.htm

I think your watch is slow and you think mine is slow in the following sense. For each of many events, I record their time t and position x while you, moving relative to me at constant velocity, observe a different time T and position X. Between two events with the same X-value, such as ticks of your pocket watch, the difference in t-times is larger than the difference in T-times by a factor gamma. Conversely, between two events with the same x-position, such as ticks of my pocket watch, the difference in T-times is larger than the difference in t-times by that same factor gamma. There is no contradiction because events with identical x-positions do not have identical X-positions. If you already know what a partial derivative is, see the first paragraph of this: http://mathpages.com/rr/s4-07/4-07.htm

Previously on News.YC: "Optimizing your wife If a man can expect to meet exactly N eligible women in his life, what strategy should he use to maximize his chances of choosing the very best one?" http://www.mathpages.com/home/kmath018/kmath018.htm Just curious, are you attempting to quantitatively rank the candidates in a spreadsheet, especially the intangibles like 'as potential mother'?

An older legendary story with the same premise: "Wren, Hooke, and Halley had discussed the problem at a coffee house following a meeting of the Royal Society in January of 1684, and Wren had offered a cash prize to whoever could provide a derivation of the shape of planetary orbits under the assumption of an inverse-square central force of attraction toward the (presumed stationary) sun. Hooke had claimed to have a proof that the paths were ellipses, but never provided it. Against this background, Halley paid a visit to Newton, who later told Abraham De Moivre about the fateful meeting. According to De Moivre: In 1684 Dr Halley came to visit him at Cambridge. After they had been some time together, the Dr asked him what he thought the curve would be that would be described by the planets supposing the force of attraction towards the sun to be reciprocal to the square of their distance from it. Sir Isaac replied immediately that it would be an ellipse. The Doctor, struck with joy and amazement, asked him how he knew it. Why, saith he, I have calculated it. Whereupon Dr Halley asked him for his calculation without any farther delay. Sir Isaac looked among his papers but could not find it, but he promised him to renew it and then to send it him…" http://www.mathpages.com/home/kmath658/kmath658.htm

The only analytical solution I could find was this: https://www.mathpages.com/home/kmath668/kmath668.htm I admit it kinda flew over my head (as I mentioned, I'm an empiricist through and through)

If you were to fall radially inwards towards a non-rotating black hole, that is not spiraling around it but "straight in", it would be the same as with plain Newton's law[1]. There's a nice graph in the midde of this[2] page that shows the difference between the proper time and the aparent time observed by the outside observer. At r = 2m you can see the aparent time goes to infinity and the quickly back again. [1]: https://physics.stackexchange.com/questions/718222/proper-time-of-fall… [2]: https://www.mathpages.com/rr/s6-04/6-04.htm

Fuller discussion at https://www.mathpages.com/home/kmath564/kmath564.htm as cited on the slide.

Nice story, however the second message he deciphered was not about Jupiter's spots, but something else. https://www.mathpages.com/home/kmath151/kmath151.htm

How to derive mathematically correct piano key widths: https://www.mathpages.com/home/kmath043.htm

You're probably already familiar with the Feynman-and-the-French-curve anecdote, but if not you may enjoy it: https://www.mathpages.com/home/kmath687/kmath687.htm

Oumuamua seemed to accelerate as it moved away from the sun. A space rock moving only due to gravity shouldn't do this, though a comet might. Gravity slingshot. Humans have used that effect themselves. There's nothing mysterious about it. https://mathpages.com/home/kmath114/kmath114.htm

https://www.mathpages.com/home/kmath077/kmath077.htm is a good overview of why trying to make the expansion explanation even work with Newtonian gravity ultimately leads you to general relativity.

Some interesting reads on the subject: - https://en.wikipedia.org/wiki/Relativity_priority_dispute#The_candidat… - https://www.mathpages.com/home/kmath650/kmath650.htm

Sure. You can start from relativity and show that for example the antisymmetric Faraday tensor (P in the link [1] below) can get you a lot of phenomena (not all the way, accelerating charges are hairy, even for Feynman, see the link [2]) To note, the reference to finding the relativistic invariant from the tensor in the first link goes back to the first edition of Landau and Lifshitz. The problem was removed from later editions because only a masochist wants to find invariants of 4x4 matrices by hand. [1] https://www.mathpages.com/home/kmath647/kmath647.htm [2] https://www.mathpages.com/home/kmath528/kmath528.htm

Here's a cool link I just found going over this exact scenario: https://www.mathpages.com/home/kmath530/kmath530.htm Interesting to note, the napkin creates a uniform gravitational field above and below it. Meaning that the force applied to an object is the same regardless of how far away it is from the napkin! That force is 2pi * G * m * rho where rho is the mass density of the napkin.

Getting a patent is an expensive business. Patentability of methods of manipulating numbers is variable around the world, but in most places its not going to work. You might have managed it in the US a few years ago, but not now. Seriously applying for a patent is not cheap. The odds are that you will spend a lot of money for no return. Getting priority for something on your CV is another matter. As others have mentioned, write up a paper and submit to arxiv.org. Incidentally, are you aware of this: http://www.mathpages.com/home/kmath505/kmath505.htm The last couple of paragraphs describe a geometric construction giving an approximation that is good enough for many purposes, and would also be the right starting point for any iterative algorithm.

A more in depth and more complete treatment: http://mathpages.com/home/kmath616/kmath616.htm

The whole website http://www.mathpages.com/home/ feels like euler project times itself

See also, the awesome Amasci site discussion of this: http://amasci.com/wing/airfoil.html http://amasci.com/wing/rotbal.html http://amasci.com/wing/whyhard.html edit: Another interesting fact is that the flawed description of how airplanes fly is so pervasive that Albert Einstein once proposed how to improve the airfoil based on his understanding of it and it was a huge flop. See the end of this article: http://www.mathpages.com/home/kmath258/kmath258.htm

Google knows: http://www.mathpages.com/home/kmath668/kmath668.htm

Then, I'm afraid, you're missing the point of the paradox. There are a few stories (improperly called "paradoxes") attributed to Zeno: http://en.wikipedia.org/wiki/Zeno%27s_paradoxes You're probably thinking of The Dichotomy. This story points out that matter must not be infinitely divisible. The paired story, The Arrow, shows that a universe of finite, indivisible pieces is also impossible. Thus, Zeno's paradox. Further, very interesting reading: http://www.mathpages.com/rr/s3-07/3-07.htm Math just gave us a way of coming up with the obvious answer, it does not describe the nature of the universe, which is what the philosophy was attempting.

Here's a quote that I think you may find interesting, as it seems to be along the lines of what you are saying: "A meta-mathematical proof of the consistency of arithmetic is not excluded by...Goedel's analysis. In point of fact, meta-mathematical proofs of the consistency of arithmetic have been constructed, notably by Gerhard Gentzen, a member of the Hilbert school, in 1936. But such proofs are in a sense pointless if, as can be demonstrated, they employ rules of inference whose own consistency is as much open to doubt as is the formal consistency of arithmetic itself. Thus, Gentzen used the so-called "principle of transfinite mathematical induction" in his proof. But the principle in effect stipulates that a formula is derivable from an infinite class of premises. Its use therefore requires the employment of nonfinitistic meta - mathematical notions, and so raises once more the question which Hilbert's original program was intended to resolve." -Ernest Nagel and James Newman Taken from https://www.mathpages.com/home/kmath347/kmath347.htm