Nevin Manimala’s Blog

I know that we’re still uncertain whether our own universe has slight negative curvature. But imagine a universe where space has an obvious negative curvature of, say, K=-1. In that universe, planets still be able to orbit stars?

I wondered if there was such thing as taking half the derivative and I googled it and found fractional calculus. I looked at it a little bit and found that it is based on Cauchy formula for repeated integration but with gamma function instead of factorials. The Cauchy formula is proven using induction which holds for natural numbers yet in fractional calculus it is used with real numbers. So I want to know how is that valid.

I’m studying electrical engineering so I’m not a mathematician which is why this might be a silly question.

I have been working with a professor on an undergraduate research project. The goal of the project was to create a program that could numerically solve differential equations. I ended up using the method of colocation, wherein I set up a linear system of equations to solve for the coefficients of the chebyshev polynomials that “solve” (as close as a polynomial can solve the equation, usually on the order of 10^-10 – 10^-16). The professor has asked me to give a 1 hr presentation on it to a group of freshmen. This is my problem, there is so much information that is crucial to why it works (chebyshev nodes and polynomials, orthagonality, fredholm alternative), and I can only assume knowledge up to cal 1. I don’t know what the best way to squeeze all of it down into an intuitive 1 hour session would be, and was wondering if you guys could give me some insight.

Thanks

If you don’t know, the peg game is a game where you have a triangle of holes all filled with pieces, one of which is empty. You jump pieces over each other into empty squares and take out the piece they jump, and the goal is to get only one peg left. This is an article showing it and how to solve it for the standard game board..

For a board like this, it’s pretty easy to determine what configurations of empty squares and pieces are possible just by checking all the possibilities. My question is this: I’m interested in a hypothetical infinitely large board, with still only one empty hole at the start. Given infinite moves, what are the kinds of patterns you can make? Are there patterns you can’t make? Which ones?

I’m not even sure how to adequately describe any given infinite board state unless it has a repeating pattern. Even if it’s not a super answerable question, if there’s any work that has been done that might be useful in talking or thinking about this kind of structure I would be very appreciative if someone could point me to it.

This has been bothering me for a while and I can’t find a good way to constructively move forward. I feel like it might not be something we can really describe very well at all. Thanks for any insights anyone can offer!