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#induction

1 post1 participant0 posts today
genehack

I got an #induction burner for Christmas, and I’ve used it several times now, for various things, and it mostly works a treat. Tonight, however, I noticed that it seems to be problematic for pasta? I think every time I’ve used it to boil some pasta, I’ve ended up with stuff stuck on the pot bottom, despite salt and oil in the water.

Is it just me?

Bruce MacDonald

"While the official cause of the fire is still under investigation by Los Angeles County Fire, Edison International Chief Executive Pedro Pizarro told The Times that induction — a rare occurrence when idle tower and power lines are reenergized — is now a leading theory into how the deadly fire was sparked."
latimes.com/california/story/2

Los Angeles Times · Edison electric tower damaged while being dismantled for Eaton fire investigationBy Salvador Hernandez
#electricity#induction#California
Paysages Mathématiques

"[Mathematical induction] is a tool that allows us to move from the finite to the infinite. This tool is always useful because it allows us to skip as many steps as we want, sparing us long, tedious, and monotonous verifications that would quickly become impractical. However, it becomes indispensable as soon as we aim at the general theorem, whose analytical verification would bring us closer and closer [...]" – Henri Poincaré (1854-1912)
#quote #mathematics #math #maths #induction

😀🚲

The 11yo is now regularly cooking their own pasta with the #induction hob, and wow is that a lot less for a parent to worry about than the hot electric stovetop or an open gas flame (it's still scalding hot water though, make sure they know how to manage it.) There is even a timer setting so it just turns off if you forget. Also, I highly recommend raising a self-feeding child 😌

Replied in thread
Akshar Varma

@BernhardWerner My favorite alternative #proof strategy for #induction proofs are #combinatorial (counting) proofs.

I suppose the standard example might be the proof of the coefficients in the binomial theorem expansion, or for the sum of binomial coefficients being powers of 2. These can be proved by induction, of course, but I'm not sure that's common given how easier it is to do a counting proof. It is also much clearer and avoids tedious algebra.

One I like is proving that the sum 1 + 2 + 3 + ⋯ + 𝑛 is 𝑛 + 1 choose 2, the binomial coefficient \(\binom{n+1}{2}\). Bijection proof, counts the same thing in two ways. The thing being counted is the number of ways of choosing two things (distinct, without repetition) from the set {0, 1, ..., 𝑛}. By definition, it is the binomial coefficient we want. The other way to count is to fix the larger number 𝑘, the remaining choices are any of the 𝑘 numbers from 0 to 𝑘 - 1. Thus, across all possible larger numbers, we get the sum from 1 to n.

An alternative alternate proof of the same, slightly more geometric is as follows: arrange dots in a triangle, 1 on row 1, 2 on row 2, and so on up to row n, with n dots. Add a phantom row of n+1 dots below. We want to add up all dots in first n rows: ∑ 𝑖. If you think of all of this as a binary tree/DAG, then every dot has two children (imagine Pascal's triangle). If you pick any two dots in the phantom row, their common ancestor is unique. So counting dots is same as picking two dots in phantom row. Which is the binomial coefficient we want.

Benjamin and Quinn's book on combinatorial proofs is amazing for interpretations of this form (I learned the first proof from it). See also: en.wikipedia.org/wiki/Combinat

en.wikipedia.orgCombinatorial proof - Wikipedia
*
Sue is Writing Solarpunk 🌞🌱

I shot this video w/Penn Environment last fall and the crew did an amazing job putting these clean energy stories together.

*Especially* given the ongoing, escalating assault on the environment, I'm proud that the things I've already done (#solar #induction #EV #native #plants) will keep on making a difference for decades to come, regardless.

"they’re investing not only in cleaner energy for themselves, but also a cleaner world for their grandchildren’s generation."

environmentamerica.org/pennsyl

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