Lie derivatives: a simple idea behind a messy calculation

I want to write about Lie derivatives. Because finding good proofs for Lie derivatives in books and on the internet is a lost cause. Because they have caused me a world of pain. Because we could all do with less pain. In all that is written below, we assume that all Lie derivatives are beingContinue reading “Lie derivatives: a simple idea behind a messy calculation”

Coming to grips with Special Relativity

Contrary to popular opinion, Special Relativity is not a more specialized, more involved part of General Relativity. It is the easier of the two Relativity theories, involving only thought experiments and Linear Algebra. However, despite having been exposed to ideas from this theory right from school, and also taking an advanced course (and doing well)Continue reading “Coming to grips with Special Relativity”

(Part of) a proof of Sard’s Theorem

I have always wanted to prove Sard’s Theorem. Now I shall stumble my way into proving a deeply unsatisfying special case of it, after a whole day of dead ends and red herrings. Consider first the special case of a smooth function . At first, I thought that the number of critical points of suchContinue reading “(Part of) a proof of Sard’s Theorem”

Furstenberg’s topological proof of the infinitude of primes

Furstenberg and Margulis won the Abel Prize today. In honor of this, I spent the better part of the evening trying to prove Furstenberg’s topological proof of the infinitude of primes. I was going down the wrong road at first, but then, after ample hints from Wikipedia and elsewhere, I was able to come upContinue reading “Furstenberg’s topological proof of the infinitude of primes”

Proving inequalities using convex functions

I have found that I am pretty bad at finding “clever factors” for Cauchy-Schwarz, whose bounds can be known from the given conditions. However, I am slowly getting comfortable with the idea of converting the expression into a convex function, and then using the Majorization Theorem. (Turkey) Let , and positive reals such that .Continue reading “Proving inequalities using convex functions”

A beautiful generalization of the Nesbitt Inequality

I want to discuss a beautiful inequality, that is a generalization of the famous Nesbitt inequality: (Romanian TST) For positive , prove that Clearly, if , then we get Nesbitt’s inequality, which states that . This is question 14 on Mildorf’s “Olympiad Inequalities”, and its solution comprises finding a factor to multiply this expression with,Continue reading “A beautiful generalization of the Nesbitt Inequality”

A small note on re-defining variables to prove inequalities

I just want to record my solution to the following problem, as it is different from the one given online. For positive real numbers, prove that This has a fairly straight forward solution using Cauchy-Schwarz inequality, which for some reason I did not think of. The way that I solved it is that I re-definedContinue reading “A small note on re-defining variables to prove inequalities”

A proof of Muirhead’s Inequality

I’ve been reading Thomas Mildorf’s Olympiad Inequalities, and trying to prove the 12 Theorems stated at the beginning. I’m recording my proof of Muirhead’s Inequality below. Although it is probably known to people working in this area, I could not find it on the internet. Muirhead’s Inequality states the following: if the sequence majorizes theContinue reading “A proof of Muirhead’s Inequality”

A simpler way to obtain smooth functions than convolutions?

Of the many mathematical concepts that I don’t understand, one of the more important ones is the convolution of functions. It is defined in the following way: Our guiding principle should be that we want to make an abelian group action (although inverses are not always present, at least when talking about integrable functions). However,Continue reading “A simpler way to obtain smooth functions than convolutions?”