## Sufficient conditions for differentiability in multi-variable calculus.

We will be focusing on sufficient conditions of differentiability of . The theorem says that if and exist and are continuous at point , then is differentiable at . We have , which we know is partially differentiable with respect to and , but may not be differentiable in general. Differentiability at point in theContinue reading “Sufficient conditions for differentiability in multi-variable calculus.”

## Lonely Runner Conjecture- II

The Lonely Runner conjecture states that each runner is lonely at some point in time. Let the speeds of the runners be , and let us prove “loneliness” for the runner with speed . As we know, the distance between and is given by the formula for and for , where stands for time. Also,Continue reading “Lonely Runner Conjecture- II”

## Of ellipses, hyperbolae and mugging

For as long as I can remember, I have had unnatural inertia in studying coordinate geometry. It seemed to be a pursuit of rote learning and regurgitating requisite formulae, which is something I detested. My refusal to “mug up” formulae cost me heavily in my engineering entrance exams, and I was rather proud of myselfContinue reading “Of ellipses, hyperbolae and mugging”

Let be a mapping. We will prove that , with equality when is injective. Note that does not have to be closed, open, or even continuous for this to be true. It can be any mapping. Let . The mapping of in is . As for , it may overlap with , we the mappingContinue reading

## Proofs of Sylow’s Theorems in Group Theory- Part 1

I will try to give a breakdown of the proof of Sylow’s theorems in group theory. These theorems can be tricky to understand, and especially retain even if you’ve understood the basic line of argument. 1. Sylow’s First Theorem- If for a prime number , , and , then there is a subgroup such thatContinue reading “Proofs of Sylow’s Theorems in Group Theory- Part 1”

Today, I will discuss this research paper by Javed Ali, Professor of Topology and Analysis, BITS Pilani. What exactly is a proximinal set? It is the set of elements in which for any , you can find the nearest point(s) to it in . More formally, for each such that . This article says aContinue reading