## |Groups|

Today we will discuss the proof of . Here, and are groups. We know , as . Let . Then . Take any . For any , find and . Then . Hence, pairs of elements ) can be found such that for any two . Hence, we can form equivalence classes which partition ,Continue reading “|Groups|”

## A new proof of Cauchy’s theorem

We will discuss a more direct proof of Cauchy’s theorem than the one given in Herstein’s “Topics in Algebra” (pg.61). Statement: If is an abelian group, and , then there is an element such that , and . We will prove this by induction. Let us assume that in every abelian group of order ,Continue reading “A new proof of Cauchy’s theorem”

Today we will discuss compactness in the metric setting. Why metric? Because metric spaces lend themselves more easily to visualisation than other spaces. Let us imagine a metric space with points scattered all over it. If we can find an infinite number of such points and construct disjoint open sets centred on them, then cannotContinue reading

## Complete metric spaces are Baire spaces- a discussion of the proof.

I refer to the proof of the statement “Every complete metric space is a Baire space.” The proof of this statement, as given in “Introduction to Banach spaces and their Geometry”, by Bernard Beauzamy, is Let be a countable set of _open_ dense subsets of complete metric space . Take any open set . WeContinue reading “Complete metric spaces are Baire spaces- a discussion of the proof.”

Today I plan to write a treatise on spaces. are normed spaces over with the p-norm, or . Say we have the space over . This just means that , where . That is a norm is proved using standard arguments (including Minkowski’s argument, which is non-trivial). Now we have a metric in  spaces: .Continue reading

## 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”

## The utility of trigonometrical substitutions

Today we will discuss the power of trigonometrical substitutions. Let us take the expression This is a math competition problem. One solution proceeds this way: let . Then as , we can write and . This is an elementary fact. But what is the reason for doing so? Now we have . Similarly, . TheContinue reading “The utility of trigonometrical substitutions”

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