## Rikki-tikka-tavi

(This is my reply to Tikka’s marriage invitation, one of the most happening woman I’ve seen back there in Pondy)

​Ola,

I’m here in Chennai, but unfortunately in a deadlock!

I wonder why August 28th is a good day to get married, because my cousin, a classmate from PU and you are getting married on the same day and unfortunately I’m going to miss all three! Smeh.

Quite a lot happening around me (this time they are happening by themselves, and I’m not making them happen) and that gives me a lot of possibilities to dramatically excuse myself from anything that comes on my way. After all, we all live everyday just to put up a new and exciting drama, don’t you think so?

Plus a fever I caught from somewhere between Tambaram and Kodambakkam to add to the flavour.

Plus the crane’s about to deliver a child to my sister – that’s the cherry on top. She’ll manage the crane, but I’m in charge of Amazon that will deliver other things while she’s in hospital and the elder one, Alana. Waiting for delivery message from both crane and Amazon now, we will get it all hopefully in a day or two.

See, I sound a lot saner now right?

Anyway, back to you! Get happy and get married for life! Plant good guava tree in your field, you can always trust a guava more than apples! Plus, I like guava better than apples!;-)

Oh, and don’t forget to panic a bit about wedding arrangements! Do some drama (the happy drama) to make that day kinda happening beyond everyone’s expectation!

May be the Oppa Tikka style!

With Love,

A fever flavoured cake with a lot of mess to sort and a cherry on top.

Correction: Grace says storks give babies and not cranes. Whatever. The rest is still from Amazon, I’m pretty sure about it.

## Hello World (Again!)

And I decided to start writing (again)!

The other day Pavi proved that I’m highly (or as she dramatically put it, “very very highly”) impulsive with her crazy line-circle-painting practical test. She also told me I’ve got a lot of ‘p’-something skill too. Well, back then I believed anything high is good, very high is very good, and I felt like a very very rich man after talking to her!

I don’t think anything else better explains why I deleted my last personal blog and I regretted it the very next day.

Here I am almost a year later, back blogging.

There will be the same old senselessness, and fact that I may be working for a while hopefully starting next week, you can also expect few mlanathafied (depressing) posts out of work-stress. I’d be meeting a lot of new people, and a few old people at a different level – it’s equally exciting and scary!

Let’s see how long this blog survives.

Bis bald. (Showing off my German skills)

## ODE Cheat Sheet: Few Relevant Definitions

It’s been a while since I wrote here, and this is an attempt to resurrect the blog, and get inspiration to contribute something for Wikibooks as well.

This is a part of the book “Ordinary Differential Equations: Cheat Sheet” that I am compiling in Wikibooks, and it will be published in five instalments before/along with finalising the book in Wikibooks. And just like Examples and Counter Examples Series (Which again I am planning to compile in Wikibooks), this will also be edited as I learn.

## Wronskian of Two Functions

### Definition

Wronskian of two functions, $y_1,y_2$ is given by $W_{y_1,y_2}(x)=\left|\begin{matrix} y_1 && y_2 \\ y_1' && y_2'\end{matrix}\right|$

### Useful Facts

If two functions $y_1,y_2$ are linearly dependent in an interval, then it’s Wronskian vanishes in that interval.

## Laplace Transforms

### Definition

$\mathcal{L}\{f(t)\}=F(s)=\int_0^\infty e^{-st}f(t)dt$

### Properties

1. $\mathcal{L}\{af + bg\} = a \mathcal{L}\{f\} + b \mathcal{L}\{g\}\,,$
2. $\mathcal{L}\{e^{at} f(t)\}(s) = F(s - a)\,$ for $s > \alpha + a$.
3. If $F(s) = \mathcal{L}\{f(t)\}$, then $\mathcal{L}\{f'(t)\} = sF(s) - f(0)$
4. Similarly, $\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f''(0)$

### Laplace Transform of Few Simple Functions

1. $\mathcal{L}\{1\} = {1 \over s}$
2. $\mathcal{L}\{e^{at}\} = {1 \over s-a}$
3. $\mathcal{L}\{\cos \omega t\} = {s \over s^2 + \omega^2}$
4. $\mathcal{L}\{\sin \omega t\} = {\omega \over s^2 + \omega^2}$
5. $\mathcal{L}\{1\} = {1 \over s}$
6. $\mathcal{L}\{t^n\} = {n! \over s^{n+1}}$

## Convolution

### Definition

$f(t)*g(t)=\int_0^t f(u)g(t-u)dt$

1. Associative
2. Commutative

## Examples and Counter Examples for Abelian Groups

Note: This is the first of “Examples and Counter Examples” series. I plan on expanding such posts in course of time, as I come across examples.

Interesting fact: These are the OEIS lists of number of distinct groups of order n: number of groups,number of abelian groups, number of non-abelian groups

### Examples

1. $\Bbb Z$
2. $\Bbb Q$
3. $\Bbb R$
4. Integers modulo n $\Bbb Z / n \Bbb Z$
5. $S_n,n\le2$, the symmetric group.
6. $A_n \subset S_n,\forall n\ge1$
7. Any group of order < 4

### Counter-Examples (or Non-Abelian Groups)

1. Symmetric groups $S_n,n\ge3$

## New kid in the block

Welcome the new kid in the block.

Almost two and a half years ago I kicked off my first WordPress blog Far Far Away, an attempt to showcase and improve my writing skills, and thankfully it was decently received in its first year before I got lazy and stopped writing regularly.

Off late I have been fascinated by the idea of “maths blogging” which is expected to improve my maths writing skills, and help me improve my skills in communicating mathematics. Thankfully WordPress.com supports $LaTeX$ (\LaTeX fails to render though).

For the record, this is my sixth or seventh blog, and second to hopefully survive my wrath and an attempt to learn writing maths. I am a maths post graduate.

I have also transferred my following tech posts from Far Far Away, an attempt to make that blog more focussed on non-maths, non-tech content:

I hope I will be able to be able to post more frequently after December 20th. Since then, I plan on at least one post a week.

The content will be mostly about (and not restricted to),