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

Properties

  1. Associative
  2. Commutative
  3. Distributive over addition

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),

  • Interesting facts in/about mathematics
  • Interesting theorems I came across
  • Interesting math.SE questions that caught my attention
  • Geek jokes
  • Interesting mathematics contest questions
  • …and hopefully compilation of definitions and important theorems, arranged topic-wise, for my own future reference.

I am also an active member in math.SE. I am more inclined to abstract side of mathematics, hope you now know what to expect here.