# Mathematics

I created a brief summary of definitions and theorems related to basic classification of Markov chain states. There is even a couple of diagrams that helped me learn some of the rules. Download, enjoy, and let me know what I can improve!

Multiplication by integrating factor is my favorite method of solving first order linear ordinary differential equations.

There’s a great new comic at Spiked Math: It’s a small world (after all). Be sure to check it out even if you are not a mathematician. Using mathematical reasoning, the comic asserts that **the number of ways you could lead your life is finite**, in other words, there is a limit to what you could do in your life. I strongly disagree, and I can **disprove** this assertion using the very same tool: mathematics :-)

Did you remember to give your mother a flower?

A Markov chain is a sequence of random variables (*states*) satisfying the Markov property: the probability of the current state depends only on the state that immediately preceded it. In other words, the past state and the future state are stochastically independent. How can we simulate such chains in Wolfram Mathematica?

There are quite a few identities concerning trigonometric functions. The most commonly used are the cosine and sine of double angles and the cosine and sine of the sum or difference of two angles. It might be perhaps surprising that these formulae can be easily derived with the help of basic complex analysis and algebra. Personally, I find it much easier to derive most of the formulae whenever I need them instead of remembering them all.

“Mathematicians attach a great importance to the elegance of their results, and this is not mere dilettantism. What is it that gives us the feeling of elegance in a solution or a demonstration?…”

Simple guide to solving separable differential equations, plus explanation how it works.

Are there more even numbers than odd numbers? How much can we take away from infinity to keep it infinite? And how much is “ten times infinity”? Is it more than “twenty times infinity”? Do these questions have a meaning at all? Let us delve into the depths of the infinite oceans once again!

Do you know what the largest natural number is? Where do the borders of the infinite realm lie? And how many monkeys does it take to write the complete works of Shakespeare? Come hither, ye finite mortal, and behold the glory of eternity that men have created in their minds!