### Finite state Markov chains

MathJax is on for this blog. This means that you can use LaTeX to type formulas:

$\int$ on a finite set $\mathcal A$ is just $\sum$

1. Tasks for June 29: resolve Exercises 1-5

Bonus problem
Let $\varphi:\mathbb R\to\mathbb R$ be a function. Prove that $\varphi$ is convex in the sense of Exercise 1 if and only if, for any $x,y\in\mathbb R$ and $\theta\in[0,1]$, we have
$$\varphi(\theta x+(1-\theta)y)\le \theta \varphi(x)+(1-\theta)\varphi(y).$$
Show also that a convex function is continuous, possesses left and right derivative at every point, and is differentiable on the complement of a countable subsets.

2. Tasks for June 30: resolve Exercises 6-9

Bonus problem
Is it possible to construct a function $\varphi:\mathbb R\to\mathbb R$ that is differentiable outside a countable set $N\subset\mathbb R$ such that $\varphi'(x)=0$ for $x\notin N$ and $\varphi(x)\to\pm\infty$ as $x\to\pm\infty$ ?

3. Tasks for July 1: resolve Exercises 10-13

Bonus problem
Let $\varphi:[0,1]\to\mathbb R$ be a differentiable function. Is it true that the derivative $\varphi':[0,1]\to\mathbb R$ is integrable in the sense of Riemann and
$$\varphi(1)-\varphi(0)=\int_0^1\varphi'(y)\,dy\, ?$$

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2. Can we say for opposing example $$\phi(x)=x^{\frac{1}{3}}?$$

3. This function is not differentiable at $x=0$.

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5. Indeed, may be $f(x)= \begin{cases} x^2\sin(\frac{1}{x^2}),& x\neq 0\\ 0, & x=0 \end{cases}$

6. Yes, this is a counterexample showing that the property is not true.

4. Tasks for July 2: resolve Exercises 14-18

Let $\varphi:[0,1]\to\mathbb R$ be a differentiable function such that the derivative $\varphi':[0,1]\to\mathbb R$ is bounded. Prove that
$$\varphi(x)=\varphi(0)+\int_0^x\varphi'(y)\,dy, \quad x\in[0,1].$$

5. Tasks for July 8: resolve Exercises 22-24 and finish the bonus problem of June 29

6. Tasks for July 9: resolve Exercises 25-27

7. Tasks for July 12 & 13: revision of the first 27 exercises

8. Tasks for July 14: complete the revision of the first 27 exercises

9. Tasks for July 15: resolve Exercises 28-30 and write down their solutions

10. Tasks for July 16: resolve Exercises 31-35 and write down their solutions

11. Tasks for July 19: resolve Exercises 36-38 and write down their solutions

12. Tasks for July 21: resolve Exercises 39-42 and write down their solutions

13. Tasks for July 22: resolve Exercises 43-45 and write down their solutions

14. Tasks for July 22: resolve Exercises 46-48 and write down their solutions