Problem 1 (yau-contest 2016)
Section: Analysis and Differential Equations

Suppose that $F$ is continuous on $[a,b]$, $F'(x)$ exists for every $x\in(a,b)$, $F'(x)$ is integrable. Prove that $F$ is absolutely continuous and

$$F(b)-F(a)=\int_a^b F'(x)\,dx.$$

Problem 2 (yau-contest 2016)
Section: Analysis and Differential Equations

Suppose that $f$ is integrable on $\mathbb{R}^n$, let $K_\delta(x)=\delta^{-n/2} e^{-\pi |x|^2/\delta}$ for each $\delta>0$. Prove that the convolution

$$ (f*K_\delta)(x)=\int_{\mathbb{R}^n} f(x-y)K_\delta(y)\,dy $$

is integrable and $\|(f*K_\delta)-f\|_{L^1(\mathbb{R}^n)}\to 0$, as $\delta\to 0$.

Problem 3 (yau-contest 2016)
Section: Analysis and Differential Equations

Prove that a bounded function on interval $I=[a,b]$ is Riemann integrable if and only if its set of discontinuities has measure zero. You may prove this by the following steps.

Define $I(c,r)=(c-r,c+r)$, $\operatorname{osc}(f,c,r)=\sup_{x,y\in J\cap I(c,r)}|f(x)-f(y)|$, $\operatorname{osc}(f,c)=\lim_{r\to 0}\operatorname{osc}(f,c,r)$. 

1) $f$ is continuous at $c\in J$ if and only if $\operatorname{osc}(f,c)=0$. 

2) For arbitrary $\epsilon>0$, $\{c\in J\mid \operatorname{osc}(f,c)\ge \epsilon\}$ is compact. 

3) If the set of discontinuities of $f$ has measure 0, then $f$ is Riemann integrable.

Problem 4 (yau-contest 2016)
Section: Analysis and Differential Equations

1) Let $f$ be the Rukowski map: $w=\tfrac{1}{2}\big(z+\tfrac{1}{z}\big)$. Show that it maps $\{z\in\mathbb{C}\mid |z|>1\}$ to $\mathbb{C}\setminus[-1,1]$, $\overline{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$. 

2) Compute the integral:

$$\int_0^{\infty} \frac{\log x}{x^2-1}\,dx.$$

Problem 5 (yau-contest 2016)
Section: Analysis and Differential Equations

Let $f$ be a doubly periodic meromorphic function over the complex plane, i.e. $f(z+1)=f(z)$, $f(z+i)=f(z)$, $z\in\mathbb{C}$, prove that the number of zeros and the number of poles are equal.

Problem 6 (yau-contest 2016)
Section: Analysis and Differential Equations

Let $A$ be a bounded self-adjoint operator over a complex Hilbert space. Prove that the spectrum of $A$ is a bounded closed subset of the real line $\mathbb{R}$.