Problem 1 (yau-contest 2012)
Section: Analysis and Differential Equations

Let $A=[a_{ij}]$ be a real symmetric $n\times n$ matrix. Define $f:\mathbb{R}^n\to\mathbb{R}$ by
$$f(x_1,\cdots,x_n)=\exp\!\left(-\tfrac12\sum_{i,j=1}^n a_{ij}x_i x_j\right).$$
Prove that $f\in L^1(\mathbb{R}^n)$ if and only if the matrix $A$ is positive definite.

Compute
$$\int_{\mathbb{R}^n} \exp\!\left(-\tfrac12\sum_{i,j=1}^n a_{ij}x_i x_j + \sum_{i=1}^n b_i x_i\right)\,dx$$
when $A$ is positive definite.

Problem 2 (yau-contest 2012)
Section: Analysis and Differential Equations

Let $V$ be a simply connected region in the complex plane and $V\neq\mathbb{C}$. Let $a,b$ be two distinct points in $V$. Let $\phi_1,\phi_2$ be two one-to-one holomorphic maps of $V$ onto itself. If $\phi_1(a)=\phi_2(a)$ and $\phi_1(b)=\phi_2(b)$, show that $\phi_1(z)=\phi_2(z)$ for all $z\in V$.

Problem 3 (yau-contest 2012)
Section: Analysis and Differential Equations

In the unit interval $[0,1]$ consider a subset
$$E=\{\,x\mid \text{in the decimal expansion of } x \text{ there is no } 4\,\}.$$
Show that $E$ is measurable and calculate its measure.

Problem 4 (yau-contest 2012)
Section: Analysis and Differential Equations

Let $1<p<\infty$. Let $L^p([0,1],dm)$ be the completion of $C[0,1]$ with the norm
$$\|f\|_p=\left(\int_0^1 |f(x)|^p\,dm\right)^{1/p},$$
where $dm$ is the Lebesgue measure. Show that
$$\lim_{\lambda\to\infty} \lambda^p\, m\big(\{x\in[0,1]\mid |f(x)|>\lambda\}\big)=0.$$

Problem 5 (yau-contest 2012)
Section: Analysis and Differential Equations

Let $\mathfrak{F}=\{e_\nu\}$, $\nu=1,2,\ldots,n$ or $\nu=1,2,\ldots$, be an orthonormal basis in an inner product space $H$. Let $E$ be the closed linear subspace spanned by $\mathfrak{F}$. For any $x\in H$ show that the following are equivalent:
1) $x\in E$;
2) $\|x\|^2 = \sum_\nu |\langle x,e_\nu\rangle|^2$;
3) $x = \sum_\nu \langle x,e_\nu\rangle e_\nu$.

Let $H=L^2[0,2\pi]$ with the inner product $\langle f,g\rangle = \frac{1}{\pi}\int_0^{2\pi} f(x)g(x)\,dx$, $\mathfrak{F}=\{\,\tfrac{1}{\sqrt{2}},\; \cos x,\; \sin x,\; \ldots,\; \cos nx,\; \sin nx,\; \ldots\,\}$ be an orthonormal basis. Show that the closed linear sub-space $E$ spanned by $\mathfrak{F}$ is $H$.

Problem 6 (yau-contest 2012)
Section: Analysis and Differential Equations

Let $\mathcal{H}=L^2[0,1]$ relative to the Lebesgue measure and define $(Kf)(s)=\int_0^s f(t)\,dt$ for each $f\in\mathcal{H}$. Show that $K$ is a compact operator without eigenvalues.