Problem 1(a) (yau-contest 2013)

We consider the wave equation $u_{tt} = \Delta u$ in $\mathbb{R}^3 \times \mathbb{R}_+$. (5 pts) A right going pulse with speed 1

$u(x, y, z, t) = 1$ for $t < x < t + 1$; $u(x, y, z, t) = 0$ else

is clearly a solution to the wave equation. However, it is a discontinuous solution, explain in which sense it is a solution to the equation.

Problem 1(b) (yau-contest 2013)

We consider the wave equation $u_{tt} = \Delta u$ in $\mathbb{R}^3 \times \mathbb{R}_+$. (5 pts) Surprisingly, one can construct smooth progressive wave solutions with speed larger than 1. In astronomy this kind of wave known as superluminal wave. Try a solution of the form

$$u(x, y, z, t) = v\!\left(\frac{x - c t}{\sqrt{c^2 - 1}},\, y,\, z\right),\quad c \in \mathbb{R}^3,\ \ |c| > 1.$$

Derive an equation for $v$ and show that there is a nontrivial solution with compact support in $(y, z)$ for any fixed $x, t$.

Problem 1(c) (yau-contest 2013)

We consider the wave equation $u_{tt} = \Delta u$ in $\mathbb{R}^3 \times \mathbb{R}_+$. (5 pts) For any $R > 0$, $0 < t < R$, show that energy

$$E(t) := \int_{|x| \le R - t} \big(|u_t(\cdot, t)|^2 + |\nabla u(\cdot, t)|^2\big)\, d\bar{x}$$

is a decreasing function.

Problem 1(c) (yau-contest 2013)

(10 pts) Show that smooth superluminal progressive wave solutions of the form

$$u(\vec{x}, t) = v(\vec{x} - \vec{c}\, t),\quad \vec{c} \in \mathbb{R}^3,\ \ |\vec{c}| > 1$$

cannot have a finite energy.

Hint: Using (c) and look at the energy of the solution in various balls.

Problem 2(a) (yau-contest 2013)

Finite time extinction and hyper-contractivity are important properties in modeling of some physical and biology systems. The essence of estimates is given by the following problem for ODE.

Assume $y(t) \ge 0$ is a $C^1$ function for $t > 0$ satisfying $y'(t) \le \alpha - \beta\, y(t)^a$ for $\alpha > 0$, $\beta > 0$, then

(10 points) For $a > 1$, $y(t)$ has the following hyper-contractive property

$$y(t) \le (\alpha/\beta)^{1/a} + \left[\frac{1}{\beta(a-1)t}\right]^{\frac{1}{a-1}},\quad \text{for } t > 0.$$

Problem 2(b) (yau-contest 2013)

Assume $y(t) \ge 0$ is a $C^1$ function for $t > 0$ satisfying $y'(t) \le \alpha - \beta\, y(t)^a$ for $\alpha > 0$, $\beta > 0$, then

(2 points) For $a = 1$, $y(t)$ decays exponentially

$$y(t) \le \alpha/\beta + y(0) e^{-\beta t}.$$

Problem 2(c) (yau-contest 2013)

Assume $y(t) \ge 0$ is a $C^1$ function for $t > 0$ satisfying $y'(t) \le \alpha - \beta\, y(t)^a$ for $\alpha > 0$, $\beta > 0$, then

(10 points) For $a < 1$, $\alpha = 0$, $y(t)$ has finite time extinction, which means that there exists $T_{ext}$ such that

$$0 < T_{ext} \le \frac{y(0)^{1-a}}{\beta(1-a)}$$

and that $y(t) = 0$ for all $t > T_{ext}$.

Problem 3(a) (yau-contest 2013)

Consider the speed $v$ of a ball (density $\rho$, radius $R$) falling through a viscous fluid (density $\rho_f$, viscosity $\mu$) with drag coefficient given by Stokes’ law $\zeta = 6\pi R\mu$:

$$\frac{4}{3}\pi R^3 \rho \frac{dv}{dt} = \frac{4}{3}\pi R^3 (\rho - \rho_f) g - \zeta v,\quad v(0) = v_0.$$

(5 points) Nondimensionalize the equation by writing, $v(t) = V\,\tilde{v}(\tilde{t})$ with $t = T\,\tilde{t}$. Select $V, T$ (characteristic scales known as terminal velocity and settling time respectively) so that all coefficients in the ODE but one are equal to 1. Your equation will have a single dimensionless parameter given by the ratio of the initial speed $v_0$ to the characteristic speed $V$.

Problem 3(b) (yau-contest 2013)

(2 points) Solve the nondimensional problem for $\tilde{v}(\tilde{t})$.

Problem 3(c) (yau-contest 2013)

(8 points) Describe the behavior of the solution if the initial speed $v_0$ is (i) faster than and (ii) slower than the characteristic speed $V$. Compute the time to reach $(v_0 + V)/2$.

Problem 4(1) (yau-contest 2013)

Let

$$V_h = \{ v : v|_{I_j} \in P^k(I_j) \quad 1 \le j \le N \}$$

where

$$I_j = (x_{j-1}, x_j), \quad 1 \le j \le N$$

with

$$x_j = jh, \quad h = \frac{1}{N}.$$

Here $P^k(I_j)$ denotes the set of polynomials of degree at most $k$ in the interval $I_j$.

Recall the $L^2$ projection of a function $u(x)$ into the space $V_h$ is defined by the unique function $u_h \in V_h$ which satisfies

$$\|u - u_h\| \le \|u - v\| \quad \forall\, v \in V_h$$

where the norm is the usual $L^2$ norm. We assume $u(x)$ has at least $k + 2$ continuous derivatives.

(5 points) Prove the error estimate

$$\|u - u_h\| \le C h^{k+1}$$

Explain how the constant $C$ depends on the derivatives of $u(x)$.

Problem 2 (yau-contest 2013)

If another function $\varphi(x)$ also has at least $k+2$ continuous derivatives, prove
$$\left|\int_0^1 (u(x) - u_h(x))\varphi(x)\,dx\right| \le C h^{2k+2}.$$
Explain how the constant $C$ depends on the derivatives of $u(x)$ and $\varphi(x)$.

Problem 5 (yau-contest 2013)

Let $G(V,E)$ be a simple graph of order $n$ and $\delta$ the minimum degree of vertices. Suppose that the degree sum of any pair of nonadjacent vertices is at least $n$ and $F\subset E$ with $|F| \le \left\lfloor \frac{\delta^2}{8} \right\rfloor$. Let $G - F$ be the graph obtained from $G$ by deleting the edges in $F$. Prove that
(1) $G - F$ is connected and
(2) $G - F$ is Hamiltonian.

Problem 6 (yau-contest 2013)

Let $(F_n)_n$ be the Fibonacci sequence. Namely, $F_0 = 0$, $F_1 = 1$, \dots, $F_{n+2} = F_{n+1} + F_n$. Establish a relation between $\begin{pmatrix}0 & 1 \\ 1 & 1\end{pmatrix}^n$ and $F_n$ and use it to design an efficient algorithm that for a given $n$ computes the $n$-th Fibonacci number $F_n$. In particular, it must be more efficient than computing $F_n$ in consecutive steps.
Give an estimate on the number of steps of your algorithm.

Hint: Not that if $m$ is even then
$$\begin{pmatrix}0 & 1 \\ 1 & 1\end{pmatrix}^m = \left( \begin{pmatrix}0 & 1 \\ 1 & 1\end{pmatrix}^{m/2} \right)^2$$
and if $m$ is odd then
$$\begin{pmatrix}0 & 1 \\ 1 & 1\end{pmatrix}^m = \begin{pmatrix}0 & 1 \\ 1 & 1\end{pmatrix}^{m-1} \cdot \begin{pmatrix}0 & 1 \\ 1 & 1\end{pmatrix}$$
and $m - 1$ is even.