Exercise 1

Prove that if $$x,y,z\in[0,\infty)$$, then $$e^{x+y+z-3}>\frac{x^3+y^3+z^3}{27}.$$

Exercise 2

Define a function $$f(\theta)$$ as $$f(\theta)=\frac{2}{\sqrt{1+\tan\theta}}+\frac{\sin\theta}{\sqrt{8-7\sin^2\theta}},\theta\in[0,\frac{\pi}{2}).$$

Find the maximum value of $$f^2(\theta)-3f(\theta)+2.$$

Exercise 3

Compute the following integral, where $$ac-b^2=1$$: $$\int_{0}^{2\pi} \frac{ d\theta}{a\cos^2\theta -2b\cos\theta \sin\theta + c\sin^2\theta}.$$

Exercise 4

Let $$\vec F =$$ and $$S=\{(x,y,z): x^2+y^2+z^2=16, z>0\}$$. Compute $$\int\int\limits_{S} (\vec \nabla \times \vec F)\cdot \vec n dS$$, where $$\vec n(x,y,z)$$ denote the outward unit normal at $$P(x,y,z)$$

Exercise 5

Let $$A = \left( \begin{array} {*{3}{cc}} 1 & 0 & 0 \\\\ 1 & 0 & 1 \\\\ 0 & 1 & 0 \end{array} \right)$$, compute $$A^{100}$$.

Exercise 6

We know that Almost sure convergence implies convergence in probability, but the opposite is generally false. Now consider a special example. Let $$\{X_n\}_{n=1}^\infty$$ be a sequence of independent random variables and $$S_n=\sum\limits_{k=1}^nS_k$$. Prove that $$S_n \overset{P}{\rightarrow} S \Leftrightarrow S_n \rightarrow S\ a.s.$$

Exercise 7

We define a random process $$M_t=E(X\vert F_t), t\in [0,T]$$, where $$X$$ is a $$F_T$$ measurable random variable. (a) Show that $$M_t$$ is a martingale. (b) Prove that for $$t\in[0,T], \ E(M_t^2)\leq\infty$$. (c) Let $$X=e^{uB_T},u\in\mathbb{R}$$, prove that solution of the following equation exist and solve the equation. $$M_t-E(M_0)-\int_0^tY_sdB_s=0.$$

Exercise 8

Suppose the equator of the Earth is a smooth perfect circle. Prove that there are two points A and B, such that the distance between A and B is greater than 100KM and A and B have the same pressure at the moment while you finish reading this question. What is the maximum possible distance (on the earth's surface) between A and B?

Exercise 9

$$A=\left(\begin{array}{ccc} 0 & 5 & 3 \\ 7 & -3 & 10 \\ -4 & -5 & 7\end{array}\right) , B=\left(\begin{array}{ccc} 7 & -3 & 10 \\ 0 & 5 & 3 \\ 20 & 25 & -35\end{array}\right)$$ Find a matrix C such that CA=B, do not use any computer software, do it by hand.

Answer: $$C=\left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -5 \end{array} \right)$$ Hint: Do not compute $$A^{-1}$$ and then do $$BA^{-1}$$. The calculation will be messy. Instead, transpose both sides. If you have any questions, please contact us at http://www.mathwit.com/contact.php.