## University math exercises

**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 =

**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.
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