[Home Squat University Mathematics magazine] NO. 405 Chinese Academy of Sciences Mathematics and Systems Science Research Institute 2015 year summer Camp analysis and Algebra questions

This paper is divided into two parts: Analysis of $5$ (total $50$), algebraic $5$ (total $50$). Exam Time: $120$ minutes

1. ($ $) to which real numbers $\al$, series $\dps{\vsm{n}\sex{\frac{1}{n}-\sin \frac{1}{n}}^\al}$ converge?

2. ($6 ' $) Set $y $ is $[0,1]$ on the $C ^1$ smooth real function, satisfies the equation $$\bex y ' (x) +y ' (x)-y (x) =0,\quad x\in (0,1), \eex$$ and $y (0) =y (1) =0$. Trial: $y (x) =0,\ x\in [0,1]$.

3. ($ $) Set $f $ is a bounded continuous real function on $\bbr^2$, defining $$\bex g (x) =\int_{\bbr} \frac{f (x,t)}{1+t^2}\rd T,\quad x\in\bbr. \eex$$ test: $g (x) $ is a continuous function on the $\bbr$.

4. ($ $) Set $f $ is $[1,\infty) $ on a continuous micro-real function, satisfying $f (1) =1$, and $$\bex F ' (x) =\frac{1}{f^2 (x) +x^2},\quad x\in (1,\infty). \eex$$ test: $\dps{\vlm{x}f (x)}$ exists and does not exceed $\dps{1+\frac{1}{4}\pi}$.

5. ($14 ' =2\times 7 ' $) set $f $ is a continuous real function on the $[0,1]$, calculate the following limits and prove your conclusion: (1). $\dps{\vlm{n}\int_0^1 x^nf (x) \rd x}$; (2). $\dps{\vlm{n}n\int_0^1 x^nf (x) \rd x}$.

6. For integer $a, b$, define $a \equiv b\ (\mod m) $ when and only if $m \mid (A-B) $ (that is, $m $ divisible $a-b$). When a positive integer $m $, what is the solution to a linear equation group? $$\bex \sedd{\ba{rrrrrrl} x&+&2y&-&z&\equiv&1\ (\mod m) \ \ 2x&-&3y&+&z&\ Equiv&4\ (\mod m) \ 4x&+&y&-&z&\equiv&9\ (\mod m) \ea} \eex$$

7. Set $\tt$ is real, $n $ is the natural number, ask $$\bex \sex{\ba{cc} e^{-i\tt}&2i\sin \tt\\ 0&e^{i\tt} \ea}^n. \eex$$

8. Set $A, B\in m_n (\BBC) $ ($n $ order complex matrix), answer the following questions and explain the reasons: (1). $AB $ is similar to $BA $? (2). $AB $ with $BA $ do you have the same characteristic polynomial? (3). $AB $ with $BA $ do you have the same minimum polynomial?

9. It is proved that the finite dimensional linear space on the real field cannot be a finite real subspace, and then the finite domain case is discussed.

10. Set $T: V\to v$ is a power 0 operator on a finite dimensional linear space on a complex domain $\bbc$ $V $ (i.e., there is a positive integer $k $, which makes $T ^k=0$), $I $ is a unit operator. The linear operator is $S $, $Q $ makes $S ^2=i+t$, $Q (i+t) =i$.

[Home Squat University Mathematics magazine] NO. 405 Chinese Academy of Sciences Mathematics and Systems Science Research Institute 2015 year summer Camp analysis and Algebra questions