University of Chinese Academy of Sciences 2016-year transfer exam questions

Reprinted from: http://www.math.org.cn/forum.php?mod=viewthread&tid=36853

September 17, 2016, the National University of Hkust held a master transfer of public Basic course examination, the test points in three directions, the exam is 90 minutes to qualify!

Mathematics: Three election II (public base section)

Analysis
one, Beg \[i=\int_0^{2\pi} \frac1{a+\cos\theta}d \theta,\quad a>1.\]

second, the complex function $f (z) $ is an integer function, and there is a positive integer $n$ and constant $r>0,m>0$, so that when $|z|>r$, there is $|f (z) |\leq m|z|^n$. Trial proof: $f (z) $ is a maximum of $n$ Polynomial or a constant of the second.

third, state the Lebesgue control convergence theorem and prove \[\lim_{n\to+\infty}\int_0^\infty\frac{\ln (x+n)}ne^{-x}\cos xd x=0.\]

to state the open mapping theorem and prove that: set $\|\cdot\|_1$ and $\|\cdot\|_2$ are two kinds of norm on linear space $x$, and make $ (x,\|\cdot\|_1) $ and $ (x,\|\cdot\|_2) $ are complete. If there is a constant $ a>0$ make the arbitrary $x\in x$, there is $\|x\|_2\leq a\|x\|_1$, there must be a constant $b>0$, so that any $x\in x$, there $\|x\|_1\leq b\|x\|_2$.

algebra

first, set $a$ and $b$ are the elements of the group $g$, the order is $m$ and $n$, $ (m,n) =1$ and $ab=ba$. Prove $ab$ 's order $mn$.

second, set $s_n$ is the $n$ symmetry group on the $\{1,2,\cdots,n\}$. Proof:
1) $S =\{\sigma|\sigma\in s_n,\sigma (1) =1\}$ is a subgroup of $s_n$;
2) $\{(1), (up), (1,3), \cdots, (1,n) \}$ A left companion set representing elements in $s$.

Thirdly, set the group $g$ function on the set $x$. $n$ for $x$ in the $g$ role of the number of tracks, to any $a\in x$, remember $\omega_a=\{ga|g\in g\}$ is $a$ in orbit, $Ga =\{g\in g|ga=a\}$ for $a$ Fixed subgroups. For any $g\in g$, remember $f (g) $ for $x$ under the action of $g$ the number of stationary points. Proof:
1) $b \in\omega_a\leftrightarrow \omega_a=\omega_b$;
2) for any $g\in g$, there is $g_{ga}=gg_ag^{-1}$;
3) $\sum_{g\in g}f (g) =n| g|$.

Four, set $r,s$ is the ring, $f: r\to S is the same state of the ring. It is proved that the homomorphism kernel $\ker f$ is the ideal of the ring $r$, and the mapping
\begin{align*}f:r/\ker f&\to s\\\overline r&\mapsto F (R) \end{align*}
is the single homomorphism of the Ring, specifically: $F: R/\ker f\to \mathrm{im} f$ is the isomorphism of the ring.

prove that the polynomial $x^2+x+1$ and $x^3+x+1$ are irreducible on the $\mathbb{z}_2$, and find out all three irreducible polynomial on the finite field $\mathbb{z}_2$.

Geometric topology

first, define a topology on the $\mathbb{r}$ set that contains $ (0,2) $ with $ (1,3) $ and contain as few open sets as possible.

second, set $x$ is a topological space, $A $ and $b$ is a subset of $x$, $\overline a$ and $\overline b$ respectively $a$ and $b$ closure. Prove $a\subset b$, $\overline a\subset Overline b$.

The $\{x_n\}$ is a series of $\mathbb{r}$ in the set of real numbers with a standard topology, wherein $x_n=\frac{( -1) ^n}n$.
1) prove that each neighborhood containing $0$ contains an opening interval of $ (-a,a) $;
2) for any $a>0$, there is $n\in \mathbb{z}^+$, so when $n\geq n$, there is $x_n\in (-a,a) $.

the curvature and torsion of $e^3$ (t) = (A\cos t,a\sin t,bt) $, where $a$ and $b$ are not constants for $0$.

v. $e^3$ the Gaussian curvature and mean curvature of the surface "(u,v) = (U\cos v,u\sin v,v) $.

System Science, cybernetics (public base part)

one, (50 points) briefly describe the following concepts and principles:
(1) dual principle;

(2) Principle of separation;

(3) minimum implementation;

(4) balance point;

(5) Progressive stability.

Second, (20 points) determine whether the following system can control:
\[\dot x = Ax + bu = \left[{\begin{array}{*{20}{c}}{-1}&1&0&0\\0&{-1}&0&0\\0&0&1& Amp;1\\0&0&0&1\end{array}} \right]x + \left[{\begin{array}{*{20}{c}}1\\0\\0\\{-1}\end{array}} \right]u .\]

three, (20 points) to determine whether the following system can be observed:
\[\left\{\begin{array}{l}\dot x = Ax + bu = \left[{\begin{array}{*{20}{c}}0&1&0\\0&0&1\\{-2}& {-4}&{-3}\end{array}} \right]x + \left[{\begin{array}{*{20}{c}}1\\0\\0\end{array}} \right]u,\\y = cx = \left[{\b Egin{array}{*{20}{c}}1&4&2\end{array}} \right]x.\end{array} \right.\]

Iv. (20 points) determine the stability of the following systems:
\[\left\{\begin{array}{l}{{\dot X}_1} = {X_2},\\{{\dot X}_2} =-{X_1}.\end{array} \right.\]

Five, (20 points) prove that the linear system can be observed in the output feedback remains unchanged.

Six, (20 points) set open area $d$ meet $0\in d\subset \mathbb{r}^n$. Consider the system $$\dot x=f (x), $$ where $f:d\to \mathbb{r}^n$ is a local Lipschitz function, and $f (0) =0$. If there is a continuous micro function $v:d\to \mathbb{r}$ satisfies
(i) when $x\in d-\{0\}$ $v (x) >0$, and $v (0) =0$,

(ii) $\dot V (x) \leq 0,x\in d$,
prove $x=0$ stable.

statistics (public base section)

One, (15 points) sequence $\{a_n\}$ satisfies the relational $a_{n+1}=a_n+\frac{n}{a_n},a_1>0$. $\lim_{n\to\infty} n (a_n-n) $ exists.


two, (15 points) set $f (x) $ in $ (a, b) $ two can be guided, and there is a constant $\alpha,\beta$, so that for $\forall X\in (A, b) $
$ $f ' (x) =\alpha f (x) +\beta F ' (x), $$ then $f (x) $ in $ (a, b) $ within infinite times can be guided.

three, (15 points) to seek power series $\sum_{n=0}^\infty \frac{n^3+2}{(n+1)!} (x-1) Convergence domain and function of ^n$.

Iv. (15 points) set $f (x) $ is a continuous function with a lower bound or upper bound on the $\mathbb{r}$ and a positive $a$ makes the $ $f (x) +a\int_{x-1}^x F (t) dt$$ a constant. Verify: $f (x) $ must be constant.

Five, (15 points) set $f (x, y) $ on the $x^2+y^2\leq 1$ has a continuous second derivative, $f _{xx}^2+2f_{xy}^2+f_{yy}^2\leq m$. If $f (0,0) =0,f_x (0,0) =f_y (0,0) = 0$, proving $$\left |\iint_{x^2+y^2\leq 1}f (x, y) dxdy\right |\leq. $$

Six, (15 points) known \[a=\left ({\begin{array}{*{20}{c}}{-\frac{{\sqrt 3}}{2}}&{-\frac{1}{2}}\\{\frac{1}{2}}&{-\ Frac{{\sqrt 3}}{2}}\end{array}} \right), \] Seek $a^{2016}$.

Seven, (15 points) Known \[a = \left ({\begin{array}{*{20}{c}}1&2\\3&4\end{array}} \right), \] and $a^n=\alpha_ni+\beta_n A$ . Beg $\alpha_n,\beta_n$.

Viii. (15 points) in $\mathbb{r}^4$, $$\alpha= (1,1,-1,1), \beta= (1,-1,1,1), \gamma= (1,0,1,1), m= (\alpha,\beta,\gamma), $$ Beg $ A set of standard orthogonal bases for m^\bot$. (Data forgot)

Nine, (15 points) known linear space $m=\{(x, y) |x-2y+z=0\}$, for $u= (a) ' $ ' orthogonal projection on $m$.

10, (15 points) set $u,v\in \mathbb{r}^n$, if $u ' U=v ' v$, prove the existence of $n$-order orthogonal matrix $q$, making $qu=v,qv=u$.

University of Chinese Academy of Sciences 2016-year transfer exam questions