SCHWEITZER 1961

Let $a$ ( $\neq e$ , the unit element) be an element of finite order of a group $G$ and let $t$ ( $\geq 2$ ) be a positive integer. Show: if the complex $A= \{ e,a,a^2, \dots , a^{t-1} \}$ is not a group, then for every positive integer $k$ ( $2 \leq k \leq t$ ) the complex $B= \{ e. a^k, a^{2k}, \dots , a^{(t-1)k} \}$ differs from $A$ . (A. 16)

Show that a ring $R$ has a unit element if and only if any $R$ -module $G$ can be written as a direct sum of $RG$ and of the trivial submodule of $G$ . (An $R$ -module is a linear space with $R$ as its scalar domain. $RG$ denotes the submodule generated by the elements of the form $rg$ ( $r \in R, g \in G$ ). The trivial submodule of $G$ consists of the elements $g$ of $G$ for which $rg=0$ holds for every $r \in R$ .) (A. 20)

Let $f(x)= x^n +a_1 x^(n-1)+ \dots + a_n$ ( $n\geq 1$ ) be an irreducible polynomial over the field $K$ . Show that every non-zero matrix commuting with the matrix $\begin{bmatrix} 0 & 1 & 0 & \dots & 0 & 0 0 & 0 & 1 & \dots & 0 & 0 \dots & \dots & \dots & \dots & \dots & \dots 0 & 0 & 0 & \dots & 0 & 1 -a_n & -a_{n-1} & -a_{n-2} & \dots & -a_2 & -a_1 \end{bmatrix}$ is invertible. (A. 4)

Let $f(x)$ be a real- or complex-value integrable function on $(0,1)$ with $\mid f(x) \mid \leq 1$ . Set $c_k = \int_0^1 f(x) e^{-2 \pi i k x} dx$ and construct the following matrices of order $n$ : $T= (t_{pq})_{p,q=0}^{n-1}, T^{*}= (t_{pq}^{*})_{p,q =0}^{n-1}$ where $t_{pq}= c_{q-p}, t^{*}= \overline {c_{p-q}}$ . Further, consider the following hyper-matrix of order $m$ : $S= \begin{bmatrix} E & T & T^2 & \dots & T^{m-2} & T^{m-1} T^{*} & E & T & \dots & T^{m-3} & T^{m-2} T^{*2} & T^{*} & E & \dots & T^{m-3} & T^{m-2} \dots & \dots & \dots & \dots & \dots & \dots T^{*m-1} & T^{*m-2} & T^{*m-3} & \dots & T^{*} & E \end{bmatrix}$ ( $S$ is a matrix of order $mn$ in the ordinary sense; E denotes the unit matrix of order $n$ ). Show that for any pair $(m , n)$ of positive integers, $S$ has only non-negative real eigenvalues. (R. 19)

Determine the functions $G$ defined on the set of all non-zero real numbers the values of which are regular matrices of order $2$ , and the functions $f$ mapping the two-dimensional real vector space $E_2$ into itself, such that for any vector $y \in E_2$ and for any regular matrix $X$ of order $2$ , $f(X_y)= G(det X)Xf(y)$ ( $det X$ denotes the determinant of $X$ ).(A. 5)

Consider a sequence $\{ a_n \}_{n=1}^{\infty}$ such that, for any convergent subsequence $\{ a_{n_k} \}$ of $\{a_n\}$ , the sequence $\{ a_{n_k +1} \}$ also is convergent and has the same limit as $\{ a_{n_k}\}$ . Prove that the sequence $\{ a_n \}$ is either convergent of has infinitely many accumulation points the set of which is dense in itself. Give an example for the second case. (A sequence $x_n \to \infty$ or $-\infty$ is considered to be convergente, too)(S. 13)

For the differential equation $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}= 2\frac{\partial^2 u}{\partial x \partial y}$ find all solutions of the form $u(x,y)=f(x)g(y)$ . (R. 14)

Let $f(x)$ be a convex function defined on the interval $[0, \frac {1}{2}]$ with $f(0)=0$ and $f(\frac{1}{2})=1$ ; Let further $f(x)$ be differentiable in $(0, \frac {1}{2})$ , and differentiable at $0$ and $\frac{1}{2}$ from the right and from the left, respectively. Finally, let $f'(0)>1$ . Extend $f(x)$ to $[0.1]$ in the following manner: let $f(x)= f(1-x)$ if $x \in (\frac {1} {2}, 1]$ . Show that the set of the points $x$ for shich the terms of the sequence $x_{n+1}=f(x_n)$ ( $x_0=x; n = 0, 1, 2, \dots$ ) are not all different is everywhere dense in $[0,1]$ ; (R. 10)

Spin a regular coin repeatedly until the heads and tails turned up both reach the number $k$ ( $k= 1, 2, \dots$ ); denote by $v_k$ the number of the necessary throws. Find the distribution of the random variable $v_k$ and the limit-distribution of the random variable $\frac {v_k -2k}{\sqrt {2k}}$ as $k \to \infty$ . (P. 10)

Given a straight line $g$ in the plane and a point $O$ on $g$ . Construct, without making use of the Parallel Axiom, the half-line perpendicular to $g$ at the point $O$ and lying in one of the half-planes defined by $g$ , under the following restrictions: The construction must be effected by use of a ruler and of a length standard (i.e. an etalon-segment) only; moreover, all lines and points of the construction must lie in the chosen half-plane. (G. 20)