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Miklós Schweitzer
2001 Miklós Schweitzer
2001 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(11)
9
1
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Miklós Schweitzer 2001 Problem 9
Let
H
H
H
be the hyperbolic plane,
I
(
H
)
I(H)
I
(
H
)
be the isometry group of
H
H
H
, and
O
∈
H
O\in H
O
∈
H
be a fixed starting point. Determine those continuous
σ
:
H
→
I
(
H
)
\sigma\colon H\rightarrow I(H)
σ
:
H
→
I
(
H
)
mappings that satisfty the following three conditions: (a)
σ
(
O
)
=
i
d
\sigma(O)=\mathrm{id}
σ
(
O
)
=
id
, and
σ
(
X
)
O
=
X
\sigma (X)O=X
σ
(
X
)
O
=
X
for all
X
∈
H
X\in H
X
∈
H
; (b) for every
X
∈
H
\
{
O
}
X\in H\backslash \{ O\}
X
∈
H
\
{
O
}
point, the
σ
(
X
)
\sigma(X)
σ
(
X
)
isometry is a paracyclic shift, i.e. every member of a system of paracycles through a common infinitely far point is left invariant; (c) for any pair
P
,
Q
∈
H
P,Q\in H
P
,
Q
∈
H
of points there exists a point
X
∈
H
X\in H
X
∈
H
such that
σ
(
X
)
P
=
Q
\sigma(X)P=Q
σ
(
X
)
P
=
Q
. Prove that the
σ
:
H
→
I
(
H
)
\sigma\colon H\rightarrow I(H)
σ
:
H
→
I
(
H
)
mappings satisfying the above conditions are differentiable with the exception of a point.
8
1
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Miklós Schweitzer 2001 Problem 8
Let
H
H
H
be a complex Hilbert space. The bounded linear operator
A
A
A
is called positive if
⟨
A
x
,
x
⟩
≥
0
\langle Ax, x\rangle \geq 0
⟨
A
x
,
x
⟩
≥
0
for all
x
∈
H
x\in H
x
∈
H
. Let
A
\sqrt A
A
be the positive square root of
A
A
A
, i.e. the uniquely determined positive operator satisfying
(
A
)
2
=
A
(\sqrt{A})^2=A
(
A
)
2
=
A
. On the set of positive operators we introduce the
A
∘
B
=
A
B
B
A\circ B=\sqrt A B\sqrt B
A
∘
B
=
A
B
B
operation. Prove that for a given pair
A
,
B
A, B
A
,
B
of positive operators the identity
(
A
∘
B
)
∘
C
=
A
∘
(
B
∘
C
)
(A\circ B)\circ C=A\circ (B\circ C)
(
A
∘
B
)
∘
C
=
A
∘
(
B
∘
C
)
holds for all positive operator
C
C
C
if and only if
A
B
=
B
A
AB=BA
A
B
=
B
A
.
7
1
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Miklós Schweitzer 2001 Problem 7
Let
e
1
,
…
,
e
n
e_1,\ldots, e_n
e
1
,
…
,
e
n
be semilines on the plane starting from a common point. Prove that if there is no
u
≢
0
u\not\equiv 0
u
≡
0
harmonic function on the whole plane that vanishes on the set
e
1
∪
⋯
∪
e
n
e_1\cup \cdots \cup e_n
e
1
∪
⋯
∪
e
n
, then there exists a pair
i
,
j
i,j
i
,
j
of indices such that no
u
≢
0
u\not\equiv 0
u
≡
0
harmonic function on the whole plane exists that vanishes on
e
i
∪
e
j
e_i\cup e_j
e
i
∪
e
j
.
6
1
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Miklós Schweitzer 2001 Problem 6
Let
I
⊂
R
I\subset \mathbb R
I
⊂
R
be a non-empty open interval,
ε
≥
0
\varepsilon\geq 0
ε
≥
0
and
f
:
I
→
R
f\colon I\rightarrow\mathbb R
f
:
I
→
R
a function satisfying the
f
(
t
x
+
(
1
−
t
)
y
)
≤
t
f
(
x
)
+
(
1
−
t
)
f
(
y
)
+
ε
t
(
1
−
t
)
∣
x
−
y
∣
f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|x-y|
f
(
t
x
+
(
1
−
t
)
y
)
≤
t
f
(
x
)
+
(
1
−
t
)
f
(
y
)
+
εt
(
1
−
t
)
∣
x
−
y
∣
inequality for all
x
,
y
∈
I
x,y\in I
x
,
y
∈
I
and
t
∈
[
0
,
1
]
t\in [0,1]
t
∈
[
0
,
1
]
. Prove that there exists a convex
g
:
I
→
R
g\colon I\rightarrow\mathbb R
g
:
I
→
R
function, such that the function
l
:
=
f
−
g
l :=f-g
l
:=
f
−
g
has the
ε
\varepsilon
ε
-Lipschitz property, that is
∣
l
(
x
)
−
l
(
y
)
∣
≤
ε
∣
x
−
y
∣
for all
x
,
y
∈
I
|l(x)-l(y)|\leq \varepsilon|x-y|\text{ for all }x,y\in I
∣
l
(
x
)
−
l
(
y
)
∣
≤
ε
∣
x
−
y
∣
for all
x
,
y
∈
I
5
1
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Miklós Schweitzer 2001 Problem 5
Prove that if the function
f
f
f
is defined on the set of positive real numbers, its values are real, and
f
f
f
satisfies the equation
f
(
x
+
y
2
)
+
f
(
2
x
y
x
+
y
)
=
f
(
x
)
+
f
(
y
)
f\left( \frac{x+y}{2}\right) + f\left(\frac{2xy}{x+y} \right) =f(x)+f(y)
f
(
2
x
+
y
)
+
f
(
x
+
y
2
x
y
)
=
f
(
x
)
+
f
(
y
)
for all positive
x
,
y
x,y
x
,
y
, then
2
f
(
x
y
)
=
f
(
x
)
+
f
(
y
)
2f(\sqrt{xy})=f(x)+f(y)
2
f
(
x
y
)
=
f
(
x
)
+
f
(
y
)
for every pair
x
,
y
x,y
x
,
y
of positive numbers.
4
1
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Miklós Schweitzer 2001 Problem 4
Find the units of
R
=
Z
[
t
]
[
t
2
−
1
]
R=\mathbb Z[t][\sqrt{t^2-1}]
R
=
Z
[
t
]
[
t
2
−
1
]
.
3
1
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Miklós Schweitzer 2001 Problem 3
How many minimal left ideals does the full matrix ring
M
n
(
K
)
M_n(K)
M
n
(
K
)
of
n
×
n
n\times n
n
×
n
matrices over a field
K
K
K
have?
2
1
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Miklós Schweitzer 2001 Problem 2
Let
α
≤
−
2
\alpha \leq -2
α
≤
−
2
be an integer. Prove that for every pair
(
β
0
,
β
1
)
(\beta_0, \beta_1)
(
β
0
,
β
1
)
of integers there exists a uniquely determined sequence
0
≤
q
0
,
…
,
q
k
<
α
2
−
α
0\leq q_0, \ldots, q_k<\alpha ^ 2 - \alpha
0
≤
q
0
,
…
,
q
k
<
α
2
−
α
of integers, such that
q
k
≠
0
q_k\neq 0
q
k
=
0
if
(
β
0
,
β
1
)
≠
(
0
,
0
)
(\beta_0, \beta 1)\neq (0,0)
(
β
0
,
β
1
)
=
(
0
,
0
)
and
β
i
=
∑
j
=
0
k
q
j
(
α
−
i
)
j
,
for
i
=
0
,
1
\beta_i=\sum_{j=0}^k q_j(\alpha - i)^j,\text{ for }i=0,1
β
i
=
j
=
0
∑
k
q
j
(
α
−
i
)
j
,
for
i
=
0
,
1
1
1
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Miklós Schweitzer 2001 Problem 1
Let
f
:
2
S
→
R
f\colon 2^S\rightarrow \mathbb R
f
:
2
S
→
R
be a function defined on the subsets of a finite set
S
S
S
. Prove that if
f
(
A
)
=
F
(
S
\
A
)
f(A)=F(S\backslash A)
f
(
A
)
=
F
(
S
\
A
)
and
max
{
f
(
A
)
,
f
(
B
)
}
≥
f
(
A
∪
B
)
\max \{ f(A), f(B)\}\geq f(A\cup B)
max
{
f
(
A
)
,
f
(
B
)}
≥
f
(
A
∪
B
)
for all subsets
A
,
B
A, B
A
,
B
of
S
S
S
, then
f
f
f
assumes at most
∣
S
∣
|S|
∣
S
∣
distinct values.
10
1
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Miklós Schweitzer 2001 Problem 10
Show that if a connected, nowhere zero sectional curvature of Riemannian manifold, where symmetric (1,1)-tensor of the Levi-Civita connection covariant derivative vanishes, then the tensor is constant times the unit tensor.(translated by j___d)
11
1
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Miklós Schweitzer 2001 Problem 11
Let
ξ
(
k
1
,
k
2
)
,
k
1
,
k
2
∈
N
\xi_{(k_1, k_2)}, k_1, k_2 \in\mathbb N
ξ
(
k
1
,
k
2
)
,
k
1
,
k
2
∈
N
be random variables uniformly bounded. Let
c
l
,
l
∈
N
c_l, l\in\mathbb N
c
l
,
l
∈
N
be a positive real strictly increasing infinite sequence such that
c
l
+
1
/
c
l
c_{l+1}/ c_l
c
l
+
1
/
c
l
is bounded. Let
d
l
=
log
(
c
l
+
1
/
c
l
)
,
l
∈
N
d_l=\log \left(c_{l+1}/c_l\right), l\in\mathbb N
d
l
=
lo
g
(
c
l
+
1
/
c
l
)
,
l
∈
N
and suppose that
D
n
=
∑
l
=
1
n
d
l
↑
∞
D_n=\sum_{l=1}^n d_l\uparrow \infty
D
n
=
∑
l
=
1
n
d
l
↑
∞
when
n
→
∞
n\to\infty
n
→
∞
Suppose there exist
C
>
0
C>0
C
>
0
and
ε
>
0
\varepsilon>0
ε
>
0
such that
∣
E
{
ξ
(
k
1
,
k
2
)
ξ
(
l
1
,
l
2
)
}
∣
≤
C
∏
i
=
1
2
{
log
+
log
+
(
c
max
{
k
i
,
l
i
}
c
min
{
k
i
,
l
i
}
)
}
−
(
1
+
ε
)
\left| \mathbb E \left\{ \xi_{(k_1,k_2)}\xi_{(l_1,l_2)}\right\}\right| \leq C\prod_{i=1}^2 \left\{ \log_+\log_+\left( \frac{c_{\max\{ k_i, l_i\}}}{c_{\min\{ k_i, l_i\}}}\right)\right\}^{-(1+\varepsilon)}
E
{
ξ
(
k
1
,
k
2
)
ξ
(
l
1
,
l
2
)
}
≤
C
i
=
1
∏
2
{
lo
g
+
lo
g
+
(
c
m
i
n
{
k
i
,
l
i
}
c
m
a
x
{
k
i
,
l
i
}
)
}
−
(
1
+
ε
)
for each
(
k
1
,
k
2
)
,
(
l
1
,
l
2
)
∈
N
2
(k_1, k_2), (l_1,l_2)\in\mathbb N^2
(
k
1
,
k
2
)
,
(
l
1
,
l
2
)
∈
N
2
(
log
+
\log_+
lo
g
+
is the positive part of the natural logarithm). Show that
lim
n
1
→
∞
n
2
→
∞
1
D
n
1
D
n
2
∑
k
1
=
1
n
1
∑
k
2
=
1
n
2
d
k
1
d
k
2
ξ
(
k
1
,
k
2
)
=
0
\lim_{\substack{n_1\to\infty \\ n_2\to\infty}} \frac{1}{D_{n_1}D_{n_2}}\sum_{k_1=1}^{n_1} \sum_{k_2=1}^{n_2} d_{k_1}d_{k_2}\xi_{(k_1,k_2)}=0
n
1
→
∞
n
2
→
∞
lim
D
n
1
D
n
2
1
k
1
=
1
∑
n
1
k
2
=
1
∑
n
2
d
k
1
d
k
2
ξ
(
k
1
,
k
2
)
=
0
almost surely.(translated by j___d)