bluuebunny/crossref_metadata_2025_split huggingface dataset:
The norm of a derivation in a 𝑊*-algebra
Authors: Zsidó László 
Published: June 2010

The norm of an inner derivation

δ
a

{\delta _a}

of a (nonnecessary separable)

W
∗

{W^ \ast }

-algebra

M
M

is shown to satisfy
\[

|

|

δ
a

|

|

=
2
inf
{

|

|

a
−
z

|

|

;
z
∈
Z
,

the center of

M
}
,

||{\delta _a}|| = 2\inf \{ ||a - z||;z \in Z,\;{\text {the center of}\; M\} ,}

\]
and some related results are obtained.
A note on a trigonometric moment problem
Authors: Young Robert M. 
Published: June 2010

A sequence

{

λ
n

}

n
=
−
∞

∞

\{ {\lambda _n}\} _{n = - \infty }^\infty

is said to be an interpolating sequence for

L
2

(
−
π
,
π
)

{L^2}( - \pi ,\pi )

if the system of equations
\[

c
n

=

∫

−
π

π

f
(
t
)

e

i

λ
n

t

d
t

(
−
∞
>
n
>
∞
)

{c_n} = \int _{ - \pi }^\pi {f(t)} {e^{i{\lambda _n}t}}dt\quad ( - \infty > n > \infty )

\]
admits a solution

f
f

in

L
2

(
−
π
,
π
)

{L^2}( - \pi ,\pi )

whenever

{

c
n

}
∈

l
2

\{ {c_n}\} \in {l^2}

. If the solution is unique then

{

λ
n

}

\{ {\lambda _n}\}

is said to be a complete interpolating sequence. It is shown that if the imaginary part of

λ
n

{\lambda _n}

is uniformly bounded and if

|

Re
⁡
(

λ
n

)
−
n

|

≤
L
>
1

/

4
(
−
∞
>
n
>
∞
)

|\operatorname {Re} ({\lambda _n}) - n| \leq L > 1/4( - \infty > n > \infty )

, then

{

λ
n

}

\{ {\lambda _n}\}

is a complete interpolating sequence and

{

e

i

λ
n

t

}

\{ {e^{i{\lambda _n}t}}\}

is a Schauder basis for

L
2

(
−
π
,
π
)

{L^2}( - \pi ,\pi )

. It is also shown that this result is sharp in the sense that the condition

|

λ
n

−
n

|

>
1

/

4

|{\lambda _n} - n| > 1/4

is not sufficient to guarantee that

{

λ
n

}

\{ {\lambda _n}\}

is an interpolating sequence.
Subordination by convex functions
Authors: Hallenbeck D. J. , Ruscheweyh Stephan 
Published: June 2010

The following theorem is proven: Let

F
(
z
)

F(z)

be convex and univalent in

Δ
=
{
z
:

|

z

|

>
1
}
,
F
(
0
)
=
1

\Delta = \{ z:|z| > 1\} ,F(0) = 1

. Let

f
(
z
)

f(z)

be analytic in

Δ
,
f
(
0
)
=
1
,

f
′

(
0
)
=
…
=

f

(
n
−
1
)

(
0
)
=
0

\Delta ,f(0) = 1,f’(0) = \ldots = {f^{(n - 1)}}(0) = 0

, and let

f
(
z
)
≺
F
(
z
)

f(z) \prec F(z)

in

Δ
\Delta

. Then for all

γ
≠
0

\gamma \ne 0

, Re

γ
⩾
0

\gamma \geqslant 0

,
\[

γ
z

−
γ

∫
0
z

τ

γ
−
1

f
(
τ
)
d
τ
≺
γ

z

−
γ

/

n

∫
0

z

1

/

n

τ

γ
−
1

F
(

τ
n

)
d
τ
.

{\gamma _z}^{ - \gamma }\int _0^z {{\tau ^{\gamma - 1}}f(\tau )d\tau \prec \gamma {z^{ - \gamma /n}}\int _0^{{z^{1/n}}} {{\tau ^{\gamma - 1}}F({\tau ^n})d\tau .} }

\]
This theorem, in combination with a method of D. Styer and D. Wright, leads to the following Corollary. Let

f
(
z
)
,
g
(
z
)

f(z),g(z)

be convex univalent in

Δ
,
f
(
0
)
=
f
(
0
)
=
g
(
0
)
=
g
(
0
)
=
0

\Delta ,f(0) = f(0) = g(0) = g(0) = 0

. Then

f
(
z
)
+
g
(
z
)

f(z) + g(z)

is starlike univalent in

Δ
\Delta

. Other applications of the theorem are concerned with the subordination of

f
(
z
)

/

z

f(z)/z

where

f
(
z
)

f(z)

belongs to certain classes of convex univalent functions.
Interpolation by transforms of discrete measures
Authors: Pigno Louis , Saeki Sadahiro 
Published: June 2010

Let

G
G

be a compact abelian group, and

Γ
\Gamma

its character group. Given

E
⊂
Γ
,

E
a

E \subset \Gamma ,{E^a}

denotes the set of all accumulation points of

E
E

in

Γ
¯

\overline \Gamma

, the Bohr compactification of

Γ
\Gamma

. In this paper it is shown that the inclusion

(

L
1

(
G
)

)
∧

|

E

⊂

(

l
1

(
G
)

)
∧

|

E

{({L^1}(G))^ \wedge }{|_E} \subset {({l^1}(G))^ \wedge }{|_E}

obtains if and only if

E
∩

E
a

=
∅

E \cap {E^a} = \emptyset

and there exists a measure

μ
ϵ
M
(
G
)

\mu \epsilon M(G)

such that

μ
^

=
1

\widehat \mu = 1

on

E
E

and

μ
^

=
0

\widehat \mu = 0

on

Γ
∩

E
a

\Gamma \cap {E^a}

.
The Rudin-Carleson theorem for vector-valued functions
Authors: Globevnik J. 
Published: June 2010

The following generalization of the Rudin-Carleson theorem is proved. Let

X
X

be a complex Banach space and let

f
:
F
→
X

f:F \to X

be a continuous function, where

F
F

is a closed subset of the unit circle in

C
C

of Lebesgue measure zero. There exists a continuous function

g
g

from the closed unit disc to

X
X

which is analytic on the open unit disc and satisfies (i)

g

|

F
=
f

g|F = f

, (ii)

max

|

z

|

⩽
1

|

|

g
(
z
)

|

|

=

max

z
ϵ
F

|

|

f
(
z
)

|

|

{\max _{|z| \leqslant 1}}||g(z)|| = {\max _{z\epsilon F}}||f(z)||

.
Solution of integral equations by product integration
Authors: Helton Jon C. 
Published: June 2010

Suppose

f
,
h

f,h

and

G
G

are functions with values in a normed complete ring. With suitable restrictions on these functions, it is established that
\[

f
(
x
)
=
h
(
x
)
+

∫
a
x

f
(
u
)
G
(
u
,
v
)

f(x) = h(x) + \int _a^x {f(u)G(u,v)}

\]
for

a
≤
x
≤
b

a \leq x \leq b

only if

∫
a
x

h
(
u
)
G

(
u
,
v
)

v

Π
x

(
1
+
G
)

\int _a^x {h(u)G{{(u,v)}_v}{\Pi ^x}} (1 + G)

exists and is

f
(
x
)
−
h
(
x
)

f(x) - h(x)

for

a
≤
x
≤
b

a \leq x \leq b

, and that
\[

f
(
x
)
=
h
(
x
)
+

∫
a
x

G
(
u
,
v
)
f
(
u
)

f(x) = h(x) + \int _a^x {G(u,v)f(u)}

\]
for

a
≤
x
≤
b

a \leq x \leq b

only if

∫
a
x

x

Π
v

(
1
+

G

)
G
(
u
,
v
)
h
(
u
)

\int _a^x {_x{\Pi ^v}} (1 + \mathcal {G})G(u,v)h(u)

exists and is

f
(
x
)
−
h
(
x
)

f(x) - h(x)

for

a
≤
x
≤
b

a \leq x \leq b

, where

G

(
s
,
r
)
=
G
(
r
,
s
)

\mathcal {G}(s,r) = G(r,s)

.
Bounds for nearly best approximations
Authors: Wegmann Rudolf 
Published: June 2010

Let

X
X

be a uniformly convex space and

ψ
\psi

be the inverse function of the modulus of convexity

δ
(
⋅
)

\delta ( \cdot )

. Assume here that

ψ
\psi

is a concave function. Let

V
V

be a linear subspace of

X
X

and let

f
f

in

X
X

be such that

|

|

f

|

|

=
1
=
min
{

|

|

f
−
v

|

|

:
v
ϵ
V
}

||f|| = 1 = \min \{ ||f - v||:v\epsilon V\}

. Then for

0
>
δ
⩽
1

0 > \delta \leqslant 1

and for

v
v

in

V
V

with

|

|

f
−
v

|

|

⩽
1
+
δ

||f - v|| \leqslant 1 + \delta

, it follows that

|

|

v

|

|

⩽
K
⋅
ψ
(
δ
)

||v|| \leqslant K \cdot \psi (\delta )

. Let

T
T

be a compact Hausdorff-space and

V
V

a finite-dimensional subspace of

C
(
T
,
X
)

C(T,X)

. When

V
V

has the interpolation property

(

P
m

)

({P_m})

with

V
=
m
⋅
dim
⁡
X

V = m \cdot \dim X

, then the same type of estimate as above holds.
𝑝th powers of distinguished subfields
Authors: Heerema Nicholas 
Published: June 2010

Let

k
⊃
s
⊃

k
0

k \supset s \supset {k_0}

be fields of characteristic

p
≠
0
,
k

/

k
0

p \ne 0,k/{k_0}

finitely generated and

s
s

a distinguished subfield. The field

k
0

(

k

(
n
)

)
=
{
x
∈
k

|

x

p
t

∈

k
0

(

k

p

n
+
t

)

for some

t
⩾
0
}

{k_0}({k^{(n)}}) = \{ x \in k|{x^{{p^t}}} \in {k_0}({k^{{p^{n + t}}}}){\text { for some }}t \geqslant 0\}

has

k
0

(

s

p
n

)

{k_0}({s^{{p^n}}})

as a distinguished subfield and is maximal in

k
k

with respect to this property. Let

k
¯

0
s

\overline k _0^s

and

k
¯

0

{\bar k_0}

be, respectively, the separably algebraic closure and the algebraic closure of

k
0

{k_0}

in

k
k

. Then

k
¯

0

=

∩
k

k
0

(

k

(
n
)

)

{\overline k _0} = { \cap _k}{k_0}({k^{(n)}})

. Also

k
¯

0
s

=

k
¯

0

\overline k _0^s = {\bar k_0}

if and only if

k
0

(

k
p

)
⊃

k
0

(

k

(
n
)

)

{k_0}({k^p}) \supset {k_0}({k^{(n)}})

for some

n
n

. For

n
n

large

k
0

(

k

(
n
)

)
=

k
¯

0

(

k

p
n

)

{k_0}({k^{(n)}}) = {\overline k _0}({k^{{p^n}}})

. The sequence

{

[

k
0

(

k

(
n
)

)
:

k
0

]
i

}
n

{\{ {[{k_0}({k^{(n)}}):{k_0}]_i}\} _n}

is decreasing, descending from

[
k
:

k
0

]
i

to

[

k
:

¯

k
0

]
i

{[k:{k_0}]_i}\;{\text {to }}{[\overline {k:} {k_0}]_i}

in a finite number of steps. Examples are given which show: (1) that

k
0

(

k

(
n
)

)

{k_0}({k^{(n)}})

may have distinguished subfields not of the form

k
0

(

s

p
n

)

{k_0}({s^{{p^n}}})

; and, (2) how to construct

k

/

k
0

k/{k_0}

so that the sequence

{
[

k
0

(

k

(
n
)

:

k
0

]
i

}

\{ [{k_0}{({k^{(n)}}:{k_0}]_i}\}

has preassigned values.
Dominated estimates in Hilbert space
Authors: Akcoglu M. A. , Miller H. D. B. 
Published: June 2010

Let

U
U

be a unitary operator on a Hilbert space

H
H

, and let

A
n

(
U
)
,
n
=
1
,
2
,
…

{A_n}(U),n = 1,2, \ldots

, be the Cesàaro means of

U
U

. It is shown that

Σ

n
=
1

∞

P
n

A
n

(
U
)

\Sigma _{n = 1}^\infty {P_n}{A_n}(U)

is bounded for every sequence of mutually orthogonal projections

P
n

,
n
=
1
,
2
,
…

{P_n},n = 1,2, \ldots

, if and only if

1
1

is not a limit point of the spectrum of

U
U

. The proof is obtained by adapting ideas of Menchoff and Burkholder to show that for any orthonormal sequence

f
n

,
n
=
0
,
±
1
,
±
2
,
…

{f_n},n = 0, \pm 1, \pm 2, \ldots

, in

H
H

, there is an orthonormal sequence

g
n

,
n
=
1
,
2
,
…

{g_n},n = 1,2, \ldots

, such that
\[

∑

k
=
1

n

|

(

f
1

+

f
2

+
⋯
+

f
k

,

g
k

)

|

2

⩾

1

36

n

(
log
⁡
n
)

2

.

\sum \limits _{k = 1}^n {|({f_1} + {f_2} + \cdots + {f_k},{g_k}){|^2} \geqslant \frac {1} {{36}}n{{(\log n)}^2}.}

\]
The existence of conjugate points for selfadjoint differential equations of even order
Authors: Lewis Roger T. 
Published: June 2010

This paper presents sufficient conditions on the coefficents of

L

2
n

y
=

Σ

k
=
0

n

(
−
1

)

n
−
k

(

p
k

y

(
n
−
k
)

)

(
n
−
k
)

{L_{2n}}y = \Sigma _{k = 0}^n{( - 1)^{n - k}}{({p_k}{y^{(n - k)}})^{(n - k)}}

which insure that

L

2
n

y
=
0

{L_{2n}}y = 0

has conjugate points

η
(
a
)

\eta (a)

for all

a
>
0

a > 0

. The main theorem implies that

(
−
1

)
n

y

(
2
n
)

+
p
y
=
0

{( - 1)^n}{y^{(2n)}} + py = 0

has conjugate points

η
(
a
)

\eta (a)

for all

a
>
0

a > 0

when

∫
∞

x
α

p
(
x
)
d
x
=
−
∞

{\smallint ^\infty }{x^\alpha }p(x)dx = - \infty

for some

α
>
2
n
−
1

\alpha > 2n - 1

with no sign restrictions on

p
(
x
)

p(x)

.
On ℎ𝑜𝑚dim𝑀𝑈_{*}(𝑋×𝑌)
Authors: O’Neill Duane 
Published: June 2010

Let

p
p

be a prime and

B

Z

/

p

B{\mathbf {Z}}/p

the classifying space for the cyclic group

Z

/

p

{\mathbf {Z}}/p

of prime order

p
p

. A finite complex

X
X

is constructed such that
\[

hom
⋅

dim

M

U
∗

M

U
∗

(
X
×
B

Z

/

p
)
>
hom
⋅

dim

M

U
∗

M

U
∗

(
X
)
+
hom
⋅

dim

M

U
∗

M

U
∗

(
B

Z

/

p
)
.

\hom \cdot {\dim _{M{U_ \ast }}}M{U_ \ast }(X \times B{\mathbf {Z}}/p) > \hom \cdot {\dim _{M{U_ \ast }}}M{U_ \ast }(X) + \hom \cdot {\dim _{M{U_ \ast }}}M{U_ \ast }(B{\mathbf {Z}}/p).

\]
Which operators are similar to partial isometries?
Authors: Fialkow L. A. 
Published: June 2010

Let

H

\mathcal {H}

denote a separable, infinite dimensional complex Hilbert space and let

L

(

H

)

\mathcal {L}(\mathcal {H})

denote the algebra of all bounded linear operators on

H

\mathcal {H}

. Let

P

=
{
T

in

L

(

H

)

|

r
(
T
)
>
1

and

T

is similar to a partial isometry with infinite rank
}

\mathcal {P} = \{ T{\text { in }}\mathcal {L}(\mathcal {H})|r(T) > 1{\text { and }}T{\text {is similar to a partial isometry with infinite rank} \}}

; let

S

=
{
S

in

L

(

H

)

|

r
(
S
)
>
1
,

range

(
S
)

is closed, and rank

(
S
)
=

nullity

(
S
)
=

corank

(
S
)
=

ℵ
0

}

\mathcal {S} = \{ S{\text { in }}\mathcal {L}(\mathcal {H})|r(S) > 1,{\text {range}}(S){\text { is closed, and rank}}(S)= {\text {nullity}}(S)= {\text {corank}}(S)={\aleph _0}\}

. It is conjectured that

P

=

S

\mathcal {P} = \mathcal {S}

and it is proved that

P

⊂

S

⊂

P

−

\mathcal {P} \subset \mathcal {S} \subset {\mathcal {P}^ - }

.
Bounds for solutions to ordinary differential equations applied to a singular Cauchy problem
Authors: Walker W. J. 
Published: June 2010

The Cauchy problem

u

t
t

−

t

2
+
ε

u

x
x

−

u
y

=
0
,
ε
>
0
,
u
(
x
,
y
,
0
)
=
α
(
x
,
y
)
,

u
t

(
x
,
y
,
0
)
=
γ
(
x
,
y
)

{u_{tt}} - {t^{2 + \varepsilon }}{u_{xx}} - {u_y} = 0,\varepsilon > 0,u(x,y,0) = \alpha (x,y),{u_t}(x,y,0) = \gamma (x,y)

, is shown to be unstable by demonstrating that there exists a sequence of solutions which increase indefinitely on a sequence of neighbourhoods of

t
=
0

t = 0

which shrink to zero, while at the same time the initial data is tending to zero. The equation

u

t
t

−

t

2
+
ε

u

x
x

−

u

y
y

−

u
y

=
0

{u_{tt}} - {t^{2 + \varepsilon }}{u_{xx}} - {u_{yy}} - {u_y} = 0

is investigated with the same initial data and in this case it is shown that the sequence of solutions remains bounded on a neighbourhood of

t
=
0

t = 0

which suggests but does not prove that the Cauchy problem for this equation is well posed. The latter result is a consequence of bounds obtained on a neighbourhood of

t
=
0

t = 0

for complex-valued solutions of the ordinary differential equation
\[

y
+
(
a
(
t
)
+
i
b
(
t
)
)
y
=
0.

y + (a(t) + ib(t))y = 0.

\]
The Hilbert transform of Schwartz distributions
Authors: Pandey J. N. 
Published: June 2010

Let

D

\mathcal {D}

be the Schwartz space of infinitely differentiable complex-valued functions defined on the real line with compact supports equipped with the usual topology. Assume

H
(

D

)

H(\mathcal {D})

to be the space of

C
∞

{C^\infty }

functions defined on the real line whose every element is the Hilbert transform of an element of

D

\mathcal {D}

. We equip the space

H
(

D

)

H(\mathcal {D})

with an appropriate topology and show that the classical Hilbert transformation

H
H

, defined by

H
f
=
P

∫

−
∞

∞

f
(
t
)

/

(
t
−
x
)
d
t

Hf = P\int _{ - \infty }^\infty {f(t)/(t - x)dt}

, is a homeomorphism from

D

\mathcal {D}

onto

H
(

D

)

H(\mathcal {D})

. The Hilbert transform

H
f

Hf

of

f
∈

D

′

f \in \mathcal {D}’

is then defined to be an element of

H
′

(

D

)

H’(\mathcal {D})

given by the relation
\[

⟨

H
f
,
φ

⟩

=

⟨

f
,
−
H
φ

⟩

∀
φ
∈
H
(

D

)
.

\left \langle {Hf,\varphi } \right \rangle = \left \langle {f, - H\varphi } \right \rangle \forall \varphi \in H(\mathcal {D}).

\]
It then follows that -

−

H
2

f

/

π
2

=
f
∀
f
∈

D

′

- {H^2}f/{\pi ^2} = f\forall f \in \mathcal {D}’

. Applications of our results in solving some singular integral equations are also discussed.
Sets of uniqueness for a certain class ℳ_{𝜖} on the dyadic group
Authors: Yoneda Kaoru 
Published: June 2010

For each sequence

ε
=

{

ε
n

}

\varepsilon = \left \{ {{\varepsilon _n}} \right \}

of real numbers which satisfies

lim

inf

n
→
∞

ε

2

n
+
1

/

ε

2
n

>
0

\lim {\inf _{n \to \infty }}{\varepsilon _{{2^{n + 1}}}}/{\varepsilon _{{2^n}}} > 0

and

ε
n

↓
0

{\varepsilon _n} \downarrow 0

as

n
→
∞

n \to \infty

, let

M

ε

{\mathfrak {M}_\varepsilon }

denote the set of all Walsh series

μ
∼

∑

k
=
0

∞

μ
^

(
k
)

w
k

(
x
)

\mu \sim \sum \nolimits _{k = 0}^\infty {\hat \mu (k){w_k}(x)}

such that

∑

k
=
0

∞

ε
k

|

μ
^

(
k
)

|

2

>
∞

\sum \nolimits _{k = 0}^\infty {{\varepsilon _k}{{\left | {\hat \mu (k)} \right |}^2} > \infty }

. We give a necessary and sufficient condition for a subset of the dyadic group to be a set of uniqueness for

M

ε

{\mathfrak {M}_\varepsilon }

.
Gel′fand-Kirillov dimension of rings of formal differential operators on affine varieties
Authors: Smith S. P. 
Published: June 2010

Let

A
A

be the coordinate ring of a smooth affine algebraic variety defined over a field

k
k

. Let

D
D

be the module of

k
k

-linear derivations on

A
A

and form

A
[
D
]

A[D]

, the ring of differential operators on

A
A

, as follows: consider

A
A

and

D
D

as subspaces of

End
k

A

{\operatorname {End}_k}A

(

A
A

acting by left multiplication on itself), and define

A
[
D
]

A[D]

to be the subalgebra generated by

A
A

and

D
D

. Let

rk
⁡
D

\operatorname {rk} D

denote the torsion-free rank of

D
D

(that is,

rk
⁡
D
=

dim
F

F

⊗
A

D

\operatorname {rk}D = {\dim _F}F{ \otimes _A}D

where

F
F

is the quotient field of

A
A

). The ring

A
[
D
]

A[D]

is a finitely generated

k
k

-algebra so its Gelfand-Kirillov dimension

GK

(
A
[
D
]
)

{\text {GK}}(A[D])

may be defined. The following is proved. Theorem.

GK

(
A
[
D
]
)
=

tr de

g

k

A
+
rk
⁡
D
=
2

tr de

g

k

A

{\text {GK}}(A[D]) = {\text {tr de}}{{\text {g}}_k}A + \operatorname {rk} D = 2{\text { tr de}}{{\text {g}}_k}A

. Actually we work in a more general setting than that just described, and although a more general result is obtained, this is the most natural and important application of the main theorem.
Totally Zippin 𝑝-groups
Authors: Megibben Charles 
Published: June 2010

If

G
G

is a

p
p

-group of limit length

λ
\lambda

, then it satisfies the

λ
\lambda

-Zippin property provided that whenever

A

/

p
λ

A
≅
G
≅
B

/

p
λ

B

A/{p^\lambda }A \cong G \cong B/{p^\lambda }B

, every isomorphism between

p
λ

A

{p^\lambda }A

and

p
λ

B

{p^\lambda }B

extends to an isomorphism between

A
A

and

B
B

. We show that if

G
G

is almost balanced in a totally projective group, then

G
G

does satisfy the

λ
\lambda

-Zippin property. This leads to the existence of a great variety of

G
G

’s that are totally Zippin in the sense that

G

/

p
α

G

G/{p^\alpha }G

satisfies the

α
\alpha

-Zippin property for all limit ordinals

α
⩽
λ
=

length of

G

\alpha \leqslant \lambda = {\text {length of }}G

. Hence totally Zippin

p
p

-groups need not be

S
S

-groups, although those of countable length turn out to be direct sums of countable groups.
Markov’s inequality for polynomials with real zeros
Authors: Borwein Peter 
Published: June 2010

Markov’s inequality asserts that

|

|

p
n
′

|

|

⩽

n
2

|

|

p
n

|

|

||{p’_n}|| \leqslant {n^2}||{p_n}||

for any polynomial

p
n

{p_n}

of degree

n
n

. (We denote the supremum norm on

[
−
1
,
1
]

[ - 1,1]

by

|

|

.

|

|

||.||

.) In the case that

p
n

{p_n}

has all real roots, none of which lie in

[
−
1
,
1
]

[ - 1,1]

, Erdös has shown that

|

|

p
n
′

|

|

⩽
e
n

|

|

p
n

|

|

/

2

||{p’_n}|| \leqslant en||{p_n}||/2

. We show that if

p
n

{p_n}

has

n
−
k

n - k

real roots, none of which lie in

[
−
1
,
1
]

[ - 1,1]

, then

|

|

p
n
′

⩽
c
n
(
k
+
1
)

|

|

p
n

|

|

||{p’_n} \leqslant cn(k + 1)||{p_n}||

, where

c
c

is independent of

n
n

and

k
k

. This extension of Markov’s and Erdös’ inequalities was conjectured by Szabados.
On the combinatorial properties of Blackwell spaces
Authors: Jasiński Jakub 
Published: June 2010

Under

MA +

¬

CH

{\text {MA + }}\neg {\text {CH}}

(Martin’s Axiom and negation of the Continuum Hypothesis) we prove that the intersection of a Blackwell space with the analytic set and the Cartesian product of a Blackwell space and a Borel set do not need to be Blackwell spaces.
The solution sets of extremal problems in 𝐻¹
Authors: Hayashi Eric 
Published: June 2010

Let

u
u

be an essentially bounded function on the unit circle

T
T

. Let

S
u

{S_u}

denote the subset of the unit sphere of

H
1

{H^1}

on which the functional

F
↦

∫
0

2
π

u
¯

(

e

i
t

)
F
(

e

i
t

)
d
t

/

2
π

F \mapsto \smallint _0^{2\pi }\bar u({e^{it}})F({e^{it}})dt/2\pi

attains its norm. A complete description of

S
u

{S_u}

is given in terms of an inner function

b
0

{b_0}

and an outer fun tion

g
0

{g_0}

in

H
2

{H^2}

for which

g
0
2

g_0^2

is an exposed point in the unit ball of

H
1

{H^1}

. An explicit description is given for the kernel of an arbitrary Toeplitz operator on

H
2

{H^2}

. The exposed points in

H
1

{H^1}

are characterized; an example is given of a strong outer function in

H
1

{H^1}

which is not exposed.
Preprojective partitions and the determinant of the Hom matrix
Authors: Igusa K. , Todorov G. 
Published: June 2010

If

Λ
\Lambda

is an artin algebra and

Λ
\Lambda

is the set of isomorphism classes of indecomposable finitely generated

Λ
\Lambda

-modules, then there is a partition

ind
⁡
Λ
=

∪

i
⩾
0

P
_

_

i

\operatorname {ind}\Lambda = { \cup _{i \geqslant 0}}{\underline {\underline P} _i}

, called the preprojective partition. We give an algorithm for computing this partition, which is given only in terms of numerical properties of the Auslander-Reiten quiver of

Λ
\Lambda

. If

Λ
\Lambda

is of finite representation type, then there are two essentially different proofs that the matrix

Hom
=
(

lengt

h

End(

N
)

/

rEnd(

N

)

Hom

Λ

⁡
(
M
,
N
)
)

\operatorname {Hom} = ({\text {lengt}}{{\text {h}}_{{\text {End(}}N)/{\text {rEnd(}}N{\text {)}}}}\operatorname {Hom}_{\Lambda }(M,N))

, where

M
,
n
∈
ind
⁡
Λ

M,n \in \operatorname {ind}\Lambda

has determinant

+
1
[

I
T
1

,

W
1

,

Z
1

]

+ 1[{\mathbf {IT1}},{\mathbf {W1}},{\mathbf {Z1}}]

. We show that the paths between the Hom matrix and the identity matrix in

GL

n

⁡
(

R

)

\operatorname {GL}_{n}(\mathbf {R})

given by these two proofs are homotopic.
A zeta-function associated with zero ternary forms
Authors: Eie Min King 
Published: June 2010

Consider the zeta-function associated with zero ternary forms defined as
\[

ξ
~

(
t
)
=

∑
x

1

|

det
x

|

t

(
Re
⁡
t
⩾
2
)
,

\tilde \xi (t) = \sum \limits _x {\frac {1} {{{{\left | {\det x} \right |}^t}}}} \quad (\operatorname {Re} t \geqslant 2),

\]
where

x
x

runs over all

SL

3

⁡
(

Z

)

\operatorname {SL}_{3}({\mathbf {Z}})

-inequivalent zero ternary forms. We shall approximate

ξ
~

(
t
)

\tilde \xi (t)

by another zeta-function which we can compute explicitly. By the approximation, we see that

ξ
~

(
2
)

\tilde \xi (2)

is very close to

2
ζ
(
2
)
ζ
(
2
)

2\zeta (2)\zeta (2)

which gives the contribution of zero ternary forms to the dimension formula of Siegel’s cusp forms of degree three (computing via Selberg Trace Formula) up to a constant multiple.
An extremal problem for polynomials with nonnegative coefficients
Authors: Milovanović Gradimir V. 
Published: June 2010

Let

W
n

{W_n}

be the set of all algebraic polynomials of exact degree

n
n

whose coefficients are all nonnegative. For the norm in

L
2

[
0
,
∞
)

{L^2}[0,\infty )

with generalized Laguerre weight function

w
(
x
)
=

x
α

e

−
x

(
α
>
−
1
)

w(x) = {x^\alpha }{e^{ - x}}\quad (\alpha > - 1)

, the extremal problem

C
n

(
α
)
=

sup

P
∈

W
n

(

‖

P
′

‖

/

‖
P
‖

)
2

{C_n}(\alpha ) = {\sup _{P \in {W_n}}}{(\left \| {P’} \right \|/\left \| P \right \|)^2}

is solved, which completes a result of A. K. Varma [7].
Extreme points in duals of complex operator spaces
Authors: Lima Åsvald , Olsen Gunnar 
Published: June 2010

We show that if

X
X

and

Y
Y

are complex Banach spaces, and

K
(
X
,
Y
)

K(X,Y)

is the space of compact linear operators from

X
X

into

Y
Y

, then

ext
⁡
B
(
K

(
X
,
Y

)
∗

)
=
ext
⁡
B
(

X

∗
∗

)
⊗
ext
⁡
B
(

Y
∗

)

\operatorname {ext}B(K{(X,Y)^ * }) = \operatorname {ext}B({X^{ * * }}) \otimes \operatorname {ext}B({Y^ * })

.
On the existence of uniformly consistent estimates
Authors: Yatracos Yannis G. 
Published: June 2010

Let

M

\mathcal {M}

be a family of probability measures on

(

X

,

A

)

(\mathfrak {X},\mathcal {A})

and

U
U

the uniform structure defined by vicinities of the form
\[

{
(
P
,
Q
)
:

sup

1
⩽
i
⩽
K

|

P
n

(

A

i
,
n

)
−

Q
n

(

A

i
,
n

)

|

>
ε
}

,

\left \{ (P, Q):\sup \limits _{1 \leqslant i \leqslant K} | P^n(A_{i,n}) - Q^n(A_{i,n}) | > \varepsilon \right \},

\]
where

P
n

{P^n}

is the product measure on

(

X

n

,

A

n

)
,

A

i
,
n

∈

A

n

,
ε
>
0
,
n
∧
K
⩾
1

({\mathfrak {X}^n},{\mathcal {A}^n}),{A_{i,n}} \in {\mathcal {A}^n},\varepsilon > 0,n \wedge K \geqslant 1

. Let

ϕ
∗

:
(

M

,
U
)
→
(

ϕ
∗

(

M

)
,
d
)

{\phi ^ * }:(\mathcal {M},U) \to ({\phi ^ * }(\mathcal {M}),d)

, where
\[

d

(

ϕ
∗

(
P
)
,

ϕ
∗

(
Q
)
)

=

|

|

P
−
Q

|

|

L
1

=
2

sup

A
∈

A

|

P
(
A
)
−
Q
(
A
)

|

.

d\left (\phi ^*(P), \phi ^*(Q) \right ) = ||P - Q||_{L_1} = 2\sup \limits _{A \in \mathcal {A}} |P(A) - Q(A)|.

\]
We consider the case where the space of measures

M
M

is

L
1

{L_1}

separable and relate the existence of uniformly consistent estimates for

ϕ
∗

(
P
)

{\phi ^ * }(P)

with uniform continuity of

ϕ
∗

{\phi ^ * }

and

L
1

{L_1}

-total boundedness of

M
M

.