bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the well-posedness of a 𝐶^{∞} Goursat problem for a partial differential operator of order greater than two
Authors: Carvalho e Silva Jaime 
Published: June 2010

We find a necessary and sufficient condition for a Goursat problem for a third order partial differential operator with constant coefficients of the form
\[

C
2

(

D
x

,

D
y

)

D
t

+

C
3

(

D
x

,

D
y

)

{C_2}({D_x},{D_y}){D_t} + {C_3}({D_x},{D_y})

\]
to be

C
∞

{C^\infty }

-well posed, showing at the same time that a necessary and sufficient condition of Hasegawa cannot be extended. The result can be generalised to operators of higher orders but leads to cumbersome conditions; nevertheless, we show that the condition of Hasegawa is also not sufficient in this case.
The atomic decomposition of Besov-Bergman-Lipschitz spaces
Authors: De Souza Geraldo Soares 
Published: June 2010

Let

b
b

denote a special atom,

b
:
[
−
π
,
π
)
→
R
,

b
(
t
)
=
1

/

2
π

b:[ - \pi ,\pi ) \to R,\;b(t) = 1/2\pi

or, for any interval

I

in

[
−
π
,
π
)

I{\text { in }}[ - \pi ,\pi )\;

b
(
t
)
=
−

|
I
|

−
1

/

p

X

R
(
t
)
+

|
I
|

−
1

/

p

X

L
(
t
)

b(t) = - {\left | I \right |^{ - 1/p}}\mathcal {X}R(t) + {\left | I \right |^{ - 1/p}}\mathcal {X}L(t)

L
L

is the left half of

I
I

,

R
R

is the right half,

|
I
|

\left | I \right |

denotes the length of

I
I

and

X

E

\mathcal {X}E

the characteristic function of

E
E

. For

1

/

2
>
p
>
∞

1/2 > p > \infty

, let

(

b
n

)

({b_n})

be special atoms and

(

c
n

)

({c_n})

a sequence of real numbers; then we define the space
\[

B
p

=

{

f
:
[
−
π
,
π
)
→
R
;
f
(
t
)
=

∑

n
=
1

∞

c
n

b
n

(
t
)
,

∑

n
=
1

∞

|

c
n

|

>
∞

}

{B^p} = \left \{ {f:[ - \pi ,\pi ) \to R;f(t) = \sum \limits _{n = 1}^\infty {{c_n}{b_n}(t),} \sum \limits _{n = 1}^\infty {\left | {{c_n}} \right | > \infty } } \right \}

\]
. We endow

B
p

{B^p}

with the norm

‖
f
‖

B
P

=

Inf

∑

n
=
1

∞

|

c
n

|

{\left \| f \right \|_{{B^P}}} = {\text {Inf}}\sum \nolimits _{n = 1}^\infty {\left | {{c_n}} \right |}

, where the infimum is taken over all possible representations of

f
f

. In the early 1960s, the following spaces were introduced, now known as Besov-Bergman-Lipschitz spaces. For

0
>
α
>
1

0 > \alpha > 1

,

1
≤
r

1 \leq r

,

s
≤
∞

s \leq \infty

, let
\[

Λ
(
α
,
r
,
s
)
=

{

f
:
[
−
π
,
π
)
→
R
,

‖
f
‖

Λ
(
α
,
r
,
s
)

=

‖
f
‖

r

+

(

∫

−
π

π

(

‖

f
(
x
+
t
)
−
f
(
x
)

‖

r

)

s

|
t
|

1
+
α
s

d
t

)

1

/

s

>
∞

}

\Lambda (\alpha ,r,s) = \left \{ {f:[ - \pi ,\pi ) \to R,{{\left \| f \right \|}_{\Lambda (\alpha ,r,s)}} = {{\left \| f \right \|}_r} + {{\left ( {\int _{ - \pi }^\pi {\frac {{{{({{\left \| {f(x + t) - f(x)} \right \|}_r})}^s}}}{{{{\left | t \right |}^{1 + \alpha s}}}}dt} } \right )}^{1/s}} > \infty } \right \}

\]
where

|

|

|

|

r

||\;|{|_r}

is the Lebesgue space

L
r

{L^r}

-norm. Now we write down the main theorem of this paper which is as follows. THEOREM

f
∈

B
P

f \in {B^P}

for

1
>
p
>
∞

1 > p > \infty

if and only if

f
∈
Λ
(
1
−
1

/

p
,
1
,
1
)

f \in \Lambda (1 - 1/p,1,1)

. Moreover, there are absolute constants

M
M

and

N
N

such that
\[

N

‖
f
‖

B
p

≤

‖
f
‖

Λ
(
1
−
1

/

p
,
1
,
1
)

≤
M

‖
f
‖

B
p

N{\left \| f \right \|_{{B^p}}} \leq {\left \| f \right \|_{\Lambda (1 - 1/p,1,1)}} \leq M{\left \| f \right \|_{{B^p}}}

\]
.
On almost everywhere convergence of Bochner-Riesz means in higher dimensions
Authors: Christ Michael 
Published: June 2010

In

R

n

{{\mathbf {R}}^n}

define

(

T

λ
,
r

f
)
(
ξ
)
=

f
^

(
ξ
)
(
1
−

|

r

−
1

ξ
2

|

)
+
λ

({T_{\lambda ,r}}f)(\xi ) = \hat f(\xi )(1 - \left | {{r^{ - 1}}{\xi ^2}} \right |)_ + ^\lambda

. If

n
≥
3

n \geq 3

,

λ
>

1
2

(
n
−
1
)

/

(
n
+
1
)

\lambda > \tfrac {1}{2}(n - 1)/(n + 1)

and

2
≤
p
>
2
n

/

(
n
−
1
−
2
λ
)

2 \leq p > 2n/(n - 1 - 2\lambda )

, then

lim

r
→
∞

T

λ
,
r

f
(
x
)
=
f
(
x
)

{\lim _{r \to \infty }}{T_{\lambda ,r}}f(x) = f(x)

a.e. for all

f
∈

L
p

(

R

n

)

f \in {L^p}({{\mathbf {R}}^n})

.
A note on the strong summability of the Riesz means of multiple Fourier series
Authors: Kuratsubo Shigehiko 
Published: June 2010

Let

S
t
α

(
x
,
f
)

S_t^\alpha (x,f)

be the Riesz means of order

α
\alpha

of an integrable function

f
(
x
)

f(x)

on

N
N

-dimensional torus

T
N

(
N
⩾
2
)

{T^N}(N \geqslant 2)

, that is,
\[

S
t
α

(
x
,
f
)
=

∑

|
m
|

2

>
t

(

1
−

|
m
|

2

t

)

α

f
^

(
m
)

e

2
π
i
m
x

.

S_t^\alpha (x,f) = \sum \limits _{{{\left | m \right |}^2} > t} {{{\left ( {1 - \frac {{{{\left | m \right |}^2}}}{t}} \right )}^\alpha }\hat f(m){e^{2\pi imx}}.}

\]
E. M. Stein has shown that if

1
>
p
⩽
2

1 > p \leqslant 2

and

α
>

α
p

\alpha > {\alpha _p}

where
\[

α
p

=

N
−
1

2

(

2
p

−
1

)

−

1

p
′

=

N
−
1

2

−

N

p
′

,

{\alpha _p} = \frac {{N - 1}}{2}\left ( {\frac {2}{p} - 1} \right ) - \frac {1}{{p’}} = \frac {{N - 1}}{2} - \frac {N}{{p’}},

\]
then for any function

f
(
x
)
∈

L
p

(

T
N

)

S
t
α

(
x
,
f
)

f(x) \in {L^p}({T^N})S_t^\alpha (x,f)

is strong summable to

f
(
x
)

f(x)

, that is,
\[

lim

T
→
∞

1
T

∫
0
T

|

S
t
α

(
x
,
f
)
−
f
(
x
)

|

2

d
t
=
0

\lim \limits _{T \to \infty } \frac {1}{T}\int _0^T {{{\left | {S_t^\alpha (x,f) - f(x)} \right |}^2}dt = 0}

\]
for almost every

x
x

. In this paper we shall show that if

1
⩽
p
⩽
2

1 \leqslant p \leqslant 2

and

−
1
>
α
>

α
p

- 1 > \alpha > {\alpha _p}

, then there exists a function

f
(
x
)
∈

L
p

(

T
N

)

f(x) \in {L^p}({T^N})

such that
\[

1
T

∫
0
T

|

S
t
α

(
x
,
f
)

|

2

d
t
−
Ω
(

T

α
p

−
α

log

−
2
τ

⁡
T
)

as

T
→
∞

\frac {1}{T} \int _0^T |S_t^\alpha (x,f)|^2 dt - \Omega (T^{\alpha _p - \alpha } \log ^{-2\tau } T) \quad \text {as $T \to \infty $}

\]
for every

x
x

and every

τ
>
1

/

p

\tau > 1/p

, in particular,
\[

lim

T
→
∞

¯

1
T

∫
0
T

|

S
t
α

(
x
,
f
)

|

2

d
t
=
∞

\overline {\lim }\limits _{T \to \infty } \frac {1}{T} \int _0^T | S_t^\alpha (x,f) |^2 dt = \infty

\]
for every

x
x

, where we can take for

f
(
x
)
,

f

σ
τ

(
x
)

f(x),{f_{\sigma \tau }}(x)

such that

f
^

σ
τ

(
m
)
=
1

/

|
m
|

σ

log
τ

|
m
|

,

|
m
|

>
1

{\hat f_{\sigma \tau }}(m) = 1/{\left | m \right |^\sigma }{\log ^\tau }\left | m \right |,\left | m \right | > 1

.
Composing functions of bounded 𝜑-variation
Authors: Ciemnoczołowski J. , Orlicz W. 
Published: June 2010

Let

F
n

{F_n}

be finite-valued functions on

(
−
∞
,
∞
)
,

F
n

(
0
)
=
0
,

n
=
1
,
2
,
…
.

( - \infty ,\infty ),{\text { }}{F_n}(0) = 0,{\text { }}n = 1,2, \ldots .

. For

x
∈

V

φ

⟨

a
,
b

⟩

x \in {\mathcal {V}_\varphi }\left \langle {a,b} \right \rangle

, the class of functions of bounded

φ
\varphi

-variation, the compositions

F
n

(
x
)

{F_n}(x)

are studied. The main result of this paper is Theorem 1 stating necessary and sufficient conditions for the sequence

var
ψ

(

F
n

(
x
)
,
a
,
b
)

{\operatorname {var} _\psi }({F_n}(x),a,b)

to be bounded for each

x
∈

V

φ

⟨

a
,
b

⟩

x \in {\mathcal {V}_\varphi }\left \langle {a,b} \right \rangle

(

ψ
\psi

denotes here another

φ
\varphi

-function).
Composition of linear fractional transformations in terms of tail sequences
Authors: Jacobsen Lisa 
Published: June 2010

We consider sequences

{

s
n

}

\left \{ {{s_n}} \right \}

of linear fractional transformations. Connected to such a sequence is another sequence

{

s
n

}

\left \{ {{s_n}} \right \}

of linear fractional transformations given by
\[

S
n

=

s
1

∘

s
2

∘
⋯
∘

s
n

,

n
=
1
,
2
,
3
,
…
.

{S_n} = {s_1} \circ {s_2} \circ \cdots \circ {s_n},\quad n = 1,2,3, \ldots .

\]
We introduce a new way of representing

s
n

{s_n}

(in terms of so-called tail sequences). This representation is established to give nice expressions for

S
n

{S_n}

. It can be seen as a generalization of the canonical form for

s
n

{s_n}

, which gives nice expressions for
\[

T
k

=

s
n

∘

s
n

∘
⋯
∘

s

n
.

⏟

(
k

terms)

{T_k} = \underbrace {{s_n} \circ {s_n} \circ \cdots \circ {s_{n.}}}_{(k{\text { terms)}}}

\]
The 𝑛th roots of solutions of linear ordinary differential equations
Authors: Harris William A. , Sibuya Yasutaka 
Published: June 2010

In this paper we shall prove the following theorem: Let

K
K

be a differential field of characteristic zero. Let

φ
\varphi

and

ψ
\psi

be elements of a differential field extension of

K
K

such that (i)

φ
≠
0

\varphi \ne 0

and

ψ
≠
0

\psi \ne 0

; (ii)

φ
\varphi

and

ψ
\psi

satisfy nontrivial linear differential equations with coefficients in

K
K

, say,

P
(
φ
)
=
0

P(\varphi ) = 0

and

Q
(
ψ
)
=
0

Q(\psi ) = 0

; (iii)

φ
=

ψ
n

\varphi = {\psi ^n}

for some positive integer

n
n

such that

n
⩾

ord

P

n \geqslant {\text { ord }}P

. Then the logarithmic derivatives of

φ
\varphi

and

ψ
\psi

are algebraic over

K
K

. (Note that

φ
′

/

φ
=
n
(

ψ
′

/

ψ
)

\varphi ’/\varphi = n(\psi ’/\psi )

.)
Ranges of joint Laplace-Fourier transforms
Authors: Grubb Daniel J. 
Published: June 2010

If

f
∈

L
1

(
[
0
,
∞
[
×

R

)

f \in {L^1}([0,\infty [ \times {\mathbf {R}})

and

f
^

(
z
,
s
)
=

∫

−
∞

∞

∫
0
∞

f
(
u
,
v
)

e

i
u
z

e

i
s
v

d
u
d
v

\hat f(z,s) = \smallint _{ - \infty }^\infty \smallint _0^\infty f(u,v){e^{iuz}}{e^{isv}}dudv

, where

z
∈
H
=
{
z
∈

C

:
Im
⩾
0
}
,
s
∈

R

z \in H = \{ z \in {\mathbf {C}}:\operatorname {Im} \geqslant 0\} ,s \in {\mathbf {R}}

, then

f
^

(
H
×

R

)
∪
{
0
}
=

f
^

(

R

×

R

)
∪
{
0
}

\hat f(H \times {\mathbf {R}}) \cup \{ 0\} = \hat f({\mathbf {R}} \times {\mathbf {R}}) \cup \{ 0\}

.
On the univalent functions starlike with respect to a boundary point
Authors: Todorov Pavel G. 
Published: June 2010

For the examined functions, we have obtained a structure formula and estimates for

|

f
(
z
)

/

(
1
−
z
)

|

|f(z)/(1 - z)|{\text { }}

and

|

arg
⁡
(
f
(
z
)

/

(
1
−
z
)
)

|

|\arg (f(z)/(1 - z))|

, the moduli of the partial sums of the coefficient series and the moduli of the coefficients.
Extreme points in 𝐶(𝐾,𝐿^{𝜑}(𝜇))
Authors: Grząślewicz Ryszard 
Published: June 2010

Let

L
ϕ

(
μ
)

{L^\phi }(\mu )

denote an Orlicz space and let

ϕ
\phi

satisfy the condition

Δ
2

{\Delta _2}

. It is shown that the extreme points of the unit ball of the space of continuous functions from a compact Hausdorff space

K
K

into

L
ϕ

(
μ
)

{L^\phi }(\mu )

with supremum norm on

C
(
K
,

L
ϕ

(
μ
)
)

C(K,{L^\phi }(\mu ))

are precisely the functions with values in the set of extreme points of the unit ball of

of

L
ϕ

(
μ
)

{\text {of }}{L^\phi }(\mu )

.
Ergodic theorems for Fourier transforms of noncommutative analogues of vector measures
Authors: Ylinen Kari 
Published: June 2010

Let

G
G

be a locally compact group and

E
E

a complex Banach space. Let

φ
:
G
→
E

\varphi :G \to E

be a function which is the Fourier transform of a weakly compact operator

Φ
:

C
∗

(
G
)
→
E

\Phi :{C^*}(G) \to E

in the sense that

Φ

∗
∗

(
ω
(
s
)
)
=
ϕ
(
s
)

{\Phi ^{**}}(\omega (s)) = \phi (s)

,

s
∈
G

s \in G

, where

ω
:
G
→

W
∗

(
G
)
⊂
L
(

H
ω

)

\omega :G \to {W^*}(G) \subset L({H_\omega })

corresponds to the universal representation of

C
∗

(
G
)

{C^ * }(G)

. It is proved that

lim
i

∫
ϕ
d

μ
i

=

Φ

∗
∗

(

p
ω

)

{\lim _i}\smallint \phi d{\mu _i} = {\Phi ^{**}}({p_\omega })

, where

p
ω

{p_\omega }

is the projection onto the space of the common fixed points of all

ω
(
s
)

\omega (s)

,

s
∈
G

s \in G

, and

(

μ
i

)

i
∈

I

{({\mu _i})_{i \in \mathcal {I}}}

is an arbitrary net in the measure algebra

M
(
G
)

M(G)

satisfying

sup

i
∈

I

‖

ω
(

μ
i

)

‖

>
∞

{\sup _{i \in \mathcal {I}}}\left \| {\omega ({\mu _i})} \right \| > \infty

,

lim
i

μ
i

(
G
)
=
1

{\lim _i}{\mu _i}(G) = 1

, and

lim
i

‖

ω
(

μ
i
∗

∗

δ
s

−

μ
i
∗

)
ξ

‖

=
0

{\lim _i}\left \| {\omega (\mu _i^* * {\delta _s} - \mu _i^*)\xi } \right \| = 0

for all

s
∈
G

s \in G

,

ξ
∈

H
ω

\xi \in {H_\omega }

. If

E
E

is a Hilbert space and

ϕ
\phi

left (resp. right) homogeneous, the second (resp. first) of the last two limit conditions may be omitted. Finally, a connection of such random fields

ϕ
\phi

to a measurability condition is established.
Symmetrization and optimal control for elliptic equations
Authors: Voas Charles , Yaniro Daniel 
Published: June 2010

We consider an optimal control problem where

u
(
x
)

u(x)

satisfies

−
div
⁡
(
H
(
x
)
∇
u
)
=
1

- \operatorname {div}(H(x)\nabla u) = 1

in

Ω
\Omega

and

H
(
x
)

H(x)

is a control. We introduce the functional

J
Ω

(
H
)
=

|

Ω

|

−
1

∫
Ω

u
(
x
)

d
x

{J_\Omega }(H) = {|\Omega |^{ - 1}}\int \limits _\Omega {u(x)} dx

and show using a symmetrization argument that if the distribution function of

H
H

is fixed, then

J
Ω

(
H
)

{J_\Omega }(H)

is largest when

Ω
\Omega

is a ball and

H
H

is radial and decreasing on radii.
An alternating procedure for operators on 𝐿_{𝑝} spaces
Authors: Akcoglu M. A. , Sucheston L. 
Published: June 2010

Let

L
p

{L_p}

be the usual Banach spaces over a

σ
\sigma

-finite measure space. If

1

>

p

>

∞

1{\text { > }}p{\text { > }}\infty

and

q
=
p

(
p
−
1

)

−
1

q = p{(p - 1)^{ - 1}}

, then

ψ
p

:

L
p

→

L
q

{\psi _p}:{L_p} \to {L_q}

denotes the duality mapping defined by the requirements that

(
f
,

ψ
p

f
)
=

‖
f
‖

p
p

=

‖
f
‖

p

‖

ψ
p

f

‖

q
,
f
∈

L
p

(f,{\psi _p}f) = \left \| f \right \|_p^p = {\left \| f \right \|_p}\left \| {{\psi _p}f} \right \|q,f \in {L_p}

. If

T
:

L
p

→

L
p

T:{L_p} \to {L_p}

is a bounded linear operator, then

M
(
T
)
:

L
p

→

L
p

M(T):{L_p} \to {L_p}

is the mapping defined by

M
(
T
)
=

ψ
q

T
∗

ψ
p

T

M(T) = {\psi _q}{T^ * }{\psi _p}T

, where

T
∗

:

L
q

→

L
q

{T^ * }:{L_q} \to {L_q}

is the adjoint of

T
T

. It is proved that if

T
n

{T_n}

is a sequence of operators on

L
p

{L_p}

such that

‖

T
n

‖

≤
1

\left \| {{T_n}} \right \| \leq 1

for all

n
n

, then

M
(

T
n

⋯

T
2

T
1

)
f

M({T_n} \cdots {T_2}{T_1})f

converges in

L
p

{L_p}

for all

f
∈

L
p

f \in {L_p}

.
For every continuous 𝑓 there is an absolutely continuous 𝑔 such that [𝑓=𝑔] is not bilaterally strongly porous
Authors: Buczolich Z. 
Published: June 2010

For any Darboux function

f
:

[

0
,
1

]

→

R

f:\left [ {0,1} \right ] \to {\mathbf {R}}

and any

0
>
δ
>
1

0 > \delta > 1

there is a point

x
∈

[

0
,
1
−
δ

]

x \in \left [ {0,1 - \delta } \right ]

and a sequence

x
n

{x_n}

such that (a)

x
n

∈

[

x
+

δ

n
+
1

,
x
+

δ
n

]

(

n
=
1
,
2
,
…

)

{x_n} \in \left [ {x + {\delta ^{n + 1}},x + {\delta ^n}} \right ]\left ( {n = 1,2, \ldots } \right )

and (b)

∑

n
=
2

∞

|

f

(

x
n

)

−
f

(

x

n
−
1

)

|

>
+
∞

\sum \nolimits _{n = 2}^\infty {\left | {f\left ( {{x_n}} \right ) - f\left ( {{x_{n - 1}}} \right )} \right |} > + \infty

. Consequently, for every

f
∈
C

[

0
,
1

]

f \in C\left [ {0,1} \right ]

there is an absolutely continuous function

g
g

such that

{

x
:
f

(
x
)

=
g

(
x
)

}

\left \{ {x:f\left ( x \right ) = g\left ( x \right )} \right \}

is not bilaterally strongly porous.
Pointwise convergence approximate identities of dilated radially decreasing kernels
Authors: Kerman R. A. 
Published: June 2010

Let

ϕ
\phi

be integrable on

R
n

{R^n}

with

∫

R
n

ϕ
(
y
)
d
y
=
1

\int _{{R^n}} {\phi (y)dy = 1}

. It is shown that
\[

lim

ε
→
0
+

(

ϕ
F
∗

f
)
(
x
)
=

lim

ε
→
0
+

ε

−
n

∫

R
n

(

x
−
y

ε

)
f
(
y
)
d
y
=
f
(
x
)

{\lim _{\varepsilon \to 0 + }}(\phi _F^*f)(x) = {\lim _{\varepsilon \to 0 + }}{\varepsilon ^{ - n}}\int _{{R^n}} {(\frac {{x - y}} {\varepsilon })f(y)dy = f(x)}

\]
a.e. on

R
n

{R^n}

, whenever the least radially decreasing majorant of

ϕ
\phi

, defined by

ψ
(
x
)
=

sup

|

y

|

⩾

|

x

|

|

ϕ
(
y
)

|

\psi (x) = {\sup _{|y| \geqslant |x|}}|\phi (y)|

, is such that

|

x

|

n

ψ
(
x
)
=

|

x

|

n

ψ
(

|

x

|

)

|x{|^n}\psi (x) = |x{|^n}\psi (|x|)

is nonincreasing in

|

x

|

|x|

when

|

x

|

|x|

is large and

(

ψ

ε
0

∗

|

f

|

)
(

x
0

)
>
∞

(\psi _{{\varepsilon _0}}^*|f|)({x_0}) > \infty

for some

x
0

∈

R
n

{x_0} \in {R^n}

and

ε
0

>
0

{\varepsilon _0} > 0

.
𝐾ⁿ-positive maps in 𝐶*-algebras
Authors: Itoh Takashi 
Published: June 2010

Let

K
n

{K^n}

be the set of

n
n

-positive maps of

B
(
H
)

B(H)

to

B
(
H
)

B(H)

. A

K
n

{K^n}

-positive map of a

C
∗

{C^ * }

-algebra

A
A

to

B
(
H
)

B(H)

is a positive linear map

ϕ
\phi

such that

∑

Tr(

ϕ

(

a
i

)

b
i
t

)
≥
0

\sum {{\text {Tr(}}\phi {\text {(}}} {a_i})b_i^t) \geq 0

for any

∑

a
i

⊗

b
i

∈
{
x
∈
A

⊗
γ

T
(
H
)

|

K
n

∋
∀

α
,
(

id

⊗
α

)(

x

)

≥

0

}

\sum {{a_i} \otimes {b_i} \in \{ x \in A{ \otimes _\gamma }T(H)|{K^n}} { \ni ^\forall }\alpha ,({\text {id}} \otimes \alpha {\text {)(}}x{\text {)}} \geq {\text {0}}\}

. It is shown that the following three statements are equivalent. (1) Every

K
n

{K^n}

-positive map of

A
A

to

B
(
H
)

B(H)

is

K

n
+
1

{K^{n + 1}}

-positive. (2) Every

K
n

{K^n}

-positive map of

A
A

to

B
(
H
)

B(H)

is completely positive. (3)

A
A

is an

n
n

-subhomogeneous

C
∗

{C^ * }

-algebra.
Unique solvability of an extended Stieltjes moment problem
Authors: Njåstad Olav 
Published: June 2010

Let

a
1

,

a
2

,
…
,

a
p

{a_1},{a_2}, \ldots ,{a_p}

be given real numbers ordered by size, and let

[
α
,
β
]

[\alpha ,\beta ]

be a real interval disjoint from the set

{

a
1

,

a
2

,
…
,

a
p

}

\{ {a_1},{a_2}, \ldots ,{a_p}\}

. Let

{

c
j

(
i
)

:
j
=
1
,
2
,
…
}

\{ c_j^{(i)}:j = 1,2, \ldots \}

, be sequences of real numbers and

c
0

{c_0}

be a real number. The extended Stieltjes moment problem is to find a distribution function

ψ
\psi

with all its points of increase in

[
α
,
β
]

[\alpha ,\beta ]

such that
\[

∫
α
β

d
ψ
(
t
)
=

c
0

,

∫
α
β

d
ψ
(
t
)

(
t
−

a
i

)

j

=

c
j

(
i
)

,

i
=
1
,
…
,
p
,

j
=
1
,
2
,
…
.

\int _\alpha ^\beta {d\psi (t) = {c_0},\quad \int _\alpha ^\beta {\frac {{d\psi (t)}}{{{{(t - {a_i})}^j}}} = c_j^{(i)},\quad i = 1, \ldots ,p,\;j = 1,2, \ldots .} }

\]
Necessary and sufficient conditions for the existence of a unique solution of the problem are given. Orthogonal

R
R

-functions and Gaussian quadrature formulas play important roles in the proof.
A remark on the existence of positive periodic solutions of superlinear parabolic problems
Authors: Esteban Maria J. 
Published: June 2010

We prove the existence of a solution for the following problem:
\[

∂
t

u
−
Δ
u
=
f
(
t
,
x
,
u
)

in (

0
,
T
)
×
Ω
,

u

>

0

in (

0
,
T
)
×
Ω
,
u
(
T
)
=
u
(
0
)

in

Ω
,

u
=
0

on (

0
,
T
)
×
∂
Ω
,

{\partial _t}u - \Delta u = f(t,x,u){\text { in (}}0,T) \times \Omega ,\;\quad u{\text { > }}0{\text { in (}}0,T) \times \Omega ,u(T) = u(0){\text { in }}\Omega ,\;\quad u = 0{\text { on (}}0,T) \times \partial \Omega ,

\]
where

Ω
\Omega

is a bounded domain of

R
N

{R^N}

and the function

f
(
t
,
x
,
⋅
)

f(t,x, \cdot )

grows more slowly than

u
α

{u^\alpha }

at

+
∞

+ \infty

, with

α

>

N

/

(
N
−
2
)

\alpha {\text { > }}N/(N - 2)

. On démontre ici l’existence d’une solution positive pour le problème parabolique périodique suivant
\[

∂
t

u
−
Δ
u
=
f
(
t
,
x
,
u
)

in (0,

T

)

×
Ω

,

u

>

0

in (0,

T
)
×
Ω
,
u
(
T
)
=
u
(
0
)

in

Ω
,

u
=
0

on (

0
,
T
)
×
∂
Ω
,

\begin {gathered} {\partial _t}u - \Delta u = f(t,x,u){\text { in (0,}}T{\text {)}} \times \Omega {\text {,}}\quad \;u{\text { > }}0{\text { in (0,}}T) \times \Omega , u(T) = u(0){\text { in }}\Omega ,\quad \;u = 0{\text { on (}}0,T) \times \partial \Omega ,\end {gathered}

\]
où

Ω
\Omega

est un domaine borné de

R
N

{R^N}

et la fonction

f
(
t
,
x
,
⋅
)

f(t,x, \cdot )

croit plus lentement que

u
α

{u^\alpha }

à l’infini, avec

α

>

N

/

(
N
−
2
)

\alpha {\text { > }}N/(N - 2)

.
The structure of bounded bilinear forms on products of 𝐶*-algebras
Authors: Ylinen Kari 
Published: June 2010

Let

A
1

{A_1}

and

A
2

{A_2}

be

C
∗

{C^ * }

-algebras and

B
:

A
1

×

A
2

→

C

B:{A_1} \times {A_2} \to {\mathbf {C}}

be a bounded bilinear form. It is proved that there exist a Hilbert space

H
H

, two Jordan morphisms

μ
i

:

A
i

→
L
(
H
)
,
i
=
1
,
2

{\mu _i}:{A_i} \to L(H),i = 1,2

, and two vectors

ξ
1

,

ξ
2

∈
H

{\xi _1},{\xi _2} \in H

such that
\[

B
(
x
,
y
)
=
(

μ
1

(
x
)

ξ
1

|

μ
2

(

y
∗

)

ξ
2

)

for all

x
∈

A
1

,
y
∈

A
2

.

B(x,y) = ({\mu _1}(x){\xi _1}\left | {{\mu _2}({y^ * }} \right .){\xi _2}){\text { for all }}x \in {A_1},y \in {A_2}.

\]
The proof depends on the Grothendieck-Pisier-Haagerup inequality and Halmos’s unitary dilation theorem. An extemely elementary proof of the latter is given.
Alternating procedures in uniformly smooth Banach spaces
Authors: Assani I. 
Published: June 2010

Let

E
E

be a uniformly smooth Banach space and

C
C

the set of real continuous strictly increasing functions

μ
\mu

on

R

+

{{\mathbf {R}}_ + }

such that

μ
(
0
)
=
0

\mu (0) = 0

. At each

μ
\mu

we can associate a unique duality map

J
μ

:
E
→

E
∗

{J_\mu }:E \to {E^ * }

such that

(

J
μ

x
,
x
)
=

‖

J
μ

x

‖

⋅

‖
x
‖

({J_\mu }x,x) = \left \| {{J_\mu }x} \right \| \cdot \left \| x \right \|

and

‖

J
μ

x

‖

=
μ

(

‖
x
‖

)

\left \| {{J_\mu }x} \right \| = \mu \left ( {\left \| x \right \|} \right )

. We prove in this note that if

T
n

{T_n}

is a sequence of linear contractions on

E
E

the sequence

T
1
∗

T
2
∗

⋯

T
n
∗

J
μ

T
n

⋯

T
2

T
1

x

T_1^ * T_2^ * \cdots T_n^ * {J_\mu }{T_n} \cdots {T_2}{T_1}x

converges strongly in

E
∗

{E^ * }

norm for all

x
x

in

E
E

. In particular if

E
∗

{E^ * }

is also uniformly smooth then for any

μ
\mu

and

ν
\nu

in

C
C

the sequence

J
ν
∗

T
1
∗

T
2
∗

⋯

T
n
∗

J
μ

T
n

⋯

T
1

x

J_\nu ^ * T_1^ * T_2^ * \cdots T_n^ * {J_\mu }{T_n} \cdots {T_1}x

converges in

E
E

norm. This generalizes a result of M. Akcoglu and L. Sucheston [1].
Construction of Hamilton sequences for certain Teichmüller mappings
Authors: Reich Edgar 
Published: June 2010

Suppose

sup
{

|

∬

|
z
|

>
1

κ
(
z
)
φ
(
z
)
d
x
d
y

|

:
φ
(
z
)

\sup \{ |\iint _{\left | z \right | > 1} {\kappa (z)\varphi (z)dxdy|:\varphi (z)}

analytic for

|
z
|

>
1
,

∬

|
z
|

>
1

|

φ
(
z
)

|

d
x
d
y
=
1
}
=

ess

sup

{

|

κ
(
z
)

|

:

|
z
|

>
1
}

\left | z \right | > 1,\iint _{\left | z \right | > 1} {|\varphi (z)|dxdy = 1\} = {\text {ess}}\sup }\{ |\kappa (z)|:\left | z \right | > 1\}

. The question of constructive determination of extremal sequences

{

φ
n

}

\{ \varphi _n \}

is considered for some classes of functions

κ
(
z
)

\kappa (z)

that arise in connection with plane quasiconformal mappings. For example, such a sequence

{

φ
n

}

\{ \varphi _n \}

is constructed explicitly for the

κ
(
z
)

\kappa (z)

that arises in connection with the affine stretch of Strebel’s chimney domain.
The number of generators of modules over polynomial rings
Authors: Lyubeznik Gennady 
Published: June 2010

Let

k
k

be an infinite field and

B
=
k
[

X
1

,
…
,

X
n

]

B = k[{X_1}, \ldots ,{X_n}]

a polynomial ring over

k
k

. Let

M
M

be a finitely generated module over

B
B

. For every prime ideal

P
⊂
B

P \subset B

let

μ
(

M
P

)

\mu ({M_P})

be the minimum number of generators of

M
P

{M_P}

, i.e.,

μ
(

M
P

)
=
dim
⁡

B
P

/

P
P

(

M
P

⊗

B
P

(

B
P

/

P
P

)
)

\mu ({M_P}) = \dim {B_P}/{P_P}({M_P}{ \otimes _{{B_P}}}({B_P}/{P_P}))

. Set

η
(
M
)
=
max
{
μ
(

M
P

)
+
dim
⁡
(
B

/

P
)

|

P
∈
Spec

B

such

that

M

P

is

not

free

}

\eta (M) = \max \{ \mu ({M_P}) + \dim (B/P)\left | {P \in \operatorname {Spec} } \right .B\;{\text {such}}\;{\text {that}}\;{M_{P\;}}{\text {is}}\;{\text {not}}\;{\text {free}}\}

. Then

M
M

can be generated by

η
(
M
)

\eta (M)

elements. This improves earlier results of A. Sathaye and N. Mohan Kumar on a conjecture of Eisenbud-Evans.