bluuebunny/crossref_metadata_2025_split huggingface dataset:
An analogue of some inequalities of P. Turán concerning algebraic polynomials having all zeros inside [-1,+1]. II
Authors: Varma A. K. 
Published: June 2010

Let

P
n

(
x
)

{P_n}(x)

be an algebraic polynomial of degree

⩽
n

\leqslant n

having all zeros inside

[
−
1
,
+
1
]

[ - 1, + 1]

; then we have
\[

∫

−
1

1

P
′

n
2

(
x
)
d
x
>

(

n
2

+

3
4

+

3

4
n

)

∫

−
1

1

P
n
2

(
x
)
d
x
.

\int _{ - 1}^1 {P’}_n^2 (x)dx > \left ( {\frac {n}{2} + \frac {3}{4} + \frac {3}{{4n}}} \right )\int _{ - 1}^1 {P_n^2(x)dx.}

\]
This bound is much sharper than found in [2]. Moreover, if

P
n

(
1
)
=

P
n

(
−
1
)
=
0

{P_n}(1) = {P_n}( - 1) = 0

, then under the above conditions we have
\[

∫

−
1

1

P
′

n
2

(
x
)
d
x
⩾

(

n
2

+

3
4

+

3

4
(
n
−
1
)

)

∫

−
1

1

P
n
2

(
x
)
d
x
,

\int _{ - 1}^1 {P’}_n^2 (x)dx \geqslant \left ( {\frac {n}{2} + \frac {3}{4} + \frac {3}{{4(n - 1)}}} \right )\int _{ - 1}^1 {P_n^2(x)dx,}

\]
equality for

P
n

(
x
)
=

(
1
−

x
2

)
m

,
n
=
2
m

{P_n}(x) = {(1 - {x^2})^m},n = 2m

.
Existence and uniqueness of solutions to abstract Volterra integral equations
Authors: Kiffe T. , Stecher M. 
Published: June 2010

The existence and uniqueness of solutions to the equation

u
(
t
)
+

∫
0
t

a
(
t
−
s
)
A
u
(
s
)

d
s
∍
f
(
t
)

u(t) + \smallint _0^ta(t - s)Au(s)\;ds \backepsilon f(t)

, where A is a maximal monotone operator, is proved under various restrictions on A and f.
An extreme point in 𝐻^{∞}(𝑈²)
Authors: Riesenberg Nathaniel R. 
Published: June 2010

In this paper an example of a function

f
∈

H
∞

(

U
2

)

f \in {H^\infty }({U^2})

with

‖
f
‖

∞

=
1

{\left \| f \right \|_\infty } = 1

and
\[

∫

T
2

log
⁡
(
1
−

|

f
∗

(
z
)

|

)
d
m
>
−
∞
,

z
∈

T
2

,

\int _{{T^2}} {\log (1 - |{f^ \ast }(z)|)dm > - \infty ,\quad z \in {T^2}} ,

\]
yet f is an extreme point in the unit ball of

H
∞

{H^\infty }

, is given. For functions

f
∈

H
∞

(

U
1

)

f \in {H^\infty }({U^1})

that
\[

∫
T

log
⁡
(
1
−

|

f
∗

(
z
)

|

)
d
m
=
−
∞
,

z
∈
T
,

\int _T {\log (1 - |{f^ \ast }(z)|)dm = - \infty ,\quad z \in T,}

\]
is both necessary and sufficient for f to be an extreme point in

H
∞

{H^\infty }

.
Vector-valued inequalities for Fourier series
Authors: De Francia José L. Rubio 
Published: June 2010

Denoting by

S
∗

{S^\ast }

the maximal partial sum operator of Fourier series, we prove that

S
∗

(

f
1

,

f
2

,
…
,

f
k

,
…
)
=
(

S
∗

f
1

,

S
∗

f
2

,
…
,

S
∗

f
k

,
…
)

{S^\ast }({f_1},{f_2}, \ldots ,{f_k}, \ldots ) = ({S^\ast }{f_1},{S^\ast }{f_2}, \ldots ,{S^\ast }{f_k}, \ldots )

is a bounded operator from

L
p

(

l
r

)

{L^p}({l^r})

to itself,

1
>
p
,
r
>
∞

1 > p,r > \infty

. Thus, we extend the theorem of Carleson and Hunt on pointwise convergence of Fourier series to the case of vector valued functions. We give also an application to the rectangular convergence of double Fourier series.
A characterization of the contact Lie algebras
Authors: Gregory Thomas B. 
Published: June 2010

We classify the simple finite-dimensional irreducible graded Lie algebras over an algebraically closed field of characteristic

p
>
5

p > 5

which have the form

L

−
2

⊕

L

−
1

⊕

L
0

⊕

L
1

⊕
⋯
⊕

L
k

,
k
⩾
3

{L_{ - 2}} \oplus {L_{ - 1}} \oplus {L_0} \oplus {L_1} \oplus \cdots \oplus {L_k},k \geqslant 3

, where

L
0

{L_0}

is classical and reductive. We show that any such Lie algebra must be a Lie algebra of the contact series of Lie algebras of Cartan type by showing how the constraints imposed by the hypotheses force the existence of a highest-weight vector in

L

−
1

{L_{ - 1}}

for the representation of

L
0

{L_0}

in

L

−
1

{L_{ - 1}}

induced by the adjoint representation of

L
L

in itself. The existence of this highest-weight vector enables us to conclude that the above-mentioned representation is restricted.

L
L

can then be determined by appeal to an earlier classification theorem.
On commuting families of nonexpansive operators
Authors: Schechtman Gideon 
Published: June 2010

For each

n
n

,

1
⩽
n
⩽
∞

1 \leqslant n \leqslant \infty

, there exist a weakly compact, convex subset

W
W

of

L
1

{L_1}

and a family

{

T
i

}

i
=
1

n

(
{

T
i

}

i
=
1

∞

in case

n
=
∞
)

\{ {T_i}\} _{i = 1}^n(\{ {T_i}\} _{i = 1}^\infty {\text { in case }}n = \infty )

of nonexpansive operators of

W
W

into

W
W

such that there is no common fixed point for the whole family but any (finite) strict subfamily admits a common fixed point.
Splitting universal bundles over flag manifolds
Authors: Stong R. E. 
Published: June 2010

Let

F

{\mathbf {F}}

be one of the fields

R

{\mathbf {R}}

,

C

{\mathbf {C}}

, or

H

{\mathbf {H}}

and correspondingly let

F

G

{\mathbf {F}}G

be

O
O

,

U
U

, or

Sp

{\text {Sp}}

, i.e. the orthogonal, unitary, or symplectic group. Over the flag manifold

F

G
(

n
1

+
⋯
+

n
k

)

/

F

G
(

n
1

)
×
⋯
×

F

G
(

n
k

)

{\mathbf {F}}G({n_1} + \cdots + {n_k})/{\mathbf {F}}G({n_1}) \times \cdots \times {\mathbf {F}}G({n_k})

one has vector bundles

γ
i

{\gamma _i}

over

F
F

of dimension

n
i

{n_i}

,

1
⩽
i
⩽
k

1 \leqslant i \leqslant k

. This paper determines all cases in which

γ
i

{\gamma _i}

decomposes nontrivially as a Whitney sum.
An infinitely divisible distribution involving modified Bessel functions
Authors: Ismail Mourad E. H. , Miller Kenneth S. 
Published: June 2010

We prove that the function
\[

(

b
a

)

μ
−
v

K
μ

(
b

x

1

/

2

)

K
v

(
a

x

1

/

2

)

K
μ

(
a

x

1

/

2

)

K
v

(
b

x

1

/

2

)

{\left ( {\frac {b} {a}} \right )^{\mu - v}}\frac {{{K_\mu }(b{x^{1/2}}){K_v}(a{x^{1/2}})}} {{{K_\mu }(a{x^{1/2}}){K_v}(b{x^{1/2}})}}

\]
is the Laplace transform of an infinitely divisible probability distribution when

v
>
μ
⩾
0

v > \mu \geqslant 0

and

b
>
a
>
0

b > a > 0

. This implies the complete monotonic ity of the function. We also establish a representation as a Stieltjes transform, which implies in particular that the function has positive real part when

x
x

lies in the right half-plane. We conjecture that
\[

(

b
a

)

μ
−
v

I
μ

(
a

x

1

/

2

)

I
v

(
b

x

1

/

2

)

I
μ

(
b

x

1

/

2

)

I
v

(
a

x

1

/

2

)

{\left ( {\frac {b} {a}} \right )^{\mu - v}}\frac {{{I_\mu }(a{x^{1/2}}){I_v}(b{x^{1/2}})}} {{{I_\mu }(b{x^{1/2}}){I_v}(a{x^{1/2}})}}

\]
also is the Laplace transform of an infinitely divisible probability distribution. It is known that in the limit as

v
→
∞

v \to \infty

, the infinite divisibility property holds for both functions.
Some applications of generalized exponentials to partial differential equations
Authors: Abramowich J. 
Published: June 2010

Using what may be considered as a natural generalization of the exponential function, some of the formalism of the theory of ordinary linear differential equations is extended to a class of linear partial differential equations among which are some important equations of mathematical physics. In

\S

1

{\text {\S }}1

we give the definitions of the generalized exponentials and derive expressions for them.

§
2

\S 2

is devoted to the study of some of the properties of the exponential in two independent variables. In

§
3

\S 3

we derive the general solutions of some key partial differential equations using the method of recursion. The last section is devoted to extending the formalism of the method of variation of parameters to a class of linear partial differential equations.
Monomial equimultiple curves in positive characteristic
Authors: Narasimhan R. 
Published: June 2010

It is known that the local equimultiple locus of a hypersurface in characteristic zero is contained in a regular hypersurface. Here we give an example of a monomial curve on a threefold in positive characteristic

p

>

2

p{\text { > }}2

which is equimultiple but not hyperplanar. As a corollary we have that any monomial curve which lies on a certain type of hypersurface (whose local equation is of a special form in its natural

p
p

-basis expression) is automatically equimultiple for the hypersurface.
Oscillation theorems for 𝑛th order nonlinear differential equations with deviating arguments
Authors: Grace S. R. , Lalli B. S. 
Published: June 2010

In this note we study the oscillatory behavior of solutions of the

n
n

th order nonlinear functional differential equation
\[

x

(
n
)

(
t
)
+
q
(
t
)
f
(
x

[

g
(
t
)

]

)
=
0
,

n

even,

{x^{(n)}}(t) + q(t)f(x\left [ {g(t)} \right ]) = 0,\quad n\;{\text {even,}}

\]
without assuming that the deviating argument is retarded or advanced. Sufficient conditions are established for all solutions of the equation to be oscillatory.
Regularity of Banach algebras generated by analytic semigroups satisfying some growth conditions
Authors: Esterle J. , Galé J. E. 
Published: June 2010

We show that if a commutative complex Banach algebra

A
A

is generated by a nonzero analytic semigroup

(

a
t

)
Re
⁡
t
>
0

({a^t})\operatorname {Re} t > 0

satisfying
\[

∫

+
∞

−
∞

log

+

‖

a

1
+
i
t

‖

1
+

t
2

d
t
>
+
∞
,

\int {\begin {array}{*{20}{c}} { + \infty } \\ { - \infty } \\ \end {array} } \frac {{{{\log }^ + }\left \| {{a^{1 + it}}} \right \|}}{{1 + {t^2}}}dt > + \infty ,

\]
, then

A
A

is regular in Shilov’s sense.
A note on the “hyperbolic” Bochner-Riesz means
Authors: Carbery Anthony 
Published: June 2010

We consider the

L
p

(

R

2

)

{L^p}({{\mathbf {R}}^2})

boundedness properties of the Fourier multiplier

m
(

ξ
1

,

ξ
2

)
=
(
1
−

ξ
1
2

ξ
2
2

)
+
α

for

α
>
0

m({\xi _1},{\xi _2}) = (1 - \xi _1^2\xi _2^2)_ + ^\alpha {\text { for }}\alpha > 0

. We prove that if

α
⩾

1
2

\alpha \geqslant \frac {1}{2}

, then

m
m

is bounded on

L
p

{L^p}

,

1
>
p
>
∞

1 > p > \infty

, and that if

α
>
0

\alpha > 0

, then

m
m

is bounded on

L
p

{L^p}

,

4
3

⩽
p
⩽
4

\frac {4}{3} \leqslant p \leqslant 4

.
Trotter’s product formula for semigroups generated by quasilinear elliptic operators
Authors: Watanabe Michiaki 
Published: June 2010

Trotter’s product formula is given for nonlinear semigroups in

L
1

(

R
N

)

{L^1}({R^N})

generated by quasilinear operators of the form

Δ
ϕ

\Delta \phi

, where

ϕ
\phi

is a suitable function: formally

exp
⁡
(
t
Δ
ϕ
)
u
=

lim

h
↓
0

{

exp
⁡
(
h
Δ

ϕ
1

)
⋯
exp
⁡
(
h
Δ

ϕ
k

)

}

[
t

/

h
]

u

\exp (t\Delta \phi )u = {\lim _{h \downarrow 0}}{\left \{ {\exp (h\Delta {\phi _1}) \cdots \exp (h\Delta {\phi _k})} \right \}^{[t/h]}}u

, where

ϕ
=

ϕ
1

+
⋯
+

ϕ
k

\phi = {\phi _1} + \cdots + {\phi _k}

. The proof is carried out by a new method for construction of a semigroup with generator

Δ
ϕ

\Delta \phi

in

L
1

(

R
N

)

{L^1}({R^N})

.
A sharp inequality for the 𝑝-center of gravity of a random variable
Authors: Cox David C. 
Published: June 2010

Let

X
X

be a real-valued random variable. For

p
>
1

p > 1

, define the

p
p

-center of gravity of

X
X

,

C
p

(
X
)

{C_p}(X)

, as the unique number

c
c

which minimizes

‖

X
−
c

‖

p

{\left \| {X - c} \right \|_p}

. This paper exhibits a more-or-less explicit expression for the best constant

γ
=

γ
p

\gamma = {\gamma _p}

in the inequality

|

E
(
X
)
−

C
p

(
X
)

|

⩽
γ

‖

X
−

C
p

(
X
)

‖

p

\left | {E(X) - {C_p}(X)} \right | \leqslant \gamma {\left \| {X - {C_p}(X)} \right \|_p}

, and presents asymptotic formulas for

γ
p

{\gamma _p}

as

p
→
1
,
2

p \to 1,2

and

+
∞

+ \infty

, respectively. The definition of

C
p

(
X
)

{C_p}(X)

is extended to variables taking values in an arbitrary Hilbert space

H
H

, and it is shown that

γ
p

{\gamma _p}

is not increased by this extension.
A partial integration formula for product integrals of unbounded operator-valued functions
Authors: Hughes Rhonda J. 
Published: June 2010

The partial integration formula for product integrals
\[

∏
y
x

e

(
A
(
s
)
+
B
(
s
)
)
d
s

=

∏
y
x

e

A
(
s
)
d
s

∏
y
x

exp

(

(

∏
s
y

e

A
(
u
)
d
u

B
(
s
)

∏
y
s

e

A
(
u
)
d
u

)

d
s

)

,

\prod \limits _y^x {{e^{(A(s) + B(s))ds}} = } \prod \limits _y^x {{e^{A(s)ds}}} \prod \limits _y^x {\exp } \left ( {\left ( {\prod \limits _s^y {{e^{A(u)du}}} B(s)\prod \limits _y^s {{e^{A(u)du}}} } \right )ds} \right ),

\]
of which the Trotter product formula is a consequence, is established for a wide class of unbounded operator-valued functions

A
(
s
)
,
B
(
s
)

A(s),B(s)

.
A simplified proof of Heinz inequality and scrutiny of its equality
Authors: Furuta Takayuki 
Published: June 2010

An operator means a bounded linear operator on a Hilbert space

H
H

. We give a simplified proof of the following inequality:
\[

(

I

1

)

|

(
T
x
,
y
)

|

2

≤
(

|

T

|

2
α

x
,
x
)
(

|

T
∗

|

2
(
1
−
α
)

y
,
y
)

({{\text {I}}_1})\quad |(Tx,y){|^2} \leq (|T{|^{2\alpha }}x,x)(|{T^*}{|^{2(1 - \alpha )}}y,y)

\]
for any operator

T
T

and for any

x
,
y
∈
H

x, y \in H

and for any real number

α
\alpha

with

0
≤
α
≤
1

0 \leq \alpha \leq 1

. In case

0
>
α
>
1

0 > \alpha > 1

, the equality in

(

I

1

)

({{\text {I}}_1})

holds iff

|

T

|

2
α

x

|T{|^{2\alpha }}x

and

T
∗

y

{T^*}y

are linearly dependent iff

T
x

Tx

and

|

T
∗

|

2
(
1
−
α
)

y

|{T^*}{|^{2(1 - \alpha )}}y

are linearly dependent.

(

I

1

)

({{\text {I}}_1})

is equivalent to
\[

(

I

2

)

|

(
T
x
,
y
)

|

≤

|

|

|

T

|

α

x

|

|

|

|

|

T
∗

|

1
−
α

y

|

|

,

({{\text {I}}_2})\quad |(Tx,y)| \leq ||\;|T{|^\alpha }x||\;||\;|{T^*}{|^{1 - \alpha }}y||,

\]
, so one might believe that the equality in

(

I

1

)

({{\text {I}}_1})

or

(

I

2

)

({{\text {I}}_2})

would hold iff

|

T

|

2
α

x

|T{|^{2\alpha }}x

and

|

T
∗

|

2
(
1
−
α
)

y

|{T^*}{|^{2(1 - \alpha )}}y

are linearly dependent or iff

|

T

|

α

x

|T{|^\alpha }x

and

|

T
∗

|

1
−
α

y

|{T^*}{|^{1 - \alpha }}y

are linearly dependent, but we can give counterexamples to these mistakes. By this fact, the form of

(

I

1

)

({{\text {I}}_1})

is more convenient than

(

I

2

)

({{\text {I}}_2})

in order to remind us of the case when the equality in

(

I

1

)

({{\text {I}}_1})

or

(

I

2

)

({{\text {I}}_2})

holds.
Some analogues of Markov and Descartes systems for right disfocality
Authors: Eloe P. W. , Henderson Johnny 
Published: June 2010

A necessary and sufficient condition for the disconjagacy of the

n
n

th order linear differential equation

y

(
n
)

+

a
1

(
x
)

y

(
n
−
1
)

+
⋯
+

a
n

(
x
)
y
=
0

{y^{(n)}} + {a_1}(x){y^{(n - 1)}} + \cdots + {a_n}(x)y = 0

on a compact interval

I
I

is that there exists a system of solutions

y
1

,
…
,

y
n

{y_1}, \ldots ,{y_n}

such that any one of the following is satisfied: (i)

W
(

y
1

,
…
,

y
k

)
>
0
,
1
≤
k
≤
n

W({y_1}, \ldots ,{y_k}) > 0,1 \leq k \leq n

, on

I
I

; (ii)

W
(

y
i

1

,
…
,

y
i

k

)
>
0
,
1
≤

i
1

>
⋯
>

i
k

≤
n
,
1
≤
k
≤
n

W({y_i}_{_1}, \ldots ,{y_i}_{_k}) > 0,1 \leq {i_1} > \cdots > {i_k} \leq n,1 \leq k \leq n

, on

I
I

; or (iii)

W
(

y
i

,

y

i
+
1

,
…
,

y
i

+
k
−
1

)
>
0
,
1
≤
i
≤
n
−
k
+
1
,
1
≤
k
≤
n

W({y_i},{y_{i + 1}}, \ldots ,{y_i}_{ + k - 1}) > 0,1 \leq i \leq n - k + 1,1 \leq k \leq n

, on

I
I

. Necessary and sufficient criteria for the right disfocality of the linear differential equation on the compact interval

I
I

are established in terms of systems of solutions satisfying conditions which are analogous to those given in (i), (ii), (iii).
An inequality for selfadjoint operators on a Hilbert space
Authors: Bernstein Herbert J. 
Published: June 2010

An elementary inequality of use in testing convergence of eigenvector calculations is proven. If

e
λ

{e_\lambda }

is a unit eigenvector corresponding to an eigenvalue

λ
\lambda

of a selfadjoint operator

A
A

on a Hilbert space

H
H

, then
\[

|

(
g
,

e
λ

)

|

2

≤

‖
g
‖

2

‖

A
g

‖

2

−

(
g
,
A
g
)

2

‖

(
A
−
λ
I
)
g

‖

2

{\left | {(g,{e_\lambda })} \right |^2} \leq \frac {{{{\left \| g \right \|}^2}{{\left \| {Ag} \right \|}^2} - {{(g,Ag)}^2}}}{{{{\left \| {(A - \lambda I)g} \right \|}^2}}}

\]
for all

g
g

in

H
H

for which

A
g
≠
λ
g

Ag \ne \lambda g

. Equality holds only when the component of

g
g

orthogonal to

e
λ

{e_\lambda }

is also an eigenvector of

A
A

.
Picard dimensions of close to rotationally invariant densities
Authors: Tada Toshimasa 
Published: June 2010

The purpose of this paper is to show that the Picard dimensions of a rotation-free density

P
P

and a general density

Q
Q

on the punctured unit disk

0
>

|
z
|

>
1

0 > \left | z \right | > 1

are equal to each other if

|

P

(
z
)

−
Q

(
z
)

|

=
O

(

|
z
|

−
2

)

\left | {P\left ( z \right ) - Q\left ( z \right )} \right | = O\left ( {{{\left | z \right |}^{ - 2}}} \right )

as

z
→
0

z \to 0

.
An inequality suggested by Littlewood
Authors: Bennett Grahame 
Published: June 2010

It is shown that
\[

∑
n

a
n
3

∑

m
=
1

n

a
m
2

∑

k
=
1

m

a
k

≤

3
2

∑
n

a
n
4

[

∑

k
=
1

n

a
k

]

2

\sum \limits _n {a_n^3} \sum \limits _{m = 1}^n {a_m^2} \sum \limits _{k = 1}^m {{a_k}} \leq \frac {3}{2}\sum \limits _n {a_n^4} {\left [ {\sum \limits _{k = 1}^n {{a_k}} } \right ]^2}

\]
for arbitrary nonnegative numbers

a
1

,

a
2

,
…

{a_1},{a_2}, \ldots

.
A simple proof of an extension of the Fuglede-Putnam theorem
Authors: Gong Wei Bang 
Published: June 2010

A simple proof is given of the Theorem. If

A
A

and

B
∗

{B^ * }

are hyponormal, then

‖

A
X
−
X
B

‖

2

≥

‖

A
∗

X
−
X

B
∗

‖

2

{\left \| {AX - XB} \right \|_2} \geq {\left \| {{A^ * }X - X{B^ * }} \right \|_2}

for every

X
X

in the Hilbert-Schmidt class.
A transformation for an 𝑛-balanced ₃Φ₂
Authors: Srivastava H. M. 
Published: June 2010

An interesting generalization of the familiar

q
q

-extension of the Pfaff-Saalschütz theorem is proved and is applied, for example, to derive a reduction formula for a certain double

q
q

-series. The main theorem (asserting the symmetry in

n
n

and

N
N

of a function

f
(
n
,
N
)

f(n,N)

defined in terms of an

n
n

-balanced basic (or

q

-

q{\text { - }}

-) hypergeometric

3

Φ
2

_3{\Phi _2}

series by equation (8)) is essentially a

q
q

-extension of Sheppard’s transformation.
Normal filters generated by a family of sets
Authors: Di Prisco Carlos Augusto , Uzcategui Carlos Enrique 
Published: June 2010

We study normal filters on the set spaces

λ
,

P

κ

(
λ
)

,

[
λ
]

κ

\lambda ,{\mathcal {P}_\kappa }\left ( \lambda \right ),{\left [ \lambda \right ]^\kappa }

, and

(
λ
)

κ

{\left ( \lambda \right )^\kappa }

. We characterize the least normal

γ
\gamma

-complete filter containing a given

γ
\gamma

-complete filter for

γ
≥

ω
1

\gamma \geq {\omega _1}

. If

F

\mathcal {F}

is a

ω
1

{\omega _1}

-complete filter on any of the set spaces mentioned, the least

ω
1

{\omega _1}

-complete normal filter containing it is the filter generated by the sets

{

x
∈
E

|

α
1

,
…
,

α
n

∈
x
→
x
∈
f

(

α
1

,
…
,

α
n

)

}

\left \{ {x \in E\left | {{\alpha _1}, \ldots ,{\alpha _n} \in x \to x \in f\left ( {{\alpha _1}, \ldots ,{\alpha _n}} \right )} \right .} \right \}

where

f
:

λ

>
ω

→

F

f:{\lambda ^{ > \omega }} \to \mathcal {F}

.
A boundary analogue of Morera’s theorem in the unit ball of 𝐶ⁿ
Authors: Grinberg Eric L. 
Published: June 2010

We show that the boundary values of holomorphic functions in the unit ball ofCn(n > 1){{\mathbf {C}}^n}(n{\text { > 1)}}are characterized by the vanishing of their contour integrals over certain closed curves.
A counterexample to an F. and M. Riesz-type theorem
Authors: Samotij Krzysztof 
Published: June 2010

A premeasure is a finitely additive complex-valued function

μ
\mu

defined on the semiring of all connected subsets of

T

{\mathbf {T}}

, continuous at

∅
\emptyset

and with

μ
(
∅
)
=
μ
(

T

)
=
0

\mu (\emptyset ) = \mu ({\mathbf {T}}) = 0

. Let

κ
\kappa

be a continuous increasing concave function on

[
0
,
2
π
]

[0,2\pi ]

with

κ
(
0
)
=
0

\kappa (0) = 0

. A conjecture from [3] saying that if the Poisson integral of a premeasure

μ
\mu

is holomorphic in the open unit disk and

Var
κ

(
μ
)
>
∞

{\operatorname {Var}_\kappa }(\mu ) > \infty

then

lim

τ
→
0

Var
κ

(

μ
τ

−
μ
)
=
0

{\lim _{\tau \to 0}}{\operatorname {Var}_\kappa }({\mu _\tau } - \mu ) = 0

is disproved, where

Var
κ

(
μ
)
=
sup

∑
j

|

μ
(

I
j

)

|

/

∑
j

κ
(

|

I
j

|

)

{\operatorname {Var} _\kappa }(\mu ) = \sup \sum \nolimits _j {|\mu ({I_j}} )|/\sum \nolimits _j {\kappa (|{I_j}|)}

(the supremum is taken over all finite partitions of

T

{\mathbf {T}}

into connected subsets

I
j

{I_j}

) and

μ
τ

{\mu _\tau }

denotes the

τ
\tau

-translation of

μ
\mu

.
Vector-valued stochastic processes. II. A Radon-Nikodým theorem for vector-valued processes with finite variation
Authors: Dinculeanu Nicolae 
Published: June 2010

Given a real-valued process

A
A

with finite variation

|
A
|

\left | A \right |

and a vector-valued process

B
B

with finite variation

|
B
|

\left | B \right |

such that for each

ω
\omega

, the Stieltjes measure

d
B

.

(
ω
)

dB{\mathbf {.}}(\omega )

is absolutely continuous with respect to

d
A
.
(
ω
)

dA.(\omega )

, there exists a vector-valued process

H
H

which, under certain separability conditions, satisfies

B
t

=

∫

[
0
,
t
]

H
s

d

A
s

{B_t} = \int _{[0,t]} {{H_s}d{A_s}}

and

|
B
|

t

=

∫

[
0
,
t
]

‖

H
s

‖

d

|
A
|

s

{\left | B \right |_t} = \int _{[0,t]} {\left \| {{H_s}} \right \|d{{\left | A \right |}_s}}

for every

t
≥
0

t \geq 0

. If, moreover,

A
A

and

B
B

are optional or predictable, then so is

H
H

.
A new proof of an inequality of Littlewood and Paley
Authors: Luecking Daniel H. 
Published: June 2010

A fairly elementary new proof is presented of the inequality

(
p
≥
2
)

(p \geq 2)

:
\[

∫

|

h
′

|

p

(

1
−

|
z
|

)

p
−
1

d
x
d
y
≤

‖
h
‖

H
p

p

,

f
∈

H
p

.

\int {{{\left | {h’} \right |}^p}{{\left ( {1 - \left | z \right |} \right )}^{p - 1}}dxdy \leq \left \| h \right \|_{{H^p}}^p} ,\quad f \in {H^p}.

\]
In addition, the inequality
\[

∫

|
h
|

p
−
s

|

h
′

|

s

(
1
−

|
z
|

)

s
−
1

d
x
d
y
≤

‖
h
‖

H
p

p

\int {{{\left | h \right |}^{p - s}}{{\left | {h’} \right |}^s}{{(1 - \left | z \right |)}^{s - 1}}dxdy \leq \left \| h \right \|_{{H^p}}^p}

\]
is shown to hold for

h
∈

H
p

,
p
>
0

h \in {H^p},p > 0

, if and only if

2
≤
s
>
p
+
2

2 \leq s > p + 2

, generalizing the known case

s
=
2

s = 2

.
Finite extensions of rings
Authors: Cortzen Barbara , Small Lance W. 
Published: June 2010

The paper concerns some cases of ring extensions

R
⊂
S

R \subset S

, where

S
S

is finitely generated as a right

R
R

-module and

R
R

is right Noetherian. In

\S

1

{\text {\S }}1

it is shown that if

R
R

is a Jacobson ring, then so is

S
S

, with the converse true in the

PI

{\text {PI}}

case. In

\S

2

{\text {\S }}2

we show that if

S
S

is semiprime

PI

{\text {PI}}

,

R
R

must also be left (as well as right) Noetherian and

S
S

is finitely generated as a left .

R
R

-module.

\S

3

{\text {\S }}3

contains a result on

E
E

-rings.
Fixed point sets of homeomorphisms of metric products
Authors: Martin John R. 
Published: June 2010

In this paper it is investigated as to when a nonempty closed subset

A
A

of a metric product

X
X

containing intervals or spheres as factors can be the fixed point set of an autohomeomorphism of

X
X

. It is shown that if

X
X

is the Hilbert cube

Q
Q

or contains either the real line

R
R

or a

(
2
n
−
1
)

(2n - 1)

-sphere

S

2
n
−
1

{S^{2n - 1}}

as a factor, then

A
A

can be any nonempty closed subset. In the case where

A
A

is in

Int
⁡
(

B

2
n
+
1

)

\operatorname {Int}({B^{2n + 1}}{\text {)}}

, the interior of the closed unit

(
2
n
+
1
)
−

ball

B

2
n
+
1

,

(2n + 1) - {\text {ball }}{B^{2n + 1}},

, a strong necessary condition is given. In particular, for

B
3

,
A

{B^3},A

can neither be a nonamphicheiral knot nor a standard closed or nonplanar bordered surface.