bluuebunny/crossref_metadata_2025 huggingface dataset:
Remarque sur l’arithmétique des 2-formes différentielles de deuxième espèce
Authors: Nejib Boulahia
Published: June 2010

We consider a complex smooth and proper surface

X
X

and a prime number

l
l

. We know that the invariant

B
2

−
ρ

{B_2} - \rho

represents the dimension of the vector space of classes of

2
2

-forms of the second kind on

X
X

. Grothendieck has observed that

B
2

−
ρ

{B_2} - \rho

is an arithmetic invariant, and this leads naturally to interpret a

2
2

-form of the second kind from an arithmetic point of view. By using Grothendieck’s techniques we associate to each class

ω
\omega

of a

2
2

-form of the second kind an element
\[

ω
1

{\omega _1}

\]
of

ω

{\omega }

-adic part of the Brauer group

H
2

(

X

et,

O

X

et

∗

)

{H^2}({X_{{\text {et,}}}}\mathcal {O}_{{X_{{\text {et}}}}}^*)

of

X
X

. In the case where

X
X

is a fibered space on a curve, if one subjects this fibration to some given conditions (Artin),

ω
l

{\omega _l}

is then also interpreted as an element of

H
1

(

Γ

et

,

i
∗

J

)

(
l
)

{H^1}{({\Gamma _{{\text {et}}}},{i^*}J)_{(l)}}

or, in the equivalent manner, as an element of the Tate-Schafarievitch group

II

I

1

(

C

(
Γ
)
,
J
)

{\text {II}}{{\text {I}}^1}({\mathbf {C}}(\Gamma ),J)

studied by Ogg, where

J
J

is the Jacobian of the generic fiber of

X
X

and

C

(
Γ
)

{\mathbf {C}}(\Gamma )

is the functions field of

Γ
\Gamma

. Reciprocally each element of the

l
l

-divisible part of

II

I

1

(

C

(
Γ
)
,
J

)

(
l
)

{\text {II}}{{\text {I}}^1}{({\mathbf {C}}(\Gamma ),J)_{(l)}}

comes from a

2
2

-form of the second kind. This correspondence permits us to deduce some consequences on

2
2

-forms of the second kind.
A superreflexive Banach space which does not admit complex structure
Authors: Szarek Stanisław J.
Published: June 2010

We construct an infinite-dimensional superreflexive real Banach space which does not admit complex structure and consequently is not isomorphic to the Cartesian square of any Banach space. We also construct a variant of Bourgain’s example of a complex Banach space with nonunique complex structure and state a number of open problems about structure of Banach spaces and their linear groups.
Cube slicing in 𝑅ⁿ
Authors: Ball Keith
Published: June 2010

We prove that every

(
n
−
1
)

(n - 1)

-dimensional section of the unit cube in

R

n

{{\mathbf {R}}^n}

has volume at most

2

\sqrt 2

. This upper bound is clearly best possible.
Some properties of ∀∃ models in the isols
Authors: McLaughlin T. G.
Published: June 2010

It is a consequence of theorems proved by Nerode [10] and Hirschfeld [7] that every countable model of

∀
∃

\forall \exists

arithmetic is isomorphic to a subsemiring of a one-generator semiring of isols. We characterize, in terms of the generators of "Nerode semirings", the contents of arbitrary semirings

R

{\mathbf {R}}

of isols that are models of

∀
∃

\forall \exists

arithmetic, and we show that all such

R

{\mathbf {R}}

are in fact models of the

ω
\omega

-true

∀
∃

\forall \exists

sentences of isol theory. We solve one of the chief problems left open in [8], and in

§
3

\S 3

we provide an example of the applied virtues of

∀
∃

\forall \exists

-correct subsemirings of the isols.
Random fixed point theorems for measurable multifunctions in Banach spaces
Authors: Papageorgiou Nikolaos S.
Published: June 2010

In this paper we prove several random fixed point theorems for measurable closed and nonclosed valued multifunctions satisfying general continuity conditions. Our work extends and sharpens earlier results by Engl, Itoh and Reich.
Non-Archimedean 𝐶^{#}(𝑋)
Authors: Domínguez Jesús M.
Published: June 2010

Let

E
E

be a nonarchimedean rank-one valued field, and

X
X

an ultraregular topological space. We consider the Gelfand subalgebra

C
#

(
X
,
E
)

{C^\# }(X,E)

of the algebra of all

E
E

-valued continuous functions on

X
X

, and the algebra

F
(
X
,
E
)

F(X,E)

consisting of those

E
E

-valued continuous functions

f
f

for which there exists a compact set

K
⊂
X

K \subset X

such that

f
(
X
−
K
)

f(X - K)

is finite. We obtain some characterizations of

C
#

(
X
,
E
)

{C^\# }(X,E)

, analogous to those obtained in the real case, which we use to find conditions that imply the equality

C
#

(
X
,
E
)
=
F
(
X
,
E
)

{C^\# }(X,E) = F(X,E)

holds.
Essential maps exist from 𝐵𝑈 to 𝐶𝑜𝑘𝑒𝑟𝐽
Authors: Feshbach Mark
Published: June 2010

We show that

[

BU,coker

J
]
≠
0

[{\text {BU,coker}}\,J] \ne 0

but that there are no infinite loop maps from BU to coker

J
J

. The proofs involve the Segal conjecture.
Positively curved bilinear forms
Authors: Sarkaria K. S. , Zoltek S. M.
Published: June 2010

A characterisation of the positivity of sectional curvature is given in terms of the angular spread of the second fundamental form.
Denting points in Bochner 𝐿^{𝑝}-spaces
Authors: Lin Bor-Luh , Lin Pei-Kee
Published: June 2010

A characterization of denting points in the unit balls of

L
p

(
μ
,
X
)

{L^p}(\mu ,X)

,

1
>
p
>
∞

1 > p > \infty

, is given. This characterization is compared to similar known results concerning strongly extreme points and extreme points.
Approximate unitary equivalence of continuous nests
Authors: Davidson Kenneth R.
Published: June 2010

A short proof is given of an important theorem of N. Andersen: All continuous nests are approximately unitarily equivalent.
Spurious Brownian motions
Authors: Csörgő Sándor
Published: June 2010

Spurious Brownian motions are characterized in

R

d

{{\mathbf {R}}^d}

,

d
⩾
2

d \geqslant 2

.
Some examples of indefinite complete complex Einstein hypersurfaces not locally symmetric
Authors: Romero Alfonso
Published: June 2010

Some examples of indefinite complete complex Einstein hypersurfaces of an indefinite complex flat space which are not locally symmetric are given.
Every exactly 2-to-1 function on the reals has an infinite set of discontinuities
Authors: Heath Jo
Published: June 2010

It has long been known that the set of discontinuities of a

2
2

-to-

1
1

function on either the closed or the open interval must be nonempty; this paper proves that the set must be infinite.
Holomorphic retracts of polyballs
Authors: Kuczumow Tadeusz
Published: June 2010

Let

T
T

be a holomorphic self-mapping in the Cartesian product

B
n

{B^n}

of

n
n

unit open balls in

C

k

{{\mathbf {C}}^k}

. In this note we give a very short proof of the fact that if

Fix
⁡
T
=
{
y
∈

B
n

:
y
=
T
y
}

\operatorname {Fix} T = \{ y \in {B^n}:y = Ty\}

is nonempty, then this set is a holomorphic retract of

B
n

{B^n}

.
Class numbers of pure fields
Authors: Mollin R. A.
Published: June 2010

Necessary and sufficient conditions are given for the class number

h

K
i

{h_{{K_i}}}

of a pure field

K
=
Q
(

m

1

/

p
i

)

 

(

for 

i
=
1
,
2
)

K = Q({m^{1/{p^i}}}){\text { }}({\text {for }}i = 1,2)

to be divisible by

p
r

{p^r}

for a given positive integer

r
r

and prime

p
p

. Moreover the divisibility of

h

K
i

{h_{{K_i}}}

by

p
p

is linked with the

p
p

-rank of the class group of the

K
(
ς
)

K(\varsigma )

and prime divisors of

m
m

, where

ς
\varsigma

is a primitive

p
p

th root of unity. Finally we prove in an easy fashion that for a given odd prime

p
p

and any natural number

t
t

there exist infinitely many non-Galois algebraic number fields (in fact pure fields) of degree

p
i

(
i
=
1
,
2
)

{p^i}(i = 1,2)

over

Q
Q

whose class numbers are all divisible by

p
t

{p^t}

.
A Riemann type theorem for unconditional convergence of operators
Authors: Kaftal Victor , Weiss Gary
Published: June 2010

We prove that if a series of bounded linear operators is compactly conditionally convergent in the strong operator topology, that is, each of its partial sums converge, in the strong operator topology to a compact operator, then the series converges in the uniform (operator norm) topology; although not necessarily absolutely. In case the operators are all mutually diagonalizable, then under the same hypothesis, the series converges absolutely uniformly.
A JC-algebra which is not the range of a positive projection on a 𝐶*-algebra
Authors: Robertson A. Guyan
Published: June 2010

An example is given of a

J
C

JC

-algebra which is not the range of any positive projection on the selfadjoint part of its generated

C
∗

{C^*}

-algebra.
Reduction of variables for minimal submanifolds
Authors: Palais Richard S. , Terng Chuu-Lian
Published: June 2010

If

G
G

is a compact Lie group and

M
M

a Riemannian

G
G

manifold, then the orbit map

∏
:
M
→
M

/

G

\prod :M \to M/G

is a stratified Riemannian submersion and the well-known "cohomogeneity method" pioneered by Hsiang and Lawson [HL] reduces the problem of finding codimension

k
k

minimal submanifolds of

M
M

to a related problem in

M

/

G

M/G

. We show that this reduction of variables technique depends only on a certain natural Riemannian geometric property of the map

∏
\prod

which we call

h
h

-projectability and which is shared by certain other naturally occurring and important classes of Riemannian submersions.
On the homotopy groups of 𝐴(𝑋)
Authors: Betley Stanisław
Published: June 2010

In this paper we will prove that if

X
X

is any space with a finite fundamental group, then Waldhausen’s algebraic

K
K

-groups of

X
X

are finitely generated. We will use Dwyer’s machinery developed in Twisted homological stability for general linear groups (Ann. of Math. 111).
A note on functionally compact spaces
Authors: Bella Angelo
Published: June 2010

In this paper we give some characterizations of functionally compact Hausdorff spaces in terms of multifunctions. Also, some applications are presented.
Power series and smooth functions equivalent to a polynomial
Authors: Kucharz Wojciech
Published: June 2010

An algebraic criterion is given for a power series in

n
n

variables over a field of characteristic 0 to be equivalent to a polynomial in

n
−
k

n - k

variables over the ring of power series in

k
k

variables. For convergent power series over the reals or complexes a geometric interpretation of the criterion is established. An analogous sufficient condition is obtained for germs of smooth functions. Most of the previously known results follow easily from the criterion.
On the stable range of uniform algebras and 𝐻^{∞}
Authors: Corach Gustavo , Suárez Fernando Daniel
Published: June 2010

We find a criterion for stable range reducibility for uniform algebras and we apply it to the algebra

H
∞

{H^\infty }

.
The parametric Gauss-Green theorem
Authors: Ortel M. , Schneider W.
Published: June 2010

A strong parametric Gauss-Green theorem, for chains homologous to zero in a plane region, follows directly from an interchange in order of integrations.
A strengthening of Leth’s uniqueness condition for sequences
Authors: Malitz Jerome
Published: June 2010

A series

∑

a
i

\sum {a_i}

of nonnegative reals summing to 1 such that

a
i

≤

∑

j
>
i

a
j

{a_i} \leq {\sum _{j > i}}{a_j}

for eacn

i
i

is uniquely characterized by the equalities of the form

∑
J

a
i

=

∑
K

a
k

{\sum _J}{a_i} = {\sum _K}{a_k}

. This characterization is an improvement of one given by Leth.
Minimal degrees of faithful characters of finite groups with a T.I. Sylow 𝑝-subgroup
Authors: Berger T. R. , Landrock P. , Michler G. O.
Published: June 2010

Using the classification of the finite simple groups we show in this article that a faithful complex character

χ
\chi

of a finite group

G
G

with a nonnormal T.I. Sylow

p
p

-subgroup

P
P

has degree

χ
(
1
)

 > 

|
P
|

−
1

\chi (1){\text { > }}\sqrt {\left | P \right |} - 1

. This result verifies a conjecture of H. S. Leonard [10].
On a theorem of Huneke concerning multiplicities
Authors: O’Carroll Liam
Published: June 2010

We consider a theorem by Huneke on multiplicities, and show that the extension surmised by Huneke is a stronger form of the Syzygy Conjecture.
Best 𝐿₁-approximation with varying weights
Authors: Kroó András
Published: June 2010

It is proved in this note that the so-called

A
A

-property is necessary in order that the finite-dimensional space

U
U

be Chebyshev in

C

(
K
)

C\left ( K \right )

with respect to the norm

‖
f
‖

=

∫
K

ω

|
f
|

\left \| f \right \| = \int _K {\omega \left | f \right |}

for every positive continuous weight

ω
\omega

. It is also shown that for each finite-dimensional subspace

U
U

there exists a positive continuous weight

ω
\omega

such that

U
U

is Chebyshev in

C

(
K
)

C\left ( K \right )

with respect to this weight

ω
\omega

.
On periodic solutions of nonlinear differential equations with singularities
Authors: Lazer A. C. , Solimini S.
Published: June 2010

Necessary and sufficient conditions for existence of periodic solutions of differential equations containing singularities are given. Our theorems apply to

u
+
1

/

u
α

=
h

(
t
)

≡
h

(

t
+
T

)

u + 1/{u^\alpha } = h\left ( t \right ) \equiv h\left ( {t + T} \right )

for all

α
>
0

\alpha > 0

and to

u
−
1

/

u
α

=
h

(
t
)

u - 1/{u^\alpha } = h\left ( t \right )

if

α
≥
1

\alpha \geq 1

, and for this case

α
≥
1

\alpha \geq 1

is an essential condition.
The analytic Radon-Nikodým property in Lebesgue Bochner function spaces
Authors: Dowling Patrick N.
Published: June 2010

Let

X
X

be a complex Banach space,

(
Ω
,
∑
,
μ
)

(\Omega ,\sum ,\mu )

a finite measure space, and

1
≤
p
>
∞

1 \leq p > \infty

. Then

L
p

(
μ
;
X
)

{L_p}(\mu ;X)

has the analytic Radon-Nikodym property if and only if

X
X

has it.
Elementarily equivalent fields with inequivalent perfect closures
Authors: Videla Carlos R.
Published: June 2010

We give a counterexample to the following conjecture due to L. V. den Dries: Let

F
,
L

F,L

be two fields of characteristic

p
p

. If

F
≡
L

F \equiv L

then

F
1

/

p
∞

≡

L
1

/

p
∞

{F^1}/{p^\infty } \equiv {L^1}/{p^\infty }

.
A note on separable Banach spaces with nonseparable dual
Authors: Hagler James N.
Published: June 2010

If

X
X

is a separable Banach space with

X
∗

{X^*}

nonseparable, then

X
X

contains a subspace

X
0

{X_0}

with a Schauder basis with

X
0
∗

X_0^*

nonseparable.
Generalizations of Paley-Wiener’s theorem for entire functions of exponential type
Authors: Saitoh Saburou
Published: June 2010

An interpretation and generalizations of the Paley-Wiener theorem for entire functions of exponential type are given in connection with the Fourier-Laplace transform.
A singular space related to the point-open game
Authors: Junnila H. J. K. , Nogura T. , Telgársky R.
Published: June 2010

The authors construct a completely regular space

X
0

{X_0}

such that Player I has a winning strategy in the point-open game

G

(

X
0

)

G\left ( {{X_0}} \right )

, but

X
0

{X_0}

has no

σ
\sigma

-closure-preserving cover by

C

{\mathbf {C}}

-scattered closed sets.
A natural formulation of quasi-interpolation by multivariate splines
Authors: Chui Charles K. , Diamond Harvey
Published: June 2010

A quasi-interpolation formula based on discrete function values is given in the form of a multivariate spline series that yields the local approximation order characterized by the Fix-Strang conditions. This formula can be considered as a partial sum of the formal Neumann series expansion of the formal interpolation operator, and hence, justifies that "quasi-interpolation" is indeed an appropriate terminology for such an approximation formula.
Certain properties of derivations
Authors: Spivack Mark
Published: June 2010

We consider two properties of implemented derivations on operator algebras, and give applications. One provides a simple test and leads to examples of nonimplemented derivations on commutative algebras. The other is stronger and yields a necessary and sufficient condition for derivations on

p
B
(
H
)

p
⊥

pB(H){p^ \bot }

to be implemented, where

H
H

is a Hilbert space and

p
p

is a projection on

H
H

. Any algebra

S
S

on

H
H

has an extension to an algebra

S
2

{S_2}

acting on

H
⊕

C

H \oplus {\mathbf {C}}

containing such an algebra. We show that any derivation

δ
\delta

on an algebra

S
S

is implemented if and only if

δ
\delta

has a bounded strongly continuous extension to

S
2

{S_2}

. If so we can construct an implementing operator explicitly.
Banach algebras with uncomplemented radical
Authors: Bachelis Gregory F. , Saeki Sadahiro
Published: June 2010

A condition is given on the invertible elements of a commutative Banach algebra

B

\mathbf {B}

which implies that the radical of

B

\mathbf {B}

is not complemented by any closed subalgebra. This condition is then applied to show that certain quotient algebras of group algebras have an uncomplemented radical.
Proof of a conjecture of Bank and Laine regarding the product of two linearly independent solutions of 𝑦”+𝐴𝑦=0
Authors: Shen Li-Chien
Published: June 2010

Let

A
A

be a transcendental entire function of order

>
1

> 1

. If

w
1

{w_1}

and

w
2

{w_2}

are two linearly independent solutions of the differential equation

y
+
A
y
=
0

y + Ay = 0

, then at least one of

w
1

,

w
2

{w_1},{w_2}

has the property that the exponent of convergence of its zeros is

>
1

> 1

.
Functions in the Fresnel class
Authors: Chang K. S. , Johnson G. W. , Skoug D. L.
Published: June 2010

Let

H
H

be a separable infinite-dimensional Hilbert space over

R

{\mathbf {R}}

. The Fresnel class

F

(
H
)

\mathcal {F}(H)

of

H
H

consists of all Fourier-Stieltjes transforms of bounded Borel measures on

H
H

. There are several results insuring that various functions of interest in connection with the Feynman integral and quantum mechanics are in

F

(
H
)

\mathcal {F}(H)

. We give a theorem which has most of these results as corollaries as well as many further corollaries involving the reproducing kernel Hilbert spaces of Gaussian stochastic processes.
𝐶^{∞}-invariants on loop spaces
Authors: Cheung Wing Sum
Published: June 2010

We compute all the

C
∞

{C^\infty }

-invariants on the spaces of immersions

C
I
∞

(

S
1

,
S
)

C_I^\infty ({S^1},S)

and

C
I
∞

(

S
1

,

R

3

)

C_I^\infty ({S^1},{{\mathbf {R}}^3})

, where

S
S

is a surface of constant curvature, by using Griffiths’ formalism on the calculus of variations via exterior differential systems.
Two-relator groups with prescribed cohomological dimension
Authors: Howie James
Published: June 2010

For each integer

n
⩾
2

n \geqslant 2

, an example is constructed of a

2
2

-generator,

2
2

-relator presentation of a group

G
(
n
)

G(n)

of cohomological dimension

n
+
1

n + 1

.
On Cournot-Nash equilibria in generalized qualitative games with an atomless measure space of agents
Authors: Khan M. Ali , Papageorgiou Nikolaos S.
Published: June 2010

We present a result on the existence of Cournot-Nash equilibria in games with an atomless measure space of players each with nonordered preferences and with strategy sets in a separable Banach space. Our result dispenses with any convexity assumption on the preference correspondence.
The preservation of atriodicity by semiconfluent mappings
Authors: Davis James F.
Published: June 2010

Grace and Vought have proven that the image of a chainable continuum under a semiconfluent mapping is atriodic. This motivates the question of whether the semiconfluent image of an atriodic continuum is atriodic. In this paper this question is answered affirmatively.
On preimage knots in 𝑆³
Authors: Soma Teruhiko
Published: June 2010

In [1], Gordon proved that a certain sequence of preimage knots in

S
3

{S^3}

is finite. In this paper, we prove that every sequence of preimage knots is finite by using the Gromov invariant for knots.
Liouvillian solutions of the differential equation 𝑦”+𝑆(𝑥)𝑦=0 with 𝑆(𝑥) binomial
Authors: Setoyanagi Minoru
Published: June 2010

If a differential equation

y
+
(
a

x
p

+
b

x
q

)
y
=
0

y + (a{x^p} + b{x^q})y = 0

with

p
>
q

p > q

has a liouvillian solution, then

p
p

is an even number

2
m

2m

and the number

s
=
(
m
+
1
)

/

(
p
−
q
)

s = (m + 1)/(p - q)

is an integer. The case

s
=
2

s = 2

occurs only if

m
=
1

m = 1

.
Simple exceptional 16-dimensional Jordan triple systems
Authors: Schwarz Thomas J. E.
Published: June 2010

We prove that every simple exceptional

16
16

-dimensional Jordan triple system is a subtriple of an exceptional Jordan algebra.
Invariance under operation 𝒜
Authors: Morgan John C. , Schilling Kenneth
Published: June 2010

The invariance under operation

A

\mathcal {A}

of the families of sets having the classical Baire property, of Lebesgue measurable sets, and of Marczewski sets is established in a unified manner.
On biduals of 𝐶*-tensor products
Authors: Quigg John C.
Published: June 2010

Huruya [4] has proven that, for

C
∗

{C^*}

-algebras

A
1

{A_1}

and

A
2

{A_2}

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A 1 circled-times upper A 2 right-parenthesis Superscript asterisk asterisk Baseline equals upper A 1 Superscript asterisk asterisk Baseline ModifyingAbove circled-times With bar upper A 2 Superscript asterisk asterisk">

(

A
1

⊗

A
2

)

∗
∗

=

A
1

∗
∗

⊗
¯

A
2

∗
∗

{({A_1} \otimes {A_2})^{**}} = A_1^{**}\overline \otimes A_2^{**}

\]
for every

A
2

{A_2}

if and only if

A
1

{A_1}

is scattered. We strengthen this by proving that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A 1 circled-times upper A 2 right-parenthesis Superscript asterisk asterisk Baseline equals upper A 1 Superscript asterisk asterisk Baseline ModifyingAbove circled-times With bar upper A 2 Superscript asterisk asterisk">

(

A
1

⊗

A
2

)

∗
∗

=

A
1

∗
∗

⊗
¯

A
2

∗
∗

{({A_1} \otimes {A_2})^{**}} = A_1^{**}\overline \otimes A_2^{**}

if and only if

A
1

{A_1}

or

A
2

{A_2}

is scattered. We discuss ramifications to representation theory and related questions regarding normal representations of

W
∗

{W^*}

-tensor products.
On the Baire class of ℐ-approximate derivatives
Authors: Łazarow Ewa
Published: June 2010

Basic properties of an

I

\mathcal {I}

-approximate derivative which is a category analogue of the approximate derivative were considered in [3]. Using the notion of balanced selection and the result of O’Malley, we shall prove that the

I

\mathcal {I}

-approximate derivative is of Baire class one.
On the realization and classification of cyclic extensions of polynomial algebras over the Steenrod algebra
Authors: Hiller Howard , Smith Larry
Published: June 2010

Suppose

R

∗

{{\mathbf {R}}^*}

is an unstable algebra over the Steenrod algebra of the form

P

∗

(

d
k

)

{{\mathbf {P}}^*}(\sqrt [k]{d})

, where

P

∗

{{\mathbf {P}}^*}

is a polynomial algebra over the Steenrod algebra. If

R

∗

{{\mathbf {R}}^*}

is integrally closed then

R

∗

=
P

(
V

)

G

X

{{\mathbf {R}}^*} = P{(V)^{{G_\mathcal {X}}}}

, where

C
⩽
G
L
(
V
)

C \leqslant GL(V)

is generated by pseudoreflections and

G

X

=
ker
⁡
{

X

:
G
→

F

p
∗

}

{G_\mathcal {X}} = \ker \{ \mathcal {X}:G \to {\mathbf {F}}_p^*\}

is a character of degree

k
k

.
Conway algebras and skein equivalence of links
Authors: Przytycki Józef H. , Traczyk Paweł
Published: June 2010

We consider a class of pairs of links which are not skein equivalent but have the same invariant in every Conway algebra.
Free products of lattice-ordered groups
Authors: Glass A. M. W.
Published: June 2010

We give a highly homogeneous representation for the free product of two nontrivial countable lattice-ordered groups and obtain, as a consequence of the method, that the free product of nontrivial lattice-ordered groups is directly indecomposable and has trivial center.
Properness of Lie algebras and enveloping algebras. I
Authors: Michaelis Walter
Published: June 2010

An associative unitary (respectively, Lie) algebra is said to be proper in case the intersection of ail of its cofinite two-sided (respectively, Lie) ideals is zero. Using the Hopf algebra structure of

U
L

UL

, it is shown that over a field of characteristic zero a Lie algebra is proper if and only if its universal enveloping algebra is proper. (In the finite-dimensional case this provides a new proof of a result of Harish-Chandra.) The analogous result for Lie

p
p

-algebras and their restricted universal enveloping algebras holds and is proved by the same technique.
Norms on enveloping algebras
Authors: Luminet Denis
Published: June 2010

Let

g

\mathfrak {g}

be a complex Lie algebra, and let

U
U

be its enveloping algebra.

U
U

is normable if and only if

g

\mathfrak {g}

is nilpotent.
On the Fekete-Szegő problem for close-to-convex functions
Authors: Koepf Wolfram
Published: June 2010

Let

S
S

be the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known result
\[

max

f
∈
S

|

a
3

−
λ

a
2
2

|

=
1
+
2

e

−
2
λ

/

1
−
λ
)

{\max _{f \in S}}{\text {|}}{a_3} - \lambda a_2^2| = 1 + 2{e^{ - 2\lambda /1 - \lambda )}}

\]
for

λ
∈
[
0
,
1
]

\lambda \in [0,1]

. We consider the corresponding problem for the family

C
C

of close-to-convex functions and get
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max Underscript f element-of upper C Endscripts vertical-bar a 3 minus lamda a 2 squared equals StartLayout Enlarged left-brace 1st Row 1st Column 3 minus 4 lamda 2nd Column a m p semicolon if lamda element-of left-bracket 0 comma 1 slash 3 right-bracket comma 2nd Row 1st Column 1 slash 3 plus 4 slash left-parenthesis 9 lamda right-parenthesis 2nd Column a m p semicolon if lamda element-of left-bracket 1 slash 3 comma 2 slash 3 right-bracket comma 3rd Row 1st Column 1 2nd Column a m p semicolon if lamda element-of left-bracket 2 slash 3 comma 1 right-bracket period EndLayout">

max

f
∈
C

|

a
3

−
λ

a
2
2

=

{

3
−
4
λ

a
m
p
;

if

λ
∈
[
0
,
1

/

3
]
,

1

/

3
+
4

/

(
9
λ
)

a
m
p
;

if

λ
∈

[1/3,2/3],

1

a
m
p
;

if

λ
∈

[2/3,1]

.

\max \limits _{f \in C} |{a_3} - \lambda a_2^2 = \left \{ {\begin {array}{*{20}{c}} {3 - 4\lambda } & {{\text {if}}\lambda \in [0,1/3],} \\ {1/3 + 4/(9\lambda )} & {{\text {if}}\lambda \in {\text {[1/3,2/3],}}} \\ 1 & {{\text {if}}\lambda \in {\text {[2/3,1]}}{\text {.}}} \\ \end {array} } \right .

\]
As an application it is shown that

|

|

a
3

|

−

|

a
2

|

|

≤
1

||{a_3}| - |{a_2}|| \leq 1

for close-to-convex functions, in contrast to the result in

S
S

\[

max

f
∈
s

||

a
3

|

−

|

a
2

|

|

=
1.029....

\max \limits _{f \in s} {\text {||}}{a_3}| - |{a_2}|| = 1.029....

\]
Sequentially compact, Franklin-Rajagopalan spaces
Authors: Nyikos P. J. , Vaughan J. E.
Published: June 2010

A locally compact

T
2

{T_2}

-space is called a Franklin-Rajagopalan space (or FR-space) provided it has a countable discrete dense subset whose complement is homeomorphic to an ordinal with the order topology. We show that (1) every sequentially compact FR-space

X
X

can be identified with a space constructed from a tower

T
T

on

ω

(

X
=
X

(
T
)

)

\omega \left ( {X = X\left ( T \right )} \right )

, and (2) for an ultrafilter

u
u

on

ω
\omega

, a sequentially compact FR-space

X

(
T
)

X\left ( T \right )

is not

u
u

-compact if and only if there exists an ultrafilter

v
v

on

ω
\omega

such that

v
⊃
T

v \supset T

, and

v
v

is below

u
u

in the Rudin-Keisler order on

ω
∗

{\omega ^ * }

. As one application of these results we show that in certain models of set theory there exists a family

T

\mathcal {T}

of towers such that

|

T

|

>

2
ω

\left | \mathcal {T} \right | > {2^\omega }

, and

∏

{

X

(
T
)

:
T
∈

T

}

\prod \left \{ {X\left ( T \right ):T \in \mathcal {T}} \right \}

is a product of sequentially compact FR-spaces which is not countably compact (a new solution to the Scarborough-Stone problem). As further applications of these results, we give consistent answers to questions of van Douwen, Stephenson, and Vaughan concerning initially

m
m

-chain compact and totally initially

m
m

-compact spaces.
The occurrence problem for mapping class groups
Authors: Stillwell John
Published: June 2010

The occurrence problem for the mapping class group

M
(
g
,
0
)

M(g,0)

of the closed orientable surface of genus

g
g

is shown to be solvable for

g
=
1

g = 1

and unsolvable for

g
>
1

g > 1

.
Unramified Whittaker functions on the metaplectic group
Authors: Flicker Yuval Z.
Published: June 2010

Kazhdan (unpublished), Shintani [Sh] and Casselman and Shalika [CS] computed explicitly the unramified Whittaker function of a quasisplit

p
p

-adic group. This is the main local ingredient used in the Rankin-Selberg-Shimura method, which yielded interesting results in the study of Euler products such as

L
(
s
,
π
⊗

π
′

)

L(s,\pi \otimes \pi ’)

by Jacquet and Shalika [JS] (here

π
,

π
′

\pi ,\pi ’

are cuspidal

G
L
(
n
,

A
F

)

GL(n,{A_F})

-modules), and

L
(
s
,
π
,
r
)

L(s,\pi ,r)

by [F] (here

π
\pi

is a cuspidal

G
L
(
n
,

A
E

)

GL(n,{A_E})

-module,

E
E

is a quadratic extension of the global field

F
F

, and

r
r

is the twisted tensor representation of the dual group of

Res

E

/

F

⁡
G
L
(
n
)
)

\operatorname {Res}_{E/F}GL(n))

. Our purpose here is to generalize Shintani’s computation [Sh] from the context of

G
L
(
n
)

GL(n)

to that of the metaplectic

r
r

-fold covering group

G
~

\tilde G

of

G
L
(
n
)

GL(n)

(see

[

F
′

,

F
K

]

[{\mathbf {F’}},{\mathbf {FK}}]

).
A note on multiplicatively prime ideals of operators
Authors: Lin Pei-Kee
Published: June 2010

An ideal

I

\mathcal {I}

of

L

(

H

)

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

is said to be multiplicatively prime if

A
X
B
∈

I

AXB \in \mathcal {I}

for all

X
∈

L

(

H

)

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

implies

A
A

or

B
B

is in

I

\mathcal {I}

. In this paper, we show there exists a nonnorm multiplicatively prime ideal other than

F

\mathcal {F}

.
Approximate innerness of positive linear maps of finite von Neumann algebras. II
Authors: Sunouchi Chôichirô , Takemoto Hideo
Published: June 2010

Let

M
M

be a

σ
\sigma

-finite, finite von Neumann algebra with a faithful, normalized normal trace Tr on

M
M

. Let

ρ
\rho

be a positive linear map of

M
M

into itself such that

ρ
(
1
)

\rho (1)

is not necessarily a projection. If

ρ
\rho

is approximately inner with respect to the norm

|

|

.

|

|

2

||.|{|_2}

induced by Tr, then

ρ
\rho

has a close connection to

∗

{*}

-homomorphisms.
On the uniqueness of a minimal norm representative of an operator in the commutant of the compressed shift
Authors: Foias Ciprian , Tannenbaum Allen
Published: June 2010

In this note we give a new criterion guaranteeing the uniqueness of a minimal norm representative of a bounded linear operator which commutes with a finite multiplicity shift. We moreover give examples which show that if the hypotheses of our theorem are violated then the minimal norm representative may not be unique.
The equivariant structure of Eilenberg-Mac Lane spaces. I. The 𝑍-torsion free case
Authors: Smith Justin R.
Published: June 2010

The purpose of this paper is to continue the work begun in [7]. That paper described an obstruction theory for topologically realizing an (equivariant) chain-complex as the equivariant chain-complex of a CW-complex. The obstructions essentially turned out to be homological

k
k

-invariants of Eilenberg-Mac Lane spaces and the key to their computation consists in developing tractable models for the chain-complexes of these spaces. The present paper constructs such a model in the

Z

{\mathbf {Z}}

-torsion free case. The model is sufficiently simple that in some cases it is possible to simply read off homological

k
k

-invariants, and thereby derive some topological results.
Total paracompactness of real GO-spaces
Authors: Balogh Zoltán T. , Bennett Harold
Published: June 2010

A topological space is said to be totally paracompact (resp. totally metacompact) if every open base of it has a locally finite (resp. pointfinite) subcover. In this paper we characterize all totally paracompact GO-spaces constructed on the real line. It turns out that in the class of GO-spaces on the real line, total paracompactness and total metacompactness are equivalent. Another consequence of our characterization is that totally metacompact GO-spaces on the real line are metrizable. Questions and partial results are given concerning total paracompactness in subspaces of real GO-spaces.
On branched covers of the 2-sphere by the 2-sphere
Authors: Gersten S. M.
Published: June 2010

A new solution of the Hurwitz problem is given for realizing branch data by a branched cover of

S
2

{S^2}

by

S
2

{S^2}

. The solution involves products of conjugacy classes in a free group.
Galois endomorphisms of the torsion subgroup of one-parameter generic formal groups
Authors: Zimmermann Karl
Published: June 2010

Let

Z

p

{{\mathbf {Z}}_p}

be the ring of

p
p

-adic integers and let

Γ
\Gamma

be a one-parameter generic formal group of finite height

h
h

defined over

Z

p

[

[

t
1

,
…
,

t

h
−
1

]

]

=
A

{{\mathbf {Z}}_p}\left [ {\left [ {{t_1}, \ldots ,{t_{h - 1}}} \right ]} \right ] = A

. Let

K
K

be the field of fractions of

A
,
G
=
Gal
⁡

(

K
¯

/

K

)

A,G = \operatorname {Gal}\left ( {\bar K/K} \right )

and

T

(
Γ
)

T\left ( \Gamma \right )

the Tate module of

Γ
\Gamma

. The purpose of this paper is to give an elementary proof that the
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m a p upper E n d Subscript upper A Baseline left-parenthesis normal upper Gamma right-parenthesis right-arrow upper E n d Subscript upper G Baseline left-parenthesis upper T left-parenthesis normal upper Gamma right-parenthesis right-parenthesis">

map
⁡

End

A

⁡

(
Γ
)

→

End

G

⁡

(

T

(
Γ
)

)

\operatorname {map}\operatorname {End}_{A}\left ( \Gamma \right ) \to \operatorname {End}_{G}\left ( {T\left ( \Gamma \right )} \right )

is a surjection.
On the global dimension of a ring modulo its nilpotent radical
Authors: Kirkman Ellen , Kuzmanovich James
Published: June 2010

For any preassigned integer

n
≥
3

n \geq 3

, a Noetherian affine PI ring

R
R

is constructed with

gl dim
⁡
R
≤
3

\operatorname {gl dim }R \leq 3

, but

gl dim
⁡

(

R

/

N

(
R
)

)

=
n

\operatorname {gl dim}\left ( {R/N\left ( R \right )} \right ) = n

. A second similar ring is constructed with

gl dim
⁡
R
≤
5

\operatorname {gl dim }R \leq 5

and

gl dim
⁡

(

R

/

N

(
R
)

)

=
∞

\operatorname {gl dim}\left ( {R/N\left ( R \right )} \right ) = \infty

.
A Tauberian theorem for Hausdorff methods
Authors: Kuttner Brian , Parameswaran Mangalam R.
Published: June 2010

Let

H
=
(
H
,
χ
)

H = (H,\chi )

be a regular Hausdorff summability method defined by the function

χ
∈

BV

[

0
,
1

]

\chi \in {\text {BV}}\left [ {0,1} \right ]

. It is shown that if

χ
\chi

is absolutely continuous on

[

0
,
1

]

\left [ {0,1} \right ]

, then the methods

H
H

and

V
⋅
H

V \cdot H

are equivalent for bounded sequences, where

V
V

belongs to a certain class of summability methods which includes the Cesàro methods

C
α

(
α

 > 

0
)

{C_\alpha }(\alpha {\text { > }}0)

, the Abel method

A
A

, and the methods

A
⋅

C
α

(
α

 > 

−
1
)

A \cdot {C_\alpha }(\alpha {\text { > }} - 1)

.
Conglomerability and finite partitions
Authors: Zame Alan
Published: June 2010

This paper gives a simplified proof of a generalization of the theorem of Schervish, Seidenfeld and Kadane on the extent of nonconglomerability of finitely additive probability measures.
Atriodic homogeneous nondegenerate continua are one-dimensional
Authors: Rogers James T.
Published: June 2010

C. L. Hagopian [3] has shown that atriodic, homogeneous, nondegenerate continua are one-dimensional. This answered a question of Maćkowiak and Tymchatyn [4]. We use a decomposition theorem to get a quick proof of this.
A fixed-point theorem for 𝑝-group actions
Authors: Jackowski Stefan
Published: June 2010

We prove Sullivan’s fixed-point conjecture for fixed-point-free actions of compact Lie groups which are extensions of a

p
p

-group by a torus. Moreover, we show that for a finite

p
p

-group

G
G

and a compact or finitely dimensional paracompact

G
G

-space

X
X

the fixed point set

X
G

{X^G}

is nonempty iff the induced homomorphism of zero-dimensional stable cohomotopy groups

π
0

(

B
G

)

→

π
0

(

E
G

×
G

X

)

{\pi ^0}\left ( {BG} \right ) \to {\pi ^0}\left ( {EG{ \times _G}X} \right )

is injective.
Some amalgam structures for Bianchi groups
Authors: Frohman Charles , Fine Benjamin
Published: June 2010

Splittings as an amalgamated sum, where one of the factors is the projective elementary matrices, are displayed for all the Bianchi groups except for the Euclidian ones. Also splittings as an HNN group, where the base is

PS

L

2

(
Z
)

{\text {PS}}{{\text {L}}_2}(Z)

, are given for all the Bianchi groups except

PS

L

2

(

O
1

)

{\text {PS}}{{\text {L}}_2}({O_1})

and

PS

L

2

(

O
3

)

{\text {PS}}{{\text {L}}_2}({O_3})

.
Path-lifting for Grothendieck toposes
Authors: Moerdijk Ieke
Published: June 2010

A general path-lifting theorem, which fails in the context of topological spaces, is shown to hold for toposes, for locales (a slight generalization of topological spaces), and hence for complete separable metric spaces. This result generalizes the known fact that any connected locally connected topos (respectively complete separable metric space) is path-connected.
Irrationality of limits of quickly convergent algebraic numbers sequences
Authors: Nabutovsky A. V.
Published: June 2010

We present criteria for irrationality of limits of convergent sequences of rational numbers, algebraic numbers of the same degree and of strictly increasing degrees. The criterion for irrationality of limits of a sequence of rational numbers has the form of an infinite system of inequalities on successive differences between elements. These inequalities are not strict. If these inequalities are satisfied, then the limit is rational if and only if all inequalities but a finite set of them are satisfied as equalities and the sequence becomes monotonous beginning from some element. So, the criterion permits to see a "border" between rationality and irrationality for some class of quickly convergent sequences.
Almost commuting matrices need not be nearly commuting
Authors: Choi Man Duen
Published: June 2010

Let

M

n

{\mathcal {M}_n}

be the collection of

n
×
n

n \times n

complex matrices with the Hilbert-space-operator norm. There exist two concrete matrices

A
,
B
∈

M

n

A,B \in {\mathcal {M}_n}

with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper A StartAbsoluteValue EndAbsoluteValue equals 1 minus 1 slash n comma StartAbsoluteValue EndAbsoluteValue upper B StartAbsoluteValue EndAbsoluteValue less-than-or-equal-to 1 comma StartAbsoluteValue EndAbsoluteValue upper A upper B minus upper B upper A StartAbsoluteValue EndAbsoluteValue less-than-or-equal-to 2 slash n">

|

|

A

|

|

=
1
−
1

/

n
,

|

|

B

|

|

≤
1
,

|

|

A
B
−
B
A

|

|

≤
2

/

n

||A|| = 1 - 1/n,||B|| \leq 1,||AB - BA|| \leq 2/n

, but

|

|

A
−
R

|

|

+

|

|

B
−
S

|

|

≥
1
−
1

/

n

||A - R|| + ||B - S|| \geq 1 - 1/n

for all commuting pairs

R
,
S
∈

M

n

R,S \in {\mathcal {M}_n}

. It is shown explicitly that there is a natural obstruction which prevents almost commuting matrices to get close to any commuting pairs.
A generalization to multifunctions of Fan’s best approximation theorem
Authors: Sehgal V. M. , Singh S. P.
Published: June 2010

We prove a theorem for set valued mappings in an approximatively compact, convex subset of a locally convex space, and then derive results due to Ky Fan and S. Reich as corollaries.
Factorization in operator algebras
Authors: Solel Baruch
Published: June 2010

Let

M
⊆
B
(
H
)

M \subseteq B(H)

be a

σ
\sigma

-finite von Neumann algebra and let

L

\mathcal {L}

be a lattice of projections associated with a unitary representation

{

W
t

}

\left \{ {{W_t}} \right \}

of a compact group with an ordered dual. (

L

\mathcal {L}

is not necessarily contained in

M
M

.) Assume that

M
M

is invariant under ad

W
t

{W_t}

for every

t
t

. Then, whenever

T
T

is an invertible operator in

M
M

that can be factored as

T
=
U
A

T = UA

where

U
U

is a unitary operator in

B
(
H
)

B(H)

and

A
A

lies in

alg
⁡

L

∩

(
alg
⁡

L

)

−
1

\operatorname {alg} \mathcal {L} \cap {(\operatorname {alg} \mathcal {L}{\text {)}}^{ - 1}}

, then

U
U

and

A
A

can be chosen in

M
M

. As corollaries we derive results about factorization with respect to CSL subalgebras and analytic subalgebras of von Neumann algebras.
Convergence and integrability of double trigonometric series with coefficients of bounded variation
Authors: Móricz Ferenc
Published: June 2010

We prove that if

c

(

j
,
k

)

→
0

c\left ( {j,k} \right ) \to 0

as

max

(

|

j

|

,

|

k

|

)

→
∞

\max \left ( {|j|,|k|} \right ) \to \infty

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript j equals negative normal infinity Overscript normal infinity Endscripts sigma-summation Underscript k equals negative normal infinity Overscript normal infinity Endscripts StartAbsoluteValue normal upper Delta 11 c left-parenthesis j comma k right-parenthesis EndAbsoluteValue greater-than normal infinity comma">

∑

j
=
−
∞

∞

∑

k
=
−
∞

∞

|

Δ

11

c

(

j
,
k

)

|

>
∞
,

\sum \limits _{j = - \infty }^\infty {\sum \limits _{k = - \infty }^\infty {\left | {{\Delta _{11}}c\left ( {j,k} \right )} \right | > \infty ,} }

\]
then the series
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript j equals negative normal infinity Overscript normal infinity Endscripts sigma-summation Underscript k equals negative normal infinity Overscript normal infinity Endscripts c left-parenthesis j comma k right-parenthesis e Superscript i left-parenthesis j x plus k y right-parenthesis">

∑

j
=
−
∞

∞

∑

k
=
−
∞

∞

c

(

j
,
k

)

e

i
(
j
x
+
k
y
)

\sum \nolimits _{j = - \infty }^\infty {\sum \nolimits _{k = - \infty }^\infty {c\left ( {j,k} \right ){e^{i(jx + ky)}}} }

converges both pointwise for every

(

x
,
y

)

∈

(

T
∖

{
0
}

)

2

\left ( {x,y} \right ) \in {\left ( {T\backslash \left \{ 0 \right \}} \right )^2}

and in the

L
p

(

T
2

)

{L^p}\left ( {{T^2}} \right )

-metric for

0
>
p
>
1

0 > p > 1

, where

T
T

is the one-dimensional torus. Both convergence statements remain valid for the three conjugate series under these same coefficient conditions.
Extremal lengths on Denjoy domains
Authors: Penner R. C.
Published: June 2010

We consider the problem of computing the extremal lengths of certain homotopy classes of curves in certain symmetric surfaces. Specifically, we concentrate on plane domains which are conformal to the Riemann sphere with a collection of slits in the real axis removed; such a conformal type is called a Denjoy domain. Using Jenkins-Strebel forms, the extremal length of any sufficiently symmetric homotopy class of curves is computed in terms of the endpoints of the slits. One can then choose a symmetric pants decomposition of the surface and invert the formulas derived, which are a set of coupled quadratic equations. In this way, one obtains a coordinatization of the space of all marked Denjoy domains of a fixed topological type.
A Krasnosel′skiĭ-type theorem involving 𝑝-arcs
Authors: Chan Jean B.
Published: June 2010

Let

p
p

be a point in

E
2

{E_2}

. A convex arc joining a pair of distinct points

x
x

and

y
y

in

E
2

{E_2}

is called a

p
p

-arc if it is contained in the simplex with vertices

x
,
y

x,y

, and

p
p

. In this paper, we prove the following Krasnosel’skii-type theorem: Let

S
S

be a compact simply connected set in

E
2

{E_2}

and let

p
p

be a point not in

S
S

. If for each three points

x

1
,

x
2

{x_{1,}}{x_2}

, and

x
3

{x_3}

of

S
S

there exists at least one point

y
∈
S

y \in S

such that

y
y

and

x
i

(

i
=
1
,
2
,
3

)

{x_i}\left ( {i = 1,2,3} \right )

can be joined by

p
p

-arcs in

S
S

, then there exists a point

k
∈
S

k \in S

such that every point

x
∈
S

x \in S

can be joined to

k
k

by some

p
p

-arc in

S
S

.
Maximal ideals in the Burnside ring of a compact Lie group
Authors: Bauer Stefan , May J. P.
Published: June 2010

A new criterion is found for deciding whether or not two maximal ideals in the Burnside ring of a compact Lie group coincide. One consequence is that certain algebraic and topological localizations in equivariant stable homotopy theory are naturally isomorphic.
Kernels of trace class operators
Authors: Brislawn Chris
Published: June 2010

Let

X
⊂

R

n

X \subset {{\mathbf {R}}^n}

and let

K
K

be a trace class operator on

L
2

(
X
)

{L^2}(X)

with corresponding kernel

K
(
x
,
y
)
∈

L
2

(
X
×
X
)

K(x,y) \in {L^2}(X \times X)

. An integral formula for tr

K
K

, proven by Duflo for continuous kernels, is generalized for arbitrary trace class kernels. This formula is shown to be equivalent to one involving the factorization of

K
K

into a product of Hilbert-Schmidt operators. The formula and its derivation yield two new necessary conditions for traceability of a Hilbert-Schmidt kernel, and these conditions are also shown to be sufficient for positive operators. The proofs make use of the boundedness of the Hardy-Littlewood maximal function on

L
2

(

R

n

)

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

.
Jordan derivations on semiprime rings
Authors: Brešar M.
Published: June 2010

I. N. Herstein has proved that any Jordan derivation on a

2
2

-torsion free prime ring is a derivation. In this paper we prove that Herstein’s result is true in

2
2

-torsion free semiprime rings. This result makes it possible for us to prove that any linear Jordan derivation on a semisimple Banach algebra is continuous, which gives an affirmative answer to the question posed by A. M. Sinclair in [5].
On a theorem of Hardy and Littlewood
Authors: Bernal Luis G.
Published: June 2010

In this paper, we give an extension of a classical theorem of Hardy and Littlewood on power series. Let

φ
\varphi

be a strictly positive function defined on some interval

(

δ
,
1

)

\left ( {\delta ,1} \right )

, satisfying a certain condition of limit. We prove that if

f

(
x
)

f\left ( x \right )

is the sum of a convergent power series for

0
>
x
>
1

0 > x > 1

with nonnegative coefficients

a
n

{a_n}

and

f

(
x
)

∼
φ

(
x
)

(

x
→
1

)

f\left ( x \right ) \sim \varphi \left ( x \right )\;\left ( {x \to 1} \right )

, then

S
n

∼
α
⋅
φ

(

x
0

1

/

n

)

(

n
→
∞

)

{S_n} \sim \alpha \cdot \varphi \left ( {x_0^{1/n}} \right )\left ( {n \to \infty } \right )

, where

S
n

=

a
0

+

a
1

+
⋯
+

a

n
,

x
0

∈

(

0
,
1

)

{S_n} = {a_0} + {a_1} + \cdots + {a_{n,\;}}{x_0} \in \left ( {0,1} \right )

and

α
\alpha

depends only upon

φ
\varphi

.
The existence of periodic solutions to nonautonomous differential inclusions
Authors: Macki Jack W. , Nistri Paolo , Zecca Pietro
Published: June 2010

For anmm-dimensional differential inclusion of the form\[x˙∈A(t)x(t)+F[t,x(t)],\dot x \in A(t)x(t) + F[t,x(t)],\]withAAandFFTT-periodic intt, we prove the existence of a nonconstant periodic solution. Our hypotheses requiremmto be odd, and requireFFto have different growth behavior for|x|\left | x \right |small and|x|\left | x \right |large (often the case in applications). The idea is to guarantee that the topological degree associated with the system has different values on two distinct neighborhoods of the origin.
Polynomial-rational bijections of 𝑅ⁿ
Authors: Kurdyka Krzysztof , Rusek Kamil
Published: June 2010

It is shown in this note that every invertible polynomial transformation of

R

n

{{\mathbf {R}}^n}

of degree two has a rational inverse defined on the whole space

R

n

{{\mathbf {R}}^n}

. The same is true for polynomial transformations of higher degrees, satisfying some differential condition which is a real analogue of Jagžev’s condition considered in [3, 4, and 6]. The proofs of these statements are based on the Bialynicki-Birula and Rosenlicht surjectivity theorem [2] and on standard properties of complex dominant polynomial mappings.
Decomposition of normal currents
Authors: Zworski Maciej
Published: June 2010

As an answer to a question of Frank Morgan, it is shown that there exist normal currents which cannot be represented as convex integrals of rectifiable currents. However, under certain additional hypotheses, such decompositions exist. Examples are given to indicate that such hypotheses are necessary.