bluuebunny/crossref_metadata_2025 huggingface dataset:
On the values at negative half-integers of the Dedekind zeta function of a real quadratic field
Authors: Eie Min King
Published: June 2010

The zeta function

ζ
(
A
,
s
)

\zeta (A,s)

associated with a narrow ideal class

A
A

for a real quadratic field can be decomposed into

∑
Q

Z
Q

(
s
)

\sum \nolimits _Q {{Z_Q}(s)}

, where

Z
Q

(
s
)

{Z_Q}(s)

is a Dirichlet series associated with a quadratic form

Q
(
x
,
y
)
=
a

x
2

+
b
x
y
+
c

y
2

Q(x,y) = a{x^2} + bxy + c{y^2}

, and the summation is over finite reduced quadratic forms associated to the narrow ideal class

A
A

. The values of

Z
Q

(
s
)

{Z_Q}(s)

at nonpositive integers were obtained by Zagier [16] and Shintani [12] via different methods. In this paper, we shall obtain the values of

Z
Q

(
s
)

{Z_Q}(s)

at negative half-integers

s
=
−
1

/

2
,
−
3

/

2
,
…
,
−
m
+
1

/

2
,
…

s = - 1/2, - 3/2, \ldots , - m + 1/2, \ldots

. The values of

Z
Q

(
s
)

{Z_Q}(s)

at nonpositive integers were also obtained by our method, and our results are consistent with those given in [16].
There can be 𝐶*-embedded dense proper subspaces in 𝛽𝜔-𝜔
Authors: van Douwen Eric K. , Kunen Kenneth , van Mill Jan
Published: June 2010

Fine and Gillman have shown that CH implies that if

X
X

is a dense proper subspace of

ω
∗

=
β
ω
−
ω

{\omega ^ * } = \beta \omega - \omega

, then

β
X
≠

ω
∗

\beta X \ne {\omega ^ * }

. Here it is shown to be consistent with

MA + c = 

ω
2

{\text {MA + c = }}{\omega _2}

that for every

p
∈

ω
∗

p \in {\omega ^ * }

we have

β
(

ω
∗

−
{
p
}
)
=

ω
∗

\beta ({\omega ^ * } - \{ p\} ) = {\omega ^ * }

and also that

ω
∗

{\omega ^ * }

has a dense subspace

X
X

with dense complement such that

β
X
=

ω
∗

\beta X = {\omega ^ * }

.
Generalized unknotting operations and tangle decompositions
Authors: Kobayashi Tsuyoshi
Published: June 2010

Suppose that a knot

K
L

{K_L}

in

S
3

{S^3}

is obtained from a knot

K
K

by an

n
n

-parallel (

n
n

-antiparallel resp.) crossing change. Let

T
T

be an incompressible,

∂
\partial

-incompressible surface properly embedded in

S
3

−

N
˙

(
K
)

{S^3} - \dot N(K)

with

∂
T

\partial T

a union of meridian loops and

χ
(
T
)
>
p
(
1
−
2
n
)

\chi (T) > p(1 - 2n)

, for some

p
p

. We show that either

T
T

is isotoped to intersect

L
L

in

≤
2
p
−
2

\leq 2p - 2

points, or there is a minimal genus Seifert surface for

K
L

{K_L}

intersecting the corresponding crossing link in two (

≤
2

\leq 2

resp.) points.
Tutte polynomials and bicycle dimension of ternary matroids
Authors: Jaeger François
Published: June 2010

Let

M
M

be a ternary matroid,

t

(

M
,
x
,
y

)

t\left ( {M,x,y} \right )

be its Tutte polynomial and

d

(
M
)

d\left ( M \right )

be the dimension of the bicycle space of any representation of

M
M

over

GF

(
3
)

{\text {GF}}\left ( 3 \right )

. We show that, for

j
=

e

2
i
π

/

3

j = {e^{2i\pi /3}}

, the modulus of the complex number

t

(

M
,
j
,

j
2

)

t\left ( {M,j,{j^2}} \right )

is equal to

(

3

)

d

(
M
)

{\left ( {\sqrt 3 } \right )^{d\left ( M \right )}}

. The proof relies on the study of the weight enumerator

W

C

(
y
)

{W_\mathcal {C}}\left ( y \right )

of the cycle space

C

\mathcal {C}

of a representation of

M
M

over

GF

(
3
)

{\text {GF}}\left ( 3 \right )

evaluated at

y
=
j

y = j

. The main tool is the concept of principal quadripartition of

C

\mathcal {C}

which allows a precise analysis of the evolution of the relevant invariants under deletion and contraction of elements. Soit

M
M

un matroïde ternaire,

t

(

M
,
x
,
y

)

t\left ( {M,x,y} \right )

son polynôme de Tutte et

d

(
M
)

d\left ( M \right )

la dimension de l’espace des bicycles d’une représentation quelconque de

M
M

sur

GF

(
3
)

{\text {GF}}\left ( 3 \right )

. Nous montrons que, pour

j
=

e

2
i
π

/

3

j = {e^{2i\pi /3}}

, le module du nombre complexe

t

(

M
,
j
,

j
2

)

t\left ( {M,j,{j^2}} \right )

est égal à

(

3

)

d

(
M
)

{\left ( {\sqrt 3 } \right )^{d\left ( M \right )}}

. La preuve s’appuie sur l’étude de l’énumérateur de poids

W

C

(
y
)

{W_\mathcal {C}}\left ( y \right )

de l’espace des cycles

C

\mathcal {C}

d’une représentation de

M
M

sur

GF

(
3
)

{\text {GF}}\left ( 3 \right )

pour la valeur

y
=
j

y = j

. L’outil essentiel est le concept de quadripartition principale de

C

\mathcal {C}

qui permet une analyse précise de l’évolution des invariants concernés relativement à la suppression ou contraction d’éléments.
On composition of four-symbol 𝛿-codes and Hadamard matrices
Authors: Yang C. H.
Published: June 2010

It is shown that key instruments for composition of four-symbol

δ
\delta

-codes are the Lagrange identity for polynomials, a certain type of quasisymmetric sequences (i.e., a set of normal or near normal sequences) and base sequences. The following is proved: If a set of base sequences for length

t
t

and a set of normal (or near normal) sequences for length

n
n

exist then four-symbol

δ
\delta

-codes of length

(

2
n
+
1

)

t

(

or 

n
t

)

\left ( {2n + 1} \right )t\left ( {{\text {or }}nt} \right )

can be composed by application of the Lagrange identity. Consequently a new infinite family of Hadamard matrices of order

4
u
w

4uw

can be constructed, where

w
w

is the order of Williamson matrices and

u
=

(

2
n
+
1

)

t

(

or 

n
t

)

u = \left ( {2n + 1} \right )t\left ( {{\text {or }}nt} \right )

. Other related topics are also discussed.
The 𝑇1 theorem for martingales
Authors: Bennett Andrew G.
Published: June 2010

The

T
1

T1

theorem of David and Journé gives necessary and sufficient conditions that a singular integral operator be bounded from

L
2

(

R
n

)

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

to

L
2

(

R
n

)

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

. In this paper, the definition of singular integral operator is extended to the setting of operators on

L
2

(
Ω
)

{L^2}(\Omega )

where

Ω
\Omega

denotes Wiener space. The main theorem is that the

T
1

T1

theorem holds in this new setting.
Zero cycles on quadric hypersurfaces
Authors: Swan Richard G.
Published: June 2010

Let

X
X

be a projective quadric hypersurface over a field of characteristic not 2. It is shown that the Chow group

A
0

(
X
)

{A_0}(X)

of

0
0

-cycles modulo rational equivalence is infinite cyclic, generated by any point of minimal degree.
Two examples of local Artinian rings
Authors: Xue Wei Min
Published: June 2010

We answer a question of D. A. Hill in the negative by providing two local artinian rings

R
R

and

S
S

such that

R
R

is right serial but the left indecomposable injective

R
R

-module is not uniserial, and that

S
S

is not right serial but the left indecomposable injective

S
S

-module is uniserial. Moreover

R
R

possesses a Morita duality but fails to have self-duality.
Operator ranges and completely bounded homomorphisms
Authors: Mathes D. Benjamin
Published: June 2010

In this paper it is shown that the set of invariant operator ranges that induce completely bounded homomorphisms is a sublattice of

Lat

1

/

2

⁡

A

\operatorname {Lat}_{1/2}\mathfrak {A}

, when

A

\mathfrak {A}

is any norm closed algebra of operators on a Hilbert space. A characterization of this sublattice is given, and several concrete examples are discussed.
The fixed-point-space dimension function for a finite group representation
Authors: Isaacs I. Martin
Published: June 2010

Given a complex representation of a finite group

G
G

, construct the integer valued function

α
\alpha

on

G
G

by setting

α
(
g
)

\alpha (g)

to be the dimension of the fixed-point-space of

g
g

in the module corresponding to the given representation. Usually,

α
\alpha

is not a generalized character of

G
G

and for trivial reasons

|

G

|

α

|G|\alpha

is always a generalized character. The main result of this paper is that

e
α

e\alpha

is always a generalized character, where

e
e

is the exponent of

G
G

.
Selfadjointness of the *-representation generated by the sum of two positive linear functionals
Authors: Inoue Atsushi
Published: June 2010

Let

ϕ
\phi

and

ψ
\psi

be positive linear functionals on a

∗
*

-algebra

A

\mathcal {A}

. When the closed

∗
*

-representations

π
ϕ

{\pi _\phi }

and

π
ψ

{\pi _\psi }

of

A

\mathcal {A}

generated by the GNS-construction for

ϕ
\phi

and

ψ
\psi

are self-adjoint, we shall show that

π

ϕ
+
ψ

{\pi _{\phi + \psi }}

is self-adjoint if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript phi plus psi Baseline left-parenthesis script upper A right-parenthesis prime Subscript w Baseline script upper D left-parenthesis pi Subscript phi plus psi Baseline right-parenthesis subset-of script upper D left-parenthesis pi Subscript phi plus psi Baseline right-parenthesis">

π

ϕ
+
ψ

(

A

)

′

w

D

(

π

ϕ
+
ψ

)

⊂

D

(

π

ϕ
+
ψ

)

{\pi _{\phi + \psi }}{\left ( \mathcal {A} \right )’}_w\mathcal {D}\left ( {{\pi _{\phi + \psi }}} \right ) \subset \mathcal {D}\left ( {{\pi _{\phi + \psi }}} \right )

; and there exists a self-adjoint extension

ρ
\rho

of

π

ϕ
+
ψ

{\pi _{\phi + \psi }}

suchthat

ρ

(

A

)

′

w

=

π

ϕ
+
ψ

(

A

)

′

w

\rho {\left ( \mathcal {A} \right )’}_w = {\pi _{\phi + \psi }}{\left ( \mathcal {A} \right )’}_w

if and only if

π

ϕ
+
ψ

(

A

)

′

w

{\pi _{\phi + \psi }}{\left ( \mathcal {A} \right )’}_w

is a von Neumann algebra.
Formation of singularities in compressible fluids in two-space dimensions
Authors: Rammaha M. A.
Published: June 2010

Classical solutions to the two-dimensional Euler equations for a polytropic ideal fluid are considered. It is shown that any local

C
1

{C^1}

-flow, regardless of the size of the initial disturbance, will develop singularities in finite-time provided the front of the initial disturbance satisfies certain conditions.
Lattice-isotopic arrangements are topologically isomorphic
Authors: Randell Richard
Published: June 2010

We prove that arrangements which are connected through a smooth family with constant intersection lattice have the same topology.
On the zeros of 𝐿’+𝐿² for certain rational functions 𝐿
Authors: Sheil-Small T.
Published: June 2010

Let

L
L

be a nonconstant rational function whose poles are real, simple with each one having a positive residue. Then, if

L
′

+

L
2

L’ + {L^2}

has no nonreal zeros,

L
L

has the form
\[

L
(
z
)
=

∑

k
=
1

n

α
k

z
−

x
k

−
a
z
+
b
,

L(z) = \sum \limits _{k = 1}^n {\frac {{{\alpha _k}}}{{z - {x_k}}} - az + b,}

\]

x
k

{x_k}

are real,

α
k

>
0

{\alpha _k} > 0

for

1
≤
k
≤
n
,
a
≥
0

1 \leq k \leq n,a \geq 0

and

b
b

is real. In particular, if

P
P

is a polynomial of degree

≥
2

\geq 2

, then

P
′

+

P
2

P’ + {P^2}

has nonreal zeros. The result is applied to entire functions in connection with zeros of the derivatives.
Selfadjointness of the momentum operator with a singular term
Authors: Watanabe Michiaki , Watanabe Shuji
Published: June 2010

Self-adjointness is shown of the momentum operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet u element-of upper H Superscript 1 Baseline left-parenthesis bold upper R Superscript 1 Baseline right-parenthesis colon u slash x element-of upper L squared left-parenthesis bold upper R Superscript 1 Baseline right-parenthesis EndSet">

{
u
∈

H
1

(

R

1

)
:
u

/

x
∈

L
2

(

R

1

)
}

\{ u \in {H^1}({{\mathbf {R}}^1}):u/x \in {L^2}({{\mathbf {R}}^1})\}

, with domain
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet u element-of upper H Superscript 1 Baseline left-parenthesis bold upper R Superscript 1 Baseline right-parenthesis colon u slash x element-of upper L squared left-parenthesis bold upper R Superscript 1 Baseline right-parenthesis EndSet">

{

u
∈

H
1

(

R

1

)
:
u

/

x
∈

L
2

(

R

1

)

}

\left \{ {u \in {H^1}({{\mathbf {R}}^1}):u/x \in {L^2}({{\mathbf {R}}^1})} \right \}

when

c
>
1

c > 1

or

c
>
−
1

c > - 1

. This operator appears in a harmonic oscillator system with the generalized commutation relations by Wigner:

i
p
=
[
x
,
H
]

ip = [x,H]

and

−
i
x
=
[
p
,
H
]

- ix = [p,H]

for the Hamiltonian

H
H

and the multiplication operator

x
x

. The proof is carried out by generation of a unitary group in terms of ip, based on the Hille-Yosida theorem and Stone’s theorem. The result is applied to the self-adjoitness of

H
=
(

p
2

+

x
2

)

/

2

H = ({p^2} + {x^2})/2

.
Vector-valued Hausdorff summability methods and ergodic theorems
Authors: Yoshimoto Takeshi
Published: June 2010

Suppose

X
X

and

Y
Y

are two general Banach spaces. Let

H
=
(

Λ

n
,
k

)

H = ({\Lambda _{n,k}})

be a general

B

[
X
,
Y
]

{\mathbf {B}}[X,Y]

-operator valued Hausdorff summability method:

Λ

n
,
k

=

(
k
n

)

Δ

n
−
k

U
k

{\Lambda _{n,k}} = (_k^n){\Delta ^{n - k}}{U_k}

for

k
≤
n

k \leq n

and

Λ

n
,
k

=

θ

X
,
Y

{\Lambda _{n,k}} = {\theta _{X,Y}}

for

k
>
n

k > n

, where

{

U
k

}

k
=
0

∞

\{ {U_k}\} _{k = 0}^\infty

is a sequence of operators in

B

[
X
,
Y
]

{\mathbf {B}}[X,Y]

and

Δ
\Delta

denotes the backward difference (operator) and

θ

X
,
Y

(
x
)
=

0
Y

{\theta _{X,Y}}(x) = {0_Y}

(the zero element in

Y
Y

) for all

x
∈

x \in

. Then some necessary and sufficient conditions are given for the mean and uniform convergence of the averages
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript k equals 0 Overscript n Endscripts left-parenthesis Subscript k Baseline Superscript n Baseline right-parenthesis normal upper Delta Superscript n minus k Baseline upper U Subscript k Baseline left-parenthesis upper T Superscript k Baseline x right-parenthesis left-parenthesis x element-of upper X comma upper T element-of bold upper B left-bracket upper X right-bracket right-parenthesis period">

∑

k
=
0

n

(
k
n

)

Δ

n
−
k

U
k

(

T
k

x
)

(
x
∈
X
,
T
∈

B

[
X
]
)
.

\sum \limits _{k = 0}^n {(_k^n){\Delta ^{n - k}}{U_k}({T^k}x)} \quad (x \in X,T \in {\mathbf {B}}[X]).

\]
The existence of almost translation invariant ultrafilters on any semigroup
Authors: Papazyan Talin
Published: June 2010

We present a short ultrafilter proof of a result which has applications in combinatorial number theory and which has previously relied on the theory of compact semigroups.
Cohomological local connectedness of decomposition spaces
Authors: Dydak Jerzy , Walsh John J.
Published: June 2010

For a map

f
:
X
→
Y

f:X \to Y

, let

H

k

[
f
]

{\mathcal {H}^k}\left [ f \right ]

denote the associated

k
k

-dimensional cohomology sheaf. The main result is that, for a proper map between locally compact metrizable spaces, if the sheaves

H

k

[
f
]

{\mathcal {H}^k}\left [ f \right ]

are locally constant and

X
X

is cohomologically locally connected, then

Y
Y

is cohomologically locally connected. The result can be viewed as a variation on a number of similar results dating to work of Vietoris. The setting for this paper is quite general and the proof is not difficult, involving a routine analysis using the Leray-Grothendieck spectral sequence. Versions of known comparable results for homotopical local connectedness can be recovered by combining the result with standard universal coefficient theorems that translate cohomological information to homological information and with a local Hurewicz theorem.
Irreducible representations of normal spaces
Authors: Rubin Leonard R.
Published: June 2010

We define the notion of irreducible polyhedral representation of a normal space making use of approximate inverse systems. This generalizes the concept of irreducible polyhedral expansions introduced in 1937 by Freudenthal for metric compacta and generalized to uniform spaces by Isbell in 1961. We show that every normal space

X
X

has an irreducible polyhedral representation whose dimension is

X
X

and whose weight is weight

(
X
)

(X)

. Approximate inverse systems were first introduced by S. Mardešić and this author. The concept generalizes that of inverse system and was essentially used in proving that each Hausdorff compactum of integral cohomological dimension

≤
n

\leq n

is the cell-like image of a Hausdorff compactum of covering dimension

≤
n

\leq n

.
A weak-star rational approximation problem connected with subnormal operators
Authors: Dudziak James
Published: June 2010

Let

μ
\mu

be a positive Borel measure on a compact subset

K
K

of the complex plane. Denote the weak-star closure in

L
∞

(
μ
)

{L^\infty }\left ( \mu \right )

of

R

(
K
)

R\left ( K \right )

by

R
∞

(

K
,
μ

)

{R^\infty }\left ( {K,\mu } \right )

. Given

f
∈

R
∞

(

K
,
μ

)

f \in {R^\infty }\left ( {K,\mu } \right )

, denote the weak-star closure in

L
∞

(
μ
)

{L^\infty }\left ( \mu \right )

of the algebra generated by

R
∞

(

K
,
μ

)

{R^\infty }\left ( {K,\mu } \right )

and the complex conjugate of

f
f

by

A
∞

(

f
,
μ

)

{A^\infty }\left ( {f,\mu } \right )

. This paper determines the structure of

A
∞

(

f
,
μ

)

{A^\infty }\left ( {f,\mu } \right )

. As a consequence, a solution is obtained to a problem concerned with minimal normal extensions of functions of a subnormal operator.
On barely 𝛼-compact spaces and remote points in 𝛽_{𝛼}𝑋_{𝑋}
Authors: Blair Robert L. , Polkowski Lech T. , Swardson Mary Anne
Published: June 2010

Let

X
X

be a Tychonoff space. It was proved by Terada that if the cellularity of

X
X

is not Ulam-measurable, then no point of

υ
X
∖
X

\upsilon X\backslash X

is a remote point of

X
X

. In this paper we generalize this result by proving that if

X
X

is barely

α
\alpha

-compact, then no point in

β
α

X
∖
X

{\beta _\alpha }X\backslash X

is an

α
\alpha

-exotic point of

X
X

. This implies, in particular, that if no cellular family in

X
X

has a

α
\alpha

-measurable cardinality, then no point of

β
α

X
∖
X

{\beta _\alpha }X\backslash X

is a remote point of

X
X

; the Terada theorem then follows as a corollary when

α
=

ω
1

\alpha = {\omega _1}

.
Examples of Poincaré duality groups
Authors: Mess Geoffrey
Published: June 2010

A closed aspherical manifold may have a fundamental group which is not residually finite and contains infinitely divisible elements.
Second order estimates in interpolation theory and applications
Authors: Milman Mario , Schonbek Tomas
Published: June 2010

We provide a new approach to the commutator theorems of Rochberg and Weiss [10], Jawerth, Rochberg, and Weiss [6], Cwikel, Jawerth, and Milman [4]. In particular our methods apply to non-linear operators of a type commonly encountered in classical harmonic analysis (maximal operators, square functions). As an application we provide an extension of a commutator inequality of Chanillo [3] for maximal operators of fractional integrals.
A generalized van Kampen-Flores theorem
Authors: Sarkaria K. S.
Published: June 2010

The

n
n

-skeleton of a

(
2
n
+
2
)

(2n + 2)

-simplex does not embed in

R

2
n

{{\mathbf {R}}^{2n}}

. This well-known result is due (independently) to van Kampen, 1932, and Flores, 1933, who proved the case

p
=
2

p = 2

of the following: Theorem. Let

p
p

be a prime, and let

s
s

and

l
l

be positive integers such that

l
(
p
−
1
)
≤
p
(
s
−
1
)

l(p - 1) \leq p(s - 1)

. Then, for any continuous map

f
f

from a

(
p
s
+
p
−
2
)

(ps + p - 2)

-dimensional simplex into

R

l

{{\mathbf {R}}^l}

, there must exist

p
p

points

{

x
1

,
…
,

x
p

}

\{ {x_1}, \ldots ,{x_p}\}

, lying in pairwise disjoint faces of dimensions

≤
s
−
1

\leq s - 1

of this simplex, such that

f
(

x
1

)
=
⋯
=
f
(

x
p

)

f({x_1}) = \cdots = f({x_p})

.
A comparison theorem for selfadjoint operators
Authors: Boumenir Amin
Published: June 2010

In this work we shall establish a result concerning the spectral theory of differential operators. Let

L
1

{L_1}

and

L
2

{L_2}

be two self-adjoint operators acting in two different Hubert spaces. Then under some conditions we shall prove that
\[

(
d

Γ
1

/

d

Γ
2

)
(

L
2

)
=

V
¯

V
′

,

(d{\Gamma _1}/d{\Gamma _2})({L_2}) = \overline V V’,

\]
where

Γ
1

(
λ
)

{\Gamma _1}(\lambda )

and

Γ
2

(
λ
)

{\Gamma _2}(\lambda )

are the spectral functions associated with

L
1

{L_1}

and

L
2

{L_2}

respectively.

V
V

is the shift operator mapping the set of generalized eigenfunctions of

L
1

{L_1}

into the set of generalized eigenfunctions of

L
2

{L_2}

, that is
\[

y
=
V
φ
,

y = V\varphi ,

\]
where

L
2

y
=
λ
y

{L_2}y = \lambda y

and

L
1

φ
=
λ
φ

{L_1}\varphi = \lambda \varphi

.
Local spectrum and generalized spectrum
Authors: Mbekhta Mostafa
Published: June 2010

This paper provides the proofs of those results announced in

[
5
,
§
5
]

[5,\S 5]

that deal with the connection between the regular set and the local resolvant set of closed operators on a Hilbert space. We also give some characterizations and properties of Cowen-Douglas operators.
Blow-up of 𝐻¹ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity
Authors: Ogawa Takayoshi , Tsutsumi Yoshio
Published: June 2010

We investigate the blow-up of solutions in

H
1

(

R

)

{H^1}({\mathbf {R}})

with negative energy for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row i u Subscript t Baseline equals minus u Subscript x x Baseline minus StartAbsoluteValue u EndAbsoluteValue Superscript 4 Baseline u comma t greater-than 0 comma x element-of bold upper R comma 2nd Row u left-parenthesis 0 comma x right-parenthesis equals u 0 left-parenthesis x right-parenthesis comma x element-of bold upper R period EndLayout period">

i

u
t

=
−

u

x
x

−

|

u

|

4

u
,

t
>
0
,
x
∈

R

,

u
(
0
,
x
)
=

u
0

(
x
)
,

x
∈

R

.

.

\begin {gathered} i{u_t} = - {u_{xx}} - |u{|^4}u,\quad t > 0,x \in {\mathbf {R}}, \hfill \\ u(0,x) = {u_0}(x),\quad x \in {\mathbf {R}}. \hfill \\ \end {gathered} .

\]
In our result we remove the weight condition

x

u
0

∈

L
2

(

R

)

x{u_0} \in {L^2}({\mathbf {R}})

, which was always assumed to show the blow-up of solutions in the previous papers.
A further generalization of the Knaster-Kuratowski-Mazurkiewicz theorem
Authors: Shioji Naoki
Published: June 2010

Granas and Dugundji obtained the following generalization of the Knaster-Kuratowski-Mazurkiewicz theorem. Let

X
X

be a subset of a topological vector space

E
E

and let

G
G

be a set-valued map from

X
X

into

E
E

such that for each finite subset

{

x
1

,
…
,

x
n

}

\{ {x_1}, \ldots ,{x_n}\}

of

X
,
c
o
{

x
1

,
…
,

x
n

}
⊂

∪

i
=
1

n

G

x
i

X,co\{ {x_1}, \ldots ,{x_n}\} \subset \cup _{i = 1}^nG{x_i}

and for each

x
∈
X
,
G
x

x \in X,Gx

is finitely closed, i.e., for any finite-dimensional subspace

L
L

of

E
,
G
x
∩
L

E,Gx \cap L

is closed in the Euclidean topology of

L
L

. Then

{
G
x
:
x
∈
X
}

\{ Gx:x \in X\}

has the finite intersection property. By relaxing, among others, the condition that

X
X

is a subset of

E
E

, we obtain a further generalization of the theorem and show some of its applications.
Vietoris-Begle theorem and spectra
Authors: Dydak Jerzy , Kozlowski George
Published: June 2010

The following generalization of the Vietoris-Begle Theorem is proved: Suppose

{

E
k

}

k
≥
1

{\left \{ {{E_k}} \right \}_{k \geq 1}}

is a CW spectrum and

f
:

X
′

→
X

f:X’ \to X

is a closed surjective map of paracompact Hausdorff spaces such that

Ind
⁡
X
=
m
>
∞

\operatorname {Ind} X = m > \infty

. If
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript asterisk Baseline colon upper E Superscript k Baseline left-parenthesis x right-parenthesis right-arrow upper E Superscript k Baseline left-parenthesis f Superscript negative 1 Baseline left-parenthesis x right-parenthesis right-parenthesis">

f
∗

:

E
k

(
x
)
→

E
k

(

f

−
1

(
x
)
)

{f^*}:{E^k}(x) \to {E^k}({f^{ - 1}}(x))

is an isomorphism for all

x
∈
X

x \in X

and

k
=

m
0

,

m
0

+
1
,
…
,

m
0

+
m

k = {m_0},{m_0} + 1, \ldots ,{m_0} + m

, then

f
∗

:

E
n

(
X
)
→

E
n

(

X
′

)

{f^*}:{E^n}(X) \to {E^n}(X’)

is an isomorphism and

f
∗

:

E

n
+
1

(
X
)
→

E

n
+
1

(

X
′

)

{f^*}:{E^{n + 1}}(X) \to {E^{n + 1}}(X’)

is a monomorphism for

n
=

m
0

+
m

n = {m_0} + m

.
A hypercyclic operator whose adjoint is also hypercyclic
Authors: Salas Héctor
Published: June 2010

An operator

T
T

acting on a Hilbert space

H
H

is hypercyclic if, for some vector

x
x

in

H
H

, the orbit

{

T
n

x
:
n
≥
0
}

\{ {T^n}x:n \geq 0\}

is dense in

H
H

. We show the existence of a hypercyclic operator—in fact, a bilateral weighted shift—whose adjoint is also hypercyclic. This answers positively a question of D. A. Herrero.
Mean growth of Bloch functions and Makarov’s law of the iterated logarithm
Authors: Bañuelos Rodrigo , Moore Charles N.
Published: June 2010

The authors construct an example of a Bloch function on the unit disc whose circular

L
2

{L^2}

means grow at the maximal possible rate but which has no lower bound in the law of the iterated logarithm for Bloch functions. This answers a question of Przytycki [4, p. 154] and Makarov [3, p. 42].
Bounds on least dilatations
Authors: Penner R. C.
Published: June 2010

We consider the collection of all pseudo-Anosov homeomorphisms on a surface of fixed topological type. To each such homeomorphism is associated a real-valued invariant, called its dilatation (which is greater than one), and we define the spectrum of the surface to be the collection of logarithms of dilatations of pseudo-Anosov maps supported on the surface. The spectrum is a natural object of study from the topological, geometric, and dynamical points of view. We are concerned in this paper with the least element of the spectrum, and explicit upper and lower bounds on this least element are derived in terms of the topological type of the surface; train tracks are the main tool used in establishing our estimates.
Markov partitions for the two-dimensional torus
Authors: Snavely Mark R.
Published: June 2010

We examine Markov partitions for hyperbolic automorphisms ofT2{\mathbb {T}^2}in the spirit of Adler, Weiss, and others and give necessary conditions on the transition matrix of a Markov partition for a given automorphism. We give necessary and sufficient conditions for partitions with two connected rectangles.
On the duals of Lebesgue-Bochner 𝐿^{𝑝} spaces
Authors: Cengiz Bahattin
Published: June 2010

Let

(
X
,

A

,
μ
)

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

be an arbitrary positive measure space. We prove that there exist an extremally disconnected (locally) compact Hausdorff space

Y
Y

and a perfect (regular) Borel measure

ν
\nu

on

Y
Y

such that

L
p

(
μ
,

E

)
≃

L
p

(
ν
,
E
)

{L^p}(\mu ,\textrm {E}) \simeq {L^p}(\nu ,E)

for all

1
≤
p
>
∞

1 \leq p > \infty

and any Banach space

E
E

. If

E
∗

{E^*}

is separable, then

L
p

(
μ
,

E

)
∗
≃

L
q

(
μ
,

E

∗

)

{L^p}(\mu ,\textrm {E})* \simeq {L^q}(\mu ,{\textrm {E}^*})

for all

1
>
p
>
∞
,

1
p

+

1
q

=
1

1 > p > \infty ,\;\frac {1}{p} + \frac {1}{q} = 1

, and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1 Baseline left-parenthesis mu comma upper E right-parenthesis asterisk asymptotically-equals upper L Superscript normal infinity Baseline left-parenthesis nu comma upper E Superscript asterisk Baseline right-parenthesis asymptotically-equals upper C left-parenthesis beta upper Y comma upper E Subscript asterisk Superscript asterisk Baseline right-parenthesis">

L
1

(
μ
,

E

)
∗
≃

L
∞

(
ν
,

E

∗

)
≃
C
(
β
Y
,

E

∗
∗

)

{L^1}(\mu ,\textrm {E})* \simeq {L^\infty }(\nu ,{\textrm {E}^*}) \simeq C(\beta Y,\textrm {E}_*^*)

, where

E
∗
∗

E_*^*

denotes

E
∗

{E^*}

endowed with the weak* topology. In particular

L
1

(
μ

)
∗

≃

L
∞

(
ν
)

{L^1}{(\mu )^*} \simeq {L^\infty }(\nu )

.
On derivations in prime rings and Banach algebras
Authors: Vukman J.
Published: June 2010

Let

R
R

be a ring with center

Z
(
R
)

Z(R)

. A mapping

F
:
R
→
R

F:R \to R

is said to be centralizing on

R
R

if

[
F
(
x
)
,
x
]
∈
Z
(
R
)

[F(x),x] \in Z(R)

holds for all

x
∈
R

x \in R

. The main purpose of this paper is to prove the following result, which generalizes a classical result of Posner: Let

R
R

be a prime ring of characteristic not 2, 3, and 5. Suppose there exists a nonzero derivation

D
:
R
→
R

D:R \to R

, such that the mapping

x
↣
[
[
D
(
x
)
,
x
]
,
x
]

x \rightarrowtail [[D(x),x],x]

is centralizing on

R
R

. In this case

R
R

is commutative. Combining this result with some well-known deep results of Sinclair and Johnson, we generalize Yood’s noncommutative extension of the Singer-Wermer theorem.
Ruled submanifolds of finite type
Authors: Dillen Franki
Published: June 2010

We show that a ruled submanifold of finite type in a Euclidean space is a cylinder on a curve of finite type or a generalized helicoid.
The numerical radius of a nilpotent operator on a Hilbert space
Authors: Haagerup Uffe , de la Harpe Pierre
Published: June 2010

Let

T
T

be a bounded linear operator of norm 1 on a Hilbert space

H
H

such that

T
n

=
0

{T^n} = 0

for some

n
≥
2

n \geq 2

. Then its numerical radius satisfies

w

(
T
)

≤
cos
⁡

π

(

n
+
1

)

w\left ( T \right ) \leq \cos \frac {\pi }{{\left ( {n + 1} \right )}}

and this bound is sharp. Moreover, if there exists a unit vector

ξ
∈
H

\xi \in H

such that

|

⟨

T
ξ

|

ξ

⟩

|

=
cos
⁡

π

(

n
+
1

)

\left | {\left \langle {T\xi |\xi } \right \rangle } \right | = \cos \frac {\pi }{{\left ( {n + 1} \right )}}

, then

T
T

has a reducing subspace of dimension

n
n

on which

T
T

is the usual

n
n

-shift. The proofs show that these facts are related to the following result of Fejer: if a trigonometric polynomial

f

(
θ
)

=

∑

k
=
−
n
+
1

n
−
1

f
k

e

i
k
θ

f\left ( \theta \right ) = \sum \nolimits _{k = - n + 1}^{n - 1} {{f_k}{e^{ik\theta }}}

is positive, one has

|

f
1

|

≤

f
0

cos
⁡

π

(

n
+
1

)

|{f_1}| \leq {f_0}\cos \frac {\pi }{{\left ( {n + 1} \right )}}

; moroever, there is essentially one polynomial for which equality holds.
Bound on the extreme zeros of orthogonal polynomials
Authors: Ismail Mourand E. H. , Li Xin
Published: June 2010

Using chain sequences we formulate a procedure to find upper (lower) bounds for the largest (smallest) zero of orthogonal polynomials in terms of their recurrence coefficients. We also apply our method to derive bounds for extreme zeros of the Laguerre, associated Laguerre, Meixner, and Meixner-Pollaczek polynomials. In addition, we consider bounds for the extreme zeros of Jacobi polynomials of degree

n
n

and parameters

a
n

{a_n}

and

b
n

{b_n}

.
The Vitali-Hahn-Saks theorem and measuroids
Authors: Saeki Sadahiro
Published: June 2010

We introduce the concept of measuroids, and prove the Uniform Boundedness Principle of Nikodym-Grothendieck, the Vitali-Hahn-Saks theorem and the Nikodym Convergence Theorem for measuroids.
Periodic potentials with minimal energy bands
Authors: Ashbaugh Mark S. , Svirsky Roman
Published: June 2010

We consider the problem of minimizing the width of the lowest band in the spectrum of Hill’s equation,

−
u
+
q

(
x
)

u
=
λ
u

- u + q\left ( x \right )u = \lambda u

on

R

\mathbb {R}

with

q

(

x
+
1

)

=
q

(
x
)

q\left ( {x + 1} \right ) = q\left ( x \right )

for all

x
∈

R

x \in \mathbb {R}

, when the potential function

q
q

is allowed to vary over a ball of radius

M
>
0

 in 

L
∞

M > 0{\text { in }}{L^\infty }

. We show that minimizing potentials

q
∗

{q_ * }

exist and that, when considered as functions on the circle, they must have exactly one well on which

q
∗

(
x
)

{q_ * }\left ( x \right )

must equal

−
M

- M

and one barrier on which

q
∗

(
x
)

{q_ * }\left ( x \right )

must equal

M
M

; these are the only values that

q
∗

{q_ * }

can assume (up to changes on sets of measure zero). That is, on the circle there is a single interval where

q
∗

(
x
)

=
M

{q_ * }\left ( x \right ) = M

and on the complementary interval

q
∗

(
x
)

=
−
M

{q_ * }\left ( x \right ) = - M

. These results can be used to solve the problem of minimizing the gap between the lowest Neumann eigenvalue and either the lowest Dirichlet eigenvalue or the second Neumann eigenvalue for the same equation restricted to the interval

[
0
,
1
]

[0,1]

.
New estimates for the collapse of the Milnor-Moore spectral sequence over a field
Authors: Jessup Barry
Published: June 2010

We use the free tensor models of Halperin and Lemaire to give a new and transparent proof of a theorem of Ginsburg’s on the collapse of the Milnor-Moore spectral under the assumption of finite L.-S. category. The method, valid over any field, provides better bounds for the collapse and these bounds are effectively computable L.-S. type invariants.
An unprovable Ramsey-type theorem
Authors: Loebl Martin , Nešetřil Jaroslav
Published: June 2010

We present a new proof of the Paris-Harrington unprovable (in PA) version of Ramsey’s theorem. This also yields a particularly short proof of the Ketonen-Solovay result on rapidly growing Ramsey functions.
Strictly positive definite functions on spheres
Authors: Xu Yuan , Cheney E. W.
Published: June 2010

A sufficient condition is given for the strict positive-definiteness of a real, continuous function on the

m
m

-dimensional sphere.
Finite correspondence of spectra in Noetherian ring extensions
Authors: Letzter Edward S.
Published: June 2010

Let

R
[
u
n
k
]
S

R[unk]S

be an embedding of associative noetherian rings such that

S
S

is finitely generated as a right

R
R

-module. There is a correspondence from the prime spectrum of

S
S

to the prime spectrum of

R
R

obtained by associating to a given prime ideal

P
P

of

S
S

the prime ideals of

R
R

minimal over

P
∩
R

P \cap R

. The prime and primitive ideal theories for several specific noncommutative noetherian rings, including group algebras, PI algebras, and enveloping algebras, depend on understanding instances of this correspondence. We prove that the correspondence has finite fibers for a class of noetherian ring extensions that unites these examples.
Characteristic conditions for a 𝑐₀-semigroup with continuity in the uniform operator topology for 𝑡>0 in Hilbert space
Authors: You Pu Hong
Published: June 2010

In this paper, we obtain necessary and sufficient conditions for a

c
0

{c_0}

-semigroup with continuity in the uniform operator topology for

t
>
0

t > 0

in Hilbert space in terms of the spectral property of its infinitesimal generator. Then we get the characteristic theorem for the compact semigroup.
On finding fields from their algebraic closure geometries
Authors: Holland Kitty L.
Published: June 2010

It is shown that if

F
1

{F_1}

and

F
2

{F_2}

are algebraically closed fields of nonzero characteristic

p
p

and

F
1

{F_1}

is not isomorphic to a subfield of

F
2

{F_2}

, then

F
1

{F_1}

does not embed in the skew field of quotients

O

F
2

{O_{{F_2}}}

of the ring of morphisms of the additive group of

F
2

{F_2}

. From this fact and results of Evans and Hrushovski, it is deduced that the algebraic closure geometries

G
(

K
1

/

F
1

)

G({K_1}/{F_1})

and

G
(

K
2

/

F
2

)

G({K_2}/{F_2})

are isomorphic if and only if

K
1

:

F
1

≃

K
2

:

F
2

{K_1}:{F_1} \simeq {K_2}:{F_2}

. It is further proved that if

F
0

{F_0}

is the prime algebraically closed field of characteristic

p
p

and

F
F

has positive transcendence degree over

F
0

{F_0}

, then

O
F

{O_F}

and

O

F
0

{O_{{F_0}}}

are not elementarily equivalent.
The axiom of choice, fixed point theorems, and inductive ordered sets
Authors: Tasković Milan R.
Published: June 2010

This paper continues the study of the inductiveness of posets in terms of fixed apexes and points. The author proves some new equivalents of the Axiom of Choice, i.e., Zorn’s lemma. These statements are of fixed apex type and fixed point type theorems. The paper includes comments about these theorems and presents new characterizations of inductiveness and quasi-inductiveness of posets in terms of fixed apexes and fixed points.
Locally finite and locally nilpotent derivations with applications to polynomial flows and polynomial morphisms
Authors: van den Essen Arno
Published: June 2010

We give a very simple proof of the fact that the Lorenz equations and the Maxwell-Bloch equations do not have a polynomial flow. We also give an algorithm to decide if a two-dimensional vector field over

R

\mathbb {R}

has a polynomial flow and how to compute the solutions (in case the vector field has a polynomial flow).
An algebraic proof for the symplectic structure of moduli space
Authors: Karshon Yael
Published: June 2010

Goldman has constructed a symplectic form on the moduli space

Hom
⁡
(
π
,
G
)

/

G

\operatorname {Hom} (\pi ,G)/G

, of flat

G
G

-bundles over a Riemann surface

S
S

whose fundamental group is

π
\pi

. The construction is in terms of the group cohomology of

π
\pi

. The proof that the form is closed, though, uses de Rham cohomology of the surface

S
S

, with local coefficients. This symplectic form is shown here to be the restriction of a tensor, that is defined on the infinite product space

G
π

{G^\pi }

. This point of view leads to a direct proof of the closedness of the form, within the language of group cohomology. The result applies to all finitely generated groups

π
\pi

whose cohomology satisfies certain conditions. Among these are the fundamental groups of compact Kähler manifolds.
Convergent free sequences in compact spaces
Authors: Juhász I. , Szentmiklóssy Z.
Published: June 2010

We prove that if

κ
\kappa

is an uncountable regular cardinal and a compact

T
2

{T_2}

space

X
X

contains a free sequence of length

κ
\kappa

, then

X
X

also contains such a sequence that is convergent. This implies that under

CH

{\text {CH}}

every nonfirst countable compact

T
2

{T_2}

space contains a convergent

ω
1

{\omega _1}

-sequence and every compact

T
2

{T_2}

space with a small diagonal is metrizable, thus answering old questions raised by the first author and M. Hušek, respectively.
Liouville equation and spherical convex polytopes
Authors: Luo Feng , Tian Gang
Published: June 2010

We study the Liouville equation

Δ
u
=
−

e

2
u

\Delta u = - {e^{2u}}

in the complex plane with prescribed singularities and obtain a necessary and sufficient condition for the existence of the solution. The proof is based on the continuity method and a uniqueness theorem.
The isoperimetric inequality for nonsimple closed curves
Authors: Gysin Liliana M.
Published: June 2010

The main purpose of this paper is the generalization to the hyperbolic and elliptic spaces of the isoperimetric inequality of Banchoff and Pohl (J. Differential Geom. 6 (1971), 175-192).
Asymptotic behaviour of measure-valued critical branching processes
Authors: Etheridge Alison M.
Published: June 2010

Measure-valued branching processes can be characterized in terms of the Laplace transform of their transition densities and this gives rise to a second order nonlinear p.d.e.—the evolution equation of the process. We write the solution to this evolution equation as a series, each of whose coefficients is expressed in terms of the linear semigroup corresponding to the spatial part of the measure-valued process. From this we obtain a simple proof that if the spatial part of the process is a recurrent (resp., transient) Markov process on a standard Borel space and the initial value of the process is an invariant measure of this spatial process, then the process has no (resp., has a unique) nontrivial limiting distribution.
Products of Poincaré domains
Authors: Stanoyevitch Alexander
Published: June 2010

A domain

Ω
⊆

R

N

\Omega \subseteq {\mathbb {R}^N}

of finite

N
N

-dimensional Lebesgue measure is a

p
p

-Poincaré domain

(
1
⩽
p
⩽
∞
)

(1 \leqslant p \leqslant \infty )

if there exists a positive constant

K
K

such that the

p
p

-Poincaré inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue u StartAbsoluteValue EndAbsoluteValue Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-slanted-equals upper K StartAbsoluteValue EndAbsoluteValue nabla u StartAbsoluteValue EndAbsoluteValue Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis">

|

|

u

|

|

L
p

(
Ω
)

⩽
K

|

|

∇
u

|

|

L
p

(
Ω
)

||u|{|_{{L^p}(\Omega )}} \leqslant K||\nabla u|{|_{{L^p}(\Omega )}}

is valid for all Sobolev functions

u
∈

W

1
,
p

(
Ω
)

u \in {W^{1,p}}(\Omega )

that integrate to zero. Define

K
p

(
Ω
)

{K_p}(\Omega )

to be the smallest such

K
K

if

Ω
\Omega

is a

p
p

-Poincaré domain and to be infinity otherwise. We obtain comparability relations between

K
p

(

Ω
1

×

Ω
2

)

{K_p}({\Omega _1} \times {\Omega _2})

and the pair

K
p

(

Ω
1

)
,

K
p

(

Ω
2

)

{K_p}({\Omega _1}),\;{K_p}({\Omega _2})

. In particular, our results show that

p
p

-Poincaré domains are closed under cartesian products (for all

p
p

), and that in case

p
p

equals

2
2

, we have
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K 2 left-parenthesis normal upper Omega 1 times normal upper Omega 2 right-parenthesis equals max left-brace right-brace comma of KK 2 left-parenthesis right-parenthesis upper Omega 1 comma of KK 2 left-parenthesis right-parenthesis upper Omega 2">

K
2

(

Ω
1

×

Ω
2

)
=
max
{

K
2

(

Ω
1

)
,

K
2

(

Ω
2

)
}

{K_2}({\Omega _1} \times {\Omega _2}) = \max \{ {K_2}({\Omega _1}),\;{K_2}({\Omega _2})\}

.
Rational elasticity of factorizations in Krull domains
Authors: Anderson D. D. , Anderson David F. , Chapman Scott T. , Smith William W.
Published: June 2010

For an atomic domain

R
R

, we define the elasticity of

R
R

as
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho left-parenthesis upper R right-parenthesis equals sup left-parenthesis m slash n vertical-bar x 1 midline-horizontal-ellipsis x Subscript m Baseline equals y 1 midline-horizontal-ellipsis y Subscript n Baseline comma for x Subscript i Baseline comma y Subscript j Baseline element-of upper R irreducibles right-brace">

ρ
(
R
)
=
sup
(
m

/

n

|

x
1

⋯

x
m

=

y
1

⋯

y
n

,

for

x
i

,

y
j

∈
R

irreducibles} 

\rho (R) = \sup (m/n|{x_1} \cdots {x_m} = {y_1} \cdots {y_n},\;{\text {for}}\;{x_i},{y_j} \in R\;{\text {irreducibles\} }}

and let

l
R

(
x
)

{l_R}(x)

and

L
R

(
x
)

{L_R}(x)

denote, respectively, the inf and sup of the lengths of factorizations of a nonzero nonunit

x
∈
R

x \in R

into the product of irreducible elements. We answer affirmatively two rationality conjectures about factorizations. First, we show that

ρ
(
R
)

\rho (R)

is rational when

R
R

is a Krull domain with finite divisor class group. Secondly, we show that when

R
R

is a Krull domain, the two limits

l
R

(

x
n

)

/

n

{l_R}({x^n})/n

and

L
R

(

x
n

)

/

n

{L_R}({x^n})/n

, as

n
n

goes to infinity, are positive rational numbers. These answer, respectively, conjectures of D. D. Anderson and D. F. Anderson, and D. F. Anderson and P. Pruis. (The second question has also been solved by A. Geroldinger and F. Halter-Koch.)
Contravariantly finite subcategories and irreducible maps
Authors: Carlson Jon F. , Happel Dieter
Published: June 2010

Let

Λ
\Lambda

be an Artin algebra. This paper presents a condition that is necessary for a subcategory of

mod
Λ

\bmod \Lambda

to be contravariantly finite. Several examples are explored in the case that

Λ
\Lambda

is the group algebra of a finite group over a field of finite characteristic.
Closures of weakened analytic groups
Authors: Stevens T. Christine
Published: June 2010

Let

(
G
,

G

)

(G,\mathcal {G})

be a topological group with dense subgroup

L
L

, and suppose that

L
L

is an analytic group in a topology

τ
\tau

that is stronger than the topology that

L
L

inherits from

G

\mathcal {G}

. It is known that

L
L

contains a

τ
\tau

-closed abelian subgroup

H
H

that completely determines the topology of

L
L

. We now prove that the

G

\mathcal {G}

-closure

H
¯

\overline H

of

H
H

similarly determines the topology of

G
G

.

(
G
,

G

)

(G,\mathcal {G})

always has a left-completion in the category of topological groups, and the properties of

H
¯

\overline H

determine whether

(
G
,

G

)

(G,\mathcal {G})

is locally compact, analytic, metrizable, left-complete, or finite dimensional. We discuss the relationship between these results and recent work of Goto.
Nef cotangent bundles of branched coverings
Authors: Spurr Michael J.
Published: June 2010

In a previous paper we characterized those

H
(
Λ
,
n
)

H(\Lambda ,n)

(compact complex surfaces constructed by Hirzebruch) that have nef cotangent bundle. In this article we extend the methods to study more general branched coverings with regard to nefness of their cotangent bundles.
Uniformly convex functions and a corresponding class of starlike functions
Authors: Rønning Frode
Published: June 2010

We investigate starlike functions

f
(
z
)
=
z
+

∑

k
=
2

∞

a
k

z
k

f(z) = z + \sum \nolimits _{k = 2}^\infty {{a_k}{z^k}}

with the property that

z

f
′

(
z
)

/

f
(
z
)

zf’(z)/f(z)

lies inside a certain parabola. These functions are closely related to a class of functions called uniformly convex and recently introduced by Goodman. We give some particular examples of functions having the required properties, and we give upper bounds on the coefficients and the modulus

|

f
(
z
)

|

|f(z)|

of the functions in the class.
Nilpotency degree of cohomology rings in characteristic two
Authors: Avrunin George S. , Carlson Jon F.
Published: June 2010

In this paper, we consider the cohomology ring of a finite

2
2

-group with coefficients in a field of characteristic two. We show that, for any positive integer

n
n

, there exists a

2
2

-group whose cohomology ring has elements of nilpotency degree

n
+
1

n + 1

and all smaller degrees.
Exact embeddings into Artinian rings
Authors: Beaulieu Raymond A. , Jensen Anders
Published: June 2010

The concept of exact embedding of a Noetherian ring into an Artinian ring discussed by Blair and Small is exploited to yield results on embeddings of various rings into Artinian rings. In particular we investigate Ore extensions, group graded rings, and filtered rings. Also, we prove that exact embeddability is a Morita invariant.
Sur un théorème spectral et son application aux noyaux lipchitziens
Authors: Hennion Hubert
Published: June 2010

The aim of this note is to point out that the Ionescu Tulcea and Marinescu theorem can be reinforced, using a Nussbaum formula for the essential spectral radius of an operator. In this stronger version, this theorem is suitable for the spectral analysis of lipschitzian, positive, not necessarily markovian kernels, Ruelle theorem follows. As an application to Probability Theory, a large deviation theorem is proved.
Eulerian operators and the Jacobian conjecture
Authors: van den Essen Arno
Published: June 2010

In this paper we introduce a new class of polynomial maps, the so-called nice polynomial maps. Using Eulerian operators we show how for these polynomial maps the main results obtained by Bass (Differential structure of étale extensions of polynomial algebras, Proc. Workshop on Commutative Algebra, MSRI, 1987) can be proved in a very simple and elementary way. Furthermore we show that every polynomial map

F
F

satisfying the Jacobian condition, det

J
F
∈

k

∗

JF \in {k^{\ast }}

, is equivalent to a nice polynomial map; more precisely the polynomial map

F

(
λ
)

(
X
)
=
F
(
X
+
λ
)
−
F
(
λ
)

{F_{(\lambda )}}(X) = F(X + \lambda ) - F(\lambda )

is nice for almost all

λ
∈

k
n

\lambda \in {k^n}

.
A counterexample to Orlik’s conjecture
Authors: Edelman Paul H. , Reiner Victor
Published: June 2010

We present a counterexample to the conjecture by Orlik that the restriction of a free hyperplane arrangement to one of its hyperplanes is free.