bluuebunny/crossref_metadata_2025 huggingface dataset:
The discrete Prüfer transformation
Authors: Bohner Martin , Došlý Ondřej
Published: July 2002

The classical Prüfer transformation has proved to be a useful tool in the study of Sturm-Liouville theory. In this paper we introduce the Prüfer transformation for self-adjoint difference equations and use it to obtain oscillation criteria and other results. We then offer an extension of this approach to the case of general symplectic systems on time scales. Time scales have been introduced in order to unify discrete and continuous analysis, and indeed our results cover as special cases both the Prüfer transformation for differential and for difference equations.
Stability and almost periodicity of asymptotically dominated semigroups of positive operators
Authors: Emel’yanov E. , Kohler U. , Räbiger F. , Wolff M.
Published: July 2002

We discuss conditions such that strong stability and strong asymptotic compactness of a (discrete or continuous) semiflow defined on a subset in the positive cone of an ordered Banach space is preserved under asymptotic domination. This is used to show that on a Banach lattice with order continuous norm strong stability and almost periodicity of a (discrete or strongly continuous) semigroup of positive operators is preserved under asymptotic domination.
0^{♯} and elementary end extensions of 𝑉_{𝜅}
Authors: Leshem Amir
Published: July 2002

In this paper we prove that if

κ
\kappa

is a cardinal in

L
[

0

♯

]

L[0^{\sharp }]

, then there is an inner model

M
M

such that

M
⊨
(

V

κ

,
∈
)

M \models (V_{\kappa },\in )

has no elementary end extension. In particular if

0

♯

0^{\sharp }

exists, then weak compactness is never downwards absolute. We complement the result with a lemma stating that any cardinal greater than

ℵ
1

\aleph _1

of uncountable cofinality in

L
[

0

♯

]

L[0^{\sharp }]

is Mahlo in every strict inner model of

L
[

0

♯

]

L[0^{\sharp }]

.
Mod 𝑝 Galois representations of solvable image
Authors: Moon Hyunsuk , Taguchi Yuichiro
Published: July 2002

It is proved that, for a number field

K
K

and a prime number

p
p

, there exist only finitely many isomorphism classes of continuous semisimple Galois representations of

K
K

into

GL

d

⁡
(

F

¯

p

)

\operatorname {GL}_{d}(\overline {\mathbb {F}}_{p})

of fixed dimension

d
d

and bounded Artin conductor outside

p
p

which have solvable images. Some auxiliary results are also proved.
Integral closure of a cubic extension and applications
Authors: Tan Sheng-Li
Published: July 2002

In this paper, we compute the integral closure of a cubic extension over a Noetherian unique factorization domain. We also present some applications to triple coverings and to rank two reflexive sheaves over an algebraic variety.
On an asymptotic behavior of elements of order 𝑝 in irreducible representations of the classical algebraic groups with large enough highest weights
Authors: Suprunenko I.
Published: July 2002

The behavior of the images of a fixed element of order

p
p

in irreducible representations of a classical algebraic group in characteristic

p
p

with highest weights large enough with respect to

p
p

and this element is investigated. More precisely, let

G
G

be a classical algebraic group of rank

r
r

over an algebraically closed field

K
K

of characteristic

p
>
2

p>2

. Assume that an element

x
∈
G

x\in G

of order

p
p

is conjugate to that of an algebraic group of the same type and rank

m
>
r

m>r

naturally embedded into

G
G

. Next, an integer function

σ
x

\sigma _x

on the set of dominant weights of

G
G

and a constant

c
x

c_x

that depend only upon

x
x

, and a polynomial

d
d

of degree one are defined. It is proved that the image of

x
x

in the irreducible representation of

G
G

with highest weight

ω
\omega

contains more than

d
(
r
−
m
)

d(r-m)

Jordan blocks of size

p
p

if

m
m

and

r
−
m

r-m

are not too small and

σ
x

(
ω
)
≥
p
−
1
+

c
x

\sigma _x(\omega )\geq p-1+c_x

.
Einstein manifolds and contact geometry
Authors: Boyer Charles , Galicki Krzysztof
Published: July 2002

We show that every K-contact Einstein manifold is Sasakian-Einstein and discuss several corollaries of this result.
Invariant subspaces for bounded operators with large localizable spectrum
Authors: Prunaru Bebe
Published: July 2002

SupposeHHis a complex Hilbert space andT∈L(H)T\in L(H)is a bounded operator. For each closed setF⊂CF\subset \mathbf {C}letHT(F)H_{T}(F)denote the corresponding spectral manifold. Letσloc(T)\sigma _{loc}(T)denote the set of all pointsλ∈σ(T)\lambda \in \sigma (T)with the property thatHT(V¯)≠0H_{T}(\overline {V})\neq 0for any open neighborhoodVVofλ.\lambda .In this paper we show that ifσloc(T)\sigma _{loc}(T)is dominating in some bounded open set, thenTThas a nontrivial invariant subspace. As a corollary, every Hilbert space operator which is a quasiaffine transform of a subdecomposable operator with large spectrum has a nontrivial invariant subspace.
Determinacy of smooth germs with real isolated line singularities
Authors: Sun Bohao , Wilson Leslie
Published: July 2002

The germ of a smooth real-valued function on Euclidean space is called a real isolated line singularity if its singular set is a nonsingular curve, its Jacobian ideal is Łojasiewicz at the singular set, and its Hessian determinant restricted to the singular set is Łojasiewicz at 0. Consider the set of all germs whose singular set contains a fixed nonsingular curve

L
L

. We prove that such a germ

f
f

is infinitely determined among all such germs with respect to composition by diffeomorphisms preserving

L
L

if, and only if, the Jacobian ideal of

f
f

contains all germs which vanish on

L
L

and are infinitely flat at 0 if, and only if,

f
f

is a real isolated line singularity.
Extinction and decay estimates for viscous Hamilton-Jacobi equations in ℝ^{ℕ}
Authors: Benachour Said , Laurençot Philippe , Schmitt Didier
Published: July 2002

We consider non-negative and integrable classical solutions to the Cauchy problem

u
t

−
Δ
u
+
|
∇
u

|
p

=
0

u_t-\Delta u+\vert \nabla u\vert ^p=0

when

p
∈
(
0
,
+
∞
)

p\in (0,+\infty )

. For

p
∈
(
0
,
N

/

(
N
+
1
)
)

p\in (0,N/(N+1))

we prove that any such solution vanishes identically after a finite time. For higher values of

p
p

temporal decay estimates are obtained.
Real analytic solutions of parabolic equations with time-measurable coefficients
Authors: Kovats Jay
Published: July 2002

We use Bernstein’s technique to show that for any fixed

t
t

, strong solutions

u
(
t
,
x
)

u(t,x)

of the uniformly parabolic equation

L
u
:=

a

i
j

(
t
)

u

x

i

x

j

−

u

t

=
0

Lu:=a^{ij}(t)u_{x_{i}x_{j}}-u_{t}=0

in

Q
Q

are real analytic in

Q
(
t
)
=
{
x
:
(
t
,
x
)
∈
Q
}

Q(t)=\{x:(t,x)\in Q\}

. Here,

Q
⊂

R

d
+
1

Q\subset \mathbb {R}^{d+1}

is a bounded domain and the coefficients

a

i
j

(
t
)

a^{ij}(t)

are measurable. We also use Bernstein’s technique to obtain interior estimates for pure second derivatives of solutions of the fully nonlinear, uniformly parabolic, concave equation

F
(

D

2

u
,
t
)
−

u

t

=
0

F(D^{2}u,t)-u_{t}=0

in

Q
Q

, where

F
F

is measurable in

t
t

.
A simple proof of Bailey’s very-well-poised ₆𝜓₆ summation
Authors: Schlosser Michael
Published: July 2002

We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauß’

2

F
1

_2F_1

summation and elementary series manipulations to give a simple proof of Dougall’s

2

H
2

_2H_2

summation. Similarly, we apply Rogers’ nonterminating

6

ϕ
5

_6\phi _5

summation and elementary series manipulations to give a simple proof of Bailey’s very-well-poised

6

ψ
6

_6\psi _6

summation. Our method of proof extends M. Jackson’s first elementary proof of Ramanujan’s

1

ψ
1

_1\psi _1

summation.
An isoperimetric comparison theorem for Schwarzschild space and other manifolds
Authors: Bray Hubert , Morgan Frank
Published: July 2002

We give a very general isoperimetric comparison theorem which, as an important special case, gives hypotheses under which the spherically symmetric

(
n
−
1
)

(n-1)

-spheres of a spherically symmetric

n
n

-manifold are isoperimetric hypersurfaces, meaning that they minimize

(
n
−
1
)

(n-1)

-dimensional area among hypersurfaces enclosing the same

n
n

-volume. This result greatly generalizes the result of Bray (Ph.D. thesis, 1997), which proved that the spherically symmetric 2-spheres of 3-dimensional Schwarzschild space (which is defined to be a totally geodesic, space-like slice of the usual

(
3
+
1
)

(3+1)

-dimensional Schwarzschild metric) are isoperimetric. We also note that this Schwarzschild result has applications to the Penrose inequality in general relativity, as described by Bray.
Sequentially independent effects
Authors: Gudder Stan , Nagy Gabriel
Published: July 2002

A quantum effect is a yes-no measurement that may be unsharp. An effect is represented by an operator

E
E

on a Hilbert space that satisfies

0
≤
E
≤
I

0\le E\le I

. We define effects

E
1

,

E
2

,
…
,

E
n

E_1,E_2,\ldots ,E_n

to be sequentially independent if the result of any sequential measurement of

E
1

,

E
2

,
…
,

E
n

E_1,E_2,\ldots ,E_n

does not depend on the order in which they are measured. We show that two effects are sequentially independent if and only if they are compatible. That is, their corresponding operators commute. We also show that three effects are sequentially independent if and only if all permutations of the product of their corresponding operators coincide. It is noted that this last condition does not imply that the three effects are mutually compatible.
Commutator inequalities associated with the polar decomposition
Authors: Kittaneh Fuad
Published: July 2002

Let

A
=
U
P

A=UP

be a polar decomposition of an

n
×
n

n\times n

complex matrix

A
A

. Then for every unitarily invariant norm

|

|

|

⋅

|

|

|

|||\cdot |||

, it is shown that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue upper U upper P minus upper P upper U StartAbsoluteValue StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue less-than-or-equal-to EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue upper A Superscript asterisk Baseline upper A minus upper A upper A Superscript asterisk Baseline StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue less-than-or-equal-to double-vertical-bar upper U upper P plus upper P upper U double-vertical-bar EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue upper U upper P minus upper P upper U StartAbsoluteValue EndAbsoluteValue EndAbsoluteValue squared comma">

|

|

|

|

U
P
−
P
U

|

2

|

|

|

≤

|

|

|

A
∗

A
−
A

A
∗

|

|

|

≤
‖
U
P
+
P
U
‖

|

|

|

U
P
−
P
U

|

|

|

,

|||\, |UP-PU|^2||| \le |||A^*A-AA^*|||\le \|UP+PU\|\,|||UP-PU|||,

\]
where

‖
⋅
‖

\|\cdot \|

denotes the operator norm. This is a quantitative version of the well-known result that

A
A

is normal if and only if

U
P
=
P
U

UP=PU

. Related inequalities involving self-commutators are also obtained.
Asymptotics for the heat equation in the exterior of a shrinking compact set in the plane via Brownian hitting times
Authors: Pinsky Ross
Published: July 2002

Let

D

r

=
{
x
∈

R

2

:

|

x

|

≤
r
}

D_{r}=\{x\in R^{2}:|x|\le r\}

and let

γ
\gamma

be a continuous, nonincreasing function on

[
0
,
∞
)

[0,\infty )

satisfying

lim

t
→
∞

γ
(
t
)
=
0

\lim _{t\to \infty }\gamma (t)=0

. Consider the heat equation in the exterior of a time-dependent shrinking disk in the plane:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column Blank 2nd Column a m p semicolon u Subscript t Baseline equals one half normal upper Delta u comma x element-of upper R squared minus upper D Subscript gamma left-parenthesis t right-parenthesis Baseline comma t greater-than 0 comma 2nd Row 1st Column Blank 2nd Column a m p semicolon u left-parenthesis x comma 0 right-parenthesis equals 0 comma x element-of upper R squared minus upper D Subscript gamma left-parenthesis t right-parenthesis Baseline comma 3rd Row 1st Column Blank 2nd Column a m p semicolon u left-parenthesis x comma t right-parenthesis equals 1 comma x element-of upper D Subscript gamma left-parenthesis t right-parenthesis Baseline comma t greater-than 0 period EndLayout">

a
m
p
;

u

t

=

1
2

Δ
u
,

x
∈

R

2

−

D

γ
(
t
)

,

t
>
0
,

a
m
p
;
u
(
x
,
0
)
=
0
,

x
∈

R

2

−

D

γ
(
t
)

,

a
m
p
;
u
(
x
,
t
)
=
1
,

x
∈

D

γ
(
t
)

,
t
>
0.

\begin{equation*} \begin {split} &u_{t}=\frac {1}{2}\Delta u, \quad x\in R^{2}-D_{\gamma (t)}, \quad t>0,\\ &u(x,0)=0,\quad x\in R^{2}-D_{\gamma (t)},\\ &u(x,t)=1,\quad x\in D_{\gamma (t)}, t>0. \end{split} \end{equation*}

If there exist constants

0
>

c

1

>

c

2

0>c_{1}>c_{2}

and a constant

k
>
0

k>0

such that

c

1

t

−
k

≤
γ
(
t
)
≤

c

2

t

−
k

c_{1}t^{-k}\le \gamma (t)\le c_{2}t^{-k}

, for sufficiently large

t
t

, then

lim

t
→
∞

u
(
x
,
t
)
=

1

1
+
2
k

\lim _{t\to \infty }u(x,t)=\frac {1}{1+2k}

. The same result is also shown to hold when

D

γ
(
t
)

D_{\gamma (t)}

is replaced by

L

γ
(
t
)

L_{\gamma (t)}

, where

L

r

=
{
(

x

1

,
0
)
∈

R

2

:

|

x

1

|

≤
r
}

L_{r}=\{(x_{1},0)\in R^{2}:|x_{1}|\le r\}

. Also, a discrepancy is noted between the asymptotics for the above forward heat equation and the corresponding backward one. The method used is probabilistic.
Strongly representable atom structures of relation algebras
Authors: Hirsch Robin , Hodkinson Ian
Published: July 2002

A relation algebra atom structure

α
\alpha

is said to be strongly representable if all atomic relation algebras with that atom structure are representable. This is equivalent to saying that the complex algebra

C
m

α

\mathfrak {Cm} \alpha

is a representable relation algebra. We show that the class of all strongly representable relation algebra atom structures is not closed under ultraproducts and is therefore not elementary. This answers a question of Maddux (1982). Our proof is based on the following construction. From an arbitrary undirected, loop-free graph

Γ
\Gamma

, we construct a relation algebra atom structure

α
(
Γ
)

\alpha (\Gamma )

and prove, for infinite

Γ
\Gamma

, that

α
(
Γ
)

\alpha (\Gamma )

is strongly representable if and only if the chromatic number of

Γ
\Gamma

is infinite. A construction of Erdös shows that there are graphs

Γ
r

\Gamma _r

(

r
>
ω

r>\omega

) with infinite chromatic number, with a non-principal ultraproduct

∏
D

Γ
r

\prod _D\Gamma _r

whose chromatic number is just two. It follows that

α
(

Γ
r

)

\alpha (\Gamma _r)

is strongly representable (each

r
>
ω

r>\omega

) but

∏
D

(
α
(

Γ
r

)
)

\prod _D(\alpha (\Gamma _r))

is not.
Octonion algebras obtained from associative algebras with involution
Authors: Petersson Holger , Racine Michel
Published: July 2002

A natural octonion algebra structure on the symmetric elements of trace 0 of central simple associative algebras of degree 3 with involution of the second kind is obtained.
New proof of the cobordism invariance of the index
Authors: Braverman Maxim
Published: July 2002

We give a simple proof of the cobordism invariance of the index of an elliptic operator. The proof is based on a study of a Witten-type deformation of an extension of the operator to a complete Riemannian manifold. One of the advantages of our approach is that it allows us to treat directly general elliptic operators which are not of Dirac type.
Subspaces with normalized tight frame wavelets in ℝ
Authors: Dai Xingde , Diao Yuanan , Gu Qing
Published: July 2002

In this paper we investigate the subspaces of

L
2

(

R

)

L^2(\mathbb R)

which have normalized tight frame wavelets that are defined by set functions on some measurable subsets of

R

\mathbb R

called Bessel sets. We show that a subspace admitting such a normalized tight frame wavelet falls into a class of subspaces called reducing subspaces. We also consider the subspaces of

L
2

(

R

)

L^2(\mathbb R)

that are generated by a Bessel set

E
E

in a special way. We present some results concerning the relation between a Bessel set

E
E

and the corresponding subspace of

L
2

(

R

)

L^2(\mathbb R)

which either has a normalized tight frame wavelet defined by the set function on

E
E

or is generated by

E
E

.
Analytic extension of differentiable functions defined in closed sets by means of continuous linear operators
Authors: Frerick Leonhard , Vogt Dietmar
Published: July 2002

In this paper we solve the following problem posed by Schmets and Valdivia: Under which conditions does there exist an extension operator from the space

E

(
F
)

{\mathscr E} (F)

of the Whitney jets on a closed set

F
⊂

R

n

F \subset {\mathbb {R}} ^n

to

E

(

R

n

)

{\mathscr E}({\mathbb {R}}^n)

so that the extended functions are real analytic outside

F
F

?
Periods of mirrors and multiple zeta values
Authors: Hoffman Michael
Published: July 2002

In a recent paper, A. Libgober showed that the multiplicative sequence{Qi(c1,…,ci)}\{Q_i(c_1,\dots ,c_i)\}of Chern classes corresponding to the power seriesQ(z)=Γ(1+z)−1Q(z)=\Gamma (1+z)^{-1}appears in a relation between the Chern classes of certain Calabi-Yau manifolds and the periods of their mirrors. We show that the polynomialsQiQ_ican be expressed in terms of multiple zeta values.
On the location of the essential spectrum of Schrödinger operators
Authors: Metafune Giorgio , Pallara Diego
Published: July 2002

We give estimates on the bottom of the essential spectrum of Schrödinger operators

−
Δ
+
V

-\Delta +V

in

L
2

(

R

N

)

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

.
Division algebras over 𝐶₂- and 𝐶₃-fields
Authors: Rowen Louis
Published: July 2002

Using elementary methods we prove a theorem of Rost, Serre, and Tignol that any division algebra of degree 4 over a

C

3

C_{3}

-field containing

−
1

\sqrt {-1}

is cyclic. Our methods also show any division algebra of degree 8 over a

C

2

C_{2}

-field containing

−
1

4

\sqrt [4 ]{-1}

is cyclic.
The extension of positive definite operator-valued functions defined on a symmetric interval of an ordered group
Authors: Bakonyi Mihály
Published: July 2002

Let

G
1

G_1

be an ordered abelian group and

a
∈

G
1

a\in G_1

. Let

G
2

G_2

be an abelian group and

f
f

an operator-valued positive definite function on

(
−
a
,
a
)
×

G
2

(-a,a)\times G_2

. We prove that

f
f

admits a positive definite extension to

G
1

×

G
2

G_1\times G_2

, generalizing in this way existing results for the case when

G
1

=

R

G_1=\mathbf {R}

and

f
f

is continuous.
Time delayed parabolic systems with coupled nonlinear boundary conditions
Authors: Pao C.
Published: July 2002

The aim of this paper is to show the existence and uniqueness of a solution for a system of time-delayed parabolic equations with coupled nonlinear boundary conditions. The time delays are of discrete type which may appear in the reaction function as well as in the boundary function. The approach to the problem is by the method of upper and lower solutions for nonquasimonotone functions.
From 𝐾(𝑛+1)_{*}(𝑋) to 𝐾(𝑛)_{*}(𝑋)
Authors: Minami Norihiko
Published: July 2002

Let

X
X

be a space of finite type. Set

q
=
2
(
p
−
1
)

q=2(p-1)

as usual, and define the mod

q
q

support of

K
(
n

)
∗

(
X
)

K(n)^*(X)

by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S left-parenthesis upper X comma upper K left-parenthesis n right-parenthesis right-parenthesis equals left-brace m element-of double-struck upper Z slash q double-struck upper Z bar circled-plus Underscript d identical-to m mod q Endscripts upper K left-parenthesis n right-parenthesis Superscript d Baseline not-equals 0 right-brace">

S
(
X
,
K
(
n
)
)
=
{
m
∈

Z

/

q

Z

∣

⨁

d
≡
m
mod
q

 
K
(
n

)
d

≠
0
}

S(X,K(n)) = \{ m \in \mathbb {Z}/q\mathbb {Z} \mid \bigoplus _{d \equiv m \bmod q}\ K(n)^d \neq 0 \}

for

n
>
0.

n>0.

Call

K
(
n

)
∗

(
X
)

K(n)^*(X)

sparse if there is no

m
∈

Z

/

q

Z

m \in \mathbb {Z}/q\mathbb {Z}

with

m
,
m
+
1
∈
S
(
X
,
K
(
n
)
)
.

m, m+1 \in S(X,K(n)).

Then we show the relation

S
(
X
,
K
(
n
)
)
⫅
S
(
X
,
K
(
n
+
1
)
)

S(X,K(n)) \subseteqq S(X,K(n+1))

for any finite type space

X
X

with

K
(
n
+
1

)
∗

(
X
)

K(n+1)^*(X)

being sparse.

As a special case, we have
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis n plus 1 right-parenthesis Superscript o d d Baseline left-parenthesis upper X right-parenthesis equals 0 long right double arrow upper K left-parenthesis n right-parenthesis Superscript o d d Baseline left-parenthesis upper X right-parenthesis equals 0 comma">

K
(
n
+
1

)

o
d
d

(
X
)
=
0
⟹
K
(
n

)

o
d
d

(
X
)
=
0
,

K(n+1)^{odd}(X) = 0 \Longrightarrow K(n)^{odd}(X) = 0,

and the main theorem of Ravenel, Wilson and Yagita is also generalized in terms of the mod

q
q

support.
The diffeomorphism type of certain 𝑆³-bundles over 𝑆⁴
Authors: Sanchez Marc , Wilhelm Frederick
Published: July 2002

In this note we show that the unit tangent bundle of

S

4

S^{4}

is diffeomorphic to the total space of a certain principal

S

3

S^{3}

-bundle over

S

4

S^{4}

, solving a problem of James and Whitehead.
Critical points of the area functional of a complex closed curve on the manifold of Kähler metrics
Authors: Castorena Abel
Published: July 2002

We consider a compact complex manifold

M
M

of dimension

n
n

that admits Kähler metrics and we assume that

C
↪
M

C\hookrightarrow M

is a closed complex curve. We denote by

K
C

(
1
)

\mathcal {KC}(1)

the space of classes of Kähler forms

[
ω
]
∈

H

1
,
1

(
M
,

R

)

[\omega ]\in H^{1,1}(M,\mathbb {R})

that define Kähler metrics of volume 1 on

M
M

and define

A

C

:

K
C

(
1
)
→

R

\mathbf {A}_{C}:\mathcal {KC}(1)\to \mathbb {R}

by

A

C

(
[
ω
]
)
=

∫

C

ω
=
area of 
C
 in the induced metric by 
ω

\mathbf {A}_{C}([\omega ])=\int _{C} \omega =\text {area of }C\text { in the induced metric by }\omega

. We show how the Riemann-Hodge bilinear relations imply that any critical point of

A

C

\mathbf {A}_{C}

is the strict global minimum and we give conditions under which there is such a critical point

[
ω
]

[\omega ]

: A positive multiple of

[
ω

]

n
−
1

∈

H

2
n
−
2

(
M
,

R

)

[\omega ]^{n-1}\in H^{2n-2}(M,\mathbb {R})

is the Poincaré dual of the homology class of

C
C

. Applying this to the Abel-Jacobi map of a curve into its Jacobian,

C
↪
J
(
C
)

C\hookrightarrow J(C)

, we obtain that the Theta metric minimizes the area of

C
C

within all Kähler metrics of volume 1 on

J
(
C
)

J(C)

.
Counting generic genus–0 curves on Hirzebruch surfaces
Authors: Spielberg Holger
Published: July 2002

Hirzebruch surfaces

F

k

F_{k}

provide an excellent example to underline the fact that in general symplectic manifolds, Gromov–Witten invariants might well count curves in the boundary components of the moduli spaces. We use this example to explain in detail that the counting argument given by Batyrev for toric manifolds does not work.
On the dual of Orlicz–Lorentz space
Authors: Hudzik H. , Kamińska A. , Mastyło M.
Published: July 2002

A description of the Köthe dual of the Orlicz–Lorentz space

Λ

φ
,
w

\Lambda _{\varphi , w}

generated by an Orlicz function

φ
\varphi

and a regular weight function

w
w

is presented. It is also shown that in the case of separable Orlicz–Lorentz spaces the regularity condition on

w
w

is necessary and sufficient for the coincidence of the Banach dual space with the described Köthe dual space.
Radial images by holomorphic mappings,
Authors: Fernández José , Pestana Domingo
Published: July 2002

Let

R

\mathcal {R}

be a nonexceptional Riemann surface, other than the punctured disk. We prove that if

f
f

is a holomorphic mapping from the unit disk

Δ
\Delta

of the complex plane into

R

\mathcal {R}

, then the set of radial images that remain bounded in the Poincaré metric of

R

\mathcal {R}

has Hausdorff dimension at least

δ
(

R

)

\delta (\mathcal {R})

, the exponent of convergence of

R

\mathcal {R}

. The result is best possible. This is a hyperbolic analog of the result of N. G. Makarov that Bloch functions are bounded on a set of radii of dimension one.
A rigidity theorem for the Clifford tori in 𝑆³
Authors: Enomoto Kazuyuki , Kitagawa Yoshihisa , Weiner Joel
Published: July 2002

Let

S
3

S^3

be the unit hypersphere in the 4-dimensional Euclidean space

R

4

\Bbb R^4

defined by

∑

i
=
1

4

x
i
2

=
1

\sum _{i=1}^4 x_i^2=1

. For each

θ
\theta

with

0
>
θ
>
π

/

2

0>\theta >\pi /2

, we denote by

M
θ

M_\theta

the Clifford torus in

S
3

S^3

given by the equations

x
1
2

+

x
2
2

=

cos
2

⁡
θ

x_1^2+x_2^2=\cos ^2\theta

and

x
3
2

+

x
4
2

=

sin
2

⁡
θ

x_3^2+x_4^2= \sin ^2\theta

. The Clifford torus

M
θ

M_\theta

is a flat Riemannian manifold equipped with the metric induced by the inclusion map

i
θ

:

M
θ

→

S
3

i_\theta \colon M_\theta \to S^3

. In this note we prove the following rigidity theorem: If

f
:

M
θ

→

S
3

f\colon M_\theta \to S^3

is an isometric embedding, then there exists an isometry

A
A

of

S
3

S^3

such that

f
=
A
∘

i
θ

f=A\circ i_\theta

. We also show no flat torus with the intrinsic diameter

≤
π

\le \pi

is embeddable in

S
3

S^3

except for a Clifford torus.
Strong F-regularity in images of regular rings
Authors: Glassbrenner Donna
Published: July 2002

We characterize strong F-regularity, a property associated with tight closure, in a large class of rings. A special case of our results is a workable criterion in complete intersection rings.
Best possibility of the Furuta inequality
Authors: Tanahashi Kôtarô
Published: July 2002

Let

0
≤
p
,
q
,
r
∈

R

,
p
+
2
r
≤
(
1
+
2
r
)
q

0\le p,q,r\in \Bbb R, p+2r\le (1+2r)q

, and

1
≤
q

1\le q

. Furuta (1987) proved that if bounded linear operators

A
,
B
∈
B
(
H
)

A,B\in B(H)

on a Hilbert space

H
H

(
dim
⁡
(
H
)
≥
2
)

(\dim (H)\ge 2)

satisfy

0
≤
B
≤
A

0\le B\le A

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A Superscript r Baseline upper B Superscript p Baseline upper A Superscript r Baseline right-parenthesis Superscript 1 slash q Baseline less-than-or-equal-to upper A Superscript left-parenthesis p plus 2 r right-parenthesis slash q">

(

A
r

B
p

A
r

)

1

/

q

≤

A

(
p
+
2
r
)

/

q

(A^r B^p A^r)^{1/q} \le A^{(p+2r)/q}

. In this paper, we prove that the range

p
+
2
r
≤
(
1
+
2
r
)
q

p+2r\le (1+2r)q

and

1
≤
q

1\le q

is best possible with respect to the Furuta inequality, that is, if

(
1
+
2
r
)
q
>
p
+
2
r

(1+2r) q>p+2r

or

0
>
q
>
1

0>q>1

, then there exist

A
,
B
∈
B
(

R

2

)

A,B\in B(\Bbb R^2)

which satisfy

0
≤
B
≤
A

0\le B\le A

but
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A Superscript r Baseline upper B Superscript p Baseline upper A Superscript r Baseline right-parenthesis Superscript 1 slash q Baseline neither-less-than-nor-equal-to upper A Superscript left-parenthesis p plus 2 r right-parenthesis slash q">

(

A
r

B
p

A
r

)

1

/

q

≰

A

(
p
+
2
r
)

/

q

(A^r B^p A^r)^{1/q}\nleq A^{(p+2r)/q}

.
Curvature and finite domination
Authors: Weiss Michael
Published: July 2002

Upper bounds obtained by Gromov on the Betti numbers of certain closed Riemannian manifolds are shown to be upper bounds on the minimum number of cells in

C
W

CW

–spaces dominating such manifolds.
𝐴-differentiability and 𝐴-analyticity
Authors: Gadea P. , Masqué J.
Published: July 2002

Let

A
A

be a finite-dimensional commutative algebra over

R

\mathbb {R}

and let

C

A

r

(
U
)

C_{A}^{r}(U)

,

C

ω

(
U
,
A
)

C^{\omega }(U,A)

and

O

A

(
U
)

\mathcal { O}_{A}(U)

be the ring of

A
A

-differentiable functions of class

C

r

,

0
≤
r
≤
∞

C^{r},\,0 \leq r \leq \infty

, the ring of real analytic mappings with values in

A
A

and the ring of

A
A

-analytic functions, respectively, defined on an open subset

U
U

of

A

n

A^{n}

. We prove two basic results concerning

A
A

-differentiability and

A
A

-analyticity:

1

s
t

1^{st}

)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript upper A Baseline left-parenthesis upper U right-parenthesis equals upper C Subscript upper A Superscript normal infinity Baseline left-parenthesis upper U right-parenthesis intersection upper C Superscript omega Baseline left-parenthesis upper U comma upper A right-parenthesis">

O

A

(
U
)
=

C

A

∞

(
U
)
⋂

C

ω

(
U
,
A
)

\mathcal { O}_{A}(U) = C^{\infty }_{A}(U) \bigcap C^{\omega }(U,A)

,

2

n
d

2^{nd}

)

O

A

(
U
)
=

C

A

∞

(
U
)

\mathcal { O}_{A}(U) = C^{\infty }_{A}(U)

if and only if

A
A

is defined over

C

\mathbb {C}

.
The 𝜇-PIP and integrability of a single function
Authors: Stefánsson Gunnar
Published: July 2002

Two examples are given that answer in the negative the following question asked by E. M. Bator: If

f
:
Ω
→

X
∗

f:\Omega \to X^*

is bounded and weakly measurable and for each

x

∗
∗

x^{**}

in

X

∗
∗

X^{**}

there is a bounded sequence

(

x
n

)

(x_n)

in

X
X

such that

x

∗
∗

f
=

lim
n

f

x
n

x^{**}f=\lim _nfx_n

a.e., does it follow that

f
f

is Pettis integrable?
Properties that characterize Gaussian periods and cyclotomic numbers
Authors: Thaine F.
Published: July 2002

Let

q
=
e
f
+
1

q=ef+1

be a prime number,

ζ
q

\zeta _q

a

q
q

-th primitive root of 1 and

η
0

,
…
,

η

e
−
1

\eta _0,\dots ,\eta _{e-1}

the periods of degree

e
e

of

Q

(

ζ
q

)

\Bbb Q(\zeta _q)

. Write

η
0

η
i

=

∑

j
=
0

e
−
1

a

i
,
j

η
j

\eta _0\eta _i=\sum _{j=0}^{e-1} a_{i,j}\eta _j

with

a

i
,
j

∈

Z

a_{i,j}\in \Bbb Z

. Several characterizations of the numbers

η
i

\eta _i

and

a

i
,
j

a_{i,j}

(or, equivalently, of the cyclotomic numbers

(
i
,
j
)

(i,j)

of order

e
e

) are given in terms of systems of equations they satisfy and a condition on the linear independence, over

Q

\Bbb Q

, of the

η
i

\eta _i

or on the irreducibility, over

Q

\Bbb Q

, of the characteristic polynomial of the matrix

[

a

i
,
j

]

0
≤
i
,
j
≤
e
−
1

[a_{i,j}]_{0\leq i,j\leq e-1}

.
Integral estimates of conformal metrics
Authors: Xu Xingwang
Published: July 2002

In this article we show that there exists a rational number

μ
\mu

, depending only on the dimension

n
(
≥
5
)

n (\ge 5)

of the manifold such that the

μ
\mu

th-power of the conformal factor is bounded in

H
2
2

H^2_2

norm in terms of volume bound and the square norm bound of the scalar curvature of the conformal metrics. Some applications are also given.
Residually nilpotent one-relator groups with nontrivial centre
Authors: McCarron James
Published: July 2002

We determine explicitly the residually nilpotent one-relator groups with nontrivial centre. We show also that, if

G
G

is a one-relator group, then

G
G

is residually nilpotent if, and only if, its central quotient

G

/

Z
(
G
)

G/Z(G)

is residually nilpotent.
Invariant points of maps between Grassmannians
Authors: Mukherjea Kalyan , Sankaran Parameswaran
Published: July 2002

We show that for a very large class of integers

l
>
k
>
n

l>k>n

and any map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper G Subscript k Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis right-arrow upper G Subscript l Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis">

f
:

G
k

(

R

n

)
→

G
l

(

R

n

)

f\colon G_k(\mathbb R^n)\to G_l(\mathbb R^n)

between Grassmannians, there is some

k
k

-plane of

R

n

\mathbb R^n

which is mapped into a subspace of itself.
Singular solutions for a class of Grusin type operators
Authors: Hanges Nicholas , Himonas A.
Published: July 2002

We construct singular solutions for a one-parameter family of partial differential equations with double characteristics and with complex lower order terms. The parameter belongs to a discrete set which is described in terms of the spectrum of a related differential operator.
Degrees of unsolvability of first order decision problems for finitely presented groups
Authors: Belegradek Oleg
Published: July 2002

We show that for any arithmetical

m
m

-degree

d

\mathbf {d}

there is a first order decision problem

P

\mathbf {P}

such that

P

\mathbf {P}

has

m
m

-degree

d

\mathbf {d}

for the free 2-step nilpotent group of rank 2. This implies a conjecture of Sacerdote.
An integral generalization of the 𝑞-binomial theorem and an application
Authors: Liu Yunkang
Published: July 2002

An integral identity which generalizes the

q
q

-binomial theorem is presented. This identity is used to determine the exponential expansion of the solution of an integro-differential equation.
The Lu Qi-Keng conjecture fails generically
Authors: Boas Harold
Published: July 2002

The bounded domains of holomorphy in 

C

n

\mathbb {C}^n

whose Bergman kernel functions are zero-free form a nowhere dense subset (with respect to a variant of the Hausdorff distance) of all bounded domains of holomorphy.
Combinatorial orthogonal expansions
Authors: de Médicis A. , Stanton D.
Published: July 2002

The linearization coefficients for a set of orthogonal polynomials are given explicitly as a weighted sum of combinatorial objects. Positivity theorems of Askey and Szwarc are corollaries of these expansions.
A note on a characteristic property based on order statistics
Authors: Rao C. , Shanbhag D.
Published: July 2002

It is shown that the extended version of the Puri-Rubin result given recently by Stadje (1994) is neither new nor the most general available in the literature.
Diffeomorphisms with persistency
Authors: Sakai Kazuhiro
Published: July 2002

The

C
1

C^1

interior of the set of all diffeomorphisms satisfying Lewowicz’s persistency is characterized as the set of all diffeomorphisms satisfying Axiom A and the strong transversality condition.
On tertiary exotic characteristic classes
Authors: Singh Neeta
Published: July 2002

We prove the existence and nontriviality of tertiary exotic characteristic classes extending the results of Peterson and Ravenel for secondary exotic classes.
A new proof of Kunen’s inconsistency
Authors: Zapletal Jindrich
Published: July 2002

Using a basic fact from Shelah’s theory of possible cofinalities, we give a new proof of Kunen’s inconsistency theorem: there is no nontrivial elementary embedding of the set-theoretical universe into itself.
Nicely generated and chaotic ideals
Authors: Brendle Jörg
Published: July 2002

We show that if the real line is the disjoint union of

κ
\kappa

meager sets such that every meager set is contained in a countable union of them, then

κ
=

ω
1

\kappa = \omega _1

. This answers a question addressed by Jacek Cichoń. We also prove two theorems saying roughly that any attempt to produce the isomorphism type of the meager ideal in the Cohen real and the random real extensions must fail. All our results hold for meager replaced by null, as well.
On the zero sets of certain entire functions
Authors: Eremenko Alexandre , Rubel L.
Published: July 2002

We consider the class

B

\mathbf B

of entire functions of the form
\[

f
=
∑

p
j

exp
⁡

g
j

,

f=\sum p_j\exp g_j,

\]
where

p
j

p_j

are polynomials and

g
j

g_j

are entire functions. We prove that the zero-set of such an

f
f

, if infinite, cannot be contained in a ray. But for every region containing the positive ray there is an example of

f
∈

B

f\in \mathbf B

with infinite zero-set which is contained in this region.
Symmetries of periodic solutions for planar potential systems
Authors: Golubitsky Martin , Mao Jian-Min , Nicol Matthew
Published: July 2002

In this article we discuss the symmetries of periodic solutions to Hamiltonian systems with two degrees of freedom in mechanical form. The possible symmetries of such periodic trajectories are generated by spatial symmetries (a finite subgroup of

\textbf {O(2)}
)

\text {\textbf {O(2)}})

, phase-shift symmetries (the circle group

\textbf {S}
1

)

\text {\textbf {S}}^1)

, and a time-reversing symmetry (associated with mechanical form). We focus on the symmetries and structures of the trajectories in configuration space (

R

2

\mathbb R^2

), showing that special properties such as self-intersections and brake orbits are consequences of symmetry.
A note on quasi-Frobenius rings
Authors: Huynh Dinh , Tung Ngo
Published: July 2002

It is shown that a semiperfect ring

R
R

is quasi-Frobenius if and only if every closed submodule of

R
(
ω
)

R(\omega )

is non-small, where

R
(
ω
)

R(\omega )

denotes the direct sum of

ω
\omega

copies of the right

R
R

-module

R
R

and

ω
\omega

is the first infinite ordinal.
Ladder operators for Szegö polynomials and related biorthogonal rational functions
Authors: Ismail Mourad , Rahman Mizan
Published: July 2002

We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szegő and for their four parameter generalization to

4

ϕ
3

{}_4\phi _3

biorthogonal rational functions on the unit circle.
The distance from the Apostol spectrum
Authors: Kordula V. , Müller V.
Published: July 2002

If

T
T

is an s-regular operator in a Banach space (i.e.

T
T

has closed range and

N
(
T
)
⊂

R

∞

(
T
)

N(T)\subset R^{\infty }(T)

) and

γ
(
T
)

\gamma (T)

is the Kato reduced minimum modulus, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts gamma left-parenthesis upper T Superscript n Baseline right-parenthesis Superscript 1 slash n Baseline equals sup left-brace right-brace colon r colon greater-than greater-than minus minus upper T times times times lamda separator separator times times times iss hyphen times times times times times times times times times times regularfor vertical-bar vertical-bar lamda r period">

lim

n
→
∞

γ
(

T

n

)

1

/

n

=
sup
{
r
:
T
−
λ

i
s
s
-
r
e
g
u
l
a
r
f
o
r

|

λ

|

>
r
}
.

\begin{equation*}\lim _{n\to \infty }\gamma (T^{n})^{1/n}=\sup \{r: T-\lambda { \operatorname {is s-regular for }}|\lambda |>r\}. \end{equation*}
Completely positive projections on a Hilbert space
Authors: Miura Yasuhide
Published: July 2002

The purpose of this paper is to prove that a completely positive projection on a Hilbert space associated with a standard form of a von Neumann algebra induces the existence of a conditional expectation of the von Neumann algebra with respect to a normal state, and we consider the application to a standard form of an injective von Neumann algebra.
A coherent family of partial functions on ℕ
Authors: Farah Ilijas
Published: July 2002

We prove that there is a family of partial functions

f
α

:

A
α

→
α

f_\alpha :A_\alpha \to \alpha

(
α
→

ω
1

,

A
α

(\alpha \to \omega _1,A_\alpha

is a tower in

P
(
ω
)

/

Fin
)

P(\omega )/\operatorname {Fin})

such that every surjection

g
:

ω
1

→
{
0
,
1
}

g:\omega _1\to \{0,1\}

is associated to a cohomologically different Hausdorff gap (see Talayco). This improves a result of Talayco.
Some converses of the strong separation theorem
Authors: Gau Hwa-Long , Wong Ngai-Ching
Published: July 2002

A convex subset

B
B

of a real locally convex space

X
X

is said to have the separation property if it can be separated from every closed convex subset

A
A

of

X
X

, which is disjoint from

B
B

, by a closed hyperplane. The strong separation theorem says that if

B
B

is weakly compact, then it has the separation property. In this paper, we present two versions of the converse and discuss an application of them. For example, we prove that a normed space is reflexive if and only if its closed unit ball has the separation property. Results in this paper can be considered as supplements of the famous theorem of James.
Derivations with Engel conditions on multilinear polynomials
Authors: Lee Pjek-Hwee , Lee Tsiu-Kwen
Published: July 2002

Let

R
R

be a prime algebra over a commutative ring

K
K

with unity and let

f
(

X

1

,
…
,

X

n

)

f(X_{1}, \ldots , X_{n})

be a multilinear polynomial over

K
K

. Suppose that

d
d

is a nonzero derivation on

R
R

such that for all

r

1

,
…
,

r

n

r_{1}, \ldots , r_{n}

in some nonzero ideal

I
I

of

R
R

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket d left-parenthesis f left-parenthesis r 1 comma ellipsis comma r Subscript n Baseline right-parenthesis right-parenthesis comma f left-parenthesis r 1 comma ellipsis comma r Subscript n Baseline right-parenthesis right-bracket Subscript k Baseline equals 0">

[

d

(

f
(

r

1

,
…
,

r

n

)

)

,
f
(

r

1

,
…
,

r

n

)

]

k

=
0

\Big [ d\big ( f(r_{1}, \ldots , r_{n})\big ), f(r_{1}, \ldots , r_{n}) \Big ]_{k} = 0

with

k
k

fixed. Then

f
(

X

1

,
…
,

X

n

)

f(X_{1}, \ldots , X_{n})

is central–valued on

R
R

except when char

R
=
2

R=2

and

R
R

satisfies the standard identity

s

4

s_{4}

in 4 variables.
Stability and dichotomy of positive semigroups on 𝐿_{𝑝}
Authors: Montgomery-Smith Stephen
Published: July 2002

A new proof of a result of Lutz Weis is given, that states that the stability of a positive strongly continuous semigroup

(

e

t
A

)

t
≥
0

(e^{tA})_{t \ge 0}

on

L
p

L_p

may be determined by the quantity

s
(
A
)

s(A)

. We also give an example to show that the dichotomy of the semigroup may not always be determined by the spectrum

σ
(
A
)

\sigma (A)

.
Multipliers of weak type on locally compact Vilenkin groups
Authors: Quek T.
Published: July 2002

Let

G
G

be a locally compact Vilenkin group with dual group

Γ
\Gamma

. We give a sufficient condition for

f
∈

L
∞

(
Γ
)

f\in L^\infty (\Gamma )

to be a multiplier of weak type

(
p
,
p
)

(p,p)

on

G
G

. Some applications of our result are given. We also prove that our result is sharp.
Remarks on the local Hopf’s lemma
Authors: Shklover Vladimir
Published: July 2002

The paper deals with the problem of extending the recent work of M.S.Baouendi and L.P.Rothschild concerning harmonic functions vanishing to infinite order in the normal direction in balls and half-spaces. Contrary to what one expects, we show that the B.-R. result extends neither to arbitrary domains nor to cases when the normal is replaced by a curve transversal to the boundary. The exact criterion when the result holds in

R

2

\mathbf {R}^2

is given.
Radial symmetry of large solutions of nonlinear elliptic equations
Authors: Taliaferro Steven
Published: July 2002

We give conditions under which all

C
2

C^2

solutions of the problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column Blank 2nd Column a m p semicolon normal upper Delta u equals f left-parenthesis StartAbsoluteValue x EndAbsoluteValue comma u right-parenthesis comma x element-of double-struck upper R Superscript n Baseline comma 3rd Column a m p semicolon limit Underscript StartAbsoluteValue x EndAbsoluteValue right-arrow normal infinity Endscripts u left-parenthesis x right-parenthesis equals normal infinity EndLayout">

a
m
p
;
Δ
u
=
f
(

|

x

|

,
u
)
,

x
∈

R

n

,
 

a
m
p
;

lim

|

x

|

→
∞

u
(
x
)
=
∞

\begin{align*} &\Delta u = f(|x|,u),\qquad x\in {\mathbb {R}}^n,\ &\lim _{|x|\to \infty } u(x) = \infty \end{align*}

are radial. We assume

f
(

|

x

|

,
u
)

f(|x|,u)

is positive when

|

x

|

|x|

and

u
u

are both large and positive. Since this problem with

f
(

|

x

|

,
u
)
=
u

f(|x|,u) = u

has non-radial solutions, we rule out this possibility by requiring that

f
(

|

x

|

,
u
)

f(|x|,u)

grow superlinearly in

u
u

when

|

x

|

|x|

and

u
u

are both large and positive. However we make no assumptions on the rate of growth of solutions.
On the higher delta invariants of a Gorenstein local ring
Authors: Yoshino Yuji
Published: July 2002

Let

(
R
,

m

)

(R , \mathfrak {m} )

be a Gorenstein complete local ring. Auslander’s higher delta invariants are denoted by

δ
R
n

(
M
)

\delta _R ^n(M)

for each module

M
M

and for each integer

n
n

. We propose a conjecture asking if

δ
R
n

(
R

/

m

ℓ

)
=
0

\delta _R ^n (R/\mathfrak {m} ^{\ell }) = 0

for any positive integers

n
n

and

ℓ
\ell

. We prove that this is true provided the associated graded ring of

R
R

has depth not less than

dim
⁡
R
−
1

\operatorname {dim} R -1

. Furthermore we show that there are only finitely many possibilities for a pair of positive integers

(
n
,
ℓ
)

(n, \ell )

for which

δ
R
n

(
R

/

m

ℓ

)
>
0

\delta _R ^n (R/ \mathfrak {m} ^{\ell }) >0

.
Danes’ Drop Theorem in locally convex spaces
Authors: Lixin Cheng , Yunchi Zhou , Fong Zhang
Published: July 2002

Danes’ Drop Theorem is generalized to locally convex spaces.
Integer-valued polynomials on Krull rings
Authors: Frisch Sophie
Published: July 2002

If

R
R

is a subring of a Krull ring

S
S

such that

R

Q

R_{Q}

is a valuation ring for every finite index

Q
=
P
∩
R

Q=P\cap R

,

P
P

in Spec

1

(
S
)

^{1}(S)

, we construct polynomials that map

R
R

into the maximal possible (for a monic polynomial of fixed degree) power of

P

S

P

PS_{P}

, for all

P
P

in Spec

1

(
S
)

^{1}(S)

simultaneously. This gives a direct sum decomposition of Int

(
R
,
S
)

(R,S)

, the

S
S

-module of polynomials with coefficients in the quotient field of

S
S

that map

R
R

into

S
S

, and a criterion when Int

(
R
,
S
)

(R,S)

has a regular basis (one consisting of 1 polynomial of each non-negative degree).
Coloring finite subsets of uncountable sets
Authors: Komjáth Péter , Shelah Saharon
Published: July 2002

It is consistent for every

1
≤
n
>
ω

1\leq n> \omega

that

2

ω

=

ω

n

2^{\omega }=\omega _{n}

and there is a function

F
:
[

ω

n

]

>
ω

→
ω

F:[\omega _{n}]^{> \omega }\to \omega

such that every finite set can be written in at most

2

n

−
1

2^{n}-1

ways as the union of two distinct monocolored sets. If GCH holds, for every such coloring there is a finite set that can be written at least

1
2

∑

i
=
1

n

(

n
+
i

n

)

(

n
i

)

\frac {1}{2}\sum ^{n}_{i=1}{\binom {n+i}{n}}{\binom {n}{i}}

ways as the union of two sets with the same color.
A Wolff-Denjoy theorem for infinitely connected Riemann surfaces
Authors: Lárusson Finnur
Published: July 2002

We generalize the classical Wolff-Denjoy theorem to certain infinitely connected Riemann surfaces. Let

X
X

be a non-parabolic Riemann surface with Martin boundary

Δ
\Delta

. Suppose each Martin function

k

y

k_{y}

,

y
∈
Δ

y\in \Delta

, extends continuously to

Δ
∖
{
y
}

\Delta \setminus \{y\}

and vanishes there. We show that if

f
f

is an endomorphism of

X
X

and the iterates of

f
f

converge to the point at infinity, then the iterates converge locally uniformly to a point in

Δ
\Delta

. As an application, we extend the Wolff-Denjoy theorem to non-elementary Gromov hyperbolic covering spaces of compact Riemann surfaces. Such covering surfaces are of independent interest. Finally, we use the theory of non-tangential boundary limits to give a version of the Wolff-Denjoy theorem that imposes certain mild restrictions on

f
f

but none on

X
X

itself.
Resonance and non-resonance in a problem of boundedness
Authors: Ortega Rafael , Tineo Antonio
Published: July 2002

This paper studies the existence of bounded solutions of a forced non-linear differential equation of arbitrary order. Necessary and sufficient conditions for the existence of such solutions are obtained. These results are inspired by classical results on the periodic problem, both in the resonant and non-resonant cases.
On the failure of close-to-normal structure type conditions and pathological Kannan maps
Authors: Smyth Michael
Published: July 2002

We consider the failure of close-to-normal structure type conditions and show that a Banach space can be renormed to fail close-to-weak normal structure exactly when it contains a norm inseparable weakly compact subset. Included is an example of a particularly pathological fixed point free Kannan map.
More nice equations for nice groups
Authors: Abhyankar Shreeram
Published: July 2002

Nice quintinomial equations are given for unramified coverings of the affine line in nonzero characteristic

p
p

with PSp

(
2
m
,
q
)

(2m,q)

and Sp

(
2
m
,
q
)

(2m,q)

as Galois groups where

m
>
2

m>2

is any integer and

q
>
1

q>1

is any power of

p
p

.