bluuebunny/crossref_metadata_2025 huggingface dataset:
𝐿𝑖𝑝𝛼 harmonic approximation on closed sets
Authors: Bonilla A. , Fariña J.
Published: July 2002

In this paper the

L
i
p
α

Lip\alpha

harmonic approximation (

0
>
α
>

1
2

0 > \alpha > \frac {1}{2}

) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on

L
i
p
α

Lip\alpha

harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in

L
i
p
α

Lip\alpha

harmonic approximation are also studied.
A simple proof of some generalized principal ideal theorems
Authors: Eisenbud David , Huneke Craig , Ulrich Bernd
Published: July 2002

Using symmetric algebras we simplify

(
(

and slightly strengthen

)
)

the Bruns-Eisenbud-Evans “generalized principal ideal theorem” on the height of order ideals of nonminimal generators in a module. We also obtain a simple proof and an extension of a result by Kwieciński, which estimates the height of certain Fitting ideals of modules having an equidimensional symmetric algebra.
Comparison of 4-class ranks of certain quadratic fields
Authors: Gerth Frank
Published: July 2002

Let

m
m

be a square-free positive integer. Let

r

4

(
K
)

r_{4}(K)

denote the 4-class rank of a quadratic field

K
K

. This paper examines how likely it is for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r 4 left-parenthesis double-struck upper Q left-parenthesis StartRoot negative m EndRoot right-parenthesis right-parenthesis equals r 4 left-parenthesis double-struck upper Q left-parenthesis StartRoot m EndRoot right-parenthesis right-parenthesis">

r

4

(

Q

(

−
m

)
)
=

r

4

(

Q

(

m

)
)

r_{4}(\mathbb {Q} (\sqrt {-m}\,)) =r_{4} (\mathbb {Q} (\sqrt {m}\,))

and for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r 4 left-parenthesis double-struck upper Q left-parenthesis StartRoot negative m EndRoot right-parenthesis right-parenthesis equals r 4 left-parenthesis double-struck upper Q left-parenthesis StartRoot m EndRoot right-parenthesis right-parenthesis plus 1">

r

4

(

Q

(

−
m

)
)
=

r

4

(

Q

(

m

)
)
+
1

r_{4} (\mathbb {Q} (\sqrt {-m}\,)) = r_{4} (\mathbb {Q} (\sqrt {m}\,)) +1

.
A remark on a theorem of J. Tits
Authors: Bennett Curtis , Shpectorov Sergey
Published: July 2002

Let

G
G

be a rank two Chevalley group and

Γ
\Gamma

be the corresponding Moufang polygon. J. Tits proved that

G
G

is the universal completion of the amalgam formed by three subgroups of

G
G

: the stabilizer

P
1

P_1

of a point

a
a

of

Γ
\Gamma

, the stabilizer

P
2

P_2

of a line

ℓ
\ell

incident with

a
a

, and the stabilizer

N
N

of an apartment

A
A

passing through

a
a

and

ℓ
\ell

. We prove a slightly stronger result, in which the exact structure of

N
N

is not required. Our result can be used in conjunction with the “weak

B
N

BN

-pair" theorem of Delgado and Stellmacher in order to identify subgroups of finite groups generated by minimal parabolics.
Decreasing flow invariant sets and fixed points of quasimonotone increasing operators
Authors: Sun Jingxian , Xu Xi’an
Published: July 2002

In this paper, we obtain some new results about the existence of multiple fixed points of a kind of quasimonotone increasing operator by the new method of decreasing flow invariant set.
A new statistic for the 3𝑥+1 problem
Authors: Gluck David , Taylor Brian
Published: July 2002

A finite

(
3
x
+
1
)

(3x+1)

–trajectory is a sequence

a
_

=

a
1

,
…
,

a
n

\underline { a}=a_1,\ldots ,a_n

of positive integers such that

a

i
+
1

=
3

a
i

+
1

a_{i+1}=3a_i+ 1

if

a
i

a_i

is odd,

a

i
+
1

=

a
i

/

2

a_{i+1}=a_i/2

if

a
i

a_i

is even,

a
i

>
1

a_i>1

if

i
>
n

i>n

and

a
n

=
1

a_n=1

. For such a sequence

a
_

\underline { a}

we define
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis a underbar right-parenthesis equals left-parenthesis a 1 a 2 plus midline-horizontal-ellipsis plus a Subscript n minus 1 Baseline a Subscript n Baseline plus a Subscript n Baseline a 1 right-parenthesis slash left-parenthesis a 1 squared plus midline-horizontal-ellipsis plus a Subscript n Superscript 2 Baseline right-parenthesis">

C
(

a
_

)
=
(

a
1

a
2

+
⋯
+

a

n
−
1

a
n

+

a
n

a
1

)

/

(

a
1
2

+
⋯
+

a
n
2

)

C(\underline { a}) = (a_1a_2+\cdots +a_{n-1}a_n+a_na_1)/(a_1^2+\cdots +a_n^2)

. We prove that

9

/

13
>
C
(

a
_

)
>
5

/

7

9/13>C(\underline { a})>5/7

if

a
1

a_1

is odd and

a
1

≥
3

a_1\ge 3

. Histograms suggest that

C
C

may have an interesting limiting distribution.
Semigroups of mappings with rigid Lipschitz constant
Authors: Llorens-Fuster Enrique
Published: July 2002

It is shown that two well-known uniformly fixed point free lipschitzian semigroups of mappings have minimal Lipschitz constant on the positive part of the unit ball of

ℓ
2

\ell _2

. This implies that a question raised by T. Kuczumow has a negative answer.
The additivity of porosity ideals
Authors: Brendle Jörg
Published: July 2002

We show that several

σ
\sigma

-ideals related to porous sets have additivity

ω
1

\omega _1

and cofinality

2
ω

2^\omega

. This answers a question addressed by Miroslav Repický.
Sequences of differential systems
Authors: Nagata Makoto
Published: July 2002

Our main result deals with certain properties of the sizes of a differential system and its dual system. It represents an improvement of part of the results in André’s G-function theory (Max-Planck-Institut, Bonn, 1989).
Any behaviour of the Mitchell ordering of normal measures is possible
Authors: Witzany Jiří
Published: July 2002

Let

U
0

,

U
1

U_0,U_1

be two normal measures on

κ
\kappa

. We say that

U
0

U_0

is in the Mitchell ordering less than

U
1

,

U_1,

U
0

⊲

U
1

,

U_0\vartriangleleft U_1,

if

U
0

∈
a
n

U
l
t

(
V
,

U
1

)
.

U_0 \in an{Ult}(V,U_1) .

The relation is well-known to be transitive and well-founded. It has been an open problem to find a model where

⊲
\vartriangleleft

embeds the four-element poset \epssymbol{proc3019e-fig-a}. We find a generic extension where all well-founded posets are embeddable. Hence there is no structural restriction on the Mitchell ordering. Moreover we show that it is possible to have two

⊲
\vartriangleleft

-incomparable measures that extend in a generic extension into two

⊲
\vartriangleleft

-comparable measures.
Proof of the Simon-Ando Theorem
Authors: Hartfiel D.
Published: July 2002

In 1961, Simon and Ando wrote a classical paper describing the convergence properties of nearly completely decomposable matrices. Basically, their work concerned a partitioned stochastic matrix e.g.
\[

A
=

[

A
1

a
m
p
;

E
1

 

E
2

a
m
p
;

A
2

]

A = \begin {bmatrix} A_1&E_1\ E_2&A_2\end {bmatrix}

\]
where

A
1

A_1

and

A
2

A_2

are square blocks whose entries are all larger than those of

E
1

E_1

and

E
2

E_2

respectively.

Setting
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Superscript k Baseline equals Start 1 By 3 Matrix 1st Row 1st Column upper A 1 Superscript left-parenthesis k right-parenthesis Baseline 2nd Column a m p semicolon upper E 1 Superscript left-parenthesis k right-parenthesis Baseline upper E 2 Superscript left-parenthesis k right-parenthesis Baseline 3rd Column a m p semicolon upper A 2 Superscript left-parenthesis k right-parenthesis Baseline EndMatrix comma">

A
k

=

[

A
1

(
k
)

a
m
p
;

E
1

(
k
)

 

E
2

(
k
)

a
m
p
;

A
2

(
k
)

]

,

A^k=\begin {bmatrix} A^{(k)}_1&E^{(k)}_1\ E^{(k)}_2&A^{(k)}_2\end {bmatrix},

\]
partitioned as in

A
A

, they observed that for some, rather short, initial sequence of iterates the main diagonal blocks tended to matrices all of whose rows are identical, e.g.

A
1

(
k
)

A^{(k)}_1

to

F
1

F_1

and

A
2

(
k
)

A^{(k)}_2

to

F
2

F_2

. After this initial sequence, subsequent iterations showed that all blocks lying in the same column as those matrices tended to a scalar multiple of them, e.g.
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript k right-arrow normal infinity Endscripts upper A Superscript k Baseline equals Start 1 By 3 Matrix 1st Row 1st Column alpha upper F 1 2nd Column a m p semicolon beta upper F 2 alpha upper F 1 3rd Column a m p semicolon beta upper F 2 EndMatrix">

lim

k
→
∞

A
k

=

[

α

F
1

a
m
p
;
β

F
2

 
α

F
1

a
m
p
;
β

F
2

]

\lim _{k\to \infty }A^k=\begin {bmatrix} \alpha F_1&\beta F_2\ \alpha F_1&\beta F_2\end {bmatrix}

\]
where

α
≥
0

\alpha \geq 0

and

β
≥
0

\beta \geq 0

.

The purpose of this paper is to give a qualitative proof of the Simon-Ando theorem.
On the unique range set of meromorphic functions
Authors: Li Ping , Yang Chung-Chun
Published: July 2002

This paper studies the unique range set of meromorphic functions and shows that there exists a finite set

S
S

such that for any two nonconstant meromorphic functions

f
f

and

g
g

the condition

E
f

(
S
)
=

E
g

(
S
)

E_f(S)=E_g(S)

implies

f
≡
g

f\equiv g

. As a special case this also answers an open question posed by Gross (1977) about entire functions and improves some results obtained recently by Yi.
On anticonformal automorphisms of Riemann surfaces with nonembeddable square
Authors: Costa Antonio
Published: July 2002

In this paper we present an example of an anticonformal automorphism whose square has prime order and is not embeddable. We prove that every embeddable automorphism of odd order of a compact Riemann surface is the square of an orientation-reversing self-homeomorphism. Finally we study whether a conformal involution is the square of an orientation-reversing automorphism.
On the classification of pyramidal central configurations
Authors: Fayçal Nelly
Published: July 2002

We present some results associated with the existence of central configurations (c.c.’s) in the classical gravitational

N
N

-body problem of Newton. We call a central configuration of five bodies, four of which are coplanar, a

p
y
r
a
m
i
d
a
l

pyramidal

central configuration (p.c.c). It can be shown that there are only three types of p.c.c.’s, admitting one or more planes of symmetry, viz. (i) the case where the planar bodies lie at the vertices of a regular trapezoid, (ii) the case where the bodies lie at the vertices of a kite-shaped quadrilateral, and (iii) the case where the bodies lie at the vertices of a rectangle. In this paper we classify all p.c.c.’s with a rectangular base and, in fact, prove that there is only one such c.c., namely, the square-based pyramid with equal masses at the corners of the square. The classification of all p.c.c.’s satisfying either (i) or (ii) will be discussed in subsequent papers.
Sobolev imbedding theorems in borderline cases
Authors: Fusco Nicola , Lions Pierre , Sbordone Carlo
Published: July 2002

An imbedding theorem is given for functions whose gradient belongs to a class slightly larger than

L
n

(
Ω
)

L^n(\Omega )

,

Ω
⊂

R

n

\Omega \subset \mathbb {R}^n

.
On the solvability of a nonlinear second-order elliptic equation at resonance
Authors: Kuo Chung-Cheng
Published: July 2002

We study the existence of solutions of the Neumann problem for semilinear second-order elliptic equations at resonance in which the nonlinear terms may grow superlinearly in one of the directions

u
→
∞

u\to \infty

and

u
→
−
∞

u\to -\infty

, and sublinearly in the other. Solvability results are obtained under assumptions either with or without a Landesman-Lazer condition. The proofs are based on degree-theoretic arguments.
Vanishing of the leading term in Harish-Chandra’s local character expansion
Authors: Huntsinger Reid
Published: July 2002

Harish-Chandra’s formula for the character

Θ
π

\Theta _\pi

of an irreducible smooth representation

π
\pi

of a reductive

p
p

-adic group

G
G

expresses

Θ
π

\Theta _\pi

near

1
1

as a linear combination of the Fourier transforms of nilpotent

G
G

-orbits in the Lie algebra of

G
G

. In this note, we prove that if

π
\pi

is tempered but not in the discrete series, then the coefficient attached to the zero nilpotent orbit vanishes.
Spectrum of positive entropy multidimensional dynamical systems with a mixed time
Authors: Kaminski B.
Published: July 2002

It is shown that if an abelian countable group

G
=

G

1

⊕

G

2

G = G_{1}\oplus G_{2}

is such that

G

2

G_{2}

is a finite group and every aperiodic positive entropy action

Φ
\Phi

of

G

1

G_{1}

on a Lebesgue probability space

(
X
,

B

,
μ
)

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

has a countable Haar spectrum in the subspace

L

0

2

(
X
,
μ
)
⊖

L

0

2

(
X
,
Π
(
Φ
)
,
μ
)

L^{2}_{0}(X,\mu )\ominus L^{2}_{0}(X,\Pi (\Phi ),\mu )

, where

Π
(
Φ
)

\Pi (\Phi )

denotes the Pinsker

σ
\sigma

- algebra of

Φ
\Phi

, then every aperiodic positive entropy action of

G
G

on

(
X
,

B

,
μ
)

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

has the same property. A positive answer to the question of J.P. Thouvenot is obtained as a corollary.
Rotational symmetry of the Hermite projection operators
Authors: Kochneff E.
Published: July 2002

We calculate an integral formula for the Hermite projection operators. We give some applications of our formula. We also give a short proof of a recent theorem of Thangavelu
Inner invariant means and conjugation operators
Authors: Takahashi Yuji
Published: July 2002

It is shown that Paschke’s result cannot be generalized to the [IN]-group setting given by Lau and Paterson. This resolves negatively a problem raised by Lau and Paterson.
Finite factorization domains
Authors: Anderson D. , Mullins Bernadette
Published: July 2002

An integral domain

R
R

is a finite factorization domain if each nonzero element of

R
R

has only finitely many divisors, up to associates. We show that a Noetherian domain

R
R

is an FFD

⇔
\Leftrightarrow

for each overring

R
′

R’

of

R
R

that is a finitely generated

R
R

-module,

U
(

R
′

)

/

U
(
R
)

U(R’)/U(R)

is finite. For

R
R

local this is also equivalent to each

R

/

[
R
:

R
′

]

R/[R:R’]

being finite. We show that a one-dimensional local domain

(
R
,
M
)

(R,M)

is an FFD

⇔
\Leftrightarrow

either

R

/

M

R/M

is finite or

R
R

is a DVR.
On the size of lemniscates of polynomials in one and several variables
Authors: Cuyt A. , Driver K. , Lubinsky D.
Published: July 2002

In the convergence theory of rational interpolation and Padé approximation, it is essential to estimate the size of the lemniscatic set

E
:=

{

z

:

|

z

|

≤
r

E:=\big \{z\,:\, |z|\le r

and

|

P
(
z
)

|

≤

ϵ

n

}

|P(z)|\le \epsilon ^{n}\big \}

, for a polynomial

P
P

of degree

≤
n

\le n

. Usually,

P
P

is taken to be monic, and either Cartan’s Lemma or potential theory is used to estimate the size of

E
E

, in terms of Hausdorff contents, planar Lebesgue measure

m

2

m_{2}

, or logarithmic capacity cap. Here we normalize

‖
P

‖

L

∞

(

|

z

|

≤
r

)

=
1

\|P\|_{L_{\infty }\bigl (|z|\le r\bigr )}=1

and show that cap

(
E
)
≤
2
r
ϵ

(E)\le 2r\epsilon

and

m

2

(
E
)
≤
π
(
2
r
ϵ

)

2

m_{2} (E)\le \pi (2r\epsilon )^{2}

are the sharp estimates for the size of

E
E

. Our main result, however, involves generalizations of this to polynomials in several variables, as measured by Lebesgue measure on

C

n

\mathbb {C}^{n}

or product capacity and Favarov’s capacity. Several of our estimates are sharp with respect to order in

r
r

and

ϵ
\epsilon

.
\undertilde{𝛿}₂¹ without sharps
Authors: Friedman Sy , Woodin W.
Published: July 2002

We show that the supremum of the lengths of

Δ
∼

2
1

{\underset {\sim }{\Delta }}^1_2

prewellorderings of the reals can be

ω

2

\omega _{2}

, with

ω

1

\omega _{1}

inaccessible to reals, assuming only the consistency of an inaccessible.
Products of quasi-measures
Authors: Grubb D.
Published: July 2002

A quasi-state is a positive functional on

C
(
X
)

C(X)

that is only assumed to be linear on singly-generated subalgebras. We consider the “iterated integral” of two quasi-states and determine when this gives a quasi-state on the product space. We also provide explicit formulas for the corresponding quasi-measures in case it does. Finally, we show the general failure of Fubini’s Theorem for quasi-states.
The indices, the nullities and the stability of totally geodesic submanifolds in the complex quadratic hypersurfaces: 𝑄_{𝑚}=𝑆𝑂(𝑚+2)/𝑆𝑂(𝑚)×𝑆𝑂(2)
Authors: Qiang Zhao
Published: July 2002

In the paper, the stability of totally geodesic submanfolds in the complex quadratic hypersurfaces:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Subscript m Baseline equals upper S upper O left-parenthesis m plus 2 right-parenthesis slash upper S upper O left-parenthesis m right-parenthesis times upper S upper O left-parenthesis 2 right-parenthesis left-parenthesis m greater-than 1 right-parenthesis">

Q
m

=
S
O
(
m
+
2
)

/

S
O
(
m
)
×
S
O
(
2
)
(
m
>
1
)

Q_m=SO(m+2)/SO(m)\times SO(2) (m>1)

is discussed, and the indices, the nullities and the Killing nullities of totally geodesic submanifolds in

Q
m

Q_m

are calculated.
Π₁¹ sets of unbounded Loeb measure
Authors: Zivaljevic Bosko
Published: July 2002

For every

Π

1

1

\Pi _{1}^{1}

and non-Borel subset

P
P

of an internal set

X
X

in a

ℵ

2

\aleph _{2}

saturated nonstandard universe there exists an internal, unbounded, non-atomic measure

μ
\mu

so that

L
(
μ
)
(
P
△
B
)

L(\mu )(P\triangle B)

is not finite for any Borel set

B
B

in

X
.

X.
A note on Fuchs’ Problem 34
Authors: Albrecht U. , Goeters H.
Published: July 2002

We investigate to what extent an abelian group

G
G

is determined by the homomorphism groups

Hom
⁡
(
G
,
B
)

\operatorname {Hom}(G,B)

where

B
B

is chosen from a set

X

\mathcal {X}

of abelian groups. In particular, we address Problem 34 in Professor Fuchs’ book which asks if

X

\mathcal {X}

can be chosen in such a way that the homomorphism groups determine

G
G

up to isomorphism. We show that there is a negative answer to this question. On the other hand, there is a set

X

\mathcal {X}

which determines the torsion-free groups of finite rank up to quasi-isomorphism.
The two-cardinals transfer property and resurrection of supercompactness 󠀼span style=󠀢color:red󠀢󠀾This article has been retracted󠀼/span󠀾
Authors: Ben-David Shai , Shelah Saharon
Published: July 2002

We show that the transfer property

(

ℵ
1

,

ℵ
0

)
→
(

λ
+

,
λ
)

(\aleph _1,\aleph _0)\to (\lambda ^+,\lambda )

for singular

λ
\lambda

does not imply (even) the existence of a non-reflecting stationary subset of

λ
+

\lambda ^+

. The result assumes the consistency of ZFC with the existence of infinitely many supercompact cardinals. We employ a technique of “resurrection of supercompactness”. Our forcing extension destroys the supercompactness of some cardinals; to show that in the extended model they still carry some of their compactness properties (such as reflection of stationary sets), we show that their supercompactness can be resurrected via a tame forcing extension.
Asymptotic behavior of 𝐶₀-semigroups in Banach spaces
Authors: Weis Lutz , Wrobel Volker
Published: July 2002

We present optimal estimates for the asymptotic behavior of strongly continuous semigroups

U
A

:
[
0
,
∞
[
→
L
(
X
)

U_A:[0,\infty [\rightarrow L(X)

in terms of growth abscissas of the resolvent function

R
(
⋅
,
A
)

R(\cdot ,A)

of the generator

A
A

. In particular we give Ljapunov’s classical stability condition a definite form for (infinite dimensional) abstract Cauchy problems: The abscissa of boundedness of

R
(
⋅
,
A
)

R(\cdot ,A)

equals the growth bound of the classical solutions of

y
′

=
A
y

y’=Ay

.
K-theory and the anti-automorphism of the Steenrod algebra
Authors: Crabb M. , Crossley M. , Hubbuck J.
Published: July 2002

We give simple proofs of some relations in the Steenrod algebra involving the powers

P

i

\mathcal {P}^i

and their duals

χ

P

i

\chi \mathcal {P}^i

and show how these relations arise from

K
K

-theory.
Cohomology of groups with metacyclic Sylow 𝑝-subgroups
Authors: Dietz Jill , Martino John , Priddy Stewart
Published: July 2002

We determine the cohomology algebras

H

∗

(
G
;

F

p

)

H^{*}(G;\mathbf {F}_{p})

for all groups

G
G

with a metacyclic Sylow

p
p

-subgroup. The complete

p
p

-local stable decomposition of the classifying space

B
G

BG

is also determined.
Countable network weight and multiplication of Borel sets
Authors: Fremlin D. , Johnson R. , Wajch E.
Published: July 2002

A space

X
X

Borel multiplies with a space

Y
Y

if each Borel set of

X
×
Y

X\times Y

is a member of the

σ
\sigma

-algebra in

X
×
Y

X\times Y

generated by Borel rectangles. We show that a regular space

X
X

Borel multiplies with every regular space if and only if

X
X

has a countable network. We give an example of a Hausdorff space with a countable network which fails to Borel multiply with any non-separable metric space. In passing, we obtain a characterization of those spaces which Borel multiply with the space of countable ordinals, and an internal necessary and sufficient condition for

X
X

to Borel multiply with every metric space.
Lipscomb’s universal space is the attractor of an infinite iterated function system
Authors: Perry J.
Published: July 2002

Lipscomb’s one-dimensional space

L
(
A
)

L(A)

on an arbitrary index set

A
A

is injected into the Tychonoff cube

I
A

I^A

. The image of

L
(
A
)

L(A)

is shown to be the attractor of an iterated function system indexed by

A
A

. This system is conjugate, under an injection, with a set of right-shift operators on Baire’s space

N
(
A
)

N(A)

regarded as a code space. This view of

L
(
A
)

L(A)

extends the fractal nature of

L
(
A
)

L(A)

initiated in a 1992 joint paper by the author and S. Lipscomb. In addition, we give a new proof that as a subspace of Hilbert’s space

l
2

(
A
)

l^2(A)

, the space

L
(
A
)

L(A)

is complete and hence is closed in

l
2

(
A
)

l^2(A)

.
Killing and subordination
Authors: Ying Jiangang
Published: July 2002

Given the one-to-one correspondence between nearly Borel right processes and non-symmetric Dirichlet forms, we prove in the present paper that the killing transform of Markov processes is equivalent to strong subordination of the respective Dirichlet forms and give a characterization of so-called bivariate smooth measures.
Generalized Aluthge transformation on 𝑝-hyponormal operators
Authors: Furuta Takayuki
Published: July 2002

We shall introduce a generalized Aluthge transformation on

p
p

- hyponormal operators and also, by using the Furuta inequality, we shall give several properties on this generalized Aluthge transformation as further extensions of some results of Aluthge.
Existence of positive solutions for singular ordinary differential equations with nonlinear boundary conditions
Authors: Bobisud L. , O’Regan Donal
Published: July 2002

We prove the existence of nonnegative solutions of the problem

(
p

y
′

)
′

/

p
+
μ
q
g
(
x
,
y
)
=
0

(py’)’/p+\mu qg(x,y)=0

,

lim

x
→
0
+

p

y
′

=
0

\lim _{x\to 0+}py’=0

,

h
(

y
′

(
1
)
)
+
y
(
1
)
=
0

h(y’(1))+y(1)=0

for a physically motivated class of nonlinearity

h
h

. The results, which are established using a “forbidden value” argument, are new even in the case of linear

h
h

.
Differentiable selection of optimal solutions in parametric linear programming
Authors: Luc Dinh , Pham Huy Dien
Published: July 2002

In the present paper we prove that if the data of a parametric linear optimization problem are smooth, the solution map admits a local smooth selection “almost” everywhere. This in particular shows that the set of points where the marginal function of the problem is nondifferentiable is nowhere dense.
Group algebras whose units satisfy a group identity
Authors: Giambruno Antonio , Sehgal Sudarshan , Valenti Angela
Published: July 2002

Let

F
G

FG

be the group algebra of a torsion group over an infinite field

F
F

. Let

U
U

be the group of units of

F
G

FG

. We prove that if

U
U

satisfies a group identity, then

F
G

FG

satisfies a polynomial identity. This confirms a conjecture of Brian Hartley.
𝑙_{∞} and interpolation between Banach lattices
Authors: Zobin Nahum , Zobin Veronica
Published: July 2002

We study the possibility of obtaining the

l

∞

l_{\infty }

-norm by an interpolation method starting from a couple of Banach lattice norms. We describe all couples of Banach lattice norms in

R

n

{\mathbb {R}}^{n}

such that the

l

∞

l_{\infty }

-norm is a strict interpolation norm between them. Further we consider the possibility of obtaining the

l

∞

l_{\infty }

-norm by any method which guarantees interpolation of not only linear operators ( = bilinear forms on

R

n

×

R

n

)

{\mathbb {R}}^{n}\times {\mathbb {R}}^{n})

but also of all polylinear forms. Here we show that either one of the initial norms has to be proportional to the

l

∞

l_{\infty }

-norm, or both have to be weighted

l

∞

l_{\infty }

-norms.
On invariants dual to the Bass numbers
Authors: Enochs Edgar , Xu Jinzhong
Published: July 2002

Let

R
R

be a commutative Noetherian ring, and let

M
M

be an

R
R

-module. In earlier papers by Bass (1963) and Roberts (1980) the Bass numbers

μ
i

(
p
,
M
)

\mu _i(p,M)

were defined for all primes

p
p

and all integers

i
≥
0

i\ge 0

by use of the minimal injective resolution of

M
M

. It is well known that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript i Baseline left-parenthesis p comma upper M right-parenthesis equals dimension Subscript k left-parenthesis p right-parenthesis Baseline upper E x t Subscript upper R Sub Subscript p Superscript i Baseline left-parenthesis k left-parenthesis p right-parenthesis comma upper M Subscript p Baseline right-parenthesis">

μ
i

(
p
,
M
)
=

dim

k
(
p
)

⁡

Ext

R
p

i

⁡
(
k
(
p
)
,

M
p

)

\mu _i(p,M)=\dim _{k(p)}\operatorname {Ext} _{R_p}^i(k(p),M_p)

. On the other hand, if

M
M

is finitely generated, the Betti numbers

β
i

(
p
,
M
)

\beta _i(p,M)

are defined by the minimal free resolution of

M
p

M_p

over the local ring

R
p

R_p

. In an earlier paper of the second author (1995), using the flat covers of modules, the invariants

π
i

(
p
,
M
)

\pi _i(p,M)

were defined by the minimal flat resolution of

M
M

over Gorenstein rings. The invariants

π
i

(
p
,
M
)

\pi _i(p,M)

were shown to be somehow dual to the Bass numbers. In this paper, we use homologies to compute these invariants and show that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript i Baseline left-parenthesis p comma upper M right-parenthesis equals dimension Subscript k left-parenthesis p right-parenthesis Baseline upper T o r Subscript i Superscript upper R Super Subscript p Baseline left-parenthesis k left-parenthesis p right-parenthesis comma upper H o m Subscript upper R Baseline left-parenthesis upper R Subscript p Baseline comma upper M right-parenthesis right-parenthesis">

π
i

(
p
,
M
)
=

dim

k
(
p
)

⁡

Tor
i

R
p

⁡
(
k
(
p
)
,

Hom
R

⁡
(

R
p

,
M
)
)

\pi _i(p,M)= \dim _{k(p)}\operatorname {Tor}_i^{R_p}(k(p), \operatorname {Hom}_R(R_p,M))

\]
for any cotorsion module

M
M

. Comparing this with the computation of the Bass numbers, we see that

Ext
\operatorname {Ext}

is replaced by

Tor
\operatorname {Tor}

and the localization

M
p

M_p

is replaced by

Hom
R

⁡
(

R
p

,
M
)

\operatorname {Hom}_R(R_p,M)

(which was called the colocalization of

M
M

at the prime ideal

p
p

by Melkersson and Schenzel).
Weak solutions of parabolic equations in non-cylindrical domains
Authors: Brown Russell , Hu Wei , Lieberman Gary
Published: July 2002

In their classical work, Ladyzhenskaya and Ural

′

’

tseva gave a definition of weak solution for parabolic equations in cylindrical domains. Their definition was broad enough to guarantee the solvability of all such problems but narrow enough to guarantee the uniqueness of these solutions. We give here some alternative definitions which are appropriate to non-cylindrical domains, and we prove the unique solvability of such problems.
Weighted Hardy-Littlewood inequality for 𝐴-harmonic tensors
Authors: Ding Shusen
Published: July 2002

In this paper we prove a local weighted integral inequality for conjugate

A
A

-harmonic tensors similar to the Hardy and Littlewood integral inequality for conjugate harmonic functions. Then by using the local weighted integral inequality, we prove a global weighted integral inequality for conjugate

A
A

-harmonic tensors in John domains.
On functions arising as potentials on spaces of homogeneous type
Authors: Gatto A. , Vági Stephen
Published: July 2002

On a space of homogeneous type we consider functions

F
F

in

L
p

L^p

,

1
>
p
>
∞

1>p>\infty

, which are potentials of order

α
\alpha

of

L
p

L^p

functions. We show that these functions belong to the class of smooth functions

C

p
,
α

C^{p,\alpha }

of Calderón-Scott. This result has applications to tangential convergence.
Generalized numerical ranges, joint positive definiteness and multiple eigenvalues
Authors: Poon Yiu
Published: July 2002

We prove a convexity theorem on a generalized numerical range that combines and generalizes the following results: 1) Friedland and Loewy’s result on the existence of a nonzero matrix with multiple first eigenvalue in subspaces of hermitian matrices, 2) Bohnenblust’s result on joint positive definiteness of hermitian matrices, 3) the Toeplitz-Hausdorff Theorem on the convexity of the classical numerical range and its various generalizations by Au-Yeung, Berger, Brickman, Halmos, Poon, Tsing and Westwick.
Commensurators of parabolic subgroups of Coxeter groups
Authors: Paris Luis
Published: July 2002

Let

(
W
,
S
)

(W,S)

be a Coxeter system, and let

X
X

be a subset of

S
S

. The subgroup of

W
W

generated by

X
X

is denoted by

W

X

W_{X}

and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of

W

X

W_{X}

in

W
W

is the subgroup of

w
w

in

W
W

such that

w

W

X

w

−
1

∩

W

X

wW_{X}w^{-1}\cap W_{X}

has finite index in both

W

X

W_{X}

and

w

W

X

w

−
1

wW_{X}w^{-1}

. The subgroup

W

X

W_{X}

can be decomposed in the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript upper X Baseline equals upper W Subscript upper X Sub Superscript 0 Baseline dot upper W Subscript upper X Sub Superscript normal infinity Baseline asymptotically-equals upper W Subscript upper X Sub Superscript 0 Baseline times upper W Subscript upper X Sub Superscript normal infinity">

W

X

=

W

X

0

⋅

W

X

∞

≃

W

X

0

×

W

X

∞

W_{X} = W_{X^{0}} \cdot W_{X^{\infty }} \simeq W_{X^{0}} \times W_{X^{\infty }}

where

W

X

0

W_{X^{0}}

is finite and all the irreducible components of

W

X

∞

W_{X^{\infty }}

are infinite. Let

Y

∞

Y^{\infty }

be the set of

t
t

in

S
S

such that

m

s
,
t

=
2

m_{s,t}=2

for all

s
∈

X

∞

s\in X^{\infty }

. We prove that the commensurator of

W

X

W_{X}

is
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript upper Y Sub Superscript normal infinity Baseline dot upper W Subscript upper X Sub Superscript normal infinity Baseline asymptotically-equals upper W Subscript upper Y Sub Superscript normal infinity Baseline times upper W Subscript upper X Sub Superscript normal infinity">

W

Y

∞

⋅

W

X

∞

≃

W

Y

∞

×

W

X

∞

W_{Y^{\infty }} \cdot W_{X^{\infty }} \simeq W_{Y^{\infty }} \times W_{X^{\infty }}

. In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and

W

X

W_{X}

is its own commensurator if and only if

X

0

=

Y

∞

X^{0}= Y^{\infty }

.
Invariant subspaces for polynomially hyponormal operators
Authors: Prunaru Bebe
Published: July 2002

We show that if

T
T

is a bounded operator on a Hilbert space such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis upper T right-parenthesis Superscript asterisk Baseline p left-parenthesis upper T right-parenthesis minus p left-parenthesis upper T right-parenthesis p left-parenthesis upper T right-parenthesis Superscript asterisk Baseline greater-than-or-equal-to 0">

p
(
T

)
∗

p
(
T
)
−
p
(
T
)
p
(
T

)
∗

≥
0

p(T)^*p(T)-p(T)p(T)^*\geq 0

for every polynomial

p
p

, then

T
T

has a nontrivial invariant subspace.
Extension and convergence theorems for families of normal maps in several complex variables
Authors: Joseph James , Kwack Myung
Published: July 2002

Let

H

(
X
,
Y
)
(

C

(
X
,
Y
)
)

\mathcal {H}(X,Y) ( \mathcal {C}(X,Y) )

represent the family of holomorphic (continuous) maps from a complex (topological) space

X
X

to a complex (topological) space

Y
Y

, and let

Y

+

=
Y
∪
{
∞
}

Y^{+} = Y \cup \{\infty \}

be the Alexandroff one–point compactification of

Y
Y

if

Y
Y

is not compact,

Y

+

=
Y

Y^{+}=Y

if

Y
Y

is compact. We say that

F

⊂

H

(
X
,
Y
)

\mathcal {F} \subset \mathcal {H}(X,Y)

is uniformly normal if

{
f
∘
φ
:
f
∈

F

\{f \circ \varphi : f \in \mathcal {F}

,

φ
∈

H

(
M
,
X
)
}

\varphi \in \mathcal {H}(M,X)\}

is relatively compact in

C

(
M
,

Y

+

)

\mathcal {C}(M,Y^{+})

(with the compact–open topology) for each complex manifold

M
M

. We show that normal maps as defined and studied by authors in various settings are, as singleton sets, uniformly normal families, and prove extension and convergence theorems for uniformly normal families. These theorems include (1) extension theorems of big Picard type for such families – defined on complex manifolds having divisors with normal crossings – which encompass results of Järvi, Kiernan, Kobayashi, and Kwack as special cases, and (2) generalizations to such families of an extension–convergence theorem due to Noguchi.
When is a 𝑝-block a 𝑞-block?
Authors: Navarro Gabriel , Willems Wolfgang
Published: July 2002

Let

p
p

and

q
q

be distinct prime numbers and let

G
G

be a finite group. If

B

p

B_{p}

is a

p
p

-block of

G
G

and

B

q

B_{q}

is a

q
q

-block, we study when the set of ordinary irreducible characters in the blocks

B

p

B_{p}

and

B

q

B_{q}

coincide.
Sublattices of the Banach envelope of weak 𝐿¹
Authors: Lotz Heinrich , Peck N.
Published: July 2002

We prove that every separable Banach lattice is lattice isometric to a closed sublattice of the Banach envelope of Weak

L

1

.

L^{1}.
Complexity of tensor products of modules and a theorem of Huneke-Wiegand
Authors: Miller Claudia
Published: July 2002

This paper concerns the notion of complexity, a measure of the growth of the Betti numbers of a module. We show that over a complete intersection

R
R

the complexity of the tensor product

M

⊗

R

N

M\otimes _{R} N

of two finitely generated modules is the sum of the complexities of each if

Tor

i

R

⁡
(
M
,
N
)
=
0

\operatorname {Tor}_{i}^{R}(M,N)=0

for

i
≥
1

i\geq 1

. One of the applications is simplification of the proofs of central results in a paper of C. Huneke and R. Wiegand on the tensor product of modules and the rigidity of Tor.
Average curvature of convex curves in 𝐻²
Authors: Bridgeman Martin
Published: July 2002

A well-known result states that, if a curve

α
\alpha

in

H
2

H^2

has geodesic curvature less than or equal to one at every point, then

α
\alpha

is embedded. The converse is obviously not true, but the embeddedness of a curve does give information about the curvature. We prove that, if

α
\alpha

is a convex embedded curve in

H
2

H^2

, then the average curvature (curvature per unit length) of

α
\alpha

, denoted

K
(
α
)

K(\alpha )

, satisfies

K
(
α
)
≤
1

K(\alpha ) \leq 1

. This bound on the average curvature is tight as

K
(
α
)
=
1

K(\alpha )=1

for

α
\alpha

a horocycle.
Weak choice principles
Authors: De la Cruz Omar , Di Prisco Carlos
Published: July 2002

We consider several weak forms of the Axiom of Choice obtained debilitating some well known choice principles. Implication relations among them and independence results are established using, in some cases, permutation models or symmetric models. The relationship between Ramsey’s Theorem and these choice principles is also explored, showing that, in this context, the known results are the best possible.
Mean exit time from convex hypersurfaces
Authors: Palmer Vicente
Published: July 2002

L. Karp and M. Pinsky proved that, for small radius

R
R

, the mean exit time function

E

R

E_{R}

of an extrinsic

R
R

-ball in a hypersurface

P

n
−
1

⊆

R

n

P^{n-1} \subseteq \mathbb {R}^{n}

is bounded from below by the corresponding function

E
~

R

\widetilde E_{R}

defined on an extrinsic

R
R

-ball in

R

n
−
1

\mathbb {R}^{n-1}

. A counterexample given by C. Mueller proves that this inequality doesn’t holds in the large. In this paper we show that, if

P
P

is convex, then the inequality holds for all radii. Moreover, we characterize the equality and show that analogous results are true in the sphere.
Lie incidence systems from projective varieties
Authors: Cohen Arjeh , Cooperstein Bruce
Published: July 2002

The homogeneous space

G

/

P

λ

G/P_{\lambda }

, where

G
G

is a simple algebraic group and

P

λ

P_{\lambda }

a parabolic subgroup corresponding to a fundamental weight

λ
\lambda

(with respect to a fixed Borel subgroup

B
B

of

G
G

in

P

λ

P_{\lambda }

), is known in at least two settings. On the one hand, it is a projective variety, embedded in the projective space corresponding to the representation with highest weight

λ
\lambda

. On the other hand, in synthetic geometry,

G

/

P

λ

G/P_{\lambda }

is furnished with certain subsets, called lines, of the form

g
B
⟨
r
⟩

P

λ

/

P

λ

gB\langle r\rangle P_{\lambda }/P_{\lambda }

where

r
r

is a preimage in

G
G

of the fundamental reflection corresponding to

λ
\lambda

and

g
∈
G

g\in G

. The result is called the Lie incidence structure on

G

/

P

λ

G/P_{\lambda }

. The lines are projective lines in the projective embedding. In this paper we investigate to what extent the projective variety data determines the Lie incidence structure.
Remarks on numerical ranges of operators in spaces with an indefinite metric
Authors: Li Chi-Kwong , Rodman Leiba
Published: July 2002

The numerical range of an operator on an indefinite inner product space (possibly infinite dimensional) is studied. In particular, operators having bounded numerical ranges are characterized, and the angle points of the numerical range and their connections with eigenvalues are described.
Convex linear combinations of sequences of monic orthogonal polynomials
Authors: Cachafeiro A. , Marcellan F.
Published: July 2002

For a sequence

{

Φ
n

}
0
∞

\{\Phi _n\}_0^\infty

of monic orthogonal polynomials (SMOP), with respect to a positive measure supported on the unit circle, we obtain necessary and sufficient conditions on a SMOP

{

Q
n

}
0
∞

\{Q_n\}_0^\infty

in order that a convex linear combination

{

R
n

}
0
∞

\{R_n\}_0^\infty

with

R
n

=
β

Φ
n

+
(
1
−
β
)

Q
n

R_n=\beta \Phi _n+(1-\beta )Q_n

be a SMOP with respect to a positive measure supported on the unit circle.
Note on compact sets of compact operators on a reflexive and separable Banach space
Authors: Galaz-Fontes Fernando
Published: July 2002

We give a criterion for a subset of the space of compact linear operators from a separable and reflexive Banach

X
X

into a Banach space

Y
Y

to be compact.
Measures with natural spectra on locally compact abelian groups
Authors: Hatori Osamu
Published: July 2002

Every bounded regular Borel measure on noncompact LCA groups is a sum of an absolutely continuous measure and a measure with natural spectrum. The set of bounded regular Borel measures with natural spectrum on a nondiscrete LCA group

G
G

whose Fourier-Stieltjes transforms vanish at infinity is closed under addition if and only if

G
G

is compact.
The maximum condition on annihilators for polynomial rings
Authors: Cedó Ferran , Herbera Dolors
Published: July 2002

For each positive integer

n
n

, we construct a commutative ring

R

{\mathcal {R}}

such that the polynomial ring

R

[

x

1

,
…
,

x

n

]

{\mathcal {R}}[x_{1},\ldots ,x_{n}]

satisfies the maximum condition on annihilators and

R

[

x

1

,
…
,

x

n
+
1

]

{\mathcal {R}}[x_{1},\ldots ,x_{n+1}]

does not. In particular, there exists a commutative Kerr ring

R

{\mathcal {R}}

such that

R

[
x
]

{\mathcal {R}}[x]

is not Kerr. This answers in the negative a question of Faith’s.
Norms of embeddings of logarithmic Bessel potential spaces
Authors: Edmunds David , Gurka Petr , Opic Bohumír
Published: July 2002

Let

Ω
\Omega

be a subset of

R

n

\mathbb {R}^{n}

with finite volume, let

ν
>
0

\nu >0

and let

Φ
\Phi

be a Young function with

Φ
(
t
)
=
exp
⁡
(
exp
⁡

t

ν

)

\Phi (t) = \exp (\exp t^{\nu })

for large

t
t

. We show that the norm on the Orlicz space

L

Φ

(
Ω
)

L_ {\Phi } (\Omega )

is equivalent to
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript 1 greater-than q greater-than normal infinity Endscripts left-parenthesis e plus log q right-parenthesis Superscript negative 1 slash nu Baseline double-vertical-bar f double-vertical-bar Subscript upper L Sub Superscript q Subscript left-parenthesis normal upper Omega right-parenthesis Baseline period">

sup

1
>
q
>
∞

(
e
+
log
⁡
q

)

−
1

/

ν

‖
f

‖

L

q

(
Ω
)

.

\begin{equation*}\sup _ {1>q>\infty } (e+\log q)^{-1/\nu } \|f\|_ {L^{q}(\Omega )}. \end{equation*}

We also obtain estimates of the norms of the embeddings of certain logarithmic Bessel potential spaces in

L

q

(
Ω
)

L^{q}(\Omega )

which are sharp in their dependences on

q
q

provided that

q
q

is large enough.
Certain invariant subspace structure of 𝐿²(𝕋²)
Authors: Ji Guoxing , Ohwada Tomoyoshi , Saito Kichi-Suke
Published: July 2002

In this note, we study certain structure of an invariant subspace

M

\mathfrak {M}

of

L

2

(

T

2

)

L^{2}(\mathbb {T}^{2})

. Considering the largest

z
z

-invariant (resp.

w
w

-invariant) subspace in the wandering subspace

M

⊖
z
w

M

\mathfrak {M} \ominus zw \mathfrak {M}

of

M

\mathfrak {M}

with respect to the shift operator

z
w

zw

, we give an alternative characterization of Beurling-type invariant subspaces. Furthermore, we consider a certain class of invariant subspaces.
Paraexponentials, Muckenhoupt weights, and resolvents of paraproducts
Authors: Pereyra María , Ward Lesley
Published: July 2002

We analyze the stability of Muckenhoupt’s

R
H

p

d

\mathbf {RH}_{\mathbf {p}}^{\mathbf {d}}

and

A

P

d

\mathbf {A}_{\mathbf {P}}^{\mathbf {d}}

classes of weights under a nonlinear operation, the

λ
\lambda

-operation. We prove that the dyadic doubling reverse Hölder classes

R
H

p

d

\mathbf {RH}_{\mathbf {p}}^{\mathbf {d}}

are not preserved under the

λ
\lambda

-operation, but the dyadic doubling

A
p

A_p

classes

A

P

d

\mathbf {A}_{\mathbf {P}}^{\mathbf {d}}

are preserved for

0
≤
λ
≤
1

0\leq \lambda \leq 1

. We give an application to the structure of resolvent sets of dyadic paraproduct operators.
The equivalence of some Bernoulli convolutions to Lebesgue measure
Authors: Mauldin R. , Simon Károly
Published: July 2002

Since the 1930’s many authors have studied the distribution

ν

λ

\nu _{\lambda }

of the random series

Y

λ

=
∑
±

λ

n

Y_{\lambda }=\sum \pm {\lambda }^n

where the signs are chosen independently with probability

(
1

/

2
,
1

/

2
)

(1/2,1/2)

and

0
>
λ
>
1

0>\lambda >1

. Solomyak recently proved that for almost every

λ
∈
[

1
2

,
1
]
,

\lambda \in [\frac {1}{2},1],

the distribution

ν

λ

\nu _{\lambda }

is absolutely continuous with respect to Lebesgue measure. In this paper we prove that

ν

λ

\nu _{\lambda }

is even equivalent to Lebesgue measure for almost all

λ
∈
[

1
2

,
1
]

\lambda \in [\frac {1}{2},1]

.
Absolute fixed point sets for multi-valued maps
Authors: McDowell Eric
Published: July 2002

The notion of a multi-valued absolute fixed point set (MAFS) will be defined and characterized in the setting of set-valued maps with images containing multiple components.
Maximal entropy permutations of even size
Authors: Geller William , Zhang Zhenhua
Published: July 2002

The entropy of a permutation is the least topological entropy of any continuous interval map having an invariant set which is shuffled according to the permutation. For each

k
k

, we identify the maximal entropy permutations of size

2
k

2k

.
An orthogonal family of polynomials on the generalized unit disk and ladder representations of 𝑈(𝑝,𝑞)
Authors: Lorch John
Published: July 2002

Inner product structures are given for realizations of the positive spin ladder representations over the generalized unit disk

D

p
,
q

=
U
(
p
,
q
)

/

K

\textbf {D}_{p,q} =U(p,q)/K

. This is accomplished by combining previous results of the author with the construction of a family of holomorphic polynomials on

D

p
,
q

\textbf {D}_{p,q}

. These polynomials, which play a crucial role in the present work, are shown to be orthogonal with respect to Lebesgue measure, and their norms are computed. The orthogonal family is then used to invert a certain integral transform, giving the desired inner product structures.
On uniqueness of 𝑝-adic meromorphic functions
Authors: Boutabaa Abdelbaki , Escassut Alain
Published: July 2002

Let

K
K

be a complete ultrametric algebraically closed field of characteristic zero, and let

M

(
K
)

{\mathcal {M}} (K)

be the field of meromorphic functions in

K
K

. For all set

S
S

in

K
K

and for all

f
∈

M

(
K
)

f\in {\mathcal {M}}(K)

we denote by

E
(
f
,
S
)

\displaystyle E(f,S)

the subset of

K

×

N

∗

K {\times } {\mathbb {N}}^{*}

:

⋃

a
∈
S

{
(
z
,
q
)
∈
K

×

N

∗

|
z

{\bigcup _{ a\in S}}\{(z,q)\in K {\times } \mathbb {N}^{*} \vert z

zero of order

q
 of
f
(
z
)
−
a
}
.

q \text { of} f(z)-a\}.

After studying unique range sets for entire functions in

K
K

in a previous article, here we consider a similar problem for meromorphic functions by showing, in particular, that, for every

n
≥
5

n\geq 5

, there exist sets

S
S

of

n
n

elements in

K
K

such that, if

f
,
g
∈

M

(
K
)

f, g\in {\mathcal {M}} (K)

have the same poles (counting multiplicities), and satisfy

E
(
f
,
S
)
=
E
(
g
,
S
)

E(f,S)=E(g,S)

, then

f
=
g

f=g

. We show how to construct such sets.
A note on Toeplitz operators on discrete groups
Authors: Xu Qingxiang , Chen Xiaoman
Published: July 2002

We study Toeplitz algebras associated to partially-ordered and quasi-partially ordered discrete groups.
A note on 𝜎-summable groups
Authors: Hill Paul
Published: July 2002

We answer questions raised by P. Danchev in a recent paper in these Proceedings. It is shown that a

σ
\sigma

-summable abelian

p
p

-group is not determined by its socle, that is, two such groups can have isometric socles without being isomorphic. It is also demonstrated that

σ
\sigma

-summability plays essentially no role in regard to the question of whether or not

V
(
G
)

/

G

V(G)/G

is totally projective, where

V
(
G
)

V(G)

denotes the group of normalized units of the group algebra

F
(
G
)

F(G)

with

F
F

being a perfect field of characteristic

p
p

.
On the hyperspace of a non-separable metric space
Authors: Costantini Camillo
Published: July 2002

We prove the following two results. 1) There exists a non-separable complete metric space whose Wijsman hypertopology is not Čech-complete. 2) There exist a non-separable metrizable space and two compatible metrics on it, such that the collections of the Borel sets generated by the relative Wijsman hypertopologies do not coincide.
Wick products of the CAR algebra
Authors: Negrin E.
Published: July 2002

The purpose of this paper is to put in a precise mathematical (algebraic) form the Wick products of the CAR algebra. We state in detail the reduction of the ordinary product of Fermi fields in terms of a finite sum of monomials in the creation and annihilation operators in which all creation operators occur to the left of all annihilation operators (Wick-ordered) and the Fock (vacuum) state of the former.