bluuebunny/crossref_metadata_2025 huggingface dataset:
On a theorem of Barbara Schmid
Authors: Smith Larry
Published: July 2002

Let

G
G

be a finite group and

ρ
:
G
↪

G
L

(
n
,

C

)

\rho \colon G\hookrightarrow \mathrm {GL} (n,\mathbb {C})

a complex representation. Barbara Schmid has shown that the algebra of invariant polynomial functions

C

[
V

]
G

\mathbb {C}[V]^G

on the vector space

V
=

C

n

V=\mathbb {C}^n

is generated by homogeneous polynomials of degree at most

β
\beta

, where

β
\beta

is the largest degree of a generator in a minimal generating set for

C

[

r
e
g

C

(
G
)

]
G

\mathbb {C}[\mathrm {reg}_{\mathbb {C}}(G)]^G

, and

r
e
g

C

(
G
)

\mathrm {reg}_{\mathbb {C}}(G)

is the complex regular representation of

G
G

. In this note we give a new proof of this result, and at the same time extend it to fields

F

\mathbb {F}

whose characteristic

p
p

is larger than

|

G

|

|G|

, the order of the group

G
G

.
The number of knot group representations is not a Vassiliev invariant
Authors: Eisermann Michael
Published: July 2002

For a finite group

G
G

and a knot

K
K

in the

3
3

-sphere, let

F
G

(
K
)

F_G(K)

be the number of representations of the knot group into

G
G

. In answer to a question of D. Altschuler we show that

F
G

F_G

is either constant or not of finite type. Moreover,

F
G

F_G

is constant if and only if

G
G

is nilpotent. We prove the following, more general boundedness theorem: If a knot invariant

F
F

is bounded by some function of the braid index, the genus, or the unknotting number, then

F
F

is either constant or not of finite type.
A composition formula in the rank two free group
Authors: González-Acuña Francisco , Ramírez Arturo
Published: July 2002

Using the fundamental group of a punctured torus, a free group

F
F

of rank two, and the fact that the natural eipmorphism from

A
u
t
F

Aut F

onto

A
u
t
(
F

/

F
′

)

Aut(F/F’)

has as kernel the group of inner automorphisms of

F
F

, we describe representatives of the conjugacy classes of generating pairs of

F
F

and give explicit relations between them.
Tightness bounds for strongly mixing random sequences
Authors: Bradley Richard
Published: July 2002

For a given strictly stationary, strongly mixing random sequence for which the distributions of the partial sums are tight, certain “tightness bounds" exist which depend only on the marginal distribution and the mixing rate.
On a characterization of the maximal ideal spaces of commutative 𝐶*-algebras in which every element is the square of another
Authors: Hatori Osamu , Miura Takeshi
Published: July 2002

A topological condition is given for a locally connected compact Hausdorff space on which every complex-valued continuous function is the square of another. The condition need not be necessary nor sufficient unless the space is locally connected.
Adjacency preserving mappings of invariant subspaces of a null system
Authors: Huang Wen-ling
Published: July 2002

In the space

I
r

I_r

of invariant

r
r

-dimensional subspaces of a null system in

(
2
r
+
1
)

(2r+1)

-dimensional projective space, W.L. Chow characterized the basic group of transformations as all the bijections

φ
:

I
r

→

I
r

\varphi :I_r\to I_r

, for which both

φ
\varphi

and

φ

−
1

\varphi ^{-1}

preserve adjacency. In the present paper we show that the two conditions

φ
:

I
r

→

I
r

\varphi :I_r\to I_r

is a surjection and

φ
\varphi

preserves adjacency are sufficient to characterize the basic group. At the end of this paper we give an application to Lie geometry.
Generic polynomials for quasi-dihedral, dihedral and modular extensions of order 16
Authors: Ledet Arne
Published: July 2002

We describe Galois extensions where the Galois group is the quasi-dihedral, dihedral or modular group of order 

16
16

, and use this description to produce generic polynomials.
The Conley index over a base
Authors: Mrozek Marian , Reineck James , Srzednicki Roman
Published: July 2002

We construct a generalization of the Conley index for flows. The new index preserves information which in the classical case is lost in the process of collapsing the exit set to a point. The new index has most of the properties of the classical index. As examples, we study a flow with a knotted orbit in

R

3

\mathrm {R}^3

, and the problem of continuing two periodic orbits which are not homotopic as loops.
A compactification of a family of determinantal Godeaux surfaces
Authors: Lee Yongnam
Published: July 2002

In this paper, we present a geometric description of the compactification of the family of determinantal Godeaux surfaces, via the study of the bicanonical pencil and using classical Prym theory. In particular, we reduce the problem of compactifying the space of bicanonical pencils of determinantal Godeaux surfaces to the compactification of the family of twisted cubic curves in

P

3

\mathbb {P}^{3}

with certain given tangent conditions.
𝐂²-saddle method and Beukers’ integral
Authors: Hata Masayoshi
Published: July 2002

We give good non-quadraticity measures for the values of logarithm at specific rational points by modifying Beukers’ double integral. The two-dimensional version of the saddle method, which we call

C

2

\mathbf {C}^{2}

-saddle method, is applied.
A regular space with a countable network and different dimensions
Authors: Delistathis George , Watson Stephen
Published: July 2002

In this paper, we construct a regular space with a countable network (even the union of countably many separable metric subspaces) in which

i
n
d

ind

and

d
i
m

dim

do not coincide under the assumption of the continuum hypothesis (CH). This gives a consistent negative answer to a question of A.V. Arhangel’skiĭ.
The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature
Authors: Morgan Frank , Hutchings Michael , Howards Hugh
Published: July 2002

We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of nonincreasing Gauss curvature consists of one or two circles, bounding a disc, the complement of a disc, or an annulus. We also provide a new isoperimetric inequality in general surfaces with boundary.
The Noetherian property in some quadratic algebras
Authors: Kramer Xenia
Published: July 2002

We introduce a new class of noncommutative rings called pseudopolynomial rings and give sufficient conditions for such a ring to be Noetherian. Pseudopolynomial rings are standard finitely presented algebras over a field with some additional restrictions on their defining relations—namely that the polynomials in a Gröbner basis for the ideal of relations must be homogeneous of degree 2—and on the Ufnarovskii graph

Γ
(
A
)

\Gamma (A)

. The class of pseudopolynomial rings properly includes the generalized skew polynomial rings introduced by M. Artin and W. Schelter. We use the graph

Γ
(
A
)

\Gamma (A)

to define a weaker notion of almost commutative, which we call almost commutative on cycles. We show as our main result that a pseudopolynomial ring which is almost commutative on cycles is Noetherian. A counterexample shows that a Noetherian pseudopolynomial ring need not be almost commutative on cycles.
Weak and Strong Density of Compositions
Authors: De Pascale Luigi , Stepanov Eugene
Published: July 2002

The convergence in various topologies of sequences of inner superposition (composition) operators acting between Lebesgue spaces and of their linear combinations is studied. In particular, the sequential density results for the linear span of such operators is proved for the weak, weak continuous and strong operator topologies.
Cyclic coverings and higher order embeddings of algebraic varieties
Authors: Bauer Thomas , Di Rocco Sandra , Szemberg Tomasz
Published: July 2002

In the present paper we study higher order embeddings in the context of cyclic coverings. Analyzing the positivity of the line bundle downstairs and its relationship with the branch divisor, we provide criteria for its pull-back to define an embedding of given order. We show that the obtained criteria are sharp. Finally, we apply them to various – sometimes seemingly unrelated–problems in algebraic geometry.
Manifolds with minimal radial curvature bounded from below and big volume
Authors: Marenich Valery
Published: July 2002

We prove that a convergence in the Gromov-Hausdorff distance of manifolds with minimal radial curvature bounded from below by 1 to the standard sphere is equivalent to a volume convergence.
Gröbner bases, H–bases and interpolation
Authors: Sauer Thomas
Published: July 2002

The paper is concerned with a construction for H–bases of polynomial ideals without relying on term orders. The main ingredient is a homogeneous reduction algorithm which orthogonalizes leading terms instead of completely canceling them. This allows for an extension of Buchberger’s algorithm to construct these H–bases algorithmically. In addition, the close connection of this approach to minimal degree interpolation, and in particular to the least interpolation scheme due to de Boor and Ron, is pointed out.
Effectively dense Boolean algebras and their applications
Authors: Nies André
Published: July 2002

A computably enumerable Boolean algebra

B

{\mathcal {B}}

is effectively dense if for each

x
∈

B

x \in {\mathcal {B}}

we can effectively determine an

F
(
x
)
≤
x

F(x)\le x

such that

x
≠
0

x \neq 0

implies

0
>
F
(
x
)
>
x

0 > F(x) > x

. We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a Boolean algebra. As an application, we also obtain an interpretation of true arithmetic in all theories of intervals of

E

{\mathcal {E}}

(the lattice of computably enumerable sets under inclusion) which are not Boolean algebras. We derive a similar result for theories of certain initial intervals

[

0

,

a

]

[{\mathbf {0}},{\mathbf {a}}]

of subrecursive degree structures, where

a

{\mathbf {a}}

is the degree of a set of relatively small complexity, for instance a set in exponential time.
Stability theory, permutations of indiscernibles, and embedded finite models
Authors: Baldwin John , Benedikt Michael
Published: July 2002

We show that the expressive power of first-order logic over finite models embedded in a model

M
M

is determined by stability-theoretic properties of

M
M

. In particular, we show that if

M
M

is stable, then every class of finite structures that can be defined by embedding the structures in

M
M

, can be defined in pure first-order logic. We also show that if

M
M

does not have the independence property, then any class of finite structures that can be defined by embedding the structures in

M
M

, can be defined in first-order logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let

I
I

be a set of indiscernibles in a model

M
M

and suppose

(
M
,
I
)

(M,I)

is elementarily equivalent to

(

M
1

,

I
1

)

(M_1,I_1)

where

M
1

M_1

is

|

I
1

|

+

|I_1|^+

-saturated. If

M
M

is stable and

(
M
,
I
)

(M,I)

is saturated, then every permutation of

I
I

extends to an automorphism of

M
M

and the theory of

(
M
,
I
)

(M,I)

is stable. Let

I
I

be a sequence of

>
>

-indiscernibles in a model

M
M

, which does not have the independence property, and suppose

(
M
,
I
)

(M,I)

is elementarily equivalent to

(

M
1

,

I
1

)

(M_1,I_1)

where

(

I
1

,
>
)

(I_1,>)

is a complete dense linear order and

M
1

M_1

is

|

I
1

|

+

|I_1|^+

-saturated. Then

(
M
,
I
)

(M,I)

-types over

I
I

are order-definable and if

(
M
,
I
)

(M,I)

is

ℵ
1

\aleph _1

-saturated, every order preserving permutation of

I
I

can be extended to a back-and-forth system.
The combinatorics of Bernstein functions
Authors: Haines Thomas
Published: July 2002

A construction of Bernstein associates to each cocharacter of a split

p
p

-adic group an element in the center of the Iwahori-Hecke algebra, which we refer to as a Bernstein function. A recent conjecture of Kottwitz predicts that Bernstein functions play an important role in the theory of bad reduction of a certain class of Shimura varieties (parahoric type). It is therefore of interest to calculate the Bernstein functions explicitly in as many cases as possible, with a view towards testing Kottwitz’ conjecture. In this paper we prove a characterization of the Bernstein function associated to a minuscule cocharacter (the case of interest for Shimura varieties). This is used to write down the Bernstein functions explicitly for some minuscule cocharacters of

G

l
n

Gl_n

; one example can be used to verify Kottwitz’ conjecture for a special class of Shimura varieties (the “Drinfeld case”). In addition, we prove some general facts concerning the support of Bernstein functions, and concerning an important set called the “

μ
\mu

-admissible” set. These facts are compatible with a conjecture of Kottwitz and Rapoport on the shape of the special fiber of a Shimura variety with parahoric type bad reduction.
Simple holonomic modules over rings of differential operators with regular coefficients of Krull dimension 2
Authors: Bavula V. , van Oystaeyen F.
Published: July 2002

Let

K
K

be an algebraically closed field of characteristic zero. Let

Λ
\Lambda

be the ring of (

K
K

-linear) differential operators with coefficients from a regular commutative affine domain of Krull dimension

2
2

which is the tensor product of two regular commutative affine domains of Krull dimension

1
1

. Simple holonomic

Λ
\Lambda

-modules are described. Let a

K
K

-algebra

D
D

be a regular affine commutative domain of Krull dimension

1
1

and

D

(
D
)

\mathcal {D} (D)

be the ring of differential operators with coefficients from

D
D

. We classify (up to irreducible elements of a certain Euclidean domain) simple

D

(
D
)

\mathcal {D}(D)

-modules (the field

K
K

is not necessarily algebraically closed).
Model category structures on chain complexes of sheaves
Authors: Hovey Mark
Published: July 2002

The unbounded derived category of a Grothendieck abelian category is the homotopy category of a Quillen model structure on the category of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently by Beke. However, in most cases of interest, such as the category of sheaves on a ringed space or the category of quasi-coherent sheaves on a nice enough scheme, the abelian category in question also has a tensor product. The injective model structure is not well-suited to the tensor product. In this paper, we consider another method for constructing a model structure. We apply it to the category of sheaves on a well-behaved ringed space. The resulting flat model structure is compatible with the tensor product and all homomorphisms of ringed spaces.
Invariants and projections of six lines in projective space
Authors: Vazzana Dana
Published: July 2002

Given six lines in

P

3

\mathbf {P}^3

, quartics through the six lines define a map from

P

3

\mathbf {P}^3

to

P

4

\mathbf {P}^4

, and the image of this map is described in terms of invariants of the six lines. The map can be interpreted as projection of the six lines, and this permits a description of the canonical model of the octic surface which is given by points which project the lines so that they are tangent to a conic. We also define polarity for sets of six lines, and discuss the above map in the case of a self-polar set of lines and in the case of six lines which form a “double-sixer” on a cubic surface.
Topological mixing in 𝐶𝐴𝑇(-1)-spaces
Authors: Charitos Charalambos , Tsapogas Georgios
Published: July 2002

If

X
X

is a proper

C
A
T

(
−
1
)

CAT\left ( -1\right )

-space and

Γ
\Gamma

a non-elementary discrete group of isometries acting properly discontinuously on

X
,

X,

it is shown that the geodesic flow on the quotient space

Y
=
X

/

Γ

Y=X/\Gamma

is topologically mixing, provided that the generalized Busemann function has zeros on the boundary

∂
X

\partial X

and the non-wandering set of the flow equals the whole quotient space of geodesics

G
Y
:=
G
X

/

Γ

GY:=GX/\,\Gamma

(the latter being redundant when

Y
Y

is compact). Applications include the proof of topological mixing for (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete

C
A
T

(
−
1
)

CAT\left ( -1\right )

-spaces by a one-ended group of isometries and (C) finite

n
n

-dimensional ideal polyhedra.
The index of a critical point for densely defined operators of type (𝑆₊)_{𝐿} in Banach spaces
Authors: Kartsatos Athanassios , Skrypnik Igor
Published: July 2002

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type(S+)L(S_{+})_{L}acting from a real, reflexive and separable Banach spaceXXintoX∗.X^{*}.This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given inlp, p>2,l^{p},~p>2,illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type(S+)).(S_{+})).Applications to nonlinear Dirichlet problems have appeared elsewhere.
Constructing division rings as module-theoretic direct limits
Authors: Bergman George
Published: July 2002

If

R
R

is an associative ring, one of several known equivalent types of data determining the structure of an arbitrary division ring

D
D

generated by a homomorphic image of

R
R

is a rule putting on all free

R
R

-modules of finite rank matroid structures (closure operators satisfying the exchange axiom) subject to certain functoriality conditions. This note gives a new description of how

D
D

may be constructed from this data. (A classical precursor of this is the construction of

Q

\mathbf Q

as a field with additive group a direct limit of copies of

Z

\mathbf Z

.) The division rings of fractions of right and left Ore rings, the universal division ring of a free ideal ring, and the concept of a specialization of division rings are then interpreted in terms of this construction.
Monoidal extensions of a Cohen-Macaulay unique factorization domain
Authors: Heinzer William , Li Aihua , Ratliff Louis , Rush David
Published: July 2002

Let

A
A

be a Noetherian Cohen-Macaulay domain,

b
b

,

c
1

c_1

,

…
\dots

,

c
g

c_g

an

A
A

-sequence,

J
J

=

(
b
,

c
1

,
…
,

c
g

)
A

(b,c_1,\dots ,c_g)A

, and

B
B

=

A
[
J

/

b
]

A[J/b]

. Then

B
B

is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets

Ass
B

⁡
(
B

/

b
B
)

\operatorname {Ass}_B(B/bB)

and

Ass
A

⁡
(
A

/

J
)

\operatorname {Ass}_A(A/J)

, and each

q
q

∈
\in

Ass
A

⁡
(
A

/

J
)

\operatorname {Ass}_A(A/J)

has height

g
+
1

g+1

. If

B
B

does not have unique factorization, then some height-one prime ideals

P
P

of

B
B

are not principal. These primes are identified in terms of

J
J

and

P
∩
A

P \cap A

, and we consider the question of how far from principal they can be. If

A
A

is integrally closed, necessary and sufficient conditions are given for

B
B

to be integrally closed, and sufficient conditions are given for

B
B

to be a UFD or a Krull domain whose class group is torsion, finite, or finite cyclic. It is shown that if

P
P

is a height-one prime ideal of

B
B

, then

P
∩
A

P \cap A

also has height one if and only if

b
b

∉
\notin

P
P

and thus

P
∩
A

P \cap A

has height one for all but finitely many of the height-one primes

P
P

of

B
B

. If

A
A

has unique factorization, a description is given of whether or not such a prime

P
P

is a principal prime ideal, or has a principal primary ideal, in terms of properties of

P
∩
A

P \cap A

. A similar description is also given for the height-one prime ideals

P
P

of

B
B

with

P
∩
A

P \cap A

of height greater than one, if the prime factors of

b
b

satisfy a mild condition. If

A
A

is a UFD and

b
b

is a power of a prime element, then

B
B

is a Krull domain with torsion class group if and only if

J
J

is primary and integrally closed, and if this holds, then

B
B

has finite cyclic class group. Also, if

J
J

is not primary, then for each height-one prime ideal

p
p

contained in at least one, but not all, prime divisors of

J
J

, it holds that the height-one prime

p
A
[
1

/

b
]
∩
B

pA[1/b] \cap B

has no principal primary ideals. This applies in particular to the Rees ring

R

{\mathbf R}

=
=

A
[
1

/

t
,
t
J
]

A[1/t, tJ]

. As an application of these results, it is shown how to construct for any finitely generated abelian group

G
G

, a monoidal transform

B
B

=

A
[
J

/

b
]

A[J/b]

such that

A
A

is a UFD,

B
B

is Cohen-Macaulay and integrally closed, and

G
G

≅
\cong

Cl
⁡
(
B
)

\operatorname {Cl}(B)

, the divisor class group of

B
B

.
Failure of normality in the box product of uncountably many real lines
Authors: Lawrence L.
Published: July 2002

We prove in ZFC that the box product of

ω
1

\omega _1

many copies of

ω
+
1

\omega +1

is neither normal nor collectionwise Hausdorff. As an addendum to the proof, we show that if the cardinality of the continuum is

2

ω
1

2^{\omega _1}

, then these properties also fail in the closed subspace consisting of all functions which assume the value

ω
\omega

on all but countably many indices.
Regularity theory and traces of 𝒜-harmonic functions
Authors: Koskela Pekka , Manfredi Juan , Villamor Enrique
Published: July 2002

In this paper we discuss two different topics concerning

A

\mathcal {A}

- harmonic functions. These are weak solutions of the partial differential equation

div
(

A

(
x
,
∇
u
)
)
=
0
,

\begin{equation*}\text {div}(\mathcal {A}(x,\nabla u))=0,\end{equation*}

where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha left-parenthesis x right-parenthesis StartAbsoluteValue xi EndAbsoluteValue Superscript p minus 1 Baseline less-than-or-equal-to mathematical left-angle script upper A left-parenthesis x comma xi right-parenthesis comma xi mathematical right-angle less-than-or-equal-to beta left-parenthesis x right-parenthesis StartAbsoluteValue xi EndAbsoluteValue Superscript p minus 1">

α
(
x
)

|

ξ

|

p
−
1

≤
⟨

A

(
x
,
ξ
)
,
ξ
⟩
≤
β
(
x
)

|

ξ

|

p
−
1

\alpha (x)|\xi |^{p-1}\le \langle \mathcal {A}(x,\xi ),\xi \rangle \le \beta (x) |\xi |^{p-1}

for some fixed

p
∈
(
1
,
∞
)

p\in (1,\infty )

, the function

β
\beta

is bounded and

α
(
x
)
>
0

\alpha (x)>0

for a.e.

x
x

. First, we present a new approach to the regularity of

A

\mathcal {A}

-harmonic functions for

p
>
n
−
1

p>n-1

. Secondly, we establish results on the existence of nontangential limits for

A

\mathcal {A}

-harmonic functions in the Sobolev space

W

1
,
q

(

B

)

W^{1,q}(\mathbb {B})

, for some

q
>
1

q>1

, where

B

\mathbb {B}

is the unit ball in

R

n

\mathbb {R}^n

. Here

q
q

is allowed to be different from

p
p

.
Uniform harmonic approximation of bounded functions
Authors: Gardiner Stephen
Published: July 2002

Let

Ω
\Omega

be an open set in

R

n

\mathbb R^n

and

E
E

be a relatively closed subset of

Ω
\Omega

. We characterize those pairs

(
Ω
,
E
)

(\Omega ,E)

which have the following property: every function which is bounded and continuous on

E
E

and harmonic on

E
0

E^0

can be uniformly approximated by functions harmonic on

Ω
\Omega

. Several related results concerning both harmonic and superharmonic approximation are also established.
Radial Solutions to a Dirichlet Problem Involving Critical Exponents when 𝑁=6
Authors: Castro Alfonso , Kurepa Alexandra
Published: July 2002

In this paper we show that, for each

λ
>
0

\lambda > 0

, the set of radially symmetric solutions to the boundary value problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column minus normal upper Delta u left-parenthesis x right-parenthesis 2nd Column a m p semicolon equals lamda u left-parenthesis x right-parenthesis plus u left-parenthesis x right-parenthesis StartAbsoluteValue u left-parenthesis x right-parenthesis EndAbsoluteValue comma 3rd Column a m p semicolon 4th Column a m p semicolon x element-of upper B colon equals StartSet x element-of upper R Superscript 6 Baseline colon double-vertical-bar x greater-than 1 double-vertical-bar EndSet comma 2nd Row 1st Column u left-parenthesis x right-parenthesis 2nd Column a m p semicolon equals 0 comma 3rd Column a m p semicolon 4th Column a m p semicolon x element-of partial-differential upper B comma EndLayout">

−
Δ
u
(
x
)

a
m
p
;
=
λ
u
(
x
)
+
u
(
x
)
|
u
(
x
)
|
,

a
m
p
;

a
m
p
;
x
∈
B
:=
{
x
∈

R
6

:
‖
x
>
1
‖
}
,

u
(
x
)

a
m
p
;
=
0
,

a
m
p
;

a
m
p
;
x
∈
∂
B
,

\begin {aligned} -\Delta u(x) &= \lambda u(x) + u(x)\vert u(x)\vert , && x\in B := \{x\in R^6\colon \|x > 1\| \},\\ u(x) &= 0, && x\in \partial B, \end {aligned}

\]
is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.
A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials
Authors: Eie Minking
Published: July 2002

Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani’s formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [Minking Eie, The special values at negative integers of Dirichlet series associated with polynomials of several variables, Proceedings of A. M. S. 119 (1993), 51–61] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.
On functions in the little Bloch space and inner functions
Authors: Rohde S.
Published: July 2002

We prove that analytic functions in the little Bloch space assume every value as a radial limit on a set of Hausdorff dimension one, unless they have radial limits on a set of positive measure. The analogue for inner functions in the little Bloch space is also proven, and characterizations of various classes of Bloch functions in terms of their level sets are given.
Real connective K-theory and the quaternion group
Authors: Bayen Dilip , Bruner Robert
Published: July 2002

Let

k
o

ko

be the real connective K-theory spectrum. We compute

k

o
∗

B
G

ko_*BG

and

k

o
∗

B
G

ko^*BG

for groups

G
G

whose Sylow 2-subgroup is quaternion of order 8. Using this we compute the coefficients

t
(
k
o

)
∗
G

t(ko)^G_*

of the

G
G

fixed points of the Tate spectrum

t
(
k
o
)

t(ko)

for

G
=
S

l
2

(
3
)

G = Sl_2(3)

and

G
=

Q
8

G = Q_8

. The results provide a counterexample to the optimistic conjecture of Greenlees and May [ J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Memoirs AMS 543 (1995)], Conj. 13.4, by showing, in particular, that

t
(
k
o

)
G

t(ko)^G

is not a wedge of Eilenberg-Mac Lane spectra, as occurs for groups of prime order.
Asymptotic Expansion for Layer Solutions of a Singularly Perturbed Reaction-Diffusion System
Authors: Lin Xiao-Biao
Published: July 2002

For a singularly perturbed

n
n

-dimensional system of reaction– diffusion equations, assuming that the 0th order solutions possess boundary and internal layers and are stable in each regular and singular region, we construct matched asymptotic expansions for formal solutions in all the regular, boundary, internal and initial layers to any desired order in

ϵ
\epsilon

. The formal solution shows that there is an invariant manifold of wave-front-like solutions that attracts other nearby solutions. We also give conditions for the wave-front-like solutions to converge slowly to stationary solutions on that manifold.
Even Linkage Classes
Authors: Nollet Scott
Published: July 2002

In this paper we generalize the

E

\mathcal {E}

and

N

\mathcal {N}

-type resolutions used by Martin-Deschamps and Perrin for curves in

P

3

\mathbb {P}^{3}

to subschemes of pure codimension in projective space, and shows that these resolutions are interchanged by the mapping cone procedure under a simple linkage. Via these resolutions, Rao’s correspondence is extended to give a bijection between even linkage classes of subschemes of pure codimension two and stable equivalence classes of reflexive sheaves

E

\mathcal {E}

satisfying

H

∗

1

(

E

)
=
0

H^{1}_{*}( \mathcal {E})=0

and

E

x

t

1

(

E

∨

,

O

)
=
0

\mathcal {E}xt^{1}( \mathcal {E}^{\vee }, \mathcal {O})=0

. Further, these resolutions are used to extend the work of Martin-Deschamps and Perrin for Cohen-Macaulay curves in

P

3

\mathbb {P}^{3}

to subschemes of pure codimension two in

P

n

\mathbb {P}^{n}

. In particular, even linkage classes of such subschemes satisfy the Lazarsfeld-Rao property and any minimal subscheme for an even linkage class links directly to a minimal subscheme for the dual class.
Reciprocity Laws in the Verlinde Formulae for the Classical Groups
Authors: Oxbury W. , Wilson S.
Published: July 2002

The Verlinde formula is computed for each of the simply-connected classical Lie groups, and it is shown that the resulting formula obeys certain reciprocity laws with respect to the exchange of the rank and the level. Some corresponding dualities between spaces of sections of theta line bundles over moduli spaces of

G
G

-bundles on curves are conjectured but not proved.
Single generator problem
Authors: Tanaka Jun-ichi
Published: July 2002

Using the Stone-Čech compactification

β

Z

\beta \mathbf Z

of integers, we introduce a free extension of an almost periodic flow. Together with some properties of outer functions, we see that, in a certain class of ergodic Hardy spaces

H
p

(
μ
)

H^p(\mu )

,

1
≤
p
≤
∞

1\le p\le \infty

, the corresponding subspaces

H
0
p

(
μ
)

H_0^p(\mu )

are all singly generated. This shows the existence of maximal weak-

∗

^*

Dirichlet algebras, different from

H
∞

H^\infty

of the disc, for which the single generator problem is settled.
Prime spectra of quantum semisimple groups
Authors: Brown K. , Goodearl K.
Published: July 2002

We study the prime ideal spaces of the quantized function algebras

R

q

[
G
]

R_{q}[G]

, for

G
G

a semisimple Lie group and

q
q

an indeterminate. Our method is to examine the structure of algebras satisfying a set of seven hypotheses, and then to demonstrate, using work of Joseph, Hodges and Levasseur, that the algebras

R

q

[
G
]

R_{q}[G]

satisfy this list of assumptions. Rings satisfying the assumptions are shown to satisfy normal separation, and therefore Jategaonkar’s strong second layer condition. For such rings much representation-theoretic information is carried by the graph of links of the prime spectrum, and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions. Homogeneity is a key feature – it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms. Bounds on the ranks of these groups are obtained in the case of

R

q

[
G
]

R_{q}[G]

. In the final section the results are specialized to the case

G
=
S

L

n

(

C

)

G= SL_{n}(\mathbb {C})

, where detailed calculations can be used to illustrate the general results. As a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine

n
n

-space satisfy our axiom scheme when the group generated by the parameters is torsionfree.
Optimal natural dualities. II: General theory
Authors: Davey B. , Priestley H.
Published: July 2002

A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set

R
R

of finitary algebraic relations yields a duality on a class of algebras

A

=

I

S

P

⁡
(

M
_

)

\mathcal {A} = \operatorname {\mathbb {I}\mathbb {S}\mathbb {P}}( \underline {M})

, those subsets

R
′

R’

of

R
R

which yield optimal dualities are characterised. Further, the manner in which the relations in

R
R

are constructed from those in

R
′

R’

is revealed in the important special case that

M
_

\underline {M}

generates a congruence-distributive variety and is such that each of its subalgebras is subdirectly irreducible. These results are obtained by studying a certain algebraic closure operator, called entailment, definable on any set of algebraic relations on

M
_

\underline {M}

. Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties

B

n

\mathbf {B}_{n}

of pseudocomplemented distributive lattices, the theory improves upon and illuminates previous results.
The ergodic theory of discrete isometry groups on manifolds of variable negative curvature
Authors: Yue Chengbo
Published: July 2002

This paper studies the ergodic theory at infinity of an arbitrary discrete isometry group

Γ
\Gamma

acting on any Hadamard manifold

H
H

of pinched variable negative curvature. Most of the results obtained by Sullivan in the constant curvature case are generalized to the case of variable curvature. We describe connections between measures supported on the limit set of

Γ
\Gamma

, dynamics of the geodesic flow and the geometry of

M
=
H

/

Γ

M=H/ \Gamma

. We explore the relationship between the growth exponent of the group, the Hausdorff dimension of the limit set and the topological entropy of the geodesic flow. The equivalence of various descriptions of an analogue of the Hopf dichotomy is proved. As applications, we settle a question of J. Feldman and M. Ratner about the horocycle flow on a finite volume surface of negative curvature and obtain an asymptotic formula for the counting function of lattice points. At the end of this paper, we apply our results to the study of some rigidity problems. More applications to Mostow rigidity for discrete subgroups of rank 1 noncompact semisimple Lie groups with infinite covolume will be published in subsequent papers by the author.
Representations of monoids arising from finite groups of Lie type
Authors: Salwa A.
Published: July 2002

A class of finite monoids

M
M

constructed from a group

G
G

of Lie type is considered. We describe the irreducible complex representations and prove the complete reducibility of the representations of

M
M

. The sandwich matrix of

M
M

is decomposed into a product of matrices corresponding to maximal parabolic subgroups of

G
G

.
Total absolute curvature and tightness of noncompact manifolds
Authors: van Gemmeren Martin
Published: July 2002

In the first part we prove an extension of the Chern-Lashof inequality for noncompact immersed manifolds with finitely many ends. For this we give a lower bound of the total absolute curvature in terms of topological invariants of the manifold. In the second part we discuss tightness properties for such immersions. Finally, we give an upper bound for the substantial codimension.
Weierstrass points on cyclic covers of the projective line
Authors: Towse Christopher
Published: July 2002

We are interested in cyclic covers of the projective line which are totally ramified at all of their branch points. We begin with curves given by an equation of the form

y

n

=
f
(
x
)

y^{n}=f(x)

, where

f
f

is a polynomial of degree

d
d

. Under a mild hypothesis, it is easy to see that all of the branch points must be Weierstrass points. Our main problem is to find the total Weierstrass weight of these points,

B
W

BW

. We obtain a lower bound for

B
W

BW

, which we show is exact if

n
n

and

d
d

are relatively prime. As a fraction of the total Weierstrass weight of all points on the curve, we get the following particularly nice asymptotic formula (as well as an interesting exact formula):

lim

d
→
∞

B
W

g

3

−
g

=

n
+
1

3
(
n
−
1

)

2

,

\begin{equation*} \lim _{d\to \infty }\frac {BW}{g^{3}-g}=\frac {n+1}{3(n-1)^{2}}, \end{equation*}

where

g
g

is the genus of the curve. In the case that

n
=
3

n=3

(cyclic trigonal curves), we are able to show in most cases that for sufficiently large primes

p
p

, the branch points and the non-branch Weierstrass points remain distinct modulo

p
p

.
An 𝜔₂-minimal Boolean algebra
Authors: Rabus Mariusz
Published: July 2002

For every linear order

L
L

we define a notion of

L
L

-minimal Boolean algebra and then give a consistent example of an

ω

2

\omega _{2}

-minimal algebra. The Stone space

X
X

of our algebra contains a point

{
∗
}

\{*\}

such that

X
−
{
∗
}

X-\{*\}

is an example of a countably tight, initially

ℵ

1

\aleph _{1}

-compact, non-compact space. This answers a question of Dow and van Douwen.
Spectral Convergence for Degenerating Sequences of Three Dimensional Hyperbolic Manifolds
Authors: Ji Lizhen
Published: July 2002

For degenerating sequences of three dimensional hyperbolic manifolds of finite volume, we prove convergence of their eigenfunctions, heat kernel and spectral measure.
Covering the integers by arithmetic sequences. II
Authors: Sun Zhi-Wei
Published: July 2002

Let

A
=
{

a

s

+

n

s

Z

}

s
=
1

k

A= \{a_{s}+n_{s}\mathbb {Z}\}^{k}_{s=1}

(

n

1

⩽
⋯
⩽

n

k

)

n_{1} \leqslant \cdots \leqslant n_{k})

be a system of arithmetic sequences where

a

1

,
⋯
,

a

k

∈

Z

a_{1}, \cdots ,a_{k}\in \mathbb {Z}

and

n

1

,
⋯
,

n

k

∈

Z

+

n_{1},\cdots ,n_{k}\in \mathbb {Z}^{+}

. For

m
∈

Z

+

m\in \mathbb {Z}^{+}

system

A
A

will be called an (exact)

m
m

-cover of

Z

\mathbb {Z}

if every integer is covered by

A
A

at least (exactly)

m
m

times. In this paper we reveal further connections between the common differences in an (exact)

m
m

-cover of

Z

\mathbb {Z}

and Egyptian fractions. Here are some typical results for those

m
m

-covers

A
A

of

Z

\mathbb {Z}

: (a) For any

m

1

,
⋯
,

m

k

∈

Z

+

m_{1},\cdots ,m_{k}\in \mathbb {Z}^{+}

there are at least

m
m

positive integers in the form

Σ

s
∈
I

m

s

/

n

s

\Sigma _{s\in I} m_{s}/n_{s}

where

I
⊆
{
1
,
⋯
,
k
}

I \subseteq \{1,\cdots ,k\}

. (b) When

n

k
−
l

>

n

k
−
l
+
1

=
⋯
=

n

k

n_{k-l}>n_{k-l+1}= \cdots =n_{k}

(

0
>
l
>
k
)

0>l>k)

, either

l
⩾

n

k

/

n

k
−
l

l \geqslant n_{k}/n_{k-l}

or

Σ

s
=
1

k
−
l

1

/

n

s

⩾
m

\Sigma ^{k-l}_{s=1}1/n_{s} \geqslant m

, and for each positive integer

λ
>

n

k

/

n

k
−
l

\lambda >n_{k}/n_{k-l}

the binomial coefficient

(

l
λ

)

\binom l{ \lambda }

can be written as the sum of some denominators

>
1

>1

of the rationals

Σ

s
∈
I

1

/

n

s

−
λ

/

n

k

,
I
⊆
{
1
,
⋯
,
k
}

\Sigma _{s\in I}1/n_{s}- \lambda /n_{k}, I \subseteq \{1,\cdots ,k\}

if

A
A

forms an exact

m
m

-cover of

Z

\mathbb {Z}

. (c) If

{

a

s

+

n

s

Z

}

s
=
1
 
s
≠
t

k

\{a_{s}+n_{s}\mathbb {Z}\}^{k}_{\substack {s=1\ s\not =t}}

is not an

m
m

-cover of

Z

\mathbb {Z}

, then

Σ

s
∈
I

1

/

n

s

,
I
⊆
{
1
,
⋯
,
k
}
∖
{
t
}

\Sigma _{s\in I}1/n_{s}, I \subseteq \{1,\cdots ,k\}\setminus \{t\}

have at least

n

t

n_{t}

distinct fractional parts and for each

r
=
0
,
1
,
⋯
,

n

t

−
1

r=0,1,\cdots ,n_{t}-1

there exist

I

1

,

I

2

⊆
{
1
,
⋯
,
k
}
∖
{
t
}

I_{1},I_{2} \subseteq \{1,\cdots ,k\}\setminus \{t\}

such that

r

/

n

t

≡

Σ

s
∈

I

1

1

/

n

s

−

Σ

s
∈

I

2

1

/

n

s

r/n_{t} \equiv \Sigma _{s\in I_{1}}1/n_{s}-\Sigma _{s\in I_{2}}1/n_{s}

(mod 1). If

A
A

forms an exact

m
m

-cover of

Z

\mathbb {Z}

with

m
=
1

m=1

or

n

1

>
⋯
>

n

k
−
l

>

n

k
−
l
+
1

=
⋯
=

n

k

n_{1}> \cdots >n_{k-l}>n_{k-l+1}= \cdots =n_{k}

(

l
>
0

l>0

) then for every

t
=
1
,
⋯
,
k

t=1, \cdots ,k

and

r
=
0
,
1
,
⋯
,

n

t

−
1

r=0,1,\cdots ,n_{t}-1

there is an

I
⊆
{
1
,
⋯
,
k
}

I \subseteq \{1,\cdots ,k\}

such that

Σ

s
∈
I

1

/

n

s

≡
r

/

n

t

\Sigma _{s\in I}1/n_{s} \equiv r/n_{t}

(mod 1).
Sharp upper bound for the first non-zero Neumann eigenvalue for bounded domains in rank-1 symmetric spaces
Authors: Aithal A. , Santhanam G.
Published: July 2002

In this paper, we prove that for a bounded domain

Ω
\Omega

in a rank-

1
1

symmetric space, the first non-zero Neumann eigenvalue

μ

1

(
Ω
)
≤

μ

1

(
B
(

r

1

)
)

\mu _{1}(\Omega )\leq \mu _{1}(B(r_{1}))

where

B
(

r

1

)

B(r_{1})

denotes the geodesic ball of radius

r

1

r_{1}

such that

v
o
l
(
Ω
)
=
v
o
l
(
B
(

r

1

)
)

\begin{equation*}vol(\Omega )=vol(B(r_{1}))\end{equation*}

and equality holds iff

Ω
=
B
(

r

1

)

\Omega =B(r_{1})

. This result generalises the works of Szego, Weinberger and Ashbaugh-Benguria for bounded domains in the spaces of constant curvature.
Krull-Schmidt fails for serial modules
Authors: Facchini Alberto
Published: July 2002

We answer a question posed by Warfield in 1975: the Krull-Schmidt Theorem does not hold for serial modules, as we show via an example. Nevertheless we prove a weak form of the Krull-Schmidt Theorem for serial modules (Theorem 1.9). And we show that the Grothendieck group of the class of serial modules of finite Goldie dimension over a fixed ring

R
R

is a free abelian group.
On the homology spectral sequence for topological Hochschild homology
Authors: Hunter Thomas
Published: July 2002

Marcel Bökstedt has computed the homotopy type of the topological Hochschild homology of

Z

/

p

\Bbb Z/p

using his definition of topological Hochschild homology for a functor with smash product. Here we show that easy conceptual proofs of his main technical result of are possible in the context of the homotopy theory of

S
S

-algebras as introduced by Elmendorf, Kriz, Mandell and May. We give algebraic arguments based on naturality properties of the topological Hochschild homology spectral sequence. In the process we demonstrate the utility of the unstable “lower” notation for the Dyer-Lashof algebra.
Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces
Authors: Lemańczyk M. , Parreau F. , Volný D.
Published: July 2002

Given an irrational rotation, in the space of real bounded variation functions it is proved that there are ergodic cocycles whose small perturbations remain ergodic; in fact, the set of ergodic cocycles has nonempty dense interior. Given a pseudo-homogeneous Banach space and an irrational rotation, we study the set of elements satisfying the mean ergodic theorem. Once such a space is not homogeneous, we prove it is not reflexive and not separable. In “natural" cases, up to

L
1

L^1

-cohomology, the only elements satisfying the mean ergodic theorem are those from the closure of trigonometric polynomials. For pseudo-homogeneous spaces admitting a Koksma’s inequality ergodicity of the corresponding cylinder flows can be deduced from spectral properties of some circle extensions. In particular this is the case of Lebesgue spectrum (in the orthocomplement of the space of eigenfunctions) for the circle extension.
Convex integral functionals
Authors: Papageorgiou Nikolaos
Published: July 2002

We study nonlinear integral functionals determined by normal convex integrands. First we obtain expressions for their convex conjugate, their

ε
\varepsilon

-subdifferential

(
ε
≥
0
)

(\varepsilon \ge 0)

and their

ε
\varepsilon

-directional derivative. Then we derive a necessary and sufficient condition for the existence of an approximate solution for the continuous infimal convolution. We also obtain general conditions which guarantee the interchangeability of the conditional expectation and subdifferential operators. Finally we examine the conditional expectation of random sets.
On The Homotopy Type of 𝐵𝐺 for Certain Finite 2-Groups 𝐺
Authors: Broto Carlos , Levi Ran
Published: July 2002

We consider the homotopy type of classifying spaces

B
G

BG

, where

G
G

is a finite

p
p

-group, and we study the question whether or not the mod

p
p

cohomology of

B
G

BG

, as an algebra over the Steenrod algebra together with the associated Bockstein spectral sequence, determine the homotopy type of

B
G

BG

. This article is devoted to producing some families of finite 2-groups where cohomological information determines the homotopy type of

B
G

BG

.
Anticanonical Rational Surfaces
Authors: Harbourne Brian
Published: July 2002

A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven over an algebraically closed field of arbitrary characteristic. Applications, to be treated in separate papers, include questions involving: points in good position, birational models of rational surfaces in projective space, and resolutions for 0-dimensional subschemes of

P

2

\mathbf {P}^{2}

defined by complete ideals.
Shadowing orbits of ordinary differential equations on invariant submanifolds
Authors: Coomes Brian
Published: July 2002

A finite time shadowing theorem for autonomous ordinary differential equations is presented. Under consideration is the case were there exists a twice continuously differentiable function 

g
g

mapping phase space into 

R

m

\mathbb {R}^{m}

with the property that for a particular regular value 

c
\boldsymbol c

of 

g
g

the submanifold 

g

−
1

(
c
)

g^{-1}(\boldsymbol c)

is invariant under the flow. The main theorem gives a condition which implies that an approximate solution lying close to 

g

−
1

(
c
)

g^{-1}(\boldsymbol c)

is uniformly close to a true solution lying in 

g

−
1

(
c
)

g^{-1}(\boldsymbol c)

. Applications of this theorem to computer generated approximate orbits are discussed.
Partition identities involving gaps and weights
Authors: Alladi Krishnaswami
Published: July 2002

We obtain interesting new identities connecting the famous partition functions of Euler, Gauss, Lebesgue, Rogers–Ramanujan and others by attaching weights to the gaps between parts. The weights are in general multiplicative. Some identities involve powers of 2 as weights and yield combinatorial information about some remarkable partition congruences modulo powers of 2.