bluuebunny/crossref_metadata_2025 huggingface dataset:
Frobenius extensions of subalgebras of Hopf algebras
Authors: Fischman D. , Montgomery S. , Schneider H.-J.
Published: July 2002

We consider when extensions

S
⊂
R

S\subset R

of subalgebras of a Hopf algebra are

β
\beta

-Frobenius, that is Frobenius of the second kind. Given a Hopf algebra

H
H

, we show that when

S
⊂
R

S\subset R

are Hopf algebras in the Yetter-Drinfeld category for

H
H

, the extension is

β
\beta

-Frobenius provided

R
R

is finite over

S
S

and the extension of biproducts

S
⋆
H
⊂
R
⋆
H

S\star H\subset R\star H

is cleft.

More generally we give conditions for an extension to be

β
\beta

-Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants.

We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras.
Disjoint paths, planarizing cycles, and spanning walks
Authors: Yu Xingxing
Published: July 2002

We study the existence of certain disjoint paths in planar graphs and generalize a theorem of Thomassen on planarizing cycles in surfaces. Results are used to prove that every 5-connected triangulation of a surface with sufficiently large representativity is hamiltonian, thus verifying a conjecture of Thomassen. We also obtain results about spanning walks in graphs embedded in a surface with large representativity.
Higher-dimensional virtual diagonals and ideal cohomology for triangular algebras
Authors: Paterson Alan , Smith Roger
Published: July 2002

We investigate the cohomology of non-self-adjoint algebras using virtual diagonals and their higher-dimensional generalizations. We show that infinite dimensional nest algebras always have non-zero second cohomology by showing that they cannot possess 2-virtual diagonals. In the case of the upper triangular atomic nest algebra we exhibit concrete modules for non-vanishing cohomology.
On the extremal structure of the unit balls of Banach spaces of weakly continuous functions and their duals
Authors: Hu Zhibao , Smith Mark
Published: July 2002

A sufficient and then a necessary condition are given for a function to be an extreme point of the unit ball of the Banach space

C
(
K
,
(
X
,
w
)
)

C(K,(X,w))

of continuous functions, under the supremum norm, from a compact Hausdorff topological space

K
K

into a Banach space

X
X

equipped with its weak topology

w
w

. Strongly extreme points of the unit ball of

C
(
K
,
(
X
,
w
)
)

C(K,(X,w))

are characterized as the norm-one functions that are uniformly strongly extreme point valued on a dense subset of

K
K

. It is shown that a variety of stronger types of extreme points (e.g. denting points) never exist in the unit ball of

C
(
K
,
(
X
,
w
)
)

C(K,(X,w))

. Lastly, some naturally arising and previously known extreme points of the unit ball of

C
(
K
,
(
X
,
w
)

)

∗

C(K,(X,w))^{*}

are shown to actually be strongly exposed points.
Curves of maximum genus in range A and stick-figures
Authors: Ballico Edoardo , Bolondi Giorgio , Ellia Philippe , Mirò-Roig Rosa
Published: July 2002

In this paper we show the existence of smooth connected space curves not contained in a surface of degree less than

m
m

, with genus maximal with respect to the degree, in half of the so-called range A. The main tool is a technique of deformation of stick-figures due to G. Fløystad.
Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers
Authors: Kleiman Steven , Ulrich Bernd
Published: July 2002

Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade

2
2

as those with a Hilbert–Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade

1
1

can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor, a symmetric syzygy matrix. In addition, we establish a correspondence, roughly speaking, between Gorenstein perfect algebras of grade

1
1

that are birational onto their image, on the one hand, and self-linked perfect ideals of grade

2
2

that have one of the self-linking elements contained in the second symbolic power, on the other hand. Finally, we provide another characterization of these ideals in terms of their symbolic Rees algebras, and we prove a criterion for these algebras to be normal.
Rumely’s local global principle for algebraic 𝑃𝒮𝒞 fields over rings
Authors: Jarden Moshe , Razon Aharon
Published: July 2002

Let

S

\mathcal {S}

be a finite set of rational primes. We denote the maximal Galois extension of

Q

\mathbb {Q}

in which all

p
∈

S

p\in \mathcal {S}

totally decompose by

N
N

. We also denote the fixed field in

N
N

of

e
e

elements

σ

1

,
…
,

σ

e

\sigma _{1},\ldots , \sigma _{e}

in the absolute Galois group

G
(

Q

)

G( \mathbb {Q})

of

Q

\mathbb {Q}

by

N
(

σ

)

N( {\boldsymbol \sigma })

. We denote the ring of integers of a given algebraic extension

M
M

of

Q

\mathbb {Q}

by

Z

M

\mathbb {Z}_{M}

. We also denote the set of all valuations of

M
M

(resp., which lie over

S
S

) by

V

M

\mathcal {V}_{M}

(resp.,

S

M

\mathcal {S}_{M}

). If

v
∈

V

M

v\in \mathcal {V}_{M}

, then

O

M
,
v

O_{M,v}

denotes the ring of integers of a Henselization of

M
M

with respect to

v
v

. We prove that for almost all

σ

∈
G
(

Q

)

e

{\boldsymbol \sigma }\in G( \mathbb {Q})^{e}

, the field

M
=
N
(

σ

)

M=N( {\boldsymbol \sigma })

satisfies the following local global principle: Let

V
V

be an affine absolutely irreducible variety defined over

M
M

. Suppose that

V
(

O

M
,
v

)
≠
∅

V(O_{M,v})\not =\varnothing

for each

v
∈

V

M

∖

S

M

v\in \mathcal {V}_{M}\backslash \mathcal {S}_{M}

and

V

s
i
m

(

O

M
,
v

)
≠
∅

V_{\mathrm {sim}}(O_{M,v})\not =\varnothing

for each

v
∈

S

M

v\in \mathcal {S}_{M}

. Then

V
(

O

M

)
≠
∅

V(O_{M})\not =\varnothing

. We also prove two approximation theorems for

M
M

.
Metric completions of ordered groups and 𝐾₀ of exchange rings
Authors: Pardo E.
Published: July 2002

We give a description of the closure of the natural affine continuous function representation of

K
0

(
R
)

K_0(R)

for any exchange ring

R
R

. This goal is achieved by extending the results of Goodearl and Handelman, about metric completions of dimension groups, to a more general class of pre-ordered groups, which includes

K
0

K_0

of exchange rings. As a consequence, the results about

K
0
+

K_0^+

of regular rings, which the author gave in an earlier paper, can be extended to a wider class of rings, which includes

C
∗

C^*

-algebras of real rank zero, among others. Also, the framework of pre-ordered groups developed here allows other potential applications.
Euler products associated to metaplectic automorphic forms on the 3-fold cover of 𝐺𝑆𝑝(4)
Authors: Goetze Thomas
Published: July 2002

If

ϕ
\phi

is a generic cubic metaplectic form on GSp(4), that is also an eigenfunction for all the Hecke operators, then corresponding to

ϕ
\phi

is an Euler product of degree 4 that has a functional equation and meromorphic continuation to the whole complex plane. This correspondence is obtained by convolving

ϕ
\phi

with the cubic

θ
\theta

-function on GL(3) in a Shimura type Rankin-Selberg integral.
Extremal vectors and invariant subspaces
Authors: Ansari Shamim , Enflo Per
Published: July 2002

For a bounded linear operator on Hilbert space we define a sequence of so-called minimal vectors in connection with invariant subspaces and show that this presents a new approach to invariant subspaces. In particular, we show that for any compact operator

K
K

some weak limit of the sequence of minimal vectors is noncyclic for all operators commuting with

K
K

and that for any normal operator

N
N

, the norm limit of the sequence of minimal vectors is noncyclic for all operators commuting with

N
N

. Thus, we give a new and more constructive proof of existence of invariant subspaces. The sequence of minimal vectors does not seem to converge in norm for an arbitrary bounded linear operator. We will prove that if

T
T

belongs to a certain class

C

{\mathcal {C}}

of operators, then the sequence of such vectors converges in norm, and that if

T
T

belongs to a subclass of

C

{\mathcal {C}}

, then the norm limit is cyclic.
Double Walsh series with coefficients of bounded variation of higher order
Authors: Chen Chang-Pao , Wu Ching-Tang
Published: July 2002

Let

D

j

k

(
x
)

D_{j}^{k}(x)

denote the Cesàro sums of order

k
k

of the Walsh functions. The estimates of

D

j

k

(
x
)

D_{j}^{k}(x)

given by Fine back in 1949 are extended to the case

k
>
2

k>2

. As a corollary, the following properties are established for the rectangular partial sums of those double Walsh series whose coefficients satisfy conditions of bounded variation of order

(
p
,
0
)
,
(
0
,
p
)

(p,0), (0,p)

, and

(
p
,
p
)

(p,p)

for some

p
≥
1

p\ge 1

: (a) regular convergence; (b) uniform convergence; (c)

L

r

L^{r}

-integrability and

L

r

L^{r}

-metric convergence for

0
>
r
>
1

/

p

0>r>1/p

; and (d) Parseval’s formula. Extensions to those with coefficients of generalized bounded variation are also derived.
The approximate functional formula for the theta function and Diophantine Gauss sums
Authors: Coutsias E. , Kazarinoff N.
Published: July 2002

We consider the polygonal lines in the complex plane

C

\Bbb {C}

whose

N
N

-th vertex is defined by

S
N

=

∑

n
=
0

N

′

exp
⁡
(
i
ω
π

n
2

)

S_N = \sum _{n=0}^{N\,’} \exp (i\omega \pi n^2)

(with

ω
∈

R

\omega \in \Bbb {R}

), where the prime means that the first and last terms in the sum are halved. By introducing the discrete curvature of the polygonal line, and by exploiting the similarity of segments of the line, for small

ω
\omega

, to Cornu spirals (C-spirals), we prove the precise renormalization formula
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column Blank 2nd Column a m p semicolon StartAbsoluteValue sigma-summation Underscript k equals 0 Overscript upper N Endscripts prime exp left-parenthesis i omega pi k squared right-parenthesis minus StartFraction exp left-parenthesis s g n left-parenthesis omega right-parenthesis i pi slash 4 right-parenthesis Over StartRoot StartAbsoluteValue omega EndAbsoluteValue EndRoot EndFraction sigma-summation Underscript k equals 0 Overscript n Endscripts prime exp left-parenthesis minus i StartFraction pi Over omega EndFraction k squared right-parenthesis EndAbsoluteValue 2nd Row 1st Column Blank 2nd Column a m p semicolon less-than-or-equal-to upper C StartAbsoluteValue StartFraction omega upper N minus n Over omega EndFraction EndAbsoluteValue comma 0 greater-than StartAbsoluteValue omega EndAbsoluteValue greater-than 1 comma EndLayout">

a
m
p
;

|

∑

k
=
0

N

′

exp
⁡
(
i
ω
π

k
2

)
−

exp
⁡
(
s
g
n
(
ω
)
i
π

/

4
)

|

ω

|

∑

k
=
0

n

′

exp
⁡
(
−
i

π
ω

k
2

)
|

a
m
p
;

≤
C

|

ω
N
−
n

ω

|

,
0
>

|

ω

|

>
1
,

\begin{equation} \begin {split} &\left | \sum _{k=0}^{N}\,’ \exp (i\omega \pi k^2) -\frac {\exp (sgn(\omega )i\pi /4)}{\sqrt {|\omega |}} \sum _{k=0}^n \,’ \exp (-i\frac {\pi }{\omega } k^2)\right |\\ &\qquad \leq C \left |\frac {\omega N - n}{\omega }\right |, 0>|\omega | >1, \end{split} \end{equation}

where

N
=
[
[
n

/

ω
]
]

N=[[n/\omega ]]

, the nearest integer to

n

/

ω

n/\omega

and

1
>
C
>
3.14

1>C>3.14

. This formula, which sharpens Hardy and Littlewood’s approximate functional formula for the theta function, generalizes to irrationals, as a Diophantine inequality, the well-known sum formula of Gauss. The geometrical meaning of the relation between the two limits is that the first sum is taken to a point of inflection of the corresponding C-spirals. The second sum replaces whole C-spirals of the first by unit vectors times scale and phase factors. The block renormalization procedure implied by this replacement is governed by the circle map

ω
→
−

1
ω

(
mod

2
)
,
ω
∈
]
−
1
,
+
1
[
∖
{
0
}
,

\begin{equation} \omega \rightarrow -\frac {1}{\omega } \pmod 2 , \omega \in ]-1,+1[ \setminus \{0\}, \end{equation}

whose orbits are analyzed by expressing

ω
\omega

as an even continued fraction.
Fox calculus, symplectic forms, and moduli spaces
Authors: Zocca Valentino
Published: July 2002

An “open pre-symplectic form” on surfaces with boundary and glueing formulae are provided to symplectically integrate the symplectic form on the deformation space of representations of the fundamental group of a Riemann surface into a reductive Lie group

G
G

.
A new degree bound for vector invariants of symmetric groups
Authors: Fleischmann P.
Published: July 2002

Let

R
R

be a commutative ring,

V
V

a finitely generated free

R
R

-module and

G
≤
G

L
R

(
V
)

G\le GL_R(V)

a finite group acting naturally on the graded symmetric algebra

A
=
S
(
V
)

A=S(V)

. Let

β
(
V
,
G
)

\beta (V,G)

denote the minimal number

m
m

, such that the ring

A
G

A^G

of invariants can be generated by finitely many elements of degree at most

m
m

.

For

G
=

Σ
n

G=\Sigma _n

and

V
(
n
,
k
)

V(n,k)

, the

k
k

-fold direct sum of the natural permutation module, one knows that

β
(
V
(
n
,
k
)
,

Σ
n

)
≤
n

\beta (V(n,k),\Sigma _n) \le n

, provided that

n
!

n!

is invertible in

R
R

. This was used by E. Noether to prove

β
(
V
,
G
)
≤

|

G

|

\beta (V,G) \le |G|

if

|

G

|

!
∈

R
∗

|G|! \in R^*

.

In this paper we prove
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta left-parenthesis upper V left-parenthesis n comma k right-parenthesis comma normal upper Sigma Subscript n Baseline right-parenthesis less-than-or-equal-to m a x left-brace n comma k left-parenthesis n minus 1 right-parenthesis right-brace">

β
(
V
(
n
,
k
)
,

Σ
n

)
≤
m
a
x
{
n
,
k
(
n
−
1
)
}

\beta (V(n,k),\Sigma _n) \le max\{n,k(n-1)\}

for arbitrary commutative rings

R
R

and show equality for

n
=

p
s

n=p^s

a prime power and

R
=

Z

R = \mathbb {Z}

or any ring with

n
⋅

1
R

=
0

n\cdot 1_R=0

. Our results imply
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta left-parenthesis upper V comma upper G right-parenthesis less-than-or-equal-to m a x left-brace StartAbsoluteValue upper G EndAbsoluteValue comma r a n k left-parenthesis upper V right-parenthesis left-parenthesis StartAbsoluteValue upper G EndAbsoluteValue minus 1 right-parenthesis right-brace">

β
(
V
,
G
)
≤
m
a
x
{

|

G

|

,
rank
⁡
(
V
)
(

|

G

|

−
1
)
}

\begin{equation*} \beta (V,G)\le max\{|G|, \operatorname {rank}(V)(|G|-1)\}\end{equation*}

for any ring with

|

G

|

∈

R
∗

|G| \in R^*

.
Green’s function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature
Authors: Bandle C. , Brillard A. , Flucher M.
Published: July 2002

We extend the method of harmonic transplantation from Euclidean domains to spaces of constant positive or negative curvature. To this end the structure of the Green’s function of the corresponding Laplace-Beltrami operator is investigated. By means of isoperimetric inequalities we derive complementary estimates for its distribution function. We apply the method of harmonic transplantation to the question of whether the best Sobolev constant for the critical exponent is attained, i.e. whether there is an extremal function for the best Sobolev constant in spaces of constant curvature. A fairly complete answer is given, based on a concentration-compactness argument and a Pohozaev identity. The result depends on the curvature.
A classification theorem for Albert algebras
Authors: Parimala R. , Sridharan R. , Thakur Maneesh
Published: July 2002

Let

k
k

be a field whose characteristic is different from 2 and 3 and let

L

/

k

L/k

be a quadratic extension. In this paper we prove that for a fixed, degree 3 central simple algebra

B
B

over

L
L

with an involution

σ
\sigma

of the second kind over

k
k

, the Jordan algebra

J
(
B
,
σ
,
u
,
μ
)

J(B,\sigma ,u,\mu )

, obtained through Tits’ second construction is determined up to isomorphism by the class of

(
u
,
μ
)

(u,\mu )

in

H
1

(
k
,
S
U
(
B
,
σ
)
)

H^1(k,SU(B,\sigma ))

, thus settling a question raised by Petersson and Racine. As a consequence, we derive a “Skolem Noether” type theorem for Albert algebras. We also show that the cohomological invariants determine the isomorphism class of

J
(
B
,
σ
,
u
,
μ
)

J(B,\sigma ,u,\mu )

, if

(
B
,
σ
)

(B,\sigma )

is fixed.
Covers of algebraic varieties III. The discriminant of a cover of degree 4 and the trigonal construction
Authors: Casnati G.
Published: July 2002

For each Gorenstein cover

ϱ
:
X
→
Y

\varrho \colon X\to Y

of degree

4
4

we define a scheme

Δ
(
X
)

\Delta (X)

and a generically finite map

Δ
(
ϱ
)
:
Δ
(
X
)
→
Y

\Delta (\varrho )\colon \Delta (X)\to Y

of degree

3
3

called the discriminant of

ϱ
\varrho

. Using this construction we deal with smooth degree

4
4

covers

ϱ
:
X
→

P

C

n

C

\varrho \colon X\to \mathbb {P}_{\mathbb {C}}^ {n}{\mathbb {C}}

with

n
≥
5

n\ge 5

. Moreover we also generalize the trigonal construction of S. Recillas.
Operations and Spectral Sequences. I
Authors: Turner James
Published: July 2002

Using methods developed by W. Singer and J. P. May, we describe a systematic approach to showing that many spectral sequences, determined by a filtration on a complex whose homology has an action of operations, possess a compatible action of the same operations. As a consequence, we obtain W. Singer’s result for Steenrod operations on Serre spectral sequence and extend A. Bahri’s action of Dyer-Lashof operations on the second quadrant Eilenberg-Moore spectral sequence.
Exact Hausdorff measure and intervals of maximum density for Cantor sets
Authors: Ayer Elizabeth , Strichartz Robert
Published: July 2002

Consider a linear Cantor set

K
K

, which is the attractor of a linear iterated function system (i.f.s.)

S

j

x
=

ρ

j

x
+

b

j

S_{j}x = \rho _{j}x+b_{j}

,

j
=
1
,
…
,
m

j = 1,\ldots ,m

, on the line satisfying the open set condition (where the open set is an interval). It is known that

K
K

has Hausdorff dimension

α
\alpha

given by the equation

∑

j
=
1

m

ρ

j

α

=
1

\sum ^{m}_{j=1} \rho ^{\alpha }_{j} = 1

, and that

H

α

(
K
)

\mathcal {H}_{\alpha }(K)

is finite and positive, where

H

α

\mathcal {H}_{\alpha }

denotes Hausdorff measure of dimension

α
\alpha

. We give an algorithm for computing

H

α

(
K
)

\mathcal {H}_{\alpha }(K)

exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When

ρ

1

=

ρ

m

\rho _{1} = \rho _{m}

(or more generally, if

log
⁡

ρ

1

\log \rho _{1}

and

log
⁡

ρ

m

\log \rho _{m}

are commensurable), the algorithm also gives an interval

I
I

that maximizes the density

d
(
I
)
=

H

α

(
K
∩
I
)

/

|

I

|

α

d(I) = \mathcal {H}_{\alpha }(K \cap I)/|I|^{\alpha }

. The Hausdorff measure

H

α

(
K
)

\mathcal {H}_{\alpha }(K)

is not a continuous function of the i.f.s. parameters. We also show that given the contraction parameters

ρ

j

\rho _{j}

, it is possible to choose the translation parameters

b

j

b_{j}

in such a way that

H

α

(
K
)
=

|

K

|

α

\mathcal {H}_{\alpha }(K) = |K|^{\alpha }

, so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs.
Operator valued weights without structure theory
Authors: Falcone Tony , Takesaki Masamichi
Published: July 2002

A result of Haagerup, generalizing a theorem of Takesaki, states the following: If

N

⊂

M

{\mathcal {N}}\subset {\mathcal {M}}

are von Neumann algebras, then there exists a faithful, normal and semi-finite (fns) operator valued weight

T
:

M

+

→

N

+

^

T \colon {\mathcal {M}}_{+} \rightarrow \widehat {{\mathcal {N}}_{+}}

if and only if there exist fns weights

φ
~

\tilde \varphi

on

M

{\mathcal {M}}

and

φ
\varphi

on

N

{\mathcal {N}}

satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript t Superscript phi Baseline left-parenthesis x right-parenthesis equals sigma Subscript t Superscript phi overTilde Baseline left-parenthesis x right-parenthesis for-all x element-of script upper N comma t element-of double-struck upper R">

σ

t

φ

(
x
)
=

σ

t

φ
~

(
x
)

∀
x
∈

N

,
t
∈

R

\sigma ^{\varphi }_{t}(x) = \sigma ^{\tilde \varphi }_{t}(x) \, \forall x \in {\mathcal {N}} , t \in \mathbb {R}

. In fact,

T
T

can be chosen such that

φ
~

=
φ
∘
T

\tilde \varphi = \varphi \circ T

;

T
T

is then uniquely determined by this condition. We present a proof of the above which does not use any structure theory.
Spectral gap estimates on compact manifolds
Authors: Oden Kevin , Sung Chiung-Jue , Wang Jiaping
Published: July 2002

For a compact Riemannian manifold with boundary, its mass gap is the difference between the first and second smallest Dirichlet eigenvalues. In this paper, taking a variational approach, we obtain an explicit lower bound estimate of the mass gap for any compact manifold in terms of geometric quantities.
Geometric groups. I
Authors: Berestovskii Valera , Plaut Conrad , Stallman Cornelius
Published: July 2002

We define a geometry on a group to be an abelian semigroup of symmetric open sets with certain properties. Examples include well-known structures such as invariant Riemannian metrics on Lie groups, hyperbolic groups, and valuations on fields. In this paper we are mostly concerned with geometries where the semigroup is isomorphic to the positive reals, which for Lie groups come from invariant Finsler metrics. We explore various aspects of these geometric groups, including a theory of covering groups for arcwise connected groups, algebraic expressions for invariant metrics and inner metrics, construction of geometries with curvature bounded below, and finding geometrically significant curves in path homotopy classes.
On the number of radially symmetric solutions to Dirichlet problems with jumping nonlinearities of superlinear order
Authors: Castro Alfonso , Kuiper Hendrik
Published: July 2002

This paper is concerned with the multiplicity of radially symmetric solutions

u
(
x
)

u(x)

to the Dirichlet problem
\[

Δ
u
+
f
(
u
)
=
h
(
x
)
+
c
ϕ
(
x
)

\Delta u+f(u)=h(x)+c\phi (x)

\]
on the unit ball

Ω
⊂

R

N

\Omega \subset \mathbf R^N

with boundary condition

u
=
0

u=0

on

∂
Ω

\partial \Omega

. Here

ϕ
(
x
)

\phi (x)

is a positive function and

f
(
u
)

f(u)

is a function that is superlinear (but of subcritical growth) for large positive

u
u

, while for large negative

u
u

we have that

f
′

(
u
)
>
μ

f’(u)>\mu

, where

μ
\mu

is the smallest positive eigenvalue for

Δ
ψ
+
μ
ψ
=
0

\Delta \psi +\mu \psi =0

in

Ω
\Omega

with

ψ
=
0

\psi =0

on

∂
Ω

\partial \Omega

. It is shown that, given any integer

k
≥
0

k\ge 0

, the value

c
c

may be chosen so large that there are

2
k
+
1

2k+1

solutions with

k
k

or less interior nodes. Existence of positive solutions is excluded for large enough values of

c
c

.
The trace space and Kauffman’s knot invariants
Authors: Liu Keqin
Published: July 2002

The traces in the construction of Kauffman’s knot invariants are studied. The trace space is determined for a semisimple finite–dimensional quantum Hopf algebra and the best lower bound of the dimension of the trace space is given for a unimodular finite–dimensional quantum Hopf algebra.
Density doubling, double-circulants, and new sphere packings
Authors: Vardy Alexander
Published: July 2002

New nonlattice sphere packings in dimensions 20, 22, and 44–47 that are denser than the best previously known sphere packings were recently discovered. We extend these results, showing that the density of many sphere packings in dimensions just below a power of 2 can be doubled using orthogonal binary codes. This produces new dense sphere packings in

R

n

\mathbb R^n

for

n
=
25
,
26
,
…
,
31

n = 25,26,\dots ,31

and

55
,
56
,
…
,
63

55,56,\dots ,63

. For

n
=
27
,
28
,
29
,
30

n = 27,28,29,30

the resulting packings are denser than any packing previously known.
Factorization in generalized power series
Authors: Berarducci Alessandro
Published: July 2002

The field of generalized power series with real coefficients and exponents in an ordered abelian divisible groupG\mathbf {G}is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ringR((G≤0))\mathbf {R}(( \mathbf {G}^{\leq 0}))consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976):∑nt−1/n+1\sum _n t^{-1/n}+1. Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. IfG=(R,+,0,≤)\mathbf {G}= ( \mathbf {R}, +, 0, \leq )we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is eitherω\omegaor of the formωωα\omega ^{\omega ^\alpha }and the series is not divisible by any monomial, then it is irreducible. To handle the general case we use a suggestion of M.-H. Mourgues, based on an idea of Gonshor, which allows us to reduce to the special caseG=R\mathbf {G}=\mathbf {R}. In the final part of the paper we study the irreducibility of series with finite support.
A Gauss-Kusmin theorem for optimal continued fractions
Authors: Dajani Karma , Kraaikamp Cor
Published: July 2002

A Gauss-Kusmin theorem for the Optimal Continued Fraction (OCF) expansion is obtained. In order to do so, first a Gauss-Kusmin theorem is derived for the natural extension of the ergodic system underlying Hurwitz’s Singular Continued Fraction (SCF) (and similarly for the continued fraction to the nearer integer (NICF)). Since the NICF, SCF and OCF are all examples of maximal

S
S

-expansions, it follows from a result of Kraaikamp that the SCF and OCF are metrically isomorphic. This isomorphism is then used to carry over the results for the SCF to any other maximal

S
S

-expansion, in particular to the OCF. Along the way, a Heilbronn-theorem is obtained for any maximal

S
S

-expansion.
Convex functions on Alexandrov surfaces
Authors: Mashiko Yukihiro
Published: July 2002

We investigate the topological structure of Alexandrov surfaces of curvature bounded below which possess convex functions. We do not assume the continuities of these functions. Nevertheless, if the convex functions satisfy a condition of local nonconstancy, then the topological structures of Alexandrov surfaces and the level sets configurations of these functions in question are determined.
Conical limit set and Poincaré exponent for iterations of rational functions
Authors: Przytycki Feliks
Published: July 2002

We contribute to the dictionary between action of Kleinian groups and iteration of rational functions on the Riemann sphere. We define the Poincaré exponent<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta left-parenthesis f comma z right-parenthesis equals inf left-brace right-brace colon greater-than-or-equal-to greater-than-or-equal-to alpha 0 colon less-than-or-equal-to less-than-or-equal-to of PP left-parenthesis right-parenthesis comma z comma alpha 0">δ(f,z)=inf{α≥0:P(z,α)≤0}\delta (f,z)=\inf \{ \alpha \ge 0: \mathcal {P}(z, \alpha ) \le 0\}, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P left-parenthesis z comma alpha right-parenthesis colon equals limit sup StartFraction 1 Over n EndFraction log sigma-summation Underscript f Superscript n Baseline left-parenthesis x right-parenthesis equals z Endscripts StartAbsoluteValue left-parenthesis f Superscript n Baseline right-parenthesis prime left-parenthesis x right-parenthesis EndAbsoluteValue Superscript negative alpha Baseline period">P(z,α):=lim supn→∞1nlog⁡∑fn(x)=z|(fn)′(x)|−α.\begin{equation*} \mathcal {P}(z, \alpha ):=\limsup _{n\to \infty }{1\over n}\log \sum _{f^n(x)=z} |(f^n)’(x)|^{- \alpha }. \end{equation*}We prove thatδ(f,z)\delta (f,z)andP(z,α)\mathcal {P}(z, \alpha )do not depend onzz, providedzzis non-exceptional.P\mathcal {P}plays the role of pressure; we prove that it coincides with the Denker-Urbański pressure ifα≤δ(f)\alpha \le \delta (f). Various notions of “conical limit set" are considered. They all have Hausdorff dimension equal toδ(f)\delta (f)which is equal to the hyperbolic dimension of the Julia set and also equal to the exponent of some conformal Patterson-Sullivan measures. In an Appendix we also discuss notions of “conical limit set" introduced recently by Urbański and by Lyubich and Minsky.
The ideal structure of some analytic crossed products
Authors: Shpigel Miron
Published: July 2002

We study the ideal structure of a class of some analytic crossed products. For an

r
r

-discrete, principal, minimal groupoid

G
G

, we consider the analytic crossed product

C
∗

(
G
,
σ
)

×
α

Z

+

C^*(G,\sigma )\times _\alpha \mathbb {Z}_+

, where

α
\alpha

is given by a cocycle

c
c

. We show that the maximal ideal space

M

\mathcal {M}

of

C
∗

(
G
,
σ
)

×
α

Z

+

C^*(G,\sigma )\times _\alpha \mathbb {Z}_+

depends on the asymptotic range of

c
c

,

R
∞

(
c
)

R_\infty (c)

; that is,

M

\mathcal {M}

is homeomorphic to

D

¯

∣

R
∞

(
c
)

\overline { \mathbb {D}}\mid R_\infty (c)

for

R
∞

(
c
)

R_\infty (c)

finite, and

M

\mathcal {M}

consists of the unique maximal ideal for

R
∞

(
c
)
=

T

R_\infty (c)= \mathbb {T}

. We also prove that

C
∗

(
G
,
σ
)

×
α

Z

+

C^*(G,\sigma )\times _\alpha \mathbb {Z}_+

is semisimple in both cases, and that

R
∞

(
c
)

R_\infty (c)

is invariant under isometric isomorphism.
Quantization of presymplectic manifolds and circle actions
Authors: da Silva Ana , Karshon Yael , Tolman Susan
Published: July 2002

We prove several versions of “quantization commutes with reduction” for circle actions on manifolds that are not symplectic. Instead, these manifolds possess a weaker structure, such as a spin

c

^c

structure. Our theorems work whenever the quantization data and the reduction data are compatible; this condition always holds if we start from a presymplectic (in particular, symplectic) manifold.
Asymptotics toward the planar rarefaction wave for viscous conservation law in two space dimensions
Authors: Nishikawa Masataka , Nishihara Kenji
Published: July 2002

This paper is concerned with the asymptotic behavior of the solution toward the planar rarefaction wave

r
(

x
t

)

r(\frac {x}{t})

connecting

u

+

u_{+}

and

u

−

u_{-}

for the scalar viscous conservation law in two space dimensions. We assume that the initial data

u

0

(
x
,
y
)

u_{0}(x,y)

tends to constant states

u

±

u_{\pm }

as

x
→
±
∞

x \rightarrow \pm \infty

, respectively. Then, the convergence rate to

r
(

x
t

)

r(\frac {x}{t})

of the solution

u
(
t
,
x
,
y
)

u(t,x,y)

is investigated without the smallness conditions of

|

u

+

−

u

−

|

|u_{+}-u_{-}|

and the initial disturbance. The proof is given by elementary

L

2

L^{2}

-energy method.
Resolutions of monomial ideals and cohomology over exterior algebras
Authors: Aramova Annetta , Avramov Luchezar , Herzog Jürgen
Published: July 2002

This paper studies the homology of finite modules over the exterior algebra

E
E

of a vector space

V
V

. To such a module

M
M

we associate an algebraic set

V
E

(
M
)
⊆
V

V_E(M)\subseteq V

, consisting of those

v
∈
V

v\in V

that have a non-minimal annihilator in

M
M

. A cohomological description of its defining ideal leads, among other things, to complementary expressions for its dimension, linked by a ‘depth formula’. Explicit results are obtained for

M
=
E

/

J

M=E/J

, when

J
J

is generated by products of elements of a basis

e
1

,
…
,

e
n

e_1,\dots ,e_n

of

V
V

. A (infinite) minimal free resolution of

E

/

J

E/J

is constructed from a (finite) minimal resolution of

S

/

I

S/I

, where

I
I

is the squarefree monomial ideal generated by ‘the same’ products of the variables in the polynomial ring

S
=
K
[

x
1

,
…
,

x
n

]

S=K[x_1,\dots ,x_n]

. It is proved that

V
E

(
E

/

J
)

V_E(E/J)

is the union of the coordinate subspaces of

V
V

, spanned by subsets of

{

e
1

,
…
,

e
n

}

\{\,e_1,\dots ,e_n\,\}

determined by the Betti numbers of

S

/

I

S/I

over

S
S

.
Direct sum decompositions of infinitely generated modules
Authors: Benson D. , Wheeler Wayne
Published: July 2002

Almost all of the basic theorems in the representation theory of finite groups have proofs that depend upon the Krull–Schmidt Theorem. Because this theorem holds only for finite-dimensional modules, however, the recent interest in infinitely generated modules raises the question of which results may hold more generally. In this paper we present an example showing that Green’s Indecomposability Theorem fails for infinitely generated modules. By developing and applying some general properties of idempotent modules, we are also able to construct explicit examples of modules for which the cancellation property fails.
Multivariate matrix refinable functions with arbitrary matrix dilation
Authors: Jiang Qingtang
Published: July 2002

Characterizations of the stability and orthonormality of a multivariate matrix refinable function

P

h
i

\mathbf {P}hi

with arbitrary matrix dilation

M
M

are provided in terms of the eigenvalue and

1
1

-eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of

Φ
\Phi

is equivalent to the order of the vanishing moment conditions of the matrix refinement mask

{

P

α

}

\{\mathbf {P}_{\alpha }\}

. The restricted transition operator associated with the matrix refinement mask

{

P

α

}

\{\mathbf {P}_{\alpha }\}

is represented by a finite matrix

(

A

M
i
−
j

)

i
,
j

({\mathcal A} _{Mi-j})_{i, j}

, with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Subscript j Baseline equals StartAbsoluteValue det left-parenthesis upper M right-parenthesis EndAbsoluteValue Superscript negative 1 Baseline sigma-summation Underscript kappa Endscripts bold upper P Subscript kappa minus j Baseline circled-times bold upper P Subscript kappa">

A

j

=

|

det
⁡
(
M
)

|

−
1

∑

κ

P

κ
−
j

⊗

P

κ

{\mathcal A} _j=|\operatorname {det}(M)|^{-1}\sum _{\kappa }\mathbf {P} _{\kappa -j}\otimes \mathbf {P}_{\kappa }

and

P

κ
−
j

⊗

P

κ

\mathbf {P} _{\kappa -j}\otimes \mathbf {P}_{\kappa }

being the Kronecker product of matrices

P

κ
−
j

\mathbf {P} _{\kappa -j}

and

P

κ

\mathbf {P}_{\kappa }

. The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function

Φ
\Phi

is given in terms of the spectral radius of the restricted transition operator to an invariant subspace. This estimate is analyzed in an example.
Classes of singular integrals along curves and surfaces
Authors: Seeger Andreas , Wainger Stephen , Wright James , Ziesler Sarah
Published: July 2002

This paper is concerned with singular convolution operators in

R

d

\mathbb {R}^{d}

,

d
≥
2

d\ge 2

, with convolution kernels supported on radial surfaces

y

d

=
Γ
(

|

y
′

|

)

y_{d}=\Gamma (|y’|)

. We show that if

Γ
(
s
)
=
log
⁡
s

\Gamma (s)=\log s

, then

L

p

L^{p}

boundedness holds if and only if

p
=
2

p=2

. This statement can be reduced to a similar statement about the multiplier

m
(
τ
,
η
)
=

|

τ

|

−
i
η

m(\tau ,\eta )=|\tau |^{-i\eta }

in

R

2

\mathbb {R}^{2}

. We also construct smooth

Γ
\Gamma

for which the corresponding operators are bounded for

p

0

>
p
≤
2

p_{0}>p\le 2

but unbounded for

p
≤

p

0

p\le p_{0}

, for given

p

0

∈
[
1
,
2
)

p_{0}\in [1,2)

. Finally we discuss some examples of singular integrals along convex curves in the plane, with odd extensions.
Isometric Extensions of zero entropy ℤ^{𝕕} Loosely Bernoulli Transformations
Authors: Johnson Aimee , Şahi̇n Ayşe
Published: July 2002

In this paper we discuss loosely Bernoulli for

Z

d

\mathbb Z^d

actions. In particular, we prove that extensions of zero entropy, ergodic, loosely Bernoulli

Z

d

\mathbb Z^d

actions are also loosely Bernoulli.
Existence and uniqueness of rectilinear slit maps
Authors: FitzGerald Carl , Weening Frederick
Published: July 2002

We consider a generalization of the parallel slit uniformization in which the angle of inclination of each image slit is assigned independently. Koebe proved that for domains of finite connectivity there is, up to a normalization, a unique rectilinear slit map achieving any given angle assignment. Koebe’s theorem is partially extended to domains of infinite connectivity. A uniqueness result is shown for domains of countable connectivity and arbitrary angle assignments, and an existence result is proved for arbitrary domains under the assumption that the angle assignment is continuous and has finite range. In order to prove the existence result a new extremal length tool, called the crossing-module, is introduced. The crossing-module allows greater freedom in the family of admissible arcs than the classical module. Several results known for the module are extended to the crossing-module. A generalization of Jenkins’

θ

{\theta }

module condition for the parallel slit problem is given for the rectilinear slit problem in terms of the crossing-module and it is shown that rectilinear slit maps satisfying this crossing-module condition exist.
The calculus of partition sequences, changing cofinalities, and a question of Woodin
Authors: Apter Arthur , Henle James , Jackson Stephen
Published: July 2002

We study in this paper polarized infinite exponent partition relations. We apply our results to constructing a model for the theory “ZF

+
+

DC

+

ω
1

+\omega _1

is the only regular, uncountable cardinal

≤

ω

ω
1

+
1

\le \omega _{\omega _1+1}

.” This gives a partial answer to a question of Woodin.
An analytic Riemann-Hilbert correspondence for semi-simple Lie groups
Authors: Smithies Laura , Taylor Joseph
Published: July 2002

Geometric Representation Theory for semi-simple Lie groups has two main sheaf theoretic models. Namely, through Beilinson-Bernstein localization theory, Harish-Chandra modules are related to holonomic sheaves ofD\mathcal Dmodules on the flag variety. Then the (algebraic) Riemann-Hilbert correspondence relates these sheaves to constructible sheaves of complex vector spaces. On the other hand, there is a parallel localization theory for globalized Harish-Chandra modules—i.e., modules over the full semi-simple group which are completions of Harish-Chandra modules. In particular, Hecht-Taylor and Smithies have developed a localization theory relating minimal globalizations of Harish-Chandra modules to group equivariant sheaves ofD\mathcal Dmodules on the flag variety. The main purpose of this paper is to develop an analytic Riemann-Hilbert correspondence relating these sheaves to constructible sheaves of complex vector spaces and to discuss the relationship between this “analytic" study of global modules and the preceding “algebraic" study of the underlying Harish-Chandra modules.
Subunit H of the V-ATPase Involved in Endocytosis Shows Homology to β-Adaptins
Authors: Geyer Matthias , Fackler Oliver T. , Peterlin B. Matija
Published: July 2002

The vacuolar ATPase (V-ATPase) is a multisubunit enzyme that facilitates the acidification of intracellular compartments in eukaryotic cells and plays an important role in receptor-mediated endocytosis, intracellular trafficking processes, and protein degradation. In this study we show that the C-terminal fragment of 350 residues of the regulatory subunit H (V1H) of the V-ATPase shares structural and functional homologies with the β-chains of adaptor protein complexes. Moreover, the fragment is similar to a region in the β-subunit of COPI coatomer complexes, which suggests the existence of a shared domain in these three different families of proteins. For β-adaptins, this fragment binds to cytoplasmic di-leucine–based sorting motifs such as in HIV-1 Nef that mediate endocytic trafficking. Expression of this fragment in cells blocks the internalization of transmembrane proteins, which depend on di-leucine–based motifs, whereas mutation of the consensus sequence GEY only partly diminishes the recognition of the sorting motif. Based on recent structural analysis, our results suggest that the di-leucine-binding domain consists of a HEAT or ARM repeat protein fold.
Maternal Glucose Concentration Influences Fetal Growth, Gestation, and Pregnancy Complications
Authors: Scholl Theresa O. , Sowers MaryFran , Chen Xinhua , Lenders Carine
Published: July 2002

Using 1990–1995 data, the authors examined the influence of post-challenge maternal glucose concentration on pregnancy outcome in 1,157 nondiabetic US gravidas. After control for potential confounding variables and comparing gravidas with lower glucose concentrations (<99 mg/dl) with the others, they found that mean birth weight increased by 50 g and 200 g with glucose concentrations of 99–130 mg/dl and >130 mg/dl, respectively. Increased maternal glucose concentration also was associated with an increased risk of large-for-gestation fetuses (p for trend < 0.001) and a decreased risk of fetal growth restriction (p for trend < 0.05). The association between glucose and gestation was inverse and significantly shortened when glucose concentrations were higher. Maternal complications increased twofold or more with high glucose concentrations and included cesarean section and clinical chorioamnionitis. Chorioamnionitis in combination with high maternal glucose concentration increased the risk of very preterm delivery almost 12-fold. These observations extend Pedersen's hypothesis—that high concentrations of maternal glucose give rise to increased nutrient transfer to the fetus and increase fetal growth, beyond the model of maternal diabetes (Acta Endocrinol 1954;16:330–42). They raise the question of whether higher, but seemingly normal maternal glucose concentration predisposes to or is a marker for placental inflammation and infection.
Automated analysis of interatomic contacts in proteins.
Authors: Sobolev V , Sorokine A , Prilusky J , Abola E E , Edelman M
Published: July 2002

MOTIVATION: New software has been designed to assist the molecular biologist in understanding the structural consequences of modifying a ligand and/or protein. RESULTS: Tools are described for the analysis of ligand-protein contacts (LPC software) and contacts of structural units (CSU software) such as helices, sheets, strands and residues. Our approach is based on a detailed analysis of interatomic contacts and interface complementarity. For any ligand or structural unit, these software automatically: (i) calculate the solvent-accessible surface of every atom; (ii) determine the contacting residues and type of interaction they undergo (hydrophobic-hydrophobic, aromatic-aromatic, etc.); (iii) indicate all putative hydrogen bonds. LPC software further predicts changes in binding strength following chemical modification of the ligand. AVAILABILITY: Both LPC and CSU can be accessed through the PDB and are integrated in the 3DB Atlas page of all PDB files. For any given file, the tools can also be accessed at http://www.pdb.bnl. gov/pdb-bin/lpc?PDB_ID= and http://www.pdb.bnl. gov/pdb-bin/csu?PDB_ID= with the four-letter PDB code added at the end in each case. Finally, LPC and CSU can be accessed at: http://sgedg.weizmann.ac.il/lpc and http://sgedg.weizmann.ac.il/csu.
LIAN 3.0: detecting linkage disequilibrium in multilocus data
Authors: Haubold Bernhard , Hudson Richard R.
Published: July 2002

Summary: LIAN is a program to test the null hypothesis of linkage equilibrium for multilocus data. LIAN incorporates both a Monte Carlo method as well as a novel algebraic method to carry out the hypothesis test. The program further returns the genetic diversity of the sample and the pairwise distances between its members.

Availability: LIAN can be accessed at http://soft.ice.mpg.de/lian

Contact: haubold@ice.mpg.de

To whom correspondence should be addressed.
Regulation of intracellular dynamics of Smad4 by its leucine‐rich nuclear export signal
Authors: Watanabe Mina , Masuyama Norihisa , Fukuda Makoto , Nishida Eisuke
Published: July 2002

Smad family proteins play a pivotal role in transmitting the transforming growth factor‐β (TGF‐β) superfamily signals from the cell surface to the nucleus. In response to ligand stimulation, Smad4 forms a complex with respective receptor‐specific Smads, and the complex translocates into the nucleus and regulates gene expression. Thus, the nuclear entry of the Smad complex is one of the key steps in signal transduction. However, little is known about regulatory mechanisms for nucleocytoplasmic transport of Smads. Here we report identification of a functional, leucine‐rich nuclear export signal (NES) in Smad4, which regulates subcellular distribution of Smad4. We then show evidence suggesting that the NES‐dependent cytoplasmic localization of Smad4 is important for ensuring optimal TGF‐β responsivenesses in transcriptional activation. Moreover, we show that the NES of Smad4 is specifically inactivated by the stimulus‐dependent hetero‐oligomerization with receptor‐specific Smads during the TGF‐β‐induced nuclear translocation of Smad4. Taken together, these results suggest an important regulatory role of the NES of Smad4 in TGF‐β signaling.
The problem of effective parameters of a mixture of two isotropic dielectrics distributed in space-time and the conservation law for wave impedance in one-dimensional wave propagation
Authors: Lurie Konstantin A.
Published: July 2002

The problem of effective parameters of a composite medium assembled from the original constituents distributed in space–time is formulated in covariant tensor form. I introduce the fourth rank tensor of material constants using Maxwel's system for a moving dielectric medium as a model example. For one–dimensional wave propagation, if a mixture is composed from two dielectrics with the same ratio ϵ/μ of permittivity ϵ to permeability μ, then the ratio
E
/
M
of an effective permittivity to an effective permeability of the mixture will preserve the value ϵ/μ. This statement may be rephrased as the conservation law for the relevant wave impedances √μ/ϵ this is similar to the law known for two–dimensional polycrystals in an analogous elliptic situation. The tensor concept developed for a dielectric medium is based on the idea of a relativistic invariance of Maxwel's system. The idea of relativistic invariance is fundamental for an adequate description of the effective parameters of any material assemblage in space–time regardless of its physical embodiment. Problems related to structural vibrations or acoustics could be completely understood on the basis of the relevant relativistic equations.
Molecular mechanisms of neurite extension
Authors: Valtorta Flavia , Leoni Chiara
Published: July 2002

The extension of neurites is a major task of developing neurons, requiring a significant metabolic effort to sustain the increase in molecular synthesis necessary for plasma membrane expansion. In addition, neurite extension involves changes in the subsets of expressed proteins and reorganization of the cytomatrix. These phenomena are driven by environmental cues which activate signal transduction processes as well as by the intrinsic genetic program of the cell. The present review summarizes some of the most recent progress made in the elucidation of the molecular mechanisms underlying these processes.