bluuebunny/crossref_metadata_2025 huggingface dataset:
On one set of orthogonal harmonic polynomials
Authors: Karachik V.
Published: July 2002

A new basis of harmonic polynomials is given. Proposed polynomials are orthogonal on the unit sphere and each term of this basis consists of monomials not present in the others.
A class of 𝑀-dilation scaling functions with regularity growing proportionally to filter support width
Authors: Shi Xianliang , Sun Qiyu
Published: July 2002

In this paper, a class of

M
M

-dilation scaling functions with regularity growing proportionally to filter support width is constructed. This answers a question proposed by Daubechies on p.338 of her book Ten Lectures on Wavelets (1992).
Existence of ergodic retraction for noncommutative semigroups in Banach spaces
Authors: Kada Osamu
Published: July 2002

We prove the existence of ergodic retraction for a noncommutative semigroup which is right Eberlein-weakly almost periodic in a uniformly convex Banach space.
On the essential self-adjointness of the general second order elliptic operators
Authors: Oleinik I.
Published: July 2002

In this paper, we give sufficient conditions for the essential self-adjointness of second order elliptic operators. It turns out that these conditions coincide with those for the Schrödinger operator on a manifold whose metric essentially depends on the principal coefficients of a given operator.
Lengths of radii under conformal maps of the unit disc
Authors: Balogh Zoltan , Bonk Mario
Published: July 2002

If

E

f

(
R
)

E_{f}(R)

is the set of endpoints of radii which have length greater than or equal to

R
>
0

R>0

under a conformal map

f
f

of the unit disc, then

cap
⁡

E

f

(
R
)
=
O
(

R

−
1

/

2

)

\operatorname {cap} E_{f}(R)=O(R^{-1/2})

as

R
→
∞

R\to \infty

for the logarithmic capacity of

E

f

(
R
)

E_{f}(R)

. The exponent

−
1

/

2

-1/2

is sharp.
Lindelöf property and absolute embeddings
Authors: Bella A. , Yaschenko I.
Published: July 2002

It is proved that a Tychonoff space is Lindelöf if and only if whenever a Tychonoff space

Y
Y

contains two disjoint closed copies

X

1

X_{1}

and

X

2

X_{2}

of

X
X

, then these copies can be separated in

Y
Y

by open sets. We also show that a Tychonoff space

X
X

is weakly

C
C

-embedded (relatively normal) in every larger Tychonoff space if and only if

X
X

is either almost compact or Lindelöf (normal almost compact or Lindelöf).
Weyl’s construction and tensor power decomposition for 𝐺₂
Authors: Huang Jing-Song , Zhu Chen-Bo
Published: July 2002

Let

V
V

be the 7-dimensional irreducible representations of

G

2

G_{2}

. We decompose the tensor power

V

⊗
n

V^{\otimes n}

into irreducible representations of

G

2

G_{2}

and obtain all irreducible representations of

G

2

G_{2}

in the decomposition. This generalizes Weyl’s work on the construction of irreducible representations and decomposition of tensor products for classical groups to the exceptional group

G

2

G_{2}

.
Positivity of polarizations of 𝑛-positive maps
Authors: Kiciński Piotr
Published: July 2002

It is shown that polarization formulas have explicit matrix representations. This enables us to prove that polarization formulas of

n
n

-positive maps between

C

∗

C^{*}

-algebras are coordinatewise positive.
𝔫-homology of generic representations for 𝔊𝔏(𝔑)
Authors: Chang Jen-Tseh , Cogdell James
Published: July 2002

We compute the

n

\mathfrak {n}

-homology for a class of representations of

G
L
(
N
,

R

)

GL(N,\mathbb {R})

and

G
L
(
N
,

C

)

GL(N,\mathbb {C})

which admit a Whittaker model. They are all completely reducible.
Harmonic maps with noncontact boundary values
Authors: Donnelly Harold
Published: July 2002

Every rank one symmetric space

M
M

, of noncompact type, admits a compactification

M
¯

\overline M

by attaching a sphere

S

n
−
1

S^{n-1}

at infinity. If

M
M

does not have constant sectional curvature, then

M
¯

−
M

\overline M-M

admits a natural contact structure. This paper presents a number of harmonic maps

h
h

, from

M
M

to

M
M

, which extend continuously to

M
¯

\overline M

, and have noncontact boundary values. If the boundary values are assumed continuously differentiable, then the contact structure must be preserved.
On ℎ-cobordisms of spherical space forms
Authors: Kwasik Sławomir , Schultz Reinhard
Published: July 2002

Given a manifold

M
M

of dimension at least 4 whose universal covering is homeomorphic to a sphere, the main result states that a compact manifold

W
W

is isomorphic to a cylinder

M
×
[
0
,
1
]

M\times [0,1]

if and only if

W
W

is homotopy equivalent to this cylinder and the boundary is isomorphic to two copies of

M
M

; this holds in the smooth, PL and topological categories. The result yields a classification of smooth, finite group actions on homotopy spheres (in dimensions

≥
5

\geq 5

) with exactly two singular points.
Composition operators: Hyperinvariant subspaces, quasi-normals and isometries
Authors: Cload Bruce
Published: July 2002

We exhibit hyperinvariant subspaces of some composition operators. We also consider quasi-normal composition operators and discuss the commutant of isometric composition operators.
A central limit theorem for Markov chains and applications to hypergroups
Authors: Gallardo Léonard
Published: July 2002

Let

(

X
n

)

(X_n)

be a homogeneous Markov chain on an unbounded Borel subset of

R

\mathbb {R}

with a drift function

d
d

which tends to a limit

m
1

m_1

at infinity. Under a very simple hypothesis on the chain we prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n Superscript negative 1 slash 2 Baseline left-parenthesis upper X Subscript n Baseline minus sigma-summation Underscript k equals 1 Overscript n Endscripts d left-parenthesis upper X Subscript k minus 1 Baseline right-parenthesis right-parenthesis">

n

−
1

/

2

(

X
n

−

∑

k
=
1

n

d
(

X

k
−
1

)
)

\displaystyle n^{-1/2} (X_n - \sum ^ n_{k=1} d(X_{k-1}))

converges in distribution to a normal law

N
(
0
,

σ
2

)

N (0, \sigma ^2)

where the variance

σ
2

\sigma ^2

depends on the asymptotic behaviour of

(

X
n

)

(X_n)

. When

d
−

m
1

d - m_1

goes to zero quickly enough and

m
1

≠
0

m_1 \neq 0

, the random centering may be replaced by

n

m
1

.

n m_1.

These results are applied to the case of random walks on some hypergroups.
The decomposition theorem for two-dimensional shifts of finite type
Authors: Johnson Aimee , Madden Kathleen
Published: July 2002

A one-dimensional shift of finite type can be described as the collection of bi-infinite “walks" along an edge graph. The Decomposition Theorem states that every conjugacy between two shifts of finite type can be broken down into a finite sequence of splittings and amalgamations of their edge graphs. When dealing with two-dimensional shifts of finite type, the appropriate edge graph description is not as clear; we turn to Nasu’s notion of a “textile system" for such a description and show that all two-dimensional shifts of finite type can be so described. We then define textile splittings and amalgamations and prove that every conjugacy between two-dimensional shifts of finite type can be broken down into a finite sequence of textile splittings, textile amalgamations, and a third operation called an inversion.
An elementary proof of the principle of local reflexivity
Authors: Martínez-Abejón Antonio
Published: July 2002

We give an elementary proof of the principle of local reflexivity.
Existence of many positive solutions of semilinear elliptic equations on an annulus
Authors: Wang Zhi-Qiang , Willem Michel
Published: July 2002

This paper is concerned with multiplicity of positive nonradial solutions of a nonlinear eigenvalue problem on an expanding annulus domain with a fixed width in

R

N

\mathbf {R}^N

with

N
≥
4

N\geq 4

. For

0
>
λ
>

π
2

0>\lambda >\pi ^2

, we show that the number of nonrotationally equivalent nonradial solutions tends to infinity as the inner radius of the domain tends to infinity.
Degenerations for modules over representation-finite algebras
Authors: Zwara Grzegorz
Published: July 2002

Let

A
A

be a representation-finite algebra. We show that a finite dimensional

A
A

-module

M
M

degenerates to another

A
A

-module

N
N

if and only if the inequalities
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension Subscript upper K Baseline upper H o m Subscript upper A Baseline left-parenthesis upper M comma upper X right-parenthesis less-than-or-equal-to dimension Subscript upper K Baseline upper H o m Subscript upper A Baseline left-parenthesis upper N comma upper X right-parenthesis">

dim

K

⁡
H
o

m

A

(
M
,
X
)
≤

dim

K

⁡
H
o

m

A

(
N
,
X
)

\dim _{K} Hom_{A}(M,X)\leq \dim _{K} Hom_{A}(N,X)

hold for all

A
A

-modules

X
X

. We prove also that if

Ext

A

1

⁡
(
X
,
X
)
=
0

\operatorname {Ext}_{A}^{1}(X,X)=0

for any indecomposable

A
A

-module

X
X

, then any degeneration of

A
A

-modules is given by a chain of short exact sequences.
On the irrationality of a certain 𝑞 series
Authors: Borwein Peter , Zhou Ping
Published: July 2002

We prove that if

q
q

is an integer greater than one,

r
r

and

s
s

are any positive rationals such that

1
+

q
m

r
−

q

2
m

s
≠
0

1+q^mr-q^{2m}s\neq 0

for all integers

m
≥
0

m\geq 0

, then
\[

∑

j
=
0

∞

1

1
+

q
j

r
−

q

2
j

s

\sum _{j=0}^\infty \frac 1{1+q^jr-q^{2j}s}

\]
is irrational and is not a Liouville number.
Counting the values taken by algebraic exponential polynomials
Authors: Everest G. , Shparlinski I.
Published: July 2002

We prove an effective mean-value theorem for the values of a non-degenerate, algebraic exponential polynomial in several variables. These objects generalise simultaneously the fundamental examples of linear recurrence sequences and sums of

S
S

-units. The proof is based on an effective, uniform estimate for the deviation of the exponential polynomial from its expected value. This estimate is also used to obtain a non-effective asymptotic formula counting the norms of these values below a fixed bound.
A necessary condition of solvability for the capillarity boundary of Monge-Ampere equations in two dimensions
Authors: Xi-Nan Ma
Published: July 2002

In this paper we consider a class of Monge-Ampere equations with a prescribed contact angle boundary value problem on a bounded strictly convex domain in two dimensions. The purpose is to give a sharp necessary condition of solvability for the above mentioned equations. This is achieved by using the maximum principle and introducing a curvilinear coordinate system for Monge-Ampere equations in two dimensions. An interesting feature of our necessary condition is the need for a certain strong restriction between the curvature of the boundary of domain and the boundary condition, which does not appear in the Dirichlet and Neumann boundary values.
Remarkable asymmetric random walks
Authors: Mattner L.
Published: July 2002

There exists an asymmetric probability measure

P
P

on the real line with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Superscript asterisk n Baseline left-parenthesis right-bracket 0 comma normal infinity left-bracket right-parenthesis plus left-parenthesis 1 slash 2 right-parenthesis upper P Superscript asterisk n Baseline left-parenthesis StartSet 0 EndSet right-parenthesis equals 1 slash 2">

P

∗
n

(
]
0
,
∞
[
)
+
(
1

/

2
)

P

∗
n

(
{
0
}
)
=
1

/

2

P^{\ast n} (]0,\infty [) + (1/2) P^{\ast n} (\{0\}) = 1/2

for every

n
∈

N

n \in \mathbf {N}

.

P
P

can be chosen absolutely continuous and

P
P

can be chosen to be concentrated on the integers. In both cases,

P
P

can be chosen to have moments of every order, but

P
P

cannot be determined by its moments.
Uniqueness of non-Archimedean entire functions sharing sets of values counting multiplicity
Authors: Cherry William , Yang Chung-Chun
Published: July 2002

A set is called a unique range set for a certain class of functions if each inverse image of that set uniquely determines a function from the given class. We show that a finite set is a unique range set, counting multiplicity, for non-Archimedean entire functions if and only if there is no non-trivial affine transformation preserving the set. Our proof uses a theorem of Berkovich to extend, to non-Archimedean entire functions, an argument used by Boutabaa, Escassut, and Haddad to prove this result for polynomials
On the quintuple product identity
Authors: Farkas Hershel , Kra Irwin
Published: July 2002

In this note we present a new proof of the quintuple product identity which is based on our study of

k

t
h

k^{th}

order theta functions with characteristics and the identities they satisfy. In this context the quintuple product identity is another example of an identity which when phrased in terms of theta functions, rather than infinite products and sums, has a simpler form and is much less mysterious.
Relative Brauer groups of discrete valued fields
Authors: Fein Burton , Schacher Murray
Published: July 2002

Let

E
E

be a non-trivial finite Galois extension of a field

K
K

. In this paper we investigate the role that valuation-theoretic properties of

E

/

K

E/K

play in determining the non-triviality of the relative Brauer group,

Br
⁡
(
E

/

K
)

\operatorname {Br} (E/K)

, of

E
E

over

K
K

. In particular, we show that when

K
K

is finitely generated of transcendence degree 1 over a

p
p

-adic field

k
k

and

q
q

is a prime dividing

[
E
:
K
]

[E:K]

, then the following conditions are equivalent: (i) the

q
q

-primary component,

Br
⁡
(
E

/

K

)

q

\operatorname {Br} (E/K)_{q}

, is non-trivial, (ii)

Br
⁡
(
E

/

K

)

q

\operatorname {Br} (E/K)_{q}

is infinite, and (iii) there exists a valuation

π
\pi

of

E
E

trivial on

k
k

such that

q
q

divides the order of the decomposition group of

E

/

K

E/K

at

π
\pi

.
A countable nondiscrete topological field without nontrivial convergent sequences
Authors: Shibakov Alexander
Published: July 2002

A construction of a space in the title is presented, answering a question asked by A. Arkhangel

′

’

skii and D. Shakhmatov.
A generalization of a theorem of Edwards
Authors: Wu Jyh-Yang
Published: July 2002

In this note we extend a theorem of Edwards on the characterization of topological manifolds for polyhedra to a more general class of stratified spaces. We show that a cone-like space

X
X

of dimension

n
≥
3

n\ge 3

is a topological manifold if and only if the base space

B

p

B_{p}

of every point

p
p

in

X
X

is a simply connected cone-like sphere.
On the product of two generalized derivations
Authors: Barraa Mohamed , Pedersen Steen
Published: July 2002

Two elements

A
A

and

B
B

in a ring

R

\mathfrak {R}

determine a generalized derivation

δ

A
,
B

\delta _{A,B}

on

R

\mathfrak {R}

by setting

δ

A
,
B

(
X
)

\delta _{A,B}(X)

=
A
X
−
X
A

=AX-XA

for any

X
X

in

R

\mathfrak {R}

. We characterize when the product

δ

C
,
D

δ

A
,
B

\delta _{C,D}\delta _{A,B}

is a generalized derivation in the cases when the ring

R

\mathfrak {R}

is the algebra of all bounded operators on a Banach space

E

\mathcal {E}

, and when

R

\mathfrak {R}

is a

C

∗

C^{*}

–algebra

A

\mathfrak {A}

. We use these characterizations to compute the commutant of the range of

δ

A
,
B

\delta _{A,B}

.
The main involutions of the metaplectic group
Authors: Kable Anthony
Published: July 2002

We determine the set of automorphisms of a metaplectic group which lift the main involution of the general linear group over an infinite field. Some basic properties of these automorphisms are also established.
The growth theorem of convex mappings on the unit ball in ℂⁿ
Authors: Hamada Hidetaka
Published: July 2002

Let

‖
⋅
‖

\Vert \cdot \Vert

be an arbitrary norm on

C

n

{\mathbb {C}}^{n}

. Let

f
f

be a normalized biholomorphic convex mapping on the unit ball in

C

n

{\mathbb {C}}^{n}

with respect to the norm

‖
⋅
‖

\Vert \cdot \Vert

. We will give an upper bound of the growth of

f
f

.
On models of 𝑈_{𝑞}(𝑠𝑙(2)) and 𝑞-Appell functions using a 𝑞-integral transformation
Authors: Sahai Vivek
Published: July 2002

We discuss few models of the quantum universal enveloping algebra of

s
l
(
2
)

sl(2)

from the special function point of view. Two sets of such models are given, one acting on the space of

1

ϕ
0

_1\phi _0

functions while the other on the space of

q
q

–Appell functions. These models are closely related through a

q
q

–integral transformation. Some interesting identities are obtained.
On the eigenvalue ratio for vibrating strings
Authors: Huang Min-Jei
Published: July 2002

For vibrating strings with concave densities or symmetric single-barrier densities, the ratio

λ
2

/

λ
1

\lambda _2/\lambda _1

of the first two eigenvalues is minimized when the density is constant; while, for vibrating strings with symmetric single-well densities, the ratio

λ
2

/

λ
1

\lambda _2/\lambda _1

is maximized when the density is constant.
Completeness of eigenvectors of group representations of operators whose Arveson spectrum is scattered
Authors: Huang Sen-Zhong
Published: July 2002

We establish the following result.

Theorem. Let

α
:
G
→

L

(
X
)

\alpha :G\to {\mathcal L}(X)

be a

σ
(
X
,

X
∗

)

\sigma (X,X_*)

integrable bounded group representation whose Arveson spectrum

Sp
⁡
(
α
)

\operatorname {Sp}(\alpha )

is scattered. Then the subspace generated by all eigenvectors of the dual representation

α
∗

\alpha ^*

is

w
∗

w^*

dense in

X
∗

.

X^*.

Moreover, the

σ
(
X
,

X
∗

)

\sigma (X,X_*)

closed subalgebra

W
α

W_\alpha

generated by the operators

α
t

\alpha _t

(

t
∈
G

t\in G

) is semisimple.

If, in addition,

X
X

does not contain any copy of

c
0

,

c_0,

then the subspace spanned by all eigenvectors of

α
\alpha

is

σ
(
X
,

X
∗

)

\sigma (X,X_*)

dense in

X
.

X.

Hence, the representation

α
\alpha

is almost periodic whenever it is strongly continuous.
Nonoscillation theorems for second order nonlinear differential equations
Authors: Wong James
Published: July 2002

We prove nonoscillation theorems for the second order Emden-Fowler equation (E):

y
+
a
(
x
)

|

y

|

γ
−
1

y
=
0

y+a(x)|y|^{\gamma -1}y=0

,

γ
>
0

\gamma >0

, where

a
(
x
)
∈
C
(
0
,
∞
)

a(x)\in C(0,\infty )

and

γ
≠
1

\gamma \not =1

. It is shown that when

x

(
γ
+
3
)

/

2
+
δ

a
(
x
)

x^{(\gamma +3)/2+\delta }a(x)

is nondecreasing for any

δ
>
0

\delta >0

and is bounded above, then (E) is nonoscillatory. This improves a well-known result of Belohorec in the sublinear case, i.e. when

0
>
γ
>
1

0>\gamma >1

and

0
>
δ
>
(
1
−
γ
)

/

2

0>\delta >(1-\gamma )/2

.
Signed quasi-measures and dimension theory
Authors: Grubb D.
Published: July 2002

A quasi-linear functional on

C
(
X
)

C(X)

is a real-valued function that is linear on each closed, singly generated subalgebra and is norm bounded. We show that if the covering dimension

dim
⁡
X
≤
1

\dim X\leq 1

, then every quasi-linear functional on

C
(
X
)

C(X)

is, in fact, linear. We do this by considering an associated set function, called a quasi-measure, and ask when such a set function can be extended to be a measure.
Sum of Cantor sets: Self-similarity and measure
Authors: Mendes Pedro
Published: July 2002

In this note it is shown that the sum of two homogeneous Cantor sets is often a uniformly contracting self-similar set and it is given a sufficient condition for such a set to be of Lebesgue measure zero (in fact, of Hausdorff dimension less than one and positive Hausdorff measure at this dimension).
Complete vector fields on (ℂ*)ⁿ
Authors: Andersén Erik
Published: July 2002

We prove necessary and sufficient conditions for a rational vector field on

(

C

∗

)
n

(\mathbb {C}^*)^n

to be complete.
The distribution of solutions of the congruence 𝑥₁𝑥₂𝑥₃…𝑥_{𝑛}≡𝑐(\mod𝑝)
Authors: Ayyad Anwar
Published: July 2002

For a cube

B

\mathcal {B}

of size

B
B

, we obtain a lower bound on

B
B

so that

B

∩
V

\mathcal {B}\cap V

is nonempty, where

V
V

is the algebraic subset of

F

p

n

\mathbb {F}_{p}^{n}

defined by

x

1

x

2

x

3

…

x

n

≡
c

(
mod

p
)
,

\begin{equation*}x_{1}x_{2}x_{3}\dots x_{n}\equiv c\pmod p ,\end{equation*}

n
n

a positive integer and

c
c

an integer not divisible by

p
p

. For

n
=
3

n=3

we obtain that

B

∩
V

\mathcal {B}\cap V

is nonempty if

B
≫

p

2
3

(
log
⁡
p

)

2
3

B\gg p^{\frac {2}{3}}(\log p)^{\frac {2}{3}}

, for

n
=
4

n=4

we obtain that

B

∩
V

\mathcal {B}\cap V

is nonempty if

B
≫

p

log
⁡
p

B\gg \sqrt {p}\log p

, and for

n
≥
5

n\ge 5

we obtain that

B

∩
V

\mathcal {B}\cap V

is nonempty if

B
≫

p

1
4

+

1

2
(
n
+
4
)

(
log
⁡
p

)

3
2

B\gg p^{\frac {1}{4}+\frac {1}{\sqrt {2(n+4)}}}(\log p)^{\frac {3}{2}}

. Using the assumption of the Grand Riemann Hypothesis we obtain

B

∩
V

\mathcal {B}\cap V

is nonempty if

B

≫

ϵ

p

2
n

+
ϵ

B\gg _{\epsilon }p^{\frac {2}{n}+\epsilon }

.
A new characterisation of the analytic Radon-Nikodym property
Authors: Shangquan Bu
Published: July 2002

We show that a separable complex Banach space

X
X

has the analytic Radon-Nikodym property if and only if there exists

1
≤
p
>
∞

1\leq p >\infty

, such that the space consisting of all

L

p

L^{p}

-bounded

X
X

-valued analytic martingales is separable.
Conormal varieties and characteristic varieties
Authors: Coutinho S. , Holland M. , Levcovitz D.
Published: July 2002

We show that the conormal variety of a quasihomogeneous hypersurface in

C

n

{\mathbb {C}}^{n}

, for

n
≥
4

n\ge 4

, whose link is a

Q

\mathbb {Q}

-homology sphere is not the characteristic variety of any

D

{\mathcal {D}}

-module.
Normalizers of the congruence subgroups of the Hecke group 𝐺₅
Authors: Lang Mong-Lung , Tan Ser-Peow
Published: July 2002

Let

λ
=
2

\lambda = 2

cos

(
π

/

5
)

(\pi /5)

and let

G
G

be the Hecke group associated to

λ
\lambda

. In this article, we show that for

τ
\tau

a prime ideal in

Z

[
λ
]

\mathbb {Z}[\lambda ]

, the congruence subgroups

G

0

(
τ
)

G_{0}(\tau )

of

G
G

are self-normalized in

P
S

L

2

(

R

)

PSL_{2}(\mathbb {R})

.
An application of the regularized Siegel-Weil formula on unitary groups to a theta lifting problem
Authors: Tan Victor
Published: July 2002

Let

U
(
2
)

U(2)

and

U
(
2
,
1
)

U(2,1)

be the pair of unitary groups over a global field

F
F

and

π
\pi

an irreducible cuspidal representation of

U
(
2
)

U(2)

which satisfies a certain

L
L

-function condition. By using a regularized Siegel-Weil formula, we can show that the global theta lifting of

π
\pi

in

U
(
2
,
1
)

U(2,1)

is non-trivial if every local factor

π

υ

\pi _{\upsilon }

of

π
\pi

has a local theta lifting (Howe lifting) in

U
(
2
,
1
)
(

F

υ

)

U(2,1)(F_{\upsilon })

.
Iwasawa invariants and class numbers of quadratic fields for the prime 3
Authors: Taya Hisao
Published: July 2002

Let

d
d

be a square-free integer with

d
≡
1

(
mod

3
)

d \equiv 1 \pmod {3}

and

d
>
0

d > 0

. Put

k

+

=

Q

(

d

)

k^{+}=\Bbb Q(\sqrt {d})

and

k

−

=

Q

(

−
3
d

)

k^{-}=\Bbb Q(\sqrt {-3d})

. For the cyclotomic

Z

3

\Bbb Z_3

-extension

k
∞

+

k^{+}_\infty

of

k

+

k^{+}

, we denote by

k
n

+

k^{+}_n

the

n
n

-th layer of

k
∞

+

k^{+}_\infty

over

k

+

k^{+}

. We prove that the

3
3

-Sylow subgroup of the ideal class group of

k
n

+

k^{+}_n

is trivial for all integers

n
≥
0

n \geq 0

if and only if the class number of

k

−

k^{-}

is not divisible by the prime

3
3

. This enables us to show that there exist infinitely many real quadratic fields in which

3
3

splits and whose Iwasawa

λ
3

\lambda _3

-invariant vanishes.
Densely hereditarily hypercyclic sequences and large hypercyclic manifolds
Authors: Bernal-González Luis
Published: July 2002

We prove in this paper that if

(

T

n

)

(T_{n})

is a hereditarily hypercyclic sequence of continuous linear mappings between two topological vector spaces

X
X

and

Y
Y

, where

Y
Y

is metrizable, then there is an infinite-dimensional linear submanifold

M
M

of

X
X

such that each non-zero vector of

M
M

is hypercyclic for

(

T

n

)

(T_{n})

. If, in addition,

X
X

is metrizable and separable and

(

T

n

)

(T_{n})

is densely hereditarily hypercyclic, then

M
M

can be chosen dense.
Boundedness of integral operators on generalized Morrey spaces and its application to Schrödinger operators
Authors: Kurata Kazuhiro , Nishigaki Seiichi , Sugano Satoko
Published: July 2002

In this paper, we study boundedness of integral operators on generalized Morrey spaces and its application to estimates in Morrey spaces for the Schrödinger operator

L
2

=
−
Δ
+
V
(
x
)
+
W
(
x
)

L_2=-\Delta +V(x)+W(x)

with nonnegative

V
∈
(
R
H

)

∞

V\in (RH)_{\infty }

(reverse Hölder class) and small perturbed potentials

W
W

.
Compact composition operators on the Smirnov class
Authors: Choa Jun , Kim Hong , Shapiro Joel
Published: July 2002

We show that a composition operator on the Smirnov class

N
+

N^+

is compact if and only if it is compact on some (equivalently: every) Hardy space

H
p

H^p

for

0
>
p
>
∞

0>p>\infty

. Along the way we show that for composition operators on

N
+

N^+

both the formally weaker notion of boundedness, and a formally stronger notion we call metric compactness, are equivalent to compactness.
Homotopically non-trivial additive subgroups of Hilbert spaces
Authors: Grabowski Janusz
Published: July 2002

We prove that for a line-free closed additive subgroup of a Hilbert space certain orthogonal projections lead to coverings of this group. This makes it possible to obtain additive subgroups which are homotopically non-trivial.
Asymptotic regularity of Daubechies’ scaling functions
Authors: Lau Ka-Sing , Sun Qiyu
Published: July 2002

Let

ϕ
N

\phi _N

,

N
≥
1

N\ge 1

, be Daubechies’ scaling function with symbol

(

1
+

e

−
i
ξ

2

)

N

Q
N

(
ξ
)

\big ({1+e^{-i\xi }\over 2}\big )^N Q_N(\xi )

, and let

s
p

(

ϕ
N

)
,
0
>
p
≤
∞

s_p(\phi _N),0>p\le \infty

, be the corresponding

L
p

L^p

Sobolev exponent. In this paper, we make a sharp estimation of

s
p

(

ϕ
N

)

s_p(\phi _N)

, and we prove that there exists a constant

C
C

independent of

N
N

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N minus StartFraction ln StartAbsoluteValue upper Q Subscript upper N Baseline left-parenthesis 2 pi slash 3 right-parenthesis EndAbsoluteValue Over ln 2 EndFraction minus StartFraction upper C Over upper N EndFraction less-than-or-equal-to s Subscript p Baseline left-parenthesis phi Subscript upper N Baseline right-parenthesis less-than-or-equal-to upper N minus StartFraction ln StartAbsoluteValue upper Q Subscript upper N Baseline left-parenthesis 2 pi slash 3 right-parenthesis EndAbsoluteValue Over ln 2 EndFraction period">

N
−

ln
⁡

|

Q
N

(
2
π

/

3
)

|

ln
⁡
2

−

C
N

≤

s
p

(

ϕ
N

)
≤
N
−

ln
⁡

|

Q
N

(
2
π

/

3
)

|

ln
⁡
2

.

N-{\ln |Q_N(2\pi /3)|\over \ln 2}-{C\over N}\le s_p(\phi _N)\le N-{\ln |Q_N(2\pi /3)|\over \ln 2}.

\]
This answers a question of Cohen and Daubeschies (Rev. Mat. Iberoamericana, 12(1996), 527-591) positively.
Geometry of a crossed product
Authors: Nikolaev Igor
Published: July 2002

We introduce a continuous dimension functionα:∙→R\alpha : \bullet \to \mathbb {R}on the Grothendieck groupK0K_0over the crossed productC∗C^*-algebraC(X)⋊ϕZC(X)\rtimes _{\phi }\mathbb {Z}. The functionα\alphahas an elegant geometry: on every minimal flowϕt\phi ^tit takes the value of the “rotation number" ofϕt\phi ^t; such a problem was posed in 1936 by A. Weil.
On the entropy of the convex hull of finite sets
Authors: Kyrezi Ioanna
Published: July 2002

We give estimates for the entropy numbers and the Gel

′

’

fand diameters of the symmetric convex hull of a finite number of points in a Banach or a Hilbert space.
On free subgroups of units of rings
Authors: Salwa A.
Published: July 2002

We prove that if

a

2

=

b

2

=
0

a^{2}=b^{2}=0

for elements

a
,
b

a,b

of a ring

R
R

of characteristic zero and

a
b

ab

is not nilpotent, then there exists

m
∈

N

m\in {\mathbf N}

such that the group generated by

1
+
m
a

1+ma

and

1
+
m
b

1+mb

is free nonabelian. This is used to prove that a noncommutative positive-definite algebra with involution over an uncountable field contains a free nonabelian subsemigroup.
Class number 3 problem for the simplest cubic fields
Authors: Byeon Dongho
Published: July 2002

We give some necessary conditions for class numbers of the simplest cubic fields to be 3 and, using Lettl’s lower bounds of residues at

s
=
1

s=1

of Dedekind zeta functions attached to cyclic cubic fields, determine all the simplest cubic fields of class number 3.
Relations between the Taylor spectrum and the Xia spectrum
Authors: Chō Muneo
Published: July 2002

Let

T

=
(

T

1

,

T

2

,
.
.
.
,

T

n

)

\mathbf { T} = (T_{1},T_{2},...,T_{n})

be a doubly commuting

n
n

-tuple of

p
p

-hyponormal operators

T

j

T_{j}

with unitary operators

U

j

U_{j}

from the polar decompositions

T

j

=

U

j

|

T

j

|

(
j
=
1
,
.
.
.
,
n
)

T_{j} = U_{j}|T_{j}| (j=1,...,n)

. Let

U

=
(

U

1

,
.
.
.
,

U

n

)

\mathbf { U} = (U_{1},...,U_{n})

and

A
=

|

T

1

|

⋯

|

T

n

|

A = |T_{1}| \cdots |T_{n}|

. In this paper, we will show relations between the Taylor spectrum

σ

T

(

T

)

\sigma _{T}(\mathbf { T})

and the Xia spectrum

σ

X

(

U

,
A
)

\sigma _{X}(\mathbf { U},A)

.
A separable space with no Schauder decomposition
Authors: Allexandrov G. , Kutzarova Denka , Plichko A.
Published: July 2002

We combine some known results to remark that there exists a separable Banach space which fails to have a Schauder decomposition. It can be chosen as a subspace of Gowers-Maurey space without any unconditional basic sequence.
Semilinear transformations
Authors: Abhyankar Shreeram
Published: July 2002

In previous papers, nice trinomial equations were given for unramified coverings of the once punctured affine line in nonzero characteristic

p
p

with the projective general group

P
G
L

(
m
,
q
)

\mathrm {PGL}(m,q)

and the general linear group

G
L

(
m
,
q
)

\mathrm {GL}(m,q)

as Galois groups where

m
>
1

m>1

is any integer and

q
>
1

q>1

is any power of

p
p

. These Galois groups were calculated over an algebraically closed ground field. Here we show that, when calculated over the prime field, as Galois groups we get the projective general semilinear group

P

Γ

L

(
m
,
q
)

\mathrm {P}\Gamma \mathrm {L}(m,q)

and the general semilinear group

Γ

L

(
m
,
q
)

\Gamma \mathrm {L}(m,q)

. We also obtain the semilinear versions of the local coverings considered in previous papers.
Large volume growth and finite topological type
Authors: Ordway D. , Stephens B. , Yang D.
Published: July 2002

It is shown in this paper that a complete noncompact

n
n

-dimen- sional Riemannian manifold with nonnegative Ricci curvature, sectional curvature bounded from below, and large volume growth is of finite topological type provided that the volume growth rate of the complement of the cone of rays from a fixed base point is less than

2
−
1

/

n

2-1/n

.
The surjectivity of the canonical homomorphism from singular homology to Cech homology
Authors: Eda Katsuya , Kawamura Kazuhiro
Published: July 2002

Let

X
X

be a locally

n
n

-connected compact metric space. Then, the canonical homomorphism from the singular homology group

H

n
+
1

(
X
)

H_{n+1}(X)

to the Čech homology group

H
ˇ

n
+
1

(
X
)

\check {H}_{n+1}(X)

is surjective. Consequently, if a compact metric space

X
X

is locally connected, then the canonical homomorphism from

H
1

(
X
)

H_1(X)

to

H
ˇ

1

(
X
)

{\check H}_1(X)

is surjective.
Semi-classical limit for random walks
Authors: Porod Ursula , Zelditch Steve
Published: July 2002

Let

(
G
,
μ
)

(G, \mu )

be a discrete symmetric random walk on a compact Lie group

G
G

with step distribution

μ
\mu

and let

T

μ

T_{\mu }

be the associated transition operator on

L
2

(
G
)

L^2(G)

. The irreducibles

V

ρ

V_{\rho }

of the left regular representation of

G
G

on

L
2

(
G
)

L^2(G)

are finite dimensional invariant subspaces for

T

μ

T_{\mu }

and the spectrum of

T

μ

T_{\mu }

is the union of the sub-spectra

σ
(

T

μ

↿

V

ρ

)

\sigma (T_{\mu }\upharpoonleft _{V_{\rho }})

on the irreducibles, which consist of real eigenvalues

{

λ

ρ
1

,
.
.
.
,

λ

ρ
dim
⁡

V

ρ

}

\{ \lambda _{\rho 1},...,\lambda _{\rho \dim V_{\rho }}\}

. Our main result is an asymptotic expansion for the spectral measures
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m Subscript rho Superscript mu Baseline left-parenthesis lamda right-parenthesis colon equals StartFraction 1 Over dimension upper V Subscript rho Baseline EndFraction sigma-summation Underscript j equals 1 Overscript dimension upper V Subscript rho Endscripts delta left-parenthesis lamda minus lamda Subscript rho j Baseline right-parenthesis">

m

ρ

μ

(
λ
)
:=

1

dim
⁡

V

ρ

∑

j
=
1

dim
⁡

V

ρ

δ
(
λ
−

λ

ρ
j

)

m_{\rho }^{\mu }(\lambda ) := \frac {1}{\dim V_{\rho }} \sum _{j=1}^{\dim V_{\rho }} \delta (\lambda - \lambda _{\rho j})

\]
along rays of representations in a positive Weyl chamber

t

+
∗

\mathbf {t}^*_+

, i.e. for sequences of representations

k
ρ

k \rho

,

k
∈

N

k\in \mathbb {N}

with

k
→
∞

k\rightarrow \infty

. As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on

U
(
n
)

U(n)

(for which

T

μ

T_{\mu }

is essentially a direct sum of Harper operators).
A Palais-Smale approach to problems in Esteban-Lions domains with holes
Authors: Wang Hwai-Chiuan
Published: July 2002

Let

Ω
⊂

R

N

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

be the upper half strip with a hole. In this paper, we show there exists a positive higher energy solution of semilinear elliptic equations in

Ω
\Omega

and describe the dynamic systems of solutions of equation

(
1
)

(1)

in various

Ω
\Omega

. We also show there exist at least two positive solutions of perturbed semilinear elliptic equations in

Ω
\Omega

.
Analytic types of plane curve singularities defined by weighted homogeneous polynomials
Authors: Kang Chunghyuk
Published: July 2002

We classify analytically isolated plane curve singularities defined by weighted homogeneous polynomials

f
(
y
,
z
)

f(y,z)

, which are not topologically equivalent to homogeneous polynomials, in an elementary way. Moreover, in preparation for the proof of the above analytic classification theorem, assuming that

g
(
y
,
z
)

g(y,z)

either satisfies the same property as the above

f
f

does or is homogeneous, then we prove easily that the weights of the above

g
g

determine the topological type of

g
g

and conversely. So, this gives another easy proof for the topological classification theorem of quasihomogenous singularities in

C

2

\mathbb {C}^{2}

, which was already known. Also, as an application, it can be shown that for a given

h
h

, where

h
(

w

1

,
…
,

w

n

)

h(w_{1},\dots ,w_{n})

is a quasihomogeneous holomorphic function with an isolated singularity at the origin or

h
(

w

1

)
=

w

1

p

h(w_{1})=w^{p}_{1}

with a positive integer

p
p

, analytic types of isolated hypersurface singularities defined by

f
+
h

f+h

are easily classified where

f
f

is defined just as above.
Mean convergence of orthogonal Fourier series of modified functions
Authors: Grigorian Martin , Kazarian Kazaros , Soria Fernando
Published: July 2002

We construct orthonormal systems (ONS) which are uniformly bounded, complete, and made up of continuous functions such that some continuous and even some arbitrarily smooth functions cannot be modified so that the Fourier series of the new function converges in the

L

p

L^{p}

-metric for any

p
>
2.

p > 2.

We prove also that if

Φ
\Phi

is a uniformly bounded ONS which is complete in all the spaces

L

[
0
,
1
]

p

,
1
≤
p
>
∞

L _ {[0,1]} ^{p} , 1 \leq p > \infty

, then there exists a rearrangement

σ
\sigma

of the natural numbers

N

\mathbf {N}

such that the system

Φ

σ

=
{

ϕ

σ
(
n
)

}

n
=
1

∞

\Phi _{\sigma }= \{ \phi _{\sigma (n)} \}_{n=1}^{\infty }

has the strong

L

p

L^{p}

-property for all

p
>
2

p>2

; that is, for every

2
≤
p
>
∞

2 \leq p > \infty

and for every

f
∈

L

[
0
,
1
]

p

f \in L _ {[0,1]} ^{p}

and

ϵ
>
0

\epsilon > 0

there exists a function

f
ϵ

∈

L

[
0
,
1
]

p

f_ \epsilon \in L _ {[0,1]} ^{p}

which coincides with

f
f

except on a set of measure less than

ϵ
\epsilon

and whose Fourier series with respect to the system

Φ

σ

\Phi _{\sigma }

converges in

L

[
0
,
1
]

p

.

L _ {[0,1]} ^{p} .
Morse theory for the Yang-Mills functional via equivariant homotopy theory
Authors: Gritsch Ursula
Published: July 2002

In this paper we show the existence of non-minimal critical points of the Yang-Mills functional over a certain family of 4-manifolds{M2g:g=0,1,2,…}\{ M_{2g} : g=0,1,2, \ldots \}with genericSU(2)SU(2)-invariant metrics using Morse and homotopy theoretic methods. These manifolds are acted on fixed point freely by the Lie groupSU(2)SU(2)with quotient a compact Riemann surface of even genus. We use a version of invariant Morse theory for the Yang-Mills functional used by Parker inA Morse theory for equivariant Yang-Mills, Duke Math. J.66-2 (1992), 337–356 and Råde inCompactness theorems for invariant connections, submitted for publication.
Local structure of Schelter-Procesi smooth orders
Authors: Le Bruyn Lieven
Published: July 2002

In this paper we give the étale local classification of Schelter-Procesi smooth orders in central simple algebras. In particular, we prove that if

Δ
\Delta

is a central simple

K
K

-algebra of dimension

n
2

n^2

, where

K
K

is a field of trancendence degree

d
d

, then there are only finitely many étale local classes of smooth orders in

Δ
\Delta

. This result is a non-commutative generalization of the fact that a smooth variety is analytically a manifold, and so has only one type of étale local behaviour.
A wall-crossing formula for the signature of symplectic quotients
Authors: Metzler David
Published: July 2002

We use symplectic cobordism, and the localization result of Ginzburg, Guillemin, and Karshon to find a wall-crossing formula for the signature of regular symplectic quotients of Hamiltonian torus actions. The formula is recursive, depending ultimately on fixed point data. In the case of a circle action, we obtain a formula for the signature of singular quotients as well. We also show how formulas for the Poincaré polynomial and the Euler characteristic (equivalent to those of Kirwan can be expressed in the same recursive manner.
𝑞-Krawtchouk polynomials as spherical functions on the Hecke algebra of type 𝐵
Authors: Koelink H.
Published: July 2002

The Hecke algebra for the hyperoctahedral group contains the Hecke algebra for the symmetric group as a subalgebra. Inducing the index representation of the subalgebra gives a Hecke algebra module, which splits multiplicity free. The corresponding zonal spherical functions are calculated in terms of

q
q

-Krawtchouk polynomials using the quantised enveloping algebra for

s
l

(
2
,

C

)

{\mathfrak {sl}}(2,\mathbb {C})

. The result covers a number of previously established interpretations of (

q
q

-)Krawtchouk polynomials on the hyperoctahedral group, finite groups of Lie type, hypergroups and the quantum

S
U
(
2
)

SU(2)

group.
Path stability and nonlinear weak ergodic theorems
Authors: Chen Yong-Zhuo
Published: July 2002

Let{fn}\{f_{n} \}be a sequence of nonlinear operators. We discuss the asymptotic properties of their inhomogeneous iteratesfn∘fn−1∘⋯∘f1f_{n} \circ f_{n-1} \circ \cdots \circ f_{1}\,in metric spaces, then apply the results to the ordered Banach spaces through projective metrics. Theorems on path stability and nonlinear weak ergodicity are obtained in this paper.
Uniform densities of regular sequences in the unit disk
Authors: Duren Peter , Schuster Alexander , Seip Kristian
Published: July 2002

The upper and lower uniform densities of some regular sequences are computed. These densities are used to determine sequences of sampling and interpolation for Bergman spaces of the unit disk.
Homology manifold bordism
Authors: Johnston Heather , Ranicki Andrew
Published: July 2002

The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact

A
N
R

ANR

homology manifolds of dimension

≥
6

\geq 6

is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.

First, we establish homology manifold transversality for submanifolds of dimension

≥
7

\geq 7

: if

f
:
M
→
P

f:M \to P

is a map from an

m
m

-dimensional homology manifold

M
M

to a space

P
P

, and

Q
⊂
P

Q \subset P

is a subspace with a topological

q
q

-block bundle neighborhood, and

m
−
q
≥
7

m-q \geq 7

, then

f
f

is homology manifold

s
s

-cobordant to a map which is transverse to

Q
Q

, with

f

−
1

(
Q
)
⊂
M

f^{-1}(Q) \subset M

an

(
m
−
q
)

(m-q)

-dimensional homology submanifold.

Second, we obtain a codimension

q
q

splitting obstruction

s
Q

(
f
)
∈
L

S

m
−
q

(
Φ
)

s_Q(f) \in LS_{m-q}(\Phi )

in the Wall

L
S

LS

-group for a simple homotopy equivalence

f
:
M
→
P

f:M \to P

from an

m
m

-dimensional homology manifold

M
M

to an

m
m

-dimensional Poincaré space

P
P

with a codimension

q
q

Poincaré subspace

Q
⊂
P

Q \subset P

with a topological normal bundle, such that

s
Q

(
f
)
=
0

s_Q(f)=0

if (and for

m
−
q
≥
7

m-q \geq 7

only if)

f
f

splits at

Q
Q

up to homology manifold

s
s

-cobordism.

Third, we obtain the multiplicative structure of the homology manifold bordism groups
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript asterisk Superscript upper H Baseline approximately-equals normal upper Omega Subscript asterisk Superscript upper T upper O upper P Baseline left-bracket upper L 0 left-parenthesis double-struck upper Z right-parenthesis right-bracket">

Ω
∗
H

≅

Ω
∗

T
O
P

[

L
0

(

Z

)
]

\Omega ^H_*\cong \Omega ^{TOP}_*[L_0(\mathbb Z)]

.
Non-special, non-canal isothermic tori with spherical lines of curvature
Authors: Bernstein Holly
Published: July 2002

This article examines isothermic surfaces smoothly immersed in Möbius space. It finds explicit examples of non-special, non-canal isothermic tori with spherical lines of curvature in two systems by analyzing Darboux transforms of Dupin tori. In addition, it characterizes the property of spherical lines of curvature in terms of differential equations on the Calapso potential of the isothermic immersion, and investigates the effect of classical transformations on this property.
On the computation of stabilized tensor functors and the relative algebraic 𝐾-theory of dual numbers
Authors: McCarthy Randy
Published: July 2002

We compute the stabilization of functors from exact categories to abelian groups derived from

n
n

-fold tensor products. Rationally, this gives a new computation for the relative algebraic

K
K

-theory of dual numbers.
The monopole equations and 𝐽-holomorphic curves on weakly convex almost Kähler 4-manifolds
Authors: Kanda Yutaka
Published: July 2002

We prove that a weakly convex almost Kähler 4-manifold contains a compact, non-constant

J
J

-holomorphic curve if the corresponding monopole invariant is not zero and if the corresponding line bundle is non-trivial.
Galois groups of some vectorial polynomials
Authors: Abhyankar Shreeram , Inglis Nicholas
Published: July 2002

Previously nice vectorial equations were constructed having various finite classical groups as Galois groups. Here such equations are constructed for the remaining classical groups. The previous equations were genus zero equations. The present equations are strong genus zero.
Cohomology of buildings and finiteness properties of ̃𝐴_{𝑛}-groups
Authors: Ramagge Jacqui , Wheeler Wayne
Published: July 2002

Borel and Serre calculated the cohomology of the building associated to a reductive group and used the result to deduce that torsion-free

S
S

-arithmetic groups are duality groups. By replacing their group-theoretic arguments with proofs relying only upon the geometry of buildings, we show that Borel and Serre’s approach can be modified to calculate the cohomology of any locally finite affine building. As an application we show that any finitely presented

A
~

n

\widetilde {A}_n

-group is a virtual duality group. A number of other finiteness conditions for

A
~

n

\widetilde {A}_n

-groups are also established.
APS boundary conditions, eta invariants and adiabatic limits
Authors: Dai Xianzhe
Published: July 2002

We prove an adiabatic limit formula for the eta invariant of a manifold with boundary. The eta invariant is defined using the Atiyah-Patodi-Singer boundary condition and the underlying manifold is fibered over a manifold with boundary. Our result extends the work of Bismut-Cheeger to manifolds with boundary.
Monge’s transport problem on a Riemannian manifold
Authors: Feldman Mikhail , McCann Robert
Published: July 2002

Monge’s problem refers to the classical problem of optimally transporting mass: given Borel probability measures

μ
+

≠

μ
−

\mu ^+ \ne \mu ^-

, find the measure-preserving map

s
:
M
⟶
M

s:M \longrightarrow M

between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold

M
M

— and assuming absolute continuity of

μ
+

\mu ^+

— an optimal map will be shown to exist. Aspects of its uniqueness are also established.
An analogue of minimal surface theory in 𝑆𝐿(𝑛,𝐂)/𝐒𝐔(𝐧)
Authors: Kokubu M. , Takahashi M. , Umehara M. , Yamada K.
Published: July 2002

We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semi-simple Lie groups (e.g.

SL
⁡
(
n
,

C

)

/

SU
⁡
(
n
)

\operatorname {SL}(n,\mathbf {C})/\operatorname {SU}(n)

), which contains minimal surfaces in

R

n

\mathbf {R}^n

and constant mean curvature

1
1

surfaces in

H

3

\mathcal {H}^3

. A Weierstrass type representation formula and a Chern-Osserman type inequality for such surfaces are given.
An estimate for weighted Hilbert transform via square functions
Authors: Petermichl S. , Pott S.
Published: July 2002

We show that the norm of the Hilbert transform as an operator on the weighted space

L
2

(
w
)

L^2(w)

is bounded by a constant multiple of the

3

/

2

3/2

power of the

A
2

A_2

constant of

w
w

, in other words by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c sup Underscript upper I Endscripts left-parenthesis mathematical left-angle omega mathematical right-angle Subscript upper I Baseline mathematical left-angle omega Superscript negative 1 Baseline mathematical right-angle Subscript upper I Baseline right-parenthesis Superscript 3 slash 2">

c

sup
I

(
⟨
ω

⟩
I

⟨

ω

−
1

⟩
I

)

3

/

2

c\, \sup _I (\langle \omega \rangle _I \langle \omega ^{-1} \rangle _I)^{3/2}

. We also give a short proof for sharp upper and lower bounds for the dyadic square function.
On the structure of 𝑃(𝑛)_{∗}𝑃((𝑛)) for 𝑝=2
Authors: Nassau Christian
Published: July 2002

We show that

P
(
n

)
∗

(
P
(
n
)
)

P(n)_\ast (P(n))

for

p
=
2

p=2

with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation

ϵ
\epsilon

nor the coproduct

Δ
\Delta

are multiplicative. As a consequence the algebra structure of

P
(
n

)
∗

(
P
(
n
)
)

P(n)_\ast (P(n))

is slightly different from what was supposed to be the case. We give formulas for

ϵ
(
x
y
)

\epsilon (xy)

and

Δ
(
x
y
)

\Delta (xy)

and show that the inversion of the formal group of

P
(
n
)

P(n)

is induced by an antimultiplicative involution

Ξ
:
P
(
n
)
→
P
(
n
)

\Xi :P(n)\rightarrow P(n)

. Some consequences for multiplicative and antimultiplicative automorphisms of

K
(
n
)

K(n)

for

p
=
2

p=2

are also discussed.