bluuebunny/crossref_metadata_2025 huggingface dataset:
Lipschitz images with fractal boundaries and their small surface wrapping
Authors: Buczolich Zoltán
Published: July 2002

Assume

E
⊂
H
⊂

R

m

E\subset H\subset \mathbf {R}^{m}

and

Φ
:
E
→

R

m

\Phi :E\to \mathbf {R}^{m}

is a Lipschitz

L
L

-mapping;

|

H

|

|H|

and

|

|

H

|

|

||H||

denote the volume and the surface area of

H
H

. We verify that there exists a figure

F
⊃
Φ
(
E
)

F\supset \Phi (E)

with

|

|

F

|

|

≤

c

L

|

|

H

|

|

||F||\leq c_{L} ||H||

, and, of course,

|

F

|

≤

c

L

|

H

|

|F|\leq c_{L} |H|

, where

c

L

c_{L}

depends only on the dimension and on

L
L

. We also give an example when

E
=
H
⊂

R

2

E=H\subset \mathbf {R}^{2}

is a square and

|

|

Φ
(
E
)

|

|

=
∞

||\Phi (E)||=\infty

; in fact, the boundary of

Φ
(
E
)

\Phi (E)

can contain a fractal of Hausdorff dimension exceeding one.
Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications
Authors: Han Y.-S.
Published: July 2002

In this paper, using the discrete Calderon reproducing formula on spaces of homogeneous type obtained by the author, we obtain the Plancherel-Pôlya type inequalities on spaces of homogeneous type. These inequalities give new characterizations of the Besov spaces

B
˙

p

α
,
q

\dot B_p^{\alpha ,q}

and the Triebel-Lizorkin spaces

F
˙

p

α
,
q

\dot F_p^{\alpha ,q}

on spaces of homogeneous type introduced earlier by the author and E. T. Sawyer and also allow us to generalize these spaces to the case where

p
,
q
≤
1

p,q\le 1

. Moreover, using these inequalities, we can easily show that the Littlewood-Paley

G
G

-function and

S
S

-function are equivalent on spaces of homogeneous type, which gives a new characterization of the Hardy spaces on spaces of homogeneous type introduced by Macias and Segovia.
A note on a question of J. Nekovár and the Birch and Swinnerton-Dyer Conjecture
Authors: Ono Ken
Published: July 2002

If

D
D

is a square-free integer, then let

E
(
D
)

E(D)

denote the elliptic curve over

Q

\mathbb {Q}

given by the equation

({1})

E
(
D
)
:
D

y

2

=
4

x

3

−
27.

\begin{equation*}E(D): Dy^{2}=4x^{3}-27. \tag {{1}} \end{equation*}

Let

L
(
E
(
D
)
,
s
)

L(E(D),s)

denote the Hasse-Weil

L
L

-function of

E
(
D
)

E(D)

, and let

L

∗

(
E
(
D
)
,
1
)

L^{*}(E(D),1)

denote the ‘algebraic part’ of the central critical value

L
(
E
(
D
)
,
1
)

L(E(D),1)

. Using a theorem of Sturm, we verify a congruence conjectured by J. Nekovář. By his work, if

S
(
3
,
E
(
D
)
)

S(3,E(D))

denotes the 3-Selmer group of

E
(
D
)

E(D)

and

D
≠
1

D\neq 1

is a square-free integer with

|

D

|

≡
1

(
mod

3
)

|D|\equiv 1\pmod 3

, then we find that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript asterisk Baseline left-parenthesis upper E left-parenthesis upper D right-parenthesis comma 1 right-parenthesis not-identical-to 0 left-parenthesis mod 3 right-parenthesis long left right double arrow upper S left-parenthesis 3 comma upper E left-parenthesis upper D right-parenthesis right-parenthesis equals 0 period">

L

∗

(
E
(
D
)
,
1
)
≢
0

(
mod

3
)
⟺
S
(
3
,
E
(
D
)
)
=
0.

\begin{equation*}L^{*}(E(D),1)\not \equiv 0 \pmod 3 \Longleftrightarrow S(3,E(D))=0. \end{equation*}
Degrees of high-dimensional subvarieties of determinantal varieties
Authors: Sethuraman B.
Published: July 2002

Let

n
=

p
a

b

n = p^ab

, where

p
p

is a prime, and

g.c.d. 
(
p
,
b
)
=
1

\text {g.c.d. }(p,b)=1

. In

P

n
2

−
1

\mathbf {P}^{n^2-1}

, let

X
r

X_r

be the variety defined by

rank

(
(

x

i
,
j

)
)
≤
n
−
r

\text {rank}\, ((x_{i,j})) \le n-r

. We show that any subvariety of

X
r

X_r

of codimension less than

p
a

r

p^ar

must have degree a multiple of

p
p

. We also show that the bounds on the codimension in our results are strict by exhibiting subvarieties of the appropriate codimension whose degrees are prime to

p
p

.
On the Poincaré series and cardinalities of finite reflection groups
Authors: Stembridge John
Published: July 2002

Let

W
W

be a crystallographic reflection group with length function

ℓ
(
⋅
)

\ell (\cdot )

. We give a short and elementary derivation of the identity
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript w element-of upper W Endscripts q Superscript script l left-parenthesis w right-parenthesis Baseline equals product left-parenthesis 1 minus q Superscript h t left-parenthesis alpha right-parenthesis plus 1 Baseline right-parenthesis slash left-parenthesis 1 minus q Superscript h t left-parenthesis alpha right-parenthesis Baseline right-parenthesis">

∑

w
∈
W

q

ℓ
(
w
)

=
∏
(
1
−

q

ht
⁡
(
α
)
+
1

)

/

(
1
−

q

ht
⁡
(
α
)

)

\sum _{w\in W}q^{\ell (w)}=\prod (1-q^{\operatorname {ht} (\alpha )+1})/(1-q^{\operatorname {ht}(\alpha )})

, where the product ranges over positive roots

α
\alpha

, and

ht
⁡
(
α
)

\operatorname {ht} (\alpha )

denotes the sum of the coordinates of

α
\alpha

with respect to the simple roots. We also prove that in the noncrystallographic case, this identity is valid in the limit

q
→
1

q\to 1

; i.e.,

|

W

|

=
∏
(
ht
⁡
(
α
)
+
1
)

/

ht
⁡
(
α
)

|W|=\prod (\operatorname {ht} (\alpha )+1)/\operatorname {ht}(\alpha )

.
When do connected spaces have nice connected preimages?
Authors: Tkachuk Vladimir
Published: July 2002

We prove that every connected Tychonoff space is an open monotone continuous image of a connected strictly

σ
\sigma

-discrete left-separated Tychonoff space. For wide classes of connected spaces it is established that they have a finer Hausdorff strictly

σ
\sigma

-discrete connected topology. Another result is that a finer Tychonoff connected strictly

σ
\sigma

-discrete topology exists for any Tychonoff topology with a countable network. We show that there are Tychonoff connected spaces with countable network which are not continuous images of connected second countable spaces. It is established also that every connected Tychonoff space

X

\mathcal {X}

is an open retract of a connected homogeneous Tychonoff space, while it is not always possible to find a finer connected homogeneous topology on

X

\mathcal {X}

.
The explicit solution of a diffusion equation with singularity
Authors: Watanabe Michiaki , Watanabe Shuji
Published: July 2002

We give the explicit solution of the Cauchy problem for the diffusion equation with a singular term:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis partial-differential slash partial-differential t right-parenthesis u equals left-parenthesis partial-differential slash partial-differential x right-parenthesis squared u minus left-parenthesis k slash x squared right-parenthesis u comma t greater-than 0 comma x element-of bold upper R Superscript 1 Baseline semicolon">

(
∂

/

∂
t
)

u
=
(
∂

/

∂
x

)
2

u
−
(
k

/

x
2

)

u

,

t
>
0

,

x
∈

R

1

;

(\partial / \partial t ) \, u = ( \partial / \partial x )^2 \, u - ( k / x^2 ) \, u \; , \quad t > 0 \; , \quad x \in \mathbf {R}^1 \; ;

\]

\[

u
(
0
,
x
)
=
f
(
x
)

,

x
∈

R

1

,

u( 0, x) = f(x) \; , \quad x \in \mathbf {R}^1 \; ,

\]
where

k
>
−
1

/

4

k > - 1/4

. We construct the solution on the basis of a generalization of the Fourier transform. We next show that the solution is expressed by an analytic semigroup, and examine smoothness of

x
↦
u
(
t
,
x
)

x \mapsto u(t, x)

and continuity of

x
↦
u
(
t
,
x
)

/

x

β

(
β
>
0
)

x \mapsto u(t, x) / x^{\beta }\left ( \beta > 0 \right )

.
The maximal normal 𝑝-subgroup of the automorphism group of an abelian 𝑝-group
Authors: Hausen Jutta , Schultz Phillip
Published: July 2002

Let

p
p

be a prime number and let

G

\,G\,

be an abelian

p
p

–group. Let

Δ
\Delta

be the maximal normal

p
p

–subgroup of

Aut
⁡
G

\operatorname {Aut}G

and

ζ
\zeta

the maximal

p
p

–subgroup of its centre. Let

t

\mathbf {t}

be the torsion radical of

E

(
G
)

{\mathcal {E}}(G)

. Then

Δ
=
(
1
+

t

)
ζ

\Delta =(1+\mathbf {t})\zeta

. The result is new for

p
=
2

p=2

and 3, and the proof is new and valid for all primes

p
p

.
Mackey weak barrelledness
Authors: Saxon Stephen , Ruiz L.
Published: July 2002

Answers to questions of Levin and Saxon (1971) and of Ferrando and López Pellicer (1991) complete a linear picture of weak barrelledness for Mackey spaces.
Polishness of the Wijsman topology revisited
Authors: Zsilinszky László
Published: July 2002

Let

X
X

be a completely metrizable space. Then the space of nonempty closed subsets of

X
X

endowed with the Wijsman topology is

α
\alpha

-favorable in the strong Choquet game. As a consequence, a short proof of the Beer-Costantini Theorem on Polishness of the Wijsman topology is given.
The Russo-Dye theorem in nest algebras
Authors: Davidson Kenneth
Published: July 2002

It is shown that the convex hull of the unitary elements of a nest algebra contains the whole unit ball if and only if both

0
+

0_+

and

I
−
⊥

I_-^\perp

are either zero or infinite rank.
The order of a meridian of a knotted Klein bottle
Authors: Yoshikawa Katsuyuki
Published: July 2002

We consider the order of a meridian (of the group) of a Klein bottle smoothly embedded in the

4
4

-sphere

S

4

S^{4}

. The order of a meridian of a Klein bottle in

S

4

S^{4}

is a non-negative even integer. Conversely, we prove that, for every non-negative even integer

n
n

, there exists a Klein bottle in

S

4

S^{4}

whose meridian has order

n
n

.
Spectral conditions guaranteeing a nontrivial solution of the abstract Cauchy problem
Authors: deLaubenfels R. , Wang S.
Published: July 2002

We characterize subsets,

Ω
\Omega

, of the complex plane, with the following property: If

A
A

has spectrum contained in

Ω
\Omega

, with polynomially bounded resolvent outside

Ω
\Omega

, then the abstract Cauchy problem corresponding to

A
A

has a nontrivial solution.
Integer sets with distinct subset sums
Authors: Frenkel P.
Published: July 2002

We give a simple, elementary new proof of a generalization of the following conjecture of Paul Erdős: the sum of the elements of a finite integer set with distinct subset sums is less than 2.
A characterization of round spheres
Authors: Koh Sung-Eun
Published: July 2002

A new characterization of geodesic spheres in the simply connected space forms in terms of higher order mean curvatures is given: An immersion of an

n
n

dimensional compact oriented manifold without boundary into

n
+
1

n+1

dimensional Euclidean space, hyperbolic space or the open half sphere is a totally umbilic immersion if, for some

r
,
r
=
2
,
…
,
n
,

r, r=2,\dots ,n,

the

(
r
−
1
)

(r-1)

-th mean curvature

H

r
−
1

H_{r-1}

does not vanish and the ratio

H

r

/

H

r
−
1

H_{r}/H_{r-1}

is constant.
Laplace transforms and generators of semigroups of operators
Authors: Peng Jigen , Chung Si-Kit
Published: July 2002

In this paper, a characterization for continuous functions on

(
0
,
∞
)

(0,\infty )

to be the Laplace transforms of

f
∈

L

∞

(
0
,
∞
)

f\in L^{\infty }(0,\infty )

is obtained. It is also shown that the vector-valued version of this characterization holds if and only if the underlying Banach space has the Radon-Nikodým property. Using these characterizations, some results, different from that of the Hille-Yosida theorem, on generators of semigroups of operators are obtained.
A maximum principle for 𝑃-harmonic maps with 𝐿^{𝑞} finite energy
Authors: Takegoshi Kensho
Published: July 2002

We show a maximum principle for

P
P

-harmonic maps with

L
q

L^q

-finite energy. As an application we can generalize a non-existence theorem for harmonic maps with finite Dirichlet integral by Schoen and Yau to those maps.
An easier proof of the maximal arcs conjecture
Authors: Ball Simeon , Blokhuis Aart
Published: July 2002

It was a long-standing conjecture in finite geometry that a Desarguesian plane of odd order contains no maximal arcs. A rather inaccessible and long proof was given recently by the authors in collaboration with Mazzocca. In this paper a new observation leads to a greatly simplified proof of the conjecture.
On a theorem of Picard
Authors: Gesztesy F. , Sticka W.
Published: July 2002

We extend Picard’s theorem on the existence of elliptic solutions of the second kind of linear homogeneousnth{n}^{\mathrm {th}}-order scalar ordinary differential equations with coefficients being elliptic functions (associated with a common period lattice) to linear homogeneous first-ordern×nn\times nsystems. In particular, the qualitative Floquet-type structure of fundamental systems of solutions in terms of elliptic and exponential functions, polynomials, and Weierstrass zeta functions of the independent variable is determined. Connections with completely integrable systems are mentioned.
Conformal flatness and self-duality of Thurston-geometries
Authors: Maier Stephan
Published: July 2002

We show which Thurston-geometries in dimensions 3 and 4 admit invariant conformally flat or half-conformally flat metrics.
On the number of invariant lines for polynomial systems
Authors: Xiang Zhang , Yanqian Ye
Published: July 2002

In this paper we will revise the mistakes in a previous paper of Zhang Xikang (Number of integral lines of polynomial systems of degree three and four, J. Nanjing Univ. Math. Biquarterly, Supplement, 1993, pp. 209–212) for the proof of the conjecture on the maximum number of invariant straight lines of cubic and quartic polynomial differential systems; and also prove the conjecture in a previous paper of the second author (Qualitative theory of polynomial differential systems, Shanghai Science-Technical Publishers, Shanghai, 1995, p. 474) for a certain special case of the

n
n

degree polynomial systems. Furthermore, we will prove that cubic and quartic differential systems have invariant straight lines along at most six and nine different directions, respectively, and also show that the maximum number of the directions can be obtained.
Real forms of a Riemann surface of even genus
Authors: Gromadzki G. , Izquierdo M.
Published: July 2002

Natanzon proved that a Riemann surface

X
X

of genus

g
≥
2

g \ge 2

has at most

2
(

g

+
1
)

2(\sqrt g+1)

conjugacy classes of symmetries, and this bound is attained for infinitely many genera

g
g

. The aim of this note is to prove that a Riemann surface of even genus

g
g

has at most four conjugacy classes of symmetries and this bound is attained for an arbitrary even

g
g

as well. An equivalent formulation in terms of algebraic curves is that a complex curve of an even genus

g
g

has at most four real forms which are not birationally equivalent.
A continuation type result for random operators
Authors: O’Regan Donal
Published: July 2002

Fixed point results of continuation type are presented for random operators. Some applications to stochastic integral equations of Volterra type are also given.
Multiplier theorems for Herz type Hardy spaces
Authors: Lu Shanzhen , Yang Dachun
Published: July 2002

In this paper, the authors establish a multiplier theorem for Herz type Hardy spaces.
Order complexes of noncomplemented lattices are nonevasive
Authors: Kozlov Dmitry
Published: July 2002

We prove the following theorem: Let

L

\mathcal {L}

be a finite lattice,

x
∈

L

¯

x\in \bar {\mathcal {L}}

. Assume

B
B

is a set of elements of

L

\mathcal {L}

which includes all complements of

x
x

and is included in the set of all upper (lower) semicomplements of

x
x

. Then

δ
(

L

∖
B

¯

)

\delta (\overline {\mathcal {L}\setminus B})

is nonevasive, in particular collapsible. This generalizes results of several previous papers, where, in different generalities, it has been proved that the mentioned complex is contractible.
A remark on Mansfield’s imprimitivity theorem
Authors: Ng Chi-Keung
Published: July 2002

We show that the Morita equivalence part of Mansfield’s Imprimitivity Theory can be obtained by Green’s Imprimitivity Theorem (and duality theory).
On the non-productivity of normality in Moore spaces
Authors: Cook H. , Reed G.
Published: July 2002

Under Martin’s Axiom and the denial of the Continuum Hypothesis, the authors give examples of normal Moore spaces whose squares are not normal.
Closures of totally geodesic immersions into locally symmetric spaces of noncompact type
Authors: Payne Tracy
Published: July 2002

It is established that if

M

1

\mathcal {M}_1

and

M

2

\mathcal {M}_2

are connected locally symmetric spaces of noncompact type where

M

2

\mathcal {M}_2

has finite volume, and

ϕ
:

M

1

→

M

2

\phi :\mathcal {M}_1 \to \mathcal {M}_2

is a totally geodesic immersion, then the closure of

ϕ
(

M

1

)

\phi (\mathcal {M}_1)

in

M

2

\mathcal {M}_2

is an immersed “algebraic” submanifold. It is also shown that if in addition, the real ranks of

M

1

\mathcal {M}_1

and

M

2

\mathcal {M}_2

are equal, then the the closure of

ϕ
(

M

1

)

\phi (\mathcal {M}_1)

in

M

2

\mathcal {M}_2

is a totally geodesic submanifold of

M

2

.

\mathcal {M}_2.

The proof is a straightforward application of Ratner’s Theorem combined with the structure theory of symmetric spaces.
Factorization of functions in generalized Nevanlinna classes
Authors: Horowitz Charles
Published: July 2002

For functions in the classical Nevanlinna class analytic projection of

log
⁡

|

f
(

e

i
θ

)

|

\log |f(e^{i \theta })|

produces

log
⁡
F
(
z
)

\log F(z)

where

F
F

is the outer part of

f
;

f;

i.e., this projection factors out the inner part of

f
f

. We show that if

log
⁡

|

f
(
z
)

|

\log |f(z)|

is area integrable with respect to certain measures on the disc, then the appropriate analytic projections of

log
⁡

|

f

|

\log |f|

factor out zeros by dividing

f
f

by a natural product which is a disc analogue of the classical Weierstrass product. This result is actually a corollary of a more general theorem of M. Andersson. Our contribution is to give a simple one complex variable proof which accentuates the connection with the Weierstrass product and other canonical objects of complex analysis.
Jacobi matrices with absolutely continuous spectrum
Authors: Janas Jan , Naboko Serguei
Published: July 2002

Let

J
J

be a Jacobi matrix defined in

l
2

l^2

as

R
e
W

Re W

, where

W
W

is a unilateral weighted shift with nonzero weights

λ
k

\lambda _k

such that

lim
k

λ
k

=
1.

\lim _k \lambda _k = 1.

Define the seqences:

ε
k

:=

λ

k
−
1

λ
k

−
1
,

\varepsilon _k:= \frac {\lambda _{k-1}}{\lambda _k} -1,

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta Subscript k Baseline colon equals StartFraction lamda Subscript k Baseline minus 1 Over lamda Subscript k Baseline EndFraction comma eta Subscript k Baseline colon equals 2 delta Subscript k Baseline plus epsilon Subscript k Baseline period">

δ
k

:=

λ
k

−
1

λ
k

,

η
k

:=
2

δ
k

+

ε
k

.

\delta _k:= \frac {\lambda _k -1}{\lambda _k}, \, \, \eta _k:= 2 \delta _k + \varepsilon _k.

If
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon Subscript k Baseline equals upper O left-parenthesis k Superscript negative alpha Baseline right-parenthesis comma eta Subscript k Baseline equals upper O left-parenthesis k Superscript negative gamma Baseline right-parenthesis comma two thirds greater-than alpha less-than-or-equal-to gamma comma alpha plus gamma greater-than 3 slash 2">

ε
k

=
O
(

k

−
α

)
,

η
k

=
O
(

k

−
γ

)
,

2
3

>
α
≤
γ
,

α
+
γ
>
3

/

2

\varepsilon _k = O(k^{-\alpha }) , \, \, \eta _k = O(k^{-\gamma }), \, \, \frac {2}{3}> \alpha \leq \gamma , \, \, \alpha + \gamma > 3/2

and

γ
>
3

/

4

\gamma > 3/4

, then

J
J

has an absolutely continuous spectrum covering

(
−
2
,
2
)

(-2,2)

. Moreover, the asymptotics of the solution

J
u
=
λ
u
,

λ
∈

R

Ju = \lambda u, \, \lambda \in \mathbb {R}

is also given.
A Dauns-Hofmann theorem for TAF-algebras
Authors: Somerset D.
Published: July 2002

Let

A
A

be a TAF-algebra,

Z
(
A
)

Z(A)

the centre of

A
,
I
d
(
A
)

A, Id(A)

the ideal lattice of

A
A

, and

M
i
r
(
A
)

Mir(A)

the space of meet-irreducible elements of

I
d
(
A
)

Id(A)

, equipped with the hull-kernel topology. It is shown that

M
i
r
(
A
)

Mir(A)

is a compact, locally compact, second countable,

T
0

T_0

-space, that

I
d
(
A
)

Id(A)

is an algebraic lattice isomorphic to the lattice of open subsets of

M
i
r
(
A
)

Mir(A)

, and that

Z
(
A
)

Z(A)

is isomorphic to the algebra of continuous, complex functions on

M
i
r
(
A
)

Mir(A)

. If

A
A

is semisimple, then

Z
(
A
)

Z(A)

is isomorphic to the algebra of continuous, complex functions on

P
r
i
m
(
A
)

Prim(A)

, the primitive ideal space of

A
A

. If

A
A

is strongly maximal, then the sum of two closed ideals of

A
A

is closed.
Asymptotics of the d’Alembertian with potential on a pseudo-Riemannian manifold
Authors: Branson Thomas , Ólafsson Gestur
Published: July 2002

Let

◻

\square

be the Laplace-d’Alembert operator on a pseudo-Riemann- ian manifold

(
M
,
g
)

(M,g)

. We derive a series expansion for the fundamental solution

G
(
x
,
y
)

G(x,y)

of

◻

+
H

\square +H

,

H
∈

C

∞

(
M
)

H\in C^{\infty }(M)

, which behaves well under various symmetric space dualities. The qualitative properties of this expansion were used in our paper in Invent. Math. 129 (1997), 63–74, to show that the property of vanishing logarithmic term for

G
(
x
,
y
)

G(x,y)

is preserved under these dualities.
Finite-dimensional right ideals in some algebras associated with a locally compact group
Authors: Filali M.
Published: July 2002

Let

G
G

be a discrete group, a commutative discrete cancellative semigroup or a locally compact abelian group. Let

U
C
(
G
)

UC(G)

be the space of bounded, uniformly continuous, complex-valued functions on

G
.

G.

With an Arens-type product, the conjugate

U
C
(
G

)

∗

UC(G)^{*}

becomes a Banach algebra. We prove, that unlike left ideals, finite-dimensional right ideals exist in

U
C
(
G

)

∗

UC(G)^{*}

if and only if

G
G

is compact.
Rules and reals
Authors: Goldstern Martin , Kojman Menachem
Published: July 2002

A “

k
k

-rule" is a sequence

A
→

=
(
(

A
n

,

B
n

)
:
n
>

N

)

\vec A=((A_n,B_n): n>\mathbb N)

of pairwise disjoint sets

B
n

B_n

, each of cardinality

≤
k

\le k

and subsets

A
n

⊆

B
n

A_n\subseteq B_n

. A subset

X
⊆

N

X\subseteq \mathbb N

(a “real”) follows a rule

A
→

\vec A

if for infinitely many

n
∈

N

n\in \mathbb N

,

X
∩

B
n

=

A
n

X\cap B_n=A_n

. Two obvious cardinal invariants arise from this definition: the least number of reals needed to follow all

k
k

-rules,

s

k

\mathfrak {s}_k

, and the least number of

k
k

-rules with no real that follows all of them,

r

k

\mathfrak {r}_k

. Call

A
→

\vec A

a bounded rule if

A
→

\vec A

is a

k
k

-rule for some

k
k

. Let

r

∞

\mathfrak {r}_\infty

be the least cardinality of a set of bounded rules with no real following all rules in the set. We prove the following:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German r Subscript normal infinity Baseline greater-than-or-equal-to max left-parenthesis c o v left-parenthesis double-struck upper K right-parenthesis comma c o v left-parenthesis double-struck upper L right-parenthesis right-parenthesis">

r

∞

≥
max
(
cov
⁡
(

K

)
,
cov
⁡
(

L

)
)

\mathfrak {r}_\infty \ge \max (\operatorname {cov}(\mathbb {K}),\operatorname {cov}(\mathbb {L}))

and

r

=

r

1

≥

r

2

=

r

k

\mathfrak {r}=\mathfrak {r}_1\ge \mathfrak {r}_2=\mathfrak {r}_k

for all

k
≥
2

k\ge 2

. However, in the Laver model,

r

2

>

b

=

r

1

\mathfrak {r}_2>\mathfrak {b}=\mathfrak {r}_1

. An application of

r

∞

\mathfrak {r}_\infty

is in Section 3: we show that below

r

∞

\mathfrak {r}_\infty

one can find proper extensions of dense independent families which preserve a pre-assigned group of automorphisms. The original motivation for discovering rules was an attempt to construct a maximal homogeneous family over

ω
\omega

. The consistency of such a family is still open.
A multiplier relation for Calderón-Zygmund operators on 𝐿¹({ℝ}ⁿ)
Authors: Bennett Jonathan
Published: July 2002

A generalised integral is used to obtain a Fourier multiplier relation for Calderón-Zygmund operators on

L
1

(

R

n

)

L^1({\mathbb R}^{n})

. In particular we conclude that an operator in our class is injective on

L
1

(

R

n

)

L^1({\mathbb R}^{n})

if it is injective on

L
2

(

R

n

)

L^2({\mathbb R}^n)

.
Remark about heat diffusion on periodic spaces
Authors: Lott John
Published: July 2002

Let

M
M

be a complete Riemannian manifold with a free cocompact

Z

k

\mathbb {Z}^k

-action. Let

k
(
t
,

m
1

,

m
2

)

k(t, m_1, m_2)

be the heat kernel on

M
M

. We compute the asymptotics of

k
(
t
,

m
1

,

m
2

)

k(t, m_1, m_2)

in the limit in which

t
→
∞

t \rightarrow \infty

and

d
(

m
1

,

m
2

)
∼

t

d(m_1, m_2) \sim \sqrt {t}

. We show that in this limit, the heat diffusion is governed by an effective Euclidean metric on

R

k

\mathbb {R}^k

coming from the Hodge inner product on

H

1

(
M

/

Z

k

;

R

)

\mathrm {H}^1(M/\mathbb {Z}^k; \mathbb {R})

.
Remarks on commuting exponentials in Banach algebras
Authors: Schmoeger Christoph
Published: July 2002

Suppose that

a
a

and

b
b

are elements of a complex unital Banach algebra such that the spectra of

a
a

and

b
b

are

2
π
i

2\pi i

-congruence-free. E.M.E. Wermuth has shown that then
\[

e
a

e
b

=

e
b

e
a

implies that

a
b
=
b
a
.

e^a e^b = e^b e^a \quad \text {implies that} \quad ab = ba.

\]
In this note we use two elementary facts concerning inner derivations on Banach algebras to give a very short proof of Wermuth’s result.
Presentation of critical modules of GK-dimension 2 over elliptic algebras
Authors: Ajitabh K. , den Bergh M.
Published: July 2002

We show that critical modules of Gelfand-Kirillov dimension 2 and multiplicity

d
d

over an elliptic algebra have (up to modules of lower GK-dimension and shifting) a presentation by

d
×
d

d\times d

-matrices of linear forms. In the language of non-commutative algebraic geometry this amounts to a generic description of “curves” of degree

d
d

in a projective quantum plane.
The unilateral shift and a norm equality for bounded linear operators
Authors: Lin C.-S.
Published: July 2002

The paper gives a necessary and sufficient condition for the norm equality

∥
S
−
T
∥=∥
S
∥
+
∥
T
∥

\parallel S-T\parallel =\parallel S\parallel +\parallel T\parallel

of bounded linear operators

S
S

and

T
T

. The invertibility of an operator which is related to the norm equality is discussed. Some new results about the unilateral shift are given.
Equations in free groups are not finitely approximable
Authors: Coulbois Thierry , Khelif Anatole
Published: July 2002

We give an equation in any free group

F
F

of rank

≥
2

\geq 2

that has a solution in any finite quotient of

F
F

, but has no solution in

F
F

.
Nontoric Hamiltonian circle actions on four-dimensional symplectic orbifolds
Authors: Singer S. , Talvacchia J. , Watson N.
Published: July 2002

We construct four-dimensional symplectic orbifolds admitting Hamiltonian circle actions with isolated fixed points, but not admitting any Hamiltonian action of a two-torus. One example is linear, and one example is compact.
Isoperimetric curves on hyperbolic surfaces
Authors: Adams Colin , Morgan Frank
Published: July 2002

Least-perimeter enclosures of prescribed area on hyperbolic surfaces are characterized.
The transversal homoclinic points are dense in the codimension-1 Hénon-like strange attractors
Authors: Cao Yongluo
Published: July 2002

We consider the codimension-1 Hénon-like strange attractors

Λ
\Lambda

. We prove that the transversal homoclinic points are dense in

Λ
\Lambda

, and that hyperbolic periodic points are dense in

Λ
\Lambda

. Moreover the hyperbolic periodic points that are heteroclinically related to the primary fixed point ( transversal intersection of stable and unstable manifolds) are dense in

Λ
\Lambda

.
On Eisenstein series and ∑_{𝑚,𝑛=-∞}^{∞}𝑞^{𝑚²+𝑚𝑛+2𝑛²}
Authors: Chan Heng Huat , Ong Yao
Published: July 2002

In this paper, we derive some new identities satisfied by the series

∑

m
,
n
=
−
∞

∞

q

m

2

+
m
n
+
2

n

2

{\sum _{m,n=-\infty }^{\infty }q^{m^{2}+mn+2n^{2}}}

using Ramanujan’s identities for

L
(
q
)

L(q)

,

M
(
q
)

M(q)

and

N
(
q
)

N(q)

. Our work is motivated by an attempt to develop a theory of elliptic functions to the septic base.
Extinguishing the distinguished logarithm problems
Authors: Finkelstein Mark , Tucker Howard , Veeh Jerry
Published: July 2002

The existence, uniqueness, and convergence properties of distinguished logarithms of multivariate characteristic functions are proved.
Hyperinvariant subspaces of operators with non-vanishing orbits
Authors: Kérchy László
Published: July 2002

It is shown that if the Banach space operator

T
T

has regular norm-sequence, its vector orbits are asymptotically non-vanishing and there exists a complete vector orbit satisfying the growth condition of non-quasianalycity, then

T
T

has infinitely many disjoint hyperinvariant subspaces.
On the dimension of almost 𝑛-dimensional spaces
Authors: Levin M. , Tymchatyn E.
Published: July 2002

Oversteegen and Tymchatyn proved that homeomorphism groups of positive dimensional Menger compacta are11-dimensional by proving that almost00-dimensional spaces are at most11-dimensional. These homeomorphism groups are almost00-dimensional and at least11-dimensional by classical results of Brechner and Bestvina. In this note we prove that almostnn-dimensional spaces forn≥1n \geq 1arenn-dimensional. As a corollary we answer in the affirmative an old question of R. Duda by proving that every hereditarily locally connected, non-degenerate, separable, metric space is11-dimensional.
Representing quasi-orders by embeddability ordering of families of topological spaces
Authors: McCluskey A. , McMaster T.
Published: July 2002

An elementary argument constructs, for each cardinal

α
\alpha

, a topological space whose subspaces, ordered by homeomorphic embeddability, can model every partial order on

α
\alpha

-many points. We show how to modify this procedure to deal also with quasi-orders (where the antisymmetry condition may fail), obtaining an initial estimate of the cardinality of the space then required.
Completely invariant Julia sets of polynomial semigroups
Authors: Stankewitz Rich
Published: July 2002

Let

G
G

be a semigroup of rational functions of degree at least two, under composition of functions. Suppose that

G
G

contains two polynomials with non-equal Julia sets. We prove that the smallest closed subset of the Riemann sphere which contains at least three points and is completely invariant under each element of

G
G

, is the sphere itself.
A one-point attractor theory for the Navier-Stokes equation on thin domains with no-slip boundary conditions
Authors: Avrin Joel
Published: July 2002

In an earlier paper related to recent results of Raugel and Sell for periodic boundary conditions, we considered the incompressible Navier-Stokes equations on 3-dimensional thin domains with zero (“no-slip”) boundary conditions and established global regularity results. We extend those results here by developing an attractor theory. We first show that under similar thinness restrictions trajectories of solutions approach each other in

L
4

L^4

-norm exponentially. Next, for constant-in-time forcing data

f
1

=

f
1

(
x
)

,

f_1=f_1\left ( x\right ) ,

we suppose that

f

(
t
)

→

f
1

f\left ( t\right ) \rightarrow f_1

in

L
2

L^2

as

t
→
+
∞
,

t\rightarrow +\infty ,

and show that if

v
v

and

w
1

w_1

solve the equations with forcing data

f
f

and

f
1

f_1

, respectively, then

‖
v

(
t
)

−

w
1

(
t
)

‖

4

→
0

\left \| v\left ( t\right ) -w_1\left ( t\right ) \right \| _4\rightarrow 0

as

t
→
+
∞
.

t\rightarrow +\infty .

For similar thinness restrictions we show that the steady-flow equations with forcing data

f
1

f_1

have a unique solution

u
s

u_s

. Under both thinness assumptions we then have that all solutions

v

(
t
)

v\left ( t\right )

converge to

u
s

u_s

in

L
4

L_4

as

t
→
+
∞

t\rightarrow +\infty

; thus we have a one-point attractor for strong solutions. In fact, we have a one-point attractor for the Leray solutions as well. Moreover, under significantly more relaxed thinness assumptions we are able to show that Leray solutions nonetheless eventually become regular.
On the local asymptotics of Faber polynomials
Authors: Pritsker Igor
Published: July 2002

We study a local asymptotics of the (generalized) Faber polynomials at the boundary of the associated domain, under certain mild smoothness conditions on the weight function and geometric conditions on the boundary. The main result exhibits how this asymptotics depends on the corners at the boundary. Its proof is based on the continuity properties of the Visser-Ostrowski quotient at the corners.
Common fixed points of commuting holomorphic maps in the unit ball of ℂⁿ
Authors: Bracci Filippo
Published: July 2002

Let

B

n

\mathbb {B}^n

be the unit ball of

C

n

\mathbb {C}^n

(

n
>
1

n>1

). We prove that if

f
,
g
∈

H
o
l

(

B

n

,

B

n

)

f,g \in \mathrm {Hol}(\mathbb {B}^n, \mathbb {B}^n)

are holomorphic self-maps of

B

n

\mathbb {B}^n

such that

f
∘
g
=
g
∘
f

f \circ g = g \circ f

, then

f
f

and

g
g

have a common fixed point (possibly at the boundary, in the sense of

K
K

-limits). Furthermore, if

f
f

and

g
g

have no fixed points in

B

n

\mathbb {B}^n

, then they have the same Wolff point, unless the restrictions of

f
f

and

g
g

to the one-dimensional complex affine subset of

B

n

\mathbb {B}^n

determined by the Wolff points of

f
f

and

g
g

are commuting hyperbolic automorphisms of that subset.
A Dixmier-Schaefer-Zhang theorem for operator algebras
Authors: Bunce L. , Saitô Kazuyuki , Wright J.
Published: July 2002

Schaefer and Zhang have recently obtained an analogue, for sequentially order continuous functionals on

C
(
K
)

C(K)

, of a much earlier theorem of Dixmier. In this note it is shown that the Schaefer-Zhang Theorem has a natural generalisation to non-commutative

C
∗

C^*

-algebras. These results are obtained as consequences of our main theorem which is concerned with affine functions on compact convex sets.
Sets of range uniqueness for classes of continuous functions
Authors: Burke Maxim , Ciesielski Krzysztof
Published: July 2002

Diamond, Pomerance and Rubel (1981) proved that there are subsets

M
M

of the complex plane such that for any two entire functions

f
f

and

g
g

if

f
[
M
]
=
g
[
M
]

f[M]=g[M]

, then

f
=
g

f=g

. Baraducci and Dikranjan showed in 1993 that the continuum hypothesis (CH) implies the existence of a similar set

M
⊂

R

M\subset \mathrm {R}

for the class

C
n

(

R

)

C_n(\mathbb {R})

of continuous nowhere constant functions from

R

\mathrm {R}

to

R

\mathrm {R}

, while it follows from the results of Burke and Ciesielski (1997) and Ciesielski and Shelah that the existence of such a set is not provable in ZFC. In this paper we will show that for several well-behaved subclasses of

C
(

R

)

C(\mathrm {R})

, including the class

D
1

D^1

of differentiable functions and the class

A
C

AC

of absolutely continuous functions, a set

M
M

with the above property can be constructed in ZFC. We will also prove the existence of a set

M
⊂

R

M\subset \mathbb {R}

with the dual property that for any

f
,
g
∈

C
n

(

R

)

f,g\in C_n(\mathrm {R})

if

f

−
1

[
M
]
=

g

−
1

[
M
]

f^{-1}[M]=g^{-1}[M]

, then

f
=
g

f=g

.
Midconvex functions in locally compact groups
Authors: Chademan A. , Mirzapour F.
Published: July 2002

The theorems of Bernstein-Doetsch, and Ostrowski, concerning the continuity of midconvex functions are extended to open subsets of locally compact and root-approximable topological groups.
On random algebraic polynomials
Authors: Farahmand K.
Published: July 2002

This paper provides asymptotic estimates for the expected number of real zeros and

K
K

-level crossings of a random algebraic polynomial of the form

a

0

(

n
−
1

0

)

1

/

2

+

a

1

(

n
−
1

1

)

1

/

2

x

a_{0}\binom {n-1 }{0}^{1/2}+ a_{1}\binom {n-1}{1}^{1/2}x

+

a

2

(

n
−
1

2

)

1

/

2

x

2

+a_{2}\binom {n-1 }{2}^{1/2}x^{2}

+
⋯
+

a

n
−
1

(

n
−
1

n
−
1

)

1

/

2

x

n
−
1

+ \cdots +a_{n-1}\binom {n-1 }{ n-1}^{1/2}x^{n-1}

, where

a

j

(
j
=
0
,
1
,
…
,
n
−
1
)

a_{j} (j=0, 1, \ldots ,n-1)

are independent standard normal random variables and

K
K

is a constant independent of

x
x

. It is shown that these asymptotic estimates are much greater than those for algebraic polynomials of the form

a

0

a_{0}

+

a

1

x

+a_{1}x

+

a

2

x

2

+
⋯
+

a

n
−
1

x

n
−
1

+a_{2}x^{2}+ \cdots +a_{n-1}x^{n-1}

.
Characterization based on conditional expectations of adjacent order statistics: A unified approach
Authors: Franco M. , Ruiz J.
Published: July 2002

In this paper, we show a unified approach to the problem of characterizing general distribution functions based on the conditional expectation between adjacent order statistics,

ξ
(
x
)
=
E
(
h
(

X

r
,
n

)
∣

X

r
+
1
,
n

=
x
)

\xi (x)=E(h(X_{r,n})\mid X_{r+1,n}=x)

or
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove xi With bar left-parenthesis x right-parenthesis equals upper E left-parenthesis h left-parenthesis upper X Subscript r plus 1 comma n Baseline right-parenthesis bar upper X Subscript r comma n Baseline equals x right-parenthesis">

ξ
¯

(
x
)
=
E
(
h
(

X

r
+
1
,
n

)
∣

X

r
,
n

=
x
)

\overline {\xi }(x)=E(h(X_{r+1,n})\mid X_{r,n}=x)

, where

h
h

is a real, continuous and strictly monotonic function. We have the explicit expression of the distribution function

F
F

from the above order mean function,

ξ
\xi

and

ξ
¯

\overline {\xi }

, and we give necessary and sufficient conditions so that any real function can be an order mean function. Our results generalize the results given for the discrete, absolutely continuous and continuous cases. Further, we show stability theorems for these characterizations.
Vanishing of a certain kind of Vassiliev invariants of 2-knots
Authors: Kamada Seiichi
Published: July 2002

In knot theory, Vassiliev’s 1-knot invariants are defined in a combinatorial way as finite type invariants. By a natural generalization of the combinatorial definition, one has a certain family of 2-knot invariants, which should be called finite type 2-knot invariants. They form a subspace of the whole space of “Vassiliev 2-knot invariants”. In this paper we prove that it is 1-dimensional.
Iteration of a class of hyperbolic meromorphic functions
Authors: Rippon P. , Stallard G.
Published: July 2002

We look at the class

B
n

B_n

which contains those transcendental meromorphic functions

f
f

for which the finite singularities of

f

−
n

f^{-n}

are in a bounded set and prove that, if

f
f

belongs to

B
n

B_n

, then there are no components of the set of normality in which

f

m
n

(
z
)
→
∞

f^{mn}(z)\to \infty

as

m
→
∞

m\to \infty

. We then consider the class

B
^

\widehat B

which contains those functions

f
f

in

B
1

B_1

for which the forward orbits of the singularities of

f

−
1

f^{-1}

stay away from the Julia set and show (a) that there is a bounded set containing the finite singularities of all the functions

f

−
n

f^{-n}

and (b) that, for points in the Julia set of

f
f

, the derivatives

(

f
n

)
′

(f^n)’

have exponential-type growth. This justifies the assertion that

B
^

\widehat B

is a class of hyperbolic functions.
Quadratic base change for 𝑝-adic 𝑆𝐿(2) as a theta correspondence I: Occurrence
Authors: Manderscheid David
Published: July 2002

The local theta correspondence is considered for reductive dual pairs

(

S
L

2

(
F
)

,

O

(
F
)

)

\left ( \mathrm {SL}_{2}\left ( F\right ) ,\mathrm {O}\left ( F\right ) \right )

where

F
F

is a

p
p

-adic field of characteristic zero and

O

\mathrm {O}

is the orthogonal group attached to a quaternary quadratic form with coefficients in

F
F

and of Witt rank one over

F
F

. It is shown that certain representations of

S
L

2

(
F
)

\mathrm {SL}_{2}\left ( F\right )

occur in the correspondence.
On real quadratic function fields of Chowla type with ideal class number one
Authors: Feng Keqin , Hu Weiqun
Published: July 2002

Let

F

q

\mathbb {F}_q

be the finite field with

q
q

elements, (2

⧸

|

q

{\not |}q

),

k
=

F

q

(
x
)

k= \mathbb {F}_q(x)

,

K
=
k
(

D

)

K=k(\sqrt {D})

where

D
=
D
(
x
)
=
A
(
x

)
2

+
a

D=D(x) =A(x)^2+a

is a square-free polynomial in

F

q

[
x
]

\mathbb {F}_q[x]

with

deg
⁡
A
(
x
)
≥
1

\deg A(x)\geq 1

and

a
∈

F

q
∗

a\in \mathbb { F}_q^*

. In this paper several equivalent conditions for the ideal class number

h
(

O
K

)

h(O_K)

to be one are presented and all such quadratic function fields with

h
(

O
K

)
=
1

h(O_K)=1

are determined.
Quadratic 𝑞-exponentials and connection coefficient problems
Authors: Ismail Mourad , Rahman Mizan , Stanton Dennis
Published: July 2002

We establish expansion formulas of

q
q

-exponential functions in terms of continuous

q
q

-ultraspherical polynomials, continuous

q
q

-Hermite polynomials and Askey-Wilson polynomials. The proofs are based on solving connection coefficient problems.
A Liouville type theorem for the Schrödinger operator
Authors: Melas Antonios
Published: July 2002

In this paper we prove that the equation

Δ
u
(
x
)
+
h
(
x
)
u
(
x
)
=
0

\Delta u(x)+h(x)u(x)=0

on a complete Riemannian manifold of dimension

n
n

without boundary and with nonnegative Ricci curvature admits no positive solution provided that

h
h

is a

C
2

C^2

function satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit sup r Superscript negative 2 Baseline inf Underscript x element-of upper B Subscript p Baseline left-parenthesis r right-parenthesis Endscripts h left-parenthesis x right-parenthesis greater-than-or-equal-to minus b Subscript n Baseline a squared">

lim sup

r
→
∞

r

−
2

inf

x
∈

B
p

(
r
)

h
(
x
)
≥
−

b
n

a
2

\limsup _{r\rightarrow \infty }r^{-2}\inf _{x\in B_p(r)}h(x) \geq -b_na^2

and

Δ
h
(
x
)
≥
−

c
n

a
2

\Delta h(x)\geq -c_na^2

where

0
≤
a
>

sup

x
∈
M

h
(
x
)

0\leq a>\sup _{x\in M}h(x)

, and

b
n

,

c
n

b_n,c_n

are constants depending only on the dimension, thus generalizing similar results in P. Li and S. T. Yau (Acta Math. 156 (1986), 153–201), J. Li (J. Funct. Anal. 100 (1991), 233–256) and E. R. Negrin (J. Funct. Anal. 127 (1995), 198–203) in all of which

h
h

is assumed to be subharmonic. We also give a generalization in case the Ricci curvature of

M
M

is not necessarily positive but its negative part has quadratic decay under the additional assumption that

h
h

is unbounded from above.
On a polynomial inequality of E. J. Remez
Authors: Dryanov D. , Rahman Q.
Published: July 2002

We prove a result which extends a well-known polynomial inequality of E. J. Remez and another one due to W. A. Markov.
On the asymptotic stability in functional differential equations
Authors: Ignatyev A.
Published: July 2002

Consider a system of functional differential equations

d
x

/

d
t
=
f
(
t
,

x

t

)

dx/dt=f(t,x_{t})

where

f
f

is the vector-valued functional. The classical asymptotic stability result for such a system calls for a positive definite functional

V
(
t
,
φ
)

V(t,\varphi )

and negative definite functional

d
V

/

d
t

{dV}/{dt}

. In applications one can construct a positive definite functional

V
V

, whose derivative is not negative definite but is less than or equal to zero. Exactly for such cases J. Hale created the effective asymptotic stability criterion if the functional

f
f

in functional differential equations is autonomous (

f
f

does not depend on

t
t

), and N. N. Krasovskii created such criterion for the case where the functional

f
f

is periodic in

t
t

. For the general case of the non-autonomous functional

f
f

  V. M. Matrosov proved that this criterion is not right even for ordinary differential equations. The goal of this paper is to prove this criterion for the case when

f
f

is almost periodic in

t
t

. This case is a particular case of the class of non-autonomous functionals.
𝐶²-perturbations of Hopf’s bifurcation points and homoclinic tangencies
Authors: Martín J. , Mora L.
Published: July 2002

In this note we show that a diffeomorphism which has a Hopf’s bifurcation point, can be

C
2

C^2

perturbed around the bifurcation point in order to get a diffeomorphism which exhibits homoclinic tangencies. In the

C
3

C^3

case this is not possible because of the typical unfolding of a Hopf’s bifurcation point.
From Nörlund matrices to Laplace representations
Authors: Sinnamon Gord
Published: July 2002

The existence of Laplace representations for functions in weighted Hardy spaces on the right half plane is established. The method uses an extension of an inequality involving Nörlund matrices and corresponding convolution operators on the line. Analogous inequalities are proved for power series representations of functions in weighted Hardy spaces on the disc.
Weak amenability of Segal algebras
Authors: Dales H. , Pandey S.
Published: July 2002

Let

G
G

be a locally compact abelian group, and let

p
∈
[
1
,
∞
)

p \in [1,\infty )

. We show that the Segal algebra

S
p

(
G
)

S_p(G)

is always weakly amenable, but that it is amenable only if

G
G

is discrete.
The converse of the Inverse-Conjugacy Theorem for unitary operators and ergodic dynamical systems
Authors: Goodson Geoffrey
Published: July 2002

We show that the converse to the main theorem of Ergodic transformations conjugate to their inverses by involutions, by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97–124), holds in the unitary category. Specifically it is shown that if

U
U

is a unitary operator defined on an

L

2

L^{2}

space which preserves real valued functions, and if

U

−
1

S
=
S
U

U^{-1}S=SU

implies

S

2

=
I

S^{2}=I

whenever

S
S

is another such operator, then

U
U

has simple spectrum. The corresponding result for measure preserving transformations is shown to be false. The counter-example we have involves Gaussian automorphisms. We show that a Gaussian automorphism is always conjugate to its inverse, so that the Inverse-Conjugacy Theorem is applicable to such maps having simple spectrum. Furthermore, there are Gaussian automorphisms having non-simple spectrum for which every conjugation of

T
T

with

T

−
1

T^{-1}

is an involution.
On the stability of approximately additive mappings
Authors: Lee Yang-Hi , Jun Kil-Woung
Published: July 2002

In this paper we prove a generalization of the stability of approximately additive mappings in the spirit of Hyers, Ulam and Rassias.
On covering translations and homeotopy groups of contractible open n-manifolds
Authors: Myers Robert
Published: July 2002

This paper gives a new proof of a result of Geoghegan and Mihalik which states that whenever a contractible open

n
n

-manifold

W
W

which is not homeomorphic to

R

n

\mathbf {R}^n

is a covering space of an

n
n

-manifold

M
M

and either

n
≥
4

n \geq 4

or

n
=
3

n=3

and

W
W

is irreducible, then the group of covering translations injects into the homeotopy group of

W
W

.
On Castelnuovo-Mumford regularity of projective curves
Authors: Bermejo Isabel , Gimenez Philippe
Published: July 2002

We give an effective method to compute the regularity of a saturated ideal

I
I

defining a projective curve that also determines in which step of a minimal graded free resolution of

I
I

the regularity is attained.
On Tate-Shafarevich groups of abelian varieties
Authors: Gonzalez-Avilés Cristian
Published: July 2002

Let

K

/

F

K/F

be a finite Galois extension of number fields with Galois group

G
G

, let

A
A

be an abelian variety defined over

F
F

, and let

Щ

(

A

/

K

)

\Shcha (A_{^{/ K}})

and

Щ

(

A

/

F

)

\Shcha (A_{^{/ F}})

denote, respectively, the Tate-Shafarevich groups of

A
A

over

K
K

and of

A
A

over

F
F

. Assuming that these groups are finite, we derive, under certain restrictions on

A
A

and

K

/

F

K/F

, a formula for the order of the subgroup of

Щ

(

A

/

K

)

\Shcha (A_{^{/ K}})

of

G
G

-invariant elements. As a corollary, we obtain a simple formula relating the orders of

Щ

(

A

/

K

)

\Shcha (A_{^{/ K}})

,

Щ

(

A

/

F

)

\Shcha (A_{^{/ F}})

and

Щ

(

A

/

F

χ

)

\Shcha (A_{^{\,/ F}}^{\chi })

when

K

/

F

K/F

is a quadratic extension and

A

χ

A^{\chi }

is the twist of

A
A

by the non-trivial character

χ
\chi

of

G
G

.