bluuebunny/crossref_metadata_2025 huggingface dataset:
Notes on a 𝐶₀-group generated by the Lévy Laplacian
Authors: Chung Dong , Ji Un Cig , Saitô Kimiaki
Published: July 2002

In this paper we shall give some results on a

C
0

C_0

-group generated by the Lévy Laplacian and operators approximating that group in the space

L

(

E

)

\mathcal { L}({\mathbf E})

of continuous linear operators defined on a certain locally convex space

E

{\mathbf E}

in

(

S

)
∗

.

(\mathcal {S})^*.
On the relation between the A-polynomial and the Jones polynomial
Authors: Gelca Răzvan
Published: July 2002

This paper shows that the noncommutative generalization of the A-polynomial of a knot, defined using Kauffman bracket skein modules, together with finitely many colored Jones polynomials, determines the remaining colored Jones polynomials of the knot. It also shows that under certain conditions, satisfied for example by the unknot and the trefoil knot, the noncommutative generalization of the A-polynomial determines all colored Jones polynomials of the knot.
On the interpolation constant for Orlicz spaces
Authors: Karlovich Alexei , Maligranda Lech
Published: July 2002

In this paper we deal with the interpolation from Lebesgue spaces

L
p

L^p

and

L
q

L^q

, into an Orlicz space

L
φ

L^\varphi

, where

1
≤
p
>
q
≤
∞

1\le p>q\le \infty

and

φ

−
1

(
t
)
=

t

1

/

p

ρ
(

t

1

/

q
−
1

/

p

)

\varphi ^{-1}(t)=t^{1/p}\rho (t^{1/q-1/p})

for some concave function

ρ
\rho

, with special attention to the interpolation constant

C
C

. For a bounded linear operator

T
T

in

L
p

L^p

and

L
q

L^q

, we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper T double-vertical-bar Subscript upper L Sub Superscript phi Subscript right-arrow upper L Sub Superscript phi Subscript Baseline less-than-or-equal-to upper C max left-brace right-brace comma double-vertical-bar double-vertical-bar upper T right-arrow right-arrow LpLp comma double-vertical-bar double-vertical-bar upper T right-arrow right-arrow LqLq comma">

‖
T

‖

L
φ

→

L
φ

≤
C
max

{

‖
T

‖

L
p

→

L
p

,
‖
T

‖

L
q

→

L
q

}

,

\|T\|_{L^\varphi \to L^\varphi } \le C\max \Big \{ \|T\|_{L^p\to L^p}, \|T\|_{L^q\to L^q} \Big \},

\]
where the interpolation constant

C
C

depends only on

p
p

and

q
q

. We give estimates for

C
C

, which imply

C
>
4

C>4

. Moreover, if either

1
>
p
>
q
≤
2

1> p>q\le 2

or

2
≤
p
>
q
>
∞

2\le p>q>\infty

, then

C
>
2

C> 2

. If

q
=
∞

q=\infty

, then

C
≤

2

1
−
1

/

p

C\le 2^{1-1/p}

, and, in particular, for the case

p
=
1

p=1

this gives the classical Orlicz interpolation theorem with the constant

C
=
1

C=1

.
Powers of ℕ*
Authors: Farah Ilijas
Published: July 2002

We prove that the Čech-Stone remainder of the integers,

N

∗

\mathbb N^*

, maps onto its square if and only if there is a nontrivial map between two of its different powers, finite or infinite. We also prove that every compact space that maps onto its own square maps onto its own countable infinite product.
A combinatorial property of cardinals
Authors: Komjáth Péter , Laczkovich Miklós
Published: July 2002

(GCH) For every cardinal

κ
≥

ω
2

\kappa \ge \omega _2

there exists

F
:
[
κ

]

≤
2

→
{
0
,
1
}

F:[\kappa ]^{\le 2} \to \{ 0,1\}

such that for every

f
:
κ
→
[
κ

]

>
ω

,
 
i
>
2

f: \kappa \to [\kappa ]^{>\omega },\ i > 2

, there are

x
,
y

x,y

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x comma t right-parenthesis equals i left-parenthesis t element-of f left-parenthesis y right-parenthesis right-parenthesis comma upper F left-parenthesis u comma y right-parenthesis equals i left-parenthesis u element-of f left-parenthesis x right-parenthesis right-parenthesis">

F
(
x
,
t
)
=
i
 
(
t
∈
f
(
y
)
)
,
 
F
(
u
,
y
)
=
i
 
(
u
∈
f
(
x
)
)

F(x,t)=i\ (t\in f(y)),\ F(u,y)=i\ (u\in f(x))

.
Direct sums of local torsion-free abelian groups
Authors: Arnold David
Published: July 2002

The category of local torsion-free abelian groups of finite rank is known to have the cancellation and

n
n

-th root properties but not the Krull-Schmidt property. It is shown that 10 is the least rank of a local torsion-free abelian group with two non-equivalent direct sum decompositions into indecomposable summands. This answers a question posed by M.C.R. Butler in the 1960’s.
On finiteness of the number of boundary slopes of immersed surfaces in 3-manifolds
Authors: Hass Joel , Wang Shicheng , Zhou Qing
Published: July 2002

For any hyperbolic 3-manifold

M
M

with totally geodesic boundary, there are finitely many boundary slopes for essential immersed surfaces of a given genus. There is a uniform bound for the number of such boundary slopes if the genus of

∂
M

\partial M

is bounded from above.
Linear systems on abelian varieties of dimension 2𝑔+1
Authors: Iyer Jaya
Published: July 2002

We show that polarisations of type

(
1
,
.
.
.
,
1
,
2
g
+
2
)

(1,...,1,2g+2)

on

g
g

-dimensional abelian varieties are never very ample, if

g
≥
3

g\geq 3

. This disproves a conjecture of Debarre, Hulek and Spandaw. We also give a criterion for non-embeddings of abelian varieties into

2
g
+
1

2g+1

-dimensional linear systems.
Covering maps that are not compositions of covering maps of lesser order
Authors: Krzempek Jerzy
Published: July 2002

In 1995 J.W. Heath asked which exactly

n
n

-to-one maps are compositions of exactly

k
k

-to-one maps with

1
>
k
>
n

1>k>n

. This paper deals with compositions of covering maps. Exactly

n
n

-to-one covering maps on locally arcwise connected continua that are not factorable into covering maps of order

≤
n
−
1

\leq n-1

are constructed for all

n
n

’s, and characterized in algebraic terms (fundamental groups). They are not proper compositions of exactly

k
k

-to-one maps, open maps, or locally one-to-one maps.
Interpretation of the deformation space of a determinantal Barlow surface via smoothings
Authors: Lee Yongnam
Published: July 2002

In this present paper, we provide an interpretation of the deformation space of a determinantal Barlow surface via smoothings.
On semiprojectivity of 𝐶*-algebras of directed graphs
Authors: Szymański Wojciech
Published: July 2002

It is shown that if

E
E

is a countable, transitive directed graph with finitely many vertices, then

C
∗

(
E
)

C^*(E)

is semiprojective.
Relative Brauer groups and 𝑚-torsion
Authors: Aljadeff Eli , Sonn Jack
Published: July 2002

Let

K
K

be a field and

B
r
(
K
)

Br(K)

its Brauer group. If

L

/

K

L/K

is a field extension, then the relative Brauer group

B
r
(
L

/

K
)

Br(L/K)

is the kernel of the restriction map

r
e

s

L

/

K

:
B
r
(
K
)
→
B
r
(
L
)

res_{L/K}:Br(K)\rightarrow Br(L)

. A subgroup of

B
r
(
K
)

Br(K)

is called an algebraic relative Brauer group if it is of the form

B
r
(
L

/

K
)

Br(L/K)

for some algebraic extension

L

/

K

L/K

. In this paper, we consider the

m
m

-torsion subgroup

B

r

m

(
K
)

Br_{m}(K)

consisting of the elements of

B
r
(
K
)

Br(K)

killed by

m
m

, where

m
m

is a positive integer, and ask whether it is an algebraic relative Brauer group. The case

K
=

Q

K=\mathbb {Q}

is already interesting: the answer is yes for

m
m

squarefree, and we do not know the answer for

m
m

arbitrary. A counterexample is given with a two-dimensional local field

K
=
k
(
(
t
)
)

K=k((t))

and

m
=
2

m=2

.
Problèmes de petites valeurs propres sur les surfaces de courbure moyenne constante
Authors: Castillon Philippe
Published: July 2002

This paper deals with the spectra of the Laplace and stability operators of a constant mean curvature surface in the hyperbolic space. In a preceding work, the author described the essential spectra of these operators, assuming that the surface is of finite total curvature. In this paper, we prove that these two operators have a finite number of eigenvalues below their essential spectra.
Convolution operators and zeros of entire functions
Authors: Cardon David
Published: July 2002

Let

G
(
z
)

G(z)

be a real entire function of order less than

2
2

with only real zeros. Then we classify certain distribution functions

F
F

such that the convolution
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper G asterisk d upper F right-parenthesis left-parenthesis z right-parenthesis equals integral Subscript negative normal infinity Superscript normal infinity Baseline upper G left-parenthesis z minus i s right-parenthesis d upper F left-parenthesis s right-parenthesis">

(
G
∗
d
F
)
(
z
)
=

∫

−
∞

∞

G
(
z
−
i
s
)

d
F
(
s
)

(G*dF)(z)=\int _{-\infty }^{\infty } G(z-is)\,dF(s)

has only real zeros.
Equivalence of domains with isomorphic semigroups of endomorphisms
Authors: Merenkov Sergei
Published: July 2002

For two bounded domains

Ω
1

,
 

Ω
2

\Omega _1,\ \Omega _2

in

C

\mathbb {C}

whose semigroups of analytic endomorphisms

E
(

Ω
1

)
,
 
E
(

Ω
2

)

E(\Omega _1), \ E(\Omega _2)

are isomorphic with an isomorphism

φ
:
 
E
(

Ω
1

)
→
E
(

Ω
2

)

\varphi :\ E(\Omega _1)\rightarrow E(\Omega _2)

, Eremenko proved in 1993 that there exists a conformal or anticonformal map

ψ
:
 

Ω
1

→

Ω
2

\psi :\ \Omega _1\rightarrow \Omega _2

such that

φ
f
=
ψ
∘
f
∘

ψ

−
1

,

\varphi f=\psi \circ f\circ \psi ^{-1},

for all

f
∈
E
(

Ω
1

)

f\in E(\Omega _1)

. In the present paper we prove an analogue of this result for the case of bounded domains in

C

n

\mathbb {C}^n

.
Normal characteristic numbers
Authors: Stong R.
Published: July 2002

This paper computes the greatest common divisor of certain normal Pontrjagin numbers and Chern numbers. These calculations are of interest in studying singularities.
Signed sums of polynomial values
Authors: Yu Hong
Published: July 2002

We give a generalization of Bleicher’s result on signed sums of

k
k

th powers. Let

f
(
x
)

f(x)

be an integral-valued polynomial of degree

k
k

satisfying the necessary condition that there exists no integer

d
>
1

d>1

dividing the values

f
(
x
)

f(x)

for all integers

x
x

. Then, for every positive integer

n
n

and every integer

l
l

, there are infinitely many integers

m
≥
l

m\ge l

and choices of

ε

i

=
±
1

\varepsilon _{i}=\pm 1

such that
\[

n
=

∑

i
=
l

m

ε

i

f
(
i
)
.

n=\sum _{i=l}^{m}\varepsilon _{i}f(i).

\]
Factorial domains
Authors: Queen Clifford
Published: July 2002

We give a simple characterization of factorial domains. We also characterize almost factorial domains and Krull domains with finite cyclic class group.
Conjugate but non inner conjugate subfactors
Authors: Choda Marie
Published: July 2002

It is shown that for each

r
∈
{
4

cos

2

⁡
(
π

/

n
)
:
n
≥
3
}
∪
[
4
,
∞
)
,

r \in \{4\cos ^{2}(\pi /n): n \ge 3\} \cup [4, \infty ),

there exist at least infinitely many subfactors of the hyperfinite II

1

_{1}

factor

R
R

with index

r
,

r,

which are pairwise conjugate but non inner conjugate. In the case that

r
r

is an integer, we have uncountably many such subfactors of

R
.

R.
Symplectic leaves and deformation quantization
Authors: Sheu Albert
Published: July 2002

In this paper, we show that for any classical simple compact Poisson Lie group

K
K

, there is no quantization of

K
K

using the quantum group

K
q

K_q

, which is both group-preserving and symplectic leaf-preserving.
On comb domains
Authors: Jenkins James
Published: July 2002

A result for comb domains is proved which is stronger than but in particular implies a conjecture of Rodin and Warschawski.
Banach algebras with unique uniform norm
Authors: Bhatt S. , Dedania H.
Published: July 2002

Commutative semisimple Banach algebras that admit exactly one uniform norm (not necessarily complete) are investigated. This unique uniform norm property is completely characterized in terms of each of spectral radius, Silov boundary, set of uniqueness, semisimple norms; and its connection with recently investigated concepts like spectral extension property, multiplicative Hahn Banach extension property and permanent radius are revealed. Several classes of Banach algebras having this property as well as those not having this property are discussed.
Class numbers and Iwasawa invariants of quadratic fields
Authors: Kraft James
Published: July 2002

Let

Q

(

−
d

)

\mathbf {Q}(\sqrt {-d})

and

Q

(

3
d

)

\mathbf {Q}(\sqrt {3d})

be quadratic fields with

d
≡

d \equiv

2 (mod 3) a positive integer. Let

λ
−

,

λ
+

\lambda ^-, \lambda ^+

be the respective Iwasawa

λ
\lambda

-invariants of the cyclotomic

Z

3

\mathbf {Z}_3

-extension of these fields. We show that if

λ
−

=
1

\lambda ^- =1

, then 3 does not divide the class number of

Q

(

3
d

)

\mathbf {Q}(\sqrt {3d})

and

λ
+

=
0

\lambda ^+ = 0

.
On a conjecture by Karlin and Szegö
Authors: Kim Deok , Kwon Kil
Published: July 2002

In 1961, Karlin and Szegö conjectured : If

{

P
n

(
x
)

}

n
=
0

∞

\{P_n(x)\}_{n=0}^\infty

is an orthogonal polynomial system and

{

P
n
′

(
x
)

}

n
=
1

∞

\{P_n’(x)\}_{n=1}^\infty

is a Sturm sequence, then

{

P
n

(
x
)

}

n
=
0

∞

\{P_n(x)\}_{n=0}^\infty

is essentially (that is, after a linear change of variable) a classical orthogonal polynomial system of Jacobi, Laguerre, or Hermite. Here, we prove that for any orthogonal polynomial system

{

P
n

(
x
)

}

n
=
0

∞

\{P_n(x)\}_{n=0}^\infty

,

{

P
n
′

(
x
)

}

n
=
1

∞

\{P_n’(x)\}_{n=1}^\infty

is always a Sturm sequence. Thus, in particular, the above conjecture by Karlin and Szegö is false.
Closed geodesics and non-differentiability of the metric in infinite-dimensional Teichmüller spaces
Authors: Zhong Li
Published: July 2002

In this paper we construct a closed geodesic in any infinite- dimensional Teichmüller space. The construction also leads to a proof of non-differentiability of the metric in infinite-dimensional Teichmüller spaces, which provides a negative answer to a problem of Goldberg.
Removability of the singular set of the heat flow of harmonic maps
Authors: Chen Yunmei , Flaminio Livio
Published: July 2002

We show that the singular set of a weak stationary solution

u
u

of the heat flow of harmonic maps between Riemannian manifolds

M
M

and

N
N

, with

N
N

compact, is removable if it has “parabolic codimension” greater than two and the initial energy

E
(

u
0

)

E(u_0)

is sufficiently small.
Polynomials and limited sets
Authors: Galindo Pablo
Published: July 2002

We prove that scalar-valued polynomials are weakly continuous on limited sets and that, as in the case of linear mappings, every

c
0

c_0

-valued polynomial maps limited sets into relatively compact ones. We also show that a scalar-valued polynomial whose derivative is limited is weakly sequentially continuous.
Composition of Blochs with bounded analytic functions
Authors: Kwon E.
Published: July 2002

If

f
f

is a holomorphic self-map of the open unit disc and

1
≤
p
>
∞

1 \leq p > \infty

, then the following are equivalent.

(
1
)

h
∘
f
∈

H

2
p

(1)\,\,\,\, h\circ f \in H^{2p}

for all Bloch functions

h
h

.
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel s u p Underscript r Endscripts integral Subscript 0 Superscript 2 pi Baseline left-parenthesis l o g StartFraction 1 Over 1 minus StartAbsoluteValue f left-parenthesis r e Superscript i theta Baseline right-parenthesis EndAbsoluteValue squared EndFraction right-parenthesis Superscript p Baseline d theta greater-than normal infinity period EndLayout">

(2)

s
u
p

r

∫

0

2
π

(
l
o
g

1

1
−
|
f
(
r

e

i
θ

)

|

2

)

p

d
θ

>
∞
.

\begin{equation*}\underset {{r} }{sup} \int _{0}^{2\pi } \left ( log \frac {1}{1 - \vert f(re^{i\theta })\vert ^{2}}\right )^{p} \,d\theta \,\, > \infty . \tag {2}\end{equation*}

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis 3 right-parenthesis EndLabel integral Subscript 0 Superscript 2 pi Baseline left-parenthesis integral Subscript 0 Superscript 1 Baseline left-parenthesis f Superscript number-sign Baseline right-parenthesis squared left-parenthesis r e Superscript i theta Baseline right-parenthesis left-parenthesis 1 minus r right-parenthesis d r right-parenthesis Superscript p Baseline d theta greater-than normal infinity comma EndLayout">

(3)

∫

0

2
π

(

∫

0

1

(

f

#

)

2

(
r

e

i
θ

)

(
1
−
r
)
d
r
)

p

d
θ
>
∞
,

\begin{equation*}\int _{0}^{2\pi } \left ( \int _{0}^{1} (f^{\#})^{2}(re^{i\theta })\, (1-r) dr \right )^{p} d\theta > \infty , \tag {3}\end{equation*}

where

f

#

f^{\#}

is the hyperbolic derivative of

f
f

:

f

#

=
|

f
′

|

/

(
1
−
|
f

|

2

)

f^{\#} = \vert f’\vert / (1-\vert f\vert ^{2})

.
Basic differential forms for actions of Lie groups
Authors: Michor Peter
Published: July 2002

A section of a Riemannian

G
G

-manifold

M
M

is a closed submanifold

Σ
\Sigma

which meets each orbit orthogonally. It is shown that the algebra of

G
G

-invariant differential forms on

M
M

which are horizontal in the sense that they kill every vector which is tangent to some orbit, is isomorphic to the algebra of those differential forms on

Σ
\Sigma

which are invariant with respect to the generalized Weyl group of

Σ
\Sigma

, under some condition.
Endomorphism rings of completely pure-injective modules
Authors: Pardo José , Asensio Pedro
Published: July 2002

Let

R
R

be a ring,

E
=
E
(

R
R

)

E=E(R_R)

its injective envelope,

S
=
End
⁡
(

E
R

)

S= \operatorname {End}(E_R)

and

J
J

the Jacobson radical of

S
S

. It is shown that if every finitely generated submodule of

E
E

embeds in a finitely presented module of projective dimension

≤
1

\le 1

, then every finitley generated right

S

/

J

S/J

-module

X
X

is canonically isomorphic to

Hom
R

⁡
(
E
,
X

⊗
S

E
)

\operatorname {Hom}_R(E,X\otimes _S E)

. This fact, together with a well-known theorem of Osofsky, allows us to prove that if, moreover,

E

/

J
E

E/JE

is completely pure-injective (a property that holds, for example, when the right pure global dimension of

R
R

is

≤
1

\le 1

and hence when

R
R

is a countable ring), then

S
S

is semiperfect and

R
R

R_R

is finite-dimensional. We obtain several applications and a characterization of right hereditary right noetherian rings.
Banach spaces in which every 𝑝-weakly summable sequence lies in the range of a vector measure
Authors: Piñeiro C.
Published: July 2002

Let

X
X

be a Banach space. For

1
>
p
>
+
∞

1>p>+\infty

we prove that the identity map

I
X

I_X

is

(
1
,
1
,
p
)

(1,1,p)

-summing if and only if the operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript asterisk Baseline element-of upper X Superscript asterisk Baseline right-arrow sigma-summation mathematical left-angle x Subscript n Baseline comma x Superscript asterisk Baseline mathematical right-angle e Subscript n Baseline element-of l Subscript q">

x
∗

∈

X
∗

→
∑
⟨

x
n

,

x
∗

⟩

e
n

∈

l
q

x^*\in X^*\to \sum \langle x_n,x^*\rangle e_n\in l_q

is nuclear for every unconditionally summable sequence

(

x
n

)

(x_n)

in

X
X

, where

q
q

is the conjugate number for

p
p

. Using this result we find a characterization of Banach spaces

X
X

in which every

p
p

-weakly summable sequence lies inside the range of an

X

∗
∗

X^{**}

-valued measure (equivalently, every

p
p

-weakly summable sequence

(

x
n

)

(x_n)

in

X
X

, satisfying that the operator

(

α
n

)
∈

l
q

→
∑

α
n

x
n

∈
X

(\alpha _n)\in l_q\to \sum \alpha _nx_n\in X

is compact, lies in the range of an

X
X

-valued measure) with bounded variation. They are those Banach spaces such that the identity operator

I

X
∗

I_{X^*}

is

(
1
,
1
,
p
)

(1,1,p)

-summing.
Bounded harmonic maps on a class of manifolds
Authors: Sung Chiung-Jue , Tam Luen-fai , Wang Jiaping
Published: July 2002

Without imposing any curvature assumptions, we show that bounded harmonic maps with image contained in a regular geodesic ball share similar behaviour at infinity with the bounded harmonic functions on the domain manifold.
On linear series on general 𝑘-gonal projective curves
Authors: Ballico E. , Keem C.
Published: July 2002

Let

X
X

be a general

k
k

-gonal curve of genus

g
g

. Here we prove a strong upper bound for the dimension of linear series on

X
X

, i.e. we prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension left-parenthesis upper W Subscript d Superscript r Baseline left-parenthesis upper X right-parenthesis right-parenthesis less-than-or-equal-to rho left-parenthesis g comma r comma d right-parenthesis plus left-parenthesis g minus 2 k plus 2 right-parenthesis colon equals g minus left-parenthesis r plus 1 right-parenthesis left-parenthesis r plus g minus d right-parenthesis plus left-parenthesis g minus 2 k plus 2 right-parenthesis">

dim
⁡
(

W
d
r

(
X
)
)
≤
ρ
(
g
,
r
,
d
)
+
(
g
−
2
k
+
2
)
:=
g
−
(
r
+
1
)
(
r
+
g
−
d
)
+
(
g
−
2
k
+
2
)

\dim (W^r_d(X))\leq \rho (g,r,d)+(g-2k+2):=g-(r+1) (r+g-d)+(g-2k+2)

.
Differences of vector-valued functions on topological groups
Authors: Basit Bolis , Pryde A.
Published: July 2002

Let

G
G

be a locally compact group equipped with right Haar measure. The right differences

△

h

φ

\triangle _{h} \varphi

of functions

φ
\varphi

on

G
G

are defined by

△

h

φ
(
t
)
=
φ
(
t
h
)
−
φ
(
t
)

\triangle _{h}\varphi (t) = \varphi (th) - \varphi (t)

for

h
,
t
∈
G

h,t \in G

. Let

φ
∈

L

∞

(
G
)

\varphi \in L^{\infty }(G)

and suppose

△

h

φ
∈

L

p

(
G
)

\triangle _{h} \varphi \in L^{p} (G)

for some

1
≤
p
>
∞

1 \leq p > \infty

and all

h
∈
G

h \in G

. We prove that

‖

△

h

φ

‖

p

\Vert \triangle _{h} \varphi \Vert _{p}

is a right uniformly continuous function of

h
h

. If

G
G

is abelian and the Beurling spectrum

s
p
(
φ
)

sp(\varphi )

does not contain the unit of the dual group

G
^

\hat {G}

, then we show

φ
∈

L

p

(
G
)

\varphi \in L^{p} (G)

. These results have analogues for functions

φ
:
G
→
X

\varphi : G\to X

, where

X
X

is a separable or reflexive Banach space. Finally, we apply our methods to vector-valued right uniformly continuous differences and to absolutely continuous elements of left Banach

G
G

-modules.
Almost linearity of 𝜀-bi-Lipschitz maps between real Banach spaces
Authors: Jun Kil-Woung , Park Dal-Won
Published: July 2002

Let

X
X

and

Y
Y

be real Banach spaces. A map

f
f

between

X
X

and

Y
Y

is called an

ϵ
\epsilon

-bi-Lipschitz map if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 minus epsilon right-parenthesis double-vertical-bar x minus y double-vertical-bar less-than-or-equal-to double-vertical-bar f left-parenthesis x right-parenthesis minus f left-parenthesis y right-parenthesis double-vertical-bar less-than-or-equal-to left-parenthesis 1 plus epsilon right-parenthesis double-vertical-bar x minus y double-vertical-bar">

(
1
−
ϵ
)
‖
x
−
y
‖
≤
‖
f
(
x
)
−
f
(
y
)
‖
≤
(
1
+
ϵ
)
‖
x
−
y
‖

(1-\epsilon )\|x-y\| \le \|f(x) -f(y)\| \le (1+\epsilon )\|x- y\|

for all

x
,
y
∈
X

x, y\in X

. In this note we show that if

f
f

is an

ϵ
\epsilon

-bi-Lipschitz map with

f
(
0
)
=
0

f(0)=0

from

X
X

onto

Y
Y

, then

f
f

is almost linear. We also show that if

f
:
X
⟶
Y

f:X\longrightarrow Y

is a surjective

ϵ
\epsilon

-bi-Lipschitz map with

f
(
0
)
=
0

f(0)=0

, then there exists a linear isomorphism

I
:
X
→
Y

I:X\to Y

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper I left-parenthesis x right-parenthesis minus f left-parenthesis x right-parenthesis double-vertical-bar less-than-or-equal-to upper E left-parenthesis epsilon comma alpha right-parenthesis left-parenthesis double-vertical-bar x double-vertical-bar Superscript alpha Baseline plus double-vertical-bar x double-vertical-bar Superscript 2 minus alpha Baseline right-parenthesis">

‖
I
(
x
)
−
f
(
x
)
‖
≤
E
(
ϵ
,
α
)
(
‖
x

‖
α

+
‖
x

‖

2
−
α

)

\|I(x)-f(x)\| \le E(\epsilon , \alpha )(\|x\|^\alpha +\|x\|^{2-\alpha })

\]
where

E
(
ϵ
,
α
)
→
0

E(\epsilon ,\alpha )\to 0

as

ϵ
→
0

\epsilon \to 0

and

0
>
α
>
1

0>\alpha >1

.
A coefficient of an asymptotic expansion of logarithms of determinants for classical elliptic pseudodifferential operators with parameters
Authors: Lee Yoonweon
Published: July 2002

For classical elliptic pseudodifferential operators

A
(
λ
)

A(\lambda )

of order

m
>
0

m>0

with parameter

λ
\lambda

of weight

χ
>
0

\chi >0

, it is known that

log
⁡

det
θ

A
(
λ
)

\log \det _\theta A(\lambda )

admits an asymptotic expansion as

λ
→
+
∞

\lambda \to +\infty

. In this paper we show, with some assumptions, that the coefficient of

λ

−
1

/

χ

\lambda ^{-1/\chi }

can be expressed by the value of a zeta function at 0 for some elliptic

ψ
DO

\psi \operatorname {DO}

on

M
×

S
1

M\times S^1

multiplied by

m
2

\frac {m}{2}

.
Direct summand conjecture and descent for flatness
Authors: Ohi Takeo
Published: July 2002

We prove the equivalence of the direct summand conjecture and descent of flatness for integral extensions.
The Euler characteristic is stable under compact perturbations
Authors: Ambrozie Calin-Grigore
Published: July 2002

We prove in the general case the stability under compact perturbations of the index (i.e. the Euler characteristic) of a Fredholm complex of Banach spaces. In particular, we obtain the corresponding stability property for Fredholm multioperators. These results are the consequence of a similar statement, concerning more general objects called Fredholm pairs.
Finite CW complexes with maximal torsion gaps
Authors: Huang Huale
Published: July 2002

We investigate some properties of finite CW complexes with maximal homotopy torsion gaps and prove a revision of the Halperin conjecture under an additional condition.
Constant-to-one extensions of shifts of finite type
Authors: Fiebig Doris
Published: July 2002

Any transitive shift of finite type has a transitive constant-to-one extension which is not of finite type.
On relative Hausdorff measures of noncompactness and relative Chebyshev radii in Banach spaces
Authors: Wisnicki Andrzej , Wosko Jacek
Published: July 2002

In this paper we prove some formulae and evaluations on relative Hausdorff measures of noncompactness and relative Chebyshev radii in various Banach spaces. We generalize the Lifschitz constant

κ
(
X
)

\kappa (X)

and introduce a function

κ
~

X

(
⋅
)

\tilde {\kappa }_X(\cdot )

.
On 𝐶*-algebras associated with locally compact groups
Authors: Bekka M. , Kaniuth E. , Lau A. , Schlichting G.
Published: July 2002

Let

G
G

be a locally compact group, and let

G

d

G_{d}

denote the same group

G
G

with the discrete topology. There are various

C

∗

-algebras

C^{*}\text {-algebras}

associated to

G
G

and

G

d

.

G_{d}.

We are concerned with the question of when these

C

∗

-algebras

C^{*}\text {-algebras}

are isomorphic. This is intimately related to amenability. The results can be reformulated in terms of Fourier and Fourier-Stieltjes algebras and of weak containment properties of unitary representations.
Souslin’s hypothesis and convergence in category
Authors: Miller Arnold
Published: July 2002

A sequence of functions

f
n

:
X
→

R

f_n\colon X\to \mathbb R

from a Baire space

X
X

to the reals

R

\mathbb R

is said to converge in category iff every subsequence has a subsequence which converges on all but a meager set. We show that if there exists a Souslin tree, then there exists a nonatomic Baire space

X
X

such that every sequence which converges in category converges everywhere on a comeager set. This answers a question of Wagner and Wilczynski who proved the converse.
On complemented subspaces of sums and products of Banach spaces
Authors: Ostrovskii M.
Published: July 2002

It is proved that there exist complemented subspaces of countable topological products (locally convex direct sums) of Banach spaces which cannot be represented as topological products (locally convex direct sums) of Banach spaces.
Characterisation of compactness through convergence in Tychonoff spaces
Authors: Petalas C. , Vidalis T.
Published: July 2002

In this article we first give a characterisation of compact spaces among

T

3.5

T_{3.5}

spaces by improving a theorem of J. Ewert. Then, with the aid of a new type of convergence, we give a characterisation of the pseudocompact and of the Lindelöf

T

3.5

T_{3.5}

spaces.
The local zeta function for the non-trivial characters associated with the singular Jordan algebras
Authors: Robinson Margaret
Published: July 2002

This paper investigates the local integrals
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z Subscript m Baseline left-parenthesis t comma chi right-parenthesis equals integral Underscript upper H Subscript m Baseline left-parenthesis upper O Subscript upper C Baseline right-parenthesis Endscripts chi left-parenthesis det left-parenthesis x right-parenthesis right-parenthesis StartAbsoluteValue det left-parenthesis x right-parenthesis EndAbsoluteValue Superscript s Baseline d x">

Z
m

(
t
,
χ
)
=

∫

H
m

(

O
C

)

χ
(
det
(
x
)
)

|

det
(
x
)

|

s

d
x

Z_m(t,\chi )=\int _{H_m(O_C)} \chi ( \det (x)) | \det (x) |^s dx

\]
where

O
C

O_C

represents the integers of a composition algebra over a non-archimedean local field

K
K

and

χ
\chi

is a non-trivial character on the units in the ring of integers of

K
K

extended to

K
∗

K^*

by setting

χ
(
π
)
=
1

\chi (\pi )=1

. The local zeta function for the trivial character is known for all composition algebras

C
C

. In this paper, we show in the quaternion case that

Z
(
t
,
χ
)
=
0

Z(t, \chi )=0

for all non-trivial characters and then compute the local zeta function in the ramified quadratic extension case for

χ
\chi

equal to the quadratic character. In this latter case,

Z
(
t
,
χ
)
=
0

Z(t, \chi )=0

for any character of order greater than

2
2

.
On pasting 𝐴_{𝑝}-weights
Authors: Schröder Bernd
Published: July 2002

Theorem 3 gives a condition when two

A
p

A_p

-weights can be “pasted” together to yield another

A
p

A_p

-weight. It is subsequently used in Example 6 to give an example that shows that a necessary condition by Gohberg, Krupnik, and Spitkovsky is not sufficient.
On a fixed point problem of Reich
Authors: Yu-Qing Chen
Published: July 2002

In this paper, we give an affirmative answer to a fixed point problem of S. Reich.
Nonresonance problems for differential inclusions in separable Banach spaces
Authors: Ding Zouhua , Kartsatos Athanassios
Published: July 2002

Let

X
X

be a real separable Banach space.  The boundary value problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis upper B right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column a m p semicolon x prime element-of upper A left-parenthesis t right-parenthesis x plus upper F left-parenthesis t comma x right-parenthesis comma t element-of script upper R Subscript plus Baseline comma 2nd Row 1st Column Blank 2nd Column a m p semicolon upper U x equals a comma EndLayout EndLayout">

(B)

a
m
p
;

x
′

∈
A
(
t
)
x
+
F
(
t
,
x
)
,
 
t
∈

R

+

,

a
m
p
;
U
x
=
a
,

\begin{equation*} \begin {split} &x’ \in A(t)x+F(t,x),~t\in \mathcal {R}_+,\\ &Ux = a, \end{split} \tag *{(B)} \end{equation*}

is studied on the infinite interval

R
+

=
[
0
,
∞
)
.

R_+=[0,\infty ).

 Here, the closed and densely defined linear operator

A
(
t
)
:
X
⊃
D
(
A
)
→
X
,
 
t
∈

R

+

,

A(t):X\supset D(A)\to X,~t\in \mathcal {R}_+,

generates an evolution operator

W
(
t
,
s
)
.

W(t,s).

 The function

F
:

R

+

×
X
→

2
X

F:\mathcal {R}_+\times X\to 2^X

is measurable in its first variable, upper semicontinuous in its second and has weakly compact and convex values.  Either

F
F

is bounded and

W
(
t
,
s
)

W(t,s)

is compact for

t
>
s
,

t > s,

or

F
F

is compact and

W
(
t
,
s
)

W(t,s)

is equicontinuous.  The mapping

U
:

C
b

(

R

+

,
X
)
→
X

U:C_b(\mathcal {R}_+,X)\to X

is a bounded linear operator and

a
∈
X

a\in X

is fixed.  The nonresonance problem is solved by using Ma’s fixed point theorem along with a recent result of Przeradzki which characterizes the compact sets in

C
b

(

R

+

,
X
)
.

C_b(\mathcal {R}_+,X).
Commutativity of automorphisms of subfactors modulo inner automorphisms
Authors: Goto Satoshi
Published: July 2002

We introduce a new algebraic invariant

χ

a

(
M
,
N
)

\chi _{a}(M,N)

of a subfactor

N
⊂
M

N \subset M

. We show that this is an abelian group and that if the subfactor is strongly amenable, then the group coincides with the relative Connes invariant

χ
(
M
,
N
)

\chi (M,N)

introduced by Y. Kawahigashi. We also show that this group is contained in the center of

Out
⁡
(
M
,
N
)

\operatorname {Out}(M,N)

in many interesting examples such as quantum

S
U
(
n

)

k

SU(n)_{k}

subfactors with level

k
k

(
k
≥
n
+
1
)

(k \geq n+1)

, but not always contained in the center. We also discuss its relation to the most general setting of the orbifold construction for subfactors.
Affine mappings of invertible operators
Authors: Harris Lawrence , Kadison Richard
Published: July 2002

The infinite-dimensional analogues of the classical general linear group appear as groups of invertible elements of Banach algebras. Mappings of these groups onto themselves that extend to affine mappings of the ambient Banach algebra are shown to be linear exactly when the Banach algebra is semi-simple. The form of such linear mappings is studied when the Banach algebra is a C*-algebra.
Integral inclusions of upper semi-continuous or lower semi-continuous type
Authors: O’Regan Donal
Published: July 2002

Topological results for set valued maps are used to establish existence results for integral inclusions of Volterra or Hammerstein type.
On the commutant of hyponormal operators
Authors: Prunaru Bebe
Published: July 2002

Let

T
T

be a pure hyponormal operator with compact self-commutator. We show that the unit ball of the commutant of

T

∗

T^{*}

is compact in the strong operator topology.
Theorem of Kuratowski-Suslin for measurable mappings. II
Authors: Wiśniewski Andrzej
Published: July 2002

The purpose of this paper is to describe these

μ
\mu

-measurable mappings on a separable complete metric space with the Borel measure

μ
\mu

, which transform every

μ
\mu

-measurable set onto a

μ
\mu

-measurable one. The obtained results are a generalization of the classical outcomes of Suslin and Kuratowski and the results from our previous paper.
On the possibilities for partitioning a cake
Authors: Barbanel Julius
Published: July 2002

We wish to consider the following type of cake division problem: There are

p
p

individuals. Each individual has available a measure that he or she uses to evaluate the sizes of pieces of cake. We wish to partition our cake into

q
q

pieces in such a way that the various evaluations that the individuals make of the sizes of the pieces satisfy certain pre-assigned equalities and inequalities. Our main result yields a quite general criterion for showing that certain such partitions exist. Following the proof, we consider various applications.
Isometries of certain operator algebras
Authors: Cole Brian , Wermer John
Published: July 2002

To a given basis

ϕ
1

,
…
,

ϕ
n

\phi _1,\dotsc ,\phi _n

on an

n
n

-dimensional Hilbert space

H

\mathcal H

, we associate the algebra

A

\mathfrak A

of all linear operators on

H

\mathcal H

having every

ϕ
j

\phi _j

as an eigenvector. So,

A

\mathfrak A

is commutative, semisimple, and

n
n

-dimensional. Given two algebras of this type,

A

\mathfrak A

and

B

\mathfrak B

, there is a natural algebraic isomorphism

τ
\tau

of

A

\mathfrak A

and

B

\mathfrak B

. We study the question: When does

τ
\tau

preserve the operator norm?
Fixed points of approximable maps
Authors: Park Sehie
Published: July 2002

We present a simple proof of the Leray-Schauder type theorem for approximable multimaps given recently by Ben-El-Mechaiekh and Idzik. We apply this theorem to obtain a Schaefer type theorem, the Birkhoff-Kellogg type theorems, a Penot type theorem for non-self-maps, and quasi-variational inequalities, all related to compact closed approximable maps.
A general coincidence theorem on contractible spaces
Authors: Park Sehie , Jeong Kwang
Published: July 2002

We obtain a general coincidence theorem for multifunctions in very large classes defined on contractible spaces. Our theorem generalizes a recent result due to Tarafdar and Yuan (1994) and many other earlier works including the Fan-Browder fixed point theorem.
Non-commutative disc algebras and their representations
Authors: Popescu Gelu
Published: July 2002

It is shown that the smallest closed subalgebra
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A l g left-parenthesis upper I Subscript script upper K Baseline comma upper V 1 comma ellipsis comma upper V Subscript n Baseline right-parenthesis subset-of script upper B left-parenthesis script upper K right-parenthesis left-parenthesis n equals 2 comma 3 comma ellipsis comma normal infinity right-parenthesis">

A
l
g
(

I

K

,

V

1

,
…
,

V

n

)
⊂

B

(

K

)

(
n
=
2
,
3
,
…
,
∞
)

\begin{equation*}Alg(I_{ \mathcal {K}} ,V_{1},\dots ,V_{n})\subset \mathcal {B} (\mathcal {K}) \qquad (n=2,3,\dots ,\infty )\end{equation*}

generated by any sequence

V

1

,
…
,

V

n

V_{1},\dots , V_{n}

of isometries on a Hilbert space

K

\mathcal {K}

such that

V

1

V

1

∗

+
⋯
+

V

n

V

n

∗

≤

I

K

V_{1}V_{1}^{*}+\cdots +V_{n}V_{n}^{*}\le I_{\mathcal {K}}

is completely isometrically isomorphic to the non-commutative “disc” algebra

A

n

\mathcal {A} _{n}

introduced in Math. Scand. 68 (1991), 292–304. We also prove that for

n
≠
m

n\ne m

the Banach algebras

A

n

\mathcal {A} _{n}

and

A

m

\mathcal {A} _{m}

are not isomorphic. In particular, we give an example of two non-isomorphic Banach algebras which are completely isometrically embedded in each other. The completely bounded (contractive) representations of the “disc” algebras

A

n

(
n
=
2
,
3
,
…
,
∞
)

\mathcal {A} _{n} (n=2,3,\dots ,\infty )

on a Hilbert space are characterized. In particular, we prove that a sequence of operators

A

1

,

A

2

,
…

A_{1},A_{2},\dots

is simultaneously similar to a contractive sequence

T

1

,

T

2

,
…

T_{1},T_{2},\dots

(i.e.,

T

1

T

1

∗

+
⋯
+

T

n

T

n

∗

≤
I

T_{1}T_{1}^{*}+\cdots +T_{n}T_{n}^{*} \le I

) if and only if it is completely polynomially bounded. The first cohomology group of

A

n

\mathcal {A} _{n}

with coefficients in

C

\mathbb {C}

is calculated, showing, in particular, that the disc algebras are not amenable. Similar results are proved for the non-commutative Hardy algebras

F

n

∞

F_{n}^{\infty }

introduced in Math. Scand. 68 (1991), 292–304. The right joint spectrum of the left creation operators on the full Fock space is also determined.
The boundedness of Riesz 𝑠-transforms of measures in ℝⁿ
Authors: Vihtilä Merja
Published: July 2002

Let

μ
\mu

be a finite nonzero Borel measure in

R

n

\mathbb {R}^{n}

satisfying

0
>

c

−
1

r

s

≤
μ
B
(
x
,
r
)
≤
c

r

s

>
∞

0 >c^{-1}r^{s}\le \mu B(x,r)\le cr^{s} >\infty

for all

x
∈
spt
⁡
μ

x\in \operatorname {spt}\mu

and

0
>
r
≤
1

0 > r\le 1

and some

c
>
0

c >0

. If the Riesz

s
s

-transform

C

s
,
μ

(
x
)
=
∫

y
−
x

|

y
−
x

|

s
+
1

d
μ
y

\begin{equation*}{\mathcal {C}}_{s,\mu }(x)=\int \frac {y-x}{|y-x|^{s+ 1}}\, d\mu y \end{equation*}

is essentially bounded, then

s
s

is an integer. We also give a related result on the

L

2

L^{2}

-boundedness.
Note on the Bradley and Ramanujan summation
Authors: Wenchang Chu
Published: July 2002

The hypergeometric series of Bradley and Ramanujan is evaluated by means of the binomial convolutions of Hagen and Rothe, which presents, alternatively, a short proof of the recent result of Bradley about Ramanujan’s enigmatic claim.
Integral equations, implicit functions, and fixed points
Authors: Burton T.
Published: July 2002

The problem is to show that (1)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V left-parenthesis t comma x right-parenthesis equals upper S left-parenthesis t comma integral Subscript 0 Superscript t Baseline upper H left-parenthesis t comma s comma x left-parenthesis s right-parenthesis right-parenthesis d s right-parenthesis">

V
(
t
,
x
)
=
S
(
t
,

∫
0
t

H
(
t
,
s
,
x
(
s
)
)

d
s
)

V(t,x) = S(t, \int _0^t H(t, s, x(s)) \, ds )

has a solution, where

V
V

defines a contraction,

V
~

\tilde V

, and

S
S

defines a compact map,

S
~

\tilde S

. A fixed point of

P
φ
=

S
~

φ
+
(
I
−

V
~

)
φ

P \varphi = \tilde S \varphi + (I - \tilde V) \varphi

would solve the problem. Such equations arise naturally in the search for a solution of

f
(
t
,
x
)
=
0

f(t, x) = 0

where

f
(
0
,
0
)
=
0

f(0,0) = 0

, but

∂
f
(
0
,
0
)

/

∂
x
=
0

\partial f(0,0) / \partial x = 0

so that the standard conditions of the implicit function theorem fail. Now

P
φ
=

S
~

φ
+
(
I
−

V
~

)
φ

P \varphi = \tilde S \varphi + ( I - \tilde V) \varphi

would be in the form for a classical fixed point theorem of Krasnoselskii if

I
−

V
~

I - \tilde V

were a contraction. But

I
−

V
~

I - \tilde V

fails to be a contraction for precisely the same reasons that the implicit function theorem fails. We verify that

I
−

V
~

I - \tilde V

has enough properties that an extension of Krasnoselskii’s theorem still holds and, hence, (1) has a solution. This substantially improves the classical implicit function theorem and proves that a general class of integral equations has a solution.
On the existence of minimal surfaces with singular boundaries
Authors: Iseri Howard
Published: July 2002

In 1931, Jesse Douglas showed that in

R

n

\mathbb {R}^{n}

, every set of

k
k

rectifiable Jordan curves, with

k
≥
2

k \ge 2

, bounds an area-minimizing minimal surface with prescribed topological type if a “condition of cohesion” is satisfied. In this paper, it is established that under similar conditions, this result can be extended to non-Jordan curves.
On spread and condensations
Authors: Arhangelskii A.
Published: July 2002

A space

X
X

has a property

P

{\mathcal {P}}

strictly if every finite power of

X
X

has

P

{\mathcal {P}}

. A condensation is a one-to-one continuous mapping onto. For Tychonoff spaces, the following results are established. If the strict spread of

X
X

is countable, then

X
X

can be condensed onto a strictly hereditarily separable space. If

s
(

C

p

(
X
)
)
≤
ω

s(C_{p}(X))\leq \omega

, then

C

p

(
X
)

C_{p}(X)

can be condensed onto a strictly hereditarily separable space, and therefore, every compact subspace of

C

p

(
X
)

C_{p}(X)

is strictly hereditarily separable. Under

(
M
A
+
¬
C
H
)

(MA+\neg CH)

, if

G
G

is a topological group such that

s
(

C

p

(
G
)
)
≤
ω

s(C_{p}(G))\leq \omega

, then

G
G

is strictly hereditarily Lindelöf and strictly hereditarily separable.
Lindelöf powers and products of function spaces
Authors: Okunev Oleg , Tamano Kenichi
Published: July 2002

We give criteria for finite and countable powers of a space similar to the Michael line being Lindelöf. As applications, we give examples related to Lindelöf property in products of spaces of Michael line type and in products of spaces of continuous functions on separable

σ
\sigma

-compact spaces.
A sum packing problem of Erdös and the Conway-Guy sequence
Authors: Bohman Tom
Published: July 2002

A set

S
S

of positive integers has distinct subset sums if the set

{

∑

x
∈
X

x
:
X
⊂
S
}

\left \{ \sum _{x \in X} x : X \subset S \right \}

has

2

|

S

|

2^{|S|}

distinct elements. Let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis n right-parenthesis equals min left-brace right-brace colon of maxmaxS colon separator separator separator equals equals vertical-bar vertical-bar SnandShas reverse-solidus hskip 2 mm distinct reverse-solidus hskip 2 mm subset reverse-solidus hskip 2 mm sums period">

f
(
n
)
=
min
{
max
S
:

|

S

|

=
n

and

S

has\hskip 2mm distinct\hskip 2mm subset \hskip 2mm sums
}
.

f(n) = \min \{ \max S: |S|=n \hskip 2mm \text {and} \hskip 2mm S \hskip 2mm \text {has\hskip 2mm distinct\hskip 2mm subset \hskip 2mm sums}\}.

\]
In 1931 Paul Erdős conjectured that

f
(
n
)
≥
c

2

n

f(n) \ge c2^{n}

for some constant

c
c

. In 1967 John Conway and Richard Guy constructed an interesting sequence of sets of integers. They conjectured that these sets have distinct subset sums and that they are close to the best possible (with respect to largest element). We prove that sets from this sequence have distinct subset sums. We also present some variations of this construction that give microscopic improvements in the best known upper bound on

f
(
n
)

f(n)

.
Multinomial expansions and the pythagorean theorem
Authors: Fryant Allan
Published: July 2002

An addition formula for homogeneous polynomials is used to obtain a generalization of the Pythagorean theorem and a new view of the multinomial expansion.
Gowers’ dichotomy for asymptotic structure
Authors: Wagner R.
Published: July 2002

In this paper Gowers’ dichotomy is extended to the context of weaker forms of unconditionality, most notably asymptotic unconditionality. A general dichotomic principle is demonstrated; a Banach space has either a subspace with some unconditionality property, or a subspace with a corresponding ‘proximity of subspaces’ property.
Crumpled laminations and manifolds of nonfinite type
Authors: Daverman R. , Tinsley F.
Published: July 2002

Using a group-theoretic construction due to Bestvina and Brady, we build

(
n
+
1
)

(n+1)

-manifolds

W
W

which admit partitions into closed, connected

n
n

-manifolds but which do not have finite homotopy type.
Each locally one-to-one map from a continuum onto a tree-like continuum is a homeomorphism
Authors: Heath Jo
Published: July 2002

In 1977 T. Maćkowiak proved that each local homeomorphism from a continuum onto a tree-like continuum is a homeomorphism. Recently, J. Rogers proved that each locally one-to-one (not necessarily open) map from a hereditarily decomposable continuum onto a tree-like continuum is a homeomorphism, and this paper removes “hereditarily decomposable" from the hypothesis of Rogers’ theorem.
On a conjecture of Danzer and Grünbaum
Authors: Katchalski Meir , Nashtir David
Published: July 2002

The main result of the paper is that if

A
A

is a family of homothetic triangles in the plane such that any 9 of them can be pierced by two points, then all members of

A
A

can be pierced by two points. This is best possible in more than one sense: (1) the number 9 cannot be replaced by 8; (2) no similar statement is true for homothetic copies (or even translates) of a symmetric convex hexagon.
An algebraic version of Demailly’s asymptotic Morse inequalities
Authors: Angelini Flavio
Published: July 2002

We give an elementary algebraic proof of some asymptotic estimates (called by Demailly asymptotic Morse inequalities) for the dimensions of cohomology groups of the difference of two ample line bundles on a smooth complex projective variety of any dimension.
On the pluricanonical map of threefolds of general type
Authors: Shin Dong-Kwan
Published: July 2002

Let

X
X

be a smooth minimal threefold of general type and let

n
n

be an integer

>
1

>1

. Assume that the image of the pluricanonical map

Φ

n

\Phi _{n}

of

X
X

is a curve. Then a simple computation shows that

n
n

is necessarily

2
2

or

3
3

. When

n
=
2

n=2

with a numerical condition or when

n
=
3

n=3

, we obtain two inequalities

χ
(

O

X

)
≤
min
{
−
1
,
2
−
2

q

1

}

\chi (\mathcal {O}_{X})\leq \text {min}\{-1,2-2q_{1}\}

and

q

1

≤

3
14

K

X

3

+
1

q_{1}\leq \dfrac {3}{14}{K_{X}}^{3}+1

, where

q

1

q_{1}

is the irregularity of

X
X

and

χ
(

O

X

)

\chi (\mathcal {O}_{X})

is the Euler characteristic of

X
X

.