bluuebunny/crossref_metadata_2025 huggingface dataset:
Eigenvalue ratios for the regular Sturm-Liouville system
Authors: Huang Yu-Ling , Law C.
Published: July 2002

Following the method of Ashbaugh-Benguria in Comm. Math. Phys. 124 (1989), 403–415; J. Differential Equations 103 (1993), 205–219, we prove an upper estimate of the arbitrary eigenvalue ratio

(

μ
m

/

μ
n

)

( \mu _m / \mu _n )

for the regular Sturm-Liouville system. This upper estimate is sharp for Neumann boundary conditions. We also discuss the sign of

μ
1

\mu _1

and include an elementary proof of a useful trigonometric inequality first given in the aforementioned articles.
Equivalent conditions involving common fixed points for maps on the unit interval
Authors: Jachymski Jacek
Published: July 2002

Let

g
g

be a continuous self-map of the unit interval

I
I

. Equivalent conditions are given to ensure that

g
g

has a common fixed point with every continuous map

f
:
I
↦
I

f:I\mapsto I

that commutes with

g
g

on a suitable subset of

I
I

. This extends a recent result of Gerald Jungck.
Cartan invariants of group algebras of finite groups
Authors: Koshitani Shigeo
Published: July 2002

We give a result on Cartan invariants of the group algebra

k
G

kG

of a finite group

G
G

over an algebraically closed field

k
k

, which implies that if the Loewy length (socle length) of the projective indecomposable

k
G

kG

-module corresponding to the trivial

k
G

kG

-module is four, then

k
k

has characteristic 2. The proof is independent of the classification of finite simple groups.
Inequalities for the Novikov-Shubin invariants
Authors: Mathai Varghese
Published: July 2002

In this paper, we prove that the Novikov-Shubin invariants satisfy a sequence of inequalities and deduce some useful consequences of this result.
An answer to A.D.Wallace’s question about countably compact cancellative semigroups
Authors: Robbie Desmond , Svetlichny Sergey
Published: July 2002

It is shown under CH that there exists a countably compact topological semigroup with two-sided cancellation which is not a topological group. “Wallace’s question” of 40 years standing is thus settled in the negative unless CH is explicitly denied. The example is a topological subsemigroup of an uncountable product of circle groups.
Boundaries of rotation sets for homeomorphisms of the 𝑛-torus
Authors: Swanson Richard , Walker Russell
Published: July 2002

We construct a

C
ω

C^\omega

diffeomorphism of the 3-torus whose rotation set is not closed. We prove that the rotation set of a homeomorphism of the

n
n

-torus contains the extreme points of its closed convex hull. Finally, we show that each pseudo-rotation set is closed for torus homeomorphisms.
Normal subgroups of 𝑃𝑆𝐿₂(𝑍[√-3])
Authors: Alperin Roger
Published: July 2002

We classify the normal subgroups of

P
S

L
2

(
Z
[

−
3

]
)

PSL_2(Z[\sqrt {-3}])

of index less than 960; they are all congruence subgroups.
The standard Cantor function is subadditive
Authors: Doboš Jozef
Published: July 2002

In this paper the subadditivity of the Cantor function

ϕ
:
[
0
,
1
]
→
[
0
,
1
]

\phi \colon [0,1]\to [0,1]

is proved.
Multiple path-valued conditional Yeh-Wiener integrals
Authors: Park Chull , Skoug David
Published: July 2002

In this paper we establish various results involving parallel line-valued conditional Yeh-Wiener integrals of the type

E
(
F
(
x
)

|

x
(

s
j

,
⋅
)
=

η
j

(
⋅
)

E(F(x)|x(s_j,\boldsymbol {\cdot })=\eta _j(\boldsymbol {\cdot })

,

j
=
1
,
…
,
n
)

j=1,\dotsc ,n)

where

0
>

s
1

>
⋯
>

s
n

0>s_1>\cdots >s_n

. We then develop a formula for converting these multiple path-valued conditional Yeh-Wiener integrals into ordinary Yeh-Wiener integrals. Next, conditional Yeh-Wiener integrals for functionals

F
F

of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x right-parenthesis equals exp left-brace integral Subscript 0 Superscript upper S Baseline integral Subscript 0 Superscript upper T Baseline phi left-parenthesis s comma t comma x left-parenthesis s comma t right-parenthesis right-parenthesis d t d s right-brace">

F
(
x
)
=
exp
⁡

{

∫
0
S

∫
0
T

ϕ
(
s
,
t
,
x
(
s
,
t
)
)

d
t

d
s
}

F(x)=\exp \left \{\int _0^S\int _0^T\phi (s,t,x(s,t))\,dt\,ds\right \}

\]
are evaluated by solving an appropriate Wiener integral equation. Finally, a Cameron-Martin translation theorem is obtained for these multiple path-valued conditional Yeh-Wiener integrals.
Some remarks on the operator of Foias and Williams
Authors: Petrovic Srdjan
Published: July 2002

In this paper we study the Foias-Williams operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis upper H Subscript g Baseline right-parenthesis equals Start 1 By 3 Matrix 1st Row 1st Column upper S Superscript asterisk Baseline 2nd Column a m p semicolon upper H Subscript g Baseline 0 3rd Column a m p semicolon upper S EndMatrix">

T
(

H

g

)
=

(

S

∗

a
m
p
;

H

g

 
0

a
m
p
;
S

)

\begin{equation*}T(H_{g})=\begin {pmatrix}S^{*} & H_{g} \ 0 & S \end{pmatrix} \end{equation*}

where

g
∈

L

∞

g\in L^{\infty }

, and

H

g

H_{g}

is a Hankel operator with symbol

g
g

. We exhibit a relationship between the similarity of

T
(

H

g

)

T(H_{g})

to a contraction and the rate of decay of

{

|

g

n

|

}

n
=
0

∞

\{|g_{n}|\}_{n=0}^{\infty }

, the absolute values of the Fourier coefficients of the symbol

g
g

.
Essential spectrum and 𝐿_2-solutions of one-dimensional Schrödinger operators
Authors: Remling Christian
Published: July 2002

In 1949, Hartman and Wintner showed that if the eigenvalue equations of a one-dimensional Schrödinger operator possess square integrable solutions, then the essential spectrum is nowhere dense. Furthermore, they conjectured that this statement could be improved and that under this condition the essential spectrum might always be void. This is shown to be false. It is proved that, on the contrary, every closed, nowhere dense set does occur as the essential spectrum of Schrödinger operators which satisfy the condition of existence of

L
2

L_2

-solutions. The proof of this theorem is based on inverse spectral theory.
The relationship between fragmentable spaces and class ℒ spaces
Authors: Yu Jian , Yuan Xian-Zhi
Published: July 2002

In this note, we show that each fragmentable space introduced by Jayne and Rogers in 1985 is of class

L

\mathcal {L}

which was introduced by Kenderov in 1984. Our example shows that a space which is of class

L

\mathcal {L}

may not be a fragmentable space.
Combinatorics of a certain ideal in the Segre coordinate ring
Authors: Brumatti Paulo , Gimenez Philippe , Simis Aron
Published: July 2002

We focus on a “fat” model of an ideal in the class of the canonical ideal of the Segre coordinate ring, looking at its Rees algebra and related arithmetical questions.
Single valuedness from weakly coercive Hamiltonians
Authors: Newcomb Richard , Su Jianzhong
Published: July 2002

We define a class of weakly coercive Hamiltonians and then demonstrate the single valuedness of the associated Hamilton-Jacobi operators (in the viscosity sense).
Extending finite group actions from surfaces to handlebodies
Authors: Reni Marco , Zimmermann Bruno
Published: July 2002

We show that every action of a finite dihedral group on a closed orientable surface

F

\mathcal F

extends to a 3-dimensional handlebody

V

\mathcal V

, with

∂

V

=

F

\partial \mathcal V=\mathcal F

. In the case of a finite abelian group

G
G

, we give necessary and sufficient conditions for a

G
G

-action on a surface to extend to a compact

3
3

-manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order

84
(
g
−
1
)

84(g-1)

of

P
S
L
(
2
,
27
)

PSL(2,27)

on a surface

F

\mathcal F

of genus

g
=
118

g=118

does not extend to any compact 3-manifold

M
M

with

∂
M
=

F

\partial M=\mathcal F

, thus resolving the only case of Hurwitz actions of type

P
S
L
(
2
,
q
)

PSL(2,q)

of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137–151).
The modular group algebra problem for metacyclic 𝑝-groups
Authors: Sandling Robert
Published: July 2002

It is shown that the isomorphism type of a metacyclic

p
p

-group is determined by its group algebra over the field

F
F

of

p
p

elements. This completes work of Bagiński. It is also shown that, if a

p
p

-group

G
G

has a cyclic commutator subgroup

G
′

G’

, then the order of the largest cyclic subgroup containing

G
′

G’

is determined by

F
G

FG

.
Heights of algebraic points lying on curves or hypersurfaces
Authors: Schmidt Wolfgang
Published: July 2002

Our first aim will be to give an explicit version of a generalization of the results of Zhang and Zagier on algebraic points

(
x
,
y
)

(x,y)

with

x
+
y
+
1
=
0

x+y+ 1 = 0

. Secondly, we will show that distinct algebraic points lying on a given curve of certain type can be distinguished in terms of some height functions. Thirdly, we will derive a bound for the number of points on such a curve whose heights are under a given bound and whose coordinates lie in a multiplicative group of given rank.
Infinite cyclic verbal subgroups of relatively free groups
Authors: Storozhev A.
Published: July 2002

We prove that there exist a relatively free group

H
H

and a word

w
(
x
,
y
)

w(x,y)

in two variables such that the verbal subgroup of

H
H

defined by

w
(
x
,
y
)

w(x,y)

is an infinite cyclic group whereas

w
(
x
,
y
)

w(x,y)

has only one nontrivial value in

H
H

.
Homotopy periodicity and coherence
Authors: Geoghegan Ross , Nicas Andrew
Published: July 2002

If

f
:

Z
→
Z

f\colon \,Z \rightarrow Z

is a periodic homotopy equivalence (

id

Z

≃

f

p

\operatorname {id}_{Z} \simeq f^{p}

) or a homotopy idempotent (

f
≃

f

2

f \simeq f^{2}

), the question arises whether this periodicity property can be achieved by a homotopy “compatible with”

f
f

. These coherence questions are answered.
Bands in lattices of operators
Authors: Huijsmans C. , Wickstead A.
Published: July 2002

We consider the lattice of regular operators on a Dedekind complete Banach lattice. We show that in general the projection onto a band generated by a lattice homomorphism need not be continuous and that the principal bands need not be closed for the operator norm. In fact it is possible to find a convergent sequence of operators all the members of which are disjoint from the limit.
Asymptotic analysis of Daubechies polynomials
Authors: Shen Jianhong , Strang Gilbert
Published: July 2002

To study wavelets and filter banks of high order, we begin with the zeros ofBp(y){\mathbf {B}}_{p}(y). This is the binomial series for(1−y)−p(1-y)^{-p}, truncated afterppterms. Its zeros give thep−1p-1zeros of the Daubechies filter inside the unit circle, byz+z−1=2−4yz+z^{-1} = 2-4y. The filter hasppadditional zeros atz=−1z = -1, and this construction makes it orthogonal and maximally flat. The dilation equation leads to orthogonal wavelets withppvanishing moments. Symmetric biorthogonal wavelets (generally better in image compression) come similarly from a subset of the zeros ofBp(y){\mathbf {B}}_{p}(y). We study the asymptotic behavior of these zeros. Matlab shows a remarkable plot forp=70p = 70. The zeros approach a limiting curve|4y(1−y)|=1|4y(1-y)| = 1in the complex plane, which is the circle|z−z−1|=2|z-z^{-1}| = 2. All zeros have|y|≤1/2|y| \le 1/2, and the rightmost zeros approachy=1/2y = 1/2(corresponding toz=±iz= \pm i) with speedp−1/2p^{- 1/2}. The curve<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue 4 y left-parenthesis 1 minus y right-parenthesis EndAbsoluteValue equals left-parenthesis 4 pi p right-parenthesis Superscript 1 slash 2 p Baseline StartAbsoluteValue 1 minus 2 y EndAbsoluteValue Superscript 1 slash p">|4y(1−y)|=(4πp)1/2p|1−2y|1/p|4y(1-y)| = {(4 \pi p)}^{{1}/{2p}} \, |1-2y|^{ 1/p}gives a very accurate approximation for finitepp. The wide dynamic range in the coefficients ofBp(y){\mathbf {B}}_{p}(y)makes the zeros difficult to compute for largepp. Rescalingyyby44allows us to reachp=80p = 80by standard codes.
On the boundary rigidity phenomenon for automorphisms of domains in ℂⁿ
Authors: Coupet Bernard , Sukhov Alexander
Published: July 2002

We prove that a piecewise smoothly bounded strictly pseudoconvex domain with a non–compact automorphism group is biholomorphic to the ball. A boundary version of the Schwarz lemma for automorphisms of such a domain is settled.
Vassiliev invariants of type two for a link
Authors: Murakami Hitoshi
Published: July 2002

We show that any type two Vassiliev invariant of a link can be expressed as a linear combination of the second coefficients of the Conway polynomials of its components and a quadratic expression of linking numbers.
Compact flat manifolds with holonomy group 𝐙₂⊕𝐙₂
Authors: Rossetti J. , Tirao P.
Published: July 2002

In this paper we construct a family of compact flat manifolds, for all dimensions

n
≥
3

n\ge 3

, with holonomy group isomorphic to

Z

2
2

\mathbf {Z}_2^2

and first Betti number zero.
On a conjecture of S. Reich
Authors: Daffer Peter , Kaneko Hideaki , Li Wu
Published: July 2002

Simeon Reich (1974) proved that the fixed point theorem for single-valued mappings proved by Boyd and Wong can be generalized to multivalued mappings which map points into compact sets. He then asked (1983) whether his theorem can be extended to multivalued mappings whose range consists of bounded closed sets. In this note, we provide an affirmative answer for a certain subclass of Boyd-Wong contractive mappings.
Sharp Hölder estimates for {̄∂} on ellipsoids and their complements via order of contact
Authors: Fleron Julian
Published: July 2002

We generalize the results of Range and Diederich et al., finding Hölder estimates for the solution of the Cauchy-Riemann equations for higher order forms on ellipsoids. We prove a dual result near the concave boundaries of complemented complex ellipsoids. In all cases the Hölder exponents are characterized in terms of the order of contact of the boundary of the domain with complex linear spaces of the appropriate dimension. Optimality is demonstrated in the convex settings, and for

(
0
,
1
)

(0,1)

forms in the concave setting. Partial results are given for complemented real ellipsoids and a method for demonstrating optimality of Hortmann’s result on complemented strictly pseudoconvex domains is given for

(
0
,
1
)

(0,1)

forms.
On compact connected sets in Banach spaces
Authors: de Blasi F. , Myjak J.
Published: July 2002

Let

E

\mathbf {E}

be a separable strictly convex Banach space of dimension at least 2. It is shown that there exists a nonempty compact connected set

X
⊂

E

X \subset \mathbf {E}

such that the nearest point mapping

p
X

:

E

→

2

E

p_X: \mathbf {E}\to 2^{ \mathbf {E}}

is not single valued on a set of points dense in

E

\mathbf {E}

. Furthermore, it is proved that most (in the sense of the Baire category) nonempty compact connected sets

X
⊂

E

X\subset \mathbf {E}

have the above property. Similar results hold for the furthest point mapping.
On rigidity of affine surfaces
Authors: Opozda Barbara
Published: July 2002

Rigidity of nondegenerate Blaschke surfaces in

R

3

\mathbf {R}^{3}

is studied. The rigidity criteria are given in terms of

∇
R

\nabla R

, where

R
R

is the curvature of the Blaschke connection

∇
\nabla

. If the rank of

∇
R

\nabla R

is 2, then the surface is rigid. If

∇
R
=
0

\nabla R=0

, it is nonrigid. In the case where the rank of

∇
R

\nabla R

is 1 there are both rigid and nonrigid surfaces. This case is discussed for various types of surfaces.
Boundary values of holomorphic semigroups
Authors: Arendt Wolfgang , El Mennaoui Omar , Hieber Matthias
Published: July 2002

The concept of boundary values of holomorphic semigroups is used to give a new proof of a result due to Hörmander, saying that the operator

i
Δ

i\Delta

generates a

C
0

C_0

-semigroup on

L
p

(

R

N

)

L^p(\mathbb R^N)

if and only if

p
=
2

p=2

. Using a recent result on Laplace transforms by Prüss one obtains by this theory also a new proof of the classical characterization theorem of holomorphic semigroups.
Wald’s equation and asymptotic bias of randomly stopped 𝑈-statistics
Authors: de la Peña Victor , Lai Tze
Published: July 2002

In this paper we make use of decoupling arguments and martingale inequalities to extend Wald’s equation for sample sums to randomly stopped de-normalized

U
U

-statistics. We also apply this result in conjunction with nonlinear renewal theory to obtain asymptotic expansions for the means of normalized

U
U

-statistics from sequential samples.
Pelczynski’s property (V*) for symmetric operator spaces
Authors: Randrianantoanina Narcisse
Published: July 2002

We show that if a rearrangement invariant Banach function space

E
E

on the positive semi-axis contains no subspace isomorphic to

c
0

c_0

then the corresponding space

E
(

M

)

E(\mathcal {M})

of

τ
\tau

-measurable operators, affiliated with an arbitrary semifinite von-Neumann algebra

M

\mathcal {M}

equipped with a distinguished faithful, normal and semifinite trace

τ
\tau

, has Pełczyński’s property (V*).
On quasispectral maximal subspaces of a class of Volterra-type operators
Authors: Drnovsek Roman
Published: July 2002

The concept of quasispectral maximal subspaces for quasinilpotent (but not nilpotent) operators was introduced by M. Omladič in 1984. As an application a class of quasinilpotent operators on

L
p

L^p

-spaces, close to the Volterra kernel operator, was studied. In the present Banach function space setting we determine all quasispectral maximal subspaces of analogues of such operators and prove that these subspaces are all the invariant bands. An example is given showing that (in general) they are not all the closed, invariant ideals of the operator.
Hypersurfaces in a sphere with constant mean curvature
Authors: Hou Zhong
Published: July 2002

Let

M
n

M^n

be a closed hypersurface of constant mean curvature immersed in the unit sphere

S

n
+
1

S^{n+1}

. Denote by

S
S

the square of the length of its second fundamental form. If

S
>
2

n
−
1

S>2\sqrt {n-1}

,

M
M

is a small hypersphere in

S

n
+
1

S^{n+1}

. We also characterize all

M
n

M^n

with

S
=
2

n
−
1

S=2\sqrt {n-1}

.
On perfect simple-injective rings
Authors: Nicholson W. , Yousif M.
Published: July 2002

Harada calls a ring

R
R

right simple-injective if every

R
R

-homomorphism with simple image from a right ideal of

R
R

to

R
R

is given by left multiplication by an element of

R
R

. In this paper we show that every left perfect, left and right simple-injective ring is quasi-Frobenius, extending a well known result of Osofsky on self-injective rings. It is also shown that if

R
R

is left perfect and right simple-injective, then

R
R

is quasi-Frobenius if and only if the second socle of

R
R

is countably generated as a left

R
R

-module, extending many recent results on self-injective rings. Examples are given to show that our results are non-trivial extensions of those on self-injective rings.
The structure of functions satisfying the law of large numbers in a class of locally convex spaces
Authors: Stolz Robert
Published: July 2002

For each function

f
f

that satisfies the law of large numbers with values in a certain class of locally convex spaces with the Radon-Nikodym property the following decomposition holds:

f
=

f
1

+

f
2

f=f_1+f_2

, where

f
1

f_1

is integrable by seminorm, and

f
2

f_2

is a Pettis integrable function which is scalarly 0.
Conjugate points in the Bott-Virasoro group and the KdV equation
Authors: Misiolek Gerard
Published: July 2002

We study the geometry of a right invariant metric on a central extension

D

^

(

S
1

)

\widehat {\mathcal {D}}(S^1)

of the diffeomorphism group of a circle (the Bott-Virasoro group) introduced by Ovsienko and Khesin. We obtain an expression for the curvature tensor of this metric and apply it to find conjugate points in

D

^

(

S
1

)

\widehat {\mathcal {D}}(S^1)

.
On the concept of 𝑃𝑖₁¹-completeness
Authors: Kechris Alexander
Published: July 2002

It is shown that two natural notions of completeness for co-analytic sets in Polish spaces, one in terms of continuous reductions and the other in terms of Borel reductions, coincide. The proof uses methods of effective descriptive set theory.
Non-existence and uniqueness results for boundary value problems for Yang-Mills connections
Authors: Isobe Takeshi
Published: July 2002

We show uniqueness results for the Dirichlet problem for Yang-Mills connections defined in

n
n

-dimensional (

n
≥
4

n\ge 4

) star-shaped domains with flat boundary values. This result also shows the non-existence result for the Dirichlet problem in dimension 4, since in 4-dimension, there exist countably many connected components of connections with prescribed Dirichlet boundary value. We also show non-existence results for the Neumann problem. Examples of non-minimal Yang-Mills connections for the Dirichlet and the Neumann problems are also given.
Isomorphisms of row and column finite matrix rings
Authors: Haefner J. , del Río A. , Simón J.
Published: July 2002

This paper investigates the ring-theoretic similarities and the categorical dissimilarities between the ring

R
F
M
(
R
)

RFM(R)

of row finite matrices and the ring

R
C
F
M
(
R
)

RCFM(R)

of row and column finite matrices. For example, we prove that two rings

R
R

and

S
S

are Morita equivalent if and only if the rings

R
C
F
M
(
R
)

RCFM(R)

and

R
C
F
M
(
S
)

RCFM(S)

are isomorphic. This resembles the result of V. P. Camillo (1984) for

R
F
M
(
R
)

RFM(R)

. We also show that the Picard groups of

R
F
M
(
R
)

RFM(R)

and

R
C
F
M
(
R
)

RCFM(R)

are isomorphic, even though the rings

R
F
M
(
R
)

RFM(R)

and

R
C
F
M
(
R
)

RCFM(R)

are never Morita equivalent.
Note on Clark’s theorem for 𝑝-adic convergence
Authors: Setoyanagi Minoru
Published: July 2002

We must read Clark’s statement under the hypothesis that the negative of each zero of the indicial polynomial is non-Liouville. In this note we shall give the example for which under the original hypothesis the statement does not hold.
A note on the Chern-Simons invariant of hyperbolic 3-manifolds
Authors: Ouyang Mingqing
Published: July 2002

In this note we study how the Chern-Simons invariant behaves when two hyperbolic 3-manifolds are glued together along incompressible thrice-punctured spheres. Such an operation produces many hyperbolic 3-manifolds with different numbers of cusps sharing the same volume and the same Chern-Simons invariant. The results in this note, combined with those of Meyerhoff and Ruberman, give an algorithm for determining the unknown constant in Neumann’s simplicial formula for the Chern-Simons invariant of hyperbolic 3-manifolds.
The classification of complete Lie algebras with commutative nilpotent radical
Authors: Cuipo Jiang , Daoji Meng
Published: July 2002

The work in this paper is a continuation of an earlier paper of the second author (Acta Math. 34 (1991), 191–202). We discuss the properties of finite-dimensional complete Lie algebras with abelian nilpotent radical over the complex field

C

\mathbf {C}

. We solve the problems of isomorphism, classification and realization of complete Lie algebras with commutative nilpotent radical.
Derivations mapping into the socle, II
Authors: Brešar Matej
Published: July 2002

Let

A

\mathcal A

be a unital Banach algebra. Several characterizations of continuous derivations of

A

\mathcal A

having only elements with finite spectrum in their ranges are obtained. In particular, if

A

\mathcal A

is semisimple, then such derivations map

A

\mathcal A

into its socle.
Actions of compact quantum groups on 𝐶*-algebras
Authors: Marciniak Marcin
Published: July 2002

In this paper we investigate a structure of the fixed point algebra under an action of compact matrix quantum group on a

C
∗

C^*

-algebra

B

\mathcal {B}

. We also show that the categories of

C

\mathcal C

-comodules in

B

\mathcal B

and inner endomorphisms restricted to the fixed point algebra coincide when the relative commutant of the fixed point algebra is trivial. Next we show a version of the Tannaka duality theorem for twisted unitary groups.
Averages of operators and their positivity
Authors: Nagisa Masaru , Wada Shuhei
Published: July 2002

Let

T
T

be a bounded linear operator on a Hilbert space. We prove that

T
T

is positive, if there exists a positive integer

N
N

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper I minus StartFraction 1 Over upper N plus 1 EndFraction sigma-summation Underscript i equals k Overscript k plus upper N Endscripts upper T Superscript i Baseline double-vertical-bar comma double-vertical-bar upper I minus StartFraction 1 Over upper N plus 2 EndFraction sigma-summation Underscript i equals k Overscript k plus upper N plus 1 Endscripts upper T Superscript i Baseline double-vertical-bar less-than-or-equal-to 1">

‖
I
−

1

N
+
1

∑

i
=
k

k
+
N

T

i

‖
,
‖
I
−

1

N
+
2

∑

i
=
k

k
+
N
+
1

T

i

‖
≤
1

\|I- {\frac {1 }{N+1 }}\sum \limits _{i=k}^{k+N} T^{i} \|, \|I- {\frac {1 }{N+2 }}\sum \limits _{i=k}^{k+N+1} T^{i} \| \le 1

\]
for any non-negative integer

k
k

. For several commuting operators, we can extend this result and get the similar statement.
On the Taylor spectrum of left-right multipliers
Authors: Harte Robin , Hernández Carlos
Published: July 2002

The Taylor spectrum of a mixed pair of multiplication operators is determined on an ultraprime Banach algebra.
Properties of subgenerators of 𝐶-regularized semigroups
Authors: Wang Sheng
Published: July 2002

We introduce two operations

∧
\wedge

,

∨
\vee

in the set

G

{\mathcal {G}}

of subgenerators of a given

C
C

- regularized semigroup and prove that

G

{\mathcal {G}}

is a complete partially ordered lattice with respect to

∧
\wedge

,

∨
\vee

and the operator inclusion

⊆
\subseteq

. Also presented are some other properties and examples for

G

.

{\mathcal {G}} .
A weak-type inequality of subharmonic functions
Authors: Choi Changsun
Published: July 2002

We prove the weak-type inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda mu left-parenthesis u plus StartAbsoluteValue v EndAbsoluteValue greater-than-or-equal-to lamda right-parenthesis less-than-or-equal-to left-parenthesis alpha plus 2 right-parenthesis integral Underscript partial-differential upper D Endscripts u d mu">

λ
μ
(
u
+

|

v

|

≥
λ
)
≤
(
α
+
2
)

∫

∂
D

u

d
μ

\lambda \mu (u+|v|\ge \lambda )\le (\alpha +2) \int _{\partial D}u\,d\mu

,

λ
>
0

\lambda >0

, between a non-negative subharmonic function

u
u

and an

H

\mathbb {H}

-valued smooth function

v
v

, defined on an open set containing the closure of a bounded domain

D
D

in a Euclidean space

R

n

\mathbb R^n

, satisfying

|

v
(
0
)

|

≤
u
(
0
)

|v(0)|\le u(0)

,

|

∇
v

|

≤

|

∇
u

|

|\nabla v|\le |\nabla u|

and

|

Δ
v

|

≤
α
Δ
u

|\Delta v|\le \alpha \Delta u

, where

α
≥
0

\alpha \ge 0

is a constant. Here

μ
\mu

is the harmonic measure on

∂
D

\partial D

with respect to 0. This inequality extends Burkholder’s inequality in which

α
=
1

\alpha =1

and

H

=

R

ν

\mathbb {H}=\mathbb {R}^\nu

, a Euclidean space.
The 6-property for simplicial complexes and a combinatorial Cartan-Hadamard theorem for manifolds
Authors: Corson J. , Trace B.
Published: July 2002

The 6-property for 2-dimensional simplicial complexes is the condition that every nontrivial circuit in the link of a vertex has length greater than or equal to six. If a compact

n
n

-manifold

M
M

has a 2-dimensional spine with the 6-property, then we show that the interior of

M
M

is covered by euclidean

n
n

-space. In dimension

n
=
3

n=3

, we show further that such a 3-manifold is Haken.
A bound for the nilpotency of a group of self homotopy equivalences
Authors: Félix Yves , Murillo Aniceto
Published: July 2002

Let

E

Ω

(
X
)

\mathcal {E}_\Omega (X)

be the group of homotopy classes of self-homotopy equivalences of

X
X

such that

Ω
f
≃
1

d

Ω
X

\Omega f\simeq 1d_{\Omega X}

. We prove that

E

Ω

(
X
)

\mathcal {E}_\Omega (X)

is a nilpotent group and that

nil
⁡

E

Ω

(
X
)
≤
cat
⁡
(
X
)
−
1

\operatorname {nil} \mathcal {E}_{\Omega }(X)\le \operatorname {cat}(X)-1

.
Inradius and integral means for Green’s functions and conformal mappings
Authors: Bañuelos Rodrigo , Carroll Tom , Housworth Elizabeth
Published: July 2002

Let

D
D

be a convex planar domain of finite inradius

R
D

R_D

. Fix the point

0
∈
D

0\in D

and suppose the disk centered at

0
0

and radius

R
D

R_D

is contained in

D
D

. Under these assumptions we prove that the symmetric decreasing rearrangement in

θ
\theta

of the Green’s function

G

D

(
0
,
ρ

e

i
θ

)

G_{D}(0, \rho e^{i\theta })

, for fixed

ρ
\rho

, is dominated by the corresponding quantity for the strip of width

2

R
D

2R_D

. From this, sharp integral mean inequalities for the Green’s function and the conformal map from the disk to the domain follow. The proof is geometric, relying on comparison estimates for the hyperbolic metric of

D
D

with that of the strip and a careful analysis of geodesics.
A note on Kamenev type theorems for second order matrix differential systems
Authors: Meng Fanwei , Wang Jizhong , Zheng Zhaowen
Published: July 2002

Some oscillation criteria are given for the second order matrix differential system

Y
+
Q
(
t
)
Y
=
0

Y+Q(t) Y=0

, where

Y
Y

and

Q
Q

are

n
×
n

n\times n

real continuous matrix functions with

Q
(
t
)

Q(t)

symmetric,

t
∈
[

t
0

,
∞
)

t\in [t_0,\infty )

. These results improve oscillation criteria recently discovered by Erbe, Kong and Ruan by using a generalized Riccati transformation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V left-parenthesis t right-parenthesis equals a left-parenthesis t right-parenthesis left-brace upper Y prime left-parenthesis t right-parenthesis upper Y Superscript negative 1 Baseline left-parenthesis t right-parenthesis plus f left-parenthesis t right-parenthesis upper I right-brace">

V
(
t
)
=
a
(
t
)
{

Y
′

(
t
)

Y

−
1

(
t
)
+
f
(
t
)
I
}

V(t)=a(t)\{Y’(t) Y^{-1}(t) +f(t)I\}

, where

I
I

is the

n
×
n

n\times n

identity matrix,

f
∈

C
1

f\in C^1

is a given function on

[

t
0

,
∞
)

[t_0,\infty )

and

a
(
t
)
=
exp
⁡
{
−
2

∫
t

f
(
s
)

d
s
}

a(t)=\exp \{-2 \int ^t f(s)\,ds\}

.
Hankel operators on the Bergman space of the unit ball
Authors: Nowak Maria
Published: July 2002

We characterize the bounded holomorphic functions

f
,
g

f, g

in the unit ball of

C

n

\mathbb {C}^{n}

for which the operator

H

g
¯

∗

H

f
¯

H^{*}_{\bar g} H_{\bar f}

is compact. For

n
=
1

n=1

the result was obtained by Axler and Gorkin in 1988 and by Zheng in 1989.
An application of the Lefschetz fixed-point theorem to non-convex differential inclusions on manifolds
Authors: Domachowski Stanisław , Pruszko Tadeusz
Published: July 2002

A selector theorem for non-convex orientor fields on closed manifolds is given and the Lefschetz fixed point theorem is used to establish an existence result for these ones.
Rank one lattices whose parabolic isometries have no rotational part
Authors: Hummel Christoph
Published: July 2002

We prove a result on certain finite index subgroups of rank one lattices which is motivated by cusp closing constructions.
Convex composite functions in Banach spaces and the primal lower-nice property
Authors: Combari C. , Alaoui A. , Levy A. , Poliquin R. , Thibault L.
Published: July 2002

Primal lower-nice functions defined on Hilbert spaces provide examples of functions that are “integrable” (i.e. of functions that are determined up to an additive constant by their subgradients). The class of primal lower-nice functions contains all convex and lower-

C
2

C^2

functions. In finite dimensions the class of primal lower-nice functions also contains the composition of a convex function with a

C
2

C^2

mapping under a constraint qualification. In Banach spaces certain convex composite functions were known to be primal lower-nice (e.g. a convex function had to be continuous relative to its domain). In this paper we weaken the assumptions and provide new examples of convex composite functions defined on a Banach space with the primal lower-nice property. One consequence of our results is the identification of new examples of integrable functions on Hilbert spaces.
Smooth rank one perturbations of selfadjoint operators
Authors: Hassi S. , de Snoo H. , Willemsma A.
Published: July 2002

Let

A
A

be a selfadjoint operator in a Hilbert space

H

\mathfrak {H}

with inner product

[
⋅
,
⋅
]

[\cdot ,\cdot ]

. The rank one perturbations of

A
A

have the form

A
+
τ
[
⋅
,
ω
]
ω

A+\tau [\cdot ,\omega ] \omega

,

τ
∈

R

\tau \in \mathbb {R}

, for some element

ω
∈

H

\omega \in \mathfrak {H}

. In this paper we consider smooth perturbations, i.e. we consider

ω
∈
dom
⁡

|

A

|

k

/

2

\omega \in \operatorname {dom} |A|^{k/2}

for some

k
∈

N

∪
{
0
}

k \in \mathbb {N} \cup \{0\}

. Function-theoretic properties of their so-called

Q
Q

-functions and operator-theoretic consequences will be studied.
A note on sequences lying in the range of a vector measure valued in the bidual
Authors: Marchena Begoña , Piñeiro Cándido
Published: July 2002

Let

X
X

be a Banach space. It is unknown if every subset

A
A

of

X
X

lying in the range of an

X

∗
∗

X^{**}

-valued measure is actually contained in the range of an

X
X

-valued measure. In this paper we solve this problem in the case when we consider only vector measures of bounded variation.
Band-sums are ribbon concordant to the connected sum
Authors: Miyazaki Katura
Published: July 2002

We show that an arbitrary band-connected sum of two or more knots are ribbon concordant to the connected sum of these knots. As an application we consider which knot can be a nontrivial band-connected sum.
Rings of continuous functions and the branch set of a covering
Authors: Mulero M.
Published: July 2002

This paper gives a characterization of the branch set of a finite coveringX→SX\to Sof a topological spaceSS, by means of finiteC(S)C(S)-subalgebrasAAofC(X)C(X)that separate points inXXand the moduleΩA/C(S)\Omega _{A/C(S)}of its Kähler differentials.
A limiting example for the local “fuzzy” sum rule in nonsmooth analysis
Authors: Vanderwerff Jon , Zhu Qiji
Published: July 2002

We show that assuming all the summand functions to be lower semicontinuous is not sufficient to ensure a (strong) fuzzy sum rule for subdifferentials in any infinite dimensional Banach space. From this we deduce that additional assumptions are also needed on functions for chain rules, multiplier rules for constrained minimization problems and Clarke-Ledyaev type mean value inequalities in the infinite dimensional setting.
Unbounded domains of normality
Authors: Anderson J. , Hinkkanen A.
Published: July 2002

Let

f
f

be a transcendental entire function of order less than 1/2. We introduce the method of “self-sustaining spread” to study the components of the set of normality of such a function. We give a new proof of the fact that any preperiodic or periodic component of the set of normality of

f
f

is bounded. We obtain the same conclusion for a wandering domain if the growth rate of

f
f

is never too small.
Completely lexsegment ideals
Authors: De Negri Emanuela , Herzog Jürgen
Published: July 2002

In this paper we study ideals which are generated by lexsegments of monomials. In contrast to initial lexsegments, the shadow of an arbitrary lexsegment is in general not again a lexsegment. An ideal generated by a lexsegment is called completely lexsegment, if all iterated shadows of the set of generators are lexsegments. We characterize all completely lexsegment ideals and describe cases in which they have a linear resolution. We also prove a persistence theorem which states that all iterated shadows of a lexsegment are again lexsegments if the first shadow has this property.
On van Mill’s example of a normed 𝑋 with 𝑋̸≈𝑋×ℝ
Authors: Marciszewski Witold
Published: July 2002

In 1987 van Mill constructed an infinite-dimensional normed space

X
X

which is not homeomorphic with the product

X
×

R

X\times \mathbb {R}

. We give a short proof of this property of van Mill’s example.
Countable linear transformations are clean
Authors: Nicholson W. , Varadarajan K.
Published: July 2002

It is shown that every linear transformation on a vector space of countable dimension is the sum of a unit and an idempotent.
The intersection of three spheres in a sphere and a new application of the Sato-Levine invariant
Authors: Ogasa Eiji
Published: July 2002

Take transverse immersions

f
:

S

1

4

⨿

f:S^{4}_{1}\amalg

S

2

4

⨿

S^{4}_{2}\amalg

S

3

4

↬

S

6

S^{4}_{3}\looparrowright S^{6}

such that (1)

f
|

S

i

4

f\vert S^{4}_{i}

is an embedding, (2)

f
(

S

i

4

)
∩
f
(

S

j

4

)
≠
∅

f(S^{4}_{i})\cap f(S^{4}_{j})\neq \varnothing

and

f
(

S

i

4

)
∩
f
(

S

j

4

)

f(S^{4}_{i})\cap f(S^{4}_{j})

is connected, and (3)

f
(

S

1

4

)
∩
f
(

S

2

4

)
∩
f
(

S

3

4

)

f(S^{4}_{1})\cap f(S^{4}_{2})\cap f(S^{4}_{3})

=
∅

=\varnothing

. Then we obtain three surface-links

L

i

L_{i}

= (
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript negative 1 Baseline left-parenthesis f left-parenthesis upper S Subscript i Superscript 4 Baseline right-parenthesis intersection f left-parenthesis upper S Subscript j Superscript 4 Baseline right-parenthesis right-parenthesis">

f

−
1

(
f
(

S

i

4

)
∩
f
(

S

j

4

)
)

f^{-1}(f(S^{4}_{i})\cap f(S^{4}_{j}))

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript negative 1 Baseline left-parenthesis f left-parenthesis upper S Subscript i Superscript 4 Baseline right-parenthesis intersection f left-parenthesis upper S Subscript k Superscript 4 Baseline right-parenthesis right-parenthesis">

f

−
1

(
f
(

S

i

4

)
∩
f
(

S

k

4

)
)

f^{-1}(f(S^{4}_{i})\cap f(S^{4}_{k}))

) in

S

i

4

S^{4}_{i}

, where

(
i
,
j
,
k
)

(i,j,k)

=(1,2,3), (2,3,1), (3,1,2). We prove that, we have the equality

β
(

L

1

)
+

\beta (L_{1})+

β
(

L

2

)
+

\beta (L_{2})+

β
(

L

3

)
=
0
,

\beta (L_{3})=0,

where

β
(

L

i

)

\beta (L_{i})

is the Sato-Levine invariant of

L

i

L_{i}

, if all

L

i

L_{i}

are semi-boundary links.
Sampling sets for Hardy spaces of the disk
Authors: Thomas Pascal
Published: July 2002

We propose two possible definitions for the notion of a sampling sequence (or set) for Hardy spaces of the disk. The first one is inspired by recent work of Bruna, Nicolau, and Øyma about interpolating sequences in the same spaces, and it yields sampling sets which do not depend on the value of

p
p

and correspond to the result proved for bounded functions (

p
=
∞

p=\infty

) by Brown, Shields and Zeller. The second notion, while formally closer to the one used for weighted Bergman spaces, leads to trivial situations only, but raises a possibly interesting problem.
On a class of Riesz-Fischer sequences
Authors: Young Robert
Published: July 2002

In this note, we give necessary and sufficient conditions for a system of complex exponentials

{

e

i

λ
n

t

}

\{e^{i\lambda _nt}\}

to form a Riesz-Fischer sequence in

L
2

(
−
A
,
A
)

L^2(-A,A)

for every positive number

A
A

. The result provides a significant strengthening of the sufficient conditions recently stated by R. M. Reid (1995).
Vector bundles with holomorphic connection over a projective manifold with tangent bundle of nonnegative degree
Authors: Biswas Indranil
Published: July 2002

For a projective manifold whose tangent bundle is of nonnegative degree, a vector bundle on it with a holomorphic connection actually admits a compatible flat holomorphic connection, if the manifold satisfies certain conditions. The conditions in question are on the Harder-Narasimhan filtration of the tangent bundle, and on the Neron-Severi group.
Products of constant curvature spaces with a Brownian independence property
Authors: Hughes H.
Published: July 2002

The time and place Brownian motion on the product of constant curvature spaces first exits a normal ball of radius

ϵ
\epsilon

centered at the starting point of the Brownian motion are considered. The asymptotic expansions, as

ϵ
\epsilon

decreases to zero, for joint moments of the first exit time and place random variables are computed with error

O
(

ϵ

10

)

O(\epsilon ^{10})

. It is shown that the first exit time and place are independent random variables only if each factor space is locally flat or of dimension three.
The Baire category theorem and the evasion number
Authors: Kada Masaru
Published: July 2002

In this paper we prove that

e

≤

cov

(

M

)

\mathfrak e\leq {\operatorname {{\textsf {cov}}}}(\mathcal M)

where

e

\mathfrak e

is the evasion number defined by Blass. This answers negatively a question asked by Brendle and Shelah.
A new proof of the Solomon-Tits theorem
Authors: Ku Chao
Published: July 2002

We give a new proof of the Solomon-Tits Theorem which asserts that the Tits building of a finite group of Lie type has the homotopy type of a bouquet of spheres.
Half-space theorems for mean curvature one surfaces in hyperbolic space
Authors: Rodriguez Lucio , Rosenberg Harold
Published: July 2002

We give conditions which oblige properly embedded constant mean curvature one surfaces in hyperbolic 3-space to intersect. Our results are inspired by the theorem that two disjoint properly immersed minimal surfaces in

R

3

\mathbf {R}^3

must be planes.
Variance and clustering
Authors: Yatracos Yannis
Published: July 2002

A measure of dissimilarity of real observations is introduced that is used to identify clusters and is based on “gaps”, and also on averages of selected subgroups of the observations. This measure is surprisingly associated with the sample variance in a way that leads to a new identity and interpretation of the notion of variance.
A characterization of uniform continuity for Volterra equations in Hilbert spaces
Authors: Lizama Carlos
Published: July 2002

We show that the norm continuity of the resolvent for a Volterra equation of scalar type is equivalent to the decay to zero of a holomorphic operator family along some imaginary axis.
An index theorem for Toeplitz operators on totally ordered groups
Authors: Adji Sriwulan , Raeburn Iain , Ströh Anton
Published: July 2002

We show that for every totally ordered group

Γ
\Gamma

and invertible function

f
∈
C
(

Γ
^

)

f\in C(\widehat \Gamma )

which does not have a logarithm, there is a representation in which the Toeplitz operator

T
f

T_f

is a Breuer-Fredholm operator with nonzero index; this representation is the GNS-representation associated to a natural unbounded trace on the Toeplitz algebra

T

(
Γ
)

\mathcal T(\Gamma )

.
High order moments of character sums
Authors: Cochrane Todd , Zheng Zhiyong
Published: July 2002

We establish the upper bound
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over p minus 1 EndFraction sigma-summation Underscript chi not-equals chi Subscript o Baseline Endscripts StartAbsoluteValue sigma-summation Underscript x equals a plus 1 Overscript a plus upper B Endscripts chi left-parenthesis x right-parenthesis EndAbsoluteValue Superscript 2 k Baseline much-less-than Subscript epsilon comma k Baseline p Superscript k minus 1 plus epsilon Baseline plus upper B Superscript k Baseline p Superscript epsilon Baseline comma">

1

p
−
1

∑

χ
≠

χ

o

|

∑

x
=
a
+
1

a
+
B

χ
(
x
)

|

2
k

≪

ϵ
,
k

p

k
−
1
+
ϵ

+

B

k

p

ϵ

,

\begin{equation*}\frac {1}{p-1} \sum _{\chi \ne \chi _{o}}\big | \sum _{x=a+1}^{a+B} \chi (x) \big |^{2k} \ll _{\epsilon ,k} p^{k-1 +\epsilon } + B^{k} p^{\epsilon }, \end{equation*}

with

p
p

a prime and

k
k

any positive integer, the sum being over all nonprincipal multiplicative characters

(
mod

p
)

\pmod p

.
Stability of the Wulff shape
Authors: Palmer Bennett
Published: July 2002

We consider the functional of a hypersurface, given by a convex elliptic integrand with a volume constraint. We show that, up to homothety and translation, the only closed, oriented, stable critical point is the Wulff shape.
Vectors of minimal norm
Authors: Spalsbury Angela
Published: July 2002

We characterize the minimal vectors (or extremal vectors) for the operator of multiplication by

(
z
−
1
)

(z-1)

on

H
2

.

H^2.

We then give an application to infinite systems of equations.
An application of Schauder’s fixed point theorem with respect to higher order BVPs
Authors: Wong Fu-Hsiang
Published: July 2002

We shall provide conditions on the functionf(t,u1,⋯,un−1)f(t,u_{1},\cdots , u_{n-1}). The higher order boundary value problem<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis left-brace BVP right-brace right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row left-parenthesis upper E right-parenthesis u Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis t right-parenthesis plus f left-parenthesis t comma u left-parenthesis t right-parenthesis comma u Superscript left-parenthesis 1 right-parenthesis Baseline left-parenthesis t right-parenthesis comma midline-horizontal-ellipsis comma u Superscript left-parenthesis n minus 2 right-parenthesis Baseline left-parenthesis t right-parenthesis right-parenthesis equals 0 normal f normal o normal r t element-of left-parenthesis 0 comma 1 right-parenthesis normal a normal n normal d n greater-than-or-equal-to 2 comma left-parenthesis upper B upper C right-parenthesis StartLayout Enlarged left-brace 1st Row u Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis 0 right-parenthesis equals 0 comma 0 less-than-or-equal-to i less-than-or-equal-to n minus 3 comma alpha u Superscript left-parenthesis n minus 2 right-parenthesis Baseline left-parenthesis 0 right-parenthesis minus beta u Superscript left-parenthesis n minus 1 right-parenthesis Baseline left-parenthesis 0 right-parenthesis equals 0 comma gamma u Superscript left-parenthesis n minus 2 right-parenthesis Baseline left-parenthesis 1 right-parenthesis plus delta u Superscript left-parenthesis n minus 1 right-parenthesis Baseline left-parenthesis 1 right-parenthesis equals 0 EndLayout EndLayout EndLayout">({BVP}){(E)  u(n)(t)+f(t,u(t),u(1)(t),⋯,u(n−2)(t))=0      for      t∈(0,1)    and      n≥2, (BC)  {u(i)(0)=0,     0≤i≤n−3, αu(n−2)(0)−βu(n−1)(0)=0, γu(n−2)(1)+δu(n−1)(1)=0\begin{equation*}\begin {cases}(E)~~ u^{(n)}(t)+ f(t, u(t),u^{(1)}(t),\cdots ,u^{(n-2)}(t))=0~~~~~\mathrm {~for~}~~~~~t\in (0,1)~~~~\mathrm {and}~~~~~~n\ge 2,\ (BC)~~ \begin {cases}u^{(i)}(0)=0,~~~~~0\le i \le n-3,\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\ \gamma u^{(n-2)}(1)+\delta u^{(n-1)}(1)=0\end{cases} \end{cases} \tag {{BVP}}\end{equation*}has at least one solution.
Class number parity for cyclotomic fields
Authors: Yoshino Ken-Ichi
Published: July 2002

We give a simple criterion for the parity of the class number of the cyclotomic field.
On the graph convergence of subdifferentials of convex functions
Authors: Combari C. , Thibault L.
Published: July 2002

This paper provides another proof of the Attouch Theorem relating the epigraphical limit of sequences of convex functions to the set limit of the graphs of the subdifferentials.
Semigroup representations, positive definite functions and abelian 𝐶*-algebras
Authors: Ressel P. , Ricker W.
Published: July 2002

It is shown that every

∗
*

-representation of a commutative semigroup

S
S

with involution via operators on a Hilbert space has an integral representation with respect to a unique, compactly supported, selfadjoint Radon spectral measure defined on the Borel sets of the character space of

S
S

. The main feature is that the proof, which is based on the theory of positive definite functions, makes no use what-so-ever (directly or indirectly) of the theory of

C
∗

C^*

-algebras or more general Banach algebra arguments. Accordingly, this integral representation theorem is used to give a new proof of the Gelfand-Naimark theorem for abelian

C
∗

C^*

-algebras.