bluuebunny/crossref_metadata_2025 huggingface dataset:
Polynomials of generators of integrated semigroups
Authors: deLaubenfels Ralph
Published: June 2010

We give general sufficient conditions on

p
p

and

A
A

, for

p
(
A
)

p(A)

to generate an exponentially bounded holomorphic

k
k

-times integrated semigroup, where

p
p

is a polynomial and

A
A

is a linear operator on a Banach space. Corollaries include the following. (1) If

i
A

iA

generates a strongly continuous group and

p
p

is a polynomial of even degree with positive leading coefficient, then

−
p
(
A
)

- p(A)

generates a strongly continuous holomorphic semigroup of angle

π
2

\frac {\pi } {2}

. (2) If

−
A

- A

generates a strongly continuous holomorphic semigroup of angle

θ
\theta

and

p
p

is an

n
n

th degree polynomial with positive leading coefficient, with

n

(

π
2

−
θ

)

>

π
2

n\left ( {\tfrac {\pi } {2} - \theta } \right ) > \tfrac {\pi } {2}

, then

−
p
(
A
)

- p(A)

generates a strongly continuous holomorphic semigroup of angle

π
2

−
n
(

π
2

−
θ
)

\tfrac {\pi } {2} - n(\tfrac {\pi } {2} - \theta )

. (3) If

(
−
A
)

( - A)

generates an exponentially bounded holomorphic

k
k

-times integrated semigroup of angle

θ
\theta

, and

p
p

and

θ
\theta

are as in (2), then

−
p
(
A
)

- p(A)

generates an exponentially bounded holomorphic

(
k
+
1
)

(k + 1)

-times integrated semigroup of angle

π
2

−
n
(

π
2

−
θ
)

\tfrac {\pi } {2} - n(\tfrac {\pi } {2} - \theta )

.
Homogeneous spacetimes of zero curvature
Authors: Duncan Della C. , Ihrig Edwin C.
Published: June 2010

In the following we show the only possible flat, connected, incomplete homogeneous spacetimes are

H

/

Δ

H / \Delta

where

H
=

{

υ
∈

R

n

|

g

(

υ
,
N

)

>
0

}

,
N

H = \left \{ {\upsilon \in {{\mathbf {R}}^n}\left | {g\left ( {\upsilon ,N} \right ) > 0} \right .} \right \},N

is a null vector, and

Δ
\Delta

is a discrete subgroup of translations.
Factorization of matrices into partial isometries
Authors: Kuo Kung Hwang , Wu Pei Yuan
Published: June 2010

In this paper, we characterize complex square matrices which are expressible as products of partial isometries and orthogonal projections. More precisely, we show that a matrix

T
T

is the product of

k
k

partial isometries

(
k
≥
1
)

(k \geq 1)

if and only if

T
T

is a contraction

(

‖
T
‖

≤
1
)

(\left \| T \right \| \leq 1)

and rank

(
1
−

T
∗

T
)
≤
k
⋅

(1 - {T^*}T) \leq k \cdot

nullity

T
T

. It follows, as a corollary, that any

n
×
n

n \times n

singular contraction is the product of

n
n

partial isometries and

n
n

is the smallest such number. On the other hand,

T
T

is the product of finitely many orthogonal projections if and only if

T
T

is unitarily equivalent to

1
⊕
S

1 \oplus S

, where

S
S

is a singular strict contraction

(

‖
S
‖

>
1
)

(\left \| S \right \| > 1)

. As contrasted to the previous case, the number of factors can be arbitrarily large.
A note on the category of the free loop space
Authors: Fadell E. , Husseini S.
Published: June 2010

A useful result in critical point theory is that the LjusternikSchnirelmann category of the space of based loops on a compact simply connected manifold

M
M

is infinite (because the cup length of

M
M

is infinite). However, the space of free loops on

M
M

may have trivial products. This note shows that, nevertheless, the space of the free loops also has infinite category.
Periodic points for homeomorphisms of hereditarily decomposable chainable continua
Authors: Ingram W. T.
Published: June 2010

In this paper it is shown that homeomorphisms of hereditarily decomposable chainable continua cannot have periodic points whose periods are not powers of two. Examples show that for each power of two there is a hereditarily decomposable chainable continuum and a homeomorphism of it which has a periodic point of period that power of two.
Compact endomorphisms of Banach algebras. II
Authors: Kamowitz Herbert , Scheinberg Stephen , Wortman Dennis
Published: June 2010

In this note we exhibit examples which show that several conjectures concerning compact endomorphisms of commutative semisimple Banach algebras prove to be false.
Traces on irregular ideals
Authors: Varga József V.
Published: June 2010

Simple answers are given to the following and related questions: For what Hilbert space operator

A
A

is it true that the smallest ideal (alternatively, the smallest norm ideal, the smallest maximal norm ideal) containing

A
A

is a norm (an intermediate norm, a principal norm) ideal? Do these ideals support a nontrivial unitary invariant positive linear functional?
On the distance between zeroes and the limit-point problem
Authors: Yan Ju Rang
Published: June 2010

For the oscillatory equation

x
(
t
)
+
q
(
t
)
x
(
t
)
=
0

x(t) + q(t)x(t) = 0

, we prove some new results on the distance between consecutive zeroes and answer the questions raised by Everitt, Giertz and Weidmann [3] and by Patula [5] respectively and show that a counterexample given by Butler and Rao [1] is incorrect.
Canonical relativized cylindric set algebras
Authors: Maddux Roger D.
Published: June 2010

For every suitable relational structure there is a canonical relativized cylindric set algebra. This construction is used to obtain a generalization of Resek’s relative representation theorem, and a stronger version of the "Stone type representation theorem" by Andréka and Thompson.
The positive fixed points of Banach lattices
Authors: Christianson Bruce
Published: June 2010

Let

Z
Z

be a Banach lattice endowed with positive cone

C
C

and an order-continuous norm

|

|

⋅

|

|

|| \cdot ||

. Let

G
G

be a left-amenable semigroup of positive linear endomorphisms of

Z
Z

. Then the positive fixed points

C
0

{C_0}

of

Z
Z

under

G
G

form a lattice cone, and their linear span

Z
0

{Z_0}

is a Banach lattice under an order-continuous norm

|

|

⋅

|

|

0

|| \cdot |{|_0}

which agrees with

|

|

⋅

|

|

|| \cdot ||

on

C
0

{C_0}

. A counterexample shows that under the given conditions

Z
0

{Z_0}

need not contain all the fixed points of

Z
Z

under

G
G

, and need not be a sublattice of

(
Z
,
C
)

(Z,C)

. The paper concludes with a discussion of some related results.
Dense barrelled subspaces of uncountable codimension
Authors: Saxon Stephen A. , Robertson Wendy J.
Published: June 2010

Let

E
E

be a Hausdorff barrelled space. If there exists a dense barrelled subspace

M
M

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis c o d i m left-parenthesis upper M right-parenthesis greater-than-or-equal-to c right-parenthesis left-bracket c o d i m left-parenthesis upper M right-parenthesis equals dimension left-parenthesis upper E right-parenthesis right-bracket">

(
codim
⁡
(
M
)
≥
c
)
[
codim
⁡
(
M
)
=
dim
⁡
(
E
)
]

(\operatorname {codim} (M) \geq c)[\operatorname {codim} (M) = \operatorname {dim} (E)]

, we say that (

M
M

is a satisfactory subspace [11]) [

E
E

is barrelledly fit], respectively. Robertson, Tweddle and Yeomans [11] proved that

E
E

has a barrelled countable enlargement (BCE) if it has a satisfactory subspace. (Trivially)

E
E

has a satisfactory subspace if

dim
⁡
(
E
)
≥
c

\dim (E) \geq c

and

E
E

is barrelledly fit. We show that

E
E

is barrelledly fit (and

dim
⁡
(
E
)
≥
c

\dim (E) \geq c

) if

E
≆
φ

E \ncong \varphi

and either (i)

E
E

is an (LF)-space, or (ii)

E
E

is an infinite-dimensional separable space and the continuum hypothesis holds. Conclusion: barrelledly fit spaces and their permanence properties arise from and advance the study of BCE’s.
A short proof and a generalization of Miranda’s existence theorem
Authors: Vrahatis Michael N.
Published: June 2010

Miranda gave in [5] an equivalent formulation of the famous Brouwer fixed point theorem. We give a short proof of Miranda’s existence theorem and then using the results obtained in this proof we give a generalization of a well-known variant of Bolzano’s existence theorem. Finally, we give a generalization of Miranda’s theorem.
Nondentable solid subsets in Banach lattices failing RNP. Applications to renormings
Authors: Werner Elisabeth
Published: June 2010

We show that for every Banach lattice

E
E

failing RNP and not containing

c
0

{c_0}

(resp. containing

c
0

{c_0}

) and for every

ε
>
0

\varepsilon > 0

there exists a solid convex closed subset

D
D

of the unit ball of

E
E

, such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d i s t left-bracket e x t left-parenthesis upper D overbar Superscript sigma left-parenthesis upper E Super Superscript asterisk asterisk Superscript comma upper E Super Superscript asterisk Superscript right-parenthesis Baseline right-parenthesis comma upper E right-bracket greater-than 1 minus epsilon left-parenthesis r e s p period greater-than one half minus epsilon right-parenthesis">

dist
⁡
[
ext
⁡
(

D
¯

σ
(

E

∗
∗

,

E
∗

)

)
,
E
]
>
1
−
ε
(
resp
.
>

1
2

−
ε
)

\operatorname {dist} [\operatorname {ext} ({\overline D ^{\sigma ({E^{ * * }},{E^ * })}}),E] > 1 - \varepsilon (\operatorname {resp} . > \tfrac {1}{2} - \varepsilon )

\]
and such that every slice of

D
D

has diameter bigger than

2
−
ε

2 - \varepsilon

. We also prove that these results are optimal. We apply them to construct rough lattice norms with almost optimal constant on non-Asplund Banach lattices.
Ordinary differential equations on closed subsets of Fréchet spaces with applications to fixed point theorems
Authors: Polewczak Jacek
Published: June 2010

The construction and convergence of an approximate solution to the initial value problem

x
′

=
f
(
t
,
x
)
,
x
(
0
)
=

x
0

x’ = f(t,x),x(0) = {x_0}

, defined on closed subsets of a Fréchet space is given. Sufficient conditions that guarantee the existence of an approximate solution are analyzed in a relation to the Nagumo boundary condition used in the Banach space case. It is indicated that the Nagumo boundary condition does not guarantee the existence of an approximate solution. Applications to fixed points are given.
The algebra of decomposable operators in direct integrals of not necessarily separable Hilbert spaces
Authors: Schaflitzel Reinhard
Published: June 2010

Considering direct integrals of not necessarily separable Hilbert spaces we examine the question whether the algebra of decomposable operators is the commutant of the algebra of diagonalizable operators. Using the continuum-hypothesis we prove this relation, if the set of square integrable vector fields is generated by a subset

Γ
0

{\Gamma _0}

such that

|

Γ
0

|

≤

|

R

|

|{\Gamma _0}| \leq |{\mathbf {R}}|

. For the general case, a counterexample is given.
Decompositions of differentiable semigroups
Authors: Holmes John P.
Published: June 2010

A differentiable semigroup is a topological semigroup

(
S
,
∗
)

(S, * )

in which

S
S

is a differentiable manifold based on a Banach space and the associative multiplication function * is continuously differentiable. If

e
e

is an idempotent element of such a semigroup we show that there is an open set

U
U

containing

e
e

so that there is a

C
1

{C^1}

retraction

Φ
\Phi

of

U
U

into the set of idempotents of

S
S

so that

Φ
(
x
)
Φ
(
y
)
=
Φ
(
x
)

\Phi (x)\Phi (y) = \Phi (x)

for

x
x

and

y
y

in

U
U

and

x
Φ
(
x
)

x\Phi (x)

is in the maximal subgroup of

S
S

determined by

Φ
(
x
)

\Phi (x)

for each

x
x

in

U
U

. This leads to a natural decomposition of

S
S

near

e
e

into the union of a collection of mutually disjoint and mutually homeomorphic local differentiable subsemigroups whose intersections with

U
U

are the point inverses under

Φ
\Phi

. In case

S
S

is the semigroup under composition of continuous linear transformations on a Banach space, in the case of a nontrivial idempotent

e
e

, the existence of

Φ
\Phi

implies that operators near an

e
e

have nontrivial invariant subspaces. A dual right handed result holds.
On hyperspaces of polyhedra
Authors: Sakai Katsuro
Published: June 2010

Let

Q
=

[
−
1
,
1

]
ω

Q = {[ - 1,1]^\omega }

be the Hilbert cube and
\[

Q
f

=

{

(

x
i

)
∈
Q

|

x
i

=
0

except for finitely many 

i

}

.

{Q_f} = \left \{ {({x_i}) \in Q|{x_i} = 0{\text {except for finitely many }}i} \right \}.

\]
For a compact connected polyhedron

X
X

with

dim
⁡
X
>
0

\dim X > 0

, the hyperspaces of (nonempty) subcompacta, subcontinua, and compact subpolyhedra of

X
X

are denoted by

2
X

,
C
(
X
)

{2^X},C(X)

, and

Pol(

X

)

{\text {Pol(}}X{\text {)}}

, respectively. And let

C

Pol

(
X
)
=
C
(
X
)
∩

Pol(

X

)

{C^{{\text {Pol}}}}(X) = C(X) \cap {\text {Pol(}}X{\text {)}}

. It is shown that the pair

(

2
X

,

Pol(

X

)

)

({2^X},{\text {Pol(}}X{\text {)}})

is homeomorphic to

(
Q
,

Q
f

)

(Q,{Q_f})

. In case

X
X

has no free arc, it is also proved that the pair

(
C
(
X
)
,

C

Pol

(
X
)
)

(C(X),{C^{{\text {Pol}}}}(X))

is homeomorphic to

(
Q
,

Q
f

)

(Q,{Q_f})

.
All types implies torsion
Authors: Parry Walter
Published: June 2010

We prove the following theorem. Given a positive integer

n
n

and a subset

A
A

of

Z

n

{{\mathbf {Z}}^n}

with the following properties: (1) for every

(

a
1

,
…
,

a
n

)

\left ( {{a_1}, \ldots ,{a_n}} \right )

in

A
A

the inequality

Σ

i
=
1

n

|

a
i

|

≥
2

\Sigma _{i = 1}^n {\left | {{a_i}} \right |} \geq 2

holds, and (2) for every

(

x
1

,
…
,

x
n

)

\left ( {{x_1}, \ldots ,{x_n}} \right )

in

R

n

{{\mathbf {R}}^n}

there exists an

(

a
1

,
…
,

a
n

)

\left ( {{a_1}, \ldots ,{a_n}} \right )

in

A
A

with

a
i

x
i

≥
0

{a_i}{x_i} \geq 0

for

i
=
1
,
…
,
n

i = 1, \ldots ,n

, there exists a subset

A
0

{A_0}

of

A
A

such that

Z

n

{{\mathbf {Z}}^n}

modulo the subgroup generated by

A
0

{A_0}

contains a nontrivial torsion element.
Sets of minimal points in 𝑙_{𝑝}
Authors: Lacruz Miguel
Published: June 2010

We show that the minimal hull of a convex set in a Banach space is not necessarily convex, even in

l
p

{l_p}

spaces (finite- or infinite-dimensional). This answers a question raised by B. Beauzamy and B. Maurey in their joint paper of 1977. We also carry out a careful study of the minimal hull and the saturation of the unit ball in

l
1

(
N
)

l_1^{(N)}

. Finally, we give a compactness theorem for the minimal hull in

l
1

{l_1}

.
A uniqueness theorem for 𝑦’=𝑓(𝑥,𝑦),𝑦(𝑥₀)=𝑦₀
Authors: Majorana Armando
Published: June 2010

Consider the initial value problem for a first-order differential equation
\[

y
′

=
f
(
x
,
y
)
,

y
(

x
0

)
=

y
0

.

y’ = f(x,y),\quad y({x_0}) = {y_0}.

\]
In this paper a new uniqueness criterion is proved. This criterion is related to the numeric equation
\[

u
=

y
0

+
(
t
−

x
0

)
f
(
t
,
u
)
.

u = {y_0} + (t - {x_0})f(t,u).

\]
It is also shown that some well-known uniqueness theorems are consequences of our result.
The envelope of holomorphy of a truncated tube
Authors: Ye Zai Fei
Published: June 2010

A counterexample is constructed to show an essential difference between local and global CR extensions in tube domains.
Spectral multiplicity on products of hyperbolic spaces
Authors: Huntley Jonathan
Published: June 2010

Using Fourier expansions and an analysis of the Rayleigh quotient, we obtain an improved upper bound for the dimension of spaces of cusp forms when the symmetric space is a product of upper half spaces. In the rank-one case, some refinements are given.
Absolute continuity of Hamiltonians with von Neumann-Wigner potentials
Authors: Behncke Horst
Published: June 2010

For separated Dirac and Schrödinger operators whose potentials have oscillatory decaying terms the asymptotics of the eigenfunctions is determined. From this the absolute continuity of the spectrum off a finite resonance set and a limiting absorption principle is derived.
On the weighted approximation of continuously differentiable functions
Authors: Nachbin Leopoldo
Published: June 2010

This note is an improvement of the available methods for getting results on the weighted approximation of continuously differentiable functions.
On the oscillation of differential equations with periodic coefficients
Authors: Philos Ch. G.
Published: June 2010

This paper is concerned with the oscillation of first-order linear delay differential equations in which the coefficients are periodic functions with a common period and the delays are constants and multiples of this period. A necessary and sufficient condition for the oscillation of all solutions is established.
Weakly infinite-dimensional product spaces
Authors: Rohm Dale M.
Published: June 2010

It is shown that the product of a weakly infinite-dimensional compactum with a

C
C

-space is weakly infinite-dimensional. Some observations on the coincidence of weak infinite-dimensionality and property

C
C

are made. The question of when a weakly infinite-dimensional space has weakly infinite-dimensional product with all zero-dimensional spaces is investigated.
Estimates for inverses of 𝑒^{𝑖𝑛𝑡} in some quotient algebras of 𝐴⁺
Authors: Zerouali Elhassan
Published: June 2010

We give estimates for the norm of

e

int

{e^{\operatorname {int} }}

in

A
+

/

I

{A^ + }/I

, where

I
I

is a closed ideal of

A
+

{A^ + }

without inner factor, provided that the hull of

I
I

satisfies suitable geometric conditions.
Combinatorial set theory and cardinal function inequalities
Authors: Hodel R. E.
Published: June 2010

Three theorems of combinatorial set theory are proven. From the first we obtain the de Groot inequality

|
X
|

≤

2

h
L
(
X
)

\left | X \right | \leq {2^{hL(X)}}

, the Ginsburg-Woods inequality

|
X
|

≤

2

e
(
X
)
Δ
(
X
)

\left | X \right | \leq {2^{e(X)\Delta (X)}}

, the Erdös-Rado Partition Theorem for

n
=
2

n = 2

, and set-theoretic versions of the Hajnal-Juhász inequalities

|
X
|

≤

2

c
(
X
)
χ
(
X
)

\left | X \right | \leq {2^{c(X)\chi (X)}}

and

|
X
|

≤

2

s
(
X
)
ψ
(
X
)

\left | X \right | \leq {2^{s(X)\psi (X)}}

. From the second we obtain a generalization of the Arhangel’skiĭ inequality

|
X
|

≤

2

L
(
X
)
χ
(
X
)

\left | X \right | \leq {2^{L(X)\chi (X)}}

. From the third we obtain the Charlesworth inequality

n
(
X
)
≤
p
s
w

(
X

)

L
(
X
)

n(X) \leq psw{(X)^{L(X)}}

and a generalization of the Burke-Hodel inequality

|

K
(
X
)

|

≤

2

e
(
X
)
p
s
w
(
X
)

\left | {K(X)} \right | \leq {2^{e(X)psw(X)}}

.
Some trace formulas for almost unperturbed Schrödinger pairs of operators
Authors: Xia Daoxing
Published: June 2010

Some trace formulas are given for an almost unperturbed Schrödinger pair of operators

{
U
,
V
}

\{ U,V\}

and a self-adjoint operator

V
0

{V_0}

, where

V
0

{V_0}

satisfies the condition that

{
U
,

V
0

}

\{ U,{V_0}\}

is a Schrödinger pair of operators and

{
V
,

V
0

}

\{ V,{V_0}\}

is almost commuting.
Classifying immersed curves
Authors: Carter J. Scott
Published: June 2010

Let a collection

γ
\gamma

of generically immersed curves be given in an oriented surface

G
G

. To each component circle, associate a Gauss word by traveling once around the circle and recording the crossing points with signs. The set of these words forms a Gauss paragraph. If

γ
1

{\gamma _1}

and

γ
2

{\gamma _2}

fill the surface

G
G

in the sense that the complementary regions are disks, then there is a homeomorphism of

G
G

taking one to the other if and only if

γ
1

{\gamma _1}

and

γ
2

{\gamma _2}

have isomorphic Gauss paragraphs. This notion of isomorphism is defined here; it ignores the choices made in defining the Gauss words.
Finiteness of the lower spectrum of Schrödinger operators with singular potentials
Authors: Donig Jörg
Published: June 2010

We assume that

q
:

R

m

→

R

(
m
≥
3
)

q:{\mathbb {R}^m} \to \mathbb {R}(m \geq 3)

is a measurable function with the property that its negative and positive parts, respectively, belong to the Kato class

K
(

R

m

)

K({\mathbb {R}^m})

and

K

l
o
c

(

R

m

)

{K_{loc}}({\mathbb {R}^m})

. We prove a conjecture by B. Simon concerning the finiteness of the lower spectrum of an s.a. realization of the Schrödinger expression

−
Δ
+
q

- \Delta + q

in

L
2

(

R

m

)

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

bounded from below.
How to compute the square root of the non-Euclidean wave operator
Authors: Kovalyov M. , Légaré M.
Published: June 2010

In this paper we derive a first-order differential operator that can serve as an alternative to the non-Euclidean wave operator to study

S1(

2
,
R
)

{\text {S1(}}2,R)

.
Two operator functions with monotone property
Authors: Furuta Takayuki
Published: June 2010

This paper proves that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis p right-parenthesis equals left-parenthesis upper B Superscript r Baseline upper A Superscript p Baseline upper B Superscript r Baseline right-parenthesis Superscript left-parenthesis 1 plus 2 r right-parenthesis slash left-parenthesis p plus 2 r right-parenthesis">

F
(
p
)
=

(

B
r

A
p

B
r

)

(
1
+
2
r
)

/

(
p
+
2
r
)

F(p) = {({B^r}{A^p}{B^r})^{(1 + 2r)/(p + 2r)}}

is an increasing function of

p
p

for

p
≧
1

p \geqq 1

and

r
≧
0

r \geqq 0

whenever

A
≧
B
≧
0

A \geqq B \geqq 0

. This result is more precise than our previous result that

A
≧
B
≧
0

A \geqq B \geqq 0

ensures
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper B Superscript r Baseline upper A Superscript p Baseline upper B Superscript r Baseline right-parenthesis Superscript left-parenthesis 1 plus 2 r right-parenthesis slash left-parenthesis p plus 2 r right-parenthesis Baseline greater-than-over-equals upper B Superscript 1 plus 2 r">

(

B
r

A
p

B
r

)

(
1
+
2
r
)

/

(
p
+
2
r
)

≧

B

1
+
2
r

{({B^r}{A^p}{B^r})^{(1 + 2r)/(p + 2r)}} \geqq {B^{1 + 2r}}

for each

p
≧
1

p \geqq 1

and

r
≧
0

r \geqq 0

. We also cite three counterexamples related to Theorem 1.
Characterization of functions in terms of rate of convergence of a quadrature process
Authors: Rahman Q. I. , Schmeisser G.
Published: June 2010

We discuss the relationship between the regularity properties of a continuous

2
π

2\pi

-periodic function

f
f

and the behavior of the remainder of the trapezoidal rule on

[
0
,
2
π
]

[0,2\pi ]

applied to

f
f

.
Characterizations of ultradistributions with compact support and decomposition by support
Authors: Chung Soon-Yeong , Kim Dohan
Published: June 2010

In this paper we prove that every nonquasianalytic ultradistribution can be uniformly majorized by the behavior of test functions only on the support and that every ultradistribution with support in the union

K
1

∪

K
2

{K_1} \cup {K_2}

of two compact sets can be decomposed as the sum of one with support in

K
1

{K_1}

and one with support in

K
2

{K_2}

, along the context of Malgrange [17].
Some trigonometric identities related to exact covers
Authors: Beebee John
Published: June 2010

Sherman K. Stein proves that if

sin
⁡
π
z
=
k

∏

i
=
1

n

sin

(

π

/

d
i

)

(

b
i

−
z

)

\sin \pi z = k\prod \limits _{i = 1}^n {\sin } \left ( {\pi /{d_i}} \right )\left ( {{b_i} - z} \right )

where the

b
i

{b_i}

are integers, the

d
i

{d_i}

are positive integers,

k
k

is a constant, then

{

(

d
i

:

b
i

)

}

\left \{ {\left ( {{d_i}:{b_i}} \right )} \right \}

is an exact cover. It is shown here that if

0
≤

b
i

>

d
i

0 \leq {b_i} > {d_i}

then

k
=
−

2

n
−
1

k = - {2^{n - 1}}

, that the converse is also true, and an analogous formula is conjectured for infinite exact covers. Many well known and lesser known trigonometric and functional identities can be derived from this result and known families of exact covers. A procedure is given for constructing exact covers by induction.
The soul at infinity in dimension 4
Authors: Walschap Gerard
Published: June 2010

It is shown that

4
4

-dimensional open manifolds with nonnegative sectional curvature whose fibers are totally geodesic are metrically rigid. For arbitrary dimension, one also concludes that the curvature is in a sense maximal at the soul.
Sofic constant-to-one extensions of subshifts of finite type
Authors: Blanchard F. , Hansel G.
Published: June 2010

Provided a constant-to-one extension of a subshift of finite type is sofic and transitive, then it is also of finite type.
Note on a theorem of Avakumović
Authors: Geluk J. L.
Published: June 2010

A short proof is given of a result due to Avakumović. More specifically the asymptotic behavior of the solution

y

(
x
)

→
0

(

x
→
∞

)

y\left ( x \right ) \to 0\left ( {x \to \infty } \right )

of the differential equation

y
=
ϕ

(
x
)

y
λ

(

λ
>
1

)

y = \phi \left ( x \right ){y^\lambda }\left ( {\lambda > 1} \right )

in case

ϕ

(

t
x

)

/

ϕ

(
x
)

→

t
σ

(

x
→
∞

)

,
σ
>
−
2

\phi \left ( {tx} \right )/\phi \left ( x \right ) \to {t^\sigma }\left ( {x \to \infty } \right ),\sigma > - 2

is given.
Homogeneous Fourier multipliers in the plane
Authors: Duoandikoetxea Javier , Moyua Adela
Published: June 2010

Given a homogeneous of degree zero function on the plane, we study conditions on the first derivative of its restriction to the unit circle in order to deduce that it is an

L
p

{L^p}

-multiplier.
Higher monotonicity properties and inequalities for zeros of Bessel functions
Authors: Nicolò-Amati Gori Laura , Laforgia Andrea , Muldoon Martin E.
Published: June 2010

L. Lorch and P. Szegö have considered the sign-regularity of the higher differences (with respect to the rank

k
k

) of the sequence

{

c

ν
k

}

\{ {c_{\nu k}}\}

of positive zeros of the Bessel function

C
ν

(
x
)

{C_\nu }(x)

. Our main purpose here is to extend one of their main results to the higher derivatives with respect to

κ
\kappa

when

c

ν
k

{c_{\nu k}}

is appropriately defined as a function of a continuous variable

κ
\kappa

rather than the discrete variable

k
k

, and the difference operator is replaced by a derivative operator. We also present some inequalities arising from these and other results.
Cohomology of local sheaves on arrangement lattices
Authors: Yuzvinsky Sergey
Published: June 2010

We apply cohomology of sheaves to arrangements of hyperplanes. In particular we prove an inequality for the depth of cohomology modules of local sheaves on the intersection lattice of an arrangement. This generalizes a result of Solomon-Terao about the cummulative property of local functors. We also prove a characterization of free arrangements by certain properties of the cohomlogy of a sheaf of derivation modules. This gives a condition on the Möbius function of the intersection lattice of a free arrangement. Using this condition we prove that certain geometric lattices cannot afford free arrangements although their Poincaré polynomials factor.
Locally convex linear topological spaces that are homeomorphic to the powers of the real line
Authors: Chigogidze A.
Published: June 2010

We give a characterization of locally convex linear topological spaces that are homeomorphic to the uncountable powers of the real line.
Fixed points of orientation reversing homeomorphisms of the plane
Authors: Kuperberg Krystyna
Published: June 2010

Let

h
h

be an orientation reversing homeomorphism of the plane onto itself. If

X
X

is a plane continuum invariant under

h
h

, then

h
h

has a fixed point in

X
X

. Furthermore, if at least one of the bounded complementary domains of

X
X

is invariant under

h
h

, then

h
h

has at least two fixed points in

X
X

.
Automorphisms of 𝑆𝐿₂ of imaginary quadratic integers
Authors: Smillie John , Vogtmann Karen
Published: June 2010

We determine the outer automorphism groups of the two-dimensional special linear and projective special linear groups of the ring of integers in an imaginary quadratic field.
No polynomial bound for the chip firing game on directed graphs
Authors: Eriksson Kimmo
Published: June 2010

Tardos has proved a polynomial bound on the length of a convergent chip firing game on an undirected graph. This paper demonstrates a game with exponential growth on a directed graph.
Subharmonicity, the distance function, and 𝑎-admissible sets
Authors: Gardiner Stephen J.
Published: June 2010

Let

Ω
\Omega

be an open subset of

R

n

{\mathbb {R}^n}

. The main result of this paper shows that a one-sided control on the curvature of

∂
Ω

\partial \Omega

is equivalent to subharmonicity of a function related to

dist
⁡
(
X
,
∂
Ω
)

\operatorname {dist}(X,\partial \Omega )

. An application is given.
An 𝑛×𝑛 matrix of linear maps of a 𝐶*-algebra
Authors: Suen Ching Yun
Published: June 2010

Every positive

n
×
n

n \times n

matrix of linear functionals on a

C
∗

{C^ * }

-algebra is completely positive. [3, Theorem 2.1] can be extended to the case of a bounded

n
×
n

n \times n

matrix of linear functionals.
On a basis for 𝐻₂(\overline𝑀_{𝑔})
Authors: González Díez Gabino
Published: June 2010

In the moduli space

M

¯

g

{\overline {\mathbf {M}} _{\mathbf {g}}}

of stable curves (Riemann surfaces with nodes) we construct a basis for the second homology group, which is dual to the standard basis for the second cohomology group. The elements of our basis are algebraic curves.
Continuous selections of solution sets to evolution equations
Authors: Staicu Vasile
Published: June 2010

We prove the existence of a continuous selection of the multivalued map

ξ
→

T

(
ξ
)

\xi \to \mathcal {T}(\xi )

, where

T

(
ξ
)

\mathcal {T}(\xi )

is the set of all weak (resp. mild) solutions of the Cauchy problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove x With dot left-parenthesis t right-parenthesis element-of upper A x left-parenthesis t right-parenthesis plus upper F left-parenthesis t comma x left-parenthesis t right-parenthesis right-parenthesis comma x left-parenthesis 0 right-parenthesis equals xi">

x
˙

(
t
)
∈
A
x
(
t
)
+
F
(
t
,
x
(
t
)
)
,

x
(
0
)
=
ξ

\dot x(t) \in Ax(t) + F(t,x(t)),\quad x(0) = \xi

\]
, assuming that

F
F

is Lipschitzian with respect to

x
x

and

−
A

- A

is a maximal monotone map (resp.

A
A

is the infinitesimal generator of a

C
0

{C_0}

-semigroup). We also establish an analog of Michael’s theorem for the solution sets of the Cauchy problem

x
˙

(
t
)
∈
F
(
t
,
x
(
t
)
)
,

x
(
0
)
=
ξ

\dot x(t) \in F(t,x(t)),\;x(0) = \xi

.
Fréchet vs. Gâteaux differentiability of Lipschitzian functions
Authors: Gieraltowska-Kedzierska Maria , Van Vleck F. S.
Published: June 2010

Examples have been given of Lipschitzian functions that are Gâteaux-differentiable everywhere, but nowhere Fréchet-differentiable. One such example has been reported, mistakenly, in several papers as having domain in

L
2

(
[
0
,
π
]
)

{L^2}([0,\pi ])

, when it should have been

L
1

(
[
0
,
π
]
)

{L^1}([0,\pi ])

. We discuss this example.
Agmon’s complex Tauberian theorem and closed orbits for hyperbolic and geodesic flows
Authors: Pollicott Mark
Published: June 2010

In this note we give an alternative proof of Sharp’s dynamical analogue of Merten’s theorem for hyperbolic flows. Our use of a tauberian theorem of Agmon also allows us to get better error terms.
On some summability factors of infinite series
Authors: Sulaiman W. T.
Published: June 2010

A new theorem concerning some summability factors of infinite series is proved. Other results, some of them known, are also deduced.
Winding number and the number of real zeros of a function
Authors: Smits Lieven , Kuyk Willem
Published: June 2010

The theorem in this paper shows that the number of real simple zeros of a function of the form

f
(
x
)
=
q
(
x
)
+
a
x
+
b
,
x
∈

R

f(x) = q(x) + ax + b,x \in \mathbb {R}

, for not too wild

q
(
x
)

q(x)

can be obtained counting the winding number of a closed plane curve about the point

(
a
,
b
)

(a,b)

.
A note on the 𝐿^{𝑝} analogue of the “zero-two” law
Authors: Berend Daniel
Published: June 2010

It was proved by R. Wittmann [2] that, given a positive linear contraction of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p Baseline left-parenthesis 1 less-than-or-equal-to p greater-than normal infinity right-parenthesis comma sup Underscript double-vertical-bar f double-vertical-bar Subscript p Baseline less-than-or-equal-to 1 Endscripts limit Underscript n right-arrow normal infinity Endscripts double-vertical-bar upper T Superscript n Baseline f minus upper T Superscript n plus 1 Baseline f double-vertical-bar">

L
p

(

1
≤
p
>
∞

)

,

sup

‖
f
‖

p

≤
1

lim

n
→
∞

‖

T
n

f
−

T

n
+
1

f

‖

{L^p}\left ( {1 \leq p > \infty } \right ),{\operatorname {sup} _{{{\left \| f \right \|}_p} \leq 1}}{\operatorname {lim} _{n \to \infty }}\left \| {{T^n}f - {T^{n + 1}}f} \right \|

is either

≥

α
p

or

0

\geq {\alpha _p}{\text {or}}0

; the (best possible) value of

α
p

{\alpha _p}

is the

l
p

{l_p}

-norm of a certain

3
×
3

3 \times 3

matrix. In this paper

α
p

{\alpha _p}

is explicitly expressed as a function of

p
p

.
Examples of ungradable algebras
Authors: Belzner Th. , Burgess W. D. , Fuller K. R. , Schulz R.
Published: June 2010

Examples are presented of finite-dimensional algebras that admit no positive grading (that is, a nontrivial grading indexed by the natural numbers). Some of these examples have finite global dimension (they are even quasihereditary), and yield a negative answer to a question of Anick and Green.
Lindelöf spaces concentrated on Bernstein subsets of the real line
Authors: Lawrence L. Brian
Published: June 2010

We show in ZFC that for each

n
n

with

n
∈
ω

n \in \omega

or

n
=
ω

n = \omega

, there is a Lindelöf space

X
X

and a separable metric space

M
M

such that for every

m
>
n

m > n

,

X
×

m

M

X \times {}^mM

is Lindelöf, whereas

X
×

n

M

X \times {}^nM

is nonnormal.
The maximal abelian subextension determines weak mixing for group extensions
Authors: Robinson E. Arthur
Published: June 2010

Weak mixing is an important property for group extensions because its absence is the principal obstruction to lifting a large number of stronger mixing properties. Whether or not an extension is weakly mixing can be determined by studying the sub-extension corresponding to the quotient by the commutator subgroup. Several conditions equivalent to weak mixing are given. In particular, an extension by any group with no abelian factors (for example any nonabelian simple group) is automatically weak mixing if it is ergodic. The proof uses spectral multiplicity theory.
Thom modules and mod 𝑝 spherical fibrations
Authors: Broto Carlos
Published: June 2010

In this paper we show that every finite Thom module over the ring of invariants of a finite nonmodular group can be realized as mod

p
p

cohomology of the Thom space of a spherical fibration.
An upper bound for the sum ∑^{𝑎+𝐻}_{𝑛=𝑎+1}𝑓(𝑛) for a certain class of functions 𝑓
Authors: Dobrowolski Edward , Williams Kenneth S.
Published: June 2010

For a certain class of functions

f
:
Z
→
C

f:Z \to C

an upper bound is obtained for the sum

∑

n
=
a
+
1

a
+
H

f

(
n
)

\sum \nolimits _{n = a + 1}^{a + H} {f\left ( n \right )}

. This bound is used to give a proof of a classical inequality due to Pólya and Vinogradov that does not require the value of the modulus of the Gauss sum and to obtain an estimate of the sum of Legendre symbols

∑

x
=
1

H

(
(

R

g
x

+
S

)

/

p

)

\sum \nolimits _{x = 1}^H {( ( {R{g^x} + S} )/p} )

, where

g
g

is a primitive root of the odd prime

p
,
1
≤
H
≤
p
−
1

p,1 \leq H \leq p - 1

and

R
S

RS

is not divisible by

p
p

.
Some Banach algebras without discontinuous derivations
Authors: Forrest Brian
Published: June 2010

It is shown that the completion of

A
(
G
)

A(G)

in either the multiplier norm or the completely bounded multiplier norm is a Banach algebra without discontinuous derivations when

G
G

is either

F
2

{F_2}

or

SL
⁡
(
2
,

R

)

\operatorname {SL}(2,\mathbb {R})

.
On supersolubility in some groups with finitely generated Fitting radical
Authors: Beidleman James C. , Smith Howard
Published: June 2010

The groups

G
G

considered in this paper have the property that every normal nonsupersoluble subgroup of

G
G

has a finite,

G
G

-invariant, nonsupersoluble image. The structure of a certain radical, defined in terms of ascendant supersoluble subgroups, is determined. The main result is a supersolubility criterion for polycyclic groups.
Centralizers of expanding maps on the circle
Authors: Arteaga Carlos
Published: June 2010

We prove here that the elements of an open and dense subset of expanding maps on the circle have trivial centralizers; i.e., the maps commute only with their own powers. Using a theorem proved in [1], we deduce that the result is also true for an open and dense subset of immersions of

S
1

{S^1}

.
Pseudo-Riemannian metrics and Hirzebruch signature
Authors: Law Peter R.
Published: June 2010

On compact, orientable,

4
k

4k

-dimensional manifolds, nonvanishing Hirzebruch signature is shown to be an obstruction to the existence of certain kinds of pseudo-Riemannian metrics.
Decidability of the existential theory of the set of natural numbers with order, divisibility, power functions, power predicates, and constants
Authors: Terrier Véronique
Published: June 2010

We construct an algorithm to test if a system of conditions of the types
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu greater-than eta comma mu slash eta comma mu equals eta Superscript a Baseline comma upper P Subscript a Baseline left-parenthesis mu right-parenthesis comma normal not-sign left-parenthesis mu greater-than eta right-parenthesis comma normal not-sign left-parenthesis mu slash eta right-parenthesis comma normal not-sign left-parenthesis mu equals eta Superscript a Baseline right-parenthesis">

μ
>
η
,
μ

/

η
,
μ
=

η
a

,

P
a

(
μ
)
,
¬
(
μ
>
η
)
,
¬
(
μ

/

η
)
,
¬
(
μ
=

η
a

)

\mu > \eta ,\mu /\eta ,\mu = {\eta ^a},{P_a}(\mu ),\neg (\mu > \eta ),\neg (\mu /\eta ),\neg (\mu = {\eta ^a})

, and

¬
(

P
a

(
μ
)
)

\neg ({P_a}(\mu ))

has a solution in natural numbers. (

a
∈
N

a \in N

, and

P
a

{P_a}

denotes the set

{

n
a

:
n
∈
N
}

\{ {n^a}:n \in N\}

.)
Finitely additive Gleason measures
Authors: Dvurečenskij Anatolij
Published: June 2010

We describe the set of all finitely additive measures which attain also infinite values on a quantum logic of a Hilbert space and which are expressible via the generalized Gleason-Lugovaja-Sherstnev formula. We prove that this set consists of those which are regular with respect to the set of all finite-dimensional subspaces. In addition, we show that this regularity does not entail the countable additivity, in general.
Uniformly continuous and weakly almost periodic functions on some topological semigroups
Authors: Dzinotyiweyi Heneri A. M.
Published: June 2010

Let

LUC
⁡
(
S
)

\operatorname {LUC} (S)

be the space of bounded left uniformly continuous functions and

WAP
⁡
(
S
)

\operatorname {WAP} (S)

the space of bounded weakly almost periodic functions on a topological semigroup

S
S

. For a large class of topological semigroups called stips that are not relatively neocompact, we show that the quotient space

LUC
⁡
(
S
)

/

WAP
⁡
(
S
)

\operatorname {LUC} (S)/\operatorname {WAP} (S)

contains an isometric linear copy of

l
∞

{l^\infty }

. The result is particularly interesting on noting that the underlying topological semigroup is not assumed to support an absolutely continuous measure—a theme studied by the author in a previous paper.
Range transformations on a Banach function algebra. IV
Authors: Hatori Osamu
Published: June 2010

Functions in

Op

(

I
D

,
Re
⁡
A
+
L

)

{\text {Op}}\left ( {{I_D},\operatorname {Re} A + L} \right )

are harmonic on

D
D

for a closed subalgebra

A
A

of

C
0

(
Y
)

{C_0}\left ( Y \right )

, an ideal

I
I

of

A
A

and a linear subspace

L
L

of finite dimension in

C
0

,
R

(
Y
)

{C_0}{,_R}\left ( Y \right )

unless the uniform closure of

I
I

is selfadjoint.
Indecomposable division algebras
Authors: Sethuraman B. A.
Published: June 2010

We present a direct construction of indecomposable division algebras of all indices

p
n

{p^n}

and exponents

p
m

(
n
≥
m
≥
2
)

{p^m}(n \geq m \geq 2)

that does not rely on generic methods.
A new Castelnuovo bound for two codimensional subvarieties of 𝑃^{𝑟}
Authors: Alzati Alberto
Published: June 2010

Let

X
X

be a smooth

n
n

-dimensional projective subvariety of

P

r

(

C

)
,
(
r
≥
3
)

{\mathbb {P}^r}(\mathbb {C}),(r \geq 3)

. For any positive integer

k
,
X

k,X

is said to be

k
k

-normal if the natural map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 0 Baseline left-parenthesis double-struck upper P Superscript r Baseline comma script upper O Subscript double-struck upper P r Baseline left-parenthesis k right-parenthesis right-parenthesis right-arrow upper H Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis k right-parenthesis right-parenthesis">

H
0

(

P

r

,

O

P

r

(
k
)
)
→

H
0

(
X
,

O

X

(
k
)
)

{H^0}({\mathbb {P}^r},{\mathcal {O}_{\mathbb {P}r}}(k)) \to {H^0}(X,{\mathcal {O}_X}(k))

is surjective. Mumford and Bayer showed that

X
X

is

k
k

-normal if

k
≥
(
n
+
1
)
(
d
−
2
)
+
1

k \geq (n + 1)(d - 2) + 1

where

d
=
deg
⁡
(
X
)

d = \deg (X)

. Better inequalities are known when

n
n

is small (Gruson-Peskine, Lazarsfeld, Ran). In this paper we consider the case

n
=
r
−
2

n = r - 2

, which is related to Hartshorne’s conjecture on complete intersections, and we show that if

k
≥
d
+
1
+
(
1

/

2
)
r
(
r
−
1
)
−
2
r

k \geq d + 1 + (1/2)r(r - 1) - 2r

then

X
X

is

k
k

-normal and

I
X

{I_X}

, the ideal sheaf of

X
X

in

P

r

{\mathbb {P}^r}

, is

(
k
+
1
)

(k + 1)

-regular. About these problems Lazarsfeld developed a technique based on generic projections of

X
X

in

P

n
+
1

{\mathbb {P}^{n + 1}}

; our proof is an application of some recent results of Ran’s (on the secants of

X
X

): we show that in our case there exists a projection such generic as Lazarsfeld requires. When

r
≥
6

r \geq 6

we also give a better inequality:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-equal-to d minus 1 plus left-parenthesis 1 slash 2 right-parenthesis r left-parenthesis r minus 1 right-parenthesis minus left-parenthesis r minus 1 right-parenthesis left-bracket left-parenthesis r plus 4 right-parenthesis slash 2 right-bracket">

k
≥
d
−
1
+
(
1

/

2
)
r
(
r
−
1
)
−
(
r
−
1
)
[
(
r
+
4
)

/

2
]

k \geq d - 1 + (1/2)r(r - 1) - (r - 1)[(r + 4)/2]

([] means: integer part); it is obtained by refining Lazarsfeld’s technique with the help of some results of ours about

k
k

-normality.
Some remarks on the structure of Mackey functors
Authors: Greenlees J. P. C. , May J. P.
Published: June 2010

All Mackey functors over a finite group

G
G

are built up by short exact sequences from Mackey functors arising from modules over the integral group rings of appropriate subquotients WH of

G
G

. The equivariant cohomology theories with coefficients in Mackey functors arising from WH-modules admit particularly simple descriptions.
Nuclear 𝐶*-algebras have amenable unitary groups
Authors: Paterson Alan L. T.
Published: June 2010

Let

A
A

be a unital

C
∗

{C^*}

-algebra with unitary group

G
G

. Give

G
G

the relative (Banach space) weak topology. Then

G
G

is a topological group, and we show that

A
A

is nuclear if and only if there exists a left invariant mean on the space of right uniformly continuous, bounded, complex-valued functions on

G
G

.
A two-generator presentation for the Picard group
Authors: Brunner A. M.
Published: June 2010

The following

2
2

-generator

5
5

-relator presentation is given for the Picard group

PSL
⁡
(
2
,
Z
(
i
)
)

\operatorname {PSL} (2,Z(i))

:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l a n g l e a comma w comma b semicolon b equals a w squared a Superscript negative 1 Baseline w Superscript negative 2 Baseline a w squared comma left-parenthesis a squared w a w Superscript negative 1 Baseline right-parenthesis squared equals left-parenthesis a w a w Superscript negative 1 Baseline right-parenthesis cubed equals left-parenthesis w b right-parenthesis squared equals left-parenthesis a b right-parenthesis squared equals b squared equals 1 mathematical right-angle period">

l
a
n
g
l
e
a
,
w
,
b
;
b
=
a

w
2

a

−
1

w

−
2

a

w
2

,
 

(

a
2

w
a

w

−
1

)
2

=

(
a
w
a

w

−
1

)
3

=

(
w
b

)
2

=

(
a
b

)
2

=

b
2

=
1
⟩
.

langle a,w,b;b = a{w^2}{a^{ - 1}}{w^{ - 2}}a{w^2},\ {({a^2}wa{w^{ - 1}})^2} = {(awa{w^{ - 1}})^3} = {(wb)^2} = {(ab)^2} = {b^2} = 1\rangle .

\]
Eigenvalues of some almost periodic functions
Authors: Egawa Jirō
Published: June 2010

Let

B
U

{B_U}

be the set of real valued functions on

R
R

which are bounded and uniformly continuous. For

f
,
g
∈

B
U

f,g \in {B_U}

, put
\[

d
(
f
,
g
)
=

sup

t
∈
R

|

f
(
t
)
−
g
(
t
)

|

.

d(f,g) = \sup \limits _{t \in R} |f(t) - g(t)|.

\]
Then

B
U

{B_U}

becomes a metric space. On

B
U

{B_U}

we define a flow

η
\eta

by

η
(
f
,
t
)
=

f
t

\eta (f,t) = {f_t}

for

(
f
,
t
)
∈

B
U

×
R

(f,t) \in {B_U} \times R

. We denote the restriction of

η
\eta

to the hull of

f
∈

B
U

f \in {B_U}

by

η
f

{\eta _f}

. If

f
f

is almost periodic, then the set of eigenvalues of

η
f

{\eta _f}

coincides with the module of

f
f

(see J. Egawa, Eigenvalues of compact minimal flows, Math. Seminar Notes (Kobe Univ.), 10 (1982), 281-291. In this paper, we extend this result to almost periodic functions with some additional properties.
Patching and analytic isomorphisms
Authors: Landsburg Steven E.
Published: June 2010

We give a particularly simple proof of the Milnor Patching Property for a class of diagrams that includes analytic isomorphism squares.
Symmetry of dichromatic links
Authors: Miyazawa Yasuyuki
Published: June 2010

Let

L
L

be a

1
1

-trivial dichromatic link in

S
3

{S^3}

and

L
~

\widetilde {L}

its covering link. A relationship between the dichromatic link polynomials, defined by Hoste and Przytycki, of

L
L

and

L
~

\widetilde {L}

is given. As an application, it is shown that the link

7
1
2

7_1^2

has no symmetries with fixed point set is either of the components.
Radon-Nikodým property in symmetric spaces of measurable operators
Authors: Xu Quan Hua
Published: June 2010

Let

E
E

be a rearrangement invariant function space on

(

0
,
∞

)

\left ( {0,\infty } \right )

with the RNP. Let

(

M
,
τ

)

\left ( {M,\tau } \right )

be a von Neumann algebra with a faithful normal semifinite trace

τ
\tau

. It is proved that the associated symmetric space

L
E

(

M
,
τ

)

{L_E}\left ( {M,\tau } \right )

of measurable operators has the RNP.
Fixed points for discontinuous quasi-monotone maps in sequence spaces
Authors: Schmidt Sabina
Published: June 2010

In [2] Hu gives a fixed point theorem for discontinuous quasimonotone increasing maps in

X
=

R

n

X = {\mathbb {R}^n}

. We will answer the question in [2] as to whether this result can be extended to

X
=

l
p

,
1
≤
p
≤
∞

X = {l^p},1 \leq p \leq \infty

.
Quadric representation of a submanifold
Authors: Dimitrić Ivko
Published: June 2010

If

x
:

M
n

→

E
m

x: M^n \to E^m

is an isometric immersion of a smooth manifold into a Euclidean space then the map

x
~

=
x

x

t

\tilde {x} = x x^{\mathrm {t}}

]> (t denotes transpose) is called the quadric representation of

M
M

.

x
~

\tilde {x}

is said to be of finite type (

k
k

-type) if it can be decomposed into a sum of finitely many

(
k
)

(k)

eigenfunctions of Laplacian from different eigenspaces. We study map

x
~

\tilde {x}

in general, especially as related to the condition of being of finite type. Certain classification results are obtained for manifolds with

1
1

-and

2
2

-type quadric representation.