bluuebunny/crossref_metadata_2025 huggingface dataset:
On a theorem of Kōmura-Koshi and of Andô-Ellis
Authors: Wong Yau Chuen
Published: June 2010

Kōmura and Koshi’s result, which states that the topology

I

\mathcal {I}

of a nuclear locally convex vector lattice

(
E
,
C
,

I

)

(E,C,\mathcal {I})

is the topology

o
(
E
,

E
′

)

o(E,E’)

of uniform convergence on all order-intervals in

E
′

E’

, is generalized to the case when

(
E
,
C
,

I

)

(E,C,\mathcal {I})

is only a locally solid space. Andô-Ellis’ theorem, concerning the duality of strict

B

\mathfrak {B}

-cones and normality in normed vector spaces, is generalized to the metrizable case.
On the range of a hyponormal derivation
Authors: Stampfli Joseph G.
Published: June 2010

The inner derivation induced by a hyponormal operator has closed range if and only if the operator has finite spectrum.
An alternate characterization of the Cantor set
Authors: Schoenfeld Alan H. , Gruenhage Gary
Published: June 2010

Let

X
X

be a compact metric space such that, up to homeomorphism,

X
X

has only two nonempty open subsets. Then

X
X

is homeomorphic to the Cantor discontinuum.
Common fixed points of commuting mappings
Authors: Gray William J. , Smith Carol M.
Published: June 2010

Let

X
X

be a dendroid and

S
S

an abelian semigroup of continuous monotone self-mappings of

X
X

. A point

x
ϵ
X

x\epsilon X

is fixed under

S
S

if

g
(
x
)
=
x

g(x) = x

for all

g
ϵ
S

g\epsilon S

. Let

f
:
X
→
X

f:X \to X

be continuous and commute with each element of

S
S

. It is shown that

f
f

and

S
S

have a common fixed point.
On the center of some finite linear groups
Authors: Blau Harvey I.
Published: June 2010

This note proves two results, one in characteristic

p
p

and the other in characteristic zero, which restrict the order of the center of some finite linear groups of degree less than a prime

p
p

which divides the group order.
On a ubiquitous cardinal
Authors: Hechler Stephen H.
Published: June 2010

We consider five combinatorial or topological structures, each with a certain associated minimal cardinal, and we show that these cardinals are always equal even though it is independent of the axioms of set theory as to just what the value of this common cardinal is. The five structures are the set of functions from

N
N

(the set of natural numbers) into

N
N

under two partial orderings, the rational numbers with respect to closed embeddings into powers of

N
N

, a certain subset of

β
N
−
N

\beta N - N

with respect to clopen decompositions, the irrationals with respect to compact decompositions, and a subclass of the Borel sets with respect to closed decompositions. The proofs presented do not require a knowledge of forcing techniques.
Two examples in proximity spaces
Authors: Sharma P. L.
Published: June 2010

Two examples of Lo-spaces are given. The first is an example of a Lo-space in which not every ultrafilter is contained in a cluster. In the Lo-space of the second example, each ultrafilter is contained in a cluster, and yet not every maximal bunch is a cluster.
On the tautness property of Alexander-Spanier cohomology
Authors: Deo Satya
Published: June 2010

Any neighborhood retract of an arbitrary topological space is tautly imbedded with respect to Alexander-Spanier cohomology.
Finite unions of ideals and modules
Authors: Quartararo Philip , Butts H. S.
Published: June 2010

We say that a commutative ring

R
R

is a

u
u

-ring provided

R
R

has the property that an ideal contained in a finite union of ideals must be contained in one of those ideals; and a

u
m

um

-ring is a ring

R
R

with the property that an

R
R

-module which is equal to a finite union of submodules must be equal to one of them. The primary purpose of this paper is to characterize

u
u

-rings and

u
m

um

-rings. We show that

R
R

is a

u
m

um

-ring if and only if the residue field

R

/

P

R/P

is infinite for each maximal ideal

P
P

of

R
R

; and

R
R

is a

u
u

-ring if and only if for each maximal ideal

P
P

of

R
R

either the residue field

R

/

P

R/P

is infinite or the quotient ring

R
p

{R_p}

is a Bézout ring.
Some commutativity results for rings with two-variable constraints
Authors: Bell H. E.
Published: June 2010

It is proved that an associative ring

R
R

has nil commutator ideal if for each

x
,

y
ϵ
R

x,\;y\epsilon R

, there is a polynomial

p
(
X
)
ϵ
X

Z

[
X
]

p(X)\epsilon X{\mathbf {Z}}[X]

for which

x
y
−
y
p
(
x
)

xy - yp(x)

is central. Two restrictions on the

p
(
X
)

p(X)

which guarantee commutativity are established.
On the radius of starlikeness of (𝑧𝑓)^{’} for 𝑓 univalent
Authors: Barnard Roger W.
Published: June 2010

Let

S
S

be the standard class of normalized univalent functions. For a given function

f
,

f
(
z
)
=
z
+

a
2

z
2

+
…

f,\;f(z) = z + {a_2}{z^2} + \ldots

, regular for

|

z

|

>
1

|z| > 1

, let

r
(
f
)

r(f)

be the radius of starlikeness of

f
f

. In 1947, R. M. Robinson considered the combination

g
f

(
z
)
=
(
z
f

)
′

/

2

{g_f}(z) = (zf)’/2

for

f
ϵ
S

f\epsilon S

. He found a lower bound of .38 for

r
(

g
f

)

r({g_f})

for all

f
ϵ
S

f\epsilon S

. He noted that the standard Koebe function

k
,

k
(
z
)
=
z

(
1
−
z

)
2

k,\;k(z) = z{(1 - z)^2}

, has its

r
(

g
k

)

r({g_k})

equal to

1

/

2

1/2

. A question that has been asked since Robinson’s paper is whether

1

/

2

1/2

is the minimum

r
(

g
f

)

r({g_f})

for all

f
f

in

S
S

. It is shown here that this is not the case by giving examples of functions

f
f

whose

r
(

g

f

)

r({g_{f}})

is less than

1

/

2

1/2

.
The Rudin-Carleson theorem for vector-valued functions
Authors: Globevnik J.
Published: June 2010

The following generalization of the Rudin-Carleson theorem is proved. Let

X
X

be a complex Banach space and let

f
:
F
→
X

f:F \to X

be a continuous function, where

F
F

is a closed subset of the unit circle in

C
C

of Lebesgue measure zero. There exists a continuous function

g
g

from the closed unit disc to

X
X

which is analytic on the open unit disc and satisfies (i)

g

|

F
=
f

g|F = f

, (ii)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max Underscript StartAbsoluteValue z EndAbsoluteValue less-than-or-slanted-equals 1 Endscripts StartAbsoluteValue EndAbsoluteValue g left-parenthesis z right-parenthesis StartAbsoluteValue EndAbsoluteValue equals max Underscript z epsilon upper F Endscripts StartAbsoluteValue EndAbsoluteValue f left-parenthesis z right-parenthesis StartAbsoluteValue EndAbsoluteValue">

max

|

z

|

⩽
1

|

|

g
(
z
)

|

|

=

max

z
ϵ
F

|

|

f
(
z
)

|

|

{\max _{|z| \leqslant 1}}||g(z)|| = {\max _{z\epsilon F}}||f(z)||

.
Weakly completely continuous elements of 𝐶*-algebras
Authors: Ylinen Kari
Published: June 2010

For a

C

∗

C^{\ast }

-algebra

A
A

and

u
ϵ
A

u\epsilon A

, the equivalence of the following three statements is proved: (i) the map

x
↦
u
x
u

x \mapsto uxu

is a compact operator on

A
A

, (ii) (resp. (iii)) the map

x
↦
u
x

x \mapsto ux

(resp.

x
↦
x
u

x \mapsto xu

) is a weakly compact operator on

A
A

. The canonical image of a dual

C

∗

{C^{\ast }}

-algebra

A
A

in its bidual

A

∗

∗

{A^{{\ast }{\ast }}}

is characterized as the set of the weakly completely continuous elements of

A

∗

∗

{A^{{\ast }{\ast }}}

.
Cones of potentials
Authors: Calvert Bruce
Published: June 2010

We construct a cone of potentials from a

T
T

-monotone operator and use this to give smoothness of supersolutions solving a minimum problem.
Degree sequences in complexes and hypergraphs
Authors: Dewdney A. K.
Published: June 2010

Given an

n
n

-complex

K
K

and a vertex

v
v

in

K
K

, the

n
n

-degree of

v
v

is the number of

n
n

-simplexes in

K
K

containing

v
v

. The set of all

n
n

-degrees in a complex

K
K

is called its

n
n

-degree sequence when arranged in nonincreasing order. The question “Which sequences of integers are

n
n

-degree sequences?” is answered in this paper. This is done by generalizing the iterative characterization for the

1
1

-dimensional (graphical) case due to V. Havel. A corollary to this general theorem yields the analogous generalization for

k
k

-graphs. The characterization of P. Erdös and T. Gallai is discussed briefly.
Decomposition of tensor products of irreducible unitary representations
Authors: Poguntke Detlev
Published: June 2010

It is shown that the tensor product of an irreducible unitary representation of a (discrete) group

G
G

and an

n
n

-dimensional

(
n
>
∞
)

(n > \infty )

unitary representation of

G
G

decomposes into at most

n
2

{n^2}

irreducible subrepresentations; the multiplicity of each irreducible constituent is not greater than

n
n

. As an application it is shown that the restriction of an irreducible unitary representation to a subgroup of finite index is a finite sum of irreducible subrepresentations.
On the convolution of a measure and a function
Authors: Berberian S. K.
Published: June 2010

Complements to a theorem of Bourbaki on the convolution of a measure and a function.
Torus-like products of 𝜆 connected continua
Authors: Hagopian Charles L.
Published: June 2010

Recently the author [5] proved that

λ
\lambda

connected continua

X
X

and

Y
Y

are arc-like if and only if the topological product

X
×
Y

X \times Y

is disklike. Here we present an analogous theorem that generalizes the result of Fort [2] and Ganea [3] that disks are not torus-like. We prove that

λ
\lambda

connected continua

X
X

and

Y
Y

are circle-like if and only if

X
×
Y

X \times Y

is torus-like.
The intersection of the subgroups of finite index in some finitely presented groups
Authors: Hirshon Ronald
Published: June 2010

We consider the intersection of the subgroups of finite index in some finitely presented non-Hopfian groups.
Characterizations of 𝐶-compact spaces
Authors: Herrington Larry L. , Long Paul E.
Published: June 2010

This paper gives characterizations of

C
C

-compact spaces, some of which make use of nets. Sufficient conditions for a space to be

C
C

-compact are given which make use of the graphs of weakly-continuous functions and a class of spaces which contains the class of Hausdorff completely normal and fully normal spaces.
Construction of Steiner quadruple systems having large numbers of nonisomorphic associated Steiner triple systems
Authors: Lindner Charles C.
Published: June 2010

If

(
Q
,
q
)

(Q,q)

is a Steiner quadruple system and

x
x

is any element in

Q
Q

it is well known that the set

Q
x

=
Q
∖
{
x
}

{Q_x} = Q\backslash \{ x\}

equipped with the collection

q
(
x
)

q(x)

of all triples

{
a
,
b
,
c
}

\{ a,b,c\}

such that

{
a
,
b
,
c
,
x
}
∈
q

\{ a,b,c,x\} \in q

is a Steiner triple system. A quadruple system

(
Q
,
q
)

(Q,q)

is said to have at least

n
n

nonisomorphic associated triple systems (NATS) provided that for at least one subset

X
X

of

Q
Q

containing

n
n

elements the triple systems

(

Q
x

,
q
(
x
)
)

({Q_x},q(x))

and

(

Q
y

,
q
(
y
)
)

({Q_y},q(y))

are nonisomorphic whenever

x
≠
y
∈
X

x \ne y \in X

. Prior to the results in this paper the maximum number of known NATS for any quadruple system was 2. The main result in this paper is the construction for each positive integer

t
t

of a quadruple system having at least

t
t

NATS.
Equality of spectra of quasi-similar hyponormal operators
Authors: Clary Stuart
Published: June 2010

It is shown that quasi-similar hyponormal operators have equal spectra.
Another proof that weak 𝑛-homogeneity implies weak (𝑛-1)-homogeneity
Authors: Roy Prabir
Published: June 2010

The assertion in the title is proved by using Ramsey’s theorem.
A basis result for Σ⁰₃ sets of reals with an application to minimal covers
Authors: Harrington Leo A. , Kechris Alexander S.
Published: June 2010

It is shown that every

Σ
3
0

\Sigma _3^0

set of reals which contains reals of arbitrarily high Turing degree in the hyperarithmetic hierarchy contains reals of every Turing degree above the degree of Kleene’s

O

\mathcal {O}

. As an application it is shown that every Turing degree above the Turing degree of Kleene’s

O

\mathcal {O}

is a minimal cover.
Restrictions of analytic functions. II
Authors: Rosenblum Marvin , Rovnyak James
Published: June 2010

An isometric expansion is derived which recaptures any

H
2

{H^2}

function from a restriction of its boundary function to a Borel set.
Nonfactorization of functions in Banach subspaces of 𝐿¹(𝐺)
Authors: Yap Leonard Y. H.
Published: June 2010

In this note we first prove a theorem on factorization of functions in certain subsets of

L
1

(
G
)

{L^1}(G)

, where

G
G

is a nondiscrete locally compact Abelian group with dual group

G
^

\hat G

. One of the corollaries of this theorem answers a question of R. Larsen concerning the algebras of functions with Fourier transforms in

L
p

(

G
^

)

{L^p}(\hat G)

. The other corollaries contain nonfactorization results which sharpen some known theorems.
Solution of integral equations by product integration
Authors: Helton Jon C.
Published: June 2010

Suppose

f
,
h

f,h

and

G
G

are functions with values in a normed complete ring. With suitable restrictions on these functions, it is established that
\[

f
(
x
)
=
h
(
x
)
+

∫
a
x

f
(
u
)
G
(
u
,
v
)

f(x) = h(x) + \int _a^x {f(u)G(u,v)}

\]
for

a
≤
x
≤
b

a \leq x \leq b

only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript a Superscript x Baseline h left-parenthesis u right-parenthesis upper G left-parenthesis u comma v right-parenthesis Subscript v Baseline normal upper Pi Superscript x left-parenthesis 1 plus upper G right-parenthesis">

∫
a
x

h
(
u
)
G

(
u
,
v
)

v

Π
x

(
1
+
G
)

\int _a^x {h(u)G{{(u,v)}_v}{\Pi ^x}} (1 + G)

exists and is

f
(
x
)
−
h
(
x
)

f(x) - h(x)

for

a
≤
x
≤
b

a \leq x \leq b

, and that
\[

f
(
x
)
=
h
(
x
)
+

∫
a
x

G
(
u
,
v
)
f
(
u
)

f(x) = h(x) + \int _a^x {G(u,v)f(u)}

\]
for

a
≤
x
≤
b

a \leq x \leq b

only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript a Superscript x Baseline Subscript x Baseline normal upper Pi Superscript v left-parenthesis 1 plus script upper G right-parenthesis upper G left-parenthesis u comma v right-parenthesis h left-parenthesis u right-parenthesis">

∫
a
x

x

Π
v

(
1
+

G

)
G
(
u
,
v
)
h
(
u
)

\int _a^x {_x{\Pi ^v}} (1 + \mathcal {G})G(u,v)h(u)

exists and is

f
(
x
)
−
h
(
x
)

f(x) - h(x)

for

a
≤
x
≤
b

a \leq x \leq b

, where

G

(
s
,
r
)
=
G
(
r
,
s
)

\mathcal {G}(s,r) = G(r,s)

.
Analytic Toeplitz operators with automorphic symbol
Authors: Abrahamse M. B.
Published: June 2010

Let

R
R

denote the annulus

{
z
:
1

/

2
>

|

z

|

>
1
}

\{ z:1/2 > |z| > 1\}

and let

π
\pi

be a holomorphic universal covering map from the unit disk onto

R
R

. It is shown that if

π
\pi

is a function of an inner function

ω
\omega

, that is, if

π
(
z
)
=
π
(
ω
(
z
)
)

\pi (z) = \pi (\omega (z))

, then

ω
\omega

is a linear fractional transformation. However, the analytic Toeplitz operator

T
π

{T_\pi }

has nontrivial reducing subspaces. These facts answer in the negative a question raised by Nordgren [10]. Let

ϕ
\phi

be the function

ϕ
(
z
)
=
π
(
z
)
−
3

/

4

\phi (z) = \pi (z) - 3/4

and let

ϕ
=
χ
F

\phi = \chi F

be the inner-outer factorization of

ϕ
\phi

. An operator

C
C

is produced which commutes with

T
ϕ

{T_\phi }

but does not commute with

T
χ

{T_\chi }

nor with

T
F

{T_F}

. This answers in the negative a question raised by Deddens and Wong [7]. The functions

π
\pi

and

ϕ
\phi

are both automorphic under the group of covering transformations for

π
\pi

and hence may be viewed as functions on the annulus

R
R

. This point of view is critical in these examples.
The double centralizer algebra as a linear space
Authors: Fontenot Robert A.
Published: June 2010

Let

A
A

be a

C

∗

{C^{\ast }}

-algebra and

M
(
A
)

M(A)

be the double centralizer algebra of

A
A

. The properties semireflexivity, nuclearity,

(

DF)

({\text {DF)}}

, and strict compactness of the unit ball are characterized in

M
(
A
)

M(A)

endowed with the strict topology.
Matrix commutators over an algebraically closed field
Authors: Gibson P. M.
Published: June 2010

Let

A
A

be an

n
n

-square matrix with zero trace over an algebraically closed field

F
F

, and let the characteristic of

F
F

not divide

n
n

. It is shown that

A
A

can be expressed as

A
=
X
Y
−
Y
X

A = XY - YX

where the eigenvalues of

X
X

and

Y
Y

may be arbitrarily specified as long as those of

X
X

are distinct.
On the structure of certain bounded linear operators
Authors: Allen G. D.
Published: June 2010

If every function

f
f

in the range of a bounded linear operator on

L
p

{L_p}

is equal to zero on a set of measure greater than a fixed number

ϵ
\epsilon

, it is shown that there is a common set of measure

ϵ
\epsilon

on which every function is zero. A decomposition theorem for such operators is proved.
Bounds for nearly best approximations
Authors: Wegmann Rudolf
Published: June 2010

Let

X
X

be a uniformly convex space and

ψ
\psi

be the inverse function of the modulus of convexity

δ
(
⋅
)

\delta ( \cdot )

. Assume here that

ψ
\psi

is a concave function. Let

V
V

be a linear subspace of

X
X

and let

f
f

in

X
X

be such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue equals 1 equals min left-brace right-brace colon minus minus times times vertical-bar vertical-bar f times times v vertical-bar vertical-bar colon times times times v epsilon upper V">

|

|

f

|

|

=
1
=
min
{

|

|

f
−
v

|

|

:
v
ϵ
V
}

||f|| = 1 = \min \{ ||f - v||:v\epsilon V\}

. Then for

0
>
δ
⩽
1

0 > \delta \leqslant 1

and for

v
v

in

V
V

with

|

|

f
−
v

|

|

⩽
1
+
δ

||f - v|| \leqslant 1 + \delta

, it follows that

|

|

v

|

|

⩽
K
⋅
ψ
(
δ
)

||v|| \leqslant K \cdot \psi (\delta )

. Let

T
T

be a compact Hausdorff-space and

V
V

a finite-dimensional subspace of

C
(
T
,
X
)

C(T,X)

. When

V
V

has the interpolation property

(

P
m

)

({P_m})

with

V
=
m
⋅
dim
⁡
X

V = m \cdot \dim X

, then the same type of estimate as above holds.
Disk-like products of 𝜆 connected continua. II
Authors: Hagopian Charles L.
Published: June 2010

R. H. Bing [3] proved that every atriodic, hereditarily decomposable, hereditarily unicoherent continuum is arc-like. Using this theorem, the author [5] showed that

λ
\lambda

connected continua

X
X

and

Y
Y

are arc-like when the topological product

X
×
Y

X \times Y

is disk-like. In this paper we consider products that have a more general mapping property. Suppose that

X
X

and

Y
Y

are

λ
\lambda

connected continua and that for each

ε
>
0

\varepsilon > 0

, there exists an

ε
\varepsilon

-map of

X
×
Y

X \times Y

into the plane. Then

X
X

is either arc-like or circle-like. Furthermore, if

X
X

is circle-like, then

Y
Y

is arc-like. Hence

X
×
Y

X \times Y

is either disk-like or annulus-like.
A refinement of Green’s theorem on the defect group of a 𝑃-block
Authors: Lam T. Y.
Published: June 2010

Let

D
D

be the defect group of a

p
p

-block of a finite group

G
G

. Let

P
,
Q

P,Q

be two

p
p

-Sylow groups of

G
G

containing

D
D

. Then there exist

x
,
y
,
z
∈

C
G

(
z
)

x,y,z \in {C_G}(z)

such that: (i)

z
z

is

p
p

-regular and

D
D

is a

p
p

-Sylow group of

C
G

(
z
)

{C_G}(z)

; (ii)

D
=

Q
x

∩
P
=
Q
∩

P
y

D = {Q^x} \cap P = Q \cap {P^y}

; and (iii)

z
=
x
y

z = xy

. This refines an earlier theorem of J. A. Green.
A theorem on nilpotent groups with restricted embeddings
Authors: Wehrhahn Karl H.
Published: June 2010

Suppose that

G
G

is a nonabelian group with a unique proper normal subgroup of some given order. It is proved that

G
G

is not contained as a normal subgroup within the Frattini subgroup of a finite

p
p

-group.
Residual solvability of an equation in nilpotent groups
Authors: Stebe Peter F.
Published: June 2010

Let

G
G

be a finitely generated nilpotent group. Let

S
1

{S_1}

and

S
2

{S_2}

be subgroups of

G
G

. Let

S
1

S
2

{S_1}{S_2}

be the set of all products

g
1

g
2

{g_1}{g_2}

, where

g
i

{g_i}

is an element of

S
i

{S_i}

. Let

g
g

be an element of

G
G

. It is shown that either

g
g

is an element of

S
1

S
2

{S_1}{S_2}

or there is a normal subgroup

N
N

of finite index in

G
G

such that

g
N

gN

does not meet

S
1

S
2

{S_1}{S_2}

. This result implies: (a) There is an algorithm to determine whether or not

g
g

is an element of

S
1

S
2

{S_1}{S_2}

. (b) Given elements

a
,
b

a,b

and

c
c

of

G
G

, there is an algorithm to determine whether there exist integers

n
n

and

m
m

such that

a
=

b
m

c
n

a = {b^m}{c^n}

. (c) Finitely generated nilpotent groups are subgroup separable (a result of K. Toh). (d) Given elements

a
a

and

b
b

of

G
G

and a subgroup

S
S

of

G
G

, there is an algorithm to determine whether or not

a
a

is an element of

S
b
S

SbS

.
On the symmetry of matrix algebras
Authors: Wichmann Josef
Published: June 2010

A

∗

^{\ast }

-algebra is called symmetric, if each element of the form

a

∗

a

{a^{\ast }}a

has nonnegative real spectrum. The study of locally compact groups with symmetric group algebras led to the following theorem: The tensoring of a Banach

∗

^{\ast }

-algebra with the

∗

^{\ast }

-algebra of all complex

n
×
n

n \times n

matrices preserves symmetry. In this note we prove, by a very simple algebraic argument, an analogue of it for arbitrary

∗

^{\ast }

-algebras.
Tietze-type theorems on monotone increasing sets
Authors: Stanek Jean Chan
Published: June 2010

The Tietze theorem on convex sets is generalized to monotone increasing sets and strictly monotone increasing sets, which include convex sets as a special case. The main theorem is that a closed connected set in

E
2

{E_2}

is monotone increasing if and only if it is locally monotone increasing. A similar result is proved for strictly monotone increasing sets.
Counting 3 by 𝑛 Latin rectangles
Authors: Bogart K. P. , Longyear J. Q.
Published: June 2010

A

k
k

by

n
n

rectangular array

A
A

is called a Latin rectangle if all the integers

1
,
2
,
…
,
n

1,2, \ldots ,n

appear in each row of

A
A

and if

k
k

distinct integers occur in each column of

A
A

. The number of

k
k

by

n
n

Latin rectangles is unknown for

k
⩾
4

k \geqslant 4

; Riordan has given a formula for the number of

3
3

by

n
n

rectangles in terms of the solutions of the derangement (or displacement) problem and the menage problem. In this paper we derive an elementary formula for the number of

3
3

by

n
n

Latin rectangles by using Möbius inversion. We include a table giving the approximate number of

3
3

by

n
n

Latin rectangles for

n
⩽
20

n \leqslant 20

. The table has exact values for

n
⩽
11

n \leqslant 11

.
The automorphisms of 𝑂⁺₄ (𝑉) in the anisotropic case
Authors: Connors Edward A.
Published: June 2010

The automorphisms of the rotation group of a quaternary anisotropic quadratic space defined over a field of characteristic not two are determined.
Connectivity of an interval
Authors: Mandelker Mark
Published: June 2010

Constructivizations are given for the classical theorem that an interval is connected in the sense that it cannot be the union of two nonvoid disjoint open subsets, and also for the theorem that an open set is the union of disjoint open intervals.
Extensions of continuous functions from dense subspaces
Authors: Blair Robert L.
Published: June 2010

Let

X
X

and

Y
Y

be topological spaces, let

S
S

be a dense subspace of

X
X

, and let

f
:
S
→
Y

f:S \to Y

be continuous. When

Y
Y

is the real line

R

{\mathbf {R}}

, the Lebesgue sets of

f
f

are used to provide necessary and sufficient conditions in order that the (bounded) function

f
f

have a continuous extension over

X
X

. These conditions yield the theorem of Taǐmanov (resp. of Engelking and of Blefko and Mrówka) which characterizes extendibility of

f
f

for

Y
Y

compact (resp. realcompact). In addition, an extension theorem of Blefko and Mrówka is sharpened for the case in which

X
X

is first countable and

Y
Y

is a closed subspace of

R

{\mathbf {R}}

.
A geometric interpretation of a classical group cohomology obstruction
Authors: Hill R. O.
Published: June 2010

For a non-Abelian group

G
G

, we show that the obstruction to the existence of an extension of

G
G

by

Π
\Pi

that induces

ϕ
:
Π
→

\phi :\Pi \to

Out

G
G

is also the

k
k

-invariant of the classifying space for

K
(
G
,
1
)

K(G,1)

-bundles.
Finite four-generated simple lattices contain all finite lattices
Authors: Poguntke Werner , Rival Ivan
Published: June 2010

Every finite lattice is embeddable in a finite, four-generated, simple lattice.
𝑝th powers of distinguished subfields
Authors: Heerema Nicholas
Published: June 2010

Let

k
⊃
s
⊃

k
0

k \supset s \supset {k_0}

be fields of characteristic

p
≠
0
,
k

/

k
0

p \ne 0,k/{k_0}

finitely generated and

s
s

a distinguished subfield. The field
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k 0 left-parenthesis k Superscript left-parenthesis n right-parenthesis Baseline right-parenthesis equals StartSet x element-of k vertical-bar x Superscript p Super Superscript t Superscript Baseline element-of k 0 left-parenthesis k Superscript p Super Superscript n plus t Superscript Baseline right-parenthesis for some t greater-than-or-slanted-equals 0 EndSet">

k
0

(

k

(
n
)

)
=
{
x
∈
k

|

x

p
t

∈

k
0

(

k

p

n
+
t

)

 for some 

t
⩾
0
}

{k_0}({k^{(n)}}) = \{ x \in k|{x^{{p^t}}} \in {k_0}({k^{{p^{n + t}}}}){\text { for some }}t \geqslant 0\}

has

k
0

(

s

p
n

)

{k_0}({s^{{p^n}}})

as a distinguished subfield and is maximal in

k
k

with respect to this property. Let

k
¯

0
s

\overline k _0^s

and

k
¯

0

{\bar k_0}

be, respectively, the separably algebraic closure and the algebraic closure of

k
0

{k_0}

in

k
k

. Then

k
¯

0

=

∩
k

k
0

(

k

(
n
)

)

{\overline k _0} = { \cap _k}{k_0}({k^{(n)}})

. Also

k
¯

0
s

=

k
¯

0

\overline k _0^s = {\bar k_0}

if and only if

k
0

(

k
p

)
⊃

k
0

(

k

(
n
)

)

{k_0}({k^p}) \supset {k_0}({k^{(n)}})

for some

n
n

. For

n
n

large
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k 0 left-parenthesis k Superscript left-parenthesis n right-parenthesis Baseline right-parenthesis equals ModifyingAbove k With bar Subscript 0 Baseline left-parenthesis k Superscript p Super Superscript n Superscript Baseline right-parenthesis">

k
0

(

k

(
n
)

)
=

k
¯

0

(

k

p
n

)

{k_0}({k^{(n)}}) = {\overline k _0}({k^{{p^n}}})

. The sequence

{

[

k
0

(

k

(
n
)

)
:

k
0

]
i

}
n

{\{ {[{k_0}({k^{(n)}}):{k_0}]_i}\} _n}

is decreasing, descending from

[
k
:

k
0

]
i

to 

[

k
:

¯

k
0

]
i

{[k:{k_0}]_i}\;{\text {to }}{[\overline {k:} {k_0}]_i}

in a finite number of steps. Examples are given which show: (1) that

k
0

(

k

(
n
)

)

{k_0}({k^{(n)}})

may have distinguished subfields not of the form

k
0

(

s

p
n

)

{k_0}({s^{{p^n}}})

; and, (2) how to construct

k

/

k
0

k/{k_0}

so that the sequence

{
[

k
0

(

k

(
n
)

:

k
0

]
i

}

\{ [{k_0}{({k^{(n)}}:{k_0}]_i}\}

has preassigned values.
A note on cocycles of unitary representations
Authors: Parry W. , Schmidt K.
Published: June 2010

Given a unitary representation

U
U

of a locally compact abelian group

G
G

, we investigate the relationship between two cocycles

a
1

,

a
2

:
V

a
1

=

a
2

+
b

{a_1},{a_2}:V{a_1} = {a_2} + b

for some unitary operator

V
V

commuting with

U
U

and some coboundary

b
b

. A necessary and sufficient condition is given in terms of canonical-finite measures defined on

G
−
1

G - 1

. These results are applied to the representation of

Z
Z

defined by the shift of a stationary Markov chain.
On 𝐻-closed spaces
Authors: Joseph James E.
Published: June 2010

A characterization of

H
H

-closed spaces in terms of projections is given along with relating properties.
A note on the construction of simply-connected 3-manifolds as branched covering spaces of 𝑆³
Authors: Birman Joan S.
Published: June 2010

Let

K
K

be a knot in

S
3

{S^3}

, and let

ω
:

π
1

(

S
3

−
K
)
→

Σ
n

\omega :{\pi _1}({S^3} - K) \to {\Sigma _n}

be a transitive representation into the symmetric group

Σ
n

{\Sigma _n}

on

n
n

letters. The pair

(
K
,
ω
)

(K,\omega )

defines a unique closed, connected orientable

3
3

-manifold

M
(
K
,
ω
)

M(K,\omega )

, which is represented as an

n
n

-sheeted covering space of

S
3

{S^3}

, branched over

K
K

. A procedure is given for representing

M
(
K
,
ω
)

M(K,\omega )

by a Heegard splitting, and a formula is given for computing the genus of that Heegard splitting of

M
(
K
,
ω
)

M(K,\omega )

. This formula is then applied to the

3
3

-sheeted irregular covering spaces studied by Hilden (Bull. Amer. Math. Soc. 80 (1974), 1243-1244) and Montesinos (Bull. Amer. Math. Soc. 80 (1974), 845-846), and, also, Tesis (Univ. de Madrid, 1971) to show that these particular covering spaces cannot yield counterexamples to the Poincaré Conjecture if the branch set has bridge number

>
4

> 4

.
A characterization of the Cauchy type
Authors: Knight Frank B.
Published: June 2010

It is proved that a random variable

X
X

is of Cauchy type if and only if

(
a
X
+
b
)

(
c
X
+
d

)

−
1

(aX + b){(cX + d)^{ - 1}}

has the same type as

X
X

for every

a
,
b
,
c
,
d

a,b,c,d

with

a
d
−
b
c
≠
0

ad - bc \ne 0

.
Dominated estimates in Hilbert space
Authors: Akcoglu M. A. , Miller H. D. B.
Published: June 2010

Let

U
U

be a unitary operator on a Hilbert space

H
H

, and let

A
n

(
U
)
,
n
=
1
,
2
,
…

{A_n}(U),n = 1,2, \ldots

, be the Cesàaro means of

U
U

. It is shown that

Σ

n
=
1

∞

P
n

A
n

(
U
)

\Sigma _{n = 1}^\infty {P_n}{A_n}(U)

is bounded for every sequence of mutually orthogonal projections

P
n

,
n
=
1
,
2
,
…

{P_n},n = 1,2, \ldots

, if and only if

1
1

is not a limit point of the spectrum of

U
U

. The proof is obtained by adapting ideas of Menchoff and Burkholder to show that for any orthonormal sequence

f
n

,
n
=
0
,
±
1
,
±
2
,
…

{f_n},n = 0, \pm 1, \pm 2, \ldots

, in

H
H

, there is an orthonormal sequence

g
n

,
n
=
1
,
2
,
…

{g_n},n = 1,2, \ldots

, such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript k equals 1 Overscript n Endscripts StartAbsoluteValue left-parenthesis f 1 plus f 2 plus midline-horizontal-ellipsis plus f Subscript k Baseline comma g Subscript k Baseline right-parenthesis EndAbsoluteValue squared greater-than-or-slanted-equals one thirty-sixth n left-parenthesis log n right-parenthesis squared period">

∑

k
=
1

n

|

(

f
1

+

f
2

+
⋯
+

f
k

,

g
k

)

|

2

⩾

1

36

n

(
log
⁡
n
)

2

.

\sum \limits _{k = 1}^n {|({f_1} + {f_2} + \cdots + {f_k},{g_k}){|^2} \geqslant \frac {1} {{36}}n{{(\log n)}^2}.}

\]
Diagonalizable normal operators
Authors: Williams J. P.
Published: June 2010

If the image

φ
(
A
)

\varphi (A)

of a normal operator

A
A

on a separable Hilbert space

K

\mathcal {K}

is a diagonal operator for some nonzero representation

φ
\varphi

of

B
(

K

)

B(\mathcal {K})

(that annihilates the compact operators), then

A
A

must itself be a diagonal operator on

K

\mathcal {K}

(with countable spectrum). This yields an “algebraic” characterization of the closure of the range of a derivation induced by a diagonal operator.
Products of sequential spaces
Authors: Tanaka Yoshio
Published: June 2010

S. P. Franklin introduced the notion of a sequential space and characterized such spaces as being precisely the quotient images of metric spaces. In this paper we investigate a necessary and sufficient condition for the product of a first countable space with a sequential space to be sequential, and we consider the property “sequential space” in

X
ω

{X^\omega }

.
Recursiveness in 𝑃¹₁ paths through 𝒪
Authors: Friedman Harvey
Published: June 2010

Kleene’s

O

\mathcal {O}

is recursive in some

Π
1
1

\Pi _1^1

path through

O

\mathcal {O}

. If every hyp set is recursive in a given

Π
1
1

\Pi _1^1

set, then

O

\mathcal {O}

is recursive in its triple jump.
Euler characteristics of complete intersections
Authors: Ewing John , Moolgavkar Suresh
Published: June 2010

We point out that a conjecture of Chen and Ogiue, regarding the Euler characteristic of complete intersections, is a simple consequence of a theorem of Hirzebruch.
Group rings with simple augmentation ideals
Authors: Bonvallet K. , Hartley B. , Passman D. S. , Smith M. K.
Published: June 2010

Group algebras of algebraically closed groups and of universal groups are shown to have simple augmentation ideals and to be primitive.
An application of representation theory to PI-algebras
Authors: Olsson Jørn B. , Regev Amitai
Published: June 2010

By realizing that the multilinear identities of degree

n
n

of a

P
I

PI

-algebra form a left ideal in the group algebra

F
[

S
n

]

F[{S_n}]

, it is possible sometimes to use the representation theory of the symmetric group

S
n

{S_n}

in the study of

T
T

-ideals and

P
I

PI

-algebras. In this note we demonstrate this method by proving: Theorem. If the codimensions of a

P
I

PI

-algebra are bounded, then they are eventually bounded by

1
1

.
Uniqueness of embeddings of certain induced modules
Authors: Lepowsky J.
Published: June 2010

A theorem of D.-N. Verma on the uniqueness of embeddings of certain algebraically induced modules is generalized from the case of Borel subalgebras of semisimple Lie algebras to the case of arbitrary parabolic subalgebras.
Conjugacy in abelian-by-cyclic groups
Authors: Boler James
Published: June 2010

It is shown that each finitely generated torsion-free abelian-by-cyclic group has solvable conjugacy problem. This is done by showing that solving the conjugacy problem for these groups is equivalent to a certain decision problem for modules over the complex group algebra of an infinite cyclic group.
The existence of conjugate points for selfadjoint differential equations of even order
Authors: Lewis Roger T.
Published: June 2010

This paper presents sufficient conditions on the coefficents of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript 2 n Baseline y equals normal upper Sigma Subscript k equals 0 Superscript n Baseline left-parenthesis negative 1 right-parenthesis Superscript n minus k Baseline left-parenthesis p Subscript k Baseline y Superscript left-parenthesis n minus k right-parenthesis Baseline right-parenthesis Superscript left-parenthesis n minus k right-parenthesis">

L

2
n

y
=

Σ

k
=
0

n

(
−
1

)

n
−
k

(

p
k

y

(
n
−
k
)

)

(
n
−
k
)

{L_{2n}}y = \Sigma _{k = 0}^n{( - 1)^{n - k}}{({p_k}{y^{(n - k)}})^{(n - k)}}

which insure that

L

2
n

y
=
0

{L_{2n}}y = 0

has conjugate points

η
(
a
)

\eta (a)

for all

a
>
0

a > 0

. The main theorem implies that

(
−
1

)
n

y

(
2
n
)

+
p
y
=
0

{( - 1)^n}{y^{(2n)}} + py = 0

has conjugate points

η
(
a
)

\eta (a)

for all

a
>
0

a > 0

when

∫
∞

x
α

p
(
x
)
d
x
=
−
∞

{\smallint ^\infty }{x^\alpha }p(x)dx = - \infty

for some

α
>
2
n
−
1

\alpha > 2n - 1

with no sign restrictions on

p
(
x
)

p(x)

.
Weighted norm inequalities for Hilbert transforms and conjugate functions of even and odd functions
Authors: Andersen Kenneth F.
Published: June 2010

It is well known that the Hilbert tranformation and the conjugate function operator restricted to even (odd) functions define bounded linear operators on weighted

L
p

{L^p}

spaces under more general conditions than is the case for the unrestricted operators. In analogy with recent results of Hunt, Muckenhoupt and Wheeden for the Hilbert transform and the conjugate function operator, we obtain necessary and sufficient conditions in order that these restricted operators should satisfy weighted weak-type inequalities and hence also necessary and sufficient conditions in order that these operators should be bounded on weighted

L
p

{L^p}

spaces for

1
>
p
>
∞

1 > p > \infty

.
Inequalities for max|𝑆_{𝑘}|/𝑏_{𝑘} where 𝑘∈𝑁^{𝑟}
Authors: Shorack Galen R. , Smythe R. T.
Published: June 2010

Lemma

1
1

presents a powerful general inequality for

max

|

S

k

|

/

b

k

\max |{S_{\mathbf {k}}}|/{b_{\mathbf {k}}}

. This is applied in multidimensional time to sums of independent random variables and martingales to yield both old and new inequalities of the Doob, Hájek-Rényi, Skorokhod and Marcinkiewicz-Zygmund types. A brief application is made to the partial sum process.
Erratum to: “Generalized relative difference sets” (Proc. Amer. Math. Soc. 25 (1970), 46–50) by Payne
Authors: Lam Clement W. H. , Payne Stanley E.
Published: June 2010

This note gives the correction of a theorem previously published. A counterexample is also given for the theorem as originally stated.
On the nonexistence of groups with extra-special commutator subgroup
Authors: Miller Michael D.
Published: June 2010

In this paper, we extend a result of Joseph and Finkelstein and show that there is no group

G
G

such that

G
′

G’

is an extra-special

p
p

-group of exponent

>
p

> p

(

p
p

odd).
First countable hyperspaces
Authors: Smithson R. E.
Published: June 2010

An example is given which shows that the space of compact subsets

K

(
X
)

\mathcal {K}(X)

of a first countable space

X
X

need not be first countable in the finite topology. Further, it is shown that if

K

(
X
)

\mathcal {K}(X)

is first countable then each compact subset of

X
X

is separable. Finally a characterization of

K

(
X
)

\mathcal {K}(X)

first countable in terms of a weak second countability condition is derived.
Notions of spanning surface equivalence
Authors: Eisner Julian R.
Published: June 2010

We show that two natural notions of spanning surface equivalence differ for minimal spanning surfaces of knots in

S
3

{S^3}

.
Polynomial identities of incidence algebras
Authors: Feinberg Robert B.
Published: June 2010

In this paper we determine the polynomial identities satisfied by incidence algebras. One of our results is logically equivalent to the Amitsur-Levitzki Theorem on the polynomial identities satisfied by

K
n

{K_n}

, the algebra of of

n
×
n

n \times n

matrices over a field

K
K

.
Conditions for the commutativity of semigroups
Authors: Kowol G.
Published: June 2010

Let

S
S

be a semigroup. Then by a theorem of Tully [7]:

S
S

is a commutative semigroup iff

a
b
=

b
n

a
m

ab = {b^n}{a^m}

for all

a
,
b
∈
S

a,b \in S

(

m
,
n
⩾
1

m,n \geqslant 1

, fixed). We prove the following:

S
S

is a commutative semigroup iff

a
b
=

b

n
(
a
,
b
)

a

m
(
a
,
b
)

ab = {b^{n(a,b)}}{a^{m(a,b)}}

for all

a
,
b
∈
S

a,b \in S

, where one of the exponents

n
(
a
,
b
)

n(a,b)

and

m
(
a
,
b
)

m(a,b)

is constant and the other is independent of

a
a

or

b
b

.
Absolute continuity characterization sets
Authors: Appling William D. L.
Published: June 2010

Suppose

U
U

is a set,

F

{\mathbf {F}}

is a field of subsets of

U
,

p

A
B

U,{\mathfrak {p}_{AB}}

is the set of all real-valued, bounded finitely additive functions on

F

{\mathbf {F}}

, and for each

ρ
\rho

in

p

A
B

,

A

ρ

{\mathfrak {p}_{AB}},{\mathcal {A}_\rho }

is the set of all elements of

p

A
B

{\mathfrak {p}_{AB}}

absolutely continuous with respect to

ρ
,

p

A
+

\rho ,\mathfrak {p}_A^ +

is the set of all nonnegative-valued elements of

p

A
B

{\mathfrak {p}_{AB}}

, and

p

B

{\mathfrak {p}_B}

is the set of all functions from

F

{\mathbf {F}}

into

exp
⁡
(

R

)

\exp ({\mathbf {R}})

with bounded range union. An extension of a previous absolute continuity characterization theorem of the author (Proc. Amer. Math. Soc. 18 (1967), 94-99) is given in the form of a characterization of those subsets

S
S

of

p

A
B

{\mathfrak {p}_{AB}}

having the property that if each of

ξ
\xi

and

μ
\mu

is in

p

A
+

\mathfrak {p}_A^ +

, then

ξ
\xi

is in

A

μ

{\mathcal {A}_\mu }

iff it is true that if

α
\alpha

is in

p

B

,

∫
U

α
(
I
)
μ
(
I
)

{\mathfrak {p}_B},{\smallint _U}\alpha (I)\mu (I)

and

∫
U

α
(
I
)
ξ
(
I
)

{\smallint _U}\alpha (I)\xi (I)

exist and the function

∫
α
μ

\smallint \alpha \mu

is in

S
S

, then

∫
α
ξ

\smallint \alpha \xi

is in

S
S

.
Two 𝑅-closed spaces revisited
Authors: Hechler Stephen H.
Published: June 2010

Recently, R. M. Stephenson has used the Continuum Hypothesis to construct two

R
R

-closed, separable regular, first countable, noncompact Hausdorff spaces. We show that the assumption of the Continuum Hypothesis can be removed by replacing a lemma used in the original construction to deal with arbitrary almost-disjoint families by the construction of a particular almost-disjoint family. We also show that while these spaces always have cardinality

c

{\mathbf {c}}

, it is at least consistent with the negation of the Continuum Hypothesis that there exist spaces with the same properties, but which have cardinality

ℵ
1

{\aleph _1}

. We conclude with some consistency results concerning relationships between open filter bases and generalizations of the notions of feeble compactness and Lindelöfness.
Absolutely stable games
Authors: Weber Robert James
Published: June 2010

Absolutely stable games, in which every monotone chain of domination reduces to direct domination, are explicitly characterized. Simple games, and

n
n

-person games in which all minimal-vital coalitions contain at least

n
−
1

n - 1

players, are seen to satisfy the characterization.
Projective extensions of Banach algebras
Authors: Magid Andy R.
Published: June 2010

It is shown that if

A
A

is a commutative Banach algebra and

B
B

a faithful

A
A

-algebra finitely generated and projective as an

A
A

-module then

B
B

can be endowed with a structure of Banach algebra extending that of

A
A

.
Compact composition operators on 𝐵(𝐷).
Authors: Swanton Donald W.
Published: June 2010

Let

D
D

be a domain in the complex plane,

ϕ
:
D
→
D

\phi :D \to D

be analytic, and

B
(
D
)

B(D)

be the uniform algebra of bounded analytic functions on

D
D

with maximal ideal space

M
M

. The composition operator

C
ϕ

(
f
)
=
f
∘
ϕ

{C_\phi }(f) = f \circ \phi

is compact if and only if the weak* and norm closures of

ϕ
(
D
)

\phi (D)

coincide if and only if whenever the Euclidean closure of

ϕ
(
D
)

\phi (D)

contains a point

λ
\lambda

of the boundary of

D
D

then each

f
∈
B
(
D
)

f \in B(D)

extends continuously from

ϕ
(
D
)

\phi (D)

to

λ
\lambda

. If

C
ϕ

{C_\phi }

is compact, then either

ϕ
\phi

fixes a point of

D
D

or else the adjoint of

C
ϕ

{C_\phi }

fixes a point of

M
M

.
Pseudocompact spaces and functionally determined uniformities
Authors: Talamo Rodolfo
Published: June 2010

A topological space is pseudocompact if and only if every admissible uniformity is functionally determined. We construct, on the discrete countable space

N
N

, an admissible (pseudo)-metric uniformity which is not functionally determined.
On ℎ𝑜𝑚dim𝑀𝑈_{*}(𝑋×𝑌)
Authors: O’Neill Duane
Published: June 2010

Let

p
p

be a prime and

B

Z

/

p

B{\mathbf {Z}}/p

the classifying space for the cyclic group

Z

/

p

{\mathbf {Z}}/p

of prime order

p
p

. A finite complex

X
X

is constructed such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="hom dot dimension Subscript upper M upper U Sub Subscript asterisk Subscript Baseline upper M upper U Subscript asterisk Baseline left-parenthesis upper X times upper B bold upper Z slash p right-parenthesis greater-than hom dot dimension Subscript upper M upper U Sub Subscript asterisk Subscript Baseline upper M upper U Subscript asterisk Baseline left-parenthesis upper X right-parenthesis plus hom dot dimension Subscript upper M upper U Sub Subscript asterisk Subscript Baseline upper M upper U Subscript asterisk Baseline left-parenthesis upper B bold upper Z slash p right-parenthesis period">

hom
⋅

dim

M

U
∗

M

U
∗

(
X
×
B

Z

/

p
)
>
hom
⋅

dim

M

U
∗

M

U
∗

(
X
)
+
hom
⋅

dim

M

U
∗

M

U
∗

(
B

Z

/

p
)
.

\hom \cdot {\dim _{M{U_ \ast }}}M{U_ \ast }(X \times B{\mathbf {Z}}/p) > \hom \cdot {\dim _{M{U_ \ast }}}M{U_ \ast }(X) + \hom \cdot {\dim _{M{U_ \ast }}}M{U_ \ast }(B{\mathbf {Z}}/p).

\]
Dynamical systems with cross-sections
Authors: Neumann Dean A.
Published: June 2010

The problem of classifying dynamical systems (flows) with global cross-sections in terms of the associated diffeomorphisms of the cross-sections is considered. Suppose that, for

i
=
1
,
2
,

ϕ
i

i = 1,2,{\phi _i}

is a

C
r

{C^r}

flow

(
r
⩾
0
)

(r \geqslant 0)

on the

C
r

{C^r}

manifold

M
i

{M_i}

that admits a global cross-section

S
i

⊆

M
i

{S_i} \subseteq {M_i}

with associated diffeomorphism (’first return map’)

d
i

{d_i}

. If rank

(

H
1

(

M
1

;

Z

)
)
=
1

({H_1}({M_1};{\mathbf {Z}})) = 1

, then

(

M
1

,

ϕ
1

)

({M_1},{\phi _1})

is

C
s

{C^s}

equivalent

(
s
⩽
r
)

(s \leqslant r)

to

(

M
2

,

ϕ
2

)

({M_2},{\phi _2})

if and only if

d
1

{d_1}

is

C
s

{C^s}

conjugate to

d
2

{d_2}

. If rank

(

H
1

(

M
1

;

Z

)
)
≠
1

({H_1}({M_1};{\mathbf {Z}})) \ne 1

and

ϕ
1

{\phi _1}

has a periodic orbit, then there are infinitely many global cross-sections

T
i

⊆

M
1

{T_i} \subseteq {M_1}

of

ϕ
1

{\phi _1}

, such that the associated diffeomorphisms are pairwise nonconjugate.
Hyperfiniteness and the Halmos-Rohlin theorem for nonsingular Abelian actions
Authors: Feldman J. , Lind D. A.
Published: June 2010

Theorem 1. Let the countable abelian group

G
G

act nonsingularly and aperiodically on Lebesgue space

(
X
,
μ
)

(X,\mu )

. Then for each finite subset

A
⊂
G

A \subset G

and

ε
>
0
∃

\varepsilon > 0\exists

finite

B
⊂
G

B \subset G

and

F
⊂
X

F \subset X

with

{
b
F
:
b
∈
B
}

\{ bF:b \in B\}

disjoint and

μ
[
(

∩

a
∈
A

B
−
a
)
F
]
>
1
−
ε

\mu [({ \cap _{a \in A}}B - a)F] > 1 - \varepsilon

. Theorem 2. Every nonsingular action of a countable abelian group on a Lebesgue space is hyperfinite.
The local resolvent set of a locally Lipschitzian transformation is open
Authors: May E. Lee
Published: June 2010

The purpose of this paper is to prove that if

p
p

is a point of a complex Banach space

H
H

at which the nonlinear transformation

T
T

on

H
H

is locally Lipschitzian, then the local resolvent set of

T
T

at

p
p

is open.
On a problem of Turan about polynomials
Authors: Pierre R. , Rahman Q. I.
Published: June 2010

It is shown that if

p
n

(
x
)

{p_n}(x)

is a polynomial of degree

n
n

whose graph on the interval

−
1
>
x
>
1

- 1 > x > 1

is contained in the unit disk then the absolute value of its second derivative cannot exceed

2
3

(
n
−
1
)
(
2

n
2

−
4
n
+
3
)

\tfrac {2}{3}(n - 1)(2{n^2} - 4n + 3)

on

[
−
1
,
1
]

[ - 1,1]

.
Which operators are similar to partial isometries?
Authors: Fialkow L. A.
Published: June 2010

Let

H

\mathcal {H}

denote a separable, infinite dimensional complex Hilbert space and let

L

(

H

)

\mathcal {L}(\mathcal {H})

denote the algebra of all bounded linear operators on

H

\mathcal {H}

. Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P equals left-brace upper T in script upper L left-parenthesis script upper H right-parenthesis vertical-bar r left-parenthesis upper T right-parenthesis greater-than 1 and upper T is similar to a partial isometry with infinite rank right-brace">

P

=
{
T

 in 

L

(

H

)

|

r
(
T
)
>
1

 and 

T

is similar to a partial isometry with infinite rank
}

\mathcal {P} = \{ T{\text { in }}\mathcal {L}(\mathcal {H})|r(T) > 1{\text { and }}T{\text {is similar to a partial isometry with infinite rank} \}}

; let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S equals left-brace upper S in script upper L left-parenthesis script upper H right-parenthesis vertical-bar r left-parenthesis upper S right-parenthesis greater-than 1 comma range left-parenthesis upper S right-parenthesis is closed comma and rank left-parenthesis upper S right-parenthesis equals nullity left-parenthesis upper S right-parenthesis equals corank left-parenthesis upper S right-parenthesis equals normal alef 0 right-brace">

S

=
{
S

 in 

L

(

H

)

|

r
(
S
)
>
1
,

range

(
S
)

 is closed, and rank

(
S
)
=

nullity

(
S
)
=

corank

(
S
)
=

ℵ
0

}

\mathcal {S} = \{ S{\text { in }}\mathcal {L}(\mathcal {H})|r(S) > 1,{\text {range}}(S){\text { is closed, and rank}}(S)= {\text {nullity}}(S)= {\text {corank}}(S)={\aleph _0}\}

. It is conjectured that

P

=

S

\mathcal {P} = \mathcal {S}

and it is proved that

P

⊂

S

⊂

P

−

\mathcal {P} \subset \mathcal {S} \subset {\mathcal {P}^ - }

.