bluuebunny/crossref_metadata_2025 huggingface dataset:
Strict topology and 𝑃-spaces
Authors: Khurana Surjit Singh , Choo Seki Alexander
Published: June 2010

For a completely regular Hausdorff space

X
X

and a normed space

E
E

, let

C
b

(
X
,

E
)

{C_b}(X,\,E)

be the space of all bounded continuous functions from

X
X

into

E
E

with strict topology

β
0

{\beta _0}

. It is proved that if

X
X

is a

P
P

-space,

(

C
b

(
X
,
E
)
,

β
0

)

({C_b}(X,E),{\beta _0})

is Mackey; if, in addition,

E
E

is complete, then

(

C
b

(
X
,
E
)
,

β
0

)

({C_b}(X,E),{\beta _0})

is strongly Mackey.
Hyperspaces of topological vector spaces: their embedding in topological vector spaces
Authors: Prem Prakash , Sertel Murat R.
Published: June 2010

Let

L
L

be a real (Hausdorff) topological vector space. The space

K

[
L
]

\mathcal {K}[L]

of nonempty compact subsets of

L
L

forms a (Hausdorff) topological semivector space with singleton origin when

K

[
L
]

\mathcal {K}[L]

is given the uniform (equivalently, the finite) hyperspace topology determined by

L
L

. Then

K

[
L
]

\mathcal {K}[L]

is locally compact iff

L
L

is so. Furthermore,

K

Q

[
L
]

\mathcal {K}\mathcal {Q}[L]

, the set of nonempty compact convex subsets of

L
L

, is the largest pointwise convex subset of

K

[
L
]

\mathcal {K}[L]

and is a cancellative topological semivector space. For any nonempty compact and convex set

X
⊂
L

X \subset L

, the collection

K

Q

[
X
]
⊂

K

Q

[
L
]

\mathcal {K}\mathcal {Q}[X] \subset \mathcal {K}\mathcal {Q}[L]

is nonempty compact and convex.

L
L

is iseomorphically embeddable in

K

Q

[
L
]

\mathcal {K}\mathcal {Q}[L]

and, in turn, there is a smallest vector space

L

\mathcal {L}

in which

K

Q

[
L
]

\mathcal {K}\mathcal {Q}[L]

is algebraically embeddable (as a cone). Furthermore, when

L
L

is locally convex,

L

\mathcal {L}

can be given a locally convex vector topology

I

\mathcal {I}

such that the algebraic embedding of

K

Q

[
L
]

\mathcal {K}\mathcal {Q}[L]

in

L

\mathcal {L}

is an iseomorphism, and then

L

\mathcal {L}

is normable iff

L
L

is so; indeed,

I

\mathcal {I}

can be so chosen that, when

L
L

is normed, the embedding of

L
L

in

K

Q

[
L
]

\mathcal {K}\mathcal {Q}[L]

and that of

K

Q

[
L
]

\mathcal {K}\mathcal {Q}[L]

in

L

\mathcal {L}

are both iseometries.
Characterizations of minimal Hausdorff spaces
Authors: Joseph James E.
Published: June 2010

Two new characterizations of minimal Hausdorff spaces are given along with some relating properties and examples.
Representing homology classes of closed orientable surfaces
Authors: Meyerson Mark D.
Published: June 2010

We prove that the primitive elements of the first homology group of a genus

g
g

torus can be represented by a simple closed curve.
Stationary points of (𝑍₂)^{𝑘}-actions
Authors: Capobianco Frank L.
Published: June 2010

This paper classifies up to bordism the manifolds with

(

Z
2

)
k

{({Z_2})^k}

-action fixing

R
P
(
n
)
,

C
P
(
n
)

RP(n),\;CP(n)

, or

S
n

{S^n}

.
The equivalence of various Lipschitz conditions on the weighted mean oscillation of a function
Authors: Lotkowski Edward P. , Wheeden Richard L.
Published: June 2010

The main result establishes the equivalence of various Lipschitz-like conditions on the weighted mean oscillation over cubes of a function. Previously known results are obtained as special cases.
Unbounded derivations of group 𝐶*-algebras
Authors: Lance Christopher , Niknam Assadollah
Published: June 2010

We describe a construction which yields unbounded derivations and strongly continuous one-parameter automorphism groups of certain group

C

∗

{C^{\ast }}

-algebras. As an application, we show that a simple

C

∗

{C^{\ast }}

-algebra can have an automorphism group which is not approximately inner.
Decomposing algebraic vector bundles on the projective line
Authors: Hanna Charles C.
Published: June 2010

V

(
R
)

\mathcal {V}(R)

denotes the category of algebraic vector bundles on

P

R
1

,

R

{\mathbf {P}}_R^1,\;R

a commutative, noetherian ring. If

K
K

is a field, it is known that any

F

∈

V

(
K
)

\mathcal {F} \in \mathcal {V}(K)

is isomorphic to a (unique) direct sum of line bundles. If

p

∈
Spec
⁡
R

\mathfrak {p} \in \operatorname {Spec} R

and

K
(

p

)

K(\mathfrak {p})

is the quotient field of

R

/

p

R/\mathfrak {p}

, any

F

∈

V

(
R
)

\mathcal {F} \in \mathcal {V}(R)

induces a bundle in

V

(
K
(

p

)
)

\mathcal {V}(K(\mathfrak {p}))

, and so a decomposition into line bundles. If the decomposition is the same for each

p

,

F

\mathfrak {p},\;\mathcal {F}

is said to be uniform. It is shown that if

R
R

is reduced, uniform vector bundles on

P

R
1

{\mathbf {P}}_R^1

are sums of tensor products of (pullbacks of) bundles on

Spec
⁡
R

\operatorname {Spec} R

with line bundles on

P

R
1

{\mathbf {P}}_R^1

.
A semidirect product decomposition for certain Hopf algebras over an algebraically closed field
Authors: Molnar Richard K.
Published: June 2010

Let

H
H

be a finite dimensional Hopf algebra over an algebraically closed field. We show that if

H
H

is commutative and the coradical

H
0

{H_0}

is a sub Hopf algebra, then the canonical inclusion

H
0

→
H

{H_0} \to H

has a Hopf algebra retract; or equivalently, if

H
H

is cocommutative and the Jacobson radical

J
(
H
)

J(H)

is a Hopf ideal, then the canonical projection

H
→
H

/

J
(
H
)

H \to H/J(H)

has a Hopf algebra section.
Largest normal neighborhoods
Authors: Ozols V.
Published: June 2010

It is well known that the largest normal neighborhood of a point in a compact Riemannian manifold is a Euclidean cell, that is, homeomorphic to the open unit ball. In this paper it is proved that this normal neighborhood is in fact

C
∞

{C^\infty }

diffeomorphic to the open unit ball. The method is to paste together a sequence of

C
∞

{C^\infty }

radial dilations which combine to engulf an open ball or all of

R

n

{{\mathbf {R}}^n}

.
Differentiability through change of variables
Authors: Bruckner A. M. , Goffman C.
Published: June 2010

A real function

f
f

on

[
0
,

1
]

[0,\;1]

can be transformed by a homeomorphism into a differentiable function with bounded derivative if and only if

f
f

is continuous and of bounded variation. This condition does not suffice for

f
f

to be transformed into a continuously differentiable function. The additional condition for this to hold is found and the theorem is proved.
Banach spaces on which nontrivial Saks spaces exist
Authors: Labuda Iwo
Published: June 2010

A necessary and sufficient condition for a Banach space to have on itself a nontrivial Saks space is given.
𝑅-automorphisms of 𝑅[𝐺] for 𝐺 abelian torsion-free
Authors: Lantz David C.
Published: June 2010

Let

R
R

be a commutative ring with identity and

G
G

a torsion-free abelian group. This note describes for a reduced

R
R

the group of

R
R

-automorphisms of the group ring

R
[
G
]

R[G]

when either

R
R

has finitely many idempotents or

G
G

has finite torsion-free rank. It also describes the

R
R

-automorphisms of

R
[
G
]

R[G]

for a general

R
R

and

G
G

finitely generated free.
When midsets are manifolds
Authors: Loveland L. D.
Published: June 2010

The midset

M
M

of two disjoint closed subsets

A
A

and

B
B

of

n
n

-space

E
n

{E^n}

is defined as the set of all points of

E
n

{E^n}

having equal distances to both

A
A

and

B
B

. Such midsets are not always manifolds, but when either

A
A

or

B
B

is a convex set it follows that

M
M

is homeomorphic to an open subset of an

(
n
−
1
)

(n - 1)

-sphere

S

n
−
1

{S^{n - 1}}

. Furthermore, in this situation

M
M

will be homeomorphic to

S

n
−
1

{S^{n - 1}}

if and only if the convex set

A
A

is bounded and lies in the interior of the convex hull

C
(
B
)

C(B)

of

B
B

. If

A
A

is a singleton set and

r
r

is the dimension of the smallest Euclidean flat

P
P

in

E
n

{E^n}

containing

A
∪
B

A \cup B

, then

P
∩
M

P \cap M

is an

(
r
−
1
)

(r - 1)

-sphere or an open

(
r
−
1
)

(r - 1)

-cell depending upon whether or not

A
A

lies in the interior (relative to

P
P

) of

C
(
B
)

C(B)

. In either case

M
=
(
P
∩
M
)
×

E

n
−
r

M = (P \cap M) \times {E^{n - r}}

. A manifold lying in a midset in

E
3

{E^3}

is always tamely embedded, as are

ε
\varepsilon

-boundaries of certain special subsets of

E
n

{E^n}

.
Dependence on dimension of a constant related to the Grötzsch ring
Authors: Anderson Glen D.
Published: June 2010

For the constant

λ
n

=

lim

a
→
0

(
mod

R

G
,
n

(
a
)
+
log
⁡
a
)

{\lambda _n} = {\lim _{a \to 0}}(\bmod {R_{G,n}}(a) + \log a)

associated with the Grötzsch extremal ring

R

G
,
n

{R_{G,n}}

in euclidean

n
n

-space, we obtain the limit

lim

n
→
∞

λ
n

1

/

n

=
e

{\lim _{n \to \infty }}\lambda _n^{1/n} = e

.
Submanifolds of a Riemannian manifold with semisymmetric metric connections
Authors: Nakao Zensho
Published: June 2010

We derive the Gauss curvature equation and the Codazzi-Mainardi equation with respect to a semisymmetric metric connection on a Riemannian manifold and the induced one on a submanifold. We then generalize the theorema egregium of Gauss.
On asymptotic values of analytic functions on Riemann surfaces
Authors: Niimura Mikio
Published: June 2010

Some generalizations of Lindelöf’s theorems on asymptotic values of bounded analytic functions are given on subregions of Riemann surfaces.
On proximinal sets of normal operators
Authors: Rogers Donald D.
Published: June 2010

It is shown that neither the set of normal operators nor the set of orthogonal projections is a proximinal subset of the space of bounded operators on an infinite-dimensional Hilbert space.
Joins and intersections of ideals of compact operators
Authors: Schweinsberg Allen
Published: June 2010

It is shown that certain well-known ideals of compact operators are the intersection of a decreasing, countable family of strictly larger ideals. Also, it is shown that if

T
1

{T_1}

and

T
2

{T_2}

are compact operators, neither of which lies in the principal ideal generated by the other, and if

I

\mathcal {I}

is an arbitrary countably generated ideal, then there exist ideals

J

1

{\mathcal {J}_1}

and

J

2

{\mathcal {J}_2}

such that

I

⊆

J

1

∨

J

2

\mathcal {I} \subseteq {\mathcal {J}_1} \vee {\mathcal {J}_2}

and

T
i

∉

J

i

,

i
=
1
,

2

{T_i} \notin {\mathcal {J}_i},\;i = 1,\,2

.
The regular group 𝐶*-algebra for real-rank one groups
Authors: Boyer Robert , Martin Robert
Published: June 2010

Let

G
G

be a connected semisimple real-rank one Lie group with finite center and let

C
ρ

∗

(
G
)

C_\rho ^{\ast }(G)

denote the regular group

C

∗

{C^{\ast }}

-algebra of

G
G

. In this paper a complete description of the structure of

C
ρ

∗

(
G
)

C_\rho ^{\ast }(G)

is obtained.
Monotone and oscillatory solution of 𝑦⁽ⁿ⁾+𝑝𝑦=0
Authors: Kim W. J.
Published: June 2010

Monotone and oscillatory behaviors of the solutions with the property that

y
(
x
)

/

x
2

→
0

y(x)/{x^2} \to 0

as

x
→
∞

x \to \infty

or

y
(
x
)

/

x
→
0

y(x)/x \to 0

as

x
→
∞

x \to \infty

are discussed. For example, it is shown that every nonoscillatory solution y, such that

y
(
x
)

/

x
→
0

y(x)/x \to 0

as

x
→
∞

x \to \infty

, monotonically tends to zero as

x
→
∞

x \to \infty

, provided n is odd,

p
≧
0

p \geqq 0

, and

∫
∞

x

n
−
1

p
(
x
)
d
x
=
∞

{\smallint ^\infty }{x^{n - 1}}p(x)dx = \infty

.
On the trace of an idempotent in a group ring
Authors: Cliff Gerald H. , Sehgal Sudarshan K.
Published: June 2010

Let KG be the group ring of a polycyclic by finite group G over a field K of characteristic zero. It is proved that if

e
=
∑
e
(
g
)
g

e = \sum e(g)g

is a nontrivial idempotent in KG then its trace

e
(
1
)

e(1)

is a rational number

r

/

s
,
(
r
,
s
)
=
1

r/s,(r,s) = 1

, with the property that for every prime divisor p of s, G has an element of order p. This result is used to prove that if R is a commutative ring of characteristic zero, without nontrivial idempotents and G is a polycyclic by finite group such that no group order

≠
1

\ne 1

is invertible in R, then RG has no nontrivial idempotents.
On the extensions to the bidual of a maximal monotone operator
Authors: Gossez Jean-Pierre
Published: June 2010

An example is given which shows that a maximal monotone operator from a Banach space X to its dual

X
∗

{X^\ast }

may have several extensions into a maximal monotone operator from

X

∗
∗

{X^{\ast \ast }}

to

X
∗

{X^\ast }

.
Contractions of convex sets
Authors: Jamison Robert E.
Published: June 2010

In this paper it is shown that, in a vector space over any ordered field, a noninfinitesimal contraction of a convex set K can be written as an intersection of translates of K.
On precompact (quasi-) uniform structures
Authors: Lambrinos P. Th.
Published: June 2010

The following are shown: (1) The subbasis theorem on precompactness does not hold even in uniform spaces. (2) The supremum of (even finitely many) precompact quasi-uniform structures is not necessarily precompact. (3) A compact quasi-uniform space is not necessarily totally bounded. These results contradict corresponding assertions in the literature.
A note on the concordance homotopy group of real projective space
Authors: Schneider H. , Wells R.
Published: June 2010

By means of the mapping torus construction the following theorem is proved Theorem. If

r
≡
3
mod
4

r \equiv 3 \bmod 4

and

r
⩾
7

r \geqslant 7

, and

P

r

{\mathcal {P}_r}

is a homotopy

P
r

{P_r}

, then there is an isomorphism

π
0

Diff
+

:

P

r

≅

π
0

Diff
+

:

P
r

{\pi _0}\;{\operatorname {Diff}^ + }:{\mathcal {P}_r} \cong {\pi _0}\;{\operatorname {Diff}^ + }:{P_r}

.
Weakly starlike meromorphic univalent functions. II
Authors: Livingston Albert E.
Published: June 2010

A function

f
(
z
)

f(z)

is said to be meromorphic weakly starlike if it has the form

f
(
z
)
=
−
ρ
z
F
(
Z
)

/

(
z
−
ρ
)
(
1
−
ρ
z
)

f(z) = - \rho zF(Z)/(z - \rho )(1 - \rho z)

for

0
>
ρ
>
1

0 > \rho > 1

where

F
(
z
)

F(z)

is a member of

Σ
∗

{\Sigma ^\ast }

, the class of meromorphic, normalized starlike univalent functions. The coefficients of the power series expansion in

|

z

|

>
ρ

|z| > \rho

of a meromorphic weakly starlike function are studied. The integral means of such functions are also discussed.
On the normal spectrum of a subnormal operator
Authors: Bunce J. W. , Deddens J. A.
Published: June 2010

In this note we present a new characterization for subnormality which is purely

C
∗

{C^ \ast }

-algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to prove that

σ
⊥

(
π
(
S
)
)
⊆

σ
⊥

(
S
)

{\sigma _ \bot }(\pi (S)) \subseteq {\sigma _ \bot }(S)

for any

∗

^ \ast

-representation

π
\pi

.
On algebras generated by invertible operators
Authors: Feintuch Avraham
Published: June 2010

Let T be an invertible operator such that the compact operators in

R
(
T
)

R(T)

are weakly dense in it. Then

T

−
1

∈
R
(
T
)

{T^{ - 1}} \in R(T)

. In particular, if T is an essentially unitary

C
0

{C_0}

operator,

T

−
1

∈
R
(
T
)

{T^{ - 1}} \in R(T)

.
On continuity of fixed points of collectively condensing maps
Authors: Cheng Martin Fan
Published: June 2010

In this paper, we prove, in two parts, the following claim. Let X be a Banach space and

Λ
\Lambda

an arbitrary topological space. Suppose that

T
:
Λ
×
X
→
X

T:\Lambda \times X \to X

is collectively condensing; then the fixed point set

S
(
λ
,
y
)

S(\lambda ,y)

has closed graph if and only if T is continuous in both

λ
\lambda

and y.
Compactification by the topologist’s sine curve
Authors: Levy Ronnie
Published: June 2010

Using a compactification of the nonnegative reals whose remainder is the topologist’s sine curve, results about growths of Stone-Čech compactifications are proved. For example, it is proved that if

β
X

\beta X

contains a nonconstant continuous image of a compact connected LOTS, then the image is contained in

υ
X

\upsilon X

. This extends a result of Peter Nyikos.
Longitude surgery on genus 1 knots
Authors: Lambert Howard
Published: June 2010

Let

l
(
K
)

l(K)

be the closed 3-manifold obtained by longitude surgery on the knot manifold K. Let C be the cube with holes obtained by removing an open regular neighborhood of a minimal spanning surface in K. The main result of this paper is that if K is of genus 1 and the longitude of K is in each term of the lower central series for

Π
1

(
C
)

{\Pi _1}(C)

, then

l
(
K
)

l(K)

is not homeomorphic to the connected sum of

S
1

×

S
2

{S^1} \times {S^2}

and a homotopy 3-sphere. In particular, this implies we cannot obtain the connected sum of

S
1

×

S
2

{S^1} \times {S^2}

and a homotopy 3-sphere by longitude surgery on any pretzel knot of genus 1.
Spin manifolds are decomposable
Authors: Stong R. E.
Published: June 2010

It is shown that in the unoriented cobordism ring every manifold with Spin structure is decomposable.
A countable self-injective ring is quasi-Frobenius
Authors: Lawrence John
Published: June 2010

A countable dimensional self-injective algebra is Artinian. There is an application to self-injective twisted group algebras.
Strongly continuous semigroups, weak solutions, and the variation of constants formula
Authors: Ball J. M.
Published: June 2010

Let A be a densely defined closed linear operator on a Banach space X, and let

f
∈

L
1

(
0
,
τ
;
X
)

f \in {L^1}(0,\tau ;X)

. A definition of weak solutions of the equation

u
˙

=
A
u
+
f
(
t
)

\dot u = Au + f(t)

is given. It is shown that a necessary and sufficient condition for the existence of unique weak solutions for every initial data in X is that A generate a strongly continuous semigroup on X, and that in this case the solution is given by the variation of constants formula.
Two notes on metric geometry
Authors: Alexander Ralph
Published: June 2010

If

μ
,
∫
d
μ
=
1

\mu ,\smallint d\mu = 1

, is a signed Borel measure on the unit ball in

E
3

{E^3}

, it is shown that

sup
μ

∫
∫

|

p
−
q

|

d
μ
(
p
)
d
μ
(
q
)
=
2

{\sup _\mu }\smallint \smallint | {p - q} |d\mu (p)d\mu (q) = 2

with no extremal measure existing. Also, a class of simplices which generalizes the notion of acute triangle is studied. The results are applied to prove inequalities for determinants of the Cayley-Menger type.
Note on the Hurwitz-Nielsen realisation problem
Authors: Maclachlan C.
Published: June 2010

This problem, which is equivalent to the problem of showing that every finite subgroup of the Teichmüller modular group of a compact surface of genus g has a fixed point in the corresponding Teichmüller space, is solved for

g
=
2

g = 2

. The generalised problem for n-punctured surfaces of genus

g
⩾
1

g \geqslant 1

and

1
⩽
n
⩽
4

1 \leqslant n \leqslant 4

is also solved.
Continuous ergodic measures on 𝑅^{∞} have disjoint powers
Authors: Kanter Marek
Published: June 2010

If

μ
\mu

is an ergodic probability measure on an infinite dimensional linear measure space and if there exists an infinite sequence of measurable linear functional on this space such that all nontrivial linear combinations have continuous distribution under

μ
\mu

, then the convolution powers of

μ
\mu

all live on disjoint sets.
The 𝑘-closure of monic and monic free ideals in a polynomial semiring
Authors: Dale Louis
Published: June 2010

The concepts of k-closure, k-boundary and weak k-ideals are introduced and necessary and sufficient conditions that an ideal be a k-ideal are given. These conditions are applied to monic and monic free k-ideals. Also, it is shown that the ascending chain condition holds for monic ideals, but not for monic free ideals, and that a semiring S is Noetherian if and only if

S
[
x
]

S[x]

satisfies the ascending chain condition for monic ideals.
A canonical form for a class of ordinary differential operators
Authors: Benzinger Harold E.
Published: June 2010

A canonical form is developed for each member of a large class of nth order linear ordinary differential operators, acting on

L
p

(
0
,
1
)

{L^p}(0,1)

. This canonical form shows how the Fourier-like behavior of eigenfunction expansions is related to structural properties of the operators.
Isoperimetric inequalities involving heat flow under linear radiation conditions
Authors: Acker Andrew
Published: June 2010

Under the assumption that a constant linear radiation condition holds on each boundary component, we show that the annulus and the strip are heat-flow minimizing with respect to area-preserving variations in one (for the annulus: the outer) boundary component.
On the support of certain symmetric stable probability measures on 𝑇𝑉𝑆
Authors: Rajput Balram S.
Published: June 2010

Let E be a LCTVS, and let

μ
\mu

be a

τ
\tau

-regular symmetric stable probability measure of type

α
∈
[
1
,
2
]

\alpha \in [1,2]

on E. Then, it is shown that the topological support

S
μ

{S_\mu }

of

μ
\mu

is a (closed) subspace of E. Further, if

μ
\mu

satisfies an additional regularity condition, then it is shown that

S
μ

{S_\mu }

is the E-closure of a Banach space contained in E. Specializing these results for

α
=
2

\alpha = 2

, simple proofs of all the known results regarding the support of centered Gaussian probability measures on LCTV spaces are obtained as corollaries.
Extremal interpolatory functions in 𝐻^{∞}
Authors: Øyma Knut
Published: June 2010

Let a Blaschke sequence

{

z
n

}

\{ {z_n}\}

and a bounded sequence

{

w
n

}

\{ {w_n}\}

be given. If we can find an f in

H
∞

{H^\infty }

such that

f
(

z
n

)
=

w
n

f({z_n}) = {w_n}

we may assume that

‖
f
‖

\left \| f \right \|

is minimal. Such an f need not be unique, but a sufficient condition for uniqueness is given. Properties of f in the case of uniqueness are studied.
An abstract existence theorem at resonance
Authors: Cesari L. , Kannan R.
Published: June 2010

By Schauder’s fixed point theorem and alternative method (bifurcation theory) an abstract existence theorem at resonance for operational equations is proved which contains as particular cases rather different existence theorems for ordinary and partial differential equations as those of Lazer and Leach and of Landesman and Lazer.
Diffeomorphisms of 3-manifolds which are homotopy equivalent to 𝑆¹
Authors: Husch L. S.
Published: June 2010

Let h be a diffeomorphism of a 3-manifold M which is homotopy equivalent to the 1-sphere. Suppose that the collection of positive iterates of h has compact closure in the space of smooth mappings of M into itself and suppose that this closed set generated by h is not a group. Necessary and sufficient conditions are given that another diffeomorphism g be topologically equivalent to h.
Aspherical generators of unoriented cobordism
Authors: Royster David C.
Published: June 2010

A new set of indecomposable generators for

M

O
∗

M{O_ \ast }

is constructed, each of which is aspherical. Two adjoint maps are used to define these generators. In the process we find a natural basis for

M

O
∗

(

Z

2

)

M{O_\ast }({{\mathbf {Z}}_2})

.
Critical edges in subdirectly irreducible lattices
Authors: Jónsson Bjarni , Rival Ivan
Published: June 2010

In studying a particular subdirectly irreducible lattice it is often important to know the location of a critical edge, that is, a quotient that is collapsed by every nontrivial congruence relation. We show that a critical edge can always be found in a rather natural location.
The nonfiniteness of Nil
Authors: Farrell F. T.
Published: June 2010

We show that Nil R is finitely generated only when it vanishes.
On 𝜋’-closure of 𝜋-homogeneous groups
Authors: Ferguson Pamela
Published: June 2010

Let

π
\pi

be a set of odd primes. It is known that

π
′

\pi ’

-closed groups are

π
\pi

-homogeneous, but that the converse does not hold in general. In this paper we prove that a finite group G which is

D
π

{D_\pi }

and

π
\pi

-homogeneous is

π
′

\pi ’

-closed.
On automorphisms of L.C. groups
Authors: Peters Justin
Published: June 2010

The left regular representation

λ
\lambda

of a locally compact group G generates a

W
∗

{W^\ast }

-algebra

R

(
λ
)

\mathcal {R}(\lambda )

, and each topological automorphism

α
~

\tilde \alpha

of G has a natural extension to an automorphism

α
~

\tilde \alpha

of

R

(
λ
)

\mathcal {R}(\lambda )

. It is proved that an automorphism

β
\beta

of

R

(
λ
)

\mathcal {R}(\lambda )

is of the form

β
=

α
~

\beta = \tilde \alpha

for

α
∈

Aut

(
G
)

\alpha \in {\operatorname {Aut}}(G)

iff

β
\beta

leaves a certain cone in

R

(
λ
)

\mathcal {R}(\lambda )

invariant.
Ideal boundaries of a Riemann surface for the equation Δ𝑢=𝑃𝑢
Authors: Schiff J. L.
Published: June 2010

For a nonnegative density P on a hyperbolic Riemann surfaces R, let

Δ
P

{\Delta ^P}

be the subset of the Royden harmonic boundary consisting of the nondensity points of P. This ideal boundary, as well as the P-harmonic boundary

δ
P

{\delta _P}

of the P-compactification of R, have been employed in the study of energy-finite solutions of

Δ
u
=
P
u

\Delta u = Pu

on R. We show that

Δ
P

{\Delta ^P}

is homeomorphic to

δ
P

−
{

s
P

}

{\delta _P} - \{ {s_P}\}

, where

s
P

{s_P}

is the P-singular point. It follows that

δ
P

{\delta _P}

fails to characterize the space

P
B
E
(
R
)

PBE(R)

in the sense that it is possible for

δ
P

{\delta _P}

to be homeomorphic to

δ
Q

{\delta _Q}

, but

P
B
E
(
R
)

PBE(R)

is not canonically isomorphic to

Q
B
E
(
R
)

QBE(R)

.
On moduli of plane domains
Authors: Guerrero Ignacio
Published: June 2010

It is well known that an arbitrary plane domain of finite connectivity can be mapped conformally onto an annulus minus a certain number of circular slits. The parameters defining such a canonical domain are studied in the context of Teichmüller theory. Let

Ω
\Omega

be a plane domain bounded by

m
⩾
3

m \geqslant 3

continua. Denote by

T
(
Ω
)

T(\Omega )

the reduced Teichmüller space of

Ω
\Omega

and by

R
(
Ω
)

R(\Omega )

the space of conformal equivalence classes of domains bounded, as

Ω
\Omega

is, by m continua. A real analytic map from

T
(
Ω
)

T(\Omega )

onto an open subset

S
(
Ω
)

S(\Omega )

of a

3
m
−
6

3m - 6

dimensional product of circles and lines is constructed. It is shown that the map

T
(
Ω
)
→
S
(
Ω
)

T(\Omega ) \to S(\Omega )

is a regular covering map. Finally, it is observed that there is a finite sheeted covering map

S
(
Ω
)
→
R
(
Ω
)

S(\Omega ) \to R(\Omega )

.
The absolute summability of Fourier series of a function of Wiener’s class by Nörlund means
Authors: Lal S. N. , Ram Siya
Published: June 2010

In this paper we prove that under suitable restrictions on

{

p
n

}

\{ {p_n}\}

and the integral modulus of continuity of a function f of Wiener’s class the Fourier series of f is summable

|

N
,

p
n

|

|N,{p_n}|

.
Similarity of certain operators in 𝑙^{𝑝}
Authors: Kantorovitz Shmuel
Published: June 2010

Let M be the multiplication operator in

l
p

,
1
⩽
p
⩽
∞

{l^p},1 \leqslant p \leqslant \infty

, i.e.,

M
:
x
=
{

x
k

}
→
{
k

x
k

}

M:x = \{ {x_k}\} \to \{ k{x_k}\}

. Let

w
=
{

w
j

}

j
=
0

∞

w = \{ {w_j}\} _{j = 0}^\infty

be a weight, i.e., a positive sequence such that

w
1

>

w
0

=
1

{w_1} > {w_0} = 1

and

w

n
+
m

⩽

w
n

w
m

{w_{n + m}} \leqslant {w_n}{w_m}

. For

ζ
∈
C

\zeta \in C

, define

N
w
ζ

N_w^\zeta

on

l
p

{l^p}

by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper N Subscript w Superscript zeta Baseline x right-parenthesis Subscript k Baseline equals sigma-summation Underscript j equals 1 Overscript k Endscripts left-parenthesis StartFraction w Subscript k Baseline Over w Subscript j Baseline EndFraction right-parenthesis StartBinomialOrMatrix zeta minus 1 plus k minus j Choose k minus j EndBinomialOrMatrix x Subscript j Baseline left-parenthesis k equals 1 comma 2 comma ellipsis right-parenthesis period">

(

N
w
ζ

x

)
k

=

∑

j
=
1

k

(

w
k

w
j

)

(

ζ
−
1
+
k
−
j

k
−
j

)

x
j

(
k
=
1
,
2
,
…
)
.

{(N_w^\zeta x)_k} = \sum \limits _{j = 1}^k {\left ( {\frac {{{w_k}}}{{{w_j}}}} \right )} \left ( {\begin {array}{*{20}{c}} {\zeta - 1 + k - j} \\ {k - j} \\ \end {array} } \right ){x_j}\quad (k = 1,2, \ldots ).

\]
Then

{

N
w
ζ

;
ζ
∈
C
}

\{ N_w^\zeta ;\zeta \in C\}

is a holomorphic group of operators, and for any function g holomorphic on the spectrum of

N
w
ζ

,
M
+
g
(

N
w
ζ

)

N_w^\zeta ,M + g(N_w^\zeta )

is similar to

M
+
g
(
1
)
I

M + g(1)I

.
On the accretivity of the inverse of an accretive relation
Authors: Gripenberg Gustaf
Published: June 2010

If X is a smooth, reflexive, real Banach space such that a relation A in

X
×
X

X \times X

is accretive iff

A

−
1

{A^{ - 1}}

is accretive, then X is isomorphic to a Hilbert space.
A generalization of a theorem of Janos and Edelstein
Authors: Park Sehie
Published: June 2010

A generalization of Edelstein’s version of a theorem of Janos and its converse are obtained: Theorem. Let X be a compact metrizable topological space, and f, g be continuous self-maps of X such that

g
f
=
f
g

gf = fg

and f is bijective. Then g is injective and

⋂
1
∞

g
n

X
=
{

x
0

}

\bigcap \nolimits _1^\infty {{g^n}X = \{ {x_0}\} }

, where

x
0

∈
X

{x_0} \in X

, iff, given

λ
,
0
>
λ
>
1

\lambda ,0 > \lambda > 1

, a homeomorphism h of X into

l
2

{l_2}

exists such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar h left-parenthesis g x right-parenthesis minus h left-parenthesis g y right-parenthesis double-vertical-bar equals lamda double-vertical-bar h left-parenthesis f x right-parenthesis minus h left-parenthesis f y right-parenthesis double-vertical-bar">

‖

h
(
g
x
)
−
h
(
g
y
)

‖

=
λ

‖

h
(
f
x
)
−
h
(
f
y
)

‖

\left \| {h(gx) - h(gy)} \right \| = \lambda \left \| {h(fx) - h(fy)} \right \|

\]
for all

x
,
y
∈
X

x,y \in X

.
Strongly rigid metrics and zero dimensionality
Authors: Martin Harold W.
Published: June 2010

A metric d is strongly rigid if and only if

d
(
x
,
y
)
≠
d
(
w
,
z
)

d(x,y) \ne d(w,z)

whenever the doubleton {x, y} is not equal to the doubleton {w, z}. It is shown that a nonempty metrizable space X admits a compatible strongly rigid metric if X has covering dimension zero and has cardinality equal to or less than that of the real line.
The Lefschetz fixed point theorem for compact groups
Authors: Knill Ronald J.
Published: June 2010

It is shown that every compact group G is a Q-simplicial space where Q is any field of characteristic zero. As a consequence it follows that G satisfies a variation of the Lefschetz fixed point theorem. It has been known for some time that the Lefschetz fixed point theorem applies to a few spaces other than just ANR spaces, especially if some care is taken to use coefficients in certain fields [2]. The case of all compact groups provides a broad class of spaces which may not have local connectivity of any order. It is shown that every compact group G satisfies the Lefschetz fixed point theorem when coefficients for the homology groups are taken in a field of characteristic zero.
Linear isotopies in 𝐸³
Authors: Starbird Michael
Published: June 2010

In this paper it is shown that if f and g are two PL embeddings of a finite complex K into

E
3

{E^3}

so that there is an orientation-preserving homeomorphism h of

E
3

{E^3}

with

h
∘
f
=
g

h \circ f = g

, then there is a triangulation T of K and a linear isotopy

h
t

:
(
K
,
T
)
→

E
3

(
t
∈
[
0
,
1
]
)

{h_t}:(K,T) \to {E^3}(t \in [0,1])

so that

h
0

=
f

{h_0} = f

and

h
1

=
g

{h_1} = g

.
Strongly annular functions with small coefficients, and related results
Authors: Bonar D. D. , Carroll F. W. , Erdős Paul
Published: June 2010

A technique of Bagemihl and Seidel is applied to two problems in annular functions. It is shown that there exists a strongly annular function with Maclaurin coefficients tending to zero, and that there exist annular functions that are far from being strongly annular.
Extremal properties of real axially symmetric harmonic functions in 𝐸³
Authors: McCoy Peter A.
Published: June 2010

The set

H

\mathcal {H}

consists of all real harmonic functions defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis r comma theta right-parenthesis equals normal upper Sigma Subscript k equals 0 Superscript normal infinity Baseline a Subscript k Baseline r Superscript k Baseline upper P Subscript k Baseline left-parenthesis cosine theta right-parenthesis">

U
(
r
,
θ
)
=

Σ

k
=
0

∞

a
k

r
k

P
k

(
cos
⁡
θ
)

U(r,\theta ) = \Sigma _{k = 0}^\infty {a_k}{r^k}{P_k}(\cos \theta )

which are regular in

Σ
\Sigma

, the open unit sphere about the origin

E
3

{E^3}

. Two problems arise concerning

H

\mathcal {H}

and a subset

H

∗

{\mathcal {H}_\ast }

whose members U have the first

n
+
1

n + 1

coefficients

a
0

,
…
,

a
n

{a_0}, \ldots ,{a_n}

specified. (1) For

U
∈

H

U \in \mathcal {H}

, determine
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis upper U right-parenthesis equals inf left-brace right-brace vertical-bar of UU left-parenthesis right-parenthesis comma r comma theta vertical-bar element-of element-of left-parenthesis right-parenthesis comma r comma theta upper Sigma">

I
(
U
)
=
inf
{
U
(
r
,
θ
)

|

(
r
,
θ
)
∈
Σ
}

I(U) = \inf \{ U(r,\theta )|(r,\theta ) \in \Sigma \}

as the limit of a monotone sequence of constants

{

λ
k

(

a
0

,
…
,

a
k

)

}

k
=
0

∞

\{ {\lambda _k}({a_0}, \ldots ,{a_k})\} _{k = 0}^\infty

which can be computed algebraically. (2) Find

U
0

∈

H

∗

{U_0} \in {\mathcal {H}_\ast }

and the constant
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Subscript n Baseline left-parenthesis a 0 comma ellipsis comma a Subscript n Baseline right-parenthesis equals upper S u p left-brace upper I left-parenthesis upper U right-parenthesis vertical-bar upper U element-of script upper H Subscript asterisk Baseline right-brace equals upper I left-parenthesis upper U 0 right-parenthesis period">

λ
n

(

a
0

,
…
,

a
n

)
=

Sup

{

I
(
U
)

|

U
∈

H

∗

}

=
I
(

U
0

)
.

{\lambda _n}({a_0}, \ldots ,{a_n}) = {\operatorname {Sup}}\left \{ {I(U)|U \in {\mathcal {H}_\ast }} \right \} = I({U_0}).

\]
These are answered by means of the Bergman Integral Operator Method and applications of the Methods of Ascent and Descent to the classical Carathéodory-Fejér problem regarding extremal properties of harmonic functions in

E
2

{E^2}

.
𝑁-compactness and weak homogeneity
Authors: Broverman S.
Published: June 2010

A characterization of those locally compact, 0-dimensional, realcompact spaces X that are N-compact is given in terms of a density condition on

β
X
−
X

\beta X - X

.
A characterization of locally connected continua by hyperspace retractions
Authors: Nadler Sam B.
Published: June 2010

Let X be a (metric) continuum. It is shown that X is locally connected if and only if there is special type of retraction from

2
X

{2^X}

onto

C
(
X
)

C(X)

, where

2
X

{2^X}

[resp.,

C
(
X
)

C(X)

] is the space of all nonempty compact subsets [resp., subcontinua] of X with the Hausdorff metric. Also, necessary and sufficient conditions are given for the “continuity of balls".
Affine complete ortholattices
Authors: Schweigert Dietmar
Published: June 2010

Every finite orthomodular lattice is affine complete.
On the von Neumann algebra of an ergodic group action
Authors: Zimmer Robert J.
Published: June 2010

We give a criterion that an ergodic action be amenable in terms of the operator algebra associated to it by the Murray-von Neumann construction.
An Abelian theorem for a class of subharmonic functions
Authors: Abi-Khuzam Faruk F.
Published: June 2010

We show that if the Riesz-mass of a subharmonic function u, of finite order

λ
\lambda

, is distributed along a ray, then regular variation (with exponent

λ
\lambda

) of the mean value of

u
(
r

e

i
θ

)

u(r{e^{i\theta }})

implies regular variation (with exponent

λ
\lambda

) of each of the

L
s

(
−
π
,
π
)

{L_s}( - \pi ,\pi )

means of

u
(
r

e

i
θ

)

u(r{e^{i\theta }})

. This result extends a known theorem of Edrei and Fuchs, but our method differs from theirs. In particular, for the case of integral orders we obtain the theorem for a much more general distribution of the Riesz-mass. A corollary, which appears to be new, on the deficiency of the value zero of entire functions with positive integral order, follows.
An inequality for a class of integral systems
Authors: Greene David E.
Published: June 2010

A new inequality of the Gronwall type is established for a class of integral systems.
Nonoscillation and eventual disconjugacy
Authors: Elias Uri
Published: June 2010

If every solution of an nth order linear differential equation has only a finite number of zeros in

[
0
,
∞
)

[0,\infty )

, it is not generally true that for sufficiently large

c
,
c
>
0

c,c > 0

, every solution has at most

n
−
1

n - 1

zeros in

[
c
,
∞
)

[c,\infty )

. Settling a known conjecture, we show that for any n, the above implication does hold for a special type of equation,

L
n

y
+
p
(
x
)
y
=
0

{L_n}y + p(x)y = 0

, where

L
n

{L_n}

is an nth order disconjugate differential operator and

p
(
x
)

p(x)

is a continuous function of a fixed sign.
Convergence of certain cosine sums in the metric spaces 𝐿
Authors: Ram Babu
Published: June 2010

We consider here the

L
1

{L^1}

convergence of Rees-Stanojević cosine sums to a cosine trigonometric series belonging to the class S defined by Sidon and deduce as corollaries some previously known results from our result.
Rota’s theorem for general functional Hilbert spaces
Authors: Ball Joseph A.
Published: June 2010

By a theorem of G.-C. Rota, every (linear) operator T on a Hilbert space with spectral radius less than one is similar to the adjoint of the unilateral shift S of infinite multiplicity restricted to an invariant subspace. This theorem is shown to be true in a rather general context, where S is multiplication by z on a Hilbert space of functions analytic on an open subset D of the complex plane, and T is an operator with spectrum contained in D. A several-variable version for an N-tuple of commuting operators with a corollary concerning complete spectral sets is also presented.
A cardinal inequality for topological spaces involving closed discrete sets
Authors: Ginsburg John , Woods R. Grant
Published: June 2010

Let X be a

T
1

{T_1}

topological space. Let

a
(
X
)
=
sup
{
α
:
X

a(X) = \sup \{ \alpha :X

has a closed discrete subspace of cardinality

α
}

\alpha \}

and

v
(
X
)
=
min
{
α
:

Δ
X

v(X) = \min \{ \alpha :{\Delta _X}

can be written as the intersection of

α
\alpha

open subsets of

X
×
X
}

X \times X\}

; here

Δ
X

{\Delta _X}

denotes the diagonal

{
(
x
,
x
)
:
x
∈
X
}

\{ (x,x):x \in X\}

of X. It is proved that

|

X

|

⩽
exp
⁡
(
a
(
X
)
v
(
X
)
)

|X| \leqslant \exp (a(X)v(X))

. If, in addition, X is Hausdorff, then X has no more than

exp
⁡
(
a
(
X
)
v
(
X
)
)

\exp (a(X)v(X))

compact subsets.
Approximate fibrations with nonfinite fibers
Authors: Ferry Steve
Published: June 2010

It is shown that every ANR fibration over

S
1

{S^1}

with finitely dominated fiber is homotopy equivalent to an approximate fibration with compact ANR total space. This yields examples of approximate fibrations with arbitrary compact pointed FANR fibers.
An alternative to the Plücker relations
Authors: Robinson Herbert A.
Published: June 2010

It is shown how to obtain a set of homogeneous, degree m polynomials in

(
m
n

)

(_m^n)

indeterminates over a field F so that the associated algebraic variety is the set of decomposable elements in the mth Grassmann space over an n-dimensional vector space over F. The same techniques are used to produce an analogous result for the tensor product of m finite dimensional vector spaces.
A second order differential equation of T. Otsuki
Authors: Utz W. R.
Published: June 2010

It is shown that an ordinary differential equation arising in the study of minimal hypersurfaces in certain Riemannian manifolds and for which one family of periodic solutions is known has, in fact, a second family of periodic solutions.
On perturbations of Fredholm operators in 𝐿_{𝑝}(𝜇)-spaces
Authors: Weis L.
Published: June 2010

Answering a question of Milman, we show that a continuous linear operator

T
:

L
p

(
μ
)
→

L
p

(
μ
)
,
1
>
p
>
2

T:{L_p}(\mu ) \to {L_p}(\mu ),1 > p > 2

, is a Fredholm perturbation iff T is strictly singular.
On nonlinear variational inequalities
Authors: Tarafdar E.
Published: June 2010

In this note we have given a direct proof of the result which states that if K is a compact convex subset of a linear Hausdorff topological space E over the reals and T is a monotone and hemicontinuous (nonlinear) mapping of K into

E
∗

{E^ \ast }

, then there is a

u
0

∈
K

{u_0} \in K

such that

(
T
(

u
0

)
,
v
−

u
0

)
⩾
0

(T({u_0}),v - {u_0}) \geqslant 0

for all

v
∈
K

v \in K

.
Weakly closed 𝑚-accretive operators
Authors: Yamamoto A. , Okazawa N.
Published: June 2010

Let A and B be weakly closed, nonlinear m-accretive (single-valued) operators in a reflexive Banach space X, and

B
n

{B_n}

be the Yosida approximation of B. Then the following condition is sufficient for the sum

A
+
B

A + B

to be also m-accretive: For each

v
∈
X
,

‖

B
n

u
n

‖

v \in X,\left \| {{B_n}{u_n}} \right \|

is bounded as n tends to infinity, where

u
n

{u_n}

is defined by the equation

u
n

+
A

u
n

+

B
n

u
n

=
v
,
n
=
1
,
2
,
…

{u_n} + A{u_n} + {B_n}{u_n} = v,n = 1,2, \ldots

. Some related conditions are also provided.
On the chromatic number of subgraphs of a given graph
Authors: Rödl V.
Published: June 2010

It is shown that for arbitrary positive integers m,n there exists a

ϕ
(
m
,
n
)

\phi (m,n)

such that if

χ
(
G
)
⩾
ϕ
(
m
,
n
)

\chi (G) \geqslant \phi (m,n)

, then G contains either a complete subgraph of size m or a subgraph H with

χ
(
H
)
=
n

\chi (H) = n

containing no

C
3

{C_3}

. This gives an answer to a problem of Erdös and Hajnal.
Quasi-pure projective and injective torsion groups
Authors: Berlinghoff W. P. , Reid J. D.
Published: June 2010

This paper characterizes quasi-pure projective (q.p.p.) and quasi-pure injective (q.p.i.) p-groups, and hence characterizes all such (abelian) torsion groups. A p-group is q.p.i. if and only if it is the direct sum of a divisible group and a torsion complete group. A nonreduced p-group is q.p.p. if and only if it is the direct sum of a divisible group and a bounded group; a reduced p-group is q.p.p. if and only if it is a direct sum of cyclic groups.
Deltas of Hochschild dimension one
Authors: Cheng Charles Ching-An
Published: June 2010

A theorem of Mitchell says that free categories have Hochschild dimension

⩽
1

\leqslant 1

. The converse is shown to be true for all deltas (i.e. small skeletal categories whose only endomorphisms are the identities).
Remarks on a theorem of Koranyi and Malliavin on the Siegel upper half plane of rank two
Authors: Johnson Kenneth D.
Published: June 2010

In [5], A. Koranyi and P. Malliavin showed that bounded functions on the Siegel upper half plane of rank two which satisfied two special elliptic differential equations were characterized by their values on the Bergman-Šilov boundary. In this paper a simple proof of this theorem is given.
Transformations into Baire 1 functions
Authors: Bruckner A. M. , Davies Roy O. , Goffman C.
Published: June 2010

A measurable f from

I
=
[
0
,
1
]

I = [0,1]

to R is equivalent to a Baire 2 function but may not be equivalent to any Baire 1 function. Gorman has obtained the following interesting contrasting facts. If f assumes finitely many values there is a homeomorphism h of I such that

f
∘
h

f \circ h

is equivalent to a Baire 1 function, but there is a measurable f which assumes countably many values which does not have this property. However, the example of Gorman is such that for some homeomorphisms h the function

f
∘
h

f \circ h

is not measurable. It is shown here that if

f
∘
h

f \circ h

is measurable, for every homeomorphism h, then there is an h for which

f
∘
h

f \circ h

is equivalent to a Baire 1 function.
On quasi-uniformities in hyperspaces
Authors: Berthiaume Gilles
Published: June 2010

Quasi-uniformities in spaces of subsets are investigated. It is shown, by means of examples, how quasi-uniformities can play a unifying role in the study of topological structures in hyperspaces. The main application is the following result: an infinite product of upper semicontinuous multi-valued mappings, with compact sets as values, is again upper semi-continuous.
Isotone functions on partially ordered linear algebras with a multiplicative diagonal map
Authors: Dai Taen Yu , DeMarr Ralph
Published: June 2010

The diagonal of the product of two triangular matrices is the product of the diagonals of each matrix. This idea is used to characterize Dedekind

σ
\sigma

-complete lattice ordered linear algebras which admit isotone functions with familiar functional and order properties as possessed by the real-valued logarithm or root function.
Groups of order 𝑝𝑞^{𝑚} with elementary abelian Sylow 𝑞-subgroups
Authors: Alonso J.
Published: June 2010

Characterization and number of the groups of order

p

q
m

p{q^m}

with elementary Abelian Sylow q-subgroups. Old methods of O. Hölder and A. E. Western are simplified and generalized.
Comparison theorems for a class of selfadjoint fourth order differential equations
Authors: Kreith Kurt
Published: June 2010

Equations of the form

(
r
(
x
)
y
)
+
p
(
x
)
y
=
0

(r(x)y) + p(x)y = 0

are considered under the assumption

r
(
x
)
>
0

r(x) > 0

and

p
(
x
)
>
0

p(x) > 0

in an interval

[
α
,
∞
)

[\alpha ,\infty )

. A polar coordinate transformation is used to study the effect of variations in the coefficients on the conjugate points of such equations.
A countably compact 𝑘’-space need not be countably 𝑏𝑖-𝑘
Authors: Olson Roy C.
Published: June 2010

An example is given of a countably compact k’-space that is not countably bi-k. Interest for this example arises from a recent paper of Michael, Olson, and Siwiec and from a 1972 paper of E. Michael, both of which discuss mapping characterizations of a range space. The construction of the example assumes the continuum hypothesis.
Four metric conditions characterizing Čech dimension zero
Authors: Broughan Kevin
Published: June 2010

If (X,d) is a metric space let

d
x

(
y
)
=
d
(
x
,
y
)

{d_x}(y) = d(x,y)

. It is proved that if each x in X has a neighbourhood P with

d
x

(
P
)

{d_x}(P)

not dense in any neighbourhood of 0 in

[
0
,
∞
)

[0,\infty )

then Ind

X
=
0

X = 0

. This metric condition characterizes metrizable spaces which have Čech dimension zero. Three other metric characterizations are given.
The domain of univalence of certain families of rational functions
Authors: Goodman A. W.
Published: June 2010

The Čakalov-Distler Theorem gives the domain of univalence for the family of all rational functions for which all the poles are simple, all the poles lie in a prescribed convex set, and all have positive residue. Here we extend this theorem to allow simple poles with negative residues. We include several open problems.
The strict dual of 𝐵*-algebras
Authors: Davenport John W.
Published: June 2010

If A is a closed two-sided ideal in the

B
∗

{B^\ast }

-algebra X, then

(
X
,

β
A

)
∗

{(X,{\beta _A})^\ast }

with the strong topology is isomorphic to

A
∗

{A^\ast }

, where

β
A

{\beta _A}

is the strict topology on X.
A note on attractors for compact sets
Authors: Solomon J. L.
Published: June 2010

Let G be a closed convex subset of a Banach space

X
,
f
:
G
→
G

X,f:G \to G

a continuous map and

M
⊂
G

M \subset G

an attractor for compact sets under f. It is shown that if M is not connected, then M has a unique invariant component which is an attractor for points; moreover, for each x in G, the set of subsequential limit points of x under f is a subset of this unique invariant component.
Duality and alternative in multi-objective optimization
Authors: Rosinger Elemer E.
Published: June 2010

Within a general framework containing multiobjective optimization, an equivalence between duality properties and alternative conditions is established for pairs of constrained optimization problems. Sufficient conditions for a Pareto type duality and a multiobjective strong duality are obtained.
Injective near-ring modules over 𝑍_{𝑛}
Authors: Meldrum John D. P.
Published: June 2010

In this note, we show that an injective unital near-ring module over

Z
n

{Z_n}

has trivial centre if

n
⩾
3

n \geqslant 3

. As a consequence there are no injective unital near-ring modules over

Z
3

{Z_3}

.