bluuebunny/crossref_metadata_2025 huggingface dataset:
Separating two compact sets by a parallelotope
Authors: Lay Steven R.
Published: June 2010

Let P and Q be compact subsets of Euclidean n-space. Paul Kirchberger has established that there exists a closed half-space M such that

P
⊂
M

P \subset M

and

Q
∩
M
=
∅

Q \cap M = \emptyset

iff for each set S consisting of

n
+
2

n + 2

or fewer points of

P
∪
Q

P \cup Q

, there exists a closed half-space

M
S

{M_S}

such that

(
P
∩
S
)
⊂

M
S

(P \cap S) \subset {M_S}

and

(
Q
∩
S
)
∩

M
S

=
∅

(Q \cap S) \cap {M_S} = \emptyset

. In this paper the problem of replacing the half-spaces by parallelotopes is considered, and the critical number of points in

P
∪
Q

P \cup Q

is shown to be

n
+
1

n + 1

. Applications of this are drawn to systems of linear inequalities and to Carathéodory’s theorem.
On a problem of L. Nachbin
Authors: Jech Thomas J.
Published: June 2010

If B is an uncountable set then there is a function

r
:
B
×
B
→

R

+

r:B \times B \to {{\mathbf {R}}_ + }

for which there is no function

t
:
B
→

R

+

t:B \to {{\mathbf {R}}_ + }

such that
\[

r
(

b
1

,

b
2

)
⩽
t
(

b
1

)
⋅
t
(

b
2

)

for all

b
1

,

b
2

∈
B
.

r({b_1},{b_2}) \leqslant t({b_1}) \cdot t({b_2})\quad {\text {for all}}\;{b_1},{b_2} \in B.

\]
A characterization of normal operators using the Hilbert-Schmidt class
Authors: Nakamoto Ritsuo
Published: June 2010

A bounded linear operator N on a Hilbert space H is normal if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper N upper X minus upper X upper N double-vertical-bar Subscript 2 Baseline equals double-vertical-bar upper N Superscript asterisk Baseline upper X minus upper X upper N Superscript asterisk Baseline double-vertical-bar Subscript 2">

‖
N
X
−
X
N

‖
2

=
‖

N
∗

X
−
X

N
∗

‖
2

\|NX - XN\|_2 = \|N^* X - X N^*\|_2

for every X in the Hilbert-Schmidt class.
Estimates for exponential sums
Authors: Smith Robert A.
Published: June 2010

If f is a polynomial over Z of degree

n
+
1

n + 1

with

n
⩾
1

n \geqslant 1

, then for each integer
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q greater-than-or-slanted-equals 1 comma StartAbsoluteValue normal upper Sigma Subscript 1 less-than-or-slanted-equals x less-than-or-slanted-equals q Baseline exp left-parenthesis 2 pi i f left-parenthesis x right-parenthesis slash q right-parenthesis EndAbsoluteValue less-than-or-slanted-equals q Superscript 1 slash 2 Baseline left-parenthesis upper D comma q right-parenthesis d Subscript n Baseline left-parenthesis q right-parenthesis">

q
⩾
1
,

|

Σ

1
⩽
x
⩽
q

exp
⁡
(
2
π
i
f
(
x
)

/

q
)

|

⩽

q

1

/

2

(
D
,
q
)

d
n

(
q
)

q \geqslant 1,|{\Sigma _{1 \leqslant x \leqslant q}}\exp (2\pi if(x)/q)| \leqslant {q^{1/2}}(D,q){d_n}(q)

, provided the discriminant D of the derivative of f does not vanish identically, where

d
n

(
q
)

{d_n}(q)

is the number of representations of q as a product of n factors.
Trace polynomial for two-generator subgroups of 𝑆𝐿(2,𝐶)
Authors: Traina Charles R.
Published: June 2010

If G is a group generated by two

2
×
2

2 \times 2

matrices A and B having determinant

+
1

+ 1

, with entries from the complex field C, it is known that the trace of any word in A and B,

W
(
A
,
B
)

W(A,B)

is a polynomial with integral coefficients in the three variables:

x
=

trace

(
A
)
,
y
=

trace

(
B
)
,
z
=

trace

(
A
B
)

x = {\text {trace}}(A),y = {\text {trace}}(B),z = {\text {trace}}(AB)

, defined as
\[

trace

W
(
A
,
B
)
=
P
(
x
,
y
,
z
)
,

{\text {trace}}\;W(A,B) = P(x,y,z),

\]
where P is determined uniquely by the conjugacy class of

W
(
A
,
B
)

W(A,B)

. The actual computation of this trace polynomial is not easily obtained. It is the purpose of this paper to derive an explicit formula for this trace polynomial, and to indicate some consequences of it.
Local completeness for eigenfunctions of regular maximal ordinary differential operators
Authors: Carlson Robert
Published: June 2010

Let

L
=

∑

m
=
0

n

A
m

(
x
)

D
m

L = \sum \nolimits _{m = 0}^n {{A_m}(x){D^m}}

be a differential operator on

⊕
k

L
2

[
0
,
1
]

{ \oplus ^k}{L^2}[0,1]

with infinitely differentiable

k
×
k

k \times k

matrix valued coefficients. Assume that

det

A
n

(
x
)
≠
0

\det \,{A_n}(x) \ne 0

for

x
∈
[
0
,
1
]

x \in [0,1]

. The domain of L is the set of k-vector valued functions f such that

f
∈

C

n
−
1

(
[
0
,
1
]
)
,

f

(
n
−
1
)

f \in {C^{n - 1}}([0,1]),{f^{(n - 1)}}

is absolutely continuous on [0, 1] and

L
f
∈

⊕
k

L
2

[
0
,
1
]

Lf \in { \oplus ^k}{L^2}[0,1]

. Let

x
0

∈
(
0
,
1
)

{x_0} \in (0,1)

. Then there is a neighborhood

U
(

x
0

)

U({x_0})

containing

x
0

{x_0}

such that the restrictions of the eigenfunctions of L to

U
(

x
0

)

U({x_0})

have dense span in

⊕
k

L
2

[
U
(

x
0

)
]

{ \oplus ^k}{L^2}[U({x_0})]

. The example

L
=

e

−
i
x

d

/

d
x

L = {e^{ - ix}}d/dx

shows that this is the best possible abstract result.
An asymptotic expansion of a beta-type integral and its application to probabilities of large deviations
Authors: Fu J. C. , Wong R.
Published: June 2010

An asymptotic expansion is obtained for an incomplete beta-type integral, which arises in the study of probabilities of large deviations. The expansion obtained yields large deviation results for binomial, quantile, and related probabilities. Our approach is based on a generalized version of Laplace’s method.
Which operators are the self-commutators of compact operators?
Authors: Fan Peng , Fong Che Kao
Published: June 2010

It is shown that, for a hermitian operator T on a Hubert space H, the following three statements are equivalent: (i)

T
=

A
∗

A
−
A

A
∗

T = {A^ \ast }A - A{A^ \ast }

for some compact operator A, (ii) there is an orthonormal basis

{

b
j

}

\{ {b_j}\}

such that

⟨
T

b
j

,

b
j

⟩
=
0

\langle T{b_j},{b_j}\rangle = 0

for all j, and (iii)

tr

T
+

=

tr

T
−

{\text {tr}}\,{T^ + } = {\text {tr}}\,{T^ - }

(possibly infinite) where

T
+

=
(

|

T

|

+
T
)

/

2

{T^ + } = (|T| + T)/2

and

T
−

=
(

|

T

|

−
T
)

/

2

{T^ - } = (|T| - T)/2

.
Norm estimates relating polynomials and their derivatives
Authors: Rubinstein Zalman
Published: June 2010

Inequalities bounding

L
p

(
p
>
0
)

{L_p}(p > 0)

norms of a polynomial by suitable norms of the derivative of the polynomial and an auxiliary function are obtained for the classes of polynomials whose zeros lie on the unit circumference or in the interior or the exterior of the unit disk. All the inequalities are best possible and generalize several known estimates in particular cases.
Linear differential equations with interval spectrum
Authors: Johnson Russell A.
Published: June 2010

Let E be the set of equations in the hull of a fixed two-dimensional, almost-periodic, linear differential equation with interval spectrum which admit a bounded solution. Among other things, we prove that E is of first category and of measure zero.
Recursively saturated models of set theory
Authors: Schlipf John Stewart
Published: June 2010

We determine when a model

M

\mathfrak {M}

of ZF can be expanded to a model

⟨

M

,

X

⟩

\langle \mathfrak {M},\mathfrak {X}\rangle

of a weak extension of Gödel Bernays:

GB

+

{\text {GB}} +

the

Δ
1
1

\Delta _1^1

comprehension axiom. For nonstandard

M

\mathfrak {M}

, the ordinal of the standard part of

M

\mathfrak {M}

must equal the inductive closure ordinal of

M

\mathfrak {M}

, and

M

\mathfrak {M}

must satisfy the axioms of ZF with replacement and separation for formulas involving predicates for all hyperelementary relations on

M

\mathfrak {M}

. We also consider expansions to models of

GB

+

Σ
1
1

{\text {GB}} + \Sigma _1^1

choice, observe that the results actually apply to more general theories of well-founded relations, and observe relationships to expansibility to models of other second order theories.
A proof of a conjecture of A. H. Stone
Authors: Dickman R. F.
Published: June 2010

In this paper we use the techniques of analytic topology to establish a conjecture of A. H. Stone: A perfectly normal, locally connected, connected space is multicoherent if and only if there exist four nonempty, closed and connected subsets

A
0

,

A
1

,

A
2

,

A
3

{A_0},{A_1},{A_2},{A_3}

of X such that

⋃

i
=
0

3

A
i

=
X

\bigcup \nolimits _{i = 0}^3 {{A_i} = X}

and the nerve of

{

A
0

,

A
1

,

A
2

,

A
3

}

\{ {A_0},{A_1},{A_2},{A_3}\}

forms a closed 4-gon, i.e.

A
i

{A_i}

meets

A

i
+
1

{A_{i + 1}}

and

A

i
−
1

{A_{i - 1}}

and no others (the suffices being taken

mod

4

\bmod \; 4

).
Bohr compactification and continuous measures
Authors: Saeki Sadahiro
Published: June 2010

Let G be an LCA group with dual

Γ
\Gamma

. As a consequence of our main result, it is shown that every continuous regular measure

μ
\mu

concentrated on a Kronecker set and with

norm

>
1

{\text {norm}} > 1

has the property that

{

|

μ
^

|

>
1
}

\{ |\hat \mu | > 1\}

is dense in the Bohr compactification of

Γ
\Gamma

.
Maximal operators associated to radial functions in 𝐿²(𝑅²)
Authors: Aguilera N. E.
Published: June 2010

Stein’s result on spherical means imply that for

n
⩾
3

n \geqslant 3

the maximal operator associated to a radial function maps

L
p

(

R

n

)

{L^p}({{\mathbf {R}}^n})

boundedly into itself for

p
>
n

/

(
n
−
1
)

p > n/(n - 1)

. In this paper we take a look at the case

p
=
n
=
2

p = n = 2

.
The epireflective hull of the category of 𝑇₁ dispersed spaces
Authors: Neumann Lara V. , Wilson R. G.
Published: June 2010

An internal characterization is given of those spaces which can be embedded in products of

T
1

{T_1}

dispersed spaces.
Weak topologies and equicontinuity
Authors: Reynolds Donald F. , Schleusner John W.
Published: June 2010

Corresponding to each family F of real-valued functions on a set X, there is a weakest topology on X for which F is equicontinuous. This equiweak topology is pseudometrizable and provides a characterization of metrizable topologies in terms of point-separating families of real-valued functions.
Special handlebody decompositions of simply connected algebraic surfaces
Authors: Mandelbaum Richard
Published: June 2010

In this article we prove that any nonsingular complete-intersection surface admits a handlebody decomposition with no 1- and 3-handles. This generalizes results of Rudolph, Harer and Akbuluf on hypersurfaces of

C

P

3

{\mathbf {C}}{{\mathbf {P}}^3}

.
A note on the minimum condition
Authors: Mann Avinoam
Published: June 2010

We characterize the minimum condition on subgroups in terms of generators.
The mean-value iteration for set-valued mappings
Authors: Kuhfittig Peter K. F.
Published: June 2010

In this note Krasnoselskii’s iteration procedure
\[

x

n
+
1

=

1
2

(
I
+
T
)

x
n

{x_{n + 1}} = \tfrac {1}{2}(I + T){x_n}

\]
is extended to certain classes of set-valued mappings.
On ordered harmonic bounded variation
Authors: Belna C. L.
Published: June 2010

An example is given of a continuous real function that is of ordered harmonic bounded variation but not of harmonic bounded variation.
Rich sets
Authors: Chong C. T.
Published: June 2010

Let

(
V
=
L

)
α

{(V = L)_\alpha }

say that every bounded subset of

α
\alpha

is an element of

L
α

{L_\alpha }

. We show that if

(
V
=
L

)
α

{(V = L)_\alpha }

, then every

X
⊆
α

X \subseteq \alpha

of order-type

α
\alpha

is rich, in the sense that every

α
\alpha

-degree above that of X is represented by a subset of X.
𝑘-spaces and products of closed images of metric spaces
Authors: Gruenhage Gary
Published: June 2010

We show that a recent theorem of Y. Tanaka giving necessary and sufficient conditions for the product of two closed images of metric spaces to be a k-space is independent of the usual axioms of set theory.
A characterization of coabsoluteness for a class of metric spaces
Authors: Gates Catherine
Published: June 2010

All regular Hausdorff topological spaces can be partitioned into classes of coabsolute spaces. It is shown that in studying the coabsolute classes, only two types of spaces need to be considered: (i) those spaces which have a dense subset of locally compact points and (ii) nowhere locally compact spaces. The absolute of a space is the disjoint union of the absolute of a space of type (i) and the absolute of a space of type (ii). A characterization of the coabsolute subclasses is given for a class of metric spaces of type (i).
Cohomological dimension of an abelian monoid
Authors: Cheng Charles Ching-an , Shapiro Jay
Published: June 2010

It is shown that the cohomological dimension of an abelian monoid is equal to that of its group reflection provided that the monoid is either finitely generated or cancellative.
Semilocal skew group rings
Authors: Okniński Jan
Published: June 2010

Semilocal skew group rings

R

∗
θ

G

R{ \ast _\theta }G

are investigated. The full characterization is given in the case of algebras over a field of characteristic zero. The relationship between semilocal skew group rings and semilocal ordinary group rings

R
[
G
]

R[G]

is considered.
Two new extremal properties of the Koebe-function
Authors: Klouth R. , Wirths K.-J.
Published: June 2010

Using essentially Löwner’s method the extremality of the Koebe-functions with respect to two coefficient problems for inverses of univalent functions is proved.
Nonnormal Dirichlet quotients and nonnormal Blaschke quotients
Authors: Yamashita Shinji
Published: June 2010

There exists a nonnormal meromorphic function

f
1

/

f
2

{f_1}/{f_2}

in

U
=
{

|

z

|

>
1
}

U = \{ |z| > 1\}

, where

f
1

{f_1}

and

f
2

{f_2}

both are holomorphic functions with finite Dirichlet integrals in U. For each

0
>
α
>
1

0 > \alpha > 1

, there exists a nonnormal meromorphic function

B
1

/

B
2

{B_1}/{B_2}

in U, where

B
1

{B_1}

and

B
2

{B_2}

both are Blaschke products with finite

α
\alpha

-weighted Dirichlet integrals in U.
Fixed points in nonconvex domains
Authors: Chandler Eric , Faulkner Gary
Published: June 2010

A lemma of Janos is used to prove that nonexpansive self-maps which “shrink” a compact set X away from its boundary in

co

¯

X

\overline {{\text {co}}} \;X

have fixed points in X. The lemma is further employed to derive a version, for nonexpansive maps on star-shaped sets, of Janos and Solomon’s fixed-point theorem for continuous maps on spaces having an attractor for compact sets.
Paracompactness in perfectly normal, locally connected, locally compact spaces
Authors: Lane Diane J.
Published: June 2010

It is shown that, under

(

MA

+
¬

CH

)

({\text {MA}} + \neg {\text {CH}})

, every perfectly normal, locally compact and locally connected space is paracompact.
𝐾-theory of Azumaya algebras
Authors: Weibel Charles A.
Published: June 2010

Quillen has defined a

K
K

-theory for symmetric monoidal categories. We show that Quillen’s groups agree with the groups

K
0

{K_0}

,

K
1

{K_1}

, and

K
2

{K_2}

defined by Bass. Finally, we compute the

K
K

-theory of the Azumaya algebras over a commutative ring.
An upper bound for the sum of large differences between prime numbers
Authors: Cook R. J.
Published: June 2010

Let

p
n

{p_n}

denote the

n
n

th prime number,

d
n

=

p

n
+
1

−

p
n

{d_n} = {p_{n + 1}} - {p_n}

. We estimate the sum

Σ

d
n

\Sigma {d_n}

taken over

p
n

⩽
x
,

d
n

>

x
μ

{p_n} \leqslant x,{d_n} > {x^\mu }

where

1

/

6
>
μ
>
5

/

9

1/6 > \mu > 5/9

.
Spectral multiplicity for tensor products of normal operators
Authors: Azoff Edward A.
Published: June 2010

Two normal operators

N
1

{N_1}

and

N
2

{N_2}

are constructed such that for any pair

m
1

{m_1}

and

m
2

{m_2}

of their respective multiplicity functions, the ’convolution’
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis m 1 asterisk m 2 right-parenthesis left-parenthesis lamda right-parenthesis identical-to normal upper Sigma left-brace m 1 left-parenthesis lamda 1 right-parenthesis dot m 2 left-parenthesis lamda 2 right-parenthesis vertical-bar lamda 1 dot lamda 2 equals lamda right-brace">

(

m
1

∗

m
2

)
(
λ
)
≡
Σ
{

m
1

(

λ
1

)
⋅

m
2

(

λ
2

)

|

λ
1

⋅

λ
2

=
λ
}

({m_1} * {m_2})(\lambda ) \equiv \Sigma \{ {m_1}({\lambda _1}) \cdot {m_2}({\lambda _2})|{\lambda _1} \cdot {\lambda _2} = \lambda \}

fails to be a multiplicity function for the tensor product

N
1

⊗

N
2

{N_1} \otimes {N_2}

.
Uniform approximation by rational modules on nowhere dense sets
Authors: Trent Tavan , Wang James Li Ming
Published: June 2010

We prove that the rational module

R

(
X
)
u
n

k
1

\mathcal {R}(X)un{k_1}

is always uniformly dense in

C
(
X
)

C(X)

if the compact set

X
X

has no interior.
Characterization of conditional expectation operators for Banach-valued functions
Authors: Landers D. , Rogge L.
Published: June 2010

Let

B
B

be a Banach space. It is well known that for every probability space

(
Ω
,

A

,
P
)

(\Omega , \mathcal {A}, P)

and every sub-

σ
\sigma

-field

A

0

⊂

A

\mathcal {A}_0 \subset \mathcal {A}

there exists a conditional expectation operator

P

A

0

X

P^{\mathcal {A}_0}X

for

B
B

-valued

P
P

-integrable functions. This operator maps the space

L
p

(
Ω
,

A

,
P
,
B
)

{L_p}(\Omega , \mathcal {A}, P, B)

into itself for each

p
⩾
1

p \geqslant 1

. The operator is linear, idempotent, constant preserving and contractive in

L
p

{L_p}

. For

B
=

R

B = {\mathbf {R}}

and

p
≠
2

p \ne 2

these conditions characterize a conditional expectation operator. It turns out that in general these properties characterize conditional expectation operators for Banach-valued functions only for

p
=
1

p = 1

and strictly convex Banach spaces.
Flat and projective character modules
Authors: Cheatham Thomas J. , Stone David R.
Published: June 2010

We characterize left coherent, left Noetherian, and left Artinian rings by the flatness, projectivity, and injectivity of the character module of certain left

R
R

-modules.
Intersections of pure subgroups in abelian groups
Authors: Boyer D. , Rangaswamy K. M.
Published: June 2010

It is shown that a subgroup

H
H

of an abelian group

G
G

is an intersection of pure subgroups of

G
G

if and only if, for all primes

p
p

and positive integers

n
n

,

p
n

g
∈
H

{p^n}g \in H

and

p

n
−
1

g
∉
H

{p^{n - 1}}g \notin H

imply that there exists

z
∈
G

z \in G

such that

p
n

z
=
0

{p^n}z = 0

and

p

n
−
1

z
∉
H

{p^{n - 1}}z \notin H

. This solves a problem posed by

L
L

. Fuchs in [2] and [3].
A quick test for nonisomorphism of one-relator groups
Authors: Horadam K. J.
Published: June 2010

A sequence of integers is read off from the presentation of a finitely generated torsion-free one-relator group with nontrivial second integral homology, without recourse to group-theoretic manipulations. This test sequence is derived from the cup coproduct on the coring of the integral homology module of the group, and reflects information about the group’s second lower central factor group. Test sequences differ only if the corresponding groups are nonisomorphic. The test process can be generalised to any one-relator group with nontrivial second integral homology.
Asymptotic normal structure and fixed points of nonexpansive mappings
Authors: Baillon J.-B. , Schöneberg R.
Published: June 2010

A mapping

f
f

defined on a subset

X
X

of a Banach space

E
E

and taking values in

E
E

is said to be nonexpansive if

|

f
(
x
)
−
f
(
y
)

|

⩽

|

x
−
y

|

\left | {f(x) - f(y)} \right | \leqslant \left | {x - y} \right |

for all

x
,
y
∈
X

x,y \in X

. In this paper we introduce a promising new geometric property of Banach spaces and show that it yields via a minor modification of known arguments a new fixed point theorem for nonexpansive mappings which includes Kirk’s famous result as well as a recent result of Karlovitz. We also discuss in detail a situation not covered by this result.
A stratifiable 𝑘_{𝑅}-space which is not a 𝑘-space
Authors: Borges Carlos R.
Published: June 2010

E. A. Michael has given an example of a

σ
\sigma

-compact, cosmic

k
R

{k_R}

-space which is not a

k
k

-space. We prove that this space is stratifiable.
Stability of periodic orbits in the theorem of Šarkovskii
Authors: Block Louis
Published: June 2010

Let

f
f

be a continuous map of a closed, bounded interval into itself. It is shown that the conclusion of the theorem of Sarkovskii holds for perturbations of

f
f

. In other words, if

f
f

has a periodic point of period

k
k

, and

g
g

is a continuous map close to

f
f

, then

g
g

has periodic points of certain periods.
A countable 𝑘-space that is not stratifiable
Authors: Foged L.
Published: June 2010

We add a point to a countable metric space to obtain a regular Fréchet space which is not stratifiable.
On a Stieltjes version of Gronwall’s inequality
Authors: Mingarelli Angelo B.
Published: June 2010

A unified formulation of Gronwall’s inequality

(

GI

)

({\text {GI}})

for certain Stieltjes integrals is presented. An advantage in using Stieltjes integrals lies in the fact that some of the "continuous" and "discrete" versions of the inequality may be obtained as a consequence of the same theorem.
A countable basis for Σ¹₂ sets and recursion theory on ℵ₁
Authors: Maass Wolfgang
Published: June 2010

Countably many

ℵ
1

{\aleph _1}

-recursively enumerable sets are constructed from which all the

ℵ
1

{\aleph _1}

-recursively enumerable sets can be generated by using countable union and countable intersection. This implies under

V
=
L

V = L

that there exists as well a countable basis for

∑
n
1

\sum _n^1

sets of reals,

n
⩾
2

n \geqslant 2

. Further under

V
=
L

V = L

the lattice

E

∗
(

ℵ
1

)

\mathcal {E}*({\aleph _1})

of

ℵ
1

{\aleph _1}

-recursively enumerable sets modulo countable sets has only

ℵ
1

{\aleph _1}

many automorphisms.
A partition theorem for perfect sets
Authors: Blass Andreas
Published: June 2010

Let

P
P

be a perfect subset of the real line, and let the

n
n

-element subsets of

P
P

be partitioned into finitely many classes, each open (or just Borel) in the natural topology on the collection of such subsets. Then

P
P

has a perfect subset whose

n
n

-element subsets lie in at most

(
n
−
1
)
!

(n - 1)!

of the classes.
Spaces dominated by two-complexes
Authors: Jajodia Sushil
Published: June 2010

We say a group

G
G

is of geometric dimension

⩽
2

\leqslant 2

if there is an aspherical

2
2

-dimensional CW-complex

P
P

with fundamental group isomorphic to

G
G

. In this note, we study the following problem: Suppose

G
G

is a group of geometric dimension

⩽
2

\leqslant 2

with associated aspherical

2
2

-dimensional CW-complex

P
P

. Suppose further that

X
X

is a CW-complex having fundamental group isomorphic to

G
G

and that

X
X

is dominated by a

2
2

-complex. If the Wall invariant

W

a

2

[
X
]
∈

K
~

0

(
Z
G
)

{\text {W}}{{\text {a}}_2}[X] \in {\tilde K_0}(ZG)

vanishes, does

X
X

have the same homotopy type as

P
∨
k

S
2

P \vee k{S^2}

where

k

S
2

k{S^2}

denotes the sum of

k
k

copies of the

2
2

-sphere

S
2

{S^2}

?
A note on the Thom isomorphism
Authors: Mahowald Mark , Ray Nigel
Published: June 2010

The generalized homology version of the Thom isomorphism theorem is exploited to give easy proofs of several recent theorems.
Deficient values of entire functions and their derivatives
Authors: Drasin David , Zhang Guang Hou , Yang Lo , Weitsman Allen
Published: June 2010

Let

f
(
z
)

f(z)

be entire and of finite order,

f

(
n
)

{f^{(n)}}

be the

n
n

th derivative, and

Δ
n

(
f
)
=
Σ
δ
(
a
,

f

(
n
)

)

{\Delta _n}(f) = \Sigma \delta (a,{f^{(n)}})

, the sum of all deficient values of

f

(
n
)

{f^{(n)}}

. The authors show that

Δ
n

(
f
)

{\Delta _n}(f)

can be strictly increasing.
Markov operators and quasi-Stonian spaces
Authors: Atalla Robert E.
Published: June 2010

Let

X
X

be a quasi-stonian space, and let

T
T

be a

σ
\sigma

-additive Markov operator on

C
(
X
)

C(X)

. Ando proved that if all

T
T

-invariant probabilities are

σ
\sigma

-additive, then

T
T

is strongly ergodic (and the space of fixed points is finite-dimensional). We prove that if the set of

σ
\sigma

-additive

T
T

-invariant probabilities is weak-* dense in the set of all

T
T

-invariant probabilities, then

T
T

is strongly ergodic. This result is easy in case

X
X

is hyperstonian. Our method of proof is to use an idea of Gordon to "hyperstonify" part of our quasi-stonian space.
On a theorem of Arhangel′skiĭ
Authors: Hung H. H.
Published: June 2010

We define a class of spaces which is more extensive than the class of BCO spaces and which counts among its members some that are not even first countable, and show that this more extensive class of spaces nevertheless intersects the class of paracompact Hausdorff spaces at precisely the class of metrizable spaces as does the class of BCO spaces, thus extending a theorem of Arhangel’skiĭ. We further show that this extension of Arhangel’skiĭ’s result has gone the farthest in the sense that any class of spaces that meets the paracompact spaces at precisely the metrizable spaces must, among the Hausdorff spaces, be smaller than the class we have defined.
An abstract fixed point theorem for nonexpansive mappings
Authors: Kirk W. A.
Published: June 2010

A class

S

\mathcal {S}

of subsets of a bounded metric space is said to be normal if each member of

S

\mathcal {S}

contains a nondiametral point. An induction proof is given for the following. Suppose

M
M

is a nonempty bounded metric space which contains a class

S

\mathcal {S}

of subsets which is countably compact, normal, stable under arbitrary intersections, and which contains the closed balls in

M
M

. Then every nonexpansive self-mapping of

M
M

has a fixed point.
Topology of certain submanifolds in the Euclidean sphere
Authors: Xin Y. L.
Published: June 2010

Using the nonexistence theorem for stable harmonic maps, we study the fundamental group of certain submanifolds in the Euclidean sphere.
On some embedding theorems for inverse semigroups
Authors: O’Carroll L.
Published: June 2010

A semilattice decomposition of an inverse semigroup has good internal mapping properties. These are used to give natural proofs of some embedding theorems, which were originally proved in a rather artificial way.
Normal classes of prime rings determined by modules
Authors: Nicholson W. K. , Watters J. F.
Published: June 2010

It is shown that primitive rings with a socle, left weakly primitive rings and prime Johnson rings constitute normal classes of rings. The technique involves characterizing each class as all rings with a faithful module of a certain type.
More on convergence in unitary matrix spaces
Authors: Arazy Jonathan
Published: June 2010

Let

E
E

be a symmetric sequence space satisfying the Radon-Riesz Property
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace StartAbsoluteValue EndAbsoluteValue x Subscript n Baseline StartAbsoluteValue EndAbsoluteValue right-arrow StartAbsoluteValue EndAbsoluteValue x StartAbsoluteValue EndAbsoluteValue and x Subscript n Baseline right-arrow x weakly right-brace right double arrow StartAbsoluteValue EndAbsoluteValue x Subscript n Baseline minus x StartAbsoluteValue EndAbsoluteValue right-arrow 0 comma">

{

|

|

x
n

|

|

→

|

|

x

|

|

and 

x
n

→
x

 weakly

}
⇒

|

|

x
n

−
x

|

|

→
0
,

\{ ||{x_n}|| \to ||x||{\text {and }}{x_n} \to x{\text { weakly}}\} \Rightarrow ||{x_n} - x|| \to 0,

\]
then the same is true for the associated unitary matrix space

C

E

C_{E}

.
𝑘-discrete differentials of certain operators on Banach spaces
Authors: Dayal S.
Published: June 2010

By observing a convex property of discrete differences, one-sided

k
k

-discrete,

k
k

-discrete Gâteaux and

k
k

-discrete Fréchet differentials are introduced. It is proved that a locally bounded

n
n

-convex function has

k
k

-discrete Fréchet differentials for

1
⩽
k
⩽
n
−
2

1 \leqslant k \leqslant n - 2

and one-sided

(
n
−
1
)

(n - 1)

-discrete differentials at every point of its domain. Various properties of discrete differentials of an

n
n

-convex function are studied. As an application of these results the author proves that an

n
n

-convex function has a strong

(
n
−
2
)

(n - 2)

-Taylor series expansion and an

(
n
−
1
)

(n - 1)

th Fréchet differential provided it has a strong

n
n

-Taylor series expansion about the point.
Distance and volume decreasing theorems for quasiconformal mappings
Authors: Petridis Nicholas C.
Published: June 2010

The method used by the author in deriving a Picard type theorem for quasiconformal mappings [Proc.. Amer. Math. Soc. 61 (1976), 265-27], improved by a proposition of S.-T. Yau [Amer. J. Math. 100 (1978), 197-203] is employed here to extend the Schwarz-Ahlfors lemma to harmonic quasiconformal mappings. The target space is not necessarily hyperbolic, not even negatively curved.
Derivatives of polynomials with positive coefficients
Authors: Varma A. K.
Published: June 2010

Let

P
n

(
x
)

{P_n}(x)

be an algebraic polynomial of degree

n
n

with positive coefficients. We set
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript n Baseline equals StartFraction StartAbsoluteValue EndAbsoluteValue upper P prime Subscript n Baseline left-parenthesis x right-parenthesis omega left-parenthesis x right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript upper L 2 left-bracket 0 comma normal infinity right-parenthesis Baseline Over StartAbsoluteValue EndAbsoluteValue upper P Subscript n Baseline left-parenthesis x right-parenthesis omega left-parenthesis x right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript upper L 2 left-bracket 0 comma normal infinity right-parenthesis Baseline EndFraction period">

I
n

=

|

|

P
n
′

(
x
)
ω
(
x
)

|

|

L
2

[
0
,
∞
)

|

|

P
n

(
x
)
ω
(
x
)

|

|

L
2

[
0
,
∞
)

.

{I_n} = \frac {{||P’_n(x)\omega (x)|{|_{{L_2}[0,\infty )}}}}{{||{P_n}(x)\omega (x)|{|_{{L_2}[0,\infty )}}}}.

\]
In this work upper bounds of

I

n

I_{n}

are investigated. We restrict ourselves here with the case

ω
(
x
)
=

x

α

/

2

e

−
x

/

2

\omega (x) = {x^{\alpha /2}}{e^{ - x/2}}

. Reslts are shown to be best possible.
Totally real submanifolds in a 6-sphere
Authors: Ejiri Norio
Published: June 2010

A

6
6

-dimensional sphere

S
6

{S^6}

has an almost complex structure induced by properties of Cayley algebra. We investigate

3
3

-dimensional totally real submanifolds in

S
6

{S^6}

and classify

3
3

-dimensional totally real submanifolds of constant sectional curvature.
Convergence of best best 𝐿_{∞}-approximations
Authors: Al-Rashed Abdallah M. , Darst Richard B.
Published: June 2010

Let

(
Ω
,

A

,
μ
)

(\Omega , \mathcal {A}, \mu )

be a probability space and let

{

B

i

}

i
=
1

∞

\{ {\mathcal {B}_i}\} _{i = 1}^\infty

be an increasing sequence of subsigma algebras of

A

\mathcal {A}

. Let

A
=

L
∞

(
Ω
,

A

,
μ
)

A = {L_\infty }(\Omega , \mathcal {A}, \mu )

, let

B
i

=

L
∞

(
Ω
,

B

i

,
μ
)

{B_i} = {L_\infty }(\Omega ,{\mathcal {B}_i},\mu )

, and let

f
∈
A

f \in A

. Let

f
i

{f_i}

denote the best best

L
∞

{L_\infty }

-approximation to

f
f

by elements of

B
i

{B_i}

. It is shown that

lim
i

f
i

(
x
)

{\lim _i}{f_i}(x)

exists a.e.
On the uniform ergodic theorem of Lin
Authors: Lloyd Stuart P.
Published: June 2010

Lin has given necessary and sufficient conditions for convergence in the uniform operator topology of

A
n

=
(
1
+
T
+
⋯
+

T

n
−
1

)

/

n

{A_n} = (1 + T + \cdots + {T^{n - 1}})/n

,

T
T

being a Banach space operator satisfying

‖

T
n

‖

/

n
→
0

\left \| {{T^n}} \right \|/n \to 0

. We prove a generalization in which the Cesàro means are replaced by any bounded sequence in the affine hull converging uniformly to invariance. In the case where

T
:

C
0

(
X
)
→

C
0

(
X
)

T:{C_0}(X) \to {C_0}(X)

is a transient Feller operator for noncompact locally compact Hausdorff space

X
X

, we show that

{

A
n

}

\{ {A_n}\}

converges strongly but never uniformly.
Lipschitz functions and spectral synthesis
Authors: Lee Sung Yung
Published: June 2010

An

S
S

-set in the circle group

T
T

is a closed subset

S
S

of

T
T

for which

j
(
S
)

¯

=
k
(
S
)

\overline {j(S)} = k(S)

. We construct a non-

S
S

-set

S
S

satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="union Underscript alpha greater-than 0 Endscripts upper L i p Subscript alpha Baseline left-parenthesis upper T right-parenthesis intersection k left-parenthesis upper S right-parenthesis subset-of ModifyingAbove j left-parenthesis upper S right-parenthesis With bar period">

⋃

α
>
0

Lip

α

(
T
)
∩
k
(
S
)
⊂

j
(
S
)

¯

.

\bigcup \limits _{\alpha > 0} {{{\operatorname {Lip} }_\alpha }(T) \cap k(S) \subset \overline {j(S)} .}

\]
Thus

Lip
α

(
T
)
∩
A
(
T
)

{\operatorname {Lip} _\alpha }(T) \cap A(T)

is not a big enough part of

A
(
T
)

A(T)

to test the synthesizability of a given closed subset of

T
T

.
On strong unicity of 𝐿₁-approximation
Authors: Kroó András
Published: June 2010

Let

C
1

{C_1}

be the space of continuous functions on

[
0
,
1
]

[0,1]

with norm

‖
f
‖

=

∫
0
1

|

f
(
x
)

|

d
x

\left \| f \right \| = \int _0^1 {|f(x)|} dx

, and let

G
⊂

C
1

G \subset {C_1}

be a finite dimensional unicity subspace, i.e. any

f
∈

C
1

f \in {C_1}

possesses a unique best approximation out of

G
G

. Consider an arbitrary

f
∈

C
1

f \in {C_1}

such that 0 is its best approximation. Then for any

0
>
δ
>

δ
0

0 > \delta > {\delta _0}

and

g
~

∈
G

\tilde g \in G

with

‖

f
−

g
~

‖

⩽

‖
f
‖

+
δ

\left \| {f - \tilde g} \right \| \leqslant \left \| f \right \| + \delta

it follows that

‖

g
~

‖

⩽
K

w
f
∗

(
δ
)

\left \| {\tilde g} \right \| \leqslant Kw_f^* (\delta )

, where

w
f
∗

(
h
)
=

(

w
f

−
1

(
h
)
h

)

−
1

w_f^*(h) = {(w_f^{ - 1}(h)h)^{ - 1}}

and

w
f

(
h
)

{w_f}(h)

denotes the modulus of continuity of

f
f

. (

−
1

- 1

is used to denote the inverse function.)
A note on 𝐿(𝑝,𝑞) spaces and Orlicz spaces with mixed norms
Authors: Milman Mario
Published: June 2010

Necessary and sufficient conditions are given for the embedding of

L
(
p
,
q
)

L(p,q)

spaces and Orlicz spaces with mixed norms.
A fixed point theory for multivalued mappings
Authors: Pasicki Lech
Published: June 2010

In the present paper several theorems are proved for the multi-valued mappings that operate on the type I and type II spaces defined in [10]. The theorems generalize the well-known results for the locally convex spaces.
Topologically unrealizable automorphisms of free groups
Authors: Stallings John R.
Published: June 2010

Let

ϕ
:
F
→
F

\phi :F \to F

be an automorphism of a finitely generated free group. It has been conjectured (I heard it from Peter Scott) that the fixed subgroup of

ϕ
\phi

is always finitely generated. This is known to be so if

ϕ
\phi

has finite order [1], or if

ϕ
\phi

is realizable by a homeomorphism of a compact

2
2

-manifold with boundary [2]. Here we give examples of automorphisms

ϕ
\phi

, no power of which is topologically realizable on any

2
2

-manifold; perhaps the simplest is the automorphism of the free group of rank 3, given by

ϕ
(
x
)
=
y

\phi (x) = y

,

ϕ
(
y
)
=
z

\phi (y) = z

,

ϕ
(
z
)
=
x
y

\phi (z) = xy

.
A negative answer to three questions on 𝐾-primitive rings
Authors: Osłowski B. J.
Published: June 2010

It is shown that three questions on

K
K

-primitive rings posed by Kezlan in

[

3

]

[{\mathbf {3}}]

have a negative answer.
Almost periodic solutions for a certain class of almost periodic systems
Authors: Seifert George
Published: June 2010

Using a result due to Medvedev [3], we obtain conditions under which systems of ordinary differential equations of the form

x
′

=
F
(
t
,
x
,
x
)
+
G
(
t
,
x
)

x’ = F(t,x,x) + G(t,x)

where

F
F

and

G
G

are almost periodic in

t
t

will have unique almost periodic solutions with certain global stability properties and module containment. These conditions are compared to conditions for the existence, but not uniqueness, for such solutions obtained by Kartsatos in [2]. Both results, our as well as Kartsatos’, are applied to a second order equation of Lienard type with almost periodic forcing.
An extremal vector-valued 𝐿^{𝑝}-function taking no extremal vectors as values
Authors: Greim Peter
Published: June 2010

We give an example of a nonseparable Banach space

V
V

and a function

x
x

on [0, 1] with values in the unit sphere of

V
V

that is an extreme point of the unit balls of all Bochner

L
p

{L^p}

-spaces

L
p

(
λ
,
V
)

{L^p}(\lambda ,V)

,

1
>
p
⩽
∞

1 > p \leqslant \infty

,

λ
\lambda

Lebesgue measure, though none of its values is an extreme point of the unit ball of

V
V

. This shows that a characterization of the extremal elements in

L
p

(
λ
,
V
)

{L^p}(\lambda ,V)

for separable

V
V

, given by J. A. Johnson, does not hold in general.
Reduction theorems for a class of semilinear equations at resonance
Authors: Bates Peter W.
Published: June 2010

In solving equations of the form

L
u
−
N
u
=
p

Lu - Nu = p

in a Hilbert space, where

L
L

is linear and

N
N

is nonlinear, the alternative method can sometimes be used to reduce the problem to one in a subspace. In this note previous reduction results are extended and at the same time the proofs are simplified. The approach is to use simple fixed point theorems in place of the traditional variational methods which are often quite delicate.
An example on strong unicity constants in trigonometric approximation
Authors: Chan K. Y. , Chen Y. M. , Liu M. C. , Ng S. M.
Published: June 2010

In this paper an explicit example is constructed to illustrate some interesting properties of the strong unicity constant in the case of approximations by trigonometric polynomials.
Relative weak convergence in semifinite von Neumann algebras
Authors: Kaftal Victor
Published: June 2010

An operator is compact relative to a semifinite von Neumann algebra, i.e., belongs to the two-sided closed ideal generated by the finite projections relative to the algebra, if and only if it maps vector sequences converging weakly relative to the algebra into strongly converging ones (generalized Hilbert condition). The generalized Wolf condition characterizes the class of almost Fredholm operators.
On a theorem of Baker, Lawrence and Zorzitto
Authors: Székelyhidi L.
Published: June 2010

The result of J. Baker, J. Lawrence and F. Zorzitto on the stability of the equation

f
(
x
+
y
)
=
f
(
x
)
f
(
y
)

f(x + y) = f(x)f(y)

is generalized by proving the following theorem: if

G
G

is a semigroup and

V
V

is a right invariant linear space of complex valued functions on

G
G

, and if

f
f

,

m
m

are complex valued functions on

G
G

for which the function

x
→
f
(
x
y
)
−
f
(
x
)
m
(
y
)

x \to f(xy) - f(x)m(y)

belongs to

V
V

for every

y
y

in

G
G

, then either

f
f

is in

V
V

or

m
m

is exponential.
Derivatives of 𝐻^{𝑝} functions
Authors: Øyma Knut , Rookshin Serge
Published: June 2010

We prove that if

{

z
n

}

\{ {z_n}\}

is uniformly separated and

f
∈

H
p

f \in {H^p}

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace f Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis z Subscript n Baseline right-parenthesis left-parenthesis 1 minus StartAbsoluteValue z Subscript n Baseline EndAbsoluteValue squared right-parenthesis Superscript k plus 1 slash p Baseline right-brace Subscript n equals 1 Superscript normal infinity Baseline element-of l Superscript p Baseline for k equals 1 comma 2 comma ellipsis">

{

f

(
k
)

(

z
n

)

(
1
−

|

z
n

|

2

)

k
+
1

/

p

}

n
=
1

∞

∈

l
p

for 

k
=
1
,
2
,
…

\{ {f^{(k)}}({z_n}){(1 - {\left | {{z_n}} \right |^2})^{k + 1/p}}\} _{n = 1}^\infty \in {l^p}\;{\text {for }}k = 1,2, \ldots

.
On coisotropic imbeddings of presymplectic manifolds
Authors: Gotay Mark J.
Published: June 2010

Existence and uniqueness theorems are proved for coisotropic imbeddings of presymplectic manifolds into symplectic manifolds.
Polynomial generators for 𝐻_{∗}(𝐵𝑆𝑈) and 𝐻_{∗}(𝐵𝑆𝑂;𝑍₂)
Authors: Kochman Stanley O.
Published: June 2010

Specific formulas are given for choosing polynomial generators of

H
∗

(
B
S
U
;
R
)

{H_* }(BSU;R)

, for various

R
R

, in terms of the canonical polynomial generators of

H
∗

(
B
U
;
R
)

{H_*}(BU;R)

. The analogous formulas for polynomial generators of

H
∗

(
B
S
O
;

Z
2

)

{H_*}(BSO;{Z_2})

are also given.
Two applications of asymptotic prime divisors
Authors: McAdam Stephen
Published: June 2010

Some recent interest has focused on the set of prime divisors of large powers of an ideal in a Noetherian ring. This note presents two results whose proofs appear to depend on knowledge of such asymptotic prime divisors.
Pseudoholomorphic functions with nonantiholomorphic characteristics
Authors: Koohara Akira
Published: June 2010

Let

κ
(
z
)
∈

C
∞

(
Ω
)

\kappa (z) \in {C^\infty }(\Omega )

and

‖
κ
‖

>
1

\left \| \kappa \right \| > 1

. Necessary and sufficient conditions for the system of equations

∂

f
¯

=
κ
(
z
)
∂
f

\partial \bar f = \kappa (z)\partial f

to be locally plentiful are given, and under them a representation of

κ
\kappa

also is given.
The asymptotic behavior of a class of nonlinear differential equations of second order
Authors: Tong Jing Cheng
Published: June 2010

Let

u
+
f
(
t
,
u
)
=
0

u + f(t,u) = 0

be a nonlinear differential equation. If there are two nonnegative continuous functions

υ
(
t
)

\upsilon (t)

,

φ
(
t
)

\varphi (t)

for

t
⩾
0

t \geqslant 0

, and a continuous function

g
(
u
)

g(u)

for

u
⩾
0

u \geqslant 0

, such that (i)

∫
1
∞

υ
(
t
)
φ
(
t
)

d
t
>
∞

\smallint _1^\infty \upsilon (t)\varphi (t)\;dt > \infty

; (ii) for

u
>
0

u > 0

,

g
(
u
)

g(u)

is positive and nondecreasing; (iii)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f left-parenthesis t comma u right-parenthesis EndAbsoluteValue greater-than upsilon left-parenthesis t right-parenthesis phi left-parenthesis t right-parenthesis g left-parenthesis StartAbsoluteValue u EndAbsoluteValue slash t right-parenthesis">

|

f
(
t
,
u
)

|

>
υ
(
t
)
φ
(
t
)
g
(

|
u
|

/

t
)

\left | {f(t,u)} \right | > \upsilon (t)\varphi (t)g(\left | u \right |/t)

for

t
⩾
1

t \geqslant 1

,

−
∞
>
u
>
∞

- \infty > u > \infty

, then the equation has solutions asymptotic to

a
+
b
t

a + bt

, where

a
a

,

b
b

are constants and

b
≠
0

b \ne 0

. Our result generalizes a theorem of D. S. Cohen [3].
The capacity of 𝐶₅ and free sets in 𝐶ₘ²
Authors: Mead D. G. , Narkiewicz W.
Published: June 2010

In a recent paper, S. K. Stein examined the problem of determining the cardinality,

τ
(

C
m
k

)

\tau (C_m^k)

, of the largest subset

S
S

of the direct product

C
m
k

C_m^k

of

k
k

copies of

C
m

{C_m}

such that distinct sums of elements of

S
S

yield distinct elements of

C
m
k

C_m^k

. In this paper we show that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau Superscript asterisk Baseline left-parenthesis upper C 5 right-parenthesis equals limit Underscript k right-arrow normal infinity Endscripts left-parenthesis tau left-parenthesis upper C 5 Superscript k Baseline right-parenthesis slash k right-parenthesis equals 2">

τ
∗

(

C
5

)
=

lim

k
→
∞

(
τ
(

C
5
k

)

/

k
)
=
2

{\tau ^* }({C_5}) = {\lim _{k \to \infty }}(\tau (C_5^k)/k) = 2

, answering a question raised by Stein. We also produce an infinite set of

m
m

’s such that

τ
(

C
m
2

)
>
2
[

log
2

m
]

\tau (C_m^2) > 2[{\log _2}m]

.
Free 𝐸-𝑚 groups and free 𝐸-𝑚 semigroups
Authors: Tamura Takayuki
Published: June 2010

A group [semigroup] is called

E
E

-

m
m

if it satisfies the identity

(
x
y

)
m

=

x
m

y
m

{(xy)^m} = {x^m}{y^m}

. In this paper the author studies a necessary and sufficient condition on

m
m

and

n
n

for the free

E
E

-

m
m

group [semigroup] to be homomorphic onto the free

E

 - 

n

E{\text { - }}n

group [semigroup].
On entire functions of affine lineage
Authors: Dahmen Wolfgang , Micchelli Charles A.
Published: June 2010

In this paper we provide a characterization of entire functions whose zero set is a union of real hyperplanes.
Finite homological dimension of 𝐵𝑃_{∗}(𝑋) for infinite complexes
Authors: Landweber Peter S.
Published: June 2010

The main result proved here is that

B

P

∗

(
E
G

×
G

X
)

{\text {B}}{{\text {P}}_* }(EG{ \times _G}X)

has finite homological dimension when

G
=

Z

p

G = {{\mathbf {Z}}_p}

and

X
X

is a finite

G

 - CW

G{\text { - CW}}

-complex. The argument uses

B

P

∗

BP

{\text {B}}{{\text {P}}_*}{\text {BP}}

-comodules.
Fixed points of subopen multifunctions
Authors: McClendon J. F.
Published: June 2010

A fixed point theorem of the Lefschetz type is proved for subopen (e.g., open-graph) multifunctions. A coincidence version is also given.
Completely metrizable spaces of embeddings
Authors: McCoy R. A.
Published: June 2010

A study is made of the complete metrizability of spaces of embeddings and spaces of homeomorphisms under the compact-open topology. It is shown that the space of continuous injections from a hemicompact metric space into a complete metric space is completely metrizable.
When total variation is additive
Authors: Cater F. S.
Published: June 2010

Let

f
f

and

g
g

be continuous functions of bounded variation on

[
0
,
1
]

[0,1]

. We use the Dini derivates of

f
f

and

g
g

to give a necessary and sufficient condition that the equation

V
(
f
+
g
)
=
V
(
f
)
+
V
(
g
)

V(f + g) = V(f) + V(g)

holds.
Oscillation theorems for a second order sublinear ordinary differential equation
Authors: Kura Takeshi
Published: June 2010

New oscillation criteria are given for the differential equation
\[

u
+
a
(
t
)

|

u

|

α

sgn
⁡
u
=
0
,

0
>
α
>
1
,

u + a(t)|u{|^\alpha }\operatorname {sgn} u = 0,\quad 0 > \alpha > 1,

\]
where

a
(
t
)

a(t)

is allowed to take on negative values for arbitrarily large

t
t

.
Decompositions formulas for a class of partial differential equations
Authors: Altin Abdullah , Young Eutiquio C.
Published: June 2010

The paper presents decomposition formulas for solutions of a class of singular partial differential equations. The equations consist of products of iterated differential operators each of which involves a real parameter. The decomposition is given in terms of solutions corresponding to each operator.
Interpolation of uniformly convex Banach spaces
Authors: Cwikel Michael , Reisner Shlomo
Published: June 2010

If

A
0

{A_0}

and

A
1

{A_1}

are a compatible couple of Banach spaces, one of which is uniformly convex, then the complex interpolation spaces

[

A
0

,

A
1

]
θ

{[{A_0},{A_1}]_\theta }

are also uniformly convex for

0
>
θ
>
1

0 > \theta > 1

. Estimates are given for the moduli of convexity and smoothness of

[

A
0

,

A
1

]
θ

{[{A_0},{A_1}]_\theta }

in terms of these moduli for

A
0

{A_0}

and

A
1

{A_1}

. In general, up to equivalence of moduli these estimates are best possible.
Nonnegative rectangular matrices having certain nonnegative 𝑊-weighted group inverses
Authors: Jain S. K.
Published: June 2010

Nonnegative rectangular matrices having nonnegative

W
W

-weighted group inverses are characterized. Our techniques suggest an interesting approach to extend the earlier known results on

λ
\lambda

-monotone square matrices to rectangular ones. We also answer a question of characterizing nonnegative matrices having a nonnegative solution

X
X

where (1)

A
=
A
X
A

A = AXA

, (2)

X
=
X
A
X

X = XAX

, (3)

(
A
X
)

(AX)

is

0
0

-symmetric, (4)

(
X
A
)

(XA)

is

0
0

-symmetric. In particular, we obtain theorems of Berman-Plemmons and Plemmons-Cline characterizing nonnegative matrices

A
A

with a nonnegative Moore-Penrose inverse. Matrices having nonnegative generalized inverses are of interest in the study of finding nonnegative best approximate solutions of linear systems. Such matrices are of considerable interest in statistics, numerical linear algebra and mathematical economics.
The bidual of 𝐶(𝑋,𝐸)
Authors: Cambern Michael , Greim Peter
Published: June 2010

In this article we obtain, under certain conditions, a characterization of the bidual of the space

C
(
X
,
E
)

C(X,E)

of continuous functions on a compact Hausdorff space

X
X

to a Banach space

E
E

. It is shown that if

X
X

is dispersed and

E
E

is arbitrary, or if

X
X

is arbitrary and

E
∗

{E^* }

has the Radon-Nikodym property, then

C

(
X
,
E

)

∗
∗

C{(X,E)^{**}}

can be represented as a space of continuous functions on a compact Hausdorff space

Z
Z

to

E

∗
∗

{E^{**}}

when the latter space is given its weak* topology.