bluuebunny/crossref_metadata_2025 huggingface dataset:
On a coefficient inequality for starlike functions
Authors: Barnard Roger W.
Published: June 2010

M. S. Robertson considered a coefficient inequality

I
I

for starlike functions that, if true, would imply a generalized Bieberbach coefficient inequality

B
B

for close to convex functions. An example is given of a starlike function whose coefficients do not satisfy coefficient inequality

I
I

.
Almost everywhere convergence for sequences of continuous functions
Authors: Schrader K. , Umamaheswaram S.
Published: June 2010

The main result in this paper is the following theorem. Theorem 1.1. Let

{
y
k
}
,
y
k
:
I
→
R

\{ yk\} ,yk:I \to R

, be a sequence of continuous real valued functions defined on the bounded interval

I
I

. Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript k j Baseline equals left-brace x colon x element-of upper I comma StartAbsoluteValue y k left-parenthesis x right-parenthesis minus y j left-parenthesis x right-parenthesis EndAbsoluteValue greater-than 0 right-brace equals union upper I Subscript k j n">

D

k
j

=
{
x
:
x
∈
I
,

|

y
k
(
x
)
−
y
j
(
x
)

|

>
0
}
=

∪
n

I

k
j
n

{D_{kj}} = \{ x:x \in I,|yk(x) - yj(x)| > 0\} = { \cup _n}{I_{kjn}}

, where each

I

k
j
n

{I_{kjn}}

is a relatively open subinterval of

I
I

and

I

k
j
n

∩

I

k
j
m

=
ϕ

{I_{kjn}} \cap {I_{kjm}} = \phi

for

n
≠
m

n \ne m

. Assume there exists a function

ϕ
,
ϕ
:
(
0
,
+
∞
)
→
(
0
,
+
∞
)

\phi ,\phi :(0, + \infty ) \to (0, + \infty )

, such that

lim

r
→

0
+

(
ϕ
(
r
)

/

r
)
=
+
∞
,
ϕ

{\lim _{r \to {0^ + }}}(\phi (r)/r) = + \infty ,\phi

is a nondecreasing function of

r
r

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n Endscripts phi left-parenthesis mu left-parenthesis upper I Subscript k j n Baseline right-parenthesis right-parenthesis sup Underscript x element-of upper I Subscript k j n Baseline Endscripts StartAbsoluteValue y Subscript k Baseline left-parenthesis x right-parenthesis minus y Subscript j Baseline left-parenthesis x right-parenthesis EndAbsoluteValue Superscript p Baseline less-than-or-equal-to upper M">

∑
n

ϕ
(
μ
(

I

k
j
n

)
)

sup

x
∈

I

k
j
n

|

y
k

(
x
)
−

y
j

(
x
)

|

p

≤
M

\sum \limits _n {\phi (\mu ({I_{kjn}}))\sup \limits _{x\in {I_{kjn}}} |{y_k}(x) - {y_j}(x){|^p} \leq M}

\]
for all

k
,
j

k,j

sufficiently large, where

μ
\mu

is Lebesgue measure and

0
>
p
>
+
∞

0 > p > + \infty

. Then

{

y
k

}

\{ {y_k}\}

contains a subsequence which converges almost everywhere to a Lebesgue measurable function

y
y

.
The use of attractive fixed points in accelerating the convergence of limit-periodic continued fractions
Authors: Gill John
Published: June 2010

A continued fraction can be interpreted as a composition of Möbius transformations. Frequently these transformations have powerful attractive fixed points which, under certain circumstances, can be used as converging factors for the continued fraction. The limit of a sequence of such fixed points can be employed as a constant converging factor.
A generalized Shilov boundary and analytic structure
Authors: Basener Richard F.
Published: June 2010

A generalization of the concept of the Shilov boundary of a uniform algebra is introduced. This makes it possible to formulate and prove several-dimensional analogues of certain well-known results which guarantee the existence of one-dimensional analytic structure when a function in the algebra is finite-to-one over a suitable part of its spectrum.
Generalized flag manifolds bound equivariantly
Authors: Pittie Harsh V.
Published: June 2010

Given a compact, connected lie group

G
G

and a maximal torus

T
T

, we give a simple, explicit construction of a

G
G

-manifold

M
M

which bounds the homogeneous space

G

/

T

G/T

equivariantly.
A proof of Bernstein’s theorem on regularly monotonic functions
Authors: McHugh J. A. M.
Published: June 2010

A function is called “regularly monotonic” if it is of class

C
∞

{C^\infty }

and each derivative is of a fixed sign (which may depend on the order of the derivative). We present a short proof of Bernstein’s theorem on the analyticity of such functions.
Application of Serrin’s kernel parametrix to the uniqueness of 𝐿₁ solutions of elliptic equations in the unit ball
Authors: Diederich J. R.
Published: June 2010

In this paper it will be established that

L
1

{L_1}

solutions of elliptic partial differential equations, with

α
\alpha

-Hölder continuous coefficients, which assume their boundary values mean continuously on the boundary of the

N
N

-dimensional unit ball are uniquely determined. An additional application of the kernel will be to establish the Fatou radial limit theorem.
Eigenvectors and maximal vectors in Boolean vector spaces
Authors: Sinzdak Ronald L.
Published: June 2010

In this paper it is shown that every idempotent, self-adjoint linear endomorphism in a finite-dimensional normed Boolean vector space has its norm as an eigenvalue. A completely algebraic proof is also given for the fact that every linear endomorphism in such a space possesses a maximal vector.
Not every near family is contained in a clan
Authors: Naimpally S. A. , Whitfield J. H. M.
Published: June 2010

In this note we solve an open problem concerning near structures and show that, unlike proximities and contiguities, not every near structure is induced by an elementary one.
Almost continuity of the Cesàro-Vietoris function
Authors: Brown Jack B.
Published: June 2010

Consider the following function due to Cesàro:

ϕ
(
0
)
=
0

\phi (0) = 0

, and if

0
>
x
≤
1

0 > x \leq 1

,
\[

ϕ
(
x
)
=
lim
sup
(

a
1

+

a
2

+
⋯
+

a
n

)

/

n
,

\phi (x) = \lim \sup ({a_1} + {a_2} + \cdots + {a_n})/n,

\]
where the

a
i

{a_i}

are given by the unique nonterminating binary expansion of

x
=
(
0.

a
1

a
2

⋯
)

x = (0.{a_1}{a_2} \cdots )

. Vietoris proved in 1921 that

ϕ
\phi

is connected (as a subset of

[
0
,
1
]
×
R

[0,1] \times R

). The purpose of this note is to alter Vietoris’s argument in order to prove that

ϕ
\phi

is actually almost continuous in the sense of Stallings, thus answering a question raised recently by B. D. Smith.
Helson sets which disobey spectral synthesis
Authors: Saeki Sadahiro
Published: June 2010

In this paper it is shown that every nondiscrete LCA group contains a compact independent Helson set which disobeys spectral synthesis.
Iterates of an absolutely continuous operator
Authors: Ghatage P.
Published: June 2010

It is proved that the powers of an absolutely continuous polynomially bounded operator, the intersection of whose spectrum with the circle has zero measure, go to zero strongly. From this we deduce a simple treatment of the Nagy-Foiaş result on completely nonunitary contractions.
An application of the separation theorem for hermitian matrices
Authors: Markham T. L.
Published: June 2010

Suppose

H
H

is an

n
×
n

n \times n

hermitian matrix over the complex field partitioned as

H
=

(

A

a
m
p
;
B

B
∗

a
m
p
;
C

)

H = \left (\begin {smallmatrix}A&B\\B*&C\end {smallmatrix}\right )

, where

C
C

is invertible. Using the separation theorem on eigenvalues of hermitian matrices, bounds are obtained for the eigenvalues of

(
H

/

C
)
=
A
−
B

C

−
1

B
∗

(H/C) = A - B{C^{ - 1}}{B^ \ast }

in terms of the eigenvalues of

H
H

and

C
C

.
Lemma on measurable cardinals
Authors: Fleissner William G.
Published: June 2010

An ordinal is moved by only finitely many measurable cardinals.
On Euler’s criterion for cubic nonresidues
Authors: Williams Kenneth S.
Published: June 2010

If

p
p

is a prime

≡
1

(
mod

3
)

\equiv 1 \pmod 3

there are integers

L
L

and

M
M

such that

4
p
=

L
2

+
27

M
2

,
L
≡
1

(
mod

3
)

4p = {L^2} + 27{M^2},L \equiv 1\pmod 3

. Indeed

L
L

and

M
2

{M^2}

are unique. If

D
D

is a cubic nonresidue

(
mod

p
)

\pmod p

it is shown how to choose the sign of

M
M

so that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Superscript left-parenthesis p minus 1 right-parenthesis slash 3 Baseline identical-to left-parenthesis upper L plus 9 upper M right-parenthesis slash left-parenthesis upper L minus 9 upper M right-parenthesis left-parenthesis mod p right-parenthesis period">

D

(
p
−
1
)

/

3

≡
(
L
+
9
M
)

/

(
L
−
9
M
)

(
mod

p
)
.

{D^{(p - 1)/3}} \equiv (L + 9M)/(L - 9M)\pmod p.

\]
The case

D
=
2

D = 2

has been treated by Emma Lehmer.
Transformation groups resembling the adjoint representation
Authors: Sullivan R. W.
Published: June 2010

If

G
G

is a compact, connected Lie group, the isotropy subgroups of the adjoint representation of

G
G

are connected and the dimension of the fixed point set of a maximal torus of

G
G

is equal to the the rank of

G
G

. Results similar to these are given when

G
G

acts differentiably on an integral cohomology sphere and has the adjoint representation as weak linear model. This is done by analyzing an induced action of the Weyl group of

G
G

.
Inequalities in dimension theory for posets
Authors: Trotter William T.
Published: June 2010

The dimension of a poset

(
X
,
P
)

(X,P)

, denoted

dim
⁡
(
X
,
P
)

\dim (X,P)

, is the minimum number of linear extensions of

P
P

whose intersection is

P
P

. It follows from Dilworth’s decomposition theorem that

dim
⁡
(
X
,
P
)
≤
width
⁡
(
X
,
P
)

\dim (X,P) \leq \operatorname {width} (X,P)

. Hiraguchi showed that

dim
⁡
(
X
,
P
)
≤

|

X

|

/

2

\dim (X,P) \leq |X|/2

. In this paper,

A
A

denotes an antichain of

(
X
,
P
)

(X,P)

and

E
E

the set of maximal elements. We then prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension left-parenthesis upper X comma upper P right-parenthesis less-than-or-equal-to StartAbsoluteValue upper X minus upper A EndAbsoluteValue semicolon dimension left-parenthesis upper X comma upper P right-parenthesis less-than-or-equal-to 1 plus w i d t h left-parenthesis upper X minus upper E right-parenthesis semicolon">

dim
⁡
(
X
,
P
)
≤

|

X
−
A

|

;
dim
⁡
(
X
,
P
)
≤
1
+
width
⁡
(
X
−
E
)
;

\dim (X,P) \leq |X - A|;\dim (X,P) \leq 1 + \operatorname {width} (X - E);

and

dim
⁡
(
X
,
P
)
≤
1
+
2
width
⁡
(
X
−
A
)

\dim (X,P) \leq 1 + 2\operatorname {width} (X - A)

. We also construct examples to show that these inequalities are sharp.
A converse Steinhaus theorem for locally compact groups
Authors: Simmons S. M.
Published: June 2010

Singular and absolutely continuous regular measures on locally compact groups are characterized in terms of the algebraic and topological structure of sets assigned positive measure.
The monad system of the finest compatible uniform structure
Authors: Henson C. Ward
Published: June 2010

The methods of nonstandard analysis are used to study the finest uniform structure compatible with the topology on a given completely regular, Hausdorff space.
Free 𝑆³-actions on simply connected eight-manifolds
Authors: Resch Richard I.
Published: June 2010

In this paper the canonical equivalence between free actions of a compact Lie group

G
G

and principal

G
G

-bundles is used to apply the theory of fiber bundles to the problem of classifying free differentiable

S
3

{S^3}

-actions. The orbit spaces that may occur are determined and a calculation of homotopy classes of maps from these spaces into the classifying space for principal

S
3

{S^3}

-bundles is made with the aid of the Postnikov system for

S
4

{S^4}

. The bundles corresponding to these classes of maps are then studied to prove that for each positive integer

k
k

there exist exactly three simply connected

8
8

-manifolds which admit free differentiable

S
3

{S^3}

-actions and have second homology group free of rank

k
k

, and that the action on each of these manifolds is unique. It is also proved that even if the second homology group of the

8
8

-manifold has torsion, it can admit at most one action.
Branching processes in simple random walk
Authors: Dwass Meyer
Published: June 2010

Let

N
(
a
)

N(a)

be the number of overcrossings of height

a
a

in a simple random walk. For

p
>
1

/

2

p > 1/2

, the process

N
(
0
)
,
N
(
1
)
,
…

N(0),N(1), \ldots

is a branching process which eventually becomes extinct. For

1

/

2
>
p
,
N
(
0
)
,
N
(
1
)
,
…

1/2 > p,N(0),N(1), \ldots

is a stationary process which is a branching process with immigration.
On 𝑟th roots in eighth-groups
Authors: Gowdy Spenser O.
Published: June 2010

Let

G
G

be an eighth-group having no relators of the form

R
≅

a
t

R \cong {a^t}

. If

X
r

=

a
n

{X^r} = {a^n}

where

a
a

is a generator and

X
X

is a word then

r
r

divides

n
n

and

X
=

a

n

/

r

X = {a^{n/r}}

.
Variational comparison theorem and perturbations of nonlinear systems
Authors: Ladde G. S.
Published: June 2010

By employing vector Lyapunov-like functions and the theory of systems of differential inequalities, a variational comparison theorem and generalized variation of constants formula are developed. Furthermore, the usefulness of the variational comparison theorem is exhibited by giving applications to stability problems.
On ergodic sequences of measures
Authors: Blum J. R. , Cogburn R.
Published: June 2010

Let

Z
Z

be the group of integers and

Z
¯

\bar Z

its Bohr compactification. A sequence of probability measures

{

μ
n

,
n
=
1
,
2
,
…
}

\{ {\mu _n},n = 1,2, \ldots \}

defined on

Z
Z

is said to be ergodic provided

μ
n

{\mu _n}

converges weakly to

μ
¯

\bar \mu

, the Haar measure on

Z
¯

\bar Z

. Let

A
n

⊂
Z
,
n
=
1
,
2
,
…

{A_n} \subset Z,n = 1,2, \ldots

and define

μ
n

{\mu _n}

by

μ
n

(
B
)
=

|

A
n

∩
B

|

/

|

A
n

|

{\mu _n}(B) = |{A_n} \cap B|/|{A_n}|

where

|

B

|

|B|

is the cardinality of

B
B

. Then it is easy to show that if

|

A
n

∩

A
n

+
k

|

/

|

A
n

|

→
1

|{A_n} \cap {A_n} + k|/|{A_n}| \to 1

for every

k
∈
Z

k \in Z

, then

μ
n

{\mu _n}

is ergodic. Let

0
≤

p
k

≤
1

0 \leq {p_k} \leq 1

. In this paper we construct (random) sequences

{

μ
n

}

\{ {\mu _n}\}

which are ergodic, and such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit left-parenthesis StartAbsoluteValue upper A Subscript n Baseline intersection upper A Subscript n Baseline plus k EndAbsoluteValue slash StartAbsoluteValue upper A Subscript n Baseline EndAbsoluteValue right-parenthesis equals p Subscript k">

lim
(

|

A
n

∩

A
n

+
k

|

/

|

A
n

|

)
=

p
k

\lim (|{A_n} \cap {A_n} + k|/|{A_n}|) = {p_k}

, for every

k
∈
Z

k \in Z

.
Extremally disconnected sets in groups
Authors: Saeki Sadahiro
Published: June 2010

It is shown that every extremally disconnected compact set in a LCA group is an

S
H

SH

-set.
Preservation of uniform asymptotic stability under perturbations
Authors: Miller R. K.
Published: June 2010

Suppose the trivial solution of the interval initial value problem for a linear, convolution, Volterra integrodifferential equation is uniformly asymptotically stable. If the kernel in this equation is integrable, then it is shown that this stability is preserved under perturbations which are nonantisipative and of order two or more.
An equation 𝑓(𝑥)=𝑘𝑥 possessing no solutions
Authors: Goebel K. , Rzymowski Witold
Published: June 2010

An example is given of a nonlinear operation

f
:
C
[
0
,
1
]
→
C
[
0
,
1
]

f:C[0,1] \to C[0,1]

having for any

k
∈
R

k \in R

no solutions of the equation

f
(
x
)
=
k
x

f(x) = kx

.
Coincidence point results for spaces with free 𝑍_{𝑝}-actions
Authors: Cohen Fred , Lusk Ewing L.
Published: June 2010

Let

X
X

support a free cyclic group action of prime order. We consider the question of determining when any map

f
:
X
→
Y

f:X \to Y

must identify two points of an orbit, and that of finding the minimum possible dimension of the union of such orbits when they exist.
Entireness of the endomorphism rings of one-dimensional formal groups
Authors: Lubin Jonathan
Published: June 2010

If, for a one-dimensional formal group of height

h
h

which is defined over the integers in a local field of characteristic zero, all the coefficients in degree less than

p
h

{p^h}

lie in an unramified extension of the

p
p

-adic numbers, then the endomorphism ring of the formal group is integrally closed.
Order in a special class of rings and a structure theorem
Authors: Abian Alexander
Published: June 2010

Below a special class of not necessarily associative or commutative rings

A
A

is considered which is characterized by the property that

A
A

has no nonzero nilpotent element and that a product of elements of

A
A

which is equal to zero remains equal to zero no matter how its factors are associated. It is shown that

(
A
,
⩽
)

(A, \leqslant )

is a partially ordered set where

x
⩽
y

x \leqslant y

if and only if

x
y
=

x
2

xy = {x^2}

. Also it is shown that

(
A
,
⩽
)

(A, \leqslant )

is infinitely distributive, i.e.,

r
sup

x
i

=
sup
r

x
i

r\sup {x_i} = \sup r{x_i}

. Finally, based on Zorn’s lemma it is shown that

A
A

is isomorphic to a subdirect product of not necessarily associative or commutative rings without zero divisors.
Gaps in the essential spectrum for second order systems
Authors: Kauffman Robert M.
Published: June 2010

Consider the equation

(

D
2

+
A
+
α
E
)
f
=
0

({D^2} + A + \alpha E)f = 0

, where

α
\alpha

is a positive real number,

E
E

is the

n
×
n

n \times n

identity matrix,

A
A

is a continuously differentiable function from

[
0
,
∞
)

[0,\infty )

to the

n
×
n

n \times n

Hermitian matrices, and

A
A

and

A
′

A’

are bounded. It is shown that, if

α
\alpha

is large with respect to

|

|

A

|

|

∞

||A|{|_\infty }

, there are small positive numbers

λ
\lambda

such that, for every solution

f
f

to the equation

(

D
2

+
A
+
α
E
)
f
=
0
,

e

−
λ
t

f
(
t
)

({D^2} + A + \alpha E)f = 0,{e^{ - \lambda t}}f(t)

is square integrable, but

e

λ
t

f
(
t
)

{e^{\lambda t}}f(t)

is not. It is also shown that, if

α
\alpha

is large with respect to

|

|

A

|

|

∞

||A|{|_\infty }

, there is a real number

λ
\lambda

close to zero such that

λ
\lambda

is in the essential spectrum of any selfadjoint operator in

L
2

{L_2}

associated with

D
2

+
A
+
α
E

{D^2} + A + \alpha E

. These results generalize the results of Hartman and Putnam, who proved these statements for the scalar case

n
=
1

n = 1

.
On the existence of totally inhomogeneous spaces
Authors: Scott Brian M.
Published: June 2010

A property of total inhomogeneity for topological spaces is defined and is shown to be stronger than that of rigidity (i.e., of having trivial autohomeomorphism group). It is further shown that compact, rigid, Hausdorff spaces are totally inhomogeneous, and that totally inhomogeneous spaces exist in profusion as dense subspaces of a class of locally compact spaces.
Critical points on closed elliptic affine subspaces
Authors: Delver Robert
Published: June 2010

The critical points of a function restricted to the solution set of a linear elliptic equation are characterized. An extension of the Lagrange multiplier method is given. Existence and the relation to elliptic eigenvalue problems are discussed.
On values of exponential sums
Authors: An Chungming
Published: June 2010

An exponential sum is defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis upper F comma phi comma alpha right-parenthesis equals sigma-summation Underscript gamma epsilon left-parenthesis upper Z slash q upper Z right-parenthesis Superscript n Baseline Endscripts exp left-parenthesis 2 pi i left-parenthesis phi upper F left-parenthesis gamma right-parenthesis plus mathematical left-angle alpha comma gamma mathematical right-angle right-parenthesis right-parenthesis">

G
(
F
,
φ
,
α
)
=

∑

γ
ϵ

(
Z

/

q
Z
)

n

exp
⁡
(
2
π
i
(
φ
F
(
γ
)
+
⟨
α
,
γ
⟩

)
)

G(F,\varphi ,\alpha ) = \sum \limits _{\gamma \epsilon {{(Z/qZ)}^n}} {\exp (2\pi i(\varphi F(\gamma ) + \langle \alpha ,\gamma \rangle } ))

\]
for

φ
=
a

/

q
ϵ
Q
,
α
ϵ

R
n

\varphi = a/q\epsilon Q,\alpha \epsilon {R^n}

, and a positive-definite form

F
(
x
)

F(x)

in

n
n

variables of degree

δ
\delta

. Its value is studied and the definition is extended to an irrational

φ
\varphi

.
Hopf algebras with nonsemisimple antipode
Authors: Taft Earl J. , Wilson Robert Lee
Published: June 2010

An example is given to show that the antipode of a finite dimensional Hopf algebra over a field of prime characteristic

p
>
2

p > 2

need not be semisimple. (For

p
=
2

p = 2

examples were previously known.) The example is a pointed irreducible Hopf algebra

H
H

(with antipode

S
S

) of dimension

p
3

{p^3}

such that

S

2
p

=
I
≠

S
2

{S^{2p}} = I \ne {S^2}

.
On perturbing bases of complex exponentials in 𝐿² (-𝜋,𝜋)
Authors: Young Robert M.
Published: June 2010

A sequence of complex exponentials

{

e

i

λ
n

t

}

\{ {e^{i{\lambda _n}t}}\}

is said to be a Riesz basis for

L
2

(
−
π
,

π
)

{L^2}( - \pi ,\;\pi )

if each function in the space has a unique representation

f
=
Σ

c
n

e

i

λ
n

t

f = \Sigma {c_n}{e^{i{\lambda _n}t}}

, with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A normal upper Sigma StartAbsoluteValue c Subscript n Baseline EndAbsoluteValue squared less-than-or-slanted-equals StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue squared less-than-or-slanted-equals upper B normal upper Sigma StartAbsoluteValue c Subscript n Baseline EndAbsoluteValue squared">

A
Σ

|

c
n

|

2

⩽

|

|

f

|

|

2

⩽
B
Σ

|

c
n

|

2

A\Sigma |{c_n}{|^2} \leqslant ||f|{|^2} \leqslant B\Sigma |{c_n}{|^2}

. It is known, for example, that if

|

λ
n

−
n

|

⩽
L
>
1

/

4
(
−
∞
>
n
>
∞
)

|{\lambda _n} - n| \leqslant L > 1/4( - \infty > n > \infty )

, then

{

e

i

λ
n

t

}

\{ {e^{i{\lambda _n}t}}\}

is a Riesz basis. In this note we show that not only the orthonormal basis

{

e

i
n
t

}

\{ {e^{int}}\}

, but any Riesz basis of complex exponentials can be suitably perturbed.
Remarks on a comparison theorem for scalar Riccati equations
Authors: Travis C. C.
Published: June 2010

A comparison theorem for scalar Riccati equations will be stated which contains a recent theorem of Stafford and Heidel.
Remarks on differential inequalities in Banach spaces
Authors: Martin Robert H.
Published: June 2010

Under certain conditions a comparison is made between solutions of a pair of initial value problems in a Banach space. This comparison includes and unifies several recent results on differential inequalities in Banach spaces.
On a nonuniform parabolic equation with mixed boundary condition
Authors: Pao C. V.
Published: June 2010

This paper discusses the existence of weak solutions for an initial boundary-value problem of a nonuniform second order parabolic equation in which the coefficient

b
(
t
,
x
)

b(t,x)

of

u
t

{u_t}

is nonnegative and the coefficient matrix

(

a

i
j

(
t
,
x
)
)

({a_{ij}}(t,x))

of the elliptic part is not necessarily positive definite. When

b
(
t
,
x
)
≡
0

b(t,x) \equiv 0

, this problem is reduced to a degenerate elliptic system. A discussion of the existence of weak solutions for the degenerate elliptic boundary-value problem from the parabolic system is included.
A 𝑊(𝑍₂) invariant for orientation preserving involutions
Authors: Alexander John P.
Published: June 2010

In this paper we calculate an invariant in

W
(

Z

2

)

W({{\mathbf {Z}}_2})

, the Witt ring of nonsingular, symmetric

Z

2

{{\mathbf {Z}}_2}

-inner product spaces, for orientation-preserving involutions on compact, closed, connected

4
n

4n

-dimensional manifolds

M
M

. This invariant with the Atiyah-Singer index theorem uniquely determines the orthogonal representation of

Z

2

{{\mathbf {Z}}_2}

on

H

2
n

(
M
;

Z

)

/

TOR

{H^{2n}}(M;{\mathbf {Z}})/\operatorname {TOR}

. We also give an example to show that this invariant detects actions that the Atiyah-Singer theorem cannot.
Factors and roots of the van der Pol polynomials
Authors: Howard F. T.
Published: June 2010

The van der Pol polynomials

V
n

(
a
)

{V_n}(a)

are defined by means of
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x cubed e Superscript x a Baseline left-bracket 6 x left-parenthesis e Superscript x Baseline plus 1 right-parenthesis minus 12 left-parenthesis e Superscript x Baseline minus 1 right-parenthesis right-bracket Superscript negative 1 Baseline equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts upper V Subscript n Baseline left-parenthesis a right-parenthesis x Superscript n Baseline slash n factorial period">

x
3

e

x
a

[
6
x
(

e
x

+
1
)
−
12
(

e
x

−
1
)

]

−
1

=

∑

n
=
0

∞

V
n

(
a
)

x
n

/

n
!
.

{x^3}{e^{xa}}{[6x({e^x} + 1) - 12({e^x} - 1)]^{ - 1}} = \sum \limits _{n = 0}^\infty {{V_n}} (a){x^n}/n!.

\]
In this paper new properties of these polynomials are derived. It is shown that neither

V

2
n

(
a
)

{V_{2n}}(a)

nor

V

2
n
+
1

(
a
)

/

(
a
−
1

/

2
)

{V_{2n + 1}}(a)/(a - 1/2)

has rational roots, and that if

n
=
2
∙

3
m

,
m
⩾
0

n = 2 \bullet {3^m},m \geqslant 0

, or

n
=

3
m

+

3
t

,
m
>
t
>
0

n = {3^m} + {3^t},m > t > 0

, or

n
=
m
(
p
−
3
)
,
p

n = m(p - 3),p

a prime number,

3
m
>
p

3m > p

, then

V
n

(
a
)

{V_n}(a)

and

V

n
+
1

(
a
)

/

(
a
−
1

/

2
)

{V_{n + 1}}(a)/(a - 1/2)

are both irreducible over the rational field. It is also shown that if

n
=

2
k

n = {2^k}

, then

V
n

(
a
)

{V_n}(a)

is irreducible over the rational field. Finally, possible factors of the van der Pol polynomials are discussed.
On the infinite dimensionality of the Dolbeault cohomology groups
Authors: Laufer Henry B.
Published: June 2010

Let

M
M

be an open subset of a Stein manifold without isolated points. Let

Ω
p

{\Omega ^p}

be the sheaf of germs of holomorphic

p
p

-forms on

M
M

. Then

H
q

(
M
,

Ω
p

)

{H^q}(M,{\Omega ^p})

is either

0
0

or else infinite dimensional.

H
q

(
M
,

S

)

{H^q}(M,\mathcal {S})

may be nonzero and finite dimensional if

M
M

is the regular points of a Stein space or if

S

\mathcal {S}

is an arbitrary coherent sheaf over

M
M

.
An equivalence theorem on generating functions
Authors: Srivastava H. M.
Published: June 2010

The present paper establishes an equivalence theorem on generating relations for a sequence of functions

{

f
n

(
z
)
}

\{ {f_n}(z)\}

defined by the Rodrigues formula (1) below. It is also shown how this theorem may be applied to a fairly large variety of special functions including, for instance, the classical orthogonal polynomials.
Birelations and Sidon sets
Authors: Drury S. W.
Published: June 2010

It is shown that a knowledge of the set of birelations with coefficients

−
1
,

0
,

1

- 1,\;0,\;1

supported on a subset of a discrete abelian group is enough to determine whether the subset has the Sidon property or not.
On compact Einstein-Kaehler manifolds
Authors: Chen Bang-yen , Ogiue Koichi
Published: June 2010

A characterization of a complex space form among compact Einstein-Kaehler manifolds is given in terms of Chern classes.
On 𝑁^{ℵ₁} and the almost-Lindelöf property
Authors: Hechler Stephen H.
Published: June 2010

In 1970, Kemperman and Maharam proved that there exists a Baire measure

μ
\mu

on

N

c

{N^{\mathbf {c}}}

(where

N
N

is the set of natural numbers) such that

N

c

{N^{\mathbf {c}}}

may be covered by a famliy of elementary open

μ
\mu

-null sets and used this to prove that

R

c

{R^{\mathbf {c}}}

(where

R
R

is the set of real numbers) does not have the “almost-Lindelöf” property. We define

K

{\mathbf {K}}

to be the smallest cardinal

κ
\kappa

for which there exists a collection of

κ
\kappa

closed subsets of

R
R

each of Lebesgue measure zero and which covers

R
R

, and we show that in the above results

c

{\mathbf {c}}

can be replaced by

K

{\mathbf {K}}

. We then note that we have shown elsewhere that it is consistent with the negation of the continuum hypothesis that

K

=

ℵ
1

{\mathbf {K}} = {\aleph _1}

, and this, therefore, implies that it is consistent with the negation of the continuum hypothesis that

R
ℵ

1

{R^\aleph }1

not be almost-Lindelöf.
Properties of weak 𝜃̄-refinable spaces
Authors: Smith J. C.
Published: June 2010

A space

X
X

is called weak

θ
¯

\overline \theta

-refinable if every open cover of

X
X

has a refinement

⋃

i
=
1

∞

G

i

\bigcup \nolimits _{i = 1}^\infty {{\mathcal {G}_i}}

satisfying (1)

G

i

=
{

G
α

:
α
ϵ

A
i

}

{\mathcal {G}_i} = \{ {G_\alpha }:\alpha \epsilon {A_i}\}

is an open collection for each

i
i

, (2) each

x
ϵ
X

x\epsilon X

has finite positive order with respect to some

G

i

{\mathcal {G}_i}

, (3) the open cover
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper G Subscript i Baseline equals union left-bracket upper G Subscript alpha Baseline colon alpha epsilon upper A Subscript i Baseline right-bracket right-brace Subscript i equals 1 Baseline Superscript normal infinity Baseline">

{

G
i

=
⋃

[

G
α

:
α
ϵ

A
i

]

}

i
=
1

∞

\{ {G_i} = \bigcup {[{G_\alpha }:\alpha \epsilon } {A_i}]\} _{i = 1}^\infty

is point finite. In this paper the author shows that the above property lies between the properties of

θ
\theta

-refinable and weak

θ
\theta

-refinable. The main result is the fact that if

X
X

is countably metacompact and satisfies property

(
δ
)

(\delta )

, every weak

θ
¯

\overline \theta

-cover of

X
X

has a countable subcover. Results concerning paracompactness, metacompactness and the star-finite property are also obtained.
Conjugate algebraic integers in an interval
Authors: Ennola Veikko
Published: June 2010

The following conjecture of R. M. Robinson is proved. If

Δ
\Delta

is a real interval of length greater than

4
4

, then for any sufficiently large

n
n

there exists an irreducible monic polynomial of degree

n
n

with integer coefficients all of whose zeros lie in

Δ
\Delta

.
Finite groups with semisimple endomorphism rings
Authors: Lyons Carter
Published: June 2010

If the near ring

E
(
G
)

E(G)

generated by the endomorphisms of a finite group

G
G

is a semisimple ring, or more generally, has square-free characteristic then it is shown that

G
G

is abelian, a direct sum of elementary

p
p

-groups.
A characterisation of Lipschitz classes on 0-dimensional groups
Authors: Bloom Walter R.
Published: June 2010

This paper is concerned with characterising, in terms of certain properties of their Fourier transforms, the Lipschitz functions of order

α
(
0
>
α
>
1
)

\alpha (0 > \alpha > 1)

defined on a locally compact metric

0
0

-dimensional Abelian group.
Transposed translation planes
Authors: Maduram D. M.
Published: June 2010

A detailed study of the transposed translation planes which are also given by dual spreads is made using the matrix representation of translation planes.
On mixed volumes of nonconvex sets
Authors: Weil Wolfgang
Published: June 2010

Mixed volumes have been extended [ill] nonconvex sets by several authors. It is shown that in any such extension at least one essential property of mixed volumes is lost.
Width-integrals of convex bodies
Authors: Lutwak Erwin
Published: June 2010

The width-integrals (Breitenintegrale) as introduced by Blaschke are examined. They are shown to satisfy a cyclic inequality similar to that satisfied by the cross-sectional measures (Quermassintegrale). Other similarities and inequalities between the width-integrals and the cross-sectional measures are shown to exist.
A note on a differential concomitant
Authors: Eiseman P. R. , Stone A. P.
Published: June 2010

If

h

{\mathbf {h}}

and

k

{\mathbf {k}}

are vector

1
1

-fotms, the vanishing of the concomitant

[

h

,

k

]

[{\mathbf {h}},\;{\mathbf {k}}]

is an integrability condition fot certain problems on manifolds. In the case that

h

=

k

{\mathbf {h}} = {\mathbf {k}}

the vanishing of the Nijenhuis tensor

[

h

,

h

]

[{\mathbf {h}},\;{\mathbf {h}}]

implies

d
(
tr
⁡

h

)

d(\operatorname {tr} {\mathbf {h}})

is a conservation law for

h

{\mathbf {h}}

, provided that

tr
⁡

h

\operatorname {tr} {\mathbf {h}}

is not constant. When the trace of

h

{\mathbf {h}}

is constant, a conservation law for

h

{\mathbf {h}}

exists if one can find a vector

1
1

-form

k

{\mathbf {k}}

with nonconstant trace such that

[

h

,

k

]
=
0

[{\mathbf {h}},\;{\mathbf {k}}] = 0

.
The genus of subfields of 𝐾(𝑛)
Authors: Dennin Joseph B.
Published: June 2010

In this paper we fix a genus

g
g

and show that the number of fields of elliptic modular functions

F
F

of genus

g
g

is finite.
Oscillations of bases for the natural numbers
Authors: Erdős Paul , Nathanson Melvyn B.
Published: June 2010

Let

A
A

be a set of positive integers. Then

A
A

is a basis if every sufficiently large integer

n
n

can be written in the form

n
=

a
i

+

a
j

n = {a_i} + {a_j}

with

a
i

,

a
j

ϵ
A

{a_i},\;{a_j}\epsilon A

. Otherwise,

A
A

is a nonbasis. In this paper we construct sets which oscillate from basis to nonbasis to basis or from nonbasis to basis to nonbasis under finite perturbations of the sets.
Artinian ideals in Noetherian rings
Authors: Lenagan T. H.
Published: June 2010

We show that in a right and left Noetherian ring two-sided ideals which are Artinian as left submodules are also Artinian as right submodules.
An invariant ideal of a group ring of a finite group. II
Authors: Hsia J. S. , Peterson Roger D.
Published: June 2010

The vanishing of the numerical invariant

ν
(
G
)

\nu (G)

of a finite group

G
G

is linked to the existence of certain central annihilators of the generic right ideal

Γ
R

(
G
)

{\Gamma _R}(G)

in the group ring

R
G

RG

. This leads to several affirmative answers of questions posed in [1]. Also, some explicit values of

ν
(
G
)

\nu (G)

are described for the class of finite nonsolvable groups having all their odd Sylow subgroups cyclic.
On the height of ideals generated by matrices
Authors: Becker Joseph
Published: June 2010

A short geometric proof of the following algebraic theorem of Buchsbaum and Rim: Let

R
R

be the reduced local ring of an analytic variety and

g
:

R
t

→

R
r

,
t
≥
r

g:{R^t} \to {R^r},t \geq r

, be a homomorphism of

R
R

modules. Then the codimension of the support of the cokernel of

g
≤
t
−
r
+
1

g \leq t - r + 1

.
Sequential weak convergence in the bidual of a Banach lattice
Authors: Gamlen John L. B.
Published: June 2010

Let

X
X

be a Banach lattice with separable dual. Then the bidual

X
X

is the closure of

X
X

under sequential monotone weak convergence.
Applications of cluster sets in minimal topological spaces
Authors: Hamlett T. R.
Published: June 2010

Given a function

f
f

from a topological space

X
X

into a topological space

Y
Y

and a point

x
ϵ
X

x\epsilon X

, the cluster set of

f
f

at

x
x

is

C

(
f
;
x
)
=
∩
{
Cl
⁡
(
f
(
U
)
)
:

U

is a neighborhood of 

x
}

\mathcal {C}(f;x) = \cap \{ \operatorname {Cl} (f(U)):\;U\;{\text {is a neighborhood of }}x\}

, where

Cl
⁡
(
U
)

\operatorname {Cl} (U)

denotes the closure of

U
U

. In this paper,

Y
Y

is taken to be a minimal topological space and

C

(
f
;
x
)

\mathcal {C}(f;x)

is used as a tool to obtain information about the continuity of

f
f

.
When is the maximal ring of quotients projective?
Authors: Handelman David
Published: June 2010

Let

R
R

be an associative ring with

1
1

, and

Q
Q

its maximal ring of right quotients. If

r
r

belongs to

R
R

, a right insulator for

r
r

in

R
R

is a finite subset of

R
,
{

r
i

}

i
=
1

m

R,\{ {r_i}\} _{i = 1}^m

, such that the right annihilator of

{
r

r
i

;
i
=
1
,
…
,
m
}

\{ r{r_i};i = 1, \ldots ,m\}

is zero. Then we have: If

Q
Q

is a projective right

R
R

-module,

Q
Q

is finitely generated; if

R
R

is nonsingular, then

Q
Q

is projective as a right

R
R

-module if and only if there exists

e
=

e
2

e = {e^2}

in

R
R

such that

e
R

eR

is injective and

e
e

has a right insulator in

R
R

; under these circumstances,

R
=
Q

R = Q

if and only if

e
e

has a left insulator in

R
R

. We prove some related results for torsionless

Q
Q

, and give an example of a prime ring

R
R

such that

Q
Q

is a cyclic projective right

R
R

-module, but

R
≠
Q

R \ne Q

.
On the Frattini subgroup of a residually finite generalized free product
Authors: Allenby R. B. J. T. , Tang C. Y.
Published: June 2010

Let

G
=

(
A
∗
B

)
H

G = {(A \ast B)_H}

be the generalized free product of the groups

A
,
B

A,B

amalgamating the subgroup

H
H

, and let

Φ
(
G
)

\Phi (G)

denote its Frattini subgroup. In support of the conjecture that

Φ
(
G
)
⊆
H

\Phi (G) \subseteq H

whenever

G
G

is resiually finite and

H
H

satisfies a nontrivial identical relation, we show, amongst several other things, that the above inequality is indeed valid if in addition at least one of the following holds: (i)

A
,
B

A,B

, each satisfies a nontrivial identical relation; (ii)

G
G

is finitely generated; (iii)

H
H

is nilpotent. In particular (i) completes earlier investigations of the second author. The methods of proof are, however, different.
Nondegenerate higher degree forms over Dedekind domains
Authors: Coleman D. B. , Cunningham Joel
Published: June 2010

It is determined which finitely generated projective modules over a Dedekind domain admit nondegenerate symmetric

r
r

-linear forms.
A summation formula and some properties of Eulerian functions
Authors: Grimson R. C.
Published: June 2010

A product formula for Eulerian functions and a general summation method are given.
Jordan derivations on rings
Authors: Cusack J. M.
Published: June 2010

I. N. Herstein has shown that every Jordan derivation on a prime ring not of charactetistic

2
2

is a derivation. This result is extended in this paper to the case of any ring in which

2
x
=
0

2x = 0

implies

x
=
0

x = 0

and which is semiprime or which has a commutator which is not a zero divisor.
Operators quasisimilar to a normal operator
Authors: Apostol Constantin
Published: June 2010

A necessary and sufficient condition for an operator to be quasisimilar to a normal operator is given.
A method of combining fixed points
Authors: Waggoner Roger
Published: June 2010

It is now well known that in the category of finite polyhedra the fixed point property is not preserved by the operations of suspension, Cartesian product, adjunction along a segment, and join. Thus far none of the examples given have involved polyhedra of dimension 2. It is shown in this paper that two fixed points

x
x

and

y
y

of a self-map of a polyhedron

K
K

can be combined in a certain way if a certain criterion is satisfied by the

f
f

-image of a path from

x
x

to

y
y

. Several corollaries follow, one of which is that if

K
K

is a finite simply connected

2
2

-polyhedron with no local separating points,

H
2

(
K
)
≠
0

{H_2}(K) \ne 0

, and

K
K

has a

2
2

-simplex

σ
\sigma

such that

π
1

(
K
−
Int
⁡
σ
,
z
)

{\pi _1}(K - \operatorname {Int} \sigma ,z)

is cyclic, then

K
K

fails to have the fixed point property. This eliminates many

2
2

-dimensional polyhedra from consideration as examples.
Locally maximal Bezout domains
Authors: Wiegand Sylvia
Published: June 2010

Let

X
X

be a finite tree. It is shown that

X
X

is order-isomorphic to the prime spectrum of a Bezout domain

R
R

such that every localization of

R
R

is a maximal valuation ring.
A relation between obstructions and functional cohomology operations
Authors: Little Robert D.
Published: June 2010

In a recent paper, P. Olum developed a formula which relates the obstruction to an extension and the induced homomorphism in cohomology. In the present paper, we develop an analogous formula which relates the obstruction to an extension and a certain functional cohomology operation. This operation is a generalization of the standard functional cohomology operation of Steenrod. Olum showed that his formula is particularly useful when the range space of the extension problem is one of the classifying spaces BU, BO, or BSp. Our formula is also useful in this context and we show that it can be used to compute the obstruction in certain situations where the Olum formula fails.
An elementary counterexample to the open mapping principle for bilinear maps
Authors: Horowitz Charles
Published: June 2010

In [2], Rudin asked whether a continuous bilinear map from the product of two Banach spaces onto a Banach space must be open at the origin; i.e., whether under such a map the image of every neighborhood of zero must contain a neighborhood of zero. Recently, Cohen [1] showed that the answer to the general question was in the negative. However, his counterexample was somewhat involved and left the issue unresolved for bilinear maps on Hilbert spaces. The purpose of this note is to show that the open mapping principle for bilinear maps, as described above, fails even in the finite dimensional case.
Bilateral random walks on compact semigroups
Authors: Mukherjea A. , Tserpes N. A.
Published: June 2010

Let

μ
\mu

be a regular Borel probability measure with support

F
F

on a compact semigroup

S
S

. Let

X
0

,

X

±
1

,

X

±
2

,
⋯

{X_0},{X_{ \pm 1}},{X_{ \pm 2}}, \cdots

be a sequence of independent random variables with values in

S
S

, having identical distribution

P
(

X
n

∈
B
)
=
μ
(
B
)

P({X_n} \in B) = \mu (B)

. The random walk

W
n

=

X

−
n

⋯

X

−
1

X
0

X
1

⋯

X
n

{W_n} = {X_{ - n}} \cdots {X_{ - 1}}{X_0}{X_1} \cdots {X_n}

is studied in this paper. Let

D
D

be the closed semigroup generated by

F
F

and let

K
K

be the kernel of

D
D

. An element

x
∈
D

x \in D

is called recurrent iff

P
x

(

W
n

∈

N
x

i

.o

.

)
=
1

{P_x}({W_n} \in {N_x}{\text {i}}{\text {.o}}{\text {.}}) = 1

for every open neighborhood

N
x

{N_x}

of

x
x

. We prove:

x
x

is essential for

W
n

{W_n}

if and only if

x
∈
K

x \in K

if and only if

x
x

is recurrent if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma upper P Subscript x Baseline left-parenthesis upper W Subscript n Baseline element-of upper N Subscript x Baseline right-parenthesis equals normal upper Sigma mu Superscript n Baseline asterisk left-bracket mu Superscript n Baseline left-parenthesis x Superscript negative 1 Baseline dot right-parenthesis right-bracket left-parenthesis upper N Subscript x Baseline right-parenthesis equals normal infinity">

Σ

P
x

(

W
n

∈

N
x

)
=
Σ

μ
n

∗
[

μ
n

(

x

−
1

⋅
)
]
(

N
x

)
=
∞

\Sigma {P_x}({W_n} \in {N_x}) = \Sigma {\mu ^n} \ast [{\mu ^n}({x^{ - 1}} \cdot )]({N_x}) = \infty

for every

N
x

{N_x}

. Moreover all states in

K
K

are recurrent positive. These results extend results of the authors for the unilateral random walks (using different methods) and recent results of Larisse for the discrete case.
Independence and additive entropy
Authors: Berg Kenneth R.
Published: June 2010

The relationship between additive entropy and independence is worked out for ergodic transformations on a Lebesgue space. Examples are given on the behavior of the deterministic part of an ergodic transformation.
Almost deterministic 𝜔-automata with existential output condition
Authors: Karpiński Marek
Published: June 2010

The theorem on reduction in the nondeterminateness degree of

ω
\omega

-automata has been formulated.
A closed graph theorem
Authors: Iyahen S. O.
Published: June 2010

A closed graph theorem is proved, implying that a Hausdorff locally convex space

E
E

need not be barrelled if every closed linear map from

E
E

into

F
F

is continuous, where

F
F

is a reflexive Fréchet or

L
F

LF

-space or a space of distributions.
A note on continuity of semigroups of maps
Authors: Chernoff Paul R.
Published: June 2010

An example is given of a separately continuous semigroup of transformations on Hilbert space which fails to be jointly continuous at

t
=
0

t = 0

.
A coefficient estimate for multivalent functions
Authors: Hallenbeck David J. , Livingston Albert E.
Published: June 2010

By making use of extreme point theory we obtain bounds on the coefficients of a class of functions, multivalent in the unit disk, closely related to the bounds conjectured by Goodman.
A characterization of discreteness for locally compact groups in terms of the Banach algebras 𝐴_{𝑝}(𝐺)
Authors: Granirer Edmond E.
Published: June 2010

The Banach algebra

A
A

is said to have the bounded power property if for any

x
∈
A

x \in A

, with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue x StartAbsoluteValue EndAbsoluteValue Subscript s p Baseline equals limit Underscript n right-arrow normal infinity Endscripts StartAbsoluteValue EndAbsoluteValue x Superscript n Baseline StartAbsoluteValue EndAbsoluteValue Superscript 1 slash n Baseline less-than-or-slanted-equals 1">

|

|

x

|

|

s
p

=

lim

n
→
∞

|

|

x
n

|

|

1

/

n

⩽
1

||x|{|_{sp}} = {\lim _{n \to \infty }}||{x^n}|{|^{1/n}} \leqslant 1

, one has

sup
n

|

|

x
n

|

|

>
∞

{\sup _n}||{x^n}|| > \infty

. It has been shown by B. M. Schreiber [9, Theorem (8.6)] that, if

G
G

is a locally compact abelian group, then the Fourier algebra

A
(
G
)
=

L
1

(
Γ

)
∧

A(G) = {L^1}{(\Gamma )^ \wedge }

has the bounded power property, if and only if

G
G

is discrete. We improve this result in the Theorem. Let

G
G

be an arbitrary locally compact group and

1
>
p
>
∞

1 > p > \infty

. Then

A
p

(
G
)

{A_p}(G)

has the bounded power property if and only if

G
G

is discrete. Our proof, even for abelian

G
G

and

p
=
2

p = 2

(then

A
2

(
G
)
=
A
(
G
)

{A_2}(G) = A(G)

is the usual Fourier algebra of

G
G

), is much simpler and entirely different from that of [9].
Some applications of Landweber-Novikov operations
Authors: Segal David M.
Published: June 2010

Previous results on the characteristic numbers of

S
p

Sp

-manifolds are extended in three different ways. I. It is shown that the primitive symplectic Pontrjagin class evaluated on a

4
(

2
j

−
1
)

4({2^j} - 1)

dimensional

S
p

Sp

-manifold always gives a number divisible by

8
8

. This forms an analogue to a well-known result of Milnor concerning

U
U

-manifolds. II. It is shown that some of the results of Floyd as well as an analogue of the previous result can be obtained for ’pseudo-symplectic’ manifolds. III. Results are generalised to

(
S
p
,
f
r
)

(Sp,fr)

manifolds.
Some sufficient conditions for quintic residuacity
Authors: Zee Yun Cheng
Published: June 2010

It is shown that for a prime

p
p

of the form

5
f
+
1

5f + 1

, a prime

q
>
5

q > 5

is a quintic residue

(
mod
p
)

(\bmod p)

if

u
≡
0
,
v
≡
k
w

u \equiv 0,v \equiv kw

or

u
≡
k
w
,
v
≡
0
(
mod
q
)

u \equiv kw,v \equiv 0(\bmod q)

, where

k
k

satisfies

k
2

≡
−
3
,
5

{k^2} \equiv - 3,5

or

−
15
(
mod
q
)

- 15(\bmod q)

.
Banach lattice structures on separable 𝐿_{𝑝} spaces
Authors: Lacey E. , Wojtaszczyk P.
Published: June 2010

A complete characterization of those lattice structures on separable

L
p

{L_p}

spaces which are Banach lattice structures under the

L
p

{L_p}

norm is given.
An imbedding theorem for metric spaces
Authors: Lipscomb Stephen Leon
Published: June 2010

A simple solution to the imbedding problem for the class of separable metric spaces has been known for a long time: (Urysohn’s Imbedding Theorem) A topological space is separable metric if, and only if, it can be imbedded in the topological product of countably many unit intervals. We see that products of the unit interval make an especially informative type of imbedding space since the finite (Lebesgue) dimensional separable metric spaces are those that can be imbedded in a finite product of intervals. The author has recently shown that this result concerning the finite case could be extended to arbitrary metric spaces if we use a topological generalization of the unit interval. This present paper shows that if we use this same generalization of the interval, then we can obtain an analogue to Urysohn’s Imbedding Theorem. Besides presenting the first unified results which simultaneously generalize both separable cases, this paper contains comparisons with existing imbedding theorems and imbedding spaces.
On the Lefschetz number for iterates of continuous mappings
Authors: Peitgen H. O.
Published: June 2010

We give an elementary proof of the

(
mod
p
)

(\bmod p)

-congruence between the Lefschetz numbers of a continuous mapping and its

p
p

th iterate,

p
p

prime, and deduce some results about periodic mappings.