bluuebunny/crossref_metadata_2025 huggingface dataset:
On the weak type (1,1) inequality for conjugate functions
Authors: Davis Burgess
Published: June 2010

A theorem of Kolmogorov states that there is a positive constant

K
K

such that if

f
~

\tilde f

is the conjugate function of an integrable real valued function

f
f

on the unit circle then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m left-brace StartAbsoluteValue f overTilde EndAbsoluteValue greater-than-over-equals lamda right-brace less-than-over-equals upper K StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue Subscript 1 Baseline slash lamda comma lamda greater-than 0">

m
{

|

f
~

|

≧
λ
}
≦
K

|

|

f

|

|

1

/

λ
,
λ
>
0

m\{ |\tilde f| \geqq \lambda \} \leqq K||f|{|_1}/\lambda ,\lambda > 0

. It is shown that the smallest possible value for

K
K

in this theorem, the so called weak type (1, 1) norm of the conjugate function operator, is
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 plus 3 Superscript negative 2 Baseline plus 5 Superscript negative 2 Baseline plus midline-horizontal-ellipsis right-parenthesis slash left-parenthesis 1 minus 3 Superscript negative 2 Baseline minus 5 Superscript negative 2 Baseline minus midline-horizontal-ellipsis right-parenthesis almost-equals 1.347">

(
1
+

3

−
2

+

5

−
2

+
⋯
)

/

(
1
−

3

−
2

−

5

−
2

−
⋯
)
≈
1.347

(1 + {3^{ - 2}} + {5^{ - 2}} + \cdots )/(1 - {3^{ - 2}} - {5^{ - 2}} - \cdots ) \approx 1.347

. This number is also shown to be the weak type (1, 1) norm of the Hilbert transform operator on functions defined on the real line. The proof uses P. Levy’s result that Brownian motion in the plane is conformally invariant.
Chain conditions on symmetric elements
Authors: Montgomery Susan
Published: June 2010

Recently Britten has proven an analog of Goldie’s theorem for the Jordan ring

S
S

of symmetric elements in a ring with involution of characteristic not 2. In this paper we first extend Britten’s theorem to the situation where

R
R

is an arbitrary ring and the Jordan ring is only an ample subring of the symmetric elements. We apply this result to show that if

S
S

has ACC on quadratic ideals, then the (Jordan) nil radical of

S
S

is nilpotent.
On generalizing alternative rings
Authors: Rodabaugh D. J.
Published: June 2010

Consider a ring

R
R

that satisfies the identity

(
x
,
x
,
x
)
=
0

(x,x,x) = 0

and any two of the three identities:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis w x comma y comma z right-parenthesis plus left-parenthesis w comma x comma left-bracket y comma z right-bracket right-parenthesis minus w left-parenthesis x comma y comma z right-parenthesis minus left-parenthesis w comma y comma z right-parenthesis x equals 0 semicolon left-parenthesis left-bracket w comma x right-bracket comma y comma z right-parenthesis plus left-parenthesis w comma x comma y z right-parenthesis minus y left-parenthesis w comma x comma z right-parenthesis minus left-parenthesis w comma x comma y right-parenthesis z equals 0 semicolon left-parenthesis w comma x dot y comma z right-parenthesis minus x dot left-parenthesis w comma y comma z right-parenthesis minus y dot left-parenthesis w comma x comma z right-parenthesis equals 0">

(
w
x
,
y
,
z
)
+
(
w
,
x
,
[
y
,
z
]
)
−
w
(
x
,
y
,
z
)
−
(
w
,
y
,
z
)
x
=
0
;
(
[
w
,
x
]
,
y
,
z
)
+
(
w
,
x
,
y
z
)
−
y
(
w
,
x
,
z
)
−
(
w
,
x
,
y
)
z
=
0
;
(
w
,
x
⋅
y
,
z
)
−
x
⋅
(
w
,
y
,
z
)
−
y
⋅
(
w
,
x
,
z
)
=
0

(wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x = 0;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z = 0;(w,x \cdot y,z) - x \cdot (w,y,z) - y \cdot (w,x,z) = 0

. In this paper, we prove that if

R
R

has characteristic prime to 6 then

R
R

semiprime with idempotent

e
e

implies

R
R

has a Peirce decomposition in which the modules multiply as they do in an alternative ring. If in addition

R
R

is prime with idempotent

e
≠
0
,
1

e \ne 0,1

then

R
R

is alternative.
Almost completely decomposable torsion free abelian groups
Authors: Lady E. L.
Published: June 2010

A finite rank torsion free abelian group

G
G

is almost completely decomposable if there exists a completely decomposable subgroup

C
C

with finite index in

G
G

. The minimum of

[
G
:
C
]

[G:C]

over all completely decomposable subgroups

C
C

of

G
G

is denoted by

i
(
G
)

i(G)

. An almost completely decomposable group

G
G

has, up to isomorphism, only finitely many summands. If

i
(
G
)

i(G)

is a prime power, then the rank 1 summands in any decomposition of

G
G

as a direct sum of indecomposable groups are uniquely determined. If

G
G

and

H
H

are almost completely decomposable groups, then the following statements are equivalent: (i)

G
⊕
L
≈
H
⊕
L

G \oplus L \approx H \oplus L

for some finite rank torsion free abelian group

L
L

. (ii)

i
(
G
)
=
i
(
H
)

i(G) = i(H)

and

H
H

contains a subgroup

G
′

G’

isomorphic to

G
G

such that

[
H
:

G
′

]

[H:G’]

is finite and prime to

i
(
G
)

i(G)

. (iii)

G
⊕
L
≈
H
⊕
L

G \oplus L \approx H \oplus L

where

L
L

is isomorphic to a completely decomposable subgroup with finite index in

G
G

.
The Hardy class of a function with slowly-growing area
Authors: Hansen Lowell J.
Published: June 2010

In this paper we show that if

f
f

is analytic on the open unit disk and if the area of

[
{

|

z

|

≤
R
}
∩

image of 

f
]

[\{ |z| \leq R\} \cap {\text {image of }}f]

grows sufficiently slowly as a function of

R
R

, then

f
f

belongs to the Hardy class

H
p

{H^p}

for all

p
p

satisfying

0
>
p
>
+
∞

0 > p > + \infty

.
A zero set for 𝐴(𝑈²)
Authors: Davis Carl Stephen
Published: June 2010

A zero set for

A
(

U
2

)

A({U^2})

is constructed using a strictly increasing function on

[
0
,
1
]

[0,1]

with a derivative that is zero almost everywhere.
On the homology of finite cyclic coverings of higher-dimensional links
Authors: Sumners D. W.
Published: June 2010

We produce an explicit formula for the betti numbers of the

k
k

-fold branched cyclic covering of a link, in terms of complex

k
k

th roots of unity which are also roots of the polynomial invariants of the link. More information is obtained when

k
k

is a prime power.
Free actions of cyclic groups of order 2ⁿ on 𝑆¹×𝑆²
Authors: Ritter Gerhard X.
Published: June 2010

In [4] Y. Tao proved that if

h
h

is a fixed point free involution of

S
1

×

S
2

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

, then

(

S
1

×

S
2

)

/

h

({S^1} \times {S^2})/h

must be homeomorphic to either

M
1

=

S
1

×

S
2

{M_1} = {S^1} \times {S^2}

, or

M
2

=

K

3

{M_2} = {{\mathbf {K}}^3}

, or

M
3

=

S
1

×

P

2

{M_3} = {S^1} \times {{\mathbf {P}}^2}

or

M
4

=

P

3

#

P

3

{M_4} = {{\mathbf {P}}^3}\# {{\mathbf {P}}^3}

. In this paper we extend this result to free actions of

Z

2
n

{Z_{{2^n}}}

on

S
1

×

S
2

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

, showing that, for

n
>
1
,
(

S
1

×

S
2

)

/

Z

2
n

n > 1,({S^1} \times {S^2})/{Z_{{2^n}}}

must be homeomorphic to either

M
1

{M_1}

or

M
2

{M_2}

.
Semihereditary polynomial rings
Authors: Camillo Victor P.
Published: June 2010

It is shown that if the ring of polynomials over a commutative ring

R
R

is semihereditary then

R
R

is von Neumann regular. This is the converse of a theorem of P. J. McCarthy.
On the extreme points of some sets of analytic functions
Authors: Milcetich John G.
Published: June 2010

Let

A
A

denote the set of analytic functions defined on the open unit disc. The extreme points of

F
=
{
f
ϵ
A
:
f
(
0
)
=
0

F = \{ f \epsilon A:f(0) = 0

and

|

Re
⁡
f
(
z
)

|

>
π

/

2

|\operatorname {Re} f(z)| > \pi /2

are determined. Also a partial characterization is given for the extreme points of

G
α

=
{
f
ϵ
A
:
f
(
0
)
=
1

{G_\alpha } = \{ f\epsilon A:f(0) = 1

and

|

arg
⁡
f
(
z
)

|

>
α
π

/

2
}
,
0
>
α
>
1

|\arg f(z)| > \alpha \pi /2\} ,0 > \alpha > 1

.
Discrete phenomena in uniqueness in the Cauchy problem
Authors: Trèves François
Published: June 2010

The concatenation method, recently applied to the study of solvability and hypoellipticity of linear partial differential equations with double characteristics, is used to show that uniqueness in the Cauchy problem, for certain degenerate hyperbolic equations, depends on whether the lower order terms (precisely, the subprincipal part) assume or not certain values belonging to a sequence of complex numbers.
On sums of Hankel operators
Authors: Fuhrmann Paul A.
Published: June 2010

Necessary and sufficient conditions are derived for the sum of two Hankel operators of closed range to have closed range. As a corollary we determine when two left invariant subspaces of

H
2

{H^2}

have positive angle.
The nonvanishing of certain character sums
Authors: Ullom S.
Published: June 2010

Let

χ
\chi

be a Dirichlet character with conductor

f
f

and

M
(
χ
)
=
Σ
a

χ
¯

(
a
)

M(\chi ) = \Sigma a\bar \chi (a)

, summation over integers

a
a

prime to

f
f

and

1
⩽
a
>
f

1 \leqslant a > f

. It is well known that the nonvanishing of the Dirichlet

L
L

-function

L
(
s
,
χ
)

L(s,\chi )

at

s
=
1

s = 1

implies

M
(
χ
)
≠
0

M(\chi ) \ne 0

for

χ
\chi

imaginary, i.e.

χ
(
−
1
)
=
−
1

\chi ( - 1) = - 1

. This article provides a purely algebraic proof that

M
(
χ
)
≠
0

M(\chi ) \ne 0

when the conductor

f
f

is a prime power and the imaginary

χ
\chi

is either a faithful character or has order a power of 2.
The minimum modulus of polynomials
Authors: Beller E. , Newman D. J.
Published: June 2010

In answer to a problem of Erdös and Littlewood we produce an

n
n

th degree polynomial,

P
(
z
)

P(z)

, with coefficients bounded by 1 satisfying

|

P
(
z
)

|

>
C

n

|P(z)| > C\sqrt n

for all

z
z

on

|

z

|

=
1

|z| = 1

(

C
C

is a positive absolute constant).
Reversing the Ahlfors’ estimate
Authors: Sommese Andrew J.
Published: June 2010

It is shown that the Ahlfors’ inequality for a complete Hermitian metric on the punctured disc with Gaussian curvature bounded below zero can be partially reversed for the averaged metric which is also complete and has the same curvature properties.
The effect of retarded actions on nonlinear oscillations
Authors: Sficas Y. G. , Staïkos V. A.
Published: June 2010

We study the oscillatory and asymptotic behavior of nonlinear retarded differential equations of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis t right-parenthesis plus left-parenthesis negative 1 right-parenthesis Superscript n plus 1 Baseline p left-parenthesis t right-parenthesis dot phi left-parenthesis x left-bracket g left-parenthesis t right-parenthesis right-bracket right-parenthesis equals 0">

x

(
n
)

(
t
)
+

(
−
1

)

n
+
1

p
(
t
)
⋅
ϕ
(
x
[
g
(
t
)
]
)
=
0

{x^{(n)}}(t) + {( - 1)^{n + 1}}p(t) \cdot \phi (x[g(t)]) = 0

under certain conditions which do not hold in the particular case

g
(
t
)
≡
t

g(t) \equiv t

of ordinary differential equations.
Exhausting an area with discs
Authors: Erdős P. , Newman D. J.
Published: June 2010

The general question is, just how much of a given area may be covered by interior disjoint discs? We simplify matters by choosing the unit square as our given area, and then give estimates (upper and lower) on how much of it may be so covered by

n
n

discs.
A generalization of Banach’s contraction principle
Authors: Ćirić Lj. B.
Published: June 2010

Let

T
:
M
→
M

T:M \to M

be a mapping of a metric space

(
M
,
d
)

(M,d)

into itself. A mapping

T
T

will be called a quasi-contraction iff
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d left-parenthesis upper T x comma upper T y right-parenthesis less-than-or-slanted-equals q max left-brace d left-parenthesis x comma y right-parenthesis semicolon d left-parenthesis x comma upper T x right-parenthesis semicolon d left-parenthesis y comma upper T y right-parenthesis semicolon d left-parenthesis x comma upper T y right-parenthesis semicolon d left-parenthesis y comma upper T x right-parenthesis right-brace">

d
(
T
x
,
T
y
)
⩽
q
max
{
d
(
x
,
y
)
;
d
(
x
,
T
x
)
;
d
(
y
,
T
y
)
;
d
(
x
,
T
y
)
;
d
(
y
,
T
x
)
}

d(Tx,Ty) \leqslant q\max \{ d(x,y);d(x,Tx);d(y,Ty);d(x,Ty);d(y,Tx)\}

for some

q
>
1

q > 1

and all

x
,
y
∈
M

x,y \in M

. In the present paper the mappings of this kind are investigated. The results presented here show that the condition of quasi-contractivity implies all conclusions of Banach’s contraction principle. Multi-valued quasi-contractions are also discussed.
Piercing points on a special arc
Authors: Rosen Harvey
Published: June 2010

We describe a cellular wild arc

A
A

in

S
3

{S^3}

that is the closure of the union of a null sequence of Alford arcs and show that if

S
S

is an arbitrary

2
2

-sphere containing

A
A

and tame modulo

A
A

, then each point of

A
A

is a piercing point of

S
S

.
Summability methods for independent identically distributed random variables
Authors: Lai Tze Leung
Published: June 2010

In this paper, we present certain theorems concerning the Cesaro

(
C
,
α
)

(C,\alpha )

, Abel

(
A
)

(A)

, Euler

(
E
,
q
)

(E,q)

and Borel

(
B
)

(B)

summability of

Σ

Y
i

\Sigma {Y_i}

, where

Y
i

=

X
i

−

X

i
−
1

,

X
0

=
0

{Y_i} = {X_i} - {X_{i - 1}},{X_0} = 0

and

X
1

,

X
2

,
⋯

{X_1},{X_2}, \cdots

are i.i.d. random variables. While the Kolmogorov strong law of large numbers and the Hartman-Wintner law of the iterated logarithm are related to

(
C
,
1
)

(C,1)

summability and involve the finiteness of, respectively, the first and second moments of

X
1

{X_1}

, their analogues for Euler and Borel summability involve different moment conditions, and the analogues for

(
C
,
α
)

(C,\alpha )

and Abel summability remain essentially the same.
Diagonal equivalence to matrices with prescribed row and column sums. II
Authors: Sinkhorn Richard
Published: June 2010

Let

A
A

be a nonnegative

m
×
n

m \times n

matrix and let

r
=
(

r
1

,
⋯
,

r
m

)

r = ({r_1}, \cdots ,{r_m})

and

c
=
(

c
1

,
⋯
,

c
n

)

c = ({c_1}, \cdots ,{c_n})

be positive vectors such that

Σ

i
=
1

m

r
i

=

Σ

j
=
1

n

c
j

\Sigma _{i = 1}^m{r_i} = \Sigma _{j = 1}^n{c_j}

. It is well known that if there exists a nonnegative

m
×
n

m \times n

matrix

B
B

with the same zero pattern as

A
A

having the

i
i

th row sum

r
i

{r_i}

and

j
j

th column sum

c
j

{c_j}

, there exist diagonal matrices

D
1

{D_1}

and

D
2

{D_2}

with positive main diagonals such that

D
1

A

D
2

{D_1}A{D_2}

has

i
i

th row sum

r
i

{r_i}

and

j
j

th column sum

c
j

{c_j}

. However the known proofs are at best cumbersome. It is shown here that this result can be obtained by considering the minimum of a certain real-valued function of

n
n

positive variables.
A singular primitive ring
Authors: Lawrence John
Published: June 2010

An example of a primitive ring with nonzero singular ideal is constructed. An example, due to B. Osofsky, of a semiprimitive ring with nonzero singular ideal is shown to be nonprimitive.
Coalgebraic coalgebras
Authors: Radford D. E.
Published: June 2010

We investigate coalgebras

C
C

over a field

k
k

such that the dual algebra

C
∗

{C^ \ast }

is an algebraic algebra (

C
C

is called coalgebraic). The study reduces to the cosemisimple and connected cases. If

C
C

is co-semisimple and coalgebraic, then

C
∗

{C^ \ast }

is of bounded degree. If

C
C

is connected, then

C
C

is coalgebraic if, and only if, every coideal is the intersection of cofinite coideals. Our main result is that if

C
C

is a coalgebra over an infinite field

k
k

and the Jacobson radical

Rad
⁡

C
∗

\operatorname {Rad} {C^ \ast }

is nil, there is an

n
n

such that

a
n

=
0

{a^n} = 0

all

a
∈
Rad
⁡

C
∗

a \in \operatorname {Rad} {C^ \ast }

. By the Nagata-Higman theorem,

Rad
⁡

C
∗

\operatorname {Rad} {C^ \ast }

is nilpotent if nil in characteristic 0.
Inertial ℎ-cobordisms with finite cyclic fundamental group
Authors: Lawson Terry C.
Published: June 2010

For

M
M

a PL

n
n

-manifold,

n
≧
5

n \geqq 5

, let

I
(
M
)

I(M)

be the subset of torsions

σ
∈
Wh
⁡
(

π
1

M
)

\sigma \in \operatorname {Wh} ({\pi _1}M)

such that the

h
h

-cobordism

W
W

constructed from

M
M

with torsion

σ
\sigma

has its other boundary component PL homeomorphic to

M
M

. We present three techniques dealing with the determination of

I
(
M
)

I(M)

and apply them when

π
1

M
=

Z
q

{\pi _1}M = {Z_q}

. We prove: (1) If

n
n

is even,

π
1

M
≃

Z
q

,
q

{\pi _1}M \simeq {Z_q},q

odd, then

I
(
M
)
=
Wh
⁡
(

π
1

M
)

I(M) = \operatorname {Wh} ({\pi _1}M)

. (2) If

n
n

is odd, then there exists

M
M

with

π
1

M
≃

Z
q

{\pi _1}M \simeq {Z_q}

such that

I
(
M
)
=
Wh
⁡
(

π
1

M
)

I(M) = \operatorname {Wh} ({\pi _1}M)

.
On 𝐿_{𝑝}-spectra of the laplacian on a Lie group with polynomial growth
Authors: Hulanicki A.
Published: June 2010

The following theorem is proved: If

G
G

is a Lie group with polynomial growth (a compact extension of a nilpotent group, e.g.) and

Δ
=

X
1
2

+
⋯
+

X
n
2

\Delta = X_1^2 + \cdots + X_n^2

, where

X
1

,
⋯
,

X
n

{X_1}, \cdots ,{X_n}

is a basis of the Lie algebra of

G
G

, then for all

p
,
1
≦
p
>
∞

p,1 \leqq p > \infty

, the operator

Δ
\Delta

has the same spectrum on all

L
p

(
G
)

{L_p}(G)

.
Characterizations of normal structure
Authors: Lim Teck Cheong
Published: June 2010

The notions of asymptotic center for a decreasing net of sets and asymptotic normal structure are defined and several characterizations of normal structure are proved. Among these, the problem of whether complete normal structure is equivalent to normal structure is answered in the affirmative.
Representations of locally convex *-algebras
Authors: Powell James D.
Published: June 2010

Conditions for a functional to be admissible on a locally convex

∗
\ast

-algebra are defined. Let

F
F

be an admissible positive Hermitian functional on a commutative locally convex

∗
\ast

-algebra; then it is shown that there exists a representation of

A
A

into a Hilbert space. Sufficient conditions for a functional

F
F

to be representable are also given.
Characterizations of 𝑤*-homomorphisms and expectations
Authors: Palmer T. W.
Published: June 2010

If

A

\mathfrak {A}

and

B

\mathfrak {B}

are

∗
\ast

-algebras a map

ϕ
:

A

→

B

\phi :\mathfrak {A} \to \mathfrak {B}

is called a Schwarz map if it is linear and satisfies the Cauchy-Schwarz inequality

ϕ

(
a

)
∗

ϕ
(
a
)
≤
ϕ
(

a
∗

a
)

\phi {(a)^ \ast }\phi (a) \leq \phi (a ^\ast a)

for all

a
∈

A

a \in \mathfrak {A}

. Under mild restrictions on

A

\mathfrak {A}

and

B

\mathfrak {B}

,

∗
\ast

-homomorphisms and expectations are characterized in terms of Schwarz maps

ϕ
:

A

→

B

\phi :\mathfrak {A} \to \mathfrak {B}

. The proofs are based on an elementary result on the multiplicative properties of Schwarz maps.
Kahane’s construction and the weak sequential completeness of 𝐿¹
Authors: Heard E. A.
Published: June 2010

If

λ
\lambda

is a normalized Lebesgue measure

(
arc length

/

2
π
)

(\text {arc length}/2\pi )

on

T
=
{

|

z

|

=
1
}

T = \{ |z| = 1\}

, the Gelfand map permits

L
∞

(
λ

)
∗

{L^\infty }{(\lambda )^ \ast }

to be identified with

M
(
X
)

M(X)

, the space of finite Baire measures on

X
X

, the maximal ideal space of

L
∞

(
λ
)

{L^\infty }(\lambda )

. The measure

m
0

{m_0}

in

M
(
X
)

M(X)

represents

λ
:
∫

X

f
^

d

m
0

=
∫

T
f
d
λ

\lambda :\int {X\hat fd{m_0} = \int {Tfd\lambda } }

for all

f
∈

L
∞

(
λ
)

f \in {L^\infty }(\lambda )

. Furthermore

μ
≪

m
0

\mu \ll {m_0}

if and only if

μ
\mu

represents some measure of the form

ϕ
d
λ
,
ϕ
∈

L
1

(
λ
)

\phi d\lambda ,\phi \in {L^1}(\lambda )

. Using this fact and a sum constructed by J. P. Kahane,

Σ

h
^

n
(
j
)

\Sigma {\hat h^{n(j)}}

when

h
^

\hat h

is an appropriate function guaranteed by Urysohn’s lemma, develops a proof that

L
1

(
λ
)

{L^1}(\lambda )

is weakly sequentially complete.
An example in fixed point theory
Authors: Johnson Gordon G.
Published: June 2010

An example is given of a homeomorphism of

R
2

{R^2}

onto

R
2

{R^2}

which has no fixed points, such that each of its iterates has a fixed point.
Global geometry of polygons. I: The theorem of Fabricius-Bjerre
Authors: Banchoff Thomas F.
Published: June 2010

Deformation methods provide a direct proof of a polygonal analogue of a theorem proved by Fabricius-Bjerre and by Halpern relating the numbers of crossings, pairs of inflections, and lines of double tangency for smooth closed plane curves.
Zero-one laws for stable measures
Authors: Dudley R. M. , Kanter Marek
Published: June 2010

For any stable measure

μ
\mu

on a vector space, every measurable linear subspace has measure 0 or 1.
Syzygies in [𝑦^{𝑝}𝑧]
Authors: Mead D. G. , Newton M. E.
Published: June 2010

For every

[

y
p

z
]

[{y^p}z]

we obtain an infinite sequence of syzygies as well as the coefficients of some of the terms in the derivatives of these syzygies.
A note on analytic measures
Authors: Muhly Paul S.
Published: June 2010

A corollary to Forelli’s generalization of the F. and M. Riesz theorem is proved. It extends Bochner’s result concerning the absolute continuity of measures on the torus whose Fourier-Stieltjes coefficients vanish outside a sector of opening less than

π
\pi

.
A note on extreme elements in 𝐴₀(𝐾,𝐸)
Authors: Sharir M.
Published: June 2010

We give a short and simple proof to a theorem of Fakhoury, characterizing extreme elements in the unit ball of

A
0

(
K
,
E
)

{A_0}(K,E)

.
𝑃𝐿 involutions on lens spaces and other 3-manifolds
Authors: Kim Paik Kee
Published: June 2010

Let

h
h

be an involution of a

3
3

-dimensional lens space

L
=
L
(
p
,
q
)

L = L(p,q)

.

h
h

is called sense preserving if

h
h

induces the identity of

H
1

(
L
)

{H_1}(L)

. The purpose of this paper is to classify the orientation preserving PL involutions of

L
L

which preserve sense and have nonempty fixed point sets for

p
p

even. It follows that, up to PL equivalences, there are exactly three PL involutions on the projective

3
3

-space

P
3

{P^3}

, and exactly seven PL involutions on

P
3

#

P
3

{P^3}\# {P^3}

.
Triple points and surgery of immersed surfaces
Authors: Banchoff Thomas F.
Published: June 2010

For a sufficiently general immersion of a smooth or polyhedral closed

2
2

-dimensional surface into Euclidean

3
3

-space, the number of triple points is congruent modulo 2 to the Euler characteristic. The approach of this paper involves elementary notions of modification of surfaces by surgery.
Pointwise in terms of weak convergence
Authors: Baxter J. R.
Published: June 2010

Let

(
Ω
,

F

,
μ
)

(\Omega ,\mathfrak {F},\mu )

be a measure space,

μ
(
Ω
)
>
∞

\mu (\Omega ) > \infty

. Let

X
n

{X_n}

be a sequence of measurable functions on

Ω
\Omega

taking values in a compact metric space

M
M

. The set of bounded stopping times

τ
\tau

for the

X
n

{X_n}

is a directed set under the obvious ordering. The following theorem is proved:

X
n

{X_n}

converges pointwise almost everywhere if and only if the generalized sequence

∫

ϕ
(

X
τ

)
d
μ

\int {\phi ({X_\tau })d\mu }

converges for every continuous function

ϕ
\phi

on

M
M

. The martingale theorem is proved as an application.
Essential maps and embeddings of annuli in nonorientable 𝑀³
Authors: Daigle R. J. , Feustel C. D.
Published: June 2010

In this note we give an example which shows that the existence of an essential map of an annulus in a nonorientable

3
3

-manifold does not guarantee the existence of an essential embedding in that manifold.
Constructive aspects of Noetherian rings
Authors: Richman Fred
Published: June 2010

If

R
R

is a discrete Noetherian ring, in the sense of Tennenbaum, such that finitely generated ideals are finitely related and detachable, then so is

R
[
x
]

R[x]

. This shows that a large class of rings enjoy the property that finitely generated ideals are detachable, and that intersections and quotients of finitely generated ideals are finitely generated.
On Dugundji’s notion of positive definiteness
Authors: Wong Chi Song
Published: June 2010

Dugundji’s notion of positive definiteness is generalized to nonnegative real-valued functions on a uniform space. Its relations with completeness and various notions of compactness are investigated. For an arbitrary uniform space

X
X

, there may be lack of the right kind of lower semicontinuous real-valued functions on

X
X

and so a further generalization of Dugundji’s notion of positive definiteness is needed for the development of the fixed point (or coincidence) theory. With such an extension, a very general fixed point theorem is obtained to include a recent result of the author, which contains, as special cases, some results of S. Banach, F.E. Browder, D. W. Boyd and J. S. W. Wong, M. Edelstein and R. Kannan.
A generalization of commutative and alternative rings. IV
Authors: Kleinfeld Erwin
Published: June 2010

We have shown previously for rings

R
R

of characteristic

≠
2
,
3

\ne 2,3

which satisfy the three identities (i)

(
x
,

y
2

,
x
)
=
y
∘
(
x
,
y
,
x
)

(x,{y^2},x) = y \circ (x,y,x)

, (ii)

(
x
,
y
,
z
)
+
(
y
,
z
,
x
)
+
(
z
,
x
,
y
)
=
0

(x,y,z) + (y,z,x) + (z,x,y) = 0

, and (iii) (

(
(
x
,
y
)
,
x
,
x
)
=
0

((x,y),x,x) = 0

, where

(
a
,
b
,
c
)
=
(
a
b
)
c
−
a
(
b
c
)
,
(
a
,
b
)
=
a
b
−
b
a

(a,b,c) = (ab)c - a(bc),(a,b) = ab - ba

, and

a
∘
b
=
a
b
+
b
a

a \circ b = ab + ba

, that under the assumption of no divisors of zero, all such

R
R

must be either associative or commutative. Here we weaken the Lie-admissible identity (ii) by assuming instead (iv) Lie-admissibility for every subring generated by two elements. It turns out that rings without divisors of zero and of characteristic

≠
2
,
3

\ne 2,3

which satisfy (i), (iii) and (iv) are either commutative or alternative. If

S
S

is a ring in which every subring generated by two elements is either commutative or associative, then identities (i), (iii) and (iv) hold in

S
S

, so that this result applies to

S
S

.
Sequential and conditional compactness in the dual of a barrelled space
Authors: Ostling Edward G.
Published: June 2010

Let

E
E

be a barrelled locally convex space and suppose

T

A

{T_{\mathcal {A}}}

is a topology on the dual

E
′

E’

of

E
E

which is admissible for the duality

(
E
,

E
′

)

(E,E’)

. It is shown that each

T

A

{T_{\mathcal {A}}}

sequentially compact subset of

E
′

E’

is

T

A

{T_{\mathcal {A}}}

conditionally compact.
Multipliers vanishing at infinity for certain compact groups
Authors: Figà-Talamanca Alessandro
Published: June 2010

We prove for certain compact groups

G
G

and

1
>
p
>
∞
,
p
≠
2

1 > p > \infty ,p \ne 2

, that there exist operators commuting with left translations on

L
p

(
G
)

{L^p}(G)

which are compact as operators on

L
2

(
G
)

{L^2}(G)

but not as operators on

L
p

(
G
)

{L^p}(G)

.
Reducing subspaces of contractions with no isometric part
Authors: Guyker James
Published: June 2010

Let

T
T

be a contraction on a Hilbert space

H
H

and suppose that there is no nonzero vector

f
f

in

H
H

such that

|

|

T
n

f

|

|

=

|

|

f

|

|

||{T^n}f|| = ||f||

for every

n
=
1
,
2
,
⋯

n = 1,2, \cdots

. In this paper, the reducing subspaces of

T
T

are characterized in terms of the range of

1
−

T
∗

T

1 - {T^ \ast }T

. As a corollary, it is shown that

T
T

is irreducible if

1
−

T
∗

T

1 - {T^ \ast }T

has

1
1

-dimensional range. In particular, if

U
U

is the simple unilateral shift, then the restriction of

U
∗

{U^ \ast }

to any invariant subspace for

U
∗

{U^ \ast }

is irreducible.
A forcing proof of the Kechris-Moschovakis constructibility theorem
Authors: Blass Andreas
Published: June 2010

We show, by forcing, that every subset of

ℵ
1

{\aleph _1}

whose codes form a

Σ
2
1

\Sigma _2^1

set of reals must be constructible.
Hyperbolic integrodifferential equations
Authors: Davis Paul L.
Published: June 2010

Hyperbolic integrodifferential equations are defined and conditions sufficient for hyperbolicity are given. The theory includes that of constant coefficient hyperbolic partial differential equations. Other examples are given.
Injective De Morgan and Kleene algebras
Authors: Cignoli Roberto
Published: June 2010

The aim of this paper is to characterize the injective objects in both the category of De Morgan algebras and the subcategory of Kleene algebras.
On saturated formations which are special relative to the strong covering-avoidance property
Authors: Beidleman J. C. , Makan A. R.
Published: June 2010

Let

F

\mathfrak {F}

be a saturated formation of finite soluble groups. Let

G
G

be a finite soluble group and

F
F

an

F

\mathfrak {F}

-projector of

G
G

. Then

F
F

is said to satisfy the strong covering-avoidance property if (i)

F
F

either covers or avoids each chief factor of

G
G

, anaaa

F
∩
L

/

F
∩
K

F \cap L/F \cap K

is a chief factor of

F
F

whenever

L

/

K

L/K

is a chief factor of

G
G

coverzd by

F
F

Let

C

F

{\mathcal {C}_\mathfrak {F}}

denote the class of all finite soluble

G
G

in which the

F

\mathfrak {F}

-projectors satisfy the strong covering-avoidance property.

C

F

{\mathcal {C}_\mathfrak {F}}

is a formation. Let

Y

F

{\mathcal {Y}_\mathfrak {F}}

be the class of groups

G
G

in which an

F

\mathfrak {F}

-normalizer is also an

F

\mathfrak {F}

-projector.

Y

F

{\mathcal {Y}_\mathfrak {F}}

aa is a formation studied by Klaus Doerk. Note that

Y

F

⊆

C

F

{\mathcal {Y}_\mathfrak {F}} \subseteq {\mathcal {C}_\mathfrak {F}}

.

F

\mathfrak {F}

is said to be

C

\mathcal {C}

-special if

C

F

=

Y

F

{\mathcal {C}_\mathfrak {F}} = {\mathcal {Y}_\mathfrak {F}}

. he puraaaaaaaaaaas note is to sdy

C

\mathcal {C}

-special formations. Two characterizations of

C

\mathcal {C}

-special formations are given. Let

i
i

be a positive integer and let

N

(
i
)

{\mathfrak {N}^{(i)}}

denote the class of finite soluble groups

G
G

whose Fitting length is at most

i
i

. Then

N

(
i
)

{\mathfrak {N}^{(i)}}

is

C

\mathcal {C}

-special. Finally, the formation

C

F

{\mathcal {C}_\mathfrak {F}}

aaaaaturated if and only if

F

\mathfrak {F}

is the class of all finite soluble groups.
On the condition 𝑐^{𝑇}𝐴⁻¹𝑏+𝑟>0, in the Lurie problem
Authors: Somolinos Alfredo S.
Published: June 2010

The problem of Lurie consists in finding NASC’s for all solutions of the system

{

x
′

=
A
x
+
b
f
(
σ
)
,

σ
′

=

c
T

x
−
r
f
(
σ
)
}

\{ x’ = Ax + bf(\sigma ),\sigma ’ = {c^T}x - rf(\sigma )\}

to tend to zero as

t
→
∞

t \to \infty

under appropriate conditions on the functions involved. When

f
(
σ
)

/

σ
>
M

f(\sigma )/\sigma > M

, for all

σ
\sigma

and a certain

M
M

, we obtain NASC’s for the system to be absolutely stable. When

f
(
σ
)

/

σ
>
M

f(\sigma )/\sigma > M

as

|

σ

|

→
∞

|\sigma | \to \infty

, we obtain conditions for ultimate uniform boundedness of the solutions of the system.
The Zariski-Lipman conjecture for homogeneous complete intersections
Authors: Hochster Melvin
Published: June 2010

A new short proof is given that if

R
R

is a homogeneous complete intersection over a field

K
K

of char 0 and

Der
K

(
R
,
R
)

{\operatorname {Der} _K}(R,R)

is

R
R

-free, then

R
R

is a polynomial ring.
Unique best nonlinear approximation in Hilbert spaces
Authors: Chui Charles K. , Smith Philip W.
Published: June 2010

Using the notion of curvature of a manifold, developed by J. R. Rice and recently studied by E. R. Rozema and the second named author, the authors prove the following result: Let

H
H

be a Hilbert space and

F
F

map

R
n

{R^n}

into

H
H

such that

F
F

is a homeomorphism onto

F

=
F
(

R
n

)

\mathfrak {F} = F({R^n})

and is twice continuously Fréchet differentiable. Then if

F
′

(
α
)
⋅

R
n

F’(\alpha ) \cdot {R^n}

is of dimension

n
n

for all

α
∈

R
n

\alpha \in {R^n}

, the manifold

F

\mathfrak {F}

has finite curvature everywhere. It follows that there is a neighborhood

U

\mathfrak {U}

of

F

\mathfrak {F}

such that each

u
∈

U

u \in \mathfrak {U}

has a unique best approximation from

F

\mathfrak {F}

. However, these results do not hold in general for uniformly smooth Banach spaces.
Pseudo-Riemannian manifolds with totally geodesic bisectors
Authors: Beem John K.
Published: June 2010

Let

M
M

be a pseudo-Riemannian manifold. Locally a distance function may be defined. The bisector of two points is the set of points equidistant from these two points. Our main result is that the bisector of two points which are not zero distance apart is a totally geodesic submanifold of

M
M

if and only if

M
M

has constant curvature.
Local sections of flows on manifolds
Authors: Chewning William C. , Owen R. Steven
Published: June 2010

We show the existence of local sections for dynamical systems on manifolds which are themselves generalized manifolds of codimension one. One implication of this is that the level surfaces of Lyapunov functions on three-manifolds are two-manifolds. This fact is used to study the geometry of stable attractors in three-manifolds.
An invariant subspace theorem
Authors: Daughtry John
Published: June 2010

If

A
Y
−
Y
A

AY - YA

has rank one for some compact

Y
Y

, then

A
A

has a nontrivial invariant subspace.
Translations in certain groups of affine motions
Authors: Scheuneman John
Published: June 2010

The purpose of this article is to prove the conjecture of L. Auslander that every nilpotent group of affine motions of

R

n

{{\mathbf {R}}^n}

that is simply transitive on

R

n

{{\mathbf {R}}^n}

has a nontrivial translation in its center. The preliminary result that every such group is unipotent is of independent interest.
On the existence of contact forms
Authors: Thurston W. P. , Winkelnkemper H. E.
Published: June 2010

Using an old theorem of Alexander, we give a short and elementary proof that every closed, orientable

3
3

-manifold has a contact form.
Heights of algebraic numbers and Szegő’s theorem
Authors: Lawton Wayne
Published: June 2010

In the present paper we derive an algorithm which yields approximations to the height of an algebraic number. The techniques are analytical and are motivated by prediction theoretic concepts.
Characterizations of the complex projective plane by curvature
Authors: Greene Robert E. , Pittie Harsh
Published: June 2010

We give short proofs to show that under various positivity assumptions on the curvature of a Kähler surface

X
X

, it is biholomorphically equivalent to

P
2

(
C
)

{P_2}(C)

. In particular, the case of

δ
\delta

-holomorphic pinching

>
1

/

2

> 1/2

(Theorem 1) is best possible and, we believe, new.
On prime ideals with generic zero 𝑥ᵢ=𝑡^{𝑛ᵢ}
Authors: Bresinsky H.
Published: June 2010

Let

n
i

,
1
≤
i
≤
r
,
r
≥
3

{n_i},1 \leq i \leq r,r \geq 3

, be natural numbers such that

(

n
1

,
⋯
,

n
r

)
=
1

({n_1}, \cdots ,{n_r}) = 1

and

n
i

=

Σ

j
=
1

r

z
j

n
j

,

z
j

{n_i} = \Sigma _{j = 1}^r{z_j}{n_j},{z_j}

. nonnegative integers, implies

z
j

=
0
,
j
≠
i

{z_j} = 0,j \ne i

, and

z
i

=
1

{z_i} = 1

. It is shown that for prime ideals with generic zero

x
i

=

t

n
i

{x_i} = {t^{{n_i}}}

and

r
≥
4

r \geq 4

, arbitrary large finite minimal sets of generators exist.
Approximating zeros of accretive operators
Authors: Reich Simeon
Published: June 2010

Let

A
A

be an

m
m

-accretive set in a reflexive Banach space

E
E

with a Gateaux differentiable norm. For positive

r
r

let

J
r

{J_r}

denote the resolvent of

A
A

. If the duality mapping of

E
E

is weakly sequentially continuous and 0 is in the range of

A
A

, then for each

x
x

in

E
E

the strong

lim

r
→
∞

J
r

x

{\lim _{r \to \infty }}{J_r}x

exists and belongs to

A

−
1

(
0
)

{A^{ - 1}}(0)

. This is an extension to a Banach space setting of a result previously known only for Hilbert space.
On the problems of determining knots by their complements and knot complements by their groups
Authors: Simon Jonathan
Published: June 2010

Modulo the conjecture that the complement of a prime knot in the

3
3

-sphere is determined by its fundamental group, we show that at most finitely many mutually inequivalent knots can have homeomorphic complements.
Partitions with congruence conditions
Authors: Robertson M. M.
Published: June 2010

Let

A
=

∪

i
=
1

q

{
a
(
i
)
+
ν
M
:
ν
=
0
,
1
,
2
,
…
}

A = \cup _{i = 1}^q\{ a(i) + \nu M:\nu = 0,1,2, \ldots \}

, where

q
,
M

q,M

and the

a
(
i
)

a(i)

are positive integers such that

a
(
1
)
>
a
(
2
)
>
⋯
>
a
(
q
)
⩽
M

a(1) > a(2) > \cdots > a(q) \leqslant M

. Asymptotic formulae are obtained for

p
(
n
,
k
,
A
)
,

p
∗

(
n
,
k
,
A
)

p(n,k,A),{p^ \ast }(n,k,A)

the number of partitions of

n
n

into

k
k

parts,

k
k

unequal parts respectively, where all the parts belong to

A
A

.
The commutator subgroup of a free topological group need not be projective
Authors: Clarke Francis
Published: June 2010

It is shown that the commutator subgroup of the free topological group on the

n
n

-sphere is, for

n
>
1

n > 1

, not projective and hence not free topological. The proof depends on the computation of the

mod
2

\bmod 2

cohomology ring of the classifying space of the commutator subgroup.
The commutants of relatively prime powers in Banach algebras
Authors: Al-Moajil Abdullah H.
Published: June 2010

Let

R
R

be a ring and

A
(
R
)
=
{
x
∈
R
:
x

A(R) = \{ x \in R:x

belongs to the second commutant of

{

x
n

,

x

n
+
1

}

\{ {x^n},{x^{n + 1}}\}

for all integers

n
>
1
}

n > 1\}

. It is shown that in a prime ring

R
,
A
(
R
)
=
R

R,A(R) = R

if and only if

R
R

has no nilpotent elements. The set

A
(
U
)

A(U)

is studied for some special

∗
\ast

-algebras. It is shown that the normal elements of a proper

∗
\ast

-algebra

U
U

belong to

A
(
U
)

A(U)

. If

U
U

is also prime then

A
(
U
)
=
{
x
∈
U
:
x

A(U) = \{ x \in U:x

belongs to the second commutant of

{

x
n

,

x

n
+
1

}

\{ {x^n},{x^{n + 1}}\}

for some

n
>
1
}

n > 1\}

. The set

A
(
B
(
H
)
)

A(B(H))

is studied, where

B
(
H
)

B(H)

is the algebra of bounded operators on a Hilbert space

H
H

. Necessary and sufficient conditions for some special types of operators to belong to

A
(
B
(
H
)
)

A(B(H))

are obtained.
Counterexamples on the fusion of involutions in finite groups
Authors: Doro Stephen
Published: June 2010

Some examples are given of simple groups

G
G

containing a conjugacy class of involutions

C

\mathcal {C}

such that if

S
S

is a Sylow

2
2

-subgroup of

G
G

, then there exist subgroups

A
⊆
B
⊆
S

A \subseteq B \subseteq S

with

|

A
:
B

|

=
2

|A:B| = 2

and

C

∩
S
⊆
B
−
A

\mathcal {C} \cap S \subseteq B - A

.
Fractional integration in Orlicz spaces
Authors: Sharpley Robert
Published: June 2010

Fractional integration and convolution results are given for Orlicz spaces using an inequality earlier developed for

Λ
α

(
X
)

{\Lambda _\alpha }(X)

spaces which generalize Lorentz

L
(
p
,
q
)

L(p,q)

spaces. The extension problem for convolution operators encountered previously by other authors is almost entirely avoided.
Triples of 2×2 matrices which generate free groups
Authors: Bachmuth S. , Mochizuki H.
Published: June 2010

A theorem is proved which guarantees that certain triples of

2
×
2

2 \times 2

matrices over the complex numbers freely generate a free group. (There is no a priori reason that these matrices lie in a two generator matrix group.)
The nonexistence of certain invariant measures
Authors: Erdős Paul , Mauldin R. Daniel
Published: June 2010

It is shown that there does not exist an uncountable group

G
G

and a nontrivial,

σ
\sigma

-finite, countably additive measure defined on all subsets of

G
G

which is left-invariant.
A deformation theorem for the Kobayashi metric
Authors: Kalka M.
Published: June 2010

Let

M
0

{M_0}

be a compact hyperbolic complex manifold. It is shown that the infinitesimal Kobayashi metric is upper semicontinuous in a

C
∞

{C^\infty }

deformation parameter

t
∈
U
⊆

R
k

t \in U \subseteq {R^k}

. This is accomplished by proving deformation theorems for holomorphic maps.
Representation of weakly additive operators
Authors: Decarlo R. A. , Saeks R.
Published: June 2010

This paper characterizes a class of nonlinear operators termed “weakly additive". A distributional kernel representation is constructed. A counterexample to a conjecture by Gersho is then given via the distributional kernel formulation.
Pseudocompactness properties
Authors: Broverman Samuel
Published: June 2010

A topological extension property is a class of Tychonoff spaces

P

\mathcal {P}

which is closed hereditary, closed under formation of topological products and contains all compact spaces. If

X
X

is Tychonoff and

P

\mathcal {P}

is an extension property, there is a space

P

X

\mathcal {P}X

such that

X
⊆

P

X
⊆
β
X
,

P

X
∈

P

X \subseteq \mathcal {P}X \subseteq \beta X,\;\mathcal {P}X \in \mathcal {P}

and if

f
:
X
→
Y

f:X \to Y

where

Y
∈

P

Y \in \mathcal {P}

then

f
f

admits a continuous extension to

P

X

\mathcal {P}X

. A space

X
X

is called

P

\mathcal {P}

-pseudocompact if

P

X
=
β
X

\mathcal {P}X = \beta X

. In this note it is shown that if

P

\mathcal {P}

is an extension property which contains the real line (e.g., the class of realcompact spaces),

X
X

is

P

\mathcal {P}

-pseudocompact and

Y
Y

is compact, then

X
×
Y

X \times Y

is

P

\mathcal {P}

-pseudocompact. An example is given of an extension property

P

\mathcal {P}

, a

P

\mathcal {P}

-pseudocompact space

X
X

and a compact space

Y
Y

such that

X
×
Y

X \times Y

is not

P

\mathcal {P}

-pseudocompact.
A note on the metrization of 𝛾-spaces
Authors: Martin Harold W.
Published: June 2010

The class of quasi Nagata spaces, which strictly includes the class of wN-spaces, is defined. A closely related class of spaces, pseudo Nagata spaces, are also defined. It is shown that a Hausdorff

γ
\gamma

-space is metrizable if and only if it is either a quasi Nagata space or a pseudo Nagata space.
Inequalities for polynomials satisfying 𝑝(𝑧)≡𝑧ⁿ𝑝(1/𝑧)
Authors: Govil N. K. , Jain V. K. , Labelle G.
Published: June 2010

If

p
(
z
)
=

Σ

v
=
0

n

a
v

z
v

p(z) = \Sigma _{v = 0}^n {{a_v}} {z^v}

is a polynomial of degree

n
n

, then it is known that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M a x Subscript StartAbsoluteValue z EndAbsoluteValue equals 1 Baseline StartAbsoluteValue p prime left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to n upper M a x Subscript StartAbsoluteValue z EndAbsoluteValue equals 1 Baseline StartAbsoluteValue p left-parenthesis z right-parenthesis EndAbsoluteValue">

Max

|

z

|

=
1

|

p
′

(
z
)

|

≤
n

Max

|

z

|

=
1

|

p
(
z
)

|

{\operatorname {Max} _{|z| = 1}}|p’(z)| \leq n{\operatorname {Max} _{|z| = 1}}|p(z)|

. In this paper we obtain the analogous inequality for a subclass of polynomials satisfying

p
(
z
)
≡

z
n

p
(
1

/

z
)

p(z) \equiv {z^n}p(1/z)

. Some other inequalities have also been obtained.
Oscillation of a forced second order nonlinear differential equation
Authors: Rankin Samuel M.
Published: June 2010

Sufficient conditions are given which insure that every solution of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis a left-parenthesis t right-parenthesis y prime right-parenthesis prime plus p left-parenthesis t right-parenthesis f left-parenthesis y right-parenthesis g left-parenthesis y prime right-parenthesis equals r left-parenthesis t right-parenthesis">

(
a
(
t
)

y
′

)
′

+
p
(
t
)
f
(
y
)
g
(

y
′

)
=
r
(
t
)

(a(t)y’)’ + p(t)f(y)g(y’) = r(t)

has arbitrarily large zeros. We seem to have a partial answer to a question posed by Kartsatos [4]. An example is given illustrating the result.
A remark on strongly exposing functionals
Authors: Lau Ka Sing
Published: June 2010

By using the concept of farthest points, we show that the set of strongly exposing functionals of a weakly compact convex subset in a Banach space

X
X

is a dense

G
δ

{G_\delta }

in

X

∗

{X^{\ast }}

. The construction also gives a new proof of existence of strongly exposed points in weakly compact convex sets.
Elliptic curves and Dedekind domains
Authors: Rosen Michael
Published: June 2010

Some results are obtained on the group of rational points on elliptic curves over infinite algebraic number fields. A certain naturally defined class of Dedekind domains, elliptic Dedekind domains, are described and it is shown that every countable abelian group can be realized as the class group of an elliptic Dedekind domain.
A characterization of Lipschitz classes on finite dimensional groups
Authors: Bloom Walter R.
Published: June 2010

An analogue of a theorem of S. N. Bernstein is developed for certain metric locally compact abelian groups. This, together with a corresponding Jackson-type theorem, gives a characterisation in terms of their Fourier transforms of the Lipschitz functions defined on a compact abelian group with finite topological dimension.
Relating group topologies by their continuous points
Authors: Sharpe Kevin J.
Published: June 2010

Let

x
x

be a point in a topological group

G
G

, and for each integer

n
n

, let

(
1

/

n
)
x

(1/n)x

be the set

{
y
:
n
y
=
x
}

\{ y:ny = x\}

in

G
G

. Then I call

x
x

a continuous point if for positive integers

n
n

, the subsets

(
1

/

n
)
x

(1/n)x

are nonvoid and eventually intersect each neighbourhood of the identity 0. I prove the following result and from it three corollaries. Let

G
G

be a divisible abelian group such that

(
1

/

n
)
0
=
{
0
}

(1/n)0 = \{ 0\}

for some integer

n
⩾
2

n \geqslant 2

. Suppose there are two group topologies

A

1

{\mathcal {A}_1}

and

A

2

{\mathcal {A}_2}

defined on

G
G

and that

G
G

is

A

2

{\mathcal {A}_2}

-locally compact and

σ
\sigma

-compact, and define

ω
2

{\omega _2}

to be the outer measure derived from the Haar measure

μ
2

{\mu _2}

on

(
G
,

A

2

)

(G,{\mathcal {A}_2})

. Also suppose that the ratio of the

A

2

{\mathcal {A}_2}

-measure of

{
n
x
:
x
∈
A
}

\{ nx:x \in A\}

to the

A

2

{\mathcal {A}_2}

-measure of

A
A

, for any

A

2

{\mathcal {A}_2}

-Borel-measurable set

A
A

(the ratio is the same for any such

A
A

with finite measure), does not exceed 1. Then for each

A

2

{\mathcal {A}_2}

-Borel-measurable set

A
A

with nonvoid

A

1

{\mathcal {A}_1}

-interior,

μ
2

(
A
)
⩾

ω
2

(

W
1

)
,

W
1

{\mu _2}(A) \geqslant {\omega _2}({W_1}),{W_1}

being the subgroup of all points in

G
G

which are

A

1

{\mathcal {A}_1}

-continuous.
On the extension of cylinder measures to 𝜏-smooth measures in linear spaces
Authors: Heidemann Dirk
Published: June 2010

We give a necessary and sufficient condition for a cylindrical probability measure in the weak

∗

^{\ast }

-dual of an arbitrary l.c.s. to extend to a

τ
\tau

-smooth Borel-measure; this is to a certain extent a “

τ
\tau

-smooth analogue” of the well-known Prohorov extension theorem (cf. [8, Lemma 3]). Finally, we give two examples marking off our result from related ones treated in the literature.
𝐿_{𝑝}(𝜇,𝑋) (1<𝑝<∞) has the Radon-Nikodým property if 𝑋 does by martingales
Authors: Turett Barry , Uhl J. J.
Published: June 2010

Using the fact that

L
p

[
0
,

1
]

(
1
>
p
>
∞
)

{L_p}[0,\;1]\;(1 > p > \infty )

has an unconditional basis, Sundaresan has shown that

L
p

(
μ
,

X
)

{L_p}(\mu ,\;X)

has the Radon-Nikodým property if

1
>
p
>
∞

1 > p > \infty

and

X
X

has the Radon-Nikodým property. In this note, Sundaresan’s theorem is proved by direct martingale methods. Then it is shown how to adapt this argument to the context of Orlicz spaces in which Sundaresan’s argument is not applicable.
Characteristic numbers for unoriented singular 𝐺-bordism
Authors: Khare S. S. , Sharma B. L.
Published: June 2010

We develop the notion of characteristic numbers for unoriented singular

G
G

-manifolds in a

G
G

-space,

G
G

being a finite group, and prove their invariance with respect to unoriented singular

G
G

-bordism.
On the Lu Qi-keng conjecture
Authors: Suita Nobuyuki , Yamada Akira
Published: June 2010

We shall give a complete answer to the Lu Qi-keng conjecture for finite Riemann surfaces. Our result is that every finite Riemann surface which is not simply-connected is never a Lu Qi-keng domain, i.e. the Bergman kernel

K
(
z
,
t
)

K(z,t)

of it has zeros for suitable

t
t

’s.