bluuebunny/crossref_metadata_2025 huggingface dataset:
On the existence of fundamental solutions of certain boundary value problems
Authors: Pathak R. S.
Published: June 2010

An existence theorem for fundamental solutions of a large class of boundary value problems involving the Bessel type differential operator

S
μ

{S_\mu }

is proved. The problem includes the Cauchy problem of the differential operator

S
μ

{S_\mu }

as a special case.
A note on generating relations for Lauricella’s function of several variables
Authors: Shukla D. P.
Published: June 2010

In this paper we have derived generating relations for Lauricella’s function by using the operator

T
k

=
x
(
k
+
x
D
)

{T_k} = x(k + xD)

and the operational relations involving this operator. Some recent results of Srivastava [3], Srivastava and Carlitz [4], Sharma and Abiodun [1] have been conveniently obtained by this method as well as some hitherto unknown results established.
Recession cones of nonconvex sets and increasing functions
Authors: Beer Gerald
Published: June 2010

In this article a local characterization theorem is given for closed sets in a linear topological space that have recession cones with nonempty interior. This theorem is then used to characterize the class of upper semicontinuous increasing functions defined on closed

E
+
d

E_ + ^d

-recessional subsets of

E
d

{E^d}

.
Unsmoothable diffeomorphisms on higher dimensional manifolds
Authors: Harrison Jenny
Published: June 2010

Examples of

C
r

{C^r}

diffeomorphisms not topologically conjugate to

C
s

{C^s}

diffeomorphisms

(
s
>
r
)

(s > r)

are constructed on manifolds of dimension

≠
1
,
4

\ne 1,4

.
A short proof of a version of Asplund’s norm averaging theorem
Authors: John K. , Zizler V.
Published: June 2010

A short proof is given of a somewhat weaker version of Asplund’s result on averaging smooth and rotund norms in Banach spaces.
Evaluation of character sums connected with elliptic curves
Authors: Williams Kenneth S.
Published: June 2010

Let p be an odd prime and let

(

⋅
p

)

(\tfrac { \cdot }{p})

be the Legendre symbol. It is shown how to evaluate the character sum

Σ

x
=
0

p
−
1

(

f
(
x
)

p

)

\Sigma _{x = 0}^{p - 1}(\tfrac {{f(x)}}{p})

, for certain quartic polynomials

f
(
x
)

f(x)

. For example, it is shown that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript x equals 0 Overscript p minus 1 Endscripts left-parenthesis StartFraction x Superscript 4 Baseline minus 8 x cubed plus 12 x squared minus 16 x plus 4 Over p EndFraction right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 2 left-parenthesis StartFraction 2 Over p EndFraction right-parenthesis x 1 minus 1 comma 2nd Column a m p semicolon if p identical-to 1 left-parenthesis mod 4 right-parenthesis comma 2nd Row 1st Column negative 1 comma 2nd Column a m p semicolon if p identical-to 3 left-parenthesis mod 4 right-parenthesis comma EndLayout">

∑

x
=
0

p
−
1

(

x
4

−
8

x
3

+
12

x
2

−
16
x
+
4

p

)

=

{

2

(

2
p

)

x
1

−
1
,

a
m
p
;

if

p
≡
1

(
mod

4
)
,

−
1
,

a
m
p
;

if

p
≡
3

(
mod

4
)
,

\sum \limits _{x = 0}^{p - 1} {\left ( {\frac {{{x^4} - 8{x^3} + 12{x^2} - 16x + 4}}{p}} \right )} = \left \{ {\begin {array}{*{20}{c}} {2\left ( {\frac {2}{p}} \right ){x_1} - 1,} \hfill & {{\text {if}}\;p \equiv 1\;\pmod 4,} \hfill \\ { - 1,} \hfill & {{\text {if}}\;p \equiv 3\;\pmod 4,} \hfill \\ \end {array} } \right .

\]
where

x
1

{x_1}

is defined for primes

p
≡
1

(
mod

4
)

p \equiv 1\;\pmod 4

by
\[

p
=

x
1
2

+

y
1
2

,

x
1

≡
−
1

(
mod

4
)
.

p = x_1^2 + y_1^2,\quad {x_1} \equiv - 1\quad \pmod 4.

\]
Embedding of a Lie algebra into Lie-admissible algebras
Authors: Myung Hyo Chul
Published: June 2010

Let A be a flexible Lie-admissible algebra over a field of characteristic

≠
\ne

2, 3. Let S be a finite-dimensional classical Lie subalgebra of

A
−

{A^ - }

which is complemented by an ideal R of

A
−

{A^ - }

. It is shown that S is a Lie algebra under the multiplication in A and is an ideal of A if and only if S contains a classical Cartan subalgebra H which is nil in A and such that

H
H
⊆
S

HH \subseteq S

and

[
H
,
R
]
=
0

[H,R] = 0

. In this case, the multiplication between S and R is determined by linear functionals on R which vanish on [R, R]. If A is finite-dimensional and of characteristic 0 then this can be applied to give a condition that a Levi-factor S of

A
−

{A^ - }

be embedded as an ideal into A and to determine the multiplication between S and the solvable radical of

A
−

{A^ - }

.
Examples for the nonlocally convex three space problem
Authors: Ribe M.
Published: June 2010

A simple way to obtain certain examples of locally bounded spaces E of the following kind is described: E is nonlocally convex but contains a locally convex subspace K such that

E

/

K

E/K

is locally convex.
Stochastic construction of new orthogonal polynomials
Authors: Lin T. F.
Published: June 2010

For

u
>
−
1
,
t
⩾
0
,
x
∈
R

u > - 1,t \geqslant 0,x \in R

, let

(
1
+
u

)

Q
(
t
)
+
t

e

−
u
t

{(1 + u)^{Q(t) + t}}{e^{ - ut}}

satisfies

d
y
(
t
)
=
u
y
(
t
)

d
Q
(
t
)

dy(t) = uy(t)\;dQ(t)

where

Q
(
t
)
=
P
(
t
)
−
t

Q(t) = P(t) - t

is the centered Poisson process with parameter

λ
=
1

\lambda = 1

, (ii)

K
n

(
t
,
x
)
,
n
⩾
0

{K_n}(t,x),n \geqslant 0

, is a system of complete orthogonal polynomials in

L
2

(
R
,
d
F
)

{L^2}(R,dF)

where

F
(
x
)

F(x)

is the distribution function of

P
(
t
)

P(t)

, (iii)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n plus 1 right-parenthesis upper K Subscript n plus 1 Baseline left-parenthesis t comma x right-parenthesis minus left-parenthesis x minus t minus n right-parenthesis upper K Subscript n Baseline left-parenthesis t comma x right-parenthesis plus t upper K Subscript n minus 1 Baseline left-parenthesis t comma x right-parenthesis equals 0 comma n greater-than-or-slanted-equals 1">

(
n
+
1
)

K

n
+
1

(
t
,
x
)
−
(
x
−
t
−
n
)

K
n

(
t
,
x
)
+
t

K

n
−
1

(
t
,
x
)
=
0
,
n
⩾
1

(n + 1){K_{n + 1}}(t,x) - (x - t - n){K_n}(t,x) + t{K_{n - 1}}(t,x) = 0,n \geqslant 1

, (iv)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t normal upper Delta squared upper K Subscript n Baseline left-parenthesis t comma x right-parenthesis plus left-parenthesis t minus x right-parenthesis normal upper Delta upper K Subscript n Baseline left-parenthesis t comma x right-parenthesis plus n upper K Subscript n Baseline left-parenthesis t comma x right-parenthesis equals 0 comma n greater-than-or-slanted-equals 0">

t

Δ
2

K
n

(
t
,
x
)
+
(
t
−
x
)
Δ

K
n

(
t
,
x
)
+
n

K
n

(
t
,
x
)
=
0
,
n
⩾
0

t{\Delta ^2}{K_n}(t,x) + (t - x)\Delta {K_n}(t,x) + n{K_n}(t,x) = 0,n \geqslant 0

.
Summability factors for Cesàro methods
Authors: Dawson David F.
Published: June 2010

It is shown that if each of r and s is a nonnegative integer and

{

f
p

}

\{ {f_p}\}

is a complex sequence such that

Σ

f
p

a
p

\Sigma {f_p}{a_p}

is Cesàro summable of order s whenever

Σ

a
p

\Sigma {a_p}

is Cesàro summable of order r, then

Σ

f
p

a
p

\Sigma {f_p}{a_p}

is Cesàro summable of order r whenever

Σ

a
p

\Sigma {a_p}

is Cesàro summable of order r.
Applications of shrinkable covers
Authors: Smith J. C.
Published: June 2010

An open cover

G

=
{

G
α

:
α
∈
A
}

\mathcal {G} = \{ {G_\alpha }:\alpha \in A\}

of a topological space X is shrinkable if there exists a closed cover

F

=
{

F
α

:
α
∈
A
}

\mathcal {F} = \{ {F_\alpha }:\alpha \in A\}

of X such that

F
α

⊆

G
α

{F_\alpha } \subseteq {G_\alpha }

for each

α
∈
A

\alpha \in A

. In this paper the author determines conditions necessary for a variety of general covers to be shrinkable. In particular it is shown that the shrinkability of special types of covers provide characterizations for normal and countably paracompact, normal spaces. The types of covers investigated are, weak

θ
¯

\bar \theta

-covers, weak

θ
¯

\bar \theta

-covers, point countable covers,

δ
θ

\delta \theta

-covers and weak

δ
θ

¯

\overline {\delta \theta }

-covers. Applications of these results are answers of unsolved problems and new results for irreducible spaces.
On 𝐾-primitive rings
Authors: Kezlan Thomas P.
Published: June 2010

Ortiz has defined a new radical for rings, called the K-radical, which in general lies strictly between the prime radical and the Jacobson radical. In this paper a simple internal characterization of K-primitive rings is given, and it is shown that among the K-primitive rings are prime Noetherian rings and prime rings which satisfy a polynomial identity. In addition an analogue of the density theorem is proved for K-primitive rings.
Intersections of commutants with closures of derivation ranges
Authors: Herrero Domingo A.
Published: June 2010

The norm closure of the set
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Subscript w Baseline left-parenthesis script upper X right-parenthesis equals union StartSet Ran left-parenthesis delta Subscript upper A Baseline right-parenthesis Superscript negative w Baseline intersection left-brace upper A right-brace Superscript prime Baseline colon upper A element-of script upper L left-parenthesis script upper X right-parenthesis EndSet">

A

w

(

X

)
=
∪

{

Ran

(

δ
A

)

−
w

∩
{
A

}
′

:
A
∈

L

(

X

)
}

{\mathcal {A}_w}(\mathcal {X}) = \cup \;\{ {\text {Ran}}{({\delta _A})^{ - w}} \cap \{ A\} ’:A \in \mathcal {L}(\mathcal {X})\}

, where

δ
A

{\delta _A}

denotes the inner derivation induced by the operator A,

Ran

(

δ
A

)

−
w

{\text {Ran}}{({\delta _A})^{ - w}}

is the weak closure of the range of

δ
A

{\delta _A}

and

{
A

}
′

\{ A\} ’

is the commutant of A, is disjoint from the open dense subset

B

(

X

)
=
{
T
∈

L

(

X

)

\mathcal {B}(\mathcal {X}) = \{ T \in \mathcal {L}(\mathcal {X})

: T has a nonzero normal eigenvalue} for every complex Banach space

X

\mathcal {X}

. For a Hilbert space

H

\mathcal {H}

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L left-parenthesis script upper H right-parenthesis equals script upper B left-parenthesis script upper H right-parenthesis union script upper A Subscript w Baseline left-parenthesis script upper H right-parenthesis Superscript minus">

L

(

H

)
=

B

(

H

)
∪

A

w

(

H

)
−

\mathcal {L}(\mathcal {H}) = \mathcal {B}(\mathcal {H}) \cup {\mathcal {A}_w}{(\mathcal {H})^ - }

, where the bar denotes norm closure.
Correction of a generalization of a theorem of Beurling and Livingston
Authors: Fowler Peter A.
Published: June 2010

A generalization of a Riesz-Fischer theorem proved by Beurling and Livingston for smooth uniformly convex Banach spaces also holds for smooth, strictly convex, reflexive Banach spaces. Theorem. Let B be a smooth, strictly convex, reflexive Banach space. Let

L
:
B
→

B
∗

L:B \to {B^\ast }

be a duality map, C a closed subspace of

B
,
h
∈
B
,
k
∈

B
∗

B,h \in B,k \in {B^\ast }

. Then

T
(
C
+
h
)
∩
(

C
⊥

+
k
)

T(C + h) \cap ({C^ \bot } + k)

is a single point. A two-dimensional counterexample shows that

T
(
C
+
h
)
∩
(

C
⊥

+
k
)
=
∅

T(C + h) \cap ({C^ \bot } + k) = \emptyset

is possible if B is not smooth, contrary to the claim of Theorem 4 of Browder, On a theorem of Beurling and Livingston, Canad. J. Math. 17 (1965), 367-372.
A two-sided H. Lewy extension phenomenon
Authors: Hunt L. R. , Kazlow M.
Published: June 2010

Let M be a

C
∞

{C^\infty }

real

(
n
+
k
)

(n + k)

-dimensional CR-manifold in

C

n

{{\mathbf {C}}^n}

. We are interested in finding conditions on M near a point

p
∈
M

p \in M

which imply that all CR-function on M extend to holomorphic functions in some fixed neighborhood of p in

C

n

{{\mathbf {C}}^n}

. Of course if M is a real hypersurface, it is known that M having eigenvalues of opposite sign in its Levi form at p will give us such an extension result. If we view the Levi form at a point on a general CR-manifold M as a quadratic map from the holomorphic tangent space to the normal space of the real tangent space in

C

n

{{\mathbf {C}}^n}

, and if this map is surjective, then we prove our CR-functions extend to holomorphic functions in an open neighborhood of the point. We also show that if the real codimension of M in

C

n

{{\mathbf {C}}^n}

is 2, and if the Levi form is totally indefinite, then the Levi form is onto

R

2

{{\mathbf {R}}^2}

as a quadratic map, and hence we have our extension theory.
Unbounded Loeb measures
Authors: Henson C. Ward
Published: June 2010

Uniqueness of the Carathéodory extension is proved for the unbounded case of Loeb’s measure space construction in nonstandard analysis and these measures are studied. Methods are developed for handling sets in the

σ
\sigma

-algebra generated by the internal sets. A quantitative theory of degrees of null measure is developed.
Weakly confluent mappings and the covering property of hyperspaces
Authors: Grispolakis J. , Tymchatyn E. D.
Published: June 2010

In this note we characterize the continua which are images of weakly confluent mappings only in terms of a covering property for hyperspaces and in terms on an embedding property. This answers some questions of B. Hughes and A. Lelek.
On the problem of partitioning {1,2,⋯,𝑛} into subsets having equal sums
Authors: Straight H. Joseph , Schillo Paul
Published: June 2010

Let N denote the set of natural numbers and let

Z
n

=
{
1
,
2
,
…
,
n
}

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

. For S a finite subset of N, let

σ
S

\sigma S

denote the sum of the elements in S. Then

σ

Z
n

=
n
(
n
+
1
)

/

2

\sigma {Z_n} = n(n + 1)/2

. Suppose

n
(
n
+
1
)
=
2
s
t

n(n + 1) = 2st

, where s and t are integers and

t
⩾
n

t \geqslant n

. We show that

Z
n

{Z_n}

can be partitioned into

T
1

∪

T
2

∪
…
∪

T
s

{T_1} \cup {T_2} \cup \ldots \cup {T_s}

such that

σ

T
i

=
t

\sigma {T_i} = t

, for

1
⩽
i
⩽
s

1 \leqslant i \leqslant s

. Such a partition is called an (s, t)-partition of

Z
n

{Z_n}

. A graph G having

n
(
n
+
1
)

/

2

n(n + 1)/2

edges is said to be path-perfect if the edge set of G can be partitioned as

E
1

∪

E
2

∪
…
∪

E
n

{E_1} \cup {E_2} \cup \ldots \cup {E_n}

so that

E
i

{E_i}

induces a path of length i, for

1
⩽
i
⩽
n

1 \leqslant i \leqslant n

. Suppose p and n are positive integers and r is an even positive integer with

p
⩾
r
+
1

p \geqslant r + 1

and

p
r
=
n
(
n
+
1
)

pr = n(n + 1)

. The existence of an (r/2, p)-partition of

Z
n

{Z_n}

is used to show the existence of an r-regular path-perfect graph G having p vertices and

n
(
n
+
1
)

/

2

n(n + 1)/2

edges.
Properties of representations of Banach algebras with approximate identities
Authors: Barnes Bruce A.
Published: June 2010

Let A be a Banach algebra with a bounded left approximate identity. The Cohen Factorization Theorem is used to prove several results concerning the representation theory of A.
A fixed-point theorem for certain operator valued maps
Authors: Brown D. R. , O’Malley M. J.
Published: June 2010

Let H be a real Hilbert space, and let

B
1

(
H
)

{B_1}(H)

denote the space of symmetric, bounded operators on H which have numerical range in [0, 1], topologized by the strong operator topology, and let L be a strongly continuous function on H into

B
1

(
H
)

{B_1}(H)

. In this paper, methods are given to locate all

z
∈
H

z \in H

which are fixed points of L in the sense that

L
(
z
)
z
=
z

L(z)z = z

. In particular, if

w
∈
H

w \in H

and if

α
\alpha

and

β
\beta

are fixed positive rational numbers with

α
∈
[

1
2

,
∞
)

\alpha \in [\tfrac {1}{2},\infty )

, a decreasing sequence of elements of

B
1

(
H
)

{B_1}(H)

is recursively defined, and converges to

Q
∈

B
1

(
H
)

Q \in {B_1}(H)

. If

α
>

1
2

\alpha > \tfrac {1}{2}

, then Q is idempotent and

z
=
Q
w

z = Qw

is a fixed point of L, and if

α
=

1
2

,
β
⩾

1
2

\alpha = \tfrac {1}{2},\beta \geqslant \tfrac {1}{2}

, then

z
=

Q
β

w

z = {Q^\beta }w

is a fixed point of L.
On an extremal problem
Authors: Nevai Paul G.
Published: June 2010

Let

X
=
(

x
1

,

x
2

,
…
,

x
N

)
,
f
:

R

→

C

X = ({x_1},{x_2}, \ldots ,{x_N}),f:{\mathbf {R}} \to {\mathbf {C}}

and let

P

n

{{\mathbf {P}}_n}

be the class of polynomials of degree at most n. The generalized Christoffel function

Λ
n

{\Lambda _n}

corresponding to the measure

d
α

d\alpha

is defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript n Baseline left-parenthesis upper X semicolon f comma upper N comma d alpha right-parenthesis equals min Underscript StartLayout 1st Row pi element-of bold upper P Subscript n minus 1 Baseline 2nd Row pi left-parenthesis x Subscript i Baseline right-parenthesis equals f left-parenthesis x Subscript i Baseline right-parenthesis 3rd Row i equals 1 comma 2 comma ellipsis comma upper N EndLayout Endscripts integral Subscript negative normal infinity Superscript normal infinity Baseline StartAbsoluteValue pi left-parenthesis t right-parenthesis EndAbsoluteValue squared d alpha left-parenthesis t right-parenthesis period">

Λ
n

(
X
;
f
,
N
,
d
α
)
=

min

π
∈

P

n
−
1

π
(

x
i

)
=
f
(

x
i

)

i
=
1
,
2
,
…
,
N

∫

−
∞

∞

|

π
(
t
)

|

2

d
α
(
t
)
.

{\Lambda _n}(X;f,N,d\alpha ) = \min \limits _{\begin {array}{*{20}{c}} {\pi \in {{\mathbf {P}}_{n - 1}}} \\ {\pi ({x_i}) = f({x_i})} \\ {i = 1,2, \ldots ,N} \\ \end {array} } \int _{ - \infty }^\infty {|\pi (t){|^2}d\alpha (t).}

\]
It is shown that if

α
\alpha

satisfies some rather weak conditions then

lim

n
→
∞

n

Λ
n

(
X
;
f
,
N
,
d
α
)

{\lim _{n \to \infty }}n{\Lambda _n}(X;f,N,d\alpha )

exists and the limit is also evaluated.
𝑃-points in random universes
Authors: Cohen Paul E.
Published: June 2010

A pathway is defined as an increasing sequence of subsets of

ω

ω

^\omega \omega

which satisfy certain closure and boundedness properties. The existence of a pathway is shown to imply the existence of a P-point in

β
N
∖
N

\beta N\backslash N

. Pathways are shown to exist in any random extension of a model of

ZFC

+

CH

{\text {ZFC}} + {\text {CH}}

.
Coherent group rings and finiteness conditions for CW-complexes
Authors: Hirschhorn Philip S.
Published: June 2010

We characterize the class of groups G that have the property that if X is any space for which

π
1

X
≅
G

{\pi _1}X \cong G

, then X is homotopy equivalent to a space with finite skeleta in the “stable range” if and only if the homotopy groups of X are finitely presented

Z
[
G
]

Z[G]

-modules in this range. This class of groups includes all finite groups, finitely generated abelian groups, finitely generated nilpotent groups, finitely generated free groups, and free products of any of these.
On monoidal transformations of coherent sheaves
Authors: Rabinowitz Joshua H.
Published: June 2010

We prove that the monoidal transformation of a coherent sheaf S is torsion-free if and only if S is locally-free.
A characterization of maximally almost periodic groups
Authors: Huang Ter Jenq
Published: June 2010

It is proved that a topological group G is maximally almost periodic if and only if G admits an action

π
\pi

on a compact Hausdorff space X such that the transformation group

(
X
,
G
,
π
)

(X,G,\pi )

is equicontinuous and effective. Using this characterization, it is proved that if H is a closed uniform subgroup of a topological group G, then G is maximally almost periodic if and only if H is maximally almost periodic. The latter gives as corollaries the results of Kuranishi, Murakami, Grosser and Moskowitz concerning maximally almost periodic groups.
The cardinality of quasiconformally nonequivalent topological 3-balls with flat boundaries is 𝔠
Authors: Näkki Raimo
Published: June 2010

The theorem mentioned in the title is proved. During the course of the proof, the failure for

n
=
3

n = 3

of the following 2-dimensional result will also be established: The boundary of a Jordan domain D in n-space is a quasiconformal

(
n
−
1
)

(n - 1)

-sphere if every quasiconformal self-mapping of D can be extended to a quasiconformal self-mapping of the whole space.
Packing and covering by translates of certain nonconvex bodies
Authors: Everett Hugh , Hickerson Dean
Published: June 2010

We develop techniques for determining the packing and covering constants for star bodies composed of cubes. In the theory of convex sets problems of tiling, packing, and covering by translates of a given set have a long history, with the main focus on the packing and covering by spheres. Only in a few cases is the densest packing or sparsest covering known, even in the case of the sphere, except, of course, when the set happens to tile Euclidean space. In a series of papers S. K. Stein [4], [5], [6], [7] and W. Hamaker [1] used algebraic techniques in the problem of tiling Euclidean space of arbitrary dimension by translates of certain star bodies composed of cubes. The present paper has two purposes. First, it establishes a “Shift Theorem” that reduces tiling, packing, and covering problems for translates of a union of cubes to translates by vectors with integer coordinates. In a sense, this reduces continuous geometric problems in Euclidean space to discrete algebraic problems in a power of the infinite cyclic group. This theorem automatically generalizes many of Stein’s results that depend on the assumption of integer translates. Second, it illustrates the Shift Theorem and a general “contribution” argument by determining the packing and covering constants for a particular star body.
Hyperspaces homeomorphic to Hilbert space
Authors: Curtis D. W.
Published: June 2010

The hyperspace

2
X

{2^X}

of a metric space X is the space of nonempty compact subsets, topologized by the Hausdorff metric. It is shown that

2
X

{2^X}

is homeomorphic to the separable Hilbert space

l
2

{l^2}

if and only if X is connected, locally connected, separable, topologically complete, and nowhere locally compact. The principal tool in the proof is Torunczyk’s mapping characterization of

l
2

{l^2}

.
On nontrivial characteristic classes of contact foliations
Authors: Ting Wei Lung
Published: June 2010

In this article we give some nontrivial realizations of secondary characteristic classes of contact foliations.
The rigidity problem for stable spaces
Authors: Kahn Donald W.
Published: June 2010

This note treats the problem of when the group of homotopy self-equivalences of a space is trivial. For stable spaces, with finitely many nonvanishing homotopy groups, we give a complete solution in an inductive sense. One of the consequences of this result is that for any stable space, with precisely two nonvanishing homotopy groups, the group of self-equivalences is nontrivial.
A 𝐺-signature theorem for 𝑉-manifolds
Authors: Liang Chao Chu
Published: June 2010

We prove an analogue of the Atiyah-Singer G-signature theorem for an isometry f on a V-manifold.
Embedding phenomena based upon decomposition theory: wild Cantor sets satisfying strong homogeneity properties
Authors: Daverman Robert J.
Published: June 2010

We point out the sharpness of earlier results of McMillan by exhibiting a map of the n-sphere

S
n

,
n
⩾
5

{S^n},n \geqslant 5

, onto itself having acyclic but non-cell-like polyhedra as its nondegenerate point inverses and for which the image of the set of nondegenerate point inverses is a Cantor set K. Of necessity, K is wildly embedded, and it has the unusual additional property that every self-homeomorphism of K extends to a self-homeomorphism of

S
n

{S^n}

.
Embedding phenomena based upon decomposition theory: an unusual cellular 2-cell
Authors: Daverman Robert J.
Published: June 2010

We construct a 2-cell D cellularly embedded in the n-sphere

S
n

,
n
⩾
5

{S^n},n \geqslant 5

, and locally flatly embedded at each of its interior points, but the boundary of which fails to be weakly flat.
A short proof of Kowalsky’s hedgehog theorem
Authors: Swardson Mary Anne
Published: June 2010

We give a simple proof of Kowalsky’s theorem that every metrizable space of weight

m

\mathfrak {m}

is embeddable in a countable product of hedgehogs with

m

\mathfrak {m}

spines.
The Noetherian property for quotient rings of infinite polynomial rings
Authors: Gilmer Robert , Heinzer William
Published: June 2010

Let X be an infinite set of indeterminates over the commutative Noetherian ring R with identity. We prove that the quotient ring of

R
[
X
]

R[X]

with respect to the multiplicative system of polynomials of unit content is also Noetherian. Moreover, we show that certain quotient rings of

R
[
X
]

R[X]

with respect to multiplicative systems of monic polynomials (where “monic” is appropriately defined) are also Noetherian.
An approximation of integrable functions by step functions with an application
Authors: Crandall M. G. , Pazy A.
Published: June 2010

Let

f
∈

L
1

(
0
,
∞
)
,
δ
>
0

f \in {L^1}(0,\infty ),\delta > 0

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper G Subscript delta Baseline f right-parenthesis left-parenthesis t right-parenthesis equals delta Superscript negative 1 Baseline integral Subscript t Superscript normal infinity Baseline e Superscript left-parenthesis t minus s right-parenthesis slash delta Baseline f left-parenthesis s right-parenthesis d s">

(

G
δ

f
)
(
t
)
=

δ

−
1

∫
t
∞

e

(
t
−
s
)

/

δ

f
(
s
)
d
s

({G_\delta }f)(t) = {\delta ^{ - 1}}\smallint _t^\infty {e^{(t - s)/\delta }}f(s)ds

. Given a partition

P
=
{
0
=

t
0

>

t
1

>
⋯
>

t
i

>

t

i
+
1

>
⋯
}

P = \{ 0 = {t_0} > {t_1} > \cdots > {t_i} > {t_{i + 1}} > \cdots \}

of

[
0
,
∞
)

[0,\infty )

where

t
i

→
∞

{t_i} \to \infty

, we approximate f by the step function

A
P

f

{A_P}f

defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript upper P Baseline f left-parenthesis t right-parenthesis equals left-parenthesis upper G Subscript delta Sub Subscript i Subscript Baseline upper G Subscript delta Sub Subscript i minus 1 Subscript Baseline midline-horizontal-ellipsis upper G Subscript delta Sub Subscript i Subscript Baseline f right-parenthesis left-parenthesis 0 right-parenthesis for t Subscript i minus 1 Baseline less-than-or-slanted-equals t greater-than t Subscript i Baseline comma">

A
P

f
(
t
)
=
(

G

δ
i

G

δ

i
−
1

⋯

G

δ
i

f
)
(
0
)

for

t

i
−
1

⩽
t
>

t
i

,

{A_P}f(t) = ({G_{{\delta _i}}}{G_{{\delta _{i - 1}}}} \cdots {G_{{\delta _i}}}f)(0)\quad {\text {for}}\;{t_{i - 1}} \leqslant t > {t_i},

\]
where

δ
i

=

t
i

−

t

i
−
1

{\delta _i} = {t_i} - {t_{i - 1}}

. The main results concern several properties of this process, with the most important one being that

A
P

f
→
f

{A_P}f \to f

in

L
1

(
0
,
∞
)

{L^1}(0,\infty )

as

μ
(
P
)
=

sup
i

δ
i

→
0

\mu (P) = {\sup _i}{\delta _i} \to 0

. An application to difference approximations of evolution problems is sketched.
Mean boundary value problems for a class of elliptic equations in 𝐸³
Authors: McCoy Peter A.
Published: June 2010

An interior Dirichlet problem for generalized axisymmetric potentials is solved constructively by means of a Riemann series expansion determined from the arithmetic means of the boundary values. The problem is generalized to a class of axisymmetric elliptic partial differential equations in

E
3

{E^3}

. The Bergman and Gilbert integral operator method is used along with the method of ascent.
The Milnor signatures of compound knots
Authors: Kearton C.
Published: June 2010

The Milnor signatures of a classical knot are related to those of its companions.
A combinatorial formula for normal Stiefel-Whitney classes
Authors: Banchoff T. , McCrory C.
Published: June 2010

Using an ordering of the vertices of a combinatorial n-manifold K, we give an explicit description of a simplicial

mod
2

\bmod 2

cycle

c

n
−
i

(
K
)

{c_{n - i}}(K)

which represents the dual of the ith normal Stiefel-Whitney class of K.
A classical variational principle for periodic Hamiltonian trajectories
Authors: Clarke Frank H.
Published: June 2010

Using only classical theorems of the calculus of variations, the existence of periodic solutions to Hamilton’s equations on a given convex energy surface is proved.
Linear maps of 𝒞*-algebras preserving the absolute value
Authors: Gardner L. Terrell
Published: June 2010

In order that a linear map of

C

∗

{\mathcal {C}^\ast }

-algebras

ϕ
:

A

→

B

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

preserve absolute values, it is necessary and sufficient that it be 2-positive and preserve zero products of positive elements: if x and y are positive in

A

\mathcal {A}

, with

x
y
=
0

xy = 0

, then

ϕ
(
x
)
ϕ
(
y
)
=
0

\phi (x)\phi (y) = 0

. The generalized Schwarz inequalities of Kadison and Choi are extended to the nonunital case.
Quotients of 𝑐₀ are almost isometric to subspaces of 𝑐₀
Authors: Alspach Dale E.
Published: June 2010

It is shown that for every

ε
>
0

\varepsilon > 0

and quotient space X of

c
0

{c_0}

there is a subspace Y of

c
0

{c_0}

such that the Banach-Mazur distance

d
(
X
,
Y
)

d(X,Y)

is less than

1
+
ε

1 + \varepsilon

. This improves a result of Johnson and Zippin.
The kernel of the Poincaré series operator
Authors: Metzger Thomas A.
Published: June 2010

By modifying a proof of Ljan, a natural basis for the kernel of the Poincaré series operator in the Bers space can be given. The basic idea behind the proof also extends to give such a basis in the case of a general Kleinian group and a discontinuous group acting on certain domains in

C

n

{{\mathbf {C}}^n}

.
The solvability of operator equations with asymptotic quasibounded nonlinearities
Authors: Milojević P. S.
Published: June 2010

We study the solvability of operator equations involving quasibounded and asymptotically quasibounded nonlinear perturbations of linear Fredholm operators.
On the vanishing of the dual Stiefel-Whitney classes of orientable manifolds
Authors: Papastavridis Stavros
Published: June 2010

We will determine for which values of n and k the dual

(
n
−
k
)

(n - k)

-Stiefel-Whitney class of all orientable n-manifolds vanishes.
On the number of automorphisms of a regular graph
Authors: Wormald Nicholas
Published: June 2010

For any connected cubic graph G with 2n points, the number of automorphisms of G divides

3
n

2
n

3n{2^n}

. This is a special case of a result which is proved for connected regular graphs in general. The result is shown to be best possible for infinitely many n in the cubic case.
A negative answer to the prime sequence question
Authors: Heitmann Raymond C.
Published: June 2010

If P is a complete intersection prime, i.e., a prime ideal generated by

h
t
(
P
)

ht(P)

elements, in a Noetherian domain, can P be generated by a prime sequence, a regular sequence whose initial segments generate prime ideals? The purpose of this article is to present an example showing that this question, the Prime Sequence Question, has a negative answer. The example is a two-dimensional integrally closed domain with a height two complete intersection prime which contains no prime elements.
Self-injective rings
Authors: Faith Carl
Published: June 2010

In 1958 Matlis proved that the study of Noetherian complete local rings could be subsumed under the study of injective modules E over a commutative ring A such that

B
=

E
n
d

A

E

B = \mathrm {End}_A E

is commutative. In this case

B
=

End
B

E

B = \text {End}_B E

, and E is said to be strongly balanced over B. The main theorem of this paper shows that the study of strongly balanced injectives over any ring, and hence the study of Monta self-dualities, is contained in the study of self-injective rings.
Hopfian Boolean algebras of power less than or equal to continuum
Authors: Loats James T.
Published: June 2010

An infinite Boolean algebra (BA) is said to be hopfian if every onto endomorphism is one-to-one. It is shown that there are no countably infinite hopfian BA’s. An atomic hopfian BA of power continuum is constructed.
Compact multipliers on Banach algebras
Authors: Friedberg Stephen H.
Published: June 2010

An elementary proof is provided to show that for a large class of Banach algebras, the compact multipliers are trivial.
Topologically equivalent measures in the Cantor space
Authors: Navarro-Bermúdez Francisco J.
Published: June 2010

The Cantor space is realized as a countable product X of two-element sets. The measures

μ
\mu

and

ν
\nu

in X are topologically equivalent if there is a homeomorphism h of X onto itself such that

μ
=
ν
h

\mu = \nu h

. Let

F

\mathcal {F}

be the family of product measures in X which are shift invariant. The members

μ
(
r
)

\mu (r)

of

F

\mathcal {F}

are in one-to-one correspondence with the real numbers r in the unit interval. The relation of topological equivalence partitions the family

F

\mathcal {F}

into classes with at most countably many measures each. A class contains only the measures

μ
(
r
)

\mu (r)

and

μ
(
1
−
r
)

\mu (1 - r)

when r is a rational or a transcendental number. Equivalently, if r is rational or transcendental and

μ
(
s
)

\mu (s)

is topologically equivalent to

μ
(
r
)

\mu (r)

then

s
=
r

s = r

or

s
=
1
−
r

s = 1 - r

.
Amalgamation and elimination of quantifiers for theories of fields
Authors: Wheeler William H.
Published: June 2010

The universal theories of integral domains and of ordered integral domains which have the amalgamation property are characterized via their existentially complete models. The results of A. Macintyre, K. McKenna, and L. van den Dries on fields and ordered fields whose complete theories permit elimination of quantifiers are then derived as easy corollaries.
On remote points in 𝜐𝑋-𝑋
Authors: Terada Toshiji
Published: June 2010

Under a certain set-theoretic assumption, a question of E. K. van Douwen is solved. More precisely, if the cellularity

c
(
X
)

c(X)

of a space X is nonmeasurable, then

υ
X
−
X

\upsilon X - X

contains no remote point of X.
Contractification of a semigroup of maps
Authors: Ko Hwei Mei , Tan Kok Keong
Published: June 2010

Let

(
X
,
τ
)

(X,\tau )

be a metrizable topological space,

P

(
τ
)

\mathcal {P}(\tau )

be the family of all metrics on X whose metric topologies are

τ
\tau

. Assume that the semigroup F of maps from X into itself, with composition as its semigroup operation, is equicontinuous under some

d
∈

P

(
τ
)

d \in \mathcal {P}(\tau )

; then we have the following results: I. There exists

d
′

∈

P

(
τ
)

d’ \in \mathcal {P}(\tau )

such that f is nonexpansive under

d
′

d’

for each

f
∈
F

f \in F

. II. If F is countable, commutative, and for each

f
∈
F

f \in F

, there is

x
f

∈
X

{x_f} \in X

such that the sequence

(

f
n

(
x
)

)

n
=
1

∞

({f^n}(x))_{n = 1}^\infty

converges to

x
f

,
∀
x
∈
X

{x_f},\forall x \in X

, then there exists

d
∈

P

(
τ
)

d \in \mathcal {P}(\tau )

such that f is contractive under

d
d

for each

f
∈
F

f \in F

. III. If there is

p
∈
X

p \in X

such that (1)

lim

n
→
∞

f
n

(
x
)
=
p
,
∀
x
∈
X

{\lim _{n \to \infty }}{f^n}(x) = p,\forall x \in X

and

∀
f
∈
F

\forall f \in F

, (2) there is a neighbourhood B of p such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript m right-arrow normal infinity Endscripts f Subscript n 1 Baseline f Subscript n 2 Baseline midline-horizontal-ellipsis f Subscript n Sub Subscript m Baseline left-parenthesis upper B right-parenthesis equals StartSet p EndSet">

lim

m
→
∞

f

n
1

f

n
2

⋯

f

n
m

(
B
)
=
{
p
}

{\lim _{m \to \infty }}{f_{{n_1}}}{f_{{n_2}}} \cdots {f_{{n_m}}}(B) = \{ p\}

for any choice of

f

n
i

∈
F
,
i
=
1
,
…
,
m

{f_{{n_i}}} \in F,i = 1, \ldots ,m

, and the limit depends on m only, then for each

λ
\lambda

with

0
>
λ
>
1

0 > \lambda > 1

, there exists

d
′

∈

P

(
τ
)

d’ \in \mathcal {P}(\tau )

such that each f in F is a Banach contraction under

d
′

d’

with Lipschitz constant

λ
\lambda

.
Nonperfect spaces with point-countable bases
Authors: Davies Peter
Published: June 2010

We construct a completely regular space of cardinality

ℵ
1

{\aleph _1}

with a point-countable base, which is not perfect. This answers a question of Fleissner and Reed. We also construct, under the hypothesis

2

ℵ
0

>

2

ℵ
1

{2^{{\aleph _0}}} > {2^{{\aleph _1}}}

, a hereditarily normal space of cardinality

ℵ
1

{\aleph _1}

with a

σ
\sigma

-disjoint base, which is not perfect.
Hyperspaces of cones and fans
Authors: Eberhart Carl , Nadler Sam B.
Published: June 2010

In this paper we investigate the structure of the hyperspaces of subcontinua of some nonlocally connected continua. It is shown that if X is either the cone over any infinite compact metric space or a fan with an infinite number of endpoints, then the space of subcontinua containing the vertex of X is homeomorphic to the Hilbert cube. The hyperspace of subcontinua of a smooth fan (i.e., a subcontinuum of the cone over the Cantor set) is completely described. Also we discuss the question of when two “nicely embedded” copies of the Hilbert cube in a hyperspace have a Hilbert cube sum. In connection with this we describe two spaces, one decomposable and one indecomposable, whose hyperspaces of continua are homeomorphic.
Examples of higher-dimensional slice knots which are not ribbon knots
Authors: Hitt L. R.
Published: June 2010

A definition for ribbon n-knot is given which reduces to the previously studied definitions for

n
=
1

n = 1

and

n
=
2

n = 2

. It is shown that for each

n
⩾
2

n \geqslant 2

there is a slice n-knot which is not a ribbon n-knot.
Sign compatible expressions for minors of the matrix 𝐼-𝐴
Authors: Hartfiel D. J.
Published: June 2010

Let

A
=
(

a

i
j

)

A = ({a_{ij}})

be an

n
×
n

n \times n

nonnegative matrix having row sums less than or equal to one. This paper shows that the ijth minor of

I
−
A

I - A

can be expressed as
\[

(
−
1

)

i
+
j

∑

Π

r
k

a

p
q

{( - 1)^{i + j}}\sum {\Pi {r_k}{a_{pq}}}

\]
where
\[

r
k

=
1
−

∑

s
=
1

n

a

k
s

{r_k} = 1 - \sum \limits _{s = 1}^n {{a_{ks}}}

\]
and each

Π

r
k

a

p
q

\Pi {r_k}{a_{pq}}

is a product of exactly

n
−
1

n - 1

numbers taken from

r
k

,

a

p
q

{r_k},{a_{pq}}

for

k
,
p
,
q
=
1
,
…
,
n

k,p,q = 1, \ldots ,n

. This theorem is then used to obtain perturbation results concerning the matrix

I
−
A

I - A

.
A uniqueness theorem for a boundary value problem
Authors: Usmani Riaz A.
Published: June 2010

In this paper it is proved that the two-point boundary value problem, namely
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d Superscript left-parenthesis 4 right-parenthesis Baseline slash d x Superscript 4 Baseline plus f right-parenthesis y equals g comma y left-parenthesis 0 right-parenthesis minus upper A 1 equals y left-parenthesis 1 right-parenthesis minus upper A 2 equals y left-parenthesis 0 right-parenthesis minus upper B 1 equals y left-parenthesis 1 right-parenthesis minus upper B 2 equals 0">

(

d

(
4
)

/

d

x
4

+
f
)
y
=
g
,
y
(
0
)
−

A
1

=
y
(
1
)
−

A
2

=
y
(
0
)
−

B
1

=
y
(
1
)
−

B
2

=
0

({d^{(4)}}/d{x^4} + f)y = g,y(0) - {A_1} = y(1) - {A_2} = y(0) - {B_1} = y(1) - {B_2} = 0

, has a unique solution provided

inf
x

f
(
x
)
=
−
η
>
−

π
4

{\inf _x}f(x) = - \eta > - {\pi ^4}

. The given boundary value problem is discretized by a finite difference scheme. This numerical approximation is proved to be a second order convergent process by establishing an error bound using the

L
2

{L_2}

-norm of a vector.
Integrals of certain univalent functions
Authors: Singh Ram , Singh Sunder
Published: June 2010

In this paper we study integrals of certain univalent functions in the unit disc

E
=
{
z
:

|

z

|

>
1
}

E = \{ z:|z| > 1\}

and extend some well-known results of Libera.
A note on invariant subspaces for finite maximal subdiagonal algebras
Authors: Saito Kichi-Suke
Published: June 2010

Let M be a von Neumann algebra with a faithful, normal, tracial state

τ
\tau

and

H
∞

{H^\infty }

be a finite, maximal, subdiagonal algebra of M. Every left- (or right-) invariant subspace with respect to

H
∞

{H^\infty }

in the noncommutative Lebesgue space

L
p

(
M
,
τ
)
,
1
⩽
p
>
∞

{L^p}(M,\tau ),1 \leqslant p > \infty

, is the closure of the space of bounded elements it contains.
A note on the asymptotic behaviour of the sum of principal radii of curvature on noncompact complete hypersurfaces
Authors: Oliker V. I.
Published: June 2010

It is shown that under certain hypotheses the following conjecture is correct: on a noncompact complete hypersurface in Euclidean space the two conditions below cannot hold simultaneously: (i) the sum of principal radii of curvature is bounded; (ii) the support function is uniformly continuous.
Gaussian measure of large balls in a Hilbert space
Authors: Hwang Chii-Ruey
Published: June 2010

Let P be a zero mean Gaussian measure in a Hilbert space. The asymptotic behavior of

P
{

‖

x
−
b

‖

2

>
ε
}

P\{ {\left \| {x - b} \right \|^2} > \varepsilon \}

as

ε
→
∞

\varepsilon \to \infty

is studied in this note.
On nowhere dense closed 𝑃-sets
Authors: Kunen Kenneth , van Mill Jan , Mills Charles F.
Published: June 2010

We show that no compact space of weight

ω
1

{\omega _1}

can be covered by nowhere dense closed P-sets. In addition, we construct a compact space of weight

ω
2

{\omega _2}

which can be covered by nowhere dense closed P-sets. As an application, we show that CH is equivalent to the statement that each small nonpseudocompact space has a remote point.
The Smith invariants of a matrix sum
Authors: Thompson Robert C.
Published: June 2010

A divisibility relation is proved relating the invariant factors of a sum of integral matrices.
On finite presentations for groups
Authors: Craggs R.
Published: June 2010

Three theorems are proved here giving canonical forms for finitely-presented groups. The first characterizes groups with balanced presentations as those groups with presentations of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle y 1 comma ellipsis comma y Subscript n Baseline vertical-bar y Subscript i Baseline equals alpha left-parenthesis y Subscript i Baseline right-parenthesis comma i equals 1 comma ellipsis comma n mathematical right-angle">

⟨

y
1

,
…
,

y
n

|

y
i

=
α
(

y
i

)
,
i
=
1
,
…
,
n
⟩

\langle {y_1}, \ldots ,{y_n}|{y_i} = \alpha ({y_i}),i = 1, \ldots ,n\rangle

\]
where

α
\alpha

is some automorphism of the free group with free basis

y
1

,
…
,

y
n

{y_1}, \ldots ,{y_n}

. Similar canonical forms, with two automorphisms rather than one, are given for arbitrary finitely-presented groups.
On 𝑝-torsion in etale cohomology and in the Brauer group
Authors: Treger Robert
Published: June 2010

If X is an affine scheme in characteristic

p
>
0

p > 0

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Br left-parenthesis upper X right-parenthesis left-parenthesis p right-parenthesis ModifyingAbove right-arrow With tilde upper H Subscript et Superscript 2 Baseline left-parenthesis upper X comma bold upper G Subscript m Baseline right-parenthesis left-parenthesis p right-parenthesis">

Br

(
X
)
(
p
)

→
~

H

et

2

(
X
,

G

m

)
(
p
)

{\text {Br}}(X)(p)\tilde \to H_{{\text {et}}}^2(X,{{\mathbf {G}}_m})(p)

and

H

et

n

(
X
,

G

m

)
(
p
)
=
0

H_{{\text {et}}}^n(X,{{\mathbf {G}}_m})(p) = 0

for

n
⩾
3

n \geqslant 3

. This gives a partial answer to the conjecture that the Brauer group of any scheme X is canonically isomorphic to the torsion part of

H

et

2

(
X
,

G

m

)

H_{{\text {et}}}^2(X,{{\mathbf {G}}_m})

. This result is then applied to prove that

Br

(
R
)
(
p
)

{\text {Br}}(R)(p)

is p-divisible where R is a commutative ring of characteristic

p
>
0

p > 0

(theorem of Knus, Ojanguren and Saltman), and also to construct examples of domains R of characteristic

p
>
0

p > 0

with large

Ker

(

Br

(
R
)
(
p
)
→

Br

(
Q
)
(
p
)
)

{\operatorname {Ker}}({\text {Br}}(R)(p) \to {\text {Br}}(Q)(p))

, where Q is the ring of fractions of R.
Equality of essential spectra of certain quasisimilar seminormal operators
Authors: Williams L. R.
Published: June 2010

Let A and B be quasisimilar seminormal operators on a separable, infinite dimensional complex Hilbert space. Several conditions which imply that A and B have equal essential spectra are presented. For example, if A and B are both biquasitriangular then A and B have equal essential spectra.
Hausdorff matrices as bounded operators over 𝑙
Authors: Rhoades B. E.
Published: June 2010

A necessary and sufficient condition is obtained for an arbitrary Hausdorff matrix to belong to

B
(
l
)

B(l)

. It is then shown that every conservative quasi-Hausdorff matrix is of type M.
A Volterra equation with square integrable solution
Authors: Staffans Olof J.
Published: June 2010

We study the asymptotic behavior of the solutions of the nonlinear Volterra integrodifferential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript t Baseline a left-parenthesis t minus s right-parenthesis g left-parenthesis x left-parenthesis s right-parenthesis right-parenthesis d s equals f left-parenthesis t right-parenthesis left-parenthesis t element-of upper R Superscript plus Baseline right-parenthesis period">

x
′

(
t
)
+

∫
0
t

a
(
t
−
s
)
g
(
x
(
s
)
)

d
s

=
f
(
t
)

(
t
∈

R
+

)
.

x’(t) + \int _0^t {a(t - s)g(x(s))\;ds\; = f(t)\quad (t \in {R^ + }).}

\]
Here

R
+

=
[
0
,
∞
)
,
a
,
g

{R^ + } = [0,\infty ),a,g

and f are given real functions, and x is the unknown solution. In particular, we give sufficient conditions which imply that x and x’ are square integrable.
A theorem of C. Ryll-Nardzewski and metrizable l.c.a. groups
Authors: Ramsey L. Thomas
Published: June 2010

Γ
\Gamma

denotes a metrizable locally compact abelian group and

Γ
¯

\bar \Gamma

its Bohr compactification. Let

γ
∈
Γ

\gamma \in \Gamma

be a cluster point of some subset E of

Γ
\Gamma

in the topology of

Γ
¯

\bar \Gamma

. Then there are two disjoint subsets of E which also cluster at

γ
\gamma

in the Bohr group topology. The proof is elementary and provides a new proof of the theorem of C. Ryll-Nardzewski on cluster points of I-sets in R. Given the continuum hypothesis, either theorem characterizes metrizability in locally compact abelian groups. One of these characterizations is shown to be equivalent to the continuum hypothesis.
A decomposition of a measure space with respect to a multiplication operator
Authors: Howland James S.
Published: June 2010

Let A be a bounded multiplication operator on

L
2

(
Ω
,
m
)

{L_2}(\Omega ,m)

, where

Ω
\Omega

is a complete separable metric space and m a Borel measure. A set of measure zero can be removed from

Ω
\Omega

so that the multiplicity function of A is equal to the cardinality of the preimage. In the proof,

Ω
\Omega

is decomposed into subsets of simple multiplicity.
On the dimension of injective Banach spaces
Authors: Argyros S.
Published: June 2010

The purpose of this note is to give an affirmative answer, assuming the generalized continuum hypothesis, to a problem of H. Rosenthal on the cardinality of the dimension on injective Banach spaces.
Homotopy and uniform homotopy. II
Authors: Calder Allan , Siegel Jerrold
Published: June 2010

An elementary proof of the Bounded Lifting Lemma is given, together with a proof that homotopy and uniform homotopy do not agree for maps into compact spaces with infinite fundamental groups even though they can agree for maps into a noncompact space with infinite fundamental group.
Abelian extensions of regular local rings
Authors: Roberts Paul
Published: June 2010

The integral closure of a regular local ring in a finite Abelian extension of its quotient field is Cohen-Macaulay, provided that the degree of the extension is not divisible by the characteristic of the residue field.