bluuebunny/crossref_metadata_2025 huggingface dataset:
Some semigroups on a manifold with boundary
Authors: Hanson T. H. McH.
Published: June 2010

In this paper,

S
S

is an abelian semigroup on an

n

{\text {n}}

-dimensional simply connected manifold with boundary whose interior is a dense, simply connected, connected Lie group. We also assume there is a vector semigroup

V
k
−

V_k^ -

in

S
S

such that the interior of

S
S

misses the boundary of

V
k
−

V_k^ -

, and such that

(
S
−
G

L
k

)

/

V
k

(S - G{L_k})/{V_k}

is a group. It is shown that if

k
=
n

k = n

, then

S
S

is iseomorphic to

V
n
−

V_n^ -

, and if

k
=
1
,
2

k = 1,2

, or

n
−
1

n - 1

, then

S
S

is iseomorphic to

V

n
−
k

×

V
k
−

{V_{n - k}} \times V_k^ -

.
An example on embedding up to homotopy type
Authors: Rees Elmer
Published: June 2010

A finite complex

K
K

is constructed with the following property.

K
∨

S
r

K \vee {S^r}

embeds in

R
n

{R^n}

up to homotopy type but

K
K

does not.
A geometric proof of Markov ergodic theorem
Authors: Yeh R. Z.
Published: June 2010

A geometric approach combined with topological results leads to a criterion for ergodic stability of Markov transformations. The matrix representation of this criterion provides an alternative proof for the well-known theorem of Markov in probability.
On the 1-1 sum of two Borel sets
Authors: Darst Richard B.
Published: June 2010

It is shown that there exists a Borel subset

B
B

of the real line and a homeomorphism

ϕ
\phi

of the real line such that the set

{
x
−
ϕ
(
x
)
;
x
∈
B
}

\{ x - \phi (x);x \in B\}

is not a Borel set.
Amenable subsemigroups of a locally compact group
Authors: Jenkins Joe W.
Published: June 2010

Let

G
G

be an amenable locally compact group and

S
S

an open subsemigroup of

G
G

. It is shown that

S
S

is amenable if, and only if, the right ideals of

S
S

have the finite intersection property.
A generalization of a theorem of P. Montel on entire functions
Authors: Yang Chung-chun
Published: June 2010

This paper investigates the functional equation

a
(
z
)

f
n

(
z
)
+
b
(
z
)

g
m

(
z
)
=
1

a(z){f^n}(z) + b(z){g^m}(z) = 1

(

a
,
b
,
f
,
g

a,b,f,g

meromorphic).
Oscillatory properties of linear third-order differential equations.
Authors: Kim W. J.
Published: June 2010

Separation theorems, distribution of zeros of solutions, and disconjugacy criteria for linear third-order differential equations are discussed. For instance, it is proved that the equation

y
′

+
p
y
+
q

y
′

+
r
y
=
0

y’ + py + qy’ + ry = 0

, where

p
∈
C
,
q
∈

C
′

p \in C,q \in C’

, and

r
∈
C

r \in C

on an interval

I
I

, is disconjugate on

I
I

if

p
p

does not change sign and if

q
≦
0
,
r
≧
0
,
q
−
2

p
′

≦
0

q \leqq 0,r \geqq 0,q - 2p’ \leqq 0

, and

r
−

q
′

+
p
≦
0

r - q’ + p \leqq 0

on

I
I

.
Regularization of singular systems of integral equations with kernels of finite double-norm on 𝐿_{∞}
Authors: Miranda Guillermo
Published: June 2010

There are known examples of linear integral transformations

T
T

of finite double-norm on

L
∞

{L_\infty }

such that neither the transformation nor any of its iterates is compact, so that Fredholm’s alternative does not hold unrestrictedly for the equation

(
I
−
λ
T
)
g
=
f

(I - \lambda T)g = f

(

λ
\lambda

a complex number,

g
,
f
∈

L
∞

g,f \in {L_\infty }

. It is also known that the alternative holds true for

|

λ

|

|\lambda |

less than the Fredholm radius of

T
T

. Using a kernel decomposition, a quantity

ω
\omega

is introduced and the equivalence of an integral transformation system with components of finite double-norm on

L
∞

{L_\infty }

, to a similar system that satisfies the Fredholm alternative for

|

λ

|

>
ω

|\lambda | > \omega

is proved. In contrast to the Fredholm radius, an easy computation for

ω
\omega

is available.
An example concerning core measures
Authors: Sidney S. J.
Published: June 2010

We shall construct a uniform algebra having a multiplicative linear functional whose space of representing measures is infinite-dimensional but nevertheless has a core measure. This answers in the negative an open question in Gamelin’s book [1, p. 106].
On a maximal ideal space separated by a peak point
Authors: Sommese Joseph E.
Published: June 2010

The purpose of this note is to answer in the negative the following question raised by Gamelin [l]: if

A
A

is a function algebra which has the property that

X
X

, the spectrum of

A
A

, is expressible as the union of two compact sets

X
1

{X_1}

and

X
2

{X_2}

which have as their intersection a peak point

p
p

of

X
X

, and if

f
∈
(
C
(
X
)

f \in (C(X)

satisfies

f

|

x
1

∈
A

|

x
1

f|{x_1} \in A|{x_1}

and

f

|

x
2

∈
A

|

x
2

f|{x_2} \in A|{x_2}

, then is

f
∈
A

f \in A

? The counterexample is obtained by the use of a construction which is applicable to general function algebras. Let

A
A

be a function algebra and

I
I

a proper closed ideal, denoting by

A
[
I
]

A[I]

the set

{
(
f
,
f
+
s
)
:
f
∈
A
,
s
∈
I
}

\{ (f,f + s):f \in A,s \in I\}

, it is shown that

A
[
I
]

A[I]

is a function algebra which has as its spectrum two copies of the spectrum of

A
A

identified along hull (

(
I
)

(I)

).
Finite operators
Authors: Williams J. P.
Published: June 2010

A bounded linear operator

A
A

on a Hilbert space

H
H

is called finite if

|

|

A
X
−
X
A
−
1

|

|

≧
1

||AX - XA - 1|| \geqq 1

for each

X
∈
B
(
H
)

X \in B(H)

. The class of finite operators is uniformly closed, contains every normal operator, every operator with a compact direct summand, and the entire

C
∗

{C^ \ast }

-algebra generated by each of its members. These results imply that the set of operators with a finite dimensional reducing subspace is not uniformly dense. It is also shown that the set of self-commutators is uniformly closed.
Handlebodies in 3-manifolds
Authors: Smythe N.
Published: June 2010

Consider a knot, link or wedge of circles which bounds a singular surface of genus

g
≧
0

g \geqq 0

within an orientable

3
3

-manifold. Then there is a handlebody in the manifold within which the

1
1

-complex bounds a singular surface of genus

g
g

. In particular, a handlebody which is contractible in an orientable

3
3

-manifold is contractible within another handlebody.
Minimal covers and arithmetical sets
Authors: Jockusch Carl G. , Soare Robert I.
Published: June 2010

If

a
a

and

b
b

are degrees of unsolvability,

a
a

is called a minimal cover of

b
b

if

b
>
a

b > a

and no degree

c
c

satisfies

b
>
c
>
a

b > c > a

. The degree

a
a

is called a minimal cover if it is a minimal cover of some degree

b
b

. We prove by a very simple argument that

0
n

{0^n}

is not a minimal cover for any

n
n

. From this result and the axiom of Borel determinateness (BD) we show that the degrees of arithmetical sets (with their usual ordering) are not elementarily equivalent to all the degrees. We also point out how this latter result can be proved without BD when the jump operation is added to the structures involved.
Graphs with a large capacity
Authors: Rosenfeld M.
Published: June 2010

A constructive method for obtaining graphs with a relatively large capacity is given. The method uses products of graphs.
Congruence relations in direct products
Authors: Fraser Grant A. , Horn Alfred
Published: June 2010

This paper studies conditions under which every congruence relation

θ
\theta

in a direct product

A
×
B

A \times B

of algebras is of the form

θ
1

×

θ
2

{\theta _1} \times {\theta _2}

, where

θ
1

{\theta _1}

and

θ
2

{\theta _2}

are congruence relations in

A
A

and

B
B

respectively. It is shown that for any equational class

K
K

, every such

θ
\theta

in every

A
×
B

A \times B

in

K
K

has this property if and only if

K
K

satisfies certain identities.
Subcartesian products of finitely many finite algebras
Authors: Shafaat Ahmad
Published: June 2010

We show that if the lattice of subquasivarieties of a quasivariety of

Ω
\Omega

-algebras has a finite maximal chain then every algebra of the quasivariety is a subcartesian product of a family of certain finitely many finite algebras.
On the multiplicity of an integral extension of a local ring
Authors: Whitman David G.
Published: June 2010

The following theorem is proved: If

R
R

is a local domain with field of quotients

F
F

and

S
S

is a local integral extension of

R
R

contained in

F
F

, then the multiplicity of

R
R

is greater than or equal to the multiplicity of

S
S

.
Complete ideals and monoidal transforms
Authors: Mattuck Arthur
Published: June 2010

It is proved that the monoidal transform of an integral noetherian scheme with respect to a sheaf

I
I

of ideals is normal if and only if high powers of

I
I

are complete. The analogous theorem for linear systems is included, and as an application, it is proved that a rational singularity is absolutely isolated.
Semisimple classes and upper-type radical classes of narings
Authors: Jenkins Terry L. , Kreiling Daryl
Published: June 2010

In the universal class of associative rings every class

M
M

of rings determines a unique upper radical class. It is shown that the same result is valid in the universal class of alternative rings but is not necessarily true in the universal class of narings. The latter class does however determine a type of upper radical.
A special class of Moufang loops
Authors: Orlik-Pflugfelder Hala
Published: June 2010

Loops satisfying the identical relation

(
x
y
)
(
z
x
)
=
[
x
(
y
z
)
]
x

(xy)(zx) = [x(yz)]x

are known as Moufang loops. In the present paper considered is a class of loops satisfying the identical relation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 right-parenthesis left-parenthesis x y right-parenthesis left-parenthesis z x Superscript lamda Baseline right-parenthesis equals left-bracket x left-parenthesis y z right-parenthesis right-bracket x Superscript lamda Baseline comma">

(
1
)

(
x
y
)
(
z

x
λ

)
=
[
x
(
y
z
)
]

x
λ

,

(1)\qquad (xy)(z{x^\lambda }) = [x(yz)]{x^\lambda },

\]
where

x
λ

{x^\lambda }

is the image of

x
x

under some mapping

λ
\lambda

of the loop into itself. Loops satisfying (1) are called

M
M

-loops. If

λ
:
x
→

x
k

,
k

\lambda :x \to {x^k},k

an integer, (1) is called an

M
k

{M_k}

-law. It is shown that every

M
M

-loop is Moufang, and every

x
δ

=

x

−
1

⋅

x
λ

{x^\delta } = {x^{ - 1}} \cdot {x^\lambda }

belongs to the nucleus. Furthermore, if a loop satisfies (1) so do all its loop-isotopes.
Dimensionality and the duals of certain locally compact groups
Authors: Schochetman I.
Published: June 2010

Let

G
G

be a separable locally compact group containing a closed normal subgroup which is type I and regularly embedded. The elements of the dual space

G
G

are known to be induced from certain closed subgroups. In this article we first determine the kernel of an induced representation and then give necessary and sufficient conditions for

G
G

to be maximally almost periodic. These results are then applied to the collection of compact extensions of abelian groups.
Contractibility of the automorphism group of a nonatomic measure space.
Authors: Keane Michael
Published: June 2010

It is shown that the group of automorphisms of a nonatomic finite measure space is contractible.
Periodic solutions of linear second order differential equations with deviating argument
Authors: Schmitt Klaus
Published: June 2010

This paper is concerned with the question of the existence of periodic solutions of periodic linear second order differential equations with deviating argument. Using a fixed point theorem for multivalued mappings and results concerning boundary value problems for such equations, we prove that the existence of periodic solutions of both types of differential inequalities implies the existence of periodic solutions. This result, in turn, is used to obtain the existence of periodic solutions of certain nonlinear differential equations with deviating argument.
On the ideal structure of the algebra of radial functions
Authors: Schwartz Alan
Published: June 2010

Let

L
L

denote the convolution Banach algebra of integrable functions defined on

R
n

{R^n}

and let

L
r

{L_r}

consist of the subalgebra of radial functions. If

I
I

is a closed ideal of

L
L

, the zero-set of

I
I

is defined by

Z
(
I
)
=
{
y

|

f
^

(
y
)
=
0

 for all 

f
∈
I
}

Z(I) = \{ y|\hat f(y) = 0{\text { for all }}f \in I\}

where

f
^

\hat f

is the Fourier transform of

f
f

. The following theorem is proved. If

I
1

{I_1}

and

I
2

{I_2}

are closed ideals of

L
r

{L_r}

such that

I
1

⊂

I
2

{I_1} \subset {I_2}

(

⊂
\subset

denotes proper inclusion) then there is a closed ideal

I
I

such that

I
1

⊂
I
⊂

I
2

{I_1} \subset I \subset {I_2}

.
Open mappings and the lack of full completeness of 𝒟’(Ω)
Authors: Harvey Charles , Harvey F. Reese
Published: June 2010

Consider a linear map

T
T

of one locally convex linear space into another which is densely defined and has a closed graph. We characterise the property that

T
T

is an open map in terms of two properties of its adjoint map

T

∗

{T^{\ast }}

. These results are used to show that if

Ω
\Omega

is an open subset of

R
n

{R_n}

for which there is a linear constant coefficient differential operator

P
P

such that

Ω
\Omega

is

P
P

-convex but not strongly

P
P

-convex, then (i)

D

′

(
Ω
)

\mathcal {D}’(\Omega )

is not fully complete, (ii) the range of the adjoint map

t

P

^tP

is closed but not bornological.
Recognizing certain factors of 𝐸⁴
Authors: Rubin Leonard R.
Published: June 2010

It has been proved that for certain peculiar decomposition spaces

Y
Y

of euclidean

3
3

-space

E
3

{E^3}

is homeomorphic to euclidean

4
4

-space,

E
4

{E^4}

. In this paper we prove that if a decomposition space

Y
Y

of

E
3

{E^3}

is generated by a trivial defining sequence whose elements are cubes with handles, and this sequence can be replaced by a toroidal defining sequence, then

Y
×

E
1

Y \times {E^1}

is homeomorphic to

E
4

{E^4}

.
A characterization of total graphs
Authors: Behzad M.
Published: June 2010

We consider “ordinary” graphs; that is, finite undirected graphs with no loops or multiple edges. The total graph

T
(
G
)

T(G)

of a graph

G
G

is that graph whose vertex set is

V
(
G
)
∪
E
(
G
)

V(G) \cup E(G)

and in which two vertices are adjacent if and only if they are adjacent or incident in

G
G

. A characterization of regular total graphs as well as some other properties of total graphs have been considered before. In this article we consider nonregular graphs and yield a method which enables us actually to determine whether or not they are total.
A combinatorial problem and congruences for the Rayleigh function
Authors: Howard F. T.
Published: June 2010

Let

z
z

be a positive integer and let

m
m

be the number of nonzero terms in the base 2 expansion of

z
z

. Define

f
(
z
,
s
)

f(z,s)

as the number of positive integers

r
≦
z

/

2

r \leqq z/2

such that the number of nonzero terms in the base 2 expansion of

r
r

plus the number of nonzero terms in the base 2 expansion of

z
−
r

z - r

is equal to

m
+
s

m + s

. We find formulas for

f
(
z
,
s
)

f(z,s)

and show how these formulas can be used in proving congruences for the Rayleigh function.
The cohomology ring of a finite group scheme
Authors: Efroymson Gustave
Published: June 2010

Let

k
k

be a field and let

A
A

be a

k
k

-algebra with additional structure so that Spec

A
A

is a finite commutative group scheme over

k
k

, (so

A
A

is a Hopf algebra). Let

H
∙

(
A
,
k
)

H^\bullet (A,k)

be the Hochschild cohomology ring. In another paper, we demonstrated that if

k
k

is a perfect field: (a)

H
∙

(
A
,
k
)

H^\bullet (A,k)

is generated by

H
1

{H^1}

and

H

sym

2

H_{\operatorname {sym} }^2

. (b) If characteristic

k
=
p
≠
2

k = p \ne 2

, then

H
∙

(
A
,
k
)

H^\bullet (A,k)

is freely generated by

H
1

{H^1}

and

H

sym

2

H_{\operatorname {sym} }^2

. (c) If characteristic

k
=
2

k = 2

, then there are subspaces

V
1

,

V
2

{V_1},{V_2}

of

H
1

{H^1}

and

V
3

{V_3}

of

H

sym

2

H_{\operatorname {sym} }^2

such that

H
∙

(
A
,
k
)

H^\bullet (A,k)

is generated by

V
1

,

V
2

,

V
3

{V_1},{V_2},{V_3}

and the only relations are

f
2

=
0

{f^2} = 0

for all

f
f

in

V
1

{V_1}

. In this paper we show that if

k
k

is arbitrary (a) and (b) still hold, and we use an example of Oort and Mumford to show that (c) does not hold for arbitrary

k
k

.
Products of Abelian subgroups
Authors: Amberg Bernhard , Scott W. R.
Published: June 2010

This paper deals with several problems concerning groups

G
=
A
B

G = AB

which are the product of two Abelian subgroups

A
A

and

B
B

and gives partial solutions under suitable finiteness conditions.
On a system of integral inequalities
Authors: Deo S. G. , Murdeshwar M. G.
Published: June 2010

The present note obtains a vector extension and a further generalization of Bihari’s Lemma on an integral inequality. The inequality proved can be used in the study of the componentwise behaviour of solutions of differential systems.
Trace-class for an arbitrary 𝐻*-algebra
Authors: Saworotnow Parfeny P. , Friedell John C.
Published: June 2010

Let

A
A

be a proper

H
∗

{H^ \ast }

-algebra and let

τ
(
A
)

\tau (A)

be the set of all products

x
y

xy

of members

x
,
y

x,y

of

A
A

. Then

τ
(
A
)

\tau (A)

is a normed algebra with respect to some norm

τ
(

)

\tau (\;)

which is related to the norm

|

|

|

|

||\;||

of

A
A

by the equality:

|

|

a

|

|

2

=
τ
(

a
∗

a
)
,
a
∈
A

||a|{|^2} = \tau (a^ \ast a),a \in A

. There is a trace tr defined on

τ
(
A
)

\tau (A)

such that

tr
⁡
(
a
)
=

∑
α

(
a

e
α

,

e
α

)

\operatorname {tr} (a) = \sum \nolimits _\alpha {(a{e_\alpha },{e_\alpha })}

for each

a
∈
τ
(
A
)

a \in \tau (A)

and each maximal family

{

e
α

}

\{ {e_\alpha }\}

of mutually orthogonal projections in

A
A

. The trace is related to the scalar product of

A
A

by the equality:

tr
⁡
(
x
y
)
=
(
x
,

y
∗

)
=
(
y
,

x
∗

)

\operatorname {tr} (xy) = (x,{y^ \ast }) = (y,{x^ \ast })

for all

x
,
y
∈
A

x,y \in A

.
Imbedding free cyclic group actions in circle group actions
Authors: Tollefson Jeffrey L.
Published: June 2010

Suppose a closed, orientable, irreducible

3
3

-manifold

M
M

admits a free cyclic group action of prime order. We consider the problem of determining when

M
M

admits an effective action of the circle group

S
O
(
2
)

SO(2)

in which the cyclic action is imbedded. The main result is that if the

Z
k

{Z_k}

action is “

Z
Z

-classified", then it is weakly equivalent to a

Z
k

{Z_k}

action imbedded in an effective action of

S
O
(
2
)

SO(2)

if and only if some homeomorphism generating the first

Z
k

{Z_k}

action is homotopic to the identity homeomorphism on

M
M

.
Span of lens spaces
Authors: Iwata Kôichi
Published: June 2010

We prove that the span (i.e., maximal number of linearly independent vector fields) of

(
2
n
+
1
)

(2n + 1)

-dimensional lens spaces

L
n

(
p
)

{L^n}(p)

, where

p
p

is an odd prime and where

n
+
1
=
m
⋅

2
t

n + 1 = m \cdot {2^t}

(

m
m

odd), is equal to the span of

(
2
n
+
1
)

(2n + 1)

-dimensional sphere if

t
+
1
≡
0
,
1
,
2

(
mod

4
)

t + 1 \equiv 0,1,2\pmod 4

or if

n
>
3
,
t
+
1
≡
3

(
mod

4
)

n > 3,t + 1 \equiv 3\pmod 4

, and

p
≧
t
+
3

p \geqq t + 3

. This result is an improvement of a theorem given by T. Yoshida.
The solvability of the word problem for certain semigroups
Authors: Yasuhara Ann
Published: June 2010

This paper establishes the solvability of the word problem for semigroups with one defining relation if that relation is of the form

A
∼
B
t
C

A \sim BtC

where (1)

A
A

and

B
t
C

BtC

are words on the generators of the semigroup but the generator

t
t

does not occur in

A
,
B

A,B

or

C
C

and (2) the length of

A
A

is greater than the

max

(
length
⁡
B
,
length
⁡
C
)

\max \;(\operatorname {length} B,\operatorname {length} C)

.
On existence of compact open normal subgroups of 0-dimensional groups
Authors: Lee D. H. , Wu T. S.
Published: June 2010

Let

G
G

be a locally compact

0
0

-dimensional group, and let

B
(
G
)

B(G)

denote the set of all elements of

G
G

whose conjugacy classes are relatively compact. We proved that the group

G
G

has an open compact normal subgroup if and only if

B
(
G
)

B(G)

is open in

G
G

.
Several theorems on boundedness and equicontinuity
Authors: Stein J. D.
Published: June 2010

This paper presents several results concerning equicontinuity of a pointwise-bounded family of linear transformations on a Banach space. The first is the following generalization of the Banach-Steinhaus Theorem: Let

{

T
α

|

α
∈
A
}

\{ {T_\alpha }|\alpha \in A\}

be a pointwise-bounded family of linear transformations from a Banach space

X
X

to a normed linear space

Y
Y

, and assume that, for each

α
∈
A
,

T
α

\alpha \in A,{T_\alpha }

is continuous on a closed subspace

S
α

{S_\alpha }

of

X
X

. Then

∃

α
1

,
⋯
,

α
n

∈
A

\exists {\alpha _1}, \cdots ,{\alpha _n} \in A

such that the family is equicontinuous on

⋂

k
=
1

n

S

α
k

\bigcap \nolimits _{k = 1}^n {{S_{\alpha k}}}

. The second theorem deals with a pointwise-bounded family of linear transformations from a Banach space

X
X

to a normed linear space with a continuous bilinear mapping into another normed linear space. The others deal with homomorphisms of Banach algebras.
An example in the Weil theory of measurable groups
Authors: Atalla Robert E.
Published: June 2010

According to a theorem of Andre Weil, if a group

G
G

possesses an invariant measure which satisfies certain conditions, in particular measurability of the map

(
x
,
y
)
→
(
x
,
x
y
)

(x,y) \to (x,xy)

of

G
×
G

G \times G

, then

G
G

has a locally bounded Hausdorff topology making

G
G

a topological group. We offer a simple counterexample to show the need for the above stated condition.
On the multiplicative behavior of regular matrices
Authors: Atalla Robert E.
Published: June 2010

Let

T
T

be a bounded linear operator on

C
(
X
)
,
X

C(X),X

compact

T
2

{T_2}

, with

T
1
=
1

T1 = 1

. We define

M
T

{M_T}

to be the subalgebra of

C
(
X
)

C(X)

consisting of

g
g

such that

T
(
f
g
)
=
T
f
T
g

T(fg) = TfTg

for all

f
f

, and give a characterization of

M
T

{M_T}

. We apply the characterization to the multiplicative behavior of regular matrices, considering these as linear operators on

C
(
β
N
∖
N
)

C(\beta N\backslash N)

. We also relate invariance properties of a matrix under suitable mappings of the integers to topological properties of its support set in

β
N
∖
N

\beta N\backslash N

, and give an example of a nonnegative multiplicative matrix whose support set is nowhere dense in

β
N
∖
N

\beta N\backslash N

.
The spectrum of a selfadjoint compression of a self-adjoint operator.
Authors: Hosack John M.
Published: June 2010

The relation between the spectrum of a selfadjoint operator and the spectrum of its compression is investigated. In particular, we show that the compression of a selfadjoint operator is essentially selfadjoint if and only if the spectrum of the closure of the compression is contained in the closed convex hull of the spectrum of the operator. Relations between two conceptions of compressions or projections of operators are also considered.
The permanent of a transitive relation
Authors: Sharp Henry
Published: June 2010

H. Minc has constructed an upper bound on the permanent of any relation on a finite set. In this paper the permanent of any transitive relation on a finite set is calculated. The work in part is based upon the interpretation of a reflexive, transitive relation as a finite topology. The relationship to (finite) Borel fields is discussed briefly. In an example it is shown how results here may be combined with Minc’s inequality to produce an improved upper bound on the permanent of any relation.
The solution to one of Ulam’s problems concerning analytic sets. II.
Authors: Mansfield R.
Published: June 2010

It is shown that the universal analytic set is not a

σ
\sigma

-combination of measurable rectangles.
𝐾₁ of projective 𝑟-spaces
Authors: Roberts Leslie G.
Published: June 2010

Let

A
A

be a commutative ring, and let

X
X

= projective

r
r

-space over

A
A

. Then we prove that

K
1

{K_1}

of the category of locally free sheaves of finite type on

X
X

is isomorphic to the direct sum of

r
+
1

r + 1

copies of

K
1

(
A
)

{K_1}(A)

.
Nonnegative expansions of polynomials
Authors: Wilson M. Wayne
Published: June 2010

Consider expanding a set of orthogonal polynomials

{

q
i

(
x
)
}

\{ {q_i}(x)\}

as a combination of orthogonal polynomials

{

p
i

(
x
)
}

\{ {p_i}(x)\}

. A sufficient condition to ensure that the expansion coefficients are nonnegative is derived. It is shown by counter-example that it is not a necessary condition.
A variation-of-parameters inequality
Authors: Lovelady David Lowell
Published: June 2010

Recent work in stability theory for linear ordinary differential equations has made much use of the inequalities which can be deduced from the variation-of-parameters formula. In this article it is shown that similar inequalities hold for nonlinear Stieltjes integral equations and hence, as a special case, for nonlinear differential equations in which the nonlinearity is of Lipschitz type.
On the nonexistence of slices for 𝑝-adic transformation groups
Authors: Franklin Lawrence M.
Published: June 2010

If a compact Lie group acts effectively on a manifold, then local cross-sections to the orbit structure can be constructed. They are called slices. We prove that if a compact group acts effectively on a manifold, but it is not a Lie group, then a slice cannot exist at every point.
On the invertibility of general Wiener-Hopf operators
Authors: Reeder John
Published: June 2010

Let

H

\mathfrak {H}

be a separable Hilbert space,

B

\mathfrak {B}

the set of bounded linear operators on

H

\mathfrak {H}

, and

P
P

an orthogonal projection on

H

\mathfrak {H}

. Denote the range of

P
P

by

R
(
P
)

R(P)

. Let

A
A

belong to

B

\mathfrak {B}

. The general Wiener-Hopf operator associated with

A
A

and

P
P

is defined by

T
P

(
A
)
=
P
A

|

R
(
P
)

{T_P}(A) = PA|R(P)

, the vertical bar denoting restriction. Let

Q
=
I
−
P

Q = I - P

. The purpose of this paper is to disprove the general conjecture that if

A
A

is an invertible element of

B

\mathfrak {B}

, then the invertibility of

T
P

(
A
)

{T_P}(A)

implies the invertibility of

T
Q

(
A
)

{T_Q}(A)

. We also disprove the conjecture in an interesting special case.
Inertial and bordism properties of spheres
Authors: Brender Allan
Published: June 2010

The

k
k

-connective bounding group

θ
n

(
k
)

{\theta ^n}(k)

and the

k
k

-connective inertial group

I
n

(
k
)

{I^n}(k)

are defined as subgroups of

θ
n

{\theta ^n}

, the group of smooth

n
n

-spheres,

n
≧
7

n \geqq 7

. It is shown

I
n

(
k
)

{I^n}(k)

is contained in

θ
n

(
k
)

{\theta ^n}(k)

. Consequently, the image of the Milnor-Novikov pairing

τ

n
,
k

{\tau _{n,k}}

is contained in

θ

n
+
k

(
k
)

{\theta ^{n + k}}(k)

when

n
≧
k
+
2

n \geqq k + 2

. It follows that

τ

7
,
3

=
0

{\tau _{7,3}} = 0

.
Sums and products of continued fractions
Authors: Cusick T. W.
Published: June 2010

It is proved that every real number is representable as a sum of two real numbers each of which has a fractional part whose continued fraction expansion contains no partial quotient less than 2, and that every real number not less than one is representable as a product of two real numbers with the same property.
Polynomial identity algebras with involution
Authors: Montgomery Susan
Published: June 2010

It has been shown by W. Baxter and W. S. Martindale that if

R
R

is an algebra with involution over a field

F
F

of characteristic not 2, and the symmetric elements of

R
R

are algebraic and satisfy a polynomial identity, then

R
R

is locally finite. This paper extends their result to an arbitrary field, giving a new proof which is independent of the characteristic of

F
F

.
An additive decomposition theorem for analytic functions
Authors: Myers Donald E.
Published: June 2010

Kraut, Busenberg, and Hall have shown that the uniqueness of the additive decomposition for functions of several variables given by Bochner is due to the asymptotic properties of the summands. In this paper, the uniqueness theorem is used to extend the class of functions possessing unique decompositions.
Ω-explosions
Authors: Palis J.
Published: June 2010

In the present paper it is shown that if a flow satisfies Smale’s Axiom

A’

{\text {A’}}

and there is a cycle on its nonwandering set

Ω
\Omega

, then the flow is not

Ω
\Omega

-stable. This is done by “blowing up” the nonwandering set with a small perturbation. It is possible, in this setting, to give a characterization of

Ω
\Omega

-stable flows when the nonwandering set is the union of a finite number of critical elements.
Sets of lattice points which contain a maximal number of edges
Authors: Clements G. F.
Published: June 2010

How should one select an

l
l

-element subset of a rectangular array of lattice points (points with integral coordinates) in

n
n

-dimensional Euclidean space so as to include the largest possible number of edges (pairs of points differing in exactly one coordinate)? It is shown that the generalized Macaulay theorem due to the author and B. Lindström contains the (known) solution.
Torsionfree projective modules
Authors: Teply Mark L.
Published: June 2010

In this paper, the following condition for a torsion theory in the sense of S. E. Dickson is examined: (

P
\text {P}

) Every nonzero torsionfree module contains a nonzero projective submodule. A special relationship between condition (

P
\text {P}

) and Goldie’s torsion theory is shown; and the rings, for which every nontrivial torsion theory satisfies condition (

P
\text {P}

), are classified.
𝐿² asymptotes for the Klein-Gordon equation
Authors: Nelson Stuart
Published: June 2010

An approximation

a
(
x
,
t
)

a(x,t)

is obtained for solutions

u
(
x
,
t
)

u(x,t)

of the Klein-Gordon equation.

a
(
x
,
t
)

a(x,t)

can be expressed in terms of the Fourier transforms of the Cauchy data and it is shown that

|

|

a
(
⋅
,
t
)
−
u
(
⋅
,
t
)

|

|

2

→
0

||a( \cdot ,t) - u( \cdot ,t)|{|_2} \to 0

as

t
→
∞

t \to \infty

. This result is applied to show how energy distributes among various conical regions.
An operator valued function space integral: A sequel to Cameron and Storvick’s paper
Authors: Johnson G. W. , Skoug D. L.
Published: June 2010

Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.
A primary decomposition for torsion modules
Authors: Alin J. S.
Published: June 2010

A definition of primary module is given and a theorem is proved characterizing rings for which each torsion module, in the sense of S. E. Dickson, decomposes as a direct sum of its primary submodules. This theorem is applied to obtain a generalization of Fuchs’ theorem on the additive group structure of Artinian rings.
Global dimension of triangular orders
Authors: Tarsy Richard B.
Published: June 2010

The triangular orders of finite global dimension in

n
×
n

n \times n

matrices over the quotient field of a DVR are found and a bound is given for their dimensions.
A note on point-countability in linearly ordered spaces
Authors: Bennett Harold R.
Published: June 2010

In this note linearly ordered topological spaces (abbreviated LOTS) with a point-countable base are examined. It is shown that a LOTS is quasi-developable if and only if it has a

σ
\sigma

-point-finite base and a LOTS with a point-countable base is paracompact. An example of a LOTS with a point-countable base that does not have a

σ
\sigma

-point-finite base is given. Conditions are given for the metrizability of a LOTS with a point-countable base and it is shown that a connected LOTS with a point-countable base is homeomorphic to a connected subset of the real line.
Mutual absolute continuity of sets of measures
Authors: Walsh Bertram
Published: June 2010

A theorem slightly stronger than the following is proved: If K is a convex set of (signed) measures that are absolutely continuous with respect to some fixed positive sigma-finite measure, then the subset consisting of those measures in K with respect to which all measures in K are absolutely continuous is the complement of a set of first category in any topology finer than the norm topology of measures. This implies, e.g., that any Banach-space-valued measure

μ
\mu

is absolutely continuous with respect to

|

⟨
μ
(
⋅
)
,

x
′

⟩

|

\left | {\langle \mu ( \cdot ),x’\rangle } \right |

for a norm-dense

G
δ

{G_\delta }

of elements

x
′

x’

of the dual of the Banach space.
A criterion for disfocality
Authors: Wong Pui-kei
Published: June 2010

A linear differential equation of the second order with coefficients holomorphic in the unit disk is considered. It is shown that if the coefficients are

H
p

{H^p}

functions such that their norms satisfy a certain inequality, then all nontrivial solutions of the equation will be disfocal in the disk.
Nonlinear and direction connections
Authors: Vilms Jaak
Published: June 2010

Nonlinear connections and direction connections are two types of connections arising in Finsler geometry. In his work on generalized sprays, P. Dazord showed that there is a relationship between these two types (nonlinear connections were called sections by him). This relationship has also been used by J. Grifone in a work on prolongation of direction connections. In this paper we examine this relationship in a general setting. In particular, we show that E. Cartan’s condition “D” is necessary and sufficient for a direction connection to define a nonlinear one. Also, we prove a nonuniqueness result for direction connections associated to a given nonlinear one.
An 𝑙₁ extremal problem for polynomials
Authors: Beller E. , Newman D. J.
Published: June 2010

Let

M

n

{\mathfrak {M}_n}

be the maximum of the

l
1

{l_1}

norm,

∑
n

|

c
k

|

\sum \nolimits ^n |{c_k}|

, of all nth degree polynomials satisfying

|

∑
n

c
k

z
k

|

≦
1

|\sum \nolimits ^n {c_k}{z^k}| \leqq 1

for

|

z

|

=
1

|z| = 1

. We prove that

M

n

{\mathfrak {M}_n}

is asymptotic to

√
n

\surd n

, by exhibiting polynomials

P
n

{P_n}

(which are partial sums of certain Fourier series), whose

l
1

{l_1}

norm is asymptotic to

√
n

\surd n

.
A characterization of regularity in topology
Authors: Wyler Oswald
Published: June 2010

We show in this paper that a topological space satisfies

T
3

{T_3}

(which we do not intend to imply

T
2

{T_2}

) if and only if convergence of filters is a continuous relation. In particular, a Hausdorff space is regular if and only if convergence of filters is a continuous mapping. We propose a new, categorically motivated, definition of continuous relations between topological spaces, and we compare it with two existing continuity concepts for relations.
The center of the free product of distributive lattices
Authors: Balbes Raymond
Published: June 2010

The object of this paper is to show that for members L and M in the class of distributive lattices with zero and unit, the center of the free product of L and M is the free product of their centers.
On the convergence of multiplicatively orthogonal series
Authors: Preston C. J.
Published: June 2010

G. Alexits and A. Sharma have recently shown that if

{

φ
n

}

n
=
1

∞

\{ {\varphi _n}\} _{n = 1}^\infty

is a uniformly bounded multiplicatively orthogonal system on a finite measure space and if

{

c
n

}

n
=
1

∞

\{ {c_n}\} _{n = 1}^\infty

is a sequence of real numbers with

∑

n
=
1

∞

c
n
2

>
∞

\sum _{n = 1}^\infty c_n^2 > \infty

, then the partial sums

∑

k
=
1

n

c

k
φ
k

\sum _{k = 1}^n{c_{k\varphi k}}

converge almost everywhere. We give here a simple proof of this result.
Quasi-differentiable functions of Banach spaces
Authors: Goodman Victor
Published: June 2010

Nonzero Fréchet differentiable functions with bounded support do not exist on certain real separable Banach spaces. As a result, the class of differentiable functions on such spaces is too small to be useful. For example, the class of differentiable functions on certain spaces does not separate disjoint closed subsets of the space. It is shown that this separation problem does not arise if Fréchet differentiability is replaced by the weaker condition of quasi-differentiability. Furthermore, it is shown that any bounded uniformly continuous function on a real separable Banach space is the uniform limit of quasi-differentiable functions.
Certain numerical radius contraction operators
Authors: Furuta Takayuki , Nakamoto Ritsuo
Published: June 2010

In this paper an operator T means a bounded linear operator on a complex Hilbert space H. The numerical radius norm

w
(
T
)

w(T)

of an operator T, is defined by

w
(
T
)
=
sup

|

(
T
x
,
x
)

|

w(T) = \sup |(Tx,x)|

for every unit vector x in H. An operator T is said to be a numerical radius contraction if

w
(
T
)
≦
1

w(T) \leqq 1

. We shall give some theorems on certain numerical radius contraction operators and related results in consequence of these theorems. Our central result is that an idempotent numerical radius contraction is a projection. Finally we prove that a periodic numerical radius contraction is the direct sum of zero and a unitary operator, that is to say, normal and partial isometric.
The graphs of semirings. II
Authors: Ratti J. S. , Lin Y.-F.
Published: June 2010

In this paper the authors continue their investigation of the connectivity of the graphs of semirings. We give some sufficient conditions under which the graph of a semiring is connected. The paper also contains some open problems.
On the convolution equation 𝑃=𝑃𝑄 of Choquet and Deny for probability measures on semigroups
Authors: Mukherjea A.
Published: June 2010

In this paper, the problem of determining regular probability measures P and Q satisfying

P
=

P
∗

Q
=

Q
∗

P

P = P ^\ast Q = Q ^\ast P

has been solved on locally compact cancellative semigroups. The results also hold in the complete metric case. The results have been used to determine the primitive idempotent measures on locally compact semigroups.
Pólya peaks and the oscillation of positive functions
Authors: Drasin David , Shea Daniel F.
Published: June 2010

A new proof is given of the existence of Pólya peaks of an (increasing) function

g
(
t
)

g(t)

. This approach yields several applications, including a characterization of those p for which g can have peaks of order p.
The radius of close-to-convexity of functions of bounded boundary rotation
Authors: Coonce H. B. , Ziegler M. R.
Published: June 2010

An analytic function whose boundary rotation is bounded by

k
π
(
k
≧
2
)

k\pi (k \geqq 2)

is shown to map a disc of radius

r
k

{r_k}

onto a close-to-convex domain, where

r
k

{r_k}

is the solution of a transcendental equation when

k
>
4

k > 4

and

r
k

=
1

{r_k} = 1

when

2
≦
k
≦
4

2 \leqq k \leqq 4

. The above value of

r
k

{r_k}

is shown to be the best possible for each k and an asymptotic expression for

r
k

{r_k}

is obtained.
A note on the compact elements of 𝐶*-algebras
Authors: Ylinen Kari
Published: June 2010

It is shown that for any

C
∗

{C^ \ast }

-algebra A there is a faithful representation

π
\pi

of A on a Hilbert space H such that, for

u
∈
A

u \in A

, the map

x
↦
u
x
u

x \mapsto uxu

is a compact operator on A if and only if

π
(
u
)

\pi (u)

is a compact operator on H.
Eventual disconjugacy of selfadjoint fourth order linear differential equations
Authors: Gustafson G. B.
Published: June 2010

At the 1969 Differential Equations Conference held at Knoxville, Tennessee, Z. Nehari asked the following question about fourth-order selfadjoint linear differential equations: “Assume that solutions of the equation have only a finite number of zeros on

t
≧
A

t \geqq A

. Does there exist a half-line on which no solution of the equation has more than three zeros?” In this paper sufficient conditions are given for the equation

(
p
(
t
)
y
)
+
(
q
(
t
)

y
′

)
′

=
0

(p(t)y) + (q(t)y’)’ = 0

to have the property that solutions possess only a finite number of zeros. This theorem is then used to construct an example which answers the above question in the negative. The example also shows that if on each half-line the equation has a solution with two consecutive double zeros, then it need not follow that there is a solution with infinitely many zeros.
Completely positive maps on 𝑈*-algebras
Authors: Paschke W. L.
Published: June 2010

Completely positive maps on

U
∗

{U^ \ast }

-algebras with identity are characterized in terms of

∗

^ \ast

-representations on Hilbert space. A result on restricted multiplicativity of such maps is established, from which it follows that completely positive maps which take unitaries to unitaries are

∗

^ \ast

-homomorphisms. It is also shown that positive maps on commutative

U
∗

{U^ \ast }

-algebras with identity are completely positive.
Sparse subsets of orthonormal systems
Authors: Price J. J.
Published: June 2010

There exist families of Walsh, Haar and trigonometric functions that have density zero and yet are complete in the sense of measure.
Injective dimension and completeness
Authors: Boratyński M.
Published: June 2010

This paper contains the proofs of the two following theorems: (1) Let

{

M
α

}

α
>
γ

{\{ {M_\alpha }\} _{\alpha > \gamma }}

be a well-ordered decreasing system of submodules of the module M such that

M
=

M
0

M = {M_0}

. If M is strongly complete and strongly Hausdorff then
\[

inj

dim

M
≦

sup

α
>
γ

inj

dim

M
α

/

M

α
+
1

.

{\text {inj}}\,\dim \,M \leqq \sup \limits _{\alpha > \gamma } \,{\text {inj}}\,\dim \,{M_\alpha }/{M_{\alpha + 1}}.

\]
(2) Let R be a commutative ring having nonzero minimal idempotent ideals

{

S
α

}

α
>
γ

{\{ {S_\alpha }\} _{\alpha > \gamma }}

and let

S
=

∐

α
>
γ

S
α

S = \coprod \nolimits _{\alpha > \gamma } {{S_\alpha }}

. An R-module is injective if and only if M=Annih

S
⊕

M
0

S \oplus {M_0}

where Annih S is injective and

M
0

{M_0}

is strongly complete and Hausdorff in the topology introduced by annihilators of the direct sums of

S
α

{S_\alpha }

.
A note on Janko’s simple group of order 175,560
Authors: Shult E.
Published: June 2010

Janko’s simple group J of order 175,560 is characterized among simple groups by the weak closure W of the involution in its centralizer. Among arbitrary finite groups, the theorem asserts that the normal closure of W is J.
On 𝑊* embedding of 𝐴𝑊*-algebras
Authors: Laison Diane
Published: June 2010

An

A

W
∗

A{W^ \ast }

-algebra N with a separating family of completely additive states and with a family

{

e
α

:
α
∈
A
}

\{ {e_\alpha }:\alpha \in A\}

of mutually orthogonal projections such that

lub
α

e
α

=
1

{\operatorname {lub} _\alpha }{e_\alpha } = 1

and

e
α

N

e
α

{e_\alpha }N{e_\alpha }

is a

W
∗

{W^ \ast }

-algebra for each

α
∈
A

\alpha \in A

is shown to have a faithful representation as a ring of operators. This gives a new and considerably shorter proof that a semifinite

A

W
∗

A{W^ \ast }

-algebra with a separating family of completely additive states has a faithful representation as a ring of operators.
Map bordism of maps
Authors: Stong R. E.
Published: June 2010

Bordism is shown to distinguish homotopic maps.
An embedding theorem for compact semilattices
Authors: Lea J. W.
Published: June 2010

It is shown that if a compact topological semilattice S is a topological lattice, then S can be embedded simultaneously algebraically and topologically in a direct product of n chains if and only if S can be algebraically embedded in a direct product of n chains.
Tychonoff’s theorem for hyperspaces
Authors: Chimenti Frank A.
Published: June 2010

If

exp
⁡
(

X
i

)
∖
{
∅
}

\exp ({X_i})\backslash \{ \emptyset \}

is equipped with a topology that preserves the topological convergence of nets of sets for every

i
∈
I

i \in I

, then the Tychonoff product of the family

{
exp
⁡
(

X
i

)
∖
{
∅
}
:
i
∈
I
}

\{ \exp ({X_i})\backslash \{ \emptyset \} :i \in I\}

is compact if and only if

X
i

{X_i}

is compact for every

i
∈
I

i \in I

. A similar result concerning sequential compactness is valid, for countable I.
The 𝑛-generator property for commutative rings
Authors: Gilmer Robert
Published: June 2010

Let

D
D

be an integral domian with identity. If for some positive integer

n
n

, each finitely generated ideal of

D
D

has a basis of

n
n

elements, then the integral closure of

D
D

is a Prüfer domain. This result generalizes to the case of commutative rings with identity that contain zero divisors.
Some examples from infinite matrix rings
Authors: Miller Robert W. , Turnidge Darrel R.
Published: June 2010

Examples are given of the following: (1) an injective Noetherian right module which is not Artinian, (2) a linearly compact Noetherian right module over a right perfect ring which is not Artinian, and (3) a (linearly compact) Artinian left module over a right perfect ring which is not Noetherian. These examples answer questions posed by J. W. Fisher and F. L Sandomierski in the Proceedings of the Utah Conference on Ring Theory held in 1971.
Differentiability of the metric projection in finite-dimensional Euclidean space
Authors: Asplund Edgar
Published: June 2010

The metric projection on a closed subset of a finite-dimensional Euclidean space is almost everywhere differentiable.
The solution of a nonlinear Gronwall inequality
Authors: Helton Burrell W.
Published: June 2010

This paper extends some of the earlier results of J. V. Herod, W. W. Schmaedeke and G. R. Sell, and B. W. Helton and shows that, under the given conditions, (1) there is a function

u
u

satisfying the inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis less-than-over-equals h left-parenthesis x right-parenthesis plus left-parenthesis upper R upper L right-parenthesis integral Subscript a Superscript x Baseline left-parenthesis f upper G plus f upper H right-parenthesis">

f
(
x
)
≦
h
(
x
)
+
(
R
L
)

∫
a
x

(
f
G
+
f
H
)

f(x) \leqq h(x) + (RL)\int _a^x {(fG + fH)}

\]
such that, if

f
f

satisfies the given inequality, then

f
(
x
)
≦
u
(
x
)

f(x) \leqq u(x)

; and (2) there is a function

u
u

satisfying the inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than f left-parenthesis x right-parenthesis less-than-over-equals k plus left-parenthesis upper R upper L right-parenthesis integral Subscript a Superscript x Baseline left-bracket left-parenthesis f upper G 1 plus f Superscript p n Baseline upper G 2 right-parenthesis plus left-parenthesis f upper H 1 plus f Superscript p n Baseline upper H 2 right-parenthesis right-bracket comma">

0
>
f
(
x
)
≦
k
+
(
R
L
)

∫
a
x

[
(
f

G
1

+

f

p
n

G
2

)
+
(
f

H
1

+

f

p
n

H
2

)
]
,

0 > f(x) \leqq k + (RL)\int _a^x {[(f{G_1} + {f^{pn}}{G_2}) + (f{H_1} + {f^{pn}}{H_2})],}

\]
where

n
n

is a positive integer and

p
=
±
1

p = \pm 1

and

p
n
≠
1

pn \ne 1

, such that, if

f
f

satisfies the given inequality, then

f
(
x
)
≦
u
(
x
)

f(x) \leqq u(x)

.