bluuebunny/crossref_metadata_2025 huggingface dataset:
The equivalence of the least upper bound property and the Hahn-Banach extension property in ordered linear spaces
Authors: To Ting-On
Published: June 2010

Let V be a partially ordered (real) linear space with the positive wedge C. It is known that V has the least upper bound property if and only if V has the Hahn-Banach extension property and C is lineally closed. In recent papers, W. E. Bonnice and R. J. Silverman proved that the Hahn-Banach extension and the least upper bound properties are equivalent. We found that their proof is valid only for a restricted class of partially ordered linear spaces. In the present paper, we supply a proof for the general case. We prove that if V contains a partially ordered linear subspace W of dimension

≧
2

\geqq 2

, whose induced wedge

K
=
W
∩
C

K = W \cap C

satisfies

K
∪
(
−
K
)
=
W

K \cup ( - K) = W

and

K
∩
(
−
K
)
=

K \cap ( - K) =

{zero vector}, then V fails to have the Hahn-Banach extension property. From this the desired result follows.
Dual necklaces of 𝑛-dimensional cubes
Authors: Kalmanson Kenneth
Published: June 2010

It has been shown that certain finite configurations, called dual necklaces, of euclidean spheres yield information on longest rectilinear circuits on the sphere centers. In the present paper, the existence of dual necklaces of

k
k

n
n

-dimensional cubes is discussed and the values of

k
k

are determined. A more general configuration of cubes, called a multidual necklace, is treated similarly.
A compactification of locally compact spaces
Authors: Lozier F. W.
Published: June 2010

Every locally compact space

X
X

has its topology determined by its 1-1 compact images and hence has a compactification

ξ
X

\xi X

obtained as the closure of the natural embedding of

X
X

in the product of those images, just as the Stone-Čech compactification

β
X

\beta X

can be obtained by embedding

X
X

in a product of intervals. The obvious question is whether

ξ
X
=
β
X

\xi X = \beta X

. In this paper we prove that

ξ
X
=
β
X

\xi X = \beta X

if

X
X

either is

0
0

-dimensional or contains an arc, and give an example in which

ξ
X
≠
β
X

\xi X \ne \beta X

.
Regular compactifications of convergence spaces
Authors: Richardson G. D. , Kent D. C.
Published: June 2010

This note gives a simple characterization for the class of convergence spaces for which regular compactifications exist and shows that each such convergence space has a largest regular compactification.
On the existence of fixed points in a totally ordered set
Authors: Metcalf F. T. , Payne T. H.
Published: June 2010

The Tarski fixed point theorem, concerning an isotone mapping of a partially ordered set into itself, is extended to mappings which are not necessarily isotone, but which must map a totally ordered set into itself. The key requirement is that the function be continuous (in a certain sense) whenever it is “decreasing". Then

a
≦
f
(
a
)
,
f
(
b
)
≦
b

a \leqq f(a),f(b) \leqq b

, and

a
≦
b
⇒

a \leqq b \Rightarrow

the existence of a fixed point of

f
f

.
The Lie algebra of the structure group of a power-associative algebra.
Authors: Scribner D. R.
Published: June 2010

For a strictly power-associative algebra

A
A

with identity let

S
S

be the span of the transitivity set of the identity under the action of the structure group. The main result of the paper is that the Lie algebra of the structure group is a subalgebra of the direct sum of the derivation algebra of

A
+

{A^ + }

and the space of left multiplications in

A
+

{A^ + }

by elements of

S
S

, and is equal to this sum if the characteristic is 0. It is also shown that

S
S

is a Jordan subalgebra of

A
+

{A^ + }

.
A necessary condition for quasitriangularity
Authors: Deddens James A.
Published: June 2010

In this note we prove that if T is a quasitriangular operator then

Λ
(
T
+
K
)
=
Π
(
T
+
K
)

\Lambda (T + K) = \Pi (T + K)

for all compact operators K.
𝐻-structures on products
Authors: Copeland Arthur H.
Published: June 2010

This note outlines a technique for determining the set of homotopy classes of H-structures on a product of H-spaces. The technique is then applied to computing the number of H-structures on

S
O
(
4
)

SO(4)

.
𝐾₁ of a commutative von Neumann regular ring
Authors: Roberts Leslie G.
Published: June 2010

Let A be a commutative regular ring (in the sense of von Neumann), and let q be an ideal in A. Then

K
1

(
A
,
q
)
=
U
(
A
,
q
)

{K_1}(A,q) = U(A,q)

.
Determination of 𝑘_{𝑛}(𝑛≥3) for global fields
Authors: Elman Richard , Lam T.-Y.
Published: June 2010

A short proof is obtained for Tate’s theorem about

k
n

{k_n}

of global fields.
Shelling 3-cells in compact triangulated 3-manifolds
Authors: Treybig L. B.
Published: June 2010

Although examples of nonshellable 3-cells are known, this paper shows that every compact 3-manifold (with or without boundary) has a triangulation in which every 3-cell which is the union of 3-simplexes of the triangulation can be shelled.
The bordism ring of manifolds with involution
Authors: Alexander J. C.
Published: June 2010

A complete set of representatives is found for the bordism ring of manifolds with involution. The multiplicative structure is expressed in terms of these representatives, and in particular, a set of ring generators can be isolated. Also the constructions involved are used to give an insight into a map of J. M. Boardman.
Minimal presentations for certain metabelian groups
Authors: Searby D. G. , Wamsley J. W.
Published: June 2010

Let G be a finite p-group,

d
(
G
)
=
dim
⁡

H
1

(
G
,
Z

/

p
Z
)

d(G) = \dim {H^1}(G,Z/pZ)

and

r
(
G
)
=
dim
⁡

H
2

(
G
,
Z

/

p
Z
)

r(G) = \dim {H^2}(G,Z/pZ)

. Then

d
(
G
)

d(G)

is the minimal number of generators of G, and we say that G is a member of a class

G

p

{\mathcal {G}_p}

of finite p-groups if G has a presentation with

d
(
G
)

d(G)

generators and

r
(
G
)

r(G)

relations. The main result is that any outer extension of a finite cyclic p-group by a finite abelian p-group belongs to

G

p

{\mathcal {G}_p}

.
Singularities of a class of meromorphic functions
Authors: Callas Nicholas P. , Thron W. J.
Published: June 2010

Estimates are obtained for the number of singular points, which are not poles, lying on the unit circle of the complex plane of a class of meromorphic functions which are represented by C-fractions.
Convex functions and harmonic maps
Authors: Gordon William B.
Published: June 2010

A subset D of a riemannian manifold Y is said to be convex supporting if every compact subset of D has a Y-open neighborhood which supports a strictly convex function. The image of a harmonic map f from a compact manifold X to Y cannot be contained in any convex supporting subset of Y unless f is constant. Also, if Y has a convex supporting covering space and

π
1

(
X
)

{\pi _1}(X)

is finite then every harmonic map from X to Y is necessarily constant. Examples of convex supporting domains and manifolds are given.
Conditions implying continuity of open and closed functions
Authors: Baggs Ivan
Published: June 2010

It is shown that every open and closed real valued function of a real variable is continuous. This result is then extended to the case where the domain space is first countable and regular for a special class of open and closed functions.
Unique factorization in partially ordered sets
Authors: Martinez Jorge
Published: June 2010

Call an ideal of a lattice L regular if it is maximal with respect to not containing some element of L. If the ideal M is maximal without

x
∈
L

x \in L

we say M is a value of x. A special ideal K is a regular ideal which is maximal without some element having only one value (namely K). Our main theorem is that the following three statements are equivalent: (1) The lattice of ideals of the lattice L is completely distributive. (2) Each regular ideal is special. (3) L is distributive, and each element

x
∈
L

x \in L

can be written uniquely as a finite join of pairwise incomparable, finitely join irreducible elements of L. By carefully generalizing our notions to partially ordered sets we get a similar theorem in this context.
The Boolean ring generated by a distributive lattice
Authors: Bernau S. J.
Published: June 2010

This note gives a simple characterization of the Boolean ring R generated by a distributive lattice L. The method is to introduce a Hausdorff topology on the Stone representation space E of L and note that R is the set of subsets of E which are compact and open with respect to this topology.
A stronger Bertrand’s postulate with an application to partitions
Authors: Dressler Robert E.
Published: June 2010

In this paper we give a stronger form of Bertrand’s postulate and use it to prove that every positive integer, except 1, 2, 4, 6, and 9, can be written as the sum of distinct odd primes.
Solution of certain matrix equations
Authors: Jones John
Published: June 2010

The main purpose of this paper is to obtain solutions of matrix equations of the following types,

A
X
−
X
B
=
C
,
X
D
X
+
A
X
+
X
B
+
C
=
0

AX - XB = C,XDX + AX + XB + C = 0

, in which case

X
X

is an unknown

n
n

by

n
n

matrix and

A
,
B
,
C
,
D

A,B,C,D

are

n
n

by

n
n

matrices having elements belonging to the field

C
C

of complex numbers. Results obtained extend those of W. E. Roth, J. E. Potter and others concerning the existence and the representation of solutions

X
X

of the above equations.
On a characterisation of matrix functions which are differences of two monotone matrix functions
Authors: Vasudeva Harkrishan
Published: June 2010

The class of matrix functions of ’bounded variation’ was introduced by O. Dobsch in a paper published in 1937 [2]. The consideration of this class of functions immediately gives rise to the consideration of those matrix functions of order n on an interval [a, b] that are representable as the difference of two monotone matrix functions on that interval. Such a difference will have high regularity properties when n is large and is therefore much more than simply a function of bounded variation. The characterization of this class was sought in the paper of Dobsch [2]. The purpose of this paper is to give a complete description of a related class: the functions defined on (—1,1) which have restrictions to any closed subinterval which are such differences.
Unexpected behavior for solutions of a second order, selfadjoint equation
Authors: DeKleine H. Arthur
Published: June 2010

Positive, continuous functions

a
(
t
)

a(t)

and

p
(
t
)

p(t)

such that

a
(
t
)
p
(
t
)
≡
t

a(t)p(t) \equiv t

and for which some solution of the selfadjoint equation

(
p

u
′

)
′

+
a
u
=
0

(pu’)’ + au = 0

satisfies

lim

sup

t
→
∞

|

u
(
t
)

|

>
0

\lim \;{\sup _{t \to \infty }}|u(t)| > 0

are shown to exist.
A sufficient condition for countable-set aposyndesis
Authors: Bennett Donald E.
Published: June 2010

In this paper a stronger form of aposyndesis is defined and continua with this property (strongly aposyndetic) are shown to be countable-set aposyndetic. Although every continuum which is a product of nondegenerate continua is countable-set aposyndetic, it is established that no product of nondegenerate continua is strongly aposyndetic.
Unitary bordism of abelian groups
Authors: Ossa Erich
Published: June 2010

It is shown that for a finite abelian group G the bordism group of unitary G-manifolds is a free

U
∗

{U_\ast }

-module on even dimensional generators.
A note on rigid substructures
Authors: Parikh R. J.
Published: June 2010

We show that a theory with a recursive set of axioms may have (nontrivial) rigid substructures and yet fail to have

Σ
1
1

\Sigma _1^1

, or

Π
1
1

\Pi _1^1

rigid substructures.
On the radical of the group algebra of a 𝑝-group over a modular field
Authors: Carns Gail L. , Chao Chong-yun
Published: June 2010

Let G be a finite p-group, K be the field of integers modulo p, KG be the group algebra of G over K and N be the radical of KG. By using the fact that the annihilator,

A
(
N
)

A(N)

, of N is one dimensional, we characterize the elements of

A
(

N
2

)

A({N^2})

. We also present relationships among the cardinality of

A
(

N
2

)

A({N^2})

, the number of maximal subgroups in G and the number of conjugate classes in G. Theorems concerning the Frattini subalgebra of N and the existence of an outer automorphism of N are also proved.
Stability of solutions of linear systems with dominant main diagonal
Authors: Kahane Charles
Published: June 2010

We establish the asymptotic stability of solutions of a first order linear system of differential equations in which the matrix characterizing the system has a dominating principal diagonal with negative entries; the domination is expressed through the condition that, numerically, each entry in the principal diagonal exceeds the sum of the absolute values of the remaining entries in the same column.
𝑃𝐿 involutions of some 3-manifolds
Authors: Myung Myung Mi
Published: June 2010

Let

h
1

{h_1}

and

h
2

{h_2}

be PL involutions of connected, oriented, closed, irreducible 3-manifolds

M
1

{M_1}

and

M
2

{M_2}

, respectively. Let

a
i

,
i
=
1
,
2

{a_i},i = 1,2

, be a fixed point of

h
i

{h_i}

such that near

a
i

{a_i}

the fixed point sets of

h
i

{h_i}

are of the same dimension. Then we obtain a PL involution

h
1

#

h
2

{h_1}\# {h_2}

on

M
1

#

M
2

{M_1}\# {M_2}

induced by

h
i

{h_i}

by taking the connected sum of

M
1

{M_1}

and

M
2

{M_2}

along neighborhoods of

a
i

{a_i}

. In this paper, we study the possibility for a PL involution h on

M
1

#

M
2

{M_1}\# {M_2}

having a 2-dimensional fixed point set

F
0

{F_0}

to be of the form

h
1

#

h
2

{h_1}\# {h_2}

, where

M
i

{M_i}

are lens spaces. It is shown that: (1) if

F
0

{F_0}

is orientable, then

M
1

=
−

M
2

{M_1} = - {M_2}

and h is the obvious involution, (2) if the fixed point set F contains a projective plane, then

M
1

=

M
2

=

a

{M_1} = {M_2} = {\text {a}}

projective 3-space, and in this case, F is the disjoint union of two projective planes and h is unique up to PL equivalences, (3) if F contains a Klein bottle K, then F is the disjoint union of a Klein bottle and two points.
A 3-dimensional Abraham-Smale example
Authors: Simon Carl P.
Published: June 2010

The author constructs an open set of non

Ω
\Omega

-stable diffeomorphisms on

T
3

{T^3}

.
Representation of the projectable and strongly projectable hulls of a lattice-ordered group
Authors: Chambless Donald A.
Published: June 2010

A representable l-group G can be embedded into a projectable (strongly projectable) l-group; an essential extension of G which is minimal with respect to being projectable (strongly projectable) is unique. In this paper these projectable and strongly projectable “hulls” of G are represented using direct limits. If G is an f-ring, or f-ring without nonzero nilpotent elements, then so are these hulls.
Normability of certain topological rings
Authors: Warner Seth
Published: June 2010

Criteria are given for the nonliability of certain topological rings. These criteria yield generalizations of known theorems on the normability of compact integral domains and topological fields.
More properties inherited by the lower radical
Authors: Rossa R. F.
Published: June 2010

Various properties possessed by a class M of not necessarily associative rings are inherited by the lower radical LM, including heredity to one-sided ideals or subrings, failure to contain nonzero noetherian rings and, in the associative case, failure to contain nonzero artinian rings.
Splitting in map groups
Authors: Robertson Lewis , Wilcox Theodore W.
Published: June 2010

Every locally compact maximally almost periodic group G has a normal vector subgroup, the centralizer of which is of finite index. This vector subgroup is nontrivial whenever the identity component of G is not compact. Furthermore, if G has relatively compact conjugacy classes, then

G
≅

R
n

×
L

G \cong {R^n} \times L

where L has a compact open normal subgroup. Several structure theorems are also obtained for cases in which splitting need not occur.
A vector measure with no derivative
Authors: Lewis D. R.
Published: June 2010

Given a nonatomic scalar measure

μ
\mu

, there is a vector valued,

μ
\mu

-continuous measure of finite variation which has no derivative with respect to

μ
\mu

, but which has the property that the closure of its range is compact and convex.
The entire functions separate 𝛽𝐶
Authors: Kelleher J. J. , Taylor B. A.
Published: June 2010

The entire functions separate the points of the Stone-Čech compactification of the complex plane.
Asymptotic estimates of the 𝑛-widths in Hilbert space
Authors: Jerome Joseph W.
Published: June 2010

Let

Ω
⊂

R
m

\Omega \subset {R^m}

be a bounded open set satisfying the restricted cone property and let R be a nonnegative selfadjoint operator on

L
2

(
Ω
)

{L^2}(\Omega )

which is the realization of a uniformly elliptic operator A of order

v
v

with suitable coefficients and principal part

a
(
x
,
ξ
)

a(x,\xi )

. Let

R

\mathcal {R}

be the ellipsoid

{
f
:
(
R
f
,
f
)
≦
1
}

\{ f:(Rf,f) \leqq 1\}

. The

L
2

{L^2}

n-widths

d
n

(

R

)

{d_n}(\mathcal {R})

satisfy

d
n

(

R

)
∼

(
c

/

n

)

v

/

2
m

{d_n}(\mathcal {R}) \sim {(c/n)^{v/2m}}

where

c
=

∫
Ω

(

∫

0
>
a
(
x
,
ξ
)
>
1

d
ξ
)

d
x

c = \smallint _\Omega {(\smallint _{0 > a(x,\xi ) > 1} {d\xi )\;dx} }

. If

B
(
u
,
v
)

B(u,v)

is a nonnegative Hermitian coercive form over a subspace

V

\mathcal {V}

of the Sobolev space

W

k
,
2

(
Ω
)

{W^{k,2}}(\Omega )

, then the n-widths of

B

=
{
f
∈

V

:
B
(
f
,
f
)
≦
1
}

\mathcal {B} = \{ f \in \mathcal {V}:B(f,f) \leqq 1\}

satisfy,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than left-parenthesis c prime right-parenthesis Superscript k slash m Baseline less-than-over-equals limit inf d Subscript n Baseline left-parenthesis script upper B right-parenthesis n Superscript k slash m Baseline less-than-over-equals limit sup d Subscript n Baseline left-parenthesis script upper B right-parenthesis n Superscript k slash m Baseline less-than-over-equals left-parenthesis c right-parenthesis Superscript k slash m">

0
>

(

c
′

)

k

/

m

≦
lim
inf

d
n

(

B

)

n

k

/

m

≦
lim
sup

d
n

(

B

)

n

k

/

m

≦

(
c

)

k

/

m

0 > {(c’)^{k/m}} \leqq \lim \inf {d_n}(\mathcal {B}){n^{k/m}} \leqq \lim \sup {d_n}(\mathcal {B}){n^{k/m}} \leqq {(c)^{k/m}}

. In some cases

c
′

=
c
=
c

c’ = c = c

where c is defined in terms of an elliptic operator of order 2k. The n-widths of

B

\mathcal {B}

in

W

j
,
2

(
Ω
)
,
0
≦
j
≦
k

{W^{j,2}}(\Omega ),0 \leqq j \leqq k

, are of order

O
(

n

−
(
k
−
j
)

/

m

)
,
n
→
∞

O({n^{ - (k - j)/m}}),n \to \infty

.
Continuous functions from a connected locally connected space into a connected space with a dispersion point
Authors: Coppin C. A.
Published: June 2010

For

T
2

{T_2}

spaces, it is shown that any continuous function from a connected locally connected space into a connected space with a dispersion point is a constant.
Tensor products of quaternion algebras
Authors: Albert A. A.
Published: June 2010

Two quaternion division algebras have a common quadratic subfield if their tensor product contains zero-divisors.
Torsion theories over semiperfect rings
Authors: Rutter Edgar A.
Published: June 2010

In this note we characterize those torsion theories over a semiperfect ring such that the class of torsion free modules is closed under projective covers and the hereditary torsion theories for which this is true of the class of torsion modules. These results are applied to determine all hereditary torsion theories over an injective cogenerator ring with the property that the torsion submodule of every module is a direct summand.
On a theorem of Rudin
Authors: Chalice Donald R.
Published: June 2010

We give short proofs of a theorem of Rudin about polynomial approximation in

R

2
+
n

{R^{2 + n}}

and a corollary of this theorem which says that any function algebra on [0, 1] generated by one complex-valued function and n real functions is all continuous functions. At the same time our proof shows that both results hold with n replaced by an arbitrary index set

Λ
\Lambda

.
Extending a complete convex metric
Authors: Dooley Robert A.
Published: June 2010

A metric D is convex if for every two points x, z there is a third point y such that

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

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

. A generalized continuum is a connected, locally compact, separable metric space. It is shown that if

M
1

{M_1}

is a space with a complete convex metric

D
1

{D_1}

and

M
2

{M_2}

is a locally connected generalized continuum whose intersection with

M
1

{M_1}

is nonempty and compact, there is a complete convex metric for

M
1

∪

M
2

{M_1} \cup {M_2}

that extends

D
1

{D_1}

. Using this result, four classes of locally connected generalized continua are characterized by the type of complete convex metric they admit.
Local triviality of fiberings
Authors: Kim Soon-kyu
Published: June 2010

We prove that a Hurewicz fibering or a Serre fibering is locally trivial if the total space is a connected separable metric ANR n-gm over a principal ideal domain and the base space is a weakly locally contractible paracompact finite dimensional space, and all fibers are homeomorphic to a space which is a connected 3-manifold with exactly one end and whose one point compactification is a 3-manifold and it has no false 3-cells, in particular a euclidean 3-space.
Some theorems on Fredholm maps
Authors: Tromba A. J.
Published: June 2010

Let

f
:
M
→
N

f:M \to N

be a

C
1

{C^1}

proper Fredholm map of index n of

C
r

{C^r}

Banach manifolds. Then under certain hypotheses one may approximate f by a

C
r

{C^r}

proper Fredholm map g of index n with g

C
1

{C^1}

properly homotopic to f. The rank theorem for Fredholm maps is stated and it is then used to obtain the invariance of domain for this class of maps and also a nonlinear version of the Fredholm alternative theorem.
A counterexample to an analogue of Artin’s conjecture
Authors: Weinberger P. J.
Published: June 2010

I construct a counterexample to a conjecture of Larry Goldstein on the density of primes which split completely in none of a set of algebraic number fields. The fields used are all Abelian over the rationals.
Nonexistence of asymptotic observables
Authors: Howland James S.
Published: June 2010

Strong asymptotic limits of all Heisenberg observables exist only for trivial Hamiltonians.
A nonlinear complementarity problem in mathematical programming in Banach space
Authors: Bazaraa M. S. , Goode J. J. , Nashed M. Z.
Published: June 2010

An existence and uniqueness theorem for the nonlinear complementarity problem over closed convex cones in a reflexive real Banach space is established, using perturbations of solutions of variational inequalities and monotonicity methods.
An application of the algebra of differentials of infinite rank
Authors: Brown William C.
Published: June 2010

Let k denote an arbitrary field and let R be an affine local domain over k. Let

(

Ω
k

(
R
)
,

δ
k
R

)

({\Omega _k}(R),\delta _k^R)

be the universal algebra of k-higher differentials over R. Let K be the quotient field of R and L the residue class field of R. If K is a separable extension of k and L is a separable algebraic extension of k, then it is shown that R is a regular local ring if and only if

Ω
k

(
R
)

{\Omega _k}(R)

is a free R-algebra. If both K and L are separable extensions of k and R has a separating residue class field, then R is a regular local ring if and only if

Ω
k

(
R
)

{\Omega _k}(R)

is a free emphR-algebra.
An ideal criterion for torsion freeness
Authors: Bridger Mark
Published: June 2010

Auslander and Bridger have shown that, under conditions somewhat weaker than finite projective dimension, the “torsion freeness” properties of a module M (e.g. being reflexive, being the kth syzygy of another module) are determined by certain arithmetic conditions on the

Ext
i

(
M
,
R
)

{\text {Ext}^i}(M,R)

. In this paper it is shown that a single ideal, the intersection of the annihilators of these modules, gives this same information. This ideal is then related to the Fitting invariants and invariant factors of M, and a computation is made of certain syzygies of a quotient of M (by a regular M-sequence).
A remark on groups with the fixed point property
Authors: Simon Barry
Published: June 2010

We prove that any group with the fixed point property actually leaves fixed points for measurable actions rather than only jointly continuous actions.
A global existence theorem for a nonautonomous differential equation in a Banach space
Authors: Lovelady David Lowell , Martin Robert H.
Published: June 2010

Suppose that X is a real or complex Banach space and that A is a continuous function from

[
0
,
∞
)
×
X

[0,\infty ) \times X

into X. Suppose also that there is a continuous real valued function

ρ
\rho

defined on

[
0
,
∞
)

[0,\infty )

such that

A
(
t
,
⋅
)
−
ρ
(
t
)
I

A(t, \cdot ) - \rho (t)I

is dissipative for each t in

[
0
,
∞
)

[0,\infty )

. In this note we show that, for each z in X, there is a unique differentiable function u from

[
0
,
∞
)

[0,\infty )

into X such that

u
(
0
)
=
z

u(0) = z

and

u
′

(
t
)
=
A
(
t
,
u
(
t
)
)

u’(t) = A(t,u(t))

for all t in

[
0
,
∞
)

[0,\infty )

. This is an improvement of previous results on this problem which require additional conditions on A.
On a general ratio ergodic theorem with weighted averages
Authors: Satō Ryōtarō
Published: June 2010

A general ratio ergodic theorem with weighted averages is shown by utilizing a method of R. V. Chacon. The theorem contains Chacon’s general ergodic theorem as a special case.
Sums of distances between points on a sphere
Authors: Stolarsky Kenneth B.
Published: June 2010

An upper bound for the sum of the

λ
\lambda

th powers of all distances determined by N points on a unit sphere is given for

0
≦
λ
≦
1

0 \leqq \lambda \leqq 1

.
Closed ideals in 𝐶(𝑋)
Authors: Nanzetta Philip , Plank Donald
Published: June 2010

The characterization of uniformly closed ideals in

C
(
X
)

C(X)

, for X compact Hausdorff, is well known. In this note, we extend this characterization to an arbitrary completely regular Hausdorff X and derive some corollaries.
Toroidal arcs are cellular
Authors: Knoblauch Tom
Published: June 2010

We prove that a toroidal, cell-like, locally connected continuum is cellular.
Knots without unknotted incompressible spanning surfaces
Authors: Lyon Herbert C.
Published: June 2010

We construct a tame knot in

S
3

{S^3}

which has no unknotted incompressible spanning surface.
The Whitney technique for Poincaré complex embeddings
Authors: Hodgson J. P. E.
Published: June 2010

The following result is proved.
A generalization of two inequalities involving means
Authors: Lawrence Scott , Segalman Daniel
Published: June 2010

Fan has proven an inequality relating the arithmetic and geometric means of

(

x
1

,
⋯
,

x
n

)

({x_1}, \cdots ,{x_n})

and

(
1
−

x
1

,
⋯
,
1
−

x
n

)

(1 - {x_1}, \cdots ,1 - {x_n})

, where

0
>

x
i

≦

1
2

,
i
=
1
,
⋯
,
n

0 > {x_i} \leqq \tfrac {1}{2},i = 1, \cdots ,n

. Levinson has generalized Fan’s inequality; his result involves functions with positive third derivatives on (0, 1). In this paper, the above condition that requires

0
>

x
i

≦

1
2

0 > {x_i} \leqq \tfrac {1}{2}

has been replaced by a condition which only weights the

x
i

{x_i}

to the left side of (0, 1) in pairs, and Levinson’s differentiability requirement has been replaced by the analogous condition on third differences.
Integrably parallelizable manifolds
Authors: Hansen Vagn Lundsgaard
Published: June 2010

A smooth manifold

M
n

{M^n}

is called integrably parallelizable if there exists an atlas for the smooth structure on

M
n

{M^n}

such that all differentials in overlap between charts are equal to the identity map of the model for

M
n

{M^n}

. We show that the class of connected, integrably parallelizable, n-dimensional smooth manifolds consists precisely of the open parallelizable manifolds and manifolds diffeomorphic to the n-torus.
Balanced and 𝑄𝐹-1 algebras
Authors: Camillo V. P. , Fuller K. R.
Published: June 2010

A ring R is QF-1 if every faithful module has the double centralizer property. It is proved that a local finite dimensional algebra is QF-1 if and only if it is QF. From this it follows that an arbitrary finite dimensional algebra has the property that every homomorphic image is QF-1 if and only if every homomorphic image is QF.
The Kobayashi distance induces the standard topology
Authors: Barth Theodore J.
Published: June 2010

The Kobayashi pseudodistance on a connected complex space is continuous with respect to the standard topology. If this pseudodistance is an actual distance, it induces the standard topology.
On the length of gaps in the essential spectrum of a generalised Dirac operator
Authors: Evans W. D.
Published: June 2010

The object of the paper is to give an upper bound for the length of the gaps that can occur in the essential spectrum of any selfadjoint operator which is generated by a generalised Dirac system of differential expressions in the Hilbert space

L
2

(
a
,
b
)

{L^2}(a,b)

. An estimate is also obtained for the limit point of the spectrum which has least absolute value.
On the characterization of Abelian 𝑊^{*}-algebras
Authors: Jenkins J. W.
Published: June 2010

In this note we present an elementary proof that an Abelian

W
∗

{W^ \ast }

-algebra is generated by the range of some real spectral measure.
The lattice of ideals of a compact semilattice
Authors: Stralka A. R.
Published: June 2010

It is shown that, if L is a compact distributive topological lattice with enough continuous join-preserving maps into I to separate points, then there is a continuous lattice homomorphism from

M

(
L
)

\mathcal {M}(L)

, the lattice of M-closed subsets of L, onto L. If

J
(
L
)

J(L)

, the set of join-irreducible elements of L, is a compact semilattice then L is iseomorphic with

M

(
J
(
L
)
)

\mathcal {M}(J(L))

.
Subfair red-and-black with a limit
Authors: Heath David C. , Pruitt William E. , Sudderth William D.
Published: June 2010

Suppose a gambler has an initial fortune in (0,1) and wishes to reach 1. It is known that, for a subfair red-and-black casino, the optimal strategy is always to bet

min
(
f
,
1
−
f
)

\min (f,1 - f)

whenever the gambler’s current fortune is f. Furthermore, the gambler should likewise play boldly if there is a house limit z which is the reciprocal of a positive integer; i.e., he should bet

min
(
f
,
1
−
f
,
z
)

\min (f,1 - f,z)

. We show that if

1

/

(
n
+
1
)
>
z
>
1

/

n

1/(n + 1) > z > 1/n

for some integer

n
≧
3

n \geqq 3

or if z is irrational and

1
3

>
z
>

1
2

\frac {1}{3} > z > \frac {1}{2}

, then bold play is not necessarily optimal.
On the automorphism group of a finite modular 𝑝-group
Authors: Davitt Richard M. , Otto Albert D.
Published: June 2010

In this paper it is shown that if G is a finite non-Abelian modular p-group

(
p
>
2
)

(p > 2)

, then the order of G divides the order of the automorphism group of G.
Quadrature-Galerkin approximations to solutions of elliptic differential equations
Authors: Schultz Martin H.
Published: June 2010

In practice the Galerkin method for solving elliptic partial differential equations yields equations involving certain integrals which cannot be evaluated analytically. Instead these integrals are approximated numerically and the resulting equations are solved to give “quadrature-Galerkin approximations” to the solution of the differential equation. Using a technique of J. Nitsche,

L
2

{L^2}

a priori error bounds are obtained for the difference between the solution of the differential equation and a class of quadrature-Galerkin approximations.
A note on the Whitehead torsion of a bundle modulo a subbundle
Authors: Anderson Douglas R.
Published: June 2010

The purpose of this note is to show how the sum theorem for Whitehead torsion due to K. W. Kwun and R. H. Szczarba, and generalized by L. C. Siebenmann, may be applied to compute the Whitehead torsion of the total space pair of a bundle pair in terms of the Whitehead torsion of the fiber pair and the Euler characteristic of the base.
Norm reduction of averaging operators
Authors: Cohen H. B. , Labbe M. A. , Wolfe J.
Published: June 2010

Suppose

ϕ
:
S
→
T

\phi :S \to T

is an irreducible map of compact Hausdorff spaces, and

μ
:
T
→
M
(
S
)

\mu :T \to M(S)

the integral representation of an averaging operator for

ϕ
\phi

. We obtain an inequality of the form

‖

μ
(
t
)

‖

≦

‖
μ
‖

−
a
(
t
)

\left \| {\mu (t)} \right \| \leqq \left \| \mu \right \| - a(t)

, where

a
(
t
)

is

a(t){\text {is}}

a positive number depending on t. From this, some results of Amir and Isbell-Semadeni on

P
λ

{P_\lambda }

spaces are shown to follow quickly and a theorem on the isomorphism of certain continuous function spaces is derived.
Dirichlet finite solutions of Δ𝑢=𝑃𝑢
Authors: Singer Ivan J.
Published: June 2010

The purpose of this paper is to give a necessary and also a sufficient condition for a Dirichlet finite harmonic function on a Riemann surface to be represented as a difference of a Dirichlet finite solution of

Δ
u
=
P
u
(
P
≧
0
)

\Delta u = Pu(P \geqq 0)

and a Dirichlet finite potential of signed measure.
Permanent groups
Authors: Beasley Leroy B. , Cummings Larry
Published: June 2010

A permanent group is a group of nonsingular matrices on which the permanent function is multiplicative. Let

A
∘
B

A \circ B

denote the Hadamard product of matrices A and B. The set of groups G of nonsingular

n
×
n

n \times n

matrices which contain the diagonal group

D

\mathcal {D}

and such that for every pair A, B of matrices in G we have

A
∘

B
T

∈

D

A \circ {B^T} \in \mathcal {D}

is denoted by

A

n

{\mathcal {A}_n}

. If the underlying field has at least three elements then

A

n

{\mathcal {A}_n}

consists of permanent groups. A partial converse is available: If a permanent group G is generated by

D

\mathcal {D}

together with a set S of elementary matrices and a set Q of permutation matrices then

G
=
H
K

G = HK

where H is the subgroup generated by Q and K is generated by

D

\mathcal {D}

and S, and

K
∈

A

n

K \in {\mathcal {A}_n}

.
A quasi-linear evolution equation and the method of Galerkin
Authors: Dickey R. W.
Published: June 2010

In this paper it is shown that under specified conditions on the initial data a certain infinite coupled system of ordinary differential equations has a solution satisfying an auxiliary convergence condition. The infinite system discussed is essentially the Galerkin expansion of the solution to a given quasi-linear wave equation. The results obtained suffice to prove the existence of a solution to this wave equation.
On the Frattini subgroups of certain generalized free products of groups
Authors: Tang C. Y.
Published: June 2010

Let

G
=
(

∏

i
∈
I

∗

A
i

)
H

G = (\prod \nolimits _{i \in I}^\ast {{A_i}{)_H}}

be the generalized free product of the groups

A
i

{A_i}

amalgamating the subgroup H. We show that if G is residually finite and the groups

A
i

{A_i}

have compatible H-filters then the Frattini subgroup

Φ
(
G
)

\Phi (G)

is contained in the maximal G-normal subgroup in H. If the groups

A
i

{A_i}

are free and H is finitely generated of infinite index in one

A
i

{A_i}

then

Φ
(
G
)
=
1

\Phi (G) = 1

. We also show that if H is simple then

Φ
(
G
)
=
1

\Phi (G) = 1

or

H
G

{H^G}

.
Free inverse semigroups
Authors: Scheiblich H. E.
Published: June 2010

At least three authors have offered proofs of the existence of a free inverse semigroup, but without describing its structure. This paper shows that if

X
X

is a nonempty set,

G
G

is the group on

X
X

, and

E
E

is a certain subsemilattice of the power set of

G
G

, then a certain collection of principal ideal isomorphisms of

E
E

is a free inverse semigroup on

X
X

.
The 𝐻^{𝑝} classes for 𝛼-convex functions
Authors: Eenigenburg P. J. , Miller S. S.
Published: June 2010

Given

α
>
0

\alpha > 0

, we determine the

H
p

{H^p}

class to which an

α
\alpha

-convex function belongs.
Oscillatory solutions for a generalized sublinear second order differential equation
Authors: Heidel J. W. , Kiguradze I. T.
Published: June 2010

A criterion is given for the existence of oscillatory solutions for equation (1) below which generalizes a recent result for the sublinear case of (1’). The present theorem is the analogue of a result of Izjumova for the generalized superlinear case.
Perspectivity in the projection lattice of an 𝐴𝑊*-algebra
Authors: Elliott George A.
Published: June 2010

Perspectivity in the projection lattice of an

A

W
∗

A{W^ \ast }

-algebra is identical with unitary equivalence. This was established by Fillmore for

W
∗

{W^ \ast }

-algebras, and a refinement of his argument is given here.
Multivalued nonexpansive mappings and Opial’s condition
Authors: Lami Dozo E.
Published: June 2010

We give relations between a condition introduced by Z. Opial which characterizes weak limits by means of the norm in some Banach spaces and approximations of the identity, in particular for systems of projections. Finally a fixed point theorem for multivalued nonexpansive mappings in a Banach space satisfying this condition is proved; this result generalizes those of J. Markin and F. Browder.
Homology of closed orbits of Anosov flows
Authors: Plante J. F.
Published: June 2010

It is shown that for a volume preserving Anosov flow on a compact manifold the closed orbits span the first homology (with real coefficients). The proof uses the notion of asymptotic cycles and results concerning the space of invariant measures for hyperbolic flows.
Direct products and sums of torsion-free Abelian groups
Authors: Murley C. E.
Published: June 2010

Let

A
A

be a finite rank, indecomposable torsion-free Abelian group whose

p
p

-ranks are less than two for all primes

p
p

. Let

G
G

be a direct product of copies of

A
A

, and

B
B

be a nonzero countable pure subgroup of

G
G

such that

B
B

is the span of the homomorphic images of

A
A

in

B
B

. Then it is shown that

B
B

is a direct sum of copies of

A
A

. This result is applied to obtain a Krull-Schmidt theorem for direct sums of groups

A
A

from a semirigid class of groups. In particular, if the groups

A
A

have rank one, then the well-known BaerKulikov-Kaplansky theorem is obtained.
Bergman operators for parabolic equations in two space variables
Authors: Colton David
Published: June 2010

An integral operator is constructed which maps analytic functions of two complex variables onto the class of real valued analytic solutions of linear second order parabolic equations in two space variables with real valued, analytic, time independent coefficients. When the solution of the parabolic equation is independent of the time variable the operator reduces to Bergman’s integral operator for elliptic equations in two independent variables.
Factorization of operators on Banach space
Authors: Embry Mary R.
Published: June 2010

In this paper it is shown that if

D
D

and

E
E

are continuous linear operators on a Banach space

X
X

, then the following are equivalent: (i)

B
B

is a right factor of

A
A

, (ii)

B
B

majorizes

A
A

and (iii) the range of

B
∗

{B^\ast }

contains the range of

A
∗

{A^\ast }

.
Trèves’ identity
Authors: Rosenbloom P. C.
Published: June 2010

We give a very simple proof of the identity which Trèves obtained for partial differential operators with constant coefficients. Our proof uses very little information about the structure of the operator and applies to a much wider class of operators. The exact scope of our method is still undetermined. The identity can be applied to obtain a priori estimates, existence theorems, and construction of fundamental solutions.
The six dimensional Gauss Bonnet integrand
Authors: Gray Alfred
Published: June 2010

Sufficient conditions are given for a compact

6
6

-dimensional Kähler manifold with nonnegative (nonpositive) curvature to have nonnegative (nonpositive) Euler characteristic.
On free abelian 𝑙-groups
Authors: Hill Paul
Published: June 2010

Let

F
F

denote the free abelian lattice ordered group over an unordered torsion free group

G
G

. Necessary and sufficient conditions are given on

G
G

in order for

F
F

to be an

l
l

-subgroup of a cardinal product of integers. The result encompasses Weinberg’s theorem that the freeness of

G
G

is sufficient. The corresponding embedding theorem for

F
F

is also established whenever

G
G

is completely decomposable and homogeneous.
Hurwitz’ theorem
Authors: Cohn J. H. E.
Published: June 2010

If

[

a
0

,

a
1

,

a
2

,
⋯
]

[{a_0},{a_1},{a_2}, \cdots ]

is the continued fraction for a real number

x
x

, and

p
n

/

q
n

{p_n}/{q_n}

the

n
n

th convergent, define

θ
n

=

q
n

|

p
n

−
x

q
n

|

{\theta _n} = {q_n}|{p_n} - x{q_n}|

. Hurwitz’ Theorem asserts that

ϕ
n

=
min
{

θ

n
−
1

,

θ
n

,

θ

n
+
1

}
>

5

−
1

/

2

{\phi _n} = \min \{ {\theta _{n - 1}},{\theta _n},{\theta _{n + 1}}\} > {5^{ - 1/2}}

whenever

ϕ
n

{\phi _n}

is defined. It is the object of this note to provide a simple proof of this fact.
The Hardy class of functions of bounded boundary rotation
Authors: Pinchuk Bernard
Published: June 2010

The Hardy class for functions of bounded boundary rotation and their derivatives is determined. In the univalent case, a more precise description of the Hardy class in terms of the behavior of the representing measure is obtained.
A representation formula for harmonic functions
Authors: Ching Chin Hung , Chui Charles K.
Published: June 2010

We give a formula to reconstruct certain entire harmonic functions from their values on some lattice points.
Perturbations of linear 𝑚-accretive operations
Authors: Okazawa Noboru
Published: June 2010

A sufficient condition is given for the sum

A
+
B

A + B

of two linear m-accretive operators A and B in a Hilbert space to be m-accretive. This condition is expressed in terms of

Re
⁡
(
A
u
,
B
u
)

\operatorname {Re} (Au,Bu)

for u in D, where D is a certain linear manifold contained in

D
(
A
+
B
)

D(A + B)

.
The compactifications to which an element of 𝐶*(𝑋) extends
Authors: Chandler Richard E. , Gellar Ralph
Published: June 2010

We first determine a necessary and sufficient condition for a function

f
∈

C
∗

(
X
)

f \in {C^\ast }(X)

, which extends to a compactification of

X
X

, to extend to a smaller compactification. We apply this result to show that when

|

β
X
∖
X

|

≦

ℵ
0

|\beta X\backslash X| \leqq {\aleph _0}

there is an

f
∈

C
∗

(
X
)

f \in {C^\ast }(X)

which extends to no compactification other than

β
X

\beta X

. Two examples show that when

ℵ
0

>

|

β
X
∖
X

|

≦
c

{\aleph _0} > |\beta X\backslash X| \leqq c

no such definite result may be obtained.
Perfect maps of symmetrizable spaces
Authors: Martin Harold W.
Published: June 2010

It is shown that if

f
:
X
→
Y

f:X \to Y

is a perfect map from a symmetrizable space

X
X

onto a space

Y
Y

, then

Y
Y

is metrizable if and only if

f
f

is a coherent map. This fact, together with certain known results, yields the following: Let

f
:
X
→
Y

f:X \to Y

be a perfect map from a Hausdorff symmetrizable space

X
X

onto a space

Y
Y

; the following are equivalent: (1)

X
X

is metrizable; (2)

f
f

is a regular map; (3)

f
f

is a coherent map; (4)

Y
Y

is metrizable.
Distortion theorems for a special class of analytic functions
Authors: Shaffer Dorothy Browne
Published: June 2010

Sharp bounds are derived for the derivative of analytic functions of class

P
α

{P_\alpha }

defined by the condition
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue p left-parenthesis z right-parenthesis minus 1 slash 2 alpha EndAbsoluteValue less-than-over-equals 1 slash 2 alpha comma 0 less-than-over-equals alpha less-than-over-equals 1 comma p left-parenthesis 0 right-parenthesis equals 1">

|

p
(
z
)
−
1

/

2
α

|

≦
1

/

2
α
,
0
≦
α
≦
1
,
p
(
0
)
=
1

|p(z) - 1/2\alpha | \leqq 1/2\alpha ,0 \leqq \alpha \leqq 1,p(0) = 1

. These results are improved for the class of functions with missing terms. Application is made to the class of functions with derivative

∈

P
α

\in {P_\alpha }

and the radius of convexity is determined for this class.
A very slowly convergent sequence of continuous functions
Authors: Rudin Walter
Published: June 2010

A sequence of continuous functions

f
n

:
[
0
,
1
]
→
(
0
,
1
]

{f_n}:[0,1] \to (0,1]

is constructed, with

lim

n
→
∞

f
n

(
x
)
=
0

{\lim _{n \to \infty }}{f_n}(x) = 0

for every

x
∈
[
0
,
1
]

x \in [0,1]

, but such that to every unbounded sequence

{

λ
n

}

\{ {\lambda _n}\}

of positive numbers corresponds a point

x
∈
[
0
,
1
]

x \in [0,1]

at which

lim

sup

n
→
∞

λ
n

f
n

(
x
)
=
∞

\lim {\sup _{n \to \infty }}{\lambda _n}{f_n}(x) = \infty

.
Restrictions of Fourier transforms of continuous measures
Authors: Wells Benjamin B.
Published: June 2010

Let

G
G

denote a compact abelian group and

Γ
\Gamma

its discrete dual. It is proved that

E
⊂
Γ

E \subset \Gamma

is Sidon if and only if the restriction to

E
E

of the algebra of Fourier transforms of continuous measures on

G
G

is all of

l
∞

(
E
)

{l_\infty }(E)

.
On the equivalence of twisted group algebras and Banach *-algebraic bundles
Authors: Busby Robert C.
Published: June 2010

The relationship between twisted group algebras and Banach

∗

^\ast

-algebraic bundles is investigated. Informally stated, the results are that bundles with Borel cross sections correspond to twisted group algebras, and “locally continuous” twisted group algebras correspond to bundles. In the separable case, these results combine to give a complete correspondence between the bundles and the “locally continuous” algebras.
Discrete Hausdorff transformations
Authors: Leibowitz Gerald
Published: June 2010

Let

K
K

be a complex valued measurable function on

(
0
,
1
]

(0,1]

such that

∫
0
1

t

−
1

/

p

|

K
(
t
)

|

d
t

\int _0^1 {{t^{ - 1/p}}|K(t)|dt}

is finite for some

p
>
1

p > 1

. Let

H
H

be the Hausdorff operator on

l
p

{l^p}

determined by the moments

μ
n

=

∫
0
1

t
n

K
(
t
)

d
t

{\mu _n} = \int _0^1 {{t^n}K(t)} dt

. Define

Ψ
(
z
)
=

∫
0
1

t
z

K
(
t
)

d
t

\Psi (z) = \int _0^1 {{t^z}K(t)} dt

. Then for each

z
z

with Re

Re
⁡
z
>
−
1

/

p
,
Ψ
(
z
)

\operatorname {Re} z > - 1/p,\Psi (z)

is an eigenvalue of

H
∗

{H^\ast }

. The spectrum of

H
H

is the union of

{
0
}

\{ 0\}

with the range of

Ψ
\Psi

on the half-plane Re

Re
⁡
z
≧
−
1

/

p

\operatorname {Re} z \geqq - 1/p

.
Elementary proofs of two theorems on the distribution of numbers 𝑛𝜃(\mod1)
Authors: Miklavc Adolf
Published: June 2010

Simple and elementary proofs of denseness and uniform distribution of the numbers

{
n
θ
}
,
θ

\{ n\theta \} ,\theta

irrational, on the unit interval.
The additive group of commutative rings generated by idempotents
Authors: Hill Paul
Published: June 2010

If

R
R

is a ring, let

R
+

{R^ + }

denote its additive group. Our purpose is to give an elementary proof that if

R
R

is a commutative ring generated by idempotents, then any subring of

R
R

generated by idempotents is pure. This yields immediately an independent proof of the following result of G. M. Bergman. If

R
R

is a commutative ring with identity and if

R
R

is generated by idempotents, then

R
+

{R^ + }

is a direct sum of cyclic groups.
Regularity of semilattice sums of rings
Authors: Janeski John , Weissglass Julian
Published: June 2010

If

R
R

is a supplementary semilattice sum of subrings

R
α

,
α
∈
Ω

{R_\alpha },\alpha \in \Omega

, then

R
R

is regular if and only if each

R
α

{R_\alpha }

is regular.
On representations of the group of Listing’s knot by subgroups of 𝑆𝐿(2,𝐶)
Authors: Whittemore Alice
Published: June 2010

The subgroups

G
G

of

S
L
(
2
,
C
)

SL(2,C)

which represent the group of Listing’s knot are characterized by the traces of

A
,
B

A,B

and

A
B

AB

, where

A
A

and

B
B

are matrices which generate

G
G

. From this it follows that there exist finitely generated nondiscrete subgroups of

S
L
(
2
,
R
)

SL(2,R)

which are not isomorphic to any Fuchsian group.
On the homomorphisms of locally compact groups
Authors: Lee D. H.
Published: June 2010

In this paper, we establish a conjugacy theorem of homomorphisms of a locally compact connected semisimple group into a locally compact group.