bluuebunny/crossref_metadata_2025 huggingface dataset:
Character semigroups of locally compact inverse semigroups
Authors: Fulp Ronald O.
Published: June 2010

We show that if

S
S

is a locally compact abelian continuous-inverse semigroup whose idempotent semigroup

E
E

satisfies a certain technical condition on its prime ideals, then the maximal subgroups of the character semigroup

S
∧

S^\wedge

of

S
S

are obtained as inverse limits of the duals of the maximal subgroups of

S
S

. It is shown that the technical conditions on

E
E

are satisfied in each of the following cases:

E
E

is compact,

E
E

is totally disconnected, or

E
E

is a chain. We then obtain necessary and sufficient conditions in order that a given inverse system of compact groups indexed by a totally disconnected semilattice

E
E

admit a compatible compact semigroup topology on their disjoint union.
A characterization of compact groups
Authors: Elliott George A.
Published: June 2010

A locally compact group is compact if and only if it is of type I and has discrete spectrum (equivalently: its

C
∗

{C^\ast }

-algebra is dual).
A remark on Mahler’s compactness theorem
Authors: Mumford David
Published: June 2010

We prove that if

G
G

is a semisimple Lie group without compact factors, then for all open sets

U
⊂
G

U \subset G

containing the unipotent elements of

G
G

and for all

C
>
0

C > 0

, the set of discrete subgroups

Γ
⊂
G

\Gamma \subset G

such that (a)

Γ
⋂
U
=
{
e
}

\Gamma \bigcap U = \{ e\}

, (b)

G

/

Γ

G/\Gamma

compact and measure

(
G

/

Γ
)
≦
C

(G/\Gamma ) \leqq C

, is compact. As an application, for any genus

g
g

and

ε
>
0

\varepsilon > 0

, the set of compact Riemann surfaces of genus

g
g

all of whose closed geodesics in the Poincaré metric have length

≧
ε

\geqq \varepsilon

, is itself compact.
An extension of the Thomson principle
Authors: Stoddart A. W. J.
Published: June 2010

The classical Thomson principle, giving lower bounds for the Dirichlet integral, is extended, with modification, to a wider class of test functions. This makes it possible to obtain simpler and better lower bounds.
Analytic functions close to mappings convex in one direction
Authors: Hengartner Walter , Schober Glenn
Published: June 2010

By analogy to the class of close-to-convex functions we define a class of analytic functions which are close to a family

Σ
\Sigma

of mappings onto domains convex in one direction. In contrast to the close-to-convex class the close-to-

Σ
\Sigma

functions are not necessarily univalent. However, we determine the radius of convexity for

Σ
\Sigma

, and this gives a lower bound for the radius of univalence of close-to-

Σ
\Sigma

functions. We next derive the coefficient estimate

|

A
n

|

≦
n

|

A
1

|

|{A_n}| \leqq n|{A_1}|

for close-to-

Σ
\Sigma

functions and conclude with an elementary distortion theorem.
Operators whose ascent is 0 or 1
Authors: Ramanujan P. B. , Patel S. M.
Published: June 2010

An operator T on a Hilbert space H is said to be of ascent 0 or 1 if the null spaces of T and

T
2

{T^2}

are equal. Let

A

\mathcal {A}

denote the collection of all operators on H which have ascent 0 or 1. The object of this note is to study some properties of operators in

A

\mathcal {A}

. The main results obtained are the following. 1. The direct sum of a collection of operators is in

A

\mathcal {A}

if and only if each of these operators is in

A

\mathcal {A}

. 2. If

T
∈

A

T \in \mathcal {A}

and T has finite descent, then the range of T is closed. 3. If

T
∈

A

T \in \mathcal {A}

and the range of T is not closed, then T is a commutator, that is, T is expressible in the form

A
B
−
B
A

AB - BA

for some operators A and B on H. 4. The set of all operators in

A

\mathcal {A}

with descent 0 or 1 is closed in the norm topology of operators. 5. If

T
∈

A

T \in \mathcal {A}

, and T has finite descent and further

T
k

{T^k}

is compact for some k, then T is a finite-dimensional operator.
Generalisation of the Muirhead-Rado inequality
Authors: Daykin D. E.
Published: June 2010

For polynomials

f
β

(
x
)

{f_\beta }(x)

of n real variables

x
=
(

x
1

,

x
2

,
⋯
,

x
n

)

x = ({x_1},{x_2}, \cdots ,{x_n})

of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript beta Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript i Endscripts sigma-summation Underscript j Endscripts x Subscript rho left-parenthesis i comma 1 right-parenthesis Superscript beta 1 e Super Subscript j Baseline 1 Superscript Baseline x Subscript rho left-parenthesis i comma 2 right-parenthesis Superscript beta 2 e Super Subscript j Baseline 2 Superscript Baseline midline-horizontal-ellipsis x Subscript rho left-parenthesis i comma n right-parenthesis Superscript beta Super Subscript n Superscript e Super Subscript j n Superscript Baseline comma">

f
β

(
x
)
=

∑
i

∑
j

x

ρ
(
i
,
1
)

β
1

e

j
1

x

ρ
(
i
,
2
)

β
2

e

j
2

⋯

x

ρ
(
i
,
n
)

β
n

e

j
n

,

{f_\beta }(x) = \sum \limits _i {\sum \limits _j {x_{\rho (i,1)}^{{\beta _1}{e_{j1}}}x_{\rho (i,2)}^{{\beta _2}{e_{j2}}} \cdots } } x_{\rho (i,n)}^{{\beta _n}{e_{jn}}},

\]
conditions are given which ensure that

f
α

(
x
)
≦

f
β

(
x
)

{f_\alpha }(x) \leqq {f_\beta }(x)

for all

x
≧
0

x \geqq 0

.
A class of comparison theorems for nonselfadjoint elliptic equations.
Authors: Kreith Kurt
Published: June 2010

Seemingly disparate comparison theorems for nonselfadjoint elliptic equations have been established by C. A. Swanson and the author. The present paper establishes a class of comparison theorems involving an arbitrary vector valued function

p
(
x
)

p(x)

. Special choices of

p
(
x
)

p(x)

yield both of the comparison theorems cited above.
A Whitehead type theorem
Authors: Haslam H. B.
Published: June 2010

Let

F

\mathfrak {F}

denote the Serre class of finite abelian groups. We consider, for example, conditions under which a map which induces an

F

\mathfrak {F}

-epimorphism in homotopy also induces an

F

\mathfrak {F}

-epimorphism in homology.
A theorem on mean-value iterations
Authors: Franks R. L. , Marzec R. P.
Published: June 2010

In this paper we consider a function which continuously maps a closed interval of the real line into itself. It is shown that a particular mean-value iterative scheme always converges to a fixed point. The result is known for functions which have a unique fixed point. This condition is not required here.
The univalence of an integral
Authors: Causey W. M.
Published: June 2010

Let

f
(
z
)

f(z)

be a normalized function, analytic and univalent in the open unit disc. It is shown that if

g
(
z
)
=

∫
0
z

(
f
(
t
)

/

t
)

α

d
t

g(z) = \int _0^z {{{(f(t)/t)}^\alpha }dt}

, then

g
g

is univalent in the open unit disc if

α
\alpha

is a complex number satisfying

0
≦

|

α

|

≦
(
√
2
−
1
)

/

4

0 \leqq |\alpha | \leqq (\surd 2 - 1)/4

.
Bounds for classical polynomials derivable by matrix methods
Authors: Brenner J. L.
Published: June 2010

For a matrix with dominant diagonal, convenient upper and lower bounds for the determinant are available. Such bounds occur in the work of Price, Ostrowski, Hoffman, Haynsworth, and Brenner. In this article, the inequalities are used to bound the classical polynomials over a range of values of the argument.
A 𝑣-integral representation for linear operators on a space of continuous vector-valued functions
Authors: Edwards J. R. , Wayment S. G.
Published: June 2010

In this note an analytic representation is given for continuous linear operators from

C
(
X
)

C(X)

into a linear normed space Y where

C
(
X
)

C(X)

is the space of continuous functions on [0, 1] with values in a linear normed space X.
Realcompactness and partitions of unity
Authors: De Marco G. , Wilson R. G.
Published: June 2010

A characterization of realcompactness in terms of locally finite open coverings and associated partitions of unity is given. Another proof of two well-known theorems of Katětov and Shirota is obtained.
A Cantor-Lebesgue theorem in two dimensions
Authors: Cooke Roger
Published: June 2010

We prove a partial extension of the classical result of Cantor and Lebesgue to two dimensions, for circular summation of trigonometric series. We assume convergence almost everywhere rather than merely on a set of positive measure. Although the theorem we prove has applications in uniqueness theorems, its purpose is to suggest what may be true in general, since the proof does not generalize to higher dimensions.
A fixed point theorem for semi-groups of mappings with a contractive iterate
Authors: Sehgal V. M. , Thomas J. W.
Published: June 2010

In a recent paper, Felix E. Browder discussed continuous self-mappings on a metric space, satisfying a functional inequality. Browder gave sufficient conditions such that the successive approximations of any point for such mappings converge to a unique fixed point. In the present paper, Browder’s result is extended to a commutative semigroup of mappings and also to single mappings that are not necessarily continuous and satisfy a weaker form of the functional inequality.
Geodesics in metrical connections
Authors: Millman Richard S.
Published: June 2010

To each connection on a Riemannian manifold we define a tensor called the Q-tensor. We prove that two metrical connections have the same geodesics if and only if their Q-tensors are equal. We then show that any manifold of dimension greater than two admits many metrical connections having the same geodesics; in particular, the Q-tensor is a strictly weaker invariant than the torsion.
Endomorphisms of finitely presented modules
Authors: Sabbagh Gabriel
Published: June 2010

It is proved that every surjective or injective endomorphism of a finitely presented left module over a right perfect ring is an isomorphism.
The strict topology and spaces with mixed topologies
Authors: Cooper J. B.
Published: June 2010

In a paper about twenty years ago, and later papers, R. C. Buck introduced a new topology, the strict topology, on spaces of continuous functions on locally compact spaces. Since then, a considerable amount of work has been done on these and similar topologies by, among others, Conway, Collins, Rubel and Shields (see references [2], [3], [4], [6], [7], [15]). In the early nineteen-fifties, the Polish mathematicians, Alexiewicz and Semadeni, considered a vector space E, on which two norms are defined, and defined a notion of convergence of sequences on E which in some sense mixed the topologies given by the norms ([1] and later papers). In 1957, Wiweger [17] showed that under natural restrictions, the space E could be given a locally convex space structure where convergent sequences were precisely the sequences considered by Alexiewicz and Semadeni. Since then, this method of mixing topologies has been studied and generalised by several mathematicians ([18], [9]). The purpose of this note is to show that the strict topology for function spaces is a special case of a mixed topology. We then intend to use the theory of mixed topologies to give quick proofs of the basic results on strict topologies. This has the advantage of giving simpler proofs and of eliminating some heavy analysis. It also allows the definition of strict topologies in a more general setting than has been considered until now and in some cases gives new results for the standard situation.
Titus’ homotopies of normal curves
Authors: Francis George K.
Published: June 2010

A regular homotopy of a plane immersion of the circle, each of whose stages is a normal curve with the exception of a finite number of stages of the homotopy presenting a nonnegative convex double point self tangency or a transverse triple point, preserves the number of topologically inequivalent extensions of the immersion to an orientation preserving immersion of the disk. Extensions to properly interior mappings of the disk are similarly investigated.
Effectively minimizing effective fixed-points
Authors: Payne T. H.
Published: June 2010

This note answers an open problem posed by H. Rogers, Jr. on p. 202 of Theory of recursive functions and effective computability by proving the following invariant form of one of his results [op. cit., p. 200, Theorem XIV]: for any fixed-point function n there exists a recursive function g such that if z is an index of an effective operator

Ψ
\Psi

, then

g
(
z
)

g(z)

is also an index of

Ψ
\Psi

, and

n
^

(
g
(
z
)
)

\hat n(g(z))

is an index of the minimum fixed-point of

Ψ
\Psi

with respect to inclusion.
The differential ideals [𝑦^{𝑝}𝑧]
Authors: Newton M. E.
Published: June 2010

In this paper we obtain Levi bases for

[

y
2

z
]

[{y^2}z]

and

[

y
3

z
]

[{y^3}z]

, and develop the associated critical weight functions.
An extension of Morita’s metrization theorem
Authors: Alexander Charles C.
Published: June 2010

Morita proved that metrizability of a

T
0

{T_0}

space is equivalent to the existence of a sequence of locally finite closed covers which satisfies a refinement condition. We introduce the definition of a cushioned pair-semidevelopment and prove that the existence of a cushioned pair-semidevelopment in a

T
0

{T_0}

space is equivalent to the metrizability of the space. In addition to Morita’s theorem, it is seen that several other well-known metrization theorems are also immediate corollaries to the new theorem.
On the immersion of an 𝑛-dimensional manifold in 𝑛+1-dimensional Euclidean space
Authors: Halpern Benjamin
Published: June 2010

Consider the subset of

n
+
1

n + 1

-dimensional Euclidean space swept out by the tangent hyperplanes drawn through the points of an immersed compact closed connected n-dimensional smooth manifold. If this is not all of the Euclidean space, then the manifold is diffeomorphic to a sphere, the immersion is an embedding, the image of the immersion is the boundary of a unique open starshaped set, and the set of points not on any tangent hyperplane is the interior of the kernel of the open starshaped set. A converse statement also holds.
𝑝-solvable groups with few automorphism classes of subgroups of order 𝑝
Authors: Gross Fletcher
Published: June 2010

This paper investigates the relationship between the p-length,

l
p

(
G
)

{l_p}(G)

, of the finite p-solvable group G and the number,

a
p

(
G
)

{a_p}(G)

, of orbits in which the subgroups of order p are permuted by the automorphism group of G. If

p
>
2

p > 2

and

a
p

(
G
)
≦
2

{a_p}(G) \leqq 2

, it is shown that

l
p

(
G
)
≦

a
p

(
G
)

{l_p}(G) \leqq {a_p}(G)

. If

p
=
2

p = 2

and

a
2

(
G
)
=
1

{a_2}(G) = 1

, it is proved that either

l
p

(
G
)
≦

a
p

(
G
)

{l_p}(G) \leqq {a_p}(G)

or

G

/

O

2
′

(
G
)

G/{O_{2’}}(G)

is a specific group of order 48.
Existence and continuous dependence for a class of nonlinear neutral-differential equations
Authors: Grimm L. J.
Published: June 2010

This paper presents existence, uniqueness, and continuous dependence theorems for solutions of initial-value problems for neutral-differential equations of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime left-parenthesis t right-parenthesis equals f left-parenthesis t comma x left-parenthesis t right-parenthesis comma x left-parenthesis g left-parenthesis t comma x right-parenthesis right-parenthesis comma x prime left-parenthesis h left-parenthesis t comma x right-parenthesis right-parenthesis right-parenthesis comma x left-parenthesis 0 right-parenthesis equals x 0 comma">

x
′

(
t
)
=
f
(
t
,
x
(
t
)
,
x
(
g
(
t
,
x
)
)
,

x
′

(
h
(
t
,
x
)
)
)
,

x
(
0
)
=

x
0

,

x’(t) = f(t,x(t),x(g(t,x)),x’(h(t,x))),\quad x(0) = {x_0},

\]
where f, g, and h are continuous functions with

g
(
0
,

x
0

)
=
h
(
0
,

x
0

)
=
0

g(0,{x_0}) = h(0,{x_0}) = 0

. The existence of a continuous solution of the functional equation

z
(
t
)
=
f
(
t
,
z
(
h
(
t
)
)
)

z(t) = f(t,z(h(t)))

is proved as a corollary.
Mappings of indecomposable continua
Authors: Bellamy David P.
Published: June 2010

We show that every compact metric continuum is a continuous image of some indecomposable compact metric continuum.
On the univalence of some classes of regular functions
Authors: Libera R. J. , Livingston A. E.
Published: June 2010

Let

F
(
z
)

F(z)

be regular in the unit disk

Δ
=
{
z
:

|

z

|

>
1
}

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

and normalized by the conditions

F
(
0
)
=
0

F(0) = 0

and

F
′

(
0
)
=
1

F’(0) = 1

and let

2
f
(
z
)
=
[
z
F
(
z
)

]
′

2f(z) = [zF(z)]’

. The paper deals with the mapping properties of

f
(
z
)

f(z)

when

F
(
z
)

F(z)

is known. For example, if

F
(
z
)

F(z)

is starlike of order

α
,
0
≤
α
>
1

\alpha , 0 \leq \alpha > 1

, then the disk in which

f
(
z
)

f(z)

is always starlike of order

β
,
α
≤
β
>
1

\beta , \alpha \leq \beta > 1

, is determined. All results are sharp.
Nonstable homotopy groups of Thom complexes
Authors: Ledden P. J.
Published: June 2010

The first nonstable homotopy group of

M
O
(
n
)
,

π

2
n

(
M
O
(
n
)
)

MO(n),{\pi _{2n}}(MO(n))

, is computed for all n, together with the corresponding Postnikov invariant. The computations give a new proof of a theorem of Mahowald on the normal bundle of an imbedding.
Semicontinuity of nullity or deficiency implies normability of the space
Authors: Hosack John M.
Published: June 2010

In this paper the upper semicontinuity of nullity and deficiency on locally convex spaces is examined. If either is semicontinuous in the topology of uniform convergence on bounded sets on

L
(
X
)

L(X)

, then X is normable. If the invertible elements in

L
(
X
)

L(X)

are open, then X is normable. The results are applied to topological algebras.
The conjugation representation and fusionless extensions
Authors: Formanek Edward
Published: June 2010

Let G be a finite group with center Z. G acts on itself by conjugation, and this induces a permutation representation P of

G

/

Z

G/Z

. It has been conjectured that every complex irreducible representation of

G

/

Z

G/Z

occurs in P. This paper gives a counterexample to this conjecture.
The compact operators are not complemented in ℬ(ℋ)
Authors: Conway John B.
Published: June 2010

If

H

\mathcal {H}

is an infinite dimensional Hilbert space then it is shown that the space of compact operators is not complemented in the space of all bounded operators.
On torsion in 𝐻-spaces of rank two
Authors: Dai Kai K.
Published: June 2010

Let X be an arcwise connected H-space that is also a finite CW complex. If X has rank two and if its

mod

2

\bmod \;2

cohomology is primitively generated, then X has no p-torsion for

p
≧
5

p \geqq 5

.
Normally closed saturated formations
Authors: Hale Mark P.
Published: June 2010

Theorems of Gaschütz and Baer concerning the Frattini subgroup, the hypercenter, and nilpotent and supersolvable subgroups are extended to normally closed saturated formations.
A convergence theorem for limitärperiodisch 𝑇-fractions of rational functions
Authors: Hag Kari
Published: June 2010

We prove that a limitärperiodisch T-fraction, which corresponds to a rational function, converges (locally uniformly) to the original function in a certain domain.
A strong maximum principle for quasilinear parabolic differential inequalities
Authors: ARENA O.
Published: June 2010

A maximum principle for

C
1

{C^1}

solutions of quasilinear parabolic differential inequalities which retains the strong conclusion of Nirenberg’s well-known result [2] is established. The case of strongly differentiable solutions rather than of class

C
1

{C^1}

is also discussed.
Isotopic closed nonconjugate braids
Authors: Murasugi K. , Thomas R. S. D.
Published: June 2010

J. S. Birman has conjectured that, when a knot is represented by a closed braid on a minimal number n of strands, the conjugacy class of the braid exhausts the set of braids in

B
n

{B_n}

closing to define the knot. Counterexamples are given to disprove the conjecture, even when it is weakened to refer only to oriented knots.
Rings with involution whose symmetric elements are regular
Authors: Lanski Charles
Published: June 2010

In this work we determine the structure of 2-torsion-free rings with involution whose nonzero symmetric elements do not annihilate one another.
Categorical semigroups
Authors: McMorris F. R. , Satyanarayana M.
Published: June 2010

The main purpose of this paper is to describe some properties of categorical semigroups, commutative semigroups which are categorical at zero, and determine the structure of commutative categorical semigroups. We also investigate whether Petrich’s tree condition, for categorical semigroups which are completely semisimple inverse semigroups, is necessary for arbitrary categorical semigroups.
Centralizers of Fourier algebra of an amenable group
Authors: Renaud P. F.
Published: June 2010

Let G be a locally compact group with Fourier algebra

A
(
G
)

A(G)

. We prove that if G is amenable then every centralizer of

A
(
G
)

A(G)

is determined by multiplication with an element of the Fourier-Stieltjes algebra of G. This result is then used to show that isometric centralizers correspond to characters of G.
Boundary behavior of starlike functions
Authors: Eenigenburg P. J.
Published: June 2010

For a starlike function f, we impose a geometric condition on the image of the open unit disc by the mapping

w
=
z

f
′

(
z
)

/

f
(
z
)

w = zf’(z)/f(z)

to insure that f be one-to-one on the closed unit disc. Applications are given to certain classes of univalent functions, including spiral-like functions.
Application of a lemma on bilinear forms to a problem in nonlinear oscillations
Authors: Lazer A. C.
Published: June 2010

In this note we give a simple condition for nondegeneracy of symmetric bilinear forms on infinite dimensional vector spaces. We apply this condition and elementary properties of Fourier series to prove a uniqueness theorem for periodic solutions of a class of second order nonlinear differential systems.
Submodules of 𝐶(𝑋)×⋯×𝐶(𝑋)
Authors: Roth B.
Published: June 2010

Let

C
(
X
)

C(X)

be the ring of continuous real-valued functions on a compact Hausdorff space X with the sup norm topology. In the present paper, the closed submodules of the

C
(
X
)

C(X)

-module

C
(
X
)
×
⋯
×
C
(
X
)

C(X) \times \cdots \times C(X)

are characterized and a necessary and sufficient condition for a submodule of

C
(
X
)
×
⋯
×
C
(
X
)

C(X) \times \cdots \times C(X)

to be closed and finitely generated is established.
Differentiable functions and rough norms on Banach spaces
Authors: Leach E. B. , Whitfield J. H. M.
Published: June 2010

The main result is that if X is a real Banach space, such that the density character of

X
∗

{X^\ast }

is greater than that of X, then there does not exist any real-valued Fréchet differentiable function on X with bounded nonempty support.
A Baire space extension
Authors: McCoy R. A.
Published: June 2010

A characterization of Baire spaces is given. Using this characterization, it is shown that every topological space is a dense subspace of some compact Baire space.
Recurrence and almost periodicity in a generative transformation group
Authors: Roberson Fred A.
Published: June 2010

A point p in a transformation group (X, T, II) is recurrent if for every neighborhood U of p there is an extensive set

E
⊂
T

E \subset T

such that

p
E
⊂
U

pE \subset U

. The point p is almost periodic if there is a syndetic set

A
⊂
T

A \subset T

such that

p
A
⊂
U

pA \subset U

. This paper proves that a recurrent point in a locally compact, generative transformation group with an equicontinuous neighborhood must also be almost periodic.
A fixed point theorem for (𝑛-2)-connected 𝑛-polyhedra
Authors: Waggoner Roger
Published: June 2010

The main result of this paper is that, for

n
≧
4

n \geqq 4

, a finite

(
n
−
2
)

(n - 2)

-connected polyhedron K of dimension n admits a fixed point free map if either

H

n
−
1

(
K
;
Q
)

{H_{n - 1}}(K;Q)

or

H
n

(
K
;
Q
)

{H_n}(K;Q)

is nonzero, where Q is the field of rational numbers. This result is obtained by first retracting K onto a subpolyhedron C of dimension n or

n
−
1

n - 1

which has no local separating points. It is then shown that C admits a map with Lefschetz number zero, and it follows from a theorem of Shi that C does not have the fixed point property. The proof involved may also be applied when K is a 3-dimensional simply connected polyhedron and the subpolyhedron C is also of dimension 3.
Fiber homotopic trivial bundles over complex projective spaces
Authors: Lam Kee Yuen
Published: June 2010

We give a more illuminating derivation of a well-known condition (obtained by M. Atiyah and J. A. Todd) for certain vector bundles over complex projective spaces to be stably fiber homotopic trivial, together with a generalization.
Distributive sublattices of a lattice
Authors: Kolibiar M.
Published: June 2010

There are given necessary and sufficient conditions for a subset of a lattice L (or of a complete lattice L) to generate a distributive sublattice of L (or a completely distributive closed sublattice of L).
On the accumulation of the zeros of a Blaschke product at a boundary point
Authors: Protas David
Published: June 2010

Let B be a Blaschke product with zeros

{

a
n

}

\{ {a_n}\}

. The series

∑

(
1
−

|

a
n

|

2

)

/

|

1
−

ζ
¯

a
n

|

γ

\sum {(1 - |{a_n}{|^2})} /|1 - \bar \zeta {a_n}{|^\gamma }

, where

γ
≧
1

\gamma \geqq 1

and

|

ζ

|

=
1

|\zeta | = 1

, has been used by P. R. Ahern, D. N. Clark, G. T. Cargo, and others in the study of the boundary behavior of B and various associated functions. In this paper the convergence of this series is shown to be equivalent to a condition on a reproducing kernel for a subspace of the Hardy space

H
2

{H^2}

. Some related conditions and corollaries are also given.
On the extremal solutions of 𝑛th-order linear differential equations
Authors: Kim W. J.
Published: June 2010

Distribution of zeros of extremal solutions of linear nth-order differential equations is discussed. Existence and nonexistence of extremal solutions with certain zero distributions are established. For instance, it is proved that every extremal solution for

[
α
,

η
1

(
α
)
]

[\alpha , {\eta _1}(\alpha )]

of the equation

y

(
n
)

+

p

n
−
1

y

(
n
−
1
)

+
⋯
+

p
0

y
=
0

{y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y = 0

has a zero of order 2 at

η
1

(
α
)

{\eta _1}(\alpha )

and has no more than

n
−
2

n - 2

zeros on

[
α
,

η
1

(
α
)
)

if

p
i

≦
0
,
i
=
0
,
1
,
⋯
,
n
−
2

[\alpha , {\eta _1}(\alpha ))\;{\text {if}}\;{p_i} \leqq 0,i = 0,1, \cdots , n - 2

.
Projecting 𝐶(𝑆) onto 𝐶₀(𝑆)
Authors: Banilower H.
Published: June 2010

If a locally compact Hausdorff space S has a denumerable discrete closed subspace N for which there exists a simultaneous extension E from

C
(
N
)

C(N)

into

C
(
S
)

C(S)

satisfying

E
(

C
0

(
N
)
)
⊂

C
0

(
S
)

E({C_0}(N)) \subset {C_0}(S)

, then

C
0

(
S
)

{C_0}(S)

is uncomplemented in

C
(
S
)

C(S)

. This holds whenever (i) S is not pseudocompact, or (ii) S is not countably compact and is a subspace of a basically disconnected space.
Annihilator ideals in the cohomology of Banach algebras
Authors: Sinclair A. M.
Published: June 2010

If A is a

C
∗

{C^\ast }

-algebra, if X is a Banach A-module, and if J is the annihilator of X in A, then the cohomology space

H

n

(
A
,

X
∗

)

{\mathcal {H}^n}(A,{X^\ast })

is isomorphic to

H

n

(
A

/

J
,

X
∗

)

{\mathcal {H}^n}(A/J,{X^\ast })

for each positive integer n.
Lattice-invariant topological properties
Authors: Sharma P. L.
Published: June 2010

The purpose of this paper is to show that an isomorphism between the lattices of all closed sets of two topological spaces induces an isomorphism on the lattices of their zero-sets. This is achieved by showing that any continuous real-valued function on a space X can be transferred to any space lattice-equivalent to X. Several topological properties are shown to be lattice-invariant.
Shift dynamical systems over finite fields
Authors: Nathanson Melvyn B.
Published: June 2010

A trajectory over the finite field

F
q

{F_q}

is a function from the integers I to

F
q

{F_q}

. The set

X
(

F
q

)

X({F_q})

of all trajectories over

F
q

{F_q}

is a topological vector space in the product topology induced by the discrete topology on

F
q

{F_q}

, and coordinatewise addition and scalar multiplication of trajectories. Let

ϕ
\phi

be a continuous linear operator on

X
(

F
q

)

X({F_q})

which commutes with the shift. If x is a trajectory over

F
q

{F_q}

, then the

ϕ
\phi

-orbit of x is the sequence of trajectories

x
,
ϕ
(
x
)
,

ϕ
2

(
x
)
,
⋯

x,\phi (x),{\phi ^2}(x), \cdots

. Suppose that

ϕ
\phi

is not a scalar multiple of the identity. Theorem. The trajectory x is periodic if and only if the

ϕ
\phi

-orbit of x is eventually periodic.
The pseudoinverse of an 𝑟-circulant matrix
Authors: Stallings W. T. , Boullion T. L.
Published: June 2010

It is shown that the Moore-Penrose pseudoinverse

C
+

{C^ + }

of an r-circulant matrix C is always the conjugate transpose of an r-circulant matrix. In addition, necessary and sufficient conditions are given for

C
+

{C^ + }

to be an s-circulant matrix. Finally, a method for calculating

C
+

{C^ + }

is given.
Existence and nonuniqueness of invariant means on ℒ^{∞}(𝒢̂)
Authors: Dunkl Charles F. , Ramirez Donald E.
Published: June 2010

For G an infinite compact group we show the existence and nonuniqueness of invariant means on the dual of the Fourier algebra. It follows that the space of weakly almost periodic functionals on the Fourier algebra is a proper closed subspace of the dual of the Fourier algebra.
Peano curves in function algebras
Authors: Eifler Larry Q.
Published: June 2010

We give a short proof of the following result which was obtained by Pełczyński. If X is an uncountable, compact metric space and if A is a function algebra on X, then there exists f in A such that

f
(
X
)

f(X)

has interior in the plane.
On characterization of Riemannian manifolds by growth of tubular neighborhoods
Authors: Grossman Nathaniel
Published: June 2010

If the area function of the tubular neighborhoods of a compact submanifold of a Riemannian manifold satisfies a certain linear differential inequality, then the codimension of the submanifold is at most the order of that inequality.
A countable connected Urysohn space containing a dispersion point
Authors: Kannan V.
Published: June 2010

Answering a question of Martin, Roy gave the first example of a countable connected Urysohn space with a dispersion point. Here we give a much simpler example of such a space.
Two new proofs of Lerch’s functional equation
Authors: Berndt Bruce C.
Published: June 2010

One bright Sunday morning I went to church, And there I met a man named Lerch. We both did sing in jubilation, For he did show me a new equation. Two simple derivations of the functional equation of
\[

∑

n
=
0

∞

exp
⁡
[
2
π
i
n
x
]

(
n
+
a
)

−
s

\sum \limits _{n = 0}^\infty {\exp [2\pi inx]{{(n + a)}^{ - s}}}

\]
are given. The original proof is due to Lerch.
Compact spaces and products of finite spaces
Authors: Harris Douglas
Published: June 2010

It is shown that the compact spaces are precisely the extension closed subspaces of products of finite spaces, where a subspace is extension closed if every open cover of the subspace extends to an open cover of the entire space. Every closed subspace is extension closed, and for Hausdorff spaces the converse also holds. A single compact space U is constructed, such that every compact space is an extension closed subspace of a product of copies of U; this parallels precisely the property possessed by the unit interval with respect to compact Hausdorff spaces.
Some examples of nonfinite diffeomorphism groups
Authors: Lawson Terry C.
Published: June 2010

An example due to Guy Hirsch is modified to give examples of manifolds N such that Diff N is not dominated by a finite complex.
On the ring of quotients of a group ring
Authors: Passman D. S.
Published: June 2010

In this paper the following extension of a result of Martha Smith is proved. Theorem. Let

K
[
G
]

K[G]

be a group ring which is an order in a ring Q. Then the center of

K
[
G
]

K[G]

is an order in the center of Q.
A finitely presented metabelian group with a free abelian derived group of infinite rank
Authors: Baumslag Gilbert
Published: June 2010

An example is given of a 3-generator, 3-relator group whose derived group is a free abelian group of infinite rank.
Euclidean neighborhoods in compact semigroups
Authors: McCharen John D.
Published: June 2010

In this paper sufficient conditions are given under which a maximal idempotent in a compact connected semigroup cannot have a two-dimensional Euclidean neighborhood.
On the Fredholm alternative for nonlinear functional equations in Banach spaces
Authors: Hess Peter
Published: June 2010

The well-known Fredholm alternative theorem for compact linear operators is carried over to a class of noncompact, asymptotically linear mappings of monotone type of a real reflexive Banach space into its dual. An application to a nonlinear elliptic boundary value problem is given.
Orders in self-injective cogenerator rings
Authors: Shock Robert C.
Published: June 2010

This note states necessary and sufficient conditions for a ring to be a right order in certain self-injective rings. A ring R is said to have the dense extension property if each R-homomorphism from a right ideal of R into R can be lifted to some dense right ideal of R. A right ideal K is rationally closed if for each

x
∈
R
−
K

x \in R - K

the set

x

−
1

K
=
{
y
∈
R
:
x
y
∈
K
}

{x^{ - 1}}K = \{ y \in R:xy \in K\}

is not a dense right ideal of R. We state a major result. Let

dim

R

\dim \, R

denote the Goldie dimension of a ring R and

Z
(
R
)

Z(R)

the right singular ideal of R. Then R is a right order in a self-injective cogenerator ring if and only if R has the dense extension property,

Z
(
R
)

Z(R)

is rationally closed and the factor ring

R

/

Z
(
R
)

R/Z(R)

is semiprime with

dim

R

/

Z
(
R
)
=
dim
⁡
R
>
∞

\dim \, R/Z(R) = \dim R > \infty

.
Some properties of special functions derived from the theory of continuous transformation groups
Authors: Das Mrinal Kanti
Published: June 2010

The theory of continuous transformation groups is utilized in the study of some properties of special functions. On constructing the continuous transformation groups corresponding to a suitably defined infinitesimal transformation, a bilateral generating relation involving Laguerre polynomials

{

L
n

(
α
)

(
x
)
}

\{ L_n^{(\alpha )}(x)\}

is obtained in

§
2

\S 2

. It is shown to be a generalisation of Brafman’s result. In the last section raising and lowering operators for

{

P
n

(
α
,
β
−
n
)

(
x
)
}

\{ P_n^{(\alpha ,\beta - n)}(x)\}

and their commutator are introduced and on showing that they generate a 3-dimensional Lie algebra, the idea of c.t. groups is employed to establish a generating relation involving

{

P
n

(
α
,
β
−
n
)

(
x
)
}

\{ P_n^{(\alpha ,\beta - n)}(x)\}

which is seen to yield a number of known results. Moreover, a bilateral generating relation involving

{

P
n

(
α
,
β
−
n
)

(
x
)
}

\{ P_n^{(\alpha ,\beta - n)}(x)\}

is obtained; this is seen to be a generalisation of a well-known relation due to Weisner.
Gauge functions and limit sets for nonautonomous ordinary differential equations
Authors: Wilson H. K.
Published: June 2010

A gauge function V for the differential equation

(

S

)

x
′

=
f
(
x
,
t
)

({\text {S}})x’ = f(x,t)

is a scalar-valued function sufficiently smooth for

d
V
(
ϕ
(
t
)
,
t
)

/

d
t

dV(\phi (t),t)/dt

to exist almost everywhere for solutions

x
=
ϕ
(
t
)

x = \phi (t)

,

t
0

≦
t
>

τ
ϕ
+

{t_0} \leqq t > \tau _\phi ^ +

, of (S). Let (S) have gauge function V that satisfies the following conditions: (1)

lim

t
→
+
∞

V
(
x
,
t
)
≡
λ
(
x
)

{\lim _{t \to + \infty }}V(x,t) \equiv \lambda (x)

exists; (2) V is continuous in x uniformly with respect to t; (3) the upper, right derivate of V with respect to (S) is nonpositive. Then, if a solution

x
=
ϕ
(
t
)

x = \phi (t)

of (S) has an

ω
\omega

-limit point P, there is a unique constant

c
(
ϕ
)

c(\phi )

such that

λ
(
p
)
=
c
(
ϕ
)

\lambda (p) = c(\phi )

. An application to second order, linear equations is given.
Second order almost linear functional differential equations—oscillation
Authors: Teufel Hugo
Published: June 2010

It is shown that all solutions of certain second order nonlinear functional differential equations are oscillatory if all solutions of an associated minorizing linear equation are oscillatory.
A note on some characterizations of Sidon sets
Authors: Blei Ron C.
Published: June 2010

A simple proof of the following theorem is presented:

E
⊂
Z

E \subset Z

is a Sidon set if and only if

A
E

(
T
)

{A_E}(T)

equals any one of the following three spaces:

U
E

(
T
)
,
∩
p
>
1

A
E
p

(
T
)

{U_E}(T), \cap p > 1A_E^p(T)

, and

C
E

(
T
,
ω
)

{C_E}(T,\omega )

.
Separability in the strict and substrict topologies
Authors: Summers W. H.
Published: June 2010

Let X denote a locally compact Hausdorff space,

C
b

(
X
)

{C_b}(X)

the space of all bounded continuous complex valued functions on X, and

β
\beta

the strict topology for

C
b

(
X
)

{C_b}(X)

. The separability of

(

C
b

(
X
)
,
β
)

({C_b}(X),\beta )

is characterized in terms of X, albeit in a more general context. This result provides a negative answer to a conjecture made by Todd, contains the classical separability theorems for continuous function spaces, and relates to the concepts of

τ
\tau

-smooth and tight functionals.
Commutants that do not dilate
Authors: Clark Douglas N.
Published: June 2010

The Lifting Theorem deals with dilation of the commutant of an operator

T
1

{T_1}

on Hilbert space. In this note, counterexamples are given to generalizations of the theorem involving N commuting operators

T
1

,

T
2

,
⋯
,

T
N

{T_1},{T_2}, \cdots ,{T_N}

.
Filter characterizations of 𝐶- and 𝐶*-embeddings
Authors: Green John William
Published: June 2010

A filter F on a space S is completely regular if the complement of each set in F is completely separated from some set in F. A characterization of the Stone-Čech compactification due to Alexandroff is used to establish the following theorem. Suppose K is a subspace of a Tychonoff space S. K is

C
∗

{C^ \ast }

-embedded in S if and only if the trace on K of every maximal completely regular filter on S intersecting K is maximal completely regular on K. A similar characterization of the C-embedded subsets of a Tychonoff space is obtained as are several related results.
Metrization of symmetric spaces and regular maps
Authors: Martin Harold W.
Published: June 2010

A symmetric d for a topological space R is said to be coherent if whenever

{
x
(
n
)
}

\{ x(n)\}

and

{
y
(
n
)
}

\{ y(n)\}

are sequences in R with

d
(
x
(
n
)
,
y
(
n
)
)
→
0

d(x(n),y(n)) \to 0

and

d
(
x
(
n
)
,
x
)
→
0

d(x(n),x) \to 0

, then

d
(
y
(
n
)
,
x
)
→
0

d(y(n),x) \to 0

. V. Niemytzki and W. A. Wilson have essentially shown that a topological space R is metrizable if and only if R is symmetrizable via a coherent symmetric. Conditions on a symmetric d which are equivalent to d being coherent are established. As a consequence, a theorem of A. Arhangel’skiĭ may be refined by showing that if

f
:
R
→
Y

f:R \to Y

is a quotient map from a metrizable space R onto a

T
0

{T_0}

-space y, then Y is metrizable if and only if f is a regular map.
Smooth structure and signature of codimension 2 embeddings
Authors: Erle Dieter
Published: June 2010

In an earlier paper (Topology 8 (1969), 99-114), the author defined a signature for codimension 2 embeddings in

S

4
k
+
1

{S^{4k + 1}}

and proved this signature to be an invariant of the topological type of the embedding. For embedded homotopy

(
4
k
−
1
)

(4k - 1)

-spheres, this signature is known to detect the smooth structure. It turns out that it determines the smooth structure also for integer homology

(
4
k
−
1
)

(4k - 1)

-spheres and, by a result of A. Durfee, for closed

(
2
k
−
2
)

(2k - 2)

connected

(
4
k
−
1
)

(4k - 1)

-manifolds with finite

(
2
k
−
1
)

(2k - 1)

-dimensional homology. As a consequence, in the above cases, the smooth structure is given by the topological type of the embedding. On the other hand, for

k
=
3
,
4
,
5
,
7
,
15

k = 3,4,5,7,15

, we exhibit examples of

(
4
k
−
1
)

(4k - 1)

manifolds embedded in a

(
4
k
+
1
)

(4k + 1)

-sphere for which the smooth structure is not determined by the signature of the embedding.
On subfields of countable codimension
Authors: Białynicki-Birula A.
Published: June 2010

In [2] the authors asked if any two real closed subfields

R
,

R
′

R, R’

of the field of complex numbers C such that

R
(
√
(
−
1
)
)
=

R
′

(
√
(
−
1
)
)
=
C

R(\surd ( - 1)) = R’(\surd ( - 1)) = C

are isomorphic. It is not difficult to see that the answer is negative. This is proved in the first part of the note. In the second we study the problem if any field which is not prime contains a proper subfield of countable (finite or infinite) codimension.
On embedding annuli in 𝑀³
Authors: Feustel C. D.
Published: June 2010

Let M be a 3-manifold. In this note we give a condition when a “nontrivial” proper map of an annulus into

M
3

{M^3}

can be replaced by a “nontrivial” proper embedding of an annulus in

M
3

{M^3}

.
The finiteness of 𝐼 when 𝑅[𝑋]/𝐼 is 𝑅-flat. II
Authors: Heinzer William , Ohm Jack
Published: June 2010

This paper supplements work of Ohm-Rush. A question which was raised by them is whether

R
[
X
]

/

I

R[X]/I

is a flat R-module implies I is locally finitely generated at primes of

R
[
X
]

R[X]

. Here R is a commutative ring with identity, X is an indeterminate, and I is an ideal of

R
[
X
]

R[X]

. It is shown that this is indeed the case, and it then follows easily that I is even locally principal at primes of

R
[
X
]

R[X]

. Ohm-Rush have also observed that a ring R with the property “

R
[
X
]

/

I

R[X]/I

is R-flat implies I is finitely generated” is necessarily an

A
(
0
)

A(0)

ring, i.e. a ring such that finitely generated flat modules are projective; and they have asked whether conversely any

A
(
0
)

A(0)

ring has this property. An example is given to show that this conjecture needs some tightening. Finally, a theorem of Ohm-Rush is applied to prove that any R with only finitely many minimal primes has the property that

R
[
X
]

/

I

R[X]/I

is R-flat implies I is finitely generated.
A splitting ring of global dimension two
Authors: Fuelberth John D. , Teply Mark L.
Published: June 2010

In this paper an example is given of a ring with left global dimension 2 having the property that the singular submodule of any R-module A is a direct summand of A. Although the example given is quite specific, the methods can be used to construct a fairly large class of these rings.
Spectra of conservative matrices
Authors: Sharma N. K.
Published: June 2010

In this paper we study the spectra of conservative matrices and show that the spectrum of any Hausdorff method is either uncountable or finite. In the latter case it is shown that the spectrum consists of either one point or two points. We obtain the snarpest possible Mercerian theorem for Euler methods. We also get some results about the location of conservative matrices with respect to the maximal group of invertible operators.
Another locally connected Hausdorff continuum not connected by ordered continua
Authors: Cornette J. L. , Lehman B.
Published: June 2010

An example is given of a locally connected Hausdorff continuum which is not connected by ordered continua.
Quadratic Jordan algebras whose elements are all invertible or nilpotent
Authors: McCrimmon Kevin
Published: June 2010

We prove that if

J

\mathfrak {J}

is a unital quadratic Jordan algebra whose elements are all either invertible or nilpotent, then modulo the nil radical

N

\mathfrak {N}

the algebra

J

/

N

\mathfrak {J}/\mathfrak {N}

is either a division algebra or the Jordan algebra determined by a traceless quadratic form in characteristic 2. We also show that if

U

\mathfrak {U}

is an associative algebra with involution whose symmetric elements are either invertible or nilpotent, then modulo its radical

U

/

ℜ

\mathfrak {U}/\Re

is a division algebra, a direct sum of anti-isomorphic division algebras, or a split quaternion algebra.
The differentiability of Riemann’s functions
Authors: Smith A.
Published: June 2010

The function

g
(
x
)
=

∑

p
=
1

∞

(
sin
⁡
π

p
2

x

/

π

p
2

)

g(x) = \sum \nolimits _{p = 1}^\infty {(\sin \pi {p^2}x/\pi {p^2})}

, thought by Riemann to be nowhere differentiable, is shown to be differentiable only at rational points expressible as the ratio of odd integers. The proof depends on properties of Gaussian sums, and these properties enable us to give a complete discussion of the possible existence of left and right derivatives at any point.
On the consistency theorem in matrix summability
Authors: Atalla Robert E.
Published: June 2010

We give a generalization of the consistency theorem for bounded convergence fields. The space c of convergent sequences is replaced by a space of more general type. Applications of the generalized consistency theorem are made to multiplicative summability theory. In particular we give conditions under which a generalized convergence field is an algebra.
On linear Volterra integral equations of convolution type
Authors: Lew John S.
Published: June 2010

Let A be the set of all complex-valued locally integrable functions defined on

[
0
,
+
∞
)

[0, + \infty )

, and let T be the topology for A determined by the seminorms

t
r

(
f
)
=

∫
0
r

|

f
(
x
)

|

d
x

{t_r}(f) = \smallint _0^r {|f(x)|dx}

for

r
=
1
,
2
,
⋯

r = 1,2, \cdots

, so that A is a topological algebra under pointwise addition, complex scalar multiplication, and Laplace convolution. Then the map

f
→

f
′

f \to f’

from each element to its quasi-inverse is a homeomorphism of (A, T) onto itself. For each f, g in A the equation

v
=
f
+

g
∗

v

v = f + g ^\ast v

has a unique solution in A which depends T-continuously on f, g, and is the T-limit of Picard approximations. The set of all f in A with

f
′

f’

in

L
1

[
0
,
+
∞
)

{L^1}[0, + \infty )

is a set of first category in (A, T) but an open subset of A with the metric

‖

f
−
g

‖

1

{\left \| {f - g} \right \|_1}

. For each series

∑

n
=
1

∞

p
n

z
n

\sum \nolimits _{n = 1}^\infty {{p_n}{z^n}}

converging in some neighborhood of

z
=
0

z = 0

, and each element f in A, the series

∑

n
=
1

∞

p
n

f

∗
n

\sum \nolimits _{n = 1}^\infty {{p_n}{f^{ \ast n}}}

converges in T to some element

p
∗

(
f
)

{p^ \ast }(f)

in A.
A characterization of spaces on which all path maps are continuous
Authors: Harley P. W.
Published: June 2010

Here it is shown that all path maps on a topological space X are continuous if and only if X is a quotient image of a locally path connected metric space. This characterization strengthens results of E. Bedford [1].
Rational functions representing all residues 𝑚𝑜𝑑𝑝. II
Authors: Mordell L. J.
Published: June 2010

Two rational functions representing all residues

mod

p

\bmod \;p

are given.