bluuebunny/crossref_metadata_2025 huggingface dataset:
A fixed point theorem for asymptotically nonexpansive mappings
Authors: Goebel K. , Kirk W. A.
Published: June 2010

Let K be a subset of a Banach space X. A mapping

F
:
K
→
K

F:K \to K

is said to be asymptotically nonexpansive if there exists a sequence

{

k
i

}

\{ {k_i}\}

of real numbers with

k
i

→
1

{k_i} \to 1

as

i
→
∞

i \to \infty

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper F Superscript i Baseline x minus upper F Superscript i Baseline y double-vertical-bar less-than-over-equals k Subscript i Baseline double-vertical-bar x minus y double-vertical-bar comma x comma y element-of upper K">

‖

F
i

x
−

F
i

y

‖

≦

k
i

‖

x
−
y

‖

,
x
,
y
∈
K

\left \| {{F^i}x - {F^i}y} \right \| \leqq {k_i}\left \| {x - y} \right \|,x,y \in K

. It is proved that if K is a non-empty, closed, convex, and bounded subset of a uniformly convex Banach space, and if

F
:
K
→
K

F:K \to K

is asymptotically nonexpansive, then F has a fixed point. This result generalizes a fixed point theorem for nonexpansive mappings proved independently by F. E. Browder, D. Göhde, and W. A. Kirk.
A dual to the bordism concept
Authors: Stong R. E.
Published: June 2010

A dual to bordism is defined by reversing arrows. For finite complexes this Grothendieckizes to give KO theory.
The location of the zeros of the derivatives of Faber polynomials
Authors: Ullman J. L.
Published: June 2010

We prove a property of Faber polynomials which supports a conjecture concerning the zeros of Faber polynomials.
Finitely generated submodules of differentiable functions
Authors: Roth B.
Published: June 2010

Suppose

[

E

m

(
Ω
)

]
p

{[{\mathcal {E}^m}(\Omega )]^p}

is the Cartesian product of the space of real-valued m-times continuously differentiable functions on an open set

Ω
\Omega

in

R
n

{R^n}

with itself p-times where m is finite and

Ω
\Omega

is connected.

[

E

m

(
Ω
)

]
p

{[{\mathcal {E}^m}(\Omega )]^p}

is a

E

m

(
Ω
)

{\mathcal {E}^m}(\Omega )

-module. The finitely generated submodules of

[

E

m

(
Ω
)

]
p

{[{\mathcal {E}^m}(\Omega )]^p}

are

im

(
F
)

{\text {im}}(F)

where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F colon left-bracket script upper E Superscript m Baseline left-parenthesis normal upper Omega right-parenthesis right-bracket Superscript q Baseline right-arrow left-bracket script upper E Superscript m Baseline left-parenthesis normal upper Omega right-parenthesis right-bracket Superscript p">

F
:

[

E

m

(
Ω
)

]
q

→

[

E

m

(
Ω
)

]
p

F:{[{\mathcal {E}^m}(\Omega )]^q} \to {[{\mathcal {E}^m}(\Omega )]^p}

is a

p
×
q

p \times q

matrix
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis f Subscript i j Baseline right-parenthesis Subscript 1 less-than-over-equals i less-than-over-equals p comma Baseline Subscript 1 less-than-over-equals j less-than-over-equals q Baseline comma f Subscript i j Baseline element-of script upper E Superscript m Baseline left-parenthesis normal upper Omega right-parenthesis">

(

f

i
j

)

1
≦
i
≦
p
,

1
≦
j
≦
q

,

f

i
j

∈

E

m

(
Ω
)

{({f_{ij}})_{1 \leqq i \leqq p,}}_{1 \leqq j \leqq q},{f_{ij}} \in {\mathcal {E}^m}(\Omega )

. In the present paper, it is shown that

im

(
F
)

{\text {im}}(F)

is closed in

[

E

m

(
Ω
)

]
p

{[{\mathcal {E}^m}(\Omega )]^p}

if and only if the rank of the matrix

(

f

i
j

(
x
)

)

1
≦
i
≦
p
,
1
≦
j
≦
q

{({f_{ij}}(x))_{1 \leqq i \leqq p,1 \leqq j \leqq q}}

is constant for

x
∈
Ω

x \in \Omega

. Applications are made to systems of division problems for distributions.
Contractible Hilbert cube manifolds
Authors: Chapman T. A.
Published: June 2010

In this note we give an example of a contractible Hilbert cube manifold which cannot be embedded as an open subset of the Hilbert cube Q so that its complement (in Q) lies in a face of the boundary of Q. This example provides a negative answer to a question raised by the author.
On topological properties of sets admitting varisolvent functions
Authors: Braess D.
Published: June 2010

Mairhuber’s theorem on Haar subspaces is generalized for the nonlinear case, where varisolvent functions are considered.
Surfaces with maximal Lipschitz-Killing curvature in the direction of mean curvature vector
Authors: Houh Chorng-shi
Published: June 2010

M
2

{M^2}

is an oriented surface in

E

2
+
N

{E^{2 + N}}

. If

M
2

{M^2}

is pseudo-umbilical, the Lipschitz-Killing curvature takes maximum in the direction of mean curvature vector. The converse is also investigated. Furthermore assuming that

M
2

{M^2}

is closed, pseudo-umbilical and its Gaussian curvature has some nonnegative lower bound,

M
2

{M^2}

is completely determined by the M-index of

M
2

{M^2}

.
Gauss’ lemma
Authors: Tang Hwa Tsang
Published: June 2010

Let

f
(
x
)

f(x)

be a polynomial in several indeterminates with coefficients in an integral domain R with quotient field K. We prove that the principal ideal generated by f in the polynomial ring

R
[
x
]

R[x]

is prime iff f is irreducible over K and

A

−
1

=
R

{A^{ - 1}} = R

where A is the content of f. We also prove that if

f
(
x
)

f(x)

is such that

A

−
1

=
R

{A^{ - 1}} = R

and

g
(
x
)

g(x)

is a primitive polynomial in the sense that only a unit of R can divide each coefficient of g, then fg will be primitive.
Characterization of rings using quasiprojective modules. III
Authors: Golan Jonathan S.
Published: June 2010

A ring

R
R

is regular [completely reducible] if and only if the character module of every left

R
R

-module is quasi-injective [quasiprojective]. Submodules of quasiprojective left

R
R

-modules over a left perfect ring

R
R

are quasiprojective if and only if singular left

R
R

-modules are injective. A splitting theorem for right perfect rings over which submodules of quasiprojective left

R
R

-modules are quasiprojective is also proven. These results continue the author’s previous work ([5] and [6]).
On the equivalence of three local theorem techniques
Authors: Hickin K. K. , Plotkin J. M.
Published: June 2010

A result formulated by P. Hall which is used in obtaining local theorems in group theory is shown to be equivalent to the projection set theorem of Kuroš and to the existence of Malcev vectors for local systems.
Topological properties of the efficient point set
Authors: Peleg Bezalel
Published: June 2010

Let y be a closed and convex subset of a Euclidean space. We prove that the set of efficient points of Y,

M
(
Y
)

M(Y)

, is contractible. Furthermore, if

M
(
Y
)

M(Y)

is closed (compact) then it is a retract of a convex closed (compact) set. Our proof relies on the Arrow-Barankin-Blackwell Theorem. A new proof is supplied for that theorem.
Noncoincidence of the strict and strong operator topologies
Authors: Shapiro Joel H.
Published: June 2010

Let E be an infinite-dimensional linear subspace of

C
(
S
)

C(S)

, the space of bounded continuous functions on a locally compact Hausdorff space S. If

μ
\mu

is a regular Borel measure on S, then each element of E may be regarded as a multiplication operator on

L
p

(
μ
)
(
1
≦
p
>
∞
)

{L^p}(\mu )(1 \leqq p > \infty )

. Our main result is that the strong operator topology this identification induces on E is properly weaker than the strict topology. For E the space of bounded analytic functions on a plane region G, and

μ
\mu

Lebesgue measure on G, this answers negatively a question raised by Rubel and Shields in [9]. In addition, our methods provide information about the absolutely p-summing properties of the strict topology on subspaces of

C
(
S
)

C(S)

, and the bounded weak star topology on conjugate Banach spaces.
Closed convex hypersurfaces with second fundamental form of constant curvature
Authors: Schneider Rolf
Published: June 2010

It is shown that the Euclidean spheres are the only closed hypersurfaces in Euclidean space on which the second fundamental form defines a (nondegenerate) Riemannian metric of constant curvature.
Inadequacy of ordinary homology theory
Authors: Lin T. Y.
Published: June 2010

Counterexamples to the homology version of Peterson’s theorem are constructed. Namely, maps are exhibited which theoretically cannot be detected by any primary or higher order homology operations.
The local rings of global dimension two
Authors: Vasconcelos Wolmer V.
Published: June 2010

A commutative local ring A of global dimension two conforms to the following description: (a) If the maximal ideal M is either principal or not finitely generated then A is a valuation domain. (b) Otherwise M is generated by a regular sequence of two elements but the ring is not necessarily noetherian. It will be noetherian if and only if it is completely integrally closed.
Topologically complete groups
Authors: Brown Lawrence G.
Published: June 2010

Topologically complete groups are characterized by the existence of a compact subgroup such that the coset space is topologically complete and metrizable. Coset spaces of topologically complete groups and extensions of one topologically complete group by another are again topologically complete. The open mapping theorem is valid for topologically complete groups.
Equivalent conditions for the 𝐿_{𝑝} convolution theorem on semisimple groups
Authors: Lipsman Ronald L.
Published: June 2010

For certain semisimple Lie groups G, it is known that convolution by an

L
p

(
G
)

{L_p}(G)

function,

1
≦
p
>
2

1 \leqq p > 2

, is a bounded operator on

L
2

(
G
)

{L_2}(G)

. This result is a consequence of the so-called “analytic continuation of the principal series” which has been carried out on these groups. However, this continuation procedure does not generalize readily to arbitrary semisimple groups. In an attempt to bypass the continuation and obtain the convolution theorem in an alternate manner, we derive in this paper several equivalent conditions for this

L
p

{L_p}

convolution theorem.
Applications of nilmanifold theory to Diophantine approximations
Authors: Brezin Jonathan
Published: June 2010

We indicate one method for applying harmonic analysis of nilmanifolds to problems of uniform distribution

mod

1

\bmod \;1

in R.
A factorable weight with zero Szegö infimum
Authors: Moore Berrien
Published: June 2010

The associated Szegö infimum of a factorable operator valued weight function need not be nonzero. An example is constructed using algebraic properties of vectorial Toeplitz operators.
Maintenance of oscillations under the effect of a periodic forcing term
Authors: Kartsatos Athanassios G.
Published: June 2010

A necessary and sufficient condition is given for the oscillation of all solutions of the differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript left-parenthesis n right-parenthesis Baseline plus upper P left-parenthesis t comma x comma x prime comma midline-horizontal-ellipsis comma x Superscript left-parenthesis n minus 1 right-parenthesis Baseline right-parenthesis equals upper Q left-parenthesis t right-parenthesis">

x

(
n
)

+
P
(
t
,
x
,

x
′

,
⋯
,

x

(
n
−
1
)

)
=
Q
(
t
)

{x^{(n)}} + P(t,x,x’, \cdots ,{x^{(n - 1)}}) = Q(t)

\]
where

x
1

P
(
t
,

x
1

,

x
2

,
⋯
,

x
n

)
>
0

{x_1}P(t,{x_1},{x_2}, \cdots ,{x_n}) > 0

for every

x
1

≠
0

{x_1} \ne 0

, and Q is a continuous periodic function. This result answers a question recently raised by J. S. W. Wong. It is also shown that a well-known sufficient condition for the existence of at least one nonoscillatory solution of the unperturbed equation guarantees, for a large class of equations, the nonexistence of bounded oscillatory solutions.
Generalized Ramsey theory for graphs. II. Small diagonal numbers
Authors: Chvátal Václav , Harary Frank
Published: June 2010

Consider a finite nonnull graph G with no loops or multiple edges and no isolated points. Its Ramsey number

r
(
G
)

r(G)

is defined as the minimum number p such that every 2-coloring of the lines of the complete graph

K
p

{K_p}

must contain a monochromatic G. This generalizes the classical diagonal Ramsey numbers

r
(
n
,
n
)
=
r
(

K
n

)

r(n,n) = r({K_n})

. We obtain the exact value of the Ramsey number of every such graph with at most four points.
Dirichlet finite biharmonic functions on the unit disk with distorted metrics
Authors: O’Malla Heppé
Published: June 2010

The Riemannian manifold

D
α

{D_\alpha }

obtained from the unit disk D by giving the distorted metric

(
1
−

|

z

|

)

−
α

|

d
z

|

{(1 - |z|)^{ - \alpha }}|dz|

does not admit any Dirichlet finite nonharmonic biharmonic function if and only if

α
≧

3
4

\alpha \geqq \tfrac {3}{4}

.
Strongly semi-aposyndetic continua are hereditarily locally connected
Authors: Vought Eldon J.
Published: June 2010

A stronger form of semi-aposyndesis is defined and used to characterize those compact, metric continua which are hereditarily locally connected.
A note on paired fibrations
Authors: Tulley McAuley Patricia
Published: June 2010

Consider pairs

(

X

,

A

)

(\mathcal {X},\mathcal {A})

where

X

=
(
X
,
p
,
B
)

\mathcal {X} = (X,p,B)

and

A

=
(
A
,
p

|

A
,
B
)

\mathcal {A} = (A,p|A,B)

are Hurewicz fibrations mapping onto B and

A
⊂
X

A \subset X

. It is proved that

(

X

,

A

)

(\mathcal {X},\mathcal {A})

is a cofibration if and only if

(

X

∪
f

Y

,

Y

)

(\mathcal {X}{ \cup _f}\mathcal {Y},\mathcal {Y})

is a strongly-paired fibration for each fibration

Y

=
(
Y
,
q
,
B
)

\mathcal {Y} = (Y,q,B)

and fiber map

f
:

A

→

Y

f:\mathcal {A} \to \mathcal {Y}

. It follows as a corollary that the notions of fiber homotopy equivalence and strong fiber homotopy equivalence [5] coincide for all Hurewicz fibrations. That

(

X

,

A

)

(\mathcal {X},\mathcal {A})

be “strongly-paired” requires more than that each lifting function for

A

\mathcal {A}

be extendable to

X

\mathcal {X}

. This and other notions of pairing are studied.
A note on embedding a partially ordered ring in a division algebra
Authors: Reynolds William H.
Published: June 2010

If H is a maximal cone of a ring A such that the subring generated by H is a commutative integral domain that satisfies a certain centrality condition in A, then there exist a maximal cone

H
′

H’

in a division ring

A
′

A’

and an order preserving monomorphism of A into

A
′

A’

, where the subring of

A
′

A’

generated by

H
′

H’

is a subfield over which

A
′

A’

is algebraic. Hypotheses are strengthened so that the main theorems of the author’s earlier paper hold for maximal cones.
On the bordism ring of complex projective space
Authors: Schochet Claude
Published: June 2010

The bordism ring

M

U
∗

(
C

P
∞

)

M{U_\ast }(C{P^\infty })

is central to the theory of formal groups as applied by D. Quillen, J. F. Adams, and others recently to complex cobordism. In the present paper, rings

E
∗

(
C

P
∞

)

{E_\ast }(C{P^\infty })

are considered, where E is an oriented ring spectrum,

R
=

π
∗

(
E
)

R = {\pi _\ast }(E)

, and

p
R
=
0

pR = 0

for a prime p. It is known that

E
∗

(
C

P
∞

)

{E_\ast }(C{P^\infty })

is freely generated as an R-module by elements

{

β
r

|

r
≧
0
}

\{ {\beta _r}|r \geqq 0\}

. The ring structure, however, is not known. It is shown that the elements

{

β

p
r

|

r
≧
0
}

\{ {\beta _{{p^r}}}|r \geqq 0\}

form a simple system of generators for

E
∗

(
C

P
∞

)

{E_\ast }(C{P^\infty })

and that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript p Sub Superscript r Superscript p Baseline identical-to s Superscript p Super Superscript r Baseline beta Subscript p Sub Superscript r Baseline mod left-parenthesis beta 1 comma midline-horizontal-ellipsis comma beta Subscript p Sub Superscript r minus 1 Subscript Baseline right-parenthesis">

β

p
r

p

≡

s

p
r

β

p
r

mod
(

β
1

,
⋯
,

β

p

r
−
1

)

\beta _{{p^r}}^p \equiv {s^{{p^r}}}{\beta _{{p^r}}}\bmod ({\beta _1}, \cdots ,{\beta _{{p^{r - 1}}}})

for an element

s
∈
R

s \in R

(which corresponds to

[
C

P

p
−
1

]

[C{P^{p - 1}}]

when

E
=
M
U

Z
p

)

E = MU{Z_p})

. This may lead to information concerning

E
∗

(
K
(
Z
,
n
)
)

{E_\ast }(K(Z,n))

.
A problem in additive number theory
Authors: Quiring Donald
Published: June 2010

For every real number

α
,
0
>
α
>
1

\alpha ,0 > \alpha > 1

, a sequence

A
=
{

a
1

,

a
2

,
⋯
}

A = \{ {a_1},{a_2}, \cdots \}

is constructed for which the density of

A
A

is

α
\alpha

and

A
A

has the following property: Given any

n
n

distinct positive integers

{

b
1

,

b
2

,
⋯
,

b
n

}

\{ {b_1},{b_2}, \cdots ,{b_n}\}

the sequence consisting of all numbers of the form

a
i

+

b
j

{a_i} + {b_j}

has density

1
−

(
1
−
α

)
n

1 - {(1 - \alpha )^n}

.
The kernel of a block of a group algebra
Authors: Michler Gerhard O.
Published: June 2010

Avoiding the theory of characters of finite groups and group algebras over fields of characteristic zero a ring theoretical proof is given for R. Brauer’s theorem which asserts that the (modular) kernel of a block of a group algebra FG of a finite group over a field F of characteristic

p
>
0

p > 0

is a p-nilpotent normal subgroup of G.
A note on enveloped measures
Authors: Sidney S. J.
Published: June 2010

There exists a uniform algebra

A

\mathfrak {A}

and a multiplicative linear functional

Φ
\Phi

on

A

\mathfrak {A}

which has an uncountable family of enveloped representing measures, no one of which is boundedly absolutely continuous with respect to any other.
Demicontinuity and hemicontinuity in Fréchet space
Authors: Junghenn Hugo D.
Published: June 2010

It is proved that the notions of demicontinuity and hemicontinuity for monotone maps from a Fréchet space into its dual are equivalent, thus generalizing a result of T. Kato.
Cyclic order and dissection order of certain arcs
Authors: Jackson S. B.
Published: June 2010

Let arc

A
A

in the conformal plane or on the sphere have local cyclic order three and cyclic order

t
t

. It can be decomposed into a finite number of subarcs of cyclic order three. Let the disection order of

A
A

be the minimum number of arcs in such a decomposition. The principal result of this paper is that the cyclic order

t
t

and dissection order

d
d

of

A
A

satisfy the relation

d
+
2
≦
t
≦
3
d

d + 2 \leqq t \leqq 3d

. In establishing this result it is proved that a necessary and sufficient condition, that an arc of local cyclic order three shall be of global cyclic order three, is that there exists a circle meeting it only at the endpoints.
A derivative in the setting of operator-valued measures
Authors: Edwards J. R. , Wayment S. G.
Published: June 2010

There has been considerable recent development concerning integration of vector-valued functions with respect to operator-valued measures, including various Radon-Nikodym type theorems. This paper is concerned with obtaining an analytic definition of derivative which will act as an inverse operator for these integration processes. A derivative is defined and, with additional conditions imposed on the measure, the two fundamental theorems of calculus are proved, i.e. that differentiation and integration are indeed inverse processes. Several examples are discussed to point out the shortcomings of this definition of derivative. Since this paper was written, some partial work has been done by D. H. Tucker and S. G. Wayment to adjust the definition to remove some of these defects. However, they were not then able to establish certain desirable theorems. For further details, the reader is referred to the Proc. Sympos. on Vector and Operator Valued Measures and Applications (Snowbird, Utah, 1972).
On the estimation of the 𝐿₂-norm of a function over a bounded subset of 𝑅ⁿ
Authors: Walker Homer F.
Published: June 2010

The objective of this paper is to present an estimate bounding the

L
2

{L_2}

-norm of a function over a bounded subset of

R
n

{R^n}

by the

L
2

{L_2}

-norms of its derivatives of arbitrary order over all of

R
n

{R^n}

and the

L
2

{L_2}

-norm of its projection onto a finite-dimensional space of functions with bounded support. The estimate essentially generalizes inequalities of Friedrichs [1, p. 284] and Lax and Phillips [2, p. 95]. An application of the estimate is made to the Fredholm theory of elliptic partial differential operators in

R
n

{R^n}

.
On finite invariant measures for Markov operators
Authors: Falkowitz M.
Published: June 2010

Two lemmas on proper vectors of convex linear combination of operators and semigroups in a Banach space are proved. They are applied to problems of invariant measures for Markov operators.
Sets of points of discontinuity
Authors: Bolstein Richard
Published: June 2010

In order that a subset

F
F

of a topological space coincide with the set of points of discontinuity of a real-valued function on the space, it is necessary that

F
F

be an

F
σ

{F_\sigma }

-set devoid of isolated points. It is shown that this condition is also sufficient if the space is “almost-resolvable", and in particular if the space is either separable, first countable, locally compact Hausdorff, or topological linear.
Realcompactifications of products of ordered spaces
Authors: McArthur William G.
Published: June 2010

The equality

υ
(
X
×
Y
)
=
υ
X
×
υ
Y

\upsilon (X \times Y) = \upsilon X \times \upsilon Y

is studied for the case when one of the factors is a linearly ordered topological space (LOTS). Among the results obtained are the following: 1. If

X
X

is any separable realcompact space and

Y
Y

is any LOTS of nonmeasurable cardinal, then

υ
(
X
×
Y
)
=
υ
X
×
υ
Y

\upsilon (X \times Y) = \upsilon X \times \upsilon Y

. 2. If

X
X

is a nonparacompact LOTS, then there is a paracompact LOTS

Y
Y

such that

υ
(
X
×
Y
)
≠
υ
X
×
υ
Y

\upsilon (X \times Y) \ne \upsilon X \times \upsilon Y

. 3. For any pair

X
,
Y

X,Y

of well-ordered spaces,

υ
(
X
×
Y
)
=
υ
X
×
υ
Y

\upsilon (X \times Y) = \upsilon X \times \upsilon Y

.
Regular 𝑃.𝐼.-rings
Authors: Armendariz E. P. , Fisher Joe W.
Published: June 2010

For a ring

R
R

which satisfies a polynomial identity we show that the following are equivalent: (1)

R
R

is von Neumann regular, (2) each two-sided ideal of

R
R

is idempotent, and (3) each simple left (right)

R
R

-module is injective. We show that a P.I.-ring

R
R

is left perfect if and only if all left

R
R

-modules have maximal submodules and

R
R

has no infinite sets of orthogonal idempotents.
The projective dimension of a compact abelian group
Authors: Nummela Eric C.
Published: June 2010

Let

X
X

be a compact group,

F
G
X

FGX

the (Graev) free topological group generated by

X
X

, and

K
K

the kernel of the canonical quotient morphism from

F
G
X

FGX

to

X
X

. Then

K
K

is a (Graev) free topological group. A corollary to the abelian analogue of this theorem is that the projective dimension of a compact abelian group, relative to the class of all continuous epimorphisms admitting sections, is exactly one.
On convex power series of a conservative Markov operator
Authors: Foguel S. R. , Weiss B.
Published: June 2010

A. Brunel proved that a conservative Markov operator,

P
P

, has a finite invariant measure if and only if every operator

Q
=

Σ

n
=
0

∞

α
n

P
n

Q = \Sigma _{n = 0}^\infty {\alpha _n}{P^n}

where

α
n

≧
0

{\alpha _n} \geqq 0

and

Σ

α
n

=
1

\Sigma {\alpha _n} = 1

is conservative. In this note we give a different proof and study related problems.
A characterization of the Browder spectrum
Authors: Salinas Norberto
Published: June 2010

Several different notions of essential spectra are known. This paper is devoted to giving an equivalent definition of the Browder essential spectrum for bounded operators on Hilbert space. Also, it is shown that the set of all operators with essentially disconnected spectrum is a uniformly dense set.
Fixed point theorems in reflexive Banach spaces
Authors: Kannan R.
Published: June 2010

In this paper fixed point theorems are established first for mappings

T
T

, mapping a closed bounded convex subset

K
K

of a reflexive Banach space into itself and satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper T x minus upper T y StartAbsoluteValue EndAbsoluteValue less-than-over-equals one half left-brace StartAbsoluteValue EndAbsoluteValue x minus upper T x StartAbsoluteValue EndAbsoluteValue plus StartAbsoluteValue EndAbsoluteValue y minus upper T y StartAbsoluteValue EndAbsoluteValue comma x comma y element-of upper K comma">

|

|

T
x
−
T
y

|

|

≦

1
2

{

|

|

x
−
T
x

|

|

+

|

|

y
−
T
y

|

|

,

x
,
y
∈
K
,

||Tx - Ty|| \leqq \tfrac {1}{2}\{ ||x - Tx|| + ||y - Ty||,\quad x,y \in K,

\]
and then an analogous result is obtained for nonexpansive mappings giving rise to a question regarding the unification of these theorems.
The Brauer group of an Amitsur field
Authors: Chang Chan Nan
Published: June 2010

In this paper we compute the entire Brauer groups of Amitsur fields (i.e. certain fields of formal power series

K
m

{K_m}

described in §1); these are interesting generalizations of the Brauer groups over the fields of local class field theory.
On Hardy’s inequality and Laplace transforms in weighted rearrangement invariant spaces
Authors: Andersen Kenneth F.
Published: June 2010

Hardy’s well-known inequality relating the norm of a function and the norm of its integral mean in the Lebesgue spaces

L
p

(
μ
)
,
d
μ
(
t
)
=

t

σ
−
1

d
t

{L^p}(\mu ),d\mu (t) = {t^{\sigma - 1}}dt

, is extended to the class of rearrangement invariant spaces

X
(
μ
)

X(\mu )

. These spaces include, for example, the

L
p

(
μ
)

{L^p}(\mu )

, the Lorentz and the Orlicz spaces. As an application, necessary and sufficient conditions are obtained for an operator related to the Laplace transform to be bounded as a linear operator between rearrangement invariant spaces of

μ
\mu

-measurable functions.
Bounds for the variational ratio of arbitrary linear systems
Authors: Wilkins J. Ernest
Published: June 2010

Melese-d’Hospital and Wilkins have recently found bounds for the error involved in replacing the maximum temperature in some geometrically simple regions, containing a nonuniformly distributed heat source, by the maximum temperature in the same region containing a uniform heat source. In this paper we present the formal mathematical reasoning in a general framework suitable for other applications.
Families of negatively curved Hermitian manifolds
Authors: Cowen Michael J.
Published: June 2010

A complex analytic family of compact hermitian manifolds has negative holomorphic sectional curvature in a neighborhood of any fibre having negative holomorphic sectional curvature.
General Wiener-Hopf operators and the numerical range of an operator
Authors: Pellegrini Victor J.
Published: June 2010

Let

H
H

be a separable Hilbert space and

A
A

a bounded operator on

H
H

. For a selfadjoint projection

P
P

on

H
H

we consider the general Wiener-Hopf operator

T
P

(
A
)
=
P

A

R
(
P
)

{T_P}(A) = P{A_{R(P)}}

where

R
(
P
)

R(P)

denotes the range of

P
P

. In this paper we study the relation between

T
P

(
A
)

{T_P}(A)

and

W
(
A
)

W(A)

, the numerical range of

A
A

. In particular we characterize those operators

A
A

such that

T
P

(
A
)

{T_P}(A)

is invertible for every

P
P

.
The commutator subgroup made abelian
Authors: Cohen Joel M.
Published: June 2010

A theorem on covering spaces is proved which yields the following information about a group

π
\pi

, its commutator subgroup

π
′

\pi ’

and their abelianizations: If

π

a
b

≅

Z

p
n

{\pi ^{ab}} \cong {Z_{{p^n}}}

, a cyclic group of order a power of the prime

p
p

, then

π

′

a
b

=
p
π

′

a
b

\pi {’^{ab}} = p\pi {’^{ab}}

. Hence if

π
\pi

is also finitely generated, then

π

′

a
b

\pi {’^{ab}}

is finite of order prime to

p
p

.
A note on Mordell’s equation 𝑦²=𝑥³+𝑘
Authors: Mohanty S. P.
Published: June 2010

In this note we prove that

lim

sup

k
→
∞

N
′

(
k
)
≧
6

\lim {\sup _{k \to \infty }}N’(k) \geqq 6

, where

N
′

(
k
)

N’(k)

is the number of integral solutions of

y
2

=

x
3

+
k

{y^2} = {x^3} + k

with

(
x
,
y
)
=
1

(x,y) = 1

.
On a generalized Volterra equation in Hilbert space
Authors: De Santis Romano M.
Published: June 2010

The objective of this paper is to establish a fundamental theorem on the existence and uniqueness of the solution of a generalized Volterra equation in Hilbert space. Some applications to the areas of abstract functional analysis and modern mathematical systems theory are pointed out.
The Sorgenfrey line is not an elastic space
Authors: Gruenhage Gary
Published: June 2010

In this paper we show that the Sorgenfrey line is an example of a paracompact monotonically normal space which is not elastic. We also give a class of examples of elastic spaces which are not linearly stratifiable.
Transformation groups of automorphisms of 𝐶(𝑋,𝐺)
Authors: Yang J. S.
Published: June 2010

If

(
X
,
T
,
π
)

(X,T,\pi )

is a transformation group with locally compact phase group

T
T

, there is a standard way to induce a transformation group on

C
(
X
,
Y
)

C(X,Y)

endowed with the compact-open topology, where

Y
Y

is a uniform space. In this paper, we consider the case where

Y
Y

is a topological group

G
G

. The reverse construction under certain conditions is also considered.
Cohomology of fibre spaces with group bundle coefficients
Authors: Mielke M. V.
Published: June 2010

The purpose of this paper is to construct a resolution of a group bundle

γ
\gamma

, a classifying spectrum for a cohomology functor

H
∗

(
;
γ
)

{H^\ast }(;\gamma )

(coefficients in

γ
\gamma

) defined on a category of fibre spaces, and to clarify and note some implications of the close relationship between these two constructions.
Random walks on compact semigroups
Authors: Mukherjea A. , Sun T. C. , Tserpes N. A.
Published: June 2010

Let

β
\beta

be a regular Borel probability measure with support

F
F

on a compact semigroup

S
S

. Let

X
1

,

X
2

,
⋯

{X_1},{X_2}, \cdots

be a sequence of independent random variables on some probability space

(
Ω
,
Σ
,
P
)

(\Omega ,\Sigma ,P)

with values in

S
S

, having identical distribution

P
(

X
n

∈
B
)
=
β
(
B
)

P({X_n} \in B) = \beta (B)

. The random walk

Z
n

=

X
1

X
2

⋯

X
n

{Z_n} = {X_1}{X_2} \cdots {X_n}

is studied in this paper. Let

D
D

be the closed semigroup generated by

F
F

. An element

x
x

in

D
D

is called recurrent iff

P
x

(

Z
n

∈

N
x

i
.
o
.
)
=
1

{P_x}({Z_n} \in {N_x}i.o.) = 1

for every open set

N
x

{N_x}

containing

x
x

. This paper characterizes the recurrence of an element

x
x

in terms of divergence of the series

∑

n
=
1

∞

β
n

(

N
x

)

\sum \nolimits _{n = 1}^\infty {{\beta ^n}} ({N_x})

for every open set

N
x

{N_x}

containing

x
x

. It also shows that the set of recurrent states of

{

Z
n

}

\{ {Z_n}\}

is precisely the kernel of

D
D

.
On the number of field topologies on an infinite field
Authors: Kiltinen John O.
Published: June 2010

K.-P. Podewski has recently proven that every countable infinite field admits

2

2

ℵ
0

{2^{{}_2{\aleph _0}}}

different field topologies. Using methods of valuation theory, it is proven that every uncountable field, and more generally, every field

F
F

of infinite transcendence degree over some subfield, admits

2

2

|

F

|

{2^{{2^{|F|}}}}

field topologies. By purely set theoretic considerations, it then follows that there are

2

2

|

F

|

{2^{{2^{|F|}}}}

field topologies on any infinite field

F
F

, no two of which are topologically isomorphic. This latter result is then generalized to any infinite commutative ring without proper zero-divisors. A further aspect of Podewski’s work on countable fields is generalized in a final theorem which states that a field

F
F

of infinite transcendence degree admits

2

2

|

F

|

{2^{{2^{|F|}}}}

field topologies which fail to be suprema of locally bounded ring topologies.
On semiprime P. I. rings
Authors: Martindale Wallace S.
Published: June 2010

The main results proved in this paper are that if

R
R

is a semiprime ring satisfying a polynomial identity then (1) the maximal right quotient ring of

R
R

is also P.I. and (2) every essential one-sided ideal of

R
R

contains an essential two-sided ideal of

R
R

.
The dilatation of harmonic quasiconformal functions
Authors: Caldwell William V.
Published: June 2010

In this paper, it is shown that the dilation of a harmonic quasiconformal function is a subharmonic function.
Local boundedness and continuity for a functional equation on topological spaces
Authors: Ng C. T.
Published: June 2010

It is known that the locally bounded solutions

f
f

of Cauchy’s functional equation

f
(
x
)
+
f
(
y
)
=
f
(
x
+
y
)

f(x) + f(y) = f(x + y)

on the reals are necessarily continuous. We shall extend this result to the functional equation

f
(
x
)
+
g
(
y
)
=
h
(
T
(
x
,
y
)
)

f(x) + g(y) = h(T(x,y))

on topological spaces.
Probability measures on semigroups
Authors: Gerl Peter
Published: June 2010

Let

S
S

be a discrete semigroup,

P
P

a probability measure on

S
S

and

s
∈
S

s \in S

with

lim

sup
n

(

P

(
n
)

(
s
)

)

1

/

n

=
1

\lim {\sup _n}{({P^{(n)}}(s))^{1/n}} = 1

. We study limit theorems for the convolution powers

P

(
n
)

{P^{(n)}}

of

P
P

implied by the above property and further the class of all semigroups with this property. Theorem 3 relates this class of semigroups to left amenable semigroups.
Interpolation sets for convolution measure algebras
Authors: Graham Colin C.
Published: June 2010

These are proved: (1) The union of two interpolation sets for a regular commutative convolution measure algebra is not necessarily an interpolation set. (2) There exists a regular commutative convolution measure algebra for which interpolation sets are not necessarily of spectral synthesis, while every singleton is a Ditkin set. (3) For every nondiscrete LCA group

G
G

, there exist compact interpolation sets for

M
(
G
)

M(G)

whose union is not an interpolation set. A tensor algebra method is used.
Subalgebras of Douglas algebras
Authors: Clancey Kevin , Cutrer Wayne
Published: June 2010

A closed subalgebra

A

\mathcal {A}

of

L
∞

{L^\infty }

is called a Douglas algebra in case

A

\mathcal {A}

is an algebra generated by

H
∞

{H^\infty }

and a set of inverses of inner functions. It is shown that if the Douglas algebra

A

\mathcal {A}

contains properly

H
∞

+
C

{H^\infty } + C

, then there is another Douglas algebra

A

′

\mathcal {A}’

such that

H
∞

+
C
⊊

A

′

⊊

A

{H^\infty } + C \subsetneq \mathcal {A}’ \subsetneq \mathcal {A}

. Some results on subalgebras are also given for algebras generated by

H
∞

{H^\infty }

and a function of the form

f

B
¯

f\overline B

, where

f
f

is in

H
∞

{H^\infty }

and

B
B

is an infinite Blaschke product.
Completeness of eigenvectors in Banach spaces
Authors: Benzinger Harold E.
Published: June 2010

We prove a general theorem on the completeness of the eigenvectors of linear operators in a Banach space. We then derive asymptotic estimates for the Green’s functions of two-point boundary value problems which allow us to apply the above theorem to a wide class of such problems in the spaces

L
p

(
0
,
1
)
,
1
≦
p
>
∞

{L^p}(0,1),1 \leqq p > \infty

.
Sufficient conditions in the calculus of variations and in the theory of optimal control
Authors: Nehari Zeev
Published: June 2010

A method is described for obtaining criteria confirming the optimality of solutions found either by integrating the Euler-Lagrange equations, or by applying the Pontrjagin necessary condition. The method is based on convexity considerations, and its success in a given problem depends on the judicious choice of certain arbitrary auxiliary functions.
Identities for series of the type Σ𝑓(𝑛)𝜇(𝑛)𝑛^{-𝑠}
Authors: Apostol Tom M.
Published: June 2010

Identities are obtained relating the series of the title with

∑

f
(
n
)
μ
(
n
)
μ

(
p
,
n
)

n

−
s

\sum {f(n)\mu (n)\mu } (p,n){n^{ - s}}

where

f
f

is completely multiplicative,

|

f
(
n
)

|

≦
1

|f(n)| \leqq 1

, and

p
p

is prime. Applications are given to vanishing subseries of

∑

μ
(
n
)

/

n

\sum {\mu (n)/n}

.
On Prüfer rings as images of Prüfer domains
Authors: Boisen Monte B. , Larsen Max D.
Published: June 2010

Only commutative rings with unity will be considered in this paper. It is shown that if

R
R

is the homomorphic image of a Prüfer domain, then

R
R

is a Prüfer ring but that the converse is not true in general. It is then shown that a Prüfer ring

R
R

is the homomorphic image of a Prüfer domain if and only if the total quotient ring of

R
R

is the homomorphic image of a Prüfer domain. A class of total quotient rings which satisfy this last condition is then presented.
Semigroup rings and semilattice sums of rings
Authors: Weissglass Julian
Published: June 2010

A generalization of the concept of a decomposition of a ring into a direct sum of ideals is introduced. The question of semisimplicity of the ring in terms of the semisimplicity of its summands is investigated. The results are applied to semigroup rings.
Open simple maps and periodic homeomorphisms
Authors: Baildon John D.
Published: June 2010

It is shown that an open map between

2
2

-manifolds without boundary that is the composition of

n
n

open simple (at most two-to-one) maps is necessarily of order 2". Since w=zz on the unit sphere is of order three, this shows an open map cannot necessarily be factored into a composition of open simple ones.
On sequences not containing a large sum-free subsequence
Authors: Choi S. L. G.
Published: June 2010

A subsequence of a sequence of integers is said to be sum-free if no integer of the subsequence is the sum of distinct integers of this same subsequence. In this paper we shall prove, provided

n
n

is sufficiently large, that there exists a sequence of

n
n

integers whose largest sum-free subsequence has at most

c
n

(
log
⁡
log
⁡
n

)

−
1

/

2

cn{(\log \log n)^{ - 1/2}}

integers, where

c
c

is an absolute constant.
The Schur multiplicator of metacyclic groups
Authors: Beyl F. Rudolf
Published: June 2010

The Schur multiplicator

H
2

G

{H_2}G

of a (finite) metacyclic group

G
G

is computed with the help of the Lyndon spectral sequence. The order of

H
2

G

{H_2}G

is a useful invariant of

G
G

. A metacyclic group with vanishing Schur multiplicator can be presented with two generators and two relators. A simple description of the totality of these groups is given and all such

p
p

-groups are classified.
On the fundamental domain of a discrete group
Authors: Nathanson Melvyn B.
Published: June 2010

Gel’fand, Graev, and Pjateckiĭ-Šapiro proposed a method to construct the fundamental domain of a discrete group acting on certain metric spaces. A counterexample is given to show that this construction sometimes fails.
Cohomology of a bounding manifold
Authors: Smith Larry , Stong R. E.
Published: June 2010

This paper characterizes those subsets of

H
∗

(
M
;
Q
)

{H^\ast }(M;Q)

which are the image of the cohomology of some manifold whose boundary is

M
M

.
Descent of projectivity for locally free modules
Authors: Wiegand Roger
Published: June 2010

Let

R
→

R
^

R \to \hat R

be the natural homomorphism from the commutative ring

R
R

into its associated von Neumann regular ring, and let

M
M

be a locally free

R
R

-module such that

R
^

⊗
M

\hat R \otimes M

is a projective

R
^

\hat R

-module. We show that if

M
M

is either countably generated or locally finitely generated, then

M
M

is projective, and we deduce that the trace of any projective ideal is projective. These results are a consequence of a more general theorem on the descent of the Mittag-Leffler condition. The “locally free” hypothesis may be weakened to “flat” if and only if

R
R

is locally perfect.
A remark on the commutativity of certain rings
Authors: Awtar Ram
Published: June 2010

In a recent paper [1] Gupta proved that a division ring satisfying the polynomial identity

x

y
2

x
=
y

x
2

y

x{y^2}x = y{x^2}y

is commutative. In this note our goal is to prove the following: If

R
R

is a semiprime ring with

x

y
2

x
−
y

x
2

y

x{y^2}x - y{x^2}y

central in

R
R

, for all

x
,
y

x,y

in

R
R

, then

R
R

is commutative.
On continuity of invariant measures
Authors: Adler Andrew
Published: June 2010

Main Theorem. Let

Φ
\Phi

be a set of transformations on a set

X
X

. The following conditions are then equivalent: (1) There is a noncontinuous finitely additive measure defined on all subsets of

X
X

and invariant under all transformations in

Φ
\Phi

. (2) There is an integer

m
m

such that for any finite subset

F
F

of

Φ
\Phi

there is a finite subset

A
F

{A_F}

of

X
X

, with no more than

m
m

elements, such that each

f
f

in

F
F

acts as a permutation on

A
F

{A_F}

.
The model theory of differential fields of characteristic 𝑝̸=0
Authors: Wood Carol
Published: June 2010

The theory of differential fields of characteristic

p
≠
0

p \ne 0

is shown to have a model companion, the theory of differentially closed fields, which is moreover the model completion of the theory of differentially perfect fields. It is also shown that the theory of differentially closed fields.is not

ω
\omega

-stable.
Radicals and bimodules
Authors: Foster D. M.
Published: June 2010

In 1964, Andrunakievič and Rjabuhin showed that the general theory of radicals of associative rings may be presented in external form in the language of modules. In this paper, we show that this theory has a natural extension to varieties of algebras where, in this case, modules are replaced by bimodules. We close with some examples and a discussion of quadratic Jordan algebras.
Exceptional values of analytic functions
Authors: Krajkiewicz P.
Published: June 2010

Let

g
(
z
)

g(z)

be an analytic function with an isolated singularity at

z
0

{z_0}

and let

f
(
z
)

f(z)

be a nonanalytic multianalytic function with a nonessential isolated singularity at

z
0

{z_0}

. It is shown that if

g
(
z
)
−
f
(
z
)

g(z) - f(z)

does not vanish in some deleted neighborhood of

z
0

{z_0}

, then

z
0

{z_0}

is a nonessential singularity of

g
(
z
)

g(z)

.
Finite Frattini factors in finitely generated soluble groups
Authors: Lennox John C.
Published: June 2010

Results of Philip Hall are used to prove that subgroups of finitely generated Abelian-by-nilpotent groups are finite if their Frattini factor groups are finite.
Finite frattini factors in finitely generated Abelian-by-polycyclic groups
Authors: Lennox John C.
Published: June 2010

A criterion for a group to be finite is used to prove that subgroups of finitely generated Abelian-by-polycyclic groups are finite if their Frattini factor groups are finite.
Allowable diagrams for purely inseparable field extensions
Authors: Kime Linda Almgren
Published: June 2010

We define a diagram associated with a purely inseparable field extension of finite exponent. We show that, under this definition, for any given field extension the shape of its diagram is unique. Thus our diagram improves the diagram defined by Sweedler in [2, p. 402]. In §2 we define an “allowable” shape for a diagram. Given any “allowable” shape for a diagram representing a finite field extension, we construct a field extension whose diagram has that shape.
Finiteness conditions for projective and injective modules
Authors: Fisher Joe W.
Published: June 2010

Does Hopkins’ theorem extend to projective modules, i.e., are projective Artinian modules Noetherian? An example is given to answer this question in the negative; however, we show that the answer is affirmative for certain large classes of projective modules. Dually, are injective Noetherian modules Artinian? Again the answer is negative; nevertheless, we provide an affirmative answer for certain classes of injective modules.
A characterization of the Jacobson radical in ternary algebras
Authors: Myung Hyo Chul
Published: June 2010

The Jacobson radical Rad

T
T

for a ternary algebra

T
T

is characterized as one of the following: (i) the set of properly quasi-invertible elements in

T
T

; (ii) the set of

x
∈
T

x \in T

such that the principal right ideal

⟨

x
T
T

⟩

\left \langle {xTT} \right \rangle

or left ideal

⟨

T
T
x

⟩

\left \langle {TTx} \right \rangle

is quasi-regular in

T
T

; (iii) the unique maximal quasi-regular ideal in

T
T

; (iv) the set of

x
∈
T

x \in T

such that Rad

T

(
x
)

=

T

(
x
)

{T^{(x)}} = {T^{(x)}}

. We also obtain ternary algebra-analogs of characterization of the radicals of certain subalgebras in an associative algebra.
On the groups of inertia of smooth manifolds
Authors: Naoum Adil G.
Published: June 2010

In this paper we study sufficient conditions for a manifold

M
n

{M^n}

to have

I
(

M
n

)
=
{
0
}

I({M^n}) = \{ 0\}

. We also prove that if

M
n

{M^n}

is a smooth manifold of dimension

n
,
n
≡
2

(
mod

8
)

n,n \equiv 2\pmod 8

, with

w
2

(

M
n

)
≠
0

{w_2}({M^n}) \ne 0

, then

I
(

M
n

)
≠
0

I({M^n}) \ne 0

.
A characterization of inner product spaces
Authors: Tapia R. A.
Published: June 2010

In this paper we define a generalized inner product on an arbitrary normed linear space and use this generalized inner product to characterize inner product spaces in the class of all normed linear spaces. We give a sharp statement of a generalized Riesz representation theorem for bounded linear functionals. This theorem should be useful in generalizing the notions of gradient methods and reproducing kernel spaces.
Anosov diffeomorphisms on nilmanifolds
Authors: Manning Anthony
Published: June 2010

The purpose of this paper is to give necessary conditions on the map induced by an Anosov diffeomorphism of a nilmanifold on its fundamental group.
Perturbations by nilpotent operators on Hilbert space
Authors: Brown Arlen , Pearcy Carl , Salinas Norberto
Published: June 2010

If

T
T

is any noncompact, bounded, linear operator on a separable Hilbert space

H
H

, then there exists a nilpotent (bounded, linear) operator

N
N

on

H
H

such that

N
+
T

N + T

is invertible.
Isometric embedding of a compact Riemannian manifold into Euclidean space
Authors: Jacobowitz Howard
Published: June 2010

An isometric immersion of an

n
n

-dimensional compact Riemannian manifold with sectional curvature always less than

λ

−
2

{\lambda ^{ - 2}}

into Euclidean space of dimension

2
n
−
1

2n - 1

can never be contained in a ball of radius

λ
\lambda

. This generalizes and includes results of Tompkins and Chern and Kuiper.
Faithful Noetherian modules
Authors: Formanek Edward
Published: June 2010

The Eakin-Nagata theorem says that if

T
T

is a commutative Noetherian ring which is finitely generated as a module over a subring

R
R

, then

R
R

is also Noetherian. This paper proves a generalization of this result. However, the main interest is that the proof is very elementary and uses little more than the definition of “Noetherian".
Ergodic properties of bounded 𝐿₁-operators
Authors: Satō Ryōtarō
Published: June 2010

Individual ergodic theorems for bounded

L
1

{L_1}

-operators are proved in §1, and the problem of existence of positive invariant functions for positive

L
1

{L_1}

-operators is considered in §2. A decomposition theorem similar to that of Sucheston [12] is proved in the last section.
Kurepa’s hypothesis and 𝜎-complete ideals
Authors: Prikry Karel
Published: June 2010

A set-theoretic combinatorial hypothesis, known as Kurepa’s hypothesis, is used to obtain results concerning the saturatedness of ideals and families of measures.
Graphs with 1-factors
Authors: Sumner David P.
Published: June 2010

In this paper it is shown that if G is a connected graph of order

2
n
(
n
>
1
)

2n(n > 1)

not containing a 1-factor, then for each k,

1
>
k
≦
n

1 > k \leqq n

, there exists an induced; connected subgraph of order 2k which also fails to possess a 1-factor. Several other sufficient conditions for a graph to contain a 1-factor are presented. In particular, it is seen that the connected even order line graphs and total graphs always contain a 1-factor.
On torsion abelian groups quasi-projective over their endomorphism rings
Authors: Fuchs László
Published: June 2010

It is shown that a torsion abelian group is quasi-projective over its endomorphism ring exactly if, for every prime p, its p-component is bounded or has an unbounded basic subgroup.