bluuebunny/crossref_metadata_2025 huggingface dataset:
A note on connected and peripherally continuous functions
Authors: Hagan Melvin R.
Published: June 2010

In this paper it is proved that under certain conditions on the domain and range spaces an open monotone connected function preserves unicoherentness and hereditary local connectedness. In addition, a monotone-light factorization theorem is proved for certain connected functions and peripherally continuous functions.
On Watson’s quintuple product identity
Authors: Subbarao M. V. , Vidyasagar M.
Published: June 2010

In 1929, in the course of proving certain results stated by Ramanujan concerning his continued fraction, G. N. Watson proved an identity involving five infinite products and an infinite series. In 1938, Watson proved another identity which again involved five products. Finally in 1961, one more quintuple product identity was established, this time by Basil Gordon. We show here that all these identities are equivalent. Also, with the help of the quintuple product identity and Jacobi’s triple product identity, we establish two new identities involving only series.
The Hardy class of a spiral-like function and its derivative
Authors: Başgöze T. , Keogh F. R.
Published: June 2010

A determination is made of the Hardy classes to which a spiral-like univalent function and its derivative belong. An estimate for the size of the Taylor coefficients is deduced.
Oscillation theorems for nonlinear second order differential equations.
Authors: Onose H.
Published: June 2010

Recently, Bobisud, Proc. Amer. Math. Soc. 23 (1969), 501-505, has discussed the oscillatory behavior of solutions of

y
+
a
(
t
)
f
(
y
)
=
0

y + a(t)f(y) = 0

. The purpose of this paper is to prove Bobisud’s theorems under weaker assumptions. The proofs are quite different from his and simpler.
Measures of 𝑁-fold symmetry for convex sets
Authors: Chui Charles K. , Parnes Milton N.
Published: June 2010

If a convex set

S
S

is

3
3

-fold symmetric about a point

0
∈
S

0 \in S

, then any

3
3

-star contained in

S
S

with vertex 0 is no smaller than any other parallel

3
3

-star contained in

S
S

. In this paper, among other results, we establish the converse. Consequently, we find two measures of

n
n

-fold symmetry, one for

n
=
2
,
3

n = 2,3

and the other for each

n
≧
2

n \geqq 2

.
Extensions of proximity functions
Authors: Mattson Don A.
Published: June 2010

Let

P
∗

(
X
)

{P^ \ast }(X)

be the algebra of bounded, real-valued proximity functions on a proximity space

(
X
,
δ
)

(X,\delta )

, where

X
X

is a dense subspace of a topological space

T
T

. In this paper we obtain several conditions which are equivalent to the following property: every member of

P
∗

(
X
)

{P^ \ast }(X)

has a continuous extension to

T
T

. Examples concerning these results are included, one of which shows that this extension property is distinct from

C
∗

{C^ \ast }

-embedding.
Commuting unilateral shifts
Authors: Deddens James A.
Published: June 2010

Examples are given to show that two commuting unilateral shifts (of dimension

∞
,
∞

\infty ,\infty

or

m
,
n

m,n

with

2
≦
m
,

n
>
∞

2 \leqq m,\;n > \infty

) need not commute with a unilateral shift of multiplicity one.
On an inequality of T. J. Willmore
Authors: Chen Bang-yen
Published: June 2010

Willmore proved that the integral of the square of mean curvature

H
H

over a closed surface

M
2

{M^2}

in

E
3

,

∫

M
2

H
2

d
V

{E^3},{\smallint _{{M^2}}}{H^2}dV

, is

≧
4
π

\geqq 4\pi

, and equal to

4
π

4\pi

when and only when

M
2

{M^2}

is a sphere in

E
3

{E^3}

. In this paper we give some generalizations of Willmore’s result.
A characterization of 𝑁-compact spaces
Authors: Chew Kim-peu
Published: June 2010

In this paper, we prove the following theorem: Theorem A. A

0
0

-dimensional space

X
X

is

N
N

-compact if and only if every clopen ultrafilter on

X
X

with the countable intersection property is fixed, where

N
N

is the space of all natural numbers. Two consequences of Theorem A are as follows: Theorem B. Suppose that

X
X

and

Y
Y

are

N
N

-compact spaces. A mapping

ϕ
\phi

from the Boolean ring

B

(
X
)

\mathfrak {B}(X)

of all clopen subsets of

X
X

onto the Boolean ring

B

(
X
)

\mathfrak {B}(X)

of all clopen subsets of

Y
Y

is an isomorphism with the property that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="intersection Underscript i equals 1 Overscript normal infinity Endscripts upper A Subscript i Baseline equals normal empty-set left-parenthesis upper A Subscript i Baseline element-of German upper B left-parenthesis upper X right-parenthesis right-parenthesis">

⋂

i
=
1

∞

A
i

=
∅
(

A
i

∈

B

(
X
)
)

\bigcap \nolimits _{i = 1}^\infty {{A_i} = \emptyset ({A_i} \in \mathfrak {B}(X))}

implies

⋂

i
=
1

∞

ϕ
(

A
i

)
=
∅

\bigcap \nolimits _{i = 1}^\infty \phi ({A_i}) = \emptyset

if and only if there exists a homeomorphism

h
h

from

X
X

onto

Y
Y

such that

ϕ
(
A
)
=
h
[
A
]

\phi (A) = h[A]

for each

A
A

in

B

(
X
)

\mathfrak {B}(X)

. Theorem C. A

0
0

-dimensional space

X
X

is

N
N

-compact if and only if the collection of all the countable clopen coverings of

X
X

is complete.
Regular-closed, Urysohn-closed and completely Hausdorff-closed spaces.
Authors: Herrlich H.
Published: June 2010

Recently M. P. Berri, J. R. Porter, and R. M. Stephenson, Jr. have given a survey on

P
P

-closed and

P
P

-minimal spaces. In the present paper the first two problems of this survey will be solved: (1) The product of two Urysohn-closed spaces need not be Urysohn-closed. (2) A completely Hausdorff-closed regular space need not be regular-closed.
A regular Lindelöf semimetric space which has no countable network
Authors: Berney E. S.
Published: June 2010

A completely regular semimetric space

M
M

is constructed which has no

σ
\sigma

-discrete network. The space

M
M

constructed has the property that every subset of

M
M

of cardinality

2

ℵ
0

{2^{{\aleph _0}}}

contains a limit point of itself; thus, assuming

2

ℵ
0

=

ℵ
1

,
M

{2^{{\aleph _0}}} = {\aleph _1},M

is Lindelöf. It is also shown from the same space

M
M

that, assuming

2

ℵ
0

=

ℵ
1

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

, there exists a regular Lindelöf semimetric space

X
X

such that

X
×
X

X \times X

is not normal (hence not Lindelöf).
On representations of selfmappings
Authors: Janoš Ludvík
Published: June 2010

It is shown in this note that every “mild” self mapping

f
:
X
→
X

f:X \to X

of a compact Hausdorff space

X
X

into itself can be represented by the product

(
Y
,
g
)
×
(
Z
,
h
)

(Y,g) \times (Z,h)

of two self mappings

g
g

and

h
h

, where

g
g

is a contraction

(

⋂
1
∞

g
n

(
Y
)
=

singleton

)

(\bigcap \nolimits _1^\infty {{g^n}(Y) = {\text {singleton}}} )

and

h
h

is a homeomorphism of

Z
Z

onto itself. Endowing the set of all selfmappings

X
X

{X^X}

with the compact-open topology, the qualifier “mild” means that the closure of the family

{

f
n

|

n
≧
1
}
⊂

X
X

\{ {f^n}|n \geqq 1\} \subset {X^X}

is compact. In case

X
X

is metrizable, some results of M. Edelstein and J. de Groot are used to linearize

(
X
,
f
)

(X,f)

in the separable Hilbert space.
𝐻-spaces and the suspension homomorphism
Authors: Haslam H. B.
Published: June 2010

If a space

X
X

is an

H
H

-space, then the homotopy suspension homomorphism is a monomorphism onto a direct factor in all dimensions. We present an example to show that the converse is false.
Completeness of Hamiltonian vector fields
Authors: Ebin David G.
Published: June 2010

We prove that under certain conditions the flow of a Hamiltonian vector field on a possibly infinite-dimensional dynamical system exists for all time.
A brief derivation of the Heisenberg commutation relations
Authors: Prosser Reese T.
Published: June 2010

The commutation relations for the canonical coordinates of a mechanical system are here derived from a suitably defined form of translational invariance.
Absolutely pure modules
Authors: Megibben Charles
Published: June 2010

A module

A
A

is shown to be absolutely pure if and only if every finite consistent system of linear equations over

A
A

has a solution in

A
A

. Noetherian, semihereditary, regular and Prüfer rings are characterized according to properties of absolutely pure modules over these rings. For example,

R
R

is Noetherian if and only if every absolutely pure

R
R

-module is injective and semihereditary if and only if the class of absolutely pure

R
R

-modules is closed under homomorphic images. If

R
R

is a Prüfer domain, then the absolutely pure

R
R

-modules are the divisible modules and

Ext
R
1

⁡
(
M
,
A
)
=
0

\operatorname {Ext} _R^1(M,A) = 0

whenever

A
A

is divisible and

M
M

is a countably generated torsion-free

R
R

-module.
Quaternions and binary quadratic forms
Authors: Rice Bart
Published: June 2010

Methods are discussed for studying binary quadratic forms by use of quaternions derived from the ternary quadratic form

f
=

x
2

−
y
z

f = {x^2} - yz

. In particular, Gauss composition of binary quadratic forms may be achieved by factoring and multiplying quaternions in a natural way.
A determinantal inequality for projectors in a unitary space
Authors: Djoković D. Ž.
Published: June 2010

Let

V
V

be a finite dimensional unitary space and

V
=

V
1

+
⋯
+

V
k

V = {V_1} + \cdots + {V_k}

a direct decomposition. Let

P
i

{P_i}

be the orthogonal projector in

V
V

with range

V
i

{V_i}

. If

A
=

P
1

+
⋯
+

P
k

A = {P_1} + \cdots + {P_k}

we prove that

0
>
det
(
A
)
≦
1

0 > \det (A) \leqq 1

and

det
(
A
)
=
1

\det (A) = 1

if and only if

V
=

V
1

+
⋯
+

V
k

V = {V_1} + \cdots + {V_k}

is an orthogonal decomposition. Let

N
i

(
1
≦
i
≦
k
)

{N_i}(1 \leqq i \leqq k)

be a normal operator in

V
V

, of rank

r
i

{r_i}

. Assume that

N
=

∑

v
=
1

k

N
i

N = \sum _{v = 1}^k{N_i}

has rank

r
=

r
1

+
⋯
+

r
k

≦
n
=
dim
⁡
V

r = {r_1} + \cdots + {r_k} \leqq n = \dim V

. If the nonzero eigenvalues of

N
N

(counting multiplicities) are the same as the nonzero eigenvalues of all

N
i

(
1
≦
i
≦
k
)

{N_i}(1 \leqq i \leqq k)

together, then

N
i

N
j

=
0

{N_i}{N_j} = 0

for

i
≠
j

i \ne j

. This generalizes a recent result of L. Brand.
Topological characterizations of pseudo-ℵ-compact spaces
Authors: Vidossich Giovanni
Published: June 2010

Pseudo-

ℵ
\aleph

-compact spaces, usually defined by cardinal bounds on uniform coverings of the fine uniformity, are characterized by cardinal bounds on locally finite open (resp. closed) coverings. This yields a new characterization of pseudo-compact spaces as well as of G. Aquaro’s paralindelöfian spaces.
On the structure of a finite solvable 𝐾-group
Authors: Kotzen Marshall
Published: June 2010

In this note we investigate the structure of a finite solvable

K
K

-group. It is proved that a finite group

G
G

is a solvable

K
K

-group if and only if

G
G

is a subdirect product of a finite collection of solvable

K
K

-groups

H
i

{H_i}

such that each

H
i

{H_i}

is isomorphic to a subgroup of

G
G

, and each

H
i

{H_i}

possesses a unique minimal normal subgroup.
Complex bordism of classifying spaces
Authors: Landweber Peter S.
Published: June 2010

It is shown that, for a finite group

G
G

, the Thom homomorphism maps the complex bordism

Ω
∗
U

(

B
G

)

\Omega _\ast ^U({B_G})

onto the homology

H
∗

(

B
G

;
Z
)

{H_\ast }({B_G};Z)

of a classifying space for

G
G

if and only if

G
G

has periodic cohomology.
Real zeros of a random sum of orthogonal polynomials
Authors: Das Minaketan
Published: June 2010

Let

c
0

,

c
1

,

c
2

,
⋯

{c_0},{c_1},{c_2}, \cdots

be a sequence of normally distributed independent random variables with mathematical expectation zero and variance unity. Let

P
k
∗

(
x
)
(
k
=
0
,
1
,
2
,
⋯
)

P_k^ \ast (x)(k = 0,1,2, \cdots )

be the normalised Legendre polynomials orthogonal with respect to the interval

(
−
1
,
1
)

( - 1,1)

. It is proved that the average number of the zeros of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 0 upper P 0 Superscript asterisk Baseline left-parenthesis x right-parenthesis plus c 1 upper P 1 Superscript asterisk Baseline left-parenthesis x right-parenthesis plus midline-horizontal-ellipsis plus c Subscript n Baseline upper P Subscript n Superscript asterisk Baseline left-parenthesis x right-parenthesis">

c
0

P
0
∗

(
x
)
+

c
1

P
1
∗

(
x
)
+
⋯
+

c
n

P
n
∗

(
x
)

{c_0}P_0^ \ast (x) + {c_1}P_1^ \ast (x) + \cdots + {c_n}P_n^ \ast (x)

in the same interval is asymptotically equal to

(
3

)

−
1

/

2

n

{(3)^{ - 1/2}}n

when

n
n

is large.
An exact solution of the nonlinear differential equation 𝑦̈+𝑝(𝑡)𝑦=𝑞ₘ(𝑡)/𝑦^{2𝑚-1}
Authors: Reid James L.
Published: June 2010

An exact solution of the nonlinear differential equation

y
¨

+
p
(
t
)
y
=

q
m

(
t
)

/

y

2
m
−
1

\ddot y + p(t)y = {q_m}(t)/{y^{2m - 1}}

is found to be

y
=

[

u
m

+
c

(
m
−
1

)

−
1

W

−
2

v

]

1

/

m

y = {[{u^m} + c{(m - 1)^{ - 1}}{W^{ - 2}}v]^{1/m}}

if

q
m

(
t
)
=
c

(
u
v

)

m
−
2

{q_m}(t) = c{(uv)^{m - 2}}

.

u
u

and

v
v

are independent solutions of

y
¨

+
p
(
t
)
y
=
0

\ddot y + p(t)y = 0

and

W
W

is their Wronskian.
Stone-Weierstrass theorems for 𝐶(𝑋) with the sequential topology
Authors: Frolík Zdeněk
Published: June 2010

The spaces

P
P

with one of the following two properties are studied: every continuous function is a Baire function with respect to any algebra

R

\mathcal {R}

of continuous functions such that

R

\mathcal {R}

projectively generates the topology, or with respect to any algebra which distinguishes the points. The former property is equivalent to the statement that of any pair of disjoint zero sets at least one is Lindelöf, the latter implies that the space is Lindelöf and is implied by analyticity. Connections with the Blackwell problem are shown.
Choices from finite sets and choices of finite subsets
Authors: Zuckerman Martin M.
Published: June 2010

In set theory without the axiom of choice we prove a consistency result involving certain “finite versions” of the axiom of choice. Assume that it is possible to select a nonempty finite subset from each nonempty set. We determine sets

Z
Z

, of integers, which have the property that

n
∈
Z

n \in Z

is a necessary and sufficient condition for the possibility of choosing an element from every

n
n

-element set. Given any nonempty set

P
P

of primes, the set

Z
p

{Z_p}

, consisting of integers which are not “linear combinations” of primes of

P
P

, is such a set

Z
Z

.
A characterisation of punctured open 3-cells
Authors: Costich O. L. , Doyle P. H. , Galewski D. E.
Published: June 2010

A proof is given using standard methods of the topology of three-dimensional manifolds of the following characterization of punctured cubes: A connected, open

3
3

-manifold

M
M

is topological

E
3

{E^3}

with

k
k

points removed if and only if every polyhedral simple closed curve in

M
M

lies in a topological cube in

M
M

and the rank of

π
2

(
M
)

{\pi _2}(M)

is

k
k

. An application is given.
Suspending homotopy 3-spheres and embedding mapping cylinders in 𝑆⁴
Authors: Lacher R. C.
Published: June 2010

A property of maps between closed

3
3

-manifolds, implied by cellularity and implying

U

V
∞

U{V^\infty }

, is that the mapping cylinder embed locally in

S
4

{S^4}

. It is not clear what topological properties are preserved under such maps. In the present note, we show that a closed

3
3

-manifold admits such a map onto

S
3

{S^3}

if and only if its suspension is

S
4

{S^4}

.
Extensions of derivations
Authors: Barr Michael , Knus Max-Albert
Published: June 2010

We show that for a class of algebras including separable algebras one can extend derivations of the center to derivations of the algebra.
Unicity of the extremum problems in 𝐻¹(𝑈ⁿ)
Authors: Yabuta Kôzô
Published: June 2010

In 1958 de Leeuw and Rudin have given a sufficient condition for a function in

H
1

(
U
)

{H^1}(U)

to be a unique solution of the extremum problem. We give in this paper a stronger sufficient condition (Theorem 1) which holds also in

n
n

-dimension. Our Theorem 1 fills up considerably the gap of de Leeuw-Rudin’s result. We give also another proof of Neuwirth-Newman’s theorem and its

n
n

-dimensional generalization.
Homogeneous inverse limit spaces with nonregular covering maps as bonding maps
Authors: Rogers James T. , Tollefson Jeffrey L.
Published: June 2010

We construct counterexamples to the conjecture that, in an inverse sequence

(
X
,
f
)

(X,f)

of closed manifolds

X
n

{X_n}

with covering maps

f
m
n

:

X
n

→

X
m

f_m^n:{X_n} \to {X_m}

as bonding maps, if the inverse limit space is homogeneous, then there exists an integer m such that (for all

n
>
m

n > m

) the covering map

f
m
n

:

X
n

→

X
m

f_m^n:{X_n} \to {X_m}

is regular.
On one sided ideals of a prime type
Authors: Koh Kwangil
Published: June 2010

The notion of prime type for right ideals is analogous to that of prime ideal in a commutative ring, and the right dimension of the ring

R
R

is defined using chains of right ideals of prime type. The meaning of zero right dimension, finite dimension for right primitive rings, and related topics are studied.
Direct product of division rings and a paper of Abian
Authors: Chacron M.
Published: June 2010

It is shown that the rings under the title admit an order-theoretical characterization as in the commutative case studied by Abian.
A characterization of bimeasurable functions in terms of universally measurable sets
Authors: Darst R. B.
Published: June 2010

The purpose of this note is to show, assuming the continuum hypothesis, that a Borel function,

f
f

, mapping a Borel subset,

D
f

{D_f}

, of a separable complete metric space,

M
1

{M_1}

, into a separable complete metric space,

M
2

{M_2}

, maps Borel subsets of

D
f

{D_f}

onto Borel subsets of

M
2

{M_2}

if, and only if,

f
f

maps universally measurable subsets of

D
f

{D_f}

onto universally measurable subsets of

M
2

{M_2}

.
“A uniqueness theorem for certain two-point boundary value problems”: A correction
Authors: Fraker Ross
Published: June 2010

The boundary value problem

x
=
f
(
t
,
x
,

x
′

)
,
x
(
a
)
=
A
,
x
(
b
)
=
B

x = f(t,x,x’),x(a) = A,x(b) = B

is shown to have at most one solution on the interval

[
a
,
b
]

[a,b]

. The function

f
(
t
,
y
,
z
)

f(t,y,z)

is such that

f
(
t
,

y
1

,

z
1

)
−
f
(
t
,

y
2

,

z
2

)
>
g
(
t
,

y
1

−

y
2

,

z
1

−

z
2

)

f(t,{y_1},{z_1}) - f(t,{y_2},{z_2}) > g(t,{y_1} - {y_2},{z_1} - {z_2})

where initial value problem solutions of

z
=
g
(
t
,
z
,

z
′

)

z = g(t,z,z’)

have a minimum interval of disconjugacy.
Convolution measure algebras with involution
Authors: Baker J. W.
Published: June 2010

Let

S
S

be a commutative, locally compact semigroup and

A
A

an

L
L

-subalgebra of

M
(
S
)

M(S)

. We obtain conditions on

A
A

to ensure that

S
S

is an inverse semigroup, a group, or an idempotent semigroup. We assume an involution on

A
A

, or equivalently on

S
S

. The proofs come readily from known facts.
The norm of a hermitian element in a Banach algebra
Authors: Sinclair A. M.
Published: June 2010

We prove that the norm of a hermitian element in a Banach algebra is equal to the spectral radius of the element.
Strictly cyclic weighted shifts
Authors: Lambert Alan
Published: June 2010

E. Nordgren recently showed that certain weighted shift operators on Hilbert space have the property that they commute with no unbounded closed densely defined linear transformations. In the present paper, it is shown that this property for a weighted shift operator is equivalent to the existence of a strictly cyclic vector for the weakly closed algebra generated by the weighted shift. After establishing some tests for the existence of such a strictly cyclic vector, several further examples are given of weighted shift operators satisfying Nordgren’s commutativity property.
The spectra of subnormal operators
Authors: Putnam C. R.
Published: June 2010

It is shown that a subnormal operator having its spectrum on a simple closed curve is necessarily normal, and that the corresponding assertion with “subnormal” replaced by “hyponormal” is in general false.
Arcs in hyperspaces which are not compact
Authors: Ward L. E.
Published: June 2010

It has been known for many years that if

X
X

is a metrizable continuum then

2
X

{2^X}

(the space of closed subsets of

X
X

) and

C
(
X
)

C(X)

(the subspace of connected members of

2
X

{2^X}

) are arcwise connected. These results are due to Borsuk and Mazurkiewicz [l] and J. L. Kelley [2], respectively. Quite recently M. M. McWaters [6] extended these theorems to the case of continua which are not necessarily metrizable, using Koch’s arc theorem for partially ordered spaces [3], [8]. In this note we prove these results for certain noncompact spaces by means of a simple generalization of Koch’s arc theorem.
Dimension-theoretic properties of completions
Authors: Wenner B. R.
Published: June 2010

In this paper we extend some previous work from situations involving countable collections of subsets to those concerning locally finite collections. An example of the results obtained here is a theorem which asserts that corresponding to any locally finite collection of finite-dimensional closed subsets of a metric space

X
X

there exists a completion of

X
X

in which taking the closure of any member of the given collection does not raise dimension. The basic technique employed in each of the proofs is similar; a topologically equivalent metric is introduced (one which is strongly dependent upon the given locally finite collection), and the desired completion is then taken with respect to this new metric.
On maximal groups of isometries
Authors: Janoš Ludvík
Published: June 2010

The purpose of this note is to introduce the concept of “Optimal Metrization” for metrizable topological spaces. Let

X
X

be such a space,

ρ
\rho

a metric on

X
X

and

K
(
ρ
)

K(\rho )

the group of all those homeomorphisms of

X
X

onto itself which preserve

ρ
\rho

. The metric

ρ
\rho

is said to be “optimal” provided there is no

ρ
∗

{\rho ^ \ast }

with

K
(

ρ
∗

)

K({\rho ^ \ast })

properly containing

K
(
ρ
)

K(\rho )

. A space having at least one optimal metric is called “optimally metrizable.” Examples of spaces which are and which are not optimally metrizable are given; it is shown that the real line

R
R

is, and that the usual metric is optimal.
Localy compact full homeomorphism groups are zero-dimensional
Authors: Keesling James
Published: June 2010

Let X be a metric space and let

G
(
X
)

G(X)

be the space of homeomorphisms of X with the compact open topology. If

G
(
X
)

G(X)

is locally compact, then it is zero-dimensional. Some examples are also given of metric spaces X for which

G
(
X
)

G(X)

is compact.
Mean convergence and compact subsets of 𝐿₁
Authors: Halpern Benjamin
Published: June 2010

It is shown that in the usual criterion for

w
w

compactness of a set

K
K

in

L
1

(
μ
)

{L_1}(\mu )

the explicit assumption that

K
K

is bounded follows from the other assumptions which are usually made if

μ
\mu

is nonatomic.
A note on two-sided ideals in 𝐶*-algebras
Authors: Bunce John
Published: June 2010

An elementary proof is given of the fact that

(
I
+
J

)
+

=

I
+

+

J
+

{(I + J)^ + } = {I^ + } + {J^ + }

for

I
I

and

J
J

closed two-sided ideals in a

C
∗

{C^ \ast }

-algebra.
Semigroups on acyclic plane continua
Authors: Wilder B. E.
Published: June 2010

It is shown that an acyclic irreducible plane continuum which admits the structure of a topological semigroup is an arc if it has an identity, and is either an arc, is trivial, or is decomposible into an arc if it satisfies

M
2

=
M

{M^2} = M

. This extends some results of Friedberg and Mahavier concerning semigroups on chainable continua.
Invariant measures on locally compact semigroups
Authors: Rigelhof Roger
Published: June 2010

The main result of this paper shows that a locally compact abelian semigroup is embeddable as an open subsemigroup of a locally compact abelian group

G
G

if and only if the translations

x
↦
x
+
y

x \mapsto x + y

are open maps and there exists a nonnegative regular measure

μ
\mu

on

S
S

such that

μ
(
U
)
=
μ
(
x
+
U
)
>
0

\mu (U) = \mu (x + U) > 0

for every open set

U
U

and

x
x

in

S
S

.
An asymptotic solution of a nonhomogeneous linear system of differential equations
Authors: Hallam Thomas G.
Published: June 2010

A complete asymptotic series expansion is found for the solutions of a nonhomogeneous linear system of differential equations whose coefficient matrix and forcing term are in

L
1

[

t
0

,
∞
)

{L^1}[{t_0},\infty )

. Related results, which require that the coefficients satisfy certain conditional integrability conditions, are shown to be reducible by a transformation to the

L
1

{L^1}

case.
A note concerning 𝐴*=𝐿₁(𝜇)
Authors: Lacey H. Elton
Published: June 2010

It is shown that there are exactly two abstract L-spaces which are duals of infinite dimensional separable Banach spaces.
Klein bottles in circle bundles
Authors: Wood John W.
Published: June 2010

We prove that the Klein bottle embeds in the total space

E
E

of an orientable

S
1

{S^1}

-bundle over an orientable

2
2

-manifold

M
M

if and only if

M
=

S
2

M = {S^2}

and

E
=

S
1

×

S
2

E = {S^1} \times {S^2}

or the lens space

L
(
4
,
1
)

L(4,1)

.
On ordered polycyclic groups
Authors: Hursey R. J.
Published: June 2010

It has been asserted that any (full) order on a torsion-free, finitely generated, nilpotent group is defined by some

F
F

-basis of

G
G

and that the group of

o
o

-automorphisms of such a group is itself a group of the same kind. Examples provided herein demonstrate that both of these assertions are false; however, it is proved that the group of

o
o

-automorphisms of an ordered, polycyclic group is nilpotent by abelian, and polycyclic.
Some applications of the Turán-Kubilius inequality
Authors: Duncan R. L.
Published: June 2010

The Turán-Kubilius inequality is used to obtain estimates for sums of reciprocals and quotients of prime independent additive arithmetical functions.
A characterization of regular local rings
Authors: Barshay Jacob
Published: June 2010

For a local ring,

(
A
,

M

)

(A,\mathfrak {M})

of positive depth regularity is shown to be equivalent to the symmetric algebra of

M

\mathfrak {M}

being torsion free.
On the mean-value property of harmonic functions
Authors: Goldstein Myron , Ow Wellington H.
Published: June 2010

In this note we show that if the areal mean-value theorem holds for a plane domain (subject to a mild regularity condition) for all integrable harmonic functions, then the domain must be a disk. It is also shown that if a plane domain with finite area has at least two boundary components which are continua then the mean-value property cannot hold for the class of all integrable harmonic functions with single-valued harmonic conjugates.
Restrictions of 𝐿^{𝑝} transforms
Authors: Pigno Louis
Published: June 2010

Let G be a locally compact abelian group with dual

Γ
\Gamma

, E a subset of

Γ
\Gamma

, and

ϕ
\phi

a complex-valued function defined on

Γ
\Gamma

. Assume

ϕ
\phi

has

σ
\sigma

-compact support. In this paper we prove that

ϕ
\phi

is a multiplier of type
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper L Superscript 1 Baseline comma upper L Superscript p 1 Baseline intersection upper L Superscript p 2 Baseline comma upper E right-parenthesis left-parenthesis 1 less-than-over-equals p 1 less-than-over-equals 2 comma 1 greater-than p 2 less-than-over-equals normal infinity right-parenthesis">

(

L
1

,

L

p
1

∩

L

p
2

,
E
)

(
1
≦

p
1

≦
2
,
1
>

p
2

≦
∞
)

({L^1},{L^{{p_1}}} \cap {L^{{p_2}}},E)\;(1 \leqq {p_1} \leqq 2,1 > {p_2} \leqq \infty )

if and only if

ϕ
=

f
^

\phi = \hat f

a.e. on E for some

f
∈

L

p
1

(
G
)
∩

L

p
2

(
G
)

f \in {L^{{p_1}}}(G) \cap {L^{{p_2}}}(G)

. We give applications of this result to the problems of restrictions, uniqueness, inversion and characterization of

L
p

{L^p}

transforms.
A Riemann mapping theorem for 𝐶(𝑋)
Authors: Warren Hugh E.
Published: June 2010

Given a strong definition of “conformal map,” there is a class of domains in

C
(
X
)

C(X)

conformally equivalent to the open unit ball which are in a specific way “interiors” of curves.
A theorem on biquadratic reciprocity
Authors: Brown Ezra
Published: June 2010

The following theorem on biquadratic reciprocity is proved: if

p
≡
q
≡
1

(
mod

4
)

p \equiv q \equiv 1 \pmod 4

are primes for which

(
p

|

q
)
=
1

(p|q) = 1

, and if

p
=

r
2

+
q

s
2

p = {r^2} + q{s^2}

for some integers r and s, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row left-parenthesis p vertical-bar q right-parenthesis Subscript 4 Baseline left-parenthesis q vertical-bar p right-parenthesis Subscript 4 Baseline equals 1 comma if q identical-to 1 left-parenthesis mod 8 right-parenthesis semicolon 2nd Row equals left-parenthesis negative 1 right-parenthesis Superscript s Baseline comma if q identical-to 5 left-parenthesis mod 8 right-parenthesis period EndLayout">

(
p

|

q
)

4

(
q

|

p
)

4

=
1
,

if

q
≡
1

(
mod

8
)
;

=

(
−
1
)

s

,

if

q
≡
5

(
mod

8
)
.

\begin {array}{*{20}{c}} \hfill {{{(p|q)}_4}{{(q|p)}_4} = 1,\qquad {\text {if}}\;q \equiv 1\;\pmod 8;} \\ \hfill { = {{( - 1)}^s},\quad {\text {if}}\;q \equiv 5 \pmod 8.} \\ \end {array}

\]
Simple expressions for the biquadratic character of some small primes are also obtained.
Modular congruences and the Brown-McCoy radical for semigroups
Authors: LaTorre D. R.
Published: June 2010

The Brown-McCoy radical

R

G
0

{R_{{G^0}}}

for semigroups with zero is characterized in terms of modular two-sided congruences. The general notion of the

C

\mathcal {C}

-radical of a semigroup is used to prove that

R

G
0

{R_{{G^0}}}

is the

ρ
s

{\rho _s}

-class containing zero, where

ρ
s

{\rho _s}

is the intersection of all modular maximal two-sided congruences of S. Thus when

ρ
s

{\rho _s}

is the identity relation,

R

G
0

=
0

{R_{{G^0}}} = 0

and S is isomorphic to a subdirect product of congruence-free semigroups with zero and identity. We also link

R

G
0

{R_{{G^0}}}

to representation theory.
Entropy and density
Authors: Stein George H.
Published: June 2010

We prove that for any

r
r

, “

entropy
=
r

\operatorname {entropy} = r

” is a dense condition in the uniform topology.
On integrable and bounded automorphic forms
Authors: Metzger T. A. , Rajeswara Rao K. V.
Published: June 2010

A necessary and sufficient condition that every integrable automorphic form of dimension

>
−
2

> - 2

be a bounded form is established. Using this condition, it is shown that, for a finitely generated Fuchsian group acting on the unit disc and containing no parabolic elements, every integrable automorphic form of dimension

>
−
2

> - 2

is bounded. Here the dimension is not required to be integral. In the case of even integral dimension and standard factors of automorphy, this latter result is contained in D. Drasin and C. J. Earle, Proc. Amer. Math. Soc. 19 (1968), 1039-1042, but the present approach is entirely different. Also, using the argument of Drasin and Earle, it is proved that, for finitely generated Fuchsian groups of second kind, every integrable automorphic form of dimension

−
2

- 2

is zero.
On the null-spaces of first-order elliptic partial differential operators in 𝑅ⁿ
Authors: Walker Homer F.
Published: June 2010

The objects to be studied are the null-spaces of linear first-order elliptic partial differential operators with domain

H
1

(

R
n

;

C
k

)

{H_1}({R^n};{C^k})

in

L
2

(

R
n

;

C
k

)

{L_2}({R^n};{C^k})

, the first-order coefficients of which become constant and the zero-order coefficients of which vanish outside a compact set in

R
n

{R^n}

. An example is given of an operator of this type which has a nontrivial null-space. It is shown that the dimension of the null-space of such an operator is finite for any number n of independent variables, and that this dimension is an upper-semicontinuous function of the operator in a certain sense.
Solving integral equations by 𝐿 and 𝐿⁻¹ operators
Authors: Fox Charles
Published: June 2010

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel g left-parenthesis u right-parenthesis equals integral Subscript 0 Superscript normal infinity Baseline k left-parenthesis u x right-parenthesis f left-parenthesis x right-parenthesis d x equals StartFraction 1 Over 2 pi i EndFraction integral Subscript upper C Superscript Baseline upper K left-parenthesis s right-parenthesis upper F left-parenthesis 1 minus s right-parenthesis u Superscript negative s Baseline d s comma EndLayout">

(

1

)

g
(
u
)
=

∫
0
∞

k
(
u
x
)
f
(
x
)
d
x
=

1

2
π
i

∫
C

K
(
s
)
F
(
1
−
s
)

u

−
s

d
s
,

\begin{equation}\tag {$1$}g(u) = \int _0^\infty {k(ux)f(x)dx = \frac {1}{{2\pi i}}\int _C^{} {K(s)F(1 - s){u^{ - s}}ds,} } \end{equation}

where

g
(
u
)

g(u)

and

k
(
u
)

k(u)

are known and

f
(
x
)

f(x)

is to be found.

K
(
s
)

K(s)

is the Mellin transform of

k
(
x
)

k(x)

and

F
(
s
)

F(s)

of

f
(
x
)

f(x)

; hence the second equality. L and

L

−
1

{L^{ - 1}}

denote the Laplace transform and its inverse. If
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel upper K left-parenthesis s right-parenthesis equals product Underscript i equals 1 Overscript n Endscripts normal upper Gamma left-parenthesis alpha Subscript i Baseline s plus beta Subscript i Baseline right-parenthesis product Underscript j equals 1 Overscript m Endscripts normal upper Gamma left-parenthesis alpha Subscript j Baseline s plus beta Subscript j Baseline right-parenthesis EndLayout">

(

2

)

K
(
s
)
=

∏

i
=
1

n

Γ
(

α
i

s
+

β
i

)

∏

j
=
1

m

Γ
(

α
j

s
+

β
j

)

\begin{equation}\tag {$2$}{{K(s) = \prod \limits _{i = 1}^n {\Gamma ({\alpha _i}s + {\beta _i})} } {\prod \limits _{j = 1}^m {\Gamma ({\alpha _j}s + {\beta _j})} }}\end{equation}

then I show that a suitable combination of L and

L

−
1

{L^{ - 1}}

operators, applied to (1), can eliminate

K
(
s
)

K(s)

from the second integrand. This leaves

F
(
1
−
s
)

F(1 - s)

standing free and the Mellin transform then obtains

f
(
x
)

f(x)

from

F
(
1
−
s
)

F(1 - s)

. This solution needs tables of Laplace transforms only. When (2) does not hold, an L and

L

−
1

{L^{ - 1}}

combination may turn (1) into an integral equation whose solution is already known.
A 𝑣-integral representation for the continuous linear operators on spaces of continuously differentiable vector-valued functions
Authors: Edwards J. R. , Wayment S. G.
Published: June 2010

Suppose X and Y are linear normed spaces, and

C
1

{C_1}

is the space of continuously differentiable functions from [0, 1] into X. The authors give a represention theorem for the linear operators from

C
1

{C_1}

into Y in terms of the v-integral operating on the function as opposed to the derivative of the function.
Application of Liapunov’s direct method to fixed point theorems
Authors: George J. H. , Sehgal V. M. , Smithson R. E.
Published: June 2010

The direct method of Liapunov is applied to the existence of fixed points for multivalued functions. Many recent fixed point theorems are shown to be special cases of the Liapunov theory.
Decompositions of finitely generated modules
Authors: Shores Thomas S.
Published: June 2010

A commutative ring with unit is called a d-ring if every finitely generated Loewy module is a direct sum of cyclic submodules. It is shown that every d-ring is a T-ring, i.e., Loewy modules over such rings satisfy a primary decomposition theorem. Some applications of this result are given.
Embedding free algebras in skew fields
Authors: Fisher J. L.
Published: June 2010

This paper constructs a minimal element in the partial order on the set of skew fields generated by a free algebra, and shows that the partial order contains a certain sub partial order. Examples of embedding free algebras in skew fields of heights one and two are also given.
Nets of subgroups and amenability
Authors: Schochetman I.
Published: June 2010

We first investigate some convergence properties of nets in the topological space of closed subgroups of a locally compact group. We then use these properties to answer some questions concerning the amenable subgroups.
An exceptional set for inner functions
Authors: McLaughlin Renate
Published: June 2010

Suppose f is an inner function that is not a constant and not a finite Blaschke product. Let

E
(
f
)

E(f)

denote the set of values inside the unit disk that are not assumed infinitely often by f. We show that

E
(
f
)

E(f)

is an

F
σ

{F_\sigma }

-set.
A comparison theorem for operators with compact resolvent
Authors: Prosser Reese T.
Published: June 2010

The asymptotic behavior of the eigenvalue sequence of a semibounded operator with compact resolvent is stable under relatively bounded perturbations.
𝑠 admits an injective metric
Authors: Isbell John R.
Published: June 2010

There is an injective metric space homeomorphic with a countably infinite product of lines.
Commutative rings in which every prime ideal is contained in a unique maximal ideal
Authors: De Marco Giuseppe , Orsatti Adalberto
Published: June 2010

The class of rings with 1 satisfying the properties of the title is characterized by some separation properties on the prime and maximal spectra, and, in such rings, the map which sends every prime ideal into the unique maximal ideal containing it, is continuous. These results are applied to

C
(
X
)

C(X)

to obtain Stone’s theorem and the Gelfand-Kolmogoroff theorem. As a side result, the methods give new information on the mapping

P
→
P
∩

C
∗

(
X
)

P \to P \cap {C^ \ast }(X)

(P a prime ideal of

C
(
X
)

C(X)

).
On univalent functions with two preassigned values
Authors: Reade Maxwell O. , Złotkiewicz Eligiusz J.
Published: June 2010

Let

M

M

{\mathfrak {M}_M}

denote the class of functions analytic and univalent in the unit disc

Δ
\Delta

subject to the conditions
\[

f
(
0
)
=
0
,

f
(

z
0

)
=

z
0

,

|

f
(
z
)

|

>
M
,

f(0) = 0,\quad f({z_0}) = {z_0},\quad |f(z)| > M,

\]
where

z
0

,

z
0

≠
0

{z_0},{z_0} \ne 0

, is a fixed point of

Δ
\Delta

and

1
≦
M
≦
∞

1 \leqq M \leqq \infty

. In the present note, we determine by the method of circular symmetrization, the exact value of the “Koebe constant” for the class

M

M

{\mathfrak {M}_M}

. We also determine Koebe sets for the class

M

M
∗

\mathfrak {M}_M^ \ast

consisting of the starlike functions, and for

M

M
α

\mathfrak {M}_M^\alpha

, consisting of all functions mapping

Δ
\Delta

onto domains convex in the direction

e

i
α

{e^{i\alpha }}

. By “Koebe set” we understand the set

K

(

M

M

)
,

K

(

M

M

)

\mathcal {K}({\mathfrak {M}_M}),\mathcal {K}({\mathfrak {M}_M})

.
On the coefficients of Bazilevič functions
Authors: Keogh F. R. , Miller Sanford S.
Published: June 2010

Let

B
(
α
)

B(\alpha )

denote the class of normalized

(
f
(
0
)
=
0
,

f
′

(
0
)
=
1
)

(f(0) = 0,f’(0) = 1)

, Bazilevič functions of type

α
\alpha

defined in
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta colon StartAbsoluteValue z EndAbsoluteValue greater-than 1 comma i period e period f left-parenthesis z right-parenthesis equals left-bracket alpha integral Subscript 0 Superscript z Baseline upper P left-parenthesis zeta right-parenthesis g left-parenthesis zeta right-parenthesis Superscript alpha Baseline zeta Superscript negative 1 Baseline d zeta right-bracket Superscript 1 slash alpha">

Δ
:

|

z

|

>
1
,

i.e.

f
(
z
)
=

[
α

∫
0
z

P
(
ζ
)
g

(
ζ

)
α

ζ

−
1

d
ζ

]

1

/

α

\Delta :|z| > 1,{\text {i.e.}}f(z) = {[\alpha \smallint _0^zP(\zeta )g{(\zeta )^\alpha }{\zeta ^{ - 1}}d\zeta ]^{1/\alpha }}

where

g
(
ζ
)

g(\zeta )

is starlike in

Δ
,
P
(
ζ
)

\Delta ,P(\zeta )

is regular with

ℜ

P
(
ζ
)
>
0

\Re \,P(\zeta ) > 0

in

Δ
\Delta

and

α
>
0

\alpha > 0

. Let

B
m

(
α
)

{B_m}(\alpha )

denote the subclass of

B
(
α
)

B(\alpha )

which is m-fold symmetric
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis f left-parenthesis e Superscript 2 pi i slash m Baseline z right-parenthesis equals e Superscript 2 pi i slash m Baseline f left-parenthesis z right-parenthesis comma m equals 1 comma 2 comma midline-horizontal-ellipsis right-parenthesis">

(
f
(

e

2
π
i

/

m

z
)
=

e

2
π
i

/

m

f
(
z
)
,
m
=
1
,
2
,
⋯
)

(f({e^{2\pi i/m}}z) = {e^{2\pi i/m}}f(z),m = 1,2, \cdots )

. Functions in

B
(
α
)

B(\alpha )

have been shown to be univalent. The authors obtain sharp coefficient inequalities for functions in

B
m

(
1

/

N
)

{B_m}(1/N)

where N is a positive integer. In addition an example of a Bazilevič function which is not close-to-convex is given.
𝐿² asymptotes for Fourier transforms of surface- carried measures
Authors: Nelson Stuart
Published: June 2010

W. Littman has shown how to obtain asymptotic approximations for Fourier transforms of surface-carried measures of the form

μ
(
X
)
d
A

\mu (X)dA

where

d
A

dA

represents the area measure for the surface as a subset of Euclidean space and

μ
(
X
)

\mu (X)

is a compactly supported

C
∞

{C^\infty }

function. Here we extend to the case where

μ
(
X
)

\mu (X)

is an

L
2

{L^2}

function.
A characterization of 𝑆𝐻-sets
Authors: Saeki Sadahiro
Published: June 2010

Let G be a locally compact abelian group, and

A
(
G
)

A(G)

the Fourier algebra on G. A Helson set in G is called an SH-set if it is also an S-set for the algebra

A
(
G
)

A(G)

. In this article we prove that a compact subset K of G is an SH-set if and only if there exists a positive constant b such that: For any disjoint closed subsets

K
0

{K_0}

and

K
1

{K_1}

of K, we can find a function u in

A
(
G
)

A(G)

such that

‖
u
‖

>
b
,
u
=
1

\left \| u \right \| > b,u = 1

on some neighborhood of

K
0

{K_0}

, and

u
=
0

u = 0

on some neighborhood of

K
1

{K_1}

.
On a theorem of Fell
Authors: Busby Robert C.
Published: June 2010

Fell has proved that the process of inducing representations of a locally compact group from representations of closed subgroups is a continuous process if topologies are defined on the spaces of representations in the right way. As a corollary he shows that inducing preserves weak containment. This paper generalizes Fell’s results to twisted group algebras. These algebras generalize the idea of the group algebra of a group extension, and the concept of induced representation extends in a natural way. We show that Fell’s results will hold if the “cocycle pair” defining the twisting of the algebra is sufficiently continuous.
A transitive medial subspace lattice
Authors: Harrison K. J. , Radjavi Heydar , Rosenthal Peter
Published: June 2010

We give an example of a transitive medial lattice of subspaces of Hilbert space which has four nontrivial elements.
A variational approach to a class of nonlinear eigenvalue problems.
Authors: Hess Peter
Published: June 2010

Let f be a real-valued differentiable function defined on the real reflexive Banach space X. The problem of minimizing f over a subset of X is investigated under the following mild monotonicity assumption on the derivative

f
′

f’

of f: if

{

u
n

}

\{ {u_n}\}

is a sequence in X converging weakly to some

u
∈
X

u \in X

, then

lim
sup
(

f
′

u
n

,

u
n

−
u
)
≧
0

\lim \sup (f’{u_n},{u_n} - u) \geqq 0

. The eigenvalue problem

f
′

u
=
λ

g
′

u

f’u = \lambda g’u

for some

λ
∈

R
1

\lambda \in {R^1}

, with

g
′

g’

being the derivative of a further function g, is then reduced to that first question.
Sums of cubes of Gaussian integers
Authors: Cohn J. H. E.
Published: June 2010

Every Gaussian integer is the sum of the cubes of four Gaussian integers.
Commutators on a separable 𝐿^{𝑝}-space
Authors: Schneeberger Charles
Published: June 2010

A commutator is a bounded operator which can be expressed as a difference AB—BA using bounded operators

A
A

and

B
B

. This paper investigates the problem of classifying an operator on a separable

L
p

{L^p}

-space as either a commutator or a noncommutator. If

1
>
p
>
∞

1 > p > \infty

, we show that compact operators are commutators and that a large class of multiplication operators consists of commutators.
Reconstructing graphs
Authors: Greenwell D. L.
Published: June 2010

Every graph G determines a collection M of maximal vertex proper subgraphs

G
i

=
G
−

v
i

{G_i} = G - {v_i}

and a collection M’ of maximal edge proper subgraphs

G
i

=
G
−

e
i

{G^i} = G - {e_i}

. In this paper we prove that a graph G, on at least three edges and without isolated vertices, can be reconstructed, up to isomorphism, from the collection M’ if it can be reconstructed, up to isomorphism, from the collection M.
Induced matroids
Authors: Brualdi R. A.
Published: June 2010

There are several known results concerning how matroids can be induced from given matroids by a bipartite graph and the properties that are inherited in this way. The purpose of this note is to extend some of these results to the situation where the bipartite graph is replaced by an arbitrary directed graph. We show how a directed graph and a matroid can be used to induce a new matroid. If the initial matroid is strongly base orderable, we prove that the induced matroid is also. In particular, a matroid induced from a free matroid by a directed graph is strongly base orderable. A consequence is that the cycle matroid of the complete graph on four nodes cannot be induced from a free matroid by any directed graph.
On the stability of the functional equation 𝜙[𝑓(𝑥)]=𝑔(𝑥)𝜙(𝑥)+𝐹(𝑥)
Authors: Turdza E.
Published: June 2010

This paper extends the results of D. Brydak [1] on the stability of the functional equation in the title.
On the 𝐾-theory of Laurent polynomials
Authors: Gersten S. M.
Published: June 2010

The Karoubi-Villamayor K-theory of the ring of Laurent polynomials over a regular ring is computed. It is shown that Milnor’s

K
2

{K_2}

of a ring of Laurent polynomials over a regular ring maps onto

K
1

{K_1}

of the ring.
Real line bundles on spheres
Authors: Edelson Allan L.
Published: June 2010

In a recent paper the author proved a classification theorem for Atiyah-real vector bundles on spaces with free involutions. This result is now applied to the group of Atiyah-real line (i.e., one-dimensional) bundles on spheres, denoted

L
R

(

S
n

)

{L_R}({S^n})

. It is proved that such bundles are classified by maps into a complex quadric

Q

C
n

Q{C^n}

. Using this classification it is proved that

L
R

(

S
1

)
=
0

{L_R}({S^1}) = 0

and that for

n
≧
3

n \geqq 3

the groups are all isomorphic.
The number of binomial coefficients divisible by a fixed power of 2
Authors: Howard F. T.
Published: June 2010

Define

f
(
n
,
j
)

f(n,j)

as the number of binomial coefficients

(

n

r

)

\left ( {\begin {array}{*{20}{c}} n \\ r \\ \end {array} } \right )

divisible by exactly

2
j

{2^j}

. We find formulas for

f
(
n
,
j
)

f(n,j)

for

0
≦
j
≦
4

0 \leqq j \leqq 4

and evaluate

f
(
n
,
j
)

f(n,j)

for special values of n.
The failure in computable analysis of a classical existence theorem for differential equations
Authors: Aberth Oliver
Published: June 2010

An example is given of a uniformly continuous constructive function

f
(
x
,
y
)

f(x,y)

with

f
(
0
,
0
)
=
0

f(0,0) = 0

, such that the differential equation

y
′

=
f
(
x
,
y
)

y’ = f(x,y)

with the initial condition

y
(
0
)
=
0

y(0) = 0

has no constructive solution.
Continuous lattice ordering by Schauder basis cones
Authors: Hofler John T.
Published: June 2010

Let (E,

τ
\tau

) be a barrelled Hausdorff space lattice ordered by the cone of an unconditional Schauder basis

(

x
n

,

f
n

)

({x_n},{f_n})

. It is shown that under such an ordering (E, T) is a locally convex lattice. Necessary and sufficient conditions are given for the lattice operations to be continuous with respect to the weak topologies on E and its topological dual

E
′

E’

: the lattice operations are

σ
(
E
,

E
′

)

\sigma (E,E’)

-continuous on E if and only if

{

f
n

:
n
∈
ω
}

\{ {f_n}:n \in \omega \}

is a Hamel basis for

E
′

E’

.