bluuebunny/crossref_metadata_2025 huggingface dataset:
Combinatorial structures and group invariant partitions
Authors: White Dennis E. , Williamson S. G.
Published: June 2010

If a group acts on a set, an action of the group is induced on the partitions of the set. A formula is developed for the number of partitions invariant under this action. The formula is extended to count combinatorial objects such as labeled rooted trees or permutations defined on the invariant partitions.
A characterization of 𝐵-slowly varying functions
Authors: Bloom Steven
Published: June 2010

A measurable function

φ
>
0

\varphi > 0

that satisfies the limit condition
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript x right-arrow normal infinity Endscripts left-parenthesis phi left-parenthesis x plus t phi left-parenthesis x right-parenthesis right-parenthesis slash phi left-parenthesis x right-parenthesis right-parenthesis equals 1">

lim

x
→
∞

(
φ
(
x
+
t
φ
(
x
)
)

/

φ
(
x
)
)
=
1

{\lim _{x \to \infty }}(\varphi (x + t\varphi (x))/\varphi (x)) = 1

for all

t
t

is said to be

B
B

-slowly varying. If

φ
\varphi

is continuous, this limit is shown to hold uniformly for

t
t

in compact sets, and an integral representation is derived.
The exact cardinality of the set of invariant means on a group
Authors: Chou Ching
Published: June 2010

The purpose of this note is to show that if

G
G

is an infinite amenable group then

G
G

has exactly

2

2

|

G

|

{2^{{2^{|G|}}}}

invariant means where

|

G

|

|G|

denotes the cardinality of

G
G

.
Spaces defined by sequences
Authors: Berner Andrew J.
Published: June 2010

The biquotient, countably biquotient, hereditarily quotient, and quotient images of

q
q

-spaces are classified. Also, the quotient images of paracompact

M
M

-spaces, and the quotient images of

M
M

-spaces are classified without any separation axioms. New definitions are given for certain familiar classes of spaces to give the definitions more uniformity.
Sequential order and spaces 𝑆_{𝑛}
Authors: Rajagopalan M.
Published: June 2010

It is shown that the space

ψ

∗

{\psi ^{\ast }}

is a sequential space of order

2
2

which does not contain a copy of

S
2

{S_2}

. This solves a problem of Franklin and J. R. Boone. It is shown that there is a sequential space of order

4
4

and not of sequential order

3
3

but which still does not contain

S
3

{S_3}

. It is shown that for strongly sequential, and also for countable spaces, the problem of Franklin and Boone has an affirmative answer. Some open problems are raised.
Computation of the unoriented cobordism ring
Authors: Brown Edgar H. , Peterson Franklin P.
Published: June 2010

This note gives a set of generators for the unoriented cobordism ring and it gives a simplification of the algebra involved in Thom’s computation of this ring.
A note on semilocal rings
Authors: Johnson Johnny A.
Published: June 2010

If

(
R
,

m
1

,
…
,

m
w

)

(R,{m_1}, \ldots ,{m_w})

is a semilocal ring whose ideal lattice is topologically complete, it is shown that: given any natural number

n
n

and any decreasing sequence

⟨

a
i

⟩

\langle {a_i}\rangle

of ideals of

R
R

, there exists a natural number

s

(
n
)

s\left ( n \right )

such that

a

s
(
n
)

⊆
(
⋂

i

a
i

)
+

m
n

{a_{s(n)}} \subseteq (\bigcap {_i{a_i}) + {m^n}}

where

m
=

⋂

i
=
1

w

m
i

m = \bigcap \nolimits _{i = 1}^w {{m_i}}

. This generalizes a well-known theorem on complete semilocal rings.
A commutative Banach algebra which factorizes but has no approximate units
Authors: Leinert Michael
Published: June 2010

It is well known that any Banach algebra having bounded approximate units factorizes. For some time it was not clear if, conversely, factorization implied the existence of bounded approximate units. This was disproved by Paschke [3], but the problem remained open for commutative Banach algebras. We give an example of a commutative semisimple Banach algebra which factorizes but has not even unbounded approximate units.
On a convexity property of the range of a maximal monotone operator
Authors: Gossez Jean-Pierre
Published: June 2010

An example is given which shows that the closure of the range of a maximal monotone operator from a (nonreflexive) Banach space into its dual is not necessarily convex.
A peculiar triangulation of the 3-sphere
Authors: Altshuler A.
Published: June 2010

A triangulation of the

3
3

-sphere with

10
10

vertices is presented, which is not directly obtainable by a generalized stellar subdivision from any

3
3

-sphere with

9
9

vertices. This answers in the affirmative a conjecture by B. Grünbaum. This example is shown to be the minimal possible.
Relative integral bases
Authors: Washington Lawrence C.
Published: June 2010

We give necessary and sufficient conditions for certain quadratic extensions of imaginary quadratic fields to have relative integral bases.
Commutativity of rings with abelian or solvable units
Authors: Nicholson W. K. , Springer H. J.
Published: June 2010

A ring is called left suitable if idempotents can be lifted modulo every left ideal. These rings include all regular and all semiperfect rings. A left suitable ring with abelian group of units is commutative if it is either semiprime or

2
2

-torsion-free. A left suitable ring with zero Jacobson radical and solvable group of units is commutative if it is

6
6

-torsion-free.
Functions with different strong and weak 𝐹-variations
Authors: Yoder Lane
Published: June 2010

This paper shows by example how different the strong

Φ
\Phi

-variation can be from the weak

Φ
\Phi

-variation. Let

Φ
\Phi

be a convex function on

[
0
,
∞
)

[0,\infty )

with

Φ
(
0
)
=
0

\Phi (0) = 0

. A continuous function

f
f

on

[
a
,
b
]

[a,b]

is of bounded strong

Φ
\Phi

-variation if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup normal upper Sigma normal upper Phi left-parenthesis StartAbsoluteValue f left-parenthesis x Subscript i Baseline right-parenthesis minus f left-parenthesis x Subscript i minus 1 Baseline right-parenthesis EndAbsoluteValue right-parenthesis greater-than normal infinity">

sup
Σ
Φ
(

|

f
(

x
i

)
−
f
(

x

i
−
1

)

|

)
>
∞

\sup \Sigma \Phi (|f({x_i}) - f({x_{i - 1}})|) > \infty

for the partitions of

[
a
,
b
]

[a,b]

. Since
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="inf normal upper Sigma normal upper Phi left-parenthesis StartAbsoluteValue f left-parenthesis x Subscript i Baseline right-parenthesis minus f left-parenthesis x Subscript i minus 1 Baseline right-parenthesis EndAbsoluteValue right-parenthesis equals 0">

inf
Σ
Φ
(

|

f
(

x
i

)
−
f
(

x

i
−
1

)

|

)
=
0

\inf \Sigma \Phi (|f({x_i}) - f({x_{i - 1}})|) = 0

if

lim

x
→
0

x

−
1

Φ
(
x
)
=
0

{\lim _{x \to 0}}{x^{ - 1}}\Phi (x) = 0

, the weak

Φ
\Phi

-variation is defined as

inf
Σ
Φ
(
ω
(
f
;

x

i
−
1

,

x
i

)
)

\inf \Sigma \Phi (\omega (f;{x_{i - 1}},{x_i}))

, where

ω
(
f
;
c
,
d
)

\omega (f;c,d)

is the oscillation of

f
f

on

[
c
,
d
]

[c,d]

. Of special interest is the case

Φ
(
x
)
=

x
p

,
p
⩾
1

\Phi (x) = {x^p},p \geqslant 1

, in terms of which strong and weak variation dimensions are defined. They are denoted by

dim

s

(
f
)

{\dim _{\text {s}}}(f)

and

dim

w

(
f
)

{\dim _{\text {w}}}(f)

, respectively. By a lemma of Goffman and Loughlin, the Hausdorff dimension

d
d

of the graph of

f
f

provides a lower bound for

dim
w

(
f
)
:
1

/

(
2
−
d
)
⩽

dim
w

(
f
)

{\dim _w}(f):1/(2 - d) \leqslant {\dim _w}(f)

. A Lipschitz condition of order a provides an upper bound for

dim
s

(
f
)
:

dim
s

(
f
)
⩽
1

/

α

{\dim _s}(f):{\dim _s}(f) \leqslant 1/\alpha

. Besicovitch and Ursell showed that

1
⩽
d
⩽
2
−
α

1 \leqslant d \leqslant 2 - \alpha

and gave examples to show that

d
d

can take on any value in this interval. It turns out that these examples provide the extreme cases for variation dimensions; i.e.,

dim
w

(
f
)
=
1

/

(
2
−
d
)

{\dim _w}(f) = 1/(2 - d)

and

dim
s

(
f
)
=
1

/

α

{\dim _s}(f) = 1/\alpha

.
Subnormal shifts with operator-valued weights
Authors: Ghatage P.
Published: June 2010

A criterion for a shift with operator-valued weights to be subnormal is given. From this a different proof of a theorem of Berger and Sarason is deduced.
Invertible composition operators on 𝐿²(𝜆)
Authors: Singh Raj Kishor
Published: June 2010

Let

C
ϕ

{C_\phi }

be a composition operator on

L
2

(
λ
)

{L^2}(\lambda )

, where

λ
\lambda

is a

σ
\sigma

-finite measure defined on the Borel subsets of a standard Borel space. In this paper a necessary and sufficient condition for the invertibility of

C
ϕ

{C_\phi }

is given in terms of invertibility of

ϕ
\phi

. Also all invertible composition operators on

L
2

(

R

)

{L^2}({\mathbf {R}})

induced by monotone continuous functions are characterised.
On totally real bisectional curvature
Authors: Houh Chorng Shi
Published: June 2010

A Kaehler manifold of dimension

⩾
3

\geqslant 3

is a complex space form if and only if it has constant totally real bisectional curvature.
Peterson-Stein formulas in the Adams spectral sequence
Authors: Krueger Warren M.
Published: June 2010

The purpose of this note is to establish Peterson-Stein formulas for second order differentials in the Adams spectral sequence.
On a problem of G. Grätzer
Authors: Katriňák Tibor
Published: June 2010

A characterization of the congruence lattice of a modular

p
p

-algebra in terms of congruence pairs is given.
Diophantine inequalities with mixed powers (𝑚𝑜𝑑 1)
Authors: Cook R. J.
Published: June 2010

A theorem of Heilbronn on the distribution of the sequence

n
2

θ

(
mod

1
)

{n^2}\theta \pmod 1

is extended to sums of mixed powers.
The need for a new classification of double hypergeometric series
Authors: Carlson B. C.
Published: June 2010

An elementary proof is given of a linear transformation which changes a particular double hypergeometric series of order two into a series of order three. A similar transformation holds for a particular series of arbitrary order in two or more variables. The change in order provides new evidence, the most compelling to date, that the order of such a series is not a fundamental property. This conclusion undermines the accepted classification of hypergeometric series in more than one variable.
Asymptotic stability for abstract nonlinear functional differential equations
Authors: Webb G. F.
Published: June 2010

The nonlinear autonomous functional differential equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove x With dot left-parenthesis t right-parenthesis equals f left-parenthesis x left-parenthesis t right-parenthesis right-parenthesis plus g left-parenthesis x Subscript t Baseline right-parenthesis comma t greater-than-or-slanted-equals 0 comma x 0 equals phi">

x
˙

(
t
)
=
f
(
x
(
t
)
)
+
g
(

x
t

)
,
t
⩾
0
,

x
0

=
ϕ

\dot x(t) = f(x(t)) + g({x_t}),t \geqslant 0,{x_0} = \phi

is investigated by means of the theory of semigroups of nonlinear operators. The properties of the semigroup associated with this equation provide stability information about the solutions.
A best constant for Zygmund’s conjugate function inequality
Authors: Bennett Colin
Published: June 2010

When the space

L

log
+

L

L{\log ^ + }L

is given the Hardy-Littlewood norm the best constant in the corresponding version of Zygmund’s conjugate function inequality is shown to be
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper K equals ContinuedFraction 1 Superscript negative 2 Baseline minus 3 Superscript negative 2 Baseline plus 5 Superscript negative 2 Baseline minus 7 Superscript negative 2 Baseline plus midline-horizontal-ellipsis Over 1 Superscript negative 2 Baseline plus 3 Superscript negative 2 Baseline plus 5 Superscript negative 2 Baseline plus 7 Superscript negative 2 Baseline plus midline-horizontal-ellipsis period">

K

=

1

−
2

−

3

−
2

+

5

−
2

−

7

−
2

+
⋯

1

−
2

+

3

−
2

+

5

−
2

+

7

−
2

+
⋯

.

{\mathbf {K}} = \frac {{{1^{ - 2}} - {3^{ - 2}} + {5^{ - 2}} - {7^{ - 2}} + \cdots }}{{{1^{ - 2}} + {3^{ - 2}} + {5^{ - 2}} + {7^{ - 2}} + \cdots }}.

\]
This complements the recent result of Burgess Davis that the best constant in Kolmogorov’s inequality is

K

−
1

{{\mathbf {K}}^{ - 1}}

.
A theorem on 𝐶-totally real minimal surface
Authors: Yamaguchi Seiichi , Kon Masahiro , Miyahara Yasushi
Published: June 2010

We prove the corresponding theorem for a

C
C

-totally real minimal surface to Yau’s theorem for a totally real minimal surface of a Kähler surface with constant holomorphic sectional curvature.
Countable paracompactness of 𝐹_{𝜎}-sets
Authors: Zenor Phillip
Published: June 2010

If each

F
σ

{F_\sigma }

-set in

X
×
Y

X \times Y

is countably paracompact, then either

X
X

is normal or no countable discrete subset of

Y
Y

has a limit point. It follows that, for each cardinal number

m

\mathfrak {m}

, there is an

m

\mathfrak {m}

-paracompact space containing a noncountably paracompact

F
σ

{F_\sigma }

-subset.
On the space of piecewise linear homeomorphisms of a manifold
Authors: Geoghegan Ross , Haver William E.
Published: June 2010

Let

M
n

{M^n}

be a compact PL manifold,

n
≠
4

n \ne 4

; if

n
=
5

n = 5

, suppose

∂
M

\partial M

is empty. Let

H
(
M
)

H(M)

be the space of homeomorphisms on

M
M

and

H

∗

(
M
)

{H^{\ast }}(M)

the elements of

H
(
M
)

H(M)

which are isotopic to PL homeomorphisms. It is shown that the space of PL homeomorphisms,

P
L
H
(
M
)

PLH(M)

, has the finite dimensional compact absorption property in

H

∗

(
M
)

{H^{\ast }}(M)

and hence that

(

H

∗

(
M
)
,
P
L
H
(
M
)
)

({H^{\ast }}(M),PLH(M))

is an

(

l
2

,

l
2
f

)

({l_2},l_2^f)

-manifold pair if and only if

H
(
M
)

H(M)

is an

l
2

{l_2}

-manifold. In particular, if

M
2

{M^2}

is a

2
2

-manifold,

(
H
(

M
2

)
,
P
L
H
(

M
2

)
)

(H({M^2}),PLH({M^2}))

is an

(

l
2

,

l
2
f

)

({l_2},l_2^f)

-manifold pair.
On quotients of moving average processes with infinite mean
Authors: Kanter Marek
Published: June 2010

In this paper it is shown that one can estimate the sum of the weights used to form a stationary moving average stochastic process based on nonnegative random variables by taking the limit in probability of suitable quotients, even when the random variables involved have infinite expectation.
A Nagumo condition for ordinary differential equations
Authors: Jackson L. K.
Published: June 2010

A condition is given which guarantees that each solution of

y

(
n
)

=
f
(
x
,
y
,

y
′

,
…
,

y

(
n
−
1
)

)

{y^{(n)}} = f(x,y,y’, \ldots ,{y^{(n - 1)}})

either extends or becomes unbounded on its maximal interval of existence.
A short proof of Schoenberg’s theorem
Authors: Ressel Paul
Published: June 2010

Using positive semidefiniteness of Laplace transforms, we give a short and simple proof of Schoenberg’s theorem characterising radially symmetric positive semidefinite functions on a Hilbert space. A slight generalisation of this theorem is also given.
A generalization of the Wiener-Lévy theorem applicable to some Volterra equations
Authors: Jordan G. S. , Wheeler Robert L.
Published: June 2010

Recently, Shea and Wainger obtained a variant of the Wiener-Lévy theorem for nonintegrable functions of the form

a
(
t
)
=
b
(
t
)
+
β
(
t
)

a(t) = b(t) + \beta (t)

, where

b
(
t
)

b(t)

is nonnegative, nonincreasing, convex and locally integrable, and

β
(
t
)
,
t
β
(
t
)
∈

L
1

(
0
,
∞
)

\beta (t),t\beta (t) \in {L^1}(0,\infty )

. It is shown here that the moment condition

t
β
(
t
)
∈

L
1

t\beta (t) \in {L^1}

may be omitted from the hypotheses of this theorem. These results are useful in the study of stability problems for some Volterra integral and integrodifferential equations.
Codimension of some subspaces in a Fréchet algebra
Authors: Christensen Jens Peter Reus
Published: June 2010

In a complete separable metrizable topological algebra, if the linear span of the set of all products of two elements has at most countable algebraic codimension, then it has finite codimension.
Extreme operators in the unit ball of 𝐿(𝐶(𝑋),𝐶(𝑌)) over the complex field
Authors: Gendler Alan
Published: June 2010

Assume that

X
X

and

Y
Y

are compact Hausdorff spaces and that

C
(
X
)

C(X)

and

C
(
Y
)

C(Y)

are the Banach spaces of continuous complex-valued functions on

X
X

and

Y
Y

, respectively.

L
(
C
(
X
)
,
C
(
Y
)
)

L(C(X),C(Y))

is the space of bounded linear operators from

C
(
X
)

C(X)

to

C
(
Y
)

C(Y)

. If

E
E

is a Banach space, then

S
(
E
)

S(E)

is the closed unit ball in

E
E

. An operator

T
T

in

S
(
L
(
C
(
X
)
,
C
(
Y
)
)
)

S(L(C(X),C(Y)))

is nice if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Superscript asterisk Baseline left-parenthesis e x t upper S left-parenthesis upper C left-parenthesis upper Y right-parenthesis Superscript asterisk Baseline right-parenthesis right-parenthesis subset-of e x t upper S left-parenthesis upper C left-parenthesis upper X right-parenthesis Superscript asterisk Baseline right-parenthesis">

T
∗

(
ext
⁡
S
(
C

(
Y

)
∗

)
)
⊂
ext
⁡
S
(
C

(
X

)
∗

)

{T^ \ast }(\operatorname {ext} S(C{(Y)^ \ast })) \subset \operatorname {ext} S(C{(X)^ \ast })

. For each

y
∈
Y
,

ε
y

y \in Y,{\varepsilon _y}

denotes point mass at

y
y

. The main theorem states that if

T
T

is extreme in

S
(
L
(
C
(
X
)
,
C
(
Y
)
)
)

S(L(C(X),C(Y)))

and

|

|

T
∗

(

ε
y

)

|

|

=
1

||{T^ \ast }({\varepsilon _y})|| = 1

for all

y
∈
Y

y \in Y

, then

T
T

is nice. Other theorems are proved by using the same techniques as in the proof of the main theorem.
A remark on irreducible spaces
Authors: Smith J. C.
Published: June 2010

A topological space

X
X

is called irreducible if every open cover of

X
X

has an open refinement which covers

X
X

minimally. In this paper we show that weak

θ
¯

\bar \theta

-refinable spaces are irreducible. A modification of the proof of this result then yields that

ℵ
1

{\aleph _1}

-compact, weak

δ
θ

¯

\overline {\delta \theta }

-refinable spaces are Lindelöf. It then follows that perfect,

ℵ
1

{\aleph _1}

-compact weak

δ
θ

\delta \theta

-refinable spaces are irreducible. A number of known results follow as corollaries.
On realizing centralizers of certain elements in the fundamental group of a 3-manifold
Authors: Feustel C. D.
Published: June 2010

The main result in this note is that if

λ
\lambda

is a simple loop in the boundary of a compact, irreducible, orientable

3
3

-manifold

M
M

and

[
λ
]
≠
1
∈

π
1

(
M
)

[\lambda ] \ne 1 \in {\pi _1}(M)

, one can represent the centralizer of

[
λ
]

[\lambda ]

in

π
1

(
M
)

{\pi _1}(M)

by a Seifert fibred submanifold of

M
M

.
The largest proper variety of lattice ordered groups
Authors: Holland W. Charles
Published: June 2010

If a lattice ordered group

G
G

satisfies any identical relation, other than those satisfied by every lattice ordered group, then

G
G

is normal valued, and hence satisfies the relation

a
b
⩽

b
2

a
2

ab \leqslant {b^2}{a^2}

for all

a
,
b
⩾
e

a,b \geqslant e

.
A characteristically nilpotent Lie algebra can be a derived algebra
Authors: Luks Eugene M.
Published: June 2010

An example is constructed of a Lie algebra whose derived algebra has only nilpotent derivations, thus answering a question of Dixmier and Lister.
On the cellularity of 𝛽𝑋-𝑋
Authors: Ginsburg John , Woods R. Grant
Published: June 2010

For a topological space

X
X

, let

c
(
X
)

c(X)

denote the cellularity of

X
X

, and let

k
(
X
)

k(X)

denote the least cardinal of a cobase for the compact subsets of

X
X

. It is shown that, if

X
X

is a completely regular Hausdorff space,

c
(
β
X
−
X
)
⩽

2

c
(
X
)
k
(
X
)

c(\beta X - X) \leqslant {2^{c(X)k(X)}}

, and examples are given to show that this inequality is sharp. It is also shown that if

X
X

is an extremally disconnected completely regular Hausdorff space for which

c
(
β
X
−
X
)
>

2

k
(
X
)

c(\beta X - X) > {2^{k(X)}}

, then

β
X
−
X

\beta X - X

contains a discrete

C
∗

{C^ \ast }

-embedded subspace of cardinality

k

(
X

)
+

k{(X)^ + }

.
The signature and arithmetic genus of certain aspherical manifolds
Authors: Farrell F. Thomas
Published: June 2010

In this paper we show that the signature and arithmetic genus of certain aspherical manifolds

M
M

vanish when the center of

π
1

M

{\pi _1}M

is nontrivial. We make the possibly technical assumption that

π
1

M

{\pi _1}M

is residually finite.
Corrigendum and addendum to: “Pseudo-matchings of a bipartite graph” (Proc. Amer. Math. Soc. 42 (1974), 28–32) by Alan Brace and Daykin
Authors: Daykin D. E.
Published: June 2010

For an undirected bipartite graph

G
G

conditions on

deg
⁡
x
+
deg
⁡
y

\deg x + \deg y

for

x
y
∉
G

xy \notin G

ensure sets of independent lines extend to hamiltonian cycles.
The sum of a digitaddition series
Authors: Stolarsky Kenneth B.
Published: June 2010

Let

B
(
x
)

B(x)

be the number of ones in the binary expansion of

x
x

. A “digitaddition series” is a sequence

y
1

>

y
2

>

y
3

>
…

{y_1} > {y_2} > {y_3} > \ldots

, where

y
1

{y_1}

is a given positive integer and

y

n
+
1

=

y
n

+
B
(

y
n

)

for

n
=
1
,
2
,
…

{y_{n + 1}} = {y_n} + B({y_n})\;{\text {for}}\;n = 1,2, \ldots

. Various questions involving the

y
m

{y_m}

are studied; in particular, the asymptotic result

y
m

∼
(
m
log
⁡
m
)

/

(
2
log
⁡
2
)

{y_m} \sim (m\log m)/(2\log 2)

is proved.
Implicitly elementary integrals
Authors: Risch Robert H.
Published: June 2010

The explicitly elementary functions of complex variables

z
1

,
…
,

z
n

{z_1}, \ldots ,{z_n}

are those functions built up from

C

(

z
1

,
…
,

z
n

)

{\mathbf {C}}({z_1}, \ldots ,{z_n})

by exponentiation, taking logarithms, and algebraic operations. The implicitly elementary functions are obtained by solving, via the implicit function theorem, for some of the variables in terms of the others, in systems of equations formed by setting a set of explicitly elementary functions equal to 0. Here we prove a 1923 conjecture of J. F. Ritt to the effect that if the indefinite integral of an explicitly elementary function is implicitly elementary, then it is explicitly elementary. The method features a geometrization of the concepts involved.
Roth’s equivalence problem in unit regular rings
Authors: Hartwig Robert E.
Published: June 2010

It is shown that for matrices over a unit regular ring
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Start 2 By 2 Matrix 1st Row 1st Column upper A 2nd Column a m p semicolon upper C 2nd Row 1st Column 0 2nd Column a m p semicolon upper D EndMatrix tilde Start 2 By 2 Matrix 1st Row 1st Column upper A 2nd Column a m p semicolon 0 2nd Row 1st Column 0 2nd Column a m p semicolon upper D EndMatrix">

[

A

a
m
p
;
C

0

a
m
p
;
D

]

∼

[

A

a
m
p
;
0

0

a
m
p
;
D

]

\left [ {\begin {array}{*{20}{c}} A & C \\ 0 & D \\ \end {array} } \right ]\sim \left [ {\begin {array}{*{20}{c}} A & 0 \\ 0 & D \\ \end {array} } \right ]

if and only if there exist solutions

X
X

and

Y
Y

to

A
X
−
Y
D
=
C

AX - YD = C

, thus providing a partial generalization to Roth’s theorem.
Locally compact subgroups of metrizable topological abelian groups
Authors: Hooper R. C.
Published: June 2010

This paper examines the question of existence of locally compact subgroups of topological abelian groups. A counterexample is given and it is shown in a certain setting that groups have a nontrivial locally compact subgroup.
Multiplier extensions other than the Mackey extension
Authors: Baggett Larry
Published: June 2010

The set of all multiplier extensions to the whole group

G
G

of a representation of a normal subgroup

N
N

is completely described, and necessary and sufficient conditions are given for equivalence of two such extensions.
Cesàro summability of the conjugate series and the double Hilbert transform
Authors: Basinger John O.
Published: June 2010

If

f
(
x
,
y
)

f(x,y)

, a

2
π

2\pi

periodic function in each variable, has a modulus of continuity

w
f

(
δ
)
=
o
(
1

/

log
⁡
(
1

/

δ
)
)

{w_f}(\delta ) = o(1/\log (1/\delta ))

then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove sigma With tilde Subscript n Baseline left-parenthesis x comma y comma f right-parenthesis minus integral Subscript 1 slash n Superscript pi Baseline integral Subscript 1 slash n Superscript pi Baseline StartFraction left-bracket f left-parenthesis x plus u comma y plus v right-parenthesis minus f left-parenthesis x minus u comma y plus v right-parenthesis minus f left-parenthesis x plus u comma y minus v right-parenthesis plus f left-parenthesis x minus u comma y minus v right-parenthesis right-bracket Over 4 tangent left-parenthesis u slash 2 right-parenthesis tangent left-parenthesis v slash 2 right-parenthesis EndFraction d u d v right-arrow 0 uniformly in left-parenthesis x comma y right-parenthesis">

σ
~

n

(
x
,
y
,
f
)
−

∫

1

/

n

π

∫

1

/

n

π

[
f
(
x
+
u
,
y
+
v
)
−
f
(
x
−
u
,
y
+
v
)
−
f
(
x
+
u
,
y
−
v
)
+
f
(
x
−
u
,
y
−
v
)
]

4
tan
⁡
(
u

/

2
)
tan
⁡
(
v

/

2
)

d
u
d
v
→
0

uniformly

in

(
x
,
y
)

{\tilde \sigma _n}(x,y,f) - \int _{1/n}^\pi {\int _{1/n}^\pi {\frac {{[f(x + u,y + v) - f(x - u,y + v) - f(x + u,y - v) + f(x - u,y - v)]}}{{4\tan (u/2)\tan (v/2)}}} } dudv \to 0\quad {\text {uniformly}}\;{\text {in}}\;(x,y)

\]
where

σ
~

n

(
x
,
y
,
f
)

{\tilde \sigma _n}(x,y,f)

is the first arithmetic mean of the conjugate series. This theorem is best possible in that

o
(
1

/

log
⁡
(
1

/

δ
)
)

o(1/\log (1/\delta ))

cannot be replaced by

O
(
1

/

log
⁡
(
1

/

δ
)
)

O(1/\log (1/\delta ))

.
A note on Walsh-Fourier series
Authors: Young Wo Sang
Published: June 2010

It is shown that the double sequence

{

λ

m
n

}

\{ {\lambda _{mn}}\}

with

λ

m
n

=
1

{\lambda _{mn}} = 1

if

n
⩽
m

n \leqslant m

and

0
0

otherwise is an

L
p

{L^p}

multiplier for the Walsh system in two dimensions only if

p
=
2

p = 2

. This result is then used to show that the one-dimensional trigonometric system and the Walsh system are nonequivalent bases of the Banach space

L
p

[
0
,

1
]

{L^p}[0,\;1]

, and hence have different

L
p

{L^p}

multipliers,

1
>
p
>
∞
,

p
≠
2

1 > p > \infty ,\;p \ne 2

.
A class of spectral sets
Authors: Warner C. Robert
Published: June 2010

The two main results are: (i) If the union and intersection of two closed sets are Ditkin sets, then each of the sets is a Ditkin set. (ii) If the union of two sets is a spectral set and their intersection is a Ditkin set, then each of the sets is a spectral set. A corollary of (i) is a generalization of a theorem due to Calderón which proved that closed polyhedral sets in

R
n

{R^n}

are Ditkin (= Calderón) sets. A corollary of (ii) establishes an analogous result for spectral sets. The proofs hold for commutative semisimple regular Banach algebras which satisfy Ditkin’s condition-that the empty set and singletons are Ditkin sets in the maximal ideal space.
A new proof of the Reidemeister-Singer theorem on stable equivalence of Heegaard splittings
Authors: Craggs Robert
Published: June 2010

A proof of the Reidemeister-Singer theorem on stable equivalence of Heegaard splittings is given. This proof makes use of the Chillingworth theorem on the preservation of simplicial collapses for subdivisions of complexes of dimension less than or equal to three, and it is based on the observation that subdivision and collapsing preserve stable equivalence.
The strong dual of a strongly nuclear space need not be nuclear
Authors: Cross Joe
Published: June 2010

A strongly nuclear sequence space is exhibited whose strong dual is normed.
On a covering property of convex sets
Authors: Groemer H.
Published: June 2010

Let

{

K
1

,

K
2

,
…
}

\{ {K_1},\;{K_2}, \ldots \}

be a class of compact convex subsets of euclidean

n
n

-space with the property that the set of their diameters is bounded. It is shown that the sets

K
i

{K_i}

can be rearranged by the application of rigid motions so as to cover the total space if and only if the sum of the volumes of all the sets

K
i

{K_i}

is infinite. Also, some statements regarding the densities of such coverings are proved.
Countable spaces having exactly one nonisolated point. I
Authors: Siwiec Frank
Published: June 2010

Some countable,

T
1

{T_1}

,

k
k

-spaces having exactly one nonisolated point are characterised by means of intrinsic properties and mapping conditions.
The number of semigroups of order 𝑛
Authors: Kleitman Daniel J. , Rothschild Bruce R. , Spencer Joel H.
Published: June 2010

The number of semigroups on

n
n

elements is counted asymptotically for large

n
n

. It is shown that “almost all” semigroups on

n
n

elements have the following property: The

n
n

elements are split into sets

A
,
B

A,B

and there is an

e
∈
B

e \in B

so that whenever

x
,
y
∈
A
,
x
y
∈
B

x,y \in A,xy \in B

, but if

x
x

or

y
y

is in

B
,
x
y
=
e

B,xy = e

.
Primitive ideals in group rings of polycyclic groups
Authors: Snider Robert L.
Published: June 2010

If

F
F

is a field which is not algebraic over a finite field and

G
G

is a polycyclic group, then all primitive ideals of the group ring

F
[
G
]

F[G]

are maximal if and only if

G
G

is nilpotent-by-finite.
Lower Frattini series in finitely generated soluble groups
Authors: Lennox John C. , Wiegold James
Published: June 2010

It is proved that some term of the lower Frattini series of a finitely generated abelian-by-polycyclic group is trivial. The analogous result for finitely generated centre-by-metabelian groups is false.
Inverse limits of topological group cohomologies
Authors: Insel Arnold J.
Published: June 2010

For second countable locally compact almost connected groups

G
G

and

A
A

, where

A
A

is abelian and

G
G

acts on

A
A

continuously, it is shown that it is possible to represent

A
A

as an inverse limit of Lie groups

{

A
n

}

\{ {A_n}\}

compatible with the action of

G
G

and such that

H

∗

(
G
,
A
)

{H^{\ast }}(G,A)

is isomorphic to

lim
n

inv
⁡

H

∗

(
G
,

A
n

)

{\lim _n}\operatorname {inv} {H^{\ast }}(G,{A_n})

, provided that

A
A

is compact or connected.
Deficiencies of the associated curves of a holomorphic curve in the projective space
Authors: Niinō Kiyoshi
Published: June 2010

Let

k

x

_kx

be the nonconstant associated holomorphic curve of rank

k
(
1
≦
k
≦
n
−
1
)

k(1 \leqq k \leqq n - 1)

of a transcendental holomorphic curve

x
:

C

→

P
n

C

x:{\mathbf {C}} \to {P_n}{\mathbf {C}}

. It is proved that if

1
≦
k
≦
n
−
2

1 \leqq k \leqq n - 2

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript j Superscript k Baseline element-of upper P Subscript l left-parenthesis k right-parenthesis minus 1 Baseline bold upper C comma j equals 1 comma ellipsis comma 2 l left-parenthesis k right-parenthesis minus 2 left-parenthesis l left-parenthesis k right-parenthesis equals left-parenthesis Subscript k plus 1 Baseline Superscript n plus 1 Baseline right-parenthesis right-parenthesis">

A
j
k

∈

P

l
(
k
)
−
1

C

,
j
=
1
,
…
,
2
l
(
k
)
−
2
(
l
(
k
)
=

(

k
+
1

n
+
1

)
)

A_j^k \in {P_{l(k) - 1}}{\mathbf {C}},j = 1, \ldots ,2l(k) - 2(l(k) = (_{k + 1}^{n + 1}))

are in general position and

⟨
k

x
,

A
j
k

⟩
≢
0

{\langle _k}x,A_j^k\rangle \not \equiv 0

for all

A
j
k

A_j^k

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript j equals 1 Overscript 2 l left-parenthesis k right-parenthesis minus 2 Endscripts delta Subscript k Baseline left-parenthesis upper A Subscript j Superscript k Baseline right-parenthesis less-than-over-equals 2 l left-parenthesis k right-parenthesis minus 3">

∑

j
=
1

2
l
(
k
)
−
2

δ
k

(

A
j
k

)
≦
2
l
(
k
)
−
3

\sum \nolimits _{j = 1}^{2l(k) - 2} {{\delta _k}(A_j^k) \leqq 2l(k) - 3}

and that in the case when
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals n minus 1 comma sigma-summation Underscript upper A Superscript n minus 1 Baseline Endscripts delta Subscript n minus 1 Baseline left-parenthesis upper A Superscript n minus 1 Baseline right-parenthesis less-than-over-equals l left-parenthesis n minus 1 right-parenthesis">

k
=
n
−
1
,

∑

A

n
−
1

δ

n
−
1

(

A

n
−
1

)
≦
l
(
n
−
1
)

k = n - 1,\sum \nolimits _{{A^{n - 1}}} {{\delta _{n - 1}}({A^{n - 1}}) \leqq l(n - 1)}

, where

{

A

n
−
1

}

\{ {A^{n - 1}}\}

is a finite subset of

P

l
(
n
−
1
)
−
1

C

{P_{l(n - 1) - 1}}{\mathbf {C}}

in general position such that

⟨

n
−
1

x
,

A

n
−
1

⟩
≢
0

{\langle _{n - 1}}x,{A^{n - 1}}\rangle \not \equiv 0

for all

A

n
−
1

{A^{n - 1}}

. These are sharp.
Absolute Schauder bases for 𝐶(𝑋) with the compact-open topology
Authors: Cook Thurlow A. , Ruckle William H.
Published: June 2010

Let

X
X

be a completely regular Hausdorff space and

C
(
X
)

C(X)

the space of all real valued continuous functions on

X
X

. We give

C
(
X
)

C(X)

the compact-open topology. The principal result of this paper is: If

C
(
X
)

C(X)

is complete and barrelled and has an absolute basis, then

C
(
X
)

C(X)

is isomorphic to a countable product of lines.
Undecidability of the theory of abelian groups with a subgroup
Authors: Baur Walter
Published: June 2010

The theory of abelian groups with an additional predicate denoting a subgroup is undecidable.
An extension theorem for functions on semigroups
Authors: Milnes Paul
Published: June 2010

For

S
S

a semitopological semigroup, a continuous function on

S
S

is said to be in

L
M
C
(
S
)

LMC(S)

if its set of right translates is relatively compact in

C
(
S
)

C(S)

for the topology of pointwise convergence on

S
S

. It is proved here that, if

S
S

is a dense subsemigroup of a topological group

G
G

, then every function in

L
M
C
(
S
)

LMC(S)

extends to a function continuous on

G
G

. This result generalizes earlier results that were arrived at independently by A. T. Lau and the present author. Some corollaries of this result are also presented.
Characteristic functions of operators similar to unitary operators
Authors: Ghatage P.
Published: June 2010

Suppose

T
T

denotes an operator which is similar to a unitary operator and

θ
T

{\theta _T}

denotes its characteristic function—considered as an operator on

H
2

(

D
T

)

{H^2}({D_T})

. It is proved that

θ
T

{\theta _T}

is a bounded operator if and only if it is the inverse of a bounded operator.
A subalgebra condition in Lie-admissible algebras
Authors: Myung Hyo Chul
Published: June 2010

Let

A
A

be a finite-dimensional, flexible, Lie-admissible algebra over a field

Φ
\Phi

of characteristic

≠
2

\ne 2

. Let

S
S

be a subalgebra of

A
−

{A^ - }

and

H
H

be a Cartan subalgebra of

S
S

. It is shown that

S
S

is a subalgebra of

A
A

if and only if

H
H
⊆
S

HH \subseteq S

.
Remarks on the cardinality of compact spaces and their Lindelöf subspaces
Authors: Hajnal A. , Juhász I.
Published: June 2010

Several applications of the Čech-Pospišil theorem are given; one of them states (under CH) that every uncountable compact space has a Lindelöf subspace of cardinality

ω
1

{\omega _1}

.
Monotone retracts and some characterizations of dendrites
Authors: Gordh G. R. , Lum Lewis
Published: June 2010

Let

M
M

be a metric continuum containing a fixed point

p
p

. The following conditions are shown to be equivalent. (i)

M
M

is a dendrite. (ii) Each subcontinuum of

M
M

is a monotone retract of

M
M

. (iii)

M
M

is arcwise connected and each subcontinuum of

M
M

containing

p
p

is a monotone retract of

M
M

.
Configuration-like spaces and the Borsuk-Ulam theorem
Authors: Cohen Fred , Lusk Ewing L.
Published: June 2010

Some extensions of the classical Borsuk-Ulam Theorem are proved by computing a bound on the homology of certain spaces similar to configuration spaces. The Bourgin-Yang Theorem and a generalization due to Munkholm are special cases of these results.
Negative tangent bundles and hyperbolic manifolds
Authors: Wong B.
Published: June 2010

We construct a family of algebraic manifolds which are hyperbolic in the sense of Kobayashi, but whose tangent bundles are not negative.
On the cohomology groups of a manifold with a nonintegrable subbundle
Authors: Kitahara Haruo , Yorozu Shinsuke
Published: June 2010

We define the cohomology groups

H
u

(
M
,

D

s
,
t

)

{H^u}(M,\;{\mathcal {D}^{s,t}})

of a manifold

M
M

with a nonintegrable subbundle

E
E

, and we give the condition for the existence of a bundle-like metric with respect to

E
E

.
One dimensional perturbations of compact operators
Authors: Vasudeva Harkrishan
Published: June 2010

Let

A
A

be a compact selfadjoint operator acting on a Hilbert space

H
H

.

P
P

denotes a one dimensional projection also acting on

H
H

. It is shown that the eigenvalues of

A
A

and

A
+
t
P
(
t
>
0
)

A + tP(t > 0)

interlace on the real axis. A converse of this result is also proved.
Analytic Toeplitz operators with automorphic symbol. II
Authors: Abrahamse M. B. , Ball Joseph A.
Published: June 2010

For

ϕ
\phi

in

H
∞

{H^\infty }

, let

T
ϕ

{T_\phi }

be the analytic Toeplitz operator with symbol

ϕ
\phi

and let

{

T
ϕ

}
′

\{ {T_\phi }\} ’

be the commutant of

T
ϕ

{T_\phi }

. Two infinite Blaschke products

ϕ
\phi

and

ψ
\psi

, are exhibited such that

{

T
ϕ

}
′

∩
{

T
ψ

}
′

\{ {T_\phi }\} ’ \cap \{ {T_\psi }\} ’

is not equal to

{

T
θ

}
′

\{ {T_\theta }\} ’

for any inner function

θ
\theta

. Also, two questions on reducing subspaces of analytic Toeplitz operators are answered.
Equicontinuity, affine mean ergodic theorem and linear equations in random normed spaces
Authors: Radu V.
Published: June 2010

The aim of this paper is to give a characterization of equicontinuous families of linear operators in random normed spaces which generalizes the normed spaces case. This characterization is used to generalize some results on iterative solutions of linear equations by using the affine mean ergodic theorem for locally convex spaces (note that every Fr 'echet space is a random normed space with the

t
t

-norm

T
=

Min

T = {\text {Min}}

).
The meaning of the Cauchy-Schwarz-Buniakovsky inequality
Authors: Zarantonello Eduardo H.
Published: June 2010

It is proved that a mapping

T
:
X
→

X

∗

T:X \to {X^{\ast }}

from a topological real vector space into its dual satisfies the inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper T y comma x mathematical right-angle less-than-or-slanted-equals mathematical left-angle upper T x comma x mathematical right-angle Superscript 1 slash 2 Baseline dot mathematical left-angle upper T y comma y mathematical right-angle Superscript 1 slash 2">

⟨
T
y
,
x
⟩
⩽

⟨
T
x
,
x

⟩

1

/

2

⋅

⟨
T
y
,
y

⟩

1

/

2

\langle Ty,x\rangle \leqslant {\langle Tx,x\rangle ^{1/2}} \cdot {\langle Ty,y\rangle ^{1/2}}

if and only if it is the restriction of a positively homogeneous subdifferential operator.
Ext à la Yoneda without the Schanuel lemma
Authors: Fritsch Rudolf
Published: June 2010

By means of a more precise treatment of the congruence relation between

n
n

-extensions we show that it is possible to avoid the Schanuel Lemma for proving some results on the functors Ext.
A generalization of a theorem by Bochner
Authors: Olesen Dorte
Published: June 2010

A theorem of Bochner states that if

μ
\mu

is a complex Borel measure on the

n
n

-dimensional torus

T

n

{{\mathbf {T}}^n}

with Fourier-coefficients that vanish outside a proper cone in

Z

n

{{\mathbf {Z}}^n}

, then

μ
\mu

is absolutely continuous with respect to Haar measure on

T

n

{{\mathbf {T}}^n}

. This result is generalized to a

C
∗

{C^ \ast }

-algebra setting using the concept of spectral subspaces for an

n
n

-parameter group of automorphisms and its dual group, in the case where the cone is the positive “octant".
On the automorphism group of compact measure hyperbolic manifolds and complex analytic bundles with compact measure hyperbolic fibres
Authors: Wong B.
Published: June 2010

We prove the automorphism groups of the compact measure hyperbolic manifolds are discrete.
Nonstandard measure theory–Hausdorff measure
Authors: Wattenberg Frank
Published: June 2010

In this paper it is shown that the Hausdorff measures

λ
t

{\lambda ^t}

for

t
∈
[
0
,
∞
)

t \in [0,\infty )

can be simultaneously represented as

∗

^\ast

finite counting measures in an appropriate nonstandard model. That is, the following theorem is proved. Theorem. Suppose X is a metric space and

ν
\nu

is an infinite positive

∗

^\ast

integer. Then there is a

∗

^\ast

finite set G such that for every standard

t
∈
[
0
,
∞
)

t \in [0,\infty )

and every

λ
t

{\lambda ^t}

-integrable Borel function,

f
:
X
→

R

f:X \to {\mathbf {R}}

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral f d lamda Superscript t Baseline equals St left-parenthesis StartFraction 1 Over nu Superscript t Baseline EndFraction sigma-summation Underscript x element-of upper G Endscripts asterisk f left-parenthesis x right-parenthesis right-parenthesis period">

∫

f

d

λ
t

=

St

(

1

ν
t

∑

x
∈
G

∗
f
(
x
)

)

.

\int {f\;d{\lambda ^t}} = {\text {St}}\left ( {\frac {1}{{{\nu ^t}}}\sum \limits _{x \in G} {*f(x)} } \right ).

\]
Relative 𝑆-invariants
Authors: Stanton Robert O.
Published: June 2010

Warfield has defined the concept of a

T
∗

{T^\ast }

-module over a discrete valuation ring and has proved a classification theorem for these modules. In this paper, the invariant S defined by the author is extended. This allows a generalization of the classification theorem of Warfield.
On a theorem of Posner
Authors: Ahmad Mansoor
Published: June 2010

Posner [1] has proved the following theorem: Let R be a prime ring

1

^{1}

and d a derivation of R such that, for all

a
∈
R
,
a
d
a
−
d
a
a
∈
Z

a \in R,ada - daa \in Z

(centre of R). Then, if d is not a zero derivation, R is commutative.
More on groups in which each element commutes with its endomorphic images
Authors: Malone J. J.
Published: June 2010

R. Faudree has given examples of nonabelian groups which have the property cited in the title. His groups are p-groups such that (i)

Z
(
G
)
=

G
′

=

G
p

=
U
(
G
)

Z(G) = G’ = {G^p} = U(G)

, (ii) each endomorphism

ϕ
\phi

(which is not an automorphism) has

(
G
)
ϕ
⩽
Z
(
G
)

(G)\phi \leqslant Z(G)

, and (iii) each automorphism is central. In this paper the necessity of these conditions is explored. It is also shown that, for p = 2, Faudree’s example does not in fact have the property cited in the title.
A converse of Taylor’s theorem for functions on Banach spaces
Authors: Dayal S.
Published: June 2010

The main results are the local representation theorems associating the local weak n-Taylor series expansion of a function defined on a Banach space to a local n-Taylor series expansion of the coefficients. These theorems are used to prove a converse of Taylor’s theorem which uses weaker hyptohesis than used by others. Another useful application of the above results is done in [2] to study a class of functions called n-convex functions.
On generating distributive sublattices of orthomodular lattices
Authors: Greechie Richard J.
Published: June 2010

A Foulis-Holland set is a nonempty subset S of an orthomodular lattice such that whenever x, y, z are distinct elements of S one of them commutes with the other two. If S is a Foulis-Holland set, then the sublattice generated by S is distributive.
Finite simple groups containing a self-centralizing element of order 6
Authors: Hayden John L. , Winter David L.
Published: June 2010

By a self-centralizing element of a group we mean an element which commutes only with its powers. In this paper we establish the following result: Theorem. Let G be a finite simple group which has a self-centralizing element of order 6. Assume that G has only one class of involutions. Then G is isomorphic to one of the groups

M

11

,

J
1

,

L
3

(
3
)
,

L
2

(
11
)
,

L
2

(
13
)

{M_{11}},{J_1},{L_3}(3),{L_2}(11),{L_2}(13)

.
Operator minimax theorems in Banach lattices
Authors: Looney Carl G.
Published: June 2010

Let

ψ
:
X
×
Y
→
(
E
,
⩽
,

‖
⋅
‖

)

\psi :X \times Y \to (E, \leqslant ,\left \| \cdot \right \|)

, where X and Y are convex, X is compact, and E is a dedekind complete Banach lattice with unit e. If each

ψ
(
⋅
,
y
)

\psi ( \cdot ,y)

is continuous

⩽
\leqslant

-concave on X,

{
ψ
(
⋅
,
y
)
:
y
∈
Y
}

\{ \psi ( \cdot ,y):y \in Y\}

is convex, and

ψ
(
X
×
Y
)

\psi (X \times Y)

is minorized in E, then

sup
inf
ψ
(
x
,
y
)
=
inf
sup
ψ
(
x
,
y
)

\sup \inf \psi (x,y) = \inf \sup \psi (x,y)

. Similar theorems are included.
On the cardinality of a topological space
Authors: Charlesworth Arthur
Published: June 2010

In recent papers, B. Šapirovskiĭ, R. Pol, and R. E. Hodel have used a transfinite construction technique of Šapirovskiĭ to provide a unified treatment of fundamental inequalities in the theory of cardinal functions. Šapirovskiĭ’s technique is used in this paper to establish an inequality whose countable version states that the continuum is an upper bound for the cardinality of any Lindelöf space having countable pseudocharacter and a point-continuum separating open cover. In addition, the unified treatment is extended to include a recent theorem of Šapirovskiĭ concerning the cardinality of

T
3

{T_3}

spaces.
Semicontinuous and irresolute images of 𝑆-closed spaces
Authors: Thompson Travis
Published: June 2010

A topological space is S-closed if and only if every semi-open cover of X has a finite subcollection whose closures cover X. The images of S-closed spaces under various mappings are investigated culminating in this main result: A Hausdorff space X is S-closed if and only if the irresolute image of X in any Hausdorff space is closed.
An example of a space which is countably compact whose square is countably paracompact but not countably compact
Authors: Parsons Lee
Published: June 2010

A subspace P of

β
N
−
N

\beta N - N

is obtained whose square is disjoint from the graph, G, of a pre-selected homeomorphism

f
:
β
N
→
β
N

f:\beta N \to \beta N

that has no fixed points. The construction is performed in such a way that, for

X
=
P
∪
N

X = P \cup N

, all countable subsets of

X
2

−
G

{X^2} - G

will have a limit point in

X
2

{X^2}

. We use the following lemma: If

K
⊂

(
β
N

)
2

−
G

K \subset {(\beta N)^2} - G

is countably infinite, then

|

cl

(
β
N
)

2

K
−
G

|

=

2
c

|{\text {cl}_{{{(\beta N)}^2}}}K - G| = {2^c}

.
On the orbit types in a circle action
Authors: Stein Elliott
Published: June 2010

An example is constructed of a smooth

S
1

{S^1}

action on

S
5

{S^5}

(or

R
5

{R^5}

) which has more orbit types than its slice representation at a fixed point.
Sums of independent random variables and the Burkholder transforms
Authors: Gabriel J.-P.
Published: June 2010

This note shows a connection between the unconditionally a.e. convergence of series with independent increments and the a.e. convergence of their Burkholder transforms. Using this result, it is then proved that the

L
1

{L_1}

-bounded condition of Burkholder is the best one in the class of martingales, which assures the a.e. convergence of their transforms.
Canonical objects in Kirillov theory on nilpotent Lie groups
Authors: Penney Richard C.
Published: June 2010

It is shown that to each element f in the dual space of the Lie algebra of a nilpotent Lie group there is a uniquely defined subgroup

K
∞

{K_\infty }

for which the representation corresponding to f is inducible from a square-integrable-modulo-its-kernel representation of

K
∞

{K_\infty }

.
Differentiability a.e. and approximate differentiability a.e
Authors: Bruckner A. M.
Published: June 2010

Let F be a finite real valued function defined on [0, 1]. We prove that F can be transformed into a function which is differentiable a.e. by a homeomorphic change of variables if and only if F is continuous on a dense set. We also show that F can be transformed into a function which is approximately differentiable a.e. if and only if each interval

I
⊂
[
0
,
1
]

I \subset [0,1]

contains a nonempty perfect set P such that

F

|

P

F|P

is continuous.