bluuebunny/crossref_metadata_2025 huggingface dataset:
Restrictions of modules to local subgroups
Authors: Burry David W. , Carlson Jon F.
Published: June 2010

The main theorem of this paper is a strengthening of the uniqueness statement in the Green correspondence for modules over certain group rings of finite groups. One consequence of this result is an extension of the work of J. L. Alperin and the first author on a module-theoretic approach to block theory. Specifically, the stronger Green correspondence is used to complete the proof of Brauer’s First Main Theorem.
A Tauberian theorem for strong Abel summability type
Authors: Sukla Indulata
Published: June 2010

In the present paper the author has defined a new method of strong Abel summability type

{
A
,
λ

}
m

{\{ A,\lambda \} _m}

and obtained a necessary and sufficient type of Tauberian conditions for

Σ

a
n

\Sigma {a_n}

to be summable

[
R
,
λ
,
k

]
m

{[R,\lambda ,k]_m}

, whenever it is summable

{
A
,
λ

}
m

{\{ A,\lambda \} _m}

. This result is analogous to one result of Flett [4].
Factorization along nest algebras
Authors: Feintuch Avraham
Published: June 2010

Let

T
T

be a positive definite operator on

H

\mathcal {H}

and

R

\mathcal {R}

a nest algebra in

B
(

H

)

B(\mathcal {H})

. A necessary and sufficient condition is given for the existence of a factorization for

T
T

of the form

T
=

A
∗

A

T = {A^* }A

with

A
A

,

A

−
1

∈

R

{A^{ - 1}} \in \mathcal {R}

.
Generators of 𝐻*(𝑀𝑆𝑂;𝑍₂) as a module over the Steenrod algebra, and the oriented cobordism ring
Authors: Papastavridis Stavros
Published: June 2010

In this paper we will describe a minimal set of

A
A

-generators of

H
∗

(
M
S
O
;

Z
2

)

{H^* }(MSO;{Z_2})

(where

A
A

is the

mod

−

2

\bmod {\mathbf { - }}2

Steenrod Algebra). The description is very much analogous to

R

{\text {R}}

. Thom’s description of generators for

H
∗

(
M
O
;

Z
2

)

{H^*}(MO;{Z_2})

(see [7]). As a corollary, we give simple cohomological criteria for a manifold to be indecomposable in the oriented cobordism. Our proof relies on work of D. J. Pengelley (see [5]).

1

^{1}
Polynomials over finite fields which commute with linear permutations
Authors: Mullen Gary L.
Published: June 2010

The polynomials over a finite field which commute with an arbitrary linear permutation of the field are characterized. This characterization is obtained through the use of a result from Pólya’s theory of enumeration.
Linear systems of real quadratic forms. II
Authors: Calabi Eugenio
Published: June 2010

Consider a mapping

F
:

R

n

→

R

3

(
n
⩾
3
)

F:{{\mathbf {R}}^n} \to {{\mathbf {R}}^3}(n \geqslant 3)

defined by an ordered triple of real-valued quadratic forms; if some linear combination of these three quadratic forms is positive definite, then the image set under the map is a convex cone.
A uniqueness theorem for superharmonic functions in 𝑅ⁿ
Authors: Schiff J. L.
Published: June 2010

Let

s
(
x
)

s(x)

be a nonnegative superharmonic function defined on the

n
n

-ball

B
n

(
y
;
r
)

{B^n}(y;r)

in

R

n

,
n
⩾
3

{{\mathbf {R}}^n}, n \geqslant 3

. If

s
(
x
)

s(x)

tends to zero "too rapidly" as

x
x

tends to a single point

ξ
\xi

on the boundary of

B
n

(
y
;
r
)

{B^n}(y;r)

, then we prove that

s
≡
0

s \equiv 0

. The same result can then be extended to domains

D
⊆

R

n

D \subseteq {{\mathbf {R}}^n}

, whose boundary

∂
D

\partial D

is locally

C
1

{C^1}

at

ξ
∈
∂
D

\xi \in \partial D

. These results generalize some earlier work of the author and Ü. Kuran for

n
=
2

n = 2

.
Property 𝐿 and asymptotic abelianness for von Neumann algebras of type 𝐼
Authors: Kawamura Shinzō
Published: June 2010

We prove a correct assertion on Property L for von Neumann algebras of type I: a type I von Neumann algebra

M
M

on a separable Hilbert space has Property L if and only if

M
M

contains no minimal projection. Furthermore, a correct proof of an assertion on asymptotic abelianness for von Neumann algebras of type I is also given.
On left thickness of subsets in semigroups
Authors: Wong James C. S.
Published: June 2010

We reformulate the concept of left thickness in a semigroup introduced by H. Junghenn in Amenability of function spaces on thick subsemigroups, Proc. Amer. Math. Soc. 75 (1979), 37-41, and obtain a number of interesting new characterisations of left thickness without assuming left amenability. Moreover, these characterisations are similar in nature to familiar characterisations of left amenability on semigroups. We also sharpen some of Junghenn’s results, paving the way for extension to locally compact semigroups.
Noetherian PI rings not module-finite over any commutative subring
Authors: Sarraillé J. J.
Published: June 2010

We construct a ring

R
R

of

3
×
3

3 \times 3

matrices over

k
[
x
,
y
,
z
]

k[x,y,z]

which is prime, affine, Noetherian, and PI, but not finitely generated as a module nor integral over any commutative subring.
Distance between normal operators
Authors: Sunder V. S.
Published: June 2010

Lidskii and Wielandt have proved independently that if

A
A

and

B
B

are selfadjoint operators on an

n
n

-dimensional space

H
H

, with eigenvalues

{

α
k

}

k
=
1

n

\{ {\alpha _k}\} _{k = 1}^n

and

{

β
k

}

k
=
1

n

\{ {\beta _k}\} _{k = 1}^n

respectively (counting multiplicity), then,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper A minus upper B double-vertical-bar greater-than-or-slanted-equals min Underscript sigma element-of upper S Subscript n Baseline Endscripts double-vertical-bar diag left-parenthesis alpha Subscript k Baseline minus beta Subscript sigma left-parenthesis k right-parenthesis Baseline right-parenthesis double-vertical-bar">

‖

A
−
B

‖

⩾

min

σ
∈

S
n

‖

diag

(

α
k

−

β

σ
(
k
)

)

‖

\left \| {A - B} \right \| \geqslant \min \limits _{\sigma \in {S_n}} \left \| {{\text {diag}}\left ( {{\alpha _k} - {\beta _{\sigma (k)}}} \right )} \right \|

\]
for any unitarily invariant norm on

L
(
H
)

L(H)

. In this note an example is given to show that this result is no longer true if

A
A

and

B
B

are only required to be normal (even unitary). It is also shown that the above inequality holds in the operator norm, if

A
A

is selfadjoint and

B
B

is skew-self-adjoint.
Undecidable theories in stationary logic
Authors: Seese Detlef , Tuschik Peter , Weese Martin
Published: June 2010

It is shown that the theories of the following classes become undecidable in stationary logic: Well orderings with one unary predicate, linear orderings, and Boolean algebras. This is done by interpreting the theory of symmetric reflexive graphs.
On the distribution of self-numbers
Authors: Zannier U.
Published: June 2010

Self-numbers are those integers which cannot be expressed as

a
+
f
(
a
)

a + f(a)

, where

f
(
a
)

f(a)

denotes the sum of the digits of

a
a

in a given scale. Here I prove that the number of self-numbers less than or equal to a large number

x
x

equals

L
x
+
O
(

log
2

x
)

Lx + O({\log ^2}x)

, where

L
L

is a strictly positive constant.
Hypercentral residuals of a join of subnormal subgroups
Authors: Whitehead Jennifer
Published: June 2010

It is proved that the hypercentral residual of a group of finite rank, generated by two subnormal subgroups, is the join of the hypercentral residuals of the generating subnormal subgroups. The theorem is subject to the condition that, when factored by their hypercentral residuals, these groups are hypercentral.
Weighted norm inequalities for the Hardy-Littlewood maximal function
Authors: Young Wo Sang
Published: June 2010

A characterization is obtained for weight functions

υ
\upsilon

for which the Hardy-Littlewood maximal operator is bounded from

L
p

(

R

n

,
w
d
x
)

{L^p}({{\mathbf {R}}^n},wdx)

to

L
p

(

R

n

,
υ
d
x
)

{L^p}({{\mathbf {R}}^n},\upsilon dx)

for some nontrivial

w
w

.
Oscillation criteria for the sublinear Schrödinger equation
Authors: Onose Hiroshi
Published: June 2010

In this paper, we propose an oscillation theorem to the sublinear Emden-Fowler equation

Δ
u
+
g
(
x
)

u
γ

=
0

\Delta u + g(x){u^\gamma } = 0

for

n
=
2

n = 2

. By using this, we will answer to the open problem of Noussair and Swanson in the case of bounded solutions.
On the Liouville theorem for harmonic maps
Authors: Choi Hyeong In
Published: June 2010

Suppose

M
M

and

N
N

are complete Riemannian manifolds;

M
M

with Ricci curvature bounded below by

−
A

- A

,

A
⩾
0

A \geqslant 0

,

N
N

with sectional curvature bounded above by a positive constant

K
K

. Let

u
:
M
→
N

u:M \to N

be a harmonic map such that

u
(
M
)
⊂

B
R

(

y
0

)

u(M) \subset {B_R}({y_0})

. If

B
R

(

y
0

)

{B_R}({y_0})

lies inside the cut locus of

y
0

{y_0}

and

R
>
π

/

2

K

R > \pi /2\sqrt K

, then the energy density

e
(
u
)

e(u)

of

u
u

is bounded by a constant depending only on

A
A

,

K
K

and

R
R

. If

A
=
0

A = 0

, then

u
u

is a constant map.
Maximal extensions of first-countable spaces
Authors: Terada Toshiji , Terasawa Jun
Published: June 2010

A first-countable space is called maximal if it is not contained as a dense subspace in a first-countable space properly. The following are shown; (1) every locally compact, first-countable space is a dense subspace of a maximal space, (2) every metrizable space is a dense subspace of a maximal space, and (3) there is a first-countable space which is not a dense subspace of any maximal space.
Collectionwise normality without large cardinals
Authors: Tall Franklin D.
Published: June 2010

It was previously known to be consistent relative to a strongly compact cardinal that locally compact perfectly normal spaces must be collectionwise normal. We obtain the same result merely by adjoining

ℵ
2

{\aleph _2}

random reals to a model of

V
=
L

V = L

.
Globalization of some local properties in Krull domains
Authors: Anderson D. D.
Published: June 2010

Let

R
R

be a Krull domain. It is shown that a nonzero locally principal ideal is invertible. This is used to show that

Cl

(
R
)

/

Pic

(
R
)

{\text {Cl}}(R)/{\text {Pic}}(R)

is torsion if and only if

Cl

(

R
M

)

{\text {Cl}}({R_M})

is torsion for each maximal ideal

M
M

of

R
R

. Here

Cl

(
R
)

{\text {Cl}}(R)

and

Pic

(
R
)

{\text {Pic}}(R)

denote the divisor class group and Picard group of

R
R

, respectively.
On semiprime rings of bounded index
Authors: Armendariz Efraim P.
Published: June 2010

A ring

R
R

is of bounded index (of nilpotency) if there is an integer

n
⩾
1

n \geqslant 1

such that

x
n

=
0

{x^n} = 0

whenever

x
∈
R

x \in R

is nilpotent. The least such positive integer is the index of

R
R

. We show that a semiprime ring

R
R

has index

⩽
n

\leqslant n

if and only if

R
R

is a subdirect product of prime rings of index

⩽
n

\leqslant n

.
On the conjugacy problem for 𝐹/𝑅’
Authors: Gupta C. K.
Published: June 2010

Let

F

/

R

F/R

be a finitely generated recursively presented group with the solvable conjugacy problem and the solvable power problem. The conjugacy problem for

F

/

R
′

F/R’

when

F

/

R

F/R

is torsion free has been solved by Remeslennikov and Sokolov (Algebra and Logic 9 (1970), 342-349). In this note we prove that

F

/

R
′

F/R’

has the solvable conjugacy problem even when

F

/

R

F/R

is not torsion free. This allows us, in particular, to conclude the solvability of the conjugacy problem for

F

/

[

F
n

,

F
n

]

F/[{F^n},{F^n}]

in terms of the conjugacy problem for

F

/

F
n

F/{F^n}

.
Pseudosemisimple rings
Authors: Mohamed Saad , Müller Bruno J.
Published: June 2010

We characterize right-pseudo-semisimple rings (i.e. rings

R
R

for which every right ideal is semisimple or isomorphic to

R
R

) which are split extensions of their right socle.
On endomorphisms of abelian topological groups
Authors: Katz Eli , Morris Sidney A.
Published: June 2010

A family

Φ
\Phi

of continuous endomorphisms of a topological group

G
G

is said to be small if for every subgroup

H
H

of

G
G

of cardinality

card

(
H
)
>

card

(
G
)

{\text {card}}(H) > {\text {card}}(G)

there exists an element

g
∈
G

g \in G

such that

Φ
g
∩
H
=
∅

\Phi g \cap H = \emptyset

. M. I. Kabenjuk [5] proved that if

G
G

is a compact connected Hausdorff abelian group of countable weight then every countable family

Φ
\Phi

of nontrivial endomorphisms of

G
G

is small. He asked if "compact" can be replaced by "complete". In this note the answer is given in the negative, but it is shown that "compact" can be replaced by "locally compact".
Strictly hypoelliptic second order differential operators
Authors: Yamamoto Kazuhiro
Published: June 2010

We shall show strict hypoellipticity of some second order differential operators which are generalized equations considered by Hörmander, Oleinik and Radkevič, using localized energy inequalities.
Mean growth and smoothness of analytic functions
Authors: Matheson A.
Published: June 2010

Let

G
α

{G_\alpha }

denote the class of functions

f
(
z
)

f(z)

analytic in the unit disk such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 1 Baseline left-parenthesis 1 minus r right-parenthesis Superscript negative alpha Baseline upper M Subscript normal infinity Baseline left-parenthesis f prime comma r right-parenthesis d r greater-than normal infinity comma">

∫
0
1

(
1
−
r
)

−
α

M
∞

(

f
′

,
r
)

d
r
>
∞
,

\int _0^1 {{{(1 - r)}^{ - \alpha }}{M_\infty }(f’,r)} dr > \infty ,

\]
for some

α
(
0
>
α
>
1
)

\alpha (0 > \alpha > 1)

. A characterization of

G
α

{G_\alpha }

is given in terms of moduli of continuity and an application is given to Riesz factorization of functions in

G
α

{G_\alpha }

.
A topological version of Slutsky’s theorem
Authors: Ressel Paul
Published: June 2010

For weak convergence of probability measures on a product of two topological spaces the convergence of the marginals is certainly necessary. If however the marginals on one of the factor spaces converge to a one-point measure, the condition becomes sufficient, too. This generalizes a well-known result of Slutsky.
The cohomology of 𝐵𝑆𝑂_{𝑛} and 𝐵𝑂_{𝑛} with integer coefficients
Authors: Brown Edgar H.
Published: June 2010

The cohomology rings of

B
S

O
n

BS{O_n}

and

B

O
n

B{O_n}

, with integer coefficients, are given by generators and relations expressed in terms of characteristic classes.
On graded rings with finiteness conditions
Authors: Camillo V. P. , Fuller K. R.
Published: June 2010

It is proved that a graded ring that is finitely graded modulo its radical is local if its initial subring is local, and that a graded artinian ring is finitely generated over its initial subring which is also artinian. These results extend theorems of Gordon and Green on artin algebras. Other results relating the structure of a graded ring to that of its initial subring are also presented.
Cyclic relations and the Goldberg coefficients in the Campbell-Baker-Hausdorff formula
Authors: Thompson Robert C.
Published: June 2010

Recursively defined coefficients for the commutator-free form of the Campbell-Baker-Hausdorff formula are shown to apply to the commutator version of this formula, and a new relation on these coefficients is derived.
Middle nucleus=center in semiprime Jordan algebras
Authors: McCrimmon Kevin , Ng Seong Nam
Published: June 2010

A. A. Albert showed that the middle nucleus and center coincide for a simple Jordan algebra finite-dimensional over a field of characteristic

≠
2

\ne 2

. E. Kleinfeld extended this to arbitrary simple Jordan algebras of characteristic

≠
2

\ne 2

. Recently this result has played a crucial role in the structure theory of E. Zelmanov. In this note we extend the result to linear Jordan algebras with no derivation-invariant trivial ideals.
The global dimension of FBN rings with enough clans
Authors: Damiano Robert F.
Published: June 2010

For an FBN ring

R
R

, a classical set of prime ideals

{

P
1

,
…
,

P
n

}

\left \{ {{P_1}, \ldots ,{P_n}} \right \}

is one for which the semiprime ideal

N
=

∩

i
=
1

n

P
i

N = \cap _{i = 1}^n{P_i}

satisfies the Artin-Rees property. A minimal classical set is called a clan. We say an FBN ring

R
R

has enough clans if each prime ideal

P
P

is an element of a clan. In this paper, we show that for such rings the Krull dimension is less than or equal to the global dimension.
On Littlewood’s conjecture for univalent functions
Authors: Hamilton D. H.
Published: June 2010

The uniform asymptotic theory of functionals on

S
S

is investigated. We prove that Littlewood’s conjecture is equivalent to the asymptotic Bieberbach conjecture of Hayman.
An example for factorization theory in Banach algebras
Authors: Dixon P. G.
Published: June 2010

It has been conjectured that, if

A
A

is a separable Banach algebra in which every element factorizes (i.e. for every

x
∈
A

x \in A

there exist

y
y

,

z
∈
A

z \in A

with

x
=
y
z
)

x = yz)

), then every pair of elements in

A
A

has a common factor. An example is given of a commutative, separable Banach algebra

A
A

where the set

M
M

of elements of

A
A

which factorize is of codimension one in

A
A

and there exists a pair of elements of

M
M

with no common factor in

A
A

.
Cone lattices of upper semicontinuous functions
Authors: Beer Gerald
Published: June 2010

Let

X
X

be a compact metric space. A well-known theorem of M. H. Stone states that if

Ω
\Omega

is a vector lattice of continuous functions on

X
X

that separates points and contains a nonzero constant function, then the uniform closure of

Ω
\Omega

is

C
(
X
)

C(X)

. In this article we generalize Stone’s sufficient conditions to the upper semicontinuous functions on

X
X

topologized in a natural way.
Completely bounded maps on 𝐶*-algebras and invariant operator ranges
Authors: Paulsen Vern I.
Published: June 2010

We give a new proof that every completely bounded map from a

C
∗

{C^* }

-algebra into

L

(

H

)

\mathcal {L}(\mathcal {H})

lies in the linear span of the completely positive maps. In addition, we obtain an equivalent reformulation of the invariant operator range problem.
Recoverability of some classes of analytic functions from their boundary values
Authors: Zayed Ahmed I.
Published: June 2010

The technique devised by D. J. Patil to recover the functions of the Hardy space

H
p

(
1
⩽
p
⩽
∞
)

{H^p}(1 \leqslant p \leqslant \infty )

from the restrictions of their boundary values to a set of positive measure on the unit circle was modified by S. E. Zarantonello in order to extend the result to

H
p

(
0
>
p
>
1
)

{H^p}(0 > p > 1)

. In this paper, we show that Zarantonello’s technique can be slightly modified to extend the result to a larger class of analytic functions in the unit disc. In particular, if

f
(
z
)

f(z)

is analytic in the unit disc and satisfies
\[

lim

r
→
1

(
1
−
r

)
β

log
⁡
M
(
r
,
f
)
=
0

for some 

β
⩾

1,

\lim \limits _{r \to 1} {(1 - r)^\beta }\log M(r,f) = 0\;{\text {for some }}\beta \geqslant {\text {1,}}

\]
then

f
(
z
)

f(z)

can be recovered from the restriction of its boundary value to an open arc.
Quantitative Helly-type theorems
Authors: Bárány Imre , Katchalski Meir , Pach János
Published: June 2010

We establish some quantitative versions of Helly’s famous theorem on convex sets in Euclidean space. We prove, for instance, that if

C

\mathcal {C}

is any finite family of convex sets in

R

d

{{\mathbf {R}}^d}

, such that the intersection of any

2
d

2d

members of

C

\mathcal {C}

has volume at least 1, then the intersection of all members belonging to

C

\mathcal {C}

is of volume

⩾

d

−

d
2

\geqslant {d^{ - {d^2}}}

. A similar theorem is true for diameter, instead of volume. A quantitative version of Steinitz’ Theorem is also proved.
Simple geodesics on Riemann surfaces
Authors: Jørgensen Troels
Published: June 2010

As observed by Myrberg, on any closed hyperbolic surface the closed geodesics cover a dense subset. Below, it is shown that the simple geodesics cannot cover such parts of a surface which are sufficiently thin.
There exist arbitrarily many different disk knots with the same exterior
Authors: Hitt L. R. , Sumners D. W.
Published: June 2010

We prove that, for

n
⩾
5

n \geqslant 5

, exteriors of disk knots of

D
n

{D^n}

in

D

n
+
2

{D^{n + 2}}

can be exteriors of arbitrarily many different disk knots.
A solution to a problem of Eilenberg concerning dimension lowering mappings
Authors: Keesling James , Wilson David C.
Published: June 2010

This paper describes a map

f
f

from a metric space

X
X

(having the same dimension at each of its points) onto a space

Y
Y

such that

dim
⁡
X
>
dim
⁡
Y
>
0

\dim X > \dim Y > 0

with the property that

dim
⁡
K
⩾
dim
⁡
f
(
K
)

\dim K \geqslant \dim f(K)

for every closed set

K
K

contained in

X
X

. This answers a question posed by Eilenberg in 1936 in The Scottish Book. This question was answered with a more complicated example by Rubin, Schori, and Walsh.
On the lattice of subalgebras of an algebra
Authors: Deneen Linda L.
Published: June 2010

Let

R
R

be a Noetherian inertial coefficient ring and let

A
A

be a finitely generated

R
R

-algebra (that is, finitely generated as an

R
R

-module) with Jacobson radical

J
(
A
)

J(A)

. Let

S
S

be a subalgebra of

A
A

with

S
+
J
(
A
)
=
A

S + J(A) = A

. We show that for every separable subalgebra

T
T

of

a
a

there is a unit a of

A
A

such that

a
T

a

−
1

⊆
S

aT{a^{ - 1}} \subseteq S

. It follows that if

S
S

is separable (hence inertial) and if

T
T

is a maximal separable subalgebra of

A
A

, then

T
T

is inertial. We also show that if

S
+
I
=
A

S + I = A

for a nil ideal

I
I

of

A
A

, then

R
R

can be taken to be an arbitrary commutative ring, and the conjugacy result still holds.
The endomorphism ring of an Artinian module whose homogeneous length is finite
Authors: Schulz Rainer
Published: June 2010

Smalø[2] showed that the index of nilpotency of the endomorphism ring of a module

M
R

{M_R}

of finite length is bounded by the number

max

{

n
A

|

A
R

simple

}

\max \left \{ {{n_A}|{A_R}\;{\text {simple}}} \right \}

, where

n
A

{n_A}

denotes the number of times

A
R

{A_R}

occurs as a factor in a composition chain of

M
R

{M_R}

. We give another proof of Smalø’s theorem which leads to an analogous result for artinian modules whose homogeneous length is finite.
Ruled function fields
Authors: Deveney James K.
Published: June 2010

Let

L
=

L
1

(

x
1

)
=

L
2

(

x
2

)
⊃
K

L = {L_1}({x_1}) = {L_2}({x_2}) \supset K

where

x
i

{x_i}

is transcendental over

L
i

{L_i}

, and

L
i

{L_i}

is a finitely generated transcendence degree 1 extension of

K
K

,

i
=
1
,
2

i = 1,2

. If the genus of

L
1

/

K
=
0

{L_1}/K = 0

, then

L
1

{L_1}

and

L
2

{L_2}

are

K
K

-isomorphic. If the genus of

L
1

/

K
>
0

{L_1}/K > 0

, then

L
1

=

L
2

{L_1} = {L_2}

and moreover

L
1

{L_1}

is invariant under all automorphisms of

L

/

K

L/K

. A criterion is also established for a subfield of a ruled field

L
L

to be ruled.
Extreme points and 𝑙₁(Γ)-spaces
Authors: Roy Nina M.
Published: June 2010

Let

X
X

be a nontrivial real Banach space and let

E
X

{E_X}

denote the set of extreme points of the closed unit ball

B
(
X
)

B(X)

. Theorem 1.

X
X

is an

l
1

(
Γ
)

{l_1}(\Gamma )

-space if and only if (i)

span

(
e
)

{\text {span}}(e)

is an

L
L

-summand in

X
X

for every

e
e

in

E
X

{E_X}

and (ii)

B
(
X
)

B(X)

is the norm closed convex hull of

E
X

{E_X}

. Theorem 2. Let

X
=

Y
∗

X = {Y^ * }

. If (i)

span

(
e
)

{\text {span}}(e)

is an

L
L

-summand in

X
X

for every

e
e

in

E
X

{E_X}

and (ii)

{

e
∈

E
X

:
e
(
y
)
=
1

}

\left \{ {e \in {E_X}:e(y) = 1} \right \}

is countable for each

y
y

in

Y
Y

with

‖
y
‖

=
1

\left \| y \right \| = 1

, then

X
X

is an

l
1

(
Γ
)

{l_1}(\Gamma )

-space.
On proximinality in 𝐿₁(𝑇×𝑆)
Authors: Holland S. M. , Light W. A. , Sulley L. J.
Published: June 2010

It is proved that if

G
G

and

H
H

are finite-dimensional subspaces of

L
1

(
S
)

{L_1}(S)

and

L
1

(
T
)

{L_1}(T)

respectively then each element of

L
1

(
T
×
S
)

{L_1}(T \times S)

has a best approximation in the subspace

L
1

(
T
)
⊗
G
+
H
⊗

L
1

(
S
)

{L_1}(T) \otimes G + H \otimes {L_1}(S)

.
Factorisation of characteristic functions on noncommutative groups
Authors: Wulfsohn Aubrey
Published: June 2010

A characteristic function, without idempotent factors, on a separable compact group is decomposed, modulo characters, as a product of indecomposable characteristic functions and an infinitely divisible characteristic function.
A projected trace and partwise isometric transformations of almost periodic systems
Authors: Vinograd Robert E.
Published: June 2010

This paper is concerned with almost periodic linear systems of ODE’s and their block diagonalization. The aim is to describe a class of all "best possible" (partwise isometric) block diagonalizing transformations and to construct two closely connected functions, a projected trace and its real part, the Liouville function, playing the same role for a subspace or a subbundle as the usual trace and its real part play in Liouville’s formula. A theorem is proved which says that the projected trace of an almost periodic system is almost periodic. For the Liouville function of a one-dimensional spectral subbundle this result was obtained by Sacker and Sell. We extend it to the case of an invariant subbundle of higher dimension. If a system admits a Whitney sum invariant decomposition (e.g., a Sacker-Sell spectral decomposition) then partwise isometric block diagonalizing transformations are possible. This class includes the transformations of Bylov-Vinograd, Coppel, Palmer, and Ellis-Johnson. A theorem is proved which says that every partwise isometric transformation preserves Liouville functions of subbundles and turns them into real parts of usual block traces. In particular, for almost periodic systems, they remain almost periodic.
A remark on linear differential systems with the same invariant subbundles
Authors: Vinograd Robert E.
Published: June 2010

Given a flow

σ
(
ω
,
t
)
=
ω
⋅
t

\sigma (\omega ,t) = \omega \cdot t

on a compact metric space

Ω
\Omega

and a continuous

n
×
n

n \times n

-matrix

A
(
ω
)

A(\omega )

, the family of ODE systems

x
˙

=
A
(
ω
⋅
t
)
x

\dot x = A(\omega \cdot t)x

defines a linear skew-product flow on

W
=
Ω
×
X

W = \Omega \times X

,

X
=

R
n

X = {R^n}

or

C
n

{C^n}

. Let

W
=
U
⊕
V

W = U \oplus V

be a Whitney sum and

P
:
W
→
W

P:W \to W

be the correspondent projector. Result: the subbundles

U
U

and

V
V

are invariant for the flows induced by

A
(
ω
)

A(\omega )

and

B
(
ω
)

B(\omega )

iff

A
(
ω
)

A(\omega )

-

B
(
ω
)

B(\omega )

commutes with

P
(
ω
)

P(\omega )

for all

ω
∈
Ω

\omega \in \Omega

.
Some characterizations of weak Radon-Nikodým sets
Authors: Saab Elias
Published: June 2010

Let

K
K

be a weak*-compact convex subset of the dual

E
∗

{E^*}

of a Banach space

E
E

. It is shown that

K
K

has the weak Radon-Nikodym property if and only if every

x

∗
∗

{x^{**}}

in

E

∗
∗

{E^{**}}

restricted to

K
K

is universally measurable if and only if every

x

∗
∗

{x^{**}}

in

E

∗
∗

{E^{**}}

restricted to any weak*-compact subset

M
M

of

K
K

has a point of continuity on (

M
M

, weak*) if and only if

K
K

is a set of complete continuity if and only if every subset of

K
K

is weak* dentable in

(
M
,
σ
(

E
∗

,

E

∗
∗

)
)

(M,\sigma ({E^*},{E^{**}}))

.
Some remarks on Brauer’s third main theorem
Authors: Juhász Arye
Published: June 2010

We consider two classes of

p
p

-blocks of a finite group

G
G

which have the property that for every block

B
B

of them and every subgroup

H
H

of

G
G

,

H
H

has only a small number of admissible blocks

b
b

with

b
G

=
B

{b^G} = B

. In this they are similar to the principal block of

G
G

. These blocks are described by means of certain modules they contain.
Decomposable positive maps on 𝐶*-algebras
Authors: Størmer Erling
Published: June 2010

It is shown that a positive linear map of a

C
∗

{C^*}

-algebra

A
A

into

B
(
H
)

B(H)

is decomposable if and only if for all

n
∈

N

n \in {\mathbf {\textrm {N}}}

whenever

(

x

i
j

)

({x_{ij}})

and

(

x

j
i

)

({x_{ji}})

belong to

M
n

(
A

)
+

{M_n}{(A)^ + }

then

(
ϕ
(

x

i
j

)
)

(\phi ({x_{ij}}))

belongs to

M
n

(
B
(
H
)

)
+

{M_n}{(B(H))^ + }

.
The dimension of peak-interpolation sets
Authors: Stout Edgar Lee
Published: June 2010

The dimension of a peak-interpolation set in the boundary of a strongly pseudoconvex domain in

C

N

{{\mathbf {C}}^N}

does not exceed

N
−
1

N - 1

.
On the ergodicity of a class of real skew product extensions of approximations
Authors: Goodson G. R.
Published: June 2010

In this paper conditions are given for real skew product extensions of cyclic

K
S

KS

approximations to be ergodic. These results are then applied to show that if

T
α

{T_\alpha }

is an irrational rotation on the unit circle, there exists an uncountable dense collection of measurable sets for which the corresponding skew product extension is ergodic.
On sets of mutually disjoint univalent meromorphic functions
Authors: Harmelin Reuven
Published: June 2010

Several conditions are established for a set of meromorphic functions in a given domain

D
D

, in the complete complex plane, to consist of mutually disjoint univalent functions in terms of a certain class of differential operators which are invariant under the Möbius group.
Characteristic exponents and some applications to differential equations
Authors: De Blasi F. S. , Boudourides M. A.
Published: June 2010

A generalization of the Lozinskiǐ logarithmic norm is introduced and some applications to stability of differential equations are given.
On heminearness spaces
Authors: Hastings Maryam Shayegan
Published: June 2010

The concept of nearness was introduced by H. Herrlich [4’], where nearness refers to collections of sets. Nearness spaces have proven to be an extremely useful generalization of a wide class of topological structures.
A random ergodic theorem in von Neumann algebras
Authors: Dang Ngoc Nghiem
Published: June 2010

E. C. Lance has recently extended the Birkhoff ergodic theorem to noncommutative dynamical systems. Using Lance’s result, we extend the random ergodic theorem of H. R. Pitt, S. M. Ulam, S. Kakutani and J. von Neumann to noncommutative context.
Norm attaining operators and simultaneously continuous retractions
Authors: Johnson Jerry , Wolfe John
Published: June 2010

A compact metric space

S
S

is constructed and it is shown that there is a bounded linear operator

T
:

L
1

[
0
,
1
]
→
C
(
S
)

T:{L^1}[0,1] \to C(S)

which cannot be approximated by a norm attaining operator. Also it is established that there does not exist a retract of

L
∞

[
0
,
1
]

{L^\infty }[0,1]

onto its unit ball which is simultaneously weak* continuous and norm uniformly continuous.
Detecting cohomologically stable mappings
Authors: Bowers Philip L.
Published: June 2010

Let

f
f

be a cohomologically stable mapping defined from a compactum

X
X

to the

(
n
+
1
)
[
−

cell

I

n
+
1

(n + 1)[ - {\text {cell}}{I^{n + 1}}

, let

π
:

I

n
+
1

→

I
n

\pi :{I^{n + 1}} \to {I^n}

be the projection, and let

A
=

I
n

×
{
1
}

A = {I^n} \times \{ 1\}

and

B
=

I
n

×
{
−
1
}

B = {I^n} \times \{ - 1\}

be opposite faces of

I

n
+
1

{I^{n + 1}}

. If

S
S

is a separator or a continuum-wise separator of

f

−
1

(
A
)

{f^{ - 1}}(A)

and

f

−
1

(
B
)

{f^{ - 1}}(B)

in

X
X

, then

π
f

|

S

\pi f |S

is cohomologically stable. This result is used to extend certain computations of cohomological dimension that are due to Walsh, who considered only the special case of the identity mapping on

I

n
+
1

{I^{n + 1}}

.
Order and dimension
Authors: Cohn Richard M.
Published: June 2010

It will be shown that the conjectured Jacobi bound for the order of differential systems cannot be valid if a very natural conjecture concerning the differential dimension of such systems is false.
Noninvertible retracts
Authors: Yanik Joe
Published: June 2010

We demonstrate that if

P
P

is a projective

R
[
X
]

R[X]

-module that is not stably extended from an

R
R

-module, then the symmetric algebra of

P
P

over

R
[
X
]

R[X]

is a retract of a polynomial ring over

R
R

, but is not an invertible

R
R

-algebra. Hence, there are noninvertible retracts over a quite general class of rings.
On the distribution of prime elements in polynomial Krull domains
Authors: Costa D. , Gallardo L. , Querré J.
Published: June 2010

Let

A
A

be a Krull domain having infinitely many height one primes. It is shown that any ideal of height two in the polynomial ring

A
[
t
]

A[t]

contains a prime element. An application to the construction of Dedekind domains with specified class groups is given, along with an example to show the necessity of assuming infinitely many height one primes.
A chaotic function with some extremal properties
Authors: Smítal J.
Published: June 2010

For a continuous function chaotic in the sense of Li and Yorke the continuum hypothesis implies the existence of a scrambled set which has among others full outer Lebesgue measure.
Closed operators and pure contractions in Hilbert space
Authors: Kaufman William E.
Published: June 2010

Some properties of the one-to-one mapping

A
→
A

(
1
−

A
∗

A

)

−
1

/

2

A \to A{(1 - {A^ * }A)^{ - 1/2}}

of the pure contractions onto the closed and densely-defined operators are proved, in particular that it commutes with adjunction and preserves normality.
A note on separating families of representations
Authors: Barnes Bruce A.
Published: June 2010

Conditions are given that imply that a separating family of unitary representations of a locally compact group

G
G

lifts to a separating family of

∗

^ *

-representations of

L
1

(
G
)

{L^1}(G)

.
Symmetric continuity of real functions
Authors: Belna C. L.
Published: June 2010

It is shown that the set of points where a real function is both symmetrically continuous and not continuous has inner measure zero but may have full outer measure.
On maximal ideals depending on some thin sets in 𝑀(𝐺)
Authors: Sato Enji
Published: June 2010

Let

M
(
G
)

M(G)

be the convolution measure algebra on the LCA group

G
G

with dual

Γ
\Gamma

. and

Δ
\Delta

the maximal ideal space of

M
(
G
)

M(G)

. For

E
⊂
G

E \subset G

a compact set, let

G
p
(
E
)

Gp(E)

be the subgroup of

G
G

generated algebraically by

E
E

.

R
(
E
)

R(E)

the measures which are carried by a countable union of translates of

G
p
(
E
)

Gp(E)

. and

P
E

{P_E}

the natural projection from

M
(
G
)

M(G)

onto

R
(
E
)

R(E)

. Also let

h
E

{h_E}

be the multiplicative linear functional

μ
↦
(

P
E

μ

)
^

(

l

)

\mu \mapsto ({P_E}\mu \hat )({\text {l}})

on

M
(
G
)

M(G)

. Then we prove that if

G
G

is an

I
I

-group, and

E
E

an

H
1

{H_1}

-set, we get

h
E

∈

Γ
¯

{h_E} \in \bar \Gamma

(i.e. the closure of

Γ
\Gamma

in

Δ
\Delta

).
A cycle is the fundamental class of an Euler space
Authors: Benedetti R. , Dedò M.
Published: June 2010

We prove that every cycle in a closed P.L. manifold

M
M

can be regarded as the fundamental class of an Euler subpolyhedron of

M
M

.
Algebraic knots are algebraically dependent
Authors: Livingston Charles , Melvin Paul
Published: June 2010

Algebraic knots are linearly dependent in

G
−

{G_ - }

, the algebraic knot concordance group. An example of a linear relation between four algebraic knots is constructed.
For any 𝑋, the product 𝑋×𝑌 is homogeneous for some 𝑌
Authors: Uspenskiĭ Vladimir V.
Published: June 2010

We prove that for every topological space

X
X

there exists a cardinal

k
k

and a nonempty subspace

Y
⊆

X
k

Y \subseteq {X^k}

such that the product

X
×
Y

X \times Y

is homogeneous. This answers a question of A. V. Arhangel’skiĭ.
A characterization of unconditioned weak sequences
Authors: Steurich Manfred
Published: June 2010

Unconditioned weak sequences are characterized in terms of the Koszulhomology and applications to the theory of Buchsbaum rings and in a special case to the resolution of the ideal generated by the weak sequence are considered.
Remark on the simplicial-cosimplicial tensor product
Authors: Fritsch Rudolf
Published: June 2010

We show that the existence of canonical representatives for the elements of the tensor product (coend) of a simplicial and a cosimplicial set depends only on the Eilenberg-Zilber property of the given cosimplicial set. Thus the second condition which is used in [5] for achieving this result is superfluous.
Completeness and the contraction principle
Authors: Borwein J. M.
Published: June 2010

We prove (something more general than) the result that a convex subset of a Banach space is closed if and only if every contraction of the space leaving the convex set invariant has a fixed point in that subset. This implies that a normed space is complete if and only if every contraction on the space has a fixed point. We also show that these results fail if "convex" is replaced by "Lipschitz-connected" or "starshaped".
Coefficient bounds for the inverse of a function with derivative in 𝒫
Authors: Libera Richard J. , Złotkiewicz Eligiusz J.
Published: June 2010

Coefficient bounds for functions with a positive real part are used in a rather novel way to find sharp bounds for the first six coefficients of a function which is inverse to a regular normalized univalent function whose derivative has a positive real part in the unit disk.
Systems of differential equations subject to mild integral conditions
Authors: Trench William F.
Published: June 2010

It is shown that solutions of a system

x
′

=
f
(
t
,
x
)

x’ = f(t,x)

approach constant vectors as

t
→
∞

t \to \infty

, under assumptions which do not require that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue f left-parenthesis t comma x right-parenthesis StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals w left-parenthesis t comma StartAbsoluteValue EndAbsoluteValue x StartAbsoluteValue EndAbsoluteValue right-parenthesis">

|

|

f
(
t
,
x
)

|

|

⩽
w
(
t
,

|

|

x

|

|

)

|| {f(t,x)} || \leqslant w(t,|| x ||)

, where

w
w

is nondecreasing in

|

|

x

|

|

|| x ||

, and which permit some or all of the integral smallness conditions on

f
f

to be stated in terms of ordinary—rather than absolute—convergence. Estimates of the order of convergence are given.
𝐿^{𝑝}-boundedness of a certain class of multipliers associated with curves on the plane. I
Authors: Ruiz Alberto
Published: June 2010

L
p

{L^p}

-unboundedness for

p
≠
2

p \ne 2

is proved in the case of multipliers which are constant along curves. In particular

y
=

x
n

y = {x^n}

is included in the range of curves.
𝐿^{𝑝}-boundedness of a certain class of multipliers associated with curves on the plane. II
Authors: Ruiz Alberto
Published: June 2010

L
p

{L^p}

-boundedness of some multiplier, almost constant along curves, is proved. The interval of

p
p

’s depends on the rate of decaying along the above curves.
A Radon-Nikodým theorem for natural cones associated with von Neumann algebras. II
Authors: Kosaki Hideki
Published: June 2010

A natural cone associated with a (

σ
\sigma

-finite) von Neumann algebra is considered. Let

ξ
0

{\xi _0}

be a cyclic and separating vector in the cone. For each vector

ξ
\xi

in the cone, there always exists a positive selfadjoint operator

t
t

affiliated with the algebra satisfying

ξ
=
t
J
t
J

ξ
0

\xi = tJtJ{\xi _0}

. Certain uniqueness results on

t
t

for a given

ξ
\xi

are also obtained.
Note on restriction of Fourier transforms
Authors: Hu Mên Ch’ang
Published: June 2010

A technique for obtaining necessary conditions on restriction of Fourier transforms is introduced.
Plurisubharmonic functions associated with uniform algebras
Authors: Kumagai Donna
Published: June 2010

We have proved earlier that a certain class of functions, which is useful in the study of analytic structure associated with a uniform algebra, is plurisubharmonic, if the algebra satisfies Basener’s conjecture. In this paper we prove the plurisubharmonicity of these functions without the hypothesis of Basener’s conjecture.
A class of maximal ideals in the lattice of topologies
Authors: Thron W. J. , Valent R. A.
Published: June 2010

In this article we investigate maximal ideals in the lattice of all topologies on a fixed set. Padmanabhan and Rao were the first to study maximal ideals in this lattice. They characterized the principal maximal ideals. Their main result was incorrect but their approach proved helpful to us. We exhibit a class of nonprincipal maximal ideals and also give an example to show that the class in question is not exhaustive.
Rigid finite-dimensional compacta whose squares are manifolds
Authors: Ancel Fredric D. , Singh S.
Published: June 2010

A space is rigid if its only self-homeomorphism is the identity. We answer questions of Jan van Mill by constructing for each

n
n

,

4
⩽
n
>
∞

4 \leqslant n > \infty

, a rigid

n
n

-dimensional compactum whose square is homogeneous because it is a manifold. Moreover, for each

n
n

,

4
⩽
n
>
∞

4 \leqslant n > \infty

, we give uncountably many topologically distinct such examples. Infinite-dimensional examples are also given.
On a certain class of 𝑀₁-spaces
Authors: Mizokami T.
Published: June 2010

Let

P

\mathcal {P}

be the class of all

M
1

{M_1}

-spaces whose every closed subset has a closure-preserving open neighborhood base. A characterization is given, and it is proved that the adjunction space

X

∪
f

Y

X{ \cup _f}Y

is an

M
1

{M_1}

-space if

X
∈

P

X \in \mathcal {P}

and

Y
Y

is an

M
1

{M_1}

-space. Moreover, it is proved that if

X
X

is a space such that for each metrizable space

Y
Y

, every closed subspace of

X
×
Y

X \times Y

is an

M
1

{M_1}

-space, then

X
∈

P

X \in \mathcal {P}

.
On the Adams conjecture
Authors: Dibag I.
Published: June 2010

We give a short elementary and self-contained proof to the Adams conjecture. We then present three equivalent formulations of the conjecture.
A note on the Cohen-Macaulay type of lines in uniform position in 𝐴ⁿ⁺¹
Authors: Brown William C.
Published: June 2010

Let

L

1

,
…
,

L

s

{\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}

be

s
s

-distinct lines in

A

k

n
+
1

{\mathbf {A}}_k^{n + 1}

passing through the origin. Assume

s
=

(
n

n
+
d

)
−
λ

s = (_n^{n + d}) - \lambda

where

n
n

,

d
⩾
2

d \geqslant 2

. If

L

1

,
…
,

L

s

{\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}

are in generic

s
s

-position, and

λ
=
0

\lambda = 0

.

1
,
…
,
n
−
1

1, \ldots ,n - 1

, then the Cohen-Macaulay type,

t
(

L

1

,
…
,

L

s

)

t({\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s})

, of

L

1

,
…
,

L

s

{\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}

is given by the following formula:

t
(

L

1

,
…
,

L

s

)
=

(

n
−
1

n
+
d
−
1

)
−
λ

t({\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}) = (_{n - 1}^{n + d - 1}) - \lambda

. This formula is known to be false for

λ
=
n

\lambda = n

. In this paper, we show that if

L

1

,
…
,

L

s

{\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}

are in uniform position, and

λ
=
n

\lambda = n

. then

t
(

L

1

,
…
,

L

s

)
=

(

n
−
1

n
+
d
−
1

)
−
n

t({\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}) = (_{\;n - 1}^{n + d - 1}) - n

.
A characterization of Warfield groups
Authors: Moore Judy H.
Published: June 2010

Local Warfield groups are generalizations of totally projective

p
p

-groups. This paper presents a characterization of local Warfield groups which is the analogue of the description of totally projective

p
p

-groups as groups with a nice composition series.
On rings of invariants with rational singularities
Authors: Peskin Barbara R.
Published: June 2010

Let

S
S

be a noetherian local

k
k

-algebra and

G
G

a finite group of

k
k

-automorphisms of

S
S

. If char

k
=
0

k = 0

and

S
S

has a rational singularity, then the invariant ring

R
=

S
G

R = {S^G}

does also. However, if char

k
≠
0

k \ne 0

, this is rarely true. We examine conditions on wild group actions in dimension two which ensure that the singularity of

R
R

is rational. In particular, we develop a criterion in terms of the minimality of

H
1

(
G
,
S
)

{H^1}(G,S)

.
Holomorphicity of a class of semigroups of measures operating on 𝐿^{𝑝}(𝐺/𝐻)
Authors: Przebinda Tomasz
Published: June 2010

In the present paper we consider the class of stable semigroups of measures on a Lie group

G
G

. This class contains the Gaussian semigroups. We prove that under certain strongly continuous representations of

G
G

acting in

L
p

(
G

/

H
)

{L^p}(G/H)

,

1
⩽
p
>
∞

1 \leqslant p > \infty

, these semigroups are holomorphic and uniformly bounded.
Inserting 𝐴_{𝑝}-weights
Authors: Neugebauer C. J.
Published: June 2010

Necessary and sufficient conditions are given for inserting a single weight

w
∈

A
p

w \in {A_p}

between

u
u

and

υ
\upsilon

, i.e.,

c
1

u
⩽
w
⩽

c
2

υ

{c_1}u \leqslant w \leqslant {c_2}\upsilon

.
A technique for characterizing Carleson measures on Bergman spaces
Authors: Luecking Daniel
Published: June 2010

A method is presented for characterizing Carleson-type measures relative to Bergman spaces. This method applies to the standard weighted and unweighted Bergman spaces on the unit ball in

C

n

{{\mathbf {C}}^n}

to yield simple proofs of the known results. It also extends these results to domains more general than balls.
Constructive decomposition of BMO functions and factorization of 𝐴_{𝑝} weights
Authors: Coifman R. , Jones Peter W. , Rubio de Francia José L.
Published: June 2010

We give elementary proofs of the decomposition of BMO functions on the line and factorization of

A
p

{A_p}

weights. Our method is an explicit version of a construction due to the third author.
One-to-one operators on function spaces
Authors: Choy Stephen T. L.
Published: June 2010

For a Banach algebra

A
A

one-to-one operators with closed range on

C
0

(
S
,
A
)

{C_0}(S,A)

are characterized in terms of the associated vector measures given by the Riesz Representation Theorems. Multiplicatively symmetric operators are also studied.
A short proof of two recently discovered independence results using recursion theoretic methods
Authors: Cichon E. A.
Published: June 2010

Recently L. A. S. Kirby and J. Paris showed that a theorem of R. L. Goodstein cannot be proved in Peano’s Arithmetic. We give an alternative short proof of their result, based only on well established results concerning recursion theoretic hierarchies of functions. A second, closely related result, due to F. S. Beckman and K. McAloon, is proved by the same means.
On weakly 𝒦-countably determined spaces of continuous functions
Authors: Argyros S. , Negrepontis S.
Published: June 2010

A compact space

K
K

is said to be Gul’ko compact if the space

C
(
K
)

C(K)

is

K

\mathcal {K}

-countably determined in the weak topology. Well-known compact sets, such as Eberlein compact sets, are Gul’ko compact. We prove here that the countable chain condition and metrizability are equivalent for Gul’ko compact sets.
Graphs orientable as distributive lattices
Authors: Duffus Dwight , Rival Ivan
Published: June 2010

There are two types of graphs commonly associated with finite (partially) ordered sets: the comparability graph and the covering graph. While the first type has been characterized, only partial descriptions of the second are known. We prove that the covering graphs of distributive lattices are precisely those graphs which are retracts of hypercubes.
Quadratic forms, rigid elements and nonreal preorders
Authors: Szymiczek Kazimierz , Yucas Joseph
Published: June 2010

A nonreal preorder of a quaternionic structure

q
:
G
×
G
→
B

q:G \times G \to B

is a subgroup

T
⊆
G

T \subseteq G

such that

−
1
∈
T

- 1 \in T

and

−
1
≠
t
∈
T

- 1 \ne t \in T

implies

D

⟨

1
,
t

⟩

⊆
T

D\left \langle {1,t} \right \rangle \subseteq T

. The basic part of

q
q

is defined to be the set

B
=

{

±
1

}

∪

{

a
∈
G

|

a

is

not

2
−

sided

rigid

}

B = \left \{ { \pm 1} \right \} \cup \left \{ {a \in G|a\;{\text {is}}\;{\text {not}}\;2 - {\text {sided}}\;{\text {rigid}}} \right \}

. A. Carson and M. Marshall have shown that if

|
G
|

>
∞

\left | G \right | > \infty

then every nontrivial nonreal preorder

T
T

must contain

B
B

. The main purpose of this note is to extend this result by replacing

|
G
|

>
∞

\left | G \right | > \infty

with

[
G
:
T
]
>
∞

[G:T] > \infty

.