bluuebunny/crossref_metadata_2025 huggingface dataset:
Liftings of functions with values in a completely regular space
Authors: Edgar G. A. , Talagrand Michel
Published: June 2010

Let T be a completely regular space, let

(
Ω
,

F

,
μ
)

(\Omega ,\mathcal {F},\mu )

be complete probability space, and let

ρ
:

L

∞

(
μ
)
→

L

∞

(
μ
)

\rho :{\mathcal {L}^\infty }(\mu ) \to {\mathcal {L}^\infty }(\mu )

be a lifting. If

f
:
Ω
→
T

f:\Omega \to T

is a Baire measurable function, must there exist a function

f
~

\tilde f

with almost all of its values in T, such that

ρ
(
h
∘
f
)
=
h
∘

f
~

\rho (h \circ f) = h \circ \tilde f

for all bounded continuous functions h on T? If T is strongly measure-compact, then the answer is “yes". If T is not measure-compact, then the answer is “no". This shows that a lifting is not always the best method for the construction of weak densities for vector measures.
A new criterion for 𝑝-valent functions
Authors: Goel R. M. , Sohi N. S.
Published: June 2010

In this paper we consider the classes

K

n
+
p
−
1

{K_{n + p - 1}}

of functions

f
(
z
)
=

z
p

+

a

p
+
1

z

p
+
1

+
⋯

f(z) = {z^p} + {a_{p + 1}}{z^{p + 1}} + \cdots

which are regular in the unit disc

E
=
{
z
:

|

z

|

>
1
}

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

and satisfying the condition
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R e left-parenthesis left-parenthesis z Superscript n Baseline f right-parenthesis Superscript left-parenthesis n plus p right-parenthesis Baseline slash left-parenthesis z Superscript n minus 1 Baseline f right-parenthesis Superscript left-parenthesis n plus p minus 1 right-parenthesis Baseline right-parenthesis greater-than left-parenthesis n plus p right-parenthesis slash 2 comma">

Re
⁡

(

(

z
n

f
)

(
n
+
p
)

/

(

z

n
−
1

f
)

(
n
+
p
−
1
)

)

>
(
n
+
p
)

/

2
,

\operatorname {Re} \left ( {{{({z^n}f)}^{(n + p)}}/{{({z^{n - 1}}f)}^{(n + p - 1)}}} \right ) > (n + p)/2,

\]
where p is a positive integer and n is any integer greater than

−
p

- p

. It is proved that

K

n
+
p

⊂

K

n
+
p
−
1

{K_{n + p}} \subset {K_{n + p - 1}}

. Since

K
0

{K_0}

is the class of p-valent functions, consequently it follows that all functions in

K

n
+
p
−
1

{K_{n + p - 1}}

are p-valent. We also obtain some special elements of

K

n
+
p
−
1

{K_{n + p - 1}}

via the Hadamard product.
Note on a theorem of Berlinskiĭ
Authors: Sestier Andrés
Published: June 2010

If a quadratic differential system has four singular points, these are elementary and the sum of their indices is 0 iff the quadrilateral with vertices at the singular points is convex; otherwise the sum of indices is 2 or

−
2

- 2

. These facts and the relative positions of the two kinds of singular points are readily proved by consideration of the pencil of isoclines of the system. The theorem is originally due to Berlinskii.
A proof of the mean ergodic theorem for nonexpansive mappings in Banach space
Authors: Hirano Norimichi
Published: June 2010

Let C be a closed convex subset of a uniformly convex Banach space. Let

T
:
C
→
C

T:C \to C

be a nonexpansive mapping. In this paper, we deal with the weak convergence of the arithmetical means of the sequence

T
n

x

{T^n}x

, and give a new proof of the mean ergodic theorem for nonexpansive mappings.
Left versus right LCM domains
Authors: Beauregard Raymond A.
Published: June 2010

It is well known that every right Bezout domain satisfying the left Ore (multiple) condition is a left Bezout domain. A similar statement for the smaller class of principal right ideal domains is a long-standing conjecture which remains unresolved. In this paper we settle the analogous question for the larger class of right LCM domains.
Smoothness and weak* sequential compactness
Authors: Hagler James , Sullivan Francis
Published: June 2010

If a Banach space E has an equivalent smooth norm, then every bounded sequence in

E
∗

{E^\ast }

has a

weak
∗

{\text {weak}^\ast }

converging subsequence. Generalizations of this result are obtained.
Perturbations of ground states of type 𝐼 𝐶*-algebras
Authors: Batty C. J. K.
Published: June 2010

It is shown that the class of irreducible representations of a type I

C
∗

{C^\ast }

-algebra A which satisfy a spectrum condition for a given dynamical system on A is unchanged if the system undergoes a sufficiently small relatively bounded perturbation. It follows that if A is also unital, then the existence of ground states is unaffected by such perturbations.
On cyclic subgroups and the conjugacy problem
Authors: Hurwitz R. Daniel
Published: June 2010

The conjugacy problem in three types of group constructions involving cyclic subgroups is discussed. First it is shown that if G has the solvable conjugacy problem and if

h
∈
G

h \in G

and

k
∈
G

k \in G

satisfy (a) h and k are not power conjugate to themselves or each other, (b) the power conjugacy problem in G with respect to h or k is solvable, and (c) the double coset solvability problem in G is solvable with respect to

⟨
h
⟩

\langle h\rangle

and

⟨
k
⟩

\langle k\rangle

, then the HNN extension

G
∗

=
⟨
G
,
t
;

t

−
1

h
t
=
k
⟩

{G^ \ast } = \langle G,t;{t^{ - 1}}ht = k\rangle

has the solvable conjugacy problem. This result is used to deduce a similar theorem for free products with amalgamation, a fact first stated by Lipschutz. Then it is shown that if A and B are groups with the solvable conjugacy problem and

h
∈
A

h \in A

and

k
∈
B

k \in B

taken with themselves satisfy the conditions above in A and B, respectively, then

⟨

A
∗

B
;
[
h
,
k
]
=
1
⟩

\langle A ^\ast B;[h,k] = 1\rangle

has the solvable conjugacy problem.
On the continuation of solutions of differential equations by vector Ljapunov functions
Authors: Hatvani L.
Published: June 2010

In this note we prove a continuation theorem applicable also when the estimate of the derivative of the vector Ljapunov function contains the phase coordinates explicitly. Our theorem combines and strengthens several earlier continuation results including a recent theorem of T. A. Burton, who conjectured that the monotonicity assumption on a function in his theorem may be dropped. By an example we show, however, that Burton’s theorem would be false without this assumption. Furthermore, applying our theorem we can replace this assumption by a much weaker one.
Hausdorff measure and Carleson thin sets
Authors: Shapiro Joel H.
Published: June 2010

A Carleson set is a closed subset of the unit circle T having measure zero, whose complement is a disjoint union of open subarcs

(

I
n

)

({I_n})

with

Σ

|

I
n

|

log
⁡

|

I
n

|

>
−
∞

\Sigma |{I_n}|\log |{I_n}| > - \infty

. Suppose H is the Hausdorff measure on T induced by the determining function h, where

h
(
t
)

/

t

h(t)/t

is strictly decreasing. We show that

H
(
E
)
=
0

H(E) = 0

for every Carleson set E if and only if

∫
0
1

h

(
t

)

−
1

d
t
=
∞

\smallint _0^1h{(t)^{ - 1}}dt = \infty

. Consequently the nonintegrability of

h

−
1

{h^{ - 1}}

is necessary and sufficient for every positive Borel measure

μ
\mu

on T with modulus of continuity

ω
μ

(
δ
)
=
O
(
h
(
δ
)
)

{\omega _\mu }(\delta ) = O(h(\delta ))

to place zero mass on every Carleson set.
Semantical completeness theorems in logic and algebra
Authors: Lifschitz Vladimir
Published: June 2010

A few well-known algebraic theorems, including the Nullstellensatz and a form of the duality theorem of Farkas-Minkowsky, are shown to be corollaries to the completeness of the hyper-resolution method in proof theory.
On the iterations of diffeomorphisms without 𝐶⁰-Ω-explosions: an example
Authors: Sawada Ken
Published: June 2010

In this note we construct a diffeomorphism f such that f has no

C
0

-

Ω

{C^0}{\text {-}}\Omega

-explosion but

f
2

{f^2}

has

Ω
\Omega

-explosion.
Continuous maps of the interval whose periodic points form a closed set
Authors: Coven Ethan M. , Hedlund G. A.
Published: June 2010

We show that for a continuous map of a closed interval to itself, if the set of periodic points is closed, then every recurrent point is periodic. If, furthermore, the set of least periods of the periodic points is finite, then every nonwandering point is periodic. This answers a question of L. Block [Proc. Amer. Math. Soc. 67 (1977), 357-360].
On odd-primary components of Lie groups
Authors: Knapp K.
Published: June 2010

The transfer map

t
:

π
s

(

P
∞

C

)
→

π
s

(

S
0

)

t:{\pi ^s}({P_\infty }{\mathbf {C}}) \to {\pi ^s}({S^0})

is represented by an element

τ
∈

π
s

−
1

(

P
∞

C

+

)

\tau \in \pi _s^{ - 1}({P_\infty }{{\mathbf {C}}^ + })

. We compute the Adams-e-invariant of

τ
\tau

and use this and the splitting of the p-localization of

S
1

∧

P
∞

C

{S^1} \wedge {P_\infty }{\mathbf {C}}

into a wedge of

(
p
−
1
)

(p - 1)

spaces to prove that for a prime

p
⩾
5

p \geqslant 5

the p-component of the element

[
G
,

L

]

[G,\mathcal {L}]

defined by a compact Lie group G in

π
∗
s

\pi _ \ast ^s

is zero in the known part of stable homotopy.
Separation axioms for interval topologies
Authors: Erné Marcel
Published: June 2010

In Theorem 1 of this note, results of Kogan [2], Kolibiar [3], Matsushima [4] and Wolk [7] concerning interval topologies are presented under a common point of view, and further characterizations of the

T

2

{{\text {T}}_2}

axiom are obtained. A sufficient order-theoretical condition for regularity of interval topologies is established in Theorem 2. In lattices, this condition turns out to be equivalent both to the

T

2

{{\text {T}}_2}

and to the

T

3

{{\text {T}}_3}

axiom. Hence, a Hausdorff interval topology of a lattice is already regular. However, an example of a poset is given where the interval topology is

T

2

{{\text {T}}_2}

but not

T

3

{{\text {T}}_3}

.
Lagrange’s theorem with 𝑁^{1/3} squares
Authors: Choi S. L. G. , Erdős Paul , Nathanson Melvyn B.
Published: June 2010

For every

N
>
1

N > 1

we construct a set A of squares such that

|

A

|

>
(
4

/

log
⁡
2
)

N

1

/

3

log
⁡
N

|A| > (4/\log 2){N^{1/3}} \log N

and every nonnegative integer

n
⩽
N

n \leqslant N

is a sum of four squares belonging to A.
The compact Hankel operators form an 𝑀-ideal in the space of Hankel operators
Authors: Luecking Daniel H.
Published: June 2010

The theorem in the title is proved and used to give a new proof that elements of

L
∞

{L^\infty }

have best approximations in

H
∞

+
C

{H^\infty } + C

.
Another characterization of BMO
Authors: Coifman R. R. , Rochberg R.
Published: June 2010

The following characterization of functions of bounded mean oscillation (BMO) is proved. f is in BMO if and only if
\[

f
=
α
log
⁡

g
∗

−
β
log
⁡

h
∗

+
b

f = \alpha \log {g^ \ast } - \beta \log {h^ \ast } + b

\]
where

g
∗

,
(

h
∗

)

{g^ \ast },({h^ \ast })

is the Hardy-Littlewood maximal function of g, (h), respectively, b is bounded and

‖
f
‖

BMO

⩽
c
(
α
+
β
+

‖
b
‖

∞

)

{\left \| f \right \|_{{\text {BMO}}}} \leqslant c(\alpha + \beta + {\left \| b \right \|_\infty })

.
An application of Yau’s maximum principle to conformally flat spaces
Authors: Goldberg S. I.
Published: June 2010

Results of M. Tani on compact conformally flat manifolds and of M. Okumura on compact hypersurfaces of Euclidean space are extended to complete spaces by an application of S.-T. Yau’s “maximum principle".
Central limit theorem for signed distributions
Authors: Hochberg Kenneth J.
Published: June 2010

This paper contains an improved version of existing generalized central limit theorems for convergence of normalized sums of independent random variables distributed by a signed measure. It is shown that under reasonable conditions, the normalized sums converge in distribution to “higher-order” analogues of the standard normal random variable, in the sense that the density of the limiting signed distribution is the fundamental solution of a higher-order parabolic partial differential equation that is a generalization of the heat equation.
Vanishing of the Rochlin invariants of some 𝑍₂-homology 3-spheres
Authors: Kawauchi Akio
Published: June 2010

A

Z
2

{Z_2}

-homology 3-sphere has the Rochlin invariant (

=
μ

= \mu

-invariant) zero if it admits an orientation-reversing, piecewise-linear autohomeomorphism of finite order.
A covering property which implies isocompactness. I
Authors: Worrell J. M. , Wicke H. H.
Published: June 2010

We define a covering property for a topological space which does not explicitly mention countability or finitude of collections although it generalizes weak

δ
θ

\delta \theta

-refinability. We prove a general theorem that implies that countably compact spaces having the covering property are compact.
Connected projections of blocking sets of 𝐼ⁿ×𝑀
Authors: Rosen Harvey
Published: June 2010

Sufficient conditions are given for each minimal blocking set K of

I
n

×
M

{I^n} \times M

to have the closure of its projection,

p
(
K
)

p(K)

, into

I
n

{I^n}

connected. The construction of some examples of almost continuous functions

f
:

I
n

→
M

f:{I^n} \to M

in the literature depends on knowing each

p
(
K
)

¯

\overline {p(K)}

is connected or perfect.
Globally stable complete minimal surfaces in 𝑅³
Authors: do Carmo M. , Da Silveira A. M.
Published: June 2010

It is proved that a globally stable complete minimal surface in

R
3

{R^3}

with finite total curvature is a plane.
Some properties of the signature of complete intersections
Authors: Libgober A.
Published: June 2010

We prove that: (1) The signature of complete intersections is a monotone function of degrees of defining equations. (2) The signature of n-dimensional complete intersection (except for

C

P

n

{\mathbf {C}}{{\mathbf {P}}^n}

) is positive for

n
≡
0

(
mod

4
)

n \equiv 0 \pmod 4

and is negative for

n
≡
2

(
mod

4
)

n \equiv 2 \pmod 4

.
A remark on cosine families
Authors: Rankin Samuel M.
Published: June 2010

Let

C
(
t
)
,
t
∈
R

C(t),t \in R

, be a strongly continuous cosine family and A its infinitesimal generator. Then the set

E

=

d
e
f

{
x
∈
X
:
C
(
t
)
x

E\stackrel {def}{=}\{ x \in X:C(t)x

is once continuously differentiable in t on R} of the Banach space X is contained in the domain of

(
−
A

)
α

{( - A)^\alpha }

for

0
⩽
α
>
1

/

2

0 \leqslant \alpha > 1/2

.
Trace-class norm multipliers
Authors: Khalil Roshdi
Published: June 2010

In this paper we study the pointwise multipliers of the space

L
2

(
X
,
ν
)

⊗
^

L
2

(
Y
,
μ
)

{L^2}(X,\nu )\hat \otimes {L^2}(Y,\mu )

for different measure spaces

(
X
,
ν
)

(X,\nu )

and

(
Y
,
μ
)

(Y,\mu )

. First we consider the case where X and Y are finite sets, then we use this to study the multipliers of

L
2

(
I
,
ν
)

⊗
^

L
2

(
I
,
ν
)

{L^2}(I,\nu )\hat \otimes {L^2}(I,\nu )

, where I is the unit interval and

ν
\nu

any Borel measure on I.
On the reflexivity of 𝐶₀(𝑁) contractions
Authors: Wu Pei Yuan
Published: June 2010

Let T be a

C
0

(
N
)

{C_0}(N)

contraction on a separable Hilbert space and let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J equals upper S left-parenthesis phi 1 right-parenthesis circled-plus upper S left-parenthesis phi 2 right-parenthesis circled-plus midline-horizontal-ellipsis circled-plus upper S left-parenthesis phi Subscript k Baseline right-parenthesis">

J
=
S
(

φ
1

)
⊕
S
(

φ
2

)
⊕
⋯
⊕
S
(

φ
k

)

J = S({\varphi _1}) \oplus S({\varphi _2}) \oplus \cdots \oplus S({\varphi _k})

be its Jordan model, where

φ
1

,

φ
2

,
…
,

φ
k

{\varphi _1},{\varphi _2}, \ldots ,{\varphi _k}

are inner functions satisfying

φ
j

|

φ

j
−
1

{\varphi _j}|{\varphi _{j - 1}}

for

j
=
2
,
3
,
…
,
k

j = 2,3, \ldots ,k

, and

S
(

φ
j

)

S({\varphi _j})

denotes the compression of the shift on

H
2

⊖

φ
j

H
2

,
j
=
1
,
2
,
…
,
k

{H^2} \ominus {\varphi _j}{H^2},j = 1,2, \ldots ,k

. In this note we show that T is reflexive if and only if

S
(

φ
1

/

φ
2

)

S({\varphi _1}/{\varphi _2})

is.
Cutting families of convex sets
Authors: Katchalski Meir , Lewis Ted
Published: June 2010

A family of convex sets in the plane admits a common transversal if there is a straight line which intersects (cuts) each member of the family. It is shown that there is a positive integer k such that for any compact convex set C in the plane and for any finite family

A

\mathcal {A}

of pairwise disjoint translates of C: If each 3-membered subfamily of

A

\mathcal {A}

admits a common transversal then there is a subfamily

B

\mathcal {B}

of

A

\mathcal {A}

such that

B

\mathcal {B}

admits a common transversal and

|

A

∖

B

|

⩽
k

|\mathcal {A}\backslash \mathcal {B}| \leqslant k

.
On unions of 𝜈-embedded sets
Authors: Tanaka Hidenori
Published: June 2010

Let A be a realcompact and C-embedded subspace of a space X and let B be a v-embedded subspace of a space X. Then

A
∪
B

A \cup B

is v-embedded in X.
Shrinking certain sliced decompositions of 𝐸ⁿ⁺¹
Authors: Daverman Robert J. , Preston D. Kriss
Published: June 2010

We set forth a connection, based on relatively elementary techniques, between the shrinkability of product decompositions of

E

n
+
1

{E^{n + 1}}

and that of sliced decompositions. In particular, if G is a decomposition of

E

n
+
1

{E^{n + 1}}

such that each decomposition element g is contained in some horizontal slice

E
n

×
{
s
}

{E^n} \times \{ s\}

and if the decomposition

G
s

{G^s}

of

E
n

{E^n}

, consisting of those subsets g of

E
n

{E^n}

for which

g
×
{
s
}
∈
G

g \times \{ s\} \in G

, expands to a shrinkable decomposition

G
s

×

E
1

{G^s} \times {E^1}

of

E
n

×

E
1

{E^n} \times {E^1}

, we show then that G itself is shrinkable.
A countably paracompact nonnormal space
Authors: Weiss W.
Published: June 2010

A countably paracompact, first countable, separable, submetrizable, locally compact space which is not normal is constructed.
𝐾₂ measures excision for 𝐾₁
Authors: Geller Susan C. , Weibel Charles A.
Published: June 2010

Let B be a commutative ring, A a subring of B, and I an ideal of B contained in A. Excision holds if

K
1

(
A
,
I
)

{K_1}(A,I)

and

K
1

(
B
,
I
)

{K_1}(B,I)

are isomorphic. We show that the obstruction to excision holding is a subquotient of

K
2

(
B

/

I
2

)

{K_2}(B/{I^2})

. We then use this obstruction to show that, if A and B are fixed, the excision problem for I has no bearing on the excision problem for ideals contained in I.
Nonexistence of coreflections and coproducts in some subcategories of semigroups
Authors: Trotter P. G.
Published: June 2010

It is shown here that there are no nontrivial free regular semigroups, free regular semigroups on a semigroup or free products of regular semigroups. Analogous results are obtained in the categories of simple, bisimple and bisimple regular semigroups.
On Bieberbach’s analysis of discrete Euclidean groups
Authors: Oliver R. K.
Published: June 2010

For a subgroup G of the euclidean group

E
n

=

O
n

⋅

R

n

{E_n} = {O_n} \cdot {{\mathbf {R}}^n}

(semidirect product) and a real number

r
>
0

r > 0

, let

G
∗

{G^ \ast }

denote the translation subgroup of G,

G
r

{G_r}

the group generated by all (A, a) in G with

‖

1
−
A

‖

>
r

\left \| {1 - A} \right \| > r

(operator norm), and

k
n

(
r
)

{k_n}(r)

the maximum number of elements of

O
n

{O_n}

with mutual distances

⩾
r

\geqslant r

relative to the metric

d
(
A
,
B
)
=

‖

A
−
B

‖

d(A,B) = \left \| {A - B} \right \|

. We give an elementary, largely geometrical proof of the following results of Bieberbach: Let G be a subgroup of

E
n

{E_n}

. (1) If G is discrete, then

G

1

/

2

{G_{1/2}}

is abelian,

G

1

/

2

◃
G

{G_{1/2}} \triangleleft G

, and

[
G
:

G

1

/

2

]
⩽

k
n

(
1

/

2
)

[G:{G_{1/2}}] \leqslant {k_n}(1/2)

. (2) G is discrete if and only if

G
⊂

O

n
−
k

×

E
k

G \subset {O_{n - k}} \times {E_k}

, where

p
2

G

{p_2}G

is discrete,

(

p
2

G

)
∗

{({p_2}G)^ \ast }

spans

R

k

{{\mathbf {R}}^k}

, and

G
∩
ker
⁡

p
2

G \cap \ker {p_2}

is finite. (Here

p
2

{p_2}

is the projection on the second factor.) (3) G is crystallographic if and only if G is discrete and

G
∗

{G^ \ast }

spans

R

n

{{\mathbf {R}}^n}

. Moreover, if G is crystallographic, then

[
G
:

G
∗

]
⩽

k
n

(
1

/

2
)

[G:{G^ \ast }] \leqslant {k_n}(1/2)

.
A classification theorem for SKT-modules
Authors: Wick Brian D.
Published: June 2010

In this paper a class of Abelian groups which includes the torsion totally projective groups, S-groups, and balanced projectives is studied. It is shown that this class of groups has a complete set of invariants.
Compacité extrémale
Authors: Bourgain Jean , Talagrand Michel
Published: June 2010

Un ensemble borné d’un espace de Banach E est faiblement compact dès qu’il est compact pour la topologie de la convergence simple sur les points extrêmaux de la boule unité du dual de E.
Cyclic vectors of weighted shifts on 𝑙^{𝑝} spaces
Authors: Yadav B. S. , Chatterjee S.
Published: June 2010

Various authors have studied the existence of cyclic vectors of weighted shifts on Banach spaces (Hilbert spaces). In this paper, the existence of cyclic vectors of weighted shifts on

l
p

{l^p}

Banach spaces is exhibited under suitable conditions on the weight sequence.
On power compact operators
Authors: Barría José
Published: June 2010

We give an operator theoretic proof of the following result of D. G. Tacon: Theorem. If

{

T
n

}

\{ {T_n}\}

is a sequence of bounded linear operators in a complex infinite dimensional Hilbert space with the property that for every bounded sequence

{

x
n

}

\{ {x_n}\}

there exists a positive integer k such that the sequence

{

T
k

x
n

}

n
=
1

∞

\{ {T_k}{x_n}\} _{n = 1}^\infty

has a convergent subsequence, then there exists k such that

T
k

{T_k}

is a compact operator.
Shortest paths through pseudorandom points in the 𝑑-cube
Authors: Steele J. Michael
Published: June 2010

A lower bound for the length of the shortest path through n points in

[
0
,
1

]
d

{[0,1]^d}

is given in terms of the discrepancy function of the n points. This bound is applied to obtain an analogue for several pseudorandom sequences to the known limit behavior of the length of the shortest path through n independent uniformly distributed random observations from

[
0
,
1

]
d

{[0,1]^d}

.
ℒ*(𝒦) and other lattices of recursively enumerable sets
Authors: Shore Richard A.
Published: June 2010

We study the direct product operation on lattices which are principal filters of

E

∗

{\mathcal {E}^ \ast }

, the lattice of r.e. sets modulo finite sets, to generate new isomorphism types of such filters and to characterize the one generated by the complete r.e. set K.
Countably compact, locally countable 𝑇₂-spaces
Authors: Vaughan J. E.
Published: June 2010

The results of this paper provide a simple method for constructing locally countable

T
2

{T_2}

-spaces (not

T
3

{T_3}

) in which every infinite closed set has cardinality

2
c

{2^c}

. The spaces are used in a variety of ways as counterexamples. One of these spaces may be considered as a countably compact version of the Katětov H-closed extension of the natural numbers.
Weyl group actions and equivariant homotopy equivalence
Authors: Kawakubo Katsuo
Published: June 2010

Let G be a compact Lie group and

G
0

{G_0}

its identity component. Then we shall show that the normal representations of the corresponding fixed point components of G-homotopy equivalent manifolds are necessarily isomorphic when

G

/

G
0

G/{G_0}

is a Weyl group of a compact connected Lie group.
On conjugacy classes of elements of finite order in compact or complex semisimple Lie groups
Authors: Djoković Dragomir Ž.
Published: June 2010

If K is a connected compact Lie group with simple Lie algebra and if k is an integer relatively prime to the order of the Weyl group W of K then the number

ν
(
K
,
k
)

\nu (K,k)

of conjugacy classes of K consisting of elements x satisfying

x
k

=
1

{x^k} = 1

is given by
\[

ν
(
K
,
k
)
=

∏

i
=
1

l

m
i

+
k

m
i

+
1

,

\nu (K,k) = \prod \limits _{i = 1}^l {\frac {{{m_i} + k}}{{{m_i} + 1}},}

\]
where l is the rank of K and

m
1

,
…
,

m
l

{m_1}, \ldots ,{m_l}

are the exponents of W. If G is the complexification of K then we have

ν
(
G
,
k
)
=
ν
(
K
,
k
)

\nu (G,k) = \nu (K,k)

without any restriction on k.
Differential algebraic group structures on the plane
Authors: Cassidy Phyllis J.
Published: June 2010

The differential algebraic group structures on the affine line and plane are classified. The additive group

G
a

{G_a}

of the coefficient field is the only differential algebraic group structure on the line. Every differential algebraic group with underlying set in the plane is unipotent and is isomorphic to a group whose law of composition is defined by the formula
\[

(

u
1

,

u
2

)
(

v
1

,

v
2

)
=
(

u
1

+

v
1

,

u
2

+

v
2

+
f
(

u
1

,

v
1

)
)
,

({u_1},{u_2})({v_1},{v_2}) = ({u_1} + {v_1},{u_2} + {v_2} + f({u_1},{v_1})),

\]
where f is a 2-cocycle of

G
a

{G_a}

into

G
a

{G_a}

.
On making rings weakly finite
Authors: Malcolmson Peter
Published: June 2010

Given an associative ring R with unit we construct the largest homomorphic image of R such that every right invertible square matrix is left invertible. The construction may be accomplished in one step. Related results are also given.
Morita equivalent semigroups of quotients
Authors: Luedeman John K.
Published: June 2010

Let S be a monoid and

S

M

_SM

and

S

N

_SN

be retracts of each other. We show that

End
S

(
M
)

{\text {End}_S}(M)

and

End
S

(
N
)

{\text {End}_S}(N)

are Morita equivalent. Using this result, we show that if A and B are Morita equivalent monoids, then their semigroups of quotients are Morita equivalent.
The Schneider-Lang theorem for functions with essential singularities
Authors: Diamond Jack
Published: June 2010

A new proof of Schwarz’s lemma for functions with a finite number of essential singularities is given. The proof is valid for p-adic as well as complex functions and is used to extend Bertrand’s version of the Schneider-Lang theorem for p-adic functions with one, common, finite singularity to functions with finitely many singularities.
The second coefficient of univalent Bieberbach-Eilenberg functions near the identity
Authors: Hummel J. A.
Published: June 2010

An asymptotic expansion is found for the maximum

|

b
2

|

|{b_2}|

, in terms of

β
=
1
−

|

b
1

|

\beta = 1 - |{b_1}|

, for functions

F
(
z
)
=

b
1

z
+

b
2

z
2

+
⋯

F(z) = {b_1}z + {b_2}{z^2} + \cdots

which are in the class of univalent Bieberbach-Eilenberg functions and which are near the identity (

b
1

{b_1}

near 1). The first two terms of this expansion are the same as the expansion of

|

b
2

|

|{b_2}|

in terms of

1
−

b
1

1 - {b_1}

for the functions which map the unit disc onto the interior of circles passing through

±
1

\pm 1

.
Remarks on Souslin properties and tree topologies
Authors: Fleissner William G.
Published: June 2010

We investigate the relation of Souslin (antichain) properties of trees and tree topologies. One result extends a result of Devlin and Shelah by proving, within ZFC, the equivalence of four properties for

ω
1

{\omega _1}

-trees-collectionwise normal, normal and collectionwise Hausdorff, property

γ
\gamma

, and antichain normal and collectionwise Hausdorff. A second result is the construction, assuming

V
=
L

V = L

, of an Aronszajn tree which is not countably metacompact. Third, we show that no tree can be a Dowker space.
Closed mappings and quasimetrics
Authors: Kofner Jacob
Published: June 2010

Closed continuous mappings with first countable images preserve quasi-metric spaces as well as nonarchimedean quasi-metric spaces and

γ
\gamma

-spaces. This is a strict generalization of analogous results on perfect mappings: there exists a closed continuous mapping of a nonarchimedean quasi-metric Moore space onto a compact metric space which is neither perfect nor boundary compact.
On 𝑆-closed spaces
Authors: Joseph James E. , Kwack Myung H.
Published: June 2010

In this paper, we initially give several new characterizations of the class of S-closed spaces, which was introduced by T. Thompson [Proc. Amer. Math. Soc. 60 (1976), 335-338]. We then employ these characterizations to produce analogues for S-closed spaces of the well-known theorem from real analysis that an upper-semicontinuous real-valued function on a closed interval assumes a maximum, and of two well-known theorems of G. Birkhoff and A. D. Wallace, which established that each upper-semicontinuous function from a compact space into a partially ordered set assumes a maximal value and that each compact space has a maximal element with respect to each upper-semicontinuous quasi order on the set. The statements in these latter analogues are then shown to characterize S-closed spaces. A “fixed set theorem” for multifunctions on S-closed spaces is also established.
The Hahn decomposition theorem
Authors: Doss Raouf
Published: June 2010

Let

(
X
,

A

,
μ
)

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

be a signed measure on the

σ
\sigma

-algebra

A

\mathcal {A}

of subsets of X. We give a very short proof of the Hahn decomposition theorem, namely, that X can be partitioned into two subsets P and N such that P is positive:

μ
(
E
)
⩾
0

\mu (E) \geqslant 0

for every

E
⊂
P

E \subset P

, and N is negative:

μ
(
E
)
⩽
0

\mu (E) \leqslant 0

for every

E
⊂
N

E \subset N

.
Instability and theories with few models
Authors: Pillay A.
Published: June 2010

Some results are obtained concerning

n
(
T
)

n(T)

, the number of countable models up to isomorphism, of a countable complete first order theory T. It is first proved that if

n
(
T
)
=
3

n(T) = 3

and T has a tight prime model, then T is unstable. Secondly, it is proved that if

n
(
T
)

n(T)

is finite and more than one, and T has few links, then T is unstable. Lastly we show that if T has an algebraic model and has few links, then

n
(
T
)

n(T)

is infinite.
A many-sorted interpolation theorem for 𝐿(𝑄)
Authors: Guichard David R.
Published: June 2010

Let L be a many-sorted relational language with

∈
\in

and consider the logic

L

ω
1

ω

(
Q
)

{L_{{\omega _1}\omega }}(Q)

, infinitary logic with a monotone quantifier. We prove a version of Feferman’s Interpolation Theorem for this logic. We then use the theorem to show that for a one-sorted language L and a countable admissible fragment

L
A

{L_A}

of

L

ω
1

ω

(
Q
)

{L_{{\omega _1}\omega }}(Q)

, any sentence which persists for end extensions is equivalent to a

Σ
\Sigma

sentence.
Automorphisms of fibrations
Authors: Dror E. , Dwyer W. G. , Kan D. M.
Published: June 2010

Let X be a simplicial set, G a simplicial group and

W
¯

G

\bar WG

the classifying complex of G. Then it is well known [1], [3] that the principal fibrations with base X and group G are classified by the components of the function complex

(

W
¯

G

)
X

{(\bar WG)^X}

. The aim of the present note is to prove the following complement to this result (1.2): Let p be a principal fibration with base X and group G, and let aut p be its simplicial group of automorphisms (which keep the base fixed). Then

W
¯

(

aut

p
)

\bar W({\operatorname {aut}}\,p)

has the homotopy type of the component of

(

W
¯

G

)
X

{(\bar WG)^X}

which (see above) corresponds to p. A similar result holds for ordinary fibrations.
Martin’s axiom implies that de Caux’s space is countably metacompact
Authors: Fleissner William G.
Published: June 2010

De Caux defined a space

S
(

L

)

S(\mathcal {L})

and, assuming

♣
\clubsuit

, showed that

S
(

L

)

S(\mathcal {L})

is normal but not countably metacompact. We assume

M
A

ω
1

\mathrm {MA}_{\omega _1}

and show that

S
(

L

)

S(\mathcal {L})

is countably metacompact.
Periodic and limit orbits and the depth of the center for piecewise monotone interval maps
Authors: Nitecki Zbigniew
Published: June 2010

For a piecewise monotone map of the interval: (a) the nonwandering points outside the closure of the periodic points are isolated in the nonwandering set; (b) the forward orbit of any such point misses all turning points; (c) the depth of the center is at most 2; and (d) all

ω
\omega

-limit points belong to the closure of the periodic points.
Character tables determine abelian Sylow 2-subgroups
Authors: Camina Alan R. , Herzog Marcel
Published: June 2010

A finite group has an abelian

S
2

{S_2}

if and only if every 2-element is 2-central.
Quotient rings of a ring and a subring which have a common right ideal
Authors: Shapiro Jay
Published: June 2010

Let R be a subring of S and let

A
⊆
R

A \subseteq R

be a right ideal of S. In this paper we show that there is a bijection between right torsion theories

τ
\tau

over S such that A is

τ
\tau

-dense and right torsion theories

σ
\sigma

over R such that S/A is

σ
\sigma

-torsion. A similar result is obtained for the left side with the bijection between torsion theories over S with SA dense and torsion theories over R with RA dense. It is also shown that the ring of quotients of R and S at these corresponding torsion theories are equal. As a corollary, when A is chosen appropriately R and S have the same right (left) maximal quotient ring.
Zero-diminishing linear transformations
Authors: Craven Thomas , Csordas George
Published: June 2010

This paper solves an open problem of Karlin. We consider linear transformations defined on the set of all polynomials with real coefficients and give a complete characterization of the transformations with the property that they maintain or reduce the number of real zeros of each polynomial.
Asymptotic behavior of solutions for a parabolic equation with nonlinear boundary conditions
Authors: Pao C. V.
Published: June 2010

This paper is concerned with the asymptotic behavior and nonexistence of global solutions for a linear parabolic equation under nonlinear boundary conditions. It is shown under certain conditions on the nonlinear boundary function that a global solution exists and converges to a steady-state solution while for another class of initial function the solution blows-up in finite time. In some special cases, threshold results for the convergence of the solution and its blowingup behavior are explicitly given.
On one-sided harmonic analysis
Authors: Henrichs Rolf Wim
Published: June 2010

It is shown that, in locally compact groups satisfying the onesided version of the Wiener property introduced by Leptin, characters of closed subgroups have always continuous positive definite extensions onto the whole group. We give a quick proof that for such groups the connected component of the identity is the direct product of a compact group and a vector group.
Best approximation in certain Douglas algebras
Authors: Younis Rahman
Published: June 2010

The main result of this paper is that if

f
∈

L
∞

f \in {L^\infty }

and S is a weak peak set for

H
∞

{H^\infty }

, then the distance from f to the Douglas algebra

{
f
∈

L
∞

:
f

|

S

∈

H
∞

|

S

}

\{ f \in {L^\infty }:f{|_S} \in {H^\infty }{|_S}\}

is attained.
A note on analytic sets
Authors: Steel John R.
Published: June 2010

We give “effective” proofs of two recent theorems on analytic sets of reals, together with counterexamples to their natural extensions.
Equivariant maps which are self homotopy equivalences
Authors: Dror E. , Dwyer W. G. , Kan D. M.
Published: June 2010

The aim of this note is (i) to give (in §2) a precise statement and proof of the (to some extent well-known) fact that the most elementary homotopy theory of “simplicial sets on which a fixed simplicial group H acts” is equivalent to the homotopy theory of “simplicial sets over the classifying complex

W
¯

H

\bar WH

", and (ii) to use this (in §1) to prove a classification theorem for simplicial sets with an H-action, which provides classifying complexes for their equivariant maps which are self homotopy equivalences.
The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality
Authors: van Douwen Eric K.
Published: June 2010

Let X be an infinite pseudocompact space. We are interested in restrictions on

κ
=

|

X

|

\kappa = |X|

and

λ
=
w
(
X
)

\lambda = w(X)

in addition to the obvious inequalities

λ
⩽

2
κ

\lambda \leqslant {2^\kappa }

and

κ
⩽

2
λ

\kappa \leqslant {2^\lambda }

and

κ
⩾

2
ω

\kappa \geqslant {2^\omega }

, valid for X without isolated points (in particular for homogeneous X). We show that if

cf

(
κ
)
=
ω

{\text {cf}}(\kappa ) = \omega

then

λ
⩽

2

>
κ

\lambda \leqslant {2^{ > \kappa }}

, and even

λ
⩽

2
μ

\lambda \leqslant {2^\mu }

for some

μ
>
κ

\mu > \kappa

if X is homogeneous. Under the Singular Cardinals Hypothesis (which is much weaker than the GCH), there are no further restrictions for X without isolated points.
𝑆-sets and 𝑆-perfect mappings
Authors: Dickman R. F. , Krystock R. L.
Published: June 2010

In this note we generalize the notion of an S-closed space to S-sets of a space. Among our characterizations of S-sets, we show that a subset A of an extremally disconnected space X is an S-set in X if and only if A is an H-set in X. We also investigate conditions under which mappings or their inverses preserve S-sets.
Finitely convergent and concentric semigroups
Authors: Satyanarayana M.
Published: June 2010

In this paper we shall study Tamura’s finitely convergent and concentric semigroups and provide some structure theorems. In particular commutative finitely convergent semigroups are completely characterized.
An additive representation for real functions on the product of a set and a lattice
Authors: Eplett W. J. R.
Published: June 2010

Given a real-valued function defined on the product of an arbitrary set and a finite lattice, a necessary and sufficient condition is obtained for the existence of an additive representation for the function in terms of functions on sublattices of the original lattice. This additive representation is of the nature of a recurrence and provides a tool for further analysis of the function. An example is given for a case where the lattice concerned is the lattice of partitions of a finite set. The main theorem of this paper generalizes a result due to Fishburn corresponding to the lattice being the lattice of subsets of a finite set.
Generalised Lipschitz class of functions and the absolute summability of Fourier series by Nörlund means
Authors: Lal S. N. , Singh K. N.
Published: June 2010

The main aim of the paper is to investigate the relationship between certain generalised Lipschitz classes of functions and to discuss the absolute Nörlund summability of Fourier series of functions of the class

L
q

{L^q}

where

2
>
q
>
∞

2 > q > \infty

.
On compact multipliers of Banach algebras
Authors: Kamowitz Herbert
Published: June 2010

We show that if the maximal ideal space of a commutative semisimple Banach algebra

B
B

contains no isolated points, then every compact multiplier is trivial.
On extending the Lebesgue integral
Authors: Foran James
Published: June 2010

In this paper continuous extensions of the Lebesgue Integral which integrate their almost everywhere approximate derivatives are considered. First, those extensions which are generated by a single function are characterized. Then the largest extension which is contained in all maximal extensions is considered and shown to contain the wide sense Denjoy Integral. Finally, properties are given which characterize this largest extension.
Dans un espace de Banach reticulé solide, la propriété de Radon-Nikodým et celle de Kreĭn-Mil′man sont équivalentes
Authors: Bourgain Jean , Talagrand Michel
Published: June 2010

Un espace de Banach réticulé solide possède la propriété de Radon-Nikodým si et seulement si tout convexe fermé borné est enveloppe convexe fermée de ses points extrémaux.
On the two-realizability of chain complexes
Authors: Jajodia Sushil
Published: June 2010

We give a sufficient condition which insures the realizability of a two-dimensional chain complex satisfying Wall’s condition by a two-dimensional CW-complex.
𝜎-coherent continua are hereditarily locally connected
Authors: Hagan M. R. , Mahavier W. S.
Published: June 2010

A

σ
\sigma

-coherent continuum is one in which every descending sequence of connected sets has a connected intersection. In this paper it is proved that such continua are hereditarily locally connnected. An example is given to show that the converse is not true.
Cartan matrices, finite groups of quaternions, and Kleinian singularities
Authors: McKay John
Published: June 2010

The eigenvectors of the Cartan matrices of affine type

A
r

{A_r}

,

D
r

{D_r}

,

E
6

{E_6}

,

E
7

{E_7}

, and

E
8

{E_8}

can be taken to be the columns of the character tables of the finite groups of quaternions.
Schur indices and the Galois group
Authors: Spiegel Eugene
Published: June 2010

In this note we show that the order of the Schur index of an irreducible representation divides the order of a certain subgroup of the Galois group of a cyclotomic extension of the ground field.
A remark on bounded strictly plurisubharmonic exhaustion functions
Authors: Range R. Michael
Published: June 2010

A more general version of the theorem of Diederich and Fornaess on the existence of bounded strictly plurisubharmonic exhaustion functions is proved by completely elementary methods.
Length estimates for holomorphic functions
Authors: Yamashita Shinji
Published: June 2010

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals a 0 plus normal upper Sigma Subscript k equals n Superscript normal infinity Baseline a Subscript k Baseline z Superscript k Baseline left-parenthesis n greater-than-or-slanted-equals 1 right-parenthesis">

f
(
z
)
=

a
0

+

Σ

k
=
n

∞

a
k

z
k

(
n
⩾
1
)

f(z) = {a_0} + \Sigma _{k = n}^\infty {a_k}{z^k}(n \geqslant 1)

be holomorphic in

U
:

|
z
|

>
1

U:\left | z \right | > 1

. MacGregor [1, Theorem 2] proved that if

l
(
r
,
f
)

l(r,f)

is the length of the outer boundary of the image

D
(
r
,
f
)

D(r,f)

of the disk:

|
z
|

>
r

\left | z \right | > r

by

f
f

, then

2
π

r
n

|

a
n

|

⩽
l
(
r
,
f
)

2\pi {r^n}\left | {{a_n}} \right | \leqslant l(r,f)

for

0
>
r
>
1

0 > r > 1

. We introduce the notion of the exact outer boundary

C
#

(
r
,
f
)

{C^\# }(r,f)

of

D
(
r
,
f
)

D(r,f)

and prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 pi r Superscript n Baseline StartAbsoluteValue a Subscript n Baseline EndAbsoluteValue less-than-or-slanted-equals l Superscript number-sign Baseline left-parenthesis r comma f right-parenthesis less-than-or-slanted-equals l left-parenthesis r comma f right-parenthesis">

2
π

r
n

|

a
n

|

⩽

l
#

(
r
,
f
)
⩽
l
(
r
,
f
)

2\pi {r^n}\left | {{a_n}} \right | \leqslant {l^\# }(r,f) \leqslant l(r,f)

for

0
>
r
>
1

0 > r > 1

, where

l
#

(
r
,
f
)

{l^\# }(r,f)

is the length of

C
#

(
r
,
f
)

{C^\# }(r,f)

. We shall make use of the estimate to obtain a criterion for

f
f

to be Bloch in

U
U

.
A remark on analytic sets with 𝜎-compact sections
Authors: Srivatsa V. V.
Published: June 2010

We show that there exists an analytic set

A
⊂

ω
ω

×

ω
ω

A \subset {\omega ^\omega } \times {\omega ^\omega }

and having

σ
\sigma

-compact vertical sections such that

A
A

contains no analytic set

B
B

with compact vertical sections and having the same projection to the first coordinate as

A
A

. This answers a question of J. R. Steel.
Scalar curvature of Lie groups
Authors: Lai Hêng Lung , Lue Huei Shyong
Published: June 2010

In this paper, we prove the following theorem: If

G
G

is a connected Lie group, then

G
G

admits left invariant metric of positive scalar curvature if and only if the universal covering space

G
~

\tilde G

of

G
G

is not homeomorphic to the Euclidean space.
The 𝐿²-norm of Maass wave functions
Authors: Smith Robert A.
Published: June 2010

Let

D
D

denote the fundamental domain for the full modular group. Suppose that

f
∈

L
2

(
D
)

f \in {L^2}(D)

satisfies the wave equation

Δ
f
=
λ
f

\Delta f = \lambda f

, where

Δ
\Delta

is the noneuclidean Laplacian, and further, assume that

f
f

is a common eigenfunction for all the Hecke operators. Then upper and lower bounds for the

L
2

{L^2}

-norm of

f
f

are determined which depend only on

λ
\lambda

and the first Fourier coefficient of

f
f

.
A linear, almost periodic equation with an almost automorphic solution
Authors: Johnson Russell A.
Published: June 2010

We construct a scalar, first order, almost periodic

ODE

(
∗
)
x
+
A
(
t
)
x
=
B
(
t
)

{\text {ODE}}( * )x + A(t)x = B(t)

which admits bounded solutions, but no almost periodic solutions. Using this equation, we give an example of a two-dimensional, almost periodic system whose projective flow admits two minimal subsets, one of which is almost automorphic but not almost periodic. Finally, we show that some equation in the hull of

(
∗
)

( * )

admits an almost automorphic, nonalmost periodic solution.
Classes of 𝐿¹-convergence of Fourier and Fourier-Stieltjes series
Authors: Stanojević Časlav V.
Published: June 2010

It is shown that the Fomin class

F

p

(
1
>
p
⩽
2
)

{\mathcal {F}_p}(1 > p \leqslant 2)

is a subclass of

C

∩

B

V

\mathcal {C} \cap \mathcal {B}\mathcal {V}

, where

C

\mathcal {C}

is the Garrett-Stanojević class and

B

V

\mathcal {B}\mathcal {V}

the class of sequences of bounded variation. Wider classes of Fourier and Fourier-Stieltjes series are found for which

a
n

lg

n
=
o
(
1
)
,
n
→
∞

{a_n}\;{\text {lg}}\;n = o(1),n \to \infty

, is a necessary and sufficient condition for

L
1

{L^1}

-convergence. For cosine series with coefficients in

B

V

\mathcal {B}\mathcal {V}

and

n
Δ

a
n

=
O
(
1
)

n\Delta {a_n} = O(1)

,

n
→
∞

n \to \infty

, necessary and sufficient integrability conditions are obtained.
Inequalities for derivatives of polynomials with restricted zeros
Authors: Máté Attila
Published: June 2010

Let

f
f

be a polynomial of degree

n
n

that has real roots all of which lie outside the interval

(
−
1
,
1
)

( - 1,1)

. P. Erdös proved that if

|
f
|

⩽
1

\left | f \right | \leqslant 1

on this interval then

|

f
′

|

>
n
e

/

2

\left | {f’} \right | > ne/2

on

[
−
1
,
1
]

[ - 1,1]

; this is much stronger than the results the inequalities of A. Markov and S. N. Bernstein would give. We will show that if only

n
−
k

n - k

roots of

f
f

are restricted as above, then

|

f
′

|

>

c
k

n

\left | {f’} \right | > {c_k}n

holds on

[
−
1
,
1
]

[ - 1,1]

with appropriate

c
k

{c_k}

. An upper estimate for the best

c
k

{c_k}

is given. Results for higher derivatives and

L
p

{L^p}

spaces are also obtained.
𝐶₁₁ contractions are reflexive. II
Authors: Wu Pei Yuan
Published: June 2010

It has been shown previously by the author that any completely nonunitary

C

11

{C_{11}}

contraction with finite defect indices is reflexive. In this note we show that this is true even without the completely nonunitary assumption.
Absolute Riesz summability of Fourier series. I
Authors: Dikshit G. D. , Rees C. S.
Published: June 2010

In this paper we prove some theorems on the absolute summability of Fourier series which connect diverse

|

C
,
γ

|

\left | {C,\gamma } \right |

results such as Bosanquet’s classical theorem (1936), Mohanty (1952), and Ray (1970) and the recent

|

R
,

exp
⁡
(

(
log
⁡
ω
)

β
+
1

)
,
γ

|

\left | {R,\;\exp ({{(\log \omega )}^{\beta + 1}}),\gamma } \right |

result of Nayak (1971). It is also shown that in some sense some of the conclusions of the paper are the best possible.
Some functions with a unique invariant mean
Authors: Talagrand Michel
Published: June 2010

In a large class of groups, we construct a function which has a unique invariant mean, but which is not Riemann-measurable.
On coverings of convex sets by translates of slabs
Authors: Groemer H.
Published: June 2010

Let

(

S
1

,

S
2

,
…
)

({S_1},{S_2}, \ldots )

be a sequence of slabs in euclidean

n
n

-dimensional space and let

t
i

{t_i}

denote the thickness of

S
i

{S_i}

. It is shown that the condition

∑

t
i

(
n
+
1
)

/

2

=
∞

\sum {t_i^{(n + 1)/2} = \infty }

implies that every convex set can be covered by translates of the slabs

S
i

{S_i}

, and that the exponent

(
n
+
1
)

/

2

(n + 1)/2

is, in a certain sense, best possible.