bluuebunny/crossref_metadata_2025 huggingface dataset:
A class of one-parameter nonlinear semigroups with differentiable approximating semigroups
Authors: Parker G. Edgar
Published: June 2010

Suppose that T is a strongly continuous semigroup of transformations on a subset C of a Banach space X. For

δ
>
0

\delta > 0

, consider
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript delta Baseline left-parenthesis t right-parenthesis equals StartSet left-parenthesis delta Superscript negative 1 Baseline integral Subscript 0 Superscript delta Baseline g Subscript x Baseline comma delta Superscript negative 1 Baseline integral Subscript 0 Superscript delta Baseline g Subscript upper T left-parenthesis t right-parenthesis x Baseline right-parenthesis colon x element-of upper C EndSet">

U
δ

(
t
)
=
{
(

δ

−
1

∫
0
δ

g
x

,

δ

−
1

∫
0
δ

g

T
(
t
)
x

)
:
x
∈
C
}

{U_\delta }(t) = \{ ({\delta ^{ - 1}}\smallint _0^\delta {g_x},{\delta ^{ - 1}}\smallint _0^\delta {g_{T(t)x}}):x \in C\}

where

g
x

g_x

denotes the trajectory of T from x. The class H of semigroups for which

U
δ

(
t
)

{U_\delta }(t)

is a function for

δ
>
0

\delta > 0

and

t
⩾
0

t \geqslant 0

contains all strongly continuous linear semigroups and Webb’s nonlinear nonexpansive example with no dense set of differentiability. If
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T element-of upper H comma upper U Subscript delta Baseline equals StartSet left-parenthesis t comma upper U Subscript delta Baseline left-parenthesis t right-parenthesis right-parenthesis colon t greater-than-or-slanted-equals 0 EndSet">

T
∈
H
,

U
δ

=
{
(
t
,

U
δ

(
t
)
)
:
t
⩾
0
}

T \in H,{U_\delta } = \{ (t,{U_\delta }(t)):t \geqslant 0\}

is a semigroup on

{

δ

−
1

∫
0
δ

g
x

:
x
∈
C
}

\{ {\delta ^{ - 1}}\smallint _0^\delta {g_x}:x \in C\}

with continuously differentiable trajectories. Also, as

{

δ
n

}

n
=
1

∞

\{ {\delta _n}\} _{n = 1}^\infty

converges to 0, the trajectories of

{

U

δ
n

}

n
=
1

∞

\{ {U_{{\delta _n}}}\} _{n = 1}^\infty

uniformly approximate the trajectories of T.
Sums of powers in large finite fields
Authors: Small Charles
Published: June 2010

If k is a positive integer, then, in any finite field with more than

(
k
−
1

)
4

{(k - 1)^4}

elements, every element is a sum of two kth powers.
On measures of column indecomposability
Authors: Hartfiel D. J.
Published: June 2010

For any given nonnegative matrix A, this paper develops a notion of the kth measure of column indecomposability of A. The behavior of this measure on products of matrices is investigated. These results are then applied in developing several results on nonhomogeneous Markov chains.
Representation of semigroups as systems of compact convex sets
Authors: Ratschek H. , Schröder G.
Published: June 2010

Under Minkowski addition and scalar multiplication the system of all compact convex subsets of

R
n

{R^n}

is an R-semigroup, i.e. a semigroup over the operator domain R of real numbers with certain conditions for the operation of R on the semigroup. Conversely, there is the question: When is an abstract R-semigroup isomorphic to a system

S

\mathfrak {S}

of compact convex subsets of a suitable locally convex space? In this paper a necessary and sufficient condition for the existence of such a representation is given. This condition remains valid if, for the representing structures

S

\mathfrak {S}

, systems of closed, bounded convex subsets with the closed Minkowski addition as addition are permitted. Finally, every R-semigroup of compact convex subsets of any locally convex space is isomorphic to a system of rectangular parallelepipeds of some vector space.
A remark on meromorphic solutions of differential equations
Authors: Strelitz Sh.
Published: June 2010

The following assertion is proven. Suppose the functions F, P, Q of the differential equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis z comma w comma w prime comma w Superscript left-parenthesis n right-parenthesis Baseline right-parenthesis equals upper P left-parenthesis z comma w right-parenthesis slash upper Q left-parenthesis z comma w right-parenthesis">

F
(
z
,
w
,

w
′

,

w

(
n
)

)
=
P
(
z
,
w
)

/

Q
(
z
,
w
)

F(z,w,w’,{w^{(n)}}) = P(z,w)/Q(z,w)

to be polynomials of all their corresponding variables. If the considered equation has a transcendental meromorphic solution in

|

z

|

>
∞

|z| > \infty

, then

Q
(
z
,
w
)

Q(z,w)

does not depend on w. An example of possible applications is stated.
Martingales in the limit and amarts
Authors: Edgar G. A. , Sucheston L.
Published: June 2010

The notion of amart is compared to that of a martingale in the limit and game fairer with time. Every real-valued amart is a martingale in the limit. More generally, a Banach space E is finite-dimensional if and only if every E-valued amart is a martingale in the limit (or a game fairer with time). Several crucial properties possessed by amarts fail both for martingales in the limit and games fairer with time: the maximal inequality, the optional stopping theorem, the optional sampling theorem, the Riesz decomposition; therefore a general theory analogous to the amart theory cannot be based on the notion of a martingale in the limit. It is also observed that either the optional sampling theorem or a weak form of the Riesz decomposition must fail for any class of sequences of random variables strictly larger than the class of amarts.
Maximal logics
Authors: Sgro Joseph
Published: June 2010

In this paper we present a general method for producing logics on various classes of models which are maximal with respect to a Łoś ultraproducts theorem. As a corollary we show that

L

Top

{\mathcal {L}^{{\text {Top}}}}

is maximal. We also show that these maximal logics satisfy the Souslin-Kleene property.
A note on a paper of J. D. Stein, Jr.: “Sequence of regular finitely additive set functions” (Trans. Amer. Math. Soc. 192 (1974), 59–66)
Authors: Khurana Surjit Singh
Published: June 2010

Among other results, it is proved that if a sequence

{

μ
n

}

\{ {\mu _n}\}

of regular measures on a Hausdorff space, with values in a normed group, is convergent to zero for all

σ
\sigma

-compact sets or all open sets, then there exists a maximal open set U such that

μ
˙

n

(
U
)
→
0
,
{

μ
˙

n

}

{\dot \mu _n}(U) \to 0,\{ {\dot \mu _n}\}

being the associated submeasures.
Remarks on subsequences, subseries and rearrangements
Authors: Rao M. Bhaskara , Rao K. P. S. Bhaskara , Rao B. V.
Published: June 2010

In this paper we shall evaluate the measure theoretic, as well as topological (Baire category theoretic) complexity of the set of convergent subsequences of a given sequence of real numbers. Similar problems regarding subseries and rearrangements of a given series are also discussed.
Convergence of probability measures on separable Banach spaces
Authors: Grinblat L. Š.
Published: June 2010

The following result follows immediately from a general theorem on the convergence of probability measures on separable Banach spaces: On the space

C
[
0
,
1
]

C[0,1]

there exists a norm

p
(
x
)

p(x)

equivalent to the ordinary norm such that if

ξ
1

(
t
)
,
…
,

ξ
n

(
t
)
,
…

{\xi _1}(t), \ldots ,{\xi _n}(t), \ldots

and

ξ
(
t
)

\xi (t)

are continuous random processes

(
0
⩽
t
⩽
1
)

(0 \leqslant t \leqslant 1)

and for any finite set of points

t
1

,
…
,

t
k

⊂
[
0
,
1
]

{t_1}, \ldots ,{t_k} \subset [0,1]

the joint distribution of

p
(

ξ
n

)
,

ξ
n

(

t
1

)
,
…
,

ξ
n

(

t
k

)

p({\xi _n}),{\xi _n}({t_1}), \ldots ,{\xi _n}({t_k})

converges to the joint distribution of

p
(
ξ
)
,
ξ
(

t
1

)
,
…
,
ξ
(

t
k

)

p(\xi ),\xi ({t_1}), \ldots ,\xi ({t_k})

then

ξ
n

(
t
)

{\xi _n}(t)

converges weakly to

ξ
(
t
)

\xi (t)

.
Tarski semigroups
Authors: Ellentuck Erik
Published: June 2010

A Tarski semigroup is an algebraic system which mirrors a fragment of the additive theory of cardinal numbers. Here we show that any two such systems have the same universal theory. We also give a simple arithmetical necessary and sufficient condition for a universal sentence to hold in a Tarski semigroup.
Modules and quadratic forms over polynomial algebras
Authors: Knus M.-A. , Ojanguren M.
Published: June 2010

Let D be a finite dimensional division algebra over a field k. Projective

D
[
x
,
y
]

D[x,y]

-modules of rank

⩾
2

\geqslant 2

are free. Projective ideals of

D
[
x
,
y
]

D[x,y]

, D a quaternionic division algebra, which are not free, are used to construct regular quadratic

k
[
x
,
y
]

k[x,y]

-modules which are not extended from k.
Isotopy of abelian quasigroups
Authors: Das Phullendu
Published: June 2010

It is proved that every abelian quasigroup possesses a class of isomorphic principal isotopes which are commutative groups. Also it is indicated how the corresponding result for topological quasigroups can be utilised to obtain results for abelian topological quasigroups from analogous results for commutative topological groups.
An extension of an operator inequality for 𝑠-numbers
Authors: Cochran James A.
Published: June 2010

If it is assumed that the s-numbers associated with a given compact operator are ordered in the usual fashion, a basic result concerning infinite series of powers of these s-numbers can be appropriately restated so as to refer solely either to the lead terms of the series or to its tail. A simple proof, based upon an interesting auxiliary result concerning stochastic matrices, is given for this useful improvement.
Sequential continuity in dyadic spaces
Authors: Marty R. H.
Published: June 2010

A characterization of metrizability in dyadic spaces in terms of sequential continuity is given. Also, it is shown by examples that invariance under projections (composition) cannot be eliminated from the hypotheses of theorems of Mazur and Engelking concerning the continuity of sequentially continuous functions.
Union of convex Hilbert cubes
Authors: Quinn J. , Wong Raymond Y. T.
Published: June 2010

We show that the finite union of Keller cubes in a Hilbert space is homeomorphic to the Hilbert cube provided every subcollection intersects in a Hilbert cube.
Necessary and sufficient type theorem for absolute Nörlund summability of conjugate series
Authors: Kumar Arun
Published: June 2010

Sufficient conditions for

|

N
,

p
n

|

|N,{p_n}|

summability of the conjugate series of Fourier series have been obtained earlier. The author obtains here a set of necessary and sufficient conditions for the result. This improves earlier results and provides a unified version of them.
Hankel operators with discontinuous symbol
Authors: Power Stephen
Published: June 2010

Douglas’s localisation theory for Toeplitz operators is used to show that there exist certain Hankel operators with discontinuous symbols which do not lie in the

C
∗

{C^\ast }

-algebra generated by the Toeplitz operators. As a simple corollary we also see that these operators do not lie in the closed linear span of the positive Hankel operators.
Nonmetrizable hereditarily Lindelöf spaces with point-countable bases from CH
Authors: Van Douwen Eric K. , Tall Franklin D. , Weiss William A. R.
Published: June 2010

A nonmetrizable hereditarily Lindelöf space with a point-countable base is obtained by using the continuum hypothesis to construct a Lusin subspace of a countable chain condition first countable nonseparable Baire space.
Independence theories and generalized zero-one laws
Authors: Stout Lawrence Neff
Published: June 2010

In this paper an abstract characterization of the properties of independent events is given with examples from topology, probability, and Baire structures. Using this notion of independence, proofs of the Hewitt-Savage and Kolmogorov zero-one laws are given which include the probabilistic case and the topological cases considered by Oxtoby, Christensen, and K. P. S. and M. Bhaskara Rao.
The smoothness of the elastic-plastic free boundary of a twisted bar
Authors: Caffarelli Luis A. , Rivière Néstor M.
Published: June 2010

We establish the regularity of the elastic-plastic free boundary of a multiply connected twisted bar. The regularity of the free boundary is determined by the regularity of the boundary of the domain.
Undecidable existential problems for addition and divisibility in algebraic number rings. II
Authors: Lipshitz Leonard
Published: June 2010

It is shown that for all algebraic number rings, except imaginary quadratic ones, the problem of deciding existential formulas involving only addition and the divisibility predicate is equivalent to the full diophantine problem for these rings.
Is the lattice of torsion classes algebraic?
Authors: Martinez Jorge
Published: June 2010

The answer is yes, if ... This note attempts to give amplification to the above statement, while at the same time arriving at a reasonable description of this lattice. The main theorem of the paper is no doubt the assertion that the lattice of torsion classes of lattice-ordered groups is completely distributive. The proof of this theorem depends on the notion of a value selector, and should not. As a consequence of this, one obtains a (local) decomposition theorem which is canonical (in every sense of the word) and always works.
The solution by iteration of nonlinear equations in Hilbert spaces
Authors: Măruşter Şt.
Published: June 2010

The weak and strong convergence of the iterates generated by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript k plus 1 Baseline equals left-parenthesis 1 minus t Subscript k Baseline right-parenthesis x Subscript k Baseline plus t Subscript k Baseline upper T x Subscript k Baseline left-parenthesis t Subscript k Baseline element-of upper R right-parenthesis">

x

k
+
1

=
(
1
−

t
k

)

x
k

+

t
k

T

x
k

(

t
k

∈
R
)

{x_{k + 1}} = (1 - {t_k}){x_k} + {t_k}T{x_k}({t_k} \in R)

to a fixed point of the mapping

T
:
C
→
C

T:C \to C

are investigated, where C is a closed convex subset of a real Hilbert space. The basic assumptions are that T has at least one fixed point in C, and that

I
−
T

I - T

is demiclosed at 0 and satisfies a certain condition of monotony. Some applications are given.
The Thom space periodicity of classifying spaces
Authors: Sjerve Denis
Published: June 2010

If G is any topological group then there exists a classifying space

B
G

{B_G}

. In this paper we shall exhibit a fiber bundle a over

B
G

{B_G}

such that the Thom complex

B
G
ω

B_G^\omega

is homeomorphic to

B
G

{B_G}

. As an application we give a new proof of the Freudenthal Suspension Theorem.
The union of two Hilbert cubes meeting in a Hilbert cube need not be a Hilbert cube
Authors: Sher R. B.
Published: June 2010

An example is given to verify the assertion of the title.
Minimal immersions of low dimensional manifolds
Authors: Little Robert D.
Published: June 2010

We show that if M is a spin n-manifold,

n
⩽
7

n \leqslant 7

, then M immerses in

R

n
+
3

{{\mathbf {R}}^{n + 3}}

.
On ordinally closed sets
Authors: Morgan John C.
Published: June 2010

Extensions of Cantor’s Intersection Theorem and Zalcwasser’s theorem on transfinite sequences of ambiguous sets of the first Baire class are given for linear sets.
Degenerations of curves in 𝑃³
Authors: Tannenbaum Allen
Published: June 2010

In this paper we prove every connected, reduced curve in

P

3

{{\mathbf {P}}^3}

of arithmetic genus 0, may be flatly smoothed. Moreover, we give a new example of a reduced singular curve in

P

3

{{\mathbf {P}}^3}

which cannot be flatly smoothed.
Approximation by polynomials in 𝑧 and another function
Authors: Preskenis Kenneth John
Published: June 2010

We present some progress in the understanding of when and why polynomials in z and a given continuous function f are uniformly dense in all continuous complex valued functions on the closed unit disk. The first theorem requires that f be an

ACL
2

{\text {ACL}^2}

-function in a neighborhood of the disk which satisfies

Re
⁡

f

z
¯

⩾

|

f
z

|

\operatorname {Re} {f_{\bar z}} \geqslant |{f_z}|

almost everywhere and

f

−
1

(
f
(
a
)
)

{f^{ - 1}}(f(a))

is countable for each a. The second theorem requires that f have a special form and satisfy

|

f

z
¯

|

>

|

f
z

|

|{f_{\bar z}}| > |{f_z}|

everywhere except at the origin. The form is that

f
=

z
¯

k

ϕ
(

|

z

|

2
k

)

f = {\bar z^k}\phi (|z{|^{2k}})

where

ϕ
\phi

is a complex valued function of a real variable satisfying

ϕ
\phi

is continuous in [0, 1],

ϕ
′

\phi ’

exists in (0, 1) and where k is a positive integer.
Maximal and minimal ring topologies
Authors: Shell Niel
Published: June 2010

An explicit description is given of a nondiscrete ring topology on the field Q of rational numbers which is strictly finer than the locally bounded topology on Q having the ring of integers as a preorder. It is observed that either there exist nonvaluable minimal ring topologies or there exist ring topologies containing no minimal ring topologies.
Residual linearity for certain nilpotent groups
Authors: Menal P.
Published: June 2010

In this note we consider relations between residual finiteness and residual linearity for a nilpotent group G. We show, amongst other things, that if the center and the commutator subgroup of G are finitely generated and G is residually linear, then G is residually finite. Indeed the property which we use on linear groups is that linear groups satisfy the minimal condition on centralizers.
Local connectedness and pseudocompactness in completely regular spaces
Authors: Hartig Donald G.
Published: June 2010

The properties of local connectedness and pseudocompactness of a completely regular space X are characterized via algebraic properties of the space

C
(
X
)

C(X)

. These characterizations are then used to prove the (well-known) theorem that

β
X

\beta X

is locally connected if and only if X is locally connected and pseudocompact.
On the Burnside ring of a compact group
Authors: Gordon Robin A.
Published: June 2010

The Burnside ring

A
(
G
)

A(G)

of an arbitrary compact group G is defined in analogy to tom Dieck’s definition for a compact Lie group. A topology is introduced on the set of conjugacy classes of closed subgroups, and a theorem of Dress on the topology of the prime spectrum of

A
(
G
)

A(G)

for profinite G is improved and extended to this general setting.
On a necessary conditon for the Erdős-Rényi law of large numbers
Authors: Steinebach Josef
Published: June 2010

For a sequence

{

X
i

}

i
=
1
,
2
,
…

{\{ {X_i}\} _{i = 1,2, \ldots }}

of independent, identically distributed random variables with existing moment-generating function

φ
(
t
)
=
E

exp
⁡
(
t

X
i

)

\varphi (t) = E\;\exp (t{X_i})

in some nondegenerate interval, Erdös and Rényi (1970) studied the maximum

D
(
N
,
K
)

D(N,K)

of the

N
−
K
+
1

N - K + 1

sample means
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Superscript negative 1 Baseline left-parenthesis upper S Subscript n plus upper K Baseline minus upper S Subscript n Baseline right-parenthesis comma 0 less-than-or-slanted-equals n less-than-or-slanted-equals upper N minus upper K">

K

−
1

(

S

n
+
K

−

S
n

)
,

0
⩽
n
⩽
N
−
K

{K^{ - 1}}({S_{n + K}} - {S_n}),\;0 \leqslant n \leqslant N - K

, where

S
0

=
0
,

S
n

=

X
1

+
⋯
+

X
n

{S_0} = 0,\;{S_n} = {X_1} + \cdots + {X_n}

. They showed that for a certain range of numbers a there exist positive constants

C
(

a

)

C({\mathbf {a}})

such that

lim

N
→
∞

D
(
N
,
[
C
(

a

)
log
⁡
N
]
)
=

a

{\lim _{N \to \infty }}D(N,[C({\mathbf {a}})\log N]) = {\mathbf {a}}

with probability one. In the present paper it is shown that the existence of the moment-generating function is also a necessary condition, i.e. that

lim

sup

N
→
∞

D
(
N
,
[
C
log
⁡
N
]
)
=
∞

\lim {\sup _{N \to \infty }}D(N,[C\log N]) = \infty

for every positive constant C, if the moment-generating function does not exist for any positive number t.
Semisimple completely distributive lattices are Boolean algebras
Authors: Lambrou M. S.
Published: June 2010

Semisimple completely distributive lattices are Boolean algebras.
Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension
Authors: Delgado V.
Published: June 2010

The existence of global solutions is proved for the Maxwell-Dirac equations, for the Thirring model (Dirac equation with vector self-interaction), for the Klein-Gordon-Dirac equations and for two Dirac equations coupled through a vector-vector interaction (Federbusch model) in one space dimension. The proof is based on charge conservation, and depends on an “a priori” estimate of

‖

‖

∞

{\left \| {} \right \|_\infty }

for the Dirac field. This estimate is obtained only on the basis of algebraical properties of the nonlinear term, and allows us to simplify the proofs of global existence. We obtain it by computing

∂
0

j
1

+

∂
1

j
0

{\partial _0}{j^1} + {\partial _1}{j^0}

, with

∂
0

j
0

+

∂
1

j
1

=
0

{\partial _0}{j^0} + {\partial _1}{j^1} = 0

being the continuity equation which expresses charge conservation. We also prove global existence for the Thirring and Federbusch models coupled in standard form with the electromagnetic field.
Bounded sets in inductive limits
Authors: Kučera J. , McKennon K.
Published: June 2010

The Dieudonné-Schwartz theorem for bounded sets in strict inductive limits does not hold for general inductive limits. A set B bounded in an inductive limit

E
=

ind

lim

E
n

E = {\operatorname {ind}}\;\lim {E_n}

of locally convex spaces may not be contained in any

E
n

{E_n}

. If, however, each

E
n

{E_n}

is closed in E, then B is contained in some

E
n

{E_n}

, but may not be bounded there.
Local contractions in metric spaces
Authors: Hu Thakyin , Kirk W. A.
Published: June 2010

It is shown that a theorem of E. Rakotch for locally contractive mappings can be deduced from Banach’s contraction mapping theorem, and a counterexample to an assertion of R. D. Holmes concerning local radial contractions is given.
There are no uniquely homogeneous spaces
Authors: Barit William , Renaud Peter
Published: June 2010

One could say a continuum is uniquely homogeneous if for each pair of points there is a unique homeomorphism taking the one point to the other. Ungar showed that such spaces are topological groups with no automorphisms. This note shows there are no such nontrivial groups.
Multiplication rules for polynomials
Authors: Nathanson Melvyn B.
Published: June 2010

It is proved that the polynomial solutions of the functional equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis z right-parenthesis upper F left-parenthesis z plus 1 slash StartRoot a EndRoot right-parenthesis equals upper F left-parenthesis StartRoot a EndRoot z squared plus left-parenthesis b slash StartRoot a EndRoot plus 1 right-parenthesis z plus c slash StartRoot a EndRoot right-parenthesis">

F
(
z
)
F
(

z
+
1

/

a

)
=
F
(

a

z
2

+
(
b

/

a

+
1
)
z
+
c

/

a

)

F(z)F( {z + 1/\sqrt a } ) = F( {\sqrt a {z^2} + (b/\sqrt a + 1)z + c/\sqrt a } )

\]
are precisely

(
a

z
2

+
b
z
+
c

)
n

{(a{z^2} + bz + c)^n}

if

b
2

−
4
a
c
≠
0

{b^2} - 4ac \ne 0

and

(

a

z
+
b

/

2

a

)
n

{(\sqrt a z + b/2\sqrt a )^n}

if

b
2

−
4
a
c
=
0

{b^2} - 4ac = 0

.
One relator groups having a finitely presented normal subgroup
Authors: Karrass A. , Solitar D.
Published: June 2010

A classification is given for one-relator groups having a finitely presented normal subgroup of infinite index.
A short proof of a Greene theorem
Authors: Wang Chung Lie
Published: June 2010

A short and simple proof of an inequality of the Gronwall type is given for a class of integral systems based upon the generalized Gronwall lemma of Sansone-Conti.
Counting eigenvalues for automorphisms of Riemann surfaces
Authors: Lawrence John , Zorzitto Frank
Published: June 2010

Let n be a prime, A, B, C disjoint sets,

f
:

Z
n

→
A
∪
B
∪
C

f:{Z_n} \to A \cup B \cup C

be such that

f
(

x
¯

)
∈
A

f(\bar x) \in A

iff

f
(
−

x
¯

)
∈
C

f( - \bar x) \in C

and

f
(

x
¯

+

y
¯

)
∉
C

f(\bar x + \bar y) \notin C

whenever

f
(

x
¯

)
,
f
(

y
¯

)
∈
A

f(\bar x),f(\bar y) \in A

. Then the cardinality of

f

−
1

[
B
]

{f^{ - 1}}[B]

tends to infinity with n. Using this, certain eigenvalues for automorphisms of the Riemann surfaces defined by the equation

y
n

=

x

m
1

(
x
−
1

)

m
2

(
x
−
z

)

m
3

{y^n} = {x^{{m_1}}}{(x - 1)^{{m_2}}}{(x - z)^{{m_3}}}

are counted.
On the boundary value problem 𝑢”+𝑢=𝛼𝑢⁻+𝑝(𝑡), 𝑢(0)=0=𝑢(𝜋)
Authors: Aguinaldo L. , Schmitt K.
Published: June 2010

In this note we consider the boundary value problem

u
+
u
=
a

u
−

+
p
(
t
)
,

u
(
0
)
=
0
=
u
(
π
)
,

α
>
0

u + u = a{u^ - } + p(t),\;u(0) = 0 = u(\pi ),\;\alpha > 0

, and show that a necessary and sufficient condition for the problem to be solvable is that

∫
0
π

p
(
s
)
sin
⁡
s

d
s

⩽
0

\smallint _0^\pi {p(s)\sin s\;ds\; \leqslant 0}

. We thus answer in the affirmative a question posed by S. Fučik.
A restriction theorem for space curves
Authors: Prestini E.
Published: June 2010

Let

α
\alpha

be a

C
k

{C^k}

curve

(
k
⩾
3
)

(k \geqslant 3)

in

R
3

{R^3}

with nonvanishing curvature and torsion. It is proved that the restriction operator

T
:
f
→

f
^

|

α

T:f \to \hat f{|_\alpha }

is bounded from

L
p

(

R
3

)

{L^p}({R^3})

to

L
q

(
α
)

{L^q}(\alpha )

if

1
⩽
p
>
15

/

13

1 \leqslant p > 15/13

and

1

/

q
>
6
(
1
−
1

/

p
)

1/q > 6(1 - 1/p)

, and that T is not bounded if

p
⩾
6

/

5

p \geqslant 6/5

or

1

/

q
>
6
(
1
−
1

/

p
)

1/q > 6(1 - 1/p)

.
A note on Volterra equations in a Hilbert space
Authors: Londen Stig-Olof , Staffans Olof J.
Published: June 2010

An existence and uniqueness result is given on the nonlinear Volterra equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis t right-parenthesis plus left-parenthesis a Superscript asterisk Baseline upper A u right-parenthesis left-parenthesis t right-parenthesis small-contains-as-member f left-parenthesis t right-parenthesis comma t element-of upper R Superscript plus Baseline comma">

u
(
t
)
+
(

a
∗

A
u
)
(
t
)
∍
f
(
t
)
,

t
∈

R
+

,

u(t) + (a ^\ast Au)(t) \backepsilon f(t),\quad t \in {R^ + },

\]
when considered in a real Hilbert space. The result extends earlier work by Barbu and Londen.
Stone’s theorem and spectral subspaces of automorphisms
Authors: Størmer Erling
Published: June 2010

It is shown that spectral subspaces of automorphisms of a von Neumann algebra can be defined by use of Stone’s theorem on unitary representations.
Maximal connected expansions of the reals
Authors: Guthrie J. A. , Stone H. E. , Wage M. L.
Published: June 2010

The question of whether there exist nontrivial maximal connected Hausdorff spaces is settled in the affirmative by showing that there is a maximal connected topology for the reals which is finer than the Euclidean topology.
Cardinal functions for 𝑘-spaces
Authors: Boone James R. , Davis Sheldon W. , Gruenhage Gary
Published: June 2010

In this paper, four cardinal functions are defined on the class of k-spaces. Some of the relationships between these cardinal functions are studied. Characterizations of various k-spaces are presented in terms of the existence of these cardinal functions. A bound for the ordinal invariant

κ
\kappa

of Arhangelskii and Franklin is established in terms of the tightness of the space. Examples are presented which exhibit the interaction between these cardinal invariants and the ordinal invariants of Arhangelskii and Franklin.
𝑝-points in iterated forcing extensions
Authors: Roitman Judy
Published: June 2010

Selective ultrafilters exist in direct iterated ccc extensions whose length has uncountable cofinality, as do p-points which are not selective. Nonselective p-points also exist e.g. in an iterated Laver or Mathias extension of length

ω
2

{\omega _2}

over a model of CH.
A factorization theorem for groups and Lie algebras
Authors: Schenkman Eugene
Published: June 2010

A proof is given for a generalization of a theorem of Mennicke that each member of a certain family of groups defined by generators and relations is finite. This leads to the following theorem on factorization of groups. Theorem. Let G be generated by abelian subgroups A, B, C, such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper A comma upper B right-bracket less-than-or-slanted-equals upper A comma left-bracket upper B comma upper C right-bracket less-than-or-slanted-equals upper B comma left-bracket upper C comma upper A right-bracket less-than-or-slanted-equals upper C">

[
A
,
B
]
⩽
A
,
[
B
,
C
]
⩽
B
,
[
C
,
A
]
⩽
C

[A,B] \leqslant A,[B,C] \leqslant B,[C,A] \leqslant C

; then the second derived group,

G
G

is nilpotent of class at most 3. Also proved is the analogue of the above theorem for Lie algebras.
A moment problem on Jordan domains
Authors: Sakai Makoto
Published: June 2010

Let

D
1

,

D
2

{D_1},{D_2}

be Jordan domains on the complex z-plane such that

∫

D
1

z
n

d
m
=

∫

D
2

z
n

d
m

{\smallint _{{D_1}}}{z^n}dm = {\smallint _{{D_2}}}{z^n}dm

for every nonnegative integer n. Here m denotes two-dimensional Lebesgue measure. Does it follow that

D
1

=

D
2

{D_1} = {D_2}

? This moment problem on Jordan domains was posed by H. S. Shapiro [2, p. 193, Problem 1]. In this paper we construct a counterexample and study conditions on

D
1

{D_1}

and

D
2

{D_2}

which imply that the above equality does not hold for some n.
On projective prime ideals in 𝐶(𝑋)
Authors: Brookshear J. Glenn
Published: June 2010

This note presents characterizations of the projective prime ideals in

C
(
X
)

C(X)

and of the hereditary and semihereditary rings of continuous functions.
Quasi-affine surfaces with 𝐺ₐ-actions
Authors: Fauntleroy Amassa , Magid Andy R.
Published: June 2010

A normal quasi-affine surface over an algebraically closed field of characteristic zero which has a

G
a

{G_a}

-action is shown to have a geometric quotient if and only if the action is without fixed points. If the surface is factorial affine, and the action without fixed points, the surface is the product of a curve and

G
a

{G_a}

.
Zones of uniform decomposition in tensor products
Authors: Feingold Alex Jay
Published: June 2010

Let

V
λ

{V_\lambda }

be a finite dimensional irreducible module for a complex semisimple Lie algebra. It is shown that the decomposition of tensor products

V
λ

⊗

V
τ

{V_\lambda } \otimes {V_\tau }

for all dominant integral weights

τ
\tau

may be derived from those for a finite set of such

τ
\tau

. An explicit choice of such a finite set (depending on

λ
\lambda

) is given.
Interchanging iterated integration
Authors: Lessner Lawrence
Published: June 2010

If

k
(
x
,
y
)

k(x,y)

is a measurable, real valued, finite a.e. function on

X
×
Y

X \times Y

, then necessary and sufficient conditions are given for the two iterated Lebesgue integrals of

k
(
x
,
y
)

k(x,y)

to be equal and finite by employing Saks’ theorem on the convergence of a sequence of finite measures and the Vitali convergence theorem. The conditions, more general than those of either Fubini’s or Tonelli’s theorems in this case, are applied to an example of a nonintegrable function to show that its iterated integrals are in fact equal and finite.
𝑡𝑎𝑛𝑥 is ergodic
Authors: Schweiger F.
Published: June 2010

It is proved that the transformation

x
↦
tan
⁡
x

x \mapsto \tan x

is ergodic on the real line with respect to Lebesgue measure.
Inner amenability and conjugation operators
Authors: Paschke William L.
Published: June 2010

It is shown that an infinite discrete group G is inner amenable if and only if the

C
∗

{C^ \ast }

-algebra generated by the unitaries on

l
2

(
G
)

{l^2}(G)

corresponding to conjugation by s

(
s
∈
G
)

(s \in G)

does not contain the projection on the point-mass at the identity.
The isoperimetric theorem for curves on minimal surfaces
Authors: Chakerian G. D.
Published: June 2010

A short proof is given for a sharpened form of the isoperimetric inequality for curves on minimal surfaces.
𝐾(𝑍/2) as a Thom spectrum
Authors: Priddy Stewart
Published: June 2010

Dyer-Lashof operations are used to give a simple proof that

K
(

Z

/

2
)

K({\mathbf {Z}}/2)

is a Thom spectrum.
Remark on the zero locus of a smooth function
Authors: Burns D.
Published: June 2010

A sufficient condition is given to insure the existence of some smooth points in the zero set of a

C
∞

{C^\infty }

function.
Avoiding self-referential statements
Authors: Smoryński C.
Published: June 2010

Recursion-theoretic proofs of metamathematical results tend to rely on a pair of effectively inseparable r.e. sets and its properties. We establish a special property for a small configuration of such pairs and derive from it some metamathematical results not previously accessible to recursion-theoretic techniques.
Maximal subgroups of prime index in a finite solvable group
Authors: Venzke Paul
Published: June 2010

In this note we show that a maximal subgroup of a finite solvable group has prime index if and only if it admits a cyclic supplement which permutes with one of its Sylow systems. In particular, a finite solvable group is supersolvable if and only if each maximal subgroup admits a cyclic supplement which permutes with a Sylow system of the maximal subgroup.
Equiconvergence of matrix transformations
Authors: Jukes K. A.
Published: June 2010

Equiconvergence of matrix transformations is related to the existence of Tauberian constants. Agnew’s result on the equiconvergence of Cesàro and Riesz means is shown to be best possible. Finally, equiconvergence of equivalent arithmetical summation methods related to the prime number theorem is investigated.
A generalization of von Neumann’s inequality to the complex ball
Authors: Drury S. W.
Published: June 2010

A necessary and sufficient condition is found for a polynomial Q of J variables to be such that

Q
(

A
1

,
…
,

A
J

)

Q({A_1}, \ldots ,{A_J})

is a contraction whenever

A
j

(
1
⩽
j
⩽
J
)

{A_j}(1 \leqslant j \leqslant J)

are commuting linear operators on complex hilbert space satisfying

Σ

j
=
1

J

A
j
∗

A
j

⩽
I

\Sigma _{j = 1}^JA_j^ \ast {A_j} \leqslant I

.
An extremal problem for quasiconformal mappings in an annulus
Authors: White Alvin M.
Published: June 2010

The following extremal problem is solved. We consider a family of continuously differentiable univalent quasiconformal mappings

w
=
f
(
z
)

w = f(z)

of the annulus

r
>

|

z

|

>
1

r > |z| > 1

onto the unit disk minus some continuum containing the origin. For a point b on a fixed circle, maximize

|

f
(
b
)

|

|f(b)|

within the family. The problem is solved by using a variational method due to Schiffer. The extremal function and the maximum are found in terms of the Weierstrass

℘
\wp

-function and the elliptic modular function.
A necessary and sufficient condition for Bloch functions
Authors: Timoney Richard M.
Published: June 2010

A necessary and sufficient condition is given for a subset

E
⊆

C

E \subseteq {\mathbf {C}}

to satisfy
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper S u p left-brace right-brace vertical-bar times times vertical-bar vertical-bar of ff prime left-parenthesis right-parenthesis z left-parenthesis right-parenthesis minus minus 1 vertical-bar vertical-bar z 2 vertical-bar element-of element-of z of ff minus minus 1 left-parenthesis right-parenthesis upper E greater-than normal infinity 2nd Row right double arrow upper S u p left-brace right-brace vertical-bar times times vertical-bar vertical-bar of ff prime left-parenthesis right-parenthesis z left-parenthesis right-parenthesis minus minus 1 vertical-bar vertical-bar z 2 vertical-bar element-of element-of zD greater-than normal infinity EndLayout">

Sup
⁡
{

|

f
′

(
z
)

|

(
1
−

|

z

|

2

)

|

z
∈

f

−
1

(
E
)
}
>
∞

⇒
Sup
⁡
{

|

f
′

(
z
)

|

(
1
−

|

z

|

2

)

|

z
∈
D
}
>
∞

\begin {array}{*{20}{c}} {\operatorname {Sup}\{ |f’(z)|(1 - |z{|^2})|z \in {f^{ - 1}}(E)\} > \infty } \\ { \Rightarrow \operatorname {Sup}\{ |f’(z)|(1 - |z{|^2})|z \in D\} > \infty } \\ \end {array}

\]
when

f
:
D
→

C

f:D \to {\mathbf {C}}

is analytic. The condition is that the complement of E should not contain large discs.
Spherically symmetric entire solutions of Δ^{𝑝}𝑢=𝑓(𝑢)
Authors: Rhee H.
Published: June 2010

A sufficient condition is given for existence of spherically symmetric entire solutions of the equation

Δ
p

u
=
f
(
u
)
,
p
⩾
2

{\Delta ^p}u = f(u),p \geqslant 2

.
Double commutants of 𝐶_{⋅0} contractions
Authors: Uchiyama Mitsuru
Published: June 2010

D. Sarason has shown that an operator in the double commutant of a contraction T of class

C
0

(
1
)

{C_0}(1)

is interpolated by a function in

H
∞

{H^\infty }

, that is,

{
T

}
′

=
{
ϕ
(
T
)
;
ϕ
∈

H
∞

}

\{ T\} ’ = \{ \phi (T);\phi \in {H^\infty }\}

, [3]. Generally, in [4] Sz.-Nagy and C. Foiaş have shown that an operator in the double commutant of a contraction T of class

C
0

(
n
)

{C_0}(n)

is interpolated by a function in

N
T

{N_T}

, that is,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet upper T EndSet equals StartSet phi 1 left-parenthesis upper T right-parenthesis Superscript negative 1 Baseline phi 2 left-parenthesis upper T right-parenthesis colon phi 2 slash phi 1 element-of upper N Subscript upper T Baseline EndSet">

{
T
}
=
{

ϕ
1

(
T

)

−
1

ϕ
2

(
T
)
:

ϕ
2

/

ϕ
1

∈

N
T

}

\{ T\} = \{ {\phi _1}{(T)^{ - 1}}{\phi _2}(T):{\phi _2}/{\phi _1} \in {N_T}\}

. In this note we shall show that an operator in the double commutant of a

C

⋅
0

{C_{ \cdot 0}}

contraction T with finite defect indices

δ

T
∗

>

δ
T

{\delta _{{T^ \ast }}} > {\delta _T}

is interpolated by a function in

H
∞

{H^\infty }

.
On the Alexander polynomials of certain three-component links
Authors: Kidwell Mark E.
Published: June 2010

Let L be a three-component link all of whose linking numbers are zero. Write the Alexander polynomial of L as
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta left-parenthesis x comma y comma z right-parenthesis equals left-parenthesis 1 minus x right-parenthesis left-parenthesis 1 minus y right-parenthesis left-parenthesis 1 minus z right-parenthesis f left-parenthesis x comma y comma z right-parenthesis">

Δ
(
x
,
y
,
z
)
=
(
1
−
x
)
(
1
−
y
)
(
1
−
z
)
f
(
x
,
y
,
z
)

\Delta (x,y,z) = (1 - x)(1 - y)(1 - z)f(x,y,z)

. Then the integer

|

f
(
1
,
1
,
1
)

|

|f(1,1,1)|

is a perfect square.
A maximum principle for compressible flow on a surface
Authors: Sibner Lesley M. , Sibner Robert J.
Published: June 2010

We show that the speed of a steady, irrotational, subsonic flow on a surface cannot attain its maximum at a point of positive Gauss curvature.
An example about normalizers in mapping class groups
Authors: Gilman Jane
Published: June 2010

J. Birman has raised the question as to whether every element of the mapping class group (the Teichmüller modular group) is in the normalizer of some element of finite order. In this paper elements of the mapping class group which are not in the normalizer of any element of finite order are constructed.
Entire functions with almost radially distributed values
Authors: Kimura Shigeru
Published: June 2010

Let

f
(
z
)

f(z)

be an entire function of finite lower order. Assume that there exist a positive number h and an unbounded sequence

{

w
n

}

n
=
1

∞

\{ {w_n}\} _{n = 1}^\infty

such that all roots of the equations

f
(
z
)
=

w
n

(
n
=
1
,
2
,
…
)

f(z) = {w_n}(n = 1,2, \ldots )

lie in

{
z
;

|

Im
⁡
z

|

>
h
}

\{ z;|\operatorname {Im} z| > h\}

. Then

f
(
z
)

f(z)

is a polynomial of degree not greater than two. The hypothesis of the finiteness of lower order of

f
(
z
)

f(z)

cannot be removed.
Sur le 17ème problème de Hilbert pour les fonctions de Nash
Authors: Bochnak Jacek
Published: June 2010

The purpose of this note is to give a more refined version of a theorem of Efroymson: If

U
⊂

R

n

U \subset {{\mathbf {R}}^n}

is defined by polynomial inequalities of the form

f
i

>
0
,
i
=
1
,
…
,
p

{f_i} > 0,i = 1, \ldots ,p

, and if g is a positive definite Nash function on U, then g is a finite sum of squares of Nash meromorphe functions on U.
Jacobi’s generating function for Jacobi polynomials
Authors: Askey Richard
Published: June 2010

An idea of Hermite is used to give a simple proof of Jacobi’s generating function for Jacobi polynomials.
A resolution of the Euler operator. I
Authors: Olver Peter J. , Shakiban Chehrzad
Published: June 2010

An exact sequence resolving the Elder operator of the calculus of variations for polynomial differential equations in one independent and one dependent variable is described. This resolution provides readily verifiable necessary and sufficient conditions for such a polynomial differential equation to be the Euler equation for some Lagrangian. An explicit construction of the Lagrangian is given.
Lie triple derivations of von Neumann algebras
Authors: Miers C. Robert
Published: June 2010

A Lie triple derivation of an associative algebra M is a linear map

L
:
M
→
M

L:M \to M

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-bracket left-bracket upper X comma upper Y right-bracket comma upper Z right-bracket equals left-bracket left-bracket upper L left-parenthesis upper X right-parenthesis comma upper Y right-bracket comma upper Z right-bracket plus left-bracket left-bracket upper X comma upper L left-parenthesis upper Y right-parenthesis right-bracket comma upper Z right-bracket plus left-bracket left-bracket upper X comma upper Y right-bracket comma upper L left-parenthesis upper Z right-parenthesis right-bracket">

L
[
[
X
,
Y
]
,
Z
]
=
[

[
L
(
X
)
,
Y
]
,
Z

]
+
[

[
X
,
L
(
Y
)
]
,
Z

]
+
[

[
X
,
Y
]
,
L
(
Z
)

]

L[[X,Y],Z] = [ {[L(X),Y],Z} ] + [ {[X,L(Y)],Z} ] + [ {[X,Y],L(Z)} ]

\]
for all

X
,
Y
,
Z
∈
M

X,Y,Z \in M

. (Here

[
X
,
Y
]
=
X
Y
−
Y
X

[X,Y] = XY - YX

and [M, M] is the linear subspace of M generated by such terms.) We show that if M is a von Neumann algebra with no central abelian summands then there exists an operator

A
∈
M

A \in M

such that

L
(
X
)
=
[
A
,
X
]
+
λ
(
X
)

L(X) = [A,X] + \lambda (X)

where

λ
:
M
→

Z
M

\lambda :M \to {Z_M}

is a linear map which annihilates brackets of operators in M.
Composition operator on 𝑙^{𝑝} and its adjoint
Authors: Singh R. K. , Komal B. S.
Published: June 2010

A necessary and sufficient condition for the invertibility of a composition operator

C
ϕ

{C_\phi }

on

l
p

{l^p}

is reported in this paper. The adjoint of

C
ϕ

{C_\phi }

is computed in the case

p
=
2

p = 2

. The necessary and sufficient conditions for unitary operators and co-isometries to be composition operators are also investigated. A study of invariant subspaces and reducing subspaces of

C
ϕ

{C_\phi }

is also made.
The canonical form of a scalar operator on a Banach space
Authors: Faulkner G. D. , Huneycutt J. E.
Published: June 2010

Let

A
=
∫
λ
d
E
(
λ
)

A = \smallint \lambda dE(\lambda )

be a scalar operator on a Banach space X. If there exists a vector

g
∈
X

g \in X

such that the closed convex hull of the range of the vector measure

μ
(
⋅
)
=
E
(
⋅
)
g

\mu ( \cdot ) = E( \cdot )g

has nonvoid interior, then A is similar to the operator

Q
f
(
λ
)
=
λ
f
(
λ
)

Qf(\lambda ) = \lambda f(\lambda )

on a quotient space of a suitably constructed

L

∞

{\mathcal {L}^\infty }

space.
Generalizations of Temple’s inequality
Authors: Harrell Evans M.
Published: June 2010

T. Kato’s little-known generalization of a classic variational inequality for eigenvalues is extended to the case of normal operators and briefly discussed.
On fixed points of nonexpansive set-valued mappings
Authors: Bose R. K. , Mukherjee R. N.
Published: June 2010

A theorem is proved concerning the existence of fixed points of nonexpansive set-valued mappings. This generalizes a result of Dotson, Jr. [1].
An implicit function theorem without differentiability
Authors: Warga J.
Published: June 2010

We combine a “global” version of the classical inverse function theorem with Schauder’s fixed point theorem to investigate the existence and continuity properties of a function

(
F
,
x
)
→
η
(
F
,
x
)

(F,x) \to \eta (F,x)

such that

η
(
F
,
x
)
=
F
(
η
(
F
,
x
)
,
x
)

\eta (F,x) = F(\eta (F,x),x)

.
The additive structure of models of arithmetic
Authors: Lipshitz Leonard , Nadel Mark
Published: June 2010

It is shown that for a model of Presburger arithmetic to have an expansion to a model of Peano arithmetic it is necessary that the model be recursively saturated. For countable models this condition is also sufficient; for uncountable models it is not.
Growth rates for monotone subsequences
Authors: del Junco A. , Steele J. Michael
Published: June 2010

The growth rate of the largest monotone subsequence of a uniformly distributed sequence is obtained. For

a
n

=
n
α
mod

1

{a_n} = n\alpha \bmod \; 1

with

α
\alpha

algebraic irrational the exponent of growth is found to be precisely the same as for a random sequence.
𝛼-recognizable semigroups
Authors: Rankin S. A. , Reis C. M. , Thierrin G.
Published: June 2010

Semigroups in which every principal congruence is of finite index are studied in this paper.
Normal and quasinormal composition operators
Authors: Whitley Robert
Published: June 2010

A bounded linear operator

C
T

{C_T}

on

L
2

(
X
,
Σ
,
m
)

{L^2}(X,\Sigma ,m)

is a composition operator if it is induced by a point mapping

T
:
X
→
X

T:X \to X

via

C
T

f
=
f
∘
T

{C_T}f = f \circ T

. Normal and quasinormal composition operators on a finite measure space are characterized:

C
T

{C_T}

is normal iff T is measure preserving and

T

−
1

(
Σ
)

{T^{ - 1}}(\Sigma )

is (essentially) all of

Σ
;

C
T

\Sigma ;{C_T}

is quasinormal iff T is measure preserving.
Characterizations of amenable Banach algebras
Authors: Lau Anthony To-Ming
Published: June 2010

In this paper we show that a Banach algebra A is amenable if and only if A has any one of the following properties: (i) whenever X is a Banach A-bimodule and Y is an A-submodule of X, then for each

f
∈

Y
∗

f \in {Y^ \ast }

such that

a
⋅
f
=
f
⋅
a
,
a
∈
A

a \cdot f = f \cdot a,a \in A

, there exists

f
~

∈

X
∗

\tilde f \in {X^ \ast }

which extends f and

a
⋅

f
~

=

f
~

⋅
a

a \cdot \tilde f = \tilde f \cdot a

for all

a
∈
A

a \in A

; (ii) whenever X is a Banach A-bimodule, there exists a bounded projection P from

X
∗

{X^ \ast }

onto {

f
∈

X
∗

;
a
⋅
f
=
f
⋅
a

f \in {X^ \ast };a \cdot f = f \cdot a

for all

a
∈
A

a \in A

} such that

T
⋅
P
=
P
⋅
T

T \cdot P = P \cdot T

for any

w
e
a
k

∗

\mathrm {weak}^*

continuous bounded linear operator T from

X
∗

{X^ \ast }

into

X
∗

{X^\ast }

commuting with the action of A on

X
∗

{X^\ast }

. The class of ultraweakly amenable von Neumann algebras with separable predual can be similarly characterized.
On a discreteness condition of the spectrum of Schrödinger operators with unbounded potential from below
Authors: Benci V. , Fortunato D.
Published: June 2010

We obtain a discreteness condition for the spectrum of the Schrödinger operator

−
Δ
+
V
(
x
)

- \Delta + V(x)

in a case in which V is not bounded from below.
The size of the set of left invariant means on an ELA semigroup
Authors: Paterson Alan L. T.
Published: June 2010

Let S be an ELA semigroup and let

m

(
S
)

\mathfrak {m}(S)

be the smallest possible cardinality of the set

{
s
∈
S
:
F
s
=
{
s
}
}

\{ s \in S:Fs = \{ s\} \}

as F ranges over the finite subsets of S. The main purpose of this note is to show that if

m

(
S
)

\mathfrak {m}(S)

is infinite, then S has exactly

2

2

m

(
S
)

{2^{{2^{\mathfrak {m}(S)}}}}

(multiplicative) left invariant means.
On multipliers of Segal algebras
Authors: Dutta M. , Tewari U. B.
Published: June 2010

Let T be a multiplier of a Segal algebra S on a locally compact abelian group G. We prove that

T
2

(
S
)

{T^2}(S)

is closed if and only if T is a product of an idempotent and an invertible multiplier. We also show that the techniques developed in the proof of this theorem can be used to obtain some other known results.