bluuebunny/crossref_metadata_2025 huggingface dataset:
The fixed point property for homeomorphisms of 1-arcwise connected continua
Authors: Mohler Lee
Published: June 2010

It is shown that continua which are arcwise connected and contain no simple closed curves have the fixed point property for homeomorphisms, answering in the affirmative a question of Bing. The proof uses measure theoretic techniques. Given a homeomorphism

h
h

of a compact metric space

X
X

onto itself, a probability measure is constructed on

X
X

which is invariant under

h
h

.
Support properties of Gaussian processes over Schwartz space
Authors: Piech M. Ann
Published: June 2010

We utilize the concept of an abstract Wiener space to prove a converse to a theorem of Minlos, thereby obtaining necessary and sufficient conditions for a Hilbert subspace of

S

′

(

R

d

)

\mathcal {S}’({{\text {R}}^d})

to support a given Gaussian process over

S

(

R

d

)

\mathcal {S}({{\text {R}}^d})

.
On the number of generators of powers of an ideal
Authors: Sally Judith D.
Published: June 2010

Let

I
I

be an ideal of a quasi-local ring. In this note we consider the question of how small—in terms of numbers of generators—the powers of the ideal

I
I

can be.
Duality theorems and theorems of the alternative
Authors: McLinden L.
Published: June 2010

It is shown, in a completely general setting, that a theorem of the alternative is logically equivalent to a duality theorem linking two constrained optimization problems.
Random shifts which preserve measure
Authors: Geman Donald , Horowitz Joseph
Published: June 2010

Given a flow

θ
g

,
g
∈
G

{\theta _g},g \in G

a group, over a probability space

(
Ω
,

F

,
P
)

(\Omega ,\mathfrak {F},P)

and a

G
G

-valued random variable

Z
Z

, we exhibit the Lebesgue decomposition of the measure

P
∘

θ
Z

−
1

P \circ \theta _Z^{ - 1}

relative to

P
P

, and give necessary and sufficient conditions for equality

(
P
∘

θ
Z

−
1

=
P
)

(P \circ \theta _Z^{ - 1} = P)

, absolute continuity

(
P
∘

θ
Z

−
1

≪
P
)

(P \circ \theta _Z^{ - 1} \ll P)

, and singularity

(
P
∘

θ
Z

−
1

⊥
P
)

(P \circ \theta _Z^{ - 1} \bot P)

in terms of the Haar measure. The proof rests on the theory of “Palm measures” as developed by Mecke and the authors. Specializing the group

G
G

, we retrieve some known results for the integers and real line, and compute the Radon-Nikodým derivatives in various cases.
Restrictions of analytic functions. III
Authors: Rosenblum Marvin , Rovnyak James
Published: June 2010

Characterizations are given of all real and all pure imaginary functions on a real Borel set which occur as restrictions of boundary functions of functions holomorphic and having positive imaginary part in the upper half-plane.
Classifying maps in fiberings of homogeneous bounded domains
Authors: Kaneyuki Soji
Published: June 2010

A homogeneous bounded domain in

C

n

{{\text {C}}^n}

admits a structure of a fiber space whose base and fibers are homogeneous bounded domains. For such a fibering there exists a universal fiber space with the same fibers, which plays an analogous role as a universal bundle does in topology. A sufficient condition is given in order that the canonical map of the base of the fibering into the classifying domain is injective. Some applications of it are also given.
Semigroups of multipliers associated with semigroups of operators
Authors: Olubummo A.
Published: June 2010

Let

G
G

be an infinite compact group with dual object

Σ
\Sigma

. Corresponding to each semigroup

T

=
{
T
(
ξ
)
;
ξ
≥
0
}

\mathcal {T} = \{ T(\xi );\xi \geq 0\}

of operators on

L
p

(
G
)
,
1
≤
p
>
∞

{L_p}(G),1 \leq p > \infty

, which commutes with right translations, there is a semigroup

E

=
{

E
ξ

(
σ
)
;
ξ
≥
0
,
σ
∈
Σ
}

\mathcal {E} = \{ {E_\xi }(\sigma );\xi \geq 0,\sigma \in \Sigma \}

of

L
p

(
G
)

{L_p}(G)

multipliers. If

T

\mathcal {T}

is strongly continuous, then

{

E
ξ

(
σ
)
;
ξ
≥
0
}

\{ {E_\xi }(\sigma );\xi \geq 0\}

is uniformly continuous for each

σ
\sigma

. Conversely a semigroup

E

\mathcal {E}

of

L
p

(
G
)

{L_p}(G)

-multipliers determines a semigroup

T

\mathcal {T}

of operators on

L
p

(
G
)

{L_p}(G)

, is strongly continuous if each

E
ξ

(
σ
)

{E_\xi }(\sigma )

is uniformly continuous; and then there exist a function

A
A

on

Σ
\Sigma

and

Σ
0

⊂
Σ

{\Sigma _0} \subset \Sigma

such that

E
ξ

(
σ
)
=

E
0

(
σ
)
exp
⁡
(
ξ

A
σ

)

{E_\xi }(\sigma ) = {E_0}(\sigma )\exp (\xi {A_\sigma })

if

σ
∈

Σ
0

\sigma \in {\Sigma _0}

and

E
ξ

(
σ
)
=
0

{E_\xi }(\sigma ) = 0

if

σ
∉

Σ
0

\sigma \notin {\Sigma _0}

.
Another view of the Weierstrass theorem
Authors: Preskenis Kenneth John
Published: June 2010

We present two theorems which conclude that polynomials in

z
z

and a given continuous function

f
f

are dense in all continuous complex valued functions on the closed unit disk. The first theorem requires that

f
f

be differentiable and satisfy

Re
⁡

f

z
¯

⩾

|

f
z

|

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

in the open disk and also that

f

−
1

(
f
(
a
)
)

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

be countable for each

a
a

in

D
D

. The second theorem requires that

f
f

be a class

C
1

{C^1}

-function in a neighborhood of the disk satisfying

|

f

z
¯

|

>

|

f
z

|

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

almost everywhere and

Re
⁡

f

z
¯

⩾

|

f
z

|

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

everywhere inside the disk.
Isometry groups of manifolds of negative curvature
Authors: Byers W.
Published: June 2010

Solvable subgroups of the isometry groups of a simply-connected manifold of negative curvature are characterized and this characterization is used to show that the isometry group of the universal Riemannian covering of a compact manifold of negative curvature is either discrete or semisimple.
A geometric property of certain plane sets
Authors: Pietz Kenneth
Published: June 2010

Suppose

K
K

is a compact subset of the plane of the form

Δ
(
0
,
1
)

¯

∖

∪

n
=
1

∞

Δ
(

p
n

,

r
n

)

\overline {\Delta (0,1)} \backslash \cup _{n = 1}^\infty \Delta ({p_n},{r_n})

where

Δ
(

p
n

,

r
n

)

¯

⊆
Δ
(
0
,
1
)

\overline {\Delta ({p_n},{r_n})} \subseteq \Delta (0,1)

for each

n
n

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal upper Delta left-parenthesis p Subscript i Baseline comma r Subscript i Baseline right-parenthesis With bar intersection ModifyingAbove normal upper Delta left-parenthesis p Subscript j Baseline comma r Subscript j Baseline right-parenthesis With bar equals normal empty-set">

Δ
(

p
i

,

r
i

)

¯

∩

Δ
(

p
j

,

r
j

)

¯

=
∅

\overline {\Delta ({p_i},{r_i})} \cap \overline {\Delta ({p_j},{r_j})} = \emptyset

for

i
≠
j

i \ne j

. Let

α
=

sup

i
⩾
1

(
(

r
i

+
1
)

/

r
i

)

\alpha = {\sup _{i \geqslant 1}}(({r_i} + 1)/{r_i})

and define the sets
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript asterisk Baseline upper K identical-to partial-differential normal upper Delta left-parenthesis 0 comma 1 right-parenthesis union left-bracket union partial-differential normal upper Delta left-parenthesis p Subscript n Baseline comma r Subscript n Baseline right-parenthesis right-bracket">

∂

∗

K
≡
∂
Δ
(
0
,
1
)
∪
[

∪

n
=
1

∞

∂
Δ
(

p
n

,

r
n

)
]

{\partial _{\ast }}K \equiv \partial \Delta (0,1) \cup [ \cup _{n = 1}^\infty \partial \Delta ({p_n},{r_n})]

and

F
(
K
)
≡
{
z
∈
K
∖

∂

∗

K
=
z

 is not a point of density of 

K
}

F(K) \equiv \{ z \in K\backslash {\partial _{\ast }}K = z{\text { is not a point of density of }}K\}

. It is proved that if

α
>
1

\alpha > 1

, then

K

1

[
F
(
K
)
]
=
0

{\mathcal {K}^1}[F(K)] = 0

, where

K

1

{\mathcal {K}^1}

denotes Hausdorff one-dimensional measure.
A note on commutators and singular integrals
Authors: Muhly Paul S.
Published: June 2010

A new approach to the analysis of a certain commutator equation is presented.
A fixed point theorem for a system of multivalued transformations
Authors: Czerwik S.
Published: June 2010

We shall prove a fixed point theorem for a system of multivalued mappings which generalizes the result obtained by the author [1, Theorem 1]. For

n
=
1

n = 1

we obtain a generalization of results of Reich [5, Theorem 5] and Nadler [3, Theorem 5], [4, Theorem 1].
Processes with infinitely many jumping particles
Authors: Sullivan Wayne G.
Published: June 2010

We give sufficient conditions for a Markov process of an infinite particle system to be specified by a formal generator which has a term for each finite subset of particles. Under stronger assumptions we show that processes of this type preserve a certain property of probability measures.
On derivation algebras of Malcev algebras and Lie triple systems
Authors: Stitzinger Ernest L.
Published: June 2010

W. H. Davenport has shown that the derivation algebra

D

(
A
)

\mathfrak {D}(A)

of a semisimple Malcev algebra

A
A

of characteristic

0
0

acts completely reducibly on

A
A

. The purpose of the present note is to characterize those Malcev algebras which have such derivation algebras as those whose radical is central and to obtain the same result for Lie triple systems. Analogous results are known to hold for standard and alternative algebras.
A comparison theorem
Authors: Guggenheimer H.
Published: June 2010

A general comparison theorem is extracted from the proof of a recent comparison theorem of Leighton and Oo Kian Ke. The result is applied to a comparison theorem for focal points.
Existence of Sidon sets in discrete FC-groups
Authors: De-Michele Leonede , Soardi Paolo M.
Published: June 2010

In this paper we prove that every infinite subset of a discrete

F
C

FC

-group contains an infinite Sidon set.
Applications of intersection theory
Authors: Goldstein Richard , Turner Edward C.
Published: June 2010

A Borsuk-Ulam type theorem for semifree actions on

S

2
n

{S^{2n}}

is obtained via a formula of Lefschetz.
A remark on Whitney’s proof of de Rham’s theorem
Authors: Dodziuk Józef
Published: June 2010

The PL differential forms are used to prove that the mapping induced on cohomology by pull-back of differential forms corresponds under the de Rham isomorphism to the pull-back of cohomology classes.
A remark on the first neighbourhood ring of a Noetherian Cohen-Macaulay local ring of dimension one
Authors: Doussouki T.
Published: June 2010

There is an isomorphism between the first neighbourhood ring of a noetherian Cohen-Macaulay local ring

A
A

of dimension one and the ring of endomorphisms of a large power of its maximal ideal.
Test modules and cogenerators
Authors: Vámos Peter
Published: June 2010

If

Hom
R

(
A
,
T
)
=
0

{\operatorname {Hom} _R}(A,T) = 0

implies that

A
=
0

A = 0

for all

R
R

-modules

A
A

, then the

R
R

-module

T
T

is a test module. The ring

R
R

is said to be a TC-ring if every test module is a cogenerator. If

S
S

is a simple module over a TC-ring then

End
R

E
(
S
)

{\operatorname {End} _R}E(S)

is a local semifir. A commutative ring

R
R

is a TC-ring if and only if

R
M

{R_M}

is a P.I.D. for all maximal ideals

M
M

of

R
R

.
Two counterexamples in semigroup theory on Hilbert space
Authors: Chernoff Paul R.
Published: June 2010

There exist

(

C
0

)

({C_0})

semigroups

T
1

(
t
)
,

T
2

(
t
)

{T_1}(t),{T_2}(t)

on Hilbert space with the following properties:

T
1

{T_1}

has a bounded generator and is uniformly bounded, but is not similar to a contraction semigroup.

T
2

{T_2}

is uniformly bounded, and there exists no scalar

α
\alpha

such that

e

−
α
t

T
2

(
t
)

{e^{ - \alpha t}}{T_2}(t)

is similar to a contraction semigroup.
Point-countability and compactness
Authors: Wicke H. H. , Worrell J. M.
Published: June 2010

We prove that if a countably compact space

X
X

has an open cover

U

=
⋃

{

V

n

:
n
>
ω
}

\mathcal {U} = \bigcup {\{ {\mathcal {V}_n}:n > \omega \} }

such that each

x
∈
X

x \in X

is in at least one but not more than countably many elements of some

V

n

{\mathcal {V}_n}

, then some finite subcollection of

U

\mathcal {U}

covers

X
X

. We apply the theorem in proving several metrization theorems for countably compact spaces and discuss consequences of weak

δ
θ

\delta \theta

-refinability, a concept implicit in the statement of the theorem.
A remark on Nelson’s best hypercontractive estimates
Authors: Simon Barry
Published: June 2010

By using a combinatorial estimate we provide a new proof of Nelson’s best hypercontractive estimates from

L
2

{L^2}

to

L
4

{L^4}

.
On a conjecture of S. Chowla
Authors: Suryanarayana D.
Published: June 2010

Let

ψ
(
x
)
=
x
−
[
x
]
−

1
2

\psi (x) = x - [x] - \tfrac {1}{2}

. It has been conjectured by S. Chowla that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Subscript n less-than-or-slanted-equals square-root x Baseline left-brace psi squared left-parenthesis x slash n right-parenthesis minus 1 slash 12 right-brace equals o left-parenthesis x Superscript 1 slash 4 plus epsilon Baseline right-parenthesis">

Σ

n
⩽
√
x

{

ψ
2

(
x

/

n
)
−
1

/

12
}
=
o
(

x

1

/

4
+
ϵ

)

{\Sigma _{n \leqslant \surd x}}\{ {\psi ^2}(x/n) - 1/12\} = o({x^{1/4 + \epsilon }})

, for every

ϵ
>
0

\epsilon > 0

. In this paper we show that this conjecture is equivalent to

Σ

n
⩽
√
x

n
2

{

ψ
2

(
x

/

n
)
−
1

/

12
}

{\Sigma _{n \leqslant \surd x}}{n^2}\{ {\psi ^2}(x/n) - 1/12\}

by proving that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n less-than-or-slanted-equals square-root x Endscripts left-brace psi squared left-parenthesis StartFraction x Over n EndFraction right-parenthesis minus one twelfth right-brace plus StartFraction 1 Over x EndFraction sigma-summation Underscript n less-than-or-slanted-equals square-root x Endscripts n squared left-brace psi squared left-parenthesis StartFraction x Over n EndFraction right-parenthesis minus one twelfth right-brace equals o left-parenthesis x Superscript 1 slash 4 Baseline right-parenthesis period">

∑

n
⩽
√
x

{

ψ
2

(

x
n

)

−

1

12

}

+

1
x

∑

n
⩽
√
x

n
2

{

ψ
2

(

x
n

)

−

1

12

}

=
o
(

x

1

/

4

)
.

\sum \limits _{n \leqslant \surd x} {\left \{ {{\psi ^2}\left ( {\tfrac {x}{n}} \right ) - \tfrac {1}{{12}}} \right \} + \tfrac {1}{x}\sum \limits _{n \leqslant \surd x} {{n^2}\left \{ {{\psi ^2}\left ( {\tfrac {x}{n}} \right ) - \tfrac {1}{{12}}} \right \} = o({x^{1/4}}).} }

\]
Commutativity of endomorphism rings of ideals. II
Authors: Alamelu S.
Published: June 2010

Let

R
R

be a commutative ring. In (1), it was proved that a ring

R
R

with noetherian total quotient ring is self-injective if and only if the endomorphism ring of every ideal is commutative. We prove here that if the ring is coherent and is its own total quotient ring, then

R
R

is self-injective if and only if

Hom
⁡
(
I
,
I
)
=
R

\operatorname {Hom} (I,I) = R

for every ideal

I
I

of

R
R

.
Counterexamples in best approximation
Authors: Poreda S. J.
Published: June 2010

Several counterexamples for approximation to continuous functions by polynomials are given. One example shows that the points of maximum deviation of a continuous real valued function on an interval from its polynomial of degree

n
n

of best uniform approximation can lie on any monotone sequence contained in that interval for infinitely many

n
n

.
A uniformly convex Banach space with a Schauder basis which is subsymmetric but not symmetric
Authors: Pujara Lakhpat R.
Published: June 2010

An example of a uniformly convex Banach space with a basis

(

x
i

)

({x_i})

is contructed such that

(

x
i

)

({x_i})

is subsymmetric but not symmetric.
Homeomorphisms of 𝐵^{𝑘}×𝑇ⁿ
Authors: Lawson Terry
Published: June 2010

An elementary proof is given that a homeomorphism of

B
k

×

T
n

rel
⁡
∂

{B^k} \times {T^n} \operatorname {rel} \partial

that is homotopic to the identity is pseudoisotopic to the identity

∀
k
,
n

\forall k,n

. The key step is to show that each homeomorphism is pseudoisotopic to any standard finite cover.
𝐾ᵢ of upper triangular matrix rings
Authors: Dennis R. Keith , Geller Susan C.
Published: June 2010

Standard techniques are used to compute

K
i

(
i
=
0
,
1
,
2
)

{K_i}(i = 0,1,2)

of generalized triangular matrix rings.
Analytic expressions for continued fractions over a vector space
Authors: Roach F. A.
Published: June 2010

In previous papers, we considered continued fractions obtained from a type of “geometric reciprocal". In this paper we develop certain “analytical” formulas for such continued fractions and, by using these formulas, obtain results analogous to certain classical theorems for ordinary continued fractions.
Anticonformal automorphisms of compact Riemann surfaces
Authors: Zarrow Robert
Published: June 2010

We show that an automorphism of prime order of a compact Riemann surface is embeddable if it is the square of an anticonformal automorphism. Also, every embeddable automorphism of odd order of a compact Riemann surface is the square of an orientation reversing selfhomeomorphism.
Generating functions for the Jacobi polynomial
Authors: Cohen M. E.
Published: June 2010

Two theorems are proved with the aid of operator and series iteration methods. Special cases appear to give new and known generating functions for the Jacobi polynomial.
One more metrization theorem
Authors: Hung H. H.
Published: June 2010

We give here a metrization theorem proved via the method of symmetrics. From our theorem follow the theorem of Stone-Arhangel’skiĭ and one in terms of a countable strongly refining sequence of open coverings.
Proof of a conjecture of Friedman
Authors: Martin Donald A.
Published: June 2010

We prove that every uncountable hyperarithmetic set has a member of each hyperdegree

⩾

0

\geqslant {\mathbf {0}}

, the hyperdegree of Kleene’s

0
0

.
A remark on a result of McKean
Authors: Tarnopolska-Weiss Marysia
Published: June 2010

The diameter of the Dirichlet polygon associated to certain discontinuous groups acting on the upper half-plane is shown to be bounded. This clarifies a point in the proof of a result of McKean.
The logarithmic center of a planar region
Authors: Hensley Douglas
Published: June 2010

Given a bounded region

S
S

in the complex plane, let

f
(
β
)
=

∫
S

log
⁡

|

z
−
β

|

d

f(\beta ) = {\smallint _S}\log |z - \beta |d

area for

β
\beta

any complex number. A logarithmic center of

S
S

is an

α
\alpha

which minimizes

f
(
β
)

f(\beta )

. When is

α
\alpha

unique? Conjecture. If

S
S

is convex then

α
\alpha

is unique. Theorem. If

S
S

is convex and symmetric about some line, then

α
\alpha

is unique.
On a fixed point theorem of contractive type
Authors: Wong Chi Song
Published: June 2010

A refinement of the proof of a recent fixed point theorem of Caristi is obtained.
Strict complex convexity
Authors: Carmignani Robert
Published: June 2010

The strictly subharmonic and strictly plurisubharmonic functions are characterized using distribution theory, namely, the distributional Laplacian and the distributional complex Hessian. Then the analogous results for strictly convex functions are given.
The Brauer group of a real curve
Authors: Demeyer F. R. , Knus M. A.
Published: June 2010

The Brauer group of a real algebraic affine curve is a finite group of exponent 2 whose rank is the number of connected real components (in the strong topology) of the curve.
The hyperoctant property in orthomodular AC-lattices
Authors: Morash Ronald P.
Published: June 2010

The complete atomic orthomodular lattice

L
L

is said to have the hyperoctant property if and only if, for every orthogonal family of atoms

{

a
α

}

\{ {a_\alpha }\}

in

L
L

with cardinality

≥
2

\geq 2

, there exists an atom

q
q

such that

q
≤

∨
α

a
α

q \leq { \vee _\alpha }{a_\alpha }

and

q
∉

a
α

q \notin {a_\alpha }

for each

α
\alpha

. The projection lattice of any separable Hilbert space has the hyperoctant property. In this paper, we show that an abstract complete atomic orthomodular lattice possessing the additional properties,

M
M

-symmetry, irreducibility, countably infinite dimension, and the angle bisection property, has the hyperoctant property. Additional remarks are made about the non-

M
M

-symmetric case.
Coherence of polynomial rings
Authors: Greenberg B. V. , Vasconcelos W. V.
Published: June 2010

The main result is that

A
[
X
]

A[X]

, the polynomial ring in any number of indeterminates over a coherent ring

A
A

of global dimension two, is coherent.
Fourier transforms of 𝐵-splines and fundamental splines for cardinal Hermite interpolations
Authors: Lee S. L.
Published: June 2010

Using the exponential Hermite Euler splines we compute the Fourier transforms of the

B
B

-splines and fundamental splines for Cardinal Hermite Interpolation, introduced by Schoenberg and Sharma and Lipow and Schoenberg respectively.
A new proof of Giesy’s theorem on non-𝐵-convexity
Authors: Sastry K. P. R. , Naidu S. Venkata Ratnam
Published: June 2010

A new proof of Giesy’s theorem on non-

B
B

-convexity is given.
Singular regular neighborhoods and local flatness in codimension one
Authors: Daverman Robert J.
Published: June 2010

For an

(
n
−
1
)

(n - 1)

-manifold

S
S

topologically embedded as a closed subset of an

n
n

-manifold

N
N

, we define what it means for

S
S

to have a singular regular neighborhood in

N
N

. The principal result demonstrates that

S
S

has a singular regular neighborhood in

N
N

if and only if the homotopy theoretic condition holds that

N
−
S

N - S

is locally simply connected (

1
1

-LC) at each point of

S
S

. Consequently,

S
S

has a singular regular neighborhood in

N
N

if and only if

S
S

is locally flatly embedded

(
n
≠
4
)

(n \ne 4)

.
Converses to measurability theorems for Yeh-Wiener space
Authors: Skoug David
Published: June 2010

In this paper we establish some relationships between Yeh-Wiener measurability and Wiener measurability of certain sets and functionals. In addition we show that an “interval” in Yeh-Wiener space is Yeh-Wiener measurable if and only if its “restriction set” in Euclidean space is Lebesgue measurable.
The class number of the field of 5ⁿth roots of unity
Authors: Washington Lawrence C.
Published: June 2010

Let

h
n
−

h_n^ -

be the relative class number of the field of

5
n

{5^n}

th roots of unity. If

l
l

is any prime number, then the

l
l

-part of

h
n
−

h_n^ -

is bounded independent of

n
n

.
On certain extremal problems for functions with positive real part
Authors: Ruscheweyh Stephan , Singh Vikramaditya
Published: June 2010

For the class

P
P

of analytic functions

p
(
z
)
,
p
(
0
)
=
1

p(z),p(0) = 1

, with positive real part in

|

z

|

>
1

|z| > 1

, a type of extremal problems is determined which can be solved already within the set

p
(
z
)
=
(
1
+
ε
z
)

/

(
1
−
ε
z
)
,

|

ε

|

=
1

p(z) = (1 + \varepsilon z)/(1 - \varepsilon z),\;|\varepsilon | = 1

. One problem of this kind is to find the largest number

ρ
(
s
,

μ
)

\rho (s,\;\mu )

such that
\[

Re
⁡
{
p
(
z
)
+
s
z

p
′

(
z
)

/

(
p
(
z
)
+
μ
)
}
>
0
,

\operatorname {Re} \{ p(z) + szp’(z)/(p(z) + \mu )\} > 0,

\]

|

z

|

⩽
ρ
(
s
,

μ
)

|z| \leqslant \rho (s,\;\mu )

, for all

p
∈
P
,

−
1
≠
μ
∈

C

,

s
>
0

p \in P,\; - 1 \ne \mu \in {\mathbf {C}},\;s > 0

. Sharp upper bounds for two other functionals over

P
P

are also given.
Hurewicz fibrations need not be locally trivial
Authors: Husch L. S.
Published: June 2010

A Hurewicz fibration,

p
:
M
→

S
1

p:M \to {S^1}

, from a closed manifold onto the

1
1

-sphere is constructed which is not locally trivial.
On the mean ergodic theorem of Sine
Authors: Lloyd Stuart P.
Published: June 2010

Robert Sine has shown that

(
1

/

n
)
(
I
+
T
+
⋯
+

T

n
−
1

)

(1/n)(I + T + \cdots + {T^{n - 1}})

, the ergodic averages, converge in the strong operator topology iff the invariant vectors of

T
T

separate the invariant vectors of the adjoint operator

T
∗

,
T

{T^ \ast },T

being any Banach space contraction. We prove a generalization in which (spectral radius of

T
T

)

≦
1

\leqq 1

replaces

|

|

T

|

|

≦
1

||T|| \leqq 1

, and any bounded averaging sequence converging uniformly to invariance replaces the ergodic averages; it is necessary to assume that such sequences exist.
Distribution of zeros and limit behavior of solutions of differential equations
Authors: Guggenheimer H.
Published: June 2010

Elementary arguments of projective geometry are used to obtain information about the limit behavior of solutions of ordinary linear differential equations if certain distributions of zeros do not occur.
Measurable tail disintegrations of the Haar integral are purely finitely additive
Authors: Dubins Lester E.
Published: June 2010

There are countably additive probability measures, P, and sub-sigma fields, relative to which P admits no proper, measurable, conditional distributions, except, possibly, those which are purely finitely additive. The usual fair, coin-tossing probability measure and the tail sigma field illustrate this phenomenon. More generally, every measurable, disintegration of the Haar integral of any compact metrizable group, G, relative to the partition,

Π
\Pi

, of G which consists of the left cosets of any dense denumerable subgroup S of G, or what comes to the same thing, relative to the sigma field of Haar-measurable subsets of G which are invariant under right translation by S, is purely finitely additive.
Injective cogenerator rings and a theorem of Tachikawa. II
Authors: Faith Carl
Published: June 2010

The main theorem states that a right injective cogenerator ring R has strongly bounded basic ring

R
0

{R_0}

, that is, every onesided ideal

≠
0

\ne 0

of

R
0

{R_0}

contains an ideal

≠
0

\ne 0

. A right injective cogenerator ring R is characterized by the condition: (1) R is semiperfect and right self-injective and (2) R has (finite) essential right socle. We show that (2) can be replaced by (2’)

R
0

{R_0}

is strongly right bounded and has finite left socle.
On the singularity of the exponential map on a Lie group
Authors: Lai Heng Lung
Published: June 2010

Let

G

\mathfrak {G}

be a connected (real or complex) Lie group with Lie algebra G. Define a conjugate point g of

G

\mathfrak {G}

as a point

g
=
exp
⁡
x

g = \exp x

for some

x
∈
G

x \in G

and

d

exp
x

d{\exp _x}

is a noninvertible linear map. We prove that

g
∈

G

g \in \mathfrak {G}

is a conjugate point if and only if

g
=
exp
⁡

x
λ

g = \exp {x_\lambda }

for at least a (complex parameter) family of elements

x
λ

(
λ
∈

C

)

{x_\lambda }(\lambda \in {\mathbf {C}})

in G.
A counterexample in nonlinear interpolation
Authors: Cwikel Michael
Published: June 2010

If

(

A
0

,

A
1

)

({A_0},{A_1})

is an interpolation pair with

A
0

⊂

A
1

{A_0} \subset {A_1}

and T is a possibly nonlinear operator which maps

A
0

{A_0}

into

A
0

{A_0}

and

A
1

{A_1}

into

A
1

{A_1}

and satisfies

‖

T
a

‖

A
0

⩽
C

‖
a
‖

A
0

{\left \| {Ta} \right \|_{{A_0}}} \leqslant C{\left \| a \right \|_{{A_0}}}

and

‖

T
b
−
T

b
′

‖

A
1

⩽
C

‖

b
−

b
′

‖

A
1

{\left \| {Tb - Tb’} \right \|_{{A_1}}} \leqslant C{\left \| {b - b’} \right \|_{{A_1}}}

for all

a
∈

A
0

a \in {A_0}

and b,

b
,

b
′

∈

A
1

b,b’ \in {A_1}

and for some constant C, then it is known that T also maps the real interpolation spaces

(

A
0

,

A
1

)

θ
,
p

{({A_0},{A_1})_{\theta ,p}}

into themselves. We give an example showing that T need not map the complex interpolation spaces

[

A
0

,

A
1

]
θ

{[{A_0},{A_1}]_\theta }

into themselves. It is also seen that quasilinear operators may fail to preserve complex interpolation spaces.
A bound for the number of automorphisms of a compact Klein surface with boundary
Authors: May Coy L.
Published: June 2010

For an integer

g
⩾
2

g \geqslant 2

, let

ν
(
g
)

\nu (g)

be the order of the largest group of automorphisms of a compact Klein surface with nonempty boundary of genus g. We show that

ν
(
g
)
⩾
4
(
g
+
1
)

\nu (g) \geqslant 4(g + 1)

for all g and that for an infinite number of values of

ν
(
g
)
=
4
(
g
+
1
)

\nu (g) = 4(g + 1)

.
Indecomposable compact perturbations of the bilateral shift
Authors: Herrero Domingo A.
Published: June 2010

Recent results of M. Radjabalipour and H. Radjavi assert that the sum of a normal operator N with spectrum on a smooth Jordan curve and a compact operator K in the Macaev ideal

S

ω

{\mathfrak {S}_\omega }

is decomposable provided the spectrum of

N
+
K

N + K

does not fill the interior of the curve. Examples are given to show that this result cannot be essentially improved by taking K in a larger ideal.
Finite groups with a standard component whose centralizer has cyclic Sylow 2-subgroups
Authors: Finkelstein Larry
Published: June 2010

Let G be a finite group with

O
(
G
)
=
1

O(G) = 1

, A a standard component of G and X the normal closure of A in G. Furthermore, assume that

C
(
A
)

C(A)

has cyclic Sylow 2-subgroups. Then conditions are given on A which imply that either

X
=
A

X = A

or

C
(
A
)

C(A)

has Sylow 2-subgroups of order 2. These results are then applied to the cases where A is isomorphic to

⋅
0

\cdot 0

or Ru, proper 2-fold covering of the Rudvalis group.
On a group that cannot be the group of a 2-knot
Authors: Murasugi Kunio
Published: June 2010

It is proved that a homomorph of the group of trefoil knot cannot be the group of a 2-knot in 4-sphere.
Poisson brackets and commutator brackets. I
Authors: Prosser Reese T.
Published: June 2010

A previous result, that for certain problems in Hamiltonian mechanics the commutator bracket is determined by the Poisson bracket, is here extended to include certain problems in Hamiltonian field theory.
The density character of unions
Authors: Comfort W. W. , Retta Teklehaimanot
Published: June 2010

We consider only completely regular, Hausdorff spaces. Responding to a question of R. Levy and R. H. McDowell [Proc. Amer. Math. Soc. 49 (1975), 426-430] we show that for

ω
⩽
γ
⩽

2

2
ω

\omega \leqslant \gamma \leqslant {2^{{2^\omega }}}

there is a separable space equal to the (appropriately topologized) disjoint union of

γ
\gamma

copies of the “Stone-Čech remainder”

β
N
∖
N

\beta N\backslash N

. More generally, denoting density character by d and weight by w, we prove this Theorem. The following statements about infinite cardinal numbers

γ
\gamma

and

α
\alpha

are equivalent: (a)

2
α

⩽

2
γ

{2^\alpha } \leqslant {2^\gamma }

and

γ
⩽

2

2
α

\gamma \leqslant {2^{{2^\alpha }}}

; (b) For every family

{

X
ξ

:
ξ
>
γ
}

\{ {X_\xi }:\xi > \gamma \}

of spaces, with

w
(

X
ξ

)
⩽

2
α

w({X_\xi }) \leqslant {2^\alpha }

for all

ξ
>
γ

\xi > \gamma

, the set-theoretic disjoint union

X
=

∪

ξ
>
γ

X
ξ

X = { \cup _{\xi > \gamma }}{X_\xi }

admits a topology such that

d
(
X
)
⩽
α

d(X) \leqslant \alpha

and each

X
ξ

{X_\xi }

is a topological subspace of X. The following observation (a special case of Theorem 3.1) suggests that it may be difficult to achieve a stronger result: If

α
⩾
ω

\alpha \geqslant \omega

and

X
0

{X_0}

and

X
1

{X_1}

denote copies of the discrete space of cardinality

α
+

{\alpha ^ + }

, then the disjoint union

X
=

X
0

∪

X
1

X = {X_0} \cup {X_1}

admits a topology (making each

X
i

{X_i}

a topological subspace) such that

d
(
X
)
⩽
α

d(X) \leqslant \alpha

.
Mappings of the interval with finitely many periodic points have zero entropy
Authors: Block Louis
Published: June 2010

It is shown that if f is a continuous map of a closed interval into itself, and f has finitely many periodic points, then the topological entropy of f is zero.
Curvature forms for Lorentz 2-manifolds
Authors: Burns John T.
Published: June 2010

As a converse to the Gauss-Bonnet theorem for Lorentz metrics on 2-manifolds, we show that if

Ω
¯

\bar \Omega

is a 2-form on the torus

T
2

{T^2}

and

∫

T
2

Ω
¯

=
0

{\smallint _{{T^2}}}\bar \Omega = 0

then

Ω
¯

\bar \Omega

is the curvature form of some Lorentz metric on

T
2

{T^2}

.
A tree argument in infinitary model theory
Authors: Harnik V. , Makkai M.
Published: June 2010

A tree argument is used to show that any counterexample to Vaught’s conjecture must have an uncountable model. A similar argument replaces the use of forcing by Burgess in a theorem on

∑
1
1

\sum _1^1

equivalence relations.
Simplicial structure of the real analytic cut locus
Authors: Buchner Michael A.
Published: June 2010

This note shows how to generalize to arbitrary dimensions the result of S.B. Myers that the cut locus in a real analytic Riemannian surface is triangulable. The basic tool is Hironaka’s theory of subanalytic sets.
Attainable sets of quasiconcave markets
Authors: Weber Robert James
Published: June 2010

The union of any finite collection of corners is the attainable set of a market with continuous, monotone increasing, quasiconcave utility functions. It follows that the attainable sets of such markets are dense in the collection of attainable sets of markets with utility functions restricted only to being upper-semicontinuous and lower-bounded.
Inseparable finite solvable groups. II
Authors: Bechtell Homer
Published: June 2010

A finite group is called inseparable if the only normal subgroups over which it splits are the group itself and the trivial subgroup. Let E be the formation of finite solvable groups with elementary abelian Sylow subgroups. This note establishes the fact that, up to isomorphism, there is exactly one nonnilpotent inseparable solvable group in which the E-residual is a metacyclic p-group.
A quick proof of Harish-Chandra’s Plancherel theorem for spherical functions on a semisimple Lie group
Authors: Rosenberg Jonathan
Published: June 2010

Some lemmas of S. Helgason and R. Gangolli, originally conceived for proving an analogue of the Paley-Wiener theorem for symmetric spaces, are used to give a quick proof of Harish-Chandra’s inversion formula and Plancherel theorem for bi-invariant functions on a semisimple Lie group. The method is elementary in that it does not require introduction of Harish-Chandra’s “Schwartz space."
A simple proof of Ramanujan’s ₁𝜓₁ sum
Authors: Ismail Mourad E. H.
Published: June 2010

We give a simple proof of the

1

ψ
1

_1{\psi _1}

sum using basic hypergeometric functions.
Peak sets for Lipschitz functions
Authors: Novinger W. P. , Oberlin D. M.
Published: June 2010

We study the peak sets for the algebras of functions analytic in the unit disc D and satisfying a Lipschitz condition on

∂
D

\partial D

.
Divisibility properties of the 𝑞-tangent numbers
Authors: Andrews George E. , Gessel Ira
Published: June 2010

The q-tangent number

T

2
n
+
1

(
q
)

{T_{2n + 1}}(q)

is shown to be divisible by

(
1
+
q
)
(
1
+

q
2

)
⋯
(
1
+

q
n

)

(1 + q)(1 + {q^2}) \cdots (1 + {q^n})

. Related divisibility questions are discussed.
Translates and multipliers of abelian difference sets
Authors: McFarland Robert L. , Rice Bart F.
Published: June 2010

It is shown that every abelian difference set has a translate which is fixed by all numerical multipliers. If an abelian difference set in a group of order v has numerical multipliers

t
1

,
…
,

t
m

{t_1}, \ldots ,{t_m}

which satisfy

gcd
(

t
1

−
1
,
…
,

t
m

−
1
,
v
)
=
1

\gcd ({t_1} - 1, \ldots ,{t_m} - 1,v) = 1

, then there is a unique translate which is fixed by all multipliers.
The perfect 𝑛th power which divides a nonzero polynomial
Authors: Chalkley Roger
Published: June 2010

In terms of rational operations, an algorithm is given to obtain the monic polynomial of maximal degree whose nth power divides a given nonzero polynomial.
A short proof of the Dawkins-Halperin theorem
Authors: Handelman David
Published: June 2010

A brief proof is presented, of the Dawkins-Halperin Theorem, that if D is a finite dimensional division algebra with centre F, then the direct limits of appropriately-sized matrix rings over D and F are isomorphic; the isomorphism can be given in a form suitable for comparing cohomology groups of D and F.
Height of exceptional character degrees
Authors: Sibley David A.
Published: June 2010

Suppose a finite group G has an abelian Sylow p-group P which is a T.I. set and whose normalizer is a Frobenius group with kernel P. It is shown that the degree of an exceptional character of G is not divisible by p. That is, exceptional characters have height 0 in the principal p-block of G.
The nonexistence of universal invariant measures
Authors: Ryll-Nardzewski C. , Telgársky R.
Published: June 2010

It is shown that for arbitrary nontrivial

σ
\sigma

-finite measure defined on all subsets of a group there are at most countably many left translations of that measure that are mutually equivalent in the sense of the absolute continuity.
Weak expectations and injectivity in operator algebras
Authors: Blackadar Bruce E.
Published: June 2010

An example is given of a noninjective von Neumann algebra M on a separable Hilbert space

H

\mathcal {H}

for which there exists a weak expectation of

L

(

H

)

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

into M. Some positive results about weak expectations are also obtained.
Global characterizations of the sphere
Authors: Stamou George
Published: June 2010

Let S be an ovaloid in Euclidean three-space

E
3

{E^3}

with Gaussian curvature

K
>
0

K > 0

and let

K

II

{K_{{\text {II}}}}

be the curvature of the second fundamental form II of S. We give some global characterizations of the sphere by the curvature

K

II

{K_{{\text {II}}}}

which generalize some results of R. Schneider [4], D. Koutroufiotis [2] and the well-known “H-Satz” theorem of H. Liebmann.
An analogue of some inequalities of P. Turán concerning algebraic polynomials having all zeros inside [-1,+1]. II
Authors: Varma A. K.
Published: June 2010

Let

P
n

(
x
)

{P_n}(x)

be an algebraic polynomial of degree

⩽
n

\leqslant n

having all zeros inside

[
−
1
,
+
1
]

[ - 1, + 1]

; then we have
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative 1 Superscript 1 Baseline upper P prime Subscript n Superscript 2 Baseline left-parenthesis x right-parenthesis d x greater-than left-parenthesis StartFraction n Over 2 EndFraction plus three fourths plus StartFraction 3 Over 4 n EndFraction right-parenthesis integral Subscript negative 1 Superscript 1 Baseline upper P Subscript n Superscript 2 Baseline left-parenthesis x right-parenthesis d x period">

∫

−
1

1

P
′

n
2

(
x
)
d
x
>

(

n
2

+

3
4

+

3

4
n

)

∫

−
1

1

P
n
2

(
x
)
d
x
.

\int _{ - 1}^1 {P’}_n^2 (x)dx > \left ( {\frac {n}{2} + \frac {3}{4} + \frac {3}{{4n}}} \right )\int _{ - 1}^1 {P_n^2(x)dx.}

\]
This bound is much sharper than found in [2]. Moreover, if

P
n

(
1
)
=

P
n

(
−
1
)
=
0

{P_n}(1) = {P_n}( - 1) = 0

, then under the above conditions we have
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative 1 Superscript 1 Baseline upper P prime Subscript n Superscript 2 Baseline left-parenthesis x right-parenthesis d x greater-than-or-slanted-equals left-parenthesis StartFraction n Over 2 EndFraction plus three fourths plus StartFraction 3 Over 4 left-parenthesis n minus 1 right-parenthesis EndFraction right-parenthesis integral Subscript negative 1 Superscript 1 Baseline upper P Subscript n Superscript 2 Baseline left-parenthesis x right-parenthesis d x comma">

∫

−
1

1

P
′

n
2

(
x
)
d
x
⩾

(

n
2

+

3
4

+

3

4
(
n
−
1
)

)

∫

−
1

1

P
n
2

(
x
)
d
x
,

\int _{ - 1}^1 {P’}_n^2 (x)dx \geqslant \left ( {\frac {n}{2} + \frac {3}{4} + \frac {3}{{4(n - 1)}}} \right )\int _{ - 1}^1 {P_n^2(x)dx,}

\]
equality for

P
n

(
x
)
=

(
1
−

x
2

)
m

,
n
=
2
m

{P_n}(x) = {(1 - {x^2})^m},n = 2m

.
Equivalences generated by families of Borel sets
Authors: Burgess John P.
Published: June 2010

The equivalence relation on the reals generated by a family of

ℵ
α

{\aleph _\alpha }

Borel sets has either

⩽

ℵ
α

\leqslant {\aleph _\alpha }

or else exactly

2

ℵ
0

{2^{{\aleph _0}}}

equivalence classes.
An inequality for generalized quadrangles
Authors: Payne Stanley E.
Published: June 2010

Let

S

\mathcal {S}

be a generalized quadrangle of order (s, t). Let X and Y be disjoint sets of pairwise noncollinear points of

S

\mathcal {S}

such that each point of X is collinear with each point of Y. If

m
=

|

X

|

m = |X|

and

n
=

|

Y

|

n = |Y|

, then

(
m
−
1
)
(
n
−
1
)
⩽

s
2

(m - 1)(n - 1) \leqslant {s^2}

. When equality holds, severe restrictions are placed on m, n, s, and t.
Closed mappings of nowhere locally compact metric spaces
Authors: Stricklen S. A.
Published: June 2010

We present a method of construction which allows us to construct many examples of nowhere first countable spaces which are images of metric spaces under closed mappings, and which allows us to prove (with a certain natural restriction on the mapping) that there is a closed mapping from a metric space M onto a space with a dense nowhere first countable subspace if and only if M is nowhere locally compact.
Integral invariant functions on the nilpotent elements of a semisimple Lie algebra
Authors: Gauger Michael A.
Published: June 2010

Let L be a semisimple Lie algebra over an algebraically closed field of characteristic zero. It is shown that there is a finitely generated ring R of integral invariant functions such that for nilpotent elements x and y of L, one has x conjugate to y if and only if

f
(
x
)
=
f
(
y
)

f(x) = f(y)

for all f in R. The result is analogous to Chevalley’s determination of conjugacy classes of semisimple elements by the ring of invariant polynomial functions.
Existence and uniqueness of solutions to abstract Volterra integral equations
Authors: Kiffe T. , Stecher M.
Published: June 2010

The existence and uniqueness of solutions to the equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript t Baseline a left-parenthesis t minus s right-parenthesis upper A u left-parenthesis s right-parenthesis d s small-contains-as-member f left-parenthesis t right-parenthesis">

u
(
t
)
+

∫
0
t

a
(
t
−
s
)
A
u
(
s
)

d
s
∍
f
(
t
)

u(t) + \smallint _0^ta(t - s)Au(s)\;ds \backepsilon f(t)

, where A is a maximal monotone operator, is proved under various restrictions on A and f.
On the duality between Asplund spaces and spaces with the Radon-Nikodým property
Authors: Sullivan Francis
Published: June 2010

A simple proof of the following result of Stegall is given. Let E be a real Banach space, then E is an Asplund space if

E
∗

{E^ \ast }

has the Radon-Nikodým property.
A universal diagram property of minimal normal extensions
Authors: Bunce John W.
Published: June 2010

We give a universal diagram property which characterizes, up to algebraic equivalence, the minimal normal extension of a subnormal operator.
Shape triviality and metric contractions
Authors: Holsztyński W.
Published: June 2010

Let (X, d) be a nonempty compact metric space such that for every

ε
>
0

\varepsilon > 0

there exists a map

f
:
X
→
X

f:X \to X

satisfying (i)

d
(
x
,
f
(
x
)
)
>
ε

d(x,f(x)) > \varepsilon

for every

x
∈
X

x \in X

, and (ii)

d
(
f
(
x
)
,
f
(
y
)
)
>
d
(
x
,
y
)

d(f(x),f(y)) > d(x,y)

for every

x
,
y
∈
X

x,y \in X

. Then, as proved in this paper, the shape of X is trivial. This improves an earlier result of K. Borsuk [1], who proved that, under the same assumptions, X is acyclic.
Submetrizable spaces and almost 𝜎-compact function spaces
Authors: McCoy R. A.
Published: June 2010

It is shown that the space of real-valued continuous functions, with the compact-open topology, defined on a Urysohn k-space X is almost

σ
\sigma

-compact if and only if X is submetrizable.
Homology equivalent finite groups are isomorphic
Authors: Culler Marc
Published: June 2010

We prove that if a homomorphism between two finite groups induces an isomorphism between their integral homology groups then it is an isomorphism.
Algebraic quotients of Bergman domains
Authors: Riera Gonzalo G.
Published: June 2010

A discrete group of automorphisms of a domain in

C

n

{{\mathbf {C}}^n}

is constructed so that the quotient is algebraic.
On topologies of maximally almost periodic groups
Authors: Huang Ter Jenq
Published: June 2010

It is proved that a necessary and sufficient condition for the group topology of any maximally almost periodic Hausdorff group to coincide with the weak topology of the group in which every complex-valued continuous almost periodic function on the group is continuous is that the group has the equivalent left and right uniform structures. The sufficiency of this condition generalizes the recent results of Glicksberg and Venkataraman concerning the group topology and the weak topology of an abelian Hausdorff group induced by the set of all continuous characters of the group.
Locally closedness of unbounded derivations in 𝐶*-algebras
Authors: Ôta Shôichi
Published: June 2010

We study a local property of unbounded derivations in

C
∗

{C^ \ast }

-algebras, introducing locally closedness for derivations. We show that a derivation is locally closed and the positive portion of the domain is closed under the square root operation if and only if for each hermitian element a in the domain the

C
∗

{C^ \ast }

-subalgebra generated by a and the identity is contained in the domain.
Continuity properties of monotone nonlinear operators in locally convex spaces
Authors: Kravvaritis Dimitrios
Published: June 2010

Let X be a real locally convex Hausdorff space,

X
∗

{X^\ast }

its dual space, and T an operator from X into

2

X
∗

{2^{{X^\ast }}}

. The main results of this paper are: (i) if T is D-maximal monotone and locally bounded at each point of

D
(
T
)

D(T)

, then T is upper demicontinuous; (ii) if X is a Fréchet space, T is monotone, and

D
(
T
)

D(T)

is open in X, then T is upper demicontinuous if and only if T is upper hemicontinuous, thus generalizing a result of [3].
A note on the combinatorial principles ♢(𝐸)
Authors: Devlin Keith J.
Published: June 2010

Shelah has proved that

♢
\diamondsuit

does not imply that

♢
(
E
)

\diamondsuit (E)

holds for every stationary set

E
⊆

ω
1

E \subseteq {\omega _1}

. We prove that, in the other direction, whenever

♢
(
E
)

\diamondsuit (E)

holds there are disjoint stationary sets

F
,
G
⊆
E

F,G \subseteq E

such that both

♢
(
F
)

\diamondsuit (F)

and

♢
(
G
)

\diamondsuit (G)

hold.
On topological isomorphisms of 𝐶₀(𝑋) and the cardinal number of 𝑋
Authors: Cengiz Babattin
Published: June 2010

In this paper it is proved that if

C
0

(
X
)

{C_0}(X)

and

C
0

(
Y
)

{C_0}(Y)

are topologically isomorphic then

|

X

|

=

|

Y

|

|X| = |Y|

.
Euler characteristics and codimensions of complete intersections
Authors: Chen Bang-Yen
Published: June 2010

Studies on relations between Euler characteristics and codimensions of complete intersections.
On groupoids defined by commutators
Authors: Kim Ki Hang , Roush Fred W.
Published: June 2010

We study matrices R, L which count the numbers of solutions of

i
x
=
j

ix = j

and

x
i
=
j

xi = j

. For slight generalizations of R, L, the relation

R
L
=
L
R

RL = LR

characterizes associativity of a groupoid. For groupoids defined by group commutators

x
y

x

−
1

y

−
1

xy{x^{ - 1}}{y^{ - 1}}

the relation

R
L
=
L
R

RL = LR

is valid. In addition one can study analogues of Green’s relations. Any

I

\mathcal {I}

-class contains at most four

H

\mathcal {H}

-classes in a commutator groupoid.
Contraction semigroups, stabilization, and the mean ergodic theorem
Authors: O’Brien Robert E.
Published: June 2010

Consider the semidynamical system,

x
˙

=
A
x
+
B
u

\dot x = Ax + Bu

, where A generates a

C
0

{C_0}

-contraction semigroup and B is bounded. If the system is controllable then it is weakly stabilizable. If in addition the semigroup is quasi-compact and B is compact, the mean ergodic theorem implies that the stability is uniform and exponential.