bluuebunny/crossref_metadata_2025 huggingface dataset:
An elementary proof about the order of the elements in a discrete group
Authors: Crombez G.
Published: June 2010

We give an elementary direct proof of the following property: if for a discrete group

G
G

some

l
p

(
G
)

{l_p}(G)

-space

(
1
>
p
>
∞
)

(1 > p > \infty )

is an algebra, then all elements of

G
G

have uniformly bounded order.
On boundary values of holomorphic functions on balls
Authors: Globevnik Josip
Published: June 2010

It is a result of Agranovski and Valski for which Nagel and Rudin, and Stout have given alternate proofs, that if

B
B

is the open unit ball in

C

n

{{\mathbf {C}}^n}

and if

f
∈
C
(
∂
B
)

f \in C(\partial B)

has the property that for every complex line

Λ
⊂

C

n

\Lambda \subset {{\mathbf {C}}^n}

,

f

|

(
Λ
∩
∂
B
)

f\left | {(\Lambda \cap \partial B)} \right .

has a continuous extension to

Λ
∩

B
¯

\Lambda \cap \bar B

which is holomorphic in

Λ
∩
B

\Lambda \cap B

, then

f
f

has a continuous extension to

B
¯

\bar B

which is holomorphic in

B
B

. In the paper we give an easier, more geometric proof of this result and then prove the local version of this result.
Paracompactness of Pixley-Roy hyperspaces. I
Authors: Tanaka Hidenori
Published: June 2010

In this paper, we will study the relations between the paracompactness of a Pixley-Roy hyperspace

F

[
X
]

\mathcal {F}\left [ X \right ]

and the paracompactness of the subspace

F

2

[
X
]

{\mathcal {F}_2}\left [ X \right ]

. We will give the affirmative answer to H. Bennett’s problem.
A space of pointwise countable type and perfect maps
Authors: Ohta Haruto
Published: June 2010

There exists a Lindelöf space, of pointwise countable type, which does not admit a perfect map onto any space in which every point is

G
δ

{G_\delta }

.
Early coefficients of the inverse of a regular convex function
Authors: Libera Richard J. , Złotkiewicz Eligiusz J.
Published: June 2010

Using known properties of functions of positive real part it is shown that the first seven coefficients of the inverse of a regular univalent function mapping the disk onto a convex region are bounded by 1.
Contractibly embedded 2-spheres in 𝑆²×𝑆²
Authors: Lee Youn W.
Published: June 2010

We construct contractibly embedded

2
2

-spheres in

S
2

×

S
2

{S^2} \times {S^2}

but not contained in a

4
4

-ball.
A note on infinite loop space multiplications
Authors: Vogt Rainer M.
Published: June 2010

A monoid

M
M

is known to be abelian iff its multiplication

M
×
M
→
M

M \times M \to M

is a homomorphism. We prove the corresponding result for homotopy-everything

H
H

-spaces, e.g. infinite loop spaces: For a homotopy-everything

H
H

space

X
X

each

n
n

-ary operation

X
n

→
X

{X^n} \to X

is a homotopy homomorphism, i.e. a homomorphism up to homotopy and all higher coherence conditions.
A result on the set-theoretic complete intersection problem
Authors: Moh T. T.
Published: June 2010

This article establishes a certain arithmetical criterion for set-theoretic complete intersection for irreducible analytic space curves.
Composition factors for indecomposable modules
Authors: Happel Dieter
Published: June 2010

Let

k
k

be a field and

A
A

be a finite-dimensional algebra over

k
k

having only a finite number of isomorphism classes of indecomposable

A
A

-modules. Let

M
M

,

N
N

be two indecomposable

A
A

-modules. Then a homomorphism

f
:
M
→
N

f:M \to N

is said to be irreducible if for every factorization

f
=
g
h

f = gh

,

g
g

is split mono or

h
h

is split epi [2]. The aim of this note is to give an elementary proof of the fact that the indecomposable

A
A

-modules are completely determined, up to isomorphism, by their composition factors if there is no chain of irreducible maps from an indecomposable module to itself. This theorem was first proved in [5] involving the theory of tilted algebras.
On the location of the zeros of the derivative of a polynomial
Authors: Conrey J. Brian , Rubel Lee A.
Published: June 2010

We show that nonreal zeros of a real polynomial

P
(
z
)

P(z)

sometimes induce nonreal zeros of

P
′

(
z
)

P’(z)

.
Topologies of pointwise convergence on families of extremal points and weak compactness
Authors: Tweddle Ian
Published: June 2010

Bourgain and Talagrand showed that a bounded subset of a Banach space is weakly relatively compact provided it is relatively countably compact for the topology of pointwise convergence on the extremal points of the closed unit ball in the dual space. We give a version of this result for quasi-complete locally convex spaces. We also consider situations where the completeness or boundedness assumptions may be relaxed.
On the minimal eigenvalue of the Laplacian operator for 𝑝-forms in conformally flat Riemannian manifolds
Authors: Perrone Domenico
Published: June 2010

Let

(
M
,
g
)

(M,g)

be a compact orientable conformally flat Riemannian manifold and

p

λ
1

^p{\lambda _1}

the minimal eigenvalue of the Laplacian operator for

p
p

-forms. We prove that if there exists a positive constant

K
K

such that

ρ
⩾
K
g

\rho \geqslant Kg

, where

ρ
\rho

is the Ricci tensor of

M
M

, then

p

λ
1

⩾
K
p
(
n
−
p
+
1
)

/

(
n
−
1
)

^p{\lambda _1} \geqslant Kp(n - p + 1)/(n - 1)

for each

p
p

,

1
⩽
p
⩽
n

/

2

1 \leqslant p \leqslant n/2

,

(
n
=
dim
⁡
M
)

(n = \dim M)

; moreover if the equality holds for some

p
p

then

M
M

is of constant curvature

σ
=
K

/

(
n
−
1
)

\sigma = K/(n - 1)

.
Every two-generator knot is prime
Authors: Norwood F. H.
Published: June 2010

Theorem. Every two-generator knot is prime. The proof gives conditions under which a free product with amalgamation cannot be generated by two elements. An example is given of a composite one-relator link.
A nonmetric indecomposable homogeneous continuum every proper subcontinuum of which is an arc
Authors: Gutek Andrzej , Hagopian Charles L.
Published: June 2010

We construct a nonmetric indecomposable homogeneous continuum with the property that each of its proper subcontinua is an arc.
Elementary proofs of some asymptotic radial uniqueness theorems
Authors: Berman Robert D.
Published: June 2010

Elementary proofs of several generalizations of Tse’s extension of an asymptotic radial uniqueness theorem of Barth and Schneider are given.
Difference equations and multipoint boundary value problems
Authors: Eloe P. W.
Published: June 2010

Let

I
=

{

a
,
a
+
1
,
…
,
b

}

I = \left \{ {a,a + 1, \ldots ,b} \right \}

be finite, let

n
⩾
1

n \geqslant 1

, and let

I
j

=

{

a
,
a
+
1
,
…
,
b
+
j

}

{I^j} = \left \{ {a,a + 1, \ldots ,b + j} \right \}

,

j
=
1
,
…
,
n

j = 1, \ldots ,n

. Let

B
B

be the set of mappings from

I
n

{I^n}

into the reals and define the linear difference operator

P
P

by (1)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P u left-parenthesis m right-parenthesis equals sigma-summation Underscript j equals 0 Overscript n Endscripts alpha Subscript j Baseline left-parenthesis m right-parenthesis u left-parenthesis m plus j right-parenthesis comma where m element-of upper I comma alpha Subscript n Baseline left-parenthesis m right-parenthesis identical-to 1 comma and alpha 0 left-parenthesis m right-parenthesis not-equals 0 on upper I period">

P
u
(
m
)
=

∑

j
=
0

n

α
j

(
m
)
u
(
m
+
j
)
,

where 

m
∈
I
,

α
n

(
m
)
≡
1
,

and 

α
0

(
m
)
≠
0

 on 

I
.

Pu(m) = \sum \limits _{j = 0}^n {{\alpha _j}(m)u(m + j),} \quad {\text {where }}m \in I,{\alpha _n}(m) \equiv 1,{\text {and }}{\alpha _0}(m) \ne 0{\text { on }}I.

\]
Existence of solutions theorems and iteration schemes that approximate solutions are given for boundary value problems of the form

P
u
(
m
)
=
f
(
m
,
u
,
E
u
,
…
,

E

n
−
1

u
)

Pu(m) = f(m,u,Eu, \ldots ,{E^{n - 1}}u)

, with boundary conditions

T
u
(
m
)
=
r

Tu(m) = r

, where

P
P

is defined by (1),

E
j

u
(
m
)
=
u
(
m
+
j
)

{E^j}u(m) = u(m + j)

,

j
=
0
,
1
,
…
,
n
−
1

j = 0,1, \ldots ,n - 1

,

f
:
I
×

R

n

→

R

f:I \times {{\mathbf {R}}^n} \to {\mathbf {R}}

is continuous, and

T
:
B
→

R

n

T:B \to {{\mathbf {R}}^n}

is a continuous linear operator. The results are based on solutions of difference inequalities and sign properties of associated Green’s functions.
Jordan-morphisms in ∗-algebras
Authors: Thomsen Klaus
Published: June 2010

As a continuation of Størmer’s work on Jordan-morphisms in

C
∗

C*

-algebras we consider Jordan-morphisms

φ
\varphi

from

∗
*

-algebras

A

\mathfrak {A}

into the

∗
*

-algebra

B
(

H

)

B(\mathcal {H})

, and assume that

φ
(

A

)

\varphi (\mathfrak {A})

is again a

∗
*

-algebra. We then establish the existence of three mutually orthogonal central projections

P
i

{P_i}

,

i
=
1
,
2
,
3

i = 1,2,3

, in

φ

(

)

′
′

\varphi {\left ( {} \right )^{\prime \prime }}

such that

P
1

+

P
2

+

P
3

=
I

{P_1} + {P_2} + {P_3} = I

and

φ
(
⋅
)

P
1

\varphi ( \cdot ){P_1}

is a morphism,

φ
(
⋅
)

P
2

\varphi ( \cdot ){P_2}

is an antimorphism.

P
3

{P_3}

is the largest projection such that

φ
(
⋅
)

P
3

\varphi ( \cdot ){P_3}

is a morphism, as well as an antimorphism. Uniqueness is also shown. The theorem improves a result of Kadison and Størmer. Our proofs are self-contained.
Splitting 𝑆⁴ on 𝑅𝑃² via the branched cover of 𝐶𝑃² over 𝑆⁴
Authors: Lawson Terry
Published: June 2010

The four sphere decomposes as a twisted double

N
2

∪
f

N
2

{N_2}{ \cup _f}{N_2}

, where

N
2

{N_2}

is the

2
2

-disk bundle over the real projective plane with Euler number 2. In this note the relationship of this splitting to the double branched cover of the complex projective plane over the four sphere as the quotient space under complex conjugation is made explicit.
The nonrealizability of modular rings of polynomial invariants by the cohomology of a topological space
Authors: Smith Larry
Published: June 2010

Let

G
>

GL

(
n
;

F

p

)

G > {\text {GL}}(n;{{\mathbf {F}}_p})

be a

p
p

-group,

p
p

an odd prime, and

R
∗

:=

F

p

[

x
1

,
…
,

x
n

]
G

{R^*}: = {{\mathbf {F}}_p}{[{x_1}, \ldots ,{x_n}]^G}

the ring of invariants. The purpose of this note is to prove that in the case where

R
∗

{R^*}

is a graded polynomial algebra, where

deg
⁡

x
1

=
⋯
=
deg
⁡

x
n

=
2

\deg {x_1} = \cdots = \deg {x_n} = 2

, then there is no space

X
X

such that

H
∗

(
X
:

F

p

)
≃

R
∗

{H^*}(X:{{\mathbf {F}}_p}) \simeq {R^*}

. This complements the work of Clark and Ewing [3] and Adams and Wilkerson [1] on the case

p
∤
[
G
;
1
]

p\nmid [G;1]

.
Order ideals of minimal generators
Authors: Evans E. Graham , Griffith Phillip
Published: June 2010

Let

R
R

be a local noetherian domain with algebraically closed residue field and let

M
M

be a finitely generated module of rank

r
r

which is not free. Then there is some minimal generator

x
x

of

M
M

such that the ideal of images of

x
x

under maps of

M
M

to

R
R

has height at most

r
r

.
Free products of abelian 𝑙-groups are cardinally indecomposable
Authors: Powell Wayne B. , Tsinakis Constantine
Published: June 2010

We show that a well-known theorem of Baer and Levi concerning the impossibility of simultaneous decomposition of a group into a free product and a direct sum has an analogue for abelian lattice ordered groups. Specifically we prove that an abelian lattice ordered group cannot be decomposed both into a free product and into a cardinal sum.
The partial differential equation 𝑢_{𝑡}+𝑓(𝑢)ₓ=-𝑐𝑢
Authors: Hattori Harumi
Published: June 2010

Lax’s solution formula for the equation

u
t

+
f

(
u

)
x

=
0

{u_t} + f{(u)_x} = 0

is extended to the equation

u
t

+
f

(
u

)
x

=
−
c
u

{u_t} + f{(u)_x} = - cu

.
The geometry of weak Radon-Nikodým sets in dual Banach spaces
Authors: Riddle Lawrence H.
Published: June 2010

Geometric characterizations in terms of trees, extreme points and dentability are presented for weak*-compact absolutely convex sets that have the Radon-Nikodym property for the Pettis integral.
Products of CW-complexes
Authors: Tanaka Yoshio
Published: June 2010

We show that Liu’s characterization for the product

K
×
L

K \times L

to be a

CW

{\text {CW}}

-complex is independent of the usual axioms of set theory.
Perfectly normal compact spaces are continuous images of 𝛽𝐍_{𝐍}
Authors: Przymusiński Teodor C.
Published: June 2010

Every perfectly normal compact space is a continuous image of

N
∗

=
β
N
∖
N

{N^ * } = \beta N\backslash N

.
Removable singularities for 𝐻^{𝑝}-functions
Authors: Järvi Pentti
Published: June 2010

Given a domain

D
D

in

C

n

{{\mathbf {C}}^n}

, a holomorphic function

f
f

on

D
D

is said to belong to

H
p

(
D
)

{H^p}(D)

,

0
>
p
>
∞

0 > p > \infty

, provided that

|

f

|

p

| f |^p

admits a harmonic majorant in

D
D

. In this note it is shown that

H
p

(
D
∖
E
)
=

H
p

(
D
)

{H^p}(D\backslash E) = {H^p}(D)

whenever

E
E

is a relatively closed polar subset of

D
D

.
Theorems of Namioka and R. E. Johnson type for upper semicontinuous and compact valued set-valued mappings
Authors: Christensen Jens Peter Reus
Published: June 2010

Some recent results of Namioka on strong continuity of weakly continuous mappings (in a dense

G
δ

{G_\delta }

set) and results of R. E. Johnson on norm separability of the range of such mappings (under conditions on the domain space) are shown to have analogues for upper semicontinuous and compact valued set-valued mappings. Some substantial improvements of known automatic continuity results for such mappings are obtained.
An extremally disconnected Dowker space
Authors: Dow Alan , van Mill Jan
Published: June 2010

We give an example of an extremally disconnected Dowker space. Our basic tool is that every

P
P

-space can be

C
∗

{C^\ast }

-embedded in an extremally disconnected compactum.
Lattices of continuous monotonic functions
Authors: Cohen Miriam , Rubin Matatyahu
Published: June 2010

Let

X
X

be a compact Hausdorff space equipped with a closed partial ordering. Let

I
I

be a linear ordering that either does not have a maximal element or does not have a minimal element. We further assume that

⟨

X
,
I

⟩

\langle {X,I} \rangle

has the Tietze extension property for order preserving continuous functions from

X
X

to

I
I

. Denote by

(
X
,
I
)

(X,I)

the lattice of order preserving continuous functions from

X
X

to

I
I

. We generalize a theorem of Kaplanski [K], and show that as a lattice alone,

M
(
X
,
I
)

M(X,I)

characterizes

X
X

as an ordered space.
Minimal splitting fields in cyclotomic extensions
Authors: Spiegel Eugene , Trojan Allan
Published: June 2010

Suppose

G
G

is a finite group of exponent

n
n

and

X
X

an irreducible character of

G
G

. In this note we give sufficient conditions for the existence of a minimal degree splitting field

L
L

with

Q
(
X
)
⊆
L
⊆
Q
(

ζ
n

)

Q(X) \subseteq L \subseteq Q({\zeta _n})

.
Relative normal complements in finite groups
Authors: Ferguson Pamela A.
Published: June 2010

(
G
,
H
,

H
0

,
π
)

(G,H,{H_0},\pi )

denotes the following configuration:

H
H

and

H
0

{H_0}

are the subgroups of the finite group

G
G

with

H
0

⊴
H

{H_0} \trianglelefteq H

is the set of primes dividing

(
H
:

H
0

)

(H:{H_0})

. For

(
G
,
H
,

H
0

,
π
)

(G,H,{H_0},\pi )

we consider conditions

(

A

)

({\text {A}})

,

(

B

0

)

({{\text {B}}_0})

, and

(

C

)

({\text {C}})

:

(

A

)

({\text {A}})

Any two

π
\pi

-elements of

H
−

H
0

H - {H_0}

which are

G
G

-conjugate are

H
H

-conjugate.

(

B

0

)

({{\text {B}}_0})

For each

π
\pi

-element

x
∈
H
−

H
0

x \in H - {H_0}

,

C
G

(
x
)
=
I
(
x
)

C
H

(
x
)

{C_G}(x) = I(x){C_H}(x)

where

I
(
x
)

I(x)

is a normal

π
′

\pi ’

-subgroup of

C
G

(
x
)

{C_G}(x)

.
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper C right-parenthesis StartAbsoluteValue left-parenthesis upper H minus upper H 0 right-parenthesis Superscript upper G comma pi Baseline EndAbsoluteValue equals left-parenthesis upper G colon upper H right-parenthesis StartAbsoluteValue upper H minus upper H 0 EndAbsoluteValue">

(

C

)

|

(
H
−

H
0

)

G
,
π

|

=
(
G
:
H
)

|

H
−

H
0

|

({\text {C}})\left | {{{(H - {H_0})}^{G,\pi }}} \right | = (G:H)\left | {H - {H_0}} \right |

. We show that if

(
G
,
H
,

H
0

,
π
)

(G,H,{H_0},\pi )

satisfies

(

B

0

)

({{\text {B}}_0})

and

(

C

)

({\text {C}})

, or

(

A

)

({\text {A}})

and

(

B

0

)

({{\text {B}}_0})

, and if

H

/

H
0

H/{H_0}

is solvable, then there is a unique relative normal complement

G
0

{G_0}

of

H
H

over

H
0

{H_0}

.
Remarks on fixed points and boundaries
Authors: Kirk W. A.
Published: June 2010

It is shown that three recent fixed point theorems of E. Chandler for nonexpansive mappings defined on compact subsets of strictly convex normed linear spaces hold in arbitrary normed linear spaces.
Metric invariance of Haar measure
Authors: Bandt Christoph
Published: June 2010

Let

d
d

be a left invariant metric for a locally compact group

G
G

. We prove that isometric subsets of

(
G
,
d
)

(G,d)

have equal Haar measure.
Automatic continuity of measurable group homomorphisms
Authors: Lewin Jonathan W.
Published: June 2010

It is well known that a measurable homomorphism from a locally compact group

G
G

to a topological group

Y
Y

must be continuous if

Y
Y

is either separable or

σ
\sigma

-compact. In this work it is shown that the above requirement on

Y
Y

can be somewhat relaxed and it is shown inter alia that a measurable homomorphism from a locally compact group to a locally compact abelian group will always be continuous. In addition, it is shown that if

H
H

is a nonopen subgroup of a locally compact group, then under a variety of circumstances, some union of cosets of

H
H

must fail to be measurable.
A converse to the Lusin-Privalov radial uniqueness theorem
Authors: Berman Robert D.
Published: June 2010

Let

E
E

be a subset of the unit circumference

C
C

. If for every nonempty open arc

A
A

of

C
C

, the set

E
E

is not both metrically dense and of second category in

A
A

, then there exists a nonconstant analytic function

f
f

on the open unit disk

Δ
\Delta

, such that

f
∗

(
η
)
=
0

{f^ * }(\eta ) = 0

,

η
∈
E

\eta \in E

, where

f
∗

{f^ * }

is the radial limit function of

f
f

.
On the realization and classification of symmetric algebras as cohomology rings
Authors: Smith Larry
Published: June 2010

In this note, we prove that every algebra of the form

E
[
σ
,
…
,

σ
n

]
⊗
P
[

ρ
1

,
…
,

ρ
n

]

E[\sigma , \ldots ,{\sigma _n}] \otimes P[{\rho _1}, \ldots ,{\rho _n}]

that is an unstable algebra over the

mod

p

\mod p

Steenrod algebra,

p
p

an odd prime, that satisfies
\[

β

σ
i

=

ρ
i

,

deg
⁡

ρ
i

≢
0
(
p
)
,

i
=
1
,
…
,
n
,

\beta {\sigma _i} = {\rho _i},\quad \deg \rho _i \not \equiv 0(p),\quad i = 1, \ldots ,n,

\]
arises as the cohomology of at least one topological space. Moreover, we show that the classification of such algebras is implicit in the work of Adams and Wilkerson and Clark and Ewing.
On almost rational co-𝐻-spaces
Authors: Henn Hans-Werner
Published: June 2010

Let

X
X

be a

0
0

-connected co-

H
H

-space whose homotopy groups

π
n

(
X
)

{\pi _n}(X)

are

Q
Q

vector spaces if

n
>
1

n > 1

and whose fundamental group

π
1

(
X
)

{\pi _1}(X)

is arbitrary. We prove that

X
X

is homotopy equivalent to a wedge of rational spheres of dimension at least two and of ordinary one-dimensional spheres.
Pseudo-algebraically closed fields over rational function fields
Authors: Jarden Moshe , Shelah Saharon
Published: June 2010

The following theorem is proved: Let

T
T

be an uncountable set of algebraically independent elements over a field

K
0

{K_0}

. Then

K
=

K
0

(
T
)

K = {K_0}(T)

is a Hilbertian field but the set of

σ
∈
G
(
K
)

\sigma \in G(K)

for which

K
~

(
σ
)

\tilde K(\sigma )

is PAC is nonmeasurable.
Similarity problem over 𝑆𝐿(𝑛,𝑍_{𝑝})
Authors: Appelgate H. , Onishi H.
Published: June 2010

The question of conjugacy separability in

S
L
(
n
,
Z
)

SL(n,Z)

leads to the conjugacy problem in

S
L
(
n
,

Z
p

)

SL(n,{Z_p})

for various primes

p
p

. We present a simple solution to the similarity problem over

S
L
(
n
,

Z
p

)

SL(n,{Z_p})

, which solves the conjugacy problem as a special case.
On lattice summing operators
Authors: Szulga Jerzy
Published: June 2010

Given a Banach space

E
E

and a Banach lattice

L
L

, necessary and sufficient conditions on

E
E

and

L
L

are given such that every lattice summing operator

T
:
E
→
L

T:E \to L

(cf. Introduction) is absolutely summing.
Collectionwise normality in screenable spaces
Authors: Rudin Mary Ellen
Published: June 2010

We show that if there is any normal screenable space which is not paracompact, there is one which is not collectionwise normal.
Some properties of Borel subgroups of real numbers
Authors: Le Gac Barthélemy
Published: June 2010

As a consequence of Souslin’s theorem, we obtain the following; if

G
G

and

H
H

both are analytic subgroups of

R

{\mathbf {R}}

such that

G
+
H
=

R

G + H = {\mathbf {R}}

and

G
∩
H
=
{
0
}

G \cap H = \{ 0\}

, then either

G
=

R

G = {\mathbf {R}}

or

G
=
{
0
}

G = \{ 0\}

. Next we obtain some measure and topological properties for uncountable proper Borel subgroups of reals. Finally, we prove that if

E
E

is a vector subspacc of

R

{\mathbf {R}}

over the rationals which admits an uncountable Borel basis, then there exists no Polish topology on

E
E

such that

E
E

is a topological group with the given Borel structure generated by the open sets.
Convolution operators of weak type (2,2) which are not of strong type (2,2)
Authors: Szwarc Ryszard
Published: June 2010

It is well known that if

G
G

is a locally compact and amenable group then the Banach spaces of operators of weak type

(
2
,
2
)

(2,2)

and of strong type

(
2
,
2
)

(2,2)

commuting with the right translations on

G
G

are the same. In contrast we show that if

G
G

is a nonabelian free group then there exists an operator of weak type

(
2
,
2
)

(2,2)

commuting with the right translations on

G
G

which is not of strong type

(
2
,
2
)

(2,2)

.
Suspensions of strong shape equivalences are CE equivalences
Authors: Hastings Harold M.
Published: June 2010

S. Ferry [7] constructed a compactum shape equivalent but not CE equivalent to the circle. We show that suspensions of shape equivalent spaces are CE equivalent.
Some inequalities of algebraic polynomials having all zeros inside [-1,1]
Authors: Varma A. K.
Published: June 2010

Let

H
n

{H_n}

be the set of all algebraic polynomials whose degree is

n
n

and whose zero are all real and lie inside

[
−
1
,
1
]

[ - 1,1]

. Then for

n
n

even we have

(
n
=
2
m
)

(n = 2m)

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative 1 Superscript 1 Baseline left-parenthesis upper P Subscript n Baseline left-parenthesis x right-parenthesis right-parenthesis squared greater-than-or-slanted-equals left-parenthesis n slash 2 plus 3 slash 4 plus 3 slash 4 left-parenthesis n minus 1 right-parenthesis 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

(
x
)
)

2

⩾
(
n

/

2
+
3

/

4
+
3

/

4
(
n
−
1
)
)

∫

−
1

1

P
n
2

(
x
)
d
x

,

\int _{ - 1}^1 {{{({P_n}(x))}^2} \geqslant (n/2 + 3/4 + 3/4(n - 1))\int _{ - 1}^1 {P_n^2(x)dx} } ,

\]
where equality holds iff

P
n

(
x
)
=

(
1
−

x
2

)
m

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

. If

n
n

is an odd positive integer, a similar inequality is valid (see (1.6) below). In the case

P
n

∈

H
n

{P_n} \in {H_n}

and subject to the condition

P
n

(
1
)
=
1

{P_n}(1) = 1

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative 1 Superscript 1 Baseline left-parenthesis upper P prime Subscript n Baseline left-parenthesis x right-parenthesis right-parenthesis Superscript 2 Baseline d x greater-than-or-slanted-equals StartFraction n Over 4 EndFraction plus one eighth plus StartFraction 1 Over 8 left-parenthesis 2 n minus 1 right-parenthesis EndFraction comma">

∫

−
1

1

(

P
′

n

(
x
)

)
2

d
x
⩾

n
4

+

1
8

+

1

8
(
2
n
−
1
)

,

\int _{ - 1}^1 {({{P’}_n}} (x){)^2}dx \geqslant \frac {n}{4} + \frac {1}{8} + \frac {1}{{8(2n - 1)}},

\]
, where equality holds for

P
n

(
x
)
=

(
(
1
+
x
)

/

2

)
n

{P_n}(x) = {((1 + x)/2)^n}

.
On the uniqueness question for Hahn-Banach extensions from the space of ℒ¹ analytic functions
Authors: Reich Edgar
Published: June 2010

Let

Ω
\Omega

be a region of the complex plane,

B

(
Ω
)

\mathcal {B}\left ( \Omega \right )

the space of analytic

L

1

{\mathcal {L}^1}

functions over

Ω
\Omega

, and

κ
∈

L

∞

(
Ω
)

\kappa \in {\mathcal {L}^\infty }\left ( \Omega \right )

. An evident necessary condition for the linear functional
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript kappa Baseline left-bracket phi right-bracket equals integral integral Underscript normal upper Omega Endscripts kappa phi d x d y left-parenthesis phi element-of script upper B left-parenthesis normal upper Omega right-parenthesis right-parenthesis">

Λ
κ

[
φ
]

=
∫

∫
Ω

κ
φ
d
x
d
y
(
φ
∈

B

(
Ω
)
)

{\Lambda _\kappa }\left [ \varphi \right ] = \smallint {\smallint _\Omega }\kappa \varphi dxdy(\varphi \in \mathcal {B}(\Omega ))

to have a unique Hahn-Banach extension from

B

(
Ω
)

\mathcal {B}(\Omega )

to

L

1

(
Ω
)

{\mathcal {L}^1}(\Omega )

is that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue normal upper Lamda Subscript kappa vertical-bar upper G Baseline StartAbsoluteValue EndAbsoluteValue equals StartAbsoluteValue EndAbsoluteValue kappa vertical-bar upper G StartAbsoluteValue EndAbsoluteValue Subscript normal infinity">

|

|

Λ

κ

|

G

|

|

=

|

|

κ

|

G

|

|

∞

|| {{\Lambda _{\kappa |G}}} || = {|| {\kappa |G} ||_\infty }

for every restriction

κ

|

G

\kappa |G

of

κ
\kappa

to a subregion

G
G

of

Ω
\Omega

. An example is constructed to show that not even a considerably stronger condition is sufficient for uniqueness of the Hahn-Banach extension. Remarks on the problem of whether

|

κ
(
z
)

|

\left | {\kappa (z)} \right |

is necessarily constant a.e. if the Hahn-Banach extension is unique indicate that this question is still open, contrary to an assertion in the literature.
Some integral formulas for hypersurfaces and a generalization of the Hilbert-Liebmann theorem
Authors: Li An Min
Published: June 2010

R. C. Reilly calculated the variations of functions of the mean curvatures for hypersurfaces in Euclidean space. In the present paper, using Reilly’s formulas, we derive some general integral formulas for hypersurfaces, which generalize the well-known Minkowski formulas, and then apply those formulas to obtain some characterizations of the hypersphere.
Critical points of one parameter families of maps of the interval
Authors: Block Louis
Published: June 2010

It is shown that some of the periodic phenomena which is well known to occur for the critical point of the quadratic family

f
s

(
x
)
=
s
x
(
1
−
x
)

{f_s}(x) = sx(1 - x)

(and other

C
1

{C^1}

families with a single critical point) occurs for each critical point in

C
1

{C^1}

families with an arbitrary (possibly infinite) number of critical points. Also, some of the same behavior occurs in families of maps (which are not necessarily differentiable) where a critical point has derivative zero on either the left or the right side. A stronger condition is obtained when the derivative on the right is zero.
Remark on continuous collections
Authors: Montgomery Deane
Published: June 2010

The result generalizes an earlier theorem of the author which showed that pointwise periodic homeomorphisms of a connected manifold are periodic.
Multiplier representations of discrete groups
Authors: Kleppner Adam
Published: June 2010

Let

σ
\sigma

be a multiplier on the discrete group

G
G

. Extending theorems of Kaniuth and Thoma to the case of multiplier representations, we determine when the left regular

σ
\sigma

representation of

G
G

has a type I subrepresentation, and when all the

σ
\sigma

representations of

G
G

are type I.
Matrix semigroups
Authors: Putcha Mohan S.
Published: June 2010

Let

S
S

be a semigroup of matrices over a field such that a power of each element lies in a subgroup (i.e., each element has a Drazin inverse within the semigroup). The main theorem of this paper is that there exist ideals

I
0

,
…
,

I
t

{I_0}, \ldots ,{I_t}

of

S
S

such that

I
0

⊆
⋯
⊆

I
t

=
S

{I_0} \subseteq \cdots \subseteq {I_t} = S

,

I
0

{I_0}

is completely simple, and each Rees factor semigroup

I
k

/

I

k
−
1

{I_k}/{I_{k - 1}}

,

k
=
1
,
…
,
t

k = 1, \ldots ,t

, is either completely

0
0

-simple or else a nilpotent semigroup. The basic technique is to study the Zariski closure of

S
S

, which is a linear algebraic semigroup.
On quotient rings of trivial extensions
Authors: Kitamura Yoshimi
Published: June 2010

Let

R
R

be a ring with identity and

M
M

a two-sided

R
R

-module. It is shown that every right quotient ring in the sense of Gabriel of the trivial extension of

R
R

by

M
M

is a trivial extension of a right quotient ring of

R
R

by a suitable two-sided module in case

R

M

_RM

is flat and finitely generated by elements which centralize with every element of

R
R

.
Generalized functions and orthogonal polynomials on the unit circle
Authors: Zayed Ahmed I.
Published: June 2010

In a recent paper, R. Morton and A. Krall introduced a new notion of orthogonality of polynomials on the real line in which a sequence of Chebyshev-type polynomials was shown to be orthogonal with respect to a moment generating linear functional. This linear functional is, in fact, a Schwartz distribution. If the polynomials are orthogonal in the classical sense, then the classical weight function can be recovered from the distributional one by means of the Fourier transform. The purpose of this paper is to study the analogue of their results for polynomials orthogonal on the unit circle.
The numerical range of a weighted shift
Authors: Stout Quentin F.
Published: June 2010

Let

T
T

be a weighted shift on a Hilbert space. We compute the numerical radius of

T
T

when

T
T

is finite, circular, Hilbert-Schmidt, periodic, or a finite perturbation of periodic. For several cases we also determine whether the numerical range is closed, completing the determination of the numerical range and answering a question of Ridge. An important step is the determination of the eigenvalues of a selfadjoint tri-diagonal matrix with zeroes on its diagonal. We give a simple formula for the eigenvalues when the matrix is finite dimensional or Hilbert-Schmidt.
On the product of stable diffeomorphisms
Authors: Sannami Atsuro
Published: June 2010

It is proved that statements (a)-(c) are equivalent, (a) Structural

(
Ω
−
)

(\Omega - )

stability conjecture. (b) If diffeomorphisms

f
f

and

g
g

are both structurally

(
Ω
−
)

(\Omega - )

stable, then so is

f
×
g

f \times g

. (c) If a diffeomorphism

f
f

is structurally

(
Ω
−
)

(\Omega - )

stable, then every diffeomorphism in a neighborhood of

f
×

f

−
1

f \times {f^{ - 1}}

has only hyperbolic periodic points.
An isomorphism theorem for valuated vector spaces
Authors: Hill Paul
Published: June 2010

The following isomorphism theorem is proved for valuated vector spaces. Let

N
N

and

N
′

N’

be nice subspaces of free valuated vector spaces

F
F

and

F
′

F’

, respectively. If

N
N

and

N
′

N’

have isomorphic basic subspaces and if the quotient spaces

F

/

N

F/N

and

F
′

/

N
′

F’/N’

are isomorphic, there exists an isomorphism from

F
F

onto

F
′

F’

that maps

N
N

onto

N
′

N’

and induces the given isomorphism on the quotient spaces. In particular,

N
N

and

N
′

N’

are isomorphic.
Infinite-dimensional Jacobi matrices associated with Julia sets
Authors: Barnsley M. F. , Geronimo J. S. , Harrington A. N.
Published: June 2010

Let

B
B

be the Julia set associated with the polynomial

T
z
=

z
N

+

k
1

z

N
−
1

+
⋯
+

k
N

Tz = {z^N} + {k_1}{z^{N - 1}} + \cdots + {k_N}

, and let

μ
\mu

be the balanced

T
T

-invariant measure on

B
B

. Assuming

B
B

is totally real, we give relations among the entries in the infinite-dimensional Jacobi matrix

J
J

whose spectral measure is

μ
\mu

. The specific example

T
z
=

z
3

−
λ
z

Tz = {z^3} - \lambda z

is given, and some of the asymptotic properties of the entries in

J
J

are presented.
The stability of the sine equation
Authors: Cholewa Piotr W.
Published: June 2010

Let

δ
\delta

be a positive real constant and let

G
G

be an abelian group (written additively) in which division by 2 is uniquely performable. Every unbounded complex-valued function

f
f

on

G
G

satisfying the inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f left-parenthesis x plus y right-parenthesis f left-parenthesis x minus y right-parenthesis minus f left-parenthesis x right-parenthesis squared plus f left-parenthesis y right-parenthesis squared EndAbsoluteValue less-than-or-slanted-equals delta for all x comma y element-of upper G">

|

f
(
x
+
y
)
f
(
x
−
y
)
−
f

(
x
)

2

+
f

(
y
)

2

|

⩽
δ

for all 

x
,
y
∈
G

\left | {f(x + y)f(x - y) - f{{(x)}^2} + f{{(y)}^2}} \right | \leqslant \delta \quad {\text {for all }}x,y \in G

\]
has to be a solution of the sine functional equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x plus y right-parenthesis f left-parenthesis x minus y right-parenthesis equals f left-parenthesis x right-parenthesis squared minus f left-parenthesis y right-parenthesis squared for all x comma y element-of upper G period">

f
(
x
+
y
)
f
(
x
−
y
)
=
f

(
x

)
2

−
f

(
y

)
2

for all 

x
,
y
∈
G
.

f(x + y)f(x - y) = f{(x)^2} - f{(y)^2}\quad {\text {for all }}x,y \in G.

\]
A note on the strong maximal function
Authors: Bagby Richard J.
Published: June 2010

Given a nonnegative measurable function

f
f

on

R
2

{R^2}

which is integrable over sets of finite measure, we construct a new function

g
g

with the same distribution function as

f
f

such that the strong maximal function of

g
g

has the same local integrability properties as its Hardy-Littlewood maximal function.
On the volume of metric balls
Authors: Croke Christopher B.
Published: June 2010

In this paper we consider metrics of the form

d

s
2

=
d

r
2

+

f
2

(
r
,
θ
)
d

θ
2

d{s^2} = d{r^2} + {f^2}(r,\theta )d{\theta ^2}

on a ball of dimension

n
⩾
3

n \geqslant 3

. We show that if the diameters (geodesics through the origin) minimize length then the volume of the ball is larger than the volume of the hemisphere of the corresponding round sphere. This relates to a conjecture first considered by Marcel Berger. We also give examples in all dimensions of radially symmetric metrics on balls of radius

π

/

2

\pi /2

having arbitrarily small volume and yet having no pair of points conjugate along a diameter.
On the nonexistence of Riemannian submersions from 𝐶𝑃(7) and 𝑄𝑃(3)
Authors: Ucci Jack
Published: June 2010

We show the nonexistence of a Serre fibration of the form

CP

(
3
)
→

CP

(
7
)
→

S
8

{\text {CP}}(3) \to {\text {CP}}(7) \to {S^8}

.
Open images of orderable spaces
Authors: Hušek M. , Kulpa W.
Published: June 2010

The class of open continuous images of orderable spaces is characterized, and it is shown that it contains all suborderable spaces. Several results concerning open images of metrizable spaces are generalized to higher cardinals, e.g., every completely linearly uniformizable space is an open continuous image of a Baire space

(

κ
λ

)
λ

{({\kappa ^\lambda })_\lambda }

.
A class of planar quadratic vector fields with a limit cycle surrounded by a saddle loop
Authors: Bamón Rodrigo
Published: June 2010

Examples of planar quadratic vector fields with a limit cycle surrounded by a saddle loop are given by means of deformations of Hamiltonian vector fields.
A remark on Horrocks’ theorem about projective 𝐴[𝑇]-modules
Authors: Nashier Budh S.
Published: June 2010

A projective module

P
P

over

A
[
T
]

A[T]

needs no more generators than

P
f

{P_f}

for a monic

f
f

, and it is nearly determined by

P
f

{P_f}

.
Combinatorial examples in universal coalgebra
Authors: Davis Robert C.
Published: June 2010

If an underlying-set functor different from the usual one is employed, the categories of multigraphs and of combinatorial systems become cotripleable over the category of sets. The category of multigraphs without loops and a common generalization are also examined.
Detecting products of elementary matrices in 𝐺𝐿₂(𝑍[√𝑑])
Authors: Tuler Randy
Published: June 2010

An elementary

n
×
n

n \times n

matrix over a ring

R
R

has 1 in each diagonal position and at most one additional nonzero element. Let

R
=

Z

[

d

]

R = {\mathbf {Z}}[\sqrt d ]

where

d
d

is an integer less than

−
4

- 4

. We give an algorithm for determining whether or not a

2
×
2

2 \times 2

invertible matrix over

R
R

is generated by elementary matrices. This is connected with the theory of integral binary quadratic forms.
Hilbert spaces induced by Hilbert space valued functions
Authors: Saitoh Saburou
Published: June 2010

Let

E
E

be an arbitrary set and

F

(
E
)

\mathcal {F}(E)

a linear space composed of all complex valued functions on

E
E

. Let

H

\mathcal {H}

be a (possibly finite-dimensional) Hilbert space with inner product

(
,

)

H

{(,)_\mathcal {H}}

. Let

h

:
E
→

H

{\mathbf {h}}:E \to \mathcal {H}

be a function and consider the linear mapping

L
L

from

H

\mathcal {H}

into

F

(
E
)

\mathcal {F}(E)

defined by

(

F

,

h

(
p
)

)

H

{({\mathbf {F}},{\mathbf {h}}(p))_\mathcal {H}}

. We let

H

~

\tilde {\mathcal {H}}

denote the range of

L
L

. Then we assert that

H

~

\tilde {\mathcal {H}}

becomes a Hilbert space with a reproducing kernel composed of functions on

E
E

, and, moreover, it is uniquely determined by the mapping

L
L

, in a sense. Furthermore, we investigate several fundamental properties for the mapping

L
L

and its inverse.
Transfers, centers, and group cohomology
Authors: Gottlieb Daniel H.
Published: June 2010

The transfer for fibrations is shown to exist for fibres with finitely generated total integral homology groups. This improvement is applied to cohomology of groups.
On the Bures distance and the Uhlmann transition probability of states on a von Neumann algebra
Authors: Kosaki Hideki
Published: June 2010

Simple expressions for the Bures distance and the Uhlmann transition probability of states on a von Neumann algebra are obtained. Based on these expressions, certain properties are immediately derived.
Some results connected with a problem of Erdős. III
Authors: Arias de Reyna J.
Published: June 2010

It is shown that if

E
E

is a subset with more than two points of the real line, then there exists a subset

S
S

of the unit interval, such that

S
S

has outer Lebesgue measure one and

S
S

is of the second Baire category and such that

S
S

does not contain a subset similar (in the sense of elementary geometry) to

E
E

. This result is related to a conjecture of P. Erdös.
A note on chains of open sets
Authors: Ginsburg John
Published: June 2010

We consider some questions concerning the nature and size of chains of open sets in Hausdorff spaces. The following results are obtained. Theorem 1. For every cardinal

κ
\kappa

there exists a space

X
X

in which all discrete subsets have cardinality at most

κ
\kappa

and which contains a chain of

(

2
κ

)
+

{({2^\kappa })^ + }

open sets. Theorem 2. If

X
X

is regular and contains a chain of

(

2
κ

)
+

{({2^\kappa })^ + }

open sets, then

X
×
X

X \times X

contains a discrete subset of cardinality

κ
+

{\kappa ^ + }

. Theorem 3. Let

M
(
X
)

M(X)

denote the set of all maximal chains of open subsets of

X
X

endowed with the Tychonoff topology. (i)

|

M
(
X
)

|

⩽

2

w

(
X
)

\left | {M(X)} \right | \leqslant {2^{{\text {w}}(X)}}

, and (ii)

ψ
(
M
(
X
)
)
⩽

w

(
X
)

\psi (M(X)) \leqslant {\text {w}}(X)

. Here

w

(
X
)

{\text {w}}(X)

denotes the weight of the space

X
X

and

ψ
(
M
(
X
)
)

\psi (M(X))

denotes the pseudocharacter of the space

M
(
X
)

M(X)

.
Cardinal functions on Pixley-Roy hyperspaces
Authors: Sakai Shozo
Published: June 2010

Some cardinal functions on the Pixley-Roy hyperspace

F

[
X
]

\mathcal {F}[X]

of a space

X
X

are determined by those on

X
X

, and conditions of

X
X

by cardinal functions from which

F

[
X
]

\mathcal {F}[X]

has or has not some properties, e.g. cosmic, paracompact, ccc, etc. are given.
Reducibilities among decision problems for HNN groups, vector addition systems and subsystems of Peano arithmetic
Authors: Anshel Michael , McAloon Kenneth
Published: June 2010

Our purpose is to exhibit reducibilities among decision problems for conjugate powers in HNN groups, reachability sets of vector addition systems and sentences in subsystems of Peano arithmetic, and show that although these problems are not primitive recursively decidable, they do admit decision procedures which are primitive recursive in the Ackermann function.
Eventual disconjugacy of a linear differential equation
Authors: Trench William F.
Published: June 2010

A sufficient condition is given for eventual disconjugacy of an

n
n

th order linear differential equation.
Remarks on sectional curvature of an indefinite metric
Authors: Nomizu Katsumi
Published: June 2010

The properties of sectional curvature for an indefinite metric are further studied following the works of R. S. Kulkarni, L. Graves and the author, S. G. Harris, and M. Dajczer and the author.
Irreducible 3-manifolds of genus 3 containing a 2-sided projective plane
Authors: Negami Seiya
Published: June 2010

It will be shown that the minimum genus over all Heegaard splittings which yield irreducible closed

3
3

-manifolds containing a

2
2

-sided projective plane but not homeomorphic to

P
2

×

S
1

{P^2} \times {S^1}

is equal to 3, with an infinite number of examples of genus 3.
The combinatorics of certain products
Authors: Rawlings Don
Published: June 2010

A combinatorial interpretation for the coefficients in the expansion of

Π
(
1
+
u

x
j

y
k

)

(
1
−
u

x
j

y
k

)

−
1

\Pi (1 + u{x^j}{y^k}){(1 - u{x^j}{y^k})^{ - 1}}

is given.
A symmetric star polyhedron that tiles but not as a fundamental domain
Authors: Szabó Sándor
Published: June 2010

In [7] S. K. Stein constructed a 10-dimensional centrally-symmetric star body whose translates tile

10
10

-space but whose translates by a lattice do not tile it. In [8] he constructed a

5
5

-dimensional star polyhedron whose translates tile

5
5

-space but whose congruent copies by a group of motions do not tile it. So there is no lattice tiling by translates of this polyhedron. In the present paper we shall construct a

5
5

-dimensional centrally-symmetric star polyhedron whose translates tile

5
5

-space but whose congruent copies by a group of motions do not tile it. Furthermore, this phenomenon occurs at an infinitude of dimensions.
Presentation of associated graded rings of Cohen-Macaulay local rings
Authors: Cho Young-Hyun
Published: June 2010

Let

(
R
,

m

)

(R,\mathfrak {m})

be a local ring and

I
I

be an

m

\mathfrak {m}

-primary ideal such that

dim
k

(
I

/

I

m

)
=
l

{\dim _k}(I/I\mathfrak {m}) = l

, where

k
=
R

/

m

k = R/\mathfrak {m}

. Denote the associated graded ring with respect to

I
,

⊕

n
=
0

∞

I
n

/

I

n
+
1

I, \oplus _{n = 0}^\infty {I^n}/{I^{n + 1}}

, by

G
I

(
R
)

{G_I}(R)

. Then

G
I

(
R
)
≃
R

/

I
[

X
1

,
…

X
l

]

/

L

{G_I}(R) \simeq R/I[{X_1}, \ldots {X_l}]/\mathcal {L}

, for some homogeneous ideal

L

\mathcal {L}

. Set

M
=
max

deg

1
⩽
i
⩽
t

f
i

M = \max {\deg _{1 \leqslant i \leqslant t}}{f_i}

, where

{

f
1

,
…
,

f
t

}

\{ {f_1}, \ldots ,{f_t}\}

is a set of homogeneous elements which form a minimal basis of

L

\mathcal {L}

. The main result in this note is that if

R
R

is a Cohen-Macaulay local ring of dimension 1 and if

G
I

(
R
)

{G_I}(R)

is free over

R

/

I

R/I

, then

M
⩽
r
(
I
)
+
1

M \leqslant r(I) + 1

, where

r
(
I
)

r(I)

is the reduction number of

I
I

. It follows that

M
⩽
e
(
R
)

M \leqslant e(R)

where

e
(
R
)

e(R)

is the multiplicity of

R
R

.
Uniqueness of Taylor’s functional calculus
Authors: Putinar Mihai
Published: June 2010

Two uniqueness results concerning Fréchet module structures over algebras of holomorphic functions defined on some complex manifolds are presented, containing as particular cases uniqueness theorems for J. L. Taylor’s analytic functional calculi for commuting

n
n

-tuples of linear continuous operators on Fréchet spaces [7], [9]. Namely, the first statement says that the Spectral Mapping Theorem insures the unicity of the functional calculus and thus it improves Zame’s unicity theorem [11, Theorem 1], while the second statement gives a unicity condition which is an analogue of the compatibility property [3, Theorem I.4.1] in spectral theory of several variables in commutative Banach algebras. As a corollary the two functional calculi constructed in [7] and [9] by J. L. Taylor coincide.
Positive martingales and their induced measures
Authors: Eisenberg Bennett , Gan Shi Xin
Published: June 2010

Conditions for the absolute continuity of probability measures are given in terms of limits of sequences of Radon-Nikodym derivatives. In the other direction, conditions for the optional stopping theorem for positive martingales are given in terms of properties of their induced measures.
𝐺-spaces with prescribed equivariant cohomology
Authors: Triantafillou Georgia
Published: June 2010

Let

G
G

be a finite group. In this note we study the question of realizing a collection of graded commutative algebras over

Q

{\mathbf {Q}}

as the cohomology algebras with rational coefficients of the fixed point sets

X
H

(
H
>
G
)

{X^H}(H > G)

of a

G
G

-space

X
X

.
On the Reidemeister moves of a classical knot
Authors: Trace Bruce
Published: June 2010

We show that under certain reasonable criteria one of the Reidemeister moves can be elminated.
Weighted group algebra as an ideal in its second dual space
Authors: Ghahramani F.
Published: June 2010

For a locally compact group

G
G

let

L
1

(
G
,
ω
λ
)

{L^1}(G,\omega \lambda )

be a weighted group algebra. We characterize compact and weakly compact multipliers on

L
1

(
G
,
ω
λ
)

{L^1}(G,\omega \lambda )

. This characterization is employed to find a necessary and sufficient condition for

L
1

(
G
,
ω
λ
)

{L^1}(G,\omega \lambda )

to be an ideal in its second dual space, where the second dual is equipped with an Arens product. In the special case where

ω
(
t
)
=
1
(
t
∈
G
)

\omega (t) = 1(t \in G)

, we deduce a result due to K. P. Wong that if

G
G

is a compact group, then

L
1

(
G
,
λ
)

{L^1}(G,\lambda )

is an ideal in its second dual space and its converse due to S. Watanabe.
On a problem of Erdős on sequences and measurable sets
Authors: Falconer K. J.
Published: June 2010

Erdös has conjectured that given a decreasing sequence of real numbers convergent to 0 there always exists a measurable set of positive measure that contains no similar copy of the sequence. We prove this conjecture if the sequence does not converge too rapidly.
Orientable products in 𝒩
Authors: Osborn Howard
Published: June 2010

It is well known that the square

W
×
W

W \times W

of any smooth closed manifold

W
W

is cobordant to an orientable manifold. This note shows more specifically that a product

U
×
V

U \times V

of smooth closed manifolds

U
U

and

V
V

is cobordant to an orientable manifold if and only if there is a smooth closed manifold

W
W

such that

U
U

and

V
V

are both products of

W
W

by orientable manifolds.
On the inertia groups of ℎ-cobordant manifolds
Authors: Frame Michael
Published: June 2010

It is shown that if

M
M

and

N
N

are

h
h

-cobordant manifolds of dimension at least eight, then their special inertia groups are equal. (G. Brumfiel has shown that special inertia groups are not homotopy type invariants.) Examples are constructed for which the inertia group is an

h
h

-cobordism invariant.