bluuebunny/crossref_metadata_2025 huggingface dataset:
Class groups of rank one semisimple monoids
Authors: Renner Lex E.
Published: July 2010

Let

M
M

be an irreducible, normal, algebraic monoid with unit group

S

l
2

(

K

)

×

K

∗

,
G

l
2

(

K

)

S{l_2}\left ( \mathcal {K} \right ) \times {\mathcal {K}^ * },G{l_2}\left ( \mathcal {K} \right )

or

P
G

l
2

(

K

)

×

K

∗

PG{l_2}\left ( \mathcal {K} \right ) \times {\mathcal {K}^ * }

. In [7] these monoids are classified numerically. In this paper we compute explicitly the divisor class group of each monoid. As a corollary we characterize the monoids with factorial coordinate algebra. All results are independent of the characteristic of

K

\mathcal {K}

.
Chain-preserving diffeomorphisms and CR equivalence
Authors: Chêng Jih Hsin
Published: July 2010

It is shown that a diffeomorphism that preserves chains between two nondegenerate CR manifolds is actually either a CR isomorphism or a conjugate CR isomorphism.
Plurisubharmonic functions outside compact sets
Authors: Cegrell Urban
Published: July 2010

We study the Monge-Ampère mass of plurisubharmonic functions defined outside a compact set.
Two mixed Hadamard type generalizations of Heinz inequality
Authors: Furuta Takayuki
Published: July 2010

We give two types of mixed Hadamard inequalities containing the terms

T
,

|
T
|

T,\left | T \right |

, and

|

T
∗

|

\left | {{T^ * }} \right |

, where

T
T

is a bounded linear operator on a complex Hilbert space. As an immediate consequence of these results, we can easily show some extensions of the Hadamard inequality and also the Heinze inequality:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk right-parenthesis StartAbsoluteValue left-parenthesis upper T x comma y right-parenthesis EndAbsoluteValue squared less-than-or-equal-to left-parenthesis StartAbsoluteValue upper T EndAbsoluteValue Superscript 2 alpha Baseline x comma x right-parenthesis left-parenthesis StartAbsoluteValue upper T Superscript asterisk Baseline EndAbsoluteValue Superscript 2 left-parenthesis 1 minus alpha right-parenthesis Baseline y comma y right-parenthesis">

(
∗
)

|

(

T
x
,
y

)

|

2

≤

(

|
T
|

2
α

x
,
x

)

(

|

T
∗

|

2

(

1
−
α

)

y
,
y

)

\left ( * \right )\quad {\left | {\left ( {Tx,y} \right )} \right |^2} \leq \left ( {{{\left | T \right |}^{2\alpha }}x,x} \right )\left ( {{{\left | {{T^ * }} \right |}^{2\left ( {1 - \alpha } \right )}}y,y} \right )

\]
for any

T
T

, any

x
,
y

x,y

in

H
H

, and any real number

α
\alpha

with

0
≤
α
≤
1

0 \leq \alpha \leq 1

. And the following conditions are equivalent in case

0
>
α
>
1

0 > \alpha > 1

: (1) the equality in (*) holds; (2)

|
T
|

2
α

x

{\left | T \right |^{2\alpha }}x

and

T
∗

y

{T^ * }y

are linearly dependent; (3)

T
x

Tx

and

|

T
∗

|

2

(

1
−
α

)

y

{\left | {{T^ * }} \right |^{2\left ( {1 - \alpha } \right )}}y

are linearly dependent. Results in this paper would remain valid for unbounded operators under slight modifications.
Representation theory of 𝑈₁(𝐻)
Authors: Boyer Robert P.
Published: July 2010

Mixed tensor representations of the inductive limit unitary group are studied according to factoriality, quasi-equivalence, and irreducibility
On semiregular points of the Martin boundary
Authors: Cornea Aurel , Loeb Peter A.
Published: July 2010

For a selfadjoint, Brelot harmonic space with Martin boundary, call "semiregular" a point

z
z

of the boundary at which the Green function has a positive upper limit; call "maximal" the filter

F

z

{\mathcal {F}_z}

converging to

z
z

along which that upper limit is attained. If such a

z
z

is minimal then any bounded harmonic function has a limit along

F

z

{\mathcal {F}_z}

and

F

z

{\mathcal {F}_z}

is coarser than the fine filter associated with

z
z

. The semiregular points form a set of harmonic measure zero.
The validity of Beurling theorems in polydiscs
Authors: Mandrekar V.
Published: July 2010

This paper gives necessary and sufficient conditions for an invariant subspace

M

\mathcal {M}

of

H
2

(

T
2

)

{H^2}({T^2})

to be of the form

q

H
2

(

T
2

)

q{H^2}({T^2})

(

q
q

inner) in terms of double commutativity of the shifts. Recent results in [8] follow directly from our work. Relation to the work in [1] is also discussed.
A characterization of nonlinear semigroups with smoothing effect
Authors: Watanabe Michiaki
Published: July 2010

Let

{
S
(
t
)
:
t
>
0
}

\{ S(t):t > 0\}

be a nonlinear semigroup of operators mapping a closed subset

C
C

of a real Banach space

X
X

into itself. Conditions are found for an accretive operator in

X
X

to be the generator of

{
S
(
t
)
:
t
>
0
}

\{ S(t):t > 0\}

with smoothing effect: For each
\[

x
∈
C
,

S
(
t
)
x
∈
V

a

.e

.

t > 0,

x \in C,\;S(t)x \in V\;{\text {a}}{\text {.e}}{\text {.}}\;{\text {t > 0,}}

\]
among other things, where

V
V

is a Banach space imbedded continuously in

X
X

. The conditions contain a Gårding-type inequality, and are shown also to be necessary if

C
C

is a closed convex subset of a "nice" Banach space

X
X

.
A Runge theorem for harmonic functions on closed subsets of Riemann surfaces
Authors: Bagby Thomas
Published: July 2010

Let

F
F

be a closed subset of a Riemann surface

Ω
\Omega

. It is shown that every function which is harmonic on a neighborhood of

F
F

can be uniformly approximated on

F
F

by functions which are harmonic on

Ω
\Omega

except for poles.
Martin boundaries of Denjoy domains
Authors: Segawa Shigeo
Published: July 2010

Let

E
(
⊂

C
^

)

E( \subset {\mathbf {\hat C}})

be a compact set in the real axis. It is shown that there exists an

E
E

with zero linear measure such that Martin compactification of the domain

C
^

−
E

{\mathbf {\hat C}} - E

is not homeomorphic to

C
^

{\mathbf {\hat C}}

. Moreover, it is shown that if for some

λ
>

1
2

\lambda > \tfrac {1}{2}

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction StartAbsoluteValue upper E Superscript c Baseline intersection left-bracket negative t comma t right-bracket EndAbsoluteValue Over t EndFraction equals upper O left-parenthesis left-parenthesis StartFraction 1 Over log t Superscript negative 1 Baseline EndFraction right-parenthesis Superscript lamda Baseline right-parenthesis left-parenthesis t right-arrow 0 right-parenthesis comma">

|

E
c

∩
[
−
t
,
t
]

|

t

=
O

(

(

1

log
⁡

t

−
1

)

λ

)

(
t
→
0
)
,

\frac {{|{E^c} \cap [ - t,t]|}}{t} = O\left ( {{{\left ( {\frac {1}{{\log {t^{ - 1}}}}} \right )}^\lambda }} \right )\quad (t \to 0),

\]
the set of minimal Martin boundary points of

C
^

−
E

{\mathbf {\hat C}} - E

’over 0’ consists of two points. This assertion is not valid for

λ
=

1
2

\lambda = \tfrac {1}{2}

.
The Franklin system as Schauder basis for 𝐿^{𝑝}_{𝜇}[0,1]
Authors: Zink Robert E.
Published: July 2010

The present work is devoted to a characterization of those spaces

L
μ
p

[
0
,
1
]
,
p
≥
1
,
μ

L_\mu ^p[0,1],p \geq 1,\mu

totally finite, for which

F

\mathcal {F}

, the Franklin system, is a Schauder basis. Because, in such cases, the measure

μ
\mu

must be absolutely continuous with respect to the Lebesgue measure,

m
m

, the necessary and sufficient condition is expressible in terms of a weight function,

W
W

, the Radon-Nikodym derivative of

μ
\mu

with respect to

m
m

. One finds that the spaces

L
μ
p

[
0
,
1
]

L_\mu ^p[0,1]

for which

F

\mathcal {F}

serves as Schauder basis are precisely those for which

W
W

satisfies the

A
p

{A_p}

condition introduced by Muckenhoupt. On the other hand, Krancberg has shown that a much less restrictive condition on

μ
\mu

is both necessary and sufficient for the Haar system,

H

\mathcal {H}

, to be a Schauder basis for

L
μ
p

[
0
,
1
]

L_\mu ^p[0,1]

. Thus, as a mildly surprising corollary of the theorem contained herein, one finds that the class of spaces

L
μ
p

[
0
,
1
]

L_\mu ^p[0,1]

, for which

H

\mathcal {H}

is a Schauder basis, properly contains the corresponding class of spaces for which the Franklin system so serves.
Polynomially moving ergodic averages
Authors: Schwartz Mark
Published: July 2010

Given an increasing sequence of positive integers

{

m
n

}

\left \{ {{m_n}} \right \}

, a non-decreasing sequence of positive integers

{

b
n

}

\left \{ {{b_n}} \right \}

, and a measurable, measure-preserving ergodic transformation

τ
\tau

on a probability space

(

Ω
,

F

,
μ

)

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

, the a.s. convergence of the moving averages
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript n Baseline left-parenthesis f right-parenthesis equals b Subscript n Superscript negative 1 Baseline sigma-summation Underscript k equals m Subscript n Baseline plus 1 Overscript m Subscript n Baseline plus b Subscript n Baseline Endscripts f left-parenthesis tau Superscript k Baseline right-parenthesis">

T
n

(
f
)

=

b
n

−
1

∑

k
=

m
n

+
1

m
n

+

b
n

f

(

τ
k

)

{T_n}\left ( f \right ) = b_n^{ - 1}\sum \nolimits _{k = {m_n} + 1}^{{m_n} + {b_n}} {f\left ( {{\tau ^k}} \right )}

is considered, for

f
∈

L
p

(
Ω
)

f \in {L_p}\left ( \Omega \right )

. A counterexample is constructed in the case of polynomial-like

{

m
n

}

\left \{ {{m_n}} \right \}

.
Homogeneous polynomials on the ball of 𝐶²
Authors: Globevnik Josip
Published: July 2010

Let

B
2

{B_2}

be the open unit ball of

C
2

{C^2}

. Let

T
=

{

(

u
,
w

)

∈
b

B
2

:
1
−
a
≤

|
u
|

2

≤
a

T = \left \{ {\left ( {u,w} \right ) \in b{B_2}:1 - a \leq {{\left | u \right |}^2} \leq a} \right .

where

1

/

2
>
a
>
1

/

2
+

3

/

4

1/2 > a > 1/2 + \sqrt 3 /4

. For each

n
∈

N

n \in {\mathbf {N}}

we construct a homogeneous polynomial

H
n

:

C
2

→

C
2

{H_n}:{C^2} \to {C^2}

of degree

n
n

such that

|

H
n

|

≤
1

\left | {{H_n}} \right | \leq 1

on

B
2

{B_2}

and

|

H
n

|

≥
c

\left | {{H_n}} \right | \geq c

on

T
T

where

c
>
0

c > 0

does not depend on

n
n

.
Horizontal holomorphic curves in 𝑆𝑝(𝑛)-flag manifolds
Authors: Yang Kichoon
Published: July 2010

A complete description of horizontal holomorphic curves in

Sp

(
n
)

{\text {Sp}}\left ( n \right )

-flag manifolds is given. For compact curves a Plücker type integral formula is derived.
The cardinality of reduced power set algebras
Authors: Taylor Alan D.
Published: July 2010

We prove a general result on the cardinality of reduced powers of structures via filters that has several consequences including the following: if

I
I

is a uniform, countably complete ideal on the real line

R

\mathcal {R}

and

B

\mathcal {B}

is the Boolean algebra of subsets of

R

\mathcal {R}

modulo

I
I

, then

|

B

|

>

2

ℵ
0

\left | \mathcal {B} \right | > {2^{{\aleph _0}}}

and if

2
ν

≤

2

ℵ
0

{2^\nu } \leq {2^{{\aleph _0}}}

for all

ν
>

2

ℵ
0

\nu > {2^{{\aleph _0}}}

then

|

B

|

=

2

2

ℵ
0

\left | \mathcal {B} \right | = {2^{{2^{{\aleph _0}}}}}

. This strengthens some results of Kunen and Pelc [7] and Prikry [8] obtained by Boolean ultrapower techniques. Our arguments are all combinatorial and some applications are included.
On the relationship of AP, RS and CEP in congruence modular varieties. II
Authors: Bergman Clifford , McKenzie Ralph
Published: July 2010

Let

V
V

be a congruence distributive variety, or a congruence modular variety whose free algebra on 2 generators is finite. If

V
V

is residually small and has the amalgamation property, then it has the congruence extension property. Several applications are presented.
A special case of positivity
Authors: Dutta S. P.
Published: July 2010

In this note we prove a special case of positivity of Serre’s Conjecture on intersection multiplicity of modules [S]. The conjecture can be stated as follows.
The longest chain among random points in Euclidean space
Authors: Bollobás Béla , Winkler Peter
Published: July 2010

Let

n
n

random points be chosen independently from the uniform distribution on the unit

k
k

-cube

[
0
,
1

]
k

{[0,1]^k}

. Order the points coordinate-wise and let

H

k

(
n
)

{{\mathbf {H}}_k}\left ( n \right )

be the cardinality of the largest chain in the resulting partially ordered set. We show that there are constants

c
1

,

c
2

,
…

{c_1},{c_2}, \ldots

such that

c
k

>
e
,

lim

k
→
∞

c
k

=
e

{c_k} > e,\;{\lim _{k \to \infty }}{c_k} = e

, and

lim

n
→
∞

H

k

(
n
)

/

n

1

/

k

=

c
k

{\lim _{n \to \infty }}{{\mathbf {H}}_k}\left ( n \right )/{n^{1/k}} = {c_k}

in probability. This generalizes results of Hammersley, Kingman and others on Ulam’s ascending subsequence problem, and settles a conjecture of Steele.
The cancellation problem for function fields
Authors: Deveney James K.
Published: July 2010

Let

L
L

be a finitely generated extension of

K
K

. We call

L
L

rigid over

K
K

if the set of

K
K

-endomorphisms of

L
L

is finite. If

L
L

is rigid over

K
K

and

x
x

is transcendental over

L
L

, then

L
L

is invariant under automorphisms of

L

(
x
)

L\left ( x \right )

over

K
K

(Theorem 2). This result is used to show that the cancellation property holds for function fields of varieties of general type in characteristic 0.
On the finiteness of the projective dimension of certain ideals
Authors: Rodicio Antonio G.
Published: July 2010

For a local noetherian ring, a subset of a minimal system of generators of the maximal ideal is a regular sequence if the ideal it generates has finite projective dimension.
The spectra of certain elliptic operators in 𝑅^{𝑁}
Authors: Brenner Terence
Published: July 2010

We give sufficient conditions for the spectra and essential spectra of certain classes of operators in

R
N

{R^N}

to be contained in an interval of the form

[
d
,
∞
)

[d,\infty )

.
Infinite tensor products of commutative subspace lattices
Authors: Wagner Bruce H.
Published: July 2010

Every infinite tensor product of commutative subspace lattices is unitarily equivalent to a certain lattice of projections on

L
2

(
X
,
ν
)

{L^2}(X,\nu )

, where

(
X
,
ν
)

(X,\nu )

is an infinite product measure space. This representation reflects the structure of the individual component lattices in that the components of the tensor product correspond to the coordinates of the product space. This result generalizes the similar representation for finite tensor products. It is then used to show that an infinite tensor product of purely atomic commutative subspace lattices must be either purely atomic or noncompact, and in the latter case the algebra of operators under which the lattice is invariant has no compact operators.
Compact homomorphisms of 𝐶*-algebras
Authors: Ghahramani F.
Published: July 2010

Suppose

A
A

is a

C
∗

{C^*}

-algebra and

B
B

is a Banach algebra such that it can be continuously imbedded in

B
(
H
)

B(H)

, the Banach algebra of bounded linear operators on some Hilbert space

H
H

. It is shown that if

θ
\theta

is a compact algebra homomorphism from

A
A

into

B
B

, then

θ
\theta

is a finite rank operator, and the range of

θ
\theta

is spanned by a finite number of idempotents. If, moreover,

B
B

is commutative, then

θ
\theta

has the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta left-parenthesis x right-parenthesis equals script upper X 1 left-parenthesis x right-parenthesis upper E 1 plus midline-horizontal-ellipsis plus script upper X Subscript k Baseline left-parenthesis x right-parenthesis upper E Subscript k">

θ
(
x
)
=

X

1

(
x
)

E
1

+
⋯
+

X

k

(
x
)

E
k

\theta (x) = {\mathcal {X}_1}(x){E_1} + \cdots + {\mathcal {X}_k}(x){E_k}

, where

E
1

,
…
,

E
k

{E_1}, \ldots ,{E_k}

are fixed mutually orthogonal idempotents in

B
B

and

X

1

,
…
,

X

k

{\mathcal {X}_1}, \ldots ,{\mathcal {X}_k}

are fixed multiplicative linear functionals on

A
A

.
The density of peak points in the Shilov boundary of a Banach function algebra
Authors: Honary Taher G.
Published: July 2010

H. G. Dales has proved in [1] that if

A
A

is a Banach function algebra on a compact metrizable space

X
X

, then

S
¯

0

(
A
,
X
)
=
Γ
(
A
,
X
)

{\bar S_0}(A,X) = \Gamma (A,X)

, where

S
0

(
A
,
X
)

{S_0}(A,X)

is the set of peak points of

A
A

(w.r.t.

X
X

) and

Γ
(
A
,
X
)

\Gamma (A,X)

is the Shilov boundary of

A
A

(w.r.t.

X
X

). Here, by considering the relation between peak sets and peak points of a Banach function algebra

A
A

and its uniform closure

A
¯

\bar A

, we present an elementary and constructive proof of the density of peak points in the Shilov boundary.
Convex sets and subharmonicity of the distance function
Authors: Parker M. J.
Published: July 2010

In this paper we show that in order for a proper closed subset

F
F

of a plane domain

D
D

to be convex it is necessary and sufficient that

dist
⁡
(
⋅
,
F
)

\operatorname {dist}( \cdot ,F)

be subharmonic in

D
D

. We also show that this result does not hold for higher dimensions.
Weighted norm inequalities and uniform algebras
Authors: Nakazi Takahiko
Published: July 2010

Generalizations of the classical conjugation operator can be defined on a general uniform algebra. In this paper the

L
2

{L^2}

weighted problems for the conjugation operators are considered and it is shown that the weights have forms similar to the classical case. The results in this paper have applications to concrete uniform algebras, for example, a polydisc algebra and a uniform algebra which consists of rational functions.
On the representation formulas for the functions in class Σ*(𝑝,𝑤₀)
Authors: Chang Yuh Lin
Published: July 2010

We prove the converse of J. Miller’s integral representation theorem for functions in the class

Σ
∗

(
p
,

w
0

)

{\Sigma ^ * }(p,{w_0})

of starlike meromorphic univalent functions. As an application we improve the bounds of the modulus. We also observe some properties of the class

ST

(
P
)

=

∪

w
0

Σ
∗

(

p
,

w
0

)

{\text {ST}}\left ( P \right ) = { \cup _{w0}}{\Sigma ^ * }\left ( {p,{w_0}} \right )

.
When do Sobolev spaces form a Hilbert scale?
Authors: Neubauer Andreas
Published: July 2010

In this paper we show that the usual Sobolev spaces

(

H
s

(
Ω
)

)

s
∈

R

{\left ( {{H^s}\left ( \Omega \right )} \right )_{s \in {\mathbf {R}}}}

are no Hilbert scale in the sense of Krein-Petunin, if

Ω
\Omega

is an open bounded subset of

R

n

{{\mathbf {R}}^n}

.
Too long shadow boundaries
Authors: Zamfirescu Tudor
Published: July 2010

It is shown that, in the sense of Baire categories, most

d
d

-dimensional convex bodies have infinitely long shadow boundaries if the light comes along one of many

(

d
−
2

)

\left ( {d - 2} \right )

-dimensional subspaces. This reveals (once again!) a striking contrast between the categorical and the measure-theoretical points of view.
Basins for general nonlinear Hénon attracting sets
Authors: Patterson Sam , Robinson Clark
Published: July 2010

When a new homoclinic intersection is created for a dissipative diffeomorphism in dimension two, there results a cascade of sinks. We show that immediately after one of these sinks

q
n

{q_n}

is formed, its basin boundary is made up of the stable manifold of the saddle periodic point

p
n

{p_n}

formed at the same time. After this sink undergoes a period doubling, there still remains a trapping region with an attracting set inside. In fact, we show that until this saddle periodic point

p
n

{p_n}

has its own homoclinic bifurcation, there is an attracting set whose boundary is made up of the stable manifold of

p
n

{p_n}

. By picking a rectangle

B
n

{B_n}

carefully, the one-parameter family of maps

f

t
n

{f_{{t^n}}}

creates these sinks and attracting sets by pulling the image

f

t
n

(

B
n

)

{f_{{t^n}}}({B_n})

across

B
n

{B_n}

and eventually forming a horseshoe in

B
n

{B_n}

. The maps,

f

t
n

{f_{{t^n}}}

on

B
n

{B_n}

, are well approximated for large

n
n

by quadratic maps equivalent to the Hénon map. We prove our results for general nonlinear Hénon maps which include not only the quadratic maps but also other nonlinear maps which also create horseshoes, including those arising from homoclinic tangencies.
Area-preserving homeomorphisms of the open disk without fixed points
Authors: Alpern Steve
Published: July 2010

D. G. Bourgin has proved that every measure-preserving orientation-preserving homeomorphism of the open two-dimensional disk

D
D

has a fixed point. He suggested that the "result is perhaps valid even if the condition of orientability preservation be dropped." We show that on the contrary there exist fixed point free homeomorphisms of

D
D

which preserve any given finite nonatomic locally positive Borel measure. Examples are also constructed in all higher dimensions.
Fine uniformity and the locally finite hyperspace topology
Authors: Naimpally S. A. , Sharma P. L.
Published: July 2010

It is shown that a uniformizable space

X
X

is normal iff the locally finite topology

e
τ

{e^\tau }

on the hyperspace

2
X

{2^X}

coincides with the topology transmitted by the fine uniformity of

X
X

. We also prove that, for

X
X

normal, the topology

e
τ

{e^\tau }

is first countable only if the set of limit points

X
′

X’

of

X
X

is countably compact. Applications of these results to pseudocompactness and Atsuji spaces are given.
Sheaves with finitely generated isomorphic stalks and homology manifolds
Authors: Dydak Jerzy , Walsh John
Published: July 2010

The setting is sheaves of modules over a commutative ring

L
L

. It is shown that on completely metrizable spaces certain sheaves having mutually isomorphic finitely generated stalks are locally constant over a dense open subset. This is used to show that a locally compact metrizable space

X
X

that is homologically locally connected with respect to a principal ideal domain

L
L

is a homology manifold over

L
L

provided it has finite cohomological dimension with respect to

L
L

and, for any two points

x
,
y
∈
X

x,y \in X

, the modules

H
k

(
X
,
X
−
{
x
}
;
L
)

{H_k}(X,X - \{ x\} ;L)

and

H
k

(
X
,
X
−
{
y
}
;
L
)

{H_k}(X,X - \{ y\} ;L)

are isomorphic and finitely generated.
𝑘-to-1 functions between graphs with finitely many discontinuities
Authors: Heath Jo
Published: July 2010

This paper classifies which pairs of graphs

G
G

and

H
H

and which integers

k
≥
2

k \geq 2

allow

k
k

-to-

1
1

functions from

G
G

onto

H
H

with only finitely many discontinuities.
Remark on a generalized quasivariational inequality
Authors: Kim Won Kyu
Published: July 2010

In this note, we prove a generalization of a generalized quasi-variational inequality of Shih and Tan in locally convex Hausdorff topological vector spaces.
Universal spaces for locally finite-dimensional Tychonoff spaces
Authors: Charalambous M. G.
Published: July 2010

We prove that the class of all locally finite-dimensional Tychonoff spaces and other related classes have universal elements.
Operator algebras and the conjugacy of transformations. II
Authors: Hadwin Don , Hoover T. B.
Published: July 2010

We prove that two automorphisms of

L
∞

{L^\infty }

-spaces are conjugate if and only if certain related operator algebras are algebraically isomorphic. This extends a result of W. Arveson by dropping the assumptions that the automorphisms are ergodic and measure-preserving.
A reduction theorem for the Zariski multiplicity conjecture
Authors: Massey David B.
Published: July 2010

We prove that the Zariski multiplicity conjecture for families of hypersurfaces of dimension

≠
2

{}\ne 2

with isolated singularities is equivalent to the conjecture for families of hypersurfaces with line singularities.
Primitive elements in free groups
Authors: Evans Martin J.
Published: July 2010

Let

F
n

{F_n}

denote the free group of rank

n
n

generated by

x
1

,

x
2

,
…
,

x
n

{x_1},{x_2}, \ldots ,{x_n}

. We say that

y
∈

F
n

y \in {F_n}

is a primitive element of

F
n

{F_n}

if it is contained in a set of free generators of

F
n

{F_n}

. In this note we construct, for each integer

n
≥
4

n \geq 4

, an

(
n
−
1
)

(n - 1)

-generator group

H
H

that has an

n
n

-generator,

2
2

-relator presentation

H
=
⟨

x
1

,
…
,

x
n

|

r
1

,

r
2

⟩

H = \langle {x_1}, \ldots ,{x_n}|{r_1},{r_2}\rangle

such that the normal closure of

{

r
1

,

r
2

}

\{ {r_1},{r_2}\}

in

F
n

{F_n}

does not contain a primitive element of

F
n

{F_n}

.
Facteurs premiers de sommes d’entiers
Authors: Tenenbaum Gérald
Published: July 2010

Let

Ω
(
n
)

\Omega (n)

denote the total number of prime factors of the positive integer

n
n

, and set for real

x
x

,

α
\alpha

, and integer

k
≥
1

k \geq 1

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript k Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript StartLayout 1st Row n less-than-or-slanted-equals x 2nd Row normal upper Omega left-parenthesis n right-parenthesis equals k EndLayout Endscripts 1 comma pi Subscript k Baseline left-parenthesis x comma alpha right-parenthesis equals sigma-summation Underscript StartLayout 1st Row n less-than-or-slanted-equals x 2nd Row normal upper Omega left-parenthesis n right-parenthesis equals k EndLayout Endscripts bold e left-parenthesis alpha n right-parenthesis comma upper E left-parenthesis x comma alpha right-parenthesis equals x Superscript negative 1 Baseline sigma-summation Underscript n less-than-or-slanted-equals x Endscripts bold e left-parenthesis alpha n right-parenthesis comma">

π
k

(
x
)
=

∑

n
⩽
x

Ω
(
n
)
=
k

1
,

π
k

(
x
,
α
)
=

∑

n
⩽
x

Ω
(
n
)
=
k

e

(
α
n
)
,

E
(
x
,
α
)
=

x

−
1

∑

n
⩽
x

e

(
α
n
)
,

\pi _k(x) = \sum \limits _{\substack {n \leqslant x \\ \Omega (n) = k}} 1, \quad \pi _k(x, \alpha ) = \sum \limits _{\substack {n \leqslant x \\ \Omega (n) = k}} \mathbf {e}(\alpha n), \quad E(x,\alpha ) = x^{-1}\sum \limits _{n \leqslant x} \mathbf {e}(\alpha n),

\]
where

e

(
t
)
:=
exp
⁡
(
2
π
i
t
)

{\mathbf {e}}(t): = \exp (2\pi it)

. We establish a best possible "independence" result of the type
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript k Baseline left-parenthesis x comma alpha right-parenthesis slash pi Subscript k Baseline left-parenthesis x right-parenthesis equals upper E left-parenthesis x comma alpha right-parenthesis plus upper O left-parenthesis delta Subscript k Baseline left-parenthesis x right-parenthesis right-parenthesis">

π
k

(
x
,
α
)

/

π
k

(
x
)
=
E
(
x
,
α
)
+
O
(

δ
k

(
x
)
)

{\pi _k}(x,\alpha )/{\pi _k}(x) = E(x,\alpha ) + O({\delta _k}(x))

\]
which is valid uniformly in

x
,
k
,
α

x,k,\alpha

, and where the error

δ
k

(
x
)

{\delta _k}(x)

tends to 0 as

x
→
+
∞

x \to + \infty

, if, and only if,

k
∼
log
⁡
log
⁡
x

k \sim \operatorname {log} \operatorname {log} x

. As an application we prove a recent conjecture of Erdös, Maier, and Sárközy concerning the remainder in their Erdös-Kac theorem for sum-sets.
Noncompact commutators in the commutant of a cyclic operator
Authors: Stafney James D.
Published: July 2010

We show that the commutant of the operator

S
⊗

(

I
+

S
∗

)

S \otimes \left ( {I + {S^*}} \right )

, where

S
S

is the shift operator, contains two operators

A
A

and

B
B

such that

A
B
−
B
A

AB - BA

is not compact operator.
Harmonic maps defined on a manifold with a degenerate metric
Authors: Hong Jia Xing
Published: July 2010

Some nonexistence and existence of harmonic maps defined on a manifold with a degenerate metric are proved.
Collisions of Lévy processes
Authors: Shieh Narn-Rueih
Published: July 2010

Let

X
,
Y

X,Y

be two independent Lévy processes in

R
R

. We describe simple conditions on the density functions of

X
,
Y

X,Y

which guarantee that the paths

X
(
⋅
)
,
Y
(
⋅
)

X( \cdot ),Y( \cdot )

will have uncountably many collisions almost surely.
On quasihereditary rings
Authors: Burgess W. D. , Fuller K. R.
Published: July 2010

The notion of a quashereditary semiprimary ring was introduced by Cline, Parshall and Scott and was studied extensively by Dlab and Ringel. It is known that if

R
R

is semiprimary of global dimension

≤
2

\leq 2

then it is quasihereditary and that there is a serial ring of global dimension 4 which is not. This paper establishes three principal results: If

R
R

is quasihereditary then the Cartan determinant conjecture is true for

R
R

(i.e.,

C
(
R
)
=
1

C(R) = 1

); a serial ring is quasihereditary iff it has a heredity ideal; in particular, every serial ring of global dimension 3 is quasihereditary; and there is a ring (a

0
0

-relations algebra in fact) of global dimension 3 which is not quasihereditary.
Quotients of bounded operators
Authors: Izumino Saichi
Published: July 2010

We define a quotient

[
B

/

A
]

[B/A]

of bounded operators

A
A

and

B
B

on a Hilbert space

H
H

with ker

A
⊂

A \subset

ker

B
B

as the mapping

A
x
→
B
x
,
x
∈
H

Ax \to Bx,x \in H

, and show explicit formulae for computing quotients which correspond to sums, products, adjoints and closures of given quotients.
Multi-states on 𝐶*-algebras
Authors: Kaplan Alexander
Published: July 2010

This paper is concerned with the study of the dual of a

C
∗

{C^ * }

-algebra as a matrix ordered space. It is shown that an

n
×
n

n \times n

matrix of linear functionals of a

C
∗

{C^ * }

-algebra, satisfying the generalized positivity condition, induces a representation of the algebra that generalizes the classical Gelfand-Naimark-Segal representation. This allows analysis of the relationship between the comparability of cyclic representations of the algebra and the matricial order structure of the dual. We consider the problem of unitary diagonalization of linear functionals and show that positive normal functionals on a matrix algebra over a semifinite von Neumann algebra can always be diagonalized.
The product of a Lindelöf space with the space of irrationals under Martin’s axiom
Authors: Alster K.
Published: July 2010

In this paper, we give an example of a Lindelöf space whose product with the space of irrationals is not Lindelöf provided that Martin’s axiom holds. Our result is an improvement of a Michael’s construction using the Continuum Hypothesis.
The Orlicz-Pettis theorem fails for Lumer’s Hardy spaces (𝐿𝐻)^{𝑝}(𝐵)
Authors: Nawrocki M.
Published: July 2010

In this paper we prove that if

n
>
1

n > 1

and

0
>
p
>
1

0 > p > 1

then the Lumer’s Hardy space

(
L
H

)
p

(

B

n

)

{(LH)^p}({{\mathbf {B}}_n})

of the unit ball

B

n

{{\mathbf {B}}_n}

in

C

n

{{\mathbf {C}}^n}

does not have the Orlicz-Pettis property.
Universal maps and surjective characterizations of completely metrizable 𝐿𝐶ⁿ-spaces
Authors: Chigogidze A. , Valov V.
Published: July 2010

We construct an

n
n

-dimensional completely metrizable

A
E
(
n
)

AE(n)

-space

P
(
n
,
τ
)

P(n,\tau )

of weight

τ
≥
ω

\tau \geq \omega

with the following property: for any at most

n
n

-dimensional completely metrizable space

Y
Y

of weight

≤
τ

\leq \tau

the set of closed embeddings

Y
→
P

(

n
,
τ

)

Y \to P\left ( {n,\tau } \right )

is dense in the space

C

(

Y
,
P

(

n
,
τ

)

)

C\left ( {Y,P\left ( {n,\tau } \right )} \right )

. It is also shown that completely metrizable

L

C
n

L{C^n}

-spaces of weight

τ
≥
ω

\tau \geq \omega

are precisely the

n
n

-invertible images of the Hilbert space

ℓ
2

(
τ
)

{\ell _2}(\tau )

.
Periodic solutions of some Liénard equations with singularities
Authors: Habets Patrick , Sanchez Luis
Published: July 2010

We consider the forced Liénard equation\[u+f(u)u′+g(t,u)=h(t)u + f(u)u’ + g(t,u) = h(t)\]together with the boundary conditions\[u(0)=u(T),u′(0)=u′(T),u(0) = u(T),\quad u’(0) = u’(T),\]whereggis continuous onR×(0,+∞){\mathbf {R}} \times (0, + \infty )and becomes infinite atu=0u = 0. We consider classical solutions as well as generalized solutions that can go into the singularityu=0u = 0. The method of approach uses upper and lower solutions and degree theory.
Homogeneity and Cantor manifolds
Authors: Krupski Paweł
Published: July 2010

Some consequences of generalized homogeneity are observed in dimension theory of metrizable spaces. In particular, if

X
X

is a connected, locally compact, metric space which is homogeneous with respect to open

0
0

-dimensional mappings and if

dim
⁡
X
=
n
≥
1
(
dim
⁡
X
=
∞
)

\dim X = n \geq 1(\dim X = \infty )

, then no subset of dimension

≤
n
−
2

\leq n - 2

(respectively, of a finite dimension) separates

X
X

. Thus, homogeneous continua are Cantor manifolds.
Unique continuation theorems for some parabolic operators
Authors: McMichael J. D. , Oberlin D. M.
Published: July 2010

We prove a unique continuation theorem for a class of differential operators containing the heat operator.
The group of measure preserving transformations of the unit interval is an absolute retract
Authors: Nhu Nguyen To
Published: July 2010

The group of measure preserving transformations of the unit interval equipped with the weak topology is an absolute retract, hence is homeomorphic to a separable Hilbert space.
Some abelian Banach algebras of operators on the matricial 𝐼𝐼₁ factor
Authors: Robertson A. Guyan
Published: July 2010

We show that, if

G
G

is an amenable discrete group, then the set of completely bounded [completely positive] multiplication operators on

V
N
(
G
)

VN(G)

is maximal abelian and norm-

1
1

complemented in various sets of bounded [positive] operators on

V
N
(
G
)

VN(G)

. Since there are many different amenable discrete groups

G
G

which generate the matricial

I

I

1

II_{1}

factor

R
R

, this shows the set of completely bounded normal operators on

R
R

contains uncountably many non-isomorphic maximal abelian subalgebras each of which is complemented by a positivity-preserving projection of norm one. Our results are closely related to [16], which considers the case of group

C
∗

{C^ * }

-algebras of general locally compact groups.
The syntactic monoid of an infix code
Authors: Petrich Mario , Thierrin Gabriel
Published: July 2010

Necessary and sufficient conditions on a monoid

M
M

are found in order that

M
M

be isomorphic to the syntactic monoid of a language

L
L

over an alphabet

X
X

having one of the following properties. In the first theorem

L
L

is a

P
L

{P_L}

-class and

P

W

(
L
)

⊆

P
L

{P_{W\left ( L \right )}} \subseteq {P_L}

where

P
L

{P_L}

is the syntactic congruence of

L
L

and

W

(
L
)

W\left ( L \right )

is the residue of

L
L

. In the second theorem

L
L

is an infix code; that is, satisfies

u
,
u
v
w
∈
L

u,uvw \in L

implying

u
=
w
=
1

u = w = 1

. In the third theorem

L
L

is an infix code satisfying a condition which amounts to the requirement that

M
M

be a nilmonoid. Various refinements of these conditions are also considered.
Small deformations of a class of compact non-Kähler manifolds
Authors: Alessandrini L. , Bassanelli G.
Published: July 2010

Small deformations of Kähler manifolds are Kähler too; we prove here that this is not true for balanced manifolds nor, more generally, for

p
p

-Kähler manifolds

(
p
>
1
)

(p > 1)

, i.e., the property of being

p
p

-Kähler is not stable under small deformations.
Extensions of *-representations
Authors: Kasparek Andreas
Published: July 2010

Let

π
\pi

be a

∗
*

-representation of a

∗
*

-algebra

A

\mathfrak {A}

. In general the strong commutant

π

(

A

)

′

s

\pi {\left ( \mathfrak {A} \right )’ }_s

and Theory weak commutant

π

(

A

)

w

\pi {\left ( \mathfrak {A} \right ) }_w

of the

O

∗

{\mathcal {O}^*}

-algebra

π

(

A

)

\pi \left ( \mathfrak {A} \right )

do not coincide. We are looking for some methods to get extensions of

π
\pi

such that the related commutants coincide or which are even selfadjoint. In §§2 and 3 we consider so-called generated extensions that are a modification of induced extensions investigated by Borchers, Yngvason [1] and Schmüdgen [7]. In §4 let

A

\mathfrak {A}

be a

∗
*

-algebra and

B

\mathfrak {B}

a subset of its hermitian part

A

h

{\mathfrak {A}_h}

such that

A

\mathfrak {A}

is generated by

B

∪

{
1
}

\mathfrak {B} \cup \left \{ 1 \right \}

as an algebra. We present a method to extend

∗
*

-representations

π
\pi

of such algebras, which is closely related with the extension of the symmetric operators

π

(
b
)

,
b
∈

B

\pi \left ( b \right ),b \in \mathfrak {B}

. In §5 we give an example that shows that the method of generated extensions is also suitable to get extensions such that the commutants of the related

O

∗

{\mathcal {O}^*}

-algebras coincide.
The structure of (exactly) 2-to-1 maps on metric compacta
Authors: Heath Jo
Published: July 2010

It is shown that the domain of a

2
2

-to-

1
1

continuous map

f
f

contains two disjoint open sets

V
V

and

V
∧

{V^ \wedge }

such that

f

(
V
)

=
f

(

V
∧

)

f\left ( V \right ) = f\left ( {{V^ \wedge }} \right )

and

f
↾
V

f \upharpoonright V

is a homeomorphism from

V
V

onto a dense open subset of the image of

f
f

. The restriction of

f
f

to

V
∪

V
∧

V \cup {V^ \wedge }

is the first "fold", and

f
f

is shown to be the union of a finite or transfinite sequence of similar folds.
A proof of the Pólya-Wiman conjecture
Authors: Kim Young-One
Published: July 2010

Let

f
(
z
)
=

e

−
α

z
2

g
(
z
)

f(z) = {e^{ - \alpha {z^2}}}g(z)

where

α
≥
0

\alpha \geq 0

and

g
g

is a real entire function of genus at most 1. It is shown that if

f
f

has only a finite number of nonreal zeros, then its derivatives, from a certain one onward, have only real zeros.
The Lie algebra associated to the lower central series of a link group and Murasugi’s conjecture
Authors: Labute John P.
Published: July 2010

The Chen-Milnor presentation can be used to determine the Lie Algebra associated to the lower central series of the fundamental group of a link in the

3
3

-sphere

S
3

{S^3}

in many interesting cases. We use this fact to obtain new and simpler proofs of unpublished results of Maeda on a conjecture of Murasugi in the sharpened form of Massey and Traldi.
Projective structures on reductive homogeneous spaces
Authors: Podestà Fabio
Published: July 2010

The aim of this work is to give a more direct and "geometric" proof of a theorem of Agaoka, that on a reductive homogeneous space

G

/

K

G/K

, every

G
G

-invariant projective structure admits a

G
G

-invariant affine connection. This connection can be chosen uniquely, subject to being torsionfree and satisfying one extra condition.
Asymptotic expansions for solutions of smooth recurrence equations
Authors: Jha Shing-Whu , Máté Attila , Nevai Paul
Published: July 2010

Let

⟨

y
n

:
n
≥
1
⟩

\langle {y_n}:n \geq 1\rangle

be a convergent sequence of reals, where for each

n
n

the tuple

⟨

y
n

,

y

n
+
1

,
…
,

y
n

+
k
,
1

/

n
⟩

\langle {y_n},{y_{n + 1}}, \ldots ,{y_n} + k,1/n\rangle

satisfies one of

r
r

equations, depending on the residue class of

n
(
mod
r
)

n(\bmod r)

, for some given

k
k

and

r
r

. Assume these equations are smooth, they have the same gradient in the first

k
+
1

k + 1

variables, and this gradient satisfies a certain nonmodularity condition. We then show that

y
n

{y_n}

has

r
r

asymptotic expansions, depending on the residue class of

n
(
mod
r
)

n(\bmod r)

, in terms of powers of

1

/

n

1/n

. This result enables us to discuss the asymptotic behavior of the recurrence coefficients associated with certain orthogonal polynomials. A key ingredient in the proof of the main result is a lemma involving exponential sums.
Sums of semiprime, 𝑧, and 𝑑𝑙-ideals in a class of 𝑓-rings
Authors: Larson Suzanne
Published: July 2010

In this paper it is shown that there is a large class of

f
f

-rings in which the sum of any two semiprime

l
l

-ideals is semiprime. This result is used to give a class of commutative

f
f

-rings with identity element in which the sum of any two

z
z

-ideals which are

l
l

-ideals is a

z
z

-ideal and the sum of any two

d
d

-ideals is a

d
d

-ideal.
On the number of subgroups of given order in a finite 𝑝-group of exponent 𝑝
Authors: Berkovich Ya. G.
Published: July 2010

A. Kulakoff [1] showed that a noncyclic

p
p

-group of order

p
m

,
p
>
2

{p^m},p > 2

, contains

1
+
p
+
k

p
2

1 + p + k{p^2}

subgroups of order

p
n

,
0
>
n
>
m

{p^n},0 > n > m

, where

k
k

is a nonnegative integer. In this note we show that for

1
>
n
>
m
−
1

1 > n > m - 1

a

p
p

-group of order

p
m

{p^m}

and exponent

p
p

contains

1
+
p
+
2

p
2

+
k

p
3

1 + p + 2{p^2} + k{p^3}

subgroups of order

p
n

{p^n}

.
Twistor spaces with Hermitian Ricci tensor
Authors: Davidov Johann , Muškarov Oleg
Published: July 2010

The twistor space

Z
Z

of an oriented Riemannian

4
4

-manifold

M
M

admits a natural

1
1

-parameter family of Riemannian metrics

h
t

{h_t}

compatible with the almost-complex structures

J
1

{J_1}

and

J
2

{J_2}

introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In the present note we describe the (real-analytic) manifolds

M
M

for which the Ricci tensor of

(

Z
,

h
t

)

\left ( {Z,{h_t}} \right )

is

J
n

{J_n}

-Hermitian,

n
=
1

or

2

n = 1\;{\text {or}}\;2

. This is used to supply examples giving a negative answer to the Blair and Ianus question of whether a compact almost-Kähler manifold with Hermitian Ricci tensor is Kählerian.
Dimension and infinite products in separable metric spaces
Authors: Kulesza John
Published: July 2010

For each pair

n
,
d

n,d

of positive integers with

n
≤
d

n \leq d

, there is a separable metric space

X

n
d

{X_{nd}}

satisfying

dim
⁡

(

X

n
d

)

=
n

\dim \left ( {{X_{nd}}} \right ) = n

and

dim
⁡

(

X

n
d

ω

)

=
d

\dim \left ( {X_{nd}^\omega } \right ) = d

.
Classification of nearly invariant subspaces of the backward shift
Authors: Hayashi Eric
Published: July 2010

Let

S
∗

{S^*}

denote the backward shift operator on the Hardy space

H
2

{H^2}

of the unit disk. A subspace

M
M

of

H
2

{H^2}

is called nearly invariant if

S
∗

h

{S^*}h

is in

M
M

whenever

h
h

belongs to

M
M

and

h
(
0
)
=
0

h(0) = 0

. In particular, the kernel of every Toeplitz operator is nearly invariant. A function theoretic characterization is given of those nearly invariant subspaces which are the kernels of Toeplitz operators, and it is shown that they can be put into one-to-one correspondence with the Cartesian product of the set of exposed points of the unit ball of

H
1

{H^1}

with the set of inner functions.
Asymptotic property of solutions of a class of third-order differential equations
Authors: Parhi N. , Das P.
Published: July 2010

It has been shown that the equation (*)
\[

y
′

+
a
(
t
)
y
+
b
(
t
)

y
′

+
c
(
t
)
y
=
0
,

y’ + a(t)y + b(t)y’ + c(t)y = 0,

\]
where

a
,
b

a,b

, and

c
c

are real-valued continuous functions on

[
α
,
∞
)

[\alpha ,\infty )

such that

a
(
t
)
≥
0
,
b
(
t
)
≤
0

a(t) \geq 0,b(t) \leq 0

, and

c
(
t
)
>
0

c(t) > 0

, admits at most one solution

y
(
t
)

y(t)

(neglecting linear dependence) with the property

y
(
t
)

y
′

(
t
)
>
0
,
y
(
t
)
y
(
t
)
>
0

y(t)y’(t) > 0,y(t)y(t) > 0

for

t
∈
[
α
,
∞
)

t \in [\alpha ,\infty )

and

lim

t
→
∞

y
(
t
)
=
0

{\lim _{t \to \infty }}y(t) = 0

, if (*) has an oscillatory solution. Further, sufficient conditions have been obtained so that (*) admits an oscillatory solution.
Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0<𝛼<\frac12
Authors: Chen Shu Ping , Triggiani Roberto
Published: July 2010

LetAA(the elastic operator) be a positive, self-adjoint operator with domainD(A)\mathcal {D}(A)in the Hilbert spaceXX, and letBB(the dissipation operator) be another positive, self-adjoint operator satisfyingρ1Aα≤B≤ρ2Aα{\rho _1}{A^\alpha } \leq B \leq {\rho _2}{A^\alpha }for some constants0>ρ1>ρ2>∞0 > {\rho _1} > {\rho _2} > \inftyand0>α≤10 > \alpha \leq 1. Consider the operator\[AB=|0amp;I−Aamp;−B|{\mathcal {A}_B} = \left | {\begin {array}{*{20}{c}} 0 & I \\ { - A} & { - B} \\ \end {array} } \right |\](corresponding to the elastic modelx¨+Bx˙+Ax=0\ddot x + B\dot x + Ax = 0written as a first order system), which (once closed) is plainly the generator of a strongly continuous semigroup of contractions on the spaceE=D(A1/2)×XE = \mathcal {D}({A^{1/2}}) \times X. In [C-T.l] [C-T.3] we showed that, for12≤α≤1\tfrac {1}{2} \leq \alpha \leq 1, such a semigroup is analytic (holomorphic) onEEon a triangular sector ofC{\mathbf {C}}containing the positive real axis and, moreover, that the property of analyticity is false for0>α>120 > \alpha > \tfrac {1}{2}, say forB=AαB = {A^\alpha }. We now complete the description ofAB{\mathcal {A}_B}in the range0>α>120 > \alpha > \tfrac {1}{2}by showing that such semigroup is in fact of Gevrey classδ>1/2α\delta > 1/2\alpha, hence differentiable onEEfor allt>0t > 0.
The influence of a small cardinal on the product of a Lindelöf space and the irrationals
Authors: Lawrence L. Brian
Published: July 2010

It is unknown whether there is in ZFC a Lindelöf space whose product with the irrationals is nonnormal. We give some necessary conditions based on the minimum cardinality of

a

≤
∗

a{ \leq ^*}

unbounded family in

ω

ω

{}^\omega \omega

.
Extensions of isometrically invariant measures on Euclidean spaces
Authors: Zakrzewski Piotr
Published: July 2010

We consider countably additive, nonnegative, extended real-valued measures that vanish on singletons. Given a group

G
G

of isometries of

R

n

{{\mathbf {R}}^n}

and a

G
G

-invariant

σ
\sigma

-finite measure on

R

n

{{\mathbf {R}}^n}

we study the problem of determining whether it has a proper

G
G

-invariant extension. We prove that it does, provided that the set of all points with uncountable

G
G

-orbits is not of measure zero. We also characterize those groups

G
G

for which every

G
G

-invariant

σ
\sigma

-finite measure on

R

n

{{\mathbf {R}}^n}

has a proper

G
G

-invariant extension.
Weakly factorial domains and groups of divisibility
Authors: Anderson D. D. , Zafrullah Muhammad
Published: July 2010

An integral domain

R
R

is said to be weakly factorial if every nonunit of

R
R

is a product of primary elements. We give several conditions equivalent to

R
R

being weakly factorial. For example, we show that the following conditions are equivalent: (1)

R
R

is weakly factorial; (2) every convex directed subgroup of the group of divisibility of

R
R

is a cardinal summand; (3) if

P
P

is a prime ideal of

R
R

minimal over a proper principal ideal (

(
x
)

\left ( x \right )

), then

P
P

has height one and

(
x
)

P

∩
R

{\left ( x \right )_P} \cap R

is principal; (4)

R
=
∩

R
P

R = \cap {R_P}

, where the intersection runs over the height-one primes of

R
R

, is locally finite, and the

t
t

-class group of

R
R

is trivial.
On products of harmonic polynomials
Authors: Schonbek Tomas
Published: July 2010

We prove that every polynomial in

n
n

variables,

n
≥
2

n \geq 2

, is a finite sum of terms, each of which is the product of two harmonic polynomials. This strengthens a result obtained by A. G Ramm in [1].
Every zero-dimensional space is cell soluble
Authors: Terada Toshiji
Published: July 2010

In his study of the question of representing a space as a retract of a homogeneous space, Arhangel’skii introduced an interesting topological property called cell solubility. He raised the following problem: Is every zero-dimensional compact space cell soluble? We will give an affirmative answer to this problem.
Isometric immersions with congruent Gauss maps
Authors: Moore John Douglas , Noronha Maria Helena
Published: July 2010

A characterization of pairs of isometric immersions which have congruent Gauss maps is given.
Embedding differential and skew polynomial rings into Artinian rings
Authors: Blair William D. , Small Lance W.
Published: July 2010

Under suitable hypotheses it is shown that a differential or skew polynomial ring can be embedded in an Artinian ring provided that its coefficient ring can be embedded in an Artinian ring.
On ranges of polynomials in finite matrix rings
Authors: Chuang Chen-Lian
Published: July 2010

Let

C
C

be a finite field and let

C
m

{C_m}

denote the ring consisting of all

m
×
m

m \times m

matrices over

C
C

. By a polynomial, we mean a polynomial in noncommuting indeterminates with coefficients in

C
C

. It is shown here that a subset

A
A

of

C
m

{C_m}

is the range of a polynomial without constant term if and only if

0
∈
A

0 \in A

and

u
A

u

−
1

⊆
A

uA{u^{ - 1}} \subseteq A

for all invertible elements

u
∈

C
m

u \in {C_m}

.