bluuebunny/crossref_metadata_2025 huggingface dataset:
Univalent functions and the Riemann mapping theorem
Authors: Garabedian P. R.
Published: June 2010

A proof of the Riemann mapping theorem is given that depends on variational formulas for univalent functions. The method of proof can be used to simplify the derivation of the ordinary differential equation for extremal univalent functions given by Schiffer in 1938.
Manifolds with involution whose fixed set has codimension four
Authors: Capobianco Frank L.
Published: June 2010

Necessary and sufficient conditions are given for an

n
n

-manifold to be cobordant to a manifold admitting an involution whose fixed set is

(
n
−
4
)

(n - 4)

-dimensional.
Central sequences in flows on 2-manifolds of finite genus
Authors: Neumann Dean A.
Published: June 2010

Let

ϕ
\phi

be a continuous flow on the metric space

X
X

and let

X
1

,

X
2

,
…

{X^1},{X^2}, \ldots

denote the “central” sequence of closed

ϕ
\phi

-invariant subsets of

X
X

obtained by iterating the process of taking nonwandering points of

ϕ
\phi

. A. Schwartz and E. Thomas have proved that, if

X
X

is an orientable

2
2

-manifold of finite genus, then this sequence can have not more than two distinct elements. We extend this result to include the nonorientable case; then this sequence can have at most three distinct elements. Analogous results are derived for the sequences obtained by iterating the processes of taking

α
\alpha

and

ω
\omega

limit sets, or closures of

α
\alpha

and

ω
\omega

limit sets.
A generalization of the little theorem of Picard
Authors: Petridis Nicholas C.
Published: June 2010

The method used by the author in cooperation with S. I. Goldberg and T. Ishihara in the study of harmonic mappings of bounded dilatation (J. Differential Geometry 10 (4) (1975)) is employed here to derive a Picard type theorem for quasiconformal mappings into spaces which are not necessarily hyperbolic. More precisely: if

N
N

is a nonpositively curved Riemannian manifold of dimension

n
n

and scalar curvature bounded away from zero, and if

M
M

is a complete locally flat Riemannian manifold of dimension

m
⩾
n

m \geqslant n

, then every harmonic

K
K

-quasiconformal mapping

f
:
M
→
N

f:M \to N

is a constant mapping. If, in addition,

M
M

and

N
N

are Kaehler manifolds then every conformal mapping,

f
:
M
→
N

f:M \to N

is of rank at most

n
−
2

n - 2

. For

m
=
n
=
2

m = n = 2

we obtain the classical “little” theorem of Picard.
A note on KC Wallman compactifications
Authors: Hajek Darrell W. , Jiménez Angel E.
Published: June 2010

In a previous paper, D. W. Hajek showed that if a space

X
X

is a

T
3

{T_3}

space and

A
A

is a compact subset of

W
X

WX

, the Wallman compactification of

X
X

, then

X
∩
A

X \cap A

is a closed subset of

X
X

. This raises the question of whether this “closed intersection” property characterizes the

T
3

{T_3}

spaces among the Hausdorff spaces. In the present paper, the authors show this conjecture is false by giving an example of a nonregular Hausdorff space whose Wallman compactification is a

KC
\operatorname {KC}

(compact closed)-space, and, hence, trivially satisfies this “closed intersection” property.
A note on the Hurewicz theorem in shape theory
Authors: Watanabe Tadashi
Published: June 2010

In this note the following is proved: For a pointed movable continuum, if the first shape group of it is a countable group, then the first Hurewicz theorem in shape theory holds. However, in general, it does not hold without the assumption of countability.
On the 𝑛-parameter concordance and isotopy theorem
Authors: Akiba Tadatoshi
Published: June 2010

J. F. P. Hudson [4] proved that “concordance” implies “isotopy". In this paper we show that the

n
n

-parameter version of Hudson’s result cannot be true without some restrictions. Assuming Millett’s result [6], we can determine a specific dimension where the

n
n

-parameter version fails. We rely on Kan fibrations and geometric techniques such as the Alexander trick.
Fourier transforms with only real zeros
Authors: Newman Charles M.
Published: June 2010

The class of even, nonnegative, finite measures

ρ
\rho

on the real line such that for any

b
>
0

b > 0

the Fourier transform of

exp
⁡
(
−
b

t
2

)
d
ρ
(
t
)

\exp ( - b{t^2})d\rho (t)

has only real zeros is completely determined. This result is then applied to the Riemann hypothesis.
An algorithm for complements of finite sets of integers
Authors: Weinstein Gerald
Published: June 2010

Let

A
k

=
{
0
,

a
2

,

a
3

,
…
,

a
k

}

{A_k} = \{ 0,{a_2},{a_3}, \ldots ,{a_k}\}

and

B
=
{
0
,

b
2

,

b
3

,
…
}

B = \{ 0,{b_2},{b_3}, \ldots \}

be sets of nonnegative integers of

k
k

elements and infinitely many elements, respectively. Suppose

B
B

has asymptotic density

x
:
d
(
B
)
=
x

x:d(B) = x

. If, for every integer

n
≧
0

n \geqq 0

, we can find

a
i

∈

A
k

,

b
j

∈
B

{a_i} \in {A_k},{b_j} \in B

such that

n
=

a
i

+

b
j

n = {a_i} + {b_j}

, then we say that

A
k

{A_k}

has a complement of density

≦
x

\leqq x

. Given

A
k

{A_k}

and

x
x

there is no known algorithm for determining if such a set

B
B

exists. We define regular complement and give an algorithm for determining if

B
B

exists when complement is replaced by regular complement. More precisely, given

A
4

{A_4}

and

x
=
1

/

3

x = 1/3

we give an algorithm for determining if

A
4

{A_4}

has a regular complement

B
B

with density

≦
1

/

3

\leqq 1/3

. We relate this result to the Conjecture. Every

A
4

{A_4}

has a complement of density

≦
1

/

3

\leqq 1/3

.
Analysis of an exponential equation with ordinal variables
Authors: Hickman J. L.
Published: June 2010

This paper is concerned with the analysis of the equation

x
y

=

y
z

{x^y} = {y^z}

, where

x
,
y
,
z

x,y,z

are variables ranging over ordinals, and where both sides of the equation are transfinite in value. The method used for this analysis consists in regarding

y
y

as a parameter and

x
x

as an independent variable, and determining necessary and sufficient conditions to be placed upon

x
x

so that the resulting equation in

z
z

has a solution. Extensive use is made of normal form, as well as results in ordinal arithmetic by both Bachmann and Sherman.
On Curvilinear cluster sets on open Riemann surfaces
Authors: Niimura Mikio
Published: June 2010

Every boundary point of the Kerékjártó-Stoïlow compactification of an open Riemann surface is the limit of a Jordan arc with this property: for every nonempty continuum in the extended complex plane there is a holomorphic function on the surface having the continuum as its cluster set along the arc.
Polynomial density in Bers spaces
Authors: Burbea Jacob
Published: June 2010

Let D be a bounded Jordan domain such that

∫

∫

D

λ
D

2
−
q

d
x

d
y

>
∞

\smallint \;\smallint {\;_D}\lambda _D^{2 - q}\;dx\;dy\; > \infty

for

q
>
1

q > 1

. Here

λ
D

(
z
)

{\lambda _D}(z)

is the Poincaré metric for D. Define

A
q
p

(
D
)

A_q^p(D)

, the Bers space, to be the Fréchet space of holomorphic functions f on D, such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f double-vertical-bar Subscript q comma p Superscript p Baseline equals integral integral Subscript upper D Baseline lamda Subscript upper D Superscript 2 minus q p Baseline StartAbsoluteValue f EndAbsoluteValue Superscript p Baseline d x d y">

‖
f
‖

q
,
p

p

=
∫

∫

D

λ
D

2
−
q
p

|

f

|

p

d
x

d
y

\left \| f \right \|_{q,p}^p = \smallint \;\smallint {\;_D}\lambda _D^{2 - qp}|f{|^p}\;dx\;dy

is finite,

0
>
p
>
∞
,
q
p
>
1

0 > p > \infty ,qp > 1

. It is well known that the polynomials are dense in

A
q
p

(
D
)

A_q^p(D)

for

q
p
⩾
2

qp \geqslant 2

. We show that they are dense in

A
q
p

(
D
)

A_q^p(D)

for

q
p
>
1

qp > 1

irrespective whether the boundary of D is rectifiable or not.
Continuity of nonlinear monotone operators
Authors: Fitzpatrick S. P.
Published: June 2010

If a Banach space E has an equivalent norm such that

weak
∗

\text {weak}^\ast

sequential convergence and norm convergence coincide on the dual unit sphere, then every monotone operator on E is single-valued and norm-norm continuous on a dense

G
δ

{G_\delta }

subset of E. In particular, this holds for reflexive spaces.
A geometric criterion for hyperbolicity of flows
Authors: Churchill R. C. , Franke John , Selgrade James
Published: June 2010

A chain recurrent set for a flow on a compact manifold is hyperbolic if and only if it is quasi-hyperbolic. This result gives an easy proof that the geodesic flow on a compact manifold of negative curvature is hyperbolic.
On the inverses of 𝑀-matrices
Authors: Murphy Brian B.
Published: June 2010

If A is the adjugate of an M-matrix, the sign of a certain class of minors of A is characterized by a simple rule. These minors, to be called ’near principal minors’, properly contain the so-called ’almost principal minors’.
Holonomy invariant cochains for foliations
Authors: Franks John M.
Published: June 2010

The notion of a holonomy invariant cochain for a foliation is developed generalizing the idea of a holonomy invariant measure. An example is given on a foliation in which every leaf has exponential growth, and it is shown that if the inclusion of every leaf is trivial on

H
1

{H_1}

for a codimension one transversally oriented foliation, then holonomy invariant cochains generate all of the one dimensional cohomology.
Simple going down in PI rings
Authors: Lestmann Phillip
Published: June 2010

In this paper we prove two generalizations of a theorem which McAdam proved for commutative rings. Theorem 1 states that if

R
⊂
S

R \subset S

is a central integral extension of PI rings, then going down for prime ideals holds between R and S if and only if going down holds in

R
⊂
R
[
s
]

R \subset R[s]

for each

s
∈
S

s \in S

. Theorem 2 gives the analogous result for going down in

C
⊂
R

C \subset R

where C is a central subring of the PI ring R. As a corollary we obtain a result of Schelter generalizing Krull’s theorem on going down for integral extensions of integrally-closed subrings.
Rational singularities of higher dimensional schemes
Authors: Viehweg Eckart
Published: June 2010

Two examples of rational singularities of schemes over an algebraically closed field of characteristic zero are given: Singularities occurring as the quotient of a regular scheme by a finite group and singularities of the type

u
2

−

v
2

−
g
(

t
1

,
…
,

t
N

)

{u^2} - {v^2} - g({t_1}, \ldots ,{t_N})

.
On the existence of ad-nilpotent elements
Authors: Benkart G. M. , Isaacs I. M.
Published: June 2010

A condition sufficient to guarantee the nilpotence of a derivation of a Lie algebra is given. It is used to obtain an elementary proof that a finite dimensional Lie algebra over an algebraically closed field of arbitrary characteristic necessarily contains an ad-nilpotent element.
Regularity of solutions to an abstract inhomogeneous linear differential equation
Authors: Webb G. F.
Published: June 2010

Let

T
(
t
)
,
t
⩾
0

T(t),t \geqslant 0

, be a strongly continuous semigroup of linear operators on a Banach space X with infinitesimal generator A satisfying

T
(
t
)
X
⊂
D
(
A
)

T(t)X \subset D(A)

for all

t
>
0

t > 0

. Let f be a function from

[
0
,
∞
)

[0,\infty )

to X of strong bounded variation. It is proved that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis t right-parenthesis equals Superscript def Baseline upper T left-parenthesis t right-parenthesis x plus integral Overscript t Baseline 0 Endscripts upper T left-parenthesis t minus s right-parenthesis f left-parenthesis s right-parenthesis d s comma x element-of upper X">

u
(
t
)

=

def

T
(
t
)
x
+

∫

t
0

T
(
t
−
s
)
f
(
s
)
d
s
,
x
∈
X

u(t){ = ^{{\text {def}}}}T(t)x + {\smallint ^{t0}}T(t - s)f(s)ds,x \in X

, is strongly differentiable and satisfies

d
u
(
t
)

/

d
t
=
A
u
(
t
)
+
f
(
t
)

du(t)/dt = Au(t) + f(t)

for all but a countable number of

t
>
0

t > 0

.
Remarks concerning the supports of solutions of pseudoparabolic equations
Authors: Rundell William , Stecher Michael
Published: June 2010

This paper deals with the existence of nontrivial solutions of the pseudoparabolic equation

u

x
x
t

−

u
t

+

u

x
x

=
0

{u_{xxt}} - {u_t} + {u_{xx}} = 0

whose supports lie in the half spaces

t
>

t
0

,
x
>

x
0

t > {t_0},x > {x_0}

. In addition, we show the nonexistence of solutions whose supports lie in bounded strips in x or t.
Conformal invariants of submanifolds
Authors: Hsiung Chuan Chih , Mugridge Larry R.
Published: June 2010

A local conformal invariant and a global conformal invariant of a submanifold immersed in a Euclidean space are derived.
On the Radon-Nikodým theorem and locally convex spaces with the Radon-Nikodým property
Authors: Chi G. Y. H.
Published: June 2010

Let F be a quasi-complete locally convex space,

(
Ω
,
Σ
,
μ
)

(\Omega ,\Sigma ,\mu )

a complete probability space, and

L
1

(
μ
;
F
)

{L^1}(\mu ;F)

the space of all strongly integrable functions

f
:
Ω
→
F

f:\Omega \to F

with the Egoroff property. If F is a Banach space, then the Radon-Nikodým theorem was proved by Rieffel. This result extends to Fréchet spaces. If F is dual nuclear, then the Lebesgue-Nikodým theorem for the strong integral has been established. However, for nonmetrizable, or nondual nuclear spaces, the Radon-Nikodým theorem is not available in general. It is shown in this article that the Radon-Nikodým theorem for the strong integral can be established for quasi-complete locally convex spaces F having the following property: (CM) For every bounded subset

B
⊂

l
N
1

{
F
}

B \subset l_N^1\{ F\}

, the space of absolutely summable sequences, there exists an absolutely convex compact metrizable subset

M
⊂
F

M \subset F

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Subscript i equals 1 Superscript normal infinity Baseline p Subscript upper M Baseline left-parenthesis x Subscript i Baseline right-parenthesis greater-than 1 comma for-all left-parenthesis x Subscript i Baseline right-parenthesis element-of upper B">

Σ

i
=
1

∞

p
M

(

x
i

)
>
1
,
∀
(

x
i

)
∈
B

\Sigma _{i = 1}^\infty {p_M}({x_i}) > 1,\forall ({x_i}) \in B

. In fact, these spaces have the Radon-Nikodým property, and they include the Montel (DF)-spaces, the strong duals of metrizable Montel spaces, the strong duals of metrizable Schwartz spaces, and the precompact duals of separable metrizable spaces. When F is dual nuclear, the Radon-Nikodým theorem reduces to the Lebesgue-Nikodým theorem. An application to probability theory is considered.
Operators commuting with a compact quasiaffinity
Authors: Gray L. J.
Published: June 2010

We present an example of a nonbiquasitriangular operator which commutes with a compact quasiaffinity.
Kähler metrics on fibered manifolds
Authors: Yang Paul C.
Published: June 2010

A Bochner type integral identity is used to derive a nonfibration theorem for complex manifolds with negative holomorphic bisectional curvature. As an application, a classification of fibered compact Kähler surfaces with nonpositive bisectional curvature is given.
The pointwise characterization of derivatives of integrals
Authors: Sarkhel D. N.
Published: June 2010

Let

σ
\sigma

denote the indefinite integral of a summable function f on

R
n

{R_n}

. We give necessary and sufficient conditions for

σ
\sigma

to possess a strong, or ordinary, or general derivative, equal to

f
(
x
)

f(x)

at a point x of approximate continuity of f. Munroe [3] states that when f is a bounded function then

f
(
x
)

f(x)

is the strong derivative of

σ
\sigma

at x if and only if x is a point of approximate continuity of f. We point out an error in the proof of the only if part of this result and show by example that this part of Munroe’s result is in fact false.
The normed space numerical index of 𝐶*-algebras
Authors: Huruya Tadashi
Published: June 2010

Given a complex

C
∗

{C^\ast }

-algebra X, we prove that the normed space numerical index

n
(
X
)

n(X)

of X is 1 or

1
2

\tfrac {1}{2}

according as X is commutative or not commutative.
A simple proof about positive harmonic functions on 𝑅ⁿ
Authors: Chung Lung Ock
Published: June 2010

A simple proof is given that positive harmonic functions on

R

n

{{\mathbf {R}}^n}

are constant.
Killing characteristic classes by surgery
Authors: Papastavridis Stavros
Published: June 2010

Let M be an n-dimensional

C
∞

{C^\infty }

manifold and let
\[

c
∈

H
∗

(
B
O
;

Z
2

)

c \in {H^\ast }(BO;{Z_2})

\]
be a characteristic class. Suppose that c as a factor annihilates all the characteristic numbers of M. We prove that if

dim
⁡
c
⩾
(
n
+
1
)

/

2

\dim c \geqslant (n + 1)/2

then M is cobordant to a manifold which has the class c zero, in that way answering in the affirmative a question raised by C. T. C. Wall. We examine the same question for more general cobordism theories, and for

Z
p

{Z_p}

characteristic classes.
Orientable line-element parallelizable manifolds
Authors: Alliston R. Michael
Published: June 2010

Examples of oriented, nonbounding, line-element parallelizable manifolds are given in all odd dimensions

n
≠
5

n \ne 5

or

2
r

−
1

{2^r} - 1

. Furthermore, these examples are indecomposable in the unoriented bordism ring, and hence represent generators of

Tor

Ω
∗

\operatorname {Tor} \;{\Omega _\ast }

, the torsion subgroup of the oriented bordism ring. It is also proven that every class of

Tor

Ω
∗

\operatorname {Tor} \;{\Omega _\ast }

admits a representative M such that

τ
(
M
)
⊕
2

\tau (M) \oplus 2

splits as a sum of line bundles over M.
A topological version of ♢
Authors: Ginsburg John
Published: June 2010

The set-theoretic principle

♢
\diamondsuit

is shown to be equivalent to the existence of universal

ω
1

{\omega _1}

-sequences in certain topological spaces. An

ω
1

{\omega _1}

-sequence

(

x
α

:
α
∈

ω
1

)

({x_\alpha }:\alpha \in {\omega _1})

in a space X is said to be universal in X if for every point

p
∈
X

p \in X

there is a stationary set

S
⊆

ω
1

S \subseteq {\omega _1}

so that the net

(

x
α

:
α
∈
S
)

({x_\alpha }:\alpha \in S)

converges to p. It is shown that the existence of universal

ω
1

{\omega _1}

-sequences in spaces of weight

⩽
c

\leqslant c

is equivalent to

♢
\diamondsuit

.
Spinor genera of unimodular 𝑍-lattices in quadratic fields
Authors: Earnest A. G.
Published: June 2010

Let L be a unimodular Z-lattice on a quadratic space V over Q,

dim
⁡
V
⩾
3

\dim V \geqslant 3

, and let

O

\mathcal {O}

be the ring of algebraic integers of the quadratic field

E
=

Q

(

m

)

E = {\mathbf {Q}}(\sqrt m )

. We explicitly calculate the number of proper spinor genera in the genus of the lattice

L

⊗

Z

O

L{ \otimes _{\mathbf {Z}}}\mathcal {O}

.
𝐻-semilocal domains and altitude 𝑅[𝑐/𝑏]
Authors: Ratliff L. J.
Published: June 2010

It is shown that altitude

R
[
u
]
=

R[u] =

altitude R holds for all u in the quotient field of a semilocal domain R such that

1

/

u

1/u

is not in the Jacobson radical of the integral closure R’ of R if and only if every height one prime ideal in R’ has depth = altitude

R
−
1

R - 1

. Also, if (R,M) is a local domain, then every height one prime ideal p in

R
[
X
]

R[X]

such that

p
⊆
(
M
,
X
)
R
[
X
]

p \subseteq (M,X)R[X]

has depth = altitude R if and only if this holds for all such prime ideals which contain a linear polynomial.
𝐹-spaces universal with respect to linear codimension
Authors: Terry Wesley E.
Published: June 2010

Rolewicz raised the question in [5] as to whether there existed a separable F-space

X
0

{X_0}

such that any other separable F-space Y is the image of

X
0

{X_0}

under a continuous linear operator. This can be equivalently phrased as the question [5, Problem II.4.3, p. 47]: Does there exist a separable F-space universal for all separable F-spaces with respect to linear codimension? Theorem 1 proves the existence of such a separable F-space. Theorem 2 generalizes this idea to larger cardinals.
Convex solutions of a Schröder equation in several variables
Authors: Hoppe Fred M.
Published: June 2010

A nonprobabilistic proof is given for the existence of the Yaglom conditional limit distribution for the subcritical multitype Galton-Watson process by using a uniqueness theorem for convex solutions of the multidimensional Schröder functional equation.
A note on D. Quillen’s paper: “Projective modules over polynomial rings” (Invent. Math. 36 (1976), 167–171)
Authors: Roitman Moshe
Published: June 2010

We give a simplified proof to the following theorem due to D. Quillen: if A is a commutative noetherian ring of global dimension

⩽
1

\leqslant 1

, then finitely generated projective modules over

A
[

T
1

,
…
,

T
n

]

A[{T_1}, \ldots ,{T_n}]

are extended from A. We prove also that if A is a commutative noetherian ring of global dimension d, then finitely generated projective modules of rank

>
d

> d

over

A
[

T
1

,
…
,

T
n

]

A[{T_1}, \ldots ,{T_n}]

are extended from A.
Second order systems with nonlinear boundary conditions
Authors: Kelley Walter G.
Published: June 2010

Some existence theorems are established for nonlinear second order systems with nonlinear two point boundary conditions. The system is assumed to satisfy certain differential inequality conditions at the boundary of a region.
A problem on Noetherian local rings of characteristic 𝑝
Authors: Goto Shiro
Published: June 2010

Let (A, m, k) be a one-dimensional Noetherian local ring of characteristic p (

p
>
0

p > 0

, a prime number) and assume that the Frobenius endomorphism F of A is finite. Further assume that the field k is algebraically closed and that it is contained in A. Let B denote A when it is regarded as an A-algebra by F. Then, if

Hom
A

⁡
(
B
,
A
)
≅
B

\operatorname {Hom}_A(B,A) \cong B

as B-modules, A is a Macaulay local ring and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r left-parenthesis upper A right-parenthesis identical-to dimension Subscript k Baseline upper E x t Subscript upper A Superscript 1 Baseline left-parenthesis k comma upper A right-parenthesis less-than-or-slanted-equals max left-brace right-brace comma separator separator ♯ Ass minus minus upper A caret 1 comma 1">

r
(
A
)
≡

dim
k

⁡

Ext
A
1

⁡
(
k
,
A
)
⩽
max
{
♯

Ass

A
^

−
1
,
1
}

r(A) \equiv \dim _k\operatorname {Ext}_A^1(k,A) \leqslant \max \{ \sharp {\text {Ass}}\hat A - 1,1\}

where Â denotes the m-adic completion of A. Thus, in case

♯

Ass

A
^

⩽
2
,
A

\sharp {\text {Ass}}{\mkern 1mu} \hat A \leqslant 2,A

is a Gorenstein local ring if and only if

Hom
A

⁡
(
B
,
A
)
≅
B

\operatorname {Hom}_A(B,A) \cong B

as B-modules. If

♯

Ass

A
^

⩾
3

\sharp {\text {Ass}}\hat A \geqslant 3

this assertion is not true and the counterexamples are given.
A note on good reduction of simple Abelian varieties
Authors: Adimoolam C.
Published: June 2010

In this note it is shown that the reduction of a simple abelian variety of dimension

⩾
2

\geqslant 2

, defined over an algebraic number field, at any finite good prime need not be simple. We give an example of a two-dimensional simple abelian variety defined over an algebraic number field whose reduction at any finite prime is isogenous either to a product of ordinary elliptic curves or to a product of supersingular elliptic curves.
A curvilinear extension of Iversen-Tsuji’s theorem for simply connected domain
Authors: Choi Un Haing
Published: June 2010

Let D be a simply connected domain with at least two boundary points in the complex plane, and t a boundary point of D. For a meromorphic function

f
(
z
)

f(z)

in D,

lim
sup

|

f
(
z
)

|

as

z
→
t

\lim \sup |f(z)|{\text {as}}\;z \to t

is given in terms of accessible boundary points and prime ends. This gives a curvilinear extension of Iversen-Tsuji’s Theorem for a simply connected domain.
Some invariant properties on summability domains
Authors: Kuan Shen Yue
Published: June 2010

Let A be an infinite matrix. Each

f
∈

c
A
′

f \in {c’_A}

has a representation

f
(
x
)
=
α

lim
A

x
+
t
(
A
x
)
+
r
x

f(x) = \alpha {\lim _A}x + t(Ax) + rx

. The purpose of this short article is to answer the following problems raised by Wilansky. 1. Does

α
\alpha

invariantly unique imply

α
⊥

{\alpha ^ \bot }

invariant? 2. Does A not-replaceable imply

α
⊥

{\alpha ^ \bot }

invariant? 3. Could a function

f
∈

c
A
′

f \in {c’_A}

with

α
\alpha

uniquely zero have a matrix representation? 4. Is the set of test functions invariant?
A Fréchet algebra example
Authors: Dors A. George
Published: June 2010

A construction is given of a Fréchet algebra whose spectrum fails to be a k-space.
On vector states and separable 𝐶*-algebras
Authors: Anderson Joel
Published: June 2010

It is proved that the set of states on a separable

C
∗

{C^\ast }

-subalgebra of the Calkin algebra may be simultaneously extended to a set of equivalent, orthogonal, pure states on the Calkin algebra.
Positive perturbations of unbounded operators
Authors: Dombrowski J.
Published: June 2010

This work studies the spectral properties of certain unbounded selfadjoint operators by considering positive perturbations of such operators and the unitary equivalence of the perturbed and unperturbed transformations. Conditions are obtained on the unitary operators implementing this equivalence which guarantee that the selfadjoint operators have an absolutely continuous part.
The five lemma for Banach spaces
Authors: Pryde A. J.
Published: June 2010

We prove a version of the five lemma which is useful for the study of boundary value problems for partial differential equations. The results are given in the category

B

\mathcal {B}

of Banach spaces and bounded linear operators, and all conditions are stated modulo an arbitrary ideal of

B

\mathcal {B}

. We also show that the results are valid in more general categories.
On rational points on conics
Authors: Macrae R. E.
Published: June 2010

The purpose of this paper is to prove the following result: let K be a finitely, separably generated extension field of transcendence degree one and genus zero over the exact constant field k. Assume that K has no k-rational points. Let L be a subfield of K that contains k. Then L has a k-rational point if and only if

[
K
:
L
]

[K:L]

is even.
Third order linear differential equations with periodic coefficients
Authors: Erbe L. H.
Published: June 2010

The third order linear differential equation

L
y
=

y
′

+

p
2

(
t
)
y
+

p
1

(
t
)

y
′

+

p
0

(
t
)
y
=
0

Ly = y’ + {p_2}(t)y + {p_1}(t)y’ + {p_0}(t)y = 0

is studied, where

p
i

(
t
)

{p_i}(t)

are continuous real-valued and periodic of period

ω
>
0

\omega > 0

. Various criteria are obtained which guarantee “partial” asymptotic stability or instability by means of effective bounds on the Floquet characteristic multipliers of

L
y
=
0

Ly = 0

.
Growth near the boundary in 𝐻²(𝜇) spaces
Authors: Kriete Thomas , Trent Tavan
Published: June 2010

Let

H
2

(
μ
)

{H^2}(\mu )

be the closure in

L
2

(
μ
)

{L^2}(\mu )

of the complex polynomials, where

μ
\mu

is a finite Borel measure supported on the closed unit disk in the complex plane. For

|

z

|

>
1

|z| > 1

, let

E
(
z
)
≡
sup

|

p
(
z
)

|

/

‖
p
‖

E(z) \equiv \sup |p(z)|/\left \| p \right \|

where the supremum is over all polynomials p whose

L
2

(
μ
)

{L^2}(\mu )

norm

‖
p
‖

\left \| p \right \|

is nonzero. An inequality is derived asymptotically relating

E
(
z
)

E(z)

(as z tends to the unit circle) to the part of

μ
\mu

supported on the unit circle. The interplay between

μ
\mu

and the growth of functions in

H
2

(
μ
)

{H^2}(\mu )

is studied in the event that

E
(
z
)
>
∞

E(z) > \infty

for

|

z

|

>
1

|z| > 1

.
On bi-quasitriangular operators
Authors: Gray L. J.
Published: June 2010

It is shown that the set of nilpotent operators T for which

T
k

{T^k}

has closed range for all k is norm dense in the set of all nilpotent operators. A consequence of this is that every bi-quasitriangular operator is a norm limit of operators which are similar to a direct sum of weighted shifts plus scalars.
Generalized Fredholm operators and the boundary of the maximal group of invertible operators
Authors: Treese G. W. , Kelly E. P.
Published: June 2010

Let V denote an infinite dimensional Banach space over the complex field and let

G
[
V
]

G[V]

denote the subset of bounded operators on V with the property that the null space has a closed complement and the range is closed, where the null space and range are proper subspaces of V. Necessary and sufficient conditions for

T
∈
G
[
V
]

T \in G[V]

to be in the boundary,

B

\mathcal {B}

, of the maximal group,

M

\mathcal {M}

, of invertible operators are determined. As a result,

B

∩
G
[
V
]

\mathcal {B} \cap G[V]

is the set of products of operators in

M

\mathcal {M}

and operators in

P

\mathcal {P}

, where

P

\mathcal {P}

is the set of projections other than the identity operator and null operator.
Quasi-nonexpansivity and two classical methods for solving nonlinear equations
Authors: Măruşter St.
Published: June 2010

Let F:

R

n

→

R

n

{{\mathbf {R}}_n} \to {{\mathbf {R}}_n}

be a vector-valued function and let

J
(
x
)

J(x)

denote the corresponding Jacobi matrix. The main result states that the functions

x
−

J

−
1

(
x
)
⋅
F
(
x
)

x - {J^{ - 1}}(x) \cdot F(x)

and

x
−
λ

J
T

(
x
)
⋅
F
(
x
)

x - \lambda {J^T}(x)\cdot F(x)

, where

λ
\lambda

is a certain positive number, are quasi-nonexpansive. This property is used for establishing the convergence of the Newton and the gradient methods in a finite-dimensional space.
Closed curves of constant torsion. II
Authors: Weiner Joel L.
Published: June 2010

In this note we show that there exist closed regular

C
3

{C^3}

space curves

α
\alpha

with curvature

κ
>
0

\kappa > 0

and nonzero constant torsion

τ
\tau

whose total torsion

∫
α

τ

d
s

\smallint _\alpha {\tau \;ds}

is arbitrarily small. In so doing, we give another proof of the existence of closed curves of nonzero constant torsion. This note shows that Conjecture 2 in [2] is incorrect since the preceding statement is equivalent to the statement that there exist closed curves of constant torsion

τ
=
1

\tau = 1

whose length is arbitrarily small.
New proofs of Chapman’s 𝐶𝐸 mapping theorem and West’s mapping cylinder theorem
Authors: Fathi A. , Marin A. , Visetti Y. M.
Published: June 2010

This note proves two theorems of the theory of Q-manifolds; the original proofs are simplified by using Bing shrinking criterion.
Holomorphic foliations and the Kobayashi metric
Authors: Duchamp T. , Kalka M.
Published: June 2010

Here we extend the notion of the Kobayashi metric from complex manifolds to holomorphic foliations of smooth manifolds. This gives a notion of a hyperbolic foliation. We prove that a hyperbolic foliation of a compact manifold is compact and Hausdorff.
On a characterization of locally compact groups of second category, assuming the continuum hypothesis
Authors: Rana Inder K.
Published: June 2010

Let G be a topological group of second category and having cardinality at most that of the continuum. Let B be some

σ
\sigma

-algebra of subsets of G such that (G, B) is a measurable group. For a probability measure P on (G, B), write

P
g

{P_g}

for the measure defined by

P
g

(
E
)
=
P
(
g
E
)
,
E
∈

B

{P_g}(E) = P(gE),E \in {\mathbf {B}}

. The aim of this paper is to prove the following: if on (G, B) there exists an inner-regular probability measure P such that

P
g

≪
μ

{P_g} \ll \mu

for every

g
∈
G

g \in G

, where

μ
\mu

is some

σ
\sigma

-finite measure on (G, B), then G is locally compact. Further if S denotes the

σ
\sigma

-algebra generated by the topology of G and m denotes a Haar measure on G, then

μ
≫
m
≫

P
g

\mu \gg m \gg {P_g}

for every

g
∈
G

g \in G

on the

σ
\sigma

-algebra

S
∩

B

S \cap {\mathbf {B}}

.
Bounded slope variation and generalized convexity
Authors: Huggins Frank N.
Published: June 2010

In this paper, the concept of bounded slope variation, that of the convexity of a function with respect to an increasing function, and the Lebesgue-Stieltjes integral are used to further generalize a theorem of F. Riesz and to give a new proof based on a weaker hypothesis that a function which has bounded slope variation with respect to an increasing function m over [a, b] can be expressed as the difference of two functions each cf which is convex with respect to m on [a, b].
Peak points, barriers and pseudoconvex boundary points
Authors: Basener Richard F.
Published: June 2010

Let x be a smooth boundary point of a domain in

C

n

{{\mathbf {C}}^n}

. It is shown that x is a limit of strictly pseudoconvex boundary points whenever there is a “plurisubharmonic barrier” for x.
The 𝐿^{𝑝} modulus of continuity and Fourier series of Lipschitz functions
Authors: Neugebauer C. J.
Published: June 2010

This paper deals with certain inequalities about the

L
p

{L^p}

modulus of continuity, and some properties of the Fourier coefficients of functions in the Lipschitz spaces

Λ
σ

p
q

\Lambda _\sigma ^{pq}

. The inequalities are similar to those recently obtained by Garsia.
Set of uniqueness on noncommutative locally compact groups
Authors: Bożejko Marek
Published: June 2010

Using the terminology of P. Eymard we adapt the notion of set of uniqueness to noncommutative case and we show that every compact and residual set in a locally compact nondiscrete group is a set of uniqueness.
The group 𝐶*-algebra of the de Sitter group
Authors: Boyer Robert , Martin Robert
Published: June 2010

Let G denote the universal-covering of the DeSitter group and

C
∗

(
G
)

{C^\ast }(G)

the group

C
∗

{C^\ast }

-algebra of G. In this paper we use the extension theory of C. Delaroche to describe the structure of

C
∗

(
G
)

{C^\ast }(G)

.
Behavior of maximally defined solutions of a nonlinear Volterra equation
Authors: Herdman Terry L.
Published: June 2010

This paper is concerned with the behavior of solutions of an n-dimensional nonlinear Volterra integral equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis t right-parenthesis equals f left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript t Baseline g left-parenthesis t comma s comma x left-parenthesis s right-parenthesis right-parenthesis d s comma t greater-than-or-slanted-equals 0 period">

x
(
t
)
=
f
(
t
)
+

∫
0
t

g
(
t
,
s
,
x
(
s
)
)
d
s
,

t
⩾
0.

x(t) = f(t) + \int _0^t {g(t,s,x(s))ds,\quad t \geqslant 0.}

\]
In particular, sufficient conditions for a solution

x
(
t
)

x(t)

on its maximal interval of existence

[
0
,
T
)

[0,T)

to possess the property that

|

x
(
t
)

|

→
+
∞

|x(t)| \to + \infty

as

t
→

T
−

t \to {T^ - }

are obtained. Thus these additional conditions give a positive answer to the problem posed by Miller [3, p. 145]. One can construct examples, satisfying the hypotheses given in [3], which provide a negative answer to this problem, see Artstein [1, Appendix A].
A continuation result for differential equations
Authors: Burton T. A.
Published: June 2010

In this note we show how the Conti-Wintner and Yoshizawa continuation results can be strengthened and combined to produce a flexible continuation theorem. That result is then applied to second and third order equations.
Decomposition of cut loci
Authors: Bishop Richard L.
Published: June 2010

If p is a point in a complete riemannian manifold, the points of the cut locus of p are designated as singular or ordinary according to whether there is just one or more minimizing geodesics from p. It is proved that the ordinary cut-points are dense in the cut locus.
Sum-free sets of integers
Authors: Abbott H. L. , Wang E. T. H.
Published: June 2010

A setSof integers is said to be sum-free ifa,b∈Sa,b \in Simpliesa+b∉Sa + b \notin S. In this paper, we investigate two new problems on sum-free sets: (1) Letf(k)f(k)denote the largest positive integer for which there exists a partition of{1,2,…,f(k)}\{ 1,2, \ldots ,f(k)\}intoksum-free sets, and leth(k)h(k)denote the largest positive integer for which there exists a partition of{1,2,…,h(k)}\{ 1,2, \ldots ,h(k)\}intoksets which are sum-freemodh(k)+1\bmod h(k) + 1. We obtain evidence to support the conjecture thatf(k)=h(k)f(k) = h(k)for allk. (2) Letg(n,k)g(n,k)denote the cardinality of a largest subset of{1,2,…,n}\{ 1,2, \ldots ,n\}that can be partitioned intoksum-free sets. We obtain upper and lower bounds forg(n,k)g(n,k). We also show thatg(n,1)=[(n+1)/2]g(n,1) = [(n + 1)/2]and indicate how one may show that for alln⩽54,g(n,2)=n−[n/5]n \leqslant 54,g(n,2) = n - [n/5].
Isomorphisms and automorphisms of universal Heffalump Lie algebras
Authors: Berman S.
Published: June 2010

The classification of a family of infinite dimensional Lie algebras is carried out, and a determination of their automorphism groups, in certain cases is supplied.
Some abstract Cauchy problems in exceptional cases
Authors: Bragg Louis R.
Published: June 2010

Let X be a Banach space and let

A
=

B
2

A = {B^2}

in which B is the infinitesimal generator of a strongly continuous group in X with dense domain

D

(
A
)

\mathcal {D}(A)

. This paper develops solutions of the abstract Euler-Poisson-Darboux problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column u Subscript t t Baseline left-parenthesis t right-parenthesis plus StartFraction 1 minus 2 m Over t EndFraction u Subscript t Baseline left-parenthesis t right-parenthesis equals upper A u left-parenthesis t right-parenthesis comma 2nd Column a m p semicolon t greater-than 0 comma m equals 1 comma 2 comma 3 comma ellipsis comma 2nd Row 1st Column double-vertical-bar u left-parenthesis t right-parenthesis minus phi double-vertical-bar right-arrow 0 as t right-arrow 0 comma 2nd Column a m p semicolon phi element-of script upper D left-parenthesis upper A Superscript r Baseline right-parenthesis comma r greater-than m comma EndLayout">

u

t
t

(
t
)
+

1
−
2
m

t

u
t

(
t
)
=
A
u
(
t
)
,

a
m
p
;

t
>
0
,

m
=
1
,
2
,
3
,
…
,

‖

u
(
t
)
−
ϕ

‖

→
0

as

t
→
0
,

a
m
p
;

ϕ
∈

D

(

A
r

)
,

r

>
m
,

\begin {array}{*{20}{c}} {{u_{tt}}(t) + \frac {{1 - 2m}}{t}{u_t}(t) = Au(t),} & {t > 0,\quad m = 1,2,3, \ldots ,} \hfill \\ {\left \| {u(t) - \phi } \right \|\to 0\;{\text {as}}\;t \to 0,} & {\phi \in \mathcal {D}({A^r}),\quad r\; > m,} \hfill \\ \end {array}

\]
and associated Cauchy problem in terms of solutions of related abstract wave problems. Connections between solutions of certain abstract hypergeometric equations play an important role in these developments. J. B. Diaz and H. Weinberger and E. K. Blum have obtained solutions of the standard Euler-Poisson-Darboux problem (i.e.

A
=

Δ
n

A = {\Delta _n}

, the Laplacian) in the exceptional cases.
Identification of certain 4-manifolds with group actions
Authors: Fintushel Ronald , Pao Peter Sie
Published: June 2010

If

M
3

{M^3}

is an orientable 3-manifold with an

S
1

{S^1}

-action or is a Seifert fiber space, then the 4-manifold obtained by surgery along singular fibers in

M
×

S
1

M \times {S^1}

can also be obtained by surgery in

V
3

×

S
1

{V^3} \times {S^1}

, where V is a manifold related to M but with fewer singular fibers. An application is given to Scharlemann’s “exotic”

(

S
3

×

S
1

#

S
2

×

S
2

)

({S^3} \times {S^1}\;\# \;{S^2} \times {S^2})

’s.
Which functions preserve Cauchy laws?
Authors: Letac Gérard
Published: June 2010

A real random variable X has a Cauchy law

C
(
a
,
b
)

C(a,b)

if its density is

b

π

−
1

[

(
x
−
a

)
2

+

b
2

]

−
1

b{\pi ^{ - 1}}{[{(x - a)^2} + {b^2}]^{ - 1}}

, with

b
>
0

b > 0

and a real. Let f be a measurable function such that

f
(
X
)

f(X)

also has a Cauchy law for any a and b. We prove that there exist

α
\alpha

real,

k
⩾
0
,
ε
=
±
1

k \geqslant 0,\varepsilon = \pm 1

and a singular positive bounded measure

μ
\mu

on R such that for almost all x of R

f
(
X
)

f(X)

has a Cauchy law when X has a Cauchy law. Furthermore, we prove that such a function preserves Lebesgue measure when

k
=
1

k = 1

, generalising a well-known Pólya and Szegö result.
The Sidon constant of a finite abelian group
Authors: Graham Colin C.
Published: June 2010

It is shown that the Helson constant of a finite abelian group, G, is exactly

(

Card

G

)

1

/

2

{({\text {Card}}\;G)^{1/2}}

.
Near compactness and separability of symmetrizable spaces
Authors: Stephenson R. M.
Published: June 2010

Although every feebly compact, Baire, semimetrizable space is separable, we prove here that for every infinite cardinal number n there exists a feebly compact, Baire, symmetrizable Hausdorff space which has no dense subset of cardinality less than n.
Maximal separable subfields
Authors: Danner Bonnie Page
Published: June 2010

If

L

/

K

L/K

is a finitely generated separable field extension of characteristic

p
≠
0

p \ne 0

and M is an intermediate field such that

L

/

M

L/M

is inseparable, it is proved there exist subfields S of M maximal with respect to the property that

L

/

S

L/S

is separable. These maximal separable subfields, denoted S-subfields for

L

/

M

L/M

, are characterized in two ways. (1) Let

L

/

S

L/S

be a separable field extension. Then S is a S-subfield for

L

/

M

L/M

if and only if

S
(

L
p

)
⊇
M

S({L^p}) \supseteq M

and S is algebraically closed in M. (2) If

L

/

S

L/S

is separable, S is a S-subfield for

L

/

M

L/M

if and only if the inseparability of

L

/

M

L/M

is equal to the transcendence degree of

M

/

S

M/S

. A S-subfield for

L

/

M

L/M

is constructed using a maximal subset of a relative p-basis for

M

/

K

M/K

which remains p-independent in L. It is proved that there is a unique S-subfield for

L

/

M

L/M

if and only if

S

/

K

S/K

is algebraic for some S.
A note on Killing torsion of manifolds by surgery
Authors: Papastavridis Stavros
Published: June 2010

In this note we prove that every manifold of dimension different than 3, oriented in a bundle theory (for most bundle theories), is cobordant to a manifold which contains not more torsion than the classifying space of the bundle theory.
Nonlinear Fredholm maps of index zero and their singularities
Authors: Plastock R. A.
Published: June 2010

Let

F
:
X
→
Y

F:X \to Y

be a

C
1

{C^1}

Fredholm map of index zero between two Banach spaces. Defining the singular set

B
=
{
x

|

F
′

(
x
)

B = \{ x|F’(x)

is not surjective}, we study the local and global effect of B on the map F. In particular it is shown that if

b
∈
B

b \in B

is isolated in B, then, for

dim
⁡
X

\dim X

and

dim
⁡
Y
⩾
3

\dim Y \geqslant 3

, F is a local homeomorphism at b. We then show that if B consists of discrete points, F is a global homeomorphism of X onto Y. A nonlinear partial differential equation is included as an illustration.
Decomposition of approximate derivatives
Authors: O’Malley Richard J.
Published: June 2010

It is shown that if

f
:
[
0
,
1
]
→
R

f:[0,1] \to R

has a finite approximate derivative

f

ap

′

{f’_{{\text {ap}}}}

everywhere in [0, 1], then there is a sequence of perfect sets

H
n

{H_n}

, whose union is [0, 1], and a sequence of differentiable functions,

h
n

{h_n}

, such that

h
n

=
f

{h_n} = f

over

H
n

{H_n}

and

h
n
′

=

f

ap

′

{h’_n} = {f’_{{\text {ap}}}}

over

H
n

{H_n}

. This result follows from a new, more general theorem relating approximate differentiability and differentiability. Applications of both theorems are given.
A general measure extension procedure
Authors: Sultan Alan
Published: June 2010

We give a very general and flexible way of producing measure extensions. We obtain as corollaries many well-known and important measure extension and integral representation theorems as well as the main theorems of several recent papers.
An abstract semilinear Volterra integrodifferential equation
Authors: Webb G. F.
Published: June 2010

The abstract semilinear Volterra integrodifferential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u prime left-parenthesis t right-parenthesis equals upper A u left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript t Baseline g left-parenthesis t minus s comma u left-parenthesis s right-parenthesis right-parenthesis d s plus f left-parenthesis t right-parenthesis comma t greater-than-or-slanted-equals 0 comma u left-parenthesis 0 right-parenthesis equals x element-of upper X comma">

u
′

(
t
)
=
A
u
(
t
)
+

∫
0
t

g
(
t
−
s
,
u
(
s
)
)
d
s
+
f
(
t
)
,

t
⩾
0
,
u
(
0
)
=
x
∈
X
,

u’(t) = Au(t) + \int _0^t {g(t - s,u(s))ds + f(t),\quad t \geqslant 0,u(0) = x \in X,}

\]
is investigated, where A is the infinitesimal generator of a semigroup of linear operators in a Banach space X and g is nonlinear and unbounded in its second place. Some results are proved concerning local existence, global existence, continuous dependence upon initial values, and asymptotic stability. The method used treats the equation in the domain of A with the graph norm employing results from linear semigroup theory concerning abstract inhomogeneous linear differential equations.
The parallelizability of the Stiefel manifolds 𝑉_{𝑘}(𝑅ⁿ): 𝑘≥3
Authors: Smith Larry
Published: June 2010

The purpose of this note is to prove the result stated in the title by constructing a Gauss map for the standard embedding

V
k

(

R

n

)
⊂

R

n
k

{V_k}({{\mathbf {R}}^n}) \subset {{\mathbf {R}}^{nk}}

.
On a theorem of E. Lukacs
Authors: Embrechts Paul
Published: June 2010

We prove that an integral transform of measures on a locally compact abelian group, which satisfies both the uniqueness and the convolution property, is closely related to the Fourier-Stieltjes transform. This extends a result obtained by Lukacs for the real line.
On the semi-canonical property in the product space 𝑋×𝐼
Authors: Okuyama A. , Yasui Y.
Published: June 2010

As one of the several properties in generalized metric spaces, the semi-canonical property has been discussed from the viewpoint of the extension of mappings. In this paper, that property will be discussed in product space

X
×
I

X \times I

and reduced to a property of X.
Invariant subgroups of groups of higher derivations
Authors: Deveney James K. , Mordeson John N.
Published: June 2010

Let L be a field of characteristic

p
>
3

p > 3

. A subgroup G of the group D of all rank

p
e

{p^e}

higher derivations on L is Galois if G is the group of all d in D having a given subfield in its field of constants. The field of constants of G is denoted as

L
G

{L^G}

. The main result states: Let

H
⊆
G

H \subseteq G

be Galois subgroups of D. Then H is an invariant subgroup of G if and only if either

L
H

=

L
G

(

L

p
r

)

{L^H} = {L^G}({L^{{p^r}}})

for some nonnegative integer r, or

L
H

⊆

L
G

(

L

p
e

)

{L^H} \subseteq {L^G}({L^{{p^e}}})

.
Absolutely continuous and locally quasi-invariant measures on locally compact semigroups
Authors: Dzinotyiweyi Henry A. M.
Published: June 2010

Let S be a locally compact semigroup and T a locally compact subsemigroup of S. It is shown that, for a large class of semigroups S, if T is ’reasonably large’, the absolutely continuous measures on T extend to measures whose translates by some points of S are absolutely continuous on S. As a consequence we prove the absolute continuity of translates of some measures related to a locally quasi-invariant measure.
Inequality between the Bergman metric and Carathéodory differential metric
Authors: Hahn Kyong T.
Published: June 2010

The author gives a short proof of an inequality between the Bergman metric and the Carathéodory differential metric on any complex manifold.
On the localization of the spectrum for systems of operators
Authors: Frunză Ştefan
Published: June 2010

Let

a
=
(

a
1

,
…
,

a
n

)

a = ({a_1}, \ldots ,{a_n})

be a commuting system of linear continuous operators on a complex Banach space X. We show that, for any

x
∈
X

x \in X

, the local analytic spectrum

σ
(
a
,
x
)

\sigma (a,x)

[1] is contained in the spectral hull of the local spectrum

sp

(
a
,
x
)

{\text {sp}}(a,x)

[4].