bluuebunny/crossref_metadata_2025 huggingface dataset:
A formula for 𝐸_{𝑊}𝑒𝑥𝑝(-2⁻¹𝑎²‖𝑥+𝑦‖²₂)
Authors: Chiang Tzuu-Shuh , Chow Yun Shyong , Lee Yuh-Jia
Published: June 2010

We prove that for a complex number

a
a

with

Re
⁡

a
2

>
−

π
2

/

4

\operatorname {Re} {a^2} > - {\pi ^2}/4

and

x
(
⋅
)
∈

L
2

[
0
,
1
]

x( \cdot ) \in {L^2}[0,1]

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript upper W Baseline left-brace exp left-parenthesis minus 2 Superscript negative 1 Baseline a squared StartAbsoluteValue EndAbsoluteValue x plus y StartAbsoluteValue EndAbsoluteValue Subscript 2 Superscript 2 Baseline right-parenthesis right-brace equals left-parenthesis hyperbolic cosine a right-parenthesis Superscript negative 1 slash 2 Baseline exp left-bracket 2 Superscript negative 1 Baseline left-parenthesis integral Subscript 0 Superscript 1 Baseline integral Subscript 0 Superscript 1 Baseline k left-parenthesis s comma t right-parenthesis x left-parenthesis s right-parenthesis x left-parenthesis t right-parenthesis d s d t minus a squared integral Subscript 0 Superscript 1 Baseline x squared left-parenthesis t right-parenthesis d t right-parenthesis right-bracket comma">

E
W

{
exp
⁡
(
−

2

−
1

a
2

|

|

x
+
y

|

|

2
2

)
}
=

(
cosh
⁡
a

)

−
1

/

2

exp
⁡

[

2

−
1

(

∫
0
1

∫
0
1

k
(
s
,
t
)
x
(
s
)
x
(
t
)
d
s
d
t

−

a
2

∫
0
1

x
2

(
t
)
d
t

)

]

,

{E_W}\{ \exp ( - {2^{ - 1}}{a^2}||x + y||_2^2)\} = {(\cosh a)^{ - 1/2}}\exp \left [ {{2^{ - 1}}\left ( {\int _0^1 {\int _0^1 {k(s,t)x(s)x(t)dsdt} - {a^2}\int _0^1 {{x^2}(t)dt} } } \right )} \right ],

\]
, where

W
W

, the standard Wiener measure on

C
[
0
,
1
]

C[0,1]

, is the distribution of

y
y

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k left-parenthesis s comma t right-parenthesis equals a cubed left-parenthesis 2 hyperbolic cosine a right-parenthesis Superscript negative 1 Baseline left-bracket hyperbolic sine left-parenthesis a left-parenthesis 1 minus StartAbsoluteValue s minus t EndAbsoluteValue right-parenthesis right-parenthesis minus hyperbolic sine left-parenthesis a left-parenthesis 1 minus StartAbsoluteValue s plus t EndAbsoluteValue right-parenthesis right-parenthesis right-bracket period">

k
(
s
,
t
)
=

a
3

(
2
cosh
⁡
a

)

−
1

[
sinh
⁡
(
a
(
1
−

|

s
−
t

|

)
)
−
sinh
⁡
(
a
(
1
−

|

s
+
t

|

)
)
]
.

k(s,t) = {a^3}{(2\cosh a)^{ - 1}}[\sinh (a(1 - |s - t|)) - \sinh (a(1 - |s + t|))].

\]
.
Asymptotic integration of a second order ordinary differential equation
Authors: Šimša Jaromír
Published: June 2010

Equation (1)

(
r
(
t
)

x
′

)
′

+
f
(
t
)
x
=
0

(r(t)x’)’ + f(t)x = 0

is regarded as a perturbation of (2)

(
r
(
t
)

y
′

)
′

+
g
(
t
)
y
=
0

(r(t)y’)’ + g(t)y = 0

, where the latter is nonoscillatory at infinity. It is shown that if a certain improper integral involving

f
−
g

f - g

converges sufficiently rapidly (but perhaps conditionally), then (1) has a solution which behaves for large

t
t

like a principal solution of (2). The proof of this result is presented in such a way that it also yields as a by-product an improvement on a recent related result of Trench.
Vertices of localized imbalance in a biased graph
Authors: Zaslavsky Thomas
Published: June 2010

A biased graph consists of a graph

Γ
\Gamma

and a subclass

B

\mathcal {B}

of the polygons of

Γ
\Gamma

, such that no theta subgraph of

Γ
\Gamma

contains exactly two members of

B

\mathcal {B}

. A subgraph is balanced when all its polygons belong to

B

\mathcal {B}

. A vertex is a balancing vertex if deleting it leaves a balanced graph. We give a construction for unbalanced biased graphs having a balancing vertex and we show that an unbalanced biased graph having more than one balancing vertex is an unbalanced series or parallel connection of balanced graphs.
Unramified double coverings of hyperelliptic surfaces. II
Authors: Farkas H. M.
Published: June 2010

In this note we illustrate how to count the fixed points of a lift of the hyperelliptic involution to a smooth unramified double cover. In this way we obtain a new proof of the assertion that only

(

2
g
+
2

2

)

\left ( {\begin {array}{*{20}{c}} {2g + 2} \\ 2 \\ \end {array} } \right )

of the double covers are hyperelliptic and classify the remaining covers in terms of

g
~

\tilde g

-hyperellipticity.
Nonexistence of equivariant degree one maps
Authors: Daccach Janey Antonio
Published: June 2010

We establish the nonexistence of equivariant maps between some classes of manifolds carrying

G
G

-free actions and spheres, whose degree is prime to

|
G
|

\left | G \right |

, the order of

G
G

, where

G
G

is a finite nontrivial group.
Zero span is a sequential strong Whitney-reversible property
Authors: Koyama Akira
Published: June 2010

The concept of span of metric spaces was introduced by Lelek [5]. Span is an important concept in regard to chainability of continua. In this paper, motivated by recent results [2, 11], we show that zero span is a sequential strong Whitney-reversible property.
On Legendrian singularities
Authors: Izumiya Shyūichi
Published: June 2010

We introduce three natural equivalence relations among Legendrian singularities and study their properties by using recent results of the theory of singularities of smooth map germs.
A unimodality result in the enumeration of subgroups of a finite abelian group
Authors: Butler Lynne M.
Published: June 2010

The number of subgroups of orderpk{p^k}in an abelian groupGGof orderpn{p^n}is a polynomial inp,α⋋(k;p)p,{\alpha _ \leftthreetimes }(k;p), determined by the typeλ\lambdaofGG. It is well known thatα⋋(k;p)=α⋋(n−k;p){\alpha _ \leftthreetimes }(k;p) = {\alpha _ \leftthreetimes }(n - k;p). Using a recent result from the theory of Hall-Littlewood symmetric functions, we prove thatα⋋(k;p){\alpha _ \leftthreetimes }(k;p), is a unimodal sequence of polynomials. That is, for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to k less-than-or-equal-to n slash 2 comma alpha Subscript lamda Baseline left-parenthesis k semicolon p right-parenthesis minus alpha Subscript lamda Baseline left-parenthesis k minus 1 semicolon p right-parenthesis">1≤k≤n/2,αλ(k;p)−αλ(k−1;p)1 \leq k \leq n/2,{\alpha _\lambda }(k;p) - {\alpha _\lambda }(k - 1;p)is a polynomial inppwith nonnegative coefficients.
Universal functions and generalized classes of functions
Authors: Cichoń J. , Morayne M.
Published: June 2010

For a class

A

\mathcal {A}

of subsets of a set

Z
Z

which is closed under countable unions we consider the families of functions
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M underbar script upper A equals left-brace f colon upper Z right-arrow left-bracket 0 comma 1 right-bracket colon left-parenthesis for-all c right-parenthesis left-parenthesis f Superscript negative 1 Baseline left-parenthesis left-parenthesis c comma 1 right-bracket right-parenthesis element-of script upper A right-parenthesis right-brace">

M
_

A

=
{
f
:
Z
→
[
0
,
1
]
:
(
∀
c
)
(

f

−
1

(
(
c
,
1
]
)
∈

A

)
}

\underline M \mathcal {A} = \{ f:Z \to [0,1]:(\forall c)({f^{ - 1}}((c,1]) \in \mathcal {A})\}

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M overbar script upper A equals left-brace f colon upper Z right-arrow left-bracket 0 comma 1 right-bracket colon left-parenthesis for-all c right-parenthesis left-parenthesis f Superscript negative 1 Baseline left-parenthesis left-bracket 0 comma c right-parenthesis right-parenthesis element-of script upper A right-parenthesis right-brace">

M
¯

A

=
{
f
:
Z
→
[
0
,
1
]
:
(
∀
c
)
(

f

−
1

(
[
0
,
c
)
)
∈

A

)
}

\overline M \mathcal {A} = \{ f:Z \to [0,1]:(\forall c)({f^{ - 1}}([0,c)) \in \mathcal {A})\}

\]
(for instance, if

Z
Z

is a topological space and

A

\mathcal {A}

is the family of all open subsets of

Z
Z

, then

M
_

A

\underline M \mathcal {A}

and

M
¯

A

\overline M \mathcal {A}

are the families of lower and upper semicontinuous functions from

Z
Z

to

[
0
,
1
]

[0,1]

, respectively). Using universal functions we show that under certain natural assumptions about

A

\mathcal {A}

there exists a function

f
∈

M
_

A

f \in \underline M \mathcal {A}

such that there is no partition

{

X
n

:
n
∈

N

}

\{ {X_n}:n \in {\mathbf {N}}\}

of

Z
Z

and a family of functions

{

h
n

:
n
∈

N

}
⊆
M

A

\{ {h_n}:n \in {\mathbf {N}}\} \subseteq M\mathcal {A}

such that

f
=

∪
n

(

h
n

|

X
n

)

f = { \cup _n}({h_n}|{X_n})

. This is a generalization of some results of this type proved by Novikov and Adian, Keldyš, and Laczkovich for the Baire hierarchy of functions. The universal functions technique we use is different from the methods of these authors.
Invariant subspaces for derivations
Authors: Kissin E. V.
Published: June 2010

In this article it is proved that most of the known sufficient conditions for a subspace from Lat

A

\mathcal {A}

to be hyperinvariant are in fact also sufficient for this subspace to be invariant for all operators from Ad

A

\mathcal {A}

.
Complemented invariant subspaces of 𝐻^{𝑝},0<𝑝<1, and the Hahn-Banach extension property
Authors: Cohn William S.
Published: June 2010

Let

0

 > 

p

 > 

1

0{\text { > }}p{\text { > }}1

and let

H
p

{H^p}

denote the usual Hardy class of functions analytic on the disc. In this note we show that an invariant subspace of

H
p

{H^p}

is complemented in

H
p

{H^p}

if and only if it has the form

B

H
p

B{H^p}

where

B
B

is a Blaschke product whose zero sequence is a Carleson sequence. We also prove that this occurs if and only if the invariant subspace has the Hahn-Banach extension property.
Lagrangian systems in the presence of singularities
Authors: Capozzi A. , Greco C. , Salvatore A.
Published: June 2010

In this paper we study dynamical systems embedded in a conservative field of forces, whose potential is "singular." We look for

T
T

-periodic solutions of these systems by variational methods.
Lusternik-Schnirelmann category and cobordism
Authors: Singh Harpreet
Published: June 2010

The Lusternik-Schnirelmann category of a manifold

M
M

is the smallest integer

k
k

that

M
M

can be covered by

k
k

open sets each of which is contractible in

M
M

. It is an upper bound for the length of nonzero products of Stiefel-Whitney classes of

M
M

. The object of this paper is to use this restriction, on the length of nonzero products, to investigate the cobordism classes of manifolds with category less than or equal to three.
Completeness theorem for singular biprobability models
Authors: Rašković Miodrag
Published: June 2010

The aim of the paper is to prove the completeness theorem for singular biprobability models. This also solves Keisler’s Problem 5.4 in the singular case (see [3]). The case of absolute continuity is considered in [6].
Rings with projective socle
Authors: Nicholson W. K. , Watters J. F.
Published: June 2010

The class of rings with projective left socle is shown to be closed under the formation of polynomial and power series extensions, direct products, and matrix rings. It is proved that a ring

R
R

has a projective left socle if and only if the right annihilator of every maximal left ideal is of the form

f
R

fR

, where

f
f

is an idempotent in

R
R

. This result is used to establish the closure properties above except for matrix rings. To prove this we characterise the rings of the title by the property of having a faithful module with projective socle, and show that if

R
R

has such a module, then so does

M
n

(
R
)

{M_n}\left ( R \right )

. In fact we obtain more than Morita invariance. Also an example is given to show that

e
R
e

eRe

, for an idempotent

e
e

in a ring

R
R

with projective socle, need not have projective socle. The same example shows that the notion is not left-right symmetric.
A structure theorem for discontinuous derivations of Banach algebras of differential functions
Authors: Ngo Viet
Published: June 2010

Let

D
:

C
n

[

0
,
1

]

→

M

D:{C^n}\left [ {0,1} \right ] \to \mathcal {M}

be a derivation from the Banach algebra of

n
n

times continuously differentiable functions on

[

0
,
1

]

\left [ {0,1} \right ]

into a Banach

C
n

[

0
,
1

]

{C^n}\left [ {0,1} \right ]

-module

M

\mathcal {M}

. If

D
D

is continuous then it is completely determined by

D

(
z
)

D\left ( z \right )

where

z

(
t
)

=
t
,
0
≤
t
≤
1

z\left ( t \right ) = t,0 \leq t \leq 1

. For the case when

D
D

is discontinuous we show that

D

(
f
)

D\left ( f \right )

is determined by

D

(
z
)

D\left ( z \right )

for all

f
f

in an ideal

T

(
D
)

2

\mathcal {T}{\left ( D \right )^2}

of

C
n

[

0
,
1

]

{C^n}\left [ {0,1} \right ]

where its closure

T

(
D
)

2

¯

\overline {\mathcal {T}{{\left ( D \right )}^2}}

is of finite codimension.
Removable singularities in the Nevanlinna class and in the Hardy classes
Authors: Riihentaus Juhani
Published: June 2010

We show that certain sets in

C

n

,
n
≥
2

{{\mathbf {C}}^n},n \geq 2

, which we call

n
n

-small, are removable singularities for holomorphic functions in the Nevanlinna class. Since our class of sets includes polar sets (in

R

2
n

{{\mathbf {R}}^{2n}}

) our result includes the previous removable singularity results for the Nevanlinna class. We give also a related result for a subclass of the Hardy class.
Generalizations of a theorem of Namioka
Authors: Stegall Charles
Published: June 2010

We prove a result concerning separate and joint continuity that contains a celebrated result of Namioka.
Invariant subspaces for algebras of linear operators and amenable locally compact groups
Authors: Lau Anthony T. M. , Wong James C. S.
Published: June 2010

Let

G
G

be a locally compact group. We prove in this paper that

G
G

is amenable if and only if the group algebra

L
1

(
G
)

{L_1}\left ( G \right )

(respectively the measure algebra

M

(
G
)

M\left ( G \right )

) satisfies a finite-dimensional invariant subspace property

T

(
n
)

T\left ( n \right )

for

n
n

-dimensional subspaces contained in a subset

X
X

of a separated locally convex space

E
E

when

L
1

(
G
)

{L_1}\left ( G \right )

(respectively

M

(
G
)

M\left ( G \right )

) is represented as continuous linear operators on

E
E

. We also prove that for any locally compact group, the Fourier algebra

A

(
G
)

A\left ( G \right )

and the Fourier Stieltjes algebra

B

(
G
)

B\left ( G \right )

always satisfy

T

(
n
)

T\left ( n \right )

for each

n
=
1
,
2
,
…

n = 1,2, \ldots

.
An internal characterization of inessential operators
Authors: Aiena Pietro
Published: June 2010

We characterize the ideal of inessential operators

I
(
E
)

I(E)

on a complex Banach space

E
E

as the largest ideal of the class

A

(
E
)

\mathcal {A}(E)

of all bounded linear operators

A
A

having the property that the restrictions

A

|
M

A\left | M \right .

of

A
A

on any closed infinite-dimensional invariant subspace

M
M

may be.
Approximation of the sphere by polytopes having few vertices
Authors: Bárány I. , Füredi Z.
Published: June 2010

How well can a polytope with

n
n

vertices approximate the unit ball

B
d

{B^d}

of the

d
d

-dimensional Euclidean space? The answer is quite well known when

d
d

is fixed and

n
n

tends to infinity. In this paper the same question is answered when

n
n

is a function of

d
d

(a polynomial in

d
d

, say) and

d
d

tends to infinity. Some applications of the results are also indicated.
Topological triviality of a family of zero-sets
Authors: Buchner Michael A. , Kucharz Wojciech
Published: June 2010

This paper gives conditions on a mapping
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F colon upper U times bold upper R Superscript k Baseline right-arrow bold upper K Superscript p Baseline left-parenthesis upper U subset-of bold upper K Superscript n Baseline open comma bold upper K equals bold upper R or bold upper C right-parenthesis">

F
:
U
×

R

k

→

K

p

(

U
⊂

K

n

open, 

K

=

R

or 

C

)

F:U \times {{\mathbf {R}}^k} \to {{\mathbf {K}}^p}\left ( {U \subset {{\mathbf {K}}^n}{\text {open, }}{\mathbf {K}} = {\mathbf {R}}\;{\text {or }}{\mathbf {C}}} \right )

such that the family

F
t

−
1

(
0
)

F_t^{ - 1}\left ( 0 \right )

is a topologically trivial family (i.e., does not change topologically as

t
∈

R

k

t \in {{\mathbf {R}}^k}

varies). As an application an easy proof is given of a counterexample to a conjecture of Thom concerning the number of topologically different realizations of a given jet.
Existence of decaying entire solutions of a class of semilinear elliptic equations
Authors: Kusano Takaŝi , Noussair Ezzat S. , Swanson Charles A.
Published: June 2010

The main result establishes the existence of a nontrivial nonnegative radial solution

u
∈

C

loc

2

(

R

N

)

u \in C_{\operatorname {loc} }^2({{\mathbf {R}}^N})

of a semilinear elliptic eigenvalue problem in

R

N

,
N
≥
3

{{\mathbf {R}}^N},N \geq 3

, such that

u
(

|

x

|

)

u(|x|)

has uniform limit zero as

|

x

|

→
∞

|x| \to \infty

. Asymptotic decay estimates and necessary conditions are obtained. Since such solutions do not exist in the space

W
0

1
,
2

(

R

N

)

W_0^{1,2}({{\mathbf {R}}^N})

, a considerable departure from standard procedures is required.
On the Jacobson radical of some endomorphism rings
Authors: Dugas Manfred
Published: June 2010

In this note we deal with a question raised by R. S. Pierce in 1963: Determine the elements of the Jacobson radical of the endomorphism ring of a primary abelian group by their action on the group. We concentrate on separable abelian

p
p

-groups and give a counterexample to a conjecture of A. D. Sands. We also show that the radical can be pinned down if the endomorphism ring is a split-extension of its ideal of all small maps.
Hyponormal powers of composition operators
Authors: Dibrell Phillip , Campbell James T.
Published: June 2010

Let

T
i

,
i
=
1
,
2

{T_i},i = 1,2

, be measurable transformations which define bounded composition operators

C

T
i

{C_{{T_i}}}

on

L
2

{L^2}

of a

σ
\sigma

-finite measure space. Denote their respective Radon-Nikodym derivatives by

h
i

,
i
=
1
,
2

{h_i},i = 1,2

. The main result of this paper is that if

h
i

∘

T
i

≤

h
j

,
i
,
j
=
1
,
2

{h_i} \circ {T_i} \leq {h_j},i,j = 1,2

, then for each of the positive integers

m
,
n
,
p

m,n,p

the operator

[

C

T
1

m

C

T
2

n

]
p

{[C_{{T_1}}^mC_{{T_2}}^n]^p}

is hyponormal. As a consequence, we see that the sufficient condition established by Harrington and Whitley for hyponormality of a composition operator is actually sufficient for all powers to be hyponormal.
Spectral manifolds of bounded 𝑆-decomposable operators
Authors: Tanahashi Kôtarô
Published: June 2010

We prove some properties of spectral manifolds of a bounded

S
S

-decomposable operator on a complex Banach space. Also, we prove a new characterization of

S
S

-decomposability.
Notes on interpolation by the real method between 𝐶(𝑇,𝐴₀) and 𝐶(𝑇,𝐴₁)
Authors: Mastyło Mieczysław
Published: June 2010

Let

A
A

be a Banach space and let

T
T

be a compact Hausdorff space. We denote by

C
(
T
,
A
)

C(T,A)

the Banach space of all

A
A

-valued continuous functions defined on

T
T

endowed with the supremum norm. We show that if

T
T

is infinite and

(

A
0

,

A
1

)

({A_0},{A_1})

is a Banach couple with

A
0

{A_0}

continuously embedded in

A
1

{A_1}

, then the interpolation space

(
C
(
T
,

A
0

)
,
C
(
T
,

A
1

)

)

φ
,
p

{(C(T,{A_0}),C(T,{A_1}))_{\varphi ,p}}

is equal to

C

(

T
,

(

A
0

,

A
1

)

φ
,
p

)

C\left ( {T,{{\left ( {{A_0},{A_1}} \right )}_{\varphi ,p}}} \right )

if and only if

A
0

{A_0}

is closed in

A
1

{A_1}

.
Positivity of stable densities
Authors: Port S. C. , Vitale R. A.
Published: June 2010

We settle two conjectures of Taylor about the positivity of the densities

p
(
t
,
x
)

p(t,x)

of a drift-free, nondegenerate, stable process on

d
d

-dimensional Euclidean space

R
d

{R^d}

starting at the origin. If

0
>
α
>
1

0 > \alpha > 1

and

p
(
1
,
0
)
=
0

p(1,0) = 0

, we show that

x
:

p
(
t
,
x
)
>
0

x:\;p(t,x) > 0

for some

t
>
0

t > 0

is an open convex cone with vertex 0 and that

p
(
t
,
x
)
>
0

p(t,x) > 0

for all

t
>
0

t > 0

for each

x
x

in this cone. If

α
=
1

\alpha = 1

we show that

p
(
t
,
x
)
>
0

p(t,x) > 0

for all

t
>
0

t > 0

and all

x
∈

R
d

x \in {R^d}

.
Stationary plans need not be uniformly adequate for leavable, Borel gambling problems
Authors: Blackwell D. , Ramakrishnan S.
Published: June 2010

There exists a leavable, Borel gambling problem with a goal, where at most three gambles are available at each fortune, where each gamble has at most two points in its support, but for which stationary plans are not uniformly adequate.
A ribbon knot group which has no free base
Authors: Yoshikawa Katsuyuki
Published: June 2010

We consider the following problem: If a group

G
G

satisfies the conditions (1)

G
G

has a finite presentation with

r
+
1

r + 1

generators and

r
r

relators, and (2) there exists an element

x
x

of

G
G

such that

G
=

⟨

⟨
x
⟩

⟩

G

G = {\left \langle {\left \langle x \right \rangle } \right \rangle ^G}

where

⟨

⟨
x
⟩

⟩

G

{\left \langle {\left \langle x \right \rangle } \right \rangle ^G}

is the normal closure of

x
x

in

G
G

, then is

G
G

an HNN (Higman-Neumann-Neumann) extension of a free group of finite rank? In this paper, we give a negative answer to the problem. Thus it follows that there exists a ribbon

n
n

-knot group

(
n
≥
2
)

(n \geq 2)

which has no free base.
Norms of integrable cusp forms
Authors: Parson L. Alayne
Published: June 2010

The norms of modular cusp forms, viewed as belonging to the Bers’ spaces of integrable and bounded forms, are estimated in terms of the Fourier coefficients of the cusp form.
Cyclic quartic fields with relative integral bases over their quadratic subfields
Authors: Spearman Blair K. , Williams Kenneth S.
Published: June 2010

Explicit conditions are given for a cyclic quartic field to have a relative integral basis over its unique quadratic subfield.
A convergence problem connected with continued fractions
Authors: Larcher Gerhard
Published: June 2010

The set
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z Subscript alpha Baseline colon equals StartSet beta StartAbsoluteValue limit Underscript n right-arrow normal infinity Endscripts EndAbsoluteValue StartAbsoluteValue q Subscript n Baseline beta EndAbsoluteValue vertical-bar equals 0 EndSet">

Z
α

:=
{
β

|

lim

n
→
∞

|

|

q
n

β

|

|

=
0
}

{Z_\alpha }: = \{ \beta |{\lim _{n \to \infty }}||{q_n}\beta || = 0\}

is considered, where

(

q
n

)

n
∈

N

{\left ( {{q_n}} \right )_{n \in {\mathbf {N}}}}

is the sequence of best approximation denominators of

α
\alpha

, and it is explicitly determined for

α
\alpha

with bounded continued fraction coefficients.
A question on invariant subspaces of Bergman spaces
Authors: Bercovici Hari
Published: June 2010

We answer a question raised by Axler and Bourdon [1] concerning the classification of finite-codimensional invariant subspaces of Bergman spaces.
Monomials in the Jones projections
Authors: Sunder V. S.
Published: June 2010

It is shown that every monomial

e
I

=

e

i
1

e

i
2

⋯

e

i
n

{e_I} = {e_{{i_1}}}{e_{{i_2}}} \cdots {e_{{i_n}}}

in the Jones projections (with parameter

τ
\tau

) satisfies

e
I

=

τ

n
(
I
)

/

2

u
I

{e_I} = {\tau ^{n(I)/2}}{u_I}

where

u
I

{u_I}

is a partial isometry and

n
(
I
)

n(I)

is an integer for which an explicit formula is given.
Transience of a pair of local martingales
Authors: Mountford T. S.
Published: June 2010

We consider the process of windings of complex Brownian motion about two points

a
a

and

b
b

in the complex plane,

{
(

θ
a

(
t
)
,

θ
b

(
t
)
)
:
t
≥
0
}

\{ ({\theta ^a}(t),{\theta ^b}(t)):t \geq 0\}

. We show that this process is transient in the sense that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript t Endscripts right-arrow normal infinity StartAbsoluteValue left-parenthesis theta Superscript a Baseline left-parenthesis t right-parenthesis comma theta Superscript b Baseline left-parenthesis t right-parenthesis right-parenthesis EndAbsoluteValue equals normal infinity">

lim
t

→
∞

|

(

θ
a

(
t
)
,

θ
b

(
t
)
)

|

=
∞

{\lim _t} \to \infty |({\theta ^a}(t),{\theta ^b}(t))| = \infty

. This extends a result found in both Lyons and McKean (1984) and McKean and Sullivan (1984). We will mostly use facts and ideas found in the former paper.
Compactifications whose remainders are retracts
Authors: Faulkner Gary D.
Published: June 2010

This paper is concerned with compactifications in which the remainder is a (neighborhood) retract. Two theorems which characterize such compactifications are proved here. One of the characterizations is in terms of singular functions and the other in terms of projections on spaces of continuous real valued functions.
Starlikeness and convexity from integral means of the derivative
Authors: Yamashita Shinji
Published: June 2010

If

f
f

is analytic in

|

z

|

>
1

|z| > 1

and normalized:

f
(
0
)
=

f
′

(
0
)
−
1
=
0

f(0) = f’(0) - 1 = 0

, then

f
f

is univalent and starlike in

|

z

|

>
1
(
f
)

|z| > 1(f)

, where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis f right-parenthesis equals sup r left-brace left-parenthesis 2 pi right-parenthesis Superscript negative 1 Baseline integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue f prime left-parenthesis r e Superscript i t Baseline right-parenthesis EndAbsoluteValue squared d t right-brace Superscript negative 1 slash 2 Baseline comma 0 less-than-or-equal-to r less-than-or-equal-to 1 period">

I
(
f
)
=
sup

r

{

(
2
π
)

−
1

∫
0

2
π

|

f
′

(
r

e

i
t

)

|

2

d
t

}

−
1

/

2

,

0
≤
r
≤
1.

I(f) = \sup \,r{\left \{ {{{(2\pi )}^{ - 1}}\int _0^{2\pi } {|f’(r{e^{it}}){|^2}dt} } \right \}^{ - 1/2}},\quad 0 \leq r \leq 1.

\]
Furthermore, there exists a normalized

f
f

such that

I
(
f
)
>
1

I(f) > 1

and that

f
′

f’

vanishes at a point on

|

z

|

=
I
(
f
)

|z| = I(f)

. If

f
f

is analytic and normalized in

|

z

|

>
1

|z| > 1

, then

f
f

is univalent and convex in

|

z

|

>
I
(
f
)

/

2

|z| > I(f)/2

.
Closed convex hypersurfaces with curvature restrictions
Authors: Schneider Rolf
Published: June 2010

Using elementary facts from the theory of convex bodies, we obtain an improved version of J. Rauch’s [5] inclusion theorem for ovaloids with comparable second fundamental forms and a generalization of a result of A. W. Goodman [4] on the position of the curvature centroids for ovaloids with restricted radii of curvature.
Integral Brown-Gitler spectra
Authors: Cohen Fred R. , Davis Donald M. , Goerss Paul G. , Mahowald Mark E.
Published: June 2010

A Thom spectrum model for integral Brown-Gitler spectra is established and shown to have a multiplicative property. This clarifies certain aspects of an earlier application to splitting

b
o
∧
b
o

bo \wedge bo

.
Isomorphisms of prime Goldie semiprincipal left ideal rings
Authors: Wolfson Kenneth G.
Published: June 2010

A prime (left) Goldie semiprincipal left ideal ring is the endomorphism ring

E
(
F
,
A
)

E(F,A)

of a free module

A
A

, of finite rank, over a (left) Ore domain

F
F

. We examine the uniqueness of the module

(
F
,
A
)

(F,A)

in the sense of determining necessary and sufficient conditions that every isomorphism of

E
(
F
,
A
)

E(F,A)

is induced by a semilinear module isomorphism of

(
F
,
A
)

(F,A)

.
Kida’s theorem for a class of nonnormal extensions
Authors: Gold Robert , Madan Manohar
Published: June 2010

Let

E
,
F

E,F

be

Z

p

{{\mathbf {Z}}_p}

-fields of CM-type such that

E

/

F

E/F

is an extension of degree

p
p

. Let

L
L

, the normal closure of

E

/

F

E/F

, be such that

Gal
⁡
(
L

/

F
)

\operatorname {Gal} (L/F)

has a normal subgroup of order

p
p

. Denote the fixed field of this group by

K
K

. We prove a Kida type formula which describes the minus part of the Iwasawa lambda invariant of

E
E

in terms of the lambda invariants of

F
F

and

K
K

.
Some applications of selection theorems to parametrization problems
Authors: Maitra A. , Rao B. V. , Srivatsa V. V.
Published: June 2010

Some results relating to parametrizations of Borel sets are deduced from selection theorems.
Fourth-order two-point boundary value problems
Authors: Yang Yi Song
Published: June 2010

We establish the uniqueness and existence theorems for a linear and a nonlinear fourth-order boundary value problem at nonresonance.
An atomic decomposition for parabolic 𝐻^{𝑝} spaces on product domains
Authors: Sato Shuichi
Published: June 2010

We obtain an atomic decomposition for two-parameter parabolic

H
p

{H^p}

spaces, showing simultaneously an integral inequality between Lusin functions and nontangential maximal functions. As its consequence, we generalize Fefferman’s weak type estimates for double singular integrals.
Covers of non-almost-finite type sofic systems
Authors: Williams Susan
Published: June 2010

An almost finite type (AFT) sofic system

S
S

has a cover which intercepts every other cover of

S
S

[BKM]. We show that if an irreducible sofic system

S
S

is not AFT, it has an infinite collection of covers such that no two are intercepted by a common cover of

S
S

.
Optimal partitioning of a measurable space
Authors: Legut Jerzy , Wilczyński Maciej
Published: June 2010

An

α
\alpha

-optimal partition of a measurable space according to

n
n

nonatomic probability measures is defined. A minmax theorem is used to find a method of obtaining the

α
\alpha

-optimal partition. An application to a problem of fair division is given.
Random nonlinear evolution inclusions in reflexive Banach spaces
Authors: Avgerinos Evgenios P. , Papageorgiou Nikolaos S.
Published: June 2010

In this paper we present two existence results for a large class of random, nonlinear, multivalued evolution equations defined in a reflexive, separable Banach space and involving an

m
m

-dissipative operator. Applications to random multivalued parabolic p.d.e.’s are presented. Our work unifies and extends earlier results of Kampé de Feriet, Gopalsamy and Bharucha-Reid, Becus and Itoh.
Sums, products and continuity of Borel maps in nonseparable metric spaces
Authors: Hansell R. W.
Published: June 2010

We give a comparatively simple proof that the complex map

(
f
,
g
)
:
T
→
X
×
Y

(f,g):T \to X \times Y

will be (Borel measurable) of class

α
+
α

\alpha + \alpha

, whenever

f
f

and

g
g

are of class

α
\alpha

, for not necessarily separable metric spaces

T
,
X

T,X

and

Y
Y

. The Borel measurability of other types of maps, such as the sum of two vector-valued maps, is easily deduced from this. A general version of this result, applicable to abstract measurable spaces, is also proven. Our second principal result shows that, if

f
:
T
×
X
→
Y

f:T \times X \to Y

is continuous in each variable separately, where

X
X

and

Y
Y

are metrizable, then

f
f

has a

σ
\sigma

-locally finite function base of closed sets, and thus will be continuous at each point of the complement of some first category subset of

T
×
X

T \times X

.
Weak amenability of Banach algebras generated by some analytic semigroups
Authors: Galé Jośe E.
Published: June 2010

In this paper it is shown that if

A
A

is a Banach algebra generated by an analytic semigroup

(

a
t

)
Re
⁡
t
>
0

({a^t})\operatorname {Re} t > 0

such tnat
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue a Superscript 1 plus i y Baseline StartAbsoluteValue EndAbsoluteValue equals upper O left-parenthesis StartAbsoluteValue y EndAbsoluteValue Superscript rho Baseline right-parenthesis left-parenthesis y element-of bold upper R right-parenthesis">

|

|

a

1
+
i
y

|

|

=
O
(

|

y

|

ρ

)

 

(
y
∈

R

)

||{a^{1 + iy}}|| = O(|y{|^\rho }){\text { }}(y \in {\mathbf {R}})

, where

0
≤
ρ
>
1

/

2

0 \leq \rho > 1/2

, then

A
A

is weakly amenable, that is, each continuous derivation from

A
A

to a commutative

A
A

-module is null.
The first diameter of 3-manifolds of positive scalar curvature
Authors: Katz Mikhail
Published: June 2010

We seek a universal upper bound for the first diameter of

3
3

-manifolds of scalar curvature

≥
+
1

\geq + 1

. We find it in the case of finite fundamental group by using a combinatorial theorem about finite trees, and in the case when

π
1

{\pi _1}

is infinite cyclic by using a weak notion of a

π
1

{\pi _1}

-equivariant Busemann function.
Hunt’s hypothesis for Lévy processes
Authors: Rao Murali
Published: June 2010

In this note, a general condition is given implying the validity of Hunt’s hypothesis

(

H

)

({\text {H}})

for Levy processes in

d
d

-dimensions.
A remark on kernels of reduction
Authors: Coleman Robert F.
Published: June 2010

We show the group of torsion points on an Abelian variety defined over an algebraic closure of the rationals,

Q
¯

{\mathbf {\bar Q}}

is generated by the kernels of reduction of all the primes of

Q
¯

{\mathbf {\bar Q}}

.
Alternative version of Shapley’s theorem on closed coverings of a simplex
Authors: Ichiishi Tatsuro
Published: June 2010

Sperner’s theorem as formulated by Ky Fan is dual to the KnasterKuratowski-Mazurkiewicz theorem. Shapley’s theorem is a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem. This paper points out that Shapley’s theorem is a generalization of Sperner’s theorem as well, by establishing an alternative version of Shapley’s theorem. Applications to the multiperson cooperative game theory are also discussed.
Phragmén-Lindelöf theorem for the minimal surface equation
Authors: Hwang Jenn-Fang
Published: June 2010

It is proved that if

u
u

satisfies the minimal surface equation in an unbounded domain

Ω
\Omega

which is properly contained in a half plane, then the growth property of

u
u

depends on

Ω
\Omega

and the boundary value of

u
u

only.
An isoperimetric theorem on the cube and the Kintchine-Kahane inequalities
Authors: Talagrand Michel
Published: June 2010

For vectors

x
1

,
…
,

x
n

{x_1}, \ldots , {x_n}

in a Banach space, we bound the deviation of

‖

∑

i
≤
n

ε
i

x
i

‖

\left \| {\sum \nolimits _{i \leq n} {{\varepsilon _i}{x_i}} } \right \|

from its median.
𝑑-final continua
Authors: Isbell John
Published: June 2010

The Hilbert cube is known to be an irreducible quotient of every perfect metric space. Its irreducible quotients are identified: all nondegenerate Peano continua no open set in which is homeomorphic with

R
R

. In compact metric spaces, every irreducible surjection

X
→
Y

X \to Y

embeds a dense

G
δ

{G_\delta }

subset of

X
X

in

Y
Y

. The Peano continua I which are strongly initial, every irreducible map from a Peano continuum to I being a homeomorphism, are identified: the dendrites the closure of whose end points is zero-dimensional.
Functions not vanishing on trivial Gleason parts of Douglas algebras
Authors: Gorkin Pamela
Published: June 2010

Let

B
B

denote a closed subalgebra of

L
∞

{L^\infty }

containing the space of bounded analytic functions. Let

M
(
B
)

M(B)

denote the maximal ideal space of

B
B

. Let

f
f

be a function in

B
B

such that

f
f

does not vanish on any Gleason part consisting of a single point. We show that if

g
g

is a function in

B
B

such that

|
g
|

≤

|
f
|

 on 

M
(
B
)

\left | g \right | \leq \left | f \right |{\text { on }}M(B)

, then

g

/

f
∈
B

g/f \in B

.
A uniqueness theorem of reflectable deformations of a Fuchsian group
Authors: Sengupta Jharna
Published: June 2010

Let

Γ
\Gamma

be a Fuchsian group of signature

(
p
,
n
,
m
;

ν
1

,

ν
2

,
…
,

ν
n

)

(p,n,m;{\nu _1},{\nu _2}, \ldots ,{\nu _n})

;

2
p
−
2
+
m
+

∑

j
=
1

n

(
1
−
1

/

ν
j

)
>
0

2p - 2 + m + \sum \nolimits _{j = 1}^n {(1 - 1/{\nu _j}) > 0}

. Let

I
1

,

I
2

,
…
,

I
m

{I_1},{I_2}, \ldots ,{I_m}

be a maximal set of inequivalent components of

Ω
∩

R
^

\Omega \cap {\mathbf {\hat R}}

;

Ω
\Omega

is the region of discontinuity and

R
^

{\mathbf {\hat R}}

is the extended real line. Let

ϕ
\phi

be a quadratic differential for

Γ
\Gamma

. Let

f
f

be a solution of the Schwarzian differential equation

S
f
=
ϕ

Sf = \phi

. If

ϕ
\phi

is reflectable,

f
f

maps each

I
j

{I_j}

into a circle

C
j

{C_j}

. For each

γ
∈
Γ

\gamma \in \Gamma

there is a Moebius transformation

X

(
γ
)

\mathcal {X}(\gamma )

such that

f
∘
γ
=

X

(
γ
)
∘
f

f \circ \gamma = \mathcal {X}(\gamma ) \circ f

. We prove that

ϕ
\phi

is determined by the homomorphism

X

\mathcal {X}

and the circles

C
1

,

C
2

,
…
,

C
m

{C_1},{C_2}, \ldots ,{C_m}

.
The fixed-point index and the fixed-point theorems of 1-set-contraction mappings
Authors: Li Guozhen
Published: June 2010

W. V. Petryshyn [1] studied the fixed-point theorem of

1
1

-set-contraction mappings. This paper gives the definition of the fixed-point index of

1
1

-set-contraction mappings and the concept of semiclosed

1
1

-set-contraction mappings; then we obtain some fixed-point theorems about it.
Virtually regular coverings
Authors: Lozano María Teresa , Safont Carmen
Published: June 2010

In this paper we introduce the concept of virtually regular covering, which include all regular branched coverings and also some irregular ones. We prove that, as in the case of regular coverings, some properties of the branching set reflect properties of the cover for virtually regular coverings of

S
3

{S^3}

, generalizing results of Kim-Tollefson, Lozano and Plotnick. As a consequence, we obtain that every virtually regular cover of

S
3

{S^3}

branched along a fracturable prime and non split link is Haken.
Powers’s binary shifts on the hyperfinite factor of type 𝐼𝐼₁
Authors: Enomoto Masatoshi , Watatani Yasuo
Published: June 2010

A unit preserving

∗
*

-endomorphism

σ
\sigma

on the hyperfinite

I

I

1

{\text {I}}{{\text {I}}_1}

factor

R
R

is called a shift if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="intersection Underscript n equals 0 Overscript normal infinity Endscripts sigma Superscript n Baseline left-parenthesis upper R right-parenthesis equals left-brace lamda Baseline 1 semicolon lamda element-of double-struck upper C right-brace">

⋂

n
=
0

∞

σ
n

(
R
)
=
{
λ
1
;
λ
∈

C

}

\bigcap \nolimits _{n = 0}^\infty {{\sigma ^n}(R) = \{ \lambda 1;\lambda \in \mathbb {C}} \}

. A shift

σ
\sigma

is called Powers’ binary shift if there is a self-adjoint unitary

u
u

such that

R
=
{

σ
n

(
u
)
;
n
∈

N

∪
{
0
}
}

R = \{ {\sigma ^n}(u);n \in \mathbb {N} \cup \{ 0\} \}

and

σ
k

(
u
)
u
=
±
u

σ
k

(
u
)

{\sigma ^k}(u)u = \pm u{\sigma ^k}(u)

for

k
∈

N

∪
{
0
}

k \in \mathbb {N} \cup \{ 0\}

. Let

q
(
σ
)

q(\sigma )

be the number

min
{
k
∈

N

;

σ
k

(
R

)
′

∩
R
≠

C

1
}

\min \{ k \in \mathbb {N};{\sigma ^k}(R)’ \cap R \ne \mathbb {C}1\}

. It is shown that the number

q
(
σ
)

q(\sigma )

is not the complete outer conjugacy invariant for Powers’ binary shifts.
A semi-Fredholm principle for periodically forced systems with homogeneous nonlinearities
Authors: Lazer A. C. , McKenna P. J.
Published: June 2010

We show that if the potential in a second-order Newtonian system of differential equations is positively homogeneous of degree two and positive semidefinite, and if the unforced system has no nontrivial

T
T

-periodic solutions

(
T
>
0
)

(T > 0)

, then for any continuous

T
T

-periodic forcing, there is at least one

T
T

-periodic solution.
The pseudo-orbit tracing property and expansiveness on the Cantor set
Authors: Shimomura Takashi
Published: June 2010

The set of all the expansive homeomorphisms with the pseudo-orbit tracing property is dense in the space of all the homeomorphisms of the Cantor set with the topology of uniform convergence. Moreover a topologically transitive (resp. mixing) homeomorphism of the Cantor set is approximated uniformly by topologically transitive (resp. mixing) expansive homeomorphisms with the pseudo-orbit tracing property.
On Tchebysheff systems
Authors: Kitahara Kazuaki
Published: June 2010

Let

u
1

,
…
,

u
n

{u_1}, \ldots ,{u_n}

be linearly independent continuously differentiable functions on the unit interval. In this paper, we obtain the following two results. One is a necessary and sufficient condition for the span of

{
1
,

u
1

,
…
,

u
n

}

\{ 1,{u_1}, \ldots ,{u_n}\}

to have a Markoff basis containing 1. The other is that any Markoff system

{

u
i

}

i
=
1

n

\{ {u_i}\} _{i = 1}^n

has a Tchebysheff extension

u

n
+
1

{u_{n + 1}}

which is continuously differentiable.
On weakly stationary sets
Authors: Feng Qi
Published: June 2010

We consider whether every weakly stationary set is stationary. We show that if

0
#

{0^\# }

does not exist then every weakly stationary set is stationary. Also there is a weakly stationary nonstationary set on

[

ω
2

]

ℵ
1

[\omega _2]^{\aleph _1}

if and only if Chang’s conjecture holds. From a

ω
1

\omega _1

-Erdös cardinal, we get a model in which

2

ℵ
0

>

ω
2

2^{{\aleph _0}} > \omega _2

and all the subsets of

2

ℵ
0

2^{\aleph _0}

of order type

ω
1

\omega _1

form a weakly stationary set which is nonstationary.
Subgroups of almost separable torsionfree abelian groups
Authors: Metelli Claudia
Published: June 2010

A characterization is given of those subgroups of an almost separable torsionfree abelian group which inherit separability from the group.
Reduction of local Lie algebra structures
Authors: Mikami Kentaro
Published: June 2010

We deal with the reduction theory of local Lie algebra structures in the function space through coadjoint equivariant momentum mappings.
A generalized Poincaré inequality for Gaussian measures
Authors: Beckner William
Published: June 2010

New inequalities are obtained which interpolate in a sharp way between the Poincaré inequality and the logarithmic Sobolev inequality for both Gaussian measure and spherical surface measure.
An extension of Rellich’s inequality
Authors: Bennett Donna Marie
Published: June 2010

In this paper we present a theorem which extends the results of an inequality originally due to Franz Rellich [4]. The theorem by Rellich establishes an inequality widely used in the spectral theory of partial differential operators. Our theorem allows for a broader range of application by extending the class of functions to which the theorem is applicable. Many authors call upon inequalities similar to the one established in our theorem in dealing with problems concerning essential self-adjointness of Schrödinger operators and other problems arising in oscillation theory of elliptic operators. In the first part of the paper we present Rellich’s inequality and discuss some problems dealing with symmetric operators on Hilbert spaces where Rellich’s inequality is a useful tool. We shall also discuss some important extensions of Rellich’s work which were established by other mathematicians. One such extension was proved by W. Allegretto [1] in dealing with elliptic equations of order

2
n

2n

. Another extension was established by U. W. Schmincke [5] in considering essential self-adjointness criteria of Schrödinger operators. Schmincke’s extension is of particular interest to us due to his elegant proof. We follow Schmincke’s method of proof. We then state and prove our generalization of Rellich’s inequality along with a useful corollary. The paper concludes with a few brief comments on our result and other work which could be done with Rellich’s inequality.
On the monotonicity of the permanent
Authors: Hwang Suk Geun
Published: June 2010

Let

Ω
n

{\Omega _n}

denote the set of all

n
×
n

n \times n

doubly stochastic matrices and let

J
n

=

[
1

/

n

]

n
×
n

{J_n} = {[1/n]_{n \times n}}

. For

A
∈

Ω
n

A \in {\Omega _n}

, if

f
A

(
t
)
=

per((1 - t)

J
n

+
t
A
)

{f_A}(t) = {\text {per((1 - t)}}{J_n} + tA)

is a nondecreasing function of

t
t

on

[
0
,
1
]

[0,1]

, we say that the monotonicity of permanent (abb. MP) holds for

A
A

. Friedland and Mine [3] proved MP for

(
n

J
n

−

I
n

)

/

(
n
−
1
)

(n{J_n} - {I_n})/(n - 1)

. In [6], Lih and Wang proposed a problem of determining whether MP holds for

J

n
1

⊗
⋯
⊗

J

n
k

,

n
i

>
0

{J_{{n_1}}} \otimes \cdots \otimes {J_{{n_k}}},{n_i} > 0

. In this note, we prove MP for

(
(
m

J
m

−

I
m

)
⊗
s

J
s

)

/

(
m
−
1
)
s

((m{J_m} - {I_m}) \otimes s{J_s})/(m - 1)s

, extending the result of Friedland and Mine, and give an affirmative answer to the Lih and Wang’s question.
Spatial critical points of solutions of a one-dimensional nonlinear parabolic problem
Authors: Turyn Lawrence
Published: June 2010

The number of spatial critical points is nonincreasing in time, for positive, analytic solutions of a scalar, nonlinear, parabolic partial differential equation in one space dimension. While proving this, we answer the question: What happens to a critical point which loses simplicity?
Stable isomorphism of hereditary 𝐶*-subalgebras and stable equivalence of open projections
Authors: Zhang Shuang
Published: June 2010

We relate the stable isomorphism of two hereditary

C
∗

{C^*}

-subalgebras to the stable equivalence of the corresponding open projections. We prove that if

A
A

is completely

σ
\sigma

-unital, then

her

(
p
)

{\operatorname {her}}(p)

and

h
e
r
(

q
)

{\operatorname {her(}}q)

generate the same closed ideal of

A
A

iff

p
⊗
1
∼
q
⊗
1

p \otimes 1 \sim q \otimes 1

in

(
A
⊗
K

)

∗
∗

{(A \otimes K)^{**}}

iff the central supports of

p
p

and

q
q

in

A

∗
∗

{A^{**}}

are the same. If, in addition,

p
⊥
q

p \bot q

, then the above three equivalent conditions are equivalent to the condition:

p
⊗
1

p \otimes 1

and

q
⊗
1

q \otimes 1

are in the same path component of open projections in

(
A
⊗
K

)

∗
∗

{(A \otimes K)^{**}}

.
Canonical domains on Riemann surfaces
Authors: Maskit Bernard
Published: June 2010

Let

S
S

be a Riemann surface of genus

g
>
0

g > 0

, and of finite topological type. Then

S
S

can be uniquely realized as a closed Riemann surface from which a finite number of disjoint points and closed circular discs have been removed. As a corollary, we obtain that the moduli space of surfaces of genus

g
g

with one hole is a topological product of the moduli space of surfaces of genus

g
g

with one puncture and an interval.
On oscillations of unbounded solutions
Authors: Győri I. , Ladas G. , Pakula L.
Published: June 2010

Consider the differential equation with deviating arguments (1)
\[

y
˙

(
t
)
+
p
[
y
(
t
−

σ
1

)
−
y
(
t
−

σ
2

)
]
=
0

\dot y(t) + p[y(t - {\sigma _1}) - y(t - {\sigma _2})] = 0

\]
where

p
,

σ
1

p,{\sigma _1}

and

σ
2

{\sigma _2}

are real numbers. We prove that every unbounded solution of (1) oscillates if and only if the characteristic equation (2)
\[

λ
+
p
(

e

−
λ

σ
1

−

e

−
λ

σ
2

)
=
0

\lambda + p({e^{ - \lambda {\sigma _1}}} - {e^{ - \lambda {\sigma _2}}}) = 0

\]
has no positive roots and 0 is a simple root of (2).
An identity in the free Lie algebra
Authors: Wigner David
Published: June 2010

An identity in the free Lie algebra is shown to imply the theorems of Friedrichs and Specht-Wever.