bluuebunny/crossref_metadata_2025 huggingface dataset:
Comfort types of ultrafilters
Authors: García-Ferreira Salvador
Published: July 2010

Let

α
\alpha

be an infinite cardinal. Comfort pre-order on

β
(
α
)
∖
α

\beta (\alpha )\backslash \alpha

is defined as follows: for

p
,
q
∈
β
(
α
)
∖
α
,

p

⩽

c

q

p,q \in \beta (\alpha )\backslash \alpha ,\;p{ \leqslant _{\text {c}}}q

if every

q
q

-compact space is

p
p

-compact. For

p
∈
U
(
α
)

p \in U(\alpha )

, we let

T

RK

(
p
)

{T_{\operatorname {RK} }}(p)

be the type of

p
p

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript c Baseline left-parenthesis p right-parenthesis equals StartSet q element-of upper U left-parenthesis alpha right-parenthesis colon q less-than-or-slanted-equals Subscript c Baseline p less-than-or-slanted-equals Subscript c Baseline q EndSet">

T

c

(
p
)
=
{
q
∈
U
(
α
)
:
q

⩽

c

p

⩽

c

q
}

{T_{\text {c}}}(p) = \{ q \in U(\alpha ):q{ \leqslant _{\text {c}}}p{ \leqslant _{\text {c}}}q\}

. Since

T

c

(
p
)

{T_{\text {c}}}(p)

is a union of types, it is natural to define
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript p Baseline equals StartAbsoluteValue StartSet upper T Subscript upper R upper K Baseline left-parenthesis q right-parenthesis colon q element-of upper T Subscript c Baseline left-parenthesis p right-parenthesis EndSet EndAbsoluteValue">

c
p

=

|

{

T

RK

(
q
)
:
q
∈

T

c

(
p
)
}

|

{c_p} = |\{ {T_{\operatorname {RK} }}(q):q \in {T_{\text {c}}}(p)\} |

. It is evident that

ω
⩽

c
p

⩽

2
α

\omega \leqslant {c_p} \leqslant {2^\alpha }

for

p
∈
U
(
α
)

p \in U(\alpha )

. We show that if

p
∈
U
(
α
)

p \in U(\alpha )

then

|

T

RK

(
p
)

|

=

|

α

α

/

p

|

|{T_{\operatorname {RK} }}(p)| = {|^\alpha }\alpha /p|

, and we use this equality to prove that

c
p

=

2
α

{c_p} = {2^\alpha }

whenever

p
p

is decomposable. We also note that if

p
p

is countably incomplete then

2
ω

⩽

c
p

⩽

2
α

{2^\omega } \leqslant {c_p} \leqslant {2^\alpha }

; if

p
p

is RK-minimal (selective) and

ω
>
α

\omega > \alpha

then

c
p

=
ω

{c_p} = \omega

and

c
p

=

2

ℵ
n

{c_p} = {2^{{\aleph _n}}}

for each

p
∈
U
(

ℵ
n

)

p \in U({\aleph _n})

and for

n
>
ω

n > \omega

. Finally, we prove that, if

α
\alpha

is a strong limit and

p
∈
U
(
α
)

p \in U(\alpha )

is indecomposable, then

β
p

(
α
)

{\beta _p}(\alpha )

is a

p
p

-compact, noninitially

α
\alpha

-compact space, where

β
p

(
α
)

{\beta _p}(\alpha )

is the

p
p

-compactification of

α
\alpha

.
A compactum that cannot be an attractor of a self-map on a manifold
Authors: Günther Bernd
Published: July 2010

A one-dimensional compactum (in fact, a certain solinoid) is constructed, such that there does not exist a self-map on a manifold having this compactum as attractor.
Regular sequences, projective dimension and criteria for regularity of local rings
Authors: Jothilingam P. , Mangayarcarassy S.
Published: July 2010

Foxby proves the following proposition (Math. Z. 132 (1973)). Let

(
R
,

m

)

(R,\mathfrak {m})

be a noetherian local ring and

M
M

any finitely generated

R
R

-module such that the projective dimension of

M

/

a

M

M/\mathfrak {a}M

is finite for all ideals

a

\mathfrak {a}

of finite projective dimension. Then either

M
M

is free or

R
R

is regular local. In this article we prove that the conclusion holds if we restrict only to ideals generated by regular sequences, with the empty sequence being interpreted as the zero ideal.
On varieties as hyperplane sections
Authors: Ballico E.
Published: July 2010

Here we extend to the the singular (but locally complete intersection) case a theorem of L’vovsky giving a condition ("

h
0

(

N
X

(
−
1
)
)
⩽
n
+
1

{h^0}({N_X}( - 1)) \leqslant n + 1

") forcing a variety

X
⊂

P

n

X \subset {{\mathbf {P}}^n}

not to be a hyperplane section (except of cones). Then we give a partial extension of this criterion to the case of subvarieties of a Grassmannian.
Sign properties of Green’s functions for a family of two-point boundary value problems
Authors: Eloe Paul W. , Ridenhour Jerry
Published: July 2010

Sign properties of and comparison theorems for Green’s functions of a family of two-point boundary value problems for a general ordinary differential operator are obtained. The results extend known results for a two-term ordinary differential operator.
Paradoxical functions on the interval
Authors: Jiménez López Víctor
Published: July 2010

In this paper it is shown that any expanding with Lipschitz derivative function

f
f

has a contradictory behaviour from the point of view of chaos in the sense of Li and Yorke. On the one hand it cannot generate scrambled sets of positive Lebesgue measure. On the other hand the two-dimensional set

Ch
⁡
(
f
)

\operatorname {Ch} (f)

including the pairs

(
x
,
y
)

(x,y)

such that

{
x
,
y
}

\{ x,y\}

is a scrambled set of

f
f

has positive measure. In fact, both the geometric structure (almost everywhere) and measure of

Ch
⁡
(
f
)

\operatorname {Ch} (f)

can be explicitly obtained.
Sur les réflexions des courbes de Lavrentiev
Authors: Zinsmeister Michel
Published: July 2010

We answer a question of Semmes: A Jordan curve

Γ
\Gamma

admitting a quasiconformal reflexion, which is "conformal" near

Γ
\Gamma

in a sense given precisely by a Carleson measure condition, is necessarily rectifiable.
The existence question in the calculus of variations: a density result
Authors: Cellina Arrigo , Mariconda Carlo
Published: July 2010

We show the existence of a dense subset

D

\mathcal {D}

of

C

(

R

)

\mathcal {C}(\mathbb {R})

such that, for

g
g

in it, the problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minimum integral Subscript 0 Superscript upper T Baseline g left-parenthesis x left-parenthesis t right-parenthesis right-parenthesis d t plus integral Subscript 0 Superscript upper T Baseline h left-parenthesis x prime left-parenthesis t right-parenthesis right-parenthesis d t comma x left-parenthesis 0 right-parenthesis equals a comma x left-parenthesis upper T right-parenthesis equals b">

minimum

∫
0
T

g
(
x
(
t
)
)
d
t
+

∫
0
T

h
(

x
′

(
t
)
)
d
t
,

x
(
0
)
=
a
,

x
(
T
)
=
b

{\text {minimum}}\;\int _0^T {g(x(t))dt + \int _0^T {h(x’(t))dt,\;x(0) = a,\;x(T) = b} }

\]
admits a solution for every lower semicontinuous

h
h

satisfying growth conditions
Examples of natural extensions of nonsingular endomorphisms
Authors: Dajani K. G. , Hawkins J. M.
Published: July 2010

We extend a nonsingular noninvertible map

T
T

on

(
X
,

B

,
μ
)

(X,\mathcal {B},\mu )

to a minimal nonsingular automorphism containing

T
T

as a factor and preserving some measure theoretic properties of

T
T

. We call the automorphism the natural extension of

T
T

. We compute some examples of natural extensions: first when

T
T

is a shift with

μ
\mu

a product measure, and then generalize the result to some nonexact examples as well.
Solvability of semilinear equations with compact perturbations of operators of monotone type
Authors: Guan Zhengyuan
Published: July 2010

The solvability of the equation

A
u
−
T
u
+
C
u
=
f

Au - Tu + Cu = f

is studied under various assumptions of monotonicity and compactness on the operators A, T, and C, which map subsets of a reflexive Banach space X into its dual space. It is nowhere assumed that X possesses a Schauder basis or that the operator T is positive definite and selfadjoint. The results extend and/or improve recent results obtained by Chen, Kartsatos and Mabry, and Kesavan.
Stable absolutely ubiquitous structures
Authors: Ivanov A. A.
Published: July 2010

It is proved that absolutely ubiquitous structures with trivial algebraic closures are monadically stable.
Solvability of the equation Δ_{𝑔}𝑢+𝑆̃𝑢^{𝜎}=𝑆𝑢 on manifolds
Authors: Tang Jun Jie
Published: July 2010

For the negative Yamabe invariant and

S
~

≤
0

\tilde S \leq 0

, we obtain that the equation

Δ
g

u
+

S
~

u
σ

=
S
u

{\Delta _g}u + \tilde S{u^\sigma } = Su

has a positive solution if and only if the supremum of the Yamabe invariant over all smooth coverings of the 0-level set of

S
~

\tilde S

is positive.
Fixed point theorems for multi-valued mappings in subsymmetrizable spaces
Authors: Nguyen Huy Viet
Published: July 2010

Three fixed point theorems for multi-valued mappings in symmetric spaces with closure operator are proved. As corollaries of these theorems we obtain the existence of fixed points of mappings in compact metric spaces. Another corollary of the last theorem extends a condition of Kannan’s theorem.
Bounded common extensions of charges
Authors: Basile A. , Bhaskara Rao K. P. S. , Shortt R. M.
Published: July 2010

Let

A

\mathcal {A}

and

B

\mathcal {B}

be fields of subsets of a set X and let

μ
:

A

→

R

\mu :\mathcal {A} \to {\mathbf {R}}

and

ν
:

B

→

R

\nu :\mathcal {B} \to {\mathbf {R}}

be consistent, bounded, finitely additive measures (i.e., charges). We give necessary and sufficient conditions for

μ
\mu

and

ν
\nu

to have a bounded common extension to

A

∨

B

\mathcal {A} \vee \mathcal {B}

. Conditions on

A

\mathcal {A}

and

B

\mathcal {B}

are given under which any bounded consistent charges

μ
\mu

and

ν
\nu

have a bounded common extension.
Strongly exposed points in Lebesgue-Bochner function spaces
Authors: Hu Zhibao , Lin Bor-Luh
Published: July 2010

It is a result of Peter Greim that if

f
f

is a strongly exposed point of the unit ball of Lebesgue-Bochner function space

L
p

(
μ
,
X
)
,

1
>
p
>
∞

{L^p}(\mu ,X),\;1 > p > \infty

, then

f
f

is a unit vector and

f
(
t
)

/

|

|

f
(
t
)

|

|

f(t)/||f(t)||

is a strongly exposed point of the unit ball of

X
X

for almost all

t
t

in the support of

f
f

. We prove that the converse is also true.
On the metrical theory of continued fractions
Authors: Nair R.
Published: July 2010

Suppose

b
k

{b_k}

denotes either

ϕ
(
k
)

\phi (k)

or

ϕ
(

p
k

)

(
k
=
1
,
2
,
…
)

\phi ({p_k})\;(k = 1,2, \ldots )

where the polynomial

ϕ
\phi

maps

N

\mathbb {N}

to

N

\mathbb {N}

and

p
k

{p_k}

denotes the

k
k

th rational prime. Suppose

(

c
k

(
x
)

)

k
=
1

∞

({c_k}(x))_{k = 1}^\infty

denotes the sequences of partial quotients of the continued function expansion of the real number

x
x

. Then for certain functions

F
:

R

⩾
0

→

R

F:{\mathbb {R}_{ \geqslant 0}} \to \mathbb {R}

we show that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript upper N right-arrow normal infinity Endscripts upper F Superscript negative 1 Baseline left-bracket StartFraction upper F left-parenthesis c Subscript b 1 Baseline left-parenthesis x right-parenthesis right-parenthesis plus midline-horizontal-ellipsis plus upper F left-parenthesis c Subscript b Sub Subscript k Subscript Baseline left-parenthesis x right-parenthesis right-parenthesis Over upper N EndFraction right-bracket equals upper F Superscript negative 1 Baseline left-bracket StartFraction 1 Over left-parenthesis log 2 right-parenthesis EndFraction integral Subscript 0 Superscript 1 Baseline StartFraction upper F left-parenthesis c 1 left-parenthesis x right-parenthesis right-parenthesis Over 1 plus x EndFraction d x right-bracket">

lim

N
→
∞

F

−
1

[

F
(

c

b
1

(
x
)
)
+
⋯
+
F
(

c

b
k

(
x
)
)

N

]

=

F

−
1

[

1

(
log
⁡
2
)

∫
0
1

F
(

c
1

(
x
)
)

1
+
x

d
x

]

\lim \limits _{N \to \infty } {F^{ - 1}}\left [ {\frac {{F({c_{{b_1}}}(x)) + \cdots + F({c_{{b_k}}}(x))}} {N}} \right ] = {F^{ - 1}}\left [ {\frac {1} {{(\log 2)}}\int _0^1 {\frac {{F({c_1}(x))}} {{1 + x}}dx} } \right ]

\]
almost everywhere with respect to Lebesgue measure. This result with

b
k

=
k

{b_k} = k

is classical and due to Ryll-Nardzewski.
On the behavior of the constant in a decoupling inequality for martingales
Authors: Hitczenko Paweł
Published: July 2010

Let

(

f
n

)

({f_n})

and

(

g
n

)

({g_n})

be two martingales with respect to the same filtration

(

F

n

)

({\mathcal {F}_n})

such that their difference sequences

(

d
n

)

({d_n})

and

(

e
n

)

({e_n})

satisfy
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis d Subscript n Baseline greater-than-or-equal-to lamda vertical-bar script upper F Subscript n minus 1 Baseline right-parenthesis equals upper P left-parenthesis e Subscript n Baseline greater-than-or-equal-to lamda vertical-bar script upper F Subscript n minus 1 Baseline right-parenthesis">

P
(

d
n

≥
λ

|

F

n
−
1

)
=
P
(

e
n

≥
λ

|

F

n
−
1

)

P({d_n} \geq \lambda |{\mathcal {F}_{n - 1}}) = P({e_n} \geq \lambda |{\mathcal {F}_{n - 1}})

\]
for all real

λ
\lambda

’s and

n
≥
1

n \geq 1

. It is known that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f Superscript asterisk Baseline double-vertical-bar Subscript p Baseline less-than-or-equal-to upper K Subscript p Baseline double-vertical-bar g Superscript asterisk Baseline double-vertical-bar Subscript p Baseline comma 1 less-than-or-equal-to p greater-than normal infinity comma">

‖

f
∗

‖

p

≤

K
p

‖

g
∗

‖

p

,

1
≤
p
>
∞
,

{\left \| {{f^ \ast }} \right \|_p} \leq {K_p}{\left \| {{g^ \ast }} \right \|_p},\quad 1 \leq p > \infty ,

\]
for some constant

K
p

{K_p}

depending only on p. We show that

K
p

=
O
(
p
)

{K_p} = O(p)

. This will be obtained via a new version of Rosenthal’s inequality which generalizes a result of Pinelis and which may be of independent interest.
Hankel operators on the Bergman space of the unit polydisc
Authors: Li Huiping
Published: July 2010

Let

D
D

be the unit polydisc in

C

n

{\mathbb {C}^n}

. Let

H
2

(
D
)

{H^2}(D)

be the Bergman space of

D
D

. In this paper, by using the integral representations of solutions to the

∂
¯

\overline \partial

-equations, we give function theoretic characterizations of functions

f
∈

L
2

(
D
)

f \in {L^2}(D)

such that the Hankel operators

H
f

{H_f}

from the Bergman space to

L
2

(
D
)

{L^2}(D)

are bounded and compact, respectively.
Some products of 𝛽-elements in the Novikov 𝐸₂ term of Moore spectra
Authors: Lin Jin Kun
Published: July 2010

In this note, we prove trivialities and nontrivialities of products of some higher-order

β

t

p
n

/

s

′

\beta _{t{p^n}/s}’

elements in the

E
2

{E_2}

terms of the Adams-Novikov spectral sequence of Moore spectra.
Anticommutant lifting and anticommuting dilation
Authors: Sebestyén Zoltán
Published: July 2010

In this paper we prove the anticommutant counterpart of the Sz.-Nagy-Foiaş commutant lifting theorem and Ando’s theorem on unitary dilation of a pair of commutative contractions. A dual extension theorem applies in the one-step procedure of our approach.
A disc-hull in 𝐶²
Authors: Alexander H.
Published: July 2010

We construct a compact set in

C

2

{{\mathbf {C}}^2}

whose disc-hull is a proper dense subset of its polynomial hull.
On Dirichlet series associated with polynomials
Authors: Carletti E. , Monti Bragadin G.
Published: July 2010

Let

P
(
X
)

P(X)

be a polynomial of degree N with complex coefficients and

d
1

,

d
2

{d_1},{d_2}

two complex numbers with real part greater then

−
1

-1

. Consider the Dirichlet series associated with the triple

(
P
(
X
)
,

d
1

,

d
2

)

(P(X),{d_1},{d_2})

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis s right-parenthesis equals sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts StartFraction upper P left-parenthesis n right-parenthesis Over left-parenthesis n plus d 1 right-parenthesis Superscript s Baseline left-parenthesis n plus d 2 right-parenthesis Superscript s Baseline EndFraction period">

L
(
s
)
=

∑

n
=
1

∞

P
(
n
)

(
n
+

d
1

)

s

(
n
+

d
2

)

s

.

L(s) = \sum \limits _{n = 1}^\infty {\frac {{P(n)}}{{{{(n + {d_1})}^s}{{(n + {d_2})}^s}}}.}

\]
In this paper we get an explicit formula for

L
(
s
)

L(s)

in terms of special functions which gives meromorphic continuation of

L
(
s
)

L(s)

with at most simple poles at

s
=
(
N
+
1
−
k
)

/

2
,
k
=
0
,
1
,
…

s = (N + 1 - k)/2,k = 0,1, \ldots

Finally we apply our explicit formula to Minakshisundaram’s zeta function of the three-dimensional sphere.
Constant codimension fixed sets of commuting involutions
Authors: Shaker R. J.
Published: July 2010

The ideals

J

∗
,
k

r

\mathcal {J}_{ \ast ,k}^r

of cobordism classes in

N

∗

{\mathcal {N}_ \ast }

, containing a representative admitting a

(

Z

2

)
k

{({{\mathbf {Z}}_2})^k}

-action with fixed point set of codimension r are determined for

2
k

>
r

{2^k} > r

.
𝜎-centred forcing and reflection of (sub)metrizability
Authors: Tall Franklin D.
Published: July 2010

By using supercompact reflection and preservation lemmas for random real forcing and

σ
\sigma

-centred forcing, we obtain a model in which every normal Moore space is submetrizable, but not every normal Moore space is metrizable.
Exact covering systems and the Gauss-Legendre multiplication formula for the gamma function
Authors: Beebee John
Published: July 2010

The Gauss-Legendre multiplication formula for the gamma function is
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 pi right-parenthesis Superscript left-parenthesis m minus 1 right-parenthesis slash 2 Baseline m Superscript 1 slash 2 minus m z Baseline normal upper Gamma left-parenthesis m z right-parenthesis equals normal upper Gamma left-parenthesis z right-parenthesis normal upper Gamma left-parenthesis z plus StartFraction 1 Over m EndFraction right-parenthesis midline-horizontal-ellipsis normal upper Gamma left-parenthesis z plus StartFraction m minus 1 Over m EndFraction right-parenthesis">

(
2
π

)

(
m
−
1
)

/

2

m

1

/

2
−
m
z

Γ
(
m
z
)
=
Γ
(
z
)
Γ
(
z
+

1
m

)
⋯
Γ
(
z
+

m
−
1

m

)

{(2\pi )^{(m - 1)/2}}{m^{1/2 - mz}}\Gamma (mz) = \Gamma (z)\Gamma (z + \tfrac {1} {m}) \cdots \Gamma (z + \tfrac {{m - 1}} {m})

. Let

{

a
i

(
mod

b
i

)
:
1
⩽
i
⩽
m
}

\{ {a_i}(\bmod {b_i}):1 \leqslant i \leqslant m\}

be an exact covering system with standardized offsets. Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma left-parenthesis z right-parenthesis equals StartFraction normal upper Gamma left-parenthesis z slash b 1 right-parenthesis Over b 1 Superscript 1 minus z slash b 1 Baseline EndFraction product Underscript i equals 2 Overscript m Endscripts StartFraction normal upper Gamma left-parenthesis left-parenthesis z plus a Subscript i Baseline right-parenthesis slash b Subscript i Baseline right-parenthesis Over b Subscript i Superscript negative z slash b Super Subscript i Superscript Baseline normal upper Gamma left-parenthesis a Subscript i Baseline slash b Subscript i Baseline right-parenthesis EndFraction period">

Γ
(
z
)
=

Γ
(
z

/

b
1

)

b
1

1
−
z

/

b
1

∏

i
=
2

m

Γ
(
(
z
+

a
i

)

/

b
i

)

b
i

−
z

/

b
i

Γ
(

a
i

/

b
i

)

.

\Gamma (z) = \frac {{\Gamma (z/{b_1})}} {{{b_1}^{1 - z/{b_1}}}}\prod \limits _{i = 2}^m {\frac {{\Gamma ((z + {a_i})/{b_i})}} {{b_i^{ - z/{b_i}}\Gamma ({a_i}/{b_i})}}}.

\]
Conversely, if the above identity holds, then

{

a
i

(
mod

b
i

)
:
1
⩽
i
⩽
m
}

\{ {a_i}(\bmod {b_i}):1 \leqslant i \leqslant m\}

is an exact covering system with standardized offsets. The Gauss-Legendre multiplication formula is a special case of this identity.
On the 𝐶*-algebra generated by the left regular representation of a locally compact group
Authors: Bédos Erik
Published: July 2010

Let

λ
\lambda

denote the left regular representation of a locally compact group

G
G

on

L
2

(
G
)

{L^2}(G)

and

C

∗

(
λ
(
G
)
)

{C^{\ast }}(\lambda (G))

the

C

∗

{C^{\ast }}

-algebra generated by

λ
(
G
)

\lambda (G)

. We show that the amenability of

G
G

and the amenability of

G
G

considered as a discrete group both may be characterized in terms of

C

∗

(
λ
(
G
)
)

{C^{\ast }}(\lambda (G))

.
Homology of Azumaya algebras
Authors: Cortiñas G. , Weibel C.
Published: July 2010

If R is a commutative k-algebra, any Azumaya R-algebra has the same Hochschild homology as R does.
Fixed algebras of residually nilpotent Lie algebras
Authors: Drensky Vesselin
Published: July 2010

Let

L
m

{L_m}

be the free Lie algebra of rank

m
>
1

m > 1

over a field

K
K

, and let

J
J

be an ideal of

L
m

{L_m}

such that

J
⊂

L
m

J \subset L_m^{}

and the algebra

L
m

/

J

{L_m}/J

is residually nilpotent. Let

G
≠
⟨
1
⟩

G \ne \langle 1\rangle

be a finite group of automorphisms of

L
m

/

J

{L_m}/J

and the order of

G
G

be invertible in

K
K

. We establish that the algebra of fixed points

(

L
m

/

J

)
G

{({L_m}/J)^G}

is not finitely generated. The class of algebras under consideration contains the free Lie algebra over an arbitrary field and the relatively free algebras in nonnilpotent varieties of Lie algebras over infinite fields of characteristic different from

2
2

and

3
3

.
The kernel of an irreducible map
Authors: Krause Henning
Published: July 2010

Let

0
→
A
→
B

→

g

C
→
0

0 \to A \to B\xrightarrow {g}C \to 0

be a short exact sequence in the category of finitely generated modules over an artin algebra. Suppose also that the map g is irreducible. Following a conjecture of Brenner, we discuss the property of the indecomposable module A to be the starting term of an almost split sequence with indecomposable middle term.
A converse to Stanley’s conjecture for 𝑆𝑙₂
Authors: Van den Bergh Michel
Published: July 2010

We prove, in the case of

Sl
2

\text {Sl}_2

, a converse to Stanley’s conjecture about Cohen-Macaulayness of invariant modules for reductive algebraic groups.
Derivations in prime near-rings
Authors: Wang Xue Kuan
Published: July 2010

Let N be a prime near-ring with center Z. The purpose of this paper is to study derivations on N. We show two main results: (1) Let N be 2-torsion-free, and let

D
1

{D_1}

and

D
2

{D_2}

be derivations on N such that

D
1

D
2

{D_1}{D_2}

is also a derivation. Then either

D
1

{D_1}

or

D
2

{D_2}

is zero if and only if

[

D
1

(
x
)
,

D
2

(
y
)
]
=
0

[{D_1}(x),{D_2}(y)] = 0

for all

x
,
y
∈
N

x,y \in N

. (2) Let n be an integer

≥
1

\geq 1

, N be n!-torsion-free, and D a derivation with

D
n

(
N
)
=
{
0
}

{D^n}(N) = \{ 0\}

. Then

D
(
Z
)
=
{
0
}

D(Z) = \{ 0\}

.
An interpolation theorem and a sharp form of a multilinear fractional integration theorem
Authors: Bak Jong-Guk
Published: July 2010

We prove a sharp interpolation theorem for Orlicz spaces with the Luxemburg norm. As a corollary we obtain a sharp form of an exponential integrability theorem, due to Grafakos, for the multilinear fractional integration operator. This generalizes a theorem of Adams.
Linear independence of iterates and meromorphic solutions of functional equations
Authors: Christensen Jens Peter Reus , Fischer Pal
Published: July 2010

It is shown that except for trivial cases a sequence generated by iteration of meromorphic functions is always linearly independent. The nonexistence of meromorphic solutions and solutions having only isolated singularities of the Feigenbaum functional equation is proven.
Generalized wavelet decompositions of bivariate functions
Authors: Chui Charles K. , Li Xin
Published: July 2010

The objective of this paper is to introduce an integral transform of wavelet-type on

L
2

(

R
2

)

{L^2}({R^2})

that can be applied to decompose the space

L
2

(

R
2

)

{L^2}({R^2})

into a direct sum of subspaces, each of which is identified as

L
2

(
R
)

{L^2}(R)

. Projections from

L
2

(

R
2

)

{L^2}({R^2})

onto these subspaces are also discussed. Moreover, wavelet expansions for functions in

L
2

(

R
2

)

{L^2}({R^2})

are derived in terms of wavelet bases of

L
2

(
R
)

{L^2}(R)

.
A generalization of the Shimizu-Leutbecher and Jørgensen inequalities to Möbius transformations in 𝑅^{𝑁}
Authors: Hersonsky Sa’ar
Published: July 2010

We give a generalization of the Shimizu-Leutbecher inequality and a partial generalization of the Jorgensen inequality to Möbius transformations in

R

N

{{\mathbf {R}}^N}

using the Clifford algebra and the Vahlen group.
On the similarity of centered operators to contractions
Authors: Petrović Srdjan
Published: July 2010

In this paper we show that every power-bounded operator-weighted shift with commuting normal weights is similar to a contraction. As an application of this result, we reduce the problem of whether every centered power-bounded operator is similar to a contraction to the analogous question about quasi-invertible centered operators.
On the Kreĭn-Rutman theorem and its applications to controllability
Authors: Phát Vũ Ngọc , Dieu Trinh Cong
Published: July 2010

This paper extends Krein-Rutman’s theorem on linear operators leaving an invariant cone in infinite-dimensional Banach spaces to multivalued convex functions. The result is applied to obtain necessary and sufficient conditions for global controllability and reachability of nonlinear discrete-time systems described by convex processes.
Semiprimitivity of group algebras of infinite simple groups of Lie type
Authors: Passman D. S.
Published: July 2010

Let G be a simple group of Lie type over an infinite locally finite field F. For any field K, we prove that the group algebra

K
[
G
]

K[G]

is semiprimitive. The argument here is a mixture of combinatorial and topological methods. Combined with earlier results, it now follows that any group algebra of an infinite locally finite simple group is semiprimitive. Furthermore, if the group is countably infinite, then the group algebra is primitive. In particular, if G is a simple group of Lie type over the field F, then

K
[
G
]

K[G]

is a primitive ring.
Central extensions of nonsymmetrizable Kac-Moody algebras over commutative algebras
Authors: Gao Yun
Published: July 2010

For a commutative algebra R over a field k of characteristic zero and a nonsymmetrizable Kac-Moody algebra

g
(
A
)

g(A)

, we prove that the Lie algebra

g
R

(
A
)
=
R

⊗
k

g
(
A
)

{g_R}(A) = R{ \otimes _k}g(A)

is centrally closed. Consequently, we get a characterization of the symmetrizability of

g
(
A
)

g(A)

by the second homology group of the Kac-Moody algebra over Laurent polynomials. Also a presentation of

g
R

(
A
)

{g_R}(A)

is given when A is of nonaffine type.
Convex bodies with similar projections
Authors: Gardner R. J. , Volčič A.
Published: July 2010

By examining an example constructed by Petty and McKinney, we show that there are pairs of centered and coaxial bodies of revolution in

E

d

,
d
≥
3

{\mathbb {E}^d}, d \geq 3

, whose projections onto each two-dimensional subspace are similar, but which are not themselves even affinely equivalent.
A remark on the spectral synthesis property for hypersurfaces of 𝑅ⁿ
Authors: Guo Kang Hui
Published: July 2010

Let M be an

(
n
−
1
)

(n - 1)

-dimensional manifold in

R
n

{R^n}

with constant relative nullity. Using an estimate established in an earlier work of the author (Canad. Math. Bull. 36 (1993), 64-73), we present a greatly simplified proof of Müller’s result on the weak spectral synthesis property of M.
Minimal surfaces with constant Kähler angle in complex projective spaces
Authors: Mo Xiao Huan
Published: July 2010

Let

ψ
:

S
2

→
C

P
n

\psi :{S^2} \to C{P^n}

be an isometric minimal immersion of the Riemann sphere

S
2

{S^2}

into

C

P
n

C{P^n}

with constant Kähler angle

θ
\theta

. In this paper, we prove that Bolton et al.’s conjecture holds if

θ
\theta

is not too close to

π
2

\frac {\pi }{2}

, that is,

ψ
\psi

is

±
\pm

holomorphic or belongs to the Veronese sequence if

|

cos
⁡
θ

|

≥

1
5

|\cos \theta | \geq \frac {1}{5}

.
On the joins of semigroup varieties with the variety of commutative semigroups
Authors: Sapir M. V. , Volkov M. V.
Published: July 2010

We show that the join of a variety of semigroups and the variety of all commutative semigroups is not finitely based, provided some weak conditions.
Zero-dimensionality of some pseudocompact groups
Authors: Dikranjan Dikran
Published: July 2010

We prove that hereditarily disconnected countably compact groups are zero-dimensional. This gives a strongly positive answer to a question of Shakhmatov. We show that hereditary or total disconnectedness yields zero-dimensionality in various classes of pseudocompact groups.
A note on subcontinua of 𝛽[0,∞)-[0,∞)
Authors: Zhu Jian-Ping
Published: July 2010

Let

M
=

⊕

n
∈
ω

I
n

M = { \oplus _{n \in \omega }}{I_n}

be the topological sum of countably many copies of the unit interval

I
I

. For any ultrafilter

u
∈

ω

∗

u \in {\omega ^{\ast }}

, we let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Superscript u Baseline equals intersection StartSet c l Subscript beta upper M Baseline left-parenthesis union StartSet upper I Subscript n Baseline colon n element-of upper A EndSet right-parenthesis colon upper A element-of u EndSet">

M
u

=
∩
{

cl

β
M

(
∪
{

I
n

:
n
∈
A
}
)
:
A
∈
u
}

{M^u} = \cap \{ {\operatorname {cl} _{\beta M}}( \cup \{ {I_n}:n \in A\} ):A \in u\}

. It is well known that

M
u

{M^u}

is a decomposable continuum with a very nice internal structure. In this paper, we show: (1) every nondegenerate subcontinuum of

β
[
0
,
∞
)
−
[
0
,
∞
)

\beta [0,\infty ) - [0,\infty )

contains a copy of

M
u

{M^u}

for some

u
∈

ω

∗

u \in {\omega ^{\ast }}

. (2) there is no nontrivial simple point in Laver’s model for the Borel conjecture. The second answers a question posed by Baldwin and Smith negatively.
Bounds for ratios of eigenvalues of the Dirichlet Laplacian
Authors: Ashbaugh Mark S. , Benguria Rafael D.
Published: July 2010

We use a doubling scheme to derive a bound for the ratio of the

2
k

{2^k}

th eigenvalue to the first for the Dirichlet Laplacian on a bounded domain

Ω
⊂

R

n

\Omega \subset {\mathbb {R}^n}

. The explicit bounds we obtain derive from the optimal bound

(

λ
2

/

λ
1

)
Ω

≤

(

λ
2

/

λ
1

)

n
−

dimensional ball

{({\lambda _2}/{\lambda _1})_\Omega } \leq {({\lambda _2}/{\lambda _1})_{n -{\text {dimensional ball}}}}

(the Payne-Pólya-Weinberger conjecture) recently proved by us.
Jordan *-derivations of standard operator algebras
Authors: Šemrl Peter
Published: July 2010

Let

H
H

be a real or complex Hilbert space,

dim
⁡
H
>
1

\dim H > 1

, and

B

(
H
)

\mathcal {B}(H)

the algebra of all bounded linear operators on

H
H

. Assume that

A

\mathcal {A}

is a standard operator algebra on

H
H

. Then every additive Jordan

∗

{\ast }

-derivation

J
:

A

→

B

(
H
)

J:\mathcal {A} \to \mathcal {B}(H)

is of the form

J
(
A
)
=
A
T
−
T

A

∗

J(A) = AT - T{A^{\ast }}

for some

T
∈

B

(
H
)

T \in \mathcal {B}(H)

.
Finite index subfactors and Hopf algebra crossed products
Authors: Szymański Wojciech
Published: July 2010

We show that if

N

⊆

M

⊆

L

⊆

K

{\mathbf {N}} \subseteq {\mathbf {M}} \subseteq {\mathbf {L}} \subseteq {\mathbf {K}}

is a Jones’s tower of type

I

I

1

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

factors satisfying

[

M

:

N

]
>
∞
,

N

′

∩

M

=

C

I
,

N

′

∩

K

[{\mathbf {M}}:{\mathbf {N}}] > \infty ,\;{\mathbf {N}}’ \cap {\mathbf {M}} = \mathbb {C}I,\;{\mathbf {N}}’ \cap {\mathbf {K}}

a factor, then

M

′

∩

K

{\mathbf {M}}’ \cap {\mathbf {K}}

bears a natural Hopf

∗

{\ast }

-algebra structure and there is an action of

M

′

∩

K

{\mathbf {M}}’ \cap {\mathbf {K}}

on

L

{\mathbf {L}}

such that the resulting crossed product is isomorphic to

K

{\mathbf {K}}

.
Local splitting theorems for Riemannian manifolds
Authors: Cai M. , Galloway G. J. , Liu Z.
Published: July 2010

In this paper we establish two local versions of the Cheeger-Gromoll Splitting Theorem. We show that if a complete Riemannian manifold

M
M

has nonnegative Ricci curvature outside a compact set

B
B

and contains a line

γ
\gamma

which does not intersect

B
B

, then the line splits in a maximal neighborhood that is contained in

M
∖
B

¯

\overline {M\backslash B}

. We use this result to give a simplified proof that

M
M

has a bounded number of ends. We also prove that if

M
M

has sectional curvature which is nonnegative (and bounded from above) in a tubular neighborhood

U
U

of a geodesic

γ
\gamma

which is a line in

U
U

, then

U
U

splits along

γ
\gamma

.
On Hosszú’s functional equation in distributions
Authors: Koh E. L.
Published: July 2010

In this paper, the Hosszú functional equation\[f(x+y−xy)+g(xy)=h(x)+k(y)f(x + y - xy) + g(xy) = h(x) + k(y)\]is reformulated in the domain of distributions. Its general solution is found which includes Fenyö’s solution as a special case.
Remarks on automorphisms of subfactors
Authors: Loi Phan H.
Published: July 2010

We establish certain properties of automorphisms on an inclusion of AFD type

II
1

{\text {II}_1}

factors with finite index and finite depth and discuss their applications to the classification problem of AFD type III subfactors, including a different proof of a result on subfactors with principal graph

D
~

n

{\tilde D_n}

.
A homological characterization of abelian 𝐵₂-groups
Authors: Rangaswamy K. M.
Published: July 2010

Assuming the Continuum Hypothesis, we show that a torsion-free abelian group G is a

B
2

{B_2}

-group if and only if

Bext
1

(
G
,
T
)
=
0
=

Bext
2

(
G
,
T
)

{\text {Bext}^1}(G,T) = 0 = {\text {Bext}^2}(G,T)

, for every torsion group T.
A characterization of nonunital operator algebras
Authors: Ruan Zhong-Jin
Published: July 2010

We give an abstract matrix norm characterization for operator algebras with contractive approximate identities by using the second dual approach. We show that if A is an

L
∞

{L^\infty }

-Banach pseudoalgebra with a contractive approximate identity, then the second dual

A

∗
∗

{A^{ \ast \ast }}

of A is a unital

L
∞

{L^\infty }

-Banach pseudoalgebra containing A as a subalgebra. It follows from the Blecher-Ruan-Sinclair characterization theorem for unital operator algebras that

A

∗
∗

{A^{ \ast \ast }}

is completely isometrically unital isomorphic to a concrete unital operator algebra on a Hilbert space. Thus A is completely isometrically isomorphic to a concrete nondegenerate operator algebra with a contractive approximate identity.
Formal fibers and complete homomorphic images
Authors: Heinzer William , Rotthaus Christel
Published: July 2010

Let

(
R
,

m

)

(R,{\mathbf {m}})

be an excellent normal local Henselian domain, and suppose that

q

{\mathbf {q}}

is a prime ideal in

R
R

of height

>
1

> 1

. We show that, if

R

/

q

R/{\mathbf {q}}

is not complete, then there are infinitely many height one prime ideals

p

⊆

q

R
^

{\mathbf {p}} \subseteq {\mathbf {q}}\hat R

of

R
^

\hat R

with

p

∩
R
=
0

{\mathbf {p}} \cap R = 0

; in particular, the dimension of the generic formal fiber of

R
R

is at least one. This result may in fact indicate that a much stronger relationship between maximal ideals in the formal fibers of an excellent Henselian local ring and its complete homomorphic images is possibly satisfied. The second half of the paper is concerned with a property of excellent normal local Henselian domains

R
R

with zero-dimensional formal fibers. We show that for such an

R
R

one has the following good property with respect to intersection: for any field

L
L

such that

Q

(
R
)
⊆
L
⊆

Q

(

R
^

)

\mathcal {Q}(R) \subseteq L \subseteq \mathcal {Q}(\hat R)

, the ring

L
∩

R
^

L \cap \hat R

is a local Noetherian domain which has completion

R
^

\hat R

.
Complemented copies of 𝑐₀ in 𝐿^{∞}(𝜇,𝐸)
Authors: Díaz Santiago
Published: July 2010

Let

E
E

be a Banach space, and let

(
Ω
,
Σ
,
μ
)

(\Omega ,\Sigma ,\mu )

be a measure space. We denote by

L
∞

(
μ
,
E
)

{L^\infty }(\mu ,E)

the Banach space of all

E
E

-valued

μ
\mu

-measurable essentially bounded functions on

Ω
\Omega

, two functions being identified if they differ only on a locally

μ
\mu

-null set. We prove that if

L
∞

(
μ
,
E
)

{L^\infty }(\mu ,E)

contains a complemented copy of

c
0

{c_0}

, then

E
E

contains a copy of

c
0

{c_0}

.
A counterexample concerning the maximum and minimum of a subharmonic function
Authors: Fryntov Alexander
Published: July 2010

For every

Δ
>
0

\Delta > 0

a function u subharmonic in the plane is constructed such that u has the order

ρ
=
1
+
Δ

\rho = 1 + \Delta

and satisfies the condition
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="min Underscript phi Endscripts u left-parenthesis r e Superscript i phi Baseline right-parenthesis slash max Underscript phi Endscripts u left-parenthesis r e Superscript i phi Baseline right-parenthesis less-than-or-equal-to minus left-parenthesis upper C plus 1 right-parenthesis for every r greater-than 0 comma">

min
φ

u
(
r

e

i
φ

)

/

max
φ

u
(
r

e

i
φ

)
≤
−
(
C
+
1
)

for every

r
>
0
,

\min \limits _\varphi u(r{e^{i\varphi }})/\max \limits _\varphi u(r{e^{i\varphi }}) \leq - (C + 1)\quad \text {for every}\,r > 0,

\]
where

C
=
C
(
ρ
)
>
0

C = C(\rho ) > 0

. This example answers a question of W. K. Hayman.
Rectifiable metric spaces: local structure and regularity of the Hausdorff measure
Authors: Kirchheim Bernd
Published: July 2010

We consider the question whether the "nice" density behaviour of Hausdorff measure on rectifiable subsets of Euclidian spaces preserves also in the general metric case. For this purpose we show the existence of a metric differential of Lipschitzian functions also in situations where the well-known theorem of Rademacher fails.
A note on minimal prime ideals
Authors: Anderson D. D.
Published: July 2010

Let R be a commutative ring and I an ideal of R. We show that if all the prime ideals minimal over I are finitely generated, then there are only finitely many prime ideals minimal over I.
A canonical ordering for free self-distributive systems
Authors: Dehornoy Patrick
Published: July 2010

We construct in free left distributive systems a natural relation that corresponds to the lexicographical ordering on sequences and proves to be a linear ordering. Applications to a word problem and to a characterization of free objects are given.
A forbidden set for embedded eigenvalues
Authors: del Río Castillo Rafael René
Published: July 2010

We study the problem of embedding eigenvalues to the spectrum of a Sturm-Liouville operator in the half axis when this spectrum is a perfect set. We prove the existence of an uncountable dense subset of the spectrum for which, by modifying the condition at the left or by locally perturbing the potential, it is not possible to add any eigenvalues.
Separation properties for self-similar sets
Authors: Schief Andreas
Published: July 2010

Given a self-similar set K in

R

s

{\mathbb {R}^s}

we prove that the strong open set condition and the open set condition are both equivalent to

H
α

(
K
)
>
0

{H^\alpha }(K) > 0

, where

α
\alpha

is the similarity dimension of K and

H
α

{H^\alpha }

denotes the Hausdorff measure of this dimension. As an application we show for the case

α
=
s

\alpha = s

that K possesses inner points iff it is not a Lebesgue null set.
Hörmander’s condition and a convolution operator generalizing Riesz potentials
Authors: Bak Jong-Guk
Published: July 2010

Under certain hypotheses including a Hörmander-type condition on the convolution kernel

K
K

we show that

K

∗

f

K{\ast }f

belongs to the space

BMO
⁡
(

R

n

)

\operatorname {BMO} ({{\mathbf {R}}^n})

whenever

f
f

belongs to the space

L

p
,
∞

(

R

n

)

{L^{p,\infty }}({{\mathbf {R}}^n})

(weak

L
p

{L^p}

) for certain

p
p

.
The inverse conductivity problem with one measurement: uniqueness for convex polyhedra
Authors: Barceló Bartolomé , Fabes Eugene , Seo Jin Keun
Published: July 2010

Let

Ω
\Omega

denote a smooth domain in

R
n

{R^n}

containing the closure of a convex polyhedron D. Set

χ
D

{\chi _D}

equal to the characteristic function of D. We find a flux g so that if u is the nonconstant solution of

div

(
(
1
+

χ
D

)
∇
u
)
=
0

\operatorname {div}\;((1 + {\chi _D})\nabla u) = 0

in

Ω
\Omega

with

∂
u

∂
n

=
g

\frac {{\partial u}}{{\partial n}} = g

on

∂
Ω

\partial \Omega

, then D is uniquely determined by the Cauchy data g and

f
≡
u

/

∂
Ω

f \equiv u/\partial \Omega

.
On the medians of gamma distributions and an equation of Ramanujan
Authors: Choi K. P.
Published: July 2010

For

n
≥
0

n \geq 0

, let

λ
(
n
)

\lambda (n)

denote the median of the

Γ
(
n
+
1
,
1
)

\Gamma (n + 1,1)

distribution. We prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n plus two thirds greater-than lamda left-parenthesis n right-parenthesis less-than-or-equal-to min left-parenthesis n plus log 2 comma n plus two thirds plus left-parenthesis 2 n plus 2 right-parenthesis Superscript negative 1 Baseline right-parenthesis">

n
+

2
3

>
λ
(
n
)
≤
min
(
n
+
log
⁡
2
,
n
+

2
3

+

(
2
n
+
2

)

−
1

)

n + \tfrac {2}{3} > \lambda (n) \leq \min (n + \log 2, n + \tfrac {2}{3} + {(2n + 2)^{ - 1}})

. These bounds are sharp. There is an intimate relationship between

λ
(
n
)

\lambda (n)

and an equation of Ramanujan. Based on this relationship, we derive the asymptotic expansion of

λ
(
n
)

\lambda (n)

as follows:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda left-parenthesis n right-parenthesis equals n plus two thirds plus StartFraction 8 Over 405 n EndFraction minus StartFraction 64 Over 5103 n squared EndFraction plus StartFraction 2 Superscript 7 Baseline dot 23 Over 3 Superscript 9 Baseline dot 5 squared n cubed EndFraction plus midline-horizontal-ellipsis period">

λ
(
n
)
=
n
+

2
3

+

8

405
n

−

64

5103

n
2

+

2
7

⋅
23

3
9

⋅

5
2

n
3

+
⋯
.

\lambda (n) = n + \frac {2}{3} + \frac {8}{{405n}} - \frac {{64}}{{5103{n^2}}} + \frac {{{2^7} \cdot 23}}{{{3^9} \cdot {5^2}{n^3}}} + \cdots .

\]
Let median

(

Z
μ

)

({Z_\mu })

denote the median of a Poisson random variable with mean

μ
\mu

, where the median is defined to be the least integer m such that

P
(

Z
μ

≤
m
)
≥

1
2

P({Z_\mu } \leq m) \geq \tfrac {1}{2}

. We show that the bounds on

λ
(
n
)

\lambda (n)

imply
\[

μ
−
log
⁡
2
≤

median

(

Z
μ

)
>
μ
+

1
3

.

\mu - \log 2 \leq {\text {median}}({Z_\mu }) > \mu + \frac {1}{3}.

\]
This proves a conjecture of Chen and Rubin. These inequalities are sharp.
Higher-order nonlinear dispersive equations
Authors: Kenig Carlos E. , Ponce Gustavo , Vega Luis
Published: July 2010

We study nonlinear dispersive equations of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript t Baseline u plus partial-differential Subscript x Superscript 2 j plus 1 Baseline u plus upper P left-parenthesis u comma partial-differential Subscript x Baseline u comma ellipsis comma partial-differential Subscript x Superscript 2 j Baseline u right-parenthesis equals 0 comma x comma t element-of double-struck upper R comma j element-of double-struck upper Z Superscript plus Baseline comma">

∂
t

u
+

∂
x

2
j
+
1

u
+
P
(
u
,

∂
x

u
,
…
,

∂
x

2
j

u
)
=
0
,

x
,
t
∈

R

,

j
∈

Z

+

,

{\partial _t}u + \partial _x^{2j + 1}u + P(u,{\partial _x}u, \ldots ,\partial _x^{2j}u) = 0,\qquad x,t \in \mathbb {R},\quad j \in {\mathbb {Z}^ + },

\]
where

P
(
⋅
)

P( \cdot )

is a polynomial having no constant or linear terms. It is shown that the associated initial value problem is locally well posed in weighted Sobolev spaces. The method of proof combines several sharp estimates for solutions of the associated linear problem and a change of dependent variable which allows us to consider data of arbitrary size.
Betti numbers of Alexandrov spaces
Authors: Koh Liang-Khoon
Published: July 2010

We obtain a uniform bound on the sum of Betti numbers for a class of complete Alexandrov spaces with narrow ends.
Picard’s theorem and Rickman’s theorem by way of Harnack’s inequality
Authors: Lewis John L.
Published: July 2010

In this note we give a very elementary proof of Picard’s Theorem and Rickman’s Theorem which uses only Harnack’s inequality.
Exponentiation is hard to avoid
Authors: Miller Chris
Published: July 2010

Let

R

\mathcal {R}

be an O-minimal expansion of the field of real numbers. If

R

\mathcal {R}

is not polynomially bounded, then the exponential function is definable (without parameters) in

R

\mathcal {R}

. If

R

\mathcal {R}

is polynomially bounded, then for every definable function

f
:

R

→

R

f:\mathbb {R} \to \mathbb {R}

, f not ultimately identically 0, there exist c,

r
∈

R

,
c
≠
0

r \in \mathbb {R},c \ne 0

, such that

x
↦

x
r

:
(
0
,
+
∞
)
→

R

x \mapsto {x^r}:(0, + \infty ) \to \mathbb {R}

is definable in

R

\mathcal {R}

and

lim

x
→
+
∞

f
(
x
)

/

x
r

=
c

{\lim _{x \to + \infty }}f(x)/{x^r} = c

.
Finitely generated groups (mod 𝑝)
Authors: Murty M. Ram
Published: July 2010

Given r mutually coprime natural numbers

a
1

,
…
,

a
r

{a_1}, \ldots ,{a_r}

greater than 1 and an irreducible polynomial

f
(
t
)

f(t)

with integer coefficients, we investigate how the structure of the group generated by

a
1

,
…
,

a
r

mod
p

{a_1}, \ldots ,{a_r} \bmod p

varies as the prime p varies subject to the condition that

f
(
t
)

f(t)

splits completely

mod

p

\bmod \;p

.
On the essential selfadjointness of Dirichlet operators on group-valued path space
Authors: Acosta Ernesto
Published: July 2010

Let G be a compact Lie group with Lie algebra

G

\mathcal {G}

. Consider the Wiener measure P on the space
\[

W
G

=
{
g
:
[
0
,
1
]
→
G
,
g

continuous

,
g
(
0
)
=
e
}

{W_G} = \{ g:[0,1] \to G,g\,{\text {continuous}},g(0) = e\}

\]
For each h in the Cameron-Martin space H over

G

\mathcal {G}

, let

∂
h

{\partial _h}

be the associated right invariant vector field over

W
G

{W_G}

and let

∂
h
∗

\partial _h^ \ast

be its adjoint with respect to P. We prove for a particular h that the space of functions on

W
G

{W_G}

generated by

C
∞

{C^\infty }

-cylindrical functions on

W
G

{W_G}

together with one Gaussian random variable is a core for the Dirichlet operator

∂
h
∗

∂
h

\partial _h^ \ast {\partial _h}

. This is the first step in proving the essential selfadjointness of the Number operator over group-valued path spaces in the natural presumed core.
Elementary proof of Furstenberg’s Diophantine result
Authors: Boshernitzan Michael D.
Published: July 2010

We present an elementary proof of a diophantine result (due to H. Furstenberg) which asserts (in a special case) that for every irrational

α
\alpha

the set

{

2
m

3
n

α

|

m
,
n
≥
0
}

\{ {2^m}{3^n}\alpha |m,n \geq 0\}

is dense modulo 1. Furstenberg’s original proof employs the theory of disjointness of topological dynamical systems.
Incompatibility of compact perturbations with the Sz. Nagy–Foias functional calculus
Authors: Davidson Kenneth R. , Zarouf Fouad
Published: July 2010

For every absolutely continuous contraction T with spectrum on the unit circle, we exhibit an

H
∞

{H^\infty }

function h and a sequence of operators

T
n

{T_n}

which are unitarily equivalent to T and differ from T by a sequence of compact operators converging to 0 in norm such that

h
(

T
n

)

h({T_n})

is never a compact perturbation of

h
(
T
)

h(T)

. When T is diagonal, it can also be arranged that

T
−

T
n

T - {T_n}

is trace class, and

T
n

{T_n}

commutes with T. Pour toute contraction absolument continue T dont le spectre rencontre le cercle unité, il existe une fonction h de

H
∞

{H^\infty }

et une suite

T
n

{T_n}

d’opérateurs unitairement equivalents à T telle que

T
−

T
n

T - {T_n}

soit compact et convergent en norme vers 0, mais

h
(

T
n

)
−
h
(
T
)

h({T_n}) - h(T)

soit non compact pour tout n. Dans le cas où T est diagonal, la suite

T
n

{T_n}

vérifie en plus

T
−

T
n

T - {T_n}

est un opérateur à trace, et

T
n

{T_n}

commute avec T.
Automorphisms of the ordered multiplicative group of positive rational numbers
Authors: Glass A. M. W. , Ribenboim Paulo
Published: July 2010

We prove Theorem. The only automorphism of the ordered multiplicative group of positive rational numbers is the trivial automorphism. We also give other examples of multiplicative groups of real positive algebraic numbers whose only order-preserving automorphisms are trivial.
Links with unlinking number one are prime
Authors: Gordon C. McA. , Luecke J.
Published: July 2010

We prove that a link with unlinking number one is prime.