bluuebunny/crossref_metadata_2025 huggingface dataset:
Some remarks on the average range of a vector measure
Authors: Freniche Francisco J.
Published: June 2010

We study some conditions on the average range of a vector measure with values in the bidual of a Banach space which imply that the range is contained in the space. We prove that Geitz’s condition is a sufficient one if the dual closed unit ball is weak-star sequentially compact. We also show how to reduce to measures with values in the bidual of

l
∞

{l^\infty }

.
Approximation by polynomials with locally geometric rates
Authors: Ivanov K. G. , Saff E. B. , Totik V.
Published: June 2010

In contrast to the behavior of best uniform polynomial approximants on

[
0
,
1
]

[0,1]

we show that if

f
∈
C
[
0
,
1
]

f \in C[0,1]

there exists a sequence of polynomials

{

P
n

}

\{ {P_n}\}

of respective degree

≤
n

\leq n

which converges uniformly to

f
f

on

[
0
,
1
]

[0,1]

and geometrically fast at each point of

[
0
,
1
]

[0,1]

where

f
f

is analytic. Moreover we describe the best possible rates of convergence at all regular points for such a sequence.
Radó’s theorem for CR-functions
Authors: Rosay Jean-Pierre , Stout Edgar Lee
Published: June 2010

We establish an analogue of a classical theorem of Radó for CR-functions on certain hypersurfaces in

C

N

{{\mathbf {C}}^{\mathbf {N}}}

.
A topology on quantum logics
Authors: Pulmannová Sylvia , Riečanová Zdena
Published: June 2010

A uniform topology

τ
M

{\tau _M}

induced by a set

M
M

of finite measures on a quantum logic

L
L

is studied. If

m
m

is a valuation on

L
L

, the topology

τ
m

{\tau _m}

induced by

{
m
}

\{ m\}

is equivalent to the topology induced by the pseudometric

ρ
(
a
,
b
)
=
m
(
a
Δ
b
)

\rho (a,b) = m(a\Delta b)

. If the set

M
M

of measures is large enough, the topology

τ
M

{\tau _M}

reflects in some sense the structure of

L
L

: if

L
L

is a continuous geometry and the measures are totally additive,

τ
M

{\tau _M}

is weaker than the order topology

τ
o

{\tau _o}

on

L
L

. If

L
L

is atomic,

τ
M

{\tau _M}

is stronger than

τ
o

{\tau _o}

. On a separable Hilbert space logic,

τ
M

{\tau _M}

coincides with the discrete topology. Some cases are found in which

τ
M

=

τ
o

{\tau _M} = {\tau _o}

.
Algebraic properties of the shift mapping
Authors: Dehornoy Patrick
Published: June 2010

We investigate the algebraic structure generated by the "shift" mapping

n
→
n
+
1

n \to n + 1

on

N

{\mathbf {N}}

under composition and another operation defined below. This allows to prove results about certain free autodistributive structures.
Existence and nonexistence of radial limits of minimal surfaces
Authors: Lancaster Kirk E.
Published: June 2010

A bounded solution of the minimal surface equation is constructed which has no radial limits at a boundary point.
Maximum modulus convexity and the location of zeros of an entire function
Authors: Abi-Khuzam Faruk F.
Published: June 2010

Let

f
f

be an entire function with non-negative Maclaurin coefficients and let

b

(
r
)

=
r

(

r

f
′

(
r
)

/

f

(
r
)

)

′

b\left ( r \right ) = r{\left ( {rf’\left ( r \right )/f\left ( r \right )} \right )’ }

. It is shown that if all the zeros of

f
f

lie in the angle

|

arg
⁡
z

|

≤
δ

\left | {\arg z} \right | \leq \delta

, where

0
>
δ
≤
π

0 > \delta \leq \pi

, then

lim

sup

r
→
∞

b

(
r
)

≥

1
4

cose

c

2

1
2

δ

\lim {\sup _{r \to \infty }}b\left ( r \right ) \geq \frac {1}{4}{\text {cose}}{{\text {c}}^2}\frac {1}{2}\delta

. In particular, we always have

lim

sup

r
→
∞

b

(
r
)

>

1
4

\lim {\sup _{r \to \infty }}b\left ( r \right ) > \frac {1}{4}

for such functions.
A note on pseudocompact groups
Authors: Fokkink Robbert
Published: June 2010

A question of W. W. Comfort and J. van Mill on pseudocompact groups is answered.
Group actions and direct sum decompositions of 𝐿^{𝑝} spaces
Authors: Nillsen Rodney
Published: June 2010

Let

G
G

be a locally compact group of measure preserving transformations on a

σ
\sigma

-finite measure space

(

X
,

B

,
m

)

\left ( {X,\mathcal {B},m} \right )

, and let

S
S

be a subset of

M
1

(
G
)

{M^1}\left ( G \right )

. Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p greater-than normal infinity comma upper I Subscript p Baseline equals left-brace f colon f element-of upper L Superscript p Baseline left-parenthesis m right-parenthesis and Subscript g Baseline f equals f comma for all g element-of upper G right-brace">

1
>
p
>
∞
,

I
p

=

{

f
:
f
∈

L
p

(
m
)

 and

 

g

f
=
f
,

for all 

g
∈
G

}

1 > p > \infty ,{I_p} = \left \{ {f:f \in {L^p}\left ( m \right ){\text { and}}{{\text { }}_g}f = f,{\text {for all }}g \in G} \right \}

, let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript p Baseline left-parenthesis upper S right-parenthesis equals StartSet f colon f element-of upper L Superscript p Baseline left-parenthesis m right-parenthesis and mu asterisk f equals f for all mu element-of upper S EndSet">

I
p

(
S
)

=

{

f
:
f
∈

L
p

(
m
)

 and 

μ
∗
f
=
f

 for all 

μ
∈
S

}

{I_p}\left ( S \right ) = \left \{ {f:f \in {L^p}\left ( m \right ){\text { and }}\mu * f = f{\text { for all }}\mu \in S} \right \}

, and let

K
p

(
S
)

{K_p}\left ( S \right )

be the closed subspace of

L
p

(
m
)

{L^p}\left ( m \right )

generated by functions of the form

μ
∗
f
−
f

\mu * f - f

, for

f
∈

L
p

(
m
)

f \in {L^p}\left ( m \right )

and

μ
∈
S

\mu \in S

. Conditions are given on

S
S

which ensure that

I
p

=

I
p

(
S
)

{I_p} = {I_p}\left ( S \right )

, and this is used to express

L
p

(
m
)

{L^p}\left ( m \right )

as a direct sum of

I
p

{I_p}

and

K
p

(
S
)

{K_p}\left ( S \right )

.
An inequality for harmonic maps of compact Kähler manifolds that implies holomorphicity
Authors: Glazebrook James F.
Published: June 2010

For harmonic maps of equidimensional compact Kähler manifolds satisfying certain conditions, a Chern class inequality is stated. If the map satisfies this inequality, it is holomorphic. The main result may be compared with a theorem of Eells and Wood for compact Riemann surfaces.
𝐿²-Dolbeault complexes on singular curves and surfaces
Authors: Haskell Peter
Published: June 2010

Give the smooth part of a singular curve or normal surface the metric induced from the ambient projective space. On this incomplete manifold the minimal

L
2

∂
¯

{L^2}\bar \partial

-complex of

(
0
,
q
)

(0,q)

-forms has finite-dimensional cohomology groups. The Euler characteristic of this cohomology equals the Todd genus of any desingularization of the singular variety.
On operators that are almost isometric on the positive cones of 𝐿^{𝑝}-spaces, 1<𝑝<+∞
Authors: Huang Sen Zhong
Published: June 2010

A linear operator

T
:

L
p

(
Ω
)

→

L
p

(

Ω
1

)

T:{L^p}\left ( \Omega \right ) \to {L^p}\left ( {{\Omega _1}} \right )

is said to be almost isometric on the positive cone (a.i.p.c.) with distortion coefficient

ε
\varepsilon

if there is an

ε
,
0
≤
ε
≤
1

\varepsilon ,0 \leq \varepsilon \leq 1

, such that

(

1
−
ε

)

‖
f
‖

≤

‖

T
f

‖

≤

‖
f
‖

\left ( {1 - \varepsilon } \right )\left \| f \right \| \leq \left \| {Tf} \right \| \leq \left \| f \right \|

for all nonnegative functions

f
f

. We prove that if

p
∈
[
2
,
∞
)

p \in [2,\infty )

, then there are continuous functions

a
p

(
⋅
)

{a_p}\left ( \cdot \right )

and

b
p

(
⋅
)

{b_p}\left ( \cdot \right )

defined on

[
0
,
1
)

[0,1)

, with

a
p

(
0
)

=

b
p

(
0
)

=
1

{a_p}\left ( 0 \right ) = {b_p}\left ( 0 \right ) = 1

, so that if

T
:

L
p

(
Ω
)

→

L
p

(

Ω
1

)

T:{L^p}\left ( \Omega \right ) \to {L^p}\left ( {{\Omega _1}} \right )

is an a.i.p.c. operator with distortion coefficient

ε
\varepsilon

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript p Baseline left-parenthesis epsilon right-parenthesis double-vertical-bar f double-vertical-bar less-than-or-equal-to double-vertical-bar upper T f double-vertical-bar less-than-or-equal-to b Subscript p Baseline left-parenthesis epsilon right-parenthesis double-vertical-bar f double-vertical-bar">

a
p

(
ε
)

‖
f
‖

≤

‖

T
f

‖

≤

b
p

(
ε
)

‖
f
‖

{a_p}\left ( \varepsilon \right )\left \| f \right \| \leq \left \| {Tf} \right \| \leq {b_p}\left ( \varepsilon \right )\left \| f \right \|

for all

f
∈

L
p

(
Ω
)

f \in {L^p}\left ( \Omega \right )

. We also prove that such functions

a
p

(
⋅
)

{a_p}\left ( \cdot \right )

and

b
p

(
⋅
)

{b_p}\left ( \cdot \right )

exist for

p
p

in the range

(

1
,
2

)

\left ( {1,2} \right )

if, in addition,

T
T

is either positive or has dense range.
On Picard’s theorem for entire quasiregular mappings
Authors: Vuorinen Matti
Published: June 2010

Several refinements of Picard’s theorem for entire functions in the complex plane have been proved by many authors in connection with the theory of Picard sets. We prove a result of this type for entire quasiregular mappings in euclidean

n
n

-space in the case when the "Picard set" consists of a sequence

a
k

{a_k}

on a ray emanating from 0 with

|

a
k

|

=

2
k

|{a_k}| = {2^k}

.
A construction for pseudocomplemented semilattices and two applications
Authors: Adams M. E. , Gould Matthew
Published: June 2010

A method is given by which pseudocomplemented semilattices can be constructed from graphs. Two consequences of the method are obtained, namely: there exist continuum-many quasivarieties of pseudocomplemented semilattices; for any non-zero cardinal

κ
\kappa

, there exist

κ
\kappa

pairwise non-isomorphic pseudocomplemented semilattices with isomorphic endomorphism monoids.
Normal forms for definite integer unimodular quadratic forms
Authors: Friedland Shmuel
Published: June 2010

In this paper we show that any two positive definite integer unimodular quadratic forms have a common sublattice of codimension 2. Moreover, any such form is equivalent to a semi-normal form with at most three eigenvalues different from 1.
On dual spaces with bounded sequences without weak*-convergent convex blocks
Authors: Schlumprecht Thomas
Published: June 2010

In this work we show that if

X
∗

{X^ * }

contains bounded sequences without weak* convergent convex blocks, then it contains an isometric copy of

L
1

(

{

0
,
1

}

ω
1

)

{L_1}\left ( {{{\left \{ {0,1} \right \}}^{{\omega _1}}}} \right )

.
The linear and quadratic Schur subgroups over the 𝑆-integers of a number field
Authors: Riehm Carl R.
Published: June 2010

Let

K
K

be an algebraic number field and let

O

\mathfrak {O}

be a ring of

S
S

-integers in

K
K

(where

S
S

is a set of primes of

K
K

containing all the archimedean primes); that is to say,

O

\mathfrak {O}

is a Dedekind domain whose field of quotients is

K
K

. In analogy with a theorem of T. Yamada in the case of a field of characteristic 0, it is shown that if

S

(

O

)

S\left ( \mathfrak {O} \right )

is the Schur subgroup of the Brauer group

B

(

O

)

B\left ( \mathfrak {O} \right )

and if

o

=

O

∩
k

\mathfrak {o} = \mathfrak {O} \cap k

, where

k
k

is any field containing the maximal abelian extension of

Q

\mathbb {Q}

in

K
K

, then

S

(

O

)

=

O

⊗
S

(

o

)

S\left ( \mathfrak {O} \right ) = \mathfrak {O} \otimes S\left ( \mathfrak {o} \right )

, i.e. the Brauer classes in

S

(

O

)

S\left ( \mathfrak {O} \right )

are those obtained from

S

(

o

)

S\left ( \mathfrak {o} \right )

by extension of the scalars to

O

\mathfrak {O}

. A similar theorem is proved as well in the case of the Schur subgroup

S

(

O

,
ω

)

S\left ( {\mathfrak {O},\omega } \right )

of the quadratic Brauer group

B

(

O

,
ω

)

B\left ( {\mathfrak {O},\omega } \right )

, where

ω
\omega

is an involution of

O

\mathfrak {O}

.
A new proof of uniqueness for multiple trigonometric series
Authors: Ash J. Marshall
Published: June 2010

Georg Cantor’s 1870 theorem that an everywhere convergent to zero trigonometric series has all its coefficients equal to zero is given a new proof. The new proof uses the first formal integral of the series, while Cantor’s proof used the second formal integral.
Uniform bounds for isoperimetric problems
Authors: Siegel Jerrold , Williams Frank
Published: June 2010

In this paper we generalize our previous joint work with Allan Calder on the width of homotopies by considering an arbitrary finite polyhedral pair

(

W
,
V

)

\left ( {W,V} \right )

rather than

(

I
,

{

0
,
1

}

)

\left ( {I,\left \{ {0,1} \right \}} \right )

. We show that given appropriate topological conditions on a Riemannian manifold

M
M

, with respect to

(

W
.
V

)

\left ( {W.V} \right )

, there are bounds,

B
q

(

a
,

(

W
,
V

)

,
M

)

{B_q}\left ( {a,\left ( {W,V} \right ),M} \right )

, such that if

F
:
K
×
W
→
M

F:K \times W \to M

is a map with

Lip

(

F

|

(

K
×
V

)

)

>
a

{\text {Lip}}\left ( {F\left | {\left ( {K \times V} \right )} \right .} \right ) > a

, then

F
F

can be deformed

rel

(

K
×
V

)

{\text {rel}}\left ( {K \times V} \right )

to

F
′

F’

with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Lip left-parenthesis upper F Superscript prime Baseline right-parenthesis greater-than upper B Subscript q Baseline left-parenthesis a comma left-parenthesis upper W comma upper V right-parenthesis comma upper M right-parenthesis plus epsilon">

Lip

(

F
′

)

>

B
q

(

a
,

(

W
,
V

)

,
M

)

+
ε

{\text {Lip}}\left ( {F’} \right ) > {B_q}\left ( {a,\left ( {W,V} \right ),M} \right ) + \varepsilon

, where

ε
>
0

\varepsilon > 0

is arbitrary and

dim
⁡

(
K
)

=
q

\dim \left ( K \right ) = q

.
Notes of the inversion of integrals. I
Authors: Kempf George R.
Published: June 2010

If

W
W

is a Picard bundle on the Jacobian

J
J

of a curve

C
C

, we have the problem of describing

W
W

globally. The theta divisor

θ
\theta

is ample on

J
J

. Thus it is possible to write

n
∗

W

{n^ * }W

as the sheaf associated to a graded

M
M

over the well-known ring

⊕

m
≥
0

Γ
(
J
,

O

J

(
m
4
θ
)
)

{ \oplus _{m \geq 0}}\Gamma (J,{\mathcal {O}_J}(m4\theta ))

. In this paper we compute the degree of generators and relations for such a module

M
M

.
Simple 𝐶*-algebras and subgroups of 𝑄
Authors: Murphy Gerald J.
Published: June 2010

A special case of a conjecture of R. Douglas is solved by an elementary argument using

K
0

{K_0}

-theory.
Remark on Walter’s inequality for Schur multipliers
Authors: Bożejko Marek
Published: June 2010

We extend and give another proof of Walter’s inequality: For a linear operator

T
∈
L
(

l
2

(
X
)
)

T \in L({l^2}(X))

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper T StartAbsoluteValue EndAbsoluteValue Subscript upper V 2 left-parenthesis upper X right-parenthesis Superscript 2 Baseline less-than-or-equal-to StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue upper T EndAbsoluteValue Subscript l Baseline StartAbsoluteValue EndAbsoluteValue Subscript upper V 2 left-parenthesis upper X right-parenthesis Baseline StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue upper T EndAbsoluteValue Subscript r Baseline StartAbsoluteValue EndAbsoluteValue Subscript upper V 2 left-parenthesis upper X right-parenthesis Baseline comma">

|

|

T

|

|

V
2

(
X
)

2

≤

|

|

|

T

|

l

|

|

V
2

(
X
)

|

|

|

T

|

r

|

|

V
2

(
X
)

,

||T||_{{V_2}(X)}^2 \leq ||\,|T{|_l}|{|_{{V_2}(X)}}||\,|T{|_r}|{|_{{V_2}(X)}},

\]
where

V
2

(
X
)

{V_2}(X)

is the Banach algebra of Schur multipliers on

L
(

l
2

(
X
)
)

L({l^2}(X))

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper T EndAbsoluteValue Subscript l Baseline equals left-parenthesis upper T upper T right-parenthesis Superscript asterisk 1 slash 2 Baseline comma StartAbsoluteValue upper T EndAbsoluteValue Subscript r Baseline equals StartAbsoluteValue upper T Superscript asterisk Baseline EndAbsoluteValue Subscript l Baseline">

|

T

|

l

=

(
T
T

)

∗
1

/

2

,

|

T

|

r

=

|

T
∗

|

l

|T{|_l} = {(TT)^{*1/2}},|T{|_r} = |{T^*}{|_l}

.
Certain positive-definite kernels
Authors: Ossiander Mina , Waymire Edward C.
Published: June 2010

In one way or another, the extension of the standard Brownian motion process

{

B
t

:
t
∈
[
0
,
∞
)
}

\{ {B_t}:t \in [0,\infty )\}

to a (Gaussian) random field

{

B
t

:

t

∈

R

+
d

}

\{ {B_t}:{\text {t}} \in {\mathbf {R}}_ + ^d\}

involves a proof of the positive semi-definiteness of the kernel used to generalize

ρ
(
s
,
t
)
=

cov(

B

s

,

B

t

) = s

∧
t

\rho (s,t) = {\text {cov(}}{{\text {B}}_s}{\text {,}}{{\text {B}}_t}{\text {) = s}} \wedge t

to multidimensional time. Simple direct analytical proofs are provided here for the cases of (i) the Lévy multiparameter Brownian motion, (ii) the Chentsov Brownian sheet, and (iii) the multiparameter fractional Brownian field.
Tucker-Ky Fan colorings
Authors: Sarkaria K. S.
Published: June 2010

The existence of a continuous

Z

2

{{\mathbf {Z}}_2}

-map from a free

m
m

-dimensional

Z

2

{{\mathbf {Z}}_2}

-simplicial complex

E
E

to the

(
m
−
1
)

(m - 1)

-dimensional antipodal sphere

S

m
−
1

{S^{m - 1}}

is characterized by means of an enumerative combinatorial criterion involving a coloring of the vertices of

E
E

. The Borsuk-Ulam theorem, 1933, and the combinatorial lemmas of Tucker, 1945, and Ky Fan, 1952, are easy consequences of this result for the case

|

E

|

=

S
m

|E| = {S^m}

.
Example of an algebra which is nontopologizable as a locally convex topological algebra
Authors: Żelazko W.
Published: June 2010

Let

X
X

be a real or complex linear space and denote by

L
(
X
)

L(X)

the algebra of all its endomorphisms. We prove that

L
(
X
)

L(X)

is topologizable as a locally convex topological algebra (with jointly continuous multiplication) if and only if it is topologizable as a Banach algebra and this holds if and only if

X
X

is of finite dimension.
A result about the Hilbert transform along curves
Authors: Saal Linda V.
Published: June 2010

Let

G
G

be a connected and simply connected, nilpotent Lie group and let

γ
:
(
−
1
,
1
)
→
G

\gamma :( - 1,1) \to G

be a (connected) analytic curve such that

γ
(
0
)
=
0

\gamma (0) = 0

. Then the Hilbert transform along

γ
\gamma

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T f left-parenthesis x right-parenthesis equals p period v period integral Underscript 0 greater-than StartAbsoluteValue t EndAbsoluteValue greater-than 1 Endscripts f left-parenthesis x gamma left-parenthesis t right-parenthesis Superscript negative 1 Baseline right-parenthesis d t slash t comma">

T
f
(
x
)
=
p
.
v
.

∫

0
>

|

t

|

>
1

f
(
x
γ

(
t
)

−
1

)
d
t

/

t

,

Tf(x) = p.v.\int _{0 > |t| > 1} {f(x\gamma {{(t)}^{ - 1}})dt/t} ,

\]
is bounded on

L
p

(
G
)
,
1
>
p
>
∞

{L^p}(G),1 > p > \infty

.
Antihook differences and some partition identities
Authors: Agarwal A. K.
Published: June 2010

Anti-hook differences are applied to give new combinatorial interpretations to three identities from Slater’s Compendium.
No continuum in 𝐸² has the TMP. I. Arcs and spheres
Authors: Loveland L. D.
Published: June 2010

A subset

X
X

of the Euclidean plane

E
2

{E^2}

is said to have the triple midset property (TMP) if, for each pair of points

x
x

and

y
y

of

X
X

, the perpendicular bisector of the segment joining

x
x

and

y
y

intersects

X
X

at exactly three points. In this paper it is proved that no arc or simple closed curve in

E
2

{E^2}

can have the TMP. In a subsequent paper these results are used to prove that no planar continuum can have the TMP.
Decay and boundedness results for a model of laminar flames with complex chemistry
Authors: Avrin Joel D.
Published: June 2010

We consider the reaction-diffusion equations modeling two-step reactions with Arrhenius kinetics on bounded spatial domains or over all of

R

n

{\mathbb {R}^n}

. After noting the existence, uniqueness, and nonnegativity of global strong solutions with virtually arbitrary nonnegative initial data, we give conditions on the initial temperature that guarantee decay of the concentrations to zero and a supremum norm bound on the temperature. In our first such result we assume that the initial temperature

T
0

{T_0}

is uniformly bounded above the two ignition temperatures. Specializing to the case of bounded spatial domains, we replace this condition by the more general requirement that the average of

T
0

{T_0}

over the domain is above both ignition temperatures. Finally, we note a boundedness result with equal diffusion coefficients that holds for arbitrary choices of the other parameters. Combining this assumption with the hypotheses, noted above, about the initial temperature, we obtain steady-state convergence results for the temperature as well as the concentrations.
Recurrent homeomorphisms on 𝑅² are periodic
Authors: Oversteegen Lex G. , Tymchatyn E. D.
Published: June 2010

A homeomorphism

f
:
(
X
,
d
)
→
(
X
,
d
)

f:(X,d) \to (X,d)

of a metric space

(
X
,
d
)

(X,d)

onto

X
X

is recurrent provided that for each

ε
>
0

\varepsilon > 0

there exists a positive integer

n
n

such that

f
n

{f^n}

is

ε
\varepsilon

-close to the identity map on

X
X

. The notion of a recurrent homeomorphism is weaker than that of an almost periodic homeomorphism. The result announced in the title generalizes the theorem of Brechner for almost periodic homeomorphisms and answers a question of R. D. Edwards.
Positive centers and the Bonnesen inequality
Authors: Gage Michael E.
Published: June 2010

A positive center of a convex curve is a point from which the function

p
(
θ
)
L
−
A
−
π
p

(
θ

)
2

p(\theta )L - A - \pi p{(\theta )^2}

is positive for all values of

θ
\theta

. The support function is

p
p

and

L
L

and

A
A

are the length and area of the curve, respectively. This paper proves that all convex curves have a positive center and gives an example which shows that the common centroids (Steiner point, etc.) are not necessarily positive centers. A strengthened version of the Bonnesen inequality is obtained and a simplified proof of the one-dimensional Firey inequality.
On dual sets generated by lacunary polynomials
Authors: Dobbertin Hans , Kasten Volker
Published: June 2010

The notion of dual sets of analytic functions has been developed by Ruscheweyh. In terms of this theory a well-known convolution theorem of Szegö states that the set of all polynomials

1
+

a
1

z
+
⋯
+

a
n

z
n

1 + {a_1}z + \cdots + {a_n}{z^n}

nonvanishing in the unit disc

D

{\mathbf {D}}

is the dual hull of

(
1
−
z

)
n

{(1 - z)^n}

. More general for

T
=
{

m
1

,
…
,

m
n

}

T = \{ {m_1}, \ldots ,{m_n}\}

let

P
^

T

{\hat P_T}

denote the set of all lacunary polynomials

1
+

a

m
1

z

m
1

+
⋯
+

a

m
n

z

m
n

1 + {a_{{m_1}}}{z^{{m_1}}} + \cdots + {a_{{m_n}}}{z^{{m_n}}}

nonvanishing in

D

{\mathbf {D}}

. In this paper we investigate whether the sets

P
^

T

{\hat P_T}

are generated in a similar way. Some necessary conditions are given, and the case

|

T

|

≤
3

|T| \leq 3

is completely solved.
On the primariness of the Banach space 𝑙_{∞}/𝐶₀
Authors: Drewnowski Lech , Roberts James W.
Published: June 2010

In a response to a question asked by Leonard and Whitfield (1983) we show that, under the Continuum Hypothesis, the Banach space

l
∞

/

C
0

{l_\infty }/{C_0}

is primary.
Level crossings of a random trigonometric polynomial
Authors: Farahmand Kambiz
Published: June 2010

This paper provides an asymptotic estimate for the expected number of

K
K

-level crossings of the random trigonometric polynomial

g
1

cos
⁡
x
+

g
2

cos
⁡
2
x
+
…
+

g
n

cos
⁡
n
x

{g_1}\cos x + {g_2}\cos 2x + \ldots + {g_n}\cos nx

, where

g
j

(
j
=
1
,
2
,
…
,
n
)

{g_j}(j = 1,2, \ldots ,n)

are independent normally distributed random variables with mean

μ
\mu

and variance one. It is shown that the result for

K
=
μ
=
0

K = \mu = 0

remains valid for any finite constant

μ
\mu

and any

K
K

such that

(

K
2

/

n
)
→
0

({K^2}/n) \to 0

as

n
→
∞

n \to \infty

.
Bond invariance of 𝐺-rings and localization
Authors: Warfield Robert B.
Published: June 2010

It is proved that if

R
R

and

S
S

are prime Noetherian rings and there exists an

(
R
,
S
)

(R,S)

-bimodule that is finitely generated and torsionfree on each side, then the intersection of the nonzero prime ideals of

R
R

is nonzero if and only if the same holds for the corresponding intersection in

S
S

. Consequently, if the right primitive ideals in a given Noetherian ring are precisely the locally closed prime ideals, then the same equivalence holds true for any finite extension ring. Another consequence of the methods used here is the following answer to a question of Braun: If the intersection of the prime ideals in a clique in a Noetherian PI ring is a prime ideal

Q
Q

, then

Q
Q

is localizable.
Second-derivative estimates for solutions of two-dimensional Monge-Ampère equations
Authors: Schulz Friedmar
Published: June 2010

Heinz-Lewy type a priori estimates are derived for the absolute values of the second derivatives of solutions

z
(
x
,
y
)
∈

C

1
,
1

(
Ω
)

z(x,y) \in {C^{1,1}}(\Omega )

of Monge-Ampère equations of the general form
\[

A
r
+
2
B
s
+
C
t
+
(
r
t
−

s
2

)
=
E

Ar + 2Bs + Ct + (rt - {s^2}) = E

\]
in the interior of the domain

Ω
\Omega

. The coefficients

A
,
B
,
C
,
E

A,B,C,E

depend in particular on the gradient of

z
(
x
,
y
)

z(x,y)

and satisfy certain structural conditions.
Uniform continuity on bounded sets and the Attouch-Wets topology
Authors: Beer Gerald , Di Concilio Anna
Published: June 2010

Let

CL(

X

)

{\text {CL(}}X{\text {)}}

be the nonempty closed subsets of a metrizable space

X
X

. If

d
d

is a compatible metric, the metrizable Attouch-Wets topology

τ

a
w

(
d
)

{\tau _{aw}}(d)

on

CL(

X

)

{\text {CL(}}X{\text {)}}

is the topology of uniform convergence of distance functionals associated with elements of

CL(

X

)

{\text {CL(}}X{\text {)}}

on bounded subsets of

X
X

. The main result of this paper shows that two compatible metrics

d
d

and

ρ
\rho

determine the same Attouch-Wets topologies if and only if they determine the same bounded sets and the same class of functions that are uniformly continuous on bounded sets.
Henstock integrable functions are Lebesgue integrable on a portion
Authors: Buczolich Zoltán
Published: June 2010

If a real function

f
f

defined on an interval

I
⊂

R

m

I \subset {{\mathbf {R}}^m}

is Henstock integrable, then one can always find a nondegenerate subinterval

J
⊂
I

J \subset I

on which

f
f

is Lebesgue integrable.
Matrix maps and the isomorphic structure of BK spaces
Authors: Connor Jeff
Published: June 2010

This note gives a characterization of BK spaces that contain isomorphic copies of

c
0

{c_0}

in terms of matrix maps and a sufficient condition for a matrix map from

l
∞

{l_\infty }

into a BK space to be a compact operator. The primary tool used in this note is the Bessaga-Pelczynski characterization of Banach spaces which contain isomorphic copies of

c
0

{c_0}

. It is shown that weakly compact matrix maps on

l
∞

{l_\infty }

are compact and that, if

E
E

is a BK space such that there is a matrix

A
A

such that

c
0

⊆

E
A

{c_0} \subseteq {E_A}

and

E
A

{E_A}

is not strongly conull, then

E
E

must contain an isomorphic copy of

c
0

{c_0}

.
Random perturbations of singular spectra
Authors: Howland James S.
Published: June 2010

The singular parts of the self-adjoint operators

T
T

and

H
=
T
+
V

H = T + V

are mutually singular for "almost every" bounded perturbation

V
V

.
The Bloch constant of bounded analytic functions on a multiply connected domain
Authors: Colonna Flavia
Published: June 2010

Let

F
F

be an analytic function on the bounded domain

R
R

, a multiply-connected region of the complex plane. The Bloch constant of

F
F

is defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript upper F Baseline equals sup Underscript StartAbsoluteValue z EndAbsoluteValue greater-than 1 Endscripts left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis StartAbsoluteValue left-parenthesis upper F ring p right-parenthesis prime left-parenthesis z right-parenthesis EndAbsoluteValue comma">

β
F

=

sup

|

z

|

>
1

(
1
−

|

z

|

2

)

|

(
F
∘
p

)
′

(
z
)

|

,

\beta _F = \sup _{|z| > 1} (1 - |z|^2)|(F \circ p)’(z)|,

\]
where

p
p

is a conformal universal cover of

R
R

with domain

Δ
\Delta

, the open unit disk. If

F
F

is bounded, then

β
F

≤

|

|

F

|

|

∞

{\beta _F} \leq ||F|{|_\infty }

, the sup-norm of

F
F

. In this paper we characterize those functions

F
F

for which

β
F

=

|

|

F

|

|

∞

{\beta _F} = ||F|{|_\infty }

in terms of the zeros of

F
F

when the boundary of

R
R

is the union of finitely many curves. We conclude this paper by showing the existence of extremal functions, and generalizing the results to bounded harmonic mappings on these domains.
A criterion for the nonporosity of a general Cantor set
Authors: Humke Paul D.
Published: June 2010

The purpose of this paper is to prove that under quite general conditions, a closed nowhere dense perfect set is non-

σ
\sigma

-porous if and only if it is nonporous.
Sums of powers of derivatives
Authors: Mařík Jan , Weil Clifford E.
Published: June 2010

In a previous paper the authors proved several theorems concerning products of positive derivatives. Here they prove analogous results for sums of powers of positive derivatives.
A note on a global stability theorem of M. W. Hirsch
Authors: Jiang Ji Fa
Published: June 2010

For a

3
3

-dimensional cooperative vector field

F
F

, we present a new global stability result which is the same as that of Hirsch [4] except that we do not assume that

Div
⁡
F
≤
0

\operatorname {Div} F \leq 0

or that the equilibrium is stable. Thus we solve the "interesting problem" pointed out by Hirsch [4].
Strong solutions of evolution equations governed by 𝑚-accretive operators and the Radon-Nikodým property
Authors: Deville Robert
Published: June 2010

We construct, in every Banach space which fails the Radon-Nikodym property, a nonlinear operator

A
A

which is

m
m

-accretive for some equivalent norm in

X
X

, such that the domain of

A
A

is not a singleton and such that the only strong solutions of the equation

u
′

+
A
u
∍
f

u’ + Au \backepsilon f

are the constant ones.
On infinitely differentiable and Gevrey vectors for representation
Authors: ter Elst A. F. M.
Published: June 2010

In the present paper we give a condition in order that the set of infinitely differentiable vectors for a representation

π
\pi

in a Banach space be equal to the set of all infinitely differentiable vectors for the restriction of

π
\pi

to a subgroup. Similar results for Gevrey vectors and analytic vectors are proved for unitary representations.
Auslander-Reiten triangles in derived categories of finite-dimensional algebras
Authors: Happel Dieter
Published: June 2010

Let

A
A

be a finite-dimensional algebra. The category

b
m
o
d
A

bmod A

of finitely generated left

A
A

-modules canonically embeds into the derived category

D
b

(
A
)

{D^b}\left ( A \right )

of bounded complexes over

b
m
o
d
A

bmod A

and the stable category

mod
_

Z

T

(
A
)

{\underline {\bmod } ^\mathbb {Z}}T\left ( A \right )

of

Z

\mathbb {Z}

-graded modules over the trivial extension algebra of

A
A

by the minimal injective cogenerator. This embedding can be extended to a full and faithful functor from

D
b

(
A
)

{D^b}\left ( A \right )

to

mod
_

Z

T

(
A
)

\underline {\bmod }^{\mathbb {Z}}T\left ( A \right )

. Using the concept of Auslander-Reiten triangles it is shown that both categories are equivalent only if

A
A

has finite global dimension.
On the evaluation of spectral projections onto absolutely continuous parts of a contraction
Authors: Rybkin Alexey V.
Published: June 2010

In terms of the characteristic function of the contraction the norms of the spectral projections onto its absolutely continuous component are evaluated. The row of examples is considered. The exposition is conducted in the framework of the Sz.-Nagy-Foias model.
On the action of Steenrod squares on polynomial algebras
Authors: Singer William M.
Published: June 2010

Let

P
s

{P_s}

be the

mod
−
2

\bmod - 2

cohomology of the elementary abelian group

(
Z

/

2
Z
)
×
⋯
×
(
Z

/

2
Z
)

(Z/2Z) \times \cdots \times (Z/2Z)

(

s
s

factors). The

mod
−
2

\bmod - 2

Steenrod algebra

A
A

acts on

P
s

{P_s}

according to well-known rules. If

A

⊂
A

{\mathbf {A}} \subset A

denotes the augmentation ideal, then we are interested in determining the image of the action

A

⊗

P
s

→

P
s

{\mathbf {A}} \otimes {P_s} \to {P_s}

: the space of elements in

P
s

{P_s}

that are hit by positive dimensional Steenrod squares. The problem is motivated by applications to cobordism theory [P1] and the homology of the Steenrod algebra [S]. Our main result, which generalizes work of Wood [W], identifies a new class of hit monomials.
The Hausdorff dimension of graphs of density continuous functions
Authors: Buczolich Zoltán , Ostaszewski Krzysztof
Published: June 2010

The density topology on the real line consists of all measurable sets whose all points are their points of Lebesgue density one. A real-valued function of a real variable that is continuous with respect to the density topology on both the domain and the range is termed density continuous. By constructing graphs of density continuous functions as invariant sets of systems of affine maps on the unit square we show that the Hausdorff dimensions of graphs of density continuous functions vary continuously between one and two.
Steenrod algebra module maps from 𝐻*(𝐵(𝑍/𝑝)ⁿ) to 𝐻*(𝐵(𝑍/𝑝)^{𝑠})
Authors: Harris John C. , Hunter Thomas J. , Shank R. James
Published: June 2010

Let

H

⊗
n

{H^{ \otimes n}}

denote the

mod
−
p

\operatorname {mod}-p

cohomology of the classifying space

B

(

Z

/

p

)
n

B{({\mathbf {Z}}/p)^n}

as a module over the Steenrod algebra

A

\mathcal {A}

. Adams, Gunawardena, and Miller have shown that the

n
×
s

n \times s

matrices with entries in

Z

/

p

{\mathbf {Z}}/p

give a basis for the space of maps

Ho

m

A

(

H

⊗
n

,

H

⊗
s

)

{\text {Ho}}{{\text {m}}_\mathcal {A}}({H^{ \otimes n}},{H^{ \otimes s}})

. For

n
n

and

s
s

relatively prime, we give a new basis for this space of maps using recent results of Campbell and Selick. The main advantage of this new basis is its compatibility with Campbell and Selick’s direct sum decomposition of

H

⊗
n

{H^{ \otimes n}}

into

(

p
n

−
1
)

({p^n} - 1)

A

\mathcal {A}

-modules. Our applications are at the prime two. We describe the unique map from

H
¯

\bar H

to

D
(
n
)

D(n)

, the algebra of Dickson invariants in

H

⊗
n

{H^{ \otimes n}}

, and we give the dimensions of the space of maps between the indecomposable summands of

H

⊗
3

{H^{ \otimes 3}}

.
Algebras of finite self-injective dimension
Authors: Hoshino Mitsuo
Published: June 2010

Let

A
A

be an artin algebra. Then

A
A

has finite self-injective dimensions on both sides if and only if every finitely generated left

A
A

-module has finite Gorenstein dimension.
Prime factors of conjugacy classes of finite solvable groups
Authors: Ferguson Pamela A.
Published: June 2010

A bound for the number of primes dividing

[
G
:
Z
(
G
)
]

[G:Z(G)]

for certain finite solvable groups

G
G

is given in terms of the maximal number of primes dividing a conjugacy class.
A certain class of triangular algebras in type 𝐼𝐼₁ hyperfinite factors
Authors: Baker Richard
Published: June 2010

Let

S
S

be the standard triangular UHF algebra in a UHF algebra

A
A

, where the rank of

A
A

is a strictly increasing sequence of positive integers. Let

M
M

be the type

I

I

1

II_{1}

hyperfinite factor defined as the weak closure of

A
A

in the tracial representation of

A
A

. Define

T
T

to be the weak closure of

S
S

in this representation. Then

T
T

is a reflexive, maximal weakly closed triangular algebra in

M
M

. Moreover,

T
T

is irreducible relative to

M
M

. We exhibit a strongly closed sublattice

L
L

of

lat
⁡
T

\operatorname {lat} T

such that

T
=
alg
⁡
L

T = \operatorname {alg} L

.
Pseudospectral operators and the pointwise ergodic theorem
Authors: Bradley R. E.
Published: June 2010

We show that for a class of operators which properly contains the normal operators on

L
2

{L_2}

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over n EndFraction sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts upper T Superscript i Baseline f right-arrow a period e period iff StartFraction 1 Over 2 Superscript n Baseline EndFraction sigma-summation Underscript i equals 0 Overscript 2 Superscript n Baseline minus 1 Endscripts upper T Superscript i Baseline f right-arrow a period e period">

1
n

∑

i
=
0

n
−
1

T
i

f
→
a
.
e
.

iff

1

2
n

∑

i
=
0

2
n

−
1

T
i

f
→
a
.
e
.

\frac {1}{n}\sum \limits _{i = 0}^{n - 1} {{T^i}f \to a.e.} {\text {iff}}\frac {1}{{{2^n}}}\sum \limits _{i = 0}^{{2^n} - 1} {{T^i}f \to a.e.}

\]
This theorem is used to give an alternate form of a theorem of Gaposhkin concerning the pointwise ergodic theorem for normal operators.
Arbitrarily slow approach to limiting behavior
Authors: Golden K. , Goldstein S.
Published: June 2010

Let

f
(

k

,
t
)
:

R

N

×
[
0
,
∞
)
→

R

f({\mathbf {k}},t):{\mathbb {R}^N} \times [0,\infty ) \to \mathbb {R}

be jointly continuous in

k

{\mathbf {k}}

and

t
t

, with

lim

t
→
∞

f
(

k

,
t
)
=
F
(

k

)

{\lim _{t \to \infty }}f({\mathbf {k}},t) = F({\mathbf {k}})

discontinuous for a dense set of

k

{\mathbf {k}}

’s. It is proven that there exists a dense set

Γ
\Gamma

of

k

{\mathbf {k}}

’s such that, for

k

∈
Γ

{\mathbf {k}} \in \Gamma

,

|

f
(

k

,
t
)
−
F
(

k

)

|

|f({\mathbf {k}},t) - F({\mathbf {k}})|

approaches 0 arbitrarily slowly, i.e., roughly speaking, more slowly than any expressible function

g
(
t
)
→
0

g(t) \to 0

. This result is applied to diffusion and conduction in quasiperiodic media and yields arbitrarily slow approaches to limiting behavior as time or volume becomes infinite. Such a slow approach is in marked contrast to the power laws widely found for random media, and, in fact, implies that there is no law whatsoever governing the asymptotics.
Irreducible outer automorphisms of a free group
Authors: Gersten S. M. , Stallings J. R.
Published: June 2010

We give sufficient conditions for all positive powers of an outer automorphism of a finitely generated free group to be irreducible, in the sense of Bestvina and Handel. We prove a conjecture of Stallings (1982), that a PV automorphism in rank

≥
3

\geq 3

has no nontrivial fixed points.
Points not as hyperplane sections of projectively normal curves
Authors: Ballico Edoardo
Published: June 2010

Here we show for many

n
,
d

n,d

(with

n
≥
6

n \geq 6

) that the general set formed by

d
d

points in

P

n

{{\mathbf {P}}^n}

is not the hyperplane section of an integral projectively normal curve in

P

n
+
1

{{\mathbf {P}}^{n + 1}}

.
On flat families of fat points
Authors: Paxia Giuseppe
Published: June 2010

The aim of this paper is to fill a gap in a proof of a theorem by B. Segre regarding the regularity of linear systems of curves in

P
2

{P^2}

. This is done by showing that some subschemes of "fat points" of

P
2

{P^2}

form a flat family over an integral quasiprojective variety of which we give an explicit construction.
Exceptional 𝑞-Askey-Wilson polynomials and continued fractions
Authors: Gupta Dharma P. , Masson David R.
Published: June 2010

Two linearly independent solutions of the three-term recurrence relation for the

q
q

-Askey-Wilson polynomials are obtained for the special cases

a
b
c
d
=

q
m

,
m
=
1
,
2
,
…

abcd = {q^m},m = 1,2, \ldots

.By obtaining the subdominant solution and employing Pincherle’s theorem, the associated continued fractions and properties of the corresponding weight functions are derived. The cases

a
b
c
d
=
q

or

q
2

abcd = q\;{\text {or}}\;{q^2}

are exceptional. They differ from the cases considered by Askey and Wilson [1 ] and are limits of a family of associated cases considered by Ismail and Rahman [5].
New complete genus zero minimal surfaces with embedded parallel ends
Authors: López Francisco J.
Published: June 2010

We construct genus zero complete minimal surfaces of finite total curvature embedded outside a compact set of

R

3

{\mathbb {R}^3}

.
𝜔! can be a nontrivial automorphism group
Authors: Roitman Judith
Published: June 2010

Under the axiom "there are

ω
1

{\omega _1}

cofinal Cohen reals," the symmetric group on

ω
\omega

is the nontrivial automorphism group of a thin-tall superatomic Boolean algebra. Certain product groups are also, under the same axiom, nontrivial automorphism groups of thin-tall superatomic Boolean algebras.
A remark on a paper of J. S. Ruan: “Invariant subspace of strictly singular operators” [Proc. Amer. Math. Soc. 108 (1990), no. 4, 931–936; MR1002160 (90g:47009)]
Authors: Fitzpatrick Patrick M. , Goldberg Seymour
Published: June 2010

We observe that a strictly singular operator is not necessarily condensing, so that the invariant subspace problem for strictly singular operators remains open.
Lebesgue’s theorem of differentiation in Fréchet lattices
Authors: Grosse-Erdmann Karl-Goswin
Published: June 2010

Lebesgue’s differentiation theorem (LDT) states that every monotonic real function is differentiable a.e. We investigate the validity of this theorem for functions with values in topological vector lattices. It is shown that a Fréchet lattice satisfies (LDT) iff it is isomorphic to a generalized echelon space, a Banach lattice satisfies (LDT) iff it is isomorphic to some

l
1

(
Γ
)

{l^1}\left ( \Gamma \right )

.
Fully indecomposable exponents of primitive matrices
Authors: Brualdi Richard A. , Liu Bo Lian
Published: June 2010

If

A
A

is a primitive matrix, then there is a smallest power of

A
A

(its fully indecomposable exponent) that is fully indecomposable, and a smallest power of

A
A

(its strict fully indecomposable exponent) starting from which all powers are fully indecomposable. We obtain bounds on these two exponents.
Solution of two conjectures in symbolic dynamics
Authors: Kim K. H. , Roush F. W.
Published: June 2010

We give an example of an inert involution on a subshift of entropy less than

log
⁡
2

\log 2

which interchanges two fixed points. This proves simple finite order generation (FOG) of Wagoner [W2] is false and answers Questions 1.10 and 1.11 [W2]. We also give an example of two near Markov subshifts which are eventually conjugate but not conjugate, and a near Markov subshift with domain a full shift not equivalent to its time reversal. The former shows sofic shift equivalence does not coincide with strong sofic shift equivalence.
Yang-Mills connections on surfaces and representations of the path group
Authors: Morrison Kent
Published: June 2010

We prove that Yang-Mills connections on a surface are characterized as those with the property that the holonomy around homotopic closed paths only depends on the oriented area between the paths. Using this we have an alternative proof for a theorem of Atiyah and Bott that the Yang-Mills connections on a compact orientable surface can be characterized by homomorphisms to the structure group from an extension of the fundamental group of the surface. In addition, for

M
=

S
2

M = {S^2}

, we obtain the results that the Yang-Mills connections on

S
2

{S^2}

are isolated and correspond with the conjugacy classes of closed geodesies through the identity in the structure group.
Bivariate monotone approximation
Authors: Anastassiou George A.
Published: June 2010

Let

f
f

be a two variable continuously differentiable real-valued function of certain order on

[
0
,
1

]
2

{[0,1]^2}

and let

L
L

be a linear differential operator involving mixed partial derivatives and suppose that

L
(
f
)
≥
0

L(f) \geq 0

. Then there exists a sequence of two-dimensional polynomials

Q

m
,
n

(
x
,
y
)

{Q_{m,n}}(x,y)

with

L
(

Q

m
,
n

)
≥
0

L({Q_{m,n}}) \geq 0

, so that

f
f

is approximated simultaneously and uniformly by

Q

m
,
n

{Q_{m,n}}

. This approximation is accomplished quantitatively by the use of a suitable two-dimensional first modulus of continuity.
About direct summands of projective modules over Laurent polynomial rings
Authors: Mandal Satya
Published: June 2010

Suppose

A
A

is a local ring and

R
=
A
[
X
,

X

−
1

]

R = A[X,{X^{ - 1}}]

is a Laurent polynomial ring. We prove that for projective

R
R

-modules

P
P

and

Q
Q

with rank

Q
>

Q >

rank

P
P

, if

Q
f

{Q_f}

is a direct summand of

P
f

{P_f}

for a doubly monic polynomial

f
f

then

Q
Q

is also a direct summand of

P
P

. We also prove the analogue of the Horrock’s theorem for Laurent polynomials rings.
Optimal control and semicontinuous viscosity solutions
Authors: Barron E. N. , Jensen R.
Published: June 2010

We study the value function for an optimal control problem with upper semicontinuous terminal data. We prove that the upper semicontinuous envelope of the value function is the unique semicontinuous viscosity solution of the Bellman equation and that it coincides with the value function obtained when using relaxed controls.
Asymptotic periodicity of solutions to a class of neutral functional-differential equations
Authors: Wu Jian Hong
Published: June 2010

In this paper, we extend a convergence result due to Takáč to continuous maps satisfying certain monotonicity properties. Applying this extension to the Poincaré map associated with the neutral equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d slash d t right-parenthesis left-bracket x left-parenthesis t right-parenthesis minus b left-parenthesis t right-parenthesis x left-parenthesis t minus r right-parenthesis right-bracket equals upper F left-bracket t comma x left-parenthesis t right-parenthesis comma x left-parenthesis t minus r right-parenthesis right-bracket">

(
d

/

d
t
)
[
x
(
t
)
−
b
(
t
)
x
(
t
−
r
)
]
=
F
[
t
,
x
(
t
)
,
x
(
t
−
r
)
]

(d/dt)[x(t) - b(t)x(t - r)] = F[t,x(t),x(t - r)]

\]
we prove that each solution of the above neutral equation tends to an

r
r

-periodic function as

t
→
∞

t \to \infty

in an oscillatory manner, where

0
≤
b
(
t
)
>
1

0 \leq b(t) > 1

is an

r
r

-periodic continuous function and

F
F

satisfies a certain order relation.
Lifting invertibles in von Neumann algebras
Authors: Rogers Robert R.
Published: June 2010

Given

B

(

H

)

\mathcal {B}(\mathcal {H})

, the algebra of bounded operators on a separable Hilbert space

H

\mathcal {H}

, and

K

\mathcal {K}

, the ideal of compact operators, it is a well-known fact that

T
T

in

B

(

H

)

\mathcal {B}(\mathcal {H})

is a Fredholm operator if and only if

π
(
T
)

\pi (T)

is invertible in

B

(

H

)

\mathcal {B}(\mathcal {H})

where

π
\pi

is the canonical quotient map. A natural question arises: When can a Fredholm operator be perturbed by a compact operator to obtain an invertible operator? Equivalently, when does the invertible element

π
(
T
)

\pi (T)

lift to an invertible operator? The answer is well known:

T
T

may be perturbed by a compact operator to obtain an invertible operator if and only if the Fredholm Index of

T
T

is 0. In this case, the perturbation may be made as small in norm as we wish. Using the generalized Fredholm index for a von Neumann algebra developed by C. L. Olsen [3], the following generalization is obtained: let

A

\mathfrak {A}

be a von Neumann algebra with norm closed ideal

F

\mathfrak {F}

and canonical quotient map

π
\pi

. Let

T
∈

A

T \in \mathfrak {A}

be such that

π
(
T
)

\pi (T)

is invertible. Then there exists

K
∈

F

K \in \mathfrak {F}

such that

T
+
K

T + K

is invertible if and only if

ind
⁡
(
T
)
=
0

\operatorname {ind} (T) = 0

.
Note on counterexamples in strong unique continuation problems
Authors: Wolff Thomas H.
Published: June 2010

There are smooth functions on

R

d

{\mathbb {R}^d}

vanishing to infinite order at a point and satisfying the differential inequality

|

Δ
u

|

≤
V

|

u

|

|\Delta u| \leq V|u|

with

V
∈

V \in

weak

L

d

/

2

{L^{d/2}}

, and with

V
∈

L
1

V \in {L^1}

if

d
=
2

d = 2

.