bluuebunny/crossref_metadata_2025 huggingface dataset:
On Hardy’s inequality in weighted rearrangement invariant spaces and applications. II
Authors: Maligranda Lech
Published: July 2010

This note is a sequel to the preceding one with the same title published in these Proceedings. The contents of the first paper are assumed to be known. References are in alphabetical order in each paper, but they, as well as the theorems, are numbered consecutively.
The structure of Galois groups of 𝐶𝑀-fields
Authors: Dodson B.
Published: July 2010

A

C
M

CM

-field

K
K

defines a triple

(
G
,
H
,
ρ
)

(G,H,\rho )

, where

G
G

is the Galois group of the Galois closure of

K
K

,

H
H

is the subgroup of

G
G

fixing

K
K

, and

ρ
∈
G

\rho \in G

is induced by complex conjugation. A "

ρ
\rho

-structure" identifies

C
M

CM

-fields when their triples are identified under the action of the group of automorphisms of

G
G

. A classification of the

ρ
\rho

-structures is given, and a general formula for the degree of the reflex field is obtained. Complete lists of

ρ
\rho

-structues and reflex fields are provided for

[
K
:

Q

]
=
2
n

[K:\mathbb {Q}] = 2n

, with

n
=
3
,
4
,
5

n = 3,4,5

and

7
7

. In addition, simple degenerate Abelian varieties of

C
M

CM

-type are constructed in every composite dimension. The collection of reflex fields is also determined for the dihedral group

G
=

D

2
n

G = {D_{2n}}

, with

n
n

odd and

H
H

of order

2
2

, and a relative class number formula is found.
Aposyndetic continua as bundle spaces
Authors: Rogers James T.
Published: July 2010

Let

S

\mathcal {S}

be the

P
P

-adic solenoid bundle, and let

η
:
X
→

S
1

\eta :X \to {S^1}

be a map of the continuum

X
X

onto

S
1

{S^1}

. The bundle space

B
B

of the induced bundle

η

−
1

S

{\eta ^{ - 1}}\mathcal {S}

is investigated. Sufficient conditions are obtained for

B
B

to be connected, to be aposyndetic, and to be homogeneous. Uncountably many aposyndetic, homogeneous, one-dimensional, nonlocally connected continua are constructed. Other classes of continua are placed into this framework.
Global analysis of two-parameter elliptic eigenvalue problems
Authors: Peitgen H.-O. , Schmitt K.
Published: July 2010

We consider the nonlinear boundary value problem

(

∗

)
L
u
+
λ
f
(
u
)
=
0

({\ast })Lu + \lambda f(u) = 0

,

x
∈
Ω
,

u
=
σ
ϕ
,

x
∈
∂
Ω

x \in \Omega ,\,u = \sigma \phi ,\,x \in \partial \Omega

, where

L
L

is a second order elliptic operator and

λ
\lambda

and

σ
\sigma

are parameters. We analyze global properties of solution continua of these problems as

λ
\lambda

and

σ
\sigma

vary. This is done by investigating particular sections, and special interest is devoted to questions of how solutions of the

σ
=
0

\sigma = 0

problem are embedded in the two-parameter family of solutions of

(

∗

)

({\ast })

. As a natural biproduct of these results we obtain (a) a new abstract method to analyze bifurcation from infinity, (b) an unfolding of the bifurcations from zero and from infinity, and (c) a new framework for the numerical computations, via numerical continuation techniques, of solutions by computing particular one-dimensional sections.
Quartic surfaces of elliptic ruled type
Authors: Umezu Yumiko
Published: July 2010

Let

X
X

be a normal quartic surface whose resolutions are birationally equivalent to elliptic ruled surfaces. We classify the singularities on

X
X

and then describe the global structure of

X
X

.
Conjugacy classes of hyperbolic matrices in 𝑆𝑙(𝑛,𝑍) and ideal classes in an order
Authors: Wallace D. I.
Published: July 2010

A bijection is proved between

Sl
⁡
(
n
,

Z

)

\operatorname {Sl} (n,{\mathbf {Z}})

-conjugacy classes of hyperbolic matrices with eigenvalues

{

λ
1

,
…
,

λ
n

}

\{ {\lambda _1}, \ldots ,{\lambda _n}\}

which are units in an

n
n

-degree number field, and narrow ideal classes of the ring

R
k

=

Z

[

λ
i

]

{R_k} = {\mathbf {Z}}[{\lambda _i}]

. A bijection between

Gl
⁡
(
n
,

Z

)

\operatorname {Gl} (n,{\mathbf {Z}})

-conjugacy classes and the wide ideal classes, which had been known, is repeated with a different proof.
Abelian subgroups of topological groups
Authors: Grosser Siegfried K. , Herfort Wolfgang N.
Published: July 2010

In [1] Šmidt’s conjecture on the existence of an infinite abelian subgroup in any infinite group is settled by counterexample. The well-known Hall-Kulatilaka Theorem asserts the existence of an infinite abelian subgroup in any infinite locally finite group. This paper discusses a topological analogue of the problem. The simultaneous consideration of a stronger condition—that centralizers of nontrivial elements be compact—turns out to be useful and, in essence, inevitable. Thus two compactness conditions that give rise to a profinite arithmetization of topological groups are added to the classical list (see, e.g., [13 or 4]).
Examples of unoriented area-minimizing surfaces
Authors: Morgan Frank
Published: July 2010

A comprehensive study is made of constructions of area-minimizing flat chains modulo two. Many have singularities. For instance, any bounded, area-minimizing submanifold of

R

n

{{\mathbf {R}}^n}

occurs as the singular set of some area-minimizing flat chain modulo two in some

R

N

{{\mathbf {R}}^N}

.
On the diagonal of an operator
Authors: Fan Peng
Published: July 2010

Characterizations of zero-diagonal operators (i.e., operators that have a diagonal whose entries consist entirely of zeros) and the norm-closure of these operators are obtained. Also included are new characterizations of trace class operators, self-commutators of bounded operators, and others.
Banach spaces which are 𝑀-ideals in their biduals
Authors: Harmand Peter , Lima Åsvald
Published: July 2010

We investigate Banach spaces

X
X

such that

X
X

is an

M
M

-ideal in

X

∗

∗

{X^{{\ast }{\ast }}}

. Subspaces, quotients and

c
0

{c_0}

-sums of spaces which are

M
M

-ideals in their biduals are again of this type. A nonreflexive space

X
X

which is an

M
M

-ideal in

X

∗

∗

{X^{{\ast }{\ast }}}

contains a copy of

c
0

{c_0}

. Recently Lima has shown that if

K
(
X
)

K(X)

is an

M
M

-ideal in

L
(
X
)

L(X)

then

X
X

is an

M
M

-ideal in

X

∗

∗

{X^{{\ast }{\ast }}}

. Here we show that if

X
X

is reflexive and

K
(
X
)

K(X)

is an

M
M

-ideal in

L
(
X
)

L(X)

, then

K

(
X

)

∗

∗

K{(X)^{{\ast }{\ast }}}

is isometric to

L
(
X
)

L(X)

, i.e.

K
(
X
)

K(X)

is an

M
M

-ideal in its bidual. Moreover, for real such spaces, we show that

K
(
X
)

K(X)

contains a proper

M
M

-ideal if and only if

X
X

or

X

∗

{X^{\ast }}

contains a proper

M
M

-ideal.
On bordism groups of immersions
Authors: Pastor Guillermo
Published: July 2010

The bordism group of immersions of oriented

n
n

-manifolds into

R

n
+
k

{{\mathbf {R}}^{n + k}}

is identified with the stable homotopy group

Π

n
+
k

s

(
MSO
⁡
(
k
)
)

\Pi _{n + k}^s(\operatorname {MSO} (k))

. We study these groups for

n
−
2
⩽
k
⩽
n

n - 2 \leqslant k \leqslant n

, and discuss the behaviour of double points and their relation with the corresponding bordism groups of embeddings.
Compact spaces and spaces of maximal complete subgraphs
Authors: Bell Murray , Ginsburg John
Published: July 2010

We consider the space

M
(
G
)

M(G)

of all maximal complete subgraphs of a graph

G
G

and, in particular, the space

M
(
P
)

M(P)

of all maximal chains of an ordered set

P
P

. The main question considered is the following: Which compact spaces can be represented as

M
(
G
)

M(G)

for some graph

G
G

or as

M
(
P
)

M(P)

for some ordered set

P
P

? The former are characterized as spaces which have a binary subbase for the closed sets which consists of clopen sets. We give an example to show that this does not include all zero-dimensional supercompact spaces. The following negative result is obtained concerning ordered sets: Let

D
D

be an uncountable discrete space and let

α
D

\alpha D

denote the one-point compactification of

D
D

. Then there is no ordered set

P
P

such that

M
(
P
)
≃
α
D

M(P) \simeq \alpha D

.
Quadratic forms of height two
Authors: Fitzgerald Robert W.
Published: July 2010

Quadratic forms of height two and leading form defined over the base field are determined over several fields. Also forms of height and degree two over an arbitrary field are classified.
Extensions of tight set functions with applications in topological measure theory
Authors: Adamski Wolfgang
Published: July 2010

Let

K

1

,

K

2

{\mathcal {K}_1},\,{\mathcal {K}_2}

be lattices of subsets of a set

X
X

with

K

1

⊂

K

2

{\mathcal {K}_1} \subset {\mathcal {K}_2}

. The main result of this paper states that every semifinite tight set function on

K

1

{\mathcal {K}_1}

can be extended to a semifinite tight set function on

K

2

{\mathcal {K}_2}

. Furthermore, conditions assuring that such an extension is uniquely determined or

σ
\sigma

-smooth at

ϕ
\phi

are given. Since a semifinite tight set function defined on a lattice

K

\mathcal {K}

[and being

σ
\sigma

-smooth at

ϕ
\phi

] can be identified with a semifinite

K

\mathcal {K}

-regular content [measure] on the algebra generated by

K

\mathcal {K}

, the general results are applied to various extension problems in abstract and topological measure theory.
Factoring compact and weakly compact operators through reflexive Banach lattices
Authors: Aliprantis C. D. , Burkinshaw O.
Published: July 2010

When does a weakly compact operator between two Banach spaces factor through a reflexive Banach lattice? This paper provides some answers to this question. One of the main results: If an operator between two Banach spaces factors through a Banach lattice with weakly compact factors, then it also factors through a reflexive Banach lattice. In particular, the square of a weakly compact operator on a Banach lattice factors through a reflexive Banach lattice. Similar results hold for compact operators. For instance, the square of a compact operator on a Banach lattice factors with compact factors through a reflexive Banach lattice.
Covers in free lattices
Authors: Freese Ralph , Nation J. B.
Published: July 2010

In this paper we study the covering relation

(
u
≻
v
)

(u \succ v)

in finitely generated free lattices. The basic result is an algorithm which, given an element

w
∈

FL

(
X
)

w \in {\text {FL}}(X)

, finds all the elements which cover or are covered by

w
w

(if any such elements exist). Using this, it is shown that covering chains in free lattices have at most five elements; in fact, all but finitely many covering chains in each free lattice contain at most three elements. Similarly, all finite intervals in

FL

(
X
)

{\text {FL}}(X)

are classified; again, with finitely many exceptions, they are all one-, two- or three-element chains.
Surgery in dimension four and noncompact 5-manifolds
Authors: Silver Daniel S.
Published: July 2010

This paper describes a precise relationship between the problems of completing surgery in dimension four and finding boundaries for noncompact

5
5

-manifolds.
The determination of the Lie algebra associated to the lower central series of a group
Authors: Labute John P.
Published: July 2010

In this paper we determine the Lie algebra associated to the lower central series of a finitely presented group in the case where the defining relators satisfy certain independence conditions. Other central series, such as the lower

p
p

-central series, are treated as well.
A reciprocity law for polynomials with Bernoulli coefficients
Authors: Fouché Willem
Published: July 2010

We study the zeros

(
mod

p
)

\pmod p

of the polynomial
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript p Baseline left-parenthesis upper X right-parenthesis equals normal upper Sigma Subscript k Baseline left-parenthesis upper B Subscript k Baseline slash k right-parenthesis left-parenthesis upper X Superscript p minus 1 minus k Baseline minus 1 right-parenthesis">

β
p

(
X
)
=

Σ
k

(

B
k

/

k
)
(

X

p
−
1
−
k

−
1
)

{\beta _p}(X) = {\Sigma _k}({B_k}/k)({X^{p - 1 - k}} - 1)

for

p
p

an odd prime, where

B
k

{B_k}

denotes the

k
k

th Bernoulli number and the summation extends over

1
⩽
k
⩽
p
−
2

1 \leqslant k \leqslant p - 2

. We establish a reciprocity law which relates the congruence

β
p

(
r
)
≡
0

(
mod

p
)

{\beta _p}(r) \equiv 0\;\pmod p

to a congruence

f
p

(
n
)
≡
0

(
mod

r
)

{f_p}(n) \equiv 0\,\pmod r

for

r
r

a prime less than

p
p

and

n
∈

Z

n \in {\mathbf {Z}}

. The polynomial

f
p

(
x
)

{f_p}(x)

is the irreducible polynomial over

Q

{\mathbf {Q}}

of the number

Tr
L

Q

(
ζ
)

⁡
ζ

\operatorname {Tr}_L^{{\mathbf {Q}}(\zeta )}\zeta

, where

ζ
\zeta

is a primitive

p
2

{p^2}

th root of unity and

L
⊂

Q

(
ζ
)

L \subset {\mathbf {Q}}(\zeta )

is the extension of degree

p
p

over

Q

{\mathbf {Q}}

. These congruences are closely related to the prime divisors of the indices

I
(
α
)
=
(

O

:

Z

[
α
]
)

I(\alpha ) = (\mathcal {O}:{\mathbf {Z}}[\alpha ])

, where

O

\mathcal {O}

is the integral closure in

L
L

and

α
∈

O

\alpha \in \mathcal {O}

is of degree

p
p

over

Q

{\mathbf {Q}}

. We establish congruences

(
mod

p
)

\pmod p

involving the numbers

I
(
α
)

I(\alpha )

and show that their prime divisors

r
≠
p

r \ne p

are closely related to the congruence

r

p
−
1

≡
1

(
mod

p
2

)

{r^{p - 1}} \equiv 1\,\pmod {p^2}

.
Stability of Godunov’s method for a class of 2×2 systems of conservation laws
Authors: LeVeque Randall J. , Temple Blake
Published: July 2010

We prove stability and convergence of the Godunov scheme for a special class of genuinely nonlinear

2
×
2

2 \times 2

systems of conservation laws. The class of systems, which was identified and studied by Temple, is a subset of the class of systems for which the shock wave curves and rarefaction wave curves coincide. None of the equations of gas dynamics fall into this class, but equations of this type do arise, for example, in the study of multicomponent chromatography. To our knowledge this is the first time that a numerical method other than the random choice method of Glimm has been shown to be stable in the variation norm for a coupled system of nonlinear conservation laws. This implies that subsequences converge to weak solutions of the Cauchy problem, although convergence for

2
×
2

2 \times 2

systems has been proved by DiPerna using the more abstract methods of compensated compactness.
A reflexivity theorem for weakly closed subspaces of operators
Authors: Bercovici Hari
Published: July 2010

It was proved in [4] that the ultraweakly closed algebras generated by certain contractions on Hilbert space have a remarkable property. This property, in conjunction with the fact that these algebras are isomorphic to

H
∞

{H^\infty }

, was used in [3] to show that such ultraweakly closed algebras are reflexive. In the present paper we prove an analogous result that does not require isomorphism with

H
∞

{H^\infty }

, and applies even to linear spaces of operators. Our result contains the reflexivity theorems of [3,2 and 9] as particular cases.
Singular integral operators of Calderón type and related operators on the energy spaces
Authors: Murai Takafumi
Published: July 2010

We show the boundedness of some singular integral operators on the energy spaces.
Best approximation and quasitriangular algebras
Authors: Feeman Timothy G.
Published: July 2010

If

P

\mathcal {P}

is a linearly ordered set of projections on a Hilbert space and

K

\mathcal {K}

is the ideal of compact operators, then

Alg

P

+

K

\operatorname {Alg}\, \mathcal {P} + \mathcal {K}

is the quasitriangular algebra associated with

P

\mathcal {P}

. We study the problem of finding best approximants in a given quasitriangular algebra to a given operator: given

T
T

and

P

\mathcal {P}

, is there an

A
A

in

Alg

P

+

K

\operatorname {Alg}\, \mathcal {P} + \mathcal {K}

such that

‖

T
−
A

‖

=
inf
{

‖

T
−
S

‖

:
S
∈
Alg

P

+

K

}

\left \| {T - A} \right \| = \inf \{ \left \| {T - S} \right \|:S \in \operatorname {Alg}\,\mathcal {P} + \mathcal {K}\}

? We prove that if

A

\mathcal {A}

is an operator subalgebra which is closed in the weak operator topology and satisfies a certain condition

Δ
\Delta

, then every operator

T
T

has a best approximant in

A

+

K

\mathcal {A} + \mathcal {K}

. We also show that if

E

\mathcal {E}

is an increasing sequence of finite rank projections converging strongly to the identity then

Alg

E

\operatorname {Alg}\,\mathcal {E}

satisfies the condition

Δ
\Delta

. Also, we show that if

T
T

is not in

Alg

E

+

K

\operatorname {Alg}\,\mathcal {E} + \mathcal {K}

then the best approximants in

Alg

E

+

K

\operatorname {Alg}\,\mathcal {E} + \mathcal {K}

to

T
T

are never unique.
Compact conformally flat hypersurfaces
Authors: do Carmo Manfredo , Dajczer Marcos , Mercuri Francesco
Published: July 2010

Roughly speaking, a conformal space is a differentiable manifold

M
n

{M^n}

in which the notion of angle of tangent vectors at a point

p
∈

M
n

p \in {M^n}

makes sense and varies differentiably with

p
p

; two such spaces are (locally) equivalent if they are related by an angle-preserving (local) diffeomorphism. A conformally flat space is a conformal space locally equivalent to the euclidean space

R
n

{R^n}

. A submanifold of a conformally flat space is said to be conformally flat if so its induced conformal structure; in particular, if the codimension is one, it is called a conformally flat hypersurface. The aim of this paper is to give a description of compact conformally flat hypersurfaces of a conformally flat space. For simplicity, assume the ambient space to be

R

n
+
1

{R^{n + 1}}

. Then, if

n
⩾
4

n \geqslant 4

, a conformally flat hypersurface

M
n

⊂

R

n
+
1

{M^n} \subset {R^{n + 1}}

can be described as follows. Diffeomorphically,

M
n

{M^n}

is a sphere

S
n

{S^n}

with

b
1

(
M
)

{b_1}(M)

handles attached, where

b
1

(
M
)

{b_1}(M)

is the first Betti number of

M
M

. Geometrically, it is made up by (perhaps infinitely many) nonumbilic submanifolds of

R

n
+
1

{R^{n + 1}}

that are foliated by complete round

(
n
−
1
)

(n - 1)

-spheres and are joined through their boundaries to the following three types of umbilic submanifolds of

R

n
+
1

{R^{n + 1}}

: (a) an open piece of an

n
n

-sphere or an

n
n

-plane bounded by round

(
n
−
1
)

(n - 1)

-sphere, (b) a round

(
n
−
1
)

(n - 1)

-sphere, (c) a point.
Sobolev space projections in strictly pseudoconvex domains
Authors: Boas Harold P.
Published: July 2010

The orthogonal projection from a Sobolev space

W
s

(
Ω
)

{W^s}(\Omega )

onto the subspace of holomorphic functions is studied. This analogue of the Bergman projection is shown to satisfy regularity estimates in higher Sobolev norms when

Ω
\Omega

is a smooth bounded strictly pseudoconvex domain in

C

n

{{\mathbf {C}}^n}

.
Nilpotent automorphism groups of Riemann surfaces
Authors: Zomorrodian Reza
Published: July 2010

The action of nilpotent groups as automorphisms of compact Riemann surfaces is investigated. It is proved that the order of a nilpotent group of automorphisms of a surface of genus

g
⩾
2

g \geqslant 2

cannot exceed

16
(
g
−
1
)

16(g - 1)

. Exact conditions of equality are obtained. This bound corresponds to a specific Fuchsian group given by the signature (0;2,4,8).
Simply transitive groups and Kähler structures on homogeneous Siegel domains
Authors: Dorfmeister Josef
Published: July 2010

We determine the Lie algebras of all simply transitive groups of automorphisms of a homogeneous Siegel domain

D
D

as modifications of standard normal

j
j

-algebras. We show that the Lie algebra of all automorphisms of

D
D

is a "complete isometry algebra in standard position". This implies that

D
D

carries a riemannian metric

g
~

\tilde g

with nonpositive sectional curvature satisfying Lie

Iso
⁡
(
D
,

g
~

)
=
Lie

Aut

D

\operatorname {Iso}(D,\tilde g) = \operatorname {Lie}\; \operatorname {Aut}\, \text {D}

. We determine all Kähler metrics

f
f

on

D
D

for which the group

Aut
⁡
(
D
,
f
)

\operatorname {Aut}(D,f)

of holomorphic isometries acts transitively. We prove that in this case

Aut
⁡
(
D
,
f
)

\operatorname {Aut}(D,f)

contains a simply transitive split solvable subgroup. The results of this paper are used to prove the fundamental conjecture for homogeneous Kähler manifolds admitting a solvable transitive group of automorphisms.
The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences
Authors: Bellow A. , Losert V.
Published: July 2010

In this paper on the weighted pointwise ergodic theorem we bring together a substantial amount of previous work by a number of mathematicians and integrate it into a systematic consistent scheme; we also bring some original contributions to the subject which extend its boundaries and suggest further avenues of research. The paper is divided into six sections. The most significant new results are contained in

§
§
5

\S \S 5

and

6
6

.
Regularization of 𝐿² norms of Lagrangian distributions
Authors: Izen Steven
Published: July 2010

Let

X
X

be a compact smooth manifold,

dim
⁡
X
=
n

\dim X = n

. Let

Λ
\Lambda

be a fixed Lagrangian submanifold of

T
∗

X

{T^\ast }X

. The space of Lagrangian distributions

I
k

(
X
,
Λ
)

{I^k}(X,\Lambda )

is contained in

L
2

(
X
)

{L^2}(X)

if

k
>
−
n

/

4

k > - n/4

. When

k
=
n

/

4

k = n/4

,

I

−
n

/

4

(
X
,
Λ
)

{I^{ - n/4}}(X,\Lambda )

just misses

L
2

(
X
)

{L^2}(X)

. A new inner product

⟨
u
,
v

⟩
R

{\langle u,v\rangle _R}

is defined on
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Superscript negative n slash 4 Baseline left-parenthesis upper X comma normal upper Lamda right-parenthesis slash upper I Superscript negative n slash 4 minus 1 Baseline left-parenthesis upper X comma normal upper Lamda right-parenthesis">

I

−
n

/

4

(
X
,
Λ
)

/

I

−
n

/

4
−
1

(
X
,
Λ
)

{I^{ - n/4}}(X,\Lambda )/{I^{ - n/4 - 1}}(X,\Lambda )

in terms of symbols. This inner product contains "

L
2

{L^2}

information" in the following sense: Slight regularizations of the Lagrangian distributions are taken, putting them in

L
2

(
X
)

{L^2}(X)

. The asymptotic behavior of the

L
2

{L^2}

inner product is examined as the regularizations approach the identity. Three different regularization schemes are presented and, in each case,

⟨
u
,
v

⟩
R

{\langle u,v\rangle _R}

is found to regulate the growth of the ordinary

L
2

{L^2}

inner product.
Lowest order invariants for real-analytic surfaces in 𝐶²
Authors: Harris Gary A.
Published: July 2010

Suppose

M
M

is a general real-analytic surface in complex euclidean two-space with complex tangent space at a point

p
p

. Further suppose

M
M

is tangent to order

k
k

at

p
p

. This paper determines a complete set of

k
k

th order local holomorphic invariants for

M
M

at

p
p

.
An upper bound for the waiting time for nonlinear degenerate parabolic equations
Authors: Chipot Michel , Sideris Thomas
Published: July 2010

An upper bound is obtained for the time when the support of the solution of some nonlinear, degenerate parabolic equations begins to spread.
Subellipticity of the ∂̄-Neumann problem on nonpseudoconvex domains
Authors: Ho Lop-Hing
Published: July 2010

Following the work of Kohn, we give a sufficient condition for subellipticity of the

∂
¯

\overline \partial

-Neumann problem for not necessarily pseudoconvex domains. We define a sequence of ideals of germs and show that if

1
1

is in any of them, then there is a subelliptic estimate. In particular, we prove subellipticity under some specific conditions for

n
−
1

n - 1

forms and for the case when the Levi-form is diagonalizable. For the necessary conditions, we use another method to prove that there is no subelliptic estimate for

q
q

forms if the Levi-form has

n
−
q
−
1

n - q - 1

positive eigenvalues and

q
q

negative eigenvalues. This was proved by Derridj. Using similar techniques, we prove a necessary condition for subellipticity for some special domains. Finally, we give a remark to Catlin’s theorem concerning the hypoellipticity of the

∂
¯

\overline \partial

-Neumann problem in the case of nonpseudoconvex domains.
Small zeros of quadratic forms
Authors: Schmidt Wolfgang M.
Published: July 2010

We give upper and lower bounds for zeros of quadratic forms in the rational, real and

p
p

-adic fields. For example, given

r
>
0

r > 0

,

s
>
0

s > 0

, there are infinitely many forms

F

\mathfrak {F}

with integer coefficients in

r
+
s

r + s

variables of the type

(
r
,
s
)

(r,s)

(i.e., equivalent over

R

{\mathbf {R}}

to

X
1
2

+
⋯
+

X
r
2

−

X

r
+
1

2

−
⋯
−

X

r
+
s

2

X_1^2 + \cdots + X_r^2 - X_{r + 1}^2 - \cdots - X_{r + s}^2

such that every nontrivial integer zero

x

{\mathbf {x}}

has

|

x

|

≫

F

r

/

2
s

|{\mathbf {x}}| \gg {F^{r/2s}}

, where

F
F

is the maximum modulus of the coefficients of

F

\mathfrak {F}

.
The traction problem for incompressible materials
Authors: Wan Y. H.
Published: July 2010

The traction problem for incompressible materials is treated as a bifurcation problem, where the applied loads are served as parameters. We take both the variational approach and the classical power series approach. The variational approach provides a natural, unified way of looking at this problem. We obtain a count of the number of equilibria together with the determination of their stability. In addition, it also lays down the foundation for the Signorini-Stoppelli type computations. We find second order sufficient conditions for the existence of power series solutions. As a consequence, the linearization stability follows, and it clarifies in some sense the role played by the linear elasticity in the context of the nonlinear elasticity theory. A systematic way of calculating the power series solution is also presented.
Propagation estimates for Schrödinger-type operators
Authors: Jensen Arne
Published: July 2010

Propagation estimates for a Schrödinger-type operator are obtained using multiple commutator techniques. A new method is given for obtaining estimates for powers of the resolvent. As an application, micro-local propagation estimates are obtained for two-body Schrödinger operators with smooth long-range potentials.
The Cauchy problem for 𝑢_{𝑡}=Δ𝑢^{𝑚} when 0<𝑚<1
Authors: Herrero Miguel A. , Pierre Michel
Published: July 2010

This paper deals with the Cauchy problem for the nonlinear diffusion equation

∂
u

/

∂
t
−
Δ
(
u

|

u

|

m
−
1

)
=
0

\partial u/\partial t - \Delta (u|u{|^{m - 1}}) = 0

on

(
0
,
∞
)
×

R

N

,
u
(
0
,
⋅
)
=

u
0

(0,\infty ) \times {{\mathbf {R}}^N},u(0, \cdot ) = {u_0}

when

0
>
m
>
1

0 > m > 1

(fast diffusion case). We prove that there exists a global time solution for any locally integrable function

u
0

{u_0}

: hence, no growth condition at infinity for

u
0

{u_0}

is required. Moreover the solution is shown to be unique in that class. Behavior at infinity of the solution and

L

loc

∞

L_{\operatorname {loc} }^\infty

-regularizing effects are also examined when

m
∈
(
max
{
(
N
−
2
)

/

N
,
0
}
,
1
)

m \in (\max \{ (N - 2)/N,0\} ,1)

.
Probabilistic square functions and a priori estimates
Authors: Bennett Andrew G.
Published: July 2010

We obtain a priori estimates for Riesz transforms and their variants, that is, estimates with bounds independent of the dimension of the space and/or the nature of the boundary. The key to our results is to give probabilistic definitions which do not depend on the geometry of the situation for the transformations in question. We then use probabilistic square functions to prove our a priori estimates.
Symmetric positive systems with boundary characteristic of constant multiplicity
Authors: Rauch Jeffrey
Published: July 2010

The theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity. No such hypothesis is needed on a neighborhood of the boundary. Both regularity theorems and mixed initial boundary value problems are discussed. Many classical ideas are sharpened in the process.
Baire sets of 𝑘-parameter words are Ramsey
Authors: Prömel Hans Jürgen , Voigt Bernd
Published: July 2010

We show that Baire sets of

k
k

-parameter words are Ramsey. This extends a result of Carlson and Simpson, A dual form of Ramsey’s theorem, Adv. in Math. 53 (1984), 265-290. Employing the method established therefore, we derive analogous results for Dowling lattices and for ascending

k
k

-parameter words.
Hyponormal operators quasisimilar to an isometry
Authors: Wu Pei Yuan
Published: July 2010

An expression for the multiplicity of an arbitrary contraction is presented. It is in terms of the isometries which can be densely intertwined to the given contraction. This is then used to obtain a generalization of a result of Sz.-Nagy and Foiaş concerning the existence of a

C

.
0

C{._0}

contraction which is a quasiaffine transform of a contraction. We then consider the problem when a hyponormal operator is quasisimilar to an isometry or, more generally, when two hyponormal contractions are quasisimilar to each other. Our main results in this respect generalize previous ones obtained by Hastings and the author. For quasinormal and certain subnormal operators, quasisimilarity or similarity to an isometry may even imply unitary equivalence.
Global solvability and regularity for ∂̄ on an annulus between two weakly pseudoconvex domains
Authors: Shaw Mei-Chi
Published: July 2010

Let

M
1

{M_1}

and

M
2

{M_2}

be two bounded pseudo-convex domains in

C

n

{{\mathbf {C}}^n}

with smooth boundaries such that

M
¯

1

⊂

M
2

{\overline M _1} \subset {M_2}

. We consider the Cauchy-Riemann operators

∂
¯

\overline \partial

on the annulus

M
=

M
2

∖

M
¯

1

M = {M_2}\backslash {\overline M _1}

. The main result of this paper is the following: Given a

∂
¯

\overline \partial

-closed

(
p
,
q
)

(p,q)

form

α
\alpha

,

0
>
q
>
n

0 > q > n

, which is

C
∞

{C^\infty }

on

M
¯

\overline M

and which is cohomologous to zero on

M
M

, there exists a

(
p
,
q
−
1
)

(p,q - 1)

form

u
u

which is

C
∞

{C^\infty }

on

M
¯

\overline M

such that

∂
¯

u
=
α

\overline \partial u = \alpha

.
On monomial algebras of finite global dimension
Authors: Anick David J.
Published: July 2010

Let

G
G

be an associative monomial

k

{\mathbf {k}}

-algebra. If

G
G

is assumed to be finitely presented, then either

G
G

contains a free subalgebra on two monomials or else

G
G

has polynomial growth. If instead

G
G

is assumed to have finite global dimension, then either

G
G

contains a free subalgebra or else

G
G

has a finite presentation and polynomial growth. Also, a graded Hopf algebra with generators in degree one and relations in degree two contains a free Hopf subalgebra if the number of relations is small enough.
Divisions of space by parallels
Authors: Alexanderson G. L. , Wetzel John E.
Published: July 2010

An arrangement of hyperplanes in

E

d

{\mathbb {E}^d}

is a "plaid" provided its hyperplanes form no multiple flats of intersection and lie in parallel families that are in general position. We develop some geometrically natural formulas for the number of

r
r

-faces that are formed by such an arrangement.
Some product formulae for nonsimply connected surgery problems
Authors: Milgram R. J. , Ranicki Andrew
Published: July 2010

For an

n
n

-dimensional normal map

f
:

M
n

→

N
n

f :{M^n} \to {N^n}

with finite fundamental group

π
1

(
N
)
=
π

{\pi _1}(N) = \pi

and PL

1
1

torsion kernel

Z
[
π
]

Z[\pi ]

-modules

K

∗

(
M
)

{K_{\ast }}(M)

the surgery obstruction

σ

∗

(
f
)
∈

L
n
h

(
Z
[
π
]
)

{\sigma _{\ast }}(f) \in L_n^h(Z[\pi ])

is expressed in terms of the projective classes
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper K Subscript asterisk Baseline left-parenthesis upper M right-parenthesis right-bracket element-of ModifyingAbove upper K With tilde Subscript 0 Baseline left-parenthesis upper Z left-bracket pi right-bracket right-parenthesis">

[

K

∗

(
M
)
]
∈

K
~

0

(
Z
[
π
]
)

[{K_{\ast }}(M)] \in {\tilde K_0}(Z[\pi ])

, assuming

K
i

(
M
)
=
0

{K_i}(M) = 0

if

n
=
2
i

n = 2i

. This expression is used to evaluate in certain cases the surgery obstruction

σ

∗

(
g
)
∈

L

m
+
n

h

(
Z
[

π
1

×
π
]
)

{\sigma _ {\ast } }(g) \in L_{m + n}^h(Z[{\pi _1} \times \pi ])

of the

(
m
+
n
)

(m + n)

-dimensional normal map

g
=
1
×
f
:

M
1

×
M
→

M
1

×
N

g = 1 \times f:{M_1} \times M \to {M_1} \times N

defined by product with an

m
m

-dimensional manifold

M
1

{M_1}

, where

π
1

=

π
1

(

M
1

)

{\pi _1} = {\pi _1}({M_1})

.
Bounds for prime solutions of some diagonal equations. II
Authors: Liu Ming Chit
Published: July 2010

Let

b
j

{b_j}

and

m
m

be certain integers. In this paper we obtain a bound for prime solutions

p
j

{p_j}

of the diagonal equations of order

k
,

b
1

p
1
k

+
⋯
+

b
s

p
s
k

=
m

k,\;{b_1}p_1^k + \cdots + {b_s}p_s^k = m

. The bound obtained is

C

(
log
⁡
B
)

2

+
C

|

m

|

1

/

k

{C^{{{(\log B)}^2}}} + C|m{|^{1/k}}

where

B
=

max
j

{
e
,

|

b
j

|

}

B = {\max _j}\{ e,|{b_j}|\}

and

C
C

are positive constants depending at most on

k
k

.
Groups presented by finite two-monadic Church-Rosser Thue systems
Authors: Avenhaus J. , Madlener K. , Otto F.
Published: July 2010

It is shown that a group

G
G

can be defined by a monoid-presentation of the form

(
Σ
;
T
)

(\Sigma ;T)

, where

T
T

is a finite two-monadic Church-Rosser Thue system over

Σ
\Sigma

, if and only if

G
G

is isomorphic to the free product of a finitely generated free group with a finite number of finite groups.
Homologie de l’espace des sections d’un fibré
Authors: Legrand Claude
Published: July 2010

For a fiber bundle with a finite cohomology dimension and

1
1

-connected base

B
B

and

1
1

-connected fiber

F
F

, we obtain the homology of the section space by an

E
1

{E^1}

-spectral sequence. In the "stable" range the

E
1

{E^1}

-terms are the homology of a product of Eilenberg-Mac Lane space of type

K
(

H

p
−
i

(
B
;

π
p

F
)
,
i
)

K({H^{p - i}}(B;{\pi _p}F),i)

(the same as those of the

E
1

{E^1}

-spectral sequences which converges to the homology of the functional space

Hom
⁡
(
B
,
F
)

\operatorname {Hom} (B,F)

[10]). The differential is the product of two operations: one appears in the

E
1

{E^1}

-spectral sequence, which converges to the homology of

Hom
⁡
(
B
,
F
)

\operatorname {Hom} (B,F)

; the second one is a "cup-product" determined by the fiber structure of the bundle. This spectral sequence is obtained by a Moore-Postnikov tower of the fiber, which generalizes Kahn’s methods [9].
Random power series generated by ergodic transformations
Authors: Halchin Judy , Petersen Karl
Published: July 2010

Generalizing classical studies of power series with sequences of independent random variables as coefficients, we study series of the forms
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g Subscript x comma phi Baseline left-parenthesis z right-parenthesis equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts phi left-parenthesis upper T Superscript n Baseline x right-parenthesis z Superscript n Baseline and f Subscript x comma phi Baseline left-parenthesis z right-parenthesis equals sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts phi left-parenthesis x right-parenthesis phi left-parenthesis upper T x right-parenthesis midline-horizontal-ellipsis phi left-parenthesis upper T Superscript n minus 1 Baseline x right-parenthesis z Superscript n Baseline comma">

g

x
,
ϕ

(
z
)
=

∑

n
=
0

∞

ϕ
(

T
n

x
)

z
n

and

f

x
,
ϕ

(
z
)
=

∑

n
=
1

∞

ϕ
(
x
)
ϕ
(
T
x
)
⋯
ϕ
(

T

n
−
1

x
)

z
n

,

{g_{x,\phi }}(z) = \sum \limits _{n = 0}^\infty {\phi ({T^n}x){z^n}\quad {\text {and}}\quad {f_{x,\phi }}(z) = \sum \limits _{n = 1}^\infty {\phi (x)\phi (Tx) \cdots \phi ({T^{n - 1}}x){z^n},} }

\]
where

T
T

is an ergodic measure-preserving transformation on a probability space

(
X
,

B

,
μ
)

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

and

ϕ
\phi

is a measurable complex-valued function which is a.e. nonzero. When

f

x
,
ϕ

{f_{x,\phi }}

is entire, its order of growth at infinity measures the speed of divergence of the ergodic averages of

log
⁡

|

ϕ

|

\log |\phi |

. We give examples to show that any order is possible for any

T
T

and that different orders are possible for fixed

ϕ
\phi

. For fixed

T
T

, the set of

ϕ
\phi

which produce infinite order is residual in the subset of

L
1

(
X
)

{L^1}(X)

consisting of those

ϕ
\phi

which are a.e. nonzero and produce entire

f

x
,
ϕ

{f_{x,\phi }}

. As in a theorem of Pólya for gap series, if

f

x
,
ϕ

{f_{x,\phi }}

is entire and has finite order, then it assumes every value infinitely many times. The functions

ϕ
∈

L
1

(
X
)

\phi \in {L^1}(X)

for which

g

x
,
ϕ

{g_{x,\phi }}

is rational a.e. are exactly the finite sums of eigenfunctions of

T
T

; their poles are all simple and are the inverses of the corresponding eigenvalues. By combining this result with a skew product construction, we can also characterize when

f

x
,
ϕ

{f_{x,\phi }}

is rational, provided that

ϕ
\phi

takes one of several particular forms.
Construction of a family of non-self-dual gauge fields
Authors: Sols Ignacio
Published: July 2010

Using the generalization of vector bundles by reflexive sheaves recently introduced by R. Hartshorne in [2] we construct a

15
15

-dimensional family of nontrivial complex gauge fields

(
U
,
E
,
∇
)

(U,E,\nabla )

which are not self-dual nor anti-self-dual. (

U
U

is an affine neighborhood in

Q
4

=
Gr
⁡
(
2
,

C

4

)

{Q_4} = \operatorname {Gr} (2,{{\mathbf {C}}^4})

of the stereographic compactification

S
4

{S^4}

of

R

4

{\mathbb {R}^4}

,

E
E

is a vector bundle on

U
U

and

∇
\nabla

is a connection on it whose curvature

ϕ
\phi

satisfies the inequalities

∗

ϕ
≠
ϕ

{}^{\ast }\phi \ne \phi

and

∗

ϕ
≠
−
ϕ

{}^{\ast }\phi \ne - \phi

.)
The intersection topology w.r.t. the real line and the countable ordinals
Authors: Reed G. M.
Published: July 2010

If

Υ
1

{\Upsilon _1}

and

Υ
2

{\Upsilon _2}

are topologies defined on the set

X
X

, then the intersection topology w.r.t.

Υ
1

{\Upsilon _1}

and

Υ
2

{\Upsilon _2}

is the topology

Υ
\Upsilon

on

X
X

such that

{

U
1

∩

U
2

|

U
1

∈

Υ
1

and

U
2

∈

Υ
2

}

\{ {U_1} \cap {U_2}|{U_1} \in {\Upsilon _1}\;{\text {and}}\;{U_2} \in {\Upsilon _2}\}

is a basis for

(
X
,
Υ
)

(X,\Upsilon )

. In this paper, the author considers spaces in the class

C

\mathcal {C}

, where

(
X
,
Υ
)
∈

C

(X,\Upsilon ) \in \mathcal {C}

iff

X
=
{

x
α

|

α
>

ω
1

}
⊆

R

X = \{ {x_\alpha }|\alpha > {\omega _1}\} \subseteq {\mathbf {R}}

,

Υ

R

{\Upsilon _{\mathbf {R}}}

is the inherited real line topology on

X
X

,

Υ

ω
1

{\Upsilon _{{\omega _1}}}

is the order topology on

X
X

of type

ω
1

{\omega _1}

, and

Υ
\Upsilon

is the intersection topology w.r.t.

Υ

R

{\Upsilon _{\mathbf {R}}}

and

Υ

ω
1

{\Upsilon _{{\omega _1}}}

. This class is shown to be a surprisingly useful tool in the study of abstract spaces. In particular, it is shown that: (1) If

X
∈

C

X \in \mathcal {C}

, then

X
X

is a completely regular, submetrizable, pseudo-normal, collectionwise Hausdorff, countably metacompact, first countable, locally countable space with a base of countable order that is neither subparacompact, metalindelöf, cometrizable, nor locally compact. (2)

(
MA
+
¬
CH
)

(\operatorname {MA} + \neg \operatorname {CH} )

If

X
∈

C

X \in \mathcal {C}

, then

X
X

is perfect. (3) There exists in ZFC an

X
∈

C

X \in \mathcal {C}

such that

X
X

is not normal. (4)

(
CH
)

(\operatorname {CH} )

There exists

X
∈

C

X \in \mathcal {C}

such that

X
X

is collectionwise normal and

ω
1

{\omega _1}

-compact but not perfect.
Shape properties of Whitney maps for hyperspaces
Authors: Kato Hisao
Published: July 2010

In this paper, some shape properties of Whitney maps for hyperspaces are investigated. In particular, the following are proved: (1) Let

X
X

be a continuum and let

H

\mathfrak {H}

be the hyperspace

2
X

{2^X}

or

C
(
X
)

C(X)

of

X
X

with the Hausdorff metric. Then if

ω
\omega

is any Whitney map for

H

\mathfrak {H}

, for any

0
⩽
s
⩽
t
⩽
ω
(
X
)

ω

−
1

(
t
)

0 \leqslant s \leqslant t \leqslant \omega (X){\omega ^{ - 1}}(t)

is an approximate strong deformation retract of

ω

−
1

(
[
s
,
t
]
)

{\omega ^{ - 1}}([s,t])

. In particular,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S h left-parenthesis omega Superscript negative 1 Baseline left-parenthesis t right-parenthesis right-parenthesis equals upper S h left-parenthesis omega Superscript negative 1 Baseline left-parenthesis left-bracket s comma t right-bracket right-parenthesis right-parenthesis">

Sh
⁡
(

ω

−
1

(
t
)
)
=
Sh
⁡
(

ω

−
1

(
[
s
,
t
]
)
)

\operatorname {Sh} ({\omega ^{ - 1}}(t)) = \operatorname {Sh} ({\omega ^{ - 1}}([s,t]))

. (2) Pointed

1
1

-movability is a Whitney property. (3) For any given

n

>
∞

{\text {n}} > \infty

, the property of (cohomological) dimension

⩽
n

\leqslant n

is a sequential strong Whitney-reversible property. (4) The property of being chainable or circle-like is a sequential strong Whitney-reversible property. (5) The property of being an FAR is a Whitney property for

1
1

-dimensional continua. Property (2) is an affirmative answer to a problem of J. T. Rogers [16, 112]. Properties (3) and (4) are affirmative answers to problems of S. B. Nadler [20, (14.57) and 21].
On 𝑅^{∞}(𝑄^{∞})-manifold bundles over CW complexes
Authors: Liem Vo Thanh
Published: July 2010

Let

Λ
∈

C

W

(

C

)
∪

C

W

(

M

)

\Lambda \in \mathcal {C}\mathcal {W}(\mathcal {C}) \cup \mathcal {C}\mathcal {W}(\mathcal {M})

be a pseudo CW complex generated either by Hausdorff compact spaces or by metric spaces. In the theory of manifolds modeled on

R
∞

{R^\infty }

or

Q
∞

{Q^\infty }

, we will prove the

Λ
\Lambda

-fiber-preserving versions of the following: Equivalences among the notions of

D
D

-sets,

D

∗

{D^{\ast }}

-sets and infinite deficient sets; relative stability theorems; relative deformation of homotopy equivalences to homeomorphisms; strong unknotting theorem for

D
D

-embeddings; and

α
\alpha

-approximation theorems.
Stratification of continuous maps of an interval
Authors: Block L. S. , Coppel W. A.
Published: July 2010

We define the motion of turbulence for a continuous map of an interval into the line and study its relation with periodic and homoclinic points. We define also strongly simple orbits and show, in particular, that they represent periodic orbits with minimum entropy. Further results are obtained for unimodal maps with negative Schwarzian, which sharpen recent results of Block and Hart.
A topological proof of the equivariant Dehn lemma
Authors: Edmonds Allan L.
Published: July 2010

An elementary topological proof is given for a completely general version of the Equivariant Dehn Lemma, in the spirit of the original proof of the nonequivariant version due to C. D. Papakyriakopolous in 1957.
Smooth maps, pullback path spaces, connections, and torsions
Authors: Chen Kuo Tsai
Published: July 2010

By generalizing the local version of the usual differential geometric notion of connections and that of torsions, a model for the pullback path space of a smooth map is constructed from the induced map of the de Rham complexes. The pullback path space serves not only as a homotopy fiber but also as a device reflecting differentiable properties of the smooth map. Applications are discussed.
Range transformations on a Banach function algebra
Authors: Hatori Osamu
Published: July 2010

We study the range transformations

Op
⁡
(

A

D
,

Re
⁡
B
)

\operatorname {Op} ({A_{D,}}\operatorname {Re} B)

and

Op
⁡
(

A
D

,
B
)

\operatorname {Op} ({A_D},B)

for Banach function algebras

A
A

and

B
B

. As a special instance, the harmonicity of functions in

Op
⁡
(

A
D

,
Re
⁡
A
)

\operatorname {Op} ({A_D},\operatorname {Re} A)

for a nontrivial function algebra

A
A

is established and is compared with previous investigations of

Op
⁡
(

A
D

,
A
)

\operatorname {Op} ({A_D},A)

and

Op
⁡
(

(
Re
⁡
A

)
I

,
(
Re
⁡
A
)
)

\operatorname {Op} ({(\operatorname {Re} A)_I},(\operatorname {Re} A))

for an interval

I
I

. In

§
2

\S 2

we present some results on

Op
⁡
(

A
D

,
B
)

\operatorname {Op} ({A_D},B)

and use them to show that functions in

Op
C

(

A
D

,
B
)

{\operatorname {Op} ^C}({A_D},B)

are analytic for certain Banach function algebras.
Cauchy problem for nonlinear hyperbolic systems of partial differential equations
Authors: Yasinovskaya Victoria
Published: July 2010

We proved the sharp Sobolev estimate for Cauchy data for the general type of hyperbolic systems of nonlinear partial differential equations, which leads to a local existence and uniqueness theorem for solutions of the Cauchy problem in Sobolev spaces.
Fine structure of the integral exponential functions below 2^{2^{𝑥}}
Authors: Dahn Bernd I.
Published: July 2010

Integral exponential functions are the members of the least class of real functions containing

1
1

, the identity function, and closed under addition, multiplication, and binary exponentiation sending

f
f

and

g
g

to

f
g

{f^g}

. This class is known to be wellordered by the relation of eventual dominance. It is shown that for each natural number

n
n

the order type of the integral exponential functions below

2

x
n

{2^{{x^n}}}

(below

x

x
n

{x^{{x^n}}}

) is exactly

ω

ω

2
n
−
1

{\omega ^{{\omega ^{2n - 1}}}}

(

ω

ω

2
n

{\omega ^{{\omega ^{2n}}}}

respectively). The proof, using iterated asymptotic expansions, contains also a new proof that integral exponential functions below

2

2
x

{2^{{2^x}}}

are wellordered.
On the factorizations of ordinary linear differential operators
Authors: Etgen G. J. , Jones G. D. , Taylor W. E.
Published: July 2010

Relations are found between the nonvanishing of certain Wronskians and disconjugacy properties of

L
n

y
+
p
y
=
0

{L_n}y + py = 0

, where

L
n

y

{L_n}y

is a disconjugate operator and

p
p

is sign definite. The results are then used to show ways in which

L
n

y
+
p
y

{L_n}y + py

can be factored.
A strong containment property for discrete amenable groups of automorphisms on 𝑊* algebras
Authors: Granirer Edmond E.
Published: July 2010

Let

G
G

be a countable group of automorphisms on a

W

∗

{W^{\ast }}

algebra

M

\mathcal {M}

and let

ϕ
0

{\phi _0}

be a

w

∗

G
δ

{w^{\ast }}{G_\delta }

point of the set of

G
G

invariant states on

M

\mathcal {M}

which belong to

w

∗

cl
⁡
Co
⁡
E

{w^{\ast }}\operatorname {cl} \operatorname {Co} E

, where

E
E

is a set of (possibly pure) states on

M

\mathcal {M}

. If

G
G

is amenable, then the cyclic representation

π

ϕ
0

{\pi _{{\phi _0}}}

corresponding to

ϕ
0

{\phi _0}

is contained in

{
⊕

π
ϕ

;
ϕ
∈
E
}

\{ \oplus {\pi _\phi };\phi \in E\}

. This property characterizes amenable groups. Related results are obtained.
Decay with a rate for noncompactly supported solutions of conservation laws
Authors: Temple Blake
Published: July 2010

We show that solutions of the Cauchy problem for systems of two conservation laws decay in the supnorm at a rate that depends only on the

L
1

{L^1}

norm of the initial data. This implies that the dissipation due to increasing entropy dominates the nonlinearities in the problem at a rate depending only on the

L
1

{L^1}

norm of the initial data. Our results apply to any BV initial data satisfying

u
0

(
±
∞
)
=
0

{u_0}( \pm \infty ) = 0

and

Sup

{

u
0

(
⋅
)
}
≪
1

{\operatorname {Sup}}\{ {u_0}( \cdot )\} \ll 1

. The problem of decay with a rate independent of the support of the initial data is central to the issue of continuous dependence in systems of conservation laws because of the scale invariance of the equations. Indeed, our result implies that the constant state is stable with respect to perturbations in

L

loc

1

L_{{\operatorname {loc}}}^1

. This is the first stability result in an

L
p

{L^p}

norm for systems of conservation laws. It is crucial that we estimate decay in the supnorm since the total variation does not decay at a rate independent of the support of the initial data. The main estimate requires an analysis of approximate characteristics for its proof. A general framework is developed for the study of approximate characteristics, and the main estimate is obtained for an arbitrary number of equations.