bluuebunny/crossref_metadata_2025 huggingface dataset:
A characterization of best Φ-approximants
Authors: Landers D. , Rogge L.
Published: June 2010

Let

T
T

be an operator from an Orlicz space

L
Φ

{L_\Phi }

into itself. It is shown in this paper that four algebraic conditions and one integration condition assure that

T
T

is the best

Φ
\Phi

-approximator, given a suitable

σ
\sigma

-lattice.
A Phragmén-Lindelöf theorem conjectured by D. J. Newman
Authors: Fuchs W. H. J.
Published: June 2010

Let

D
D

be a region of the complex plane,

∞
∈
∂
D

\infty \in \partial D

. If

f
(
z
)

f(z)

is holomorphic in

D
D

, write

M
(
r
)
=

sup

|

z

|

=
r
,

z
∈
D

|

f
(
z
)

|

M(r) = {\sup _{|z| = r,\,z \in D}}|f(z)|

. Theorem 1. If

f
(
z
)

f(z)

is holomorphic in

D
D

and

lim

sup

z
→
ζ
,

z
∈
D

|

f
(
z
)

|

⩽
1

\lim {\sup _{z \to \zeta ,\,z \in D}}|f(z)| \leqslant 1

for

ζ
∈
∂
D

\zeta \in \partial D

,

ζ
≠
∞

\zeta \ne \infty

, then one of the following holds (a)

|

f
(
z
)

|

>
1
(
z
∈
D
)

|f(z)| > 1(z \in D)

, (b)

f
(
z
)

f(z)

has a pole at

∞
\infty

, (c)

log

M
(
r
)

/

log
⁡
r
→
∞

\log \,M(r)/\log r \to \infty

as

r
→
∞

r \to \infty

. If

M
(
r
)

/

r
→
0
(
r
→
∞
)

M(r)/r \to 0(r \to \infty )

, then (a) must hold.
A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves
Authors: Neumann Walter D.
Published: June 2010

Any graph-manifold can be obtained by plumbing according to some plumbing graph

Γ
\Gamma

. A calculus for plumbing which includes normal forms for such graphs is developed. This is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves. For instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity.
Geodesic rigidity in compact nonpositively curved manifolds
Authors: Eberlein Patrick
Published: June 2010

Our goal is to find geometric properties that are shared by homotopically equivalent compact Riemannian manifolds of sectional curvature

K
⩽
0

K \leqslant 0

. In this paper we consider mainly properties of free homotopy classes of closed curves. Each free homotopy class is represented by at least one smooth periodic geodesic, and the nonpositive curvature condition implies that any two periodic geodesic representatives are connected by a flat totally geodesic homotopy of periodic geodesic representatives. By imposing certain geometric conditions on these periodic geodesic representatives we define and study three types of free homotopy classes: Clifford, bounded and rank

1
1

. Let

M
M

,

M
′

M\prime

denote compact Riemannian manifolds with

K
⩽
0

K \leqslant 0

, and let

θ
:

π
1

(
M
,

m
)
→

π
1

(
M
′
,

m
′
)

\theta :{\pi _1}(M,\,m) \to {\pi _1}(M\prime ,\,m\prime )

be an isomorphism. Let

θ
\theta

also denote the induced bijection on free homotopy classes. Theorem A. The free homotopy class

[
α
]

[\alpha ]

in

M
M

is, respectively, Clifford, bounded or rank

1
1

if and only if the class

θ
[
α
]

\theta [\alpha ]

in

M
′

M\prime

is of the same type. Theorem B. If

M
M

,

M
′

M\prime

have dimension

3
3

and do not have a rank

1
1

free homotopy class then they have diffeomorphic finite covers of the form

S
1

×

M
2

{S^1} \times {M^2}

. The proofs of Theorems A and B use the fact that

θ
\theta

is induced by a homotopy equivalence

f
:
(
M
,

m
)
→
(
M
′
,

m
′
)

f:(M,\,m) \to (M\prime ,\,m\prime )

. Theorem C. The manifold

M
M

satisfies the Visibility axiom if and only if

M
′

M\prime

satisfies the Visibility axiom.
On (𝐾_{∗}(𝑍/𝑝²𝑍) and related homology groups
Authors: Evens Leonard , Friedlander Eric M.
Published: June 2010

It is shown that, for

p
⩾
5

p \geqslant 5

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R equals bold upper Z slash p squared bold upper Z comma upper K 3 left-parenthesis upper R right-parenthesis equals bold upper Z slash p squared bold upper Z plus bold upper Z slash left-parenthesis p squared minus 1 right-parenthesis bold upper Z">

R
=

Z

/

p
2

Z

,

K
3

(
R
)
=

Z

/

p
2

Z

+

Z

/

(

p
2

−
1
)

Z

R = {\mathbf {Z}} / {p^2}{\mathbf {Z}},\,{K_3}(R) = {\mathbf {Z}} / {p^2}{\mathbf {Z}} + {\mathbf {Z}} / ({p^2} - 1){\mathbf {Z}}

\]
and

K
4

(
R
)
=
0

{K_4}(R) = 0

. Similar calculations are made for

R
R

the ring of dual numbers over

Z

/

p

Z

{\mathbf {Z}} / p{\mathbf {Z}}

. The calculation reduces to finding homology groups of

Sl
⁡
(
R
)

\operatorname {Sl} (R)

. A key tool is the spectral sequence of the group extension of

Sl
⁡
(
n
,

p
2

)

\operatorname {Sl} (n,\,{p^2})

over

Sl
⁡
(
n
,

p
)

\operatorname {Sl} (n,\,p)

. The terms of this spectral sequence depend in turn on the homology of

Gl
⁡
(
n
,

p
)

\operatorname {Gl} (n,\,p)

with coefficients various multilinear modules. Calculation of the differentials uses the Charlap-Vasquez description of

d
2

{d^2}

.
The group of automorphisms of a class of finite 𝑝-groups
Authors: Juhász Arye
Published: June 2010

Let

G
G

be a finite

p
p

-group and denote by

K
i

(
G
)

{K_i}(G)

the members of the lower central series of

G
G

. We call

G
G

of type

(
m
,

n
)

(m,\,n)

if (a)

G
G

has nilpotency class

m
−
1

m - 1

, (b)

G

/

K
2

(
G
)
≅

Z

p
n

×

Z

p
n

G/{K_2}(G) \cong {{\mathbf {Z}}_{{p^n}}} \times {{\mathbf {Z}}_{{p^n}}}

and

K
i

(
G
)

/

K

i
+
1

(
G
)
≅

Z

p
n

{K_i}(G)/{K_{i + 1}}(G) \cong {{\mathbf {Z}}_{{p^n}}}

for every

i
i

,

2
⩽
i
⩽
n
−
1

2 \leqslant i \leqslant n - 1

. In this work we describe the structure of

Aut
⁡
(
G
)

\operatorname {Aut} (G)

and certain relations between

Out
⁡
(
G
)

\operatorname {Out} (G)

and

G
G

.
Spectral properties of orthogonal polynomials on unbounded sets
Authors: Chihara T. S.
Published: June 2010

We consider orthogonal polynomials when the three term recurrence formula for the monic polynomials has unbounded coefficients. We obtain information relative to three questions: Under what conditions on the coefficients will the derived set of the spectrum have a finite infimum

σ
\sigma

? If

σ
\sigma

is finite, when will there be at most finitely many spectral points smaller than

σ
\sigma

; and when will the distribution function be continuous at

σ
\sigma

?
Topological invariant means on the von Neumann algebra 𝑉𝑁(𝐺)
Authors: Chou Ching
Published: June 2010

Let

V
N
(
G
)

VN(G)

be the von Neumann algebra generated by the left regular representation of a locally compact group

G
G

,

A
(
G
)

A(G)

the Fourier algebra of

G
G

and

T
I
M
(

G
^

)

TIM(\hat G)

the set of topological invariant means on

V
N
(
G
)

VN(G)

. Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F 1 equals left-brace script upper O element-of left-parenthesis l Superscript normal infinity Baseline right-parenthesis Superscript asterisk Baseline right-brace colon script upper O greater-than-or-slanted-equals 0 comma StartAbsoluteValue EndAbsoluteValue script upper O StartAbsoluteValue EndAbsoluteValue equals 1">

F

1

=
{

O

∈

(

l
∞

)
∗

}
:

O

⩾
0
,

|

|

O

|

|

=
1

{\mathcal {F}_1} = \{ \mathcal {O} \in {({l^\infty })^ \ast }\} :\mathcal {O} \geqslant 0,\;||\mathcal {O}|| = 1

and

O

(
f
)
=
0

if

f
∈

l
∞

\mathcal {O}(f) = 0\;{\text {if}}\;f \in {l^\infty }

and

f
(
n
)
→
0
}

f(n) \to 0\}

. We show that if

G
G

is nondiscrete then there exists a linear isometry

Λ
\Lambda

of

(

l
∞

)
∗

{({l^\infty })^ \ast }

into

V
N

(
G

)
∗

VN{(G)^ \ast }

such that

Λ
(

F

1

)
⊂
T
I
M
(

G
^

)

\Lambda ({\mathcal {F}_1}) \subset TIM(\hat G)

. When

G
G

is further assumed to be second countable then

F

1

{\mathcal {F}_1}

can be embedded into some predescribed subsets of

T
I
M
(

G
^

)

TIM(\hat G)

. To prove these embedding theorems for second countable groups we need the existence of a sequence of means

{

u
n

}

\{ {u_n}\}

in

A
(
G
)

A(G)

such that their supports in

V
N
(
G
)

VN(G)

are mutually orthogonal and

|

|

u

u
n

−

u
n

|

|

→
0

if

u

||u{u_n} - {u_n}|| \to 0\;{\text {if}}\;u

is a mean in

A
(
G
)

A(G)

. Let

F
(

G
^

)

F(\hat G)

be the space of all

T
∈
V
N
(
G
)

T \in VN(G)

such that

m
(
T
)

m(T)

is a constant as

m
m

runs through

T
I
M
(

G
^

)

TIM(\hat G)

and let

W
(

G
^

)

W(\hat G)

be the space of weakly almost periodic elements in

V
N
(
G
)

VN(G)

. We show that the following conditions are equivalent: (i)

G
G

is discrete, (ii)

F
(

G
^

)

F(\hat G)

is an algebra and (iii)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A left-parenthesis upper G right-parenthesis dot upper V upper N left-parenthesis upper G right-parenthesis right-parenthesis intersection upper F left-parenthesis ModifyingAbove upper G With caret right-parenthesis subset-of upper W left-parenthesis ModifyingAbove upper G With caret right-parenthesis">

(
A
(
G
)
⋅
V
N
(
G
)
)
∩
F
(

G
^

)
⊂
W
(

G
^

)

(A(G) \cdot VN(G)) \cap F(\hat G) \subset W(\hat G)

.
Multiplicative properties of power maps. II
Authors: McGibbon C. A.
Published: June 2010

The notions of

A
n

A_n

-maps and

C
n

C_n

-forms can be regarded as crude approximations to the concepts of homomorphisms and commutativity, respectively. These approximations are used to study power maps on connected Lie groups and their localizations. For such groups the power map

x
↦

x
n

x \mapsto {x^n}

is known to be an

A
2

A_2

-map if and only if

n
n

is a solution to a certain quadratic congruence. In this paper,

A
3

A_3

-power maps are studied. For the Lie group Sp(l) it is shown that the

A
3

A_3

-powers coincide with solutions which are common to the quadratic congruence, mentioned earlier, and another cubic congruence. Other Lie groups, when localized so as to become homotopy commutative, are also shown to have infinitely many

A
3

A_3

-powers. The proofs reflect the combinatorial nature of the obstructions involved.
Independent families in complete Boolean algebras
Authors: Balcar B. , Franěk F.
Published: June 2010

We present a proof (without any set-theoretical assumptions) that every infinite complete Boolean algebra includes a free subalgebra of the same cardinality. It follows that the set of all ultrafilters on an infinite complete Boolean algebra

B
B

has power

2

|

B

|

2^{|B|}

.
Polynomials on affine manifolds
Authors: Fried David
Published: June 2010

For a closed affine manifold

M
M

of dimension

m
m

the developing map defines an open subset

D
(

M
~

)
⊂

R

m

D(\tilde M) \subset {{\mathbf {R}}^m}

. We show that

D
(

M
~

)

D(\tilde M)

cannot lie between parallel hyperplanes. When

m
≤
3

m \le 3

we show that any nonconstant polynomial

p
:

R

m

→

R

p:{{\mathbf {R}}^m} \to {\mathbf {R}}

is unbounded on

D
(

M
~

)

D(\tilde M)

. If

D
(

M
~

)

D(\tilde M)

lies in a half-space we show

M
M

has zero Euler characteristic. Under various special conditions on

M
M

we show that

M
M

has no nonconstant functions given by polynomials in affine coordinates.
Locally strange hyperbolic sets
Authors: Jones Lowell
Published: June 2010

The purpose of this paper is to present a very general method of constructing basic sets having complicated local homeomorphism types.
Classical solutions of the Hamilton-Jacobi-Bellman equation for uniformly elliptic operators
Authors: Evans Lawrence C.
Published: June 2010

We prove under appropriate hypotheses that the Hamilton-JacobiBellman dynamic programming equation with uniformly elliptic operators,

max

1
⩽
k
⩽
m

{

L
k

u
−

f
k

}
=
0

{\max _{1 \leqslant k \leqslant m}}\{{L^k}u - {f^k}\} = 0

, has a classical solution

u
∈

C

2
,
β

u \in {C^{2,\beta }}

, for some (small) Hölder exponent

β
>
0

\beta > 0

.
The spectrum of a Riemannian manifold with a unit Killing vector field
Authors: Bleecker David D.
Published: June 2010

Let

(
P
,
g
)

(P,g)

be a compact, connected,

C
∞

{C^\infty }

Riemannian

(
n
+
1
)

(n + 1)

-manifold

(
n
⩾
1
)

(n \geqslant 1)

with a unit Killing vector field with dual

1
1

-form

η
\eta

. For

t
>
0

t > 0

, let

g

t

=

t

−
1

g
+
(

t

n

−

t

−
1

)
η
⊗
η

{g_{t}} = {t^{ - 1}}g + (t^{n}-t^{-1})\eta \otimes \eta

, a family of metrics of fixed volume element on

P
P

. Let

λ
1

(
t
)

{\lambda _1}(t)

be the first nonzero eigenvalue of the Laplace operator on

C
∞

(
P
)

{C^\infty }(P)

of the metric

g
t

{g_t}

. We prove that if

d
η

d\eta

is nowhere zero, then

λ
1

(
t
)
→
∞

{\lambda _1}(t) \to \infty

as

t
→
∞

t \to \infty

. Using this construction, we find that, for every dimension greater than two, there are infinitely many topologically distinct compact manifolds for which

λ
1

{\lambda _1}

is unbounded on the space of fixed-volume metrics.
Branched coverings. II
Authors: Stong R. E.
Published: June 2010

This paper improves the analysis of the possible cobordism classes

[
M
]
−
(
deg

ϕ
)
[
N
]

[M] - (\deg \;\phi )[N]

for

ϕ
:
M
→
N

\phi : M \to N

a smooth branched covering of closed oriented smooth manifolds. It is assumed that the branch set is a codimension

2
2

submanifold.
A correction and some additions to: “Fundamental solutions for differential equations associated with the number operator”
Authors: Lee Yuh Jia
Published: June 2010

Let

(
H
,
B
)

(H,B)

be an abstract Wiener pair and

N

\mathfrak {N}

the operator defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper N u left-parenthesis x right-parenthesis equals minus trace Subscript upper H Baseline upper D squared u left-parenthesis x right-parenthesis plus left-parenthesis x comma upper D u left-parenthesis x right-parenthesis right-parenthesis">

N

u
(
x
)
=
−

trace

H

D
2

u
(
x
)
+
(
x
,
D
u
(
x
)
)

\mathfrak {N}u(x) = - {\text {trace}}_H{D^2}u(x) + (x,Du(x))

, where

x
∈
B

x \in B

and

(
⋅
,
⋅
)

(\cdot , \cdot )

denotes the

B
-

B

∗

B\text {-}B^{\ast }

pairing. In this paper, we point out a mistake in the previous paper concerning the existence of fundamental solutions of

N

k

{\mathfrak {N}^k}

and intend to make a correction. For this purpose, we study the fundamental solution of the operator

(

N

+
λ
I

)
k

(
λ
>
0
)

{(\mathfrak {N} + \lambda I)^k}\,(\lambda > 0)

and investigate its behavior as

λ
→
0

\lambda \to 0

. We show that there exists a family

{

Q
λ

(
x
,
d
y
)
}

\{{Q_\lambda }(x,dy)\}

of measures which serves as the fundamental solution of

(

N

+
λ
I

)
k

{(\mathfrak {N} + \lambda I)^k}

and, for a suitable function

f
f

, we prove that the solution of

N

k

u
=
f

{\mathfrak {N}^k}u = f

can be represented by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis x right-parenthesis equals limit Underscript lamda right-arrow 0 Endscripts integral Underscript upper B Endscripts f left-parenthesis y right-parenthesis upper Q Subscript lamda Baseline left-parenthesis x comma d y right-parenthesis plus upper C">

u
(
x
)
=

lim

λ
→
0

∫
B

f
(
y
)

Q
λ

(
x
,
d
y
)
+
C

u(x) = {\lim _{\lambda \to 0}}\int _B f(y){Q_\lambda }(x,dy) + C

, where

C
C

is a constant.
Are primitive words universal for infinite symmetric groups?
Authors: Silberger D. M.
Published: June 2010

Let

W
=
W
(

x
1

,
…
,

x
j

)

W = W({x_1}, \ldots ,{x_j})

be any word in the

j
j

free generators

x
1

,
…
,

x
j

{x_1}, \ldots ,{x_j}

, and suppose that

W
W

cannot be expressed in the form

W
=

V
k

W = {V^k}

for

V
V

a word and for

k
k

an integer with

|
k
|

≠
1

\left | k \right | \ne 1

. We ask whether the equation

f
=
W

f = W

has a solution

(

x
1

,
…
,

x
j

)
=
(

a

1

,
…
,

a

j

)
∈

G

j

({x_1}, \ldots ,{x_j}) = (a_{1}, \ldots , a_{j}) \in G^{j}

whenever

G
G

is an infinite symmetric group and

f
f

is an element in

G
G

. We establish an affirmative answer in the case that

W
(
x
,
y
)
=

x
m

y
n

W(x,y) = {x^m}{y^n}

for

m
m

and

n
n

nonzero integers.
Nonfactorization theorems in weighted Bergman and Hardy spaces on the unit ball of 𝐶ⁿ (𝑛>1)
Authors: Gowda M. Seetharama
Published: June 2010

Let

A

p
,
α

(
B
)
,

A

q
,
α

(
B
)

{A^{p,\alpha }}(B),{A^{q,\alpha }}(B)

and

A

l
,
α

(
B
)

{A^{l,\alpha }}(B)

be weighted Bergman spaces on the unit ball of

C

n

(
n
>
1
)

{{\text {C}}^{n}}\,(n > 1)

. We prove: Theorem 1. If

1

/

l
=
1

/

p
+
1

/

q

1/l = 1/p + 1/q

then

A

p
,
α

(
B
)
⋅

A

q
,
α

(
B
)

{A^{p,\alpha }}(B) \cdot {A^{q,\alpha }}(B)

is of first category in

A

l
,
α

(
B
)

{A^{l,\alpha }}(B)

. Theorem 2. Theorem 1 holds for Hardy spaces in place of weighted Bergman spaces. We also show that Theorems 1 and 2 hold for the polydisc

U
n

{U^n}

in place of

B
B

.
Systems of fixed point sets
Authors: Elmendorf A. D.
Published: June 2010

Let

G
G

be a compact Lie group. A canonical method is given for constructing a

G
G

-space from homotopy theoretic information about its fixed point sets. The construction is a special case of the categorical bar construction. Applications include easy constructions of certain classifying spaces, as well as

G
G

-Eilenberg-Mac Lane spaces and Postnikov towers.
Linear superpositions with mappings which lower dimension
Authors: Sternfeld Y.
Published: June 2010

It is shown that for every

n
n

-dimensional compact metric space

X
X

, there exist

2
n
+
1

2n + 1

functions

{

φ
j

}

j
=
1

2
n
+
1

\{ {\varphi _j}\}_{j = 1}^{2n + 1}

in

C
(
X
)

C(X)

and

n
n

mappings

{

ψ
i

}

i
=
1

n

\{ {\psi _i}\}_{i = 1}^n

on

X
X

with

1
1

-dimensional range each, with the following property: for every

0
⩽
k
⩽
n

0 \leqslant k \leqslant n

, every

k
k

tuple

{

ψ

i
l

}

l
=
1

k

\{ {\psi _{i_l}}\}_{l = 1}^k

of the

ψ
i

{\psi _i}

’s and every

2
(
n
−
k
)
+
1

2(n - k) + 1

tuple

{

φ

j
m

}

m
=
1

2
(
n
−
k
)
+
1

\{ {\varphi _{{j_m}}}\}_{m = 1}^{2(n - k) + 1}

of the

φ
j

{\varphi _j}

’s, each

f
∈
C
(
X
)

f \in C(X)

can be represented as
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals normal upper Sigma Subscript l equals 1 Superscript k Baseline g Subscript l Baseline left-parenthesis psi Subscript i Sub Subscript l Subscript Baseline left-parenthesis x right-parenthesis right-parenthesis plus normal upper Sigma Subscript m equals 1 Superscript 2 left-parenthesis n minus k right-parenthesis plus 1 Baseline h Subscript m Baseline left-parenthesis phi Subscript j Sub Subscript m Subscript Baseline left-parenthesis x right-parenthesis right-parenthesis">

f
(
x
)
=

Σ

l
=
1

k

g
l

(

ψ

i
l

(
x
)
)
+

Σ

m
=
1

2
(
n
−
k
)
+
1

h
m

(

φ

j
m

(
x
)
)

f(x) = \Sigma _{l = 1}^k{g_l}({\psi _{{i_l}}}(x)) + \Sigma _{m = 1}^{2(n - k) + 1}{h_m}({\varphi _{{j_m}}}(x))

, with

g
l

∈
C
(

ψ

i
l

(
X
)
)

{g_l} \in C({\psi _{{i_l}}}(X))

and

h
m

∈
C
(
R
)

{h_m} \in C(R)

. It is also shown that in many cases the number

2
(
n
−
k
)
+
1

2(n - k) + 1

is the smallest possible.
Independence results on the global structure of the Turing degrees
Authors: Groszek Marcia J. , Slaman Theodore A.
Published: June 2010

From

CON(ZFC)
\text {CON(ZFC)}

we obtain: 1.

CON
(
ZFC
+

2
ω

\text {CON}(\text {ZFC} + 2^\omega

is arbitrarily large

+
+

there is a locally finite upper semilattice of size

ω
2

{\omega _2}

which cannot be embedded into the Turing degrees as an upper semilattice). 2.

CON
(
ZFC
+

2
ω

\text {CON}(\text {ZFC} + 2^\omega

is arbitrarily large

+
+

there is a maximal independent set of Turing degrees of size

ω
1

{\omega _1}

).
Semilinear parabolic problems define semiflows on 𝐶^{𝑘} spaces
Authors: Mora Xavier
Published: June 2010

Linear parabolic problems of a general class are proved to determine analytic semigroups on certain closed subspaces of

C
k

(

Ω
¯

)

{C^k}(\overline \Omega )

(

k
k

integer);

C
k

(

Ω
¯

)

{C^k}(\overline \Omega )

denotes the space of functions whose derivatives or order

⩽
k

\leqslant k

are bounded and uniformly continuous, with the usual supremum norm; the closed subspaces where the semigroups are obtained, denoted by

C
^

k

(

Ω
¯

)

{\hat C^k}(\overline \Omega )

, are determined by the boundary conditions and a possible condition at infinity. One also obtains certain embedding relations concerning the fractional power spaces associated to these semigroups. Usually, results of this type are based upon the theory of solution of elliptic problems, while this work uses the corresponding theory for parabolic problems. The preceding results are applied to show that certain semilinear parabolic problems define semiflows on spaces of the type

C
^

k

(

Ω
¯

)

{\hat C^k}(\overline \Omega )

.
Persistently finite theories with hyperarithmetic models
Authors: Millar Terrence
Published: June 2010

Nerode asked if there could be a complete decidable theory with only finitely many countable models up to isomorphism, such that not all of the countable models were decidable. Morley, Lachlan, and Peretyatkin produced examples of such theories. However, all the countable models of those theories were decidable in

0
′

0’

. The question then arose whether all countable models of such theories had to be, for example, arithmetic. In this paper we provide a negative answer to that question by showing that there are such examples with countable models of arbitrarily high hyperarithmetic degree. It is not difficult to show that any countable model of a hyperarithmetic theory which has only finitely many countable models must be decidable in some hyperarithmetic degree.
Homology cobordisms, link concordances, and hyperbolic 3-manifolds
Authors: Myers Robert
Published: June 2010

Let

M
0
3

M_0^3

and

M
1
3

M_1^3

be compact, oriented

3
3

-manifolds. They are homology cobordant (respectively relative homology cobordant) if

∂

M
1
3

=
∅

(

resp.

∂

M
1
3

≠
∅
)

\partial M_1^3 = \emptyset \;({\text {resp.}}\;\partial M_1^3 \ne \emptyset )

and there is a smooth, compact oriented

4
4

-manifold

W
4

{W^4}

such that

∂

W
4

=

M
0
3

−

M
1
3

\partial {W^4} = M_0^3 - M_1^3

(resp.
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential upper W Superscript 4 Baseline equals upper M 0 cubed minus upper M 1 cubed right-parenthesis union left-parenthesis upper M Subscript i Superscript 3 Baseline times left-bracket 0 comma 1 right-bracket right-parenthesis">

∂

W
4

=

M
0
3

−

M
1
3

)
∪
(

M
i
3

×
[
0
,
1
]
)

\partial {W^4} = M_0^3 - M_1^3) \cup (M_i^3 \times [0,1])

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript asterisk Baseline left-parenthesis upper M Subscript i Superscript 3 Baseline semicolon bold upper Z right-parenthesis right-arrow upper H Subscript asterisk Baseline left-parenthesis upper W Superscript 4 Baseline semicolon bold upper Z right-parenthesis">

H

∗

(

M
i
3

;

Z

)
→

H

∗

(

W
4

;

Z

)

{H_{\ast }}(M_i^3;{\mathbf {Z}}) \to {H_{\ast }}({W^4};{\mathbf {Z}})

are isomorphisms,

i
=
0
,
1

i = 0,1

. Theorem. Every closed, oriented

3
3

-manifold is homology cobordant to a hyperbolic

3
3

-manifold. Theorem. Every compact, oriented

3
3

-manifold whose boundary is nonempty and contains no

2
2

-spheres is relative homology cobordant to a hyperbolic

3
3

-manifold. Two oriented links

L
0

{L_0}

and

L
1

{L_1}

in a

3
3

-manifold

M
3

{M^3}

are concordant if there is a set

A
2

{A^2}

of smooth, disjoint, oriented annuli in

M
×
[
0
,
1
]

M \times [0,1]

such that

∂

A
2

=

L
0

−

L
1

\partial {A^2} = {L_0} - {L_1}

, where

L

i

⊆

M
3

×
{
i
}
,
i
=
0
,
1

{L_{i}} \subseteq \;{M^3} \times \{ i\} ,i = 0,1

. Theorem. Every link in a compact, oriented

3
3

-manifold

M
3

{M^3}

whose boundary contains no

2
2

-spheres is concordant to a link whose exterior is hyperbolic. Corollary. Every knot in

S
3

{S^3}

is concordant to a knot whose exterior is hyperbolic.
Sign changes in harmonic analysis on reductive groups
Authors: Kottwitz Robert E.
Published: June 2010

Let

G
G

be a connected reductive group over a field

F
F

. In this note the author constructs an element

e
(
G
)

e(G)

of the Brauer group of

F
F

. The square of this element is trivial. For a local field,

e
(
G
)

e(G)

may be regarded as an element of

{
±
1
}

\{ \pm 1\}

and is needed for harmonic analysis on reductive groups over that field. For a global field there is a product formula.
𝐼^{𝑋}, the hyperspace of fuzzy sets, a natural nontopological fuzzy topological space
Authors: Lowen R.
Published: June 2010

Let

X
X

be a uniform topological space, then on the family

I
X

{I^X}

(resp.

Φ
(
X
)

\Phi (X)

) of all nonzero functions (resp. nonzero uppersemicontinuous functions) from

X
X

to the unit interval

I
I

, a fuzzy uniform topology is constructed such that

2
X

{2^X}

(resp.

F

(
X
)

\mathcal {F}(X)

), the family of all nonvoid (resp. nonvoid closed) subsets of

X
X

equipped with the Hausdorff-Bourbaki structure is isomorphically injected in

I
X

{I^X}

(resp.

Φ
(
X
)

\Phi (X)

). The main result of this paper is a complete description of convergence in

I
X

{I^X}

, by means of a notion of degree of incidence of members of

I
X

{I^X}

. Immediate consequences are that first it can be shown that this notion of convergence refines some particular useful notions of convergence of fuzzy sets used in applications, and that second it follows from its construction and properties that for each ordinary uniform topological space

X
X

there exists a natural nontopological fuzzy uniform topology on

I
X

{I^X}

.
The divisor classes of the hypersurface 𝑧^{𝑝^{𝑚}}=𝐺(𝑥₁,⋯,𝑥_{𝑛}) in characteristic 𝑝>0
Authors: Lang Jeffrey
Published: June 2010

In this article we use P. Samuel’s purely inseparable descent techniques to study the divisor class groups of normal affine hypersurfaces of the form

z
p

=
G
(

x
1

,
…
,

x
n

)

{z^p} = G({x_1},\ldots ,{x_n})

and develop an inductive procedure for studying those of the form

z

p
m

=
G

{z^{p^m}} = G

. We obtain results concerning the order and type of these groups and apply this theory to some specific examples.
Stable complete constant mean curvature surfaces in 𝑅³ and 𝐻³
Authors: Mori Hiroshi
Published: June 2010

We construct some

1
1

-parameter families of complete rotation surfaces with constant mean curvature in the hyperbolic

3
3

-space

H
3

{H^3}

of constant sectional curvature

−
1

-1

, and show that some of them are stable for the variational problem of area together with oriented volume, and that a complete connected, oriented surface with constant mean curvature in the Euclidean

3
3

-space

R
3

{R^3}

which is stable for the variational problem is a plane.
On first countable, countably compact spaces. I. (𝜔₁,𝜔*₁)-gaps
Authors: Nyikos Peter J. , Vaughan Jerry E.
Published: June 2010

This paper is concerned with the

(

ω
1

,

ω
1

∗

)

({\omega _1},\omega _1^{\ast })

-gaps of F. Hausdorff and the topological spaces defined from them by Eric van Douwen. We construct special gaps in order that the associated gap spaces will have interesting topological properties. For example, the gap spaces we construct show that in certain models of set theory, there exist countably compact, first countable, separable, nonnormal

T
2

{T_2}

-spaces.
Global solvability on two-step compact nilmanifolds
Authors: Cygan Jacek M. , Richardson Leonard F.
Published: June 2010

We apply the methods of representation theory of nilpotent Lie groups to study the convergence of Fourier series of smooth global solutions to first order invariant partial differential equations

D
f
=
g

Df = g

in

C
∞

{C^\infty }

of a two-step compact nilmanifold. We show that, under algebraically well-defined conditions on

D
D

in the complexified Lie algebra, smooth infinite-dimensional irreducible solutions, when they exist, satisfy estimates strong enough to guarantee uniform convergence of the irreducible (or primary) Fourier series to a smooth global solution. Such strong estimates are not possible on multidimensional tori.
Quotients by 𝐶* and 𝑆𝐿(2,𝐶) actions
Authors: Białynicki-Birula Andrzej , Sommese Andrew John
Published: June 2010

Let

ρ
:

C

∗

×
X
→
X

\rho :{{\mathbf {C}}^{\ast }} \times X \to X

be a meromorphic action of

C

∗

{{\mathbf {C}}^{\ast }}

on a compact normal analytic space. We completely classify

C

∗

{{\mathbf {C}}^{\ast }}

-invariant open

U
⊆
X

U \subseteq X

with a compact analytic space

U

/

T

U/T

as a geometric quotient for a wide variety of actions, including all algebraic actions. As one application, we settle affirmatively a conjecture of

D
\text {D}

. Mumford on compact geometric quotients by

SL(2

,

C

)

{\text {SL(2}},{\mathbf {C}})

of Zariski open sets of

(

P

C

1

)
n

{({\mathbf {P}}_{\mathbf {C}}^1)^n}

.
Toeplitz operators on bounded symmetric domains
Authors: Upmeier Harald
Published: June 2010

In this paper Jordan algebraic methods are applied to study Toeplitz operators on the Hardy space

H
2

(
S
)

{H^2}(S)

associated with the Shilov boundary

S
S

of a bounded symmetric domain

D
D

in

C

n

{{\mathbf {C}}^n}

of arbitrary rank. The Jordan triple system

Z
≈

C

n

Z \approx {{\mathbf {C}}^n}

associated with

D
D

is used to determine the relationship between Toeplitz operators and differential operators. Further, it is shown that each Jordan triple idempotent

e
∈
Z

e \in Z

induces an irreducible representation ("

e
e

-symbol") of the

C

∗

{C^{\ast } }

-algebra

T

\mathcal {T}

generated by all Toeplitz operators

T
f

{T_f}

with continuous symbol function

f
f

.
Global invariants for measured foliations
Authors: Hurder Steven
Published: June 2010

New exotic invariants for measured foliations are constructed, which we call the

μ
\mu

-classes of a pair

(

F

,
μ
)

(\mathcal {F},\mu )

. The dependence of the

μ
\mu

-classes on the geometry of the foliation

F

\mathcal {F}

is examined, and the dynamics of a foliation is shown to determine the

μ
\mu

-classes in many cases. We use the

μ
\mu

-classes to study the classifying space

B

Γ

S

L
q

B{\Gamma _{S{L_q}}}

of foliations with a transverse invariant volume form, and we show the homotopy groups of

B

Γ

S

L
q

B{\Gamma _{S{L_q}}}

are uncountably generated starting in degrees

q
+
3

q + 3

. New invariants for groups of volume preserving diffeomorphisms also arise from the

μ
\mu

-classes; these invariants are nontrivial and related to the geometric aspects of the group action. Relations between the

μ
\mu

-classes and the secondary classes of a foliation are exhibited.
A topological group having no homeomorphisms other than translations
Authors: van Mill Jan
Published: June 2010

We give an example of a (separable metric) connected and locally connected topological group, the only autohomeomorphisms of which are group translations.
Multi-invariant sets on tori
Authors: Berend Daniel
Published: June 2010

Given a compact metric group

G
G

, we are interested in those semigroups

Σ
\Sigma

of continuous endomorphisms of

G
G

, possessing the following property: The only infinite, closed,

Σ
\Sigma

-invariant subset of

G
G

is

G
G

itself. Generalizing a one-dimensional result of Furstenberg, we give here a full characterization—for the case of finitedimensional tori—of those commutative semigroups with the aforementioned property.
Real vs. complex rational Chebyshev approximation on an interval
Authors: Trefethen Lloyd N. , Gutknecht Martin H.
Published: June 2010

If

f
∈
C
[
−
1
,
1
]

f \in C[ - 1,1]

is real-valued, let

E

r

(
f
)

{E^{r}(f)}

and

E

c

(
f
)

{E^{c}(f)}

be the errors in best approximation to

f
f

in the supremum norm by rational functions of type

(
m
,
n
)

(m,n)

with real and complex coefficients, respectively. It has recently been observed that

E
c

(
f
)
>

E
r

(
f
)

{E^c}(f) > {E^r}(f)

can occur for any

n
⩾
1

n \geqslant 1

, but for no

n
⩾
1

n \geqslant 1

is it known whether

γ

m
n

=

inf
f

E
c

(
f
)

/

E

r

(
f
)

{\gamma _{mn}} = \inf _f\,{E^c}(f)/{E^{r}(f)}

is zero or strictly positive. Here we show that both are possible:

γ

01

>
0

{\gamma _{01}} > 0

, but

γ

m
n

=
0

{\gamma _{mn}} = 0

for

n
⩾
m
+
3

n \geqslant m + 3

. Related results are obtained for approximation on regions in the plane.
Semidirect products and reduction in mechanics
Authors: Marsden Jerrold E. , Raţiu Tudor , Weinstein Alan
Published: June 2010

This paper shows how to reduce a Hamiltonian system on the cotangent bundle of a Lie group to a Hamiltonian system in the dual of the Lie algebra of a semidirect product. The procedure simplifies, unifies, and extends work of Greene, Guillemin, Holm, Holmes, Kupershmidt, Marsden, Morrison, Ratiu, Sternberg and others. The heavy top, compressible fluids, magnetohydrodynamics, elasticity, the Maxwell-Vlasov equations and multifluid plasmas are presented as examples. Starting with Lagrangian variables, our method explains in a direct way why semidirect products occur so frequently in examples. It also provides a framework for the systematic introduction of Clebsch, or canonical, variables.
𝑀-structure in the Banach algebra of operators on 𝐶₀(Ω)
Authors: Flinn P. H. , Smith R. R.
Published: June 2010

The

M
M

-ideals in

B
(

C
0

(
Ω
)
)

B({C_0}(\Omega ))

, the space of continuous linear operators on

C
0

(
Ω
)

{C_0}(\Omega )

, are determined where

Ω
\Omega

is a locally compact Hausdorff countably paracompact space. A one-to-one correspondence between

M
M

-ideals in

B
(

C
0

(
Ω
)
)

B({C_0}(\Omega ))

, open subsets of the Stone-Čech compactification of

Ω
\Omega

, and lower semicontinuous Hermitian projections in

B

(

C
0

(
Ω
)

)

∗
∗

B{({C_0}(\Omega ))^{\ast \ast }}

is established.
A two weight weak type inequality for fractional integrals
Authors: Sawyer Eric
Published: June 2010

For

1
>
p
⩽
q
>
∞
,
0
>
α
>
n

1 > p \leqslant q > \infty ,0 > \alpha > n

and

w
(
x
)
,
υ
(
x
)

w(x),\upsilon (x)

nonnegative weight functions on

R
n

{R^n}

we show that the weak type inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript left-brace upper T Subscript alpha Baseline f greater-than lamda right-brace Endscripts w left-parenthesis x right-parenthesis d x less-than-or-slanted-equals upper A lamda Superscript negative q Baseline left-parenthesis integral StartAbsoluteValue f left-parenthesis x right-parenthesis EndAbsoluteValue Superscript p Baseline upsilon left-parenthesis x right-parenthesis d x right-parenthesis Superscript q slash p">

∫

{

T
α

f
>
λ
}

w
(
x
)

d
x
⩽
A

λ

−
q

(
∫

|

f
(
x
)

|

p

υ
(
x
)

d
x
)

q

/

p

\int _{\{ {T_\alpha }f > \lambda \} }\,w(x)\;dx \leqslant A{\lambda ^{ - q}}{\left ( \int |f(x){|^p}\;\upsilon (x)\;dx \right )^{q/p}}

\]
holds for all

f
⩾
0

f \geqslant 0

if and only if
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper Q Endscripts left-bracket upper T Subscript alpha Baseline left-parenthesis chi Subscript upper Q Baseline w right-parenthesis left-parenthesis x right-parenthesis right-bracket Superscript p prime Baseline upsilon left-parenthesis x right-parenthesis Superscript 1 minus p prime Baseline d x less-than-or-slanted-equals upper B left-parenthesis integral Underscript upper Q Endscripts w right-parenthesis Superscript p prime slash q prime Baseline greater-than normal infinity">

∫
Q

[

T
α

(

χ
Q

w
)

(
x
)

]

p
′

υ
(
x

)

1
−

p
′

d
x
⩽
B

(

∫
Q

w
)

p
′

/

q
′

>
∞

\int _Q\,[{T_\alpha }({\chi _Q}w)\,(x)]^{p’}\upsilon (x)^{1 - p’}\,dx \leqslant B\left ( \int _Qw \right )^{p’/q’} > \infty

\]
for all cubes

Q
Q

in

R
n

{R^n}

. Here

T
α

{T_\alpha }

denotes the fractional integral of order

α
,

T
α

f
(
x
)
=
∫

|

x
−
y

|

α
−
n

f
(
y
)

d
y

\alpha ,{T_\alpha }f(x) = \int |x - y{|^{\alpha - n}}f(y)\,dy

. More generally we can replace

T
α

{T_\alpha }

by any suitable convolution operator with radial kernel decreasing in

|

x

|

|x|

.
On the paths of symmetric stable processes
Authors: Davis Burgess
Published: June 2010

It is shown that if

X
(
t
)
,
t
⩾
0

X(t), t \geqslant 0

, is a symmetric stable process of index

α
,
0
>
α
>
2

\alpha , 0 > \alpha > 2

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript t Endscripts limit inf Underscript h down-arrow 0 Endscripts left-parenthesis upper X left-parenthesis t plus h right-parenthesis minus upper X left-parenthesis t right-parenthesis right-parenthesis h Superscript negative 1 slash alpha Baseline equals normal infinity">

sup
t

lim

inf

h
↓
0

(
X
(
t
+
h
)
−
X
(
t
)
)

h

−
1

/

α

=
∞

\sup _t \lim \inf _{h \downarrow 0} (X(t + h) - X(t))h^{-1/\alpha } = \infty

a.s. This settles a question of Fristedt about strictly stable subordinators.
On Block’s condition for simple periodic orbits of functions on an interval
Authors: Ho Chung-Wu
Published: June 2010

Recently, L. Block has shown that for any mapping

f
f

of an interval, whether

f
f

has a periodic point whose period contains an odd factor greater than

1
1

depends entirely on the periodic orbits of

f
f

whose periods are powers of

2
2

. In this paper the author shows that Block’s result is a special case of a more general phenomenon.
The generalized Zahorski class structure of symmetric derivatives
Authors: Larson Lee
Published: June 2010

A generalized Zahorski class structure is demonstrated for symmetric derivatives. A monotonicity theorem is proved and a condition sufficient to ensure that a symmetric derivative has the Darboux property is presented.
Noncommutative topological dynamics. II
Authors: Avitzour Daniel
Published: June 2010

This part deals with almost periodic and weakly mixing

C
∗

{C^ \ast }

-flows, and with disjointness and weak disjointness of

C
∗

{C^ \ast }

-flows (flows on

C
∗

{C^ \ast }

-algebras). The main result is a generalization to

C
∗

{C^ \ast }

-flows of Keynes and Robertson’s characterization of minimal weakly mixing flows. Examples are discussed exhibiting anomalous behaviour of disjointness in the

C
∗

{C^ \ast }

-flow case.
Knots prime on many strings
Authors: Bleiler Steven A.
Published: June 2010

A study is made of the factorization of prime knots into tangles. Several infinite families of knots which do not factor into prime tangles are examined, and a new characterization of knot primality is developed.
Stable viscosities and shock profiles for systems of conservation laws
Authors: Pego Robert L.
Published: June 2010

Wide classes of high order "viscosity" terms are determined, for which small amplitude shock wave solutions of a nonlinear hyperbolic system of conservation laws

u
t

+
f

(
u

)
x

=
0

{u_t} + f{(u)_x} = 0

are realized as limits of traveling wave solutions of a dissipative system
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline plus f left-parenthesis u right-parenthesis Subscript x Baseline equals nu left-parenthesis upper D 1 u Subscript x Baseline right-parenthesis Subscript x Baseline plus midline-horizontal-ellipsis plus nu Superscript n Baseline left-parenthesis upper D Subscript n Baseline u Superscript left-parenthesis n right-parenthesis Baseline right-parenthesis Subscript x">

u
t

+
f

(
u

)
x

=
ν

(

D
1

u
x

)
x

+
⋯
+

ν
n

(

D
n

u

(
n
)

)
x

{u_t} + f{(u)_x} = \nu {({D_1}{u_x})_x} + \cdots + {\nu ^n}{({D_n}{u^{(n)}})_x}

. The set of such "admissible" viscosities includes those for which the dissipative system satisfies a linearized stability condition previously investigated in the case

n
=
1

n = 1

by A. Majda and the author. When

n
=
1

n = 1

we also establish admissibility criteria for singular viscosity matrices

D
1

(
u
)

{D_1}(u)

, and apply our results to the compressible Navier-Stokes equations with viscosity and heat conduction, determining minimal conditions on the equation of state which ensure the existence of the "shock layer" for weak shocks.
An integral version of the Brown-Gitler spectrum
Authors: Shimamoto Don H.
Published: June 2010

In this paper, certain spectra

B
1

(
k
)

{B_1}(k)

are studied whose behavior qualifies them as being integral versions of the Brown-Gitler spectra

B
(
k
)

B(k)

. The bulk of our work emphasizes the similarities between

B
1

(
k
)

{B_1}(k)

and

B
(
k
)

B(k)

, shown mainly using the techniques of Brown and Gitler. In analyzing the homotopy type of

B
1

(
k
)

{B_1}(k)

, we provide a free resolution over the Steenrod algebra for its cohomology and study its Adams spectral sequence. We also list conditions which characterize it at the prime

2
2

. The paper begins, however, on a somewhat different topic, namely, the construction of a configuration space model for

Ω
2

(

S
3

⟨
3
⟩

)

{\Omega ^2}({S^3}\left \langle 3 \right \rangle )

and other related spaces.
On stable blocks of Auslander-algebras
Authors: Riedtmann Christine
Published: June 2010

The Auslander-algebra

E
Λ

{E_\Lambda }

of an algebra

Λ
\Lambda

of finite representation type is the endomorphism algebra of the direct sum

M
=
⊕

M
i

M = \oplus {M_i}

of one copy of each indecomposable

Λ
\Lambda

-module. A stable block of

E
Λ

{E_\Lambda }

is a connected direct factor of the residue algebra of

E
Λ

{E_\Lambda }

modulo the two-sided ideal generated by the projections of

M
M

to the

M
i

{M_i}

’s that are not stable under

D
T
r

DTr

. This paper describes the stable blocks whose quiver is a stable translation-quiver of class

A
n

{A_n}

or

D
n

{D_n}

.
The law of exponential decay for expanding transformations of the unit interval into itself
Authors: Jabłoński M.
Published: June 2010

Let

T
:
[
0
,
1
]
→
[
0
,
1
]

T:[0,1] \to [0,1]

be an expanding map of the unit interval and let

ξ
ε

(
x
)

{\xi _\varepsilon }(x)

be the smallest integer

n
n

for which

T
n

(
x
)
∈
[
0
,
ε
]

{T^n}(x) \in [0,\varepsilon ]

; that is, it is the random variable given by the formula
\[

ξ
ε

(
x
)
=
min
{
n
:

T
n

(
x
)
⩽
ε
}
.

{\xi _\varepsilon }(x) = \min \{ n:{T^n}\;(x) \leqslant \varepsilon \}.

\]
It is shown that for any

z
⩾
0

z \geqslant 0

and for any integrable function

f
:
[
0
,
1
]
→

R
+

f:[0,1] \to {R^ + }

the measure

μ
f

{\mu _f}

(where

μ
\mu

is Lebesgue measure and

μ
f

{\mu _f}

is defined by

d

μ
f

=
f
d
μ

d{\mu _f} = fd\mu

) of the set of points

x
x

for which

ξ
ε

(
x
)
⩽
z

/

ε

{\xi _\varepsilon }(x) \leqslant z/\varepsilon

tends to an exponential function of

z
z

as

ε
\varepsilon

tends to zero.
The heat equation with a singular potential
Authors: Baras Pierre , Goldstein Jerome A.
Published: June 2010

Of concern is the singular problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential u slash partial-differential t equals normal upper Delta u plus left-parenthesis c slash StartAbsoluteValue x EndAbsoluteValue squared right-parenthesis u plus f left-parenthesis t comma x right-parenthesis comma u left-parenthesis x comma 0 right-parenthesis equals u 0 left-parenthesis x right-parenthesis">

∂
u

/

∂
t
=
Δ
u
+
(
c

/

|

x

|

2

)

u
+
f
(
t
,
x
)
,
u
(
x
,
0
)
=

u

0

(
x
)

\partial u/\partial t = \Delta u + (c/|x{|^2})\,u + f(t,x), u(x,0) = u_{0}(x)

, and its generalizations. Here

c
⩾
0
,
x
∈

R

N

,
t
>
0

c \geqslant 0,x \in {{\mathbf {R}}^N},t > 0

, and

f
f

and

u
0

{u_0}

are nonnegative and not both identically zero. There is a dimension dependent constant

C

∗

(
N
)

{C_{\ast } }(N)

such that the problem has no solution for

c
>

C

∗

(
N
)

c > {C_{\ast } }(N)

. For

c
⩽

C

∗

(
N
)

c \leqslant {C_{\ast } }(N)

necessary and sufficient conditions are found for

f
f

and

u
0

{u_0}

so that a nonnegative solution exists.
𝐿²-cohomology of noncompact surfaces
Authors: DeBaun David R.
Published: June 2010

This paper is motivated by the question of whether nonzero

L
2

{L^2}

-harmonic differentials exist on coverings of a Riemann surface of genus

⩾
2

\geqslant 2

. Our approach will be via an analogue of the de Rham theorem. Some results concerning the invariance of

L
2

{L^2}

-homology and the intersection number of

L
2

{L^2}

-cycles are demonstrated. A growth estimate for triangulations of planar coverings of the two-holed torus is derived. Finally, the equivalence between the existence of

L
2

{L^2}

-harmonic one-cycles and the transience of random walks on a planar surface is shown.
On the arithmetic of projective coordinate systems
Authors: Herrmann Christian
Published: June 2010

A complete list of subdirectly irreducible modular (Arguesian) lattices generated by a frame of order

n
≥
4

(
n
≥
3
)

n \geq 4\;(n \geq 3)

is given. Also, it is shown that a modular lattice variety containing the rational projective geometries cannot be both finitely based and generated by its finite dimensional members.
Rank change on adjoining real powers to Hardy fields
Authors: Rosenlicht Maxwell
Published: June 2010

This paper concerns asymptotic approximations and expansions in cases where the usual Poincaré power series in

1

/

x

1/x

do not suffice because there may be more than one comparability class of functions that are very large or very small. The attempt to find asymptotic approximations in terms of real powers of given representatives of the comparability classes fails in general, but the situation can be saved by the adjunction of suitable real power products of the original functions, at the possible cost of an increase in the number of comparability classes.
On bases in the disc algebra
Authors: Bourgain J.
Published: June 2010

It is shown that the disc algebra has no Besselian basis. In fact, concrete minorations on certain Lebesgue functions are obtained. A consequence is the nonisomorphism of the disc algebra and the space of uniformly convergent Fourier series on the circle.
Asymptotic expansions of traces for certain convolution operators
Authors: Roccaforte Raymond
Published: June 2010

A version of Szegö’s theorem in Euclidean space gives the first two terms of the asymptotics asα→∞\alpha \to \inftyof the determinant of convolution operators onL2(αΩ){L_2}(\alpha \,\Omega ), whereΩ\Omegais a bounded subset ofRn{{\mathbf {R}}^n}with smooth boundary. In this paper the more general problem of the asymptotics of traces of certain analytic functions of the operators is considered and the next term in the expansion is obtained.
The interfaces of one-dimensional flows in porous media
Authors: Vázquez Juan L.
Published: June 2010

The solutions of the equation

u
t

=

(

u
m

)

x

x

{u_t} = {({u^m})_{x\,x}}

for

x
∈

R

,
0
>
t
>
T
,
m
>
1

x \in {\mathbf {R}},0 > t > T,m > 1

, where

u
(
x
,
0
)

u(x,0)

is a nonnegative Borel measure that vanishes for

x
>
0

x > 0

(and satisfies a growth condition at

−
∞

- \infty

), exhibit a finite, monotone, continuous interface

x
=
ζ
(
t
)

x = \zeta (t)

that bounds to the right the region where

u
>
0

u > 0

. We perform a detailed study of

ζ
\zeta

: initial behaviour, waiting time, behaviour as

t
→
∞

t \to \infty

. For certain initial data the solutions blow up in a finite time

T

∗

{T^{\ast }}

: we calculate

T

∗

{T^{\ast }}

in terms of

u
(
x
,
0
)

u(x,0)

and describe the behaviour of

ζ
\zeta

as

t

↑

T

∗

t\, \uparrow \,{T^{\ast }}

.