bluuebunny/crossref_metadata_2025 huggingface dataset:
Pseudo-Chern classes of an almost pseudo-Hermitian manifold
Authors: Matsushita Yasuo
Published: June 2010

For an almost pseudo-Hermitian manifold, pseudo-Chern classes are defined on its complexified tangent bundle with the pseudo-Hermitian structure as represented by certain

ad

(
U
(
p
,

q
)
)

{\text {ad}}(U(p,\,q))

-invariant forms on the manifold. It is shown that such a manifold always admits an almost Hermitian structure, and hence that Chern classes are also defined on the complexified tangent bundle with such an almost Hermitian structure. A relation between the pseudo-Chern classes and the Chern classes is established. From the relation, the pseudo-Chern classes are considered as the characteristic classes which measure how the almost pseudo-Hermitian structure deviates from an almost Hermitian structure.
Positive forms and dilations
Authors: Szymański Wacław
Published: June 2010

By using the quadratic form and unbounded operator theory a new approach to the general dilation theory is presented. The boundedness condition is explained in terms of the Friedrichs extension of symmetric operators. Unbounded dilations are introduced and discussed. Applications are given to various problems involving positive definite functions.
Thue equations with few coefficients
Authors: Schmidt Wolfgang M.
Published: June 2010

Let

F
(
x
,

y
)

F(x,\,y)

be a binary form of degree

r
⩾
3

r \geqslant 3

with integer coefficients, and irreducible over the rationals. Suppose that only

s
+
1

s + 1

of the

r
+
1

r + 1

coefficients of

F
F

are nonzero. Then the Thue equations

F
(
x
,

y
)
=
1

F(x,\,y) = 1

has

≪

(
r
s

)

1

/

2

\ll {(rs)^{1/2}}

solutions. More generally, the inequality

|

F
(
x
,

y
)

|

⩽
h

|F(x,\,y)| \leqslant h

has

≪

(
r
s

)

1

/

2

h

2

/

r

(
1
+
log
⁡

h

1

/

r

)

\ll {(rs)^{1/2}}{h^{2/r}}(1 + \log {h^{1/r}})

solutions.
A formula for the resolvent of (-Δ)^{𝑚}+𝑀^{2𝑚}_{𝑞} with applications to trace class
Authors: Takáč Peter
Published: June 2010

We derive a formula for the resolvent of the elliptic operator

H
=

(
−
Δ

)
m

+

M
q

2
m

H = {( - \Delta )^m} + M_q^{2m}

on

L
2

(

R

N

)

{L_2}({\mathbb {R}^N})

in terms of bounded integral operators

S
λ

{S_\lambda }

and

T
λ

{T_\lambda }

whose kernels we know explicitly. We use this formula to specify the domain of the operator

A
λ

=
(
H
+
λ
I
)

M
p

{A_\lambda } = (H + \lambda I){M_p}

on

L
2

(

R

N

)

{L_2}({\mathbb {R}^N})

, and to estimate the Hilbert-Schmidt norm of its inverse

A
λ

−
1

A_\lambda ^{ - 1}

, for

λ
⩾
0

\lambda \geqslant 0

. Finally we exploit the last two results to prove a trace class criterion for an integral operator

K
K

on

L
2

(

R

N

)

{L_2}({\mathbb {R}^N})

.
Nil 𝐾-theory maps to cyclic homology
Authors: Weibel Charles A.
Published: June 2010

Algebraic

K
K

-theory breaks into two pieces: nil

K
K

-theory and Karoubi-Villamayor

K
K

-theory. Karoubi has constructed Chern classes from the latter groups into cyclic homology. We construct maps from nil

K
K

-theory to cyclic homology which are compatible with Karoubi’s maps, but with a degree shift. Several recent results show that in characteristic zero our map is often an isomorphism.
Hyperarithmetical index sets in recursion theory
Authors: Lempp Steffen
Published: June 2010

We define a family of properties on hyperhypersimple sets and show that they yield index sets at each level of the hyperarithmetical hierarchy. An extension yields a

Π
1
1

\Pi _1^1

-complete index set. We also classify the index set of quasimaximal sets, of coinfinite r.e. sets not having an atomless superset, and of r.e. sets major in a fixed nonrecursive r.e. set.
Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups
Authors: Corwin L. , Greenleaf F. P. , Grélaud G.
Published: June 2010

Let

K
K

be a Lie subgroup of the connected, simply connected nilpotent Lie group

G
G

, and let

k

\mathfrak {k}

,

g

\mathfrak {g}

be the corresponding Lie algebras. Suppose that

σ
\sigma

is an irreducible unitary representation of

K
K

. We give an explicit direct integral decomposition of

Ind

k
→
G

σ

{\operatorname {Ind} _{k \to G}}\sigma

into irreducibles. The description uses the Kirillov orbit picture, which gives a bijection between

G
∧

G^\wedge

and the coadjoint orbits in

g

∗

{\mathfrak {g}^{\ast }}

(and similarly for

K
∧

,

k

∗

K^\wedge ,\,{\mathfrak {k}^{\ast }}

). Let

P
:

k

∗

→

g

∗

P:{\mathfrak {k}^{\ast }} \to {\mathfrak {g}^{\ast }}

be the canonical projection, let

O

σ

⊂

k

∗

{\mathcal {O}_\sigma } \subset {\mathfrak {k}^{\ast }}

be the orbit corresponding to

σ
\sigma

, and, for

π
∈

G
∧

\pi \in G^\wedge

, let

O

π

⊂

g

∗

{\mathcal {O}_\pi } \subset {\mathfrak {g}^{\ast }}

be the corresponding orbit. The main result of the paper says essentially that

π
∈

G
∧

\pi \in G^\wedge

appears in the direct integral iff

P

−
1

(

O

σ

)

{P^{ - 1}}({\mathcal {O}_\sigma })

meets

O

π

{\mathcal {O}_\pi }

; the multiplicity of

π
\pi

is the number of

Ad

∗

(
K
)

{\operatorname {Ad} ^{\ast }}(K)

-orbits in

O

π

∩

P

−
1

(

O

σ

)

{\mathcal {O}_\pi } \cap {P^{ - 1}}({\mathcal {O}_\sigma })

. There is also a natural description of the measure class in the integral.
New combinatorial interpretations of Ramanujan’s partition congruences mod 5,7 and 11
Authors: Garvan F. G.
Published: June 2010

Let

p
(
n
)

p(n)

denote the number of unrestricted partitions of

n
n

. The congruences referred to in the title are

p
(
5
n
+
4
)

p(5n + 4)

,

p
(
7
n
+
5
)

p(7n + 5)

and

p
(
11
n
+
6
)
≡
0

p(11n + 6) \equiv 0

(

mod
5

\bmod 5

,

7
7

and

11
11

, respectively). Dyson conjectured and Atkin and Swinnerton-Dyer proved combinatorial results which imply the congruences

mod
5

\bmod 5

and

7
7

. These are in terms of the rank of partitions. Dyson also conjectured the existence of a "crank" which would likewise imply the congruence

mod
11

\bmod 11

. In this paper we give a crank which not only gives a combinatorial interpretation of the congruence

mod
11

\bmod 11

but also gives new combinatorial interpretations of the congruences

mod
5

\bmod 5

and

7
7

. However, our crank is not quite what Dyson asked for; it is in terms of certain restricted triples of partitions, rather than in terms of ordinary partitions alone. Our results and those of Dyson, Atkin and Swinnerton-Dyer are closely related to two unproved identities that appear in Ramanujan’s "lost" notebook. We prove the first identity and show how the second is equivalent to the main theorem in Atkin and Swinnerton-Dyer’s paper. We note that all of Dyson’s conjectures

mod
5

\bmod 5

are encapsulated in this second identity. We give a number of relations for the crank of vector partitions

mod
5

\bmod 5

and

7
7

, as well as some new inequalities for the rank of ordinary partitions

mod
5

\bmod 5

and

7
7

. Our methods are elementary relying for the most part on classical identities of Euler and Jacobi.
Sharp distortion theorems for quasiconformal mappings
Authors: Anderson G. D. , Vamanamurthy M. K. , Vuorinen M.
Published: June 2010

Continuing their earlier work on distortion theory, the authors prove some dimension-free distortion theorems for

K
K

-quasiconformal mappings in

R
n

{R^n}

. For example, one of the present results is the following sharp variant of the Schwarz lemma: If

f
f

is a

K
K

-quasiconformal self-mapping of the unit ball

B
n

{B^n}

,

n
⩾
2

n \geqslant 2

, with

f
(
0
)
=
0

f(0) = 0

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 Superscript 1 minus upper K squared Baseline StartAbsoluteValue x EndAbsoluteValue Superscript upper K Baseline less-than-or-slanted-equals StartAbsoluteValue f left-parenthesis x right-parenthesis EndAbsoluteValue less-than-or-slanted-equals 4 Superscript 1 minus 1 slash upper K squared Baseline StartAbsoluteValue x EndAbsoluteValue Superscript 1 slash upper K">

4

1
−

K
2

|

x

|

K

⩽

|

f
(
x
)

|

⩽

4

1
−
1

/

K
2

|

x

|

1

/

K

{4^{1 - {K^2}}}|x{|^K} \leqslant |f(x)| \leqslant {4^{1 - 1/{K^2}}}|x{|^{1/K}}

for all

x
x

in

B
n

{B^n}

.
Coexistence theorems of steady states for predator-prey interacting systems
Authors: Li Lige
Published: June 2010

In this paper we give necessary and sufficient conditions for the existence of positive solutions of steady states for predator-prey systems under Dirichlet boundary conditions on

Ω
⋐

R

n

\Omega \Subset {{\mathbf {R}}^n}

. We show that the positive coexistence of predatorprey densities is completely determined by the "marginal density," the unique density of prey or predator while the other one is absent, i.e. the

(

u
0

,

0
)

({u_0},\,0)

or

(
0
,

ν
0

)

(0,\,{\nu _0})

. More specifically, the situation of coexistence is determined by the spectral behavior of certain operators related to these marginal densities and is also completely determined by the stability properties of these marginal densities. The main results are Theorems 1 and 4.2.
Longtime dynamics of a conductive fluid in the presence of a strong magnetic field
Authors: Bardos C. , Sulem C. , Sulem P.-L.
Published: June 2010

We prove existence in the large of localized solutions to the MHD equations for an ideal conducting fluid subject to a strong magnetic field. We show that, for large time, the dynamics may reduce to linear Alfven waves.
Isometries between function spaces
Authors: Jarosz Krzysztof , Pathak Vijay D.
Published: June 2010

Surjective isometries between some classical function spaces are investigated. We give a simple technical scheme which verifies whether any such isometry is given by a homeomorphism between corresponding Hausdorff compact spaces. In particular the answer is positive for the

C
1

(
X
)

{C^1}(X)

,

AC
⁡
[
0
,
1
]

\operatorname {AC} [0,1]

,

Lip
α

(
X
)

{\operatorname {Lip} _\alpha }(X)

and

lip
α

(
X
)

{\operatorname {lip} _\alpha }(X)

spaces provided with various natural norms.
Local projective resolutions and translation functors for Kac-Moody algebras
Authors: Neidhardt Wayne
Published: June 2010

Let

g

\mathfrak {g}

be a Kac-Moody algebra defined by a not necessarily symmetrizable generalized Cartan matrix. We define translation functors and use them to show that the multiplicities

(
M
(

w
1

⋅
λ
)
:
L
(

w
2

⋅
λ
)
)

(M({w_1} \cdot \lambda ):L({w_2} \cdot \lambda ))

are independent of the dominant integral weight

λ
\lambda

, depending only on the elements of the Weyl group. In order to define the translation functors, we introduce the notion of local projective resolutions and use them to develop the machinery of homological algebra in certain categories of

g

\mathfrak {g}

-modules.
Operator theoretical realization of some geometric notions
Authors: Lin Qing
Published: June 2010

This paper studies the realization of certain geometric constructions in Cowen-Douglas operator class. Through this realization, some operator theoretical phenomena are easily seen from the corresponding geometric phenomena. In particular, we use this technique to solve the first-order equivalence problem and introduce a new operation among certain operators.
A problem in convexity leading to the analysis of two functional equations
Authors: Ryff John V.
Published: June 2010

Transformation semigroups can often be studied effectively by examining their orbit structure. If the class of transformations has a special quality, such as convexity, it is generally reflected in the orbits. This work is concerned with such a circumstance. The goal is to examine the behavior of transformations on extreme points of orbits through the construction of a class of extreme operators. The construction leads naturally to the study of two functional equations which are analyzed in detail. Information about solutions is obtained through different

L
2

{L^2}

-methods depending on whether or not two basic parameters are rational or irrational. In two cases all solutions are classified. In a third an example of a spanning set of solutions is obtained. Techniques of harmonic analysis and ergodic theory are used to study the functional equations.
Finite order solutions of second order linear differential equations
Authors: Gundersen Gary G.
Published: June 2010

We consider the differential equation

f
+
A
(
z
)

f
′

+
B
(
z
)
f
=
0

f + A(z)f’ + B(z)f = 0

where

A
(
z
)

A(z)

and

B
(
z
)

B(z)

are entire functions. We will find conditions on

A
(
z
)

A(z)

and

B
(
z
)

B(z)

which will guarantee that every solution

f
≢
0

f\not \equiv 0

of the equation will have infinite order. We will also find conditions on

A
(
z
)

A(z)

and

B
(
z
)

B(z)

which will guarantee that any finite order solution

f
≢
0

f\not \equiv 0

of the equation will not have zero as a Borel exceptional value. We will also show that if

A
(
z
)

A(z)

and

B
(
z
)

B(z)

satisfy certain growth conditions, then any finite order solution of the equation will satisfy certain other growth conditions. Related results are also proven. Several examples are given to complement the theory.
The automorphism group of a shift of finite type
Authors: Boyle Mike , Lind Douglas , Rudolph Daniel
Published: June 2010

Let

(

X
T

,

σ
T

)

({X_T},{\sigma _T})

be a shift of finite type, and

G
=
aut
⁡
(

σ
T

)

G = \operatorname {aut} ({\sigma _T})

denote the group of homeomorphisms of

X
T

{X_T}

commuting with

σ
T

{\sigma _T}

. We investigate the algebraic properties of the countable group

G
G

and the dynamics of its action on

X
T

{X_T}

and associated spaces. Using "marker" constructions, we show

G
G

contains many groups, such as the free group on two generators. However,

G
G

is residually finite, so does not contain divisible groups or the infinite symmetric group. The doubly exponential growth rate of the number of automorphisms depending on

n
n

coordinates leads to a new and nontrivial topological invariant of

σ
T

{\sigma _T}

whose exact value is not known. We prove that, modulo a few points of low period,

G
G

acts transitively on the set of points with least

σ
T

{\sigma _T}

-period

n
n

. Using

p
p

-adic analysis, we generalize to most finite type shifts a result of Boyle and Krieger that the gyration function of a full shift has infinite order. The action of

G
G

on the dimension group of

σ
T

{\sigma _T}

is investigated. We show there are no proper infinite compact

G
G

-invariant sets. We give a complete characterization of the

G
G

-orbit closure of a continuous probability measure, and deduce that the only continuous

G
G

-invariant measure is that of maximal entropy. Examples, questions, and problems complement our analysis, and we conclude with a brief survey of some remaining open problems.
Quasiconformal mappings and chord-arc curves
Authors: Semmes Stephen W.
Published: June 2010

Given a quasiconformal mapping

ρ
\rho

on the plane, what conditions on its dilatation

μ
\mu

guarantee that

ρ
(

R

)

\rho ({\mathbf {R}})

is rectifiable and

ρ

|

R

\rho {|_{\mathbf {R}}}

is locally absolutely continuous? We show in this paper that if

μ
\mu

satisfies certain quadratic Carleson measure conditions, with small norm, then

ρ
(

R

)

\rho ({\mathbf {R}})

is a chord-arc curve with small constant, and

ρ
(
x
)
=
ρ
(
0
)
+

∫
0
x

e

a
(
t
)

d
t

\rho (x) = \rho (0) + \int _0^x {{e^{a(t)}}dt}

for

x
∈

R

x \in {\mathbf {R}}

, with

a
∈
BMO

a \in \operatorname {BMO}

having small norm. Conversely, given any such map from

R

→

C

{\mathbf {R}} \to {\mathbf {C}}

, we show that it has an extension to

C

{\mathbf {C}}

with the right kind of dilatation. Similar results hold with

R

{\mathbf {R}}

replaced by a chord-arc curve. Examples are given that show that it would be hard to improve these results. Applications are given to conformal welding and the theorem of Coifman and Meyer on the real analyticity of the Riemann mapping on the manifold of chord-arc curves.
An approach to homotopy classification of links
Authors: Levine J. P.
Published: June 2010

A reformulation and refinement of the

μ
¯

\overline \mu

-invariants of Milnor are used to give a homotopy classification of

4
4

component links and suggest a possible general homotopy classification. The main idea is to use the (reduced) group of a link and its "geometric" automorphisms to define the precise indeterminacy of these invariants.
The cohomology representation of an action of 𝐶_{𝑝} on a surface
Authors: Symonds Peter
Published: June 2010

When a finite group

G
G

acts on a surface

S
S

, then

H
1

(
S
;

Z

)

{H^1}(S;\,{\mathbf {Z}})

posseses naturally the structure of a

Z

G

{\mathbf {Z}}G

-module with invariant symplectic inner product. In the case of a cyclic group of odd prime order we describe explicitly this symplectic inner product space in terms of the fixed-point data of the action.
Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators
Authors: Andersen K. F. , Sawyer E. T.
Published: June 2010

The weight functions

u
(
x
)

u(x)

for which

R
α

{R_\alpha }

, the Riemann-Liouville fractional integral operator of order

α
>
0

\alpha > 0

, is bounded from

L
p

(

u
p

d
x
)

{L^p}({u^p}\,dx)

to

L
q

(

u
q

d
x
)

{L^q}({u^q}\,dx)

,

1
>
p
>
1

/

α

1 > p > 1/\alpha

,

1

/

q
=
1

/

p
−
α

1/q = 1/p - \alpha

, are characterized. Further, given

p
p

,

q
q

with

1

/

q
⩾
1

/

p
−
α

1/q \geqslant 1/p - \alpha

, the weight functions

u
>
0

u > 0

a.e. (resp.

v
>
∞

v > \infty

a.e.) for which there is

v
>
∞

v > \infty

a.e. (resp.

u
>
0

u > 0

a.e.) so that

R
α

{R_\alpha }

is bounded from

L
p

(

v
p

d
x
)

{L^p}({v^p}\,dx)

to

L
q

(

u
q

d
x
)

{L^q}({u^q}\,dx)

are characterized. Analogous results are obtained for the Weyl fractional integral. The method involves the use of complex interpolation of analytic families of operators to obtain similar results for fractional "one-sided" maximal function operators which are of independent interest.
The Bergman spaces, the Bloch space, and Gleason’s problem
Authors: Zhu Ke He
Published: June 2010

Suppose

f
f

is a holomorphic function on the open unit ball

B
n

{B_n}

of

C

n

{{\mathbf {C}}^n}

. For

1
⩽
p
>
∞

1 \leqslant p > \infty

and

m
>
0

m > 0

an integer, we show that

f
f

is in

L
p

(

B
n

,

d
V
)

{L^p}({B_n},\,dV)

(with

d
V

dV

the volume measure) iff all the functions

∂
m

f

/

∂

z

α

(

|

α

|

=
m
)

{\partial ^m}f/\partial {z^{\alpha \,}}\;(|\alpha |\, = m)

are in

L
p

(

B
n

,

d
V
)

{L^p}({B_n},\,dV)

. We also prove that

f
f

is in the Bloch space of

B
n

{B_n}

iff all the functions

∂
m

f

/

∂

z
α

(

|

α

|

=
m
)

{\partial ^m}f/\partial {z^\alpha }\;(|\alpha |\, = m)

are bounded on

B
n

{B_n}

. The corresponding result for the little Bloch space of

B
n

{B_n}

is established as well. We will solve Gleason’s problem for the Bergman spaces and the Bloch space of

B
n

{B_n}

before proving the results stated above. The approach here is functional analytic. We make extensive use of the reproducing kernels of

B
n

{B_n}

. The corresponding results for the polydisc in

C

n

{{\mathbf {C}}^n}

are indicated without detailed proof.
Geodesics and conformal transformations of Heisenberg-Reiter spaces
Authors: Torres Lopera J. F.
Published: June 2010

Generalized Heisenberg groups, in the sense of Reiter, can be endowed with left-invariant metrics whose geodesies and curvature are obtained. Using these curvature data it is also proved that on their nilmanifolds (compact or not), every conformal transformation is in fact an isometry. A large family of nonisometric examples is given.
Alexander modules of links with all linking numbers zero
Authors: Platt M. L.
Published: June 2010

In this paper we characterize the Alexander modules of links resulting from a surgical modification on the trivial link of any number of components. Using the presentation matrix obtained, we derive some properties of the Alexander polynomials of such links.
Tensor products of composition algebras, Albert forms and some exceptional simple Lie algebras
Authors: Allison B. N.
Published: June 2010

In this paper, we study algebras with involution that are isomorphic after base field extension to the tensor product of two composition algebras. To any such algebra

(

A

,

−
)

(\mathcal {A},\, - )

, we associate a quadratic form

Q
Q

called the Albert form of

(

A

,

−
)

(\mathcal {A},\, - )

. The Albert form is used to give necessary and sufficient conditions for two such algebras to be isotopic. Using a Lie algebra construction of Kantor, we are then able to give a description of the isomorphism classes of Lie algebras of index

F

4
,
1

21

F_{4,1}^{21}

,

2

E

6
,
1

29

{}^2E_{6,1}^{29}

,

E

7
,
1

48

E_{7,1}^{48}

and

E

8
,
1

91

E_{8,1}^{91}

. That description is used to obtain a classification of the indicated Lie algebras over

R

(
(

T
1

,
…
,

T
n

)
)
,

n
⩽
3

{\mathbf {R}}(({T_1}, \ldots ,{T_n})),\;n \leqslant 3

.
Tauberian theorems and stability of one-parameter semigroups
Authors: Arendt W. , Batty C. J. K.
Published: June 2010

The main result is the following stability theorem: Let

T

=

(
T
(
t
)

)

t
⩾
0

\mathcal {T} = {(T(t))_{t \geqslant 0}}

be a bounded

C
0

{C_0}

-semigroup on a reflexive space

X
X

. Denote by

A
A

the generator of

T

\mathcal {T}

and by

σ
(
A
)

\sigma (A)

the spectrum of

A
A

. If

σ
(
A
)
∩
i

R

\sigma (A) \cap i{\mathbf {R}}

is countable and no eigenvalue of

A
A

lies on the imaginary axis, then

lim

t
→
∞

T
(
t
)
x
=
0

{\lim _{t \to \infty }}T(t)x = 0

for all

x
∈
X

x \in X

.
Multiple fibers on rational elliptic surfaces
Authors: Harbourne Brian , Lang William E.
Published: June 2010

Our main result, Theorem (0.1), classifies multiple fibers on rational elliptic surfaces over algebraically closed fields of arbitrary characteristic. One result of this is the existence in positive characteristics of tame multiple fibers of additive type for several of the Kodaira fiber-types for which no examples were previously known.
Some sharp inequalities for martingale transforms
Authors: Choi K. P.
Published: June 2010

Two sharp inequalities for martingale transforms are proved. These results extend some earlier work of Burkholder. The inequalities are then extended to stochastic integrals and differentially subordinate martingales.
Homogeneous continua in Euclidean (𝑛+1)-space which contain an 𝑛-cube are locally connected
Authors: Prajs Janusz R.
Published: June 2010

We prove that each homogeneous continuum which topologically contains an

n
n

-dimensional unit cube and lies in

(
n
+
1
)

(n + 1)

-dimensional Euclidean space is locally connected.
Stable extensions of homeomorphisms on the pseudo-arc
Authors: Kennedy Judy
Published: June 2010

We prove the following: Theorem. If

P
′

P’

is a proper subcontinuum of the pseudoarc

P
,

h
′

P,\,h’

is a homeomorphism from

P
′

P’

onto itself, and

Θ
\Theta

is an open set in

P
P

that contains

P
′

P’

, then there is a homeomorphism

h
h

from

P
P

onto itself such that

h

|

P
′

=

h
′

h|P’ = h’

and

h
(
x
)
=
x

h(x) = x

for

x
∉
Θ

x \notin \Theta

.
A fractal-like algebraic splitting of the classifying space for vector bundles
Authors: Giambalvo V. , Pengelley David J. , Ravenel Douglas C.
Published: June 2010

The connected covers of the classifying space

B
O

BO

induce a decreasing filtration

{

B
n

}

\{ {B_n}\}

of

H

∗

(
B
O
;

Z

/

2
)

{H_{\ast }}(BO;\,Z/2)

by sub-Hopf algebras over the Steenrod algebra

A
A

. We describe a multiplicative grading on

H

∗

(
B
O
;

Z

/

2
)

{H_{\ast }}(BO;\,Z/2)

inducing a direct sum splitting of

B
n

{B_n}

over

A
n

{A_n}

, where

{

A
n

}

\{ {A_n}\}

is the usual (increasing) filtration of

A
A

. The pieces in the splittings are finite, and the grading extends that of

H

∗

Ω
2

S
3

{H_{\ast }}{\Omega ^2}{S^3}

which splits it into Brown-Gitler modules. We also apply the grading to the Thomifications

{

M
n

}

\{ {M_n}\}

of

{

B
n

}

\{ {B_n}\}

, where it induces splittings of the corresponding cobordism modules over the entire Steenrod algebra. These generalize algebraically the previously known topological splittings of the connective cobordism spectra

M
O

MO

,

M
S
O

MSO

and

M

S
p
i
n

M\,Spin

.
Weighted norm inequalities for potential operators
Authors: Schechter Martin
Published: June 2010

We give sufficient conditions for inequalities of the form\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis integral left-parenthesis integral upper G left-parenthesis x minus y right-parenthesis f left-parenthesis y right-parenthesis d mu left-parenthesis y right-parenthesis right-parenthesis Superscript q Baseline d omega left-parenthesis x right-parenthesis right-parenthesis Superscript 1 slash q Baseline less-than-or-slanted-equals upper C left-parenthesis integral StartAbsoluteValue f left-parenthesis y right-parenthesis EndAbsoluteValue Superscript p Baseline d nu left-parenthesis y right-parenthesis right-parenthesis Superscript 1 slash p">(∫(∫G(x−y)f(y)dμ(y))qdω(x))1/q⩽C(∫|f(y)|pdν(y))1/p{\left ( {\int {{{\left ( {\int {G(x - y)f(y)\,d\mu (y)} } \right )}^q}\,d\omega (x)} } \right )^{1/q}}\, \leqslant C{\left ( {\int {|f(y){|^p}d\nu (y)} } \right )^{1/p}}\]to hold for measurable functionsff. We determine the dependence of the constantCCon the measuresμ\mu,ν\nu,ω\omegaand give some applications.
Spectrum reducing extension for one operator on a Banach space
Authors: Read C. J.
Published: June 2010

In this paper we show that, given an operator

T
T

on a Banach space

X
X

, there is an extension

Y
Y

of

X
X

such that

T
T

extends in a natural way to an operator

T
∼

{T^ \sim }

on

Y
Y

, and the spectrum of

T
∼

{T^ \sim }

is the approximate point spectrum of

T
T

. This answers a question posed by Bollobás, and contributes to a theory investigated by Shilov, Arens, Bollobás, etc. The unusual transfinite construction is similar to that which we used earlier to find an inverse producing extension for a commutative unital Banach algebra which eliminates the residual spectrum of one element. We also give a counterexample, consisting of a Banach algebra

L
L

containing elements

g
1

{g_1}

and

g
2

{g_2}

such that in no extension

L
′

L’

of

L
L

are the residual spectra of

g
1

{g_1}

and

g

2

{g_{_2}}

eliminated simultaneously.
Explicit formula for weighted scalar nonlinear hyperbolic conservation laws
Authors: LeFloch Philippe , Nédélec Jean-Claude
Published: June 2010

We prove a uniqueness and existence theorem for the entropy weak solution of nonlinear hyperbolic conservation laws of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential Over partial-differential t EndFraction left-parenthesis r u right-parenthesis plus StartFraction partial-differential Over partial-differential x EndFraction left-parenthesis r f left-parenthesis u right-parenthesis right-parenthesis equals 0 comma">

∂

∂
t

(
r
u
)
+

∂

∂
x

(
r
f
(
u
)
)
=
0
,

\frac {\partial } {{\partial t}}(ru) + \frac {\partial } {{\partial x}}(rf(u)) = 0,

\]
with initial data and boundary condition. The scalar function

u
=
u
(
x
,

t
)

u = u(x,\,t)

,

x
>
0

x > 0

,

t
>
0

t > 0

, is the unknown, the function

f
=
f
(
u
)

f = f(u)

is assumed to be strictly convex with inf

f
(
⋅
)
=
0

f( \cdot ) = 0

and the weight function

r
=
r
(
x
)

r = r(x)

,

x
>
0

x > 0

, to be positive (for example,

r
(
x
)
=

x
α

r(x) = {x^\alpha }

, with an arbitrary real

α
\alpha

). We give an explicit formula, which generalizes a result of P. D. Lax. In particular, a free boundary problem for the flux

r
(
⋅
)
f
(
u
(
⋅
,
⋅
)
)

r( \cdot )f(u( \cdot , \cdot ))

at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to B. L. Keyfitz.
Wiener’s criterion for parabolic equations with variable coefficients and its consequences
Authors: Garofalo Nicola , Lanconelli Ermanno
Published: June 2010

In a bounded set in

R

n
+
1

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

we study the problem of the regularity of boundary points for the Dirichlet problem for a parabolic operator with smooth coefficients. We give a geometric characterization, modelled on Wiener’s criterion for Laplace’s equation, of those boundary points that are regular. We also present some important consequences. Here is the main one: a point is regular for a variable coefficient operator if and only if it is regular for the constant coefficient operator obtained by freezing the coefficients at that point.
Trace Paley-Wiener theorem in the twisted case
Authors: Rogawski J. D.
Published: June 2010

A version of the trace Paley-Wiener theorem for a reductive

p
p

-adic group in the context of twisted harmonic analysis with respect to an outer automorphism is proved.
Locally bounded sets of holomorphic mappings
Authors: Bonet José , Galindo Pablo , García Domingo , Maestre Manuel
Published: June 2010

Several results and examples about locally bounded sets of holomorphic mappings defined on certain classes of locally convex spaces (Baire spaces,

(
D
F
)

(DF)

-spaces,

C
(
X
)

C(X)

-spaces) are presented. Their relation with the classification of locally convex spaces according to holomorphic analogues of barrelled and bornological properties of the linear theory is considered.
Decidability and invariant classes for degree structures
Authors: Lerman Manuel , Shore Richard A.
Published: June 2010

We present a decision procedure for the

∀
∃

\forall \exists

-theory of

D

[

0

,

0

′

]

\mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ]

, the Turing degrees below

0

′

{\mathbf {0}}’

. The two main ingredients are a new extension of embeddings result and a strengthening of the initial segments results below

0

′

{\mathbf {0}}’

of [Le1]. First, given any finite subuppersemilattice

U
U

of

D

[

0

,

0

′

]

\mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ]

with top element

0

′

{\mathbf {0}}’

and an isomorphism type

V
V

of a poset extending

U
U

consistently with its structure as an usl such that

V
V

and

U
U

have the same top element and

V
V

is an end extension of

U
−
{

0

′

}

U - \{ {\mathbf {0}}’ \}

, we construct an extension of

U
U

inside

D

[

0

,

0

′

]

\mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ]

isomorphic to

V
V

. Second, we obtain an initial segment

W
W

of

D

[

0

,

0

′

]

\mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ]

which is isomorphic to

U
−
{

0

′

}

U - \{ {\mathbf {0}}’ \}

such that

W
∪
{

0

′

}

W \cup \{ {\mathbf {0}}’ \}

is a subusl of

D

\mathcal {D}

. The decision procedure follows easily from these results. As a corollary to the

∀
∃

\forall \exists

-decision procedure for

D

\mathcal {D}

, we show that no degree

a

>

0

{\mathbf {a}} > {\mathbf {0}}

is definable by any

∃
∀

\exists \forall

-formula of degree theory. As a start on restricting the formulas which could possibly define the various jump classes we classify the generalized jump classes which are invariant for any

∀
\forall

or

∃
\exists

-formula. The analysis again uses the decision procedure for the

∀
∃

\forall \exists

-theory of

D

\mathcal {D}

. A similar analysis is carried out for the high/low hierarchy using the decision procedure for the

∀
∃

\forall \exists

-theory of

D

[

0

,

0

′

]

\mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ]

. (A jump class

C

\mathcal {C}

is

σ
\sigma

-invariant if

σ
(

a

)

\sigma ({\mathbf {a}})

holds for every

a

{\mathbf {a}}

in

C

\mathcal {C}

.)
Continuous cohomology and real homotopy type
Authors: Brown Edgar H. , Szczarba Robert H.
Published: June 2010

Various aspects of homotopy theory in the category of simplicial spaces are developed. Topics covered include continuous cohomology, continuous de Rham cohomology, the Kan extension condition, the homotopy relation, fibrations, the Serre spectral sequence, real homotopy type and its relation to graded commutative differential algebras over the reals.
Orbital parameters for induced and restricted representations
Authors: Lipsman Ronald L.
Published: June 2010

General formulas for the spectral decomposition of both induced and restricted representations are laid out for the case of connected Lie groups

H
⊂
G

H \subset G

. The formulas—which detail the actual spectrum, the multiplicits, and the spectral measure—are in terms of the usual parameters in the so-called orbit method. A proof of these formulas is given in the nilpotent situation. The proof is much simpler than a previously obtained proof using nilpotent algebraic geometry. It is also capable of generalization to nonnilpotent groups. With that in mind, many new examples are presented for semisimple and symmetric homogeneous spaces. Also, a start is made in the case of exponential solvable homogeneous spaces with the treatment of both normal and conormal subgroups.
Weakly almost periodic flows
Authors: Ellis R. , Nerurkar M.
Published: June 2010

The notion of the enveloping semigroup of a flow is applied to some situations in ergodic theory. In particular, weakly almost periodic functions on groups are studied and Moore’s ergodic theorem is proved.
On hypersurfaces of hyperbolic space infinitesimally supported by horospheres
Authors: Currier Robert J.
Published: June 2010

This paper is concerned with complete, smooth immersed hypersurfaces of hyperbolic space that are infinitesimally supported by horospheres. This latter condition may be restated as requiring that all eigenvalues of the second fundamental form, with respect to a particular unit normal field, be at least one. The following alternative must hold: either there is a point where all the eigenvalues of the second fundamental form are strictly greater than one, in which case the hypersurface is compact, imbedded and diffeomorphic to a sphere; or, the second fundamental form at every point has

1
1

as an eigenvalue, in which case the hypersurface is isometric to Euclidean space and is imbedded in hyperbolic space as a horosphere.
Generalizations of the stacked bases theorem
Authors: Hill Paul , Megibben Charles
Published: June 2010

LetHHbe a subgroup of the free abelian groupGG. In order for there to exist a basis{xi}i∈I{\{ {x_i}\} _{i \in I}}ofGGfor whichH=⊕i∈I⟨nixi⟩H = { \oplus _{i \in I}}\langle {n_i}{x_i}\ranglefor suitable nonnegative integersni{n_i}, it is obviously necessary forG/HG/Hto be a direct sum of cyclic groups. In the 1950’s, Kaplansky raised the question of whether this condition onG/HG/His sufficient for the existence of such a basis. J. Cohen and H. Gluck demonstrated in 1970 that the answer is "yes"; their result is known as the stacked bases theorem, and it extends the classical and well-known invariant factor theorem for finitely generated abelian groups. In this paper, we develop a theory that contains and, in fact, generalizes in several directions the stacked bases theorem. Our work includes a complete classification, using numerical invariants, of the various free resolutions of any abelian group.
Compact abelian prime actions on von Neumann algebras
Authors: Thomsen Klaus
Published: June 2010

We classify the compact abelian actions on semifinite injective von Neumann algebras with factor fixed point algebra. The method uses that the (nonzero) eigenspaces of such an action contain unitaries which give rise to a classifying invariant.
Optimal stopping of two-parameter processes on nonstandard probability spaces
Authors: Dalang Robert C.
Published: June 2010

We prove the existence of optimal stopping points for upper semicontinuous two-parameter processes defined on filtered nonstandard (Loeb) probability spaces that satisfy a classical conditional independence hypothesis. The proof is obtained via a lifting theorem for elements of the convex set of randomized stopping points, which shows in particular that extremal elements of this set are ordinary stopping points.
Harmonic measure and radial projection
Authors: Marshall Donald E. , Sundberg Carl
Published: June 2010

Among all curves in the closed unit disk that meet every radius, there is one,

γ
0

{\gamma _0}

, whose harmonic measure at the origin is minimal. We give an explicit description of

γ
0

{\gamma _0}

and compute its harmonic measure. We also give a quadratically convergent algorithm to compute the harmonic measure of one side of a rectangle at its center.
Finitely presented lattices: canonical forms and the covering relation
Authors: Freese Ralph
Published: June 2010

A canonical form for elements of a lattice freely generated by a partial lattice is given. This form agrees with Whitman’s canonical form for free lattices when the partial lattice is an antichain. The connection between this canonical form and the arithmetic of the lattice is given. For example, it is shown that every element of a finitely presented lattice has only finitely many minimal join representations and that every join representation can be refined to one of these. An algorithm is given which decides if a given element of a finitely presented lattice has a cover and finds them if it does. An example is given of a nontrivial, finitely presented lattice with no cover at all.
Differential-difference operators associated to reflection groups
Authors: Dunkl Charles F.
Published: June 2010

There is a theory of spherical harmonics for measures invariant under a finite reflection group. The measures are products of powers of linear functions, whose zero-sets are the mirrors of the reflections in the group, times the rotation-invariant measure on the unit sphere in

R

n

{{\mathbf {R}}^n}

. A commutative set of differential-difference operators, each homogeneous of degree

−
1

-1

, is the analogue of the set of first-order partial derivatives in the ordinary theory of spherical harmonics. In the case of

R

2

{{\mathbf {R}}^2}

and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions.
A new algebraic approach to microlocalization of filtered rings
Authors: Asensio Maria Jesus , Van den Bergh Michel , Van Oystaeyen Freddy
Published: June 2010

Using the construction of the Rees ring associated to a filtered ring we provide a description of the microlocalization of the filtered ring by using only purely algebraic techniques. The method yields an easy approach towards the study of exactness properties of the microlocalization functor. Every microlocalization at a regular multiplicative Ore set in the associated graded ring can be obtained as the completion of a localization at an Ore set of the filtered ring.
Generators for the bordism algebra of immersions
Authors: Aguilar M. A.
Published: June 2010

Let us denote by

I
(
n
,
k
)

I(n,k)

the group of bordism classes of immersions of closed smooth

n
n

-manifolds in closed smooth

(
n
+
k
)

(n + k)

-manifolds

(
k
>
0
)

(k > 0)

. We can make

I
(

∗

,
k
)

I({\ast },k)

into a graded algebra over the unoriented bordism ring. This algebra is polynomial. In this paper we give two sets of immersions which are polynomial generators.
Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities
Authors: Lazer A. C. , McKenna P. J.
Published: June 2010

We give conditions for the existence, uniqueness, and asymptotic stability of periodic solutions of a second-order differential equation with piecewise linear restoring and

2
π

2\pi

-periodic forcing where the range of the derivative of the restoring term possibly contains the square of an integer. With suitable restrictions on the restoring and forcing in the undamped case, we give a necessary and sufficient condition.
Remarks on classical invariant theory
Authors: Howe Roger
Published: June 2010

A uniform formulation, applying to all classical groups simultaneously, of the First Fundamental Theory of Classical Invariant Theory is given in terms of the Weyl algebra. The formulation also allows skew-symmetric as well as symmetric variables. Examples illustrate the scope of this formulation.
Hamburger moment problems and orthogonal polynomials
Authors: Chihara T. S.
Published: June 2010

We consider a sequence of orthogonal polynomials given by the classical three term recurrence relation. We address the problem of deciding the determinacy or indeterminacy of the associated Hamburger moment problem on the basis of the behavior of the coefficients in the three term recurrence relation. Comparisons are made with other criteria in the literature. The efficacy of the criteria obtained is illustrated by application to many specific examples of orthogonal polynomials.
A characterization of nonchaotic continuous maps of the interval stable under small perturbations
Authors: Preiss D. , Smítal J.
Published: June 2010

Recent results of the second author show that every continuous map of the interval to itself either has every trajectory approximable by cycles (sometimes this is possible even in the case when the trajectory is not asymptotically periodic) or is

ε
\varepsilon

-chaotic for some

ε
>
0

\varepsilon > 0

. In certain cases, the first property is stable under small perturbations. This means that a perturbed map can be chaotic but the chaos must be small whenever the perturbation is small. In other words, there are nonchaotic maps without "chaos explosions". In the paper we give a characterization of these maps along with some consequences. Using the main result it is possible to prove that generically the nonchaotic maps are stable.
A method of “alternating corrections” for the numerical solution of two-point boundary values problems
Authors: Pope David A.
Published: June 2010

In this paper a method of “alternating corrections” is defined and analyzed for the numerical solution of the two-point boundary value problem

y
=
f
(
x
,

y
)

y
(
0
)
=
a

y
(
1
)
=
b
.

\begin{align} y = f(x,\,y) \\ y(0) = a \\ y(1) = b. \\ \end{align}

The case where the first derivative does not enter explicitly into the differential equation is chosen for simplicity of treatment. The alternating corrections method can easily be modified to treat the more general case. The function

f
(
x
,
y
)

f(x,y)

is assumed to have continuous second derivatives, but the differential equation may, of course, be non-linear. The method to be described is essentially a relaxation technique suitable for an automatic digital computer. The main feature of the method is that most of the “correcting” is done in the early stages of the computation, using a small number of points; thus a rough approximation to the solution is obtained quickly. This approximation can then be made more accurate in the later stages of the computation, as the number of points is increased. In Section 1 the method is described. Section 2 gives a rigorous truncation and stability analysis. Section 3 contains the proof of the convergence of the method giving an estimate of the rate of convergence, and in Section 4 some experimental results obtained on a digital computer are examined.
Two theorem tables of matrix algebra
Authors: Best Gilbert C.
Published: June 2010

It is shown in this paper how it is possible, by a slight modification of the mode of presentation, to state many of the theorems of matrix algebra in table form. Due to the large number of theorems presented the tables give only the statement of each theorem. It is not considered feasible to also attempt to indicate sources and proofs.
A simple experimental computer with negative bias
Authors: Łazarkiewicz A. , Balasiński W.
Published: June 2010

This paper presents the technical data, logic, and control organization of a simple experimental computer operating in the minus-two system. The principles for composing the instructions from elementary operations, the characteristics of minus-two computer arithmetic, and the logic of the arithmetic unit are briefly explained. The paper is divided into the following sections: 1, Introduction; 2, Fundamental Logical Circuits; 3, The Memory; 4, Block Diagram; 5, The Control Unit; 6, Coordination with Teleprinter; 7, Instructions; 8, Minus-Two System and Range of Numbers in the Computer; 9, The Adding Unit; 10, The Sign and Overflow Register; 11, Multiplication; 12, Acknowledgments.
Iterated square root expansions for the inverse cosine and inverse hyperbolic cosine
Authors: Thacher Henry C.
Published: June 2010

Let

R
1

=

2
+
2
x

{R_1} = \sqrt {2 + 2x}

,

R

k
+
1

=

|

2
+

R
k

|

{R_{k + 1}} = \sqrt {|{2 + {R_k}}|}

. Then

2
k

|

2
−

R
k

|

{2^k}\sqrt {|{2 - {R_k}}|}

and

2
k

{

|

6
−
2

3
+
3

R
k

|

}

1

/

2

{2^k}{\{ |{6 - 2\sqrt {3 + 3{R_k}} }|\} ^{1/2}}

both converge to

arccos

x

\arccos \,x

if

|

x

|

≦
1

|x| \leqq 1

and to

arccosh

x

\operatorname {arccosh}\,x

if

1
≦
x
>
∞

1 \leqq x > \infty

. Truncation errors for the two expressions are of the order of 2

−
2
k

^{-2k}

and 2

−
4
k

^{-4k}

, respectively.
Euler’s constant to 1271 places
Authors: Knuth Donald E.
Published: June 2010

The value of Euler’s or Mascheroni’s constant
\[

γ

=

lim

n
→
∞

(
1

+

1
2

+

⋯

+

(

1

/

n

)

−

ln

n
)

\gamma \,=\,\underset {n\to \infty }{\lim }\,\,(1\,+\,\tfrac {1}{2}\,+\,\cdots \,+\,({1}/{n})\,-\,\ln \,n)

\]
has now been determined to 1271 decimal places, thus extending the previously known value of 328 places. A calculation of partial quotients and best rational approximations to

γ
\gamma

was also made.
A note on the relative merits of Padé and Maehly’s diagonal convergents in computing 𝑒^{𝑥}
Authors: Sankar R. , Malini V.
Published: June 2010

Methods for calculating functions to a high degree of accuracy have assumed increased importance following the advent of the computers. It has been found that rational approximations require fewer operations on a computer than the older polynomial approximations. Among the known methods those due to Padé [1] and Maehly [2] are perhaps the most important. In this paper we have analyzed these methods as applied to the exponential function. It is observed that Maehly’s method is superior to the Padé method in the sense of yielding better accuracy over a given range on the real axis for a given order of approximation. Maehly’s formulas for computing

e
x

{e^x}

correct to eight decimal places have been worked out.
Asymptotic computation of the repeated integrals of the error function complement
Authors: Wilcox W. R.
Published: June 2010

Previously, the complementary error function and its repeated integrals were given only for small values of the argument. Several new calculation techniques are derived which permit evaluation for the complete range of the argument. Some new values of these functions for large values of the argument are calculated. These values are plotted in such a manner that approximate values can easily be found for all values of x.
On the round-off error in the method of successive over-relaxation
Authors: Lynn M. Stuart
Published: June 2010

The asymptotic behavior of the round-off error, which accumulates when the well-known iterative method of (point) successive over-relaxation is used to solve a large-scale system of linear equations, is examined by means of a statistical model. The local round-off errors are treated as independent random variables and expressions for the mean and variance of the accumulated round-off error are obtained, as the number of iterations tends to infinity.
Chebyshev approximations for the complete elliptic integrals 𝐾 and 𝐸
Authors: Cody W. J.
Published: June 2010

Chebyshev approximations of the Hastings form are given for the complete elliptic integrals

K
K

and

E
E

. Maximal errors range from

4
×

10

−
5

4 \times {10^{ - 5}}

down to

4
×

10

−
18

4 \times {10^{ - 18}}

.
On the nonsingularity of real matrices
Authors: Hoffman A. J.
Published: June 2010

By exploiting the theory of linear inequalities, new bounds for the real eigenvalues of a real matrix are derived, along with sufficient conditions for matrix games to be completely mixed, for determinants to be positive, etc. The simple observation on which the derivation of new results and the unification of old results are based is that the typical conditions of diagonal dominance which insure the nonsingularity of matrices are essentially systems of linear inequalities on the rows of the matrices.