bluuebunny/crossref_metadata_2025 huggingface dataset:
Generating combinatorial complexes of polyhedral type
Authors: Schulte Egon
Published: June 2010

The paper describes a method for generating combinatorial complexes of polyhedral type. Building blocks

B

{\mathbf {B}}

are implanted into the maximal simplices of a simplicial complex

C

{\mathbf {C}}

, on which a group operates as a combinatorial reflection group. Of particular interest is the case where

B

{\mathbf {B}}

is a polyhedral block and

C

{\mathbf {C}}

the barycentric subdivision of a regular incidence-polytope

K

{\mathbf {K}}

together with the action of the automorphism group of

K

{\mathbf {K}}

.
Folds and cusps in Banach spaces with applications to nonlinear partial differential equations. II
Authors: Berger M. S. , Church P. T. , Timourian J. G.
Published: June 2010

Earlier the authors have given abstract properties characterizing the fold and cusp maps on Banach spaces, and these results are applied here to the study of specific nonlinear elliptic boundary value problems. Functional analysis methods are used, specifically, weak solutions in Sobolev spaces. One problem studied is the inhomogeneous nonlinear Dirichlet problem
\[

Δ
u
+
λ
u
−

u
3

=
g

on

Ω
,

u

|

∂
Ω
=
0
,

\Delta u + \lambda u - {u^3} = g\quad {\text {on}}\;\Omega ,\qquad u|\partial \Omega = 0,

\]
where

Ω
⊂

R

n

(
n
⩽
4
)

\Omega \subset {{\mathbf {R}}^n}(n \leqslant 4)

is a bounded domain. Another is a nonlinear elliptic system, the von Kármán equations for the buckling of a thin planar elastic plate when compressive forces are applied to its edge.
Isometry groups of Riemannian solvmanifolds
Authors: Gordon Carolyn S. , Wilson Edward N.
Published: June 2010

A simply connected solvable Lie group

R
R

together with a left-invariant Riemannian metric

g
g

is called a (simply connected) Riemannian solvmanifold. Two Riemannian solvmanifolds

(
R
,

g
)

(R,\,g)

and

(

R
′

,

g
′

)

(R’ ,\,g’ )

may be isometric even when

R
R

and

R
′

R’

are not isomorphic. This article addresses the problems of (i) finding the "nicest" realization

(
R
,

g
)

(R,\,g)

of a given solvmanifold, (ii) describing the embedding of

R
R

in the full isometry group

I
(
R
,

g
)

I(R,\,g)

, and (iii) testing whether two given solvmanifolds are isometric. The paper also classifies all connected transitive groups of isometries of symmetric spaces of noncompact type and partially describes the transitive subgroups of

I
(
R
,

g
)

I(R,\,g)

for arbitrary solvmanifolds

(
R
,

g
)

(R,\,g)

.
Bonnesen-style inequalities for Minkowski relative geometry
Authors: Sangwine-Yager J. R.
Published: June 2010

Two Bonnesen-style inequalities are obtained for the relative inradius of one convex body with respect to another in

n
n

-dimensional space. Both reduce to the known planar inequality; one sharpens the relative isoperimetric inequality, the other states that a quadratic polynomial is negative at the inradius. Circumradius inequalities follow.
Infinitely many periodic solutions for the equation: 𝑢_{𝑡𝑡}-𝑢ₓₓ±|𝑢|^{𝑝-1}𝑢=𝑓(𝑥,𝑡). II
Authors: Tanaka Kazunaga
Published: June 2010

Existence of forced vibrations of nonlinear wave equation:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column u Subscript t t Baseline minus u Subscript x x Baseline plus-or-minus StartAbsoluteValue u EndAbsoluteValue Superscript p minus 1 Baseline u equals f left-parenthesis x comma t right-parenthesis comma 2nd Column a m p semicolon left-parenthesis x comma t right-parenthesis element-of left-parenthesis 0 comma pi right-parenthesis times bold upper R comma 2nd Row 1st Column u left-parenthesis 0 comma t right-parenthesis equals u left-parenthesis pi comma t right-parenthesis equals 0 comma 2nd Column a m p semicolon t element-of bold upper R comma 3rd Row 1st Column u left-parenthesis x comma t plus 2 pi right-parenthesis equals u left-parenthesis x comma t right-parenthesis comma 2nd Column a m p semicolon left-parenthesis x comma t right-parenthesis element-of left-parenthesis 0 comma pi right-parenthesis times bold upper R comma EndLayout">

u

t
t

−

u

x
x

±

|

u

|

p
−
1

u
=
f
(
x
,

t
)
,

a
m
p
;

(
x
,

t
)
∈
(
0
,

π
)
×

R

,

u
(
0
,

t
)
=
u
(
π
,

t
)
=
0
,

a
m
p
;

t
∈

R

,

u
(
x
,

t
+
2
π
)
=
u
(
x
,

t
)
,

a
m
p
;

(
x
,

t
)
∈
(
0
,

π
)
×

R

,

\begin {array}{*{20}{c}} {{u_{tt}} - {u_{xx}} \pm |u{|^{p - 1}}u = f(x,\,t),} \hfill & {(x,\,t) \in (0,\,\pi ) \times {\mathbf {R}},} \hfill \\ {u(0,\,t) = u(\pi ,\,t) = 0,} \hfill & {t \in {\mathbf {R}},} \hfill \\ {u(x,\,t + 2\pi ) = u(x,\,t),} \hfill & {(x,\,t) \in (0,\,\pi ) \times {\mathbf {R}},} \hfill \\ \end {array}

\]
is considered. For all

p
∈
(
1
,

∞
)

p \in (1,\,\infty )

and

f
(
x
,

t
)
∈

L

(
p
+
1
)

/

p

f(x,\,t) \in {L^{(p + 1)/p}}

, existence of infinitely many periodic solutions is proved. This improves the results of the author [29, 30]. We use variational methods to show the existence result. Minimax arguments and energy estimates for the corresponding functional play an essential role in the proof.
Borel measures and Hausdorff distance
Authors: Beer Gerald , Villar Luzviminda
Published: June 2010

In this article we study the restriction of Borel measures defined on a metric space

X
X

to the nonempty closed subsets

CL
⁡
(
X
)

\operatorname {CL} (X)

of

X
X

, topologized by Hausdorff distance. We show that a

σ
\sigma

-finite Radon measure is a Borel function on

CL
⁡
(
X
)

\operatorname {CL} (X)

, and characterize those

X
X

for which each outer regular Radon measure on

X
X

is semicontinuous when restricted to

CL
⁡
(
X
)

\operatorname {CL} (X)

. A number of density theorems for Radon measures are also presented.
On the local boundedness of singular integral operators
Authors: Leckband Mark
Published: June 2010

The class of singular integral operators whose kernels satisfy the usual smoothness conditions is studied. Let such an operator be denoted by

K
K

. We establish necessary conditions that imply

K
K

has local (weighted)

L
p

{L^p}

norm inequalities. The underlying principle is as follows. If

χ
Q

{\chi _Q}

is the characteristic function of a fixed cube

Q
Q

of

R
n

{R^n}

, or all of

R
n

{R^n}

, then

K

χ
Q

K{\chi _Q}

and (the adjoint of

K
K

)

K

∗

χ
Q

{K^{\ast }}{\chi _Q}

determine the boundedness properties of

K
K

for functions supported in a proper fraction of

Q
Q

.
The Brauer group of graded continuous trace 𝐶*-algebras
Authors: Parker Ellen Maycock
Published: June 2010

Let

X
X

be a locally compact Hausdorff space. The graded Morita equivalence classes of separable,

Z

2

{{\mathbf {Z}}_2}

-graded, continuous trace

C

∗

{C^{\ast }}

-algebras which have spectrum

X
X

form a group,

GBr
∞

(
X
)

{\operatorname {GBr} ^\infty }(X)

, the infinite-dimensional graded Brauer group of

X
X

. Techniques from algebraic topology are used to prove that

GBr
∞

(
X
)

{\operatorname {GBr} ^\infty }(X)

is isomorphic via an isomorphism

w
w

to the direct sum
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper H With ˇ Superscript 1 Baseline left-parenthesis upper X semicolon bold upper Z underbar Subscript 2 Baseline right-parenthesis circled-plus ModifyingAbove upper H With ˇ cubed left-parenthesis upper X semicolon bold upper Z underbar right-parenthesis">

H
ˇ

1

(
X
;

Z

_

2

)
⊕

H
ˇ

3

(
X
;

Z

_

)

\check {H}^1(X; \underline {\mathbf {Z}}_2) \oplus \check {H}^3 (X; \underline {\mathbf {Z}})

. The group

GBr
∞

(
X
)

{\operatorname {GBr} ^\infty }(X)

includes as a subgroup the ungraded continuous trace

C

∗

{C^{\ast }}

-algebras, and the Dixmier-Douady invariant of such an ungraded

C

∗

{C^{\ast }}

-algebra is its image in

H
ˇ

3

(
X
;

Z

_

)

\check {H}^3 (X; \underline {\mathbf {Z}})

under

w
w

.
The Morse index theorem where the ends are submanifolds
Authors: Kalish Diane
Published: June 2010

In this paper the Morse Index Theorem is proven in the case where submanifolds

P
P

and

Q
Q

are at the endpoints of a geodesic,

γ
\gamma

. At

γ
\gamma

, the index of the Hessian of the energy function defined on paths joining

P
P

and

Q
Q

is computed using

P
P

-focal points, and a calculation at the endpoint of

γ
\gamma

, involving the second fundamental form of

Q
Q

.
Construction of an inner function in the little Bloch space
Authors: Stephenson Kenneth
Published: June 2010

An explicit construction using Riemann surfaces and Brownian motion is given for an inner function in the unit disc which is not a finite Blaschke product yet belongs to the little Bloch space

B

0

{\mathcal {B}_0}

. In addition to showing how an inner function can meet the geometric conditions for

B

0

{\mathcal {B}_0}

, this example settles an open question concerning the finite ranges of inner functions: the values which it takes only finitely often are dense in the disc.
Templates and train tracks
Authors: Frank George
Published: June 2010

Within the context of Smale flows on compact manifolds, this article deals with a relationship between abstract templates, branched

1
1

-manifolds (train tracks), and laminations representing unstable separatrices of basic sets. We show that an abstract template, the richest in information of the above three entities, determines a member of each of the remaining two groups, and partial determinations in other directions are developed. As a result of this relationship, an obstruction to the realization of certain abstract templates in nonsingular Smale flows on homology

3
3

-spheres is raised.
Normal derivative for bounded domains with general boundary
Authors: Gong Guang Lu , Qian Min Ping , Silverstein Martin L.
Published: June 2010

Let

D
D

be a general bounded domain in the Euclidean space

R
n

{R^n}

. A Brownian motion which enters from and returns to the boundary symmetrically is used to define the normal derivative as a functional for

f
f

with

f
f

,

∇
f

\nabla f

and

Δ
f

\Delta f

all in

L
2

{L^2}

on

D
D

. The corresponding Neumann condition (normal derivative

=
0

= 0

) is an honest boundary condition for the

L
2

{L^2}

generator of reflected Brownian notion on

D
D

. A conditioning argument shows that for

D
D

and

f
f

sufficiently smooth this general definition of the normal derivative agrees with the usual one.
Rigidity theorems for right angled reflection groups
Authors: Rosas Ennis
Published: June 2010

Let

Γ
\Gamma

be a right angled reflection group. Let

M
M

and

M
′

M’

be Coxeter manifolds. Then any

Γ
\Gamma

-map

f
:
M
→

M
′

f:M \to M’

is

Γ
\Gamma

-homotopic to a homeomorphism
Engulfing and subgroup separability for hyperbolic groups
Authors: Long D. D.
Published: June 2010

If a group is subgroup separable, otherwise known as locally extended residually finite or LERF, one can pass from immersions to embeddings in some finite covering space. We show that a certain ’engulfing’ property gives subgroup separability for a large and useful class of subgroups of hyperbolic

3
3

-manifold groups.
Carleson measures and multipliers of Dirichlet-type spaces
Authors: Kerman Ron , Sawyer Eric
Published: June 2010

A function

ρ
\rho

from

[
0
,

1
]

[0,\,1]

onto itself is a Dirichlet weight if it is increasing,

ρ
⩽
0

\rho \leqslant 0

and

lim

x
→
0
+

x

/

ρ
(
x
)
=
0

{\lim _{x \to 0 + }}x/\rho (x) = 0

. The corresponding Dirichlet-type space,

D
ρ

{D_\rho }

, consists of those bounded holomorphic functions on

U
=
{
z
∈

C

:

|

z

|

>
1
}

U = \{ z \in {\mathbf {C}}:\,|z| > 1\}

such that

|

f
′

(
z
)

|

2

ρ
(
1
−

|

z

|

)

|f’(z){|^2}\rho (1 - |z|)

is integrable with respect to Lebesgue measure on

U
U

. We characterize in terms of a Carleson-type maximal operator the functions in the set of pointwise multipliers of

D
ρ

{D_\rho }

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis upper D Subscript rho Baseline right-parenthesis equals left-brace g colon upper U right-arrow bold upper C colon g f element-of upper D Subscript rho Baseline comma for-all f element-of upper D Subscript rho Baseline right-brace">

M
(

D
ρ

)
=
{
g
:

U
→

C

:
g
f
∈

D
ρ

,
∀
f
∈

D
ρ

}

M({D_\rho }) = \{ g:\,U \to {\mathbf {C}}:gf \in {D_\rho },\forall f \in {D_\rho }\}

.
On the canonical rings of some Horikawa surfaces. I
Authors: Iliev Valentin
Published: June 2010

This paper is devoted to finding necessary and sufficient conditions for a graded ring to be the canonical ring of a minimal surface of general type with

K
2

=
2

p
g

−
3

{K^2} = 2{p_g} - 3

,

p
g

⩾
3

{p_g} \geqslant 3

, and such that its canonical linear system has one base point.
A Diophantine problem on elliptic curves
Authors: Tubbs Robert
Published: June 2010

This paper examines simultaneous diophantine approximations to coordinates of certain points on a product of elliptic curves. Specifically, let

℘
(
z
)

\wp (z)

be a Weierstrass elliptic function with algebraic invariants and complex multiplication. Suppose that

β
\beta

is cubic over the "field of multiplications" of

℘
(
z
)

\wp (z)

and that

u
∈

C

u \in \mathbb {C}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta equals left-parenthesis normal script upper P left-parenthesis u right-parenthesis comma normal script upper P left-parenthesis beta u right-parenthesis comma normal script upper P left-parenthesis beta squared u right-parenthesis right-parenthesis">

ζ
=
(
℘
(
u
)
,

℘
(
β
u
)
,

℘
(

β
2

u
)
)

\zeta = (\wp (u),\,\wp (\beta u),\,\wp ({\beta ^2}u))

is defined. We study approximations to

ζ
\zeta

by points which lie on curves defined over

Z

\mathbb {Z}

.
Boundary behavior of invariant Green’s potentials on the unit ball in 𝐶ⁿ
Authors: Hahn K. T. , Singman David
Published: June 2010

Let

p
(
z
)
=

∫
B

G
(
z
,

w
)

d
μ
(
w
)

p(z) = \int _B {G(z,\,w)\,d\mu (w)}

be an invariant Green’s potential on the unit ball

B
B

in

C

n

(
n
⩾
1
)

{{\mathbf {C}}^n}\;(n \geqslant 1)

, where

G
G

is the invariant Green’s function and

μ
\mu

is a positive measure with

∫
B

(
1
−

|

w

|

2

)

n

d
μ
(
w
)
>
∞

\int _B {{{(1 - |w{|^2})}^n}\,d\mu (w) > \infty }

. In this paper, a necessary and sufficient condition on a subset

E
E

of

B
B

such that for every invariant Green’s potential

p
p

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript z right-arrow e Endscripts inf left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis Superscript n Baseline p left-parenthesis z right-parenthesis equals 0 comma e equals left-parenthesis 1 comma 0 comma ellipsis comma 0 right-parenthesis element-of partial-differential upper B comma z element-of upper E comma">

lim

z
→
e

inf

(
1
−

|

z

|

2

)
n

p
(
z
)
=
0
,

e
=
(
1
,

0
,

…
,

0
)

∈
∂
B
,

z
∈
E
,

\lim \limits _{z \to e} \,\inf {(1 - |z{|^2})^n}p(z) = 0,\qquad e = (1,\,0,\, \ldots ,\,0)\; \in \partial B,\;z \in E,

\]
is given. The condition is that the capacity of the sets

E
∩
{
z
∈
B

|

|

z
−
e

|

>
ε
}

E \cap \{ z \in B|\;|z - e| > \varepsilon \}

,

ε
>
0

\varepsilon > 0

, is bounded away from

0
0

. The result obtained here generalizes Luecking’s result, see [L], on the unit disc in

C

{\mathbf {C}}

.
The box product of countably many copies of the rationals is consistently paracompact
Authors: Lawrence L. Brian
Published: June 2010

By proving the theorem stated in the title, we show that local compactness in the factor spaces is not necessary for paracompactness in the box product.
Borel orderings
Authors: Harrington Leo , Marker David , Shelah Saharon
Published: June 2010

We show that any Borel linear order can be embedded in an order preserving way into

2
α

{2^\alpha }

for some countable ordinal

α
\alpha

and that any thin Borel partial order can be written as a union of countably many Borel chains.
Banach spaces with separable duals
Authors: Zippin M.
Published: June 2010

It is proved that every Banach space with a separable dual embeds into a space with a shrinking basis. It follows that every separable reflexive space can be embedded in a reflexive space with a basis.
A characterization of the weakly continuous polynomials in the method of compensated compactness
Authors: Rogers Robert C. , Temple Blake
Published: June 2010

We present a sufficient condition for weak continuity in the method of compensated compactness. The condition links weak continuity to the structure of the wave cone and the characteristic set for polynomials of degree greater than two. The condition applies to all the classical examples of weakly continuous functions and generalizes the Quadratic Theorem and the Wedge Product Theorem. In fact, the condition reduces to the Legendre-Hadamard Necessary Condition when the polynomial is quadratic, and also whenever a certain orthogonality condition is satisfied. The condition is derived by isolating conditions under which the quadratic theorem can be iterated.
Generalizations of Cauchy’s summation theorem for Schur functions
Authors: Andrews G. E. , Goulden I. P. , Jackson D. M.
Published: June 2010

Cauchy’s summation theorem for Schur functions is generalized, and a number of related results are given. The result is applied to a combinatorial problem involving products of pairs of permuations, by appeal to properties of the group algebra of the symmetric group.
Remarks on approximation methods in degree theory
Authors: Kryszewski W. , Przeradzki B. , Wereński S.
Published: June 2010

An approximative approach to a generalized theory of the topological mapping degree is presented. Some new wide classes of operators acting in Banach spaces, which include

A
A

-proper mappings of Petryshyn, are introduced and studied from the viewpoint of the homotopic properties of the topological degree. The results are applied in some existence aspects of abstract nonlinear equations.
Determination of all imaginary cyclic quartic fields with class number 2
Authors: Hardy Kenneth , Hudson Richard H. , Richman David , Williams Kenneth S.
Published: June 2010

It is proved that there are exactly

8
8

imaginary cyclic quartic fields with class number

2
2

.
Excessive measures and the existence of right semigroups and processes
Authors: Steffens J.
Published: June 2010

Given a resolvent

(

U
α

)

({U_\alpha })

on a Lusin space

(
E
,

E

)

(E,\mathfrak {E})

, the paper gives necessary and sufficient conditions in terms of the excessive measures that ensure the existence of a right process, resp. a right continuous semigroup, on

(
E
,

E

)

(E,\mathfrak {E})

with resolvent

(

U
α

)

({U_\alpha })

. Furthermore, a notion of nonbranch points with respect to

(

U
α

)

({U_\alpha })

is introduced—also in terms of the excessive measures—and various characterizations are given. They show, in particular, the equivalence of this definition with those introduced and discussed by Engelbert and Wittmann.
Two-dimensional Riemann problem for a single conservation law
Authors: Zhang Tong , Zheng Yu Xi
Published: June 2010

The entropy solutions to the partial differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis partial-differential slash partial-differential t right-parenthesis u left-parenthesis t comma x comma y right-parenthesis plus left-parenthesis partial-differential slash partial-differential x right-parenthesis f left-parenthesis u left-parenthesis t comma x comma y right-parenthesis right-parenthesis plus left-parenthesis partial-differential slash partial-differential y right-parenthesis g left-parenthesis u left-parenthesis t comma x comma y right-parenthesis right-parenthesis equals 0 comma">

(
∂

/

∂
t
)
u
(
t
,
x
,
y
)
+
(
∂

/

∂
x
)
f
(
u
(
t
,
x
,
y
)
)
+
(
∂

/

∂
y
)
g
(
u
(
t
,
x
,
y
)
)
=
0
,

(\partial /\partial t)u(t,x,y) + (\partial /\partial x)f(u(t,x,y)) + (\partial /\partial y)g(u(t,x,y)) = 0,

\]
with initial data constant in each quadrant of the

(
x
,
y
)

(x,y)

plane, have been constructed and are piecewise smooth under the condition
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis u right-parenthesis not-equals 0 comma g left-parenthesis u right-parenthesis not-equals 0 comma left-parenthesis f left-parenthesis u right-parenthesis slash g left-parenthesis u right-parenthesis right-parenthesis prime not-equals 0">

f
(
u
)
≠
0
,
g
(
u
)
≠
0
,
(
f
(
u
)

/

g
(
u
)
)
′
≠
0

f(u) \ne 0, g(u) \ne 0, (f(u)/g(u))\prime \ne 0

. This problem generalizes to several space dimensions the important Riemann problem for equations in one-space dimension. Although existence and uniqueness of solutions are well known, little is known about the qualitative behavior of solutions. It is this with which we are concerned here.
On invariants of graphs with applications to knot theory
Authors: Murasugi Kunio
Published: June 2010

To each weighted graph

Γ
\Gamma

, two invariants, a polynomial

P
Γ

(
x
,
y
,
z
)

{P_\Gamma }(x,y,z)

and the signature

σ
(
Γ
)

\sigma (\Gamma )

, are defined. The various partial degress of

P
Γ

(
x
,
y
,
z
)

{P_\Gamma }(x,y,z)

and

σ
(
Γ
)

\sigma (\Gamma )

are expressed in terms of maximal spanning graphs of

Γ
\Gamma

. Furthermore, one unexpected property of Tutte’s dichromate is proved. These results are applied to knots or links in

S
3

{S^3}

.
Cauchy-Szegő maps, invariant differential operators and some representations of 𝑆𝑈(𝑛+1,1)
Authors: Meaney Christopher
Published: June 2010

Fix an integer

n
>
1

n > 1

. Let

G
G

be the semisimple Lie group

SU

(
n
+
1
,
1
)

{\text {SU}}(n + 1,1)

and

K
K

be the subgroup

S(U

(
n
+
1
)
×

U

(
1
)
)

{\text {S(U}}(n + 1) \times {\text {U}}(1))

. For each finite dimensional representation

(
τ
,

H

τ

)

(\tau ,{\mathcal {H}_\tau })

of

K
K

there is the space of smooth

τ
\tau

-covariant functions on

G
G

, denoted by

C
∞

(
G
,
τ
)

{C^\infty }(G,\tau )

and equipped with the action of

G
G

by right translation. Now take

(
τ
,

H

τ

)

(\tau ,{\mathcal {H}_\tau })

to be

(

τ

p
,
p

,

H

p
,
p

)

({\tau _{p,p}},{\mathcal {H}_{p,p}})

, the representation of

K
K

on the space of harmonic polynomials on

C

n
+
1

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

which are bihomogeneous of degree

(
p
,
p
)

(p,p)

. For a real number

ν
\nu

there is the corresponding spherical principal series representation of

G
G

, denoted by

(

π
ν

,

I

1
,
ν

)

({\pi _\nu },{{\mathbf {I}}_{1,\nu }})

. In this paper we show that, as a

(

g

,
K
)

(\mathfrak {g},K)

-module, the irreducible quotient of

I

1
,
1
−
n
−
2
p

{{\mathbf {I}}_{1,1 - n - 2p}}

can be realized as the space of the

K
K

-finite elements of the kernel of a certain invariant first order differential operator acting on

C
∞

(
G
,

τ

p
,
p

)

{C^\infty }(G,{\tau _{p,p}})

. Johnson and Wallach had shown that these representations are not square-integrable. Thus, some exceptional representations of

G
G

are realized in a manner similar to Schmid’s realization of the discrete series. The kernels of the differential operators which we use here are the intersection of kernels of some Schmid operators and quotient maps, which we call Cauchy-Szegö maps, a generalization the Szegö maps used by Knapp and Wallach. We also identify this representation of

G
G

with an end of complementary series representation.
Cohomology equations and commutators of germs of contact diffeomorphisms
Authors: Banyaga Augustin , de la Llave Rafael , Wayne C. Eugene
Published: June 2010

We study the group of germs of contact diffeomorphisms at a fixed point. We prove that the abelianization of this group is isomorphic to the multiplicative group of real positive numbers. The principal ingredient in this proof is a version of the Sternberg linearization theorem in which the conjugating diffeomorphism preserves the contact structure.
Deviations of trajectory averages and the defect in Pesin’s formula for Anosov diffeomorphisms
Authors: Orey Steven , Pelikan Stephan
Published: June 2010

A large deviation theorem at the Donsker Varadhan level three is obtained for the convergence of trajectory averages of Anosov diffeomorphisms. It is possible to provide an explicit description of the rate function.
Quantization of curvature of harmonic two-spheres in Grassmann manifolds
Authors: Zheng Yunbo
Published: June 2010

Various pinching theorems for curvature of minimal two-spheres in Grassmann manifolds have been proved. In particular, we show that when the curvature is large, then the minimal map from

S
2

{S^2}

into

G
(
m
,
N
)

G(m,N)

must be either holomorphic or antiholomorphic. Also, minimal two-spheres of curvature

κ
⩾
2

\kappa \geqslant 2

in

G
(
2
,
4
)

G(2,4)

have been classified.
Double shock fronts for hyperbolic systems of conservation laws in multidimensional space
Authors: Bui An Ton , Li De Ning
Published: June 2010

The existence of a unique double shock front for hyperbolic systems of conservation laws in several space variables is established, extending an earlier result of Metivier. An example of a double shock wave arising from physical applications is given.
Propagation of 𝐿^{𝑞}_{𝑘}-smoothness for solutions of the Euler equation
Authors: Ponce Gustavo
Published: June 2010

The motion of an ideal incompressible fluid is described by a system of partial differential equations known as the Euler equation. Considering the initial value problem for this equation, we prove that in a classical solution the

L
k
q

L_k^q

-regularity of the data propagates along the fluid lines. Our method consists of combining properties of the

ε
\varepsilon

-approximate solution with

L
q

{L^q}

-energy estimates and simple results of classical singular integral operators. In particular, for the two-dimensional case we present an elementary proof.
On a problem of S. Mazur
Authors: Székelyhidi László
Published: June 2010

In this work a generalization of Mazur’s problem concerning the continuity of linear functionals is given.
The intrinsic conformal structure and Gauss map of a light-like hypersurface in Minkowski space
Authors: Kossowski Marek
Published: June 2010

We begin by pointing out two subtleties in the global properties of hypersurfaces in Minkowski space which inherit a uniformly degenerate metric (i.e., the existence of global space-like sections and the notion of an icon; see Appendices 1 and 2). We then construct a Gauss map for such hypersurfaces and an intrinsic invariant. This leads us to results concerning light-like hypersurfaces which parallel known results concerning surfaces in Euclidean space.
On the Kummer congruences and the stable homotopy of 𝐵U
Authors: Baker Andrew , Clarke Francis , Ray Nigel , Schwartz Lionel
Published: June 2010

We study the torsion-free part of the stable homotopy groups of the space

B
U

BU

, by considering upper and lower bounds. The upper bound is furnished by the ring

P

K

∗

(
B
U
)

P{K_{\ast }}(BU)

of coaction primitives into which

π

∗

S

(
B
U
)

\pi _{\ast }^S(BU)

is mapped by the complex

K
K

-theoretic Hurewicz homomorphism
\[

π

∗

S

(
B
U
)
→
P

K

∗

(
B
U
)
.

\pi _{\ast }^S(BU) \to P{K_{\ast }}(BU).

\]
We characterize

P

K

∗

(
B
U
)

P{K_{\ast }}(BU)

in terms of symmetric numerical polynomials and describe systematic families of elements by utilizing the classical Kummer congruences among the Bernoulli numbers. For a lower bound we choose the ring of those framed bordism classes which may be represented by singular hypersurfaces in

B
U

BU

. From among these we define families of classes constructed from regular neighborhoods of embeddings of iterated Thom complexes in Euclidean space. Employing techniques of duality theory, we deduce that these two families correspond, except possibly in the lowest dimensions, under the Hurewicz homomorphism, which thus provides a link between the algebra and the geometry. In the course of this work we greatly extend certain

e
e

-invariant calculations of J. F. Adams.
On the dissipative evolution equations associated with the Zakharov-Shabat system with a quadratic spectral parameter
Authors: Lee Jyh-Hao
Published: June 2010

In this paper we derive some results for the Zakharov-Shabat system of the form

d
m

/

d
x
=

z
2

[
J
,
m
]
+
(
z
Q
+
P
)
m

dm/dx = {z^2}[J,m] + (zQ + P)m

;

J
J

is diagonal and skew-Hermitian

[
8
,
10
,
12
]

[8,10,12]

. Following the idea of R. Beals and R. R. Coifman, we estimate the wedge products of the columns of

m
m

by

L
2

{L^2}

-norm of the potential

(
Q
,
P
)

[
4
]

(Q,P)\,[4]

. By this result we have the global existence of the dissipative evolution equations associated with this spectral problem if the generic initial data

(
Q
(
x
,
0
)
,

P
(
x
,
0
)
)
=
(

Q
0

,

P
0

)

(Q(x,0),\,P(x,0)) = ({Q_0},{P_0})

is of Schwartz class.
Topological types of finitely-𝐶⁰-𝐾-determined map-germs
Authors: Nishimura Takashi
Published: June 2010

In this article, we investigate the following two problems Problem 1. Is finite-

C
0

-

K

{C^0}{\text {-}}K

-determinacy a topological invariant among analytic map-germs? Problem 2. Do the topological types of all finitely-

C
0

-

K

{C^0}{\text {-}}K

-determined map-germs have topological moduli, i.e. do they have infinitely many topological types with the cardinal number of continuum? Problem

1
1

is solved affirmatively in the complex case. Problem

2
2

is solved negatively in the complex case; and affirmatively in the real case.
Scattering for the Yang-Mills equations
Authors: Baez John C.
Published: June 2010

We construct wave and scattering operators for the Yang-Mills equations on Minkowski space,

M

0

≅

R

4

{{\mathbf {M}}_0} \cong {{\mathbf {R}}^4}

. Sufficiently regular solutions of the Yang-Mills equations on

M

0

{{\mathbf {M}}_0}

are known to extend uniquely to solutions of the corresponding equations on the universal cover of its conformal compactification,

M

~

≅

R

×

S
3

\tilde {\mathbf {M}} \cong {\mathbf {R}} \times {S^3}

. Moreover, the boundary of

M

0

{{\mathbf {M}}_0}

as embedded in

M

~

\tilde {\mathbf {M}}

is the union of "lightcones at future and past infinity",

C
±

{C_ \pm }

. We construct wave operators

W
±

{W_ \pm }

as continuous maps from a space

X

{\mathbf {X}}

of time-zero Cauchy data for the Yang-Mills equations to Hilbert spaces

H

(

C
±

)

{\mathbf {H}}({C_ \pm })

of Goursat data on

C
±

{C_ \pm }

. The scattering operator is then a homeomorphism

S
:

X

→

X

S:{\mathbf {X}} \to {\mathbf {X}}

such that

U

W
+

=

W
−

S

U{W_ + } = {W_ - }S

, where

U
:

H

(

C
+

)
→

H

(

C
−

)

U:{\mathbf {H}}({C_ + }) \to {\mathbf {H}}({C_ - })

is the linear isomorphism arising from a conformal transformation of

M

~

\tilde {\mathbf {M}}

mapping

C
−

{C_ - }

onto

C
+

{C_ + }

. The maps

W
±

{W_ \pm }

and

S
S

are shown to be smooth in a certain sense.
The spectrum of the Schrödinger operator
Authors: Schechter Martin
Published: June 2010

We describe the negative spectrum of the Schrödinger operator with a singular potential. We determine the exact value of the bottom of the spectrum and estimate it from above and below. We describe the dependence of a crucial constant on the eigenvalue parameter and discuss some of its properties. We show how recent results of others are simple consequences of a theorem proved by the author in 1972.
Semicharacteristics, bordism, and free group actions
Authors: Davis James F. , Milgram R. James
Published: June 2010

In this paper we give characteristic class formulae for all semicharacteristic classes of all compact, closed manifolds with finite fundamental groups. These invariants are identified with elements in certain odd

L
L

-groups, and exactly which elements occur is specified. An appendix calculates the cohomology of the model groups needed. A second appendix determines the structure of the

L
L

-groups needed.
Pixley-Roy hyperspaces of 𝜔-graphs
Authors: Mashburn J. D.
Published: June 2010

The techniques developed by Wage and Norden are used to show that the Pixley-Roy hyperspaces of any two

ω
\omega

-graphs are homeomorphic. The Pixley-Roy hyperspaces of several subsets of

R

n

{{\mathbf {R}}^n}

are also shown to be homeomorphic.
The structure of an even liaison class
Authors: Bolondi Giorgio , Migliore Juan C.
Published: June 2010

We describe a structure called the Lazarsfeld-Rao property for even liaison classes in projective space. This property holds for many even liaison classes of curves in

P

3

{{\mathbf {P}}^3}

. We give a procedure for showing that an even liaison class in codimension

2
2

possesses this property, and we prove it for a family of even liaison classes in codimension

2
2

in any

P

n

,

n
⩾
3

{{\mathbf {P}}^n},\;n \geqslant 3

. However, we conjecture that it in fact holds for every even liaison class in codimension

2
2

, so we want to give consequences for an even liaison class that possesses this property. The main element in describing this structure is the notion of a basic double link. The Lazarsfeld-Rao property says that there exist minimal elements of the even liaison class and that any element of the even liaison class can be deformed to a curve obtained by a sequence of basic double links beginning with any minimal element. We show that there is a unique standard type of sequence for any given element of the even liaison class. As a result, we can express the even liaison class as a disjoint union of irreducible nonempty families parameterized by certain finite sequences of integers. The standard numerical invariants of the elements of any family can be computed from the associated sequence of integers. We apply this to surfaces in

P

4

{{\mathbf {P}}^4}

. Our main tool for these results is a deformation technique related to liaison in codimension

2
2

. We also study Schwartau’s procedure of Liaison Addition in codimension

2
2

from the point of view of vector bundles. Using this, we give a different sort of structure for an even liaison class with the Lazarsfeld-Rao property.
A converse to the mean value property on homogeneous trees
Authors: Picardello Massimo A. , Woess Wolfgang
Published: June 2010

The homogeneous tree

T

{\mathbf {T}}

of degree

q
+
1

(
q
≥
2
)

q + 1\quad (q \geq 2)

may be considered as a discrete analogue of the open unit disc

D

{\mathbf {D}}

. On

D

{\mathbf {D}}

, every harmonic function satisfies the mean value property (MVP) at every point. Conversely, positive functions on

D

{\mathbf {D}}

having the MVP with respect to a ball with specified radius at each point of

D

{\mathbf {D}}

are harmonic under certain assumptions concerning the radius function: results of this type are due to J. R. Baxter, W. Veech and others. Here we consider harmonic functions on

T

{\mathbf {T}}

with respect to a natural choice of a discrete Laplacian: the analogous MVP is true in this setting. We present a Lipschitz-type condition on the radius function (which now has integer values and refers to the discrete metric of

T

{\mathbf {T}}

) under which harmonicity holds for positive functions whose value at each point is the mean of its values over the ball of the radius assigned to this point. The method is based upon our previous results concerning the geometrical realization of Martin boundaries of certain transition operators as the space of ends of the underlying graph.
Torsion points of generic formal groups
Authors: Rosen Michael , Zimmermann Karl
Published: June 2010

Let

F
F

be a generic formal group of height

h
h

defined over

A
=

Z

p

[
[

t
1

,
…
,

t

h
−
1

]
]

A = {{\mathbf {Z}}_p}[[{t_1}, \ldots ,{t_{h - 1}}]]

. Let

K
K

be the quotient field of

A
A

. We show the natural map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript n Baseline colon Gal left-parenthesis upper K left-parenthesis kernel left-bracket p Superscript n Baseline right-bracket right-parenthesis slash upper K right-parenthesis right-arrow upper G upper L Subscript h Baseline left-parenthesis bold upper Z slash p Superscript n Baseline bold upper Z right-parenthesis">

ρ
n

:

Gal

(
K
(
ker
⁡
[

p
n

]
)

/

K
)
→
G

L
h

(

Z

/

p
n

Z

)

{\rho _n}:{\text {Gal}}(K(\operatorname {ker} [{p^n}])/K) \to G{L_h}({\mathbf {Z}}/{p^n}{\mathbf {Z}})

isomorphisms for all

n
≥
1

n \ge 1

provided

p
≠
2

p \ne 2

.
Spreading of singularities at the boundary in semilinear hyperbolic mixed problems. II. Crossing and self-spreading
Authors: Williams Mark
Published: June 2010

The creation of anomalous singularities in solutions to nonlinear hyperbolic equations due to crossing or self-spreading in free space is by now rather well understood. In this paper we study how anomalous singularities are produced in mixed problems for semilinear wave equations

◻

u
=
f
(
u
)

\square u = f(u)

on the half-space

R

+

n
+
1

,
u
∈

H

loc

s

,
s
>
(
n
+
1
)

/

2

{\mathbf {R}}_ + ^{n + 1},u \in H_{{\operatorname {loc}}}^s,s > (n + 1)/2

, due to crossing and self-spreading at boundary points. Several phenomena appear in the problems considered here which distinguish spreading at the boundary from spreading in free space: (1) Anomalous singularities of strength

∼
2
s
−
n

/

2

\sim 2s - n/2

can arise when incoming singularity-bearing rays cross or self-spread at a point on the boundary. A consequence of this, announced in [14], is that the analogue of Beals’s

3
s

3s

theorem fails for reflection in second-order mixed problems. Although

H
r

{H^r}

regularity for

r
>∼
3
s
−
n

r > \sim 3s -n

propagates along null bicharacteristics in free space, for

r
>∼
2
s
−
n

/

2

r > \sim 2s -n/2

it does not in general reflect. (2) For nonlinear wave equations in free space, anomalous singular support is never produced by the interaction of fewer than three bicharacteristics, unless self-spreading occurs. However, anomalous singularities can arise when a pair of rays cross at a boundary point. (3) Suppose

◻

u
=

u
2

\square u = {u^2}

and

u
∈

C
∞

u \in {C^\infty }

on the boundary. For certain choices of initial data, anomalous singularities of strength

∼
2
s
−
n

/

2

\sim 2s - n/2

arise at the boundary from three sources: interactions of incoming rays with incoming rays, incoming rays with reflected rays, and reflected rays with reflected rays. Singularities produced by the incoming-reflected interactions differ in sign from and are strictly weaker than the other two types, so some cancellations occur. As the incoming rays approach being gliding rays, the difference in strength decreases and hence the cancellations become increasingly significant.
Primary cycles on the circle
Authors: Block Louis , Coven Ethan M. , Jonker Leo , Misiurewicz Michał
Published: June 2010

We consider cycles, i.e., periodic orbits, of continuous degree one maps of the circle. One cycle forces another if every such map that has a cycle which looks like the first also has a cycle which looks like the second. We characterize primary cycles, i.e., those which force no other cycle of the same period.
Specializations of finitely generated subgroups of abelian varieties
Authors: Masser D. W.
Published: June 2010

Given a generic Mordell-Weil group over a function field, we can specialize it down to a number field. It has been known for some time that the resulting homomorphism of groups is injective "infinitely often". We prove that this is in fact true "almost always", in a sense that is quantitatively nearly best possible.
Some families of isoparametric hypersurfaces and rigidity in a complex hyperbolic space
Authors: Vernon Micheal H.
Published: June 2010

The geometric notion of equivalence for submanifolds in a chosen ambient space is that of congruence. In this study, a certain type of isoparametric hypersurface of a complex hyperbolic space form is shown to have a rigid immersion by utilizing the congruences of a Lorentzian hyperbolic space form that lies as an

S
1

{S^1}

-fiber bundle over the complex hyperbolic space. Several families of isoparametric hypersurfaces (namely tubes and horospheres) are constructed whose immersions are rigid.
The uniform bound problem for local birational nonsingular morphisms
Authors: Johnston Bernard
Published: June 2010

It is known that any factorization of a local birational morphism

f
:
Spec

S
→
Spec

R

f:\operatorname {Spec}\;S \to \operatorname {Spec}\;R

of nonsingular (affine) schemes of arbitrary dimension via other nonsingular schemes must be finite in length. This fact generalizes the classical Local Factorization Theorem of Zariski and Abhyankar, which states that there is a unique such factorization, that given by quadratic transformations, in the surface case. A much stronger generalization is given here, namely, that there exists a uniform bound on the lengths of all such factorizations, provided that

R
R

is excellent. This bound is explicitly calculated for some concrete extensions and examples are given to show that this is the strongest generalization possible in some sense.
The transfer ideal of quadratic forms and a Hasse norm theorem mod squares
Authors: Leep David B. , Wadsworth Adrian R.
Published: June 2010

Any finite degree field extension

K

/

F

K/F

determines an ideal

T

K

/

F

{\mathcal {T}_{K/F}}

of the Witt ring

W
F

WF

of

F
F

, called the transfer ideal, which is the image of any nonzero transfer map

W
K
→
W
F

WK \to WF

. The ideal

T

K

/

F

{\mathcal {T}_{K/F}}

is computed for certain field extensions, concentrating on the case where

K
K

has the form

F

(

a
1

,
…
,

a
n

)

F\left ({\sqrt {{a_1}} , \ldots ,\sqrt {{a_n}} } \right )

,

a
i

∈
F

{a_i} \in F

. When

F
F

and

K
K

are global fields, we investigate whether there is a local global principle for membership in

T

K

/

F

{\mathcal {T}_{K/F}}

. This is shown to be equivalent to the existence of a "Hasse norm theorem mod squares," i.e., a local global principle for the image of the norm map

N

K

/

F

:

K
∗

/

K

∗
2

→

F
∗

/

F

∗
2

{N_{K/F}}: {K^\ast }/{K^{\ast 2}} \to {F^\ast }/{F^{\ast 2}}

. It is shown that such a Hasse norm theorem holds whenever

K
=
F
(

a
1

,
…
,

a
n

)

K = F(\sqrt {a_1},\ldots ,\sqrt {a_n})

, although it does not always hold for more general extensions of global fields, even some Galois extensions with group

Z

/

2

Z

×

Z

/

4

Z

\mathbb {Z}/2\mathbb {Z} \times \mathbb {Z}/4\mathbb {Z}

.
Nonuniqueness for solutions of the Korteweg-de Vries equation
Authors: Cohen Amy , Kappeler Thomas
Published: June 2010

Variants of the inverse scattering method give examples of nonuniqueness for the Cauchy problem for

KdV

{\text {KdV}}

. One example gives a nontrivial

C
∞

{C^\infty }

solution

u
u

in a domain

{
(
x
,
t
)
:
0
>
t
>
H
(
x
)
}

\{ (x,t):0 > t > H(x)\}

for a positive nondecreasing function

H
H

, such that

u
u

vanishes to all orders as

t
↓
0

t \downarrow 0

. This solution decays rapidly as

x
→
+
∞

x \to + \infty

, but cannot be well behaved as

x
x

moves left. A different example of nonuniqueness is given in the quadrant

x
≥
0
,
t
≥
0

x \geq 0,t \geq 0

, with nonzero initial data.
Braids, link polynomials and a new algebra
Authors: Birman Joan S. , Wenzl Hans
Published: June 2010

A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on

2
2

parameters. The decomposition of the corresponding algebras into irreducible components is given and it is shown how they are related to Jones’ algebras and to Brauer’s centralizer algebras.
Area integral estimates for caloric functions
Authors: Brown Russell M.
Published: June 2010

We study the relationship between the area integral and the parabolic maximal function of solutions to the heat equation in domains whose boundary satisfies a

(

1
2

,
1

)

\left ({\frac {1}{2},1}\right )

mixed Lipschitz condition. Our main result states that the area integral and the parabolic maximal function are equivalent in

L
p

(
μ
)

{L^p}(\mu )

,

0
>
p
>
∞

0 > p > \infty

. The measure

μ
\mu

must satisfy Muckenhoupt’s

A
∞

{A_\infty }

-condition with respect to caloric measure. We also give a Fatou theorem which shows that the existence of parabolic limits is a.e. (with respect to caloric measure) equivalent to the finiteness of the area integral.
On numerical integration of ordinary differential equations
Authors: Nordsieck Arnold
Published: June 2010

A reliable efficient general-purpose method for automatic digital computer integration of systems of ordinary differential equations is described. The method operates with the current values of the higher derivatives of a polynomial approximating the solution. It is thoroughly stable under all circumstances, incorporates automatic starting and automatic choice and revision of elementary interval size, approximately minimizes the amount of computation for a specified accuracy of solution, and applies to any system of differential equations with derivatives continuous or piecewise continuous with finite jumps. ILLIAC library subroutine #F7, University of Illinois Digital Computer Laboratory, is a digital computer program applying this method.
Variation principles for an arbitrary operator. III
Authors: Delves L. M.
Published: June 2010

The methods, described in two previous papers, for generating variation principles for the matrix elements of Hermitian operators are extended here in several ways. The method is first extended to cover inhomogeneous equations. A defect of the original formulation, that it involved two trial functions, is removed by rewriting the principle so that one only appears. Finally, variation-iteration schemes are proposed.
Numerical calculation of integrals with strongly oscillating integrand
Authors: van de Vooren A. I. , van Linde H. J.
Published: June 2010

In this paper a method is presented for evaluating\[∫0Nf(x)eiwxdx where ωN=p⋅2π, p integer.\int _0^N {f(x){e^{iwx}}} dx{\text { where }}\omega N = p \cdot 2\pi ,{\text { }}p{\text { integer}}{\text {.}}\]The idea is to approximatef(x)f(x)instead of the whole integrand by aid of polynomials. The Romberg-Stiefel algorithm has been extended to this case. The new method is complementary to the usual Romberg-Stiefel algorithm in the sense that it is more advantageous for larger values ofω\omega. An expression for the remainder term is also included. Results for the real part are exact iff(x)f(x)is of at most 7th degree and for the imaginary part iff(x)f(x)is of at most 8th degree.
On the evaluation of the incomplete gamma function
Authors: Takenaga Roy
Published: June 2010

The accurate evaluation of the

χ
2

{\chi ^2}

distribution for high degrees of freedom by the usual methods is very difficult (even with a digital computer) because the series to be evaluated would become unbearably long. Also, when a series becomes long, more precision in the numbers used is required in order to offset the effects of round-off errors. On a computer this would mean the use of multiple precision. Accurate tables can be, and have been, prepared by use of the Cornish-Fisher approximation. Comparison of the table values with the values obtained by the method in the writer’s paper show that these tables have an accuracy of about six significant figures. For practical purposes there seems to be no lack of

χ
2

{\chi ^2}

tables for high degrees of freedom. The method in the writer’s paper is still useful in checking on the accuracy of tables computed by approximate methods or in producing tables with more significant figures. With single precision it can produce tables of seven figure accuracy at a speed far better than could be by the usual accurate methods. Some unique and useful tables can be produced using this method.
Computation of tangent, Euler, and Bernoulli numbers
Authors: Knuth Donald E. , Buckholtz Thomas J.
Published: June 2010

Some elementary methods are described which may be used to calculate tangent numbers, Euler numbers, and Bernoulli numbers much more easily and rapidly on electronic computers than the traditional recurrence relations which have been used for over a century. These methods have been used to prepare an accompanying table which extends the existing tables of these numbers. Some theorems about the periodicity of the tangent numbers, which were suggested by the tables, are also proved.
On the construction of discrete approximations to linear differential expressions
Authors: Ballester C. , Pereyra V.
Published: June 2010

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