bluuebunny/crossref_metadata_2025 huggingface dataset:
Extremal problems for polynomials with exponential weights
Authors: Mhaskar H. N. , Saff E. B.
Published: June 2010

For the extremal problem:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript n comma r Baseline left-parenthesis alpha right-parenthesis colon equals min parallel-to exp left-parenthesis minus StartAbsoluteValue x EndAbsoluteValue Superscript alpha Baseline right-parenthesis left-parenthesis x Superscript n Baseline plus midline-horizontal-ellipsis right-parenthesis parallel-to comma alpha greater-than 0 comma">

E

n
,
r

(
α
)
:=
min
∥
exp
⁡
(
−

|

x

|

α

)

(

x
n

+
⋯
)

∥

L
r

,

α
>
0
,

{E_{n,r}}(\alpha ): = \min \parallel \exp ( - |x{|^\alpha })\,({x^n} + \cdots ){\parallel _{{L^r}}}, \qquad \alpha > 0,

\]
where

L
r

(
0
>
r
⩽
∞
)

{L^r}\,(0 > r \leqslant \infty )

denotes the usual integral norm over

R

{\mathbf {R}}

, and the minimum is taken over all monic polynomials of degree

n
n

, we describe the asymptotic form of the error

E

n
,
r

(
α
)

(

as

n
→
∞
)

{E_{n,r}}(\alpha )\;({\text {as}}\;n \to \infty )

as well as the limiting distribution of the zeros of the corresponding extremal polynomials. The case

r
=
2

r = 2

yields new information regarding the polynomials

{

p
n

(
α
;
x
)
=

γ
n

(
α
)

x
n

+
⋯
}

\{ {p_n}(\alpha ;x) = {\gamma _n}(\alpha )\,{x^n} + \cdots \}

which are orthonormal on

R

{\mathbf {R}}

with respect to

exp
⁡
(
−
2

|

x

|

α

)

\exp ( - 2|x{|^\alpha })

. In particular, it is shown that a conjecture of Freud concerning the leading coefficients

γ
n

(
α
)

{\gamma _n}(\alpha )

is true in a Cesàro sense. Furthermore a contracted zero distribution theorem is proved which, unlike a previous result of Ullman, does not require the truth of the Freud’s conjecture. For

r
=
∞
,
α
>
0

r = \infty ,\alpha > 0

we also prove that, if

deg
⁡

P
n

(
x
)
⩽
n

\deg {P_n}(x) \leqslant n

, the norm

∥
exp
⁡
(
−

|

x

|

α

)

P
n

(
x
)

∥

L
∞

\parallel \exp ( - |x|^{\alpha })\,{P_n}(x)\parallel _{{L^\infty }}

is attained on the finite interval
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket minus left-parenthesis n slash lamda Subscript alpha Baseline right-parenthesis Superscript 1 slash alpha Baseline comma left-parenthesis n slash lamda Subscript alpha Baseline right-parenthesis Superscript 1 slash alpha Baseline right-bracket comma where lamda Subscript alpha Baseline equals normal upper Gamma left-parenthesis alpha right-parenthesis slash 2 Superscript alpha minus 2 Baseline left-brace normal upper Gamma left-parenthesis alpha slash 2 right-parenthesis right-brace squared period">

[

−

(
n

/

λ
α

)

1

/

α

,

(
n

/

λ
α

)

1

/

α

]

,

where

λ
α

=
Γ
(
α
)

/

2

α
−
2

{
Γ
(
α

/

2
)

}
2

.

\left [ { - {{(n/{\lambda _\alpha })}^{1/\alpha }},{{(n/{\lambda _\alpha })}^{1/\alpha }}} \right ],\quad {\text {where}}\;{\lambda _\alpha } = \Gamma (\alpha )/{2^{\alpha - 2}}{\{ \Gamma (\alpha /2)\} ^2}.

\]
Extensions of Nikolskii-type inequalities are also given.
On the representation of order continuous operators by random measures
Authors: Weis L.
Published: June 2010

Using the representation

T
f
(
y
)
=
∫
f

d

v
y

Tf(y) = \smallint f\;d{v_y}

, where

(

v
y

)

({v_y})

is a random measure, we characterize some interesting bands in the lattice of all order-continuous operators on a space of measurable functions. For example, an operator

T
T

is (lattice-)orthogonal to all integral operators (i.e. all

v
y

{v_y}

are singular) or belongs to the band generated by all Riesz homomorphisms (i.e. all

v
y

{v_y}

are atomic) if and only if

T
T

satisfies certain properties which are modeled after the Riesz homomorphism property [31] and continuity with respect to convergence in measure. On the other hand, for operators orthogonal to all Riesz homomorphisms (i.e. all

v
y

{v_y}

are diffuse) we give characterizations analogous to the characterizations of Dunford and Pettis, and Buhvalov for integral operators. The latter results are related to Enflo operators, to a result of J. Bourgain on Dunford-Pettis operators and martingale representations of operators.
Embeddings of Harish-Chandra modules, 𝔫-homology and the composition series problem: the case of real rank one
Authors: Collingwood David H.
Published: June 2010

Let

G
G

be a connected semisimple matrix group of real rank one. Fix a minimal parabolic subgroup

P
=
M
A
N

P = MAN

and form the (normalized) principal series representations

I
P
G

(
U
)

I_P^G(U)

. In the case of regular infinitesimal character, we explicitly determine (in terms of Langlands’ classification) all irreducible submodules and quotients of

I
P
G

(
U
)

I_P^G(U)

. As a corollary, all embeddings of an irreducible Harish-Chandra module into principal series are computed. The number of such embeddings is always less than or equal to three. These computations are equivalent to the determination of zero

n

{\mathfrak {n}}

-homology.
Structure sets vanish for certain bundles over Seifert manifolds
Authors: Stark Christopher W.
Published: June 2010

Let

M

n
+
3

{M^{n + 3}}

be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold

K
3

{K^3}

with an effective circle action of hyperbolic type. Assume that the fiber

N
n

{N^n}

in this bundle is a closed orientable manifold with Noetherian integral group ring, with vanishing projective class and Whitehead groups, and such that the structure set

S

TOP

(

N
n

×

D
k

,
∂
)

{S_{\text {TOP}}}\,({N^n} \times {D^k},\partial )

of topological surgery vanishes for sufficiently large

k
k

. Then the projective class and Whitehead groups of

M
M

vanish and

S

TOP

(

M

n
+
3

×

D
k

,
∂
)
=
0

{S_{\text {TOP}}}\,({M^{n + 3}}\, \times \, {D^k},\partial ) = 0

if

n
+
k
⩾
3

n + k \geqslant 3

or if

K
3

{K^3}

is closed and

n
=
2

n = 2

. The

UNil
\text {UNil}

groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that

UNil

j

{\text {UNil}}_j

vanishes in spite of the failure of Cappell’s sufficient condition.
The Dieudonné property on 𝐶(𝐾,𝐸)
Authors: Bombal Fernando , Cembranos Pilar
Published: June 2010

In this paper we prove that if

E
E

is a Banach space with separable dual, then the space

C
(
K
,
E
)

C(K,E)

of all continuous

E
E

-valued functions on a compact Hausdorff topological space

K
K

has the Dieudonné property.
The behavior under projection of dilating sets in a covering space
Authors: Randol Burton
Published: June 2010

Let

M
M

be a compact Riemannian manifold with covering space

S
S

, and suppose

d

μ
r

(
0
>
r
>
∞
)

d{\mu _r}\;(0 > r > \infty )

is a family of Borel probability measures on

S
S

, all of which arise from some fixed measure by

r
r

-homotheties of

S
S

about some point, followed by renormalization of the resulting measure. In this paper we study the ergodic properties, as a function of

r
r

, of the corresponding family of projected measures on

M
M

in the Euclidean and hyperbolic cases. A typical example arises by considering the behavior of a dilating family of spheres under projection.
On the structure of real transitive Lie algebras
Authors: Conn Jack F.
Published: June 2010

In this paper, we examine some of the ways in which abstract algebraic objects in a transitive Lie algebra

L
L

are expressed geometrically in the action of each transitive Lie pseudogroup

Γ
\Gamma

associated to

L
L

. We relate those chain decompositions of

Γ
\Gamma

which result from considering

Γ
\Gamma

-invariant foliations to Jordan-Hölder sequences (in the sense of Cartan and Guillemin) for

L
L

. Local coordinates are constructed which display the nature of the partial differential equations defining

Γ
\Gamma

; in particular, locally homogeneous pseudocomplex structures (also called

CR

{\text {CR}}

-structures) are associated to the nonabelian quotients of complex type in a Jordan-Hölder sequence for

L
L

.
Sous-espaces bien disposés de 𝐿¹-applications
Authors: Godefroy Gilles
Published: June 2010

RÉsumÉ. On montre que le quotient d’un espaceL1{L^1}par un sous-espace fermé dont la boule unité est fermée dansL0{L^0}est faiblement séquentiellement complet; cette situation se présente dans de nombreux cas concrets, tels que le quotientL1/H1{L^1}/{H^1}. On applique le résultat général dans diverses situations: duaux de certaines algères uniformes, analyse harmonique, fonctions de plusieurs variables complexes. On montre ensuite comment peuvent s’appliquer les métheodes deMM-structure; on considère aussi de nouvelles classes d’uniques préduaux. A titre d’exemples, on montre: (1) Le caractère f.s.c. d’espacesCE(G)∗{\mathcal {C}_E}{(G)^\ast }, pour de "gros" sous-ensemblesEEdu groupe dualΓ=G^\Gamma = \hat G. (2) Le caractère f.s.c. d’espacesL1/H1{L^1}/{H^1}mutli-dimensionnels, tels queL1/H1(Dn){L^1}/{H^1}({D^n})etL1/H1(Bn){L^1}/{H^1}({B^n}). (3) L’unicité du prédual pour certaines sous-algèbres ultrafaiblement fermées non-autoadjointes deL(H)\mathcal {L}(H). One shows that the quotient of anL1{L^1}-space by a closed subspace, whose unit ball is closed inL0{L^0}, is weakly sequentially complete. This situation occurs in many natural cases, likeL1/H1{L^1}/{H^1}. This result is applied in several situations: uniform algebras, harmonic analysis, functions of several complex variables. One shows how to applyMM-structure theory; several new classes of unique preduals are also obtained. As an example, one shows: (1) IfEEis a "big" subset of the dual groupΓ=G^\Gamma = \hat G, thenCE(G)∗{\mathcal {C}_E}{(G)^\ast }is w.s.c. (2) The spacesL1/H1(Dn){L^1}/{H^1}({D^n})andL1/H1(Bn){L^1}/{H^1}({B^n})are w.s.c. (3) Several classes ofω∗{\omega ^\ast }-closed non-self-adjoint subalgebras ofL(H)\mathcal {L}(H)have unique preduals.
The evolution of random graphs
Authors: Bollobás Béla
Published: June 2010

According to a fundamental result of Erdös and Rényi, the structure of a random graphGM{G_M}changes suddenly whenM∼n/2M \sim n/2: ifM=⌊cn⌋M = \left \lfloor {cn} \right \rfloorandc>12c > \frac {1}{2}then a.e. random graph of ordernnand sinceMMis such that its largest component hasO(log⁡n)O(\log n)vertices, but forc>12c > \frac {1}{2}a.e.GM{G_M}has a giant component: a component of order(1−αc+o(1))n(1-{\alpha _c}+o(1))nwhereαc>1{\alpha _c} > 1. The aim of this paper is to examine in detail the structure of a random graphGM{G_M}whenMMis close ton/2n/2. Among others it is proved that ifM=n/2+sM = n/2 + s,s=o(n)s = o(n)ands≥(log⁡n)1/2n2/3s \geq {(\log n)^{1/2}}{n^{2/3}}then the giant component has(4+o(1))s(4 + o(1))svertices. Furthermore, rather precise estimates are given for the order of therrth largest component for every fixedrr.
Stochastic representation and singularities of solutions of second order equations with semidefinite characteristic form
Authors: Amano Kazuo
Published: June 2010

In the theory of partial differential equations, there is no explicit representation of solutions for general degenerate elliptic-parabolic equations. However, Stroock and Varadhan [15] have obtained a stochastic representation for such a wider class of equations in

L
∞

{L^\infty }

space. In this paper we establish, by using Stroock and Varadhan’s stochastic representation, a method which enables us to construct solutions with singularities of second order equations with semidefinite characteristic form. Our theorems are not probabilistic paraphrases of the results obtained in the theory of partial differential equations. In fact, each assumption of the theorems is much weaker than any assumption of corresponding known results.
The Selberg trace formula. V. Questions of trace class
Authors: Osborne M. Scott , Warner Garth
Published: June 2010

The purpose of this paper is to develop criteria which will ensure that the

K
K

-finite elements of

C
c
∞

(
G
)

C_c^\infty (G)

are represented on

L

dis

2

(
G

/

Γ
)

L_{{\text {dis}}}^2(G/\Gamma )

by trace class operators.
Minimal periodic orbits for continuous maps of the interval
Authors: Alsedà Lluís , Llibre Jaume , Serra Rafel
Published: June 2010

For continuous maps of the interval into itself, Sarkovskii’s Theorem gives the notion of minimal periodic orbit. We complete the characterization of the behavior of minimal periodic orbits. Also, we show for unimodal maps that the min-max essentially describes the behavior of minimal periodic orbits.
Radial functions and invariant convolution operators
Authors: Meaney Christopher
Published: June 2010

For

1
>
p
>
2

1 > p > 2

and

n
>
1

n > 1

, let

A
p

(

R

n

)

{A_p}({{\mathbf {R}}^n})

denote the Figà-Talamanca-Herz algebra, consisting of functions of the form

(
∗
)

( \ast )

\[

∑

k
=
0

∞

f
k

∗

g
k

\sum \limits _{k = 0}^\infty {{f_k} \ast {g_k}}

\]
with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript k Endscripts StartAbsoluteValue EndAbsoluteValue f Subscript k Baseline StartAbsoluteValue EndAbsoluteValue Subscript p Baseline dot StartAbsoluteValue EndAbsoluteValue g Subscript k Baseline StartAbsoluteValue EndAbsoluteValue Subscript p prime Baseline greater-than normal infinity">

∑
k

|

|

f
k

|

|

p

⋅

|

|

g
k

|

|

p
′

>
∞

\sum \nolimits _k {||{f_k}|{|_p}\cdot ||{g_k}|{|_{p\prime }} > \infty }

. We show that if

2
n

/

(
n
+
1
)
>
p
>
2

2n/(n + 1) > p > 2

, then the subalgebra of radial functions in

A
p

(

R

n

)

{A_p}({{\mathbf {R}}^n})

is strictly larger than the subspace of functions with expansions

(
∗
)

( \ast )

subject to the additional condition that

f
k

{f_k}

and

g
k

{g_k}

are radial for all

k
k

. This is a partial answer to a question of Eymard and is a consequence of results of Herz and Fefferman. We arrive at the statement above after examining a more abstract situation. Namely, we fix

G
∈
[
F
I
A

]

B

−

G \in [FIA]_{B}^{ - }

and consider

B

A
p

(
G
)

^B{A_p}(G)

the subalgebra of

B
B

-invariant elements of

A
p

(
G
)

{A_p}(G)

. In particular, we show that the dual of

B

A
p

(
G
)

^B{A_p}(G)

is equal to the space of bounded, right-translation invariant operators on

L

p

(
G
)

{L^{p}}(G)

which commute with the action of

B
B

.
Finite subgroups of formal 𝐴-modules over 𝔭-adic integer rings
Authors: Nakamura Tetsuo
Published: June 2010

Let

B
⊃
A

B \supset A

be

p

\mathfrak {p}

-adic integer rings such that

A

/

Z
p

A/{Z_p}

is finite and

B

/

A

B/A

is unramified. Generalizing a result of Fontaine on finite commutative

p
p

-group schemes, we show that galois homomorphisms of finite subgroups of one-dimensional formal

A
A

-modules over

B
B

are given by power series.
Exact sequences in stable homotopy pair theory
Authors: Hardie K. A. , Jansen A. V.
Published: June 2010

A cylinder-web diagram with associated diagonal sequences is described in stable homotopy pair theory. The diagram may be used to compute stable homotopy pair groups and also stable track groups of two-cell complexes. For the stable Hopf class

η
\eta

the stable homotopy pair groups

G
k

(
η
,
η
)
(
k
⩽
8
)

{G_k}(\eta ,\eta )(k \leqslant 8)

are computed together with some of the additive structure of the stable homotopy ring of the complex projective plane.
Graphs of tangles
Authors: Gómez-Larrañaga J. C.
Published: June 2010

We prove that under necessary conditions a graph of tangles is a prime link. For this we generalize the result that the sum of

2
2

-string prime

L
L

-tangles is a prime link. Some applications are found. We explore Property

L

{\text {L}}

for tangles in order to prove primeness of knots.
Projections on tensor product spaces
Authors: Halton E. J. , Light W. A.
Published: June 2010

(
S
,
Σ
,
μ
)
,
(
T
,
Θ
,
υ
)

(S,\Sigma ,\mu ),(T,\Theta ,\upsilon )

are finite, nonatomic measure spaces.

G
G

and

H
H

are finite-dimensional subspaces of

L
1

(
S
)

{L_1}(S)

and

L
1

(
T
)

{L_1}(T)

respectively. Both

G
G

and

H
H

contain the constant functions. It is shown that the relative projection constant of

L
1

(
S
)
⊗
H
+
G
⊗

L
1

(
T
)

{L_1}(S) \otimes H + G \otimes {L_1}(T)

in

L
1

(
S
×
T
)

{L_1}(S \times T)

is at least

3
3

.
The algebra of the finite Fourier transform and coding theory
Authors: Tolimieri R.
Published: June 2010

The role of the finite Fourier transform in the theory of error correcting codes has been explored in a recent text by Richard Blahut. In this work we study how the finite Fourier transform relates to certain polynomial identities involving weight enumerator polynomials of linear codes. These include the generalized MacWilliams identities and theorems originally due to

R

{\text {R}}

. Gleason concerning polynomial algebras containing weight enumerator polynomials. The Heisenberg group model of the finite Fourier transform provides certain algebras of classical theta functions which will be applied to reprove Gleason’s results.
Anosov diffeomorphisms and expanding immersions. I
Authors: Jones Lowell
Published: June 2010

The purpose of this paper is to develop a theory for representing Anosov diffeomorphisms by expanding immersions on compact branched manifolds. This theory was motivated by R. F. Williams’ study of expanding attractors [15,17].
The splittability and triviality of 3-bridge links
Authors: Negami Seiya , Okita Kazuo
Published: June 2010

A method to simplify

3
3

-bridge projections of links and knots, called a wave move, is discussed in general situation and it is shown what kind of properties of

3
3

-bridge links and knots can be recognized from their projections by wave moves. In particular, it will be proved that every

3
3

-bridge projection of a splittable link or a trivial knot can be transformed into a disconnected one or a hexagon, respectively, by a finite sequence of wave moves. As its translation via the concept of

2
2

-fold branched coverings of

S
3

{S^3}

, it follows that every genus

2
2

Heegaard diagram of

S
2

×

S
2

#
L
(
p
,
q
)

{S^2} \times {S^2}\# L(p,q)

or

S
3

{S^3}

can be transformed into one of specific standard forms by a finite sequence of operations also called wave moves.
Points fixes d’applications holomorphes dans un domaine borné convexe de 𝐶ⁿ
Authors: Vigué Jean-Pierre
Published: June 2010

Let

D
D

be a bounded convex domain in

C

n

{{\mathbf {C}}^n}

. We prove that the set

V
V

of fixed points of a holomorphic map

f
:
D
→
D

f:D \to D

is a complex submanifold of

D
D

and, if

V
V

is not empty,

V
V

is a holomorphic retract of

D
D

.
Fixed points and conjugacy classes of regular elliptic elements in 𝑆𝑝(3,𝑍)
Authors: Eie Min King , Lin Chung Yuan
Published: June 2010

In this paper, we obtain

13
13

isolated fixed points (up to a

Sp
⁡
(
3
,

Z

)

\operatorname {Sp}(3,{\mathbf {Z}})

-equivalence) and

86
86

conjugacy classes of regular elliptic elements in

Sp
⁡
(
3
,

Z

)

\operatorname {Sp}(3,{\mathbf {Z}})

. Hence the contributions from regular elliptic conjugacy classes in

Sp
⁡
(
3
,

Z

)

\operatorname {Sp}(3,{\mathbf {Z}})

to the dimension formula computed via the Selberg trace formula can be computed explictly by the main theorem of [

4

{\mathbf {4}}

or

5

{\mathbf {5}}

].
The Cauchy integral, Calderón commutators, and conjugations of singular integrals in 𝑅ⁿ
Authors: Murray Margaret A. M.
Published: June 2010

We consider the Cauchy integral and Hilbert transform for Lipschitz domains in the Clifford algebra based on

R
n

{R^n}

. The Hilbert transform is shown to be the generating function for the Calderón commutators in

R
n

{R^n}

. We make use of an intrinsic characterization of these commutators to obtain

L
2

{L^2}

estimates. These estimates are used to show the analyticity of the Hilbert transform and of the conjugation of singular integral operators by bi-Lipschitz changes of variable in

R
n

{R^n}

.
Error bounds for Glimm difference approximations for scalar conservation laws
Authors: Hoff David , Smoller Joel
Published: June 2010

We derive error bounds for the Glimm difference approximation to the solution of a genuinely nonlinear scalar conservation law with

BV
\text {BV}

initial data. We show that the

L
1

{L^1}

error is bounded by

O
(
Δ

x

1

/

6

|

log
⁡
Δ
x

|

)

O(\Delta {x^{1/6}}|\log \Delta x|)

in the general case, and by

O
(
Δ

x

1

/

2

|

log
⁡
Δ
x

|

)

O(\Delta {x^{1/2}}|\log \Delta x|)

for a generic class of piecewise constant data.
On the proper Steenrod homotopy groups, and proper embeddings of planes into 3-manifolds
Authors: Brin Matthew G. , Thickstun T. L.
Published: June 2010

Standard algebraic invariants of proper homotopy type are discussed. These do not naturally fit into long exact sequences. Groups of proper homotopy classes of proper maps of Euclidean spaces and open annuli which do naturally form a long exact sequence are defined, and a diagram relating these groups to the standard algebraic invariants of proper homotopy type is given. The structures defined are used to compare several notions of essentiality for proper maps. Some results and examples are given for proper maps of spaces into manifolds of dimension

2
2

and

3
3

. These results are used to add information to a theorem of Brown and Feustel about properly embedding planes in

3
3

-manifolds.
Directed graphs and traveling waves
Authors: Terman David
Published: June 2010

The existence of traveling wave solutions for equations of the form

u
t

=

u

x
x

+
F
′
(
u
)

{u_t} = {u_{xx}} + F\prime (u)

is considered. All that is assumed about

F
F

is that it is sufficiently smooth,

lim

|

u

|

→
∞

F
(
u
)
=
−
∞

{\lim _{|u| \to \infty }}F(u) = - \infty

,

F
F

has only a finite number of critical points, each of which is nondegenerate, and if

A
A

and

B
B

are distinct critical points of

F
F

, then

F
(
A
)
≠
F
(
B
)

F(A) \ne F(B)

. The results demonstrate that, for a given function

F
F

, there may exist zero, exactly one, a finite number, or an infinite number of waves which connect two fixed, stable rest points. The main technique is to identify the phase planes, which arise naturally from the problem, with an array of integers. While the phase planes may be very complicated, the arrays of integers are always quite simple to analyze. Using the arrays of integers one is able to construct a directed graph; each path in the directed graph indicates a possible ordering, starting with the fastest, of which waves must exist. For a large class of functions

F
F

one is then able to use the directed graphs in order to determine how many waves connect two stable rest points.
Free boundary regularity for surfaces minimizing 𝐴𝑟𝑒𝑎(𝑆)+𝑐𝐴𝑟𝑒𝑎(∂𝑆)
Authors: Cook Edith A.
Published: June 2010

In

R

n

{{\mathbf {R}}^n}

, fix a hyperplane

Z
Z

and

a

(
k
−
1
)

a\;(k - 1)

-dimensional surface

F
F

lying to one side of

Z
Z

with boundary in

Z
Z

. We prove the existence of

S
S

and

B
B

minimizing

Area
⁡
(
S
)
+
c
Area
⁡
(
B
)

\operatorname {Area}(S) + c\operatorname {Area}(B)

among all

k
k

-dimensional

S
S

having boundary

F
∪
B

F \cup B

, where

B
B

is a free boundary constrained to lie in

Z
Z

. We prove that except possibly on a set of Hausdorff dimension

k
−
2

k - 2

,

S
S

is locally a

C

1
,
α

{C^{1,\alpha }}

manifold with

C

1
,
α

{C^{1,\alpha }}

boundary

B
B

for

0
>
α
>
1

/

2

0 > \alpha > 1/2

. If

k
=
n
−
1

k = n - 1

,

C

1
,
α

{C^{1,\alpha }}

is replaced by real analytic.
On minimal surfaces in a Kähler manifold of constant holomorphic sectional curvature
Authors: Wolfson Jon G.
Published: June 2010

This paper studies minimal surfaces in Kähler manifolds of constant holomorphic sectional curvature using the technique of the moving frame. In particular, we provide a classification of the minimal two-spheres in

C

P
n

{\mathbf {C}}{P^n}

, complex projective

n
n

-space, equipped with the Fubini-Study metric. This classification can be described as follows: To each holomorphic curve in

C

P
n

{\mathbf {C}}{P^n}

classically there is associated a particular framing of

C

n
+
1

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

called the Frenet frame. Each element of the Frenet frame induces a minimal surface in

C

P
n

{\mathbf {C}}{P^n}

. The classification theorem states that all minimal surfaces of topological type of the two-sphere occur in this manner. The theorem is proved using holomorphic differentials that occur naturally on minimal surfaces in Kähler manifolds of constant holomorphic sectional curvature together with the Riemann-Roch Theorem.
Unstable ground state of nonlinear Klein-Gordon equations
Authors: Shatah Jalal
Published: June 2010

In this paper we prove the instability of the ground state, i.e. least energy steady-state solution of nonlinear Klein-Gordon equations with space dimension

n
⩾
3

n \geqslant 3

.
Variants of the maximal double Hilbert transform
Authors: Prestini Elena
Published: June 2010

We prove the boundedness on

L
p

(

T
2

)

{L_p}({T^2})

,

1
>
p
>
∞

1 > p > \infty

, of two variants of the double Hilbert transform and maximal double Hilbert transform. They have an application to a problem of almost everywhere convergence of double Fourier series.
Structure and dimension of global branches of solutions to multiparameter nonlinear equations
Authors: Ize J. , Massabò I. , Pejsachowicz J. , Vignoli A.
Published: June 2010

This paper is concerned with the topological dimension of global branches of solutions appearing in different problems of Nonlinear Analysis, in particular multiparameter (including infinite dimensional) continuation and bifurcation problems. By considering an extension of the notion of essential maps defined on sets and using elementary point set topology, we are able to unify and extend, in a selfcontained fashion, most of the recent results on such problems. Our theory applies whenever any generalized degree theory with the boundary dependence property may be used, but with no need of algebraic structures. Our applications to continuation and bifurcation follow from the nontriviality of a local invariant, in the stable homotopy group of a sphere, and give information on the local dimension and behavior of the sets of solutions, of bifurcation points and of continuation points.
Projective modules in the category 𝒪_{𝒮}: Loewy series
Authors: Irving Ronald S.
Published: June 2010

Let

g

\mathfrak {g}

be a complex, semisimple Lie algebra with a parabolic subalgebra

p

S

{\mathfrak {p}_S}

. The Loewy lengths and Loewy series of generalized Verma modules and of their projective covers in

O

S

{\mathcal {O}_S}

are studied with primary emphasis on the case in which

p

S

{\mathfrak {p}_S}

is a Borel subalgebra and

O

S

{\mathcal {O}_S}

is the category

O

\mathcal {O}

. An examination of the change in Loewy length of modules under translation leads to the calculation of Loewy length for Verma modules and for self-dual projectives in

O

\mathcal {O}

, assuming the Kazhdan-Lusztig conjecture (in an equivalent formulation due to Vogan). In turn, it is shown that the Loewy length results imply Vogan’s statement, and lead to the determination of Loewy length for the self-dual projectives and certain generalized Verma modules in

O

S

{\mathcal {O}_S}

. Under the stronger assumption of Jantzen’s conjecture, the radical and socle series are computed for self-dual projectives in

O

\mathcal {O}

. An analogous result is formulated for self-dual projectives in

O

S

{\mathcal {O}_S}

and proved in certain cases.
A commutator theorem and weighted BMO
Authors: Bloom Steven
Published: June 2010

The main result of this paper is a commutator theorem: If

μ
\mu

and

λ
\lambda

are

A
p

{A_p}

weights, then the commutator

H
H

,

M
b

{M_b}

is a bounded operator from

L
p

(
μ
)

{L^p}(\mu )

into

L
p

(
λ
)

{L^p}(\lambda )

if and only if

b
∈

BMO

(
μ

λ

−
1

)

1

/

p

b \in {\operatorname {BMO} _{{{(\mu {\lambda ^{ - 1}})}^{1/p}}}}

. The proof relies heavily on a weighted sharp function theorem. Along the way, several other applications of this theorem are derived, including a doubly-weighted

L
p

{L^p}

estimate for BMO. Finally, the commutator theorem is used to obtain vector-valued weighted norm inequalities for the Hilbert transform.
Representation of set valued operators
Authors: Papageorgiou Nikolaos S.
Published: June 2010

In this paper we prove representation theorems for set valued additive operators acting on the spaces

L
X
1

(
X
=

separable Banach space)

L_X^1(X = {\text {separable Banach space)}}

,

L
1

{L^1}

and

L
∞

{L^\infty }

. Those results generalize well-known ones for single valued operators and among them the celebrated Dunford-Pettis theorem. The properties of these representing integrals are studied. We also have a differentiability result for multifunctions analogous to the one that says that an absolutely continuous function from a closed interval into a Banach space with the Radon-Nikodým property is almost everywhere differentiable and also it is the primitive of its strong derivative. Finally we have a necessary and sufficient condition for the set of integrable selectors of a multifunction to be

w
w

-compact in

L
X
1

L_X^1

. This result is a new very general result about

w
w

-compactness in the Lebesgue-Bochner space

L
X
1

L_X^1

.
Embeddings of locally connected compacta
Authors: Venema Gerard A.
Published: June 2010

Let

X
X

be a

k
k

-dimensional compactum and

f
:
X
→

M
n

f:X \to {M^n}

a map into a piecewise linear

n
n

-manifold,

n
⩾
k
+
3

n \geqslant k + 3

. The main result of this paper asserts that if

X
X

is locally

(
2
k
−
n
)

(2k - n)

-connected and

f
f

is

(
2
k
−
n
+
1
)

(2k - n + 1)

-connected, then

f
f

is homotopic to a CE equivalence. In particular, every

k
k

-dimensional,

r
r

-connected, locally

r
r

-connected compactum is CE equivalent to a compact subset of

R

2
k
−
r

{{\mathbf {R}}^{2k - r}}

as long as

r
⩽
k
−
3

r \leqslant k - 3

.
Noncommutative algebras of dimension three over integral schemes
Authors: Miranda Rick , Teicher Mina
Published: June 2010

In this article we describe the algebraic data which is equivalent to giving an associative, noncommutative algebra

O

X

{\mathcal {O}_X}

over an integral

k
k

-scheme

Y
Y

(where

k
k

is an algebraically closed field of characteristic

≠
3

\ne 3

), which is locally free of rank

3
3

. The description allows us to conclude that, essentially, all such are locally upper triangular

2
×
2

2 \times 2

matrices, with degenerations of a restricted form allowed.
A canonical partition of the periodic orbits of chaotic maps
Authors: Alligood Kathleen T.
Published: June 2010

We show that the periodic orbits of an area-contracting horseshoe map can be partitioned into subsets of orbits of minimum period

k
,

2
k
,

4
k
,

8
k
…

k,\;2k,\;4k,\;8k \ldots

, for some positive integer

k
k

. This partition is natural in the following sense: for any parametrized area-contracting map which forms a horseshoe, the orbits in one subset of the partition are contained in a single component of orbits in the full parameter space. Furthermore, prior to the formation of the horseshoe, this component contains attracting orbits of minimum period

2
m

k

{2^m}k

, for each nonnegative integer

m
m

.
Self-maps of loop spaces. II
Authors: Campbell H. E. A. , Cohen F. R. , Peterson F. P. , Selick P. S.
Published: June 2010

We study under what conditions on a finite CW complex

X
X

is

Q
(
X
)

Q(X)

atomic.
Derivatives of mappings with applications to nonlinear differential equations
Authors: Schechter Martin
Published: June 2010

We present a new definition of differentiation for mappings of sets in topological vector spaces. Complete flexibility is allowed in choosing the topology with which the derivative is taken. We determine the largest space on which the derivative can act. Our definition includes all others hitherto given, and the basic theorems of calculus hold for it. Applications are considered here and elsewhere.
Regularization for 𝑛th-order linear boundary value problems using 𝑚th-order differential operators
Authors: Kouba D. A. , Locker John
Published: June 2010

Let

X
X

and

Y
Y

denote real Hilbert spaces, and let

L
:

X
→
Y

L:\,X \to Y

be a closed densely-defined linear operator having closed range. Given an element

y
∈
Y

y \in Y

, we determine least squares solutions of the linear equation

L
x
=
y

Lx = y

by using the method of regularization. Let

Z
Z

be a third Hilbert space, and let

T
:

X
→
Z

T:\,X \to Z

be a linear operator with

D

(
L
)
⊆

D

(
T
)

\mathcal {D}(L) \subseteq \mathcal {D}(T)

. Under suitable conditions on

L
L

and

T
T

and for each

α
≠
0

\alpha \ne 0

, we show that there exists a unique element

x
α

∈

D

(
L
)

{x_\alpha } \in \mathcal {D}(L)

which minimizes the functional

G
α

(
x
)
=

‖

L
x
−
y

‖

2

+

α
2

‖

T
x

‖

2

{G_\alpha }(x) = {\left \| {Lx - y} \right \|^2} + {\alpha ^2}{\left \| {Tx} \right \|^2}

, and the

x
α

{x_\alpha }

converge to a least squares solution

x
0

{x_0}

of

L
x
=
y

Lx = y

as

α
→
0

\alpha \to 0

. We apply our results to the special case where

L
L

is an

n
n

th-order differential operator in

X
=

L
2

[
a
,
b
]

X = {L^2}[a,b]

, and we regularize using for

T
T

an

m
m

th-order differential operator in

L
2

[
a
,
b
]

{L^2}[a,b]

with

m
≤
n

m \le n

. Using an approximating space of Hermite splines, we construct numerical solutions to

L
x
=
y

Lx = y

by the method of continuous least squares and the method of discrete least squares.
The normal closure of the coproduct of rings over a division ring
Authors: Martindale Wallace S.
Published: June 2010

Let

R
=

R
1

∐

R
2

R = {R_1}\coprod {R_2}

be the coproduct of

Δ
\Delta

-rings

R
1

{R_1}

and

R
2

{R_2}

with 1 over a division ring

Δ
,

R
1

≠
Δ
,

R
2

≠
Δ

\Delta ,\qquad {R_1} \ne \Delta ,\qquad {R_2} \ne \Delta

, with at least one of the dimensions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R Subscript i Baseline colon normal upper Delta right-parenthesis Subscript r Baseline comma left-parenthesis upper R Subscript i Baseline colon normal upper Delta right-parenthesis Subscript l Baseline comma i equals 1 comma 2">

(

R
i

:
Δ

)
r

,

(

R
i

:
Δ

)
l

,

i
=
1
,

2

{({R_i}:\Delta )_r},\,{({R_i}:\Delta )_l},\,i = 1,\,2

, greater than 2. If

R
1

{R_1}

and

R
2

{R_2}

are weakly

1
1

-finite (i.e., one-sided inverses are two-sided) then it is shown that every

X
X

-inner automorphism of

R
R

(in the sense of Kharchenko) is inner, unless

R
1

,

R
2

{R_1},\,{R_2}

satisfy one of the following conditions: (I) each

R
i

{R_i}

is primary (i.e.,

R
i

=
Δ
+
T
,

T
2

=
0

{R_i} = \Delta + T,\,{T^2} = 0

), (II) one

R
i

{R_i}

is primary and the other is

2
2

-dimensional, (III) char.

Δ
=
2

\Delta = 2

, one

R
i

{R_i}

is not a domain, and one

R
i

{R_i}

is

2
2

-dimensional. This generalizes a recent joint result with Lichtman (where each

R
i

{R_i}

was a domain) and an earlier joint result with Montgomery (where each

R
i

{R_i}

was a domain and

Δ
\Delta

was a field).
A simple theory of differential calculus in locally convex spaces
Authors: Graff Richard A.
Published: June 2010

A theory of differential calculus for nonlinear maps between general locally convex spaces is developed. All convergence notions are topological, and only familiarity with basic results from point set topology, differential calculus in Banach spaces, and locally convex space theory is assumed. The chain rule for continuous

k
k

th order differentiability, smoothness of inverse functions, and function space continuity properties of higher order derivatives are examined. It is shown that this theory extends the classical Fréchet theory of differential calculus for maps between Banach spaces.
Some results concerning the boundary zero sets of general analytic functions
Authors: Berman Robert D.
Published: June 2010

Two results concerning the boundary zero sets of analytic functions on the unit disk

Δ
\Delta

are proved. First we consider nonconstant analytic functions

f
f

on

Δ
\Delta

for which the radial limit function

f

∗

{f^{\ast }}

is defined at each point of the unit circumference

C
C

. We show that a subset

E
E

of

C
C

is the zero set of

f

∗

{f^{\ast }}

for some such function

f
f

if and only if it is a

G

δ

{\mathcal {G}_\delta }

that is not metrically dense in any open arc of

C
C

. We then give a precise version of an asymptotic radial uniqueness theorem and its converse. The constructions given in the proofs of each of these theorems employ an approximation theorem of Arakeljan.
Clarke’s gradients and epsilon-subgradients in Banach spaces
Authors: Treiman Jay S.
Published: June 2010

A new characterization of Clarke’s normal cone to a closed set in a Banach space is given. The normal cone is characterized in terms of weak-star limits of epsilon normals. A similar characterization of Clarke’s generalized gradients is also presented. Restrictions must be placed on the Banach spaces to make the formulas valid.
Martingale transforms and complex uniform convexity
Authors: Bourgain J. , Davis W. J.
Published: June 2010

Martingale transforms and Calderon-Zygmund singular integral operators are bounded as operators from

L
2

(

L
1

)

{L_2}({L_1})

to

L
2

(

L
q

)

{L_2}({L_q})

when

0
>
q
>
1

0 > q > 1

. If

Y
Y

is a reflexive subspace of

L
1

{L_1}

then

L
1

/

Y

{L_1}/Y

can be renormed to be

2
2

-complex uniformly convex. A new proof of the cotype 2 property of

L
1

/

H
1

{L_1}/{H_1}

is given.
On the ampleness of homogeneous vector bundles
Authors: Snow Dennis M.
Published: June 2010

A formula is proved which expresses the ampleness of a homogeneous vector bundle over

G

/

P

G/P

in terms of the distance of the weights of the representation of

P
P

to certain dominant weights of

G
G

.
Fixed points of topologically stable flows
Authors: Hurley Mike
Published: June 2010

This paper concerns certain necessary conditions for a flow to be topologically stable (in the sense of P. Walters). In particular, it is shown that under fairly general conditions one can conclude that a topologically stable flow has a finite number of fixed points, and each of these is isolated in the chain recurrent set of the flow.
Global boundedness for a delay-differential equation
Authors: Luckhaus Stephan
Published: June 2010

The inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis partial-differential Subscript t Baseline u minus normal upper Delta u right-parenthesis left-parenthesis t comma x right-parenthesis less-than-or-equal-to u left-parenthesis t comma x right-parenthesis left-parenthesis 1 minus u left-parenthesis t minus tau comma x right-parenthesis right-parenthesis">

(

∂
t

u
−
Δ
u
)
(
t
,

x
)

≤

u
(
t
,

x
)
(
1
−
u
(
t
−
τ
,

x
)
)

({\partial _t}u - \Delta u)(t,\,x)\qquad \leq \qquad u(t,\,x)(1 - u(t - \tau ,\,x))

is investigated. It is shown that nonnegative solutions of the Dirichlet problem in a bounded interval remain bounded as time goes to infinity, whereas in a more dimensional domain, in general, this holds only if the delay is not too large.
On the generalized Nakayama conjecture and the Cartan determinant problem
Authors: Fuller K. R. , Zimmermann-Huisgen B.
Published: June 2010

For Artin algebras allowing certain filtered module categories, the Generalized Nakayama Conjecture is shown to be true; our result covers all positively graded Artin algebras and those whose radical cube is zero. For the corresponding class of left artinian rings we prove that finite global dimension forces the determinant of the Cartan matrix to be 1.
On quintic surfaces of general type
Authors: Yang Jin Gen
Published: June 2010

The study of quintic surfaces is of special interest because

5
5

is the lowest degree of surfaces of general type. The aim of this paper is to give a classification of the quintic surfaces of general type over the complex number field

C

{\mathbf {C}}

. We show that if

S
S

is an irreducible quintic surface of general type; then it must be normal, and it has only elliptic double or triple points as essential singularities. Then we classify all such surfaces in terms of the classification of the elliptic double and triple points. We give many examples in order to verify the existence of various types of quintic surfaces of general type. We also make a study of the double or triple covering of a quintic surface over

P

2

{{\mathbf {P}}^2}

obtained by the projection from a triple or double point on the surface. This reduces the classification of the surfaces to the classification of branch loci satisfying certain conditions. Finally we derive some properties of the Hilbert schemes of some types of quintic surfaces.
Definable sets in ordered structures. II
Authors: Knight Julia F. , Pillay Anand , Steinhorn Charles
Published: June 2010

It is proved that any

0
0

-minimal structure

M
M

(in which the underlying order is dense) is strongly

0
0

-minimal (namely, every

N
N

elementarily equivalent to

M
M

is

0
0

-minimal). It is simultaneously proved that if

M
M

is

0
0

-minimal, then every definable set of

n
n

-tuples of

M
M

has finitely many "definably connected components."
Closed geodesics on a Riemann surface with application to the Markov spectrum
Authors: Beardon A. F. , Lehner J. , Sheingorn M.
Published: June 2010

This paper determines those Riemann surfaces on which each nonsimple closed geodesic has a parabolic intersection—that is, an intersection in the form of a loop enclosing a puncture or a deleted disk. An application is made characterizing the simple closed geodesic on

H

/

Γ
(
3
)

H/\Gamma (3)

in terms of the Markov spectrum. The thrust of the situation is this: If we call loops about punctures or deleted disks boundary curves, then if the surface has "little" topology, each nonsimple closed geodesic must contain a boundary curve. But if there is "enough" topology, there are nonsimple closed geodesics not containing boundary curves.
The divergence theorem
Authors: Pfeffer W. F.
Published: June 2010

We define a well-behaved multidimensional Riemann type integral such that the divergence of any vector field continuous in a compact interval and differentiable in its interior is integrable, and the integral equals the flux of the vector field out of the interval.
Inequalities for the ergodic maximal function and convergence of the averages in weighted 𝐿^{𝑝}-spaces
Authors: Martín-Reyes F. J.
Published: June 2010

This paper is concerned with the characterization of those positive functions

w
w

such that Hopf’s averages associated to an invertible measure preserving transformation

T
T

and a positive function

g
g

converge almost everywhere for every

f
∈

L
p

(
w

d
μ
)

f \in {L^p}(w\,d\mu )

. We also study mean convergence when

g
g

satisfies a "doubling condition" over orbits. In order to do this, we first characterize the pairs of positive functions

(
u
,
w
)

(u,w)

such that the ergodic maximal operator associated to

T
T

and

g
g

is of weak or strong type with respect to the measures

w

d
μ

w\,d\mu

and

u

d
μ

u\,d\mu

.
Desingularizations of plane vector fields
Authors: Cano F.
Published: June 2010

The singularities of a plane vector field can be reduced under quadratic blowing ups. We describe a control method for the singularities of the vector field which works for ground fields of any characteristic and which has no essential obstruction for generalizing to higher dimensional cases.
A parametrix for step-two hypoelliptic diffusion equations
Authors: Taylor Thomas
Published: June 2010

In this paper I construct a parametrix for the hypoelliptic diffusion equations

(
∂

/

∂
t
−
L
)
u
=
0

(\partial /\partial t - L)u = 0

, where

L
=

∑

a
=
1

n

g
a
2

L = \sum \nolimits _{a = 1}^n {g_a^2}

and where the

g
a

{g_a}

are vector fields which satisfy the property that they, together with all of the commutators

[

g

a
,

g
b

]

[{g_{a,}}{g_b}]

for

a
>
b

a > b

, are at each point linearly independent and span the tangent space.
On a conormal module of smooth set theoretic complete intersections
Authors: Boratyński M.
Published: June 2010

We prove that

V
⊂

A

k
n

V \subset {\mathbf {A}}_k^n

(

V
V

-smooth) is a set-theoretic complete intersection (stci) if and only if

V
V

imbedded as a zero section of its normal bundle is a stci, we give a characterization of smooth codimension

2
2

stci of index

≤
4

\leq 4

in terms of their conormal modules.
Chebyshev rank in 𝐿₁-approximation
Authors: Kroó András
Published: June 2010

Let

C
ω

(
K
)

{C_\omega }(K)

denote the space of continuous functions endowed with the norm

∫
K

ω

|
f
|

=

‖
f
‖

ω

,
ω
>
0

{\smallint _K}\omega \left | f \right | = {\left \| f \right \|_\omega },\omega > 0

. In this paper we characterize the subspaces

U
n

⊂

C
ω

(
K
)

{U_n} \subset {C_\omega }(K)

having Chebyshev rank at most

k

(
0
≤
k
≤
n
−
1
)

k\;(0 \leq k \leq n - 1)

with respect to all bounded positive weights

ω
\omega

. Various applications of main results are also presented.
Inequalities for some maximal functions. II
Authors: Cowling M. , Mauceri G.
Published: June 2010

Let

S
S

be a smooth compact hypersurface in

R

n

{{\mathbf {R}}^n}

, and let

μ
\mu

be a measure on

S
S

, absolutely continuous with respect to surface measure. For

t
t

in

R

+

,

μ
t

{{\mathbf {R}}^ + },{\mu _t}

denotes the dilate of

μ
\mu

by

t
t

, normalised to have the same total variation as

μ
\mu

: for

f
f

in

S

(

R

n

)
,

μ
#

f

\mathcal {S}({{\mathbf {R}}^n}),{\mu ^\# }f

denotes the maximal function

sup

t
>
0

|

μ
t

∗
f

|

{\sup _{t > 0}}|{\mu _t}\ast f|

. We seek conditions on

μ
\mu

which guarantee that the a priori estimate
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar mu Superscript number-sign Baseline f double-vertical-bar Subscript p Baseline less-than-or-equal-to upper C double-vertical-bar f double-vertical-bar Subscript p Baseline comma f element-of upper S left-parenthesis bold upper R Superscript n Baseline right-parenthesis comma">

‖

μ
#

f
‖

p

≤
C

‖
f
‖

p

,

f
∈
S
(

R

n

)
,

\left \| \mu ^\# f\right \|_p \leq C\left \| f \right \|_p, \quad f \in S(\mathbf {R}^n),

\]
holds; this estimate entails that the sublinear operator

μ
#

{\mu ^\# }

extends to a bounded operator on the Lebesgue space

L
p

(

R

n

)

{L^p}({{\mathbf {R}}^n})

. Our methods generalise E. M. Stein’s treatment of the "spherical maximal function" [5]: a study of "Riesz operators",

g
g

-functions, and analytic families of measures reduces the problem to that of obtaining decay estimates for the Fourier transform of

μ
\mu

. These depend on the geometry of

S
S

and the relation between

μ
\mu

and surface measure on

S
S

. In particular, we find that there are natural geometric maximal operators limited on

L
p

(

R

n

)

{L^p}({{\mathbf {R}}^n})

if and only if

p
∈
(
q
,
∞
]
;
q

p \in (q,\infty ];q

is some number in

(
1
,
∞
)

(1,\infty )

, and may be greater than

2
2

. This answers a question of S. Wainger posed by Stein [6]>.
Equivariant intersection forms, knots in 𝑆⁴, and rotations in 2-spheres
Authors: Plotnick Steven P.
Published: June 2010

We consider the problem of distinguishing the homotopy types of certain pairs of nonsimply-connected four-manifolds, which have identical three-skeleta and intersection pairings, by the equivariant isometry classes of the intersection pairings on their universal covers. As applications of our calculations, we: (i) construct distinct homology four-spheres with the same three-skeleta, (ii) generalize a theorem of Gordon to show that any nontrivial fibered knot in

S
4

{S^4}

with odd order monodromy is not determined by its complement, and (iii) give a more constructive proof of a theorem of Hendriks concerning rotations in two-spheres embedded in threemanifolds.