bluuebunny/crossref_metadata_2025 huggingface dataset:
Arithmetic of elliptic curves upon quadratic extension
Authors: Kramer Kenneth
Published: June 2010

This paper is a study of variations in the rank of the Mordell-Weil group of an elliptic curve

E
E

defined over a number field

F
F

as one passes to quadratic extensions

K
K

of

F
F

. Let

S
(
K
)

S(K)

be the Selmer group for multiplication by

2
2

on

E
(
K
)

E(K)

. In analogy with genus theory, we describe

S
(
K
)

S(K)

in terms of various objects defined over

F
F

and the local norm indices
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i Subscript upsilon Baseline equals dimension Subscript bold upper F 2 Baseline upper E left-parenthesis upper F Subscript upsilon Baseline right-parenthesis slash Norm left-brace upper E left-parenthesis upper K Subscript w Baseline right-parenthesis right-brace">

i
υ

=

dim

F

2

E
(

F
υ

)

/

Norm
{
E
(

K
w

)
}

{i_\upsilon } = {\dim _{{{\mathbf {F}}_2}}}E({F_\upsilon })/\text {Norm} \{ E({K_w})\}

for each completion

F
υ

{F_\upsilon }

of

F
F

. In particular we show that

dim
⁡
S
(
K
)
+
dim
⁡
E

(
K

)
2

\dim S(K) + \dim E{(K)_2}

has the same parity as

Σ

i
υ

\Sigma {i_\upsilon }

. We compute

i
υ

{i_\upsilon }

when

E
E

has good or multiplicative reduction modulo

υ
\upsilon

. Assuming that the

2
2

-primary component of the Tate-Shafarevitch group

Ш

(
K
)

\Sha (K)

is finite, as conjectured, we obtain the parity of rank

E
(
K
)

E(K)

. For semistable elliptic curves defined over

Q

{\mathbf {Q}}

and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of Birch and Swinnerton-Dyer.
Which curves over 𝑍 have points with coordinates in a discrete ordered ring?
Authors: van den Dries Lou
Published: June 2010

A criterion is given for curves defined over

Z

{\mathbf {Z}}

to have an infinite point in a discrete ordered ring. Using this, one can decide effectively whether a given polynomial in

Z

[
X
,
Y
]

{\mathbf {Z}}[X,Y]

has a zero in a model for the axioms of open induction. Riemann-Roch for curves over

Q

{\mathbf {Q}}

is the main tool used.
A tangential convergence for bounded harmonic functions on a rank one symmetric space
Authors: Cygan Jacek
Published: June 2010

Let

u
u

be a bounded harmonic function on a noncompact rank one symmetric space

M
=
G

/

K
≈

N
−

A
,

N
−

A
K

M = G/K \approx {N^ - }A,{N^ - }AK

being a fixed Iwasawa decomposition of

G
G

. We prove that if for an

a
0

∈
A

{a_0} \in A

there exists a limit

u
(
n

a
0

)
≡

c
0

u(n{a_0}) \equiv {c_0}

, as

n
∈

N
−

n \in {N^ - }

goes to infinity, then for any

a
∈
A

a \in A

,

u
(
n
a
)
=

c
0

u(na) = {c_0}

. For
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M equals upper S upper U left-parenthesis n comma 1 right-parenthesis slash upper S left-parenthesis upper U left-parenthesis n right-parenthesis times upper U left-parenthesis 1 right-parenthesis right-parenthesis equals upper B Superscript n">

M
=
S
U
(
n
,
1
)

/

S
(
U
(
n
)
×
U
(
1
)
)
=

B
n

M = SU(n,1)/S(U(n) \times U(1)) = {B^n}

, the unit ball in

C

n

{{\mathbf {C}}^n}

with the Bergman metric, this is a result of Hulanicki and Ricci, and in this case it reads (via the Cayley transformation) as a theorem on convergence of a bounded harmonic function to a boundary value at a fixed boundary point, along appropriate, tangent to

∂

B
n

\partial {B^n}

, surfaces.
A faithful Hille-Yosida theorem for finite-dimensional evolutions
Authors: Freedman M. A.
Published: June 2010

As a natural generalization of the classical Hille-Yosida theorem to evolution operators, necessary and sufficient conditions are found for an evolution

U
U

acting in

R
N

{R^N}

so that for each

s
⩾
t

s \geqslant t

,

U
(
s
,
t
)

U(s,t)

can be uniquely represented as a product integral

∏
t
s

[
I
+
V

]

−
1

\prod _t^s{[I + V]^{ - 1}}

for some additive, accretive generator

V
V

. Under these conditions, we further show that

U
(
ξ
,
ζ
)

U(\xi ,\zeta )

is differentiable a.e.
Standard 3-components of type 𝑆𝑝(6,2)
Authors: Finkelstein Larry , Frohardt Daniel
Published: June 2010

It is shown that if

G
G

is a finite simple group with a standard

3
3

-component of type

Sp

(
6
,
2
)

{\text {Sp}}(6,2)

and

G
G

satisfies certain

2
2

-local and

3
3

-local conditions then either

G
G

is isomorphic to

Sp

(
8
,
2
)

{\text {Sp}}(8,2)

or

G
G

is isomorphic to

F
4

(
2
)

{F_4}(2)

.
Neighborhood fixed pendant vertices
Authors: Anacker S. E. , Robertson G. N.
Published: June 2010

If

x
x

is pendant in

G
G

, then

x
∗

{x^ \ast }

denotes the unique vertex of

G
G

adjacent to

x
x

. Such an

x
x

is said to be neighborhood-fixed whenever

x
∗

{x^ \ast }

is fixed by

A
(
G
−
x
)

A(G - x)

. It is shown that if

G
G

is not a tree and has a pendant vertex, but no *-fixed pendant vertex, then there is a subgraph

G
#

{G^\# }

of

G
G

such that for some

y
∈
V
(

G
#

)

y \in V({G^\# })

,

O
(
A

(

G
#

)
y

)
⩾
t
!

O(A{({G^\# })_y}) \geqslant t!

where

t
t

is the maximum number of edges in a tree rooted in

G
#

{G^\# }

.
Time-ordered operators. I. Foundations for an alternative view of reality
Authors: Gill Tepper L.
Published: June 2010

The purpose of this paper is to present the proper framework for the mathematical foundations of time-ordered operators. We introduce a new mathematical process which we call the chronological process. This process generalizes the notion of a limit and allows us to recapture many properties lost in time-ordering. We then construct time-ordered integrals and evolution operators. We show that under reasonable assumptions, the time-ordered sum of two generators of contraction semigroups is a generator. This result resolves a question that has been debated in physics for forty years.
Nonseparable approximate equivalence
Authors: Hadwin Donald W.
Published: June 2010

This paper extends Voiculescu’s theorem on approximate equivalence to the case of nonseparable representations of nonseparable

C
∗

{C^ \ast }

-algebras. The main result states that two representations

f
f

and

g
g

are approximately equivalent if and only if

rank

f
(
x
)
=

rank

g
(
x
)

{\text {rank}}f(x) = {\text {rank}}g(x)

for every

x
x

. For representations of separable

C
∗

{C^ \ast }

-algebras a multiplicity theory is developed that characterizes approximate equivalence. Thus for a separable

C
∗

{C^ \ast }

-algebra, the space of representations modulo approximate equivalence can be identified with a class of cardinal-valued functions on the primitive ideal space of the algebra. Nonseparable extensions of Voiculescu’s reflexivity theorem for subalgebras of the Calkin algebra are also obtained.
𝐶*-extreme points
Authors: Hopenwasser Alan , Moore Robert L. , Paulsen V. I.
Published: June 2010

Let

A

\mathcal {A}

be a

C
∗

{C^ \ast }

-algebra and let

S

\mathcal {S}

be a subset of

A

\mathcal {A}

.

S

\mathcal {S}

is

C
∗

{C^ \ast }

-convex if whenever

T
1

,

T
2

,
…
,

T
n

{T_1},{T_2}, \ldots ,{T_n}

are in

S

\mathcal {S}

and

A
1

,
…
,

A
n

{A_1}, \ldots ,{A_n}

are in

A

\mathcal {A}

with

∑

i
=
1

n

A
i
∗

A
i

=
I

\sum \nolimits _{i = 1}^n {A_i^ \ast {A_i} = I}

, then

∑

i
=
1

n

A
i
∗

T
i

A
i

\sum \nolimits _{i = 1}^n {A_i^ \ast {T_i}{A_i}}

is in

S

\mathcal {S}

. An element

T
T

in

S

\mathcal {S}

is called

C
∗

{C^ \ast }

-extreme in

S

\mathcal {S}

if whenever

T
=

∑

i
=
1

n

A
i
∗

T
i

A
i

T = \sum \nolimits _{i = 1}^n {A_i^ \ast {T_i}{A_i}}

with

T
i

{T_i}

and

A
i

{A_i}

as above and with

A
i

{A_i}

invertible, then

T
i

{T_i}

is unitarily equivalent to

T
T

for each

i
i

. We investigate the linear extreme points and

C
∗

{C^ \ast }

-extreme points for three sets: first, the unit ball of operators in Hilbert space; next, the set of

2
×
2

2 \times 2

matrices with numerical radius bounded by

1
1

; and last, the unit interval of positive operators on Hilbert space. In particular we find that for the second set, the linear and

C
∗

{C^ \ast }

-extreme points are different.
The discontinuous initial value problem of a reacting gas flow
Authors: Ying Lung An , Wang Ching Hua
Published: June 2010

We show that the local solvability of the solution of a reacting gas flow system

(
1.1
)
−
(
1.4
)

(1.1) - (1.4)

with initial values

(
1.5
)

(1.5)

, which has a large jump at the point

x
′

=
0

x’ = 0

and the structure of the solution near the origin

(
0
,
0
)

(0,0)

are identical to those of the Riemann problem of the homogeneous system corresponding to the Cauchy problem

(
1.1
)
−
(
1.5
)

(1.1) - (1.5)

.
Nonweakly compact operators from order-Cauchy complete 𝐶(𝑆) lattices, with application to Baire classes
Authors: Dashiell Frederick K.
Published: June 2010

This paper is concerned with the connection between weak compactness properties in the duals of certain Banach spaces of type

C
(
S
)

C(S)

and order properties in the vector lattice

C
(
S
)

C(S)

. The weak compactness property of principal interest here is the condition that every nonweakly compact operator from

C
(
S
)

C(S)

into a Banach space must restrict to an isomorphism on some copy of

l
∞

{l^\infty }

in

C
(
S
)

C(S)

. (This implies Grothendieck’s property that every

w
∗

{w^ \ast }

-convergent sequence in

C

(
S

)
∗

C{(S)^ \ast }

is weakly convergent.) The related vector lattice property studied here is order-Cauchy completeness, a weak type of completeness property weaker than

σ
\sigma

-completeness and weaker than the interposition property of Seever. An apphcation of our results is a proof that all Baire classes (of fixed order) of bounded functions generated by a vector lattice of functions are Banach spaces satisfying Grothendieck’s property. Another application extends previous results on weak convergence of sequences of finitely additive measures defined on certain fields of sets.
Liapounoff’s theorem for nonatomic, finitely-additive, bounded, finite-dimensional, vector-valued measures
Authors: Armstrong Thomas E. , Prikry Karel
Published: June 2010

Liapounoff’s theorem states that if

(
X
,
Σ
)

(X,\Sigma )

is a measurable space and

μ
:
Σ
→

R

d

\mu :\Sigma \to {{\mathbf {R}}^d}

is nonatomic, bounded, and countably additive, then

R

(
μ
)
=
{
μ
(
A
)
:
A
∈
Σ
}

\mathcal {R}(\mu ) = \{ \mu (A):A \in \Sigma \}

is compact and convex. When

Σ
\Sigma

is replaced by a

σ
\sigma

-complete Boolean algebra or an

F
F

-algebra (to be defined) and

μ
\mu

is allowed to be only finitely additive,

R

(
μ
)

\mathcal {R}(\mu )

is still convex. If

Σ
\Sigma

is any Boolean algebra supporting nontrivial, nonatomic, finitely-additive measures and

Z
Z

is a zonoid, there exists a nonatomic measure on

Σ
\Sigma

with range dense in

Z
Z

. A wide variety of pathology is examined which indicates that ranges of finitely-additive, nonatomic, finite-dimensional, vector-valued measures are fairly arbitrary.
Global Warfield groups
Authors: Hunter Roger , Richman Fred
Published: June 2010

A global Warfield group is a summand of a simply presented abelian group. The theory of global Warfield groups encompasses both the theory of totally projective

p
p

-groups, which includes the classical Ulm-Zippin theory of countable

p
p

-groups, and the theory of completely decomposable torsion-free groups. This paper develops the central results of the theory including existence and uniqueness theorems. In addition it is shown that every decomposition basis of a global Warfield group has a nice subordinate with simply presented torsion cokernel, and that every global Warfield group is a direct sum of a group of countable torsion-free rank and a simply presented group.
𝐿^{𝑝} norms of certain kernels of the 𝑁-dimensional torus
Authors: Colzani L. , Soardi P. M.
Published: June 2010

In this paper we study a class of kernels

F
R

{F_R}

which generalize the Bochner-Riesz kernels on the

N
N

-dimensional torus. Our main result consists in upper estimates for the

L
p

{L^p}

norms of

F
R

{F_R}

as

R
R

tends to infinity. As a consequence we prove a convergence theorem for means of functions belonging to suitable Besov spaces.
Arborescent structures. II. Interpretability in the theory of trees
Authors: Schmerl James H.
Published: June 2010

The first-order theory of arborescent structures is shown to be completely faithfully interpretable in the first-order theory of trees. It follows from this interpretation that Vaught’s conjecture is true for arborescent structures, the theory of arborescent structures is decidable, and every

ℵ
0

{\aleph _0}

-categorical arborescent structure has a decidable theory.
The second conjugate algebra of the Fourier algebra of a locally compact group
Authors: Lau Anthony To Ming
Published: June 2010

Let

G
G

be a locally compact group and let

V
N
(
G
)

VN(G)

denote the von Neumann algebra generated by the left translations of

G
G

on

L
2

(
G
)

{L_2}(G)

. Then

V
N

(
G

)

∗

VN{(G)^{\ast }}

, when regarded as the second conjugate space of the Fourier algebra of

G
G

, is a Banach algebra with the Arens product. We prove among other things that when

G
G

is amenable,

V
N

(
G

)

∗

VN{(G)^{\ast }}

is neither commutative nor semisimple unless

G
G

is finite. We study in detail the class of maximal regular left ideals in

V
N

(
G

)

∗

VN{(G)^{\ast }}

. We also show that if

G
1

{G_1}

and

G
2

{G_2}

are discrete groups, then

G
1

{G_1}

and

G
2

{G_2}

are isomorphic if and only if

V
N

(

G
1

)

∗

VN{({G_1})^{\ast }}

and

V
N

(

G
2

)

∗

VN{({G_2})^{\ast }}

are isometric order isomorphic.
Anneaux de valuation discrète complets non commutatifs
Authors: Vidal Robert
Published: June 2010

On construit, grace au concept de bimodule au sens de M. Artin, une bonne généralisation non commutative de la notion d’anneau de valuation discrète complet; puis on étudie la validité du théorème de structure de I. S. Cohen en égale caractéristique. Lorsque l’anneau de valuation possède un corps de représentants qui est un corps local (commutatif) à corps résiduel de caractéristique nulle et algébriquement clos, on donne une classification des déviations de commutativité engendrées par une dérivation continue de ce corps local. Enfin, on propose une méthode générale de construction d’anneaux sans corps de Cohen en égale caractéristique et l’article se termine par des problèmes ouverts dans cette théorie.
Essential spectra of elementary operators
Authors: Fialkow L. A.
Published: June 2010

This paper describes the essential spectrum and index function of the operator

X
→
A
X
B

X \to AXB

, where

A
A

,

B
B

, and

X
X

are Hilbert space operators. Analogous results are given for the restriction of this operator to a norm ideal and partial analogues are given for sums of such operators and for the case when the operators act on a Banach space.
Markov processes with Lipschitz semigroups
Authors: Bass Richard
Published: June 2010

For

f
f

a function on a metric space, let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L i p f equals sup Underscript x not-equals y Endscripts StartAbsoluteValue f left-parenthesis x right-parenthesis minus f left-parenthesis y right-parenthesis EndAbsoluteValue slash d left-parenthesis x comma y right-parenthesis comma">

Lip
⁡
f
=

sup

x
≠
y

|

f
(
x
)
−
f
(
y
)

|

/

d
(
x
,

y
)
,

\operatorname {Lip} f = \sup \limits _{x \ne y} |f(x) - f(y)|/d(x,\,y),

\]
and say that a semigroup

P
t

{P_t}

is Lipschitz if

Lip
⁡
(

P
t

f
)
⩽

e

K
t

Lip
⁡
f

\operatorname {Lip} ({P_t}f) \leqslant {e^{Kt}}\operatorname {Lip} f

for all

f
f

,

t
t

, where

K
K

is a constant. If one has two Lipschitz semigroups, then, with some additional assumptions, the sum of their infinitesimal generators will also generate a Lipschitz semigroup. Furthermore a sequence of uniformly Lipschitz semigroups has a subsequence which converges in the strong operator topology. Examples of Markov processes with Lipschitz semigroups include all diffusions on the real line which are on natural scale whose speed measures satisfy mild conditions, as well as some jump processes. One thus gets Markov processes whose generators are certain integro-differential operators. One can also interpret the results as giving some smoothness conditions for the solutions of certain parabolic partial differential equations.
Embedding processes in Brownian motion in 𝑅ⁿ
Authors: Falkner Neil
Published: June 2010

We give a potential-theoretic characterization of the right-continuous processes which can be embedded in Brownian motion in

R

n

{{\mathbf {R}}^n}

by means of an increasing family of standard stopping times. In general it is necessary to use a Brownian motion process whose filtration is richer than the natural one.
Self-maps of flag manifolds
Authors: Glover Henry H. , Homer William D.
Published: June 2010

Rationally, a map between flag manifolds is seen to be determined up to homotopy by the homomorphism it induces on cohomology. Two algebraic results for cohomology endomorphisms then serve (a) to determine those flag manifolds which have (nontrivial) self-maps that factor through a complex projective space, and (b) for a special class of flag manifolds, to classify the self-maps of their rationalizations up to homotopy.
Differentiable group actions on homotopy spheres. II. Ultrasemifree actions
Authors: Schultz Reinhard
Published: June 2010

A conceptually simple but very useful class of topological or differentiable transformation groups is given by semifree actions, for which the group acts freely off the fixed point set. In this paper, the slightly more general notion of an ultrasemifree action is introduced, and it is shown that the existing machinery for studying semifree actions on spheres may be adapted to study ultrasemifree actions equally well. Some examples and applications are given to illustrate how ultrasemifree actions (i) may be used to study questions not answerable using semifree actions alone, and (ii) provide examples of unusual smooth group actions on spheres with no semifree counterparts.
A Poisson summation formula for integrals over quadratic surfaces
Authors: Strichartz Robert S.
Published: June 2010

Let

S
(
t
)

S(t)

denote Lebesgue measure on the sphere of radius

t
>
0

t > 0

in

R

n

{{\mathbf {R}}^n}

, and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript k Baseline left-parenthesis t right-parenthesis equals left-parenthesis StartFraction partial-differential Over partial-differential t EndFraction StartFraction 1 Over t EndFraction right-parenthesis Superscript k Baseline upper S left-parenthesis t right-parenthesis period">

S
k

(
t
)
=

(

∂

∂
t

1
t

)

k

S
(
t
)
.

{S_k}(t) = {\left ( {\frac {\partial } {{\partial t}}\quad \frac {1} {t}} \right )^k}S(t).

\]
Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P sigma-summation Underscript k Endscripts equals upper S Subscript k Baseline left-parenthesis 0 right-parenthesis plus 2 sigma-summation Underscript m equals 1 Overscript normal infinity Endscripts upper S Subscript k Baseline left-parenthesis m right-parenthesis">

P

∑
k

=

S
k

(
0
)
+
2

∑

m
=
1

∞

S
k

(
m
)

P{\sum _k} = {S_k}(0) + 2\sum _{m = 1}^\infty {S_k}(m)

. Theorem. If

n
n

is odd and

j
j

and

k
k

are nonnegative integers with

j
+
k
=
(
n
−
1
)

/

2

j + k = (n - 1) / 2

, then the Fourier transform of

P

∑
j

P{\sum _j}

is

(
2
π

)

j
−
k

P

∑
k

{(2\pi )^{j - k}}P{\sum _k}

. There is an analogous, although slightly different, identity involving integrals over hyperboloids in odd dimensions. These results were inspired by recent work of M. Vergne.
Degeneracy theorems for holomorphic mappings between algebraic varieties
Authors: Molzon Robert
Published: June 2010

Degeneracy theorems are proved for holomorphic mappings from affine algebraic manifolds to projective algebraic manifolds of equal dimensions. A mapping is degenerate if it satisfies a growth estimate and omits a set of

κ
\kappa

-plane sections of positive capacity; the capacity being defined in terms of a singular integral. The capacity is a more delicate method of measuring the size of a set of

κ
\kappa

-plane sections than Hausdorff measure and arises naturally by considering the singular integrals in the First Main Theorem of Nevanlinna.
Maximal positive boundary value problems as limits of singular perturbation problems
Authors: Bardos Claude , Rauch Jeffrey
Published: June 2010

We study three types of singular perturbations of a symmetric positive system of partial differential equations on a domain

Ω
⊂

R

n

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

. In all cases the limiting behavior is given by the solution of a maximal positive boundary value problem in the sense of Friedrichs. The perturbation is either a second order elliptic term or a term large on the complement of

Ω
\Omega

. The first corresponds to a sort of viscosity and the second to physical systems with vastly different properties in

Ω
\Omega

and outside

Ω
\Omega

. The results show that in the limit of zero viscosity or infinitely large difference the behavior is described by a maximal positive boundary value problem in

Ω
\Omega

. The boundary condition is determined in a simple way from the system and the singular terms.
Hopf manifolds and spectral geometry
Authors: Tsukada Kazumi
Published: June 2010

We characterize Hopf manifolds in the class of Hermitian manifolds by the spectra of the real Laplacians and the complex Laplacians.
The structure of quasinormal operators and the double commutant property
Authors: Conway John B. , Wu Pei Yuan
Published: June 2010

In this paper a characterization of those quasinormal operators having the double commutant property is obtained. That is, a necessary and sufficient condition is given that a quasinormal operator

T
T

satisfy the equation

{
T
}
=

A

(
T
)

\{ T\} = \mathcal {A}(T)

, the weakly closed algebra generated by

T
T

and

1
1

. In particular, it is shown that every pure quasinormal operator has the double commutant property. In addition two new representation theorems for certain quasinormal operators are established. The first of these represents a pure quasinormal operator

T
T

as multiplication by

z
z

on a subspace of an

L
2

{L^2}

space whenever there is a vector

f
f

such that

{

|

T

|

k

T
j

f
:

k
,

j
⩾
0
}

\{ |T{|^k}{T^j}f:\,k,\,j \geqslant 0\}

has dense linear span. The second representation theorem applies to those pure quasinormal operators

T
T

such that

T

∗

T

{T^{\ast }}T

is invertible. The second of these representation theorems will be used to determine which quasinormal operators have the double commutant property.
Torsion-groups of abelian coverings of links
Authors: Mayberry John P. , Murasugi Kunio
Published: June 2010

IfMMis an abelian branched covering ofS3{S^3}along a linkLL, the order ofH1(M){H_1}(M)can be expressed in terms of (i) the Alexander polynomials ofLLand of its sublinks, and (ii) a "redundancy" function characteristic of the monodromy-group. In 1954, the first author thus generalized a result of Fox (forLLa knot, in which case the monodromy-group is cyclic and the redundancy trivial); we now prove earlier conjectures and give a simple interpretation of the redundancy. Cyclic coverings of links are discussed as simple special cases. We also prove that the Poincaré conjecture is valid for the above-specified family of33-manifoldsMM. We state related results for unbranched coverings.
Tangent 2-fields on even-dimensional nonorientable manifolds
Authors: Pollina Benedict J.
Published: June 2010

This paper uses the Postnikov decomposition of a nonsimple fibration to describe the obstructions to a tangent

2
2

-field on an even-dimensional nonorientable manifold.
The sums of powers theorem for commuting block maps
Authors: Rhodes Frank
Published: June 2010

A block map is a map

f
:

{
0
,

1

}
n

→
{
0
,

1
}

f:{\{ 0,\,1\} ^n} \to \{ 0,\,1\}

for some

n
⩾
1

n \geqslant 1

. A block map

f
f

induces an endomorphism

f
∞

{f_\infty }

of the full

2
2

-shift

(
X
,

σ
)

(X,\,\sigma )

. Composition of block maps is defined in such a way that

(
f
∘
g

)
∞

=

f
∞

∘

g
∞

{(f \circ g)_\infty } = {f_\infty } \circ {g_\infty }

. In this paper some recent results concerning the set

{
g

|

g
∘
f
=
f
∘
g
}

\{ g|g \circ f = f \circ g\}

for certain types of block maps

f
f

are extended.
On the construction and distribution of a local martingale with a given absolute value
Authors: Perkins Edwin
Published: June 2010

A local martingale is constructed on an appropriate Loeb space whose absolute value equals a given nonnegative local submartingale. Nonstandard analysis is used to reduce the problem to the discrete time setting where the original construction of D. Gilat is fairly simple. This approach has the advantage of allowing explicit computations. In particular, the distribution of the local martingale is described in terms of the Doob-Meyer decomposition of the original local submartingale.
Algebras generated by a subnormal operator
Authors: Olin Robert F. , Thomson James E.
Published: June 2010

We use the notion of generalized Toeplitz operators to obtain some basic results concerning the

C

∗

{C^{\ast }}

-algebra generated by a subnormal operator. We apply these results to problems concerning the intersection of

C

∗

(
S
)

{C^{\ast }}(S)

with rationally closed algebras generated by

S
S

. In particular, we prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk Baseline left-parenthesis upper S right-parenthesis intersection script upper W left-parenthesis upper S right-parenthesis equals StartSet f left-parenthesis upper S right-parenthesis colon f element-of upper R left-parenthesis sigma Subscript script upper W left-parenthesis upper S right-parenthesis Baseline left-parenthesis upper S right-parenthesis right-parenthesis EndSet">

C

∗

(
S
)
∩

W

(
S
)
=
{
f
(
S
)
:
f
∈
R
(

σ

W

(
S
)

(
S
)
)
}

{C^{\ast }}(S) \cap \mathcal {W}(S) = \{ f(S):f \in R({\sigma _{\mathcal {W}(S)}}(S))\}

. The spectral inclusion property for generalized Toeplitz operators with symbols in

P
∞

(
μ
)
+
C
(
σ
(
N
)
)

{P^\infty }(\mu ) + C(\sigma (N))

is also considered.
Classification of oriented equivariant spherical fibrations
Authors: Waner Stefan
Published: June 2010

Classifying spaces for oriented equivariant spherical fibrations are constructed, and the notion of an equivariant

S
F

SF

-fibration is introduced. It is shown that equivariant

S
F

SF

-fibrations are naturally oriented in

R
O
(
G
)

RO(G)

-graded equivariant singular cohomology.
Symmetric skew balanced starters and complete balanced Howell rotations
Authors: Du Ding Zhu , Hwang F. K.
Published: June 2010

Symmetric skew balanced starters on

n
n

elements have been previously constructed for

n
=
4
k
+
3

n = 4k + 3

a prime power and

8
k
+
5

8k + 5

a prime power. In this paper we give an approach for the general case

n
=

2
m

k
+
1

n = {2^m}k + 1

a prime power with

k
k

odd. In particular we show how this approach works for

m
=
2

m = 2

and

3
3

. Furthermore, we prove that for

n
n

of the general form and

k
>
9
⋅

2

3
m

k > 9 \cdot {2^{3m}}

, then a symmetric skew balanced starter always exists. It is known that a symmetric skew balanced starter on

n
n

elements,

n
n

odd, can be used to construct complete balanced Howell rotations (balanced Room squares) for

n
n

players and

2
(
n
+
1
)

2(n + 1)

players, and in the case that

n
n

is congruent to

3
3

modulo

4
4

, also for

n
+
1

n + 1

players.
The Budan-Fourier theorem and Hermite-Birkhoff spline interpolation
Authors: Goodman T. N. T. , Lee S. L.
Published: June 2010

We extend the classical Budan-Fourier theorem to Hermite-Birkhoff splines, that is splines whose knots are determined by a finite incidence matrix. This is then applied to problems of interpolation by Hermite-Birkhoff splines, where the nodes of interpolation are also determined by a finite incidence matrix. For specified knots and nodes in a finite interval, conditions are examined under which there is a unique interpolating spline for any interpolation data. For knots and nodes spaced periodically on the real line, conditions are examined under which there is a unique interpolating spline of power growth for data of power growth.
On weighted norm inequalities for the Hilbert transform of functions with moments zero
Authors: Adams Ernst
Published: June 2010

Let

f
~

\tilde f

denote the Hilbert transform of

f
f

, i.e.
\[

f
~

(
x
)
=

p

.
v

.

∫

f
(
t
)

x
−
t

d
t

\tilde f(x) = {\rm {p}}{\rm {.v}}{\rm {.}}\int {\frac {{f(t)}}{{x - t}}dt}

\]
and let

1
>
p
>
∞

1 > p > \infty

. A weight function

w
w

is shown to satisfy
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral vertical-bar ModifyingAbove f With tilde left-parenthesis x right-parenthesis vertical-bar Superscript p Baseline w left-parenthesis x right-parenthesis d x less-than-or-equal-to upper C integral StartAbsoluteValue f left-parenthesis x right-parenthesis EndAbsoluteValue Superscript p Baseline w left-parenthesis x right-parenthesis d x">

∫

|

f
~

(
x
)

|

p

w
(
x
)
d
x
≤
C

∫

|

f
(
x
)

|

p

w
(
x
)
d
x

\int {|\tilde f(x)} {|^p}w(x)dx \le C{\int {|f(x)|} ^p}w(x)dx

\]
for all

f
f

with the first

N
N

moments zero, if and only if it is of the form

w
(
x
)
=

|

q
(
x
)

|

p

U
(
x
)

w(x) = |q(x){|^p}U(x)

, where

q
q

is a polynomial of degree at most

N
N

and

U
∈

A
p

U \in {A_p}

.
Coextensions of regular semigroups by rectangular bands. II
Authors: Meakin John , Nambooripad K. S. S.
Published: June 2010

A construction of all coextensions of a regular semigroup

S
S

by rectangular bands is obtained. The construction is analogous to Hall’s construction of orthodox semigroups as spined products of bands and inverse semigroups and reduces to that construction when

S
S

is inverse. The results are specialized to provide a construction of the category of all normal coextensions of a regular semigroup.
On group 𝐶*-algebras of bounded representation dimension
Authors: Raeburn Iain
Published: June 2010

We consider the structure of group

C

∗

C^{\ast }

-algebras whose irreducible representations have bounded dimension. We give some general results, including a description of the topology on the spectrum, and then calculate explicitly the

C

∗

C^{\ast }

-algebras of two specific groups, one of them a nonsymmorphic space group.
The asymptotic expansion for the trace of the heat kernel on a generalized surface of revolution
Authors: Lue Ping Charng
Published: June 2010

Let

M
M

be a smooth compact Riemannian manifold without boundary. Let

I
I

be an open interval. Let

h
(
r
)

h(r)

be a smooth positive function. Let

g
g

be the metric on

M
M

. Consider the fundamental solution

E
(
x
,
y
,

r
1

,

r
2

;
t
)

E(x,y,{r_1},{r_2};t)

of the heat equation on

M
×
I

M \times I

with metric

h
2

(
r
)
g
+
d
r
⊗
d
r

{h^2}(r)g + dr \otimes dr

(when

E
E

exists globally we call it the heat kernel on

M
×
I

M \times I

). The coefficients of the asymptotic expansion of the trace

E
E

are studied and expressed in terms of corresponding coefficients on the basis

M
M

. It is fulfilled by means of constructing a parametrix for

E
E

which is different from a parametrix in the standard form. One important result is that each of the former coefficients is a linear combination of the latter coefficients.
Exact dynamical systems and the Frobenius-Perron operator
Authors: Lasota A. , Yorke James A.
Published: June 2010

Conditions are investigated which guarantee exactness for measurable maps on measure spaces. The main application is to certain piecewise continuous maps

T
T

on

[
0
,
1
]

[0,1]

for which

T
′

(
0
)
>
1

T’(0) > 1

. We assume

[
0
,
1
]

[0,1]

can be broken into intervals on which

T
T

is continuous and convex and at the left end of these intervals

T
=
0

T = 0

and

d
t

/

d
x
>
0

dt/dx > 0

. Such maps have an invariant absolutely continuous density which is exact.
Circle actions and fundamental groups for homology 4-spheres
Authors: Plotnick Steven
Published: June 2010

We generalize work of Fintushel and Pao to give a topological classification of smooth circle actions on oriented

4
4

-manifolds

Σ
\Sigma

satisfying

H
1

(
Σ
)
=
0

{H_1}(\Sigma ) = 0

. We then use these ideas to construct infinite families of homology

4
4

-spheres that do not admit effective circle actions, and whose fundamental groups cannot be

3
3

-manifold groups.
Recognition of linear actions on spheres
Authors: Illman Sören
Published: June 2010

Let

G
G

be a finite group acting smoothly on a homotopy sphere

Σ
m

\Sigma ^m

. We wish to establish necessary and sufficient conditions for the given

G
G

-action on

Σ
\Sigma

to be topologically equivalent to a linear action. That is, we want to be able to decide whether or not there exists a

G
G

-homeomorphism

γ
:
Σ
→

S
m

(
ρ
)

\gamma :\Sigma \to S^m(\rho )

, where

S
m

(
ρ
)
⊂

R

m
+
1

(
ρ
)

{S^m}(\rho ) \subset {\mathbf {R}^{m + 1}}(\rho )

denotes the unit sphere in an orthogonal representation space

R

m
+
1

(
ρ
)

\mathbf {R}^{m + 1}(\rho )

for

G
G

. In order for a

G
G

-action on

Σ
\Sigma

to be topologically equivalent to a linear action it is clearly necessary that: (i) For each subgroup

H
H

of

G
G

the fixed-point set

Σ
H

\Sigma ^H

is homeomorphic to a sphere, or empty. (ii) For any subgroups

H
H

and

H
⊊

H
i

,

1
≤
i
≤
k

H \subsetneq {H_i},\,1 \leq i \leq k

, of

G
G

the pair

(

Σ

H

,

∪

i
=
1

k

Σ

H

i

)

(\Sigma ^{H},\,\cup _{i=1}^{k}\Sigma ^{H_{i}})

is homeomorphic to a standard pair

(

S

n

,

∪

i
=
1

k

S

i

n

i

)

(S^{n},\,\cup _{i=1}^{k}S_{i}^{n_{i}})

, where each

S
i

n
i

,

1
≤
i
≤
k

S_i^{{n_i}},\,1 \le i \le k

, is a standard

n
i

n_i

-subsphere of

S
n

S^n

. In this paper we consider the case where the fixed-point set

Σ
G

\Sigma ^G

is nonempty and all other fixed-point sets have dimension at least 5. In giving efficient sufficient conditions we do not need the full strength of condition (ii). We only need: (ii)

∗

^{\ast }

For any subgroups

H
H

and

H
⊊

H
i

,

1
≤
i
≤
p

H \subsetneq {H_i},\,1 \leq i \leq p

, of

G
G

such that

dim

Σ

H
i

=

dim

Σ
H

−
2

{\operatorname {dim}}\,{\Sigma ^{{H_i}}} = {\operatorname {dim}}\,{\Sigma ^H} - 2

, the pair

Σ

H

,

∪

i
=
1

p

Σ

H

i

)

\Sigma ^{H},\,\cup _{i=1}^{p}\Sigma ^{H_{i}})

is homeomorphic to a standard pair

(

S
n

,

∪

i
=
1

p

S
i

n
−
2

)

({S^n},\, \cup _{i = 1}^pS_i^{n - 2})

, where each

S
i

n
−
2

,

1
≤
i
≤
p

S_i^{n - 2},\,1 \le i \le p

, is a standard

(
n
−
2
)

(n-2)

-subsphere of

S
n

S^n

. Our main results are then that, in the case when

G
G

is abelian, conditions (i) and (ii)

∗

^{\ast }

are necessary and sufficient for a given

G
G

-action on

Σ
\Sigma

to be topologically equivalent to a linear action, and in the case of an action of an arbitrary finite group the same holds under the additional assumption that any simultaneous codimension 1 and 2 fixed-point situation is simple. Our results generalize, for actions of finite groups, a well-known theorem of Connell, Montgomery and Yang, and are the first to also cover the case where codimension 2 fixed-point situations occur.
Eisenstein series of weight 3/2. I
Authors: Pei Ting Yi
Published: June 2010

We prove that in the space of elliptic modular forms with weight 3/2, the orthogonal complement of the subspace of cusp forms with respect to the Petersson inner product is generated by Eisenstein series in some special cases.
On some open problems of P. Turán concerning Birkhoff interpolation
Authors: Varma A. K.
Published: June 2010

In 1974 P. Turán (see [6]) raised many interesting open problems in Approximation Theory some of which are on Birkhoff Interpolation. The object of this paper is to answer some of these problems (XXXVI-XXXIX). We obtain some new quadrature formulas where function values and second derivatives are only prescribed on the zeros of
\[

π
n

(
x
)
=

c
n

∫

−
1

x

P

n
−
1

(
t
)

d
t

{\pi _n}(x) = {c_n}\int _{ - 1}^x {{P_{n - 1}}(t)\,dt}

\]
,

P
n

(
x
)

{P_n}(x)

being Legendre polynomial of degree

n
n

.
Mixed Hodge structures
Authors: El Zein Fouad
Published: June 2010

The theory of Mixed Hodge Structures (M.H.S.) on the cohomology of an algebraic variety

X
X

over complex numbers was found by Deligne in 1970. The case when

X
X

is a Normal Crossing Divisor is fundamental. When the variety

X
X

is embedded in a smooth ambient space we get the Mixed Hodge Structure using standard exact sequences in topology. This technique uses resolution of singularities one time for a complete variety and

2
2

times for a quasi-projective one. As applications to the study of local cohomology we give the spectral sequence to the Mixed Hodge Structure on cohomology with support on a subspace

Y
Y

.
Dimension of stratifiable spaces
Authors: Oka Shinpei
Published: June 2010

We define a subclass, denoted by

E

M
3

E{M_3}

, of the class of stratifiable spaces, and obtain several dimension theoretical results for

E

M
3

E{M_3}

including the coincidence theorem for dim and Ind. The class

E

M
3

E{M_3}

is countably productive, hereditary, preserved under closed maps and, furthermore, the largest subclass of stratifiable spaces in which a harmonious dimension theory can be established.
Evolution generated by semilinear dissipative plus compact operators
Authors: Schechter Eric
Published: June 2010

Existence results and sharp continuous dependence results are given for an evolution equation in an arbitrary Banach space. The right-hand side of the equation consists of a linear dissipative term plus a continuous dissipative term plus a compact term.
Reality of the zeros of an entire function and its derivatives
Authors: Hellerstein Simon , Shen Li Chien , Williamson Jack
Published: June 2010

In 1914 Pólya raised the problem of classifying the entire functions which together with all their derivatives have only real zeros. In earlier work Hellerstein and Williamson settled this problem for entire functions which are real on the real axis. We complete the classification in all cases and show that it is sufficient to consider the function and its first two derivatives.
Weak-star convergence in the dual of the continuous functions on the 𝑛-cube, 1≤𝑛≤∞
Authors: Darst Richard B. , Honargohar Zorabi
Published: June 2010

Let

n
n

be a positive integer and let

J
=

×

j
=
1

n

[
0
,
1

]
j

J = \times _{j = 1}^n{[0,1]_j}

denote the

n
n

-cube. Let

C

=

C

(
J
)

\mathbf {C} = \mathbf {C}(J)

denote the (sup norm) space of continuous (real-valued) functions defined on

J
J

, and let

M

\mathfrak {M}

denote the (variation norm) space of (real-valued) signed Borel measures defined on the Borel subsets of

J
J

. Let

⟨

μ
l

⟩

\left \langle {{\mu _l}} \right \rangle

be a sequence of elements of

M

\mathfrak {M}

. Necessary and sufficient conditions are given in order that

li

m

l

∫
f

d

μ
l

{\text {li}}{{\text {m}}_l}\int f \,d{\mu _l}

exists for every

f
∈

C

f \in \mathbf {C}

. After considering a finite dimensional case, the infinite dimensional case is entertained.
Sufficient conditions for the generalized problem of Bolza
Authors: Zeidan Vera
Published: June 2010

This paper presents sufficient conditions for strong local optimality in the generalized problem of Bolza. These conditions represent a unification, in the sense that they can be applied to both the calculus of variations and to optimal control problems, as well as problems with nonsmooth data. Also, this paper brings to light a new point of view concerning the Jacobi condition in the classical calculus of variations, showing that it can be considered as a condition which guarantees the existence of a canonical transformation which transforms the original Hamiltonian to a locally concave-convex Hamiltonian.
𝐿^{𝑝} multipliers with weight 𝑥^{𝑘𝑝-1}
Authors: Muckenhoupt Benjamin , Young Wo Sang
Published: June 2010

Let

k
k

be a positive integer and

1
>
p
>
∞

1 > p > \infty

. It is shown that if

T
T

is a multiplier operator on

L
p

{L^p}

of the line with weight

|

x

|

k
p
−
1

|x{|^{kp-1}}

, then

T
f

Tf

equals a constant times

f
f

almost everywhere. This does not extend to the periodic case since

m
(
j
)
=
1

/

j
,
j
≠
0

m(j) = 1/j, j \ne 0

, is a multiplier sequence for

L
p

{L^p}

of the circle with weight

|

x

|

k
p
−
1

|x{|^{kp-1}}

. A necessary and sufficient condition is derived for a sequence

m
(
j
)

m(j)

to be a multiplier on

L
2

{L^2}

of the circle with weight

|

x

|

2
k
−
1

|x{|^{2k - 1}}

.
Schrödinger operators with rapidly oscillating central potentials
Authors: White Denis A. W.
Published: June 2010

Spectral and scattering theory is discussed for the Schrödinger operators

H
=
−
Δ
+
V

H = - \Delta + V

and

H
0

=
−
Δ

{H_0} = - \Delta

when the potential

V
V

is central and may be rapidly oscillating and unbounded. A spectral representation for

H
H

is obtained along with the spectral properties of

H
H

. The existence and completeness of the modified wave operators is also demonstrated. Then a condition on

V
V

is derived which is both necessary and sufficient for the Møller wave operators to exist and be complete. This last result disproves a recent conjecture of Mochizuki and Uchiyama.
A geometric interpretation of the Chern classes
Authors: Sivera Villanueva R.
Published: June 2010

Let

f
ξ

:
M
→
B
U

{f_\xi }: M \to BU

be a classifying map of the stable complex bundle

ξ
\xi

over the weakly complex manifold

M
M

. If

τ
\tau

is the stable right homotopical inverse of the infinite loop spaces map

η
:
Q
B
U
(
1
)
→
B
U

\eta :QBU(1) \to BU

, we define

f
ξ
′

=
τ
⋅

f
ξ

f_\xi ’ = \tau \cdot {f_\xi }

and we prove that the Chern classes

c
k

(
ξ
)

{c_k}(\xi )

are

f
ξ

′
∗

(

h
k

∗

(

t
k

)
)

f_\xi ^{\prime \ast }(h_k^{\ast }(t_k))

, where

h
k

{h_k}

is given by the stable splitting of

Q
B
U
(
1
)

QBU(1)

and

t
k

{t_k}

is the Thom class of the bundle

γ

(
k
)

=
E

Σ
k

X

Σ
k

γ
k

{\gamma ^{(k)}} = E{\Sigma _k}{X_{{\Sigma _k}}}{\gamma ^k}

. Also, we associate to

f
′

f’

an immersion

g
:
N
→
M

g:N \to M

and we prove that

c
k

(
ξ
)

{c_k}(\xi )

is the dual of the image of the fundamental class of the

k
k

-tuple points manifold of the immersion

g
,

g
k

∗

(
[

N
k

]
)

g,g_k^{\ast }([{N_k}])

.
On the dimension of the 𝑙ⁿ_{𝑝}-subspaces of Banach spaces, for 1≤𝑝<2
Authors: Pisier Gilles
Published: June 2010

We give an estimate relating the stable type

p
p

constant of a Banach space

X
X

with the dimension of the

l
p
n

l_p^n

-subspaces of

X
X

. Precisely, let

C
C

be this constant and assume

1
>
p
>
2

1 > p > 2

. We show that, for each

ε
>
0
,
X

\varepsilon > 0,X

must contain a subspace

(
1
+
ε
)

(1 + \varepsilon )

-isomorphic to

l
p
k

l_p^k

, for every

k
k

less than

δ
(
ε
)

C

p
′

\delta (\varepsilon ){C^{p’}}

where

δ
(
ε
)
>
0

\delta (\varepsilon ) > 0

is a number depending only on

p
p

and

ε
\varepsilon

.
A general sufficiency theorem for nonsmooth nonlinear programming
Authors: Chaney R. W.
Published: June 2010

Second-order conditions are given which are sufficient to guarantee that a given point be a local minimizer for a real-valued locally Lipschitzian function over a closed set in

n
n

-dimensional real Euclidean space. These conditions are expressed in terms of the generalized gradients of Clarke. The conditions provide a very general and unified framework into which many previous first- and second-order theorems fit.
Sufficient conditions for smoothing codimension one foliations
Authors: Ennis Christopher
Published: June 2010

Let

M
M

be a compact

C
∞

{C^\infty }

manifold. Let

X
X

be a

C
0

{C^0}

nonsingular vector field on

M
M

, having unique integral curves

(
p
,
t
)

(p,t)

through

p
∈
M

p \in M

. For

f
:
M
→

R

f: M \to {\mathbf {R}}

continuous, call

X
f
(
p
)
=
d
f
(
p
,
t
)

/

d
t
|

t
=
0

\left . Xf(p) = df(p,t)/dt\right |_{t = 0}

whenever defined. Similarly, call

X
k

f
(
p
)
=
X
(

X

k
−
1

f
)
(
p
)

{X^k}f(p)=X(X^{k-1}f)(p)

. For

0
⩽
r
>
k

0 \leqslant r > k

, a

C
r

{C^r}

foliation

F

\mathcal {F}

of

M
M

is said to be

C
k

{C^k}

smoothable if there exist a

C
k

{C^k}

foliation

G

\mathcal {G}

, which

C
r

{C^r}

approximates

F

\mathcal {F}

, and a homeomorphism

h
:
M
→
M

h:M \to M

such that

h
h

takes leaves of

F

\mathcal {F}

onto leaves of

G

\mathcal {G}

. Definition. A transversely oriented Lyapunov foliation is a pair

(

F

,
X
)

(\mathcal {F},X)

consisting of a

C
0

{C^0}

codimension one foliation

F

\mathcal {F}

of

M
M

and a

C
0

{C^0}

nonsingular, uniquely integrable vector field

X
X

on

M
M

, such that there is a covering of

M
M

by neighborhoods

{

W
i

}

\{{W_i}\}

,

0
⩽
i
⩽
N

0 \leqslant i \leqslant N

, on which

F

\mathcal {F}

is described as level sets of continuous functions

f
i

:

W
i

→

R

{f_i}:{W_i} \to {\mathbf {R}}

for which

X

f
i

(
p
)

X{f_i}(p)

is continuous and strictly positive. We prove the following theorems. Theorem 1. Every

C
0

{C^0}

transversely oriented Lyapunov foliation

(

F

,
X
)

(\mathcal {F},X)

is

C
1

{C^1}

smoothable to a

C
1

{C^1}

transversely oriented Lyapunov foliation

(

G

,
X
)

(\mathcal {G},X)

. Theorem 2. If

(

F

,
X
)

(\mathcal {F},X)

is a

C
0

{C^0}

transversely oriented Lyapunov foliation, with

X
∈

C

k
−
1

X \in {C^{k - 1}}

and

X
j

f
i

(
p
)

{X^j}{f_i}(p)

continuous for

1
⩽
j
⩽
k

1 \leqslant j \leqslant k

and

0
⩽
i
⩽
N

0 \leqslant i \leqslant N

, then

(

F

,
X
)

(\mathcal {F},X)

is

C
k

{C^k}

smoothable to a

C
k

{C^k}

transversely oriented Lyapunov foliation

(

G

,
X
)

(\mathcal {G},X)

. The proofs of the above theorems depend on a fairly deep result in analysis due to F. Wesley Wilson, Jr. With only elementary arguments we obtain the

C
k

{C^k}

version of Theorem 1. Theorem 3. If

(

F

,
X
)

(\mathcal {F},X)

is a

C

k
−
1

(
k
⩾
2
)

{C^{k - 1}}\;(k \geqslant 2)

transversely oriented Lyapunov foliation, with

X
∈

C

k
−
1

X \in {C^{k - 1}}

and

X
k

f
i

(
p
)

{X^k}{f_i}(p)

is continuous, then

(

F

,
X
)

(\mathcal {F},X)

is

C
k

{C^k}

smoothable to a

C
k

{C^k}

transversely oriented Lyapunov foliation

(

G

,
X
)

(\mathcal {G},X)

.
Convergence of functions: equi-semicontinuity
Authors: Dolecki Szymon , Salinetti Gabriella , Wets Roger J.-B.
Published: June 2010

We study the relationship between various types of convergence for extended real-valued functionals engendered by the associated convergence of their epigraphs; pointwise convergence being treated as a special case. A condition of equi-semicontinuity is introduced and shown to be necessary and sufficient to allow the passage from one type of convergence to another. A number of compactness criteria are obtained for families of semicontinuous functions; in the process we give a new derivation of the Arzelá-Ascoli Theorem.
Topological semiconjugacy of piecewise monotone maps of the interval
Authors: Byers Bill
Published: June 2010

This paper establishes a topological semiconjugacy between two piecewise monotone maps of the interval which have the same kneading sequences and do not map one turning point into another, whenever itineraries under the second map are given uniquely by their invariant coordinate. Various examples are given and consequences obtained.
Smooth type 𝐼𝐼𝐼 diffeomorphisms of manifolds
Authors: Hawkins Jane
Published: June 2010

In this paper we prove that every smooth paracompact connected manifold of dimension

⩾
3

\geqslant 3

admits a smooth type

III

λ

{\text {III}}_\lambda

diffeomorphism for every

0
⩽
λ
⩽
1

0 \leqslant \lambda \leqslant 1

. (Herman proved the result for

λ
=
1

\lambda = 1

in [7].) The result follows from a theorem which gives sufficient conditions for the existence of smooth ergodic real line extensions of diffeomorphisms of manifolds.