bluuebunny/crossref_metadata_2025 huggingface dataset:
On the Sobczyk-Hammer decomposition of additive set functions
Authors: Siebe Wilfried
Published: June 2010

It is observed that continuity for charges is equivalent to the absence of two-valued minorants. This characterization forms the basis of a new short proof within a functional-analytic context of a decomposition theorem by A. Sobczyk and P. C. Hammer [5] for charges on a field

A

\mathfrak {A}

into a continuous part and a part which can be written as a sum of at most two-valued charges on

A

\mathfrak {A}

. A counterexample shows that in general the decomposition of a charge into a nonatomic part and a part which has no nonnull nonatomic minorant is not unique.
On the integral means of derivatives of the atomic function
Authors: Mateljević Miodrag , Pavlović Miroslav
Published: June 2010

In this note we give upper and lower estimates on integral means of the atomic function and its derivatives over a circle of radius

r
r

as

r
r

approaches 1. From this we derive some known and new results.
Lower bounds for the unknotting numbers of certain torus knots
Authors: Yamamoto Makoto
Published: June 2010

In this paper we shall show that the unknotting numbers of the

(
l
,
2
k
l
±
1
)

(l,2kl \pm 1)

-torus knots are at least

(
k
(

l
2

−
1
)
−
2
)

/

2

(k({l^2} - 1) - 2)/2

for

l
l

odd and

(
k

l
2

−
2
)

/

2

(k{l^2} - 2)/2

for

l
l

even, where

l
l

is an integer greater than one and

k
k

is a positive integer.
Extending expansive homeomorphisms
Authors: Wine J. D.
Published: June 2010

In previous work on extensions of expansive homeomorphisms there has always been a finiteness condition placed on the remainder. In this note we give a necessary and sufficient condition for a self-homeomorphism to be expansive by restrictions on orbits, and then use this to give a sufficient condition for the homeomorphic extension of an expansive homeomorphism to be expansive. Two examples are given wherein the homeomorphic extensions of expansive homeomorphism are examined relative to the conditions of the theorem.
On quaternionic James numbers and almost-quaternion substructures on the sphere
Authors: Önder Turgut
Published: June 2010

In this paper a theorem about the relation between the divisibility of orders of obstructions to cross sectioning symplectic Stiefel manifolds and quaternionic James numbers is proved. As an application of this, the existence problem of almost-quaternion

k
k

-substructures on the sphere

S
n

{S^n}

is solved for all

n
n

and

k
k

except for the case

n
=
4
m
−
3

n = 4m - 3

,

k
=
m
−
1

k = m - 1

for some

m
⩾
1

m \geqslant 1

.
The recovery of some abelian groups from their socles
Authors: Hill Paul
Published: June 2010

The first main result of this paper is that summable isotype subgroups inherit total projectivity. This means: (

∗
\ast

) if an isotype subgroup of a totally projective group has a free socle (viewed as a valuated vector space) then the socle completely determines the subgroup up to isomorphism. The next major result is that (

∗
\ast

) does not generalize to the case where the socle is that of a totally projective group of length exceeding

Ω
\Omega

, nor does (

∗
\ast

) generalize to the case where the socle is that of an

S
S

-group of length

Ω
\Omega

. Finally, it is shown that if, in addition to

H
H

being isotype in a d.s.c. group

G
G

, it is also known that

K

/

H

K/H

is divisible where

K
K

is the closure of

H
H

in

G
G

relative to the

p
Ω

{p^\Omega }

-topology, then (

∗
\ast

) again prevails when the socle is that of an

S
S

-group.
Height one differential ideals in polynomial rings
Authors: Chen Matthew S.
Published: June 2010

Let

D
D

be a derivation of a polynomial ring

k
[

X
1

,
…
,

X
n

]

k[{X_1}, \ldots ,{X_n}]

with

k
k

a field of characteristic 0 and

D
k
=
{
0
}

Dk = \{ 0\}

. If infinitely many principal prime ideals

(
f
)

(f)

satisfy

D
f
∈
(
f
)

Df \in (f)

, then every maximal ideal contains such an

(
f
)

(f)

.
Unstable weak attractors
Authors: Knight Ronald A.
Published: June 2010

Our objective in this paper is to continue the process of classification and characterization of weak attractors initiated by the author in an earlier paper. In particular, we obtain additional characterizations of those weak attractors which are saddle sets and bilateral weak attractors.
Strongly ergodic sequences of integers and the individual ergodic theorem
Authors: Blum J. R. , Reich J. I.
Published: June 2010

Let

S
=
{

k
1

,

k
2

,
…
}

S = \{ {k_1},{k_2}, \ldots \}

be an increasing sequence of positive integers. We call

S
S

strongly ergodic if for every measure preserving transformation

T
T

on a probability space

(
Ω
,

F

,
P
)

(\Omega ,\mathcal {F},P)

and every

f
∈

L
1

(
Ω
)

f \in {L_1}(\Omega )

we have
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts left-parenthesis 1 slash n right-parenthesis sigma-summation Underscript j equals 1 Overscript n Endscripts f left-parenthesis upper T Superscript k j Baseline omega right-parenthesis equals upper P f left-parenthesis omega right-parenthesis">

lim

n
→
∞

(
1

/

n
)

∑

j
=
1

n

f
(

T

k
j

ω
)
=
P
f
(
ω
)

{\lim _{n \to \infty }}(1/n)\sum \nolimits _{j = 1}^n {f({T^{kj}}\omega ) = Pf(\omega )}

a.e. where

P
f

Pf

is the appropriate limit guaranteed by the individual ergodic theorem. We give sufficient conditions for a sequence

S
S

to be strongly ergodic and provide examples.
Uniqueness of solutions of partial differential equations when the initial surface is characteristic at a point
Authors: Korbly Letitia J.
Published: June 2010

Uniqueness in the Cauchy problem for hyperbolic operators degenerate at a point on the initial surface depends on values of the coefficients of the lower order terms. If the operator

P
P

is doubly characteristic at the origin with respect to the

t
=
0

t = 0

line,

P
P

has uniqueness for functions which are smooth enough if the coefficient of the

D
t

{D_t}

term does not lie in a certain discrete set of numbers.
Inner amenability and fullness
Authors: Choda Marie
Published: June 2010

Let

G
G

be a countable group which is not inner amenable. Then the II

1

_{1}

-factor

M
M

is full in the following cases: (1)

M
M

is given by the group measure space construction from a triple

(
X
,
μ
,
G
)

(X,\mu ,G)

with respect to a strongly ergodic measure preserving action of

G
G

on a probability space

(
X
,
μ
)

(X,\mu )

. (2)

M
M

is the crossed product of a full II

1

_{1}

-factor by

G
G

with respect to an action.
Dickson’s pseudodeterminant without matrices
Authors: Grove Larry C.
Published: June 2010

The Dickson pseudodeterminant for orthogonal transformations at characteristic 2 is developed without using its formula in terms of matrix entries.
On productive classes of function rings
Authors: Bankston Paul
Published: June 2010

No nontrivial

P
P

-class ("

P
P

" for "productive") of rings of continuous real-valued functions can be category equivalent to any elementary

P
P

-class of finitary universal algebras.
Small transitive lattices
Authors: Hadwin D. W. , Longstaff W. E. , Rosenthal Peter
Published: June 2010

Partial results are obtained on the problem of determining the smallest lattice of subspaces of a Hilbert space with the property that the only operators leaving all the subspaces invariant are the multiples of the identity.
The Baire class of approximate symmetric derivates
Authors: Larson Lee
Published: June 2010

It is shown that all approximate symmetric derivatives of measurable functions are in Baire class one. Further, if

f
f

is a measurable function which is finite a.e., then its upper and lower approximate symmetric derivates are in Baire class three.
Neighborhood contractible spaces
Authors: Martin John R.
Published: June 2010

In this paper the concept of a neighborhood contractible space (or NC-space) is introduced. The class of NC-spaces is a proper subclass of the class of locally contractible compacta and it is shown that several classes of spaces, including the class of ANR-spaces. belong to the class of NC-spaces.
Geometric realization of a finite subgroup of 𝜋₀𝜖(𝑀). II
Authors: Lee Kyung Bai
Published: June 2010

Let

M
M

be a closed aspherical manifold with a virtually

2
2

-step nilpotent fundamental group. Then any finite group

G
G

of homotopy classes of self-homotopy equivalences of

M
M

can be realized as an effective group of self-homeomorphisms of

M
M

if and only if there exists a group extension

E
E

of

π
\pi

by

G
G

realizing

G
→

Out

π
1

M

G \to {\operatorname {Out }}{\pi _1}M

so that

C
E

(
π
)

{C_E}(\pi )

, the centralizer of

π
\pi

in

E
E

, is torsion-free. If this is the case, the action

(
G
,
M
)

(G,M)

is equivalent to an affine action

(
G
,

M
′

)

(G,M’)

on a complete affinely flat manifold homeomorphic to

M
M

. This generalizes the same result for flat Riemannian manifolds.
Oriented toral knot lattices
Authors: Walker Russell B.
Published: June 2010

The common intersections between two sets of toral knots, having differing toral knot types and differing sequences of orientations, are removed. The remaining arc-segments are reconnected in a prescribed manner. In closed form, the number of "resulting loops" is provided.
Rational tilings by 𝑛-dimensional crosses
Authors: Szabó Sándor
Published: June 2010

Consider the set of closed unit cubes whose edges are parallel to the coordinate unit vectors

e

1

,
…
,

e

n

{{\mathbf {e}}_1}, \ldots ,{{\mathbf {e}}_n}

and whose centers are

i

e

j

i{{\mathbf {e}}_j}

,

0
⩽

|

i

|

⩽
k

0 \leqslant |i| \leqslant k

, in

n
n

-dimensional Euclidean space. The union of these cubes is called a cross. This cross consists of

2
k
n
+
1

2kn + 1

cubes; a central cube together with

2
n

2n

arms of length

k
k

. A family of translates of a cross whose union is

n
n

-dimensional Euclidean space and whose interiors are disjoint is a tiling. Denote the set of translation vectors by

L

{\mathbf {L}}

. If the vector set

L

{\mathbf {L}}

is a vector lattice, then we say that the tiling is a lattice tiling. If every vector of

L

{\mathbf {L}}

has rational coordinates, then we say that the tiling is a rational tiling, and, similarly, if every vector of

L

{\mathbf {L}}

has integer coordinates, then we say that the tiling is an integer tiling. Is there a noninteger tiling by crosses? In this paper we shall prove that if there is an integer lattice tiling by crosses, if

2
k
n
+
1

2kn + 1

is not a prime, and if

p
>
k

p > k

for every prime divisor

p
p

of

2
k
n
+
1

2kn + 1

, then there is a rational noninteger lattice tiling by crosses and there is an integer nonlattice tiling by crosses. We will illustrate this in the case of a cross with arms of length 2 in

55
55

-dimensional Euclidean space. Throughout, the techniques are algebraic.
Rough and strongly rough norms on Banach spaces
Authors: Godini G.
Published: June 2010

We give equivalent conditions for a norm on a Banach space

X
X

to be rough using the sets

A
(
x
)

A(x)

defined for each

x
∈
X

x \in X

by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis x right-parenthesis equals left-brace f element-of upper X Superscript asterisk Baseline StartAbsoluteValue f left-parenthesis x right-parenthesis equals EndAbsoluteValue StartAbsoluteValue x EndAbsoluteValue StartAbsoluteValue comma EndAbsoluteValue StartAbsoluteValue f EndAbsoluteValue vertical-bar equals 1 right-brace">

A
(
x
)
=
{

f
∈

X
∗

|

f
(
x
)
=

|

|

x

|

|

,

|

|

f

|

|

=
1
}

A(x) = \{ {f \in {X^ * }|f(x) = || x ||,|| f || = 1\}}

. This enables us to obtain unitary characterizations for rough and strongly rough norms.
𝐿^{𝑝}-behavior of power series with positive coefficients and Hardy spaces
Authors: Mateljević Miodrag , Pavlović Miroslav
Published: June 2010

For the power series

f
(
x
)
=

∑
1
∞

a
n

x
n

f(x) = \sum \nolimits _1^\infty {{a_n}{x^n}}

with

a
n

⩾
0

{a_n} \geqslant 0

, certain weighted

L
p

{L^p}

-norms of

f
f

on

[
0
,
1
]

[0,1]

are estimated from above and below in terms of the coefficients

a
n

{a_n}

. Some consequences of this are obtained. For example, some known results concerning Hardy spaces may be extended to a wider class of spaces.
𝐻*(𝑀𝑂⟨8⟩;𝑍/2) is an extended 𝐴*₂-coalgebra
Authors: Pengelley David J.
Published: June 2010

We show that

H
∗

(
M
O

⟨
8
⟩

;
Z

/

2
)

{H^ * }(MO\left \langle 8 \right \rangle ;Z/2)

is an extended

A
2
∗

A_2^ *

-coalgebra, where

A
2
∗

A_2^ *

is the subalgebra of the Steenrod algebra generated by

{

S

q

1

,

S

q

2

,

S

q

4

}

\left \{ {{\text {S}}{{\text {q}}^1},{\text {S}}{{\text {q}}^2},{\text {S}}{{\text {q}}^4}} \right \}

. The method yields an analogous result for

H
∗

(
M

Spin

;
Z

/

2
)

{H^ * }(M {\operatorname {Spin}}; Z/2)

.
𝑋-inner automorphisms of filtered algebras. II
Authors: Montgomery Susan
Published: June 2010

We continue earlier work and compute the

X
X

-inner automorphisms of the ring of differential polynomials in one variable over an arbitrary domain. This is then applied to iterated Ore extensions. We also show that the ring of generic matrices has no nonidentity automorphisms which fix the center.
Numbers of solutions of congruences: Poincaré series for strongly nondegenerate forms
Authors: Goldman Jay R.
Published: June 2010

Let

p
p

be a fixed prime,

f
(

x
1

,
…
,

x
k

)

f({x_1}, \ldots ,{x_k})

a polynomial over

Z

p

{{\mathbf {Z}}_p}

, the

p
p

-adic integers.

c
n

{c_n}

the number of solutions of

f
=
0

f = 0

over

Z

/

p
n

Z

{\mathbf {Z}}/{p^n}{\mathbf {Z}}

and

P
f

(
t
)
=

∑

i
=
0

∞

c
i

t
i

{P_f}(t) = \sum \nolimits _{i = 0}^\infty {{c_i}{t^i}}

the Poincaré series. A general approach to computing

c
n

{c_n}

and

P
f

(
t
)

{P_f}(t)

is given and explicit formulas are derived for strongly nondegenerate forms.
Uniform 𝜎-additivity in spaces of Bochner or Pettis integrable functions over a locally compact group
Authors: Dinculeanu Nicolae
Published: June 2010

If

G
G

is an abelian locally compact group with Haar measure

μ
\mu

,

E
E

is a Banach space and

K
⊂

L
E
1

(
μ
)

K \subset L_E^1(\mu )

, we give necessary and sufficient conditions for the set

{

f

(
⋅
)

|
f
|

d
μ
;
f
∈
K

}

\left \{ {{f_{( \cdot )}}\left | f \right |d\mu ;f \in K} \right \}

to be uniformly

σ
\sigma

-additive in terms of uniform convergence on

K
K

, for the topology

σ
(

L
E
1

,

L

E
′

∞

)

\sigma (L_E^1,L_{E’}^\infty )

of convolution and translation operators. In case

E
=
R

E = R

, this gives a new characterization of relatively weakly compact sets

K
⊂

L
1

K \subset {L^1}

.
ℵ₀-categorical distributive lattices of finite breadth
Authors: Schmerl James H.
Published: June 2010

Every

ℵ
0

{\aleph _0}

-categorical distributive lattice of finite breadth has a finitely axiomatizable theory. This result extends the analogous result for partially ordered sets of finite width.
The Cartan matrix of a group algebra modulo any power of its radical
Authors: Landrock Peter
Published: June 2010

We prove that the Cartan matrix of a group algebra

F
[
G
]

F[G]

modulo any power of its radical

J
J

is dual symmetric, provided

F
F

is a splitting field of

F
[
G
]

/

J

F[G]/J

. This eases the process of determining the Loewy series of the projective indecomposable

F
[
G
]

F[G]

-modules.
The equivalence of high subgroups
Authors: Hill Paul
Published: June 2010

Two subgroups of a group

G
G

are called equivalent if there is an automorphism of

G
G

that maps one of the subgroups onto the other. Suppose that

G
G

is a

p
p

-primary abelian group and that

λ
\lambda

is an ordinal. A subgroup

H
H

of

G
G

is

p
λ

{p^\lambda }

-high in

G
G

if

H
H

is maximal in

G
G

with respect to having zero intersection with

p
λ

G

{p^\lambda }G

. Under certain conditions on the quotient group

G

/

p
λ

G

G/{p^\lambda }G

slightly weaker than total projectivity, it is shown, for a given

λ
\lambda

, that any two

p
λ

{p^\lambda }

-high subgroups of

G
G

are equivalent. In particular, if

G

/

p
ω

G

G/{p^\omega }G

is

p

ω
+
1

{p^{\omega + 1}}

-projective, the

p
ω

{p^\omega }

-high subgroups of

G
G

are all equivalent.
Asymptotic behavior of solutions of retarded differential equations
Authors: Ladas G. , Sficas Y. G. , Stavroulakis I. P.
Published: June 2010

In this paper we obtain sufficient conditions under which every solution of the retarded differential equation
\[

(
1
)

x
′

(
t
)
+
p
(
t
)
x
(
t
−
τ
)
=
0
,

t
⩾

t
0

,

(1)\quad x’(t) + p(t)x(t - \tau ) = 0,\quad t \geqslant {t_0},

\]
, where

τ
\tau

is a nonnegative constant, and

p
(
t
)
>
0

p(t) > 0

, is a continuous function, tends to zero as

t
→
∞

t \to \infty

. Also, under milder conditions, we prove that every oscillatory solution of (1) tends to zero as

t
→
∞

t \to \infty

. More precisely the following theorems have been established. Theorem 1. Assume that

∫

t
0

∞

p
(
t
)
d
t
=
+
∞

\int _{{t_0}}^\infty {p(t)dt = + \infty }

and

lim

t
→
∞

∫

t
−
τ

t

p
(
s
)
d
s
>
π

/

2

{\lim _{t \to \infty }}\int _{t - \tau }^t {p(s)ds > \pi /2}

or

lim

sup

t
→
∞

∫

t
−
τ

t

p
(
s
)
d
s
>
1

\lim {\sup _{t \to \infty }}\int _{t - \tau }^t {p(s)ds > 1}

. Then every solution of (1) tends to zero as

t
→
∞

t \to \infty

. Theorem 2. Assume that

lim

sup

t
→
∞

∫

t
→
τ

t

p
(
s
)
d
s
>
1

\lim {\sup _{t \to \infty }}\int _{t \to \tau }^t {p(s)ds > 1}

. Then every oscillatory solution of (1) tends to zero as

t
→
∞

t \to \infty

.
Conditions for some polygonal functions to be Bazilevič
Authors: Case B. A. , Quine J. R.
Published: June 2010

Univalent functions in the disc whose image is a particular eight-sided polygonal region determined by two parameters are studied. Whether such a function is Bazilevič is determined in terms of the two parameters, and the set of real

α
\alpha

’s is specified such that the function is

(
α
,
β
)

(\alpha ,\beta )

Bazilevič for some

β
\beta

. For any interval

[

a
,
b

]

\left [ {a,b} \right ]

where

1
>
a
⩽
3
⩽
b

1 > a \leqslant 3 \leqslant b

, a function of this type which is

(
α
,
0
)

(\alpha ,0)

Bazilevič precisely when

α
\alpha

is in this interval is found. Examples are given of non-Bazilevič functions with polygonal images and Bazilevič functions which are

(
α
,
0
)

(\alpha ,0)

Bazilevič for a single value

α
\alpha

.
On the normal structure coefficient and the bounded sequence coefficient
Authors: Lim Teck-Cheong
Published: June 2010

The two notions of normal structure coefficient and bounded sequence coefficient introduced by Bynum are shown to be the same. A lower bound for the normal structure coefficient in

L
p

{L^p}

,

p
>
2

p > 2

, is also given.
Tensor products of preclosed operators on 𝐶*-algebras
Authors: Wu Liang Sen
Published: June 2010

In this paper, we prove the following result: If

A
1

{A_1}

,

A
2

{A_2}

are

C
∗

{C^ * }

-algebras, and

T
1

{T_1}

,

T
2

{T_2}

are preclosed operators on

A
1

{A_1}

,

A
2

{A_2}

respectively, then

T
1

⊗

T
2

{T_1} \otimes {T_2}

is preclosed on

A
1

⊗

min

A
2

{A_1}{ \otimes _{\min }}{A_2}

. Furthermore, we show that the injective

C
∗

{C^ * }

-cross norm

|

|

⋅

|

|

min

|| \cdot |{|_{\min }}

is reflexive on the algebraic tensor product

A
1

⊗

A
2

{A_1} \otimes {A_2}

.
Removable singularity sets for analytic functions having modulus with bounded Laplace mass
Authors: Cegrell Urban
Published: June 2010

We prove that certain closed sets are removable singularity sets for analytic functions having modulus with bounded Laplace mass. As a special case, we find that every function which is analytic and with modulus having a harmonic majorant outside an analytic set extends analytically across this set.
On the necessity of the Hörmander condition for multipliers on 𝐻^{𝑝}(𝑅ⁿ)
Authors: Daly James E.
Published: June 2010

In this paper we prove that a class of multiplier operators on

H

p

(

R

n

)

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

, that send atoms to molecules boundedly, must satisfy a Hörmander condition. This provides a partial converse to a theorem of Taibleson and Weiss.
An injective metrization for collapsible polyhedra
Authors: Mai Jie Hua , Tang Yun
Published: June 2010

In this paper we prove that any finite collapsible polyhedron is injectively metrizable.
An algorithm for checking property P for knots with complements of Heegaard genus 2
Authors: Osborne R. P.
Published: June 2010

One of the most fundamental questions about knots is: If we know the topological type of the complement of a knot, is the knot determined? In this paper we give an algorithm for deciding if certain knots called tunnel number one knots are determined by their complements. This algorithm turns out to be practical and efficient in that it can be used on knots with ten crossings without the aid of a computer and one can expect to be able to handle knots with, say, twenty crossings with the aid of a desk-top computer.
The spectral decomposition of a product of automorphic forms
Authors: Moreno C. J.
Published: June 2010

The spectral theory of Roelke and Selberg provides a decomposition of the space of square integrable automorphic forms for the group

S
L
(
2
)

SL(2)

in terms of eigenfunctions of the non-Euclidean Laplacian and of the Hecke operators. The main result of the paper uses the Roelke-Selberg theory to give an interpretation of the

L
L

-functions of Rankin type as "multiplicity factors" in the decomposition of the product of a nonholomorphic Eisenstein series and a cusp form.
Measures associated to Chern, Levine and Nirenberg norms
Authors: Kalka Morris
Published: June 2010

We define an intrinsic outer measure on complex submanifolds of a given complex manifold. This is modeled on the norm on the real homology defined by Chern, Levine and Nirenberg.
With respect to tail sigma fields, standard measures possess measurable disintegrations
Authors: Dubins Lester E. , Heath David
Published: June 2010

Let

P
P

be a countably additive probability on a standard space, and let

A

\mathcal {A}

be a tail subfield. Though no disintegration of

P
P

with respect to

A

\mathcal {A}

that is countably additive need exist, there always is one which is finitely additive.
Local ergodicity of nonpositive contractions on 𝐶(𝑋)
Authors: Atalla Robert E.
Published: June 2010

Let

T
T

be an operator on

C
(
X
)

C(X)

,

X
X

compact, with

‖
T
‖

⩽
1

\left \| T \right \| \leqslant 1

, and suppose

T
T

has a nowhere vanishing invariant function

ψ

−
1

{\psi ^{ - 1}}

. The operator

R
R

defined by

R
f
=
T
(
f

ψ

−
1

)
ψ

Rf = T(f{\psi ^{ - 1}})\psi

is (a) "locally" a Markov operator, and (b) (locally) strongly ergodic iff

T
T

is. This is used to prove Sine’s local strong ergodicity theorem without assuming that

T
T

is positive.
A note on a maximal function of C. Fefferman and Stein
Authors: Chanillo Sagun , Wheeden Richard L.
Published: June 2010

We extend the class of weight functions for which a known inequality for the maximal function

T

λ
,
r

(
f
)
(
x
)

{T_{\lambda ,r}}(f)(x)

is valid.
Embedding 𝑙_{𝑝}^{𝑛^{𝛼}} in 𝑙ⁿ_{𝑝,𝑞}
Authors: Carothers N. L. , Flinn P. H.
Published: June 2010

It is shown that for

1
>
p
>
∞

1 > p > \infty

,

1
⩽
q
⩽
∞

1 \leqslant q \leqslant \infty

and

0
>
α
>
1

0 > \alpha > 1

, there is a constant

C
=
C
(
p
,
q
,
α
)
>
∞

C = C(p,q,\alpha ) > \infty

such that

l
p
k

l_p^k

is

C
C

-isomorphic to a subspace of

l

p
,
q

n

l_{p,q}^n

where

k
=
O
(

n
α

)

k = O({n^\alpha })

.
Forced differences between terms of subsequences of integer sequences
Authors: Gilpin Michael , Shelton Robert
Published: June 2010

Let

a
1

,

a
2

,
…

{a_1},{a_2}, \ldots

be a sequence of integers and let

D
=

{

d
1

,
…
,

d
k

}

D = \left \{ {{d_1}, \ldots ,{d_k}} \right \}

be a fixed finite set of integers. For each positive integer

n
n

we investigate the problem of choosing maximal subsequences

a

i
1

,
…
,

a

i
t

{a_{{i_1}}}, \ldots ,{a_{{i_t}}}

from

a
1

,
…
,

a
n

{a_1}, \ldots ,{a_n}

such that

|

a

i
α

−

a

i
β

|

∉
D

\left | {{a_{{i_\alpha }}} - {a_{{i_\beta }}}} \right | \notin D

for

α
≠
β

\alpha \ne \beta

. An asymptotic form for

t
t

, the maximum length of such subsequences, is derived in the special case

a
i

=
i

{a_i} = i

.
Minimal ideals in quadratic Jordan algebras
Authors: Ng Seong Nam , McCrimmon Kevin
Published: June 2010

In associative and alternative algebras a minimal ideal is either trivial or simple. This is not known for quadratic Jordan algebras. In the present note we show that a minimal ideal is either trivial or

D

\mathcal {D}

-simple (possesses no proper ideals invariant under all inner derivations induced from the ambient algebra). In particular, the heart of any quadratic Jordan algebra is either trivial or

D

\mathcal {D}

-simple. Hearts have recently played an important role in Zelmanov’s theory of prime Jordan algebras.
A note concerning Fox’s paper on Fenchel’s conjecture
Authors: Chau T. C.
Published: June 2010

The object of this note is to point out and correct an error in the paper [2] of Fox, purporting to prove Fenchel’s conjecture that a finitely generated, infinite Fuchsian group has a torsion-free normal subgroup of finite index. Fox divided the proof of the main result in his paper into four cases and an error occurs in the proofs of Cases (III) and (IV). A direct proof of Case (III) was given. While a direct proof of Case (IV) can be given, the author shows that it also follows indirectly from the result in the paper [1] of Burns and Solitar.
Existence of solutions of a nonlinear differential equation
Authors: Cesari L. , Kannan R.
Published: June 2010

A criterion is proved for the existence of at least one solution to the equation

u
+
u
=
g
(
u
)
+
h

u + u = g(u) + h

with

u
(
0
)
=
u
(
π
)
=
0

u(0) = u(\pi ) = 0

, where

h
∈

L
2

[
0
,
π
]

h \in {L_2}[0,\pi ]

and

g
g

is continuous monotone nonincreasing.
The singular integral characterization of 𝐻^{𝑝} on simple martingales
Authors: Uchiyama Akihito
Published: June 2010

The singular integral characterization of

H
1

{H^1}

on simple martingales was given by S. Janson. We show that his result cannot be extended to

H
p

{H^p}

if

p
(
>
0
)

p( > 0)

is very small.
Conformal inequivalence of annuli and the first-order theory of subgroups of 𝑃𝑆𝐿(2,𝑅)
Authors: Rubel Lee A.
Published: June 2010

An algebraic proof is given of the classical fact that two different concentric circular annuli

A
(
r
)

A(r)

and

A
(
s
)

A(s)

are conformally inequivalent, where

A
(
r
)
=
{
z
∈

C

:
1
>

|
z
|

>
r
}

A(r) = \{ z \in {\mathbf {C}}:1 > \left | z \right | > r\}

. Indeed, it is shown that the covering groups of these annuli are not elementarily equivalent in the context of

PSL

(
2
,

R

)

{\text {PSL}}(2,{\mathbf {R}})

. Considering the universal covering surface as

U
U

, the upper half-plane, the covering group of a bounded plane domain is naturally contained in

PSL

(
2
,

R

)

{\text {PSL}}(2,{\mathbf {R}})

as the group of covering transformations.
The nontriviality of the first rational homology group of some connected invariant subsets of periodic transformations
Authors: Assadi Amir , Burghelea Dan
Published: June 2010

This note was inspired by some results of P. A. Smith [S]. One proves that for any periodic map of a manifold

M
M

and any codimension two invariant submanifold

P
P

of

M
M

containing part of the stationary point set, connected invariant subsets of the complement of

P
P

must carry nontrivial one-dimensional rational cycles, provided that

M
M

satisfies some simple homological conditions (Theorem A). This fact has interesting consequences in transformation group theory.
On regular-invariance of continuity
Authors: Ori R. G. , Rajagopalan M.
Published: June 2010

Let

(
X
,
σ
)

(X,\sigma )

be a given topological space. A compression

τ
\tau

of

σ
\sigma

is regular-invariant if and only if for every regular space

Y
Y

the

σ
\sigma

-continuous functions into

Y
Y

are also

τ
\tau

-continuous.

σ
\sigma

is regular minimal if no proper compression of

σ
\sigma

is regular-invariant. J. A. Guthrie and H. E. Stone posed the problem of whether every semiregular space is regular minimal. We answer this question in the negative.
Lagrange identity for polynomials and 𝛿-codes of lengths 7𝑡 and 13𝑡
Authors: Yang C. H.
Published: June 2010

It is known that application of the Lagrange identity for polynomials (see [1]) is the key to composing four-symbol

δ
\delta

-codes of length

(
2
s
+
1
)
t

(2s + 1)t

for

s
=

2
a

10
b

26
c

s = {2^a}{10^b}{26^c}

and odd

t
⩽
59

t \leqslant 59

or

t
=

2
d

10
e

26
f

+
1

t = {2^d}{10^e}{26^f} + 1

, where

a
a

,

b
b

,

c
c

,

d
d

,

e
e

and

f
f

are nonnegative integers. Applications of the Lagrange identity also lead to constructions of four-symbol

δ
\delta

-codes of length

u
u

for

u
=
7
t

u = 7t

or

13
t

13t

. Consequently, new families of Hadamard matrices of orders

4
u
w

4uw

and

20
u
w

20uw

can be constructed, where

w
w

is the order of Williamson matrices. Related topics on zero correlation codes are also discussed.
A short proof of the algebraic Weierstrass preparation theorem
Authors: Gersten S. M.
Published: June 2010

The contraction mapping principle yields a short proof of the algebraic Weierstrass Preparation Theorem.
Local units modulo circular units
Authors: Coleman R.
Published: June 2010

In his paper, Some Modules in the Theory of Cyclotomic Fields [2], Iwasawa obtained the remarkable theorem that the quotient of the

p
p

-adic cyclotomic units by the completion of the circular units is isomorphic to the quotient of the group ring by the Stickleberger ideal. He then used this to deduce some interesting global results, the most striking of which is an explanation of the plus part of the analytic class number formula under the assumption that the class group at the first layer is cyclic, together with a regularity assumption. In this note, we will show how with the results in our paper, Division Values in Local Fields [1], it is now possible to give a substantially simpler proof of the above theorem. We also describe, briefly, how one can obtain various global results of Iwasawa from this Theorem, which are not included in either Lang’s [3], or Washington’s [4] books.
Rational curve with one cusp
Authors: Yoshihara Hisao
Published: June 2010

In this paper we shall give new examples of plane curves

C
−
{
P
}
≅

A

1

C - \{ P\} \cong {{\mathbf {A}}^1}

satisfying

C
−
{
P
}
≅

A

1

C - \{ P\} \cong {{\mathbf {A}}^1}

for some point

P
∈
C

P \in C

by using automorphisms of open surfaces.
Finite generation of Noetherian graded rings
Authors: Goto Shiro , Yamagishi Kikumichi
Published: June 2010

Let

H
H

be an additive abelian group. Then a commutative ring

A
A

is said to be

H
H

-graded if there is given a family

{

A
h

}

h
∈
H

{\{ {A_h}\} _{h \in H}}

of subgroups of

A
A

such that

A
=

⊕

h
∈
H

A
h

A = { \oplus _{h \in H}}{A_h}

and

A
h

A
g

⊂

A

h
+
g

{A_h}{A_g} \subset {A_{h + g}}

for all

h
h

,

g
∈
H

g \in H

. In this note it is proved, provided

H
H

is finitely generated, that an

H
H

-graded ring

A
A

is Noetherian if and only if the ring

A
0

{A_0}

is Noetherian and the

A
0

{A_0}

-algebra

A
A

is finitely generated. This is not true in general unless

H
H

is finitely generated. A counterexample is given.
On 𝑛-widths of certain functional classes defined by linear differential operators
Authors: Makovoz Y.
Published: June 2010

Let

A
=

D
r

+

∑

k
=
0

r
−
1

a
k

(
t
)

D
k

A = {D^r} + \sum \nolimits _{k = 0}^{r - 1} {{a_k}(t){D^k}}

,

a
k

∈

C
k

{a_k} \in {C^k}

, be a differential operator and let

W
p

(
A
)

{W_p}(A)

be the class of functions

x
(
t
)

x(t)

for which

‖

A
x

‖

⩽
1

\left \| {Ax} \right \| \leqslant 1

in

L
p

[
0
,
1
]

{L_p}[0,1]

. We prove that the asymptotic behavior of the Kolmogorov widths

d
n

(

W
p

(
A
)
,

L
q

)

{d_n}({W_p}(A),{L_q})

,

1
⩽
p

1 \leqslant p

,

q
⩽
∞

q \leqslant \infty

, when

n
→
∞

n \to \infty

does not depend on

a
k

{a_k}

.
On the Alexander polynomial of a cyclically periodic knot
Authors: Hillman Jonathan A.
Published: June 2010

We show that a theorem of Burde on the Alexander polynomial of a cyclically periodic knot

K
K

, such that

Δ
2

(
K
)
=
1

{\Delta _2}(K) = 1

, may be extended to all cyclically periodic knots.
Support points of the unit ball of 𝐻^{𝑝} (1≤𝑝≤∞)
Authors: Abu-Muhanna Yusuf
Published: June 2010

The following results are obtained for the

H
p

{H^p}

class, over the open unit disc, whenever

1
⩽
p
⩽
∞

1 \leqslant p \leqslant \infty

. (1)

f
f

is a support point of the unit ball of

H
p

{H^p}

, whenever

1
⩽
p
>
∞

1 \leqslant p > \infty

, if and only if

|

|

f

|

|

p

=
1

{|| f ||_p} = 1

and

f
f

is of the form

f
(
z
)
=

[
Q
(
z
)

]

2

/

p

⋅
W
(
z
)

f(z) = {[Q(z)]^{2/p}} \cdot W(z)

where

W
(
z
)

W(z)

is a function analytic in the closed unit disc and nonvanishing on its boundary and

Q
(
z
)

Q(z)

is either a nonzero constant or a polynomial with all of its zeros on the boundary of the unit disc. (2)

f
f

is a support point of the unit ball of

H
∞

{H^\infty }

if and only if

f
f

is a finite Blaschke product.
On the existence and boundary behavior of solutions to a class of nonlinear Dirichlet problems
Authors: Kenig Carlos E. , Ni Wei-Ming
Published: June 2010

In this paper, we first extend the well-known method of super- and sub-solutions for elliptic boundary value problems to

L
∞

{L^\infty }

-boundary functions. Then we apply this method to investigate the solvability and the boundary behavior of solutions to some nonlinear elliptic equations, some Fatou-type results are obtained.
Monadic properties of uniformities
Authors: Coyle D. G. , Szabo M. E.
Published: June 2010

We introduce the nonstandard concept of a monadic set and characterize diagonal and covering uniformities in terms of such sets. This formulation relates the two fundamental aspects of uniformities directly and obviates the need for bases. We illustrate the utility of our approach by showing that it leads to significant simplifications of the standard proofs characterizing pseudo-metrizable uniform and uniformizable topological spaces.
Semiperfect FPF rings
Authors: Page S. S.
Published: June 2010

In this paper we derive some of the structure of semiperfect FPF rings. A ring is right FPF if every f.g. faithful right module is a generator. For semiperfect right and left FPF rings we show that if all one sided zero divisors are two sided zero divisors, then the classical and maximal quotient rings coincide (all four of them) and are self-injective. We show that if the intersection of the powers of the Jacobson radical is zero, then right and left regular elements are regular. Also, we show right FPF semiperfect rings contain the singular submodule of their injective hulls and that every finitely generated module contained in the injective hull and containing the ring is isomorphic to the ring.
Cyclic algebras of small exponent
Authors: Tignol J.-P.
Published: June 2010

We prove that every cyclic algebra of exponent

n
n

and degree

m
n

mn

over a field containing a primitive

n
n

th root of unity is similar to a tensor product of at most

m
m

symbols of degree

n
n

.
Extreme points in function spaces
Authors: Werner Dirk
Published: June 2010

We show that the extreme points of the unit ball of

C
(
K
,

L
1

(
μ
)
)

C(K,{L^1}(\mu ))

(

K
K

compact Hausdorff,

(
Ω
,
Σ
,
μ
)

(\Omega ,\Sigma ,\mu )

arbitrary) are precisely the functions with extremal values. The result is applied to characterize the extreme points of the unit ball of certain spaces of compact operators.
On a problem of Banach
Authors: Pelc Andrzej , Prikry Karel
Published: June 2010

Assuming the continuum hypothesis, we obtain a translation invariant version of the following result of E. Grzegorek: There are two countably generated

σ
\sigma

-algebras on the interval [0, 1] such that both carry a nonatomic countably additive probability measure, but the

σ
\sigma

-algebra generated by their union does not carry any such measure.
The Riemann problem for nonconvex scalar conservation laws and Hamilton-Jacobi equations
Authors: Osher Stanley
Published: June 2010

We present a closed form expression for the viscosity solution to the Riemann problem for any scalar nonconvex conservation law. We then define an analogous problem for any scalar nonconvex Hamilton-Jacobi equation and obtain its (even simpler) solution. Extensions to two (and by inference, higher) space dimensional problems, when the initial discontinuity lies on a hyperplane, are also given.
Differentiability criteria and harmonic functions on 𝐵ⁿ
Authors: Ahern Patrick R. , Johnson Kenneth D.
Published: June 2010

Let

B
n

{B^n}

be the unit ball in

F

n

{{\mathbf {F}}^n}

where

F

{\mathbf {F}}

is either

R

{\mathbf {R}}

,

C

{\mathbf {C}}

or

H

{\mathbf {H}}

. The space

B
n

{B^n}

is a classical rank one symmetric space with respect to the action of a Lie group

G
G

. Suppose

f
f

is a

G
G

-harmonic function on

B
n

{B^n}

all of whose derivatives of order

⩽
k

\leqslant k

are bounded. Our main result obtains restrictions on

f
f

depending on

k
k

.
On Noetherianness of Nash rings
Authors: Mora Fulvio , Raimondo Mario
Published: June 2010

We introduce a class of rings, called Nash Rings, which generalize the notation of rings of Nash functions. Let

k
k

be any field,

X
X

be a normal algebraic variety in

k
n

{k^n}

, and

U
⊂
X

U \subset X

. A Nash ring

D
D

is the algebraic closure of

Γ
(
X
,

O

X

)

\Gamma (X,{\mathcal {O}_X})

in a suitable domain

B
B

such that

U
U

is contained in the maximal spectrum of

B
B

and

Γ
(
X
,

O

X

)

\Gamma (X,{\mathcal {O}_X})

is analytically isomorphic to

B
B

at each

x
∈
U

x \in U

. We show that

D
D

is a ring of fractions of the integral closure of

Γ
(
X
,

O

X

)

\Gamma (X,{\mathcal {O}_X})

in

B
B

. Moreover, if

k
k

is algebraically nonclosed and if every algebraic subvariety

V
⊂
X

V \subset X

intersects

U
U

in a finite number of connected components (in the topology induced by

B
B

), then

D
D

is noetherian.
Mullin’s sequence of primes is not monotonic
Authors: Naur Thorkil
Published: June 2010

The sequence of primes defined by

p
1

=
2

{p_1} = 2

and

p

n
+
1

=
(

largest

prime

factor

of

p
1

⋅

p
2

⋯

p
n

+
1
)

{p_{n + 1}} = ({\text {largest}}\;{\text {prime}}\;{\text {factor}}\;{\text {of}}\;{p_1}\cdot {p_2} \cdots {p_n} + 1)

is not monotone increasing. We present the first eleven primes of the sequence and observe that

p

10

>

p
9

{p_{10}} > {p_9}

.
On the generalized potential problem for a surface of revolution
Authors: Fabrikant V. I. , Sankar T. S. , Swamy M. N. S.
Published: June 2010

The following generalized potential problem is considered: given an arbitrary potential distribution on a surface of revolution, the charge density is to be determined. The problem is called generalized due to the assumption that the point charge potential diminishes with the distance

R
R

as

R

−
1
−
v

{R^{ - 1 - v}}

,

|
v
|

>
1

\left | v \right | > 1

. The particular case of

v
=
0

v = 0

corresponds to the electrostatic potential problem. A closed form solution to the problem is obtained for a certain class of surfaces of revolution due to a special integral representation of the kernel of the governing integral equation. Three examples are considered: the uniform potential distribution over a spherical cap, the case of an earthed conducting spherical cap in a uniform external field acting in two different directions, namely along the

0
z

0z

axis and the

0
x

0x

axis, respectively. The expressions for the charge density distributions are determined for each of the examples. The general results presented in this paper may also be applied to the solution of the mathematically identitical problems in hydrodynamics, thermal conductivity, etc.
A stronger metric for closed operators in Hilbert space
Authors: Kaufman William E.
Published: June 2010

There is available, for the closed and densely-defined operators in a Hilbert space, a metric which is stronger than the gap metric while sharing many of its properties. On the bounded operators it is equivalent to the metric generated by the usual operator-norm.
A technique for proving uniformity
Authors: Becker Howard
Published: June 2010

We introduce a general technique for showing that if various properties hold, then they hold uniformly, assuming some determinacy. Uniform means in the boldface sense, that is, there are continuous functions taking indices to indices. For example, if

Γ
\Gamma

is a parametrized pointclass then all closure properties of

Δ
\Delta

are uniform, and the separation property for

Γ
\Gamma

implies uniform separation.
On countable compactness and sequential compactness
Authors: Zhou Hao Xuan
Published: June 2010

If a countably compact

T
3

{T_3}

space

X
X

can be expressed as a union of less then

c
c

many first countable subspaces, then MA implies that

X
X

is sequentially compact. Also MA implies that every countably compact space of size

>
c

> c

is sequentially compact. However, there is a model of ZFC in which

ω
1

>
c

{\omega _1} > c

and there is a countably compact, separable

T
2

{T_2}

space of size

ω
1

{\omega _1}

, which is not sequentially compact.
Resolving acyclic images of nonorientable three-manifolds
Authors: Repovš Dušan , Lacher R. C.
Published: June 2010

We show that every

1
−

LC

1 - {\text {LC}}

Z

2

{{\mathbf {Z}}_2}

-homology

3
3

-manifold (without boundary) which is an almost

1
1

-acyclic (over

Z

2

{{\mathbf {Z}}_2}

) proper image of a nonorientable

3
3

-manifold

M
M

(without boundary) is a resolvable generalized

3
3

-manifold. The analogous result for the case when

M
M

is orientable was recently proved by J. L. Bryant and R. C. Lacher.
Countable abelian groups with a discrete norm are free
Authors: Lawrence John
Published: June 2010

It is proved that a countable Abelian group with a norm which induces the discrete topology is free.
Into isomorphisms of spaces of continuous functions
Authors: Jarosz Krzysztof
Published: June 2010

If

X
X

and

Y
Y

are locally compact Hausdorff spaces and

T
T

is a linear map from an extremely regular subspace of

C
0

(
X
)

{C_0}\left ( X \right )

into

C
0

(
Y
)

{C_0}\left ( Y \right )

such that

|

|

T

|

|

|

|

T

−
1

|

|

>
2

|| T |||| {{T^{ - 1}}} || > 2

, then

X
X

is a continuous image of a subset of

Y
Y

.
Asymptotic behavior of systems of linear ordinary differential equations
Authors: Tomastik E. C.
Published: June 2010

Conditions will be placed on the

m
×
m

m \times m

matrices

G
(
t
)

G(t)

and

G
i

(
t
)

{G_i}(t)

to assure that for any integer

k
=
1
,
…
,
n

k = 1, \ldots ,n

, the linear differential system
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime Subscript i Baseline equals upper G Subscript i Baseline left-parenthesis t right-parenthesis x Subscript i plus 1 Baseline comma i equals 1 comma ellipsis comma n minus 1 comma x prime Subscript n Baseline equals upper G left-parenthesis t right-parenthesis x 1 comma">

x
i
′

=

G
i

(
t
)

x

i
+
1

,

i
=
1
,
…
,
n
−
1
,

x
n
′

=
G
(
t
)

x
1

,

{x’_i} = {G_i}(t){x_{i + 1}},\quad i = 1, \ldots ,n - 1,\quad {x’_n} = G(t){x_1},

\]
where the

x
i

{x_i}

are

m
×
m

m \times m

matrices, has a solution

(

x
1

,
…
,

x
n

)

({x_1}, \ldots ,{x_n})

with the property that

x
k

(
t
)
→
I

{x_k}(t) \to I

(the identity matrix) and if

k
>
n

k > n

,

x
i

(
t
)
→
0

{x_i}(t) \to 0

,

i
=
k
+
1
,
…
,
n

i = k + 1, \ldots ,n

, as

t
→
∞

t \to \infty

. Furthermore, important bounds on the

x
i

(
t
)

{x_i}(t)

will be given. Some of these conditions will require that

∫
a
∞

|
G
|

>
∞

\int _a^\infty {\left | G \right | > \infty }

while others will not. Corollaries will be given for special cases such as

(
R
(
t
)
x
)
=
G
(
t
)
x

(R(t)x) = G(t)x

. No selfadjointness conditions are assumed; however, the results are new even in the selfadjoint case.
An infinite-dimensional pre-Hilbert space not homeomorphic to its own square
Authors: Pol Roman
Published: June 2010

Given an arbitrary infinite-dimensional separable complete linear metric space

X
X

, there exists a direct sum decomposition

X
=

V
0

⊕

V
1

X = {V_0} \oplus {V_1}

such that each summand

V
i

{V_i}

intersects every linearly independent Cantor set in

X
X

(this decomposition can be considered as a linear analogue to the classical Bernstein’s decomposition into totally imperfect sets). Theorem. Each summand

V
V

of such a decomposition is not homeomorphic to its own square, and if

T
:
V
→
V

T:V \to V

is a linear bounded operator, then either the kernel or the range of

T
T

is finite-dimensional. In the case of

X
=

l
2

X = {l_2}

this provides an example of a space

V
V

with the properties stated in the title, which answers a well-known question, cf. Arhangelskiĭ [A, Problem 21] and Geoghegan [G, Problem (LS 12)].
Cross products of strongly Morita equivalent 𝐶*-algebras
Authors: Curto Raúl E. , Muhly Paul S. , Williams Dana P.
Published: June 2010

Suppose that a locally compact group

G
G

acts on strongly Morita equivalent

C
∗

{C^ * }

-algebras

A
A

and

B
B

and let

A
⋊
G

A \rtimes G

and

B
⋊
G

B \rtimes G

denote the corresponding crossed products. We present conditions which imply that

A
⋊
G

A \rtimes G

and

B
⋊
G

B \rtimes G

are also strongly Morita equivalent and we apply our result to improve upon known theorems concerning strong Morita equivalence between certain transformation group

C
∗

{C^ * }

-algebras.
The very well poised ₆𝜓₆. II
Authors: Askey Richard
Published: June 2010

A proof of Bailey’s sum of the very well poised

6

ψ
6

_6{\psi _6}

series is obtained from a simple difference equation and special cases that are easy to evaluate.
Some function spaces of CW type
Authors: Kahn Peter J.
Published: June 2010

J. Milnor’s result on the CW type of certain function spaces

map

(

X
,
Y

)

{\operatorname {map}}\left ( {X,Y} \right )

is extended to allow the case in which

X
X

has a finite

k
k

-skeleton and

π
i

Y
=
0

{\pi _i}Y = 0

,

i
>
k

i > k

. One conclusion is that the self-equivalence monoid of any Postnikov stage of a finite complex has CW type. Another is that the monoid of pointed self-equivalences of a

K

(

π
,
1

)

K\left ( {\pi ,1} \right )

manifold has contractible components when

π
\pi

is finitely-generated.
Fibrations and homotopy type
Authors: Noakes J. L.
Published: June 2010

We prove the results announced in [13]. These allow us to distinguish homotopy types of spherical fibrations using fibre homotopy invariants.
Oscillation of linear second-order differential systems
Authors: Kwong Man Kam , Kaper Hans G. , Akiyama Kazuo , Mingarelli Angelo B.
Published: June 2010

This article is concerned with the oscillatory behavior at infinity of the solution

y
:
[
a
,
∞
)
→

R

n

y:[a,\infty ) \to {{\mathbf {R}}^n}

of a system of second-order differential equations,

y

(
t
)

+
Q

(
t
)

y

(
t
)

=
0

y\left ( t \right ) + Q\left ( t \right )y\left ( t \right ) = 0

,

t
∈
[
a
,
∞
)

t \in [a,\infty )

;

Q
Q

is a continuous matrix-valued function on

[
a
,
∞
)

[a,\infty )

whose values are real symmetric matrices of order

n
n

; it is assumed that the largest eigenvalue of the matrix

∫
a
t

Q

(
s
)

d
s

\int _a^t {Q\left ( s \right )ds}

tends to infinity as

t
→
∞

t \to \infty

. Various sufficient conditions are given which guarantee oscillatory behavior at infinity; these conditions generalize those of Mingarelli [C.R. Math. Rep. Acad. Sci. Canada 2 (1980), 287-290, and Proc. Amer. Math. Soc. 82 (1981), 593-598].
On the Hans Lewy extension phenomenon in higher codimension
Authors: Hill C. D. , Taiani G.
Published: June 2010

In this work the authors extend the results of their paper entitled Families of analytic discs in

C

n

{{\mathbf {C}}^n}

with boundaries on a prescribed

C
R

CR

submanifold. It is proven that a nongeneric CR manifold

M
M

whose Lewy form is not identically zero can be extended to a manifold

M
~

\tilde M

of one higher dimension, which is foliated by analytic discs. Moreover this result is used to prove that a sufficiently smooth CR function

f
f

on

M
M

extends to a function

f
~

\tilde f

which is CR on

M
~

\tilde M

.