bluuebunny/crossref_metadata_2025 huggingface dataset:
Spectroscopic elucidation of uncoupled transition energies in the major photosynthetic light-harvesting complex, LHCII
Authors: Schlau-Cohen Gabriela S. , Calhoun Tessa R. , Ginsberg Naomi S. , Ballottari Matteo , Bassi Roberto , Fleming Graham R.
Published: July 2010

Electrostatic couplings between chromophores in photosynthetic pigment–protein complexes, and interactions of pigments with the surrounding protein environment, produce a complicated energy landscape of delocalized excited states. The resultant electronic structure absorbs light and gives rise to energy transfer steps that direct the excitation toward a site of charge separation with near unity quantum efficiency. Knowledge of the transition energies of the uncoupled chromophores is required to describe how the wave functions of the individual pigments combine to form this manifold of delocalized excited states that effectively harvests light energy. In an investigation of the major light-harvesting complex of photosystem II (LHCII), we develop a method based on polarized 2D electronic spectroscopy to experimentally access the energies of the S
0
–S
1
transitions in the chromophore site basis. Rotating the linear polarization of the incident laser pulses reveals previously hidden off-diagonal features. We exploit the polarization dependence of energy transfer peaks to find the angles between the excited state transition dipole moments. We show that these angles provide a spectroscopic method to directly inform on the relationship between the delocalized excitons and the individual chlorophylls through the site energies of the uncoupled chromophores.
High-throughput engineering and analysis of peptide binding to class II MHC
Authors: Jiang Wei , Boder Eric T.
Published: July 2010

Class II major histocompatibility complex (MHC-II) proteins govern stimulation of adaptive immunity by presenting antigenic peptides to CD4+ T lymphocytes. Many allelic variants of MHC-II exist with implications in peptide presentation and immunity; thus, high-throughput experimental tools for rapid and quantitative analysis of peptide binding to MHC-II are needed. Here, we present an expression system wherein peptide and MHC-II are codisplayed on the surface of yeast in an intracellular association-dependent manner and assayed by flow cytometry. Accordingly, the relative binding of different peptides and/or MHC-II variants can be assayed by genetically manipulating either partner, enabling the application of directed evolution approaches for high-throughput characterization or engineering. We demonstrate the application of this tool to map the side-chain preference for peptides binding to HLA-DR1 and to evolve novel HLA-DR1 mutants with altered peptide-binding specificity.
An extension theorem for obtaining measures on uncountable product spaces
Authors: Elliott E. O.
Published: July 2010

Several theorems are known for extending consistent families of measures to an inverse limit or product space [1]. In this paper the notion of a consistent family of measures is generalized so that, as with general product measures [2], the spaces are not required to be of unit measure or even

σ
\sigma

-finite. The general extension problem may be separated into two parts, from finite to countable product spaces and from countable to uncountable product spaces. The first of these is discussed in [3]. The present paper concentrates on the second. The ultimate virtual identity of sets is defined and used as a key part of the generalization and nilsets similar to those of general product measures [2] are introduced to assure the measurability of the fundamental covering family. To exemplify the extension process, it is applied to product measures to obtain a general product measure. The paper is presented in terms of outer measures and Carathéodory measurability; however, some of the implications in terms of measure algebras should be obvious.
The semilattices with distinguished endomorphisms which are equationally compact
Authors: Bulman-Fleming Sydney , Fleischer Isidore , Keimel Klaus
Published: July 2010

We consider universal algebras

(
S
;
{
∧
}
∪
E
)

(S;\{ \wedge \} \cup E)

in which E is a set of endomorphisms of the semilattice

(
S
;
∧
)

(S; \wedge )

. It is proved in this paper that such an algebra is equationally compact iff (i) every nonempty subset of S has an infimum, (ii) every up-directed subset of S has a supremum, (iii) for every

s
∈
S

s \in S

and every up-directed family

(

d
i

)

({d_i})

in S the equality

s
∧
∨

d
i

=
∨
s
∧

d
i

s \wedge \vee {d_i} = \vee s \wedge {d_i}

holds, (iv) for each

f
∈
E
,
f
(
∧

s
i

)
=
∧
f
(

s
i

)

f \in E,f( \wedge {s_i}) = \wedge f({s_i})

holds for every family

(

s
i

)

({s_i})

in S, and (v) for each

f
∈
E
,
f
(
∨

d
i

)
=
∨
f
(

d
i

)

f \in E,f( \vee {d_i}) = \vee f({d_i})

holds for every up-directed family

(

d
i

)

({d_i})

in S. In addition, it is shown that every equationally compact algebra of this type is a retract (algebraic) of a compact, Hausdorff, 0-dimensional topological one. These results reduce to known ones for semilattices without additional structure.
Indecomposable decompositions and the minimal direct summand containing the nilpotents
Authors: Birkenmeier G. F.
Published: July 2010

It is well known that an indecomposable right ideal decomposition of a ring is not necessarily unique. In this paper we show that the reduced right ideals of such a decomposition are unique up to isomorphism and the remainder of the decomposition forms the unique MDSN. In the main theorem we use triangular matrices to prove that a ring with an indecomposable decomposition is basically composed of a nilpotent ring, a ring (containing a unity) with an indecomposable decomposition which equals its MDSN, and a direct sum of indecomposable reduced rings with unity.
The conormal module of an almost complete intersection
Authors: Kunz Ernst
Published: July 2010

The conormal module of an ideal I in a commutative ring S is the

S

/

I

S/I

-module

I

/

I
2

I/{I^2}

. Assume S is a regular noetherian ring and I a prime ideal, which is locally everywhere a complete intersection or an almost complete intersection (i.e. needs one generator more than in the complete intersection case). In this situation necessary and sufficient conditions for

I

/

I
2

I/{I^2}

being torsion free are given. Moreover the torsion of

I

/

I
2

I/{I^2}

is expressed in terms of Kähler differentials of

S

/

I

S/I

.
Two counterexamples involving inner functions
Authors: Stephenson Kenneth
Published: July 2010

Two questions about the existence of bounded analytic functions with prescribed behavior are answered using a construction technique of McLaughlin and Piranian. If A is a relatively closed, countable subset of the unit disc, the construction gives (I) Function

f
∈

H
∞

f \in {H^\infty }

so that the inner factor of

f
−
α

f - \alpha

is a finite Blaschke product if and only if

α
∈
A

\alpha \in A

. (II) Inner function

ϕ
\phi

so that

ϕ
−
α

\phi - \alpha

has a discrete singular inner factor if and only if

α
∈
A

\alpha \in A

.
Existence of fixed points of nonexpansive mappings in certain Banach lattices
Authors: Soardi Paolo M.
Published: July 2010

A fixed point theorem for nonexpansive mappings in dual Banach spaces is proved. Applications in certain Banach lattices are given.
A nonlinear theorem of ergodic type
Authors: Feathers G. , Pace J. Wayne , Dotson W. G.
Published: July 2010

A theorem is proved concerning the relationship between a certain collection of mappings and a fixed point for that collection of mappings. The conditions on the mappings are very similar to ones in a paper by W. F. Eberlein.
A characterization of 𝑍-separating algebras
Authors: Hegde Shankar
Published: July 2010

Let A be a uniformly closed point separating algebra of bounded real valued functions on a set X, containing the constant functions. A is called z-separating if whenever

Z
1

,

Z
2

{Z_1},{Z_2}

are disjoint zero sets of members of A there is some

f
∈
A

f \in A

with

f
(

Z
1

)
=
0

f({Z_1}) = 0

and

f
(

Z
2

)
=
1

f({Z_2}) = 1

. We prove that A is z-separating if and only if A consists of precisely those bounded real valued functions f on X for which

f

−
1

(
C
)

{f^{ - 1}}(C)

is a zero set of some member of A for every closed set C of real line.
Universal Lusin measurability and subfamily summable families in abelian topological groups
Authors: Graves William H.
Published: July 2010

It is proved that if G is a Hausdorff abelian topological group with respect to topologies

α
⊆
β

\alpha \subseteq \beta

such that

β
\beta

is complete and the identity map of

(
G
,
α
)

(G,\alpha )

onto

(
G
,
β
)

(G,\beta )

is universally Lusin measurable, then the subfamily summable families are the same for

α
\alpha

and

β
\beta

.
Spectral estimates and oscillations of singular differential operators
Authors: Allegretto Walter
Published: July 2010

Several estimates are established for the lower spectrum of singular second order elliptic operators. These estimates are obtained by considering the oscillation properties of the operators involved.
A Volterra equation with nonintegrable resolvent
Authors: Gripenberg Gustaf
Published: July 2010

A counterexample, which shows that the resolvent associated with a nonnegative, nonincreasing function need not be integrable, is constructed.
Fixed point theorems for multivalued approximable mappings
Authors: Milojević P. S.
Published: July 2010

In this paper we introduce several classes of multivalued approximable mappings and develop the fixed point theory for these mappings acting in a cone. As an important special case we have the theory of k-ball-contractive perturbations of strongly pseudo-contractive and accretive mappings.
Loosely Bernoulli Cartesian products
Authors: Swanson Laif
Published: July 2010

For any totally ergodic loosely Bernoulli automorphism T, a class

S

(
T
)

\mathcal {S}(T)

of loosely Bernoulli automorphisms is constructed. Each class

S

(
T
)

\mathcal {S}(T)

includes zero entropy automorphisms which do not have discrete spectrum, and has the property that for

S
∈

S

(
T
)

S \in \mathcal {S}(T)

, the Cartesian product

T
×
S

T \times S

is loosely Bernoulli.
Derivations and the trace-class operators
Authors: Weber R. E.
Published: July 2010

Let

R

(

Δ
A

)

\mathcal {R}({\Delta _A})

represent the range of the derivation generated by

A
∈

B

(

H

)

A \in \mathcal {B}(\mathcal {H})

. It is shown that for each

n
⩾
2
,

tr

(

T
n

)
=
0

n \geqslant 2,{\text {tr}}({T^n}) = 0

for any trace-class operator

T
∈
{
A

}
′

T \in \{ A\} ’

which is either (a) the weak limit of a sequence in

R

(

Δ
A

)

\mathcal {R}({\Delta _A})

or (b) a finite rank operator in the weak closure of

R

(

Δ
A

)

\mathcal {R}({\Delta _A})

. From this it follows that if

K
∈
{
A

}
′

K \in \{ A\} ’

is a compact operator in the weak closure of

R

(

Δ
A

)

\mathcal {R}({\Delta _A})

, then K is quasinilpotent.
A multiplier theorem for 𝐻¹(𝑅ⁿ)
Authors: Oberlin Daniel M.
Published: July 2010

We prove the following theorem. Theorem. Let m be a nonnegative measurable function on

[
0
,
∞
)

[0,\infty )

. For

n
⩾
2

n \geqslant 2

, the two conditions below are equivalent: (a)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript upper R Sub Superscript n Baseline StartAbsoluteValue ModifyingAbove f With caret left-parenthesis x right-parenthesis EndAbsoluteValue m left-parenthesis StartAbsoluteValue x EndAbsoluteValue right-parenthesis d x greater-than normal infinity">

∫

R
n

|

f
^

(
x
)

|

m
(

|

x

|

)

d
x
>
∞

\smallint {\;_{{R^n}}}|\hat f(x)|m(|x|)\;dx > \infty

for each

f
∈

H
1

(

R
n

)

f \in {H^1}({R^n})

, (b)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup left-brace right-brace colon times times 2 times times left-parenthesis right-parenthesis minus minus n 1 k integral integral 2 k 2 plus plus k 1 of mm left-parenthesis right-parenthesis r separator d separator r colon greater-than greater-than greater-than minus minus infinity k infinity greater-than normal infinity">

sup
{

2

(
n
−
1
)
k

∫

2
k

2

k
+
1

m
(
r
)

d
r
:
−
∞
>
k
>
∞
}
>
∞

\sup \{ {2^{(n - 1)k}}\smallint _{{2^k}}^{{2^{k + 1}}}m(r)\;dr: - \infty > k > \infty \} > \infty

.
A note on an inverse scattering problem
Authors: Ahner John F. , Kahane Charles S.
Published: July 2010

Here it is shown that the inverse scattering problem of a plane wave incident off a spherically symmetric medium is an improperly posed problem and that if a continuous index of refraction exists, then it is unique.
A counterexample to a “theorem” on 𝐿_{𝑛} sets
Authors: Baildon John D.
Published: July 2010

An example is given of a closed connected set in

E
r

E^r

whose points of local nonconvexity can be decomposed into two convex subsets, but which is not arcwise connected and hence is not an

L
n

L_n

set. This contradicts a result by Valentine to which Stavrakas and Jamison have given a second proof. It is also shown that if the set of points of local nonconvexity of a closed connected set S in

E
r

E^r

can be decomposed into n compact subsets which are convex relative to S, then S is an

L

2
n
+
1

L_{2n+1}

set.
𝐶𝑇₀ is stronger than 𝐶𝑇₀!
Authors: Lifschitz Vladimir
Published: July 2010

CT
0

{\text {CT}_0}

! is the result of adding the uniqueness condition to the antecedent of

CT
0

{\text {CT}_0}

.

H
A

+

CT
0

{\mathbf {HA}} + {\text {CT}_0}

is shown to be essentially stronger than

H
A

+

CT
0

{\mathbf {HA}} + {\text {CT}_0}

!.
The geometry and the Laplace operator on the exterior 2-forms on a compact Riemannian manifold
Authors: Tsagas Gr. , Kockinos C.
Published: July 2010

A compact, orientable, Riemannian manifold of dimension n is considered, with the Laplace operator acting on the exterior 2-forms of the manifold. Examining the spectrum,

Sp
2

{\text {Sp}^2}

, of the Laplace operator acting on 2-forms, the question is raised whether

Sp
2

{\text {Sp}^2}

exerts an influence on the geometry of the Riemannian manifold. To answer this question, after some preliminaries, two compact, orientable, equispectral, i.e., having the same

Sp
2

{\text {Sp}^2}

, Riemannian manifolds are considered in §3. (We note, in particular, that equispectral implies that the two manifolds are equidimensional.) Assuming further that the second Riemannian manifold has constant sectional curvature, the paper exhibits all the dimensions, commencing with 2, for which the two Riemannian equispectral manifolds have the same constant sectional curvature. In particular, this implies that for certain dimensions, which are explicitly stated, the Euclidean n-sphere is completely characterized by the spectrum,

Sp
2

{\text {Sp}^2}

, of the Laplacian on exterior 2-forms. Next, two compact, orientable, equispectral, Einsteinian manifolds are considered. (Again, equispectral implies equidimensional.) Assuming that the second Einsteinian manifold is of constant sectional curvature, the paper exhibits all the dimensions for which the two Einsteinian equispectral manifolds have equal constant sectional curvature. In particular, taking the second manifold to be the standard Euclidean sphere, the paper classifies Einsteinian manifolds, which are equispectral to the sphere, by calculating all the dimensions for which the Einsteinian manifold is isometric to the sphere. In short, if one of the Einsteinian manifolds is the sphere, then for certain dimensions, equispectral implies isometric. In §4, compact, equispectral, Kählerian manifolds are considered, and additional conditions are examined which determine their geometry. Studying two compact, equispectral, Kählerian manifolds, and again assuming that one of the manifolds is of real, constant, holomorphic, sectional curvature, the paper exhibits all the dimensions for which the two manifolds have equal real, constant, holomorphic, sectional curvatures. As a particular case, the paper classifies all the dimensions for which complex projective space, with Fubini-Study metric, is completely characterized by the spectrum,

Sp
2

{\text {Sp}^2}

, of the Laplacian acting on exterior 2-forms. The calculations were performed by utilizing an electronic computer.
A counterexample in 𝑙₂-manifold theory
Authors: Liem Vo Thanh
Published: July 2010

There is a space X such that

X
×
X

X \times X

is

l
2

{l_2}

, but X is not

l
2

{l_2}

.
Orbits of paths under hyperbolic toral automorphisms
Authors: Mañé Ricardo
Published: July 2010

A hyperbolic toral automorphism is a map

f
:

T
n

↩

f:{T^n} \hookleftarrow

such that has a linear lifting

L
:

R

n

↩

L:{{\mathbf {R}}^n} \hookleftarrow

without eigenvalues of modulus 1. In this note we prove that the orbit under f of a rectifiable nonconstant path

γ
:
[
a
,
b
]
→

T
n

\gamma :[a,b] \to {T^n}

contains a coset of a toral subgroup invariant under same power of f. For

C
2

{C^2}

paths the same result was proved by J. Franks. For

C
0

{C^0}

arcs S.G. Hancock proved that it is false.
On 𝐾-semimetric spaces
Authors: Burke Dennis K.
Published: July 2010

An example is constructed of a separable Moore space that does not possess a compatible K-semimetric.
Compact-open versus 𝑘-compact-open
Authors: Borges Carlos R.
Published: July 2010

We obtain various examples of k-spaces, one of which is first countable and another is compact, such that the space of continuous functions from a compact metric space to any of these spaces, with the compact-open topology, is not a k-space. We also improve some results on products of k-spaces.
A general Ramsey product theorem
Authors: Graham R. L. , Spencer J. H.
Published: July 2010

Call a family

F

\mathcal {F}

of subsets of a set U Ramsey if no partition of U into finitely many parts can split every

F
∈

F

F \in \mathcal {F}

. We show that under very general conditions an arbitrary collection of Ramsey families in fact has a much stronger uniform Ramsey property.
One-variable equational compactness in partially distributive semilattices with pseudocomplementation
Authors: Bulman-Fleming Sydney , Fleischer Isidore
Published: July 2010

A universal algebra A is called one-variable equationally compact if every system of equations with constants in A involving a single variable x, every finite subsystem of which has a solution in A, has itself a solution in A. The one-variable equationally compact semilattices with pseudocomplementation

⟨
S
;
∧

,
∗

,
0
⟩

\langle S; \wedge {,^ \ast },0\rangle

which satisfy the partial distributive law
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x logical-and left-parenthesis y logical-and z right-parenthesis Superscript asterisk Baseline equals left-parenthesis x logical-and y Superscript asterisk Baseline right-parenthesis logical-or left-parenthesis x logical-and z Superscript asterisk Baseline right-parenthesis">

x
∧

(
y
∧
z

)
∗

=
(
x
∧

y
∗

)
∨
(
x
∧

z
∗

)

x \wedge {(y \wedge z)^ \ast } = (x \wedge {y^ \ast }) \vee (x \wedge {z^ \ast })

are characterized, and as a consequence we are able to describe the one-variable compact Stone semilattices. Similar considerations yield a characterization of the one-variable equationally compact Stone algebras, extending a well known result for distributive lattices.
The Greenspan bound for the order of differential systems
Authors: Cohn Richard M.
Published: July 2010

Let S be a system of ordinary differential polynomials in indeterminates

y
1

,
…
,

y
n

{y_1}, \ldots ,{y_n}

and of order at most

r
i

{r_i}

in

y
i

,
1
⩽
i
⩽
n

{y_i},1 \leqslant i \leqslant n

. It was shown by J. F. Ritt that if

M

\mathfrak {M}

is a component of S of differential dimension 0, then the order of

M

\mathfrak {M}

is at most

r
1

+
…
+

r
n

{r_1} + \ldots + {r_n}

. B. Greenspan improved this bound in the case that every component of S has differential dimension 0. (His work was carried out for difference equations, but is easily transferred to the differential case.) It is shown that the Greenspan bound is valid without this restriction.
Frobenius extensions of QF-3 rings
Authors: Kitamura Yoshimi
Published: July 2010

We investigate the inheritance of QF-3 property for ring extensions, mainly, for Frobenius extensions. Let A be a ring with identity. It is proved that a group ring

A
[
G
]

A[G]

of A with a finite group G is left QF-3 iff A is left QF-3 and that in case A is a. G-Galois extension of the fixed subring

A
G

{A^G}

relative to a finite group G of ring automorphism of A, A is left QF-3 iff

A
G

{A^G}

is left QF-3.
Conformal transformations and Clifford algebras
Authors: Lounesto Pertti , Latvamaa Esko
Published: July 2010

A spinor representation for the conformal group of the real orthogonal space

R

p
,
q

{R^{p,q}}

is given. First, the real orthogonal space

R

p
,
q

{R^{p,q}}

is compactified by adjoining a (closed) isotropic cone at infinity. Then the nonlinear conformal transformations are linearized by regarding the conformal group as a factor group of a larger orthogonal group. Finally, the spin covering group of this larger orthogonal group is realized in the Clifford algebra

R

1
+
p
,
q

{R_{1 + p,q}}

containing the Clifford algebra

R

p
,
q

{R_{p,q}}

on the orthogonal space

R

p
,
q

{R^{p,q}}

. Explicit formulas for orthogonal transformations, translations, dilatations and special conformal transformations are given in Clifford language.
Algebras with unconditional orthogonal bases
Authors: Husain Taqdir , Watson Saleem
Published: July 2010

This is a continuation of our study of topological algebras with orthogonal Schauder bases. In the previous paper, the structure of closed ideals was determined and it was shown that the closed-maximal ideal space is homeomorphic with the discrete space of positive integers. Here it is shown that the space of all maximal ideals equipped with the hull-kernel topology is homeomorphic with the Stone-Čech compactification of natural numbers. Among other results it is also proved that the intersection of dense maximal ideals is isomorphic with the topological dual of the algebra under certain conditions.
Local algebraicity of some analytic hypersurface
Authors: Adkins William A.
Published: July 2010

It is proved that an analytic hypersurface germ

(
X
,
0
)
⊆
(

C

n
+
1

,
0
)

(X,0) \subseteq ({{\mathbf {C}}^{n + 1}},0)

, with nonsingular normalization, whose only singularities outside the origin are normal crossings of two n-manifolds is isomorphic to a germ of an algebraic variety at 0. As a corollary we find that weakly normal surfaces

V
⊆

C

3

V \subseteq {{\mathbf {C}}^3}

with nonsingular normalization are locally algebraic.
A construction of ergodic BMO functions
Authors: Petersen Karl
Published: July 2010

One can explicitly construct unbounded functions having bounded mean oscillation (and in fact vanishing mean oscillation) with respect to any given ergodic measure-preserving transformation of a Lebesgue space.
Convex domains and Kobayashi hyperbolicity
Authors: Barth Theodore J.
Published: July 2010

A geometrically convex domain in

C

n

{{\mathbf {C}}^n}

is Kobayashi hyperbolic if and only if it contains no complex affine lines. This contrasts with an example of a nonhyperbolic pseudoconvex domain in

C

2

{{\mathbf {C}}^2}

containing no (nonconstant) entire holomorphic curves.
Approximating maps and a Stone-Weierstrass theorem for 𝐶*-algebras
Authors: Bunce John W.
Published: July 2010

Let A be a

C
∗

{C^ \ast }

-algebra with identity and B a

C
∗

{C^ \ast }

-subalgebra of A which separates the pure states of A. We give an easy proof of the fact that, assuming there is a sequence of norm one linear maps

L
n

:
A
→
B

{L_n}:A \to B

such that

L
n

(
b
)

{L_n}(b)

converges weakly to b for each b in B, B must equal A. As corollaries we prove that if B separates the pure states of A, then

B
=
A

B = A

if B is nuclear, or if

B
=

C
r
∗

(

F
2

)

B = C_r^ \ast ({F_2})

and

A
⊆
V
N
(

F
2

)

A \subseteq VN({F_2})

, where

F
2

{F_2}

is the free group on two generators.
A measure-theoretic proof of the Stone-Weierstrass approximation theorem
Authors: Khurana Surjit Singh
Published: July 2010

Using the Daniell integral, a simple proof of the Stone-Weierstrass theorem is obtained.
On the integrability of the maximal ergodic function
Authors: Đă̇ng-Ngọc Nghiêm
Published: July 2010

Let

G
=

R

d

G = {{\mathbf {R}}^d}

or

Z

d

{{\mathbf {Z}}^d}

and consider an ergodic measure-preserving action of G on a probability space

(
X
,

A

,
P
)

(X,\mathfrak {A},P)

, let

f
∈

L
1

(
X
,
P
)

f \in {L^1}(X,P)

and Mf be its maximal ergodic function. Our purpose is to prove the converse of the following theorem of N. Wiener: if

|

f

|

log
+

|

f

|

|f|{\log ^ + }|f|

is integrable then Mf is integrable. For the particular case

G
=

Z

G = {\mathbf {Z}}

this result was already obtained by D. Ornstein whose proof is based on induced transformations and seems to be specific to Z, our proof is based on a result of E. M. Stein on the Hardy-Littlewood maximal function on

R

d

{{\mathbf {R}}^d}

and its analogue on

Z

d

{{\mathbf {Z}}^d}

.
On dominated extensions in function algebras
Authors: Globevnik J.
Published: July 2010

The Bishop-Gamelin interpolation theorem asserts that given a compact Hausdorff space K, a closed subspace A of

C
(
K
)

C(K)

, a positive continuous function p on K and a closed set

F
⊂
K

F \subset K

such that every measure in the annihilator of A vanishes on F, every function

f
∈
C
(
F
)

f \in C(F)

satisfying

|

f
(
s
)

|

⩽
p
(
s
)
(
s
∈
F
)

|f(s)| \leqslant p(s)(s \in F)

extends to a function

f
~

∈
A

\tilde f \in A

satisfying

|

f
~

(
z
)

|

⩽
p
(
z
)
(
z
∈
K
)

|\tilde f(z)| \leqslant p(z)(z \in K)

. In the paper we consider a special case where the theorem is extended to the situation when the dominating function is nonnegative.
On the boundary of reduced Teichmüller space
Authors: Wason Judith C.
Published: July 2010

Quasifuchsian groups used in the description of the boundary of Teichmüller space do not have analogs for reduced Teichmüller space.
Quotients of 𝐶^{𝑚}-{0} by diagonal 𝐶*-actions
Authors: Takijima Kunio
Published: July 2010

Let

q
1

,
…
,

q
m

{q_1}, \ldots ,{q_m}

be positive integers with

(

q
1

,
…
,

q
m

)
=
1

({q_1}, \ldots ,{q_m}) = 1

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho colon bold upper C Superscript asterisk Baseline times bold upper C Superscript m Baseline right-arrow bold upper C Superscript m Baseline comma rho left-parenthesis t comma z 1 comma ellipsis comma z Subscript m Baseline right-parenthesis equals left-parenthesis t Superscript q 1 Baseline z 1 comma ellipsis comma t Superscript q Super Subscript m Superscript Baseline z Subscript m Baseline right-parenthesis">

ρ
:

C

∗

×

C

m

→

C

m

,
ρ
(
t
,

z
1

,
…
,

z
m

)
=
(

t

q
1

z
1

,
…
,

t

q
m

z
m

)

\rho :{{\mathbf {C}}^ \ast } \times {{\mathbf {C}}^m} \to {{\mathbf {C}}^m},\rho (t,{z_1}, \ldots ,{z_m}) = ({t^{{q_1}}}{z_1}, \ldots ,{t^{{q_m}}}{z_m})

the diagonal

C

∗

{{\mathbf {C}}^ \ast }

-action on

C

m

{{\mathbf {C}}^m}

. Then the orbit space

C

m

−
{
0
}

/

C

∗

{{\mathbf {C}}^m} - \{ 0\} /{{\mathbf {C}}^ \ast }

is a normal analytic space. In this paper, we shall show that

C

m

−
{
0
}

/

C

∗

{{\mathbf {C}}^m} - \{ 0\} /{{\mathbf {C}}^ \ast }

has only rational singularities and, if

δ
(

q
1

,
…
,

q
m

)
⩽
m
−
3

\delta ({q_1}, \ldots ,{q_m}) \leqslant m - 3

and

m
⩾
3
,

C

m

−
{
0
}

/

C

∗

m \geqslant 3,{{\mathbf {C}}^m} - \{ 0\} /{{\mathbf {C}}^ \ast }

is rigid, where

δ
(

q
1

,
…
,

q
m

)

\delta ({q_1}, \ldots ,{q_m})

is the positive integer defined by

q
1

,
…
,

q
m

{q_1}, \ldots ,{q_m}

.
A remark on the F. and M. Riesz theorem
Authors: Pigno Louis , Smith Brent
Published: July 2010

Let

μ
\mu

be a measure of analytic type on the unit circle. We give a short direct proof that

μ
2

{\mu ^2}

is absolutely continuous with respect to Lebesgue measure; our method also gives a convolution product version of some related several variable results.
On a certain 𝐶*-crossed product inside a 𝑊*-crossed product
Authors: Olesen Dorte , Pedersen Gert K.
Published: July 2010

To each

W
∗

{W^ \ast }

-dynamical system

(

M

,
G
,
α
)

(\mathfrak {M},G,\alpha )

corresponds canonically a

C
∗

{C^ \ast }

-dynamical system

(

M

c

,
G
,
α

|

M

c

)

({\mathfrak {M}^c},G,\alpha |{\mathfrak {M}^c})

. We show that the

C
∗

{C^ \ast }

-crossed product

G

×
α

M

c

G{ \times _\alpha }{\mathfrak {M}^c}

can be identified with a certain

C
∗

{C^ \ast }

-subalgebra of the

W
∗

{W^ \ast }

-crossed product

G

×
α

M

G{ \times _\alpha }\mathfrak {M}

.
A characterization of the range of a bounded linear transformation in Hilbert space
Authors: Golightly George O.
Published: July 2010

It is a theorem of Smul’jan and Mac Nerney that for B a bounded linear transformation from a complete complex inner product space

{
S
,
(
⋅
,
⋅
)
}

\{S,(\cdot , \cdot )\}

to S, with adjoint transformation

B
∗

,
B
(
S
)

{B^ \ast },B(S)

is the set of all z in S for which there is a nonnegative number b such that for all x in

S
,

|

(
z
,
x
)

|

2

⩽
b

‖

B
∗

x
‖

2

S,|(z,x){|^2} \leqslant b\left \|{B^ \ast }x\right \|{^2}

, in which case if w is that point of

(
ker
⁡
B

)
⊥

{(\ker B)^ \bot }

such that

B
w
=
z

Bw = z

then the least such b is

‖
w
‖

2

\left \|w\right \|{^2}

. This paper provides another description of

B
(
S
)

B(S)

and formula for

‖
w
‖

2

\left \|w\right \|{^2}

.
The Zygmund condition for Bloch functions in the ball in 𝐶ⁿ
Authors: Cima Joseph A. , Weinstock Barnet M.
Published: July 2010

In this paper we prove the equivalence of the Bloch condition for a holomorphic function f on the ball

B
n

{B_n}

with the Zygmund second difference condition for a suitable primitive F of f.
A remark on complemented subspaces of unitary matrix spaces
Authors: Arazy Jonathan
Published: July 2010

Theorem A. Let P be a bounded projection in a unitary matrix space

C
E

{C_E}

. Then either

P

C
E

P{C_E}

or

(
I
−
P
)

C
E

(I - P){C_E}

contains a subspace which is isomorphic to

C
E

{C_E}

and complemented in

C
E

{C_E}

.
Slicing convex bodies—bounds for slice area in terms of the body’s covariance
Authors: Hensley Douglas
Published: July 2010

Let Q be a zero-symmetric convex set in

R

N

{{\mathbf {R}}^N}

with volume 1 and covariance matrix

V
2

I
d

N
×
N

V^2 \mathrm {Id}_{N \times N}

. Let P be a K-dimensional vector subspace of

R

n

,
K
>
N

{{\mathbf {R}}^n},K > N

, and let

J
=
N
−
K

J = N - K

. Then there exist constants

C
1

(
J
)

{C_1}(J)

and

C
2

(
J
)

{C_2}(J)

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V Superscript negative upper J Baseline upper C 1 left-parenthesis upper J right-parenthesis less-than-or-slanted-equals normal v normal o normal l Subscript upper K Baseline left-parenthesis upper P intersection upper Q right-parenthesis less-than-or-slanted-equals upper V Superscript negative upper J Baseline upper C 2 left-parenthesis upper J right-parenthesis period">

V

−
J

C
1

(
J
)
⩽

v
o
l

K

(
P
∩
Q
)
⩽

V

−
J

C
2

(
J
)
.

{V^{ - J}}{C_1}(J) \leqslant \mathrm {vol}_K(P \cap Q) \leqslant V^{-J}{C_2}(J).

\]
The lower bound has applications to Diophantine equations.
Decomposition of recursively enumerable degrees
Authors: Lachlan A. H.
Published: July 2010

It is shown that any nonzero recursively enumerable degree can be expressed as the join of two distinct such degrees having a greatest lower bound.
On the distribution of Alexander polynomials of alternating knots and links
Authors: Kinoshita Shin’ichi
Published: July 2010

In this paper we will be concerned with the distribution of the reduced Alexander polynomials of alternating knots and links. Our theorems are quite different in style from those of R. H. Crowell, K. Murasugi and R. Riley, which intend to characterize the reduced Alexander polynomials of alternating knots and links.
Planar polynomial foliations
Authors: Schecter Stephen , Singer Michael F.
Published: July 2010

Let

P
(
x
,
y
)

P(x,y)

and

Q
(
x
,
y
)

Q(x,y)

be two real polynomials of degree

⩽
n

\leqslant n

with no common real zeros. The solution curves of the vector field

x
˙

=
P
(
x
,
y
)
,

y
˙

=
Q
(
x
,
y
)

\dot x = P(x,y),\dot y = Q(x,y)

give a foliation of the plane. The leaf space

L

\mathcal {L}

of this foliation may not be a hausdorff space: there may be leaves L,

L
′

∈

L

L’ \in \mathcal {L}

which cannot be separated by open sets. We show that the number of such leaves is at most 2n and construct an example, for each even

n
⩾
4

n \geqslant 4

, of a planar polynomial foliation of degree n whose leaf space contains

2
n
−
4

2n - 4

such leaves.
Having a small weight is determined by the small subspaces
Authors: Hajnal A. , Juhász I.
Published: July 2010

We show that for every cardinal

κ
>
ω

\kappa > \omega

and an arbitrary topological space X if we have

w
(
Y
)
>
κ

w(Y) > \kappa

whenever

Y
⊂
X

Y \subset X

and

|

Y

|

⩽
κ

|Y| \leqslant \kappa

then

w
(
X
)
>
κ

w(X) > \kappa

as well. M. G. Tkačenko proved this for

T
3

{T_3}

spaces in [2]. We also prove an analogous statement for the

π
\pi

-weight if

κ
\kappa

is regular.
Pure descent for the module of Zariski differentials
Authors: Platte Erich
Published: July 2010

It will be shown that for any given pure extension

A
→
B

A \to B

of noetherian

k
k

-algebras, with

k
k

being a field of characteristic zero, and for any prime ideal

p

⊆
A

\mathfrak {p} \subseteq A

the Zariski-Lipman conjecture for

A

p

{A_\mathfrak {p}}

is solvable, if

B
B

is a locally factorial domain for which the finite differential module is reflexive. We will also discuss an embedding property with respect to the module of Zariski differentials of

A

p

{A_\mathfrak {p}}

.
𝑆-groups revisited
Authors: Hunter Roger , Walker Elbert
Published: July 2010

We provide a new characterization of

S
S

-groups which is used to develop central results of the theory and, in particular, to show that summands of

S
S

-groups are

S
S

-groups.
On the Collatz 3𝑛+1 algorithm
Authors: Garner Lynn E.
Published: July 2010

The number theoretic function

s
(
n
)
=

1
2

n

s(n) = \tfrac {1} {2}n

if

n
n

is even,

s
(
n
)
=
3
n
+
1

s(n) = 3n + 1

if

n
n

is odd, generates for each

n
n

a Collatz sequence

{

s
k

(
n
)

}

k
=
0

∞

\{ {{s^k}(n)} \}_{k = 0}^\infty

,

s
0

(
n
)
=
n

{s^0}(n) = n

,

s
k

(
n
)
=
s
(

s

k
−
1

(
n
)
)

{s^k}(n) = s({s^{k - 1}}(n))

. It is shown that if a Collatz sequence enters a cycle other than the

4
,
2
,
1
,
4
,
…

4,2,1,4, \ldots

cycle, then the cycle must have many thousands of terms.
A ring with arithmetical congruence lattice not preserved by any Pixley function
Authors: Korec Ivan
Published: July 2010

A ring

(
A
;
+
,
⋅
)

(A; + , \cdot )

is constructed such that the congruence lattice

L
A

{L_A}

of the ring

(
A
;
+
,
⋅
)

(A; + , \cdot )

is distributive, the elements of

L
A

{L_A}

are pairwise permutable and there is no

L
A

{L_A}

-compatible function

p
p

on

A
A

such that
\[

p
(
a
,
b
,
b
)
=
p
(
a
,
b
,
a
)
=
p
(
b
,
b
,
a
)
=
a

for

all

a
,
b
∈
A
.

p(a,b,b) = p(a,b,a) = p(b,b,a) = a\quad {\text {for}}\;{\text {all}}\;a,b \in A.

\]
(1)
Decomposing overrings
Authors: Cortzen B. , Small L. W. , Stafford J. T.
Published: July 2010

We show that if

S
⊃
R

S \supset R

are rings such that

S
R

{S_R}

is projective, then

R
R

{R_R}

is a direct summand of

S
R

{S_R}

if and only if

S
R

{S_R}

is faithfully projective (this condition holds, in particular, if

S
R

{S_R}

is free).
Enlarging a free subgroup of a symmetric group freely
Authors: Haimo Franklin , Tretkoff Marvin
Published: July 2010

Free subgroups, the elements of which have ample cycles in their cycle decompositions, are enlarged to free subgroups of an infinite symmetric group by the specific construction of one more free generator.
On the dual of a certain operator ideal
Authors: Racher Gerhard
Published: July 2010

For complex Banach spaces

E
E

and

F
F

and a real number

1
>
p
>
∞

1 > p > \infty

let

S
p

(
E
,
F
)

{S^p}(E,F)

denote the operator ideal obtained by complex interpolation between the nuclear and the compact operators. If

E
E

and

F
F

are reflexive and one of them has the approximation property the dual of

S
p

(
E
,
F
)

{S^p}(E,F)

is shown to be

S

p
′

(

E
′

,

F
′

)

{S^{p’}}(E’,F’)

,

p
′

p’

conjugate to

p
p

.
Subordination by univalent functions
Authors: Singh Sunder , Singh Ram
Published: July 2010

Let

K
K

be the class of functions

f
(
z
)
=
z
+

a
2

z
2

+
⋯

f(z) = z + {a_2}{z^2} + \cdots

, which are regular and univalently convex in

|
z
|

>
1

\left | z \right | > 1

. In this paper we establish certain subordination relations between an arbitrary member

f
f

of

K
K

, its partial sums and the functions

(
λ

/

z
)

∫
0
z

f
(
t
)
d
t

(\lambda /z)\int _0^z {f(t)dt}

and

μ

∫
0
z

t

−
1

f
(
t
)
d
t

\mu \int _0^z {{t^{ - 1}}f(t)dt}

. The well-known result that

z

/

2

z/2

is subordinate to

f
(
z
)

f(z)

in

|
z
|

>
1

\left | z \right | > 1

for every

f
f

belonging to

K
K

follows as a particular case from our results. We also improve certain results of Robinson regarding subordination by univalent functions. A sufficient condition for a univalent function to be convex of order

α
\alpha

is also given.
A proof of the boundary theorem
Authors: Davidson Kenneth R.
Published: July 2010

This note contains a simple proof of the following theorem of Arveson: If

A

\mathcal {A}

is an irreducible subspace of

B

(
H
)

\mathcal {B}(H)

, then the identity map

ϕ
0

(
A
)
=
A

{\phi _0}(A) = A

on

A

\mathcal {A}

has a unique completely positive extension to

B

(
H
)

\mathcal {B}(H)

if and only if the quotient map

q
q

by the compact operators is not completely isometric on

S

=
[

A

+

A

∗

]

\mathcal {S} = [\mathcal {A} + \mathcal {A}^*]

.
An abstract linear Volterra equation with a nonconvolution kernel
Authors: Kiffe T.
Published: July 2010

This paper is concerned with the existence and uniqueness of solutions to the equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript t Baseline a left-parenthesis t comma tau right-parenthesis upper A x left-parenthesis tau right-parenthesis d tau equals f left-parenthesis t right-parenthesis">

x
(
t
)
+

∫
0
t

a
(
t
,
τ
)
A
x
(
τ
)
d
τ
=
f
(
t
)

x(t) + \int _0^t {a(t,\tau )Ax(\tau )d\tau = f(t)}

where

A
A

is an unbounded, positive, selfadjoint operator on a Hilbert space. A representation is given for the solution of this equation.
Spirallike nonanalytic functions
Authors: Al-Amiri Hassoon , Mocanu Petru T.
Published: July 2010

Let

f
(
z
)
=
u
(
x
,
y
)
+
i
υ
(
x
,
y
)

f(z) = u(x,y) + i\upsilon (x,y)

be a complex function defined in the unit disc

U
.
f

U.f

is said to belong to the class

C
1

(
U
)

{C^1}(U)

if the functions

u
(
x
,
y
)

u(x,y)

and

υ
(
x
,
y
)

\upsilon (x,y)

have continuous first order partial derivatives in

U
U

. We determine sufficient conditions for functions in the class

C
1

(
U
)

{C^1}(U)

to be univalent and to map

U
U

onto spirallike domains. These conditions are similar to those in the analytic case as given by Spaček and Rakhmanov.
The Goffman-Liu area and Plateau’s problem
Authors: Bindschadler David , Nishiura Togo
Published: July 2010

Goffman and Liu defined a lower semicontinuous area

A
(
f
)

A(f)

for linearly continuous maps

f
f

from the disk into

R

n

{{\mathbf {R}}^n}

, showed that

A
(
f
)

A(f)

is the Lebesgue area when

f
f

is continuous and thereby extended the notion of area to some discontinuous maps. With the aid of a simple retraction of the punctured disk onto its boundary, a canonical linearly continuous map

f
Γ

{f_\Gamma }

of the disk is associated with each continuous closed curve

Γ
\Gamma

defined on the boundary of the disk. When

Γ
\Gamma

is a Jordan curve, it is shown that the discontinuous map

f
Γ

{f_\Gamma }

has the property that

A
(

f
Γ

)
=
A
(
σ
)

A({f_\Gamma }) = A(\sigma )

where

σ
\sigma

is a continuous map of least area spanning

Γ
\Gamma

from the classical Plateau problem. Finally, the corresponding least area problem in the class of linearly continuous maps is shown to be trivial, that is, the least area is zero in the class of linearly continuous maps.
The order of entire functions with radially distributed zeros
Authors: Abi-Khuzam Faruk F.
Published: July 2010

It is shown that an entire function with radially distributed zeros has finite order

λ
\lambda

if it has finite lower order

μ
\mu

. It is then shown that functions with real negative zeros only are extremal for the problem of maximizing the Nevanlinna characteristic in the class of entire functions satisfying

λ
−
μ
>
1

\lambda - \mu > 1

.
Central sequences associated with a state
Authors: Tsui Sze Kai
Published: July 2010

Central sequences associated with a state are defined and used to derive a characterization of the factor state in question. This characterization is used to study the factor state extension problem. One of the affirmative results obtained in this paper is as follows. Let

A

1

\mathcal {A}_1

, be a finite dimensional sub-

C
∗

{C^ * }

*-algebra of

A

\mathcal {A}

. Then every factor state on the relative commutant of

A

1

\mathcal {A}_1

, in

A

\mathcal {A}

extends to a factor state on

A

\mathcal {A}

.
Pettis integration
Authors: Geitz Robert F.
Published: July 2010

The functions that are Pettis integrable with respect to perfect measures are characterized.
Schwarz reflection principles for solutions of parabolic equations
Authors: Colton David
Published: July 2010

A reflection principle is obtained for solutions of the heat equation defined in a cylindrical domain of the form

Ω
×
(
0
,
T
)

\Omega \times (0,T)

where

Ω
\Omega

is a ball in

R

n

{{\mathbf {R}}^n}

and the solution vanishes on

∂
Ω
×
(
0
,
T
)

\partial \Omega \times (0,T)

. It is shown that the domain of dependence of the solution at a point outside the cylinder

Ω
×
(
0
,
T
)

\Omega \times (0,T)

is a line segment contained inside

Ω
×
(
0
,
T
)

\Omega \times (0,T)

. In the case

n
=
2

n = 2

this result is generalized to the case of analytic solutions of parabolic equations with analytic coefficients defined in an arbitrary bounded simply connected cylinder

D
×
(
0
,
T
)

D \times (0,T)

where the solution vanishes on a portion of

∂
D
×
(
0
,
T
)

\partial D \times (0,T)

.
Multiples of Weierstrass points as special divisors
Authors: Lax R. F.
Published: July 2010

Complex spacss

W

n
r

\mathcal {W}_n^r

of Weierstrass points are isomorphic to the intersection, on the

n
n

th symmetric product of the universal curve over the Teichmüller space, of complex spaces

G

n
r

\mathcal {G}_n^r

of special divisors with the diagonal

Δ
n

{\Delta _n}

consisting of divisors which are multiples of a point. The tangent space at a point of this intersection is described and it is shown that

G

n
1

−

G

n
2

\mathcal {G}_n^1 - \mathcal {G}_n^2

and

Δ
n

{\Delta _n}

intersect transversally.
Open centralizers and the continuity of group representations
Authors: Schochet Claude , Schreiber Bertram
Published: July 2010

Let

G
G

be a locally compact group,

π
:
G
→

L

(

L
2

(
G
)
)

\pi :G \to \mathfrak {L}({L^2}(G))

the right regular representation of

G
G

, and

G
c

=
{
x
∈
G
:

{G^c} = \{ x \in G:

: the function

g
⇝
π
(
g
x

g

−
1

)

g \rightsquigarrow \pi (gx{g^{ - 1}})

is norm continuous}. This note is devoted to the study of

G
c

{G^c}

. In particular, the compactly generated groups for which

G
=

G
c

G = {G^c}

are characterized.
Comments on notes of Stanojević et al
Authors: Day Mahlon M.
Published: July 2010

Some further inequalities which characterize inner-product spaces among normed linear spaces are derived by averaging characters of locally compact groups.
Ergodic undefinability in set theory and recursion theory
Authors: Mundici Daniele
Published: July 2010

Let

T
T

be a measure preserving ergodic transformation of a compact Abelian group

G
G

with normalized Haar measure

m
m

on the collection

B

\mathcal {B}

of Borel sets; call

g
∈
G

g \in G

generic w.r.t. a set

B
∈

B

B \in \mathcal {B}

iff, upon action by

T
T

,

g
g

is to stay in

B
B

with limit frequency equal to

m
(
B
)

m(B)

. We study the definability of generic elements in Zermelo-Fraenkel set theory with Global Choice (ZFGC, which is a very good conservative extension of ZFC), and in higher recursion theory. We prove

(
1
)

(1)

the set of those

g
∈
G

g \in G

which are generic w.r.t. all ZFGC-definable Borel subsets of

G
G

is not ZFGC-definable, and

(
2
)

(2)

"being generic w.r.t. all hyperarithmetical properties of dyadic sequences" is not itself a hyperarithmetical property of dyadic sequences.
A continuous version of the Borsuk-Ulam theorem
Authors: Jaworowski Jan
Published: July 2010

Let

p
:
E
→
B

p:E \to B

be an

n
n

-sphere bundle,

q
:
V
→
B

q:V \to B

be an

R

n

{{\mathbf {R}}^n}

-bundle and

f
:
E
→
V

f:E \to V

be a fibre preserving map over a paracompact space

B
B

. Let

p
¯

:

E
¯

→
B

\overline p :\overline E \to B

be the projectivized bundle obtained from

p
p

by the antipodal identification and let

A
¯

f

{\overline A _f}

be the subset of

E
¯

\overline E

consisting of pairs

{
e
,
−
e
}

\{ e, - e\}

such that

f
e
=
f
(
−
e
)

fe = f( - e)

. If the cohomology dimension

d
d

of

B
B

is finite then the map

(

p
¯

|

A
¯

f

)
∗

(\bar {p} | \overline {A}_f)^*

is injective for a continuous cohomology theory

H
∗

{H^*}

. Moreover, if the

j
j

th Stiefel-Whitney class of

q
q

is zero for

1
⩽
j
⩽
r

1 \leqslant j \leqslant r

then

(

p
¯

|

A
¯

f

)
∗

(\bar {p} | \overline {A}_f)^*

is injective in degrees

i
⩾
d
−
r

i \geqslant d - r

. If all the Stiefel-Whitney classes of

q
q

are zero then

(

p
¯

|

A
¯

f

)
∗

(\bar {p} | \overline {A}_f)^*

is injective in every degree.
Inductively perfect maps and tri-quotient maps
Authors: Michael E.
Published: July 2010

It is proved that every tri-quotient map

f
:
X
→
Y

f:X \to Y

from a metric space

X
X

onto a countable regular space

Y
Y

, with each

f

−
1

(
y
)

{f^{ - 1}}(y)

completely metrizable, is inductively perfect. It is not known to what extent all the hypotheses in this result are necessary, and that leads to some open questions regarding simple compactness properties of mappings between separable metric spaces.
A mod 𝑝 Whitehead theorem
Authors: Schiffman Stephen J.
Published: July 2010

A mod

p
p

Whitehead theorem is proved which is the relative version of a basic result of localization theory. It is applied to give a family of fibrations which are also cofibrations.
More counterexamples to Coleman’s conjecture
Authors: Pixton Dennis
Published: July 2010

For any

m
,
n
⩾
2

m,n \geqslant 2

we construct a smooth vector field with a topologically hyperbolic equilibrium of type

(
m
,
n
)

(m,n)

which is not locally topologically conjugate to a linear vector field. This refutes Coleman’s conjecture in all cases not covered by previous work of Neumann, Walker, and Wilson.
Cut points of 𝑋 and the hyperspace of subcontinua 𝐶(𝑋)
Authors: Nishiura Togo , Rhee Choon Jai
Published: July 2010

Let

X
X

be a nondegenerate metric continuum and

p
0

{p_0}

a point with

X
=

X
1

∪

X
2

X = {X_1} \cup {X_2}

,

{

p
0

}
=

X
1

∩

X
2

\{ {p_0}\} = {X_1} \cap {X_2}

,

X
1

{X_1}

and

X
2

{X_2}

continua. Denote by

C
(
X
)

C(X)

,

C
(

X
1

)

C({X_1})

and

C
(

X
2

)

C({X_2})

the hyperspaces of nonempty subcontinua of

X
X

,

X
1

{X_1}

and

X
2

{X_2}

respectively. Theorem.

C
(
X
)

C(X)

is contractible if and only if

C
(

X
1

)

C({X_1})

and

C
(

X
2

)

C({X_2})

are contractible and either

X
1

{X_1}

or

X
2

{X_2}

is contractible im kleinen at

p
0

{p_0}

(a modification of connected im kleinen at

p
0

{p_0}

). Theorem. Let

X
1

{X_1}

and

X
2

{X_2}

satisfy Kelley’s condition

K
K

. Then

C
(
X
)

C(X)

is contractible when and only when either

X
1

{X_1}

or

X
2

{X_2}

is connected im kleinen at

p
0

{p_0}

. Examples are given.
Links with free groups are trivial
Authors: Massey William S. , Traldi Lorenzo
Published: July 2010

In this note we extend a well-known consequence of Dehn’s Lemma to include tame links in the three-sphere.