bluuebunny/crossref_metadata_2025 huggingface dataset:
Some smooth maps with infinitely many hyperbolic periodic points
Authors: Franks John M.
Published: July 2010

If a smooth map of the two-disk to itself has only hyperbolic periodic points and has no source or sink whose period is a power of two then it has infinitely many periodic points. This and similar results are proved.
Fields generated by linear combinations of roots of unity
Authors: Evans R. J. , Isaacs I. M.
Published: July 2010

It is shown that a linear combination of roots of unity with rational coefficients generates a large subfield of the field generated by the set of roots of unity involved, except when certain partial sums vanish. Some related results about polygons with all sides and angles rational are also proved.
Cluster values of bounded analytic functions
Authors: Gamelin T. W.
Published: July 2010

Let D be a bounded domain in the complex plane, and let

ζ
\zeta

belong to the topological boundary

∂
D

\partial D

of D. We prove two theorems concerning the cluster set

Cl

(
f
,
ζ
)

{\text {Cl}}(f,\zeta )

of a bounded analytic function f on D. The first theorem asserts that values in

Cl

(
f
,
ζ
)
∖
f
(

Ш

ζ

)

{\text {Cl}}(f,\zeta )\backslash f(\Sha _\zeta )

are assumed infinitely often in every neighborhood of

ζ
\zeta

, with the exception of those lying in a set of zero analytic capacity. The second asserts that all values in

Cl

(
f
,
ζ
)
∖
f
(

M

ζ

∩

supp

λ
)

{\text {Cl}}(f,\zeta )\backslash f({\mathfrak {M}_\zeta } \cap {\text {supp}}\;\lambda )

are assumed infinitely often in every neighborhood of

ζ
\zeta

, with the exception of those lying in a set of zero logarithmic capacity. Here

M

ζ

{\mathfrak {M}_\zeta }

is the fiber of the maximal ideal space

M

(
D
)

\mathfrak {M}(D)

of

H
∞

(
D
)

{H^\infty }(D)

lying over

ζ
\zeta

,

Ш

ζ

\Sha _\zeta

is the Shilov boundary of the fiber algebra, and

λ
\lambda

is the harmonic measure on

M

(
D
)

\mathfrak {M}(D)

.
Nonsmoothable, unstable group actions
Authors: Pixton Dennis
Published: July 2010

For

k
>
1

k > 1

there is a nonempty open set of

C
1

{C^1}

actions of

Z

k

{{\mathbf {Z}}^k}

on

S
1

{S^1}

, no element of which is either topologically conjugate to a

C
2

{C^2}

action or structurally stable. The

C
1

{C^1}

closure of this set contains all

C
2

{C^2}

actions which have compact orbits, so no such action is structurally stable in the space of

C
1

{C^1}

actions.
The asymptotic behavior of a Volterra-renewal equation
Authors: Ney Peter
Published: July 2010

Theorem. Assume that the functions

x
(
⋅
)
,
h
(
⋅
)
,
G
(
⋅
)

x( \cdot ),h( \cdot ),G( \cdot )

satisfy: (i)

0
⩽
x
(
t
)
,
t
∈
[
0
,
∞
)
;
x
(
t
)
→
0

0 \leqslant x(t),t \in [0,\infty );x(t) \to 0

as

t
→
∞
;
x

t \to \infty ;x

bounded, measurable; (ii)

0
⩽
h
(
s
)
;
h
(
s
)

0 \leqslant h(s);h(s)

Lipschitz continuous for

s
∈
I

s \in I

, where I is a closed interval containing the range of

x
;
h
(
0
)
=
0
,

h
′

(
0
+
)
=
1
,
h
(
0
+
)
>
0

x;h(0) = 0,h’(0 + ) = 1,h(0 + ) > 0

; (iii) G a probability distribution on

(
0
,
∞
)

(0,\infty )

having nontrivial absolutely continuous component and finite second moment. Let

H
x
(
t
)
=

∫
0
t

h
[
x
(
t
−
y
)
]
d
G
(
y
)

Hx(t) = \smallint _0^th[x(t - y)]dG(y)

. If

0
⩽
(
x
−
H
x
)
(
t
)
=
o
(

t

−
2

)

0 \leqslant (x - Hx)(t) = o({t^{ - 2}})

, with strict inequality on the left on a set of positive measure, then

x
(
t
)
∼
γ

/

t
,
t
→
∞

x(t) \sim \gamma /t,t \to \infty

, where

γ
\gamma

is a constant depending only on h and G. The condition

o
(

t

−
2

)

o({t^{ - 2}})

is close to best possible, and cannot, e.g., be replaced by

O
(

t

−
2

)

O({t^{ - 2}})

.
Knots with infinitely many minimal spanning surfaces
Authors: Eisner Julian R.
Published: July 2010

We show that if

k
1

{k_1}

and

k
2

{k_2}

are nonfibered knots, then the composite knot

K
=

k
1

#

k
2

K = {k_1}\# {k_2}

has an infinite collection of minimal spanning surfaces, no two of which are isotopic by an isotopy which leaves the knot K fixed. This result is then applied to show that whether or not a knot has a unique minimal spanning surface can depend on what definition of spanning surface equivalence is used.
Function fields with isomorphic Galois groups
Authors: Bond Robert J.
Published: July 2010

Let K be a local field or a global field of characteristic p. Let

G
K

{G_K}

be the Galois group of the separable closure of K over K. In the local case we show that

G
K

{G_K}

, considered as an abstract profinite group, determines the characteristic of K and the number of elements in the residue class field. In the global case we show that

G
K

{G_K}

determines the number of elements in the constant field of K as well as the zeta function, genus and class number of K. Let

K
′

K’

be another global field of characteristic p and assume we have

λ
:

G
K

→

G

K
′

\lambda :{G_K} \to {G_{K’}}

, an isomorphism of profinite groups. Then K and

K
′

K’

have the same constant field, zeta function, genus and class number. We also prove that the idele class groups and divisor class groups of K and

K
′

K’

are isomorphic. If E is a finite extension of k, the constant field of K and

K
′

K’

, we show that the E-rational points of the Jacobian varieties of K and

K
′

K’

are isomorphic as

G
(
E

/

k
)

G(E/k)

-modules. If

K
=

K
′

K = K’

and

K
¯

=

k
¯

K

\bar K = \bar kK

where

k
¯

\bar k

is the algebraic closure of k, we prove that

λ
(

G

K
¯

)
=

G

K
¯

\lambda ({G_{\bar K}}) = {G_{\bar K}}

and the induced automorphism of

G
(

K
¯

/

K
)

G(\bar K/K)

is the identity.
Integral representations of invariant measures
Authors: Maitra Ashok
Published: July 2010

In this paper we prove, under suitable conditions, several representation theorems for invariant measures arising out of the action of a family of measurable transformations

J

\mathcal {J}

on a measurable space

(
X
,

A

)

(X,\mathcal {A})

. Our results unify and extend results of Farrell and Varadarajan on the representation of invariant measures.
Analysis on the Heisenberg manifold
Authors: Tolimieri Richard
Published: July 2010

A study of the function theory on the Heisenberg manifold in terms of theta functions. Subject to an explicit error, a

C
∞

{C^\infty }

-function is written as an infinite sum, with theta functions of different degrees and characteristics playing the same role as exponentials do in the abelian theory.
Distribution of eigenvalues in the presence of higher order turning points
Authors: Leung Anthony
Published: July 2010

This article is concerned with the eigenvalue problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis x right-parenthesis minus lamda squared p left-parenthesis x right-parenthesis u left-parenthesis x right-parenthesis equals 0 comma u left-parenthesis x right-parenthesis element-of upper L 2 left-parenthesis negative normal infinity comma normal infinity right-parenthesis">

u
(
x
)
−

λ
2

p
(
x
)
u
(
x
)
=
0
,
u
(
x
)
∈

L
2

(
−
∞
,
∞
)

u(x) - {\lambda ^2}p(x)u(x) = 0,u(x) \in {L_2}( - \infty ,\infty )

, where

p
(
x
)

p(x)

is real, analytic and possesses zeroes of arbitrary orders. Under certain conditions on

p
(
x
)

p(x)

, approximate formulas for the eigenvalues are found. The problem considered is of interest in the study of particle scattering and wave mechanics. The formula is analogous to the quantum rule of Bohr-Sommerfeld.
Best possible approximation constants
Authors: Fink A. M.
Published: July 2010

We study inequalities between the norm of the best approximating polynomial and the nth derivative of the function. These inequalities are then related to inequalities that have been considered elsewhere in different contexts.
A note on limits of unitarily equivalent operators
Authors: Fialkow Lawrence A.
Published: July 2010

Let

U

(

H

)

\mathcal {U}(\mathcal {H})

denote the set of all unitary operators on a separable complex Hilbert space

H

\mathcal {H}

. If T is a bounded linear operator on

H

\mathcal {H}

, let

π
T

{\pi _T}

denote the mapping of

U

(

H

)

\mathcal {U}(\mathcal {H})

onto

U

(
T
)

\mathcal {U}(T)

given by conjugation. It is proved that if T is normal or isometric, then there exists a locally defined continuous cross-section for

π
T

{\pi _T}

if and only if the spectrum of T is finite. Examples of nonnormal operators with local cross-sections are given.
Completely unstable flows on 2-manifolds
Authors: Neumann Dean A.
Published: July 2010

Completely unstable flows on 2-manifolds are classified under both topological and

C
r

{C^r}

-equivalence

(
1
⩽
r
⩽
∞
)

(1 \leqslant r \leqslant \infty )

, in terms of the corresponding orbit spaces.
Generalized Hankel conjugate transformations on rearrangement invariant spaces
Authors: Kerman R. A.
Published: July 2010

The boundedness properties of the generalized Hankel conjugate transformations

H
λ

{H_\lambda }

on certain weighted Lebesgue spaces are studied. These are used to establish a boundedness criterion for the

H
λ

{H_\lambda }

on the more general class of rearrangement invariant spaces. The positive operators in terms of which the criterion is given are used to construct pairs of spaces between which the

H
λ

{H_\lambda }

are continuous; in particular, a natural analogue of a well-known result of Zygmund concerning the classical conjugate function operator is obtained for the

H
λ

{H_\lambda }

.
Anisotropic 𝐻^{𝑝} real interpolation, and fractional Riesz potentials
Authors: Madych W. R.
Published: July 2010

We observe that the anisotropic variants of

H
p

{H^p}

interpolate by the real method in the usual manner. Using this fact we show that the corresponding fractional Riesz potentials and related operators perform an embedding in

H
p

,
p
>
0

{H^p},p > 0

, analogous to the one for

L
p

,
p
>
1

{L^p},p > 1

. We also state a theorem concerning the mapping properties of

f
→
h
∗
f

f \to h \ast f

, where h is in

B
α

1
,
∞

B_\alpha ^{1,\infty }

, which hold only for a restricted range of p.
Products of sequentially compact spaces and the 𝑉-process
Authors: Rajagopalan M. , Woods R. Grant
Published: July 2010

In this paper we produce a family of sequentially compact, locally compact,

T
2

{T_2}

first countable, scattered and separable spaces whose product is not countably compact and thus answer a problem of C. T. Scarborough and A. H. Stone [11] in the negative. We do this using the continuum hypothesis. We also produce a completely regular,

T
2

{T_2}

, sequentially compact space K which is not p-compact for any

p
∈
β
N
−
N

p \in \beta N - N

.
The Huygens property for the heat equation
Authors: Widder D. V.
Published: July 2010

This note summarizes several criteria which guarantee that a solution of the heat equation should also have the semigroup property described in equation (1.2) below. In particular, it corrects a mistake in an earlier proof of one of these.
The Mackey Borel structure on the spectrum of an approximately finite-dimensional separable 𝐶*-algebra
Authors: Elliott George A.
Published: July 2010

It is shown that the Mackey Borel structures on the spectra of any two approximately finite-dimensional separable

C
∗

{C^\ast }

-algebras not of type I are isomorphic.
Local and global factorizations of matrix-valued functions
Authors: Clancey K. F. , Gohberg I.
Published: July 2010

Let C be a simple closed Liapounov contour in the complex plane and A an invertible

n
×
n

n \times n

matrix-valued function on C with bounded measurable entries. There is a well-known concept of factorization of the matrix function A relative to the Lebesgue space

L
p

(
C
)

{L_p}(C)

. The notion of local factorization of A relative to

L
p

{L_p}

at a point

t
0

{t_0}

in C is introduced. It is shown that A admits a factorization relative to

L
p

(
C
)

{L_p}(C)

if and only if A admits a local factorization relative to

L
p

{L_p}

at each point

t
0

{t_0}

in C. Several problems connected with local factorizations relative to

L
p

{L_p}

are raised.
Quasi-multipliers
Authors: McKennon Kelly
Published: July 2010

A quasi-multiplier m on an algebra A is a bilinear mapping from

A
×
A

A \times A

into itself such that

m
(
a
x
,
y
b
)
=
a
m
(
x
,
y
)
b

m(ax,yb) = am(x,y)b

for all

a
,
x
,
y
,
b
∈
A

a,x,y,b \in A

. An introduction to the theory of quasi-multipliers on Banach algebras with minimal approximate identities is given and applications to

C
∗

{C^\ast }

-algebras and group algebras are developed.
Packing and covering constants for certain families of trees. II
Authors: Meir A. , Moon J. W.
Published: July 2010

In an earlier paper we considered the problem of determining the packing and covering constants for families of trees whose generating function y satisfied a relation

y
=
x
ϕ
(
y
)

y = x\phi (y)

for some power series

ϕ
\phi

in y. In the present paper we consider the problem for some families of trees whose generating functions satisfy a more complicated relation.
Systems of nonlinear Volterra equations with positive definite kernels
Authors: Staffans Olof J.
Published: July 2010

We study the boundedness and the asymptotic behavior of the solutions of a nonlinear,

R

n

{{\mathbf {R}}^n}

-valued Volterra equation with a positive definite kernel, generalizing earlier scalar results.
The Kobayashi pseudometric on algebraic manifolds of general type and in deformations of complex manifolds
Authors: Wright Marcus
Published: July 2010

This paper deals with regularity properties of the infinitesimal form of the Kobayashi pseudo-distance. This form is shown to be upper semicontinuous in the parameters of a deformation of a complex manifold. The method of proof involves the use of a parametrized version of the Newlander-Nirenberg Theorem together with a theorem of Royden on extending regular mappings from polydiscs into complex manifolds. Various consequences and improvements of this result are discussed; for example, if the manifold is compact hyperbolic the infinitesimal Kobayashi metric is continuous on the union of the holomorphic tangent bundles of the fibers of the deformation. This result leads to the fact that the coarse moduli space of a compact hyperbolic manifold is Hausdorff. Finally, the infinitesimal form is studied for a class of algebraic manifolds which contains algebraic manifolds of general type. It is shown that the form is continuous on the tangent bundle of a manifold in this class. Many members of this class are not hyperbolic.
On the spectra of the restrictions of an operator
Authors: Herrero Domingo A.
Published: July 2010

Let T be a bounded linear operator from a complex Banach space

X

\mathfrak {X}

into itself and let

M

\mathfrak {M}

be a closed invariant subspace of T. Let

T

|

M

T|\mathfrak {M}

denote the restriction of T to

M

\mathfrak {M}

and let

σ
\sigma

denote the spectrum of an operator. The main results say that: (1) If

X

\mathfrak {X}

is the closed linear span of a family

{

M

v

}

\{ {\mathfrak {M}_v}\}

of invariant subspaces, then every component of

σ
(
T
)

\sigma (T)

intersects the closure of the set

∪
v

σ
(
T

|

M

v

)

{ \cup _v}\sigma (T|{\mathfrak {M}_v})

and every point of

σ
(
T
)
∖

∪
v

σ
(
T

|

M

v

)

\sigma (T)\backslash { \cup _v}\sigma (T|{\mathfrak {M}_v})

is an approximate eigenvalue of T. (2) If

X

\mathfrak {X}

is the closed linear span of a finite family

{

M

1

,
…
,

M

n

}

\{ {\mathfrak {M}_1}, \ldots ,{\mathfrak {M}_n}\}

of invariant subspaces, and the spectra

σ
(
T

|

M

j

)
,
j
=
1
,
2
,
…
,
n

\sigma (T|{\mathfrak {M}_j}),j = 1,2, \ldots ,n

, are pairwise disjoint, then

X

\mathfrak {X}

is actually equal to the algebraic direct sum of the

M

j

{\mathfrak {M}_j}

’s, the

M

j

{\mathfrak {M}_j}

’s are hyperinvariant subspaces of T and

σ
(
T
)
=

∪

j
=
1

n

σ
(
T

|

M

j

)

\sigma (T) = \cup _{j = 1}^n\sigma (T|{\mathfrak {M}_j})

. This last result is sharp in a certain specified sense. The results of (1) have a “dual version”

(

1
′

)

(1’)

; (1) and

(

1
′

)

(1’)

are applied to analyze the spectrum of an operator having a chain of invariant subspaces which is “piecewise well-ordered by inclusion", extending in several ways recent results of J. D. Stafney on the spectra of lower triangular matrices.
Closed convex invariant subsets of 𝐿_{𝑝}(𝐺)
Authors: Lau Anthony To Ming
Published: July 2010

Let G be a locally compact group. We characterize in this paper closed convex subsets K of

L
p

(
G
)
,
1
⩽
p
>
∞

{L_p}(G),1 \leqslant p > \infty

, that are invariant under all left or all right translations. We prove, among other things, that

K
=
{
0
}

K = \{ 0\}

is the only nonempty compact (weakly compact) convex invariant subset of

L
p

(
G
)

(

L
1

(
G
)
)

{L_p}(G)\;({L_1}(G))

. We also characterize affine continuous mappings from

P
1

(
G
)

{P_1}(G)

into a bounded closed invariant subset of

L
p

(
G
)

{L_p}(G)

which commute with translations, where

P
1

(
G
)

{P_1}(G)

denotes the set of nonnegative functions in

L
1

(
G
)

{L_1}(G)

of norm one. Our results have a number of applications to multipliers from

L
q

(
G
)

{L_q}(G)

into

L
p

(
G
)

{L_p}(G)

.
The diameter of orbits of compact groups of isometries; Newman’s theorem for noncompact manifolds
Authors: Hoffman David
Published: July 2010

The diameter of orbits of a compact isometry group G of a Riemannian manifold M cannot be uniformly small. If the sectional curvature of M is bounded above by

b
2

{b^2}

(b real or pure imaginary), then explicit bounds are found for

D
(
M
)

D(M)

, where

D
(
M
)

D(M)

is defined to be the largest number such that: If every orbit G has diameter less than

D
(
M
)

D(M)

, then G acts trivially on M. These bounds depend only on b and the injectivity radius of M. The proofs involve an investigation of various types of convex sets and an estimate for distance contraction of the exponential map on a manifold with bounded curvature.
On curves on formal groups
Authors: Morris Robert A. , Pareigis Bodo
Published: July 2010

The structure of the group of curves on a formal group is determined when the formal group is on a truncated power series algebra over a commutative ring. The resulting curve functor is faithful but not full. Applications to the Lie algebra of the formal group are given.
Knotting a 𝑘-connected closed 𝑃𝐿 𝑚-manifold in 𝐸^{2𝑚-𝑘}
Authors: Vrabec Jože
Published: July 2010

Embeddings of a k-connected closed PL m-manifold

(
0
⩽
k
⩽
m
−
3
)

(0 \leqslant k \leqslant m - 3)

in

(
2
m
−
k
)

(2m - k)

-dimensional euclidean space are classified up to isotopy. Thus this paper completes the results stated, and partly proved, in J. F. P. Hudson’s Piecewis linear topology.
𝐿_{∞}_{𝜆}-equivalence, isomorphism and potential isomorphism
Authors: Nadel Mark , Stavi Jonathan
Published: July 2010

It is well known that two structures are

L

∞
ω

{L_{\infty \omega }}

-equivalent iff they are potentially isomorphic [that is, isomorphic in some (Cohen) extension of the universe]. We prove that no characterization of

L

∞
λ

{L_{\infty \lambda }}

-equivalence along these lines is possible (at least for successor cardinals

λ
\lambda

) and the potential-isomorphism relation that naturally comes to mind in connection with

L

∞
λ

{L_{\infty \lambda }}

is often not even transitive and never characterizes

≡

∞
λ

{ \equiv _{\infty \lambda }}

for

λ
>
ω

\lambda > \omega

. A major part of the work is the construction of

κ
+

{\kappa ^ + }

-like linear orderings (also Boolean algebras) A, B such that

N

κ
+

(

A

,

B

)

{N_{{\kappa ^ + }}}({\mathbf {A}},{\mathbf {B}})

, where

N
λ

(

A

,

B

)

{N_\lambda }({\mathbf {A}},{\mathbf {B}})

means: A and B are nonisomorphic

L

∞
λ

{L_{\infty \lambda }}

-equivalent structures of cardinality

λ
\lambda

.
Simple Lie algebras of toral rank one
Authors: Wilson Robert Lee
Published: July 2010

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic

p
>
7

p > 7

. Let L have Cartan decomposition

L
=
H
+

∑

γ
∈
Γ

L
γ

L = H + {\sum _{\gamma \in \Gamma }}{L_\gamma }

. If

Γ
\Gamma

generates a cyclic group then L is isomorphic to

sl

(
2
,
F
)

{\text {sl}}(2,F)

or to one of the simple Lie algebras of generalized Cartan type

W
(
1
:

n

)

W(1:{\mathbf {n}})

or

H

(
2
:

n

:
Φ

)

(
2
)

H{(2:{\mathbf {n}}:\Phi )^{(2)}}

.
A nonlinear semigroup for a functional differential equation
Authors: Brewer Dennis W.
Published: July 2010

A representation theorem is obtained for solutions of the nonlinear functional differential equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel u prime left-parenthesis t right-parenthesis equals upper F left-parenthesis u Subscript t Baseline right-parenthesis comma t greater-than-or-slanted-equals 0 comma u left-parenthesis t right-parenthesis equals phi left-parenthesis t right-parenthesis comma t less-than-or-slanted-equals 0 comma EndLayout">

(

1

)

u
′

(
t
)
=
F
(

u
t

)
,
t
⩾
0
,

u
(
t
)
=
ϕ
(
t
)
,
t
⩽
0
,

\begin{equation}\tag {$1$} u’(t) = F({u_t}), t \geqslant 0,\quad u(t) = \phi (t), t \leqslant 0,\end{equation}

as a semigroup of nonlinear operators on a space of initial data X of “fading memory type.” Equation (1) is studied in the abstract setting of a Banach space E. The nonlinear functional F is a uniformly Lipschitz continuous mapping from X to E. The semigroup is constructed by transforming (1) to an abstract Cauchy problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis upper C upper P right-parenthesis EndLabel w prime left-parenthesis t right-parenthesis plus upper A w left-parenthesis t right-parenthesis equals 0 comma w left-parenthesis 0 right-parenthesis equals phi comma EndLayout">

(

C
P

)

w
′

(
t
)
+
A
w
(
t
)
=
0
,

w
(
0
)
=
ϕ
,

\begin{equation}\tag {$CP$} w’(t) + Aw(t) = 0,\quad w(0) = \phi ,\end{equation}

in the space X and applying a generation theorem of M. Crandall and T. Liggett to the operator A in X. The case when (1) is a nonlinear Volterra integrodifferential equation of infinite delay is given special consideration. The semigroup representation is used to obtain finite difference approximations for solutions of (CP) and to study the continuity of solutions of (1) with respect to perturbations of F and

ϕ
\phi

.
The classification of flat solvmanifolds
Authors: Morgan Alexander
Published: July 2010

This paper contains a complete algebraic characterization of the fundamental groups of flat solvmanifolds. This characterization is in terms of finite integral representations of free abelian groups and the associated cohomology. A classification of compact flat solvmanifolds follows, and a list of all compact flat solvmanifolds of dimensions 3, 4, and 5 (except the 5-dimensional with first betti number 1) is given. Some theorems on the classification of noncompact flat solvmanifolds have also been obtained. These give full results in some cases, partial results in others. For example, the odd order holonomy group case is completely settled.
Invariant means on the continuous bounded functions
Authors: Rosenblatt Joseph
Published: July 2010

Let G be a noncompact nondiscrete

σ
\sigma

-compact locally compact metric group. A Baire category argument gives measurable sets

{

A
γ

:
γ
∈
Γ
}

\{ {A_\gamma }:\gamma \in \Gamma \}

of finite measure with card

(
Γ
)
=
c

(\Gamma ) = c

which are independent on the open sets. One approximates

{

A
γ

:
γ
∈
Γ
}

\{ {A_\gamma }:\gamma \in \Gamma \}

by arrays of continuous bounded functions with compact support and then scatters these arrays to construct functions

{

f
γ

:
γ
∈
Γ
}

\{ {f_\gamma }:\gamma \in \Gamma \}

in

CB

(
G
)

{\text {CB}}(G)

with a certain independence property. If G is also amenable as a discrete group, the existence of these independent functions shows that on

CB

(
G
)

{\text {CB}}(G)

there are

2
c

{2^c}

mutually singular elements of LIM each of which is singular to TLIM.
The conjugacy problem for boundary loops in 3-manifolds
Authors: Evans Benny D.
Published: July 2010

A geometric solution of the word problem for fundamental groups of compact, orientable, irreducible, sufficiently large 3-manifolds has been given by F. Waldhausen. We present here a solution of a restricted version of the conjugacy problem for this same class of 3-manifolds; however, the conjugacy problem for 3-manifolds remains in general unsolved. The main results is that there is an algorithm that will determine for any two loops

L
1

,

L
2

{L_1},{L_2}

in the boundary of a compact, orientable, irreducible sufficiently large 3-manifold M if

L
1

{L_1}

, is freely homotopic in M to

L
2

{L_2}

.
The product of nonplanar complexes does not imbed in 4-space
Authors: Ummel Brian R.
Published: July 2010

We prove that if

K
1

{K_1}

and

K
2

{K_2}

are nonplanar simplicial complexes, then

K
1

×

K
2

{K_1}\, \times \, {K_2}

does not imbed in

R

4

{{\textbf {R}}^4}

.
Translation planes of order 𝑞²: asymptotic estimates
Authors: Ebert Gary L.
Published: July 2010

R. H. Bruck has pointed out the one-to-one correspondence between the isomorphism classes of certain translation planes, called subregular, and the equivalence classes of disjoint circles in a finite miquelian inversive plane

I
P
(
q
)

IP(q)

. The problem of determining the number of isomorphism classes of translation planes is old and difficult. Let q be an odd prime-power. In this paper, a study of sets of disjoint circles in

I
P
(
q
)

IP(q)

enables the author to find an asymptotic estimate of the number of isomorphism classes of translation planes of order

q
2

{q^2}

which are subregular of index 3 or 4. It is conjectured (and proved for

n
⩽
3

n \leqslant 3

) that, given a set of n disjoint circles in

I
P
(
q
)

IP(q)

, the numbers of circles disjoint from each of the given n circles is asymptotic to

q
3

/

2
n

{q^3}/{2^n}

. This conjecture, if true, would allow one to estimate the number of subregular translation planes of order

q
2

{q^2}

with any positive index.
The semigroup of varieties of Brouwerian semilattices
Authors: Köhler Peter
Published: July 2010

It is shown that the semigroup of varieties of Brouwerian semilattices is free.
Complete universal locally finite groups
Authors: Hickin Ken
Published: July 2010

This paper will partly strengthen a recent application of model theory to the construction of sets of pairwise nonembeddable universal locally finite groups [8]. Our result is Theorem. There is a set

U

\mathcal {U}

of

2

ℵ
1

{2^{{\aleph _1}}}

universal locally finite groups of order

ℵ
1

{\aleph _1}

with the following properties: 0.1. If

U
≠
V
∈

U

U \ne V \in \mathcal {U}

and A and B are uncountable sugroups of U and V, then A and B are not isomorphic. Let A be an uncountable subgroup of

U
∈

U

U \in \mathcal {U}

. 0.2. A does not belong to any proper variety of groups, and 0.3. A is not isomorphic to any of its proper subgroups. 0.4. Every

U
∈

U

U \in \mathcal {U}

is a complete group (every automorphism of U is inner).
Taming and the Poincaré conjecture
Authors: Thickstun T. L.
Published: July 2010

L. Glaser and L. Siebenmann have shown that the double suspension of a homotopy 3-sphere is homeomorphic to the 5-sphere. This result, together with a well-known characterization of

S
3

{S^3}

due to R. H. Bing, is used to establish a relationship between the Poincaré conjecture and two conjectures concerned with taming embeddings in higher dimensions. One of the two conjectures, each of which implies the Poincaré conjecture, states, in effect, that a codimension two sphere is tame if it is tame “modulo” a tame disk contained in it.
(𝐸³/𝑋)×𝐸¹≈𝐸⁴ (𝑋, a cell-like set): an alternative proof
Authors: Cannon J. W.
Published: July 2010

The author gives an alternative proof that a cell-like closed-0dimensional decomposition of

E
3

{E^3}

is an

E
4

{E^4}

factor. The argument is essentially 2-dimensional. The 3- and 4-dimensional topology employed is truly minimal.
Projective modules over subrings of 𝑘[𝑋,𝑌]
Authors: Anderson David F.
Published: July 2010

In this paper we study projective modules over subrings of

k
[
X
,
Y
]

k[X,Y]

. Conditions are given for projective modules to decompose into free

⊕
\oplus

rank 1 modules. Our main result is that if k is an algebraically closed field and A a subring of

B
=
k
[
X
,
Y
]

B = k[X,Y]

with

A
⊂
B

A \subset B

integral and

sing

(
A
)

{\text {sing}}(A)

finite, then all f.g. projective A-modules have the form free

⊕
\oplus

rank 1. We also give several examples of subrings of

k
[
X
,
Y
]

k[X,Y]

which have indecomposable projective modules of rank 2.
The asymptotic behaviour of certain integral functions
Authors: Fenton P. C.
Published: July 2010

Let

f
(
z
)

f(z)

be an integral function satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript Superscript normal infinity Baseline left-brace log m left-parenthesis r comma f right-parenthesis minus cosine pi rho log upper M left-parenthesis r comma f right-parenthesis right-brace Superscript plus Baseline StartFraction d r Over r Superscript rho plus 1 Baseline EndFraction greater-than normal infinity">

∫

∞

{
log

m
(
r
,
f
)

−

cos

π
ρ

log

M
(
r
,
f
)

}
+

d
r

r

ρ
+
1

>

∞

{\int _{}^\infty \{\log \,m(r,f)\, - \,\cos \,\pi \rho \,\log \,M(r,f)\} ^ + }\frac {{dr}}{{{r^{\rho + 1}}}}\, > \,\infty

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than limit Underscript ModifyingAbove r right-arrow normal infinity With bar Endscripts StartFraction log upper M left-parenthesis r comma f right-parenthesis Over r Superscript rho Baseline EndFraction greater-than normal infinity">

0

>

lim

r

→
∞

¯

log

M
(
r
,
f
)

r
ρ

>

∞

0\, > \,\lim \limits _{\overline {r\, \to \infty } } \,\frac {{\log \,M(r,f)}}{{{r^\rho }}}\, > \,\infty

\]
for some

ρ
:

0

>

ρ

>

1

\rho :\,0\, > \,\rho \, > \,1

. It is shown that such functions have regular asymptotic behaviour outside a set of circles with centres

ζ
i

{\zeta _i}

and radii

t
i

{t_i}

for which
\[

∑

i
=
1

∞

t
i

|

ζ
i

|

>
∞

\sum \limits _{i = 1}^\infty {\frac {{{t_i}}}{{\left | {{\zeta _i}} \right |}}} > \infty

\]
.
Twist spinning revisited
Authors: Goldsmith Deborah L. , Kauffman Louis H.
Published: July 2010

This paper contains several applications of the following theorem: The 1-twist spin

L
1

(
k
)

{L_1}(k)

of any knot

k
⊂

S

n
−
1

k \subset {S^{n - 1}}

is interchangeable with the standard unknotted

(
n
−
2
)

(n - 2)

-sphere K in

S
n

{S^n}

by means of a homeomorphism of triples
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h colon left-parenthesis upper S Superscript n Baseline comma upper K comma upper L 1 left-parenthesis k right-parenthesis right-parenthesis right-arrow left-parenthesis upper S Superscript n Baseline comma upper L 1 left-parenthesis k right-parenthesis comma upper K right-parenthesis">

h
:
(

S
n

,
K
,

L
1

(
k
)
)
→
(

S
n

,

L
1

(
k
)
,
K
)

h:({S^n},K,{L_1}(k)) \to ({S^n},{L_1}(k),K)

which reverses the orientation of

S
n

{S^n}

, and preserves the orientations of K and

L
1

(
k
)

{L_1}(k)

. One of these applications is Zeeman’s Theorem about twist spun knots; another is a proof of a conjecture of R. H. Fox about certain manifolds which have the same fundamental group. We also prove that the iterated twist spun knot

L

a
,
b

(
k
)
⊂

S

n
+
1

{L_{a,b}}(k) \subset {S^{n + 1}}

is fiber equivalent to one of

L

0
,
g

(
k
)

{L_{0,g}}(k)

or

L

g
,
g

(
k
)

{L_{g,g}}(k)

where

g
=

g.c.d.

(
a
,
b
)

g = {\text {g.c.d.}}(a,b)

.
Completeness theorems, incompleteness theorems and models of arithmetic
Authors: McAloon Kenneth
Published: July 2010

Let

A

\mathcal {A}

be a consistent extension of Peano arithmetic and let

A

n
0

\mathcal {A}_n^0

denote the set of

Π
n
0

\Pi _n^0

consequences of

A

\mathcal {A}

. Employing incompleteness theorems to generate independent formulas and completeness theorems to construct models, we build nonstandard models of

A

n
+
2

0

\mathcal {A}_{n + 2}^0

in which the standard integers are

Δ

n
+
1

0

\Delta _{n + 1}^0

-definable. We thus pinpoint induction axioms which are not provable in

A

n
+
2

0

\mathcal {A}_{n + 2}^0

; in particular, we show that (parameter free)

Δ
1
0

\Delta _1^0

-induction is not provable in Primitive Recursive Arithmetic. Also, we give a solution of a problem of Gaifman on the existence of roots of diophantine equations in end extensions and answer questions about existentially complete models of

A

2
0

\mathcal {A}_2^0

. Furthermore, it is shown that the proof of the Gödel Completeness Theorem cannot be formalized in

A

2
0

\mathcal {A}_2^0

and that the MacDowell-Specker Theorem fails for all truncated theories

A

n
0

\mathcal {A}_n^0

.
Heisenberg manifolds and theta functions
Authors: Tolimieri R.
Published: July 2010

The algebraic structure of the

(
2
n
+
1
)

(2n + 1)

-dimensional Heisenberg group naturally induces a special class of differential operators whose solutions

(
D
f
=
0
)

(Df = 0)

are related to classical theta function theory.
A chain functor for bordism
Authors: Kochman Stanley O.
Published: July 2010

Chains of differential graded abelian monoids are defined for bordism and cobordism theories. These chains are used to define matric Massey products and can be filtered so as to define the Adams spectral sequence. From this point of view, we prove three basic theorems which show how Massey products behave in the Adams spectral sequence.
Lexicographic partial order
Authors: Crapo Henry
Published: July 2010

Given a (partially) ordered set P with the descending chain condition, and an ordered set Q, the set

Q
P

{Q^P}

of functions from P to Q has a natural lexicographic order, given by

f
⩽
g

f \leqslant g

if and only if

f
(
y
)
>
g
(
y
)

f(y) > g(y)

for all minimal elements of the set

{
x
;
f
(
x
)
≠
g
(
x
)
}

\{ x;f(x) \ne g(x)\}

where the functions differ. We show that if Q is a complete lattice, so also is the set

Q
P

{Q^P}

, in the lexicographic order. The same holds for the set

Hom

(
P
,
Q
)

{\operatorname {Hom}}(P,Q)

of order-preserving functions, and for the set

Op

(
P
)

{\text {Op}}(P)

of increasing order-preserving functions on the set P. However, the set

Cl

(
P
)

{\text {Cl}}(P)

of closure operators on P is not necessarily a lattice even if P is a complete lattice.
Projective varieties of low codimension in characteristic 𝑝>0
Authors: Speiser Robert
Published: July 2010

Let X be an s-dimensional closed Cohen-Macaulay subvariety of projective n-space, over an algebraically closed field of characteristic

p
>
0

p > 0

. Assume

s
⩾

1
2

(
n
+
1
)

s \geqslant \tfrac {1}{2}(n + 1)

. Then (1) every stratified vector bundle on X is trivial; (2) X is simply connected. Assertion (1) generalizes Gieseker’s result for projective space, while (2) is a strengthened analogue of results of Barth and Ogus in characteristic zero.
A probable Hasse principle for pencils of quadrics
Authors: Waterhouse William C.
Published: July 2010

Let k be a global field,

char

(
k
)
≠
2

{\text {char}}(k) \ne 2

. Although pencils of quadrics over k may fail to satisfy a local-to-global equivalence principle, the failures are exceptional in the precise sense of having limiting probability zero. The proof uses the classification of pairs of quadratic forms. It also requires knowing that a square class in a finite extension usually comes from k when it does so locally; the Galois-theoretic criterion for this is determined.
Regular Hjelmslev planes. II
Authors: Jungnickel Dieter
Published: July 2010

In this paper we introduce the notion of an x-partition for a Hjelmslev-matrix (H-matrix). This allows us to prove a new composition theorem for H-matrices. We obtain the existence of

(

t
,

r

)

({t,\,r})

-H-matrices and hence of regular

(

t
,

r

)

({t,\,r})

-H-planes for infinitely many series of invariants which were not yet known. In fact, many of these invariants were not even known to occur as the invariants of any H-plane at all (whether regular or not).
Maxima of random algebraic curves
Authors: Das M. , Bhatt S. S.
Published: July 2010

Let

X
1

,

X
2

,
…
,

X
n

{X_1},{X_2}, \ldots ,{X_n}

be a sequence of independent and identically distributed random variables with common characteristic function

exp

(
−

|
Z
|

α

)

{\exp }( - {\left | Z \right |^\alpha })

where

0
>
α
⩽
2

0 > \alpha \leqslant 2

, and

P
(
x
)
=

∑
1
n

X
k

x
k

P(x) = \sum \nolimits _1^n {{X_k}{x^k}}

. Then we show that the numbers

M
n

{M_n}

of maxima of the curves

y
=
P
(
x
)

y = P(x)

have expectation

E

M
n

∼
c
log
⁡
n

E{M_n} \sim c \log n

, as

n
→
∞

n \to \infty

, where

c
=
c
(
α
)
=

c
1

(
α
)
+

c
2

(
α
)

c = c(\alpha ) = {c_1}(\alpha ) + {c_2}(\alpha )

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row c 1 left-parenthesis alpha right-parenthesis equals StartFraction 1 Over pi squared alpha squared EndFraction integral Subscript negative normal infinity Superscript normal infinity Baseline d v log integral Subscript 0 Superscript normal infinity Baseline left-bracket StartFraction StartAbsoluteValue v minus y EndAbsoluteValue Superscript alpha Baseline Over StartAbsoluteValue v minus 1 EndAbsoluteValue Superscript alpha Baseline EndFraction right-bracket exp left-parenthesis negative y right-parenthesis d y comma c 2 left-parenthesis alpha right-parenthesis equals StartFraction 1 Over pi squared alpha squared EndFraction integral Subscript negative normal infinity Superscript normal infinity Baseline log integral Subscript 0 Superscript normal infinity Baseline left-brace left-parenthesis StartFraction StartAbsoluteValue v minus z EndAbsoluteValue Superscript alpha Baseline Over StartAbsoluteValue v minus alpha minus 1 EndAbsoluteValue Superscript alpha Baseline EndFraction right-parenthesis StartFraction z Superscript alpha Baseline Over normal upper Gamma left-parenthesis 1 plus alpha right-parenthesis EndFraction exp left-parenthesis negative z right-parenthesis d z right-brace d v period EndLayout">

c
1

(
α
)

=

1

π
2

α
2

∫

−
∞

∞

d
v

log

∫
0
∞

[

|

v

−

y

|

α

|

v

−

1

|

α

]

exp
⁡
(
−
y
)

d
y
,

c
2

(
α
)

=

1

π
2

α
2

∫

−
∞

∞

log

∫
0
∞

{

(

|

v

−

z

|

α

|

v

−

α

−

1

|

α

)

z
α

Γ
(
1

+

α
)

exp

(
−
z
)

d
z

}

d
v
.

\begin {array}{*{20}{c}} {{c_1}(\alpha )\, = \,\frac {1} {{{\pi ^2}{\alpha ^2}}}\int _{ - \infty }^\infty {dv\,\log \,\int _0^\infty {\left [ {\frac {{{{\left | {v\, - \,y} \right |}^\alpha }}} {{{{\left | {v\, - \,1} \right |}^\alpha }}}} \right ]\,\exp ( - y)\,dy,} } } {{c_2}(\alpha )\, = \,\frac {1} {{{\pi ^2}{\alpha ^2}}}\int _{ - \infty }^\infty {\log \,\int _0^\infty {\left \{ {\left ( {\frac {{{{\left | {v\, - \,z} \right |}^\alpha }}} {{{{\left | {v\, - \,\alpha \, - \,1} \right |}^\alpha }}}} \right )\,\frac {{{z^\alpha }}} {{\Gamma (1\, + \,\alpha )}}\,\exp \,( - z)\,dz} \right \}} \,dv.} } \end {array}

\]
The invariant Π⁰_{𝛼} separation principle
Authors: Miller Douglas E.
Published: July 2010

We “invariantize” the classical theory of alternated unions to obtain new separation results in both invariant descriptive set theory and in infinitary logic. Application is made to the theory of definitions of countable models.
A theorem of Ahlfors for hyperbolic spaces
Authors: Chen Su Shing
Published: July 2010

L. Ahlfors has proved that if the Dirichlet fundamental polyhedron of a Kleinian group G in the unit ball

B
3

{B^3}

has finitely many sides, then the normalized Lebesgue measure of

L
(
G
)

L(G)

is either zero or one. We generalize this theorem and a theorem of Beardon and Maskit to the n-dimensional case.
On the stable decomposition of Ω²𝑆^{𝑟+2}
Authors: Brown E. H. , Peterson F. P.
Published: July 2010

In this paper we show that

Ω
2

S

r
+
2

{\Omega ^2}{S^{r + 2}}

is stably homotopy equivalent to a wedge of suspensions of other spaces

C
k
1

C_k^1

, and that

C
k
1

C_k^1

is homotopy 2-equivalent to the Brown-Gitler spectrum.
Induced representations of 𝐶*-algebras and complete positivity
Authors: Bennett James G.
Published: July 2010

It is shown that

∗

^{*}

representations may be induced from one

C

∗

{C^{\ast }}

-algebra B to another

C

∗

{C^{\ast }}

-algebra A via a vector space equipped with a completely positive B-valued inner product and a

∗

^{*}

representation of A. Theorems are proved on induction in stages, on continuity of the inducing process and on completely positive linear maps of finite dimensional

C

∗

{C^{\ast }}

-algebras and of group algebras.
On the boundary behaviour of functions orthogonal to invariant subspaces
Authors: Leung Kar Koi
Published: July 2010

In this paper we study the relationship between the boundary behaviour of functions in the orthogonal subspace of the invariant subspace

φ

H
2

\varphi {H^2}

and the boundary behaviour of the inner function

φ
\varphi

itself.
Two-dimensional 𝜖-isometries
Authors: Bourgin D. G.
Published: July 2010

An affirmative answer to the antipodal

ε
\varepsilon

-isometry conjecture is established for 2-dimensional Banach spaces.
The Hanf number of the first order theory of Banach spaces
Authors: Shelah Saharon , Stern Jacques
Published: July 2010

In this paper, we discuss the possibility of developing a nice i.e. first order theory for Banach spaces: the restrictions on the set of sentences for recent compactness arguments applied to Banach spaces as well as for other model-theoretic results are both natural and necessary; without them we essentially get a second order logic with quantification over countable sets. Especially, the Hanf number for sets of sentences of the first order theory of Banach spaces is exactly the Hanf number for the second order logic of binary relations (with the second order quantifiers ranging over countable sets).
On the exceptional central simple non-Lie Mal’cev algebras
Authors: Carlsson Renate
Published: July 2010

Malcev algebras belong to the class of binary Lie algebras. Any Lie algebra is a Malcev algebra. In this paper we show that for each seven-dimensional central simple non-Lie Malcev algebra any finite dimensional Malcev module is completely reducible also for positive characteristics. This contrasts with each modular semisimple Lie algebra. As a consequence we get that the classical structure theory for characteristic zero is valid also in the modular case if semisimplicity is replaced by

G
1

{G_1}

-separability. The Wedderburn principal theorem is proved for Malcev algebras.
𝑅-separation of variables for the four-dimensional flat space Laplace and Hamilton-Jacobi equations
Authors: Kalnins E. G. , Miller Willard
Published: July 2010

All A-separable orthogonal coordinate systems for the complex equations

Σ

i
=
1

4

∂

i
i

Ψ
=
0

\Sigma _{i = 1}^4 {{\partial _{ii}}\Psi = 0}

and

Σ

i
=
1

4

(

∂
i

W
)

2

=
0

\Sigma _{i = 1}^4 {{{({\partial _i}W)}^2} = 0}

are classified and it is shown that these equations separate in exactly the same systems.
Algebraic structures for ⨁∑_{𝑛≥1}𝐿²(𝑍/𝑛) compatible with the finite Fourier transform
Authors: Auslander L. , Tolimieri R.
Published: July 2010

Let

Z

/

n

{Z / n}

denote the integers

mod

n

\bmod \,n

and let

F

n

{\mathcal {F}_n}

denote the finite Fourier transform on

L
2

(

Z

/

n

)

{L^2}({Z / n})

. We let

⊕
Σ

F

n

=
F

\oplus \Sigma {{\mathcal {F}_n}} = F

operate on

⊕
Σ

L
2

(

Z

/

n

)

\oplus \Sigma {L^2}({Z / n})

and show that

⊕
Σ

L
2

(

Z

/

n

)

\oplus \Sigma {L^2}({Z / n})

can be given a graded algebra structure (with no zero divisors) such that

F

(
f
g
)
=

F

(
f
)

F

(
g
)

\mathcal {F}(fg) = \mathcal {F}(f)\mathcal {F}(g)

. We do this by establishing a natural isomorphism with the algebra of theta functions with period i. In addition, we find all algebra structures on

⊕
Σ

L
2

(

Z

/

n

)

\oplus \Sigma {L^2}({Z / n})

satisfying the above condition.
An asymptotic double commutant theorem for 𝐶*-algebras
Authors: Hadwin Donald W.
Published: July 2010

An asymptotic version of von Neumann’s double commutant theorem is proved in which

C

∗

{C^{\ast }}

-algebras play the role of von Neumann algebras. This theorem is used to investigate asymptotic versions of similarity, reflexivity, and reductivity. It is shown that every nonseparable, norm closed, commutative, strongly reductive algebra is selfadjoint. Applications are made to the study of operators that are similar to normal (subnormal) operators. In particular, if T is similar to a normal (subnormal) operator and

π
\pi

is a representation of the

C

∗

{C^{\ast }}

-algebra generated by t, then

π
(
T
)

\pi (T)

is similar to a normal (subnormal) operator.
Some 3-manifolds which admit Klein bottles
Authors: Kim Paik Kee
Published: July 2010

Consider a closed, orientable, irreducible 3-manifold M with

|

π
1

(
M
)

|

>
∞

\left | {{\pi _1}(M)} \right | > \infty

, in which a Klein bottle can be embedded. We present a classification of the spaces M and show that, if

π
1

(
M
)

{\pi _1}(M)

is cyclic, then M is homeomorphic to a lens space. Note that all surfaces of even genus can be embedded in each space M. We also classify all free involutions on lens spaces whose orbit spaces contain Klein bottles.
Unitary invariance in algebraic algebras
Authors: Lanski Charles
Published: July 2010

A structure theorem is obtained for subspaces invariant under conjugation by the unitary group of a prime algebraic algebra over an infinite field. For an invariant subalgebra W, it is shown that either W is central, W contains an ideal, or the ring satisfies the standard identity of degree eight. Also, for prime algebras not satisfying such an identity, the unitary group is not solvable.
On the algebraic criteria for local Pareto optima. II
Authors: Wan Yieh Hei
Published: July 2010

Constrained vector optimization problems in the large are studied in this paper. Fixing any constraints so that the feasible set is a manifold with corners, we prove that the set of local Pareto optima for typical vector-valued functions admit Whitney prestratifications. Furthermore, these prestratifications persist under small perturbation of vector-valued functions. The main tools used here are a variant of Thom’s transversality theorem and the stratification theory of semialgebraic sets.
Measurable parametrizations and selections
Authors: Cenzer Douglas , Mauldin R. Daniel
Published: July 2010

Let W be a Borel subset of

I
×
I

I \times I

(where

I
=
[
0
,
1
]

I = [0,1]

) such that, for each x,

W
x

=
{
y
:

(
x
,
y
)
∈
W
}

{W_x} = \{ y:\,(x,y) \in W\}

is uncountable. It is shown that there is a map, g, of

I
×
I

I \times I

onto W such that (1) for each x,

g
(
x
,
⋅
)

g(x, \cdot )

is a Borel isomorphism of I onto

W
x

{W_x}

and (2) both g and

g

−
1

{g^{ - 1}}

are

S
(
I
×
I
)

S(I \times I)

-measurable maps. Here, if X is a topological space,

S
(
X
)

S(X)

is the smallest family containing the open subsets of X which is closed under operation (A) and complementation. Notice that

S
(
X
)

S(X)

is a subfamily of the universally or absolutely measurable subsets of X. This result answers a problem of A. H. Stone. This result improves a theorem of Wesley and as a corollary a selection theorem is obtained which extends the measurable selection theorem of von Neumann. We also show an analogous result holds if W is only assumed to be analytic.
A remark on zeta functions
Authors: Igusa Jun-ichi
Published: July 2010

In the adelic definition of the zeta function by Tate and Iwasawa, especially in the form given by Weil, one uses all Schwartz-Bruhat functions as “test functions"; we have found that such an adelic zeta function relative to Q contains the Dedekind zeta function of any finite normal extension of Q and that the normality assumption can be removed if Artin’s conjecture is true.
A unified approach to measurable and continuous selections
Authors: Mägerl G.
Published: July 2010

An abstract selection theorem is presented which contains as special cases-among others-the measurable selection theorem of Kuratowski and Ryll-Nardzewski, as well as the continuous selection theorem of Michael.