bluuebunny/crossref_metadata_2025 huggingface dataset:
Dual Lie elements and a derivation for the cofree coassociative coalgebra
Authors: Griffing Gary
Published: June 2010

We construct a derivation D in the Hopf algebra TcV, the cofree coassociative coalgebra on a vector space V. We then define the subspace of TcV consisting of dual Lie elements, which is analogous to the subspace of the Hopf algebra TV, the free associative algebra on V, consisting of Lie elements. Thereafter, we formulate a dual Dynkin-Specht-Wever theorem. Using our map D, we then give very short proofs of both the dual Dynkin-Specht-Wever and dual Friedrichs’ theorems, each of which characterizes the space of dual Lie elements in TcV at characteristic 0.
Monotonicity properties of Lorentz spaces
Authors: Hudzik Henryk , Kamińska Anna
Published: June 2010

Criteria for uniform monotonicity, local uniform monotonicity and strict monotonicity of Lorentz spaces are given.
Contractive projections in nonatomic function spaces
Authors: Randrianantoanina Beata
Published: June 2010

We prove that there are no 1-complemented subspaces of finite codimension in separable rearrangement-invariant nonatomic function spaces not isometric to

L
2

{L_2}

.
Remarks about 𝛾-sets and Borel-dense sets
Authors: Recław Ireneusz
Published: June 2010

We show, assuming Martin’s Axiom, that every set of cardinality the continuum containing a Borel-dense set of cardinality less than the continuum is a

γ
\gamma

-set but is not a hereditarily

γ
\gamma

-set. This answers a question of D. H. Fremlin and J. Jasinski.
Compacta with dense ambiguous loci of metric projections and antiprojections
Authors: Zhivkov N. V.
Published: June 2010

In every strictly convexifiable Banach space X with

dim
⁡
X
≥
2

\dim X \geq 2

there exists a dense

G
δ

{G_\delta }

set of compacta

A

\mathcal {A}

in the Hausdorff set topology such that with respect to an arbitrary equivalent strictly convex norm in X both the metric projection and the metric antiprojection generated by any member of

A

\mathcal {A}

are densely multivalued.
Functions on noncompact Lie groups with positive Fourier transforms
Authors: Kawazoe Takeshi
Published: June 2010

Let G be a homogeneous group with the graded Lie algebra or a noncompact semisimple Lie group with finite center. We define the Fourier transform

f
^

\hat f

of f as a family of operators
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove f With caret left-parenthesis pi right-parenthesis equals integral Underscript upper G Endscripts f left-parenthesis x right-parenthesis pi left-parenthesis x right-parenthesis d x left-parenthesis pi element-of ModifyingAbove upper G With caret right-parenthesis">

f
^

(
π
)
=

∫
G

f
(
x
)
π
(
x
)
d
x
(
π
∈

G
^

)

\hat f(\pi ) = {\smallint _G}f(x)\pi (x)dx(\pi \in \hat G)

, and we say that

f
^

\hat f

is positive if all

f
^

(
π
)

\hat f(\pi )

are positive. Then, we construct an integrable function f on G with positive

f
^

\hat f

and the restriction of f to any ball centered at the origin of G is square-integrable, however, f is not square-integrable on G.
Equivariant, almost homeomorphic maps between 𝑆¹ and 𝑆²
Authors: Soma Teruhiko
Published: June 2010

Let

Π
\Pi

be a Fuchsian group isomorphic to a non-trivial, closed surface group, and let

M
=

H

3

/

Γ

M = {\mathbb {H}^3}/\Gamma

be a hyperbolic 3-manifold admitting an isomorphism

ρ
:
Π
→
Γ

\rho :\Pi \to \Gamma

. Under certain assumptions, Cannon-Thurston and Minsky showed that there exists a

ρ
\rho

-equivariant, surjective, continuous map

f
:

S
∞
1

→

S
∞
2

f:S_\infty ^1 \to S_\infty ^2

. In this paper, we prove that there exist zero-measure sets

Λ
1

{\Lambda ^1}

in

S
∞
1

S_\infty ^1

and

Λ
2

{\Lambda ^2}

in

S
∞
2

S_\infty ^2

such that the restriction
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f vertical-bar Subscript upper S Sub Subscript normal infinity Sub Superscript 1 Subscript minus normal upper Lamda Sub Superscript 1 Subscript Baseline colon upper S Subscript normal infinity Superscript 1 Baseline minus normal upper Lamda Superscript 1 Baseline right-arrow upper S Subscript normal infinity Superscript 2 Baseline minus normal upper Lamda squared">

f

|

S
∞
1

−

Λ
1

:

S
∞
1

−

Λ
1

→

S
∞
2

−

Λ
2

f{|_{S_\infty ^1 - {\Lambda ^1}}}:S_\infty ^1 - {\Lambda ^1} \to S_\infty ^2 - {\Lambda ^2}

is a homeomorphism.
The error term of holomorphic mappings in Nevanlinna theory
Authors: Ye Zhuan
Published: June 2010

We construct a holomorphic mapping from

C

m

{\mathbb {C}^m}

to

P

n

{\mathbb {P}^n}

for any m and n with

m
≥
n
≥
1

m \geq n \geq 1

which shows Cherry-Lang-Wong’s upper bound of the error term of the Second Main Theorem in Nevanlinna theory is essentially the best possible. Thus a question of Serge Lang is answered affirmatively in higher dimensions.
New formulae for the Bernoulli and Euler polynomials at rational arguments
Authors: Cvijović Djurdje , Klinowski Jacek
Published: June 2010

We prove theorems on the values of the Bernoulli polynomials

B
n

(
x
)

{B_n}(x)

with

n
=
2
,
3
,
4
,
…

n = 2,3,4, \ldots

, and the Euler polynomials

E
n

(
x
)

{E_n}(x)

with

n
=
1
,
2
,
3
,
…

n = 1,2,3, \ldots

for

0
≤
x
≤
1

0 \leq x \leq 1

where x is rational.

B
n

(
x
)

{B_n}(x)

and

E
n

(
x
)

{E_n}(x)

are expressible in terms of a finite combination of trigonometric functions and the Hurwitz zeta function

ζ
(
z
,
α
)

\zeta (z,\alpha )

. The well-known argument-addition formulae and reflection property of

B
n

(
x
)

{B_n}(x)

and

E
n

(
x
)

{E_n}(x)

, extend this result to any rational argument. We also deduce new relations concerning the finite sums of the Hurwitz zeta function and sum some classical trigonometric series.
Remarks on the topology of folds
Authors: Kikuchi Shigeki , Saeki Osamu
Published: June 2010

We give some necessary conditions for a closed manifold to admit a smooth map into an Euclidean space with only fold singular points.
On some ideals of nest algebras
Authors: Hou Cheng Jun , Han De Guang
Published: June 2010

The purpose of this paper is to characterize the relations of some ideals of a nest algebra such as the Jacobson radical

R

N

{R_\mathcal {N}}

, Larson strong radical

R

N

∞

R_\mathcal {N}^\infty

, and ideal

G

\mathcal {G}

, and to describe all the ideals between

R

N

{R_\mathcal {N}}

and

G

\mathcal {G}

.
Algebraic and triangular 𝑛-hyponormal operators
Authors: Ko Eungil
Published: June 2010

In this paper we shall prove that if an operator

T
∈

L

(

⊕
1
n

H

)

T \in \mathcal {L}( \oplus _1^n{\mathbf {H}})

is a finite triangular operator matrix with hyponormal operators on main diagonal, then T is subscalar. As corollaries we get the following: (1) Every algebraic operator is subscalar. (2) Every operator on a finite-dimensional complex space is subscalar. (3) Every triangular n-hyponormal operator is subscalar.
The boundary integral equation method in plane elasticity
Authors: Constanda Christian
Published: June 2010

The boundary integral equation method in terms of real variables is applied to solve the interior and exterior Dirichlet and Neumann problems of plane elasticity. In the exterior case, a special far-field pattern for the displacements is considered, without which the classical scheme fails to work. The connection between the results obtained by means of this technique and those of the direct method is indicated.
Uniqueness of positive solutions of nonlinear second-order equations
Authors: Dalmasso Robert
Published: June 2010

In this paper we study the uniqueness question of positive solutions of the two-point boundary value problem:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis t right-parenthesis plus f left-parenthesis StartAbsoluteValue t EndAbsoluteValue comma u left-parenthesis t right-parenthesis right-parenthesis equals 0 comma negative upper R greater-than t greater-than upper R comma u left-parenthesis plus-or-minus upper R right-parenthesis equals 0">

u
(
t
)
+
f
(

|

t

|

,
u
(
t
)
)
=
0
,
−
R
>
t
>
R
,
u
(
±
R
)
=
0

u(t) + f(|t|,u(t)) = 0, - R > t > R,u( \pm R) = 0

where

R
>
0

R > 0

is fixed and

f
:
[
0
,
R
]
×
[
0
,
∞
)
→

R

f:[0,R] \times [0,\infty ) \to \mathbb {R}

is in

C
1

(
[
0
,
R
]
×
[
0
,
∞
)
)

{C^1}([0,R] \times [0,\infty ))

. A uniqueness result is proved when f satisfies some appropriate conditions. Some examples illustrating our theorem are also given.
Normalizing elements in PI rings
Authors: Guralnick Robert M. , Robson J. Chris , Small Lance W.
Published: June 2010

This paper explores the question: If R is a prime PI ring and a an element such that

a
R
⊇
R
a

aR \supseteq Ra

, is it true that

a
R
=
R
a

aR = Ra

?
Almost periodicity in semiflows
Authors: Seifert George
Published: June 2010

For a semiflow on a complete metric space we show that if a semiorbit has compact closure and its positive limit set consists of stable points, the orbit is asymptotically almost periodic. Under stronger hypotheses which are more tractable in applications, the same conclusion holds. An application to a scalar delay-differential equation is given.
On the compactness of the evolution operator generated by certain nonlinear Ω-accretive operators in general Banach spaces
Authors: Kartsatos Athanassios G.
Published: June 2010

A sufficient condition is given for the compactness of the evolution operator

U
(
t
,
s
)

U(t,s)

generated by a family of nonlinear

ω
\omega

-accretive operators

A
(
t
)

A(t)

. This family

A
(
t
)

A(t)

satisfies a time-dependence condition which is not covered by the results of Calvert and the author. It is also shown that the main part of this sufficient condition is necessary.
Sums and products of Hilbert spaces
Authors: Castillo Jesús M. F.
Published: June 2010

Let

H
H

be a Hilbert space. We prove that the locally convex sum

⊕
I

H

{ \oplus _I}H

is a subspace of the product

H
J

{H^J}

if and only if

I
I

is countable,

H
H

is infinite dimensional, and card

J
≥

2

χ
0

J \geq {2^{{\chi _0}}}

.
Convolutions with kernels having singularities on a sphere
Authors: Strichartz Robert S.
Published: June 2010

We prove that convolution with

(
1
−

|

x

|

2

)
+

−
α

(1 - |x{|^2})_ + ^{ - \alpha }

and related convolutions are bounded from

L
p

{L^p}

to

L
q

{L^q}

for certain values of p and q. There is a unique choice of p which maximizes the measure of smoothing

1

/

p
−
1

/

q

1/p - 1/q

, in contrast with fractional integration where

1

/

p
−
1

/

q

1/p - 1/q

is constant. We apply the results to obtain a priori estimates for solutions of the wave equation in which we sacrifice one derivative but gain more smoothing than in Sobolev’s inequality.
Recursive functions modulo 𝐶𝑂-𝑟-maximal sets
Authors: Lerman Manuel
Published: June 2010

Define the equivalence relation

∼
A

{ \sim _A}

on the set of recursive functions of one variable by

f

∼
A

g

f\sim _A g

if and only if

f
(
x
)
=
g
(
x
)

f(x) = g(x)

for all but finitely many

x
∈

A
¯

x \in \bar A

, where

A
¯

\bar A

is an r-cohesive set, to obtain the structure

R

/

A
¯

\mathcal {R}/\bar A

. Then the recursive functions modulo such an equivalence relation form a semiring with no zero divisors. It is shown that if A is r-maximal, then the structure obtained above is not a nonstandard model for arithmetic, a result due to Feferman, Scott, and Tennenbaum. Furthermore, if A and B are maximal sets, then a necessary and sufficient condition for

R

/

A
¯

\mathcal {R}/\bar A

and

R

/

B
¯

\mathcal {R}/\bar B

to be elementarily equivalent is obtained. It is also shown that many different elementary theories can be obtained for

R

/

A
¯

\mathcal {R}/\bar A

by proper choice of

A
¯

\bar A

.
Existence and stability of a class of nonlinear Volterra integral equations
Authors: Grossman Stanley I.
Published: June 2010

In this paper we study the problem of existence and uniqueness to solutions of the nonlinear Volterra integral equation

x
=
f
+

a
1

g
1

(
x
)
+
⋯
+

a
n

g
n

(
x
)

x = f + {a_1}{g_1}(x) + \cdots + {a_n}{g_n}(x)

, where the

a
i

{a_i}

are continuous linear operators mapping a Fréchet space

F

\mathcal {F}

into itself and the

g
i

{g_i}

are nonlinear operators in that space. Solutions are sought which lie in a Banach subspace of

F

\mathcal {F}

having a stronger topology. The equations are studied first when the

g
i

{g_i}

are of the form

g
i

(
x
)
=
x
+

h
i

(
x
)

{g_i}(x) = x + {h_i}(x)

where

h
i

(
x
)

{h_i}(x)

is “small", and then when the

g
i

{g_i}

are slope restricted. This generalizes certain results in recent papers by Miller, Nohel, Wong, Sandberg, and Beneš.
Monotone approximation by algebraic polynomials
Authors: Lorentz G. G. , Zeller K. L.
Published: June 2010

A given real continuous function f on [a, b] is approximated by polynomials

P
n

{P_n}

of degree n that are subject to certain restrictions. Let

1
≦

k
1

>
⋯
>

k
p

≦
n

1 \leqq {k_1} > \cdots > {k_p} \leqq n

be given integers,

ε
i

=
±
1

{\varepsilon _i} = \pm 1

, given signs. It is assumed that

P
n

(

k
i

)

(
x
)

P_n^{({k_i})}(x)

has the sign of

ε
i

,
i
=
1
,
…
,
p
,
a
≦
x
≦
b

{\varepsilon _i},i = 1, \ldots ,p,a \leqq x \leqq b

. Theorems are obtained which describe the polynomials of best approximation, and (for

p
=
1

p = 1

) establish their uniqueness. Relations to Birkhoff interpolation problems are of importance. Another tool are the sets A, where

|

f
(
x
)
−

P
n

(
x
)

|

|f(x) - {P_n}(x)|

attains its maximum, and the sets

B
i

{B_i}

with

P
n

(

k
i

)

(
x
)
=
0

P_n^{({k_i})}(x) = 0

. Conditions are discussed which these sets must satisfy for a polynomial

P
n

{P_n}

of best approximation for f. Numbers of the points of sets A,

B
i

{B_i}

are studied, the possibility of certain extreme situations established. For example, if

p
=
1
,

k
1

=
1
,
n
=
2
q
+
1

p = 1,{k_1} = 1,n = 2q + 1

, it is possible that

|

A

|

=
3
,

|

B

|

=
n

|A| = 3,|B| = n

.
Invariant states
Authors: Herman Richard H.
Published: June 2010

States of a

C
∗

{C^ \ast }

-algebra invariant under the action of a group of automorphisms of the

C
∗

{C^ \ast }

-algebra are considered. It is shown that “clustering” states in the same part are equal and thus the same is true of extremal invariant states under suitable conditions. The central decomposition of an invariant state is considered and it is shown that the central measure is mixing if and only if the state satisfies a strong notion of clustering. Under transitivity of the central measure and some reasonable restrictions, the central decomposition is the ergodic decomposition of the state with respect to the isotropy subgroup.
Inner product modules arising from compact automorphism groups of von Neumann algebras
Authors: Paschke William L.
Published: June 2010

Let M be a von Neumann algebra of operators on a separable Hilbert space H, and G a compact, strong-operator continuous group of

∗

^\ast

-automorphisms of M. The action of G on M gives rise to a faithful, ultraweakly continuous conditional expectation of M on the subalgebra

N
=
{
A
∈
M
:
g
(
A
)
=
A
∀
g
∈
G
}

N = \{ A \in M:g(A) = A\forall g \in G\}

, which in turn makes M into an inner product module over N. The inner product module M may be “completed” to yield a self-dual inner product module

M
¯

\bar M

over N; our most general result states that the

W
∗

{W^\ast }

-algebra

A
(

M
¯

)

A(\bar M)

of bounded N-module maps of

M
¯

\bar M

into itself is isomorphic to a restriction of the crossed product

M
×
G

M \times G

of M by G. When G is compact abelian, we give conditions for

A
(

M
¯

)

A(\bar M)

and

M
×
G

M \times G

to be isomorphic and show, among other things, that if G acts faithfully on M, then

M
×
G

M \times G

is a factor if and only if N is a factor. As an example, we discuss certain compact abelian automorphism groups of group von Neumann algebras.
Admissible and singular translates of measures on vector spaces
Authors: Gleit Alan , Zinn Joel
Published: June 2010

We provide a general setting for studying admissible and singular translates of measures on linear spaces. We apply our results to measures on

D
[
0
,
1
]

D[0,1]

. Further, we show that in many cases convex, balanced, bounded, and complete subsets of the admissible translates are compact. In addition, we generalize Sudakov’s theorem on the characterization of certain quasi-invariant sets to separable reflexive spaces which have the Central Limit Property.
Pointwise and norm convergence of a class of biorthogonal expansions
Authors: Benzinger Harold E.
Published: June 2010

Let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace u Subscript k Baseline left-parenthesis x right-parenthesis right-brace comma left-brace v Subscript k Baseline left-parenthesis x right-parenthesis right-brace comma k equals 0 comma plus-or-minus 1 comma ellipsis comma 0 less-than-or-slanted-equals x less-than-or-slanted-equals 1">{uk(x)},{vk(x)},k=0,±1,…,0⩽x⩽1\{ {u_k}(x)\} ,\{ {v_k}(x)\} ,k = 0, \pm 1, \ldots ,0 \leqslant x \leqslant 1, be sequences of functions inL∞(0,1){L^\infty }(0,1), such that(uk,vj)=δkj({u_k},{v_j}) = {\delta _{kj}}. Letϕk(x)=exp2kπix{\phi _k}(x) = \exp \;2k\pi ix. It is shown that if for a givenp,1>p>∞1 > p > \infty, the sequence{uk}\{ {u_k}\}is complete inLp(0,1){L^p}(0,1), and{vk}\{ {v_k}\}is complete inLq(0,1),pq=p+q{L^q}(0,1),pq = p + q, and if theuk{u_k}’s,vj{v_j}’s are asymptotically related to theϕk{\phi _k}’s, in a sense to be made precise, then{uk}\{ {u_k}\}is a basis forLp(0,1){L^p}(0,1), equivalent to the basis{ϕk}\{ {\phi _k}\}, and for everyfinLp(0,1){L^p}(0,1)a.e. This result is then applied to the eigenfunction expansions of a large class of ordinary differential operators.
Inequalities for polynomials on the unit interval
Authors: Rahman Q. I. , Schmeisser G.
Published: June 2010

Let

p
n

(
z
)
=

∑

k
=
0

n

a
k

z
k

{p_n}(z) = \sum \nolimits _{k = 0}^n {{a_k}{z^k}}

be a polynomial of degree at most n with real coefficients. Generalizing certain results of I. Schur related to the well-known inequalities of Chebyshev and Markov we prove that if

p
n

(
z
)

{p_n}(z)

has at most

n
−
1

n - 1

distinct zeros in

(
−
1
,
1
)

( - 1,1)

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row StartAbsoluteValue a Subscript n Baseline EndAbsoluteValue less-than-or-slanted-equals 2 Superscript n minus 1 Baseline left-parenthesis cosine StartFraction pi Over 4 n EndFraction right-parenthesis Superscript 2 n Baseline max Underscript negative 1 less-than-or-slanted-equals x less-than-or-slanted-equals 1 Endscripts StartAbsoluteValue p Subscript n Baseline left-parenthesis x right-parenthesis EndAbsoluteValue comma 2nd Row max Underscript negative 1 less-than-or-slanted-equals x less-than-or-slanted-equals 1 Endscripts StartAbsoluteValue p prime Subscript n Baseline left-parenthesis x right-parenthesis EndAbsoluteValue less-than-or-slanted-equals left-parenthesis n cosine StartFraction pi Over 4 n EndFraction right-parenthesis squared max Underscript negative 1 less-than-or-slanted-equals x less-than-or-slanted-equals 1 Endscripts StartAbsoluteValue p Subscript n Baseline left-parenthesis x right-parenthesis EndAbsoluteValue period EndLayout">

|

a
n

|

⩽

2

n
−
1

(

cos
⁡

π

4
n

)

2
n

max

−
1
⩽
x
⩽
1

|

p
n

(
x
)

|

,

max

−
1
⩽
x
⩽
1

|

p
′

n

(
x
)

|

⩽

(

n
cos
⁡

π

4
n

)

2

max

−
1
⩽
x
⩽
1

|

p
n

(
x
)

|

.

\begin {array}{*{20}{c}} {|{a_n}| \leqslant {2^{n - 1}}{{\left ( {\cos \frac {\pi }{{4n}}} \right )}^{2n}}\max \limits _{ - 1 \leqslant x \leqslant 1} |{p_n}(x)|,} \\ {\max \limits _{ - 1 \leqslant x \leqslant 1} |{{p’}_n}(x)| \leqslant {{\left ( {n\cos \frac {\pi }{{4n}}} \right )}^2}\max \limits _{ - 1 \leqslant x \leqslant 1} |{p_n}(x)|.} \\ \end {array}

\]
A class of infinitely connected domains and the corona
Authors: Deeb W. M.
Published: June 2010

Let D be a bounded domain in the complex plane. Let

H
∞

(
D
)

{H^\infty }(D)

be the Banach algebra of bounded analytic functions on D. The corona problem asks whether D is

weak
∗

\text {weak}^\ast

dense in the space

M

(
D
)

\mathfrak {M}(D)

of maximal ideals of

H
∞

(
D
)

{H^\infty }(D)

. Carleson [3] proved that the open unit disc

Δ
0

{\Delta _0}

is dense in

M

(

Δ
0

)

\mathfrak {M}({\Delta _0})

. Stout [9] extended Carleson’s result to finitely connected domains. Behrens [2] found a class of infinitely connected domains for which the corona problem has an affirmative answer. In this paper we will use Behrens’ idea to extend the results to more general domains. See [11] for further extensions and applications of these techniques.
𝐷-domains and the corona
Authors: Deeb W. M. , Wilken D. R.
Published: June 2010

Let D be a bounded domain in the complex plane C. Let

H
∞

(
D
)

{H^\infty }(D)

denote the usual Banach algebra of bounded analytic functions on D. The Corona Conjecture asserts that D is

weak
∗

\text {weak}^\ast

dense in the space

M

(
D
)

\mathfrak {M}(D)

of maximal ideals of

H
∞

(
D
)

{H^\infty }(D)

. In [2] Carleson proved that the unit disk

Δ
0

{\Delta _0}

is dense in

M

(

Δ
0

)

\mathfrak {M}({\Delta _0})

. In [7] Stout extended Carleson’s result to finitely connected domains. In [4] Gamelin showed that the problem is local. In [1] Behrens reduced the problem to very special types of infinitely connected domains and established the conjecture for a large class of such domains. In this paper we extract some of the crucial ingredients of Behrens’ methods and extend his results to a broader class of infinitely connected domains.
On the degree of convergence of piecewise polynomial approximation on optimal meshes
Authors: Burchard H. G.
Published: June 2010

The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F-spaces

N

0

p
,
n

{\mathbf {N}}_0^{p,n}

(introduced for this purpose in [2]). We obtain two o-results and use pairs of inequalities of Bernstein- and Jackson-type to prove several direct and converse theorems. For f in

N

0

p
,
n

{\mathbf {N}}_0^{p,n}

we define a derivative

D

n
,
σ

f

{D^{n,\sigma }}f

in

L
σ

,
σ
=

(
n
+

p

−
1

)

−
1

{L^\sigma },\sigma = {(n + {p^{ - 1}})^{ - 1}}

, which agrees with

D
n

f

{D^n}f

for smooth f, and prove several properties of

D

n
,
σ

{D^{n,\sigma }}

.
On one-dimensional coupled Dirac equations
Authors: Glassey R. T.
Published: June 2010

The Cauchy Problem for Dirac equations coupled through scalar and Fermi interactions is considered in one space dimension. Global solutions of finite energy are shown to exist, provided that either the magnitude of the coupling constant or the

L
2

(

R
1

)

{L_2}({R^1})

-norm of the initial data is suitably restricted.
Functional calculus and positive-definite functions
Authors: Graham Colin C.
Published: June 2010

For a LCA group G with dual group Ĝ, let

D
(
G
)
=
D
(

G
^

)

D(G) = D(\hat G)

denote the convex (not closed) hull of

{
⟨
x
,
γ
⟩
:
x
∈
G
,
γ
∈

G
^

}

\{ \langle x,\gamma \rangle :x \in G,\gamma \in \hat G\}

. The set

D
(
G
)

D(G)

is the natural domain for functions that operate by composition from the class,

P

D
1

(

G
^

)

P{D_1}(\hat G)

, of Fourier-Stieltjes transforms of probability measures on G to

B
(

G
^

)

B(\hat G)

, the class of all Fourier-Stieltjes transforms on Ĝ. Little is known about the behavior of F on the boundary of

D
(
G
)

D(G)

. In §1, we show (1) if F operators from

P

D
1

(
G
)

P{D_1}(G)

to

B
(
G
)

B(G)

and G is compact, then

K
(
z
)
=

lim

t
→

1
−

F
(
t
z
)

K(z) = {\lim _{t \to {1^ - }}}F(tz)

exists for all

z
∈
D
(
G
)

z \in D(G)

and K operates from

P

D
1

(

G
^

)

P{D_1}(\hat G)

to

B
(

G
^

)

B(\hat G)

; (2) if F operates from

P

D
1

(

G
^

)

P{D_1}(\hat G)

to

P
D
(

G
^

)
=

∪

r
>
0

r
P

D
1

(

G
^

)

PD(\hat G) = { \cup _{r > 0}}rP{D_1}(\hat G)

and G is compact, then K operates from

P

D
1

(

G
^

)

P{D_1}(\hat G)

to

P
D
(

G
^

)

PD(\hat G)

, and so also does

F
−
K

F - K

; (3) if

G
=

D

q

,
q
⩾
2

G = {{\mathbf {D}}_q},q \geqslant 2

, and F operates from

P

D
1

(

G
^

)

P{D_1}(\hat G)

to

B
(

G
^

)

B(\hat G)

, then

F
=
K

F = K

on

D
(
G
)
∩
{
z
:

|

z

|

>
1
}

D(G) \cap \{ z:|z| > 1\}

. This third result is shown to be sharp for compact groups of bounded order. In §2, an example is given that fills a gap in the theory of functions operating from

P

D
1

(

G
^

)

P{D_1}(\hat G)

to

B
(

G
^

)

B(\hat G)

. In §3 we show that most Riesz products and all continuous measures on K-sets have a property that is very useful in proving symbolic calculus theorems. Applications of this are indicated. Some open questions are given in §4.
On tails and domains of attraction of stable measures in Banach spaces
Authors: Araujo Aloisio , Giné Evarist
Published: June 2010

The exact tail behavior of stable measures in Banach spaces and measures in their domains of attraction is given. Conditions for a p.m. to be in the domain of attraction of a stable p.m. of order

α
\alpha

are derived which are sufficient in type p spaces,

p
>
α

p > \alpha

, and necessary in general. This paper also contains a short proof of the Lévy-Khinchin formula in Banach spaces.
Distribution of zeros of orthogonal polynomials
Authors: Nevai Paul G.
Published: June 2010

The purpose of the paper is to investigate distribution of zeros of orthogonal polynomials given by a three term recurrence relation.
Maximal subspaces of Besov spaces invariant under multiplication by characters
Authors: Johnson R.
Published: June 2010

Unlike the familiar

L
p

{L^p}

spaces, neither the homogeneous Besov spaces nor the

H
p

{H^p}

spaces,

0

>

p

>

1

0\, > \,p\, > \,\,1

, are closed under multiplication by the functions

x

→

e

i

⟨

x
,
h

⟩

x\, \to \,{e^{i\left \langle {x,h} \right \rangle }}

. We determine the maximal subspace of these spaces which are closed under multiplication by these functions, which are the characters of

R
n

{R^n}

.
On the zeros of Jacobi polynomials 𝑃_{𝑛}^{(𝛼_{𝑛},𝛽_{𝑛})}(𝑥)
Authors: Moak D. S. , Saff E. B. , Varga R. S.
Published: June 2010

If

r
n

{r_n}

and

s
n

{s_n}

denote, respectively, the smallest and largest zeros of the Jacobi polynomial

P
n

(

α
n

,

β
n

)

P_n^{({\alpha _n},{\beta _n})}

, where

α
n

>

1

{\alpha _n}\, > \,1

,

β
n

−

1

{\beta _n}\, - \,1

, and if

lim

n
→
∞

α
n

/

(
2
n

+

α
n

+

β
n

+

1
)

=

a

{\lim _{n \to \infty }}\,{\alpha _n}/(2n\, + \,{\alpha _n}\, + \,{\beta _n}\, + \,1)\, = \,a

and if

lim

n
→
∞

β
n

/

(
2
n

+

α
n

+

β
n

+

1
)

=

b

{\lim _{n \to \infty }}{\beta _n}/(2n\, + \,{\alpha _n}\, + \,{\beta _n}\, + \,1)\, = \,b

, then the numbers

r

a
,
b

{r_{a,b}}

and

s

a
,
b

{s_{a,b}}

are determined where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts r Subscript n Baseline equals r Subscript a comma b Baseline comma limit Underscript n right-arrow normal infinity Endscripts s Subscript n Baseline equals s Subscript a comma b Baseline">

lim

n
→
∞

r

n

=

r

a
,
b

,

lim

n
→
∞

s

n

=

s

a
,
b

\lim \limits _{n \to \infty } \,{r_{n\,}}\, = \,{r_{a,b}},\lim \limits _{n \to \infty } \,{s_{n\,}}\, = \,{s_{a,b}}

\]
. Furthermore, the zeros of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper P Subscript n Superscript left-parenthesis alpha Super Subscript n Superscript comma beta Super Subscript n Superscript right-parenthesis Baseline left-parenthesis x right-parenthesis right-brace Subscript n equals 0 Superscript normal infinity">

{

P
n

(

α
n

,

β
n

)

(
x
)

}

n
=
0

∞

\{ P_n^{({\alpha _n},{\beta _n})}(x)\} _{n = 0}^\infty

are dense in

[

r

a
,
b

,

s

a
,
b

]

[{r_{a,b}},{s_{a,b}}]

.
Rotundity in Lebesgue-Bochner function spaces
Authors: Smith Mark A. , Turett Barry
Published: June 2010

This paper concerns the isometric theory of the Lebesgue-Bochner function space

L
p

(
μ
,

X
)

{L^p}(\mu ,\,X)

where

1
>
p
>
∞

1 > p > \infty

. Specifically, the question of whether a geometrical property lifts from X to

L
p

(
μ
,

X
)

{L^p}\,(\mu ,\,X)

is examined. Positive results are obtained for the properties local uniform rotundity, weak uniform rotundity, uniform rotundity in each direction, midpoint local uniform rotundity, and B-convexity. However, it is shown that the Radon-Riesz property does not lift from X to

L
p

(
μ
,

X
)

{L^p}\,(\mu ,\,X)

. Consequently, Lebesgue-Bochner function spaces with the Radon-Riesz property are examined more closely.
Homeomorphisms of 𝑆³ leaving a Heegaard surface invariant
Authors: Powell Jerome
Published: June 2010

We find a finite set of generators for the group

H

g

{\mathcal {H} _g}

of isotopy classes of orientation-preserving homeomorphisms of the 3-sphere

S
3

{S^3}

which leave a Heegaard surface T of genus g in

S
3

{S^3}

invariant. We also show that every element of the group

H

g

{\mathcal {H} _g}

can be represented by a deformation of the surface T in

S
3

{S^3}

of a very special type: during the deformation the surface T is the boundary of the regular neighborhood of a graph embedded in a fixed 2-sphere. The only exception occurs when a subset of the graph contained in a disc on the 2-sphere is “flipped over."
Diophantine sets over algebraic integer rings. II
Authors: Denef J.
Published: June 2010

We prove that Z is diophantine over the ring of algebraic integers in any totally real number field or quadratic extension of a totally real number field.
Constructing Smale diffeomorphisms on compact surfaces
Authors: Batterson Steve
Published: June 2010

A necessary condition for an isotopy class on a compact surface to admit a Smale diffeomorphism whose dynamics are a specified set of subshifts of finite type is that the Euler characteristic of the manifold be equal to a sum and difference of certain numbers obtained from the matrices representing the subshifts. In this paper it is shown that this condition is sufficient up to a finite power of the subshifts.
On maximally elliptic singularities
Authors: Yau Stephen Shing Toung
Published: June 2010

Let p be the unique singularity of a normal two-dimensional Stein space V. Let m be the maximal ideal in

V

O

p

_V{\mathcal {O}_p}

, the local ring of germs of holomorphic functions at p. We first define the maximal ideal cycle which serves to identify the maximal ideal. We give an “upper” estimate for maximal ideal cycle in terms of the canonical divisor which is computable via the topological information, i.e., the weighted dual graph of the singularity. Let

M
→
V

M \to V

be a resolution of V. It is known that

h

=

dim

H
1

(
M
,

O

)

h\, = \,\dim \,{H^1}(M,\,\mathcal {O})

is independent of resolution. Rational singularities in the sense of M. Artin are equivalent to

h

=

0

h\, = \,0

. Minimally elliptic singularity in the sense of Laufer is equivalent to saying that

h

=

1

h\, = \,1

and

V

O

p

_V{\mathcal {O}_p}

is Gorenstein. In this paper we develop a theory for a general class of weakly elliptic singularities which satisfy a maximality condition. Maximally elliptic singularities may have h arbitrarily large. Also minimally elliptic singlarities are maximally elliptic singularities. We prove that maximally elliptic singularities are Gorenstein singularities. We are able to identify the maximal ideal. Therefore, the important invariants of the singularities (such as multiplicity) are extracted from the topological information. For weakly elliptic singularities we introduce a new concept called “elliptic sequence". This elliptic sequence is defined purely topologically, i.e., it can be computed explicitly via the intersection matrix. We prove that —K, where K is the canonical divisor, is equal to the summation of the elliptic sequence. Moreover, the analytic data

dim

H
1

(
M
,

O

)

\dim \,{H^1}(M,\,\mathcal {O})

is bounded by the topological data, the length of elliptic sequence. We also prove that

h

=

2

h\, = \,2

and

V

O

p

_V{\mathcal {O}_p}

Gorenstein implies that the singularity is weakly elliptic.
The lattice of 𝑙-group varieties
Authors: Smith J. E.
Published: June 2010

For any type of abstract algebra, a variety is an equationally defined class of such algebras. Recently, attempts have been made to study varieties of lattice-ordered groups (l-groups). Martinez has shown that the set L of all l-group varieties forms a lattice under set inclusion with a compatible associative multiplication. Certain varieties

S

p

{\mathcal {S}_p}

(p prime) have been proved by Scrimger to be minimal nonabelian varieties in L. In the present paper, it is shown that these varieties can be used to produce varieties minimal with respect to properly containing various other varieties in L. Also discussed are the relations among the

S

n

(
n

∈

N
)

{\mathcal {S}_n}\,(n\, \in \,N)

, and it is established that all infinite collections of the

S

n

{\mathcal {S}_n}

have the same least upper bound in L. Martinez has also classified l-groups using torsion classes, a generalization of the idea of varieties. It is proved here that L is not a sublattice of T, the lattice of torsion classes.
On the cohomology of real Grassmanians
Authors: Hiller Howard L.
Published: June 2010

LetGk(Rn+k){G_k}({\textbf {R}^{n + k}})denote the grassman manifold ofk-planes in real(n+k)(n\, +\, k)-space andw1∈H1(Gk(Rn+k);Z2){w_1}\, \in \, {H^1}({G_k}({\textbf {R}^{n + k}});\,{\textbf {Z}_2})the first Stiefel-Whitney class of the universal bundle. Using Schubert calculus techniques and the cohomology of flag manifolds we estimate the height ofw1{w_1}in the cohomology ring. We then apply this to improve earlier lower bounds on the Lusternik-Schnirelmann category of real grassmanians.
Convolution equations in spaces of infinite-dimensional entire functions of exponential and related types
Authors: Colombeau J.-F. , Perrot B.
Published: June 2010

We prove results of existence and approximation of the solutions of the convolution equations in spaces of entire functions of exponential type on infinite dimensional spaces. In particular we obtain: let E be a complex, quasi-complete and dual nuclear locally convex space and

Ω
\Omega

a convex balanced open subset of E; let

H

(
Ω
)

\mathcal {H} (\Omega )

be the space of the holomorphic functions on

Ω
\Omega

, equipped with the compact open topology and

H

′

(
Ω
)

\mathcal {H}’(\Omega )

its strong dual; let

F

H

′

(
Ω
)

\mathcal {F} \mathcal {H}’(\Omega )

denote the image of

H

′

(
Ω
)

\mathcal {H}’(\Omega )

through the Fourier-Borel transform

F

\mathcal {F}

; equip this space

F

H

′

(
Ω
)

\mathcal {F} \mathcal {H}’(\Omega )

with the image of the topology of

H

′

(
Ω
)

\mathcal {H}’(\Omega )

via the map

F

\mathcal {F}

. Then, “every nonzero convolution operator on

F

H

′

(
Ω
)

\mathcal {F} \mathcal {H}’(\Omega )

is surjective” and “every solution of the homogeneous equation is limit of exponential-polynomial solutions". Our results are more generally valid when E is a Schwartz bornological vector space with the approximation property. Previous results in Fréchet-Schwartz and Silva spaces are thus extended to domains that are not Fréchet or D.F.-spaces.
Minimal excessive measures and functions
Authors: Dynkin E. B.
Published: June 2010

Let H be a class of measures or functions. An element h of H is minimal if the relation

h

=

h
1

+

h
2

h\, = \,{h_1}\, + \,{h_2}

,

h
1

{h_1}

,

h
2

∈
H

{h_2} \in H

implies that

h
1

{h_1}

,

h
2

{h_2}

are proportional to h. We give a limit procedure for computing minimal excessive measures for an arbitrary Markov semigroup

T
t

{T_t}

in a standard Borel space E. Analogous results for excessive functions are obtained assuming that an excessive measure

γ
\gamma

on E exists such that

T
t

f

=
0

{T_t}f\, = 0

if

f

=

0

f\, = \,0

γ
\gamma

-a.e. In the Appendix, we prove that each excessive element can be decomposed into minimal elements and that such a decomposition is unique.
The Witt ring of a space of orderings
Authors: Marshall Murray
Published: June 2010

The theory of “space of orderings” generalizes the reduced theory of quadratic forms over fields (or, more generally, over semilocal rings). The category of spaces of orderings is equivalent to a certain category of “abstract Witt rings". In the particular case of the space of orderings of a formally real field K, the corresponding abstract Witt ring is just the reduced Witt ring of K. In this paper it is proved that if

X

=

(
X
,

G
)

X\, = \,(X,\,G)

is any space of orderings with Witt ring W(X), and

X

→

Z

X\, \to \,Z

is any continuous function, then g is represented by an element of W(X) if and only if

Σ

σ

∈

V

g
(
σ
)

≡

0

mod

|
V
|

{\Sigma _{\sigma \, \in \,V}}g(\sigma )\, \equiv \,0\,\bmod \,\left | V \right |

holds for all finite fans

V
⊆
X

V \subseteq X

. This generalizes a recent field theoretic result of Becker and Bröcker. Following the proof of this, applications are given to the computation of the stability index of X, and to the representation of continuous functions

g
:

X

→

±
1

g:\,X\, \to \, \pm 1

by elements of G.
The group of rational solutions of 𝑦²=𝑥(𝑥-1)(𝑥-𝑡²-𝑐)
Authors: Schwartz Charles F.
Published: June 2010

In this paper, we show that the Mordell-Weil group of the Weierstrass equation

y
2

=

x
(
x

−

1
)
(
x

−

t
2

−

c
)
,

c
≠

0
,

1

{y^2}\, = \,x(x\, - \,1)(x\, - \,{t^2}\, - \,c),\,c \ne \,0,\,1

(i.e., the group of solutions (x,y), with

x
,

y

∈

C

(
t
)

x,\,y\, \in \,{\textbf {C}}(t)

) is generated by its elements of order 2, together with one element of infinite order, which is exhibited.
Some examples of sequence entropy as an isomorphism invariant
Authors: Dekking F. M.
Published: June 2010

With certain geometrically diverging sequences A and the shift T on dynamical systems arising from substitutions we associate a Markov shift S such that the A-entropy of T equals the usual entropy of S. We present examples to demonstrate the following results. Sequence entropy can distinguish between an invertible ergodic transformation and its inverse. A-entropy does not depend monotonically on A. The variational principle for topological sequence entropy need not hold.
Some categorical equivalences for 𝐸-unitary inverse semigroups
Authors: Petrich Mario
Published: June 2010

The structure of E-unitary inverse semigroups has been described by McAlister and by Reilly and the author. The parameters in the first structure theorem may be made into a category, and the same holds for the parameters in the second structure theorem. We prove that each of these categories is equivalent to the category of E-unitary inverse semigroups and their homomorphisms. We also provide functors between the two first-mentioned categories which are naturally equivalent to the composition of the functors figuring in the categorical equivalence referred to above.
Stability theorems for holomorphic foliations
Authors: Duchamp T. , Kalka M.
Published: June 2010

Here we investigate topological stability in the space of holomorphic foliations on a compact manifold. We show that under certain conditions nearby holomorphic foliations are topologically equivalent. We then present examples of foliations which are stable as holomorphic foliations but unstable as smooth foliations.
Median algebra
Authors: Isbell John R.
Published: June 2010

A study of algebras with a ternary operation

(
x
,

y
,

z
)

(x,\,y,\,z)

satisfying some identities, equivalent to embeddability in a lattice with

(
x
,

y
,

z
)

(x,\,y,\,z)

realized as, simultaneously,

(
x

∧

(
y

∨

z
)
)

∨

(
y

∧

z
)

(x\, \wedge \,(y\, \vee \,z))\, \vee \,(y\, \wedge \,z)

and

(
x

∨

(
y

∧

z
)
)

∧

(
y

∨

z
)

(x\, \vee \,(y\, \wedge \,z))\, \wedge \,(y\, \vee \,z)

. This is weaker than embeddability in a modular lattice, where those expressions coincide for all x, y, and z, but much of the theory survives the extension. For actual embedding in a modular lattice, some necessary conditions are found, and the investigation is carried much further in a special, geometrically described class of examples ("2-cells"). In distributive lattices

(
x
,

y
,

z
)

(x,\,y,\,z)

reduces to the median

(
x

∧

y
)

∨

(
x

∧

z
)

∨

(
y

∧

z
)

(x\, \wedge \,y)\, \vee \,(x\, \wedge \,z)\, \vee \,(y\, \wedge \,z)

, previously studied by G. Birkhoff and S. Kiss. It is shown that Birkhoff and Kiss found a basis for the laws; indeed, their algebras are embeddable in distributive lattices, i.e. in powers of the 2-element lattice. Their theory is much further developed and is connected into an explicit Pontrjagin-type duality.
Nonexistence of continuous selections of the metric projection for a class of weak Chebyshev spaces
Authors: Sommer Manfred
Published: June 2010

The class

B

\mathfrak {B}

of all those n-dimensional weak Chebyshev subspaces of

C

[
a
,

b
]

C\,[a,\,b]

whose elements have no zero intervals is considered. It is shown that there does not exist any continuous selection of the metric projection for G if there is a nonzero g in G having at least

n

+

1

n\, + \,1

distinct zeros. Together with a recent result of Nürnberger-Sommer, the following characterization of continuous selections for

B

\mathfrak {B}

is valid: There exists a continuous selection of the metric projection for G in

B

\mathfrak {B}

if and only if each nonzero g in G has at most n distinct zeros.
On the ranges of analytic functions
Authors: Hwang J. S.
Published: June 2010

Following Doob, we say that a function

f
(
z
)

f(z)

analytic in the unit disk U has the property

K
(
ρ
)

K(\rho )\,

if

f
(
0
)

=

0

f(0)\, = \,0

and for some

arc

A

\operatorname {arc} \,A\,

on the unit circle whose measure

|
A
|

⩾

2
ρ

>

0

\left | A \right |\, \geqslant \,2\rho \, > \,0

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit inf Underscript i right-arrow normal infinity Endscripts StartAbsoluteValue f left-parenthesis upper P Subscript i Baseline right-parenthesis EndAbsoluteValue greater-than-or-slanted-equals 1 where upper P Subscript i Baseline right-arrow upper P element-of upper A and upper P Subscript i Baseline element-of upper U period">

lim

inf

i
→
∞

|

f
(

P
i

)

|

⩾

1

where

P
i

→

P

∈

A

and

P
i

∈

U
.

\lim \,\inf \limits _{i \to \infty } \,\left | {f({P_i})} \right |\, \geqslant \,1\,{\text {where}}\,{P_i}\, \to \,P\, \in \,A\,{\text {and}}\,{P_i}\, \in \,U.

\]
We recently have solved a problem of Doob by showing that there is an integer

N
(
ρ
)

N(\rho )

such that no function with the property

K
(
ρ
)

K(\rho )

can satisfy
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue right-parenthesis StartAbsoluteValue f prime Subscript n Baseline left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-slanted-equals 1 slash n for z element-of upper U comma where n greater-than upper N left-parenthesis rho right-parenthesis period">

(
1

−

|
z
|

)

|

f
n

′

(
z
)

|

⩽

1

/

n

for

z

∈

U
,

where

n

>

N
(
ρ
)
.

(1\, - \,\left | z \right |)\left | {{f_n}’ (z)} \right |\, \leqslant \,1/n\,{\text {for}}\,z\, \in \,U,\,{\text {where}}\,n\, > \,N(\rho ).

\]
The function
\[

f
n

(
z
)

=

1

+

(
1

−

z
n

)

/

n
2

,

{f_n}(z)\, = \,1\, + \,(1\, - \,{z^n})/{n^2},

\]
shows that the condition

f
n

(
0
)

=

0

{f_n}(0)\, = \,0

is necessary and cannot be replaced by

f
n

(
0
)

=

r

e

i
α

{f_n}(0)\, = \,r{e^{i\alpha }}

, for

r

>

1

r\, > \,1

. Naturally, we may ask whether this can be replaced by

f
n

(
0
)

=

r

e

i
α

{f_n}(0)\, = \,r{e^{i\alpha }}

, for

r

>

1

r\, > \,1

? The answer turns out to be yes, when

n

>

N

(
r
,

ρ
)

n\, > \,N\,(r,\,\rho )

, where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis r comma rho right-parenthesis geometrically-equals left-parenthesis 1 slash left-parenthesis 1 minus r right-parenthesis right-parenthesis log left-parenthesis 1 slash left-parenthesis 1 minus cosine rho right-parenthesis right-parenthesis period">

N
(
r
,

ρ
)

≑

(
1

/

(
1

−

r
)
)
log
⁡
(
1

/

(
1

−

cos
⁡
ρ
)
)
.

N(r,\,\rho )\,\doteqdot \,(1/(1\, - \,r))\log (1/(1\, - \,\cos \rho )).

\]
.
Topological spaces with prescribed nonconstant continuous mappings
Authors: Trnková Věra
Published: June 2010

Given a

T
1

{T_1}

-space Y and a

T
3

{T_3}

-space V, consider

T
3

{T_3}

-spaces X such that X has a closed covering by spaces homeomorphic to V and any continuous mapping

f
:

X
→
Y

f:\,X \to Y

is constant. All such spaces and all their continuous mappings are shown to form a very comprehensive category, containing, e.g., a proper class of spaces without nonconstant, nonidentical mappings or containing a space X, for every monoid M, such that all the nonconstant continuous mappings of X into itself are closed under composition and form a monoid isomorphic to M. The category of paracompact connected spaces, having a closed covering by a given totally disconnected paracompact space, has, e.g., analogous properties. Categories of metrizable spaces are also investigated.
Undecidability and definability for the theory of global fields
Authors: Rumely R. S.
Published: June 2010

We prove that the theory of global fields is essentially undecidable, using predicates based on Hasse’s Norm Theorem to define valuations. Polynomial rings or the natural numbers are uniformly defined in all global fields, as well as Gödel functions encoding finite sequences of elements.
Deformation theory and the tame fundamental group
Authors: Harbater David
Published: June 2010

Let U be a curve of genus g with

n

+

1

n\, + \,1

points deleted, defined over an algebraically closed field of characteristic

p

⩾

0

p\, \geqslant \,0

. Then there exists a bijection between the Galois finite étale covers of U having degree prime to p, and the finite

p
′

p’

-groups on

n

+

2
g

n\, + \,2g

generators. This fact has been proven using analytic considerations; here we construct such a bijection algebraically. We do this by algebraizing an analytic construction of covers which uses Hurwitz families. The process of algebraization relies on a deformation theorem, which we prove using Artin’s Algebraization Theorem, and which allows the patching of local families into global families. That our construction provides the desired bijection is afterwards verified analytically.
Nonstandard analysis and lattice statistical mechanics: a variational principle
Authors: Hurd A. E.
Published: June 2010

Using nonstandard methods we construct a configuration space appropriate for the statistical mechanics of lattice systems with infinitely many particles and infinite volumes. Nonstandard representations of generalized equilibrium measures are obtained, yielding as a consequence a simple proof of the existence of standard equilibrium measures. As another application we establish an extension for generalized equilibrium measures of the basic variational principle of Landord and Ruelle. The same methods are applicable to continuous systems, and will be presented in a later paper.
Multivariate rearrangements and Banach function spaces with mixed norms
Authors: Blozinski A. P.
Published: June 2010

Multivariate nonincreasing rearrangement and averaging functions are defined for functions defined over product spaces. An investigation is made of Banach function spaces with mixed norms and using multivariate rearrangements. Particular emphasis is given to the

L
(
P
,
Q
;
∗
)

L(P,Q;\ast )

spaces. These are Banach function spaces which are in terms of mixed norms, multivariate rearrangements and the Lorentz

L
(
p
,
g
)

L(p,g)

spaces. Embedding theorems are given for the various function spaces. Several well-known theorems are extended to the

L
(
P
,
Q
;
∗
)

L(P,Q;\ast )

spaces. Principal among these are the Strong Type (Riesz-Thorin) Interpolation Theorem and the Convolution (Young’s inequality) Theorem.