bluuebunny/crossref_metadata_2025 huggingface dataset:
On the representation of lattices by modules
Authors: Hutchinson George
Published: July 2010

For a commutative ring

R
R

with unit, a lattice

L
L

is “representable by

R
R

-modules” if

L
L

is embeddable in the lattice of submodules of some unitary left

R
R

-module. A procedure is given for generating an infinite first-order axiomatization of the class of all lattices representable by

R
R

-modules. Each axiom is a universal Horn formula for lattices. The procedure for generating the axioms is closely related to the ring structure, and is “effective” in the sense that many nontrivial axioms can be obtained by moderate amounts of computation.
Similarity of quadratic forms and isomorphism of their function fields
Authors: Wadsworth Adrian R.
Published: July 2010

This paper considers the question: Given anisotropic quadratic forms

Q
Q

and

Q
′

Q’

over a field

K
K

(char

K
≠
2

K \ne 2

), if their function fields are isomorphic must

Q
Q

and

Q
′

Q’

be similar? It is proved that the answer is yes if

Q
Q

is a Pfister form or the pure part of a Pfister form, or a

4
4

-dimensional form. The argument for Pfister forms and their pure parts does not generalize because these are the only anisotropic forms which attain maximal Witt index over their function fields. To handle the

4
4

-dimensional case the following theorem is proved: If

Q
Q

and

Q
′

Q’

are two

4
4

-dimensional forms over

K
K

with the same determinant

d
d

, then

Q
Q

and

Q
′

Q’

are similar over

K
K

iff they are similar over

K
[

d

]

K[\sqrt d ]

. The example of Pfister neighbors suggests that quadratic forms arguments are unlikely to settle the original question for other kinds of forms.
A generalization of Jarník’s theorem on Diophantine approximations to Ridout type numbers
Authors: Borosh I. , Fraenkel A. S.
Published: July 2010

Let s be a positive integer,

c
>
1
,

μ
0

,
…
,

μ
s

c > 1,{\mu _0}, \ldots ,{\mu _s}

reals in [0, 1],

σ
=

Σ

i
=
0

s

μ
i

\sigma = \Sigma _{i = 0}^s\;{\mu _i}

, and t the number of nonzero

μ
i

{\mu _i}

. Let

Π
i

(
i
=
0
,
…
,
s
)

{\Pi _i}\;(i = 0, \ldots ,s)

be

s
+
1

s + 1

disjoint sets of primes and S the set of all

(
s
+
1
)

(s + 1)

-tuples of integers

(

p
0

,
…
,

p
s

)

({p_0}, \ldots ,{p_s})

satisfying

p
0

>
0
,

p
i

=

p
i
∗

p
i
′

{p_0} > 0,{p_i} = p_i^\ast {p’_i}

, where the

p
i
∗

p_i^\ast

are integers satisfying

|

p
i
∗

|

≤
c

|

p
i

|

μ
i

|p_i^\ast | \leq c|{p_i}{|^{{\mu _i}}}

, and all prime factors of

p
i
′

{p’_i}

are in

Π
i

,
i
=
0
,
…
,
s

{\Pi _i},i = 0, \ldots ,s

. Let

λ
>
0

\lambda > 0

if

t
=
0
,
λ
>
σ

/

min
(
s
,
t
)

t = 0,\lambda > \sigma /\min (s,t)

otherwise,

E
λ

{E_\lambda }

the set of all real s-tuples

(

α
1

,
…
,

α
s

)

({\alpha _1}, \ldots ,{\alpha _s})

satisfying

|

α
i

−

p
i

/

p
0

|

>

p
0

−
λ

(
i
=
1
,
…
,
s
)

|{\alpha _i} - {p_i}/{p_0}| > p_0^{ - \lambda }\;(i = 1, \ldots ,s)

for an infinite number of

(

p
0

,
…
,

p
s

)
∈
S

({p_0}, \ldots ,{p_s}) \in S

. The main result is that the Hausdorff dimension of

E
λ

{E_\lambda }

is

σ

/

λ

\sigma /\lambda

. Related results are obtained when also lower bounds are placed on the

p
i
∗

p_i^\ast

. The case

s
=
1

s = 1

was settled previously (Proc. London Math. Soc. 15 (1965), 458-470). The case

μ
i

=
1

(
i
=
0
,
…
,
s
)

{\mu _i} = 1\;(i = 0, \ldots ,s)

gives a well-known theorem of Jarník (Math. Z. 33 (1931), 505-543).
The subgroups of a tree product of groups
Authors: Fischer J.
Published: July 2010

Let

G
=

Π
∗

(

A
i

;

U

j
k

=

U

k
j

)

G = {\Pi ^ \ast }({A_i};{U_{jk}} = {U_{kj}})

be a tree product with

H
H

a subgroup of

G
G

. By extending the technique of using a rewriting process we show that

H
H

is an HNN group whose base is a tree product with vertices of the form

x

A
i

x

−
1

∩
H

x{A_i}{x^{ - 1}} \cap H

. The associated subgroups are contained in vertices of the base, and both the associated subgroups of

H
H

and the edges of its base are of the form

y

U

j
k

y

−
1

∩
H

y{U_{jk}}{y^{ - 1}} \cap H

. The

x
x

and

y
y

are certain double coset representatives for

G
mod
(
H
,

A
i

)

G\bmod (H,{A_i})

and

G
mod
(
H
,

U

j
k

)

G\bmod (H,{U_{jk}})

, respectively, and the elements defined by the free part of

H
H

are specified. More precise information about

H
H

is given when

H
H

is either indecomposable or

H
H

satisfies a nontrivial law. Introducing direct tree products, we use our subgroup theorem to prove that if each edge of

G
G

is contained in the center of its two vertices then the cartesian subgoup of

G
G

is a free group. We also use our subgroup theorem in proving that if each edge of

G
G

is a finitely generated subgroup of finite index in both of its vertices and some edge is a proper subgroup of both its vertices then

G
G

is a finite extension of a free group iff the orders of the

A
i

{A_i}

are uniformly bounded.
Functions of vanishing mean oscillation
Authors: Sarason Donald
Published: July 2010

A function of bounded mean oscillation is said to have vanishing mean oscillation if, roughly speaking, its mean oscillation is locally small, in a uniform sense. In the present paper the class of functions of vanishing mean oscillation is characterized in several ways. This class is then applied to answer two questions in analysis, one involving stationary stochastic processes satisfying the strong mixing condition, the other involving the algebra

H
∞

+
C

{H^\infty } + C

.
Product integral techniques for abstract hyperbolic partial differential equations
Authors: Spellmann J. W.
Published: July 2010

Explicit and implicit product integral techniques are used to represent a solution

U
U

to the abstract system:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U 12 left-parenthesis x comma y right-parenthesis equals upper A upper U left-parenthesis x comma y right-parenthesis semicolon upper U left-parenthesis x comma 0 right-parenthesis equals p equals upper U left-parenthesis 0 comma y right-parenthesis">

U

12

(
x
,
y
)
=
A
U
(
x
,
y
)
;
U
(
x
,
0
)
=
p
=
U
(
0
,
y
)

{U_{12}}(x,y) = AU(x,y);U(x,0) = p = U(0,y)

. The coefficient

A
A

is a closed linear transformation defined on a dense subspace

D
(
A
)

D(A)

of the Banach space

X
X

and the point

p
p

in

D
(
A
)

D(A)

satisfies the condition that

|

|

A
i

p

|

|

>

S
i

(
i
!

)

3

/

2

||{A^i}p|| > {S^i}{(i!)^{3/2}}

for all integers

i
⩾
0

i \geqslant 0

and some

S
>
0

S > 0

. The implicit technique is developed under the additional assumption that

A
A

generates a strongly continuous semigroup of bounded linear transformations on

X
X

. Both methods provide representations for the

J
0

{J_0}

Bessel function.
Unions of Hilbert cubes
Authors: Wong Raymond Y. T. , Kroonenberg Nelly
Published: July 2010

This paper gives a partial solution to the problem whether the union of two Hilbert cubes is a Hilbert cube if the intersection is a Hilbert cube and a Z-set in one of them. Our results imply West’s Intermediate Sum Theorem on Hilbert cube factors. Also a technique is developed to obtain Z-sets as limits of Z-sets.
The uniqueness of the one-dimensional paraboson field
Authors: Robbins Steven
Published: July 2010

A paraboson analog of the one-dimensional boson field is discussed and a uniqueness result similar to a result of Putnam is obtained. It is shown that the paraboson operators must be unbounded.
The Boolean space of orderings of a field
Authors: Craven Thomas C.
Published: July 2010

It has been pointed out by Knebusch, Rosenberg and Ware that the set

X
X

of all orderings on a formally real field can be topologized to make a Boolean space (compact, Hausdorff and totally disconnected). They have called the sets of orderings

W
(
a
)
=
{
>

 in 

X

|

a
>
0
}

W(a) = \{ > {\text { in }}X|a > 0\}

the Harrison subbasis of

X
X

. This subbasis is closed under symmetric difference and complementation. In this paper it is proved that, given any Boolean space

X
X

, there exists a formally real field

F
F

such that

X
X

is homeomorphic to the space of orderings on

F
F

. Also, an example is given of a Boolean space and a basis of clopen sets closed under symmetric difference and complementation which cannot be the Harrison subbasis of any formally real field.
The structure of inseparable field extensions
Authors: Waterhouse William C.
Published: July 2010

The goal of this paper is to introduce some structural ideas into the hitherto chaotic subject of infinite inseparable field extensions. The basic discovery is that the theory is closely related to the well-developed study of primary abelian groups. This analogy undoubtedly has implications beyond those included here. We consider only modular extensions, which are the inseparable equivalent of galois extensions. §§2 and 3 develop the theory of pure independence, basic subfields, and tensor products of simple extensions. The following sections are devoted to Ulm invariants and their computation; the existence of nonzero invariants of arbitrary index is proved by means of a theorem which furnishes an actual connection between primary groups and inseparable fields. The final section displays some complications in the field extensions not occurring in abelian groups.
Finite groups with Sylow 2-subgroups of class two. I
Authors: Gilman Robert , Gorenstein Daniel
Published: July 2010

In this paper we classify finite simple groups whose Sylow

2
2

-subgroups have nilpotency class two.
On Kōmura’s closed-graph theorem
Authors: Powell Michael H.
Published: July 2010

Let

(
α
)

(\alpha )

be a property of separated locally convex spaces. Call a locally convex space

E
[

J

]

E[\mathcal {J}]

an

(

α
¯

)

(\bar \alpha )

-space if

J

\mathcal {J}

is the final topology defined by

{

u
i

:

E
i

[

J

i

]
→
E

}

i
∈
I

{\{ {u_i}:{E_i}[{\mathcal {J}_i}] \to E\} _{i \in I}}

, where each

E
i

[

J

i

]

{E_i}[{\mathcal {J}_i}]

is an

(
α
)

(\alpha )

-space. Then, for each locally convex space

E
[

J

]

E[\mathcal {J}]

, there is a weakest

(

α
¯

)

(\bar \alpha )

-topology on E stronger that

J

\mathcal {J}

, denoted

J

α
¯

{\mathcal {J}^{\bar \alpha }}

. Kōmura’s closed-graph theorem states that the following statements about a locally convex space

E
[

J

]

E[\mathcal {J}]

are equivalent: (1) For every

(
α
)

(\alpha )

-space F and every closed linear map

u
:
F
→
E
[

J

]

u: F \to E[\mathcal {J}]

, u is continuous. (2) For every separated locally convex topology

J

0

{\mathcal {J}_0}

on E, weaker than

J

\mathcal {J}

, we have

J

⊂

J

0

α
¯

\mathcal {J} \subset \mathcal {J}_0^{\bar \alpha }

. Much of this paper is devoted to amplifying Kōmura’s theorem in special cases, some well-known, others not. An entire class of special cases, generalizing Adasch’s theory of infra-(s) spaces, is established by considering a certain class of functors, defined on the category of locally convex spaces, each functor yielding various notions of “completeness” in the dual space.
On 𝐽-convexity and some ergodic super-properties of Banach spaces
Authors: Brunel Antoine , Sucheston Louis
Published: July 2010

Given two Banach spaces

F

|

|

F||

and

X

|

|

|

|

X||\,||

, write

F

 fr 

X

 iff

F{\text { fr }}X{\text { iff}}

for each finite-dimensional subspace

F
′

F’

of

F
F

and each number

ε
>
0

\varepsilon > 0

, there is an isomorphism

V
V

of

F
′

F’

into

X
X

such that

|

|

x

|

−

|

|

V
x

|

|

|

≤
ε

||x| - ||Vx||| \leq \varepsilon

for each

x
x

in the unit ball of

F
′

F’

. Given a property

P

{\mathbf {P}}

of Banach spaces,

X
X

is called super-

P

 iff 

F

 fr 

X

{\mathbf {P}}{\text { iff }}F{\text { fr }}X

implies

F
F

is

P

{\mathbf {P}}

. Ergodicity and stability were defined in our articles On

B
B

-convex Banach spaces, Math. Systems Theory 7 (1974), 294-299, and C. R. Acad. Sci. Paris Ser. A 275 (1972), 993, where it is shown that super-ergodicity and super-stability are equivalent to super-reflexivity introduced by R. C. James [Canad. J. Math. 24 (1972), 896-904].

Q
Q

-ergodicity is defined, and it is proved that super-

Q
Q

-ergodicity is another property equivalent with super-reflexivity. A new proof is given of the theorem that

J
J

-spaces are reflexive [Schaffer-Sundaresan, Math. Ann. 184 (1970), 163-168]. It is shown that if a Banach space

X
X

is

B
B

-convex, then each bounded sequence in

X
X

contains a subsequence

(

y
n

)

({y_n})

such that the Cesàro averages of

(
−
1

)
i

y
i

{( - 1)^i}{y_i}

converge to zero.
On the generation of one-relator groups
Authors: Pride Stephen J.
Published: July 2010

This paper is concerned with obtaining information about the Nielsen equivalence classes and

T
T

-systems of certain two-generator HNN groups, and in particular of certain two-generator one-relator groups. The theorems presented here extend results of the author appearing in the Proceedings of the Second International Conference on the Theory of Groups. In particular it is shown here that if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals mathematical left-angle a comma t semicolon a Superscript alpha 1 Baseline upper E Subscript r Superscript negative 1 Baseline a Superscript beta 1 Baseline upper E Subscript r Baseline midline-horizontal-ellipsis a Superscript a Super Subscript s Superscript Baseline upper E Subscript r Superscript negative 1 Baseline a Superscript upper B Super Subscript s Superscript Baseline upper E Subscript r Baseline right-parenthesis">

G
=
⟨
a
,
t
;

a

α
1

E
r

−
1

a

β
1

E
r

⋯

a

a
s

E
r

−
1

a

B
s

E
r

)

G = \langle a,t;{a^{{\alpha _1}}}E_r^{ - 1}{a^{{\beta _1}}}{E_r} \cdots {a^{{a_s}}}E_r^{ - 1}{a^{{B_s}}}{E_r})

where the

α
j

{\alpha _j}

are positive, the

β
i

{\beta _i}

are nonzero,

E
r

{E_r}

has the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket a Superscript element-of Super Subscript 1 Superscript Baseline comma left-bracket a Superscript element-of Super Subscript 2 Superscript Baseline comma left-bracket midline-horizontal-ellipsis comma left-bracket a Superscript element-of Super Subscript r Superscript Baseline comma t right-bracket midline-horizontal-ellipsis right-bracket right-bracket right-bracket">

[

a

∈
1

,
[

a

∈
2

,
[
⋯
,
[

a

∈
r

,
t
]
⋯
]
]
]

[{a^{{ \in _1}}},[{a^{{ \in _2}}},[ \cdots ,[{a^{{ \in _r}}},t] \cdots ]]]

with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue element-of Subscript 1 Baseline EndAbsoluteValue equals StartAbsoluteValue element-of Subscript 2 Baseline EndAbsoluteValue equals midline-horizontal-ellipsis equals StartAbsoluteValue element-of Subscript r Baseline EndAbsoluteValue equals 1">

|

∈
1

|

=

|

∈
2

|

=
⋯
=

|

∈
r

|

=
1

|{ \in _1}| = |{ \in _2}| = \cdots = |{ \in _r}| = 1

, then in a large number of cases

G
G

has one Nielsen equivalence class. Similar results are also obtained for certain groups with more than one relator. A fair proportion of the paper is given to developing a method for reducing pairs of elements in HNN groups. This method has some of the features of Nielsen’s reduction theorem for free groups. One other interesting result obtained here is that a one-relator group with torsion which has one

T
T

-system is Hopfian. The early part of the paper is discursive. It contains most of the known results concerning

T
T

-systems of one-relator groups, and highlights several open problems, some of which have been raised by other authors.
On differential rings of entire functions
Authors: Cayford A. H. , Straus E. G.
Published: July 2010

Consider an entire function

f
f

which is a solution of the differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket c 0 left-parenthesis z right-parenthesis plus c 1 left-parenthesis z right-parenthesis upper D plus ellipsis plus c Subscript m Baseline left-parenthesis z right-parenthesis upper D Superscript m Baseline right-bracket left-parenthesis f Superscript n Baseline right-parenthesis equals upper P left-parenthesis f comma f prime comma ellipsis comma f Superscript left-parenthesis k right-parenthesis Baseline right-parenthesis">

[

c
0

(
z
)
+

c
1

(
z
)
D
+
…
+

c
m

(
z
)

D
m

]
(

f
n

)
=
P
(
f
,

f
′

,
…
,

f

(
k
)

)

[{c_0}(z) + {c_1}(z)D + \ldots + {c_m}(z){D^m}]({f^n}) = P(f,f’, \ldots ,{f^{(k)}})

\]
where

c
i

(
z
)

{c_i}(z)

are entire functions in a differential ring

R
R

and

P
P

is a polynomial in a differential field related to

R
R

. We prove the following THEOREM. If

f
f

satisfies the equation above then

f
f

is of finite type in case

R
=

C

R = {\mathbf {C}}

and of finite exponential order in case

R
=

C

[
z
]

R = {\mathbf {C}}[z]

. We use this result to prove a conjecture made in [2] that entire functions of order

ρ
>
s

\rho > s

, all of whose derivatives at

s
s

points are integers in an imaginary quadratic number field, must be solutions of linear differential equations with constant coefficients and therefore of order

⩽
1

\leqslant 1

.
The zeros of holomorphic functions in strictly pseudoconvex domains
Authors: Gruman Lawrence
Published: July 2010

We determine a sufficient condition on a positive divisor in certain strictly pseudoconvex domains in

C

n

{{\mathbf {C}}^n}

such that there exists a function in the Nevanlinna class which determines the divisor.
On the variety of manifolds without conjugate points
Authors: Gulliver Robert
Published: July 2010

The longest geodesic segment in a convex ball of a riemannian manifold, where the convexity is ensured by an upper bound on sectional curvatures, is the diameter. This and related results are demonstrated and applied to show that there exist manifolds with sectional curvatures of both signs but with-out conjugate points.
Analytic continuation, envelopes of holomorphy, and projective and direct limit spaces
Authors: Carmignani Robert
Published: July 2010

For a Riemann domain

Ω
\Omega

, a connected complex manifold where

n
(
n
=
d
i
m
e
n
s
i
o
n
)

n(n = dimension)

globally defined functions form a local system of coordinates at every point, and an arbitrary holomorphic function

f
f

in

Ω
\Omega

, the “Riemann surface”

Ω
f

{\Omega _f}

, a maximal holomorphic extension Riemann domain for

f
f

, is formed from the direct limit of a sequence of Riemann domains. Projective limits are used to construct an envelope of holomorphy for

Ω
\Omega

, a maximal holomorphic extension Riemann domain for all holomorphic functions in

Ω
\Omega

, which is shown to be the projective limit space of the “Riemann surfaces”

Ω
f

{\Omega _f}

. Then it is shown that the generalized notion of envelope of holomorphy of an arbitrary subset of a Riemann domain can also be characterized in a natural way as the projective limit space of a family of “Riemann surfaces".
Convergence and divergence of series conjugate to a convergent multiple Fourier series
Authors: Ash J. Marshall , Gluck Lawrence
Published: July 2010

In this note we consider to what extent the classical theorems of Plessner and Kuttner comparing the set of convergence of a trigonometric series with that of the conjugate trigonometric series can be generalized to higher dimensions. We show that if a function belongs to

L
p

,
p
>
1

{L^p},p > 1

, of the

2
2

-torus, then the convergence (= unrestricted rectangular convergence) of the Fourier series on a set implies its three conjugate functions converge almost everywhere on that set. That this theorem approaches the best possible may be seen from two examples which show that the dimension may not be increased to 3, nor the required power of integrability be decreased to 1. We also construct a continuous function having a boundedly divergent Fourier series of power series type and an a.e. circularly convergent double Fourier series whose

y
y

-conjugate diverges circularly a.e. Our

L
p

{L^p}

result depends on a theorem of L. Gogöladze (our proof is included for the reader’s convenience), work of J. M. Ash and G. Welland on

(
C
,
1
,
0
)

(C,1,0)

summability, and on a result deducing the boundedness of certain partial linear means from convergence of those partial means. The construction of the counterexamples utilizes examples given by C. Fefferman, J. Marcinkiewicz, A. Zygmund, D. Menšov, and the present authors’ earlier work.
Interpolation and uniqueness results for entire functions
Authors: Child James D.
Published: July 2010

Let

K
[
Ω
]

K[\Omega ]

denote the collection of entire functions of exponential type whose Borel transforms are analytic on

Ω
c

{\Omega ^c}

(the complement of the simply connected domain taken relative to the sphere). Let

f
f

be in

K
[
Ω
]

K[\Omega ]

and set
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript n Baseline left-parenthesis f right-parenthesis equals left-parenthesis 2 pi i right-parenthesis Superscript negative 1 Baseline integral normal upper Gamma g Subscript n Baseline left-parenthesis lamda right-parenthesis upper F left-parenthesis lamda right-parenthesis d lamda left-parenthesis n equals 0 comma 1 comma ellipsis right-parenthesis">

L
n

(
f
)
=

(
2
π
i

)

−
1

∫
Γ

g
n

(
λ
)
F
(
λ
)
d
λ
(
n
=
0
,
1
,
…
)

{L_n}(f) = {(2\pi i)^{ - 1}}\smallint \Gamma {g_n}(\lambda )F(\lambda )d\lambda (n = 0,1, \ldots )

where

F
F

is the Borel transform of

f
,
Γ
⊂
Ω

f,\Gamma \subset \Omega

is a simple closed contour chosen so that

F
F

is analytic outside and on

Γ
\Gamma

and each

g
n

{g_n}

is in

H
(
Ω
)

H(\Omega )

(the collection of functions analytic on

Ω
\Omega

). In what follows read ’the sequence of linear functionals

{

L
n

(
f
)
}

\{ {L_n}(f)\}

’ wherever the sequence of functions ’

{

g
n

}

\{ {g_n}\}

’ appears. Let

T
T

denote a continuous linear operator from

H
(
Ω
)

H(\Omega )

to

H
(
Λ
)

H(\Lambda )

where

Λ
\Lambda

is also a simply connected domain. The topologies on

H
(
Ω
)

H(\Omega )

and

H
(
Λ
)

H(\Lambda )

are those of uniform convergence on compact subsets of

Ω
\Omega

(resp.

Λ
\Lambda

). The purpose of this paper is to consider uniqueness preserving operators, i.e., operators

T
T

which have the property that

K
[
Λ
]

K[\Lambda ]

is a uniqueness class for

{
T
(

g
n

)
}

\{ T({g_n})\}

whenever

K
[
Ω
]

K[\Omega ]

is a uniqueness class for

{

g
n

}

\{ {g_n}\}

, and to examine interpolation preserving operators, i.e., operators

T
T

which have the property that

K
[
Λ
]

K[\Lambda ]

interpolates the sequence of complex numbers

{

b
n

}

\{ {b_n}\}

relative to

{
T
(

g
n

)
}

\{ T({g_n})\}

whenever

K
[
Ω
]

K[\Omega ]

interpolates

{

b
n

}

\{ {b_n}\}

relatives to

{

g
n

}

\{ {g_n}\}

. Once some classes of uniqueness preserving operators and some classes of interpolation preserving operators have been found, we proceed to obtain new uniqueness and interpolation results from our knowledge of these operators and from previously known uniqueness and interpolation results. Operators which multiply by analytic functions and some differential operators are considered. Composition operators are studied and the results are used to extend the interpolation results for sequences of functions of the form

{

[
W
(
ζ
)

]
n

}

\{ {[W(\zeta )]^n}\}

where

W
W

is analytic and univalent on a simply connected domain.
On the structure of 𝑆 and 𝐶(𝑆) for 𝑆 dyadic
Authors: Hagler James
Published: July 2010

A dyadic space S is defined to be a continuous image of

{
0
,
1

}

m

{\{ 0,1\} ^\mathfrak {m}}

for some infinite cardinal number

m

\mathfrak {m}

. We deduce Banach space properties of

C
(
S
)

C(S)

and topological properties of S. For example, under certain cardinality restrictions on

m

\mathfrak {m}

, we show: Every dyadic space of topological weight

m

\mathfrak {m}

contains a closed subset homeomorphic to

{
0
,
1

}

m

{\{ 0,1\} ^\mathfrak {m}}

. Every Banach space X isomorphic to an

m

\mathfrak {m}

dimensional subspace of

C
(
S
)

C(S)

(for S dyadic) contains a subspace isomorphic to

l
1

(
Γ
)

{l^1}(\Gamma )

where

Γ
\Gamma

has cardinality

m

\mathfrak {m}

.
Inclusion maps of 3-manifolds which induce monomorphisms of fundamental groups
Authors: Vrabec Jože
Published: July 2010

The main result is the following “duality” theorem. Let M be a 3-manifold, P a compact and connected polyhedral 3-submanifold of

∫
M

\int M

, and X a compact and connected polyhedron in

∫
P

\int P

. If

π
1

(
X
)
→

π
1

(
P
)

{\pi _1}(X) \to {\pi _1}(P)

is onto, then

π
1

(
M
−
P
)
→

π
1

(
M
−
X
)

{\pi _1}(M - P) \to {\pi _1}(M - X)

is one-to-one. Some related results are proved, for instance: we can allow P to be noncompact if also X satisfies a certain noncompactness condition: if M lies in a 3-manifold W with

H
1

(
W
)
=
0

{H_1}(W) = 0

, then the condition that

π
1

(
X
)
→

π
1

(
P
)

{\pi _1}(X) \to {\pi _1}(P)

is onto can be replaced by the weaker one that

H
1

(
X
)
→

H
1

(
P
)

{H_1}(X) \to {H_1}(P)

is onto.
Polar sets and Palm measures in the theory of flows
Authors: Geman Donald , Horowitz Joseph
Published: July 2010

Given a flow

(

θ
t

)
,
t

({\theta _t}),t

real, over a probability space

Ω
\Omega

, we prove that certain measures on

Ω
\Omega

(viewed as the state space of the flow) decompose uniquely into a Palm measure

Q
Q

which charges no “polar set” and a measure supported by a polar set. Considering the continuous and discrete parts of the additive functional corresponding to

Q
Q

, we find that

Q
Q

further decomposes into a measure charging no “semipolar set” and a measure supported by one. As a consequence, Palm measures are exactly those which neglect sets which the flow neglects, and polar sets are exactly those neglected by every Palm measure. Finally, we characterize various properties, such as predictability and continuity, of an additive functional in terms of its Palm measure. These results further illuminate the role played by supermartingales in the theory of flows, as pointed by J. de Sam Lazaro and P. A. Meyer.
On stable noetherian rings
Authors: Papp Zoltán
Published: July 2010

A ring R is called stable if every localizing subcategory of

R

M

_R{\text {M}}

is closed under taking injective envelopes. In this paper the stable noetherian rings are characterized in terms of the idempotent kernel functors of

R

M

_R{\text {M}}

(O. Goldman [5]). The stable noetherian rings, the classical rings (Riley [11]) and the noetherian rings “with sufficiently many two-sided ideals” (Gabriel [4]) are compared and their relationships are studied. The close similarity between the commutative noetherian rings and the stable noetherian rings is also pointed out in the results.
Automorphisms of commutative rings
Authors: Kreimer H. F.
Published: July 2010

Let

B
B

be a commutative ring with 1, let

G
G

be a finite group of automorphisms of

B
B

, and let

A
A

be the subring of

G
G

-invariant elements of

B
B

. For any separable

A
A

-subalgebra

A
′

A’

of

B
B

, the following assertions are proved: (1)

A
′

A’

is a finitely generated, protective

A
A

-module; (2) for each prime ideal

p
p

of

A
A

, the rank of

A
p
′

{A’_p}

over

A
p

{A_p}

does not exceed the order of

G
G

; (3) there is a finite group

H
H

of automorphisms of

B
B

such that

A
′

A’

is the subring of

H
H

-invariant elements of

B
B

. If, in addition,

A
′

A’

is

G
G

-stable, then every automorphism of

A
′

A’

over

A
A

is the restriction of an automorphism of

B
B

, and

Hom
A

(

A
′

,

A
′

)

{\operatorname {Hom} _A}(A’,A’)

is generated as a left

A
′

A’

-module by those automorphisms of

A
′

A’

which are the restrictions of elements of

G
G

.
On rearrangements of Vilenkin-Fourier series which preserve almost everywhere convergence
Authors: Gosselin J. A. , Young W. S.
Published: July 2010

It is known that the partial sums of Vilenkin-Fourier series of

L
q

{L^q}

functions

(
q
>
1
)

(q > 1)

converge a.e. In this paper we establish the

L
2

{L^2}

result for a class of rearrangements of the Vilenkin-Fourier series, and the

L
q

{L^q}

result

(
1
>
q
>
2
)

(1 > q > 2)

for a subclass of rearrangements. In the case of the Walsh-Fourier series, these classes include the Kaczmarz rearrangement studied by L. A. Balashov. The

L
2

{L^2}

result for the Kaczmarz rearrangement was first proved by K. H. Moon. The techniques of proof involve a modification of the Carleson-Hunt method and estimates on maximal functions of the Hardy-Littlewood type that arise from these rearrangements.
Infinite convolutions on locally compact Abelian groups and additive functions
Authors: Hartman Philip
Published: July 2010

Let

μ
1

,

μ
2

,
…

{\mu _1},{\mu _2}, \ldots

be regular probability measures on a locally compact Abelian group G such that

μ
=

μ
1

∗

μ
2

∗
⋯
=
lim

μ
1

∗
⋯
∗

μ
n

\mu = {\mu _1} \ast {\mu _2} \ast \cdots = \lim {\mu _1} \ast \cdots \ast {\mu _n}

exists (and is a probability measure). For arbitrary G, we derive analogues of the Lévy theorem on the existence of an atom for

μ
\mu

and of the “pure theorems” of Jessen, Wintner and van Kampen (dealing with discrete

μ
1

,

μ
2

,
…

{\mu _1},{\mu _2}, \ldots

) in the case

G
=

R
d

G = {R^d}

. These results are applied to the asymptotic distribution

μ
\mu

of an additive function

f
:

Z
+

→
G

f:{Z_ + } \to G

after generalizing the Erdös-Wintner result

(
G
=

R
1

)

(G = {R^1})

which implies that

μ
\mu

is an infinite convolution of discrete probability measures.
Ergodic theorems for the asymmetric simple exclusion process
Authors: Liggett Thomas M.
Published: July 2010

Consider the infinite particle system on the integers with the simple exclusion interaction and one-particle motion determined by

p
(
x
,
x
+
1
)
=
p

p(x,x + 1) = p

and

p
(
x
,
x
−
1
)
=
q

p(x,x - 1) = q

for

x
∈
Z

x \in Z

, where

p
+
q
=
1

p + q = 1

and

p
>
q

p > q

. If

μ
\mu

is the initial distribution of the system, let

μ
t

{\mu _t}

be the distribution at time t. The main results determine the limiting behavior of

μ
t

{\mu _t}

as

t
→
∞

t \to \infty

for simple choices of

μ
\mu

. For example, it is shown that if

μ
\mu

is the pointmass on the configuration in which all sites to the left of the origin are occupied, while those to the right are vacant, then the system converges as

t
→
∞

t \to \infty

to the product measure on

{
0
,
1

}
Z

{\{ 0,1\} ^Z}

with density

1

/

2

1/2

. For the proof, an auxiliary process is introduced which is of interest in its own right. It is a process on the positive integers in which particles move according to the simple exclusion process, but with the additional feature that there can be creation and destruction of particles at one. Ergodic theorems are proved for this process also.
The stability problem in shape, and a Whitehead theorem in pro-homotopy
Authors: Edwards David A. , Geoghegan Ross
Published: July 2010

Theorem 3.1 is a Whitehead theorem in pro-homotopy for finite-dimensional pro-complexes. This is used to obtain necessary and sufficient algebraic conditions for a finite-dimensional tower of complexes to be pro-homotopy equivalent to a complex (§4) and for a finite-dimensional compact metric space to be pointed shape equivalent to an absolute neighborhood retract (§5).
Partition theorems related to some identities of Rogers and Watson
Authors: Connor Willard G.
Published: July 2010

This paper proves two general partition theorems and several special cases of each with both of the general theorems based on four q-series identities originally due to L. J. Rogers and G. N. Watson. One of the most interesting special cases proves that the number of partitions of an integer n into parts where even parts may not be repeated, and where odd parts occur only if an adjacent even part occurs is equal to the number of partitions of n into parts

≡
±
2
,
±
3
,
±
4
,
±
5
,
±
6
,
±
7

(
mod

2
)
0

\equiv \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 7 \pmod 20

. The companion theorem proves that the number of partitions of an integer n into parts where even parts may not be repeated, where odd parts

>
1

> 1

occur only if an adjacent even part occurs, and where 1’s occur arbitrarily is equal to the number of partitions of n into parts

≡
±
1
,
±
2
,
±
5
,
±
6
,
±
8
,
±
9

(
mod

2
)
0

\equiv \pm 1, \pm 2, \pm 5, \pm 6, \pm 8, \pm 9 \pmod 20

.
Teichmüller spaces and representability of functors
Authors: Engber Michael
Published: July 2010

The Teichmëller space of compact Riemann surfaces with punctures is exhibited as the object representing a certain functor. This extends the work of Grothendieck, who treated the unpunctured case. The relationship between the two cases is exploited to obtain specific results on the connection between the two Teichmüller spaces.
The theory of countable analytical sets
Authors: Kechris Alexander S.
Published: July 2010

The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd

n
n

a largest countable

Π
n
1

\Pi _n^1

set of reals,

C

n

{\mathcal {C}_n}

(this is also true for

n
n

even, replacing

Π
n
1

\Pi _n^1

by

Σ
n
1

\Sigma _n^1

and has been established earlier by Solovay for

n
=
2

n = 2

and by Moschovakis and the author for all even

n
>
2

n > 2

). The internal structure of the sets

C

n

{\mathcal {C}_n}

is then investigated in detail, the point of departure being the fact that each

C

n

{\mathcal {C}_n}

is a set of

Δ
n
1

\Delta _n^1

-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases,

ω
\omega

-models of analysis, higher-level analogs of the constructible universe, inductive definability, etc.
Newman’s theorem in the Riemannian category
Authors: Mann L. N. , Sicks J. L.
Published: July 2010

In 1931 M. H. A. Newman proved that if

M
M

is a connected topological manifold with metric

d
d

, there exists a number

ε
>
0

\varepsilon > 0

, depending only upon

M
M

and

d
d

, such that every compact Lie group

G
G

acting effectively on

M
M

has at least one orbit of diameter at least

ε
\varepsilon

. Aside from isolated results nothing appears to be known about

ε
\varepsilon

. In order to learn more about the invariant

ε
\varepsilon

, attention is restricted here to groups of isometries on a Riemannian manifold. It is found that the invariant

ε
\varepsilon

of

M
M

is connected with the notion of convexity introduced by J. H. C. Whitehead in 1932.
Localization and sheaf reflectors
Authors: Lambek J. , Rattray B. A.
Published: July 2010

Given a triple

(
S
,
η
,
μ
)

(S,\eta ,\mu )

on a category

A

\mathcal {A}

with equalizers, one can form a new triple whose functor

Q
Q

is the equalizer of

η
S

\eta S

and

S
η

S\eta

. Fakir has studied conditions for

Q
Q

to be idempotent, that is, to determine a reflective subcategory of

A

\mathcal {A}

. Here we regard

S
S

as the composition of an adjoint pair of functors and give several new such conditions. As one application we construct a reflector in an elementary topos

A

\mathcal {A}

from an injective object

I
I

, taking

S
=

I

I

(
−
)

S = {I^{{I^{( - )}}}}

. We show that this reflector preserves finite limits and that the sheaf reflector for a topology in

A

\mathcal {A}

can be obtained in this way. We also show that sheaf reflectors in functor categories can be obtained from a triple of the form

S
=

I

(
−
,
I
)

,
I

S = {I^{( - ,I)}},I

injective, which we studied in a previous paper. We deduce that the opposite of a sheaf subcategory of a functor category is tripleable over Sets.
A property for inverses in a partially ordered linear algebra
Authors: Dai Taen Yu , DeMarr Ralph
Published: July 2010

We consider a Dedekind

σ
\sigma

-complete partially ordered linear algebra A which has the following property: if

x
∈
A

x \in A

and

1
⩽
x

1 \leqslant x

, then

−
u
⩽

x

−
1

- u \leqslant {x^{ - 1}}

, where

u
=

u
2

u = {u^2}

. This property is used to show that A must be commutative. We also show that A is the direct sum of two algebras, each of which behaves like an algebra of real-valued functions.
Cell-like closed-0-dimensional decompositions of 𝑅³ are 𝑅⁴ factors
Authors: Edwards Robert D. , Miller Richard T.
Published: July 2010

It is proved that the product of a cell-like closed-0-dimensional upper semicontinuous decomposition of

R
3

{R^3}

with a line is

R
4

{R^4}

. This establishes at once this feature for all the various dogbone-inspired decompositions of

R
3

{R^3}

. The proof makes use of an observation of L. Rubin that the universal cover of a wedge of circles admits a 1-1 immersion into the wedge crossed with

R
1

{R^1}

.
Mean convergence of generalized Walsh-Fourier series
Authors: Young Wo Sang
Published: July 2010

Paley proved that Walsh-Fourier series converges in

L
p

(
1
>
p
>
∞
)

{L^p}(1 > p > \infty )

. We generalize Paley’s result to Fourier series with respect to characters of countable direct products of finite cyclic groups of arbitrary orders.
Adjoint abelian operators on 𝐿^{𝑝} and 𝐶(𝐾)
Authors: Fleming Richard J. , Jamison James E.
Published: July 2010

An operator A on a Banach space X is said to be adjoint abelian if there is a semi-inner product

[
⋅
,
⋅
]

[ \cdot , \cdot ]

consistent with the norm on X such that

[
A
x
,
y
]
=
[
x
,
A
y
]

[Ax,y] = [x,Ay]

for all

x
,
y
∈
X

x,y \in X

. In this paper we show that every adjoint abelian operator on

C
(
K
)

C(K)

or

L
p

(
Ω
,
Σ
,
μ
)

(
1
>
p
>
∞
,
p
≠
2
)

{L^p}(\Omega ,\Sigma ,\mu )\;(1 > p > \infty ,p \ne 2)

is a multiple of an isometry whose square is the identity and hence is of the form

A
x
(
⋅
)
=
λ
α
(
⋅
)
(
x
∘
ϕ
)
(
⋅
)

Ax( \cdot ) = \lambda \alpha ( \cdot )(x \circ \phi )( \cdot )

where

α
\alpha

is a scalar valued function with

α
(
⋅
)
α
∘
ϕ
(
⋅
)
=
1

\alpha ( \cdot )\alpha \circ \phi ( \cdot ) = 1

and

ϕ
\phi

is a homeomorphism of K (or a set isomorphism in case of

L
p

(
Ω
,
Σ
,
μ
)
)

{L^p}(\Omega ,\Sigma ,\mu ))

with

ϕ
∘
ϕ
=

\phi \circ \phi =

identity (essentially).
Boundary value problems for second order differential equations in convex subsets of a Banach space
Authors: Schmitt Klaus , Volkmann Peter
Published: July 2010

Let E be a real Banach space, C a closed, convex subset of E and

f
:
[
0
,
1
]
×
E
×
E
→
E

f:[0,1] \times E \times E \to E

be continuous. Let

u
0

,

u
1

∈
C

{u_0},{u_1} \in C

and consider the boundary value problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel u equals f left-parenthesis t comma u comma u Superscript prime Baseline right-parenthesis comma u left-parenthesis 0 right-parenthesis equals u 0 comma u left-parenthesis 1 right-parenthesis equals u 1 period EndLayout">

(

∗

)

u
=
f
(
t
,
u
,

u
′

)
,

u
(
0
)
=

u
0

,

u
(
1
)
=

u
1

.

\begin{equation}\tag {$\ast $} u = f(t,u,u’),\quad u(0) = {u_0},\quad u(1) = {u_1}.\end{equation}

We establish sufficient conditions in order that

(
∗
)

(\ast )

have a solution

u
:
[
0
,
1
]
→
C

u:[0,1] \to C

.
Lie group representations and harmonic polynomials of a matrix variable
Authors: Ton That Tuong
Published: July 2010

The first part of this paper deals with problems concerning the symmetric algebra of complex-valued polynomial functions on the complex vector space of n by k matrices. In this context, a generalization of the so-called “classical separation of variables theorem” for the symmetric algebra is obtained. The second part is devoted to the study of certain linear representations, on the above linear space (the symmetric algebra) and its subspaces, of the complex general linear group of order k and of its subgroups, namely, the unitary group, and the real and complex special orthogonal groups. The results of the first part lead to generalizations of several well-known theorems in the theory of group representations. The above representation, of the real special orthogonal group, which arises from the right action of this group on the underlying vector space (of the symmetric algebra) of matrices, possesses interesting properties when restricted to the Stiefel manifold. The latter is defined as the orbit (under the action of the real special orthogonal group) of the n by k matrix formed by the first n row vectors of the canonical basis of the k-dimensional real Euclidean space. Thus the last part of this paper is involved with questions in harmonic analysis on this Stiefel manifold. In particular, an interesting orthogonal decomposition of the complex Hilbert space consisting of all square-integrable functions on the Stiefel manifold is also obtained.
Piecewise monotone interpolation and approximation with Muntz polynomials
Authors: Passow Eli , Raymon Louis , Shisha Oved
Published: July 2010

The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of “Muntz polynomials” on a real interval is demonstrated: (i) approximation of a continuous function by a “copositive” Muntz polynomial; (ii) approximation of a continuous function by a “comonotone” Muntz polynomial; (iii) approximation of a continuous function with a monotone kth difference by a Muntz polynomial with a monotone kth derivative; (iv) interpolation by piecewise monotone Muntz polynomials—i. e., polynomials that are monotone on each of the intervals determined by the points of interpolation. The strong interrelationship of these problems is shown implicitly in the proofs.
Equivariant bordism and Smith theory. IV
Authors: Stong R. E.
Published: July 2010

This paper analyzes two types of characteristic numbers defined for manifolds with

Z
4

{Z_4}

action, showing their relation and that neither suffices to detect

Z
4

{Z_4}

equivariant bordism. This extends work of Bix who had given examples not detected by one type of number.
On the trivial extension of equivalence relations on analytic spaces
Authors: Takijima Kunio , Suzuki Tetsutaro
Published: July 2010

In this paper, we shall consider the problem: let X be a (reduced) analytic space and A a nowhere dense analytic set in X. And let R be a proper equivalence relation on A such that the quotient space

A

/

R

A/R

is an analytic space, and

R
~

\tilde R

the trivial extension of R to X. Then, is

X

/

R
~

X/\tilde R

an analytic space? To this, we have three sufficient conditions. Moreover, using this result we shall extend Satz 1 of H. Kerner [8].
Classification theory of abelian groups. I. Balanced projectives
Authors: Warfield R. B.
Published: July 2010

We introduce in this paper a class of Abelian groups which includes the torsion totally projective groups and those torsion-free groups which are direct sums of groups of rank one. Characterizations of the groups in this class are given, and a complete classification theorem, in terms of additive numerical invariants, is proved.
On bounded functions satisfying averaging conditions. II
Authors: Cahill Rotraut Goubau
Published: July 2010

Let

S
(
f
)

S(f)

denote the subspace of

L
∞

(

R
n

)

{L^\infty }({R^n})

consisting of those real valued functions f for which
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript r right-arrow 0 Endscripts StartFraction 1 Over StartAbsoluteValue upper B left-parenthesis x comma r right-parenthesis EndAbsoluteValue EndFraction integral Underscript upper B left-parenthesis x comma r right-parenthesis Endscripts f left-parenthesis y right-parenthesis d y equals f left-parenthesis x right-parenthesis">

lim

r
→
0

1

|

B
(
x
,
r
)

|

∫

B
(
x
,
r
)

f
(
y
)
d
y
=
f
(
x
)

\lim \limits _{r \to 0} \frac {1}{{|B(x,r)|}} {\int } _{B(x,r)}f(y)dy = f(x)

\]
for all x in

R
n

{R^n}

and let

L
(
f
)

L(f)

be the subspace of

S
(
f
)

S(f)

consisting of the approximately continuous functions. A number of results concerning the existence of functions in

S
(
f
)

S(f)

and

L
(
f
)

L(f)

with special properties are obtained. The extreme points of the unit balls of both spaces are characterized and it is shown that

L
(
f
)

L(f)

is not a dual space. As a preliminary step, it is shown that if E is a

G
δ

{G_\delta }

set of measure 0 in

R
n

{R^n}

, then the complement of E can be decomposed into a collection of closed sets in a particularly useful way.
𝑇 measure of Cartesian product sets. II
Authors: Ernst Lawrence R.
Published: July 2010

It is proven that there exists a subset A of Euclidean 2-space such that the 2-dimensional T measure of the Cartesian product of an interval of unit length and A is less than the 1-dimensional T measure of A. In a previous paper it was shown that there exists a subset of Euclidean 2-space such that the reverse inequality holds. T measure is the first measure of its type for which it has been shown that both of these relations are possible.
The generalized Fredholm operators
Authors: Yang Kung Wei
Published: July 2010

Let X, Y be Banach spaces over either the real field or the complex field. A continuous linear operator will be called a generalized Fredholm operator if

T
(
X
)

T(X)

is closed in Y, and Ker T and Coker T are reflexive Banach spaces. A theory similar to the classical Fredholm theory exists for the generalized Fredholm operators; and the similarity brings out the correspondence: Reflexive Banach spaces

↔
\leftrightarrow

finite-dimensional spaces, weakly compact operators

↔
\leftrightarrow

compact operators, generalized Fredholm operators

↔
\leftrightarrow

Fredholm operators, Tauberian operators with closed range

↔
\leftrightarrow

semi-Fredholm operators.
Sufficient conditions for an operator-valued Feynman-Kac formula
Authors: Grady Michael Dale
Published: July 2010

Let E be a locally compact, second countable Hausdorff space and let

X
(
t
)

X(t)

be a Markov process with state space E. Sufficient conditions are given for the existence of a solution to the initial value problem,

∂
u

/

∂
t
=
A
u
+
V
(
x
)
⋅
u
,
u
(
0
)
=
f

\partial u/\partial t = Au + V(x) \cdot u,u(0) = f

, where A is the infinitesimal generator of the process X on a certain Banach space and for each

x
∈
E
,
V
(
x
)

x \in E,V(x)

is the infinitesimal generator of a

C
0

{C_0}

contraction semigroup on another Banach space.