bluuebunny/crossref_metadata_2025 huggingface dataset:
Explicit class field theory for rational function fields
Authors: Hayes D. R.
Published: July 2010

Developing an idea of Carlitz, I show how one can describe explicitly the maximal abelian extension of the rational function field over

F

q

{{\mathbf {F}}_q}

(the finite field of q elements) and the action of the idèle class group via the reciprocity law homomorphism. The theory is closely analogous to the classical theory of cyclotomic extensions of the rational numbers.
Canonical forms and principal systems for general disconjugate equations
Authors: Trench William F.
Published: July 2010

It is shown that the disconjugate equation (1)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L x identical-to left-parenthesis 1 slash beta Subscript n Baseline right-parenthesis left-parenthesis d slash d t right-parenthesis dot left-parenthesis 1 slash beta Subscript n minus 1 Baseline right-parenthesis midline-horizontal-ellipsis left-parenthesis d slash d t right-parenthesis left-parenthesis 1 slash beta 1 right-parenthesis left-parenthesis d slash d t right-parenthesis left-parenthesis x slash beta 0 right-parenthesis equals 0">

L
x
≡
(
1

/

β
n

)
(
d

/

d
t
)
⋅
(
1

/

β

n
−
1

)
⋯
(
d

/

d
t
)
(
1

/

β
1

)
(
d

/

d
t
)
(
x

/

β
0

)
=
0

Lx \equiv (1/{\beta _n})(d/dt) \cdot (1/{\beta _{n - 1}}) \cdots (d/dt)(1/{\beta _1})(d/dt)(x/{\beta _0}) = 0

, a

>
t
>
b

> t > b

, where

β
i

>
0

{\beta _i} > 0

, and (2)

β
i

∈
C
(
a
,
b
)

{\beta _i} \in C(a,b)

, can be written in essentially unique canonical forms so that

∫
b

β
i

d
t
=
∞
(

∫
a

β
i

d
t
=
∞
)

{\smallint ^b}{\beta _i}dt = \infty ({\smallint _a}{\beta _i}dt = \infty )

for

1
≤
i
≤
n
−
1

1 \leq i \leq n - 1

. From this it follows easily that (1) has solutions

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

which are positive in (a, b) near

b
(
a
)

b(a)

and satisfy
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript t right-arrow b Endscripts minus x Subscript i Baseline left-parenthesis t right-parenthesis slash x Subscript j Baseline left-parenthesis t right-parenthesis equals 0 left-parenthesis limit Underscript t right-arrow a Endscripts plus x Subscript i Baseline left-parenthesis t right-parenthesis slash x Subscript j Baseline left-parenthesis t right-parenthesis equals normal infinity right-parenthesis">

lim

t
→
b

−

x
i

(
t
)

/

x
j

(
t
)
=
0
(

lim

t
→
a

+

x
i

(
t
)

/

x
j

(
t
)
=
∞
)

{\lim _{t \to b}} - {x_i}(t)/{x_j}(t) = 0({\lim _{t \to a}} + {x_i}(t)/{x_j}(t) = \infty )

for

1
≤
i
>
j
≤
n

1 \leq i > j \leq n

. Necessary and sufficient conditions are given for (1) to have solutions

y
1

,
…
,

y
n

{y_1}, \ldots ,{y_n}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript t right-arrow b Endscripts minus y Subscript i Baseline left-parenthesis t right-parenthesis slash y Subscript j Baseline left-parenthesis t right-parenthesis equals limit Underscript t right-arrow a Endscripts plus y Subscript j Baseline left-parenthesis t right-parenthesis slash y Subscript i Baseline left-parenthesis t right-parenthesis equals 0">

lim

t
→
b

−

y
i

(
t
)

/

y
j

(
t
)
=

lim

t
→
a

+

y
j

(
t
)

/

y
i

(
t
)
=
0

{\lim _{t \to b}} - {y_i}(t)/{y_j}(t) = {\lim _{t \to a}} + {y_j}(t)/{y_i}(t) = 0

for

1
≤
i
>
j
≤
n

1 \leq i > j \leq n

. Using different methods, P. Hartman, A. Yu. Levin and D. Willett have investigated the existence of fundamental systems for (1) with these properties under assumptions which imply the stronger condition

(

2
′

)

β
i

∈

C

(
n
−
i
)

(
a
,
b
)

(2’){\beta _i} \in {C^{(n - i)}}(a,b)

.
A partition property characterizing cardinals hyperinaccessible of finite type
Authors: Schmerl James H.
Published: July 2010

Let

P

(
n
,
α
)

{\mathbf {P}}(n,\alpha )

be the class of infinite cardinals which have the following property: Suppose for each

ν
>
κ

\nu > \kappa

that

C
ν

{C_\nu }

is a partition of

[
κ

]
n

{[\kappa ]^n}

and card

(

C
ν

)
>
κ

({C_\nu }) > \kappa

; then there is

X
⊂
κ

X \subset \kappa

of length

α
\alpha

such that for each

ν
>
κ

\nu > \kappa

, the set

X
−
(
ν
+
1
)

X - (\nu + 1)

is

C
ν

{C_\nu }

-homogeneous. In this paper the classes

P

(
n
,
α
)

{\mathbf {P}}(n,\alpha )

are studied and a nearly complete characterization of them is given. A principal result is that

P

(
n
+
2
,
n
+
5
)

{\mathbf {P}}(n + 2,n + 5)

is the class of cardinals which are hyperinaccessible of type n.
Convex hulls and extreme points of some families of univalent functions
Authors: Hallenbeck D. J.
Published: July 2010

The closed convex hull and extreme points are obtained for the functions which are convex, starlike, and close-to-convex and in addition are real on(−1,1)( - 1,1). We also obtain this result for the functions which are convex in the direction of the imaginary axis and real on(−1,1)( - 1,1). Integral representations are given for the hulls of these families in terms of probability measures on suitable sets. We also obtain such a representation for the functionsf(z)f(z)analytic in the unit disk, normalized and satisfyingRe⁡f′(z)>α\operatorname {Re} f’(z) > \alphaforα>1\alpha > 1. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to some function satisfyingRe⁡f′(z)>α\operatorname {Re} f’(z) > \alpha.
Global residues and intersections on a complex manifold
Authors: King James R.
Published: July 2010

This paper is the study of a class of forms

η
\eta

on a complex manifold V which are smooth on

V
−
W

V - W

and have poles of kernel type on a complex submanifold W of codimension d; such a form is one whose pull-back to the monoidal transform of V along W has a logarithmic pole. A global existence theorem is proved which asserts that any smooth form

φ
\varphi

on W of filtration s (no (p, q) components with

p
>
s

p > s

) is the residue of a form

η
\eta

of filtration

s
+
d

s + d

such that

d
η

d\eta

is smooth on V. This result is used to construct global kernels for

∂
¯

\bar \partial

which establish similar global existence theorems for W with singularities. We then establish formulas connecting intersection and wedge product on the d-cohomology theory of Dolbeault which preserve the Hodge filtration. A number of results are also proved on the integrability of

f
∗

η

{f^\ast }\eta

where f is a rather general holomorphic map.
The geometry of flat Banach spaces
Authors: Harrell R. E. , Karlovitz L. A.
Published: July 2010

A Banach space is flat if the girth of its unit ball is 4 and if the girth is achieved by some curve. (Equivalently, its unit ball can be circumnavigated along a centrally symmetric path whose length is 4.) Some basic geometric properties of flat Banach spaces are given. In particular, the term flat is justified.
Isometries of *-invariant subspaces
Authors: Lubin Arthur
Published: July 2010

We consider families of increasing

∗

^\ast

-invariant subspaces of

H
2

(
D
)

{H^2}(D)

, and from these we construct canonical isometrics from certain

L
2

{L^2}

spaces to

H
2

{H^2}

. We give necessary and sufficient conditions for these maps to be unitary, and discuss the relevance to a problem concerning a concrete model theory for a certain class of operators.
Semigroups with midunits
Authors: Ault Janet E.
Published: July 2010

A semigroup has a midunit u if

a
u
b
=
a
b

aub = ab

for all a and b. It is the purpose of this paper to explore semigroups with midunits, and to find an analog to the group of units. In addition, two constructions will be given which produce some semigroups with midunits, and an abstract characterization will be made of certain of these semigroups.
Pairs of compacta and trivial shape
Authors: Mardešić Sibe
Published: July 2010

Let (X, Y, A),

(

X
′

,

Y
′

,

A
′

)

(X’,Y’,A’)

be triples of compact Hausdorff spaces. Using ANR-systems the following is proved:

sh

Y
=

sh

Y
′

=
0
,

sh

(
X
,
Y
)
=

sh

(

X
′

,

Y
′

)

{\text {sh}}\;Y = {\text {sh}}\;Y’ = 0,{\text {sh}}(X,Y) = {\text {sh}}(X’,Y’)

and

sh

A
=

sh

A
′

{\text {sh}}\;A = {\text {sh}}\;A’

imply

sh

(
X
,
Y
,
A
)
=

sh

(

X
′

,

Y
′

,

A
′

)

{\text {sh}}(X,Y,A) = {\text {sh}}(X’,Y’,A’)

. All results concerning the shape of decomposition spaces and addition properties of FAR’s, due to K. Borsuk and T.A. Chapman, follow readily from this theorem. In particular,

sh

X
=

sh

X
′

=
0

{\text {sh}}\;X = {\text {sh}}\;X’ = 0

and

sh

A
=

sh

A
′

{\text {sh}}\;A = {\text {sh}}\;A’

imply

sh

(
X
,
A
)
=

sh

(

X
′

,

A
′

)

{\text {sh}}(X,A) = {\text {sh}}(X’,A’)

, which in view of an example of Borsuk shows that for compact metric pairs the ANR-system approach to shapes differs from the Borsuk approach.
Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients
Authors: Gossez Jean-Pierre
Published: July 2010

Variational boundary value problems for quasilinear elliptic systems in divergence form are studied in the case where the nonlinearities are nonpolynomial. Monotonicity methods are used to derive several existence theorems which generalize the basic results of Browder and Leray-Lions. Some features of the mappings of monotone type which arise here are that they act in nonreflexive Banach spaces, that they are unbounded and not everywhere defined, and that their inverse is also unbounded and not everywhere defined.
Invariant measures and growth conditions
Authors: Rosenblatt Joseph Max
Published: July 2010

Let G be a finitely-generated group acting on a set X and let A be a nonempty subset of X. If G has polynomial growth then there exists a finitely-additive G-invariant positive extended real-valued measure

μ
\mu

defined on all subsets of X such that

μ
(
A
)
=
1

\mu (A) = 1

. When G is solvable, it has polynomial growth if and only if it does not contain a free subsemigroup on two generators. If G contains a free subsemigroup S on two generators, then G has exponential growth and there does not exist a measure

μ
\mu

as above with G acting on itself by multiplication and

A
=
S

A = S

.
Pointwise differentiability and absolute continuity
Authors: Bagby Thomas , Ziemer William P.
Published: July 2010

This paper is concerned with the relationships between

L
p

{L_p}

differentiability and Sobolev functions. It is shown that if f is a Sobolev function with weak derivatives up to order k in

L
p

{L_p}

, and

0
≤
l
≤
k

0 \leq l \leq k

, then f has an

L
p

{L_p}

derivative of order l everywhere except for a set which is small in the sense of an appropriate capacity. It is also shown that if a function has an

L
p

{L_p}

derivative everywhere except for a set small in capacity and if these derivatives are in

L
p

{L_p}

, then the function is a Sobolev function. A similar analysis is applied to determine general conditions under which the Gauss-Green theorem is valid.
Instability and nonexistence of global solutions to nonlinear wave equations of the form 𝑃𝑢_{𝑡𝑡}=-𝐴𝑢+ℱ(𝓊)
Authors: Levine Howard A.
Published: July 2010

For the equation in the title, let P and A be positive semidefinite operators (with P strictly positive) defined on a dense subdomain

D
⊆
H

D \subseteq H

, a Hilbert space. Let D be equipped with a Hilbert space norm and let the imbedding be continuous. Let

F

:
D
→
H

\mathcal {F}:D \to H

be a continuously differentiable gradient operator with associated potential function

G

\mathcal {G}

. Assume that

(
x
,

F

(
x
)
)
≥
2
(
2
α
+
1
)

G

(
x
)

(x,\mathcal {F}(x)) \geq 2(2\alpha + 1)\mathcal {G}(x)

for all

x
∈
D

x \in D

and some

α
>
0

\alpha > 0

. Let

E
(
0
)
=

1
2

[
(

u
0

,
A

u
0

)
+
(

v
0

,
P

v
0

)
]

E(0) = \tfrac {1}{2}[({u_0},A{u_0}) + ({v_0},P{v_0})]

where

u
0

=
u
(
0
)
,

v
0

=

u
t

(
0
)

{u_0} = u(0),{v_0} = {u_t}(0)

and

u
:
[
0
,
T
)
→
D

u:[0,T) \to D

be a solution to the equation in the title. The following statements hold: If

G

(

u
0

)
>
E
(
0
)

\mathcal {G}({u_0}) > E(0)

, then

lim

t
→

T
−

(
u
,
P
u
)
=
+
∞

{\lim _{t \to {T^ - }}}(u,Pu) = + \infty

for some

T
>
∞

T > \infty

. If
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis u 0 comma upper P v 0 right-parenthesis greater-than 0 comma 0 greater-than upper E left-parenthesis 0 right-parenthesis minus script upper G left-parenthesis u 0 right-parenthesis greater-than alpha left-parenthesis u 0 comma upper P v 0 right-parenthesis squared slash 4 left-parenthesis 2 alpha plus 1 right-parenthesis left-parenthesis u 0 comma upper P u 0 right-parenthesis">

(

u
0

,
P

v
0

)
>
0
,
0
>
E
(
0
)
−

G

(

u
0

)
>
α

(

u
0

,
P

v
0

)
2

/

4
(
2
α
+
1
)
(

u
0

,
P

u
0

)

({u_0},P{v_0}) > 0,0 > E(0) - \mathcal {G}({u_0}) > \alpha {({u_0},P{v_0})^2}/4(2\alpha + 1)({u_0},P{u_0})

and if u exists on

[
0
,
∞
)

[0,\infty )

, then (u,Pu) grows at least exponentially. If

(

u
0

,
P

v
0

)
>
0

({u_0},P{v_0}) > 0

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha left-parenthesis u 0 comma upper P v 0 right-parenthesis squared slash 4 left-parenthesis 2 alpha plus 1 right-parenthesis left-parenthesis u 0 comma upper P u 0 right-parenthesis less-than-or-equal-to upper E left-parenthesis 0 right-parenthesis minus script upper G left-parenthesis u 0 right-parenthesis greater-than one half left-parenthesis u 0 comma upper P v 0 right-parenthesis squared slash left-parenthesis u 0 comma upper P u 0 right-parenthesis">

α

(

u
0

,
P

v
0

)
2

/

4
(
2
α
+
1
)
(

u
0

,
P

u
0

)
≤
E
(
0
)
−

G

(

u
0

)
>

1
2

(

u
0

,
P

v
0

)
2

/

(

u
0

,
P

u
0

)

\alpha {({u_0},P{v_0})^2}/4(2\alpha + 1)({u_0},P{u_0}) \leq E(0) - \mathcal {G}({u_0}) > \tfrac {1}{2}{({u_0},P{v_0})^2}/({u_0},P{u_0})

and if the solution exists on

[
0
,
∞
)

[0,\infty )

, then (u,Pu) grows at least as fast as

t
2

{t^2}

. A number of examples are given.
Somewhere locally flat codimension one manifolds with 1-𝑈𝐿𝐶 complements are locally flat
Authors: Price T. M. , Seebeck C. L.
Published: July 2010

The purpose of this paper is to prove a taming theorem for a codimension one manifold that is locally flat at some point and has 1-ULC complement. We also prove that any two sufficiently close locally flat embeddings of a codimension one manifold are ambient isotopic. Since this paper was first submitted, R. Daverman has shown that, given any point on a codimension one manifold with 1-ULC complement, some neighborhood of that point lies on a codimension one sphere that is locally flat at some points and has 1-ULC complement. Hence the two papers combined prove that a codimension one manifold is locally flat if and only if its complement is 1-ULC.
Irreducible congruences over 𝐺𝐹(2)
Authors: Hanneken C. B.
Published: July 2010

In characterizing and determining the number of conjugate sets of irreducible congruences of degree m belonging to

G
F
(
p
)

GF(p)

relative to the group

G
(
p
)

G(p)

of linear fractional transformations with coefficients belonging to the same field, the case

p
=
2

p = 2

has been consistently excluded from considerations. In this paper we consider the special case

p
=
2

p = 2

and determine the number of conjugate sets of m-ic congruences belonging to

G
F
(
2
)

GF(2)

relative to

G
(
2
)

G(2)

.
Maximal quotients of semiprime PI-algebras
Authors: Rowen Louis Halle
Published: July 2010

J. Fisher [3] initiated the study of maximal quotient rings of semiprime PI-rings by noting that the singular ideal of any semiprime Pi-ring R is 0; hence there is a von Neumann regular maximal quotient ring

Q
(
R
)

Q(R)

of R. In this paper we characterize

Q
(
R
)

Q(R)

in terms of essential ideals of C = cent R. This permits immediate reduction of many facets of

Q
(
R
)

Q(R)

to the commutative case, yielding some new results and some rapid proofs of known results. Direct product decompositions of

Q
(
R
)

Q(R)

are given, and

Q
(
R
)

Q(R)

turns out to have an involution when R has an involution.
Lower semicontinuity of integral functionals
Authors: Berkovitz Leonard D.
Published: July 2010

It is shown that the integral functional

I
(
y
,
z
)
=

∫
G

f
(
t
,
y
(
t
)
,
z
(
t
)
)
d
μ

I(y,z) = {\smallint _G}f(t,y(t),z(t))d\mu

is lower semicontinuous on its domain with respect to the joint strong convergence of

y
k

→
y

{y_k} \to y

in

L
p

(
G
)

{L_p}(G)

and the weak convergence of

z
k

→
z

{z_k} \to z

in

L
p

(
G
)

{L_p}(G)

, where

1
≤
p
≤
∞

1 \leq p \leq \infty

and

1
≤
q
≤
∞

1 \leq q \leq \infty

, under the following conditions. The function

f
:
(
t
,
x
,
w
)
→
f
(
t
,
x
,
w
)

f:(t,x,w) \to f(t,x,w)

is measurable in t for fixed (x, w), is continuous in (x, w) for a.e. t, and is convex in w for fixed (t, x).
On a Wedderburn principal theorem for the flexible algebras
Authors: Chaffer Robert A.
Published: July 2010

A strictly power-associative algebra A over a field K is said to have a Wedderburn decomposition if there is a subalgebra S of A such that

A
=
S
+
N

A = S + N

, where N is the nil radical of A, and

S
=
A
−
N

S = A - N

. A Wedderburn principal theorem for a class of algebras is a theorem which asserts that the algebras A, in the class, with

A
−
N

A - N

separable have Wedderburn decompositions. It is known that there is no such theorem for the class of noncommutative Jordan algebras. A partial result in this direction is the following theorem. Theorem. Let A be a strictly power-associative, flexible algebra over a field F with characteristic not 2 or 3, with

A
−
N

A - N

separable and such that

A
=

A
1

⊕

A
2

⊕
⋯
⊕

A
n

A = {A_1} \oplus {A_2} \oplus \cdots \oplus {A_n}

where each

A
i

{A_i}

. has

A
i

−

N
i

{A_i} - {N_i}

simple and has more than two pairwise orthogonal idempotents. Then

A
=
S
+
N

A = S + N

where S is a subalgebra of A.
Conjugate points, triangular matrices, and Riccati equations
Authors: Nehari Zeev
Published: July 2010

Let

A
A

be a real continuous

n
×
n

n \times n

matrix on an interval

Γ
\Gamma

, and let the

n
n

-vector

x
x

be a solution of the differential equation

x
′

=
A
x

x’ = Ax

on

Γ
\Gamma

. If

[
α
,
β
]
∈
Γ
,
β

[\alpha ,\beta ] \in \Gamma ,\beta

is called a conjugate point of

α
\alpha

if the equation has a nontrivial solution vector

x
=
(

x
1

,
…
,

x
n

)

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

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x 1 left-parenthesis alpha right-parenthesis equals ellipsis equals x Subscript k Baseline left-parenthesis alpha right-parenthesis equals x Subscript k plus 1 Baseline left-parenthesis beta right-parenthesis equals ellipsis equals x Subscript n Baseline left-parenthesis beta right-parenthesis equals 0">

x
1

(
α
)
=
…
=

x
k

(
α
)
=

x

k
+
1

(
β
)
=
…
=

x
n

(
β
)
=
0

{x_1}(\alpha ) = \ldots = {x_k}(\alpha ) = {x_{k + 1}}(\beta ) = \ldots = {x_n}(\beta ) = 0

for some

k
∈
[
1
,
n
−
1
]

k \in [1,n - 1]

. It is shown that the absence on

(

t
1

,

t
2

)

({t_1},{t_2})

of a point conjugate to

t
1

{t_1}

with respect to the equation

x
′

=
A
x

x’ = Ax

is equivalent to the existence on

(

t
1

,

t
2

)

({t_1},{t_2})

of a continuous matrix solution

L
L

of the nonlinear differential equation

L
′

=

[
L

A
∗

L

−
1

]

τ
0

L

L’ = {[L{A^ \ast }{L^{ - 1}}]_{{\tau _0}}}L

with the initial condition

L
(

t
1

)
=
I

L({t_1}) = I

, where

[
B

]

τ
0

{[B]_{{\tau _0}}}

denotes the matrix obtained from the

n
×
n

n \times n

matrix

B
B

by replacing the elements on and above the main diagonal by zeros. This nonlinear equation—which may be regarded as a generalization of the Riccati equation, to which it reduces for

n
=
2

n = 2

—can be used to derive criteria for the presence or absence of conjugate points on a given interval.
Ultrafilter mappings and their Dedekind cuts
Authors: Blass Andreas
Published: July 2010

Let D be an ultrafilter on the set N of natural numbers. To each function

p
:
N
→
N

p:N \to N

and each ultrafilter E that is mapped to D by p, we associate a Dedekind cut in the ultrapower D-prod N. We characterize, in terms of rather simple closure conditions, the cuts obtainable in this manner when various restrictions are imposed on E and p. These results imply existence theorems, some known and some new, for various special kinds of ultrafilters and maps.
The category of generalized Lie groups
Authors: Chen Su Shing , Yoh Richard W.
Published: July 2010

We consider the category

Γ
\Gamma

of generalized Lie groups. A generalized Lie group is a topological group

G
G

such that the set

L
G
=
H
o
m
(

R

,
G
)

LG = Hom({\mathbf {R}},G)

of continuous homomorphisms from the reals

R

{\mathbf {R}}

into

G
G

has certain Lie algebra and locally convex topological vector space structures. The full subcategory

Γ
r

{\Gamma ^r}

of

r
r

-bounded (

r
r

positive real number) generalized Lie groups is shown to be left complete. The class of locally compact groups is contained in

Γ
\Gamma

. Various properties of generalized Lie groups

G
G

and their locally convex topological Lie algebras

L
G
=
H
o
m
(

R

,
G
)

LG = Hom({\mathbf {R}},G)

are investigated.
On dynamical systems with the specification property
Authors: Sigmund Karl
Published: July 2010

A continuous transformation T of a compact metric space X satisfies the specification property if one can approximate distinct pieces of orbits by single periodic orbits with a certain uniformity. There are many examples of such transformations which have recently been studied in ergodic theory and statistical mechanics. This paper investigates the relation between Tinvariant measures and the frequencies of T-orbits. In particular, it is shown that every invariant measure (and even every closed connected subset of such measures) has generic points, but that the set of all generic points is of first category in X. This generalizes number theoretic results concerning decimal expansions and normal numbers.
Inequalities for polynomials with a prescribed zero
Authors: Giroux A. , Rahman Q. I.
Published: July 2010

Inequalities for the derivative and for the maximum modulus on a larger circle of a polynomial with a given zero on the unit circle are obtained in terms of its degree and maximum modulus on the unit circle; examples are given to show that these are sharp with respect to the degree (best constants are not known). Inequalities for

L
p

{L^p}

norms, in particular

L
2

{L^2}

norms, are also derived. Also certain functions of exponential type are considered and similar inequalities are obtained for them. Finally, the problem of estimating

P
n

(
r
)

{P_n}(r)

(with

0
>
r
>
1

0 > r > 1

) given

P
n

(
1
)
=
0

{P_n}(1) = 0

is taken up.
Equivariant endomorphisms of the space of convex bodies
Authors: Schneider Rolf
Published: July 2010

We consider maps of the set of convex bodies in d-dimensional Euclidean space into itself which are linear with respect to Minkowski addition, continuous with respect to Hausdorff metric, and which commute with rigid motions. Examples constructed by means of different methods show that there are various nontrivial maps of this type. The main object of the paper is to find some reasonable additional assumptions which suffice to single out certain special maps, namely suitable combinations of dilatations and reflections, and of rotations if

d
=
2

d = 2

. For instance, we determine all maps which, besides having the properties mentioned above, commute with affine maps, or are surjective, or preserve the volume. The method of proof consists in an application of spherical harmonics, together with some convexity arguments.
Regular self-injective rings with a polynomial identity
Authors: Armendariz Efraim P. , Steinberg Stuart A.
Published: July 2010

This paper studies maximal quotient rings of semiprime P. I.-rings; such rings are regular, self-injective and satisfy a polynomial identity. We show that the center of a regular self-injective ring is regular self-injective; this enables us to establish that the center of the maximal quotient ring of a semiprime P. I.-ring R is the maximal quotient ring of the center of R, as well as some other relationships. We give two decompositions of a regular self-injective ring with a polynomial identity which enable us to show that such rings are biregular and are finitely generated projective modules over their center.
The oriented bordism of 𝑍_{2^{𝑘}} actions
Authors: Wheeler E. R.
Published: July 2010

Let

R
2

{R_2}

be the subring of the rationals given by

R
2

=
Z
[
1

/

2
]

{R_2} = Z[1/2]

. It is shown that for

G
=

Z

2
k

G = {Z_{{2^k}}}

the bordism group of orientation preserving

G
G

actions on oriented manifolds tensored with

R
2

{R_2}

is a free

Ω
∗

⊗

R
2

{\Omega _ \ast } \otimes {R_2}

module on even dimensional generators (where

Ω
∗

{\Omega _ \ast }

is the oriented bordism ring).
Groups, semilattices and inverse semigroups. I, II
Authors: McAlister D. B.
Published: July 2010

An inverse semigroup S is called proper if the equations

e
a
=
e
=

e
2

ea = e = {e^2}

together imply

a
2

=
a

{a^2} = a

for each a,

a
,
e
∈
S

a,e \in S

. In this paper a construction is given for a large class of proper inverse semigroups in terms of groups and partially ordered sets; the semigroups in this class are called P-semigroups. It is shown that every inverse semigroup divides a P-semigroup in the sense that it is the image, under an idempotent separating homomorphism, of a full subsemigroup of a P-semigroup. Explicit divisions of this type are given for

ω
\omega

-bisimple semigroups, proper bisimple inverse semigroups, semilattices of groups and Brandt semigroups.
Limit theorems for variational sums
Authors: Hudson William N. , Tucker Howard G.
Published: July 2010

Limit theorems in the sense of a.s. convergence, convergence in

L
1

{L_1}

-norm and convergence in distribution are proved for variational series. In the first two cases, if g is a bounded, nonnegative continuous function satisfying an additional assumption at zero, and if

{
X
(
t
)
,
0
≤
t
≤
T
}

\{ X(t),0 \leq t \leq T\}

is a stochastically continuous stochastic process with independent increments, with no Gaussian component and whose trend term is of bounded variation, then the sequence of variational sums of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Subscript k equals 1 Superscript n Baseline g left-parenthesis upper X left-parenthesis t Subscript n k Baseline right-parenthesis minus upper X left-parenthesis t Subscript n comma k minus 1 Baseline right-parenthesis right-parenthesis">

Σ

k
=
1

n

g
(
X
(

t

n
k

)
−
X
(

t

n
,
k
−
1

)
)

\Sigma _{k = 1}^ng(X({t_{nk}}) - X({t_{n,k - 1}}))

is shown to converge with probability one and in

L
1

{L_1}

-norm. Also, under the basic assumption that the distribution of the centered sum of independent random variables from an infinitesimal system converges to a (necessarily) infinitely divisible limit distribution, necessary and sufficient conditions are obtained for the joint distribution of the appropriately centered sums of the positive parts and of the negative parts of these random variables to converge to a bivariate infinitely divisible distribution. A characterization of all such limit distributions is obtained. An application is made of this result, using the first theorem, to stochastic processes with (not necessarily stationary) independent increments and with a Gaussian component.
On the tensor product of 𝑊* algebras
Authors: Renshaw Bruce B.
Published: July 2010

We develop the algebra underlying the reduction theory of von Neumann in the language and spirit of Sakai’s abstract

W
∗

{W^ \ast }

algebras, and using the maximum spectrum of an abelian von Neumann algebra rather than a measure-theoretic surrogate. We are thus enabled to obtain the basic fact of the von Neumann theory as a special case of a weaker general decomposition theorem, valid without separability or type restrictions, and adapted to comparison with Wright’s theory in the finite case.
Fixed point theorems for certain classes of semigroups of mappings
Authors: Kiang Mo Tak
Published: July 2010

Fixed point theorems for commuting semigroups of self-mappings are considered in this paper. A generalization of the classical Markov-Kakutani theorem is first given. This is followed by a fixed point theorem for commutative semigroups of continuous asymptotically-nonexpansive self-mappings on a weakly compact, convex subset of a strictly convex Banach space.
Class numbers of real quadratic number fields
Authors: Brown Ezra
Published: July 2010

This article is a study of congruence conditions, modulo powers of two, on class number of real quadratic number fields

Q
(

d

)

Q(\sqrt d )

, for which d has at most three distinct prime divisors. Techniques used are those associated with Gaussian composition of binary quadratic forms.
Subspaces of the nonstandard hull of a normed space
Authors: Henson C. Ward , Moore L. C.
Published: July 2010

Normed spaces which are isomorphic to subspaces of the nonstandard hull of a given normed space are characterized. As a consequence it is shown that a normed space is B-convex if and only if the nonstandard hull contains no subspace isomorphic to

l
1

{l_1}

and a Banach space is super-reflexive if and only if the nonstandard hull is reflexive. Also, embeddings of second dual spaces into the nonstandard hull are studied. In particular, it is shown that the second dual space of a normed space E is isometric to a complemented subspace of the nonstandard hull of E.
Behnke-Stein theorem for analytic spaces
Authors: Silva Alessandro
Published: July 2010

The notion of

q
q

-Runge pair is extended to reduced complex analytic spaces. A necessary and sufficient condition for a pair of

n
n

-dimensional analytic spaces to be an

n
n

-Runge pair is proved and it is shown that this result extends a Behnke-Stein theorem when

n
=
1

n = 1

. A topological property of

q
q

-Runge pairs of spaces is also proved.
An algebraic property of the Čech cohomology groups which prevents local connectivity and movability
Authors: Keesling James
Published: July 2010

Let C denote the category of compact Hausdorff spaces and

H
:
C
→
H
C

H:C \to HC

be the homotopy functor. Let

S
:
C
→
S
C

S:C \to SC

be the functor of shape in the sense of Holsztyński for the projection functor H. Let X be a continuum and

H
n

(
X
)

{H^n}(X)

denote n-dimensional Čech cohomology with integer coefficients. Let

A
x

=

char

H
1

(
X
)

{A_x} = {\text {char}}\;{H^1}(X)

be the character group of

H
1

(
X
)

{H^1}(X)

considering

H
1

(
X
)

{H^1}(X)

as a discrete group. In this paper it is shown that there is a shape morphism

F
∈

Mor

S
C

(
X
,

A
X

)

F \in {\text {Mor}_{SC}}(X,{A_X})

such that

F
∗

:

H
1

(

A
X

)
→

H
1

(
X
)

{F^\ast }:{H^1}({A_X}) \to {H^1}(X)

is an isomorphism. It follows from the results of a previous paper by the author that there is a continuous mapping

f
:
X
→

A
X

f:X \to {A_X}

such that

S
(
f
)
=
F

S(f) = F

and thus that

f
∗

:

H
1

(

A
X

)
→

H
1

(
X
)

{f^\ast }:{H^1}({A_X}) \to {H^1}(X)

is an isomorphism. This result is applied to show that if X is locally connected, then

H
1

(
X
)

{H^1}(X)

has property L. Examples are given to show that X may be locally connected and

H
n

(
X
)

{H^n}(X)

not have property L for

n
>
1

n > 1

. The result is also applied to compact connected topological groups. In the last section of the paper it is shown that if X is compact and movable, then for every integer n,

H
n

(
X
)

/

Tor

H
n

(
X
)

{H^n}(X)/{\operatorname {Tor}}\;{H^n}(X)

has property L. This result allows us to construct peano continua which are nonmovable. An example is given to show that

H
n

(
X
)

{H^n}(X)

itself may not have property L even if X is a finite-dimensional movable continuum.
Spaces homeomorphic to (2^{𝑎})ₐ. II
Authors: Hung H. H. , Negrepontis S.
Published: July 2010

Topological characterizations and properties of the spaces

(

2
α

)
α

{({2^\alpha })_\alpha }

, where

α
\alpha

is an infinite regular cardinal, are studied; the principal interest lying in the significance that these spaces have in questions of existence of ultrafilters (or of elements of the Stone-Čech compactification of spaces) with special properties. The main results are (a) the characterization theorem of the spaces

(

2
α

)
α

{({2^\alpha })_\alpha }

in terms of a simple set of conditions, and (b) the

α
\alpha

-Baire category property of

(

2
α

)
α

{({2^\alpha })_\alpha }

and the stability of the class of spaces homeomorphic to

(

2
α

)
α

{({2^\alpha })_\alpha }

(or to

(

α
α

)
α

{({\alpha ^\alpha })_\alpha }

) when taking intersections of at most

α
\alpha

open and dense subsets of

(

2
α

)
α

{({2^\alpha })_\alpha }

. Among the applications of these results are the following. Assuming

α
+

=

2
α

{\alpha ^ + } = {2^\alpha }

, the class of spaces homeomorphic to

(

2

(

α
+

)

)

α
+

{({2^{({\alpha ^ + })}})_{{\alpha ^ + }}}

includes the space of uniform ultrafilters on

α
\alpha

with the

P

α
+

{P_{{\alpha ^ + }}}

-topology

(
U
(
α
)

)

α
+

{(U(\alpha ))_{{\alpha ^ + }}}

, its subspaces of good ultrafilters and/or Rudin-Keisler minimal ultrafilters. Assuming

ω
+

=

2
ω

{\omega ^ + } = {2^\omega }

(or in some cases only Martin’s axiom), the class of spaces homeomorphic to

(

2

(

ω
+

)

)

ω
+

{({2^{({\omega ^ + })}})_{{\omega ^ + }}}

includes the following: The space

(
β
X
∖
X

)

ω
+

{(\beta X\backslash X)_{{\omega ^ + }}}

where X is a noncompact locally compact realcompact space such that

|

C
(
X
)

|

≤

2
ω

|C(X)| \leq {2^\omega }

and its subspaces of

P

ω
+

{P_{{\omega ^ + }}}

-points of

β
X
∖
X

\beta X\backslash X

and/or (if X is in addition a metric space without isolated elements) the remote points. In particular the existence of good and/or Rudin-Keisler minimal ultrafilters and the existence of P-points and/or remote points follows always from a Baire category type of argument.
Recapturing 𝐻²-functions on a polydisc
Authors: Patil D. J.
Published: July 2010

Let

U
2

{U^2}

be the unit polydisc and

T
2

{T^2}

its distinguished boundary. If

E
⊂

T
2

E \subset {T^2}

is a set of positive measure and the restriction to E of a function f in

H
2

(

U
2

)

{H^2}({U^2})

is given then an algorithm to recapture f is developed.
Representing measures and topological type of finite bordered Riemann surfaces
Authors: Nash David
Published: July 2010

A finite bordered Riemann surface

R

\mathcal {R}

with s boundary components and interior genus g has first Betti number

r
=
2
g
+
s
−
1

r = 2g + s - 1

. Let a be any interior point of

R

\mathcal {R}

and

e
a

{e_a}

denote evaluation at a on the usual hypo-Dirichlet algebra associated with

R

\mathcal {R}

. We establish some connections between the topological and, more strongly, the conformal type of

R

\mathcal {R}

and the geometry of

M

a

{\mathfrak {M}_a}

the set of representing measures for

e
a

{e_a}

. For example, we show that if

M

a

{\mathfrak {M}_a}

has an isolated extreme point, then

R

\mathcal {R}

must be a planar surface. Several questions posed by Sarason are answered through exhausting the possibilities for the case

r
=
2

r = 2

.
A bounded difference property for classes of Banach-valued functions
Authors: Veith Wilbur P.
Published: July 2010

Let

A
(
G
,
E
)

A(G,E)

denote the set of functions f from a Hausdorff topological group G to a Banach space E such that the range of f is relatively compact in E and

ϕ
∘
f

\phi \circ f

is in

A
(
G
,
C
)

A(G,C)

for each

ϕ
\phi

in the dual of E, where

A
(
G
,
C
)

A(G,C)

is a translation-invariant

C
∗

{C^\ast }

algebra of bounded, continuous, complex-valued functions on G with respect to the supremum norm and complex conjugation.

A
(
G
,
E
)

A(G,E)

has the bounded difference property if whenever

F
:
G
→
E

F:G \to E

is a bounded function such that

Δ
t

F
(
x
)
=
F
(
t
x
)
−
F
(
x
)

{\Delta _t}F(x) = F(tx) - F(x)

is in

A
(
G
,
E
)

A(G,E)

for each t in G, then F is also an element of

A
(
G
,
E
)

A(G,E)

. A condition on

A
(
G
,
C
)

A(G,C)

and a condition on E are given under which

A
(
G
,
E
)

A(G,E)

has the bounded difference property. The condition on

A
(
G
,
C
)

A(G,C)

is satisfied by both the class of almost periodic functions and the class of almost automorphic functions.
Norm inequalities for the Littlewood-Paley function 𝑔*_{𝜆}
Authors: Muckenhoupt Benjamin , Wheeden Richard L.
Published: July 2010

Weighted norm inequalities for

L
p

{L^p}

and

H
p

{H^p}

are derived for the Littlewood-Paley function

g
λ
∗

g_\lambda ^ \ast

. New results concerning the boundedness of this function are obtained, by a different method of proof, even in the unweighted case. The proof exhibits a connection between

g
λ
∗

g_\lambda ^\ast

and a maximal function for harmonic functions which was introduced by C. Fefferman and E. M. Stein. A new and simpler way to determine the behavior of this maximal function is given.
Interpolating sequences in polydisks
Authors: Kronstadt Eric P.
Published: July 2010

Let

D
n

{D^n}

be the unit polydisk in

C

n

{{\mathbf {C}}^n}

,

A
A

be a uniform algebra,

H
∞

(

D
n

,
A
)

{H^\infty }({D^n},A)

, the space of bounded analytic

A
A

valued functions on

D
n

{D^n}

,

l
∞

A

{l^\infty }A

, the space of bounded sequences of elements in

A
A

. A sequence,

S
=
{

a
i

}

i
=
1

∞

⊂

D
n

S = \{ {a_i}\} _{i = 1}^\infty \subset {D^n}

will be called an interpolating sequence with respect to

A
A

if the map

T
:

H
∞

(

D
n

,
A
)
→

l
∞

A

T:{H^\infty }({D^n},A) \to {l^\infty }A

given by

T
(
f
)
=
{
f
(

a
i

)

}

i
=
1

∞

T(f) = \{ f({a_i})\} _{i = 1}^\infty

is surjective. In 1958, L. Carleson showed that for

n
=
1
,
S

n = 1,S

is interpolating w.r.t.

C

{\mathbf {C}}

iff

S
S

satisfies a certain zero-one interpolation property called uniform separation. We generalize this result to cases where

n
>
1

n > 1

and

A
≠

C

A \ne {\mathbf {C}}

. In particular, we show that if

S
⊂

D
n

S \subset {D^n}

is uniformly separated and

S
⊂

W
1

×

W
2

×
⋯
×

W
n

S \subset {W_1} \times {W_2} \times \cdots \times {W_n}

(where each

W
j

{W_j}

is a region in

D
D

lying between two circular arcs which intersect twice on the boundary of

D
D

) then

S
S

is an interpolating sequence w.r.t. any uniform algebra. If

S
⊂

D
n

S \subset {D^n}

is uniformly separated and

S
⊂
D
×

W
2

×
⋯
×

W
n

S \subset D \times {W_2} \times \cdots \times {W_n}

then

S
S

is interpolating w.r.t.

C

{\mathbf {C}}

. Other examples and generalizations of interpolating sequences are discussed.
Products of initially 𝑚-compact spaces
Authors: Stephenson R. M. , Vaughan J. E.
Published: July 2010

The main purpose of this paper is to give several theorems and examples which we hope will be of use in the solution of the following problem. For an infinite cardinal number

m

\mathfrak {m}

, is initial

m

\mathfrak {m}

-compactness preserved by products? We also give some results concerning properties of Stone-Čech compactifications of discrete spaces.
The norm of the 𝐿^{𝑝}-Fourier transform on unimodular groups
Authors: Russo Bernard
Published: July 2010

We discuss sharpness in the Hausdorff Young theorem for unimodular groups. First the functions on unimodular locally compact groups for which equality holds in the Hausdorff Young theorem are determined. Then it is shown that the Hausdorff Young theorem is not sharp on any unimodular group which contains the real line as a direct summand, or any unimodular group which contains an Abelian normal subgroup with compact quotient as a semidirect summand. A key tool in the proof of the latter statement is a Hausdorff Young theorem for integral operators, which is of independent interest. Whether the Hausdorff Young theorem is sharp on a particular connected unimodular group is an interesting open question which was previously considered in the literature only for groups which were compact or locally compact Abelian.
Nil and power-central polynomials in rings
Authors: Leron Uri
Published: July 2010

A polynomial in noncommuting variables is vanishing, nil or central in a ring,

R
R

, if its value under every substitution from

R
R

is 0, nilpotent or a central element of

R
R

, respectively. THEOREM. If

R
R

has no nonvanishing multilinear nil polynomials then neither has the matrix ring

R
n

{R_n}

. THEOREM. Let

R
R

be a ring satisfying a polynomial identity modulo its nil radical

N
N

, and let

f
f

be a multilinear polynomial. If

f
f

is nil in

R
R

then

f
f

is vanishing in

R

/

N

R/N

. Applied to the polynomial

x
y
−
y
x

xy - yx

, this establishes the validity of a conjecture of Herstein’s, in the presence of polynomial identity. THEOREM. Let

m
m

be a positive integer and let

F
F

be a field containing no

m
m

th roots of unity other than 1. If

f
f

is a multilinear polynomial such that for some

n
>
2

f
m

n > 2{f^m}

is central in

F
n

{F_n}

, then

f
f

is central in

F
n

{F_n}

. This is related to the (non)existence of noncrossed products among

p
2

{p^2}

-dimensional central division rings.
Fatou properties of monotone seminorms on Riesz spaces
Authors: Dodds Theresa K. Y. Chow
Published: July 2010

A monotone seminorm

ρ
\rho

on a Riesz space

L
L

is called

σ
\sigma

-Fatou if

ρ
(

u
n

)
↑
ρ
(
u
)

\rho ({u_n}) \uparrow \rho (u)

holds for every

u
∈

L
+

u \in {L^ + }

and sequence

{

u
n

}

\{ {u_n}\}

in

L
L

satisfying

0
≤

u
n

↑
u

0 \leq {u_n} \uparrow u

. A monotone seminorm

ρ
\rho

on

L
L

is called strong Fatou if

ρ
(

u
v

)
↑
ρ
(
u
)

\rho ({u_v}) \uparrow \rho (u)

holds for every

u
∈

L
+

u \in {L^ + }

and directed system

{

u
v

}

\{ {u_v}\}

in

L
L

satisfying

0
≤

u
v

↑
u

0 \leq {u_v} \uparrow u

. In this paper we determine those Riesz spaces

L
L

which have the property that, for any monotone seminorm

ρ
\rho

on

L
L

, the largest strong Fatou seminorm

ρ
m

{\rho _m}

majorized by

ρ
\rho

is of the form:

ρ
m

(
f
)
=
inf
{

sup
v

ρ
(

u
v

)
:
0
≤

u
v

↑

|

f

|

}

{\rho _m}(f) = \inf \{ {\sup _v}\rho ({u_v}):0 \leq {u_v} \uparrow |f|\}

]> for

f
∈
L

f \in L

. We discuss, in a Riesz space

L
L

, the condition that a monotone seminorm

ρ
\rho

as well as its Lorentz seminorm

ρ
L

{\rho _L}

is

σ
\sigma

-Fatou in terms of the order and relative uniform topologies on

L
L

. A parallel discussion is also given for outer measures on Boolean algebras.
Selection theorems and the reduction principle
Authors: Maitra A. , Rao B. V.
Published: July 2010

In this paper we attempt a unification of several selection theorems in the literature. It is proved that the existence of “nice” selectors in a certain class of selection problems is essentially equivalent to the fact that certain families of sets satisfy a weak version of Kuratowski’s reduction principle. Various special cases are discussed.
Modular constructions for combinatorial geometries
Authors: Brylawski Tom
Published: July 2010

R. Stanley, in an investigation of modular flats in geometries (Algebra Universalis 1-2 (1971), 214-217), proved that the characteristic polynomial

χ
(
x
)

\chi (x)

of a modular flat

x
x

divides the characteristic polynomial

χ
(
G
)

\chi (G)

of a geometry

G
G

. In this paper we identify the quotient: THEOREM. If

x
x

is a modular flat of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G comma chi left-parenthesis upper G right-parenthesis slash chi left-parenthesis x right-parenthesis equals chi left-parenthesis ModifyingAbove upper T Subscript x Baseline With bar left-parenthesis upper G right-parenthesis right-parenthesis slash left-parenthesis lamda minus 1 right-parenthesis">

G
,
χ
(
G
)

/

χ
(
x
)
=
χ
(

T
x

¯

(
G
)
)

/

(
λ
−
1
)

G,\chi (G)/\chi (x) = \chi (\overline {{T_x}} (G))/(\lambda - 1)

, where

T
x

¯

(
G
)

\overline {{T_x}} (G)

is the complete Brown truncation of

G
G

by

x
x

. (The lattice of

T
x

¯

(
G
)

\overline {{T_x}} (G)

consists of all flats containing

x
x

and all flats disjoint from

x
x

, with the induced order from

G
G

.) We give many characterizations of modular flats in terms of their lattice properties as well as by means of a short-circuit axiom and a modular version of the MacLane-Steinitz exchange axiom. Modular flats are shown to have many of the useful properties of points and distributive flats (separators) in addition to being much more prevalent. The theorem relating the chromatic polynomials of two graphs and the polynomial of their vertex join across a common clique generalizes to geometries: THEOREM. Given geometries

G
G

and

H
H

, if

x
x

is a modular flat of

G
G

as well as a subgeometry of

H
H

, then there exists a geometry

P
=

P
x

(
G
,
H
)

P = {P_x}(G,H)

which is a pushout in the category of injective strong maps and such that

χ
(
P
)
=
χ
(
G
)
χ
(
H
)

/

χ
(
x
)

\chi (P) = \chi (G)\chi (H)/\chi (x)

. The closed set structure, rank function, independent sets, and lattice properties of

P
P

are characterized. After proving a modular extension theorem we give applications of our results to Crapo’s single element extension theorem, Crapo’s join operation, chain groups, unimodular geometries, transversal geometries, and graphs.
Polynomials and the limit point condition
Authors: Kauffman Robert M.
Published: July 2010

An

n
n

th order, possibly nonselfadjoint, ordinary differential expression

L
L

is said to be in the limit point condition if the maximal operator

L
M

{L_M}

in

L
2

[
0
,
∞
)

{L_2}[0,\infty )

is an

n
n

-dimensional extension of the minimal operator

L
0

{L_0}

. If range

L
0

{L_0}

is closed, this definition is equivalent to the assertion that nullity

L
M

+
nullity

(

L
+

)
M

=
n

{L_M} + \text {nullity} {({L^ + })_M} = n

, where

L
+

{L^ + }

is the formal adjoint of

L
L

. It also implies that any operator

T
T

such that

L
0

⊆
T
⊆

L
M

{L_0} \subseteq T \subseteq {L_M}

is the restriction of

L
M

{L_M}

to a set of functions described by a boundary condition at zero. In this paper, we discuss the question of when differential expressions involving complex polynomials in selfadjoint expressions are in the limit point condition.
Manifolds with no periodic homeomorphisms
Authors: Bloomberg Edward M.
Published: July 2010

An analysis of the ends of the universal cover of the connected sum of two aspherical manifolds leads to the construction of a class of closed manifolds with no nontrivial periodic homeomorphisms.
On the dimension of varieties of special divisors
Authors: Lax R. F.
Published: July 2010

Let

T
g

{T_g}

denote the Teichmüller space and let

V
V

denote the universal family of Teichmüller surfaces of genus

g
g

Let

V

T
g

(
n
)

V_{{T_g}}^{(n)}

denote the

n
n

th symmetric product of

V
V

over

T
g

{T_g}

and let

J
J

denote the family of Jacobians over

T
g

{T_g}

. Let

f
:

V

T
g

(
n
)

→
J

f:V_{{T_g}}^{(n)} \to \text {J}

be the natural relativization over

T
g

{T_g}

of the classical map defined by integrating holomorphic differentials. Let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u colon f Superscript asterisk Baseline normal upper Omega Subscript upper J slash upper T Sub Subscript g Subscript Superscript 1 Baseline right-arrow normal upper Omega Subscript upper V Sub Subscript upper T Sub Sub Subscript g Sub Subscript slash upper T Sub Sub Subscript g Sub Subscript Sub Superscript left-parenthesis n right-parenthesis Subscript Superscript 1">

u
:

f
∗

Ω

J

/

T
g

1

→

Ω

V

T
g

/

T
g

(
n
)

1

u:{f^\ast }\Omega _{\text {J} /{T_g}}^1 \to \Omega _{V_{{T_g}/{T_g}}^{(n)}}^1

\]
be the map induced by

f
f

. We define

G
n
r

G_n^r

to be the analytic subspace of

V

T
g

(
n
)

V_{{T_g}}^{(n)}

defined by the vanishing of

∧

n
−
r
+
1

u

{ \wedge ^{n - r + 1}}u

. Put

τ
=
(
r
+
1
)
(
n
−
r
)
−
r
g

\tau = (r + 1)(n - r) - rg

. We show that

G
n
1

−

G
n
2

G_n^1 - G_n^2

, if nonempty, is smooth of pure dimension

3
g
−
3
+
τ
+
1

3g - 3 + \tau + 1

. From this result, we may conclude that, for a generic curve

X
X

, the fiber of

G
n
1

−

G
n
2

G_n^1 - G_n^2

over the module point of

X
X

, if nonempty, is smooth of pure dimension

τ
+
1

\tau + 1

, a classical assertion. Variational formulas due to Schiffer and Spencer and Rauch are employed in the study of

G
n
r

G_n^r

.