bluuebunny/crossref_metadata_2025 huggingface dataset:
Construction of automorphic forms and integrals
Authors: Niebur Douglas
Published: July 2010

It is well known that modular forms of positive dimension have Fourier coefficients given by certain infinite series involving Kloostermann sums and the modified Bessel function of the first kind. In this paper a functional equation which characterizes all such Fourier series is found. It is also shown that these Fourier series have a construction similar to that of Poincaré series of negative dimension.
Mayer-Vietoris sequences and Brauer groups of nonnormal domains
Authors: Childs L. N.
Published: July 2010

Let R be a Noetherian domain with finite integral closure

R
¯

\bar R

. We study the map from the Brauer group of

R
,
B
(
R
)

R,B(R)

, to

B
(

R
¯

)

B(\bar R)

: first, by embedding

B
(
R
)

B(R)

into the Čech etale cohomology group

H
2

(
R
,
U
)

{H^2}(R,U)

and using a Mayer-Vietoris sequence for Čech cohomology of commutative rings; second, via Milnor’s theorem from algebraic K-theory. We apply our results to show, i.e., that if R is a domain with quotient field K a global field, then the map from

B
(
R
)

B(R)

to

B
(
K
)

B(K)

is 1-1.
A generalization of the 𝑐𝑜𝑠𝜋𝜌 theorem
Authors: Baernstein Albert
Published: July 2010

Let f be an entire function, and let

β
\beta

and

λ
\lambda

be positive numbers with

β
≤
π

\beta \leq \pi

and

β
λ
>
π

\beta \lambda > \pi

. Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis r right-parenthesis equals StartSet theta colon log StartAbsoluteValue f left-parenthesis r e Superscript i theta Baseline right-parenthesis EndAbsoluteValue greater-than cosine beta lamda log upper M left-parenthesis r right-parenthesis EndSet">

E
(
r
)
=
{
θ
:
log
⁡

|

f
(
r

e

i
θ

)

|

>
cos
⁡
β
λ
log
⁡
M
(
r
)
}

E(r) = \{ \theta :\log |f(r{e^{i\theta }})| > \cos \beta \lambda \log M(r)\}

. It is proved that either there exist arbitrarily large values of r for which

E
(
r
)

E(r)

contains an interval of length at least

2
β

2\beta

, or else

lim

r
→
∞

r

−
λ

log
⁡
M
(
r
,
f
)

{\lim _{r \to \infty }}{r^{ - \lambda }}\log M(r,f)

exists and is positive or infinite. For

β
=
π

\beta = \pi

this is Kjellberg’s refinement of the cos

π
ρ

\pi \rho

theorem.
Analytic domination with quadratic form type estimates and nondegeneracy of ground states in quantum field theory
Authors: Sloan Alan D.
Published: July 2010

We present a theorem concerning the analytic domination by a semi-bounded selfadjoint operator H of another linear operator A which requires only the quadratic form type estimates
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper H Superscript negative 1 slash 2 Baseline left-parenthesis left-parenthesis ad upper A right-parenthesis Superscript n Baseline upper H right-parenthesis upper H Superscript negative 1 slash 2 Baseline u double-vertical-bar less-than-or-equal-to c Subscript n Baseline double-vertical-bar u double-vertical-bar">

‖

H

−
1

/

2

(

(

ad

A
)

n

H
)

H

−
1

/

2

u

‖

≤

c
n

‖
u
‖

\left \| {{H^{ - 1/2}}({{({\text {ad}}\;A)}^n}H){H^{ - 1/2}}u} \right \| \leq {c_n}\left \| u \right \|

\]
instead of the norm estimates
\[

‖

(

ad

A
)

n

H
u

‖

≤

c
n

‖

H
u

‖

\left \| {{{({\text {ad}}\;A)}^n}Hu} \right \| \leq {c_n}\left \| {Hu} \right \|

\]
usually required for this type of theorem. We call the new estimates “quadratic form type", since they are sometimes equivalent to
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue left-parenthesis left-parenthesis ad upper A right-parenthesis Superscript n Baseline upper H u comma u right-parenthesis EndAbsoluteValue less-than-or-equal-to c Subscript n Baseline StartAbsoluteValue left-parenthesis upper H u comma u right-parenthesis EndAbsoluteValue period">

|

(

(

ad

A

)
n

H
u
,
u
)

|

≤

c
n

|

(
H
u
,
u
)

|

.

|({({\text {ad}}\;A)^n}Hu,u)| \leq {c_n}|(Hu,u)|.

\]
The theorem is then applied with H the Hamiltonian for the spatially cutoff boson field model with real, bounded below, even ordered polynomial self-interaction in one space dimension and

A
=
π
(
g
)

A = \pi (g)

, the conjugate momentum to the free field. When the underlying Hilbert space of this model is represented as

L
2

(
Q
,
d
q
)

{L^2}(Q,dq)

where dq is a probability measure on Q, the spectrum of the von Neumann algebra generated by bounded functions of certain field operators, then

e

−
t
H

{e^{ - tH}}

maximizes support in the sense that

e

−
t
H

f

{e^{ - tH}}f

is nonzero almost everywhere whenever f is not identically zero.
Algebras over absolutely flat commutative rings
Authors: Wehlen Joseph A.
Published: July 2010

Let A be a finitely generated algebra over an absolutely flat commutative ring. Using sheaf-theoretic techniques, it is shown that the weak Hochschild dimension of A is equal to the supremum of the Hochschild dimension of

A
x

{A_x}

for x in the decomposition space of R. Using this fact, relations are obtained among the weak Hochschild dimension of A and the weak global dimensions of A and

A
e

{A^e}

. It is also shown that a central separable algebra is a biregular ring which is finitely generated over its center. A result of S. Eilenberg concerning the separability of A modulo its Jacobson radical is extended. Finally, it is shown that every homomorphic image of an algebra of weak Hochschild dimension 1 is a type of triangular matrix algebra.
Riesz points of the spectrum of an element in a semisimple Banach algebra
Authors: Pearlman Lynn D.
Published: July 2010

Let A be a semisimple Banach algebra with unit element and let

S
A

{S_A}

denote the socle of A. For an element y in A, let

L
y

[

R
y

]

{L_y}[{R_y}]

denote the operator of left [right] multiplication by y on A. The operational calculus and A. E. Taylor’s theory of the ascent

α
(
T
)

\alpha (T)

and descent

δ
(
T
)

\delta (T)

of an operator T on A are used to show that the following conditions on a number

λ
\lambda

in the spectrum of an element x in A are all equivalent. (1)

λ
\lambda

is a pole of the resolvent mapping

z
→

(
z
−
x

)

−
1

z \to {(z - x)^{ - 1}}

and the spectral idempotent f, for x at

λ
\lambda

is in

S
A

{S_A}

. (2)

λ
−
x
−
c

\lambda - x - c

is invertible in A for some c in the closure of

S
A

{S_A}

such that

c
x
=
x
c

cx = xc

. (3)

λ
−
x

\lambda - x

is invertible modulo the closure of

S
A

{S_A}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than alpha left-parenthesis upper L Subscript left-parenthesis lamda minus x right-parenthesis Baseline right-parenthesis equals delta left-parenthesis upper L Subscript left-parenthesis lamda minus x right-parenthesis Baseline right-parenthesis greater-than normal infinity">

0
>
α
(

L

(
λ
−
x
)

)
=
δ
(

L

(
λ
−
x
)

)
>
∞

0 > \alpha ({L_{(\lambda - x)}}) = \delta ({L_{(\lambda - x)}}) > \infty

. (4)

λ
−
x

\lambda - x

is invertible modulo the closure of

S
A

{S_A}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than alpha left-parenthesis upper R Subscript left-parenthesis lamda minus x right-parenthesis Baseline right-parenthesis equals delta left-parenthesis upper R Subscript left-parenthesis lamda minus x right-parenthesis Baseline right-parenthesis equals alpha left-parenthesis upper L Subscript left-parenthesis lamda minus x right-parenthesis Baseline right-parenthesis equals delta left-parenthesis upper L Subscript left-parenthesis lamda minus x right-parenthesis Baseline right-parenthesis greater-than normal infinity">

0
>
α
(

R

(
λ
−
x
)

)
=
δ
(

R

(
λ
−
x
)

)
=
α
(

L

(
λ
−
x
)

)
=
δ
(

L

(
λ
−
x
)

)
>
∞

0 > \alpha ({R_{(\lambda - x)}}) = \delta ({R_{(\lambda - x)}}) = \alpha ({L_{(\lambda - x)}}) = \delta ({L_{(\lambda - x)}}) > \infty

. Such numbers

λ
\lambda

are called Riesz points. An element x is called a Riesz element of A if it is topologically nilpotent modulo the closure of

S
A

{S_A}

. It is shown that x is a Riesz element if and only if every nonzero number in the spectrum of x is a Riesz point.
On totally real submanifolds
Authors: Chen Bang-yen , Ogiue Koichi
Published: July 2010

Complex analytic submanifolds and totally real submanifolds are two typical classes among all submanifolds of an almost Hermitian manifold. In this paper, some characterizations of totally real submanifolds are given. Moreover some classifications of totally real submanifolds in complex space forms are obtained.
Homeomorphisms of a certain cube with holes
Authors: Myers Donald
Published: July 2010

For some manifolds the group of isotopy classes of self-homeomorphisms is known. In this paper this group is computed for a well-known cube with two holes. Two related manifolds are defined and the groups of isotopy classes on these manifolds are given without proof. One of these cubes with holes is such that every homeomorphism is isotopic to the identity.
Primitive satisfaction and equational problems for lattices and other algebras
Authors: Baker Kirby A.
Published: July 2010

This paper presents a general method of solving equational problems in all equational classes of algebras whose congruence lattices are distributive, such as those consisting of lattices, relation algebras, cylindric algebras, orthomodular lattices, lattice-ordered rings, lattice-ordered groups, Heyting algebras, other lattice-ordered algebras, implication algebras, arithmetic rings, and arithmetical algebras.
Standard polynomials in matrix algebras
Authors: Rowen Louis H.
Published: July 2010

Let

M
n

(
F
)

{M_n}(F)

be an

n
×
n

n \times n

matrix ring with entries in the field F, and let

S
k

(

X
1

,
…
,

X
k

)

{S_k}({X_1}, \ldots ,{X_k})

be the standard polynomial in k variables. Amitsur-Levitzki have shown that

S

2
n

(

X
1

,
…
,

X

2
n

)

{S_{2n}}({X_1}, \ldots ,{X_{2n}})

vanishes for all specializations of

X
1

,
…
,

X

2
n

{X_1}, \ldots ,{X_{2n}}

to elements of

M
n

(
F
)

{M_n}(F)

. Now, with respect to the transpose, let

M
n
−

(
F
)

M_n^ - (F)

be the set of antisymmetric elements and let

M
n
+

(
F
)

M_n^ + (F)

be the set of symmetric elements. Kostant has shown using Lie group theory that for n even

S

2
n
−
2

(

X
1

,
…
,

X

2
n
−
2

)

{S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})

vanishes for all specializations of

X
1

,
…
,

X

2
n
−
2

{X_1}, \ldots ,{X_{2n - 2}}

to elements of

M
n
−

(
F
)

M_n^ - (F)

. By strictly elementary methods we have obtained the following strengthening of Kostant’s theorem:

S

2
n
−
2

(

X
1

,
…
,

X

2
n
−
2

)

{S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})

vanishes for all specializations of

X
1

,
…
,

X

2
n
−
2

{X_1}, \ldots ,{X_{2n - 2}}

to elements of

M
n
−

(
F
)

M_n^ - (F)

, for all n.

S

2
n
−
1

(

X
1

,
…
,

X

2
n
−
1

)

{S_{2n - 1}}({X_1}, \ldots ,{X_{2n - 1}})

vanishes for all specializations of

X
1

,
…
,

X

2
n
−
2

{X_1}, \ldots ,{X_{2n - 2}}

to elements of

M
n
−

(
F
)

M_n^ - (F)

and of

X

2
n
−
1

{X_{2n - 1}}

to an element of

M
n
+

(
F
)

M_n^ + (F)

, for all n.

S

2
n
−
2

(

X
1

,
…
,

X

2
n
−
2

)

{S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})

vanishes for all specializations of

X
1

,
…
,

X

2
n
−
3

{X_1}, \ldots ,{X_{2n - 3}}

to elements of

M
n
−

(
F
)

M_n^ - (F)

and of

X

2
n
−
2

{X_{2n - 2}}

to an element of

M
n
+

(
F
)

M_n^ + (F)

, for n odd. These are the best possible results if F has characteristic 0; a complete analysis of the problem is also given if F has characteristic 2.
Product of ring varieties and attainability
Authors: Iskander Awad A.
Published: July 2010

The class of all rings that are Everett extensions of a ring in a variety

U

\mathfrak {U}

by a ring in a variety

B

\mathfrak {B}

is a variety

U

⋅

B

\mathfrak {U} \cdot \mathfrak {B}

. With respect to this operation the set of all ring varieties is a partially ordered groupoid (under inclusion), that is not associative. A variety is idempotent iff it is the variety of all rings, or generated by a finite number of finite fields. No families of polynomial identities other than those equivalent to

x
=
x

x = x

or

x
=
y

x = y

are attainable on the class of all rings or on the class of all commutative rings.
Goldie-like conditions on Jordan matrix rings
Authors: Britten Daniel J.
Published: July 2010

In this paper Goldie-like conditions are put on a Jordan matrix ring

J
=
H
(

R
n

,

γ
a

)

J = H({R_n},{\gamma _a})

which are necessary and sufficient for R to be a

∗
\ast

-prime Goldie ring or a Cayley-Dickson ring. Existing theory is then used to obtain a Jordan ring of quotients for J.
Semi-isotopies and the lattice of inner ideals of certain quadratic Jordan algebras
Authors: Katz Jerome M.
Published: July 2010

The concept of isotopy plays an extremely important role in the structure theory of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals. We take Koecher’s characterization of isotopy and use it as the basis of a definition of semi-isotopy. It is clear that semi-isotopies induce, in a natural way, automorphisms of the lattice of inner ideals. We concern ourselves with the converse problem; namely, if

η
\eta

is a semilinear bijection of a quadratic Jordan algebra such that

η
\eta

induces an automorphism of the lattice of inner ideals, is

η
\eta

necessarily a semi-isotopy? We answer the above question in the affirmative for a large class of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals (said class includes all such algebras of capacity at least three over fields of characteristic unequal to two). Moreover, we prove that the only such maps which induce the identity automorphism on the lattice are the scalar multiplications.
Some 𝑝-groups of maximal class
Authors: Miech R. J.
Published: July 2010

This paper deals with the construction of some p-groups of maximal class.
On certain convex sets in the space of locally schlicht functions
Authors: Kim Y. J. , Merkes E. P.
Published: July 2010

Let

H
=
H

(
∗

,
[
+
]
)

H = H{(^ \ast },[ + ])

denote the real linear space of locally schlicht normalized functions in

|

z

|

>
1

|z| > 1

as defined by Hornich. Let K and C respectively be the classes of convex functions and the close-to-convex functions. If

M
⊂
H

M \subset H

there is a closed nonempty convex set in the

α
β

\alpha \beta

-plane such that for

(
α
,
β
)

(\alpha ,\beta )

in this set

α
∗

f
[
+
]

β
∗

g
∈
C

{\alpha ^ \ast }f[ + ]{\beta ^ \ast }g \in C

(in K) whenever f,

g
∈
M

g \in M

. This planar convex set is explicitly given when M is the class K, the class C, and for other classes. Some consequences of these results are that K and C are convex sets in H and that the extreme points of C are not in K.
Polyanalytic functions with exceptional values
Authors: Krajkiewicz P.
Published: July 2010

Let

f
(
z
)
=

∑

k
=
0

n

z
¯

k

f
k

(
z
)

f(z) = \sum \nolimits _{k = 0}^n {{{\bar z}^k}{f_k}(z)}

where the functions

f
0

,

f
1

,
⋯
,

f
n

{f_0},{f_1}, \cdots ,{f_n}

are analytic on some annular neighborhood A of the point

∞
\infty

and

f
n

≡
1

{f_n} \equiv 1

on A and z denotes the complex conjugate of z. If f does not vanish on A it is shown that the functions

f
0

,

f
1

,
⋯
,

f

n
−
1

{f_0},{f_1}, \cdots ,{f_{n - 1}}

have a nonessential isolated singularity at the point infinity.
Parabolic equations associated with the number operator
Authors: Piech M. Ann
Published: July 2010

We study existence and uniqueness of solutions of the Cauchy problem for

u
˙

=
N
u

\dot u = Nu

where N is the number operator on abstract Wiener space.
On structure spaces of ideals in rings of continuous functions
Authors: Rudd David
Published: July 2010

A ring of continuous functions is a ring of the form

C
(
X
)

C(X)

, the ring of all continuous real-valued functions on a completely regular Hausdorff space X. With each ideal I of

C
(
X
)

C(X)

, we associate certain subalgebras of

C
(
X
)

C(X)

, and discuss their structure spaces. We give necessary and sufficient conditions for two ideals in rings of continuous functions to have homeomorphic structure spaces.
Finite extensions of minimal transformation groups
Authors: Sacker Robert J. , Sell George R.
Published: July 2010

In this paper we shall study homomorphisms

p
:
W
→
Y

p:W \to Y

on minimal transformation groups. We shall prove, in the case that W and Y are metrizable, that W is a finite (N-to-1) extension of Y if and only if W is an N-fold covering space of Y and p is a covering map. This result places no further restrictions on the acting group. We shall then use this characterization to investigate the question of lifting an equicontinuous structure from Y to W. We show that, under very weak restrictions on the acting group, this lifting is always possible when W is a finite extension of Y.
Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups
Authors: Kutz Ronald E.
Published: July 2010

Theorem. Let R be a commutative Noetherian ring with identity. Let M =

M
=
(

c

i
j

)

M = ({c_{ij}})

be an s by s symmetric matrix with entries in R. Let I the be ideal of

t
+
1

t + 1

by

t
+
1

t + 1

minors of M. Suppose that the grade of I is as large as possible, namely, gr

I
=
g
=
s
(
s
+
1
)

/

2
−
s
t
+
t
(
t
−
1
)

/

2

I = g = s(s + 1)/2 - st + t(t - 1)/2

. Then I is a perfect ideal, so that

R

/

I

R/I

is Cohen Macaulay if R is. Let G be a linear algebraic group acting rationally on

R
=
K
[

x
1

,
…
,

x
n

]

R = K[{x_1}, \ldots ,{x_n}]

. Hochster has conjectured that if G is reductive, then

R
G

{R^G}

is Cohen-Macaulay, where

R
G

{R^G}

denotes the ring of invariants of the action of G. The above theorem provides a special case of this conjecture. For

G
=
O
(
t
,
K
)

G = O(t,K)

, the orthogonal group, and K a field of characteristic zero, the above yields: Corollary. For R and G as above,

R
G

{R^G}

is Cohen-Macaulay for an appropriate action of G. In order to obtain these results it was necessary to prove a more general form of the theorem stated above, which in turn yields a more general form of the corollary.
New criteria for freeness in abelian groups. II
Authors: Hill Paul
Published: July 2010

A new criterion is established for an abelian group to be free. The criterion is in terms of an ascending chain of free subgroups and is dependent upon a new class of torsion-free groups. The result leads to the construction, for each positive integer n, of a group

G
n

{G_n}

of cardinality

ℵ
n

{\aleph _n}

that is not free but is

ℵ
n

{\aleph _n}

-free. A conjecture in infinitary logic concerning free abelian groups is also verified.
The 𝑝-adic hull of abelian groups
Authors: Mader A.
Published: July 2010

In this paper we define “p-adic hull” for p-reduced groups K. The p-adic hull

K
P

{K^P}

of K is a module over the ring P of p-adic integers containing K and satisfying certain additional properties. The notion is investigated and then used to prove some known and some new theorems on

Ext
⁡
(
K
,
T
)

\operatorname {Ext}(K,T)

and

Hom
⁡
(
K
,
T
)

\operatorname {Hom}(K,T)

for K torsion-free and T a reduced p-group.
An isomorphism and isometry theorem for a class of linear functionals
Authors: Appling William D. L.
Published: July 2010

Suppose

U
U

is a set,

F

{\mathbf {F}}

is a field of subsets of

U
U

and

p

A
B

{\mathfrak {p}_{AB}}

is the set of all real-valued, finitely additive functions defined on

F

{\mathbf {F}}

. Two principal notions are considered in this paper. The first of these is that of a subset of

p

A
B

{\mathfrak {p}_{AB}}

, defined by certain closure properties and called a

C
C

-set. The second is that of a collection

C

\mathcal {C}

of linear transformations from

p

A
B

{\mathfrak {p}_{AB}}

into

p

A
B

{\mathfrak {p}_{AB}}

with special boundedness properties. Given a

C
C

-set

M
M

which is a linear space, an isometric isomorphism is established from the dual of

M
M

onto the set of all elements of

C

\mathcal {C}

with range a subset of

M
M

. As a corollary it is demonstrated that the above-mentioned isomorphism and isometry theorem, together with a previous representation theorem of the author (J. London Math. Soc. 44 (1969), pp. 385-396), imply an analogue of a dual representation theorem of Edwards and Wayment (Trans. Amer. Math. Soc. 154 (1971), pp. 251-265). Finally, a “pseudo-representation theorem” for the dual of

p

A
B

{\mathfrak {p}_{AB}}

is demonstrated.
Almost complex structures on complex projective spaces
Authors: Thomas Alan
Published: July 2010

In this paper we classify the almost complex structures on a complex projective space as roots of a certain polynomial equation.
Global dimension of tiled orders over commutative noetherian domains
Authors: Jategaonkar Vasanti A.
Published: July 2010

Let R be a commutative noetherian domain and

Λ
=
(

Λ

i
j

)
⊆

M
n

(
R
)

\Lambda = ({\Lambda _{ij}}) \subseteq {M_n}(R)

be a tiled R-order. The main result of this paper is the following Theorem. Let gl

dim
⁡
R
=
d
>
∞

\dim R = d > \infty

and

Λ
\Lambda

a triangular tiled R-order (i.e.,

Λ

i
j

=
R

{\Lambda _{ij}} = R

whenever

i
≤
j

i \leq j

). Then the following three conditions are equivalent: (1) gl

dim
⁡
Λ
>
∞

\dim \Lambda > \infty

; (2)

Λ

i
,
i
−
1

=
R

{\Lambda _{i,i - 1}} = R

or gl

dim

(
R

/

Λ

i
,
i
−
1

)
>
∞

\dim \;(R/{\Lambda _{i,i - 1}}) > \infty

, whenever

2
≤
i
≤
n

2 \leq i \leq n

; (3) gl

dim
⁡
Λ
≤
d
(
n
−
1
)

\dim \Lambda \leq d(n - 1)

. If

d
=
1

d = 1

or 2 then the upper bound in the above theorem is best possible. We give a sufficient condition for an arbitrary tiled R-order

Λ
\Lambda

to be of finite global dimension.
Subgroups of groups of central type
Authors: Timmer Kathleen M.
Published: July 2010

Let

λ
\lambda

be a linear character on the center Z of a finite group Z of a finite group H, such that (1)

λ
H

=

∑

i
=
1

p

ϕ
i

(
1
)

ϕ
i

{\lambda ^H} = \sum \nolimits _{i = 1}^p {{\phi _i}(1){\phi _i}}

where the

ϕ
i

{\phi _i}

’s are inequivalent irreducible characters on H of the same degree, and (2) if

∑

i
=
1

p

m
i

ϕ
i

(
x
)
=
0

\sum \nolimits _{i = 1}^p {{m_i}{\phi _i}(x) = 0}

for some

x
∈
H

x \in H

and nonnegative integers

m
i

{m_i}

, then either

ϕ
i

(
x
)
=
0

{\phi _i}(x) = 0

for all i or

m
i

=

m
j

{m_i} = {m_j}

for all i, j. The object of the paper is to describe finite groups which satisfy conditions (1) and (2) in terms of the multiplication of the group. If S is a p Sylow subgroup of the group H, and

R
=
S
⋅
Z

R = S \cdot Z

, then H satisfies conditions (1) and (2) if and only if (a)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace x element-of upper H colon x Superscript negative 1 Baseline h Superscript negative 1 Baseline x h element-of upper Z right double arrow lamda left-parenthesis x Superscript negative 1 Baseline h Superscript negative 1 Baseline x h right-parenthesis equals 1 comma h element-of upper H right-brace slash upper Z">

{
x
∈
H
:

x

−
1

h

−
1

x
h
∈
Z
⇒
λ
(

x

−
1

h

−
1

x
h
)
=
1
,
h
∈
H
}

/

Z

\{ x \in H:{x^{ - 1}}{h^{ - 1}}xh \in Z \Rightarrow \lambda ({x^{ - 1}}{h^{ - 1}}xh) = 1,h \in H\} /Z

consists of elements of order a power of p in

H

/

Z

H/Z

, and these elements form p conjugacy classes of

H

/

Z

H/Z

, and (b) the elements of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace x element-of upper R colon x Superscript negative 1 Baseline r Superscript negative 1 Baseline x r element-of upper Z right double arrow lamda left-parenthesis x Superscript negative 1 Baseline r Superscript negative 1 Baseline x r right-parenthesis equals 1 comma r element-of upper R right-brace slash upper Z">

{
x
∈
R
:

x

−
1

r

−
1

x
r
∈
Z
⇒
λ
(

x

−
1

r

−
1

x
r
)
=
1
,
r
∈
R
}

/

Z

\{ x \in R:{x^{ - 1}}{r^{ - 1}}xr \in Z \Rightarrow \lambda ({x^{ - 1}}{r^{ - 1}}xr) = 1,r \in R\} /Z

form p conjugacy classes of

R

/

Z

R/Z

.
Multiplier transformations on compact Lie groups and algebras
Authors: Strichartz Robert S.
Published: July 2010

Let G be a semisimple compact Lie group and

T
f
=
∑
ϕ
(
m
)

d

m
χ
m

∗
f

Tf = \sum \phi (m){d_{m \chi m}} \ast f

a bi-invariant operator on

L
2

(
G
)

{L^2}(G)

, where

χ
m

{\chi _m}

and

d
m

{d_m}

are the characters and dimensions of the irreducible representations of G, which are indexed by a lattice of points m in the Lie algebra

G

\mathfrak {G}

in a natural way. If

Φ
\Phi

is a bounded ad-invariant function on

G

\mathfrak {G}

and

(

∗

)

ϕ

(

m

) = 

Φ

(

m

 + 

β

)

or

\begin{equation}\tag {$\ast $} \phi {\text {(}}m{\text {) = }}\Phi {\text {(}}m{\text { + }}\beta {\text {)}}\quad {\text {or}}\end{equation}

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk asterisk right-parenthesis EndLabel phi left-parenthesis m right-parenthesis equals integral Underscript upper G Endscripts normal upper Phi left-parenthesis m plus beta minus ad g beta right-parenthesis d g EndLayout">

(

∗
∗

)

ϕ

(

m

) = 

∫
G

Φ
(
m
+
β
−

ad

g
β
)
d
g

\begin{equation}\tag {$\ast \ast $} \phi {\text {(}}m{\text {) = }}\int _G {\Phi (m + \beta - {\text {ad}}\;g\beta )dg} \end{equation}

β
\beta

being half the sum of the positive roots, then various properties of T are related to properties of the Fourier multiplier transformation on

G

\mathfrak {G}

with multiplier

Φ
\Phi

. These properties include boundedness on

L
1

{L^1}

, uniform boundedness on

L
p

{L^p}

of a family of operators, and, in the special case

G
=

SO

(
3
)

G = {\text {SO}}(3)

, boundedness in

L
p

{L^p}

for ad-invariant functions with

1
≤
p
>
3

/

2

1 \leq p > 3/2

.
Analytic centers and analytic diameters of planar continua
Authors: Minsker Steven
Published: July 2010

This paper contains some basic results about analytic centers and analytic diameters, concepts which arise in Vitushkin’s work on rational approximation. We use Carathéodory’s theorem to calculate

β
(
K
,
z
)

\beta (K,z)

in the case in which K is a continuum in the complex plane. This leads to the result that, if

g
:
Ω
(
K
)
→
Δ
(
0
;
1
)

g:\Omega (K) \to \Delta (0;1)

is the normalized Riemann map, then

β
(
g
,
0
)

/

γ
(
K
)

\beta (g,0)/\gamma (K)

is the unique analytic center of K and

β
(
K
)
=
γ
(
K
)

\beta (K) = \gamma (K)

. We also give two proofs of the fact that

β
(
g
,
0
)

/

γ
(
K
)
∈

co

(
K
)

\beta (g,0)/\gamma (K) \in {\text {co}}\;(K)

. We use Bieberbach’s and Pick’s theorems to obtain more information about the geometric location of the analytic center. Finally, we obtain inequalities for

β
(
E
,
z
)

\beta (E,z)

for arbitrary bounded planar sets E.
Fundamental constants for rational functions
Authors: Poreda S. J. , Saff E. B. , Shapiro G. S.
Published: July 2010

Suppose R is a rational function with n poles all of which lie inside

Γ
\Gamma

, a closed Jordan curve. Lower bounds for the uniform norm of the difference

R
−
p

R - p

on

Γ
\Gamma

, where p is any polynomial, are obtained (in terms of the norm of R on

Γ
\Gamma

). In some cases these bounds are independent of

Γ
\Gamma

as well as R and p. Some related results are also given.
Analytic capacity, Hölder conditions and 𝜏-spikes
Authors: O’Farrell Anthony G.
Published: July 2010

We consider the uniform algebra

R
(
X
)

R(X)

, for compact

X
⊂

C

X \subset {\text {C}}

, in relation to the condition
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript p plus alpha Baseline equals normal upper Sigma 1 Superscript normal infinity Baseline 2 Superscript left-parenthesis p plus alpha plus 1 right-parenthesis n Baseline gamma left-parenthesis upper A Subscript n Baseline left-parenthesis x right-parenthesis minus upper X right-parenthesis greater-than plus normal infinity">

I

p
+
α

=

Σ
1
∞

2

(
p
+
α
+
1
)
n

γ
(

A
n

(
x
)
∖
X
)
>
+
∞

{I_{p + \alpha }} = \Sigma _1^\infty {2^{(p + \alpha + 1)n}}\gamma ({A_n}(x)\backslash X) > + \infty

, where

0
≤
p
∈

Z

,
0
>
α
>
1
,
γ

0 \leq p \in {\mathbf {Z}},0 > \alpha > 1,\gamma

is analytic capacity, and

A
n

(
x
)

{A_n}(x)

is the annulus

{
z
∈

C

:

2

−
n
−
1

>

|

z
−
x

|

>

2

−
n

}

\{ z \in {\text {C}}:{2^{ - n - 1}} > |z - x| > {2^{ - n}}\}

. We introduce the notion of

τ
\tau

-spike for

τ
>
0

\tau > 0

, and show that

I

p
+
α

=
+
∞

{I_{p + \alpha }} = + \infty

implies

x
x

is a

p
+
α

p + \alpha

-spike. If

X
˚

\mathring {X}

satisfies a cone condition at

x
x

, and

I

p
+
α

>
+
∞

{I_{p + \alpha }} > + \infty

, we show that the

p
p

th derivatives of the functions in

R
(
X
)

R(X)

satisfy a uniform Hölder condition at

x
x

for nontangential approach. The structure of the set of non-

τ
\tau

-spikes is examined and the results are applied to rational approximation. A geometric question is settled.
Commutative twisted group algebras
Authors: Smith Harvey A.
Published: July 2010

A twisted group algebra

L
1

(
A
,
G
;
T
,
α
)

{L^1}(A,G;T,\alpha )

is commutative iff A and G are, T is trivial and

α
\alpha

is symmetric:

α
(
γ
,
g
)
=
α
(
g
,
γ
)

\alpha (\gamma ,g) = \alpha (g,\gamma )

. The maximal ideal space

L
1

(
A
,
G
;
α
)

∅
^

{L^1}(A,G;\alpha )\hat \emptyset

of a commutative twisted group algebra is a principal

G

∅
^

G\hat \emptyset

bundle over

A

∅
^

A\hat \emptyset

. A class of principal

G

∅
^

G\hat \emptyset

bundles over second countable locally compact M is defined which is in 1-1 correspondence with the (isomorphism classes of)

C
∞

(
M
)

{C_\infty }(M)

-valued commutative twisted group algebras on G. If G is finite only locally trivial bundles can be such duals, but in general the duals need not be locally trivial.
Entire vectors and holomorphic extension of representations. II
Authors: Penney Richard
Published: July 2010

Let G be a connected, simply connected Lie group and let

G
c

{G_c}

be its complexification. Let U be a unitary representation of G. The space of vectors v at which U is holomorphically extendible to

G
c

{G_c}

is denoted

H

∞
ω

(
U
)

\mathcal {H}_\infty ^\omega (U)

. In [9] we characterized those U for which

H

∞
ω

\mathcal {H}_\infty ^\omega

is dense. In the present work we study

H

∞
ω

\mathcal {H}_\infty ^\omega

as a topological vector space, proving e.g., that

H

∞
ω

\mathcal {H}_\infty ^\omega

is a Montel space if U is irreducible and G is nilpotent. We prove a representation theorem for

(

H

∞
ω

)
′

(\mathcal {H}_\infty ^\omega )’

which yields a Bergman kernel type theorem for G. As an application we give a necessary and sufficient condition for the set of holomorphic functions on certain solvmanifolds to separate points.
General theory of the factorization of ordinary linear differential operators
Authors: Zettl Anton
Published: July 2010

The problem of factoring the general ordinary linear differential operator

L
y
=

y

(
n
)

+

p

n
−
1

y

(
n
−
1
)

+
⋯
+

p
0

y

Ly = {y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y

into products of lower order factors is studied. The factors are characterized completely in terms of solutions of the equation

L
y
=
0

Ly = 0

and its adjoint equation

L
∗

y
=
0

{L^ \ast }y = 0

. The special case when L is formally selfadjoint of order

n
=
2
k

n = 2k

and the factors are of order k and adjoint to each other reduces to a well-known result of Rellich and Heinz:

L
=

Q
∗

Q

L = {Q^ \ast }Q

if and only if there exist solutions

y
1

,
⋯
,

y
k

{y_1}, \cdots ,{y_k}

of

L
y
=
0

Ly = 0

satisfying

W
(

y
1

,
⋯
,

y
k

)
≠
0

W({y_1}, \cdots ,{y_k}) \ne 0

and

[

y
i

;

y
j

]
=
0

[{y_i};{y_j}] = 0

for

i
,
j
=
1
,
⋯
,
k

i,j = 1, \cdots ,k

; where [ ; ] is the Lagrange bilinear form of L.
A probabilistic approach to 𝐻^{𝑝}(𝑅^{𝑑})
Authors: Stroock D. , Varadhan S. R. S.
Published: July 2010

The relationship between

H
p

(

R
d

)
,
1
≤
p
>
∞

{H^p}({R^d}),1 \leq p > \infty

, and the integrability of certain functionals of Brownian motion is established using the connection between probabilistic and analytic notions of functions with bounded mean oscillation. An application of this relationship is given in the derivation of an interpolation theorem for operators taking

H
1

(

R
d

)

{H^1}({R^d})

to

L
1

(

R
d

)

{L^1}({R^d})

.
Contracting spaces of maps on the countable direct limit of a space
Authors: Heisey Richard E.
Published: July 2010

We give conditions sufficient to imply the contractibility of the space of maps, with compact-open topology, on the countable direct limit of a space. Applying these conditions we obtain the following: Let F be the conjugate of a separable infinite-dimensional Banach space with bounded weak-

∗

^\ast

topology, or the countable direct limit of the real line. Then there is a contraction of the space of maps on F which simultaneously contracts the subspaces of open maps, embeddings, closed embeddings, and homeomorphisms. Corollaries of our work are that any homeomorphism on F, F as above, is invertibly isotopic to the identity, and the general linear group of the countable direct limit of lines is contractible.
Spectral orders, uniform integrability and Lebesgue’s dominated convergence theorem
Authors: Chong Kong Ming
Published: July 2010

Using the ’spectral’ order relations

≺
\prec

and

≺≺
\prec \prec

introduced by Hardy, Littlewood and Pólya, we characterize the uniform integrability of a family of integrable functions. We also prove an extension and a ’converse’ of the classical Lebesgue’s dominated convergence theorem in terms of the ’spectral’ orders

≺
\prec

and

≺≺
\prec \prec

.
On the constructibility of prime characteristic periodic associative and Jordan rings
Authors: Loustau J. A.
Published: July 2010

The object of this paper is to show that any periodic associative ring of prime characteristic can be embedded in a periodic associative ring of prime characteristic which is constructible from a relatively complemented, distributive lattic and a family of periodic fields. Further, it will be proved that any periodic Jordan ring of prime characteristic is also embeddable in a periodic Jordan ring which is constructible from a lattice of the above type and a family of periodic Jordan rings of a symmetric bilinear form.
A multiplier theorem for Fourier transforms
Authors: McCall James D.
Published: July 2010

A function f analytic in the upper half-plane

Π
+

{\Pi ^ + }

is said to be of class

E
p

(

Π
+

)
(
0
>
p
>
∞
)

{E_p}({\Pi ^ + })(0 > p > \infty )

if there exists a constant C such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative normal infinity Superscript normal infinity Baseline StartAbsoluteValue f left-parenthesis x plus i y right-parenthesis EndAbsoluteValue Superscript p Baseline d x less-than-or-equal-to upper C greater-than normal infinity">

∫

−
∞

∞

|

f
(
x
+
i
y
)

|

p

d
x
≤
C
>
∞

\smallint _{ - \infty }^\infty |f(x + iy){|^p}dx \leq C > \infty

for all

y
>
0

y > 0

. These classes are an extension of the

H
p

{H_p}

spaces of the unit disc U. For f belonging to

E
p

(

Π
+

)
(
0
>
p
≤
2
)

{E_p}({\Pi ^ + })(0 > p \leq 2)

, there exists a Fourier transform f with the property that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals 2 left-parenthesis pi right-parenthesis Superscript negative 1 Baseline integral Subscript 0 Superscript normal infinity Baseline ModifyingAbove f With caret left-parenthesis t right-parenthesis e Superscript i z t Baseline d t">

f
(
z
)
=
2

(
π

)

−
1

∫
0
∞

f
^

(
t
)

e

i
z
t

d
t

f(z) = 2{(\pi )^{ - 1}}\smallint _0^\infty \hat f(t){e^{izt}}dt

. This makes it possible to give a definition for the multiplication of

E
p

(

Π
+

)
(
0
>
p
≤
2
)

{E_p}({\Pi ^ + })(0 > p \leq 2)

into

L
q

(
0
,
∞
)

{L_q}(0,\infty )

that is analogous to the multiplication of

H
p

(
U
)

{H_p}(U)

into

l
q

{l_q}

. In this paper, we consider the case

0
>
p
>
1

0 > p > 1

and

p
≤
q

p \leq q

and derive a necessary and sufficient condition for multiplying

E
p

(

Π
+

)

{E_p}({\Pi ^ + })

into

L
q

(
0
,
∞
)

{L_q}(0,\infty )

.
Equisingular deformations of plane algebroid curves
Authors: Wahl Jonathan M.
Published: July 2010

We construct a formal versal equisingular deformation of a plane algebroid curve (in characteristic zero), and show it is smoothly embedded in the whole deformation space of the singularity. Closer analysis relates equisingular deformations of the curve to locally trivial deformations of a certain (nonreduced) projective curve. Finally, we prove that algebraic

π
1

{\pi _1}

of the complement of a plane algebroid curve remains constant during formal equisingular deformation.
On deformations of homomorphisms of locally compact groups
Authors: Lee Dong Hoon
Published: July 2010

The rigidity of homomorphisms of compactly generated locally compact groups into Lie groups is investigated.
The Fredholm spectrum of the sum and product of two operators
Authors: Shapiro Jack , Snow Morris
Published: July 2010

Let

C
(
X
)

C(X)

denote the set of closed operators with dense domain on a Banach space X, and

L
(
X
)

L(X)

the set of all bounded linear operators on X. Let

Φ

(
X
)

{\mathbf {\Phi }}(X)

denote the set of all Fredholm operators on X, and

σ

Φ

(
A
)

{\sigma _{\mathbf {\Phi }}}(A)

the set of all complex numbers

λ

{\mathbf {\lambda }}

such that

(

λ

−
A
)
∉

Φ

(
X
)

({\mathbf {\lambda }} - A) \notin {\mathbf {\Phi }}(X)

. In this paper we establish conditions under which
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript bold upper Phi Baseline left-parenthesis upper A plus upper B right-parenthesis subset-of-or-equal-to sigma Subscript bold upper Phi Baseline left-parenthesis upper A right-parenthesis plus sigma Subscript bold upper Phi Baseline left-parenthesis upper B right-parenthesis comma sigma Subscript bold upper Phi Baseline left-parenthesis ModifyingAbove upper B upper A With bar right-parenthesis subset-of-or-equal-to sigma Subscript bold upper Phi Baseline left-parenthesis upper A right-parenthesis dot sigma Subscript bold upper Phi Baseline left-parenthesis upper B right-parenthesis">

σ

Φ

(
A
+
B
)
⊆

σ

Φ

(
A
)
+

σ

Φ

(
B
)
,

σ

Φ

(

B
A

¯

)
⊆

σ

Φ

(
A
)
⋅

σ

Φ

(
B
)

{\sigma _{\mathbf {\Phi }}}(A + B) \subseteq {\sigma _{\mathbf {\Phi }}}(A) + {\sigma _{\mathbf {\Phi }}}(B),{\sigma _{\mathbf {\Phi }}}(\overline {BA} ) \subseteq {\sigma _{\mathbf {\Phi }}}(A) \cdot {\sigma _{\mathbf {\Phi }}}(B)

, and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript normal upper Phi Baseline left-parenthesis upper A upper B right-parenthesis subset-of-or-equal-to sigma Subscript normal upper Phi Baseline left-parenthesis upper A right-parenthesis sigma Subscript normal upper Phi Baseline left-parenthesis upper B right-parenthesis">

σ
Φ

(
A
B
)
⊆

σ
Φ

(
A
)

σ
Φ

(
B
)

{\sigma _\Phi }(AB) \subseteq {\sigma _\Phi }(A){\sigma _\Phi }(B)

.
A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact mappings
Authors: Petryshyn W. V. , Fitzpatrick P. M.
Published: July 2010

We define and study the properties of a topological degree for ultimately compact, multivalued vector fields defined on the closures of open subsets of certain locally convex topological vector spaces. In addition to compact mappings, the class of ultimately compact mappings includes condensing mappings, generalized condensing mappings, perturbations of compact mappings by certain Lipschitz-type mappings, and others. Using this degree we obtain fixed point theorems and mapping theorems.
𝑃𝐼-algebras satisfying identities of degree 3
Authors: Klein Abraham A.
Published: July 2010

A method of classification of PI-algebras over fields of characteristic 0 is described and applied to algebras satisfying polynomial identities of degree 3. Two algebras satisfying the same identities of degree 3 are considered in the same class. For the degree 3 all the possible classes are obtained. In each case the identities of degree 4 that can be deduced from those of degree 3 have been obtained by means of a computer. These computations have made it possible to obtain-except for three cases-all the identities of higher degrees. It turns out that except for a finite number of cases an algebra satisfying an identity of degree 3 is either nilpotent of order 4, or commutative of order 4, namely the product of 4 elements of the algebra is a symmetric function of its factors.
Solvable groups and quadratic forms
Authors: Tolimieri Richard
Published: July 2010

A solvable Lie group and a unitary representation are constructed from a given binary quadratic form. The multiplicity of this representation is related to the arithmetic of the form.
A homotopy theory of pro-spaces
Authors: Grossman Jerrold W.
Published: July 2010

The category of towers of spaces,

…
→

X

s
+
1

→

X
s

→
…
→

X
0

\ldots \to {X_{s + 1}} \to {X_s} \to \ldots \to {X_0}

, viewed as pro-spaces, appears to be useful in the study of the relation between homology and homotopy of nonsimply connected spaces. We show that this category admits the structure of a closed model category, in the sense of Quillen; notions of fibration, cofibration, and weak equivalence are defined and shown to satisfy fundamental properties that the corresponding notions satisfy in the category of spaces. This enables one to develop a “homotopy theory” for pro-spaces.
Binomial enumeration on dissects
Authors: Henle Michael
Published: July 2010

The Mullin-Rota theory of binomial enumeration is generalized to an abstract context and applied to rook polynomials, order invariants of posets, Tutte invariants of combinatorial geometries, cycle indices and symmetric functions.
On the Frattini subgroups of generalized free products and the embedding of amalgams
Authors: Allenby R. B. J. T. , Tang C. Y.
Published: July 2010

In this paper we shall prove a basic relation between the Frattini subgroup of the generalized free product of an amalgam

A

=
(
A
,
B
;
H
)

\mathfrak {A} = (A,B;H)

and the embedding of

A

\mathfrak {A}

into nonisomorphic groups, namely, if

A

\mathfrak {A}

can be embedded into two non-isomorphic groups

G
1

=
⟨
A
,
B
⟩

{G_1} = \langle A,B\rangle

and

G
2

=
⟨
A
,
B
⟩

{G_2} = \langle A,B\rangle

then the Frattini subgroup of

G
=

(
A
∗
B

)
H

G = {(A \ast B)_H}

is contained in

H
H

. We apply this result to various cases. In particular, we show that if

A
,
B

A,B

are locally solvable and

H
H

is infinite cyclic then

Φ
(
G
)

\Phi (G)

is contained in

H
H

.
𝑘-regular elements in semisimple algebraic groups
Authors: Andre Peter P.
Published: July 2010

In this paper, Steinberg’s concept of a regular element in a semisimple algebraic group defined over an algebraically closed field is generalized to the concept of a

k
k

-regular element in a semisimple algebraic group defined over an arbitrary field of characteristic zero. The existence of semisimple and unipotent

k
k

-regular elements in a semisimple algebraic group defined over a field of characteristic zero is proved. The structure of all

k
k

-regular unipotent elements is given. The number of minimal parabolic subgroups containing a

k
k

-regular element is given. The number of conjugacy classes of

R
R

-regular unipotent elements is given, where

R
R

is the real field. The number of conjugacy classes of

Q
p

{Q_p}

-regular unipotent elements is shown to be finite, where

Q
p

{Q_p}

is the field of

p
p

-adic numbers.
On a class of new inequalities
Authors: Shum Daniel T.
Published: July 2010

Inequalities of considerable interest are associated with the names of Beesack, Benson, Boyd, Calvert, Das, Hardy, Hua, Opial, Wong and Yang. In this note an elementary method used in a recent paper by Benson will be further investigated. The resultant new class of inequalities will bring a great number of inequalities—such as inequalities of Hardy’s and those of Opial’s—under one roof, so to speak.
Subcategories of uniform spaces
Authors: Rice Michael D.
Published: July 2010

The problem of embedding a topological space as a closed subspace of a product of members from a given family has received considerable attention in the past twenty years, while the corresponding problem in uniform spaces has been largely ignored. In this paper we initiate the study of the closed uniform subspaces of products of metric spaces. In §1 we introduce the functor

m
m

, which is used in §2 to characterize the closed subspaces of products of metric spaces and separable metric spaces, and the closed subspaces of powers of the open unit interval

(
0
,
1
)

(0, 1)

. In §3 we obtain various descriptions of the functor

d
d

which associates to each uniform space a closed subspace of a product of metric spaces and establish the equation

m
d
=
d
m

md = dm

. This leads to a characterzation of the completeness of

e
u
X

euX

, the uniform space generated by the countable

u
u

-uniform covers, in terms of the completeness of

u
X

uX

and a countable intersection property on Cauchy filters.
Fuchsian manifolds
Authors: Chen Su Shing
Published: July 2010

Recently Eberlein and O’Neill have investigated Riemannian manifolds of negative sectional curvature. For visibility manifolds, they have obtained a classification into three types: parabolic, axial and fuchsian. Fundamental groups of fuchsian manifolds of finite type will be investigated. The main theorem is that isometry groups of certain (not necessarily compact) fuchsian manifolds are finite. Fundamental groups of fuchsian manifolds of finite type are not amenable. The spectral radius of the random matrix of the fundamental group of a compact Riemannian manifold of negative sectional curvature is less than one.
On the Alder polynomials and a new generalization of the Rogers-Ramanujan identities
Authors: Andrews George E.
Published: July 2010

In 1954, H. L. Alder obtained certain analytic generalizations of the Rogers-Ramanujan identities that involved a sequence of polynomials subsequently named the Alder polynomials. The object here is to obtain a new representation of the Alder polynomials from which a new generalization of the Rogers-Ramanujan identities is easily deduced.
𝐻-closed extensions. II
Authors: Porter Jack R. , Votaw Charles
Published: July 2010

The internal structure and external properties (in terms of other

H
H

-closed extensions) of the Fomin extension

σ
X

\sigma X

of a Hausdorff space

X
X

are investigated. The relationship between

σ
X

\sigma X

and the Stone-Čech compactification of the absolute of

X
X

is developed and used to prove that a

σ
X

\sigma X

-closed subset of

σ
X
∖
X

\sigma X\backslash X

is compact and to show the existence of a Tychonoff space

Y
Y

such that

σ
X
∖
X

\sigma X\backslash X

is homeomorphic to

β
Y
∖
Y

\beta Y\backslash Y

. The sequential closure of

X
X

in

σ
X

\sigma X

is shown to be

X
X

. It is known that

σ
X

\sigma X

is not necessarily projectively larger than any other strict

H
H

-closed extension of

X
X

; a necessary and sufficient condition is developed to determine when a

H
H

-closed extension of

X
X

is projectively smaller then

σ
X

\sigma X

. A theorem by Magill is extended by showing that the sets of

θ
\theta

-isomorphism classes of

H
H

-closed extensions of locally

H
H

-closed spaces

X
X

and

Z
Z

are lattice isomorphic if and only if

σ
X
∖
X

\sigma X\backslash X

and

σ
Z
∖
Z

\sigma Z\backslash Z

are homeomorphic. Harris has characterized those simple Hausdorff extensions of

X
X

which are subextensions of the Katětov extension. Characterizations of Hausdorff (not necessarily simple) extensions of

X
X

which are subextensions of

H
H

-closed extensions

θ
\theta

-isomorphic and

S
S

-equivalent to the Katětov extension are presented.