bluuebunny/crossref_metadata_2025 huggingface dataset:
Duality theories for metabelian Lie algebras
Authors: Gauger Michael A.
Published: July 2010

This paper is concerned with duality theories for metabelian (2-step nilpotent) Lie algebras. A duality theory associates to each metabelian Lie algebra N with cod

N
2

=
g

{N^2} = g

, another such algebra

N
D

{N_D}

satisfying

(

N
D

)
D

≅
N
,

N
1

≅

N
2

{({N_D})_D} \cong N,{N_1} \cong {N_2}

if and only if

(

N
1

)
D

≅

(

N
2

)
D

{({N_1})_D} \cong {({N_2})_D}

, and if

dim

N
=
g
+
p

\dim \,N = g + p

then

dim

N
D

=
g
+

(
2
g

)
−
p

\dim \,{N_D} = g + (_2^g) - p

. The obvious benefit of such a theory lies in its reduction of the classification problem.
On Sylow 2-subgroups with no normal Abelian subgroups of rank 3, in finite fusion-simple groups
Authors: Patterson Anne R.
Published: July 2010

Let T be any finite 2-group which has a normal four-group but has no normal Abelian subgroup of rank 3, and assume T is not the dihedral group of order 8. If T is a Sylow 2-subgroup of a finite fusion-simple group G, it follows (Thompson) from Glauberman’s

Z
∗

{Z^ \ast }

-theorem that T has exactly one normal four-group, say W. This paper establishes what isomorphism types of T can so occur under the hypothesis that

N

G

(
T
)
=
T

C

G

(
T
)

{{\mathbf {N}}_G}(T) = T{{\mathbf {C}}_G}(T)

and the three nonidentity elements of W are not all G-conjugate. All T arrived at in this paper are known to so occur. The reason for this hypothesis is that the similar situation for T with a normal four-group and no normal Abelian subgroup of rank 3, where T is a Sylow 2-subgroup of a finite simple group G but without the above hypothesis, had been analyzed earlier by the author (under her maiden name, MacWilliams; Trans. Amer. Math. Soc. 150 (1970), 345-408).
Complex approximation for vector-valued functions with an application to boundary behaviour
Authors: Brown Leon , Gauthier P. M. , Seidel W.
Published: July 2010

This paper deals with the qualitative theory of uniform approximation by holomorphic functions. The first theorem is an extension to vector-valued mappings of N. U. Arakélian’s theorem on uniform holomorphic approximation on closed sets. Our second theorem is on asymptotic approximation and yields, as in the scalar case, applications to cluster sets.
On Fourier transforms
Authors: Nasim C.
Published: July 2010

If

f
(
x
)

f(x)

and

g
(
x
)

g(x)

satisfy the equations
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis x right-parenthesis equals StartFraction d Over d x EndFraction integral Subscript 0 Superscript normal infinity Baseline StartFraction 1 Over t EndFraction f left-parenthesis t right-parenthesis k 1 left-parenthesis x t right-parenthesis d t comma f left-parenthesis x right-parenthesis equals StartFraction d Over d x EndFraction integral Subscript 0 Superscript normal infinity Baseline StartFraction 1 Over t EndFraction g left-parenthesis t right-parenthesis k 1 left-parenthesis x t right-parenthesis d t comma">

g
(
x
)
=

d

d
x

∫
0
∞

1
t

f
(
t
)

k
1

(
x
t
)
d
t
,

f
(
x
)
=

d

d
x

∫
0
∞

1
t

g
(
t
)

k
1

(
x
t
)
d
t
,

g(x) = \frac {d}{{dx}}\int _0^\infty \frac {1}{t}f(t){k_1}(xt)dt,\quad f(x) = \frac {d}{{dx}}\int _0^\infty \frac {1}{t}g(t){k_1}(xt)dt,

\]
then we call f and g a pair of

k
1

{k_1}

-transforms, where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k 1 equals StartFraction 1 Over 2 pi i EndFraction integral Subscript 1 slash 2 minus i normal infinity Superscript 1 slash 2 plus i normal infinity Baseline StartFraction upper K left-parenthesis s right-parenthesis Over 1 minus s EndFraction x Superscript 1 minus s Baseline d s period">

k
1

=

1

2
π
i

∫

1

/

2
−
i
∞

1

/

2
+
i
∞

K
(
s
)

1
−
s

x

1
−
s

d
s
.

k_1 = \frac {1}{2\pi i} \int _{1/2 - i\infty }^{1/2 + i\infty } \frac {K(s)}{1 - s} x^{1-s}\,ds.

\]
In this paper alternative sets of conditions are established for f and g to be

k
1

{k_1}

-transform provided

K
(
s
)

K(s)

is decomposable in a special way. These conditions involve simpler functions, which replace the kernel

k
1

(
x
)

{k_1}(x)

. Results are proved for the function spaces

L
2

{L^2}

. The necessary and sufficient conditions are established for the two functions to be self-reciprocal. Conditions are given for generating pairs of transforms for a given kernel. Two examples are given at the end to illustrate the methods and the advantage of the results.
Spectra of polar factors of hyponormal operators
Authors: Putnam C. R.
Published: July 2010

An investigation is made of the interdependence and properties of the spectrum of a hyponormal operator T and of the spectra, and absolutely continuous spectra, of the factors in a polar factorization of T when the latter exists.
An internal characterization of paracompact 𝑝-spaces
Authors: Stoltenberg R. A.
Published: July 2010

The purpose of this paper is to characterize paracompact p-spaces in terms of spaces with refining sequences

mod

k

\bmod \;k

. A space X has a refining sequence

mod

k

\bmod \;k

if there exists a sequence

{

G

n

|

n
∈
N
}

\{ {\mathcal {G}_n}|n \in N\}

of open covers for X such that

∩

n
=
1

∞

St

(
C
,

G

n

)
=

P
C
1

\cap _{n = 1}^\infty {\text {St}}(C,{\mathcal {G}_n}) = P_C^1

is compact for each compact subset C of X and

{ St

(
C
,

G

n

)
−

|

n
∈
N
}

{\text {\{ St}}{(C,{\mathcal {G}_n})^ - }|n \in N\}

is a neighborhood base for

P
C
1

P_C^1

. If

P
C
1

=
C

P_C^1 = C

for each compact subset C of X then X is metrizable. On the other hand if we restrict the set C to the family of finite subsets of X in the above definition then we have a characterization for strict p-spaces. Moreover, in this case, if

P
C
1

=
C

P_C^1 = C

for all such sets then X is developable. Thus the concept of a refining sequence

mod

k

\bmod \;k

is natural and it is helpful in understanding paracompact p-spaces.
Fixed point iterations using infinite matrices
Authors: Rhoades B. E.
Published: July 2010

Let E be a closed, bounded, convex subset of a Banach space

X
,
f
:
E
→
E

X,f:E \to E

. Consider the iteration scheme defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x overbar Subscript 0 Baseline equals x 0 element-of upper E comma x overbar Subscript n plus 1 Baseline equals f left-parenthesis x Subscript n Baseline right-parenthesis comma x Subscript n Baseline equals normal upper Sigma Subscript k equals 0 Superscript n Baseline a Subscript n k Baseline x overbar Subscript k Baseline comma n greater-than-or-equal-to 1">

x
¯

0

=

x
0

∈
E
,

x
¯

n
+
1

=
f
(

x
n

)
,

x
n

=

Σ

k
=
0

n

a

n
k

x
¯

k

,

n
≥
1

{\bar x_0} = {x_0} \in E,{\bar x_{n + 1}} = f({x_n}),{x_n} = \Sigma _{k = 0}^n{a_{nk}}{\bar x_k},\;n \geq 1

, where A is a regular weighted mean matrix. For particular spaces X and functions f we show that this iterative scheme converges to a fixed point of f.
Involutions preserving an 𝑆𝑈 structure
Authors: Rowlett R. J.
Published: July 2010

Bordism theories

S

U
∗

(

Z
2

,
a
l
l
)

S{U_ \ast }({Z_2},all)

for SU-manifolds with involution and

S

U
∗

(

Z
2

,
f
r
e
e
)

S{U_ \ast }({Z_2},free)

for SU-manifolds with free involution are defined. The latter is studied by use of the SU-bordism spectral sequence of

B

Z
2

B{Z_2}

, and the orders of the spheres

S

4
n
+
3

{S^{4n + 3}}

with antipodal action are determined. It is shown that

S

U

2
k

(

Z
2

,
f
r
e
e
)
→
S

U

2
k

(

Z
2

,
a
l
l
)

S{U_{2k}}({Z_2},free) \to S{U_{2k}}({Z_2},all)

is monic, and that an element of

S

U

2
k

(

Z
2

,
a
l
l
)

S{U_{2k}}({Z_2},all)

bounds as a unitary involution if and only if it is a multiple of the nonzero class

α
∈
S

U
1

\alpha \in S{U_1}

.
On the zeros of Dirichlet 𝐿-functions. I
Authors: Fujii Akio
Published: July 2010

A mean value theorem for

arg
⁡
L
(
1

/

2
+
i
(
t
+
h
)
,
χ
)
−
arg
⁡
L
(
1

/

2
+
i
t
,
χ
)

\arg L(1/2 + i(t + h), \chi ) - \arg L(1/2 + it, \chi )

is established. This yields mean estimates for the number of zeros of

L
(
s
,
χ
)

L(s, \chi )

in small boxes.
Injective modules and localization in noncommutative Noetherian rings
Authors: Jategaonkar Arun Vinayak
Published: July 2010

Let

S

\mathfrak {S}

be a semiprime ideal in a right Noetherian ring R and

C

(

S

)
=
{
c
∈
R

|

[
c
+

S

\mathcal {C}(\mathfrak {S}) = \{ c \in R|[c + \mathfrak {S}

regular in

R

/

S

}

R/\mathfrak {S}\}

. We investigate the following two conditions:

(

A

)

C

(

S

)

({\text {A}})\;\mathcal {C}(\mathfrak {S})

is a right Ore set in R.

(

B

)

C

(

S

)

({\text {B}})\;\mathcal {C}(\mathfrak {S})

is a right Ore set in R and the right ideals of

R

S

{R_{\mathfrak {S}}}

, the classical right quotient ring of R w.r.t.

C

(

S

)

\mathcal {C}(\mathfrak {S})

are closed in the

J
(

R

S

)

J({R_{\mathfrak {S}}})

-adic topology. The main results show that conditions (A) and (B) can be characterized in terms of the injective hull of the right R-module

R

/

S

R/\mathfrak {S}

. The J-adic completion of a semilocal right Noetherian ring is also considered.
Global dimension of tiled orders over a discrete valuation ring
Authors: Jategaonkar Vasanti A.
Published: July 2010

Let R be a discrete valuation ring with maximal ideal

m

\mathfrak {m}

and the quotient field K. Let

Λ
=
(

m

λ

i
j

)
⊆

M
n

(
K
)

\Lambda = ({\mathfrak {m}^{{\lambda _{ij}}}}) \subseteq {M_n}(K)

be a tiled R-order, where

λ

i
j

∈

Z

{\lambda _{ij}} \in {\mathbf {Z}}

and

λ

i
i

=
0

{\lambda _{ii}} = 0

for

1
≤
i
≤
n

1 \leq i \leq n

. The following results are proved. Theorem 1. There are, up to conjugation, only finitely many tiled R-orders in

M
n

(
K
)

{M_n}(K)

of finite global dimension. Theorem 2. Tiled R-orders in

M
n

(
K
)

{M_n}(K)

of finite global dimension satisfy DCC. Theorem 3. Let

Λ
⊆

M
n

(
R
)

\Lambda \subseteq {M_n}(R)

and let

Γ
\Gamma

be obtained from

Λ
\Lambda

by replacing the entries above the main diagonal by arbitrary entries from R. If

Γ
\Gamma

is a ring and if gl

dim

Λ
>
∞

\dim \;\Lambda > \infty

, then gl

dim

Γ
>
∞

\dim \;\Gamma > \infty

. Theorem 4. Let

Λ
\Lambda

be a tiled R-order in

M
4

(
K
)

{M_4}(K)

. Then gl

dim

Λ
>
∞

\dim \;\Lambda > \infty

if and only if

Λ
\Lambda

is conjugate to a triangular tiled R-order of finite global dimension or is conjugate to the tiled R-order

Γ
=
(

m

λ

i
j

)
⊆

M
4

(
R
)

\Gamma = ({\mathfrak {m}^{{\lambda _{ij}}}}) \subseteq {M_4}(R)

, where

γ

i
i

=

γ

1
i

=
0

{\gamma _{ii}} = {\gamma _{1i}} = 0

for all i, and

γ

i
j

=
1

{\gamma _{ij}} = 1

otherwise.
Series inequalities involving convex functions
Authors: Imoru Christopher O.
Published: July 2010

The object of this paper is to extend some recent generalizations of Petrović’s inequality by Vasić and others. We shall also use our technique to obtain some results which have interesting applications in the theory of Fourier series as well as the theory of approximations.
Location of the zeros of polynomials with a prescribed norm
Authors: Rahman Q. I. , Schmeisser G.
Published: July 2010

For monic polynomials

f
n

(
z
)

{f_n}(z)

of degree n with prescribed

L
p

{L^p}

norm

(
1
≤
p
≤
∞
)

(1 \leq p \leq \infty )

on the unit circle or supremum norm on the unit interval we determine bounded regions in the complex plane containing at least

k
(
1
≤
k
≤
n
)

k(1 \leq k \leq n)

zeros. We deduce our results from some new inequalities which are similar to an inequality of Vicente Gonçalves and relate the zeros of a polynomial to its norm.
Further results on prime entire functions
Authors: Gross Fred , Yang Chung Chun
Published: July 2010

Let H denote the set of all the entire functions

f
(
z
)

f(z)

of the form:

f
(
z
)
≡
h
(
z
)

e

p
(
z
)

+
k
(
z
)

f(z) \equiv h(z){e^{p(z)}} + k(z)

where

p
(
z
)

p(z)

is a nonconstant polynomial of degree m, and

h
(

≢

0
)

h(\nequiv \;0)

,

k
(

≢

k(\nequiv

constant) are two entire functions of order less than m. In this paper, a necessary and sufficient condition for a function in H to be a prime is established. Several generalizations of known results follow. Some sufficient conditions for primeness of various subclasses of H are derived. The methods used in the proofs are based on Nevanlinna’s theory of meromorphic functions and some elementary facts about algebraic functions.
On the completion of Hausdorff locally solid Riesz spaces
Authors: Aliprantis Charalambos D.
Published: July 2010

In this paper we consider Hausdorff locally solid Riesz spaces

(
L
,
τ
)

(L,\tau )

and we denote by

(

L
^

,

τ
^

)

(\hat L,\hat \tau )

the Hausdorff topological completion of

(
L
,
τ
)

(L,\tau )

. It is proved that

(

L
^

,

τ
^

)

(\hat L,\hat \tau )

is a Hausdorff locally solid Riesz space containing L as a Riesz subspace. We study the properties of

(
L
,
τ
)

(L,\tau )

which are inherited by

(

L
^

,

τ
^

)

(\hat L,\hat \tau )

.
The inertial aspects of Stein’s condition 𝐻-𝐶*𝐻𝐶≫𝑂
Authors: Cain Bryan E.
Published: July 2010

To each bounded operator C on the complex Hilbert space

H

\mathcal {H}

we associate the vector space

K

C

{\mathcal {K}_C}

consisting of those

x
∈

H

x \in \mathcal {H}

for which

C
n

x
→
0

{C^n}x \to 0

as

n
→
∞

n \to \infty

. We let

α
(
C
)

\alpha (C)

denote the dimension of the closure of

K

C

{\mathcal {K}_C}

and we set

β
(
C
)
=
dim
⁡
(

K

C
⊥

)

\beta (C) = \dim (\mathcal {K}_C^ \bot )

. Our main theorem states that if H is Hermitian and if

H
−

C
∗

H
C

H - {C^ \ast }HC

is positive and invertible then

α
(
C
)
≤
π
(
H
)
,
β
(
C
)
=
ν
(
H
)

\alpha (C) \leq \pi (H),\beta (C) = \nu (H)

, and

β
(
C
)
≥
δ
(
H
)

\beta (C) \geq \delta (H)

where

(
π
(
H
)
,
ν
(
H
)
,
δ
(
H
)
)

(\pi (H),\nu (H),\delta (H))

is the inertia of H. (That is,

π
(
H
)
=
dim

(

Range

E
[
(
0
,
∞
)
]
)

\pi (H) = \dim \;({\text {Range}}\;E[(0,\infty )])

) where E is the spectral measure of H;

ν
(
H
)
=
π
(
−
H
)

\nu (H) = \pi ( - H)

; and

δ
(
H
)
=
dim
⁡
(

Ker

H
)

\delta (H) = \dim ({\operatorname {Ker}}\;H)

.) We also show (l) that in general no stronger conclusion is possible, (2) that, unlike previous inertia theorems, our theorem allows 1 to lie in

σ
(
C
)

\sigma (C)

, the spectrum of C, and (3) that the main inertial results associated with the hypothesis that

Re
⁡
(
H
A
)

\operatorname {Re} (HA)

is positive and invertible can be derived from our theorem. Our theorems (1) characterize C in the extreme cases that either

π
(
H
)
=
0

\pi (H) = 0

or

ν
(
H
)
=
0

\nu (H) = 0

, and (2) prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha left-parenthesis upper C right-parenthesis equals pi left-parenthesis upper H right-parenthesis comma beta left-parenthesis upper C right-parenthesis equals nu left-parenthesis upper H right-parenthesis comma delta left-parenthesis upper H right-parenthesis equals 0">

α
(
C
)
=
π
(
H
)
,
β
(
C
)
=
ν
(
H
)
,
δ
(
H
)
=
0

\alpha (C) = \pi (H),\beta (C) = \nu (H),\delta (H) = 0

if either

1
∉
σ
(
C
)

1 \notin \sigma (C)

or

β
(
C
)
>
∞

\beta (C) > \infty

.
Extremal problems of distance geometry related to energy integrals
Authors: Alexander Ralph , Stolarsky Kenneth B.
Published: July 2010

Let K be a compact set,

M

\mathcal {M}

a prescribed family of (possibly signed) Borel measures of total mass one supported by K, and f a continuous real-valued function on

K
×
K

K \times K

. We study the problem of determining for which

μ
∈

M

\mu \in \mathcal {M}

(if any) the energy integral
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis upper K comma mu right-parenthesis equals integral Underscript upper K Endscripts integral Underscript upper K Endscripts f left-parenthesis x comma y right-parenthesis d mu left-parenthesis x right-parenthesis d mu left-parenthesis y right-parenthesis">

I
(
K
,
μ
)
=

∫
K

∫
K

f
(
x
,
y
)
d
μ
(
x
)
d
μ
(
y
)

I(K,\mu ) = \smallint _K {\smallint _K {f(x,y)d\mu (x)d\mu (y)} }

is maximal, and what this maximum is. The more symmetry K has, the more we can say; our results are best when K is a sphere. In particular, when

M

\mathcal {M}

is atomic we obtain good upper bounds for the sums of powers of all

(
2
n

)

(_2^n)

distances determined by n points on the surface of a sphere. We make use of results from Schoenberg’s theory of metric embedding, and of techniques devised by Pólya and Szegö for the calculation of transfinite diameters.
Killing vector fields and harmonic forms
Authors: Wright Edward T.
Published: July 2010

The paper is concerned with harmonic

(
p
,
q
)

(p,q)

-forms on compact Kähler manifolds which admit Killing vector fields with discrete zero sets. Let

h

p
,
q

{h^{p,q}}

denote the dimension of the space of harmonic

(
p
,
q
)

(p,q)

-forms. The main theorem states that

h

p
,
q

=
0
,
p
≠
q

{h^{p,q}} = 0,p \ne q

.
Free products of topological groups which are 𝑘_{𝜔}-spaces
Authors: Ordman Edward T.
Published: July 2010

Let G and H be topological groups and

G
∗
H

G \ast H

their free product topologized in the manner due to Graev. The topological space

G
∗
H

G \ast H

is studied, largely by means of its compact subsets. It is established that if G and H are

k
ω

{k_\omega }

-spaces (respectively: countable CW-complexes) then so is

G
∗
H

G \ast H

. These results extend to countably infinite free products. If G and H are

k
ω

{k_\omega }

-spaces,

G
∗
H

G \ast H

is neither locally compact nor metrizable, provided G is nondiscrete and H is nontrivial. Incomplete results are obtained about the fundamental group

π
(
G
∗
H
)

\pi (G \ast H)

. If

G
1

{G_1}

and

H
1

{H_1}

are quotients (continuous open homomorphic images) of G and H, then

G
1

∗

H
1

{G_1} \ast {H_1}

is a quotient of

G
∗
H

G \ast H

.
𝑆𝑈(𝑛) actions on differentiable manifolds with vanishing first and second integral Pontrjagin classes
Authors: Grove Edward A.
Published: July 2010

In this paper we determine the connected component of the identity of the isotropy subgroups of a given action of

SU

(
n
)

{\text {SU}}(n)

on a connected manifold whose first and second integral Pontrjagin classes are zero and whose dimension is less than

n
2

−
8
n

/

3
−
1

{n^2} - 8n/3 - 1

.
On the asymptotic distribution of eigenvalues for semi-elliptic operators
Authors: Tsutsumi Akira , Wang Chung Lie
Published: July 2010

This paper is focused on the asymptotic distribution of eigenvalues for semielliptic operators under weaker smoothness assumptions on coefficients of operators than those of F. E. Browder [3] and Y. Kannai [8] by applying the method of Maruo-Tanabe [9].
Any infinite-dimensional Fréchet space homeomorphic with its countable product is topologically a Hilbert space
Authors: Terry Wesley E.
Published: July 2010

In this paper we will prove that any infinite-dimensional Fréchet space homeomorphic with its own countable product is topologically a Hilbert space. This will be done in two parts. First we will prove the result for infinite-dimensional Banach spaces, and then we will show that the result for Fréchet spaces follows as a corollary.
Semicellularity, decompositions and mappings in manifolds
Authors: Coram Donald
Published: July 2010

If X is an arbitrary compact set in a manifold, we give algebraic criteria on X and on its embedding to determine that X has an arbitrarily small, closed neighborhood each component of which is a p-connected, piecewise linear manifold which collapses to a q-dimensional subpolyhedron from some p and q. This property generalizes cellularity. The criteria are in terms of UV properties and Alexander-Spanier cohomology. These criteria are then applied to decide when the components of a given compact set in a manifold are elements of a decomposition such that the quotient space is the n-sphere. Conversely, algebraic criteria are given for the point inverses of a map between manifolds to have arbitrarily small neighborhoods of the type mentioned above; these criteria are considerably weaker than for an arbitrary compact set.
The homotopy type of the space of diffeomorphisms. I
Authors: Burghelea Dan , Lashof Richard
Published: July 2010

A new proof is given of the unpublished results of Morlet on the relation between the homeomorphism group and the diffeomorphism group of a smooth manifold. In particular, the result
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D i f f left-parenthesis upper D Superscript n Baseline comma partial-differential right-parenthesis asymptotically-equals normal upper Omega Superscript n plus 1 Baseline left-parenthesis Top Subscript n Baseline slash upper O Subscript n Baseline right-parenthesis">

Diff

(

D
n

,
∂
)
≃

Ω

n
+
1

(

Top
n

/

O
n

)

{\operatorname {Diff}}({D^n},\partial ) \simeq {\Omega ^{n + 1}}({\text {Top}_n}/{O_n})

is obtained. The main technique is fibrewise smoothing.
Group algebras whose simple modules are injective
Authors: Farkas Daniel R. , Snider Robert L.
Published: July 2010

Let F be either a field of char 0 with all roots of unity or a field of char

p
>
0

p > 0

. Let G be a countable group. Then all simple

F
[
G
]

F[G]

-modules are injective if and only if G is locally finite with no elements of order char F and G has an abelian subgroup of finite index.
Interpolation in a classical Hilbert space of entire functions
Authors: Young Robert M.
Published: July 2010

Let H denote the Paley-Wiener space of entire functions of exponential type

π
\pi

which belong to

L
2

(
−
∞
,
∞
)

{L^2}( - \infty ,\infty )

on the real axis. A sequence

{

λ
n

}

\{ {\lambda _n}\}

of distinct complex numbers will be called an interpolating sequence for H if

T
H
⊃

l
2

TH \supset {l^2}

, where T is the mapping defined by

T
f
=
{
f
(

λ
n

)
}

Tf = \{ f({\lambda _n})\}

. If in addition

{

λ
n

}

\{ {\lambda _n}\}

is a set of uniqueness for H, then

{

λ
n

}

\{ {\lambda _n}\}

is called a complete interpolating sequence. The following results are established. If

Re
⁡
(

λ

n
+
1

)
−
Re
⁡
(

λ
n

)
≥
γ
>
1

\operatorname {Re} ({\lambda _{n + 1}}) - \operatorname {Re} ({\lambda _n}) \geq \gamma > 1

and if the imaginary part of

λ
n

{\lambda _n}

is sufficiently small, then

{

λ
n

}

\{ {\lambda _n}\}

is an interpolating sequence. If
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper R e left-parenthesis lamda Subscript n Baseline right-parenthesis minus n EndAbsoluteValue less-than-or-equal-to upper L less-than-or-equal-to left-parenthesis log 2 right-parenthesis slash pi left-parenthesis negative normal infinity greater-than n greater-than normal infinity right-parenthesis">

|

Re
⁡
(

λ
n

)
−
n

|

≤
L
≤
(
log
⁡
2
)

/

π

(
−
∞
>
n
>
∞
)

|\operatorname {Re} ({\lambda _n}) - n| \leq L \leq (\log 2)/\pi \;( - \infty > n > \infty )

and if the imaginary part of

λ
n

{\lambda _n}

is uniformly bounded, then

{

λ
n

}

\{ {\lambda _n}\}

is a complete interpolating sequence and

{

e

i

λ
n

t

}

\{ {e^{i{\lambda _n}t}}\}

is a basis for

L
2

(
−
π
,
π
)

{L^2}( - \pi ,\pi )

. These results are used to investigate interpolating sequences in several related spaces of entire functions of exponential type.
Expansion of entire functions of several complex variables having finite growth
Authors: Kamthan P. K. , Gupta Manjul
Published: July 2010

We consider the space

χ
\chi

of entire functions of two complex variables having a finite nonzero order point and type, equip it with the natural locally convex topology, such that

χ
\chi

becomes a Fréchet space. Apart from finding the characterization of continuous linear functionals, linear transformations on

χ
\chi

, we have obtained the necessary and sufficient conditions for a double sequence in

χ
\chi

to be a proper base.
Volumes of images of varieties in projective space and in Grassmannians
Authors: Alexander H.
Published: July 2010

If V is a complex analytic subvariety of pure dimension k in the unit ball in

C

n

{{\mathbf {C}}^n}

which does not contain the origin, then the 2k-volume of V equals the measure computed with multiplicity of the set of

(
n
−
k
)

(n - k)

-complex subspaces through the origin which meet V. The measure of this set computed without multiplicity is a smaller quantity which is nevertheless bounded below by a number depending only on the distance from V to the origin. As an application we characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. The complex analysis which we shall need will be developed in the context of uniform algebras.
Unitary measures on LCA groups
Authors: Corwin Lawrence
Published: July 2010

A unitary measure on a locally compact Abelian (LCA) group G is a complex measure whose Fourier transform is of absolute value 1 everywhere. The problem of finding all such measures is known to be closely related to that of finding all invertible measures on G. In this paper, we find all unitary measures when G is the circle or a discrete group. If G is a torsion-free discrete group, the characterization generalizes a theorem of Bohr.
Analytic equivalence among simply connected domains in 𝐶(𝑋)
Authors: Warren Hugh E.
Published: July 2010

This work considers analytic equivalence within the analytic function theory for commutative Banach algebras which was introduced by E. R. Lorch. Necessary conditions of a geometric nature are given for simply connected domains in

C
(
X
)

C(X)

. These show that there are a great many equivalence classes. In some important cases, as when one domain is the unit ball, the given conditions are also sufficient. The main technique is the association of a simply connected domain in

C
(
X
)

C(X)

with a family of Riemann surfaces over the plane.
A density property and applications
Authors: O’Malley Richard J.
Published: July 2010

An unexpected metric density property of a certain type of

F
σ

{F_\sigma }

set is proven. An immediate application is a characterization of monotone functions similar to a well-known result by Zygmund. Several corollaries of this characterization are given as well as a simple proof of a theorem due to Tolstoff.
A proof that ℋ² and 𝒯² are distinct measures
Authors: Ernst Lawrence R.
Published: July 2010

It is proven that there exists a subset E of

R

3

{{\mathbf {R}}^3}

such that the two-dimensional

J

\mathcal {J}

measure of E is less than its two-dimensional Hausdorff measure. E is the image under the usual isomorphism of

R

×

R

2

{\mathbf {R}} \times {{\mathbf {R}}^2}

onto

R

3

{{\mathbf {R}}^3}

of the Cartesian product of

{
x
:
−
4
≤
x
≤
4
}

\{ x: - 4 \leq x \leq 4\}

and a Cantor type subset of

R

2

{{\mathbf {R}}^2}

; the latter term in this product is the intersection of a decreasing sequence, every member of which is the union of certain closed circular disks.
Density of parts of algebras on the plane
Authors: O’Farrell Anthony G.
Published: July 2010

We study the Gleason parts of a uniform algebra A on a compact subset of the plane, where it is assumed that for each point

x
∈

C

x \in {\text {C}}

the functions in A which are analytic in a neighborhood of x are uniformly dense in A. We prove that a part neighborhood N of a nonpeak point x for A satisfies a density condition of Wiener type at
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x colon normal upper Sigma Subscript n equals 1 Superscript plus normal infinity Baseline 2 Superscript n Baseline upper C left-parenthesis upper A Subscript n Baseline left-parenthesis x right-parenthesis minus upper N right-parenthesis greater-than plus normal infinity">

x
:

Σ

n
=
1

+
∞

2
n

C
(

A
n

(
x
)
∖
N
)
>
+
∞

x:\Sigma _{n = 1}^{ + \infty }{2^n}C({A_n}(x)\backslash N) > + \infty

, and if A admits a pth order bounded point derivation at x, then N satisfies a stronger density condition:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Subscript n equals 1 Superscript plus normal infinity Baseline 2 Superscript left-parenthesis p plus 1 right-parenthesis n Baseline upper C left-parenthesis upper A Subscript n Baseline left-parenthesis x right-parenthesis minus upper N right-parenthesis greater-than plus normal infinity">

Σ

n
=
1

+
∞

2

(
p
+
1
)
n

C
(

A
n

(
x
)
∖
N
)
>
+
∞

\Sigma _{n = 1}^{ + \infty }{2^{(p + 1)n}}C({A_n}(x)\backslash N) > + \infty

. Here C is Newtonian capacity and

A
n

(
x
)

{A_n}(x)

is

{
z
∈

C

:

2

−
n
−
1

≤

|

z
−
x

|

≤

2

−
n

}

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

. These results strengthen and extend Browder’s metric density theorem. The relation with potential theory is examined, and analogous results for the algebra

H
∞

(
U
)

{H^\infty }(U)

are obtained as corollaries.
Compact sets definable by free 3-manifolds
Authors: Row W. H.
Published: July 2010

Shape conditions are given that force a compactum (i.e., a compact metric space) embedded in the interior of a nonclosed, piecewise-linear 3-manifold to have arbitrarily close, compact, polyhedral neighborhoods each component of which is a 3-manifold with free fundamental group (i.e., to be definable by free 3-manifolds). For compact, connected ANR’s these conditions reduce to the criterion of having a free fundamental group. Additional conditions are given that insure definability by handlebodies or cubes-with-handles. An embedding of Menger’s universal 1-dimensional curve in Euclidean 3-space is shown to have the property that all tame surfaces, separating in 3-space a fixed pair of points, cannot be adjusted (by a small space homeomorphism) to intersect the embedded curve in a 0-dimensional set.
The undecidability of the word problems for projective geometries and modular lattices
Authors: Lipshitz L.
Published: July 2010

We show that the restricted word problems for finite-dimensional projective geometries and finite modular lattices and the word problem for modular lattices are undecidable.
Semirings and 𝑇₁ compactifications. I
Authors: Harris Douglas
Published: July 2010

With each infinite cardinal

ω
μ

{\omega _\mu }

is associated a topological semiring

F

μ

{{\mathbf {F}}_\mu }

, whose underlying space is finite complement topology on the set of all ordinals less than

ω
μ

{\omega _\mu }

, and whose operations are the natural sum and natural product defined by Hessenberg. The theory of the semirings

C
μ

(
X
)

{C_\mu }(X)

of maps from a space X into

F

μ

{{\mathbf {F}}_\mu }

is developed in close analogy with the theory of the ring

C
(
X
)

C(X)

of continuous real-valued functions; the analogy is not on the surface alone, but may be pursued in great detail. With each semiring a structure space is associated; the structure space of

C
μ

(
X
)

{C_\mu }(X)

for sufficiently large

ω
μ

{\omega _\mu }

will be the Wallman compactification of X. The classes of

ω
μ

{\omega _\mu }

-entire and

ω
μ

{\omega _\mu }

-total spaces, which are respectively analogues of realcompact and pseudocompact spaces, are examined, and an

ω
μ

{\omega _\mu }

-entire extension analogous to the Hewitt realcompactification is constructed with the property (not possessed by the Wallman compactification) that every map between spaces has a unique extension to their

ω
μ

{\omega _\mu }

-entire extensions. The semiring of functions of compact-small support is considered, and shown to be related to the locally compact-small spaces in the same way that the ring of functions of compact support is related to locally compact spaces.
Formal groups and Hopf algebras over discrete rings
Authors: Morris Robert A. , Pareigis Bodo
Published: July 2010

A theory of formal schemes and groups over abitrary rings is presented. The flat formal schemes in this theory have coalgebras of distributions which behave in the usual way. Frobenius and Verschiebung maps are studied.
The group of 𝑃𝐿-homeomorphisms of a compact 𝑃𝐿-manifold is an 1^{𝑓}₂-manifold
Authors: Keesling James , Wilson David C.
Published: July 2010

In this paper it is shown that if

M
M

is a compact PL-manifold and

H

P
L

(
M
)

{H_{PL}}(M)

is the group of PL-homeomorphisms of

M
M

onto itself, then

H

P
L

(
M
)

{H_{PL}}(M)

is an

l
2
f

l_2^f

-manifold. Here

l
2

{l_2}

is the Hilbert space of all real-valued square-summable sequences and

l
2
f

=
{
(

x
i

)
∈

l
2

:

x
i

=
0

l_2^f = \{ ({x_i}) \in {l_2}:{x_i} = 0

for almost all

i
i

.
Transversally parallelizable foliations of codimension two
Authors: Conlon Lawrence
Published: July 2010

We study framed foliations such that the framing of the normal bundle can be chosen to be invariant under the linear holonomy of each leaf. In codimension one there is a strong structure theory for such foliations due, e.g., to Novikov, Sacksteder, Rosenberg, Moussu. An analogous theory is developed here for the case of codimension two.
Smooth partitions of unity on manifolds
Authors: Lloyd John
Published: July 2010

This paper continues the study of the smoothness properties of (real) topological linear spaces. First, the smoothness results previously obtained about various important classes of locally convex spaces, such as Schwartz spaces, are improved. Then, following the ideas of Bonic and Frampton, we use these results to give sufficient conditions for the existence of smooth partitions of unity on manifolds modelled on topological linear spaces.
Sequence of regular finitely additive set functions
Authors: Stein J. D.
Published: July 2010

The purpose of this paper is to deduce versions of Phillips’ lemma and the Vitali-Hahn-Saks theorem, with weaker conditions placed on the set functions and convergence conditions than is usually required.
Rational approximation on product sets
Authors: Bekken Otto B.
Published: July 2010

Our object here is to study pointwise bounded limits, decomposition of orthogonal measures and distance estimates for

R
(

K
1

×

K
2

)

R({K_1} \times {K_2})

where

K
1

{K_1}

and

K
2

{K_2}

are compact sets in the complex plane.
Examples of nonsolvable partial differential equations
Authors: Rubinstein Robert
Published: July 2010

Two examples of nonsolvable partial differential operators with multiple characteristics are presented. They illustrate the possibility that certain terms in the principal part may play no role in determining the solvability properties of the operator. This situation cannot occur for simple characteristics, where solvability is determined by the principal part.
Isolated singularities for solutions of the nonlinear stationary Navier-Stokes equations
Authors: Shapiro Victor L.
Published: July 2010

The notion for (u, p) to be a distribution solution of the nonlinear stationary Navier-Stokes equations in an open set is defined, and a theorem concerning the removability of isolated singularities for distribution solutions in the punctured open ball

B
(
0
,

r
0

)
−
{
0
}

B(0,{r_0}) - \{ 0\}

is established. This result is then applied to the classical situation to obtain a new theorem for the removability of isolated singularities. In particular, in two dimensions this gives a better than expected result when compared with the theory of removable isolated singularities for harmonic functions.
The continuity of Arens’ product on the Stone-Čech compactification of semigroups
Authors: Macri Nicholas
Published: July 2010

A discrete semigroup is said to have the compact semigroup property (c.s.p.) [the compact semi-semigroup property (c.s.s.p.)] if the multiplication Arens’ product, on its Stone-Čech compactification, is jointly [separately]

w
∗

{w^ \ast }

-continuous. We obtain an algebraic characterization of those semigroups which have c.s.p. by characterizing algebraically their almost periodic subsets. We show that a semigroup has c.s.p. if and only if each of its subsets is almost periodic. This characterization is employed to prove that for a cancellation semigroup to have c.s.p., it is necessary and sufficient that each of its countable subsets be almost periodic. We answer in the negative a heretofore open question—is c.s.p. equivalent to c.s.s.p.
On the construction of split-face topologies
Authors: Gleit Alan
Published: July 2010

We give a general theorem to facilitate the construction of interesting examples of split-face topologies of compact, convex sets.
Convolution operators on 𝐺-holomorphic functions in infinite dimensions
Authors: Boland Philip J. , Dineen Seán
Published: July 2010

For a complex vector space E, let

H
G

(
E
)

{H_G}(E)

denote the space of G (Gateaux)-holomorphic functions on

E

(
f
:
E
→
C

E\;(f:E \to C

is G-holomorphic if the restriction of f to every finite dimensional subspace of E is holomorphic in the usual sense). The most natural topology on

H
G

(
E
)

{H_G}(E)

is that of uniform convergence on finite dimensional compact subsets of E. A convolution operator A on

H
G

(
E
)

{H_G}(E)

is a continuous linear mapping

A
:

H
G

(
E
)
→

H
G

(
E
)

A:{H_G}(E) \to {H_G}(E)

such that A commutes with translations. The concept of a convolution operator generalizes that of a differential operator with constant coefficients. We prove that if A is a convolution operator on

H
G

(
E
)

{H_G}(E)

, then the kernel of A is the closed linear span of the exponential polynomials contained in the kernel. In addition, we show that any nonzero convolution operator on

H
G

(
E
)

{H_G}(E)

is a surjective mapping.
On homeomorphisms of infinite-dimensional bundles. I
Authors: Wong Raymond Y. T.
Published: July 2010

In this paper we present several aspects of homeomorphism theory in the setting of fibre bundles modeled on separable infinite-dimensional Hilbert (Fréchet) spaces. We study (homotopic) negligibility of subsets, separation of sets, characterization of subsets of infinite-deficiency and extending homeomorphisms; in an essential way they generalize previously known results for manifolds. An important tool is a lemma concerning the lifting of a map to the total space of a bundle whose image misses a certain closed subset presented as obstruction; from this we are able to obtain a result characterizing all subsets of infinite deficiency (for bundles) by their restriction to each fibre. Other results then follow more or less routinely by employing the rather standard methods of infinite-dimensional topology.
Prime and search computability, characterized as definability in certain sublanguages of constructible 𝐿_{𝜔}_{1,𝜔}
Authors: Gordon Carl E.
Published: July 2010

The prime computable (respectively, search computable) relations of an arbitrary mathematical structure are shown to be those relations R such that both R and its complement are definable by disjunctions of recursively enumerable sets of quantifier free (respectively, existential) formulas of the first order language for the structure. The prime and search computable functions are also characterized in terms of recursive sequences of terms and formulas of this language.
A theory of Stiefel harmonics
Authors: Gelbart Stephen S.
Published: July 2010

An explicit theory of special functions is developed for the homogeneous space

S
O
(
n
)

/

S
O
(
n
−
m
)

SO(n)/SO(n - m)

generalizing the classical theory of spherical harmonics. This theory is applied to describe the decomposition of the Fourier operator on

n
×
m

n \times m

matrix space in terms of operator valued Bessel functions of matrix argument. Underlying these results is a hitherto unnoticed relation between certain irreducible representations of

S
O
(
n
)

SO(n)

and the polynomial representations of

G
L
(
m
,

C

)

GL(m,{\mathbf {C}})

.
Cloverleaf representations of simply connected 3-manifolds
Authors: Moise Edwin E.
Published: July 2010

Let

M
M

be a triangulated

3
3

-manifold satisfying the hypothesis of the Poincaré Conjecture. In the present paper it is shown that there is a finite linear graph

K
1

{K_1}

in the

3
3

-sphere, with exactly two components, and a finite linear graph

K
2

{K_2}

in

M
M

, such that when the components of the graphs

K
i

{K_i}

are regarded as points, the resulting hyperspaces are homeomorphic.

K
2

{K_2}

satisfies certain conditions which imply that each component of

K
2

{K_2}

is contractible in

M
M

. Thus the conclusion of the theorem proved here is equivalent to the hypothesis of the Poincaré Conjecture.
𝐾-groups generated by 𝐾-spaces
Authors: Nummela Eric C.
Published: July 2010

A

K
K

-group

G
G

with identity

e
e

is said to be generated by the

K
K

-space

X
X

if

X
X

is a subspace of

G
G

containing

e
,
X

e,X

algebraically generates

G
G

, and the canonical morphism from the Graev free

K
K

-group over

(
X
,
e
)

(X,e)

on-to

G
G

is a quotient morphism. An internal characterization of the topology of such a group

G
G

is obtained, as well as a sufficient condition that a subgroup

H
H

of

G
G

be generated by a subspace

Y
Y

of

H
H

. Several illuminating examples are provided.
A 5-𝑟 uniqueness theorem
Authors: Engle Jessie Ann
Published: July 2010

A Borel-regular Carathéodory outer measure

Λ
\Lambda

on a separable metric space

X
X

, where

Λ
\Lambda

is invariant with respect to a family

H
H

of homeomorphisms from

X
X

onto

X
X

, is unique if

Λ
\Lambda

satisfies a

5
5

-

r
r

condition at one point in

X
X

and if

H
H

satisfies Condition I, a condition much weaker than, but related to, the invariance of distance under

H
H

.
A sufficient condition for hyperbolicity of partial differential operators with constant coefficient principal part
Authors: Dunn Joseph L.
Published: July 2010

Let

P
P

be a differential operator with principal part

P
m

{P_m}

, and suppose that

P
m

{P_m}

has constant coefficients and is hyperbolic. It is shown that the condition for hyperbolicity of

P
P

when

P
P

has constant coefficients, namely, that

P
P

is weaker than

P
m

{P_m}

is also a sufficient condition for hyperbolicity in the case where

P
P

does not have constant coefficients. Some generalizations are also made to the case where

P
P

is a square matrix of differential operators.
Associated and perspective simplexes
Authors: Gerber Leon
Published: July 2010

A set of

n
+
1

n + 1

lines in

n
n

-space such that any

(

n

−
2
)

({\text {n}} - 2)

-dimensional flat which meets

n
n

of the lines also meets the remaining line is said to be an associated set of lines. Two Simplexes are associated if the joins of corresponding vertices are associated. A simple criterion is given for simplexes to be associated and an analogous one for Simplexes to be perspective. These are used to give a brief proof of the following generalization of the theorem of Pappus. Let

A

\mathcal {A}

and

B

\mathcal {B}

be

n
n

-simplexes and let

p
p

be a permutation on the vertices of

B

\mathcal {B}

. If

A

\mathcal {A}

and

B

\mathcal {B}

are associated (respectively perspective) and

A

\mathcal {A}

and

B

p

\mathcal {B}p

are associated (perspective) then

A

\mathcal {A}

and

B

p
k

\mathcal {B}{p^k}

are associated (perspective) for any integer

k
k

. Very short proofs are given of extensions to

n
n

-dimensions of many theorems from Neuberg’s famous Memoir sur le Tétraèdre, such as: the altitudes of a simplex are associated.
Local Jordan algebras
Authors: Camburn Marvin E.
Published: July 2010

A local Jordan algebra

J

\mathfrak {J}

is a unital quadratic Jordan algebra in which

Rad
⁡

J

\operatorname {Rad} \mathfrak {J}

is a maximal ideal,

J

/

Rad
⁡

J

\mathfrak {J}/\operatorname {Rad} \mathfrak {J}

satisfies the DCC, and

∩
k

Rad
⁡

J

(
k
)

=
0

{ \cap _k}\operatorname {Rad} {\mathfrak {J}^{(k)}} = 0

where

K

(
n
+
1
)

=

U
K

(
n
)

K

(
n
)

{K^{(n + 1)}} = {U_K}(n){K^{(n)}}

. We show that the completion of a local Jordan algebra is also local Jordan, and if

J

\mathfrak {J}

is a complete local Jordan algebra over a field of characteristic not 2, then either (1)

J

\mathfrak {J}

is a complete completely primary Jordan algebra, (2)

J

≅

J

1

⊕

J

2

⊕
S

\mathfrak {J} \cong {\mathfrak {J}_1} \oplus {\mathfrak {J}_2} \oplus S

where each

J

i

{\mathfrak {J}_i}

is a completely primary local Jordan algebra, or (3)

J

≅

H

(

D
n

,

J
a

)

\mathfrak {J} \cong \mathfrak {H}({D_n},{J_a})

where

(
D
,
j
)

(D,j)

is either a not associative alternative algebra with involution or a complete semilocal associative algebra with involution.
Integration of ordinary linear differential equations by Laplace-Stieltjes transforms
Authors: D’Archangelo James , Hartman Philip
Published: July 2010

Let

R
R

be a constant

N
×
N

N \times N

matrix and

g
(
t
)

g(t)

an

N
×
N

N \times N

matrix of functions representable as absolutely convergent Laplace-Stieltjes transforms for

t
>
0

t > 0

. The paper gives sufficient conditions for certain solutions of the system

y
′

=
(
R
+
g
(
t
)
)
y

y’ = (R + g(t))y

to be expressed as Laplace-Stieltjes transforms or as linear combinations of such transforms with coefficients which are powers of

t
t

.
Sequential convergence in the order duals of certain classes of Riesz spaces
Authors: Dodds P. G.
Published: July 2010

Several results of Hahn-Vitali-Saks type are given for sequences in the order dual of an Archimedean Riesz space with separating order dual. The class of Riesz spaces considered contains those which are Dedekind

σ
\sigma

-complete, or have the projection property or have an interpolation property introduced by G. L. Seever. The results depend on recent work of O. Burkinshaw and some results of uniform boundedness type.
Presentations of 𝑛-knots
Authors: Kearton C.
Published: July 2010

The method of critical level embeddings is used to generalize the technique of knot presentations from the classical case to the case of

n
n

-knots. For

n
>
3

n > 3

, it is shown that an

n
n

-knot with algebraically simple complement has a correspondingly simple presentation.
The factorization and representation of lattices
Authors: Markowsky George
Published: July 2010

For a complete lattice

L
L

, in which every element is a join of completely join-irreducibles and a meet of completely meet-irreducibles (we say

L
L

is a jm-lattice) we define the poset of irreducibles

P
(
L
)

P(L)

to be the poset (of height one)

J
(
L
)
∪
M
(
L
)
(
J
(
L
)

J(L) \cup M(L)(J(L)

is the set of completely join-irreducibles and

M
(
L
)

M(L)

is the set of completely meet-irreducibles) ordered as follows:

a

>

P
(
L
)

b

a{ > _{P(L)}}b

if and only if

a
∈
J
(
L
)
,
b
∈
M
(
L
)

a \in J(L),b \in M(L)

, and

a
⪇

L

b

a \nleqslant { _L}b

. For a jm-lattice

L
L

, the automorphism groups of

L
L

and

P
(
L
)

P(L)

are isomorphic,

L
L

can be reconstructed from

P
(
L
)

P(L)

, and the irreducible factorization of

L
L

can be gotten from the components of

P
(
L
)

P(L)

. In fact, we can give a simple characterization of the center of a jm-lattice in terms of its separators (or unions of connected components of

P
(
L
)

P(L)

). Thus

P
(
L
)

P(L)

extends many of the properties of the poset of join-irreducibles of a finite distributive lattice to the class of all jm-lattices. We characterize those posets of height 1 which are

P
(
L
)

P(L)

for some jm-lattice

L
L

. We also characterize those posets of height 1 which are

P
(
L
)

P(L)

for a completely distributive jm-lattice, as well as those posets which are

P
(
L
)

P(L)

for some geometric lattice

L
L

. More generally, if

L
L

is a complete lattice, many of the above arguments apply if we use “join-spanning” and “meet-spanning” subsets of

L
L

, instead of

J
(
L
)

J(L)

and

M
(
L
)

M(L)

. If

L
L

is an arbitrary lattice, the same arguments apply to “join-generating” and “meet-generating” subsets of

L
L

.
Right orders in full linear rings
Authors: O’Meara K. C.
Published: July 2010

In this paper a right orderRRin an infinite dimensional full linear ring is characterized as a ring satisfying: (1)RRis meet-irreducible (with zero right singular ideal) and contains uniform right ideals; (2) the closed right ideals ofRRare right annihilator ideals, and each such right ideal is essentially finitely generated; (3)RRpossesses a reducing pair (i.e. a pair(β1,β2)({\beta _1},{\beta _2})of elements for whichβ1R,β2R{\beta _1}R,{\beta _2}Randβ1r+β2r\beta _1^r + \beta _2^rare large right ideals ofRR); (4) for eacha∈Ra \in Rwithal=0,aR{a^l} = 0,aRcontains a regular element ofRR. A second characterization ofRRis also given. This is in terms of the right annihilator ideals ofRRwhich have the same (uniform) dimension asRR{R_R}. The problem of characterizing right orders in (infinite dimensional) full linear rings was posed by Carl Faith. The Goldie theorems settled the finite dimensional case.