bluuebunny/crossref_metadata_2025 huggingface dataset:
On 𝐿^{𝑝} norms and the equimeasurability of functions
Authors: Andersen Kenneth F.
Published: June 2010

For measurable functions

f
f

and

g
g

, necessary and sufficient conditions are given for the equality of certain

L
p

{L^p}

norms of

f
f

and

g
g

to imply that

f
f

and

g
g

are equimeasurable.
Dirichlet problems for singular elliptic equations
Authors: Lo Chi Yeung
Published: June 2010

Boundary value problems are formulated for the equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk right-parenthesis upper L left-bracket u right-bracket equals sigma-summation Underscript i comma j equals 1 Overscript n Endscripts a Subscript i j Baseline StartFraction partial-differential squared u Over partial-differential x Subscript i Baseline partial-differential x Subscript j Baseline EndFraction plus sigma-summation Underscript i equals 1 Overscript n minus 1 Endscripts b Subscript i Baseline StartFraction partial-differential u Over partial-differential x Subscript i Baseline EndFraction plus StartFraction h Over x Subscript n Baseline EndFraction StartFraction partial-differential u Over partial-differential x Subscript n Baseline EndFraction plus c u equals f">

(
∗
)

L
[
u
]
=

∑

i
,
j
=
1

n

a

i
j

∂
2

u

∂

x
i

∂

x
j

+

∑

i
=
1

n
−
1

b
i

∂
u

∂

x
i

+

h

x
n

∂
u

∂

x
n

+
c
u
=
f

( \ast )\quad L[u] = \sum \limits _{i,j = 1}^n {{a_{ij}}\frac {{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}}} + \sum \limits _{i = 1}^{n - 1} {{b_i}\frac {{\partial u}}{{\partial {x_i}}}} + \frac {h}{{{x_n}}}\frac {{\partial u}}{{\partial {x_n}}} + cu = f

\]
in a bounded domain

G
G

in

E
n

{E_n}

with boundary

∂
G
=

S
1

∪

S
2

\partial G = {S_1} \cup {S_2}

where

S
1

{S_1}

is in

x
n

=
0

{x_n} = 0

and

S
2

{S_2}

is in

x
n

>
0

{x_n} > 0

. A uniqueness theorem is established for

(
∗
)

( \ast )

when boundary data is only given on

S
2

{S_2}

for
\[

h
(

x
1

,
⋯
,

x

n
−
1

,
0
)
≧
1
;

h({x_1}, \cdots ,{x_{n - 1}},0) \geqq 1;

\]
; whereas an existence and uniqueness theorem for the Dirichlet problem is proved for

h
(

x
1

,

x
2

,
⋯
,

x

n
−
1

,
0
)
>
1

h({x_1},{x_2}, \cdots ,{x_{n - 1}},0) > 1

.
A Kneser theorem for Volterra integral equations
Authors: Kelley Walter G.
Published: June 2010

A connectedness result is obtained for the space of solutions of any one of a class of Volterra integral equations.
Automorphisms of commutative Banach algebras
Authors: Johnson B. E.
Published: June 2010

This paper presents a new proof of the theorem of Kamowitz and Scheinberg which states that if

α
\alpha

is an element of infinite order of the automorphism group of a commutative semisimple Banach algebra then the spectrum of

α
\alpha

contains all complex numbers of absolute value 1. The proof depends on the fact that the only closed translation invariant subalgebras of

l
∞

(
−
∞
,
+
∞
)

{l^\infty }( - \infty , + \infty )

(pointwise multiplication) for which the restriction of the shift has a complex number of absolute value 1 in its resolvent set are certain spaces of periodic sequences.
Beurling generalized prime number systems in which the Chebyshev inequalities fail
Authors: Hall R. S.
Published: June 2010

It is proved that there exist systems of generalized primes in which the asymptotic distribution of integers is

N
(
x
)
=
A
x
+
O
(
x
⋅

log

−
γ

x
)

N(x) = Ax + O(x \cdot {\log ^{ - \gamma }}x)

with

A
>
0

A > 0

and

γ
∈
[
0
,
1
)

\gamma \in [0,1)

and in which the Chebyshev inequalities
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit inf Underscript x right-arrow normal infinity Endscripts StartFraction pi left-parenthesis x right-parenthesis log x Over x EndFraction greater-than 0 comma limit sup Underscript x right-arrow normal infinity Endscripts StartFraction pi left-parenthesis x right-parenthesis log x Over x EndFraction greater-than normal infinity">

lim

inf

x
→
∞

π
(
x
)
log
⁡
x

x

>
0
,

lim

sup

x
→
∞

π
(
x
)
log
⁡
x

x

>
∞

\lim \inf \limits _{x \to \infty } \frac {{\pi (x)\log x}}{x} > 0,\quad \lim \sup \limits _{x \to \infty } \frac {{\pi (x)\log x}}{x} > \infty

\]
do not hold.
On compact operators in the weak closure of the range of a derivation
Authors: Kim Hong W.
Published: June 2010

It is shown that if

K
K

is a compact operator which commutes with a bounded operator

A
A

on a Hilbert space

H
H

and if

K
K

is contained in the weak closure of the range of the derivation induced by

A
A

, then

K
K

is quasinilpotent.
More noneuclidian 𝑃𝐼𝐷’s and Dedekind domains with prescribed class group
Authors: Eakin Paul , Heinzer W.
Published: June 2010

Let

Z
Z

denote the integers,

Q
Q

the rationals,

X
X

an indeterminate and

G
G

a finitely generated abelian group. Then there is a Dedekind domain

D
D

such that

Z
[
X
]
⊂
D
⫋
Q
[
X
]

Z[X] \subset D \varsubsetneqq Q[X]

, and

D
D

has class group

G
G

. If

G
=
0

G = 0

then

D
D

is a noneuclidian PID.
Structure of semiprime P.I. rings
Authors: Fisher Joe W.
Published: June 2010

In this paper we make an investigation into the structure of semiprime polynomial identity rings which is culminated by showing that each such ring

R
R

has a unique maximal left quotient ring

Q
Q

such that (1)

Q
Q

is von Neumann regular with unity and (2) every regular element in

R
R

is invertible in

Q
Q

.
On simple injective rings
Authors: Elliger Sigurd
Published: June 2010

It is proved that a simple ring which is injective on both sides must be artinian. This answers a question asked by C. Faith in the negative.
On finitely generated subgroups which are of finite index in generalized free products
Authors: Karrass A. , Solitar D.
Published: June 2010

Let

G
=
(
A
∗
B
;

U
)

G = (A \ast B;\;U)

be the free product of A and B with the subgroup U amalgamated. Various conditions are given which imply that every finitely generated subgroup H containing a (nontrivial) normal subgroup of G has finite index in G (in such a case we say G has the f.g.c.n. property). In particular, if A is a noncyclic free group and U is cyclic, then G has the f.g.c.n. property. We use this last result to give a combinatorial proof that Fuchsian groups have the f.g.c.n. property; this was first proved by Greenberg using non-Euclidean geometry.
Normal subgroups of groups which are products of two Abelian subgroups
Authors: Knop Larry E.
Published: June 2010

It is shown that if a group

G
=
A
B

G = AB

, where

A
A

and

B
B

are Abelian subgroups of

G
,
A
≠
B

G,A \ne B

, and either

A
A

or

B
B

satisfies the maximum condition, then there is a normal subgroup

N
N

of

G
G

,

N
≠
G

N \ne G

, such that

N
N

contains either

A
A

or

B
B

.
Convolutions and absolute continuity
Authors: Lind D. A.
Published: June 2010

We show that if

E
E

is a subset of the circle with positive Lebesgue measure, and

g
g

is integrable on almost every translate of

E
E

, then

g
g

is integrable on the whole circle. A generalization of this fact leads to a characterization of positive measures with nonvanishing absolutely continuous part.
A general theorem for decomposition of linear random processes
Authors: Hebert D. J.
Published: June 2010

Let

E
E

and

F
F

be locally convex spaces in duality and let

f
f

be a linear random process indexed by

F
F

such that the corresponding cylindrical measure is a Radon measure. It is shown without any assumptions of metrizability or countability that there is an equivalent process with continuous linear trajectories.
A class of univalent functions
Authors: Caplinger T. R. , Causey W. M.
Published: June 2010

A sharp coefficient estimate is obtained for a class

D
(
α
)

D(\alpha )

of functions univalent in the open unit disc. The radius of convexity and an arclength result are also determined for the class.
Classification of bounded solutions of a linear nonhomogeneous differential equation
Authors: Hallam Thomas G.
Published: June 2010

An elementary criterion, depending only upon the initial data of a solution, is formulated to determine the boundedness of solutions of a nonhomogeneous linear system of ordinary differential equations. The associated homogeneous linear differential equation is required to be either conditionally stable or conditionally asymptotically stable.
Some remarks on absolute continuity on groups
Authors: Glicksberg I.
Published: June 2010

The Tam-Larsen result that Radon measures with separable orbits are absolutely continuous is extended somewhat. A consequence is a characterization among finite measures of the absolutely continuous ones as those whose orbits are equi-regular.
Type II 𝑊-* algebras are not normal
Authors: Willig Paul
Published: June 2010

If

A

\mathcal {A}

is a

W
W

-

∗
\ast

algebra of type II on separable Hilbert space

H
H

, then

A

\mathcal {A}

is not normal.
Compact 𝒢-souslin sets are 𝒢_{𝛿}’s
Authors: Braude Eric John
Published: June 2010

The result of the title generalizes and places in a new set-theoretical context the well-known theorem of Halmos that every compact Baire set is a

G

δ

{\mathcal {G}_\delta }

. Šneĭder’s result of 1968 that every perfectly normal compact space with

G

\mathcal {G}

-Souslin diagonal is metrizable, can now be seen to be true without the perfect normality condition.
The congruence lemma for [𝑦^{𝑝}𝑧]
Authors: Newton Mervin E.
Published: June 2010

A sufficient condition for membership in

[

y
p

z
]

[{y^p}z]

, based on the weight and signature of a term, is given.
An approximation theory for oscillations of differential equations
Authors: Gregory John
Published: June 2010

In an earlier paper we gave an approximation theory of focal points and focal intervals. The fundamental purpose of this paper is to show that the ideas and methods of that paper can be used to give an approximation theory for oscillation for linear selfadjoint differential (and integral-differential) equations. These methods follow from the theory of quadratic forms given by Hestenes. In §1 we give the preliminaries needed for this paper. In §2 we define oscillation for quadratic control problems and discuss their connection with differential equations. In §3 we give our main approximation results relating the oscillation points

λ
n

(

σ
0

)
(
n
=
i
,
2
,
3
,
⋯
)

{\lambda _n}({\sigma _0})(n = i,2,3, \cdots )

for a given problem to the oscillation points

λ
n

(
σ
)
(
n
=
1
,
2
,
3
,
⋯
)

{\lambda _n}(\sigma )(n = 1,2,3, \cdots )

of an approximating problem. The element

σ
\sigma

belongs to a metric space

(
Σ
,
d
)

(\Sigma ,d)

. The main result is to show that the

n
n

th oscillation point

λ
n

(
σ
)

{\lambda _n}(\sigma )

is a continuous function of

σ
(
n
=
i
,
2
,
3
,
⋯
)

\sigma (n = i,2,3, \cdots )

. For completeness in §4 we present an example for fourth order differential equations where the approximation is by discrete problems. Thus oscillation points can be “easily” computed by numerical algorithms.
A note on the extension of contractive mappings
Authors: Solomon J. L.
Published: June 2010

In this note a question posed by J. D. Bryant and L. F. Guseman, Jr. is answered. It is shown that for dense subsets of finite dimensional Banach spaces, the iterative test being conclusive and the contractive extension property are not equivalent. If

(
X
,
d
)

(X,d)

is a complete locally compact and chainable-metric space, a characterization of these dense subsets for which the iterative test is conclusive is given.
Isotopy equivalence classes of normal arcs in 𝐹×𝐼
Authors: Feustel C. D.
Published: June 2010

Let

F
F

be a compact

2
2

-manifold and

I
I

the closed unit interval. Let

α
\alpha

and

β
\beta

be arcs embedded in

F
×
I

F \times I

such that

α
\alpha

and

β
\beta

meet the boundary of

F
×
I

F \times I

in the boundary of

α
\alpha

and

β
\beta

respectively. Then we give necessary and sufficient conditions for the existence of an ambient isotopy, constant on the boundary of

F
×
I

F \times I

, moving

α
\alpha

to

β
\beta

. We also obtain ambient isotopies of families of arcs properly embedded in

F
×
I

F \times I

.
A generalization of the dog bone space to 𝐸ⁿ
Authors: Eaton W. T.
Published: June 2010

In this paper we construct an upper semicontinuous decomposition of

E
n

(
n
≧
3
)

{E^n}(n \geqq 3)

into points and tame arcs such that the associated decomposition space is distinct from

E
n

{E^n}

.
The finiteness of 𝐼 when 𝑅[𝑋]/𝐼 is 𝑅-projective
Authors: Brewer J. W. , Montgomery P. R.
Published: June 2010

This paper is concerned with the relationship between

R
[
X
]

/

I

R[X]/I

being a projective

R
R

-module and

I
I

being a finitely generated ideal of

R
[
X
]

R[X]

. It is shown that if

R
[
X
]

/

I

R[X]/I

is

R
R

-free, then

I
=
f
R
[
X
]
,
f

I = fR[X],f

a monic polynomial of

R
[
X
]

R[X]

. Also,

R
[
X
]

/

I

R[X]/I

is a finitely generated projective

R
R

-module if and only if

R
[
X
]

/

I

R[X]/I

is a finitely generated

R
R

-module and

I
=
f
R
[
X
]

I = fR[X]

for some

f
∈
R
[
X
]

f \in R[X]

. When

R
[
X
]

/

I

R[X]/I

is projective,

I
I

is a finitely generated ideal if and only if

I
I

is a principal ideal. Finally, an example is given to show that

R
[
X
]

/

I

R[X]/I

can be projective without

I
I

being finitely generated.
The cobordism of involutions on orientable manifolds
Authors: Rowlett R. J.
Published: June 2010

This note calculates the cobordism of smooth, oriented manifolds with orientation-reversing involution, and also the

2
2

-torsion in the cobordism of such manifolds with orientation-preserving involution. In particular, it is shown that the forgetful homomorphism maps both groups monomorphically into the cobordism of unoriented manifolds with involution.
A short proof of the martingale convergence theorem
Authors: Lamb Charles W.
Published: June 2010

The martingale convergence theorem is first proved for uniformly integrable martingales by a standard application of Doob’s maximal inequality. A simple truncation argument is then given which reduces the proof of the

L
1

{L^1}

-bounded martingale theorem to the uniformly integrable case. A similar method is used to prove Burkholder’s martingale transform convergence theorem.
Coherence of one polynomial ring
Authors: Vasconcelos Wolmer V.
Published: June 2010

The main result is that

A
[
x
]

A[x]

, the polynomial ring in one variable over the domain

A
A

of global dimension two, is coherent. As an application, the quotient rings of

A
[
x
]

A[x]

which have finite global dimension are examined.
Groups with normal subgroups possessing subnormal complements
Authors: Toh K. H.
Published: June 2010

J. Wiegold has characterized groups in which every normal subgroup is a direct factor as the restricted direct products of simple groups. In this paper, it is proved that for a group

G
G

to have the structure above, it is sufficient that every normal subgroup of

G
G

has a subnormal complement in

G
G

.
A remark on 𝐶_{𝜎} spaces
Authors: Reich Simeon
Published: June 2010

We give a simple new proof of the following result, conjectured by Effros and proved by Fakhoury: Let

E
E

be a

C
σ

{C_\sigma }

space and

Z
Z

the set of extreme points of the unit ball of

E
∗

{E^ \ast }

. Then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z union StartSet 0 EndSet equals left-brace p element-of upper E Superscript asterisk Baseline colon mathematical left-angle f g h comma p mathematical right-angle equals mathematical left-angle f comma p mathematical right-angle mathematical left-angle g comma p mathematical right-angle mathematical left-angle h comma p mathematical right-angle">

Z
∪
{
0
}
=
{
p
∈

E
∗

:

⟨

f
g
h
,
p

⟩

=

⟨

f
,
p

⟩

⟨

g
,
p

⟩

⟨

h
,
p

⟩

Z \cup \{ 0\} = \{ p \in {E^ \ast }:\left \langle {fgh,p} \right \rangle = \left \langle {f,p} \right \rangle \left \langle {g,p} \right \rangle \left \langle {h,p} \right \rangle

for all

f
,
g
,
h

f,g,h

in

E
}

E\}

.
The norm of a derivation in a 𝑊*-algebra
Authors: Zsidó László
Published: June 2010

The norm of an inner derivation

δ
a

{\delta _a}

of a (nonnecessary separable)

W
∗

{W^ \ast }

-algebra

M
M

is shown to satisfy
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue delta Subscript a Baseline StartAbsoluteValue EndAbsoluteValue equals 2 inf left-brace StartAbsoluteValue EndAbsoluteValue a minus z StartAbsoluteValue EndAbsoluteValue semicolon z element-of upper Z comma the center of upper M right-brace comma">

|

|

δ
a

|

|

=
2
inf
{

|

|

a
−
z

|

|

;
z
∈
Z
,

the center of

M
}
,

||{\delta _a}|| = 2\inf \{ ||a - z||;z \in Z,\;{\text {the center of}\; M\} ,}

\]
and some related results are obtained.
Quasi-unmixed local rings and quasi-subspaces
Authors: Sawtelle Peter G.
Published: June 2010

The concept of a quasi-subspace is defined so that it plays a role relative to quasi-unmixedness analogous to that of subspace to unmixedness. This definition is used to characterize quasi-unmixed local rings.
Weak solutions with unbounded variation
Authors: Ballou Donald P.
Published: June 2010

To solve a quasilinear system of hyperbolic partial differential equations with given initial data, the usual procedure is to approximate the initial data, solve the resulting problems, and show that the variation of the approximating solutions is uniformly bounded. A limiting process then can be used. This paper shows that, for simple systems, the variation of the solution need not be finite.
The number of continua
Authors: Lozier F. W. , Marty R. H.
Published: June 2010

It is shown there are precisely

2
n

{2^n}

topologically distinct continua of weight

n
n

and power

m
m

where

p
≦
n
≦
m

p \leqq n \leqq m

and

p
p

is the smallest cardinal for which there is a continuum of power

m
m

and weight

p
p

. In particular, there are precisely

2
m

{2^m}

topologically distinct continua of power

m
m

.
𝐶-separated sets in certain metric spaces
Authors: Dickman R. F. , McCoy R. A. , Rubin L. R.
Published: June 2010

A space

X
X

has Property

C

{\text {C}}

provided that for every separated closed subset

A
A

of

X
X

there exist disjoint closed and connected sets

C
1

{C_1}

and

C
2

{C_2}

in

X
X

each of which intersects

A
A

and which contain

A
A

in their union. This property has been used in characterizing unicoherence. A metric space

X
X

has Property

S

{\text {S}}

if for each

ε
>
0
,
X

\varepsilon > 0,X

is the union of a finite number of connected sets each of diameter less than

ε
\varepsilon

. In this paper a sufficient condition for a space to have Property

C

{\text {C}}

is established and used to show that separable Hilbert space has Property

C

{\text {C}}

and that every connected metric space having Property

S

{\text {S}}

has Property

C

{\text {C}}

. It follows from the latter result that every separable, locally connected, connected, rimcompact metric space has Property

C

{\text {C}}

. An example is given of a unicoherent, connected, uniformly locally connected, locally arcwise connected, separable metric space that does not have Property

C

{\text {C}}

.
On contractive mappings
Authors: Geraghty Michael A.
Published: June 2010

The Cauchy condition for convergence of a contractive iteration in a complete metric space is replaced by an equivalent functional condition. Many generalizations of the Banach-Neumann contractive mapping principle follow.
A filter characterization of regular Baire spaces
Authors: McCoy R. A.
Published: June 2010

A filter characterization of regular Baire spaces is given.
Uniform approximation by solutions of elliptic equations
Authors: Weinstock Barnet M.
Published: June 2010

The space

H
A

(
K
)

{H_A}(K)

of continuous functions on a compact set

K
K

in Euclidean space which can be uniformly approximated by solutions of the elliptic, constant-coefficient partial differential equation

A
f
=
0

Af = 0

is studied. In particular, it is shown that

H
A

(
K
)

{H_A}(K)

is local, in the same sense as in the theory of rational approximation in the complex plane. Simultaneous approximation of functions and their derivatives is also considered.
Conjugacy separating representations of free groups
Authors: Wehrfritz B. A. F.
Published: June 2010

If

G
G

is a free group and

g
g

is an element of

G
G

we show that there exists a residually finite (commutative) integral domain

R
R

and a faithful matrix representation

ρ
\rho

of

G
G

over

R
R

of finite degree such that the conjugacy class of

g
ρ

g\rho

in

G
ρ

G\rho

is closed in the topology induced on

G
ρ

G\rho

by the Zariski topology on the full matrix algebra. It follows that free groups are conjugacy separable, a result obtained by a number of authors, see [1], [5] and [6].
On going down for simple overrings
Authors: Dobbs David E.
Published: June 2010

Let

R
R

be an integral domain with quotient field

K
K

. If

R
R

is Noetherian: then the Krull dimension of

R
R

is at most

1
⇔

1 \Leftrightarrow

for all overrings

S
S

of

R
R

,

R
⊂
S

R \subset S

satisfies going down.

R
R

is Dedekind

(resp

., PID)

⇔
R

{\text {(resp}}{\text {., PID)}} \Leftrightarrow R

is Krull (resp., UFD) and, for all

u
∈
K
,
R
⊂
R
[
u
]

u \in K,R \subset R[u]

satisfies going down.

R
R

is Prüfer

⇔
R

\Leftrightarrow R

is integrally closed, every intersection of two principal ideals of

R
R

is finitely generated, and

R
⊂
R
[
u
]

R \subset R[u]

satisfies going down for all

u
∈
K

u \in K

.
Injective objects in the category of 𝑝-rings
Authors: Haines David C.
Published: June 2010

A p-ring (or generalized Boolean ring) P is a ring of fixed prime characteristic p in which

a
p

=
a

{a^p} = a

for all a in P. In this paper P is partially ordered by a relation which is a generalization of the usual Boolean order. A subset S of P is then called quasiorthogonal if

a
b
(
a
−
b
)
=
0

ab(a - b) = 0

for all a, b in S. It is shown that P is injective in the category of p-rings if and only if every quasiorthogonal subset has a supremum under this partial order.
Translation invariant sigma algebras on groups
Authors: Lind D. A.
Published: June 2010

The translation invariant sigma subalgebras of the Baire sets of a locally compact abelian group are characterized as those periodic under some closed subgroup.
Two lifting theorems
Authors: Lloyd Stuart P.
Published: June 2010

It is assumed that the measure algebra involved has cardinality

2

ℵ
0

{2^{{\aleph _0}}}

, and it is assumed further that

2

ℵ
0

=

ℵ
1

{2^{{\aleph _0}}} = {\aleph _1}

. Then liftings exist when the

σ
\sigma

-field is not necessarily complete, and strong Borel liftings exist in the locally compact

σ
\sigma

-compact metric case.
On a theorem of a Pełczyński
Authors: Gamlen John L. B.
Published: June 2010

If

Y
Y

is a weakly complete Banach space, and

X
X

is a Banach space with separable dual, then every continuous linear operator from

C
X

(
K
)

{C_X}(K)

to

Y
Y

must be weakly compact. Here

C
X

(
K
)

{C_X}(K)

denotes the space of continuous functions on the compact Hausdorff space

K
K

, having values in

X
X

.
Central localizations of regular rings
Authors: Armendariz E. P. , Fisher Joe W. , Steinberg Stuart A.
Published: June 2010

In this paper we show that a ring

R
R

is von Neumann regular (or a

V
V

-ring) if and only if every central localization of

R
R

at a maximal ideal of its center is von Neumann regular (or a

V
V

-ring). Strongly regular rings are characterized by the property that all central localizations at maximal ideals of the center are division rings. Also we consider whether regular PI-rings can be characterized by the property that all central localizations at maximal ideals of the center are simple.
Proving Kneser’s theorem for finite groups by another 𝑒-transform
Authors: Lee R. A.
Published: June 2010

Although neither the result nor the

e
e

-transformation is new, a new order for the successive transformations is prescribed. From this follow some interesting properties which in turn imply the result.
An extension of the Hausdorff-Young theorem
Authors: Young Robert M.
Published: June 2010

Using the Riesz-Thorin interpolation theorem, we show that if

1
>
p
>
2

1 > p > 2

and

f
f

belongs to

L
p

(
−
π
,
π
)

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

, then

{

f
^

(

z
n

)
}

\{ \hat f({z_n})\}

belongs to

l
q

(
q
=
p

/

(
p
−
1
)
)

{l^q}(q = p/(p - 1))

for a very general class of complex sequences

{

z
n

}

\{ {z_n}\}

. We also obtain a convergence criterion for a related class of exponential sums.
Formal Taylor series and complementary invariant subspaces
Authors: Herrero Domingo A.
Published: June 2010

A class of operators (which includes the unilateral weighted shifts and the noninvertible bilateral weighted shifts on Hilbert spaces), with the property that every element of the commutant of the operator is canonically associated to a formal Taylor series, is characterized. Let

T
T

be one of such operators, then the following result is true:

T
T

has no nontrivial complementary invariant subspaces, no roots and no logarithm. This result can be partially extended to the case when “commutant” is replaced by “double commutant", then:

T
T

has no nontrivial complementary hyperinvariant subspaces.
Absolute summability matrices that are stronger than the identity mapping
Authors: Fridy J. A.
Published: June 2010

The main result gives a simple column-sum property which implies that the matrix

A
A

maps

l
A

{l_A}

properly into

l
1

{l^1}

, i.e.,

l
1

⫋

A

−
1

[

l
1

]

{l^1} \subsetneqq {A^{ - 1}}[{l^1}]

. Also, the means of Nörlund, Euler-Knopp, Taylor, and Hausdorff are investigated as mappings of

l
1

{l^1}

into itself.
Commutative regular rings without prime model extensions
Authors: Saracino D. , Weispfenning V.
Published: June 2010

It is known that the theory

K
K

of commutative regular rings with identity has a model completion

K
′

K’

. We show that there exists a countable model of

K
K

which has no prime extension to a model of

K
′

K’

.
3-variétés qui ne sont pas des revêtements cycliques ramifiés sur 𝑆³
Authors: Montesinos José M.
Published: June 2010

It is shown that if

F
g

{F_g}

is a compact, orientable surface of genus

g
g

, then

S
1

×

F
g

{S^1} \times {F_g}

is not a cyclic branched covering of

S
3

{S^3}

.
Genera in abelian extensions
Authors: Gold Robert
Published: June 2010

The principal genus in an abelian extension is characterized as the kernel of norm residue symbols at the ramified primes.
Well-distributed sequences with respect to systems of convex sets
Authors: Niederreiter H.
Published: June 2010

A theorem of W. M. Schmidt concerning the existence of sequences which are extremely well distributed with respect to suitable convex sets is generalized. We prove the existence of sequences which are simultaneously well distributed with respect to suitable systems of convex sets. The proof depends on combinatorial results dealing with the distribution of sequences in finite and countable sets.
Horizontal chords of the graph of a continuous function
Authors: Herzberg Uwe
Published: June 2010

The continuous function

f
f

, defined on

[
a
,
b
]

[a,b]

, changing sign exactly

n
n

times on

(
a
,
b
)

(a,b)

, and

f
(
a
)
=
f
(
b
)
=
0

f(a) = f(b) = 0

, has horizontal chords of every length

h
>
H

h > H

with endpoints in

(
a
,
b
)

(a,b)

. We determine the largest such

H
H

as a function of

n
n

.
The dimension of the ring of coefficients in a polynomial ring
Authors: Arnold Jimmy T.
Published: June 2010

A
A

and

B
B

are commutative rings with identity. We say that

A
A

and

B
B

are stably equivalent provided there exists a positive integer

n
n

such that the polynomial rings

A
[

X
1

,
⋯
,

X
n

]

A[{X_1}, \cdots ,{X_n}]

and

B
[

Y
1

,
⋯
,

Y
n

]

B[{Y_1}, \cdots ,{Y_n}]

are isomorphic. If

A
A

and

B
B

are stably equivalent, then they have equal Krull dimension.
A characterization of Steinitz group rings
Authors: Allen Paul J. , Neggers Joseph
Published: June 2010

A ring

R
R

with an identity is a (right) Steinitz ring provided any linearly independent subset of a free (right)

R
R

-module can be extended to a basis for the module by adjoining elements from any given basis. In this paper, we characterize those group rings which are Steinitz rings by the following: Theorem. The group ring

R
[
G
]

R[G]

is a Steinitz ring if and only if

R
R

is a Steinitz ring and either (1) char

R
=

p
i

R = {p^i}

and

G
G

is a finite

p
p

-group or (2) char

R
=
0

R = 0

and

G
=
1

G = 1

.
Strong uniform distributions and ergodic theorems
Authors: Blum J. R. , Hahn L.-S.
Published: June 2010

Let

G
G

and

H
H

be locally compact

σ
\sigma

-compact abelian groups,

A

\mathcal {A}

a mapping from

G
G

to

H
H

, and

{

μ
n

}

n
=
1

∞

\{ {\mu _n}\} _{n = 1}^\infty

a sequence of measures on

G
G

. We define the notions: “

A

\mathcal {A}

is a uniform distribution with respect ot

{

μ
n

}

\{ {\mu _n}\}

” and “

A

\mathcal {A}

is a strong uniform distribution". We give a number of examples of these notions and derive some general individual ergodic theorems for measure-preserving transformations with discrete spectrum.
Existence and representation of solutions of parabolic equations
Authors: Eklund Neil A.
Published: June 2010

Let

L
L

be a linear, second order parabolic operator in divergence form and let

Q
Q

be a bounded cylindrical domain in

E

n
+
1

{E^{n + 1}}

. Let

∂
p

Q

{\partial _p}Q

denote the parabolic boundary of

Q
Q

. To each continuous function

f
f

on

∂
p

Q

{\partial _p}Q

there is a unique solution

u
u

of the boundary value problem

L
u
=
0

Lu = 0

in

Q
,
u
=
f

Q,u = f

on

∂
p

Q

{\partial _p}Q

. Moreover, for the given

L
L

and

Q
Q

, to each

(
x
,
t
)
∈
Q

(x,t) \in Q

there is a unique nonnegative measure

μ

(
x
,
t
)

{\mu _{(x,t)}}

with support on

∂
p

Q

{\partial _p}Q

such that the solution of the boundary value problem is given by

u
(
x
,
t
)
=

∫

∂
p

Q

f
d

μ

(
x
,
t
)

u(x,t) = \int _{{\partial _p}Q} {fd{\mu _{(x,t)}}}

.
A sufficient condition for the insertion of a continuous function
Authors: Lane Ernest P.
Published: June 2010

A sufficient condition for the insertion of a continuous function between two comparable real-valued functions is given. Four insertion theorems are obtained as corollaries.
A cohomological characterization of preimages of nonplanar, circle-like continua
Authors: Rogers James T.
Published: June 2010

Let

G
G

be an additively written Abelian group, and let

P
=
{

p
1

,

p
2

,

p
3

,
⋯
}

P = \{ {p_1},{p_2},{p_3}, \cdots \}

be a sequence of positive integers. An element

g
g

in

G
G

is said to have infinite

P
P

-height if (1)

g
≠
0

g \ne 0

, (2) each

p
i

>
1

{p_i} > 1

, and (3) for each positive integer

n
n

, there is an element

h
h

in

G
G

such that

(

p
1

p
2

⋯

p
n

)
h
=
g

({p_1}{p_2} \cdots {p_n})h = g

. The purpose of this paper is to prove the following Theorem. If

X
X

is a continuum, then the following are equivalent: (1)

H
1

(
X
)

{H^1}(X)

contains an element of infinite

P
P

-height, for some sequence

P
P

of positive integers; (2)

X
X

can be mapped onto a solenoid; (3)

X
X

can be mapped onto a nonplanar, circle-like continuum. Here

H
1

(
X
)

{H^1}(X)

is Alexander-Čech cohomology with integral coefficients.
Steinitz classes of cyclic extensions of degree 𝑙^{𝑟}
Authors: Long Robert
Published: June 2010

Let

k
k

be an algebraic number field and

l
l

an odd prime. The set of Steinitz classes of tamely ramified cyclic extensions

K

/

k

K/k

of degree

l
r

{l^r}

is calculated and shown to be a subgroup of the ideal class group of

k
k

.
A real place on the real number field is trivial
Authors: Garner Lynn E.
Published: June 2010

This paper proves that any real place from the real number system must be trivial. The result extends to a real place on any archimedean real closed field.
An extension of the Denjoy integral
Authors: Foran James
Published: June 2010

The purpose of this paper is to exhibit an additive class of continuous functions such that each function of the class satisfies Lusin’s condition

(

N

)

({\text {N}})

. Restricting this class to those functions which are approximately derivable almost everywhere produces a class of primitives which is more extensive than the class of ACG functions, the primitives for the wide sense Denjoy integral.
A note on the zeros of Faber polynomials
Authors: Goodman A. W.
Published: June 2010

By an elementary counterexample we show that a conjecture about the zeros of the Faber polynomials is false.
A weighted norm inequality for Vilenkin-Fourier series
Authors: Gosselin John A.
Published: June 2010

Various operators related to the Hardy-Littlewood maximal function have been shown to satisfy a strong type

(
p
,
p
)

(p,p)

condition,

1
>
p
>
∞

1 > p > \infty

, for weighted

L
p

{L^p}

spaces providing the weight function satisfies the

A
p

Ap

condition of B. Muckenhoupt. In particular this result for the maximal partial sum operator for trigonometric series was established by R. Hunt and W. S. Young. In this note a result similar to that of Hunt and Young is established for Vilenkin-Fourier series, which include Walsh series as a special case.
A note on a trigonometric moment problem
Authors: Young Robert M.
Published: June 2010

A sequence

{

λ
n

}

n
=
−
∞

∞

\{ {\lambda _n}\} _{n = - \infty }^\infty

is said to be an interpolating sequence for

L
2

(
−
π
,
π
)

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

if the system of equations
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript n Baseline equals integral Subscript negative pi Superscript pi Baseline f left-parenthesis t right-parenthesis e Superscript i lamda Super Subscript n Superscript t Baseline d t left-parenthesis negative normal infinity greater-than n greater-than normal infinity right-parenthesis">

c
n

=

∫

−
π

π

f
(
t
)

e

i

λ
n

t

d
t

(
−
∞
>
n
>
∞
)

{c_n} = \int _{ - \pi }^\pi {f(t)} {e^{i{\lambda _n}t}}dt\quad ( - \infty > n > \infty )

\]
admits a solution

f
f

in

L
2

(
−
π
,
π
)

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

whenever

{

c
n

}
∈

l
2

\{ {c_n}\} \in {l^2}

. If the solution is unique then

{

λ
n

}

\{ {\lambda _n}\}

is said to be a complete interpolating sequence. It is shown that if the imaginary part of

λ
n

{\lambda _n}

is uniformly bounded and 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 greater-than 1 slash 4 left-parenthesis negative normal infinity greater-than n greater-than normal infinity right-parenthesis">

|

Re
⁡
(

λ
n

)
−
n

|

≤
L
>
1

/

4
(
−
∞
>
n
>
∞
)

|\operatorname {Re} ({\lambda _n}) - n| \leq L > 1/4( - \infty > n > \infty )

, then

{

λ
n

}

\{ {\lambda _n}\}

is a complete interpolating sequence and

{

e

i

λ
n

t

}

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

is a Schauder basis for

L
2

(
−
π
,
π
)

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

. It is also shown that this result is sharp in the sense that the condition

|

λ
n

−
n

|

>
1

/

4

|{\lambda _n} - n| > 1/4

is not sufficient to guarantee that

{

λ
n

}

\{ {\lambda _n}\}

is an interpolating sequence.
A note on Jones’ function 𝐾
Authors: Rosasco John
Published: June 2010

For each point

x
x

of a continuum

M
M

, F. B. Jones [5, Theorem 2] defines

K
(
x
)

K(x)

to be the closed set consisting of all points

y
y

of

M
M

such that

M
M

is not aposyndetic at

x
x

with respect to

y
y

. Suppose

M
M

is a plane continuum and for any positive real number

ϵ
\epsilon

there are at most a finite number of complementary domains of

M
M

of diameter greater than

ϵ
\epsilon

. In this paper it is proved that for each point

x
x

of

M
M

, the set

K
(
x
)

K(x)

is connected.
Univalent functions with negative coefficients
Authors: Silverman Herb
Published: June 2010

Coefficient, distortion, covering, and coefficient inequalities are determined for univalent functions with negative coefficients that are starlike of order

α
\alpha

and convex of order

α
\alpha

. Extreme points for these classes are also determined.
Existence theorems for Urysohn’s integral equation
Authors: Joshi M.
Published: June 2010

The theory of abstract Hammerstein operators is applied to obtain existence theorems for Urysohn’s integral equation.
Compact operators in the algebra generated by essentially unitary 𝐶₀ operators
Authors: Nordgren Eric A.
Published: June 2010

It will be shown that the compact operators in the weakly closed algebra generated by an essentially unitary

C
0

{C_0}

contraction are weakly dense in the algebra. The result implies the extension of a double dual theorem of Kriete, Moore and Page and yields a partial answer to a question on reductive algebras raised by Rosenthal.
Products of Steiner’s quasi-proximity spaces
Authors: Hayashi E.
Published: June 2010

E. F. Steiner introduced a quasi-proximity

δ
\delta

satisfying

A
δ
B
iff

{
x
}
δ
B

A\delta B \operatorname {iff}\; \{ x\} \delta B

for some

x
x

of

A
A

. The purpose of this paper is to describe the Tychonoff product of topologies in terms of Steiner’s quasi-proximities. Whenever

(

X
a

,

δ
a

)

({X_a},{\delta _a})

is the Steiner quasi-proximity space, the product proximity on

X
=
Π

X
a

X = \Pi {X_a}

can be given, by using the concept of finite coverings, as the smallest proximity on

X
X

which makes each projection

δ
\delta

-continuous.
On matrix approximation
Authors: Friedland Shmuel
Published: June 2010

In this paper we give an algebraic characterization of the best approximants to a given matrix

A
A

from a real line spanned by a matrix

B
B

. The distance

|

|

A
−
α
B

|

|

||A - \alpha B||

is taken to be the spectral norm of

A
−
α
B

A - \alpha B

.
Π₂¹ sets and Π₂¹ singletons
Authors: Harrington Leo
Published: June 2010

The following are equivalent: (a) every real is constructible; (b) every nonempty

∏
2
1

\prod _2^1

set of reals contains a

∏
2
1

\prod _2^1

singleton. (Implication

(

a

)
⇒
(

b

)

({\text {a}}) \Rightarrow ({\text {b}})

is due solely to H. Friedman.)
A three and five color theorem
Authors: Bernhart Frank R.
Published: June 2010

Let

f
f

be a face of a plane graph

G
G

. The Three and Five Color Theorem proved here states that the vertices of

G
G

can be colored with five colors, and using at most three colors on the boundary of

f
f

. With this result the well-known Five Color Theorem for planar graphs can be strengthened, and a relative coloring conjecture of Kainen can be settled except for a single case which happens to be a paraphrase of the Four Color Conjecture. Some conjectures are presented which are intermediate in strength to the Four Color Conjecture and the Three and Five Color Theorem.
Vanishing homology over nilpotent groups
Authors: Dwyer William G.
Published: June 2010

Let

π
\pi

be a nilpotent group and let

M
M

be a

π
\pi

-module. Under certain finiteness assumptions we prove that the twisted homology groups

H
i

(
π
,
M
)

{H_i}(\pi ,M)

vanish for all positive

i
i

whenever

H
0

(
π
,
M
)
=
0

{H_0}(\pi ,M) = 0

.
On the solvability of groups of central type
Authors: Yellen Jay
Published: June 2010

Let

G
G

be a finite group with center

Z
Z

8 and irreducible complex character

x
x

so that

x

(
1

)
2

=
[
G
:
Z
]

x{(1)^2} = [G:Z]

, If the

2
2

-Sylow subgroup of

G

/

Z

G/Z

has order 16 or less then

G
G

is solvable.
Subordination by convex functions
Authors: Hallenbeck D. J. , Ruscheweyh Stephan
Published: June 2010

The following theorem is proven: Let

F
(
z
)

F(z)

be convex and univalent in

Δ
=
{
z
:

|

z

|

>
1
}
,
F
(
0
)
=
1

\Delta = \{ z:|z| > 1\} ,F(0) = 1

. Let

f
(
z
)

f(z)

be analytic in
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta comma f left-parenthesis 0 right-parenthesis equals 1 comma f prime left-parenthesis 0 right-parenthesis equals ellipsis equals f Superscript left-parenthesis n minus 1 right-parenthesis Baseline left-parenthesis 0 right-parenthesis equals 0">

Δ
,
f
(
0
)
=
1
,

f
′

(
0
)
=
…
=

f

(
n
−
1
)

(
0
)
=
0

\Delta ,f(0) = 1,f’(0) = \ldots = {f^{(n - 1)}}(0) = 0

, and let

f
(
z
)
≺
F
(
z
)

f(z) \prec F(z)

in

Δ
\Delta

. Then for all

γ
≠
0

\gamma \ne 0

, Re

γ
⩾
0

\gamma \geqslant 0

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma Subscript z Baseline Superscript negative gamma Baseline integral Subscript 0 Superscript z Baseline tau Superscript gamma minus 1 Baseline f left-parenthesis tau right-parenthesis d tau precedes gamma z Superscript negative gamma slash n Baseline integral Subscript 0 Superscript z Superscript 1 slash n Baseline Baseline tau Superscript gamma minus 1 Baseline upper F left-parenthesis tau Superscript n Baseline right-parenthesis d tau period">

γ
z

−
γ

∫
0
z

τ

γ
−
1

f
(
τ
)
d
τ
≺
γ

z

−
γ

/

n

∫
0

z

1

/

n

τ

γ
−
1

F
(

τ
n

)
d
τ
.

{\gamma _z}^{ - \gamma }\int _0^z {{\tau ^{\gamma - 1}}f(\tau )d\tau \prec \gamma {z^{ - \gamma /n}}\int _0^{{z^{1/n}}} {{\tau ^{\gamma - 1}}F({\tau ^n})d\tau .} }

\]
This theorem, in combination with a method of D. Styer and D. Wright, leads to the following Corollary. Let

f
(
z
)
,
g
(
z
)

f(z),g(z)

be convex univalent in

Δ
,
f
(
0
)
=
f
(
0
)
=
g
(
0
)
=
g
(
0
)
=
0

\Delta ,f(0) = f(0) = g(0) = g(0) = 0

. Then

f
(
z
)
+
g
(
z
)

f(z) + g(z)

is starlike univalent in

Δ
\Delta

. Other applications of the theorem are concerned with the subordination of

f
(
z
)

/

z

f(z)/z

where

f
(
z
)

f(z)

belongs to certain classes of convex univalent functions.
A factorization theorem in 𝐻¹(𝑈³)
Authors: Miles Joseph
Published: June 2010

It is shown that there exists

f
ϵ

H
1

(

U
3

)

f\epsilon {H^1}({U^3})

which is not the product of two functions in

H
2

(

U
3

)

{H^2}({U^3})

. This partially answers a question posed by W. Rudin.
On the growth of solutions in the oscillatory case
Authors: Kauffman Robert M.
Published: June 2010

Suppose that

A
A

is a bounded continuously differentiable function from

[
0
,
∞
)

[0,\infty )

to the real

n
×
n

n \times n

Hermitian matrices such that, for every

ε
>
0

\varepsilon > 0

and every

λ
>
0

\lambda > 0

, there is an

a
a

(depending on

ε
\varepsilon

and

λ
\lambda

) such that

D
2

−
A
−
ε
E

{D^2} - A - \varepsilon E

and

D
2

+

A
′

/

λ
−
ε
E

{D^2} + A’/\lambda - \varepsilon E

are disconjugate on

[
a
,
∞
)

[a,\infty )

, where

E
E

is the

n
×
n

n \times n

identity matrix. It follows from the result of this paper that no solution of

(

D
2

+
A
)
f
=
0

({D^2} + A)f = 0

can either grow or decay exponentially.
An intermediate value property
Authors: Gamlen John L. , Muldowney James S.
Published: June 2010

An intermediate value property is shown to hold for monotone perturbations of maps which have this property. Applications are given to initial value problems and boundary value problems for functional differential equations.
Relatively uniform Banach lattices
Authors: Wirth Andrew
Published: June 2010

Sequential relative uniform and norm convergence agree in a Banach lattice, if and only if it is equivalent to an

M
M

space.
Hyponormal operators with infinite essential spectrum
Authors: Kim Hong W.
Published: June 2010

It is shown that if

T
T

is an essentially hyponormal operator (i.e., the image of

T
∗

T
−
T

T
∗

{T^ \ast }T - T{T^ \ast }

in the Calkin algebra is a positive element) in

L

(

H

)

\mathfrak {L}(\mathcal {H})

, and if the left essential spectrum of

T
T

is infinite, then

R

(

δ
T

)
−

+
{

T
∗

}
′

R{({\delta _T})^ - } + \{ {T^ \ast }\} ’

is not norm dense in

L

(

H

)

\mathfrak {L}(\mathcal {H})

, where

R

(

δ
T

)
−

R{({\delta _T})^ - }

denotes the norm closure of the range of derivation induced by

T
T

, and

{

T
∗

}
′

\{ {T^ \ast }\} ’

denotes the commutant of

T
∗

{T^ \ast }

.
𝑅₃-quasi-uniform spaces and topological homeomorphism groups
Authors: Seyedin Massood
Published: June 2010

It is well known that if

X
X

is a completely regular space and

G
G

is a homeomorphism group of

X
X

onto itself such that

G
G

is equicontinuous with respect to a compatible uniformity of

X
X

, then

G
G

is a topological group under the topology of pointwise convergence. In this paper, we obtain a generalization of the above result by means of

R
3

{R_3}

-quasi-uniformities.
On homogeneous hereditarily unicoherent continua
Authors: Gordh G. R.
Published: June 2010

Let

M

\mathfrak {M}

denote the class of all hereditarily unicoherent Hausdorff continua in which each indecomposable subcontinuum is irreducible. It is shown that if the continuum

M
M

in

M

\mathfrak {M}

is decomposable, then the set of weak terminal points of

M
M

is a nonempty, proper subset. The following generalization of a theorem of F. Burton Jones is an immediate corollary: if the continuum

M
M

in

M

\mathfrak {M}

is homogeneous, then

M
M

is indecomposable. As an application, it is proved that if

X
X

is a homogenous, hereditarily unicoherent Hausdorff continuum which is an image of an ordered compactum, then

X
X

is an indecomposable metrizable continuum.
Nielsen numbers of maps of tori
Authors: Brooks Robin B. S. , Brown Robert F. , Pak Jingyal , Taylor Douglas H.
Published: June 2010

The main result states that if

f
:
X
→
X

f:X \to X

is any map on a

k
k

-dimensional torus

X
X

, then the Nielsen number and Lefschetz number of

f
f

are related by the formula

N
(
f
)
=

|

L
(
f
)

|

N(f) = |L(f)|

. Thus, on the torus, the Lefschetz number gives information, not just on the existence of fixed points, but on the number of fixed points as well. No other compact Lie group has this property. The main result, when applied to certain types of maps on compact Lie groups, produces new information on the fixed point theory of such maps.
Reductions of ideals in Prüfer domains
Authors: Hays James H.
Published: June 2010

All rings under consideration are Prüfer domains or valuation domains. We characterize the set of basic ideals and the set of

C
C

-ideals in an arbitrary valuation ring. Basic ideals were introduced in 1954 by Northcott and Rees. The concept of a

C
C

-ideal is, in a sense, directly opposite to that of a basic ideal. We then prove that a necessary and sufficient condition for every ideal in a domain

D
D

to be basic is that

D
D

be a one-dimensional Prüfer domain.
A commutative Noetherian Hopf algebra over a field is finitely generated
Authors: Molnar Richard K.
Published: June 2010

Let

k
k

be an arbitrary field and

H
H

a commutative Hopf algebra over

k
k

. We give a short proof of the fact that

H
H

is Noetherian if and only if

H
H

is finitely generated as a

k
k

-algebra.
Sometimes only square matrices can be diagonalized
Authors: Levy Lawrence S.
Published: June 2010

It is proved that every SQUARE matrix over a serial ring is equivalent to some diagonal matrix, even though there are rectangular matrices over these rings which cannot be diagonalized.
Interpolation by transforms of discrete measures
Authors: Pigno Louis , Saeki Sadahiro
Published: June 2010

Let

G
G

be a compact abelian group, and

Γ
\Gamma

its character group. Given

E
⊂
Γ
,

E
a

E \subset \Gamma ,{E^a}

denotes the set of all accumulation points of

E
E

in

Γ
¯

\overline \Gamma

, the Bohr compactification of

Γ
\Gamma

. In this paper it is shown that the inclusion
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper L Superscript 1 Baseline left-parenthesis upper G right-parenthesis right-parenthesis Superscript logical-and Baseline StartAbsoluteValue subset-of left-parenthesis l Superscript 1 Baseline left-parenthesis upper G right-parenthesis right-parenthesis Superscript logical-and Baseline EndAbsoluteValue Subscript upper E Baseline Subscript upper E">

(

L
1

(
G
)

)
∧

|

E

⊂

(

l
1

(
G
)

)
∧

|

E

{({L^1}(G))^ \wedge }{|_E} \subset {({l^1}(G))^ \wedge }{|_E}

obtains if and only if

E
∩

E
a

=
∅

E \cap {E^a} = \emptyset

and there exists a measure

μ
ϵ
M
(
G
)

\mu \epsilon M(G)

such that

μ
^

=
1

\widehat \mu = 1

on

E
E

and

μ
^

=
0

\widehat \mu = 0

on

Γ
∩

E
a

\Gamma \cap {E^a}

.