bluuebunny/crossref_metadata_2025 huggingface dataset:
𝐿^{𝑝}-estimates for matrix coefficients of irreducible representations of compact groups
Authors: Giulini Saverio , Travaglini Giancarlo
Published: June 2010

The following result is proved. Theorem. Let G be a compact connected semisimple Lie group. For any

p
>
0

p > 0

there exist two positive numbers

A
p

{A_p}

and

B
p

{B_p}

such that (up to equivalence) for any continuous irreducible unitary representation

π
\pi

of G there exists a matrix coefficient

a
π

{a_\pi }

of

π
\pi

such that
\[

A
p

>

d
π

∫
G

|

a
π

|

p

>

B
p

{A_p} > {d_\pi }\int _G {|{a_\pi }{|^p} > {B_p}}

\]
where

d
π

{d_\pi }

is the degree of

π
\pi

. As an application we show the nonexistence of infinite local

Λ
q

{\Lambda _q}

-sets.
Concerning Daniljuk’s existence theorem for free boundary value problems with the Bernoulli condition
Authors: Acker Andrew
Published: June 2010

We show that a variational method proposed by Ī. Ī. Daniljuk for proving the existence of free boundaries satisfying the Bernoulli condition is not valid under the condition given by Daniljuk.
Mad families and ultrafilters
Authors: Weese Martin
Published: June 2010

For each almost disjoint family X let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis upper X right-parenthesis equals StartSet a subset-of-or-equal-to omega colon card StartSet s element-of upper X colon s minus a is finite EndSet equals 2 Superscript omega Baseline EndSet comma upper I left-parenthesis upper X right-parenthesis equals StartSet a subset-of-or-equal-to omega colon card StartSet s element-of upper X colon card left-parenthesis s intersection a right-parenthesis equals omega EndSet equals 2 Superscript omega Baseline EndSet">

F
(
X
)
=
{
a
⊆
ω
:

card

{
s
∈
X
:
s
∖
a

is

finite

}
=

2
ω

}
,
I
(
X
)
=
{
a
⊆
ω
:

card

{
s
∈
X
:

card

(
s
∩
a
)
=
ω
}
=

2
ω

}

F(X) = \{ a \subseteq \omega :{\text {card}}\{ s \in X:s\backslash a\;{\text {is}}\;{\text {finite}}\} = {2^\omega }\} ,I(X) = \{ a \subseteq \omega :{\text {card}}\;\{ s \in X:{\text {card}}\;(s \cap a) = \omega \} = {2^\omega }\}

. Assuming

P
(

2
ω

)

P({2^\omega })

we show that for each nonprincipal ultrafilter p there exist a maximal almost disjoint family X and an almost disjoint family Y with

F
(
X
)
=
I
(
Y
)
=
p

F(X) = I(Y) = p

.
Desingularizing maps of corank one
Authors: Blank S. J. , Curley C.
Published: June 2010

In 1960, A. Haefliger presented necessary and sufficient conditions for factoring a smooth map from a surface into the plane through an immersion into

R
3

{R^3}

. Here, necessary and sufficient conditions are given for factoring a map between manifolds of dimension n,

n
⩾
2

n \geqslant 2

, through an immersion into a line bundle over the range. In addition, conditions are given for factoring such a map through a submersion from a line bundle over the domain.
On bialgebras which are simple Hopf modules
Authors: Radford David E.
Published: June 2010

This paper gives a module characterization of commutative or cocommutative Hopf algebras over a field.
Weak continuity of invariant measures for a class of piecewise monotonic transformations
Authors: Boyarsky Abraham , Cooper Sarah
Published: June 2010

Let

τ
a

:
[
0
,
1
]
→
[
0
,
1
]
,
0
>
a
⩽

1
2

{\tau _a}:[0,1] \to [0,1],0 > a \leqslant \frac {1}{2}

, denote the class of transformations defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau Subscript a Baseline left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 2 x comma 2nd Column a m p semicolon 0 less-than-or-slanted-equals x less-than-or-slanted-equals one half comma 2nd Row 1st Column left-parenthesis 2 minus a right-parenthesis minus 2 left-parenthesis 1 minus a right-parenthesis x comma 2nd Column a m p semicolon one half less-than-or-slanted-equals x less-than-or-slanted-equals 1 period EndLayout">

τ
a

(
x
)
=

{

2
x
,

a
m
p
;

0
⩽
x
⩽

1
2

,

(
2
−
a
)
−
2
(
1
−
a
)
x
,

a
m
p
;

1
2

⩽
x
⩽
1.

{\tau _a}(x) = \left \{ {\begin {array}{*{20}{c}} {2x,} \hfill & {0 \leqslant x \leqslant \frac {1}{2},} \hfill \\ {(2 - a) - 2(1 - a)x,} \hfill & {\frac {1}{2} \leqslant x \leqslant 1.} \hfill \\ \end {array} } \right .

\]
For

a
∗

⩽
a
>

1
2

{a^ \ast } \leqslant a > \frac {1}{2}

, the transformation

τ
a

{\tau _a}

admits unique absolutely continuous invariant measures

μ
a

{\mu _a}

whose density function

f
a

(
x
)

{f_a}(x)

assumes the value 0 on an interval which expands as

a
↑

1
2

a \uparrow \frac {1}{2}

. Furthermore, the measures

μ
a

{\mu _a}

converge weakly to the measure

μ

1

/

2

=

1
2

δ

1

/

2

+

1
2

δ
1

{\mu _{1/2}} = \frac {1}{2}{\delta _{1/2}} + \frac {1}{2}{\delta _1}

as

a
↑

1
2

a \uparrow \frac {1}{2}

, where

μ

1

/

2

{\mu _{1/2}}

is an invariant measure under

τ
a

{\tau _a}

, and

δ
y

{\delta _y}

denotes the Dirac measure at the point y.
Mixed norm 𝑛-widths
Authors: de Boor C. , DeVore R. , Höllig K.
Published: June 2010

Recently, the Soviet mathematicians R. Ismagilov [4], E. Gluskin [3] and B. Kashin [5] have obtained some deep and surprising results on n-widths for Sobolev spaces in the mixed norm case. In this note, we will give a new and simpler proof of Gluskin’s result and show its connection with a certain classical combinatorial problem.
Trace and the regular ring of a finite 𝐴𝑊*-algebra
Authors: Berberian S. K.
Published: June 2010

A finite

AW
∗

\text {AW}^*

-algebra is of type I if and only if its maximal ring of quotients has a center-valued trace. In particular, a center-valued trace need not be extendible to the maximal (or classical) ring of quotients.
Hyperfinite-dimensional subspaces of the nonstandard hull of 𝑐₀
Authors: Moore L. C.
Published: June 2010

Let

c
^

0

{\hat c_0}

be the nonstandard hull of the Banach space

c
0

{c_0}

formed with respect to an

ℵ
1

{\aleph _1}

-saturated extension. Then

c
^

0

{\hat c_0}

is not isometrically isomorphic to any hyperfinite-dimensional subspace of

c
^

0

{\hat c_0}

and hence not to any hyperfinite-dimensional Banach space. This gives a negative answer to the question posed by Ward Henson: “Does every Banach space have a nonstandard hull which is isometrically isomorphic to a hyperfinite-dimensional Banach space?” As a consequence of the result, no ultrapower of

c
0

{c_0}

is isometrically isomorphic to an ultraproduct of finite-dimensional Banach spaces.
Convex and starlike meromorphic functions
Authors: Miller James
Published: June 2010

In this paper we study meromorphic univalent functions with a simple pole at p which map the unit disk onto the exterior of convex sets or the exterior of starlike sets. A question raised by Libera and Livingston concerning the equality of two classes of starlike meromorphic univalent functions is answered. We show that the set of points omitted by the class of convex functions is

{
−
p

/

(
1
+

p
2

)
}

\{ - p/(1 + {p^2})\}

. The coefficients of convex and starlike meromorphic functions are also studied.
A Runge theorem for solutions of the heat equation
Authors: Diaz R.
Published: June 2010

Let

Ω
1

{\Omega _1}

and

Ω
2

{\Omega _2}

be open sets in

R
n

{R^n}

such that

Ω
1

⊂

Ω
2

{\Omega _1} \subset {\Omega _2}

. Every solution of the heat equation on

Ω
1

{\Omega _1}

admits approximation on the compact subsets of

Ω
1

{\Omega _1}

by functions which satisfy the heat equation throughout

Ω
2

{\Omega _2}

if and only if this topological condition is met: For every hyperplane

π
\pi

in

R
n

{R^n}

orthogonal to the time axis, every compact component of

π
∖

Ω
1

\pi \backslash {\Omega _1}

contains a compact component of

π
∖

Ω
2

\pi \backslash {\Omega _2}

.
Existence theorems on unbounded sets in Banach spaces
Authors: Hirano Norimichi , Takahashi Wataru
Published: June 2010

In this paper, we give a necessary and sufficient condition under which a variational inequality defined on unbounded sets in a Banach space has a solution. Furthermore, we establish a necessary and sufficient condition under which the minimax equality on unbounded sets is true.
On a criterion for the univalence of holomorphic functions
Authors: Lau Kee Wai , Liu Ming Chit
Published: June 2010

A sufficient condition for the univalence of holomorphic functions in the unit disc is given in terms of

|

f

/

f
′

|

|f/f’|

.
On operator ranges
Authors: Wadhwa Bhushan L.
Published: June 2010

If f is any vector-valued bounded function defined on open set D of the complex plane, and T is any bounded linear operator on a Hilbert space such that

σ
R

(

T
∗

)

{\sigma _R}({T^ \ast })

is empty and if

(
T
−
z
I

)
2

f
(
z
)
=
x

{(T - zI)^2}f(z) = x

for all z in D then f is analytic.
Prediction 𝑛 units of time ahead
Authors: Nakazi Takahiko , Takahashi Katutoshi
Published: June 2010

The purpose of this note is to give a simple expression in terms of w of the quantities
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript n Baseline left-parenthesis w right-parenthesis equals inf Underscript f Endscripts integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue 1 plus e Superscript i n theta Baseline f EndAbsoluteValue squared w d theta slash 2 pi left-parenthesis n equals 0 comma 1 comma 2 comma ellipsis right-parenthesis comma">

ρ
n

(
w
)
=

inf
f

∫
0

2
π

|

1
+

e

i
n
θ

f

|

2

w

d
θ

/

2
π

(
n
=
0
,
1
,
2
,
…
)
,

{\rho _n}(w) = \inf \limits _f \int _0^{2\pi } {|1 + {e^{in\theta }}f{|^2}w\,d\theta /2\pi } \quad (n = 0,1,2, \ldots ),

\]
where f ranges over the analytic trigonometric polynomials with mean value zero and w is nonnegative and summable on the circle.
Some remarks on projective Stiefel manifolds, immersions of projective spaces and spheres
Authors: Smith Larry
Published: June 2010

Let

M
m

{M^m}

be a closed smooth manifold,

M
[
u
n
k
]

R

2
m

M[unk]{{\mathbf {R}}^{2m}}

an immersion and

M
~

m

↓
π

M
m

{\tilde M^m}{ \downarrow ^\pi }{M^m}

a double covering. For m odd we show that the normal bundle

ν
~

↓

M
~

\tilde \nu \downarrow \tilde M

of the immersion

M
~

↓
π

M
[
u
n
k
]

R

2
m

\tilde M{ \downarrow ^\pi }M[unk]{{\mathbf {R}}^{2m}}

is independent of

φ
\varphi

and applying this to

M
=

R

P
(
m
)

M = {\mathbf {R}}P(m)

reobtain the result of E. H. Brown that a symmetric immersion

S
m

[
u
n
k
]

R

2
m

{S^m}[unk]{{\mathbf {R}}^{2m}}

is regularly homotopic to an embedding iff

m
=

2
p

−
1

m = {2^p} - 1

.
Linear vector fields on 𝐺̃_{𝑘}(𝐑ⁿ)
Authors: Leite Maria Luiza , de Miatello Isabel Dotti
Published: June 2010

We determine the maximal number of linearly independent vector fields on the grassmannian of oriented k-subspaces of

R
n

{R^n}

, which are induced by linear transformations of

Λ
k

(

R
n

)

{\Lambda ^k}({R^n})

.
Invariant sets for classes of matrices of zeros and ones
Authors: Brualdi Richard A. , Ross Jeffrey A.
Published: June 2010

Let

A

(
R
,
S
)

\mathfrak {A}(R,S)

denote the class of all

m
×
n

m \times n

matrices of 0’s and 1’s with row sum vector R and column sum vector S. A set

I
×
J
(
I
⊆
{
1
,
…
,
m
}
,
J
⊆
{
1
,
…
,
n
}
)

I \times J(I \subseteq \{ 1, \ldots ,m\} ,J \subseteq \{ 1, \ldots ,n\} )

is said to be invariant if each matrix in

A

(
R
,
S
)

\mathfrak {A}(R,S)

contains the same number of 1’s in the positions

I
×
J

I \times J

. We prove that if there are no invariant singletons, then an invariant set

I
×
J

I \times J

satisfies

I
=
∅
,
I
=
{
1
,
…
,
m
}
,
J
=
∅

I = \emptyset ,I = \{ 1, \ldots ,m\} ,J = \emptyset

or

J
=
{
1
,
…
,
n
}

J = \{ 1, \ldots ,n\}

.
A new proof of a result of Levitzki
Authors: Klein Abraham A.
Published: June 2010

A very short proof of the result which states that a nil ring of bounded index has a nonzero nilpotent ideal is given and it is shown that the same method of proof yields a more general result.
Quotients of valuated vector spaces
Authors: Hill Paul
Published: June 2010

This paper has three purposes. The first is to present a direct example of a quotient of an injective space that is not itself injective. The second is to demonstrate that Theorem 3 in [2] is false. An

s
s

-dense subspace

S
S

of a free space

F
F

is not free by virtue of

F

/

S

F/S

having only values that are cofinal with

ω
\omega

. Finally, we furnish a valid proof, independent of the aforementioned Theorem 3, that a subspace

V
V

of a free space is free if

V
V

satisfies Fuchs’ countability condition.
Another graded algebra with a nonrational Hilbert series
Authors: Kobayashi Yuji
Published: June 2010

Recently J. B. Shearer has constructed a graded algebra with a nonrational Hilbert series, a counterexample to Govorov’s conjecture. In this note we give a simpler example.
A lower bound for the fundamental frequency of a convex region
Authors: Protter M. H.
Published: June 2010

A lower bound for the first eigenvalue of the Laplace operator is obtained in terms of the radius of the largest ball which can be inscribed in a convex region in

R
n

{R^n}

,

n
⩾
2

n \geqslant 2

.
On the fixed-point theory for local 𝑘-pseudocontractions
Authors: Morales Claudio
Published: June 2010

Two commonly used boundary conditions which imply existence of fixed points for local strong pseudocontractions in Banach spaces are compared, a previous fixed-point theorem for this class of mappings is improved, and an almost fixed-point result is obtained for local pseudocontractions.
A Sturm theorem for partial differential equations of mixed type
Authors: Kreith Kurt
Published: June 2010

A generalization of the Sturm comparison theorem is given for differential equations of mixed type. The results constitute a generalization of Sturmian theorems used in the study of hyperbolic initial boundary value problems.
On the weak Radon-Nikodým property
Authors: Ghoussoub Nassif , Saab Elias
Published: June 2010

It is shown that in a Banach lattice the notions of the Radon-Nikodym property and the weak Randon-Nikodym property coincide. A problem of H. P. Lotz is solved affirmatively, namely, if the dual

Y
∗

{Y^ * }

of a Banach space

Y
Y

is complemented in a Banach lattice

E
E

and if

l
1

{l^1}

does not embed into

Y
Y

then

Y
∗

{Y^ * }

has the Radon-Nikodym property.
An elementary proof of the Stone-Weierstrass theorem
Authors: Brosowski Bruno , Deutsch Frank
Published: June 2010

An elementary proof of the Stone-Weierstrass theorem is given.
Simple homotopy types for (𝐺,𝑚)-complexes
Authors: Dyer Micheal N.
Published: June 2010

Let

G
G

be a finite group. We use the fact that each element of the Whitehead group Wh

(
G
)

(G)

may be represented by at most a

2
×
2

2 \times 2

(nonsingular) matrix to deduce results about when simple homotopy type and homotopy type agree. As examples, we give complete descriptions of the simple homotopy types for

(

Z
m

×

Z
n

,
2
)

({Z_m} \times {Z_n},2)

-complexes, provided

S

K
1

(
Z
(

Z
m

×

Z
n

)
)
=
0

S{K_1}(Z({Z_m} \times {Z_n})) = 0

.
Product spaces in locales
Authors: Isbell John
Published: June 2010

The embedding of sober spaces in locales preserves products

X
×
Y

X \times Y

where

X
X

is quasi-locally compact. A completely regular space

X
X

has this preservation property, for all

Y
Y

, if and only if

X
X

is a complemented sublocale of a compact space. Equivalently, every closed subset of

X
X

is locally compact somewhere.
On insertion of a continuous function
Authors: Powderly M.
Published: June 2010

Recently E. P. Lane proved that if space

X
X

has the weak

C

{\mathbf {C}}

-insertion property and satisfies another condition, then

X
X

has the strong

C

{\mathbf {C}}

-insertion property. This paper establishes the converse of this result.
A remarkable simple closed curve: revisited
Authors: Harrold O. G.
Published: June 2010

It is shown that the pathology of R. H. Fox’s remarkable simple closed curve is in a sense explained below more complicated than that of some examples of the well-known Fox-Artin paper.
Further divisibility properties of the 𝑞-tangent numbers
Authors: Foata Dominique
Published: June 2010

The

q
q

-tangent number

T

2
n
+
1

(
q
)

{T_{2n + 1}}(q)

is shown to be divisible by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 plus q right-parenthesis Superscript a left-parenthesis n comma 1 right-parenthesis Baseline left-parenthesis 1 plus q squared right-parenthesis Superscript a left-parenthesis n comma 2 right-parenthesis Baseline midline-horizontal-ellipsis left-parenthesis 1 plus q Superscript n Baseline right-parenthesis Superscript a left-parenthesis n comma n right-parenthesis">

(
1
+
q

)

a
(
n
,
1
)

(
1
+

q
2

)

a
(
n
,
2
)

⋯

(
1
+

q
n

)

a
(
n
,
n
)

{(1 + q)^{a(n,1)}}{(1 + {q^2})^{a(n,2)}} \cdots {(1 + {q^n})^{a(n,n)}}

, where the

a
(
n
,
i
)

a(n,i)

’s are positive integers having the maximal property that

a
(
n
,
1
)
+
a
(
n
,
2
)
+
⋯
+
a
(
n
,
n
)
=
2
n

a(n,1) + a(n,2) + \cdots + a(n,n) = 2n

whenever

n
n

is a power of 2.
A limit on the Loewy length of the endomorphism ring of a module of finite length
Authors: Smalø Sverre O.
Published: June 2010

Let

M
M

be a module of finite length over an arbitrary ring

A
A

. Let

S
1

,

S
2

,
…
,

S
m

{S_1},{S_2}, \ldots ,{S_m}

be the nonisomorphic simple composition factors of

M
M

and let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper M right-bracket equals n 1 left-bracket upper S 1 right-bracket plus n 2 left-bracket upper S 2 right-bracket plus midline-horizontal-ellipsis plus n Subscript m Baseline left-bracket upper S Subscript m Baseline right-bracket">

[
M
]
=

n
1

[

S
1

]
+

n
2

[

S
2

]
+
⋯
+

n
m

[

S
m

]

[M] = {n_1}[{S_1}] + {n_2}[{S_2}] + \cdots + {n_m}[{S_m}]

denote that

S
i

{S_i}

occurs

n
i

{n_i}

times in a composition series for

M
M

. As a generalization of Schur’s lemma we have the following wellknown result: If

M
M

is nonzero and all the

n
i

{n_i}

are equal to one then the Loewy length of the endomorphism ring of

M
M

is one. We will generalize this fact showing that in general the Loewy length of the endomorphism ring of

M
M

is less than or equal to the maximum of the

n
i

{n_i}

,

i
=
1
,
…
,
m

i = 1, \ldots ,m

.
Relations in hyperreflection groups
Authors: Ellers Erich W.
Published: June 2010

Hyperreflection groups

G
m

{G_m}

are generalizations of groups generated by reflections. A hyperreflection group is generated by hyperreflections if

dim
⁡
V

\dim V

is finite. A hyperreflection is a simple mapping

σ
\sigma

such that

det
σ
=
γ

\det \sigma = \gamma

, where

γ
m

=
1

{\gamma ^m} = 1

. If the field of scalars is commutative, the order of

σ
\sigma

is

m
m

. Our main result states that every relation between hyperreflections and their inverses is a consequence of relations of lengths 2, 4, and

m
m

. The most interesting special case occurs for

m
=
2

m = 2

. Then our result refers to relations between reflections.
Group rings whose torsion units form a subgroup
Authors: Polcino Milies César
Published: June 2010

Denote by

T
U
(

Z

G
)

TU({\mathbf {Z}}G)

the set of units of finite order of the integral group ring of a group

G
G

. We determine the class of all groups

G
G

such that

T
U
(

Z

G
)

TU({\mathbf {Z}}G)

is a subgroup and study how this property relates to certain properties of the unit groups.
Groups of finite weight
Authors: Rhemtulla A. H.
Published: June 2010

If

N
N

is a group and

E
E

is a group of operators on

N
N

then write

d
E

(
N
)

{d_E}(N)

for the minimum number of elements needed to generate

N
N

as an

E
E

-group. It is shown that if

N
N

is a normal subgroup of

E
E

and

E
E

acts on

N
N

by conjugation, then

d
E

(
N
)
=

d
E

(
N

/

N
′

)

{d_E}(N) = {d_E}(N/N’)

if

d
E

(
N
)

{d_E}(N)

is finite and there does not exist an infinite descending series of

E
E

-normal subgroups

N
′

=

C
0

>

C
1

>
⋯

N’ = {C_0} > {C_1} > \cdots

with each

C
i

/

C

i
+
1

{C_i}/{C_{i + 1}}

perfect. Both these conditions are, in general, necessary.
A counterexample to conjectures of Papakyriakopoulos and Swarup
Authors: McCool James
Published: June 2010

In [5] Swarup states a group theoretic conjecture

P2

{\text {P2}}

and shows that

P1

⇒

P2

⇒

{\text {P1}} \Rightarrow {\text {P2}} \Rightarrow

the Poincaré conjecture, where

P1

{\text {P1}}

is a conjecture of Papakyriakopoulos [3]. We give a counterexample to conjecture

P2

{\text {P2}}

.
On the limiting set of stochastic products 𝑥𝐴₁⋯𝐴_{𝑘}
Authors: Hartfiel D. J.
Published: June 2010

In this paper we study the sequential limiting set

L
L

of the sequence

R
1

,
…
,

R
k

=
{
x

A
1

…

A
k

|

x

{R_1}, \ldots ,{R_k} = \{ x{A_1} \ldots {A_k}|x

is a stochastic vector, and

A
i

{A_i}

a stochastic matrix with

P
⩽

A
i

⩽
Q

P \leqslant {A_i} \leqslant Q

, for nonnegative matrices

P
P

and

Q
}
,
…

Q\} , \ldots

. We characterize

L
L

and provide some work on approximating

L
L

.
Generalized cyclotomic periods
Authors: Evans Ronald J.
Published: June 2010

Let

n
n

and

q
q

be relatively prime integers with

n
>
1

n > 1

, and set

N
N

equal to twice the product of the distinct prime factors of

n
n

. Let

t
(
n
)

t(n)

denote the order of

q
q

(
mod
n
)

(\bmod n)

. Write

η
=

∑

υ
=
0

t
(
n
)
−
1

a
υ

ζ
n

q
υ

\eta = \sum _{\upsilon = 0}^{t(n) - 1}{a_\upsilon }\zeta _n^{{q^\upsilon }}

where

ζ
n

=
exp
⁡
(
2
π
i

/

n
)

{\zeta _n} = \exp (2\pi i/n)

. If

a
υ

=
1

{a_\upsilon } = 1

for all

υ
\upsilon

, then

η
\eta

is Kummer’s cyclotomic period, and if

a
υ

=
exp
⁡
(
2
π
i
υ

/

t
(
n
)
)

{a_\upsilon } = \exp (2\pi i\upsilon /t(n))

for each

υ
\upsilon

, then

η
\eta

is a type of Lagrange resolvent. For certain classes of

a
υ

∈

Q

(

ζ
n
N

)

{a_\upsilon } \in {\mathbf {Q}}(\zeta _n^N)

, necessary and sufficient conditions for the vanishing of

η
\eta

are given, and the degree of

η
\eta

over

Q

{\mathbf {Q}}

is determined.
Fixed subalgebra of a commutative Frobenius algebra
Authors: Azumaya Gorô
Published: June 2010

Let

B
B

be a finite-dimensional commutative algebra generated by a single element, and let

A
=
B
⊗
B

A = B \otimes B

. We prove that the fixed subalgebra of

A
A

under the involution

b
1

⊗

b
2

↦

b
2

⊗

b
1

{b_1} \otimes {b_2} \mapsto {b_2} \otimes {b_1}

is Frobenius if and only if either the characteristic of

B
B

is different from 2 or

B
B

is separable.
A complex that resolves the ideal of minors having 𝑛-1 columns in common
Authors: Andrade J. F. , Simis A.
Published: June 2010

Let

A
A

be an

n
×
m

n \times m

matrix

(
m
⩾
n
)

(m \geqslant n)

with entries in a noetherian ring

R
R

, let

J
J

be the ideal of

R
R

generated by the

n
×
n

n \times n

minors that are formed with

n
−
1

n - 1

fixed columns of

A
A

. Necessary and sufficient conditions are given in order that a suitably defined complex be a free resolution of the

R
R

-module

R

/

J

R/J

. The complex is closely related to the complexes of Buchsbaum-Rim [2, §2] and it is perhaps worthwhile mentioning that a special case of the present result was obtained by Buchsbaum-Eisenbud in a different context [1, Theorem 8.1].
Composition operators isolated in the uniform operator topology
Authors: Berkson Earl
Published: June 2010

It is is shown that

ϕ
\phi

is an analytic map of the disc

|
z
|

>
1

\left | z \right | > 1

into itself such that

ϕ
\phi

has radial limits of modulus 1 on a set of positive measure, then for

1
⩽
p
>
∞

1 \leqslant p > \infty

the corresponding composition operator on

H
p

{H^p}

is isolated in the topological space of composition operators on

H
p

{H^p}

(with the uniform operator topology).
An extension of the Fuglede-Putnam theorem to subnormal operators using a Hilbert-Schmidt norm inequality
Authors: Furuta Takayuki
Published: June 2010

We prove that if

A
A

and

B
∗

{B^ * }

are subnormal operators acting on a Hubert space, then for every bounded linear operator

X
X

, the Hilbert-Schmidt norm of

A
X
−
X
B

AX - XB

is greater than or equal to the Hilbert-Schmidt norm of

A
∗

X
−
X

B
∗

{A^ * }X - X{B^ * }

. In particular,

A
X
=
X
B

AX = XB

implies

A
∗

X
=
X

B
∗

{A^ * }X = X{B^ * }

. In addition, if we assume

X
X

is a Hilbert-Schmidt operator, we can relax the subnormality conditions to hyponormality and still retain the inequality.
A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space
Authors: Takahashi Wataru
Published: June 2010

We prove a nonlinear ergodic theorem for noncommutative semigroups of nonexpansive mappings in a Hilbert space. Furthermore, we give a necessary and sufficient condition for a noncommutative semigroup to have a fixed point.
On the Dunford-Pettis property
Authors: Bourgain J.
Published: June 2010

It is shown that the Banach spaces

C

L
1

{C_{{L^1}}}

and

L
C
1

L_C^1

and their duals have the Dunford-Pettis property. Some other more local problems are also solved.
On the product of a Riesz set and a small 𝑝 set
Authors: Yamaguchi Hiroshi
Published: June 2010

Let

Z
+

{Z^ + }

be the semigroup consisting of all nonnegative integers. By a famous theorem of Bochner,

Z
+

×

Z
+

{Z^ + } \times {Z^ + }

is a Riesz set in

Z
⊕
Z

Z \oplus Z

. In this paper, we prove that the product set of a Riesz set and a small

p
p

set is a small

p
p

set.
Stable summation methods for a class of singular Sturm-Liouville expansions
Authors: Diamond Harvey , Kon Mark , Raphael Louise
Published: June 2010

Given the Sturm-Liouville eigenfunction expansion of an

L
2

{L_2}

function

f
(
x
)

f(x)

, summability theory provides means for recovering the value of

f
(

x
0

)

f({x_0})

at points

x
0

{x_0}

where

f
f

is sufficiently regular. If the coefficients in the expansion are perturbed slightly (in the

L
2

{L_2}

norm), a stable summation method will recover from the perturbed expansion a good approximation to

f
(

x
0

)

f({x_0})

. In this paper we develop stable summation methods for expansions in eigenfunctions of the singular Sturm-Liouville system
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u minus q left-parenthesis x right-parenthesis u equals minus lamda u comma u left-parenthesis 0 right-parenthesis cosine beta plus u prime left-parenthesis 0 right-parenthesis sine beta equals 0 comma u left-parenthesis normal infinity right-parenthesis greater-than normal infinity">

u
−
q
(
x
)
u
=
−
λ
u
,
u
(
0
)
cos
⁡
β
+

u
′

(
0
)
sin
⁡
β
=
0
,
u
(
∞
)
>
∞

u - q(x)u = - \lambda u,u(0)\cos \beta + u’(0)\sin \beta = 0,u(\infty ) > \infty

; where

q
(
x
)

q(x)

and continuous. Given a summability method known to work at

x
0

{x_0}

for a particular expansion, our results say that if the summation parameter is appropriately scaled with the

L
2

{L_2}

error in the perturbed expansion, a stable summation method is obtained. We obtain a sharp scaling requirement for guaranteeing stability. We apply our results to Riesz and Stieltjes summability.
Transformation of a class of nonselfadjoint eigenvalue problems
Authors: Rao D. R. K. S. , Murty K. N.
Published: June 2010

A class of non-self-adjoint boundary value problems possessing countably many real eigenvalues can be made self-adjoint by means of a nonsingular transformation. A set of criteria for such problems to be self-adjoint is derived.
Arens regularity and the algebra of double multipliers
Authors: Tomiuk B. J.
Published: June 2010

We show how Arens regularity of a w.c.c. Banach algebra

A
A

with a bounded approximate identity is connected with the algebra of double multipliers on

A
A

.
The first eigenvalue of the Laplacian for plane domains
Authors: Croke Christopher B.
Published: June 2010

We prove an improved lower bound for the first eigenvalue of the Laplacian of a connected plane domain in terms of its inradius and connectivity.
Characterization of the set-theoretical geometric realization in the non-Euclidean case
Authors: Ruiz Salguero Carlos , Ruiz Salguero Roberto
Published: June 2010

A helpful feature in Milnor’s geometric realization [1] of a simplicial set

X
X

is that each equivalence class admits one and only one element (optimal pair) of the form

(

x
¯

,

t
¯

)

(\bar x,\bar t)

where

x
¯

\bar x

is nondegenerate and

t
¯

\bar t

is an interior point. This realization and several other aspects of Algebraic Topology admit generalizations (R. Ruiz [3]) changing the cosimplicial topological space of the

Δ
n

{\Delta ^n}

by a general one, say

Y
Y

. This paper is devoted to establishing conditions on

Y
Y

which guarantee the existence of such pairs on

R
Y

(
X
)

{R_Y}(X)

for every simplicial set

X
X

.

R
Y

{R_Y}

denotes the new realization via

Y
Y

.)
Fixed point sets of 𝐿𝐶^{∞}, 𝐶^{∞} continua
Authors: Martin John R.
Published: June 2010

A space

X
X

is said to have the complete invariance property (CIP) if every nonempty closed subset of

X
X

is the fixed point set of some self-map of

X
X

. An example is given to show that there is an

L

C
∞

L{C^\infty }

,

C
∞

{C^\infty }

continuum

X
X

which does not have CIP. Moreover,

X
X

is a wedge of two

L

C
∞

L{C^\infty }

,

C
∞

{C^\infty }

continua each of which has CIP.
Construction of spaces with a 𝜎-minimal base
Authors: Burke Dennis K.
Published: June 2010

Using product spaces, a method is given for constructing and recognizing certain spaces with a

σ
\sigma

-minimal base. This technique shows that every topological space can be embedded as a closed subspace of a space with a

σ
\sigma

-minimal base; hence a

σ
\sigma

-minimal base, by itself, does not imply any nontrivial closed hereditary topological property. It is also shown that any space

Y
Y

can be expressed as the open perfect image of some space with a

σ
\sigma

-minimal base. Examples are given, illustrating a surprisingly large class of product spaces with a

σ
\sigma

-minimal base.
A remark on the multiplicity of the discrete spectrum of congruence groups
Authors: Randol Burton
Published: June 2010

It is shown that the discrete spectrum of principal congruence groups of high level can contain eigenvalues of large multiplicity.
Minimal positive 2-spanning sets of vectors
Authors: Marcus Daniel A.
Published: June 2010

Let

(

υ
1

,
…
,

υ
n

)

({\upsilon _1}, \ldots ,{\upsilon _n})

be a sequence in an

m
m

-dimensional vector space

V
V

over an ordered field such that, for each

i
i

,

{

υ
j

:
j
≠
i

}

\left \{ {{\upsilon _j}:j \ne i} \right \}

positively spans

V
V

. It is shown that if

(

υ
1

,
…
,

υ
n

)

({\upsilon _1}, \ldots ,{\upsilon _n})

is minimal with this property, then
\[

n
⩽

{

m
(
m
+
1
)

/

2
+
5

4
m

if

m
⩾
5

if

m
⩽
5

n \leqslant \left \{ {_{m(m + 1)/2 + 5}^{4m}} \right .\quad _{{\text {if}}\;m \geqslant 5}^{{\text {if}}\;m \leqslant 5}

\]
and all cases are determined in which

n
=
4
m

n = 4m

,

m
⩽
4

m \leqslant 4

. An application to convex polytopes is given.
Monotonicity conjecture on permanents of doubly stochastic matrices
Authors: Lih Ko Wei , Wang Edward T. H.
Published: June 2010

A stronger version of the van der Waerden permanent conjecture asserts that if

J
n

{J_n}

denotes the

n
×
n

n \times n

matrix all of whose entries are

1

/

n

1/n

and

A
A

is any fixed matrix on the boundary of the set of

n
×
n

n \times n

doubly stochastic matrices, then

per

(
λ
A
+
(
1
−
λ
)

J
n

)

{\text {per}}(\lambda A + (1 - \lambda ){J_n})

as a function of

λ
\lambda

is nondecreasing in the interval

[
0
,
1
]

[0,1]

. In this paper, we elucidate the relation of this assertion to some other conjectures known to be stronger than van der Waerden’s. We also show that this assertion is true when

n
=
3

n = 3

and in the case when, up to permutations of rows and columns, either (i)

A
=

J
s

⊕

J
t

A = {J_s} \oplus {J_t}

,

0
>
s

0 > s

,

t
t

,

s
+
t
=
n

s + t = n

or (ii)

A
=

[

0

a
m
p
;
Y

Y
T

a
m
p
;
Z

]

A = \left [\begin {smallmatrix} 0 & Y \\ Y^T & Z\end {smallmatrix} \right ]

where 0 is an

s
×
s

s \times s

zero matrix,

Y
Y

is

s
×
t

s \times t

with all entries equal to

1

/

t

1/t

, and

Z
Z

is

t
×
t

t \times t

with all entries equal to

(
t
−
s
)

/

t
2

(t - s)/{t^2}

,

0
>
s
⩽
t

0 > s \leqslant t

,

s
+
t
=
n

s + t = n

.
The zero divisor problem for a class of torsion-free groups
Authors: Lichtman A. I.
Published: June 2010

Let

G
G

be a group,

N
N

be a central torsion-free subgroup of

G
G

and let

R
R

be an arbitrary field. Then

R
(
G
)

R(G)

contains no nilpotent elements provided that

R
(
G

/

N
)

R(G/N)

contains no nilpotent elements. When

G
G

is torsion-free the conditions of the theorem imply that

R
G

RG

is a domain; this generalizes Passman’s theorem in [1].
Unibranched prime ideals and going down in PI rings
Authors: Lestmann Phillip
Published: June 2010

The purpose of this paper is to answer the question of whether going down is equivalent to unibranchedness of prime ideals in integral extensions of prime

PI

{\text {PI}}

rings. We show by example that, in general, the answer is no; and we find an additional condition which, together with going down, implies prime ideals of

ht

>
1

{\text {ht}} > 1

are unibranched.
A remark on an example of R. A. Johnson
Authors: Sell George R.
Published: June 2010

In [3] Johnson constructs an example of a second-order linear differential equation with almost periodic coefficients and with an almost automorphic behavior which we describe as Property J. In this paper we give a necessary and sufficient condition that a second order linear differential equation has Property J.
On monogenic operators and measures
Authors: Lipecki Z. , Plachky D.
Published: June 2010

The notion of a monogenic operator between linear lattices, generalizing that of a monogenic measure, is introduced and investigated. The decomposition of an operator into its monogenic and antimonogenic parts is established. Products of monogenic measures are also considered.
On a problem of Doob about the fine topology and normal functions
Authors: Hwang J. S.
Published: June 2010

Early in the sixties, Joseph L. Doob proved that if

f
(
z
)

f(z)

is a normal function in a disk, then every angular cluster value at a boundary point is also a fine cluster value at that point. He then asked whether or not the converse of this theorem is true. It was pointed out by the referee that Choquet was the first one to show that the converse of the above theorem is false. His example was published by Brelot and Doob (see [2, p. 404]). Recently, the referee suggests the question as to whether a bounded analytic function can actually have an angular limit but not a fine limit. We construct a Blaschke product which has the fine cluster value 0 and the angular limit 1 at the same boundary point. This answers both questions of Doob and the referee.
Lie groups which admit flat left invariant metrics
Authors: Herring John R. , O’Sullivan John J.
Published: June 2010

Let

G
G

be a Lie group which admits a flat left invariant metric. We show that any nonflat left invariant metric on

G
G

has conjugate points and we describe how some of the conjugate points arise.
A proof of the Burkholder theorem for martingale transforms
Authors: Shintani T.
Published: June 2010

If

g
g

is the transform of an

L
1

{L^1}

-bounded martingale

f
f

under a predictable sequence

υ
\upsilon

satisfying

sup
n

|

υ
n

|

>
∞

{\text {sup}_n}\left | {{\upsilon _n}} \right | > \infty

almost everywhere, then a proof of the convergence of

g
g

is given using an approximation of

f
f

by a martingale of bounded variation.
On countably generated 𝑧-ideals of 𝐶(𝑋) for first countable spaces
Authors: Le Donne Attilio
Published: June 2010

In paper [L], a question asked in [D] has been answered first by proving that: if

X
X

is normal and first countable, then every countably generated

z
z

-ideal of

C
(
X
)

C(X)

is pure; then, by giving an example of a nonpure countably generated

z
z

-ideal of

C
(
X
)

C(X)

in a

σ
\sigma

-compact (hence normal) but not first countable space

X
X

. In this paper a class

C

\mathcal {C}

of topological spaces

X
X

whose

C
(
X
)

C(X)

has a nonpure countably generated

z
z

-ideal is constructed; it is proved that

C

\mathcal {C}

contains a space

X
X

which is first countable. So it is proved that in the proposition above the hypotheses "normal" and "first countable" are both essential. Finally in

§
4

\S 4

I prove, as announced in [L], that if

X
X

is a locally compact normal space, then every countably generated

z
z

-ideal of

C
(
X
)

C(X)

is pure. For the terminology and notations see [GJ], [D], [L].
On the dimension of cubic 𝜇-spaces
Authors: Mizokami T.
Published: June 2010

Let

X
X

be the countable product of special

σ
\sigma

-metric spaces defined below. Then it is proved that

X
⩽
n

X \leqslant n

if and only if there exists a

σ
\sigma

-closure-preserving open base

W

\mathcal {W}

for

X
X

such that

Ind

B
(
W
)
⩽
n
−
1

{\text {Ind}}\;B(W) \leqslant n - 1

for every

W
∈

W

W \in \mathcal {W}

.
A pathological flow in the torus
Authors: Beck Anatole
Published: June 2010

In a recent book (Continuous flows in the plane, Springer Grundlehren, vol. 201, 1974) the author of this article comments that there exist flows in the plane which have the property that some of their orbits have positive planar measure. Indeed, there are flows with the property that the complement of a countable set of orbits has measure 0. This fact does not appear in the literature, and the author has been asked to provide a proof. He proves instead the existence of a continuous flow in the torus in which the complement of a single orbit has measure 0. Then, lifting the torus to the plane gives the other example.
Regularity conditions and intersecting hypergraphs
Authors: Frankl Peter
Published: June 2010

Let

(

F

,
X
)

(\mathcal {F},X)

be a hypergraph with a transitive group of automorphisms. Suppose further that any four edges of

F

\mathcal {F}

intersect nontrivially. Denoting

|
X
|

\left | X \right |

by

n
n

we prove

|

F

|

=
O
(

2
n

)

\left | \mathcal {F} \right | = O({2^n})

. We show as well that it is not sufficient to suppose regularity instead of the transitivity of

Aut
⁡
(

F

)

\operatorname {Aut} (\mathcal {F})

.
A short proof that a compact ordered space cannot be mapped onto a nonmetric product
Authors: Bula W. , Dębski W. , Kulpa W.
Published: June 2010

In this note we give an elementary proof of a theorem of Treybig: if a product

X
×
Y

X \times Y

of two infinite Hausdorff spaces is a continuous image of a compact linearly ordered space, then

X
×
Y

X \times Y

is metrizable.
Two results on fixed rings
Authors: Pascaud J.-L.
Published: June 2010

Let

R
R

be a semiprime ring,

G
G

a finite group of automorphisms of

R
R

and

B
B

the algebra of the group. (A) If

R
R

is left primitive and

B
B

is

G
G

-simple then the fixed subring

R
G

{R^G}

is left primitive. (B) If

B
B

is semiprime and

R
G

{R^G}

is a left Goldie ring, then

R
R

can be embedded in a free left

R
G

{R^G}

-module of finite rank. Consequently if

R
G

{R^G}

is left Noetherian,

R
R

is left Noetherian.
Density relative to a torsion theory
Authors: Bland Paul , Riley Stephen
Published: June 2010

If

(
ℑ
,

F

)

(\Im , \mathcal {F})

is a torsion theory on Mod

R
R

, then a ring

B
B

of biendomorphisms of a

ℑ
\Im

-cocritical module is topologized. Moreover, if a certain factor module of

R
R

is quasi-projective, a ring monomorphism

φ
:
R
→
B

\varphi :R \to B

is found such that

φ
(
R
)

\varphi (R)

is topologically dense in

B
B

. This is done in such a way that when

(
ℑ
,

F

)

(\Im , \mathcal {F})

is the torsion theory in which every module is torsion free, the Jacobson density theorem is recovered.
A result related to a theorem by Pianigiani
Authors: Boyarsky Abraham , Haddad Gabriel
Published: June 2010

Let

τ
:
J
→
J

\tau :J \to J

be a piecewise

C
2

{C^2}

map, where

J
J

is an interval, satisfying

|

τ
′

|

>
1

\left | {\tau ’} \right | > 1

. An upper bound for the number of independent absolutely continuous measures invariant under

τ
\tau

is presented.
On measures of analytic type
Authors: Pigno Louis , Smith Brent
Published: June 2010

The purpose of this note is to show how the characterization of an absolutely continuous or singular measure of R. Doss can be used to prove the Helson-Lowdenslager Theorem for measures of analytic type. Our methods were inspired by the work of P. Cohen and H. Davenport on the

L
1

{L^1}

norm of exponential sums.
An oscillation condition for differential equations of arbitrary order
Authors: Trench William F.
Published: June 2010

In separate papers, D. L. Lovelady has related oscillation of solutions of certain linear differential equations of odd order

⩾
3

\geqslant 3

and even order

⩾
4

\geqslant 4

to oscillation of an associated second order equation. This paper presents a unified proof of Lovelady’s results for equations of arbitrary order

⩾
3

\geqslant 3

. The results are somewhat more detailed and the equations need not be linear.
Level curves of functions of bounded characteristic
Authors: Barth Karl F. , Clunie James G.
Published: June 2010

The authors construct a meromorphic function of bounded characteristic whose level set for 1 contains a sequence of closed Jordan curves which surround the origin and which converge to all of the unit circle. Some sufficient conditions are given for each level set of a meromorphic function of bounded characteristic to end at points.
A Tauberian problem for a Volterra integral operator
Authors: Gripenberg Gustaf
Published: June 2010

The following question is studied: For which (nonintegrable) kernels

A
A

does

lim

t
→
∞

∫
0
t

A
(
t
−
s
)
x
(
s
)
d
s
=
0

{\lim _{t \to \infty }}\int _0^tA(t - s)x(s)ds = 0

imply that

lim

t
→
∞

x
(
t
)
=
0

{\lim _{t \to \infty }}x(t) = 0

when

x
x

is bounded and satisfies a Tauberian condition.
Approximating fixed points of some mappings
Authors: Bose Ramendra Krishna , Mukherjee Rathindra Nath
Published: June 2010

In a uniformly convex Banach space, Senter and Dotson, Jr., have given conditions under which certain types of iterates of a quasi-nonexpansive mapping converge to a fixed point of the mapping. Here we consider two types of mappings, one considered by Ray and the other considered by Goebel, Kirk and Shimi, and prove some results concerning the approximations of fixed points of such mappings. A result of Kannan is obtained as a particular case of our result under relaxed conditions.
Admissible kernels for starshaped sets
Authors: Breen Marilyn
Published: June 2010

Steven Lay has posed the following interesting question: If

D
D

is a convex subset of

R

d

{{\mathbf {R}}^d}

, then is there a starshaped set

S
≠
D

S \ne D

in

R

d

{{\mathbf {R}}^d}

whose kernel is

D
D

? Thus the problem is that of characterizing those convex sets which are admissible as the kernel of some nonconvex starshaped set in

R

d

{{\mathbf {R}}^d}

. Here Lay’s problem is investigated for closed sets, and the following results are obtained: If

D
D

is a nonempty closed convex subset of

R

2

{{\mathbf {R}}^2}

, then

D
D

is the kernel of some planar set

S
≠
D

S \ne D

if and only if

D
D

contains no line. If

D
D

is a compact convex set in

R

d

{{\mathbf {R}}^d}

, then there is a compact set

S
≠
D

S \ne D

in

R

d

{{\mathbf {R}}^d}

whose kernel is

D
D

.
An infinite family in 𝜋_{∗}𝑆⁰ derived from Mahowald’s 𝜂ⱼ family
Authors: Bruner Robert R.
Published: June 2010

Combining the relationship due to D. S. Kahn between

∪
i

{ \cup _i}

operations in homotopy and Steenrod operations in the

E
2

{E_2}

term of the Adams spectral sequence with Mahowald’s result that

h
1

h
j

{h_1}{h_j}

is a permanent cycle for

j
⩾
4

j \geqslant 4

, we show that

h
2

h
j
2

{h_2}h_j^2

is also a permanent cycle for

j
⩾
5

j \geqslant 5

. This gives another infinite family of nonzero elements in the stable homotopy of spheres. Properties of the

∪
i

{ \cup _i}

homotopy operations further imply that these elements generate

Z
2

{Z_2}

direct summands.
Krull dimension and reflexivity in some Noetherian rings
Authors: Haghany A. , Sarath B.
Published: June 2010

In this paper we study Noetherian prime rings

R
R

which satisfy the formula

|
R
|

=
sup
{

|

I

∗
∗

/

I

|

:
I

\left | R \right | = \sup \{ \left | {{I^{* *}}/I} \right |:I

is an essential left ideal of

R
}
+
2

R\} + 2

, where

|

|

|\;|

denotes left Krull dimension. If further

Q

/

R

Q/R

is

|
R
|

−
1

\left | R \right | - 1

unmixed, where

Q
Q

is the simple Artinian quotient ring of

R
R

, we characterize

R
R

using torsion theories cogenerated by the injective hulls of

|
R
|

−
1

\left | R \right | - 1

dimensional critical modules. Also equivalent statements are established, linking homological properties with dimension theory, for

R
R

-modules to be reflexive.
A counterexample to the unimodular conjecture on finitely generated dimension groups
Authors: Riedel Norbert
Published: June 2010

We give a series of examples of simple finitely generated dimension groups which cannot be obtained as the inductive limit of a system
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Z Superscript r Baseline right-arrow Overscript upper A 1 Endscripts bold upper Z Superscript r Baseline right-arrow Overscript upper A 2 Endscripts midline-horizontal-ellipsis bold upper Z Superscript r Baseline right-arrow Overscript upper A Subscript n Baseline Endscripts midline-horizontal-ellipsis comma">

Z

r

→

A
1

Z

r

→

A
2

⋯

Z

r

→

A
n

⋯
,

\mathbf {Z}^r \stackrel {A_1}{\rightarrow } \mathbf {Z}^r \stackrel {A_2}{\rightarrow } \cdots \mathbf {Z}^r \stackrel {A_n}{\rightarrow } \cdots ,

\]
where each

A

n

A_{n}

is a unimodular matrix whose entries are nonnegative integers.
The Eisenbud-Evans generalized principal ideal theorem and determinantal ideals
Authors: Bruns Winfried
Published: June 2010

In [2] Eisenbud and Evans gave an important generalization of Krull’s Principal Ideal Theorem. However, their proof, using maximal Cohen-Macaulay modules, may have limited the validity of their theorem to a proper subclass of all local rings. (Hochster proved the existence of maximal Cohen-Macaulay modules for local rings which contain a field, cf. [4]). In the first section we present a proof which is simpler and guarantees the Generalized Principal Ideal Theorem for all local rings. The main result of the second section was conjectured in [2]. Under a hypothesis typically being satisfied for the most important fitting invariant of a module, it improves the Eagon-Northcott bound [1] on the height of a determinantai ideal considerably. Finally we will discuss the implications of a recent theorem of Fairings [3] on determinantal ideals.
An extension theorem for characters
Authors: Gagola Stephen M.
Published: June 2010

If

N
N

is a normal subgroup of the finite group

G
G

and

ψ
\psi

is an irreducible complex character of

N
N

that is invariant in

G
G

, then

ψ
\psi

is extendible to a character of

G
G

if

(

|

G
:
N

|

,

|
N
|

/

ψ
(
1
)
)
=
1

(\left | {G:N} \right |,\left | N \right |/\psi (1)) = 1
On simple reducible Lie algebras of depth two
Authors: Gregory Thomas B.
Published: June 2010

We show that under certain circumstances simple finite-dimensional reducible graded Lie algebras of the form

L

−
2

⊕

L

−
1

⊕

L
0

⊕

L
1

⊕
⋯
⊕

L
k

{L_{ - 2}} \oplus {L_{ - 1}} \oplus {L_0} \oplus {L_1} \oplus \cdots \oplus {L_k}

can be given irreducible transitive gradations of the form

M

−
1

⊕

M
0

⊕
⋯
⊕

M

[
k

/

2
]

{M_{ - 1}} \oplus {M_0} \oplus \cdots \oplus {M_{[k/2]}}

.
Recurrent and Poisson stable flows
Authors: Knight Ronald A.
Published: June 2010

The purpose of this paper is to demonstrate that the families of Poisson stable flows and recurrent flows coincide whenever the phase space is locally compact.
Boundary value problems for second order nonlinear matrix differential equations
Authors: Shreve Warren E.
Published: June 2010

Existence of a solution to

X
=
F
(
t
,
X
,

X
′

)

X = F(t,X,X’)

,

X
(
a
)
=
A

X(a) = A

,

X
(
b
)
=
B

X(b) = B

, is shown under certain conditions using a degree-theoretic result similar to one of Bebernes where

X
,
F
,
A
,
B

X,F,A,B

are

n
×
n

n \times n

matrices.