bluuebunny/crossref_metadata_2025_split huggingface dataset:
Oscillation of first-order nonlinear differential equations with deviating arguments
Authors: Kitamura Yuichi , Kusano Takaŝi 
Published: June 2010

This paper is devoted to the study of the oscillatory behavior of solutions of the first-order nonlinear functional differential equations
\[

(

A

)

x
′

(
t
)
=

∑

i
=
1

N

q
i

(
t
)

f
i

(
x
(

g
i

(
t
)
)
)

+
F
(
t
,
x
(
t
)
,
x
(

g
1

(
t
)
)
,
…
,
x
(

g
N

(
t
)
)
)
,

\begin {array}{*{20}{c}} x’(t) = \sum \limits _{i = 1}^N {{q_i}(t){f_i}(x({g_i}(t)))} \\ + F(t,x(t),x({g_1}(t)), \ldots ,x({g_N}(t))),\tag {$A$} \\ \end {array}

\]

\[

(

B

)

x
′

(
t
)
+

∑

i
=
1

N

q
i

(
t
)

f
i

(
x
(

g
i

(
t
)
)
)

+
F
(
t
,
x
(
t
)
,
x
(

g
1

(
t
)
)
,
…
,
x
(

g
N

(
t
)
)
)
=
0.

\begin {array}{*{20}{c}} x’(t) + \sum \limits _{i = 1}^N {{q_i}(t){f_i}(x({g_i}(t)))} \\ + F(t,x(t),x({g_1}(t)), \ldots ,x({g_N}(t))) = 0.\tag {$B$} \\ \end {array}

\]
First, without assuming that the deviating arguments

g
i

(
t
)
,
1
⩽
i
⩽
N

{g_i}(t),1 \leqslant i \leqslant N

, are retarded or advanced, sufficient conditions are established for all solutions of (A) and (B) to be oscillatory. Secondly, a characterization of oscillation of all solutions is obtained for equation (A) with

F
≡
0

F \equiv 0

and

g
i

(
t
)
>
t
,
1
⩽
i
⩽
N

{g_i}(t) > t,1 \leqslant i \leqslant N

, as well as for equation (B) with

F
≡
0

F \equiv 0

and

g
i

(
t
)
>
t
,
1
⩽
i
⩽
N

{g_i}(t) > t,1 \leqslant i \leqslant N

.
Ergodic projections of continuous and discrete semigroups
Authors: Shaw Sen-Yen 
Published: June 2010

Let X be a Banach space. Let

{
T
(
t
)
;
t
>
0
}

\{ T(t);t > 0\}

be a uniformly bounded semigroup of operators on X, which converges strongly to P, known to be a projection, as t goes to 0. If A is its generator and

X
0

{X_0}

[resp.,

X
t

,
t
>
0

{X_t},t > 0

] is the set of x for which
\[

P
0

x
≡

lim

t
→
∞

t

−
1

∫
0
t

T
(
τ
)
x

d
τ

[

resp

.
,

P
t

x
≡

lim

n
→
∞

n

−
1

∑

i
=
0

n
−
1

T
(
i
t
)
x

]

{P_0}x \equiv \lim \limits _{t \to \infty } {t^{ - 1}}\int _0^t {T(\tau )x\;d\tau \;\left [ {{\text {resp}}.,\;{P_t}x \equiv \lim \limits _{n \to \infty } {n^{ - 1}}\sum \limits _{i = 0}^{n - 1} {T(it)x} } \right ]}

\]
exists, then, for each

t
⩾
0
,

P
t

t \geqslant 0,{P_t}

is a bounded projection in

X
t

{X_t}

; when

t
=
0
,

X
0

=
N
(
A
)
⊕

R
(
A
)

¯

⊕
N
(
P
)
,

R
(

P
0

)
=
N
(
A
)

t = 0,{X_0} = N(A) \oplus \overline {R(A)} \oplus N(P),\;R({P_0}) = N(A)

and

N
(

P
0

)
=

R
(
A
)

¯

⊕
N
(
P
)

N({P_0}) = \overline {R(A)} \oplus N(P)

; when

t
>
0

t > 0

, then
\[

X
t

=
N
(
T
(
t
)
−
I
)
⊕

R
(
T
(
t
)
−
I
)

¯

,

{X_t} = N(T(t) - I) \oplus \overline {R(T(t) - I)} ,

\]

R
(

P
t

)
=
N
(
T
(
t
)
−
I
)

R({P_t}) = N(T(t) - I)

and

N
(

P
t

)
=

R
(
T
(
t
)
−
I
)

¯

;

X
t

=
X

N({P_t}) = \overline {R(T(t) - I)} ;\;{X_t} = X

for all

t
⩾
0

t \geqslant 0

if X is reflexive. Some results on relations among the projections

P
t

,
t
⩾
0

{P_t},t \geqslant 0

, are obtained. In particular, we have

P
t

=

P
0

{P_t} = {P_0}

for all sufficiently small t if A is bounded.
A simplicity theorem for amoebas over random reals
Authors: Abramson Fred G. 
Published: June 2010

Let M be a countable standard transitive model of ZFC,

A

\mathcal {A}

be an amoeba over M, and r be a random real over M. Theorem. (a) There is no infinite set of reals X contained in the complement of

A

\mathcal {A}

with

X
∈
M
[
r
]
;
(
b
)

I
f

{

x
n

|

n
>
ω
}
∈
M
[
r
]

X \in M[r];(b)\;If\;\{ {x_n}|n > \omega \} \in M[r]

is a sequence of distinct reals, then for all large enough n,

{

x
i

|

2
n

⩽
i
>

2

n
+
1

}
∩

A

≠
∅

\{ {x_i}|{2^n} \leqslant i > {2^{n + 1}}\} \cap \mathcal {A} \ne \emptyset

.
The extension of 𝐻^{𝑝}-functions from certain hypersurfaces of a polydisc
Authors: Zarantonello Sergio E. 
Published: June 2010

Let E be a subvariety of the open unit polydisc

U
n

,
n
⩾
2

{U^n},n \geqslant 2

, of pure dimension

n
−
1

n - 1

, satisfying the following conditions. There exists an annular domain

Q
n

=
{
(

z
1

,
…
,

z
n

)
∈

C

n

:
r
>

|

z
i

|

>
1
,
1
⩽
i
⩽
n
}

{Q^n} = \{ ({z_1}, \ldots ,{z_n}) \in {{\mathbf {C}}^n}:r > |{z_i}| > 1,1 \leqslant i \leqslant n\}

, a continuous function

η
:
[
r
,
1
)
→
[
r
,
1
)

\eta :[r,1) \to [r,1)

, and a

δ
>
0

\delta > 0

, such that (i)

|

z
n

|

⩽
η
(
(

|

z
1

|

+
⋯
+

|

z

n
−
1

|

)

/

(
n
−
1
)
)

|{z_n}| \leqslant \eta ((|{z_1}| + \cdots + |{z_{n - 1}}|)/(n - 1))

whenever

(

z
1

,
…
,

z
n

)
∈
E
∩

Q
n

({z_1}, \ldots ,{z_n}) \in E \cap {Q^n}

, (ii)

|

α
−
β

|

⩾
δ

|\alpha - \beta | \geqslant \delta

whenever

1
⩽
j
⩽
n

1 \leqslant j \leqslant n

and

(

ζ
′

,
α
,
ζ
)
≠
(

ζ
′

,
β
,
ζ
)

(\zeta ’,\alpha ,\zeta ) \ne (\zeta ’,\beta ,\zeta )

are both in

(

Q

j
−
1

×
U
×

Q

n
−
j

)
∩
E

({Q^{j - 1}} \times U \times {Q^{n - j}}) \cap E

. Theorem. Let

0
>
p
>
∞

0 > p > \infty

, let g be holomorphic on E and let u be the real part of a holomorphic function on E. If

|

g
(
z
)

|

p

⩽
u
(
z
)

|g(z){|^p} \leqslant u(z)

for all

z
∈
E

z \in E

, then g can be extended to a function in the Hardy space

H
p

(

U
n

)

{H^p}({U^n})

.
Retracts in metric spaces
Authors: Pasicki Lech 
Published: June 2010

In this paper we define S-contractibility and two classes of spaces connected with this notion. A space X is said to be S-contractible provided that S is a function

S
:
X
×
⟨
0
,
1
⟩
×
X
(
x
,
α
,
y
)
↦

S
x

(
α
,
y
)
∈
X

S:X \times \langle 0,1\rangle \times X (x,\alpha ,y) \mapsto {S_x}(\alpha ,y) \in X

that is continuous in

α
\alpha

and y, and for every

x
,
y
∈
X
,

S
x

(
0
,
y
)
=
y
,

S
x

(
1
,
y
)
=
x

x,y \in X,{S_x}(0,y) = y,{S_x}(1,y) = x

. This notion is close to equiconnectedness, which can be defined as follows. A space X is equiconnected if there exists a map S such that X is S-contractible and

S
x

(
α
,
x
)
=
x

{S_x}(\alpha ,x) = x

for all

x
∈
X

x \in X

and

α
∈
I

\alpha \in I

(cf. [4]). The results we obtain in the theory of retracts are close to those that are known for equiconnected spaces. Also the thickness of the neighborhood that can be retracted on a set in a metric space is estimated, which enables to prove a theorem belonging to fixed point theory.
A remark on Carleson’s characterization of BMO
Authors: Uchiyama Akihito 
Published: June 2010

L. Carleson showed that if

φ
∈
BMO
(

R
n

)

\varphi \in \text {BMO}(R ^n)

and supp

φ
\varphi

is compact, then

φ
\varphi

can be written in the form

φ
(
x
)
=

Σ

k
=
1

∞

∫

P

t
k

(
y
)

(
x
−
y
)

b
k

(
y
)
d
y
+

b
0

(
x
)

\varphi (x) = \Sigma _{k = 1}^\infty \smallint {P_{{t_k}(y)}}(x - y){b_k}(y)dy + {b_0}(x)

where

Σ

k
=
0

∞

‖

b
k

‖

∞

⩽
C

‖
φ
‖

BMO

,

t
k

(
y
)
>
0

\Sigma _{k = 0}^\infty {\left \| {{b_k}} \right \|_\infty } \leqslant C{\left \| \varphi \right \|_{{\text {BMO}}}},{t_k}(y) > 0

and

P
t

(
x
)
=

C
n

t

(

|

x

|

2

+

t
2

)

−
(
n
+
1
)

/

2

{P_t}(x) = {C_n}t{(|x{|^2} + {t^2})^{ - (n + 1)/2}}

is the Poisson kernel. We show that we can take

b
2

=

b
3

=
⋯
=
0

{b_2} = {b_3} = \cdots = 0

. This can be generalized on the space of homogeneous type with certain assumptions.
An integro-differential equation
Authors: Burton T. A. 
Published: June 2010

The vector equation
\[

x
′

(
t
)
=
A
(
t
)
x
(
t
)
+

∫
0
t

C
(
t
,
s
)
D
(
x
(
s
)
)
x
(
s
)
d
s
+
F
(
t
)

x’(t) = A(t)x(t) + \int _0^t {C(t,s)D(x(s))x(s)ds + F(t)}

\]
is considered in which A is not necessarily a stable matrix, but

A
(
t
)
+
G
(
t
,
t
)
D
(
0
)

A(t) + G(t,t)D(0)

is stable where G is an antiderivative of C with respect to t. Stability and boundedness results are then obtained. We also point out that boundedness results of Levin for the scalar equation

u
′

(
t
)
=
−

∫
0
t

a
(
t
−
s
)
g
(
u
(
s
)
)
d
s

u’(t) = - \int _0^t {a(t - s)g(u(s))ds}

can be extended to a vector system

x
′

(
t
)
=
−

∫
0
t

H
(
t
,
s
)
x
(
s
)
d
s

x’(t) = - \int _0^t {H(t,s)x(s)ds}

.
Balayage defined by the nonnegative convex functions
Authors: Fischer P. , Holbrook J. A. R. 
Published: June 2010

We study the Choquet order induced on measures on a linear space by the cone of nonnegative convex functions. We are concerned mainly with discrete measures, and the following result is typical. Let

x
1

,
…
,

x
r

,

y
1

,
…
,

y
n

{x_1}, \ldots ,{x_r},{y_1}, \ldots ,{y_n}

, where

r
⩽
n

r \leqslant n

, be points in

R

d

{{\mathbf {R}}^d}

. Then
\[

∑
1
r

f
(

x
k

)
⩽

∑
1
n

f
(

y
k

)

\sum \limits _1^r {f({x_k}) \leqslant } \sum \limits _1^n {f({y_k})}

\]
for all nonnegative, continuous, convex functions f if, and only if, there exists a doubly stochastic matrix M such that
\[

x
j

=

∑

k
=
1

n

m

j
k

y
k

(
j
=
1
,
…
,
r
)
.

{x_j} = \sum \limits _{k = 1}^n {{m_{jk}}{y_k}\quad (j = 1, \ldots ,r).}

\]
In the case

d
=
1

d = 1

, this result may be found in the work of L. Mirsky; our methods allow us to place such results in a general setting.
A short proof for a.e. convergence of generalized conditional expectations
Authors: Landers D. , Rogge L. 
Published: June 2010

Let

L
s

(
μ
)

{L_s}(\mu )

be the space of real valued random variables with

μ
(

|

f

|

s

)
>
∞
,
1
>
s
>
∞

\mu (|f{|^s}) > \infty ,1 > s > \infty

. Let

C
⊂

L
s

(
μ
)

C \subset {L_s}(\mu )

be a closed convex set. For each

f
∈

L
s

(
μ
)

f \in {L_s}(\mu )

there exists a unique element

μ
s

(
f

|

C
)

{\mu _s}(f|C)

with

‖

f
−

μ
s

(
f

|

C
)

‖

s

⩽

‖

f
−
c

‖

s

{\left \| {f - {\mu _s}(f|C)} \right \|_s} \leqslant {\left \| {f - c} \right \|_s}

for every

c
∈
C

c \in C

. Let

C
n

{C_n}

be a decreasing or increasing sequence of closed convex lattices converging to the closed convex lattice

C
∞

{C_\infty }

. We show that

μ
s

(
f

|

C
n

)
→

μ
s

(
f

|

C
∞

)
μ

{\mu _s}(f|{C_n}) \to {\mu _s}(f|{C_\infty })\mu

-a.e. for every

f
∈

L
s

(
μ
)

f \in {L_s}(\mu )

. This result contains the results of a.e. convergence of prediction sequences of Ando-Amemiya and the result of Brunk and Johansen of a.e. convergence of conditional expectations given

σ
\sigma

-lattices.
On finite groups containing three 𝐶𝐶-subgroups
Authors: Arad Zvi , Ferguson Pamela 
Published: June 2010

A finite group G has a self-centralization system of type

(
2

|

A
1

|

,
4

|

A
2

|

,
4

|

A
3

|

)

(2|{A_1}|,4|{A_2}|,4|{A_3}|)

if G contains three nonconjugate CC-subgroups

A
1

,

A
2

,

A
3

{A_1},{A_2},{A_3}

, such that

|

N
G

(

A
1

)

|

=
2

|

A
1

|

,

|

N
G

(

A
2

)

|

=
4

|

A
2

|

,

|

N
G

(

A
3

)

|

=
4

|

A
3

|

|{N_G}({A_1})| = 2|{A_1}|,|{N_G}({A_2})| = 4|{A_2}|,|{N_G}({A_3})| = 4|{A_3}|

. The authors prove that if a finite group G has a self-centralization system of type

(
2

|

A
1

|

,
4

|

A
2

|

,
4

|

A
3

|

)

(2|{A_1}|,4|{A_2}|,4|{A_3}|)

and

|

G

|

⩽
3

|

A
1

|

2

|

A
2

|

2

|

A
3

|

2

|G| \leqslant 3|{A_1}{|^2}|{A_2}{|^2}|{A_3}{|^2}

, then G has a nilpotent normal subgroup N such that G/N is isomorphic to

S
z
(
q
)

Sz(q)

for suitable q.
Divided differences and systems of nonharmonic Fourier series
Authors: Ullrich David 
Published: June 2010

Suppose that

ω

n
,
0

,

ω

n
,
1

,
…
,

ω

n
,
k

{\omega _{n,0}},{\omega _{n,1}}, \ldots ,{\omega _{n,k}}

are distinct complex numbers with

|

n
−

ω

n
,
j

|

⩽
δ

|n - {\omega _{n,j}}| \leqslant \delta

for all

n
∈

Z

,
j
=
0
,
1
,
…
,
k

n \in {\mathbf {Z}},j = 0,1, \ldots ,k

. We show that if

δ
>
0

\delta > 0

is small enough then, given complex numbers

c

n
,
j

(
n
∈

Z

,
j
=
0
,
1
,
…
,
k
)

{c_{n,j}}(n \in {\mathbf {Z}},j = 0,1, \ldots ,k)

there exists

f
∈

L
2

(
−
(
k
+
1
)
π
,
(
k
+
1
)
π
)

f \in {L^2}( - (k + 1)\pi ,(k + 1)\pi )

with
\[

∫

−
(
k
+
1
)
π

(
k
+
1
)
π

f
(
t
)

e

−
i
t

ω

n
,
j

d
t
=

c

n
,
j

for

n
∈

Z

,
j
=
0
,
1
,
…
,
k

\int _{ - (k + 1)\pi }^{(k + 1)\pi } {f(t){e^{ - it{\omega _{n,j}}}}} dt = {c_{n,j}}\quad {\text {for}}\;n \in {\mathbf {Z}},j = 0,1, \ldots ,k

\]
if and only if certain “divided differences” involving the

c

n
,
j

{c_{n,j}}

’s and the

ω

n
,
j

{\omega _{n,j}}

’s are square summable. This extends a classical theorem of Paley and Wiener, which is equivalent to the case

k
=
0

k = 0

above.
Tame measures on certain compact sets
Authors: Lee Hsuan Pei 
Published: June 2010

A finite complex Borel measure

μ
\mu

on a compact subset

X
⊂

C

n

X \subset {{\mathbf {C}}^n}

is called tame if there exist finite measures

σ
1

,
…
,

σ
n

{\sigma _1}, \ldots ,{\sigma _n}

on X with
\[

∫
X

ϕ
d
μ
=

∫
X

∑
1
n

∂
ϕ

∂

z
¯

j

d

σ
j

\int _X \phi d\mu = \int _X {\sum \limits _1^n {\frac {{\partial \phi }}{{\partial {{\bar z}_j}}}d{\sigma _j}} }

\]
for every

ϕ
∈

C
0
∞

(

C

n

)

\phi \in C_0^\infty ({{\mathbf {C}}^n})

. We define

X
T

=
{
(

z
1

,

z
2

)
:

|

z
1

|

2

+

|

z
2

|

2

=
1

{X_T} = \{ ({z_1},{z_2}):|{z_1}{|^2} + |{z_2}{|^2} = 1

and

z
1

∈
T
}

{z_1} \in T\}

, where T is a compact subset of

{

|

z
1

|

>
1
}

\{ |{z_1}| > 1\}

in

C

1

{{\mathbf {C}}^1}

. It is shown in this paper that tame measures form a weak-

∗

^ \ast

dense subset of

R

(

X
T

)
⊥

R{({X_T})^ \bot }

. It follows then, with the help of a theorem by Weinstock, that

R
(

X
T

)

R({X_T})

is a local algebra.
The stability of the cosine equation
Authors: Baker John A. 
Published: June 2010

If

δ
>
0

\delta > 0

, G is an abelian group and f is a complex-valued function defined on G such that

|

f
(
x
+
y
)
+
f
(
x
−
y
)
−
2
f
(
x
)
f
(
y
)

|

⩽
δ

|f(x + y) + f(x - y) - 2f(x)f(y)| \leqslant \delta

for all

x
,
y
∈
G

x,y \in G

, then either

|

f
(
x
)

|

⩽
(
1
+

1
+
2
δ

)

/

2

|f(x)| \leqslant (1 + \sqrt {1 + 2\delta } )/2

for all

x
∈
G

x \in G

or

f
(
x
+
y
)
+
f
(
x
−
y
)
=
2
f
(
x
)
f
(
y
)

f(x + y) + f(x - y) = 2f(x)f(y)

for all

x
,
y
∈
G

x,y \in G

.
Mad families and ultrafilters
Authors: Weese Martin 
Published: June 2010

For each almost disjoint family X let

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

.
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
\[

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.
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
\[

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.
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

(
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

[
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

.
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

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.
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
\[

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 distribution function in the Morrey space
Authors: Álvarez Alonso Josefina 
Published: June 2010

For

1
⩽
p
⩽
∞

1 \leqslant p \leqslant \infty

, we consider

p
p

-integrable functions on a finite cube

Q
0

{Q_0}

in

R

n

{{\mathbf {R}}^n}

, satisfying
\[

(

1

|

Q

|

∫
Q

|

f
(
x
)
−

f
Q

|

p

d
x

)

1

/

p

⩽
C
φ
(

|

Q

|

)

{\left ( {\frac {1} {{|Q|}}\int _Q {|f(x) - {f_Q}{|^p}dx} } \right )^{1/p}} \leqslant C\varphi (|Q|)

\]
for every parallel subcube

Q
Q

of

Q
0

{Q_0}

, where

|

Q

|

|Q|

denotes the volume of

Q
Q

,

f
Q

{f_Q}

is the mean value of

f
f

over

Q
Q

and

φ
(
t
)

\varphi (t)

is a nonnegative function defined in

(
0
,
∞
)

(0,\infty )

, such that

φ
(
t
)

\varphi (t)

is nonincreasing near zero,

φ
(
t
)
→
∞

\varphi (t) \to \infty

as

t
→
0

t \to 0

, and

t

φ
p

(
t
)

t{\varphi ^p}(t)

is nondecreasing near zero. The constant

C
C

does not depend on

Q
Q

. Let

g
g

be a nonnegative

p
p

-integrable function

g
:
(
0
,
1
)
→

R

g:(0,1) \to {\mathbf {R}}

such that

g
g

is nonincreasing and

g
(
t
)
→
∞

g(t) \to \infty

as

t
→
0

t \to 0

. We prove here that there exist a cube

Q
0

{Q_0}

and a function

f
f

satisfying condition

(
1
)

(1)

for every parallel subcube

Q
Q

of

Q
0

{Q_0}

, such that

δ
f

(
λ
)
⩾

C
1

δ
g

(
λ
)

{\delta _f}(\lambda ) \geqslant {C_1}{\delta _g}(\lambda )

for

λ
⩾

λ
0

\lambda \geqslant {\lambda _0}

,

C
1

>
0

{C_1} > 0

, where

δ
(
λ
)

\delta (\lambda )

denotes the distribution function.
Derivations on commutative Banach algebras
Authors: Khosravi A. 
Published: June 2010

Let

A
A

be a commutative Banach algebra with radical

R
R

and

D
D

be a derivation on

A
A

. If

K
=
{
x
∈
R
:

for every

n
⩾
1
,

D
n

x
∈
R
}

K = \{ x \in R:\;{\text {for every }}n \geqslant 1,\;{D^n}x \in R\}

, then

D
A
⊆
R

DA \subseteq R

if and only if

K
K

is closed.
Strict local rings
Authors: Herzog J. 
Published: June 2010

In this paper we introduce the notion of a strict local ring. A local Cohen-Macaulay ring

(
B
,
m
)

(B,m)

is called strict if, whenever a local ring

(
A
,
n
)

(A,n)

specializes by a regular sequence to

B
B

, then the associated graded ring

g

r

n

(
A
)

{\text {g}}{{\text {r}}_n}(A)

is Cohen-Macaulay. We show that an artinian graded algebra

B
B

is strict if for the graded cotangent module we have

T
1

(
B

/

k
,
B

)
r

=
0

for

ν
>
−
1

{T^1}{(B/k,B)_r} = 0{\text {for }}\nu > - 1

. Various examples are considered where this condition holds. In particular, with this method we reprove a result of J. Sally [6].
Glidebending of general caps: an infinitesimal treatment
Authors: Kann Edgar 
Published: June 2010

We prove that two

C
3

{C^3}

isometric surfaces

S
′

S’

and

S
S

with (not necessarily planar) boundary, immersed in

E
3

{E^3}

, with

K
>
0

K > 0

, having their spherical images in the same open hemisphere, and which are oriented so that the mean curvatures have the same sign, are congruent if they satisfy the glidebending boundary condition: Let

{
n
:
e
⋅
n
>
0

\{ n:e \cdot n > 0

,

e
=

const

. vector}

e = {\text {const}}{\text {. vector\} }}

be the open hemisphere. Then

e
⋅

X
′

=
e
⋅
X

e \cdot X’ = e \cdot X

at corresponding boundary points, where

X
′

X’

and

X
X

are position vectors of

S
′

S’

and

S
S

. The method uses the fact that the surface

1
2

(
X
−

X
′

)

\tfrac {1} {2}(X - X’)

is an infinitesimal bending field for the mean surface

1
2

(
X
+

X
′

)

\tfrac {1} {2}(X + X’)

and is elementary in that it uses the rotation vector of classical infinitesimal bending theory but no integral formulas, maximum principles for elliptic operators or index theorems. The surfaces considered need not have a simple projection on a plane, be convex in the large (an example is given) nor be simply connected. We use the method to prove the finite and infinitesimal rigidity of general caps and caps under glidebending.