bluuebunny/crossref_metadata_2025_split huggingface dataset:
On Kannan maps
Authors: Wong Chi Song 
Published: June 2010

Let

K
K

be a (nonempty) weakly compact convex subset of a Banach space

B
B

. Let

T
T

be a self map on

K
K

such that for all

x
,
y

x,y

in

K
,

|

|

T
(
x
)
−
T
(
y
)

|

|

⩽
(

|

|

x
−
T
(
x
)

|

|

+

|

|

y
−
T
(
y
)

|

|

)

/

2

K,||T(x) - T(y)|| \leqslant (||x - T(x)|| + ||y - T(y)||)/2

. It is proved without the continuity of

T
T

and Zorn’s lemma that

T
T

has a fixed point if and only if

inf
{

|

|

x
−
T
(
x
)

|

|

:
x
∈
K
}
=
0

\inf \{ ||x - T(x)||:x \in K\} = 0

. A characterization of the existence of fixed points for such

T
T

is obtained in terms of close-to-normal structure. As consequences, the following results are obtained: (i)

T
T

has a unique fixed point if

B
B

is locally uniformly convex or more generally if

B
B

has the property

A

{\mathbf {A}}

: For any sequence

{

x
n

}

\{ {x_n}\}

in

B
,
{

x
n

}

B,\{ {x_n}\}

converges to a point

x
x

in

B
B

if it converges weakly to

x
x

and

{

|

|

x

|

|

}

\{ ||x||\}

converges to

|

|

x

|

|

||x||

; (ii)

T
T

has a unique fixed point if

B
B

is separable.
The zeros of Jensen polynomials are simple
Authors: Csordas George , Williamson Jack 
Published: June 2010

An entire function

f
(
z
)
=

Σ

k
=
0

∞

a
k

z

k
+
m

/

k
!

f(z) = \Sigma _{k = 0}^\infty {a_k}{z^{k + m}}/k!

is said to be in the class

L

−

P

\mathcal {L} - \mathcal {P}

(Laguerre-Pólya) if it can be represented in the form
\[

f
(
z
)
=
c

z
m

e

−
α

z
2

+
β
z

∐
n

(
1
−
z

/

z
n

)

e

z

/

z
n

,

f(z) = c{z^m}{e^{ - \alpha {z^2} + \beta z}}\coprod \limits _n {(1 - z/{z_n}){e^{z/{z_n}}}} {\text { }},

\]
where

α
≥
0
,
c
,
β

\alpha \geq 0,c,\beta

and

z
n

{z_n}

are real, and

Σ
n

z
n

−
2

>
∞

{\Sigma _n}z_n^{ - 2} > \infty

. A well-known result of Jensen asserts that the associated (Jensen) polynomials
\[

g
n

(
x
)
=

∑

k
=
0

n

(

n

k

)

a
k

x
k

{g_n}(x) = \sum \limits _{k = 0}^n {\left ( {\begin {array}{*{20}{c}} n \\ k \\ \end {array} } \right ){a_k}{x^k}}

\]
have only real zeros. Here we present an elementary proof of this fact; we also show that the zeros of

g
n

(
x
)

{g_n}(x)

are simple.
Identities for conjugation in the Steenrod algebra
Authors: Straffin Philip D. 
Published: June 2010

Let

χ
\chi

be the canonical conjugation in the Steenrod algebra

A

2

{\mathcal {A}_2}

. I prove the identity
\[

S

q

2
n

+
χ
(
S

q

2
n

)
=
S

q

2

n
−
1

χ
(
S

q

2

n
−
1

)

S{q^{{2^n}}} + \chi (S{q^{{2^n}}}) = S{q^{{2^{n - 1}}}}\chi (S{q^{{2^{n - 1}}}})

\]
and generalizations of this identity both in

A

2

{\mathcal {A}_2}

and in

A

p

{\mathcal {A}_p}

where

p
p

is an odd prime.
On nonisomorphic Room squares
Authors: Dinitz J. H. , Stinson D. R. 
Published: June 2010

Let

NR

(
s
)

{\text {NR}}(s)

denote the number of nonisomorphic Room squares of side

s
s

. We prove that for

s
s

sufficiently large,

NR

(
s
)
⩾
exp
⁡
(
c

s
2

)

{\text {NR}}(s) \geqslant \exp (c{s^2})

for some absolute constant

c
c

. More precisely,

NR

(
s
)
⩾
.19
exp
⁡
(
.04

s
2

)

{\text {NR}}(s) \geqslant .19\exp (.04{s^2})

for

s
⩾
153

odd

s \geqslant 153\;{\operatorname {odd}}

; and

NR

(
s
)
⩾
.19
exp
⁡
(
.09

s
2

)

{\text {NR}}(s) \geqslant .19\exp (.09{s^2})

for

s
⩾
1001

odd

s \geqslant 1001{\text { odd }}

.
A note on the stable homotopy groups of 𝑀𝑆𝑝(𝑛)
Authors: Imaoka Mitsunori 
Published: June 2010

The kernel of the epimorphism

π

8
n
+
3

S

(
M
S
p
(
n
)
)
→

π

4
n
+
3

(
M
S
p
)

\pi _{8n + 3}^S(MSp(n)) \to {\pi _{4n + 3}}(MSp)

is a cyclic group of order

8
m
(
n
+
1
)

8m(n + 1)

for some integer

m
(
n
+
1
)

m(n + 1)

defined using the characteristic numbers of the symplectic cobordism classes, and this epimorphism splits for some

n
n

.
Shifted plane partitions of trapezoidal shape
Authors: Proctor Robert A. 
Published: June 2010

The number of shifted plane partitions contained in the shifted shape

[
p
+
q
−
1
,
p
+
q
−
3
,
…
,
p
−
q
+
1
]

[p + q - 1,p + q - 3, \ldots ,p - q + 1]

with part size bounded by

m
m

is shown to be equal to the number of ordinary plane partitions contained in the shape

(
p
,
p
,
…
,
p
)

(p,p, \ldots ,p)

(
q

rows

)

(q{\text { rows}})

with part size bounded by

m
m

. The proof uses known combinatorial descriptions of finite-dimensional representations of semisimple Lie algebras. A separate simpler argument shows that the number of chains of cardinality

k
k

in the poset underlying the shifted plane partitions is equal to the number of chains of cardinality

k
k

in the poset underlying the ordinary plane partitions. The first result can also be formulated as an equality of chain counts for a pair of posets. The pair of posets is obtained by taking order ideals in the other pair of posets.
Phragmén-Lindelöf theorem in a cohomological form
Authors: Lin Ching Her 
Published: June 2010

The main result of this paper is as follows. Given functons

ϕ
1

(
ε
)
,
…
,

ϕ
ν

(
ε
)

{\phi _1}(\varepsilon ), \ldots ,{\phi _\nu }(\varepsilon )

which are holomorphic in sectors

S
1

,
…
,

S
p

{S_1}, \ldots ,{S_p}

, respectively, where

S
1

∪
⋯
∪

S
ν

=
{
ε
:

|

arg
⁡
ε

|

>
π

/

2
α
,
0

|

ε

|

>
ρ
}

{S_1} \cup \cdots \cup {S_\nu } = \{ \varepsilon :|\arg \varepsilon | > \pi /2\alpha ,0|\varepsilon | > \rho \}

for

α
>
1

\alpha > 1

,

ρ
>
0

\rho > 0

, set

ϕ

j
k

=

ϕ
j

−

ϕ
k

{\phi _{jk}} = {\phi _j} - {\phi _k}

if

S
j

∩

S
k

≠
∅

{S_j} \cap {S_k} \ne \emptyset

. Then

{

ϕ

j
k

}

\{ {\phi _{jk}}\}

satisfy cocycle conditions

ϕ

j
k

+

ϕ

k
l

=

ϕ

j
l

{\phi _{jk}} + {\phi _{kl}} = {\phi _{jl}}

whenever

S
j

∩

S
k

∩

S
l

≠
∅

{S_j} \cap {S_k} \cap {S_l} \ne \emptyset

. In addition to the conditions

|

ϕ
1

|

>

M
0

|{\phi _1}| > {M_0}

and

|

ϕ
ν

|

>

M
0

|{\phi _\nu }| > {M_0}

on the two rays of the boundary

(

i

.e

.

arg
⁡
ε
=
π

/

2
α
)

({\text {i}}{\text {.e}}{\text {. }}\arg \varepsilon = \pi /2\alpha )

, and

|

ϕ
j

(
ε
)

|

⩽
A
exp
⁡
(
c

/

|

ε

|

)

|{\phi _j}(\varepsilon )| \leqslant A\exp (c/|\varepsilon |)

in

S
j

{S_j}

for some positive numbers

A
A

and

c
c

,

j
=
1
,
2
,
…
,
ν

j = 1,2, \ldots ,\nu

, if the

{

ϕ
j

}

\{ {\phi _j}\}

satisfy the conditions

{

ϕ

j
k

}
>

M
0

\{ {\phi _{jk}}\} > {M_0}

on

S
j

∩

S
k

(
≠
∅
)

{S_j} \cap {S_k}( \ne \emptyset )

, then we get

|

ϕ
j

|

>
M

|{\phi _j}| > M

on

S
S

,

j
=
1
,
2
,
…
,
ν

j = 1,2, \ldots ,\nu

. (From the cohomological point of view, we can get global results for

ϕ
j

{\phi _j}

,oce the local data on cocycles is known.)
Stability of isometries on Banach spaces
Authors: Gevirtz Julian 
Published: June 2010

Let

X
X

and

Y
Y

be Banach spaces. A mapping

f
:
X
→
Y

f:X \to Y

is called an

ε
\varepsilon

-isometry if

|

‖

f
(

x
0

)
−
f
(

x
1

)

‖

−

‖

x
0

−

x
1

‖

|

⩽
ε

|\left \| {f({x_0}) - f({x_1})} \right \| - \left \| {{x_0} - {x_1}} \right \|| \leqslant \varepsilon

for all

x
0

,

x
1

∈
X

{x_0},{x_1} \in X

. It is shown that there exist constants

A
A

and

B
B

such that if

f
:
X
→
Y

f:X \to Y

is a surjective

ε
\varepsilon

-isometry, then

‖

f
(
(

x
0

+

x
1

)

/

2
)
−
(
f
(

x
0

)
+
f
(

x
1

)
)

/

2

‖

⩽
A

(
ε

‖

x
0

−

x
1

‖

)

1

/

2

+
B
ε

\left \| {f(({x_0} + {x_1})/2) - (f({x_0}) + f({x_1}))/2} \right \| \leqslant A{(\varepsilon \left \| {{x_0} - {x_1}} \right \|)^{1/2}} + B\varepsilon

for all

x
0

,

x
1

∈
X

{x_0},{x_1} \in X

. This, together with a result of Peter M. Gruber, is used to show that if

f
:
X
→
Y

f:X \to Y

is a surjective

ε
\varepsilon

-isometry, then there exists a surjective isometry

I
:
X
→
Y

I:X \to Y

for which

‖

f
(
x
)
−
I
(
x
)

‖

⩽
5
ε

\left \| {f(x) - I(x)} \right \| \leqslant 5\varepsilon

, thus answering a question of Hyers and Ulam about the stability of isometries on Banach spaces.
Immersions of highly connected manifolds
Authors: Wiegmann Reinhard 
Published: June 2010

If

[

M
n

↬

R

2
n
−
k

]

[{M^n}\looparrowright {{\mathbf {R}}^{2n - k}}]

denotes the set of regular homotopy classes of immersions,

M
n

{M^n}

a

k
k

-connected compact manifold, we show by a direct geometric construction the correspondence

[

M
n

↬

R

2
n
−
k

]
→
[

S
n

↬

R

2
n
−
k

]

[{M^n}\looparrowright {{\bf {R}}^{2n - k}}] \to [{S^n}\looparrowright {{\bf {R}}^{2n - k}}]

for

0
≤
2
k
≤
n
−
2

0 \le 2k \le n - 2

.
A formula for Ramanujan’s tau function
Authors: Ewell John A. 
Published: June 2010

A formula for Ramanujan’s tau function

τ
\tau

, defined by

∑
1
∞

τ
(
n
)

x
n

=

x

∏
1
∞

(
1
−

x
n

)

24

(

|
x
|

>
1
)

\sum \nolimits _1^\infty {\tau (n){x^n} = } x\prod _1^\infty {(1 - {x^n})^{24}}(\left | x \right | > 1)

, is presented. The author then observes that some of the known congruence properties of

τ
\tau

are immediate consequences of this formula representation.
Extreme points of a class of subordinate functions
Authors: Sheil-Small T. 
Published: June 2010

It is shown that, if

F

(
z
)

F\left ( z \right )

is subordinate to

H

(

(

1
+
z

)

/

(

1
−
z

)

)

H\left ( {\left ( {1 + z} \right ) / \left ( {1 - z} \right )} \right )

in the unit disc, where

H

(
w
)

H\left ( w \right )

is a quadratic polynomial for which

H
′

(
w
)

≠
0

H’\left ( w \right ) \ne 0

in the right half-plane, then
\[

F

(
z
)

=

∫
T

H

(

1
+
z

e

−
i
t

1
−
z

e

−
i
t

)

d
μ

F\left ( z \right ) = \int \limits _T {H\left ( {\frac {{1 + z{e^{ - it}}}}{{1 - z{e^{ - it}}}}} \right )d\mu }

\]
for a suitable probability measure

μ
\mu

on the unit circle

T
T

.
Constructing prime-field planar configurations
Authors: Gordon Gary 
Published: June 2010

An infinite class of planar configurations is constructed with distinct prime-field characteristic sets (i.e., configurations represented over a finite set of prime fields but over fields of no other characteristic). It is shown that if

p
p

is sufficiently large, then every subset of

k
k

primes between

p
p

and

f
(
p
,
k
)

f(p,k)

forms such a set (where

f
(
p
,
k
)
=

2

[
(

p

−
A

k

3

/

2

)

/

B

k

3

/

2

]

f(p,k) = {2^{[(\sqrt p - A{k^{3/2}})/B{k^{3/2}}]}}

for constants

A
A

and

B
B

). In particular, for every positive integer

k
k

, there exist infinitely many planar matroid configurations

C

i
,
k

{C_{i,k}}

with

|

χ

p
f

(

C

i
,
k

)

|

=
k

\left | {{\chi _{pf}}({C_{i,k}})} \right | = k

(where

χ

p
f

(
C
)

{\chi _{pf}}(C)

denotes the prime-field characteristic set of

C
C

). We also give a result concerning cofinite prime-field characteristic sets.
Convergence of Fourier series expansion related to free groups
Authors: Kuhn G. 
Published: June 2010

In

[
0
,
π
]

[0,\pi ]

we consider the complete orthogonal system

P
n

{P_n}

associated to the weight function

ψ
=
r
(
2
r
−
1
)

π

−
1

s
i

n

2

θ

(

r
2

−
(
2
r
−
1
)

c
o

s

2

θ

)

−
1

\psi = r(2r - 1){\pi ^{ - 1}}{\rm {si}}{{\rm {n}}^2}\theta {({r^2} - (2r - 1){\rm {co}}{{\rm {s}}^2}\theta )^{ - 1}}

and we study mean and pointwise convergence of series expansions with respect to the system

P
n

{P_n}

in

L
p

(
[
0
,
π
]
,
d
ψ
)

{L^p}([0,\pi ],d\psi )

. This weight function, and the corresponding system

P
n

{P_n}

arise from the study of Gelfand transforms of radial functions on a finitely generated free group

F
r

{F_r}

and our results can be interpreted in terms of multipliers theory on

F
r

{F_r}

.
Characterizing the topology of infinite-dimensional 𝜎-compact manifolds
Authors: Mogilski Jerzy 
Published: June 2010

A metric space

(
X
,
d
)

(X,d)

, which is a countable union of finite-dimensional compacta, is a manifold modelled on the space

l
2
f

=
{
(

x
i

)
∈

l
2

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

all but finitely many

x
i

=
0
}

{x_i} = 0\}

iff

X
X

is an ANR and the following condition holds: given

ε
>
0

\varepsilon > 0

, a pair of finite-dimensional compacta

(
A
,
B
)

(A,B)

and a map

f
:
A
→
X

f:A \to X

such that

f

|

B

f|B

is an embedding, there is an embedding

g
:
A
→
X

g:A \to X

such that

g

|

B
=
f

|

B

g\left | {B = f} \right |B

and

d
(
f
(
x
)
,
g
(
x
)
)
>
ε

d(f(x),g(x)) > \varepsilon

for all

x
∈
A

x \in A

. An analogous condition characterizes manifolds modelled on the space

Σ
=
{
(

x
i

)
∈

l
2

:

∑

i
=
1

∞

(
i

x
i

)
2

>
∞
}

\Sigma = \{ ({x_i}) \in {l_2}:\sum _{i = 1}^\infty {(i{x_i})^2} > \infty \}

.
On the multiplicities of the powers of a Banach space operator
Authors: Herrero Domingo A. 
Published: June 2010

The multiplicities of the powers of a bounded linear operator

T
T

, acting on a complex separable infinite-dimensional Banach space

X

\mathfrak {X}

, satisfy the inequalities
\[

(
∗
∗
)

μ
(

T
n

)
⩽
μ
(

T

h
n

)
⩽
h
μ
(

T
n

)

for

all

h
,
n
⩾
1.

( * * )\qquad \mu ({T^n}) \leqslant \mu ({T^{hn}}) \leqslant h\mu ({T^n})\quad {\text {for}}\;{\text {all}}\;h,n \geqslant 1.

\]
Nothing else can be said, in general, because simple examples show that for each sequence

{

μ
n

}

n
=
1

∞

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

, satisfying the inequalities

(
∗
∗
)

( * * )

, there exists

T
T

acting on

X

\mathfrak {X}

such that

μ
(

T
n

)
=

μ
n

\mu ({T^n}) = {\mu _n}

for all

n
⩾
1

n \geqslant 1

.
On the Riesz-transforms along surfaces in 𝑅³
Authors: El Kohen A. 
Published: June 2010

For

x
=
(

x
1

,

x
2

,

x
3

)

x = ({x_1},{x_2},{x_3})

in

R
3

{R^3}

,

t
=
(

t
1

,

t
2

)

t = ({t_1},{t_2})

in

R
2

{R^2}

,

|

t

|

=

(

t
1
2

+

t
2
2

)

1

/

2

|t| = {(t_1^2 + t_2^2)^{1/2}}

and

a
>
0

a > 0

, we define
\[

R
a

f
(
x
)
=

∫

R
2

f
(

x
1

−

t
1

,

x
2

−

t
2

,

x
3

−

|

t

|

a

)

t
1

|

t

|

3

d
t
.

{R_a}f(x) = \int _{{R^2}} {f({x_1} - {t_1},{x_2} - {t_2},{x_3} - |t{|^a})\frac {{{t_1}}}{{|t{|^3}}}dt.}

\]
The transformation

R
a

{R_a}

can be thought of as a Riesz-transform along the surface

(

t
1

,

t
2

,

|

t

|

a

)

({t_1},{t_2},|t{|^a})

in

R
3

{R^3}

. Our purpose here is to show that, for all

a
>
0

a > 0

, the operator

R
a

{R_a}

is bounded on

L
p

{L^p}

for

6

/

5
>
p
>
6

6/5 > p > 6

. Singular integrals along lower dimensional nonaffine varieties have been studied by several authors. See for example [3, 5, 6]. In the proof of our result, we use techniques similar to the ones used in [1, 2].
Pettis integrability and the equality of the norms of the weak* integral and the Dunford integral
Authors: Bator Elizabeth M. 
Published: June 2010

If

(
Ω
,
∑
,
μ
)

(\Omega ,\sum ,\mu )

is a perfect finite measure space and

X
X

is a Banach space, then it is shown that

X
∗

{X^ * }

has the

μ
\mu

-Pettis Integral Property if and only if
\[

‖

(

wea

k

∗

)
−

∫
Ω

f
d
μ

‖

=

‖

(

Dunford

)
−

∫
Ω

f
d
μ

‖

\left \| {({\text {wea}}{{\text {k}}^ * }) - \int \limits _\Omega {fd\mu } } \right \| = \left \| {({\text {Dunford}}) - \int \limits _\Omega {fd\mu } } \right \|

\]
for every bounded weakly measurable function

f
:
Ω
→

X
∗

f:\Omega \to {X^ * }

.
Weight spaces of Lie algebra modules
Authors: Smith Martha K. 
Published: June 2010

Let

V
V

be a finite-dimensional module for the finite-dimensional Lie algebra

L
L

over a field of characteristic zero. If

V
λ

=
{
v
∈
V

|

all

x
∈
L
,

[
x
−
λ
(
x
)

]
i

v
=
0

for some

}

{V^\lambda } = \{ v \in V|\;{\text {all }}x \in L,{[x - \lambda (x)]^i}v = 0\;{\text {for some }}\}

is nonzero, then

λ
∈

L
∗

\lambda \in {L^*}

and is a character of

L
L

. Moreover, the corresponding eigenspace

{
v
∈
V

|

all

x
∈
L
,
x
v
=
λ
(
x
)
v
}

\{ v \in V|{\text {all }}x \in L,xv = \lambda (x)v\}

is nonzero and

V
λ

{V^\lambda }

is an

L
L

submodule of

V
V

.
Generalization of two results of the theory of uniform distribution
Authors: Proĭnov Petko D. 
Published: June 2010

For a sequence

x
1

,
…
,

x
N

{x_1}, \ldots ,{x_N}

of points in

[
0
,
1
]

[0,1]

and a sequence

p
1

,
…
,

p
N

(

p
1

+

p
2

+
⋯
+

p
N

=
1
)

{p_1}, \ldots ,{p_N}({p_1} + {p_2} + \cdots + {p_N} = 1)

of nonnegative numbers, define the distribution function
\[

g
(
x
)
=
x
−

∑

x
k

>
x

p
k

.

g(x) = x - \sum \limits _{{x_k} > x} {{p_k}.}

\]
Let

φ
\varphi

be an increasing function on

[
0
,
1
]

[0,1]

and

φ
(
0
)
=
0

\varphi (0) = 0

. The main result of the paper is
\[

F
(

D
N

)
⩽

∫
0
1

φ

(

|

g
(
x
)

|

)

d
x
⩽
φ
(

D
N

)
,

F({D_N}) \leqslant \int _0^1 {\varphi \left ( {\left | {g(x)} \right |} \right )} dx \leqslant \varphi ({D_N}),

\]
where

D
N

{D_N}

is the supremum norm of

g
g

on

[
0
,
1
]

[0,1]

and

F
F

is the antiderivative of

φ
\varphi

with

F
(
0
)
=
0

F(0) = 0

. This result generalizes and improves an estimate of Niederreiter [1] for the

L
2

{L^2}

discrepancy of the sequence

x
1

,
…
,

x
N

{x_1}, \ldots ,{x_N}

. Applying the above inequality we also obtain a new criterion for uniform distribution modulo one.
Algebraic numbers and topologically equivalent measures in the Cantor set
Authors: Huang K. J. 
Published: June 2010

It is known that the transcendental and rational numbers in the unit interval are not binomial numbers. In this article we will show that the algebraic integers of degree 2 are not binomial numbers either. Therefore two shift invariant measures

u
(
s
)
,
u
(
r
)

u(s),u(r)

with

r
r

being an algebraic integer of degree 2 in the unit interval are topologically equivalent if and only if

s
=
r

s = r

or

s
=
1
−
r

s = 1 - r

. We also show that for each positive integer

n

> 2

n{\text { > 2}}

, there are algebraic integers and fractionals of degree

n
n

in the unit interval that are binomial numbers.
Integral representation without additivity
Authors: Schmeidler David 
Published: June 2010

Let

I
I

be a norm-continuous functional on the space

B
B

of bounded

Σ
\Sigma

-measurable real valued functions on a set

S
S

, where

Σ
\Sigma

is an algebra of subsets of

S
S

. Define a set function

v
v

on

Σ
\Sigma

by:

v
(
E
)

v (E)

equals the value of

I
I

at the indicator function of

E
E

. For each

a
a

in

B
B

let
\[

J
(
a
)
=

∫

−
∞

0

(
v
(
a
≥
α
)
−
v
(
S
)
)
d
α
+

∫
0
∞

v
(
a
≥
α
)
d
α
.

J(a) = \int _{ - \infty }^0 {(v (a \geq \alpha ) - v (S))d\alpha + \int _0^\infty {v (a \geq \alpha )d\alpha .} }

\]
Then

I
=
J

I = J

on

B
B

if and only if

I
(
b
+
c
)
=
I
(
b
)
+
I
(
c
)

I(b + c) = I(b) + I(c)

whenever

(
b
(
s
)
−
b
(
t
)
)
(
c
(
s
)
−
c
(
t
)
)
⩾
0

(b(s) - b(t))(c(s) - c(t)) \geqslant 0

for all

s
s

and

t
t

in

S
S

.
On the indecomposable elements of the bar construction
Authors: Hain Richard M. 
Published: June 2010

An explicit formula for a canonical splitting

s
:
Q

B

(

E

⋅

)
→

B

(

E

⋅

)

s:Q\mathcal {B}({\mathcal {E}^ \cdot }) \to \mathcal {B}({\mathcal {E}^ \cdot })

of the projection

B

(

E

⋅

)
→
Q

B

(

E

⋅

)

\mathcal {B}({\mathcal {E}^ \cdot }) \to Q\mathcal {B}({\mathcal {E}^ \cdot })

of the bar construction on a commutative d.g. algebra

E

⋅

{\mathcal {E}^\cdot }

onto its indecomposables is given. We prove that

s
s

induces a d.g. algebra isomorphism

Λ
(
Q

B

(

E

⋅

)
)
→

B

(

E

⋅

)

\Lambda (Q\mathcal {B}({\mathcal {E}^\cdot })) \to \mathcal {B}({\mathcal {E}^\cdot })

and that

H
(
Q

B

(

E

⋅

)
)

H(Q\mathcal {B}({\mathcal {E}^\cdot }))

is isomorphic with

Q
H
(

B

(

E

⋅

)
)

QH(\mathcal {B}({\mathcal {E}^\cdot }))

.
Morrey space
Authors: Zorko Cristina T. 
Published: June 2010

For

1
≤
p
>
∞

1 \leq p > \infty

,

Ω
\Omega

an open and bounded subset of

R
n

{R^n}

, and a nonincreasing and nonnegative function

φ
\varphi

defined in

(
0
,

ρ
0

]

(0,{\rho _0}]

,

ρ
0

=
diam
⁡
Ω

{\rho _0} = \operatorname {diam} \Omega

, we introduce the space

M

φ
,
0

p

(
Ω
)

\mathcal {M}_{\varphi ,0}^p(\Omega )

of locally integrable functions satisfying
\[

inf

c
∈
C

{

∫

B
(

x
0

,
ρ
)
∩
Ω

|

f
(
x
)
−
c

|

p

d
x

}

≤
A

|

B
(

x
0

,
ρ
)

|

φ
p

(
ρ
)

{\inf _{c \in C}}\left \{ {\int \limits _{B({x_0},\rho ) \cap \Omega } {|f(x) - c{|^p}dx} } \right \} \leq A|B({x_0},\rho )|{\varphi ^p}(\rho )

\]
for every

x
0

∈
Ω

{x_0} \in \Omega

and

0
>
ρ
≤

ρ
0

0 > \rho \leq {\rho _0}

, where

|

B
(

x
0

,
ρ
)

|

|B({x_0},\rho )|

denotes the volume of the ball centered in

x
0

{x_0}

and radius

ρ
\rho

. The constant

A
>
0

A > 0

does not depend on

(

x
0

,
ρ
)

({x_0},\rho )

. (i) We list some results on the structure, regularity, and density properties of the space so defined. (ii)

M

φ
,
0

p

\mathcal {M}_{\varphi ,0}^p

is represented as the dual of an atomic space.
A mean ergodic theorem in Banach spaces
Authors: Yoshimoto Takeshi 
Published: June 2010

Let

Γ
=

{

U
t

:
t
∈
Λ

}

(

Λ
=

Z

+

−

{
0
}

or

R

+

−

{
0
}

)

\Gamma = \left \{ {{U_t}:t \in \Lambda } \right \}\left ( {\Lambda = {{\mathbf {Z}}^ + } - \left \{ 0 \right \}{\text {or }}{{\mathbf {R}}^ + } - \left \{ 0 \right \}} \right )

be a commuting family of nonexpansive affine operators in a Banach space

X
X

satisfying the following conditions: (i) there is a function

M

(

x

|
Γ

)

≥
0

M\left ( {x\left | \Gamma \right .} \right ) \geq 0

defined on

X
X

such that
\[

‖

U

t
+
s

x
−

U
t

U
s

x

‖

≤
M

(

x

|
Γ

)

(

s
,
t
∈
Λ
,
x
∈
X

)

,

\left \| {{U_{t + s}}x - {U_t}{U_s}x} \right \| \leq M\left ( {x\left | \Gamma \right .} \right )\quad \left ( {s,t \in \Lambda ,x \in X} \right ),

\]
(ii)
\[

sup

{

t

−
1

‖

U
t

x

‖

:
t
∈
Λ
,
t
≥
1

}

=
K

(

x

|
Γ

)

>
∞

(

x
∈
X

)

.

\sup \left \{ {{t^{ - 1}}\left \| {{U_t}x} \right \|:t \in \Lambda ,t \geq 1} \right \} = K\left ( {x\left | \Gamma \right .} \right ) > \infty \left ( {x \in X} \right ).

\]
Then it is proved that if

{

t

−
1

U
t

x
:
t
∈
Λ

}

\left \{ {{t^{ - 1}}{U_t}x:t \in \Lambda } \right \}

is relatively compact for

x
∈
X

x \in X

, the limit

X
−

lim

t
→
∞

t

−
1

U
t

x
=

x
¯

X -\lim _{t \to \infty }{t^{ - 1}}{U_t}x = \bar x

exists in

X
X

and

U
t

x

¯

=

x
¯

(

t
∈
Λ

)

\overline {{U_t}x} = \overline x \left ( {t \in \Lambda } \right )

.
A Steinhaus type theorem
Authors: Natarajan P. N. 
Published: June 2010

The sequence space

Λ
r

,
r
≥
1

{\Lambda _r},r \geq 1

being a fixed integer, is defined as
\[

Λ
r

=

{

x
=

{

x
k

}

∈

l
∞

,

x
k

∈
K
,
k
=
0
,
1
,
2
,
…
,

|

x

k
+
r

−

x
k

|

→
0
,
k
→
∞

}

,

{\Lambda _r} = \left \{ {x = \left \{ {{x_k}} \right \} \in {l_\infty },{x_k} \in K,k = 0,1,2, \ldots ,\left | {{x_{k + r}} - {x_k}} \right | \to 0,k \to \infty } \right \},

\]
where

K
K

is a complete, nontrivially valued field and

l
∞

{l_\infty }

is the space of bounded sequences with entries in

K
K

. In this paper, it is proved that given a regular matrix

A
=
(

a

n
k

)
,

a

n
k

∈
K
=

R

A = ({a_{nk}}),{a_{nk}} \in K = {\mathbf {R}}

or

C

{\mathbf {C}}

, there exists a sequence in

Λ
r

−

∪

i
=
1

r
−
1

Λ
i

{\Lambda _r} - \cup _{i = 1}^{r - 1}{\Lambda _i}

which is not

A
A

-summable. This is an improvement of the well-known Steinhaus theorem. It is, however, shown that this result fails to hold when

K
K

is a complete, nontrivially valued, nonarchimedean field, whereas it is known that the Steinhaus theorem continues to hold.
Stability of solutions of linear delay differential equations
Authors: Kulenović M. R. S. , Ladas G. , Meimaridou A. 
Published: June 2010

Consider the linear differential equation
\[

(
1
)

x
˙

(
t
)
=

∑

i
=
1

n

p
i

(
t
)
x
(
t
−

τ
i

)

=
0
,

t
⩾

t
0

,

(1)\quad \dot x(t) = \sum \limits _{i = 1}^n {{p_i}(t)x(t - {\tau _i})} = 0,\quad t \geqslant {t_0},

\]
where

p
i

∈
C
(
[

t
0

,
∞
)
,

R

)

{p_i} \in C([{t_0},\infty ),{\mathbf {R}})

and

τ
i

⩾
0

{\tau _i} \geqslant 0

for

i
=
1
,
2
,
…
,
n

i = 1,2, \ldots ,n

. By investigating the asymptotic behavior first of the nonoscillatory solutions of (1) and then of the oscillatory solutions we are led to new sufficient conditions for the asymptotic stability of the trivial solution of (1). When the coefficients of (1) are all of the same sign, we obtain a comparison result which shows that the nonoscillatory solutions of (1) dominate the growth of the oscillatory solutions.