bluuebunny/crossref_metadata_2025_split huggingface dataset:
The 𝑙₁-completion of a metric combinatorial ∞-manifold
Authors: Sakai Katsuro 
Published: June 2010

Let

K
K

be a simplicial complex. The realization

|
K
|

\left | K \right |

of

K
K

admits the metric
\[

d
1

(
x
,
y
)
=

∑

υ
∈

K
0

|

x
(
υ
)
−
y
(
υ
)

|

,

{d_1}(x,y) = \sum \limits _{\upsilon \in {K^0}} {\left | {x(\upsilon ) - y(\upsilon )} \right |,}

\]
where

x
(
υ
)

x(\upsilon )

and

y
(
υ
)
,
υ
∈

K
0

y(\upsilon ),\upsilon \in {K^0}

, are the barycentric coordinates of

x
x

and

y
y

respectively. The completion of the metric space

(

|
K
|

,

d
1

)

(\left | K \right |,{d_1})

is called the

l
1

{l_1}

-completion and is denoted by

|

K

|

¯

l
1

{\overline {|K|} ^{{l_1}}}

. In this paper, we prove that

|

K

|

¯

l
1

{\overline {|K|} ^{{l_1}}}

is an

l
2

{l_2}

-manifold if and only if

K
K

is a combinatorial

∞
\infty

-manifold.
Denting points in tensor products of Banach spaces
Authors: Werner Dirk 
Published: June 2010

Let dent

A
A

denote the set of denting points of a subset

A
A

of some Banach space. We prove
\[

dent cl co(

K
⊗
L
)
=

dent

K
⊗

dent

L

{\text {dent cl co(}}K \otimes L) = {\text {dent }}K \otimes {\text {dent }}L

\]
for closed, bounded, absolutely convex subsets

K
K

and

L
L

of Banach spaces

X
X

and

Y
Y

. Here the closure refers to the completion of

X
⊗
Y

X \otimes Y

w.r.t. some reasonable crossnorm.
Small perturbations and the eigenvalues of the Laplacian on large bounded domains
Authors: Kirsch Werner 
Published: June 2010

Denote by

Δ
L
D

\Delta _L^D

the Laplacian on a hypercube in

R

d

{{\mathbf {R}}^d}

with side length

π
L

\pi L

. Also denote by

N

(

λ
,
A

)

N\left ( {\lambda ,A} \right )

the number of eigenvalues of the operator

A
A

below

λ
\lambda

. If

V
≥
0

V \geq 0

is a bounded function of compact support, (

V
>
0

V > 0

on a set of positive measure) then

N

(

−

Δ
L
D

,
λ

)

−
N

(

−

Δ
L
D

+
V
,
λ

)

N\left ( { - \Delta _L^D,\lambda } \right ) - N\left ( { - \Delta _L^D + V,\lambda } \right )

is not bounded as

L
→
∞

L \to \infty

for dimension

d
>
1

d > 1

.
On an inequality of Osgood, Phillips and Sarnak
Authors: Widom Harold 
Published: June 2010

In their work [4,5] on isospectral families for the Laplacian Osgood, Phillips and Sarnak needed and proved the following inequality:For any real-valued functionϕ\phibelonging to the Sobolev spaceH1/2(=W1/2,2){H_{1/2}}( = {W^{1/2,2}})of the unit circle and satisfying the side condition∫eϕeiθdθ=0\int {{e^\phi }{e^{i\theta }}d\theta = 0}one has\[log⁡12π∫eϕdθ−12π∫ϕdθ≤12∑k=1∞k|ϕ^(k)|2\log \frac {1} {{2\pi }}\int {{e^\phi }d\theta - \frac {1} {{2\pi }}\int \phi \,d\theta \leq \frac {1} {2}\sum \limits _{k = 1}^\infty {k|\hat \phi } (k){|^2}}\]whereϕ^(k)\hat \phi (k)is thekkth Fourier coefficient ofϕ\phi. Without the side condition the factor12\frac {1} {2}does not appear on the right side and the inequality is the first Lebedev-Milin inequality [1, §5.1]. The purpose of this note is to show that these are the first two of a series of inequalities which follow quickly from some theorems of G. Szegö on Toeplitz determinants.
The integrability of generalized Garrett-Stanojević sums
Authors: Singh Niranjan , Sharma K. M. 
Published: June 2010

In this paper we have defined the generalized Garrett-Stanojević cosine sums
\[

h
n

(
x
)

=

∑

p
=
0

n

S
p

r
−
1

Δ
r

a
p

{h_n}\left ( x \right ) = \sum \limits _{p = 0}^n {S_p^{r - 1}{\Delta ^r}{a_p}}

\]
and have proved that under suitable conditions

h
n

→
h

{h_n} \to h

in the

L
1

{L^1}

-norm, where

h

(
x
)

=

a
0

/

2
+

∑

n
=
1

∞

a
n

cos
⁡
n
x

h\left ( x \right ) = {a_0}/2 + \sum \nolimits _{n = 1}^\infty {{a_n}\cos nx}

. If

r
=
1

r = 1

, then

h
n

(
x
)

{h_n}\left ( x \right )

reduces to the modified cosine sums introduced by Rees and Stanojević.
A note on extreme points of subordination classes
Authors: Hallenbeck D. J. 
Published: June 2010

Let

s
(
F
)

s(F)

denote the set of functions subordinate to a univalent function

F
F

in

Δ
\Delta

in the unit disc. Let

B
B

denote the set of functions

ϕ
(
z
)

\phi (z)

analytic in

Δ
\Delta

satisfying

|

ϕ
(
x
)

|

>
1

|{\phi (x)}| > 1

and

ϕ
(
0
)
=
0

\phi (0) = 0

. Let

D
=
F
(
Δ
)

D = F(\Delta )

and

λ
(
w
,
∂
D
)

\lambda (w,\partial D)

denote the distance between

w
w

and

∂
D

\partial D

(boundary of

D
D

). We prove that if

ϕ
\phi

is an extreme point of

B
B

then

∫
0

2
π

log
⁡
λ
(
F
(
ϕ
(

e

i
t

)
)
,
∂
D
)
d
t
=
−
∞

\int _0^{2\pi } {\log \lambda (F(\phi ({e^{it}})),\partial D)dt = - \infty }

. As a corollary we prove that if

F
∘
ϕ

F \circ \phi

is an extreme point of

s
(
F
)

s(F)

then

∫
0

2
π

log
⁡
λ
(
F
(
ϕ
(

e

i
t

)
)
,
∂
D
)
d
t
=
−
∞

\int _0^{2\pi } {\log \lambda (F(\phi ({e^{it}})),\partial D)dt = - \infty }

.
Singular integrals in product domains and the method of rotations
Authors: Krug Donald 
Published: June 2010

Singular integrals with kernels of the form

K
(
x
,
y
)

K(x,y)

where

K
K

satisfies conditions to be a bounded singular integral operator in each of its variables have been much studied lately. In this paper we use the classical method of rotations to give a proof that kernels of the form

K
(
x
,
y
)
=
Ω
(
x
,
y
)

/

|

x

|

n

|

y

|

m

K(x,y) = \Omega (x,y)/|x{|^n}|y{|^m}

correspond to bounded singular integral operators.
Simultaneous systems of representatives for finite families of finite sets
Authors: Jia Xing De 
Published: June 2010

Let

h
≥
2

h \geq 2

and

k
≥
1

k \geq 1

. It is proved that if

S

=
{

S
i

}

i
=
1

s

\mathcal {S} = \{ {S_i}\} _{i = 1}^s

and

T

=
{

T
j

}

j
=
1

t

\mathcal {T} = \{ {T_j}\} _{j = 1}^t

are two families of nonempty, pairwise disjoint sets such that

|

S
i

|

≤
h
,

|

T
j

|

≤
k

|{S_i}| \leq h,|{T_j}| \leq k

and

S
i

⊈

T
j

{S_i} \nsubseteq {T_j}

for all

i
i

and

j
j

, then the number

N
(

S

,

T

)

N(\mathcal {S},\mathcal {T})

of the sets

X
X

such that

X
X

is a minimal system of representatives for

S

\mathcal {S}

and

X
X

is simultaneously a system of representatives for

T

\mathcal {T}

that satisfies

N
(

S

,

T

)
≤

h
s

(
1
−
(
h
−
r
)

/

h

q
+
1

)
t

N(\mathcal {S},\mathcal {T}) \leq {h^s}{(1 - (h - r)/{h^{q + 1}})^t}

, where

k
=
q
(
h
−
1
)
+
r

k = q(h - 1) + r

with

0
≤
r
≤
h
−
2

0 \leq r \leq h - 2

. This was conjectured by M. B. Nathanson [3] in 1985.
Inverses of generators
Authors: deLaubenfels Ralph 
Published: June 2010

Let

A
A

be a (possibly unbounded) linear operator on a Banach space

X
X

that generates a bounded holomorphic semigroup of angle

θ
(
0
>
θ
≤
π

/

2
)

\theta (0 > \theta \leq \pi /2)

. We show that, if the range of

A
A

is dense, then

A
A

is one-to-one, and

A

−
1

{A^{ - 1}}

(defined on the range of

A
A

) generates a bounded holomorphic semigroup of angle

θ
\theta

, given by
\[

e

z

A

−
1

=
∫

e

−
w

(
w
A
+
z
)

−
1

d
w

2
π
i

,

{e^{z{A^{ - 1}}}} = \int {{e^{ - w}}{{(wA + z)}^{ - 1}}\frac {{dw}}{{2\pi i}},}

\]
over an appropriate curve. When

X
X

is reflexive, it is sufficient that

A
A

be one-to-one.
Commensurate sequences of characters
Authors: Pełczyński A. 
Published: June 2010

If

(

a
j

)

({a_j})

and

(

b
j

)

({b_j})

are sequences of characters on compact abelian groups

S
S

and

T
T

respectively such that for every sequence of scalars

(

α
j

)

|

|

∑

α
j

a
j

|

|

∞

≍

|

|

∑

α
j

b
j

|

|

∞

({\alpha _j})||\sum {\alpha _j}{a_j}|{|_\infty } \asymp ||\sum {\alpha _j}{b_j}|{|_\infty }

tnen for every

1
≤
p
>
∞

1 \leq p > \infty

and every sequence

(

x
j

)

({x_j})

of elements of an arbitrary Banach space

X
X

\[

∫
S

‖

∑

x
j

a
j

‖

p

d
s
≍

∫
T

‖

∑

x
j

b

‖

p

d
t
.

{\int _S {\left \| {\sum {{x_j}{a_j}} } \right \|} ^p}ds \asymp {\int _T {\left \| {\sum {{x_j}b} } \right \|} ^p}dt.

\]
This result generalizes a result of Pisier [Pi 1] for Sidon sets. For topological Sidon sets on

R

{\mathbf {R}}

a slightly stronger result holds.
Some consequences of the standard polynomial
Authors: Chang Qing 
Published: June 2010

The standard polynomial of degree

m
m

is the polynomial

∑

{

sign(

ρ

)

x

ρ
(
1
)

x

ρ
(
2
)

⋯

x

ρ
(
m
)

|

ρ
∈

S
m

}

\sum {\{ {\text {sign(}}\rho {\text {)}}{x_{\rho (1)}}{x_{\rho (2)}} \cdots {x_{\rho (m)}}|\rho \in {S_m}} \}

, where

S
m

{S_m}

is the symmetric group on

m
m

letters. We show that the polynomial
\[

∑

{

sign(

ρ
σ

)

x

ρ
(
1
)

y

σ
(
1
)

x

ρ
(
2
)

y

σ
(
2
)

⋯

x

ρ
(
m
)

y

σ
(
m
)

|

ρ
,
σ
∈

S
m

}

\sum {\{ {\text {sign(}}\rho \sigma {\text {)}}{x_{\rho (1)}}{y_{\sigma (1)}}{x_{\rho (2)}}{y_{\sigma (2)}} \cdots {x_{\rho (m)}}{y_{\sigma (m)}}|\rho ,\sigma \in {S_m}\} }

\]
is a consequence of the standard polynomial of degree

m
m

. We also show that certain polynomials of the form

∑
{

sign(

ρ

)

x

ρ
(
1
)

x

ρ
(
2
)

⋯

x

ρ
(
n
)

|

ρ
∈
Q
}

\sum \{ {\text {sign(}}\rho {\text {)}}{x_{\rho (1)}}{x_{\rho (2)}} \cdots {x_{\rho (n)}}|\rho \in Q\}

, where

n
≥
m

n \geq m

and

Q
Q

is a suitable subset of

S
n

{S_n}

, are consequences of the standard polynomial of degree

m
m

.
An inequality for some nonnormal operators
Authors: Furuta Takayuki 
Published: June 2010

An inequality of use in testing convergence of eigenvector calculations is improved. If

e
λ

{e_\lambda }

is a unit eigenvector corresponding to an eigenvalue

λ
\lambda

of a dominant operator

A
A

on a Hilbert space

H
H

, then
\[

|

(
g
,

e
λ

)

|

2

≤

|

|

g

|

|

2

|

|

A
g

|

|

2

−

|

(
g
,
A
g
)

|

2

|

|

(
A
−
λ
I
)
g

|

|

2

|(g,{e_\lambda }){|^2} \leq \frac {{||g|{|^2}||Ag|{|^2} - |(g,Ag){|^2}}}{{||(A - \lambda I)g|{|^2}}}

\]
for all

g
g

in

H
H

for which

A
g
≠
λ
g

Ag \ne \lambda g

. The equality holds if and only if the component of

g
g

orthogonal to

e
λ

{e_\lambda }

is also an eigenvector of

A
A

. This result is an improvement of Bernstein’s result for selfadjoint operators.
A note on Roe’s characterization of the sine function
Authors: Howard Ralph 
Published: June 2010

Let

f

(
n
)

n
=
0
,
±
1
,
±
2
,
…

{f^{(n)}}n = 0, \pm 1, \pm 2, \ldots

be a sequence of complex valued functions on the real line with

(
d

/

d
x
)

f

(
n
)

=

f

(
n
+
1
)

(d/dx){f^{(n)}} = {f^{(n + 1)}}

and satisfying inequalities

|

f

(
n
)

(
x
)

|

≤

M
n

(
1
+

|

x

|

)
k

|{f^{(n)}}(x)| \leq {M_n}{(1 + |x|)^k}

where as

n
→
∞

n \to \infty

the growth conditions

lim
_

M
n

(
1
+
ε

)

−
n

=
0

\underline {\lim } {M_n}{(1 + \varepsilon )^{ - n}} = 0

and

lim
_

M

−
n

(
1
+
ε

)

−
n

=
0

\underline \lim {M_{ - n}}{(1 + \varepsilon )^{ - n}} = 0

hold for all

ε
>
0

\varepsilon > 0

. Then

f

(
0
)

(
x
)
=
p
(
x
)

e

i
x

+
q
(
x
)

e

−
i
x

{f^{(0)}}(x) = p(x){e^{ix}} + q(x){e^{ - ix}}

where

p
p

and

q
q

are polynomials of degree at most

k
k

.
Finiteness of index and total scalar curvature for minimal hypersurfaces
Authors: Tysk Johan 
Published: June 2010

Let

M
n

,
n
≥
3

{M^n},n \geq 3

, be an oriented minimally immersed complete hypersurface in Euclidean space. We show that for

n
=
3
,
4
,
5
,

or

6

n = 3,4,5,{\text { or }}6

, the index of

M
n

{M^n}

is finite if and only if the total scalar curvature of

M
n

{M^n}

is finite, provided that the volume growth of

M
n

{M^n}

is bounded by a constant times

r
n

{r^n}

, where

r
r

is the Euclidean distance function. We also note that this result does not hold for

n
≥
8

n \geq 8

. Moreover, we show that the index of

M
n

{M^n}

is bounded by a constant multiple of the total scalar curvature for all

n
≥
3

n \geq 3

, without any assumptions on the volume growth of

M
n

{M^n}

.
Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater
Authors: Martín-Reyes F. J. , Sawyer E. 
Published: June 2010

A simple characterization is given for two-weight norm inequalities for generalized Hardy operators

T
φ

f
(
x
)
=

∫
0
x

φ
(

t
x

)
f
(
t
)
d
t

{T_\varphi }f(x) = \smallint _0^x\varphi (\tfrac {t}{x})f(t)dt

, where

φ
:
(
0
,
1
)
→
(
0
,
∞
)

\varphi :(0,1) \to (0,\infty )

is nonincreasing and satisfies

φ
(
a
b
)
≤
D
[
φ
(
a
)
+
φ
(
b
)
]

\varphi (ab) \leq D[\varphi (a) + \varphi (b)]

for

0
>
a
,
b
>
1

0 > a,b > 1

. Included in particular are the Riemann-Liouville fractional integrals.
Isometries of spaces of weak * continuous functions
Authors: Cambern Michael , Jarosz Krzysztof 
Published: June 2010

If

X
X

is a compact Hausdorff space and

E
∗

{E^*}

a Banach dual, we denote by

C
(
X
,
(

E
∗

,

σ
∗

)
)

C(X,({E^*},{\sigma ^*}))

the Banach space of continuous functions from

X
X

to

E
∗

{E^*}

when the latter space is given its weak * topology. It is shown that if

E
1
∗

,

E
2
∗

E_1^*,E_2^*

have trivial centralizers and satisfy a topological condition introduced by Namioka and Phelps, then, given an isometry

T
T

mapping

C
(

X
1
∗

,
(

E
1

,

σ
∗

)
)

C(X_1^*,({E_1},{\sigma ^*}))

onto

C
(

X
2

,
(

E
2
∗

,

σ
∗

)
)

C({X_2},(E_2^*,{\sigma ^*}))

and given

F
∈
C
(

X
1

,
(

E
1
∗

,

σ
∗

)
)

F \in C({X_1},(E_1^*,{\sigma ^*}))

, there exists a dense

G
δ

{G_\delta }

in

X
2

{X_2}

on which
\[

(
∗
)

(
T
F
)
(
x
)
=
U
(
x
)
F
∘
Φ
(
x
)
,

(*)\quad (TF)(x) = U(x)F \circ \Phi (x),

\]
where

Φ
\Phi

is a homeomorphism of

X
2

{X_2}

onto

X
1

{X_1}

and

U
(
⋅
)

U( \cdot )

an operator-valued function independent of

F
F

. If the

E
i
∗

E_i^*

are separable then the

U
(
x
)

U(x)

are surjective isometries and

x
→
U
(
x
)

x \to U(x)

is continuous. When the

X
i

{X_i}

are metric the representation (*) holds on all of

X
2

{X_2}

.
The Cauchy transform on bounded domains
Authors: Anderson J. M. , Hinkkanen A. 
Published: June 2010

Suppose that

f
f

is in

L
2

(
Δ
)

{L^2}(\Delta )

where

Δ
\Delta

is the unit disk, and that

f
=
0

f = 0

outside

Δ
\Delta

. We show that then the Cauchy transform

C

f

\mathcal {C}\,f

of

f
f

, when restricted to

Δ
\Delta

, satisfies

|

|

C

f

|

|

2

≤
(
2

/

α
)

|

|

f

|

|

2

||\mathcal {C}\,f|{|_2} \leq (2/\alpha )||f|{|_2}

, where

α
≈
2.4048

\alpha \approx 2.4048

is the smallest positive zero of the Bessel function

J
0

{J_0}

. This inequality is sharp.
Weighted weak-type (1,1) inequalities for rough operators
Authors: Hofmann Steve 
Published: June 2010

Let

Ω
\Omega

be homogeneous of degree 0, have mean value 0 on the circle, and belong to

L
q

(

S
1

)

,
1
>
q
≤
∞

{L^q}\left ( {{S^1}} \right ),1 > q \leq \infty

. Then the two-dimensional operator defined by
\[

T
f

(
x
)

=

pv

∫

Ω

(
y
)

|
y
|

−
2

f

(

x
−
y

)

d
y

Tf\left ( x \right ) = {\text {pv}}\int {\Omega \left ( y \right ){{\left | y \right |}^{ - 2}}} f\left ( {x - y} \right )dy

\]
is shown to be of weak-type

(
1
,
1
)

(1,1)

with respect to the weighted measures

|
x
|

α

d
x

{\left | x \right |^\alpha }dx

, if

−
2
+
1

/

q
>
α
>
0

- 2 + 1/q > \alpha > 0

. Under the weaker assumption that

Ω
\Omega

belongs to

L

log
+

L

(

S
1

)

L\,{\log ^ + }L\left ( {{S^1}} \right )

, the same result holds if

−
1
>
α
>
0

- 1 > \alpha > 0

. Similar results are also obtained for the related maximal operator
\[

M
Ω

f

(
x
)

=

sup

r
>
0

r

−
2

∫

|

y

|

>
r

|

Ω

(
y
)

f

(

x
−
y

)

|

d
y
.

{M_\Omega }f\left ( x \right ) = \sup \limits _{r > 0} {r^{ - 2}}\int _{|y| > r} {\left | {\Omega \left ( y \right )f\left ( {x - y} \right )} \right |dy.}

\]
On rings for which homogeneous maps are linear
Authors: Fuchs P. , Maxson C. J. , Pilz G. 
Published: June 2010

Let

R
R

be the collection of all rings

R
R

such that for every

R
R

-module

G
G

, the centralizer near-ring

M
R

(
G
)
=
{
f
:
G
→
G

|

f
(
r
x
)
=
r
f
(
x
)
,
r
∈
R
,
x
∈
G
}

{M_R}(G) = \{ f:G \to G|f(rx) = rf(x),r \in R,x \in G\}

is a ring. We show

R
∈
R

R \in R

if and only if

M
R

(
G
)
=

En

d

R

(
G
)

{M_R}(G) = {\text {En}}{{\text {d}}_R}(G)

for each

R
R

-module

G
G

. Further information about

R
R

is collected and the Artinian rings in

R
R

are completely characterized.
On solutions of linear differential equations with real zeros; proof of a conjecture of Hellerstein and Rossi
Authors: Brüggemann Franz 
Published: June 2010

We prove the following conjecture that is due to Hellerstein and Rossi: Let

{

w
1

,
…
,

w
n

}

\left \{ {{w_1}, \ldots ,{w_n}} \right \}

be a fundamental system of
\[

L
w
=

w

(
n
)

+

a

n
−
1

(
z
)

w

(
n
−
1
)

+
⋯
+

a
0

(
z
)
w
≡
0

Lw = {w^{(n)}} + {a_{n - 1}}(z){w^{(n - 1)}} + \cdots + {a_0}(z)w \equiv 0

\]
with polynomials

a
j

(
z
)
(
0
≤
j
≤
n
−
1
)

{a_j}(z)(0 \leq j \leq n - 1)

. If each

w
k

(
1
≤
k
≤
n
)

{w_k}(1 \leq k \leq n)

has only finitely many nonreal zeros, then there exists a polynomial

q
(
z
)

q(z)

such that

u
k

:=
exp
⁡
(
q
(
z
)
)

w
k

(
1
≤
k
≤
n
)

{u_k}: = \exp (q(z)){w_k}(1 \leq k \leq n)

form a fundamental system of a homogeneous linear differential equation with constant coefficients.
A geometric interpretation of Segal’s inequality ‖𝑒^{𝑋+𝑌}‖≤‖𝑒^{𝑋/2}𝑒^{𝑌}𝑒^{𝑋/2}‖
Authors: Corach G. , Porta H. , Recht L. 
Published: June 2010

It is shown that the exponential mapping of the manifold of positive elements of a

C
∗

{C^*}

-algebra (provided with its natural connection) increases distances (when measured in the natural Finsler structure). The proof relies on Segal’s inequality

|

|

e

X
+
Y

|

|

≤

|

|

e

X

/

2

e
Y

e

X

/

2

|

|

||{e^{X + Y}}|| \leq ||{e^{X/2}}{e^Y}{e^{X/2}}||

, valid for all symmetric

X
,
Y

X,Y

in any

C
∗

{C^*}

-algebra. In turn, this geometric inequality implies Segal’s’ inequality.
Oscillation criteria for second order nonlinear differential equations with integrable coefficients
Authors: Wong James S. W. 
Published: June 2010

Consider the second order nonlinear differential equation

y
+
a

(
t
)

f

(
y
)

=
0

y + a\left ( t \right )f\left ( y \right ) = 0

where

a

(
t
)

∈
C
[
0
,
∞
)
,
f

(
y
)

∈

C
1

(

−
∞
,
∞

)

,

f
′

(
y
)

≥
0

a\left ( t \right ) \in C[0,\infty ),f\left ( y \right ) \in {C^1}\left ( { - \infty ,\infty } \right ),f’\left ( y \right ) \geq 0

, and

y
f

(
y
)

>
0

yf\left ( y \right ) > 0

for

y

≠

0

{\text {y}} \ne {\text {0}}

. Furthermore,

f

(
y
)

f\left ( y \right )

also satisfies either a superlinear or a sublinear condition, which covers the prototype nonlinear function

f

(
y
)

=

|

y

|

γ

sgn
⁡
y

f\left ( y \right ) = |y{|^\gamma }\operatorname {sgn} y

with

γ

> 1

\gamma {\text { > 1}}

and

0
>
γ
>
1

0 > \gamma > 1

respectively. The coefficient

a

(
t
)

a\left ( t \right )

is allowed to be negative for arbitrarily large values of

t
t

and is integrable in the sense that the improper interval

∫
t
∞

a

(
s
)

d
s

=
A

(
t
)

\int _t^\infty {a\left ( s \right )ds} = A\left ( t \right )

exists for each

t
≥
0

t \geq 0

. Oscillation criteria involving integrals of

A

(
t
)

A\left ( t \right )

due to Coles and Butler for the superlinear and sublinear cases are shown to remain valid without the additional hypothesis that

A

(
t
)

≥
0

A\left ( t \right ) \geq 0

.
On a problem of Nirenberg concerning expanding maps in Hilbert space
Authors: Szczepański Janusz 
Published: June 2010

Let

H

{\mathbf {H}}

be a Hilbert space and

f
:

H

→

H

f:{\mathbf {H}} \to {\mathbf {H}}

a continuous map which is expanding (i.e.,

|

|

f
(

x

)
−
f
(

y

)

|

|

≥

|

|

x

−

y

|

|

||f({\mathbf {x}}) - f({\mathbf {y}})|| \geq ||{\mathbf {x}} - {\mathbf {y}}||

for all

x

,

y

∈

H

{\mathbf {x}},{\mathbf {y}} \in {\mathbf {H}}

) and such that

f
(

H

)

f({\mathbf {H}})

has nonempty interior. Are these conditions sufficient to ensure that

f
f

is onto? This question was stated by Nirenberg in 1974. In this paper we give a partial negative answer to this problem; namely, we present an example of a map

F
:

H

→

H

F:{\mathbf {H}} \to {\mathbf {H}}

which is not onto, continuous,

F
(

H

)

F({\mathbf {H}})

has nonempty interior, and for every

x

,

y

∈

H

{\mathbf {x}},{\mathbf {y}} \in {\mathbf {H}}

there is

n
0

∈

N

{n_0} \in \mathbb {N}

(depending on

x

{\mathbf {x}}

and

y

{\mathbf {y}}

) such that for every

n
≥

n
0

n \geq {n_0}

\[

|

|

F
n

(

x

)
−

F
n

(

y

)

|

|

≥

c

n
−
m

|

|

x

−

y

|

|

||{F^n}({\mathbf {x}}) - {F^n}({\mathbf {y}})|| \geq {c^{n - m}}||{\mathbf {x}} - {\mathbf {y}}||

\]
where

F
n

{F^n}

is the

n
n

th iterate of the map

F
,
c

F,c

is a constant greater than 2, and

m
m

is an integer depending on

x

{\mathbf {x}}

and

y

{\mathbf {y}}

. Our example satisfies

|

|

F
(

x

)

|

|

=
c

|

|

x

|

|

||F({\mathbf {x}})|| = c||{\mathbf {x}}||

for all

x

∈

H

{\mathbf {x}} \in {\mathbf {H}}

. We show that no map with the above properties exists in the finite-dimensional case.
Asymptotic behavior of a linear delay difference equation
Authors: Driver R. D. , Ladas G. , Vlahos P. N. 
Published: June 2010

Consider the linear delay difference equations\[xn+1−xn=∑j=1maj(xn−kj−xn−lj),n=0,1,2,…{x_{n + 1}} - {x_n} = \sum \limits _{j = 1}^m {{a_j}({x_{n - {k_j}}} - {x_{n - {l_j}}}),\quad n = 0,1,2, \ldots }\]and\[yn+1−yn=∑j=1kbjyn−j,n=0,1,2,…,{y_{n + 1}} - {y_n} = \sum \limits _{j = 1}^k {{b_j}{y_{n - j}},\quad n = 0,1,2, \ldots ,}\]where the coefficientsaj{a_j}andbj{b_j}are real andkj{k_j}andlj{l_j}are nonnegative integers. In this note we describe, in terms of the initial conditions, the asymptotic behavior of solutions of these equations in several cases when the characteristic equation has a dominant real root. Some of the results extend to systems of equations.
Refinements of Ky Fan’s inequality
Authors: Alzer Horst 
Published: June 2010

We prove the inequalities
\[

A
n
′

/

G
n
′

⩽
(
1
−

G
n
′

)

/

(
1
−

A
n
′

)
⩽

A
n

/

G
n

A_n’ /G_n’ \leqslant (1 - G_n’ )/(1 - A_n’ ) \leqslant {A_n}/{G_n}

\]
and
\[

A
n
′

/

G
n
′

⩽
(
1
−

G
n

)

/

(
1
−

A
n

)
⩽

A
n

/

G
n

,

A_n’ /G_n’ \leqslant (1 - {G_n})/(1 - {A_n}) \leqslant {A_n}/{G_n},

\]
where

A
n

{A_n}

and

G
n

{G_n}

(respectively,

A
n
′

A_n’

and

G
n
′

G_n’

) denote the unweighted arithmetic and geometric means of

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

(respectively,

1
−

x
1

,
…
,

1
−

x
n

1 - {x_1}, \ldots ,\;1 - {x_n}

) with

x
i

∈
(
0
,

1
2

]
(
i
=
1
,
…
,
n
;
n
⩾
2

{x_i} \in (0,\tfrac {1} {2}](i = 1, \ldots ,n;n \geqslant 2

. Further we show that the ratios

(
1
−

G
n
′

)

/

(
1
−

A
n
′

)

(1 - G_n’ )/(1 - A_n’)

and

(
1
−

G
n

)

/

(
1
−

A
n

)

(1 - {G_n})/(1 - {A_n})

can be compared if and only if

n
=
2

n = 2

.
Some deformations of the Hopf foliation are also Kähler
Authors: Scofield Paul D. 
Published: June 2010

Fix

α
=
(

α
0

,
…
,

α
n

)
∈

R

˚

n
+
1

\alpha = ({\alpha _0}, \ldots ,{\alpha _n}) \in \mathring {\mathbf {R}}^{n + 1}

. The trajectories of the flow on

S

2
n
+
1

⊂

C

n
+
1

{{\mathbf {S}}^{2n + 1}} \subset {{\mathbf {C}}^{n + 1}}

given by
\[

ϕ
t

:
(

z
0

,
…
,

z
n

)
↦
(

z
0

e

i

α
0

t

,
…
,

z
n

e

i

α
n

t

)

{\phi _t}:({z_0}, \ldots ,{z_n}) \mapsto ({z_0}{e^{i{\alpha _0}t}}, \ldots ,{z_n}{e^{i{\alpha _n}t}})

\]
constitute the leaves of a

2
n

2n

-codimensional (nonsingular) foliation of

S

2
n
+
1

{{\mathbf {S}}^{2n + 1}}

. We use (locally defined) branches of the logarithm to give this foliation a (global) transverse Kähler structure.