bluuebunny/crossref_metadata_2025_split huggingface dataset:
Spherical harmonics generating bounded biharmonics
Authors: Beaver Bradley , Sario Leo , Wang Cecilia 
Published: June 2010

Let

H
2

B
(
R
)

{H^2}B(R)

be the family of bounded nonharmonic biharmonic functions on a Riemannian manifold

R
R

. On the punctured Euclidean

N
N

-space

E
0
N

=
{
x
=
(

x
1

,
…
,

x
N

)

|

0
>

|

x

|

>
∞
}

E_0^N = \{ x = ({x^1}, \ldots ,{x^N})|0 > |x| > \infty \}

,

H
2

B

{H^2}B

is void for

N
>
3

N > 3

, whereas for

N
=
2
,
3

N = 2,3

, it is generated by certain fundamental spherical harmonics. It is also known that

H
2

B

{H^2}B

remains void on the Riemannian manifold

E
α
N

E_\alpha ^N

,

N
>
3

N > 3

, obtained by endowing

E
0
N

E_0^N

with the non-Euclidean metric

d

s
α

=

r
α

|

d
x

|

d{s_\alpha } = {r^\alpha }|dx|

,

α
∈
R

\alpha \in R

. The purpose of the present paper is to show that the fundamental spherical harmonics continue generating

H
2

B
(

E
α
3

)

{H^2}B(E_\alpha ^3)

, despite the distorting metric

d

s
α

d{s_\alpha }

. An analogous result holds for

E
α
2

E_\alpha ^2

.
On holomorphic functions satisfying 𝑓(𝑧)(1-𝑧²)≤1 in the unit disc
Authors: Wirths Karl-Joachim 
Published: June 2010

Let

f
f

be holomorphic in

D
=
{

z
|

|
z
|

>
1
}

D = \{ \left . z \right |\left | z \right | > 1\}

,

|

f
(
z
)

|

(
1
−

|
z
|

2

)
⩽
1

\left | {f(z)} \right |(1 - {\left | z \right |^2}) \leqslant 1

in

D
D

,

lim
¯

|
z
|

→
1

|

f
(
z
)

|

(
1
−

|
z
|

2

)
>
1

{\overline {\lim } _{\left | z \right | \to 1}}\left | {f(z)} \right |(1 - {\left | z \right |^2}) > 1

and

L
(
f
)
:=
{

z
|

|

f
(
z
)

|

(
1
−

|
z
|

2

)
=
1
}

L(f): = \{ \left . z \right |\left | {f(z)} \right |(1 - {\left | z \right |^2}) = 1\}

. It is shown that the set

L
(
f
)

L(f)

consists of one simple closed curve

γ
\gamma

and a finite number of points in the bounded component of

C

∖
γ

{\mathbf {C}}\backslash \gamma

if

L
(
f
)

L(f)

is an infinite set.
Nomographic functions are nowhere dense
Authors: Buck R. Creighton 
Published: June 2010

A function

f
f

of

n
n

variables is nomographic if it can be represented in the format
\[

f
(

x
1

,
…
,

x
n

)
=
h
(

ϕ
1

(

x
1

)
+
⋯
+

ϕ
n

(

x
n

)
)

f({x_1}, \ldots ,{x_n}) = h({\phi _1}({x_1}) + \cdots + {\phi _n}({x_n}))

\]
where the

ϕ
i

{\phi _i}

and

h
h

are continuous. Every continuous function of

n
n

variables has a representation as a sum of not more than

2
n
+
1

2n + 1

nomographic functions [textbf9]. This paser gives a constructive proof that the nomographic functions form a nowhere dense subset of the space

C
[

I
n

]

C[{I^n}]

.
Perturbations preserving asymptotics of spectrum with remainder
Authors: Ramm A. G. 
Published: June 2010

Let

A
A

be a compact linear operator on a Hilbert space

H
H

,

s
n

(
A
)
=

λ
n

1

/

2

(

A
∗

A
)

{s_n}(A) = \lambda _n^{1/2}({A^* }A)

,

Q
Q

be a linear operator,

|

Q
f

|

⩽
c

|

A
f

|

a

|

f

|

1
−
a

|Qf| \leqslant c|Af{|^a}|f{|^{1 - a}}

,

a
>
0

a > 0

,

c
>
0

c > 0

,

∀
f
∈
H

\forall f \in H

. Let
\[

s
n

(
A
)
=
c

n

−
r

{
1
+
O
(

n

−
q

)
}
,

r
,
q
>
0
,
B
=
A
(
I
+
Q
)
.

{s_n}(A) = c{n^{ - r}}\{ 1 + O({n^{ - q}})\} ,\quad r,q > 0,B = A(I + Q).

\]
Then
\[

s
n

(
B
)
=

s
n

(
A
)
{
1
+
O
(

n

−
γ

)
}
,

γ
=
min

(

q
,

r
a

1
+
r
a

)

.

{s_n}(B) = {s_n}(A)\{ 1 + O({n^{ - \gamma }})\} ,\quad \gamma = \min \left ( {q,\frac {{ra}} {{1 + ra}}} \right ).

\]
Some applications of this result to the spectral theory of elliptic operators are given.
Certain Schur-Hadamard multipliers in the spaces 𝐶_{𝑝}
Authors: Arazy Jonathan 
Published: June 2010

Let

f
f

be a continuously differentiable function on

[
−
1
,
1
]

[ - 1,1]

satisfying

|

f
′

(
t
)

|

⩽
C

|
t
|

α

\left | {f’(t)} \right | \leqslant C{\left | t \right |^\alpha }

for some

0
>
C

0 > C

,

α
>
∞

\alpha > \infty

and all

−
1
⩽
t
⩽
1

- 1 \leqslant t \leqslant 1

, and let

λ
=
(

λ
i

)
∈

l
r

\lambda = ({\lambda _i}) \in {l_r}

satisfy

−
1
⩽

λ
i

⩽
1

- 1 \leqslant {\lambda _i} \leqslant 1

for all

i
i

. Then
\[

a

f
,
λ

=

(

f
(

λ
i

)
−
f
(

λ
j

)

λ
i

−

λ
j

)

{a_{f,\lambda }} = \left ( {\frac {{f({\lambda _i}) - f({\lambda _j})}} {{{\lambda _i} - {\lambda _j}}}} \right )

\]
is a Schur-Hadamard multiplier from

C

p
1

{C_{{p_1}}}

into

C

p
2

{C_{{p_2}}}

for all

p
1

{p_1}

,

p
2

{p_2}

satisfying

1
⩽

p
2

⩽
2
⩽

p
1

⩽
∞

1 \leqslant {p_2} \leqslant 2 \leqslant {p_1} \leqslant \infty

and

p
2

−
1

⩽

p
1

−
1

+
α

/

r

p_2^{ - 1} \leqslant p_1^{ - 1} + \alpha /r

.
Distance estimates for von Neumann algebras
Authors: Rosenoer Shlomo 
Published: June 2010

It is shown that for certain von Neumann algebras

A

\mathcal {A}

, there is a constant

C
C

such that
\[

dist

(
T
,

A

)
⩽
C

sup

P
∈

lat

A

‖

P
⊥

T
P

‖

for

all

T

in

B

(

H

)

.

{\text {dist}}(T, \mathcal {A}) \leqslant C\sup \limits _{P \in {\text {lat}}\mathcal {A}} \left \| {{P^ \bot }TP} \right \|\quad {\text {for}}\;{\text {all}}\;T\;{\text {in}}\;\mathcal {B}{\text {(}}\mathcal {H}{\text {)}}{\text {.}}

\]
On factorizations of selfadjoint ordinary differential operators
Authors: Granata Antonio 
Published: June 2010

Consider an ordinary linear differential operator

L
L

, of order

n
⩾
1

n \geqslant 1

, represented by

L
u
≡

a
n

(
t
)

u

(
n
)

+
⋯
+

a
0

(
t
)
u

∀
u
∈

C
n

(
a
,
b
)

Lu \equiv {a_n}(t){u^{(n)}} + \cdots + {a_0}(t)u\;\forall u \in {C^n}(a,b)

, with real-valued coefficients

a
k

∈

C
k

(
a
,
b
)

{a_k} \in {C^k}(a,b)

,

0
⩽
k
⩽
n

0 \leqslant k \leqslant n

,

a
n

≠
0

{a_n} \ne 0

on

(
a
,
b
)

(a,b)

. According to a classical result, if

L
L

is formally selfadjoint on

(
a
,
b
)

(a,b)

then it has a factorization of the type

L
u
≡

p
n

[

p

n
−
1

(
⋯
(

p
1

(

p
0

u

)
′

)
′

⋯

)
′

]
′

∀
u
∈

C
n

(
a
,
b
)

Lu \equiv {p_n}[{p_{n - 1}}( \cdots ({p_1}({p_0}u)’)’ \cdots )’]’\forall u \in {C^n}(a,b)

, where the

p
k

{p_k}

’s are sufficiently-smooth and everywhere nonzero functions on

(
a
,
b
)

(a,b)

such that

p
k

=

p

n
−
k

{p_k} = {p_{n - k}}

(
k
=
0
,
…
,
n
)

(k = 0, \ldots ,n)

. In this note we shall examine this result critically and show by means of counterexamples that the different classical proofs are either merely local or purely heuristic. A proof, which is both rigorous and global, is inferred from recent results on canonical factorizations of disconjugate operators. In addition, information is obtained on the behavior of the

p
k

{p_k}

’s at the endpoints of

(
a
,
b
)

(a,b)

which may prove useful in applications.
Discrete generalized Cesàro operators
Authors: Rhaly H. C. 
Published: June 2010

For

|
λ
|

⩽
1

\left | \lambda \right | \leqslant 1

,

A
λ
∗

A_\lambda ^*

is the operator defined formally on the Hardy space

H
2

{H^2}

by
\[

(

A
λ
∗

f
)
(
z
)
=
−

(
λ
−
z

)

−
1

∫
λ
z

f
(
s
)

d
s
,

|
z
|

>
1.

(A_\lambda ^*f)(z) = - {(\lambda - z)^{ - 1}}\int _\lambda ^z {f(s)\,} ds,\quad \left | z \right | > 1.

\]
If

λ
=
1

\lambda = 1

, then the usual identification of

H
2

{H^2}

with

l
2

{l^2}

takes

A
1

{A_1}

onto the discrete Cesàro operator. Here we answer questions about boundedness, spectra, unitary equivalence, compactness, and subnormality for the operators

A
λ

{A_\lambda }

.
Generic existence of a solution for a differential equation in a scale of Banach spaces
Authors: Benavides Tomás Domínguez 
Published: June 2010

Let

{

X
s

:
α
⩽
s
⩽
β

}

\left \{ {{X_s}:\alpha \leqslant s \leqslant \beta } \right \}

be a scale of Banach spaces,

J
J

a real interval,

U
U

an open subset of

J
×

X
s

J \times {X_s}

for some

s
s

. In this paper we prove that the existence of solutions for
\[

x
′

=
A
(
t
)
x
+
f
(
t
,
x
)
,

x
(

t
0

)
=

x
0

,

x’ = A(t)x + f(t,x),\quad x({t_0}) = {x_0},

\]
is a generic property, when

A
(
t
)

A(t)

is an operator satisfying
\[

|

A
(
t
)

|

L
(

X

s
′

;

X
s

)

⩽
M

(

s
′

−
s

)

−
1

(
M
>
0

independent

of

s
,

s
′

,
t
)

{\left | {A(t)} \right |_{L({X_{s’}};{X_s})}} \leqslant M{(s’ - s)^{ - 1}}\quad (M > 0\;{\text {independent}}\;{\text {of}}\;s,s’,t)

\]
in the scale

{

X
s

}

\left \{ {{X_s}} \right \}

and

f
:
J
×
U
→

X
β

f:J \times U \to {X_\beta }

is continuous.
A generalization of a theorem of Ayoub and Chowla
Authors: Redmond Don 
Published: June 2010

Let

X

1

\mathcal {X}1

and

X

2

\mathcal {X}2

be characters modulo

q
1

{q_1}

and

q
2

{q_2}

, respectively, where

q
1

{q_1}

and

q
2

{q_2}

are positive integers. Let
\[

f
(
n
)
=

∑

d

|

n

X

1
(
d
)

X

2
(
n

/

d
)
.

f(n) = \sum \limits _{d|n} \mathcal {X}1 (d)\mathcal {X}2(n/d).

\]
In this paper we shall give an estimate for the sum
\[

∑

n
⩽
x

f
(
n
)

log
⁡
(
x

/

n
)
.

\sum \limits _{n \leqslant x} {f(n)} \log (x/n).

\]
Inequalities of Gauss-Bonnet type for a convex domain
Authors: Dekster B. V. 
Published: June 2010

Let

N
N

be a compact convex

n
n

-dimensional Riemannian manifold with a boundary

∂
N

\partial N

having normal curvatures

⩾
κ
>
0

\geqslant \kappa > 0

. Suppose the sectional curvature

>
−

κ
2

> - {\kappa ^2}

in

N
N

. Let

H
H

be the integral mean curvature of

∂
N

\partial N

,

V
V

be the volume of

N
N

,

k

s
c

{k_{sc}}

be the scalar curvature and

k
¯

R

(
p
)

{\bar k_R}(p)

,

p
∈
N

p \in N

, be the maximum Ricci curvature at

p
p

. Then
\[

H
⩾

n
−
2

2

κ
2

V
−

1

2
(
n
−
1
)

∫
N

k

s
c

d
V

,

H
⩾
(
n
−
2
)

κ
2

V
−

1

n
−
1

∫
N

k
¯

R

d
V
.

H \geqslant \frac {{n - 2}} {2}{\kappa ^2}V - \frac {1} {{2(n - 1)}}\int _N {{k_{sc}}\;dV} ,\quad H \geqslant (n - 2){\kappa ^2}V - \frac {1} {{n - 1}}\int _N {{{\bar k}_R}\;dV.}

\]
Let

N
−

{N_ - }

denote

N
N

with nonpositive sectional curvature. Let

G
G

be the integral Gauss curvature of

∂

N
−

\partial {N_ - }

. Then

G
⩾
−

κ

n
−
2

∫

N
−

k
¯

R

d
V

G \geqslant - {\kappa ^{n - 2}}\int _{N - } {{{\bar k}_R}\;dV}

. These three estimates are sharp. For a ball in

3
3

-dimensional hyperbolic space, the ratio of the right-hand part of each estimate to its left-hand part (i.e.

V
(

κ
2

+
3
)

/

2
H

V({\kappa ^2} + 3)/2H

,

V
(

κ
2

+
1
)

/

H

V({\kappa ^2} + 1)/H

and

2
κ
V

/

G

2\kappa V/G

respectively) approaches 1 as the

radius

→
∞

{\operatorname {radius}} \to \infty

. The same ratios for the estimates
\[

H
⩾
−

1

2
(
n
−
1
)

∫
N

k

s
c

d
V

and

H
⩾
−

1

n
−
1

∫
N

k
¯

R

d
V

H \geqslant - \frac {1} {{2(n - 1)}}\int _N {{k_{sc}}\;} dV\quad {\text {and}}\quad H \geqslant - \frac {1} {{n - 1}}\int _N {{{\bar k}_R}\;dV}

\]
(rougher ones but without

κ
\kappa

) approach

3
4

\tfrac {3} {4}

and

1
2

\tfrac {1} {2}

respectively.
On a generalized moment problem
Authors: Hwang J. S. 
Published: June 2010

The well-known Müntz-Szász theorem asserts that the sequence of powers

x

n
p

{x^{{n_p}}}

is complete on

[
a
,
b
]

[a,b]

, where

a
⩾
0

a \geqslant 0

, if and only if (1)
\[

(
1
)

∑

p
=
1

∞

1

n
p

=
∞
,

where

0
>

n
1

>

n
2

>
⋯
.

(1)\quad \sum \limits _{p = 1}^\infty {\frac {1} {{{n_p}}} = \infty ,\quad {\text {where}}\;0 > {n_1} > {n_2} > \cdots .}

\]
Let

f
(
x
)

f(x)

be absolutely continuous,

|

f
′

(
x
)

|

⩾
k
>
0

\left | {f’(x)} \right | \geqslant k > 0

, and

f
(
a
)
f
(
b
)
⩾
0

f(a)f(b) \geqslant 0

. We prove that under the assumption (1) the sequence

{

f

(
x
)

n
p

}

\left \{ {f{{(x)}^{{n_p}}}} \right \}

is complete on

[
a
,
b
]

[a,b]

if and only if

f
(
x
)

f(x)

is monotone on

[
a
,
b
]

[a,b]

.
On the operator ranges of analytic functions
Authors: Hwang J. S. 
Published: June 2010

Following Doob, we say that a function

f
(
z
)

f(z)

analytic in the unit disk

U
U

has the property

K
(
ρ
)

K(\rho )

if

f
(
0
)
=
0

f(0) = 0

and for some arc

γ
\gamma

on the unit circle whose measure

|
γ
|

⩾
2
ρ
>
0

\left | \gamma \right | \geqslant 2\rho > 0

,
\[

lim

inf

j
→
∞

|

f
(

z
j

)

|

⩾
1

where

z
j

→
z
∈
γ

and

z
j

∈
U
.

\lim \inf \limits _{j \to \infty } \left | {f({z_j})} \right | \geqslant 1\quad {\text {where}}\;{z_j} \to z \in \gamma \;{\text {and}}\;{z_j} \in U.

\]
Let

H
H

be a Hilbert space over the complex field,

A
A

an operator whose spectrum is included in

U
U

,

|

|

A

|

|

|| A ||

the operator norm of

A
A

, and

f
(
A
)

f(A)

the usual Riesz-Dunford operator. We prove that there is no function with the property

K
(
ρ
)

K(\rho )

satisfying
\[

(
1
−

|

|

A

|

|

)

|

|

f
′

(
A
)

|

|

⩽
1

/

n

for

all

|

|

A

|

|

>
1
,

(1 - || A ||)|| {f’(A)} || \leqslant 1/n\quad {\text {for}}\;{\text {all}}\;|| A || > 1,

\]
where

n
>
N
(
ρ
)
=
log
⁡
(
1

/

(
1
−
cos
⁡
ρ
)
)

n > N(\rho ) = \log (1/(1 - \cos \rho ))

. We also show that if

f
f

has the property

K
(
ρ
)

K(\rho )

then the operator range of

f
(
A
)

f(A)

covers a ball of radius

k
(
ρ
)
=

3

/

(
4
N
(
ρ
)
)

k(\rho ) = \sqrt 3 /(4N(\rho ))

. These two results generalize our previous solutions of two long open problems of Doob [1]. Finally, we prove that the operator range of any

4
4

-fold univalent function is not convex. This extends our solution to Ky Fan’s Problem [4].
Proof of Scott’s conjecture
Authors: Svrtan D. 
Published: June 2010

We give a proof of Conjecture 7 in [2, p. 155] first stated in 1881 by R. F. Scott [4]. It reads as follows: Conjecture 7 (R. F. Scott). Let

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

and

y
1

,
…
,

y
n

{y_1}, \ldots ,{y_n}

be the distinct roots of

x
n

−
1
=
0

{x^n} - 1 = 0

and

y
n

+
1
=
0

{y^n} + 1 = 0

, respectively. Let

A
A

be the

n
×
n

n \times n

matrix whose

(
i
,
j
)

(i,j)

entry is

1

/

(

x
i

−

y
i

)
,
i
,
j
=
1
,
…
,
n

1/({x_i} - {y_i}),i,j = 1, \ldots ,n

. Then
\[

|

per
⁡
(
A
)

|

=

{

n

(
1
⋅
3
⋅
5
⋯
(
n
−
2
)
)

2

/

2

n

,

a
m
p
;

if

n

is

odd
,

0
,

a
m
p
;

if

n

is

even
⁡

.

\left | {\operatorname {per} (A)} \right | = \left \{ {\begin {array}{*{20}{c}} {n{{(1 \cdot 3 \cdot 5 \cdots (n - 2))}^2}/{2^{n}}}, & {{\text {if}}\;n\;{\text {is }}\operatorname {odd} ,} \\ {0,} & {{\text {if}}\;n\;{\text {is}}\;\operatorname {even} {\text {.}}} \\ \end {array} } \right .

\]
Actually, our proof gives more, namely an explicit expression for

per

(
A
)

{\operatorname {per}}(A)

(see Theorem 2.1).
Principal homogeneous spaces over Hensel rings
Authors: Strano Rosario 
Published: June 2010

We prove that if

(
A
,

a
_

)

(A,\underline {a})

is a Hensel couple and

G
G

is an affine, smooth group scheme over

A
A

then

H

e
t

1

(
A
,
G
)
=

H

e
t

1

(
A

/

a
_

,
G

/

a
_

G
)

H_{\mathrm {et}}^1 (A, G) = H_{\mathrm {et}}^1 (A/\underline {a}, G/\underline {a} G)

.
Length of ray-images under conformal maps
Authors: Karunakaran V. 
Published: June 2010

Let

w
=
f
(
z
)

w = f(z)

be regular and univalent in

|

z

|

>
1

|z| > 1

with

f
(
0
)
=
0

f(0) = 0

. Suppose that

f
f

maps the unit disc onto a domain

D
D

. Let

l
(
r
,
θ
)

l(r,\theta )

be the length of the image curve of the ray joining

z
=
0

z = 0

to

z
=
r

e

i
θ

z = r{e^{i\theta }}

in

D
D

and

A
(
r
)
=
Sup
⁡
[

|

f
(
r

e

i
θ

)

|

−
1

l
(
r
,
θ
)
]

A(r) = \operatorname {Sup}[{\left | {f(r{e^{i\theta }})} \right |^{ - 1}}l(r,\theta )]

where the supremum is taken over all starlike functions. In this paper we show that

A
(
r
)
⩽
(
1
+
r
)

A(r) \leqslant (1 + r)

.
𝐹_{𝑝} classes and hypergeometric series
Authors: Moak Daniel S. 
Published: June 2010

For

Re
⁡
(
p
)
⩾
0

\operatorname {Re} (p) \geqslant 0

, let

F
p

=

{

f
:
f
(
z
)
=
f

(
1
−
x
z

)

−
p

d
μ
(
x
)
,

|
z
|

>
1
,

μ

{F_p} = {\text { \{ }}f:f(z) = f{(1 - xz)^{ - p}}d\mu (x),\;\left | z \right | > 1,\;\mu

a probability measure on

|
x
|

=
1
}

\left | x \right | = 1\}

, and let

F
p

⋅

F
q

=

{

f
⋅
g
:
f

in
⁡

F
p

,
g

in

F
q

}

{F_p} \cdot {F_q} = \left \{ {f \cdot g:f\;\operatorname {in} {F_p},g\;{\text {in}}\;{F_q}} \right \}

. Brickman, Hallenbeck, MacGregor and Wilken proved that

p
>
0

p > 0

and

q
>
0

q > 0

, then

F
p

⋅

F
p

⊆

F

p
+
q

{F_p} \cdot {F_p} \subseteq {F_{p + q}}

. Kent Pearce recently proved a converse result: if

F
p

⋅

F
q

⊆

F

p
+
q

{F_p} \cdot {F_q} \subseteq {F_{p + q}}

, then

p
>
0

p > 0

and

q
>
0

q > 0

, or

p
=
q
=
1
+
i
t

p = q = 1 + it

for some real

t
t

. The case

p
=
q
=
1
+
i
t

p = q = 1 + it

,

t
≠
0

t \ne 0

, will be excluded. Consequently a full converse of the above theorem holds.
Fourier coefficients of continuous functions on compact groups
Authors: Heiman Barbara 
Published: June 2010

Let

G
G

be an infinite compact group with dual object

Σ
\Sigma

. Letting

K

σ

{\mathcal {K}_\sigma }

be the representation space for

σ
∈
Σ

\sigma \in \Sigma

,

E

2

(
Σ
)

{\mathcal {E}^2}(\Sigma )

is the set

{
A
=
(

A
σ

)
∈
Π

B

(

K

σ

)
:

‖
A
‖

2
2

=

∑
σ

d
σ

Tr
⁡
(

A
σ

A

σ
∗

)
>
∞
}

\{ A = ({A^\sigma }) \in \Pi \mathcal {B}({\mathcal {K}_\sigma }):\left \| A \right \|_2^2 = {\sum _\sigma }{d_\sigma }\operatorname {Tr} ({A^\sigma }{A^{\sigma *}}) > \infty \}

. For

A
∈

E

2

(
Σ
)

A \in {\mathcal {E}^2}(\Sigma )

, we show that there is a function

f
f

in

C
(
G
)

C(G)

such that

‖
f
‖

∞

⩽
C

‖
A
‖

2

{\left \| f \right \|_\infty } \leqslant C{\left \| A \right \|_2}

and

Tr
⁡
(

f
^

(
σ
)

f
^

(
σ

)
∗

)
⩾
Tr
⁡
(

A
σ

A

σ
∗

)

\operatorname {Tr} (\hat f(\sigma )\hat f{(\sigma )^*}) \geqslant \operatorname {Tr} ({A^\sigma }{A^{\sigma *}})

for every

σ
∈
Σ

\sigma \in \Sigma

.
A transcendence measure for some special values of elliptic functions
Authors: Tubbs Robert 
Published: June 2010

Among T. Schneider’s results is the following: Let

℘
(
z
)

\wp (z)

be the Weierstrass elliptic function with algebraic invariants. If

℘
(
u
)

\wp (u)

and

β
\beta

are both algebraic,

β
∉

K
τ

\beta \notin {K_\tau }

, then

℘
(
β
u
)

\wp (\beta u)

is transcendental. In this paper we provide a transcendence measure for this value. Let

P
(
X
)

P(X)

be a nonzero polynomial, with integral coefficients, of degree

d
d

and height

h
h

, and put

t
=
d
+
log
⁡
h

t = d + \log h

. Then there is an effectively computable constant

C
C

, which does not depend on

P
(
X
)

P(X)

. such that: (A) If

℘
(
z
)

\wp (z)

has complex multiplication then

log
⁡

|

P
(
℘
(
β
u
)
)

|

>
−
C

d
2

t
2

(
log
⁡
t

)
4

\log \left | {P(\wp (\beta u))} \right | > - C{d^2}{t^2}{(\log t)^4}

. (B) If

℘
(
z
)

\wp (z)

does not have complex multiplication then

log
⁡

|

P
(
℘
(
β
u
)
)

|

>
−
C

d
6

t
2

(
log
⁡
t

)

14

\log \left | {P(\wp (\beta u))} \right | > - C{d^6}{t^2}{(\log t)^{14}}

.
Analytic generators and the KMS condition
Authors: De Cannière Jean 
Published: June 2010

The KMS condition at arbitrary

β
\beta

for a state

ω
\omega

of a

C
∗

{C^ * }

-dynamical system

(
A
,

R

,
α
)

(A,{\mathbf {R}},\alpha )

is stated in terms of the analytic continuation of the correlation functions

t
↦
ω
(

x
∗

α
t

(
x
)
)
(
x
∈
A
)

in

−
i
(

or

+
i
)

t \mapsto \omega ({x^ * }{\alpha _t}(x))(x \in A){\text {in}} - i({\text {or}} + i)

. The precise formulation involves the analytic generator of

α
\alpha

(or its inverse).
A relation between pointwise convergence of functions and convergence of functionals
Authors: Brézis Haïm , Lieb Elliott 
Published: June 2010

We show that if

{

f
n

}

\left \{ {{f_n}} \right \}

is a sequence of uniformly

L
p

{L^p}

-bounded functions on a measure space, and if

f
n

→
f

{f_n} \to f

pointwise a.e., then

lim

n
→
∞

{

‖

f
n

‖

p
p

−

‖

f
n

−
f

‖

p
p

}

=

‖
f
‖

p
p

{\lim _{n \to \infty }}\left \{ {\left \| {{f_n}} \right \|_p^p - \left \| {{f_n} - f} \right \|_p^p} \right \} = \left \| f \right \|_p^p

for all

0
>
p
>
∞

0 > p > \infty

. This result is also generalized in Theorem 2 to some functionals other than the

L
p

{L^p}

norm, namely

∫

|

j
(

f
n

)
−
j
(

f
n

−
f
)
−
j
(
f
)

|

→
0

\int \left | {j({f_n}) - j({f_n} - f) - j(f)} \right | \to 0

for suitable

j
:

C

→

C

j:{\mathbf {C}} \to {\mathbf {C}}

and a suitable sequence

{

f
n

}

\left \{ {{f_n}} \right \}

. A brief discussion is given of the usefulness of this result in variational problems.
Inequalities for the derivative of a polynomial
Authors: Aziz Abdul 
Published: June 2010

Let

P
(
z
)
=

∑

j
=
0

n

a
j

z
j

=
c

∏

j
=
1

n

(
z
−

z
j

)

P(z) = \sum \nolimits _{j = 0}^n {{a_j}{z^j} = c\prod \nolimits _{j = 1}^n {(z - {z_j})} }

be a polynomial of degree

n
n

and

P
′

(
z
)

P’(z)

its derivative. In this paper we consider the problem of estimating the maximum of

|

P
′

(
z
)

|

\left | {P’(z)} \right |

on

|
z
|

=
1

\left | z \right | = 1

under some assumptions on the zeros or on the coefficients of

P
(
z
)

P(z)

and obtain certain sharp results.
A tower of spectra that realizes a chain complex
Authors: Suárez Pedro A. 
Published: June 2010

This paper presents the construction of a tower of spectra

Y
j

{Y_j}

with

k
k

-invariants coming from the relations

S

q

1

(
X

S

q

2

j
+
1

)

S

q

1

(
X

S

q

2
j

)
=
0

{\text {S}}{{\text {q}}^1}(X{\text {S}}{{\text {q}}^{{2^{j + 1}}}}){\text {S}}{{\text {q}}^1}(X{\text {S}}{{\text {q}}^{{2^j}}}) = 0

in

A

/

A

S

q

1

A / A{\text {S}}{{\text {q}}^1}

, for

0
⩽
j
⩽
5

0 \leqslant j \leqslant 5

and

A
A

= Steenrod algebra

mod

2

\mod 2

, such that

Y
5

{Y_5}

has prescribed homotopy groups:

π
n

(

Y
5

)
=
Z

{\pi _n}({Y_5}) = Z

(integers) if

n
=

2

j
+
1

−
2

n = {2^{j + 1}} - 2

, and zero otherwise.
Local uncertainty inequalities for Fourier series
Authors: Price John F. , Racki Paul C. 
Published: June 2010

Necessary and sufficient conditions are given on

α
\alpha

,

β
\beta

and

t
t

for there to exist a constant

K
K

such that
\[

(

∑

n
∈
E

|

f
^

(
n
)

|

2

)

1

/

2

⩽
K

|
E
|

α

‖

f

|
x
|

β

‖

t

{\left ( {{{\sum \limits _{n \in E} {\left | {\hat f(n)} \right |} }^2}} \right )^{1/2}} \leqslant K{\left | E \right |^\alpha }{\left \| {f{{\left | x \right |}^\beta }} \right \|_t}

\]
for all

f
∈

L
1

(

T
d

)

f \in {L^1}({T^d})

and finite

E
⊂

Z
d

E \subset {Z^d}

.
Sturmian theorems for second order systems
Authors: Allegretto W. 
Published: June 2010

Sturmian theorem are established for weakly coupled elliptic systems generated in a bounded domain by the expressions

l
1

u
→

=
−
Δ

u
→

+
A

u
→

,

l
2

w
→

=
−
Δ

w
→

+
B

w
→

{l_1}\vec u = - \Delta \vec u + A\vec u,{l_2}\vec w = - \Delta \vec w + B\vec w

, and Dirichlet boundary conditions. Here

Δ
\Delta

denotes the Laplace operator, and

A
,
B

A,B

are

m
×
m

m \times m

matrices. We do not assume that

A
,
B

A,B

are symmetric, but instead essentially require

B
B

irreducible and

b

i
j

⩽
0

if

i
≠
j

{b_{ij}} \leqslant 0{\text { if }}i \ne j

. Estimates on the real eigenvalue of

l
2

{l_2}

, with a positive eigenvector are then obtained as applications. Our results are motivated by recent theorems for ordinary differential equations established by Ahmad, Lazer and Dannan.
On Lipschitz functions of normal operators
Authors: Kittaneh Fuad 
Published: June 2010

It is shown that if

N
N

and

M
M

are normal operators on a separable, complex Hilbert space

H
H

, and

f
f

is a Lipschitz function on

Ω
=
σ
(
N
)
∪
σ
(
M
)

\Omega = \sigma (N) \cup \sigma (M)

(i.e.,

|

f
(
z
)
−
f
(
w
)

|

⩽
k

|

z
−
w

|

\left | {f(z) - f(w)} \right | \leqslant k\left | {z - w} \right |

for some positive constant

k
k

and all

z
,
w
∈
Ω
)

z,w \in \Omega )

, then

‖

f
(
N
)
X
−
X
f
(
M
)

‖

2

⩽
k

‖

N
X
−
X
M

‖

2

{\left \| {f(N)X - Xf(M)} \right \|_2} \leqslant k{\left \| {NX - XM} \right \|_2}

for any operator

X
X

on

H
H

. In particular,

‖

f
(
N
)
−
f
(
M
)

‖

2

⩽
k

‖

N
−
M

‖

2

{\left \| {f(N) - f(M)} \right \|_2} \leqslant k{\left \| {N - M} \right \|_2}

.