bluuebunny/crossref_metadata_2025_split huggingface dataset:
Primarité de 𝐿^{𝑝}(𝑋)
Authors: Capon Michèle 
Published: June 2010

Soit

X
X

un espace de Banach à base symétrique. Nous étudions les opérateurs de

L
p

(
X
)

{L^p}(X)

dans lui-même, en leur associant une "matrice" d’opérateurs de

L
p

{L^p}

. Cette technique nous permet de démontrer que pour tout

p
p

réel dans

[
1
,
+
∞
[
,

l’espace

L
p

(
X
)

[1, + \infty [,{\text {l’espace}}\ L^p(X)

est primaire.
Borel functions of bounded class
Authors: Fremlin D. H. , Hansell R. W. , Junnila H. J. K. 
Published: June 2010

Let

X
X

and

Y
Y

be metric spaces and

f
:
X
→
Y

f:X \to Y

a Borel measurable function. Does

f
f

have to be of bounded class, i.e. are the sets

f

−
1

[
H
]

{f^{ - 1}}[ H ]

, for open

H
⊆
Y

H \subseteq Y

, of bounded Baire class in

X
?

X?

This is an old problem of A. H. Stone. Positive answers have been given under a variety of extra hypotheses and special axioms. Here we show that (i) unless something similar to a measurable cardinal exists, then

f
f

is of bounded class and (ii) if

f
f

is actually a Borel isomorphism, then

f

(

and

f

−
1

)

f\,({\text {and}}\ {f^{ - 1}})

are of bounded class.
Existence of infinitely many solutions for a forward backward heat equation
Authors: Höllig Klaus 
Published: June 2010

Let

ϕ
\phi

be a piecewise linear function which satisfies the condition

s
ϕ
(
s
)
⩾
c

s
2

,
c
>
0
,
s
∈

R

s\phi (s) \geqslant c{s^2},c > 0,s \in {\mathbf {R}}

, and which is monotone decreasing on an interval

(
a
,
b
)
⊂

R

+

(a,b) \subset {{\mathbf {R}}_ + }

. It is shown that for

f
∈

C
2

[
0
,
1
]

f \in {C^2}[0,1]

, with

max

f
′

>
a

\max f^\prime > a

, there exists a

T
>
0

T > 0

such that the initial boundary value problem
\[

u
t

=
ϕ

(

u
x

)
x

,

u
x

(
0
,
t
)
=

u
x

(
1
,
t
)
=
0
,

u
(
⋅
,
0
)
=
f
,

{u_t} = \phi \,{({u_x})_x},\qquad {u_x}(0,t) = {u_x}(1,t) = 0,\qquad u( \cdot ,0) = f,

\]
has infinitely many solutions

u
u

satisfying

∥

u

∥

α

,
∥

u
x

∥

∞

,
∥

u
t

∥

2

⩽
c
(
f
,
ϕ
)

\parallel \;u\;{\parallel _{\alpha }},\parallel \;{u_x}{\parallel _{\infty }},\parallel \;{u_t}{\parallel _{2}} \leqslant c(f,\phi )

on

[
0
,
1
]
×
[
0
,
T
]

[0,1] \times [0,T]

.
An application of asymptotic techniques to certain problems of spectral and scattering theory of Stark-like Hamiltonians
Authors: Ben-Artzi Matania 
Published: June 2010

Let

L
0

=
−
Δ
+
V
(

x
1

)
,
L
=

L
0

+

V
p

(
x
)

{L_0} = - \Delta + V({x_1}),L = {L_0} + {V_p}(x)

be selfadjoint in

L
2

(

R
n

)

{L^2}({R^n})

. Here

V
,

V
p

V,{V_p}

are real functions,

V
(

x
1

)

V({x_1})

depends only on the first coordinate. Existence of the wave-operators

W
±

(
L
,

L
0

)
=
s
-

lim

t
→
±
∞

exp
⁡
(
i
t
L
)
exp
⁡
(
−
i
t

L
0

)

{W_ \pm }\,(L,{L_0}) = s \text {-} {\lim _{t \to \pm \infty }}\,\exp (itL)\exp ( - it{L_0})

is proved, using the stationary phase method. For this, an asymptotic technique is applied to the study of

−

d
2

/

d

t
2

+
V
(
t
)

-{d^2}/d{t^2} + V(t)

in

L
2

(
R
)

{L^2}(R)

. Its absolute continuity is proved as well as a suitable eigenfunction expansion.

V
V

is a "Stark-like" potential. In particular, the cases

V
(

x
1

)
=
(
−
sgn
⁡

x
1

)

|

x
1

|

α

,
0
>
α
⩽
2

V({x_1}) = ( - \operatorname {sgn}{x_1})|{x_1}\,{|^\alpha },0 > \alpha \leqslant 2

, are included.

V
p

{V_p}

may be taken as the sum of an

L
2

{L^2}

-function and a function satisfying growth conditions in the

+

x
1

+ {x_1}

direction.

V
p

(
x
)
=

|

x

|

−
1

{V_p}(x) = |x|^{ - 1}

is included.
Invariant subspaces on Riemann surfaces of Parreau-Widom type
Authors: Hayashi Mikihiro 
Published: June 2010

In this paper we generalize Beurling’s invariant subspace theorem to the Hardy classes on a Riemann surface with infinite handles. The problem is to classify all closed (

weak

∗

\text {weak}^{\ast }

closed, if

p
=
∞

p = \infty

)

H
∞

(
d
χ
)

{H^\infty }(d\chi )

-submodules, say

m

\mathfrak {m}

, of

L
p

(
d
χ
)

{L^p}(d\chi )

,

1
⩽
p
⩽
∞

1 \leqslant p \leqslant \infty

, where

d
χ

d\chi

is the harmonic measure on the Martin boundary of a Riemann surface

R
R

, and

H
∞

(
d
χ
)

{H^\infty }(d\chi )

is the set of boundary functions of all bounded analytic functions on

R
R

. Our main result is stated roughly as follows. Let

R
R

be of Parreau-Widom type, that is, the space

H
∞

(
R
,
γ
)

{H^\infty }(R,\gamma )

of bounded analytic sections contains a nonzero element for every complex flat line bundle

γ
∈
π

(
R

)

∗

\gamma \in \pi {(R)^{\ast }}

. We may assume, without loss of generality, that the Green’s function of

R
R

vanishes at the infinity. Set

m
∞

(
γ
)
=
sup
{

|

f
(

O

)

|

:
f
∈

H
∞

(
R
,
γ
)
,

|

f

|

⩽
1
}

{m^\infty }(\gamma ) = \sup \{ |f({\mathbf {O}})|:f \in {H^\infty }(R,\gamma ),|f| \leqslant 1\}

for a fixed point

O

{\mathbf {O}}

of

R
R

. Then, a necessary and sufficient condition in order that every such an

m

\mathfrak {m}

takes either the form

m

=

C
E

L
p

(
d
χ
)

\mathfrak {m} = {C_E}{L^p}(d\chi )

, where

C
E

{C_E}

is the characteristic function of a set

E
E

, or the form

m

=
q

H
p

(
d
χ
,
γ
)

\mathfrak {m} = q{H^p}(d\chi ,\gamma )

, where

|

q

|

=
1

|q| = 1

a.e. and

γ
\gamma

is some element of

π

(
R

)

∗

\pi {(R)^{\ast }}

is that

m
∞

(
γ
)

{m^\infty }(\gamma )

is continuous for the variable

γ
∈
π

(
R

)

∗

\gamma \in \pi {(R)^{\ast }}

.
Maximal abelian subalgebras of von Neumann algebras and representations of equivalence relations
Authors: Sutherland Colin E. 
Published: June 2010

We associate to each pair

(

M

,

A

)

(\mathcal {M},\mathcal {A})

, (with

M

\mathcal {M}

a von Neumann algebra, and

A

\mathcal {A}

a maximal abelian subalgebra) a representation

α
\alpha

of the Takesaki equivalence relation

R

(

M

,

A

)

\mathcal {R}\,(\mathcal {M},\mathcal {A})

of

(

M

,

A

)

(\mathcal {M},\mathcal {A})

as automorphisms of a

I

∞

{{\text {I}}_\infty }

factor. Conversely each such representation

α
\alpha

of

R

\mathcal {R}

on

(
X
,
μ
)

(X,\mu )

as automorphisms of

B

(

H

)

\mathcal {B}\,(\mathcal {H})

determines a von Neumann algebra-abelian subalgebra pair

S
′

(

R

,
α
)
=
(

N

,

B

)

S’\,(\mathcal {R},\alpha ) = (\mathcal {N},\mathcal {B})

where

N

\mathcal {N}

is the commutant of the algebra of "self-intertwiners" for

α
\alpha

and

B

=

L
∞

(
X
,
μ
)
⊗
1

\mathcal {B} = {L^\infty }(X,\mu ) \otimes 1

on

L
2

(
X
,
μ
)
⊗

H

{L^2}(X,\mu ) \otimes \mathcal {H}

. The main concern is the assignments

(

M

,

A

)
→

T

(

M

,

A

)
=
(

R

(

M

,

A

)
,
α
)

(\mathcal {M},\mathcal {A}) \to \mathcal {T}\;(\mathcal {M},\mathcal {A}) = (\mathcal {R}\,(\mathcal {M},\mathcal {A}),\alpha )

and

(

R

,
α
)
→

S
′

(

R

,
α
)

(\mathcal {R},\alpha ) \to S’(\mathcal {R},\alpha )

, and in particular, the extent to which they are inverse to each other—this occurs if

R

\mathcal {R}

is countable nonsingular and

α
\alpha

is (conjugation by) a projective square-integrable representation (cf. [8]), or if

A

\mathcal {A}

is a Cartan subalgebra (cf. [5]), among other cases. A partial dictionary between the representations

(

R

,
α
)

(\mathcal {R},\alpha )

and pairs

(

M

,

A

)

(\mathcal {M},\mathcal {A})

is given—thus if

R

\mathcal {R}

is countable nonsingular and

α
\alpha

is what we term replete,

S
′

(

R

,
α
)

S’(\mathcal {R},\alpha )

is injective whenever

R

\mathcal {R}

is amenable, and a complete Galois theory generalizing that for crossed products by discrete groups is available. We also show how to construct various pathological examples such as a singular maximal abelian subalgebra

A

⊆

M

\mathcal {A} \subseteq \mathcal {M}

for which the Takesaki equivalence relation

R

(

M

,

A

)

\mathcal {R}\,(\mathcal {M},\mathcal {A})

is nontrivial.
The number of factorizations of numbers less than 𝑥 into divisors greater than 𝑦
Authors: Hensley Douglas 
Published: June 2010

Let

A
(
x
,

y
)

A(x,\,y)

be the number in the title. There is a function

h
:
[
0
,

∞
)
→
[
0
,

2
]

h:[0,\,\infty ) \to [0,\,2]

, decreasing and convex, with

h
(
0
)
=
2

h(0) = 2

and

lim

r
→
∞

⁡
h
(
r
)
=
0

\operatorname {lim}_{r \to \infty }h(r) = 0

, such that if

r
=
log
⁡
y

/

log

x

r = \operatorname {log} y/\sqrt {\operatorname {log} \,x}

then as

x
→
∞

x \to \infty

with

r
r

fixed,
\[

A
(
x
,
y
)
=

C
(
r
)
x

exp
⁡

(

h
(
r
)

log

x

)

(
log

x
)

3

/

4

(

1
+
O

(
log

x
)

−
1

/

4

)

A(x,y) = \frac {{C(r)x\,\operatorname {exp} \left ( {h(r)\sqrt {\operatorname {log} \,x} } \right )}}{{{{(\operatorname {log} \,x)}^{3/4}}}}\left ( {1 + O{{(\operatorname {log} \,x)}^{ - 1/4}}} \right )

\]
. The estimate is uniform on intervals

0
>
r
≤

R
0

0 > r \leq {R_0}

. As corollaries we have for

log

y
=
θ

(
log

x

)

1

/

4

\operatorname {log} \,y = \theta {(\operatorname {log} \,x)^{1/4}}

,
\[

lim

x
→
∞

A
(
x
,

y
)

A
(
x
,

1
)

/

y

=

e

θ
2

/

2

\lim \limits _{x \to \infty } \,\frac {{A(x,\,y)}} {{A(x,\,1)/y}} = {e^{{\theta ^2}/2}}

\]
,and if

log

y
=
o

(
log

x

)

1

/

4

\operatorname {log} \,y = o {(\operatorname {log} \,x)^{1/4}}

then

A
(
x
,

y
)
≈
A
(
x
,

1
)

/

y

A(x,\,y) \approx A(x,\,1)/y

.
On specializations of curves. I
Authors: Nobile A. 
Published: June 2010

The following is proved: Given a family of projective reduced curves

X
→
T

X \to T

(

T
T

irreducible), if

X
t

{X_t}

(the general curve) is integral and

X
0

{X_0}

is a special curve (having irreducible components

X
1

,
…
,

X
r

{X_1}, \ldots ,{X_r}

), then

∑

i
=
1

r

g
i

(

X
i

)
⩽
g
(

X
t

)

\sum \nolimits _{i = 1}^r {{g_i}({X_i}) \leqslant g({X_t})}

, where

g
(
Z
)
=

g(Z) =

geometric genus of

Z
Z

. Conversely, if

A
A

is a reduced plane projective curve, of degree

n
n

with irreducible components

X
1

,
…
,

X
r

{X_1}, \ldots ,{X_r}

, and

g
g

satisfies

∑

i
=
1

r

g
i

(

X
i

)
⩽
g
⩽

1
2

(
n
−
1
)
(
n
−
2
)

\sum \nolimits _{i = 1}^r {{g_i}({X_i}) \leqslant g \leqslant \frac {1} {2}(n - 1)(n - 2)}

, then a family of plane curves

X
→
T

X \to T

(with

T
T

integral) exists, where for some

t
0

∈
T
,

X

t
0

=
Z

{t_0} \in T,{X_{{t_0}}} = Z

and for

t
t

generic,

X
t

{X_t}

is integral and has only nodes as singularities. Results of this type appear in an old paper by G. Albanese, but the exposition is rather obscure.
Necessary and sufficient conditions for oscillations of higher order delay differential equations
Authors: Ladas G. , Sficas Y. G. , Stavroulakis I. P. 
Published: June 2010

Consider the

n

th

n{\text {th}}

order delay differential equation (1)
\[

x

(
n
)

(
t
)
+

(
−
1

)

n
+
1

∑

i
=
0

k

p
i

x
(
t
−

τ
i

)
=
0
,

t
≥

t
0

,

{x^{(n)}}(t) + {( - 1)^{n + 1}}\sum \limits _{i = 0}^k {{p_i}x(t - {\tau _i}) = 0, \qquad t \geq {t_0}},

\]
where the coefficients and the delays are constants such that

0
=

τ
0

>

τ

1

>
⋯
>

τ
k

;

p
0

≥
0
,

p
i

>
0
,
i
=
1
,
2
,
…
,
k
;
k
≥
1

0 = {\tau _0} > {\tau _{1}}\, > \cdots > {\tau _k};{p_0}\, \geq 0,{p_i} > 0,i = 1,2,\ldots ,k;k \geq 1

and

n
≥
1

n \geq 1

. The characteristic equation of (1) is (2)
\[

λ
n

+

(
−
1

)

n
+
1

∑

i
=
0

k

p
i

e

−
λ

τ
i

=
0

.

{\lambda ^n} + {( - 1)^{n + 1}}\;\sum \limits _{i = 0}^k {{p_i}{e^{ - \lambda {\tau _i}}} = 0}.

\]
We prove the following theorem. Theorem. (i) For

n
n

odd every solution of (1) oscillates if and only if (2) has no real roots. (ii) For

n
n

even every bounded solution of (1) oscillates if and only if (2) has no real roots in

(
−
∞
,
0
]

( - \infty ,0]

. The above results have straightforward extensions for advanced differential equations.
A quasilinear hyperbolic integro-differential equation related to a nonlinear string
Authors: Heard Melvin L. 
Published: June 2010

We discuss global existence, boundedness and regularity of solutions to the integrodifferential equation
\[

u

t

t

(
t
,
x
)
+
c

(
t
)

u
t

(
t
,
x
)
−
M

(

∫

−
∞

+
∞

|

u
x

(
t
,
s
)

|

2

d
s

)

u

x

x

(
t
,
x
)
+
u

(
t
,
x
)

=
h

(
t
,
x
,
u

(
t
,
x
)
)
,

0
≤
t
>
∞
,
x
∈

R

,

u

(
0
,
x
)
=

u
0

(
x
)
,

u
t

(
0
,
x
)
=

u
1

(
x
)
,

x
∈

R

.

\begin {array}{*{20}{c}} {{u_{t\,t}}\,(t,x) + c\,(t)\,{u_t}(t,x) - M\;\left ( {\int _{ - \infty }^{ + \infty } {|{u_x}(t,s){|^2}\;ds} } \right )\;{u_{x\,x}}(t,x) + u\,(t,x)} \\ { = h\,(t,x,u\,(t,x)), \qquad 0 \leq t > \infty ,x \in {\mathbf {R}},} \\ {u\,(0,x) = {u_0}(x),\quad {u_t}(0,x) = {u_1}(x), \qquad x \in {\mathbf {R}}.} \\ \end {array}

\]
This type of equation occurs in the study of the nonlinear behavior of elastic strings. We show that if the initial data

u
0

(
x
)
,

u
1

(
x
)

{u_0}\,(x),{u_1}\,(x)

is small in a suitable sense, and if the damping coefficient

c

(
t
)

c\,(t)

grows sufficiently fast, then the above equation possesses a globally defined classical solution for forcing terms

h

(
t
,
x
,
u
)

h\,(t,x,u)

which are sublinear in

u
u

. In the nonlinearity we require that

M
∈

C
1

[
0
,
∞
)

M \in {C^1}\,[0,\infty )

and, in addition, satisfies

M
(
λ
)
≥

m
0

>
0

M( \lambda ) \geq {m_0} > 0

for all

λ
≥
0

\lambda \geq 0

.
Regularity properties of solutions to the basic problem in the calculus of variations
Authors: Clarke F. H. , Vinter R. B. 
Published: June 2010

This paper concerns the basic problem in the calculus of variations: minimize a functional

J
J

defined by
\[

J
(
x
)
=

∫
a
b

L
(
t
,
x
(
t
)
,

x
˙

(
t
)
)

d
t

J(x) = \int _a^b {L(t,x(t),\dot x(t))\;dt}

\]
over a class of arcs

x
x

whose values at

a
a

and

b
b

have been specified. Existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs, conditions which must be strengthened in order that the standard necessary conditions apply. The question arises: What necessary conditions hold merely under hypotheses of existence theory, say the classical Tonelli conditions? It is shown that, given a solution

x
x

, there exists a relatively open subset

Ω
\Omega

of

[
a
,
b
]

[a,b]

, of full measure, on which

x
x

is locally Lipschitz and satisfies a form of the Euler-Lagrange equation. The main theorem, of which this is a corollary, can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions. Three such classes are identified, and results of Bernstein, Tonelli, and Morrey are extended. One of these classes is of a novel nature, and its study implies the new result that when

L
L

is independent of

t
t

, the solution has essentially bounded derivative.
Functional equations for character series associated with 𝑛×𝑛 matrices
Authors: Formanek Edward 
Published: June 2010

Let

A
A

be either the ring of invariants or the trace ring of

r
r

generic

n
×
n

n \times n

matrices. Then

A
A

has a character series

χ
(
A
)

\chi (A)

which is a symmetric rational function of commuting variables

x
1

,
…
,

x
r

{x_1}, \ldots ,{x_r}

. The main result is that if

r
≥

n
2

r \geq {n^2}

, then

χ
(
A
)

\chi (A)

satisfies the functional equation
\[

χ
(
A
)
(

x
1

−
1

,
…
,

x
r

−
1

)
=

(
−
1

)
d

(

x
1

⋯

x
r

)

n
2

χ
(
A
)
(

x
1

,
…
,

x
r

)

\chi (A)(x_1^{ - 1}, \ldots ,x_r^{ - 1}) = {( - 1)^d}{({x_1} \cdots {x_r})^{{n^2}}}\chi (A)({x_1}, \ldots ,{x_r})

\]
, where

d
d

is the Krull dimension of

A
A

.
Weak type estimates for Bochner-Riesz spherical summation multipliers
Authors: Chanillo Sagun , Muckenhoupt Benjamin 
Published: June 2010

We consider the Bochner-Riesz multiplier
\[

T
δ

f

^

(
ξ
)
=

(
1
−

|
ξ
|

2

)
δ

+

f
^

(
ξ
)
,

δ
>
0
,

\widehat {{T_\delta }f}(\xi ) = {(1 - {\left | \xi \right |^2})^\delta } + \hat f(\xi ),\qquad \delta > 0,

\]
where

^

\widehat {}

denotes the Fourier transform. It is shown that the multiplier operator

T
δ

{T_\delta }

is weak type

(

p
0

,

p
0

)

({p_0},\,{p_0})

acting on

L

p
0

(

R

n

)

{L^{p0}}({{\mathbf {R}}^n})

radial functions, where

p
0

{p_0}

is the critical value

2
n

/

(
n
+
1
+
2
δ
)

2n/(n + 1 + 2\delta )

.
Weighted inequalities for the one-sided Hardy-Littlewood maximal functions
Authors: Sawyer E. 
Published: June 2010

Let

M
+

f
(
x
)
=

sup

h
>
0

(
1

/

h
)

∫
x

x
+
h

|

f
(
t
)

|

d
t

{M^ + }f(x) = {\sup _{h > 0}}(1/h)\int _x^{x + h} {|f(t)|\,dt}

denote the one-sided maximal function of Hardy and Littlewood. For

w
(
x
)
⩾
0

w(x) \geqslant 0

on

R
R

and

1
>
p
>
∞

1 > p > \infty

, we show that

M
+

{M^ + }

is bounded on

L
p

(
w
)

{L^p}(w)

if and only if

w
w

satisfies the one-sided

A
p

{A_p}

condition:
\[

(

A
p
+

)

[

1
h

∫

a
−
h

a

w
(
x
)
d
x

]

[

1
h

∫
a

a
+
h

w

(
x
)

−
1

/

(
p
−
1
)

d
x

]

p
−
1

⩽
C

\left ( {A_p^ + } \right )\qquad \left [ {\frac {1} {h}\int _{a - h}^a {w(x)dx} } \right ]{\left [ {\frac {1} {h}\int _a^{a + h} {w{{(x)}^{ - 1/(p - 1)}}dx} } \right ]^{p - 1}} \leqslant C

\]
for all real

a
a

and positive

h
h

. If in addition

v
(
x
)
⩾
0

v(x) \geqslant 0

and

σ
=

v

−
1

/

(
p
−
1
)

\sigma = {v^{ - 1/(p - 1)}}

,then

M
+

{M^ + }

is bounded from

L
p

(
v
)

{L^p}(v)

to

L
p

(
w
)

{L^p}(w)

if and only if
\[

∫
I

[

M
+

(

χ
I

σ
)
]

p

w
⩽
C

∫
I

σ
>
∞

\int _I {{{[{M^ + }({\chi _I}\sigma )]}^p}w \leqslant C\int _I {\sigma > \infty } }

\]
for all intervals

I
=
(
a
,
b
)

I = (a,b)

such that

∫

−
∞

a

w
>
0

\int _{ - \infty }^a {w > 0}

. The corresponding weak type inequality is also characterized. Further properties of

A
p
+

A_p^ +

weights, such as

A
p
+

⇒

A

p
−
ε

+

A_p^ + \Rightarrow A_{p - \varepsilon }^ +

and

A
p
+

=
(

A
1
+

)

(

A
1
−

)

1
−
p

A_p^ + = (A_1^ + ){(A_1^ - )^{1 - p}}

, are established.
Application of group cohomology to space constructions
Authors: Igodt Paul , Lee Kyung Bai 
Published: June 2010

From a short exact sequence of crossed modules

1
→
K
→
H
→

H
¯

→
1

1 \to K \to H \to \bar H \to 1

and a

2
2

-cocycle

(
ϕ
,

h
)
∈

Z
2

(
G
;

H
)

(\phi ,\,h) \in {Z^2}(G;\,H)

, a

4
4

-term cohomology exact sequence

H

a
b

1

(
G
;
Z
)
→

H

(

ϕ
¯

,

h
¯

)

1

(
G
;

H
¯

,

Z
¯

)

→

δ

⋃
{

H
ψ
2

(
G
;
K
)
:

ψ

o
u
t

=

ϕ

o
u
t

}
→

H

a
b

2

(
G
;

Z
)

H_{ab}^1(G;Z) \to H_{(\bar {\phi }, \bar {h})}^1 (G; \bar {H}, \bar {Z}) \stackrel {\delta }{\to } \bigcup \{ H_\psi ^2 (G;K) : \psi _{\mathrm {out}} = \phi _{\mathrm {out}} \} \to H_{ab}^2(G;\,Z)

is obtained. Here the first and the last term are the ordinary (=abelian) cohomology groups, and

Z
Z

is the center of the crossed module

H
H

. The second term is shown to be in one-to-one correspondence with certain geometric constructions, called Seifert fiber space construction. Therefore, it follows that, if both the end terms vanish, the geometric construction exists and is unique.
Weighted norm estimates for Sobolev spaces
Authors: Schechter Martin 
Published: June 2010

We give sufficient conditions for estimates of the form\[∫|u(x)|qdμ(x)⩽C‖u‖s,p1,u∈Hs,p,{\int {\left | {u(x)} \right |} ^q}d\mu (x) \leqslant C\left \| u \right \|_{s,p}^1,\qquad u \in {H^{s,p}},\]to hold, whereμ(x)\mu (x)is a measure and‖u‖s,p{\left \| u \right \|_{s,p}}is the norm of the Sobolev spaceHs,p{H^{s,p}}. Ifdμ=dxd\mu = dx, this reduces to the usual Sobolev inequality. The general form has much wider applications in both linear and nonlinear partial differential equations. An application is given in the last section.
Decay rates of Fourier transforms of curves
Authors: Marshall B. P. 
Published: June 2010

Let

d
μ

d\mu

be a smooth measure on a nondegenerate curve in

R

n

{{\mathbf {R}}^n}

. This paper examines the decay rate of spherical averages of its Fourier transform

d
μ

^

\widehat {d\mu }

. Thus estimates of the following form are considered:
\[

(

∫

∑
r

|

d
μ

^

(
ξ

|

p

d
ξ

)

1

/

p

⩽
C

r

−
σ

|

|

f

|

|

{\left ( {\int _{{\sum _r}} {|\widehat {d\mu }(\xi {|^p}d\xi } } \right )^{1/p}} \leqslant C{r^{ - \sigma }}||f||

\]
where

∑
r

=
{
ξ
∈

R

n

:

|

ξ

|

=
r
}

{\sum _r} = \{ \xi \in {{\mathbf {R}}^n}:|\xi | = r\}

.
Algebras on the disk and doubly commuting multiplication operators
Authors: Axler Sheldon , Gorkin Pamela 
Published: June 2010

We prove that a bounded analytic function

f
f

on the unit disk is in the little Bloch space if and only if the uniformly closed algebra on the disk generated by

H
∞

{H^\infty }

and

f
¯

\overline f

does not contain the complex conjugate of any interpolating Blaschke product. A version of this result is then used to prove that if

f
f

and

g
g

are bounded analytic functions on the unit disk such that the commutator

T
f

T
g

∗

−

T
g

∗

T
f

{T_f}T_g^{\ast } - T_g^{\ast }{T_f}

(here

T
f

{T_f}

denotes the operator of multiplication by

f
f

on the Bergman space of the disk) is compact, then

(
1
−

|

z

|

2

)
min
{

|

f
′

(
z
)

|

,

|

g
′

(
z
)

|

}
→
0

(1 - |z{|^2})\min \{ |f’ (z)|,\;|g’ (z)|\} \to 0

as

|

z

|

↑
1

|z| \uparrow 1

.
Tables for the evaluation of ∫₀^{∞}𝑥^{𝛽}𝑒^{-𝑥}𝑓(𝑥)𝑑𝑥 by Gauss-Laguerre quadrature
Authors: Concus P. , Cassatt D. , Jaehnig G. , Melby E. 
Published: June 2010

Tables of abscissae and weight coefficients to fifteen places are presented for the Gauss-Laguerre quadrature formula

∫
0
∞

x
β

e

−
x

f
(
x
)
d
x
∼

∑

k
=
1

n

H
k

f
(

a
k

)

\int _0^\infty {{x^\beta }{e^{ - x}}f(x)dx \sim \sum \limits _{k = 1}^n {{H_k}f({a_k})} }

for

β
=
−

1
4

,
−

1
2

,

and

−

3
4

\beta = - \tfrac {1}{4}, - \tfrac {1}{2},{\text {and}} - \tfrac {3}{4}

and n = 1(1)15.
An extension of the Hille-Hardy formula
Authors: Srivastava H. M. 
Published: June 2010

While attempting to give extensions of the well-known Hille-Hardy formula for the generalized Laguerre polynomials

{

L
n

(
α
)

(
x
)
}

\{ {L_n}^{(\alpha )}(x)\}

defined by
\[

(
1
−
t

)

−
1
−
α

exp
⁡

[

−

x
t

1
−
t

]

=

∑

n
=
0

∞

L
n

(
α
)

(
x
)

t
n

{(1 - t)^{ - 1 - \alpha }}\exp \left [ { - \frac {{xt}} {{1 - t}}} \right ] = \sum \limits _{n = 0}^\infty {{L_n}^{(\alpha )}} (x){t^n}

\]
, the author applies here certain operational techniques and the method of finite mathematical induction to derive several bilinear generating functions associated with various classes of generalized hypergeometric polynomials. It is observed that the earlier works of Brafman [2], [3], [4], Chaundy [5], Meixner [12], Weisner [16], and others quoted in the literature, are only specialized or limiting forms of the results presented here.
The integral analogue of the Leibniz rule
Authors: Osler Thomas J. 
Published: June 2010

This paper demonstrates that the classical Leibniz rule for the derivative of the product of two functions
\[

D
N

u
υ
=

∑

k
=
0

N

(
k
N

)

D

N
−
k

u

D
k

υ

{D^N}u\upsilon = \sum \limits _{k = 0}^N {(_k^N)} {D^{N - k}}u{D^k}\upsilon

\]
has the integral analog
\[

D
α

u
υ
=

∫

−
∞

∞

(
ω
α

)

D

α
−
ω

u

D
ω

υ
d
ω
.

{D^\alpha }u\upsilon = \int _{ - \infty }^\infty {(_\omega ^\alpha )} {D^{\alpha - \omega }}u{D^\omega }\upsilon d\omega .{\text { }}

\]
The derivatives occurring are “fractional derivatives.” Various generalizations of the integral are given, and their relationship to Parseval’s formula from the theory of Fourier integrals is revealed. Finally, several definite integrals are evaluated using our results.
The distribution of small gaps between successive primes
Authors: Brent Richard P. 
Published: June 2010

For

r
≧
1

r \geqq 1

and large N, a well-known conjecture of Hardy and Littlewood implies that the number of primes

p
≦
N

p \leqq N

such that

p
+
2
r

p + 2r

is the least prime greater than p is asymptotic to
\[

∫
2
N

(

∑

k
=
1

r

A

r
,
k

(
log
⁡
x
)

k
+
1

)

d
x
,

\int _2^N {\left ( {\sum \limits _{k = 1}^r {\frac {{{A_{r,k}}}}{{{{(\log x)}^{k + 1}}}}} } \right )} \;dx,

\]
where the

A

r
,
k

{A_{r,k}}

are certain constants. We describe a method for computing these constants. Related constants are given to 10D for

r
=
1
(
1
)
40

r = 1(1)40

, and some empirical evidence supporting the conjecture is mentioned.