bluuebunny/crossref_metadata_2025_split huggingface dataset:
Facteurs premiers de sommes d’entiers
Authors: Tenenbaum Gérald 
Published: July 2010

Let

Ω
(
n
)

\Omega (n)

denote the total number of prime factors of the positive integer

n
n

, and set for real

x
x

,

α
\alpha

, and integer

k
≥
1

k \geq 1

,
\[

π
k

(
x
)
=

∑

n
⩽
x

Ω
(
n
)
=
k

1
,

π
k

(
x
,
α
)
=

∑

n
⩽
x

Ω
(
n
)
=
k

e

(
α
n
)
,

E
(
x
,
α
)
=

x

−
1

∑

n
⩽
x

e

(
α
n
)
,

\pi _k(x) = \sum \limits _{\substack {n \leqslant x \\ \Omega (n) = k}} 1, \quad \pi _k(x, \alpha ) = \sum \limits _{\substack {n \leqslant x \\ \Omega (n) = k}} \mathbf {e}(\alpha n), \quad E(x,\alpha ) = x^{-1}\sum \limits _{n \leqslant x} \mathbf {e}(\alpha n),

\]
where

e

(
t
)
:=
exp
⁡
(
2
π
i
t
)

{\mathbf {e}}(t): = \exp (2\pi it)

. We establish a best possible "independence" result of the type
\[

π
k

(
x
,
α
)

/

π
k

(
x
)
=
E
(
x
,
α
)
+
O
(

δ
k

(
x
)
)

{\pi _k}(x,\alpha )/{\pi _k}(x) = E(x,\alpha ) + O({\delta _k}(x))

\]
which is valid uniformly in

x
,
k
,
α

x,k,\alpha

, and where the error

δ
k

(
x
)

{\delta _k}(x)

tends to 0 as

x
→
+
∞

x \to + \infty

, if, and only if,

k
∼
log
⁡
log
⁡
x

k \sim \operatorname {log} \operatorname {log} x

. As an application we prove a recent conjecture of Erdös, Maier, and Sárközy concerning the remainder in their Erdös-Kac theorem for sum-sets.
A multivariate generalization of a theorem of R. H. Farrell
Authors: Plachky D. 
Published: July 2010

Let

(
Ω
,

S

,
μ
)

(\Omega ,\mathfrak {S},\mu )

denote a finite measure space and

T
j

:
Ω
→
[

a
j

,

b
j

]
(

S

,

B

∩
[

a
j

,

b
j

]
)

{T_j}:\Omega \to [{a_j},{b_j}](\mathfrak {S},\mathfrak {B} \cap [{a_j},{b_j}])

-measurable functions,

j
=
1
,
…
,
n

j = 1, \ldots ,n

. Then the algebra of functions generated by

1
,

T
1

,
…
,

T
n

1,{T_1}, \ldots ,{T_n}

is a dense subset of

L

1

(
Ω
,

S

,
P
)

{\mathcal {L}_1}(\Omega ,\mathfrak {S},P)

if and only if for any

A
∈

S

A \in \mathfrak {S}

there exists some

B
∈

(

T
1

,
…
,

T
n

)

−
1

(

B

n

∩
[

a
1

,

b
1

]
×
⋯
×
[

a
n

,

b
n

]
)

B \in {({T_1}, \ldots ,{T_n})^{ - 1}}({\mathfrak {B}^n} \cap [{a_1},{b_1}] \times \cdots \times [{a_n},{b_n}])

, such that

μ
(
A
Δ
B
)
=
0

\mu (A\Delta B) = 0

is valid. In particular, this condition is satisfied if

(
Ω
,

S

)

(\Omega ,\mathfrak {S})

is a Blackwell space and

(

T
1

,
…
,

T
n

)
:
Ω
→
[

a
1

,

b
1

]
×
⋯
×
[

a
n

,

b
n

]

({T_1}, \ldots ,{T_n}):\Omega \to [{a_1},{b_1}] \times \cdots \times [{a_n},{b_n}]

is in addition one-to-one.
Discretization in the method of averaging
Authors: Fečkan Michal 
Published: July 2010

Let

f
:
R
×

R

m
¯

×
R
→

R

m
¯

,
f
=
f
(
ε
,
x
,
t
)

f:R \times {R^{\overline m }} \times R \to {R^{\overline m }},f = f(\varepsilon ,x,t)

be a

C
2

{C^2}

-mapping

1
1

-periodic in

t
t

having the form

f
(
0
,
x
,
t
)
=
A
x
+
o
(

|

x

|

)

f(0,x,t) = Ax + o(|x|)

as

x
→
0

x \to 0

where

A
∈

L

(

R

m
¯

)

A \in \mathcal {L}({R^{\overline m }})

has no eigenvalues with zero real parts. We study the relation between local stable manifolds of the equation
\[

x
′

=
ε
f
(
ε
,
x
,
t
)
,
ε
>
0

is

small

x’ = \varepsilon f(\varepsilon ,x,t),\varepsilon > 0{\text {is}}\;{\text {small}}

\]
and of its discretization
\[

x

n
+
1

=

x
n

+
(
ε

/

m
)
f
(
ε
,

x
n

,

t
n

)
,

t

n
+
1

=

t
n

+
1

/

m
,

{x_{n + 1}} = {x_n} + (\varepsilon /m)f(\varepsilon ,{x_n},{t_n}),{t_{n + 1}} = {t_n} + 1/m,

\]
where

m
∈
{
1
,
2
,
…
}
=

N

m \in \{ 1,2, \ldots \} = \mathcal {N}

. We show behavior of these manifolds of the discretization for the following cases: (a)

m
→
∞
,
ε
→

ε
¯

>
0

m \to \infty ,\varepsilon \to \overline \varepsilon > 0

, (b)

m
→
∞
,
ε
→
0

m \to \infty ,\varepsilon \to 0

, (c)

m
→
k
∈

N

,
ε
→
0

m \to k \in \mathcal {N},\varepsilon \to 0

.
Weak compactness in 𝐿¹(𝜇,𝑋)
Authors: Ülger A. 
Published: July 2010

Let

(
Ω
,
Σ
,
μ
)

(\Omega ,\Sigma ,\mu )

be a probability space,

X
X

a Banach space, and

L
1

(
μ
,
X
)

{L^1}(\mu ,X)

the Banach space of Bochner integrable functions

f
:
Ω
→
X

f:\Omega \to X

. Let

W
=
{
f
∈

L
1

(
μ
,
X
)
:

for a

.e

.

ω

in

Ω
,

|

|

f
(
ω
)

|

|

≤
1
}

W = \{ f \in {L^1}(\mu ,X):{\text { for a}}{\text {.e}}{\text {. }}\omega {\text { in }}\Omega ,||f(\omega )|| \leq 1\}

. In this paper we characterize the rwc (relatively weakly compact) subsets of

L
1

(
μ
,
X
)

{L^1}(\mu ,X)

. The main results are as follows: Theorem A. A subset

H
H

of

W
W

is rwc iff given any sequence

(

f
n

)

({f_n})

in

H
H

there exists a sequence

(

f
~

n

)

({\tilde f_n})

, with

f
~

n

∈
Co
⁡
(

f
n

,

f

n
+
1

,
…
)

{\tilde f_n} \in \operatorname {Co}({f_n},{f_{n + 1}}, \ldots )

such that, for a.e.

ω
\omega

in

Ω
\Omega

, the sequence

(

f
~

n

(
ω
)
)

({\tilde f_n}(\omega ))

converges weakly in

X
X

. Theorem B. A subset

A
A

of

L
1

(
μ
,
X
)

{L^1}(\mu ,X)

is rwc iff given any

ε
>
0

\varepsilon > 0

there exist an integer

N
N

and a rwc subset

H
H

of NW such that

A
⊆
H
+
ε
B
(
0
)

A \subseteq H + \varepsilon B(0)

, where

B
(
0
)

B(0)

is the unit ball of

L
1

(
μ
,
X
)

{L^1}(\mu ,X)

.
Density of the polynomials in the Hardy space of certain slit domains
Authors: Akeroyd John 
Published: July 2010

In this article we construct a Jordan arc

Γ
\Gamma

in the complex plane, with endpoints 0 and 1, such that the polynomials are dense in the Hardy space

H
2

(

D

/

Γ
)
;

D

:=
{
z
∈

C

:

|

z

|

>
1
}

{H^2}(\mathbb {D}/\Gamma );\mathbb {D}: = \{ z \in \mathbb {C}:|z| > 1\}

.
Asymptotic behavior of almost-orbits of nonlinear semigroups of non-Lipschitzian mappings in Hilbert spaces
Authors: Tan Kok-Keong , Xu Hong Kun 
Published: July 2010

Let

C
C

be a nonempty closed convex subset of a Hilbert space

H
H

,

F

=
{
T
(
t
)
:
t
⩾
0
}

\mathcal {F} = \{ T(t):t \geqslant 0\}

be a continuous nonlinear asymptotically nonexpansive semigroup acting on

C
C

with a nonempty fixed point set

F
(

F

)

F(\mathcal {F})

, and

u
:
[
0
,
∞
)
→
C

u:[0,\infty ) \to C

be an almost-orbit of

F

\mathcal {F}

. Then

{
u
(
t
)
}

\{ u(t)\}

almost converges weakly to a fixed point of

F

\mathcal {F}

, i.e., there exists an element

y
y

in

F
(

F

)

F(\mathcal {F})

such that
\[

weak-

lim

1
t

∫
0
t

u
(
r
+
h
)
d
r
=
y

uniformly for

h
⩾
0.

{\text {weak-}}\lim \frac {1} {t}\int _0^t {u(r + h)dr = y\quad {\text {uniformly for }}h \geqslant 0.}

\]
This implies that

{
u
(
t
)
}

\{ u(t)\}

converges weakly to a fixed point of

F

\mathcal {F}

if and only if

{
u
(
t
+
h
)
−
u
(
t
)
}

\{ u(t + h) - u(t)\}

converges weakly to zero as

t
t

tends to infinity for each

h
⩾
0

h \geqslant 0

.
Periodic 𝐿²-solutions of an integrodifferential equation in a Hilbert space
Authors: Staffans Olof J. 
Published: July 2010

Let

A
A

be a closed, densely defined operator in a Hilbert space

X
X

, and let

μ
,

ν

\mu ,\;\nu

, and

η
\eta

be finite, scalar-valued measures on

R

{\mathbf {R}}

. Consider the abstract integrodifferential equation
\[

∫

R

d

d
t

u
(
t
−
s
)
μ
(
d
s
)
+

∫

R

u
(
t
−
s
)
ν
(
d
s
)
+

∫

R

A
u
(
t
−
s
)
η
(
d
s
)
=
f
(
t
)
,

t
∈

R

,

\int _{\mathbf {R}} {\frac {d} {{dt}}u(t - s)\mu (ds) + \int _{\mathbf {R}} {u(t - s)\nu (ds) + \int _{\mathbf {R}} {Au(t - s)\eta (ds) = f(t),\qquad t \in {\mathbf {R}},} } }

\]
where

f
f

is a

2
π

2\pi

-periodic

L
2

{L^2}

function with values in

X
X

. We give necessary and sufficient conditions for this equation to have a mild

2
π

2\pi

-periodic

L
2

{L^2}

-solution with values in

X
X

for all

f
f

, as well as necessary and sufficient conditions for it to have a strong solution for all

f
f

. Furthermore, we give necessary and sufficient conditions for the operator mapping

f
f

into the periodic solution

u
u

to be compact. These results are applied to prove existence of periodic solutions of a nonlinear equation.
Coefficients de Fourier de fonctions à variation bornée
Authors: Wu Jie 
Published: July 2010

Let

f
:

R

→

C

f:\mathbb {R} \to \mathbb {C}

be a function of period

2
π

2\pi

and of bounded variation on

[
0
,
2
π
]

[0,2\pi ]

with the total variation

V
(
f
)

V(f)

. Its Fourier coefficients are denoted by

f
^

(
n
)

\hat f(n)

. In this short note, we give a very simple proof of the known result

|

f
^

(
n
)

|

⩽
V
(
f
)

/

2
π

|

n

|

(
n
∈

Z

,

n
≠
0
)

|\hat f(n)| \leqslant V(f)/2\pi |n|\;(n \in \mathbb {Z},\;n \ne 0)

.
Ratios of regulators in extensions of number fields
Authors: Costa Antone , Friedman Eduardo 
Published: July 2010

Let

L

/

K

L/K

be an extension of number fields. Then
\[

Reg
⁡
(
L
)

/

Reg
⁡
(
K
)
>

c

[
L
:

Q

]

(
log
⁡

|

D
L

|

)
m

,

\operatorname {Reg} (L)/\operatorname {Reg} (K) > {c_{[L:{\mathbf {Q}}]}}{(\log |{D_L}|)^m},

\]
where Reg denotes the regulator,

D
L

{D_L}

is the absolute discriminant of

L
L

, and

c

[
L
:

Q

]

>
0

{c_{[L:{\mathbf {Q}}]}} > 0

depends only on the degree of

L
L

. The nonnegative integer

m
=
m
(
L

/

K
)

m = m(L/K)

is positive if

L

/

K

L/K

does not belong to certain precisely defined infinite families of extensions, analogous to CM fields, along which

Reg
⁡
(
L
)

/

Reg
⁡
(
K
)

\operatorname {Reg} (L)/\operatorname {Reg} (K)

is constant. This generalizes some inequalities due to Remak and Silverman, who assumed that

K
K

is the rational field

Q

{\mathbf {Q}}

, and modifies those of Bergé-Martinet, who dealt with a general extension

L

/

K

L/K

but used its relative discriminant where we use the absolute one.
Multiplicative perturbations of linear Volterra equations
Authors: Rhandi Abdelaziz 
Published: July 2010

We prove that the following problems are essentially equivalent:
\[

[
VO
]

C
T

a
m
p
;

u
(
t
)
=
x
+

∫
0
t

a
(
t
−
s
)
C
T
u
(
s
)

d
s
,

[
VO
]

T
C

a
m
p
;

v
(
t
)
=
y
+

∫
0
t

a
(
t
−
s
)
T
C
v
(
s
)

d
s
,

\begin {array}{*{20}{c}} {{{[\operatorname {VO} ]}_{CT}}} & {\qquad \qquad u(t) = x + \int _0^t {a(t - s)CTu(s)\,ds,} } \\ {{{[\operatorname {VO} ]}_{TC}}} & {\qquad \qquad v(t) = y + \int _0^t {a(t - s)TCv(s)\,ds,} } \\ \end {array}

\]
where

T
T

is an unbounded closed linear operator in a Banach space

X
X

with dense domain

D
(
T
)
,

C

D(T),\;C

is a bounded linear operator on

X
X

, and

a
∈

L

loc

1

(
[
0
,
∞
)
,

R

)

a \in L_{\operatorname {loc} }^1([0,\infty ),\mathbb {R})

, which is exponentially bounded. We give some applications of our abstract theorem to second-order differential operators on the line.
Oscillation criteria for Hamiltonian matrix difference systems
Authors: Erbe L. H. , Yan Peng Xiang 
Published: July 2010

We obtain some oscillation criteria for the Hamiltonian difference system
\[

{

Δ
Y
(
t
)
=
B
(
t
)
Y
(
t
+
1
)
+
C
(
t
)
Z
(
t
)
,

Δ
Z
(
t
)
=
−
A
(
t
)
Y
(
t
+
1
)
−

B

∗

(
t
)
Z
(
t
)
,

\left \{ \begin {gathered} \Delta Y(t) = B(t)Y(t + 1) + C(t)Z(t), \hfill \\ \Delta Z(t) = - A(t)Y(t + 1) - {B^{\ast }}(t)Z(t), \hfill \\ \end {gathered} \right .

\]
where

A
,
B
,
C
,
Y
,
Z

A,B,C,Y,Z

are

d
×
d

d \times d

matrix functions. As a corollary, we establish the validity of an earlier conjecture for a second-order matrix difference system.
A weighted inequality for the Kakeya maximal operator
Authors: Vargas A. M. 
Published: July 2010

Let

K

δ

{\mathcal {K}_\delta }

be the Kakeya Maximal Operator defined as the supremum of averages over parallelepipeds of eccentricity

δ
\delta

. We show that

K

δ

{\mathcal {K}_\delta }

satisfies

|

|

K

δ

f

|

|

L
p

(
ω
)

⩽

C

n
,
p

(
1

/

δ

)

n

/

p
−
1

(
log
⁡
(
1

/

δ
)

)

α
n

|

|

f

|

|

L
p

(

K

δ

ω
)

||{\mathcal {K}_\delta }f|{|_{{L^p}(\omega )}} \leqslant {C_{n,p}}{(1/\delta )^{n/p - 1}}{(\log (1/\delta ))^{{\alpha _n}}}||f|{|_{{L^p}({\mathcal {K}_\delta }\omega )}}

for all

p
⩽
(
n
+
1
)

/

2

p \leqslant (n + 1)/2

with some constants

C

n
,
p

,

α
n

{C_{n,p}},\;{\alpha _n}

, independent of

f
f

and the weight

ω
\omega

.
A cohomological class of vector bundles
Authors: Miró-Roig Rosa M. 
Published: July 2010

The goal of this paper is to give a cohomological characterization of

F

n
,
t

{F_{n,t}}

, where

F

n
,
t

:=
Ker
⁡
(
(
n
+
t
;
n
)

O

P

n

(
−
t
)
→

O

P

n

)

{F_{n,t}}: = \operatorname {Ker} ((n + t;n){\mathcal {O}_{{{\mathbf {P}}^n}}}( - t) \to {\mathcal {O}_{{{\mathbf {P}}^n}}})

.
Characterization of classical type orthogonal polynomials
Authors: Kwon K. H. , Littlejohn L. L. , Lee J. K. , Yoo B. H. 
Published: July 2010

We characterize the classical type orthogonal polynomials

{

P
n

(
x
)

}
0
∞

\{ {P_n}(x)\} _0^\infty

satisfying a fourth-order differential equation of type
\[

∑

i
=
0

4

ℓ
i

(
x
)

y

(
i
)

(
x
)
=

λ
n

y
(
x
)

\sum \limits _{i = 0}^4 {{\ell _i}(x){y^{(i)}}(x) = {\lambda _n}y(x)}

\]
where

ℓ
i

(
x
)

{\ell _i}(x)

are polynomials of degree

⩽
i

\leqslant i

and

λ
n

{\lambda _n}

is a constant. They are only the orthogonal polynomials satisfying an orthogonality of the form
\[

⟨

τ
2

,

P
m

P
n

⟩
+
⟨

τ
1

,

P
m
′

P
n
′

⟩
+
⟨

τ
0

,

P
m

P
n

⟩
=
0

for

m
≠
n

\langle {\tau _2},P_m^{}P_n^{}\rangle + \langle {\tau _1},P_m’P_n’\rangle + \langle {\tau _0},{P_m}{P_n}\rangle = 0\quad {\text {for}}\;m \ne n

\]
where

τ
0

,

τ
1

{\tau _0},{\tau _1}

, and

τ
2

{\tau _2}

are moment functionals.
The conjugation operator on 𝐴_{𝑞}(𝐺)
Authors: Gupta Sanjiv Kumar , Madan Shobha , Tewari U. B. 
Published: July 2010

Let G be a compact abelian group and

Γ
\Gamma

its dual. For

1
≤
q
>
∞

1 \leq q > \infty

, the space

A
q

(
G
)

{A_q}(G)

is defined as
\[

A
q

(
G
)
=
{
f

|

f
∈

L
1

(
G
)
,

f
^

∈

l
q

(
Γ
)
}

{A_q}(G) = \{ f|f \in {L^1}(G),\quad \hat f \in {l_q}(\Gamma )\}

\]
with the norm

‖
f
‖

A
q

=

‖
f
‖

L
1

+

‖

f
^

‖

l
q

{\left \| f \right \|_{{A_q}}} = {\left \| f \right \|_{{L^1}}} + {\left \| {\hat f} \right \|_{{l_q}}}

. We prove: Let G be a compact, connected abelian group and P any fixed order on

Γ
\Gamma

. If

q
>
2

q > 2

and

ϕ
\phi

is a Young’s function, then the conjugation operator H does not extend to a bounded operator from

A
q

(
G
)

{A_q}(G)

to the Orlicz space

L
ϕ

(
G
)

{L^\phi }(G)

.
A geometric approach to the multivariate Müntz problem
Authors: Kroó András 
Published: July 2010

For a countable set

Ω
⊂

R

n

\Omega \subset {\mathbb {R}^n}

denote by

P
(
Ω
)

P(\Omega )

the space of polynomials spanned by

x
ω

,
ω
∈
Ω
(
x
=
(

x
1

,
…
,

x
n

)
∈

R

n

,
ω
=
(

ω
1

,
…
,

ω
n

)
∈
Ω
,

x
ω

=

∏

i
=
1

n

x
i

ω
i

)

{x^\omega }, \omega \in \Omega (x = ({x_1}, \ldots ,{x_n}) \in {\mathbb {R}^n}, \omega = ({\omega _1}, \ldots ,{\omega _n}) \in \Omega , {x^\omega } = \prod _{i = 1}^nx_i^{{\omega _i}})

. In this paper we investigate the question of the density of

P
(
Ω
)

P(\Omega )

in

C
(
K
)

C(K)

, the space of real valued continuous functions endowed with the supremum norm on compact set

K
⊂

R

n

K \subset {\mathbb {R}^n}

. In case

n
=
1

n = 1

the classical theorem of Müntz gives an elegant necessary and sufficient condition for density. This problem (closely related to the distribution of zeros of Fourier transforms) is much more complex in the multivariate setting. We shall present an extension of Müntz’ condition to the case

n
>
1

n > 1

which will suffice for density. This, in particular, will enable us to construct "optimally sparse" lattice point sets

Ω
\Omega

for which density holds.
The conjugacy problem for HNN extensions with infinite cyclic associated groups
Authors: Horadam K. J. , Farr G. E. 
Published: July 2010

Under suitable recursive conditions, the conjugacy problem for HNN extensions of the form

⟨
G
,

t
i

,
i
∈
I
:

x

k
i

=

t
i

−
1

x

l
i

t
i

,

k
i

,

l
i

⩾
1
,

i
∈
I
⟩

\langle G,{t_i},i \in I:{x^{{k_i}}} = t_i^{ - 1}{x^{{l_i}}}{t_i},\;{k_i},\;{l_i} \geqslant 1,\;i \in I\rangle

is solvable if and only if the inverse subsemigroup generated by

{
(

k
i

,

l
i

)
,

i
∈
I
}

\{ ({k_i},{l_i}),\;i \in I\}

has solvable extended word problem in the semigroup
\[

S
=
(

N

×

N

,

(
a
,
b
)
(
c
,
d
)
=
(
a
c

/

gcd
(
b
,
c
)
,

b
d

/

gcd
(
b
,
c
)
)
)
.

S = (\mathbb {N} \times \mathbb {N},\;(a,b)(c,d) = (ac/\gcd (b,c),\;bd/\gcd (b,c))).

\]
Furthermore,

S
S

is isomorphic to the direct sum of countably many copies of the bicyclic semigroup, which has a central place in the theory of inverse semigroups. This new approach to the conjugacy problem is used to determine several classes of HNN extensions with infinitely many stable letters and solvable conjugacy problem.
A noniterative 2-adic statement of the 3𝑁+1 conjecture
Authors: Bernstein Daniel J. 
Published: July 2010

Associated with the

3
N
+
1

3N + 1

problem is a permutation

Φ
\Phi

of the 2-adic integers. The

3
N
+
1

3N + 1

conjecture is equivalent to the conjecture that 3Q is an integer if

Φ
(
Q
)

\Phi (Q)

is a positive integer. We state a new definition of

Φ
\Phi

. To wit: Q and

N
=
Φ
(
Q
)

N = \Phi (Q)

are linked by the equations

Q
=

2

d
0

+

2

d
1

+
⋯

Q = {2^{{d_0}}} + {2^{{d_1}}} + \cdots

and

N
=
(
−
1

/

3
)

2

d
0

+
(
−
1

/

9
)

2

d
1

+
(
−
1

/

27
)

2

d
2

+
⋯

N = ( - 1/3){2^{{d_0}}} + ( - 1/9){2^{{d_1}}} + ( - 1/27){2^{{d_2}}} + \cdots

with

0
≤

d
0

>

d
1

>
⋯

0 \leq {d_0} > {d_1} > \cdots

. We list four applications of this definition.
On the failure of the Urysohn-Menger sum formula for cohomological dimension
Authors: Dranišnikov A. N. , Repovš D. , Ščepin E. V. 
Published: July 2010

We prove that the classical Urysohn-Menger sum formula,

dim
⁡
(
A
∪
B
)
⩽
dim
⁡
A
+
dim
⁡
B
+
1

\dim (A \cup B) \leqslant \dim A + \dim B + 1

, which is also known to be true for cohomological dimension over the integers (and some other abelian groups), does not hold for cohomological dimension over an arbitrary abelian group of coefficients. In particular, we prove that there exist subsets

A
,

B
⊂

R

4

A,\;B \subset {\mathbb {R}^4}

such that

4
=

dim

Q

/

Z

(
A
∪
B
)
>

dim

Q

/

Z

A
+

dim

Q

/

Z

B
+
1
=
3

4 = {\dim _{\mathbb {Q}/\mathbb {Z}}}(A \cup B) > {\dim _{\mathbb {Q}/\mathbb {Z}}}A + {\dim _{\mathbb {Q}/\mathbb {Z}}}B + 1 = 3

.
Geometric properties of some subspaces of 𝑉𝑁(𝐺)
Authors: Belanger Alain , Forrest Brian E. 
Published: July 2010

Let G be a locally compact group. We show that any one of the spaces

U
C
B
(

G
^

)
,
W
A
P
(

G
^

)
,
A
P
(

G
^

)

UCB(\hat G),WAP(\hat G),AP(\hat G)

, and

C
δ
∗

(
G
)

C_\delta ^ \ast (G)

is Asplund if and only if the group G is finite. We also show that any one of the spaces

V
N
(
G
)
,
U
C
B
(

G
^

)

VN(G),UCB(\hat G)

, and

C
δ
∗

(
G
)

C_\delta ^\ast (G)

has the DPP if and only if the group G has an abelian subgroup of finite index.
On the number of generators of modules over polynomial rings
Authors: Li Hongnian 
Published: July 2010

In this paper we prove the following Theorem. Let

B
=
A
[

X
1

,
⋅
,

X
n

]

B = A[{X_1}, \cdot ,{X_n}]

, where A is a universally catenary equidimensional ring. Let M be a finitely generated B-module of rank r. Denote by d the Krull dimension of A, by

μ
(
M
)

\mu (M)

the minimal number of generators of M, and by

I
M

{I_M}

the (radical) ideal which defines the set of primes of B at which M is not locally free. Assume that
\[

μ
(
M

/

I
M

M
)
≤
η

a
n
d

η
≥
max
{
d
+
r
,
dim
⁡
B

/

I
M

+
r
+
1
}
,

\mu (M/{I_M}M) \leq \eta \;and\;\eta \geq \max \{ d + r,\dim B/{I_M} + r + 1\} ,

\]
where

η
\eta

is a positive integer. Then

μ
(
M
)
≤
η

\mu (M) \leq \eta

. This improves a result of R. G. López, On the number of generators of modules over polynomial affine rings, Math. Z. 208 (1991), 11-21.
Values of 𝐿-functions at the critical point
Authors: Mai Liem 
Published: July 2010

For a discriminant D of a binary quadratic form, we study the average value of

L
(
s
,

ε
D

)

L(s,{\varepsilon _D})

at the critical point

1
2

\frac {1}{2}

where

ε
D

{\varepsilon _D}

is defined by W. Kohnen and D. Zagier:
\[

ε
D

(
n
)
=

∑

g
>
0

g

|

δ
,

g
2

|

n

(
δ

/

g
,
n

/

g
2

)
=
1

(

D
0

g

−
2

n

)

g

{\varepsilon _D}(n) = \sum \limits _{\begin {array}{*{20}{c}} {g > 0} \\ {g|\delta ,{g^2}|n} \\ {(\delta /g,n/{g^2}) = 1} \\ \end {array} } {\left ( {\frac {{{D_0}}}{{{g^{ - 2}}n}}} \right )} \,g

\]
for

n
∈

N

n \in \mathbb {N}

and

D
=

D
0

δ
2

,

D
0

D = {D_0}{\delta ^2},{D_0}

a fundamental discriminant and

δ
∈

N

\delta \in \mathbb {N}

. When

D
=

D
0

,
L
(
s
,

ε

D
0

)

D = {D_0},L(s,{\varepsilon _{{D_0}}})

is the Dirichlet series

L
(
s
,
(

D
0

⋅

)
)

L(s,(\frac {{{D_0}}}{ \cdot }))

. We derive an asymptotic formula for

∑
D

L
(

1
2

,

ε
D

)

\sum \nolimits _D {L(\frac {1}{2},{\varepsilon _D})}

, where the sum runs over all discriminants

D
∈
(
0
,
Y
]

D \in (0,Y]

or

[
−
Y
,
0
)

[ - Y,0)

.
On Minakshisundaram-Pleijel zeta functions of spheres
Authors: Carletti E. , Monti Bragadin G. 
Published: July 2010

The aim of this paper is to show that the Minakshisundaram-Pleijel zeta function

Z
k

(
s
)

{Z_k}(s)

of k-dimensional sphere

S

k

,
k
≥
2

{\mathbb {S}^k},k \geq 2

(defined in

ℜ
e
(
s
)
>

k
2

\Re e(s) > \frac {k}{2}

by
\[

Z
k

(
s
)
=

∑

n
=
1

∞

P
k

(
n
)

[
n
(
n
+
k
−
1
)
]

s

{Z_k}(s) = \sum \limits _{n = 1}^\infty {\frac {{{P_k}(n)}}{{{{[n(n + k - 1)]}^s}}}}

\]
with

(
k
−
1
)
!

P
k

(
n
)
=

R

(
n
+
1
,
k
−
2
)
(
2
n
+
k
−
1
)

(k - 1)!{P_k}(n) = \mathcal {R}(n + 1,k - 2)(2n + k - 1)

where the "rising factorial"

R

(
x
,
n
)
=
x
(
x
+
1
)
⋯
(
x
+
n
−
1
)

\mathcal {R}(x,n) = x(x + 1) \cdots (x + n - 1)

is defined for real number x and n nonnegative integer) can be put in the form
\[

(
k
−
1
)
!

Z
k

(
s
)
=

∑

l
=
0

∞

(
−
1
)

l

(

k
−
1

2

)

2
l

(

−
s

l

)

∑

j
=
0

k
−
1

B

k
,

j

ζ
(
2
s
+
2
l
−
j
,

k
+
1

2

)

(k - 1)!{Z_k}(s) = {\sum \limits _{l = 0}^\infty {{{( - 1)}^l}\left ( {\frac {{k - 1}}{2}} \right )} ^{2l}}\left ( {\begin {array}{*{20}{c}} { - s} \\ l \\ \end {array} } \right )\sum \limits _{j = 0}^{k - 1} {{B_{k,}}_j\zeta (2s + 2l - j,\frac {{k + 1}}{2})}

\]
where

B

k
,
j

{B_{k,j}}

are explicitly computed. The above formula allows us to find explicitly the residue of

Z
k

(
s
)

{Z_k}(s)

at the pole

s
=

k
2

−
n
,
n
∈

N

s = \frac {k}{2} - n,n \in \mathbb {N}

,
\[

1

(
k
−
1
)
!

∑

h
=
0

k
2

−
1

∑

l
+
h
=
n

l
≥
0

(
−
1
)

l

(

k
−
1

2

)

2
l

(

n
−

k
2

l

)

B

k
,
k
−
2
h
−
1

.

\frac {1}{{(k - 1)!}}\sum \limits _{h = 0}^{\frac {k}{2} - 1} {{{\sum \limits _{\begin {array}{*{20}{c}} {l + h = n} \\ {l \geq 0} \\ \end {array} } {{{( - 1)}^l}\left ( {\frac {{k - 1}}{2}} \right )} }^{2l}}\left ( {\begin {array}{*{20}{c}} {n - \frac {k}{2}} \\ l \\ \end {array} } \right )} {B_{k,k - 2h - 1}}.

\]
In passing, we also obtain apparently new relations among the Stirling numbers.
Putnam’s inequality for 𝑝-hyponormal operators
Authors: Chō Muneo , Itoh Masuo 
Published: July 2010

The purpose of this paper is to show the following: Let

0
>
p
>

1
2

0 > p > \frac {1}{2}

. If T is a p-hyponormal operator on a Hilbert space, then
\[

‖

(

T
∗

T
)

p

−

(
T

T
∗

)

p

‖

≤

p
π

∬

σ
(
T
)

ρ

2
p
−
1

d
ρ
d
θ

.

\left \| {{{({T^ \ast }T)}^p} - {{(T{T^ \ast })}^p}} \right \| \leq \frac {p}{\pi }\iint _{\sigma (T)} {{\rho ^{2p - 1}}d\rho d\theta }.

\]
That is, Putnam’s inequality holds for p-hyponormal operators.
Uniform asymptotic stability via Liapunov-Razumikhin technique
Authors: Liu James H. 
Published: July 2010

The Liapunov-Razumikhin technique is applied to obtain the uniform asymptotic stability for linear integrodifferential equations in Hilbert spaces,
\[

x
′

(
t
)
=
A

[

x
(
t
)
+

∫
#
t

F
(
t
−
s
)
x
(
s
)

d
s

]

,

t
≥

t
0

≥
0
(
#
=
0

or

−
∞
)
,

x’(t) = A\left [ {x(t) + \int _\# ^t {F(t - s)x(s)\,ds} } \right ],\quad t \geq {t_0} \geq 0(\# = 0\;{\text {or}} - \infty ),

\]
which occur in viscoelasticity and in heat conduction for materials with memory.