bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the uniqueness question for Hahn-Banach extensions from the space of ℒ¹ analytic functions
Authors: Reich Edgar 
Published: June 2010

Let

Ω
\Omega

be a region of the complex plane,

B

(
Ω
)

\mathcal {B}\left ( \Omega \right )

the space of analytic

L

1

{\mathcal {L}^1}

functions over

Ω
\Omega

, and

κ
∈

L

∞

(
Ω
)

\kappa \in {\mathcal {L}^\infty }\left ( \Omega \right )

. An evident necessary condition for the linear functional

Λ
κ

[
φ
]

=
∫

∫
Ω

κ
φ
d
x
d
y
(
φ
∈

B

(
Ω
)
)

{\Lambda _\kappa }\left [ \varphi \right ] = \smallint {\smallint _\Omega }\kappa \varphi dxdy(\varphi \in \mathcal {B}(\Omega ))

to have a unique Hahn-Banach extension from

B

(
Ω
)

\mathcal {B}(\Omega )

to

L

1

(
Ω
)

{\mathcal {L}^1}(\Omega )

is that

|

|

Λ

κ

|

G

|

|

=

|

|

κ

|

G

|

|

∞

|| {{\Lambda _{\kappa |G}}} || = {|| {\kappa |G} ||_\infty }

for every restriction

κ

|

G

\kappa |G

of

κ
\kappa

to a subregion

G
G

of

Ω
\Omega

. An example is constructed to show that not even a considerably stronger condition is sufficient for uniqueness of the Hahn-Banach extension. Remarks on the problem of whether

|

κ
(
z
)

|

\left | {\kappa (z)} \right |

is necessarily constant a.e. if the Hahn-Banach extension is unique indicate that this question is still open, contrary to an assertion in the literature.
The stability of the sine equation
Authors: Cholewa Piotr W. 
Published: June 2010

Let

δ
\delta

be a positive real constant and let

G
G

be an abelian group (written additively) in which division by 2 is uniquely performable. Every unbounded complex-valued function

f
f

on

G
G

satisfying the inequality
\[

|

f
(
x
+
y
)
f
(
x
−
y
)
−
f

(
x
)

2

+
f

(
y
)

2

|

⩽
δ

for all

x
,
y
∈
G

\left | {f(x + y)f(x - y) - f{{(x)}^2} + f{{(y)}^2}} \right | \leqslant \delta \quad {\text {for all }}x,y \in G

\]
has to be a solution of the sine functional equation
\[

f
(
x
+
y
)
f
(
x
−
y
)
=
f

(
x

)
2

−
f

(
y

)
2

for all

x
,
y
∈
G
.

f(x + y)f(x - y) = f{(x)^2} - f{(y)^2}\quad {\text {for all }}x,y \in G.

\]
A two weight inequality for the fractional integral when 𝑝=𝑛/𝛼
Authors: Harboure Eleonor , Macías Roberto A. , Segovia Carlos 
Published: June 2010

Let

I
α

{I_\alpha }

be the fractional integral operator defined as
\[

I
α

f
(
x
)
=
∫

f
(
y
)

|

x
−
y

|

α
−
n

d
y
.

{I_\alpha }f(x) = \int {f(y){{\left | {x - y} \right |}^{\alpha - n}}dy.}

\]
Given a weight

w
w

(resp.

υ
\upsilon

), necessary and sufficient conditions are given for the existence of a nontrivial weight

υ
\upsilon

(resp.

w
w

) such that
\[

‖

υ

χ
B

‖

∞

1

|
B
|

∫
B

|

I
α

f
(
x
)
−

m
B

(

I
α

f
)

|

d
x
⩽
C

(

∫

|
f
|

n

/

α

w

)

α

/

n

{\left \| {\upsilon {\chi _B}} \right \|_\infty }\frac {1}{{\left | B \right |}}\int _B {\left | {{I_\alpha }f(x) - {m_B}({I_\alpha }f)} \right |} dx \leqslant C{\left ( {\int {{{\left | f \right |}^{n/\alpha }}w} } \right )^{\alpha /n}}

\]
holds for any ball

B
B

such that

‖

v

χ
B

‖

∞

>
0

{\left \| {v{\chi _B}} \right \|_\infty } > 0

.
Higher Whitehead groups of certain bundles over Seifert manifolds
Authors: Nicas A. J. , Stark C. W. 
Published: June 2010

Vanishing results for

W

h

j

(

π
1

M
)
⊗
R
(
R
=

Z

,

Q

,

or

Z

[
1

/

2
]
)

{\text {W}}{{\text {h}}_j}({\pi _1}M) \otimes R(R = {\mathbf {Z}},{\mathbf {Q}},{\text { or }}{\mathbf {Z}}[1/2])

are obtained when

M
M

is a closed aspherical manifold which is the total space of a bundle over an insufficiently large Seifert manifold with infinite fundamental group of hyperbolic type. Allowable fibers include Riemannian flat manifolds and closed aspherical manifolds with poly-

Z

{\mathbf {Z}}

fundamental groups. Corollaries. concern the homotopy groups of the group

TOP

(
M
)

{\text {TOP}}(M)

of self-homeomorphisms of

M
M

.
Inversion and representation for the Poisson-Laguerre transform
Authors: Haimo Deborah Tepper 
Published: June 2010

The Poisson-Laguerre transform of a function

ϕ
\phi

is given by
\[

u
(
n
,
t
)
=

∑

m
=
0

∞

g
(
n
,
m
;
t
)
ϕ
(
m
)

m
!

Γ
(
m
+
α
+
1
)

u(n,t) = \sum \limits _{m = 0}^\infty {g(n,m;t)\phi (m)\frac {{m!}}{{\Gamma (m + \alpha + 1)}}}

\]
where

g
g

, defined by
\[

g
(
n
,
m
;
t
)
=

Γ
(
n
+
m
+
α
+
1
)

n
!
m
!

t

m
+
m

(
1
+
t
)

n
+
m
+
α
+
1

⋅
2

F
1

(

−
n
,
−
m
;
−
n
−
m
−
α
;
1
−

1

t
2

)

,

g(n,m;t) = \frac {{\Gamma (n + m + \alpha + 1)}}{{n!m!}}\frac {{{t^{m + m}}}}{{{{(1 + t)}^{n + m + \alpha + 1}}}}{ \cdot _2}{F_1}\left ( { - n, - m; - n - m - \alpha ;1 - \frac {1}{{{t^2}}}} \right ),

\]
s the associated function of the source solution

g
(
n
;
t
)
=
g
(
n
,
0
;
t
)

g(n;t) = g(n,0;t)

of the Laguerre difference heat equation
\[

∇
n

u
(
n
,
t
)
=

u
t

(
n
,
t
)
,

{\nabla _n}u(n,t) = {u_t}(n,t),

\]
with
\[

∇
n

f
(
n
)
=
(
n
+
1
)
f
(
n
+
1
)
=
(
2
n
+
α
+
1
)
f
(
n
)
+
(
n
+
α
)
f
(
n
−
1
)
.

{\nabla _n}f(n) = (n + 1)f(n + 1) = (2n + \alpha + 1)f(n) + (n + \alpha )f(n - 1).

\]
A simple algorithm for the inversion of the transform

(
∗
)

(*)

is derived. For

m
=
0

m = 0

, the transform

(
∗
)

(*)

is basically a power series so that the inversion algorithm leads to a useful identity involving binomial coefficients. In addition, a subclass of functions is characterized that is representable by a Poisson-Laguerre transform

(
∗
)

(*)

.
Topological equivalence in the space of integrable vector-valued functions
Authors: Gutman Semion 
Published: June 2010

The Banach space

L
1

(
0
,
T
;
X
)

{L^1}(0,T;X)

is retopologized by

|

|

|

f

|

|

|

=
max

|

|

∫
a
b

f
d
t

|

|

|||f||| = \max ||\int _a^bfdt||

,

0
⩽
a
⩽
b
⩽
T

0 \leqslant a \leqslant b \leqslant T

, where

|

|

.

|

|

||.||

is the norm in the given Banach space

X
X

. It is shown here that this topology coincides with the usual weak topology of

L
1

(
0
,
T
;
X
)

{L^1}(0,T;X)

on a wide class of weakly compact subsets.
An upper bound for the permanent of a 3-dimensional (0,1)-matrix
Authors: Dow Stephen J. , Gibson Peter M. 
Published: June 2010

Let

A
=
(

a

i
j
k

)

A = ({a_{ijk}})

be a

3
3

-dimensional matrix of order

n
n

. The permanent of

A
A

is defined by
\[

per
⁡
A
=

∑

σ
,
τ
∈

S
n

∏

i
=
1

n

a

i
σ
(
i
)
τ
(
i
)

,

\operatorname {per} A = \sum \limits _{\sigma ,\tau \in {S_n}} {\prod \limits _{i = 1}^n {{a_{i\sigma (i)\tau (i)}},} }

\]
where

S
n

{S_n}

is the symmetric group on

{
1
,
2
,
…
,
n
}

\{ 1,2, \ldots ,n\}

. Suppose that

A
A

is a (0,1)-matrix and that

r
i

=

∑

j
,
k
=
1

n

a

i
j
k

for

i
=
1
,
2
,
…
,
n

{r_i} = \sum \nolimits _{j,k = 1}^n {{a_{ijk}}} {\text { for }}i = 1,2, \ldots ,n

. In this paper it is shown that per

A
≤

∏

i
=
1

n

r
i

!

1

/

r
i

.

A \leq \prod \nolimits _{i = 1}^n {{r_i}{!^{1/{r_i}}}.}

A similar bound is then obtained for a second function, the

2
2

-permanent of a

3
3

-dimensional matrix, that is another analogue of the permanent of an ordinary (

2
2

-dimensional) matrix.
The Jantzen filtration of a certain class of Verma modules
Authors: Ku Jong Min 
Published: June 2010

Let

G
=

N
+

⊕
H
⊕

N
−

G = {N_ + } \oplus H \oplus {N_ - }

be a Kac-Moody Lie algebra. For each

M
M

in the category

O

\mathcal {O}

of

G
G

-modules, there is a filtration

(

M
i

)

i
≥
0

{({M_i})_{i \geq 0}}

by

G
G

-submodules of

M
M

naturally associated with the set

{

υ
∈
M

|

N
+

υ

=
0

}

\left \{ {\upsilon \in M\left | {{N_ + }\upsilon } \right . = 0} \right \}

. If

G
G

is symmetrizable and

M
M

is a Verma module,

M
i

=

M
i

{M_i} = {M^i}

for all

i
i

if and only if

[

M
:
L
(
μ
)

]

=
dim
⁡

Hom
G

⁡
(
M
(
μ
)
,
M
)

\left [ {M:L(\mu )} \right ] = \dim \operatorname {Hom}_G(M(\mu ),M)

for all

μ
∈

H
∗

\mu \in {H^ * }

where

(

M
i

)

i
≥
0

{({M^i})_{i \geq 0}}

is the Jantzen filtration of

M
M

. The main tools used are the nondegenerate form on each

M
i

/

M

i
+
1

{M^i}/{M^{i + 1}}

together with the

Γ
\Gamma

-operator of

G
G

.
On analytic slit mappings in the class Σ
Authors: Abu-Muhanna Yusuf , Leung Y. J. 
Published: June 2010

We show that any analytic slit mapping is a support point in

Σ
\Sigma

. As an application, we use Schiffer’s theory to show that a rational slit mapping must be of the form

z

+

a
0

+

a
1

/

(
z
−

z
1

)

z{\text { + }}{a_0} + {a_1}/(z - {z_1})

.
Weighted norm inequalities for the Fourier transform on certain totally disconnected groups
Authors: Quek T. S. 
Published: June 2010

Let

G
G

be a locally compact totally disconnected Abelian group with dual group

Γ
\Gamma

. Let

U
U

and

V
V

be nonnegative measurable functions on

Γ
\Gamma

and

G
G

, respectively. In this paper we give, in terms of

U
U

and

V
V

, a necessary condition and some sufficient conditions for the inequality

|

|

f
^

U

|

|

q

≤
C

|

|

f
V

|

|

p

||\hat fU|{|_q} \leq C||fV|{|_p}

to hold for all

f
f

in

L
1

(
G
)

{L_1}\left ( G \right )

, where

f
^

\hat f

denotes the Fourier transform of

f
f

and

1
>
p
≤
q
>
∞

1 > p \leq q > \infty

. If

U
U

and

V
V

are both radial, we give a necessary and sufficient condition for the above norm inequality to hold for all

f
f

in

L
1

(
G
)

{L_1}\left ( G \right )

.
Some new multipliers of Fourier series
Authors: Buntinas Martin 
Published: June 2010

Let

L
1

{L^1}

be the space of all complex-valued

2
π

2\pi

-periodic integrable functions

f
f

and let

L
1

^

\widehat {{L^1}}

be the space of sequences of Fourier coefficients

f
^

\hat f

. A sequence

λ
\lambda

is an

(

L
1

^

→

L
1

^

)

\left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right )

multiplier if

λ
⋅

f
^

=

(

λ

(
n
)

f
^

(
n
)

)

\lambda \cdot \hat f = \left ( {\lambda \left ( n \right )\hat f\left ( n \right )} \right )

belongs to

L
1

^

\widehat {{L^1}}

for every

f
f

in

L
1

{L^1}

. The space of even sequences of bounded variation is defined by

b
υ
=

{

λ

|

λ
n

=

λ

−
n

,

∑

k
=
0

∞

|

Δ

λ
k

|

+

sup

n

|

λ
n

|

>
∞

}

b\upsilon = \left \{ {\lambda \left | {{\lambda _n} = {\lambda _{ - n}},\sum \nolimits _{k = 0}^\infty {\left | {\Delta {\lambda _k}} \right |} + {{\sup }_n}\left | {{\lambda _n}} \right | > \infty } \right .} \right \}

, where

Δ

λ
k

=

λ
k

−

λ

k
+
1

\Delta {\lambda _k} = {\lambda _k} - {\lambda _{k + 1}}

and the space of even bounded quasiconvex sequences is defined by

q
=

{

λ

|

λ
n

=

λ

−
n

,

∑

k
=
1

∞

k

|

Δ
2

λ
k

|

+

sup

n

|

λ
n

|

>
∞

}

q = \left \{ {\lambda \left | {{\lambda _n} = {\lambda _{ - n}},\sum \nolimits _{k = 1}^\infty {k\left | {{\Delta ^2}{\lambda _k}} \right | + {{\sup }_n}\left | {{\lambda _n}} \right | > \infty } } \right .} \right \}

, where

Δ
2

λ
k

=
Δ

λ
k

−
Δ

λ

k
+
1

{\Delta ^2}{\lambda _k} = \Delta {\lambda _k} - \Delta {\lambda _{k + 1}}

. It is well known that

q
⊂
b
υ

q \subset b\upsilon

and

q
⊂

(

L
1

^

→

L
1

^

)

q \subset \left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right )

but

b
υ
⊄

(

L
1

^

→

L
1

^

)

b\upsilon \not \subset \left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right )

. This result is significantly improved by finding an increasing family of sequence spaces

d

υ
p

d{\upsilon _p}

between

q
q

and

b
υ

b\upsilon

which are

(

L
1

^

→

L
1

^

)

\left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right )

multipliers. Since the

(

L
1

^

→

L
1

^

)

\left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right )

multipliers are the

2
π

2\pi

-periodic measures, this result gives sufficient conditions for a sequence to be the Fourier coefficients of a measure.
On a family of elliptic surfaces with Mordell-Weil rank 4
Authors: Schwartz Charles F. 
Published: June 2010

In this paper, we find bases for the Mordell-Weil groups of a family of elliptic surfaces. In particular, let

E

(
a
,
b
)

→
B

{E_{(a,b)}} \to B

be the elliptic surface given by
\[

y
2

=
4

[

x
3

−

∑

i
=
0

2

a
i

u
i

x
+

∑

j
=
0

3

b
j

u
j

]

.

{y^2} = 4\left [ {{x^3} - \sum \limits _{i = 0}^2 {{a_i}{u^i}x + } \sum \limits _{j = 0}^3 {{b_j}{u^j}} } \right ].

\]
If the elliptic surface has Mordell-Weil rank 4 over

C

{\mathbf {C}}

, then we find a basis

{

σ
i

=
(

x
i

,

y
i

)

|

1
≤
i
≤
4
}

\{ {\sigma _i} = ({x_i},{y_i})|1 \leq i \leq 4\}

with

x
i

{x_i}

and

y
i

{y_i}

, linear in

u
u

. We do this by finding a parametrization of this family of elliptic surfaces; furthermore, if the parameters are rational numbers, then the Mordell-Weil group is rational over

Q

{\bf {Q}}
Extensions of certain compact operators on vector-valued continuous functions
Authors: Khurana Surjit Singh 
Published: June 2010

For any compact Hausdorff spaces

X
,
Y

X,Y

with

φ
:

X
→
Y

\varphi :\;X \to Y

a continuous onto mapping,

E
,
F

E,F

, Hausdorff locally convex spaces with

F
F

complete,

C
(
X
,
E
)

(
C
(
Y
,
E
)
)

C(X,E)\;(C(Y,E))

all

E
E

-valued continuous functions on

X
(
Y
)

X(Y)

, and

L
:
C
(
Y
,
E
)
→
F

L:C(Y,E) \to F

a

T

\mathcal {T}

-compact continuous operator

(
σ
(
F
,

F
′

)
≤

T

≤
τ
(
F
,

F
′

)
)

(\sigma (F,F’) \leq \mathcal {T} \leq \tau (F,F’))

, it is proved there exists a

T

\mathcal {T}

-compact continuous operator

L
0

:
C
(
X
,
E
)
→
F

{L_0}:C(X,E) \to F

such that

L
0

(
f
∘
φ
)
=
L
(
f
)

{L_0}(f \circ \varphi ) = L(f)

for every

f
∈
C
(
Y
,
E
)

f \in C(Y,E)

.
A remark on angular complex dilatations of quasiconformal mappings
Authors: Fehlmann Richard 
Published: June 2010

By a theorem of Ortel an angular complex dilatation

κ
\kappa

is extremal iff it is Teichmüllèr (quadratic differential with finite norm) or if it satisfies an integral condition involving the angular limits

λ
x

(
ϑ
)

{\lambda _x}(\vartheta )

. We show that this second case occurs iff

λ
x

(
ϑ
)

{\lambda _x}(\vartheta )

can be given explicitly at a certain point

x
x

, namely by

λ
x

(
ϑ
)
=

‖
κ
‖

∞

e

2
i

(
ϑ
−

ϑ
0

)

{\lambda _x}(\vartheta ) = {\left \| \kappa \right \|_\infty }{e^{2i}}(\vartheta - {\vartheta _0})

. Moreover, we investigate this statement under the weaker condition of angularity when the uniformity part in its definition is dropped.
Operator theoretic characterizations of [𝐼𝑁]-groups and inner amenability
Authors: Lau Anthony To Ming , Paterson Alan L. T. 
Published: June 2010

Let

G
G

be a locally compact group and

p
∈
[
1
,
∞
]

p \in [1,\infty ]

. Let

π
p

{\pi _p}

be the isometric representation of

G
G

on

L
p

(
G
)

{L_p}(G)

given by

π
p

(
x
)
f
(
t
)
=
f
(

x

−
1

t
x
)
Δ

(
x

)

1

/

p

{\pi _p}(x)f(t) = f({x^{ - 1}}tx)\Delta {(x)^{1/p}}

. Let

A

p
′

{\mathcal {A}’_p}

be the commutant of

A

p

{\mathcal {A}_p}

in

B
(

L
p

(
G
)
)

B({L_p}(G))

. In this paper we determine those

G
G

for which: (*)

A

p
′

{\mathcal {A}’_p}

contains a nonzero compact operator. We prove, among other things, that if

p
∈
[
1
,
∞
)

p \in [1,\infty )

, then (*) holds if and only if

G
∈
[
I
N
]

G \in [IN]

, and that if

p
=
∞

p = \infty

, then (*) holds if and only if

G
G

is inner amenable.
Some stationary subsets of 𝒫(𝜆)
Authors: Donder Hans-Dieter , Koepke Peter , Levinski Jean-Pierre 
Published: June 2010

Let

κ
\kappa

and

λ
\lambda

be uncountable cardinals such that

κ
≤
λ

\kappa \leq \lambda

, and set

S
(
κ
,
λ
)
=

{

X
∈

P

κ

(
λ
)

|

|

X
∩
κ

|

>

|

X

|

}

S(\kappa ,\lambda ) = \left \{ {X \in {\mathcal {P}_\kappa }(\lambda )|\;|X \cap \kappa | > |X|} \right \}

. We determine the consistency strength of the statement "

(

∃
λ
≥
κ

)

\left ( {\exists \lambda \geq \kappa } \right )

(

(
S
(
κ
,
λ
)

(S(\kappa ,\lambda )

is stationary in

P

κ

(
λ
)

{\mathcal {P}_\kappa }(\lambda )

)" using a new type of partition cardinals. In addition, we show that the property "

S
(
κ
,

κ
+

)

S(\kappa ,{\kappa ^ + })

is stationary in

P

κ

(

κ
+

)

{\mathcal {P}_\kappa }({\kappa ^ + })

" is much stronger.
On the monodromy group of everywhere tangent lines to the octic surface in 𝑃³
Authors: D’Souza Harry 
Published: June 2010

Let

S
0

{S_0}

be an octic surface in

P

3

,
G
=
G
(
1
,
3
)

{{\mathbf {P}}^3},G = G(1,3)

= Grassmannian of lines in

P

3

{{\mathbf {P}}^3}

, and

J

=
{
(
x
,
l
)

|

x
∈
l
∩

S
0

}
⊂

S
0

×
G

{\mathbf {J}} = \{ (x,l)|x \in l \cap {S_0}\} \subset {S_0} \times G

. Then

dim
⁡

J

=
5

\dim {\mathbf {J}} = 5

. Let

L

=

{
l

|

l

is

e
v
e
r
y
w
h
e
r
e

tangent to

S
0

}
−

⊂
G

{\mathbf {L}} = {\{ l|l{\text { is }}everywhere{\text { tangent to }}{S_0}\} ^ - } \subset G

. Let

π
2

:

S
0

×
G
→
G

{\pi _2}:{S_0} \times G \to G

be the projection onto the second factor. We denote its restriction to

J

{\mathbf {J}}

also by

π
2

{\pi _2}

. Then the locus of everywhere tangent lines is

π
2

(

L

)

{\pi _2}({\mathbf {L}})

. In this article we show that the monodromy group of these lines is the full symmetric group.
Fixed points for discontinuous quasi-monotone maps in 𝑅ⁿ
Authors: Hu Shou Chuan 
Published: June 2010

Let

K
n

{K_n}

be the unit cube in

R
n

{R^n}

and

f
=
(

f
1

,

f
2

,
…
,

f
n

)
:

K
n

→

R
n

f = ({f_1},{f_2}, \ldots ,{f_n}):{K_n} \to {R^n}

. It is known that

f
f

has maximal and minimal fixed points in

K
n

{K^n}

if

f
:

K
n

→

K
n

f:{K_n} \to {K_n}

and

f
f

is monotone increasing. In this paper, a weaker condition, namely quasi-monotonicity, is considered and it is proved that the above mentioned conclusion is still true if

f
f

is either quasi-monotone and
\[

lim

inf

t
→
0

[

f
i

(
x
+
t

e
i

)
−

f
i

(
x
)
]

t

≠
−
∞
,

\lim \inf \limits _{t \to 0} \frac {{[{f_i}(x + t{e_i}) - {f_i}(x)]}}{t} \ne - \infty ,

\]
or

−
f

- f

is quasi-monotone and
\[

lim

sup

t
→
0

[

f
i

(
x
+
t

e
i

)
−

f
i

(
x
)
]

t

≠
+
∞

\lim \sup \limits _{t \to 0} \frac {{[{f_i}(x + t{e_i}) - {f_i}(x)]}}{t} \ne + \infty

\]
.
Zeros of diagonal equations over finite fields
Authors: Wan Da Qing 
Published: June 2010

Let

N
N

be the number of solutions

(

x
1

,
…
,

x
n

)

({x_1}, \ldots ,{x_n})

of the equation (1)
\[

(
1
)

c
1

x
1

d
1

+

c
2

x
2

d
2

+
⋯
+

c
n

x
n

d
n

=
c

(1)\quad {c_1}x_1^{{d_1}} + {c_2}x_2^{{d_2}} + \cdots + {c_n}x_n^{{d_n}} = c

\]
over the finite field

F
q

{F_q}

, where

d
i

|

(
q
−
1
)
,

c
i

∈

F
q
∗

(
i
=
1
,
…
,
n
)

{d_i}|(q - 1),{c_i} \in F_q^*(i = 1, \ldots ,n)

, and

c
∈

F
q

c \in {F_q}

. If
\[

1

d
1

+

1

d
2

+
⋯
+

1

d
n

>
b
≥
1

\frac {1} {{{d_1}}} + \frac {1} {{{d_2}}} + \cdots + \frac {1} {{{d_n}}} > b \geq 1

\]
for some positive integer

b
b

, we prove that

q
b

|

N

{q^b}|N

. This result is an improvement of the theorem that

p

|

N

p|N

obtained by B. Morlaye [7] and also by J. R. Joly [3].
An asymptotic expansion for the expected number of real zeros of a random polynomial
Authors: Wilkins J. Ernest 
Published: June 2010

Let

ν
n

{\nu _n}

be the expected number of real zeros of a polynomial of degree

n
n

whose coefficients are independent random variables, normally distributed with mean 0 and variance 1. We find an asymptotic expansion for

ν
n

{\nu _n}

of the form
\[

ν
n

=

2
π

log
⁡
(
n
+
1
)
+

∑

p
=
0

∞

A
p

(
n
+
1
)

−
p

\nu _n = \frac {2}{\pi } \log (n + 1) + \sum \limits _{p = 0}^\infty {{A_p}{{(n + 1)}^{ - p}}}

\]
in which

A
0

=
0.625735818
,

A
1

=
0
,

A
2

=
−
0.24261274
,

A
3

=
0
,

A
4

=
−
0.08794067
,

A
5

=
0

{A_0} = 0.625735818,{A_1} = 0,{A_2} = - 0.24261274,{A_3} = 0,{A_4} = - 0.08794067,{A_5} = 0

. The numerical values of

ν
n

{\nu _n}

calculated from this expansion, using only the first four, or six, coefficients, agree with previously tabulated seven decimal place values

(
1
≤
n
≤
100
)

(1 \leq n \leq 100)

with an error of at most

10

−
7

{10^{ - 7}}

when

n
≥
30

n \geq 30

, or

n
≥
8

n \geq 8

.
The derivative of Bazilevič functions
Authors: London R. R. , Thomas D. K. 
Published: June 2010

For

α

>

0

\alpha {\text { > }}0

, let

B
1

(
α
)

{B_1}(\alpha )

be the class of normalised analytic functions

f
f

defined in the open unit disc

D
D

such that

Re(

f
(
z
)

/

z

)

α
−
1

f
′

(
z
)

>

0

{\text {Re(}}f(z)/z{{\text {)}}^{\alpha - 1}}f’(z){\text { > }}0

for

z
∈
D

z \in D

. Sharp upper and lower bounds are obtained for

|

z

f
′

(
z
)

/

f
(
z
)

|

\left | {zf’(z)/f(z)} \right |

when

f
∈

B
1

(
α
)

f \in {B_1}(\alpha )

.
𝐿² boundedness of highly oscillatory integrals on product domains
Authors: Prestini Elena 
Published: June 2010

We prove

L
2

{L^2}

boundedness of the oscillatory singular integral
\[

T
f
(
x
,
y
)
=

∬

D
y

exp
⁡
(
2
π
i
N
(
y
)

x
′

)

x
′

y
′

f
(
x
−

x
′

,
y
−

y
′

)
d

x
′

d

y
′

Tf(x,y) = \iint \limits _{{D_y}} {\frac {{\operatorname {exp} (2\pi iN(y)x’)}}{{x’y’}}}f(x - x’,y - y’)dx’dy’

\]
where

N
(
y
)

N(y)

is an arbitrary integer-valued

L
∞

{L^\infty }

function and where nothing is assumed on the dependency upon

y
y

of the domain of integration

D
y

{D_y}

. We also prove

L
2

{L^2}

boundedness of the corresponding maximal opertaor. Operators of this kind appear in a problem of a.e. convergence of double Fourier series.
The Zalcman conjecture for close-to-convex functions
Authors: Ma Wan Cang 
Published: June 2010

Let

S
S

be the class of functions

f
(
z
)
=
z
+
⋯

f(z) = z + \cdots

analytic and univalent in the unit disk

D
D

. For

f
(
z
)
=
z
+

a
2

z
2

+
⋯
∈
S

f(z) = z + {a_2}{z^2} + \cdots \in S

, Zalcman conjectured that

|

a
n
2

−

a

2
n
−
1

|

≤

(
n
−
1

)
2

(
n
=
2
,
3
,
…
)

|a_n^2 - {a_{2n - 1}}|\; \leq \;{(n - 1)^2}(n = 2,3, \ldots )

. This conjecture is verified for

n
≥
4

n \geq 4

and close-to-convex functions.
Unique solutions for a class of discontinuous differential equations
Authors: Bressan Alberto 
Published: June 2010

This paper is concerned with the Cauchy Problem
\[

x
˙

(
t
)

=
f

(

t
,
x

(
t
)

)

,

x

(

t
0

)

=

x
0

∈

R

n

,

\dot x\left ( t \right ) = f\left ( {t,x\left ( t \right )} \right ),\quad x\left ( {{t_0}} \right ) = {x_0} \in {\mathbb {R}^n},

\]
where the vector field

f
f

may be discontinuous with respect to both variables

t
,
x

t,x

. If the total variation of

f
f

along certain directions is locally finite, we prove the existence of a unique solution, depending continuously on the initial data.
On a natural connection between the entropy spaces and Hardy space 𝑅𝑒𝐻¹
Authors: Dąbrowski Romuald 
Published: June 2010

In 1983 B. Korenblum [7, 8] introduced a class of Banach function spaces associated with the notion of entropy (we will call these spaces and their norms entropy spaces and entropy norms, respectively). Entropy spaces were used in [8] as a tool for proving a new convergence test for Fourier series which includes classical tests of Dirichlet-Jordan and Dini-Lipschitz. In this paper we construct natural linear operators from the entropy spaces to Hardy space

Re

H
1

{\text {Re}}{H^1}

[5, 6]. In fact, these operators define multiplier type bounded embeddings of entropy spaces to

Re

H
1

{\text {Re}}{H^1}

. As a corollary we obtain a growth condition for Fourier coefficients of a continuous periodic function of bounded entropy norm (as announced in [4]). In particular, we show that if

f
f

is a continuous periodic function of bounded Shannon entropy norm (resp. of bounded Lipschitz entropy norm with exponent

q
,
0
>
q
>
1

q,0 > q > 1

), and

{

a
n

}

n
∈

Z

{\left \{ {{a_n}} \right \}_{n \in {\mathbf {Z}}}}

are the Fourier coefficients of

f
f

, then

∑

n
≠
0

|

a
n

(

log(|n|)/n

)

|

>
∞

\sum \nolimits _{n \ne 0} {|{a_n}({\text {log(|n|)/n}})| > \infty }

(resp.

∑

n
≠
0

|

a
n

|

/

|n|

)
q

>
∞

\sum \nolimits _{n \ne 0} {|{a_n}|/{\text {|n|}}{)^q} > \infty }

). In §4 we study the relationship between the dual spaces of entropy spaces and space B.M.O. of functions of bounded mean oscillation. In §5 we conjecture that

Re
⁡

H
1

\operatorname {Re} {H^1}

is a direct limit of the entropy spaces in an appropriate sense.