bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the solvability of the equation ∑ⁿᵢ₌₁𝑥ᵢ/𝑑ᵢ≡0(𝑚𝑜𝑑1) and its application
Authors: Sun Qi , Wan Da Qing 
Published: June 2010

In this paper, we obtain a necessary and sufficient condition under which the equation of the title is unsolvable. More precisely, for the equation
\[

x
1

d
1

+

x
2

d
2

+
⋯
+

x
n

d
n

≡
0

(
mod
1
)
,

x
i

integral

,

1

≤

x
i

>

d
i

(
1
≤
i
≤
n
)
,

\frac {{{x_1}}}{{{d_1}}} + \frac {{{x_2}}}{{{d_2}}} + \cdots + \frac {{{x_n}}}{{{d_n}}} \equiv 0\quad (\bmod 1),\quad {x_i}{\text { integral}},{\text {1}} \leq {x_i} > {d_i}(1 \leq i \leq n),

\]
where

d
1

,
…
,

d
n

{d_1}, \ldots ,{d_n}

are fixed positive integers, we prove the following result: The above equation is unsolvable if and only if 1. For some

d
i

,
(

d
i

,

d
1

d
2

⋯

d
n

/

d
i

)
=
1

{d_i},({d_i},{d_1}{d_2} \cdots {d_n}/{d_i}) = 1

, or 2. If

d

i
1

,
…
,

d

i
k

(
1
≤
i
>
⋯
>

i
k

≤
n
)

{d_{{i_1}}}, \ldots ,{d_{{i_k}}}(1 \leq i > \cdots > {i_k} \leq n)

is the set of all even integers among

{

d
1

,
…
,

d
n

}

\left \{ {{d_1}, \ldots ,{d_n}} \right \}

, then

2
∤
k
,

d

i
1

/

2
,
…
,

d

i
k

/

2

2\nmid k,{d_{{i_1}}}/2, \ldots ,{d_{{i_k}}}/2

are pairwise prime, and

d

i
j

{d_{{i_j}}}

is prime to any odd number in

{

d
1

,
…
,

d
n

}
(
j
=
1
,
…
,
k
)

\{ {d_1}, \ldots ,{d_n}\} (j = 1, \ldots ,k)

.
On extreme points of families described by subordination
Authors: Younis Rahman 
Published: June 2010

Let

s

(
F
)

s\left ( F \right )

denote the set of analytic functions in

D
=
{
z
:

|

z

|

>
1
}

D = \{ z:|z| > 1\}

subordinate to an analytic function

F
F

. It is shown that if

F
F

is a polynomial then the extreme points of the closed convex hull of

s
(
F
)
⊂
{
F
∘
ϕ
:
ϕ
∈

extreme

points

of

B
(

H
0
∞

)
}

s(F) \subset \{ F \circ \phi :\phi \in {\text {extreme}}\;{\text {points}}\;{\text {of}}\;B(H_0^\infty )\}

. Also if

F
(
z
)
=

(
(
z
−
α
)

/

(
1
−

α
¯

z
)

)
n

,

|

α

|

>
1

F(z) = {((z - \alpha )/(1 - \bar \alpha z))^n},|\alpha | > 1

and

n
n

is a positive integer then the extreme points of the closed convex hull of

s
(
F
)
=
{
F
∘
ϕ
:
ϕ
∈

extreme

points

of

B
(

H
0
∞

)
}

s(F) = \{ F \circ \phi :\phi \in {\text {extreme}}\;{\text {points}}\;{\text {of}}\;B(H_0^\infty )\}

. An analogue of Ryff’s theorem, and other results related to subordination in Bergman spaces have been obtained.
Rees algebras and mixed multiplicities
Authors: Verma J. K. 
Published: June 2010

Let

(
R
,
m
)

(R,m)

be a local ring of positive dimension

d
d

and

I
I

and

J
J

two

m
m

-primary ideals of

R
R

. Let

T
T

denote the Rees algebra

R
[
J
t
]

R[Jt]

localized at the maximal homogeneous ideal

(
m
,
J
t
)

(m,Jt)

. It is proved that
\[

e
(
(
I
,
J
t
)
T
=

e
0

(
I

|

J
)
+

e
1

(
I

|

J
)
+
⋯
+

e

d
−
1

(
I

|

J
)
,

e((I,Jt)T = {e_0}(I|J) + {e_1}(I|J) + \cdots + {e_{d - 1}}(I|J),

\]
where

e
i

(
I

|

J
)
,
i
=
0
,
1
,
…
,
d
−
1

{e_i}(I|J),i = 0,1, \ldots ,d - 1

are the first

d
d

mixed multiplicities of

I
I

and

J
J

. A formula due to Huneke and Sally concerning the multiplicity of the Rees algebra (of a complete zero-dimensional ideal of a two dimensional regular local ring) at its maximal homogeneous ideal is recovered.
On complementation of vector-valued Hardy spaces
Authors: Hensgen Wolfgang 
Published: June 2010

Let

X
X

be a complex Banach space and

1
>
p
>
∞

1 > p > \infty

.

H
p

(
X
)

{H^p}(X)

resp.

h
p

(
X
)

{h^p}(X)

denote the Hardy spaces of

X
X

-valued analytic resp. harmonic functions on the disc.

L
p

(
X
)

{L^p}(X)

is the Lebesgue-Bochner space of

X
X

-valued integrable functions on the circle and

H

p

(
X
)

{{\mathbf {H}}^p}(X)

its Hardy-type subspace

{
f
∈

L
p

(
X
)
:

f
^

(
n
)
=
0
∀
n
>
0
}

\{ f \in {L^p}(X):\hat f(n) = 0\forall n > 0\}

. It is proved that the following four conditions are equivalent:

H
p

(
X
)

{H^p}(X)

is complemented in

h
p

(
X
)

{h^p}(X)

; the canonical analytic (or Riesz) projection is a bounded operator

h
p

(
X
)
→

H
p

(
X
)
;

H

p

(
X
)

{h^p}(X) \to {H^p}(X);{{\mathbf {H}}^p}(X)

is complemented in

L
p

(
X
)

{L^p}(X)

; analytic projection is a bounded operator

L
p

(
X
)
→

H

p

(
X
)

{L^p}(X) \to {{\mathbf {H}}^p}(X)

. It is well known that the last condition, in turn, is equivalent to the UMD property of

X
X

.
Metacyclic 𝑝-algebras
Authors: Kang Ming Chang 
Published: June 2010

Let

K
K

be a field of char

K
=
p
>
0
,
m
,
r

K = p > 0,m,r

any positive integers with

(
p
,
m
)
=
1

(p,m) = 1

, and

L
L

a metacyclic extension of

K
K

with degree

p
r

m

{p^r}m

, i.e.

Gal(

L

/

K
)
=

⟨

σ
,
τ
:

σ

p
r

=

τ
m

=
1
,
τ
σ

τ

−
1

=

σ
e

⟩

{\text {Gal(}}L/K) = \left \langle {\sigma ,\tau :{\sigma ^{{p^r}}} = {\tau ^m} = 1,\tau \sigma {\tau ^{ - 1}} = {\sigma ^e}} \right \rangle

for some integer

e
e

. If

A
A

is a central simple

K
K

-algebra of degree

p
r

{p^r}

and is split by

L
L

, then

A
A

is a cyclic algebra. For

r
=
1

r = 1

, the theorem has been proved by A. A. Albert [1].
Convolution operators on groups and multiplier theorems for Hermite and Laguerre expansions
Authors: Długosz Jolanta 
Published: June 2010

Using harmonic analysis on nilpotent Lie groups the following theorem is proved. Let a sequence

{

a

n

}

\{ {a_{\text {n}}}\}

be defined by a function

K
∈

C
N

(

R

+

)

K \in {C^N}({{\mathbf {R}}_ + })

such that

sup

λ
>
0

|

K

(
j
)

(
λ
)

λ
j

|

>
∞
,
j
=
0
,
1
,
…
,
N

{\sup _{\lambda > 0}}|{K^{(j)}}(\lambda ){\lambda ^j}| > \infty ,j = 0,1, \ldots ,N

, for

N
N

sufficiently large, putting

a

n

=
K
(

|

n

|

+
m

/

2
)

{a_{\text {n}}} = K(|{\text {n}}| + m/2)

. Let

φ

n

{\varphi _{\text {n}}}

be either Hermite or Laguerre functions. Then the operator
\[

∑

n

(
f
,

φ

n

)

φ

n

→

∑

n

a

n

(
f
,

φ

n

)

φ

n

\sum \limits _{\text {n}} {(f,{\varphi _{\text {n}}}){\varphi _{\text {n}}} \to \sum \limits _{\text {n}} {{a_{\text {n}}}(f,{\varphi _{\text {n}}}){\varphi _{\text {n}}}} }

\]
is bounded on

L
p

(

R

m

)

{L^p}\left ( {{\mathbb {R}^m}} \right )

or

L
p

(

R

+
m

)

{L^p}(\mathbb {R}_ + ^m)

respectively,

1
>
p
>
∞

1 > p > \infty

.
Exponential dichotomies and Fredholm operators
Authors: Palmer Kenneth J. 
Published: June 2010

It is shown that if the operator

(

L
x

)

(
t
)

=

x
˙

(
t
)

−
A

(
t
)

x

(
t
)

\left ( {Lx} \right )\left ( t \right ) = \dot x\left ( t \right ) - A\left ( t \right )x\left ( t \right )

is semi-Fredholm, then the differential equation

x
˙

=
A

(
t
)

x

\dot x = A\left ( t \right )x

has an exponential dichotomy on both

[
0
,
∞
)

[0,\infty )

and

(
−
∞
,
0
]

( - \infty ,0]

. This gives a converse to an earlier result.
A necessary and sufficient condition for convergence in law of random sums of random variables under nonrandom centering
Authors: Finkelstein Mark , Tucker Howard G. 
Published: June 2010

Let

{

X
n

}

\{X_n\}

be a sequence of independent, identically distributed (i.i.d.) random variables with common mean

μ
≠
0

\mu \ne 0

and variance

σ
2

>
0

\sigma ^2 > 0

. Let

{

S
n

}

\{S_n\}

be a sequence of nonnegative integer-valued random variables such that for each

n
n

the random variables

S
n

S_n

,

X
1

X_1

,

X
2

X_2

, … are independent. Then

(

X
1

+
⋯
+

X

S
n

−
n
μ
)

/

n

σ
2

→

L

(X_1 + \dots + X_{S_n} - n \mu ) / \sqrt {n\sigma ^2} \stackrel {\mathcal {L}}{\rightarrow } {}

(some)

Z
Z

if and only if

(

S
n

−
n
)

/

n

→

L

(S_n - n)/\sqrt {n} \stackrel {\mathcal {L}}{\rightarrow } {}

(some)

U
U

, in which case the distribution of

Z
Z

is that of

X
+
Y

X+Y

, where

X
X

and

Y
Y

are independent random variables,

X
X

being

N

(
0
,
1
)

\mathcal {N}(0, 1)

and

Y
Y

having the same distribution as

μ
U

/

σ

\mu U/\sigma

.
The relationships of spans of convex continua in 𝑅ⁿ
Authors: West Thelma 
Published: June 2010

It has been conjectured that

σ
∗

(
X
)
≥

1
2

σ
0
∗

(
X
)

{\sigma ^ * }(X) \geq \tfrac {1}{2}\sigma _0^ * (X)

for each nonempty connected metric space

X
X

. In this paper we show that

σ
∗

(
X
)
≥

13

4

σ
0
∗

(
X
)

{\sigma ^ * }(X) \geq \tfrac {{\sqrt {13} }}{4}\sigma _0^ * (X)

for each convex continuum

X
X

in

R
n

{R^n}

. We also show that under certain conditions a lower bound for the ratio

σ
∗

(
X
)

/

σ
0
∗

(
X
)

{\sigma ^ * }(X)/\sigma _0^ * (X)

is larger than

13

4

\tfrac {{\sqrt {13} }}{4}

. It has also been conjectured that

σ
∗

(
X
)
≥
σ
(
X
)

/

2

{\sigma ^ * }(X) \geq \sigma (X)/2

and that

σ
0
∗

(
X
)
≥

σ
0

(
X
)

/

2

\sigma _0^ * (X) \geq {\sigma _0}(X)/2

for each nonempty connected metric space

X
X

. We show that these two inequalities hold when

X
X

is a convex continuum in

R
n

{R^n}

.
On the Diophantine equation ∑ⁿᵢ₌₁𝑥ᵢ/𝑑ᵢ≡0 (mod 1)
Authors: Sun Qi , Wan Da Qing 
Published: June 2010

Let

d
1

,
…
,

d
n

{d_1}, \ldots ,{d_n}

be

n
n

positive integers. The purpose of this note is to study the number of solutions and the least solutions of the following diophantine equation:
\[

(
1
)

x
1

d
1

+
⋯
+

x
n

d
n

≡
0
(
mod
1
)
,

1
≤

x
i

≤

d
i

−
1
,

(1)\quad \frac {{{x_1}}}{{{d_1}}} + \cdots + \frac {{{x_n}}}{{{d_n}}} \equiv 0(\bmod 1),\quad 1 \leq {x_i} \leq {d_i} - 1,

\]
which arises from diagonal hypersurfaces over a finite field. In particular, we determine all the

d
i

{d_i}

’s for which (1) has a unique solution.
A comparison result for the oscillation of delay differential equations
Authors: Ladas G. , Qian C. , Yan J. 
Published: June 2010

We obtain a comparison result for the oscillation of all solutions of the equation\[y˙(t)+∑i=1nqi(t)y(t−σi(t))=0\dot y(t) + \sum \limits _{i = 1}^n {{q_i}(t)y(t - {\sigma _i}(t)) = 0}\]in terms of the oscillation of all solutions of the equation\[x˙(t)+∑i=1npi(t)x(t−τi(t))=0\dot x(t) + \sum \limits _{i = 1}^n {{p_i}(t)x(t - {\tau _i}(t)) = 0}\]under appropriate hypotheses on the asymptotic behavior of the quotientspi(t)/qi(t){p_i}(t)/{q_i}(t)andτi(t)/σi(t){\tau _i}(t)/{\sigma _i}(t)fori=1,2,…,ni = 1,2, \ldots ,n. An application to the oscillation of the nonautonomous delay-logistic equation is given.
An inequality of Araki-Lieb-Thirring (von Neumann algebra case)
Authors: Kosaki Hideki 
Published: June 2010

For a trace

τ
\tau

on a semifinite von Neumann algebra we will prove

τ
(

(

b

1

/

2

a

b

1

/

2

)

r
p

)
≤
τ
(

(

b

r

/

2

a
r

b

r

/

2

)
p

)

\tau ({({b^{1/2}}a{b^{1/2}})^{rp}}) \leq \tau ({({b^{r/2}}{a^r}{b^{r/2}})^p})

. Here,

r
≥
1
,
p
>
0

r \geq 1,p > 0

, and

a
,
b

a,b

are positive operators.
Presentations for 3-dimensional special linear groups over integer rings
Authors: Conder Marston , Robertson Edmund , Williams Peter 
Published: June 2010

The following

2
2

-generator

6
6

-relator presentation is obtained for the

3
3

-dimensional special linear group

SL
⁡
(
3
,

Z

k

)

\operatorname {SL}(3,{\mathbb {Z}_k})

for each odd integer

k
>
1

k > 1

:
\[

SL
⁡
(
3
,

Z

k

)
=
⟨
x
,
y

|

x
3

=

y
3

=

(
x
y

)
6

=

(

x

−
1

y

x

−
1

y

−
1

x
y

)
2

=

(
x

y

−
1

x
y
x

y

−
1

x

−
1

y

−
1

)
k

=

(

(
x

y

−
1

x
y
x

y

−
1

x

−
1

y

−
1

)

(
k
−
1
)

/

2

x
y

)
4

=
1
⟩
.

\operatorname {SL}(3,{\mathbb {Z}_k}) = \langle x,y|{x^3} = {y^3} = {(xy)^6} = {({x^{ - 1}}y{x^{ - 1}}{y^{ - 1}}xy)^2} = {(x{y^{ - 1}}xyx{y^{ - 1}}{x^{ - 1}}{y^{ - 1}})^k} = {({(x{y^{ - 1}}xyx{y^{ - 1}}{x^{ - 1}}{y^{ - 1}})^{(k - 1)/2}}xy)^4} = 1\rangle .

\]
Alternative presentations for these groups and other groups associated with them are also given.
On the cup product for groups
Authors: Dyer M. N. , Leahy J. V. , Mooney L. J. 
Published: June 2010

For

G
G

any group and trivial coefficients in

R
R

, a commutative ring, the authors analyze the kernel of the cup product

∪
:

H
1

(

G
,
R

)

⊗

H
1

(

G
,
R

)

→

H
2

(

G
,
R

)

\cup :{H^1}\left ( {G,R} \right ) \otimes {H^1}\left ( {G,R} \right ) \to {H^2}\left ( {G,R} \right )

by splicing it together with part of the six-term Hom-Ext exact sequence obtained from

0
→

I
2

/

I
3

→
I

/

I
3

→
I

/

I
2

→
0

0 \to {I^2}/{I^3} \to I/{I^3} \to I/{I^2} \to 0

.
On Opial’s inequality for 𝑓⁽ⁿ⁾
Authors: Fink A. M. 
Published: June 2010

We prove inequalities of the type
\[

∫
0
h

|

f

(
i
)

(
x
)

f

(
j
)

(
x
)

|

d
x
≤
C
(
n
,
i
,
j
,
p
)

h

2
n
−
i
−
j
+
1
−
2

/

p

(

∫
0
h

|

f

(
n
)

(
x
)

|

p

d
x

)

2

/

p

{\int _0^h {|{f^{(i)}}(x){f^{(j)}}(x)|dx \leq C(n,i,j,p){h^{2n - i - j + 1 - 2/p}}\left ( {\int _0^h {|{f^{(n)}}(x){|^p}dx} } \right )} ^{2/p}}

\]
when

f
(
0
)
=

f
′

(
0
)
=
⋯
=

f

(
n
−
1
)

(
0
)
=
0

f(0) = f’(0) = \cdots = {f^{(n - 1)}}(0) = 0

. We assume that

f

(
n
−
1
)

{f^{(n - 1)}}

is absolutely continuous and

f

(
n
)

∈

L
p

(
0
,
h
)

{f^{(n)}} \in {L_p}(0,h)

, with

p
≥
1
,
n
≥
2

p \geq 1,n \geq 2

, and

0
≤
i
≤
j
≤
n
−
1

0 \leq i \leq j \leq n - 1

.
Some remarks of drop property
Authors: Lin Pei-Kee 
Published: June 2010

Let

C
C

be a proper closed convex set.

C
C

is said to have the drop property if for any nonempty closed set

A
A

disjoint with

C
C

, there is

a
∈
A

a \in A

such that

co(

a
,
C

)

∩

A =

{
a
}

{\text {co(}}a,C{\text {)}} \cap {\text {A = }}\left \{ a \right \}

. We show that if

X
X

contains a noncompact set with the drop property, then

X
X

is reflexive. Moreover, we prove that if

C
C

is a noncompact closed convex subset of a reflexive Banach space, then

C
C

has the drop property if and only if

C
C

satisfies the following conditions: (i) the interior of

C
C

is nonempty; (ii)

C
C

does not have any asymptote, and the boundary of

C
C

does not contain any ray; and (iii) every support point

x
x

of

C
C

is a point of continuity.
A fixed point theorem in non-Archimedean vector spaces
Authors: Petalas C. , Vidalis T. 
Published: June 2010

A mapping

T
T

defined on a normed linear space

X
X

and taking values in

X
X

is said to be contractive (nonexpansive) if whenever

x
x

and

y
y

are distinct points in

X
,

|

|

T
x
−
T
y

|

|

>

|

|

x
−
y

|

|

(

|

|

T
x
−
T
y

|

|

⩽

|

|

x
−
y

|

|

)

X,\,||Tx - Ty|| > ||x - y||\;(||Tx - Ty|| \leqslant ||x - y||)

. In this paper we prove that every contractive mapping on a spherically complete non-Archimedean normed space has a unique fixed point.
An Engel condition with derivation
Authors: Lanski Charles 
Published: June 2010

Let

R
R

be a prime ring,

L
L

a noncommutative Lie ideal of

R
R

, and

D
D

a nonzero derivation of

R
R

. If for each

x
∈
L

x \in L

,

[
D
(
x
)
,
x

]
k

=
[
[
⋯
[
D
(
x
)
,
x
]
,
x
]
,
…
,
x
]
=
0

{[D(x),x]_k} = [[ \cdots [D(x),x],x], \ldots ,x] = 0

with

k
k

fixed, then

char
⁡
(
R
)
=
2

\operatorname {char} (R) = 2

and

R
⊆

M
2

(
F
)

R \subseteq {M_2}(F)

for

F
F

a field.
Potential space estimates for Green potentials in convex domains
Authors: Fromm Stephen J. 
Published: June 2010

Weak type

(
1
,
1
)

(1,1)

bounds are demonstrated for the operators
\[

f
↦
∫

∇
x

∇
x

G
(
x
,
y
)
f
(
y
)
d
y

and

f
↦
∫

∇
x

∇
y

G
(
x
,
y
)
f
(
y
)
d
y
,

f \mapsto \int {{\nabla _x}{\nabla _x}G(x,y)f(y)dy} \quad {\text {and}}\quad f \mapsto \int {{\nabla _x}{\nabla _y}G(x,y)f(y)dy,}

\]
where

G
G

is the Green operator for the Dirichlet problem for the Poisson equation on a bounded convex domain in

R

n

{\mathbb {R}^n}

. These results are used to investigate smoothing properties of the Green operator in potential spaces. An application is given to the restriction of the potential space to the boundary of the domain.
A function in the Dirichlet space such that its Fourier series diverges almost everywhere
Authors: De Souza Geraldo Soares , Sampson G. 
Published: June 2010

An analytic function

F
F

on the disc belongs to

B
B

if

|

|

F

|

|

B

=

∫
0
1

∫
0

2
π

|

F
′

(
r

e

i
θ

)

|

d
θ

d
r
>
∞

||F|{|_B} = \int _0^1 {\int _0^{2\pi } {|{F’}(r{e^{i\theta }})|d\theta \,dr > \infty } }

. Notice that

B
⫋

H
1

⫋

L
1

B \varsubsetneqq {H^1} \varsubsetneqq {L^1}

, where

H
1

{H^1}

is the Hardy space of all analytic functions

F
F

so that
\[

|

|

F

|

|

H
1

=

sup

0
>
r
>
1

∫
0

2
π

|

F
(
r

e

i
θ

)

|

d
θ
>
∞
,

||F|{|_{{H^1}}} = \sup \limits _{0 > r > 1} \int _0^{2\pi } {|F(r{e^{i\theta }})|\,d\theta > \infty ,}

\]

L
1

{L^1}

is the Lebesgue space of integrable functions on

[
0
,
2
π
]

[0,2\pi ]

, and the inclusion

H
1

⫋

L
1

{H^1} \varsubsetneqq {L^1}

is taken in the sense of boundary values, that is,

F
∈

H
1

⇒

lim

r
→
1

−
ℜ
F
(
r

e

i
θ

)
∈

L
1

F \in {H^1} \Rightarrow {\lim _{r \to 1}} - \Re F(r{e^{i\theta }}) \in {L^1}

. Kolmogorov in 1923 showed that there exists an

f
f

in

L
1

{L^1}

so that its Fourier series diverges almost everywhere. In 1953 Sunouchi showed that there exists an

f
f

in

H
1

{H^1}

with an almost everywhere divergent Fourier series. The purpose of this note is to announce. Theorem 1. There exists an

f
∈
B

f \in B

, whose Fourier series diverges a.e. This problem was mentioned to the first author by Professor Guido Weiss.
Equivariant maps for homology representations
Authors: Dotzel Ronald M. 
Published: June 2010

If Y is a homotopy representation of the finite group G of order n and X is a finite G-CW complex such that, for each subgroup H of G,

H
∗

(

X
H

;

Z

n

)
=

H
∗

(

Y
H

;

Z

n

)

{H_ \ast }({X^H};{\mathbb {Z}_n}) = {H_ \ast }({Y^H};{\mathbb {Z}_n})

then there exists a G-map

f
:
X
→
Y

f:X \to Y

such that

f
∗
H

:

H
∗

(

X
H

;

Z

n

)
→

H
∗

(

Y
H

;

Z

n

)

f_ \ast ^H:{H_ \ast }({X^H};{\mathbb {Z}_n}) \to {H_ \ast }({Y^H};{\mathbb {Z}_n})

is an isomorphism for each subgroup H.
The weak stability of the positive face in 𝐿¹
Authors: Hu Zhibao 
Published: June 2010

Let F be the positive face of the unit ball of

L
1

[
0
,
1
]

{L^1}[0,1]

. We show that F is weakly stable in the sense that the midpoint map

Φ

1

/

2

:
F
×
F
→
F

{\Phi _{1/2}}:F \times F \to F

, with

Φ

1

/

2

(
f
,
g
)
=

1
2

(
f
+
g
)

{\Phi _{1/2}}(f,g) = \frac {1}{2}(f + g)

, is open with respect to the weak topology. This weak stability of the set F is the reason behind the fact that the notions of "huskable" and "strongly regular" operators coincide for operators from

L
1

[
0
,
1
]

{L^1}[0,1]

to a Banach space X. We prove this stability by showing that if

f
1

,

f
2

∈
F
,
λ
∈
(
0
,
1
)
,
ε
>
0

{f_1},{f_2} \in F,\lambda \in (0,1),\varepsilon > 0

and

δ
≥
max
{
2
ε

/

λ
,
2
ε

/

(
1
−
λ
)
}

\delta \geq \max \{ 2\varepsilon /\lambda ,2\varepsilon /(1 - \lambda )\}

, then
\[

λ

V

P
,
δ

(

f
1

)
+
(
1
−
λ
)

V

P
,
δ

(

f
2

)
⊃

V

P
,
ε

[
λ

f
1

+
(
1
−
λ
)

f
2

]
,

\lambda {V_{P,\delta }}({f_1}) + (1 - \lambda ){V_{P,\delta }}({f_2}) \supset {V_{P,\varepsilon }}[\lambda {f_1} + (1 - \lambda ){f_2}],

\]
where

P
=
{

A
1

,
…
,

A
n

}

P = \{ {A_1}, \ldots ,{A_n}\}

is a finite positive partition of [0, 1] and
\[

V

P
,
ε

(
f
)
=

{

g
∈
F
:

∑

i
=
1

n

|

∫

A
i

(
f
−
g
)
(
t
)
d
μ
(
t
)

|

≤
ε

}

{V_{P,\varepsilon }}(f) = \left \{ {g \in F:\sum \limits _{i = 1}^n {\left | {\int _{{A_i}} {(f - g)(t)d\mu (t)} } \right | \leq \varepsilon } } \right \}

\]
for any f in F. We construct an example showing that for any

0
>
λ
>
1

0 > \lambda > 1

there are functions

f
1

{f_1}

and

f
2

{f_2}

in F such that if

0
>
ε
>
2
min
{
λ
,
1
−
λ
}

0 > \varepsilon > 2\min \{ \lambda ,1 - \lambda \}

and

0
≤
δ
>
max
{
ε

/

λ
,
ε

/

(
1
−
λ
)
}

0 \leq \delta > \max \{ \varepsilon /\lambda ,\varepsilon /(1 - \lambda )\}

, then
\[

λ

V

P
,
δ

(

f
1

)
+
(
1
−
λ
)

V

P
,
δ

(

f
2

)
⊅

V

P
,
ε

(
λ

f
1

+
(
1
−
λ
)

f
2

)
.

\lambda {V_{P,\delta }}({f_1}) + (1 - \lambda ){V_{P,\delta }}({f_2})\not \supset {V_{P,\varepsilon }}(\lambda {f_1} + (1 - \lambda ){f_2}).

\]
Thus the "formula" that

λ

V

p
,
ε

(

f
1

)
+
(
1
−
λ
)

V

p
,
ε

(

f
2

)
=

V

p
,
ε

(
λ

f
1

+
(
1
−
λ
)

f
2

)

\lambda {V_{p,\varepsilon }}({f_1}) + (1 - \lambda ){V_{p,\varepsilon }}({f_2}) = {V_{p,\varepsilon }}(\lambda {f_1} + (1 - \lambda ){f_2})

given by Ghoussoub et al. in Mem. Amer. Math. Soc., vol. 70, no. 378, which is used there to establish the weak stability of F, is false.
On a sequence transformation with integral coefficients for Euler’s constant
Authors: Elsner C. 
Published: June 2010

Let

γ
\gamma

denote Euler’s constant, and let
\[

s
n

=

(

1
+

1
2

+
⋯
+

1

n
−
1

)

−
log
⁡
n

(
n
≥
2
)
.

{s_n} = \left ( {1 + \frac {1}{2} + \cdots + \frac {1}{{n - 1}}} \right ) - \log n\quad (n \geq 2).

\]
We prove by Ser’s formula for the remainder

γ
−

s
n

\gamma - {s_n}

that for all integers

n
≥
1

n \geq 1

and

τ
≥
2

\tau \geq 2

there are integers

μ

n
,
0
,

μ

n
,
1

,
…
,

μ

n
,
n

{\mu _{n,0,}}{\mu _{n,1}}, \ldots ,{\mu _{n,n}}

such that
\[

μ

n
,
0

s
τ

+

μ

n
,
1

s

τ
+
1

+
⋯
+

μ

n
,
n

s

τ
+
n

=
γ
+

O
τ

(

(
n
(
n
+
1
)
(
n
+
2
)
∙
⋯
∙
(
n
+
τ
)

)

−
1

)
,

{\mu _{n,0}}{s_\tau } + {\mu _{n,1}}{s_{\tau + 1}} + \cdots + {\mu _{n,n}}{s_{\tau + n}} = \gamma + {O_\tau }({(n(n + 1)(n + 2) \bullet \cdots \bullet (n + \tau ))^{ - 1}}),

\]
where the constant in

O
τ

{O_\tau }

depends only on

τ
\tau

. The coefficients

μ

n
,
k

{\mu _{n,k}}

are explicitly given and are bounded by

2

3
n
+
τ
−
1

{2^{3n + \tau - 1}}

.
A lower bound for the class numbers of abelian algebraic number fields with odd degree
Authors: Le Mao Hua 
Published: June 2010

Let

Δ
K

,

h
K

,

R
K

{\Delta _K},{h_K},{R_K}

denote the discriminant, the class number, and the regulator of the Abelian algebraic number field

K
=

Q

(
α
)

K = \mathbb {Q}(\alpha )

with degree d, respectively. In this note we prove that if

d
>
1
,
2
∤
d

d > 1,2\nmid d

, and the defining polynomial of

α
\alpha

has exactly

r
1

{r_1}

real zeros and

r
2

{r_2}

pairs of complex zeros, then

h
K

>
w

|

Δ
K

|

/

2

r
1

(
2
π

)

r
2

33

R
K

log
⁡
4

|

Δ
K

|

{h_K} > w\sqrt {|{\Delta _K}|} /{2^{{r_1}}}{(2\pi )^{{r_2}}}33{R_K}\log 4|{\Delta _K}|

, where w is the number of roots of unity in K.
The Hardy-Littlewood theorem on fractional integration for Laguerre series
Authors: Kanjin Yūichi , Sato Enji 
Published: June 2010

The Hardy-Littlewood theorem on fractional integration for Fourier series says that if

I
σ

g
∼

∑

n
≠
0

|

n

|

−
σ

g
^

(
n
)

e

int

{I_\sigma }g \sim \sum \nolimits _{n \ne 0} {|n{|^{ - \sigma }}\hat g} (n){e^{\operatorname {int} }}

, then

I
σ

{I_\sigma }

is bounded from

L
p

{L^p}

to

L
q

{L^q}

, where

1
>
p
>
q
>
∞
,

1
q

=

1
p

−
σ

1 > p > q > \infty ,\frac {1}{q} = \frac {1}{p} - \sigma

. We shall establish an analogue of this theorem for the Laguerre function system

{

(

n
!

Γ
(
n
+
α
+
1
)

)

1
2

L
n
α

(
x
)

e

−

x
2

x

α
2

}

n
=
0

∞

\left \{ {{{\left ( {\frac {{n!}}{{\Gamma (n + \alpha + 1)}}} \right )}^{\frac {1}{2}}}L_n^\alpha (x){e^{ - \frac {x}{2}}}{x^{\frac {\alpha }{2}}}} \right \}_{n = 0}^\infty

.