bluuebunny/crossref_metadata_2025_split huggingface dataset:
A note on the central limit theorem for square-integrable processes
Authors: Hahn Marjorie G. 
Published: June 2010

A method is given for constructing sample-continuous processes which do not satisfy the central limit theorem in

C
[
0
,
1
]

C[0,1]

. Let

{
X
(
t
)
:
t
∈
[
0
,
1
]
}

\{ X(t):t \in [0,1]\}

be a stochastic process. Using our method we characterize all possible nonnegative functions f for which the condition
\[

E
(
X
(
t
)
−
X
(
s
)

)
2

⩽
f
(

|

t
−
s

|

)

E( X(t) - X(s) )^2 \leqslant f( | t - s | )

\]
alone is sufficient to imply that

X
(
t
)

X(t)

satisfies the central limit theorem in

C
[
0
,
1
]

C[0,1]

.
A remark on directional contractions
Authors: Kirk W. A. , Ray William O. 
Published: June 2010

Let X be a Banach space and D a convex subset of X. A mapping

T
:
D
→
X

T:D \to X

is called a directional contraction if there exists a constant

α
∈
(
0
,
1
)

\alpha \in (0,1)

such that corresponding to each

x
,
y
∈
D

x,y \in D

there exists

ε
=
ε
(
x
,
y
)
∈
(
0
,
1
]

\varepsilon = \varepsilon (x,y) \in (0,1]

for which

‖

T
(
x
+
ε
(
y
−
x
)
)
−
T
(
x
)

‖

⩽
α
ε

‖

x
−
y

‖

\left \| {T(x + \varepsilon (y - x)) - T(x)} \right \| \leqslant \alpha \varepsilon \left \| {x - y} \right \|

. Tests for lipschitzianness are obtained which yield the fact that if a closed mapping is a directional contraction, then it must be a global contraction, and sufficient conditions are given under which a nonclosed directional contraction

T
:
D
→
D

T:D \to D

always has a fixed point.
Landau-Kolmogorov inequalities for semigroups and groups
Authors: Certain Melinda W. , Kurtz Thomas G. 
Published: June 2010

An elementary functional analytic argument is given showing how inequalities of the form

‖

f

(
k
)

‖

n

⩽

K

n
,
k

‖
f
‖

n
−
k

‖

f

(
n
)

‖

k

{\left \| {{f^{(k)}}} \right \|^n} \leqslant {K_{n,k}}{\left \| f \right \|^{n - k}}{\left \| {{f^{(n)}}} \right \|^k}

, where f is a real, n-times differentiable function and

‖
⋅
‖

\left \| \cdot \right \|

denotes the sup norm on

(
0
,
∞
)

(0,\infty )

(or

(
−
∞
,
∞
)

( - \infty ,\infty )

), yield corresponding inequalities,

|

A
k

x

|

n

⩽

K

n
,
k

|

x

|

n
−
k

|

A
n

x

|

k

|{A^k}x{|^n} \leqslant {K_{n,k}}|x{|^{n - k}}|{A^n}x{|^k}

, for generators of linear contraction semigroups (or groups) on arbitrary Banach spaces with norm

|

⋅

|

| \cdot |

. Since Landau, Kolmogorov, Schoenberg and Cavaretta have established the function inequalities with the best possible constants, this argument gives the generator inequalities with the best possible constants for general Banach spaces extending work of Kallman and Rota, Hille and others. Questions concerning the best possible constants for specific Banach spaces remain open.
Successive derivatives of entire functions
Authors: Hellerstein Simon , Williamson Jack 
Published: June 2010

We show that if f is a real entire function which has, along with each of its derivatives, only real nonpositive zeros, then either

f
(
z
)
=
c

e

σ
z

,
c

f(z) = c{e^{\sigma z}},c

and

σ
\sigma

real constants, or
\[

f
(
z
)
=
c

z
m

e

σ
z

∏
n

(

1
+

z

|

z
n

|

)

f(z) = c{z^m}{e^{\sigma z}}\prod \limits _n {\left ( {1 + \frac {z}{{|{z_n}|}}} \right )}

\]
where

σ
⩾
0

\sigma \geqslant 0

and

∑
n

|

z
n

|

−
1

>
∞

\sum \nolimits _n {|{z_n}{|^{ - 1}} > \infty }

.
Note on Brewer’s character sum
Authors: Williams Kenneth S. 
Published: June 2010

A very short proof is given of the result
\[

∑

x
=
0

p
−
1

(

(
x
+
2
)
(

x
2

−
2
)

p

)

=
0
,

\sum \limits _{x = 0}^{p - 1} {\left ( {\frac {{(x + 2)({x^2} - 2)}}{p}} \right )} = 0,

\]
if

p
≡
5

p \equiv 5

or

7

(
mod

8
)

7 \pmod 8

.
Nice sets of multi-indices
Authors: Madych W. R. , Szeptycki P. 
Published: June 2010

Finite sets, A, of n-tuples for which

(

Σ

α
∈
A

(

∏

j
=
1

n

|

x
j

|

α
j

)

)

−
p

,
p
>
0

{({\Sigma _{\alpha \in A}}(\prod _{j = 1}^n|{x_j}{|^{{\alpha _j}}}))^{ - p}},p > 0

, is integrable over

R
n

{R^n}

are given a simple characterization. Applications to certain Fourier multiplier theorems are mentioned.
On a theorem of Duren, Shapiro and Shields
Authors: Yamashita Shinji 
Published: June 2010

We shall show that, an extension of the theorem of Duren, Shapiro and Shields on the univalence of a function f holomorphic in the unit disk D, still remains of significance, if the power

α
∈
(
0
,
1
)

\alpha \in (0,1)

in
\[

sup

z
∈
D

(
1
−

|

z

|

2

)
α

|

f
(
z
)

/

f
′

(
z
)

|

{\sup _{z \in D}}{(1 - |z{|^2})^\alpha }|f(z)/f’(z)|

\]
is small enough.
Evaluation of character sums connected with elliptic curves
Authors: Williams Kenneth S. 
Published: June 2010

Let p be an odd prime and let

(

⋅
p

)

(\tfrac { \cdot }{p})

be the Legendre symbol. It is shown how to evaluate the character sum

Σ

x
=
0

p
−
1

(

f
(
x
)

p

)

\Sigma _{x = 0}^{p - 1}(\tfrac {{f(x)}}{p})

, for certain quartic polynomials

f
(
x
)

f(x)

. For example, it is shown that
\[

∑

x
=
0

p
−
1

(

x
4

−
8

x
3

+
12

x
2

−
16
x
+
4

p

)

=

{

2

(

2
p

)

x
1

−
1
,

a
m
p
;

if

p
≡
1

(
mod

4
)
,

−
1
,

a
m
p
;

if

p
≡
3

(
mod

4
)
,

\sum \limits _{x = 0}^{p - 1} {\left ( {\frac {{{x^4} - 8{x^3} + 12{x^2} - 16x + 4}}{p}} \right )} = \left \{ {\begin {array}{*{20}{c}} {2\left ( {\frac {2}{p}} \right ){x_1} - 1,} \hfill & {{\text {if}}\;p \equiv 1\;\pmod 4,} \hfill \\ { - 1,} \hfill & {{\text {if}}\;p \equiv 3\;\pmod 4,} \hfill \\ \end {array} } \right .

\]
where

x
1

{x_1}

is defined for primes

p
≡
1

(
mod

4
)

p \equiv 1\;\pmod 4

by
\[

p
=

x
1
2

+

y
1
2

,

x
1

≡
−
1

(
mod

4
)
.

p = x_1^2 + y_1^2,\quad {x_1} \equiv - 1\quad \pmod 4.

\]
Stochastic construction of new orthogonal polynomials
Authors: Lin T. F. 
Published: June 2010

For

u
>
−
1
,
t
⩾
0
,
x
∈
R

u > - 1,t \geqslant 0,x \in R

, let

(
1
+
u

)

Q
(
t
)
+
t

e

−
u
t

{(1 + u)^{Q(t) + t}}{e^{ - ut}}

satisfies

d
y
(
t
)
=
u
y
(
t
)

d
Q
(
t
)

dy(t) = uy(t)\;dQ(t)

where

Q
(
t
)
=
P
(
t
)
−
t

Q(t) = P(t) - t

is the centered Poisson process with parameter

λ
=
1

\lambda = 1

, (ii)

K
n

(
t
,
x
)
,
n
⩾
0

{K_n}(t,x),n \geqslant 0

, is a system of complete orthogonal polynomials in

L
2

(
R
,
d
F
)

{L^2}(R,dF)

where

F
(
x
)

F(x)

is the distribution function of

P
(
t
)

P(t)

, (iii)

(
n
+
1
)

K

n
+
1

(
t
,
x
)
−
(
x
−
t
−
n
)

K
n

(
t
,
x
)
+
t

K

n
−
1

(
t
,
x
)
=
0
,
n
⩾
1

(n + 1){K_{n + 1}}(t,x) - (x - t - n){K_n}(t,x) + t{K_{n - 1}}(t,x) = 0,n \geqslant 1

, (iv)

t

Δ
2

K
n

(
t
,
x
)
+
(
t
−
x
)
Δ

K
n

(
t
,
x
)
+
n

K
n

(
t
,
x
)
=
0
,
n
⩾
0

t{\Delta ^2}{K_n}(t,x) + (t - x)\Delta {K_n}(t,x) + n{K_n}(t,x) = 0,n \geqslant 0

.
Intersections of commutants with closures of derivation ranges
Authors: Herrero Domingo A. 
Published: June 2010

The norm closure of the set

A

w

(

X

)
=
∪

{

Ran

(

δ
A

)

−
w

∩
{
A

}
′

:
A
∈

L

(

X

)
}

{\mathcal {A}_w}(\mathcal {X}) = \cup \;\{ {\text {Ran}}{({\delta _A})^{ - w}} \cap \{ A\} ’:A \in \mathcal {L}(\mathcal {X})\}

, where

δ
A

{\delta _A}

denotes the inner derivation induced by the operator A,

Ran

(

δ
A

)

−
w

{\text {Ran}}{({\delta _A})^{ - w}}

is the weak closure of the range of

δ
A

{\delta _A}

and

{
A

}
′

\{ A\} ’

is the commutant of A, is disjoint from the open dense subset

B

(

X

)
=
{
T
∈

L

(

X

)

\mathcal {B}(\mathcal {X}) = \{ T \in \mathcal {L}(\mathcal {X})

: T has a nonzero normal eigenvalue} for every complex Banach space

X

\mathcal {X}

. For a Hilbert space

H

\mathcal {H}

,

L

(

H

)
=

B

(

H

)
∪

A

w

(

H

)
−

\mathcal {L}(\mathcal {H}) = \mathcal {B}(\mathcal {H}) \cup {\mathcal {A}_w}{(\mathcal {H})^ - }

, where the bar denotes norm closure.
On an extremal problem
Authors: Nevai Paul G. 
Published: June 2010

Let

X
=
(

x
1

,

x
2

,
…
,

x
N

)
,
f
:

R

→

C

X = ({x_1},{x_2}, \ldots ,{x_N}),f:{\mathbf {R}} \to {\mathbf {C}}

and let

P

n

{{\mathbf {P}}_n}

be the class of polynomials of degree at most n. The generalized Christoffel function

Λ
n

{\Lambda _n}

corresponding to the measure

d
α

d\alpha

is defined by
\[

Λ
n

(
X
;
f
,
N
,
d
α
)
=

min

π
∈

P

n
−
1

π
(

x
i

)
=
f
(

x
i

)

i
=
1
,
2
,
…
,
N

∫

−
∞

∞

|

π
(
t
)

|

2

d
α
(
t
)
.

{\Lambda _n}(X;f,N,d\alpha ) = \min \limits _{\begin {array}{*{20}{c}} {\pi \in {{\mathbf {P}}_{n - 1}}} \\ {\pi ({x_i}) = f({x_i})} \\ {i = 1,2, \ldots ,N} \\ \end {array} } \int _{ - \infty }^\infty {|\pi (t){|^2}d\alpha (t).}

\]
It is shown that if

α
\alpha

satisfies some rather weak conditions then

lim

n
→
∞

n

Λ
n

(
X
;
f
,
N
,
d
α
)

{\lim _{n \to \infty }}n{\Lambda _n}(X;f,N,d\alpha )

exists and the limit is also evaluated.
An approximation of integrable functions by step functions with an application
Authors: Crandall M. G. , Pazy A. 
Published: June 2010

Let

f
∈

L
1

(
0
,
∞
)
,
δ
>
0

f \in {L^1}(0,\infty ),\delta > 0

and

(

G
δ

f
)
(
t
)
=

δ

−
1

∫
t
∞

e

(
t
−
s
)

/

δ

f
(
s
)
d
s

({G_\delta }f)(t) = {\delta ^{ - 1}}\smallint _t^\infty {e^{(t - s)/\delta }}f(s)ds

. Given a partition

P
=
{
0
=

t
0

>

t
1

>
⋯
>

t
i

>

t

i
+
1

>
⋯
}

P = \{ 0 = {t_0} > {t_1} > \cdots > {t_i} > {t_{i + 1}} > \cdots \}

of

[
0
,
∞
)

[0,\infty )

where

t
i

→
∞

{t_i} \to \infty

, we approximate f by the step function

A
P

f

{A_P}f

defined by
\[

A
P

f
(
t
)
=
(

G

δ
i

G

δ

i
−
1

⋯

G

δ
i

f
)
(
0
)

for

t

i
−
1

⩽
t
>

t
i

,

{A_P}f(t) = ({G_{{\delta _i}}}{G_{{\delta _{i - 1}}}} \cdots {G_{{\delta _i}}}f)(0)\quad {\text {for}}\;{t_{i - 1}} \leqslant t > {t_i},

\]
where

δ
i

=

t
i

−

t

i
−
1

{\delta _i} = {t_i} - {t_{i - 1}}

. The main results concern several properties of this process, with the most important one being that

A
P

f
→
f

{A_P}f \to f

in

L
1

(
0
,
∞
)

{L^1}(0,\infty )

as

μ
(
P
)
=

sup
i

δ
i

→
0

\mu (P) = {\sup _i}{\delta _i} \to 0

. An application to difference approximations of evolution problems is sketched.
Contractification of a semigroup of maps
Authors: Ko Hwei Mei , Tan Kok Keong 
Published: June 2010

Let

(
X
,
τ
)

(X,\tau )

be a metrizable topological space,

P

(
τ
)

\mathcal {P}(\tau )

be the family of all metrics on X whose metric topologies are

τ
\tau

. Assume that the semigroup F of maps from X into itself, with composition as its semigroup operation, is equicontinuous under some

d
∈

P

(
τ
)

d \in \mathcal {P}(\tau )

; then we have the following results: I. There exists

d
′

∈

P

(
τ
)

d’ \in \mathcal {P}(\tau )

such that f is nonexpansive under

d
′

d’

for each

f
∈
F

f \in F

. II. If F is countable, commutative, and for each

f
∈
F

f \in F

, there is

x
f

∈
X

{x_f} \in X

such that the sequence

(

f
n

(
x
)

)

n
=
1

∞

({f^n}(x))_{n = 1}^\infty

converges to

x
f

,
∀
x
∈
X

{x_f},\forall x \in X

, then there exists

d
∈

P

(
τ
)

d \in \mathcal {P}(\tau )

such that f is contractive under

d
d

for each

f
∈
F

f \in F

. III. If there is

p
∈
X

p \in X

such that (1)

lim

n
→
∞

f
n

(
x
)
=
p
,
∀
x
∈
X

{\lim _{n \to \infty }}{f^n}(x) = p,\forall x \in X

and

∀
f
∈
F

\forall f \in F

, (2) there is a neighbourhood B of p such that

lim

m
→
∞

f

n
1

f

n
2

⋯

f

n
m

(
B
)
=
{
p
}

{\lim _{m \to \infty }}{f_{{n_1}}}{f_{{n_2}}} \cdots {f_{{n_m}}}(B) = \{ p\}

for any choice of

f

n
i

∈
F
,
i
=
1
,
…
,
m

{f_{{n_i}}} \in F,i = 1, \ldots ,m

, and the limit depends on m only, then for each

λ
\lambda

with

0
>
λ
>
1

0 > \lambda > 1

, there exists

d
′

∈

P

(
τ
)

d’ \in \mathcal {P}(\tau )

such that each f in F is a Banach contraction under

d
′

d’

with Lipschitz constant

λ
\lambda

.
A uniqueness theorem for a boundary value problem
Authors: Usmani Riaz A. 
Published: June 2010

In this paper it is proved that the two-point boundary value problem, namely

(

d

(
4
)

/

d

x
4

+
f
)
y
=
g
,
y
(
0
)
−

A
1

=
y
(
1
)
−

A
2

=
y
(
0
)
−

B
1

=
y
(
1
)
−

B
2

=
0

({d^{(4)}}/d{x^4} + f)y = g,y(0) - {A_1} = y(1) - {A_2} = y(0) - {B_1} = y(1) - {B_2} = 0

, has a unique solution provided

inf
x

f
(
x
)
=
−
η
>
−

π
4

{\inf _x}f(x) = - \eta > - {\pi ^4}

. The given boundary value problem is discretized by a finite difference scheme. This numerical approximation is proved to be a second order convergent process by establishing an error bound using the

L
2

{L_2}

-norm of a vector.
On finite presentations for groups
Authors: Craggs R. 
Published: June 2010

Three theorems are proved here giving canonical forms for finitely-presented groups. The first characterizes groups with balanced presentations as those groups with presentations of the form
\[

⟨

y
1

,
…
,

y
n

|

y
i

=
α
(

y
i

)
,
i
=
1
,
…
,
n
⟩

\langle {y_1}, \ldots ,{y_n}|{y_i} = \alpha ({y_i}),i = 1, \ldots ,n\rangle

\]
where

α
\alpha

is some automorphism of the free group with free basis

y
1

,
…
,

y
n

{y_1}, \ldots ,{y_n}

. Similar canonical forms, with two automorphisms rather than one, are given for arbitrary finitely-presented groups.
On 𝑝-torsion in etale cohomology and in the Brauer group
Authors: Treger Robert 
Published: June 2010

If X is an affine scheme in characteristic

p
>
0

p > 0

, then

Br

(
X
)
(
p
)

→
~

H

et

2

(
X
,

G

m

)
(
p
)

{\text {Br}}(X)(p)\tilde \to H_{{\text {et}}}^2(X,{{\mathbf {G}}_m})(p)

and

H

et

n

(
X
,

G

m

)
(
p
)
=
0

H_{{\text {et}}}^n(X,{{\mathbf {G}}_m})(p) = 0

for

n
⩾
3

n \geqslant 3

. This gives a partial answer to the conjecture that the Brauer group of any scheme X is canonically isomorphic to the torsion part of

H

et

2

(
X
,

G

m

)

H_{{\text {et}}}^2(X,{{\mathbf {G}}_m})

. This result is then applied to prove that

Br

(
R
)
(
p
)

{\text {Br}}(R)(p)

is p-divisible where R is a commutative ring of characteristic

p
>
0

p > 0

(theorem of Knus, Ojanguren and Saltman), and also to construct examples of domains R of characteristic

p
>
0

p > 0

with large

Ker

(

Br

(
R
)
(
p
)
→

Br

(
Q
)
(
p
)
)

{\operatorname {Ker}}({\text {Br}}(R)(p) \to {\text {Br}}(Q)(p))

, where Q is the ring of fractions of R.
A Volterra equation with square integrable solution
Authors: Staffans Olof J. 
Published: June 2010

We study the asymptotic behavior of the solutions of the nonlinear Volterra integrodifferential equation
\[

x
′

(
t
)
+

∫
0
t

a
(
t
−
s
)
g
(
x
(
s
)
)

d
s

=
f
(
t
)

(
t
∈

R
+

)
.

x’(t) + \int _0^t {a(t - s)g(x(s))\;ds\; = f(t)\quad (t \in {R^ + }).}

\]
Here

R
+

=
[
0
,
∞
)
,
a
,
g

{R^ + } = [0,\infty ),a,g

and f are given real functions, and x is the unknown solution. In particular, we give sufficient conditions which imply that x and x’ are square integrable.
A new criterion for 𝑝-valent functions
Authors: Goel R. M. , Sohi N. S. 
Published: June 2010

In this paper we consider the classes

K

n
+
p
−
1

{K_{n + p - 1}}

of functions

f
(
z
)
=

z
p

+

a

p
+
1

z

p
+
1

+
⋯

f(z) = {z^p} + {a_{p + 1}}{z^{p + 1}} + \cdots

which are regular in the unit disc

E
=
{
z
:

|

z

|

>
1
}

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

and satisfying the condition
\[

Re
⁡

(

(

z
n

f
)

(
n
+
p
)

/

(

z

n
−
1

f
)

(
n
+
p
−
1
)

)

>
(
n
+
p
)

/

2
,

\operatorname {Re} \left ( {{{({z^n}f)}^{(n + p)}}/{{({z^{n - 1}}f)}^{(n + p - 1)}}} \right ) > (n + p)/2,

\]
where p is a positive integer and n is any integer greater than

−
p

- p

. It is proved that

K

n
+
p

⊂

K

n
+
p
−
1

{K_{n + p}} \subset {K_{n + p - 1}}

. Since

K
0

{K_0}

is the class of p-valent functions, consequently it follows that all functions in

K

n
+
p
−
1

{K_{n + p - 1}}

are p-valent. We also obtain some special elements of

K

n
+
p
−
1

{K_{n + p - 1}}

via the Hadamard product.
On the reflexivity of 𝐶₀(𝑁) contractions
Authors: Wu Pei Yuan 
Published: June 2010

Let T be a

C
0

(
N
)

{C_0}(N)

contraction on a separable Hilbert space and let

J
=
S
(

φ
1

)
⊕
S
(

φ
2

)
⊕
⋯
⊕
S
(

φ
k

)

J = S({\varphi _1}) \oplus S({\varphi _2}) \oplus \cdots \oplus S({\varphi _k})

be its Jordan model, where

φ
1

,

φ
2

,
…
,

φ
k

{\varphi _1},{\varphi _2}, \ldots ,{\varphi _k}

are inner functions satisfying

φ
j

|

φ

j
−
1

{\varphi _j}|{\varphi _{j - 1}}

for

j
=
2
,
3
,
…
,
k

j = 2,3, \ldots ,k

, and

S
(

φ
j

)

S({\varphi _j})

denotes the compression of the shift on

H
2

⊖

φ
j

H
2

,
j
=
1
,
2
,
…
,
k

{H^2} \ominus {\varphi _j}{H^2},j = 1,2, \ldots ,k

. In this note we show that T is reflexive if and only if

S
(

φ
1

/

φ
2

)

S({\varphi _1}/{\varphi _2})

is.
Length estimates for holomorphic functions
Authors: Yamashita Shinji 
Published: June 2010

Let

f
(
z
)
=

a
0

+

Σ

k
=
n

∞

a
k

z
k

(
n
⩾
1
)

f(z) = {a_0} + \Sigma _{k = n}^\infty {a_k}{z^k}(n \geqslant 1)

be holomorphic in

U
:

|
z
|

>
1

U:\left | z \right | > 1

. MacGregor [1, Theorem 2] proved that if

l
(
r
,
f
)

l(r,f)

is the length of the outer boundary of the image

D
(
r
,
f
)

D(r,f)

of the disk:

|
z
|

>
r

\left | z \right | > r

by

f
f

, then

2
π

r
n

|

a
n

|

⩽
l
(
r
,
f
)

2\pi {r^n}\left | {{a_n}} \right | \leqslant l(r,f)

for

0
>
r
>
1

0 > r > 1

. We introduce the notion of the exact outer boundary

C
#

(
r
,
f
)

{C^\# }(r,f)

of

D
(
r
,
f
)

D(r,f)

and prove that

2
π

r
n

|

a
n

|

⩽

l
#

(
r
,
f
)
⩽
l
(
r
,
f
)

2\pi {r^n}\left | {{a_n}} \right | \leqslant {l^\# }(r,f) \leqslant l(r,f)

for

0
>
r
>
1

0 > r > 1

, where

l
#

(
r
,
f
)

{l^\# }(r,f)

is the length of

C
#

(
r
,
f
)

{C^\# }(r,f)

. We shall make use of the estimate to obtain a criterion for

f
f

to be Bloch in

U
U

.
Some conditions for 𝑛-convex functions
Authors: Cross G. E. 
Published: June 2010

It is shown that if

F
F

is a function defined and continuous on

[
a
,
b
]

[a,b]

such that (where

n
n

is an even integer): (a)
\[

F

(
r
)

(
x
)
=

lim

h
→
0

(

r
!

h
r

)

(

F
(
x
+
h
)
−
F
(
x
)
−

∑

k
=
1

r
−
1

h
k

F

(
k
)

(
x
)

k
!

)

{F_{(r)}}(x) = \lim \limits _{h \to 0} \left ( {\frac {{r!}} {{{h^r}}}} \right )\left ( {F(x + h) - F(x) - \sum \limits _{k = 1}^{r - 1} {\frac {{{h^k}{F_{(k)}}(x)}} {{k!}}} } \right )

\]
exists and is finite in

(
a
,
b
)

(a,b)

for

1
⩽
r
⩽
n
−
2

1 \leqslant r \leqslant n - 2

; (b) for

x
∈
(
a
,
b
)
∖
E

x \in (a,b)\backslash E

, where

E
E

is countable,

F
F

is

n
n

-smooth, i.e.,
\[

lim

h
→
0

(

n
!

h

n
−
1

)

[

F
(
x
+
h
)
+
F
(
x
−
h
)

2

−

∑

k
=
0

n

/

2
−
1

h

2
k

D

2
k

F
(
x
)

(
2
k
)
!

]

=
0

\lim \limits _{h \to 0} \left ( {\frac {{n!}} {{{h^{n - 1}}}}} \right )\left [ {\frac {{F(x + h) + F(x - h)}} {2} - \sum \limits _{k = 0}^{n/2 - 1} {\frac {{{h^{2k}}{D^{2k}}F(x)}} {{(2k)!}}} } \right ] = 0

\]
where

D

2
k

F
(
x
)

{D^{2k}}F(x)

denotes the (symmetric) de la Vallée-Poussin derivative; (c)

D
¯

n

F
(
x
)
⩾
0

{\bar D^n}F(x) \geqslant 0

a.e. in

(
a
,
b
)

(a,b)

; (d)

D
¯

n

F
(
x
)
>
−
∞

{\bar D^n}F(x) > - \infty

for

x
∈
(
a
,
b
)
∖
S

x \in (a,b)\backslash S

where

S
S

is countable and

F
(
x
)

F(x)

is

n
n

-smooth in

S
S

; then

F
(
x
)

F(x)

is

n
n

-convex in

[
a
,
b
]

[a,b]

. The same result holds for

n
n

odd. This is an improvement on the known result when

S
S

is a scattered set.
On a conjecture for oscillation of second-order ordinary differential systems
Authors: Mingarelli Angelo B. 
Published: June 2010

We present here some results pertaining to the oscillatory behavior at infinity of the vector differential equation
\[

y
+
Q
(
t
)
y
=
0
,

t
∈
[
0
,
∞
)

y + Q(t)y = 0,\quad t \in [0,\infty )

\]
, where

Q
(
t
)

Q(t)

is a real continuous

n
×
n

n \times n

symmetric matrix function. It has been conjectured (cf., e.g. [6]) that the criterion
\[

lim

t
→
∞

λ
1

{

∫
0
t

Q
(
s
)

d
s

}

=
∞

\lim \limits _{t \to \infty } {\lambda _1}\left \{ {\int _0^t {Q(s)\;ds} } \right \} = \infty

\]
where

λ
1

(
⋅
)

{\lambda _1}( \cdot )

denotes the maximum eigenvalue of the matrix concerned, implies oscillation. We show that this is so under the tacit assumption
\[

lim

inf

t
→
∞

t

−
1

tr

{

∫
0
t

Q
(
s
)

d
s

}

>
−
∞

\lim \inf \limits _{t \to \infty } {t^{ - 1}}{\text {tr}}\left \{ {\int _0^t {Q(s)\;ds} } \right \} > - \infty

\]
where

tr

(
⋅
)

{\text {tr}}( \cdot )

represents the trace of the matrix under consideration.
Interpolation spaces and unitary representations
Authors: Goodman Roe 
Published: June 2010

Let

G
G

be a Lie group,

π
\pi

a unitary representation of

G
G

on a Hilbert space

H

(
π
)

\mathcal {H}(\pi )

, and

H

k

(
π
)

{\mathcal {H}^k}(\pi )

the subspace of

C
k

{C^k}

vectors for

π
\pi

. By quadratic interpolation there is a continuous scale

H

s

(
π
)

{\mathcal {H}^s}(\pi )

,

s
>
0

s > 0

, of

G
G

-invariant Hilbert spaces. When

G
=
H
⋅
K

G = H \cdot K

is a semidirect product of closed subgroups, then it is proved that

H

s

(
π
)
=

H

s

(

π
|

H

)
∩

H

s

(

π
|

K

)

{\mathcal {H}^s}(\pi ) = {\mathcal {H}^s}({\left . \pi \right |_H}) \cap {\mathcal {H}^s}({\left . \pi \right |_K})

for

s
>
0

s > 0

. For solvable

G
G

this gives a characterisation of

H

s

(
π
)

{\mathcal {H}^s}(\pi )

in terms of smoothness along one-parameter subgroups, and an elliptic regularity result.
View-obstruction problems. II
Authors: Cusick T. W. 
Published: June 2010

Let

S
n

{S^n}

denote the region

0
>

x
i

>
∞
(
i
=
1
,
2
,
…
,
n
)

0 > {x_i} > \infty (i = 1,2, \ldots ,n)

of

n
n

-dimensional Euclidean space

E
n

{E^n}

. Suppose

C
C

is a closed convex body in

E
n

{E^n}

which contains the origin as an interior point. Define

α
C

\alpha C

for each real number

α
⩾
0

\alpha \geqslant 0

to be the magnification of

C
C

by the factor

α
\alpha

and define

C
+
(

m
1

,
…
,

m
n

)

C + ({m_1}, \ldots ,{m_n})

for each point

(

m
1

,
…
,

m
n

)

({m_1}, \ldots ,{m_n})

in

E
n

{E^n}

to be the translation of

C
C

by the vector

(

m
1

,
…
,

m
n

)

({m_1}, \ldots ,{m_n})

. Define the point set

Δ
(
C
,
α
)

\Delta (C,\alpha )

by

Δ
(
C
,
α
)
=
{
α
C
+
(

m
1

+

1
2

,
…
,

m
n

+

1
2

)
:

m
1

,
…
,

m
n

\Delta (C,\alpha ) = \{ \alpha C + ({m_1} + \frac {1} {2}, \ldots ,{m_n} + \frac {1} {2}):{m_1}, \ldots ,{m_n}

nonnegative integers}. The view-obstruction problem for

C
C

is the problem of finding the constant

K
(
C
)

K(C)

defined to be the lower bound of those

α
\alpha

such that any half-line

L
L

given by

x
i

=

a
i

t
(
i
=
1
,
2
,
…
,
n
)

{x_i} = {a_i}t(i = 1,2, \ldots ,n)

, where the

a
i

(
1
⩽
i
⩽
n
)

{a_i}(1 \leqslant i \leqslant n)

are positive real numbers, and the parameter

t
t

runs through

[
0
,
∞
)

[0,\infty )

, intersects

Δ
(
C
,
α
)

\Delta (C,\alpha )

. The paper considers the case where

C
C

is the

n
n

-dimensional cube with side 1, and in this case the constant

K
(
C
)

K(C)

is known for

n
⩽
3

n \leqslant 3

. The paper gives a new proof for the case

n
=
3

n = 3

. Unlike earlier proofs, this one could be extended to study the cases with

n
⩾
4

n \geqslant 4

.
Oscillation theorems for nonlinear second order differential equations with damped term
Authors: Yeh Cheh Chih 
Published: June 2010

Some new integral criteria for the oscillation of the nonlinear second order differential equation with damped term

y
(
t
)
+
p
(
t
)

y
′

(
t
)
+
q
(
t
)
f
(
y
(
t
)
)
=
0

y(t) + p(t)y’(t) + q(t)f(y(t)) = 0

are given.
On the first factor of the class number of a cyclotomic field
Authors: Feng Ke Qin 
Published: June 2010

Let

p
p

be an odd prime.

h
1

(
p
)

{h_1}(p)

is the first factor of the class number of field

Q
(

ζ
p

)

Q({\zeta _p})

. We proved that
\[

h
1

(
p
)
⩽

{

2
p

(

p
−
1

8

(

2

l

/

2

+
1
)

4

/

l

)

(
p
−
1
)

/

4

,

if

l

is even,

2
p

(

p
−
1

8

(

2
l

−
1
)

2

/

l

)

(
p
−
1
)

/

4

,

if

l

is odd

.

{h_1}(p) \leqslant \left \{ \begin {gathered} 2p{\left ( {\frac {{p - 1}} {{8{{({2^{l/2}} + 1)}^{4/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is even,}} \hfill \\ 2p{\left ( {\frac {{p - 1}} {{8{{({2^l} - 1)}^{2/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is odd}}{\text {.}} \hfill \\ \end {gathered} \right .

\]
From that we obtain

h
1

(
p
)
⩽
2
p

(
(
p
−
1
)

/

31.997158
…

)

(
p
−
1
)

/

4

{h_1}(p) \leqslant 2p{((p - 1)/31.997158 \ldots )^{(p - 1)/4}}

which is better than Carlitz’s and Metsänkyla’s results. For the fields

Q
(

ζ

2
n

)

Q({\zeta _{{2^n}}})

and

Q
(

ζ

p
n

)
(
n
⩾
2
)

Q({\zeta _{{p^n}}})(n \geqslant 2)

, we get the similar results.