bluuebunny/crossref_metadata_2025_split huggingface dataset:
Starshaped unions and nonempty intersections of convex sets in 𝑅^{𝑑}
Authors: Breen Marilyn 
Published: June 2010

Let

G

\mathcal {G}

be a nonempty family of compact convex sets in

R
d

,
d
≥
1

{R^d},d \geq 1

. Then every subfamily of

G

\mathcal {G}

consisting of

d
+
1

d + 1

or fewer sets has a starshaped union if and only if

∩
{
G
:
G

in

G

}
≠
∅

\cap \{ G:G{\text { in }}\mathcal {G}\} \ne \emptyset

.
Wiener transformation on functions with bounded averages
Authors: Chen Yong Zhuo , Lau Ka-Sing 
Published: June 2010

The Wiener transformation (integrated Fourier transformation) on the space

B
2

=
{
f
:

‖
f
‖

=

sup

T
≥
1

(

1

2
T

∫

−
T

T

|
f
|

2

)

1

/

2

>
∞
}

{B^2} = \{ f:\left \| f \right \| = {\sup _{T \geq 1}}{(\frac {1}{{2T}}{\int _{ - T}^T {\left | f \right |} ^2})^{1/2}} > \infty \}

is studied.
On the fixed point index of iterates of planar homeomorphisms
Authors: Brown Morton 
Published: June 2010

If

f
f

is an orientation preserving homeomorphism of the plane with an isolated fixed point at the origin 0 and

index(

f
,
0

) =

p

{\text {index(}}f,0{\text {) = }}p

, then

index(

f
n

,0)

{\text {index(}}{f^n}{\text {,0)}}

is always well defined provided that

p
≠
1

p \ne 1

. In this case, for each

n
≠
0

n \ne 0

,

index(

f
n

,0) = index(

f
,
o

) =

p

{\text {index(}}{f^n}{\text {,0) = index(}}f,o{\text {) = }}p

. If

index(

f
,
0

) = 1

{\text {index(}}f,0{\text {) = 1}}

, then there is an integer

p
p

(possibly

p
=
1

p = 1

) such that for those values of

n
n

for which

index(

f
n

,0)

{\text {index(}}{f^n}{\text {,0)}}

is defined (i.e 0 is an isolated fixed point of

f
n

{f^n}

),

index(

f
n

,0) = 1

{\text {index(}}{f^n}{\text {,0) = 1}}

or

index(

f
n

,0) =

p

{\text {index(}}{f^n}{\text {,0) = }}p

.
Matrix generation of Pythagorean 𝑛-tuples
Authors: Cass Daniel , Arpaia Pasquale J. 
Published: June 2010

We construct, for each

n
(
4
≤
n
≤
9
)

n(4 \leq n \leq 9)

, a matrix

A
n

{A_n}

which generates all the primitive Pythagorean

n
n

-tuples

(

x
1

,
…
,

x
n

)

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

with

x
n

>
1

{x_n} > 1

\[

(
1
)

x
1
2

+
⋯
+

x

n
−
1

2

=

x
n
2

,

gcd
(

x
1

,
…
,

x
n

)
=
1

(1)\quad x_1^2 + \cdots + x_{n - 1}^2 = x_n^2,\quad \gcd ({x_1}, \ldots ,{x_n}) = 1

\]
from the single

n
n

-tuple

(
1
,
0
,
…
,
0
,
1
)

(1,0, \ldots ,0,1)

. Once a particular

n
n

-tuple is generated, one permutes the first

n
−
1

n - 1

coordinates and/or changes some of their signs, and applies

A
n

{A_n}

to obtain another

n
n

-tuple. This extends a result of Barning which presents an appropriate matrix

A
3

{A_3}

for the Pythagorean triples. One cannot so generate the Pythagorean

n
n

-tuples if

n
≥
10

n \geq 10

; in fact we show the Pythagorean

n
n

-tuples fall into at least

[
(
n
+
6
)

/

8
]

[(n + 6)/8]

distinct orbits under the automorphism group of (1).
Real analytic boundary regularity of the Cauchy kernel on convex domains
Authors: Chen So-Chin 
Published: June 2010

It is well-known that in one complex variable the Cauchy integral preserves real analyticity near the boundary. In this paper we show that the same conclusion also holds on convex domains with real analytic boundary in higher dimension, where the Cauchy kernel is given by the Cauchy-Fantappiè form of order zero generated by the (l.0)-form

C

(

ξ
,
z

)

C\left ( {\xi ,z} \right )

,
\[

C

(

ξ
,
z

)

=

(

∑

j
=
1

n

∂
r

∂

ξ
j

(
ξ
)

d

ξ
j

)

(

∑

j
=
1

n

∂
r

∂

ξ
j

(
ξ
)

(

ξ
j

−

z
j

)

)

−
1

,

C\left ( {\xi ,z} \right ) = \left ( {\sum \limits _{j = 1}^n {\frac {{\partial r}}{{\partial {\xi _j}}}} \left ( \xi \right )d{\xi _j}} \right ){\left ( {\sum \limits _{j = 1}^n {\frac {{\partial r}}{{\partial {\xi _j}}}} \left ( \xi \right )\left ( {{\xi _j} - {z_j}} \right )} \right )^{ - 1}},

\]
where

r

(
ξ
)

r\left ( \xi \right )

is the defining function of the domain.
The Hausdorff dimension of the graphs of continuous self-affine functions
Authors: Urbański M. 
Published: June 2010

The exact formula for the Hausdorff dimension of the graph of a continuous self-affine function is obtained. The Hausdorff dimension of some class of Borel probability measures is computed. The Hausdorff measures corresponding to the functions

φ
c

(
t
)
=

t

H
D
(

graph

(
f
)
)

exp
⁡
(
c

log
⁡
1

/

t
log
⁡
log
⁡
log
⁡
1

/

t

{\varphi _c}(t) = {t^{HD({\text {graph}}(f))}}\exp (c\sqrt {\log 1/t\log \log \log 1/t}

are studied.
A note on the number of primes in short intervals
Authors: Goldston D. A. , Gonek S. M. 
Published: June 2010

Let

J

(

β
,
T

)

=

∫
1

T
β

(

∑

x
>

p
k

≤
x
+
x

/

T

log
⁡
p
−
x

/

T

)

2

d
x

/

x
2

J\left ( {\beta ,T} \right ) = \int _1^{{T^\beta }} {{{\left ( {\sum \nolimits _{x > {p^k} \leq x + x/T} {\log p - x/T} } \right )}^2}dx/{x^2}}

, where the sum is over prime powers. H. L. Montgomery has shown that on the Riemann hypothesis, there is a positive constant

C
0

{C_0}

such that for each

β
≥
1
,
J

(

β
,
T

)

≤

C
0

β

log
2

T

/

T

\beta \geq 1,J\left ( {\beta ,T} \right ) \leq {C_0}\beta {\log ^2}T/T

, provided that

T
T

is sufficiently large. Here we prove a slightly stronger result from which we deduce a lower bound of the same order.
Two generalizations of Titchmarsh’s convolution theorem
Authors: Doss Raouf 
Published: June 2010

Titchmarsh’s convolution theorem states that if the functions

f
,
g

f,g

vanish on

(
−
∞
,
0
)

( - \infty ,0)

and if the convolution

f
∗
g
(
t
)
=
0

f * g(t) = 0

on an interval

(
0
,
T
)

(0,T)

, then there are two numbers

α
,
β
≥
0

\alpha ,\beta \geq 0

such that

α
+
β
=
T
,
f
=
0

\alpha + \beta = T,f = 0

a.e. on

(
0
,
α
)

(0,\alpha )

, and

g
=
0

g = 0

a.e. on

(
0
,
β
)

(0,\beta )

.

T
T

may be infinite. For the case

T
=
∞

T = \infty

we prove that if

f
∗
g
=
0

f * g = 0

on

R
R

and one of the two functions

f
,
g

f,g

is 0 on

(
−
∞
,
0
)

( - \infty ,0)

, then either

f
f

or

g
g

is 0 a.e. on

R
R

. Next we consider the integro-differential-difference equation

f
∗
g
(
t
)
+
∑

λ

p
σ

f

(
p
)

(
t
−

a

p
σ

)
=
0

f * g(t) + \sum {{\lambda _{p\sigma }}{f^{(p)}}(t - {a_{p\sigma }}) = 0}

for

t
t

in

(
0
,
T
)

(0,T)

, where

a

ρ
σ

≥
0
,

λ

p
σ

{a_{\rho \sigma }} \geq 0,{\lambda _{p\sigma }}

are constants. Conclusions similar to Titchmarsh’s hold with the additional information that

α
≥
T
−

a

ρ
σ

\alpha \geq T - {a_{\rho \sigma }}

whenever

λ

ρ
σ

≠
0

{\lambda _{\rho \sigma }} \ne 0

.
Semiderivations of prime rings
Authors: Brešar Matej 
Published: June 2010

A semiderivation of a ring

R
R

is an additive mapping

f
:
R
→
R

f:R \to R

together with a function

g
:
R
→
R

g:R \to R

such that

f
(
x
y
)
=
f
(
x
)
g
(
y
)
+
x
f
(
y
)
=
f
(
x
)
y
+
g
(
x
)
f
(
y
)

f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y)

and

f
(
g
(
x
)
)
=
g
(
f
(
x
)
)

f(g(x)) = g(f(x))

for all

x
,
y
∈
R

x,y \in R

. We prove that the only semiderivations of prime rings are derivations and mappings of the form

f
(
x
)
=
λ
(
x
−
g
(
x
)
)

f(x) = \lambda (x - g(x))

, where

g
g

is an endomorphism and

λ
\lambda

is an element in the extended centroid.
On the equation 𝑑𝑖𝑣(|∇𝑢|^{𝑝-2}∇𝑢)+𝜆|𝑢|^{𝑝-2}𝑢=0
Authors: Lindqvist Peter 
Published: June 2010

The first eigenvalue

λ
=

λ
1

\lambda = {\lambda _1}

for the equation

div
⁡
(

|

∇
u

|

p
−
2

∇
u

) +

λ

|

u

|

p
−
2

u
=
0

\operatorname {div} ({\text {|}}\nabla u{{\text {|}}^{p - 2}}\nabla u{\text {) + }}\lambda {\text {|}}u{{\text {|}}^{p - 2}}u = 0

is simple in any bounded domain. (Through the nonlinear counterpart to the Rayleigh quotient

λ
1

{\lambda _1}

is related to the Poincaré inequality.)
On the characterization of 𝐻^{𝑝}(𝑅ⁿ) in terms of Fourier multipliers
Authors: Uchiyama Akihito 
Published: June 2010

Let

p
∈
(
0
,
1
]

p \in (0,1]

, let

f
∈

L
2

(

R
n

)
∩

H
p

(

R
n

)

f \in {L^2}({R^n}) \cap {H^p}({R^n})

and let

θ
∈
C
(

R
n

∖

{
0
}

)

\theta \in C({R^n}\backslash \left \{ 0 \right \})

be homogeneous of degree zero. We will give one sufficient condition in order for

f
f

and

θ
\theta

to satisfy
\[

|

|

f

|

|

H
p

≤
C

{

|

|

f

|

|

L
p

+

|

|

F

−
1

{

θ
(
ξ
)

F

f
(
ξ
)

}

|

|

L
p

}

.

||f|{|_{{H^p}}} \leq C\left \{ {||f|{|_{{L^p}}} + ||{\mathcal {F}^{ - 1}}\left \{ {\theta (\xi )\mathcal {F}f(\xi )} \right \}|{|_{{L^p}}}} \right \}.

\]
On eigenvalue problems of 𝑝-Laplacian with Neumann boundary conditions
Authors: Huang Yin Xi 
Published: June 2010

We study the nonlinear eigenvalue problem
\[

−

Δ
p

u
=
λ
m
(
x
)

|

u

|

p
−
2

u

in

Ω
,

∂
u

∂
n

=
0

on

∂
Ω
,

where

p
>
1
,
λ
∈

R

.

- {\Delta _p}u = \lambda m(x)|u{|^{p - 2}}u\,\quad {\text {in}}\,\Omega ,\quad \,\frac {{\partial u}} {{\partial n}} = 0\,\quad {\text {on}}\,\partial \Omega ,\,\;{\text {where}}\,p > 1,\lambda \in {\mathbf {R}}.

\]
For

∫
Ω

m
(
x
)
>
0

\int _\Omega {m(x) > 0}

, we prove that the first positive eigenvalue

λ
1

{\lambda _1}

exists and is simple and unique, in the sense that it is the only eigenvalue with a positive eigenfunction. In the case

∫
Ω

m
(
x
)
=
0

\int _\Omega {m(x) = 0}

, we prove that

λ
0

=
0

{\lambda _0} = 0

is the only eigenvalue with a positive eigenfunction.
Hardy inequalities
Authors: Wannebo Andreas 
Published: June 2010

Here the following Hardy inequalities are studied
\[

∑

k
=
0

m
−
1

∫

|

∇
k

u

|

p

d

∂
Ω

(
x
)

t

d

∂
Ω

(
x
)

(
m
−
k
)
p

d
x
≤

A
Ω

∫

|

∇
m

u

|

p

d

∂
Ω

(
x
)

t

d
x

\sum \limits _{k = 0}^{m - 1} {\int {\frac {{|{\nabla ^k}u{|^p}{d_{\partial \Omega }}{{(x)}^t}}}{{{d_{\partial \Omega }}{{(x)}^{(m - k)p}}}}dx \leq {A_\Omega }\int {|{\nabla ^m}u{|^p}{d_{\partial \Omega }}{{(x)}^t}dx} } }

\]
for

u
∈

C
0
∞

(
Ω
)

u \in C_0^\infty (\Omega )

, an open proper subset of

R

N

{{\mathbf {R}}^N}

and

t
>

t
0

t > {t_0}

, some small positive

t
0

{t_0}

. This inequality has previously been shown to hold for bounded Lipschitz domains. The question discussed is, How general can

Ω
\Omega

be and allow the same inequality? A sufficient condition is given in the form of a local Maz’ja capacity condition. In

R

2

{{\mathbf {R}}^2}

, or generally if

p
>
N
−
1

p > N - 1

, this is satisfied for any

Ω
\Omega

which is deformable to a point. Furthermore, if

p
>
N

p > N

the condition is satisfied for all

Ω
\Omega

.
Convolution in the harmonic Hardy class ℎ^{𝑝} with 0<𝑝<1
Authors: Pavlović Miroslav 
Published: June 2010

It is proved that if

u
∈

h
p

,
0
>
p
>
1

u \in {h^p},0 > p > 1

, and

v
∈

h
q

,
q
≥
p

v \in {h^q},q \geq p

, then
\[

M
q

(
u
∗
v
,
r
)
=
0

(

(
1
−
r
)

1
−
1

/

p

)

,
r
→
1
−
,

{M_q}(u * v,r) = 0\left ( {{{(1 - r)}^{1 - 1/p}}} \right ),r \to 1 - ,

\]
where

u
∗
v

u*v

stands for the convolution of

u
u

and

v
v

.
Higher-dimensional shift equivalence and strong shift equivalence are the same over the integers
Authors: Wagoner J. B. 
Published: June 2010

Let

R
S
(
Λ
)

RS(\Lambda )

and

S
(
Λ
)

S(\Lambda )

denote, respectively, the spaces of strong shift equivalences and shift equivalences over a subset

Λ
\Lambda

of a ring which is closed under addition and multiplication. For example, let

Λ
\Lambda

be the integers

Z
Z

or the nonnegative integers

Z
+

{Z^ + }

. For any principal ideal domain

Λ
\Lambda

, we prove that the continuous map

R
S

(
Λ
)

→
S

(
Λ
)

RS\left ( \Lambda \right ) \to S\left ( \Lambda \right )

is a homotopy equivalence. The methods also show that any inert automorphism, i.e., an element in the kernel of

π
1

(

R
S

(

Z
+

)

,
A

)

→

π
1

(

S

(

Z
+

)

,
A

)

{\pi _1}\left ( {RS\left ( {{Z^ + }} \right ),A} \right ) \to {\pi _1}\left ( {S\left ( {{Z^ + }} \right ),A} \right )

can be represented by a closed loop in

R
S

(

Z
+

)

RS\left ( {{Z^ + }} \right )

which in

S
R

(
Z
)

SR\left ( Z \right )

is spanned by a triangulated

2
2

-disc supporting a positive

1
1

-cocycle. These cocycles are used in work of Kim-Roush that leads to a counterexample to Williams’ lifting problem for automorphisms of finite subsystems of subshifts of finite type.
Centralizers of immersions of the circle
Authors: Arteaga Carlos 
Published: June 2010

We prove here that for every element

f
f

of an open and dense subset of immersions of the circle

S
1

{S^1}

, either the centralizer

Z

(
f
)

Z\left ( f \right )

of

f
f

is trivial (i.e.

f
f

only commutes with its own powers) or

f
f

is topologically conjugate to a map

f
n

:

S
1

→

S
1

{f_n}:{S^1} \to {S^1}

given by

f
n

(
z
)

=

z
n

{f_n}\left ( z \right ) = {z^n}

and, in this case, if

h
h

is the conjugacy between

f
f

and

f
n

{f_n}

then

Z

(
f
)

Z\left ( f \right )

is a subgroup of

{

h

−
1

∘
ω

f
m

∘
h
;
m
∈

N

and

ω

n
−
1

=
1

}

\left \{ {{h^{ - 1}} \circ \omega {f_m} \circ h;m \in {\mathbf {N}}{\text { and }}{\omega ^{n - 1}} = 1} \right \}

.
Periodic solutions for nonlinear evolution equations in a Banach space
Authors: Vrabie Ioan I. 
Published: June 2010

We prove an existence result for

T
T

-periodic mild solutions to nonlinear evolution equations of the form
\[

u
′

(
t
)
+
A
u
(
t
)
∍
F
(
t
,
u
(
t
)
)
,

t
∈

R
+

.

u’(t) + Au(t) \backepsilon F(t,u(t)),\quad t \in {R_ + }.

\]
Here

(
X
,

|

|

⋅

|

|

)

(X,|| \cdot ||)

is a real Banach space,

A
:
D
(
A
)
⊂
X
→

2
X

A:D(A) \subset X \to {2^X}

is an operator with

A
−
a
I

A - aI

m
m

-accretive for some

a
>
0

a > 0

and such that

−
A

- A

. generates a compact semigroup, while

F
:

R
+

×

D
(
A
)

¯

→
X

F:{R_ + } \times \overline {D(A)} \to X

is a Carathéodory mapping which is

T
T

-periodic with respect to its first argument and satisfies
\[

lim

r
→
+
∞

1
r

sup

{

|

|

F
(
t
,
v
)

|

|

;
t
∈

R
+

,
v
∈

D
(
A
)

¯

,

|

|

v

|

|

≤
r

}

>
a
.

\lim \limits _{r \to + \infty } \tfrac {1}{r}\sup \left \{ {||F(t,v)||;t \in {R_ + },v \in \overline {D(A)} ,||v|| \leq r} \right \} > a.

\]
. As a consequence, we obtain an existence theorem for

T
T

-periodic solutions to the porous medium equation.
The numerical analysis of implicit Runge-Kutta methods for a certain nonlinear integro-differential equation
Authors: Yuan Wei , Tang Tao 
Published: June 2010

In this paper and in an earlier 1987 paper, the mathematical theory and numerical methods for the nonlinear integro-differential equation
\[

u
′

(
t
)
+
p
(
t
)
u
(
t
)
+

∫
0
t

k
(
t

,
s
)
u
(
t
−
s
)
u
(
s
)

d
s
=
q
(
t
)
,

0
≤
t
≤
T
,

u
(
0
)
=

u
0

\begin {array}{*{20}{c}} {u’(t) + p(t)u(t) + \int _0^t {k(t} ,s)u(t - s)u(s)\,ds = q(t),\quad 0 \leq t \leq T,} \hfill \\ {u(0) = {u_0}} \hfill \\ \end {array}

\]
are considered. Equations of this type occur as model equations for describing turbulent diffusion. Previously, the existence and uniqueness properties of the solutions of the model equation were solved completely, and a class of implicit Runge-Kutta methods with m stages for the approximate solution of the model equation was introduced. In this paper, we give a further numerical analysis of these methods. It is proved that the implicit Runge-Kutta methods with n stages are of optimal approximation order

p
=
2
m

p = 2m

. Some computational examples are given.
Generalized noninterpolatory rules for Cauchy principal value integrals
Authors: Rabinowitz Philip 
Published: June 2010

Consider the Cauchy principal value integral
\[

I
(
k
f
;
λ
)
=
∮
k
(
x
)

f
(
x
)

x
−
λ

d
x
,

−
1
>
λ
>
1.

I(kf;\lambda ) = \oint k(x)\frac {{f(x)}}{{x - \lambda }}\,dx,\quad - 1 > \lambda > 1.

\]
If we approximate

f
(
x
)

f(x)

by

∑

j
=
0

N

a
j

p
j

(
x
;
w
)

\sum _{j = 0}^N\;{a_j}{p_j}(x;w)

where

{

p
j

}

\{ {p_j}\}

is a sequence of orthonormal polynomials with respect to an admissible weight function w and

a
j

=
(
f
,

p
j

)

{a_j} = (f,{p_j})

, then an approximation to

I
(
k
f
;
λ
)

I(kf;\lambda )

is given by

∑

j
=
0

N

a
j

I
(
k

p
j

;
λ
)

\sum _{j = 0}^N\;{a_j}I(k{p_j};\lambda )

. If, in turn, we approximate

a
j

{a_j}

by

a

j
m

=

∑

i
=
1

m

w

i
m

f
(

x

i
m

)

p
j

(

x

i
m

)

{a_{jm}} = \sum _{i = 1}^m\;{w_{im}}f({x_{im}}){p_j}({x_{im}})

, then we get a double sequence of approximations

{

Q
m
N

(
f
;
λ
)
}

\{ Q_m^N(f;\lambda )\}

to

I
(
k
f
;
λ
)

I(kf;\lambda )

. We study the convergence of this sequence by relating it to the sequence of approximations associated with

I
(
w
f
;
λ
)

I(wf;\lambda )

which has been investigated previously.
Integer relations among algebraic numbers
Authors: Just Bettina 
Published: June 2010

A vector

m
=
(

m
1

,
…
,

m
n

)
∈

Z

n

∖
{
0
}

m = ({m_1}, \ldots ,{m_n}) \in {{\mathbf {Z}}^n}\backslash \{ 0\}

is called an integer relation for the real numbers

α
1

,
…
,

α
n

{\alpha _1}, \ldots ,{\alpha _n}

, if

∑

α
i

m
i

=
0

\sum {\alpha _i}{m_i} = 0

holds. We present an algorithm that, when given algebraic numbers

α
1

,
…
,

α
n

{\alpha _1}, \ldots ,{\alpha _n}

and a parameter

ε
\varepsilon

, either finds an integer relation for

α
1

,
…
,

α
n

{\alpha _1}, \ldots ,{\alpha _n}

or proves that no relation of Euclidean length shorter than

1

/

ε

1/\varepsilon

exists. Each algebraic number is assumed to be given by its minimal polynomial and by a sufficiently precise rational approximation. Our algorithm uses the Lenstra-Lenstra-Lovász lattice basis reduction technique. It performs
\[

poly

(

log
⁡
1

/

ε
,
n
,
log
⁡

max
i

height

(

α
i

)
,
[

Q

(

α
1

,
…
,

α
n

)
:

Q

]

)

{\operatorname {poly}}\left ( {\log 1/\varepsilon ,n,\log \max \limits _i {\text {height}}({\alpha _i}),[{\mathbf {Q}}({\alpha _1}, \ldots ,{\alpha _n}):{\mathbf {Q}}]} \right )

\]
bit operations. The straightforward algorithm that works with a primitive element of the field extension

Q

(

α
1

,
…
,

α
n

)

{\mathbf {Q}}({\alpha _1}, \ldots ,{\alpha _n})

of Q would take
\[

poly

(

n
,
log
⁡

max
i

height

(

α
i

)
,

∏

i
=
1

n

degree

(

α
i

)

)

{\operatorname {poly}}\left ( {n,\log \max \limits _i {\text {height}}({\alpha _i}),\prod \limits _{i = 1}^n {{\text {degree}}({\alpha _i})} } \right )

\]
bit operations. In order to prove the correctness of the algorithm, we show a lower bound for

|

∑

α
1

m
i

|

\left | {\sum {\alpha _1}{m_i}} \right |

if m is not an integer relation for

α
1

,
…
,

α
n

{\alpha _1}, \ldots ,{\alpha _n}

, which may be interesting in its own right.
On estimates for the weights in Gaussian quadrature in the ultraspherical case
Authors: Förster Klaus-Jürgen , Petras Knut 
Published: June 2010

In this paper the Christoffel numbers

a

v
,
n

(
λ
)
G

a_{v,n}^{(\lambda )G}

for ultraspherical weight functions

w
λ

{w_\lambda }

,

w
λ

(
x
)
=

(
1
−

x
2

)

λ
−
1

/

2

{w_\lambda }(x) = {(1 - {x^2})^{\lambda - 1/2}}

, are investigated. Using only elementary functions, we state new inequalities, monotonicity properties and asymptotic approximations, which improve several known results. In particular, denoting by

θ

v
,
n

(
λ
)

\theta _{v,n}^{(\lambda )}

the trigonometric representation of the Gaussian nodes, we obtain for

λ
∈
[
0
,
1
]

\lambda \in [0,1]

the inequalities
\[

π

n
+
λ

sin

2
λ

θ

v
,
n

(
λ
)

{

1
−

λ
(
1
−
λ
)

2

(
n
+
λ
)

2

sin

2

θ

v
,
n

(
λ
)

}

≤

a

v
,
n

(
λ
)
G

≤

π

n
+
λ

sin

2
λ

θ

v
,
n

(
λ
)

\begin {array}{*{20}{c}} {\frac {\pi }{{n + \lambda }}{{\sin }^{2\lambda }}\theta _{v,n}^{(\lambda )}\left \{ {1 - \frac {{\lambda (1 - \lambda )}}{{2{{(n + \lambda )}^2}{{\sin }^2}\theta _{v,n}^{(\lambda )}}}} \right \}} \\ { \leq a_{v,n}^{(\lambda )G} \leq \frac {\pi }{{n + \lambda }}\;{{\sin }^{2\lambda }}\theta _{v,n}^{(\lambda )}} \\ \end {array}

\]
and similar results for

λ
∉
(
0
,
1
)

\lambda \notin (0,1)

. Furthermore, assuming that

θ

v
,
n

(
λ
)

\theta _{v,n}^{(\lambda )}

remains in a fixed closed interval, lying in the interior of

(
0
,
π
)

(0,\pi )

as

n
→
∞

n \to \infty

, we show that, for every fixed

λ
>
−
1

/

2

\lambda > - 1/2

,
\[

a

v
,
n

(
λ
)
G

=

π

n
+
λ

sin

2
λ

θ

v
,
n

(
λ
)

{

1
−

λ
(
1
−
λ
)

2

(
n
+
λ
)

2

sin

2

θ

v
,
n

(
λ
)

−

λ
(
1
−
λ
)

[
3
(
λ
+
1
)
(
λ
−
2
)
+
4

sin

2

θ

v
,
n

(
λ
)

]

8

(
n
+
λ
)

4

sin

4

θ

v
,
n

(
λ
)

}

+
O
(

n

−
7

)
.

a_{v,n}^{(\lambda )G} = \frac {\pi }{{n + \lambda }}\;{\sin ^{2\lambda }}\theta _{v,n}^{(\lambda )}\left \{ {1 - \frac {{\lambda (1 - \lambda )}}{{2{{(n + \lambda )}^2}{{\sin }^2}\theta _{v,n}^{(\lambda )}}} - \frac {{\lambda (1 - \lambda )\;[3(\lambda + 1)(\lambda - 2) + 4{{\sin }^2}\theta _{v,n}^{(\lambda )}]}}{{8{{(n + \lambda )}^4}{{\sin }^4}\theta _{v,n}^{(\lambda )}}}} \right \} + O({n^{ - 7}}).

\]
Numerical evaluation of Cauchy principal value integrals with singular integrands
Authors: Rabinowitz Philip 
Published: June 2010

Convergence results are proved for sequences of interpolatory integration rules for Cauchy principal value integrals of the form
\[

∮
k
(
x
)
(
f
(
x
)

/

(
x
−
λ
)
)

d
x
,

−
1
>
λ
>
1
,

\oint k(x)(f(x)/(x - \lambda ))\,dx,\quad - 1 > \lambda > 1,

\]
when

f
(
x
)

f(x)

is singular at a point

ξ
≠
λ

\xi \ne \lambda

and the singularity is ignored or avoided.
Plana’s summation formula for ∑^{∞}_{𝑚=1,3,⋯}𝑚⁻²sin(𝑚𝛼),𝑚⁻³cos(𝑚𝛼),𝑚⁻²𝐴^{𝑚},𝑚⁻³𝐴^{𝑚}
Authors: Dempsey Kevin M. , Liu Dajin , Dempsey John P. 
Published: June 2010

The four series
\[

∑
0
∞

sin
⁡
(
2
k
+
1
)
α

/

(
2
k
+
1
)

2

,

∑
0
∞

cos
⁡
(
2
k
+
1
)
α

/

(
2
k
+
1
)

3

,

∑
0
∞

A

2
k
+
1

/

(
2
k
+
1
)

2

,

and

∑
0
∞

A

2
k
+
1

/

(
2
k
+
1
)

3

\begin {array}{*{20}{c}} {\sum \limits _0^\infty {\sin (2k + 1)\alpha /{{(2k + 1)}^2},} \quad \sum \limits _0^\infty {\cos (2k + 1)\alpha /{{(2k + 1)}^3},} } \\ {\sum \limits _0^\infty {{A^{2k + 1}}/{{(2k + 1)}^2},\quad {\text {and}}\quad \sum \limits _0^\infty {{A^{2k + 1}}/{{(2k + 1)}^3}} } } \\ \end {array}

\]
are very slowly convergent for

0
≤
α
≤
π

0 \leq \alpha \leq \pi

and as

A
→

1
−

A \to {1^ - }

. Direct summation involves thousands of terms to get the accuracy desired. Plana’s summation formula along with Romberg’s method of integration significantly and consistently improves the convergence and accuracy for the above series.
A stabilized Galerkin method for a third-order evolutionary problem
Authors: Fairweather Graeme , Sanz-Serna J. M. , Christie I. 
Published: June 2010

The periodic initial value problem for the partial differential equation

u
t

+

u

x
x
x

+
β

(

u
2

)
x

+

γ
2

(

u
2

)

x
x

+
ε

u

x
x

−
δ

u

t
x

=
0

{u_t} + {u_{xxx}} + \beta {({u^2})_x} + \frac {\gamma }{2}{({u^2})_{xx}} + \varepsilon {u_{xx}} - \delta {u_{tx}} = 0

,

ε
\varepsilon

,

δ
>
0

\delta > 0

, arises in fluidization models. The numerical integration of the problem is a difficult task in that many "reasonable" finite difference and finite element methods give rise to unstable discretizations. We show how to modify the standard Galerkin technique in order to stabilize it. Optimal-order error estimates are derived and the results of numerical experiments are presented. The stabilization technique suggested in the paper can be interpreted as rewriting the problem in Sobolev form and would also be useful for other equations involving terms of the form

u
t

−
δ

u

t
x

{u_t} - \delta {u_{tx}}

.