bluuebunny/crossref_metadata_2025_split huggingface dataset:
On 𝑘-free sequences of integers
Authors: Wagstaff Samuel S. 
Published: June 2010

Let

A

(
k
)

(
n
)

{A^{(k)}}(n)

denote the cardinality of the largest subsequence of

0
,
1
,
2
,
⋯
,
n
−
1

0,1,2, \cdots ,n - 1

, which contains no

k
k

numbers in arithmetical progression. (Such a sequence is called

k
k

-free.)

A

(
k
)

(
n
)

{A^{(k)}}(n)

is computed (on an IBM 360/65) for

3
≦
k
≦
8

3 \leqq k \leqq 8

, and various values of

n
n

to about 50. The results support the old conjecture that for all

k
k

, the limit

τ

(
k
)

=

lim

n
→
∞

(

A

(
k
)

(
n
)
)

/

n
=
0

{\tau ^{(k)}} = {\lim _{n \to \infty }}({A^{(k)}}(n))/n = 0

. The results

τ

(
5
)

>
.649
,

τ

(
6
)

>
.721
,

τ

(
7
)

>
.776

{\tau ^{(5)}} > .649,{\tau ^{(6)}} > .721,{\tau ^{(7)}} > .776

, and

τ

(
8
)

>
.8071

{\tau ^{(8)}} > .8071

are obtained. Several cases of a (disproved) conjecture of G. Szekeres are verified, including

A

(
5
)

(
94
)
=
64

{A^{(5)}}(94) = 64

.
On the nonexistence of simplex integration rules for infinite integrals
Authors: Davis P. J. , Rabinowitz P. 
Published: June 2010

It is shown that there do not exist integration rules of the form
\[

∫
0
∞

f
(
x
)

d
x
=

∑

i
=
1

n

w

i

f
(

x
i

)
+

C
n

f

(
m
)

(
ξ
)
,

0
>
ξ
>
∞
.

\int _0^\infty {f(x)} dx = \sum \limits _{i = 1}^n {{w_{_i}}f({x_i}) + {C_n}{f^{(m)}}(\xi ),\quad 0 > \xi > \infty .}

\]
An elliptic integral identity
Authors: Wrigge H. S. 
Published: June 2010

The identity
\[

K
(
τ
)
=

1

(
2
π
)

1

/

2

∫

−
∞

∞

∫

−
∞

∞

exp
⁡

[

−

1
2

(

x
4

−
2
(
2

τ
2

−
1
)

x
2

y
2

+

y
4

)

]

d
x

d
y
,

K(\tau ) = \frac {1}{{{{(2\pi )}^{1/2}}}}\int _{ - \infty }^\infty {\int _{ - \infty }^\infty {\exp \left [ { - \frac {1}{2}({x^4} - 2(2{\tau ^2} - 1){x^2}{y^2} + {y^4})} \right ]dx\;dy,} }

\]
where

K
(
τ
)

K(\tau )

is the complete elliptic integral of the first kind, is used to prove that

K
(
√
2
−
1
)
=

π

3

/

2

(
2
+
√
2

)

1

/

2

/

4
Γ

(

5
8

)

Γ

(

7
8

)

K(\surd 2 - 1) = {\pi ^{3/2}}{(2 + \surd 2)^{1/2}}/4\Gamma \left ( {\frac {5}{8}} \right )\Gamma \left ( {\frac {7}{8}} \right )

.
Computational techniques based on the Lanczos representation
Authors: Lyness J. N. 
Published: June 2010

In his bookDiscourse on Fourier Series, Lanczos deals in some detail with representations off(x)f(x)of the typef(x)=hp−1(x)+gp(x)f(x) = {h_{p - 1}}(x) + {g_p}(x)wherehp−1(x){h_{p - 1}}(x)is a polynomial of degreep−1p - 1andgp(x){g_p}(x)has the property that its full range Fourier coefficients converge at the rater−p{r^{ - p}}. In Part I, some properties ofhp(x){h_p}(x)and of the series{hp(x)}1∞\{ {h_p}(x)\} _1^\inftyare described. These properties are used here to provide criteria for the convergence or divergence of the Euler-Maclaurin series, in the case whenf(x)f(x)is an analytic function. The similarities and differences between this series and the Lidstone and other two-point series are briefly mentioned. In Part II, the Lanczos representation is employed to derive an approximate representationF(x)F(x)for an analytic functionf(x)f(x)on the interval [0, 1] is derived. This has the form\[F(x)=∑q=1p−1λq−1Bq(x)/q!+2∑r=0m/2(μrcos⁡2πrx+νrsin⁡2πrx)F(x) = \sum \limits _{q = 1}^{p - 1} {{\lambda _{q - 1}}{B_q}(x)/q! + 2\sum \limits _{r = 0}^{m/2} {({\mu _r}\cos 2\pi rx + {\nu _r}\sin 2\pi rx)} }\]and requires for its determination the values of the derivativesf(q−1)(1)−f(q−1)(0)(q=1,2,⋯p−1){f^{(q - 1)}}(1) - {f^{(q - 1)}}(0)\;(q = 1,2, \cdots p - 1)and the regularly spaced function valuesf(j/m)(j=0,1,⋯,m)f(j/m)\;(j = 0,1, \cdots ,m). It involves replacinggp(x)g_p(x)by a discrete Fourier expansion based on trapezoidal rule approximations to its Fourier coefficients. This representation is a powerful one. The drawback is that it requires derivatives. Most of Part II is devoted to the effect of using only approximate derivatives. It is shown that when these are successively less accurate with increasing order (the sort of behaviour encountered using finite difference formula), then the representation is still powerful and reliable. In a computational context the only penalty for using inaccurate derivatives is that a larger value ofmmay—or may not—be required to attain a specific accuracy.
Some problems in optimally stable Lagrangian differentiation
Authors: Salzer Herbert E. 
Published: June 2010

In many practical problems in numerical differentiation of a function

f
(
x
)

f(x)

that is known, observed, measured, or found experimentally to limited accuracy, the computing error is often much more significant than the truncating error. In numerical differentiation of the n-point Lagrangian interpolation polynomial, i.e.,

f

(
k
)

(
x
)
∼

Σ

i
=
1

n

L
i

n
(
k
)

(
x
)
f
(

x
i

)

{f^{(k)}}(x) \sim \Sigma _{i = 1}^nL_i^{n(k)}(x)f({x_i})

, a criterion for optimal stability is minimization of

Σ

i
=
1

n

|

L
i

n
(
k
)

(
x
)

|

\Sigma _{i = 1}^n|L_i^{n(k)}(x)|

. Let

L
≡
L
(
n
,
k
,

x
1

,
…
,

x
n

;
x

or

x
0

)
=

Σ

i
=
1

n

|

L
i

n
(
k
)

(
x

or

x
0

)

|

L \equiv L(n,k,{x_1}, \ldots ,{x_n};x\;{\text {or}}\;{x_0}) = \Sigma _{i = 1}^n|L_i^{n(k)}(x\;{\text {or}}\;{x_0})|

. For

x
i

{x_i}

and fixed

x
=

x
0

x = {x_0}

in

[
−
1
,
1
]

[ - 1,1]

, one problem is to find the n

x
i

{x_i}

’s to give

L
0

≡

L
0

(
n
,
k
,

x
0

)
=
min
L

{L_0} \equiv {L_0}(n,k,{x_0}) = \min L

. When the truncation error is negligible for any

x
0

{x_0}

within

[
−
1
,
1
]

[ - 1,1]

, a second problem is to find

x
0

=

x
∗

{x_0} = {x^\ast }

to obtain

L
∗

≡

L
∗

(
n
,
k
)
=
min

L
0

=
min
min
L

{L^\ast } \equiv {L^\ast }(n,k) = \min {L_0} = \min \min L

. A third much simpler problem, for

x
i

{x_i}

equally spaced,

x
1

=
−
1
,

x
n

=
1

{x_1} = - 1,{x_n} = 1

, is to find

x
¯

\bar x

to give

L
¯

≡

L
¯

(
n
,
k
)
=
min
L

\bar L \equiv \bar L(n,k) = \min L

. For lower values of n, some results were obtained on

L
0

{L_0}

and

L
∗

{L^\ast }

when

k
=
1

k = 1

, and on

L
¯

\bar L

when

k
=
1

k = 1

and 2 by direct calculation from available tables of

L
i

n
(
k
)

(
x
)

L_i^{n(k)}(x)

. The relation of

L
0

,

L
∗

{L_0},{L^\ast }

and

L
¯

\bar L

to equally spaced points, Chebyshev points, Chebyshev polynomials

T
m

(
x
)

{T_m}(x)

for

m
⩽
n
−
1

m \leqslant n - 1

, minimax solutions, and central difference formulas, considering also larger values of n, is indicated sketchily.
The resultant iteration for determining the stability of a polynomial
Authors: Howland James Lucien 
Published: June 2010

The square root iteration

x

i
+
1

=
(

x
i

+

x
i

−
1

)

/

2

{x_{i + 1}} = ({x_i} + x_i^{ - 1})/2

is known to be convergent to

+
1

+ 1

from starting values

x
0

{x_0}

in the open right half-plane, and to

−
1

- 1

from starting values in the open left half-plane. The resultant of the quadratic

z
2

−
2
w
z
+
1

{z^2} - 2wz + 1

and a given polynomial

P
(
z
)

P(z)

, of proper degree n, is a polynomial

R
(
w
)

R(w)

whose zeros, expressed in terms of the zeros

z
i

{z_i}

of

P
(
z
)

P(z)

, are just

w
i

=
(

z
i

+

z
i

−
1

)

/

2

{w_i} = ({z_i} + z_i^{ - 1})/2

. The construction of

R
(
w
)

R(w)

is thus equivalent to n independent applications of the square root iteration, with unknown starting values

z
i

{z_i}

. Repetition of this construction generates a sequence of resultants whose zeros are independently convergent to either

+
1

+ 1

or

−
1

- 1

according as the initial zero of

P
(
z
)

P(z)

lies in the open right or left half-plane. When the given polynomial, and each of the resultants, is written as a linear combination of the

n
+
1

n + 1

polynomials

(
z
+
1

)
n

,

(
z
+
1

)

n
−
1

(
z
−
1
)
,

(
z
+
1

)

n
−
2

(
z
−
1

)
2

,
…
,
(
z
+
1
)

(
z
−
1

)

n
−
1

,

(
z
−
1

)
n

{(z + 1)^n},{(z + 1)^{n - 1}}(z - 1),{(z + 1)^{n - 2}}{(z - 1)^2}, \ldots ,(z + 1){(z - 1)^{n - 1}},{(z - 1)^n}

. the sequence of resultants is ultimately convergent to one of the elements

(
z
+
1

)

n
−
p

(
z
−
1

)
p

{(z + 1)^{n - p}}{(z - 1)^p}

of this basis, whence it follows that

P
(
z
)

P(z)

has p zeros with positive real part and

n
−
p

n - p

zeros with negative real part, and the Hurwitz stability problem for

P
(
z
)

P(z)

is solved. By an application of the principle of argument it is, in general, possible to determine p at a finite stage of the resultant iteration. When formulated in terms of this basis, the resultant iteration becomes formally identical with the root-squaring process. This fact may be exploited to establish the properties of the resultant iteration, and to show that root-squaring may be applied to solve the Schur stability problem for a given polynomial. Although precise results are not available, numerical results and the general properties of this iteration suggest that, for the purpose at hand, it is unaffected by the progressive deterioration of condition that sometimes occurs in other applications of root-squaring.
Some interior estimates for semidiscrete Galerkin approximations for parabolic equations
Authors: Thomée Vidar 
Published: June 2010

Consider a solution u of the parabolic equation
\[

u
t

+
A
u
=
f

in

Ω
×
[
0
,
T
]
,

{u_t} + Au = f\quad {\text {in}}\quad \Omega \times [0,T],

\]
where A is a second order elliptic differential operator. Let

S
h

{S_h}

; h small denote a family of finite element subspaces of

H
1

(
Ω
)

{H^1}(\Omega )

which permits approximation of a smooth function to order

O
(

h
r

)

O({h^r})

. Let

Ω
0

⊂
Ω

{\Omega _0} \subset \Omega

and assume that

u
h

:
[
0
,
T
]
→

S
h

{u_h}:[0,T] \to {S_h}

is an approximate solution which satisfies the semidiscrete interior equation
\[

(

u

h
,
t

,
χ
)
+
A
(

u
h

,
χ
)
=
(
f
,
χ
)

∀
χ
∈

S
h
0

(

Ω
0

)
=
{
χ
∈

S
h

,

supp

χ
⊂

Ω
0

}
,

({u_{h,t}},\chi ) + A({u_h},\chi ) = (f,\chi )\quad \forall \chi \in S_h^0({\Omega _0}) = \{ \chi \in {S_h},{\text {supp}}\chi \subset {\Omega _0}\} ,

\]
where

A
(
⋅
,
⋅
)

A( \cdot , \cdot )

denotes the bilinear form on

H
1

(
Ω
)

{H^1}(\Omega )

associated with A. It is shown that if the finite element spaces are based on uniform partitions in a specific sense in

Ω
0

{\Omega _0}

, then difference quotients of

u
h

{u_h}

may be used to approximate derivatives of u in the interior of

Ω
0

{\Omega _0}

to order

O
(

h
r

)

O({h^r})

provided certain weak global error estimates for

u
h

−
u

{u_h} - u

to this order are available. This generalizes results proved for elliptic problems by Nitsche and Schatz [9) and Bramble, Nitsche and Schatz [1].
An error estimate for Stenger’s quadrature formula
Authors: Beighton S. , Noble B. 
Published: June 2010

The basis of this paper is the quadrature formula
\[

∫

−
1

1

f
(
x
)

d
x
≈
log
⁡
q

∑

m
=
−
M

M

2

q
m

(
1
+

q
m

)

2

f

(

q
m

−
1

q
m

+
1

)

,

\int _{ - 1}^1 {f(x)\,dx \approx \log q\sum \limits _{m = - M}^M {\frac {{2{q^m}}}{{{{(1 + {q^m})}^2}}}f\left ( {\frac {{{q^m} - 1}}{{{q^m} + 1}}} \right )} ,}

\]
where

q
=
exp
⁡
(
2
h
)

q = \exp (2h)

, h being a chosen step length. This formula has been derived from the Trapezoidal Rule formula by F. Stenger. An explicit form of the error is given for the case where the integrand has a factor of the form

(
1
−
x

)
α

(
1
+
x

)
β

,
α
,
β
>
−
1

{(1 - x)^\alpha }{(1 + x)^\beta },\alpha ,\beta > - 1

. Application is made to the evaluation of Cauchy principal value integrals with endpoint singularities and an appropriate error form is derived.
Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces
Authors: Véron Laurent 
Published: June 2010

Let

∂
Φ

\partial \Phi

be the subdifferential of some lower semicontinuous convex function

Φ
\Phi

of a real Hilbert space H,

f
∈

L
2

(
0
,
T
;
H
)

f \in {L^2}(0,T;H)

and

u
n

{u_n}

a continouous piecewise linear approximate solution of

d
u

/

d
t
+
∂
Φ
(
u
)
∋
f

du/dt + \partial \Phi (u) \ni f

, obtained by an implicit scheme. If

u
0

∈
Dom
⁡
(
Φ
)

{u_0} \in \operatorname {Dom} (\Phi )

, then

d

u
n

/

d
t

d{u_n}/dt

converges to

d
u

/

d
t

du/dt

in

L
2

(
0
,
T
;
H
)

{L^2}(0,T;H)

. Moreover, if

u
0

∈

Dom
⁡
(
∂
Φ
)

¯

{u_0} \in \overline {\operatorname {Dom} (\partial \Phi )}

, we construct a step function

η
n

(
t
)

{\eta _n}(t)

approximating t such that

lim

n
→
+
∞

∫
0
T

η
n

|

d

u
n

/

d
t
−
d
u

/

d
t

|

2

d
t
=
0

{\lim _{n \to + \infty }}\smallint _0^T{\eta _n}|d{u_n}/dt - du/dt{|^2}\;dt = 0

. When

Φ
\Phi

is inf-compact and when the sequence of approximation of f is weakly convergent to f, then

u
n

{u_n}

converges to u in

C
(
[
0
,
T
]
;
H
)

C([0,T];H)

and

η
n

d

u
n

/

d
t

{\eta _n}d{u_n}/dt

is weakly convergent to

t
d
u

/

d
t

tdu/dt

.
Linear multistep methods for stable differential equations 𝑦̈=𝐴𝑦+𝐵(𝑡)𝑦̇+𝑐(𝑡)
Authors: Gekeler Eckart 
Published: June 2010

The approximation of

y

.
.

=
A
y
+
B
(
t
)

y
.

+
c
(
t
)

{y^{..}} = Ay + B(t){y^.} + c(t)

by linear multistep methods is studied. It is supposed that the matrix A is real symmetric and negative semidefinite, that the multistep method has an interval of absolute stability

[
−
s
,
0
]

[ - s,0]

, and that

h
2

‖
A
‖

⩽
s

{h^2}\left \| A \right \| \leqslant s

where h is the time step. A priori error bounds are derived which show that the exponential multiplication factor is of the form

exp
⁡
{

Γ
s

|

|

|

B

|

|

|

n

(
n
h
)
}

\exp \{ {\Gamma _s}|||B|||_{n}(nh)\}

,

|

|

|

B

|

|

|

n

=

max

0
⩽
t
⩽
n
h

‖

B
(
t
)

‖

|||B|||_{n} = {\max _{0 \leqslant t \leqslant nh}}\left \| {B(t)} \right \|

.
Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials
Authors: Smith William E. , Sloan Ian H. , Opie Alex H. 
Published: June 2010

The paper discusses both theoretical properties and practical implementation of product integration rules of the form
\[

∫

−
∞

∞

k
(
x
)
f
(
x
)

d
x
≈

∑

i
=
1

n

w

n
i

f
(

x

n
i

)
,

\int _{ - \infty }^\infty {k(x)f(x)\,dx \approx \sum \limits _{i = 1}^n {{w_{ni}}f({x_{ni}}),} }

\]
where f is continuous, k is absolutely integrable, the nodes

{

x

n
i

}

\{ {x_{ni}}\}

are roots of the Hermite polynomials

H
n

(
x
)

{H_n}(x)

, and the weights

{

w

n
i

}

\{ {w_{ni}}\}

are chosen so that the rule is exact if f is any polynomial of degree

>
n

> n

. Convergence of the rule to the exact integral as

n
→
∞

n \to \infty

is proved for a wide class of functions f and k (including singular or oscillatory functions k), and rates of convergence are estimated. The rules are shown to have the property of asymptotic positivity, and as a consequence exhibit good numerical stability. Numerical calculations for some practical cases are presented, which show the method to be computationally effective for integrands (including highly oscillatory ones) that decay suitably at infinity. Applications of the method to integration over

[
0
,
∞
)

[0,\infty )

are also discussed.
Class groups of complex quadratic fields
Authors: Schoof R. J. 
Published: June 2010

We present 75 new examples of complex quadratic fields that have 5-rank of their class groups

⩾
3

\geqslant 3

. Only one of these fields has 5-rank of its class group

>
3

> 3

: The field

Q

(

−
258559351511807

)

{\mathbf {Q}}(\sqrt { - 258559351511807} )

has a class group isomorphic to
\[

C
(
5
)
×
C
(
5
)
×
C
(
5
)
×
C
(
5
)
×
C
(
2
)
×
C
(
11828
)
.

C(5) \times C(5) \times C(5) \times C(5) \times C(2) \times C(11828).

\]
The fields were obtained by applying ideas of J. F. Mestre to the 5-isogeny

X
1

(
11
)
→

X
0

(
11
)

{X_1}(11) \to {X_0}(11)

.
Calculation of the moments and the moment generating function for the reciprocal gamma distribution
Authors: Fransén Arne , Wrigge Staffan 
Published: June 2010

In this paper we consider the distribution

G
(
x
)
=

F

−
1

∫
0
x

(
Γ
(
t
)

)

−
1

d
t

G(x) = {F^{ - 1}}\smallint _0^x{(\Gamma (t))^{ - 1}}\;dt

. The aim of the investigation is twofold: first,to find numerical values of characteristics such as moments, variance, skewness, kurtosis,etc.; second, to study analytically and numerically the moment generating function

φ
(
t
)
=

∫
0
∞

e

−
t
x

/

Γ
(
x
)

d
x

\varphi (t) = \smallint _0^\infty {e^{ - tx}}/\Gamma (x)\;dx

. Furthermore, we also make a generalization of the reciprocal gamma distribution, and study some of its properties.
A Monte Carlo factoring algorithm with linear storage
Authors: Schnorr C.-P. , Lenstra H. W. 
Published: June 2010

We present an algorithm which will factor an integer n quite efficiently if the class number

h
(
−
n
)

h( - n)

is free of large prime divisors. The running time

T
(
n
)

T(n)

(number of compositions in the class group) satisfies

prob
⁡
[
T
(
m
)
⩽

n

1

/

2
r

]
≳

(
r
−
2

)

−
(
r
−
2
)

\operatorname {prob}[T(m) \leqslant {n^{1/2r}}] \gtrsim {(r - 2)^{ - (r - 2)}}

for random

m
∈
[
n

/

2
,
n
]

m \in [n/2,n]

and

r
⩾
2

r \geqslant 2

. So far it is unpredictable which numbers will be factored fast. Running the algorithm on all discriminants - ns with

s
⩽

r
r

s \leqslant {r^r}

and

r
=

ln
⁡
n

/

ln
⁡
ln
⁡
n

r = \sqrt {\ln n/\ln \ln n}

, every composite integer n will be factored in

o
(
exp
⁡

ln
⁡
n
ln
⁡
ln
⁡
n

)

o(\exp \sqrt {\ln n\ln \ln n} )

bit operations. The method requires an amount of storage space which is proportional to the length of the input n. In our analysis we assume a lower bound on the frequency of class numbers

h
(
−
m
)

h( - m)

,

m
⩽
n

m \leqslant n

, which are free of large prime divisors.
On Mordell’s equation 𝑦²-𝑘=𝑥³: a problem of Stolarsky
Authors: Steiner Ray P. 
Published: June 2010

On page 1 of his book Algebraic Numbers and Diophantine Approximation, K. B. Stolarsky posed the problem of solving the equationy2+999=x3{y^2} + 999 = {x^3}in positive integers. In the present paper we refine some techniques of Ellison and Pethö and show that the complete set of integer solutions of Stolarsky’s equation is\[x=10,amp;y=±1,x=12,amp;y=±27,x=40,amp;y=±251,x=147,amp;y=±1782,x=174,amp;y=±2295,\begin {array}{*{20}{c}} {x = 10,} \hfill & {y = \pm 1,} \hfill \\ {x = 12,} \hfill & {y = \pm 27,} \hfill \\ {x = 40,} \hfill & {y = \pm 251,} \hfill \\ {x = 147,} \hfill & {y = \pm 1782,} \hfill \\ {x = 174,} \hfill & {y = \pm 2295,} \hfill \\ \end {array}\]and\[x=22480,y=±3370501.x = 22480,\quad y = \pm 3370501.\]
Convergence rates for regularized solutions
Authors: Lukas Mark A. 
Published: June 2010

Given a first-kind integral equation
\[

K

u
(
x
)
=

∫
0
1

K
(
x
,
t
)
u
(
t
)

d
t
=
f
(
x
)

\mathcal {K}u(x) = \int _0^1 {K(x,t)u(t)\,dt = f(x)}

\]
with discrete noisy data

d
i

=
f
(

x
i

)
+

ε
i

{d_i} = f({x_i}) + {\varepsilon _i}

,

i
=
1
,
2
,
…
,
n

i = 1,2, \ldots ,n

, let

u

n
α

{u_{n\alpha }}

be the minimizer in a Hilbert space W of the regularization functional

(
1

/

n
)
∑

(

K

u
(

x
i

)
−

d
i

)
2

+
α

‖
u
‖

W
2

(1/n)\sum {(\mathcal {K}} u({x_i}) - {d_i}{)^2} + \alpha \left \| u \right \|_W^2

. It is shown that in any one of a wide class of norms, which includes

‖
⋅
‖

W

{\left \| \cdot \right \|_W}

, if

α
→
0

\alpha \to 0

in a certain way as

n
→
∞

n \to \infty

, then

u

n
α

{u_{n\alpha }}

converges to the true solution

u
0

{u_0}

. Convergence rates are obtained and are used to estimate the optimal regularization parameter

α
\alpha

.
Convergence results for piecewise linear quadratures for Cauchy principal value integrals
Authors: Rabinowitz Philip 
Published: June 2010

Conditions on k and f are given for the pointwise and uniform convergence to the Cauchy principal value integral
\[

∫

k
(
x
)
f
(
x
)

x
−
λ

d
x

,

−
1
>
λ
>
1
,

\int {\frac {{k(x)f(x)}}{{x - \lambda }}\,dx} ,\quad - 1 > \lambda > 1,

\]
of a sequence of integrals of piecewise linear approximations to

f
(
x
)

f(x)

or

g
λ

(
x
)
=
(
f
(
x
)
=
f
(
λ
)
)

/

(
x
−
λ
)

{g_\lambda }(x) = (f(x) = f(\lambda ))/(x - \lambda )

. The important special case,

k
(
x
)
=

(
1
−
x

)
α

(
1
+
x

)
β

k(x) = {(1 - x)^\alpha }{(1 + x)^\beta }

, is considered in detail.
An elliptic integral identity
Authors: Glasser M. L. 
Published: June 2010

The definite integral
\[

∫
0
∞

[

(

x
2

+

a
2

)

1

/

2

−
a

x
2

+

a
2

]

1

/

2

K

[

(

x
2

+

b
2

)

1

/

2

−
b

(

x
2

+

b
2

)

1

/

2

+
b

]

d
x

(

x
2

+

b
2

)

1

/

2

+
b

{\int _0^\infty {\left [ {\frac {{{{({x^2} + {a^2})}^{1/2}} - a}}{{{x^2} + {a^2}}}} \right ]} ^{1/2}}K\left [ {\frac {{{{({x^2} + {b^2})}^{1/2}} - b}}{{{{({x^2} + {b^2})}^{1/2}} + b}}} \right ]\frac {{dx}}{{{{({x^2} + {b^2})}^{1/2}} + b}}

\]
is evaluated in closed form.
Phenoclustering: online mining of cross-species phenotypes
Authors: Groth Philip , Kalev Ivan , Kirov Ivaylo , Traikov Borislav , Leser Ulf , Weiss Bertram 
Published: June 2010

Abstract
Summary: Recently, several methods for analyzing phenotype data have been published, but only few are able to cope with data sets generated in different studies, with different methods, or for different species. We developed an online system in which more than 300 000 phenotypes from a wide variety of sources and screening methods can be analyzed together. Clusters of similar phenotypes are visualized as networks of highly similar phenotypes, inducing gene groups useful for functional analysis. This system is part of PhenomicDB, providing the world's largest cross-species phenotype data collection with a tool to mine its wealth of information.
Availability: Freely available at http://www.phenomicdb.de
Contact: bertram.weiss@bayerhealthcare.com
Supplementary information: Supplementary data are available at Bioinformatics online.
Assemble: an interactive graphical tool to analyze and build RNA architectures at the 2D and 3D levels
Authors: Jossinet Fabrice , Ludwig Thomas E. , Westhof Eric 
Published: June 2010

Abstract
Summary: Assemble is an intuitive graphical interface to analyze, manipulate and build complex 3D RNA architectures. It provides several advanced and unique features within the framework of a semi-automated modeling process that can be performed by homology and ab initio with or without electron density maps. Those include the interactive editing of a secondary structure and a searchable, embedded library of annotated tertiary structures. Assemble helps users with performing recurrent and otherwise tedious tasks in structural RNA research.
Availability and Implementation: Assemble is released under an open-source license (MIT license) and is freely available at http://bioinformatics.org/assemble. It is implemented in the Java language and runs on MacOSX, Linux and Windows operating systems.
Contact: f.jossinet@ibmc-cnrs.unistra.fr
CplexA: a Mathematica package to study macromolecular-assembly control of gene expression
Authors: Vilar Jose M.G. , Saiz Leonor 
Published: June 2010

Abstract
Summary: Macromolecular assembly coordinates essential cellular processes, such as gene regulation and signal transduction. A major challenge for conventional computational methods to study these processes is tackling the exponential increase of the number of configurational states with the number of components. CplexA is a Mathematica package that uses functional programming to efficiently compute probabilities and average properties over such exponentially large number of states from the energetics of the interactions. The package is particularly suited to study gene expression at complex promoters controlled by multiple, local and distal, DNA binding sites for transcription factors.
Availability: CplexA is freely available together with documentation at http://sourceforge.net/projects/cplexa/
Contact: j.vilar@ikerbasque.org; lsaiz@ucdavis.edu
Supplementary information: Supplementary data are available at Bioinformatics online.