bluuebunny/crossref_metadata_2025 huggingface dataset:
Numerical solution of nonlinear differential equations with algebraic constraints. I. Convergence results for backward differentiation formulas
Authors: Lötstedt Per , Petzold Linda
Published: June 2010

In this paper we investigate the behavior of numerical ODE methods for the solution of systems of differential equations coupled with algebraic constraints. Systems of this form arise frequently in the modelling of problems from physics and engineering; we study some particular examples from electrical networks, fluid dynamics and constrained mechanical systems. We show that backward differentiation formulas converge with the expected order of accuracy for these systems.
Rate of convergence of Shepard’s global interpolation formula
Authors: Farwig Reinhard
Published: June 2010

Given any data points

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

in

R

s

{{\mathbf {R}}^s}

and values

S
p
q

S_p^q

of a function f, Shepard’s global interpolation formula reads as follows:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript p Superscript 0 Baseline f left-parenthesis x right-parenthesis equals sigma-summation Underscript i Endscripts f left-parenthesis x Subscript i Baseline right-parenthesis w Subscript i Baseline left-parenthesis x right-parenthesis comma w Subscript i Baseline left-parenthesis x right-parenthesis equals StartAbsoluteValue x minus x Subscript i Baseline EndAbsoluteValue Superscript negative p Baseline slash sigma-summation Underscript j Endscripts StartAbsoluteValue x minus x Subscript j Baseline EndAbsoluteValue Superscript negative p Baseline comma">

S
p
0

f
(
x
)
=

∑
i

f
(

x
i

)

w
i

(
x
)
,

w
i

(
x
)
=

|

x
−

x
i

|

−
p

/

∑
j

|

x
−

x
j

|

−
p

,

S_p^0f(x) = \sum \limits _i {f({x_i}){w_i}(x),\quad {w_i}(x) = |x - {x_i}{|^{ - p}}/\sum \limits _j {|x - {x_j}{|^{ - p}},} }

\]
where

f
(

x
1

)
,
…
,
f
(

x
n

)

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

denotes the Euclidean norm in

|

⋅

|

| \cdot |

. This interpolation scheme is stable, but if

R

s

{{\mathbf {R}}^s}

, the gradient of the interpolating function vanishes in all data points. The interpolation operator

p
>
1

p > 1

is defined by replacing the values

S
p
q

S_p^q

in

f
(

x
i

)

f({x_i})

by Taylor polynomials of f of degree

S
p
0

f

S_p^0f

. In this paper, we investigate the approximating power of

q
∈

N

q \in {\mathbf {N}}

for all values of p, q and s.
Effect of improved multiplication efficiency on exponentiation algorithms derived from addition chains
Authors: McCarthy D. P.
Published: June 2010

The interaction between the efficiency of the basic multiplication algorithm and the addition chain used to compute

x
n

{x^n}

is studied. We conclude that either repeated multiplication by x or repeated squaring should be used and the provenance of each technique is established.
On the zeros of the Riemann zeta function in the critical strip. IV
Authors: van de Lune J. , te Riele H. J. J. , Winter D. T.
Published: June 2010

Very extensive computations are reported which extend and, partly, check previous computations concerning the location of the complex zeros of the Riemann zeta function. The results imply the truth of the Riemann hypothesis for the first 1,500,000,001 zeros of the form

σ
+
i
t

\sigma + it

in the critical strip with

0
>
t
>
545
,
439
,
823.215

0 > t > 545,439,823.215

, i.e., all these zeros have real part

σ
=
1

/

2

\sigma = 1/2

. Moreover, all these zeros are simple. Various tables are given with statistical data concerning the numbers and first occurrences of Gram blocks of various types; the numbers of Gram intervals containing m zeros, for

m
=
0
,
1
,
2
,
3

m = 0,1,2,3

and 4; and the numbers of exceptions to "Rosser’s rule" of various types (including some formerly unobserved types). Graphs of the function

Z
(
t
)

Z(t)

are given near five rarely occurring exceptions to Rosser’s rule, near the first Gram block of length 9, near the closest observed pair of zeros of the Riemann zeta function, and near the largest (positive and negative) found values of

Z
(
t
)

Z(t)

at Gram points. Finally, a number of references are given to various number-theoretical implications.
Integers with digits 0 or 1
Authors: Lehmer D. H. , Mahler K. , van der Poorten A. J.
Published: June 2010

Let

g
⩾
2

g \geqslant 2

be a given integer and

L

\mathcal {L}

the set of nonnegative integers which may be expressed in base g employing only the digits 0 or 1. Given an integer

k
>
1

k > 1

, we study congruences

l
≡
a

(
mod

k
)

l \equiv a\;\pmod k

,

l
∈

L

l \in \mathcal {L}

and show that such a congruence either has infinitely many solutions, or no solutions in

L

\mathcal {L}

. There is a simple criterion to distinguish the two cases. The casual reader will be intrigued by our subsequent discussion of techniques for obtaining the smallest nontrivial solution of the cited congruence.
On the congruence 2^{𝑛-𝑘}≡1(𝑚𝑜𝑑𝑛)
Authors: Shen Mok-Kong
Published: June 2010

It is shown that there are infinitely many positive integers k such that the congruence

2

n
−
k

≡
1

(
mod

n
)

{2^{n - k}} \equiv 1\;\pmod n

has infinitely many solutions n.
Two triads of squares
Authors: Lagrange J. , Leech J.
Published: June 2010

The thirteen points (0,0),

(
±

a
i

,
0
)

( \pm {a_i},0)

,

i
=
1
,
2
,
3

i = 1,2,3

,

(
0
,
±

b
j

)

(0, \pm {b_j})

,

j
=
1
,
2
,
3

j = 1,2,3

, will be at integer distances from one another if the two triads

a
1
2

a_1^2

,

a
2
2

a_2^2

,

a
3
2

a_3^2

,

b
1
2

b_1^2

,

b
2
2

b_2^2

,

b
3
2

b_3^2

are such that the nine sums

a
i
2

+

b
j
2

a_i^2 + b_j^2

are all perfect squares. Infinite families of solutions are derived from solutions of

{
m
,
n

}
2

=
{
p
,
q
}
{
r
,
s
}

{\{ m,n\} ^2} = \{ p,q\} \{ r,s\}

, where

{
m
,
n
}
=
(

m
2

−

n
2

)

/

2
m
n

\{ m,n\} = ({m^2} - {n^2})/2mn

, etc. Additional numerical examples are given. Two solutions are given in which one of the triads is extended to a tetrad.
On solving singular integral equations via a hyperbolic tangent quadrature rule
Authors: Venturino Ezio
Published: June 2010

We propose a scheme for solving singular integral equations based on a "hyperbolic tangent" quadrature rule. The integral equation is reduced to a system of linear equations, after quadrature and collocation. The matrix of the system is shown to be nonsingular for every choice of the number of quadrature nodes by producing a lower bound for its determinant.
A stability analysis of incomplete 𝐿𝑈 factorizations
Authors: Elman Howard C.
Published: June 2010

The combination of iterative methods with preconditionings based on incomplete LU factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations. In this paper, we show that there are some settings in which the incomplete LU preconditioners are not effective, and we demonstrate that their poor performance is due to numerical instability. Our analysis consists of an analytic and numerical study of a sample two-dimensional non-self-adjoint elliptic problem discretized by several finite-difference schemes.
Quadrature formulas for functions with poles near the interval of integration
Authors: Monegato Giovanni
Published: June 2010

In this paper, we examine the construction of quadrature rules of interpolatory type, using only real function values, for functions with complex conjugate pairs of poles near the interval of integration.
Absorbing boundary conditions for difference approximations to the multidimensional wave equation
Authors: Higdon Robert L.
Published: June 2010

We consider the problem of constructing absorbing boundary conditions for the multi-dimensional wave equation. Here we work directly with a difference approximation to the equation, rather than first finding analytical boundary conditions and then discretizing the analytical conditions. This approach yields some simple and effective discrete conditions. These discrete conditions are consistent with analytical conditions that are perfectly absorbing at certain nonzero angles of incidence. This fact leads to a simple and general canonical form for analytical absorbing boundary conditions. The use of this form has theoretical and practical advantages.
Supra-convergent schemes on irregular grids
Authors: Kreiss H.-O. , Manteuffel T. A. , Swartz B. , Wendroff B. , White A. B.
Published: June 2010

As Tikhonov and Samarskiĭ showed for

k
=
2

k = 2

, it is not essential that kth-order compact difference schemes be centered at the arithmetic mean of the stencil’s points to yield second-order convergence (although it does suffice). For stable schemes and even k, the main point is seen when the kth difference quotient is set equal to the value of the kth derivative at the middle point of the stencil; the proof is particularly transparent for

k
=
2

k = 2

. For any k, in fact, there is a

⌊

k

/

2

⌋

\left \lfloor {k/2} \right \rfloor

-parameter family of symmetric averages of the values of the kth derivative at the points of the stencil which, when similarly used, yield second-order convergence. The result extends to stable compact schemes for equations with lower-order terms under general boundary conditions. Although the extension of Numerov’s tridiagonal scheme (approximating

D
2

y
=
f

{D^2}y = f

with third-order truncation error) yields fourth-order convergence on meshes consisting of a bounded number of pieces in which the mesh size changes monotonically, it yields only third-order convergence to quintic polynomials on any three-periodic mesh with unequal adjacent mesh sizes and fixed adjacent mesh ratios. A result of some independent interest is appended (and applied): it characterizes, simply, those functions of k variables which possess the property that their average value, as one translates over one period of an arbitrary periodic sequence of arguments, is zero; i.e., those bounded functions whose average value, as one translates over arbitrary finite sequences of arguments, goes to zero as the length of the sequences increases.
Integral equations on the half-line: a modified finite-section approximation
Authors: Sloan I. H. , Spence A.
Published: June 2010

We consider the approximate solution of integral equations of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y left-parenthesis t right-parenthesis minus integral Subscript 0 Superscript normal infinity Baseline k left-parenthesis t comma s right-parenthesis y left-parenthesis s right-parenthesis d s equals f left-parenthesis t right-parenthesis">

y
(
t
)
−

∫
0
∞

k
(
t
,
s
)
y
(
s
)

d
s
=
f
(
t
)

y(t) - \smallint _0^\infty k(t,s) y(s)\,ds = f(t)

, where the conditions on

k
(
t
,
s
)

k(t,s)

are such that kernels of the Wiener-Hopf form

k
(
t
,
s
)
=
κ
(
t
−
s
)

k(t,s) = \kappa (t - s)

are included in the analysis. The finite-section approximation, in which the infinite integral is replaced by

∫
0
β

\smallint _0^\beta

for some

β
>
0

\beta > 0

, yields an approximate solution

y
β

(
t
)

{y_\beta }(t)

that is known, under very general conditions, to converge to

y
(
t
)

y(t)

as

β
→
∞

\beta \to \infty

with t fixed. However, the convergence is uniform only on finite intervals, and the approximation is typically very poor for

t
>
β

t > \beta

. Under the assumption that f has a limit at infinity, we here introduce a modified finite-section approximation with improved approximation properties, and prove that the new approximate solution converges uniformly to y as

β
→
∞

\beta \to \infty

. A numerical example illustrates the improvement.
On computing Gauss-Kronrod quadrature formulae
Authors: Caliò Franca , Gautschi Walter , Marchetti Elena
Published: June 2010

We discuss the use of Newton’s method for computing Gauss-Kronrod quadrature formulae from modified moments. The underlying nonlinear maps are analyzed from the point of view of numerical condition. A method is indicated of computing the polynomial whose zeros are the Kronrod nodes. Examples include Gauss-Kronrod formulae for integrals with a logarithmic and algebraic singularity at one endpoint. Pertinent numerical results are tabulated in the supplements section at the end of this issue.
An estimate of goodness of cubatures for the unit circle in 𝑅²
Authors: Maeztu J. I.
Published: June 2010

The Sarma-Eberlein estimate

s
E

{s_E}

is an estimate of goodness of cubature formulae for n-cubes defined as the integral of the square of the formula truncation error, over a function space provided with a measure. In this paper, cubature formulae for the unit circle in

R

2

{{\mathbf {R}}^2}

are considered and an estimate of the above type is constructed with the desirable property of being compatible with the symmetry group of the circle.
The primality of 𝑅1031
Authors: Williams H. C. , Dubner Harvey
Published: June 2010

A description is given of a technique for proving

R
1031

(
=
(

10

1031

−
1
)

/

9
)

R1031\;( = ({10^{1031}} - 1)/9)

a prime.
Special units in real cyclic sextic fields
Authors: Gras Marie-Nicole
Published: June 2010

We study the real cyclic sextic fields generated by a root w of

(
X
−
1

)
6

−
(

t
2

+
108
)

(

X
2

+
X

)
2

{(X - 1)^6} - ({t^2} + 108){({X^2} + X)^2}

,

t
∈

Z

−
{
0
,
±
6
,
±
26
}

t \in {\mathbf {Z}} - \{ 0, \pm 6, \pm 26\}

. We show that, when

t
2

+
108

{t^2} + 108

is square-free (except for powers of 2 and 3), and

t
≠
0

t \ne 0

,

±
10

\pm 10

,

±
54

\pm 54

, then w is a generator of the module of relative units. The details of the proofs are given in [3].
On computing isomorphisms of equation orders
Authors: Pohst M.
Published: June 2010

A number-geometric method for computing isomorphisms of algebraic number fields (respectively,

Z

\mathbb {Z}

-orders of such fields) is developed. Its main advantage is its easy implementation and moderate computation time.
New congruences for the Bernoulli numbers
Authors: Tanner Jonathan W. , Wagstaff Samuel S.
Published: June 2010

We prove a new congruence for computing Bernoulli numbers modulo a prime. Since it is similar to Vandiver’s congruences but has fewer terms, it may be used to test primes for regularity efficiently. We have programmed this test on a CYBER 205 computer. Fermat’s "Last Theorem" has been proved for all exponents up to 150000.
Steady state computations for wave propagation problems
Authors: Engquist Björn , Gustafsson Bertil
Published: June 2010

The behavior of difference approximations of hyperbolic partial differential equations as time

t
→
∞

t \to \infty

is studied. The rate of convergence to steady state is analyzed theoretically and expe imentally for the advection equation and the linearized Euler equations. The choice of difference formulas and boundary conditions strongly influences the rate of convergence in practical steady state calculations. In particular it is shown that upwind difference methods and characteristic boundary conditions have very attractive convergence properties.
TVB uniformly high-order schemes for conservation laws
Authors: Shu Chi-Wang
Published: June 2010

In the computation of conservation laws

u
t

+
f

(
u

)
x

=
0

{u_t} + f{(u)_x} = 0

, TVD (total-variation-diminishing) schemes have been very successful. But there is a severe disadvantage of all TVD schemes: They must degenerate locally to first-order accuracy at nonsonic critical points. In this paper we describe a procedure to obtain TVB (total-variation-bounded) schemes which are of uniformly high-order accuracy in space including at critical points. Together with a TVD high-order time discretization (discussed in a separate paper), we may have globally high-order in space and time TVB schemes. Numerical examples are provided to illustrate these schemes.
Quadrature formulae for Cauchy principal value integrals of oscillatory kind
Authors: Okecha G. E.
Published: June 2010

The problem considered is that of evaluating numerically an integral of the form

f

−
1

1

e

i
ω
x

f
(
x
)

d
x

f_{ - 1}^1\;{e^{i\omega x}}f(x)\,dx

, where f has one simple pole in the interval

[
−
1
,
1
]

[ - 1,1]

. Modified forms of the Lagrangian interpolation formula, taking account of the simple pole are obtained, and form the bases for the numerical quadrature rules obtained. Further modification to deal with the case when an abscissa in the interpolation formula is coincident with the pole is also considered. An error bound is provided and some numerical examples are given to illustrate the formulae developed.
A note on elliptic curves over finite fields
Authors: Rück Hans-Georg
Published: June 2010

Let E be an elliptic curve over a finite field k and let

E
(
k
)

E(k)

be the group of k-rational points on E. We evaluate all the possible groups

E
(
k
)

E(k)

where E runs through all the elliptic curves over a given fixed finite field k.
On the characterization of 𝑞-superlinear convergence of quasi-Newton methods for constrained optimization
Authors: Stoer J. , Tapia R. A.
Published: June 2010

In this paper we present a short, straightforward and self-contained derivation of the Boggs-Tolle-Wang characterization of those quasi-Newton methods for equality-constrained optimization which produce iterates which are q-superlinearly convergent.
Some inequalities for continued fractions
Authors: Dudley R. M.
Published: June 2010

For some continued fractions

Q
=

b
0

+

a
1

/

(

b
1

+
⋯
)

Q = {b_0} + {a_1}/({b_1} + \cdots )

with mth convergent

Q
m

{Q_m}

, it is shown that relative errors are monotone in some arguments. If all the entries

a
j

{a_j}

and

b
j

{b_j}

in Q are positive, then the relative error

|

Q
m

/

Q
−
1

|

|{Q_m}/Q - 1|

is bounded by

|

Q
m

/

Q

m
+
1

−
1

|

|{Q_m}/{Q_{m + 1}} - 1|

, which is nonincreasing in the partial denominator

b
j

{b_j}

for each

j
⩾
0

j \geqslant 0

, as is

|

Q
m

/

Q
−
1

|

|{Q_m}/Q - 1|

for

j
⩽
m
+
1

j \leqslant m + 1

. If

b
j

⩾
1

{b_j} \geqslant 1

for all

j
⩾
1

j \geqslant 1

,

b
0

⩾
0

{b_0} \geqslant 0

, and

a
j

=

(
−
1

)

j
+
1

c
j

{a_j} = {( - 1)^{j + 1}}{c_j}

where

c
j

⩾
0

{c_j} \geqslant 0

and for j even,

c
j

>
1

{c_j} > 1

, then

|

Q
m

/

Q
−
1

|

|{Q_m}/Q - 1|

is bounded by

|

Q
m

/

Q

m
+
2

−
1

|

|{Q_m}/{Q_{m + 2}} - 1|

, and both are nonincreasing in

b
j

{b_j}

for even

j
⩽
m
+
2

j \leqslant m + 2

. These facts apply to continued fractions of Euler, Gauss and Laplace used in computing Poisson, binomial and normal probabilities, respectively, giving monotonicity of relative errors as functions of the variables in suitable ranges.
Calculation of the class numbers of imaginary cyclic quartic fields
Authors: Hardy Kenneth , Hudson R. H. , Richman D. , Williams Kenneth S. , Holtz N. M.
Published: June 2010

Any imaginary cyclic quartic field can be expressed uniquely in the form

K
=
Q
(

A
(
D
+
B

D

)

)

K = Q(\sqrt {A(D + B\sqrt D )} )

, where A is squarefree, odd and negative,

D
=

B
2

+

C
2

D = {B^2} + {C^2}

is squarefree,

B
>
0
,
C
>
0

B > 0,C > 0

, and

(
A
,
D
)
=
1

(A,D) = 1

. Explicit formulae for the discriminant and conductor of K are given in terms of A, B, C, D. The calculation of tables of the class numbers

h
(
K
)

h(K)

of such fields K is described.
Product integration-collocation methods for noncompact integral operator equations
Authors: Chandler G. A. , Graham I. G.
Published: June 2010

We discuss the numerical solution of a class of second-kind integral equations in which the integral operator is not compact. Such equations arise, for example, when boundary integral methods are applied to potential problems in a two-dimensional domain with corners in the boundary. We are able to prove the optimal orders of convergence for the usual collocation and product integration methods on graded meshes, provided some simple modifications are made to the underlying basis functions. These are sufficient to ensure stability, but do not damage the rate of convergence. Numerical experiments show that such modifications are necessary in certain circumstances.
A note on piecewise linear and multilinear table interpolation in many dimensions
Authors: Weiser Alan , Zarantonello Sergio E.
Published: June 2010

This note is concerned with N-dimensional rectangular table interpolation, where N is relatively large (4 to 10). Two interpolants are considered: a piecewise multilinear generalization of piecewise bilinear interpolation on rectangles, and a piecewise linear generalization of piecewise linear interpolation on triangles. We show that the two interpolants have similar approximation properties, but the piecewise linear interpolant is much cheaper to evaluate.
The twentieth Fermat number is composite
Authors: Young Jeff , Buell Duncan A.
Published: June 2010

The twentieth Fermat number,

F

20

=

2

2

20

+
1

{F_{20}} = {2^{{2^{20}}}} + 1

, has been proven composite by machine computation.
An adaptive finite element method for linear elliptic problems
Authors: Eriksson Kenneth , Johnson Claes
Published: June 2010

We propose an adaptive finite element method for linear elliptic problems based on an optimal maximum norm error estimate. The algorithm produces a sequence of successively refined meshes with a final mesh on which a given error tolerance is satisfied. In each step the refinement to be made is determined by locally estimating the size of certain derivatives of the exact solution through computed finite element solutions. We analyze and justify the algorithm in a model case.
Efficient algorithms for the evaluation of the eigenvalues of (block) banded Toeplitz matrices
Authors: Bini D. , Pan V.
Published: June 2010

Let A be an

n
×
n

n \times n

banded block Toeplitz matrix of bandwidth k with

m
×
m

m \times m

blocks having entries in a field F. We present algorithms for computing

p
(
λ
)
=
det
(
A
−
λ
I
)

p(\lambda ) = \det (A - \lambda I)

as well as the ratio

p
(
λ
)

/

p
′

(
λ
)

p(\lambda )/p’(\lambda )

, where

p
′

(
λ
)

p’(\lambda )

is the first derivative of

p
(
λ
)

p(\lambda )

with respect to

λ
\lambda

, in roughly

(
3

/

2
)

k
2

log
⁡
n
+
O
(

k
3

)

(3/2){k^2}\log n + O({k^3})

block multiplications. If the field F supports FFT, then the cost is reduced to

O
(
(

m
2

k
log
⁡
k
+

m
3

k
)
log
⁡
n
+

k
3

m
3

)

O(({m^2}k\log k + {m^3}k)\log n + {k^3}{m^3})

scalar multiplications. The algorithms generalize an algorithm given by W. Trench for computing

p
(
λ
)

p(\lambda )

in the case

m
=
1

m = 1

in roughly

k
log
⁡
n
+
O
(

k
3

)

k\log n + O({k^3})

multiplications and rely on powering a companion matrix associated with the linear recurrence relation representing the original problem.
Error bounds for linear recurrence relations
Authors: Olver F. W. J.
Published: June 2010

Recurrence relations of the form
\[

a
r

p

r
+
1

=

b
r

p
r

+

c
r

p

r
−
1

{a_r}{p_{r + 1}} = {b_r}{p_r} + {c_r}{p_{r - 1}}

\]
are examined in two cases: (A) oscillatory systems, for which

b
r
2

+
4

a
r

c
r

>
0

b_r^2 + 4{a_r}{c_r} > 0

; (B) monotonic systems, for which

b
r
2

+
4

a
r

c
r

≥
0

b_r^2 + 4{a_r}{c_r} \geq 0

. In both cases, a posteriori methods are supplied for constructing strict and realistic error bounds in

O
(
r
)

O(r)

arithmetic operations. A priori bounds, also requiring

O
(
r
)

O(r)

arithmetic operations, are supplied in Case B. Several illustrative numerical examples are included.
Connection between Gaussian periods and cyclic units
Authors: Lehmer Emma
Published: June 2010

This paper finds that all known parametric families of units in real quadratic, cubic, quartic and sextic fields with prime conductor are linear combinations of Gaussian periods and exhibits these combinations. This approach is used to find new units in the real quintic field for prime conductors

p
=

n
4

+
5

n
3

+
15

n
2

+
25
n
+
25

p = {n^4} + 5{n^3} + 15{n^2} + 25n + 25
Are there odd amicable numbers not divisible by three?
Authors: Battiato S. , Borho W.
Published: June 2010

A conjecture of Bratley and McKay, according to which odd amicable numbers should be divisible by three, is disproved by some counterexamples.
The construction of preconditioners for elliptic problems by substructuring. III
Authors: Bramble James H. , Pasciak Joseph E. , Schatz Alfred H.
Published: June 2010

In earlier parts of this series of papers, we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems. The resulting algorithms are well suited for implementation on computers with parallel architecture. In this paper, we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners. The iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns. These preconditioners involve an incomplete Chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work. Theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods.
Generation of finite difference formulas on arbitrarily spaced grids
Authors: Fornberg Bengt
Published: June 2010

Simple recursions are derived for calculating the weights in compact finite difference formulas for any order of derivative and to any order of accuracy on one-dimensional grids with arbitrary spacing. Tables are included for some special cases (of equispaced grids).
An efficient linear scheme to approximate parabolic free boundary problems: error estimates and implementation
Authors: Nochetto Ricardo H. , Verdi Claudio
Published: June 2010

This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; two-phase Stefan problems and porous medium equations are included. The algorithm consists of approximating at each time step a linear elliptic partial differential equation by piecewise linear finite elements and then making an element-by-element algebraic correction to account for the nonlinearity. Several energy error estimates are derived for the physical unknowns; a sharp rate of convergence of

O
(

h

1

/

2

)

O({h^{1/2}})

is our main result. The crucial point in implementing the scheme is the efficient resolution of linear systems involved. This topic is discussed, and the results of several numerical experiments are shown.
The representation of lattice quadrature rules as multiple sums
Authors: Sloan Ian H. , Lyness James N.
Published: June 2010

We provide a classification of lattice rules. Applying elementary group theory, we assign to each s-dimensional lattice rule a rank m and a set of positive integer invariants

n
1

,

n
2

,
…
,

n
s

{n_1},{n_2}, \ldots ,{n_s}

. The number

ν
(
Q
)

\nu (Q)

of abscissas required by the rule is the product

n
1

n
2

⋯

n
s

{n_1}{n_2} \cdots {n_s}

, and the rule may be expressed in a canonical form with m independent summations. Under this classification an N-point number-theoretic rule in the sense of Korobov and Conroy is a rank

m
=
1

m = 1

rule having invariants N, 1, 1,..., 1, and the product trapezoidal rule using

n
s

{n^s}

points is a rank

m
=
s

m = s

rule having invariants n, n,..., n. Besides providing a canonical form, we give some of the properties of copy rules and of projections into lower dimensions.
Cosine methods for nonlinear second-order hyperbolic equations
Authors: Bales Laurence A. , Dougalis Vassilios A.
Published: June 2010

We construct and analyze efficient, high-order accurate methods for approximating the smooth solutions of a class of nonlinear, second-order hyperbolic equations. The methods are based on Galerkin type discretizations in space and on a class of fourth-order accurate two-step schemes in time generated by rational approximations to the cosine. Extrapolation from previous values in the coefficients of the nonlinear terms and use of preconditioned iterative techniques yield schemes whose implementation requires solving a number of linear systems at each time step with the same operator.

L
2

{L^2}

optimal-order error estimates are proved.
Nonlinear filters for efficient shock computation
Authors: Engquist Björn , Lötstedt Per , Sjögreen Björn
Published: June 2010

A new type of methods for the numerical approximation of hyperbolic conservation laws with discontinuous solution is introduced. The methods are based on standard finite difference schemes. The difference solution is processed with a nonlinear conservation form filter at every time level to eliminate spurious oscillations near shocks. It is proved that the filter can control the total variation of the solution and also produce sharp discrete shocks. The method is simpler and faster than many other high resolution schemes for shock calculations. Numerical examples in one and two space dimensions are presented.
Nonconforming finite elements for the Stokes problem
Authors: Crouzeix Michel , Falk Richard S.
Published: June 2010

A new stability result is obtained for the approximation of the stationary Stokes problem by nonconforming piecewise cubic approximations to the velocities and a discontinuous piecewise quadratic approximation to the pressure. The basic result is that for most reasonable meshes, these elements form a stable pair without the addition of quartic bubble functions (which had previously been added to insure stability).
Design of a large helical system for magnetic fusion experiments
Authors: Garabedian P. R.
Published: June 2010

Computer codes are described that solve the partial differential equations of ideal magnetohydrodynamics describing toroidal equilibrium of a plasma. These numerical methods are applied to find a helical coil winding law for stellarator experiments designed so that the parallel current is relatively low.
A family of generalized Jacobi polynomials
Authors: Locher F.
Published: June 2010

The family of orthogonal polynomials corresponding to a generalized Jacobi weight function was considered by Wheeler and Gautschi who derived recurrence relations, both for the related Chebyshev moments and for the associated orthogonal polynomials. We obtain an explicit representation of these polynomials, from which the recurrence relation can be derived.
Some pseudoprimes and related numbers having special forms
Authors: McDaniel Wayne L.
Published: June 2010

We give an example of a pseudoprime which is itself of the form

2
n

−
2

{2^n} - 2

, answering a question posed by A. Rotkiewicz, show that Lehmer’s example of an even pseudoprime having three prime factors is not unique, and answer a question of Benkoski concerning the solutions of

2

n
−
2

≡
1

(
mod

n
)

{2^{n - 2}} \equiv 1\;\pmod n

.
Uniform error estimates of Galerkin methods for monotone Abel-Volterra integral equations on the half-line
Authors: Eggermont P. P. B.
Published: June 2010

We consider Galerkin methods for monotone Abel-Volterra integral equations of the second kind on the half-line. The

L
2

{L^2}

theory follows from Kolodner’s theory of monotone Hammerstein, equations. We derive the

L
∞

{L^\infty }

theory from the

L
2

{L^2}

theory by relating the

L
2

{L^2}

- and

L
∞

{L^\infty }

-spectra of operators of the form

x
→
b
∗
(
a
x
)

x \to b \ast (ax)

to one another. Here

∗
\ast

denotes convolution, and

b
∈

L
1

b \in {L^1}

and

a
∈

L
∞

a \in {L^\infty }

. As an extra condition we need

b
(
t
)
=
O
(

t

−
α
−
1

)

b(t) = O({t^{ - \alpha - 1}})

, with

α
>
0

\alpha > 0

. We also prove the discrete analogue. In particular, we verify that the Galerkin matrix satisfies the "discrete" conditions.
On the convergence of a time discretization scheme for the Navier-Stokes equations
Authors: Geveci T.
Published: June 2010

A linearized version of the implicit Euler scheme is considered for the approximation of the solutions to the Navier-Stokes equations in a two-dimensional domain. The rate of convergence in the

H
1

{H^1}

-norm is established.
Estimates of eigenvalues for iterative methods
Authors: Golub Gene H. , Kent Mark D.
Published: June 2010

We describe a procedure for determining estimates of the eigenvalues of operators used in various iterative methods for the solution of linear systems of equations. We also show how to determine upper and lower bounds for the error in the approximate solution of linear equations, using essentially the same information as that needed for the eigenvalue calculations. The methods described depend strongly upon the theory of moments and Gauss quadrature.
Stability and convergence of spectral methods for hyperbolic initial-boundary value problems
Authors: Dutt P.
Published: June 2010

In this paper we present a modified version of the pseudospectral method for solving initial-boundary value systems of hyperbolic partial differential equations. We are able to avoid problems of instability by regularizing the boundary conditions. We prove the stability and convergence of our proposed scheme and obtain error estimates.
Some properties of rank-2 lattice rules
Authors: Lyness J. N. , Sloan I. H.
Published: June 2010

A rank-2 lattice rule is a quadrature rule for the (unit) s-dimensional hypercube, of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q f equals left-parenthesis 1 slash n 1 n 2 right-parenthesis sigma-summation Underscript j 1 equals 1 Overscript n 1 Endscripts sigma-summation Underscript j 2 equals 1 Overscript n 2 Endscripts f overbar left-parenthesis j 1 bold z 1 slash n 1 plus j 2 bold z 2 slash n 2 right-parenthesis comma">

Q
f
=
(
1

/

n
1

n
2

)

∑

j
1

=
1

n
1

∑

j
2

=
1

n
2

f
¯

(

j
1

z

1

/

n
1

+

j
2

z

2

/

n
2

)
,

Qf = (1/{n_1}{n_2})\sum \limits _{{j_1} = 1}^{{n_1}} {\sum \limits _{{j_2} = 1}^{{n_2}} {\bar f({j_1}{{\mathbf {z}}_1}/{n_1} + {j_2}{{\mathbf {z}}_2}/{n_2}),} }

\]
which cannot be re-expressed in an analogous form with a single sum. Here

f
¯

\bar f

is a periodic extension of f, and

z

1

{{\mathbf {z}}_1}

,

z

2

{{\mathbf {z}}_2}

are integer vectors. In this paper we discuss these rules in detail; in particular, we categorize a special subclass, whose leading one- and two-dimensional projections contain the maximum feasible number of abscissas. We show that rules of this subclass can be expressed uniquely in a simple tricycle form.
The second-order sharpening of blurred smooth borders
Authors: Swartz Blair
Published: June 2010

The problem is to approximate, with local second-order accuracy, the smooth boundary separating a black and a white region in the plane, given discretely located gray values associated with a blurring of that border. "Second-order", here, is with respect to the size h of the scale of the prescribed blurring. The locally determined approximations are line segments. The algorithms discussed here can result in second-order accuracy, but they may not in certain geometric circumstances. Typical local curvature estimates based on adjacent line segments do not converge, but an atypical one does. Consideration of a class of scaled blurrings leads to a type of blurring of borders which is particularly easy for a computer to undo locally, yielding a line which is locally second-order accurate. Some extensions to three (and more) dimensions are appended.
The weighted particle method for convection-diffusion equations. II. The anisotropic case
Authors: Degond P. , Mas-Gallic S.
Published: June 2010

This paper is devoted to the presentation and the analysis of a new particle method for convection-diffusion equations. The method has been presented in detail in the first part of this paper for an isotropic diffusion operator. This part is concerned with the extension of the method to anisotropic diffusion operators. The consistency and the accuracy of the method require much more complex conditions on the cutoff functions than in the isotropic case. After detailing these conditions, we give several examples of cutoff functions which can be used for practical computations. A detailed error analysis is then performed.
Runge-Kutta methods applied to fully implicit differential-algebraic equations of index 1
Authors: Kværnø Anne
Published: June 2010

In this paper we study the order of Runge-Kutta methods applied to differential-algebraic equations of index one. We derive general order conditions for the local order

k
L

{k_L}

, and give a convergence result, which shows that the order

k
G

{k_G}

of the global error satisfies

k
G

≥

k
L

−
1

{k_G} \geq {k_L} - 1

. We also describe some numerical experiments, which are in agreement with our results.
Gauss quadrature rules for the evaluation of 2𝜋^{-1/2}∫₀^{∞}exp(-𝑥²)𝑓(𝑥)𝑑𝑥
Authors: Galant David
Published: June 2010

Gauss quadrature rules for evaluating integrals of the form
\[

2

π

−
1

/

2

−

∫
0
∞

exp
⁡
(
−

x
2

)
f
(
x
)
d
x

2{\pi ^{ - 1/2}} - \smallint _0^\infty \exp ( - {x^2})f(x)dx

\]
have been calculated to

20S

{\text {20S}}

for one to twenty nodes. The coefficients for the threeterm recurrence relation of the first twenty orthogonal polynomials associated with the weight function exp

(
−

x
2

)

( - {x^2})

on the interval

[
0
,
∞
)

[0, \infty )

are also tabulated to

20

S

20{\text {S}}

.
An algorithm for the exact reduction of a matrix to Frobenius form using modular arithmetic. II
Authors: Howell Jo Ann
Published: June 2010

Part I contained a description of the single-modulus algorithm for reducing a matrix to Frobenius form, obtaining exact integral factors of the characteristic polynomial. Part II contains a description of the multiple-modulus algorithm. Since different moduli may yield different factorizations, an algorithm is given for determining which factorizations are not correct factorizations over the integers of the characteristic polynomial. Part II also contains a discussion of the selection of the moduli and numerical examples.
Activation of liver X receptors with T0901317 attenuates cardiac hypertrophy in vivo
Authors: Kuipers Irma , Li Jiang , Vreeswijk‐Baudoin Inge , Koster Johan , van der Harst Pim , Silljé Herman H.W. , Kuipers Folkert , van Veldhuisen Dirk J. , van Gilst Wiek H. , de Boer Rudolf A.
Published: June 2010

Aims

Liver X receptor (LXR) is a nuclear receptor regulating cholesterol metabolism. Liver X receptor has also been shown to exert anti‐proliferative and anti‐inflammatory properties. In this study, we evaluated the effect of LXR activation on cardiac hypertrophy in vitro and in vivo.

Methods and results

Treatment with the synthetic LXR agonist T0901317 (T09) attenuated the hypertrophic response of cultured cardiomyocytes to endothelin‐1 almost to control levels. siRNA interference showed that this effect was indeed LXR specific. To corroborate these findings in vivo, abdominal aortic constriction (AC) was used as a pressure overload model to induce cardiac hypertrophy in wild‐type and LXR‐α‐deficient (LXR‐α−/−) mice. In wild‐type mice, T09 treatment resulted in a decrease of cardiac wall thickening 4 and 7 weeks after AC. Also, after 7 weeks of AC, mean arterial blood pressure and left ventricular weight/body weight (LVW/BW) ratios were decreased in T09 treated mice. These effects were not observed in LXR‐α−/− mice, indicating that the beneficial effect of LXR activation on cardiac hypertrophy is attributable to the LXR‐α isoform. T09 induced robust cardiac expression of metabolic genes which are downstream of LXR‐α, such as SREBP‐1c, ABCA1, and ABCG1.

Conclusion

Together these results indicate that LXR exerts salutary effects in cardiac hypertrophy, possibly via metabolic remodelling.
The association between a deep-sea gastropod Pedicularia sicula (Caenogastropoda: Pediculariidae) and its coral host Errina dabneyi (Hydrozoa: Stylasteridae) in the Azores
Authors: Braga-Henriques Andreia , Carreiro-Silva Marina , Porteiro Filipe M. , de Matos Valentina , Sampaio Íris , Ocaña Oscar , Ávila Sérgio P.
Published: June 2010

Braga-Henriques, A., Carreiro-Silva, M., Porteiro, F. M., de Matos, V., Sampaio, Í., Ocaña, O., and Ávila, S., P. 2011. The association between a deep-sea gastropod Pedicularia sicula (Caenogastropoda: Pediculariidae) and its coral host Errina dabneyi (Hydrozoa: Stylasteridae) in the Azores. – ICES Journal of Marine Science, 68: 399–407. The Azores region harbours the richest communities of stylasterid corals in the Northeast Atlantic area. Of the nine deep-water species found there, Errina dabneyi seems to be the most abundant species; it is commonly collected as bycatch from longline fishing. E. dabneyi host Pedicularia gastropods on their branches, and a detailed study of shell shape and morphometry at different growth stages, complemented by shell characterization through scanning electron microscopy, allows the individuals to be identified as Pedicularia sicula. The incidence of this species on E. dabneyi was high (69.8%), with abundances ranging between 1 and 223 individuals per colony. The pediculariids exhibited a high degree of plasticity and produced evident traces on the stylasterid skeletons at their fixation points, suggesting that they are ectoparasites and not predators of E. dabneyi. The stylasterid colonies also hosted a rich associated fauna dominated by suspension-feeders using the coral as substratum and for protection.
Brachioradial pruritus successfully treated with gabapentin
Authors: YILMAZ Sadik , CEYHAN Ali Murat , BAYSAL AKKAYA Vahide
Published: June 2010

Brachioradial pruritus (BRP) is a mysterious entity characterized by localized pruritus of the dorsolateral aspect of the arm. The precise etiology of BRP remains unknown, but sun exposure and/or cervical spine lesions seem to be triggering or at least aggravating factors. Many treatment alternatives including non‐steroidal anti‐inflammatory drugs, topical capsaicin, topical corticosteroids, photoprotection, carbamazepine and acupuncture have been used with different success rates. Recently, gabapentin, an antiepileptic agent, has been reported to be an effective therapeutic agent in BRP. Herein, we report a 64‐year‐old man with BRP who showed good response to gabapentin therapy.
Necrotizing fasciitis caused by Staphylococcus epidermidis in a neonate with extremely low birthweight
Authors: PEKER Erdal , KIRIMI Ercan , TUNCER Oguz , CEYLAN Abdullah , CAGAN Eren , DOGAN Murat
Published: June 2010

Necrotizing fasciitis (NF) is a severe life‐threatening soft tissue infection characterized by rapidly spreading necrosis of the fascia and the subcutaneous tissue. The initial skin presentation ranges from minimal rash to cellulites. The lesions subsequently spread rapidly. Even with appropriate medical and surgical therapy, the mortality rate in NF is more than 50%. In cases of neonates, mortality rate is higher than that ratio. High index of suspicion, prompt aggressive surgery, appropriate antibiotics and supportive care are the mainstays of management in the newborn infant with NF. Herein, we report a case of invasive and mortal NF caused by Staphylococcus epidermidis in an infant with extremely low birthweight.
MEK/ERK and p38 MAPK regulate chondrogenesis of rat bone marrow mesenchymal stem cells through delicate interaction with TGF‐β1/Smads pathway
Authors: Li J. , Zhao Z. , Liu J. , Huang N. , Long D. , Wang J. , Li X. , Liu Y.
Published: June 2010

Objectives:  This study was carried out to reveal functions and mechanisms of MEK/ERK and p38 pathways in chondrogenesis of rat bone marrow mesenchymal stem cells (BMSCs), and to investigate further any interactions between the mitogen‐activated protein kinase (MAPK) and transforming growth factor‐β1 (TGF‐β1)/Smads pathway in the process.

Materials and methods:  Chondrogenic differentiation of rat BMSCs was initiated in micromass culture, in the presence of TGF‐β1, for 2 weeks. ERK1/2 and p38 kinase activities were investigated by Western Blot analysis. Specific MAPK inhibitors PD98059 and SB20350 were employed to investigate regulatory effects of MEK/ERK and p38 signals on gene expression of chondrocyte‐specific markers, and TGF‐β1 downstream pathways of Smad2/3.

Results:  ERK1/2 was phosphorylated in a rapid but transient manner, whereas p38 was activated in a slow and sustained way. The two MAPK subtypes played opposing roles in mediating transcription of cartilage‐specific genes for Col2α and aggrecan. TGF‐β1‐stimulated gene expression of chondrogenic regulators, Sox9, Runx2 and Ihh, was also affected by activity of PD98059 and SB203580, to different degrees. However, influences of MAPK inhibitors on gene expression were relatively minor when not treated with TGF‐β1. In addition, gene transcription of Smad2/3 was significantly upregulated by TGF‐β1, but was regulated more subtly by treatment with MAPK inhibitors.

Conclusions:  MAPK subtypes seemed to regulate chondrogenesis with a delicate balance, interacting with the TGF‐β1/Smads signalling pathway.
Spontaneous immortalization of neural crest‐derived corneal progenitor cells after chromosomal aberration
Authors: Brandl C. , Kaesbauer J. , Weber B. H. F. , Morsczeck C.
Published: June 2010

Objectives:  In a previous study, we have reported the existence of neural crest‐derived stem cell‐like cells originating from the corneal limbus of juvenile mice (termed murine corneal cells, MCCs). To yield a sufficient number of MCCs, for example, for cell‐therapy approaches, here we have investigated MCCs’ ability for extensive proliferation, and we have evaluated their stem cell qualities and genetic stability after large‐scale culture.

Materials and methods:  MCCs were established from corneal limbal tissue of juvenile mice. To determine their cell proliferation and self‐renewing potential, MTT tests and an estimation of colony forming unit efficiency were carried out. Multipotency of cell differentiation was examined by applying adipogenic and osteogenic differentiation protocols. Moreover, karyotyping was performed and expression of stem cell markers and cell cycle‐associated genes was analysed.

Results:  MCCs, as primary cells, could be cultured for more than 60 passages. We observed increased cell proliferation and high number of colony forming units (CFUs) after extensive culture. Interestingly, there were no changes in expression of MCC markers. Furthermore, cell differentiation potentials remained comparable with MCCs at early passages. However, karyotyping revealed numeric chromosomal aberrations at higher passages. Moreover, tumour suppressor genes such as p16 and p21 were found to be down‐regulated after large‐scale cell culture.

Conclusions:  MCCs immortalize spontaneously after extensive cell culture, but still demonstrate stem cell‐like qualities.
Mach–Zehnder interference microscopy optically records electrically stimulated cellular activity in unstained nerve cells
Authors: KAUL R.A. , MAHLMANN D.M. , LOOSEN P.
Published: June 2010

Dual‐beam white light interference microscopy monitors changes in the optical density of the investigated object with high sensitivity. We report on the recording of dynamic changes in a neuron's optical density evoked by extracellular electrical stimulation. These recorded changes were analysed and unambiguously connected to the investigated object, an invertebrate neuron of the pond snail Lymnaea stagnalis. The results provide evidence for the method's applicability in visualizing cellular dynamics purely by evaluating changes in a cell's optical properties.
Determining the age of individual Lepeophtheirus salmonis (Krøyer, 1837) copepodids by measuring stored lipid volume; proof of principle
Authors: COOK P.F. , MCBEATH S.J. , BRICKNELL I.R. , BRON J.E.
Published: June 2010

Confocal microscopy has facilitated measurement of stained lipid volume in Lepeophtheirus salmonis copepodid larvae. Quantity of lipid, location and morphology of vesicles may allow an estimate of age and viability