bluuebunny/crossref_metadata_2025 huggingface dataset:
Quadrature methods for integral equations of the second kind over infinite intervals
Authors: Sloan Ian H.
Published: June 2010

Convergence results are proved for a class of quadrature methods for integral equations of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y left-parenthesis t right-parenthesis equals f left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript normal infinity Baseline k left-parenthesis t comma s right-parenthesis y left-parenthesis s right-parenthesis d s">

y
(
t
)
=
f
(
t
)
+

∫
0
∞

k
(
t
,
s
)
y
(
s
)

d
s

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

. An important special case is the Nyström method, in which the integral term is approximated by an ordinary quadrature rule. For all of the methods considered here, the rate of convergence is the same, apart from a constant factor, as that of the quadrature approximation to the integral term.
Subgroups of finite index in a free product with amalgamated subgroup
Authors: Stothers W. W.
Published: June 2010

Let G be a free product of finitely many finite groups with amalgamated subgroup. Using coset diagrams, a recurrence relation is obtained for the number of subgroups, and of free subgroups, of each finite index in G. In the latter case, an asymptotic formula is derived. When the amalgamated subgroup is central, the relation takes a simpler form.
The Lagrange multiplier method for Dirichlet’s problem
Authors: Bramble James H.
Published: June 2010

The Lagrange multiplier method of Babuška for the approximate solution of Dirichlet’s problem for second order elliptic equations is reformulated. Based on this formulation, new estimates for the error in the solution and the boundary flux are given. Efficient methods for the solution of the approximate problem are discussed.
On a dimensional reduction method. II. Some approximation-theoretic results
Authors: Vogelius M. , Babuška I.
Published: June 2010

This paper is the second in a series of three that analyze a method of dimensional reduction. It contains some results for approximation of functions on the interval

[
−
1
,
1
]

[ - 1,1]

with elements from the null-space of

P
N

{P^N}

,

N
⩾
1

N \geqslant 1

, where P is a second-order ordinary differential operator. A special case of this is approximation by polynomials. The one-dimensional results are used as a tool to prove similar versions in several dimensions. These multi-dimensional results are directly related to the approximate method of dimensional reduction that was introduced in [13], and they lead to statements about the convergence properties of this approach. The third paper, which analyzes the adaptive aspects of the method, is forthcoming.
Asymptotic expansions for a class of elliptic difference schemes
Authors: Starius Göran
Published: June 2010

In this paper, we derive an asymptotic expansion of the global error for Kreiss’ difference scheme for the Dirichlet problem for Poisson’s equation. This scheme, combined with a deferred correction procedure or the Richardson extrapolation technique, yields a method of accuracy at least

O
(

h

6.5

)

O({h^{6.5}})

in

L
2

{L_2}

, where h is the mesh length.
A sparse quasi-Newton update derived variationally with a nondiagonally weighted Frobenius norm
Authors: Toint Ph. L.
Published: June 2010

The problem of symmetric sparse updating is considered from a variational point of view and a new class of sparse symmetric quasi-Newton updating formulae is derived. This class results from the use of a nondiagonally weighted Frobenius norm. The computation of the update involves only one positive definite and symmetric linear system that has the same sparsity pattern as the problem itself.
An asymptotic formula for a type of singular oscillatory integrals
Authors: Hsu L. C. , Chou Y. S.
Published: June 2010

This paper offers a general expansion formula for oscillatory integrals of the form

∫
0
1

x

−
α

f
(
x
,
{
N
x
}
)

d
x

\smallint _0^1{x^{ - \alpha }}f(x,\{ Nx\} )\,dx

in which N is a large parameter, Nx denotes the fractional part of Nx, and

α
\alpha

is a fixed real number in

0
>
α
>
1

0 > \alpha > 1

. Our formula is expressed in terms of some ordinary integrals with integrands containing periodic Bernoulli functions and the generalized Riemann zeta function.
On a method of asymptotic evaluation of multiple integrals
Authors: Wong R. , McClure J. P.
Published: June 2010

In this paper, some of the formal arguments given by Jones and Kline [J. Math. Phys., v. 37, 1958, pp. 1-28] are made rigorous. In particular, the reduction procedure of a multiple oscillatory integral to a one-dimensional Fourier transform is justified, and a Taylor-type theorem with remainder is proved for the Dirac

δ
\delta

-function. The analyticity condition of Jones and Kline is now replaced by infinite differentiability. Connections with the asymptotic expansions of Jeanquartier and Malgrange are also discussed.
Primes differing by a fixed integer
Authors: Leavitt W. G. , Mullin Albert A.
Published: June 2010

It is shown that the equation

(
∗
)

(
n
−
1

)
2

−
σ
(
n
)
ϕ
(
n
)
=

m
2

( \ast )\;{(n - 1)^2} - \sigma (n)\phi (n) = {m^2}

is always solvable by

n
=

p
1

p
2

n = {p_1}{p_2}

where

p
1

,

p
2

{p_1},{p_2}

are primes differing by the integer m. This is called the "Standard" solution of

(
∗
)

( \ast )

and an m for which this is the only solution is called a "

∗

^\ast

-number". While there are an infinite number of non

∗

^\ast

-numbers there are many (almost certainly infinitely many)

∗

^\ast

-numbers, including

m
=
2

m = 2

(the twin prime case). A procedure for calculating all non

∗

^\ast

-numbers less than a given bound L is devised and a table is given for

L
=
1000

L = 1000

.
On the quasi-optimality in 𝐿_{∞} of the 𝐻̇¹-projection into finite element spaces
Authors: Schatz A. H. , Wahlbin L. B.
Published: June 2010

The

H
˙

1

{\dot {H}^1}

-projection into finite element spaces based on quasi-uniform partitions of a bounded smooth domain in

R
N

{R^N}

,

N
⩾
2

N \geqslant 2

arbitrary, is shown to be stable in the maximum norm (or, in the case of piecewise linear or bilinear functions, almost stable). It is not assumed that the mesh-domains coincide with the basic domain.
A superconvergent finite element method for the Korteweg-de Vries equation
Authors: Arnold Douglas N. , Winther Ragnar
Published: June 2010

An unconditionally stable fully discrete finite element method for the Korteweg-de Vries equation is presented. In addition to satisfying optimal order global estimates, it is shown that this method is superconvergent at the nodes. The algorithm is derived from the conservative method proposed by the second author by the introduction of a small time-independent forcing term into the discrete equations. This term is a form of the quasiprojection which was first employed in the analysis of superconvergence phenomena for parabolic problems. However, in the present work, unlike in the parabolic case, the quasiprojection is used as perturbation of the discrete equations and does not affect the choice of initial values.
A collocation-Galerkin method for a first order hyperbolic equation with space and time dependent coefficient
Authors: Archer David , Díaz Julio César
Published: June 2010

A collocation-Galerkin scheme is proposed for an initial-boundary value problem for a first order hyperbolic equation in one space dimension. The Galerkin equations satisfied by the approximating solution are obtained from a weak-weak formulation of the initial-boundary value problem. The collocation points are taken to be affine images of the roots of the Jacobian polynomials of degree

r
−
1

r - 1

on [0, 1] with respect to the weight function

x
(
1
−
x
)

x(1 - x)

. Optimal

L
∞

(

L
2

)

{L^\infty }({L^2})

-norm estimates of the error are derived.
Fixed point iteration with inexact function values
Authors: Alfeld Peter
Published: June 2010

In many iterative schemes, the precision of each step depends on the computational effort spent on that step. A method of specifying a suitable amount of computation at each step is described. The approach is adaptive and aimed at minimizing the overall computational cost subject to attaining a final iterate that satisfies a suitable error criterion. General and particular cost functions are considered, and a numerical example is given.
Composite exponential approximations
Authors: Iserles Arieh
Published: June 2010

The Composite Exponential Approximations (CEA) arise in a natural way when one investigates the stability and order properties of a combination of several methods for the numerical solution of ordinary differential equations, sequentially implemented with different step-lengths. Some general results on the order, acceptability and exponential fitting properties of CEA are derived. The composite Padé approximations and N-approximations are explored in detail.
Asymptotic error estimates for Gaussian quadrature formulas
Authors: Chen T. H. Charles
Published: June 2010

This paper gives derivative-free asymptotic error estimates for the Gaussian quadrature formula with the nonnegative weight function

w
(
x
)

w(x)

belonging to a certain class. Numerical examples are presented.
Determination of principal factors in 𝒬(√𝒟) and 𝒬(\root3\of𝒟)
Authors: Williams H. C.
Published: June 2010

Let

l
=
2

l = 2

or 3 and let D be a positive l-power-free integer. Also, let R be the product of all the rational primes which completely ramify in

K
=

Q

(

D

1

/

l

)

K = \mathcal {Q}({D^{1/l}})

. The integer d is a principal factor of the discriminant of K if

d
=
N
(
α
)

d = N(\alpha )

, where

α
\alpha

is an algebraic integer of K and

d

|

R

l
−
1

d|{R^{l - 1}}

. In this paper algorithms for finding these principal factors are described. Special attention is given to the case of

l
=
3

l = 3

, where it is shown that Voronoi’s continued fraction algorithm can be used to find principal factors. Some results of a computer search for principal factors for all

Q

(

D
3

)

\mathcal {Q}(\sqrt [3]{D})

with

2
⩽
D
⩽
15000

2 \leqslant D \leqslant 15000

are also presented.
On effective computation of fundamental units. II
Authors: Pohst Michael , Weiler Peter , Zassenhaus Hans
Published: June 2010

The new method for efficient computation of fundamental units of an algebraic number field originated by two of die authors in part I is used to develop a powerful computer program. Besides the description of this program part II contains a complete list of fundamental units of algebraic number fields of degree

n
=
3
,
4
,
5
,
6

n = 3,4,5,6

and small absolute discriminant.
Analysis of some mixed finite element methods related to reduced integration
Authors: Johnson Claes , Pitkäranta Juhani
Published: June 2010

We prove error estimates for the following two mixed finite element methods related to reduced integration: A method for Stokes’ problem using rectangular elements with piecewise bilinear approximations for the velocities and piecewise constants for the pressure, and one method for a plate problem using bilinear approximations for transversal displacement and rotations and piecewise constants for the shear stress. The main idea of the proof in the case of Stokes’ problem is to combine a weak Babuška-Brezzi type stability estimate for the pressure with a superapproximability property for the velocities. A similar technique is used in the case of the plate problem.
The numerical evaluation of very oscillatory infinite integrals by extrapolation
Authors: Sidi Avram
Published: June 2010

Recently the author has given two modifications of a nonlinear extrapolation method due to Levin and Sidi, which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity. In this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity. The new method is applied to a number of complicated integrals, among them the solution to a problem in viscoelasticity. Some convergence results for this method are presented.
An algorithm for solving a certain class of Diophantine equations. I
Authors: Hilliker David Lee
Published: June 2010

A class of Diophantine equations is defined and an algorithm for solving each equation in this class is developed. The methods consist of techniques for the computation of an upper bound for the absolute value of each solution. The computability of these bounds is guaranteed. Typically, these bounds are well within the range of computer programming and so they constitute a practical method for computing all solutions to the Diophantine equation in question. As a first application, a bound for a cubic equation is computed. As a second application, a set of quartic equations is studied. Methods are developed for deriving various sets of conditions on the coefficients in such equations under which a bound exists and can be computed.
Evaluation of the integral ∫₀^{𝑝}𝑢ⁿ𝑒^{-𝑢²}(𝑢+𝑥)⁻¹𝑑𝑢
Authors: Nagaraja K. S. , Verma G. R.
Published: June 2010

The representations of the above integral, in a power series form for small values of x and in an asymptotic form for large values of x, are given for integer values of n. In view of the usefulness of this integral, tabulated values are also presented for a wide range of values of x and p, and for a few values of n.
A 𝑝+1 method of factoring
Authors: Williams H. C.
Published: June 2010

Let N have a prime divisor p such that

p
+
1

p + 1

has only small prime divisors. A method is described which will allow for the determination of p, given N. This method is analogous to the

p
−
1

p - 1

method of factoring which was described in 1974 by Pollard. The results of testing this method on a large number of composite numbers are also presented.
On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations
Authors: Baker Garth A. , Dougalis Vassilios A. , Karakashian Ohannes A.
Published: June 2010

We consider approximating the solution of the initial and boundary value problem for the Navier-Stokes equations in bounded two- and three-dimensional domains using a nonstandard Galerkin (finite element) method for the space discretization and the third order accurate, three-step backward differentiation method (coupled with extrapolation for the nonlinear terms) for the time stepping. The resulting scheme requires the solution of one linear system per time step plus the solution of five linear systems for the computation of the required initial conditions; all these linear systems have the same matrix. The resulting approximations of the velocity are shown to have optimal rate of convergence in

L
2

{L^2}

under suitable restrictions on the discretization parameters of the problem and the size of the solution in an appropriate function space.
Numerical solution of systems of ordinary differential equations with the Tau method: an error analysis
Authors: Freilich J. H. , Ortiz E. L.
Published: June 2010

The recursive formulation of the Tau method is extended to the case of systems of ordinary differential equations, and an error analysis is given. Upper and lower error bounds are given in one of the examples considered. The asymptotic behavior of the error compares in this case with that of the best approximant by algebraic polynomials for each of the components of the vector solution.
On the zeros of the Riemann zeta function in the critical strip. II
Authors: Brent R. P. , van de Lune J. , te Riele H. J. J. , Winter D. T.
Published: June 2010

We describe extensive computations which show that Riemann’s zeta function

ζ
(
s
)

\zeta (s)

has exactly 200,000,001 zeros of the form

σ
+

\sigma +

in the region

0
>
t
>
81
,
702
,
130.19

0 > t > 81,702,130.19

; all these zeros are simple and lie on the line

σ
=

1
2

\sigma = \frac {1}{2}

. (This extends a similar result for the first 81,000,001 zeros, established by Brent in Math. Comp., v. 33, 1979, pp. 1361-1372.) Counts of the numbers of Gram blocks of various types and the failures of "Rosser’s rule" are given.
Implicit, time-dependent variable grid finite difference methods for the approximation of a linear waterflood
Authors: Douglas Jim , Wheeler Mary Fanett
Published: June 2010

An implicit, time-dependent variable grid finite difference method based on the addition of an artificial diffusivity is introduced and analyzed for approximating the solution of a scalar conservation law in a single space variable. No relation between the grids at successive time steps is required for convergence. Two adaptive grid selection procedures are shown to be covered by the analysis. Analogous results are also established for an implicit upwinding procedure.
Block Runge-Kutta methods for the numerical integration of initial value problems in ordinary differential equations. I. The nonstiff case
Authors: Cash J. R.
Published: June 2010

Block Runge-Kutta formulae suitable for the approximate numerical integration of initial value problems for first order systems of ordinary differential equations are derived. Considered in detail are the problems of varying both order and stepsize automatically. This leads to a class of variable order block explicit Runge-Kutta formulae for the integration of nonstiff problems and a class of variable order block implicit formulae suitable for stiff problems. The central idea is similar to one due to C. W. Gear in developing Runge-Kutta starters for linear multistep methods. Some numerical results are given to illustrate the algorithms developed for both the stiff and nonstiff cases and comparisons with standard Runge-Kutta methods are made.
Block Runge-Kutta methods for the numerical integration of initial value problems in ordinary differential equations. II. The stiff case
Authors: Cash J. R.
Published: June 2010

The approach described in the first part of this paper is extended to include diagonally implicit Runge-Kutta (DIRK) formulae. The algorithms developed are suitable for the numerical integration of stiff differential systems, and their efficiency is illustrated by means of some numerical examples.
Modified multilag methods for Volterra functional equations
Authors: Wolkenfelt P. H. M.
Published: June 2010

Linear multistep methods for ordinary differential equations in conjunction with a family of computationally efficient quadrature rules are employed to define a class of so-called multilag methods for the solution of Volterra integral and integro-differential equations. In addition, modified multilag methods are proposed which have the property that the stability behavior is independent of the choice of the quadrature rules. High order convergence of the methods is established. In particular, a special class of high order convergent methods is presented for the efficient solution of first-kind Volterra equations. Numerical experiments are reported.
A remarkable property of definite integrals
Authors: Glasser M. L.
Published: June 2010

We show that the integral of

F
(
u
)

F(u)

is the same as the integral of

F
(
x
)

F(x)

, both with respect to x, for a variety of functions u.
On the integral ∫₀^{𝜋/2}𝑙𝑜𝑔ⁿ𝑐𝑜𝑠𝑥𝑙𝑜𝑔^{𝑝}𝑠𝑖𝑛𝑥𝑑𝑥
Authors: Kölbig K. S.
Published: June 2010

A formula is derived for the integral in the title which allows easy evaluation by formula manipulation on a computer.
Applications of a computer implementation of Poincaré’s theorem on fundamental polyhedra
Authors: Riley Robert
Published: June 2010

Poincaré’s Theorem asserts that a group

Γ
\Gamma

of isometries of hyperbolic space

H

\mathbb {H}

is discrete if its generators act suitably on the boundary of some polyhedron in

H

\mathbb {H}

, and when this happens a presentation of

Γ
\Gamma

can be derived from this action. We explain methods for deducing the precise hypotheses of the theorem from calculation in

Γ
\Gamma

when

Γ
\Gamma

is "algorithmically defined", and we describe a file of Fortran programs that use these methods for groups

Γ
\Gamma

acting on the upper half space model of hyperbolic 3-space

H

3

{\mathbb {H}^3}

. We exhibit one modest example of the application of these programs, and we summarize computations of repesentations of groups

PSL

(
2
,

O

)

{\text {PSL}}(2,\mathcal {O})

where

O

\mathcal {O}

is an order in a complex quadratic number field.
The discriminant of a quadratic extension of an algebraic field
Authors: Vaughan Theresa P.
Published: June 2010

Let F be an algebraic field, and K an extension of F of degree 2. We describe a method for computing the relative discriminant D for K over F. We work out the details for the case when F is quadratic and give tables which yield D very easily. We also apply the method to one type of cubic field F, and give tables for it.
Boundary conditions for a fourth order hyperbolic difference scheme
Authors: Sloan D. M.
Published: June 2010

Oliger [6] has used a stable time-averaged boundary condition with a fourth order leap-frog scheme for the numerical solution of hyperbolic partial differential equations. Gary [3] generalized the time-averaged boundary condition by including a scalar parameter. This paper examines the stability and accuracy of the more general boundary condition. The limit of the stability interval is found for the parameter, and it is shown that the parameter should be given a value close to this limit in order to minimize the boundary errors. Numerical experiments are described which support the theoretical predictions.
On the simplified hybrid-combined method
Authors: Li Zi Cai , Liang Guo Ping
Published: June 2010

In order to solve the boundary value problems of elliptic equations, especially with singularities and unbounded domains, the simplified hybrid-combined method, which is equivalent to the coupling method of Zienkiewicz et al. [15], is presented. This is a combination of the Ritz-Galerkin and the finite element methods. Its optimal error estimates are proved in this paper, and the solution strategy of its algebraic equation system is discussed.
Parameters for integrating periodic functions of several variables
Authors: Haber Seymour
Published: June 2010

A number-theoretical method for numerical integration of periodic functions of several variables was developed some years ago. This paper presents lists of numerical parameters to be used in implementing that method. The parameters define quadrature formulas for functions of 2, 3, ..., 8 variables; error bounds for those formulas are also tabulated. The derivation of the parameters and error bounds is described.
On the linear independence of multivariate 𝐵-splines. II. Complete configurations
Authors: Dahmen Wolfgang A. , Micchelli Charles A.
Published: June 2010

The first part of this paper is concerned with global characterizations of both the multivariate B-spline and the multivariate truncated power function as smooth piecewise polynomials. In the second part of the paper we establish combinatorial criteria for the linear independence of multivariate B-splines corresponding to certain configurations of knot sets.
Representations of intervals and optimal error bounds
Authors: Rall L. B.
Published: June 2010

Classical methods of numerical analysis lead to approximate values and error bounds for computed results, while interval analysis provides lower and upper bounds for the exact values. Equivalence of these approaches is investigated on the basis of a general definition of an error function, and a way is given to find optimal approximation points

y
∗

{y^\ast }

in an interval I and the corresponding minimum error bound

ε
∗

{\varepsilon ^\ast }

. Conversely, it is shown that

y
∗

{y^\ast }

,

ε
∗

{\varepsilon ^\ast }

can be used as alternative coordinates for the representation of the interval I, and the rules for interval arithmetic can be formulated in the resulting system. In particular, the arithmetic, harmonic, and geometric means of the endpoints of an interval turn out to be the optimal approximation points corresponding to absolute error, relative error, and relative precision, respectively. The rules for interval arithmetic are given explicitly in the resulting coordinate systems. It is also shown that calculations performed in interval arithmetic generally lead to smaller intervals than predicted by certain a priori bounds.
Numerical approximations to nonlinear conservation laws with locally varying time and space grids
Authors: Osher Stanley , Sanders Richard
Published: June 2010

An explicit time differencing technique is introduced to approximate nonlinear conservation laws. This differencing technique links together an arbitrary number of space regimes containing fine and coarse time increments. Previous stability requirements, i.e. the CFL condition, placed a global bound on the size of the time increments. For scalar, monotone, approximations in one space dimension, using this variable step time differencing, convergence to the correct physical solution is proven given only a local CFL condition.
Generalized Chebyshev interpolation and its application to automatic quadrature
Authors: Hasegawa Takemitsu , Torii Tatsuo , Ninomiya Ichizo
Published: June 2010

A generalized Chebyshev interpolation procedure increasing a fixed number of sample points at a time is developed and analyzed. It is incorporated into an efficient automatic quadrature scheme of Clenshaw-Curtis type. Numerical examples indicate that the present method is efficient not only for well-behaved functions but for those with discontinuous low order derivatives by virtue of adequate error estimation as well as saving of sample points.
Some integrals relating to the 𝐼ₑ-function
Authors: Okui Shigehiko
Published: June 2010

Various integrals relating to the

I
e

{I_e}

-function
\[

I
e

(
k
,
z
)
=

∫
0
z

e

−
x

I
0

(
k
x
)

d
x
,

{I_e}(k,z) = \int _0^z {{e^{ - x}}{I_0}(kx)\;dx,}

\]
which finds a wide variety of applications in the fields of statistical communication theory and noise analysis, are evaluated in closed form.
Nonlinear transformations for accelerating the convergence of 𝑀-dimensional series
Authors: Albertsen N. C. , Jacobsen G. , Sørensen S. B.
Published: June 2010

Two transformations for accelerating the convergence of infinite M-dimensional series are presented. They are generalizations of Shanks’ e-transformation and Levin’s one-dimensional transformations, respectively.
On the zeros of the Riemann zeta function in the critical strip. III
Authors: van de Lune J. , te Riele H. J. J.
Published: June 2010

We describe extensive computations which show that Riemann’s zeta function

ζ
(
s
)

\zeta (s)

has exactly 300,000,001 zeros of the form

σ
+
i
t

\sigma + it

in the region

0
>
t
>
119
,
590
,
809.282

0 > t > 119,590,809.282

. All these zeros are simple and lie on the line

σ
=

1
2

\sigma = \frac {1}{2}

. (This extends a similar result for the first 200,000,001 zeros, established by Brent, van de Lune, te Riele and Winter in Math. Comp., v. 39, 1982, pp. 681-688.) Counts of the numbers of Gram blocks of various types and the failures of "Rosser’s rule" are given, together with some graphs of the function

Z
(
t
)

Z(t)

near the first observed failures of Rosser’s rule.
On a sequence arising in series for 𝜋
Authors: Newman Morris , Shanks Daniel
Published: June 2010

In a recent investigation of dihedral quartic fields [6] a rational sequence

{

a
n

}

\{ {a_n}\}

was encountered. We show that these

a
n

{a_n}

are positive integers and that they satisfy surprising congruences modulo a prime p. They generate unknown p-adic numbers and may therefore be compared with the cubic recurrences in [1], where the corresponding p-adic numbers are known completely [2]. Other unsolved problems are presented. The growth of the

a
n

{a_n}

is examined and a new algorithm for computing

a
n

{a_n}

is given. An appendix by D. Zagier, which carries the investigation further, is added.
On generating new amicable pairs from given amicable pairs
Authors: te Riele Herman J. J.
Published: June 2010

Methods are given for constructing new amicable pairs from given amicable pairs. By applying these methods to 1575 "mother" pairs known to the author, 1782 new amicable pairs were generated, so that the average "offspring" of these mother pairs is greater than 1. By a second, much more restricted application of the methods to the 400 smallest "daughter" pairs, 88 new "granddaughter" pairs were generated.
Odd triperfect numbers
Authors: Kishore Masao
Published: June 2010

We prove that an odd triperfect number has at least ten distinct prime factors.
Divisors in residue classes
Authors: Lenstra H. W.
Published: June 2010

In this paper the following result is proved. Let r, s and n be integers satisfying

0
⩽
r
>
s
>
n

0 \leqslant r > s > n

,

s
>

n

1

/

3

s > {n^{1/3}}

,

gcd
(
r
,
s
)
=
1

\gcd (r,s) = 1

. Then there exist at most 11 positive divisors of n that are congruent to r modulo s. Moreover, there exists an efficient algorithm for determining all these divisors. The bound 11 is obtained by means of a combinatorial model related to coding theory. It is not known whether 11 is best possible; in any case it cannot be replaced by 5. Nor is it known whether similar results are true for significantly smaller values of

log
⁡
s

/

log
⁡
n

\log s/\log n

. The algorithm treated in the paper has applications in computational number theory.
On the convergence of Galerkin approximation schemes for second-order hyperbolic equations in energy and negative norms
Authors: Geveci Tunc
Published: June 2010

Given certain semidiscrete and single step fully discrete Galerkin approximations to the solution of an initial-boundary value problem for a second-order hyperbolic equation,

H
1

{H^1}

and

L
2

{L^2}

error estimates are obtained. These estimates are valid simultaneously when the approximation to the initial data is taken to be the projection onto the approximating space with respect to the inner product which induces the energy norm that is naturally associated with the problem. The

L
2

{L^2}

-estimate is obtained as a by-product of the analysis of convergence in certain negative norms. Estimates are also obtained for the convergence of higher-order time derivatives in the presence of sufficiently smooth data.
A difference method for a singular boundary value problem of second order
Authors: Weinmüller Ewa
Published: June 2010

The standard three-point discretization applied to the numerical solution of linear boundary value problems for second order systems with a singularity at the origin is investigated. A number of numerical examples illustrating the theoretical results are presented.
Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems
Authors: Saad Youcef
Published: June 2010

The present paper deals with the problem of computing a few of the eigenvalues with largest (or smallest) real parts, of a large sparse nonsymmetric matrix. We present a general acceleration technique based on Chebyshev polynomials and discuss its practical application to Arnoldi’s method and the subspace iteration method. The resulting algorithms are compared with the classical ones in a few experiments which exhibit a sharp superiority of the Arnoldi-Chebyshev approach.
Fourier series coefficients for powers of the Jacobian elliptic functions
Authors: Kiper Ayşe
Published: June 2010

The Fourier series expansion coefficients for the Jacobian elliptic functions

sn
m

(
u
,
k
)

{\text {sn}^m}(u,k)

,

cn
m

(
u
,
k
)

{\text {cn}^m}(u,k)

and

dn
m

(
u
,
k
)

{\text {dn}^m}(u,k)

, with

m
⩾
1

m \geqslant 1

, are studied. Two-term recurrence formulae are obtained and some of the coefficients are tabulated.
Primitive 𝛼-abundant numbers
Authors: Cohen Graeme L.
Published: June 2010

A number N is primitive

α
\alpha

-abundant if

σ
(
M
)

/

M
>
α
⩽
σ
(
N
)

/

N

\sigma (M)/M > \alpha \leqslant \sigma (N)/N

for all proper divisors M of N. In this paper, we tabulate, for

1
>
α
⩽
5.4

1 > \alpha \leqslant 5.4

, all such N for which

σ
(
N
)

/

N

\sigma (N)/N

is greatest. We show that, if N is primitive

α
\alpha

-abundant and

α
>
1.6

\alpha > 1.6

, then

σ
(
N
)

/

N
>
α
+
min
{

2
5

,
3
α

/

2

e

5
α

/

9

}

\sigma (N)/N > \alpha + \min \{ \frac {2}{5},3\alpha /2{e^{5\alpha /9}}\}

.
The expectation of success using a Monte Carlo factoring method—some statistics on quadratic class numbers
Authors: Buell Duncan A.
Published: June 2010

A method has been proposed for factoring an integer N by using the structure of the class groups of quadratic fields of radicand

−
k
N

- kN

for various small multipliers k. We discuss the method and an implementation of the method, and various theoretical questions which have an impact on the practical use of the method in factoring. Some of the theoretical questions relate to the nature of class numbers and class groups; we present extensive statistical results on the class numbers and class groups of imaginary quadratic fields.
Splitting of quartic polynomials
Authors: Adams William W.
Published: June 2010

For integers r, s, t, u define the recursion
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis n plus 4 right-parenthesis equals r upper A left-parenthesis n plus 3 right-parenthesis minus s upper A left-parenthesis n plus 2 right-parenthesis plus t upper A left-parenthesis n plus 1 right-parenthesis minus u upper A left-parenthesis n right-parenthesis">

A
(
n
+
4
)
=
r
A
(
n
+
3
)
−
s
A
(
n
+
2
)
+
t
A
(
n
+
1
)
−
u
A
(
n
)

A(n + 4) = rA(n + 3) - sA(n + 2) + tA(n + 1) - uA(n)

where the initial conditions are set up in such a way that

A
(
n
)
=

α
n

+

β
n

+

γ
n

+

δ
n

A(n) = {\alpha ^n} + {\beta ^n} + {\gamma ^n} + {\delta ^n}

where

α
,
β
,
γ
,
δ

\alpha ,\beta ,\gamma ,\delta

are the roots of the associated polynomial

f
(
x
)
=

x
4

−
r

x
3

+
s

x
2

−
t
x
+
u

f(x) = {x^4} - r{x^3} + s{x^2} - tx + u

In this paper a detailed deterministic procedure using the

A
(
n
)

A(n)

for finding how

f
(
x
)

f(x)

splits modulo a prime integer p is given. This gives for p not dividing the discriminant of

f
(
x
)

f(x)

the splitting of p in the field obtained by adjoining a root of

f
(
x
)

f(x)

to the rational numbers. There is an interesting connection between the results here for reciprocal polynomials and some work of D. Shanks.
Semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time-dependent coefficients
Authors: Bales Laurence A.
Published: June 2010

L
2

{L^2}

norm error estimates are proved for finite element approximations to the solutions of initial boundary value problems for second order hyperbolic partial differential equations with time-dependent coefficients. Optimal order rates of convergence are shown for semidiscrete and single step fully discrete schemes using specially constructed initial data. The initial data are designed so that the data used for the fully discrete equation is reasonable to compute and so that the optimal order estimates can be proved.
The lattices of six-dimensional Euclidean space
Authors: Plesken W. , Hanrath W.
Published: June 2010

The lattices of full rank of the six-dimensional Euclidean space are classified according to their automorphism groups (Bravais classification). We find 826 types of such lattices.
Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series
Authors: Gautschi Walter , Milovanović Gradimir V.
Published: June 2010

Polynomials

π
k

(
⋅
)
=

π
k

(
⋅
;
d
λ
)

{\pi _k}( \cdot ) = {\pi _k}( \cdot ;d\lambda )

,

k
=
0
,
1
,
2
,
…

k = 0,1,2, \ldots

, are constructed which are orthogonal with respect to the weight distributions

d
λ
(
t
)
=

(
t

/

(

e
t

−
1
)

)
r

d
t

d\lambda (t) = {(t/({e^t} - 1))^r}\;dt

and

d
λ
(
t
)
=

(
1

/

(

e
t

+
1
)

)
r

d
t

d\lambda (t) = {(1/({e^t} + 1))^r}\;dt

,

r
=
1
,
2

r = 1,2

, on

(
0
,
∞
)

(0,\infty )

. Moment-related methods being inadequate, a discretized Stieltjes procedure is used to generate the coefficients

α
k

,

β
k

{\alpha _k},{\beta _k}

in the recursion formula
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript k plus 1 Baseline left-parenthesis t right-parenthesis equals left-parenthesis t minus alpha Subscript k Baseline right-parenthesis pi Subscript k Baseline left-parenthesis t right-parenthesis minus beta Subscript k Baseline pi Subscript k minus 1 Baseline left-parenthesis t right-parenthesis">

π

k
+
1

(
t
)
=
(
t
−

α
k

)

π
k

(
t
)
−

β
k

π

k
−
1

(
t
)

{\pi _{k + 1}}(t) = (t - {\alpha _k}){\pi _k}(t) - {\beta _k}{\pi _{k - 1}}(t)

,

k
=
0
,
1
,
2
,
…

k = 0,1,2, \ldots

,

π
0

(
t
)
=
1

{\pi _0}(t) = 1

,

π

−
1

(
t
)
=
0

{\pi _{ - 1}}(t) = 0

. The discretization is effected by the Gauss-Laguerre and a composite Fejér quadrature rule, respectively. Numerical values of

α
k

,

β
k

{\alpha _k},{\beta _k}

, as well as associated error constants, are provided for

0
⩽
k
⩽
39

0 \leqslant k \leqslant 39

. These allow the construction of Gaussian quadrature formulae, including error terms, with up to 40 points. Examples of n-point formulae,

n
=
5
(
5
)
40

n = 5(5)40

, are provided in the supplements section at the end of this issue. Such quadrature formulae may prove useful in solid state physics calculations and can also be applied to sum slowly convergent series.
Computing an arithmetic constant related to the ring of Gaussian integers
Authors: Gramain F. , Weber M.
Published: June 2010

We compute the analogue for

Z

[
i
]

{\mathbf {Z}}[i]

of Euler’s constant, that is

δ
=

lim

n
→
+
∞

δ
n

\delta = {\lim _{n \to + \infty }}{\delta _n}

, where

δ
n

=
(

Σ

2
⩽
k
⩽
n

1

/

π

r
k
2

)
−
log
⁡
n

{\delta _n} = ({\Sigma _{2 \leqslant k \leqslant n}}1/\pi r_k^2) - \log n

. For this purpose we give an estimate for
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r Subscript k Baseline equals min left-brace r greater-than-or-slanted-equals 0 semicolon there exists z element-of bold upper C such that card left-parenthesis bold upper Z left-bracket i right-bracket intersection ModifyingAbove upper D With bar left-parenthesis z comma r right-parenthesis right-parenthesis greater-than-or-slanted-equals k right-brace comma">

r
k

=
min

{

r
⩾
0
;

there exists

z
∈

C

such that card

(

Z

[
i
]
∩

D
¯

(
z
,
r
)
)
⩾
k

}

,

{r_k} = \min \left \{ {r \geqslant 0;{\text {there exists}}\;z \in {\mathbf {C}}\;{\text {such that card}}({\mathbf {Z}}[i] \cap \bar D(z,r)) \geqslant k} \right \},

\]
and we compute a great number of values of

r
k

{r_k}

.
A note on the Diophantine equation 𝑥³+𝑦³+𝑧³=3
Authors: Cassels J. W. S.
Published: June 2010

Any integral solution of the title equation has

x
≡
y
≡
z

x \equiv y \equiv z

(9).
Mixed finite element methods for quasilinear second-order elliptic problems
Authors: Milner F. A.
Published: June 2010

A mixed finite element method is developed to approximate the solution of a quasilinear second-order elliptic partial differential equation. The existence and uniqueness of the approximation are demonstrated and optimal rate error estimates are derived.
On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3
Authors: Buhler Joe P. , Gross Benedict H. , Zagier Don B.
Published: June 2010

The elliptic curve

y
2

=
4

x
3

−
28
x
+
25

{y^2} = 4{x^3} - 28x + 25

has rank 3 over Q. Assuming the Weil-Taniyama conjecture for this curve, we show that its L-series

L
(
s
)

L(s)

has a triple zero at

s
=
1

s = 1

and compute

lim

s
→
1

L
(
s
)

/

(
s
−
1

)
3

{\lim _{s \to 1}}L(s)/{(s - 1)^3}

to 28 decimal places; its value agrees with the product of the regulator and real period, in accordance with the Birch-Swinnerton-Dyer conjecture if III is trivial.
Error bounds for finite-difference approximations for a class of nonlinear parabolic systems
Authors: Hoff David , Smoller Joel
Published: June 2010

In this paper we establish error bounds for a finite-difference approximation to solutions of certain parabolic systems of the form

v
t

+
f

(
v

)
x

=
ε

v

x
x

{v_t} + f{(v)_x} = \varepsilon {v_{xx}}

. We assume that the Cauchy data is of class BV, and we show that the sup norm of the error is bounded by

O
(
Δ
x

|

ln
⁡
Δ
x

|

)

O(\Delta x|\ln \Delta x|)

at positive times.
On the asymptotic convergence of collocation methods with spline functions of even degree
Authors: Saranen J. , Wendland W. L.
Published: June 2010

We investigate the collocation of linear one-dimensional strongly elliptic integro-differential or, more generally, pseudo-differential equations on closed curves by even-degree polynomial splines. The equations are collocated at the respective midpoints subject to uniform nodal grids of the even-degree B-splines. We prove quasioptimal and optimal order asymptotic error estimates in a scale of Sobolev spaces. The results apply, in particular, to boundary element methods used for numerical computations in engineering applications. The equations considered include Fredholm integral equations of the second and the first kind, singular integral equations involving Cauchy kernels, and integro-differential equations having convolutional or constant coefficient principal parts, respectively. The error analysis is based on an equivalence between the collocation and certain variational methods with different degree splines as trial and as test functions. We further need to restrict our operators essentially to pseudo-differential operators having convolutional principal part. This allows an explicit Fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing Babuška’s stability condition based on strong ellipticity. Our asymptotic error estimates extend partly those obtained by D. N. Arnold and W. L. Wendland from the case of odd-degree splines to the case of even-degree splines.
Remark on a lemma: “On a method of asymptotic evaluation of multiple integrals” [Math. Comp. 37 (1981), no. 156, 509–521; MR0628712 (83b:41040)] by R. Wong and J. P. McClure
Authors: Meinardus Günter
Published: June 2010

A short proof is presented for a formula arising in the above-mentioned paper.
Computation of the class number and class group of a complex cubic field
Authors: Dueck G. , Williams H. C.
Published: June 2010

Let h and G be, respectively, the class number and the class group of a complex cubic field of discriminant

Δ
\Delta

. A method is described which makes use of recent ideas of Lenstra and Schoof to develop fast algorithms for finding h and G. Under certain Riemann hypotheses it is shown that these algorithms will compute h in

O
(

|

Δ

|

1

/

5
+
ε

)

O(|\Delta {|^{1/5 + \varepsilon }})

elementary operations and G in

O
(

|

Δ

|

1

/

4
+
ε

)

O(|\Delta {|^{1/4 + \varepsilon }})

elementary operations. Finally, the results of running some computer programs to determine h and G for all pure cubic fields

Q

(

D
3

)

\mathcal {Q}(\sqrt [3]{D})

, with

2
⩽
D
>
30
,
000

2 \leqslant D > 30,000

, are summarized.
Accuracy bounds for semidiscretizations of hyperbolic problems
Authors: Jeltsch Rolf , Strack Klaus-Günther
Published: June 2010

Bounds are given for the error constant of stable finite-difference methods for first-order hyperbolic equations in one space dimension, which use r downwind and s upwind points in the discretization of the space derivatives, and which are of optimal order

p
=
min
(
r
+
s
,
2
r
+
2
,
2
s
)

p = \min (r + s,2r + 2,2s)

. It is known that this order can be obtained by interpolatory methods. Examples show, however, that their error constants can be improved.
An improved version of Marti’s method for solving ill-posed linear integral equations
Authors: Engl Heinz W. , Neubauer Andreas
Published: June 2010

We propose an algorithm for solving linear integral equations of the first kind that can be viewed as a variant of Marti’s method; as opposed to that method, our algorithm leads to optimal convergence rates (also with noisy data).
On computing the discriminant of an algebraic number field
Authors: Vaughan Theresa P.
Published: June 2010

Let

f
(
x
)

f(x)

be a monic irreducible polynomial in

Z

[
x
]

{\mathbf {Z}}[x]

, and r a root of

f
(
x
)

f(x)

in C. Let K be the field

Q
(
r
)

{\mathbf {Q(r)}}

and

R

\mathcal {R}

the ring of integers in K. Then for some

k
∈

Z

k \in {\mathbf {Z}}

,

disc

r

=

k
2

disc

R

\operatorname {disc}\,{\mathbf {r}} = {k^2}\,\operatorname {disc}\,\mathcal {R}

. In this paper we give constructive methods for (a) deciding if a prime p divides k, and (b) if

p

|

k

p|k

, finding a polynomial

g
(
x
)
∈

Z

[
x
]

g(x) \in {\mathbf {Z}}[x]

so that

g
(
x
)

≢

0

(
mod

p
)

g(x)\nequiv 0\;\pmod p

but

g
(

r

)

/

p
∈

R

g({\mathbf {r}})/p \in \mathcal {R}

.
An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation
Authors: Johnson C. , Pitkäranta J.
Published: June 2010

We prove

L
p

{L_p}

stability and error estimates for the discontinuous Galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain. Using finite element analysis techniques, we obtain

L
2

{L_2}

estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials.

L
p

{L_p}

estimates for

p
≠
2

p \ne 2

are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree. The latter estimates are proved by combining finite difference and finite element analysis techniques.
Boundary integral solutions of the heat equation
Authors: McIntyre E. A.
Published: June 2010

The Boundary Integral Method (BIM) has recently become quite popular because of its ability to provide cheap numerical solutions to the Laplace equation. This paper describes an attempt to apply a similar approach to the (time-dependent) heat equation in two space variables.
Computing volumes of polyhedra
Authors: Allgower Eugene L. , Schmidt Phillip H.
Published: June 2010

In this note we give two simple methods for calculating the volume of any closed bounded polyhedron in

R

n

{{\mathbf {R}}^n}

having an orientable boundary which is triangulated into a set of

(
n
−
1
)

(n - 1)

-dimensional simplices. The formulas given require only coordinates of the vertices of the polyhedron.
Numerical solution of large sets of algebraic nonlinear equations
Authors: Toint Ph. L.
Published: June 2010

This paper describes the application of the partitioned updating quasi-Newton methods for the solution of high-dimensional systems of algebraic nonlinear equations. This concept was introduced and successfully tested in nonlinear optimization of partially separable functions (see [6]). Here its application to the case of nonlinear equations is explored. Nonlinear systems of this nature arise in many large-scale applications, including finite elements and econometry. It is shown that the method presents some advantages in efficiency over competing algorithms, and that use of the partially separable structure of the system can lead to significant improvements also in the more classical discrete Newton method.
Finite element technique for optimal pressure recovery from stream function formulation of viscous flows
Authors: Cayco M. E. , Nicolaides R. A.
Published: June 2010

Following a general analysis of convergence for the finite element solution of the stream function formulation of the Navier-Stokes equation in bounded regions of the plane, an algorithm for pressure recovery is presented. This algorithm, which is easy to implement, is then analyzed and conditions ensuring optimality of the approximation are given. An application is made to a standard conforming cubic macroelement.
On a large time-step high resolution scheme
Authors: Harten Ami
Published: June 2010

This paper presents a class of new second-order accurate

(
2
K
+
3
)

(2K + 3)

-point explicit schemes for the computation of weak solutions of hyperbolic conservation laws, that are total-variation-diminishing under a CFL restriction of K. These highly nonlinear schemes are obtained by applying a nonoscillatory first-order accurate

(
2
K
+
1
)

(2K + 1)

-point scheme to a modified flux. The so-derived second-order accurate schemes achieve high resolution, while retaining the robustness of the original first-order accurate scheme.