bluuebunny/crossref_metadata_2025 huggingface dataset:
Cyclic large sets of Steiner triple systems of order 15
Authors: Phelps Kevin T.
Published: June 2010

We examine a class of large sets of Steiner triple systems of order 15 having an automorphism consisting of two fixed points and a 13-cycle. We exhibit all members of this class: there are 256 nonisomorphic systems. We examined these members for initial configurations which could lead to a large set of Steiner quadruple systems of order 16 and established that no large set exists having three fixed points and a 13-cycle.
Two-step Runge-Kutta methods and hyperbolic partial differential equations
Authors: Renaut R. A.
Published: June 2010

The purpose of this study is the design of efficient methods for the solution of an ordinary differential system of equations arising from the semidiscretization of a hyperbolic partial differential equation. Jameson recently introduced the use of one-step Runge-Kutta methods for the numerical solution of the Euler equations. Improvements in efficiency up to 80
Catalan’s equation 𝑥^{𝑝}-𝑦^{𝑞}=1 and related congruences
Authors: Aaltonen M. , Inkeri K.
Published: June 2010

We investigate solutions of Catalan’s equation

x
p

−

y
q

=
1

{x^p} - {y^q} = 1

in nonzero integers x, y, p, q. By use of class numbers and congruences

p
q

≡
p

(
mod

q
2

)

{p^q} \equiv p\;\pmod {q^2}

we show the impossibility of the equation for a large number of prime exponents p, q. Both theoretical and computer results are included. We also study lower bounds on possible, hitherto unknown, solutions

x
,
y
>
2

x,y > 2

; we especially wish to communicate the bound

x
,
y
≥

10

500

x,y \geq {10^{500}}

.
A Galerkin method for the forward-backward heat equation
Authors: Aziz A. K. , Liu J.-L.
Published: June 2010

In this paper a new variational method is proposed for the numerical approximation of the solution of the forward-backward heat equation. The approach consists of first reducing the second-order problem to an equivalent first-order system, and then using a finite element procedure with continuous elements in both space and time for the numerical approximation. Under suitable regularity assumptions, error estimates and the results of some numerical experiments are presented.
A table of elliptic integrals: one quadratic factor
Authors: Carlson B. C.
Published: June 2010

Integration in terms of real quantities is accomplished for 33 integrands that are rational except for the square root of a cubic or quartic polynomial with exactly one pair of conjugate complex zeros. Formulas are provided by which 45 more integrals of the same type can be expressed in terms of real quantities with the help of earlier papers. Neither limit of integration is assumed to be a singular point of the integrand. All the integrals are reduced to R-functions, for which Fortran programs are available. Most of the integrals are not listed in other tables.
Uniform error estimates of operational quadrature methods for nonlinear convolution equations on the half-line
Authors: Eggermont P. P. B. , Lubich Ch.
Published: June 2010

We study uniform error estimates of operational quadrature methods for nonlinear convolution equations on the half-line. Equations of this kind arise in control engineering and diffusion problems. The essential ingredients are the stability of the operational quadrature method in an

L
2

{L^2}

setting, which is inherited from the continuous equation by its very construction, and a theorem that says that the behavior of the linearized equations is the same in all

L
p

{L^p}

spaces

(
1
≤
p
≤
∞
)

(1 \leq p \leq \infty )

.
Inversive congruential pseudorandom numbers avoid the planes
Authors: Eichenauer-Herrmann Jürgen
Published: June 2010

Nonlinear congruential pseudorandom number generators based on inversions have recently been introduced and analyzed. These generators do not show the lattice structure of the widely used linear congruential method. In the present paper it is proved that the points formed by d consecutive pseudorandom numbers of an inversive congruential generator with prime modulus possess an even stronger property: Any hyperplane in d-space contains at most d of these points, that is to say, the hyperplane spanned by d arbitrary points of an inversive congruential generator contains no further points. This feature makes the inversive congruential method particularly attractive for simulation problems where linear structures within the generated points should be avoided.
Contractivity-preserving implicit linear multistep methods
Authors: Lenferink H. W. J.
Published: June 2010

We investigate contractivity properties of implicit linear multistep methods in the numerical solution of ordinary differential equations. The emphasis is on nonlinear and linear systems

d

d
t

U
(
t
)
=
f
(
t
,
U
(
t
)
)

\frac {d}{{dt}}U(t) = f(t,U(t))

, where f satisfies a so-called circle condition in an arbitrary norm. The results for the two types of systems turn out to be closely related. We construct optimal multistep methods of given order and stepnumber, which allow the use of a maximal stepsize.
Finding isomorphisms between finite fields
Authors: Lenstra H. W.
Published: June 2010

We show that an isomorphism between two explicitly given finite fields of the same cardinality can be exhibited in deterministic polynomial time.
Finite difference discretizations of some initial and boundary value problems with interface
Authors: Akrivis Georgios D. , Dougalis Vassilios A.
Published: June 2010

We analyze the discretization of initial and boundary value problems with a stationary interface in one space dimension for the heat equation, the Schrödinger equation, and the wave equation by finite difference methods. Extending the concept of the elliptic projection, well known from the analysis of Galerkin finite element methods, to our finite difference case, we prove second-order error estimates in space and time in thel2{l^2}norm.
Some estimates for a weighted 𝐿² projection
Authors: Bramble James H. , Xu Jinchao
Published: June 2010

This paper is devoted to the error estimates for some weighted

L
2

{L^2}

projections. Nearly optimal estimates are obtained. These estimates can be applied to the analysis of the usual multigrid method, multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities.
The CFL condition for spectral approximations to hyperbolic initial-boundary value problems
Authors: Gottlieb David , Tadmor Eitan
Published: June 2010

We study the stability of spectral approximations to scalar hyperbolic initial-boundary value problems with variable coefficients. Time is discretized by explicit multi-level or Runge-Kutta methods of order

≤
3

\leq 3

(forward Euler time-differencing is included), and we study spatial discretizations by spectral and pseudospectral approximations associated with the general family of Jacobi polynomials. We prove that these fully explicit spectral approximations are stable provided their time step,

Δ
t

\Delta t

, is restricted by the CFL-like condition

Δ
t
>

Const

∙

N

−
2

\Delta t > {\text {Const}} \bullet {N^{ - 2}}

, where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of appropriate

L
2

{L^2}

-weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. Our result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral approximations in the nonperiodic case.
On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics
Authors: Gunzburger Max D. , Meir Amnon J. , Peterson Janet S.
Published: June 2010

We consider the equations of stationary, incompressible magnetohydrodynamics posed in a bounded domain in three dimensions and treat the full, coupled system of equations with inhomogeneous boundary conditions. Under certain conditions on the data, we show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. We discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution.
Superconvergence and extrapolation for mixed finite element methods on rectangular domains
Authors: Wang Jun Ping
Published: June 2010

Asymptotic expansions for the RT (Raviart-Thomas) mixed finite element approximation by the lowest-order rectangular element associated with a second-order elliptic equation on a rectangular domain are derived. Superconvergence for the vector field along the Gauss lines is obtained as a result of the expansion. A procedure of postprocessed extrapolation is presented for the scalar field, as well as procedures of pure Richardson extrapolation for both the vector and the scalar fields.
Generation of elements with small modular squares and provably fast integer factoring algorithms
Authors: Vallée Brigitte
Published: June 2010

Finding small modular squares, when the modulus is a large composite number of unknown factorization, is almost certainly a computationally hard problem. This problem arises in a natural way when factoring the modulus by the use of congruences of squares. We study here, with the help of lattices, the set of elements whose squares

mod

n

\bmod \, n

are small enough, less than

O
(

n

2

/

3

)

O({n^{2/3}})

. We obtain a precise description of the gaps between such elements, and we develop two polynomial-time algorithms that find elements with small modular squares. The first is a randomized algorithm that generates such elements in a near uniform way. We use it to derive a class of integer factorization algorithms, the fastest of which provides the best rigorously established probabilistic complexity bound for integer factorization algorithms. The second algorithm is deterministic and often finds, amongst the neighbors of a given point, the nearest one that has a small modular square.
Uniform convergence results for Cauchy principal value integrals
Authors: Rabinowitz Philip
Published: June 2010

A general uniform convergence theorem for numerical integration of Cauchy principal value integrals is proved. Seven special instances of this theorem are given as corollaries.
A quasi-Monte Carlo method for the Boltzmann equation
Authors: Lécot Christian
Published: June 2010

A new quasi-Monte Carlo method for solving the Boltzmann equation in a simplified case is described. The analysis is restricted to a spatially homogeneous and isotropic gas; in addition, the molecular model only involves isotropic scattering. The scheme makes use of particles and combines an Euler scheme with numerical integrations. The sequence which is used for the quadratures must possess some symmetry properties which prescribe energy conservation for colliding particles. The error of the method is estimated by means of the discrepancy of the sequence which performs the quadratures. An algorithm for generating convenient sequences is proposed. In an example, where an exact solution is known, the computation of effective errors is included.
A Hamiltonian approximation to simulate solitary waves of the Korteweg-de Vries equation
Authors: Huang Ming You
Published: June 2010

Given the Hamiltonian nature and conservation laws of the Korteweg-de Vries equation, the simulation of the solitary waves of this equation by numerical methods should be effected in such a way as to maintain the Hamiltonian nature of the problem. A semidiscrete finite element approximation of Petrov-Galerkin type, proposed by R. Winther, is analyzed here. It is shown that this approximation is a finite Hamiltonian system, and as a consequence, the energy integral
\[

I
(
u
)
=

∫
0
1

(

u
x
2

2

+

u
3

)

d
x

I(u) = \int _0^1 {\left ( {\frac {{u_x^2}}{2} + {u^3}} \right )\;dx}

\]
is exactly conserved by this method. In addition, there is a discussion of error estimates and superconvergence properties of the method, in which there is no perturbation term but instead a suitable choice of initial data. A single-step fully discrete scheme, and some numerical results, are presented.
An automatic quadrature for Cauchy principal value integrals
Authors: Hasegawa Takemitsu , Torii Tatsuo
Published: June 2010

An automatic quadrature is presented for computing Cauchy principal value integrals

Q
(
f
;
c
)
=

₯

a
b

⁡
f
(
t
)

/

(
t
−
c
)

d
t
,
a
>
c
>
b

Q(f;c) = \fint _a^bf(t)/(t-c)\,dt, a > c > b

, for smooth functions

f
(
t
)

f(t)

. After subtracting out the singularity, we approximate the function

f
(
t
)

f(t)

by a sum of Chebyshev polynomials whose coefficients are computed using the FFT. The evaluations of

Q
(
f
;
c
)

Q(f;c)

for a set of values of c in (a, b) are efficiently accomplished with the same number of function evaluations. Numerical examples are also given.
Note on a polynomial of Emma Lehmer
Authors: Darmon Henri
Published: June 2010

Recently, Emma Lehmer constructed a parametric family of units in real quintic fields of prime conductor

p
=

t
4

+
5

t
3

+
15

t
2

+
25
t
+
25

p = {t^4} + 5{t^3} + 15{t^2} + 25t + 25

as translates of Gaussian periods. Later, Schoof and Washington showed that these units were fundamental units. In this note, we observe that Lehmer’s family comes from the covering of modular curves

X
1

(
25
)
→

X
0

(
25
)

{X_1}(25) \to {X_0}(25)

. This gives a conceptual explanation for the existence of Lehmer’s units: they are modular units (which have been studied extensively). By relating Lehmer’s construction with ours, one finds expressions for certain Gauss sums as values of modular units on

X
1

(
25
)

{X_1}(25)

.
A convergence analysis for nonsymmetric Lanczos algorithms
Authors: Ye Qiang
Published: June 2010

A convergence analysis for the nonsymmetric Lanczos algorithm is presented. By using a tridiagonal structure of the algorithm, some identities concerning Ritz values and Ritz vectors are established and used to derive approximation bounds. In particular, the analysis implies the classical results for the symmetric Lanczos algorithm.
On the continuity in 𝐵𝑉(Ω) of the 𝐿²-projection into finite element spaces
Authors: Cockburn Bernardo
Published: June 2010

We show how to obtain continuity in the

BV

(
Ω
)

{\text {BV}}(\Omega )

-seminorm of the

L
2

{L^2}

-projection of

u
∈

BV

(
Ω
)

u \in {\text {BV}}(\Omega )

into a large class of finite element spaces.
Optimal-order error estimates for the finite element approximation of the solution of a nonconvex variational problem
Authors: Collins Charles , Luskin Mitchell
Published: June 2010

Nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials. The solution of these variational problems must be described in terms of a microstructure rather than in terms of a deformation. Moreover, the numerical approximation of the deformation gradient often does not converge strongly as the mesh is refined. Nevertheless, the probability distribution of the deformation gradients near each material point does converge. Recently we introduced a metric to analyze this convergence. In this paper, we give an optimal-order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier.
A tabulation of oriented links
Authors: Doll Helmut , Hoste Jim
Published: June 2010

In this paper we enumerate all prime, nonsplit, oriented, classical links having two or more components and nine or fewer crossings. Our list is complete up to diffeomorphism of

S
3

{S^3}

and complete reorientation of the link. (That is, reorienting every component of the link.) Previously, only tables of nonoriented links have been compiled. Furthermore, we list, in the case of alternating links, all possible minimal diagrams of each link up to orientation. We also include the skein polynomials of each link. Our methods are direct generalizations of those used by Dowker and Thistlethwaite to enumerate knots. We rely heavily on the HOMFLY and Kauffman polynomials to distinguish inequivalent links. In a few cases these invariants will not suffice and other link invariants are employed. Our table is generated "from scratch" rather than by introducing orientations into already existing nonoriented tables. This provides a check on Conway’s table in the range mentioned above.
Nonconforming finite element methods for the equations of linear elasticity
Authors: Falk Richard S.
Published: June 2010

In the adaptation of nonconforming finite element methods to the equations of elasticity with traction boundary conditions, the main difficulty in the analysis is to prove that an appropriate discrete version of Korn’s second inequality is valid. Such a result is shown to hold for nonconforming piecewise quadratic and cubic finite elements and to be false for nonconforming piecewise linears. Optimal-order error estimates, uniform for Poisson ratio

ν
∈
[
0
,
1

/

2
)

\nu \in [0,1/2)

, are then derived for the corresponding

P
2

{P_2}

and

P
3

{P_3}

methods. This contrasts with the use of

C
0

{C^0}

finite elements, where there is a deterioration in the convergence rate as

ν
→
1

/

2

\nu \to 1/2

for piecewise polynomials of degree

≤
3

\leq 3

. Modifications of the continuous methods and the nonconforming linear method which also give uniform optimal-order error estimates are discussed.
The prime factors of Wendt’s binomial circulant determinant
Authors: Fee Greg , Granville Andrew
Published: June 2010

Wendt’s binomial circulant determinant,Wm{W_m}, is the determinant of anmbymcirculant matrix of integers, with (i, j)th entry(m|i−j|)\left ( {\begin {array}{*{20}{c}} m \\ {|i - j|} \\ \end {array} } \right )whenever 2 dividesmbut 3 does not. We explain how we found the prime factors ofWm{W_m}for each evenm≤200m \leq 200by implementing a new method for computations in algebraic number fields that uses only modular arithmetic. As a consequence we prove that ifpandq=mp+1q = mp + 1are odd primes, 3 does not dividem, andm≤200m \leq 200, then the first case of Fermat’s Last Theorem is true for exponentp.
Approximation of some diffusion evolution equations in unbounded domains by Hermite functions
Authors: Funaro Daniele , Kavian Otared
Published: June 2010

Spectral and pseudospectral approximations of the heat equation are analyzed. The solution is represented in a suitable basis constructed with Hermite polynomials. Stability and convergence estimates are given and numerical tests are discussed.
Numerical approximations of algebraic Riccati equations for abstract systems modelled by analytic semigroups, and applications
Authors: Lasiecka I. , Triggiani R.
Published: June 2010

This paper provides a numerical approximation theory of algebraic Riccati operator equations with unbounded coefficient operatorsAandB, such as arise in the study of optimal quadratic cost problems over the time interval[0,∞][0,\infty ]for the abstract dynamicsy˙=Ay+Bu\dot y = Ay + Bu. Here,Ais the generator of a strongly continuous analytic semigroup, andBis an unbounded operator with any degree of unboundedness less than that ofA. Convergence results are provided for the Riccati operators, as well as for all the other relevant quantities which enter into the dynamic optimization problem. The present numerical theory is the counterpart of a known continuous theory. Several examples of partial differential equations with boundary/point control, where all the required assumptions are verified, illustrate the theory. They include parabolic equations withL2{L_2}-Dirichlet control, as well as plate equations with a strong degree of damping and point control.
Structural properties for two classes of combined random number generators
Authors: L’Ecuyer Pierre , Tezuka Shu
Published: June 2010

We analyze a class of combined random number generators recently proposed by L’Ecuyer, which combines a set of linear congruential generators (LCG’s) with distinct prime moduli. We show that the geometrical behavior of the vectors of points produced by the combined generator can be approximated by the lattice structure of an associated LCG, whose modulus is the product of the moduli of the individual components. The approximation is good if these individual moduli are near each other and if the dimension of the vectors is large enough. The associated LCG is also exactly equivalent to a slightly different combined generator of the form suggested by Wichmann and Hill. We give illustrations, for which we examine the approximation error and assess the quality of the lattice structure of the associated LCG.
A search for aliquot cycles below 10¹⁰
Authors: Moews David , Moews Paul C.
Published: June 2010

A search for aliquotk-cycles below1010{10^{10}}withk≥3k \geq 3is described. Two new 4-cycles are exhibited. Six new 4-cycles not below1010{10^{10}}are also exhibited.
Complete solutions to families of quartic Thue equations
Authors: Pethő Attila
Published: June 2010

Using a method due to E. Thomas, we prove that if|a|>9.9⋅1027|a| > 9.9 \cdot {10^{27}}then the Diophantine equations\[x4−ax3y−x2y2+axy3+y4=1{x^4} - a{x^3}y - {x^2}{y^2} + ax{y^3} + {y^4} = 1\]and\[x4−ax3y−3x2y2+axy3+y4=±1{x^4} - a{x^3}y - 3{x^2}{y^2} + ax{y^3} + {y^4} = \pm 1\]have exactly twelve solutions, namely<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x comma y right-parenthesis equals left-parenthesis 0 comma plus-or-minus 1 right-parenthesis comma left-parenthesis plus-or-minus 1 comma 0 right-parenthesis comma left-parenthesis plus-or-minus 1 comma plus-or-minus 1 right-parenthesis comma left-parenthesis minus-or-plus 1 comma plus-or-minus 1 right-parenthesis comma left-parenthesis plus-or-minus a comma plus-or-minus 1 right-parenthesis comma left-parenthesis plus-or-minus 1 comma minus-or-plus a right-parenthesis">(x,y)=(0,±1),(±1,0),(±1,±1),(∓1,±1),(±a,±1),(±1,∓a)(x,y) = (0, \pm 1), ( \pm 1,0), ( \pm 1, \pm 1), ( \mp 1, \pm 1), ( \pm a, \pm 1), ( \pm 1, \mp a)and eight solutions,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x comma y right-parenthesis equals left-parenthesis 0 comma plus-or-minus 1 right-parenthesis comma left-parenthesis plus-or-minus 1 comma 0 right-parenthesis comma left-parenthesis plus-or-minus 1 comma plus-or-minus 1 right-parenthesis comma left-parenthesis plus-or-minus 1 comma minus-or-plus 1 right-parenthesis">(x,y)=(0,±1),(±1,0),(±1,±1),(±1,∓1)(x,y) = (0, \pm 1), ( \pm 1,0), ( \pm 1, \pm 1), ( \pm 1, \mp 1), respectively.
Ideal 9th-order multigrades and Letac’s elliptic curve
Authors: Smyth C. J.
Published: June 2010

By showing that the elliptic curve

(

x
2

−
13
)
(

y
2

−
13
)
=
48

({x^2} - 13)({y^2} - 13) = 48

has infinitely many rational points, we prove that Letac’s construction produces infinitely many genuinely different ideal 9th-order multigrades. We give one (not very small) new example, and, by finding the Mordell-Weil group of the curve, show how to find all examples obtainable by Letac’s method.
Solving a specific Thue-Mahler equation
Authors: Tzanakis N. , de Weger B. M. M.
Published: June 2010

The diophantine equation

x
3

−
3
x

y
2

−

y
3

=
±

3

n
0

17

n
1

19

n
2

{x^3} - 3x{y^2} - {y^3} = \pm {3^{{n_0}}}{17^{{n_1}}}{19^{{n_2}}}

is completely solved as follows. First, a large upper bound for the variables is obtained from the theory of linear forms in p-adic and real logarithms of algebraic numbers. Then this bound is reduced to a manageable size by p-adic and real computational diophantine approximation, based on the

L
3

{L^3}

-algorithm. Finally the complete list of solutions is found in a sieving process. The method is in principle applicable to any Thue-Mahler equation, as the authors will show in a forthcoming paper.
Transitive Steiner and Kirkman triple systems of order 27
Authors: Colbourn Charles J. , Magliveras Spyros S. , Mathon Rudolf A.
Published: June 2010

There are 71 Steiner triple systems of order 27 whose automorphism groups are point-transitive, and there are 248 transitive Kirkman triple systems of order 27. Computational methods used to find these designs are outlined. The designs and some of their properties are presented.
Stability results for one-step discretized collocation methods in the numerical treatment of Volterra integral equations
Authors: Crisci M. R. , Russo E. , Vecchio A.
Published: June 2010

This paper is concerned with the stability analysis of the discretized collocation method for the second-kind Volterra integral equation with degenerate kernel. A fixed-order recurrence relation with variable coefficients is derived, and local stability conditions are given independent of the discretization. Local stability and stability with respect to an isolated perturbation of some methods are proved. The reliability of the derived stability conditions is shown by numerical experiments.
Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension
Authors: Glassey R. T.
Published: June 2010

An energy-preserving, linearly implicit finite difference scheme is presented for approximating solutions to the periodic Cauchy problem for the one-dimensional Zakharov system of two nonlinear partial differential equations. First-order convergence estimates are obtained in a standard "energy" norm in terms of the initial errors and the usual discretization errors.
Levels of positive definite ternary quadratic forms
Authors: Lehman J. Larry
Published: June 2010

The level N and squarefree character q of a positive definite ternary quadratic form are defined so that its associated modular form has level N and character

χ
q

{\chi _q}

. We define à collection of correspondences between classes of quadratic forms having the same level and different discriminants. This makes practical a method for finding representatives of all classes of ternary forms having a given level. We also give a formula for the number of genera of ternary forms with a given level and character.
Computations of Siegel modular forms of genus two
Authors: Skoruppa Nils-Peter
Published: June 2010

We explain the basic notions and theorems for doing computations in the theory of Siegel modular forms of degree two, on the full modular group and of even weight. This synopsis concludes with a handy and computationally realistic algorithm for tabulating the Fourier coefficients of such forms and the Euler factors of their Spinor zeta functions. In the second part of this paper we present and discuss some of the results of actual computations which we performed following this algorithm. We point out two (theoretically) striking phenomena that are implied by the results of these computations.
The Monte Carlo algorithm with a pseudorandom generator
Authors: Traub J. F. , Woźniakowski H.
Published: June 2010

We analyze the Monte Carlo algorithm for the approximation of multivariate integrals when a pseudorandom generator is used. We establish lower and upper bounds on the error of such algorithms. We prove that as long as a pseudorandom generator is capable of producing only finitely many points, the Monte Carlo algorithm with such a pseudorandom generator fails for

L
2

{L_2}

or continuous functions. It also fails for Lipschitz functions if the number of points does not depend on the number of variables. This is the case if a linear congruential generator is used with one initial seed. On the other hand, if a linear congruential generator of period m is used for each component, with independent uniformly distributed initial seeds, then the Monte Carlo algorithm with such a pseudorandom generator using n function values behaves as for the uniform distribution and its expected error is roughly

n

−
1

/

2

{n^{ - 1/2}}

as long as the number n of function values is less than

m
2

{m^2}

.
Embedded diagonally implicit Runge-Kutta algorithms on parallel computers
Authors: van der Houwen P. J. , Sommeijer B. P. , Couzy W.
Published: June 2010

This paper investigates diagonally implicit Runge-Kutta methods in which the implicit relations can be solved in parallel and are singly diagonal-implicit on each processor. The algorithms are based on diagonally implicit iteration of fully implicit Runge-Kutta methods of high order. The iteration scheme is chosen in such a way that the resulting algorithm is

A
(
α
)

A(\alpha )

-stable or

L
(
α
)

L(\alpha )

-stable with

α
\alpha

equal or very close to

π

/

2

\pi /2

. In this way, highly stable, singly diagonal-implicit Runge-Kutta methods of orders up to 10 can be constructed. Because of the iterative nature of the methods, embedded formulas of lower orders are automatically available, allowing a strategy for step and order variation.
On the numerical evaluation of Legendre’s chi-function
Authors: Boersma J. , Dempsey J. P.
Published: June 2010

Legendre’s chi-function,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi Subscript n Baseline left-parenthesis z right-parenthesis equals normal upper Sigma Subscript k equals 0 Superscript normal infinity Baseline z Superscript 2 k plus 1 Baseline slash left-parenthesis 2 k plus 1 right-parenthesis Superscript n">

χ
n

(
z
)
=

Σ

k
=
0

∞

z

2
k
+
1

/

(
2
k
+
1

)
n

{\chi _n}(z) = \Sigma _{k = 0}^\infty {z^{2k + 1}}/{(2k + 1)^n}

, is reexpanded in a power series in powers of

log
⁡
z

\log z

. The expansion obtained is well suited for the computation of

χ
n

(
z
)

{\chi _n}(z)

in the two cases of real z close to 1, and

z
=

e

i
α

,
α
∈

R

z = {e^{i\alpha }},\alpha \in \mathbb {R}

. For

n
=
2

n = 2

and

n
=
3

n = 3

, the present computational procedure is shown to be superior to the procedure recently proposed by Dempsey, Liu, and Dempsey, which uses Plana’s summation formula.
Irregular primes to one million
Authors: Buhler J. P. , Crandall R. E. , Sompolski R. W.
Published: June 2010

Using "fast" algorithms for power series inversion (based on the fast Fourier transform and multisectioning of power series), we have calculated all irregular primes up to one million, including their indices of irregularity and associated irregular pairs. Using this data, we verified that Fermat’s "Last Theorem" and Vandiver’s conjecture are true for these primes. Two primes with index of irregularity five were already known; we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six.
A table of elliptic integrals: two quadratic factors
Authors: Carlson B. C.
Published: June 2010

Thirteen integrands that are rational except for the square root of a quartic polynomial with two pairs of conjugate complex zeros are integrated in terms of R-functions of real variables. In contrast with previous tables, the formulas hold for all real intervals of integration for which the integrals exist (possibly as Cauchy principal values). This is achieved by using Landen’s transformation and the duplication theorem. In an appendix, an elliptic integral of the third kind with a restricted complex parameter is transformed to make the parameter real. Also, a degenerate integral of the first kind is separated into real and imaginary parts.
Some singular moduli for 𝑄(√3)
Authors: Cohn Harvey , Deutsch Jesse
Published: June 2010

In an earlier paper in this journal, the authors derived the equations which transform the Hilbert modular function field for

Q

(

3

)

{\mathbf {Q}}(\sqrt 3 )

when the arguments are multiplied by

(
1
+

3

,
1
−

3

)

(1 + \sqrt 3 ,1 - \sqrt 3 )

. These equations define a complex

V
2

{V_2}

, but we concentrate on special diagonal curves on which the values of some of the singular moduli can be evaluated numerically by using the "PSOS" algorithm. In this way the ring class fields can be evaluated for the forms

ξ
2

+

2
t

A

η
2

{\xi ^2} + {2^t}A{\eta ^2}

, where

A
=
1
,
2
,
3
,
6

A = 1,2,3,6

and

t
>
0

t > 0

. These last results are based partly on conjectures supported here by numerical evidence.
Cyclotomic invariants for primes to one million
Authors: Ernvall R. , Metsänkylä T.
Published: June 2010

Our recent computation of cyclotomic invariants for primes between 125000 and 150000 was extended to one million. No new phenomena appear.
On Bessel functions and rate of convergence of zeros of Lommel polynomials
Authors: Feinsilver P. , Schott R.
Published: June 2010

In this note, we solve an open problem of Flajolet and Schott concerning the rate of convergence of the zeros of Lommel polynomials to the zeros of a Bessel function (considered as a function of the order). The average case analysis of dynamic data structures was the initial motivation for this investigation. The Maple program, whose results are reported at the end, illustrates that our result is quite good.
Error estimates for a nonlinear degenerate parabolic problem
Authors: Lesaint P. , Pousin J.
Published: June 2010

In this paper we are dealing with a partial differential equation of parabolic type, which degenerates on one side of the domain. This equation may be viewed either as a model of particle diffusion in plasma physics, or as a simplified model of a viscous boundary layer in two dimensions. Known results for the existence and uniqueness of the weak solution are first recalled. A finite difference implicit scheme is then defined, and error bounds are derived, taking into account the low degree of smoothness of the exact solution. An iterative algorithm for the computation of the numerical solution at each time step is shown to be convergent.
Finite element approximation to initial-boundary value problems of the semiconductor device equations with magnetic influence
Authors: Zhu Jiang
Published: June 2010

We shall consider Zlámal’s approach to the nonstationary equations of the semiconductor device theory under magnetic fields, with mixed boundary conditions. Owing to the reduced smoothness of the electric potential

ψ
\psi

and carrier densities n and p caused by considering the mixed boundary conditions, we must use a nonstandard analysis for this procedure. Existence as well as uniqueness of the approximate solution is proved. The convergence rates obtained in this paper are slower than those previously obtained for pure Dirichlet or Neumann boundary conditions.
On the relation between two local convergence theories of least-change secant update methods
Authors: Martínez José Mario
Published: June 2010

In this paper, we show that the main results of the local convergence theory for least-change secant update methods of Dennis and Walker (SIAM J. Numer. Anal. 18 (1981), 949-987) can be proved using the theory introduced recently by Martinez (Math. Comp. 55 (1990), 143-167). In addition, we exhibit two generalizations of well-known methods whose local convergence can be easily proved using Martínez’s theory.
Gaussian cubature and bivariate polynomial interpolation
Authors: Xu Yuan
Published: June 2010

Gaussian cubature is used to study bivariate polynomial interpolation based on the common zeros of quasi-orthogonal polynomials.
A practical analysis of the elliptic curve factoring algorithm
Authors: Silverman Robert D. , Wagstaff Samuel S.
Published: June 2010

Much asymptotic analysis has been devoted to factoring algorithms. We present a practical analysis of the complexity of the elliptic curve algorithm, suggesting optimal parameter selection and run-time guidelines. The parameter selection is aided by a novel use of Bayesian statistical decision techniques as applied to random algorithms. We discuss how frequently the elliptic curve algorithm succeeds in practice and compare it with the quadratic sieve algorithm.
Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems
Authors: Eriksson Kenneth , Johnson Claes
Published: June 2010

Adaptive finite element methods for stationary convection-diffusion problems are designed and analyzed. The underlying discretization scheme is the Shock-capturing Streamline Diffusion method. The adaptive algorithms proposed are based on a posteriori error estimates for this method leading to reliable methods in the sense that the desired error control is guaranteed. A priori error estimates are used to show that the algorithms are efficient in a certain sense.
A quadrature formula involving zeros of Bessel functions
Authors: Frappier Clément , Olivier Patrick
Published: June 2010

An exact quadrature formula for entire functions of exponential type is obtained. The nodes of the formula are zeros of the Bessel function of the first kind of order

α
\alpha

. It generalizes and refines a known quadrature formula related to the sampling theorem. The uniqueness of the nodes is studied.
Two formulas for numerical indefinite integration
Authors: Haber Seymour
Published: June 2010

We derive two formulas for approximating the indefinite integral over a finite interval. The approximation error is

O
(

c

−
c

m

)

O({c^{ - c\sqrt m }})

uniformly, where m is the number of integrand evaluations. The integrand is required to be analytic in the interior of the integration interval, but may be singular at the endpoints. Some sample calculations indicate that the actual convergence rate accords with the error bound derived.
A method of fractional steps for scalar conservation laws without the CFL condition
Authors: Holden Helge , Risebro Nils Henrik
Published: June 2010

We present a numerical method for the n-dimensional initial value problem for the scalar conservation law
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline comma t right-parenthesis Subscript t Baseline plus sigma-summation Underscript i equals 1 Overscript n Endscripts f Subscript i Baseline left-parenthesis u right-parenthesis Subscript x 1 Baseline equals 0 comma u left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline comma 0 right-parenthesis equals u 0 left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis">

u

(

x
1

,
…
,

x
n

,
t

)
t

+

∑

i
=
1

n

f
i

(
u

)

x
1

=
0
,
u
(

x
1

,
…
,

x
n

,
0
)
=

u
0

(

x
1

,
…
,

x
n

)

u{({x_1}, \ldots ,{x_n},t)_t} + \sum _{i = 1}^n{f_i}{(u)_{{x_1}}} = 0, u({x_1}, \ldots ,{x_n},0) = {u_0}({x_1}, \ldots ,{x_n})

. Our method is based on the use of dimensional splitting and Dafermos’s method to solve the one-dimensional equations. This method is unconditionally stable in the sense that the time step is not limited by the space discretization. Furthermore, we show that this method produces a subsequence which converges to the weak entropy solution as both the time and space discretization go to zero. Finally, two numerical examples are discussed.
Runge-Kutta methods for parabolic equations and convolution quadrature
Authors: Lubich Ch. , Ostermann A.
Published: June 2010

We study the approximation properties of Runge-Kutta time discretizations of linear and semilinear parabolic equations, including incompressible Navier-Stokes equations. We derive asymptotically sharp error bounds and relate the temporal order of convergence, which is generally noninteger, to spatial regularity and the type of boundary conditions. The analysis relies on an interpretation of Runge-Kutta methods as convolution quadratures. In a different context, these can be used as efficient computational methods for the approximation of convolution integrals and integral equations. They use the Laplace transform of the convolution kernel via a discrete operational calculus.
Finite volume solutions of convection-diffusion test problems
Authors: Mackenzie J. A. , Morton K. W.
Published: June 2010

The cell-vertex formulation of the finite volume method has been developed and widely used to model inviscid flows in aerodynamics: more recently, one of us has proposed an extension for viscous flows. The purpose of the present paper is two-fold: first we have applied this scheme to a well-known convection-diffusion model problem, involving flow round a

180
∘

180^\circ

bend, which highlights some of the issues concerning the application of the boundary conditions in such cell-based schemes. The results are remarkably good when the boundary conditions are applied in an appropriate manner. In our efforts to explain the high quality of the results we were led to a detailed analysis of the corresponding one-dimensional problem. Our second purpose is thus to gather together various approaches to the analysis of this problem and to draw attention to the supra-convergence phenomena enjoyed by the proposed methods.
Total variation and error estimates for spectral viscosity approximations
Authors: Tadmor Eitan
Published: June 2010

We study the behavior of spectral viscosity approximations to non-linear scalar conservation laws. We show how the spectral viscosity method compromises between the total-variation bounded viscosity approximations— which are restricted to first-order accuracy—and the spectrally accurate, yet unstable, Fourier method. In particular, we prove that the spectral viscosity method is

L
1

{L^1}

-stable and hence total-variation bounded. Moreover, the spectral viscosity solutions are shown to be

Lip
+

{\text {Lip}^ + }

-stable, in agreement with Oleinik’s E-entropy condition. This essentially nonoscillatory behavior of the spectral viscosity method implies convergence to the exact entropy solution, and we provide convergence rate estimates of both global and local types.
Quadrature of integrands with a logarithmic singularity
Authors: Crow John A.
Published: June 2010

A quadrature rule is presented that is exact for integrands of the form

π
(
ξ
)
+
ϖ
(
ξ
)
log
⁡
ξ

\pi (\xi ) + \varpi (\xi )\log \xi

on the interval (0, 1), where

π
\pi

and

ϖ
\varpi

are polynomials. The computed weights and abscissae are given for one- through seven-point rules. In particular, the four-point rule is exact for integral operators with log-arithmically singular kernel on a cubic B-spline basis, and it is expected these results shall prove useful for numerical applications of weighted-residual finite element methods.
On the coefficients of the minimal polynomials of Gaussian periods
Authors: Gupta S. , Zagier D.
Published: June 2010

Let l be a prime number and m a divisor of

l
−
1

l - 1

. Then the Gauss period

ω
=
ζ
+

ζ
λ

+

ζ

λ
2

+
⋯
+

ζ

λ

m
−
1

\omega = \zeta + {\zeta ^\lambda } + {\zeta ^{{\lambda ^2}}} + \cdots + {\zeta ^{{\lambda ^{m - 1}}}}

where

ζ
=

e

2
π
i

/

l

\zeta = {e^{2\pi i/l}}

and

λ
\lambda

is a primitive mth root of unity modulo l, generates a subfield K of

Q

(
ζ
)

\mathbb {Q}(\zeta )

of degree

(
l
−
1
)

/

m

(l - 1)/m

. In this paper we study the reciprocal minimal polynomial

F

l
,
m

(
X
)
=

N

K

/

Q

(
1
−
ω
X
)

{F_{l,m}}(X) = {N_{K/\mathbb {Q}}}(1 - \omega X)

of

ω
\omega

over

Q

\mathbb {Q}

. It will be shown that for fixed m and every N we have
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript l comma m Baseline left-parenthesis upper X right-parenthesis identical-to left-parenthesis upper B Subscript m Baseline left-parenthesis upper X right-parenthesis Superscript l Baseline slash left-parenthesis 1 equals m upper X right-parenthesis right-parenthesis Superscript 1 slash m Baseline left-parenthesis mod upper X Superscript upper N Baseline right-parenthesis">

F

l
,
m

(
X
)
≡

(

B
m

(
X

)
l

/

(
1
=
m
X
)

)

1

/

m

(
mod

X
N

)

{F_{l,m}}(X) \equiv {({B_m}{(X)^l}/(1 = mX))^{1/m}}\;\pmod {{X^N}}

for all but finitely many "exceptional primes" l (depending on m and N), where

B
m

(
X
)
∈

Z

[
[
X
]
]

{B_m}(X) \in \mathbb {Z}[[X]]

is a power series depending only on m . A method of computation of this set of exceptional primes is presented. The generalization of the results to the case of composite l is also discussed.
Statistical independence of a new class of inversive congruential pseudorandom numbers
Authors: Eichenauer-Herrmann Jürgen
Published: June 2010

Linear congruential pseudorandom numbers show several undesirable regularities which can render them useless for certain stochastic simulations. This was the motivation for important recent developments in nonlinear congruential methods for generating uniform pseudorandom numbers. It is particularly promising to achieve nonlinearity by employing the operation of multiplicative inversion with respect to a prime modulus. In the present paper a new class of such inversive congruential generators is introduced and analyzed. It is shown that they have excellent statistical independence properties and model true random numbers very closely. The methods of proof rely heavily on Weil-Stepanov bounds for rational exponential sums.
𝐿¹-stability of stationary discrete shocks
Authors: Liu Jian-Guo , Xin Zhou Ping
Published: June 2010

The nonlinear stability in the

L
p

{L^p}

-norm,

p
≥
1

p \geq 1

, of stationary weak discrete shocks for the Lax-Friedrichs scheme approximating general

m
×
m

m \times m

systems of nonlinear hyperbolic conservation laws is proved, provided that the summations of the initial perturbations equal zero. The result is proved by using both a weighted estimate and characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme.
Operational quadrature methods for Wiener-Hopf integral equations
Authors: Eggermont P. P. B. , Lubich Ch.
Published: June 2010

We study the numerical solution of Wiener-Hopf integral equations by a class of quadrature methods which lead to discrete Wiener-Hopf equations, with quadrature weights constructed from the Fourier transform of the kernel (or rather, from the Laplace transforms of the kernel halves). As the analytical theory of Wiener-Hopf equations is likewise based on the Fourier transform of the kernel, this approach enables us to obtain results on solvability and stability and error estimates for the discretization. The discrete Wiener-Hopf equations are solved by using an approximate Wiener-Hopf factorization obtained with FFT. Numerical experiments with the Milne equation of radiative transfer are included.
On the computation of unit groups and class groups of totally complex quartic fields
Authors: Pohst M. , von Schmettow J. Graf
Published: June 2010

We describe the computation of the unit group and the class group of the 81322 totally complex quartic fields with discriminant less than one million. 45.6
A least squares Petrov-Galerkin finite element method for the stationary Navier-Stokes equations
Authors: Zhou Tian Xiao , Feng Min Fu
Published: June 2010

In this paper, a Galerkin/least squares-type finite element method is proposed and analyzed for the stationary Navier-Stokes equations. The method is consistent and stable for any combination of discrete velocity and pressure spaces (without requiring a Babuška-Brezzi stability condition). The existence, uniqueness and convergence (at optimal rate) of the discrete solution is proved in the case of sufficient viscosity (or small data).
On the distribution of 𝑘-dimensional vectors for simple and combined Tausworthe sequences
Authors: Couture Raymond , L’Ecuyer Pierre , Tezuka Shu
Published: June 2010

The lattice structure of conventional linear congruential random number generators (LCGs), over integers, is well known. In this paper, we study LCGs in the field of formal Laurent series, with coefficients in the Galois field

F

2

{\mathbb {F}_2}

. The state of the generator (a Laurent series) evolves according to a linear recursion and can be mapped to a number between 0 and 1, producing what we call a LS2 sequence. In particular, the sequences produced by simple or combined Tausworthe generators are special cases of LS2 sequences. By analyzing the lattice structure of the LCG, we obtain a precise description of how all the k-dimensional vectors formed by successive values in the LS2 sequence are distributed in the unit hypercube. More specifically, for any partition of the k-dimensional hypercube into

2

k
l

{2^{kl}}

identical subcubes, we can quickly compute a table giving the exact number of subcubes that contain exactly n points, for each integer n. We give numerical examples and discuss the practical implications of our results.