bluuebunny/crossref_metadata_2025 huggingface dataset:
Further convergence results for the weighted Galerkin method of numerical solution of Cauchy-type singular integral equations
Authors: Ioakimidis N. I.
Published: June 2010

The convergence of the weighted Galerkin method (based on Chebyshev or Jacobi polynomials) for the direct numerical solution of one-dimensional, real Cauchy-type singular integral equations of the first and of the second kind on a finite interval is proved under sufficiently weak continuity assumptions for the kernels and the right-side functions of the integral equations.
Sinc function quadrature rules for the Fourier integral
Authors: Lund John
Published: June 2010

In this paper a numerical algorithm is proposed for the computation of the Fourier Transform. The quadrature rule developed is based on the Whittaker Cardinal Function expansion of the integrand and a certain Conformal Map. The error of the method is analyzed and numerical results are reported which confirm the accuracy of the quadrature rule.
Common zeros of two Bessel functions. II. Approximations and tables
Authors: Benton T. C.
Published: June 2010

In [1] it was shown that two Bessel functions

J
ν

(
x
)

{J_\nu }(x)

,

J
μ

(
x
)

{J_\mu }(x)

could have two zeros which were common to both functions, and a computer program was made which takes approximate values of

ν
\nu

,

μ
\mu

and

j

ν
,
k

=

j

μ
,
h

{j_{\nu ,k}} = {j_{\mu ,h}}

and

j

ν
,
k
+
n

=

j

μ
,
h
+
m

{j_{\nu , k + n}} = {j_{\mu ,h + m}}

and from them computes the exact values. Here it will be shown how to find the necessary approximate values to initiate the computation. A table of the smaller ratios m : n with the orders of the functions less than one hundred is given.
A rapid method of evaluating the regulator and class number of a pure cubic field
Authors: Williams H. C. , Dueck G. W. , Schmid B. K.
Published: June 2010

Let

K

=

Q

(
θ
)

\mathcal {K} = \mathcal {Q}(\theta )

be the algebraic number field formed by adjoining

θ
\theta

to the rationals

Q

\mathcal {Q}

. Let R and h be, respectively, the regulator and class number of

K

\mathcal {K}

. Shanks has described a method of evaluating R for

Q

(

D

)

\mathcal {Q}(\sqrt D )

, where D is a positive integer. His technique improved the speed of the usual continued fraction algorithm for finding R by allowing one to proceed almost directly from the nth to the mth step, where m is approximately 2n, in the continued fraction expansion of

D

\sqrt D

. This paper shows how Shanks’ idea can be extended to the Voronoi algorithm, which is used to find R in cubic fields of negative discriminant. It also discusses at length an algorithm for finding R and h for pure cubic fields

Q

(

D
3

)

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

, D an integer. Under a certain generalized Riemann Hypothesis the ideas developed here will provide a new method which will find R and h in

O
(

D

2

/

5
+
ε

)

O({D^{2/5 + \varepsilon }})

operations. When h is small, this is an improvement over the

O
(
D

/

h
)

O(D/h)

operations required by Voronoi’s algorithm to find R. For example, with

D
=
200171999

D = 200171999

, it required only 5 minutes for an AMDAHL 470/V7 computer to find that

R
=
518594546.969083

R = 518594546.969083

and

h
=
1

h = 1

. This same calculation would require about 8 days of computer time if it used only the standard Voronoi algorithm.
Local convergence of difference Newton-like methods
Authors: Ypma T. J.
Published: June 2010

Using affine invariant terms, we give a local convergence analysis of difference Newton-like methods for solving the nonlinear equation

F
(
x
)
=
0

F(x) = 0

. The convergence conditions are weaker than those standardly required for methods of this class. The technique and results are valid for all currently known difference Newton-like methods which require evaluation of all components of F at the same point. Radius of convergence and rate of convergence results for particular difference Newton-like methods may easily be derived from the results reported here.
Analytical calculation of a class of integrals containing exponential and trigonometric functions
Authors: Massidda Vittorio
Published: June 2010

It is shown how to evaluate analytically integrals from 0 to

2
π

2\pi

of functions of the type

f
(
ϕ
)

g
(
ϕ
)
exp
⁡
{
G
(
ϕ
)
}

f(\phi )\;g(\phi )\exp \{ G(\phi )\}

, where

g
(
ϕ
)

g(\phi )

and

G
(
ϕ
)

G(\phi )

are linear combinations of powers of

sin
⁡
ϕ

\sin \phi

and

cos
⁡
ϕ

\cos \phi

, and

f
(
ϕ
)

f(\phi )

is such that its Fourier coefficients can be evaluated analytically. The result is expressed in terms of modified Bessel functions.
Modulus and phase of the reduced logarithmic derivative of the Hankel function
Authors: Cruz Andrés , Sesma Javier
Published: June 2010

The modulus and phase of the reduced logarithmic derivative of the Hankel function
\[

z

H
v

(
1
)

′

(
z
)

/

H
v

(
1
)

(
z
)

zH_v^{(1)}{}’ (z)/H_v^{(1)}(z)

\]
for complex variable z and real order v, are investigated. Special attention is paid to the location of saddle points and their trajectories as v varies.
New integer factorizations
Authors: Naur Thorkil
Published: June 2010

New factorizations of Fibonacci numbers, Lucas numbers, and numbers of the form

2
n

±
1

{2^n} \pm 1

are presented together with the strategy (a combination of known factorization methods) used to obtain them.
Irreducibility testing and factorization of polynomials
Authors: Adleman Leonard M. , Odlyzko Andrew M.
Published: June 2010

It is shown that under certain hypotheses, irreducibility testing and factorization of polynomials with integer coefficients are polynomial time reducible to primality testing and factorization of integers, respectively. Combined with recently discovered fast primality tests, this yields an almost polynomial time irreducibility algorithm. The assertions of irreducibility produced by this algorithm are always certain and yield short proofs of irreducibility.
On optimal shooting intervals
Authors: Mattheij R. M. M. , Staarink G. W. M.
Published: June 2010

We develop an adaptive multiple shooting strategy, which is nearly optimal with respect to cpu time. Since the costs of integration are the most important components in this, we investigate in some detail how the gridpoints are chosen by an adaptive integration routine. We use this information to find out where the shooting points have to be selected. We also show that our final strategy is stable in the sense that rounding errors can be kept below a given tolerance. Finally we pay attention to the question how the need for memory can be minimized.
Chebyshev quadrature rules for a new class of weight functions
Authors: Byrd Paul F. , Stalla Lawrence
Published: June 2010

Proof is given that the weight functions

w
(
x
,
p
)
=
1

/

[
π
(
p
+
x
)

x
(
1
−
x
)

]

w(x,p) = 1/[\pi (p + x)\sqrt {x(1 - x)} ]

on (0, 1) admit Chebyshev quadratures for any fixed

p
⩾
1

p \geqslant 1

, and every N. For the particular cases when

p
=
1

p = 1

and

p
=
2

p = 2

, the nodes are tabulated to ten decimal places for N-point rules with

N
=
2
,
4
,
6
,
8

N = 2,4,6,8

, and 12. Numerical tables are also given for a coefficient in the expression of the error term.
A note on the Diophantine equation 𝑥ⁿ+𝑦ⁿ+𝑧ⁿ=3
Authors: Scarowsky Manny , Boyarsky Abraham
Published: June 2010

In this note solutions for the Diophantine equation
\[

x
3

+

y
3

+

z
3

=
3

{x^3} + {y^3} + {z^3} = 3

\]
are sought along planes

x
+
y
+
z
=
3
m

x + y + z = 3m

,

m
∈

Z

m \in {\mathbf {Z}}

. This was done for

|

m

|

⩽
50000

|m| \leqslant 50000

, and no new solutions were found.
On zeros of Mellin transforms of 𝑆𝐿₂(𝑍) cusp forms
Authors: Ferguson H. R. P. , Major R. D. , Powell K. E. , Throolin H. G.
Published: June 2010

We compute zeros of Mellin transforms of modular cusp forms for

S

L
2

(

Z

)

S{L_2}({\mathbf {Z}})

. Such Mellin transforms are eigenforms of Hecke operators. We recall that, for all weights k and all dimensions of cusp forms, the Mellin transforms of cusp forms have infinitely many zeros of the form

k

/

2
+
t

−
1

k/2 + t\sqrt { - 1}

, i.e., infinitely many zeros on the critical line. A new basis theorem for the space of cusp forms is given which, together with the Selberg trace formula, renders practicable the explicit computations of the algebraic Fourier coefficients of cusp eigenforms required for the computations of the zeros. The first forty of these Mellin transforms corresponding to cusp eigenforms of weight

k
⩽
50

k \leqslant 50

and dimension

⩽
4

\leqslant 4

are computed for the sections of the critical strips,

σ
+
t

−
1

\sigma + t\sqrt { - 1}

,

k
−
1
>
2
σ
>
k
+
1

k - 1 > 2\sigma > k + 1

,

−
40
⩽
t
⩽
40

- 40 \leqslant t \leqslant 40

. The first few zeros lie on the respective critical lines

k

/

2
+
t

−
1

k/2 + t\sqrt { - 1}

and are simple. A measure argument, depending upon the Riemann hypothesis for finite fields, is given which shows that Hasse-Weil L-functions (including the above) lie among Dirichlet series which do satisfy Riemann hypotheses (but which need not have functional equations nor analytic continuations).
On quartic fields of signature one with small discriminant. II
Authors: Godwin H. J.
Published: June 2010

Tables of quartic fields, having two real and two complex conjugates, and with discriminants between

−
7776

-7776

and 0, are given.
Two approximations of solutions of Hamilton-Jacobi equations
Authors: Crandall M. G. , Lions P.-L.
Published: June 2010

Equations of Hamilton-Jacobi type arise in many areas of application, including the calculus of variations, control theory and differential games. The associated initial-value problems almost never have global-time classical solutions, and one must deal with suitable generalized solutions. The correct class of generalized solutions has only recently been established by the authors. This article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates. Analogous results are proved by similar means for the method of vanishing viscosity.
Methods for the numerical solution of the nonlinear Schroedinger equation
Authors: Sanz-Serna J. M.
Published: June 2010

Optimal

L
2

{L^2}

rates of convergence are established for several fully-discrete schemes for the numerical solution of the nonlinear Schroedinger equation. Both finite differences and finite elements are considered for the discretization in space, while the integration in time is treated either by the leap-frog technique or by a modified Crank-Nicolson procedure, which generalizes the one suggested by Delfour, Fortin and Payne and possesses two useful conserved quantities.
On common zeros of Legendre’s associated functions
Authors: Lacroix Norbert H. J.
Published: June 2010

In this paper it is proved that any two given Legendre associated functions

P
n
m

(
μ
)

P_n^m(\mu )

and

P
n
s

(
μ
)

P_n^s(\mu )

, where

n
⩾
1

n \geqslant 1

is an integer and where one of the integers m or s may be 0 (and

m
≠
±
s

m \ne \pm s

), have either no zero in common or exactly one common zero, namely

μ
=
0

\mu = 0

. An auxiliary result states that the

n
−
m

n - m

zeros of

P
n
m

P_n^m

known to lie in the open interval

(
−
1
,
1
)

( - 1,1)

lie in fact in the open interval

(
−
c
,
c
)

( - c,c)

, where

±
c

\pm c

are the two zeros of

n
(
n
+
1
)
−

m
2

/

(
1
−

μ
2

)

n(n + 1) - {m^2}/(1 - {\mu ^2})

which is one of the coefficients in the Legendre associated equation satisfied by

P
n
m

P_n^m

. Some monotonicity behavior of

P
n
m

P_n^m

is simultaneously described. The proof of the main result is based on properties of Prüfer polar coordinates.
Generalized Bernoulli numbers and 𝑚-regular primes
Authors: Hao Fred H. , Parry Charles J.
Published: June 2010

A primepis defined to bem-regular ifpdoes not divide the class number of a certain abelian number field. Several different characterizations are given for a prime to bem-regular, including a description in terms of the generalized Bernoulli numbers. A summary is given of two computations which determine them-regularity orm-irregularity of primespfor certain values ofmandp.
A new scalar potential formulation of the magnetostatic field problem
Authors: Pasciak Joseph E.
Published: June 2010

A new method for approximating magnetostatic field problems is given in this paper. The new method approximates the scalar potential for the magnetic intensity and is based on a volume integral formulation. The derivation of the new computational method uses the spectral properties of the relevant integral operator. The corresponding algorithm is similar to that obtained from coupled differential and boundary integral approaches. Convergence and stability theorems are proven. Finally, convergence results in actual computations are compared with results for the usual volume integral method used in GFUN3D.
Spectral properties for the magnetization integral operator
Authors: Friedman Mark J. , Pasciak Joseph E.
Published: June 2010

We analyze the spectrum of a certain singular integral operator on the space

(

L
2

(
Ω
)

)
3

{({L^2}(\Omega ))^3}

where

Ω
\Omega

is contained in three dimensional Euclidean space and has a Lipschitz continuous boundary. This operator arises in the integral formulation of the magnetostatic field problem. We decompose

(

L
2

(
Ω
)

)
3

{({L^2}(\Omega ))^3}

into invariant subspaces: in one where the operator is the zero map; in one, the identity map; and in one where the operator is positive definite and bounded. These results give rise to the formulation of new efficient numerical techniques for approximating nonlinear magnetostatic field problems [5], [6], [12].
Differential-algebraic systems as differential equations on manifolds
Authors: Rheinboldt Werner C.
Published: June 2010

Based on the theory of differential equations on manifolds, existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications. Both the autonomous and nonautonomous case are considered. Moreover, a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower-dimensional manifolds. This class includes systems for which the application of ODE-solvers is known to lead to difficulties. Finally, some solution approach based on continuation techniques is outlined.
An algorithm for the second immanant
Authors: Grone Robert , Merris Russell
Published: June 2010

Let

χ
\chi

be an irreducible character of the symmetric group

S
n

{S_n}

. For

A
=
(

a

i
j

)

A = ({a_{ij}})

an n-by-n matrix, define the immanant of A corresponding to

χ
\chi

by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d left-parenthesis upper A right-parenthesis equals sigma-summation Underscript sigma element-of upper S Subscript n Baseline Endscripts chi left-parenthesis sigma right-parenthesis product Underscript t equals 1 Overscript n Endscripts a Subscript t sigma left-parenthesis t right-parenthesis Baseline period">

d
(
A
)
=

∑

σ
∈

S
n

χ
(
σ
)

∏

t
=
1

n

a

t
σ
(
t
)

.

d(A) = \sum \limits _{\sigma \in {S_n}} {\chi (\sigma )\prod \limits _{t = 1}^n {{a_{t\sigma (t)}}.} }

\]
The article contains an algorithm for computing

d
(
A
)

d(A)

when

χ
\chi

corresponds to the partition (2,

1

n
−
2

{1^{n - 2}}

).
Second-order conservative schemes and the entropy condition
Authors: Schonbek Maria E.
Published: June 2010

We consider numerical approximations to solutions of systems of hyperbolic conservation laws of the form

∂
u

/

∂
t
+
∂
f
(
u
)

/

∂
x
=
0

\partial u/\partial t + \partial f(u)/\partial x = 0

,

u
∈

R

n

u \in {{\mathbf {R}}^n}

and

f
:

R
n

→

R
n

f:{R^n} \to {R^n}

smooth. We show that conservative three-point second-order accurate methods cannot satisfy a local entropy inequality.
Global estimates for mixed methods for second order elliptic equations
Authors: Douglas Jim , Roberts Jean E.
Published: June 2010

Global error estimates in

L
2

(
Ω
)

{L^2}(\Omega )

,

L
∞

(
Ω
)

{L^\infty }(\Omega )

, and

H

−
s

(
Ω
)

{H^{ - s}}(\Omega )

,

Ω
\Omega

in

R

2

{{\mathbf {R}}^2}

or

R

3

{{\mathbf {R}}^3}

, are derived for a mixed finite element method for the Dirichlet problem for the elliptic operator

L
p
=
−
div
⁡
(
a

g
r
a
d

p
+

b

p
)
+
c
p

Lp = - \operatorname {div}(a\;{\mathbf {grad}}\;p + {\mathbf {b}}p) + cp

based on the Raviart-Thomas-Nedelec space

V

h

×

W
h

⊂

H

(
div
;
Ω
)
×

L
2

(
Ω
)

{{\mathbf {V}}_h} \times {W_h} \subset {\mathbf {H}}(\operatorname {div};\Omega ) \times {L^2}(\Omega )

. Optimal order estimates are obtained for the approximation of p and the associated velocity field

u

=
−
(
a

g
r
a
d

p
+

b

p
)

{\mathbf {u}} = - (a\;{\mathbf {grad}}\;p + {\mathbf {b}}p)

in

L
2

(
Ω
)

{L^2}(\Omega )

and

H

−
s

(
Ω
)

{H^{ - s}}(\Omega )

,

0
⩽
s
⩽
k
+
1

0 \leqslant s \leqslant k + 1

, and, if

Ω
⊂

R

2

\Omega \subset {{\mathbf {R}}^2}

for p in

L
∞

(
Ω
)

{L^\infty }(\Omega )

.
Application of quadrature rules for Cauchy-type integrals to the generalized Poincaré-Bertrand formula
Authors: Ioakimidis N. I.
Published: June 2010

The classical Poincaré-Bertrand transposition formula for the inversion of the order of integration in repeated Cauchy-type integrals is generalized in accordance with a new interpretation of Cauchy-type integrals. Next, the Gauss-Jacobi quadrature rule is applied, in a particular case of the generalized Poincaré-Bertrand formula, to both members of this formula and it is proved that this formula still remains valid (after the approximation of the integrals by quadrature sums). Two simple applications of this result, one concerning the convergence of a quadrature rule for repeated Cauchy-type integrals, and the other the numerical solution of singular integral equations, are made. Further generalizations and applications of the present results follow easily.
Implementing the continued fraction factoring algorithm on parallel machines
Authors: Wunderlich Marvin C.
Published: June 2010

An implementation is described of the continued fraction factoring algorithm on the DAP parallel processor located in Queen Mary College in London. The DAP has 4096 parallel processors each containing 16K bits of memory and the suggested implementation incorporates the early abort strategy and the large prime variation.
Improved methods for calculating vectors of short length in a lattice, including a complexity analysis
Authors: Fincke U. , Pohst M.
Published: June 2010

The standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of

Z

m

{{\mathbf {Z}}^m}

in a suitable box. However, it suffices to consider those

x

∈

Z

m

{\mathbf {x}} \in {{\mathbf {Z}}^m}

which lie in a suitable ellipsoid having a much smaller volume than the box. We show in this paper that searching through that ellipsoid is in many cases much more efficient. If combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions. Several randomly constructed numerical examples illustrate the superiority of our new method over the known ones.
Analysis of the finite element variable penalty method for Stokes equations
Authors: Kheshgi Haroon , Luskin Mitchell
Published: June 2010

We give an error analysis of the finite element variable penalty method for Stokes equations. It is shown that the variable penalty method is of higher order than the standard penalty method.
Optimal integration for functions of bounded variation
Authors: Traub J. F. , Lee D.
Published: June 2010

The unique optimal information and the unique optimal linear algorithm are obtained for the integration of functions of bounded variation.
On monotone and convex spline interpolation
Authors: Costantini Paolo
Published: June 2010

This paper is concerned with the problem of existence of monotone and/or convex splines, having degree n and order of continuity k, which interpolate to a set of data at the knots. The interpolating splines are obtained by using Bernstein polynomials of suitable continuous piecewise linear functions; they satisfy the inequality

k
⩽
n
−
k

k \leqslant n - k

. The theorems presented here are useful in developing algorithms for the construction of shape-preserving splines interpolating arbitrary sets of data points. Earlier results of McAllister, Passow and Roulier can be deduced from those given in this paper.
On the number of false witnesses for a composite number
Authors: Erdős Paul , Pomerance Carl
Published: June 2010

Ifais not a multiple ofnandan−1≢1modn{a^{n - 1}}\;\nequiv \;1\bmod \,n, thennmust be composite andais called a "witness" forn. LetF(n)F(n)denote the number of "false witnesses" forn, that is, the number ofamodna\bmod nwithan−1≡1modn{a^{n - 1}} \equiv 1\bmod n. Considered here is the normal and average size ofF(n)F(n)forncomposite. Also considered is the situation for the more stringent Euler and strong pseudoprime tests.
Weight distributions of some irreducible cyclic codes
Authors: Segal Robert , Ward Robert L.
Published: June 2010

The theory of weight distributions of irreducible cyclic codes over a finite field has been extensively developed by R. J. McEliece and others. We apply that theory to compute the weight enumerators of some binary codes which have hitherto not been possible. In so doing, we correct an error by McEliece and describe his process in somewhat more detail.
Artificial boundary conditions for the linear advection diffusion equation
Authors: Halpern Laurence
Published: June 2010

A family of artificial boundary conditions for the linear advection diffusion equation with small viscosity is developed. Well-posedness for the associated initial boundary value problem is analyzed. The error produced by truncating the domain is estimated. Numerical results are presented.
Intermediate boundary conditions for time-split methods applied to hyperbolic partial differential equations
Authors: LeVeque Randall J.
Published: June 2010

When time-split or fractional step methods are used to solve partial differential equations numerically, nonphysical intermediate solutions are introduced for which boundary data must often be specified. Here the appropriate boundary conditions are derived for splittings of hyperbolic problems into subproblems with disparate wave speeds. Numerical experiments are performed for the one-dimensional shallow water equations, a quasilinear system with inflow-outflow boundaries. Stability of the initial-boundary value problem is demonstrated for boundary conditions of the type derived here.
The construction of preconditioners for elliptic problems by substructuring. I
Authors: Bramble J. H. , Pasciak J. E. , Schatz A. H.
Published: June 2010

We consider the problem of solving the algebraic system of equations which arise from the discretization of symmetric elliptic boundary value problems via finite element methods. A new class of preconditioners for these discrete systems is developed based on substructuring (also known as domain decomposition). The resulting preconditioned algorithms are well suited to emerging parallel computing architectures. The proposed methods are applicable to problems on general domains involving differential operators with rather general coefficients. A basic theory for the analysis of the condition number of the preconditioned system (which determines the iterative convergence rate of the algorithm) is given. Techniques for applying the theory and algorithms to problems with irregular geometry are discussed and the results of extensive numerical experiments are reported.
Two formulas for the general multivariate polynomial which interpolates a regular grid on a simplex
Authors: Olmsted Coert
Published: June 2010

Two formulas are exhibited for the multivariate Lagrange shape polynomials which interpolate a regular grid on a simplex in

R

n

{\mathbb {R}^n}

.
Difference schemes for degenerate parabolic equations
Authors: Socolovsky E. A.
Published: June 2010

Diagonal dominant implicit-difference schemes approximating a porous media type class of multidimensional nonlinear equations are shown to generate semigroups in an approximate

L
1

{L^1}

-space, and the rate of convergence to the semigroup solution in

L
1

{L^1}

is given. The numerical schemes proposed by Berger et al. in [4] are described and a proof of convergence for the fully discrete algorithms is outlined. Numerical experiments are discussed.
A Shamanskiĭ-like acceleration scheme for nonlinear equations at singular roots
Authors: Kelley C. T.
Published: June 2010

A variation of the Shamanskii method is used to obtain a superlinearly convergent method for a class of nonlinear equations having singular Fréchet derivative at the root. The cost of a superlinear step is one derivative evaluation and two function evaluations.
Class groups of quadratic fields. II
Authors: Buell Duncan A.
Published: June 2010

A computation has been made of the noncyclic class groups of imaginary quadratic fields

Q
(

−
D

)

Q(\sqrt { - D} )

for even and odd discriminants

−
D

- D

from 0 to

−
25000000

- 25000000

. Among the results are that 95% of the class groups are cyclic, and that

−
11203620

- 11203620

and

−
18397407

- 18397407

are the first discriminants of imaginary quadratic fields for which the class group has rank three in the 5-Sylow subgroup. The latter was known to be of rank three; this computation demonstrates that it is the first odd discriminant of 5-rank three or more.
Primes at a glance
Authors: Guy R. K. , Lacampagne C. B. , Selfridge J. L.
Published: June 2010

Let

N
=
B
−
L

N = B - L

,

B
⩾

|

L

|

B \geqslant |L|

,

gcd
(
B
,
L
)
=
1

\gcd (B,L) = 1

,

p

|

B
L

p|BL

for all primes

p
⩽

N

p \leqslant \sqrt N

. Then N is 0, 1 or a prime. Writing N in this form suggests a primality and a squarefreeness test. If we also require that when the prime

q

|

B
L

q|BL

and

p
>
q

p > q

then

p

|

B
L

p|BL

, we say that

B
−
L

B - L

is a presentation of N. We list all presentations found for any N. We believe our list is complete.
Two-step methods and bi-orthogonality
Authors: Iserles A. , Nørsett S. P.
Published: June 2010

We study order and zero-stability of two-step methods of Obrechkoff type for ordinary differential equations. A relation between order and properties of mth degree polynomials orthogonal to

x

μ
i

{x^{{\mu _i}}}

,

1
⩽
i
⩽
m

1 \leqslant i \leqslant m

, where

−
1
>

μ
1

>

μ
2

>
⋯
>

μ
m

- 1 > {\mu _1} > {\mu _2} > \cdots > {\mu _m}

, is established. These polynomials are investigated, focusing on their explicit form, Rodrigues-type formulae and loci of their zeros.
Bi-cyclide and flat-ring cyclide coordinate surfaces: correction of two expressions
Authors: Kuchel Philip W. , Bulliman Brian T. , Fackerell Edward D.
Published: June 2010

Bi-cyclide and flat-ring cyclide coordinates are three-dimensional rotational coordinate systems based on conformal transformations using the Jacobian elliptic function sn. We have checked the previously published formulae of these systems (P. Moon and D. E. Spencer.Field Theory Handbook, Springer-Verlag, Berlin, 1971). In both cases the expression for the rotation-cyclide surfaces was incorrect: thus we present rederivations. The expressions were verified with the symbolic-algebraic computation package MACSYMA.
An optimal-order error estimate for the discontinuous Galerkin method
Authors: Richter Gerard R.
Published: June 2010

In this paper a new approach is developed for analyzing the discontinuous Galerkin method for hyperbolic equations. For a model problem in

R
2

{R^2}

, the method is shown to converge at a rate

O
(

h

n
+
1

)

O({h^{n + 1}})

when applied with nth degree polynomial approximations over a semiuniform triangulation, assuming sufficient regularity in the solution.
Weak uniform distribution for divisor functions. I
Authors: Rayner Francis J.
Published: June 2010

Narkiewicz (reference [3, pp. 204-205]) has proposed an algorithm for determining the moduli with respect to which a given arithmetic function (of suitable type) has weak uniform distribution. The class of functions to which this algorithm applies includes the divisor functions

σ
i

{\sigma _i}

. The present paper gives an improvement to the algorithm for odd values of i, which makes computation feasible for values of i up to 200. The results of calculations for odd values of i in the range

1
≤
i
≤
199

1 \leq i \leq 199

are reported.
Superconvergence of a collocation-type method for simple turning points of Hammerstein equations
Authors: Kumar Sunil
Published: June 2010

In this paper a simple turning point (

y
=

y
c

y = {y^c}

,

λ
=

λ
c

\lambda = {\lambda ^c}

) of the parameter-dependent Hammerstein equation
\[
<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 lamda integral Subscript a Superscript b Baseline k left-parenthesis t comma s right-parenthesis g left-parenthesis s comma y left-parenthesis s right-parenthesis right-parenthesis d s comma t element-of left-bracket a comma b right-bracket comma">

y
(
t
)
=
f
(
t
)
+
λ

∫
a
b

k
(
t
,
s
)
g
(
s
,
y
(
s
)
)

d
s
,

t
∈
[
a
,
b
]
,

y(t) = f(t) + \lambda \int _a^b {k(t,s)g(s,y(s))\;ds,\quad t \in [a,b],}

\]
is approximated numerically in the following way. A simple turning point (

z
=

z
c

z = {z^c}

,

λ
=

λ
c

\lambda = {\lambda ^c}

) of an equivalent equation for

z
(
t
)
:=
λ
g
(
t
,
y
(
t
)
)

z(t):=\lambda g(t,y(t))

is computed first. This is done by solving a discretized version of a certain system of equations which has (

z
c

{z^c}

,

λ
c

{\lambda ^c}

) as part of an isolated solution. The particular discretization used here is standard piecewise polynomial collocation. Finally, an approximation to

y
c

{y^c}

is obtained by use of the (exact) equation
\[
<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 a Superscript b Baseline k left-parenthesis t comma s right-parenthesis z left-parenthesis s right-parenthesis d s comma t element-of left-bracket a comma b right-bracket period">

y
(
t
)
=
f
(
t
)
+

∫
a
b

k
(
t
,
s
)
z
(
s
)

d
s
,

t
∈
[
a
,
b
]
.

y(t) = f(t) + \int _a^b {k(t,s)z(s)\;ds,\quad t \in [a,b].}

\]
The main result of the paper is that, under suitable conditions, the approximations to

y
c

{y^c}

and

λ
c

{\lambda ^c}

are both superconvergent, that is, they both converge to their respective exact values at a faster rate than the collocation approximation (of

z
c

{z^c}

) does to

z
c

{z^c}

.
On the enumeration of one-factorizations of complete graphs containing prescribed automorphism groups
Authors: Seah E. , Stinson D. R.
Published: June 2010

In this paper we use orderly algorithms to enumerate (perfect) one-factorizations of complete graphs, the automorphism groups of which contain certain prescribed subgroups. We showed that, for the complete graph

K

12

{K_{12}}

, excluding those one-factorizations containing exactly one automorphism of six disjoint cycles of length two, there are precisely 56391 nonisomorphic one-factorizations of

K

12

{K_{12}}

with nontrivial automorphism groups. We also determined that there are precisely 21 perfect one-factorizations of

K

14

{K_{14}}

that have nontrivial automorphism groups.
Some computational results on a problem concerning powerful numbers
Authors: Stephens A. J. , Williams H. C.
Published: June 2010

Let D be a positive square-free integer and let

X
+
Y

D

X + Y\sqrt D

be the fundamental unit in the order with Z-basis

{
1
,

D

}

\{ 1,\sqrt D \}

. An algorithm, which is of time complexity

O
(

D

1

/

4
+
ε

)

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

for any positive

ε
\varepsilon

, is developed for determining whether or not

D

|

Y

D|Y

. Results are presented for a computer run of this algorithm on all

D
>

10
8

D > {10^8}

. The conjecture of Ankeny, Artin and Chowla is verified for all primes

≡
1

(
mod

4
)

\equiv 1\,\pmod 4

less than

10
9

{10^9}

.
An iterative finite element method for approximating the biharmonic equation
Authors: Monk P. B.
Published: June 2010

A mixed finite element method for the biharmonic model of the simply supported and clamped plate is analyzed and error estimates are obtained. We show that the discrete problem may be solved efficiently by using the conjugate gradient method and a sequence of Dirichlet problems for Poisson’s equation.
An analysis of a uniformly convergent finite difference/finite element scheme for a model singular-perturbation problem
Authors: Gartland Eugene C.
Published: June 2010

Uniform

O

(

h
2

)

\mathcal {O}({h^2})

convergence is proved for the El-Mistikawy-Werle discretization of the problem

−
ε
u
+
a

u
′

+
b
u
=
f

- \varepsilon u+ au’+ bu = f

on (0,1),

u
(
0
)
=
A

u(0) = A

,

u
(
1
)
=
B

u(1) = B

, subject only to the conditions

a
,
b
,
f
∈

W

2
,
∞

[
0
,
1
]

a,b,f \in {\mathcal {W}^{2,\infty }}[0,1]

and

a
(
x
)
>
0
,
0
≤
x
≤
1

a(x) > 0, 0 \leq x \leq 1

. The principal tools used are a certain representation result for the solutions of such problems that is due to the author [Math. Comp., v. 48, 1987, pp. 551-564] and the general stability results of Niederdrenk and Yserentant [Numer. Math., v. 41, 1983, pp. 223-253]. Global uniform

O

(
h
)

\mathcal {O}(h)

convergence is proved under slightly weaker assumptions for an equivalent Petrov-Galerkin formulation.
On the convergence of collocation methods for Symm’s integral equation on open curves
Authors: Costabel M. , Ervin V. J. , Stephan E. P.
Published: June 2010

Recently, Costabel and Stephan in [8] presented convergence proofs for collocation with piecewise linear trial functions for Symm’s integral equation on plane closed curves with corners. In this article we prove the convergence of the above collocation method in the case of open curves. We derive asymptotic error estimates in Sobolev norms and analyze the effect of graded meshes. Numerical experiments based on the implementation of [6] show experimental orders of convergence which confirm our theoretical results on the asymptotic rates of convergence.
The Stieltjes function—definition and properties
Authors: Bohman Jan , Fröberg Carl-Erik
Published: June 2010

Close to the singular point

s
=
1

s = 1

the zeta function can be represented as a Laurent series in

(
s
−
1
)

(s - 1)

. The coefficients in this series are called the Stieltjes constants, and the first ones were computed already 100 years ago. In order to investigate their somewhat unexpected behavior we have defined a related function which we call the Stieltjes function, and examined its properties.
A computer algorithm for determining the Hausdorff dimension of certain fractals
Authors: Garnett Lucy
Published: June 2010

A fractal is a set which has nonintegral Hausdorff dimension. Computation of the dimension directly from the definition would be very time-consuming on a computer. However, the dimension can be computed using Newton’s method if there exists a self-expanding map on the set. This technique is applied to compute the dimension of the Julia set of the quadratic mapping

z
→

z
2

+
c

z \to {z^2} + c

for small real values of c.
On prime divisors of binomial coefficients
Authors: Goetgheluck Pierre
Published: June 2010

This paper, using computational and theoretical methods, deals with prime divisors of binomial coefficients: Geometric distribution and number of distinct prime divisors are studied. We give a numerical result on a conjecture by Erdős on square divisors of binomial coefficients.
Polynomials with Galois groups 𝐴𝑢𝑡(𝑀₂₂),𝑀₂₂, and 𝑃𝑆𝐿₃(𝐹₄)⋅2₂ over 𝑄
Authors: Malle Gunter
Published: June 2010

In this paper the construction of infinite families of polynomials with Galois groups

Aut
⁡
(

M

22

)

\operatorname {Aut} ({M_{22}})

,

M

22

{M_{22}}

and

PSL
3

(

F

4

)

⋅

2

{\text {PSL}_3}({\mathbb {F}_4})\;\cdot \;2

over

Q

\mathbb {Q}

is achieved. The determination of these polynomials leads to a system of nonlinear algebraic equations in 22 unknowns. The solutions belonging to the Galois extensions with the desired Galois groups are computed with a p-modular version of the Buchberger algorithm. The application of this method, which is described in some detail, turns out to be feasible even for relatively large systems of nonlinear equations.
Parameter determination for complex number-theoretic transforms using cyclotomic polynomials
Authors: Creutzburg R. , Tasche M.
Published: June 2010

Some new results for finding all convenient moduli m for a complex number-theoretic transform with given transform length n and given primitive nth root of unity modulo m are presented. The main result is based on the prime factorization for values of cyclotomic polynomials in the ring of Gaussian integers.
On some numerical schemes for transonic flow problems
Authors: Mostrel Marco Mosché
Published: June 2010

New second-order accurate finite difference approximations for a class of nonlinear PDE’s of mixed type, which includes the 2D Low Frequency Transonic Small Disturbance equation (TSD) and the 2D Full Potential equation (FP), are presented. For the TSD equation, the scheme is implemented via a time splitting algorithm; the inclusion of flux limiters keeps the total variation nonincreasing and eliminates spurious oscillations near shocks. Global Linear Stability, Total Variation Diminishing and Entropy Stability results are proven. Numerical results for the flow over a thin airfoil are presented. Current techniques used to solve the TSD equation may easily be extended to second-order accuracy by this method. For the FP equation, the new scheme requires no subsonic/supersonic switching and no numerical flux biasing. Global Linear Stability for all values of the Mach number is proven.
Nonnegativity-, monotonicity-, or convexity-preserving cubic and quintic Hermite interpolation
Authors: Dougherty Randall L. , Edelman Alan S. , Hyman James M.
Published: June 2010

The Hermite polynomials are simple, effective interpolants of discrete data. These interpolants can preserve local positivity, monotonicity, and convexity of the data if we restrict their derivatives to satisfy constraints at the data points. This paper describes the conditions that must be satisfied for cubic and quintic Hermite interpolants to preserve these properties when they exist in the discrete data. We construct algorithms to ensure that these constraints are satisfied and give numerical examples to illustrate the effectiveness of the algorithms on locally smooth and rough data.
Convergence of a random particle method to solutions of the Kolmogorov equation 𝑢_{𝑡}=𝜈𝑢ₓₓ+𝑢(1-𝑢)
Authors: Puckett Elbridge Gerry
Published: June 2010

We study a random particle method for solving the reaction-diffusion equation

u
t

=
ν

u

x
x

+
f
(
u
)

{u_t} = \nu {u_{xx}} + f(u)

which is a one-dimensional analogue of the random vortex method. It is a fractional step method in which

u
t

=
ν

u

x
x

{u_t} = \nu {u_{xx}}

is solved by random walking the particles while

u
t

=
f
(
u
)

{u_t} = f(u)

is solved with a numerical ordinary differential equation solver such as Euler’s method. We prove that the method converges when

f
(
u
)
=
u
(
1
−
u
)

f(u) = u(1 - u)

, i.e. the Kolmogorov equation, and that when the time step

Δ
t

\Delta t

is

O
(

N
4

−
1

)

O({\sqrt [4]{N}^{ - 1}})

the rate of convergence is like

ln
⁡
N
⋅

N
4

−
1

\ln N \cdot \,{\sqrt [4]{N}^{ - 1}}

where N denotes the number of particles. Furthermore, we show that this rate of convergence is uniform as the diffusion coefficient

ν
\nu

tends to 0. Thus, travelling waves with arbitrarily steep wavefronts may be modeled without an increase in the computational cost. We also present the results of numerical experiments including the use of second-order time discretization and second-order operator splitting and use these results to estimate the expected value and standard deviation of the error.
Newton interpolation in Fejér and Chebyshev points
Authors: Fischer Bernd , Reichel Lothar
Published: June 2010

Let

Γ
\Gamma

be a Jordan curve in the complex plane, and let

Ω
\Omega

be the compact set bounded by

Γ
\Gamma

. Let f denote a function analytic on

Ω
\Omega

. We consider the approximation of f on

Ω
\Omega

by a polynomial p of degree less than n that interpolates f in n points on

Γ
\Gamma

. A convenient way to compute such a polynomial is provided by the Newton interpolation formula. This formula allows the addition of one interpolation point at a time until an interpolation polynomial p is obtained which approximates f sufficiently accurately. We choose the sets of interpolation points to be subsets of sets of Fejér points. The interpolation points are ordered using van der Corput’s sequence, which ensures that p converges uniformly and maximally to f on

Ω
\Omega

as n increases. We show that p is fairly insensitive to perturbations of f if

Γ
\Gamma

is smooth and is scaled to have capacity one. If

Γ
\Gamma

is an interval, then the Fejér points become Chebyshev points. This special case is also considered. A further application of the interpolation scheme is the computation of an analytic continuation of f in the exterior of

Γ
\Gamma

.
Numerical approximation of a wave equation with unilateral constraints
Authors: Schatzman Michelle , Bercovier Michel
Published: June 2010

The system

u

t
t

−

u

x
x

∍
f

{u_{tt}} - {u_{xx}} \backepsilon f

,

x
∈
(
0
,
L
)
×
(
0
,
T
)

x \in (0,L) \times (0,T)

, with initial data

u
(
x
,
0
)
=

u
0

(
x
)

u(x,0) = {u_0}(x)

,

u
t

(
x
,
0
)
=

u
1

(
x
)

{u_t}(x,0) = {u_1}(x)

almost everywhere on (0, L) and boundary conditions

u
(
0
,
t
)
=
0

u(0,t) = 0

, for all

t
≥
0

t \geq 0

, and the unilateral condition
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript x Baseline left-parenthesis upper L comma t right-parenthesis greater-than-or-equal-to 0 comma u left-parenthesis upper L comma t right-parenthesis greater-than-or-equal-to k 0 comma left-parenthesis u left-parenthesis upper L comma t right-parenthesis minus k 0 right-parenthesis u Subscript x Baseline left-parenthesis upper L comma t right-parenthesis equals 0">

u
x

(
L
,
t
)
≥
0
,
u
(
L
,
t
)
≥

k
0

,
(
u
(
L
,
t
)
−

k
0

)

u
x

(
L
,
t
)
=
0

{u_x}(L,t) \geq 0,u(L,t) \geq {k_0},(u(L,t) - {k_0}){u_x}(L,t) = 0

\]
models the longitudinal vibrations of a rod, whose motion is limited by a rigid obstacle at one end. A new variational formulation is given; existence and uniqueness are proved. Finite elements and finite difference schemes are given, and their convergence is proved. Numerical experiments are reported; the characteristic schemes perform better in terms of accuracy, and the subcharacteristic schemes look better.
Algorithms for piecewise polynomials and splines with free knots
Authors: Meinardus G. , Nürnberger G. , Sommer M. , Strauss H.
Published: June 2010

We describe an algorithm for computing points

a
=

x
0

>

x
1

>
⋯
>

x
k

>

x

k
+
1

=
b

a = {x_0} > {x_1} > \cdots > {x_k} > {x_{k + 1}} = b

which solve certain nonlinear systems

d
(

x

i
−
1

,

x
i

)
=
d
(

x
i

,

x

i
+
1

)

d({x_{i - 1}},{x_i}) = d({x_i},{x_{i + 1}})

,

i
=
1
,
…
,
k

i = 1, \ldots ,k

. In contrast to Newton-type methods, the algorithm converges when starting with arbitrary points. The method is applied to compute best piecewise polynomial approximations with free knots. The advantage is that in the starting phase only simple expressions have to be evaluated instead of computing best polynomial approximations. We finally discuss the relation to the computation of good spline approximations with free knots.
Error estimates for semidiscrete finite element methods for parabolic integro-differential equations
Authors: Thomée Vidar , Zhang Nai Ying
Published: June 2010

The purpose of this paper is to attempt to carry over known results for spatially discrete finite element methods for linear parabolic equations to integro-differential equations of parabolic type with an integral kernel consisting of a partial differential operator of order

β
≤
2

\beta \leq 2

. It is shown first that this is possible without restrictions when the exact solution is smooth. In the case of a homogeneous equation with nonsmooth initial data

v
,
v
∈

L
2

v,v \in {L_2}

, optimal

O
(

h
r

)

O({h^r})

convergence for positive time is possible in general only if

r
≤
4
−
β

r \leq 4 - \beta

. This depends on the fact that the exact solution is then only in

H

4
−
β

{H^{4 - \beta }}

.
Computation of independent units in number fields by Dirichlet’s method
Authors: Buchmann Johannes , Pethő Attila
Published: June 2010

Using the basis reduction algorithm of A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovász [8] and an idea of Buchmann [4], we describe a method for computing maximal systems of independent units in arbitrary number fields. The tables in the supplements section display such systems for the fields

Q

(

D
n

)

{\mathbf {Q}}(\sqrt [n]{D})

where

6
≤
n
≤
11

6 \leq n \leq 11

.
Noninterpolatory integration rules for Cauchy principal value integrals
Authors: Rabinowitz P. , Lubinsky D. S.
Published: June 2010

Let

w
(
x
)

w(x)

be an admissible weight on

[
−
1
,
1
]

[ - 1,1]

and let

{

p
n

(
x
)

}
0
∞

\{ {p_n}(x)\} _0^\infty

be its associated sequence of orthonormal polynomials. We study the convergence of noninterpolatory integration rules for approximating Cauchy principal value integrals
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis f semicolon lamda right-parenthesis colon equals contour-integral w left-parenthesis x right-parenthesis StartFraction f left-parenthesis x right-parenthesis Over x minus lamda EndFraction d x comma lamda element-of left-parenthesis negative 1 comma 1 right-parenthesis period">

I
(
f
;
λ
)
:=
∮
w
(
x
)

f
(
x
)

x
−
λ

d
x
,

λ
∈
(
−
1
,
1
)
.

I(f;\lambda ):=\oint w(x)\frac {{f(x)}}{{x - \lambda }}\,dx,\quad \lambda \in ( - 1,1).

\]
This requires investigation of the convergence of the expansion
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis f semicolon lamda right-parenthesis tilde sigma-summation Underscript k equals 0 Overscript normal infinity Endscripts left-parenthesis f comma p Subscript k Baseline right-parenthesis q Subscript k Baseline left-parenthesis lamda right-parenthesis comma lamda element-of left-parenthesis negative 1 comma 1 right-parenthesis comma">

I
(
f
;
λ
)
∼

∑

k
=
0

∞

(
f
,

p
k

)

q
k

(
λ
)
,

λ
∈
(
−
1
,
1
)
,

I(f;\lambda ) \sim \sum \limits _{k = 0}^\infty {(f,{p_k}){q_k}(\lambda ),\quad \lambda \in ( - 1,1),}

\]
in terms of the functions of the second kind

{

q
k

(
λ
)

}
0
∞

\{ {q_k}(\lambda )\} _0^\infty

associated with w, where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis f comma p Subscript k Baseline right-parenthesis colon equals integral Subscript negative 1 Superscript 1 Baseline w left-parenthesis x right-parenthesis f left-parenthesis x right-parenthesis p Subscript k Baseline left-parenthesis x right-parenthesis d x and q Subscript k Baseline left-parenthesis lamda right-parenthesis colon equals contour-integral w left-parenthesis x right-parenthesis StartFraction p Subscript k Baseline left-parenthesis x right-parenthesis Over x minus lamda EndFraction d x comma">

(
f
,

p
k

)
:=

∫

−
1

1

w
(
x
)
f
(
x
)

p
k

(
x
)

d
x

and

q
k

(
λ

)
:=
∮
w
(
x
)

p
k

(
x
)

x
−
λ

d
x
,

(f,{p_k}):=\int _{ - 1}^1 {w(x)f(x){p_k}(x)\,dx\quad {\text {and}}\quad {q_k}(\lambda } ):=\oint w(x)\frac {{{p_k}(x)}}{{x - \lambda }}\,dx,

\]

k
=
0
,
1
,
2
,
…
,
λ
∈
(
−
1
,
1
)

k = 0,1,2, \ldots ,\lambda \in ( - 1,1)

.