bluuebunny/crossref_metadata_2025 huggingface dataset:
Some inequalities for elementary mean values
Authors: Meyer Burnett
Published: June 2010

Upper and lower bounds for the difference between the arithmetic and harmonic means of n positive numbers are obtained in terms of n and the largest and smallest of the numbers. Also, results of S. H. Tung [2], are used to obtain upper and lower bounds for the elementary mean values

M
p

{M_p}

of Hardy, Littlewood, and Pólya.
Explicit estimates for 𝜃(𝑥;3,𝑙) and 𝜓(𝑥;3,𝑙)
Authors: McCurley Kevin S.
Published: June 2010

Let

θ
(
x
;
3
,
l
)

\theta (x;3,l)

be the sum of the logarithms of the primes not exceeding x that are congruent to l modulo 3, where l is 1 or 2. By the prime number theorem for arithmetic progressions,

θ
(
x
;
3
,
l
)
∼
x

/

2

\theta (x;3,l) \sim x/2

as

x
→
∞

x \to \infty

. Using information concerning zeros of Dirichlet L-functions, we prove explicit numerical bounds for

θ
(
x
;
3
,
l
)

\theta (x;3,l)

of the form

|

θ
(
x
;
3
,
l
)
−
x

/

2

|

>
e
x

|\theta (x;3,l) - x/2| > ex

,

x
⩾

x
0

(
ε
)

x \geqslant {x_0}(\varepsilon )

.
Accurate approximation of eigenvalues and zeros of selected eigenfunctions of regular Sturm-Liouville problems
Authors: Gartland Eugene C.
Published: June 2010

A method for simultaneously approximating to high accuracy the corresponding eigenvalue and zeros of the

(
n
+
1
)

(n + 1)

st eigenfunction of a regular Sturm-Liouville eigenvalue problem is presented. It is based upon equilibrating the minimum eigenvalues of several problems on subintervals that form a partition of the orginal interval. The method is easily derived from classical mini-max variational principles. The equilibration is accomplished iteratively using an approximate Newton Method. Numerical results are given.
The convergence rate of a multigrid method with Gauss-Seidel relaxation for the Poisson equation
Authors: Braess Dietrich
Published: June 2010

The convergence rate of a multigrid method for the numerical solution of the Poisson equation on a uniform grid is estimated. The results are independent of the shape of the domain as long as it is convex and polygonal. On the other hand, pollution effects become apparent when the domain contains reentrant corners. To estimate the smoothing of the Gauss-Seidel relaxation, the smoothness is measured by comparing the energy norm with a (weaker) discrete seminorm.
Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. I. The continuous problem
Authors: Bendali A.
Published: June 2010

A general finite element method is applied to compute the skin currents flowing on a perfectly conducting surface when it is illuminated by a time-harmonic incident electromagnetic wave. In this paper, we introduce and study the framework in which the continuous problem can be stated in order to make possible the numerical analysis which will follow in a second part.
Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. II. The discrete problem
Authors: Bendali A.
Published: June 2010

With the help of curved and mixed finite elements, we introduce an approximate surface on which the discrete problem is defined and construct surface currents and charges which approximate the solution of the continuous problem studied in a previous part. We study the existence and uniqueness of the solution of the discrete problem and give estimates for the error between currents, charges, corresponding fields and their calculated approximations.
On the congruence 2ⁿ⁻²≡1(𝑚𝑜𝑑𝑛)
Authors: Rotkiewicz A.
Published: June 2010

There exist infinitely many positive integers n such that

2

n
−
2

≡
1

(
mod

n
)

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

.
Best approximation of positive power series
Authors: Shawyer B. L. R.
Published: June 2010

This paper extends work of Fiedler, Jurkat and the present author to series of the form

Σ

a
n

x
n

\Sigma \,{a_n}{x^n}

where

{

a
n

}

\{ {a_n}\}

is a moment sequence and

0
>
x
>
1

0 > x > 1

. In the cases where it is possible to calculate it exactly, we find the best

L
p

{L^p}

approximation to the sum of the series and the actual terms of the matrices involved. We have an advantage over accelerators commonly used for accelerating convergence in that we know explicitly the errors in our calculations.
Pisot numbers in the neighborhood of a limit point. II
Authors: Boyd David W.
Published: June 2010

Let S denote the set of real algebraic integers greater than one, all of whose other conjugates lie within the unit circle. In an earlier paper, we introduced the notion of "width" of a limit point

α
\alpha

of S and showed that, if the width of

α
\alpha

is smaller than 1.28... then there is an algorithm for determining all members of S in a neighborhood of

α
\alpha

. Recently, we introduced the "derived tree" in order to deal with limit points of greater width. Here, we apply these ideas to the study of the limit point

α
3

{\alpha _3}

, the zero of

z
4

−
2

z
3

+
z
−
1

{z^4} - 2{z^3} + z - 1

outside the unit circle. We determine the smallest neighborhood

θ
1

>

α
3

>

θ
2

{\theta _1} > {\alpha _3} > {\theta _2}

of

α
3

{\alpha _3}

in which all elements of S other than

α
3

{\alpha _3}

satisfy one of the equations

z
n

(

z
4

−
2

z
3

+
z
−
1
)
±
A
(
z
)
=
0

{z^n}({z^4} - 2{z^3} + z - 1) \pm A(z) = 0

, where

A
(
z
)

A(z)

is one of

z
3

−

z
2

+
1

{z^3} - {z^2} + 1

,

z
3

−
z
+
1

{z^3} - z + 1

or

z
4

−

z
3

+
z
−
1

{z^4} - {z^3} + z - 1

. The endpoints

θ
1

{\theta _1}

, and

θ
2

{\theta _2}

are elements of S of degrees 23 and 42, respectively.
Least squares methods for elliptic systems
Authors: Aziz A. K. , Kellogg R. B. , Stephens A. B.
Published: June 2010

A weighted least squares method is given for the numerical solution of elliptic partial differential equations of Agmon-Douglis-Nirenberg type and an error analysis is provided. Some examples are given.
Difference methods for problems with different time scales
Authors: Scheid Robert E.
Published: June 2010

We consider the use of difference methods for weakly nonlinear systems of ordinary differential equations with rapidly oscillating solutions and develop a general approach which depends only on the smoothness of the coefficients and the nonlinearities. In particular, one is led to a strategy which is suitable for the detection and resolution of turning points in such systems. A computational example is presented.
Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems
Authors: Goldberg Moshe , Tadmor Eitan
Published: June 2010

New convenient stability criteria are provided in this paper for a large class of finite-difference approximations to initial-boundary value problems associated with the hyperbolic system

u

t

=
A

u

x

+
B

u

+

f

{{\mathbf {u}}_t} = A{{\mathbf {u}}_x} + B{\mathbf {u}} + {\mathbf {f}}

in the quarter plane

x
⩾
0

x \geqslant 0

,

t
⩾
0

t \geqslant 0

. Using the new criteria, stability is easily established for numerous combinations of well-known basic schemes and boundary conditions, thus generalizing many special cases studied in the recent literature.
The convergence of Galerkin approximation schemes for second-order hyperbolic equations with dissipation
Authors: Kok Barbara , Geveci Tunc
Published: June 2010

In this paper we consider certain semidiscrete and fully discrete Galerkin approximations to the solution of an initial-boundary value problem for a second-order hyperbolic equation with a dissipative term. Estimates are obtained in the energy and negative norms associated with the problem, yielding in particular

H
1

{H^1}

- and

L
2

{L^2}

-error estimates. The approximation to the initial data is taken, in this case, as the projection with respect to the energy inner product, onto the approximating space. We also obtain estimates for higher-order time derivatives.
On the differential-difference properties of the extended Jacobi polynomials
Authors: Lewanowicz S.
Published: June 2010

We discuss differential-difference properties of the extended Jacobi polynomials
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript n Baseline left-parenthesis x right-parenthesis equals Subscript p plus 2 Baseline upper F Subscript q Baseline left-parenthesis negative n comma n plus lamda comma a Subscript p Baseline semicolon b Subscript q Baseline semicolon x right-parenthesis left-parenthesis n equals 0 comma 1 comma ellipsis right-parenthesis period">

P
n

(
x
)

=

p
+
2

F
q

(
−
n
,
n
+
λ
,

a
p

;

b
q

;
x
)

(
n
=
0
,
1
,
…
)
.

{P_n}(x){ = _{p + 2}}{F_q}( - n,n + \lambda ,{a_p};{b_q};x)\quad (n = 0,1, \ldots ).

\]
The point of departure is a corrected and reformulated version of a differential-difference equation satisfied by the polynomials

P
n

(
x
)

{P_n}(x)

, which was derived by Wimp (Math. Comp., v. 29, 1975, pp. 577-581).
Computing 𝜋(𝑥): the Meissel-Lehmer method
Authors: Lagarias J. C. , Miller V. S. , Odlyzko A. M.
Published: June 2010

E. D. F. Meissel, a German astronomer, found in the 1870’s a method for computing individual values of

π
(
x
)

\pi (x)

, the counting function for the number of primes

⩽
x

\leqslant x

. His method was based on recurrences for partial sieving functions, and he used it to compute

π
(

10
9

)

\pi ({10^9})

. D. H. Lehmer simplified and extended Meissel’s method. We present further refinements of the Meissel-Lehmer method which incorporate some new sieving techniques. We give an asymptotic running time analysis of the resulting algorithm, showing that for every

ε
>
0

\varepsilon > 0

it computes

π
(
x
)

\pi (x)

using at most

O
(

x

2

/

3
+
ε

)

O({x^{2/3 + \varepsilon }})

arithmetic operations and using at most

O
(

x

1

/

3
+
ε

)

O({x^{1/3 + \varepsilon }})

storage locations on a Random Access Machine (RAM) using words of length

[

log
2

x
]
+
1

[{\log _2}x] + 1

bits. The algorithm can be further speeded up using parallel processors. We show that there is an algorithm which, when given M RAM parallel processors, computes

π
(
x
)

\pi (x)

in time at most

O
(

M

−
1

x

2

/

3
+
ε

)

O({M^{ - 1}}{x^{2/3 + \varepsilon }})

using at most

O
(

x

1

/

3
+
ε

)

O({x^{1/3 + \varepsilon }})

storage locations on each parallel processor, provided

M
⩽

x

1

/

3

M \leqslant {x^{1/3}}

. A variant of the algorithm was implemented and used to compute

π
(
4
×

10

16

)

\pi (4 \times {10^{16}})

.
Cosine methods for second-order hyperbolic equations with time-dependent coefficients
Authors: Bales Laurence A. , Dougalis Vassilios A. , Serbin Steven M.
Published: June 2010

We analyze efficient, high-order accurate methods for the approximation of the solutions of linear, second-order hyperbolic equations with time-dependent coefficients. The methods are based on Galerkin-type discretizations in space and on a class of fourth-order accurate, two-step, cosine time-stepping schemes. Preconditioned iterative techniques are used to solve linear systems with the same operator at each time step. The schemes are supplemented by single-step high-order starting procedures and need no evaluations of derivatives of operators.

L
2

{L^2}

-optimal error estimates are proved throughout.
High-order local rate of convergence by mesh-refinement in the finite element method
Authors: Eriksson Kenneth
Published: June 2010

We seek approximations of the solution u of the Neumann problem for the equation

L
u
=
f

Lu = f

in

Ω
\Omega

with special emphasis on high-order accuracy at a given point

x
0

∈

Ω
¯

{x_0} \in \bar \Omega

. Here

Ω
\Omega

is a bounded domain in

R
N

(
N
⩾
2
)

{R^N}(N \geqslant 2)

with smooth boundary, and L is a second-order, uniformly elliptic, differential operator with smooth coefficients. An approximate solution

u
h

{u_h}

is determined by the standard Galerkin method in a space of continuous piecewise polynomials of degree at most

r
−
1

r - 1

on a partition

Δ
h

(

x
0

,
α
)

{\Delta _h}({x_0},\alpha )

of

Ω
\Omega

. Here h is a global mesh-size parameter, and

α
\alpha

is the degree of a certain systematic refinement of the mesh around the given point

x
0

{x_0}

, where larger

α
\alpha

’s mean finer mesh, and

α
=
0

\alpha = 0

corresponds to the quasi-uniform case with no refinement. It is proved that, for suitable (sufficiently large)

α
\alpha

’s the high-order error estimate

(
u
−

u
h

)
(

x
0

)
=
O
(

h

2
r
−
2

)

(u - {u_h})({x_0}) = O({h^{2r - 2}})

holds. A corresponding estimate with the same order of convergence is obtained for the first-order derivatives of

u
−

u
h

u - {u_h}

. These estimates are sharp in the sense that the required degree of refinement in each case is essentially the same as is needed for the local approximation to this order near

x
0

{x_0}

. For the estimates to hold, it is sufficient that the exact solution u have derivatives to the rth order which are bounded close to

x
0

{x_0}

and square integrable in the rest of

Ω
\Omega

. The proof of this uses high-order negative-norm estimates of

u
−

u
h

u - {u_h}

. The number of elements in the considered partitions is of the same order as in the corresponding quasi-uniform ones. Applications of the results to other types of boundary value problems are indicated.
On Shapiro’s cyclic inequality for 𝑁=13
Authors: Troesch B. A.
Published: June 2010

A cyclic sum

S
(

x

)
=
Σ

x
i

/

(

x

i
+
1

+

x

i
+
2

)

S({\mathbf {x}}) = \Sigma \;{x_i}/({x_{i + 1}} + {x_{i + 2}})

is formed with the N components of a vector x, where

x

N
+
1

=

x
1

{x_{N + 1}} = {x_1}

,

x

N
+
2

=

x
2

{x_{N + 2}} = {x_2}

, and where all denominators are positive and all numerators are nonnegative. It is known that there exist vectors x for which

S
(

x

)
>
N

/

2

S({\mathbf {x}}) > N/2

if

N
⩾
14

N \geqslant 14

and even, and if

N
⩾
25

N \geqslant 25

. It has been proved that the inequality

S
(

x

)
⩾
N

/

2

S({\mathbf {x}}) \geqslant N/2

holds for

N
⩽
12

N \leqslant 12

. Although it has been conjectured repeatedly that the inequality also holds for odd N between 13 and 23. this has apparently not yet been proved. Here we will confirm that the inequality indeed holds for

N
=
13

N = 13

.
Finite difference approximations of generalized solutions
Authors: Süli Endre , Jovanović Boško , Ivanović Lav
Published: June 2010

We consider finite difference schemes approximating the Dirichlet problem for the Poisson equation. We provide scales of error estimates in discrete Sobolev-like norms assuming that the generalized solution belongs to a nonnegative order Sobolev space.
Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems
Authors: Spijker M. N.
Published: June 2010

This paper deals with the analysis of general one-step methods for the numerical solution of initial (-boundary) value problems for stiff ordinary and partial differential equations. Restrictions on the stepsize are derived that are necessary and sufficient for the rate of error growth in these methods to be of moderate size. These restrictions are related to disks contained in the stability region of the method, and the errors are measured with arbitrary norms (not necessarily generated by an inner product). The theory is illustrated in the numerical solution of a diffusion-convection problem where the error growth is measured with the maximum norm.
The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes
Authors: Brunner Hermann
Published: June 2010

Since the solution of a second-kind Volterra integral equation with weakly singular kernel has, in general, unbounded derivatives at the left endpoint of the interval of integration, its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates. In this paper we investigate the convergence rates with respect to graded meshes, and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation.
Estimation of the error in the reduced basis method solution of nonlinear equations
Authors: Porsching T. A.
Published: June 2010

The reduced basis method is a projection technique for approximating the solution curve of a finite system of nonlinear algebraic equations by the solution curve of a related system that is typically of much lower dimension. In this paper, the reduced basis error is shown to be dominated by an approximation error. This, in turn, leads to error estimates for projection onto specific subspaces; for example, subspaces related to Taylor, Lagrange and discrete least-squares approximation.
A note on asymptotic evaluation of some Hankel transforms
Authors: Frenzen C. L. , Wong R.
Published: June 2010

Asymptotic behavior of the integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript f Baseline left-parenthesis w right-parenthesis equals integral Subscript 0 Superscript normal infinity Baseline e Superscript minus x squared Baseline upper J 0 left-parenthesis w x right-parenthesis f left-parenthesis x squared right-parenthesis x d x">

I
f

(
w
)
=

∫
0
∞

e

−

x
2

J
0

(
w
x
)
f
(

x
2

)

x

d
x

{I_f}(w) = \int _0^\infty {{e^{ - {x^2}}}{J_0}(wx)f({x^2})\;x\;dx}

\]
is investigated, where

J
0

(
x
)

{J_0}(x)

is the Bessel function of the first kind and w is a large positive parameter. It is shown that

I
f

(
w
)

{I_f}(w)

decays exponentially like

e

−
γ

w
2

{e^{ - \gamma {w^2}}}

,

γ
>
0

\gamma > 0

, when

f
(
z
)

f(z)

is an entire function subject to a suitable growth condition. A complete asymptotic expansion is obtained when

f
(
z
)

f(z)

is a meromorphic function satisfying the same growth condition. Similar results are given when

f
(
z
)

f(z)

has some specific branch point singularities.
Convergence of upwind schemes for a stationary shock
Authors: Lorenz Jens
Published: June 2010

A nonlinear first-order boundary value problem with discontinuous solutions is considered. It arises in the study of gasflow through a duct and allows, in general, for multiple solutions. New convergence results for three difference schemes are presented and the sharpness of numerical layers is established. For the EO-scheme, stability of a physically correct solution with respect to time evolution is shown.
A class of cubic splines obtained through minimum conditions
Authors: Bini D. , Capovani M.
Published: June 2010

A class of cubic spline minimizing some special functional is investigated. This class is determined by the solution of a quadratic programming problem in which the minimizing function depends linearly on a parameter

α
>
2

\alpha > 2

. For

α
=
1

/

2

\alpha = 1/2

natural splines are obtained. For

α
=
−
1

\alpha = - 1

the spline minimizing the mean value of the third derivative is obtained. It is shown that this spline has the best convergence order.
Computing when multiplications cost nothing
Authors: Newman D. J.
Published: June 2010

A (rather strange) computer is considered which costs 1 cent to perform each addition but costs nothing to perform a multiplication. It is shown that the addition chain from 1 to n cost maximally

(
Log
⁡
n

)

1

/

2
+
o
(
1
)

{(\operatorname {Log} n)^{1/2 + o(1)}}

rather than the classical

∼
Log
⁡
n

\sim \operatorname {Log} n

.
A note on class-number one in certain real quadratic and pure cubic fields
Authors: Tennenhouse M. , Williams H. C.
Published: June 2010

Let p be any odd prime and let

h
(
p
)

h(p)

be the class number of the real quadratic field

Q

(

p

)

\mathcal {Q}(\sqrt p )

. The results of a computer run to determine the density of the field

Q

(

p

)

\mathcal {Q}(\sqrt p )

with

h
(
p
)
=
1

h(p) = 1

and

p
>

10
8

p > {10^8}

are presented. Similar results are given for pure cubic fields

Q

(

p
3

)

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

with

p
>

10
6

p > {10^6}

.
The fast adaptive composite grid (FAC) method for elliptic equations
Authors: McCormick S. , Thomas J.
Published: June 2010

The fast adaptive composite grid (FAC) method is a systematic process for solving differential boundary value problems. FAC uses global and local uniform grids both to define the composite grid problem and to interact for its fast solution. It can with little added cost substantially improve accuracy of the coarse grid solution and is very suitable for vector and parallel computation. This paper develops both the theoretical and practical aspects of FAC as it applies to elliptic problems.
On two methods for approximating minimal surfaces in parametric form
Authors: Tsuchiya Takuya
Published: June 2010

Two methods for approximating minimal surfaces in parametric form are considered. One minimizes the area of the surface, and the other the energy of the surface. The convergence of the algorithm of the first method is proved. The application of the second method to the approximation of conformal maps is examined. Several examples of computations are given.
Convex interpolation by splines of arbitrary degree
Authors: Mettke Holger
Published: June 2010

An algorithm is described for computing an interpolation spline of arbitrary but fixed degree which preserves the convexity of the given data set. Necessary and sufficient conditions for the solvability of the problem, some special cases and error estimations are given.
Effective irrationality measures for certain algebraic numbers
Authors: Easton David
Published: June 2010

A result of Chudnovsky concerning rational approximation to certain algebraic numbers is reworked to provide a quantitative result in which all constants are explicitly given. More particularly, Padé approximants to the function

(
1
−
x

)

1

/

3

{(1 - x)^{1/3}}

are employed to show, for certain integers a and b, that
\[

|

(
a

/

b
)

1

/

3

−
p

/

q

|

>
c

q

−
κ

when

q
>
0.

\left | {{{(a/b)}^{1/3}} - p/q} \right | > c{q^{ - \kappa }}\quad {\text {when}}\,q > 0.

\]
Here, c and k are given as functions of a and b only.
On nonlocal monotone difference schemes for scalar conservation laws
Authors: Lucier Bradley J.
Published: June 2010

We provide error analyses for explicit, implicit, and semi-implicit monotone finite-difference schemes on uniform meshes with nonlocal numerical fluxes. We are motivated by finite-difference discretizations of certain long-wave (Sobolev) regularizations of the conservation laws that explicitly add a dispersive term as well as a nonlinear dissipative term. We also develop certain relationships between dispersion and stability in finite-difference schemes. Specifically, we find that discretization and explicit dispersion have identical effects on the amount of artificial dissipation necessary for stability.
A modification of the Weeks method for numerical inversion of the Laplace transform
Authors: Lyness J. N. , Giunta G.
Published: June 2010

The Weeks method for calculating numerical values of

f
(
t
)

f(t)

from its Laplace transform

F
(
s
)

F(s)

involves the numerical evaluation of a set of contour integrals. In this paper, we describe a powerful modification, by means of which results of comparable accuracy can be obtained using approximately half the number of complex function values. This and other possible minor modifications are discussed.
Computation of all the amicable pairs below 10¹⁰
Authors: te Riele H. J. J.
Published: June 2010

An efficient exhaustive numerical search method for amicable pairs is described. With the aid of this method all 1427 amicable pairs with smaller member below

10

10

{10^{10}}

have been computed, more than 800 pairs being new. This extends previous exhaustive work below

10
8

{10^8}

by H. Cohen. In three appendices (contained in the supplements section of this issue), various statistics are given, including an ordered list of all the gcd’s of the 1427 amicable pairs below

10

10

{10^{10}}

(which may be useful in further amicable pair research). Suggested by the numerical results, a theorem of Borho and Hoffmann for constructing APs has been extended.
Construction of variable-stepsize multistep formulas
Authors: Skeel Robert D.
Published: June 2010

A systematic way of extending a general fixed-stepsize multistep formula to a minimum storage variable-stepsize formula has been discovered that encompasses fixed-coefficient (interpolatory), variable-coefficient (variable step), and fixed leading coefficient as special cases. In particular, it is shown that the "interpolatory" stepsize changing technique of Nordsieck leads to a truly variable-stepsize multistep formula (which has implications for local error estimation and formula changing), and it is shown that the "variable-step" stepsize changing technique applicable to the Adams and backward-differentiation formulas has a reasonable generalization to the general multistep formula. In fact, it is shown how to construct a variable-order family of variable-coefficient formulas. Finally, it is observed that the first Dahlquist barrier does not apply to adaptable multistep methods if storage rather than stepnumber is the key consideration.
The numerical solution of second-order boundary value problems on nonuniform meshes
Authors: Manteuffel Thomas A. , White Andrew B.
Published: June 2010

In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes. A technique of reducing centered-difference approximations of first-order systems to discretizations of the underlying scalar equation is developed. We treat both vertex-centered and cell-centered difference schemes and indicate how these results apply to partial differential equations on Cartesian product grids.
Properties of the polynomials associated with the Jacobi polynomials
Authors: Lewanowicz S.
Published: June 2010

Power forms and Jacobi polynomial forms are found for the polynomials

W
n

(
α
,
β
)

W_n^{(\alpha ,\beta )}

associated with Jacobi polynomials. Also, some differential-difference equations and evaluations of certain integrals involving

W
n

(
α
,
β
)

W_n^{(\alpha ,\beta )}

are given.
Characterizing pseudoprimes for third-order linear recurrences
Authors: Adams William W.
Published: June 2010

This paper continues the work begun by D. Shanks and myself in [1] where certain cubic recurrences were used to give a very strong primality test. A complete characterization of the pseudoprimes for this test is given in terms of the periods of the corresponding sequences. Then these results are used to produce various types of pseudoprimes. A discussion of open problems is included.
Solving bivariate quadratic congruences in random polynomial time
Authors: Adleman Leonard M. , Estes Dennis R. , McCurley Kevin S.
Published: June 2010

It has been known for some time that solving

x
2

≡
a

(
mod
n
)

{x^2} \equiv a\;(\bmod n)

is as difficult as factoring n, at least in the sense that the two problems are random polynomial time equivalent. By contrast, solving a bivariate quadratic congruence

x
2

−
k

y
2

≡
m

(
mod
n
)

{x^2} - k{y^2} \equiv m\;(\bmod n)

can usually be done in random polynomial time even if the factorization of n is unknown. This was first proved by Pollard and Schnorr in 1985 under the assumption of the Piltz conjecture for Dirichlet L-functions. We now prove the result without assuming any unproved hypothesis.
A table of fundamental pairs of units in totally real cubic fields
Authors: Cusick T. W. , Schoenfeld Lowell
Published: June 2010

We apply a method of Cusick [5] to tabulate data on the first 250 totally real cubic fields F having discriminant

D
⩽
6
,
885

D \leqslant 6,885

. Apart from D, we list the class number H and the regulator R of F. Also given are the integer coefficients A,B,C of a defining polynomial

g
(
x
)
=

x
3

−
A

x
2

+
B
x
−
C

g(x) = {x^3} - A{x^2} + Bx - C

, its index I, and its largest zero

R
0

{R_0}

. For

j
=
1
,
2

j = 1,2

, we also tabulate both the integer coefficients

X
j

,

Y
j

,

Z
j

{X_j},{Y_j},{Z_j}

for the two units

E
j

=
(

X
j

+

R
0

Y
j

+

R
0
2

Z
j

)

/

I

{E_j} = ({X_j} + {R_0}{Y_j} + R_0^2{Z_j})/I

with norm

+
1

+ 1

, forming a fundamental pair, as well as the

E
j

{E_j}

and the integers

F
j

=

trace

(

E
j
2

)

{F_j} = {\text {trace}}(E_j^2)

.
Hecke operators and the fundamental domain for 𝑆𝐿(3,𝑍)
Authors: Gordon Daniel , Grenier Douglas , Terras Audrey
Published: June 2010

We report on a detailed study of the fundamental domain for the special linear group

SL

(
3
,

Z

)

{\text {SL}}(3,{\mathbf {Z}})

of

3
×
3

3 \times 3

integral matrices with determinant one. Graphs of points coming from the action of Hecke operators are considered.
Computation of the Néron-Tate height on elliptic curves
Authors: Tschöpe Heinz M. , Zimmer Horst G.
Published: June 2010

Using Néron’s reduction theory and a method of Tate, we develop a procedure for calculating the local and global Néron-Tate height on an elliptic curve over the rationals. The procedure is illustrated by means of two examples of Silverman and is then applied to calculate the global Néron-Tate height of a series of rank-one curves of Bremner-Cassels and of a series of rank-two curves of Selmer. In the latter case, the regulator is also computed, and a conjecture of S. Lang is investigated numerically. In dealing with the arithmetic of elliptic curves E over a global field K, the task arises of computing the Néron-Tate height on the group

E
(
K
)

E(K)

of rational points of E over K. Solving this task in an efficient manner is important, for instance, in view of calculations concerning the Birch and Swinnerton-Dyer conjecture (see [2]) or of the conjectures of Serge Lang [6]. The purpose of this note is to suggest a procedure for performing the necessary calculations.
On the asymptotic evaluation of ∫^{𝜋/2}₀𝐽²₀(𝜆𝑠𝑖𝑛𝑥)𝑑𝑥
Authors: Stoyanov Basil J. , Farrell Richard A.
Published: June 2010

The asymptotic behavior of the integral
\[

I
(
λ
)
=

∫
0

π

/

2

J
0
2

(
λ
sin
⁡
x
)

d
x

I(\lambda ) = \int _0^{\pi /2} {J_0^2(\lambda \sin x)\,dx}

\]
is investigated, where

J
0

(
x
)

{J_0}(x)

is the zeroth-order Bessel function of the first kind and

λ
\lambda

is a large positive parameter. A practical analytical expression of the integral at large

λ
\lambda

is obtained and the leading term is

(
ln
⁡
λ
)

/

(
λ
π
)

(\ln \lambda )/(\lambda \pi )

.
The F-E-M test for convergence of nonconforming finite elements
Authors: Shi Zhong Ci
Published: June 2010

A new convergence test, the F-E-M-Test, is established for the method of nonconforming finite elements. The F-E-M-Test is simple to apply, it checks only the local properties of shape functions along each interface or on each element. The test is valid for a wide class of nonconforming elements in practical applications.
The boundary element numerical method for two-dimensional linear quadratic elliptic problems. I. Neumann control
Authors: Chen Goong , Tsai Ying-Liang
Published: June 2010

For two-dimensional distributed control systems governed by the Laplace equation, the boundary element method is an efficient numerical method to solve problems whose quadratic cost involves boundary integrals only. In this paper we formulate a duality-boundary integral equation scheme and use piecewise constant boundary elements to approximate the problem. This method involves discretization of the boundary curve only and it can conveniently handle the compatibility constraint due to the Neumann data. Convergence and optimal error estimates

O

(
h
)

\mathcal {O}(h)

have been proved. Numerical data for the case of a disk are computed to illustrate the theory.
Eigenvalue finite difference approximations for regular and singular Sturm-Liouville problems
Authors: Nassif Nabil R.
Published: June 2010

This paper includes two parts. In the first part, general error estimates for "stable" eigenvalue approximations are obtained. These are practical in the sense that they are based on the discretization error of the difference formula over the eigenspace associated with the isolated eigenvalue under consideration. Verification of these general estimates are carried out on two difference schemes: that of Numerov to solve the Schrödinger singular equation and that of the central difference formula for regular Sturm-Liouville problems. In the second part, a sufficient condition for obtaining a "stable" difference scheme is derived. Such a condition (condition (N) of Theorem 2.1) leads to a simple "by hand" verification, when one selects a difference scheme to compute eigenvalues of a differential operator. This condition is checked for one- and two-dimensional problems.
Perfect multiple error-correcting arithmetic codes
Authors: Gordon Daniel M.
Published: June 2010

An arithmetic code is a subgroup of

Z

r
n

±
1

{{\mathbf {Z}}_{{r^n} \pm 1}}

, with the arithmetic distance
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d left-parenthesis x comma y right-parenthesis equals min Underscript t Endscripts x minus y identical-to normal upper Sigma Subscript i equals 1 Superscript t Baseline c Subscript i Baseline r Superscript n left-parenthesis i right-parenthesis Baseline left-parenthesis mod r Superscript n Baseline plus-or-minus 1 right-parenthesis">

d
(
x
,
y
)
=

min
t

x
−
y
≡

Σ

i
=
1

t

c
i

r

n
(
i
)

(
mod

r
n

±
1
)

d(x,y) = {\min _t}x - y \equiv \Sigma _{i = 1}^t{c_i}{r^{n(i)}}\;(\bmod {r^n} \pm 1)

, for

|

c
i

|

>
r

|{c_i}| > r

,

n
(
i
)
⩾
0

n(i) \geqslant 0

for

1
⩽
i
⩽
t

1 \leqslant i \leqslant t

. A perfect e-error-correcting code is one from which all

x
∈

Z

r
n

±
1

x \in {{\mathbf {Z}}_{{r^n} \pm 1}}

, are within distance e of exactly one codeword. Necessary and sufficient (assuming the Generalized Riemann Hypothesis) conditions for the existence of infinitely many perfect single error-correcting codes for a given r are known. In this paper some conditions for the existence of perfect multiple error-correcting codes are given, as well as the results of a computer search for examples.
A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems
Authors: Bramble James H. , Pasciak Joseph E.
Published: June 2010

This paper provides a preconditioned iterative technique for the solution of saddle point problems. These problems typically arise in the numerical approximation of partial differential equations by Lagrange multiplier techniques and/or mixed methods. The saddle point problem is reformulated as a symmetric positive definite system, which is then solved by conjugate gradient iteration. Applications to the equations of elasticity and Stokes are discussed and the results of numerical experiments are given.
On the convergence of difference approximations to scalar conservation laws
Authors: Osher Stanley , Tadmor Eitan
Published: June 2010

We present a unified treatment of explicit in time, two-level, second-order resolution (SOR), total-variation diminishing (TVD), approximations to scalar conservation laws. The schemes are assumed only to have conservation form and incremental form. We introduce a modified flux and a viscosity coefficient and obtain results in terms of the latter. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is anEscheme on monotone (actually more general) data, hence at most only first-order accurate in general. Convergence for TVD-SOR schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.
Critical point approximation through exact regularization
Authors: Fernández Cara Enrique , Moreno Carlos
Published: June 2010

We present several iterative methods for finding the critical points and/or the minima of a functional which is essentially the difference between two convex functions. The underlying idea relies upon partial and exact regularization of the functional, which allows us to preserve the local feature in a large number of applications, as well as to obtain some convergence results. These methods are further applied to some differential problems of the semilinear elliptic type arising in plasma physics and fluid mechanics.
A hybrid algorithm for solving sparse nonlinear systems of equations
Authors: Dennis J. E. , Li Guang Ye
Published: June 2010

This paper presents a hybrid algorithm for solving sparse nonlinear systems of equations. The algorithm is based on dividing the columns of the Jacobian into two parts and using different algorithms on each part. The hybrid algorithm incorporates advantages of both component algorithms by exploiting the special structure of the Jacobian to obtain a good approximation to the Jacobian, using as little effort as possible. A Kantorovich-type analysis and a locally q-superlinear convergence result for this algorithm are given.
A comparison of some Taylor and Chebyshev series
Authors: Scraton R. E.
Published: June 2010

A function is approximated in the interval

−
1
≤
x
≤
1

- 1 \leq x \leq 1

by (i) a Taylor series in x; (ii) a Taylor series in

y
=
(
x
+
λ
)

/

(
1
+
λ
x
)

y = (x + \lambda )/(1 + \lambda x)

; (iii) a Chebyshev series in x; and (iv) a Chebyshev series in

z
=
(
x
+
μ
)

/

(
1
+
μ
x
)

z = (x + \mu )/(1 + \mu x)

. The convergence of all four series is discussed, and a method is given for finding the values of

λ
\lambda

and

μ
\mu

which optimize convergence. Methods are also given for transforming one of the above series into another, some of which provide effective methods for acceleration of convergence. The application of the theory to even and odd functions is also discussed.
The computation of 𝜋 to 29,360,000 decimal digits using Borweins’ quartically convergent algorithm
Authors: Bailey David H.
Published: June 2010

In a recent work [6], Borwein and Borwein derived a class of algorithms based on the theory of elliptic integrals that yield very rapidly convergent approximations to elementary constants. The author has implemented Borweins’ quartically convergent algorithm for

1

/

π

1/\pi

, using a prime modulus transform multi-precision technique, to compute over 29,360,000 digits of the decimal expansion of

π
\pi

. The result was checked by using a different algorithm, also due to the Borweins, that converges quadratically to

π
\pi

. These computations were performed as a system test of the Cray-2 operated by the Numerical Aerodynamical Simulation (NAS) Program at NASA Ames Research Center. The calculations were made possible by the very large memory of the Cray-2. Until recently, the largest computation of the decimal expansion of

π
\pi

was due to Kanada and Tamura [12] of the University of Tokyo. In 1983 they computed approximately 16 million digits on a Hitachi S-810 computer. Late in 1985 Gosper [9] reported computing 17 million digits using a Symbolics workstation. Since the computation described in this paper was performed, Kanada has reported extending the computation of

π
\pi

to over 134 million digits (January 1987). This paper describes the algorithms and techniques used in the author’s computation, both for converging to

π
\pi

and for performing the required multi-precision arithmetic. The results of statistical analyses of the computed decimal expansion are also included.
The subgroup structure of the Mathieu group 𝑀₁₂
Authors: Buekenhout Francis , Rees Sarah
Published: June 2010

We present a full description of the lattice of subgroups of the Mathieu group

M

12

{M_{12}}

.
The spectrum and the stability of the Chebyshev collocation operator for transonic flow
Authors: Fishelov Dalia
Published: June 2010

The extension of spectral methods to the small disturbance equation of transonic flow is considered. It is shown that the real parts of the eigenvalues of its spatial operator are nonpositive. Two schemes are considered; the first is spectral in the x and y variables, while the second is spectral in x and of second order in y. Stability for the second scheme is proved. Similar results hold for the two-dimensional heat equation.
Fricke’s two-valued modular equations
Authors: Cohn Harvey
Published: June 2010

The modular equation of order b is a polynomial relation between

j
(
z
)

j(z)

and

j
(
z

/

b
)

j(z/b)

, which has astronomically large coefficients even for small values of b. Fricke showed that a two-valued relation exists for 37 small values of b. This relation would have much smaller coefficients and would also be convenient for finding singular moduli. Although Fricke produced no two-valued relations explicitly (no doubt because of the tedious amount of algebraic manipulation), they are now found by use of MACSYMA. For 31 cases ranging from

b
=
2

b = 2

to 49, Fricke provided the equations necessary to generate the relations (with two corrections required). The remaining six cases (of order 39, 41, 47, 50, 59, 71) require an extension of Fricke’s methods, using the discriminant function, theta functions, and power series approximations.
Incompressible finite element methods for Navier-Stokes equations with nonstandard boundary conditions in 𝑅³
Authors: Girault V.
Published: June 2010

This paper is devoted to the steady state, incompressible Navier-Stokes equations with nonstandard boundary conditions of the form

u

⋅

n

=
0

{\mathbf {u}} \cdot {\mathbf {n}} = 0

,

c
u
r
l

u

×

n

=

0

\mathbf {curl}\;{\mathbf {u}} \times {\mathbf {n}} = {\mathbf {0}}

, either on the entire boundary or mixed with the standard boundary condition

u

=

0

{\mathbf {u}} = {\mathbf {0}}

on part of the boundary. The problem is expressed in terms of vector potential, vorticity and pressure. The vorticity and vector potential are approximated with curl-conforming finite elements and the pressure with standard continuous finite elements. The error estimates yield nearly optimal results for the purely nonstandard problem.
On secant updates for use in general constrained optimization
Authors: Tapia Richard
Published: June 2010

In this paper we present two new classes of successive quadratic programming (SQP) secant methods for the equality-constrained optimization problem. One class of methods uses the SQP augmented Lagrangian formulation, while the other class uses the SQP Lagrangian formulation. We demonstrate, under the standard assumptions, that in both cases the BFGS and DFP versions of the algorithm are locally q-superlinearly convergent. To our knowledge this is the first time that either local or q-superlinear convergence has been established for an SQP Lagrangian secant method which uses either the BFGS or DFP updating philosophy and assumes no more than the standard assumptions. Since the standard assumptions do not require positive definiteness of the Hessian of the Lagrangian at the solution, it is no surprise that our BFGS and DFP updates possess the hereditary positive definiteness property only on a proper subspace.
Bivariate interpolation with quadratic box splines
Authors: Dæhlen Morten , Lyche Tom
Published: June 2010

Existence and uniqueness results are given for interpolation with translates of a bivariate, three-directional,

C
0

{C^0}

-quadratic box spline over a finite polygonal region. A Hermite interpolation problem for a slightly more general box spline is also considered.
An algebraic study of Gauss-Kronrod quadrature formulae for Jacobi weight functions
Authors: Gautschi Walter , Notaris Sotirios E.
Published: June 2010

We study Gauss-Kronrod quadrature formulae for the Jacobi weight function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis t right-parenthesis equals left-parenthesis 1 minus t right-parenthesis Superscript alpha Baseline left-parenthesis 1 plus t right-parenthesis Superscript beta">

w

(
α
,
β
)

(
t
)
=

(
1
−
t

)
α

(
1
+
t

)
β

{w^{(\alpha ,\beta )}}(t) = {(1 - t)^\alpha }{(1 + t)^\beta }

and its special case

α
=
β
=
λ
−

1
2

\alpha = \beta = \lambda - \frac {1}{2}

of the Gegenbauer weight function. We are interested in delineating regions in the

(
α
,
β
)

(\alpha ,\beta )

-plane, resp. intervals in

λ
\lambda

, for which the quadrature rule has (a) the interlacing property, i.e., the Gauss nodes and the Kronrod nodes interlace; (b) all nodes contained in

(
−
1
,
1
)

( - 1,1)

; (c) all weights positive; (d) only real nodes (not necessarily satisfying (a) and/or (b)). We determine the respective regions numerically for

n
=
1
(
1
)
20
(
4
)
40

n = 1(1)20(4)40

in the Gegenbauer case, and for

n
=
1
(
1
)
10

n = 1(1)10

in the Jacobi case, where n is the number of Gauss nodes. Algebraic criteria, in particular the vanishing of appropriate resultants and discriminants, are used to determine the boundaries of the regions identifying properties (a) and (d). The regions for properties (b) and (c) are found more directly. A number of conjectures are suggested by the numerical results. Finally, the Gauss-Kronrod formula for the weight

w

(
α
,
1

/

2
)

{w^{(\alpha ,1/2)}}

is obtained from the one for the weight

w

(
α
,
α
)

{w^{(\alpha ,\alpha )}}

, and similarly, the Gauss-Kronrod formula with an odd number of Gauss nodes for the weight function

w
(
t
)
=

|

t

|

γ

(
1
−

t
2

)
α

w(t) = |t{|^\gamma }{(1 - {t^2})^\alpha }

is derived from the Gauss-Kronrod formula for the weight

w

(
α
,
(
1
+
γ
)

/

2
)

{w^{(\alpha ,(1 + \gamma )/2)}}

.
On the period length of pseudorandom vector sequences generated by matrix generators
Authors: Eichenauer-Herrmann Jürgen , Grothe Holger , Lehn Jürgen
Published: June 2010

In Tahmi [5], Niederreiter [4], Afflerbach and Grothe [1], and Grothe [2] linear recursive congruential matrix generators for generating r-dimensional pseudorandom vectors are analyzed. In particular, conditions are established which ensure that the period length equals

p
r

−
1

{p^r} - 1

for any nonzero starting vector in case of a prime modulus p. For a modulus of the form

p
α

{p^\alpha }

,

α
≥
2

\alpha \geq 2

and p prime, this paper describes a simple method for constructing matrix generators having the maximal possible period length

(

p
r

−
1
)
⋅

p

α
−
1

({p^r} - 1) \cdot {p^{\alpha - 1}}

for any starting vector which is nonzero modulo p.
Galerkin/Runge-Kutta discretizations for parabolic equations with time-dependent coefficients
Authors: Keeling Stephen L.
Published: June 2010

A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for linear parabolic initial-boundary value problems with time-dependent coefficients. Unlike any classical counterpart, this class offers arbitrarily high order of convergence while significantly avoiding what has been called order reduction. In support of this claim, error estimates are proved and computational results are presented. Additionally, since the time stepping equations involve coefficient matrices changing at each time step, a preconditioned iterative technique is used to solve the linear systems only approximately. Nevertheless, the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time-independent coefficients. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low-order method.
Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems
Authors: Babuška I. , Osborn J. E.
Published: June 2010

Refined estimates for finite element or, more generally, Galerkin approximations of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems are presented. More specifically, refined results on the asymptotic behavior of the eigenvalue and eigenvector errors are proved. Both simple and multiple eigenvalues are treated.