bluuebunny/crossref_metadata_2025 huggingface dataset:
Small class numbers and extreme values of 𝐿-functions of quadratic fields
Authors: Buell Duncan A.
Published: June 2010

The table of class numbers h of imaginary quadratic fields described in [1] was placed on magnetic tape. This tape was then processed to find the occurrences of

h
⩽
125

h \leqslant 125

and to find the successive extreme values of the Dirichlet L-functions

L
(
1
,

χ

−
D

)

L(1,{\chi _{ - D}})

,

χ

−
D

{\chi _{ - D}}

the Kronecker symbol of the field

Q
(

−
D

)

Q(\sqrt { - D} )

of discriminant - D. A comparison was made between the observed extrema and the bounds obtained for the L-functions by Littlewood [5] assuming Riemann hypotheses.
Seventeen primes in arithmetic progression
Authors: Weintraub Sol
Published: June 2010

Two sets of primes in arithmetic progression are listed. One is a set of 17 primes and the second is a set of six consecutive primes.
Anomalous convergence of a continued fraction for ratios of Kummer functions
Authors: Gautschi Walter
Published: June 2010

We exhibit a phenomenon of apparent convergence to the wrong limit in connection with a continued fraction of Perron for ratios of Kummer functions. The phenomenon is further illustrated in the special cases of Bessel functions and incomplete gamma functions.
Some classes of generating functions for the Laguerre and Hermite polynomials
Authors: Cohen M. E.
Published: June 2010

In the first half of the article, we present two theorems which give, as special cases, a number of new classes of generating functions for the Laguerre polynomial. These formulae extend the recent results of Carlitz [2] and others. The latter part of our work deals with two theorems involving new generating functions for the Hermite and generalized Hermite polynomials, thus generalizing some well-known expansions. The method of proof adopted in this paper differs from that of previous workers.
On a boundary extrapolation theorem by Kreiss
Authors: Goldberg Moshe
Published: June 2010

A hardly known and very important result of Kreiss is proven explicitly: Outflow boundary extrapolation, which complements stable dissipative schemes for linear hyperbolic initial value problems, maintains stability. In view of this result, the Lax-Wendroff and the Gottlieb-Turkel schemes are applied to a test problem. As expected from the rate-of-convergence theory by Gustafsson, global order of accuracy is preserved if outflow boundary computations employ extrapolation of (local) accuracy of the same order.
Application of method of collocation on lines for solving nonlinear hyperbolic problems
Authors: Houstis E. N.
Published: June 2010

A collocation on lines procedure based on piecewise polynomials is applied to initial/boundary value problems for nonlinear hyperbolic partial differential equations. Optimal order a priori estimates are obtained for the error of approximation. The Crank-Nicholson discretization in time is studied and convergence rates of the collocation-Crank-Nicholson procedure are established. Finally, the superconvergence is verified at particular points for linear hyperbolic problems.
On maximal finite irreducible subgroups of 𝐺𝐿(𝑛,𝑍). I. The five and seven dimensional cases
Authors: Plesken Wilhelm , Pohst Michael
Published: June 2010

General methods for the determination of maximal finite absolutely irreducible subgroups of

G
L
(
n
,

Z

)

GL(n,{\mathbf {Z}})

are described. For

n
=
5
,
7

n = 5,7

all these groups are computed up to Z-equivalence.
A continued fraction expansion for a generalization of Dawson’s integral
Authors: Dijkstra D.
Published: June 2010

A continued fraction expansion for a generalization of Dawson’s integral is presented. An exact formula for the truncation error in terms of the confluent hypergeometric function is derived. The expansion is shown to have good convergence properties for both small and large values of the argument.
Computations of the Hill functions of higher order
Authors: Jerri A. J.
Published: June 2010

In this note, we express the hill function

ϕ
n

(
x
)

{\phi _n}(x)

of an order n as a Fourier cosine series which is of simple form that allows proving the function’s basic properties. For the hill functions of higher order

(
15
>
n
>
50
)

(15 > n > 50)

the form of the coefficients makes the series "essentially" self-truncating. For such high order hill functions, this truncated series (with thirty terms) computes the hill function with the same accuracy as the method of Legendre polynomials with local coordinates, but without the latter required

n
2

{n^2}

coefficients which are to be computed in advance. The preliminary time analysis indicates that the time for the two methods starts to be the same at

n
∼
15

n \sim 15

, changes slightly for the cosine series for

n
>
15

n > 15

and varies roughly as

n
3

{n^3}

for the localized Legendre polynomial method. In comparison with the most recent efficient methods which require a storage of order n, this note’s method required a storage of the order 25-40 for

n
>
60

n > 60

, executed with almost the same speed and accuracy and stayed stable as long as the above methods did.
Computation of the bivariate normal integral
Authors: Drezner Z.
Published: June 2010

This paper presents a simple and efficient computation for the bivariate normal integral based on direct computation of the double integral by the Gauss quadrature method.
𝐴-stability and dominating pairs
Authors: Iserles Arieh
Published: June 2010

It is considered whether linear combinations of A-acceptable exponential approximations preserve the A-stability, when the coefficients of the linear combination are selected in order to achieve exponential fitting. Various pairs of exponential approximations are discussed and the satisfactory pairs are characterized.
An iterative process for nonlinear monotonic nonexpansive operators in Hilbert space
Authors: Dotson W. G.
Published: June 2010

The following theorem is proved: Suppose H is a complex Hilbert space, and

T
:
H
→
H

T:H \to H

is a monotonic, nonexpansive operator on H, and

f
∈
H

f \in H

. Define

S
:
H
→
H

S:H \to H

by

S
u
=
−
T
u
+
f

Su = - Tu + f

for all

u
∈
H

u \in H

. Suppose

0
⩽

t
n

⩽
1

0 \leqslant {t_n} \leqslant 1

for all

n
=
1
,
2
,
3
,
…
,

n = 1,2,3, \ldots ,

and

Σ

n
=
1

∞

t
n

(
1
−

t
n

)

\Sigma _{n = 1}^\infty \;{t_n}(1 - {t_n})

diverges. Then the iterative process

V

n
+
1

=
(
1
−

t
n

)

V
n

+

t
n

S

V
n

{V_{n + 1}} = (1 - {t_n}){V_n} + {t_n}S{V_n}

converges to the unique solution

u
=
p

u = p

of the equation

u
+
T
u
=
f

u + Tu = f

.
On the fluctuations of Littlewood for primes of the form 4𝑛̸=1
Authors: Bays Carter , Hudson Richard H.
Published: June 2010

Let

π

b
,
c

(
x
)

{\pi _{b,c}}(x)

denote the number of primes

⩽
x

\leqslant x

which are

≡
c

(
mod

b
)

\equiv c\;\pmod b

. Among the first 950,000,000 integers there are only a few thousand integers n with

π

4
,
3

(
n
)
>

π

4
,
1

(
n
)

{\pi _{4,3}}(n) > {\pi _{4,1}}(n)

. The authors find three new widely spaced regions containing hundreds of millions of such integers; the density of these integers and the spacing of the regions is of some importance because of their intimate connection with the truth or falsity of the analogue of the Riemann hypothesis for

L
(
s
)

L(s)

. The discovery that the majority of all Integers n less than

2
×

10

10

2 \times {10^{10}}

with

π

4
,
3

(
n
)
>

π

4
,
1

(
n
)

{\pi _{4,3}}(n) > {\pi _{4,1}}(n)

are the 410,000,000 (consecutive) integers lying between 18,540,000,000 and 18,950,000,000 is a major surprise; results are carefully corroborated and some of the implications are discussed.
Details of the first region of integers 𝑥 with 𝜋_{3,2}(𝑥)<𝜋_{3,1}(𝑥)
Authors: Bays Carter , Hudson Richard H.
Published: June 2010

Since the time of Chebyshev [4] there has been interest in the magnitude of the smallest integer x with

π

3
,
2

(
x
)
>

π

3
,
1

(
x
)

{\pi _{3,2}}(x) > {\pi _{3,1}}(x)

, where

π

b
,
c

(
x
)

{\pi _{b,c}}(x)

denotes the number of positive primes

⩽
x

\leqslant x

and

≡
c

(
mod

b
)

\equiv c\;\pmod b

. The authors have recently reached this threshold with the discovery that

π

3
,
2

(
608981813029
)
−

π

3
,
1

(
608981813029
)
=
−
1

{\pi _{3,2}}(608981813029) - {\pi _{3,1}}(608981813029) = - 1

. This paper includes a detailed numerical and graphical description of values of

π

3
,
2

(
x
)
−

π

3
,
1

(
x
)

{\pi _{3,2}}(x) - {\pi _{3,1}}(x)

in the vicinity of this long sought number.
A generalization of the simple continued fraction algorithm
Authors: Vaughan Theresa P.
Published: June 2010

In this paper we present a generalization of the continued fraction algorithm, based on a geometric and matrix-theoretic approach. We first give a geometric representation in the plane

R
2

{R^2}

, of the simple continued fraction algorithm, described in terms of geometric and arithmetic properties of

2
×
2

2 \times 2

matrices with nonnegative integer entries and determinant 1. The algorithm of this paper is then derived as a natural generalization of the situation in

R
2

{R^2}

. We describe a computational procedure for our algorithm, and give several examples.
Unicity of best mean approximation by second order splines with variable knots
Authors: Barrow D. L. , Chui C. K. , Smith P. W. , Ward J. D.
Published: June 2010

Let

S
N
2

S_N^2

denote the nonlinear manifold of second order splines defined on [0, 1] having at most

N
N

interior knots, counting multiplicities. We consider the ques tion of unicity of best approximations to a function

f
f

by elements of

S
N
2

S_N^2

. Approximation relative to the

L
2

[
0
,
1
]

{L_2}[0,1]

norm is treated first, with the results then extended to the best

L
1

{L_1}

and best one-sided

L
1

{L_1}

approximation problems. The conclusions in each case are essentially the same, and can be summarized as follows: a sufficiently smooth function

f
f

satisfying

f
>
0

f > 0

has a unique best approximant from

S
N
2

S_N^2

provided either

log
⁡
f

\log f

is concave, or

N
N

is sufficiently large,

N
⩾

N
0

(
f
)

N \geqslant {N_0}(f)

; for any

N
N

, there is a smooth function

f
f

, with

f
>
0

f > 0

, having at least two best approximants. A principal tool in the analysis is the finite dimensional topological degree of a mapping.
Cyclotomic invariants and 𝐸-irregular primes
Authors: Ernvall R. , Metsänkylä T.
Published: June 2010

We prove some general results about the Iwasawa invariants

λ
−

{\lambda ^ - }

and

μ
−

{\mu ^ - }

of the 4pth cyclotomic field (p an odd prime), and determine the values of these invariants for

p
>

10
4

p > {10^4}

. The properties of

λ
−

{\lambda ^ - }

and

μ
−

{\mu ^ - }

are closely connected with the E-irregularity (i.e. the irregularity with respect to the Euler numbers) of p. A list of all E-irregular primes less than

10
4

{10^4}

, computed by the first author, is included and analyzed.
A note on the operator compact implicit method for the wave equation
Authors: Ciment Melvyn , Leventhal Stephen H.
Published: June 2010

In a previous paper a fourth order compact implicit scheme was presented for the second order wave equation. A very efficient factorization technique was developed when only second order terms were present. In this note we implement the operator compact implicit spatial discretization method for the second order wave equation when first order terms are present. The resulting algorithm is completely analogous to the compact implicit algorithm when lower order terms were not present. For this more general operator compact implicit spatial approximation the same factorization as in our previous paper is developed.
An extension of Olver’s method for the numerical solution of linear recurrence relations
Authors: Cash J. R.
Published: June 2010

An algorithm is developed for computing the solution of a class of linear recurrence relations of order greater than two when unstable error propagation prevents the required solution being found by direct forward recurrence. By abandoning an appropriate number of initial conditions the original problem may be replaced by an inexact but well-conditioned boundary value problem, and in certain circumstances the solution of this new problem is a good approximation to the required solution of the original problem. The required solution of this reposed problem is generated using an algorithm based on Gaussian elimination, and a technique developed by Olver is extended to estimate automatically the truncation error of the proposed algorithm.
Pisot and Salem numbers in intervals of the real line
Authors: Boyd David W.
Published: June 2010

Based on the work of Dufresnoy and Pisot, we develop an algorithm for determining all the Pisot numbers in an interval of the real line, provided this number is finite. We apply the algorithm to the problem of determining small Salem numbers by Salem’s construction, and to the proof that certain Pisot sequences satisfy no linear recurrence relation.
A necessary condition for the stability of a difference approximation to a hyperbolic system of partial differential equations
Authors: Burns Anne M.
Published: June 2010

We are interested in the boundary conditions for a difference approximation to a hyperbolic system of partial differential equations

u
t

=
A

u
x

{u_t} = A{u_x}

,

u
(
x
,
0
)
=
F
(
x
)

u(x,0) = F(x)

,

K
u
(
0
,
t
)
=
0

Ku(0,t) = 0

in the quarter plane

x
⩾
0

x \geqslant 0

,

t
⩾
0

t \geqslant 0

. We consider approximations of the type:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript j Baseline left-parenthesis t plus normal upper Delta t right-parenthesis equals sigma-summation Underscript k equals negative r Overscript p Endscripts upper C Subscript k Baseline u Subscript j plus k Baseline left-parenthesis t right-parenthesis comma j equals 1 comma 2 comma ellipsis comma">

u
j

(
t
+
Δ
t
)
=

∑

k
=
−
r

p

C
k

u

j
+
k

(
t
)
,

j
=
1
,
2
,
…
,

{u_j}(t + \Delta t) = \sum \limits _{k = - r}^p {{C_k}{u_{j + k}}(t),\quad j = 1,2, \ldots ,}

\]

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript j Baseline plus sigma-summation Underscript k equals 1 Overscript s Endscripts alpha Subscript j k Baseline u Subscript k Baseline left-parenthesis t plus normal upper Delta t right-parenthesis equals 0 comma j equals negative r plus 1 comma ellipsis comma 0 period">

u
j

+

∑

k
=
1

s

α

j
k

u
k

(
t
+
Δ
t
)
=
0
,

j
=
−
r
+
1
,
…
,
0.

{u_j} + \sum \limits _{k = 1}^s {{\alpha _{jk}}{u_k}(t + \Delta t) = 0,\quad j = - r + 1, \ldots ,0.}

\]
If N is the null space of K and E is the "negative" eigenspace of A, then the system of partial differential equations is well-posed if and only if

K
∩
E
=

{
0
}

K \cap E = \left \{ 0 \right \}

and Rank

K
=

K =

the number of negative eigenvalues of A. In direct analogy to this, we prove that for a difference scheme of the above type with

r
=
p
=
1

r = p = 1

,

K
′
=
I
+

Σ

k
=
1

s

α
k

K\prime = I + \Sigma _{k = 1}^s\;{\alpha _k}

and

N
′
=

N\prime =

null space of

K
′

K\prime

, a necessary condition for stability is

N
′
∩
E
=

{
0
}

N\prime \cap E = \left \{ 0 \right \}

. If, in addition, a condition proven by S. J. Osher to be sufficient for stability is not satisfied, then Rank

K
=

K =

the number of negative eigenvalues of A is also necessary for stability. We then generalize this result to the case

r
>
1

r > 1

,

p
>
1

p > 1

. Together these conditions imply that "extrapolation" on "negative" eigenvectors leads to instability; "extrapolation" on "positive" eigenvectors is "almost necessary. "Extrapolation" on "positive" eigenvectors and not on "negative" eigenvectors is sufficient for stability.
Calculating the best approximate solution of an operator equation
Authors: Wolkowicz H. , Zlobec S.
Published: June 2010

This paper furnishes two classes of methods for calculating the best approximate solution of an operator equation in Banach spaces, where the operator is bounded, linear and has closed range. The best approximate solution can be calculated by an iterative method in Banach spaces stated in terms of an operator parameter. Specifying the parameter yields some new and some old iterative techniques. Another approach is to extend the classical approximation theory of Kantorovich for equations with invertible operators to the singular case. The best approximate solution is now obtained as the limit of the best approximate solutions of simpler equations, usually systems of linear algebraic equations. In particular, a Galerkin-type method is formulated and its convergence to the best approximate solution is established. The methods of this paper can also be used for calculating the best least squares solution in Hilbert spaces or the true solution in the case of an invertible operator.
A sum of binomial coefficients
Authors: Takács Lajos
Published: June 2010

An explicit expression is derived for the sum of the

(
k
+
1
)

(k + 1)

st binomial coefficients in the nth,

(
n
−
m
)

(n - m)

th,

(
n
−
2
m
)

(n - 2m)

th,... row of the arithmetic triangle.
On the surface area of the ellipsoid
Authors: Keller Stuart R.
Published: June 2010

An expression is derived for the surface area of an ellipsoid in the form of a convergent series. The derivation is based upon an n-point Gauss-Chebyshev numerical quadrature. The rate of convergence and accuracy of the formula are demonstrated by computing the surface area of several ellipsoids.
Tables of Weyl fractional integrals for the Airy function
Authors: Balazs N. L. , Pauli H.-C. , Dabbousi O. B.
Published: June 2010

Functions related to the Weyl fractional integral of the Airy function are tabulated for the 16 lowest orders and for a range of parameters of practical interest. This range, coupled with the asymptotic relations given, covers the entire real axis.
Dyadotropic polynomials. II
Authors: Cohn Harvey
Published: June 2010

A computer search is made to test whether independent units of dyadotropic biquadratic fields (constructed parametrically) are always fundamental. A few exceptions are found. Because of the exponential rate of growth of parameters, p-adic methods are desirable. The feasibility of dyadotropic normed polynomials is also considered for general cases of degree 2 and special cases of degree 6.
Implementation aspects of band Lanczos algorithms for computation of eigenvalues of large sparse symmetric matrices
Authors: Ruhe Axel
Published: June 2010

A band Lanczos algorithm for the iterative computation of eigenvalues and eigenvectors of a large sparse symmetric matrix is described and tested on numerical examples. It starts with a p dimensional subspace, and computes an orthonormal basis for the Krylov spaces of A, generated from this starting subspace, in which A is represented by a

2
p
+
1

2p + 1

band matrix, whose eigenvalues can be computed. Special emphasis is given to devising an implementation that gives a satisfactory numerical orthogonality, with a simple program and few arithmetic operations.
Bernoulli related polynomials and numbers
Authors: Charalambides Ch. A.
Published: June 2010

The polynomials

φ
n

(
x
;
a
,
b
)

{\varphi _n}(x;a,b)

of degree n defined by the equations
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript a Baseline phi Subscript n Baseline left-parenthesis x semicolon a comma b right-parenthesis equals StartFraction left-parenthesis x right-parenthesis Subscript n minus 1 comma b Baseline Over b Superscript n minus 1 Baseline dot left-parenthesis n minus 1 right-parenthesis factorial EndFraction and normal upper Delta Subscript b Baseline phi Subscript n Baseline left-parenthesis x semicolon a comma b right-parenthesis equals phi Subscript n minus 1 Baseline left-parenthesis x semicolon a comma b right-parenthesis">

Δ
a

φ
n

(
x
;
a
,
b
)
=

(
x
)

n
−
1
,
b

b

n
−
1

⋅
(
n
−
1
)
!

and

Δ
b

φ
n

(
x
;
a
,
b
)
=

φ

n
−
1

(
x
;
a
,
b
)

{\Delta _a}{\varphi _n}(x;a,b) = \frac {{{{(x)}_{n - 1,b}}}}{{{b^{n - 1}} \cdot (n - 1)!}}\quad {\text {and}}\quad {\Delta _b}{\varphi _n}(x;a,b) = {\varphi _{n - 1}}(x;a,b)

\]
where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x right-parenthesis Subscript n comma b Baseline equals x left-parenthesis x minus b right-parenthesis left-parenthesis x minus 2 b right-parenthesis midline-horizontal-ellipsis left-parenthesis x minus n b plus b right-parenthesis">

(
x

)

n
,
b

=
x
(
x
−
b
)
(
x
−
2
b
)
⋯
(
x
−
n
b
+
b
)

{(x)_{n,b}} = x(x - b)(x - 2b) \cdots (x - nb + b)

is the generalized factorial and

Δ
a

f
(
x
)
=
f
(
x
+
a
)
−
f
(
x
)

{\Delta _a}f(x) = f(x + a) - f(x)

, are the subject of this paper. A representation of these polynomials as a sum of generalized factorials is given. The coefficients,

B
(
n
,
s
)

B(n,s)

,

s
=
a

/

b

s = a/b

, of this representation are given explicitly or by a recurrence relation. The generating functions of

φ
n

(
x
;
a
,
b
)

{\varphi _n}(x;a,b)

and

B
(
n
,
s
)

B(n,s)

are obtained. The limits of

φ
n

(
x
;
a
,
b
)

{\varphi _n}(x;a,b)

as

a
→
1

a \to 1

,

b
→
0

b \to 0

or

a
→
0

a \to 0

,

b
→
1

b \to 1

and the limits of

B
(
n
,
s
)

B(n,s)

as

s
→
±
∞

s \to \pm \infty

or

s
→
0

s \to 0

are shown to be the Bernoulli polynomials and numbers of the first and second kind, respectively. Finally, the generalized factorial moments of a discrete rectangular distribution are obtained in terms of

B
(
n
,
s
)

B(n,s)

in a form similar to that giving its usual moments in terms of the Bernoulli numbers.
Quadratic fields with 3-rank equal to 4
Authors: Diaz y Diaz F. , Shanks Daniel , Williams H. C.
Published: June 2010

In [2] there is reference to 119 known imaginary quadratic fields that have 3-rank

r
⩾
4

r \geqslant 4

. We examine these fields and determine the exact values of r. Their associated real fields and the distribution of their 3-Sylow subgroups are also studied. Some of the class groups are recorded since they are of special interest. These include examples having an infinite class field tower and only one ramified prime, and others having an infinite tower because of two different components of their class groups.
On some theoretical and practical aspects of multigrid methods
Authors: Nicolaides R. A.
Published: June 2010

A description and explanation of a simple multigrid algorithm for solving finite element systems is given. Numerical results from an implementation are reported for a number of elliptic equations, including cases with singular coefficients and indefinite equations. The method shows the high efficiency, essentially independent of the grid spacing, predicted by the theory.
Numerical conformal mapping
Authors: Chakravarthy Sukumar , Anderson Dale
Published: June 2010

A numerical procedure to determine the discrete conformal mapping of an arbitrary simply connected region onto the open unit disk is described. The method is fast and directly provides an estimate of the global error due to the discretization of the mapping. The intimate relationship between the Riemann Mapping Theorem and the numerical method of construction of the conformal mapping is brought out in the presentation. The procedure is an interesting example of a method of construction that is directly based on a theorem guaranteeing existence and uniqueness.
Computing the fast Fourier transform on a vector computer
Authors: Korn David G. , Lambiotte Jules J.
Published: June 2010

Two algorithms are presented for performing a Fast Fourier Transform on a vector computer and are compared on the Control Data Corporation STAR-100. The relative merits of the two algorithms are shown to depend upon whether only a few or many independent transforms are desired. A theorem is proved which shows that a set of independent transforms can be computed by performing a partial transformation on a single vector. The results of this theorem also apply to nonvector machines and have reduced the average time per transform by a factor of two on the CDC 6600 computer.
Operational evaluation of certain infinite Bessel function integrals
Authors: Babb Stanley E. , Cafky James W.
Published: June 2010

Some infinite integrals, primarily over trigonometric functions, are operationally evaluated by two extensions of the Weber-Schafheitlin approach.
On high precision methods for computing integrals involving Bessel functions
Authors: Gabutti Bruno
Published: June 2010

The technique of Bakhvalov and Vasil’eva for evaluating Fourier integrals is generalized to integrals involving exponential and Bessel functions.
Solution of Nathanson’s exponential congruence
Authors: Wagstaff Samuel S.
Published: June 2010

The exponential congruence

5
n

≡
2

(
mod

3
n

)

{5^n} \equiv 2\;\pmod {3^n}

has no solution

n
>
1

n > 1

. This result is proved by using a theorem of van der Poorten to produce an upper bound for the size of such solutions n which is within range of machine verification, and then checking that no n below this bound satisfies the congruence.
On the SHASTA FCT algorithm for the equation ∂𝜌/∂𝑡+(∂/∂𝑥)(𝑣(𝜌)𝜌)=0
Authors: Ikeda Tsutomu , Nakagawa Tomoyasu
Published: June 2010

In recent years, Boris, Book and Hain have proposed a family of finite difference methods called FCT techniques for the Cauchy problem of the continuity equation. The purpose of this paper is to study the stability and convergence about the SHASTA FCT algorithm, which is one of the basic schemes among many FCT techniques, though not in its original form but a slightly modified one for our technical reason. (Our numerical experiments indicate less distinction between the algorithm dealt with here and the original SHASTA FCT one in terms of reproduction of sharp discontinuities.) The main results are Theorems 1 and 2 concerning the

L
∞

{L^\infty }

-stability and the

L

loc

1

L_{{\operatorname {loc}}}^1

-convergence, respectively.
A note on class-number one in pure cubic fields
Authors: Williams H. C. , Shanks Daniel
Published: June 2010

We examine a subset of the pure cubic fields wherein individual fields appear to have a probability of having class-number one approximately equal to3/53/5. We also suggest more elaborate but more efficient algorithms that could be used to extend the data.
Negative norm estimates and superconvergence in Galerkin methods for parabolic problems
Authors: Thomée Vidar
Published: June 2010

Negative norm error estimates for semidiscrete Galerkin-finite element methods for parabolic problems are derived from known such estimates for elliptic problems and applied to prove superconvergence of certain procedures for evaluating point values of the exact solution and its derivatives.
The shooting method applied to a cyclic inequality
Authors: Troesch B. A.
Published: June 2010

It is known that the cyclic sum
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript n Baseline left-parenthesis bold x right-parenthesis equals sigma-summation Underscript i equals 1 Overscript i equals n Endscripts x Subscript i Baseline slash left-parenthesis x Subscript i plus 1 Baseline plus x Subscript i plus 2 Baseline right-parenthesis">

S
n

(

x

)
=

∑

i
=
1

i
=
n

x
i

/

(

x

i
+
1

+

x

i
+
2

)

{S_n}({\mathbf {x}}) = \sum \limits _{i = 1}^{i = n} {{x_i}/({x_{i + 1}} + {x_{i + 2}})}

\]
where

x

n
+
1

=

x
1

{x_{n + 1}} = {x_1}

,

x

n
+
2

=

x
2

{x_{n + 2}} = {x_2}

,

x
i

⩾
0

{x_i} \geqslant 0

,

(

x

i
+
1

+

x

i
+
2

)
>
0

({x_{i + 1}} + {x_{i + 2}}) > 0

, can be made smaller than

n

/

2

n/2

for

n
⩾
24

n \geqslant 24

. The value of

λ
(
n
)
=
lim
inf

(
n
→
∞
)

S
n

/

n

\lambda (n) = \lim \inf \;(n \to \infty )\;{S_n}/n

is investigated by the shooting method for two-point boundary value problems, and the analytical result

lim
(
n
→
∞
)

λ
(
n
)
⩽
0.49457

\lim (n \to \infty )\;\lambda (n) \leqslant 0.49457

is proved. The inherent difficulty in a straight-forward minimization of

S
n

(

x

)

{S_n}({\mathbf {x}})

is mentioned.
Solution of linear equations with rational Toeplitz matrices
Authors: Dickinson Bradley W.
Published: June 2010

We associate a sequence of Toeplitz matrices with the rational formal power series

T
(
z
)

T(z)

. An algorithm for solving linear equations with a Toeplitz matrix from this sequence is given. The algorithm requires

O
(
n
)

O(n)

operations to solve a set of n equations, for n sufficiently large.
Generating random variables with a 𝑡-distribution
Authors: Marsaglia George
Published: June 2010

Let RNOR and REXP represent normal and exponential random variables produced by computer subroutines. Then this simple algorithm may be used to generate a random variable T with

t
n

{t_n}

density

c
(
1
+

t
2

/

n

)

−
1

/

2
n
−
1

/

2

c(1 + {t^2}/n)^{- 1/2 n - 1/2}

, for any

n
>
2

n > 2

: Generate

A
=

R
N
O
R

,
B
=

A
2

/

(
n
−
2
)

A = \mathrm {RNOR}, B = A^2/(n - 2)

and

C
=

R
E
X
P

/

1

/

2
n
−
1
)

C = \mathrm {REXP}/ 1/2 n - 1)

until

e

−
B
−
C

⩽
1
−
B

e^{ - B - C} \leqslant 1 - B

, then exit with

T
=
A
[
(
1
−
2

/

n
)
(
1
−
B
)

]

−
1

/

2

T = A[(1 - 2/n)(1 - B)]^{-1/2}

.
Very high accuracy Chebyshev expansions for the basic trigonometric functions
Authors: Schonfelder J. L.
Published: June 2010

Chebyshev expansion coefficients, accurate to forty decimal places, for the functions sine, cosine, and tangent, are tabulated. The methods used to generate the expansions are outlined and the ways in which accuracy of the tabulated coefficients were checked are noted.
Generalized OCI schemes for boundary layer problems
Authors: Berger Alan E. , Solomon Jay M. , Ciment Melvyn , Leventhal Stephen H. , Weinberg Bernard C.
Published: June 2010

A family of tridiagonal formally fourth-order difference schemes is developed for a class of singular perturbation problems. These schemes have no cell Reynolds number limitation and satisfy a discrete maximum principle. Error estimates and numerical results for this family of methods are given, and are compared with those for several other schemes.
A cardinal function method of solution of the equation Δ𝑢=𝑢-𝑢³
Authors: Lundin L. R.
Published: June 2010

The steady-state form of the Klein-Gordon equation is given by

(
∗

)

(^\ast )

\[

Δ
u
=
u
−

u
3

,

u
=
u
(
X
)
,

X
∈

R
3

.

\Delta u = u - {u^3},\quad u = u(X),\quad X \in {R^3}.

\]
For solutions which are spherically symmetric,

(
∗

)

(^\ast )

takes the form

u
¨

+
2

u
˙

/

r
=
u
−

u
3

\ddot u + 2\dot u/r = u - {u^3}

,

u
=
u
(
r
)

u = u(r)

, where r is the distance from the origin in

R
3

{R^3}

. The function

y
=
r
u

y = ru

satisfies

(
∗

∗

)

{(^\ast }^\ast )

\[

y
¨

=
y
−

y
3

/

r
2

.

\ddot y = y - {y^3}/{r^2}.

\]
It is known that

(
∗

∗

)

{(^\ast }^\ast )

has solutions

{

y
n

}

n
=
0

∞

\{ {y_n}\} _{n = 0}^\infty

, where

y
n

{y_n}

has exactly n zeros in

(
0
,
∞
)

(0,\infty )

, and where

y
(
0
)
=
y
(
∞
)
=
0

y(0) = y(\infty ) = 0

. In this paper, an approximation is obtained for the solution

y
0

{y_0}

by minimizing a certain functional over a class of functions of the form
\[

∑

k
=
−
m

m

a
k

sinc

[

r
−
k

h
m

h
m

]

.

\sum \limits _{k = - m}^m {{a_k}\;} {\operatorname {sinc}}\left [ {\frac {{r - k{h_m}}}{{{h_m}}}} \right ].

\]
It is shown that the norm of the error is

O
(

m

3

/

8

exp
⁡
(
−
α

m

1

/

2

)
)

O({m^{3/8}}\exp ( - \alpha m^{1/2}))

as

m
→
∞

m \to \infty

, where

α
\alpha

is positive.
Numerical quadrature and nonlinear sequence transformations; unified rules for efficient computation of integrals with algebraic and logarithmic endpoint singularities
Authors: Sidi Avram
Published: June 2010

Some nonlinear transformations for accelerating the convergence of infinite sequences due to Levin are reviewed, and new results of practical importance in applications are given. Using these results, the transformations of Levin are modified and used to obtain new numerical integration formulas for weight functions with algebraic and logarithmic endpoint singularities, which are simpler to compute and practically as efficient as the corresponding Gaussian formulas. They also have the additional advantage that different weight functions of a certain type can have the same set of abscissas associated with them. It is shown that the formulas obtained are of interpolatory type. Furthermore, for some cases it is proved that the abscissas are in the interval of integration, although numerical results indicate that this is so in all cases and that the weights are all positive. Several numerical examples that illustrate the high accuracy and convenience of the new formulas are appended.
Exponential laws for fractional differences
Authors: Isaacs Godfrey L.
Published: June 2010

In Math. Comp., v. 28, 1974, pp. 185-202, Diaz and Osler gave the following (formal) definition for

Δ
˙

α

f
(
z
)

{\dot \Delta ^\alpha }f(z)

, the

α
\alpha

th fractional difference of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis colon ModifyingAbove normal upper Delta With dot Superscript alpha Baseline f left-parenthesis z right-parenthesis equals normal upper Sigma Subscript p equals 0 Superscript normal infinity Baseline upper A Subscript p Superscript negative alpha minus 1 Baseline f left-parenthesis z plus alpha minus p right-parenthesis">

f
(
z
)
:

Δ
˙

α

f
(
z
)
=

Σ

p
=
0

∞

A
p

−
α
−
1

f
(
z
+
α
−
p
)

f(z):{\dot \Delta ^\alpha }f(z) = \Sigma _{p = 0}^\infty A_p^{ - \alpha - 1}f(z + \alpha - p)

. They derived formulas and applications involving this difference. They asked whether their differences satisfied an exponent law and what the relation was between their differences and others, such as
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Superscript alpha Baseline f left-parenthesis z right-parenthesis equals normal upper Sigma Subscript p equals 0 Superscript normal infinity Baseline upper A Subscript p Superscript negative alpha minus 1 Baseline f left-parenthesis z plus p right-parenthesis">

Δ
α

f
(
z
)
=

Σ

p
=
0

∞

A
p

−
α
−
1

f
(
z
+
p
)

{\Delta ^\alpha }f(z) = \Sigma _{p = 0}^\infty A_p^{ - \alpha - 1}f(z + p)

. In this paper an exponent law for their differences is established and a relation found between the two differences mentioned above. Applications of these results are given.
A special class of Bell polynomials
Authors: Howard F. T.
Published: June 2010

We examine the integersV(n,k)V(n,k)defined by means of\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k factorial sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts upper V left-parenthesis n comma k right-parenthesis x Superscript n Baseline slash n factorial equals left-bracket x left-parenthesis e Superscript x Baseline plus 1 right-parenthesis minus 2 left-parenthesis e Superscript x Baseline minus 1 right-parenthesis right-bracket Superscript k Baseline comma">k!∑n=0∞V(n,k)xn/n!=[x(ex+1)−2(ex−1)]k,k!\sum \limits _{n = 0}^\infty {V(n,k){x^n}/n! = {{[x({e^x} + 1) - 2({e^x} - 1)]}^k},}\]and, in particular, we show how these integers are related to the Bernoulli, Genocchi and van der Pol numbers, and the numbers generated by the reciprocal ofex−x−1{e^x} - x - 1. We prove that theV(n,k)V(n,k)are also related to the numbersW(n,k)W(n,k)defined by\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k factorial sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts upper W left-parenthesis n comma k right-parenthesis x Superscript n Baseline slash n factorial equals left-bracket left-parenthesis x minus 2 right-parenthesis left-parenthesis e Superscript x Baseline minus 1 right-parenthesis right-bracket Superscript k">k!∑n=0∞W(n,k)xn/n!=[(x−2)(ex−1)]kk!\sum \limits _{n = 0}^\infty {W(n,k){x^n}/n! = {{[(x - 2)({e^x} - 1)]}^k}}\]in much the same way the associated Stirling numbers are related to the Stirling numbers. Finally, we examine, more generally, the Bell polynomialsBn,k(a1,a2,3−α,4−α,5−α,…){B_{n,k}}({a_1},{a_2},3 - \alpha ,4 - \alpha ,5 - \alpha , \ldots )and show how the methods of this paper can be used to prove several formulas involving the Bernoulli and Stirling numbers.
Elliptic curves of conductor 11
Authors: Agrawal M. K. , Coates J. H. , Hunt D. C. , van der Poorten A. J.
Published: June 2010

We determine all elliptic curves defined over Q of conductor 11. Firstly, we reduce the problem to one of solving a diophantine equation, namely a certain Thue-Mahler equation. Then we apply recent sharp inequalities for linear forms in the logarithms of algebraic numbers to bound solutions of that equation. Finally, some straightforward computations yield all solutions of the diophantine equation. Our results are in accordance with the conjecture of Taniyama-Weil for conductor 11.
A finite element method for first-order hyperbolic systems
Authors: Luskin Mitchell
Published: June 2010

A new finite element method is proposed for the numerical solution of a class of initial-boundary value problems for first-order hyperbolic systems in one space dimension. An application of our procedure to a system modeling gas flow in a pipe is discussed. Asymptotic error estimates are derived in the

L
2

{L^2}

norm in space.
Local stability conditions for the Babuška method of Lagrange multipliers
Authors: Pitkäranta Juhani
Published: June 2010

We consider the so-called Babuška method of finite elements with Lagrange multipliers for numerically solving the problem

Δ
u
=
f

\Delta u = f

in

Ω
\Omega

,

u
=
g

u = g

on

∂
Ω

\partial \Omega

,

Ω
⊂

R
n

\Omega \subset {R^n}

,

n
⩾
2

n \geqslant 2

. We state a number of local conditions from which we prove the uniform stability of the Lagrange multiplier method in terms of a weighted, mesh-dependent norm. The stability conditions given weaken the conditions known so far and allow mesh refinements on the boundary. As an application, we introduce a class of finite element schemes, for which the stability conditions are satisfied, and we show that the convergence rate of these schemes is of optimal order.
Numerical stability of nested dissection orderings
Authors: Anand Indu Mati
Published: June 2010

Rigorous bounds on rounding errors for sparse positive definite matrices are obtained. When used for nested dissection orderings of finite element matrices, the analysis furnishes bounds which are stronger than those for band orderings.
On some trigonometric integrals
Authors: Fettis Henry E.
Published: June 2010

Expressions are obtained for the integrals
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript lamda Superscript left-parenthesis p right-parenthesis Baseline equals integral Subscript 0 Superscript pi slash 2 Baseline left-parenthesis StartFraction sine lamda theta Over sine theta EndFraction right-parenthesis Superscript p Baseline d theta comma upper J Subscript lamda Superscript left-parenthesis p right-parenthesis Baseline equals integral Subscript 0 Superscript pi slash 2 Baseline left-parenthesis StartFraction 1 minus cosine lamda theta Over sine theta EndFraction right-parenthesis Superscript p Baseline d theta">

I
λ

(
p
)

=

∫
0

π

/

2

(

sin
⁡
λ
θ

sin
⁡
θ

)

p

d
θ
,

J
λ

(
p
)

=

∫
0

π

/

2

(

1
−
cos
⁡
λ
θ

sin
⁡
θ

)

p

d
θ

I_\lambda ^{(p)} = \int _0^{\pi /2}{\left ( {\frac {{\sin \lambda \theta }}{{\sin \theta }}} \right )^p}d\theta ,\quad J_\lambda ^{(p)} = \int _0^{\pi /2}{\left ( {\frac {{1 - \cos \lambda \theta }}{{\sin \theta }}} \right )^p}d\theta

\]
for arbitrary real values of "

λ
\lambda

", and

p
=
1
,
2

p = 1,2

.
On accelerating the convergence of infinite double series and integrals
Authors: Levin David
Published: June 2010

The generalization of Shanks’ e-transformation to double series is discussed and a class of nonlinear transformations, the

[
A

/

S

]
R

{[A/S]_R}

transformations, for accelerating the convergence of infinite double series is presented. It is constructed so as to sum exactly infinite double series whose terms satisfy certain finite linear double difference equations; in that sense it is a generalization of Shanks’ e-transformation or its equivalent Wynn’s

ε
\varepsilon

-algorithm. A generalization of the

[
A

/

S

]
R

{[A/S]_R}

transformation to N-dimensional series is also presented and their application to power series is discussed and exemplified. Some transformations for accelerating the convergence of infinite double integrals are also obtained, generalizing the confluent

ε
\varepsilon

-algorithm of Wynn and the G-transformation of Gray, Atchison, and McWilliams for infinite 1-D integrals.
Note on irreducibility testing
Authors: Brillhart John
Published: June 2010

An effective method is developed for deducing the irreducibility of a given polynomial with integer coefficients from a single occurrence of a prime value of that polynomial.
Improving the speed of calculating the regulator of certain pure cubic fields
Authors: Williams H. C.
Published: June 2010

To calculate R, the regulator of a pure cubic field

Q
(

D
)

3

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

, a complete period of Voronoi’s continued fraction algorithm over

Q
(

D
)

3

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

is usually generated. In this paper it is shown how, in certain pure cubic fields, R can be determined by generating only about one third of this period. These results were used on a computer to find R and then the class number for all pure cubic fields

Q
(

p
)

3

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

, where p is a prime,

p
≡
−
1

(
mod

3
)

p \equiv - 1\;\pmod 3

, and

p
>
2
×

10
5

p > 2 \times {10^5}

. Graphs illustrating the distribution of such cubic fields with class number one are presented.
Collocation approximation to eigenvalues of an ordinary differential equation: numerical illustrations
Authors: de Boor Carl , Swartz Blair
Published: June 2010

We display the numerical results associated with the collocation of three eigenvalue problems using from one to four Gauss points per partition interval in order to document the sharpness of the error bounds we have previously obtained. The ordinary differential operators involved are real with constant coefficients; two of the problems have an eigenvalue whose ascent exceeds one. We propose an explanation for the observed manner in which a set of simple approximate eigenvalues can approach a single multiple eigenvalue.
Local piecewise polynomial projection methods for an O.D.E. which give high-order convergence at knots
Authors: de Boor Carl , Swartz Blair
Published: June 2010

Local projection methods which yield

C

(
m
−
1
)

{C^{(m - 1)}}

piecewise polynomials of order

m
+
k

m + k

as approximate solutions of a boundary value problem for an mth order ordinary differential equation are determined by the k linear functional at which the residual error in each partition interval is required to vanish on. We develop a condition on these k functionals which implies breakpoint superconvergence (of derivatives of order less than m) for the approximating piecewise polynomials. The same order of superconvergence is associated with eigenvalue problems. A discrete connection between two particular projectors yielding

O

(

|

Δ

|

2
k

)

\mathcal {O}(|\Delta {|^{2k}})

superconvergence, namely (a) collocation at the k Gauss-Legendre points in each partition interval and (b) "essential least-squares" (i.e., local moment methods), is made by asking that this same order of superconvergence result when using collocation at

k
−
r

k - r

points per interval and simultaneous local orthogonality of the residual to polynomials of order r; the

k
−
r

k - r

points then necessarily form a subset of the k Gauss-Legendre points.
Uniform expansions for a class of finite difference schemes for elliptic boundary value problems
Authors: Munz Harry
Published: June 2010

For a class of finite difference schemes for the Dirichlet problem on a bounded region

Ω
⊂

R

n

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

, the existence of uniform expansions of the approximate solution for meshlength

h
→
0

h \to 0

is shown. The results also improve error bounds which Pereyra, Proskurowski, and Widlund obtained with respect to certain discrete

L
2

{L_2}

-norms.
Optimum accelerated overrelaxation method in a special case
Authors: Avdelas G. , Hadjidimos A.
Published: June 2010

In this paper we give the optimum parameters for the Accelerated Overrelaxation (AOR) method in the special case where the matrix coefficient of the linear system, which is solved, is consistently ordered with nonvanishing diagonal elements. Under certain assumptions, concerning the eigenvalues of the corresponding Jacobi matrix, it is shown that the optimum AOR method gives better convergence rates than the optimum SOR does, while in the remaining cases the optimum AOR method coincides with the optimum SOR one.
Computation of Pólya polynomials of primitive permutation groups
Authors: Land Rudolf
Published: June 2010

An almost complete list of Pólya polynomials of all primitive permutation groups up to degree 20 has been computed. The number-theoretical interpretation of Pólya polynomials and van der Waerden’s test make this a good tool to find safe conjectures for determining the group of an equation. This work was encouraged and supported by Professor W. Jehne, Universität zu Köln.
A population of linear, second order, elliptic partial differential equations on rectangular domains. I, II
Authors: Rice John R. , Houstis Elias N. , Dyksen Wayne R.
Published: June 2010

We present a population of 56 linear, two-dimensional elliptic partial differential equations (PDEs) suitable for evaluating numerical methods and software. Forty-two of the PDEs are parametrized which allows much larger populations to be made; 189 specific cases are presented here along with solutions (some are only approximate). Many of the PDEs are artificially created so as to exhibit various mathematical behaviors of interest; the others are taken from "real world" problems in various ways. The population has been structured by introducing measures of complexity of the operator, boundary conditions, solution and problem. The PDEs are first presented in mathematical terms along with contour plots of the 189 specific solutions. Machine-readable descriptions are given in Part 2; many of the PDEs involve lengthy expressions and about a dozen involve extensive tabulations of approximate solutions.
Uniqueness of the optimal nodes of quadrature formulae
Authors: Bojanov Borislav D.
Published: June 2010

We prove the uniqueness of the quadrature formula with minimal error in the space

W
~

q
r

[
a
,
b
]
,
1
>
q
>
∞

\tilde W_q^r[a,b],1 > q > \infty

, of

(
b
−
a
)

(b - a)

-periodic differentiable functions among all quadratures with n free nodes

{

x
k

}
1
n

\{ {x_k}\} _1^n

,

a
=

x
1

>
⋯
>

x
n

>
b

a = {x_1} > \cdots > {x_n} > b

, of fixed multiplicities

{

v
k

}
1
n

\{ {v_k}\} _1^n

, respectively. As a corollary, we get that the equidistant nodes are optimal in

W
~

q
r

[
a
,
b
]

\tilde W_q^r[a,b]

for

1
⩽
q
⩽
∞

1 \leqslant q \leqslant \infty

if

v
1

=
⋯
=

v
n

{v_1} = \cdots = {v_n}

.
The Jacobi-Perron algorithm in integer form
Authors: Hendy M. D. , Jeans N. S.
Published: June 2010

We present an alternative expression of the Jacobi-Perron algorithm on a set of

n
−
1

n - 1

independent numbers of an algebraic number field of degree n, where computation of real valued (nonrational) numbers is avoided. In some instances this saves the need to compute with high levels of precision. We also demonstrate a necessary and sufficient condition for the algorithm to cycle. The paper is accompanied by several numerical examples.