bluuebunny/crossref_metadata_2025 huggingface dataset:
An algorithm for the exact reduction of a matrix to Frobenius form using modular arithmetic. I
Authors: Howell Jo Ann
Published: June 2010

This paper is in two parts. Part I contains a description of the Danilewski algorithm for reducing a matrix to Frobenius form using rational arithmetic. This algorithm is modified for use over the field of integers modulo p. The modified algorithm yields exact integral factors of the characteristic polynomial. A description of the single-modulus algorithm is given. Part II contains a description of the multiple-modulus algorithm. Since different moduli may yield different factorizations, an algorithm is given for determining which factorizations are not correct factorizations over the integers of the characteristic polynomial.
Rational approximants defined from double power series
Authors: Chisholm J. S. R.
Published: June 2010

Rational approximants are defined from double power series in variables x and y, and it is shown that these approximants have the following properties: (i) they possess symmetry between x and y; (ii) they are in general unique; (iii) if

x
=
0

x = 0

or

y
=
0

y = 0

, they reduce to diagonal Padé approximants; (iv) their definition is invariant under the group of transformations

x
=
A
u

/

(
1
−
B
u
)
,
y
=
A
v

/

(
1
−
C
v
)

x = Au/(1 - Bu),y = Av/(1 - Cv)

with

A
≠
0

A \ne 0

; (v) an approximant formed from the reciprocal series is the reciprocal of the corresponding original approximant. Possible variations, extensions and generalisations of these results are discussed.
Reduction formulas for multiple series
Authors: Glasser M. L.
Published: June 2010

A simple procedure is given for reducing broad classes of multiple series to single series. Examples are given for double series.
Some factorizations of 10ⁿ±1
Authors: Angell I. O. , Godwin H. J.
Published: June 2010

Factorizations of

10
n

+
1

{10^n} + 1

and/or

10
n

−
1

{10^n} - 1

are given for a number of values of n.
On the number of nonzeros added when Gaussian elimination is performed on sparse random matrices
Authors: Duff I. S.
Published: June 2010

This paper studies the fill-in properties of Gaussian elimination on sparse random matrices. A theoretical study using the concept of random graphs yields formulae from which the fill-in can be evaluated. The predictions are examined for matrices of various orders and densities and are found to be in close agreement with experimental results. A possible consequence of the results of this paper relating to error analysis for sparse systems is given in the concluding section.
Numerical solution of Plateau’s problem by a finite element method
Authors: Hinata Masahiro , Shimasaki Masaaki , Kiyono Takeshi
Published: June 2010

This paper concerns the application of a finite element method to the numerical solution of a nonrestricted form of the Plateau problem, as well as to a free boundary problem of Plateau type. The solutions obtained here are examined for several examples and are considered to be sufficiently accurate. It is also observed that the hysteresis effect, which is a feature of a nonlinear problem, appears in this problem.
Vector-valued approximation and its application to fitting exponential decay curves
Authors: Belford Geneva G.
Published: June 2010

This paper deals with characterization of best approximations to vector-valued functions. The approximations are themselves vector-valued functions with components depending nonlinearly on the approximation parameters. The constraint is imposed that certain of the parameters should be identical for all components. An application to exponential approximation is discussed in some detail.
Sums of distinct primes from congruence classes modulo 12
Authors: Dressler Robert E. , Mąkowski Andrzej , Parker Thomas
Published: June 2010

It is shown that every integer greater than 1969, 1349, 1387, 1475 is a sum of distinct primes of the form

12
k
+
1
,
12
k
+
5
,
12
k
+
7
,
12
k
+
11

12k + 1,12k + 5,12k + 7,12k + 11

, respectively. Furthermore, these lower bounds are best possible.
Confluent expansions for functions of two variables
Authors: Deshpande V. L.
Published: June 2010

In a recent paper, J. L. Fields established four theorems giving confluent expansions for functions of one variable. In the present paper, we extended one theorem of Fields for functions of two variables. The usefulness of the theorem is illustrated by obtaining known and hitherto unknown transformations for Appell functions and Horn functions.
Numerical computation of a generalized exponential integral function
Authors: Breig W. F. , Crosbie A. L.
Published: June 2010

Series expansions and recurrence relations suitable for numerical computation are developed for the generalized exponential integral functions. Tables of these functions are presented in the microfiche section of this issue.
Convergence rates of parabolic difference schemes for non-smooth data
Authors: Thomée Vidar , Wahlbin Lars
Published: June 2010

Consider the approximate solution of the initial-value problem for a parabolic system by means of a parabolic finite difference scheme of accuracy

μ
\mu

. The main result of the present paper is essentially that for positive time and v in

W
1

s

{W_1}^s

with

1
>
s
≤
μ

1 > s \leq \mu

, the error in the maximum norm is

O
(

h
s

)

O({h^s})

for small mesh-widths h.
A quadratic partition of primes ≡1 (𝑚𝑜𝑑7)
Authors: Williams Kenneth S.
Published: June 2010

The solutions of a quadratic partition of primes

p
≡
1

(
mod

7
)

p \equiv 1 \pmod 7

, in terms of which the author and P. A. Leonard have given the cyclotomic numbers of order seven and also necessary and sufficient conditions for 2, 3, 5 and 7 to be seventh powers

(
mod

p
)

\pmod p

, are obtained for all such primes

>
1000

> 1000

.
The simplest cubic fields
Authors: Shanks Daniel
Published: June 2010

The cyclic cubic fields generated by

x
3

=
a

x
2

+
(
a
+
3
)
x
+
1

{x^3} = a{x^2} + (a + 3)x + 1

are studied in detail. The regulators are relatively small and are known at once. The class numbers are always of the form

A
2

+
3

B
2

{A^2} + 3{B^2}

, are relatively large and easy to compute. The class groups are usually easy to determine since one has the theorem that if m is divisible only by

primes

≡
2

(
mod

3
)

{\text {primes}} \equiv 2\pmod 3

, then the m-rank of the class group is even. Fields with different 3-ranks are treated separately.
Factorization tables for 𝑥ⁿ-1 over 𝐺𝐹(𝑞)
Authors: Beard Jacob T. B. , West Karen I.
Published: June 2010

These tables give the complete factorization of

x
n

−
1

{x^n} - 1

over

GF

(
q
)
,
q
=

p
a

,
2
⩽
n
⩽
d

{\text {GF}}(q),q = {p^a},2 \leqslant n \leqslant d

as below, together with the Euler

Φ
\Phi

-function of

x
n

−
1

{x^n} - 1

whenever

Φ
(

x
n

−
1
)
>

10
8

\Phi ({x^n} - 1) > {10^8}

.
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column q equals 2 semicolon d equals 32 2nd Column a m p semicolon q equals 3 semicolon d equals 27 3rd Column a m p semicolon q equals 11 semicolon d equals 15 2nd Row 1st Column q equals 2 squared semicolon d equals 16 2nd Column a m p semicolon q equals 3 squared semicolon d equals 15 3rd Column a m p semicolon q equals 13 semicolon d equals 15 3rd Row 1st Column q equals 2 cubed semicolon d equals 16 2nd Column a m p semicolon q equals 5 semicolon d equals 25 comma n not-equals 23 Superscript dagger Baseline 3rd Column a m p semicolon q equals 17 semicolon d equals 15 4th Row 1st Column q equals 2 Superscript 4 Baseline semicolon d equals 16 2nd Column a m p semicolon q equals 5 squared semicolon d equals 10 3rd Column a m p semicolon q equals 19 semicolon d equals 12 5th Row 1st Column q equals 2 Superscript 5 Baseline semicolon d equals 12 2nd Column a m p semicolon q equals 7 semicolon d equals 15 3rd Column a m p semicolon q equals 23 semicolon d equals 10 EndLayout">

q
=
2
;
d
=
32

a
m
p
;

q
=
3
;
d
=
27

a
m
p
;

q
=
11
;
d
=
15

q
=

2
2

;
d
=
16

a
m
p
;

q
=

3
2

;
d
=
15

a
m
p
;

q
=
13
;
d
=
15

q
=

2
3

;
d
=
16

a
m
p
;

q
=
5
;
d
=
25
,
n
≠

23

†

a
m
p
;

q
=
17
;
d
=
15

q
=

2
4

;
d
=
16

a
m
p
;

q
=

5
2

;
d
=
10

a
m
p
;

q
=
19
;
d
=
12

q
=

2
5

;
d
=
12

a
m
p
;

q
=
7
;
d
=
15

a
m
p
;

q
=
23
;
d
=
10

\begin {array}{*{20}{c}} {q = 2;d = 32} \hfill & {q = 3;d = 27} \hfill & {q = 11;d = 15} \hfill \\ {q = {2^2};d = 16} \hfill & {q = {3^2};d = 15} \hfill & {q = 13;d = 15} \hfill \\ {q = {2^3};d = 16} \hfill & {q = 5;d = 25,n \ne {{23}^\dagger }} \hfill & {q = 17;d = 15} \hfill \\ {q = {2^4};d = 16} \hfill & {q = {5^2};d = 10} \hfill & {q = 19;d = 12} \hfill \\ {q = {2^5};d = 12} \hfill & {q = 7;d = 15} \hfill & {q = 23;d = 10} \hfill \\ \end {array}

\]
Exponential fitting of matricial multistep methods for ordinary differential equations
Authors: Sarkany E. F. , Liniger W.
Published: June 2010

We study a class of explicit or implicit multistep integration formulas for solving

N
×
N

N \times N

systems of ordinary differential equations. The coefficients of these formulas are diagonal matrices of order N, depending on a diagonal matrix of parameters Q of the same order. By definition, the formulas considered here are exact with respect to

y
′

=
−
D
y
+
ϕ
(
x
,
y
)

y’ = - Dy + \phi (x,y)

provided

Q
=
h
D
,
h

Q = hD,h

is the integration step, and

ϕ
\phi

belongs to a certain class of polynomials in the independent variable x. For arbitrary step number

k
⩾
1

k \geqslant 1

, the coefficients of the formulas are given explicitly as functions of Q. The present formulas are generalizations of the Adams methods

(
Q
=
0
)

(Q = 0)

and of the backward differentiation formulas

(
Q
=
+
∞
)

(Q = + \infty )

. For arbitrary Q they are fitted exponentially at Q in a matricial sense. The implicit formulas are unconditionally fixed-h stable. We give two different algorithmic implementations of the methods in question. The first is based on implicit formulas alone and utilizes the Newton-Raphson method; it is well suited for stiff problems. The second implementation is a predictor-corrector approach. An error analysis is carried out for arbitrarily large Q. Finally, results of numerical test calculations are presented.
Interior estimates for Ritz-Galerkin methods
Authors: Nitsche Joachim A. , Schatz Alfred H.
Published: June 2010

Interior a priori error estimates in Sobolev norms are derived from interior Ritz-Galerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on both uniform and nonuniform meshes. It is shown that the error in an interior domain

Ω
\Omega

can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error in a weaker norm over a slightly larger domain which measures the effects from outside of the domain

Ω
\Omega

. Additional results are given in the case when the subspaces are defined on a uniform mesh. Applications to specific boundary value problems are given.
An observation concerning Ritz-Galerkin methods with indefinite bilinear forms
Authors: Schatz Alfred H.
Published: June 2010

Existence, uniqueness and error estimates for Ritz-Galerkin methods are discussed in the case where the associated bilinear form satisfies a Gårding type inequality, i.e., it is indefinite in a certain way. An application to the finite element method is given.
On Tikhonov’s method for ill-posed problems
Authors: Franklin Joel N.
Published: June 2010

For Tikhonov’s regularization of ill-posed linear integral equations, numerical accuracy is estimated by a modulus of convergence, for which upper and lower bounds are obtained. Applications are made to the backward heat equation, to harmonic continuation, and to numerical differentiation.
An algorithm for finding the blocks of a permutation group
Authors: Atkinson M. D.
Published: June 2010

An efficient algorithm for finding the blocks of imprimitivity of a group from generating permutations is described and justified.
On Laplace transforms near the origin
Authors: Wong R.
Published: June 2010

Let

f
(
t
)

f(t)

be locally integrable on

[
0
,
∞
)

[0,\infty )

and let

L
{
f
}
(
s
)

L\{ f\} (s)

denote the Laplace transform of

f
(
t
)

f(t)

. In this note, we prove that if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis t right-parenthesis tilde t Superscript negative beta Baseline normal upper Sigma Subscript n equals 0 Superscript normal infinity Baseline a Subscript n Baseline left-parenthesis log t right-parenthesis Superscript negative n">

f
(
t
)
∼

t

−
β

Σ

n
=
0

∞

a
n

(
log
⁡
t

)

−
n

f(t) \sim {t^{ - \beta }}\Sigma _{n = 0}^\infty {a_n}{(\log t)^{ - n}}

as

t
→
∞

t \to \infty

, where

0
⩽
Re
⁡
β
>
1

0 \leqslant \operatorname {Re} \beta > 1

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-brace f right-brace left-parenthesis s right-parenthesis tilde s Superscript beta minus 1 Baseline normal upper Sigma Subscript n equals 0 Superscript normal infinity Baseline c Subscript n Baseline left-parenthesis log 1 slash s right-parenthesis Superscript negative n">

L
{
f
}
(
s
)
∼

s

β
−
1

Σ

n
=
0

∞

c
n

(
log
⁡
1

/

s

)

−
n

L\{ f\} (s) \sim {s^{\beta - 1}}\Sigma _{n = 0}^\infty {c_n}{(\log 1/s)^{ - n}}

as

s
→
0

s \to 0

in

|

arg
⁡
s

|

⩽
π

/

2
−
Δ

|\arg s| \leqslant \pi /2 - \Delta

, the

c
n

{c_n}

being constants.
A note on primality testing using Lucas sequences
Authors: Morrison Michael A.
Published: June 2010

For an odd integer

N
>
1

N > 1

, thought to be prime, a test is given which uses Lucas sequences and which can establish that any prime divisors of N are

≡
±
1

\equiv \pm 1

modulo the factored portion of

N
+
1

N + 1

.
Approximation methods for nonlinear problems with application to two-point boundary value problems
Authors: Keller H. B.
Published: June 2010

General nonlinear problems in the abstract form

F
(
χ
)
=
0

F(\chi ) = 0

and corresponding families of approximating problems in the form

F
h

(

χ
h

)
=
0

{F_h}({\chi _h}) = 0

are considered (in an appropriate Banach space setting). The relation between "isolation" and "stability" of solutions is briefly studied. The main result shows, essentially, that, if the nonlinear problem has an isolated solution and the approximating family has stable Lipschitz continuous linearizations, then the approximating problem has a stable solution which is close to the exact solution. Error estimates are obtained and Newton’s method is shown to converge quadratically. These results are then used to justify a broad class of difference schemes (resembling linear multistep methods) for general nonlinear two-point boundary value problems.
Congruence subgroups of the modular group
Authors: Newman Morris
Published: June 2010

The congruence subgroups of the classical modular group which can be defined as the automorphs modulo q of some fixed matrix are studied, and their genera determined.
Polynomial expansions
Authors: Fields Jerry L. , Ismail Mourad E. H.
Published: June 2010

The expansion of arbitrary power series in various classes of polynomial sets is considered. Some applications are also given.
GAP—a PIC-type fluid code
Authors: Marder B. M.
Published: June 2010

GAP, a PIC-type fluid code for computing compressible flows, is described and demonstrated. While retaining some features of PIC, it is felt that the GAP approach is conceptually and operationally simpler.
Three elliptic curves with rank at least seven
Authors: Penney David E. , Pomerance Carl
Published: June 2010

Three rational elliptic curves whose ranks are at least 7 are exhibited. The arithmetical details are given for one of the curves, namely

y
2

=

x
3

+
1692602

x
2

−
530052723915
x

{y^2} = {x^3} + 1692602{x^2} - 530052723915x

.
Multiply schemes and shuffling
Authors: Rosenblatt M.
Published: June 2010

Multiply schemes are used as a model of a linear congruential scheme. It is suggested how the properties of linear congruential schemes as pseudo-random number generators might be improved by shuffling. Asymptotic frequencies of pairs and triples from multiply schemes are obtained.
Restricted range approximation and its application to digital filter design
Authors: Lewis James T.
Published: June 2010

The multiple exchange algorithm for restricted range approximation is discussed. Efficient formulas are derived for the numerical implementation of the method. Discretization effects are analyzed mathematically. The method is applied to a certain problem arising in digital filter design.
Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function
Authors: Temme N. M.
Published: June 2010

New asymptotic expansions are derived for the incomplete gamma functions and the incomplete beta function. In each case the expansion contains the complementary error function and an asymptotic series. The expansions are uniformly valid with respect to certain domains of the parameters.
Polynomial evaluation schemes
Authors: Newbery A. C. R.
Published: June 2010

An attempt is made to define a polynomial evaluation algorithm that is more resistant to accumulated round-off error than the schemes of Horner and Clenshaw under conditions of floating-point arithmetic. An algorithm is presented which generally compares favorably with both. Some suggestions are made, which could plausibly lead to substantial further improvements.
Coupled sound and heat flow and the method of least squares
Authors: Carasso Alfred
Published: June 2010

We construct and analyze a least-squares procedure for approximately solving the initial-value problem for the linearized equations of coupled sound and heat flow, in a bounded domainΩ\OmegainRN{R^N}, with homogeneous Dirichlet boundary conditions. The method is based on Crank-Nicolson time differencing. To approximately solve the resulting system of boundary value problems at each time step, a least-squares method is devised, using trial functions which need not satisfy the homogeneous boundary conditions. Certain unknown normal derivatives of the solution enter the boundary integrals. By using suitable weights, these unknown derivatives can be set equal to zero without impairing theO(k2)O({k^2})accuracy of the Crank-Nicolson scheme. However, one must use smoother trial functions to obtain this accuracy.
Seven-point difference schemes for hyperbolic equations
Authors: Livne Avishai
Published: June 2010

A necessary and sufficient condition is given for all hyperbolic difference schemes that use up to nine mesh points to be of second-order accuracy. We also construct a new difference scheme for two-dimensional hyperbolic systems of conservation laws. The scheme is of second-order accuracy and requires knowledge of only seven mesh points. A stability condition is obtained and is utilized in numerical computations.
Conversion of continued fractions into power series
Authors: Zajta A. J. , Pandikow W.
Published: June 2010

In Section 1, continued fractions of the special form

(

1

)

[
u
n
k
]

\begin{equation}\tag {$1$} [unk]\end{equation}

are considered, and a general formula is given for the coefficients of the power series corresponding to (1). In Section 2, the problems of programming the computation of coefficients are discussed. The continued fraction (1), or its variants, has been studied by many distinguished mathematicians. The problem of converting (1) into a power series has also been considered, and a number of partial results are known. A detailed account can be found in Perron [1]. In this paper, we will not make use of the techniques that are generally applied in the theory of continued fractions. Instead, our approach employs some simple combinatorial and probabilistic arguments. Nevertheless, the methods are quite elementary and can be understood even by those who are unfamiliar with probability theory.
Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions
Authors: Schett Alois
Published: June 2010

Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions and tables for the first fifteen leading terms are given. Relations of these coefficients with the randomization distributions are shown.
Nonelliptic approximation of a class of partial differential equations with Neumann boundary condition
Authors: Girault V.
Published: June 2010

This paper is devoted to the numerical resolution of a class of linear partial differential equations with an inhomogeneous Neumann boundary condition. A first order quadrilateral finite element method is used, together with a one-point integration formula. The resulting scheme is simple and widely used but its theory is not classical, in a sense described as "nonelliptic". An important boundary value theorem is derived, in order to handle the Neumann condition. An error bound shows that the scheme is of order one.
Class groups of the quadratic fields found by F. Diaz y Diaz
Authors: Shanks Daniel
Published: June 2010

F. Diaz y Diaz has discovered 99 discriminants d between

−
3321607

- 3321607

and

−
60638515

- 60638515

inclusive for which

Q
(

d

)

Q(\sqrt d )

have a 3-rank

r
3

=
3

{r_3} = 3

. These 99 imaginary quadratic fields are analyzed here and the class groups are given and discussed for all those of special interest. In 98 cases, the associated real quadratic fields have

r
3

=
2

{r_3} = 2

, but for

d
=
44806173
=
3
⋅
14935391
,
Q
(

d

)

d = 44806173 = 3 \cdot 14935391,Q(\sqrt d )

has a class group

C
(
3
)
×
C
(
3
)
×
C
(
3
)

C(3) \times C(3) \times C(3)

; and this is now the smallest known d for which a real quadratic field has

r
3

=
3

{r_3} = 3

.
Properties of the sequences 3⋅2ⁿ+1
Authors: Golomb Solomon W.
Published: June 2010

For applications to fast finite field transforms, one is interested in the arithmetic of

G
F
(
p
)

GF(p)

, where the order of the multiplicative group,

φ
(
p
)
=
p
−
1

\varphi (p) = p - 1

, is divisible by a high power of 2, and where the multiplicative order of 2 modulo p is large. Primes of the form

3
∙

2
n

+
1

3 \bullet {2^n} + 1

appear well-suited to these objectives. Results are obtained on the divisibility properties of the numbers

A
n

=
3
∙

2
n

+
1

{A_n} = 3 \bullet {2^n} + 1

, and on the exponent of 2 modulo

A
n

{A_n}

when

A
n

{A_n}

is prime. Generalizations to various related types of sequences are also considered.
Asymptotic normality in Monte Carlo integration
Authors: Okamoto Masashi
Published: June 2010

To estimate a multiple integral of a function over the unit cube, Haber proposed two Monte Carlo estimators

J
1
′

J’_1

and

J
2
′

J’_2

based on 2N and 4N observations, respectively, of the function. He also considered estimators

D
1
2

D_1^2

and

D
2
2

D_2^2

of the variances of

J
1
′

J’_1

and

J
2
′

J’_2

, respectively. This paper shows that all these estimators are asymptotically normally distributed as N tends to infinity.
A finite element method for the stationary Stokes equations using trial functions which do not have to satisfy 𝑑𝑖𝑣𝜈=0
Authors: Falk Richard S.
Published: June 2010

By adding a term to the variational equations, we derive a new finite element method for the stationary Stokes equations which eliminates the

div

υ
=
0

{\operatorname {div}}\upsilon = 0

restriction on the trial functions. The method is described using continuous piecewise linear functions, and the optimal

O
(
h
)

O(h)

order of convergence estimate is derived for the error in the

H
1

(
Ω
)

{H^1}(\Omega )

norm.
An averaging method for the stiff highly oscillatory problem
Authors: Miranker W. L. , Wahba G.
Published: June 2010

We show how to replace the point functionals of numerical analysis with types of stable functionals of highly oscillatory solutions of differential equations. This replacement leads to the development of effective numerical methods for the stiff highly oscillatory problem.
Projection methods with different trial and test spaces
Authors: Mock M. S.
Published: June 2010

We consider finite element projection methods for linear partial differential equations, in which the spaces of trial functions and test functions may be different. In addition to approximation and smoothness properties, conditions implying equality of dimensions and uniform coerciveness are required, the most important of which resembles a strong form of an inverse assumption. Our results provide a mechanism for the difference in the rate of convergence of Galerkin procedures with cubic splines and Hermite cubics, applied to first order symmetric hyperbolic problems [13].
On solving weakly singular Volterra equations of the first kind with Galerkin approximations
Authors: Bownds John M.
Published: June 2010

The basic linear, Volterra integral equation of the first kind with a weakly singular kernel is solved via a Galerkin approximation. It is shown that the approximate solution is a sum with the first term being the solution of Abel’s equation and the remaining terms computable as components of the solution of an initial-value problem. The method represents a significant decrease in the normal number of computations required to solve the integral equation.
Sharper bounds for the Chebyshev functions 𝜃(𝑥) and 𝜓(𝑥). II
Authors: Schoenfeld Lowell
Published: June 2010

In this paper, bounds given in the first part of the paper are strengthened. In addition, it is shown that the interval

(
x
,
x
+
x

/

16597
)

(x,x + x/16597)

contains a prime for all

x
⩾
2
,
010
,
760

x \geqslant 2,010,760

; and explicit bounds for the Chebyshev functions are given under the assumption of the Riemann hypothesis.
On the distribution of pseudo-random numbers generated by the linear congruential method. III
Authors: Niederreiter Harald
Published: June 2010

The discrepancy of a sequence of pseudo-random numbers generated by the linear congruential method, both homogeneous and inhomogeneous, is estimated for parts of the period that are somewhat larger than the square root of the modulus. The analogous problem for an arbitrary linear congruential generator modulo a prime is also considered, the result being particularly interesting for maximal period sequences. It is shown that the discrepancy estimates in this paper are best possible apart from logarithmic factors.
Improvement by iteration for compact operator equations
Authors: Sloan Ian H.
Published: June 2010

The equation

y
=
f
+
K
y

y = f + Ky

is considered in a separable Hilbert space H, with K assumed compact and linear. It is shown that every approximation to y of the form

y

1
n

=

Σ
n

a

n
i

u
i

{y_{1n}} = {\Sigma ^n}{a_{ni}}{u_i}

(where {

u
i

{u_i}

} is a given complete set in H, and the

a

n
i

,
1
⩽
i
⩽
n

{a_{ni}},1 \leqslant i \leqslant n

, are arbitrary numbers) is less accurate than the best approximation of the form

y

2
n

=
f
+

Σ
n

b

n
i

K

u
i

{y_{2n}} = f + {\Sigma ^n}{b_{ni}}K{u_i}

, if n is sufficiently large. Specifically it is shown that if

y

1
n

{y_{1n}}

is chosen optimally (i.e. if the coefficients

a

n
i

{a_{ni}}

are chosen to minimize

‖

y
−

y

1
n

‖

\left \| {y - {y_{1n}}} \right \|

), and if

y

2
n

{y_{2n}}

is chosen to be the first iterate of

y

1
n

{y_{1n}}

, i.e.

y

2
n

=
f
+
K

y

1
n

{y_{2n}} = f + K{y_{1n}}

, then

‖

y
−

y

2
n

‖

⩽

α
n

‖

y
−

y

1
n

‖

\left \| {y - {y_{2n}}} \right \| \leqslant {\alpha _n}\left \| {y - {y_{1n}}} \right \|

, with

α
n

→
0

{\alpha _n} \to 0

. A similar result is also obtained, provided the homogeneous equation

x
=
K
x

x = Kx

has no nontrivial solution, if instead

y

1
n

{y_{1n}}

is chosen to be the approximate solution by the Galerkin or Galerkin-Petrov method. A generalization of the first result to the approximate forms

y

3
n

,

y

4
n

,
…

{y_{3n}},{y_{4n}}, \ldots

obtained by further iteration is also shown to be valid, if the range of K is dense in H.
On the smoothness of best 𝐿₂ approximants from nonlinear spline manifolds
Authors: Chui Charles K. , Smith Philip W. , Ward Joseph D.
Published: June 2010

Let

S
n
k

S_n^k

be the nonlinear spline manifold of order k and with n - k interior variable knots. We prove that all best

L
2

[
0
,
1
]

{L_2}[0,1]

approximants from

S
n
k

S_n^k

to a continuous function on [0, 1] are also continuous there. We also prove that there exists a

C
∞

[
0
,
1
]

{C^\infty }[0,1]

function with no

C
2

[
0
,
1
]

{C^2}[0,1]

best

L
2

[
0
,
1
]

{L_2}[0,1]

approximants from

S
n
k

S_n^k

.
A discrete least squares method
Authors: Sammon Peter H.
Published: June 2010

We consider a discrete least squares approximation to the solution of a two-point boundary value problem for a 2mth order elliptic operator. We describe the approximation space of piecewise polynomials and devise a Gaussian quadrature rule that is suitable for replacing the integrals in the usual least squares method. We then show that if the quadrature rule is of sufficient accuracy, the optimal order of convergence is obtained.
High order fast Laplace solvers for the Dirichlet problem on general regions
Authors: Pereyra Victor , Proskurowski Wlodzimierz , Widlund Olof
Published: June 2010

Highly accurate finite difference schemes are developed for Laplace’s equation with the Dirichlet boundary condition on general bounded regions in

R
n

{R^n}

. A second order accurate scheme is combined with a deferred correction or Richardson extrapolation method to increase the accuracy. The Dirichlet condition is approximated by a method suggested by Heinz-Otto Kreiss. A convergence proof of his, previously not published, is given which shows that, for the interval size h, one of the methods has an accuracy of at least

O
(

h

5.5

)

O({h^{5.5}})

in

L
2

{L_2}

. The linear systems of algebraic equations are solved by a capacitance matrix method. The results of our numerical experiments show that highly accurate solutions are obtained with only a slight additional use of computer time when compared to the results obtained by second order accurate methods.
Estimates of the speed of convergence of continued fraction expansions of functions
Authors: Field David A.
Published: June 2010

The best recently published methods of obtaining a priori and a posteriori truncation error estimates of continued fractions are reviewed. The applicability of these methods is discussed and a numerical example with tables of their actual estimates is provided.
Uniform convergence of Galerkin’s method for splines on highly nonuniform meshes
Authors: Natterer Frank
Published: June 2010

Different sets of conditions for an estimate of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar y minus y Superscript pi Baseline double-vertical-bar Subscript upper L Sub Subscript normal infinity Subscript left-parenthesis a comma b right-parenthesis Baseline less-than-or-slanted-equals upper C max Underscript i Endscripts h Subscript i Superscript r plus 1 Baseline double-vertical-bar y Superscript left-parenthesis r plus 1 right-parenthesis Baseline double-vertical-bar Subscript upper L Sub Subscript normal infinity Subscript left-parenthesis upper I Sub Subscript i Subscript right-parenthesis">

‖

y
−

y
π

‖

L
∞

(
a
,
b
)

⩽
C

max
i

h
i

r
+
1

‖

y

(
r
+
1
)

‖

L
∞

(

I
i

)

{\left \| {y - {y^\pi }} \right \|_{{L_\infty }(a,b)}} \leqslant C\max \limits _i h_i^{r + 1}{\left \| {{y^{(r + 1)}}} \right \|_{{L_\infty }({I_i})}}

\]
to hold are given. Here,

y
π

{y^\pi }

is the Galerkin approximation to the solution y of a boundary value problem for an ordinary differential equation, the trial functions being polynomials of degree

⩽
r

\leqslant r

on the subintervals

I
i

=
[

x
i

,

x

i
+
1

]

{I_i} = [{x_i},{x_{i + 1}}]

of length

h
i

{h_i}

. The sequence of subdivisions

π
:

x
0

>

x
1

>
⋯
>

x
n

\pi :{x_0} > {x_1} > \cdots > {x_n}

need not be quasi-uniform.
On the imaginary bicyclic biquadratic fields with class-number 2
Authors: Buell D. A. , Williams H. C. , Williams K. S.
Published: June 2010

Assuming that the list of imaginary quadratic number fields of class-number 4 is complete, a determination is made of all imaginary bicyclic biquadratic number fields of class-number 2.
Corrections to Dickson’s table of three dimensional division algebras over 𝐹₅
Authors: Long F. W.
Published: June 2010

In 1906 Dickson published a list of noncommutative division algebras of dimension three over the field of finite elements. Recently, Kaplansky has found a mistake in this list; and I used a computer to check all the entries in the list. I then found an algebra missing from the list and used the computer again to make a complete search for all such algebras.
Chebyshev polynomials in the numerical solution of differential equations
Authors: Morris A. G. , Horner T. S.
Published: June 2010

Amongst satisfactory techniques for the numerical solution of differential equations, the use of Chebyshev series is often avoided because of the tedious nature of the calculations. A systematic application of the Chebyshev method is given for certain fourth order boundary value problems in which the derivatives have polynomial coefficients. Numerical results for various problems using the Chebyshev method are superior to those obtained by alternative methods.
On maximal finite irreducible subgroups of 𝐺𝐿(𝑛,𝑍). II. The six dimensional case
Authors: Plesken Wilhelm , Pohst Michael
Published: June 2010

All maximal finite absolutely irreducible subgroups of

G
L
(
6
,

Z

)

GL(6,{\mathbf {Z}})

are determined up to Z-equivalence.
Single step Galerkin approximations for parabolic problems
Authors: Baker Garth A. , Bramble James H. , Thomée Vidar
Published: June 2010

In this paper we construct and analyze classes of single step methods of arbitrary order for homogeneous linear initial boundary value problems for parabolic equations with time-independent coefficients. The spatial discretization is done by means of general Galerkin-type methods.
On projective representations of finite wreath products
Authors: Durbin John R. , Farmer K. Bolling
Published: June 2010

The theory of induced projective representations is applied to finite wreath products, yielding algorithms which add to the collection of groups for which projective representations can be computed systematically. For finite Abelian and Abelian-wreath-cyclic groups, the factor sets are determined explicitly by establishing a one-to-one correspondence between certain lower triangular matrices and the inequivalent factor sets of these two classes of groups. This correspondence is used to determine the number and degrees of the inequivalent, irreducible projective representations.
Error estimates for a stiff differential equation procedure
Authors: Sacks-Davis R.
Published: June 2010

For numerical procedures which solve stiff systems of ordinary differential equations there are problems associated with estimating the local error. In this paper an analysis based on the linear model

y
′
=
A
y

y\prime = Ay

is carried out for a particular method based on second derivative formulas. It is shown that there exists an error estimate based on a comparison between predicted and corrected values which is both reliable and efficient.
Gauss interpolation formulas and totally positive kernels
Authors: Barrow David L.
Published: June 2010

This paper simplifies and generalizes an earlier result of the author’s on Gauss interpolation formulas for the one-dimensional heat equation. Such formulas approximate a function at a point

(

x
∗

,

t
∗

)

({x^\ast },{t^\ast })

in terms of a linear combination of its values on an initial-boundary curve in the (x, t) plane. The formulas are characterized by the requirement that they be exact for as many basis functions as possible. The basis functions are generated from a Tchebycheff system on the line

t
=
0

t = 0

by an integral kernel

K
(
x
,
y
,
t
)

K(x,y,t)

, in analogy with the way heat polynomials are generated from the monomials

x
i

{x^i}

by the fundamental solution to the heat equation. The total positivity properties of

K
(
x
,
y
,
t
)

K(x,y,t)

together with the theory of topological degree are used to establish the existence of the formulas.
An effective number geometric method of computing the fundamental units of an algebraic number field
Authors: Pohst Michael , Zassenhaus Hans
Published: June 2010

The Minkowski method of unit search is applied to particular types of parallelotopes permitting to discover algebraic integers of bounded norm in a given algebraic number field of degree n at will by solving successively

2
n

2n

linear inequalities for one unknown each. Application is made to the unit search for all totally real number fields of minimal discriminant for

n
⩽
7

n \leqslant 7

.
On the 𝑝-divisibility of the Fermat quotients
Authors: Johnson Wells
Published: June 2010

Upper bounds for the power of p which divides the Fermat quotient

q
a

=
(

a

p
−
1

−
1
)

/

p

{q_a} = ({a^{p - 1}} - 1)/p

are obtained, and conditions are given which imply that

q
a

≢

0

{q_a}\nequiv \;0

(
mod

p
)

\pmod p

. The results are in terms of the number of steps in a simple algorithm which determines the semiorder of a

(
mod

p
)

\pmod p

.
To compute the optimal interpolation formula
Authors: Gaffney P. W.
Published: June 2010

The purpose of this paper is to explain how to compute the function

Ω
\Omega

which interpolates values of a function of one variable

f
(
x
)

f(x)

at n distinct points

x
1

>

x
2

>
⋯
>

x

n
−
1

>

x
n

{x_1} > {x_2} > \cdots > {x_{n - 1}} > {x_n}

and which, whenever

‖

f

(
k
)

‖

∞

\left \|{f^{(k)}}\right \|_\infty

is bounded and the value of the bound is unknown, provides the smallest possible value of

B
(
x
)

B(x)

in the error bound
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f left-parenthesis x right-parenthesis minus normal upper Omega left-parenthesis x right-parenthesis EndAbsoluteValue less-than-or-slanted-equals upper B left-parenthesis x right-parenthesis double-vertical-bar f Superscript left-parenthesis k right-parenthesis Baseline double-vertical-bar Subscript normal infinity period">

|

f
(
x
)
−
Ω
(
x
)

|

⩽
B
(
x
)

‖

f

(
k
)

‖

∞
.

|f(x) - \Omega (x)| \leqslant B(x)\left \|{f^{(k)}}\right \|_{\infty .}

\]
Revision of a derivative-free quasi-Newton method
Authors: Greenstadt John
Published: June 2010

A derivative-free Quasi-Newton (DFQN) method previously published [J. Greenstadt, Math. Comp., v. 26, 1972, pp. 145-166] has been revised and simplified. The main modification has the effect of keeping all the successive approximants to the Hessian matrix positive-definite. This, coupled with some improvements in the line search, has enhanced the performance of the method considerably. The results of numerical trials on many of the "standard" test functions are displayed, in addition to comparisons with two other methods. These indicate that the present DFQN method is not too far behind that of Gill, Murray and Pitfield, the most efficient one presently known.
On the “3𝑥+1” problem
Authors: Crandall R. E.
Published: June 2010

It is an open conjecture that for any positive odd integer m the function
\[

C
(
m
)
=
(
3
m
+
1
)

/

2

e
(
m
)

,

C(m) = (3m + 1)/{2^{e(m)}},

\]
where

e
(
m
)

e(m)

is chosen so that

C
(
m
)

C(m)

is again an odd integer, satisfies

C
h

(
m
)
=
1

{C^h}(m) = 1

for some h. Here we show that the number of

m
⩽
x

m \leqslant x

which satisfy the conjecture is at least

x
c

{x^c}

for a positive constant c. A connection between the validity of the conjecture and the diophantine equation

2
x

−

3
y

=
p

{2^x} - {3^y} = p

is established. It is shown that if the conjecture fails due to an occurrence

m
=

C
k

(
m
)

m = {C^k}(m)

, then k is greater than 17985. Finally, an analogous "

q
x
+
r

qx + r

" problem is settled for certain pairs

(
q
,
r
)
≠
(
3
,
1
)

(q,r) \ne (3,1)

.
A note on congruent numbers
Authors: Godwin H. J.
Published: June 2010

The list of known congruent numbers less than 1000 is extended by a proof that 19 primes of the form

8
k
+
7

8k + 7

are congruent.
Computation of the 2-rank of pure cubic fields
Authors: Eisenbeis H. , Frey G. , Ommerborn B.
Published: June 2010

For

k
∈

Z

∖
{
0
}

k \in {\mathbf {Z}}\backslash \{ 0\}

there is a close connection between a certain subgroup of the Selmer group of the elliptic curve given by:

y
2

=

x
3

+
k

{y^2} = {x^3} + k

, and the group of elements of order 2 of the class group

Cl

(
k
)

{\text {Cl}}(k)

of

Q

(

k
3

)

{\mathbf {Q}}(\sqrt [3]{k})

denoted by

Cl
2

(
k
)

{\text {Cl}_2}(k)

(cf. [4]). In the following paper we give some consequences of this fact, that make the computation of

Cl
2

(
k
)

{\text {Cl}_2}(k)

considerably easier. For

k
>
10

000

k > 10\,000

we compute

Cl
2

(
k
)

{\text {Cl}_2}(k)

by methods developed in [2], and by using [1] we get the structure of the 2-primary part of

Cl

(
k
)

{\text {Cl}}(k)

with the exception of 39 cases.
Odd integers 𝑁 with five distinct prime factors for which 2-10⁻¹²<𝜎(𝑁)/𝑁<2+10⁻¹²
Authors: Kishore Masao
Published: June 2010

We make a table of odd integers N with five distinct prime factors for which

2
−

10

−
12

>
σ
(
N
)

/

N
>
2
+

10

−
12

2 - {10^{ - 12}} > \sigma (N)/N > 2 + {10^{ - 12}}

, and show that for such

N

|

σ
(
N
)

/

N
−
2

|

>

10

−
14

N\;|\sigma (N)/N - 2| > {10^{ - 14}}

. Using this inequality, we prove that there are no odd perfect numbers, no quasiperfect numbers and no odd almost perfect numbers with five distinct prime factors. We also make a table of odd primitive abundant numbers N with five distinct prime factors for which

2
>
σ
(
N
)

/

N
>
2
+
2

/

10

10

2 > \sigma (N)/N > 2 + 2/{10^{10}}

.
Convergence of vortex methods for Euler’s equations
Authors: Hald Ole , del Prete Vincenza Mauceri
Published: June 2010

A numerical method for approximating the flow of a two dimensional incompressible, inviscid fluid is examined. It is proved that for a short time interval Chorin’s vortex method converges superlinearly toward the solution of Euler’s equations, which govern the flow. The length of the time interval depends upon the smoothness of the flow and of the particular cutoff. The theory is supported by numerical experiments. These suggest that the vortex method may even be a second order method.
Minimum norm symmetric quasi-Newton updates restricted to subspaces
Authors: Schnabel Robert B.
Published: June 2010

The Davidon-Fletcher-Powell and Broyden-Fletcher-Goldfarb-Shanno updates have been the two most successful quasi-Newton updates for a variety of applications. One reason offered in explanation is that they constitute, in an appropriate norm and metric, the minimum norm change to the matrix, or its inverse, being approximated which preserves symmetry and obeys the quasi-Newton equation. Recent methods have reason to consider updates restricted to certain subspaces. In this paper we derive the general minimum norm symmetric quasi-Newton updates restricted to such subspaces. In the same appropriate norm and metric, the minimum norm change update to the matrix or its inverse is shown to be, respectively, the rank-two update which is a particular projection of the DFP or BFGS onto this subspace.
Chebyshev expansions for the error and related functions
Authors: Schonfelder J. L.
Published: June 2010

New 30 decimal place Chebyshev expansions, which can be used to obtain approximations up to that accuracy, for the error function and its complement are presented. Bilinear and biquadratic mappings of the independent variable have been employed, in one case to improve convergence and in the other to increase the basic approximation range. The practical effects of these mappings are discussed, and the method used to generate the tabulated expansions is outlined. The expansions here presented form the basis of the routines in the S15 chapter of the NAG library.
A proof of convergence and an error bound for the method of bisection in 𝑅ⁿ
Authors: Kearfott Baker
Published: June 2010

Let

S
=
⟨

X
0

,
.
.
,

X
m

⟩

S = \langle {X_0},..,{X_m}\rangle

be an m-simplex in

R

n

{{\mathbf {R}}^n}

. We define "bisection" of S as follows. We find the longest edge

⟨

X
i

,

X
j

⟩

\langle {X_i},{X_j}\rangle

of S, calculate its midpoint

M
=
(

X
i

+

X
j

)

/

2

M = ({X_i} + {X_j})/2

, and define two new m-simplexes

S
1

{S_1}

and

S
2

{S_2}

by replacing

X
i

{X_i}

by M or

X
j

{X_j}

by M. Suppose we bisect

S
1

{S_1}

and

S
2

{S_2}

, and continue the process for p iterations. It is shown that the diameters of the resulting Simplexes are no greater then

(

3

/

2

)

⌊

p

/

m

⌋

{(\sqrt 3 /2)^{\left \lfloor {p/m} \right \rfloor }}

times the diameter of the original simplex, where

⌊

p

/

m

⌋

\left \lfloor {p/m} \right \rfloor

is the largest integer less than or equal to

p

/

m

p/m

.
A dispersion analysis for difference schemes: tables of generalized Airy functions
Authors: Chin R. C. Y. , Hedstrom G. W.
Published: June 2010

This paper contains graphs and tables of the function
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A i Subscript p comma q Baseline left-parenthesis alpha comma x right-parenthesis equals integral Subscript negative normal infinity Superscript normal infinity Baseline left-parenthesis 2 pi right-parenthesis Superscript negative 1 Baseline exp left-brace i y Superscript p Baseline slash p minus alpha y Superscript q Baseline slash q plus i x y right-brace d y">

A

i

p
,
q

(
α
,
x
)
=

∫

−
∞

∞

(
2
π
)

−
1

exp
⁡
{
i

y
p

/

p
−
α

y
q

/

q
+
i
x
y
}

d
y

A{i_{p,q}}(\alpha ,x) = \int _{ - \infty }^\infty {{{(2\pi )}^{ - 1}}\exp \{ i{y^p}/p - \alpha {y^q}/q + ixy\} \;dy}

\]
and its indefinite integral for

p
=
3
,
5
,
7

p = 3,5,7

, for

q
=
2
,
4
,
6

q = 2,4,6

, and for several values of

α
\alpha

with

α
⩾
0

\alpha \geqslant 0

. It is shown how these tables should influence the choice of an artificial viscosity for a difference scheme for a linear hyperbolic equation.
Table of the cyclotomic class numbers ℎ*(𝑝) and their factors for 200<𝑝<521
Authors: Lehmer D. H. , Masley J. M.
Published: June 2010

This table gives the values of the "first factor"

h
∗

(
p
)

{h^\ast }(p)

and their factorizations for all primes

p
,
200
>
p
>
521

p,200 > p > 521

. This extends similar data by M. Newman [Math. Comp., v. 24, 1970, pp. 215-219], and Schrutka [Berlin Akad. Abn., 1964]. The two methods used to compute these data are described.
Limiting precision in differential equation solvers. II. Sources of trouble and starting a code
Authors: Shampine L. F.
Published: June 2010

The reasons a class of codes for solving ordinary differential equations might want to use an extremely small step size are investigated. For this class the likelihood of precision difficulties is evaluated and remedies examined. The investigation suggests a way of selecting automatically an initial step size which should be reliably on scale.
An asymptotic approximation for a type of Fourier integral
Authors: Schmidt Paul W.
Published: June 2010

A uniform asymptotic approximation which can be used for all

q
h
⩾
0

qh \geqslant 0

is developed for the Fourier integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis h right-parenthesis equals integral Subscript q Superscript z Baseline StartFraction f left-parenthesis StartRoot y squared minus q squared right-parenthesis EndRoot Over left-parenthesis y squared minus q squared right-parenthesis Superscript 1 slash 2 Baseline EndFraction sine y h d y">

I
(
h
)
=

∫
q
z

f
(

y
2

−

q
2

)

(

y
2

−

q
2

)

1

/

2

sin
⁡
y
h

d
y

I(h) = \int _q^z {\frac {{f(\sqrt {{y^2} - {q^2})} }}{{{{({y^2} - {q^2})}^{1/2}}}}\sin yh\,dy}

\]
under the assumptions that

h
z
>>
1

hz > > 1

, that the first

L
+
2

L + 2

derivatives of

f
(
y
)

f(y)

are continuous for

0
⩽
y
⩽

(

z
2

−

q
2

)

1

/

2

0 \leqslant y \leqslant {({z^2} - {q^2})^{1/2}}

, and that the first

2
L
+
2

2L + 2

derivatives of

f
(
y
)

f(y)

are continuous at

y
=
0

y = 0

. The approximation is especially convenient when

z
>>
q

z > > q

.