bluuebunny/crossref_metadata_2025 huggingface dataset:
On the sparse and symmetric least-change secant update
Authors: Steihaug Trond
Published: June 2010

To find the sparse and symmetric n by n least-change secant update we have to solve a consistent linear system of n equations in n unknowns, where the coefficient matrix is symmetric and positive semidefinite. We give bounds on the eigenvalues of the coefficient matrix and show that the preconditioned conjugate gradient method is a very efficient method for solving the linear equation. By solving the linear system only approximately, we generate a family of sparse and symmetric updates with a residual in the secant equation. We address the question of how accurate a solution is needed not to impede the convergence of quasi-Newton methods using the approximate least-change update. We show that the quasi-Newton methods are locally and superlinearly convergent after one or more preconditioned conjugate gradient iterations.
Numerical solution of two transcendental equations
Authors: Misici Luciano
Published: June 2010

This paper deals with the study of the transcendental equations:

sin
⁡
(
s
+
v
)

/

(
s
+
v
)
=
±
sin
⁡
(
s
−
v
)

/

(
s
−
v
)

\sin (s + v)/(s + v) = \pm \sin (s - v)/(s - v)

, where

v
=

(

s
2

−

γ
2

)

1

/

2

v = {({s^2} - {\gamma ^2})^{1/2}}

. These equations are obtained in the study of some boundary value problems for a modified biharmonic equation using the Papkovich-Fadle series. Some numerical solutions obtained with an iterative procedure are given.
Further inequalities for the gamma function
Authors: Laforgia Andrea
Published: June 2010

For

λ
>
0

\lambda > 0

and

k
⩾
0

k \geqslant 0

we present a method which permits us to obtain inequalities of the type
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis k plus alpha right-parenthesis Superscript lamda minus 1 Baseline greater-than normal upper Gamma left-parenthesis k plus lamda right-parenthesis slash normal upper Gamma left-parenthesis k plus 1 right-parenthesis greater-than left-parenthesis k plus beta right-parenthesis Superscript lamda minus 1">

(
k
+
α

)

λ
−
1

>
Γ
(
k
+
λ
)

/

Γ
(
k
+
1
)
>

(
k
+
β

)

λ
−
1

{(k + \alpha )^{\lambda - 1}} > \Gamma (k + \lambda )/\Gamma (k + 1) > {(k + \beta )^{\lambda - 1}}

, with the usual notation for the gamma function, where

α
\alpha

and

β
\beta

are independent of k. Some examples are also given which improve well-known inequalities. Finally, we are also able to show in some cases that the values

α
\alpha

and

β
\beta

in the inequalities that we obtain cannot be improved.
A complete axiomatization of computer arithmetic
Authors: Mansfield Richard
Published: June 2010

We define an axiom system for rounded arithmetic to be complete if we can recover from any model of the axioms the exact algebra from whence it came. A complete set of axioms is given for rounded addition and multiplication.
Hensel and Newton methods in valuation rings
Authors: von zur Gathen Joachim
Published: June 2010

We give a computational description of Hensel’s method for lifting approximate factorizations of polynomials. The general setting of valuation rings provides the framework for this and the other results of the paper. We describe a Newton method for solving algebraic and differential equations. Finally, we discuss a fast algorithm for factoring polynomials via computing short vectors in modules.
The mean values of totally real algebraic integers
Authors: Smyth C. J.
Published: June 2010

Let

M
p

(
α
)

{M_p}(\alpha )

be the pth root of the mean absolute values of the pth powers of a totally real algebraic integer

α
\alpha

. For each fixed

p
>
0

p > 0

we study the set

M

p

{\mathfrak {M}_p}

of such

M
p

(
α
)

{M_p}(\alpha )

. We show that its structure is as follows: on the nonnegative real line it consists of some isolated points, followed by a small interval in which its structure is as yet undetermined. Beyond this small interval, it is everywhere dense.
Sharp convergence rates for nonlinear product formulas
Authors: Schechter Eric
Published: June 2010

Nonlinear versions of the Lie-Trotter product formula
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="exp left-bracket t left-parenthesis upper A plus upper B right-parenthesis right-bracket equals limit Underscript n right-arrow normal infinity Endscripts left-bracket exp left-parenthesis left-parenthesis t slash n right-parenthesis upper A right-parenthesis exp left-parenthesis left-parenthesis t slash n right-parenthesis upper B right-parenthesis right-bracket Superscript n">

exp
⁡
[
t
(
A
+
B
)
]
=

lim

n
→
∞

[
exp
⁡
(
(
t

/

n
)
A
)
exp
⁡
(
(
t

/

n
)
B
)

]
n

\exp [t(A + B)] = {\lim _{n \to \infty }}{[\exp ((t/n)A)\exp ((t/n)B)]^n}

and related formulas are given in this paper. The convergence rates are optimal. The results are applicable to some nonlinear partial differential equations.
Finite element formulation of the general magnetostatic problem in the space of solenoidal vector functions
Authors: Friedman Mark J.
Published: June 2010

A new finite element method for the solution of the general magnetostatic problem is formulated and analyzed. The space of trial functions consists of solenoidal piecewise polynomial vector functions. We start with an integral formulation, in terms of the flux density, in the domain occupied by magnetic material. Using the properties [10] of the spectrum of the relevant singular integral operator we derive a weak formulation involving an integral operator on the boundary only. Thus the resulting finite element matrix consists of a sparse part corresponding to the interior of the iron domain and a full part corresponding to the boundary. Using the method of monotone operators, existence and uniqueness of the solution of the weak formulation as well as its discretization are proven. Error estimates are derived with the special emphasis on the case when magnetic permeability is large. Finally, solution of the problem by successive iteration is analyzed.
On the asymptotic behavior of scaled singular value and 𝑄𝑅 decompositions
Authors: Stewart G. W.
Published: June 2010

Asymptotic expressions are derived for the singular value decompositon of a matrix some of whose columns approach zero. Expressions are also derived for the QR factorization of a matrix some of whose rows approach zero. The expressions give insight into the method of weights for approximating the solutions of constrained least squares problems.
Accurate computation of divided differences of the exponential function
Authors: McCurdy A. , Ng K. C. , Parlett B. N.
Published: June 2010

The traditional recurrence for the computation of exponential divided differences, along with a new method based on the properties of the exponential function, are studied in detail in this paper. Our results show that it is possible to combine these two methods to compute exponential divided differences accurately. A hybrid algorithm is presented for which our error bound grows quite slowly with the order of the divided difference.
Composite Hermite-Birkhoff quadrature formulas of Gaussian type
Authors: Dyn Nira
Published: June 2010

We show how to combine incidence matrices, which admit Hermite-Birkhoff quadrature formulas of Gaussian type for any positive measure, in such a way that the resulting matrix also admits Gaussian type quadratures for any positive measure. Moreover, the uniqueness property and the extremal property of the formulas corresponding to the submatrices are transferred to the formula admitted by the composed matrix.
Analysis of some finite elements for the Stokes problem
Authors: Bernardi Christine , Raugel Geneviève
Published: June 2010

We study some finite elements which are used in the approximation of the Stokes problem, so as to obtain error estimates of optimal order. Nous étudions deux éléments finis utilisés pour l’approximation du problème de Stokes et obtenons des estimations d’erreur d’ordre optimal.
A rapid robust rootfinder
Authors: Shrager Richard I.
Published: June 2010

A numerical algorithm is presented for solving one nonlinear equation in one real variable. Given

F
X

{F_X}

with brackets A and B, i.e.,

sign
⁡
(

F
A

)
=
−
sign
⁡
(

F
B

)
≠
0

\operatorname {sign}({F_A}) = - \operatorname {sign}({F_B}) \ne 0

, the algorithm finds a zero of F,

A
>
X
>
B

A > X > B

. Alternately, a crossover pair is found, i.e., (X, Y):

sign
⁡
(

F
X

)
=
−
sign
⁡
(

F
Y

)
≠
0

\operatorname {sign}({F_X}) = - \operatorname {sign}({F_Y}) \ne 0

where there is no floating-point number in the system between X and Y. This feature allows full use of machine precision. Optionally, a tolerance

TOL

>
0

{\text {TOL}} > 0

may be given, to permit termination when

|

Y
−
X

|

>

TOL

|Y - X| > {\text {TOL}}

. The method, once rapid convergence sets in, is alternation of one linear interpolation or extrapolation with one inverse quadratic interpolation. The resulting asymptotic convergence rate is competitive with other methods that refine both brackets and do not require

d
F

/

d
X

dF/dX

. Other merits of the algorithm are: robust calculation, efficient three-point interpolation, and superior behavior in bad cases. The algorithm is tested and compared with others.
On the Diophantine equation 1+2^{𝑎}=3^{𝑏}5^{𝑐}+2^{𝑑}3^{𝑒}5^{𝑓}
Authors: Alex Leo J.
Published: June 2010

In this paper the Diophantine equation

1
+

2
a

=

3
b

5
c

+

2
d

3
e

5
f

1 + {2^a} = {3^b}{5^c} + {2^d}{3^e}{5^f}

, where a, b, c, d, e and f are nonnegative integers, is solved. The related equations

1
+

3
a

=

2
b

5
c

+

2
d

3
e

5
f

1 + {3^a} = {2^b}{5^c} + {2^d}{3^e}{5^f}

and

1
+

5
a

=

2
b

3
c

+

2
d

3
e

5
f

1 + {5^a} = {2^b}{3^c} + {2^d}{3^e}{5^f}

are also solved. This work is related to and extends recent work of L. L. Foster, J. L. Brenner, and the author.
Variable step size predictor-corrector schemes for second kind Volterra integral equations
Authors: Jones H. M. , McKee S.
Published: June 2010

In this paper a family of implicit multistep methods for the solution of Volterra integral equations is derived. These methods together with an explicit Euler predictor permit the use of a variable step size when solving integral equations. Means of controlling the error and stability by varying the step size and the order of the method are described. Extensive numerical results are presented.
On polynomial approximation in the complex plane with application to conformal mapping
Authors: Reichel Lothar
Published: June 2010

We consider the selection of polynomial bases for polynomial approximation of analytic functions on bounded simply connected regions in the complex plane. While a monomial basis may be very ill-conditioned, we show that a basis of Lagrange polynomials with Fejér points as nodes is well-conditioned. Numerical examples, where we compute polynomial approximations of conformal mappings, conclude the paper.
On totally real cubic fields
Authors: Ennola Veikko , Turunen Reino
Published: June 2010

The authors have constructed a table of the 26440 nonconjugate totally real cubic number fields of discriminant

D
>
500000

D > 500000

thereby extending the existing table of fields with

D
>
100000

D > 100000

by I. O. Angell [1]. Serious defects in Angell’s table are pointed out. For each field, running number, discriminant, coefficients of a generating polynomial, integral basis, class number, and a fundamental pair of units are listed. The article contains statistics about the following subjects: distribution of class numbers; fields in which every norm-positive unit is totally positive; nonconjugate fields with the same discriminant; fields with noncyclic class group. The fields are tabulated by means of a method due to Davenport and Heilbronn [7], [8] which leads to a unique normalized generating polynomial. The given units are chosen so that the fundamental parallelogram of the unit lattice determined by the corresponding vectors in the logarithmic space is reduced.
Long arithmetic progressions of primes: some old, some new
Authors: Pritchard Paul A.
Published: June 2010

The results are reported of an extensive search with a computer for "long" arithmetic progressions of primes. Such progressions with minimum last term are now known for all lengths up to and including nineteen.
Linear multistep methods for Volterra integral and integro-differential equations
Authors: van der Houwen P. J. , te Riele H. J. J.
Published: June 2010

A general class of linear multistep methods is presented for numerically solving first- and second-kind Volterra integral equations, and Volterra integro-differential equations. These so-called VLM methods, which include the well-known direct quadrature methods, allow for a unified treatment of the problems of consistency and convergence, and have an analogue in linear multistep methods for ODEs, as treated in any textbook on computational methods in ordinary differential equations. General consistency and convergence results are presented (and proved in an Appendix), together with results of numerical experiments which support the theory.
Fractional linear multistep methods for Abel-Volterra integral equations of the second kind
Authors: Lubich Ch.
Published: June 2010

Fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular Volterra integral equations. The proposed methods are convergent of the order of the underlying multistep method, also in the generic case of solutions which are not smooth at the origin. The stability properties (stability region, A-stability,

A
(
α
)

A(\alpha )

-stability) are closely related to those of the underlying linear multistep method.
A geometric proof of total positivity for spline interpolation
Authors: de Boor C. , DeVore R.
Published: June 2010

Simple geometric proofs are given for the total positivity of the B-spline collocation matrix and the variation diminishing property of the B-spline representation of a spline.
The error norm of certain Gaussian quadrature formulae
Authors: Akrivis G.
Published: June 2010

We consider Gauss quadrature formulae

Q
n

{Q_n}

,

n
∈

N

n \in {\mathbf {N}}

, approximating the integral

I
(
f
)
:=

∫

−
1

1

w
(
x
)
f
(
x
)

d
x

I(f): = \smallint _{ - 1}^1w(x)f(x)\;dx

,

w
=
W

/

p
i

w = W/{p_i}

,

i
=
1
,
2

i = 1,2

, with

W
(
x
)
=

(
1
−
x

)
α

(
1
+
x

)
β

W(x) = {(1 - x)^\alpha }{(1 + x)^\beta }

,

α
,
β
=
±
1

/

2

\alpha ,\beta = \pm 1/2

and

p
1

(
x
)
=
1
+

a
2

+
2
a
x

{p_1}(x) = 1 + {a^2} + 2ax

,

p
2

(
x
)
=
(
2
b
+
1
)

x
2

+

b
2

{p_2}(x) = (2b + 1){x^2} + {b^2}

,

b
>
0

b > 0

. In certain spaces of analytic functions the error functional

R
n

:=
I
−

Q
n

{R_n}: = I - {Q_n}

is continuous. In [1] and [2] estimates for

‖

R
n

‖

\left \| {{R_n}} \right \|

are given for a wide class of weight functions. Here, for a restricted class of weight functions, we calculate the norm of

R
n

{R_n}

explicitly.
Further extensions of a Legendre function integral
Authors: Fettis Henry E.
Published: June 2010

The integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript z Superscript 1 Baseline left-parenthesis StartFraction 1 minus t Over 2 EndFraction right-parenthesis Superscript beta minus 1 Baseline left-parenthesis StartFraction 1 minus t Over 1 plus t EndFraction right-parenthesis Superscript mu slash 2 Baseline ln left-parenthesis StartFraction 1 minus t Over 2 EndFraction right-parenthesis upper P Subscript nu minus 1 Superscript mu Baseline left-parenthesis t right-parenthesis d t">

∫
z
1

(

1
−
t

2

)

β
−
1

(

1
−
t

1
+
t

)

μ

/

2

ln
⁡

(

1
−
t

2

)

P

ν
−
1

μ

(
t
)

d
t

\int _z^1 {{{\left ( {\frac {{1 - t}}{2}} \right )}^{\beta - 1}}{{\left ( {\frac {{1 - t}}{{1 + t}}} \right )}^{\mu /2}}\ln \left ( {\frac {{1 - t}}{2}} \right )P_{\nu - 1}^\mu (t)\;dt}

\]
is evaluated as a hypergeometric function for arbitrary values of

ν
\nu

,

μ
\mu

,

−
1
⩽
z
⩽
1

- 1 \leqslant z \leqslant 1

, and

Re
⁡
(
β
)
>
0

\operatorname {Re} (\beta ) > 0

.
Evaluation of Howland-type integrals involving 𝑡𝑎𝑛ℎ and 𝑐𝑜𝑡ℎ functions
Authors: Ling Chih-Bing
Published: June 2010

Four Howland-type integrals involving tanh and coth functions in their integrands are evaluated in this paper. The results are tabulated to 20D.
Breeding amicable numbers in abundance
Authors: Borho W. , Hoffmann H.
Published: June 2010

We give some new methods for the constructive search for amicable number pairs. Our numerical experiments using these methods produced a total of 3501 new amicable pairs of a very special form. They provide some experimental evidence for the infinity of such pairs.
Class numbers of cyclotomic function fields
Authors: Ireland K. F. , Small R. D.
Published: June 2010

Using results of Galovich and Rosen the plus and minus factors for the class number of the cyclotomic function field associated with irreducibles of degree three and four over the field with three elements are computed. As a consequence it is shown that the analogue to a result of Kummer on the p-divisibility of these factors is false.
An iterative method for elliptic problems on regions partitioned into substructures
Authors: Bramble J. H. , Pasciak J. E. , Schatz A. H.
Published: June 2010

Some new preconditioners for discretizations of elliptic boundary problems are studied. With these preconditioners, the domain under consideration is broken into subdomains and preconditioners are defined which only require the solution of matrix problems on the subdomains. Analytic estimates are given which guarantee that under appropriate hypotheses, the preconditioned iterative procedure converges to the solution of the discrete equations with a rate per iteration that is independent of the number of unknowns. Numerical examples are presented which illustrate the theoretically predicted iterative convergence rates.
Complexity of indefinite elliptic problems
Authors: Werschulz Arthur G.
Published: June 2010

This paper deals with the approximate solution of a linear regularly-elliptic 2mth-order boundary-value problem

L
u
=
f

Lu = f

, with

f
∈

H
r

(
Ω
)

f \in {H^r}(\Omega )

for

r
≥
−
m

r \geq - m

. Suppose that the problem is indefinite, i.e., the variational form of the problem involves a weaklycoercive bilinear form. Of particular interest is the quality of the finite element method (FEM) of degree k using n inner products of f. The error of the approximation is measured in the Sobolev l-norm

(
0
≤
l
≤
m
)

(0 \leq l \leq m)

; we assume that

k
≥
2
m
−
1
−
l

k \geq 2m - 1 - l

. We assume that an a priori bound is known for either the Sobolev r-norm or for the Sobolev r-seminorm of f. We first consider the normed case. We find that the FEM has minimal error if and only if

k
≥
2
m
−
1
+
r

k \geq 2m - 1 + r

. Regardless of the values of k and r, there exists a linear combination (called the spline algorithm) of the inner products used by the FEM which does have minimal error. For the seminormed case, we give a very restrictive condition which is necessary and sufficient for the error of the FEM to have a bound which is independent of f. When this condition holds, we find that the FEM has minimal error if and only if

k
≥
2
m
−
1
+
r

k \geq 2m - 1 + r

. However, we once again find that the spline algorithm (using the same inner products as does the FEM) has minimal error, no matter what values k and r have and regardless of whether the FEM has uniformly bounded error. We also show that the inner products used by the FEM is the best set of linear functionals to use.
Fast primality tests for numbers less than 50⋅10⁹
Authors: Kurtz G. C. , Shanks Daniel , Williams H. C.
Published: June 2010

Consider the doubly infinite set of sequences

A
(
n
)

A(n)

given by
\[

A
(
n
+
3
)
=
r
A
(
n
+
2
)
−
s
A
(
n
+
1
)
+
A
(
n
)

A(n + 3) = rA(n + 2) - sA(n + 1) + A(n)

\]
with

A
(
−
1
)
=
s

A( - 1) = s

,

A
(
0
)
=
3

A(0) = 3

,

A
(
1
)
=
r

A(1) = r

. For a given pair r,s, the "signature" of n is defined to be the sextet
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis negative n minus 1 right-parenthesis comma upper A left-parenthesis negative n right-parenthesis comma upper A left-parenthesis negative n plus 1 right-parenthesis comma upper A left-parenthesis n minus 1 right-parenthesis comma upper A left-parenthesis n right-parenthesis comma upper A left-parenthesis n plus 1 right-parenthesis comma">

A
(
−
n
−
1
)
,
A
(
−
n
)
,
A
(
−
n
+
1
)
,
A
(
n
−
1
)
,
A
(
n
)
,
A
(
n
+
1
)
,

A( - n - 1),A( - n),A( - n + 1),A(n - 1),A(n),A(n + 1),

\]
each reduced modulo n. Primes have only three types of signatures, depending on how they split in the cubic field generated by

x
3

−
r

x
2

+
s
x
−
1

{x^3} - r{x^2} + sx - 1

. An "acceptable" composite is a composite integer which has the same type of signature as a prime; such integers are very rare. In this paper, a description is given of the results of a computer search for all acceptable composites

⩽
50
⋅

10
9

\leqslant 50 \cdot {10^9}

in the Perrin sequence

(
r
=
0
,
s
=
−
1
)

(r = 0,s = - 1)

. Also, some numbers which are acceptable composites for both the Perrin sequence and the sequence with

r
=
1

r = 1

,

s
=
0

s = 0

are presented.
The convergence of accelerated overrelaxation iterations
Authors: Hughes-Hallett A. J.
Published: June 2010

Accelerated Overrelaxation Iterations extrapolate the standard Successive Overrelaxation Iterations. This paper provides conditions for the convergence of the Accelerated Overrelaxation process; and a conditionally optimal version is derived for solving an arbitrary real linear equation system.
Some new aspects of rational interpolation
Authors: Schneider Claus , Werner Wilhelm
Published: June 2010

A new algorithm for rational interpolation based on the barycentric formula is developed; the barycentric representation of the rational interpolation function possesses various advantages in comparison with other representations such as continued fractions: it provides, e.g., information concerning the existence and location of poles of the interpolant.
What is the complexity of related elliptic, parabolic, and hyperbolic problems?
Authors: Werschulz Arthur G.
Published: June 2010

Traub and Woźniakowski have dealt with the complexity of some simple partial differential equations. They chose three model problems and showed that the parabolic problem considered had significantly lower complexity than the elliptic problem, which in turn had significantly lower complexity than the hyperbolic problem considered. They asked whether this is true in general. We show that this is not the case by proving that if L is a reasonably well-behaved elliptic operator, then the steady-state heat equation

L
u
=
f

Lu = f

, the heat equation

∂
t

u
+
L
u
=
f

{\partial _t}u + Lu = f

, and the wave equation

∂

t
t

u
+
L
u
=
f

{\partial _{tt}}u + Lu = f

all have roughly the same worst-case complexity for f in the unit ball of a certain Sobolev space of smoothness r.
Stability with large step sizes for multistep discretizations of stiff ordinary differential equations
Authors: Majda George
Published: June 2010

In this paper we consider a large set of variable coefficient linear systems of ordinary differential equations which possess two different time scales, a slow one and a fast one. A small parameter

ε
\varepsilon

characterizes the stiffness of these systems. We approximate a system of ODE’s in this set by a general class of multistep discretizations which includes both one-leg and linear multistep methods. We determine sufficient conditions under which each solution of a multistep method is uniformly bounded, with a bound which is independent of the stiffness of the system of ODE’s, when the step size resolves the slow time scale but not the fast one. We call this property stability with large step sizes. The theory presented in this paper lets us compare properties of one-leg methods and linear multistep methods when they approximate variable coefficient systems of stiff ODE’s. In particular, we show that one-leg methods have better stability properties with large step sizes than their linear multistep counterparts. This observation is consistent with results obtained by Dahlquist and Lindberg [11], Nevanlinna and Liniger [33] and van Veldhuizen [42]. Our theory also allows us to relate the concept of D-stability (van Veldhuizen [42]) to the usual notions of stability and stability domains and to the propagation of errors for multistep methods which use large step sizes.
Products of prime powers in binary recurrence sequences. I. The hyperbolic case, with an application to the generalized Ramanujan-Nagell equation
Authors: Pethö A. , de Weger B. M. M.
Published: June 2010

We show how the Gelfond-Baker theory and diophantine approximation techniques can be applied to solve explicitly the diophantine equationGn=wp1m1⋯ptmt{G_n} = wp_1^{{m_1}} \cdots p_t^{{m_t}}(where{Gn}n=0∞\{ {G_n}\} _{n = 0}^\inftyis a binary recurrence sequence with positive discriminant), for arbitrary values of the parameters. We apply this to the equationx2+k=p1z1⋯ptzt{x^2} + k = p_1^{{z_1}} \cdots p_t^{{z_t}}, which is a generalization of the Ramanujan-Nagell equationx2+7=2z{x^2} + 7 = {2^z}. We present algorithms to reduce upper bounds for the solutions of these equations. The algorithms are easy to translate into computer programs. We present an example which shows that in practice the method works well.
On principal ideal testing in totally complex quartic fields and the determination of certain cyclotomic constants
Authors: Buchmann Johannes , Williams H. C.
Published: June 2010

Let

L

\mathcal {L}

be any totally complex quartic field. Two algorithms are described for determining whether or not any given ideal in

L

\mathcal {L}

is principal. One of these algorithms is very efficient in practice, but its complexity is difficult to analyze; the other algorithm is computationally more elaborate but, in this case, a complexity analysis can be provided. These ideas are applied to the problem of determining the cyclotomic numbers of order 5 for a prime

p
≡
1

(
mod

5
)

p \equiv 1\;\pmod 5

. Given any quadratic (or quintic) nonresidue of p, it is shown that these cyclotomic numbers can be efficiently computed in

O
(

(
log
⁡
p

)
3

)

O({(\log p)^3})

binary operations.
Class numbers of quadratic fields determined by solvability of Diophantine equations
Authors: Mollin R. A.
Published: June 2010

In the literature there has been considerable attention given to the exploration of relationships between certain diophantine equations and class numbers of quadratic fields. In this paper we provide criteria for the insolvability of certain diophantine equations. This result is then used to determine when related real quadratic fields have class number bigger than 1. Moreover, based on criteria which we find for the solvability of a certain class of diophantine equations, we are able to determine when the class number of related imaginary quadratic fields is divisible by a given integer.
Speeding the Pollard and elliptic curve methods of factorization
Authors: Montgomery Peter L.
Published: June 2010

Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations modulo N and occasionally taking GCDs with N. These began with Pollard’s

p
−
1

p - 1

and Monte Carlo methods. More recently, Williams published a

p
+
1

p + 1

method, and Lenstra discovered an elliptic curve method (ECM). We present ways to speed all of these. One improvement uses two tables during the second phases of

p
±
1

p \pm 1

and ECM, looking for a match. Polynomial preconditioning lets us search a fixed table of size n with

n

/

2
+
o
(
n
)

n/2 + o(n)

multiplications. A parametrization of elliptic curves lets Step 1 of ECM compute the x-coordinate of nP from that of P in about 9.3

log
2

{\log _2}

n multiplications for arbitrary P.
Numerical values of Goldberg’s coefficients in the series for 𝑙𝑜𝑔(𝑒^{𝑥}𝑒^{𝑦})
Authors: Newman Morris , Thompson Robert C.
Published: June 2010

The coefficients of K. Goldberg in the infinite series for

log
⁡
(

e
x

e
y

)

\log ({e^x}{e^y})

for noncommuting x and y are computed as far as words of length twenty.
On the sign of the difference 𝜋(𝑥)-𝑙𝑖(𝑥)
Authors: te Riele Herman J. J.
Published: June 2010

Following a method of Sherman Lehman we show that between

6.62
×

10

370

6.62 \times {10^{370}}

and

6.69
×

10

370

6.69 \times {10^{370}}

there are more than

10

180

{10^{180}}

successive integers x for which

π
(
x
)
−

li

(
x
)
>
0

\pi (x) - {\text {li}}(x) > 0

. This brings down Sherman Lehman’s bound on the smallest number x for which

π
(
x
)
−

li

(
x
)
>
0

\pi (x) - {\text {li}}(x) > 0

, namely from

1.65
×

10

1165

1.65 \times {10^{1165}}

to

6.69
×

10

370

6.69 \times {10^{370}}

. Our result is based on the knowledge of the truth of the Riemann hypothesis for the complex zeros

β
+
i
γ

\beta + i\gamma

of the Riemann zeta function which satisfy

|

γ

|

>
450
,
000

|\gamma | > 450,000

, and on the knowledge of the first 15,000 complex zeros to about 28 digits and the next 35,000 to about 14 digits.
On the parallel generation of the residues for the continued fraction factoring algorithm
Authors: Williams H. C. , Wunderlich M. C.
Published: June 2010

In order to implement the continued fraction algorithm on a highly parallel computer, like the Massively Parallel Processor, it is necessary to be able to compute certain numbers which occur at widely-spaced intervals within the continued fraction expansion of

N

\sqrt N

. where N is the number to be factored. In this paper several properties of the continued fraction expansion of a quadratic irrational are developed. These results are then applied to the development of a very simple algorithm for finding the widely-spaced numbers referred to above.
TVB boundary treatment for numerical solutions of conservation laws
Authors: Shu Chi-Wang
Published: June 2010

In the computation of hyperbolic conservation laws

u
t

+
f

(
u

)
x

=
0

{u_t} + f{(u)_x} = 0

, TVD (total-variation-diminishing) and TVB (total-variation-bounded) schemes have been very successful for initial value problems. But most of the existing boundary treatments are only proved to be linearly stable, hence the combined initial-boundary scheme may not be TVB. In this paper we describe a procedure of boundary treatment which uses the original high-order scheme up to the boundary, plus extrapolation and upwind treatment at the boundary. The resulting scheme is proved to be TVB for the scalar nonlinear case and for linear systems.
Some plane curvature approximations
Authors: Mjolsness R. C. , Swartz Blair
Published: June 2010

Second-order accurate approximations to the curvature function along a sufficiently smooth plane curve are presented, the curve being given in finite form (and thus, approximately) by

N
+
2

N + 2

points taken along its full length. The curvature estimates are continuous and invariant under translation and rotation, and they are based on local information—so are easy to implement computationally. In particular, second-order accurate estimates of surface tension forces halfway between immediate neighbors in the curve’s mesh can thereby be made for hydrodynamic simulations. The construction makes use of any of the common techniques one might contemplate for using the information present in three adjacent points (of the

N
+
2

N + 2

points) in order to estimate the curve’s curvature near those three points. It may do this because each of these techniques yields a number which is, to within second order in the distances between the three points, the value of the true curvature function at the same place, namely, at the arithmetic mean of the location of the three points as measured along the curve. The asymptotic form, displaying all terms through the second order, of error estimates for these techniques is provided, along with comparison of gross properties and numerical examples. Finally, continuous, locally second-order accurate, global approximation to the curvature function is obtained by interpolation of successive local estimates between the locations of successive means. A related result is given for the simpler but analogous situation concerning the nth-order difference quotient of a function of one variable. The broken line interpolant of successive nth difference quotients, between the successive mean values of their stencil points, provides a continuous, locally second-order accurate, global approximation to the nth derivative. It also coincides, between two successive stencil means, with the nth derivative of the polynomial interpolant of the

n
+
2

n + 2

data points associated with the two successive stencils.
High-order schemes and entropy condition for nonlinear hyperbolic systems of conservation laws
Authors: Vila J.-P.
Published: June 2010

A systematic procedure for constructing explicit, quasi second-order approximations to strictly hyperbolic systems of conservation laws is presented. These new schemes are obtained by correcting first-order schemes. We prove that limit solutions satisfy the entropy inequality. In the scalar case, we prove convergence to the unique entropy-satisfying solution if the initial scheme is Total Variation Decreasing (i.e., TVD) and consistent with the entropy condition. Finally, we slightly modify Harten’s high-order schemes such that they obey the previous conditions and thus converge towards the "entropy" solution.
On the 3-Sylow subgroup of the class group of quadratic fields
Authors: Llorente Pascual , Quer Jordi
Published: June 2010

In this paper we give a large number of quadratic fields

Q

(

d

)

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

whose class group

H
(
d
)

H(d)

has 3-Sylow subgroup

H
3

(
d
)

{H_3}(d)

with rank

r
3

(
d
)
>
1

{r_3}(d) > 1

. They include 20 discriminants

d
>
0

d > 0

with

r
3

(
d
)
=
5

{r_3}(d) = 5

. The associate real fields have

r
3

(

d
′

)
=
4

{r_3}(d’ ) = 4

. The distribution of the

H
3

(
d
)

{H_3}(d)

is studied and its relative frequency is compared with the heuristic conjectures of Cohen and Lenstra. Some of the

H
(
d
)

H(d)

are recorded since they are of special interest. Some related topics are also considered.
Hyperelliptic integrals and multiple hypergeometric series
Authors: Loiseau Jean-Francis , Codaccioni Jean-Pierre , Caboz Régis
Published: June 2010

We consider the complete hyperelliptic integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J left-parenthesis a right-parenthesis equals integral Subscript alpha left-parenthesis a right-parenthesis Superscript beta left-parenthesis a right-parenthesis Baseline StartFraction d x Over StartRoot a minus upper P Subscript n Baseline left-parenthesis x right-parenthesis EndRoot EndFraction comma">

J
(
a
)
=

∫

α
(
a
)

β
(
a
)

d
x

a
−

P
n

(
x
)

,

J(a) = \int _{\alpha (a)}^{\beta (a)} {\frac {{dx}}{{\sqrt {a - {P_n}(x)} }},}

\]
where

a
>
0

a > 0

and
\[

P
n

(
x
)
=

∑

k
=
2

n

λ
k

x
k

,

{P_n}(x) = \sum \limits _{k = 2}^n {{\lambda _k}{x^k},}

\]
with

λ
2

>
0
,
[
α
,
β
]

{\lambda _2} > 0,[\alpha ,\beta ]

being the connected component of

{
x

|

P
n

(
x
)
≤
a
}

\{ x|{P_n}(x) \leq a\}

containing the origin. Using a recent result concerning the Taylor expansion of the

δ
\delta

-Dirac function, we write

J
(
a
)

J(a)

as a power series of a parameter involving a and the

λ
k

{\lambda _k}

’s. We prove this series to be a sum of multiple hypergeometric series which reduces to a single term when the number of odd monomial terms in

P
n

{P_n}

is less than or equal to one. The region of convergence is then studied and a few particular cases are detailed.
On a new definition of the fractional difference
Authors: Gray Henry L. , Zhang Nien Fan
Published: June 2010

A new definition of the fractional difference is introduced. Many properties based on this definition are established including an extensive exponential law and the important Leibniz rule. The results are then applied to solving second-order linear difference equations.
On the infrastructure of the principal ideal class of an algebraic number field of unit rank one
Authors: Buchmann Johannes , Williams H. C.
Published: June 2010

Let R be the regulator and let D be the absolute value of the discriminant of an order

O

\mathcal {O}

of an algebraic number field of unit rank 1. It is shown how the infrastructure idea of Shanks can be used to decrease the number of binary operations needed to compute R from the best known

O
(
R

D
ε

)

O(R{D^\varepsilon })

for most continued fraction methods to

O
(

R

1

/

2

D
ε

)

O({R^{1/2}}{D^\varepsilon })

. These ideas can also be applied to significantly decrease the number of operations needed to determine whether or not any fractional ideal of

O

\mathcal {O}

is principal.
The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems
Authors: Bramble James H. , Pasciak Joseph E. , Xu Jinchao
Published: June 2010

We prove some new estimates for the convergence of multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems. We provide results for the so-called ’symmetric’ multigrid schemes. We show that for the variable

V

\mathcal {V}

-cycle and the

W

\mathcal {W}

-cycle schemes, multigrid algorithms with any amount of smoothing on the finest grid converge at a rate that is independent of the number of levels or unknowns, provided that the initial grid is sufficiently fine. We show that the

V

\mathcal {V}

-cycle algorithm also converges (under appropriate assumptions on the coarsest grid) but at a rate which may deteriorate as the number of levels increases. This deterioration for the

V

\mathcal {V}

-cycle may occur even in the case of full elliptic regularity. Finally, the results of numerical experiments are given which illustrate the convergence behavior suggested by the theory.
Convergence of a higher-order vortex method for two-dimensional Euler equations
Authors: Chiu C. , Nicolaides R. A.
Published: June 2010

There has been considerable interest recently in the convergence properties of point vortex methods. In this paper, we define a vortex method using vortex multipoles and obtain error estimates for it. In the case of a nonuniform mesh, the rate of convergence of the dipolar algorithm is shown to be of higher order of accuracy than obtained with the simple vortex methods.
Convergence properties and numerical approximation of the solution of the Mindlin plate bending problem
Authors: Pierre Roger
Published: June 2010

We study the behavior of the solution of the Mindlin problem when the thickness becomes small, paying particular attention to the shear stress. We also propose a modification of a known scheme that allows the use of linear finite element approximation and we prove optimal error bounds.
Asymptotic expansions for the discretization error of least squares solutions of linear boundary value problems
Authors: Böhmer Klaus , Locker John
Published: June 2010

For determining least squares solutions of linear boundary value problems, the method of regularization provides uniquely solvable boundary value problems, which are solved with difference methods. The determination of the coefficients in an asymptotic expansion of the discretization error in powers of the regularization and discretization parameters

α
\alpha

and h, respectively, is an ill-posed problem. We present here an asymptotic expansion of this type and discuss the numerical implications for Richardson extrapolation, thereby establishing for the first time methods of arbitrarily high order.
A table of elliptic integrals of the third kind
Authors: Carlson B. C.
Published: June 2010

As many as 72 elliptic integrals of the third kind in previous tables are unified by evaluation in terms of R-functions instead of Legendre’s integrals. The present table includes only integrals having integrands with real singular points. In addition to 31 integrals of the third kind, most of them unavailable elsewhere, 10 integrals of the first and second kinds from an earlier table are listed again in new notation. In contrast to conventional tables, the interval of integration is not required to begin or end at a singular point of the integrand. Fortran codes for the standard R-functions

R
C

{R_C}

and

R
J

{R_J}

, revised to include their Cauchy principal values, are listed in a Supplement.
Some infinite product identities
Authors: Blecksmith Richard , Brillhart John , Gerst Irving
Published: June 2010

In this paper we derive the power series expansions of four infinite products of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript n element-of upper S 1 Endscripts left-parenthesis 1 minus x Superscript n Baseline right-parenthesis product Underscript n element-of upper S 2 Endscripts left-parenthesis 1 plus x Superscript n Baseline right-parenthesis comma">

∏

n
∈

S
1

(
1
−

x
n

)

∏

n
∈

S
2

(
1
+

x
n

)
,

\prod \limits _{n \in {S_1}} {(1 - {x^n})\;\prod \limits _{n \in {S_2}} {(1 + {x^n}),} }

\]
where the index sets

S
1

{S_1}

and

S
2

{S_2}

are specified with respect to a modulus (Theorems 1, 3, and 4). We also establish a useful formula for expanding the product of two Jacobi triple products (Theorem 2). Finally, we give nonexistence results for identities of two forms.
Weak uniform distribution for divisor functions. II
Authors: Rayner Francis J.
Published: June 2010

The author’s methods (reference [4]) are developed further to apply to divisor functions

σ
i

{\sigma _i}

for even values of i. The results of calculations for even values of i in the range

4
≤
i
≤
50

4 \leq i \leq 50

are tabulated.
Convergence of a generalized pulse-spectrum technique (GPST) for inverse problems of 1-D diffusion equations in space-time domain
Authors: Liu X. Y. , Chen Y. M.
Published: June 2010

The problem of convergence of a special form of the generalized pulse-spectrum technique (GPST) for solving inverse problems of one-dimensional diffusion equations in space-time domain is considered. Under the assumptions that a Tikhonov regularized solution exists and the derivative operator of the regularized forward problem at the regularized solution is invertible, the iterative solutions of this special GPST converge to the Tikhonov regularized solution in C norm if the initial guess is close enough to the Tikhonov regularized solution and the rate of convergence is at least linear.
A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations
Authors: Funaro D. , Gottlieb D.
Published: June 2010

A new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested. This method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions. Stability and convergence results are proven for the Chebyshev approximation of linear scalar hyperbolic equations. The eigenvalues of this method applied to parabolic equations are shown to be real and negative.
An optimal-order multigrid method for 𝑃1 nonconforming finite elements
Authors: Brenner Susanne C.
Published: June 2010

An optimal-order multigrid method for solving second-order elliptic boundary value problems using P1 nonconforming finite elements is developed.
On quartic Thue equations with trivial solutions
Authors: Stroeker R. J.
Published: June 2010

Let K be a quartic number field with negative absolute discriminant and let

L

=

Q

(

d

)

{\mathbf {L}} = {\mathbf {Q}}(\sqrt d )

be its real quadratic subfield, with

d
≡
3

(
mod

4
)

d \equiv 3\;\pmod 4

. Moreover, assume K to be embedded in the reals. Further, let

ξ
>
1

\xi > 1

generate the subgroup of units relative to L in the group of positive units of K. Under certain conditions, which can be explicitly checked, and for suitable linear forms

X
(
u
,
v
)

X(u,v)

and

Y
(
u
,
v
)

Y(u,v)

with coefficients in

Z

[

d

]

{\mathbf {Z}}[\sqrt d ]

, the diophantine equation
\[

Norm

K

/

Q

(
X
(
u
,
v
)
+
Y
(
u
,
v
)

ξ
2

)
=
1
,

{\text {Norm}_{{\mathbf {K}}/{\mathbf {Q}}}}(X(u,v) + Y(u,v){\xi ^2}) = 1,

\]
which is a quartic Thue equation in the indeterminates u and v, has only trivial solutions, that is, solutions given by

X
Y
=
0

XY = 0

. Information on a substantial number of equations of this type and their associated number fields is incorporated in a few tables.
On the computation of finite invariant sets of mappings
Authors: Gelman Alex , Rheinboldt Werner C.
Published: June 2010

This paper suggests a new computational method for determining closed curves that are invariant under a given mapping. Unlike other authors, we discretize not only the curve but also the mapping itself. This allows us to avoid completely the computational difficulties connected with the numerical solution of large linear systems. The method uses simple recurrence formulas, which greatly reduce the execution times.
Incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data
Authors: Bramble James H. , Pasciak Joseph E. , Sammon Peter H. , Thomée Vidar
Published: June 2010

Backward difference methods for the discretization of parabolic boundary value problems are considered in this paper. In particular, we analyze the case when the backward difference equations are only solved ’approximately’ by a preconditioned iteration. We provide an analysis which shows that these methods remain stable and accurate if a suitable number of iterations (often independent of the spatial discretization and time step size) are used. Results are provided for the smooth as well as non-smooth initial data cases. Finally, the results of numerical experiments illustrating the algorithms’ performance on model problems are given.
On the mean iteration (𝑎,𝑏)←((𝑎+3𝑏)/4,(√𝑎𝑏+𝑏)/2)
Authors: Borwein J. M. , Borwein P. B.
Published: June 2010

The iterative process
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript n plus 1 Baseline equals left-parenthesis a Subscript n Baseline plus 3 b Subscript n Baseline right-parenthesis slash 4 comma b Subscript n plus 1 Baseline equals left-parenthesis StartRoot a Subscript n Baseline b Subscript n Baseline EndRoot plus b Subscript n Baseline right-parenthesis slash 2">

a

n
+
1

=
(

a
n

+
3

b
n

)

/

4
,

b

n
+
1

=
(

a
n

b
n

+

b
n

)

/

2

{a_{n + 1}} = ({a_n} + 3{b_n})/4,\quad {b_{n + 1}} = (\sqrt {{a_n}{b_n}} + {b_n})/2

\]
is studied in detail. The limit of this quadratically converging process is explicitly identified, as are the uniformizing parameters. The role of symbolic computation, in discovering these nontrivial identifications, is highlighted.
On the numerical solution of the regularized Birkhoff equations
Authors: Börgers Christoph
Published: June 2010

The Birkhoff equations for the evolution of vortex sheets are regularized in a way proposed by Krasny. The convergence of numerical approximations to a fixed regularization is studied theoretically and numerically. The numerical test problem is a two-dimensional inviscid jet.
On the dimension of bivariate superspline spaces
Authors: Chui Charles K. , He Tian Xiao
Published: June 2010

A bivariate piecewise polynomial function of total degree d on some grid partition

Δ
\Delta

that has rth order continuous partial derivatives everywhere may have higher-order partial derivatives at the vertices of the grid partition. In finite element considerations and in the construction of vertex splines, it happens that only those functions with continuous partial derivatives of order higher than r at the vertices are needed to give the same full approximation order as the entire space of piecewise polynomials. This is certainly the case for

d
≥
4
r
+
1

d \geq 4r + 1

. Such piecewise polynomial functions are called supersplines. This paper is devoted to the study of the dimension of certain superspline spaces. Since an exact dimension would have to depend on the geometric structure of the partition

Δ
\Delta

, we will give only upper and lower bounds. We will show, however, that the lower bound value is sharp for all quasi-crosscut partitions; and under suitable assumptions on r and d, the upper and lower bounds agree on both type-1 and type-2 arbitrary triangulations. In addition, a dimension criterion which guarantees that the lower bound gives the actual dimension is given.