bluuebunny/crossref_metadata_2025 huggingface dataset:
A method of virtual displacements for the degenerate discrete 𝑙₁ approximation problem
Authors: Fraser W. , Bennett J. M.
Published: June 2010

Given the system of equations
\[

∑

j
=
1

n

a

i
j

x
j

=

b
i

,

i
=
1
,
…
,
m
,

\sum \limits _{j = 1}^n {{a_{ij}}{x_j} = {b_i},\quad i = 1, \ldots ,m,}

\]
let

A
i

=
(

a

i
1

,
…
,

a

i
n

)

{A_i} = ({a_{i1}}, \ldots ,{a_{in}})

. It is known that if the matrix

A
=
(

a

i
j

)

A = ({a_{ij}})

has rank

k
⩽
n

k \leqslant n

, then there is a point X which provides a minimum of
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R left-parenthesis upper X right-parenthesis equals sigma-summation Underscript i equals 1 Overscript m Endscripts StartAbsoluteValue r Subscript i Baseline left-parenthesis upper X right-parenthesis EndAbsoluteValue equals sigma-summation Underscript i equals 1 Overscript m Endscripts StartAbsoluteValue left-parenthesis upper A Subscript i Baseline comma upper X right-parenthesis minus b Subscript i Baseline EndAbsoluteValue">

R
(
X
)
=

∑

i
=
1

m

|

r
i

(
X
)

|

=

∑

i
=
1

m

|

(

A
i

,
X
)
−

b
i

|

R(X) = \sum \limits _{i = 1}^m {|{r_i}(X)| = } \sum \limits _{i = 1}^m {|({A_i},X) - {b_i}|}

\]
such that

r
i

(
X
)
=
0

{r_i}(X) = 0

for at least k values of the index i. If

r
i

(
X
)
=
0

{r_i}(X) = 0

for exactly k values of the index i, the point or vertex is called ordinary, while if

r
i

(
X
)
=
0

{r_i}(X) = 0

for more than k values of i, the vertex is termed degenerate. A necessary and sufficient condition to determine if X minimizes R is valid if X is an ordinary vertex but not if X is degenerate. A degeneracy at X can be removed by applying perturbations to an appropriate number of the

b
i

{b_i}

so that X becomes an ordinary vertex of a modified problem. By noting that the test uses only values of the

A
i

{A_i}

, it is possible to avoid actual introduction of the perturbations to the

b
i

{b_i}

with a resulting substantial improvement of the efficiency of the computation.
On a method of solving a class of Diophantine equations
Authors: Grinstead Charles M.
Published: June 2010

An elementary method for solving simultaneous Diophantine equations is given. This method will in general lead quickly to a solution-free region on the order of

1
>
x
>

10

10

50

1 > x > {10^{{{10}^{50}}}}

. The method is illustrated by applying it to a set of Diophantine equations.
On an integral summable to 2𝜉(𝑠)/(𝑠(𝑠-1))
Authors: Walker P. L.
Published: June 2010

Let

ψ
(
x
)
=

Σ

n
=
1

∞

e

−

n
2

π
x

\psi (x) = \Sigma _{n = 1}^\infty {e^{ - {n^2}\pi x}}

, and

χ
(
u
)
=

e

u

/

2

(
1
+
2
ψ
(

e

2
u

)
)

\chi (u) = {e^{u/2}}(1 + 2\psi ({e^{2u}}))

. The divergent integral

2

∫
0
∞

χ
(
u
)
cos

2\smallint _0^\infty \chi (u)\cos

tu du is shown to be summable for certain complex values of t to the function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 xi left-parenthesis s right-parenthesis slash s left-parenthesis s minus 1 right-parenthesis equals pi Superscript negative s slash 2 Baseline normal upper Gamma left-parenthesis 1 slash 2 s right-parenthesis zeta left-parenthesis s right-parenthesis">

2
ξ
(
s
)

/

s
(
s
−
1
)
=

π

−
s

/

2

Γ
(
1

/

2
s
)
ζ
(
s
)

2\xi (s)/s(s - 1) = {\pi ^{ - s/2}}\Gamma (1/2s)\zeta (s)

, where

s
=
1

/

2
+
i
t

s = 1/2 + it

, and

ζ
(
s
)

\zeta (s)

is the zeta-function of Riemann. The values of a resulting approximation to

2
ξ
(
s
)

/

s
(
s
−
1
)

2\xi (s)/s(s - 1)

are computed and its zeros located.
Further evaluation of Howland integrals
Authors: Ling Chih Bing
Published: June 2010

The purpose of this paper is to further evaluate two Howland integrals to 25D when their index is an even integer.
Deciding associativity for partial multiplication tables of order 3
Authors: Bunting Paul W. , van Leeuwen Jan , Tamari Dov
Published: June 2010

The associativity problem for the class of finite multiplication tables is known to be undecidable, even for quite narrow infinite subclasses of tables. We present criteria which can be used to decide associativity in many cases, although any effective method based on such criteria must eventually fail on a table of some size (as otherwise decidability for the general class would follow). By means of an extensive computer search we have been able to use the criteria successfully to solve the associativity problem for all tables of order up to 3. We find 24,733 associative tables and 237,411 nonassociative tables, and present some further statistics about how "deep" we had to search to establish the nonassociativity of a table. We also prove that there are tables of order 3 for which no "one-mountain" theorem holds (which was known previously only for order 6 examples). Our methods make use of efficient data-representations and techniques of heuristic and adaptive programming.
The inverse Sturm-Liouville problem and the Rayleigh-Ritz method
Authors: Hald Ole H.
Published: June 2010

In this paper we present an algorithm for solving the inverse Sturm-Liouville problem with symmetric potential and Dirichlet boundary conditions. The algorithm is based on the Rayleigh-Ritz method for calculating the eigenvalues of a two point boundary value problem, and reduces the inverse problem for the differential equation to a nonstandard discrete inverse eigenvalue problem. It is proved that the solution of the discrete problem converges to the solution of the continuous problem. Finally, we establish the stability of the method and give numerical examples.
A numerical accuracy consideration in polynomial deflation
Authors: O’Neill C. J. , Downs T.
Published: June 2010

In a recent paper, Peters and Wilkinson described a composite deflation method which provides an accurate technique for the determination of the roots of a polynomial where these roots are widely spaced. By examples involving deflation by linear factors they demonstrated that the method was much more accurate than forward or backward deflation. In addition, they stated that the method could also be used when deflating by a quadratic factor or a polynomial of higher degree. In this short note it is shown that composite deflation by quadratic factors can lead to severe rounding error where two successive deflations by linear factors produce a perfectly accurate result.
On the class number of relative quadratic fields
Authors: Parry Charles J.
Published: June 2010

We describe class number relations between certain pairs of algebraic number fields, both of the same degree over the rational numbers. This relationship is discussed in great detail when the common degree is equal to 4. Some numerical results are given in this case.
Energy-conserving norms for the solution of hyperbolic systems of partial differential equations
Authors: Gunzburger Max D. , Plemmons Robert J.
Published: June 2010

The problem of finding an energy conserving norm for the solution of the hyperbolic system of partial differential equations

∂
u

/

∂
t
=
A
∂
u

/

∂
x

\partial u/\partial t = A\partial u/\partial x

, subject to boundary conditions, is reduced to the problem of characterizing those matrices appearing in the boundary conditions which satisfy two specific matrix equations. Necessary and sufficient conditions on the coefficient matrix A and the matrices appearing in boundary conditions are derived for an energy conserving norm to exist. Thus, these conditions serve as tests on a given system which determine whether or not the solution will have its energy conserved in some norm. In addition, some examples of specific systems and boundary conditions are provided.
The affine scale invariance of minimization algorithms
Authors: Lyness J. N.
Published: June 2010

Let

f
(
x
)

f(x)

be a general objective function and let

f
¯

(
x
)
=
h
+
m
f
(
A
x
+
d
)

\bar f(x) = h + mf(Ax + d)

. An analytic estimation of the minimum of one would resemble an analytic estimation of the other in all nontrivial respects. However, the use of a minimization algorithm on either might or might not lead to apparently unrelated sequences of calculations. This paper is devoted to providing a general theory for the affine scale invariance of algorithms. Key elements in this theory are groups of transformations T whose elements relate

f
¯

(
x
)

\bar f(x)

and

f
(
x
)

f(x)

given above. The statement that a specified algorithm is scale invariant with respect to a specified group T is defined. The scale invariance properties of several well-known algorithms are discussed.
Note on the Kelvin phase functions
Authors: Philip J. R.
Published: June 2010

Both theoretical consistency and practical convenience demand that the values of the Kelvin phase functions

θ
n

(
x
)

{\theta _n}(x)

and

ϕ
n

(
x
)

{\phi _n}(x)

be defined uniquely and be well-ordered in n. This is achieved by taking, for

n
⩾
0

n \geqslant 0

,

θ
n

(
0
)
=
−

ϕ
n

(
0
)
=
3
n
π

/

4

{\theta _n}(0) = - {\phi _n}(0) = 3n\pi /4

. The necessary amendments to extant tables are indicated.
Calculation of the regulator of 𝑄(√𝐷) by use of the nearest integer continued fraction algorithm
Authors: Williams H. C. , Buhr P. A.
Published: June 2010

A computational method for determining the regulator of a real quadratic field

Q
(

D

)

Q(\sqrt D )

is described. This method makes use of the properties of the nearest integer continued fraction of

D

\sqrt D

and is about 25
Multistep-Galerkin methods for hyperbolic equations
Authors: Dougalis Vassilios A.
Published: June 2010

Multistep methods for first- and second-order ordinary differential equations are used for the full discretizations of standard Galerkin approximations to the initial-periodic boundary value problem for first-order linear hyperbolic equations in one space dimension and to the initial-boundary value problem for second-order linear selfadjoint hyperbolic equations in many space dimensions.

L
2

{L^2}

-error bounds of optimal order in space and time are achieved for large classes of such multistep methods.
Two conjectures of B. R. Santos concerning totitives
Authors: Kopetzky H. G. , Schwarz W.
Published: June 2010

Recently B. R. Santos conjectured that 12 is the largest integer n with the following property:

(
∗
)

(\ast )

\[

{

If

m

∈

[
1
,
n
]

and

n

are

relatively

prime,

then

n
+
m

is

a

prime

number

.

\left \{ {\begin {array}{*{20}{c}} {\begin {array}{*{20}{c}} {{\text {If}}\;m\; \in \;[1,n]\;{\text {and}}\;n\;{\text {are}}\;{\text {relatively}}\;{\text {prime,}}\;{\text {then}}} \hfill \\ {n + m\;{\text {is}}\;{\text {a}}\;{\text {prime}}\;{\text {number}}{\text {.}}} \hfill \\ \end {array} } \hfill \\ \end {array} } \right .

\]
Using deep numerical estimates of Rosser and Schoenfeld for the number

π
(
x
)

\pi (x)

of primes less than x, it is proved that the conjecture of Santos is true. The same result holds, if in addition it is assumed in

(
∗
)

(\ast )

that m is a prime.
Extensions of von Neumann’s method for generating random variables
Authors: Monahan John F.
Published: June 2010

Von Neumann’s method of generating random variables with the exponential distribution and Forsythe’s method for obtaining distributions with densities of the form

e

−
G
(
x
)

{e^{ - G(x)}}

are generalized to apply to certain power series representations. The flexibility of the power series methods is illustrated by algorithms for the Cauchy and geometric distributions.
An asymptotic expansion for the upper percentage points of the 𝜒²-distribution
Authors: Fettis Henry E.
Published: June 2010

An asymptotic development is given for estimating the value of the variable

χ
\chi

for which the

χ
2

{\chi ^2}

-distribution
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q left-parenthesis chi squared comma v right-parenthesis equals StartFraction 1 Over normal upper Gamma left-parenthesis v slash 2 right-parenthesis EndFraction integral Subscript chi squared slash 2 Superscript normal infinity Baseline t Superscript v slash 2 minus 1 Baseline e Superscript negative t Baseline d t">

Q
(

χ
2

,
v
)
=

1

Γ
(
v

/

2
)

∫

χ
2

/

2

∞

t

v

/

2
−
1

e

−
t

d
t

Q({\chi ^2},v) = \frac {1}{{\Gamma (v/2)}}\int _{{\chi ^2}/2}^\infty {t^{v/2 - 1}}{e^{ - t}}dt

\]
assumes a preassigned value

α
\alpha

, in the region where the quantity

η
=
−
ln
⁡
[
Γ
(
v

/

2
)
α
]

\eta = - \ln [\Gamma (v/2)\alpha ]

satisfies
\[

η
>>
ln
⁡
η
.

\eta > > \ln \eta .

\]
This development generalizes a similar one given by Blair and coauthors [2] for the case

v
=
1

v = 1

. It is also shown how the estimates thus obtained may be used in conjunction with various iterative schemes to give more accurate values.
A Goldbach conjecture using twin primes
Authors: Zwillinger Dan
Published: June 2010

The numbers

2
N
=
2
(
2
)
1000000

2N = 2(2)1000000

are checked to determine if they can be written as the sum of two twin primes. Thirty-three numbers are found that cannot be so represented; they are all less than 5000. The largest number in the range

2
N
=
2
(
2
)
500000

2N = 2(2)500000

that can be written as the sum of two twin primes in only one way is

2
N
=
24098

2N = 24098

.
Adaptive numerical differentiation
Authors: Stepleman R. S. , Winarsky N. D.
Published: June 2010

It is well known that the calculation of an accurate approximate derivative

f
′
(
x
)

f\prime (x)

of a nontabular function

f
(
x
)

f(x)

on a finite-precision computer by the formula

d
(
h
)
=
(
f
(
x
+
h
)
−
f
(
x
−
h
)
)

/

2
h

d(h) = (f(x + h) - f(x - h))/2h

is a delicate task. If h is too large, truncation errors cause poor answers, while if h is too small, cancellation and other "rounding" errors cause poor answers. We will show that by using simple results on the nature of the asymptotic convergence of

d
(
h
)

d(h)

to

f
′

f\prime

, a reliable numerical method can be obtained which can yield efficiently the theoretical maximum number of accurate digits for the given machine precision.
Cyclic-sixteen class fields for 𝑄((-𝑝)^{1/2}) by modular arithmetic
Authors: Cohn Harvey
Published: June 2010

Numerical experiments result in the construction of cyclic-sixteen class fields for

Q

(
−
p

)

1

/

2

{\mathbf {Q}}{( - p)^{1/2}}

, p prime

>
2000

> 2000

, by radicals involving quadratic and biquadratic parameters. These fields are characterized by rational factorization properties modulo a variable prime; but it suffices to use only three primes selected and checked by computer to verify the class field, if earlier work (jointly with Cooke) on the cyclic-eight class field is utilized.
Arithmetic progressions consisting only of primes
Authors: Grosswald Emil , Hagis Peter
Published: June 2010

LetNm(x){N_m}(x)denote the number of arithmetic progressions consisting ofmprimes with largest member not exceedingx.Nm(x){N_m}(x)has been tabulated for3⩽m⩽103 \leqslant m \leqslant 10and selected values ofxbetween 1000 and 50000, and the results are compared here with those obtained by (heuristic) asymptotic approximations toNm(x){N_m}(x).
Stable and entropy satisfying approximations for transonic flow calculations
Authors: Engquist Björn , Osher Stanley
Published: June 2010

Finite difference approximations for the small disturbance equation of transonic flow are developed and analyzed. New schemes of the Cole-Murman type are presented for which nonlinear stability is proved. The Cole-Murman scheme may have entropy violating expansion shocks as solutions. In the new schemes the switch between the subsonic and supersonic domains is designed such that these nonphysical shocks are guaranteed not to occur. Results from numerical calculations are given which illustrate these conclusions.
On the roles of “stability” and “convergence” in semidiscrete projection methods for initial-value problems
Authors: Parter Seymour V.
Published: June 2010

Consider the initial value problem (1.1)
\[

d

d
t

u
(
t
)
=
A
u
(
t
)
+
f
(
t
)
,

t
>
0
,

\frac {d}{{dt}}u(t) = Au(t) + f(t),\quad t > 0,

\]
(1.2)
\[

u
(
0
)
=

u
0

,

u(0) = {u_0},

\]
where A is a linear operator taking

D
(
A
)
⊂
X

D(A) \subset X

into X, where X is a Banach space. Consider also semidiscrete numerical methods of the form: find

U
N

(
t
)
:
[
0
,
T
]
→

X
N

{U_N}(t):[0,T] \to {X_N}

such that

(
1.1
′
)

(1.1\prime )

\[

d

U
N

d
t

=

A
N

U
N

+

P
N

f
,

\frac {{d{U_N}}}{{dt}} = {A_N}{U_N} + {P_N}f,

\]

(
1.2
′
)

(1.2\prime )

\[

U
N

(
0
)
=

U
N
0

∈

X
N

,

{U_N}(0) = U_N^0 \in {X_N},

\]
where

X
N

{X_N}

is a finite dimensional subspace and

P
N

{P_N}

is a projector onto

X
N

{X_N}

. The study of such numerical methods may be related to the approximation of semigroups and Laplace transform methods making use of the resolvent operators
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A minus lamda upper I right-parenthesis Superscript negative 1 Baseline comma left-parenthesis upper A Subscript upper N Baseline minus lamda upper I Subscript upper N Baseline right-parenthesis Superscript negative 1">

(
A
−
λ
I

)

−
1

,

(

A
N

−
λ

I
N

)

−
1

{(A - \lambda I)^{ - 1}},{({A_N} - \lambda {I_N})^{ - 1}}

. The basic results require stability or weak stability and give convergence rates of the same order as in the steady state problems.
Majorizing sequences and error bounds for iterative methods
Authors: Miel George J.
Published: June 2010

Given a sequence

{

x
n

}

n
=
0

∞

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

in a Banach space, it is well known that if there is a sequence

{

t
n

}

n
=
0

∞

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

such that

‖

x

n
+
1

−

x
n

‖

⩽

t

n
+
1

−

t
n

\left \| {{x_{n + 1}} - {x_n}} \right \| \leqslant {t_{n + 1}} - {t_n}

and

lim

t
n

=

t
∗

>
∞

\lim {t_n} = {t^\ast } > \infty

, then

{

x
n

}

n
=
0

∞

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

converges to some

x
∗

{x^\ast }

and the error bounds

‖

x
∗

−

x
n

‖

⩽

t
∗

−

t
n

\left \| {{x^\ast } - {x_n}} \right \| \leqslant {t^\ast } - {t_n}

hold. It is shown that certain stronger hypotheses imply sharper error bounds,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar x Superscript asterisk Baseline minus x Subscript n Baseline double-vertical-bar less-than-or-slanted-equals StartFraction t Superscript asterisk Baseline minus t Subscript n Baseline Over left-parenthesis t Subscript n Baseline minus t Subscript n minus 1 Baseline right-parenthesis Superscript mu Baseline EndFraction double-vertical-bar x Subscript n Baseline minus x Subscript n minus 1 Baseline double-vertical-bar Superscript mu Baseline less-than-or-slanted-equals StartFraction t Superscript asterisk Baseline minus t Subscript n Baseline Over left-parenthesis t 1 minus t 0 right-parenthesis Superscript mu Baseline EndFraction double-vertical-bar x 1 minus x 0 double-vertical-bar Superscript mu Baseline comma mu greater-than-or-slanted-equals 0 period">

‖

x
∗

−

x
n

‖

⩽

t
∗

−

t
n

(

t
n

−

t

n
−
1

)

μ

‖

x
n

−

x

n
−
1

‖

μ

⩽

t
∗

−

t
n

(

t
1

−

t
0

)

μ

‖

x
1

−

x
0

‖

μ

,

μ
⩾
0.

\left \| {{x^\ast } - {x_n}} \right \| \leqslant \frac {{{t^\ast } - {t_n}}}{{{{({t_n} - {t_{n - 1}})}^\mu }}}{\left \| {{x_n} - {x_{n - 1}}} \right \|^\mu } \leqslant \frac {{{t^\ast } - {t_n}}}{{{{({t_1} - {t_0})}^\mu }}}{\left \| {{x_1} - {x_0}} \right \|^\mu },\quad \mu \geqslant 0.

\]
Representative applications to infinite series and to iterates of types

x
n

=
G

x

n
−
1

{x_n} = G{x_{n - 1}}

and

x
n

=
H
(

x
n

,

x

n
−
1

)

{x_n} = H({x_n},{x_{n - 1}})

are given for

μ
=
1

\mu = 1

. Error estimates with

0
⩽
μ
⩽
2

0 \leqslant \mu \leqslant 2

are shown to be valid and optimal for Newton iterates under the hypotheses of the Kantorovich theorem. The unified convergence theory of Rheinboldt is used to derive error bounds with

0
⩽
μ
⩽
1

0 \leqslant \mu \leqslant 1

for a class of Newton-type methods, and these bounds are shown to be optimal for a subclass of methods. Practical limitations of the error bounds are described.
On maximal finite irreducible subgroups of 𝐺𝐿(𝑛,𝑍). III. The nine-dimensional case
Authors: Plesken Wilhelm , Pohst Michael
Published: June 2010

All maximal finite absolutely irreducible subgroups of

G
L
(
9
,

Z

)

GL(9,{\mathbf {Z}})

are determined up to conjugacy in

G
L
(
9
,

Z

)

GL(9,{\mathbf {Z}})

.
On maximal finite irreducible subgroups of 𝐺𝐿(𝑛,𝑍). IV. Remarks on even dimensions with applications to 𝑛=8
Authors: Plesken Wilhelm , Pohst Michael
Published: June 2010

The general methods for the determination of maximal finite absolutely irreducible subgroups of

G
L
(
n
,

Z

)

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

developed in Part I of this series of papers [6] are refined for even n. Applications are made to

n
=
8

n = 8

in view of Part V [7], where a complete classification is obtained.
Some new algorithms for high-precision computation of Euler’s constant
Authors: Brent Richard P. , McMillan Edwin M.
Published: June 2010

We describe several new algorithms for the high-precision computation of Euler’s constant

γ
=
0.577
…

\gamma = 0.577 \ldots

Using one of the algorithms, which is based on an identity involving Bessel functions,

γ
\gamma

has been computed to 30,100 decimal places. By computing their regular continued fractions we show that, if

γ
\gamma

or

exp
⁡
(
γ
)

\exp (\gamma )

is of the form

P

/

Q

P/Q

for integers P and Q, then

|

Q

|

>

10

15000

|Q| > {10^{15000}}

.
Convergence of multigrid iterations applied to difference equations
Authors: Hackbusch Wolfgang
Published: June 2010

Convergence proofs for the multi-grid iteration are known for the case of finite element equations and for the case of some difference schemes discretizing boundary value problems in a rectangular region. In the present paper we give criteria of convergence that apply to general difference schemes for boundary value problems in Lipschitzian regions. Furthermore, convergence is proved for the multi-grid algorithm with Gauss-Seidel’s iteration as smoothing procedure.
Error analysis of some techniques for updating orthogonal decompositions
Authors: Paige C. C.
Published: June 2010

We consider accurate and efficient methods for updating the result of the transformation

C
=
B
Q

C = BQ

, Q orthogonal, of a given matrix B when Q is available. Adding or deleting a row, or adding a column of B leads to a continuation of the original transformation, and as such is numerically stable. In particular, we discuss a well-known method for updating when a column of B is deleted, and show that it is as numerically stable as the problem allows. The results extend to two-sided transformations of the form

C
=

Z
T

B
Q

C = {Z^T}BQ

. The methods and analyses are independent of the form or rank of B and C, and so are widely applicable.
On variable-metric methods for sparse Hessians
Authors: Shanno D. F.
Published: June 2010

The relationship between variable-metric methods derived by norm minimization and those derived by symmetrization of rank-one updates for sparse systems is studied, and an analogue of Dennis’s nonsparse symmetrization formula derived. A new method of using norm minimization to produce a sparse analogue of any nonsparse variable-metric method is proposed. The sparse BFGS generated by this method is tested against the sparse PSB and variable-memory conjugate gradient methods, with computational experience uniformly favoring the sparse BFGS.
On the convergence of an algorithm computing minimum-norm solutions of ill-posed problems
Authors: Marti J. T.
Published: June 2010

The paper studies a finite element algorithm giving approximations to the minimum-norm solution of ill-posed problems of the form

A
f
=
g

Af = g

, where A is a bounded linear operator from one Hubert space to another. It is shown that the algorithm is norm convergent in the general case and an error bound is derived for the case where g is in the range of

A

A
∗

A{A^\ast }

. As an example, the method has been applied to the problem of evaluating the second derivative f of a function g numerically.
High-precision values of the gamma function and of some related coefficients
Authors: Fransén Arne , Wrigge Staffan
Published: June 2010

In this paper we determine numerical values to 80D of the coefficients in the Taylor series expansion

Γ
m

(
s
+
x
)
=

Σ
0
∞

g
k

(
m
,
s
)

x
k

{\Gamma ^m}(s + x) = \Sigma _0^\infty {g_k}(m,s){x^k}

for certain values of m and s and use these values to calculate

Γ
(
p

/

q
)

(
p
,
q
=
1
,
2
,
…
,
10
;

p
>
q
)

\Gamma (p/q)\;(p,q = 1,2, \ldots ,10;\;p > q)

and

min

x
>
0

Γ
(
x
)

{\min _{x > 0}}\Gamma (x)

to 80D. Finally, we obtain a high-precision value of the integral

∫
0
∞

(
Γ
(
x
)

)

−
1

d
x

\smallint _0^\infty {(\Gamma (x))^{ - 1}}\;dx

.
Efficient higher order single step methods for parabolic problems. I
Authors: Bramble James H. , Sammon Peter H.
Published: June 2010

Some efficient, high order methods are discussed for approximating the solution of an initial boundary value problem for a homogeneous parabolic equation with time dependent coefficients. The methods are based on Galerkin-type approximations in the spacial variables and single step methods in the time variable. The equations defining the time-stepping procedure are solved only approximately, however. A preconditioned iterative technique is used for this purpose. The resulting algorithm is shown to produce optimal order approximations using only the order of work required by the single step method applied to the parabolic problem with time independent coefficients.
Compact, implicit difference schemes for a differential equation’s side conditions
Authors: Swartz Blair
Published: June 2010

Lynch and Rice have recently derived compact, implicit (averaged-operator) difference schemes for the approximate solution of an mth order linear ordinary differential equation under m separated side conditions. We construct here a simpler form for a compact, implicit difference scheme which approximates a more general side condition. We relax the order of polynomial exactness required for such approximate side conditions. We prove appropriate convergence rates of the approximate solution (and its first

m
−
1

m - 1

divided differences) to (those of) the solution, even, of multi-interval differential equations. Appropriate, here, means kth order convergence for schemes whose interior equations are exact for polynomials of order

k
+
m

k + m

and whose approximation of a side condition of order l is exact for polynomials of order

k
+
l

k + l

. We also prove the feasibility of shooting (and of multiple shooting) based on initial divided differences. The simplicity of the proofs is based upon the simplicity of form of the approximating side conditions, together with the crucial stability result of Lynch and Rice for their interior difference equations under divided-difference initial data.
Recursive algorithms for the matrix Padé problem
Authors: Bultheel Adhemar
Published: June 2010

A matrix triangularization interpretation is given for the recursive algorithms computing the Padé approximants along a certain path in the Padé table, which makes it possible to unify all known algorithms in this field [5], [6]. For the normal Padé table, all these results carry over to the matrix Padé problem in a straightforward way. Additional features, resulting from the noncommutativity are investigated. A generalization of the Trench-Zohar algorithm and related recursions are studied in greater detail.
The convergence and partial convergence of alternating series
Authors: Philip J. R.
Published: June 2010

The alternating series is

Σ

n
=
1

∞

j
n

f
(
n
)
=
[
j
]
f

\Sigma _{n = 1}^\infty {j_n}f(n) = [j]f

, with f a single-signed monotonic function of the real variable x. The

j
n

{j_n}

are

±
1

\pm 1

, their sign fixed by repetition of the ’template’ [j] of finite length 2p. [j] constitutes a difference scheme of ’differential order’ D, which can be determined. The principal theorem is that

[
j
]
f

[j]f

is ’partially convergent’ if and only if

lim

x
→
∞

f

(
D
−
1
)

(
x
)

{\lim _{x \to \infty }}{f^{(D - 1)}}(x)

is bounded. A series is partially convergent when the limit as

M
→
∞

M \to \infty

of the sum of 2pM terms exists. For [j] ’pure’, the improved Euler-Maclaurin expansion (IEM) gives the compact representation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis upper A right-parenthesis EndLabel upper S Superscript left-parenthesis p right-parenthesis Baseline tilde minus StartFraction mu left-parenthesis upper D right-parenthesis Over 2 p EndFraction sigma-summation Underscript r equals 0 Overscript normal infinity Endscripts left-parenthesis 2 p right-parenthesis Superscript 2 r Baseline StartFraction upper B Subscript 2 r Baseline left-parenthesis 1 slash 2 right-parenthesis Over left-parenthesis 2 r factorial right-parenthesis EndFraction f Superscript left-parenthesis 2 r plus upper D minus 1 right-parenthesis Baseline left-parenthesis theta Subscript r Baseline right-parenthesis comma 1 minus p less-than-or-slanted-equals theta Subscript r Baseline less-than-or-slanted-equals p period EndLayout">

(A)

S

(
p
)

∼
−

μ
(
D
)

2
p

∑

r
=
0

∞

(
2
p

)

2
r

B

2
r

(
1

/

2
)

(
2
r
!
)

f

(
2
r
+
D
−
1
)

(

θ
r

)
,

1
−
p
⩽

θ
r

⩽
p
.

\begin{equation}\tag {A} S^{(p)} \sim - \frac {\mu (D)}{2p} \sum _{r = 0}^\infty (2p)^{2r} \frac {B_{2r}(1/2)}{(2r!)} f^{(2r+D-1)} (\theta _r),\quad 1 - p \leqslant \theta _r \leqslant p. \end{equation}

S

(
p
)

S^{(p)}

is the sum,

μ
(
D
)

\mu (D)

is the Dth ’template moment’, and the

B

2
r

{B_{2r}}

are Bernoulli numbers. Efficient means for practical summation of these series follow also from IEM. In illustration, 10 alternating series with D ranging from 1 to 3 are summed using IEM. It is found that the leading term of (A) with

θ
0

=
1

/

2

\theta _0 = 1/2

gives a simple but effective estimate of sums. The paper also gives a comparison with Euler’s transformation in the case

p
=
1

p = 1

and discusses sums to N terms with

N

/

2
p

N/2p

nonintegral and finite but large.
On an algorithm for finding a base and a strong generating set for a group given by generating permutations
Authors: Leon Jeffrey S.
Published: June 2010

This paper deals with the problem of finding a base and strong generating set for the group generated by a given set of permutations. The concepts of base and strong generating set were introduced by Sims [5], [6] and provide the most effective tool for computing with permutation groups of high degree. One algorithm, originally proposed by Sims [7], is described in detail; its behavior on a number of groups is studied, and the influence of certain parameters on its performance is investigated. Another algorithm, developed by the author, is given, and it is shown how the two algorithms may be combined to yield an exceptionally fast and effective method.
On determination of best-possible constants in integral inequalities involving derivatives
Authors: Neta Beny
Published: June 2010

This paper is concerned with the numerical approximation of the best possible constants

γ

n
,
k

{\gamma _{n,k}}

in the inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper F Superscript left-parenthesis k right-parenthesis Baseline double-vertical-bar squared less-than-or-slanted-equals gamma Subscript n comma k Superscript negative 1 Baseline left-brace double-vertical-bar upper F double-vertical-bar squared plus double-vertical-bar upper F Superscript left-parenthesis n right-parenthesis Baseline double-vertical-bar squared right-brace comma">

‖

F

(
k
)

‖

2

⩽

γ

n
,
k

−
1

{

‖
F
‖

2

+

‖

F

(
n
)

‖

2

}

,

{\left \| {{F^{(k)}}} \right \|^2} \leqslant \gamma _{n,k}^{ - 1}\;\left \{ {{{\left \| F \right \|}^2} + {{\left \| {{F^{(n)}}} \right \|}^2}} \right \},

\]
where
\[

‖
F
‖

2

=

∫
0
∞

|

F
(
x
)

|

2

d
x
.

{\left \| F \right \|^2} = \int _0^\infty |F(x){|^2}\;dx.

\]
A list of all constants

γ

n
,
k

{\gamma _{n,k}}

for

n
⩽
10

n \leqslant 10

is given.
Arithmetic in a finite field
Authors: Willett Michael
Published: June 2010

An algorithm for realizing finite field arithmetic is presented. The relationship between linear recursions and polynomial arithmetic (modulo a fixed polynomial) over Zp is exploited to reduce the storage and computation requirements of the algorithm. A primitive normal polynomial is used to simplify the calculation of multiplicative inverses.
Estimates away from a discontinuity for dissipative Galerkin methods for hyperbolic equations
Authors: Layton William J.
Published: June 2010

We consider the approximate solution of the initial value problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential u Over partial-differential t EndFraction equals StartFraction partial-differential u Over partial-differential x EndFraction comma u left-parenthesis x comma 0 right-parenthesis equals v left-parenthesis x right-parenthesis comma">

∂
u

∂
t

=

∂
u

∂
x

,

u
(
x
,
0
)
=
v
(
x
)
,

\frac {{\partial u}}{{\partial t}} = \frac {{\partial u}}{{\partial x}},\quad u(x,0) = v(x),

\]
by a dissipative Galerkin method. When v is taken to have a jump discontinuity at zero, that discontinuity will propagate along

x
+
t
=
0

x + t = 0

, in the true solution u. Estimates in

L
2

{L_2}

and

L
∞

{L_\infty }

of the pollution effects of the discontinuity are found. These estimates show those effects to decay exponentially in

h

−
1

{h^{ - 1}}

in regions a fixed distance d from the discontinuity and exponentially in d for fixed h.
Stability of numerical schemes solving quasilinear wave equations
Authors: le Roux A. Y.
Published: June 2010

A generalization of the Riemann invariants for quasi-linear wave equations of the type

∂
2

w

/

∂

t
2

=
∂
f
(
∂
w

/

∂
x
)

/

∂
x

{\partial ^2}w/\partial {t^2} = \partial f(\partial w/\partial x)/\partial x

, which includes the shock curves, is proposed and is used to solve the Riemann problem. Three numerical schemes, whose accuracy is of order one (the Lax-Friedrichs scheme and two extensions of the upstreaming scheme), are constructed by

L
2

{L^2}

-projection, onto piecewise constant functions, of the solutions of a set of Riemann problems. They are stable in the

L
∞

{L^\infty }

-norm for a class of wave equations, including a nonlinear model of extensible strings, which are not genuinely nonlinear. The problem with boundary conditions is detailed, as is its treatment, by the numerical schemes.
The vortex method with finite elements
Authors: Bardos Claude , Bercovier Michel , Pironneau Olivier
Published: June 2010

This work shows that the method of charcteristics is well suited for the numerical solution of first order hyperbolic partial differential equations whose coefficients are approximated by functions piecewise constant on a finite element triangulation of the domain of integration. We apply this method to the numerical solution of Euler’s equation and prove convergence when the time step and the mesh size tend to zero. The proof is based upon the results of regularity given by Kato and Wolibner and on

L
∞

{L^\infty }

estimates for the solution of the Dirichlet problem given by Nitsche. The method obtained belongs to the family of vortex methods usually studied in a finite difference context.
Quadrature methods for stiff ordinary differential systems
Authors: Iserles A.
Published: June 2010

The quadrature methods are based upon a substitution of an explicit A-stable first approximation into a generalized convolution formula. They are A-stable, explicit, and of arbitrarily high order. The generalized convolution formula is derived and its order-raising properties examined. Two families of explicit A-stable first approximations are developed, generalizing results of Lawson and Nørsett. Various aspects of the numerical implementation are discussed. Numerical results supplement the paper and exemplify the various merits and weaknesses of the quadrature methods.
Galerkin methods for singular integral equations
Authors: Thomas K. S.
Published: June 2010

The approximate solution of a singular integral equation by Galerkin’s method is studied. We discuss the theoretical aspects of such problems and give error bounds for the approximate solution.
Discontinuous Galerkin methods for ordinary differential equations
Authors: Delfour M. , Hager W. , Trochu F.
Published: June 2010

A class of Galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed. For polynomials of degree k, these methods lead to a family of one-step schemes generating approximations up to order

2
k
+
2

2k + 2

for the solution of an ordinary differential equation.
On odd perfect, quasiperfect, and odd almost perfect numbers
Authors: Kishore Masao
Published: June 2010

We establish upper bounds for the six smallest prime factors of odd perfect, quasiperfect, and odd almost perfect numbers.
Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II
Authors: Goldberg Moshe , Tadmor Eitan
Published: June 2010

Convenient stability criteria are obtained for difference approximations to initial-boundary value problems associated with the hyperbolic system

u

t

=
A

u

x

+
B

u

+

f

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

in the quarter plane

x
⩾
0

x \geqslant 0

,

t
⩾
0

t \geqslant 0

. The approximations consist of arbitrary basic schemes and a wide class of boundary conditions. The new criteria are given in terms of the outflow part of the boundary conditions and are independent of the basic scheme. The results easily imply that a number of well-known boundary treatments, when used in combination with arbitrary stable basic schemes, always maintain stability. Consequently, many special cases studied in recent literature are generalized.
Some results concerning Voronoĭ’s continued fraction over 𝑄(\root3\of{𝐷})
Authors: Williams H. C.
Published: June 2010

Let D be a cube-free integer and let

ε
0

{\varepsilon _0}

be the fundamental unit of the pure cubic field

Q

(

D
3

)

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

. It is well known that Voronoi’s algorithm can be used to determine

ε
0

{\varepsilon _0}

. In this work several results concerning Voronoi’s algorithm in

Q

(

D
3

)

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

are derived and it is shown how these results can be used to increase the speed of calculating

ε
0

{\varepsilon _0}

for many values of D. Among these D values are those such that

D
(
>
3
)

D( > 3)

is not a prime

≡
8

(
mod

9
)

\equiv 8\; \pmod 9

and the class number of

Q

(

D
3

)

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

is not divisible by 3. A frequency table of all class numbers not divisible by 3 for all

Q

(

D
3

)

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

with

D
>
2
×

10
5

D > 2 \times {10^5}

is also presented.
Note on irreducible diagonally dominant matrices and the convergence of the AOR iterative method
Authors: Martins M. Madalena
Published: June 2010

Considering the linear systems

A
x
=
b

Ax = b

, where the matrix A is irreducible and diagonally dominant, we obtain bounds for the spectral radius of the

L

r
,
ω

{L_{r,\omega }}

matrix of the AOR method and we achieve the convergence conditions given in [2] by a different method. If A is strictly diagonally dominant, we get larger intervals for the parameter

ω
\omega

of the SOR method, and we improve the results of Theorems 5, 6 of [3] for the AOR method.
Optimal numerical differentiation using three function evaluations
Authors: Ash J. Marshall , Jones Roger L.
Published: June 2010

Approximation of

f
′

(
x
)

f’ (x)

by a difference quotient of the form
\[

h

−
1

[

a
1

f
(
x
+

b
1

h
)
+

a
2

f
(
x
+

b
2

h
)
+

a
3

f
(
x
+

b
3

h
)
]

{h^{ - 1}}[{a_1}f(x + {b_1}h) + {a_2}f(x + {b_2}h) + {a_3}f(x + {b_3}h)]

\]
is found to be optimized for a wide class of real-valued functions by the surprisingly asymmetric choice of

b

=
(

b
1

,

b
2

,

b
3

)
=
(
1

/

3

−
1
,
1

/

3

,
1

/

3

+
1
)

{\mathbf {b}} = ({b_1},{b_2},{b_3}) = (1/\sqrt 3 - 1,1/\sqrt 3 ,1/\sqrt 3 + 1)

. The nearly optimal choice of

b

=
(
−
2
,
3
,
6
)

{\mathbf {b}} = ( - 2,3,6)

is also discussed.
On algorithmic equivalence in linear fractional programming
Authors: Mond B.
Published: June 2010

This paper compares the effectiveness of three published algorithms for solving the linear fractional programming problem.
Lowest order squared rectangles and squares with the largest element not on the boundary
Authors: Duijvestijn A. J. W. , Leeuw P.
Published: June 2010

The lowest order squared rectangles and squares with the largest element not on the boundary are presented.
Addendum and corrigendum to: “High-precision values of the gamma function and of some related coefficients” [Math. Comp. 34 (1980), no. 150, 553–566; MR 81f:65004] by Fransén and S. Wrigge
Authors: Fransén Arne
Published: June 2010

I present corrected values for the Tables IV and V together with an extension of the Tables I-V. I also present an 80D value of my constant F.
Rates of convergence for viscous splitting of the Navier-Stokes equations
Authors: Beale J. Thomas , Majda Andrew
Published: June 2010

Viscous splitting algorithms are the underlying design principle for many numerical algorithms which solve the Navier-Stokes equations at high Reynolds number. In this work, error estimates for splitting algorithms are developed which are uniform in the viscosity

ν
\nu

as it becomes small for either two- or three-dimensional fluid flow in all of space. In particular, it is proved that standard viscous splitting converges uniformly at the rate

C
ν
Δ
t

C\nu \Delta t

, Strang-type splitting converges at the rate

C
ν

(
Δ
t

)
2

C\nu {(\Delta t)^2}

, and also that solutions of the Navier-Stokes and Euler equations differ by

C
ν

C\nu

in this case. Here C depends only on the time interval and the smoothness of the initial data. The subtlety in the analysis occurs in proving these estimates for fixed large time intervals for solutions of the Navier-Stokes equations in two space dimensions. The authors derive a new long-time estimate for the two-dimensional Navier-Stokes equations to achieve this. The results in three space dimensions are valid for appropriate short time intervals; this is consistent with the existing mathematical theory.
Modular functions arising from some finite groups
Authors: Queen Larissa
Published: June 2010

In [2] Conway and Norton have assigned a "Thompson series" of the form
\[

q

−
1

+

H
1

q
+

H
2

q
2

+
…

{q^{ - 1}} + {H_1}q + {H_2}{q^2} + \ldots

\]
to each element m of the Fischer-Griess "Monster" group M and conjectured that these functions are Hauptmoduls for certain genus-zero modular groups. We have found, for a large number of values of N, all the genus-zero groups between

Γ
0

(
N
)

{\Gamma _0}(N)

and

P
S
L
(
2
,
R
)

PSL(2,R)

that have Hauptmoduls of the above form, and this provides the necessary verification that the series assigned in [2] to particular elements of M really are such Hauptmoduls. (Atkin and Fong [1] have recently verified that

H
n

(
m
)

{H_n}(m)

really is a character of M for all n.) We compute Thompson series for various finite groups and discuss the differences between these groups and M. We find that the resulting Thompson series are all Hauptmoduls for suitable genus-zero subgroups of

P
S
L
(
2
,
R
)

PSL(2,R)

.
Inequalities for certain hypergeometric functions
Authors: Joshi C. M. , Arya J. P.
Published: June 2010

Theorems on two-sided inequalities for Gauss and Kummer’s hypergeometric functions as given by Buschman have been improved. Complex analogues of the said inequalities have been developed and it is pointed out that a similar analysis gives extensions of Luke’s, Flett’s, and Carlson’s theorems.
Direct secant updates of matrix factorizations
Authors: Dennis J. E. , Marwil Earl S.
Published: June 2010

This paper presents a new context for using the sparse Broyden update method to solve systems of nonlinear equations. The setting for this work is that a Newton-like algorithm is assumed to be available which incorporates a workable strategy for improving poor initial guesses and providing a satisfactory Jacobian matrix approximation whenever required. The total cost of obtaining each Jacobian matrix, or the cost of factoring it to solve for the Newton step, is assumed to be sufficiently high to make it attractive to keep the same Jacobian approximation for several steps. This paper suggests the extremely convenient and apparently effective technique of applying the sparse Broyden update directly to the matrix factors in the iterations between reevaluations in the hope that fewer fresh factorizations will be required. The strategy is shown to be locally and q-superlinearly convergent, and some encouraging numerical results are presented.
Polynomial type Padé approximants
Authors: Németh Géza , Zimányi Magda
Published: June 2010

Some results are established giving conditions on

f
(
x
)

f(x)

so that its main diagonal Padé approximation

R
n

(
x
)

{R_n}(x)

is of the form

P
n

(
x
)

/

P
n

(
−
x
)

{P_n}(x)/{P_n}( - x)

, where

P
n

(
x
)

{P_n}(x)

is a polynomial in x of degree n. A number of applications to special functions are presented. Numerical computations are given for the gamma function using the "bignum" arithmetical facilities of formula manipulation languages REDUCE2, FORMAC.
Six new factors of Fermat numbers
Authors: Gostin Gary B. , McLaughlin Philip B.
Published: June 2010

A new prime factor is given for each of the Fermat numbers

F

29

{F_{29}}

,

F

36

{F_{36}}

,

F

99

{F_{99}}

,

F

147

{F_{147}}

,

F

150

{F_{150}}

, and

F

201

{F_{201}}

. A summary of search limits and recent results is included.
Correction of Sturm-Liouville eigenvalue estimates
Authors: Paine J.
Published: June 2010

The error in the Sturm-Liouville eigenvalue estimates obtained by replacing the coefficient function with a piecewise constant interpolate is not uniform. In this paper we present a method for correcting these estimates to obtain a uniform approximation of all eigenvalues.
Eigenvalue problems on infinite intervals
Authors: Markowich Peter A.
Published: June 2010

This paper is concerned with eigenvalue problems for boundary value problems of ordinary differential equations posed on an infinite interval. Problems of that kind occur for example in fluid mechanics when the stability of laminar flows is investigated. Characterizations of eigenvalues and spectral subspaces are given, and the convergence of approximating problems, which are derived by reducing the infinite interval to a finite but large one and by imposing additional boundary conditions at the far end, is proved. Exponential convergence is shown for a large class of problems.
On the real cubic fields
Authors: Llorente P. , Oneto A. V.
Published: June 2010

In this paper the authors announce a table of the 4753 totally real nonconjugate nonabelian cubic fields with discriminant less than 100000. Each field is given by its discriminant, the coefficients of a generating polynomial and the index of this polynomial over the field. A basis of the integers of the field is also given. Some differences with other tables are pointed out.
On the approximate calculation of double integrals
Authors: Levin Moshe
Published: June 2010

Cubature formulas are obtained which are optimal or asymptotically optimal on given sets of functions. These formulas consist of line integrals which may be evaluated by optimal or asymptotically optimal quadrature formulas. The advantage of these formulas over the optimal and asymptotically optimal cubature formulas with rectangular-lattices of knots is shown.
Truncation error bounds for limit periodic continued fractions
Authors: Thron W. J. , Waadeland Haakon
Published: June 2010

We derive a truncation error bound for limit periodic continued fractions

K
(

a
n

/

l
)

K({a_n}/l)

which, at least asymptotically, is best possible by comparsion with periodic continued fractions. For a given limit periodic continued fraction the bound is easy to compute.