bluuebunny/crossref_metadata_2025 huggingface dataset:
Growth of partial sums of divergent series
Authors: Boas R. P.
Published: June 2010

Let

Σ
f
(
n
)

\Sigma f(n)

be a divergent series of decreasing positive terms, with partial sums

s
n

{s_n}

, where f decreases sufficiently smoothly; let

φ
(
x
)
=

∫
1
x

f
(
t
)
d
t

\varphi (x) = \smallint _1^xf(t)dt

and let

ψ
\psi

be the inverse of

φ
\varphi

. Let

n
A

{n_A}

be the smallest integer n such that

s
n

⩾
A

{s_n} \geqslant A

but

s

n
−
1

>
A
(
A
=
2
,
3
,
…
)

{s_{n - 1}} > A(A = 2,3, \ldots )

; let

γ
=
lim
{

Σ
1
n

f
(
k
)
−
φ
(
n
)
}

\gamma = \lim \{ \Sigma _1^nf(k) - \varphi (n)\}

be the analog of Euler’s constant; let

m
=
[
ψ
(
A
−
γ
)
]

m = [\psi (A - \gamma )]

. Call

ω
\omega

a Comtet function for

Σ
f
(
n
)

\Sigma f(n)

if

n
A

=
m

{n_A} = m

when the fractional part of

ψ
(
A
−
γ
)

\psi (A - \gamma )

is less than

ω
(
A
)

\omega (A)

and

n
A

=
m
+
1

{n_A} = m + 1

when the fractional part of

ψ
(
A
−
γ
)

\psi (A - \gamma )

is greater than

ω
(
A
)

\omega (A)

. It has been conjectured that

ω
(
A
)
=
1

/

2

\omega (A) = 1/2

is a Comtet function for

Σ
1

/

n

\Sigma 1/n

. It is shown that in general there is a Comtet function of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega left-parenthesis upper A right-parenthesis equals one half plus one twenty-fourth left-brace StartAbsoluteValue f prime left-parenthesis m right-parenthesis EndAbsoluteValue slash f left-parenthesis m right-parenthesis right-brace left-parenthesis 1 plus o left-parenthesis 1 right-parenthesis right-parenthesis period">

ω
(
A
)
=

1
2

+

1
24

{

|

f
′
(
m
)

|

/

f
(
m
)
}

(
1
+
o
(
1
)
)
.

\omega (A) = \frac {1}{2} + \frac {1}{24} \left \{ |f\prime (m)|/f(m) \right \} (1 + o(1)).

\]
For

Σ
1

/

n

\Sigma 1/n

there is a Comtet function of the form

1

/

2
+
1

/

(
24
)

{
1

/

(
48

m
2

)
}

(
1
+
o
(
1
)
)

1/2 + 1/(24) \left \{ 1/(48m^2) \right \} (1 + o(1))

. Some numerical results are presented.
Taylor series methods for the solution of Volterra integral and integro-differential equations
Authors: Goldfine Alan
Published: June 2010

Algorithms based on the use of Taylor series are developed for the numerical solution of Volterra integral and integro-differential equations of arbitrary order. It is shown that these algorithms are uniformly convergent, bounds are obtained for the truncation error, and an asymptotic error analysis is provided for the integral equation case. The various problems of computer implementation are discussed, and the results of certain experiments suggested by the theory are presented.
Recurrence formula of the Taylor series expansion coefficients of the Jacobian elliptic functions
Authors: Schett Alois
Published: June 2010

A general recurrence formula permitting calculation of the Taylor series expansion coefficients of the Jacobian elliptic functions and the number of permutations of n natural numbers with a given run up or peak is given and its application is demonstrated.
Computer methods for sampling from Student’s 𝑡 distribution
Authors: Kinderman A. J. , Monahan J. F. , Ramage J. G.
Published: June 2010

Several new algorithms for generating deviates from the t family for the degrees of freedom parameter

⩾
1

\geqslant 1

are presented. Both acceptance-rejection and probability mixing procedures are developed. The new algorithms outperform traditional methods for generating deviates from the t family. Recommendations are made concerning choosing an algorithm suited to its application.
Analysis of a mixed finite element method for elasto-plastic plates
Authors: Brezzi F. , Johnson C. , Mercier B.
Published: June 2010

We consider a mixed finite element method for finding approximations of the displacement and moments in a thin elastic-perfectly plastic plate. Under some weak assumptions concerning the regularity of the exact solution, we prove an error estimate for the moments.
Algorithms for computing shape preserving spline interpolations to data
Authors: McAllister David F. , Passow Eli , Roulier John A.
Published: June 2010

Algorithms are presented for computing a smooth piecewise polynomial interpolation which preserves the monotonicity and/or convexity of the data.
On sparse and symmetric matrix updating subject to a linear equation
Authors: Toint Ph. L.
Published: June 2010

A procedure for symmetric matrix updating subject to a linear equation and retaining any sparsity present in the original matrix is derived. The main feature of this procedure is the reduction of the problem to the solution of an n dimensional sparse system of linear equations. The matrix of this system is shown to be symmetric and positive definite. The method depends on the Frobenius matrix norm. Comments are made on the difficulties of extending the technique so that it uses more general norms, the main points being shown by a numerical example.
Real quadratic fields with class numbers divisible by five
Authors: Parry Charles J.
Published: June 2010

Conditions are given for a real quadratic field to have class number divisible by five. If 5 does not dividem, then a necessary condition for 5 to divide the class number of the real quadratic field with conductormor 5mis that 5 divide the class number of a certain cyclic biquadratic field with conductor 5m. Conversely, if 5 divides the class number of the cyclic field, then either one of the quadratic fields has class number divisible by 5 or one of their fundamental units satisfies a certain congruence condition modulo 25.
Comments on: “A comparison of global methods for linear two-point boundary value problems” (Math. Comp. 29 (1975), no. 132, 1007–1019) by R. D. Russell and J. M. Varah
Authors: de Boor Carl , Swartz Blair
Published: June 2010

A more careful count of the operations involved in solving the linear system associated with collocation of a two-point boundary value problem using rough splines reverses results recently reported by others in this journal. In addition, it is observed that the use of the technique of "condensation of parameters" can decrease the computer storage required. Furthermore, the use of a particular highly localized basis can also reduce the setup time when the mesh is irregular. Finally, operation counts are roughly estimated for the solution of certain linear systems associated with two competing collocation methods; namely, collocation with smooth splines and collocation of the equivalent first order system with continuous piecewise polynomials.
Correction in the dominant space: a numerical technique for a certain class of stiff initial value problems
Authors: Alfeld P. , Lambert J. D.
Published: June 2010

Consider a stiff linear initial value problem

y
′
=
A
(
x
)
y
+
g
(
x
)

y\prime = A(x)y + g(x)

, where the eigenvalues of

A
(
x
)

A(x)

may be separated into two sets, one of which dominates the other. The dominant eigenvalues and corresponding right and left eigenvectors may be computed by the power method. A technique is proposed which consists of taking one forward step by a conventional multistep method and then making a correction entirely in the subspace spanned by the eigenvectors corresponding to the dominant eigenvalues. A number of alternative corrections are proposed and discussed. It is shown that the technique is stable provided that the product of the steplength and each of the subdominant eigenvalues lies within the region of absolute stability of the multistep method. The application of the technique to nonlinear problems is discussed, and numerical results are reported.
Positivity of the weights of extended Gauss-Legendre quadrature rules
Authors: Monegato Giovanni
Published: June 2010

We show that the weights of extended Gauss-Legendre quadrature rules are all positive.
Some remarks on the construction of extended Gaussian quadrature rules
Authors: Monegato Giovanni
Published: June 2010

We recall some results from a paper by Szegö on a class of polynomials which are related to extended Gaussian quadrature rules. We show that a very efficient algorithm, for the computation of the abscissas of the rules in question, was already described in that paper. We also point out that this method extends to rules for integrals with an ultraspherical-type weight function. A bound for the error of some of the above rules is also given.
On the observed rate of convergence of an iterative method applied to a model elliptic difference equation
Authors: Nicolaides R. A.
Published: June 2010

A proof is given of the fact that the rate of convergence of a multiple grid type of algorithm is

O
(

h

1

/

2

)

O({h^{1/2}})

in the case of a model elliptic difference equation.
A Clough-Tocher type element useful for fourth order problems over nonpolygonal domains
Authors: Mansfield Lois
Published: June 2010

A new macro-element useful for finite element approximations to fourth order elliptic problems with curved boundaries is presented. An error analysis is given which shows that these elements achieve

O
(

h
2

)

O({h^2})

accuracy in energy.
On computing the discrete Fourier transform
Authors: Winograd S.
Published: June 2010

A new algorithm for computing the Discrete Fourier Transform is described. The algorithm is based on a recent result in complexity theory which enables us to derive efficient algorithms for convolution. These algorithms are then used to obtain the new Discrete Fourier Transform algorithm.
On the computation of modified Bessel function ratios
Authors: Gautschi Walter , Slavik Josef
Published: June 2010

A detailed comparison is made between a continued fraction of Gauss, and one of Perron, for the evaluation of ratios of modified Bessel functions

I
v

(
x
)

/

I

v
−
1

(
x
)
,
x
>
0

{I_v}(x)/{I_{v - 1}}(x),x > 0

,

v
>
0

v > 0

. It will be shown that Perron’s continued fraction has remarkable advantages over Gauss’ continued fraction, particularly when

x
>>
v

x > > v

.
An implicit two-point numerical integration formula for linear and nonlinear stiff systems of ordinary differential equations
Authors: Fatunla Simeon Ola.
Published: June 2010

In [1], the author proposed a semi-implicit one-step integration formula which effectively copes with linear systems of ordinary differential equations with widely varying eigenvalues. The integration algorithm is based on a local representation of the theoretical solution to the initial value problem by a linear combination of exponential functions. The resultant integration formula is of order four. Unfortunately, this algorithm cannot cope with nonlinear stiff systems of ordinary differential equations. In this paper, the author extends the concept adopted in [1] to construct an implicit two-point formula which can effectively cope with nonlinear stiff systems. The resultant integration formula is of order five and it is L-stable and convergent.
Some results on the global inversion of bilinear and quadratic isoparametric finite element transformations
Authors: Frey A. E. , Hall C. A. , Porsching T. A.
Published: June 2010

This paper contains sufficient conditions under which a map whose domain is a compact set is a bijection onto a given set. Relative to certain isoparametric finite element maps, one set of conditions involves the nonvanishing of the Jacobian; another the notion of overspill. An algorithm based on elimination is given for the numerical inversion of these maps.
A remark on the Hensel factorization method
Authors: Zassenhaus Hans
Published: June 2010

In response to a remark of David Yun a new version of "On Hensel factorization I" (J. Number Theory, v. 1, 1969) is given which sets in evidence the quadratic convergence behavior of the method using a minimum of bookkeeping.
Implementing second-derivative multistep methods using the Nordsieck polynomial representation
Authors: Gupta G. K.
Published: June 2010

A polynomial representation for the second-derivative linear multistep methods for solving ordinary differential equations is presented. This representation leads to an implementation of the second-derivative methods using the Nordsieck polynomial representation. Possible advantages of such an implementation are then discussed.
Some primes with interesting digit patterns
Authors: Williams H. C.
Published: June 2010

Several tables of prime numbers whose forms are generalizations of the form

(

10
n

−
1
)

/

9

({10^n} - 1)/9

of the repunit numbers are presented. The repunit number

(

10

317

−
1
)

/

9

({10^{317}} - 1)/9

is shown to be a prime.
Improved error estimates for numerical solutions of symmetric integral equations
Authors: Rakotch E.
Published: June 2010

The most widely employed method for a numerical solution of a symmetric integral equation with kernel

K
(
x
,
t
)

K(x,t)

in interval

I
≡
[
a
,
b
]

I \equiv [a,b]

is the replacement of the original problem by the sequence of eigenproblems
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Superscript left-parenthesis n right-parenthesis Baseline y Subscript i Superscript left-parenthesis n right-parenthesis Baseline equals mu Subscript i n Baseline y Subscript i Superscript left-parenthesis n right-parenthesis Baseline comma upper K Superscript left-parenthesis n right-parenthesis Baseline identical-to left-brace w Subscript j n Baseline upper K left-parenthesis x Subscript i n Baseline comma x Subscript j n Baseline right-parenthesis right-brace comma i equals 1 comma ellipsis comma n comma">

K

(
n
)

y
i

(
n
)

=

μ

i
n

y
i

(
n
)

,

K

(
n
)

≡
{

w

j
n

K
(

x

i
n

,

x

j
n

)
}
,

i
=
1
,
…
,
n
,

{K^{(n)}}y_i^{(n)} = {\mu _{in}}y_i^{(n)},\quad {K^{(n)}} \equiv \{ {w_{jn}}K({x_{in}},{x_{jn}})\} ,\quad i = 1, \ldots ,n,

\]
with

w

j
n

>
0

{w_{jn}} > 0

and

x

j
n

∈
I
,
j
=
1
,
…
,
n

{x_{jn}} \in I,j = 1, \ldots ,n

. The eigenvectors

y
i

(
n
)

y_i^{(n)}

are further used to obtain an approximation, with improved error estimates, of the numerical eigensolution for some

N
>
n

N > n

, with no necessity of computing

μ

i
N

{\mu _{iN}}

and

y
i

(
N
)

,
i
=
1
,
…
,
N

y_i^{(N)},i = 1, \ldots ,N

, and of constructing another matrix.
Vector groups and the equality problem for vector addition systems
Authors: Anshel Michael
Published: June 2010

Our purpose is to demonstrate that results concerning the equality problem for vector addition systems, may be uséd to establish the decidability and undecidability of decision problems associated with the class of HNN extensions of the infinite cyclic group. We call these groups ’vector groups.’
Discrete variable methods for a boundary value problem with engineering applications
Authors: Usmani Riaz A.
Published: June 2010

In this paper we develop numerical techniques of order 2, 4 and 6 for the solution of a fourth order linear equation. A priori error bound is obtained for the fourth order method to prove the convergence of the finite difference scheme. A sufficient condition guaranteeing the uniqueness of the solution of the boundary value problem is also given. Numerical illustrations are tabulated and results compared with the classical Runge-Kutta method.
The method of envelopes
Authors: Miranker W. L. , van Veldhuizen M.
Published: June 2010

The differential equation
\[

d
x

d
t

=

A
ε

x
+
g
(
t
,
x
)

\frac {{dx}}{{dt}} = \frac {A}{\varepsilon }x + g(t,x)

\]
where

A
=

[

0

a
m
p
;

−
1

1

a
m
p
;
0

]

A = \left [ {\begin {array}{*{20}{c}} 0 & { - 1} \\ 1 & 0 \\ \end {array} } \right ]

and

ε
>
0

\varepsilon > 0

is a small parameter is a model for the stiff highly oscillatory problem. In this paper we discuss a new method for obtaining numerical approximations to the solution of the initial value problem for this differential equation. As

ε
→
0

\varepsilon \to 0

, the asymptotic theory for this initial value problem yields an approximation to the solution which develops on two time scales, a fast time t and a slow time

τ
=
t

/

ε

\tau = t/\varepsilon

. We redevelop this asymptotic theory in such a form that the approximation consists of a series of simple functions of

τ
\tau

, called carriers. (This series may be thought of as a Fourier series.) The coefficients of the terms of this series are functions of t. They are called envelopes and they modulate the carriers. Our computational method consists of determining numerical approximations to a finite collection of these envelopes. One of the principal merits of our method is its accuracy for the nonlinear problem.
Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. I
Authors: Goldberg Moshe , Tadmor Eitan
Published: June 2010

Easily checkable sufficient stability criteria are obtained for explicit dissipative approximations to mixed initial-boundary value problems associated with the system

u
t

=
A

u
x

{u_t} = A{u_x}

in the quarter plane

x
⩾
0

x \geqslant 0

,

t
⩾
0

t \geqslant 0

. The criteria are given entirely in terms of the boundary conditions for the outflow unknowns. The results imply that certain well-known boundary conditions, when used in combination with any (stable) dissipative scheme, always maintain stability.
On the convergence of difference approximations to nonlinear contraction semigroups in Hilbert spaces
Authors: Nevanlinna Olavi
Published: June 2010

Convergence properties of the difference schemes (S)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Superscript negative 1 Baseline sigma-summation Underscript j equals 0 Overscript k Endscripts alpha Subscript j Baseline u Subscript n plus j Baseline plus sigma-summation Underscript j equals 0 Overscript k Endscripts beta Subscript j Baseline upper A u Subscript n plus j Baseline equals 0 comma n greater-than-or-slanted-equals 0 comma">

h

−
1

∑

j
=
0

k

α
j

u

n
+
j

+

∑

j
=
0

k

β
j

A

u

n
+
j

=
0
,

n
⩾
0
,

{h^{ - 1}}\sum \limits _{j = 0}^k {{\alpha _j}{u_{n + j}}} + \sum \limits _{j = 0}^k {{\beta _j}A{u_{n + j}}} = 0,\quad n \geqslant 0,

\]
, for evolution equations (E)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d u left-parenthesis t right-parenthesis Over d t EndFraction plus upper A u left-parenthesis t right-parenthesis equals 0 comma t greater-than-or-slanted-equals 0 semicolon u left-parenthesis 0 right-parenthesis equals u 0 element-of ModifyingAbove upper D left-parenthesis upper A right-parenthesis With bar">

d
u
(
t
)

d
t

+
A
u
(
t
)
=
0
,

t
⩾
0
;

u
(
0
)
=

u
0

∈

D
(
A
)

¯

\frac {{du(t)}}{{dt}} + Au(t) = 0,\quad t \geqslant 0;\quad u(0) = {u_0} \in \overline {D(A)}

\]
are studied. Here A is a nonlinear, maximally monotone operator in a real Hilbert space. It is shown, in particular, that if the scheme (S) is consistent and stable for the test equation

x
′
=
λ
x

x\prime = \lambda x

for

λ
∈

C

−
K

\lambda \in {\text {C}} - K

, where K is a compact subset of the right half-plane, then (S) is convergent as

h
↓
0

h \downarrow 0

, with suitable initial values, for (E), on compact intervals [0, T]. Moreover, the convergence is uniform on the half-axis

t
⩾
0

t \geqslant 0

, if the solution

u
(
t
)

u(t)

tends strongly to a constant as

t
→
∞

t \to \infty

. We also show that under weaker stability conditions one can construct conditionally convergent methods.
On multigrid convergence in the indefinite case
Authors: Nicolaides R. A.
Published: June 2010

Previous results of the author on the convergence of the multigrid method for positive definite elliptic problems are generalized to cover the indefinite case.
Maximum norm estimates in the finite element method on plane polygonal domains. II. Refinements
Authors: Schatz A. H. , Wahlbin L. B.
Published: June 2010

The finite element method is considered when applied to a model Dirichlet problem on a plane polygonal domain. Local error estimates are given for the case when the finite element partitions are refined in a systematic fashion near corners.
Eigenvalue approximation by the finite element method: the method of Lagrange multipliers
Authors: Kolata William G.
Published: June 2010

The purpose of this paper is to investigate the application of the finite element method of Lagrange multipliers to the problem of approximating the eigenvalues of a selfadjoint elliptic operator satisfying Dirichlet boundary conditions. Although the Lagrange multiplier method is not a Rayleigh-Ritz-Galerkin approximation scheme, it is shown that at least asymptotically the Lagrange multiplier method has some of the properties of such a scheme. In particular, the approximate eigenvalues are greater than or equal to the exact eigenvalues and can be computed from a nonnegative definite matrix problem. It is also shown that the known estimates for the eigenvalue error are optimal.
The effective choice of the smoothing norm in regularization
Authors: Cullum Jane
Published: June 2010

We consider ill-posed problems of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript 1 Baseline upper K left-parenthesis t comma s right-parenthesis f left-parenthesis s right-parenthesis d s comma 0 less-than-or-slanted-equals t less-than-or-slanted-equals 1 comma">

g
(
t
)
=

∫
0
1

K
(
t
,
s
)
f
(
s
)
d
s
,

0
⩽
t
⩽
1
,

g(t) = \int _0^1 {K(t,s)f(s)ds,\quad 0 \leqslant t \leqslant 1,}

\]
where g and K are given, and we must compute f. The Tikhonov regularization procedure replaces (1) by a one-parameter family of minimization problems-Minimize

(

‖

K
f
−
g

‖

2

+
α
Ω
(
f
)
)

({\left \| {Kf - g} \right \|^2} + \alpha \Omega (f))

-where

Ω
\Omega

is a smoothing norm chosen by the user. We demonstrate by example that the choice of

Ω
\Omega

is not simply a matter of convenience. We then show how this choice affects the convergence rate, and the condition of the problems generated by the regularization. An appropriate choice for

Ω
\Omega

depends upon the character of the compactness of K and upon the smoothness of the desired solution.
A search for large twin prime pairs
Authors: Crandall R. E. , Penk M. A.
Published: June 2010

Two methods are discussed for finding large integers m such that

m
−
1

m - 1

and

m
+
1

m + 1

are both primes. Eight such numbers m of magnitudes 22, 22, 32, 64, 136, 154, 203, and 303 digits are listed; together with primitive roots (for

m
+
1

m + 1

) and Lucas-Lehmer parameters (for

m
−
1

m - 1

). The Hardy-Littlewood twin prime conjecture is supported by a statistical test involving the generation of 249 twin prime pairs in the 50-to-54 digit region.
The Euclidean condition in pure cubic and complex quartic fields
Authors: Cioffari Vincent G.
Published: June 2010

In this paper we prove that a field

Q

(
3

d

)

Q{(^3}\sqrt d )

is euclidean with respect to the ordinary norm if and only if

d
=
2
,
3

d = 2,3

or 10. We also prove that certain fields of the form

Q

(
4

−
d

)
,
d
>
0

Q{(^4}\sqrt { - d} ),d > 0

, are or are not euclidean.
An inductive algorithm to construct finite lattices
Authors: Kyuno Shoji
Published: June 2010

G. Birkhoff [1] proposed the following problem: Enumerate all finite lattices which are uniquely determined (up to isomorphism) by their diagram, considered as a graph. It is not known how many lattices of order n exist, except when the value of n is quite small. The aim of this note is to give an algorithm to construct inductively all finite lattices of order n. Using this algorithm, we have found that there exist 222 lattices for

n
=
8

n = 8

and 1078 lattices for

n
=
9

n = 9

. All lattices of order

n
⩽
8

n \leqslant 8

are shown at the end of this note.
An improved version of the reduction to scalar CDS method for the numerical solution of separably stiff initial value problems
Authors: Alfeld Peter
Published: June 2010

In [1] the Reduction to Scalar CDS method for the solution of separably stiff initial value problems is proposed. In this paper an improved version is given that is equivalent for linear problems but considerably superior for nonlinear problems. A naturally arising numerical example is given, for which the old version fails, yet the new version yields very good results. The disadvantage of the new version is that in the case of several dominant eigenvalues

s
>
1

s > 1

, say, a system of s nonlinear equations has to be solved, whereas the old version gives rise to s uncoupled nonlinear equations.
A Legendre polynomial integral
Authors: Blue James L.
Published: June 2010

Let

{

P
n

(
x
)
}

\{ {P_n}(x)\}

be the usual Legendre polynomials. The following integral is apparently new.
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 1 Baseline upper P Subscript n Baseline left-parenthesis 2 x minus 1 right-parenthesis log StartFraction 1 Over x EndFraction d x equals StartFraction left-parenthesis negative 1 right-parenthesis Superscript n Baseline Over n left-parenthesis n plus 1 right-parenthesis EndFraction for n greater-than-or-slanted-equals 1 period">

∫
0
1

P
n

(
2
x
−
1
)
log
⁡

1
x

d
x
=

(
−
1
)

n

n
(
n
+
1
)

for

n
⩾
1.

\int _0^1{P_n}(2x - 1)\log \frac {1}{x}dx = \frac {{{{( - 1)}^n}}}{{n(n + 1)}}\quad {\text {for}}\;n \geqslant 1.

\]
It has an application in the construction of Gauss quadrature formulas on (0, 1) with weight function

log
⁡
(
1

/

x
)

\log (1/x)

.
On the preceding paper “A Legendre polynomial integral” by James L. Blue
Authors: Gautschi Walter
Published: June 2010

The modified moments of the distribution

d
σ
(
x
)
=

x
α

ln
⁡
(
1

/

x
)

d
x

d\sigma (x) = {x^\alpha }\ln (1/x)\;dx

on [0, 1], with respect to the shifted Legendre polynomials, are explicitly evaluated.
The dilogarithm function of a real argument
Authors: Morris Robert
Published: June 2010

This paper is a user’s guide to the dilogarithm function
\[

L

i
2

(
z
)
=
−

∫
0
z

log
⁡
(
1
−
z
)

z

d
z

L{i_2}(z) = - \int _0^z {\frac {{\log (1 - z)}}{z}} \;dz

\]
of a real argument. It is intended for those who are primarily interested in the values of the dilogarithm rather than in its functional relationships. The paper is deliberately written in the style of the book Computer Approximations by Hart, Cheney et al.
A note on the integral ∫^{∞}₀ 𝑡^{2𝛼-1}(1+𝑡²) ^{1-𝛼-𝛽}𝐽_{𝜈}(𝑥√(1+𝑡²))𝑑𝑡
Authors: Glasser M. L.
Published: June 2010

The integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript v Superscript alpha comma beta Baseline left-parenthesis x right-parenthesis equals integral Subscript 0 Superscript normal infinity Baseline t Superscript 2 alpha minus 1 Baseline left-parenthesis 1 plus t squared right-parenthesis Superscript 1 minus alpha minus beta Baseline upper J Subscript v Baseline left-parenthesis x StartRoot 1 plus t squared right-parenthesis EndRoot d t">

I
v

α
,
β

(
x
)
=

∫
0
∞

t

2
α
−
1

(
1
+

t
2

)

1
−
α
−
β

J
v

(
x

1
+

t
2

)

d
t

I_v^{\alpha ,\beta }(x) = \int _0^\infty {{t^{2\alpha - 1}}{{(1 + {t^2})}^{1 - \alpha - \beta }}{J_v}(x} \sqrt {1 + {t^2})} \;dt

\]
is expressed in terms of Bessel and related functions for various values of the parameters by summing the hypergeometric series representation given by Schmidt.
The nonabelian simple groups 𝐺, 𝐺<10⁶—minimal generating pairs
Authors: McKay John , Young Kiang Chuen
Published: June 2010

Minimal (k, m, n) generating pairs and their associated presentations are defined for all nonabelian simple groups G,

|

G

|

>

10
6

|G| > {10^6}

, excepting the family

PSL

(
2
,
q
)

{\text {PSL}}(2,q)

. A complete set of minimal (2, m, n) generating permutations of minimal degree is tabulated for these G. The set is complete in the sense that any minimal generating pair for G will satisfy the same presentation as exactly one of the listed pairs.
Semidiscretization in time for parabolic problems
Authors: Le Roux Marie-Noëlle
Published: June 2010

We study the error to the discretization in time of a parabolic evolution equation by a single-step method or by a multistep method when the initial condition is not regular.
Stable evaluation of polynomials in time 𝑙𝑜𝑔𝑛
Authors: Kusterer Roland , Reimer Manfred
Published: June 2010

An algorithm is investigated which evaluates real polynomials of degree n in time

log
⁡
n

\log n

at asymptotically minimum costs. The algorithm is considerably stable with respect to round-off.
On computing Artin 𝐿-functions in the critical strip
Authors: Lagarias J. C. , Odlyzko A. M.
Published: June 2010

This paper gives a method for computing values of certain nonabelian Artin L-functions in the complex plane. These Artin L-functions are attached to irreducible characters of degree 2 of Galois groups of certain normal extensions K of Q. These fields K are the ones for which

G
=

Gal

(
K

/

Q

)

G = {\operatorname {Gal}}(K/{\mathbf {Q}})

has an abelian subgroup A of index 2, whose fixed field

Q

(

d

)

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

is complex, and such that there is a

σ
∈
G
−
A

\sigma \in G - A

for which

σ
a

σ

−
1

=

a

−
1

\sigma a{\sigma ^{ - 1}} = {a^{ - 1}}

for all

a
∈
A

a \in A

. The key property proved here is that these particular Artin L-functions are Hecke (abelian) L-functions attached to ring class characters of the imaginary quadratic field

Q

(

d

)

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

and, therefore, can be expressed as linear combinations of Epstein zeta functions of positive definite binary quadratic forms. Such Epstein zeta functions have rapidly convergent expansions in terms of incomplete gamma functions. In the special case

K
=

Q

(

−
3

,

a

1

/

3

)

K = {\mathbf {Q}}(\sqrt { - 3} ,{a^{1/3}})

, where

a
>
0

a > 0

is cube-free, the Artin L-function attached to the unique irreducible character of degree 2 of

Gal

(
K

/

Q

)
≅

S
3

{\operatorname {Gal}}(K/{\mathbf {Q}}) \cong {S_3}

is the quotient of the Dedekind zeta function of the pure cubic field

L
=

Q

(

a

1

/

3

)

L = {\mathbf {Q}}({a^{1/3}})

by the Riemann zeta function. For functions of this latter form, representations as linear combinations of Epstein zeta functions were worked out by Dedekind in 1879. For

a
=
2
,
3
,
6

a = 2,3,6

and 12, such representations are used to show that all of the zeroes

ρ
=
σ
+
i
t

\rho = \sigma + it

of these L-functions with

0
>
σ
>
1

0 > \sigma > 1

and

|

t

|

⩽
15

|t| \leqslant 15

are simple and lie on the critical line

σ
=
1

/

2

\sigma = 1/2

. These methods currently cannot be used to compute values of L-functions with

Im
⁡
(
s
)

\operatorname {Im} (s)

much larger than 15, but approaches to overcome these deficiencies are discussed in the final section.
A special class of explicit linear multistep methods as basic methods for the correction in the dominant space technique
Authors: Alfeld Peter
Published: June 2010

A class of explicit linear multistep methods is suggested as basic methods for the CDS schemes introduced in [3]. These schemes are designed for the numerical solution of certain stiff ordinary differential equations, and operate with dominant eigenvalues, and the corresponding eigenvectors, of the Jacobian. The motivation, and the stability analysis for CDS schemes assumes that the eigensystem is constant. Here methods are introduced that perform particularly well if the eigensystem is not constant. In a certain sense the methods introduced here can be considered explicit approximations to the well-known implicit backward-differentiation formulas used by Gear [6] for the stiff option of his o.d.e. solver.
Recurrence relations for computing with modified divided differences
Authors: Krogh Fred T.
Published: June 2010

Modified divided differences (MDD) provide a good way of representing a polynomial passing through points with unequally spaced abscissas. This note gives recurrence relations for computing coefficients in either the monomial or Chebyshev basis from the MDD coefficients, and for computing the MDD coefficients for either the differentiated or the integrated polynomial. The latter operation is likely to be useful if MDD are used in a method for solving stiff differential equations.
An application of the finite element approximation method to find the complex zeros of the modified Bessel function 𝐾_{𝑛}(𝑧)
Authors: Leung K. V. , Ghaderpanah S. S.
Published: June 2010

Using a finite element approximation, an iterative optimization scheme is described to find the z zeros of

K
n

(
z
)

{K_n}(z)

for fixed order n. Two computer programs have been implemented to find the complex zeros with a computational accuracy of either 13 or 27 significant digits. The optimization scheme described in the paper may also be readily applied to find real and complex zeros of an arbitrary function with real and complex coefficients. Neither its accuracy nor its efficiency is affected by the number of the roots of the function.
New primes of the form 𝑘⋅2ⁿ+1
Authors: Baillie Robert
Published: June 2010

All primes of the form

k
⋅

2
n

+
1

k \cdot {2^n} + 1

for k odd,

1
⩽
k
>
150

1 \leqslant k > 150

,

1
⩽
n
⩽
1500

1 \leqslant n \leqslant 1500

, have now been computed. Those not previously published are given here. Numbers with

151
⩽
k
⩽
999

151 \leqslant k \leqslant 999

and

n
⩽
600

n \leqslant 600

were also tested. Three new factors of Fermat numbers and a large pair of twin primes were found.
Some primes of the form (𝑎ⁿ-1)/(𝑎-1)
Authors: Williams H. C. , Seah E.
Published: June 2010

A table of primes of the form

(

a
n

−
1
)

/

(
a
−
1
)

({a^n} - 1)/(a - 1)

for values of a and n such that

3
⩽
a
⩽
12

3 \leqslant a \leqslant 12

,

2
⩽
n
⩽
1000

2 \leqslant n \leqslant 1000

is presented. A description is given of the techniques used to obtain this table, and some numbers such as

(

10

1031

−
1
)

/

9

({10^{1031}} - 1)/9

which are pseudoprime but whose primality is not yet rigorously established are also discussed.
Sets of integers with nonlong arithmetic progressions generated by the greedy algorithm
Authors: Gerver Joseph L. , Ramsey L. Thomas
Published: June 2010

Let

S
k

{S_k}

be the set of positive integers containing no arithmetic progression of k terms, generated by the greedy algorithm. A heuristic formula, supported by computational evidence, is derived for the asymptotic density of

S
k

{S_k}

in the case where k is composite. This formula, with a couple of additional assumptions, is shown to imply that the greedy algorithm would not maximize

Σ

n
∈
S

1

/

n

{\Sigma _{n \in S}}1/n

over all S with no arithmetic progression of k terms. Finally it is proved, without relying on any conjecture, that for all

ε
>
0

\varepsilon > 0

, the number of elements of

S
k

{S_k}

which are less than n is greater than

(
1
−
ε
)

2
n

(1 - \varepsilon )\sqrt {2n}

for sufficiently large n.
A conservative finite element method for the Korteweg-de Vries equation
Authors: Winther Ragnar
Published: June 2010

A finite element method for the 1-periodic Korteweg-de Vries equation
\[

u
t

+
2
u

u
x

+

u

x
x
x

=
0

{u_t} + 2u{u_x} + {u_{xxx}} = 0

\]
is analyzed. We consider first a semidiscrete method (i.e., discretization only in the space variable), and then we analyze some unconditionally stable fully discrete methods. In a special case, the fully discrete methods reduce to twelve point finite difference schemes (three time levels) which have second order accuracy both in the space and time variable.
Misstatements in: The calculus of finite differences [Macmillan, London, 1933; Zbl 8, 18] by L. M. Milne-Thomson
Authors: Salzer Herbert E.
Published: June 2010

Milne-Thomson’s "proofs" of the interpolating property of the continued fraction employing reciprocal differences, as well as the symmetry property of reciprocal differences, are shown to be faulty. He also makes a similar misleading, though correct, statement in connection with divided differences.
Computation of steady shocks by second-order finite-difference schemes
Authors: Karlsen Lasse K.
Published: June 2010

The computational stability of steady shocks which satisfy the entropy condition is considered for the scalar conservation law
\[

∂
u

∂
t

+

∂

∂
x

(

1
2

u
2

)

=
0.

\frac {{\partial u}}{{\partial t}} + \frac {\partial }{{\partial x}}\left ( {\frac {1}{2}{u^2}} \right ) = 0.

\]
It is shown that the computation of the pure initial value problem by Lax-Wendroff type schemes approaches a steady state if the initial data satisfies a specified condition, and that this condition is always satisfied for the corresponding initial-boundary value problem with a finite number of grid points. The effect of machine accuracy on the influence of the boundaries on the error near the shock is also discussed.
Polynomial approximation of functions in Sobolev spaces
Authors: Dupont Todd , Scott Ridgway
Published: June 2010

Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.
An incomplete factorization technique for positive definite linear systems
Authors: Manteuffel T. A.
Published: June 2010

This paper describes a technique for solving the large sparse symmetric linear systems that arise from the application of finite element methods. The technique combines an incomplete factorization method called the shifted incomplete Cholesky factorization with the method of generalized conjugate gradients. The shifted incomplete Cholesky factorization produces a splitting of the matrix A that is dependent upon a parameter

α
\alpha

. It is shown that if A is positive definite, then there is some

α
\alpha

for which this splitting is possible and that this splitting is at least as good as the Jacobi splitting. The method is shown to be more efficient on a set of test problems than either direct methods or explicit iteration schemes.
On Stieltjes’ continued fraction for the gamma function
Authors: Char Bruce W.
Published: June 2010

The first forty-one coefficients of a continued fraction for

ln
⁡
Γ
(
z
)
+
z
−
(
z
−
1

/

2
)
ln
⁡
z
−
ln
⁡

2
π

\ln \Gamma (z) + z - (z - 1/2) \ln z - \ln \sqrt {2\pi }

, are given. The computation, based on Wall’s algorithm for converting a function’s power series representation to a continued fraction representation, was run on the algebraic manipulation system MACSYMA.

∗
∗

^{\ast \ast }
Existence theorems for transforms over finite rings with applications to 2-D convolution
Authors: Maher David P.
Published: June 2010

An existence theorem for Fourier-like transforms over arbitrary finite commutative rings is proven in a simple fashion. Corollaries for the case of residue class rings over the integers and extensions of those rings follow directly. The theory is applied to construct very fast algorithms for the computation of two-dimensional convolutions over the integers

mod

M

\bmod \, M

.
Root-finding by fitting rational functions
Authors: Larkin F. M.
Published: June 2010

A tabular, recursive method is presented for the computation of a sequence of abscissae designed to converge to a simple zero of an analytic function. The key to the method is an efficient means for evaluating the zeros of a sequence of rational functions, having linear numerators, fitted to information previously computed. Regional and asymptotic convergence properties of the method are described. Conditions sufficient to ensure convergence are derived, and it is shown that asymptotically quadratic convergence can be achieved, at the cost of only a moderate amount of "overhead" computation.
Analysis of mixed methods using mesh dependent norms
Authors: Babuška I. , Osborn J. , Pitkäranta J.
Published: June 2010

This paper analyzes mixed methods for the biharmonic problem by means of new families of mesh dependent norms which are introduced and studied. More specifically, several mixed methods are shown to be stable with respect to these norms and, as a consequence, error estimates are obtained in a simple and direct manner.
Spectral and pseudospectral methods for advection equations
Authors: Pasciak Joseph E.
Published: June 2010

Spectral and pseudo spectral methods for advection equations are investigated. A basic framework is given which allows the application of techniques used in finite element analysis to spectral methods with trigonometric polynomials. Error estimates for semidiscrete spectral and pseudo spectral as well as fully discrete explicit pseudo spectral methods are given. The approximation schemes are shown to converge with infinite order.
Variational crimes and 𝐿^{∞} error estimates in the finite element method
Authors: Goldstein Charles I.
Published: June 2010

In order to numerically solve a second-order linear elliptic boundary value problem in a bounded domain, using the finite element method, it is often necessary in practice to violate certain assumptions of the standard variational formulation. Two of these "variational crimes" will be emphasized here and it will be shown that optimal

L
∞

{L^\infty }

error estimates still hold. The first "crime" occurs when a nonconforming finite element method is employed, so that smoothness requirements are violated at interelement boundaries. The second "crime" occurs when numerical integration is employed, so that the bilinear form is perturbed. In both cases, the "patch test" is crucial to the proof of

L
∞

{L^\infty }

estimates, just as it was in the case of mean-square estimates.
Additive methods for the numerical solution of ordinary differential equations
Authors: Cooper G. J. , Sayfy A.
Published: June 2010

Consider a system of differential equations

x
′
=
f
(
x
)

x\prime = f(x)

. Most methods for the numerical solution of such a system may be characterized by a pair of matrices (A, B) and make no special use of any structure inherent in the system. In this article, methods which are characterized by a triple of matrices

(
A
;

B
1

,

B
2

)

(A;{B_1},{B_2})

are considered. These methods are applied in an additive fashion to a decomposition

f
=

f
1

+

f
2

f = {f_1} + {f_2}

and some methods have pronounced advantages when one term of the decomposition is linear. This article obtains algebraic conditions which give the order of convergence of such methods. Some simple examples are displayed.
Five-diagonal sixth-order methods for two-point boundary value problems involving fourth-order differential equations
Authors: Katti C. P.
Published: June 2010

We present a sixth order finite difference method for the two-point boundary value problem

y

(
4
)

+
f
(
x
,
y
)
=
0

{y^{(4)}} + f(x,y) = 0

,

y
(
a
)
=

A
0

y(a) = {A_0}

,

y
(
b
)
=

B
0

y(b) = {B_0}

,

y
′
(
a
)
=

A
1

y\prime (a) = {A_1}

,

y
′
(
b
)
=

B
1

y\prime (b) = {B_1}

. In the case of linear differential equations, our difference scheme leads to five-diagonal linear systems.
The spectral transformation Lánczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems
Authors: Ericsson Thomas , Ruhe Axel
Published: June 2010

A new algorithm is developed which computes a specified number of eigenvalues in any part of the spectrum of a generalized symmetric matrix eigenvalue problem. It uses a linear system routine (factorization and solution) as a tool for applying the Lanczos algorithm to a shifted and inverted problem. The algorithm determines a sequence of shifts and checks that all eigenvalues get computed in the intervals between them. It is shown that for each shift several eigenvectors will converge after very few steps of the Lanczos algorithm, and the most effective combination of shifts and Lanczos runs is determined for different sizes and sparsity properties of the matrices. For large problems the operation counts are about five times smaller than for traditional subspace iteration methods. Tests on a numerical example, arising from a finite element computation of a nuclear power piping system, are reported, and it is shown how the performance predicted bears out in a practical situation.
On some orthogonal polynomial integrals
Authors: Gatteschi Luigi
Published: June 2010

The modified moments of the weight functions

w
(
x
)
=

x
ρ

(
1
−
x

)
a

ln
⁡
(
1

/

x
)

w(x) = {x^\rho }{(1 - x)^a}\ln (1/x)

, on [0, 1], with respect to the shifted Jacobi polynomials
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript n Superscript asterisk left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis x right-parenthesis equals upper P Subscript n Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis 2 x minus 1 right-parenthesis">

P
n

∗
(
α
,
β
)

(
x
)
=

P
n

(
α
,
β
)

(
2
x
−
1
)

P_n^{\ast (\alpha ,\beta )}(x) = P_n^{(\alpha ,\beta )}(2x - 1)

, and

w
p

(
x
)
=

x
ρ

e

−
x

(
ln
⁡
x

)
p

{w_p}(x) = {x^\rho }{e^{ - x}}{(\ln x)^p}

,

p
=
1
,
2

p = 1,2

, on

[
0
,
∞
)

[0,\infty )

, with respect to the generalized La guerre polynomials

L
n

(
α
)

(
x
)

L_n^{(\alpha )}(x)

, are explicitly evaluated.
The stability of pseudospectral-Chebyshev methods
Authors: Gottlieb David
Published: June 2010

The stability of pseudospectral-Chebyshev methods is demonstrated for parabolic and hyperbolic problems with variable coefficients. The choice of collocation points is discussed. Numerical examples are given for the case of variable coefficient hyperbolic equations.
Multiple grid methods for the solution of Fredholm integral equations of the second kind
Authors: Hemker P. W. , Schippers H.
Published: June 2010

In this paper multiple grid methods are applied for the fast solution of the large nonsparse systems of equations that arise from the discretization of Fredholm integral equations of the second kind. Various multiple grid schemes, both with Nyström and with direct interpolation, are considered. For these iterative methods, the rates of convergence are derived using the collectively compact operator theory by Anselone and Atkinson. Estimates for the asymptotic computational complexity are given, which show that the multiple grid schemes result in

O

(

N
2

)

\mathcal {O}({N^2})

arithmetic operations.
Partial fraction evaluation by an escalation technique
Authors: Mahoney J. F.
Published: June 2010

An escalation method for performing partial fraction expansions is presented for the case that the complete list of zeros of the denominator of the proper rational function is known. Expressions for the number of divisions and multiplications required are developed. The new method requires fewer such arithmetic operations than do other known methods. A numerical example is provided.
A large prime gap
Authors: Weintraub Sol
Published: June 2010

A prime gap of 654 (653 consecutive composites) is found near

1.1
×

10

16

{1.1 \times 10^{16}}

.
Gunderson’s function in Fermat’s last theorem
Authors: Shanks Daniel , Williams H. C.
Published: June 2010

We study Gunderson’s function which gives a bound on the first case of Fermat’s last theorem, assuming that the generalized Wieferich criterion is valid for the first n prime bases. We note two unexpected phenomena.
Hyperperfect numbers with three different prime factors
Authors: te Riele Herman J. J.
Published: June 2010

The existence of hyperperfect numbers with more than two different prime factors is shown by five examples.
One-sided difference approximations for nonlinear conservation laws
Authors: Engquist Björn , Osher Stanley
Published: June 2010

We analyze one-sided or upwind finite difference approximations to hyperbolic partial differential equations and, in particular, nonlinear conservation laws. Second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions. We show that no such stable approximation of order higher than two is possible. These one-sided schemes have desirable properties for shock calculations. We show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance. New and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles, existence, uniqueness, and stability of discrete shocks. Numerical examples are given.