bluuebunny/crossref_metadata_2025 huggingface dataset:
Some relations between the values of a function and its first derivative at 𝑛 abscissa points
Authors: Huddleston Robert E.
Published: June 2010

For a polynomial, P, of degree

2
n
−
2

2n - 2

, there exists a relation between the values of P and the values of its first derivative,

P
′

P’

, at the n abscissa points

x
1

,
⋯
,

x
n

{x_1}, \cdots ,{x_n}

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i equals 1 Overscript n Endscripts left-bracket a Subscript i Baseline upper P left-parenthesis x Subscript i Baseline right-parenthesis plus b Subscript i Baseline upper P prime left-parenthesis x Subscript i Baseline right-parenthesis right-bracket equals 0 period">

∑

i
=
1

n

[

a
i

P
(

x
i

)
+

b
i

P
′

(

x
i

)
]
=
0.

\sum \limits _{i = 1}^n {[{a_i}P({x_i}) + {b_i}P’({x_i})] = 0.}

\]
Replacing P by a differentiable function y yields
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i equals 1 Overscript n Endscripts left-bracket a Subscript i Baseline y left-parenthesis x Subscript i Baseline right-parenthesis plus b Subscript i Baseline y prime left-parenthesis x Subscript i Baseline right-parenthesis right-bracket equals upper E left-parenthesis y comma x right-parenthesis period">

∑

i
=
1

n

[

a
i

y
(

x
i

)
+

b
i

y
′

(

x
i

)
]
=
E
(
y
,
x
)
.

\sum \limits _{i = 1}^n {[{a_i}y({x_i}) + {b_i}y’({x_i})] = E(y,x).}

\]
These relations are obtained and the error function

E
(
y
,
x
)

E(y,x)

is given explicitly.
Comparing error estimators for Runge-Kutta methods
Authors: Shampine L. F. , Watts H. A.
Published: June 2010

A way is proposed to compare local error estimators. This is applied to the major estimators for fourth-order Runge-Kutta procedures. An estimator which leads to a production code 18
Contractive difference schemes for symmetric hyperbolic systems
Authors: Brenner Philip , Thomée Vidar
Published: June 2010

Consider the initial-value problem for a constant-coefficient symmetric hyperbolic system with initial-values vanishing in a half-space. Consider also a finite difference operator consistent with the system. Conditions are given in terms of the orders of dissipation and accuracy which ensure that the solution of the discrete problem tends to zero exponentially with the mesh-width in half-spaces where the solution of the continuous problem vanishes.
On the Feit-Thompson conjecture
Authors: Stephens N. M.
Published: June 2010

A counterexample is found to the conjecture that

(

p
q

−
1
)

/

(
p
−
1
)

({p^q} - 1)/(p - 1)

and

(

q
p

−
1
)

/

(
q
−
1
)

({q^p} - 1)/(q - 1)

are coprime where p, q are primes.
Pseudo-random numbers. The exact distribution of pairs
Authors: Dieter U.
Published: June 2010

Pseudo-random numbers are usually generated by linear congruential methods. Starting with an integer

y
0

{y_0}

, a sequence

{

y
i

}

\{ {y_i}\}

is constructed by

y

i
+
1

≡
a

y
i

+
r

(
mod

m
)
,
m
,
a
,
r

{y_{i + 1}} \equiv a{y_i} + r \pmod m, m, a, r

being integers. The derived fractions

x
i

≡

y
i

/

m

{x_i} \equiv {y_i}/m

are taken as samples from the uniform distribution on [0, 1). In this paper it is shown that the joint probability distribution of pairs

x
i

,

x

i
+
s

{x_i},{x_{i + s}}

can be calculated exactly. Explicit calculations show that this distribution is surprisingly near to the uniform distribution for most ’reasonable’ generators. The best approximation to the uniform distribution on the unit-square is achieved if the continued fraction for

a
s

{a^s}

and m (or

a
s

{a^s}

and m/f) is long.
Obtaining cubatures for rectangles and other planar regions by using orthogonal polynomials
Authors: Franke Richard
Published: June 2010

A. H. Stroud has recently shown the existence of cubature formulas for planar regions which use

m
2

{m^2}

points and have polynomial precision

2
m
−
1

2m - 1

. In this paper, the author gives sufficient conditions for the existence of formulaa using fewer than

m
2

{m^2}

points, and having polynomial precision

2
m
−
1

2m - 1

. An algorithm is given for computing such formulas, and is shown to be useful in a more general setting than given in the theorem. Numerical examples are given, both in terms of previously known and new cubature formulas.
On the error in the numerical integration of Chebyshev polynomials
Authors: Nicholson D. , Rabinowitz P. , Richter-Dyn Nira , Zeilberger D.
Published: June 2010

A general method is described to compute the exact error in the numerical integration of a given polynomial by certain types of integration rules. This method is applied to get exact errors in the integration of certain Chebyshev polynomials of the first kind by Gauss and Lobatto rule and asymptotic errors in the integration of Chebyshev polynomials of both kinds by Gauss, Lobatto and Radau rules.
Roundoff error analysis of the fast Fourier transform
Authors: Ramos George U.
Published: June 2010

This paper presents an analysis of roundoff errors occurring in the floating-point computation of the fast Fourier transform. Upper bounds are derived for the ratios of the root-mean-square (RMS) and maximum roundoff errors in the output data to the RMS value of the output data for both single and multidimensional transformations. These bounds are compared experimentally with actual roundoff errors.
On ignoring the singularity in numerical quadrature
Authors: Miller R. K.
Published: June 2010

This paper studies the convergence of numerical quadratures of singular integrands. The singularities are ignored in the sense that whenever a singularity occurs the integrand is redefined to be zero. Several convergence theorems are proved under the assumption that the integrand can be dominated near each singularity by a monotone, integrable function.
Matrix representations of nonlinear equation iterations—Application to parallel computation
Authors: Rice John R.
Published: June 2010

A matrix representation of iterative methods is presented which includes almost all those based on polynomial methods. A simple lemma and corollaries are established which show that the order of convergence of the iteration is the spectral radius of the matrix representation. A number of old and new methods, particularly those adapted to parallel computation, are analyzed using this representation.
Third and fourth order accurate schemes for hyperbolic equations of conservation law form
Authors: Zwas Gideon , Abarbanel Saul
Published: June 2010

It is shown that for quasi-linear hyperbolic systems of the conservation formWt=−Fx=−AWx{W_t} = - {F_x} = - A{W_x}, it is possible to build up relatively simple finite-difference numerical schemes accurate to 3rd and 4th order provided that the matrixAsatisfies commutativity relations with its partial-derivative-matrices. These schemes generalize the Lax-Wendroff 2nd order scheme, and are written down explicitly. As found by Strang [8] odd order schemes are linearly unstable, unless modified by adding a term containing the next higher space derivative or, alternatively, by rewriting the zeroth term as an average of the correct order. Thus stabilized, the schemes, both odd and even, can be made to meet the C.F.L. (Courant-Friedrichs-Lewy) criterion of the Courant-number being less or equal to unity. Numerical calculations were made with a 3rd order and a 4th order scheme for scalar equations with continuous and discontinuous solutions. The results are compared with analytic solutions and the predicted improvement is verified. The computation reported on here was carried out on the CDC-3400 computer at the Tel Aviv University computation center.
The Dirichlet problem for a class of elliptic difference equations
Authors: McAllister G. T.
Published: June 2010

Under suitable assumptions on the order of nonlinearity we prove existence and uniqueness theorems for difference Dirichlet problems of divergence type. We also show that the discrete solutions converge to a solution of the continuous problem. We do not assume that our equation comes from a variational problem. Some of our results are constructive or allow for the application of constructive methods.
A finite-difference method for parabolic differential equations with mixed derivatives
Authors: Kowalski Jan Krzysztof
Published: June 2010

In a recent paper, P. Jamet constructed a positive difference operator for a parabolic differential operator whose coefficients are singular on the boundary, and proved the existence of a unique solution of the boundary-value problem for the differential equation using discrete barriers. In the present paper, Jamet’s results are extended to the parabolic operator with mixed derivatives.
Miniaturized tables of Bessel functions. III
Authors: Luke Yudell L.
Published: June 2010

After the manner of our previous studies, coefficients for the expansion of

J
ν

(
z
)

{J_\nu }(z)

and

Y
ν

(
z
)

{Y_\nu }(z)

in double series of Chebyshev polynomials are presented. For

J
ν

(
z
)

{J_\nu }(z)

the ranges are (1)

0
>
z
≦
8
,
0
≦
ν
≦
4

0 > z \leqq 8,0 \leqq \nu \leqq 4

, (2)

0
>
z
≦
8
,
4
≦
ν
≦
8

0 > z \leqq 8,4 \leqq \nu \leqq 8

. For

J
ν

(
z
)
+
i

Y
ν

(
z
)

{J_\nu }(z) + i{Y_\nu }(z)

, the ranges are

z
≧
5

z \geqq 5

and

0
≦
ν
≦
1

0 \leqq \nu \leqq 1

. The coefficients are given with sufficient accuracy to enable the evaluation of the Bessel functions to at least 20 decimals.
Monotonicity in Romberg quadrature
Authors: Ström Torsten
Published: June 2010

Monotonicity of one or more derivatives of the integrand is shown to imply a corresponding property of the approximating Romberg scheme. This is of importance in connection with error estimation by majorants [6]. The monotonicity properties are derived from an elementary study of the kernel functions involved. A possible explanation is given of the monotonicity which frequently occurs in applications where nothing is presupposed about the signs of derivatives of the integrand. Finally, a nonlinear addition to Havie’s scheme is suggested.
Tridiagonalization of completely nonnegative matrices
Authors: Rainey J. W. , Habetler G. J.
Published: June 2010

Let

M
=
[

m

i
j

]

i
,
j
=
1

n

M = [{m_{ij}}]_{i,j = 1}^n

be completely nonnegative (CNN), i.e., every minor of

M
M

is nonnegative. Two methods for reducing the eigenvalue problem for

M
M

to that of a CNN, tridiagonal matrix,

T
=
[

t

i
j

]

T = [{t_{ij}}]

(

t

i
j

=
0

{t_{ij}} = 0

when

|

i
−
j

|

>
1
)

|i - j| > 1)

), are presented in this paper. In the particular case that

M
M

is nonsingular it is shown for one of the methods that there exists a CNN nonsingular

S
S

such that

S
M
=
T
S

SM = TS

.
Uniform approximation of vector-valued functions with a constraint
Authors: Belford Geneva G.
Published: June 2010

This paper deals with existence and characterization of best approximations to vector-valued functions. The approximations are themselves vector-valued functions with components taken from a linear space, but the constraint is imposed that certain of the approximation parameters should be identical for all components.
The condition of orthogonal polynomials
Authors: Gautschi Walter
Published: June 2010

An estimate is given for the condition number of the coordinate map associating to each polynomial its coefficients with respect to a system of orthogonal polynomials.
On the solution of the equations arising from collocation with cubic 𝐵-splines
Authors: Sincovec Richard F.
Published: June 2010

An iterative technique for solving the equations arising from collocation with cubic

B
B

-splines in solving second-order nonlinear boundary-value problems is defined and shown to converge.
High precision evaluation of the irregular Coulomb wave functions
Authors: Strecok A. J. , Gregory J. A.
Published: June 2010

This tutorial paper presents practical methods for accurately evaluating irregular Coulomb wave functions. Rational approximations to

G
0

(
η
,
ρ
)

{G_0}(\eta ,\rho )

and

G
0
′

(
η
,
ρ
)

{G’_0}(\eta ,\rho )

are developed along line segments in the

(
η
,
ρ
)

(\eta ,\rho )

plane to provide useful initial values for the associated differential equation. These approximations, designed for IBM System 360 Fortran double precision, yield results to at least 13 significant decimal places.
von Neumann’s comparison method for random sampling from the normal and other distributions
Authors: Forsythe George E.
Published: June 2010

The author presents a generalization he worked out in 1950 of von Neumann’s method of generating random samples from the exponential distribution by comparisons of uniform random numbers on (0,1). It is shown how to generate samples from any distribution whose probability density function is piecewise both absolutely continuous and monotonic on

(
−
∞
,
∞
)

( - \infty ,\infty )

. A special case delivers normal deviates at an average cost of only 4.036 uniform deviates each. This seems more efficient than the Center-Tail method of Dieter and Ahrens, which uses a related, but different, method of generalizing the von Neumann idea to the normal distribution.
Density bounds for the sum of divisors function
Authors: Wall Charles R. , Crews Phillip L. , Johnson Donald B.
Published: June 2010

Upper and lower bounds are presented for the density of the integers

n
n

for which

σ
(
n
)

/

n
≧
x

\sigma (n)/n \geqq x

, where

σ
(
n
)

\sigma (n)

is the sum of divisors of

n
n

, and

1
≦
x
≦
5

1 \leqq x \leqq 5

.
Numerical quadrature by the 𝜖-algorithm
Authors: Kahaner David K.
Published: June 2010

Modifications of Romberg’s integration method are applicable to functions with endpoint singularities. Several authors have reached different conclusions on the usefulness of these schemes due to the potential difficulties in calculating certain exponents in the asymptotic error expansion. In this paper, we consider a method based on the

ϵ
\epsilon

-algorithm which does not require the user to supply these parameters. Tests show this luxury produces a method substantially better than the unmodified Romberg’s method, but not as good as the modified procedure which assumes all the exponents are known. It is possible to design a scheme which incorporates the best of both methods and allows the user to decide how much information he wishes to provide. The algorithm is stable numerically in quadrature applications and converges for functions with endpoint singularities of an algebraic or logarithmic nature.
An error bound for quadratures
Authors: Hetherington J. H.
Published: June 2010

A definition of a length scale is given for analytic functions. Quadrature error bounds based on this length scale can be used to compare different orders and types of quadrature rules.
Semidiscrete least-squares methods for a parabolic boundary value problem
Authors: Bramble James H. , Thomée Vidar
Published: June 2010

In this paper some approximate methods for solving the initial-boundary value problem for the heat equation in a cylinder under homogeneous boundary conditions are analyzed. The methods consist in discretizing with respect to time and solving approximately the resulting elliptic problem for fixed time by least squares methods. The approximate solutions will belong to a finite-dimensional subspace of functions in space which will not be required to satisfy the homogeneous boundary conditions.
On the Gaussian primes on the line 𝐼𝑚(𝑋)=1
Authors: Wunderlich M. C.
Published: June 2010

This paper contains a summary table of the author’s computation of the Gaussian primes of the form

a
+
i

a + i

. For the values

x
=
1000
,
10000
,
100000
,
180000

x = 1000, 10000, 100000, 180000

, and 500000 (500000) 14000000, the following values are tabulated:

G
(
x
)

G(x)

, the numbers of Gaussisn primes

a
+
i

a + i

with

a
≦
14000000
;

π
1

(
x
)

a \leqq 14000000;{\pi _1}(x)

, the number of primes

≦
x

\leqq x

congruent to

1
m
o
d
4

1 mod 4

;

π
3

(
x
)

{\pi _3}(x)

, the number of primes

≦
x

\leqq x

congruent to

3
m
o
d
4

3 mod 4

; and

G
(
x
)

/

π
3

(
x
)

G(x)/{\pi _3}(x)

.
Chebyshev approximations for the psi function
Authors: Cody W. J. , Strecok Anthony J. , Thacher Henry C.
Published: June 2010

Rational Chebyshev approximations to the psi (digamma) function are presented for

.5
≦
x
≦
3.0

.5 \leqq x \leqq 3.0

, and

3.0
≦
x

3.0 \leqq x

. Maximum relative errors range down to the order of

10

−
20

{10^{ - 20}}

.
Simplified proofs of error estimates for the least squares method for Dirichlet’s problem
Authors: Baker Garth A.
Published: June 2010

Recently, Bramble and Schatz have proposed a projection method for approximating the solution of Dirichlet’s problem. Error estimates are derived by the authors using arguments based on certain interpolation theorems for linear operators on Hilbert spaces. It is shown here that simpler and shorter methods can be used to obtain these error estimates.
A posteriori error bounds for numerical solutions of the neutron transport equation
Authors: Madsen Niel K.
Published: June 2010

The theory and application of a method for computing rigorous a posteriori error bounds for numerical solutions to time-independent neutron transport problems are presented. The bounds are obtained for the

L
2

{L_2}

and

L
1

{L_1}

norms of the error function.
Chebyshev polynomials corresponding to a semi-infinite interval and an exponential weight factor
Authors: Kammler David W.
Published: June 2010

An algorithm is presented for the computation of the n zeros of the polynomial

q
n

{q_n}

having the property that

q
n

(
t
)
exp

(
−
t
)

{q_n}(t)\exp \;( - t)

alternates n times, at the maximum value 1, on

[
0
,
+
∞
)

[0, + \infty )

. Numerical values of the zeros and extremal points are given for

n
≦
10

n \leqq 10

.
The first occurrence of large gaps between successive primes
Authors: Brent Richard P.
Published: June 2010

A table of the first occurrence of a string of

2
r
−
1

2r - 1

composite numbers between two primes is given for

r
=
158
(
1
)
267
,
269
,
270
,
273
,
275
,
276
,
281
,
282
,
291
,
294

r = 158(1)267,269,270,273,275,276,281,282,291,294

and 301. All such strings between primes less than

2.6
×

10

12

2.6 \times 10^{12}

have been accounted for. The computation supports some conjectures on the distribution of these strings.
Approximate solution of the differential equation 𝑦^{”}=𝑓(𝑥,𝑦) with spline functions
Authors: Micula Gh.
Published: June 2010

An approximate spline is constructed for the solution of Cauchy’s problem regarding a second-order differential equation. The existence, uniqueness and convergence of the approximate spline solution are investigated.
A new error analysis for a cubic spline approximate solution of a class of Volterra integro-differential equations
Authors: Guzek Joseph A. , Kemper Gene A.
Published: June 2010

In this paper a third-order numerical method is considered which utilizes a twice continuously differentiable third degree spline to approximate the solution of
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row ModifyingAbove x With dot left-parenthesis t right-parenthesis equals upper F left-parenthesis t comma x left-parenthesis t right-parenthesis comma integral Subscript a Superscript t Baseline upper K left-parenthesis t comma u comma x left-parenthesis u right-parenthesis right-parenthesis d u right-parenthesis comma 2nd Row x left-parenthesis a right-parenthesis equals x 0 comma EndLayout">

x
˙

(
t
)
=
F

(

t
,
x
(
t
)
,

∫
a
t

K
(
t
,
u
,
x
(
u
)
)

d
u

)

,

x
(
a
)
=

x
0

,

\begin {array}{*{20}{c}} {\dot x(t) = F\left ( {t,x(t),\int _a^t {K(t,u,x(u))\;du} } \right ),} \hfill \\ {x(a) = {x_0},} \hfill \\ \end {array}

\]
at discrete points in the interval [a, b]. The error analysis uses a technique usually associated with linear multistep methods.
Some integrals involving associated Legendre functions
Authors: Samaddar S. N.
Published: June 2010

Calculations of some uncommon integrals involving Legendre functions and their derivatives, which may not be readily evaluated using known results, are presented. Some results show a special type of orthogonality relation in a certain sense. A few of these integrals find their applications in diffraction or scattering problems.
A new algorithm for the Chebyshev solution of overdetermined linear systems
Authors: Boggs Paul T.
Published: June 2010

Let

x
(
p
)

x(p)

be the point which minimizes the residual of a linear system in the

l
p

{l_p}

norm. It is known that under certain conditions

x
(
p
)
→

x
∗

x(p) \to {x^\ast }

, the Chebyshev or

l
∞

{l_\infty }

solution, as

p
→
∞

p \to \infty

. A differential equation describing

x
(
p
)

x(p)

is derived from which an iterative scheme is devised. A convergence analysis is given and numerical results are presented.
A note on congruent numbers
Authors: Alter Ronald , Curtz Thaddeus B.
Published: June 2010

An integer a is called a congruent number if and only if there are positive integer solutions to the system of equations
\[

x
2

+
a

y
2

=

z
2

and

x
2

−
a

y
2

=

t
2

.

{x^2} + a{y^2} = {z^2}\quad {\text {and}}\quad {x^2} - a{y^2} = {t^2}.

\]
In this note congruent numbers are discussed and a table of known square-free congruent numbers less than 1000 is exhibited.
A new factorization technique using quadratic forms
Authors: Lehmer D. H. , Lehmer Emma
Published: June 2010

The paper presents a practical method for factoring an arbitrary N by representing N or

λ
N

\lambda N

by one of at most three quadratic forms:

λ
N
=

x
2

−
D

y
2

,
λ
=
1
,
−
1
,
2
,
D
=
−
1
,
±
2
,
±
3
,
±
6

\lambda N = {x^2} - D{y^2},\lambda = 1, - 1,2,D = - 1, \pm 2, \pm 3, \pm 6

. These three forms appropriate to N, together with inequalities for y, are given for all N prime to 6. Presently available sieving facilities make the method quite effective and economical for numbers N having 20 to 25 digits. Four examples arising from aliquot series are discussed in detail.
Irregular prime divisors of the Bernoulli numbers
Authors: Johnson Wells
Published: June 2010

Ifpis an irregular prime,p>8000p > 8000, then the indices 2nfor which the Bernoulli quotientsB2n/2n{B_{2n}}/2nare divisible byp2{p^2}are completely characterized. In particular, it is always true that2n>p2n > pand thatB2n/2n≢(B2n+p−1/2n+p−1)(modp2){B_{2n}}/2n\;\nequiv ({B_{2n + p - 1}}/2n + p - 1)\pmod {p^2}if(p,2n)is an irregular pair. As a result, we obtain another verification that the cyclotomic invariantsμp{\mu _p}of Iwasawa all vanish for primesp>8000p > 8000.
Algebraic-numerical method for the slightly perturbed harmonic oscillator
Authors: Nadeau A. , Guyard J. , Feix M. R.
Published: June 2010

The solution of slightly perturbed harmonic oscillators can easily be obtained in the form of a series given by Poisson’s method. However, this perturbation method leads to secular terms unbounded for large time (the time unit being the fundamental period of the harmonic oscillator), which prevent the use of finite series. The analytical elimination of such terms was first solved by Poincaré and, more recently, generalized by Krylov and Bogoliubov. Unfortunately, these methods are very difficult to handle and are not easily carried out for high orders. A numerical reinitialization method is combined here with the Poisson perturbation treatment to avoid the growth of secular terms and therefore to get the solution at any time. The advantages of such a method is that the analytical work can be carried to high orders keeping the step of numerical integration to a relatively large value (compared to a purely numerical method). This algorithm has been tested on the Mathieu equation. A method for the computation of the eigenvalues of this equation is given. By properly selecting the order of the perturbation and the time step of reinitialization, we can recover, at any order, all the effects of the slight perturbation (including all the unstable zones). Consequently, such a method is a useful intermediate between purely analytical and purely numerical algorithms.
A nonexistence theorem for explicit 𝐴-stable methods
Authors: Nevanlinna Olavi , Sipilä Aarne H.
Published: June 2010

It is proved that there are no A-stable explicit methods in a general class of "linear" methods. The class contains, for example, Runge-Kutta methods, linear multistep methods, predictor-corrector formulas, cyclic multistep methods and linear multistep methods with higher derivatives.
The application of invariant imbedding to the solution of linear two-point boundary value problems on an infinite interval
Authors: Alspaugh Dale W.
Published: June 2010

Linear two-point boundary value problems defined on an infinite domain are converted to initial-value problems using invariant imbedding. The resulting Riccati equations are integrated numerically until the desired accuracy is obtained. Several criteria for determining the appropriate length of integration are presented. Several example problems are presented.
Existence questions for the problem of Chebyshev approximation by interpolating rationals
Authors: Taylor G. D. , Williams J.
Published: June 2010

This paper considers a problem of Chebyshev approximation by interpolating rationals. Examples are given which show that best approximations may not exist. Sufficient conditions for existence are established, some of which can easily be checked in practice. Illustrative examples are also presented.
Estimates near plane portions of the boundary for discrete elliptic boundary problems
Authors: Johnson C. G. L.
Published: June 2010

We consider an elliptic difference operator together with certain boundary difference operators near a plane portion of the boundary parallel to some coordinate direction. We prove discrete analogues of known estimates in

L
p

{L_p}

and Schauder norms for elliptic boundary problems. The discrete estimates are then used to prove results about convergence near plane portions of the boundary of difference quotients of solutions

u
h

{u_h}

of a discrete elliptic problem to the derivatives of the solution u of the corresponding continuous problem, when it is known that

u
h

{u_h}

converges to u in the maximum norm or in a discrete

L
p

{L_p}

norm as h tends to zero.
The character table of Fischer’s simple group, 𝑀(23)
Authors: Hunt David C.
Published: June 2010

This paper describes the calculation of the character table of

M
(
23
)

M(23)

, the sporadic simple group discovered by B. Fischer [1].
Finite sum representations for partial derivatives of special functions with respect to parameters
Authors: Buschman R. G.
Published: June 2010

The Mellin transformation is used as a method for discovery of cases where the partial derivatives with respect to parameters for certain Whittaker and Bessel functions can be expressed in terms of finite sums involving these functions. These results are easily generalized to the G-function, from which, by specialization, formulas involving hypergeometric and other functions can be obtained.
Reduction of the pseudoinverse of a Hermitian persymmetric matrix
Authors: Goldstein Marvin J.
Published: June 2010

When the pseudoinverse of a Hermitian persymmetric matrix is computed, both computer time and storage can be reduced by taking advantage of the special structure of the matrix.
A comparison of some numerical methods for two-point boundary value problems
Authors: Varah James M.
Published: June 2010

In this paper we discuss and compare two useful variable mesh schemes for linear second-order two-point boundary value problems: the midpoint rule and collocation with cubic Hermite functions. We analyze the stability of the block-tridiagonal factorization for solving the linear systems, compare the amount of computer time required, and test the methods on some particular numerical problems.
Computation of the ideal class group of certain complex quartic fields
Authors: Lakein Richard B.
Published: June 2010

The ideal class group of quartic fields

K
=
F
(

μ

)

K = F(\sqrt \mu )

, where

F
=

Q

(
i
)

F = {\mathbf {Q}}(i)

, is calculated by a method adapted from the method of cycles of reduced ideals for real quadratic fields. The class number is found in this way for 5000 fields

K
=
F
(

π

)
,
π
≡
±
1
mod
4
,
π

K = F(\sqrt \pi ),\pi \equiv \pm 1 \bmod 4,\pi

a prime of F. A tabulation of the distribution of class numbers shows a striking similarity to that for real quadratic fields with prime discriminant. Also, two fields were found with noncyclic ideal class group

C
(
3
)
×
C
(
3
)

C(3) \times C(3)

.
Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind
Authors: Nashed M. Z. , Wahba Grace
Published: June 2010

We consider approximations{xn}\{ {x_n}\}obtained by moment discretization to (i) the minimalL2{\mathcal {L}_2}-norm solution ofKx=y\mathcal {K}x = ywhereK\mathcal {K}is a Hilbert-Schmidt integral operator onL2{\mathcal {L}_2}, and to (ii) the least squares solution of minimalL2{\mathcal {L}_2}-norm of the same equation whenyis not in the rangeR(K)\mathcal {R}(\mathcal {K})ofK\mathcal {K}. In case (i), ify∈R(K)y \in \mathcal {R}(\mathcal {K}), thenxn→K†y{x_n} \to {\mathcal {K}^\dagger }y, whereK†{\mathcal {K}^\dagger }is the generalized inverse ofK\mathcal {K}, and‖xn‖→∞\left \| {{x_n}} \right \| \to \inftyotherwise. Rates of convergence are given in this case if furtherK†y∈K∗(L2){\mathcal {K}^\dagger }y \in {\mathcal {K}^\ast }({\mathcal {L}_2}), whereK∗{\mathcal {K}^\ast }is the adjoint ofK\mathcal {K}, and the Hilbert-Schmidt kernel ofKK∗\mathcal {K}{\mathcal {K}^\ast }satisfies certain smoothness conditions. In case (ii), ify∈R(K)⊕R(K)⊥y \in \mathcal {R}(\mathcal {K}) \oplus \mathcal {R}{(\mathcal {K})^ \bot }, thenxn→K†y{x_n} \to {\mathcal {K}^\dagger }y, and‖xn‖→∞\left \| {{x_n}} \right \| \to \inftyotherwise. If furtherK†y∈K∗K(L2){\mathcal {K}^\dagger }y \in {\mathcal {K}^\ast }\mathcal {K}({\mathcal {L}_2}), then rates of convergence are given in terms of the smoothness properties of the Hilbert-Schmidt kernel of(KK∗)2{(\mathcal {K}{\mathcal {K}^\ast })^2}. Some of these results are generalized to a class of linear operator equations on abstract Hilbert spaces.
Irregularities in the distribution of primes and twin primes
Authors: Brent Richard P.
Published: June 2010

The maxima and minima of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper L left-parenthesis x right-parenthesis mathematical right-angle minus pi left-parenthesis x right-parenthesis comma mathematical left-angle upper R left-parenthesis x right-parenthesis mathematical right-angle minus pi left-parenthesis x right-parenthesis">

⟨
L
(
x
)
⟩
−
π
(
x
)
,
⟨
R
(
x
)
⟩
−
π
(
x
)

\langle L(x)\rangle - \pi (x),\langle R(x)\rangle - \pi (x)

, and

⟨

L
2

(
x
)
⟩
−

π
2

(
x
)

\langle {L_2}(x)\rangle - {\pi _2}(x)

in various intervals up to

x
=
8
×

10

10

x = 8 \times {10^{10}}

are tabulated. Here

π
(
x
)

\pi (x)

and

π
2

(
x
)

{\pi _2}(x)

are respectively the number of primes and twin primes not exceeding

x
,
L
(
x
)

x,L(x)

is the logarithmic integral,

R
(
x
)

R(x)

is Riemann’s approximation to

π
(
x
)

\pi (x)

, and

L
2

(
x
)

{L_2}(x)

is the Hardy-Littlewood approximation to

π
2

(
x
)

{\pi _2}(x)

. The computation of the sum of inverses of twin primes less than

8
×

10

10

8 \times {10^{10}}

gives a probable value

1.9021604
±
5
×

10

−
7

1.9021604 \pm 5 \times {10^{ - 7}}

for Brun’s constant.
Explicit-implicit schemes for the numerical solution of nonlinear hyperbolic systems
Authors: McGuire G. R. , Morris J. Ll.
Published: June 2010

A class of methods, comprising combinations of explicit and implicit methods, for solving systems of conservation laws in one space dimension is developed. The explicit methods of McGuire and Morris [5] are combined with the implicit methods of McGuire and Morris [11] in a manner similar to that for creating Hopscotch methods (Gourlay [13]). The stability properties of these explicit-implicit methods is investigated and the results of some numerical experiments are presented. Extensions of these methods to systems of conservation laws in two space dimensions are also briefly discussed.
Adaptive integration and improper integrals
Authors: Haber Seymour
Published: June 2010

Let R be the class of all functions that are properly Riemann-integrable on [0, 1], and let IR be the class of all functions that are properly Riemann-integrable on [a, 1] for all

a
>
0

a > 0

and for which
\[

lim

a
→

0
+

∫
a
1

f
(
x
)

d
x

\lim \limits _{a \to {0^+}} \int _a^1 {f(x)\;dx}

\]
exists and is finite. There are computational schemes that produce a convergent sequence of approximations to the integral of any function in R; the trapezoid rule is one. In this paper, it is shown that there is no computational scheme that uses only evaluations of the integrand, that is similarly effective for IR.
Irregular primes and cyclotomic invariants
Authors: Johnson Wells
Published: June 2010

The table of irregular primes less than 30000 has been computed and deposited in the UMT file. The fraction of irregular primes in this range is 0.3924, close to the heuristic prediction of

1
−

e

−
1

/

2

1 - {e^{ - 1/2}}

. Fermat’s Last Theorem has been verified for all prime exponents

p
>
30000

p > 30000

, and the cyclotomic invariants

μ
p

,

λ
p

{\mu _p},{\lambda _p}

, and

ν
p

{\nu _p}

of Iwasawa have been completely determined for these primes. The computations show that for p in this range,

μ
p

=
0

{\mu _p} = 0

and the invariants

λ
p

{\lambda _p}

and

ν
p

{\nu _p}

both equal the index of irregularity of p.
Discrete rational 𝐿_{𝑝} approximation
Authors: Wolfe Jerry M.
Published: June 2010

In this paper, the problem of approximating a function defined on a finite subset of the real line by a family of generalized rational functions whose numerator and denominator spaces satisfy the Haar conditions on some closed interval [a, b] containing the finite set is considered. The pointwise closure of the family restricted to the finite set is explicitly determined. The representation obtained is used to analyze the convergence of best approximations on discrete subsets of [a, b] to best approximations over the whole interval (as the discrete subsets become dense) in the case that the function approximated is continuous on [a, b] and the rational family consists of quotients of algebraic polynomials. It is found that the convergence is uniform over [a, b] if the function approximated is a so-called normal point. Only

L
p

{L_p}

norms with

1
⩽
p
>
∞

1 \leqslant p > \infty

are employed.
On detecting a periodic event by means of periodic observations. I
Authors: Burr Stefan A.
Published: June 2010

The following situation is considered. A certain event is observable only intermittently according to some period. An observer attempts to detect an occurrence of the event by making observations according to some other period. We wish to determine the probability

P
(
s
)

P(s)

that the event is first detected on the sth observation. The period of the event may be known or unknown. This paper discusses the number-theoretic calculations necessary to solve the problem; computational aspects will be discussed in a future paper.
On the stability of the Ritz-Galerkin method for Hammerstein equations
Authors: Hertling Jörg , Şchiop Alexandru I.
Published: June 2010

For the numerical treatment of Hammerstein equations by variational methods which has been considered by Hertling, we establish the stability in the sense of Mikhlin, Stetter and Tucker.
The convergence rate for difference approximations to mixed initial boundary value problems
Authors: Gustafsson Bertil
Published: June 2010

The convergence rate for difference approximations to mixed initial boundary value problems for hyperbolic systems is treated. Assuming that the approximation at the boundary has one-order lower accuracy than at inner points, conditions are given such that the overall accuracy of the solution is kept at the higher order.
Zeros of 𝑝-adic 𝐿-functions
Authors: Wagstaff Samuel S.
Published: June 2010

The p-adic coefficients and zeros of certain formal power series defined by Iwasawa have been calculated modulo various powers of p. Using these results and Iwasawa’s formula for the p-adic L-function

L
p

(
s
;
χ
)

{L_p}(s;\chi )

of Kubota and Leopoldt, several p-adic places of the zero of

L
p

(
s
;
χ
)

{L_p}(s;\chi )

were computed for the irregular primes

p
⩽
157

p \leqslant 157

.
Tables of reductions of symmetrized inner products (“inner plethysms”) of ordinary irreducible representations of symmetric groups
Authors: Esper N.
Published: June 2010

Decompositions of symmetrized inner products

[
α
]
⊡
[
β
]

[\alpha ] \boxdot [\beta ]

of ordinary irreducible representations

[
α
]

[\alpha ]

of symmetric groups

S
n

{S_n}

and

[
β
]

[\beta ]

of

S
m

{S_m}

were evaluated on a CDC 6400. Tables were obtained for

2
⩽
n
⩽
10

2 \leqslant n \leqslant 10

and

2
⩽
m
⩽
5

2 \leqslant m \leqslant 5

as well as for

m
=
6

m = 6

and

2
⩽
n
⩽
7

2 \leqslant n \leqslant 7

.
On generalized Gaussian quadrature
Authors: Luke Yudell L. , Ting Bing Yuan , Kemp Marilyn J.
Published: June 2010

A general interpolation formula is derived such that when it is multiplied by a weight function and integrated, the result becomes a generalized Gaussian quadrature scheme which allows for an arbitrary number of preassigned nodes. The errors in the interpolation and quadrature formulas are studied. All of this generalizes previous results of the first-named author where no preassigned nodes were permitted.
Error estimates for a finite element approximation of a minimal surface
Authors: Johnson Claes , Thomée Vidar
Published: June 2010

A finite element approximation of the minimal surface problem for a strictly convex bounded plane domain

Ω
\Omega

is considered. The approximating functions are continuous and piecewise linear on a triangulation of

Ω
\Omega

. Error estimates of the form

O
(
h
)

O(h)

in the

H
1

{H^1}

norm and

O
(

h
2

)

O({h^2})

in the

L
p

{L_p}

-norm

(
p
>
2
)

(p > 2)

are proved, where h denotes the maximal side in the triangulation.
A note on 1-class groups of number fields
Authors: Gerth Frank
Published: June 2010

Let F be a number field and K a cyclic extension of degree l over F, where l is a rational prime. The l-class group of K is analyzed as a

Gal

(
K

/

F
)

{\operatorname {Gal}}(K/F)

-module in the case where the l-class group of F is trivial. The resulting structure theorem is used to compute the structure of the 3-class groups of certain cyclic cubic fields that are discussed in a paper of D. Shanks.
Differential-difference properties of hypergeometric polynomials
Authors: Wimp Jet
Published: June 2010

We develop differential-difference properties of a class of hypergeometric polynomials which are a generalization of the Jacobi polynomials. The formulas are analogous to known formulas for the classical orthogonal polynomials.
Computation of the ideal class group of certain complex quartic fields. II
Authors: Lakein Richard B.
Published: June 2010

For quartic fields

K
=

F
3

(

π

)

K = {F_3}(\sqrt \pi )

, where

F
3

=
Q
(
ρ
)

{F_3} = Q(\rho )

and

π
≡
1
mod
4

\pi \equiv 1 \bmod 4

is a prime of

F
3

{F_3}

, the ideal class group is calculated by the same method used previously for quadratic extensions of

F
1

=
Q
(
i
)

{F_1} = Q(i)

, but using Hurwitz’ complex continued fraction over

Q
(
ρ
)

Q(\rho )

. The class number was found for 10000 such fields, and the previous computation over

F
1

{F_1}

was extended to 10000 cases. The distribution of class numbers is the same for 10000 fields of each type: real quadratic, quadratic over

F
1

{F_1}

, quadratic over

F
3

{F_3}

. Many fields were found with non-cyclic class group, including the first known real quadratics with groups

5
×
5

5 \times 5

and

7
×
7

7 \times 7

. Further properties of the continued fractions are also discussed.
Linear Chebyshev approximation in the complex plane using Lawson’s algorithm
Authors: Ellacott S. , Williams Jack
Published: June 2010

In this paper we discuss the application of Lawson’s algorithm for computing best linear Chebyshev approximations to complex-valued functions. Some numerical examples are also presented.
A note on diophantine equation 𝑌²+𝑘=𝑋⁵
Authors: Blass Josef
Published: June 2010

We show that if the class number of the quadratic field

A
(

−
k

)

A(\sqrt { - k} )

is not divisible by 5, and if k is not congruent to 7 modulo 8, then the equation

Y
2

+
k
=

X
5

{Y^2} + k = {X^5}

has no solutions in rational integers X, Y with the exception of

k
=
1
,
19
,
341

k = 1,19,341

.
Zeros of Hurwitz zeta functions
Authors: Spira Robert
Published: June 2010

All complex zeros of each Hurwitz zeta function are shown to lie in a vertical strip. Trivial real zeros analogous to those for the Riemann zeta function are found. Zeros of two particular Hurwitz zeta functions are calculated.
Complex roots of 𝑠𝑖𝑛 𝑧=𝑎𝑧, 𝑐𝑜𝑠 𝑧=𝑎𝑧, and 𝑐𝑜𝑠ℎ 𝑧=𝑎𝑧
Authors: Fettis Henry E.
Published: June 2010

Values of the first five complex roots of the equations

sin
⁡
z
=
a
z
,
cos
⁡
z
=
a
z

\sin z = az,\cos z = az

and

cosh
⁡
z
=
a
z

\cosh z = az

are given to 10S, for

a
=
10
,
5
,
2
,
1.6
,
1.2
,
1
(
.1
)
.3

a = 10,5,2,1.6,1.2,1(.1).3

, and selected values below.
On Cartesian products of good lattices
Authors: Zaremba S. K.
Published: June 2010

Good lattices yield a powerful method of computing multiple integrals. Asymptotically, a lattice generated by one good lattice point is much more efficient than a Cartesian product of such lattices. However, if the number of dimensions is large, this does not always apply to the case when the number of points remains within reasonable limits. Examples of such products of two or three lattices being more efficient than good lattices generated by single lattice points are systematically presented. Additional symmetries of Cartesian products of lattices offer a further advantage when the integrand has to be symmetrized beforehand.
A bound on the 𝐿_{∞}-norm of 𝐿₂-approximation by splines in terms of a global mesh ratio
Authors: de Boor Carl
Published: June 2010

Let

L
k

f

{L_k}f

denote the least-squares approximation to

f
∈

L

1

f \in {{\mathbf {L}}_1}

by splines of order k with knot sequence

t

=
(

t
i

)
1

n
+
k

{\mathbf {t}} = ({t_i})_1^{n + k}

. In connection with their work on Galerkin’s method for solving differential equations, Douglas, Dupont and Wahlbin have shown that the norm

‖

L
k

‖

∞

{\left \| {{L_k}} \right \|_\infty }

, of

L
k

{L_k}

as a map on

L

∞

{{\mathbf {L}}_\infty }

can be bounded as follows,
\[

‖

L
k

‖

∞

⩽

const
k

M

t

,

{\left \| {{L_k}} \right \|_\infty } \leqslant {\operatorname {const}_k}{M_{\mathbf {t}}},

\]
with

M

t

{M_{\mathbf {t}}}

a global mesh ratio, given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript bold t Baseline colon equals max Underscript i Endscripts normal upper Delta t Subscript i Baseline slash min left-brace right-brace vertical-bar times times upper Delta ti vertical-bar greater-than greater-than times times upper Delta ti 0 period">

M

t

:=

max
i

Δ

t
i

/

min

{
Δ

t
i

|

Δ

t
i

>
0
}
.

{M_{\mathbf {t}}}: = \max \limits _i \;\Delta {t_i}/\min \,\{ \Delta {t_i}|\Delta {t_i} > 0\}.

\]
Using their very nice idea together with some facts about B-splines, it is shown here that even
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper L Subscript k Baseline double-vertical-bar Subscript normal infinity Baseline less-than-or-slanted-equals c o n s t Subscript k Baseline left-parenthesis upper M Subscript bold t Superscript left-parenthesis k right-parenthesis Baseline right-parenthesis Superscript 1 slash 2">

‖

L
k

‖
∞

⩽

const
k

⁡
(

M

t

(
k
)

)

1

/

2

\| L_k \|_\infty \leqslant \operatorname {const}_k(M_{\mathbf {t}}^{(k)})^{1/2}

\]
with the smaller global mesh ratio

M

t

(
k
)

M_{\mathbf {t}}^{(k)}

given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript bold t Superscript left-parenthesis k right-parenthesis Baseline colon equals max Underscript i comma j Endscripts left-parenthesis t Subscript i plus k Baseline minus t Subscript i Baseline right-parenthesis slash t Subscript j plus k Baseline minus t Subscript j Baseline right-parenthesis period">

M

t

(
k
)

:=

max

i
,
j

(

t

i
+
k

−

t
i

)

/

t

j
+
k

−

t
j

)
.

M_{\mathbf {t}}^{(k)}: = \max \limits _{i,j} ({t_{i + k}} - {t_i})/{t_{j + k}} - {t_j}).

\]
A mesh independent bound for

L

2

{{\mathbf {L}}_2}

-approximation by continuous piecewise polynomials is also given.
Groups of square-free order, an algorithm
Authors: Alonso J.
Published: June 2010

An abstract definition of the groups of square-free order is given that leads naturally to a programmable computation of their number. O. Hölder’s alternative description of the groups of square-free order is incidentally derived.
Asymptotic behavior of vector recurrences with applications
Authors: Feldstein Alan , Traub J. F.
Published: June 2010

The behavior of the vector recurrence

y

n
+
1

=
M

y

n

+

w

n
+
1

{{\mathbf {y}}_{n + 1}} = M{{\mathbf {y}}_n} + {{\mathbf {w}}_{n + 1}}

is studied under very weak assumptions. Let

λ
(
M
)

\lambda (M)

denote the spectral radius of M and let

λ
(
M
)
⩾
1

\lambda (M) \geqslant 1

. Then if the

w

n

{{\mathbf {w}}_n}

are bounded in norm and a certain subspace hypothesis holds, the root order of the

y

n

{{\mathbf {y}}_n}

is shown to be

λ
(
M
)

\lambda (M)

. If one additional hypothesis on the dimension of the principal Jordan blocks of M holds, then the quotient order of the

y

n

{{\mathbf {y}}_n}

is also

λ
(
M
)

\lambda (M)

. The behavior of the homogeneous recurrence is studied for all values of

λ
(
M
)

\lambda (M)

. These results are applied to the analysis of (1) Nonlinear iteration with application to iteration with memory and to parallel iteration algorithms. (2) Order and efficiency of composite iteration.
Some stable methods for calculating inertia and solving symmetric linear systems
Authors: Bunch James R. , Kaufman Linda
Published: June 2010

Several decompositions of symmetric matrices for calculating inertia and solving systems of linear equations are discussed. New partial pivoting strategies for decomposing symmetric matrices are introduced and analyzed.
Odd perfect numbers not divisible by 3 are divisible by at least ten distinct primes
Authors: Kishore Masao
Published: June 2010

Hagis and McDaniel have shown that the largest prime factor of an odd perfect number N is at least 100111, and Pomerance has shown that the second largest prime factor is at least 139. Using these facts together with the method we develop, we show that if

3
∤
N

3\nmid N

, N is divisible by at least ten distinct primes.
On the stability of Galerkin methods for initial-boundary value problems for hyperbolic systems
Authors: Gunzburger Max D.
Published: June 2010

The stability of approximating the solution of mixed initial-boundary value problems for hyperbolic systems by semidiscrete Galerkin methods is studied. It is shown that a particular straightforward Galerkin method yields an unstable approximation, and that this numerical instability is caused by an improper treatment of the boundary. Stable schemes are then presented, one of which differs from the unstable scheme only insofar as the treatment of the boundary is concerned. These stable schemes make use of a particular matrix which symmetrizes the differential system. It is therefore shown that the use of this matrix is crucial to the stability of the computations as well as for obtaining a priori bounds on the energy of the continuous system. This symmetrizing matrix is also related to the diagonalizing matrix for the system of hyperbolic equations and to the Lyapunov matrix for the system of ordinary differential equations resulting from the application of Galerkin’s method.