bluuebunny/crossref_metadata_2025 huggingface dataset:
An analysis of a uniformly accurate difference method for a singular perturbation problem
Authors: Berger Alan E. , Solomon Jay M. , Ciment Melvyn
Published: June 2010

It will be proven that an exponential tridiagonal difference scheme, when applied with a uniform mesh of size h to:

ε

u

x
x

+
b
(
x
)

u
x

=
f
(
x
)

\varepsilon {u_{xx}} + b(x){u_x} = f(x)

for

0
>
x
>
1
,
b
>
0

0 > x > 1,b > 0

, b and f smooth,

ε
\varepsilon

in (0, 1], and

u
(
0
)

u(0)

and

u
(
1
)

u(1)

given, is uniformly second-order accurate (i.e., the maximum of the errors at the grid points is bounded by

C

h
2

C{h^2}

with the constant C independent of h and

ε
\varepsilon

). This scheme was derived by El-Mistikawy and Werle by a

C
1

{C^1}

patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point. The behavior of the approximate solution in between the grid points will be analyzed, and some numerical results will also be given.
Numerical stability for some equations of gas dynamics
Authors: le Roux A. Y.
Published: June 2010

The isentropic gas dynamics equations in Eulerian coordinates are expressed by means of the density

ρ
\rho

and the momentum

q
=
ρ
u

q = \rho u

, instead of the velocity u, in order to get domains bounded and invariant in the

(
ρ
,
q
)

(\rho ,q)

-plane, for a wide class of pressure laws

p
(
ρ
)

p(\rho )

and in the monodimensional case. A numerical scheme of the transport-projection type is proposed, which builds an approximate solution valued in such a domain. Since the characteristic speeds are bounded in this set, the stability condition is easily fulfilled and then estimates in the

L
∞

{L^\infty }

-norm are derived at any time step. Similar results are extended to the model involving friction and topographical terms, and for a simplified model of supersonic flows. The nonapplication of this study to the gas dynamics in Lagrangian coordinates is shown.
Implicit schemes and 𝐿𝑈 decompositions
Authors: Jameson A. , Turkel E.
Published: June 2010

Implicit methods for hyperbolic equations are analyzed by constructing LU factorizations. It is shown that the solution of the resulting tridiagonal systems in one dimension is well conditioned if and only if the LU factors are diagonally dominant. Stable implicit methods that have diagonally dominant factors are constructed for hyperbolic equations in n space dimensions. Only two factors are required even in three space dimensions. Acceleration to a steady state is analyzed. When the multidimensional backward Euler method is used with large time steps, it is shown that the scheme approximates a Newton-Raphson iteration procedure.
On the 𝐵𝑁 stability of the Runge-Kutta methods
Authors: Piazza Giuseppe
Published: June 2010

In this note sufficient conditions that let Runge-Kutta s stages methods of at least order s be BN stable are given.
Semigroups, antiautomorphisms, and involutions: a computer solution to an open problem. I
Authors: Winker S. K. , Wos L. , Lusk E. L.
Published: June 2010

An antiautomorphism H of a semigroup S is a 1-1 mapping of S onto itself such that

H
(
x
y
)
=
H
(
y
)
H
(
x
)

H(xy) = H(y)H(x)

for all x, y in S. An antiautomorphism H is an involution if

H
2

(
x
)
=
x

{H^2}(x) = x

for all x in S. In this paper the following question is answered: Does there exist a finite semigroup with antiautomorphism but no involution? This question, suggested by I. Kaplansky, was answered in the affirmative with the aid of an automated theorem-proving program. More precisely, there are exactly four such semigroups of order seven and none of smaller order. The program was a completely general one, and did not calculate the solution directly, but rather rendered invaluable assistance to the mathematicians investigating the question by helping to generate and examine various models. A detailed discussion of the approach is presented, with the intention of demonstrating the usefulness of a theorem prover in carrying out certain types of mathematical research.
On the distribution of pseudoprimes
Authors: Pomerance Carl
Published: June 2010

LetP(x)\mathcal {P}(x)denote the pseudoprime counting function. With\[L(x)=exp⁡{log⁡xlog⁡log⁡log⁡x/log⁡log⁡x},L(x) = \exp \{ \log x\log \log \log x/\log \log x\} ,\]we proveP(x)⩽x∙L(x)−1/2\mathcal {P}(x) \leqslant x \bullet L{(x)^{ - 1/2}}for largex, an improvement on the 1956 work of Erdös. We conjecture thatP(x)=x∙L(x)−1+o(1)\mathcal {P}(x) = x \bullet L{(x)^{ - 1 + o(1)}}.
Numerical stability of the Halley-iteration for the solution of a system of nonlinear equations
Authors: Cuyt Annie A. M.
Published: June 2010

Let

F
:

R

q

→

R

q

F:{{\mathbf {R}}^q} \to {{\mathbf {R}}^q}

and

x
∗

{x^ \ast }

a simple root in

R

q

{{\mathbf {R}}^q}

of the system of nonlinear equations

F
(
x
)
=
0

F(x) = 0

. Abstract Padé approximants (APA) and abstract Rational approximants (ARA) for the operator F have been introduced in [2] and [3]. The adjective “abstract” refers to the use of abstract polynomials [5] for the construction of the rational operators. The APA and ARA have been used for the solution of a system of nonlinear equations in [4]. Of particular interest was the following third order iterative procedure:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript i plus 1 Baseline equals x Subscript i Baseline plus StartFraction a Subscript i Superscript 2 Baseline Over a Subscript i Baseline plus one half upper F Subscript i Superscript prime minus 1 Baseline upper F Subscript i Superscript Baseline a Subscript i Superscript 2 Baseline EndFraction comma">

x

i
+
1

=

x
i

+

a
i
2

a
i

+

1
2

F
i

′

−
1

F
i

a
i
2

,

{x_{i + 1}} = {x_i} + \frac {{a_i^2}}{{{a_i} + \frac {1}{2}F_i^{’- 1}F_i^{}a_i^2}},

\]
with

F
i
′

F_i’

the 1st Fréchet-derivative of F in

x
1

,

a
i

=
−

F
i

′

−
1

F
i

{x_1},{a_i} = - F_i^{’- 1}{F_i}

the Newton-correction where

F
i

=
F
(

x
i

)
,

F
i

{F_i} = F({x_i}),F_i^{}

the 2nd Fréchet-derivative of F in

x
i

{x_i}

where

F
i

a
i
2

F_i^{}a_i^2

is the bilinear operator

F
i

F_i^{}

evaluated in

(

a
i

,

a
i

)

({a_i},{a_i})

, and componentwise multiplication and division in

R

q

{{\mathbf {R}}^q}

. For

q
=
1

q = 1

this technique is known as the Halley-iteration [6, p. 91]. In this paper the numerical stability [7] of the Halley-iteration for the case

q
⩾
1

q \geqslant 1

is investigated and illustrated by a numerical example.
An explicit modular equation in two variables and Hilbert’s twelfth problem
Authors: Cohn Harvey
Published: June 2010

The Hilbert modular function field over

Q

(
√
2
)

{\mathbf {Q}}(\surd 2)

has generators satisfying modular equations when the arguments are multiplied by factors of norm two. These equations are found by machine use of Fourier series and are further used to show computationally that Weber’s ring class field theory for rationals has an illustration of Hecke’s type for

Q

(
√
2
)

{\mathbf {Q}}(\surd 2)

. This has bearing on Hubert’s twelfth problem, the generation of algebraic fields by transcendental functions.
An algorithm for the solution of certain differential-difference equations of advanced type
Authors: Greaves G.
Published: June 2010

The differential-difference problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d Over d s EndFraction left-brace s upper J left-parenthesis s right-parenthesis right-brace plus upper J left-parenthesis s plus 1 right-parenthesis equals 0 left-parenthesis upper R e left-parenthesis s right-parenthesis greater-than 0 right-parenthesis semicolon s upper J left-parenthesis s right-parenthesis right-arrow 1 as s right-arrow 0 comma">

d

d
s

{

s
J
(
s
)

}
+
J
(
s
+
1
)
=
0

(
Re
⁡
(
s
)
>
0
)
;

s
J
(
s
)
→
1

as

s
→
0
,

\frac {d}{{ds}}\;\{ {sJ(s)} \} + J(s + 1) = 0\;(\operatorname {Re} (s) > 0);\quad sJ(s) \to 1\;{\text {as}}\;s \to 0,

\]
which arises in the theory of numbers, is studied and an algorithm developed for the simultaneous numerical approximation of the derivatives

J

(
n
)

(
s
)

{J^{(n)}}(s)

. An error bound is established for the process described and its numerical performance and stability are discussed.
An algorithm for finding a minimal Weierstrass equation for an elliptic curve
Authors: Laska Michael
Published: June 2010

Let E be an elliptic curve defined over an algebraic number field K and assume that some Weierstrass equation for E over K is given. Then an algorithm is described which yields a global minimal Weierstrass equation for E over K provided such a global minimal Weierstrass equation does exist.
Optimal error properties of finite element methods for second order elliptic Dirichlet problems
Authors: Werschulz Arthur G.
Published: June 2010

We use the informational approach of Traub and Woźniakowski [9] to study the variational form of the second order elliptic Dirichlet problem

L
u
=
f

Lu = f

on

Ω
⊂

R

N

\Omega \subset {{\mathbf {R}}^N}

. For

f
∈

H
r

(
Ω
)

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

, where

r
⩾
−
1

r \geqslant - 1

, a quasi-uniform finite element method using n linear functionals

∫
Ω

f

ψ
i

{\smallint _\Omega }f{\psi _i}

has

H
1

(
Ω
)

{H^1}(\Omega )

-norm error

Θ
(

n

−
(
r
+
1
)

/

N

)

\Theta ({n^{ - (r + 1)/N}})

. We prove that it is asymptotically optimal among all methods using any information consisting of any n linear functionals. An analogous result holds if L is of order 2m: if

f
∈

H
r

(
Ω
)

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

, where

r
⩾
−
m

r \geqslant - m

, then there is a finite element method whose

H
α

(
Ω
)

{H^\alpha }(\Omega )

-norm error is

Θ
(

n

−
(
2
m
+
r
−
α
)

/

N

)

\Theta ({n^{ - (2m + r - \alpha )/N}})

for

0
⩽
α
⩽
m

0 \leqslant \alpha \leqslant m

, and this is asymptotically optimal; thus, the optimal error improves as m increases. If the integrals

∫
Ω

f

ψ
i

{\smallint _\Omega }f{\psi _i}

are approximated by using n evaluations of f, then there is a finite element method with quadrature with

H
1

(
Ω
)

{H^1}(\Omega )

-norm error

O
(

n

−
r

/

N

)

O({n^{ - r/N}})

where

r
>
N

/

2

r > N/2

. We show that when

N
=
1

N = 1

, there is no method using n function evaluations whose error is better than

Ω
(

n

−
r

)

\Omega ({n^{ - r}})

; thus for

N
=
1

N = 1

, the finite element method with quadrature is asymptotically optimal among all methods using n evaluations of f.
A note on the stable decompostion of skew-symmetric matrices
Authors: Bunch James R.
Published: June 2010

Computationally stable decompositions for skew-symmetric matrices, which take advantage of the skew-symmetry in order to halve the work and storage, are presented for solving linear systems of equations.
A Chebyshev polynomial rate-of-convergence theorem for Stieltjes functions
Authors: Boyd John P.
Published: June 2010

The theorem proved here extends the author’s previous work on Chebyshev series [4] by showing that if

f
(
x
)

f(x)

is a member of the class of so-called "Stieltjes functions" whose asymptotic power series

Σ

a
n

x
n

\Sigma {a_n}{x^n}

about

x
=
0

x = 0

is such that
\[

lim

n
→
∞

¯

log
⁡

|

a
n

|

n
log
⁡
n

=
r
,

\overline {\lim \limits _{n \to \infty } } \frac {{\log |{a_n}|}}{{n\log n}} = r,

\]
then the coefficients of the series of shifted Chebyshev polynomials on

x
∈
[
0
,
a
]
,
Σ

b
n

T
n
∗

(
x

/

a
)

x \in [0,a],\Sigma {b_n}T_n^\ast (x/a)

, satisfy the inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 2 Over r plus 2 EndFraction greater-than-or-slanted-equals limit Underscript n right-arrow normal infinity Endscripts Overscript bar Endscripts StartFraction log StartAbsoluteValue left-parenthesis log StartAbsoluteValue b Subscript n Baseline EndAbsoluteValue right-parenthesis EndAbsoluteValue Over log n EndFraction greater-than-or-slanted-equals 1 minus StartFraction r Over 2 EndFraction period">

2

r
+
2

⩾

lim

n
→
∞

¯

log
⁡

|

(
log
⁡

|

b
n

|

)

|

log
⁡
n

⩾
1
−

r
2

.

\frac {2}{{r + 2}} \geqslant \overline {\lim \limits _{n \to \infty } } \frac {{\log |(\log |{b_n}|)|}}{{\log n}} \geqslant 1 - \frac {r}{2}.

\]
There is an intriguing relationship between this theorem and a similar rate-of-convergence theorem for Padé approximants of Stieltjes functions which is discussed below.
On the zeros of a special sequence of polynomials
Authors: Mahler K.
Published: June 2010

The zeros of the polynomials

P
n

(
z
)
=

Σ

k
=
0

n
−
1

z

2
k

{P_n}(z) = \Sigma _{k = 0}^{n - 1}{z^{{2^k}}}

are studied and a number of open questions are discussed.
The integer points on three related elliptic curves
Authors: Bremner Andrew , Morton Patrick
Published: June 2010

The integer points on the three elliptic curves

y
2

=
4
c

x
3

+
13

{y^2} = 4c{x^3} + 13

,

c
=
1
,
3
,
9

c = 1,3,9

are found, with an application to coding theory. It is also shown that there are precisely three nonisomorphic cubic extensions of the rationals with discriminant

−

3
5

⋅
13

- {3^5} \cdot 13

.
Approximation of the spectrum of an operator given by the magnetohydrodynamic stability of a plasma
Authors: Jaccard Yves , Evéquoz Hugo
Published: June 2010

The study of the magnetohydrodynamic (MHD) stability of a plasma in a toroidal configuration leads to a problem of computing the spectrum of a noncompact selfadjoint operator T. The spectrum of T will be approximated by the eigenvalues of

T
h

{T_h}

, a Galerkin approximation of T. We present a two-dimensional model problem with two components containing most difficulties arising in the physical problem. We give subspaces and prove sufficient conditions for obtaining convergence using partial regularity of T.
Evaluation of Fourier integrals using 𝐵-splines
Authors: Lax M. , Agrawal G. P.
Published: June 2010

Finite Fourier integrals of functions possessing jumps in value, in the first or in the second derivative, are shown to be evaluated more efficiently, and more accurately, using a continuous Fourier transform (CFT) method than the discrete transform method used by the fast Fourier transform (FFT) algorithm. A B-spline fit is made to the input function, and the Fourier transform of the set of B-splines is performed analytically for a possibly nonuniform mesh. Several applications of the CFT method are made to compare its performance with the FFT method. The use of a 256-point FFT yields errors of order

10

−
2

{10^{ - 2}}

, whereas the same information used by the CFT algorithm yields errors of order

10

−
7

{10^{ - 7}}

—the machine accuracy available in single precision. Comparable accuracy is obtainable from the FFT over the limited original domain if more than 20,000 points are used.
An acceleration method for the power series of entire functions of order 1
Authors: Gabutti B. , Lyness J. N.
Published: June 2010

When

f
(
z
)

f(z)

is given by a known power series expansion, it is possible to construct the power series expansion for

f
(
z
;
p
)
=

e

−
p
z

f
(
z
)

f(z;p) = {e^{ - pz}}f(z)

. We define

p

opt

{p_{{\text {opt}}}}

to be the value of p for which the expansion for

f
(
z
;
p
)

f(z;p)

converges most rapidly. When

f
(
z
)

f(z)

is an entire function of order 1, we show that

p

opt

{p_{{\text {opt}}}}

is uniquely defined and may be characterized in terms of the set of singularities

z
i

=
1

/

σ
i

{z_i} = 1/{\sigma _i}

of an associated function

h
(
z
)

h(z)

. Specifically, it is the center of the smallest circle in the complex plane which contains all points

σ
i

{\sigma _i}

.
Rational Chebyshev approximations for the Bessel functions 𝐽₀(𝑥), 𝐽₁(𝑥), 𝑌₀(𝑥), 𝑌₁(𝑥)
Authors: Wills C. A. , Blair J. M. , Ragde P. L.
Published: June 2010

This report presents near-minimax rational approximations for the Bessel functions

J
0

(
x
)

{J_0}(x)

,

J
1

(
x
)

{J_1}(x)

,

Y
0

(
x
)

{Y_0}(x)

, and

Y
1

(
x
)

{Y_1}(x)

for the complete range of x, with relative errors ranging down to

10

−
23

{10^{ - 23}}

. The first thirty zeros of each function are listed to 35D. The tabulated zeros and the McMahon asymptotic formulae may be used to construct an algorithm which retains relative accuracy in the neighborhood of zeros.
Closed expressions for ∫¹₀𝑡⁻¹𝑙𝑜𝑔ⁿ⁻¹𝑡log^{𝑝}(1-𝑡)𝑑𝑡
Authors: Kölbig K. S.
Published: June 2010

Closed expressions for the integral

∫
0
1

t

−
1

log

n
−
1

t

log
p

(
1
−
t
)

d
t

\smallint _0^1{t^{ - 1}}{\log ^{n - 1}}t{\log ^p}(1 - t)\;dt

, whose general form is given elsewhere, are listed for

n
=
1
(
1
)
9

n = 1(1)9

,

p
=
1
(
1
)
9

p = 1(1)9

. A formula is derived which allows an easy evaluation of these expressions by formula manipulation on a computer.
Computing in permutation and matrix groups. II. Backtrack algorithm
Authors: Butler Gregory
Published: June 2010

This is the second paper in a series which discusses computation in permutation and matrix groups of very large order. The essential aspects of a backtrack algorithm which searches these groups are presented. We then uniformly describe algorithms for computing centralizers, intersections, and set stabilizers, as well as an algorithm which determines whether two elements are conjugate.
Class number computations of real abelian number fields
Authors: van der Linden F. J.
Published: June 2010

In this paper we describe the calculation of the class numbers of most real abelian number fields of conductor⩽200\leqslant 200. The technique is due to J. M. Masley and makes use of discriminant bounds of A. M. Odlyzko. In several cases we have to assume the generalized Riemann hypothesis.
Collocation methods for boundary value problems on “long” intervals
Authors: Markowich Peter A. , Ringhofer Christian A.
Published: June 2010

This paper deals with the numerical solution of boundary value problems of ordinary differential equations posed on infinite intervals. We cut the infinite interval at a finite, large enough point and insert additional, so-called asymptotic boundary conditions at the far (right) end and then solve the resulting two-point boundary value problem by an A-stable symmetric collocation method. Problems arise, because standard theory predicts the use of many grid points as the length of the interval increases. Using the exponential decay of the ’infinite’ solution, an ’asymptotic’ a priori mesh-size sequence which increases exponentially, and which therefore only employs a reasonable number of meshpoints, is developed and stability, as the length of the interval tends to infinity, is shown. We also show that the condition number of the collocation equations is asymptotically proportional to the number of meshpoints employed when using this exponentially graded mesh. Using k-stage collocation at Gaussian points and requiring an accuracy

O
(
ε
)

O(\varepsilon )

at the knots implies that the number of meshpoints is

O
(

ε

−
1

/

2
k

)

O({\varepsilon ^{ - 1/2k}})

as

ε
→
0

\varepsilon \to 0

.
A posteriori improvements for interpolating periodic splines
Authors: Lucas Thomas R.
Published: June 2010

A method of a posteriori improvements of interpolating periodic splines of order 2r and their derivatives over a uniform mesh is developed using polynomial-type correction terms. These improvements enhance the order of convergence by several powers of the step size h and are convenient and inexpensive to implement. The polynomials are specified in closed form using the Bernoulli numbers. That the first of these is related to the Bernoulli polynomial of degree 2r is due to Swartz [10], but no general development beyond the first has previously been made. These polynomials are multiplied by high order derivatives of the function evaluated at the mesh points. Some recent results by Lucas [8] are used to accurately estimate these values. Some numerical results are given which correspond closely with the predictions of the theory.
An explicit quasi-Newton update for sparse optimization calculations
Authors: Lucia Angelo
Published: June 2010

A new quasi-Newton updating formula for sparse optimization calculations is presented. It makes combined use of a simple strategy for fixing symmetry and a Schubert correction to the upper triangle of a permuted Hessian approximation. Interesting properties of this new update are that it is closed form and that it does not satisfy the secant condition at every iteration of the calculations. Some numerical results are given that show that this update compares favorably with the sparse PSB update and appears to have a superlinear rate of convergence.
On the smallest 𝑘 such that all 𝑘⋅2ⁿ+1 are composite
Authors: Jaeschke G.
Published: June 2010

In this note we present some computational results which restrict the least odd value of k such that

k
⋅

2
n

+
1

k \cdot {2^n} + 1

is composite for all

n
⩾
1

n \geqslant 1

to one of 91 numbers between 3061 and 78557,inclusive. Further, we give the computational results of a relaxed problem and prove for any positive integer r the existence of infinitely many odd integers k such that

k
⋅

2
r

+
1

k\cdot {2^r} + 1

is prime but

k
⋅

2
v

+
1

k\cdot {2^v} + 1

is not prime for

v
>
r

v > r

.
Frequency fitting of rational approximations to the exponential functions
Authors: Iserles A. , Nørsett S. P.
Published: June 2010

Rational approximations to the exponential function are considered. Let

R
=
P

/

Q

R = P/Q

,

deg
⁡
P
=
deg
⁡
Q
=
n

\deg P = \deg Q = n

,

R
(
z
)
=
exp
⁡
(
z
)
+

O

(

z

2
n
−
1

)

R(z) = \exp (z) + \mathcal {O}({z^{2n - 1}})

and

R
(
±
i
T
)
=
exp
⁡
(
±
i
T
)

R( \pm iT) = \exp ( \pm iT)

for a given positive number T. We show that this approximation is A-acceptable if and only if T belongs to one of intervals, whose endpoints are related to zeros of certain Bessel functions. The existence of this type of approximation and its connection to diagonal Padé approximations is studied. Approximations which interpolate the exponential on the imaginary axis are important in the numerical analysis of highly-oscillatory ordinary differential systems.
Three summation criteria for Fermat’s last theorem
Authors: Schwindt H.
Published: June 2010

This paper extends the search for solutions of the congruences
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript 1 Overscript left-bracket p slash 6 right-bracket Endscripts StartFraction 1 Over i EndFraction identical-to 0 comma sigma-summation Underscript 1 Overscript left-bracket p slash 6 right-bracket Endscripts StartFraction 1 Over i squared EndFraction identical-to 0 and sigma-summation Underscript left-bracket p slash 6 right-bracket plus 1 Overscript left-bracket p slash 5 right-bracket Endscripts StartFraction 1 Over i EndFraction identical-to 0 left-parenthesis mod p right-parenthesis">

∑
1

[
p

/

6
]

1
i

≡
0
,

∑
1

[
p

/

6
]

1

i
2

≡
0

and

∑

[
p

/

6
]
+
1

[
p

/

5
]

1
i

≡
0

(
mod

p
)

\sum \limits _1^{[p/6]} {\frac {1}{i} \equiv 0,} \quad \sum \limits _1^{[p/6]} {\frac {1}{{{i^2}}} \equiv 0} \quad {\text {and}}\quad \sum \limits _{[p/6] + 1}^{[p/5]} {\frac {1}{i} \equiv 0\;\pmod p}

\]
to the limit

p
>
600000

p > 600000

. The only solutions found were

p
=
61

p = 61

in the first case, in the second

p
=
205129

p = 205129

, and in the third case

p
=
109

p = 109

and

p
=
491

p = 491

.
Gauss-Kronrod integration rules for Cauchy principal value integrals
Authors: Rabinowitz Philip
Published: June 2010

Kronrod extensions to two classes of Gauss and Lobatto integration rules for the evaluation of Cauchy principal value integrals are derived. Since in one frequently occurring case, the Kronrod extension involves evaluating the derivative of the integrand, a new extension is introduced using

n
+
2

n + 2

points which requires only values of the integrand. However, this new rule does not exist for all n, and when it does, several significant figures are lost in its use.
The unconditional instability of inflow-dependent boundary conditions in difference approximations to hyperbolic systems
Authors: Tadmor Eitan
Published: June 2010

It is well known that for a mixed initial-boundary hyberbolic system to be well-defined it is necessary to impose additional boundary conditions only on the inflow eigenspace of the problem. We prove the discrete analogue of the above concerning difference approximations to such a system; that is, imposing numerical boundary conditions which are at least zeroth-order accurate with an inflow part of the interior equations leads to unconditional instability.
A modified finite difference scheme for the Stefan problem
Authors: White R. E.
Published: June 2010

In this paper we describe an algorithm which can be used to approximate the solution to the enthalpy formulation of the Stefan problem. We allow the thermal properties to have a space and temperature dependence. The algorithm is not explicit in the time variable and, hence, the stability condition on

Δ
t

\Delta t

is not too severe. A proof of convergence is given and two numerical examples are presented.
On the asymptotic convergence of collocation methods
Authors: Arnold Douglas N. , Wendland Wolfgang L.
Published: June 2010

We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines. The analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications. Equations to which the analysis is applied include Fredholm integral equations of the second kind, certain first kind Fredholm equations, singular integral equations involving Cauchy kernels, a variety of integro-differential equations, and two-point boundary value problems for ordinary differential equations. The error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard Galerkin methods. We compare the collocation method with a standard Galerkin method using splines of the same degree, showing that the Galerkin method is quasioptimal in a Sobolev space of lower index and furnishes optimal order approximation for a range of Sobolev indices containing and extending below that for the collocation method, and so the standard Galerkin method achieves higher rates of convergence.
A linear scheme for the numerical solution of nonlinear quasistationary magnetic fields
Authors: Zlámal Miloš
Published: June 2010

The computation of nonlinear quasistationary two-dimensional magnetic fields leads to the following problem. There exists a bounded domain

Ω
\Omega

and an open nonempty set

R
⊂
Ω

R \subset \Omega

. We are looking for the magnetic vector potential

u
(

x
1

,

x
2

,
t
)

u({x_1},{x_2},t)

which satisfies: (1) a certain nonlinear parabolic equation and an initial condition in R, (2) a nonlinear elliptic equation in

S
=
Ω
−

R
¯

S = \Omega - \bar R

, (3) a boundary conditon on

∂
Ω

\partial \Omega

and the condition that u as well as its conormal derivative are continuous across

Γ
=
∂
R
∩
∂
S

\Gamma = \partial R \cap \partial S

. This problem is formulated in an abstract variational way. We construct an approximate solution discretized in space by a generalized Galerkin method and by a one-step method in time. The resulting scheme is unconditionally stable and linear. A strong convergence of the approximate solution is proved without any regularity assumptions for the exact solution. We also derive an error bound for the solution of the two-dimensional magnetic field equations under the assumption that the exact solution is sufficiently smooth.
Numerical methods for a model for compressible miscible displacement in porous media
Authors: Douglas Jim , Roberts Jean E.
Published: June 2010

A nonlinear parabolic system is derived to describe compressible miscible displacement in a porous medium. The system is consistent with the usual model for incompressible miscible displacement. Two finite element procedures are introduced to approximate the concentration of one of the fluids and the pressure of the mixture. The concentration is treated by a Galerkin method in both procedures, while the pressure is treated by either a Galerkin method or by a parabolic mixed finite element method. Optimal order estimates in

L
2

{L^2}

and essentially optimal order estimates in

L
∞

{L^\infty }

are derived for the errors in the approximate solutions for both methods.
An efficient algorithm for bifurcation problems of variational inequalities
Authors: Mittelmann H. D.
Published: June 2010

For a class of variational inequalities on a Hilbert space H bifurcating solutions exist and may be characterized as critical points of a functional with respect to the intersection of the level surfaces of another functional and a closed convex subset K of H. In a recent paper [13] we have used a gradient-projection type algorithm to obtain the solutions for discretizations of the variational inequalities. A related but Newton-based method is given here. Global and asymptotically quadratic convergence is proved. Numerical results show that it may be used very efficiently in following the bifurcating branches and that it compares favorably with several other algorithms. The method is also attractive for a class of nonlinear eigenvalue problems

(
K
=
H
)

(K = H)

for which it reduces to a generalized Rayleigh-quotient iteration. So some results are included for the path following in turning-point problems.
Block AOR iteration for nonsymmetric matrices
Authors: Papatheodorou Theodore S.
Published: June 2010

We consider a class of matrices that are of interest to numerical applications and are large, sparse, but not symmetric or diagonally dominant. We give a criterion for the existence of (and we actually construct) the inverse matrix in terms of powers of a "small" matrix. We use this criterion to find that the spectral radius of the Jacobi iteration matrix, corresponding to a block tridiagonal partition, is in general

⩾
1

\geqslant 1

. We also derive conditions (that are satisfied in cases of interest to applications) for the Jacobi matrix to have spectral radius = 1. We introduce convergent "block AOR" iterative schemes such as extrapolated Jacobi and extrapolated Gauss-Seidel schemes with optimum (under) relaxation parameter

ω
=
.5

\omega = .5

. A numerical example pertaining to the solution of Poisson’s equation is given, as a demonstration of some of our hypotheses and results. A comparison with SOR, applied to the 5-point finite difference method, is also included.
The maximum of a quasismooth function
Authors: Wilkins J. Ernest , Hatcher Theodore R.
Published: June 2010

Let Z be the class of real-valued functions, defined and continuous on the closed interval

I
=
[
−
1
,
1
]

I = [ - 1,1]

, such that

f
(
±
1
)
=
0

f( \pm 1) = 0

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f left-parenthesis xi right-parenthesis minus 2 f left-brace left-parenthesis xi plus eta right-parenthesis slash 2 right-brace plus f left-parenthesis eta right-parenthesis EndAbsoluteValue less-than-or-slanted-equals StartAbsoluteValue xi minus eta EndAbsoluteValue">

|

f
(
ξ
)
−
2
f
{
(
ξ
+
η
)

/

2
}
+
f
(
η
)

|

⩽

|

ξ
−
η

|

|f(\xi ) - 2f\{ (\xi + \eta )/2\} + f(\eta )| \leqslant |\xi - \eta |

for all

ξ
\xi

and

η
\eta

in I. Let

K
=

sup

f
∈
Z

max

x
∈
I

|

f
(
x
)

|

K = {\sup _{f \in Z}}{\max _{x \in I}}|f(x)|

. We will prove that

13

/

10
⩽
K
⩽
1014

/

779
>
1.301669

13/10 \leqslant K \leqslant 1014/779 > 1.301669

.
Solutions of the Diophantine equation 𝐴⁴+𝐵⁴=𝐶⁴+𝐷⁴
Authors: Zajta Aurel J.
Published: June 2010

A survey is presented of the more important solution methods of the equation of the title. When space permits, a brief description of the methods and numerical examples are also given. The paper concludes with an incomplete list of 218 primitive nontrivial solutions in rational integers not exceeding

10
6

{10^6}

.
Factors of Fermat numbers and large primes of the form 𝑘⋅2ⁿ+1
Authors: Keller Wilfrid
Published: June 2010

A new factor is given for each of the Fermat numbersF52,F931,F6835{F_{52}},{F_{931}},{F_{6835}}, andF9448{F_{9448}}. In addition, a factor ofF75{F_{75}}discovered by Gary Gostin is presented. The current status for allFm{F_m}is shown in a table. Primes of the formk⋅2n+1,kk \cdot {2^n} + 1,kodd, are listed for31⩽k⩽14931 \leqslant k \leqslant 149,1500>n⩽40001500 > n \leqslant 4000, and for151⩽k⩽199151 \leqslant k \leqslant 199,1000>n⩽40001000 > n \leqslant 4000. Some primes for even larger values ofnare included, the largest one being5⋅213165+15 \cdot {2^{13165}} + 1. Also, a survey of several related questions is given. In particular, values ofksuch thatk⋅2n+1k\cdot {2^n} + 1is composite for everynare considered, as well as odd values ofhsuch that3h⋅2n±13h\cdot {2^n} \pm 1never yields a twin prime pair.
Sets of 𝑛 squares of which any 𝑛-1 have their sum square
Authors: Lagrange Jean
Published: June 2010

A systematic method is given for calculating sets of n squares of which any

n
−
1

n - 1

have their sum square. A particular method is developed for

n
=
4

n = 4

. Tables give the smallest solution for each

n
⩽
8

n \leqslant 8

and other small solutions for

n
⩽
5

n \leqslant 5

.
Cyclotomy with short periods
Authors: Lehmer D. H. , Lehmer Emma
Published: June 2010

This paper develops cyclotomy for periods of lengths 2, 3 and 4 for moduli which are primes and products of two primes.
The Lanczos algorithm with partial reorthogonalization
Authors: Simon Horst D.
Published: June 2010

The Lanczos algorithm is becoming accepted as a powerful tool for finding the eigenvalues and for solving linear systems of equations. Any practical implementation of the algorithm suffers however from roundoff errors, which usually cause the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, full reorthogonalization (FRO) and selective orthogonalization (SO) have been used in the past as a remedy. Here partial reorthogonalization (PRO) is proposed as a new method for maintaining semiorthogonality among the Lanczos vectors. PRO is based on a simple recurrence, which allows us to monitor the loss of orthogonality among the Lanczos vectors directly without computing the inner products. Based on the information from the recurrence, reorthogonalizations occur only when necessary. Thus substantial savings are made as compared to FRO. In some numerical examples we apply the Lanczos algorithm with PRO to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the Lanczos vectors. The results obtained compare favorably with the conjugate gradient method.
Approximation of complex harmonic functions by complex harmonic splines
Authors: Chen Han Lin , Hvaring Tron
Published: June 2010

In this paper, a class of complex harmonic spline functions (C.H.S.) are defined on the unit discU. We use the C.H.S. to approximate the complex harmonic function onU, showing that C.H.S. may be represented by elementary functions. If the maximum step tends to zero and the mesh ratio is bounded, then C.H.S. converge uniformly to the interpolated functionFon the closed discŪ. If the interpolated functionFis a conformal mapping, then the C.H.S. is a quasi-conformal mapping.
A termination criterion for iterative methods used to find the zeros of polynomials
Authors: Igarashi Masao
Published: June 2010

A new criterion for terminating iterations when searching for polynomial zeros is described. This method does not depend on the number of digits in the mantissa; moreover, it can be used to determine the accuracy of the resulting zeros. Examples are included.
Diophantine equations in partitions
Authors: Gupta Hansraj
Published: June 2010

Given positive integers

r
1

,

r
2

,

r
3

,
…
,

r
j

{r_1},{r_2},{r_3}, \ldots ,{r_j}

such that
\[

r
1

>

r
2

>

r
3

>
⋯
>

r
j

>
m
;

m
>
1
;

{r_1} > {r_2} > {r_3} > \cdots > {r_j} > m;\quad m > 1;

\]
we find the number

P
(
n
,
m
;
R
)

P(n,m;R)

of partitions of a given positive integer n into parts belonging to the set R of residue classes
\[

r
1

(
mod

m
)
,

r
2

(
mod

m
)
,
…
,

r
j

(
mod

m
)
.

{r_1}\pmod m,\quad {r_2}\pmod m, \ldots ,{r_j}\pmod m.

\]
This leads to an identity which is more general though less elegant then the well-known Rogers-Ramanujan identities.
On the Diophantine equation ∑𝑋ᵢ=∏𝑋ᵢ
Authors: Brown M. L.
Published: June 2010

The diophantine equation

X
1

+
⋯
+

X
k

=

X
1

⋯

X
k

{X_1} + \cdots + {X_k} = {X_1} \cdots {X_k}

has at least one solution in positive integers for

k
⩾
2

k \geqslant 2

. The set of integers k for which this is the only solution are investigated; in particular, this set is conjectured to be a known finite sequence.
Preconditioning by incomplete block cyclic reduction
Authors: Rodrigue Garry , Wolitzer Donald
Published: June 2010

Iterative methods for solving linear systems arising from the discretization of elliptic/parabolic partial differential equations require the use of preconditioners to gain increased rates of convergence. Preconditioners arising from incomplete factorizations have been shown to be very effective. However, the recursiveness of these methods can offset these gains somewhat on a vector processor. In this paper, an incomplete factorization based on block cyclic reduction is developed. It is shown that under block diagonal dominance conditions the off-diagonal terms decay quadratically, yielding more effective algorithms.
A note on the moment generating function for the reciprocal gamma distribution
Authors: Wrigge Staffan
Published: June 2010

In this note we consider the function

φ
(
t
)
=

∫
0
∞

e

−
t
x

/

Γ
(
x
)

d
x

\varphi (t) = \smallint _0^\infty {e^{ - tx}}/\Gamma (x)\, dx

and use the Euler-Maclaurin expansion with the step-length

h
=
1

/

4

h = 1/4

to obtain some useful (from a numerical point of view) formulae. Numerical values of

φ
(
t
)

\varphi (t)

correct to 11D are given for

t
=
0.0
(
0.1
)
5.0

t = 0.0(0.1)5.0

.
An analogue of the nearest integer continued fraction for certain cubic irrationalities
Authors: Williams H. C. , Dueck G. W.
Published: June 2010

Let

θ
\theta

be any irrational and define

N
e
(
θ
)

Ne(\theta )

to be that integer such that

|

θ
−
N
e
(
θ
)

|

>

1
2

|\theta - Ne(\theta )|\; > \frac {1}{2}

. Put

ρ
0

=
θ

{\rho _0} = \theta

,

r
0

=
N
e
(

ρ
0

)

{r_0} = Ne({\rho _0})

,

ρ

k
+
1

=
1

/

(

r
k

−

ρ
k

)

{\rho _{k + 1}} = 1/({r_k} - {\rho _k})

,

r

k
+
1

=
N
e
(

ρ

k
+
1

)

{r_{k + 1}} = Ne({\rho _{k + 1}})

. Then the r’s here are the partial quotients of the nearest integer continued fraction (NICF) expansion of

θ
\theta

. When D is a positive nonsquare integer, and

θ
=

D

\theta = \sqrt D

, this expansion is periodic. It can be used to find the regulator of

Q

(

D

)

\mathcal {Q}(\sqrt D )

in less than 75 percent of the time needed by the usual continued fraction algorithm. A geometric interpretation of this algorithm is given and this is used to extend the NICF to a nearest integer analogue of the Voronoi Continued Fraction, which is used to find the regulator of a cubic field

F

\mathcal {F}

with negative discriminant

Δ
\Delta

. This new algorithm (NIVCF) is periodic and can be used to find the regulator of

F

\mathcal {F}

. If

I
>

|

Δ

|

/

148

4

I > \sqrt [4]{{|\Delta |/148}}

, the NIVCF algorithm can be used to find any algebraic integer

α
\alpha

of

F

\mathcal {F}

such that

N
(
α
)
=
I

N(\alpha ) = I

. Numerical results suggest that the NIVCF algorithm finds the regulator of

F

=

Q

(

D
3

)

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

in about 80 percent of the time needed by Voronoi’s algorithm.
Collocation for singular perturbation problems. II. Linear first order systems without turning points
Authors: Ascher U. , Weiss R.
Published: June 2010

We consider singularly perturbed linear boundary value problems for ODE’s with variable coefficients, but without turning points. Convergence results are obtained for collocation schemes based on Gauss and Lobatto points, showing that highly accurate numerical solutions for these problems can be obtained at a very reasonable cost using such schemes, provided that appropriate meshes are used. The implementation of the numerical schemes and the practical construction of corresponding meshes are discussed. These results extend those of a previous paper which deals with systems with constant coefficients.
Approximation properties of quadrature methods for Volterra integral equations of the first kind
Authors: Eggermont P. P. B.
Published: June 2010

We present a unifying analysis of quadrature methods for Volterra integral equations of the first kind that are zero-stable and have an asymptotic repetition factor. We show that such methods are essentially collocation-projection methods with underlying subspaces that have nice approximation properties, and which are stable as projection methods. This is used to derive asymptotically optimal error estimates under minimal smoothness conditions. The class of quadrature methods covered includes the cyclic linear multistep and the reducible quadrature methods, but not (really) Runge-Kutta methods.
Approximations for the Bessel and Struve functions
Authors: Newman J. N.
Published: June 2010

Polynomials and rational-fraction approximations with minimax accuracy are presented for the Bessel functions of the first and second kind of orders zero and one, and for the Struve functions of the same order. The accuracy of these approximations is consistent with typical single-precision computations. The results for the Bessel functions improve upon those in Abramowitz and Stegun [1].
More trigonometric integrals
Authors: Fettis Henry E.
Published: June 2010

Integrals of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript pi slash 2 Baseline e Superscript i p theta Baseline cosine Superscript q Baseline theta d theta comma integral Subscript 0 Superscript pi slash 2 Baseline e Superscript i p theta Baseline sine Superscript q Baseline theta d theta">

∫
0

π

/

2

e

i
p
θ

cos

q

θ

d
θ
,

∫
0

π

/

2

e

i
p
θ

sin

q

θ

d
θ

\int _0^{\pi /2} {{e^{ip\theta }}{{\cos }^q}\theta \,d\theta ,\quad \int _0^{\pi /2} {{e^{ip\theta }}{{\sin }^q}\theta \,d\theta } }

\]
(p real,

Re
⁡
(
q
)
>
−
1

\operatorname {Re} (q) > - 1

) are expressed in terms of Gamma and hypergeometric functions for integer and noninteger values of q and p. The results include those of [2] as special cases.
Generation of permutations following Lehmer and Howell
Authors: Spoletini Enrico
Published: June 2010

This paper proves two formulas used to obtain, by an arithmetic method, both the next permutation with respect to a given one and the mth permutation, in lexicographic direct ordering.
Products and sums of powers of binomial coefficients mod 𝑝 and solutions of certain quaternary Diophantine systems
Authors: Hudson Richard H.
Published: June 2010

In this paper we prove that certain products and sums of powers of binomial coefficients modulo

p
=
q
f
+
1

p = qf + 1

,

q
=

a
2

+

b
2

q = {a^2} + {b^2}

, are determined by the parameters x occurring in distinct solutions of the quaternary quadratic partition
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 16 p Superscript alpha Baseline equals x squared plus 2 q u squared plus 2 q v squared plus q w squared comma left-parenthesis x comma u comma v comma w comma p right-parenthesis equals 1 comma 2nd Row x w equals a v squared minus 2 b u v minus a u squared comma x identical-to 4 left-parenthesis mod q right-parenthesis comma alpha greater-than-or-slanted-equals 1 period EndLayout">

16

p
α

=

x
2

+
2
q

u
2

+
2
q

v
2

+
q

w
2

,

(
x
,
u
,
v
,
w
,
p
)
=
1
,

x
w
=
a

v
2

−
2
b
u
v
−
a

u
2

,

x
≡
4

(
mod

q
)
,
α
⩾
1.

\begin {array}{*{20}{c}} {16{p^\alpha } = {x^2} + 2q{u^2} + 2q{v^2} + q{w^2},\quad (x,u,v,w,p) = 1,} \\ {xw = a{v^2} - 2buv - a{u^2},\quad x \equiv 4\pmod q,\alpha \geqslant 1.} \\ \end {array}

\]
The number of distinct solutions of this partition depends heavily on the class number of the imaginary cyclic quartic field
\[

K
=
Q

(

i

2
q
+
2
a

q

)

,

K = Q\left ( {i\sqrt {2q + 2a\sqrt q } } \right ),

\]
as well as on the number of roots of unity in K and on the way that p splits into prime ideals in the ring of integers of the field

Q
(

e

2
π
i
p

/

q

)

Q({e^{2\pi ip/q}})

. Let the four cosets of the subgroup A of quartic residues be given by

c
j

=

2
j

A
,
j
=
0
,
1
,
2
,
3

{c_j} = {2^j}A,j = 0,1,2,3

, and let
\[

s
j

=

1
q

∑

t
∈

c
j

t
,

j
=
0
,
1
,
2
,
3.

{s_j} = \frac {1}{q}\sum \limits _{t \in {c_j}} {t,\quad j = 0,1,2,3.}

\]
Let

s
m

{s_m}

and

s
n

{s_n}

denote the smallest and next smallest of the

s
j

{s_j}

respectively. We give new, and unexpectedly simple determinations of

Π

k
∈

c
n

k
f
!

{\Pi _{k \in {c_n}}}kf!

and

Π

k
∈

c

n
+
2

k
f
!

{\Pi _{k \in {c_{n + 2}}}}kf!

, in terms of the parameters x in the above partition of

16

p
α

16{p^\alpha }

, in the complicated case that arises when the class number of K is

>
1

> 1

and

s
m

≠

s
n

{s_m} \ne {s_n}

.
Entropy condition satisfying approximations for the full potential equation of transonic flow
Authors: Osher Stanley , Hafez Mohamed , Whitlow Woodrow
Published: June 2010

We shall present a new class of conservative difference approximations for the steady full potential equation. They are, in general, easier to program than the usual density biasing algorithms, and in fact, differ only slightly from them. We prove rigorously that these new schemes satisfy a new discrete "entropy inequality", which rules out expansion shocks, and that they have sharp, steady, discrete shocks. A key tool in our analysis is the construction of an "entropy inequality" for the full potential equation itself. We conclude by presenting results of some numerical experiments using our new schemes.
A look-ahead Lánczos algorithm for unsymmetric matrices
Authors: Parlett Beresford N. , Taylor Derek R. , Liu Zhishun A.
Published: June 2010

The two-sided Lanczos algorithm sometimes suffers from serious breakdowns. These occur when the associated moment matrix does not permit triangular factorization. We modify the algorithm slightly so that it corresponds to using a

2
×
2

2 \times 2

pivot in triangular factorization whenever a

1
×
1

1 \times 1

pivot would be dangerous. The likelihood of breakdown is greatly reduced. The price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a

2
×
2

2 \times 2

pivot is used. Experiments with several versions of the algorithm on a variety of matrices are described, including some large problems arising in the study of plasma instability.
A globally convergent method for simultaneously finding polynomial roots
Authors: Pasquini L. , Trigiante D.
Published: June 2010

A new method for the simultaneous approximation of all the roots of a polynomial is given. The method converges for almost every initial approximation, the set of the exceptional starting points being a closed set of measure zero, at least if all the polynomial roots are real and simple. The method exhibits quadratic convergence not only to simple, but also to multiple roots.
Estimates for the Chebyshev function 𝜓(𝑥)-𝜃(𝑥)
Authors: Costa Pereira N.
Published: June 2010

A simple approximation for the difference

ψ
(
x
)
−
θ
(
x
)

\psi (x) - \theta (x)

is established by elementary methods. This approximation is used to obtain several estimates for

ψ
(
x
)
−
θ
(
x
)

\psi (x) - \theta (x)

which are sharper than those previously given in the literature.
Odd triperfect numbers are divisible by eleven distinct prime factors
Authors: Kishore Masao
Published: June 2010

We prove that an odd triperfect number has at least eleven distinct prime factors.
Finite element methods of optimal order for problems with singular data
Authors: Eriksson Kenneth
Published: June 2010

An adapted finite element method is proposed for a class of elliptic problems with singular data. The idea is to subtract the main singularity from the solution and to solve for the remainder using suitable mesh-refinements. Optimal order error estimates are proved.
On the steady states of finitely many chemical cells
Authors: Bigge J. , Bohl E.
Published: June 2010

Mathematical models of the form (1) and (2) for diffusion-reaction phenomena are discussed. The occurrence of bifurcation points in the discrete case (2) is explained via a simple two-dimensional model.