bluuebunny/crossref_metadata_2025 huggingface dataset:
Uniform error estimates for certain narrow Lagrange finite elements
Authors: Shenk N. Al
Published: June 2010

Error estimates of Dupont and Scott are used to derive uniform error estimates for Lagrange finite elements in

ℜ
n

(
n
≥
2
)

{\Re ^n}\;(n \geq 2)

under the following conditions: (1) The elements can be arbitrarily narrow in any coordinate direction such that a sufficient number of interpolation points are grouped on lines parallel to that coordinate axis, and (2) the space of approximating functions

F
T

{F_T}

in each element T must include the space of polynomials of degree

≤
m
−
1

\leq m - 1

for some

m
≥
1
+
n

/

2

m \geq 1 + n/2

. If n is odd, this does not cover elements of lowest degree that are normally considered with the shape regularity requirement that the ratio of their outer and inner diameters be bounded. For example, if

n
=
3

n = 3

, the usual requirement with shape regularity is that each

F
T

{F_T}

contain all first-degree polynomials. The result of this paper requires that each

F
T

{F_T}

contain all quadratic polynomials, and consequently does not apply to linear (Courant) elements in tetrahedrons or trilinear (tensor) elements in rectangular boxes. Counterexamples in these two cases are included.
An error estimate of the least squares finite element method for the Stokes problem in three dimensions
Authors: Chang Ching Lung
Published: June 2010

In this paper we are concerned with the Stokes problem in three dimensions (see recent works of the author and B. N. Jiang for the two-dimensional case). It is a linear system of four PDEs with velocity

u
_

\underline u

and pressure p as unknowns. With the additional variable

ω
_

=

curl

u
_

\underline \omega = {\operatorname {curl}}\underline u

, the second-order problem is reduced to a first-order system. Considering the compatibility condition

div
⁡

ω
_

=
0

\operatorname {div} \underline \omega = 0

, we have a system with eight first-order equations and seven unknowns. A least squares method is applied to this extended system, and also to the corresponding boundary conditions. The analysis based on works of Agmon, Douglis, and Nirenberg; Wendland; Zienkiewicz, Owen, and Niles; etc. shows that this method is stable in the h-version. For instance, if we choose continuous piecewise polynomials to approximate

u
_

,

ω
_

\underline u ,\underline \omega

, and p, this method achieves optimal rates of convergence in the

H
1

{H^1}

-norms.
A quadrature formula for entire functions of exponential type
Authors: Rahman Qazi Ibadur , Schmeisser Gerhard
Published: June 2010

A quadrature formula over a semi-infinite interval for entire functions of exponential type is established. An alternative approach to a known expansion formula and an extension of it are also presented.
Acceleration of convergence of a family of logarithmically convergent sequences
Authors: Van Tuyl Andrew H.
Published: June 2010

The asymptotic behavior of several sequence transformations is investigated as

n
→
∞

n \to \infty

when applied to a certain family of logarithmically convergent sequences. The transformations considered are the iterations of the transformations

e
1

(
s
)

(

A
n

)

e_1^{(s)}({A_n})

of Shanks and

W
n

{W_n}

of Lubkin, the

θ
\theta

-algorithm of Brezinski, the Levin u-and v-transforms, and generalizations of the

ρ
\rho

-algorithm and the Neville table. Computational results are given for both real and complex sequences.
Approximation of analytic functions: a method of enhanced convergence
Authors: Bruno Oscar P. , Reitich Fernando
Published: June 2010

We deal with a method of enhanced convergence for the approximation of analytic functions. This method introduces conformal transformations in the approximation problems, in order to help extract the values of a given analytic function from its Taylor expansion around a point. An instance of this method, based on the Euler transform, has long been known; recently we introduced more general versions of it in connection with certain problems in wave scattering. In §2 we present a general discussion of this approach. As is known in the case of the Euler transform, conformal transformations can enlarge the region of convergence of power series and can enhance substantially the convergence rates inside the circles of convergence. We show that conformal maps can also produce a rather dramatic improvement in the conditioning of Padé approximation. This improvement, which we discuss theoretically for Stieltjes-type functions, is most notorious in cases of very poorly conditioned Padé problems. In many instances, an application of enhanced convergence in conjunction with Padé approximation leads to results which are many orders of magnitude more accurate than those obtained by either classical Padé approximants or the summation of a truncated enhanced series.
Approximate zeros of quadratically convergent algorithms
Authors: Chen Pengyuan
Published: June 2010

Smale’s condition for a point to be an approximate zero of a function for Newton’s method is extended to the general quadratically convergent iterative algorithm. It is shown in which way the bound in the condition is affected by the characteristics of the algorithm. This puts the original condition of Smale for Newton’s method in a more general perspective. The results are also discussed in the light of numerical evidence.
A generalized sampling theorem for locally compact abelian groups
Authors: Faridani Adel
Published: June 2010

We present a sampling theorem for locally compact abelian groups. The sampling sets are finite unions of cosets of a closed subgroup. This generalizes the well-known case of nonequidistant but periodic sampling on the real line. For nonbandlimited functions an

L
1

{L_1}

-type estimate for the aliasing error is given. We discuss the application of the theorem to a class of sampling sets in

R

s

{{\mathbf {R}}^s}

, give a general algorithm for a computer implementation, present a detailed description of the implementation for the s-dimensional torus group, and point out connections to lattice rules for numerical integration.
Table of primitive binary polynomials. II
Authors: Živković Miodrag
Published: June 2010

For those

n
>
5000

n > 5000

, for which the factorization of

2
n

−
1

{2^n} - 1

is known, the first primitive trinomial (if such exists) and a randomly generated primitive 5- and 7-nomial of degree n in

G

F
(
2
)

{\text {G}}F(2)

are given, if the respective entry is absent from the previously published table.
Resolvent estimates for elliptic finite element operators in one dimension
Authors: Crouzeix M. , Larsson S. , Thomée V.
Published: June 2010

We prove the analyticity (uniform in h) of the semigroups generated on

L
p

(
0
,
1
)
,
1
≤
p
≤
∞

{L_p}(0,1),1 \leq p \leq \infty

, by finite element analogues

A
h

{A_h}

of a one-dimensional second-order elliptic operator A under Dirichlet boundary conditions. This is accomplished by showing the appropriate estimates for the resolvents by means of energy arguments. The results are applied to prove stability and optimal-order error bounds for numerical solutions of the associated parabolic problem for both smooth and nonsmooth data.
A new criterion for the first case of Fermat’s last theorem
Authors: Dilcher Karl , Skula Ladislav
Published: June 2010

It is shown that if the first case of Fermat’s last theorem fails for an odd prime l, then the sums of reciprocals modulo l,

s
(
k
,
N
)
=
∑
1

/

j

(
k
l

/

N
>
j
>
(
k
+
1
)
l

/

N
)

s(k,N) = \sum 1/j\;(kl/N > j > (k + 1)l/N)

are congruent to zero

mod

l

\bmod \;l

for all integers N and k with

1
≤
N
≤
46

1 \leq N \leq 46

and

0
≤
k
≤
N
−
1

0 \leq k \leq N - 1

. This is equivalent to

B

l
−
1

(
k

/

N
)
−

B

l
−
1

≡
0

(
mod

l
)

{B_{l - 1}}(k/N) - {B_{l - 1}} \equiv 0 \pmod l

, where

B
n

{B_n}

and

B
n

(
x
)

{B_n}(x)

are the nth Bernoulli number and polynomial, respectively. The work can be considered as a result on Kummer’s system of congruences.
Determination of all nonquadratic imaginary cyclic number fields of 2-power degrees with ideal class groups of exponents ≤2
Authors: Louboutin Stéphane
Published: June 2010

We determine all nonquadratic imaginary cyclic number fieldsKof 2-power degrees with ideal class groups of exponents≤2\leq 2, i.e., with ideal class groups such that the square of each ideal class is the principal class, i.e., such that the ideal class groups are isomorphic to some(Z/2Z)m,m≥0{({\mathbf {Z}}/2{\mathbf {Z}})^m},m \geq 0. There are 38 such number fields: 33 of them are quartic ones (see Theorem 13), 4 of them are octic ones (see Theorem 12), and 1 of them has degree 16 (see Theorem 11).
On the absolute Mahler measure of polynomials having all zeros in a sector
Authors: Rhin Georges , Smyth Christopher
Published: June 2010

Let

α
\alpha

be an algebraic integer of degree d, not 0 or a root of unity, all of whose conjugates

α
i

{\alpha _i}

, are confined to a sector

|

arg
⁡
z

|

≤
θ

|\arg z| \leq \theta

. We compute the greatest lower bound

c
(
θ
)

c(\theta )

of the absolute Mahler measure

(

∏

i
=
1

d

max
(
1
,

|

α
i

|

)

)

1

/

d

(\prod \nolimits _{i = 1}^d {\max (1,|{\alpha _i}|){)^{1/d}}}

of

α
\alpha

, for

θ
\theta

belonging to nine subintervals of

[
0
,
2
π

/

3
]

[0,2\pi /3]

. In particular, we show that

c
(
π

/

2
)
=
1.12933793

c(\pi /2) = 1.12933793

, from which it follows that any integer

α
≠
1

\alpha \ne 1

and

α
≠

e

±
i
π

/

3

\alpha \ne {e^{ \pm i\pi /3}}

all of whose conjugates have positive real part has absolute Mahler measure at least

c
(
π

/

2
)

c(\pi /2)

. This value is achieved for

α
\alpha

satisfying

α
+
1

/

α
=

β
0
2

\alpha + 1/\alpha = \beta _0^2

, where

β
0

=
1.3247
…

{\beta _0} = 1.3247 \ldots

is the smallest Pisot number (the real root of

β
0
3

=

β
0

+
1

\beta _0^3 = {\beta _0} + 1

).
Rabin-Miller primality test: composite numbers which pass it
Authors: Arnault F.
Published: June 2010

The Rabin-Miller primality test is a probabilistic test which can be found in several algebraic computing systems (such as Pari, Maple, ScratchPad) because it is very easy to implement and, with a reasonable amount of computing, indicates whether a number is composite or "probably prime" with a very low probability of error. In this paper, we compute composite numbers which are strong pseudoprimes to several chosen bases. Because these bases are those used by the ScratchPad implementation of the test, we obtain, by a method which differs from a recent one by Jaeschke, composite numbers which are found to be "probably prime" by this test.
The Faber polynomials for annular sectors
Authors: Coleman John P. , Myers Nick J.
Published: June 2010

A conformal mapping of the exterior of the unit circle to the exterior of a region of the complex plane determines the Faber polynomials for that region. These polynomials are of interest in providing near-optimal polynomial approximations in a variety of contexts, including the construction of semiiterative methods for linear equations. The relevant conformal map for an annular sector

{
z
:
R
≤

|

z

|

≤
1
,
θ
≤

|

arg
⁡
z

|

≤
π
}

\{ z:R \leq |z| \leq 1,\theta \leq |\arg z| \leq \pi \}

, with

0
>
θ
≤
π

0 > \theta \leq \pi

, is derived here and a recurrence relation is established for the coefficients of its Laurent expansion about the point at infinity. The recursive evaluation of scaled Faber polynomials is formulated in such a way that an algebraic manipulation package may be used to generate explicit expressions for their coefficients, in terms of two parameters which are determined by the interior angle of the annular sector and the ratio of its radii. Properties of the coefficients of the scaled Faber polynomials are established, and those for polynomials of degree

≤
15

\leq 15

are tabulated in a Supplement at the end of this issue. A simple closed form is obtained for the coefficients of the Faber series for 1/z. Known results for an interval, a circular arc, and a circular sector are reproduced as special cases.
Imprimitive ninth-degree number fields with small discriminants
Authors: Diaz y Diaz F. , Olivier M.
Published: June 2010

We present tables of ninth-degree nonprimitive (i.e., containing a cubic subfield) number fields. Each table corresponds to one signature, except for fields with signature (3,3), for which we give two different tables depending on the signature of the cubic subfield. Details related to the computation of the tables are given, as well as information about the CPU time used, the number of polynomials that we deal with, etc. For each field in the tables, we give its discriminant, the discriminant of its cubic subfields, the relative polynomial generating the field over one of its cubic subfields, the corresponding (irreducible) polynomial over

Q

\mathbb {Q}

, and the Galois group of the Galois closure. Fields having interesting properties are studied in more detail, especially those associated with sextic number fields having a class group divisible by 3.
Converting approximate error bounds into exact ones
Authors: Ziv Abraham
Published: June 2010

In order to produce error bounds quickly and easily, people often apply to error bounds linearized propagation rules. This is done instead of a precise error analysis. The payoff: Estimates so produced are not guaranteed to be true bounds. One can at most hope that they are good approximations of true bounds. This paper discusses a way to convert such approximate error bounds into true bounds. This is done by dividing the approximate bound by

1
−
δ

1 - \delta

, with a small

δ
\delta

. Both the approximate bound and

δ
\delta

are produced by the same linearized error analysis. This method makes it possible both to simplify the error analyses and to sharpen the bounds in an interesting class of numerical algorithms. In particular it seems to be ideal for the derivation of tight, true error bounds for simple and accurate algorithms, like those used in subroutines for the evaluation of elementary mathematical functions (EXP, LOG, SIN, etc.), for instance. The main subject of this paper is forward a priori error analysis. However, the method may be fitted to other types of error analysis too. In fact the outlines of a forward a posteriori error analysis theory and of running error analysis are given also. In the course of proofs a new methodology is applied for the representation of propagated error bounds. This methodology promotes easy derivation of sharp, helpful inequalities. Several examples of forward a priori error analysis and one of a posteriori error analysis and running error analysis are included.
Numerical schemes for conservation laws via Hamilton-Jacobi equations
Authors: Corrias L. , Falcone M. , Natalini R.
Published: June 2010

We present some difference approximation schemes which converge to the entropy solution of a scalar conservation law having a convex flux. The numerical methods described here take their origin from approximation schemes for Hamilton-Jacobi-Bellman equations related to optimal control problems and exhibit several interesting features: the convergence result still holds for quite arbitrary time steps, the main assumption for convergence can be interpreted as a discrete analogue of Oleinik’s entropy condition, numerical diffusion around the shocks is very limited. Some tests are included in order to compare the performances of these methods with other classical methods (Godunov, TVD).
Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions
Authors: Eckhoff Knut S.
Published: June 2010

Knowledge of a truncated Fourier series expansion for a

2
π

2\pi

-periodic function of finite regularity, which is assumed to be piecewise smooth in each period, is used to accurately reconstruct the corresponding function. An algebraic equation of degree M is constructed for the M singularity locations in each period for the function in question. The M coefficients in this algebraic equation are obtained by solving an algebraic system of M equations determined by the coefficients in the known truncated expansion. If discontinuities in the derivatives of the function are considered, in addition to discontinuities in the function itself, that algebraic system will be nonlinear with respect to the M unknown coefficients. The degree of the algebraic system will depend on the desired order of accuracy for the reconstruction, i.e., a higher degree will normally lead to a more accurate determination of the singularity locations. By solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations, we are able to reconstruct the

2
π

2\pi

-periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiable up to the specified order.
Pseudorandom vector generation by the multiple-recursive matrix method
Authors: Niederreiter Harald
Published: June 2010

Pseudorandom vectors are of importance for parallelized simulation methods. In this paper we carry out an in-depth analysis of the multiple-recursive matrix method for the generation of uniform pseudorandom vectors which was introduced in an earlier paper of the author. We study, in particular, the periodicity properties, the lattice structure, and the behavior under the serial test for sequences of pseudorandom vectors generated by this method.
On a new factorization algorithm for polynomials over finite fields
Authors: Niederreiter Harald , Göttfert Rainer
Published: June 2010

A new deterministic factorization algorithm for polynomials over finite fields was recently developed by the first author. The bottleneck in this algorithm is the last stage in which the irreducible factors of the polynomial are derived from the solutions of a system of linear equations. An efficient approach to the last stage was designed by the second author for the case of finite fields of characteristic 2. In this paper, we describe a different approach to the last stage which works for arbitrary fields of positive characteristic. In particular, we obtain in this way an acceleration of the factorization algorithm of the first author which makes this algorithm polynomial time for fixed characteristic.
Density bounds for the 3𝑥+1 problem. I. Tree-search method
Authors: Applegate David , Lagarias Jeffrey C.
Published: June 2010

The

3
x
+
1

3x + 1

function

T
(
x
)

T(x)

takes the values

(
3
x
+
1
)

/

2

(3x + 1)/2

if x is odd and

x

/

2

x/2

if x is even. Let a be any integer with

a

≢

0

(
mod

3
)

a \nequiv 0\; \pmod 3

. If

n
k

(
a
)

{n_k}(a)

counts the number of n with

T

(
k
)

(
n
)
=
a

{T^{(k)}}(n) = a

, then for all sufficiently large k,

(
1.302

)
k

≤

n
k

(
a
)
≤

(
1.359

)
k

{(1.302)^k} \leq {n_k}(a) \leq {(1.359)^k}

. If

π
a

(
x
)

{\pi _a}(x)

counts the number of n with

|

n

|

≤
x

|n| \leq x

which eventually reach a under iteration by T, then for sufficiently large x,

π
a

(
x
)
≥

x

.65

{\pi _a}(x) \geq {x^{.65}}

. The proofs are computer-intensive.
Density bounds for the 3𝑥+1 problem. II. Krasikov inequalities
Authors: Applegate David , Lagarias Jeffrey C.
Published: June 2010

The

3
x
+
1

3x + 1

function

T
(
x
)

T(x)

takes the values

(
3
x
+
1
)

/

2

(3x + 1)/2

if x is odd and x/2 if x is even. Let a be any integer with

a

≢

0

(
mod

3
)

a \nequiv 0\; \pmod 3

. If

π
a

(
x
)

{\pi _a}(x)

counts the number of n with

|

n

|

≤
x

|n| \leq x

which eventually reach a under iteration by T, then for all sufficiently large x,

π
a

(
x
)
≥

x

.81

{\pi _a}(x) \geq {x^{.81}}

. The proof is based on solving nonlinear programming problems constructed using difference inequalities of Krasikov.
Adaptive search in quasi-Monte Carlo optimization
Authors: Biester Christian , Grabner Peter J. , Larcher Gerhard , Tichy Robert F.
Published: June 2010

Motivated by a linear time-complexity result for an adaptive Monte Carlo algorithm, we propose and analyze an adaptive deterministic algorithm. We restrict a grid search to nested subregions that promise to provide improvement of the current solution, and we obtain an exponential rate of convergence in function evaluations. For proving the main result we restrict ourselves to functions on hypercubes. In a final section we outline how to extend the method to the general case and give some numerical examples.
Numerical evaluation of surface integrals in three dimensions
Authors: Chien David
Published: June 2010

In this paper, we consider the evaluation of surface integrals over piecewise smooth surfaces in three dimensions. The method consists in first replacing a parametrization for the surface and the integrand function by piecewise polynomial interpolants of them, and second, using a numerical integration scheme for the resulting integral. The order of convergence is higher than would be expected based on the underlying interpolation theory.
Discrete piecewise monotonic approximation by a strictly convex distance function
Authors: Demetriou I. C.
Published: June 2010

Theory and algorithms are presented for the following smoothing problem. We are given n measurements of a real-valued function that have been altered by random errors caused by the deriving process. For a given integer k, some efficient algorithms are developed that approximate the data by minimizing the sum of strictly convex functions of the errors in such a way that the approximated values are made up of at most k monotonic sections. If

k
=
1

k = 1

, then the problem can be solved by a special strictly convex programming calculation. If

k
>
1

k > 1

, then there are

O
(

n
k

)

O({n^k})

possible choices of the monotonic sections, so that it is impossible to test each one separately. A characterization theorem is derived that allows dynamic programming to be used for dividing the data into optimal disjoint sections of adjacent data, where each section requires a single monotonic calculation. It is remarkable that the theorem reduces the work for a global minimum to

O
(
n
)

O(n)

monotonic calculations to subranges of data and

O
(
k

s
2

)

O(k{s^2})

computer operations, where

s
−
2

s - 2

is the number of sign changes in the sequence of the first divided differences of the data. Further, certain monotonicity properties of the extrema of best approximations with k and

k
−
1

k - 1

, and with k and

k
−
2

k - 2

monotonic sections make the calculation quite efficient. A Fortran 77 program has been written and some numerical results illustrate the performance of the smoothing technique in a variety of data sets.
Factors of generalized Fermat numbers
Authors: Dubner Harvey , Keller Wilfrid
Published: June 2010

Generalized Fermat numbers have the form

F

b
,
m

=

b

2
m

+
1

{F_{b,m}} = {b^{{2^m}}} + 1

. Their odd prime factors are of the form

k
⋅

2
n

+
1

k \cdot {2^n} + 1

, k odd,

n
>
m

n > m

. It is shown that each prime is a factor of some

F

b
,
m

{F_{b,m}}

for approximately

1

/

k

1/k

bases b, independent of n. Divisors of generalized Fermat numbers of base 6, base 10, and base 12 are tabulated. Three new factors of standard Fermat numbers are included.
The largest prime dividing the maximal order of an element of 𝑆_{𝑛}
Authors: Grantham Jon
Published: June 2010

We define

g
(
n
)

g(n)

to be the maximal order of an element of the symmetric group on n elements. Results about the prime factorization of

g
(
n
)

g(n)

allow a reduction of the upper bound on the largest prime divisor of

g
(
n
)

g(n)

to

1.328

n
log
⁡
n

1.328\sqrt {n\log n}

.
Analysis of Coppersmith’s block Wiedemann algorithm for the parallel solution of sparse linear systems
Authors: Kaltofen Erich
Published: June 2010

By using projections by a block of vectors in place of a single vector it is possible to parallelize the outer loop of iterative methods for solving sparse linear systems. We analyze such a scheme proposed by Coppersmith for Wiedemann’s coordinate recurrence algorithm, which is based in part on the Krylov subspace approach. We prove that by use of certain randomizations on the input system the parallel speed up is roughly by the number of vectors in the blocks when using as many processors. Our analysis is valid for fields of entries that have sufficiently large cardinality. Our analysis also deals with an arising subproblem of solving a singular block Toeplitz system by use of the theory of Toeplitz-like matrices.
Precise solution of Laplace’s equation
Authors: Shi Zhixin , Hassard Brian
Published: June 2010

An approximate method is described for solving Laplace’s equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace StartLayout 1st Row normal upper Delta u equals 0 2nd Row u vertical-bar Subscript partial-differential normal upper Omega Baseline equals g EndLayout StartLayout 1st Row in normal upper Omega equals left-parenthesis 0 comma 1 right-parenthesis times left-parenthesis 0 comma 1 right-parenthesis comma 2nd Row on partial-differential normal upper Omega EndLayout">

{

Δ
u
=
0

u

|

∂
Ω

=
g

in

Ω
=
(
0
,
1
)
×
(
0
,
1
)
,

on

∂
Ω

\left \{ {\begin {array}{*{20}{c}} {\Delta u = 0} \\ {u{|_{\partial \Omega }} = g} \\ \end {array} \quad \begin {array}{*{20}{c}} {{\text {in}}\;\Omega = (0,1) \times (0,1),} \hfill \\ {{\text {on}}\;\partial \Omega } \hfill \\ \end {array} } \right .

\]
precisely in the sense of Aberth’s 1988 monograph. The algorithm uses singularity extraction, Fourier series methods, Taylor series methods, and interval analysis to construct an approximation

U
(
x
,
y
)

U(x,y)

to the solution

u
(
x
,
y
)

u(x,y)

at points in the square, and a uniform bound on the error

|

U
(
x
,
y
)
−
u
(
x
,
y
)

|

|U(x,y) - u(x,y)|

. The algorithm applies to problems in which the boundary data g is specified in terms of four elementary functions. The boundary data may be discontinuous at the corners.
The sharpness of Kuznetsov’s 𝑂(√Δ𝑥)𝐿¹-error estimate for monotone difference schemes
Authors: Tang Tao , Teng Zhen Huan
Published: June 2010

We derive a lower error bound for monotone difference schemes to the solution of the linear advection equation with BV initial data. A rigorous analysis shows that for any monotone difference scheme the lower

L
1

{L^1}

-error bound is

O
(

Δ
x

)

O(\sqrt {\Delta x} )

, where

Δ
x

\Delta x

is the spatial stepsize.
A posteriori error estimates for boundary element methods
Authors: Carstensen Carsten , Stephan Ernst P.
Published: June 2010

This paper deals with a general framework for a posteriori error estimates in boundary element methods which is specified for three examples, namely Symm’s integral equation, an integral equation with a hypersingular operator, and a boundary integral equation for a transmission problem. Based on these estimates, an analog of Eriksson and Johnson’s adaptive finite element method is proposed for the h-version of the Galerkin boundary element method for integral equations of the first kind. The efficiency of the approach is shown by numerical experiments which yield almost optimal convergence rates even in the presence of singularities.
The twenty-second Fermat number is composite
Authors: Crandall R. , Doenias J. , Norrie C. , Young J.
Published: June 2010

We have shown by machine proof that

F

22

=

2
2

22

+
1

{F_{22}} = {2^2}^{^{22}} + 1

is composite. In addition, we reenacted Young and Buell’s 1988 resolution of

F

20

{F_{20}}

as composite, finding agreement with their final Selfridge-Hurwitz residues. We also resolved the character of all extant cofactors of

F
n

,
n
≤
22

{F_n}, n \leq 22

, finding no new primes, and ruling out prime powers.
Faster computation of the first factor of the class number of 𝑄(𝜁_{𝑝})
Authors: Jha Vijay
Published: June 2010

We describe two fast methods for computing the first factor of the class number of the cyclotomic field

Q

(

ζ
p

)

\mathbb {Q}({\zeta _p})

in

O

(

p
2

log
5

p
)

\mathcal {O}({p^2}{\log ^5}p)

and

O

(

p
2

log
⁡
p
)

\mathcal {O}({p^2}\log p)

steps of elementary arithmetic operations on the numbers of size p, respectively. The first is deterministic, while the second holds under the GRH. This is an improvement over the previous method of Lehmer and Masley, which has complexity

O

(

p

3.81

)

\mathcal {O}({p^{3.81}})

.
An analytic method for convergence acceleration of certain hypergeometric series
Authors: Lewanowicz Stanisław , Paszkowski Stefan
Published: June 2010

A method is presented for convergence acceleration of the generalized hypergeometric series

3

F
2

_3{F_2}

with the argument

±
1

\pm 1

, using analytic properties of their terms. Iterated transformation of the series is performed analytically, which results in obtaining new fast converging expansions for some special functions and mathematical constants.
Locking effects in the finite element approximation of plate models
Authors: Suri Manil , Babuška Ivo , Schwab Christoph
Published: June 2010

We analyze the robustness of various standard finite element schemes for a hierarchy of plate models and obtain asymptotic convergence estimates that are uniform in terms of the thickness d. We identify h version schemes that show locking, i.e., for which the asymptotic convergence rate deteriorates as

d
→
0

d \to 0

, and also show that the p version is free of locking. In order to isolate locking effects from boundary layer effects (which also arise as

d
→
0

d \to 0

), our analysis is carried out for the periodic case, which is free of boundary layers. We analyze in detail the lowest model of the hierarchy, the well-known Reissner-Mindlin model, and show that the locking and robustness of finite element schemes for higher models of the hierarchy are essentially identical to the Riessner-Mindlin case.
Computing the Hilbert transform on the real line
Authors: Weideman J. A. C.
Published: June 2010

We introduce a new method for computing the Hilbert transform on the real line. It is a collocation method, based on an expansion in rational eigenfunctions of the Hilbert transform operator, and implemented through the Fast Fourier Transform. An error analysis is given, and convergence rates for some simple classes of functions are established. Numerical tests indicate that the method compares favorably with existing methods.
Half-step modular equations
Authors: Cohn Harvey
Published: June 2010

The classical modular equations relating Klein-Weber’s

j
(
τ
)

j(\tau )

to

j
(
b
τ
)

j(b\tau )

can be computed as the composition of two "half-step" equations relating

j
m

(
τ
)

{j_m}(\tau )

and

j
m

(
τ

b

)

{j_m}(\tau \sqrt b )

, where

j
m

{j_m}

is an extended modular function (corresponding to

τ
→
τ
+

m

,
τ
→
−
1

/

τ

\tau \to \tau + \sqrt m ,\tau \to - 1/\tau

, et al.). The half-step equations are easily constructed and manipulated in computer algebra. The cases computed here are b prime,

m
=
a

m = a

(or ab),

gcd
(
a
,
b
)
=
1
,
a
b

|

30

\gcd (a,b) = 1,ab|30

. This includes many cases where the property of "normal parametrization" occurs, which is of interest in class field theory. Extended modular functions have found recent application in group character theory but they arose in the present context as traces at

∞
\infty

of Hilbert modular equations.
Error estimators for nonconforming finite element approximations of the Stokes problem
Authors: Dari Enzo , Durán Ricardo , Padra Claudio
Published: June 2010

In this paper we define and analyze a posteriori error estimators for nonconforming approximations of the Stokes equations. We prove that these estimators are equivalent to an appropriate norm of the error. For the case of piecewise linear elements we define two estimators. Both of them are easy to compute, but the second is simpler because it can be computed using only the right-hand side and the approximate velocity. We show how the first estimator can be generalized to higher-order elements. Finally, we present several numerical examples in which one of our estimators is used for adaptive refinement.
On the Gibbs phenomenon. IV. Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function
Authors: Gottlieb David , Shu Chi-Wang
Published: June 2010

We continue our investigation of overcoming the Gibbs phenomenon, i.e., to obtain exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we are given the first N Gegenbauer expansion coefficients, based on the Gegenbauer polynomials

C
k
μ

(
x
)

C_k^\mu (x)

with the weight function

(
1
−

x
2

)

μ
−
1

/

2

{(1 - {x^2})^{\mu - 1/2}}

for any constant

μ
≥
0

\mu \geq 0

, of an

L
1

{L_1}

function

f
(
x
)

f(x)

, we can construct an exponentially convergent approximation to the point values of

f
(
x
)

f(x)

in any subinterval in which the function is analytic. The proof covers the cases of Chebyshev or Legendre partial sums, which are most common in applications.
An algorithm for nondominant solutions of linear second-order inhomogeneous difference equations
Authors: Hasegawa Takemitsu , Torii Tatsuo
Published: June 2010

An algorithm is given for computing a weighted sum of a nondominant solution of a linear second-order inhomogeneous difference equation to a prescribed accuracy by estimating the truncation error. The present method is an extension of both the stable numerical method due to Olver and Sookne and a summation technique due to Deuflhard for computing minimal solutions of a homogeneous difference equation. The method is illustrated by numerical examples.
The 𝐾-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates
Authors: Tran Thanh
Published: June 2010

Superconvergence in the

L
2

{L^2}

-norm for the Galerkin approximation of the integral equation

L
u
=
f

Lu = f

is studied, where L is a strongly elliptic pseudodifferential operator on a smooth, closed or open curve. Let

u
h

{u_h}

be the Galerkin approximation to u. By using the K-operator, an operator that averages the values of

u
h

{u_h}

, we will construct a better approximation than

u
h

{u_h}

itself. That better approximation is a legacy of the highest order of convergence in negative norms. For Symm’s equation on a slit the same order of convergence can be recovered if the mesh is suitably graded.
Primitive divisors of Lucas and Lehmer sequences
Authors: Voutier Paul M.
Published: June 2010

Stewart reduced the problem of determining all Lucas and Lehmer sequences whosenth element does not have a primitive divisor to solving certain Thue equations. Using the method of Tzanakis and de Weger for solving Thue equations, we determine such sequences forn≤30n \leq 30. Further computations lead us to conjecture that, forn>30n > 30, thenth element of such sequences always has a primitive divisor.
On the convergence rate of the cell discretization algorithm for solving elliptic problems
Authors: Cayco Maria , Foster Leslie , Swann Howard
Published: June 2010

Error estimates for the cell discretization algorithm are obtained for polynomial bases used to approximate both

H
k

(
Ω
)

{H^k}(\Omega )

and analytic solutions to selfadjoint elliptic problems. The polynomial implementation of this algorithm can be viewed as a nonconforming version of the h-p finite element method that also can produce the continuous approximations of the h-p method. The examples provided by our experiments provide discontinuous approximations that have errors similar to the finite element results.
On a conjecture of Crandall concerning the 𝑞𝑥+1 problem
Authors: Franco Zachary , Pomerance Carl
Published: June 2010

R. E. Crandall has conjectured that for any odd integer

q
>
3

q > 3

, there is a positive integer m whose orbit in the "

q
x
+
1

qx + 1

problem" does not contain 1. We show that this is true for almost all odd numbers q, in the sense of asymptotic density.
Upper semicontinuity of attractors for linear multistep methods approximating sectorial evolution equations
Authors: Hill Adrian T. , Süli Endre
Published: June 2010

This paper sets out a theoretical framework for approximating the attractor

A

\mathcal {A}

of a semigroup

S
(
t
)

S(t)

defined on a Banach space X by a q-step semidiscretization in time with constant steplength k. Using the one-step theory of Hale, Lin and Raugel, sufficient conditions are established for the existence of a family of attractors

{

A

k

}
⊂

X
q

\{ {\mathcal {A}_k}\} \subset {X^q}

, for the discrete semigroups

S
k
n

S_k^n

defined by the numerical method. The convergence properties of this family are also considered. Full details of the theory are exemplified in the context of strictly

A
(
α
)

A(\alpha )

-stable linear multistep approximations of an abstract dissipative sectorial evolution equation.
A polyhedral method for solving sparse polynomial systems
Authors: Huber Birkett , Sturmfels Bernd
Published: June 2010

A continuation method is presented for computing all isolated roots of a semimixed sparse system of polynomial equations. We introduce mixed subdivisions of Newton polytopes, and we apply them to give a new proof and algorithm for Bernstein’s theorem on the expected number of roots. This results in a numerical homotopy with the optimal number of paths to be followed. In this homotopy there is one starting system for each cell of the mixed subdivision, and the roots of these starting systems are obtained by an easy combinatorial construction.
A recursive method to calculate the number of solutions of quadratic equations over finite fields
Authors: Iyanaga Kenichi
Published: June 2010

The number

S
m

(
α
)

{S_m}(\alpha )

of solutions of the quadratic equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x 1 squared plus x 2 squared plus midline-horizontal-ellipsis plus x Subscript m Superscript 2 Baseline equals alpha left-parenthesis x Subscript i Superscript 2 Baseline not-equals plus-or-minus x Subscript j Superscript 2 Baseline for i not-equals j right-parenthesis">

x
1
2

+

x
2
2

+
⋯
+

x
m
2

=
α

(

x
i
2

≠
±

x
j
2

for

i
≠
j
)

x_1^2 + x_2^2 + \cdots + x_m^2 = \alpha \quad (x_i^2 \ne \pm x_j^2\quad {\text {for}}\;i \ne j)

\]
for given m, with

α
\alpha

and

x
i

{x_i}

belonging to a finite field, is studied and a recursive method to compute

S
m

(
α
)

{S_m}(\alpha )

is established.
Consistent structures of invariant quadrature rules for the 𝑛-simplex
Authors: Maeztu J. I. , Sáinz de la Maza E.
Published: June 2010

In this paper we develop a technique to obtain, in a systematic way, the consistency conditions for the n-dimensional simplex

T
n

{T_n}

for any dimension n and degree of precision d. The introduction of a convenient basis of invariant polynomials provides a powerful tool to analyze and obtain consistent structures. We also present tables listing the optimal consistent structures for dimensions

n
=
2
,
…
,
8

n = 2, \ldots ,8

and degree of precision up to

d
=
23

d = 23

. This paper is devoted only to structures. No quadrature rules are presented here.
Twenty-two primes in arithmetic progression
Authors: Pritchard Paul A. , Moran Andrew , Thyssen Anthony
Published: June 2010

Some newly-discovered arithmetic progressions of primes are presented, including five of length twenty-one and one of length twenty-two.
On multivariate Lagrange interpolation
Authors: Sauer Thomas , Xu Yuan
Published: June 2010

Lagrange interpolation by polynomials in several variables is studied through a finite difference approach. We establish an interpolation formula analogous to that of Newton and a remainder formula, both of them in terms of finite differences. We prove that the finite difference admits an integral representation involving simplex spline functions. In particular, this provides a remainder formula for Lagrange interpolation of degree n of a function f, which is a sum of integrals of certain

(
n
+
1
)

(n + 1)

st directional derivatives of f multiplied by simplex spline functions. We also provide two algorithms for the computation of Lagrange interpolants which use only addition, scalar multiplication, and point evaluation of polynomials.
Interior maximum-norm estimates for finite element methods. II
Authors: Schatz A. H. , Wahlbin L. B.
Published: June 2010

We consider bilinear forms

A
(
∙
,
∙
)

A( \bullet , \bullet )

connected with second-order elliptic problems and assume that for

u
h

{u_h}

in a finite element space

S
h

{S_h}

, we have

A
(
u
−

u
h

,
χ
)
=
F
(
χ
)

A(u - {u_h},\chi ) = F(\chi )

for

χ
\chi

in

S
h

{S_h}

with local compact support. We give local estimates for

u
−

u
h

u - {u_h}

in

L
∞

{L_\infty }

and

W
∞
1

W_\infty ^1

of the type "local best approximation plus weak outside influences plus the local size of F".
On the implementation of mixed methods as nonconforming methods for second-order elliptic problems
Authors: Arbogast Todd , Chen Zhangxin
Published: June 2010

In this paper we show that mixed finite element methods for a fairly general second-order elliptic problem with variable coefficients can be given a nonmixed formulation. (Lower-order terms are treated, so our results apply also to parabolic equations.) We define an approximation method by incorporating some projection operators within a standard Galerkin method, which we call a projection finite element method. It is shown that for a given mixed method, if the projection method’s finite element space

M
h

{M_h}

satisfies three conditions, then the two approximation methods are equivalent. These three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator. We then construct appropriate nonconforming spaces

M
h

{M_h}

for the known triangular and rectangular elements. The lowest-order Raviart-Thomas mixed solution on rectangular finite elements in

R

2

{\mathbb {R}^2}

and

R

3

{\mathbb {R}^3}

, on simplices, or on prisms, is then implemented as a nonconforming method modified in a simple and computationally trivial manner. This new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution. A rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal-order multigrid methods for the solution of the linear system.
Voronoĭ-algorithm expansion of two families with period length going to infinity
Authors: Adam Brigitte
Published: June 2010

We consider families of orders of complex cubic fields introduced recently by Levesque and Rhin and find the Voronoï-algorithm expansions and the fundamental units. We compare with the Jacobi-Perron algorithm expansions.
Multiquadric prewavelets on nonequally spaced knots in one dimension
Authors: Buhmann M. D.
Published: June 2010

In this paper, we identify univariate prewavelets on spaces spanned by translates of multiquadric functions and other radial basis functions with nonequally spaced centers (or "knots"). Although the multiquadric function and its relations are our prime examples, the theory is sufficiently broad to admit prewavelets from other radial basis function spaces as well.
Multiamicable numbers
Authors: Cohen Graeme L. , Gretton Stephen F. , Hagis Peter
Published: June 2010

Multiamicable numbers are a natural generalization of amicable numbers: two numbers form a multiamicable pair if the sum of the proper divisors of each is a multiple of the other. Many other generalizations have been considered in the past. This paper reviews those earlier generalizations and gives examples and properties of multiamicable pairs. It includes a proof that the set of all multiamicable numbers has density 0.
On the 𝑙-adic Iwasawa 𝜆-invariant in a 𝑝-extension
Authors: Friedman Eduardo , Sands Jonathan W.
Published: June 2010

For distinct primes l and p, the Iwasawa invariant

λ
l
−

\lambda _l^ -

stabilizes in the cyclotomic

Z

p

{\mathbb {Z}_p}

-tower over a complex abelian base field. We calculate this stable invariant for

p
=
3

p = 3

and various l and K. Our motivation was to search for a formula of Riemann-Hurwitz type for

λ
l
−

\lambda _l^ -

that might hold in a p-extension. From our numerical results, it is clear that no formula of such a simple kind can hold. In the course of our calculations, we develop a method of computing

λ
l
−

\lambda _l^ -

for an arbitrary complex abelian field and, for

p
=
3

p = 3

, we make effective Washington’s theorem on the stabilization of the l-part of the class group in the cyclotomic

Z

p

{\mathbb {Z}_p}

-extension. A new proof of this theorem is given in the appendix.
Parity of class numbers and Witt equivalence of quartic fields
Authors: Jakubec Stanislav , Marko František , Szymiczek Kazimierz
Published: June 2010

We show that 27 out of the 29 Witt equivalence classes of quartic number fields can be represented by fields of class number 1. It is known that the remaining two classes contain solely fields of even class numbers. We show that these two classes can be represented by fields of class number 2.
A dual finite element approach for stresses of elasto-perfectly plastic bodies
Authors: Neittaanmäki P. , Rivkind V. , Serëgin G.
Published: June 2010

Primal and dual approaches are introduced for the elasto-perfectly plastic problems. We prove theorems for approximating the stresses of elastic-perfectly plastic bodies.
Summation by parts, projections, and stability. II
Authors: Olsson Pelle
Published: June 2010

In this paper we prove strict stability of high-order finite difference approximations of parabolic and symmetric hyperbolic systems of partial differential equations on bounded, curvilinear domains in two space dimensions. The boundary need not be smooth. We also show how to derive strict stability estimates for inhomogeneous boundary conditions.
48 more solutions of Martin Davis’s quaternary quartic equation
Authors: Shanks Daniel , Wagstaff Samuel S.
Published: June 2010

We find 48 more solutions to a Diophantine equation investigated by Martin Davis. Before our work, only two solutions were known. Construction of the new solutions required the factorization of several large integers. Because the equation relates to Hilbert’s Tenth Problem it is desirable to know if it has only finitely many solutions. An elaborate argument is given for the conjecture that the equation has infinitely many solutions in integers.
Quintuples with square triplets
Authors: Wagon Stan
Published: June 2010

By applying modern computing power to an 1848 formula of C. Gill, we find five positive integers, every three of which sum to a square.
On Yokoi’s conjecture
Authors: Zhang Ming Yao
Published: June 2010

In this paper we obtain a lower bound for those discriminants of real quadratic fields

Q

(

D

)

\mathbb {Q}(\sqrt D )

with

D
=

m
2

+
4

D = {m^2} + 4

and

h
(
D
)
=
1

h(D) = 1

.
Fast Gaussian elimination with partial pivoting for matrices with displacement structure
Authors: Gohberg I. , Kailath T. , Olshevsky V.
Published: June 2010

Fast

O
(

n
2

)

O({n^2})

implementation of Gaussian elimination with partial pivoting is designed for matrices possessing Cauchy-like displacement structure. We show how Toeplitz-like, Toeplitz-plus-Hankel-like and Vandermonde-like matrices can be transformed into Cauchy-like matrices by using Discrete Fourier, Cosine or Sine Transform matrices. In particular this allows us to propose a new fast

O
(

n
2

)

O({n^2})

Toeplitz solver GKO, which incorporates partial pivoting. A large set of numerical examples showed that GKO demonstrated stable numerical behavior and can be recommended for solving linear systems, especially with nonsymmetric, indefinite and ill-conditioned positive definite Toeplitz matrices. It is also useful for block Toeplitz and mosaic Toeplitz (Toeplitz-block) matrices. The algorithms proposed in this paper suggest an attractive alternative to look-ahead approaches, where one has to jump over ill-conditioned leading submatrices, which in the worst case requires

O
(

n
3

)

O({n^3})

operations.