bluuebunny/crossref_metadata_2025 huggingface dataset:
A linearly implicit finite-difference scheme for the one-dimensional porous medium equation
Authors: Hoff David
Published: June 2010

We present and analyze a linearly implicit finite-difference scheme for computing approximate solutions and interface curves for the porous medium equation in one space variable. Our scheme requires only that linear, tridiagonal systems of equations be solved at each time step. We derive error bounds for the approximate interface curves as well as for the approximate solutions under the rather mild mesh condition

Δ
t

/

Δ
x
⩽

constant

\Delta t/\Delta x \leqslant {\text {constant}}

.
On the rate of convergence for the approximation of nonlinear problems
Authors: Descloux J. , Rappaz J. , Scholz R.
Published: June 2010

This paper shows how to obtain from estimates on linear problems error bounds in various norms for the approximation of nonlinear problems. The theory developed in this paper is applied to finite element methods for approximating the problem

−
Δ
u
=
λ

e
u

- \Delta u = \lambda {e^u}

and the Navier-Stokes equations.
Stability of finite-difference models containing two boundaries or interfaces
Authors: Trefethen Lloyd N.
Published: June 2010

It is known that the stability of finite-difference models of hyperbolic initial-boundary value problems is connected with the propagation and reflection of parasitic waves. Here the waves point of view is applied to models containing two boundaries or interfaces, where repeated reflection of trapped wave packets is a potential new source of instability. Our analysis accounts for various known instability phenomena in a unified way and leads to several new results, three of which are as follows. (1) Dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface. (2) Algebraic "GKS instabilities" can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present. (3) "GKS-stability" and "P-stability" can be established in certain problems by showing that the numerical reflection coefficient matrices have norm less than one.
Aitken sequences and generalized Fibonacci numbers
Authors: McCabe J. H. , Phillips G. M.
Published: June 2010

Consider the sequence(vn)({v_n})generated byvn+1=avn−bvn−1{v_{n + 1}} = a{v_n} - b{v_{n - 1}},n⩾2n \geqslant 2, wherev1=1{v_1} = 1,v2=a{v_2} = a, withaandbreal, of which the Fibonacci sequence is a special case. It is shown that if Aitken acceleration is used on the sequence(xn)({x_n})defined byxn=vn+1/vn{x_n} = {v_{n + 1}}/{v_n}, the resulting sequence is a subsequence of(xn)({x_n}). Second, if Newton’s method and the secant method are used (with suitable starting values) to solve the equationx2−ax+b=0{x^2} - ax + b = 0, then the sequences obtained from both of those methods are also subsequences of the original sequence.
On cyclic cubic fields
Authors: Ennola Veikko , Turunen Reino
Published: June 2010

A table of class numbers and units in cyclic cubic fields with conductor

>
4000

> 4000

has been given by Marie-Nicole Gras [6]. The authors have constructed an extended table for conductor

>
16000

> 16000

. The article comprises lists of fields with totally positive fundamental units and fields in which the class group has a Sylow p-subgroup which is not elementary abelian. We also give statistics about the distribution of class numbers.
A counterexample to a conjecture of Mahler on best 𝑝-adic Diophantine approximation constants
Authors: Deanin Alice A.
Published: June 2010

In 1940, Mahler proposed a conjecture regarding the value of best P-adic Diophantine approximation constants. In this paper, a computational technique which tests the conjecture for any particular P is described. A computer search verified the conjecture for all

P
⩽
101

P \leqslant 101

, except 83. The case P = 83 is discussed. A counterexample is given.
A moving mesh numerical method for hyperbolic conservation laws
Authors: Lucier Bradley J.
Published: June 2010

We show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in

L
1

(

R

)

{L^1}({\mathbf {R}})

to within

O
(

N

−
2

)

O({N^{ - 2}})

by a piecewise linear function with

O
(
N
)

O(N)

nodes; the nodes are moved according to the method of characteristics. We also show that a previous method of Dafermos, which uses piecewise constant approximations, is accurate to

O
(

N

−
1

)

O({N^{ - 1}})

. These numerical methods for conservation laws are the first to have proven convergence rates of greater than

O
(

N

−
1

/

2

)

O({N^{ - 1/2}})

.
Convergence of product integration rules for functions with interior and endpoint singularities over bounded and unbounded intervals
Authors: Lubinsky D. S. , Sidi Avram
Published: June 2010

The convergence of product integration rules, based on Gaussian quadrature points, is investigated for functions with interior and endpoint singularities over bounded and unbounded intervals. The investigation is based on a new convergence result for Lagrangian interpolation and Gaussian quadrature of singular integrands.
An efficient algorithm for obtaining the volume of a special kind of pyramid and application to convex polyhedra
Authors: Speevak Ted
Published: June 2010

An efficient method is given for obtaining the volume of a pyramid of species

n
−
2

n - 2

whose base is a convex polygon. The pyramid is "transformed" into a simplex whose volume is computed directly. A refinement is provided to the Cohen-Hickey method for determining volumes of convex polyhedra.
On weighted Chebyshev-type quadrature formulas
Authors: Förster Klaus-Jürgen , Ostermeyer Georg-Peter
Published: June 2010

A weighted quadrature formula is of Chebyshev type if it has equal coefficients and real (but not necessarily distinct) nodes. For a given weight function we study the set

T
(
n
,
d
)

T(n,d)

consisting of all Chebyshev-type formulas with n nodes and at least degree d. It is shown that in nonempty

T
(
n
,
d
)

T(n,d)

there exist two special formulas having "extremal" properties. This result is used to prove uniqueness and further results for E-optimal Chebyshev-type formulas. For the weight function

w
≡
1

w \equiv 1

, numerical investigations are carried out for

n
⩽
25

n \leqslant 25

.
A conjecture of Frobenius and the sporadic simple groups. II
Authors: Yamaki Hiroyoshi
Published: June 2010

A conjecture of Frobenius which has been reduced to the classification of finite simple groups is verified for the sporadic simple groups.
Construction of elliptic curves with large rank
Authors: Kretschmer Thomas J.
Published: June 2010

We describe an algorithm for finding elliptic curves over Q with large rank and nontrivial torsion group. In particular, an example of a curve of rank exactly 10 with a point of order 2 is given. This method seems to suggest that the rank may be large independently of the torsion group.
A lower bound for the class number of certain cubic number fields
Authors: Lettl Günter
Published: June 2010

Let K be a cyclic number field with generating polynomial
\[

X
3

−

a
−
3

2

X
2

−

a
+
3

2

X
−
1

{X^3} - \frac {{a - 3}}{2}{X^2} - \frac {{a + 3}}{2}X - 1

\]
and conductor m. We will derive a lower bound for the class number of these fields and list all such fields with prime conductor

m
=
(

a
2

+
27
)

/

4

m = ({a^2} + 27)/4

or

m
=
(
1
+
27

b
2

)

/

4

m = (1 + 27{b^2})/4

and small class number.
On Runge-Kutta methods for parabolic problems with time-dependent coefficients
Authors: Karakashian Ohannes A.
Published: June 2010

Galerkin fully discrete approximations for parabolic equations with time-dependent coefficients are analyzed. The schemes are based on implicit Runge-Kutta methods, and are coupled with preconditioned iterative methods to approximately solve the resulting systems of linear equations. It is shown that for certain classes of Runge-Kutta methods, the fully discrete equations exhibit parallel features that can be exploited to reduce the final execution time to that of a low-order method.
Some asymptotic properties of Padé approximants to 𝑒^{-𝑥}
Authors: Underhill C.
Published: June 2010

The method of matched asymptotic expansions is used to analyze the asymptotic behavior of the real zeros of, and error incurred by, Padé approximants to

e

−
x

{e^{ - x}}

. These approximants are of interest because of their application in solving systems of ordinary differential equations arising from mathematical models of physical processes, for example, the heat equation.
Symmetric FFTs
Authors: Swarztrauber Paul N.
Published: June 2010

In this paper, we examine the FFT of sequences

x
n

=

x

N
+
n

{x_n} = {x_{N + n}}

with period N that satisfy certain symmetric relations. It is known that one can take advantage of these relations in order to reduce the computing time required by the FFT. For example, the time required for the FFT of a real sequence is about half that required for the FFT of a complex sequence. Also, the time required for the FFT of a real even sequence

x
n

=

x

N
−
n

{x_n} = {x_{N - n}}

requires about half that required for a real sequence. We first define a class of five symmetries that are shown to be "closed" in the sense that if

x
n

{x_n}

has any one of the symmetries, then no additional symmetries are generated in the course of the FFT. Symmetric FFTs are developed that take advantage of these intermediate symmetries. They do not require the traditional pre- and postprocessing associated with symmetric FFTs and, as a consequence, they are somewhat more efficient and general than existing symmetric FFTs.
Convergence of the vortex filament method
Authors: Greengard Claude
Published: June 2010

Fully discrete convergence estimates have previously been given for the three-dimensional vortex method proposed by Beale and Majda. It is shown in this paper that vortex filament methods of the kind used in practice converge, provided smooth vortex structures consisting of closed filaments are appropriately discretized, and the stretching of the discrete filaments is computed sufficiently accurately. The error estimates obtained are those of the previous theory.
Products of prime powers in binary recurrence sequences. II. The elliptic case, with an application to a mixed quadratic-exponential equation
Authors: de Weger B. M. M.
Published: June 2010

In Part I the diophantine equation

G
n

=
w

p
1

m
1

⋯

p
t

m
t

{G_n} = wp_1^{{m_1}} \cdots p_t^{{m_t}}

was studied, where

{

G
n

}

n
=
0

∞

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

is a linear binary recurrence sequence with positive discriminant. In this second part we extend this to negative discriminants. We use the p-adic and complex Gelfond-Baker theory to find explicit upper bounds for the solutions of the equation. We give algorithms to reduce those bounds, based on diophantine approximation techniques. Thus we have a method to solve the equation completely for arbitrary values of the parameters. We give an application to a quadratic-exponential equation.
Some remarks on Cohen-Lenstra heuristics
Authors: Washington Lawrence C.
Published: June 2010

Cohen and Lenstra have given a heuristic model which predicts the fraction of imaginary quadratic fields with class number divisible by a given odd prime p and of those whose class groups have a given p-rank. We show that these numbers also arise by considering the proportion of matrices in

GL
n

(

Z

/

p

Z

)

{\text {GL}_n}({\mathbf {Z}}/p{\mathbf {Z}})

with 1 as an eigenvalue and those whose 1-eigenspaces have a given dimension, then letting

n
→
∞

n \to \infty

. In the last section we discuss some relations with elliptic curves.
Class groups of number fields: numerical heuristics
Authors: Cohen H. , Martinet J.
Published: June 2010

Extending previous work of H. W. Lenstra, Jr. and the first author, we give quantitative conjectures for the statistical behavior of class groups and class numbers for every type of field of degree less than or equal to four (given the signature and the Galois group of the Galois closure). The theoretical justifications for these conjectures will appear elsewhere, but the agreement with the existing tables is quite good.
Very short primality proofs
Authors: Pomerance Carl
Published: June 2010

It is shown that every prime p has a proof of its primality of length

O
(
log
⁡
p
)

O(\log p)

multiplications modulo p.
Class numbers of the simplest cubic fields
Authors: Washington Lawrence C.
Published: June 2010

Using the "simplest cubic fields" of D. Shanks, we give a modified proof and an extension of a result of Uchida, showing how to obtain cyclic cubic fields with class number divisible by n, for any n. Using 2-descents on elliptic curves, we obtain precise information on the 2-Sylow subgroups of the class groups of these fields. A theorem of H. Heilbronn associates a set of quartic fields to the class group. We show how to obtain these fields via these elliptic curves.
Effective primality tests for some integers of the forms 𝐴5ⁿ-1 and 𝐴7ⁿ-1
Authors: Williams H. C.
Published: June 2010

It is shown how polynomial time prime tests, which are both fast and deterministic, can be developed for many numbers of the formArn−1(r=5,7;A>rn)A{r^n} - 1\;(r = 5,7;A > {r^n}). These tests, like the Lucas-Lehmer test for the primality of the Mersenne numbers, are derived by using the properties of the Lucas functions. We exemplify these ideas by using numbers of the form2⋅10n−12 \cdot {10^n} - 1.
Quasi-optimal estimates for finite element approximations using Orlicz norms
Authors: Durán Ricardo G.
Published: June 2010

We consider the approximation by linear finite elements of the solution of the Dirichlet problem

−
Δ
u
=
f

- \Delta u = f

. We obtain a relation between the error in the infinite norm and the error in some Orlicz spaces. As a consequence, we get quasi-optimal uniform estimates when u has second derivatives in the Orlicz space associated with the exponential function. This estimate contains, in particular, the case where f belongs to

L
∞

{L^\infty }

and the boundary of the domain is regular. We also show that optimal order estimates are valid for the error in this Orlicz space provided that u be regular enough.
Piecewise cubic curve-fitting algorithm
Authors: Yan Zheng
Published: June 2010

We give a piecewise cubic curve fitting algorithm which preserves monotonicity of the data. The algorithm has a higher order of convergence than the Fritsch-Carlson algorithm and is simpler than the Eisenstat-Jackson-Lewis algorithm.
Asymptotic expansions of integrals of certain rapidly oscillating functions
Authors: Banerjee U. , Lardy L. J. , Lutoborski A.
Published: June 2010

An expansion in terms of powers of

m

−
1

{m^{ - 1}}

is given for integrals of the form

∫
0
1

f
(
x
)

w
¯

(
m
x
)

d
x

\smallint _0^1f(x)\bar w(mx)\,dx

, where m is a positive integer,

w
¯

(
m
x
)

\bar w(mx)

is an integrable rapidly oscillating function with period

m

−
1

{m^{ - 1}}

, and

f
(
x
)

f(x)

is a smooth function.
Some summation formulas for the series ₃𝐹₂(1)
Authors: Lavoie J.-L.
Published: June 2010

Summation formulas, contiguous to Watson’s and Whipple’s theorems in the theory of the generalized hypergeometric series, are obtained. Certain limiting cases of these results are given.
A method for computing the Iwasawa 𝜆-invariant
Authors: Ernvall R. , Metsänkylä T.
Published: June 2010

We present a method for computing the minus-part of the Iwasawa

λ
\lambda

-invariant of an Abelian field K. Applying this method, we have computed

λ
−

{\lambda ^ - }

for several odd primes p when K runs through a large number of quadratic extensions of the pth cyclotomic field. A report on these computations and an analysis of the results is included.
New convergence estimates for multigrid algorithms
Authors: Bramble James H. , Pasciak Joseph E.
Published: June 2010

In this paper, new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems. Our theory relates the convergence of multigrid algorithms to a "regularity and approximation" parameter

α
∈
(
0
,
1
]

\alpha \in (0,1]

and the number of relaxations m. We show that for the symmetric and nonsymmetric

V

\mathcal {V}

cycles, the multigrid iteration converges for any positive m at a rate which deteriorates no worse than

1
−
c

j

−
(
1
−
α
)

/

α

1 - c{j^{ - (1 - \alpha )/\alpha }}

, where j is the number of grid levels. We then define a generalized

V

\mathcal {V}

cycle algorithm which involves exponentially increasing (for example, doubling) the number of smoothings on successively coarser grids. We show that the resulting symmetric and nonsymmetric multigrid iterations converge for any

α
\alpha

with rates that are independent of the mesh size. The theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods.
A third-order boundary condition for the exterior Stokes problem in three dimensions
Authors: Guirguis Georges H.
Published: June 2010

We approximate the Stokes operator on an exterior domain in three dimensions by a truncated problem on a finite subdomain. A third-order artificial boundary condition is introduced. We discuss the approximating behavior of the truncated problem and its discretization in a finite element space. Combined errors arising from truncation and discretization are considered.
Convergence theorem for difference approximations of hyperbolic quasilinear initial-boundary value problems
Authors: Michelson Daniel
Published: June 2010

Dissipative difference approximations to multi-dimensional hyperbolic quasi-linear initial-boundary value problems are considered. The difference approximation is assumed to be consistent with the differential problem and its linearization should be stable in

l
2

{l_2}

. A formal asymptotic expansion to the difference solution is constructed. This expansion includes boundary and initial layers. It is proved that the expansion indeed approximates the difference solution to the required order. As a result, the difference solution converges to the differential one as the mesh size h tends to 0.
𝑃-stable Obrechkoff methods with minimal phase-lag for periodic initial value problems
Authors: Anantha Krishnaiah U.
Published: June 2010

In this paper P-stable methods of

O
(

h
6

)

O({h^6})

and

O
(

h
8

)

O({h^8})

with minimal phase-lag (frequency distortion) are derived. Numerical results for both linear and nonlinear problems are presented.
A table of elliptic integrals of the second kind
Authors: Carlson B. C.
Published: June 2010

By evaluating elliptic integrals in terms of standard R-functions instead of Legendre’s integrals, many (in one case 144) formulas in previous tables are unified. The present table includes only integrals of the first and second kinds having integrands with real singular points. The 216 integrals of this type listed in Gradshteyn and Ryzhik’s table constitute a small fraction of the special cases of 13 integrals evaluated here. The interval of integration is not required, as it is in previous tables, to begin or end at a singular point of the integrand. Fortran codes for the standard R-functions are included in a Supplement.
Asymptotic expansions of multiple integrals of rapidly oscillating functions
Authors: Iwaniec T. , Lutoborski A.
Published: June 2010

Expansions of multiple integrals
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript a 1 Superscript b 1 Baseline midline-horizontal-ellipsis integral Subscript a Subscript n Baseline Superscript b Subscript n Baseline w left-parenthesis sigma 1 x 1 comma ellipsis comma sigma Subscript n Baseline x Subscript n Baseline right-parenthesis g left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis d x 1 midline-horizontal-ellipsis d x Subscript n Baseline comma">

∫

a
1

b
1

⋯

∫

a
n

b
n

w
(

σ
1

x
1

,
…
,

σ
n

x
n

)
g
(

x
1

,
…
,

x
n

)

d

x
1

⋯
d

x
n

,

\int _{{a_1}}^{{b_1}} \cdots \int _{{a_n}}^{{b_n}} {w({\sigma _1}{x_1}, \ldots ,{\sigma _n}{x_n})g({x_1}, \ldots ,{x_n})\;d{x_1} \cdots d{x_n},}

\]
where w is a function on

R

n

{{\mathbf {R}}^n}

which is

(

b
k

−

a
k

)

({b_k} - {a_k})

-periodic in the kth variable,

k
=
1
,
…
,
n

k = 1, \ldots ,n

, and g is smooth, are given in terms of negative powers of the integers

σ
1

,
…
,

σ
n

{\sigma _1}, \ldots ,{\sigma _n}

. Estimates of the remainder term in the expansion are also given.
The sum of like powers of the zeros of the Riemann zeta function
Authors: Lehmer D. H.
Published: June 2010

In this paper we discuss a method of evaluating the sum

σ
r

=
∑

ρ

−
r

{\sigma _r} = \sum {{\rho ^{ - r}}}

where r is an integer greater than 1 and the sum is taken over all the complex zeros of

ζ
(
s
)

\zeta (s)

, the Riemann zeta function. The method requires the coefficients of the Maclaurin expansion of the entire function

f
(
s
)
=
(
s
−
1
)
ζ
(
s
)

f(s) = (s - 1)\zeta (s)

. These are obtained from a limit theorem of Sitaramachandrarao by the use of the Euler-Maclaurin summation formula. The sum

σ
r

{\sigma _r}

is then obtained from the logarithmic derivative of the function

f
(
s
)

f(s)

. A table of

σ
r

{\sigma _r}

is given to 30 decimals for

r
=
2
(
1
)
26

r = 2(1)26

.
Numerical results on the transcendence of constants involving 𝜋,𝑒, and Euler’s constant
Authors: Bailey David H.
Published: June 2010

Let

x
=
(

x
1

,

x
2

,
…
,

x
n

)

x = ({x_1},{x_2}, \ldots ,{x_n})

be a vector of real numbers. x is said to possess an integer relation if there exist integers

a
i

{a_i}

such that

a
1

x
1

+

a
2

x
2

+
⋯
+

a
n

x
n

=
0

{a_1}{x_1} + {a_2}{x_2} + \cdots + {a_n}{x_n} = 0

. Recently, Ferguson and Forcade discovered practical algorithms [7], [8], [9] which, for any n, either find a relation if one exists or else establish bounds within which no relation can exist. One obvious application of these algorithms is to determine whether or not a given computed real number satisfies any algebraic equation with integer coefficients (where the sizes of the coefficients are within some bound). The recursive form of the Ferguson-Forcade algorithm has been implemented with multiprecision arithmetic on the Cray-2 supercomputer at NASA Ames Research Center. The resulting computer program has been used to probe the question of whether or not certain constants involving

π
\pi

, e, and

γ
\gamma

satisfy any simple polynomial equations. These computations established that the following constants cannot satisfy any algebraic equation of degree eight or less with integer coefficients whose Euclidean norm is

10
9

{10^9}

or less:

e

/

π

e/\pi

,

e
+
π

e + \pi

,

log
e

π

{\log _e}\pi

,

γ
\gamma

,

e
γ

{e^\gamma }

,

γ

/

e

\gamma /e

,

γ

/

π

\gamma /\pi

, and

log
e

γ

{\log _e}\gamma

. Stronger results were obtained in several cases. These computations thus lend credence to the conjecture that each of the above mathematical constants is transcendental.
On optimal GFSR pseudorandom number generators
Authors: Tezuka Shu
Published: June 2010

It is shown that in

t
(
≥
4
)

t( \geq 4)

dimensions no optimal GFSR generators exist.
Quintic polynomials and real cyclotomic fields with large class numbers
Authors: Schoof René , Washington Lawrence C.
Published: June 2010

We study a family of quintic polynomials discoverd by Emma Lehmer. We show that the roots are fundamental units for the corresponding quintic fields. These fields have large class numbers and several examples are calculated. As a consequence, we show that for the prime

p
=
641491

p = 641491

the class number of the maximal real subfield of the pth cyclotomic field is divisible by the prime 1566401. In an appendix we give a characterization of the "simplest" quadratic, cubic and quartic fields.
An explicit modular equation in two variables for 𝑄(√3)
Authors: Cohn Harvey , Deutsch Jesse
Published: June 2010

A system of modular equations of norm 2 had been found for the Hilbert modular function field of

Q

(

2

)

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

in an earlier issue of this journal. Here an analogous system is found for

Q

(

3

)

{\mathbf {Q}}(\sqrt 3 )

but with the help of MACSYMA. There are special difficulties in the fact that two spaces of Hilbert modular functions exist for

Q

(

3

)

{\mathbf {Q}}(\sqrt 3 )

that can be interchanged by the modular equations. The equations are also a remarkable example of hidden symmetries in the algebraic manifold

V

2

{{\mathbf {V}}_2}

which is defined in

C

4

{{\mathbf {C}}^4}

by the modular equation.
On totally real cubic fields with discriminant 𝐷<10⁷
Authors: Llorente Pascual , Quer Jordi
Published: June 2010

The authors have constructed a table of the 592923 nonconjugate totally real cubic number fields of discriminant

D
>

10
7

D > {10^7}

, thereby extending the existing table of fields with

D
>
5
×

10
5

D > 5 \times {10^5}

constructed by Ennola and Turunen [4]. Each field is given by its discriminant and the coefficients of a generating polynomial. The method used is an improved version of the method developed in [8]. The article contains an exposition of the modified method, statistics and examples. The decomposition of the rational primes is studied and the relative frequency of each type of decomposition is compared with the corresponding density given by Davenport and Heilbronn [2].
Convergence results for invariant curve algorithms
Authors: van Veldhuizen M.
Published: June 2010

In this paper a convergence result for the algorithm described by Kevrekidis et al. [7] is given. It is shown that this algorithm for the approximation of an invariant curve converges provided the curve is attracting. The approximation error is estimated. Numerical examples for three different algorithms in this class and a closely related one illustrate the theory.
Spectral methods and a maximum principle
Authors: Canuto Claudio
Published: June 2010

Various spectral Chebyshev approximations of a model boundary layer problem for both a Helmholtz and an advection-diffusion operator are considered. It is assumed that simultaneously the boundary layer width tends to zero and the resolution power of the numerical method tends to infinity. The behavior of the spectral solutions in the frequency space and in the physical space is investigated. Error estimates are derived.
A third-order accurate variation nonexpansive difference scheme for single nonlinear conservation laws
Authors: Sanders Richard
Published: June 2010

It was widely believed that all variation nonexpansive finite difference schemes for single conservation laws must reduce to first-order at extreme points of the approximation. It is shown here that this belief is in fact false. A third-order scheme, which at worst may reduce to second order at extreme points, is developed and analyzed. Moreover, extensive numerical experiments indicate that the third-order scheme introduced here yields superior approximations when compared with other variation nonexpansive difference schemes.
A fast algorithm for linear complex Chebyshev approximations
Authors: Tang Ping Tak Peter
Published: June 2010

We propose a new algorithm for finding best minimax polynomial approximations in the complex plane. The algorithm is the first satisfactory generalization of the well-known Remez algorithm for real approximations. Among all available algorithms, ours is the only quadratically convergent one. Numerical examples are presented to illustrate rapid convergence.
The 𝑝-version of the finite element method for constraint boundary conditions
Authors: Babuška I. , Suri Manil
Published: June 2010

The paper addresses the implementation of general constraint boundary conditions for a system of equations by the p-version of the finite element method. By constraint boundary conditions we mean conditions where some relation between the components is prescribed at the boundary. Optimal error bounds are proven.
Sinc-Nyström method for numerical solution of one-dimensional Cauchy singular integral equation given on a smooth arc in the complex plane
Authors: Bialecki Bernard , Stenger Frank
Published: June 2010

We develop a numerical method based on Sinc functions to obtain an approximate solution of a one-dimensional Cauchy singular integral equation (CSIE) over an arbitrary, smooth, open arc L of finite length in the complex plane. At the outset, we reduce the CSIE to a Fredholm integral equation of the second kind via a regularization procedure. We then obtain an approximate solution to the Fredholm integral equation by means of Nyström’s method based on a Sinc quadrature rule. We approximate the matrix and right-hand side of the resulting linear system by an efficient method of computing the Cauchy principal value integrals. The error of an N-point approximation converges to zero at the rate

O
(

e

−
c

N

1

/

2

)

O({e^{ - c{N^{1/2}}}})

, as

N
→
∞

N \to \infty

, provided that the coefficients of the CSIE are analytic in a region D containing the arc L and satisfy a Lipschitz condition in D.
On the resolution of inhomogeneous norm form equations in two dominating variables
Authors: Gaál István
Published: June 2010

Applying Baker’s well-known method and the reduction procedure described by Baker and Davenport, we give a numerical algorithm for finding all solutions of inhomogeneous Thue equations of type\[NK/Q(x+αy+λ)=1{N_{K/Q}}(x + \alpha y + \lambda ) = 1\]in the variablesx,y∈Zx,y \in Zandλ∈ZK\lambda \in {Z_K}with⌈λ⌉>(max|x|,|y|))1/2\lceil \lambda \rceil > (\max |x|,|y|){)^{1/2}}, whereK=Q(α)K = Q(\alpha )is a totally real cubic field.
Factoring with cyclotomic polynomials
Authors: Bach Eric , Shallit Jeffrey
Published: June 2010

This paper discusses some new integer factoring methods involving cyclotomic polynomials. There are several polynomials

f
(
X
)

f(X)

known to have the following property: given a multiple of

f
(
p
)

f(p)

, we can quickly split any composite number that has p as a prime divisor. For example—taking

f
(
X
)

f(X)

to be

X
−
1

X - 1

—a multiple of

p
−
1

p - 1

will suffice to easily factor any multiple of p, using an algorithm of Pollard. Other methods (due to Guy, Williams, and Judd) make use of

X
+
1

X + 1

,

X
2

+
1

{X^2} + 1

, and

X
2

±
X
+
1

{X^2} \pm X + 1

. We show that one may take f to be

Φ
k

{\Phi _k}

, the kth cyclotomic polynomial. In contrast to the ad hoc methods used previously, we give a universal construction based on algebraic number theory that subsumes all the above results. Assuming generalized Riemann hypotheses, the expected time to factor N (given a multiple E of

Φ
k

(
p
)

{\Phi _k}(p)

) is bounded by a polynomial in k,

log
⁡
E

\log E

, and

log
⁡
N

\log N

.
First occurrence prime gaps
Authors: Young Jeff , Potler Aaron
Published: June 2010

An ongoing search for first occurrence prime gaps continues.
Comment on A.-P. Calderón’s paper: “On an inverse boundary value problem” [in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), 65–73, Soc. Brasil. Mat., Rio de Janeiro, 1980; MR0590275 (81k:35160)]
Authors: Isaacson David , Isaacson Eli L.
Published: June 2010

Calderón determined a method to approximate the conductivity

σ
\sigma

of a conducting body in

R
n

{R^n}

(for

n
≥
2

n \geq 2

) based on measurements of boundary data. The approximation is good in the

L
∞

{L_\infty }

norm provided that the conductivity is a small perturbation from a constant. We calculate the approximation exactly for the case of homogeneous concentric conducting disks in

R
2

{R^2}

with different conductivities. Here, the difference in the conductivities is the perturbation. We show that the approximation yields precise information about the spatial variation of

σ
\sigma

, even when the perturbation is large. This ability to distinguish spatial regions with different conductivities is important for clinical monitoring applications.
The exact determination of rectangle discrepancy for linear congruential pseudorandom numbers
Authors: Afflerbach Lothar , Weilbächer Rainer
Published: June 2010

Up to now, the rectangle discrepancy of linear congruential pseudorandom number generators could be exactly calculated only in some simple cases for a small number of generated points. Here an algorithm for the exact determination of the two-dimensional rectangle discrepancy is presented which is practicable for large generators and requires less computation time. The algorithm is based on special properties of linear congruential generators.
Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems
Authors: Hundsdorfer W. H. , Verwer J. G.
Published: June 2010

In this paper an analysis will be presented for the ADI (alternating direction implicit) method of Peaceman and Rachford applied to initial-boundary value problems for partial differential equations in two space dimensions. We shall use the method of lines approach. Motivated by developments in the field of stiff nonlinear ordinary differential equations, our analysis will focus on problems where the semidiscrete system, obtained after discretization in space, satisfies a one-sided Lipschitz condition with a constant independent of the grid spacing. For such problems, unconditional stability and convergence results will be derived.
Elliptic pseudoprimes
Authors: Miyamoto I. , Murty M. Ram
Published: June 2010

LetEbe an elliptic curve overQwith complex multiplication by an order in an imaginary quadratic field. Letψn{\psi _n}denote thenth division polynomial, and letPbe a rational point ofEof infinite order. A natural numbernis called anelliptic pseudoprimeifn|ψn+1(P)n|{\psi _{n + 1}}(P)andnis composite. LetN(x)N(x)denote the number of elliptic pseudoprimes up tox. We show thatN(x)≪x(log⁡log⁡x)7/2/(log⁡x)3/2N(x) \ll x{(\log \log x)^{7/2}}/{(\log x)^{3/2}}. More generally, ifP1,…,Pr{P_1}, \ldots ,{P_r}arerindependent rational points ofEwhich have infinite order, andΓ\Gammais the subgroup generated by them, denote byNΓ(x){N_\Gamma }(x)the number of compositen≤xn \leq xsatisfyingn|ψn+1(Pi)n|{\psi _{n + 1}}({P_i}),1≤i≤r1 \leq i \leq r. Forr≥2r \geq 2, we prove<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Subscript normal upper Gamma Baseline left-parenthesis x right-parenthesis much-less-than x exp left-parenthesis minus c StartRoot left-parenthesis log x right-parenthesis left-parenthesis log log x right-parenthesis right-parenthesis EndRoot">NΓ(x)≪xexp⁡(−c(log⁡x)(log⁡log⁡x)){N_\Gamma }(x) \ll x\exp ( - c\sqrt {(\log x)(\log \log x))}for some positive constantc.
Natural continuous extensions of Runge-Kutta methods for Volterra integral equations of the second kind and their applications
Authors: Bellen A. , Jackiewicz Z. , Vermiglio R. , Zennaro M.
Published: June 2010

We consider a very general class of Runge-Kutta methods for the numerical solution of Volterra integral equations of the second kind, which includes as special cases all the more important methods which have been considered in the literature. The main purpose of this paper is to define and prove the existence of the Natural Continuous Extensions (NCE’s) of Runge-Kutta methods, i.e., piecewise polynomial functions which extend the approximation at the grid points to the whole interval of integration. The particular properties required of the NCE’s allow us to construct the tail approximations, which are quite efficient in terms of kernel evaluations.
On the convergence of product formulas for the evaluation of certain two-dimensional Cauchy principal value integrals
Authors: Mastroianni Giuseppe
Published: June 2010

The convergence of product rules of interpolatory type for the numerical evaluation of certain two-dimensional Cauchy principal value integrals is proved. Some estimates of the remainder are established for several classes of functions. Earlier results are generalized and improved.
Collocation methods for nonlinear Volterra integro-differential equations with infinite delay
Authors: Brunner Hermann
Published: June 2010

In this paper we study the numerical solution of nonlinear Volterra integro-differential equations with infinite delay by spline collocation and related Runge-Kutta type methods. The kernel function in these equations is of the form

k
(
t
,
s
,
y
(
t
)
,
y
(
s
)
)

k(t,s,y(t),y(s))

, with a representative example given by Volterra’s population equation, where we have
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k left-parenthesis t comma s comma y left-parenthesis t right-parenthesis comma y left-parenthesis s right-parenthesis right-parenthesis equals a left-parenthesis t minus s right-parenthesis dot upper G left-parenthesis y left-parenthesis t right-parenthesis comma y left-parenthesis s right-parenthesis right-parenthesis">

k
(
t
,
s
,
y
(
t
)
,
y
(
s
)
)
=
a
(
t
−
s
)
⋅
G
(
y
(
t
)
,
y
(
s
)
)

k(t,s,y(t),y(s)) = a(t - s) \cdot G(y(t),y(s))

.
Uniform enclosure of high order of boundary value problems by monotone discretization
Authors: Grossmann Ch. , Roos H.-G.
Published: June 2010

In the investigation of boundary value problems the construction of a two-sided inclusion of the solution can be as important as a numerical approximation of the solution itself. In the present paper we analyze a monotone discretization technique of higher order based upon piecewise interpolation and shifting such that bounding upper and lower solutions are obtained. The monotone discretization under consideration takes advantage of the property of the operator to be of monotone kind.
Rational Chebyshev approximations for Fermi-Dirac integrals of orders -1/2, 1/2 and 3/2
Authors: Cody W. J. , Thacher, Jr. Henry C.
Published: June 2010

Rational Chebyshev approximations are given for the complete Fermi-Dirac integrals of orders

−

1
2

- \frac {1} {2}

1
2

\frac {1} {2}

and

3
2

\frac {3} {2}

. Maximal relative errors vary with the function and interval considered, but generally range down to

10

−
9

{10^{ - 9}}

or less.
On a Diophantine equation related to perfect codes
Authors: Alter Ronald
Published: June 2010

A necessary condition for the existence of perfect double Hamming-error-correcting codes on q symbols, for q a prime power, is that the Diophantine equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartBinomialOrMatrix n Choose 0 EndBinomialOrMatrix plus StartBinomialOrMatrix n Choose 1 EndBinomialOrMatrix left-parenthesis q minus 1 right-parenthesis plus StartBinomialOrMatrix n Choose 2 EndBinomialOrMatrix left-parenthesis q minus 1 right-parenthesis squared equals q Superscript k">

(

n

0

)

+

(

n

1

)

(
q
−
1
)
+

(

n

2

)

(
q
−
1

)
2

=

q
k

\left ( {\begin {array}{*{20}{c}} n \\ 0 \\ \end {array} } \right ) + \left ( {\begin {array}{*{20}{c}} n \\ 1 \\ \end {array} } \right )(q - 1) + \left ( {\begin {array}{*{20}{c}} n \\ 2 \\ \end {array} } \right ){(q - 1)^2} = {q^k}

\]
have a nontrivial solution in positive integers. In this paper this equation is considered for all q, and by applying Newton’s method for approximating the roots of a polynomial, it is established that it has no nontrivial solutions for all n, odd k, and q of the form

q
=
2

s
2

q = 2{s^2}

.
A vector implementation of the Fast Fourier Transform algorithm
Authors: Fornberg Bengt
Published: June 2010

A recent article in this journal by D. G. Korn and J. J. Lambiotte, Jr. discusses implementations of the FFT algorithm on the CDC STAR-100 vector computer. The ’Pease’-algorithm is recommended in cases when only a few transforms can be performed simultaneously. We show how the use of a different algorithm and of trigonometric tables will lead to more than three times faster execution times. The times for large transforms increase only about 39
Type-insensitive ODE codes based on implicit 𝐴-stable formulas
Authors: Shampine L. F.
Published: June 2010

A special concept of stiffness is appropriate for implicit A -stable formulas. It is possible to recognize this kind of stiffness economically and reliably using information readily available during the integration of an ODE. Using this development, a variety of effective ODE solvers could be made insensitive to the type of problem, i.e. the code would automatically recognize and alter automatically its algorithm at any step depending on whether the problem is stiff there.
Mining metabolic pathways through gene expression
Authors: Hancock Timothy , Takigawa Ichigaku , Mamitsuka Hiroshi
Published: June 2010

Motivation: An observed metabolic response is the result of the coordinated activation and interaction between multiple genetic pathways. However, the complex structure of metabolism has meant that a compete understanding of which pathways are required to produce an observed metabolic response is not fully understood. In this article, we propose an approach that can identify the genetic pathways which dictate the response of metabolic network to specific experimental conditions.

Results: Our approach is a combination of probabilistic models for pathway ranking, clustering and classification. First, we use a non-parametric pathway extraction method to identify the most highly correlated paths through the metabolic network. We then extract the defining structure within these top-ranked pathways using both Markov clustering and classification algorithms. Furthermore, we define detailed node and edge annotations, which enable us to track each pathway, not only with respect to its genetic dependencies, but also allow for an analysis of the interacting reactions, compounds and KEGG sub-networks. We show that our approach identifies biologically meaningful pathways within two microarray expression datasets using entire KEGG metabolic networks.

Availability and implementation: An R package containing a full implementation of our proposed method is currently available from http://www.bic.kyoto-u.ac.jp/pathway/timhancock

Contact:  timhancock@kuicr.kyoto-u.ac.jp

Supplementary information:  Supplementary data are available at Bioinformatics online.
Use of annual catch limits to avoid stock depletion in the Bering Sea and Aleutian Islands management area (Northeast Pacific)
Authors: DiCosimo Jane , Methot Richard D. , Ormseth Olav A.
Published: June 2010

DiCosimo, J., Methot, R. D., and Ormseth, O. A. 2010. Use of annual catch limits to avoid stock depletion in the Bering Sea and Aleutian Islands management area (Northeast Pacific). – ICES Journal of Marine Science, 67: 1861–1865. In total, 41 fish stocks in US ocean waters continue to be fished at unsustainable levels, and 46 fish stocks are overfished. In 2006, the US Congress required the implementation of annual catch limits (ACLs) and accountability measures by 2010 to prevent overfishing, and by 2011 to recover overfished stocks. These requirements were modelled on the existing management system for Northeast Pacific groundfish, where more than 20 fish stocks and assemblages have been managed sustainably for 30 years. Science-based overfishing levels and acceptable biological catches (ABCs) have been implemented for each stock or assemblage, with buffers between the two to avoid overfishing. Total allowable catches are set at or below the acceptable biological catch. Suballocations of quotas by season, area, and gear type, along with in-season fishery closures based on extensive observer coverage and vessel monitoring, ensure that quotas are not exceeded. To comply with ACL requirements, the North Pacific Fishery Management Council has defined ABC as an ACL. We demonstrate the effectiveness of ACLs for successful management of Northeast Pacific groundfish, suggesting that their use in other US fisheries might reduce the risk of overfishing and enhance the recovery of overfished stocks.
Metastatic cutaneous squamous cell carcinoma treated successfully with surgery, radiotherapy and S‐1/cisplatin chemotherapy
Authors: MATSUSHITA Shigeto , KAWAI Kazuhiro , TADA Ko‐ichi , MERA Kentaro , KUBO Hidemichi , IBUSUKI Atsuko , YOSHII Noriko , KANEKURA Takuro
Published: June 2010

Primary cutaneous squamous cell carcinoma (SCC) is a malignant tumor that arises from keratinizing cells of the epidermis or its appendages. We present a patient with cutaneous SCC on the left instep with metastases to multiple lymph nodes in the para‐aortic, iliac and groin region. We chose a combination of surgery and concurrent chemoradiotherapy. The chemotherapeutic agent S‐1/cisplatin was selected based on results of the histoculture drug response assay. The patient responded dramatically to this multidisciplinary treatment and complete remission was achieved.
High‐resolution cytogenetic map for the African malaria vector Anopheles gambiae
Authors: George P. , Sharakhova M. V. , Sharakhov I. V.
Published: June 2010

Cytogenetic and physical maps are indispensible for precise assembly of genome sequences, functional characterization of chromosomal regions, and population genetic and taxonomic studies. We have created a new cytogenetic map for Anopheles gambiae by using a high‐pressure squash technique that increases overall band clarity. To link chromosomal regions to the genome sequence, we attached genome coordinates, based on 302 markers of bacterial artificial chromosome, cDNA clones, and PCR‐amplified gene fragments, to the chromosomal bands and interbands at approximately a 0.5–1 Mb interval. In addition, we placed the breakpoints of seven common polymorphic inversions on the map and described the chromosomal landmarks for the arm and inversion identification. The map's increased resolution can be used to further enhance physical mapping, improve genome assembly, and stimulate epigenomic studies of malaria vectors.
Elucidation of the role of activin in organogenesis using a multiple organ induction system with amphibian and mouse undifferentiated cellsin vitro
Authors: Asashima Makoto , Michiue Tatsuo , Kurisaki Akira
Published: June 2010

Studies performed over the last century have clarified the mechanisms of organ and tissue formation. Mesoderm formation is one of the most important events in early body pattern determination during embryogenesis. In 1988, we found that activin A has mesoderm‐inducing activity. As activin A could induce dorsal mesoderm formation, unlike fibroblast growth factor and bone morphogenetic protein, this factor was thought to be the molecular entity of the Spemann‐Mangold organizer. Subsequently, the mechanisms of early embryogenesis have been clarified using molecular biological techniques, resulting in the identification of many genes that are involved in organ and tissue development. This finding that activin A could induce dorsal mesoderm formation spurred research into the application of agents that induce organs and tissuesin vitro. In this regard, we have shown that many organ types can be induced by activin Ain vitro. Moreover, we have found that other types of organs can be induced by changing the conditions of treatment. To date, more than 20 different types of tissues and organs have been successfully induced fromXenopusundifferentiated cellsin vitro. In recent years, we have applied these protocols to mouse embryonic stem cells, and we have successfully induced several tissues, such as the pancreas and cardiomyocytes. We are also investigating how the pluripotency of undifferentiated stem cells is regulated. In this review, we summarize the current knowledge regarding activin as a mesoderm‐inducing factor and its application for the induction of tissues and organs from undifferentiated cells. Moreover, we provide some examples ofin vitrotissue differentiation from mouse embryonic stem cells, which may prove useful in regenerative medicine.