bluuebunny/crossref_metadata_2025 huggingface dataset:
Numerical identification of a spatially varying diffusion coefficient
Authors: Richter Gerard R.
Published: June 2010

We consider the inverse problem of identifying a spatially varying diffusion coefficient on the basis of an observed solution to the forward problem. Under appropriate conditions, this inverse problem can be solved as a first order hyperbolic problem in the unknown coefficient. We provide a modified upwind difference scheme for this hyperbolic problem and prove that its convergence rate is

O
(
h
)

O(h)

when certain conditions are met.
A mesh refinement method for 𝐴𝑥=𝜆𝐵𝑥
Authors: McCormick Stephen F.
Published: June 2010

The aim of this paper is to introduce a simple but efficient mesh refinement strategy for use with inverse iteration for finding one or a few solutions of an ordinary or partial differential eigenproblem of the form

A
x
=
λ
B
x

Ax = \lambda Bx

. The focus is upon the case where A and B are symmetric and B is positive definite, although the approaches have a very broad application. A discussion of the combined use of mesh refinement and a correction scheme multigrid technique is also provided. The methods are illustrated by numerical results from experiments with two-point boundary value problems.
Algorithms for computing the ℎ-range of the postage stamp problem
Authors: Mossige Svein
Published: June 2010

New algorithms, based on a very efficient method to compute the h-range, have been used to extend known tables of the extremal h-range, to complete the solution in the case

k
=
3

k = 3

, and to find a lower bound for the extremal 2-range.
Improving convergence rate in the method of successive approximations
Authors: Pennline James A.
Published: June 2010

An application of the method of successive approximations for obtaining the solution of a nonlinear integral equation arising from a two-point boundary value problem is illustrated. In particular, we show sufficient conditions under which the convergence rate of the sequence can be improved.
Computation of integrals with oscillatory and singular integrands
Authors: Ting Bing Yuan , Luke Yudell L.
Published: June 2010

This paper is concerned with evaluation of integrals whose integrands are oscillatory and contain singularities at the endpoints of the interval of integration. A typical form is

G
(
θ
)
=

∫
a
b

w
(
x
)

e

i
θ
x

f
(
x
)

d
x

G(\theta ) = \smallint _a^bw(x){e^{i\theta x}}f(x)\,dx

, where a and b can be finite or infinite,

θ
\theta

is a parameter which is usually large,

f
(
x
)

f(x)

is analytic in the range of integration, and the singularities are encompassed in the weight function

w
(
x
)

w(x)

. We suppose that

f
(
x
)

f(x)

can be expanded in series of polynomials which are orthogonal over the interval of integration with respect to

w
(
x
)

w(x)

. There are two such expansions for

f
(
x
)

f(x)

. One is an infinite series which follows from the usual orthogonality property. The other is a polynomial approximation plus a remainder. The relations between the coefficients in these representations are detailed and methods for the evaluation of these are analyzed. Error analyses are provided. A numerical example is given to illustrate the effectiveness of the schemes developed.
An algorithm for determining invertible quadratic isoparametric finite element transformations
Authors: Field David A.
Published: June 2010

This paper derives an algorithm which determines the invertibility of arbitrary two-dimensional quadratic isoparametric finite element transformations. Theorems verifying the algorithm and guiding the construction of invertible transformations are proven.
Interpolation on uniform meshes by the translates of one function and related attenuation factors
Authors: Locher F.
Published: June 2010

The exact Fourier coefficients

c
j

(

P
n

f
)

{c_j}({P_n}f)

are proportional to the discrete Fourier coefficients

d
j

(
n
)

(
f
)

d_j^{(n)}(f)

if

P
n

{P_n}

is a translation invariant operator which depends only on the values of f on an equidistant mesh of width

2
π

/

n

2\pi /n

. The proportionality factors which depend only on

P
n

{P_n}

but not on f are called attenuation factors and have been calculated for several operators

P
n

{P_n}

of spline type. Here we analyze first the interpolation problem which is produced by the functions

σ
(
∙
−
2
π
j

/

n
)
,
j
=
0
,
…
,
n
−
1

\sigma ( \bullet - 2\pi j/n),j = 0, \ldots ,n - 1

, where

σ
\sigma

is a suitable

2
π

2\pi

-periodic generating function. It is essential that the associated interpolation matrix is of discrete convolution type. Thus, we can derive conditions guaranteeing the unique solvability of the interpolation problem and representations of the interpolating function. Then the attenuation factors may be expressed in terms of the Fourier coefficients of

σ
\sigma

. We point especially to the case where

σ
\sigma

is a reproducing kernel in a suitable Hilbert space. Here we get attenuation factors of a new type which are generated by interpolation with analytic functions.
Approximation results for orthogonal polynomials in Sobolev spaces
Authors: Canuto C. , Quarteroni A.
Published: June 2010

We analyze the approximation properties of some interpolation operators and some

L
ω
2

L_\omega ^2

-orthogonal projection operators related to systems of polynomials which are orthonormal with respect to a weight function

ω
(

x
1

,
…
,

x
d

)

\omega ({x_1}, \ldots ,{x_d})

,

d
⩾
1

d \geqslant 1

. The error estimates for the Legendre system and the Chebyshev system of the first kind are given in the norms of the Sobolev spaces

H
ω
s

H_\omega ^s

. These results are useful in the numerical analysis of the approximation of partial differential equations by spectral methods.
On estimating the largest eigenvalue with the Lanczos algorithm
Authors: Parlett B. N. , Simon H. , Stringer L. M.
Published: June 2010

The Lanczos algorithm applied to a positive definite matrix produces good approximations to the eigenvalues at the extreme ends of the spectrum after a few iterations. In this note we utilize this behavior and develop a simple algorithm which computes the largest eigenvalue. The algorithm is especially economical if the order of the matrix is large and the accuracy requirements are low. The phenomenon of misconvergence is discussed. Some simple extensions of the algorithm are also indicated. Finally, some numerical examples and a comparison with the power method are given.
Polynomial formulation of second derivative multistep methods
Authors: Kovvali S. , Gupta G. K.
Published: June 2010

Following the work of Enright [3] there has been interest in studying second derivative methods for solving stiff ordinary differential equations. Successful implementations of second derivative methods have been reported by Enright [3], Sacks-Davis [9], [10] and Addison[l]. Wallace and Gupta [13] have suggested a polynomial formulation of the usual first-derivative multistep methods. Recently Skeel [11] has shown the equivalence of several formulations of multistep methods. The work of Wallace and Gupta [13] was extended to second derivative methods by Gupta [8]. The present work includes results obtained regarding the stability and truncation error of second derivative methods using the polynomial formulation.
Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations
Authors: Levin David
Published: June 2010

A collocation procedure for efficient integration of rapidly oscillatory functions is presented. The integration problem is transformed into a certain O.D.E. problem, and this is solved by a collocation technique. The method is also extended to two-dimensional integration, and some numerical results are appended showing the efficiency of the method in handling difficult cases of rapid irregular oscillations.
The computation of a certain metric invariant of an algebraic number field
Authors: Brunotte Horst
Published: June 2010

LetFbe an algebraic number field and denote byN(a)N(a)the absolute norm and bya~\tilde {a}the maximum of the absolute values of the conjugates of the elementaofF. DefinecF{c_F}to be the best possible constant with the property: For everya∈Fa \in Fthere exists a unituofFsuch thatua~⩽cFN(a)1/[F:Q]\widetilde {ua} \leqslant {c_F}N{(a)^{1/[F:{\mathbf {Q}}]}}. An algorithm for the computation ofcF{c_F}is described and some examples are given.
On the evaluation of double integrals
Authors: Levin Moshe
Published: June 2010

A cubature formula consisting of line integrals which is optimal on a set of functions satisfying given boundary conditions is obtained. The line integrals of this formula may be evaluated by optimal quadrature formulas. The advantage of this formula over the optimal cubature formula with a rectangular lattice of knots is shown. This approach to optimal cubatures was stimulated by the idea of blending [1], [2].
Error estimates for the multidimensional two-phase Stefan problem
Authors: Jerome Joseph W. , Rose Michael E.
Published: June 2010

In this paper we derive rates of convergence for regularizations of the multidimensional two-phase Stefan problem and use the regularized problems to define backward-difference in time and

C
0

{C^0}

piecewise-linear in space Galerkin approximations. We find an

L
2

{L^2}

rate of convergence of order

ε

\sqrt \varepsilon

in the

ε
\varepsilon

-regularization and an

L
2

{L^2}

rate of convergence of order

(

h
2

/

ε
+
Δ
t

/

ε

)

({h^2}/\varepsilon + \Delta t/\sqrt \varepsilon )

in the Galerkin estimates which leads to the natural choices

ε
∼

h

4

/

3

\varepsilon \sim {h^{4/3}}

,

Δ
t
∼

h

4

/

3

\Delta t \sim {h^{4/3}}

, and a resulting

O
(

h

2

/

3

)

L
2

O({h^{2/3}})\;{L^2}

rate of convergence of the numerical scheme to the solution of the differential equation. An essentially

O
(
h
)

O(h)

rate is demonstrated when

ε
=
0

\varepsilon = 0

and

Δ
t
∼

h
2

\Delta t \sim {h^2}

in our Galerkin scheme under a boundedness hypothesis on the Galerkin approximations. The latter result is consistent with computational experience.
A method for interpolating scattered data based upon a minimum norm network
Authors: Nielson Gregory M.
Published: June 2010

A method for interpolating scattered data is described. Given

(

x
i

,

y
i

,

z
i

)
,
i
=
1
,
…
,
N

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

, a bivariate function S with continuous first order partial derivatives is defined which has the property that

S
(

x
i

,

y
i

)
=

z
i

,
i
=
1
,
…
,
N

S({x_i},{y_i}) = {z_i},i = 1, \ldots ,N

. The method is based upon a triangulation of the domain and a curve network which has certain minimum pseudonorm properties. Algorithms and examples are included.
A general orthogonalization technique with applications to time series analysis and signal processing
Authors: Cybenko George
Published: June 2010

A new orthogonalization technique is presented for computing theQRfactorization of a generaln×pn \times pmatrix of full rankp(n⩾p)p\,(n \geqslant p). The method is based on the use of projections to solve increasingly larger subproblems recursively and has anO(np2)O(n{p^2})operation count for general matrices. The technique is readily adaptable to solving linear least-squares problems. If the initial matrix has a circulant structure the algorithm simplifies significantly and gives the so-called lattice algorithm for solving linear prediction problems. From this point of view it is seen that the lattice algorithm is really an efficient way of solving specially structured least-squares problems by orthogonalization as opposed to solving the normal equations by fast Toeplitz algorithms.
Numerical methods for flows through porous media. I
Authors: Rose Michael E.
Published: June 2010

The degenerate parabolic equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential u Over partial-differential t EndFraction equals nabla dot left-parenthesis StartAbsoluteValue u EndAbsoluteValue Superscript nu Baseline nabla u right-parenthesis comma nu greater-than-or-slanted-equals 1 comma">

∂
u

∂
t

=
∇
⋅

(

|

u

|

ν

∇
u

)

,

ν
⩾
1
,

\frac {{\partial u}}{{\partial t}} = \nabla \cdot \left ( {|u{|^\nu }\nabla u} \right ),\quad \nu \geqslant 1,

\]
has been used to model the flow of gas through a porous medium. Error estimates for continuous and discrete time finite element procedures to approximate the solution of this equation are proved, and several new regularity results are given.
The numerical solution of equality constrained quadratic programming problems
Authors: Dyn Nira , Ferguson Warren E.
Published: June 2010

This paper proves that a large class of iterative schemes can be used to solve a certain constrained minimization problem. The constrained minimization problem considered involves the minimization of a quadratic functional subject to linear equality constraints. Among this class of convergent iterative schemes are generalizations of the relaxed Jacobi, Gauss-Seidel, and symmetric Gauss-Seidel schemes.
The accurate numerical solution of highly oscillatory ordinary differential equations
Authors: Scheid Robert E.
Published: June 2010

An asymptotic theory for weakly nonlinear, highly oscillatory systems of ordinary differential equations leads to methods which are suitable for accurate computation with large time steps. The theory is developed for systems of the form\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row bold upper Z prime equals left-parenthesis upper A left-parenthesis t right-parenthesis slash epsilon right-parenthesis bold upper Z plus bold upper H left-parenthesis bold upper Z comma t right-parenthesis comma 2nd Row bold upper Z left-parenthesis 0 comma epsilon right-parenthesis equals bold upper Z 0 comma 0 greater-than t greater-than upper T comma 0 greater-than epsilon much-less-than 1 comma EndLayout">Z′=(A(t)/ε)Z+H(Z,t),Z(0,ε)=Z0,0>t>T,0>ε≪1,\begin {array}{*{20}{c}} {{\mathbf {Z}}’= (A(t)/\varepsilon ){\mathbf {Z}} + {\mathbf {H}}({\mathbf {Z}},t),} \hfill \\ {{\mathbf {Z}}(0,\varepsilon ) = {{\mathbf {Z}}_0},\quad 0 > t > T,0 > \varepsilon \ll 1,} \hfill \\ \end {array}\]where the diagonal matrixA(t)A(t)has smooth, purely imaginary eigenvalues and the components ofH(Z,t){\mathbf {H}}({\mathbf {Z}},t)are polynomial in the components ofZwith smootht-dependent coefficients. Computational examples are presented.
Analysis of mixed finite elements methods for the Stokes problem: a unified approach
Authors: Stenberg Rolf
Published: June 2010

We develop a method for the analysis of mixed finite element methods for the Stokes problem in the velocity-pressure formulation. A technical "macroelement condition", which is sufficient for the classical Babuška-Brezzi inequality to be valid, is introduced. Using this condition,we are able to verify the stability, and optimal order of convergence, of several known mixed finite element methods.
A series expansion for the first positive zero of the Bessel functions
Authors: Piessens R.
Published: June 2010

It is shown that the first positive zero

j

v
,
l

{j_{v,l}}

of the Bessel function

J
v

(
x
)

{J_v}(x)

is given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j Subscript v comma l Baseline equals 2 left-parenthesis v plus 1 right-parenthesis Superscript 1 slash 2 Baseline left-bracket 1 plus StartFraction left-parenthesis v plus 1 right-parenthesis Over 4 EndFraction minus StartFraction 7 left-parenthesis v plus 1 right-parenthesis squared Over 96 EndFraction plus StartFraction 49 left-parenthesis v plus 1 right-parenthesis cubed Over 1152 EndFraction minus StartFraction 8363 left-parenthesis v plus 1 right-parenthesis Superscript 4 Baseline Over 276480 EndFraction plus midline-horizontal-ellipsis right-bracket">

j

v
,
l

=
2

(
v
+
1

)

1

/

2

[

1
+

(
v
+
1
)

4

−

7

(
v
+
1
)

2

96

+

49

(
v
+
1
)

3

1152

−

8363

(
v
+
1
)

4

276480

+
⋯

]

{j_{v,l}} = 2{(v + 1)^{1/2}}\left [ {1 + \frac {{(v + 1)}}{4} - \frac {{7{{(v + 1)}^2}}}{{96}} + \frac {{49{{(v + 1)}^3}}}{{1152}} - \frac {{8363{{(v + 1)}^4}}}{{276480}} + \cdots } \right ]

\]
for

−
1
>
v
>
0

- 1 > v > 0

.
Primality testing and Jacobi sums
Authors: Cohen H. , Lenstra H. W.
Published: June 2010

We present a theoretically and algorithmically simplified version of a primality testing algorithm that was recently invented by Adleman and Rumely. The new algorithm performs well in practice. It is the first primality test in existence that can routinely handle numbers of hundreds of decimal digits.
Polynomial interpolation: Lagrange versus Newton
Authors: Werner Wilhelm
Published: June 2010

We show that the Lagrangian form of the interpolating polynomial can be calculated with the same number of arithmetic operations as the Newtonian form.
The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme
Authors: Tadmor Eitan
Published: June 2010

We study the Lax-Friedrichs scheme, approximating the scalar, genuinely nonlinear conservation law

u
t

+

f
x

(
u
)
=
0

{u_t} + {f_x}(u) = 0

, where

f
(
u
)

f(u)

is, say, strictly convex,

f
¨

⩾

a
˙

∗

>
0

\ddot f \geqslant {\dot a_ \ast } > 0

. We show that the divided differences of the numerical solution at time t do not exceed

2

(
t

a
˙

∗

)

−
1

2{(t{\dot a_ \ast })^{ - 1}}

. This one-sided Lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case; in particular, it is independent of the initial amplitude, in sharp contrast to linear problems. It guarantees the entropy compactness of the scheme in this case, as well as providing a quantitative insight into the large-time behavior of the numerical computation.
Rational approximations for the Fresnel integrals
Authors: Heald Mark A.
Published: June 2010

A class of simple rational polynomial approximations for the Fresnel integrals is given with maximum errors from

1.7
×

10

−
3

1.7 \times {10^{ - 3}}

down to

4
×

10

−
8

4 \times {10^{ - 8}}

. The domain

[
0
,
∞
]

[0,\infty ]

is not subdivided. The format is particularly convenient for programmable hand calculators and microcomputer subroutines.
Modular multiplication without trial division
Authors: Montgomery Peter L.
Published: June 2010

Let

N
>
1

N > 1

. We present a method for multiplying two integers (called N-residues) modulo N while avoiding division by N. N-residues are represented in a nonstandard way, so this method is useful only if several computations are done modulo one N. The addition and subtraction algorithms are unchanged.
Stability of interfaces with mesh refinement
Authors: Berger Marsha J.
Published: June 2010

We study the stability of mesh refinement in space and time for several different interface equations and finite-difference approximations. First, we derive a root condition which implies stability for the initial-boundary value problem for this type of interface. From the root condition, we prove the stability of several interface equations using the maximum principle. In some cases, the final verification steps can be done analytically; in other cases, a simple computer program has been written to check the condition for values of a parameter along the boundary of the unit circle. Using this method, we prove stability for Lax-Wendroff with all the interface conditions considered, and for Leapfrog with interpolation interface conditions when the fine and coarse grids overlap.
A stable algorithm for updating triangular factors under a rank one change
Authors: Fletcher R. , Matthews S. P. J.
Published: June 2010

An algorithm is presented for updating the LU factors of an

n
×
n

n \times n

matrix A, when A is changed by a matrix of rank one. The algorithm is based on the repeated use of triangular operations, and stability is obtained by allowing both row and column pivoting. The cost of the algorithm is approximately proportional to the maximum permitted depth for the pivot search. For well-conditioned matrices a maximum depth of 3 is sufficient to ensure stability. For substantially rank deficient matrices it is theoretically possible that pivots of any depth may be required, but in practice we find that a value of 5 is adequate. We suggest a pivot strategy, based on minimizing a growth bound, which penalizes deep pivots and imposes a maximum depth of pivot through a default value. On well-behaved problems the asymptotic cost of the update is observed to be approximately

2.6

n
2

2.6{n^2}

compared with

8

n
2

8{n^2}

(or worse) for updating orthogonal factors. Given the accuracy obtained by the new algorithm, we feel that there are many applications in which the lower cost of triangular factors can be exploited. Comparison with ab initio factorization indicates that for

n
⩾
10

n \geqslant 10

updating triangular factors is advantageous.
On real cyclic sextic fields
Authors: Ennola V. , Mäki S. , Turunen R.
Published: June 2010

A table of units and class numbers of real cyclic sextic fields with conductorf6⩽2021{f_6} \leqslant 2021has been given by the second author [13]. We first fill in the gaps in [13] and then construct an extended table for2021>f6>40002021 > {f_6} > 4000. The article contains results about Galois module structure of the unit group, relative norms of the units, and ideal classes of the subfields becoming principal in the sextic field. The connection with Leopoldt’s theory [11] is described. A parametric family of fields containing exceptional units [14] is constructed. We give statistics referring to class numbers of fields with prime conductor, the appearance of units of different types if the relative class number is>1> 1, Leopoldt’s unit index, and the signature rank of the unit group.
Finite element solution of the fundamental equations of semiconductor devices. I
Authors: Zlámal Miloš
Published: June 2010

We investigate the nonstationary equations of the semiconductor device theory consisting of a Poisson equation for the electric potential

ψ
\psi

and of two highly nonlinear continuity equations for carrier densities n and p. We use simplicial elements with linear polynomials and four-node two-dimensional and eight-node three-dimensional isoparametric elements. There are constructed finite element solutions such that the current densities

J

n

{{\mathbf {J}}_n}

,

J

p

{{\mathbf {J}}_p}

and the electric field strength

‖

∇
ψ

‖

\left \| {\nabla \psi } \right \|

are constant on each element. Two schemes are proposed: one is nonlinear, the other is partly linear. The schemes preserve the property of the exact solution (corresponding to the physical meaning) that the carrier densities n and p are positive. Existence of the solution is proved in both cases, unicity in the second case. A subsequent paper II will be devoted to problems of stability and convergence.
Approximate solution of boundary value problems on infinite intervals by collocation methods
Authors: Schmeiser Christian
Published: June 2010

The numerical solution of boundary value problems for ordinary differential equations on infinite intervals is considered. The infinite interval is cut at a finite, large enough point and additional boundary conditions are posed there. For the solution of the resulting problem, A-stable symmetric collocation methods are employed. Using the behavior of the solution of the "infinite" problem, meshes are defined which avoid an unreasonably high number of meshpoints. Stability and convergence of the resulting schemes are shown.
Newton’s method for the matrix square root
Authors: Higham Nicholas J.
Published: June 2010

One approach to computing a square root of a matrix A is to apply Newton’s method to the quadratic matrix equation

F
(
X
)
≡

X
2

−
A
=
0

F(X) \equiv {X^2} - A = 0

. Two widely-quoted matrix square root iterations obtained by rewriting this Newton iteration are shown to have excellent mathematical convergence properties. However, by means of a perturbation analysis and supportive numerical examples, it is shown that these simplified iterations are numerically unstable. A further variant of Newton’s method for the matrix square root, recently proposed in the literature, is shown to be, for practical purposes, numerically stable.
Tables of the roots of the associated Legendre function with respect to the degree
Authors: Bauer Helmut F.
Published: June 2010

Tables of the numerical value of the roots with respect to the degree of the Legendre functions of the first kind and those of the first derivative are presented.
284457 Steiner triple systems of order 19 contain a subsystem of order 9
Authors: Stinson D. R. , Seah E.
Published: June 2010

In this paper, we enumerate the (nonisomorphic) Steiner triple systems of order 19 which contain a subsystem of order 9. The number of these designs is precisely 284457. We also determine which of these designs also contain at least one subsystem of order 7, and how many. Exactly 13529 of them contain at least one subsystem of order 7.
Discontinuous polynomial approximations in the theory of one-step, hybrid and multistep methods for nonlinear ordinary differential equations
Authors: Delfour M. C. , Dubeau F.
Published: June 2010

This paper studies the approximation of the solution of nonlinear ordinary differential equations by (discontinuous) piecewise polynomials of degree K and traces at the nodes of discretization. A mesh-dependent variational framework underlying this discontinuous approximation is derived. Several families of one-step, hybrid and multistep schemes are obtained. It is shown that the convergence rate in the

L
2

{L^2}

-norm is

K
+
1

K + 1

. The nodal-convergence rate can go up to

2
K
+
2

2K + 2

, depending on the particular scheme under consideration. The mesh-dependent variational framework introduced here is of special interest in the approximation of the solution of optimal control problems governed by differential equations.
Explicit, nearly optimal, linear rational approximation with preassigned poles
Authors: Stenger Frank
Published: June 2010

This paper gives explicit rational functions for interpolating and approximating functions on the intervals

[
−
1
,
1
]

[ - 1,1]

,

[
0
,
∞
]

[0,\infty ]

, and

[
−
∞
,
∞
]

[ - \infty ,\infty ]

. The rational functions are linear in the functions to be approximated, and they have preassigned poles. The error of approximation of these rationals is nearly as small as the error of best rational approximation with numerator and denominator polynomials of the same degrees. Regions of analyticity are described, which make it possible to tell a priori the accuracy which we can expect from this type of rational approximation.
Counting sums of two squares: the Meissel-Lehmer method
Authors: Shiu P.
Published: June 2010

In 1870, E. D. F. Meissel developed a method for computing the individual values of the prime-counting function, and, in 1959, D. H. Lehmer simplified and extended Meissel’s method. Let

W
(
x
)

W(x)

count the numbers not exceeding x that are sums of two squares. We develop a variant of the Meissel-Lehmer method for

W
(
x
)

W(x)

and use it to calculate

W
(

10

12

)

W({10^{12}})

.
A double integral containing the modified Bessel function: asymptotics and computation
Authors: Temme N. M.
Published: June 2010

A two-dimensional integral containing

exp
⁡
(
−
u
−
t
)

I
0

(
2

u
t

)

\exp ( - u - t){I_0}(2\sqrt {ut} )

is considered.

I
0

(
z
)

{I_0}(z)

is the modified Bessel function and the integral is taken over the rectangle

0
⩽
u
⩽
x

0 \leqslant u \leqslant x

,

0
⩽
t
⩽
y

0 \leqslant t \leqslant y

. The integral is difficult to compute when x and y are large, especially when x and y are almost equal. Computer programs based on existing series expansions are inefficient in this case. A representation in terms of the error function (normal distribution function) is discussed, from which more efficient algorithms can be constructed.
Number-theoretic transforms of prescribed length
Authors: Creutzburg R. , Tasche M.
Published: June 2010

A new constructive method for finding all convenient moduli m for a number-theoretic transform with given length N and given primitive Nth root of unity modulo m is presented. This method is based on the prime factorization of cyclotomic polynomials with integer-valued argument or on the primitive divisors of integers. Many known results can be obtained as simple corollaries.
The computation of the fundamental unit of totally complex quartic orders
Authors: Buchmann Johannes
Published: June 2010

We describe an efficient algorithm for the computation of the regulator and a fundamental unit of an arbitrary totally complex quartic order. We analyze its complexity and we present tables with computational results for the orders

Z

[
4

−
d

]

{\mathbf {Z}}{[^4}\sqrt { - d} ]

,

1
⩽
d
⩽
500

1 \leqslant d \leqslant 500

.
The construction of preconditioners for elliptic problems by substructuring. II
Authors: Bramble J. H. , Pasciak J. E. , Schatz A. H.
Published: June 2010

We give a method for constructing preconditioners for the discrete systems arising in the approximation of solutions of elliptic boundary value problems. These preconditioners are based on domain decomposition techniques and lead to algorithms which are well suited for parallel computing environments. The method presented in this paper leads to a preconditioned system with condition number proportional to

d

/

h

d/h

where d is the subdomain size and h is the mesh size. These techniques are applied to singularly perturbed problems and problems in three dimensions. The results of numerical experiments illustrating the performance of the method on problems in two and three dimensions are given.
On the computation of all imaginary quadratic fields of class number one
Authors: Cherubini Juergen M. , Wallisser Rolf V.
Published: June 2010

Let d be the discriminant of an imaginary quadratic field with class number one. If

d
⩽
−

10
4

d \leqslant - {10^4}

it is easy to show, using an idea from Stark, that

h
(
12
d
)
⩽
2

|

d

|

h(12d) \leqslant 2\sqrt {|d|}

,

h
(
24
d
)
⩽
2

|

d

|

h(24d) \leqslant 2\sqrt {|d|}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue h left-parenthesis 24 d right-parenthesis ln left-parenthesis 5 plus 2 StartRoot 6 EndRoot right-parenthesis minus 2 h left-parenthesis 12 d right-parenthesis ln left-parenthesis 2 plus StartRoot 3 EndRoot right-parenthesis EndAbsoluteValue greater-than 50 exp left-parenthesis negative pi slash 24 dot StartRoot StartAbsoluteValue d EndAbsoluteValue EndRoot right-parenthesis">

|

h
(
24
d
)
ln
⁡
(
5
+
2

6

)
−
2
h
(
12
d
)
ln
⁡
(
2
+

3

)

|

>
50
exp
⁡
(
−
π

/

24
⋅

|

d

|

)

|h(24d)\ln (5 + 2\sqrt 6 ) - 2h(12d)\ln (2 + \sqrt 3 )| > 50\exp ( - \pi /24 \cdot \sqrt {|d|} )

. This linear form is estimated for large

|

d

|

|d|

from below with the aid of the quantitative version of Schneider’s

α
β

{\alpha ^\beta }

-theorem by Mignotte and Waldschmidt. In the "medium large" region

2
⋅

10
4

⩽

|

d

|

⩽

10

34

2 \cdot {10^4} \leqslant |d| \leqslant {10^{34}}

it is shown by computing the beginning of the continued fraction of

ln
⁡
(
5
+
2

6

)

/

ln
⁡
(
2
+

3

)

\ln (5 + 2\sqrt 6 )/\ln (2 + \sqrt 3 )

that the above relations cannot hold.
Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data
Authors: Johnson Claes , Larsson Stig , Thomée Vidar , Wahlbin Lars B.
Published: June 2010

We consider time-continuous spatially discrete approximations by the Galerkin finite element method of initial-boundary value problems for semilinear parabolic equations with nonsmooth or incompatible initial data. We find that the numerical solution enjoys a gain in accuracy at positive time of essentially two orders relative to the initial regularity, as a result of the smoothing property of the parabolic evolution operator. For higher-order elements the restriction to two orders is in contrast to known optimal order results in the linear case.
On the discretization in time of semilinear parabolic equations with nonsmooth initial data
Authors: Crouzeix Michel , Thomée Vidar
Published: June 2010

Single-step discretization methods are considered for equations of the form

u
t

+
A
u
=
f
(
t
,
u
)

{u_t} + Au = f(t,u)

, where A is a linear positive definite operator in a Hilbert space H. It is shown that if the method is consistent with the differential equation then the convergence is essentially of first order in the stepsize, even if the initial data v are only in H, but also that, in contrast to the situation in the linear homogeneous case, higher-order convergence is not possible in general without further assumptions on v.
Convergence of a vortex in cell method for the two-dimensional Euler equations
Authors: Cottet G.-H.
Published: June 2010

We describe a Vortex In Cell method in which the assignment function used to compute vorticity values at the grid points from particles is twice differentiable, while the velocity need only be continuous. We prove an error estimate for the velocity in terms of the meshsize, the interparticle distance and the size of the computational domain.
On the convergence of collocation methods for boundary integral equations on polygons
Authors: Costabel Martin , Stephan Ernst P.
Published: June 2010

The integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions. Convergence proofs and error estimates for these approximation methods have been only available in the following cases: Fredholm integral equations of the second kind [4], [7], one-dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves [2], [3], [17], [26]—[29], and some special results on the classical Neumann integral equation of potential theory for polygonal plane domains [5], [8], [9]. Here we give convergence proofs for collocation with piecewise linear trial functions for Neumann’s integral equation and Symm’s integral equation on plane curves with corners. We derive asymptotic error estimates in Sobolev norms and analyze the effect of graded meshes.
Numerical solution of stochastic differential equations with constant diffusion coefficients
Authors: Chang Chien Cheng
Published: June 2010

We present Runge-Kutta methods of high accuracy for stochastic differential equations with constant diffusion coefficients. We analyze

L
2

{L_2}

convergence of these methods and present convergence proofs. For scalar equations a second-order method is derived, and for systems a method of order one-and-one-half is derived. We further consider a variance reduction technique based on Hermite expansions for evaluating expectations of functions of sample solutions. Numerical examples in two dimensions are presented.
Upper semicontinuity of attractors for approximations of semigroups and partial differential equations
Authors: Hale Jack K. , Lin Xiao-Biao , Raugel Geneviève
Published: June 2010

Suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite-dimensional system. Conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined. Applications are given to parabolic and hyperbolic partial differential equations.
Integer sets with distinct subset-sums
Authors: Lunnon W. F.
Published: June 2010

In Section 1 we introduce the problem of finding minimal-height sets of n natural numbers with distinct subset-sums (SSD), and in Section 2 review the well-known Conway-Guy sequence u, conjectured to yield a minimal SSD set for every n. We go on (Section 3) to prove that u certainly cannot be improved upon by any "greedy" sequence, to verify numerically (Section 4) that it does yield SSD sets for

n
>
80

n > 80

, and (Section 5) by direct search to show that these are minimal for

n
⩽
8

n \leqslant 8

. There is a brief interlude (Section 6) on the problem of decoding the subset from its sum. In Section 7 generalizations of u are constructed which are asymptotically smaller: Defining the Limit Ratio of a sequence w to be

α
=

lim

n
→
∞

w
n

/

2

n
−
1

\alpha = {\lim _{n \to \infty }}{w_n}/{2^{n - 1}}

, the Atkinson-Negro-Santoro sequence v (known to give SSD sets) has

α
=
0.6334

\alpha = 0.6334

, Conway-Guy (conjectured to) has

α
=
0.4703

\alpha = 0.4703

, and our best generalization has

α
=
0.4419

\alpha = 0.4419

. We also (Section 8) discuss when such sequences have the same

α
\alpha

, and (Section 9) how

α
\alpha

may efficiently be computed to high accuracy.
Inverse-average-type finite element discretizations of selfadjoint second-order elliptic problems
Authors: Markowich Peter A. , Zlámal Miloš A.
Published: June 2010

This paper is concerned with the analysis of a class of "special purpose" piecewise linear finite element discretizations of selfadjoint second-order elliptic boundary value problems. The discretization differs from standard finite element methods by inverse-average-type approximations (along element sides) of the coefficient functiona(x)a(x)in the operator−div⁡(a(x)gradu)- \operatorname {div}(a(x)\,{\operatorname {grad}}\,u). The derivation of the discretization is based on approximating the flux densityJ=agraduJ = a\,{\operatorname {grad}}{\mkern 1mu} uby constants on each element. In many cases the flux density is well behaved (moderately varying) even ifa(x)a(x)andu(x)u(x)are fast varying. Discretization methods of this type have been used successfully in semiconductor device simulation for many years; however, except in the one-dimensional case, the mathematical understanding of these methods was rather limited. We analyze the stiffness matrix and prove that—under a rather mild restriction on the mesh—it is a diagonally dominant Stieltjes matrix. Most importantly, we derive an estimate which asserts that the piecewise linear interpolant of the solutionuis approximated to order 1 by the finite element solution in theH1{H^1}-norm. The estimate depends only on the mesh width and on derivatives of the flux density and of a possibly occurring inhomogeneity.
Accurate boundary conditions for exterior problems in gas dynamics
Authors: Hagstrom Thomas , Hariharan S. I.
Published: June 2010

The numerical solution of exterior problems is typically accomplished by introducing an artificial, far field boundary and solving the equations on a truncated domain. For hyperbolic systems, boundary conditions at this boundary are often derived by imposing a principle of no reflection. However, waves with spherical symmetry in gas dynamics satisfy equations where incoming and outgoing Riemann variables are coupled. This suggests that ’natural’ reflections may be important. We propose a reflecting boundary condition based on an asymptotic solution of the far field equations. We obtain nonlinear energy estimates for the truncated problem and present numerical experiments to validate our theory.
A lower bound for the counting function of Lucas pseudoprimes
Authors: Erdős P. , Kiss P. , Sárközy A.
Published: June 2010

We show that there is an absolute constantcsuch that, for any nondegenerate Lucas sequence, the number of Lucas pseudoprimes not exceedingxis greater thanexp⁡{(log⁡x)c}\exp \{ {(\log x)^c}\}ifxis sufficiently large.
Computing heights on elliptic curves
Authors: Silverman Joseph H.
Published: June 2010

We describe how to compute the canonical height of points on elliptic curves. Tate has given a rapidly converging series for Archimedean local heights over R. We describe a modified version of Tate’s series which also converges over C, and give an efficient procedure for calculating local heights at non-Archimedean places. In this way we can calculate heights over number fields having complex embeddings. We also give explicit estimates for the tail of our series, and present several examples.
The Hellan-Herrmann-Johnson method: some new error estimates and postprocessing
Authors: Comodi M. I.
Published: June 2010

We analyze the behavior of the mixed Hellan-Herrmann-Johnson method for solving the biharmonic problem

Δ
2

ψ
=
f

{\Delta ^2}\psi = f

. We show a superconvergence result for the distance between

ψ
h

{\psi ^h}

(the approximation of the displacement) and

P
h

ψ

{P_h}\psi

(where

P
h

{P_h}

is a suitable projection operator). If the discrete equations are solved (as is usually done) by using interelement Lagrange multipliers, our superconvergence result allows us to prove the convergence, in suitable norms, of the Lagrange multipliers to the normal derivative of the displacement, and to construct a new approximation of

∇
ψ

\nabla \psi

which converges to

∇
ψ

\nabla \psi

faster than

∇

ψ
h

\nabla {\psi ^h}

.
Comparison of numerical methods for the calculation of two-dimensional turbulence
Authors: Browning G. L. , Kreiss H.-O.
Published: June 2010

The estimate derived by Henshaw, Kreiss, and Reyna for the smallest scale present in solutions of the two-dimensional incompressible Navier-Stokes equations is employed to obtain convergent pseudospectral approximations. These solutions are then compared with those obtained by a number of commonly used numerical methods. If the viscosity term is deleted and the energy in high wave numbers removed by setting the amplitudes of all wave numbers above a certain point in the spectrum to zero, the "chopped" solution differs considerably from the convergent solution, even at early times. In the case that the regular viscosity is replaced by a hyperviscosity term, i.e., the square of the Laplacian, we also derive an estimate for the smallest scale present. If the coefficient of hyperviscosity is chosen so that the spectrum of the hyperviscosity solution disappears at the same point as for the regular viscosity solution, the hyperviscosity solution is also completely different from the convergent solution. If we "tune" the hyperviscosity coefficient, then the solutions are similar in amplitude or phase, but not both. The solution obtained by a second-order difference method with twice the number of points as the pseudospectral model, or a fourth-order difference method with the same number of points as the pseudospectral model, is essentially identical to the convergent solution. This is reasonable since most of the energy of the solution is contained in the lower part of the spectrum.
An absolutely stabilized finite element method for the Stokes problem
Authors: Douglas Jim , Wang Jun Ping
Published: June 2010

An absolutely stabilized finite element formulation for the Stokes problem is presented in this paper. This new formulation, which is nonsymmetric but stable without employment of any stability constant, can be regarded as a modification of the formulation proposed recently by Hughes and Franca in [8]. Optimal error estimates in

L
2

{L^2}

-norm for the new stabilized finite element approximation of both the velocity and the pressure fields are established, as well as one in

H
1

{H^1}

-norm for the velocity field.
Reciprocal polynomials having small measure. II
Authors: Boyd David W.
Published: June 2010

We consider here polynomials with integer coefficients having measure at most 1.3. In the first paper of this series we determined all such polynomials of degree at most 16, and all such polynomials of height 1 and degree at most 26. In this paper we extend these results to all polynomials of degree at most 20, and all polynomials of height 1 and degree at most 32. We observe some curious statistics concerning the number of roots outside the unit circle for the polynomials investigated here.
The cohomology of the groups of order 32
Authors: Rusin David J.
Published: June 2010

We have calculated the

mod
-
2

\bmod \text {-}2

cohomology rings of all the groups of 32 elements. This paper describes the methods of calculation; the computer routines used can be adapted to assist in the calculation of the modular cohomology rings of other finite groups. We also describe the results of the calculations; the data we have collected provide a substantial increase in the supply of completed calculations in group cohomology, and so we take this opportunity to compare known results and open conjectures.
Error estimates for spatially discrete approximations of semilinear parabolic equations with initial data of low regularity
Authors: Crouzeix M. , Thomée V. , Wahlbin L. B.
Published: June 2010

Semidiscrete finite element methods for a semilinear parabolic equation in

R
d

{R^d}

,

d
≤
3

d \leq 3

, were considered by Johnson, Larsson, Thomée, and Wahlbin. With h the discretization parameter, it was proved that, for compatible and bounded initial data in

H
α

{H^\alpha }

, the convergence rate is essentially

O
(

h

2
+
α

)

O({h^{2 + \alpha }})

for t positive, and for

α
=
0

\alpha = 0

this was seen to be best possible. Here we shall show that for

0
≤
α
>
2

0 \leq \alpha > 2

the convergence rate is, in fact, essentially

O
(

h

2
+
2
α

)

O({h^{2 + 2\alpha }})

, which is sharp.
Efficient multiple-precision evaluation of elementary functions
Authors: Smith David M.
Published: June 2010

Let

M
(
t
)

M(t)

denote the time required to multiply two t-digit numbers using base b arithmetic. Methods are presented for computing the elementary functions in

O
(

t

1

/

3

M
(
t
)
)

O({t^{1/3}}M(t))

time.
Improved condition number for spectral methods
Authors: Heinrichs Wilhelm
Published: June 2010

For the known spectral methods (Galerkin, Tau, Collocation) the condition number behaves like

O
(

N
4

)

O({N^4})

(N: maximal degree of polynomials). We introduce a spectral method with an

O
(

N
2

)

O({N^2})

condition number. The advantages with respect to propagation of rounding errors and preconditioning are demonstrated. A direct solver for constant coefficient problems is given. Extensions to variable coefficient problems and first-order problems are discussed. Numerical results are presented, showing the effectiveness of our methods.
The probability that a random probable prime is composite
Authors: Kim Su Hee , Pomerance Carl
Published: June 2010

Consider a procedure which (1) chooses a random odd number

n
≤
x

n \leq x

, (2) chooses a random number b,

1
>
b
>
n
−
1

1 > b > n - 1

, and (3) accepts n if

b

n
−
1

≡
1

(
mod

n
)

{b^{n - 1}} \equiv 1\;\pmod n

. Let

P
(
x
)

P(x)

denote the probability that this procedure accepts a composite number. It is known from work of Erdös and the second author that

P
(
x
)
→
0

P(x) \to 0

as

x
→
∞

x \to \infty

. In this paper, explicit inequalities are established for

P
(
x
)

P(x)

. For example, it is shown that

P
(

10

100

)
>
2.77
×

10

−
8

P({10^{100}}) > 2.77 \times {10^{ - 8}}

and that

P
(
x
)
≤

(
log
⁡
x

)

−
197

P(x) \leq {(\log x)^{ - 197}}

for

x
≥

10

10

5

x \geq {10^{{{10}^5}}}

.
Complex variable and regularization methods of inversion of the Laplace transform
Authors: Ang D. D. , Lund John , Stenger Frank
Published: June 2010

In this paper three methods are derived for approximating f, given its Laplace transform g on

(
0
,
∞
)

(0,\infty )

, i.e.,

∫
0
∞

f
(
t
)
exp
⁡
(
−
s
t
)

d
t
=
g
(
s
)

\smallint _0^\infty {f(t)\exp ( - st)\,dt = g(s)}

. Assuming that

g
∈

L
2

(
0
,
∞
)

g \in {L^2}(0,\infty )

, the first method is based on a Sinc-like rational approximation of g, the second on a Sinc solution of the integral equation

∫
0
∞

f
(
t
)
exp
⁡
(
−
s
t
)

d
t
=
g
(
s
)

\smallint _0^\infty {f(t)\exp ( - st)\,dt = g(s)}

via standard regularization, and the third method is based on first converting

∫
0
∞

f
(
t
)
exp
⁡
(
−
s
t
)

d
t
=
g
(
s
)

\smallint _0^\infty {f(t)\exp ( - st){\mkern 1mu} dt = g(s)}

to a convolution integral over

R

\mathbb {R}

, and then finding a Sinc approximation to f via the application of a special regularization procedure to solve the Fourier transform problem. We also obtain bounds on the error of approximation, which depend on both the method of approximation and the regularization parameter.
𝑝-adic computation of real quadratic class numbers
Authors: Buchmann J. , Sands J. W. , Williams H. C.
Published: June 2010

Let

K

\mathcal {K}

be any real quadratic field and let

h

K

{h_\mathcal {K}}

be the class number of

K

\mathcal {K}

. A method utilizing the p-adic class number formula for

K

\mathcal {K}

is described for evaluating

h

K

{h_\mathcal {K}}

. The technique was programmed for a micro VAX II computer and run on all fields

K

\mathcal {K}

with radicand

>

10
6

> {10^6}

.