bluuebunny/crossref_metadata_2025 huggingface dataset:
A partition formula for the integer coefficients of the theta function nome
Authors: Ferguson Helaman Rolfe Pratt , Nielsen Dale E. , Cook Grant
Published: June 2010

In elliptic function theory, the nome q can be given as a power series in

ε
\varepsilon

with integer coefficients,

q
=

Σ

n
⩾
0

δ
n

ε

4
n
+
1

q = {\Sigma _{n \geqslant 0}}{\delta _n}{\varepsilon ^{4n + 1}}

. Heretofore, the first 14 coefficients were calculated with considerable difficulty. In this paper, an explicit and general formula involving partitions is given for all the

δ
n

{\delta _n}

. A table of the first 59 of these integers is given. The table is of number-theoretical interest as well as useful for calculating complete and incomplete elliptic integrals.
Two new factors of Fermat numbers
Authors: Hallyburton John C. , Brillhart John
Published: June 2010

A new prime factor is given for each of the Fermat numbers

F

12

{F_{12}}

and

F

13

{F_{13}}

(none was previously known for

F

13

{F_{13}}

). The factoring method used and its machine implementation are discussed. A short table of factors and a current status list are also included.
The second largest prime factor of an odd perfect number
Authors: Pomerance Carl
Published: June 2010

Recently Hagis and McDaniel have studied the largest prime factor of an odd perfect number. Using their results, we begin the study here of the second largest prime factor. We show it is at least 139. We apply this result to show that any odd perfect number not divisible by eight distinct primes must be divisible by 5 or 7.
Models of difference schemes for 𝑢_{𝑡}+𝑢ₓ=0 by partial differential equations
Authors: Hedstrom G. W.
Published: June 2010

It is well known that difference schemes for hyperbolic equations display dispersion of waves. For a general dissipative difference scheme, we present a dispersive wave equation and show that the dispersions are essentially the same when the initial data is a step function.
Stability of multistep methods for delay differential equations
Authors: Wiederholt Lawrence F.
Published: June 2010

The absolute and relative stability of linear multistep methods for a finite step size is studied for delay differential equations. The differential equations are assumed linear and the delays a constant integer multiple of the step size. Computable conditions for stability are developed for scalar equations. Plots of the stability regions for several common multistep methods are included. For the integration methods considered, the stability regions for delay differential equations are significantly different from the stability regions for ordinary differential equations.
Finite-difference approximations to singular Sturm-Liouville eigenvalue problems
Authors: Reddien G. W.
Published: June 2010

A modification of the central-difference method is given which greatly improves the convergence when applied to a certain class of singular eigenvalue problems, including the Klein-Gordon equation. The singularity given special treatment is at the finite end.
An analysis of the finite element method using Lagrange multipliers for the stationary Stokes equations
Authors: Falk Richard S.
Published: June 2010

An error analysis is presented for the approximation of the stationary Stokes equations by a finite element method using Lagrange multipliers.
On asymptotic properties of aliquot sequences
Authors: Erdős P.
Published: June 2010

Put
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s Superscript left-parenthesis 1 right-parenthesis Baseline left-parenthesis n right-parenthesis equals sigma left-parenthesis n right-parenthesis minus n comma sigma left-parenthesis n right-parenthesis equals normal upper Sigma Subscript d slash n Baseline d">

s

(
1
)

(
n
)
=
σ
(
n
)
−
n
,
σ
(
n
)
=

Σ

d

/

n

d

{s^{(1)}}(n) = \sigma (n) - n,\sigma (n) = {\Sigma _{d/n}}d

.
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s Superscript k Baseline left-parenthesis n right-parenthesis equals s Superscript left-parenthesis 1 right-parenthesis Baseline left-parenthesis s Superscript left-parenthesis k minus 1 right-parenthesis Baseline left-parenthesis n right-parenthesis right-parenthesis">

s
k

(
n
)
=

s

(
1
)

(

s

(
k
−
1
)

(
n
)
)

{s^k}(n) = {s^{(1)}}({s^{(k - 1)}}(n))

. In this note we prove that for every k the density of integers satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s Superscript k Baseline left-parenthesis n right-parenthesis equals left-parenthesis 1 plus sigma left-parenthesis 1 right-parenthesis right-parenthesis n left-parenthesis left-parenthesis sigma left-parenthesis n right-parenthesis minus n right-parenthesis slash n right-parenthesis Superscript k">

s
k

(
n
)
=
(
1
+
σ
(
1
)
)
n

(
(
σ
(
n
)
−
n
)

/

n

)
k

{s^k}(n) = (1 + \sigma (1))n{((\sigma (n) - n)/n)^k}

\]
is 1. Several unsolved problems are stated.
Some new high-order multistep formulae for solving stiff equations
Authors: Gupta G. K.
Published: June 2010

Several new multistep formulae of orders up to 9 for solving stiff ordinary differential equations are presented. Results of numerical testing of these new formulae and some formulae presented in earlier papers and the stiff formulae used by Gear are included.
Angstl’s mechanism for checking wellformedness of parenthesis-freeformulae
Authors: Bauer F. L.
Published: June 2010

In 1950, H. Angstl invented a mechanical device, the design of which was the functional basis of a later construction of a relay calculator for logical formulae in polish notation. The principles of this device are illustrated.
Identifying differential equations by Galerkin’s method
Authors: Mosevich Jack W.
Published: June 2010

A numerical technique based on Galerkin’s method is presented for computing unknown parameters or functions occurring in a differential equation whose solution is known. Under certain conditions a solution can be shown to exist to the integral equation formulation of this problem. It is also shown that the resulting nonlinear system is nonsingular.
The optimal algorithm to evaluate 𝑥ⁿ using elementary multiplication methods
Authors: McCarthy D. P.
Published: June 2010

The optimality of the binary algorithm to evaluate

x
n

{x^n}

is established where x is an integer or a completely dense polynomial modulo m, n is a positive integer, and the multiplications are done using a simple improvement on the naive algorithm.
An approximation for ∫^{∞}ᵪ𝑒^{-𝑡2/2}𝑡^{𝑝}𝑑𝑡, 𝜒>0, 𝑝 real
Authors: DiDonato A. R.
Published: June 2010

A new approximation is given for

∫
x
∞

e

−

t
2

/

2

t
p

d
t

\smallint _x^\infty {e^{ - {t^2}/2}{t^p}dt}

,

x
>
0

x > 0

, p real, which extends an earlier approximation of Boyd’s for

p
=
0

p = 0

.
Some observations on primality testing
Authors: Williams H. C. , Holte R.
Published: June 2010

Let N be an integer which is to be tested for primality. Previous methods of ascertaining the primality of N make use of factors of

N
±
1

N \pm 1

,

N
2

±
N
+
1

{N^2} \pm N + 1

, and

N
2

+
1

{N^2} + 1

in order to increase the size of any possible prime divisor of N until it is impossible for N to be the product of two or more primes. These methods usually 2 work as long as

N
>

K
2

N > {K^2}

, where K is

1

/

12

1/12

of the product of the known prime power factors of

N
±
1

N \pm 1

,

N
2

±
N
+
1

{N^2} \pm N + 1

, and

N
2

+
1

{N^2} + 1

. In this paper a technique is described which, when used in conjunction with these methods, will often determine the pri mality of N when

N
>
l

K
3

N > l{K^3}

and l is small.
Some extremal 2-bases
Authors: Riddell J. , Chan C.
Published: June 2010

By means of a computer search, some extremal additive bases have been constructed which have heretofore been unknown.
A further correspondence property of 𝑀 fractions
Authors: McCabe J. H.
Published: June 2010

It is shown that when all the coefficients, after the first, of one of the two corresponding series of an M fraction are zero then the M fraction has a symmetrical form.
On multiple node Gaussian quadrature formulae
Authors: Barrow David L.
Published: June 2010

Let

μ
1

,
…
,

μ
k

{\mu _1}, \ldots ,{\mu _k}

be odd positive integers and

n
=

Σ

i
=
1

k

(

μ
i

+
1
)

n = \Sigma _{i = 1}^k({\mu _i} + 1)

. Let

{

μ
i

}

i
=
1

n

\{ {\mu _i}\} _{i = 1}^n

be an extended Tchebycheff system on

[
a
,
b
]

[a,b]

. Let L be a positive linear functional on

U
≡

span

(
{

μ
i

}
)

U \equiv {\operatorname {span}}(\{ {\mu _i}\} )

. We prove that L has a unique representation in the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis p right-parenthesis equals sigma-summation Underscript i equals 1 Overscript k Endscripts sigma-summation Underscript j equals 0 Overscript mu Subscript i Baseline minus 1 Endscripts a Subscript i j Baseline p Superscript left-parenthesis j right-parenthesis Baseline left-parenthesis t Subscript i Baseline right-parenthesis comma a greater-than t 1 greater-than midline-horizontal-ellipsis greater-than t Subscript k Baseline greater-than b comma">

L
(
p
)
=

∑

i
=
1

k

∑

j
=
0

μ
i

−
1

a

i
j

p

(
j
)

(

t
i

)
,

a
>

t
1

>
⋯
>

t
k

>
b
,

L(p) = \sum \limits _{i = 1}^k {\sum \limits _{j = 0}^{{\mu _i} - 1} {{a_{ij}}{p^{(j)}}({t_i}),\quad a > {t_1} > \cdots > {t_k} > b,} }

\]
for all

p
∈
U

p \in U

. The proof uses the topological degree of a mapping

F
:

D
¯

⊂

R
k

→

R
k

F:\overline D \subset {R^k} \to {R^k}

. The result is proved by showing that the equation

F
(

t
_

)
=
0

F(\underline {t}) = 0

has a unique solution, which in turn is proved by showing that F has degree 1 and that for any solution

t
_

\underline {t}

to the equation

F
(

t
_

)
=
0

F(\underline {t}) = 0

,

det
F
′
(

t
_

)
>
0

\det F\prime (\underline {t}) > 0

. We also give extensions to the cases when the

{

u
i

}

\{ {u_i}\}

are a periodic extended Tchebycheff system and when L is a nonnegative linear functional.
Tabulation of constants for full grade 𝐼_{𝑀𝑁} approximants
Authors: Zakian V. , Edwards M. J.
Published: June 2010

For

f
:
[
0
,
∞
)
→
R

f:[0,\infty ) \to R

the

I

M
N

{I_{MN}}

approximant of

f
(
t
)

f(t)

is
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript upper M upper N Baseline left-parenthesis f comma t right-parenthesis equals integral Subscript 0 Superscript normal infinity Baseline f left-parenthesis x t right-parenthesis sigma-summation Underscript i equals 1 Overscript upper N Endscripts upper K Subscript i Baseline e Superscript minus alpha Super Subscript i Superscript x Baseline d x comma">

I

M
N

(
f
,
t
)
=

∫
0
∞

f
(
x
t
)

∑

i
=
1

N

K
i

e

−

α
i

x

d
x
,

{I_{MN}}(f,t) = \int _0^\infty {f(xt)\sum \limits _{i = 1}^N {{K_i}{e^{ - {\alpha _i}x}}dx,} }

\]
where

α
i

{\alpha _i}

,

K
i

{K_i}

are defined constants. Under appropriate conditions on f,

I

M
N

{I_{MN}}

approximants of full grade are capable of giving good approximation both for small and large t. These and other properties of full grade

I

M
N

{I_{MN}}

approximants make them particularly useful in a wide range of practical applications. The constants

α
i

{\alpha _i}

,

K
i

{K_i}

of full grade

I

M
N

{I_{MN}}

approximants are generated by partial fraction decompositions of certain Padé approximants to

e

−
z

{e^{ - z}}

. The purpose of this paper is firstly to tabulate the constants

α
i

{\alpha _i}

,

K
i

{K_i}

for all full grade

I

M
N

{I_{MN}}

approximants for

1
⩽
N
⩽
10

1 \leqslant N \leqslant 10

; secondly, to give accurate estimates of their precision; and thirdly, to describe the methods of tabulation and estimation in sufficient detail to allow the results of this paper to be extended readily.
Stability analysis of WONDY (a hydrocode based on the artificial viscosity method of von Neumann and Richtmyer) for a special case of Maxwell’s law
Authors: Hicks D. L.
Published: June 2010

The artificial viscosity method of von Neumann and Richtmyer was originally designed and analyzed for stability in the case when the material was an ideal gas. Recently a hydrocode (WONDY) based on the von Neumann-Richtmyer scheme was used in calculating wave propagation problems in materials obeying a form of Maxwell’s material law; signs of an unstable difference scheme appeared. A stability analysis shows that the timestep restrictions required for stability in certain cases can be more stringent for material laws of the Maxwell type than they are for material laws of the ideal gas type.
The comparison of numerical methods for solving polynomial equations
Authors: Galántai Aurél
Published: June 2010

In this paper we compare the Turán process [5]-[6] with the Lehmer-Schur method [2]. We prove that the latter is better.
The nonabelian simple groups 𝐺, ∣𝐺∣<10⁶—maximal subgroups
Authors: Fischer J. , McKay J.
Published: June 2010

The maximal subgroups of all the simple groups (except

L
(
2
,
q
)

L(2,q)

) of order up to one million are given to within conjugacy. Permutation characters on the cosets of the maximal subgroups are given, as are orbit lengths (whenever practical).
Complete characterization of multistep methods with an interval of periodicity for solving 𝑦′′=𝑓(𝑥,𝑦)
Authors: Jeltsch Rolf
Published: June 2010

Linear multistep methods for the second order differential equation

y
=
−

λ
2

y

y = - {\lambda ^2}y

,

λ
\lambda

real, are said to have an interval of periodicity if for a fixed

λ
\lambda

and a stepsize sufficiently small the numerical solution neither explodes nor decays. We give a very simple necessary and sufficient condition under which a linear multistep method has an interval of periodicity. This condition is then applied to multistep methods with an optimal error order.
Minkowski reduction of integral matrices
Authors: Donaldson John L.
Published: June 2010

In 1905 Hermann Minkowski introduced his theory of reduction of positive definite quadratic forms. Recently, Hans J. Zassenhaus has suggested that this theory can be applied to the problem of row reduction of matrices of integers. Computational investigations have shown that for matrices with more columns than rows, the number of steps required for reduction decreases drastically. In this paper it is proved that as the number of columns increases, the probability that a matrix is Minkowski reduced approaches one. This fact is the motivation behind the introduction of a modified version of Minkowski reduction, resulting in a reduction procedure more suitable for computation.
Semiexplicit 𝐴-stable Runge-Kutta methods
Authors: Cooper G. J. , Sayfy A.
Published: June 2010

An

s
−
1

s - 1

stage semiexplicit Runge-Kutta method is represented by an

s
×
s

s \times s

real lower triangular matrix where the number of implicit stages is given by the number of nonzero diagonal elements. It is shown that the maximum order attainable is s when

s
⩽
5

s \leqslant 5

. Necessary and sufficient conditions for A-stability are derived and it is shown that there must be

s
−
1

s - 1

implicit stages if the order is s and

s
⩽
5

s \leqslant 5

. Examples are given for

s
⩽
4

s \leqslant 4

where all the nonzero diagonal elements are equal. Additional problems arise when

s
>
4

s > 4

; but when

s
=
6

s = 6

, an A-stable method of order 5 is obtained. This method has five nonzero diagonal elements, and these elements are equal. Finally, a six stage A-stable method of order six is given. Again, this method has five nonzero (and equal) diagonal elements.
The Hankel power sum matrix inverse and the Bernoulli continued fraction
Authors: Frame J. S.
Published: June 2010

The

m
×
m

m \times m

Hankel power sum matrix

W
=
V

V
T

W = V{V^T}

(where V is the

m
×
n

m \times n

Vandermonde matrix) has (i, j)-entry

S

i
+
j
−
2

(
n
)

{S_{i + j - 2}}(n)

, where

S
p

(
n
)
=

Σ

k
=
1

n

k
p

{S_p}(n) = \Sigma _{k = 1}^n{k^p}

. In solving a statistical problem on curve fitting it was required to determine

f
(
m
)

f(m)

so that for

n
>
f
(
m
)

n > f(m)

all eigenvalues of

W

−
1

{W^{ - 1}}

would be less than 1. It is proved, after calcu lating

W

−
1

{W^{ - 1}}

by first factoring W into easily invertible factors, that

f
(
m
)
=
(
13

m
2

−
5
)

/

8

f(m) = (13{m^2} - 5)/8

suffices. As by-products of the proof, close approximations are given for the Hilbert determinant, and a convergent continued fraction with mth partial denominator

m

−
1

+

(
m
+
1

)

−
1

{m^{ - 1}} + {(m + 1)^{ - 1}}

is found for the divergent Bernoulli number series

Σ

B

2
k

(
2
x

)

2
k

\Sigma {B_{2k}}{(2x)^{2k}}

.
The cyclotomic numbers of order sixteen
Authors: Evans Ronald J. , Hill Jay Roderick
Published: June 2010

A complete table of 408 formulas for cyclotomic numbers of order sixteen is presented. Each number is expressed as a linear combination of parameters of quartic, octic, and bioctic Jacobi sums. Recent applications of these formulas are discussed.
Numerical approximation of a Cauchy problem for a parabolic partial differential equation
Authors: Ewing Richard E. , Falk Richard S.
Published: June 2010

A procedure for the numerical approximation of the Cauchy problem for the following linear parabolic partial differential equation is defined:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row u Subscript t Baseline minus left-parenthesis p left-parenthesis x right-parenthesis u Subscript x Baseline right-parenthesis Subscript x Baseline plus q left-parenthesis x right-parenthesis u equals 0 comma 0 greater-than x greater-than 1 comma 0 greater-than t less-than-or-slanted-equals upper T semicolon u left-parenthesis 0 comma t right-parenthesis equals f 1 left-parenthesis t right-parenthesis comma 2nd Row 0 greater-than t less-than-or-slanted-equals upper T semicolon u left-parenthesis 1 comma t right-parenthesis equals f 2 left-parenthesis t right-parenthesis comma 0 greater-than t less-than-or-slanted-equals upper T semicolon p left-parenthesis 0 right-parenthesis u Subscript x Baseline left-parenthesis 0 comma t right-parenthesis equals g left-parenthesis t right-parenthesis comma 3rd Row 0 greater-than t 0 less-than-or-slanted-equals t less-than-or-slanted-equals upper T semicolon StartAbsoluteValue u left-parenthesis x comma t right-parenthesis EndAbsoluteValue less-than-or-slanted-equals upper M comma 0 less-than-or-slanted-equals x less-than-or-slanted-equals 1 comma 0 less-than-or-slanted-equals t less-than-or-slanted-equals upper T period EndLayout">

u
t

−

(
p
(
x
)

u
x

)

x

+
q
(
x
)
u
=
0
,

0
>
x
>
1
,
0
>
t
⩽
T
;

u
(
0
,
t
)
=

f
1

(
t
)
,

0
>
t
⩽
T
;

u
(
1
,
t
)
=

f
2

(
t
)
,

0
>
t
⩽
T
;

p
(
0
)

u
x

(
0
,
t
)
=
g
(
t
)
,

0
>

t
0

⩽
t
⩽
T
;

|

u
(
x
,
t
)

|

⩽
M
,

0
⩽
x
⩽
1
,
0
⩽
t
⩽
T
.

\begin {array}{*{20}{c}} {{u_t} - {{(p(x){u_x})}_x} + q(x)u = 0,\quad 0 > x > 1,0 > t \leqslant T;\quad u(0,t) = {f_1}(t),} \hfill \\ {0 > t \leqslant T;\quad u(1,t) = {f_2}(t),\quad 0 > t \leqslant T;\quad p(0){u_x}(0,t) = g(t),} \hfill \\ {0 > {t_0} \leqslant t \leqslant T;\quad |u(x,t)| \leqslant M,\quad 0 \leqslant x \leqslant 1,0 \leqslant t \leqslant T.} \hfill \\ \end {array}

\]
The procedure involves Galerkin-type numerical methods for related parabolic initial boundary-value problems and a linear programming problem. Explicit a priori error estimates are presented for the entire discrete procedure when the data

f
1

{f_1}

,

f
2

{f_2}

, and g are known only approximately.
Nonlinear curve-fitting in the 𝐿₁ and 𝐿_{∞} norms
Authors: Shrager Richard L. , Hill Edward
Published: June 2010

In extending the Levenberg-Marquardt

L
2

{L_2}

method for nonlinear curve-fitting to the

L
1

{L_1}

and

L
∞

{L_\infty }

norms, the following problems arise, but are dealt with successfully: (1) Trial parameters are generated by linear programming, which can be time-consuming. (2) Trial parameters are not uniquely specified in some cases. (3) There are intervals of the search parameter for which the trial parameters remain constant. (4) In

L
1

{L_1}

, the trial parameters are discontinuous with respect to the search parameter. It is shown that linear constraints on the parameters are easily included in the computations. Finally, some numerical results are presented.
Inductive formulae for general sum operations
Authors: Minoli Daniel
Published: June 2010

In this note we report some computer generated formulae for the sum of powers of numbers with nonunitary increments; these reduce to the well-known cases when the increment is one.
Convergence of a block-by-block method for nonlinear Volterra integro-differential equations
Authors: Makroglou Athena
Published: June 2010

The theory of a block-by-block method for solving Volterra integral equations is extended to nonsingular Volterra integro-differential equations. Convergence is proved and a rate of convergence is found. The convergence results obtained are analogous to those obtained by Weiss [12] for Volterra integral equations. Several numerical examples are included.
Iterative refinement implies numerical stability for Gaussian elimination
Authors: Skeel Robert D.
Published: June 2010

Because of scaling problems, Gaussian elimination with pivoting is not always as accurate as one might reasonably expect. It is shown that even a single iteration of iterative refinement in single precision is enough to make Gaussian elimination stable in a very strong sense. Also, it is shown that without iterative refinement row pivoting is inferior to column pivoting in situations where the norm of the residual is important.
The 25th and 26th Mersenne primes
Authors: Noll Curt , Nickel Laura
Published: June 2010

The 25th and 26th Mersenne primes are

2

21701

−
1

{2^{21701}} - 1

and

2

23209

−
1

{2^{23209}} - 1

, respectively. Their primality was determined with an implementation of the Lucas-Lehmer test on a CDC Cyber 174 computer. The 25th and 26th even perfect numbers are

(

2

21701

−
1
)

2

21700

({2^{21701}} - 1)\;{2^{21700}}

and

(

2

23209

−
1
)

2

23208

({2^{23209}} - 1)\;{2^{23208}}

, respectively.
On polynomial factorization over finite fields
Authors: Gunji Hiroshi , Arnon Dennis
Published: June 2010

Let

f
(
x
)

f(x)

be a polynomial over a finite field F. An algorithm for determining the degrees of the factors of

f
(
x
)

f(x)

is presented. As in the Berlekamp algorithm (1968) for determining the factors of

f
(
x
)

f(x)

, the Frobenius endomorphism on

F
[
x
]

/

(
f
(
x
)
)

F[x]/(f(x))

plays a central role. Little-known theorems of Schwarz (1956) and Cesàro (1888) provide the basis for the algorithm we present. New and stream-lined proofs of both theorems are provided.
Numerical analysis of spectral properties of coupled oscillator Schrödinger operators. I. Single and double well anharmonic oscillators
Authors: Isaacson D. , Isaacson E. L. , Marchesin D. , Paes-Leme P. J.
Published: June 2010

We describe several methods for computing many eigenvalues and eigenfunctions of a single anharmonic oscillator Schrödinger operator whose potential may have one or two minima. One of the methods requires the solution of an ill-conditioned generalized eigenvalue problem. This method has the virtue of using a bounded amount of work to achieve a given accuracy in both the single and double well regions. We give rigorous bounds, and we prove that the approximations converge faster than any inverse power of the size of the matrices needed to compute them. We present the results of our computations for the

g
:

ϕ
4

:
1

g:{\phi ^4}{:_1}

theory. These results indicate that the methods actually converge exponentially fast. We conjecture why this is so.
Integer points on 𝑦²=𝑥³-7𝑥+10
Authors: Bremner Andrew , Tzanakis Nicholas
Published: June 2010

The 26 integer solutions of

y
2

=

x
3

−
7
x
+
10

{y^2} = {x^3} - 7x + 10

are found and an error in a published table of fundamental units is corrected.
On the sharpness of certain local estimates for 𝐻̊¹ projections into finite element spaces: influence of a re-entrant corner
Authors: Wahlbin Lars B.
Published: June 2010

In a plane polygonal domain with a reentrant corner, consider a homogeneous Dirichlet problem for Poisson’s equation

−
Δ
u
=
f

- \Delta u = f

with f smooth and the corresponding Galerkin finite element solutions in a family of piecewise polynomial spaces based on quasi-uniform (uniformly regular) triangulations with the diameter of each element comparable to h,

0
>
h
⩽
1

0 > h \leqslant 1

. Assuming that u has a singularity of the type

|

x
−

v
M

|

β

|x - {v_M}{|^\beta }

at the vertex

v
M

{v_M}

of maximal angle

π

/

β

\pi /\beta

, we show: (i) For any subdomain A and any s, the error measured in

H

−
s

(
A
)

{H^{ - s}}(A)

is not better than

O
(

h

2
β

)

O({h^{2\beta }})

. (ii)On annular strips of points of distance of order d from

v
M

{v_M}

, the pointwise error is not better than

O
(

h

2
β

d

−
β

)

O({h^{2\beta }}{d^{ - \beta }})

.
Numerical solution of stiff and singularly perturbed boundary value problems with a segmented-adaptive formulation of the tau method
Authors: Onumanyi P. , Ortiz E. L.
Published: June 2010

This paper concerns the application of Ortiz’ recursive formulation of the Tau method to the construction of piecewise polynomial approximations to the solution of linear and nonlinear boundary value problems for ordinary differential equations. A practical error estimation technique, related to the concept of correction in Zadunaisky’s sense, is considered and used in the design of an adaptive approach to the Tau method. It proves efficient in the numerical treatment of problems with rapid functional variations, stiff and singularly perturbed problems. A technique of increased accuracy at matching points of segmented Tau approximants is also discussed and successfully applied to several problems. Numerical examples show that, for a given degree of approximation, our segmented Tau approximant gives an accuracy comparable to that of the best segmented approximation of the exact solution by means of algebraic polynomials.
Conjugate gradient-like algorithms for solving nonsymmetric linear systems
Authors: Saad Youcef , Schultz Martin H.
Published: June 2010

This paper presents a unified formulation of a class of the conjugate gradient-like algorithms for solving nonsymmetric linear systems. The common framework is the Petrov-Galerkin method on Krylov subspaces. We discuss some practical points concerning the methods and point out some of the interrelations between them.
2000000 Steiner triple systems of order 19
Authors: Stinson D. R. , Ferch H.
Published: June 2010

Using a hill-climbing algorithm, we construct 2117600 Steiner triple systems of order 19. These are tested for isomorphism by means of invariants, and 2111276 are shown to be nonisomorphic.
An analysis of a superconvergence result for a singularly perturbed boundary value problem
Authors: O’Riordan Eugene , Stynes Martin
Published: June 2010

We give a new proof that the El-Mistikawy and Werle finite-difference scheme is uniformly second-order accurate for a nonselfadjoint singularly perturbed boundary value problem. To do this, we use exponential finite elements and a discretized Green’s function. The proof is direct, gives the nodal errors explicitly in integral form, and involves much less computation than in previous proofs of the result.
Natural continuous extensions of Runge-Kutta methods
Authors: Zennaro M.
Published: June 2010

The present paper develops a theory of Natural Continuous Extensions (NCEs) for the discrete approximate solution of an ODE given by a Runge-Kutta process. These NCEs are defined in such a way that the continuous solutions furnished by the one-step collocation methods are included.
A remark on a theorem of W. E. H. Berwick
Authors: Tzanakis Nicholas
Published: June 2010

We indicate and fill a gap in a theorem of W. E. H. Berwick concerning the computation of the fundamental units in a semireal biquadratic field.
Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson’s equation
Authors: Falk R. S. , Monk P. B.
Published: June 2010

Logarithmic convexity type continuous dependence results for discrete harmonic functions defined as solutions of the standard

C
0

{C^0}

piecewise-linear approximation to Laplace’s equation are proved. Using this result, error estimates for a regularization method for approximating the Cauchy problem for Poisson’s equation on a rectangle are obtained. Numerical results are presented.
Well-posedness of one-way wave equations and absorbing boundary conditions
Authors: Trefethen Lloyd N. , Halpern Laurence
Published: June 2010

A one-way wave equation is a partial differential equation which, in some approximate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one. Such equations are used in geophysics, in underwater acoustics, and as numerical "absorbing boundary conditions". Their construction can be reduced to the approximation of

1
−

s
2

\sqrt {1 - {s^2}}

on

[
−
1
,
1
]

[ - 1,1]

by a rational function

r
(
s
)
=

p
m

(
s
)

/

q
n

(
s
)

r(s) = {p_m}(s)/{q_n}(s)

. This paper characterizes those rational functions r for which the corresponding one-way wave equation is well posed, both as a partial differential equation and as an absorbing boundary condition for the wave equation. We find that if

r
(
s
)

r(s)

interpolates

1
−

s
2

\sqrt {1 - {s^2}}

at sufficiently many points in

(
−
1
,
1
)

( - 1,1)

, then well-posedness is assured. It follows that absorbing boundary conditions based on Padé approximation are well posed if and only if (m, n) lies in one of two distinct diagonals in the Padé table, the two proposed by Engquist and Majda. Analogous results also hold for one-way wave equations derived by Chebyshev or least-squares approximation.
The cyclotomic numbers of order fifteen
Authors: Buck Nicholas , Smith Lones , Spearman Blair K. , Williams Kenneth S.
Published: June 2010

Explicit formulae are obtained for the cyclotomic numbers of order 15.
Application of symbolic manipulation to Hecke transformations of modular forms in two variables
Authors: Cohn Harvey , Deutsch Jesse
Published: June 2010

The Hecke transformation of modular forms in several variables generates nonsymmetric modular forms out of symmetric forms. This is useful since symmetric forms arise out of Eisenstein series and are easy to construct, while nonsymmetric forms are much harder to construct. A symbolic manipulation system is required because of the magnitude of the Fourier expansions. This process is carried out for Hilbert modular functions over

Q

(

2

)

\mathbb {Q}(\sqrt 2 )

.
Cyclotomic resultants
Authors: Lehmer D. H. , Lehmer Emma
Published: June 2010

This paper examines the eth power character of the divisors of two cyclotomic period polynomials of degree

e
1

{e_1}

and

e
2

{e_2}

. The special cases

e
1

=
2

{e_1} = 2

and

e
2

=
3
,
4

{e_2} = 3,4

, are considered in detail. As corollaries one finds new conditions for cubic and quartic residuacity. The computational method consists in representing cyclotomic numbers in the form

c
1

ζ
+

c
2

ζ
2

+
⋯
+

c

p
−
1

ζ

p
−
1

{c_1}\zeta + {c_2}{\zeta ^2} + \cdots + {c_{p - 1}}{\zeta ^{p - 1}}

, where

ζ
=

e

2
π
i

/

p

\zeta = {e^{2\pi i/p}}

. Multiplication is reduced to addition and subtraction, which are carried out in a multi-precision system.
Convergence analysis for a nonsymmetric Galerkin method for a class of singular boundary value problems in one space dimension
Authors: Eriksson Kenneth , Nie Yi Yong
Published: June 2010

For the method and problems under consideration we estimate the error in the maximum norm as well as at individual nodal points. In order to obtain full superconvergence at all nodal points we have to introduce local mesh refinements, even though the exact solution is smooth for the given class of problems.
Abstract generalized bisection and a cost bound
Authors: Kearfott R. Baker
Published: June 2010

The purpose of this paper is to study desirable properties of binary search algorithms for isolating all solutions to nonlinear systems of equations

F
(
X
)
=
0

F(X) = 0

within a given compact domain

D

∈

R

n

{\mathbf {D}} \in {{\mathbf {R}}^n}

. We devise a general framework such that any algorithm fitting into the general framework will always isolate all solutions

Z
∈

D

Z \in {\mathbf {D}}

such that

F
(
Z
)
=
0

F(Z) = 0

; this framework contains a new idea for handling the occurrence of roots on boundaries. We then present and prove a bound on the total amount of computation which is valid for any algorithm in the class. Finally, we define a specific prototypical algorithm valid for F satisfying certain natural smoothness properties; we show that it satisfies the hypotheses for the general framework. This algorithm is based on "bisection" of generalized rectangles, the Kantorovich theorem, and second-order Taylor type models for F. It is meant to provide further guidelines for the development of effective heuristics, etc., for actual implementations.
On weight functions admitting Chebyshev quadrature
Authors: Förster Klaus-Jürgen
Published: June 2010

In this paper we prove the existence of Chebyshev quadrature for three new weight functions which are quite different from the two known examples given by Ullman [15] and Byrd and Stalla [2]. In particular, we indicate a simple method to construct weight functions for which there exist infinitely many Chebyshev quadrature rules.
Backward differentiation approximations of nonlinear differential/algebraic systems
Authors: Brenan Kathryn E. , Engquist Björn E.
Published: June 2010

Finite difference approximations of dynamical systems modelled by nonlinear, semiexplicit, differential/algebraic equations are analyzed. Convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints. The appropriate asymptotic convergence rates and the leading error terms are determined.
Another surprising property of 42. A remark on a paper by P. S. Wang: “Factoring multivariate polynomials over algebraic number fields” [Math. Comp. 30 (1976), no. 134, 324–336; MR0568283 (58 #27887a)]
Authors: Abbott J. A. , Davenport J. H.
Published: June 2010

We give a counterexample to a bound quoted in Wang’s paper on polynomial factorization over algebraic number fields. We also give an alternative to that bound which seems not to have been published before.
A curiosity concerning the representation of integers in noninteger bases
Authors: Stewart G. W.
Published: June 2010

In this note it is shown how an integer x can be represented uniquely in a noninteger basis provided the "digits" of the representation are allowed to be nonintegers. It is then shown that the integer parts of these "digits" contain all the information necessary to recover x.
Sharp maximum norm error estimates for finite element approximations of the Stokes problem in 2-D
Authors: Durán R. , Nochetto R. H. , Wang Jun Ping
Published: June 2010

This paper deals with finite element approximations of the Stokes equations in a plane bounded domain

Ω
\Omega

, using the so-called velocity-pressure mixed formulation. Quasi-optimal error estimates in the maximum norm are derived for the velocity, its gradient and the pressure fields. The analysis relies on abstract properties which are in turn a consequence of the eixstence of a local projection operator

Π
h

{\Pi _h}

satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript normal upper Omega Endscripts d i v left-parenthesis bold v minus normal upper Pi Subscript h Baseline bold v right-parenthesis q d bold x equals 0 comma for-all bold v element-of left-bracket upper H 0 Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis right-bracket squared comma for-all q element-of upper M Subscript h Baseline comma">

∫
Ω

div
⁡
(

v

−

Π
h

v

)
q

d

x

=
0
,

∀

v

∈

[

H
0
1

(
Ω
)
]

2

,
∀
q
∈

M
h

,

\int _\Omega \operatorname {div}({\mathbf {v}} - {\Pi _h}{\mathbf {v}})q\;d{\mathbf {x}} = 0,\quad \forall {\mathbf {v}} \in {{[H_0^1(\Omega )]}^2},\forall q \in {M_h},

\]
where

M
h

{M_h}

is the finite element space associated with the pressure. Several examples for which this operator can be constructed locally illustrate the theory.
A family of Gauss-Kronrod quadrature formulae
Authors: Gautschi Walter , Rivlin Theodore J.
Published: June 2010

We show, for each

n
≥
1

n \geq 1

, that the

(
2
n
+
1
)

(2n + 1)

-point Kronrod extension of the n-point Gaussian quadrature formula for the measure
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d sigma Subscript gamma Baseline left-parenthesis t right-parenthesis equals left-parenthesis 1 plus gamma right-parenthesis squared left-parenthesis 1 minus t squared right-parenthesis Superscript 1 slash 2 Baseline d t slash left-parenthesis left-parenthesis 1 plus gamma right-parenthesis squared minus 4 gamma t squared right-parenthesis comma negative 1 greater-than gamma less-than-or-equal-to 1 comma">

d

σ
γ

(
t
)
=

(
1
+
γ

)
2

(
1
−

t
2

)

1

/

2

d
t

/

(

(
1
+
γ

)
2

−
4
γ

t
2

)
,

−
1
>
γ
≤
1
,

d{\sigma _\gamma }(t) = {(1 + \gamma )^2}{(1 - {t^2})^{1/2}}dt/({(1 + \gamma )^2} - 4\gamma {t^2}),\quad - 1 > \gamma \leq 1,

\]
has the properties that its

n
+
1

n + 1

Kronrod nodes interlace with the n Gauss nodes and all its

2
n
+
1

2n + 1

weights are positive. We also produce explicit formulae for the weights.
On four-dimensional terminal quotient singularities
Authors: Mori Shigefumi , Morrison David R. , Morrison Ian
Published: June 2010

We report on an investigation of four-dimensional terminal cyclic quotient singularities which are not Gorenstein. (For simplicity, we focus on quotients by cyclic groups of prime order.) An enumeration, using a computer, of all such singularities for primes

>
1600

> 1600

led us to conjecture a structure theorem for these singularities (which is rather more complicated than the known structure theorem in dimension three). We discuss this conjecture and our evidence for it; we also discuss properties of the anticanonical and antibicanonical linear systems of these singularities.
Computation of real quadratic fields with class number one
Authors: Stephens A. J. , Williams H. C.
Published: June 2010

A rapid method for determining whether the real quadratic field

K

=

Q

(

D

)

\mathcal {K} = \mathcal {Q}(\sqrt D )

has class number one is described. The method makes use of the infrastructure idea of Shanks to determine the regulator of

K

\mathcal {K}

and then uses the Generalized Riemann Hypothesis to rapidly estimate

L
(
1
,
χ
)

L(1,\chi )

to the accuracy needed for determining whether or not the class number of

K

\mathcal {K}

is one. The results of running this algorithm on a computer for all prime values of D up to

10
9

{10^9}

are also presented, together with further results on runs on intervals of size

10
7

{10^7}

starting at

10
i

(
i
=
9
,
10
,
…
,
16
)

{10^i}\,(i = 9,10, \ldots ,16)

.
Infinite sets of primes with fast primality tests and quick generation of large primes
Authors: Pintz János , Steiger William L. , Szemerédi Endre
Published: June 2010

Infinite sets P and Q of primes are described,

P
⊂
Q

P \subset Q

. For any natural number n it can be decided if

n
∈
P

n \in P

in (deterministic) time

O
(

(
log
⁡
n

)
9

)

O({(\log n)^9})

. This answers affirmatively the question of whether there exists an infinite set of primes whose membership can be tested in polynomial time, and is a main result of the paper. Also, for every

n
∈
Q

n \in Q

, we show how to randomly produce a proof of the primality of n. The expected time is that needed for

1

1
2

1\frac {1}{2}

exponentiations

mod
n

\bmod n

. We also show how to randomly generate k-digit integers which will be in Q with probability proportional to

k

−
1

{k^{ - 1}}

. Combined with the fast verification of

n
∈
Q

n \in Q

just mentioned, this gives an

O
(

k
4

)

O({k^4})

expected time algorithm to generate and certify primes in a given range and is probably the fastest method to generate large certified primes known to belong to an infinite subset. Finally, it is important that P and Q are relatively dense (at least

c

n

2

/

3

/

log
⁡
n

c{n^{2/3}}/\log n

elements less than n). Elements of Q in a given range may be generated quickly, but it would be costly for an adversary to search Q in this range, a property that could be useful in cryptography.
Determination of the 𝐷^{1/2}-norm of the SOR iterative matrix for the unsymmetric case
Authors: Evans D. J. , Li C.
Published: June 2010

This paper is concerned with the determination of the Jordan canonical form and

D

1

/

2

{D^{1/2}}

-norm of the SOR iterative matrix derived from the coefficient matrix A having the form
\[

A
=

(

D
1

a
m
p
;

−
H

H
T

a
m
p
;

D
2

)

A = \left ( {\begin {array}{*{20}{c}} {D1} \hfill & { - H} \hfill \\ {{H^T}} \hfill & {{D_2}} \hfill \\ \end {array} } \right )

\]
with

D
1

{D_1}

and

D
2

{D_2}

symmetric and positive definite. The theoretical results show that the Jordan form is not diagonal, but has only q principal vectors of grade 2 and that the

D

1

/

2

{D^{1/2}}

-norm of

L

ω
b

{\mathcal {L}_{{\omega _b}}}

(

ω
b

{\omega _b}

, the optimum parameter) is less than unity if and only if

μ
¯

=
ρ
(
B
)

\bar \mu = \rho (B)

, the spectral radius of the associated Jacobi iterative matrix, is less than unity. Here q is the multiplicity of the eigenvalue

i

μ
¯

i\bar \mu

of B.
On the computation of unit groups and class groups of totally real quartic fields
Authors: Buchmann J. , Pohst M. , von Schmettow J.
Published: June 2010

In this paper we describe the computation of a system of fundamental units and of the class group for each totally real quartic field

F

\mathcal {F}

of discriminant less than

10
6

{10^6}

. Generating equations, integral bases, and the Galois groups for all those fields were recently given by Buchmann and Ford.
Incomplete hyperelliptic integrals and hypergeometric series
Authors: Loiseau J.-F. , Codaccioni J.-P. , Caboz R.
Published: June 2010

We consider the incomplete hyperelliptic integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis a comma upper X right-parenthesis equals integral Subscript 0 Superscript upper X Baseline StartFraction d x Over StartRoot a minus lamda 2 x squared minus lamda Subscript n Baseline x Superscript n Baseline EndRoot EndFraction">

H
(
a
,
X
)
=

∫
0
X

d
x

a
−

λ
2

x
2

−

λ
n

x
n

H(a,X) = \int _0^X {\frac {{dx}}{{\sqrt {a - {\lambda _2}{x^2} - {\lambda _n}{x^n}} }}}

\]
with

a
>
0

a > 0

,

λ
2

>
0

{\lambda _2} > 0

,

n
>
2

n > 2

, where X belongs to the connected component of

{
x

|

λ
2

x
2

+

λ
n

x
n

>
a
}

\{ x|{\lambda _2}{x^2} + {\lambda _n}{x^n} > a\}

containing the origin. Continuing previous work on the complete hyperelliptic integral, we express in this paper

H
(
a
,
X
)

H(a,X)

as a convergent series of hypergeometric type. A brief survey of some applications to algebraic equations and mechanics is then given.
Computation of Galois group elements of a polynomial equation
Authors: Cockayne E. J.
Published: June 2010

This note demonstrates the use of the computer for constructing elements of the Galois group over the rationals of a polynomial equation with rational coefficients.
Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions
Authors: Raugel Geneviève , Sell George R.
Published: June 2010

We examine the Navier-Stokes equations (NS) on a thin

3
3

-dimensional domain

Ω
ε

=

Q
2

×
(
0
,
ε
)

{\Omega _\varepsilon } = {Q_2} \times (0,\varepsilon )

, where

Q
2

{Q_2}

is a suitable bounded domain in

R

2

{\mathbb {R}^2}

and

ε
\varepsilon

is a small, positive, real parameter. We consider these equations with various homogeneous boundary conditions, especially spatially periodic boundary conditions. We show that there are large sets

R

(
ε
)

\mathcal {R}(\varepsilon )

in

H
1

(

Ω
ε

)

{H^1}({\Omega _\varepsilon })

and

S

(
ε
)

\mathcal {S}(\varepsilon )

in
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript 1 comma normal infinity Baseline left-parenthesis left-parenthesis 0 comma normal infinity right-parenthesis comma upper L squared left-parenthesis normal upper Omega Subscript epsilon Baseline right-parenthesis right-parenthesis">

W

1
,
∞

(
(
0
,
∞
)
,

L
2

(

Ω
ε

)
)

{W^{1,\infty }}((0,\infty ),{L^2}({\Omega _\varepsilon }))

such that if

U
0

∈

R

(
ε
)

{U_0} \in \mathcal {R}(\varepsilon )

and

F
∈

S

(
ε
)

F \in \mathcal {S}(\varepsilon )

, then (NS) has a strong solution

U
(
t
)

U(t)

that remains in

H
1

(

Ω
ε

)

{H^1}({\Omega _\varepsilon })

for all

t
≥
0

t \geq 0

and in

H
2

(

Ω
ε

)

{H^2}({\Omega _\varepsilon })

for all

t
>
0

t > 0

. We show that the set of strong solutions of (NS) has a local attractor

A

ε

{\mathfrak {A}_\varepsilon }

in

H
1

(

Ω
ε

)

{H^1}({\Omega _\varepsilon })

, which is compact in

H
2

(

Ω
ε

)

{H^2}({\Omega _\varepsilon })

. Furthermore, this local attractor

A

ε

{\mathfrak {A}_\varepsilon }

turns out to be the global attractor for all the weak solutions (in the sense of Leray) of (NS). We also show that, under reasonable assumptions,

A

ε

{\mathfrak {A}_\varepsilon }

is upper semicontinuous at

ε
=
0

\varepsilon = 0

.
The unconditional basic sequence problem
Authors: Gowers W. T. , Maurey B.
Published: June 2010

We construct a Banach space that does not contain any infinite unconditional basic sequence and investigate further properties of this space. For example, it has no subspace that can be written as a topological direct sum of two infinite-dimensional spaces. This property implies that every operator on the space is a strictly singular perturbation of a multiple of the identity. In particular, it is either strictly singular or Fredholm with index zero. This implies that the space is not isomorphic to any proper subspace.
On a problem of Erdős and Lovász. II. 𝑛(𝑟)=𝑂(𝑟)
Authors: Kahn Jeff
Published: June 2010

We prove a linear upper bound on the function

n
(
r
)
=

min

{

|

H

|

:

H

n(r) = {\text {min}}\{ \left | {\mathcal {H}} \right |:\mathcal {H}

an

r
r

-uniform, intersecting hypergraph with

τ
(

H

)
=
r
}

\tau (\mathcal {H}) = r\}

, thus settling a longstanding problem of Erdös and Lovász.
Linear decision trees, subspace arrangements and Möbius functions
Authors: Björner Anders , Lovász László
Published: June 2010

Topological methods are described for estimating the size and depth of decision trees where a linear test is performed at each node. The methods are applied, among others, to the questions of deciding by a linear decision tree whether givennnreal numbers (1) somekkof them are equal, or (2) somekkof them are unequal. We show that the minimum depth of a linear decision tree for these problems is at least (1)max{n−1,nlog3(n/3k)}{\text {max}}\{ n - 1,\quad n\;{\text {lo}}{{\text {g}}_3}(n/3k)\}, and (2)max{n−1,nlog3(k−1)−k+1}{\text {max}}\{ n - 1,\quad n\;{\text {lo}}{{\text {g}}_3}(k - 1) - k + 1\}. Our main lower bound for the size of linear decision trees for polyhedraPPinRn{{\mathbf {R}}^n}is given by the sum of Betti numbers for the complementRn∖P{{\mathbf {R}}^n}\backslash P. The applications of this general topological bound involve the computation of the Möbius function of intersection lattices of certain subspace arrangements. In particular, this leads to computing various expressions for the Möbius function of posets of partitions with restricted block sizes. Some of these formulas have topological meaning. For instance, we derive a formula for the Euler characteristic of the subset ofRn{{\mathbf {R}}^n}of points with nokkcoordinates equal in terms of the roots of the truncated exponential∑i>kxi/i!\sum \nolimits _{i > k} {{x^i}} /i!.