bluuebunny/crossref_metadata_2025 huggingface dataset:
A high-order difference method for differential equations
Authors: Lynch Robert E. , Rice John R.
Published: June 2010

This paper analyzes a high-accuracy approximation to the mth-order linear ordinary differential equation

M
u
=
f

Mu = f

. At mesh points, U is the estimate of u; and U satisfies

M
n

U
=

I
n

f

{M_n}U = {I_n}f

, where

M
n

U

{M_n}U

is a linear combination of values of U at

m
+
1

m + 1

stencil points (adjacent mesh points) and

I
n

f

{I_n}f

is a linear combination of values of f at J auxiliary points, which are between the first and last stencil points. The coefficients of

M
n

{M_n}

,

I
n

{I_n}

are obtained “locally” by solving a small linear system for each group of stencil points in order to make the approximation exact on a linear space S of dimension

L
+
1

L + 1

. For separated two-point boundary value problems, U is the solution of an n-by-n linear system with full bandwidth

m
+
1

m + 1

. For S a space of polynomials, existence and uniqueness are established, and the discretization error is

O
(

h

L
+
1
−
m

)

O({h^{L + 1 - m}})

the first

m
−
1

m - 1

divided differences of U tend to those of u at this rate. For a general set of auxiliary points one has

L
=
J
+
m

L = J + m

; but special auxiliary points, which depend upon M and the stencil points, allow larger L, up to

L
=
2
J
+
m

L = 2J + m

. Comparison of operation counts for this method and five other common schemes shows that this method is among the most efficient for given convergence rate. A brief selection from extensive experiments is presented which supports the theoretical results and the practicality of the method.
The groups of order 𝑝⁶ (𝑝 an odd prime)
Authors: James Rodney
Published: June 2010

A complete list of the groups of order

p
6

{p^6}

, where p denotes an odd prime number, is given using P. Hall’s concept of isoclinism.
A factor of 𝐹₁₇
Authors: Gostin Gary B.
Published: June 2010

A prime factor is given for

F

17

{F_{17}}

. The method of factoring and its machine implementation are given.
A method of skipping the transient phase in the solution of separably stiff ordinary initial value problems
Authors: Alfeld Peter
Published: June 2010

Stiff systems of ordinary differential equations are characterized by an initial phase in which the solution changes rapidly. Often there is no interest in reproducing this transient phase. A method is proposed for modifying the initial value if the system of differential equations is separably stiff, i.e. is characterized by the occurrence of a few (typically one) large negative real eigenvalues which dominate the others. The modified system does not possess a transient phase, and in the constant coefficient linear case its solution does not differ from that of the original one in the nonstiff components.
The construction of Jacobi and periodic Jacobi matrices with prescribed spectra
Authors: Ferguson Warren E.
Published: June 2010

The spectral properties of Jacobi and periodic Jacobi matrices are analyzed and algorithms for the construction of Jacobi and periodic Jacobi matrices with prescribed spectra are presented. Numerical evidence demonstrates that these algorithms are of practical utility. These algorithms have been used in studies of the periodic Toda lattice, and might also be used in studies of inverse eigenvalue problems for Sturm-Liouville equations and Hill’s equation.
On factoring a class of complex symmetric matrices without pivoting
Authors: Serbin Steven M.
Published: June 2010

Let

A

=

B

+
i

C

\mathcal {A} = \mathcal {B} + i\mathcal {C}

be a complex, symmetric

n
×
n

n \times n

matrix with

B

\mathcal {B}

and

C

\mathcal {C}

each real, symmetric and positive definite. We show that the LINPACK diagonal pivoting decomposition

U

−
1

A

(

U

−
1

)
T

=

D

{\mathcal {U}^{ - 1}}\mathcal {A}{({\mathcal {U}^{ - 1}})^T} = \mathcal {D}

proceeds without the necessity for pivoting. In particular, when

B

\mathcal {B}

and

C

\mathcal {C}

are band matrices, bandwidth is preserved.
Chebyshev approximation of (1+2𝑥)𝑒𝑥𝑝(𝑥²)𝑒𝑟𝑓𝑐𝑥 in 0≤𝑥<∞
Authors: Shepherd M. M. , Laframboise J. G.
Published: June 2010

We have obtained a single Chebyshev expansion of the function

f
(
x
)
=
(
1
+
2
x
)
exp
⁡
(

x
2

)

erfc

x

f(x) = (1 + 2x)\exp ({x^2}){\text {erfc}}\,x

in

0
⩽
x
>
∞

0 \leqslant x > \infty

, accurate to 22 decimal digits. The presence of the factors

(
1
+
2
x
)
exp
⁡
(

x
2

)

(1 + 2x)\exp ({x^2})

causes

f
(
x
)

f(x)

to be of order unity throughout this range, ensuring that the use of

f
(
x
)

f(x)

for approximating erfc x will give uniform relative accuracy for all values of x
On Fermat’s quotient, base two
Authors: Lehmer D. H.
Published: June 2010

This paper extends the search for solutions of the congruence
\[

2

p
−
1

−
1
≡
0

(
mod

p
2

)

{2^{p - 1}} - 1 \equiv 0\quad \pmod {p^2}

\]
to the limit

p
>
6
⋅

10
9

p > 6 \cdot {10^9}

. No solution, except the well-known

p
=
1093

p = 1093

and

p
=
3511

p = 3511

, was found.
Unitary hyperperfect numbers
Authors: Hagis Peter
Published: June 2010

Unitary hyperperfect numbers are generalized unitary perfect numbers. In this paper a list of such numbers is given, and some results concerning them are proved.
A class of Steiner triple systems of order 21 and associated Kirkman systems
Authors: Mathon Rudolf A. , Phelps Kevin T. , Rosa Alexander
Published: June 2010

We examine a class of Steiner triple systems or order 21 with an automorphism consisting of three disjoint cycles of length 7. We exhibit explicitly all members of this class: they number 95 including the 7 cyclic systems. We then examine resolvability of the obtained systems; only 6 of the 95 are resolvable yielding a total of 30 nonisomorphic Kirkman triple systems of order 21. We also list several invariants of the systems and investigate their further properties.
Stability of pseudospectral and finite-difference methods for variable coefficient problems
Authors: Gottlieb David , Orszag Steven A. , Turkel Eli
Published: June 2010

It is shown that pseudospectral approximation to a special class of variable coefficient one-dimensional wave equations is stable and convergent even though the wave speed changes sign within the domain. Computer experiments indicate similar results are valid for more general problems. Similarly, computer results indicate that the leapfrog finite-difference scheme is stable even though the wave speed changes sign within the domain. However, both schemes can be asymptotically unstable in time when a fixed spatial mesh is used.
Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function 𝑠𝑛(𝑥,𝑘)
Authors: Fransén Arne
Published: June 2010

Two conjectures are posed for the coefficients introduced by Alois Schett in the Taylor series expansion of the Jacobian elliptic function

sn

(
x
,
k
)

{\text {sn}}(x,k)

. The first conjecture is furnished with a proof revealing a procedure which might be useful when calculating further coefficients. Some of the coefficients are tabulated.
Succinct proofs of primality for the factors of some Fermat numbers
Authors: Brent Richard P.
Published: June 2010

We give short and easily verified proofs of primality for the factors of the Fermat numbers

F
5

,

F
6

,

F
7

{F_5},{F_6},{F_7}

and

F
8

{F_8}

.
Absorbing boundary conditions for the discretization schemes of the one-dimensional wave equation
Authors: Halpern Laurence
Published: June 2010

When computing a partial differential equation, it is often necessary to introduce artificial boundaries. Here we explain a systematic method to obtain boundary conditions for the wave equation in one dimension, fitting to the discretization scheme and stable. Moreover, we give error estimates on the reflected part.
An error estimate for Stenger’s quadrature formula
Authors: Beighton S. , Noble B.
Published: June 2010

The basis of this paper is the quadrature formula
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative 1 Superscript 1 Baseline f left-parenthesis x right-parenthesis d x almost-equals log q sigma-summation Underscript m equals negative upper M Overscript upper M Endscripts StartFraction 2 q Superscript m Baseline Over left-parenthesis 1 plus q Superscript m Baseline right-parenthesis squared EndFraction f left-parenthesis StartFraction q Superscript m Baseline minus 1 Over q Superscript m Baseline plus 1 EndFraction right-parenthesis comma">

∫

−
1

1

f
(
x
)

d
x
≈
log
⁡
q

∑

m
=
−
M

M

2

q
m

(
1
+

q
m

)

2

f

(

q
m

−
1

q
m

+
1

)

,

\int _{ - 1}^1 {f(x)\,dx \approx \log q\sum \limits _{m = - M}^M {\frac {{2{q^m}}}{{{{(1 + {q^m})}^2}}}f\left ( {\frac {{{q^m} - 1}}{{{q^m} + 1}}} \right )} ,}

\]
where

q
=
exp
⁡
(
2
h
)

q = \exp (2h)

, h being a chosen step length. This formula has been derived from the Trapezoidal Rule formula by F. Stenger. An explicit form of the error is given for the case where the integrand has a factor of the form

(
1
−
x

)
α

(
1
+
x

)
β

,
α
,
β
>
−
1

{(1 - x)^\alpha }{(1 + x)^\beta },\alpha ,\beta > - 1

. Application is made to the evaluation of Cauchy principal value integrals with endpoint singularities and an appropriate error form is derived.
Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces
Authors: Véron Laurent
Published: June 2010

Let

∂
Φ

\partial \Phi

be the subdifferential of some lower semicontinuous convex function

Φ
\Phi

of a real Hilbert space H,

f
∈

L
2

(
0
,
T
;
H
)

f \in {L^2}(0,T;H)

and

u
n

{u_n}

a continouous piecewise linear approximate solution of

d
u

/

d
t
+
∂
Φ
(
u
)
∋
f

du/dt + \partial \Phi (u) \ni f

, obtained by an implicit scheme. If

u
0

∈
Dom
⁡
(
Φ
)

{u_0} \in \operatorname {Dom} (\Phi )

, then

d

u
n

/

d
t

d{u_n}/dt

converges to

d
u

/

d
t

du/dt

in

L
2

(
0
,
T
;
H
)

{L^2}(0,T;H)

. Moreover, if

u
0

∈

Dom
⁡
(
∂
Φ
)

¯

{u_0} \in \overline {\operatorname {Dom} (\partial \Phi )}

, we construct a step function

η
n

(
t
)

{\eta _n}(t)

approximating t such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow plus normal infinity Endscripts integral Subscript 0 Superscript upper T Baseline eta Subscript n Baseline StartAbsoluteValue d u Subscript n Baseline slash d t minus d u slash d t EndAbsoluteValue squared d t equals 0">

lim

n
→
+
∞

∫
0
T

η
n

|

d

u
n

/

d
t
−
d
u

/

d
t

|

2

d
t
=
0

{\lim _{n \to + \infty }}\smallint _0^T{\eta _n}|d{u_n}/dt - du/dt{|^2}\;dt = 0

. When

Φ
\Phi

is inf-compact and when the sequence of approximation of f is weakly convergent to f, then

u
n

{u_n}

converges to u in

C
(
[
0
,
T
]
;
H
)

C([0,T];H)

and

η
n

d

u
n

/

d
t

{\eta _n}d{u_n}/dt

is weakly convergent to

t
d
u

/

d
t

tdu/dt

.
Linear multistep methods for stable differential equations 𝑦̈=𝐴𝑦+𝐵(𝑡)𝑦̇+𝑐(𝑡)
Authors: Gekeler Eckart
Published: June 2010

The approximation of

y

.
.

=
A
y
+
B
(
t
)

y
.

+
c
(
t
)

{y^{..}} = Ay + B(t){y^.} + c(t)

by linear multistep methods is studied. It is supposed that the matrix A is real symmetric and negative semidefinite, that the multistep method has an interval of absolute stability

[
−
s
,
0
]

[ - s,0]

, and that

h
2

‖
A
‖

⩽
s

{h^2}\left \| A \right \| \leqslant s

where h is the time step. A priori error bounds are derived which show that the exponential multiplication factor is of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="exp left-brace normal upper Gamma Subscript s Baseline StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue upper B EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript n Baseline left-parenthesis n h right-parenthesis right-brace">

exp
⁡
{

Γ
s

|

|

|

B

|

|

|

n

(
n
h
)
}

\exp \{ {\Gamma _s}|||B|||_{n}(nh)\}

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue upper B EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript n Baseline equals max Underscript 0 less-than-or-slanted-equals t less-than-or-slanted-equals n h Endscripts double-vertical-bar upper B left-parenthesis t right-parenthesis double-vertical-bar">

|

|

|

B

|

|

|

n

=

max

0
⩽
t
⩽
n
h

‖

B
(
t
)

‖

|||B|||_{n} = {\max _{0 \leqslant t \leqslant nh}}\left \| {B(t)} \right \|

.
A generalized Lánczos scheme
Authors: van der Vorst H. A.
Published: June 2010

It is shown in this paper how the Lanczos algorithm can be generalized so that it applies to both symmetric and skew-symmetric matrices and corresponding generalized eigenvalue problems.
On the computation of certain integrals containing the modified Bessel function 𝐼₀(𝜉)
Authors: Lassey Keith R.
Published: June 2010

Efficient stratagems are developed for numerically evaluating one- and two-dimensional integrals over x, y with integrand

exp
⁡
(
−
x
−
y
)

I
0

(
2

p
x
y

)

\exp ( - x - y){I_0}(2\sqrt {pxy} )

. The integrals are expressed in terms of convergent series, which exhibit the correct limiting behavior, and which can be evaluated recursively. The performances of these stratagems are compared with numerical integration.
Class number formulae of Dirichlet type
Authors: Hudson Richard H. , Williams Kenneth S.
Published: June 2010

Applying a theorem of Johnson and Mitchell, some new class number formulae are derived.
Stability theory of difference approximations for multidimensional initial-boundary value problems
Authors: Michelson Daniel
Published: June 2010

A stability theory is developed for dissipative difference approximations to multidimensional initial-boundary value problems. The original differential problem should be strictly hyperbolic and the difference problem consistent with the differential one. An algebra of pseudo-difference operators is built and later used to prove the stability of the difference approximation with variable coefficients. In addition, stability of the Cauchy problem for weakly dissipative difference approximations with variable coefficients is proved.
Convergence of Galerkin approximations for the Korteweg-de Vries equation
Authors: Baker Garth A. , Dougalis Vassilios A. , Karakashian Ohannes A.
Published: June 2010

Standard Galerkin approximations, using smooth splines on a uniform mesh, to 1-periodic solutions of the Korteweg-de Vries equation are analyzed. Optimal rate of convergence estimates are obtained for both semidiscrete and second order in time fully discrete schemes. At each time level, the resulting system of nonlinear equations can be solved by Newton’s method. It is shown that if a proper extrapolation is used as a starting value, then only one step of the Newton iteration is required.
Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials
Authors: Smith William E. , Sloan Ian H. , Opie Alex H.
Published: June 2010

The paper discusses both theoretical properties and practical implementation of product integration rules of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative normal infinity Superscript normal infinity Baseline k left-parenthesis x right-parenthesis f left-parenthesis x right-parenthesis d x almost-equals sigma-summation Underscript i equals 1 Overscript n Endscripts w Subscript n i Baseline f left-parenthesis x Subscript n i Baseline right-parenthesis comma">

∫

−
∞

∞

k
(
x
)
f
(
x
)

d
x
≈

∑

i
=
1

n

w

n
i

f
(

x

n
i

)
,

\int _{ - \infty }^\infty {k(x)f(x)\,dx \approx \sum \limits _{i = 1}^n {{w_{ni}}f({x_{ni}}),} }

\]
where f is continuous, k is absolutely integrable, the nodes

{

x

n
i

}

\{ {x_{ni}}\}

are roots of the Hermite polynomials

H
n

(
x
)

{H_n}(x)

, and the weights

{

w

n
i

}

\{ {w_{ni}}\}

are chosen so that the rule is exact if f is any polynomial of degree

>
n

> n

. Convergence of the rule to the exact integral as

n
→
∞

n \to \infty

is proved for a wide class of functions f and k (including singular or oscillatory functions k), and rates of convergence are estimated. The rules are shown to have the property of asymptotic positivity, and as a consequence exhibit good numerical stability. Numerical calculations for some practical cases are presented, which show the method to be computationally effective for integrands (including highly oscillatory ones) that decay suitably at infinity. Applications of the method to integration over

[
0
,
∞
)

[0,\infty )

are also discussed.
Error estimates for the numerical identification of a variable coefficient
Authors: Falk Richard S.
Published: June 2010

Error estimates are derived for the approximate identification of an unknown transmissivity coefficient in a partial differential equation describing a model problem in groundwater flow. The approximation scheme considered determines the coefficient by least squares fitting of the observed pressure data.
The determination of the value of Rado’s noncomputable function Σ(𝑘) for four-state Turing machines
Authors: Brady Allen H.
Published: June 2010

The well-defined but noncomputable functions

Σ
(
k
)

\Sigma (k)

and

S
(
k
)

S(k)

given by T. Rado as the "score" and "shift number" for the k-state Turing machine "Busy Beaver Game" were previously known only for

k
⩽
3

k \leqslant 3

. The largest known lower bounds yielding the relations

Σ
(
4
)
⩾
13

\Sigma (4) \geqslant 13

and

S
(
4
)
⩾
107

S(4) \geqslant 107

, reported by this author, supported the conjecture that these lower bounds are the actual particular values of the functions for

k
=
4

k = 4

. The four-state case has previously been reduced to solving the blank input tape halting problem of only 5,820 individual machines. In this final stage of the

k
=
4

k = 4

case, one appears to move into a heuristic level of higher order where it is necessary to treat each machine as representing a distinct theorem. The remaining set consists of two primary classes in which a machine and its tape are viewed as the representation of a growing string of cellular automata. The proof techniques, embodied in programs, are entirely heuristic, while the inductive proofs, once established by the computer, are completely rigorous and become the key to the proof of the new and original mathematical results:

Σ
(
4
)
=
13

\Sigma (4) = 13

and

S
(
4
)
=
107

S(4) = 107

.
Irregular sets of integers generated by the greedy algorithm
Authors: Gerver Joseph L.
Published: June 2010

The greedy algorithm was used to generate sets of positive integers containing no subset of the form

{
x
,
x
+
y
,
x
+
2
y
}

\{ x,x + y,x + 2y\}

,

{
x
,
x
+
y
,
x
+
3
y
}

\{ x,x + y,x + 3y\}

,

{
x
,
x
+
2
y
,
x
+
3
y
}

\{ x,x + 2y,x + 3y\}

,

{
x
,
x
+
3
y
,
x
+
4
y
}

\{ x,x + 3y,x + 4y\}

,

{
x
,
x
+
3
y
,
x
+
5
y
}

\{ x,x + 3y,x + 5y\}

, and

{
x
,
x
+
y
,
x
+
2
y
,
x
+
3
y
}

\{ x,x + y,x + 2y,x + 3y\}

, respectively. All of these sets have peaks of density in roughly geometric progression.
Class groups of complex quadratic fields
Authors: Schoof R. J.
Published: June 2010

We present 75 new examples of complex quadratic fields that have 5-rank of their class groups

⩾
3

\geqslant 3

. Only one of these fields has 5-rank of its class group

>
3

> 3

: The field

Q

(

−
258559351511807

)

{\mathbf {Q}}(\sqrt { - 258559351511807} )

has a class group isomorphic to
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis 5 right-parenthesis times upper C left-parenthesis 5 right-parenthesis times upper C left-parenthesis 5 right-parenthesis times upper C left-parenthesis 5 right-parenthesis times upper C left-parenthesis 2 right-parenthesis times upper C left-parenthesis 11828 right-parenthesis period">

C
(
5
)
×
C
(
5
)
×
C
(
5
)
×
C
(
5
)
×
C
(
2
)
×
C
(
11828
)
.

C(5) \times C(5) \times C(5) \times C(5) \times C(2) \times C(11828).

\]
The fields were obtained by applying ideas of J. F. Mestre to the 5-isogeny

X
1

(
11
)
→

X
0

(
11
)

{X_1}(11) \to {X_0}(11)

.
A spline-trigonometric Galerkin method and an exponentially convergent boundary integral method
Authors: Arnold Douglas N.
Published: June 2010

We consider a Galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions. We analyze the method applied to the integral equation of the first kind arising from a single layer potential formulation of the Dirichlet problem in the interior or exterior of an analytic plane curve. In constrast to ordinary spline Galerkin methods, we show that the method is stable, and so provides quasioptimal approximation, in a large family of Hilbert spaces including all the Sobolev spaces of negative order. As a consequence we prove that the approximate solution to the Dirichlet problem and all its derivatives converge pointwise with exponential rate.
A sinc approximation for the indefinite integral
Authors: Kearfott Ralph Baker
Published: June 2010

A method for computing

∫
0
x

f
(
t
)

d
t
,
x
=
(
0
,
1
)

\smallint _0^xf(t)\;dt,x = (0,1)

is outlined, where

f
(
t
)

f(t)

may have singularities at

t
=
0

t = 0

and

t
=
1

t = 1

. The method depends on the approximation properties of Whittaker cardinal, or sinc function expansions; the general technique may be used for semi-infinite or infinite intervals in addition to (0,1). Tables of numerical results are given.
A lower bound for the norm of the theta operator
Authors: Parson L. Alayne
Published: June 2010

The Poincaré theta operator maps the space of holomorphic functions with period one onto the space of cusp forms for a finitely generated Fuchsian group. It is easy to show that the norm of the operator does not exceed one. In the case of the classical modular group and weight six, it is now shown that the norm is bounded below by .927.
On the existence of fields governing the 2-invariants of the classgroup of 𝑄(√𝑑𝑝) as 𝑝 varies
Authors: Cohn H. , Lagarias J. C.
Published: June 2010

This paper formulates general conjectures relating the structure of the 2-classgroup

C
2

(
d
p
)

{C_2}(dp)

associated to

Q

(

d
p

)

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

to the splitting of the ideal (p) in certain algebraic number fields. Here

d

≢

2

d\,\nequiv \,2

(
mod

4
)

\pmod 4

is a fixed integer and p varies over primes. The conjectures assert that there exists an algebraic number field

Ω
j

(
d
)

{\Omega _j}(d)

such that the Artin symbol

[
(

Ω
j

(
d
)

/

Q

)

/

(
p
)
]

[({\Omega _j}(d)/{\mathbf {Q}})/(p)]

determines the first j 2-invariants of the group

C
2

(
d
p
)

{C_2}(dp)

, i.e. it determines

C
2

(
d
p
)

/

C
2

(
d
p

)

2
j

{C_2}(dp)/{C_2}{(dp)^{{2^j}}}

. These conjectures imply that the set of primes p for which

C
2

(
d
p
)

{C_2}(dp)

has a given set of 2-invariants has a natural density which is a rational number. Existing results prove the conjectures whenever

j
=
1

j = 1

or 2 and also for an infinite set of d with

j
=
3

j = 3

. The smallest open case is

j
=
3

j = 3

,

d
=
−
21

d = - 21

. This paper presents evidence concerning these conjectures for

d
=
−
4

d = - 4

, 8 and

−
21

- 21

. Numerical evidence is given that

Ω
3

(
−
21
)

{\Omega _3}( - 21)

exists, and that natural densities which are rational numbers exist for the sets of primes with

2
j

|

h
(
d
p
)

{2^j}|h(dp)

for

d
=
−
4

d = - 4

and 8, for

1
⩽
j
⩽
7

1 \leqslant j \leqslant 7

. A search for the hypothetical field

Ω
4

(
−
4
)

{\Omega _4}( - 4)

ruled out the simplest candidate fields:

Ω
4

(
−
4
)

{\Omega _4}( - 4)

is not a normal extension of Q of degree 16 ramifying only at (2).
An analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems
Authors: Weiss Richard
Published: June 2010

Stability and convergence results are derived for the box and trapezoidal schemes applied to boundary value problems for linear singularly perturbed first order systems of o.d.e.’s without turning points.
Does increased regularity lower complexity?
Authors: Werschulz Arthur G.
Published: June 2010

Intuitively, the more regular a problem, the easier it should be to solve. Examples drawn from ordinary and partial differential equations, as well as from approximation, support the intuition. Traub and Woźniakowski conjectured that this is always the case. In this paper, we study linear problems. We prove a weak form of the conjecture, and show that this weak form cannot be strengthened. To do this, we consider what happens to the optimal error when regularity is increased. If regularity is measured by a Sobolev norm, increasing the regularity improves the optimal error, which allows us to establish the conjecture in the normed case. On the other hand, if regularity is measured by a Sobolev seminorm, it is no longer true that increasing the regularity improves the optimal error. However, a "shifted" version of this statement holds, which enables us to establish the conjecture in the seminormed case.
A note on an algorithm for interpolating rationals
Authors: Huang You Gun , Williams Jack
Published: June 2010

This note considers an aspect of the simultaneous exchange algorithm for Chebyshev approximation by interpolating rationals. An example has been given by Dunham claiming to show that this algorithm may fail to produce an interpolating rational approximant. In this note we show that his example cannot occur as a step of the exchange algorithm. A second example is given where Dunham’s objection to the algorithm is valid.
On the equation 𝑌²=𝑋(𝑋²+𝑝)
Authors: Bremner A. , Cassels J. W. S.
Published: June 2010

Generators are found for the group of rational points on the title curve for all primes

p
≡
5

(
mod

8
)

p \equiv 5\;\pmod 8

less than 1,000. The rank is always 1 in accordance with conjectures of Selmer and Mordell. Some of the generators are rather large.
Calculation of the moments and the moment generating function for the reciprocal gamma distribution
Authors: Fransén Arne , Wrigge Staffan
Published: June 2010

In this paper we consider the distribution
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis x right-parenthesis equals upper F Superscript negative 1 Baseline integral Subscript 0 Superscript x Baseline left-parenthesis normal upper Gamma left-parenthesis t right-parenthesis right-parenthesis Superscript negative 1 Baseline d t">

G
(
x
)
=

F

−
1

∫
0
x

(
Γ
(
t
)

)

−
1

d
t

G(x) = {F^{ - 1}}\smallint _0^x{(\Gamma (t))^{ - 1}}\;dt

. The aim of the investigation is twofold: first,to find numerical values of characteristics such as moments, variance, skewness, kurtosis,etc.; second, to study analytically and numerically the moment generating function

φ
(
t
)
=

∫
0
∞

e

−
t
x

/

Γ
(
x
)

d
x

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

. Furthermore, we also make a generalization of the reciprocal gamma distribution, and study some of its properties.
Optimal estimation of Jacobian and Hessian matrices that arise in finite difference calculations
Authors: Goldfarb D. , Toint Ph. L.
Published: June 2010

In this paper, the problem of estimating Jacobian and Hessian matrices arising in the finite difference approximation of partial differential equations is considered. Using the notion of computational molecule or stencil, schemes are developed that require the minimum number of differences to estimate these matrices. A procedure applicable to more complicated structures is also given.
Higher order multigrid methods
Authors: Schaffer Steve
Published: June 2010

This paper is concerned with the treatment of higher order multi-grid techniques for obtaining accurate finite difference approximations to partial differential equations. The three basic techniques considered are a multi-grid process involving smoothing via higher order difference approximations, iterated defect corrections with multi-grid used as an inner loop equation solver, and tau-extrapolation. Efficient versions of each of these three basic schemes are developed and analyzed by local mode analysis and numerical experiments. The numerical tests focus on fourth and sixth order discretizations of Poisson’s equations and demonstrate that the three methods performed similarly yet substantially better than the usual multi-grid method, even when the right-hand side lacked sufficient smoothness.
A nonaveraging set of integers with a large sum of reciprocals
Authors: Wróblewski J.
Published: June 2010

A set of integers is constructed with no three elements in arithmetic progression and with a rather large sum of reciprocals.
A Monte Carlo factoring algorithm with linear storage
Authors: Schnorr C.-P. , Lenstra H. W.
Published: June 2010

We present an algorithm which will factor an integer n quite efficiently if the class number

h
(
−
n
)

h( - n)

is free of large prime divisors. The running time

T
(
n
)

T(n)

(number of compositions in the class group) satisfies
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p r o b left-bracket upper T left-parenthesis m right-parenthesis less-than-or-slanted-equals n Superscript 1 slash 2 r Baseline right-bracket greater-than-or-equivalent-to left-parenthesis r minus 2 right-parenthesis Superscript minus left-parenthesis r minus 2 right-parenthesis">

prob
⁡
[
T
(
m
)
⩽

n

1

/

2
r

]
≳

(
r
−
2

)

−
(
r
−
2
)

\operatorname {prob}[T(m) \leqslant {n^{1/2r}}] \gtrsim {(r - 2)^{ - (r - 2)}}

for random

m
∈
[
n

/

2
,
n
]

m \in [n/2,n]

and

r
⩾
2

r \geqslant 2

. So far it is unpredictable which numbers will be factored fast. Running the algorithm on all discriminants - ns with

s
⩽

r
r

s \leqslant {r^r}

and

r
=

ln
⁡
n

/

ln
⁡
ln
⁡
n

r = \sqrt {\ln n/\ln \ln n}

, every composite integer n will be factored in

o
(
exp
⁡

ln
⁡
n
ln
⁡
ln
⁡
n

)

o(\exp \sqrt {\ln n\ln \ln n} )

bit operations. The method requires an amount of storage space which is proportional to the length of the input n. In our analysis we assume a lower bound on the frequency of class numbers

h
(
−
m
)

h( - m)

,

m
⩽
n

m \leqslant n

, which are free of large prime divisors.
Gauss quadratures and Jacobi matrices for weight functions not of one sign
Authors: Kautsky J. , Elhay S.
Published: June 2010

Construction of Gauss quadratures with prescribed knots via Jacobi matrices is extended to the case where not all orthogonal polynomials exist due to the weight function changing sign. An algorithm is described and is demonstrated by calculating the knots of Kronrod schemes and other Gauss quadratures with prescribed knots.
Partial fraction evaluation and incomplete decomposition of a rational function whose denominator contains a repeated polynomial factor
Authors: Mahoney J. F.
Published: June 2010

Attention is directed to those proper rational functions whose denominators may be expressed as the product of an Nth degree polynomial raised to the Kth power and another polynomial of degree M. A method is presented for decomposing such a rational function into the sum of the K partial fraction terms which proceed from the repeated polynomial plus a proper rational function which completes the equality. Use is made of an extended version of Horner’s scheme. Two numerical examples and an operations count are presented. The method is free of complex arithmetic provided that all of the coefficients of the entering polynomials are real.
Computing self-intersections of closed geodesics on finite-sheeted covers of the modular surface
Authors: Lehner J. , Sheingorn M.
Published: June 2010

An algorithm is given for deciding whether a closed geodesic on a finite-sheeted cover of the modular surface has self-intersections; if it does, the algorithm gives them in the order they occur in traversing the geodesic. The following general result on geodesics is proved: any closed geodesic on a Riemann surface R can be lifted to a simple closed geodesic on some finite-sheeted cover of R. In the last two sections the connection with the stabilizer (under the modular group) of a Markov quadratic irrationality is discussed.
Some periodic continued fractions with long periods
Authors: Patterson C. D. , Williams H. C.
Published: June 2010

Let

p
(
D
)

p(D)

be the period length of the continued fraction for

D

\sqrt D

. Under the extended Riemann Hypothesis for

Q

(

D

)

\mathcal {Q}(\sqrt D )

one would expect that

p
(
D
)
=
O
(

D

1

/

2

log
⁡
log
⁡
D
)

p(D) = O({D^{1/2}}\log \log D)

. In order to test this it is necessary to find values of D for which

p
(
D
)

p(D)

is large. This, in turn, requires that we be able to find solutions to large sets of simultaneous linear congruences. The University of Manitoba Sieve Unit (UMSU), a machine similar to D. H. Lehmer’s DLS-127, was used to find such values of D. For example, if

D
=
46257585588439

D = 46257585588439

, then

p
(
D
)
=
25679652

p(D) = 25679652

25679652. Some results are also obtained for the Voronoi continued fraction for

3

D

^3\sqrt D

.
A multigrid version of a simple finite element method for the Stokes problem
Authors: Pitkäranta Juhani , Saarinen Tuomo
Published: June 2010

We consider a finite element method for the Stokes problem on a rectangular domain based on piecewise bilinear velocities and piecewise constant pressures on a uniform rectangular grid. It is shown that by a simple stabilization strategy the method can be implemented in a convergent multigrid procedure.
Divided differences, shift transformations and Larkin’s root finding method
Authors: Neumaier A. , Schäfer A.
Published: June 2010

For a one-dimensional complex-valued function f this paper deals with iterative root finding methods using divided differences of f. Assuming that f is given in a Newtonian representation we show how Horner-like transformations ("shift transformations") yield the divided differences needed in each iteration step. In particular, we consider an iteration method given by Larkin [5] and derive an equivalent version of this method fitting into this context. Monotonic convergence to real roots of real polynomials is investigated. Both "shift-" and "nonshift versions" of several root finding methods are tested and compared with respect to their numerical behavior.
The construction of unramified cyclic quartic extensions of 𝑄(√𝑚)
Authors: Vaughan Theresa P.
Published: June 2010

We give an elementary general method for constructing fields K satisfying

[
K
:
Q
]
=
8

[K:Q] = 8

, the Galois group of K over Q is dihedral, and K is unramified over one of its quadratic subfields. Given an integer m, we describe all such fields K which contain

Q
(

m

)

Q(\sqrt m )

. The description is specific and is given in terms of the arithmetic of the quadratic subfields of K.
The maximal modulus of an algebraic integer
Authors: Boyd David W.
Published: June 2010

The maximal modulus of an algebraic integer is the absolute value of its largest conjugate. We compute the minimum of the maximal modulus of all algebraic integers of degree d which are not roots of unity, for d at most 12. The computations suggest that the minimum is never attained for a reciprocal algebraic integer. The truth of this conjecture would show that the conjecture of Schinzel and Zassenhaus follows from a theorem of Smyth. We further test our conjecture by computing the minimum of the maximal modulus of all reciprocal algebraic integers of degree d which are not roots of unity, for d at most 16. Our computations strongly suggest that the best constant in the conjecture of Schinzel and Zassenhaus is 1.5

log
⁡

θ
0

\log {\theta _0}

, where

θ
0

{\theta _0}

is the smallest P.V. number. They also shed some light on a recent conjecture of Lind concerning the Perron numbers.
Sets in which 𝑥𝑦+𝑘 is always a square
Authors: Brown Ezra
Published: June 2010

APk{P_k}-set of sizenis a set{x1,…,xn}\{ {x_1}, \ldots ,{x_n}\}of distinct positive integers such thatxixj+k{x_i}{x_j} + kis a perfect square, wheneveri≠ji \ne j; aPk{P_k}-setXcan be extended if there existsy∉Xy \notin Xsuch thatX∪{y}X \cup \{ y\}is still aPk{P_k}-set. The most famous result onPk{P_k}-sets is due to Baker and Davenport, who proved that theP1{P_1}-set 1, 3, 8, 120 cannot be extended. In this paper, we show, among other things, that ifk≡2(mod4)k \equiv 2\;\pmod 4, then there does not exist aPk{P_k}-set of size 4, and that theP−1{P_{ - 1}}-set 1, 2, 5 cannot be extended.
On the definiteness of Gauss-Kronrod integration rules
Authors: Rabinowitz Philip
Published: June 2010

The nondefiniteness of the Kronrod extension of the Gauss-Gegenbauer integration rule with weight function

w
(
x
;
μ
)
=

(
1
−

x
2

)

μ
−
1

/

2

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

,

0
>
μ
>
1

0 > \mu > 1

, is shown when there are more than three abscissas.
More quadratically converging algorithms for 𝜋
Authors: Borwein J. M. , Borwein P. B.
Published: June 2010

We present a quadratically converging algorithm for

π
\pi

based on a formula of Legendre’s for complete elliptic integrals of modulus

sin
⁡
(
π

/

12
)

\sin (\pi /12)

and the arithmetic-geometric mean iteration of Gauss and Legendre. Precise asymptotics are provided which show this algorithm to be (marginally) the most efficient developed to date. As such it provides a natural computational check for the recent large-scale calculations of

π
\pi

.
Use of a computer scan to prove 𝑄(√2+√2) and 𝑄(√3+√2) are Euclidean
Authors: Cohn Harvey , Deutsch Jesse
Published: June 2010

The fields in the title are shown to be norm-Euclidean by a computer scan of the unit 4-cube representing coordinates of a field element translated by integers. The method is to subdivide this cube into sufficiently many small boxes so the norm is less than unity in each box, when referred to an appropriate "neighboring" integer.
Computing the irreducible characters of the group 𝐺𝐿₆(2)
Authors: Darafsheh M. R.
Published: June 2010

All the sixty ordinary irreducible characters of the group of six by six nonsingular matrices over a field with two elements are found. To do this we use methods of Steinberg and also characters induced from certain subgroups which makes it possible to calculate the whole character table by hand.
A generalization of Swan’s theorem
Authors: Fredricksen Harold M. , Hales Alfred W. , Sweet Melvin M.
Published: June 2010

Let f and g denote polynomials over the two-element field. In this paper we show that the parity of the number of irreducible factors of

x
n

f
+
g

{x^n}f + g

is a periodic function of n, with period dividing eight times the period of the polynomial

f
2

(
x
(
g

/

f

)
′

−
n
(
g

/

f
)
)

{f^2}(x(g/f)’- n(g/f))

. This can be considered a generalization of Swan’s trinomial theorem [3].
On Mordell’s equation 𝑦²-𝑘=𝑥³: a problem of Stolarsky
Authors: Steiner Ray P.
Published: June 2010

On page 1 of his book Algebraic Numbers and Diophantine Approximation, K. B. Stolarsky posed the problem of solving the equationy2+999=x3{y^2} + 999 = {x^3}in positive integers. In the present paper we refine some techniques of Ellison and Pethö and show that the complete set of integer solutions of Stolarsky’s equation is\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column x equals 10 comma 2nd Column a m p semicolon y equals plus-or-minus 1 comma 2nd Row 1st Column x equals 12 comma 2nd Column a m p semicolon y equals plus-or-minus 27 comma 3rd Row 1st Column x equals 40 comma 2nd Column a m p semicolon y equals plus-or-minus 251 comma 4th Row 1st Column x equals 147 comma 2nd Column a m p semicolon y equals plus-or-minus 1782 comma 5th Row 1st Column x equals 174 comma 2nd Column a m p semicolon y equals plus-or-minus 2295 comma EndLayout">x=10,amp;y=±1,x=12,amp;y=±27,x=40,amp;y=±251,x=147,amp;y=±1782,x=174,amp;y=±2295,\begin {array}{*{20}{c}} {x = 10,} \hfill & {y = \pm 1,} \hfill \\ {x = 12,} \hfill & {y = \pm 27,} \hfill \\ {x = 40,} \hfill & {y = \pm 251,} \hfill \\ {x = 147,} \hfill & {y = \pm 1782,} \hfill \\ {x = 174,} \hfill & {y = \pm 2295,} \hfill \\ \end {array}\]and\[x=22480,y=±3370501.x = 22480,\quad y = \pm 3370501.\]
The convergence of multilevel methods for solving finite-element equations in the presence of singularities
Authors: Yserentant Harry
Published: June 2010

The known convergence proofs for multi-level methods assume the quasi-uniformity of the family of domain triangulations used. Such triangulations are not suitable for problems with singularities caused by re-entrant corners and abrupt changes in the boundary conditions. In this paper it is shown that families of properly refined grids yield the same convergence behavior of multi-level methods for such singular problems as quasi-uniform subdivisions do for

H
2

{H^2}

-regular problems.
On gamma function inequalities
Authors: Bustoz Joaquin , Ismail Mourad E. H.
Published: June 2010

We show that certain functions involving quotients of gamma functions are completely monotonic. This leads to inequalities involving gamma functions. We also establish the infinite divisibility of several probability distributions whose Laplace transforms involve quotients of gamma functions.
Elliptic curve cryptosystems
Authors: Koblitz Neal
Published: June 2010

We discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field. These elliptic curve cryptosystems may be more secure, because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem, especially over

GF

(

2
n

)

{\text {GF}}({2^n})

. We discuss the question of primitive points on an elliptic curve modulo p, and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point.
Numerical absorbing boundary conditions for the wave equation
Authors: Higdon Robert L.
Published: June 2010

We develop a theory of difference approximations to absorbing boundary conditions for the scalar wave equation in several space dimensions. This generalizes the work of the author described in [8]. The theory is based on a representation of analytical absorbing boundary conditions proven in [8]. These conditions are defined by compositions of first-order, one-dimensional differential operators. Here the operators are discretized individually, and their composition is used as a discretization of the boundary condition. The analysis of stability and reflection properties reduces to separate studies of the individual factors. A representation of the discrete boundary conditions makes it possible to perform the analysis geometrically, with little explicit calculation.
Asymptotic theory of filtering for linear operator equations with discrete noisy data
Authors: Groetsch C. W. , Vogel C. R.
Published: June 2010

We consider Fredholm integral equations of the first kind with continuous kernels in which the data is discretely sampled and contaminated by white noise. A sufficient condition for the convergence of general filtering methods applied to such equations is derived. The condition in essence relates the decay rate of the singular values of the integral operator to the shape of the filter function used in the regularization method. Specific illustrations of the condition are given for Tikhonov regularization, the truncated singular value decomposition, and Landweber iteration.
Integer squares with constant second difference
Authors: Buell Duncan A.
Published: June 2010

The problem addressed is this: Do there exist nonconsecutive integers

n
0

,

n
1

,

n
2

,
…
,

{n_0},{n_1},{n_2}, \ldots ,

, such that the second differences of the squares of the

n
i

{n_i}

, are constant? Specifically, can that constant be equal to 2? A complete characterization of sequences of length four can be given. The question of whether or not sequences of length five exist is still open but the existence or nonexistence of such sequences can be described in a more algorithmic way than the simple statement of the problem.
A simple homotopy method for determining all isolated solutions to polynomial systems
Authors: Zulehner Walter
Published: June 2010

A new homotopy method for solving systems of polynomial equations is presented. The homotopy equation is extremely simple: It is linear with respect to the homotopy parameter and only one auxiliary parameter is needed to regularize the problem. Within some limits, an arbitrary starting problem can be chosen, as long as its solution set is known. No restrictions on the polynomial systems are made. A few numerical tests are reported which show the influence of the auxiliary parameter, resp. the starting problem, upon the computational cost of the method.
A fast algorithm for the multiplication of generalized Hilbert matrices with vectors
Authors: Gerasoulis A.
Published: June 2010

The existence of a fast algorithm with an

O
A

(
n

(
log
⁡
n

)
2

)

{O_A}(n{(\log n)^2})

time complexity for the multiplication of generalized Hilbert matrices with vectors is shown. These matrices are defined by

(

B
p

)

i
,
j

=
1

/

(

t
i

−

s
j

)
p

{({B_p})_{i,j}} = 1/{({t_i} - {s_j})^p}

,

i
,
j
=
1
,
…
,
n

i,j = 1, \ldots ,n

,

p
=
1
,
…
,
q

p = 1, \ldots ,q

,

q
≪
n

q \ll n

, where

t
i

{t_i}

and

s
i

{s_i}

are distinct points in the complex plane and

t
i

≠

s
j

{t_i} \ne {s_j}

,

i
,
j
=
1
,
…
,
n

i,j = 1, \ldots ,n

. The major contribution of the paper is the stable implementation of the algorithm for two important sets of points, the Chebyshev points and the nth roots of unity. Such points have applications in the numerical approximation of Cauchy singular integral equations. The time complexity of the algorithm, for these special sets of points, reduces to

O
A

(
n
log
⁡
n
)

{O_A}(n\log n)

.
Asymptotic expansion of ∫^{𝜋/2}₀𝐽²_{𝜈}(𝜆𝑐𝑜𝑠𝜃)𝑑𝜃
Authors: Wong R.
Published: June 2010

An asymptotic expansion is obtained, as

λ
→
+
∞

\lambda \to + \infty

, for the integral
\[

I
(
λ
)
=

∫
0

π

/

2

J
v
2

(
λ
cos
⁡
θ
)

d
θ
,

I(\lambda ) = \int _0^{\pi /2} {J_v^2(\lambda \cos \theta )\;d\theta ,}

\]
where

J
v

(
t
)

{J_v}(t)

is the Bessel function of the first kind and

v
>
−

1
2

v > - \tfrac {1}{2}

. This integral arises in studies of crystallography and diffraction theory. We show in particular that

I
(
λ
)
∼
ln
⁡
λ

/

λ
π

I(\lambda ) \sim \ln \lambda /\lambda \pi

.
Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers
Authors: Kannan R. , Lenstra A. K. , Lovász L.
Published: June 2010

We show that the binary expansions of algebraic numbers do not form secure pseudorandom sequences; given sufficiently many initial bits of an algebraic number, its minimal polynomial can be reconstructed, and therefore the further bits of the algebraic number can be computed. This also enables us to devise a simple algorithm to factor polynomials with rational coefficients. All algorithms work in polynomial time.
Convergence rates for regularized solutions
Authors: Lukas Mark A.
Published: June 2010

Given a first-kind integral equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper K u left-parenthesis x right-parenthesis equals integral Subscript 0 Superscript 1 Baseline upper K left-parenthesis x comma t right-parenthesis u left-parenthesis t right-parenthesis d t equals f left-parenthesis x right-parenthesis">

K

u
(
x
)
=

∫
0
1

K
(
x
,
t
)
u
(
t
)

d
t
=
f
(
x
)

\mathcal {K}u(x) = \int _0^1 {K(x,t)u(t)\,dt = f(x)}

\]
with discrete noisy data

d
i

=
f
(

x
i

)
+

ε
i

{d_i} = f({x_i}) + {\varepsilon _i}

,

i
=
1
,
2
,
…
,
n

i = 1,2, \ldots ,n

, let

u

n
α

{u_{n\alpha }}

be the minimizer in a Hilbert space W of the regularization functional
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 slash n right-parenthesis sigma-summation left-parenthesis script upper K u left-parenthesis x Subscript i Baseline right-parenthesis minus d Subscript i Baseline right-parenthesis Superscript 2 Baseline plus alpha double-vertical-bar u double-vertical-bar Subscript upper W Superscript 2">

(
1

/

n
)
∑

(

K

u
(

x
i

)
−

d
i

)
2

+
α

‖
u
‖

W
2

(1/n)\sum {(\mathcal {K}} u({x_i}) - {d_i}{)^2} + \alpha \left \| u \right \|_W^2

. It is shown that in any one of a wide class of norms, which includes

‖
⋅
‖

W

{\left \| \cdot \right \|_W}

, if

α
→
0

\alpha \to 0

in a certain way as

n
→
∞

n \to \infty

, then

u

n
α

{u_{n\alpha }}

converges to the true solution

u
0

{u_0}

. Convergence rates are obtained and are used to estimate the optimal regularization parameter

α
\alpha

.