bluuebunny/crossref_metadata_2025 huggingface dataset:
Finite element approximation of the 𝑝-Laplacian
Authors: Barrett John W. , Liu W. B.
Published: June 2010

In this paper we consider the continuous piecewise linear finite element approximation of the following problem: Given

p
∈
(
1
,
∞
)

p \in (1,\infty )

, f, and g, find u such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus nabla dot left-parenthesis StartAbsoluteValue nabla u EndAbsoluteValue Superscript p minus 2 Baseline nabla u right-parenthesis equals f in normal upper Omega subset-of double-struck upper R squared comma u equals g on partial-differential normal upper Omega period">

−
∇
⋅
(

|

∇
u

|

p
−
2

∇
u
)
=
f

in

Ω
⊂

R

2

,

u
=
g

on

∂
Ω
.

- \nabla \cdot (|\nabla u{|^{p - 2}}\nabla u) = f\quad {\text {in}}\;\Omega \subset {\mathbb {R}^2},\quad u = g\quad {\text {on}}\;\partial \Omega .

\]
The finite element approximation is defined over

Ω
h

{\Omega ^h}

, a union of regular triangles, yielding a polygonal approximation to

Ω
\Omega

. For sufficiently regular solutions u, achievable for a subclass of data f, g, and

Ω
\Omega

, we prove optimal error bounds for this approximation in the norm

W

1
,
q

(

Ω
h

)
,
q
=
p

{W^{1,q}}({\Omega ^h}),q = p

for

p
>
2

p > 2

and

q
∈
[
1
,
2
]

q \in [1,2]

for

p
>
2

p > 2

, under the additional assumption that

Ω
h

⊆
Ω

{\Omega ^h} \subseteq \Omega

. Numerical results demonstrating these bounds are also presented.
Analysis of the finite element variational crimes in the numerical approximation of transonic flow
Authors: Berger Harald , Feistauer Miloslav
Published: June 2010

The paper presents a detailed theory of the finite element approximations of two-dimensional transonic potential flow. We consider the boundary value problem for the full potential equation in a general bounded domainΩ\Omegawith mixed Dirichlet-Neumann boundary conditions. In the discretization of the problem we proceed as usual in practice: the domainΩ\Omegais approximated by a polygonal domain, conforming piecewise linear triangular elements are used, and the integrals are evaluated by numerical quadratures. Using a new version of entropy compactification of transonic flow and the theory of finite element variational crimes for nonlinear elliptic problems, we prove the convergence of approximate solutions to the exact physical solution of the continuous problem, provided its existence can be shown.
Asymptotically optimal error bounds for quadrature rules of given degree
Authors: Brass H.
Published: June 2010

If the quadrature rule Q is applied to the function f, then the error can in many situations be bounded by

ρ
s

(
Q
)

‖

f

(
s
)

‖

∞

{\rho _s}(Q){\left \| {{f^{(s)}}} \right \|_\infty }

, where

ρ
s

(
Q
)

{\rho _s}(Q)

is independent of f. We obtain the asymptotics of these numbers for the Gaussian method

Q
n

G

(
n
=
1
,
2
,
…
)

Q_n^{\text {G}}\;(n = 1,2, \ldots )

with very general weight functions and show that

ρ
s

(

Q
n

G

)

{\rho _s}(Q_n^{\text {G}})

is (asymptotically) an upper bound for

ρ
s

(
Q
)

{\rho _s}(Q)

, if Q is any quadrature rule with the same degree as

Q
n

G

Q_n^{\text {G}}

.
On strong pseudoprimes to several bases
Authors: Jaeschke Gerhard
Published: June 2010

With

ψ
k

{\psi _k}

denoting the smallest strong pseudoprime to all of the first k primes taken as bases we determine the exact values for

ψ
5

,

ψ
6

,

ψ
7

,

ψ
8

{\psi _5},{\psi _6},{\psi _7},{\psi _8}

and give upper bounds for

ψ
9

,

ψ

10

,

ψ

11

{\psi _9},{\psi _{10}},{\psi _{11}}

. We discuss the methods and underlying facts for obtaining these results.
Continued fractions and linear recurrences
Authors: Lenstra H. W. , Shallit J. O.
Published: June 2010

We prove that the numerators and denominators of the convergents to a real irrational number

θ
\theta

satisfy a linear recurrence with constant coefficients if and only if

θ
\theta

is a quadratic irrational. The proof uses the Hadamard Quotient Theorem of A. van der Poorten.
Iterative and semi-iterative methods for computing stationary probability vectors of Markov operators
Authors: Marek Ivo , Szyld Daniel B.
Published: June 2010

Iterative and semi-iterative methods for computing stationary probability vectors of Markov-type operators are proposed and their convergence properties are analyzed. The methods studied apply to certain classes of problems in infinite-dimensional spaces as well as to classical

n
×
n

n \times n

stochastic matrices.
Generalized Fibonacci and Lucas sequences and rootfinding methods
Authors: Muskat Joseph B.
Published: June 2010

Consider the sequences{un}\{ {u_n}\}and{vn}\{ {v_n}\}generated byun+1=pun−qun−1{u_{n + 1}} = p{u_n} - q{u_{n - 1}}andvn+1=pvn−qvn−1,n≥1{v_{n + 1}} = p{v_n} - q{v_{n - 1}},n \geq 1, whereu0=0,u1=1,v0=2,v1=p{u_0} = 0,{u_1} = 1,{v_0} = 2,{v_1} = p, withpandqreal and nonzero. The Fibonacci sequence and the Lucas sequence are special cases of{un}\{ {u_n}\}and{vn}\{ {v_n}\}, respectively. Definern=un+d/un,Rn=vn+d/vn{r_n} = {u_{n + d}}/{u_n},{R_n} = {v_{n + d}}/{v_n}, wheredis a positive integer. McCabe and Phillips showed that ford=1d = 1, applying one step of Aitken acceleration to any appropriate triple of elements of{rn}\{ {r_n}\}yields another element of{rn}\{ {r_n}\}. They also proved ford=1d = 1that if a step of the Newton-Raphson method or the secant method is applied to elements of{rn}\{ {r_n}\}in solving the characteristic equationx2−px+q=0{x^2} - px + q = 0, then the result is an element of{rn}\{ {r_n}\}. The above results are obtained ford>1d > 1. It is shown that if any of the above methods is applied to elements of{Rn}\{ {R_n}\}, then the result is an element of{rn}\{ {r_n}\}. The application of certain higher-order iterative procedures, such as Halley’s method, to elements of{rn}\{ {r_n}\}and{Rn}\{ {R_n}\}is also investigated.
Second-order absorbing boundary conditions for the wave equation in a rectangular domain
Authors: Sheen Dongwoo
Published: June 2010

We study finite element methods for the wave equation in a rectangular domain with a second-order absorbing boundary condition imposed on the boundary. For this problem there seems to be no known finite element method, although many finite difference methods have been proposed. A third-order energy, however, will be introduced which will be utilized to reduce our original second-order problem to a first-order symmetric dissipative hyperbolic system. Then, for this first-order system a weak formulation will be given and continuous-time and discrete-time Galerkin procedures will be investigated. Error estimates will also be given.
Finite element approximations of nonlinear elastic waves
Authors: Makridakis Charalambos G.
Published: June 2010

In this paper we study finite element methods for a class of problems of nonlinear elastodynamics. We discretize the equations in space using Galerkin methods. For the temporal discretization, the construction of our schemes is based on rational approximations of

cos
⁡
x

\cos x

and

e
x

{e^x}

. We analyze semidiscrete as well as second- and fourth-order accurate in time fully discrete methods for the approximation of the solution of the problem and prove optimal-order

L
2

{L^2}

error estimates. For some schemes a Taylor-type technique is used so that only linear systems of equations need be solved at each time step. In the proofs we need various estimates for a nonlinear elliptic projection, the proofs of which are also established in the paper.
The relative class numbers of imaginary cyclic fields of degrees 4,6,8, and 10
Authors: Girstmair Kurt
Published: June 2010

We express the relative class number of an imaginary abelian number fieldKof prime power conductor as a sort of Maillet determinant. Thereby we obtain explicit relative class number formulas for fieldsKof conductorp,p≥3p \geq 3prime, and degree2d=[K:Q]≤102d = [K:\mathbb {Q}] \leq 10, in terms of sums of 2d-power residues. In particular, tables are given forp≤10000p \leq 10000.
Accurate and efficient reconstruction of discontinuous functions from truncated series expansions
Authors: Eckhoff Knut S.
Published: June 2010

Knowledge of a truncated Fourier series expansion for a discontinuous

2
π

2\pi

-periodic function, or a truncated Chebyshev series expansion for a discontinuous nonperiodic function defined on the interval

[
−
1
,
1
]

[-1, 1]

, is used in this paper to accurately and efficiently reconstruct the corresponding discontinuous function. First an algebraic equation of degree M for the M locations of discontinuities in each period for a periodic function, or in the interval

(
−
1
,
1
)

(-1, 1)

for a nonperiodic function, is constructed. The M coefficients in that algebraic equation of degree M are obtained by solving a linear algebraic system of equations determined by the coefficients in the known truncated expansion. By solving an additional linear algebraic system for the M jumps of the function at the calculated discontinuity locations, we are able to reconstruct the discontinuous function as a linear combination of step functions and a continuous function.
Spectral methods for periodic initial value problems with nonsmooth data
Authors: Dutt Pravir , Singh A. K.
Published: June 2010

In this paper we consider hyperbolic initial value problems subject to periodic boundary conditions with nonsmooth data. We show that if we filter the data and solve the problem by the Galerkin-Collocation method, recently proposed by us, then we can recover pointwise values with spectral accuracy, provided that the actual solution is piecewise smooth. For this we have to perform a local smoothing of the computed solution.
Statistical evidence for small generating sets
Authors: Bach Eric , Huelsbergen Lorenz
Published: June 2010

For an integer n, let

G
(
n
)

G(n)

denote the smallest x such that the primes

≤
x

\leq x

generate the multiplicative group modulo n. We offer heuristic arguments and numerical data supporting the idea that
\[

G
(
n
)
≤

(
log
⁡
2

)

−
1

log
⁡
n
log
⁡
log
⁡
n

G(n) \leq {(\log 2)^{ - 1}}\log n\log \log n

\]
asymptotically. We believe that the coefficient

1

/

log
⁡
2

1/\log 2

is optimal. Finally, we show the average value of

G
(
n
)

G(n)

for

n
≤
N

n \leq N

is at least
\[

(
1
+
o
(
1
)
)
log
⁡
log
⁡
N
log
⁡
log
⁡
log
⁡
N
,

(1 + o(1))\log \log N\log \log \log N,

\]
and give a heuristic argument that this is also an upper bound. This work gives additional evidence, independent of the ERH, that primality testing can be done in deterministic polynomial time; if our bound on

G
(
n
)

G(n)

is correct, there is a deterministic primality test using

O

(
log
⁡
n

)
2

O{(\log n)^2}

multiplications modulo n.
How branching properties determine modular equations
Authors: Cohn Harvey
Published: June 2010

If a prime p is decomposed as

x
2

+
4

y
2

{x^2} + 4{y^2}

, the power

2
m

|

|

y

{2^m}||y

can be determined by an algorithm of polynomial efficiency based on use of singular moduli from the modular equation of order 2. The properties of the modular functions required in this algorithm are simple branching and parametrization properties, which in turn define the modular functions and equations (essentially uniquely). The well-known equations of "Klein’s Icosahedron" and their Hecke analogues come into play here, and to some extent they can be uniquely characterized in this fashion. The extraneous cases which arise are in some sense interesting analogues of modular equations.
Numerical computations concerning the ERH
Authors: Rumely Robert
Published: June 2010

This paper describes a computation which established the ERH to height 10000 for all primitive Dirichlet L-series with conductor

Q
≤
13

Q \leq 13

, and to height 2500 for all

Q
≤
72

Q \leq 72

, all composite

Q
≤
112

Q \leq 112

, and other moduli. The computations were based on Euler-Maclaurin summation. Care was taken to obtain mathematically rigorous results: the zeros were first located within

10

−
12

{10^{ - 12}}

, then rigorously separated using an interval arithmetic package. A generalized Turing Criterion was used to show there were no zeros off the critical line. Statistics about the spacings between zeros were compiled to test the Pair Correlation Conjecture and GUE hypothesis.
A fundamental modular identity and some applications
Authors: Blecksmith Richard , Brillhart John , Gerst Irving
Published: June 2010

We prove a six-parameter identity whose terms have the form

x
α

T
(

k
1

,

l
1

)
T
(

k
2

,

l
2

)

{x^\alpha }T({k_1},{l_1})T({k_2},{l_2})

, where

T
(
k
,
l
)
=

∑

−
∞

∞

x

k

n
2

+
l

n

T(k,l) = \sum \nolimits _{ - \infty }^\infty {{x^{k{n^2} + l\,n}}}

. This identity is then used to give a new proof of the familiar Ramanujan identity
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis x right-parenthesis upper G left-parenthesis x Superscript 11 Baseline right-parenthesis minus x squared upper G left-parenthesis x right-parenthesis upper H left-parenthesis x Superscript 11 Baseline right-parenthesis equals 1">

H
(
x
)
G
(

x

11

)
−

x
2

G
(
x
)
H
(

x

11

)
=
1

H(x)G({x^{11}}) - {x^2}G(x)H({x^{11}}) = 1

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis x right-parenthesis equals product Underscript n equals 0 Overscript normal infinity Endscripts left-bracket left-parenthesis 1 minus x Superscript 5 n plus 1 Baseline right-parenthesis left-parenthesis 1 minus x Superscript 5 n plus 4 Baseline right-parenthesis right-bracket Superscript negative 1">

G
(
x
)
=

∏

n
=
0

∞

[
(
1
−

x

5
n
+
1

)
(
1
−

x

5
n
+
4

)
]

−
1

G(x) = \prod \nolimits _{n = 0}^\infty {{{[(1 - {x^{5n + 1}})(1 - {x^{5n + 4}})]}^{ - 1}}}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis x right-parenthesis equals product Underscript n equals 0 Overscript normal infinity Endscripts left-bracket left-parenthesis 1 minus x Superscript 5 n plus 2 Baseline right-parenthesis left-parenthesis 1 minus x Superscript 5 n plus 3 Baseline right-parenthesis right-bracket Superscript negative 1">

H
(
x
)
=

∏

n
=
0

∞

[
(
1
−

x

5
n
+
2

)
(
1
−

x

5
n
+
3

)
]

−
1

H(x) = \prod \nolimits _{n = 0}^\infty {{{[(1 - {x^{5n + 2}})(1 - {x^{5n + 3}})]}^{ - 1}}}

. Two other identities, called "balanced

Q
2

{Q^2}

identities", are also established through its use.
Johann Faulhaber and sums of powers
Authors: Knuth Donald E.
Published: June 2010

Early 17th-century mathematical publications of Johann Faulhaber contain some remarkable theorems, such as the fact that the r-fold summation of

1
m

,

2
m

,
…
,

n
m

{1^m},{2^m}, \ldots ,{n^m}

is a polynomial in

n
(
n
+
r
)

n(n + r)

when m is a positive odd number. The present paper explores a computation-based approach by which Faulhaber may well have discovered such results, and solves a 360-year-old riddle that Faulhaber presented to his readers. It also shows that similar results hold when we express the sums in terms of central factorial powers instead of ordinary powers. Faulhaber’s coefficients can moreover be generalized to noninteger exponents, obtaining asymptotic series for

1
α

+

2
α

+
⋯
+

n
α

{1^\alpha } + {2^\alpha } + \cdots + {n^\alpha }

in powers of

n

−
1

(
n
+
1

)

−
1

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

.
Elliptic curves and primality proving
Authors: Atkin A. O. L. , Morain F.
Published: June 2010

The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. Problema, numeros primos a compositis dignoscendi, hosque in factores suos primos resolvendi, ad gravissima ac utilissima totius arithmeticae pertinere, et geometrarum tum veterum tum recentiorum industriam ac sagacitatem occupavisse, tam notum est, ut de hac re copiose loqui superfluum foret.
An upper bound in Goldbach’s problem
Authors: Deshouillers Jean-Marc , Granville Andrew , Narkiewicz Władysław , Pomerance Carl
Published: June 2010

It is clear that the number of distinct representations of a number n as the sum of two primes is at most the number of primes in the interval

[
n

/

2
,
n
−
2
]

[n/2,n - 2]

. We show that 210 is the largest value of n for which this upper bound is attained.
On solving the Diophantine equation 𝑥³+𝑦³+𝑧³=𝑘 on a vector computer
Authors: Heath-Brown D. R. , Lioen W. M. , te Riele H. J. J.
Published: June 2010

Recently, the first author has proposed a new algorithm for solving the Diophantine equation

x
3

+

y
3

+

z
3

=
k

{x^3} + {y^3} + {z^3} = k

, where k is a given nonzero integer. In this paper we present the detailed versions of this algorithm for some values of k given below, and we describe how we have optimized and run the algorithm on a Cyber 205 vector computer. A vectorized version of the Euclidean algorithm was written, which is capable of solving the equations

w
i

x
i

≡
1
mod

n
i

,
i
=
1
,
2
,
…

{w_i}{x_i} \equiv 1 \bmod {n_i},i = 1,2, \ldots

, at vector speed. Moreover, the basic double-precision arithmetic operations

(
+
,
−
,
×
,

/

)

( + , - , \times ,/)

were vectorized. The following cases were implemented and run:

k
=
3
,
30
,
2
,
20
,
39

k = 3,30,2,20,39

, and 42. For

k
=
3

k = 3

a range was searched which includes the cube

|

x

|

,

|

y

|

,

|

z

|

≤

10
8

|x|,|y|,|z| \leq {10^8}

; this considerably extends an earlier search in the cube

|

x

|

,

|

y

|

,

|

z

|

≤

2

16

|x|,|y|,|z| \leq {2^{16}}

. No solutions were found apart from the known ones (1, 1, 1) and

(
4
,
4
,
−
5
)

(4,4, - 5)

. For

k
=
30

k = 30

, which probably is, with

k
=
3

k = 3

, the case which has attracted most attention in the literature, no solution was found. It is the smallest case for which no solution is known and for which one has not been able to find a proof that no solution exists. For

k
=
2

k = 2

a parametric form solution is known, but we also found one which does not belong to this parametric form, viz.,

(
1214928
,
3480205
,
−
3528875
)

(1214928,3480205, - 3528875)

. For

k
=
20

k = 20

, several new large solutions were found in addition to several known ones; this case served as a (partial) check of the correctness of our program. Finally, for

k
=
39

k = 39

we found the first solution:

(
134476
,
117367
,
−
159380
)

(134476,117367, - 159380)

. Hence, this case can be dropped from the list of values of k

(
k
=
30
,
33
,
39
,
42
,
…
)

(k = 30,33,39,42, \ldots )

for which no solution is known (yet).
The Carmichael numbers up to 10¹⁵
Authors: Pinch R. G. E.
Published: June 2010

There are 105212 Carmichael numbers up to

10

15

{10^{15}}

: we describe the calculations. The numbers were generated by a back-tracking search for possible prime factorizations, and the computations checked by searching selected ranges of integers directly using a sieving technique, together with a "large-prime variation".
A subexponential algorithm for discrete logarithms over all finite fields
Authors: Adleman Leonard M. , DeMarrais Jonathan
Published: June 2010

There are numerous subexponential algorithms for computing discrete logarithms over certain classes of finite fields. However, there appears to be no published subexponential algorithm for computing discrete logarithms over all finite fields. We present such an algorithm and a heuristic argument that there exists ac∈ℜ>0c \in {\Re _{ > 0}}such that for all sufficiently large prime powerspn{p^n}, the algorithm computes discrete logarithms overGF(pn){\text {GF}}({p^n})within expected time:ec(log⁡(pn)log⁡log⁡(pn))1/2{e^{c{{(\log ({p^n})\log \log ({p^n}))}^{1/2}}}}.
Discrete weighted transforms and large-integer arithmetic
Authors: Crandall Richard , Fagin Barry
Published: June 2010

It is well known that Discrete Fourier Transform (DFT) techniques may be used to multiply large integers. We introduce the concept of Discrete Weighted Transforms (DWTs) which, in certain situations, substantially improve the speed of multiplication by obviating costly zero-padding of digits. In particular, when arithmetic is to be performed modulo Fermat Numbers

2

2
m

+
1

{2^{{2^m}}} + 1

, or Mersenne Numbers

2
q

−
1

{2^q} - 1

, weighted transforms effectively reduce FFT run lengths. We indicate how these ideas can be applied to enhance known algorithms for general multiplication, division, and factorization of large integers.
Solving homogeneous linear equations over 𝐺𝐹(2) via block Wiedemann algorithm
Authors: Coppersmith Don
Published: June 2010

We propose a method of solving large sparse systems of homogeneous linear equations over

G
F
(
2
)

GF(2)

, the field with two elements. We modify an algorithm due to Wiedemann. A block version of the algorithm allows us to perform 32 matrix-vector operations for the cost of one. The resulting algorithm is competitive with structured Gaussian elimination in terms of time and has much lower space requirements. It may be useful in the last stage of integer factorization.
Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalue problem
Authors: Bai Zhaojun
Published: June 2010

This paper presents an error analysis of the Lanczos algorithm in finite-precision arithmetic for solving the standard nonsymmetric eigenvalue problem, if no breakdown occurs. An analog of Paige’s theory on the relationship between the loss of orthogonality among the Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos algorithm is discussed. The theory developed illustrates that in the nonsymmetric Lanczos scheme, if Ritz values are well conditioned, then the loss of biorthogonality among the computed Lanczos vectors implies the convergence of a group of Ritz triplets in terms of small residuals. Numerical experimental results confirm this observation.
Consistency estimates for a double-layer potential and application to the numerical analysis of the boundary-element approximation of acoustic scattering by a penetrable object
Authors: Bendali A. , Souilah M.
Published: June 2010

The consistency of the approximation of a double-layer potential when using a boundary-element method and approximating the curved boundary by a polygonal curve in two-dimensional problems or by a polyhedral surface in three-dimensional ones is investigated. The results are applied to the numerical analysis of the approximation of a model problem: the diffraction of a time-harmonic acoustic wave by a penetrable object.
Why nonconservative schemes converge to wrong solutions: error analysis
Authors: Hou Thomas Y. , LeFloch Philippe G.
Published: June 2010

This paper attempts to give a qualitative and quantitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws. We show that these schemes converge strongly in

L

loc

1

L_{\operatorname {loc}}^1

norm to the solution of an inhomogeneous conservation law containing a Borel measure source term. Moreover, we analyze the properties of this Borel measure, and derive a sharp estimate for the

L
1

{L^1}

error between the limit function given by the scheme and the correct solution. In general, the measure source term is of the order of the entropy dissipation measure associated with the scheme. In certain cases, the error can be small for short times, which makes it difficult to detect numerically. But generically, such an error will grow in time, and this would lead to a large error for large-time calculations. Finally, we show that a local correction of any high-order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution.
Polynomial approximations of functions with endpoint singularities and product integration formulas
Authors: Mastroianni Giuseppe , Monegato Giovanni
Published: June 2010

Several problems of mathematical physics lead to Fredholm integral equations of the second kind where the kernels are either weakly or strongly singular and the known terms are smooth. These equations have solutions which are smooth in the whole interval of integration except at the endpoints where they have mild singularities. In this paper we derive new pointwise and uniform polynomial approximation error estimates for that type of function. These estimates are then used to obtain bounds for the remainder terms of interpolatory product rules, based on the zeros of classical Jacobi orthogonal polynomials, that have been proposed for the discretization of integrals of the form
\[

∫

−
1

1

k
(
x
,
y
)
f
(
x
)
d
x
,

\int _{ - 1}^1 {k(x,y)f(x)dx,}

\]
appearing in the integral equations mentioned above.
On the stability of variable stepsize rational approximations of holomorphic semigroups
Authors: Palencia C.
Published: June 2010

We consider variable stepsize time approximations of holomorphic semigroups on general Banach spaces. For strongly

A

(
θ
)

{\text {A}}(\theta )

-acceptable rational functions a general stability theorem is proved, which does not impose any constraint on the ratios between stepsizes.
𝐵-convergence properties of multistep Runge-Kutta methods
Authors: Li Shou Fu
Published: June 2010

By using the theory of B-convergence for general linear methods to the special case of multistep Runge-Kutta methods, a series of B-convergence results for multistep Runge-Kutta methods is obtained, and it is proved that the family of algebraically stable r-step s-stage multistep Runge-Kutta methods with parameters

α
1

,

α
2

,
…
,

α
r

{\alpha _1},{\alpha _2}, \ldots ,{\alpha _r}

presented by Burrage in 1987 is optimally Bconvergent of order at least s, and B-convergent of order

s
+
1

s + 1

, provided that

r
≥
s

r \geq s

and

α
j

>
0
,
j
=
1
,
2
,
…
,
r

{\alpha _j} > 0,j = 1,2, \ldots ,r

. Furthermore, this family of methods is optimally B-convergent of order

s
+
1

s + 1

if some other additional conditions are satisfied.
On the existence of Turyn sequences of length less than 43
Authors: Edmondson Genet M. , Seberry Jennifer , Anderson Malcolm R.
Published: June 2010

Some theoretical results and computational algorithms are described which verify previous calculations, prove some theoretical nonexistence results, and totally enumerate all inequivalent Turyn sequences of length less than 43, that is, the longer sequence has length less than 43.
Computation of the zeros of 𝑝-adic 𝐿-functions. II
Authors: Ernvall R. , Metsänkylä T.
Published: June 2010

The authors have carried out a computational study of the zeros of Kubota-Leopoldt p-adic L-functions. Results of this study have appeared recently in a previous article. The present paper is a sequel to that article, dealing with the computation of the zeros under certain conditions that complicate the original situation.
Remarks on mixed finite element methods for problems with rough coefficients
Authors: Falk Richard S. , Osborn John E.
Published: June 2010

This paper considers the finite element approximation of elliptic boundary value problems in divergence form with rough coefficients. The solution of such problems will, in general, be rough, and it is well known that the usual (Ritz or displacement) finite element method will be inaccurate in general. The purpose of the paper is to help clarify the issue of whether the use of mixed variational principles leads to finite element schemes, i.e., to mixed methods, that are more accurate than the Ritz or displacement method for such problems. For one-dimensional problems, it is well known that certain mixed methods are more accurate and robust than the Ritz method for problems with rough coefficients. Our results for two-dimensional problems are mostly of a negative character. Through an examination of examples, we show that certain standard mixed methods fail to provide accurate approximations for problems with rough coefficients except in some special situations.
Divergence of continued fractions related to hypergeometric series
Authors: Lorentzen Lisa
Published: June 2010

Let

K
(

a
n

/

b
n

)

K({a_n}/{b_n})

be a limit periodic continued fraction of elliptic type; i.e.,

a
n

→
a

{a_n} \to a

and

b
n

→
b

{b_n} \to b

, where

a

/

(
b
+
w
)

a/(b + w)

is an elliptic linear fractional transformation of w. We show that if

∑

|

a
n

−
a

|

>
∞

\sum {|{a_n} - a| > \infty }

and

∑

|

b
n

−
b

|

>
∞

\sum {|{b_n} - b| > \infty }

, then

K
(

a
n

/

b
n

)

K({a_n}/{b_n})

diverges. This generalizes the well-known Stern-Stolz Theorem. The Gauss continued fraction (related to hypergeometric functions) is used as an example. We also give an example where

a
n

−
a
=

O

(

n

−
1

)

{a_n} - a = \mathcal {O}({n^{ - 1}})

and

b
n

=
b
=
1

{b_n} = b = 1

and

K
(

a
n

/

b
n

)

K({a_n}/{b_n})

converges. The divergence result is also generalized further.
Error bounds for Gauss-Kronrod quadrature formulae
Authors: Ehrich Sven
Published: June 2010

The Gauss-Kronrod quadrature formula

Q

2
n
+
1

G
K

Q_{2n + 1}^{GK}

, is used for a practical estimate of the error

R
n
G

R_n^G

of an approximate integration using the Gaussian quadrature formula

Q
n
G

Q_n^G

. Studying an often-used theoretical quality measure, for

Q

2
n
+
1

G
K

Q_{2n + 1}^{GK}

we prove best presently known bounds for the error constants
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript s Baseline left-parenthesis upper R Subscript 2 n plus 1 Superscript upper G upper K Baseline right-parenthesis equals sup Underscript double-vertical-bar f Superscript left-parenthesis s right-parenthesis Baseline double-vertical-bar Subscript normal infinity Baseline less-than-or-equal-to 1 Endscripts StartAbsoluteValue upper R Subscript 2 n plus 1 Superscript upper G upper K Baseline left-bracket f right-bracket EndAbsoluteValue">

c
s

(

R

2
n
+
1

G
K

)
=

sup

‖

f

(
s
)

‖

∞

≤
1

|

R

2
n
+
1

G
K

[
f
]

|

{c_s}(R_{2n + 1}^{GK}) = \sup \limits _{{{\left \| {{f^{(s)}}} \right \|}_\infty } \leq 1} |R_{2n + 1}^{GK}[f]|

\]
in the case

s
=
3
n
+
2
+
κ
,
κ
=

⌊

n
+
1

2

⌋

−

⌊

n
2

⌋

s = 3n + 2 + \kappa ,\kappa = \left \lfloor {\frac {{n + 1}}{2}} \right \rfloor - \left \lfloor {\frac {n}{2}} \right \rfloor

. A comparison with the Gaussian quadrature formula

Q

2
n
+
1

G

Q_{2n + 1}^G

shows that there exist quadrature formulae using the same number of nodes but having considerably better error constants.
On the spline collocation method for the single-layer heat operator equation
Authors: Hamina Martti , Saranen Jukka
Published: June 2010

We consider a boundary element collocation method for the heat equation. As trial functions we use the tensor products of continuous piecewise linear splines with collocation at the nodal points. Convergence and stability is proved in the case where the spatial domain is a disc. Moreover, practical implementation is discussed in some detail. Numerical experiments confirm our results.
On perturbations of matrix pencils with real spectra
Authors: Li Ren Cang
Published: June 2010

Perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil

A
−
λ
B

A - \lambda B

with real spectrum are developed. It is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices. The applications of these bounds to the spectral variations of definite pencils are conducted in such a way that extra attention is paid to their peculiarities so as to derive more sophisticated perturbation bounds. Our results for generalized eigenvalues are counterparts of some celebrated theorems for the spectral variations of Hermitian matrices such as the Weyl-Lidskii theorem and the Hoffman-Wielandt theorem; and those for generalized eigenspaces are counterparts of the celebrated Davis-Kahan

sin
⁡
θ
,
sin
⁡
2
θ

\sin \theta ,\sin 2\theta

theorems for the eigenspace variations of Hermitian matrices. The paper consists of two parts. Part I is for generalized eigenvalue perturbations, while Part II deals with generalized eigenspace perturbations.
A finite difference method for symmetric positive differential equations
Authors: Liu Jinn Liang
Published: June 2010

A finite difference method is developed for solving symmetric positive differential equations in the sense of Friedrichs. The method is applicable to partial differential equations of mixed type with more general boundary conditions. The method is shown to have a convergence rate of

O
(

h

1

/

2

)

O({h^{1/2}})

, h being the size of mesh grid. Some numerical results are presented for a model problem of forward-backward heat equations.
A new formula for box splines on three-directional meshes
Authors: Neuman Edward
Published: June 2010

A new formula for s-variate box splines on three-directional meshes is derived. An application to evaluation of a certain multiple integral is included.
The 3-modular characters of the Rudvalis sporadic simple group and its covering group
Authors: Hiss Gerhard
Published: June 2010

The decomposition numbers of the Rudvalis sporadic simple group and of its double covering group modulo 3 are completely determined. The results were obtained using the computer algebra system MOC developed by Jansen, Lux, Parker, and the author, but the proofs are given in conventional form and can be verified by hand with the ordinary character table.
The sigma-SOR algorithm and the optimal strategy for the utilization of the SOR iterative method
Authors: Woźnicki Zbigniew I.
Published: June 2010

The paper describes, discusses, and numerically illustrates the method for obtaining a priori estimates of the optimum relaxation factor in the SOR iteration method. The computational strategy of this method uses the so-called Sigma-SOR algorithm based on the theoretical result proven in the paper. The method presented is especially efficient for problems with slowly convergent iteration process and in this case is strongly competitive with adaptive procedures used for determining dynamically the optimum relaxation factor during the course of the SOR solution.
A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations
Authors: Verfürth R.
Published: June 2010

We give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems. In a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual. In a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite-dimensional approximation of the residual. This quantity can easily be evaluated, and for many practical applications sharp explicit upper and lower bounds are readily obtained. The general results are then applied to finite element discretizations of scalar quasi-linear elliptic partial differential equations of 2nd order, the eigenvalue problem for scalar linear elliptic operators of 2nd order, and the stationary incompressible Navier-Stokes equations. They immediately yield a posteriori error estimates, which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution.
On the periods of generalized Fibonacci recurrences
Authors: Brent Richard P.
Published: June 2010

We give a simple condition for a linear recurrence

(
mod

2
w

)

\pmod {2^w}

of degree r to have the maximal possible period

2

w
−
1

(

2
r

−
1
)

{2^{w - 1}}({2^r} - 1)

. It follows that the period is maximal in the cases of interest for pseudorandom number generation, i.e., for three-term linear recurrences defined by trinomials which are primitive

(
mod

2
)

\pmod 2

and of degree

r
>
2

r > 2

. We consider the enumeration of certain exceptional polynomials which do not give maximal period, and list all such polynomials of degree less than 15.
A special extension of Wieferich’s criterion
Authors: Cikánek Petr
Published: June 2010

The following theorem is proved in this paper: "If the first case of Fermat’s Last Theorem does not hold for sufficiently large prime l, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript x Endscripts x Superscript l minus 2 Baseline left-bracket StartFraction k l Over upper N EndFraction greater-than x greater-than StartFraction left-parenthesis k plus 1 right-parenthesis l Over upper N EndFraction right-bracket identical-to 0 left-parenthesis mod l right-parenthesis">

∑
x

x

l
−
2

[

k
l

N

>
x
>

(
k
+
1
)
l

N

]

≡
0

(
mod

l
)

\sum \limits _x {{x^{l - 2}}} \left [ {\frac {{kl}}{N} > x > \frac {{(k + 1)l}}{N}} \right ] \equiv 0\quad \pmod l

\]
for all pairs of positive integers

N
,
k
,
N
≤
94
,
0
≤
k
≤
N
−
1

N, k,N \leq 94, 0 \leq k \leq N - 1

." The proof of this theorem is based on a recent paper of Skula and uses computer techniques.
On the stability analysis of boundary conditions for the wave equation by energy methods. I. The homogeneous case
Authors: Ha-Duong T. , Joly P.
Published: June 2010

We reconsider the stability theory of boundary conditions for the wave equation from the point of view of energy techniques. We study, for the case of the homogeneous half-space, a large class of boundary conditions including the so-called absorbing conditions. We show that the results of strong stability in the sense of Kreiss, studied from the point of view of the modal analysis by Trefethen and Halpern, always correspond to the decay in time of a particular energy. This result leads to the derivation of new estimates for the solution of the associated mixed problem.
The structure of the projective indecomposable modules of 3̂𝑀₂₂ in characteristic 2
Authors: Lempken W. , Staszewski R.
Published: June 2010

This paper presents the socle series of the projective indecomposable modules for the triple cover

3
^

M

22

\hat 3{M_{22}}

in characteristic 2. The results are obtained by computational means; the methods as well as the constructive approach are explained.
Factoring polynomials over finite fields using differential equations and normal bases
Authors: Niederreiter Harald
Published: June 2010

The deterministic factorization algorithm for polynomials over finite fields that was recently introduced by the author is based on a new type of linearization of the factorization problem. The main ingredients are differential equations in rational function fields and normal bases of field extensions. For finite fields of characteristic 2, it is known that this algorithm has several advantages over the classical Berlekamp algorithm. We develop the algorithm in a general framework, and we show that it is feasible for arbitrary finite fields, in the sense that the linearization can be achieved in polynomial time.
Some remarks on the 𝑎𝑏𝑐-conjecture
Authors: Browkin J. , Brzeziński J.
Published: June 2010

Let

r
(
x
)

r(x)

be the product of all distinct primes dividing a nonzero integer x . The abc-conjecture says that if a, b, c are nonzero relatively prime integers such that

a
+
b
+
c
=
0

a + b + c = 0

, then the biggest limit point of the numbers
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction log max left-parenthesis StartAbsoluteValue a EndAbsoluteValue comma StartAbsoluteValue b EndAbsoluteValue comma StartAbsoluteValue c EndAbsoluteValue right-parenthesis Over log r left-parenthesis a b c right-parenthesis EndFraction">

log
⁡
max
(

|

a

|

,

|

b

|

,

|

c

|

)

log
⁡
r
(
a
b
c
)

\frac {{\log \max (|a|,|b|,|c|)}}{{\log r(abc)}}

\]
equals 1. We show that in a natural anologue of this conjecture for

n
≥
3

n \geq 3

integers, the largest limit point should be replaced by at least

2
n
−
5

2n - 5

. We present an algorithm leading to numerous examples of triples a, b, c for which the above quotients strongly deviate from the conjectural value 1.
Rank-one Drinfel’d modules on elliptic curves
Authors: Dummit D. S. , Hayes David
Published: June 2010

The sgn-normalized rank-one Drinfeld modules

ϕ
\phi

associated with all elliptic curves E over

F

q

{\mathbb {F}_q}

for

4
≤
q
≤
13

4 \leq q \leq 13

are computed in explicit form. (Such

ϕ
\phi

for

q
>
4

q > 4

were computed previously.) These computations verify a conjecture of Dorman on the norm of

j
(
ϕ
)
=

a

q
+
1

j(\phi ) = {a^{q + 1}}

and also suggest some interesting new properties of

ϕ
\phi

. We prove Dorman’s conjecture in the ramified case. We also prove the formula

deg
⁡
N
(
a
)
=
q
(

h
k

−
1
+
q
)

\deg N(a) = q({h_k} - 1 + q)

, where

N
(
a
)

N(a)

is the norm of a and

h
k

{h_k}

is the class number of

k
=

F

q

(
E
)

k = {\mathbb {F}_q}(E)

. We describe a remarkable conjectural property of the trace of a in even characteristic that holds in all the examples.
An acceleration of the Niederreiter factorization algorithm in characteristic 2
Authors: Göttfert Rainer
Published: June 2010

A new deterministic factorization algorithm for polynomials over finite fields was recently developed by Niederreiter. The bottleneck in this algorithm is the last stage in which the irreducible factors of the polynomial are derived from the solutions of a system of linear equations. In this paper, we consider finite fields of characteristic 2, and we show that in this case there is a more efficient approach to the last stage of the Niederreiter algorithm, which speeds up the algorithm considerably.
The determination of the imaginary abelian number fields with class number one
Authors: Yamamura Ken
Published: June 2010

In this paper, we determine all the imaginary abelian number fields with class number one. There exist exactly 172 imaginary abelian number fields with class number one. The maximal conductor of these fields is10921=67⋅16310921 = 67 \cdot 163, which is the conductor of the biquadratic number fieldQ(−67,−163){\mathbf {Q}}(\sqrt { - 67} ,\sqrt { - 163} ).
Polar generation of random variates with the 𝑡-distribution
Authors: Bailey Ralph W.
Published: June 2010

The "polar" method of Box and Muller uses two independent uniform variates in order to generate two independent normal variates. It can be adapted so that two variates from Student’s t-distribution with parameter

ν
\nu

are generated, though the two variates are now not independent. An algorithm based on the polar method is exact, inexpensive, and valid for all

ν
>
0

\nu > 0

.
On a theorem of C. Posse concerning Gaussian quadrature of Chebyshev type
Authors: Förster Klaus-Jürgen
Published: June 2010

We consider (n, m) Chebyshev formulae of algebraic degree m using n nodes. The aim of this short note is to show that by a simple algebraic method C. Posse’s theorem concerning Gaussian quadrature of Chebyshev type can be improved. Furthermore, we given an application of this method to Gauss-Kronrod quadrature of Chebyshev type.
A table of quintic number fields
Authors: Schwarz A. , Pohst M. , Diaz y Diaz F.
Published: June 2010

All algebraic number fields F of degree 5 and absolute discriminant less than

2
×

10
7

2 \times {10^7}

(totally real fields), respectively

5
×

10
6

5 \times {10^6}

(other signatures) are determined. We describe the methods which we applied and list significant data.
Long-time numerical solution of a parabolic equation with memory
Authors: Thomée Vidar , Wahlbin Lars B.
Published: June 2010

Long-time stability and convergence properties of two time-discretization methods for an integro-differential equation of parabolic type are studied. The methods are based on the standard backward Euler and second-order backward differencing methods. The memory term is approximated by a quadrature rule, with emphasis on such rules with reduced computational memory requirements. Discretization of the spatial partial differential operators by the finite element method is also considered.
Algorithms for optimal discontinuous piecewise linear and constant 𝐿₂ fits to continuous functions with adjustable nodes in one and two dimensions
Authors: Baines M. J.
Published: June 2010

In this paper a direct variational approach (with nonstandard variations) is used to generate algorithms to determine optimal discontinuous piecewise linear and piecewise constant

L
2

{L_2}

fits to a continuous function of one or two variables with adjustable nodes. In the one-variable case the algorithm is fast and robust, the mesh cannot tangle, and the resulting fits are continuous a.e. In the two-variable case, on an adjustable triangular grid, the algorithm is less robust but gives good results for particular functions possessing a single steep feature. The extension to higher dimensions is straightforward.
The 𝑘-dimensional distribution of combined GFSR sequences
Authors: Tezuka Shu
Published: June 2010

We develop an efficient method for analysis of the k-dimensional distribution of combinations of several GFSR sequences by bitwise exclusive-or operations. First, we introduce the notion of a resolution-wise lattice structure for GFSR sequences, and show that by applying a theorem of Couture to this type of lattice, we obtain a precise description of k-dimensional distribution of combined GFSR sequences in the same way as for combined Tausworthe sequences. Finally, we apply this method to the combination of two different Twisted GFSR generators, which were recently proposed by Matsumoto and Kurita, and investigate the order of equidistribution of the combined sequence.
A characterization of positive quadrature formulae
Authors: Xu Yuan
Published: June 2010

A positive quadrature formula with n nodes which is exact for polynomials of degree

2
n
−
r
−
1
,
0
≤
r
≤
n

2n - r - 1,0 \leq r \leq n

, is based on the zeros of certain quasi-orthogonal polynomials of degree n. We show that the quasi-orthogonal polynomials that lead to the positive quadrature formulae can all be expressed as characteristic polynomials of a symmetric tridiagonal matrix with positive subdiagonal entries. As a consequence, for a fixed n, every positive quadrature formula is a Gaussian quadrature formula for some nonnegative measure.
Computing irreducible representations of supersolvable groups
Authors: Baum Ulrich , Clausen Michael
Published: June 2010

Recently, it has been shown that the ordinary irreducible representations of a supersolvable group G of order n given by a power-commutator presentation can be constructed in time

O
(

n
2

log
⁡
n
)

O({n^2}\log n)

. We present an improved algorithm with running time

O
(
n
log
⁡
n
)

O(n\log n)

.
Error estimates for a finite element method for the drift-diffusion semiconductor device equations: the zero diffusion case
Authors: Cockburn Bernardo , Triandaf Ioana
Published: June 2010

In this paper new error estimates for an explicit finite element method for numerically solving the so-called zero-diffusion unipolar model (a one-dimensional simplified version of the drift-diffusion semiconductor device equations) are obtained. The method, studied in a previous paper, combines a mixed finite element method using a continuous piecewise-linear approximation of the electric field, with an explicit upwinding finite element method using a piecewise-constant approximation of the electron concentration. By using a suitable extension of Kuznetsov approximation theory for scalar conservation laws, it is proved that, under proper hypotheses on the data, the

L
∞

(

L
1

)

{L^\infty }({L^1})

-error between the approximate and exact electron concentrations of the zero-diffusion unipolar model is of order

Δ

x

1

/

2

\Delta {x^{1/2}}

. These estimates are sharp.
Carmichael’s conjecture on the Euler function is valid below 10^{10,000,000}
Authors: Schlafly Aaron , Wagon Stan
Published: June 2010

Carmichael’s conjecture states that if

ϕ
(
x
)
=
n

\phi (x) = n

, then

ϕ
(
y
)
=
n

\phi (y) = n

for some

y
≠
x

y \ne x

(

ϕ
\phi

is Euler’s totient function). We show that the conjecture is valid for all x under

10

10
,
900
,
000

{10^{10,900,000}}

. The main new idea is the application of a prime-certification technique that allows us to very quickly certify the primality of the thousands of large numbers that must divide a counterexample.