bluuebunny/crossref_metadata_2025 huggingface dataset:
Positivity of the weights of extended Clenshaw-Curtis quadrature rules
Authors: Hasegawa Takemitsu , Sugiura Hirosi , Torii Tatsuo
Published: June 2010

We prove that some extended Clenshaw-Curtis quadrature rules have all weights positive. We also present extended Filippi rules of open type having all weights positive. Conjectures on the possibility of other positive quadrature rules embedded in Clenshaw-Curtis or Filippi rule are suggested.
Fourier analysis of multigrid methods for general systems of PDEs
Authors: Lötstedt Per , Gustafsson Bertil
Published: June 2010

Most iteration methods for solving boundary value problems can be viewed as approximations of a time-dependent differential equation. In this paper we show that the multigrid method has the effect of increasing the time-step for the smooth part of the solution leading back to an increase of the convergence rate. For the nonsmooth part the convergence is an effect of damping. Fourier analysis is used to find the relation between the convergence rate for multigrid methods and singlegrid methods. The analysis is performed for general partial differential equations and an arbitrary number of grids. The difference in the behavior of the iterations between first- and second-order equations is discussed. The theoretical results are confirmed in simple numerical experiments.
Constructing integral lattices with prescribed minimum. II
Authors: Plesken W. , Pohst M.
Published: June 2010

Integral laminated lattices with minimum 4 which are generated by vectors of minimum length are constructed systematically together with their automorphism groups. All lattices obtained lie in the Leech lattice.
On bivariate polynomial factorization over finite fields
Authors: Shparlinski Igor E.
Published: June 2010

This paper shows that a recently proposed approach of D. Q. Wan to bivariate factorization over finite fields, the univariate factoring algorithm of V. Shoup, and the new bound of this paper for the average number of irreducible divisors of polynomials of a given degree over a finite field can be used to design a bivariate factoring algorithm that is polynomial for "almost all" bivariate polynomials.
The structure of the projective indecomposable modules of the Suzuki group 𝑆𝑧(8) in characteristic 2
Authors: Schneider Gerhard J. A.
Published: June 2010

This paper describes the socle series of the projective indecomposable modules and of tensor products of simple modules for the simple group

Sz

(
8
)

{\text {Sz}}(8)

in characteristic 2. The results have been obtained by computational means and the various steps are described. The main algorithm was modified to allow for parallel execution on a network of workstations. This made possible the effective handling of modules of degree 4030.
New solutions of 𝑎^{𝑝-1}≡1 (mod 𝑝²)
Authors: Montgomery Peter L.
Published: June 2010

We tabulate solutions of

a

p
−
1

≡
1

(
mod

p
2

)

{a^{p - 1}} \equiv 1\;\pmod {p^2}

where

2
≤
a
≤
99

2 \leq a \leq 99

and where p is an odd prime,

p
>

2

32

p > {2^{32}}

.
Iterated absolute values of differences of consecutive primes
Authors: Odlyzko Andrew M.
Published: June 2010

Let

d
0

(
n
)
=

p
n

{d_0}(n) = {p_n}

, the nth prime, for

n
≥
1

n \geq 1

, and let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript k plus 1 Baseline left-parenthesis n right-parenthesis equals StartAbsoluteValue d Subscript k Baseline left-parenthesis n right-parenthesis minus d Subscript k Baseline left-parenthesis n plus 1 right-parenthesis EndAbsoluteValue">

d

k
+
1

(
n
)
=

|

d
k

(
n
)
−

d
k

(
n
+
1
)

|

{d_{k + 1}}(n) = |{d_k}(n) - {d_k}(n + 1)|

for

k
≥
0
,
n
≥
1

k \geq 0,n \geq 1

. A well-known conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that

d
k

(
1
)
=
1

{d_k}(1) = 1

for all

k
≥
1

k \geq 1

. This paper reports on a computation that verified this conjecture for

k
≤
π
(

10

13

)
≈
3
×

10

11

k \leq \pi ({10^{13}}) \approx 3 \times {10^{11}}

. It also discusses the evidence and the heuristics about this conjecture. It is very likely that similar conjectures are also valid for many other integer sequences.
How was 𝐹₆ factored?
Authors: Williams H. C.
Published: June 2010

In 1880 at the age of 82 Fortuné Landry factored the 20 digit number

F
6

=

2

64

+
1

{F_6} = {2^{64}} + 1

. How did he do it? Landry himself never described how he factored

F
6

{F_6}

; however, he did leave enough clues in his work and letters to provide some indication of the ideas with which he was working. In this paper we present a likely reconstruction of Landry’s technique.
Convergence of second-order schemes for isentropic gas dynamics
Authors: Chen Gui Qiang , Liu Jian-Guo
Published: June 2010

Convergence of a second-order shock-capturing scheme for the system of isentropic gas dynamics with

L
∞

{L^\infty }

initial data is established by analyzing the entropy dissipation measures. This scheme is modified from the classical MUSCL scheme to treat the vacuum problem in gas fluids and to capture local entropy near shock waves. Convergence of this scheme for the piston problem is also discussed.
Spectral viscosity approximations to multidimensional scalar conservation laws
Authors: Chen Gui Qiang , Du Qiang , Tadmor Eitan
Published: June 2010

We study the spectral viscosity (SV) method in the context of multidimensional scalar conservation laws with periodic boundary conditions. We show that the spectral viscosity, which is sufficiently small to retain the formal spectral accuracy of the underlying Fourier approximation, is large enough to enforce the correct amount of entropy dissipation (which is otherwise missing in the standard Fourier method). Moreover, we prove that because of the presence of the spectral viscosity, the truncation error in this case becomes spectrally small, independent of whether the underlying solution is smooth or not. Consequently, the SV approximation remains uniformly bounded and converges to a measure-valued solution satisfying the entropy condition, that is, the unique entropy solution. We also show that the SV solution has a bounded total variation, provided that the total variation of the initial data is bounded, thus confirming its strong convergence to the entropy solution. We obtain an

L
1

{L^1}

convergence rate of the usual optimal order one-half.
Generalized repunit primes
Authors: Dubner Harvey
Published: June 2010

Generalized repunits have the form

(

b
n

−
1
)

/

(
b
−
1
)

({b^n} - 1)/(b - 1)

. A table of generalized repunit primes and probable primes is presented for b up to 99 and large values of n.
Arithmetical study of a certain ternary recurrence sequence and related questions
Authors: Mignotte M. , Tzanakis N.
Published: June 2010

The complete solution in(n,y1,y2)∈Z3(n,{y_1},{y_2}) \in {{\mathbf {Z}}^3}of the Diophantine equation\[bn=±2y13y2{b_n} = \pm {2^{{y_1}}}{3^{{y_2}}}\]is given, where(bn)n∈Z{({b_n})_{n \in {\mathbf {Z}}}}is Berstel’s recurrence sequence defined by\[b0=b1=0,b2=1,bn+3=2bn+2−4bn+1+4bn.{b_0} = {b_1} = 0,\quad {b_2} = 1,\quad {b_{n + 3}} = 2{b_{n + 2}} - 4{b_{n + 1}} + 4{b_n}.\]
Which integers are representable as the product of the sum of three integers with the sum of their reciprocals?
Authors: Bremner Andrew , Guy Richard K. , Nowakowski Richard J.
Published: June 2010

For example, the integer 564 is so representable by the three integers 122 44200 50100028778116351171 17995213613513491867, -3460 69586 84255 04865 64589 22621 88752 08971 30654 24460 and 74807 1910153025 27837945836017146464 94820 59055 28060.
Average case error estimates for the strong probable prime test
Authors: Damgård Ivan , Landrock Peter , Pomerance Carl
Published: June 2010

Consider a procedure that chooses k-bit odd numbers independently and from the uniform distribution, subjects each number to t independent iterations of the strong probable prime test (Miller-Rabin test) with randomly chosen bases, and outputs the first number found that passes all t tests. Let

p

k
,
t

{p_{k,t}}

denote the probability that this procedure returns a composite number. We obtain numerical upper bounds for

p

k
,
t

{p_{k,t}}

for various choices of k, t and obtain clean explicit functions that bound

p

k
,
t

{p_{k,t}}

for certain infinite classes of k, t. For example, we show
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript 100 comma 10 Baseline less-than-or-equal-to 2 Superscript negative 44 Baseline comma p Subscript 300 comma 5 Baseline less-than-or-equal-to 2 Superscript negative 60 Baseline comma p Subscript 600 comma 1 Baseline less-than-or-equal-to 2 Superscript negative 75">

p

100
,
10

≤

2

−
44

,

p

300
,
5

≤

2

−
60

,

p

600
,
1

≤

2

−
75

{p_{100,10}} \leq {2^{ - 44}},{p_{300,5}} \leq {2^{ - 60}},{p_{600,1}} \leq {2^{ - 75}}

, and

p

k
,
1

≤

k
2

4

2
−

k

{p_{k,1}} \leq {k^2}{4^{2 - \sqrt k }}

for all

k
≥
2

k \geq 2

. In addition, we characterize the worst-case numbers with unusually many "false witnesses" and give an upper bound on their distribution that is probably close to best possible.
The Lehmer project
Authors: Lehmer D. H. , Lehmer Emma
Published: June 2010

It is shown that cyclic differences of cyclotomic periods can be useful in finding units in cyclic extensions of the rationals of degree less than or equal to 6. The polynomials for these differences are simpler than the polynomials for the corresponding periods. The cyclic differences depend on the choice of a primitive root; the question is raised as to which choices of primitive root yield units.
Fast Fourier transforms for symmetric groups: theory and implementation
Authors: Clausen Michael , Baum Ulrich
Published: June 2010

Recently, it has been proved that a Fourier transform for the symmetric group

S
n

{S_n}

based on Young’s seminormal form can be evaluated in less than

0.5
(

n
3

+

n
2

)
n
!

0.5({n^3} + {n^2})n!

arithmetic operations. We look at this algorithm in more detail and show that it allows an efficient software implementation using appropriate data structures. We also describe a similarly efficient algorithm for the inverse Fourier transform. We compare the time and memory requirements of our program to those of other existing implementations.
Image processing through reaction combined with nonlinear diffusion
Authors: Cottet G.-H. , Germain L.
Published: June 2010

We propose a method based on nonlinear diffusion and reaction for edge detection and contrast enhancement in image processing. We prove that the mathematical model is well posed and show numerically that the processed image can be observed on the asymptotic state of its solution. We illustrate the methods on test images and show on medical images how it can help to draw contours and detect one-dimensional coherent signals.
Quadrature formulas based on rational interpolation
Authors: Van Assche Walter , Vanherwegen Ingrid
Published: June 2010

We consider quadrature formulas based on interpolation using the basis functions

1

/

(
1
+

t
k

x
)

(
k
=
1
,
2
,
3
,
…
)

1/(1 + {t_k}x)\quad (k = 1,2,3, \ldots )

on

[
−
1
,
1
]

[ - 1,1]

, where

t
k

{t_k}

are parameters on the interval

(
−
1
,
1
)

( - 1,1)

. We investigate two types of quadratures: quadrature formulas of maximum accuracy which correctly integrate as many basis functions as possible (Gaussian quadrature), and quadrature formulas whose nodes are the zeros of the orthogonal functions obtained by orthogonalizing the system of basis functions (orthogonal quadrature). We show that both approaches involve orthogonal polynomials with modified weights which depend on the number of quadrature nodes. The asymptotic distribution of the nodes is obtained as well as various interlacing properties and monotonicity results for the nodes.
Order barriers and characterizations for continuous mono-implicit Runge-Kutta schemes
Authors: Muir Paul , Owren Brynjulf
Published: June 2010

The mono-implicit Runge-Kutta (MIRK) schemes, a subset of the family of implicit Runge-Kutta (IRK) schemes, were originally proposed for the numerical solution of initial value ODEs more than fifteen years ago. During the last decade, a considerable amount of attention has been given to the use of these schemes in the numerical solution of boundary value ODE problems, where their efficient implementation suggests that they may provide a worthwhile alternative to the widely used collocation schemes. Recent work in this area has seen the development of some software packages for boundary value ODEs based on these schemes. Unfortunately, these schemes lead to algorithms which provide only a discrete solution approximation at a set of mesh points over the problem interval, while the collocation schemes provide a natural continuous solution approximation. The availability of a continuous solution is important not only to the user of the software but also within the code itself, for example, in estimation of errors, defect control, mesh selection, and the provision of initial solution estimates for new meshes. An approach for the construction of a continuous solution approximation based on the MIRK schemes is suggested by recent work in the area of continuous extensions for explicit Runge-Kutta schemes for initial value ODEs. In this paper, we describe our work in the investigation of continuous versions of the MIRK schemes: (i) we give some lower bounds relating the stage order to the minimal number of stages for general continuous IRK schemes, (ii) we establish lower bounds on the number of stages needed to derive continuous MIRK schemes of orders 1 through 6, and (iii) we provide characterizations of these schemes having a minimal number of stages for each of these orders.
Bounds on the lattice rule criterion 𝑅
Authors: Joe Stephen
Published: June 2010

Lattice rules are used for the numerical integration of periodic functions over the s-dimensional unit cube. They are normally classified according to their ’rank’; in simple terms, the rank of a lattice rule is the minimum number of sums required to write it down. One criterion for measuring the ’goodness’ of a lattice rule is the quantity R which is the quadrature error for a certain test function. Bounds on R exist for rank-1 and rank-2 lattice rules, but not for lattice rules of higher rank. For

1
≤
m
≤
s

1 \leq m \leq s

, we shall look at certain rank-m rules and obtain bounds on R for them. These rank-m rules have

n
m

r

{n^m}r

quadrature points, where n and r are relatively prime numbers. In order to obtain these bounds, we make use of a result which shows that R may be considered to be the quadrature error obtained when a modified lattice rule with only r quadrature points is applied to a modified test function. Some numerical results are given.
The decomposition numbers of the Hecke algebra of type 𝐸*₆
Authors: Geck Meinolf
Published: June 2010

Let

E
6

(
q
)

{E_6}(q)

be the Chevalley group of type

E
6

{E_6}

, over a finite field with q elements, l be a prime not dividing q, and

H
R

(
q
)

{H_R}(q)

be the endomorphism ring of the permutation representation (over a valuation ring R with residue class field of characteristic l) of

E
6

(
q
)

{E_6}(q)

on the cosets of a standard Borel subgroup

B
(
q
)

B(q)

. Then the l-modular decomposition matrix

D
l

{D_l}

of the algebra

H
R

(
q
)

{H_R}(q)

is a submatrix of the l-modular decomposition matrix of the finite group

E
6

(
q
)

{E_6}(q)

. In this paper we determine the matrices

D
l

{D_l}

, for all l, q as above. For this purpose, we consider the generic Hecke algebra H associated with the finite Weyl group of type

E
6

{E_6}

over the ring

A
=

Z

[
v
,

v

−
1

]

A = \mathbb {Z}[v,{v^{ - 1}}]

of Laurent polynomials in an indeterminate v, and calculate the decomposition matrices of H which are associated with specializations of v to roots of unity over

Q

\mathbb {Q}

or values in a finite field. The computations were done by using the computer algebra systems MAPLE and GAP.
Ambiguous classes in quadratic fields
Authors: Mollin R. A.
Published: June 2010

We provide sufficient conditions for the class group of a quadratic field (with positive or negative discriminant) to be generated by ambiguous ideals. This investigation was motivated by a recent result of F. Halter-Koch, which we show is false.
Explicit primality criteria for ℎ⋅2^{𝑘}±1
Authors: Bosma Wieb
Published: June 2010

Algorithms are described to obtain explicit primality criteria for integers of the form

h
⋅

2
k

±
1

h \cdot {2^k} \pm 1

(in particular with h divisible by 3) that generalize classical tests for

2
k

±
1

{2^k} \pm 1

in a well-defined finite sense. Numerical evidence (including all cases with

h
>

10
5

h > {10^5}

) seems to indicate that these finite generalizations exist for every h, unless

h
=

4
m

−
1

h = {4^m} - 1

for some m, in which case it is proved they cannot exist.
Algebraic numbers close to both 0 and 1
Authors: Zagier D.
Published: June 2010

A recent theorem of Zhang asserts that
\[

H
(
α
)
+
H
(
1
−
α
)
≥
C

H(\alpha ) + H(1 - \alpha ) \geq C

\]
for all algebraic numbers

α
≠
0
,
1
,
(
1
±

−
3

)

/

2

\alpha \ne 0,1, (1 \pm \sqrt { - 3} )/2

, and some constant

C
>
0

C > 0

. An elementary proof of this, with a sharp value for the constant, is given (the optimal value of C is

1
2

log
⁡
(

1
2

(
1
+

5

)
)
=
0
,
2406
…

\tfrac {1}{2}\log (\tfrac {1}{2}(1 + \sqrt 5 )) = 0,2406 \ldots

, attained for eight values of

α
\alpha

) and generalizations to other curves are discussed.
Estimates of the least prime factor of a binomial coefficient
Authors: Erdős P. , Lacampagne C. B. , Selfridge J. L.
Published: June 2010

We estimate the least prime factor p of the binomial coefficient

(

k
N

)

\left ( {_k^N} \right )

for

k
≥
2

k \geq 2

. The conjecture that

p
≤
max
(
N

/

k
,
29
)

p \leq \max (N/k,29)

is supported by considerable numerical evidence. Call a binomial coefficient good if

p
>
k

p > k

. For

1
≤
i
≤
k

1 \leq i \leq k

write

N
−
k
+
i
=

a
i

b
i

N - k + i = {a_i}{b_i}

, where

b
i

{b_i}

contains just those prime factors

>
k

> k

, and define the deficiency of a good binomial coefficient as the number of i for which

b
i

=
1

{b_i} = 1

. Let

g
(
k
)

g(k)

be the least integer

N
>
k
+
1

N > k + 1

such that

(

k
N

)

\left ( {_k^N} \right )

is good. The bound

g
(
k
)
>
c

k
2

/

ln
⁡
k

g(k) > c{k^2}/\ln k

is proved. We conjecture that our list of 17 binomial coefficients with deficiency

>
1

> 1

is complete, and it seems that the number with deficiency 1 is finite. All

(

k
N

)

\left ( {_k^N} \right )

with positive deficiency and

k
≤
101

k \leq 101

are listed.
Numerical results for sums of five and seven biquadrates and consequences for sums of 19 biquadrates
Authors: Deshouillers J.-M. , Dress F.
Published: June 2010

We describe the algorithms which allowed us to show that all the integers congruent to 50 modulo 80 that lie in the interval

(
0.3651
×

10

12

,
1.0400
×

10

12

)

(0.3651 \times {10^{12}},1.0400 \times {10^{12}})

are sums of five biquadrates, and that all the integers congruent to 67 modulo 80 that lie in the interval

(
0.3651
×

10

12

,
9.5956
×

10

18

)

(0.3651 \times {10^{12}},9.5956 \times {10^{18}})

are sums of seven biquadrates. We finally describe some ascent lemmas that we use to deduce from the previous results that every integer not exceeding

10

448

{10^{448}}

is a sum of 19 biquadrates.
Periods of cusp forms and elliptic curves over imaginary quadratic fields
Authors: Cremona J. E. , Whitley E.
Published: June 2010

In this paper we explore the arithmetic correspondence between, on the one hand, (isogeny classes of) elliptic curves E defined over an imaginary quadratic field K of class number one, and on the other hand, rational newforms F of weight two for the congruence subgroups

Γ
0

(

n

)

{\Gamma _0}(\mathfrak {n})

, where n is an ideal in the ring of integers R of K. This continues work of the first author and forms part of the Ph.D. thesis of the second author. In each case we compute numerically the value of the L-series

L
(
F
,
s
)

L(F,s)

at

s
=
1

s = 1

and compare with the value of

L
(
E
,
1
)

L(E,1)

which is predicted by the Birch-Swinnerton-Dyer conjecture, finding agreement to several decimal places. In particular, we find that

L
(
F
,
1
)
=
0

L(F,1) = 0

whenever

E
(
K
)

E(K)

has a point of infinite order. Several examples are given in detail from the extensive tables computed by the authors.
Computation of the trivariate normal integral
Authors: Drezner Zvi
Published: June 2010

We propose a simple and efficient way to calculate trivariate normal probabilities. The algorithm is based on a formula for the partial derivative of the trivariate probability with respect to a correlation coefficient.
Generalizations of Dixon’s theorem on the sum of a ₃𝐹₂
Authors: Lavoie J.-L. , Grondin F. , Rathie A. K. , Arora K.
Published: June 2010

Twenty-three formulas, closely related to Dixon’s theorem in the theory of the generalized hypergeometric series, are obtained. Twenty-six limiting cases are also deduced.
Computation of topological degree using interval arithmetic, and applications
Authors: Aberth Oliver
Published: June 2010

A method is described for computing the topological degree of a mapping from

R
n

{R^n}

into

R
n

{R^n}

defined by n functions of n variables on a region specified as a product of n intervals, a generalized box B. The method is an adaptation of Kearfott’s method to boxes, and begins by checking the signs of the n functions on the boundary of B with interval arithmetic. On the basis of this check, a portion,

B

(
1
)

{B^{(1)}}

, of the boundary of B is designated for further investigation, and one of the n functions defining the mapping is dropped. The signs of the remaining functions are checked on the boundary of

B

(
1
)

{B^{(1)}}

. Again a portion,

B

(
2
)

{B^{(2)}}

, of the boundary of

B

(
1
)

{B^{(1)}}

is designated for further investigation, and another of the functions is dropped. On the nth cycle of the process, the topological degree finally is evaluated by determining the signs of a single function on a collection of isolated points, comprising the boundary of a region

B

(
n
−
1
)

{B^{(n - 1)}}

. When the topological degree is nonzero, there is at least one point inside B where the n functions are simultaneously zero. To locate such a point, the familiar bisection method for functions

f
(
x
)

f(x)

defined over an interval [a, b], using sign changes of

f
(
x
)

f(x)

, is easily generalized to apply to n functions defined over boxes, using the topological degree. For this application we actually use the topological degree

mod

2

\bmod \;2

, the crossing parity, because its computation is easier. If the n functions have all partial derivatives in the box B, with a nonzero Jacobian at any point where the functions are simultaneously zero, then all such points inside B can be located by another method, which also uses the crossing parity.
Deciphering singularities by discrete methods
Authors: Tourigny Yves , Grinfeld Michael
Published: June 2010

We consider the problem of estimating numerically the parameters of singularities of solutions of differential equations. We propose a novel approach which is based on discretizing the governing equation and "time-stepping" in the complex domain. Some applications to ordinary and partial differential equations are discussed.
Sinc-collocation method with orthogonalization for singular Poisson-like problems
Authors: Yin Guang Yan
Published: June 2010

This paper uses the Sine-collocation method to solve singular Poisson-like problems (a first- or higher-order partial derivative of the exact solution is unbounded on the boundary). A linear system is obtained which is the same as that obtained by using the Sinc-Galerkin method. With a smart choice of the stepsize and the number of the gridpoints, the orthogonalization technique is successfully applied to solve the linear system obtained, and a numerical approximation is obtained with an exponential accuracy

O
(
exp
⁡
(
−
c

N

1
2

)
)

O(\exp ( - c{N^{\frac {1}{2}}}))

, where N is a truncation parameter and c is a constant independent of N.
An inductive schema for computing conjugacy classes in permutation groups
Authors: Butler Greg
Published: June 2010

An approach to computing the conjugacy classes of elements of large permutation groups is presented in detail, and a prototype implementation is described. The approach builds on recent efficient algorithms for computing conjugacy classes of p-groups, and for computing Sylow subgroups of large permutation groups. Classes of elements of composite order are determined by recursively analyzing quotients of centralizers of p-elements.
Simple perfect squared squares and 2×1 squared rectangles of order 25
Authors: Duijvestijn A. J. W.
Published: June 2010

In this note tables of all simple perfect squared squares and simple

2
×
1

2 \times 1

perfect squared rectangles of order 25 are presented.
Maximum principle on the entropy and second-order kinetic schemes
Authors: Khobalatte Brahim , Perthame Benoît
Published: June 2010

We consider kinetic schemes for the multidimensional inviscid gas dynamics equations (compressible Euler equations). We prove that the discrete maximum principle holds for the specific entropy. This fixes the choice of the equilibrium functions necessary for kinetic schemes. We use this property to perform a second-order oscillation-free scheme, where only one slope limitation (for three conserved quantities in 1D) is necessary. Numerical results exhibit stability and strong convergence of the scheme.
On the computation of impasse points of quasilinear differential-algebraic equations
Authors: Rabier Patrick J. , Rheinboldt Werner C.
Published: June 2010

We present computational algorithms for the calculation of impasse points and higher-order singularities in quasi-linear differential-algebraic equations. Our method combines a reduction step, transforming the DAE into a singular ODE, with an augmentation procedure inspired by numerical bifurcation theory. Singularities are characterized by the vanishing of a scalar quantity that may be monitored along any trajectory. Two numerical examples with physical relevance are given.
On unit groups and class groups of quartic fields of signature (2,1)
Authors: Buchmann J. , Pohst M. , von Schmettow J. Graf
Published: June 2010

This is the third and last paper of a series, now completing the description of the unit group and class group of all quartic number fields F of discriminant

d
F

{d_F}

with

|

d
F

|

>

10
6

|{d_F}| > {10^6}

.
A quick way of obtaining an approximate solution to a Sturm-Liouville problem
Authors: Chambers Ll. G.
Published: June 2010

Use is made of the generalized consistency equation and the boundary conditions to obtain a quick approximate solution to a Sturm-Liouville problem.
Explicit inversive congruential pseudorandom numbers with power of two modulus
Authors: Eichenauer-Herrmann Jürgen , Ickstadt Katja
Published: June 2010

An explicit version of the inversive congruential method with power of two modulus for generating uniform pseudorandom numbers is introduced. Statistical independence properties of the generated sequences are studied by means of the serial test. The method of proof relies on a detailed analysis of certain exponential sums.
The numerical evaluation of a 2-D Cauchy principal value integral arising in boundary integral equation methods
Authors: Monegato Giovanni
Published: June 2010

In this paper we consider the problem of computing 2-D Cauchy principal value integrals of the form
\[

₯

S

F
(

P
0

;
P
)

d
P
,

P
0

∈
S
,

{\fint _S}F({P_0};P)\,dP,\qquad {P_0} \in S,

\]
where S is either a rectangle or a triangle, and

F
(

P
0

;
P
)

F({P_0};P)

is integrable over S, except at the point

P
0

{P_0}

where it has a second-order pole. Using polar coordinates, the integral is first reduced to the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript theta 1 Superscript theta 2 Baseline left-bracket u n k right-bracket Subscript 0 Superscript upper R left-parenthesis theta right-parenthesis Baseline StartFraction f left-parenthesis r comma theta right-parenthesis Over r EndFraction d r d theta comma">

∫

θ
1

θ
2

[
u
n
k

]
0

R
(
θ
)

f
(
r
,
θ
)

r

d
r

d
θ
,

\int _{{\theta _1}}^{{\theta _2}} {[unk]_0^{R(\theta )}\frac {{f(r,\theta )}}{r}dr\, d \theta ,}

\]
where

[
u
n
k
]

[unk]

denotes the finite part of the (divergent) integral. Then ad hoc products of one-dimensional quadrature rules of Gaussian type are constructed, and corresponding convergence results derived. Some numerical tests are also presented.
Recurrence formulas for multivariate orthogonal polynomials
Authors: Xu Yuan
Published: June 2010

In this paper, necessary and sufficient conditions are given so that multivariate orthogonal polynomials can be generated by a recurrence formula. As a consequence, orthogonal polynomials of total degree n in d variables that have

dim
⁡

Π
n
d

\dim \Pi _n^d

common zeros can now be constructed recursively. The result is important to the construction of Gaussian cubature formulas.
Iterated collocation methods for Volterra integral equations with delay arguments
Authors: Brunner Hermann
Published: June 2010

In this paper we give a complete analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for Volterra integral equations with constant delay. This analysis includes continuous collocation-based Volterra-Runge-Kutta methods as well as iterated collocation methods and their discretizations.
Irreducible finite integral matrix groups of degree 8 and 10
Authors: Souvignier Bernd
Published: June 2010

The lattices of eight- and ten-dimensional Euclidean space with irreducible automorphism group or, equivalently, the conjugacy classes of these groups inGLn(Z)\mathrm {GL}_n(\mathbb {Z})forn=8,10n = 8,10, are classified in this paper. The number of types is 52 in the casen=8n = 8, and 47 in the casen=10n = 10. As a consequence of this classification one has 26, resp. 46, conjugacy classes of maximal finite irreducible subgroups ofGL8(Z)\mathrm {GL}_8(\mathbb {Z}), resp.GL10(Z)\mathrm {GL}_{10}(\mathbb {Z}). In particular, each such group is absolutely irreducible, and therefore each of the maximal finite groups of degree 8 turns up in earlier lists of classifications.
Wavelet calculus and finite difference operators
Authors: McCormick Kent , Wells Raymond O.
Published: June 2010

This paper shows that the naturally induced discrete differentiation operators induced from a wavelet-Galerkin finite-dimensional approximation to a standard function space approximates differentiation with an error of orderO(h2d+2){\text {O}}({h^{2d + 2}}), wheredis the degree of the wavelet system. The degree of a wavelet system is defined as one less than the degree of the lowest-order nonvanishing moment of the fundamental wavelet. We consider in this paper compactly supported wavelets of the type introduced by Daubechies in 1988. The induced differentiation operators are described in terms of connection coefficients which are intrinsically defined functional invariants of the wavelet system (defined asL2{L^2}inner products of derivatives of wavelet basis functions with the basis functions themselves). These connection coefficients can be explicitly computed without quadrature and they themselves have key moment-vanishing properties proved in this paper which are dependent upon the degree of the wavelet system. This is the basis for the proof of the principal results concerning the degree of approximation of the differentiation operator by the wavelet-Galerkin discrete differentiation operator.
A remark concerning 𝑚-divisibility and the discrete logarithm in the divisor class group of curves
Authors: Frey Gerhard , Rück Hans-Georg
Published: June 2010

The aim of this paper is to show that the computation of the discrete logarithm in the m-torsion part of the divisor class group of a curve X over a finite field

k
0

{k_0}

(with

char

(

k
0

)

{\operatorname {char}}({k_0})

prime to m), or over a local field k with residue field

k
0

{k_0}

, can be reduced to the computation of the discrete logarithm in

k
0

(

ζ
m

)
∗

{k_0}{({\zeta _m})^ \ast }

. For this purpose we use a variant of the (tame) Tate pairing for Abelian varieties over local fields. In the same way the problem to determine all linear combinations of a finite set of elements in the divisor class group of a curve over k or

k
0

{k_0}

which are divisible by m is reduced to the computation of the discrete logarithm in

k
0

(

ζ
m

)
∗

{k_0}{({\zeta _m})^ \ast }

.
Some error estimates for the numerical approximation of surface integrals
Authors: Georg Kurt , Tausch Johannes
Published: June 2010

Recently, the first author introduced a new approach to the numerical quadrature of surface integrals in the context of boundary element methods. It is assumed that a global parametrization m of the surface is only indirectly given (e.g., via an iterative method) and that m is not accessible analytically. Of particular interest are parametrizations which are based on automatic triangulations of surfaces. In order to avoid an explicit reference to the partial derivatives of m, modified trapezoidal and midpoint rules were introduced. The present paper discusses some error estimates for these methods. The estimates are surprisingly difficult since

O

(

h
3

)

\mathcal {O}({h^3})

-terms have to be shown to cancel; this does not occur in the expansion of the standard rules.
On the lattice structure of certain linear congruential sequences related to AWC/SWB generators
Authors: Couture Raymond , L’Ecuyer Pierre
Published: June 2010

We analyze the lattice structure of certain types of linear congruential generators (LCGs), which include close approximations to the add-with-carry and subtract-with-borrow (AWC/SWB) random number generators introduced by Marsaglia and Zaman, and also to combinations of the latter with ordinary LCGs. It follows from our results that all these generators have an unfavorable lattice structure in large dimensions.
On a cell entropy inequality for discontinuous Galerkin methods
Authors: Jiang Guang Shan , Shu Chi-Wang
Published: June 2010

We prove a cell entropy inequality for a class of high-order discontinuous Galerkin finite element methods approximating conservation laws, which implies convergence for the one-dimensional scalar convex case.
Finding finite 𝐵₂-sequences with larger 𝑚-𝑎^{1/2}_{𝑚}
Authors: Zhang Zhen Xiang
Published: June 2010

A sequence of positive integers

a
1

>

a
2

>
⋯
>

a
m

{a_1} > {a_2} > \cdots > {a_m}

is called a (finite)

B
2

{B_2}

-sequence, or a (finite) Sidon sequence, if the pairwise differences are all distinct. Let
\[

K
(
m
)
=
max
(
m
−

a
m

1

/

2

)
,

K(m) = \max (m - a_m^{1/2}),

\]
where the maximum is taken over all m-element

B
2

{B_2}

-sequences. Erdős and Turán ask if

K
(
m
)
=
O
(
1
)

K(m) = O(1)

. In this paper we give an algorithm, based on the Bose-Chowla theorem on finite fields, for finding a lower bound of

K
(
p
)

K(p)

and a p-element

B
2

{B_2}

-sequence with

p
−

a
p

1

/

2

p - a_p^{1/2}

equal to this bound, taking

O
(

p
3

log
2

p
K
(
p
)
)

O({p^3}{\log ^2}pK(p))

bit operations and requiring

O
(
p
log
⁡
p
)

O(p\log p)

storage, where p is a prime. A search for lower bounds of

K
(
p
)

K(p)

for

p
≤

p

145

p \leq {p_{145}}

is given, especially

K
(

p

145

)
>
10.279

K({p_{145}}) > 10.279

, where

p
i

{p_i}

is the ith prime.
On the numerical integration of Walsh series by number-theoretic methods
Authors: Larcher Gerhard , Traunfellner Claudia
Published: June 2010

In analogy to the theory of good lattice points for the numerical integration of rapidly converging Fourier series, a theory for the fast numerical integration of Walsh series is developed. The basis for this theory is provided by a class of very well-distributed point sets in the s-dimensional unit cube, the so-called (t, m, s)-nets.
On the shape of tetrahedra from bisection
Authors: Liu Anwei , Joe Barry
Published: June 2010

We present a procedure for bisecting a tetrahedron T successively into an infinite sequence of tetrahedral meshes

T

0

,

T

1

,

T

2

,
…

{\mathcal {T}^0},{\mathcal {T}^1},{\mathcal {T}^2}, \ldots

, which has the following properties: (1) Each mesh

T

n

{\mathcal {T}^n}

is conforming. (2) There are a finite number of classes of similar tetrahedra in all the

T

n

,
n
≥
0

{\mathcal {T}^n},n \geq 0

. (3) For any tetrahedron

T

i
n

{\mathbf {T}}_i^n

in

T

n

,
η
(

T

i
n

)
≥

c
1

η
(

T

)

{\mathcal {T}^n},\eta ({\mathbf {T}}_i^n) \geq {c_1}\eta ({\mathbf {T}})

, where

η
\eta

is a tetrahedron shape measure and

c
1

{c_1}

is a constant. (4)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta left-parenthesis bold upper T Subscript i Superscript n Baseline right-parenthesis less-than-or-equal-to c 2 left-parenthesis 1 slash 2 right-parenthesis Superscript n slash 3 Baseline delta left-parenthesis bold upper T right-parenthesis">

δ
(

T

i
n

)
≤

c
2

(
1

/

2

)

n

/

3

δ
(

T

)

\delta ({\mathbf {T}}_i^n) \leq {c_2}{(1/2)^{n/3}}\delta ({\mathbf {T}})

, where

δ
(

T
′

)

\delta ({\mathbf {T’}})

denotes the diameter of tetrahedron

T
′

{\mathbf {T’}}

and

c
2

{c_2}

is a constant. Estimates of

c
1

{c_1}

and

c
2

{c_2}

are provided. Properties (2) and (3) extend similar results of Stynes and Adler, and of Rosenberg and Stenger, respectively, for the 2-D case. The diameter bound in property (4) is better than one given by Kearfott.
A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries
Authors: Bramble James H. , King J. Thomas
Published: June 2010

In this paper we consider a simple finite element method on an approximate polygonal domain using linear elements. The Dirichlet data are transferred in a natural way and the resulting linear system can be solved using multigrid techniques. Our analysis takes into account the change in domain and data transfer, and optimal-error estimates are obtained that are robust in the regularity of the boundary data provided they are at least square integrable. It is proved that the natural extension of our finite element approximation to the original domain is optimal-order accurate.
On generalized inversive congruential pseudorandom numbers
Authors: Eichenauer-Herrmann Jürgen
Published: June 2010

The inversive congruential method with prime modulus for generating uniform pseudorandom numbers has several very promising properties. Very recently, a generalization for composite moduli has been introduced. In the present paper it is shown that the generated sequences have very attractive statistical independence properties.
Chebyshev-type quadrature and partial sums of the exponential series
Authors: Kuijlaars Arno
Published: June 2010

Chebyshev-type quadrature for the weight functions
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w Subscript a Baseline left-parenthesis t right-parenthesis equals StartFraction 1 minus a t Over pi StartRoot 1 minus t squared EndRoot EndFraction comma negative 1 greater-than t greater-than 1 comma negative 1 greater-than a greater-than 1 comma">

w
a

(
t
)
=

1
−
a
t

π

1
−

t
2

,

−
1
>
t
>
1
,

−
1
>
a
>
1
,

{w_a}(t) = \frac {{1 - at}}{{\pi \sqrt {1 - {t^2}} }},\quad - 1 > t > 1,\quad - 1 > a > 1,

\]
is related to a problem concerning partial sums of the exponential series, namely the problem to extend the nth partial sum to a polynomial of degree 2N having all zeros on the circle

|

z

|

=

|

a

|

N

|z| = |a|N

. Using this connection, we show that the minimal number N of nodes needed for Chebyshev-type quadrature of degree n for

w
a

(
t
)

{w_a}(t)

satisfies an inequality

C
1

n
≤
N
≤

C
2

n

{C_1}n \leq N \leq {C_2}n

with positive constants

C
1

,

C
2

{C_1},{C_2}

. As an application we prove that the minimal number N of nodes for Chebyshev-type quadrature of degree n on a torus embedded in

R

3

{{\mathbf {R}}^3}

satisfies an inequality

C
1

n
2

≤
N
≤

C
2

n
2

{C_1}{n^2} \leq N \leq {C_2}{n^2}

.
Optimal-order nonnested multigrid methods for solving finite element equations. III. On degenerate meshes
Authors: Zhang Shangyou
Published: June 2010

In this paper, we consider several model problems where finite element triangular meshes with arbitrarily small angles (high aspect ratios) are utilized to deal with anisotropy, interfaces, or singular perturbations. The constant-rate (independent of the number of unknowns, the smallest angle, the interface discontinuity, the singular-perturbation parameter, etc.) convergence of some special nonnested multigrid methods for solving the finite element systems on such degenerate meshes will be proved. Numerical data are provided to support the analysis in each case.
Polynomial roots from companion matrix eigenvalues
Authors: Edelman Alan , Murakami H.
Published: June 2010

In classical linear algebra, the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition. We derive a first-order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm. Our error analysis expands on work by Van Dooren and Dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully. Regarding the stronger concept of a small componentwise backward error, our analysis predicts a small such error on a test suite of eight random polynomials suggested by Toh and Trefethen. However, we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice. We extend our results to polynomial matrices, where the result is essentially the same, but the geometry becomes more complicated.
Convergence of product integration rules over (0,∞) for functions with weak singularities at the origin
Authors: Mastroianni G. , Monegato G.
Published: June 2010

In this paper we consider integrals of the form\[∫0∞e−xK(x,y)f(x)dx,\int _0^\infty {{e^{ - x}}K(x,y)f(x)dx,}\]with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper C Superscript p Baseline left-bracket 0 comma normal infinity right-parenthesis intersection upper C Superscript q Baseline left-parenthesis 0 comma normal infinity right-parenthesis comma q greater-than-or-equal-to p greater-than-or-equal-to 0">f∈Cp[0,∞)∩Cq(0,∞),q≥p≥0f \in {C^p}[0,\infty ) \cap {C^q}(0,\infty ),q \geq p \geq 0, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript i Baseline f Superscript left-parenthesis p plus i right-parenthesis Baseline left-parenthesis x right-parenthesis element-of upper C left-bracket 0 comma normal infinity right-parenthesis comma i equals 1 comma ellipsis comma q minus p">xif(p+i)(x)∈C[0,∞),i=1,…,q−p{x^i}{f^{(p + i)}}(x) \in C[0,\infty ),i = 1, \ldots ,q - p, whenq>pq > p. They appear for instance in certain Wiener-Hopf integral equations and are of interest if one wants to solve these by a Nyström method. To discretize the integral above, we propose to use a product rule of interpolatory type based on the zeros of Laguerre polynomials. For this rule we derive (weighted) uniform convergence estimates and present some numerical examples.
The first nontrivial genus of positive definite ternary forms
Authors: Kaplansky Irving
Published: June 2010

The first nontrivial genus of positive ternary forms has discriminant 7. The paper presents all known results concerning this genus, including some computations.
Closed-form summation of some trigonometric series
Authors: Cvijović Djurdje , Klinowski Jacek
Published: June 2010

The problem of numerical evaluation of the classical trigonometric series
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript nu Baseline left-parenthesis alpha right-parenthesis equals sigma-summation Underscript k equals 0 Overscript normal infinity Endscripts StartFraction sine left-parenthesis 2 k plus 1 right-parenthesis alpha Over left-parenthesis 2 k plus 1 right-parenthesis Superscript nu Baseline EndFraction and upper C Subscript nu Baseline left-parenthesis alpha right-parenthesis equals sigma-summation Underscript k equals 0 Overscript normal infinity Endscripts StartFraction cosine left-parenthesis 2 k plus 1 right-parenthesis alpha Over left-parenthesis 2 k plus 1 right-parenthesis Superscript nu Baseline EndFraction comma">

S
ν

(
α
)
=

∑

k
=
0

∞

sin
⁡
(
2
k
+
1
)
α

(
2
k
+
1
)

ν

and

C
ν

(
α
)
=

∑

k
=
0

∞

cos
⁡
(
2
k
+
1
)
α

(
2
k
+
1
)

ν

,

{S_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\sin (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}}\quad {\text {and}}\quad } {C_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\cos (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}},}

\]
where

ν
>
1

\nu > 1

in the case of

S

2
n

(
α
)

{S_{2n}}(\alpha )

and

C

2
n
+
1

(
α
)

{C_{2n + 1}}(\alpha )

with

n
=
1
,
2
,
3
,
…

n = 1,2,3, \ldots

has been recently addressed by Dempsey, Liu, and Dempsey; Boersma and Dempsey; and by Gautschi. We show that, when

α
\alpha

is equal to a rational multiple of

2
π

2\pi

, these series can in the general case be summed in closed form in terms of known constants and special functions. General formulae giving

C
ν

(
α
)

{C_\nu }(\alpha )

and

S
ν

(
α
)

{S_\nu }(\alpha )

in terms of the generalized Riemann zeta function and the cosine and sine functions, respectively, are derived. Some simpler variants of these formulae are obtained, and various special results are established.
On the sharpness of 𝐿²-error estimates of 𝐻¹₀-projections onto subspaces of piecewise, high-order polynomials
Authors: Han Weimin , Jensen Søren
Published: June 2010

In a plane polygonal domain, consider a Poisson problem

−
Δ
u
=
f

- \Delta u = f

with homogeneous Dirichlet boundary condition and the p-version finite element solutions of this. We give various upper and lower bounds for the error measured in

L
2

{L^2}

. In the case of a single element (i.e., a convex domain), we reduce the question of sharpness of these estimates to the behavior of a certain inf-sup constant, which is numerically determined, and a likely sharp estimate is then conjectured. This is confirmed during a series of numerical experiments also for the case of a reentrant corner. For a one-dimensional analogue problem (of rotational symmetry), sharp

L
2

{L^2}

-error estimates are proven directly and via an extension of the classical duality argument. Here, we give sharp

L
∞

{L^\infty }

-error estimates in some weighted and unweighted norms also.