bluuebunny/crossref_metadata_2025 huggingface dataset:
Generalized Euler constants for arithmetical progressions
Authors: Dilcher Karl
Published: June 2010

The work of Lehmer and Briggs on Euler constants in arithmetical progressions is extended to the generalized Euler constants that arise in the Laurent expansion of

ζ
(
s
)

\zeta (s)

about

s
=
1

s = 1

. The results are applied to the summation of several classes of slowly converging series. A table of the constants is provided.
Clothoid spline transition spirals
Authors: Meek D. S. , Walton D. J.
Published: June 2010

Highway and railway designers use clothoid splines (planar

G
2

{G^2}

curves consisting of straight line segments, circular arcs, and clothoid segments) as center lines in route location. This paper considers the problem of finding a clothoid spline transition spiral which joins two given points and matches given curvatures and unit tangents at the two points. Conditions are given for the existence and uniqueness of the clothoid spline transition spirals, and algorithms for finding them are outlined.
Enumeration of small nonisomorphic 1-rotational twofold triple systems
Authors: Chee Yeow Meng , Royle Gordon F.
Published: June 2010

In this paper, twofold triple systems of order v are enumerated for all

υ
≤
19

\upsilon \leq 19

.
Runge-Kutta methods for partial differential equations and fractional orders of convergence
Authors: Ostermann A. , Roche M.
Published: June 2010

We apply Runge-Kutta methods to linear partial differential equations of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline left-parenthesis x comma t right-parenthesis equals script upper L left-parenthesis x comma partial-differential right-parenthesis u left-parenthesis x comma t right-parenthesis plus f left-parenthesis x comma t right-parenthesis">

u
t

(
x
,
t
)
=

L

(
x
,
∂
)
u
(
x
,
t
)
+
f
(
x
,
t
)

{u_t}(x,t) = \mathcal {L}(x,\partial )u(x,t) + f(x,t)

. Under appropriate assumptions on the eigenvalues of the operator

L

\mathcal {L}

and the (generalized) Fourier coefficients of f, we give a sharp lower bound for the order of convergence of these methods. We further show that this order is, in general, fractional and that it depends on the

L
r

{L^r}

-norm used to estimate the global error. The analysis also applies to systems arising from spatial discretization of partial differential equations by finite differences or finite element techniques. Numerical examples illustrate the results.
Detection of singularities using segment approximation
Authors: Grothmann R. , Mhaskar H. N.
Published: June 2010

We discuss best segment approximation (with free knots) by polynomials to piecewise analytic functions on a real interval. It is shown that, if the degree of the polynomials tends to infinity and the number of knots is the same as the number of singularities of the function, then the optimal knots converge geometrically fast to the singularities. When the degree is held fixed and the number of knots tends to infinity, we study the asymptotic distribution of the optimal knots.
On fully discrete Galerkin approximations for partial integro-differential equations of parabolic type
Authors: Zhang Nai Ying
Published: June 2010

The subject of this work is the application of fully discrete Galerkin finite element methods to initial-boundary value problems for linear partial integro-differential equations of parabolic type. We investigate numerical schemes based on the Padé discretization with respect to time and associated with certain quadrature formulas to approximate the integral term. A preliminary error estimate is established, which contains a term related to the quadrature rule to be specified. In particular, we consider quadrature rules with sparse quadrature points so as to limit the storage requirements, without sacrificing the order of overall convergence. For the backward Euler scheme, the Crank-Nicolson scheme, and a third-order (1,2) Padé-type scheme, the specific quadrature rules analyzed are based on the rectangular, the trapezoidal, and Simpson’s rule. For all the schemes studied, optimal-order error estimates are obtained in the case that the solution of the problem is smooth enough. Since this is important for our error analysis, we also discuss the regularity of the exact solutions of our equations. High-order regularity results with respect to both space and time are given for the solution of problems with smooth enough data.
On the remainder of Gaussian quadrature formulas for Bernstein-Szegő weight functions
Authors: Peherstorfer F.
Published: June 2010

We give an explicit expression for the kernel of the error functional for Gaussian quadrature formulas with respect to weight functions of Bernstein-Szegö type, i.e., weight functions of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 minus x right-parenthesis Superscript alpha Baseline left-parenthesis 1 plus x right-parenthesis Superscript beta Baseline slash rho left-parenthesis x right-parenthesis comma x element-of left-parenthesis negative 1 comma 1 right-parenthesis">

(
1
−
x

)
α

(
1
+
x

)
β

/

ρ
(
x
)
,

x
∈
(
−
1
,
1
)

{(1 - x)^\alpha }{(1 + x)^\beta }/\rho (x),\quad x \in ( - 1,1)

, where

α
,
β
∈
{
−

1
2

,

1
2

}

\alpha ,\beta \in \{ - \tfrac {1}{2},\tfrac {1}{2}\}

and

ρ
\rho

is a polynomial of arbitrary degree which is positive on

[
−
1
,
1
]

[ - 1,1]

. With the help of this result the norm of the error functional can easily be calculated explicitly for a wide subclass of these weight functions.
Calculation of Fibonacci polynomials for GFSR sequences with low discrepancies
Authors: Tezuka Shu , Fushimi Masanori
Published: June 2010

Fibonacci polynomials are defined in the context of the two-dimensional discrepancy of Tausworthe pseudorandom sequences as an analogue to Fibonacci numbers, which give the best figure of merit for the two-dimensional discrepancy of linear congruential sequences. We conduct an exhaustive search for the Fibonacci polynomials of degree less than 32 whose associated Tausworthe sequences can be easily implemented and very quickly generated.
Orthogonal spline collocation Laplace-modified and alternating-direction methods for parabolic problems on rectangles
Authors: Bialecki Bernard , Fernandes Ryan I.
Published: June 2010

A complete stability and convergence analysis is given for two- and three-level, piecewise Hermite bicubic orthogonal spline collocation, Laplace-modified and alternating-direction schemes for the approximate solution of linear parabolic problems on rectangles. It is shown that the schemes are unconditionally stable and of optimal-order accuracy in space and time.
Enumeration of quartic fields of small discriminant
Authors: Buchmann Johannes , Ford David , Pohst Michael
Published: June 2010

With the mixed-type case now completed, all algebraic number fields of degree 4 with absolute discriminant

>

10
6

> {10^6}

have been enumerated. Methods from the totally real and totally complex cases were used without major modification. Isomorphism of fields was determined by a method similar to one of Lenstra. The

T
2

{T_2}

criterion of Pohst was applied to reduce the number of redundant examples.
A theory of secant preconditioners
Authors: Martínez José Mario
Published: June 2010

In this paper we analyze the use of structured quasi-Newton formulae as preconditioners of iterative linear methods when the inexact-Newton approach is employed for solving nonlinear systems of equations. We prove that superlinear convergence and bounded work per iteration is obtained if the preconditioners satisfy a Dennis-Moré condition. We develop a theory of Least-Change Secant Update preconditioners and we present an application concerning a structured BFGS preconditioner.
Good parameters for a class of node sets in quasi-Monte Carlo integration
Authors: Hansen Tom , Mullen Gary L. , Niederreiter Harald
Published: June 2010

For

2
≤
s
≤
12

2 \leq s \leq 12

we determine good parameters in a general construction of node sets for s-dimensional quasi-Monte Carlo integration recently introduced by the third author. Some of the parameters represent optimal choices in this construction and lead to improvements on node sets obtained by earlier techniques.
Checking the Goldbach conjecture up to 4⋅10¹¹
Authors: Sinisalo Matti K.
Published: June 2010

One of the most studied problems in additive number theory, Goldbach’s conjecture, states that every even integer greater than or equal to 4 can be expressed as a sum of two primes. In this paper checking of this conjecture up to

4
⋅

10

11

4 \cdot {10^{11}}

by the IBM 3083 mainframe with vector processor is reported.
An extension of the theorem on primitive divisors in algebraic number fields
Authors: Schinzel A.
Published: June 2010

The theorem about primitive divisors in algebraic number fields is generalized in the following manner. Let A, B be algebraic integers,

(
A
,
B
)
=
1
,
A
B
≠
0

(A,B) = 1, AB \ne 0

, A/B not a root of unity, and

ζ
k

{\zeta _k}

a primitive root of unity of order k. For all sufficiently large n, the number

A
n

−

ζ
k

B
n

{A^n} - {\zeta _k}{B^n}

has a prime ideal factor that does not divide

A
m

−

ζ
k
j

B
m

{A^m} - \zeta _k^j{B^m}

for arbitrary

m
>
n

m > n

and

j
>
k

j > k

.
On enclosing simple roots of nonlinear equations
Authors: Alefeld G. , Potra F. A. , Shi Yixun
Published: June 2010

In this paper we present two efficient algorithms for enclosing a simple root of the nonlinear equation

f
(
x
)
=
0

f(x) = 0

in the interval [a, b]. They improve recent methods of Alefeld and Potra by achieving higher efficiency indices and avoiding the solution of a quadratic equation per iteration. The efficiency indices of our methods are 1.5537... and 1.618... , respectively. We show that our second method is an optimal algorithm in some sense. Our numerical experiments show that the two methods of the present paper compare well with the above methods of Alefeld and Potra as well as efficient solvers of Dekker, Brent, and Le. The second method in this paper has the best behavior of all, especially when the termination tolerance is small.
Circulant preconditioners for Toeplitz matrices with piecewise continuous generating functions
Authors: Yeung Man-Chung , Chan Raymond H.
Published: June 2010

We consider the solution of n-by-n Toeplitz systems

T
n

x
=
b

{T_n}x = b

by preconditioned conjugate gradient methods. The preconditioner

C
n

{C_n}

is the T. Chan circulant preconditioner, which is defined to be the circulant matrix that minimizes

‖

B
n

−

T
n

‖

F

{\left \| {{B_n} - {T_n}} \right \|_F}

over all circulant matrices

B
n

{B_n}

. For Toeplitz matrices generated by positive

2
π

2\pi

-periodic continuous functions, we have shown earlier that the spectrum of the preconditioned system

C
n

−
1

T
n

C_n^{ - 1}{T_n}

is clustered around 1 and hence the convergence rate of the preconditioned system is superlinear. However, in this paper, we show that if instead the generating function is only piecewise continuous, then for all

ε
\varepsilon

sufficiently small, there are

O
(
log
⁡
n
)

O(\log n)

eigenvalues of

C
n

−
1

T
n

C_n^{ - 1}{T_n}

that lie outside the interval

(
1
−
ε
,
1
+
ε
)

(1 - \varepsilon ,1 + \varepsilon )

. In particular, the spectrum of

C
n

−
1

T
n

C_n^{ - 1}{T_n}

cannot be clustered around 1. Numerical examples are given to verify that the convergence rate of the method is no longer superlinear in general.
Irregular primes and cyclotomic invariants to four million
Authors: Buhler J. , Crandall R. , Ernvall R. , Metsänkylä T.
Published: June 2010

Recent computations of irregular primes, and associated cyclotomic invariants, were extended to all primes below four million using an enhanced multisectioning/convolution method. Fermat’s "Last Theorem" and Vandiver’s conjecture were found to be true for those primes, and the cyclotomic invariants behaved as expected. There is exactly one prime less than four million whose index of irregularity is equal to seven.
On Fourier coefficients of Maass waveforms for 𝑃𝑆𝐿(2,𝑍)
Authors: Hejhal D. A. , Arno S.
Published: June 2010

In this paper, we use machine experiments to test the validity of the Sato-Tate conjecture for Maass waveforms on

P
S
L

(
2
,
 
m
a
t
h
b
b

Z

)
∖
H

\mathrm {PSL}(2,\ mathbb{Z})\backslash H

. We also elaborate on Stark’s iterative method for calculating the Fourier coefficients of such forms.
On computing factors of cyclotomic polynomials
Authors: Brent Richard P.
Published: June 2010

For odd square-free

n
>
1

n > 1

the cyclotomic polynomial

Φ
n

(
x
)

{\Phi _n}(x)

satisfies the identity of Gauss,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 normal upper Phi Subscript n Baseline left-parenthesis x right-parenthesis equals upper A Subscript n Superscript 2 Baseline minus left-parenthesis negative 1 right-parenthesis Superscript left-parenthesis n minus 1 right-parenthesis slash 2 Baseline n upper B Subscript n Superscript 2 Baseline period">

4

Φ
n

(
x
)
=

A
n
2

−

(
−
1

)

(
n
−
1
)

/

2

n

B
n
2

.

4{\Phi _n}(x) = A_n^2 - {( - 1)^{(n - 1)/2}}nB_n^2.

\]
A similar identity of Aurifeuille, Le Lasseur, and Lucas is
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi Subscript n Baseline left-parenthesis left-parenthesis negative 1 right-parenthesis Superscript left-parenthesis n minus 1 right-parenthesis slash 2 Baseline x right-parenthesis equals upper C Subscript n Superscript 2 Baseline minus n x upper D Subscript n Superscript 2">

Φ
n

(

(
−
1

)

(
n
−
1
)

/

2

x
)
=

C
n
2

−
n
x

D
n
2

{\Phi _n}({( - 1)^{(n - 1)/2}}x) = C_n^2 - nxD_n^2

\]
or, in the case that n is even and square-free,
\[

±

Φ

n

/

2

(
−

x
2

)
=

C
n
2

−
n
x

D
n
2

.

\pm {\Phi _{n/2}}( - {x^2}) = C_n^2 - nxD_n^2.

\]
Here,

A
n

(
x
)
,
…
,

D
n

(
x
)

{A_n}(x), \ldots ,{D_n}(x)

are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require

O
(

n
2

)

O({n^2})

arithmetic operations and work over the integers. We also give explicit formulae and generating functions for

A
n

(
x
)
,
…
,

D
n

(
x
)

{A_n}(x), \ldots ,{D_n}(x)

, and illustrate the application to integer factorization with some numerical examples.
A breakdown-free variation of the nonsymmetric Lanczos algorithms
Authors: Ye Qiang
Published: June 2010

The nonsymmetric Lanczos tridiagonalization algorithm is essentially the Gram-Schmidt biorthogonalization method for generating biorthogonal bases of a pair of Krylov subspaces. It suffers from breakdown and instability when a pivot at some step is zero or nearly zero, which is often the result of mismatch of the two Krylov subspaces. In this paper, we propose to modify one of the two Krylov subspaces by introducing a "new-start" vector when a pivot is small. This new-start vector generates another Krylov subspace, which we add to the old one in an appropriate way so that the Gram-Schmidt method for the modified subspaces yields a recurrence similar to the Lanczos algorithm. Our method enforces the pivots to be above a certain threshold and can handle both exact breakdown and near-breakdown. In particular, we recover look-ahead Lanczos algorithms and Arnoldi’s algorithm as two special cases. We also discuss theoretical and practical issues concerning the new-start procedure and present a convergence analysis as well as some numerical examples.
A table of primitive binary polynomials
Authors: Živković Miodrag
Published: June 2010

For thosen>5000n > 5000for which the factorization of2n−1{2^n} - 1is known, the first primitive trinomial (if such exists) and a randomly generated primitive 5- and 7-nomial of degreeninGF(2){\text {GF}}(2)are given.
On Faber polynomials generated by an 𝑚-star
Authors: Bartolomeo J. , He Matthew
Published: June 2010

In this paper, we study the Faber polynomials

F
n

(
z
)

{F_n}(z)

generated by a regular m-star

(
m
=
3
,
4
,
…
)

(m = 3,4, \ldots )

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript m Baseline equals left-brace x omega Superscript k Baseline semicolon 0 less-than-or-equal-to x less-than-or-equal-to 4 Superscript 1 slash m Baseline comma k equals 0 comma 1 comma ellipsis comma m minus 1 comma omega Superscript m Baseline equals 1 right-brace period">

S
m

=
{

x

ω
k

;
0
≤
x
≤

4

1

/

m

,
k
=
0
,
1
,
…
,
m
−
1
,

ω
m

=
1

}
.

{S_m} = \{ {x{\omega ^k};0 \leq x \leq {4^{1/m}},k = 0,1, \ldots ,m - 1,{\omega ^m} = 1} \}.

\]
An explicit and precise expression for

F
n

(
z
)

{F_n}(z)

is obtained by computing the coefficients via a Cauchy integral formula. The location and limiting distribution of zeros of

F
n

(
z
)

{F_n}(z)

are explored. We also find a class of second-order hypergeometric differential equations satisfied by

F
n

(
z
)

{F_n}(z)

. Our results extend some classical results of Chebyshev polynomials for a segment

[
−
2
,
2
]

[ - 2,2]

in the case when

m
=
2

m = 2

.
Backward error analysis of cyclic reduction for the solution of tridiagonal systems
Authors: Amodio Pierluigi , Mazzia Francesca
Published: June 2010

Tridiagonal systems play a fundamental role in matrix computation. In particular, in recent years parallel algorithms for the solution of tridiagonal systems have been developed. Among these, the cyclic reduction algorithm is particularly interesting. Here the stability of the cyclic reduction method is studied under the assumption of diagonal dominance. A backward error analysis is made, yielding a representation of the error matrix for the factorization and for the solution of the linear system. The results are compared with those for LU factorization.
Improved lower bounds for the discrepancy of inversive congruential pseudorandom numbers
Authors: Eichenauer-Herrmann Jürgen
Published: June 2010

The inversive congruential method with prime modulus for generating uniform pseudorandom numbers is studied. Lower bounds for the discrepancy of k-tuples of successive pseudorandom numbers are established, which improve earlier results of Niederreiter. Moreover, the present proof is substantially simpler than the earlier one.
Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind
Authors: Kaneko Hideaki , Xu Yuesheng
Published: June 2010

In this paper we establish Gauss-type quadrature formulas for weakly singular integrals. An application of the quadrature scheme is given to obtain numerical solutions of the weakly singular Fredholm integral equation of the second kind. We call this method a discrete product-integration method since the weights involved in the standard product-integration method are computed numerically.
Quadratic residue covers for certain real quadratic fields
Authors: Mollin R. A. , Williams H. C.
Published: June 2010

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript n Baseline left-parenthesis a comma b right-parenthesis equals left-parenthesis b a Superscript n Baseline plus left-parenthesis a minus 1 right-parenthesis slash b right-parenthesis squared plus 4 a Superscript n">

Δ
n

(
a
,
b
)
=

(
b

a
n

+
(
a
−
1
)

/

b

)
2

+
4

a
n

{\Delta _n}(a,b) = {(b{a^n} + (a - 1)/b)^2} + 4{a^n}

with

n
≥
1

n \geq 1

and

b

|

a
−
1

b|a - 1

. If

C

\mathcal {C}

is a finite set of primes such that for each

n
≥
1

n \geq 1

there exists some

q
∈

C

q \in \mathcal {C}

for which the Legendre symbol

(

Δ
n

(
a
,
b
)

/

q
)
≠
−
1

({\Delta _n}(a,b)/q) \ne - 1

, we call

C

\mathcal {C}

a quadratic residue cover (QRC) for the quadratic fields

K
n

(
a
,
b
)
=
Q
(

Δ
n

(
a
,
b
)
)

{K_n}(a,b) = Q(\sqrt {{\Delta _n}(a,b))}

. It is shown how the existence of a QRC for any a, b can be used to determine lower bounds on the class number of

K
n

(
a
,
b
)

{K_n}(a,b)

when

Δ
n

(
a
,
b
)

{\Delta _n}(a,b)

is the discriminant of

K
n

(
a
,
b
)

{K_n}(a,b)

. Also, QRCs are computed for all

1
≤
a
,
b
≤
10000

1 \leq a,b \leq 10000

.
Heuristics on class groups: some good primes are not too good
Authors: Cohen Henri , Martinet Jacques
Published: June 2010

We correct some too optimistic predictions in an earlier paper of ours.
Error estimate in an isoparametric finite element eigenvalue problem
Authors: Lebaud M.-P.
Published: June 2010

The aim of this paper is to obtain an eigenvalue approximation for elliptic operators defined on a domain

Ω
\Omega

with the help of isoparametric finite elements of degree k. We prove that

λ
−

λ
h

=
O
(

h

2
k

)

\lambda - {\lambda _h} = O({h^{2k}})

provided the boundary of

Ω
\Omega

is well approximated, which is the same estimate as the one obtained in the case of conforming finite elements.
An error estimate for finite volume methods for multidimensional conservation laws
Authors: Cockburn Bernardo , Coquel Frédéric , LeFloch Philippe
Published: June 2010

In this paper, an

L
∞

(

L
1

)

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

-error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained. These methods can be formally high-order accurate and are defined on general triangulations. The error is proven to be of order

h

1

/

4

{h^{1/4}}

, where h represents the "size" of the mesh, via an extension of Kuznetsov approximation theory for which no estimate of the total variation and of the modulus of continuity in time are needed. The result is new even for the finite volume method constructed from monotone numerical flux functions.
Improved rounding for spline coefficients and knots
Authors: Grosse Eric , Hobby John D.
Published: June 2010

When representing the coefficients and knots of a spline using only small integers, independently rounding each infinite-precision value is not the best strategy. We show how to build an affine model for the error expanded about the optimal full-precision free-knot or parameterized spline, then use the Lovász basis reduction algorithm to select a better rounding. The technique could be used for other situations in which a quadratic error model can be computed.
Numerical evaluation of some trigonometric series
Authors: McLean W.
Published: June 2010

We present a method for the accurate numerical evaluation of a family of trigonometric series arising in the design of special-purpose quadrature rules for boundary element methods. The series converge rather slowly, but can be expressed in terms of Fourier-Chebyshev series that converge rapidly.
Recognizing units in number fields
Authors: Ge Guoqiang
Published: June 2010

We present a deterministic polynomial-time algorithm that decides whether a power product

∏

i
=
1

k

γ
i

n
i

\prod \nolimits _{i = 1}^k {\gamma _i^{{n_i}}}

is a unit in the ring of integers of K, where K is a number field,

γ
i

{\gamma _i}

are nonzero elements of K and

n
i

{n_i}

are rational integers. The main algorithm is based on the factor refinement method for ideals, which might be of independent interest.
A framework for block ILU factorizations using block-size reduction
Authors: Chan Tony F. , Vassilevski Panayot S.
Published: June 2010

We propose a block ILU factorization technique for block tridiagonal matrices that need not necessarily be M-matrices. The technique explores reduction by a coarse-vector restriction of the block size of the approximate Schur complements computed throughout the factorization process. Then on the basis of the Sherman-Morrison-Woodbury formula these are easily inverted. We prove the existence of the proposed factorization techniques in the case of (nonsymmetric, in general) M-matrices. For block tridiagonal matrices with positive definite symmetric part we show the existence of a limit version of the factorization (exact inverses of the reduced matrices are needed). The theory is illustrated with numerical tests.
The semigroup stability of the difference approximations for initial-boundary value problems
Authors: Wu Lixin
Published: June 2010

For semidiscrete approximations and one-step fully discretized approximations of the initial-boundary value problem for linear hyperbolic equations with diagonalizable coefficient matrices, we prove that the Kreiss condition is a sufficient condition for the semigroup stability (or

l
2

{l_2}

stability). Also, we show that the stability of a fully discretized approximation generated by a locally stable Runge-Kutta method is determined by the stability of the semidiscrete approximation.
Negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous 𝐿₂ Dirichlet boundary data
Authors: Bales L. , Lasiecka I.
Published: June 2010

This paper treats time and space finite element approximations of the solution to the nonhomogeneous wave equation with

L
2

{L_2}

boundary terms and smooth right-hand side. For the case of

L
2

{L_2}

boundary data, the rates of convergence in negative norms are derived. In the case of smooth forcing term and zero boundary data, optimal rates of convergence in "positive" norms are provided.
Strong convergence of numerical solutions to degenerate variational problems
Authors: Nicolaides R. A. , Walkington Noel J.
Published: June 2010

Numerical approximations of strongly degenerate variational problems of the form

J
(
u
)
=

∫
0
1

F
(

u
′

)
+

(
u
−
f

)
2

J(u) = \smallint _0^1F(u’ ) + {(u - f)^2}

are considered, where F is assumed convex but may have intervals where

F
=
0

F = 0

. It is shown that, in spite of the degeneracy, natural numerical approximations still converge in

W

1
,
p

{W^{1,p}}

. Rates in weaker norms and the connection with nonconvex variational problems are also considered.
New factors of Fermat numbers
Authors: Gostin Gary B.
Published: June 2010

Forty-six new factors of Fermat numbers are given, along with a summary of the search limits.
Pointwise a posteriori error estimates for elliptic problems on highly graded meshes
Authors: Nochetto Ricardo H.
Published: June 2010

Pointwise a posteriori error estimates are derived for linear second-order elliptic problems over general polygonal domains in 2D. The analysis carries over regardless of convexity, accounting even for slit domains, and applies to highly graded unstructured meshes as well. A key ingredient is a new asymptotic a priori estimate for regularized Green’s functions. The estimators lead always to upper bounds for the error in the maximum norm, along with lower bounds under very mild regularity and nondegeneracy assumptions. The effect of both point and line singularities is examined. Three popular local estimators for the energy norm are shown to be equivalent, when suitably interpreted, to those introduced here.
A quadrature formula with zeros of Bessel functions as nodes
Authors: Grozev Georgi R. , Rahman Qazi I.
Published: June 2010

A quadrature formula for entire functions of exponential type wherein the nodes are the zeros of the Bessel function of the first kind was recently obtained by C. Frappier and P. Olivier. Here the condition imposed on the function is relaxed. Some applications of the formula are also given.
On the primality of 𝑛!±1 and 2⋅3⋅5⋯𝑝±1
Authors: Caldwell Chris K.
Published: June 2010

For each prime p let p# be the product of the primes less than or equal to p. Using a new type of microcomputer coprocessor, we have found five new primes of the form

n
!
−
1

n! - 1

, two new primes of the form

p
#
+
1

p\# + 1

, seven new primes of the form

p
#
−
1

p\# - 1

, and greatly extended the search bounds for primes of the form

n
!
±
1

n! \pm 1

and

p
#
±
1

p\# \pm 1

.
Deformations of vector fields and Hamiltonian vector fields on the plane
Authors: van den Hijligenberg Nico , Kotchetkov Youri , Post Gerhard
Published: June 2010

For the Lie algebras

L
1

(
H
(
2
)
)

{L_1}(H(2))

and

L
1

(
W
(
2
)
)

{L_1}(W(2))

, we study their infinitesimal deformations and the corresponding global ones. We show that, as in the case of

L
1

(
W
(
1
)
)

{L_1}(W(1))

, each integrable infinitesimal deformation of

L
1

(
H
(
2
)
)

{L_1}(H(2))

and

L
1

(
W
(
2
)
)

{L_1}(W(2))

can be represented by a 2-cocycle that defines a global deformation by means of a trivial extension. We also illustrate that all deformations of

L
1

(
H
(
2
)
)

{L_1}(H(2))

arise as restrictions of deformations of

L
1

(
W
(
2
)
)

{L_1}(W(2))

.
A mixed finite element method for a strongly nonlinear second-order elliptic problem
Authors: Milner F. A. , Park E.-J.
Published: June 2010

The approximation of the solution of the first boundary value problem for a strongly nonlinear second-order elliptic problem in divergence form by the mixed finite element method is considered. Existence and uniqueness of the approximation are proved and optimal error estimates in

L
2

{L^2}

are established for both the scalar and vector functions approximated by the method. Error estimates are also derived in

L
q

,
2
≤
q
≤
+
∞

{L^q},2 \leq q \leq + \infty

.
Conjugacy classes of Γ(2) and spectral rigidity
Authors: Phillips Ralph
Published: June 2010

The free groupΓ(2)\Gamma (2)is generated byA=(12,01)A = (1\;2,0\;1)andB=(10,−21)B = (1\;0, - 2\;1), and setting<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi Subscript left-parenthesis xi comma eta right-parenthesis Baseline left-parenthesis upper A right-parenthesis equals exp left-parenthesis 2 pi i xi right-parenthesis comma chi Subscript left-parenthesis xi comma eta right-parenthesis Baseline left-parenthesis upper B right-parenthesis equals exp left-parenthesis 2 pi i eta right-parenthesis">χ(ξ,η)(A)=exp⁡(2πiξ),χ(ξ,η)(B)=exp⁡(2πiη){\chi _{(\xi ,\eta )}}(A) = \exp (2\pi i\xi ),{\chi _{(\xi ,\eta )}}(B) = \exp (2\pi i\eta )defines a unitary character onΓ(2)\Gamma (2)for0≤ξ,η>10 \leq \xi ,\eta > 1. A program is devised to compute\[μ(tr)=∑χ(ξ,η)(conj.class),\mu ({\text {tr}}) = \sum {{\chi _{(\xi ,\eta )}}({\text {conj}}. {\text {class}}),}\]summed over all primitive conjugacy classes ofΓ(2)\Gamma (2)of trace tr. Combined with a Luo-Sarnak theorem, this yields lower bounds for the spectral variance for a large sampling of characters in the range0>ξ,η>10 > \xi ,\eta > 1. The results indicate that the Berry conjecture for spectral rigidity does not hold for this set of classically chaotic systems. The program is also used to compute\[θ(x)=∑ln⁡(N({γ})),\theta (x) = \sum {\ln (N(\{ \gamma \} )),}\]summed over all primitive conjugacy classes ofΓ(2)\Gamma (2)of normN({γ})≤xN(\{ \gamma \} ) \leq x. The functionθ(x)\theta (x)is asymptotic tox, and the remainder can be written as|θ(x)−x|=xβ|\theta (x) - x| = {x^\beta }. The values ofβ(x)\beta (x)are computed for all traces between 3202 and 4802 (herex=tr2−2x = {\text {tr}^2} - 2). Theβ\beta’s cluster around 0.6, attaining a maximum of2/32/3. Finally, it is proved that the remainderθ(x)−x\theta (x) - xhas a negative bias by showing that the mean normalized remainder converges to a negative limit.
On a method for obtaining approximate solutions to Sturm-Liouville problems
Authors: Abrate Serge
Published: June 2010

A method presented recently to find approximate solutions to a Sturm-Liouville problem is shown to be a special case of a classical weighted residual method. The choice of approximation functions and convergence of the method are also examined.
Interpolation between Sobolev spaces in Lipschitz domains with an application to multigrid theory
Authors: Bramble James H.
Published: June 2010

In this paper we describe an interpolation result for the Sobolev spaces

H
0
s

(
Ω
)

H_0^s(\Omega )

where

Ω
\Omega

is a bounded domain with a Lipschitz boundary. This result is applied to derive discrete norm estimates related to multilevel preconditioners and multigrid methods in the finite element method. The estimates are valid for operators of order 2m with Dirichlet boundary conditions.
A finite element model for the time-dependent Joule heating problem
Authors: Elliott Charles M. , Larsson Stig
Published: June 2010

We study a spatially semidiscrete and a completely discrete finite element model for a nonlinear system consisting of an elliptic and a parabolic partial differential equation describing the electric heating of a conducting body. We prove error bounds of optimal order under minimal regularity assumptions when the number of spatial variables

d
≤
3

d \leq 3

. We establish the existence of solutions with the required regularity over arbitrarily long intervals of time when

d
≤
2

d \leq 2

.
Hermite interpolation by Pythagorean hodograph quintics
Authors: Farouki R. T. , Neff C. A.
Published: June 2010

The Pythagorean hodograph (PH) curves are polynomial parametric curves

{
x
(
t
)
,
y
(
t
)
}

\{ x(t),y(t)\}

whose hodograph (derivative) components satisfy the Pythagorean condition

x
′

2

(
t
)
+

y
′

2

(
t
)
≡

σ
2

(
t
)

x’{}^2(t) + y’{}^2(t) \equiv {\sigma ^2}(t)

for some polynomial

σ
(
t
)

\sigma (t)

. Thus, unlike polynomial curves in general, PH curves have arc lengths and offset curves that admit exact rational representations. The lowest-order PH curves that are sufficiently flexible for general interpolation/approximation problems are the quintics. While the PH quintics are capable of matching arbitrary first-order Hermite data, the solution procedure is not straightforward and furthermore does not yield a unique result—there are always four distinct interpolants (of which only one, in general, has acceptable "shape" characteristics). We show that formulating PH quintics as complex-valued functions of a real parameter leads to a compact Hermite interpolation algorithm and facilitates an identification of the "good" interpolant (in terms of minimizing the absolute rotation number). This algorithm establishes the PH quintics as a viable medium for the design or approximation of free-form curves, and allows a one-for-one substitution of PH quintics in lieu of the widely-used "ordinary" cubics.
Low-rank modification of the unsymmetric Lanczos algorithm
Authors: Huckle Thomas
Published: June 2010

The unsymmetric Lanczos algorithm is an important method for eigenvalue estimation and for solving linear equations. Unfortunately, the algorithm may break down without providing useful information; this is referred to as a serious breakdown in the literature. Here, we introduce a low-rank modification of the original matrix A in the case of a serious breakdown. This modification can be used to cure a serious breakdown as long as we have orthogonality of the already computed Lanczos vectors. We can switch to a new rank-1 modified matrix

A
~

=
A
+
a

b
T

\tilde A = A + a{b^T}

such that - the Lanczos algorithm has no serious breakdown in this step when applied on

A
~

\tilde A

, - the already computed variables in the Lanczos algorithm for A and

A
~

\tilde A

coincide, - using a Lanczos-based iterative solver, e.g. BCG or QMR, with start vectors

x
0

=
0

{x_0} = 0

and

v
1

=
f

{v_1} = f

, we have

A

−
1

f
=

A
~

−
1

f

{A^{ - 1}}f = {\tilde A^{ - 1}}f

, and thus by continuing the Lanczos algorithm with

A
~

\tilde A

we automatically get the desired solution

A

−
1

f

{A^{ - 1}}f

. Also, if the Lanczos vectors have lost their orthogonality, we show theoretically and by numerical examples that the modified Lanczos method has the same convergence behavior as the Lanczos method without breakdown. Thus, in the case of a serious breakdown we only have to compute the new rank-1 modified matrix

A
~

\tilde A

and step further in the original algorithm now using

A
~

\tilde A

.
Convergence analysis for a generalized Richardson extrapolation process with an application to the 𝑑⁽¹⁾-transformation on convergent and divergent logarithmic sequences
Authors: Sidi Avram
Published: June 2010

In an earlier work by the author the Generalized Richardson Extrapolation Process (GREP) was introduced and some of its convergence and stability properties were discussed. In a more recent work by the author a special case of GREP, which we now call

GREP

(
1
)

{\text {GREP}^{(1)}}

, was considered and its properties were reviewed with emphasis on oscillatory sequences. In the first part of the present work we give a detailed convergence and stability analysis of

GREP

(
1
)

{\text {GREP}^{(1)}}

as it applies to a large class of logarithmic sequences, both convergent and divergent. In particular, we prove several theorems concerning the columns and the diagonals of the corresponding extrapolation table. These theorems are very realistic in the sense that they explain the remarkable efficiency of

GREP

(
1
)

{\text {GREP}^{(1)}}

in a very precise manner. In the second part we apply this analysis to the Levin-Sidi

d

(
1
)

{d^{(1)}}

-transformation, as the latter is used with a new strategy to accelerate the convergence of infinite series that converge logarithmically, or to sum the divergent extensions of such series. This is made possible by the observation that, when the proper analogy is drawn, the

d

(
1
)

{d^{(1)}}

-transformation is, in fact, a

GREP

(
1
)

{\text {GREP}^{(1)}}

. We append numerical examples that demonstrate the theory.
Leptospira: a spirochaete with a hybrid outer membrane
Authors: Haake David A. , Matsunaga James
Published: June 2010

Leptospira is a genus of spirochaetes that includes organisms with a variety of lifestyles ranging from aquatic saprophytes to invasive pathogens. Adaptation to a wide variety of environmental conditions has required leptospires to acquire a large genome and a complex outer membrane with features that are unique among bacteria. The most abundant surface‐exposed outer membrane proteins are lipoproteins that are integrated into the lipid bilayer by amino‐terminal fatty acids. In contrast to many spirochaetes, the leptospiral outer membrane also includes lipopolysaccharide and many homologues of well‐known beta‐barrel transmembrane outer membrane proteins. Research on leptospiral transmembrane outer membrane proteins has lagged behind studies of lipoproteins because of their aberrant behaviour by Triton X‐114 detergent fractionation. For this reason, transmembrane outer membrane proteins are best characterized by assessing membrane integration and surface exposure. Not surprisingly, some outer membrane proteins that mediate host–pathogen interactions are strongly regulated by conditions found in mammalian host tissues. For example, the leptospiral immunoglobulin‐like (Lig) repeat proteins are dramatically induced by osmolarity and mediate interactions with host extracellular matrix proteins. Development of molecular genetic tools are making it possible to finally understand the roles of these and other outer membrane proteins in mechanisms of leptospiral pathogenesis.
Molecular identification of three distinct Polerovirus species and a recombinant Cucurbit aphid‐borne yellows virus strain infecting cucurbit crops in Taiwan
Authors: Knierim D. , Deng T. C. , Tsai W. S. , Green S. K. , Kenyon L.
Published: June 2010

To better understand the genetic diversity of cucurbit‐infecting poleroviruses in Taiwan, a survey was conducted in 2008 and 2009 as a prelude to screening germplasm for resistance. Out of 102 cucurbit samples showing yellowing symptoms collected across eight counties, 29 were identified as infected by polerovirus(es) by reverse transcription polymerase chain reaction (RT‐PCR) using a set of universal polerovirus primers. Sequence analysis of a 1·4‐kb PCR product spanning part of the RNA‐dependent RNA polymerase (RdRp) gene, the intergenic region (IR) and the complete coat protein (CP) gene of 13 of the 29 samples revealed the presence of three distinct Polerovirus species, namely Cucurbit aphid‐borne yellows virus (CABYV), Melon aphid‐borne yellows virus (MABYV) and Suakwa aphid‐borne yellows virus (SABYV). In addition to the common strain of CABYV (CABYV‐C) a recombinant strain of CABYV (CABYV‐R) was identified. The recombinant strain probably arose by recombination in the IR, a recombination hot‐spot of poleroviruses, between ancestors of MABYV and CABYV‐C. RT‐PCR protocols based on sets of specific primers were developed to distinguish between the different cucurbit‐infecting polerovirus species and the two CABYV strains. These findings are discussed in relation to strategies for breeding for durable resistance against poleroviruses.
Serum uric acid level is associated with cardiac hypertrophy in renal transplant recipients
Authors: Caliskan Yasar , Gorgulu Numan , Yelken Berna , Akturk Faruk , Yazici Halil , Turkmen Aydin , Sever Mehmet Sukru
Published: June 2010

Caliskan Y, Gorgulu N, Yelken B, Akturk F, Yazici H, Turkmen A, Sever MS. Serum uric acid level is associated with cardiac hypertrophy in renal transplant recipients. Clin Transplant 2011: 25: 368–374. © 2010 John Wiley & Sons A/S.

Abstract:  Background:  Serum uric acid (UA) level as a significant and independent risk factor for cardiovascular disease, and the link between this marker and left ventricular hypertrophy (LVH) in renal transplant recipients remains to be clarified.

Methods:  A total of 141 renal transplant recipients (83 men), between ages of 18 and 69 (mean age 37 ± 11), were included in this single center study. In addition to demographic, clinical, and laboratory parameters, serum UA concentrations were evaluated. LVH was determined by two‐dimensional and M‐mode echocardiography.

Results:  Serum UA levels were significantly higher (6.14 ± 1.15 mg/dL) in patients with LVH (n = 54) when compared to patients (n = 87) who did not have this abnormality (5.29 ± 1.43 mg/dL) (p = 0.006). Serum UA levels were significantly correlated with septal wall thickness, LV posterior wall thickness, LV mass index (LVMI), and pulmonary arterial pressure. Multiple linear regression analysis revealed that UA predicted LVMI (r2 = 0.150, β = 0.369, p = 0.001). However, serum creatinine (β = 0.060, p = 0.593) and age (β = 0.146, p = 0.175) were not predictors of LVMI.

Conclusion:  High serum UA levels are associated with LVH in renal transplant recipients, which underlines the importance of treating hyperuricemia.
Type 1 and type IV hypersensitivity to nickel
Authors: Walsh Maeve L , Smith Victoria H , King Clodagh M
Published: June 2010

We report a case of type 1 and type IV contact hypersensitivity to nickel. Very few cases of immediate contact urticaria to nickel have been reported. The mechanism behind this is not fully understood, although it has been postulated that nickel may act as a mast cell discharger on a non‐immunological basis. Nickel is found in varying concentrations in stainless steels; its biological availability varies depending on the alloy composition. Our patient has adapted her lifestyle to minimize her contact with household appliances containing nickel. If medical or dental treatment using stainless steel equipment is required, we recommend a short course of oral corticosteroids and antihistamines.
Sustained low hepatitis B viral load predicts good outcome after curative resection in patients with hepatocellular carcinoma
Authors: An Ho Jung , Jang Jeong Won , Bae Si Hyun , Choi Jong Young , Cho Se Hyun , Yoon Seung Kew , Han June Yeol , Lee Keun Ho , Kim Dong Goo , Jung Eun Sun
Published: June 2010

Background and Aim:  Little is known about the role of hepatitis B virus (HBV) factors in the long‐term prognosis of hepatocellular carcinoma (HCC) after resection. The objective of the present study was to identify the changing patterns of HBV levels and its effect on outcome after resection.

Methods:  This study recruited 188 patients with HBV‐related HCC who underwent curative resection. Among the 188 patients, 115 were alive without recurrence at 12 months, and had serial measurements of viral levels.

Results:  The mean age was 53 years and the mean follow‐up period was 48.5 months. With multivariate analysis, tumor size > 5 cm (P = 0.047), Child‐Pugh class B (P = 0.017), vascular invasion (P = 0.028), and HBV DNA > 104 copies/mL at the time of resection (P = 0.003) were independently predictive of HCC recurrence for the entire population. For the 115 patients with serial measurements of viral levels, tumor size > 5 cm, HBV DNA > 104 copies/mL at resection, and the absence of sustained HBV DNA level < 104 copies/mL, the presence of cirrhosis, and elevated aminotransferase levels (> 40 IU/L) were marginally or significantly associated with HCC recurrence and overall survival. However, on multivariate analysis, sustained HBV DNA level < 104 copies/mL was the only factor for both low recurrence (P = 0.002; odds ratio [OR] 3.13; 95% confidence interval [CI] 1.55–6.35) and longer survival (P = 0.002; OR 3.76; 95% CI 1.61–8.78).

Conclusions:  A high HBV replication state is among the most important predictors of adverse outcome after resection of HBV‐related HCC. The sustained suppression of HBV below 104 copies/mL is a strong protective factor for long‐term recurrence‐free and overall survival.
Father involvement and child well‐being
Authors: Wilson Katherine R , Prior Margot R
Published: June 2010

Children's development is influenced by the quantity and quality of father involvement (FI) in their care. High levels of FI are associated with significant and highly desirable outcomes for children and families. In order to foster increased positive FI, dissemination of evidence‐based knowledge regarding its importance and value is needed. Evidence summarised here suggests that fathers and mothers alike need to be educated about the fact that positive paternal involvement is highly beneficial to children.
Infantile hypertrophic pyloric stenosis: Does size really matter?
Authors: Hsu Peter , Klimek Jan , Nanan Ralph
Published: June 2010

Pyloric stenosis is a common infantile disorder typically occurring between 2 weeks and 8 weeks of age. Presentation outside this range and in premature infants is uncommon and often atypical. We present three cases, a pair of premature twins and a 5.5‐month‐old infant with pyloric stenosis. We highlight that despite their difference in size and weight, ultrasound examination remains a reliable diagnostic tool. However, pyloric measurements should be interpreted in combination with the other dynamic features on ultrasound.
Differing causal roles for lateral occipital cortex and occipital face area in invariant shape recognition
Authors: Silvanto J. , Schwarzkopf D. S. , Gilaie‐Dotan S. , Rees G.
Published: June 2010

The human extrastriate visual cortex contains functionally distinct regions where neuronal populations exhibit signals that are selective for objects. How such regions might play a causal role in underpinning our ability to recognize objects across different viewpoints remains uncertain. Here, we tested whether two extrastriate areas, the lateral occipital (LO) region and occipital face area (OFA), contained neuronal populations that play a causal role in recognizing two‐dimensional shapes across different rotations. We used visual priming to modulate the rotation‐sensitive activity of neuronal populations in these areas. State‐dependent transcranial magnetic stimulation (TMS) was applied after the presentation of a shape and immediately before a subsequent probe shape to which participants had to respond. We found that TMS applied to both the LO region and OFA modulated rotation‐invariant shape priming but, whereas the LO region was modulated by TMS for small rotations, the OFA was modulated for larger rotations. Importantly, our results demonstrate that a node in the face‐sensitive network, the OFA, participates in causally relevant encoding of non‐face stimuli.
To IPO or Not To IPO: Risks, Uncertainty and the Decision to Go Public
Authors: Latham Scott , Braun Michael R.
Published: June 2010

We employ three views of agency theory in unison to investigate managerial risk‐taking in uncertain markets. Using 124 firms that filed to go public toward the end of the technology boom (2000–2002), we explore the influence of CEO ownership on the decision to continue or withdraw an initial public offering (IPO) in deteriorating public equity markets. We find an inverse U‐shaped relationship between CEOs' equity participation and the decision to proceed with a public offering: the probability of IPO cancellation in weak capital markets increases as CEOs hold too little or too much ownership. Our results also indicate that a firm's debt levels are positively linked with the IPO decision. CEO ownership and leverage intensity interact to influence the decision to take a firm public.