bluuebunny/crossref_metadata_2025 huggingface dataset:
The exponential of iteration of 𝑒^{𝑥}-1
Authors: Walker Peter L.
Published: June 2010

We study the mapping properties of a nonconstant entire solution of the equation
\[

f
(
z
+
1
)
=

e

f
(
z
)

−
1.

f(z + 1) = {e^{f(z)}} - 1.

\]
𝐿(𝑋,𝐶(𝐾)) as a dual space
Authors: Rao T. S. S. R. K.
Published: June 2010

We exhibit a class of Banach spaces

X
X

, with

X
∗

{X^*}

having nontrivial centralizer for which the space of operators

L
(
X
,
C
(
K
)
)

L(X,C(K))

is a dual space implies that

K
K

is hyperstonian.
Jones and 𝑄 polynomials for 2-bridge knots and links
Authors: Kanenobu Taizo
Published: June 2010

It is known that the

Q
Q

polynomial of a

2
2

-bridge knot or link can be obtained from the Jones polynomial. We construct arbitrarily many

2
2

-bridge knots or links with the same

Q
Q

polynomial but distinct Jones polynomials.
Ultraweakly closed algebras and preannihilators
Authors: MacDonald Gordon W.
Published: June 2010

We give an alternate description of algebras in the class of ultra-weakly closed subspaces of

B

(

H

)

\mathcal {B}(\mathcal {H})

via the preannihilator. We then apply this result to show that proper ultraweakly closed algebras of bounded operators on an infinite-dimensional Hilbert space

H

\mathcal {H}

have infinite codimension. We also use this alternate description of algebras to say something the structure of rank-one operators in unicellular algebras.
Isometrically invariant extensions of Lebesgue measure
Authors: Ciesielski Krzysztof
Published: June 2010

The purpose of this note is to give a very short prove of the theorem thta every isometrically invariant measure extending Lebesgue measure on

R

n

{{\mathbf {R}}^n}

has a proper isometrically invariant extension, i.e., that there is no maximal isometrically invariant extension of Lebesgue measure on

R

n

{{\mathbf {R}}^n}

.
Analyticity of almost everywhere differentiable functions
Authors: Howard Eric J.
Published: June 2010

We develop a partitioning lemma (see Lemma 5) for superadditive set functions satisfying certain continuity conditions. This leads to a relatively simple proof of two theorems of A. S. Besicovitch on when a function of a complex variable that is continuous and differentiable outside of small exceptional sets is analytic (or almost everywhere equal to an analytic function).
Topological transitivity of compact actions on 𝐶*-algebras
Authors: Lee Sa Ge , Jang Sun Young
Published: June 2010

We give some conditions under which minimality is equivalent to the other notions in noncommutative ergodic theory and prove that in a compact

C

∗

{{\mathbf {C}}^*}

-dynamical system topological transitivity is invariant under exterior equivalence.
A remark on a paper by C. Fefferman: “The uncertainty principle” [Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206; MR0707957 (85f:35001)]
Authors: Chiarenza Filippo , Frasca Michele
Published: June 2010

We give a simplified proof of an imbedding theorem by C. Fefferman [3].
Crickets, zippers and the Bers universal Teichmüller space
Authors: Astala K. , Gehring F. W.
Published: June 2010

This paper contains a simple characterization of the conformal mappings

f
f

of the unit disk

U
U

whose Schwarzians lie in the closure of the Bers universal Teichmüller space. A second characterization is given and three examples are studied for the case where

f
(
U
)

f(U)

is the complement of a quasiarc.
Derivatives of Hardy functions
Authors: Choe Boo Rim
Published: June 2010

Let

B
B

be the open unit ball of

C
n

{C^n}

, and set

S
=
∂
B

S = \partial B

. It is shown that if

φ
∈

L
p

(
S
)

,
φ
>
0

\varphi \in {L^p}\left ( S \right ),\varphi > 0

, is a lower semicontinuous function on

S
S

and

1

/

q
>
1
+
1

/

p

1/q > 1 + 1/p

, then, for a given

ε
>
0

\varepsilon > 0

, there exists a function

f
∈

H
p

(
B
)

f \in {H^p}\left ( B \right )

with

f

(
0
)

=
0

f\left ( 0 \right ) = 0

such that

|

f
∗

|

=
φ

\left | {{f^ * }} \right | = \varphi

almost everywhere on

S
S

and

∫
B

|

∇
f

|

q

d
V
>
ε

\int _B {{{\left | {\nabla f} \right |}^q}dV > \varepsilon }

where

V
V

denotes the normalized volume measure on

B
B

.
Square function estimates and the 𝑇(𝑏) theorem
Authors: Semmes Stephen
Published: June 2010

A very simple proof is given of the sort of square function estimates provided by the

T
(
b
)

T(b)

theorem. The approach is based on the bilinear methods of Coifman and Meyer. Among the applications is an easy proof of the boundedness of the Cauchy integral operator on Lipschitz graphs.
On the divisor of involutions in an elliptic modular surface
Authors: Hewitt P. R.
Published: June 2010

Let

E
→
X

E \to X

be an elliptic modular surface and

S
S

the tangential ruled surface of a projective embedding of

X
X

. The divisor that collects the involutions of the elliptic fibers of

E
E

is precisely the branch locus of

E
→
S

E \to S

(at least generically). In this paper, we present two theorems that characterize this divisor in terms of the action of the group of modular automorphisms. These results extend work of D. Burns [1].
Inverse systems of abstract Lebesgue spaces
Authors: Christianson Bruce
Published: June 2010

We show that inverse limits exist in the category of

L
1

{L^1}

spaces and positive linear contractions between them. This result generalizes the wellknown classical results for inverse systems of Choquet Simplexes and of

L
L

-balls, but our proof is simple and more purely geometrical. The result finds physical application in the study of random fields.
Asymptotic depth and connectedness in projective schemes
Authors: Brodmann M.
Published: June 2010

Let

I
⊆

m

I \subseteq \mathfrak {m}

be an ideal of a local noetherian ring

(

R
,

m

)

\left ( {R,\mathfrak {m}} \right )

. Consider the exceptional fiber

π

−
1

(

V

(
1
)

)

{\pi ^{ - 1}}\left ( {V\left ( 1 \right )} \right )

of the blowing-up morphism
\[

π
:
Proj
⁡

(

⊕

n
≥
0

I
n

)

→
Spec
⁡

(
R
)

\pi :\operatorname {Proj}\left ( {{ \oplus _{n \geq 0}}{I^n}} \right ) \to \operatorname {Spec}\left ( R \right )

\]
and the special fiber

π

−
1

(

m

)

{\pi ^{ - 1}}\left ( \mathfrak {m} \right )

. We show that the complement set
\[

π

−
1

(

V

(
I
)

)

−

π

−
1

(

m

)

{\pi ^{ - 1}}\left ( {V\left ( I \right )} \right ) - {\pi ^{ - 1}}\left ( \mathfrak {m} \right )

\]
is highly connected if the asymptotic depth of the higher conormal modules

I
n

/

I

n
+
1

{I^n}/{I^{n + 1}}

is large.
Fixed points of unitary 𝑍/𝑝^{𝑠}-manifolds
Authors: Waner Stefan , Wu Yihren
Published: June 2010

Let

G
=

Z

/

p
s

G = {\mathbf {Z}}/{p^s}

(

p
p

an odd prime). We show that restricting the local representations in a unitary

G
G

-manifold

M
M

with isolated fixed points results in severe restrictions on the number of fixed points (counted with the sign of their orientation), paralleling results obtained by Conner and Floyd in the case

G
=

Z

/

p

G = {\mathbf {Z}}/p

. Specifically, the number of noncancelling fixed points is either zero or divisible by

p
n

{p^n}

, where

n
→
∞

n \to \infty

as the dimension of

M
→
∞

M \to \infty

. This result also parallels phenomena in framed

G
G

-manifolds, as discussed by the first author in a previous paper.
Divisibility by 2 of Stirling-like numbers
Authors: Davis Donald M.
Published: June 2010

We give a characterization of functions of the form

f
(
n
)
=
ν
(
n
−
E
)

f(n) = \nu (n - E)

, where

ν
(
−
)

\nu ( - )

denotes the exponent of 2, and

E
E

is a

2
2

-adic integer. We show that it applies to the restriction to even or odd integers of the function

f
(
n
)
=
ν
(
a

5
n

+
b

3
n

+
c
)

f(n) = \nu (a{5^n} + b{3^n} + c)

, with mild restrictions on

a
,
b

a,b

, and

c
c

. This function is closely related to divisibility of certain Stirling numbers of the second kind.
The numerical analysis of implicit Runge-Kutta methods for a certain nonlinear integro-differential equation
Authors: Yuan Wei , Tang Tao
Published: June 2010

In this paper and in an earlier 1987 paper, the mathematical theory and numerical methods for the nonlinear integro-differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row u prime left-parenthesis t right-parenthesis plus p left-parenthesis t right-parenthesis u left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript t Baseline k left-parenthesis t comma s right-parenthesis u left-parenthesis t minus s right-parenthesis u left-parenthesis s right-parenthesis d s equals q left-parenthesis t right-parenthesis comma 0 less-than-or-equal-to t less-than-or-equal-to upper T comma 2nd Row u left-parenthesis 0 right-parenthesis equals u 0 EndLayout">

u
′

(
t
)
+
p
(
t
)
u
(
t
)
+

∫
0
t

k
(
t

,
s
)
u
(
t
−
s
)
u
(
s
)

d
s
=
q
(
t
)
,

0
≤
t
≤
T
,

u
(
0
)
=

u
0

\begin {array}{*{20}{c}} {u’(t) + p(t)u(t) + \int _0^t {k(t} ,s)u(t - s)u(s)\,ds = q(t),\quad 0 \leq t \leq T,} \hfill \\ {u(0) = {u_0}} \hfill \\ \end {array}

\]
are considered. Equations of this type occur as model equations for describing turbulent diffusion. Previously, the existence and uniqueness properties of the solutions of the model equation were solved completely, and a class of implicit Runge-Kutta methods with m stages for the approximate solution of the model equation was introduced. In this paper, we give a further numerical analysis of these methods. It is proved that the implicit Runge-Kutta methods with n stages are of optimal approximation order

p
=
2
m

p = 2m

. Some computational examples are given.
Accurate calculation of functions used in a model of the nematic behavior of self-assembling systems
Authors: Berger Alan E.
Published: June 2010

An algorithm used to evaluate double sums arising in a model describing the nematic phase behavior of surfactant solutions is demonstrated to yield approximations accurate to within a tenth of a percent. When direct summation would converge slowly, an asymptotic result is employed based on a double application of the Euler-Maclaurin sum formula.
Error and stability analysis of boundary methods for elliptic problems with interfaces
Authors: Li Zi Cai , Mathon Rudolf
Published: June 2010

In boundary methods, piecewise particular solutions are employed to solve a given elliptic equation within subdomains of some region of interest. A boundary approximation is then obtained by satisfying the interior and exterior boundary conditions in a least squares sense. In this paper, we examine convergence, derive error norm bounds for approximate solutions and conduct a stability analysis of the associated algebraic problem. The aim of this analysis is to help choosing good partitions of subdomains. Finally, numerical experiments are carried out for a typical interface problem, demonstrating that very accurate solutions can be obtained while at the same time keeping small the condition numbers of the associated coefficient matrices.
Generalized noninterpolatory rules for Cauchy principal value integrals
Authors: Rabinowitz Philip
Published: June 2010

Consider the Cauchy principal value integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis k f semicolon lamda right-parenthesis equals contour-integral k left-parenthesis x right-parenthesis StartFraction f left-parenthesis x right-parenthesis Over x minus lamda EndFraction d x comma negative 1 greater-than lamda greater-than 1 period">

I
(
k
f
;
λ
)
=
∮
k
(
x
)

f
(
x
)

x
−
λ

d
x
,

−
1
>
λ
>
1.

I(kf;\lambda ) = \oint k(x)\frac {{f(x)}}{{x - \lambda }}\,dx,\quad - 1 > \lambda > 1.

\]
If we approximate

f
(
x
)

f(x)

by

∑

j
=
0

N

a
j

p
j

(
x
;
w
)

\sum _{j = 0}^N\;{a_j}{p_j}(x;w)

where

{

p
j

}

\{ {p_j}\}

is a sequence of orthonormal polynomials with respect to an admissible weight function w and

a
j

=
(
f
,

p
j

)

{a_j} = (f,{p_j})

, then an approximation to

I
(
k
f
;
λ
)

I(kf;\lambda )

is given by

∑

j
=
0

N

a
j

I
(
k

p
j

;
λ
)

\sum _{j = 0}^N\;{a_j}I(k{p_j};\lambda )

. If, in turn, we approximate

a
j

{a_j}

by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript j m Baseline equals sigma-summation Underscript i equals 1 Overscript m Endscripts w Subscript i m Baseline f left-parenthesis x Subscript i m Baseline right-parenthesis p Subscript j Baseline left-parenthesis x Subscript i m Baseline right-parenthesis">

a

j
m

=

∑

i
=
1

m

w

i
m

f
(

x

i
m

)

p
j

(

x

i
m

)

{a_{jm}} = \sum _{i = 1}^m\;{w_{im}}f({x_{im}}){p_j}({x_{im}})

, then we get a double sequence of approximations

{

Q
m
N

(
f
;
λ
)
}

\{ Q_m^N(f;\lambda )\}

to

I
(
k
f
;
λ
)

I(kf;\lambda )

. We study the convergence of this sequence by relating it to the sequence of approximations associated with

I
(
w
f
;
λ
)

I(wf;\lambda )

which has been investigated previously.
Bivariate 𝐶¹ quadratic finite elements and vertex splines
Authors: Chui Charles K. , He Tian Xiao
Published: June 2010

Following work of Heindl and of Powell and Sabin, each triangle of an arbitrary (regular) triangulation

Δ
\Delta

of a polygonal region

Ω
\Omega

in

R

2

{\mathbb {R}^2}

is subdivided into twelve triangles, using the three medians, yielding the refinement

Δ
^

\hat \Delta

of

Δ
\Delta

, so that

C
1

{C^1}

quadratic finite elements can be constructed. In this paper, we derive the Bézier nets of these elements in terms of the parameters that describe function and first partial derivative values at the vertices and values of the normal derivatives at the midpoints of the edges of

Δ
\Delta

. Consequently, bivariate

C
1

{C^1}

quadratic (generalized) vertex splines on

Δ
\Delta

have an explicit formulation. Here, a generalized vertex spline is one which is a piecewise polynomial on the refined grid partition

Δ
^

\hat \Delta

and has support that contains at most one vertex of the original partition

Δ
\Delta

in its interior. The collection of all

C
1

{C^1}

quadratic generalized vertex splines on

Δ
\Delta

so constructed is shown to form a basis of

S
2
1

(

Δ
^

)

S_2^1(\hat \Delta )

, the vector space of all functions on

C
1

(
Ω
)

{C^1}(\Omega )

whose restrictions to each triangular cell of the partition

Δ
^

\hat \Delta

are quadratic polynomials. A subspace with the basis given by appropriately chosen generalized vertex splines with exactly one vertex of

Δ
\Delta

in the interior of their supports, that reproduces all quadratic polynomials, is identified, and hence, has approximation order three. Quasi-interpolation formulas using this subspace are obtained. In addition, a constructive procedure that yields a locally supported basis of yet another subspace with dimension given by the number of vertices of

Δ
\Delta

, that has approximation order three, is given.
On an integer’s infinitary divisors
Authors: Cohen Graeme L.
Published: June 2010

The notions of unitary divisor and biunitary divisor are extended in a natural fashion to givek-ary divisors, for any natural numberk. We show that we may sensibly allowkto increase indefinitely, and this leads toinfinitary divisors. The infinitary divisors of an integer are described in full, and applications to the obvious analogues of the classical perfect and amicable numbers and aliquot sequences are given.
Multiplicative congruential random number generators with modulus 2^{𝛽}: an exhaustive analysis for 𝛽=32 and a partial analysis for 𝛽=48
Authors: Fishman George S.
Published: June 2010

This paper presents the results of a search to find optimal maximal period multipliers for multiplicative congruential random number generators with moduli

2

32

{2^{32}}

and

2

48

{2^{48}}

. Here a multiplier is said to be optimal if the distance between adjacent parallel hyperplanes on which k-tuples lie does not exceed the minimal achievable distance by more than 25 percent for

k
=
2
,
…
,
6

k = 2, \ldots ,6

. This criterion is considerably more stringent than prevailing standards of acceptability and leads to a total of only 132 multipliers out of the more than 536 million candidate multipliers that exist for modulus

2

32

{2^{32}}

and to only 42 multipliers in a sample of about 67.1 million tested among the more than

351
×

10

11

351 \times {10^{11}}

candidate multipliers for modulus

2

48

{2^{48}}

. Section 1 reviews the basic properties of multiplicative congruential generators and

§
2

\S 2

describes worst case performance measures. These include the maximal distance between adjacent parallel hyperplanes, the minimal number of parallel hyperplanes, the minimal distance between k-tuples and the discrepancy. For modulus

2

32

{2^{32}}

,

§
3

\S 3

presents the ten best multipliers and compares their performances with those of two multipliers that have been recommended in the literature. Comparisons using packing measures in the space of k-tuples and in the dual space are also made. For modulus

2

48

{2^{48}}

,

§
4

\S 4

also presents analogous results for the five best multipliers and for two multipliers suggested in the literature.
Integer relations among algebraic numbers
Authors: Just Bettina
Published: June 2010

A vector

m
=
(

m
1

,
…
,

m
n

)
∈

Z

n

∖
{
0
}

m = ({m_1}, \ldots ,{m_n}) \in {{\mathbf {Z}}^n}\backslash \{ 0\}

is called an integer relation for the real numbers

α
1

,
…
,

α
n

{\alpha _1}, \ldots ,{\alpha _n}

, if

∑

α
i

m
i

=
0

\sum {\alpha _i}{m_i} = 0

holds. We present an algorithm that, when given algebraic numbers

α
1

,
…
,

α
n

{\alpha _1}, \ldots ,{\alpha _n}

and a parameter

ε
\varepsilon

, either finds an integer relation for

α
1

,
…
,

α
n

{\alpha _1}, \ldots ,{\alpha _n}

or proves that no relation of Euclidean length shorter than

1

/

ε

1/\varepsilon

exists. Each algebraic number is assumed to be given by its minimal polynomial and by a sufficiently precise rational approximation. Our algorithm uses the Lenstra-Lenstra-Lovász lattice basis reduction technique. It performs
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p o l y left-parenthesis log 1 slash epsilon comma n comma log max Underscript i Endscripts height left-parenthesis alpha Subscript i Baseline right-parenthesis comma left-bracket bold upper Q left-parenthesis alpha 1 comma ellipsis comma alpha Subscript n Baseline right-parenthesis colon bold upper Q right-bracket right-parenthesis">

poly

(

log
⁡
1

/

ε
,
n
,
log
⁡

max
i

height

(

α
i

)
,
[

Q

(

α
1

,
…
,

α
n

)
:

Q

]

)

{\operatorname {poly}}\left ( {\log 1/\varepsilon ,n,\log \max \limits _i {\text {height}}({\alpha _i}),[{\mathbf {Q}}({\alpha _1}, \ldots ,{\alpha _n}):{\mathbf {Q}}]} \right )

\]
bit operations. The straightforward algorithm that works with a primitive element of the field extension

Q

(

α
1

,
…
,

α
n

)

{\mathbf {Q}}({\alpha _1}, \ldots ,{\alpha _n})

of Q would take
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p o l y left-parenthesis n comma log max Underscript i Endscripts height left-parenthesis alpha Subscript i Baseline right-parenthesis comma product Underscript i equals 1 Overscript n Endscripts degree left-parenthesis alpha Subscript i Baseline right-parenthesis right-parenthesis">

poly

(

n
,
log
⁡

max
i

height

(

α
i

)
,

∏

i
=
1

n

degree

(

α
i

)

)

{\operatorname {poly}}\left ( {n,\log \max \limits _i {\text {height}}({\alpha _i}),\prod \limits _{i = 1}^n {{\text {degree}}({\alpha _i})} } \right )

\]
bit operations. In order to prove the correctness of the algorithm, we show a lower bound for

|

∑

α
1

m
i

|

\left | {\sum {\alpha _1}{m_i}} \right |

if m is not an integer relation for

α
1

,
…
,

α
n

{\alpha _1}, \ldots ,{\alpha _n}

, which may be interesting in its own right.
Multivariate interpolation and conditionally positive definite functions. II
Authors: Madych W. R. , Nelson S. A.
Published: June 2010

We continue an earlier study of certain spaces that provide a variational framework for multivariate interpolation. Using the Fourier transform to analyze these spaces, we obtain error estimates of arbitrarily high order for a class of interpolation methods that includes multiquadrics.
A finite element method for time-dependent convection-diffusion equations
Authors: Richter Gerard R.
Published: June 2010

We present a finite element method for time-dependent convection-diffusion equations. The method is explicit and is applicable with piecewise polynomials of degree

n
≥
2

n \geq 2

. In the limit of zero diffusion, it reduces to a recently analyzed finite element method for hyperbolic equations. Near optimal error estimates are derived. Numerical results are given.
Coupling finite element and spectral methods: first results
Authors: Bernardi Christine , Debit Naïma , Maday Yvon
Published: June 2010

A Poisson equation on a rectangular domain is solved by coupling two methods: the domain is divided in two squares; a finite element approximation is used on the first square and a spectral discretization is used on the second. Two kinds of matching conditions on the interface are presented and compared; in both cases, error estimates are proved.
Nonnegative and skew-symmetric perturbations of a matrix with positive inverse
Authors: Buffoni Giuseppe
Published: June 2010

Let A be a nonsingular matrix with positive inverse and B a non-negative matrix. Let the inverse of

A
+
v
B

A + vB

be positive for

0
≤
v
>

v
∗

>
+
∞

0 \leq v > {v^ \ast } > + \infty

and at least one of its entries be equal to zero for

v
=

v
∗

v = {v^ \ast }

; an algorithm to compute

v
∗

{v^ \ast }

is described in this paper. Furthermore, it is shown that if

A
+

A

T

A + {A^{\text {T}}}

is positive definite, then the inverse of

A
+
v
(
B
−

B

T

)

A + v(B - {B^{\text {T}}})

is positive for

0
≤
v
>

v
∗

0 \leq v > {v^ \ast }

.
Weighted inf-sup condition and pointwise error estimates for the Stokes problem
Authors: Durán Ricardo G. , Nochetto Ricardo H.
Published: June 2010

Convergence of mixed finite element approximations to the Stokes problem in the primitive variables is examined in maximum norm. Quasioptimal pointwise error estimates are derived for discrete spaces satisfying a weighted inf-sup condition similar to the Babuška -Brezzi condition. The usual techniques employed to prove the inf-sup condition in energy norm can be easily extended to the present situation, thus providing several examples to our abstract framework. The popular Taylor-Hood finite element is the most relevant one.
Lattice rules for multiple integration and discrepancy
Authors: Niederreiter Harald , Sloan Ian H.
Published: June 2010

Upper and lower bounds for the discrepancy of nodes in lattice rules for multidimensional numerical integration are established. In this way the applicability of lattice rules is extended to nonperiodic integrands.
The collocation method for first-kind boundary integral equations on polygonal regions
Authors: Yan Yi
Published: June 2010

In this paper the collocation method for first-kind boundary integral equations, by using piecewise constant trial functions with uniform mesh, is shown to be equivalent to a projection method for second-kind Fredholm equations. In a certain sense this projection is an interpolation projection. By introducing this technique of analysis, we particularly consider the case of polygonal boundaries. We give asymptotic error estimates in

L
2

{L_2}

norm on the boundaries, and some superconvergence results for the single layer potential.
Rayleigh quotient iteration for nonsymmetric matrices
Authors: Batterson Steve , Smillie John
Published: June 2010

Rayleigh quotient iteration is an iterative algorithm for the calculation of approximate eigenvectors of a matrix. Given a matrix, the algorithm supplies a function whose iteration of an initial vector,v0{v_0}, produces a sequence of vectors,vn{v_n}. If the matrix is symmetric, then for almost any choice ofv0{v_0}the sequence will converge to an eigenvector at an eventually cubic rate. In this paper we show that there exist open sets of real matrices, each of which possesses an open set of initial vectors for which the algorithm will not converge to an eigenspace. The proof employs techniques from dynamical systems and bifurcation theory.
Convergence of extended Lagrange interpolation
Authors: Criscuolo Giuliana , Mastroianni Giuseppe , Occorsio Donatella
Published: June 2010

The authors give a procedure to construct extended interpolation formulae and prove some uniform convergence theorems.
Convergence properties of a class of product formulas for weakly singular integral equations
Authors: Criscuolo Giuliana , Mastroianni Giuseppe , Monegato Giovanni
Published: June 2010

We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the type
\[

∫

−
1

1

K
(
x
,
y
)
f
(
x
)

d
x
,

−
1
≤
y
≤
1
,

\int _{ - 1}^1 {K(x,y)f(x)\,dx,} \quad - 1 \leq y \leq 1,

\]
where the kernel

K
(
x
,
y
)

K(x,y)

is weakly singular and the function

f
(
x
)

f(x)

has singularities only at the endpoints

±
1

\pm 1

. In particular, when

K
(
x
,
y
)
=
log
⁡

|

x
−
y

|

K(x,y) = \log |x - y|

,

K
(
x
,
y
)
=

|

x
−
y

|

v

K(x,y) = |x - y{|^v}

,

v
>
−
1

v > - 1

, and

f
(
x
)

f(x)

has algebraic singularities of the form

(
1
±
x

)
σ

{(1 \pm x)^\sigma }

,

σ
>
−
1

\sigma > - 1

, we prove that the uniform rate of convergence of the rules is

O
(

m

−
2
−
2
σ

log
2

m
)

O({m^{ - 2 - 2\sigma }}{\log ^2}m)

in the case of the first kernel, and

O
(

m

−
2
−
2
σ
−
2
v

log
⁡
m
)

O({m^{ - 2 - 2\sigma - 2v}}\log m)

if

v
≤
0

v \leq 0

, or

O
(

m

−
2
−
2
σ

log
⁡
m
)

O({m^{ - 2 - 2\sigma }}\log m)

if

v
>
0

v > 0

, for the second, where m is the number of points in the quadrature rule.
A characterization theorem for the discrete best monotonic approximation problem
Authors: Demetriou I. C.
Published: June 2010

A characterization theorem is derived that motivates a procedure for generating discrete best monotonic approximations to n sequential data values, when a strictly convex objective function is used in the calculation. The procedure is highly useful in the discrete nonlinear optimization calculation that produces the best piecewise monotonic approximations to the data.
Boundedness of dispersive difference schemes
Authors: Estep Donald , Loss Michael , Rauch Jeffrey
Published: June 2010

The pointwise behavior of dispersive difference schemes for the simple wave equation in one dimension is analyzed. If the initial data are in certain Besov spaces, the scheme is shown to be pointwise unbounded. Boundedness is shown when the initial data are of bounded variation.
Local convergence theory of inexact Newton methods based on structured least change updates
Authors: Martínez José Mario
Published: June 2010

In this paper we introduce a local convergence theory for Least Change Secant Update methods. This theory includes most known methods of this class, as well as some new interesting quasi-Newton methods. Further, we prove that this class of LCSU updates may be used to generate iterative linear methods to solve the Newton linear equation in the Inexact-Newton context. Convergence at a q-superlinear rate (or at an "ideal" linear rate, in the sense of Dennis-Walker) of the Inexact Newton methods generated in this way is proved, independently of the number of iterations used in the linear iterative subalgorithm. We apply the new theory to some particular methods.
On computations with dense structured matrices
Authors: Pan Victor
Published: June 2010

We reduce several computations with Hilbert and Vandermonde type matrices to matrix computations of the Hankel-Toeplitz type (and vice versa). This unifies various known algorithms for computations with dense structured matrices and enables us to extend any progress in computations with matrices of one class to the computations with other classes of matrices. In particular, this enables us to compute the inverses and the determinants of

n
×
n

n \times n

matrices of Vandermonde and Hilbert types for the cost of

O
(
n

log
2

n
)

O(n{\log ^2}n)

arithmetic operations. (Previously, such results were only known for the more narrow class of Vandermonde and generalized Hilbert matrices.)
Linear combinations of orthogonal polynomials generating positive quadrature formulas
Authors: Peherstorfer Franz
Published: June 2010

Let

p
k

(
x
)
=

x
k

+
⋯

{p_k}(x) = {x^k} + \cdots

,

k
∈

N

0

k \in {{\mathbf {N}}_0}

, be the polynomials orthogonal on

[
−
1
,
+
1
]

[ - 1, + 1]

with respect to the positive measure

d
σ

d\sigma

. We give sufficient conditions on the real numbers

μ
j

{\mu _j}

,

j
=
0
,
…
,
m

j = 0, \ldots ,m

, such that the linear combination of orthogonal polynomials

∑

j
=
0

m

μ
j

p

n
−
j

\sum _{j = 0}^m{\mu _j}{p_{n - j}}

has n simple zeros in

(
−
1
,
+
1
)

( - 1, + 1)

and that the interpolatory quadrature formula whose nodes are the zeros of

∑

j
=
0

m

μ
j

p

n
−
j

\sum _{j = 0}^m{\mu _j}{p_{n - j}}

has positive weights.
Viscous splitting for the unbounded problem of the Navier-Stokes equations
Authors: Ying Lung-An
Published: June 2010

The viscous splitting for the exterior initial-boundary value problems of the Navier-Stokes equations is considered. It is proved that the approximate solutions are uniformly bounded in the space

L
∞

(
0
,
T
;

H

s
+
1

(
Ω
)
)

{L^\infty }(0,T;{H^{s + 1}}(\Omega ))

,

s
>

3
2

s > \frac {3}{2}

, and converge with a rate of

O
(
k
)

O(k)

in the space

L
∞

(
0
,
T
;

H
1

(
Ω
)
)

{L^\infty }(0,T;{H^1}(\Omega ))

, where k is the length of the time steps.
Optimal-order nonnested multigrid methods for solving finite element equations. I. On quasi-uniform meshes
Authors: Zhang Shangyou
Published: June 2010

We prove that the multigrid method works with optimal computational order even when the multiple meshes are not nested. When a coarse mesh is not a submesh of the finer one, the coarse-level correction usually does not have the

a
(
⋅
,
⋅
)

a( \cdot , \cdot )

projection property and does amplify the iterative error in some components. Nevertheless, the low-frequency components of the error can still be caught by the coarse-level correction. Since the (amplified) high-frequency errors will be damped out by the fine-level smoothing efficiently, the optimal work order of the standard multigrid method can still be maintained. However, unlike the case of nested meshes, a nonnested multigrid method with one smoothing does not converge in general, no matter whether it is a V-cycle or a W-cycle method. It is shown numerically that the convergence rates of nonnested multigrid methods are not necessarily worse than those of nested ones. Since nonnested multigrid methods accept quite arbitrarily related meshes, we may then combine the efficiencies of adaptive refinements and of multigrid algorithms.
Explicit bounds for primality testing and related problems
Authors: Bach Eric
Published: June 2010

Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omits a number that is

O
(

log
2

m
)

O({\log ^2}m)

. This has been generalized by Lagarias, Montgomery, and Odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields. This paper gives a different proof of this theorem, in which explicit constants are supplied. The bounds imply that if the ERH holds, a composite number m has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most

2

log
2

m

2{\log ^2}m

.
On the computation of a table of complex cubic fields with discriminant 𝐷>-10⁶
Authors: Fung Gilbert W. , Williams H. C.
Published: June 2010

A method for finding all the nonisomorphic complex cubic fields with discriminant

D
>
−

10
6

D > - {10^6}

is described. Three different methods were used to find the class number of each of these fields. The speed of these techniques is discussed and several tables illustrating the computational results are presented. These include tables of the distribution of the fields and the class numbers and the class group structures of these fields.
A deterministic algorithm for solving 𝑛=𝑓𝑢²+𝑔𝑣² in coprime integers 𝑢 and 𝑣
Authors: Hardy Kenneth , Muskat Joseph B. , Williams Kenneth S.
Published: June 2010

We give a deterministic algorithm for finding all primitive representations of a natural number n in the form

f

u
2

+
g

v
2

f{u^2} + g{v^2}

, where f and g are given positive coprime integers, and

n
≥
f
+
g
+
1

n \geq f + g + 1

,

(
n
,
f
g
)
=
1

(n,fg) = 1

. The running time of this algorithm is at most
\[

O

(

n

1

/

4

(
log
⁡
n

)
3

(
log
⁡
log
⁡
n
)
(
log
⁡
log
⁡
log
⁡
n
)
)
,

\mathcal {O}({n^{1/4}}{(\log n)^3}(\log \log n)(\log \log \log n)),

\]
uniformly in f and g.
On base and Turyn sequences
Authors: Koukouvinos C. , Kounias S. , Sotirakoglou K.
Published: June 2010

Base sequences of lengths

n
+
1

n + 1

,

n
+
1

n + 1

, n, n are constructed for all decompositions of

4
n
+
2

4n + 2

into four squares for

n
=
19
,
…
,
24

n = 19, \ldots ,24

. The construction is achieved through an algorithm which is also presented. It is proved through an exhaustive search that Turyn sequences do not exist for

n
=
18
,
…
,
27

n = 18, \ldots ,27

; since Turyn sequences cannot exist for

n
=
28

n = 28

or 29, the first unsettled case is

n
=
30

n = 30

.
Error estimates arising from certain pseudorandom sequences in a quasirandom search method
Authors: Mitchell Ricardo A.
Published: June 2010

In this paper we apply number-theoretic results to estimate the dispersion, a measure of denseness for sequences in a bounded set, of the Halton and Hammersley sequences in the hypercube

I
s

=

[
0
,
1

]
s

{I^s} = {[0,1]^s}

. It is seen that they attain the minimal order of magnitude for the dispersion.
Largest known twin primes
Authors: Parady B. K. , Smith Joel F. , Zarantonello Sergio E.
Published: June 2010

663777
⋅

2

7650

±
1
,
571305
⋅

2

7701

±
1

663777 \cdot {2^{7650}} \pm 1,571305 \cdot {2^{7701}} \pm 1

and

1706595
⋅

2

11235

±
1

1706595 \cdot {2^{11235}} \pm 1

are twin primes.
Numerical evaluation of Cauchy principal value integrals with singular integrands
Authors: Rabinowitz Philip
Published: June 2010

Convergence results are proved for sequences of interpolatory integration rules for Cauchy principal value integrals of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="contour-integral k left-parenthesis x right-parenthesis left-parenthesis f left-parenthesis x right-parenthesis slash left-parenthesis x minus lamda right-parenthesis right-parenthesis d x comma negative 1 greater-than lamda greater-than 1 comma">

∮
k
(
x
)
(
f
(
x
)

/

(
x
−
λ
)
)

d
x
,

−
1
>
λ
>
1
,

\oint k(x)(f(x)/(x - \lambda ))\,dx,\quad - 1 > \lambda > 1,

\]
when

f
(
x
)

f(x)

is singular at a point

ξ
≠
λ

\xi \ne \lambda

and the singularity is ignored or avoided.
Some identities involving harmonic numbers
Authors: Spiess Jürgen
Published: June 2010

Let

H
n

{H_n}

denote the nth harmonic number. Explicit formulas for sums of the form

∑

a
k

H
k

\sum {a_k}{H_k}

or

∑

a
k

H
k

H

n
−
k

\sum {a_k}{H_k}{H_{n - k}}

are derived, where the

a
k

{a_k}

are simple functions of k. These identities are generalized in a natural way by means of generating functions.
A sinc-Hunter quadrature rule for Cauchy principal value integrals
Authors: Bialecki Bernard
Published: June 2010

A Sinc function approach is used to derive a new Hunter type quadrature rule for the evaluation of Cauchy principal value integrals of certain analytic functions. Integration over a general arc in the complex plane is considered. Special treatment is given to integrals over the interval

(
−
1
,
1
)

( - 1,1)

. It is shown that the quadrature error is of order

O
(

e

−
c

N

)

O({e^{ - c\sqrt N }})

, where N is the number of nodes used, and where c is a positive constant which is independent of N. An application of the rule to the approximate solution of Cauchy singular integral equations is also discussed. Numerical examples are included to illustrate the performance of the rule.
A nonconforming multigrid method for the stationary Stokes equations
Authors: Brenner Susanne C.
Published: June 2010

An optimal-order W-cycle multigrid method for solving the stationary Stokes equations is developed, using P1 nonconforming divergence-free finite elements.
Plana’s summation formula for ∑^{∞}_{𝑚=1,3,⋯}𝑚⁻²sin(𝑚𝛼),𝑚⁻³cos(𝑚𝛼),𝑚⁻²𝐴^{𝑚},𝑚⁻³𝐴^{𝑚}
Authors: Dempsey Kevin M. , Liu Dajin , Dempsey John P.
Published: June 2010

The four series
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row sigma-summation Underscript 0 Overscript normal infinity Endscripts sine left-parenthesis 2 k plus 1 right-parenthesis alpha slash left-parenthesis 2 k plus 1 right-parenthesis squared comma sigma-summation Underscript 0 Overscript normal infinity Endscripts cosine left-parenthesis 2 k plus 1 right-parenthesis alpha slash left-parenthesis 2 k plus 1 right-parenthesis cubed comma 2nd Row sigma-summation Underscript 0 Overscript normal infinity Endscripts upper A Superscript 2 k plus 1 Baseline slash left-parenthesis 2 k plus 1 right-parenthesis squared comma and sigma-summation Underscript 0 Overscript normal infinity Endscripts upper A Superscript 2 k plus 1 Baseline slash left-parenthesis 2 k plus 1 right-parenthesis cubed EndLayout">

∑
0
∞

sin
⁡
(
2
k
+
1
)
α

/

(
2
k
+
1
)

2

,

∑
0
∞

cos
⁡
(
2
k
+
1
)
α

/

(
2
k
+
1
)

3

,

∑
0
∞

A

2
k
+
1

/

(
2
k
+
1
)

2

,

and

∑
0
∞

A

2
k
+
1

/

(
2
k
+
1
)

3

\begin {array}{*{20}{c}} {\sum \limits _0^\infty {\sin (2k + 1)\alpha /{{(2k + 1)}^2},} \quad \sum \limits _0^\infty {\cos (2k + 1)\alpha /{{(2k + 1)}^3},} } \\ {\sum \limits _0^\infty {{A^{2k + 1}}/{{(2k + 1)}^2},\quad {\text {and}}\quad \sum \limits _0^\infty {{A^{2k + 1}}/{{(2k + 1)}^3}} } } \\ \end {array}

\]
are very slowly convergent for

0
≤
α
≤
π

0 \leq \alpha \leq \pi

and as

A
→

1
−

A \to {1^ - }

. Direct summation involves thousands of terms to get the accuracy desired. Plana’s summation formula along with Romberg’s method of integration significantly and consistently improves the convergence and accuracy for the above series.
A stabilized Galerkin method for a third-order evolutionary problem
Authors: Fairweather Graeme , Sanz-Serna J. M. , Christie I.
Published: June 2010

The periodic initial value problem for the partial differential equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline plus u Subscript x x x Baseline plus beta left-parenthesis u squared right-parenthesis Subscript x Baseline plus StartFraction gamma Over 2 EndFraction left-parenthesis u squared right-parenthesis Subscript x x Baseline plus epsilon u Subscript x x Baseline minus delta u Subscript t x Baseline equals 0">

u
t

+

u

x
x
x

+
β

(

u
2

)
x

+

γ
2

(

u
2

)

x
x

+
ε

u

x
x

−
δ

u

t
x

=
0

{u_t} + {u_{xxx}} + \beta {({u^2})_x} + \frac {\gamma }{2}{({u^2})_{xx}} + \varepsilon {u_{xx}} - \delta {u_{tx}} = 0

,

ε
\varepsilon

,

δ
>
0

\delta > 0

, arises in fluidization models. The numerical integration of the problem is a difficult task in that many "reasonable" finite difference and finite element methods give rise to unstable discretizations. We show how to modify the standard Galerkin technique in order to stabilize it. Optimal-order error estimates are derived and the results of numerical experiments are presented. The stabilization technique suggested in the paper can be interpreted as rewriting the problem in Sobolev form and would also be useful for other equations involving terms of the form

u
t

−
δ

u

t
x

{u_t} - \delta {u_{tx}}

.
Pseudoprimes for higher-order linear recurrence sequences
Authors: Gurak S.
Published: June 2010

With the advent of high-speed computing, there is a rekindled interest in the problem of determining when a given whole number

N
>
1

N > 1

is prime or composite. While complex algorithms have been developed to settle this for 200-digit numbers in a matter of minutes with a supercomputer, there is a need for simpler, more practical algorithms for dealing with numbers of a more modest size. Such practical tests for primality have recently been given (running in deterministic linear time) in terms of pseudoprimes for certain second- or third-order linear recurrence sequences. Here, a powerful general theory is described to characterize pseudoprimes for higher-order recurrence sequences. This characterization leads to a broadening and strengthening of practical primality tests based on such pseudoprimes.
Extending Waring’s conjecture to 471,600,000
Authors: Kubina Jeffrey M. , Wunderlich Marvin C.
Published: June 2010

Two computer techniques are described that the authors used to confirm Waring’s Conjecture to 471,600,000, thus extending R. M. Stemmler’s result of 200,000 computed in 1964. M. C. Wunderlich increased the result to 175,600,000 in August 1988, and in the following October, J. M. Kubina, using a method suggested by Jean-Marc Deshouillers, raised it to 471,600,000.
Frobenius maps of abelian varieties and finding roots of unity in finite fields
Authors: Pila J.
Published: June 2010

We give a generalization to Abelian varieties over finite fields of the algorithm of Schoof for elliptic curves. Schoof showed that for an elliptic curveEoverFq{{\mathbf {F}}_q}, given by a Weierstrass equation, one can compute the number ofFq{{\mathbf {F}}_q}-rational points ofEin timeO((log⁡q)9)O({(\log q)^9}). Our result is the following. LetAbe an Abelian variety overFq{{\mathbf {F}}_q}. Then one can compute the characteristic polynomial of the Frobenius endomorphism ofAin timeO((log⁡q)Δ)O({(\log q)^\Delta }), whereΔ\Deltaand the implied constant depend only on the dimension of the embedding space ofA, the number of equations definingAand the addition law, and their degrees. The method, generalizing that of Schoof, is to use the machinery developed by Weil to prove the Riemann hypothesis for Abelian varieties. By means of this theory, the calculation is reduced to ideal-theoretic computations in a ring of polynomials in several variables overFq{{\mathbf {F}}_q}. As applications we show how to count the rational points on the reductions modulo primespof a fixed curve overQin time polynomial in logp; we show also that, for a fixed primel, we can compute thelth roots of unity modp, when they exist, in polynomial time in logp. This generalizes Schoof’s application of his algorithm to find square roots of a fixed integerxmodp.
The difference between the Weil height and the canonical height on elliptic curves
Authors: Silverman Joseph H.
Published: June 2010

Estimates for the difference of the Weil height and the canonical height of points on elliptic curves are used for many purposes, both theoretical and computational. In this note we give an explicit estimate for this difference in terms of the j-invariant and discriminant of the elliptic curve. The method of proof, suggested by Serge Lang, is to use the decomposition of the canonical height into a sum of local heights. We illustrate one use for our estimate by computing generators for the Mordell-Weil group in three examples.
A numerical method for evaluating zeros of solutions of second-order linear differential equations
Authors: Spigler Renato , Vianello Marco
Published: June 2010

A numerical algorithm for computing real zeros of solutions of 2nd-order linear differential equations

y
+
q
(
x
)
y
=
0

y+ q(x)y = 0

in the oscillatory case on a half line is studied. The method applies to the class

q
(
x
)
=
a
+
b

/

x
+
O
(

x

−
p

)

q(x) = a + b/x + O({x^{ - p}})

, with

a
>
0

a > 0

,

b
∈

R

b \in {\mathbf {R}}

,

p
>
1

p > 1

. This procedure is based on a certain nonlinear 3rd-order equation (Kummer’s equation) which plays a role in the theory of transformations of 2nd-order differential equations into each other, and was earlier introduced by F. W. J. Olver in 1950 to compute zeros of cylinder functions. A rigorous asymptotic and numerical analysis is developed by combining Borůvka’s approach to the study of Kummer’s equation and Olver’s original idea. Numerical examples are presented.
Regularization and the general Gauss-Markov linear model
Authors: Zha Hong Yuan , Hansen Per Christian
Published: June 2010

If the coefficient matrix in the general Gauss-Markov linear model is ill-conditioned, then the solution is very sensitive to perturbations. For such problems, we propose to add Tikhonov regularization to the model, and we show that this actually stabilizes the solution and decreases its variance. We also give a numerically stable algorithm for computing the regularized solution efficiently.
Optimal-order nonnested multigrid methods for solving finite element equations. II. On nonquasiuniform meshes
Authors: Zhang Shangyou
Published: June 2010

Nonnested multigrid methods are proved to be optimal-order solvers for finite element equations arising from elliptic problems in the presence of singularities caused by re-entrant corners and abrupt changes in the boundary conditions, where the multilevel grids are appropriately refined near singularities and are not necessarily nested. Therefore, optimal and realistic finer grids (compared with nested local refinements) could be used because of the freedom in generating nonnested multilevel grids.
The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs
Authors: Atkinson Kendall E. , Sloan Ian H.
Published: June 2010

Consider solving the Dirichlet problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column normal upper Delta u left-parenthesis upper P right-parenthesis equals 0 comma 2nd Column a m p semicolon upper P element-of double-struck upper R squared minus upper S comma 2nd Row 1st Column u left-parenthesis upper P right-parenthesis equals h left-parenthesis upper P right-parenthesis comma 2nd Column a m p semicolon upper P element-of upper S comma 3rd Row 1st Column sup StartAbsoluteValue u left-parenthesis upper P right-parenthesis EndAbsoluteValue greater-than normal infinity comma 2nd Column a m p semicolon 4th Row 1st Column upper P element-of double-struck upper R squared 2nd Column a m p semicolon EndLayout">

Δ
u
(
P
)
=
0
,

a
m
p
;

P
∈

R

2

∖
S
,

u
(
P
)
=
h
(
P
)
,

a
m
p
;

P
∈
S
,

sup

|

u
(
P
)

|

>
∞
,

a
m
p
;

P
∈

R

2

a
m
p
;

\begin {array}{*{20}{c}} {\Delta u(P) = 0,} \hfill & {P \in {\mathbb {R}^2}\backslash S,} \hfill \\ {u(P) = h(P),} \hfill & {P \in S,} \hfill \\ {\sup |u(P)| > \infty ,} \hfill & {} \hfill \\ {P \in {\mathbb {R}^2}} \hfill & {} \hfill \\ \end {array}

\]
with S a smooth open curve in the plane. We use single-layer potentials to construct a solution

u
(
P
)

u(P)

. This leads to the solution of equations of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper S Endscripts g left-parenthesis upper Q right-parenthesis log StartAbsoluteValue upper P minus upper Q EndAbsoluteValue d upper S left-parenthesis upper Q right-parenthesis equals h left-parenthesis upper P right-parenthesis comma upper P element-of upper S period">

∫
S

g
(
Q
)
log
⁡

|

P
−
Q

|

d
S
(
Q
)
=
h
(
P
)
,

P
∈
S
.

\int _S {g(Q)\log |P - Q|dS(Q) = h(P),\quad P \in S.}

\]
This equation is reformulated using a special change of variable, leading to a new first-kind equation with a smooth solution function. This new equation is split into a principal part, which is explicitly invertible, and a compact perturbation. Then a discrete Galerkin method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. A complete convergence analysis is given; numerical examples conclude the paper.