bluuebunny/crossref_metadata_2025 huggingface dataset:
Semistability at ∞, ∞-ended groups and group cohomology
Authors: Mihalik Michael L.
Published: June 2010

A finitely presented group

G
G

, is semistable at

∞
\infty

if for some (equivalently any) finite complex

X
X

, with

π
1

(
X
)
=
G

{\pi _1}(X) = G

, any two proper maps

r
,

s
:
[
0
,

∞
)
→

X
~

r,\,s:[0,\,\infty ) \to \tilde X

(

≡
\equiv

the universal cover of

X
X

) that determine the same end of

X
~

\tilde X

are properly homotopic in

X
~

\tilde X

. If

G
G

is semistable at

∞
\infty

, then

H
2

(
G
;

Z
G
)

{H^2}(G;\,ZG)

is free abelian.

0
0

- and

2
2

-ended groups are all semistable at

∞
\infty

. Theorem. If

G
=
A

∗

C

B

G = A{{\ast }_C}B

where

C
C

is finite and

A
A

and

B
B

are finitely presented, semistable at

∞
\infty

groups, then

G
G

is semistable at

∞
\infty

. Theorem. If

α
:
C
→
D

\alpha :C \to D

is an isomorphism between finite subgroups of the finitely presented semistable at

∞
\infty

group

H
H

, then the resulting

H
N
N

HNN

extension is semistable at

∞
\infty

. Combining these results with the accessibility theorem of M. Dunwoody gives Theorem. If all finitely presented

1
1

-ended groups are semistable at

∞
\infty

, then all finitely presented groups are semistable at

∞
\infty

.
Invariance results for delay and Volterra equations in fractional order Sobolev spaces
Authors: Kappel F. , Kunisch K.
Published: June 2010

Invariance of the trajectories of infinite delay- and Volterra-type equations in fractional order Sobolev spaces are derived under minimal assumptions on the problem data. Properties of fractional order Sobolev spaces defined over intervals are summarized.
Riemann problems for nonstrictly hyperbolic 2×2 systems of conservation laws
Authors: Schaeffer David G. , Shearer Michael
Published: June 2010

The Riemann problem is solved for

2
×
2

2 \times 2

systems of hyperbolic conservation laws having quadratic flux functions. Equations with quadratic flux functions arise from neglecting higher order nonlinear terms in hyperbolic systems that fail to be strictly hyperbolic everywhere. Such equations divide into four classes, three of which are considered in this paper. The solution of the Riemann problem is complicated, with new types of shock waves, and new singularities in the dependence of the solution on the initial data. Several ideas are introduced to help organize and clarify the new phenomena.
On the elliptic equations Δ𝑢=𝐾(𝑥)𝑢^{𝜎} and Δ𝑢=𝐾(𝑥)𝑒^{2𝑢}
Authors: Cheng Kuo-Shung , Lin Jenn-Tsann
Published: June 2010

We give some nonexistence results for the equations

Δ
u
=
K
(
x
)

u
σ

\Delta u = K(x){u^\sigma }

and

Δ
u
=
K
(
x
)

e

2
u

\Delta u = K(x){e^{2u}}

for

K
(
x
)
⩾
0

K(x) \geqslant 0

.
Decompositions of Banach lattices into direct sums
Authors: Casazza P. G. , Kalton N. J. , Tzafriri L.
Published: June 2010

We consider the problem of decomposing a Banach lattice

Z
Z

as a direct sum

Z
=
X
⊕
Y

Z = X \oplus Y

where

X
X

and

Y
Y

are complemented subspaces satisfying a condition of incomparability (e.g. every operator from

Y
Y

to

X
X

is strictly singular). We treat both the atomic and nonatomic cases. In particular we answer a question of Wojtaszczyk by showing that

L
1

⊕

L
2

{L_1} \oplus {L_2}

has unique structure as a nonatomic Banach lattice.
Ensembles de Riesz
Authors: Tardivel Valérie
Published: June 2010

Let

G
G

be an abelian countable discrete group. We show that there exists no positive characterization of Riesz subsets of

G
G

, by proving that the Riesz subsets of

G
G

form a coanalytic non-Borel subset of

2
G

{2^G}

.
On the behavior of harmonic functions near a boundary point
Authors: Ramey Wade , Ullrich David
Published: June 2010

Several results on the behavior of harmonic functions at an individual boundary point are obtained. The results apply to positive harmonic functions as well as to Poisson integrals of functions in BMO.
On the second fundamental theorem of Nevanlinna
Authors: Fernández Arias Arturo
Published: June 2010

It is shown that a condition on the size of the exceptional set in the second fundamental theorem of Nevanlinna cannot be improved. The method is based on a construction of Hayman and also makes use of a quantitative version of a result of F. Nevanlinna about the growth of the characteristic function of a meromorophic function omitting a finite number of points
A truncated Gauss-Kuz′min law
Authors: Hensley Doug
Published: June 2010

The transformations

T
n

{T_n}

which map

x
∈
[
0
,

1
)

x \in [0,\,1)

onto

0
0

(if

x
⩽
1

/

(
n
+
1
)

x \leqslant 1/(n + 1)

), and to

{
1

/

x
}

\{ 1/x\}

otherwise, are truncated versions of the continued fraction transformation

T
:
x
→
{
1

/

x
}

T:x \to \{ 1/x\}

(but

0
→
0

0 \to 0

). An analog to the Gauss-Kuzmin result is obtained for these

T
n

{T_n}

, and is used to show that the Lebesgue measure of

T
n

−
k

{
0
}

T_n^{ - k}\{ 0\}

approaches

1
1

exponentially. From this fact is obtained a new proof that the ratios

ν

/

k

\nu /k

, where

ν
\nu

denotes any solution of

ν
2

≡
−
1
mod
k

{\nu ^2} \equiv - 1\bmod k

, are uniformly distributed

mod
1

\bmod 1

in the sense of Weyl.
Mixed norm estimates for certain means
Authors: Börjeson Lennart
Published: June 2010

We obtain estimates of the mean
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript x Superscript gamma Baseline left-parenthesis t right-parenthesis equals upper C Subscript gamma Baseline integral Underscript StartAbsoluteValue y EndAbsoluteValue greater-than 1 Endscripts left-parenthesis 1 minus StartAbsoluteValue y EndAbsoluteValue squared right-parenthesis Superscript gamma Baseline f left-parenthesis x minus t y right-parenthesis d y">

F
x
γ

(
t
)
=

C
γ

∫

|

y

|

>
1

(
1
−

|

y

|

2

)

γ

f
(
x
−
t
y
)

d
y

F_x^\gamma (t) = {C_\gamma }\int _{|y| > 1} {{{(1 - |y{|^2})}^\gamma }f(x - ty)\,dy}

\]
in mixed Lebesgue and Sobolev spaces. They generalize earlier estimates of the spherical mean
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript x Superscript negative 1 Baseline left-parenthesis t right-parenthesis equals upper C integral Underscript upper S Superscript n minus 1 Baseline Endscripts f left-parenthesis x minus t y right-parenthesis d upper S left-parenthesis y right-parenthesis">

F
x

−
1

(
t
)
=
C

∫

S

n
−
1

f
(
x
−
t
y
)

d
S
(
y
)

F_x^{ - 1}(t) = C\;\int _{{S^{n - 1}}} {f(x - ty)\,dS(y)}

and of solutions of the wave equation

Δ
x

u
=

∂
2

u

/

∂

t
2

{\Delta _x}u = {\partial ^2}u/\partial {t^2}

.
Deficient values and angular distribution of entire functions
Authors: Yang Lo
Published: June 2010

Let

f
(
z
)

f(z)

be an entire function of positive and finite order

μ
\mu

. If

f
(
z
)

f(z)

has a finite number of Borel directions of order

⩾
μ

\geqslant \mu

, then the sum of numbers of finite nonzero deficient values of

f
(
z
)

f(z)

and all its primitives does not exceed

2
μ

2\mu

. The proof is based on several lemmas and application of harmonic measure.
The central limit theorem for empirical processes under local conditions: the case of Radon infinitely divisible limits without Gaussian component
Authors: Andersen Niels T. , Giné Evarist , Zinn Joel
Published: June 2010

Weak convergence results are obtained for empirical processes indexed by classes

F

\mathcal {F}

of functions in the case of infinitely divisible purely Poisson (in particular, stable) Radon limits, under conditions on the local modulus of the processes

{
f
(
X
)
:

f
∈

F

}

\{ f(X):\,f \in \mathcal {F}\}

("bracketing" conditions). They extend (and slightly improve upon) a central limit theorem of Marcus and Pisier (1984) for Lipschitzian processes. The law of the iterated logarithm is also considered. The examples include Marcinkiewicz type laws of large numbers for weighted empirical processes and for the dual-bounded-Lipschitz distance between a probability in

R

{\mathbf {R}}

and its associated empirical measures.
On the vanishing of homology and cohomology groups of associative algebras
Authors: Farnsteiner Rolf
Published: June 2010

This paper establishes sufficient conditions for the vanishing of the homology and cohomology groups of an associative algebra with coefficients in a two-sided module.
Outer functions in function algebras on the bidisc
Authors: Hedenmalm Håkan
Published: June 2010

Let

f
f

be a function in the bidisc algebra

A
(

D

2

)

A({{\mathbf {D}}^2})

whose zero set

Z
(
f
)

Z(f)

is contained in

{
1
}
×

D

¯

\{ 1\} \times \overline {\mathbf {D}}

. We show that the closure of the ideal generated by

f
f

coincides with the ideal of functions vanishing on

Z
(
f
)

Z(f)

if and only if

f
(
⋅
,

α
)

f( \cdot ,\,\alpha )

is an outer function for all

α
∈

D

¯

\alpha \in \overline {\mathbf {D}}

, and

f
(
1
,

⋅
)

f(1,\, \cdot )

either vanishes identically or is an outer function. Similar results are obtained for a few other function algebras on

D

2

{{\mathbf {D}}^2}

as well.
Representations of anisotropic unitary groups
Authors: James Donald G.
Published: June 2010

Let

S
U
(
f
)

SU(f)

be the special unitary group of an anisotropic hermitian form

f
f

over a field

k
k

. Assume

f
f

represents only one norm class in

k
k

. The representations

α
:

S
U
(
f
)
→
S
L
(
n
,

R
)

\alpha :\,SU(f) \to SL(n,\,R)

are characterized when

R
R

is a commutative ring with

2
2

not a zero divisor and

n
=
dim
⁡
f
⩾
3

n = \dim f \geqslant 3

with

n
≠
4
,

6

n \ne 4,\,6

. In particular, a partial classification of the normal subgroups of

S
U
(
f
)

SU(f)

is given when

k
k

is the function field

C

(
X
)

{\mathbf {C}}(X)

.
Sieving the positive integers by small primes
Authors: Goldston D. A. , McCurley Kevin S.
Published: June 2010

Let

Q
Q

be a set of primes that has relative density

δ
\delta

among the primes, and let

ϕ
(
x
,

y
,

Q
)

\phi (x,\,y,\,Q)

be the number of positive integers

⩽
x

\leqslant x

that have no prime factor

⩽
y

\leqslant y

from the set

Q
Q

. Standard sieve methods do not seem to give an asymptotic formula for

ϕ
(
x
,

y
,

Q
)

\phi (x,\,y,\,Q)

in the case that

1
2

⩽
δ
>
1

\tfrac {1}{2} \leqslant \delta > 1

. We use a method of Hildebrand to prove that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis x comma y comma upper Q right-parenthesis tilde x f left-parenthesis u right-parenthesis product Underscript StartLayout 1st Row p greater-than y 2nd Row p element-of upper Q EndLayout Endscripts left-parenthesis 1 minus StartFraction 1 Over p EndFraction right-parenthesis">

ϕ
(
x
,
y
,
Q
)
∼
x
f
(
u
)

∏

p
>
y

p
∈
Q

(

1
−

1
p

)

\phi (x,y,Q) \sim x f(u) \prod _{\substack {p > y\\p \in Q}} {\left ( {1 - \frac {1}{p}} \right )}

\]
as

x
→
∞

x \to \infty

, where

u
=

log
⁡
x

log
⁡
y

u = \frac {{\log x}}{{\log y}}

and

f
(
u
)

f(u)

is defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Superscript delta Baseline f left-parenthesis u right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartFraction e Superscript gamma Super Superscript delta Superscript Baseline Over normal upper Gamma left-parenthesis 1 minus delta right-parenthesis EndFraction comma 2nd Column a m p semicolon 0 greater-than u less-than-or-slanted-equals 1 comma 2nd Row 1st Column StartFraction e Superscript gamma Super Superscript delta Superscript Baseline Over normal upper Gamma left-parenthesis 1 minus delta right-parenthesis EndFraction plus delta integral Subscript 0 Superscript u minus 1 Baseline f left-parenthesis t right-parenthesis left-parenthesis 1 plus t right-parenthesis Superscript delta minus 1 Baseline d t comma 2nd Column a m p semicolon u greater-than 1 period EndLayout">

u
δ

f
(
u
)
=

{

e

γ
δ

Γ
(
1
−
δ
)

,

a
m
p
;

0
>
u
⩽
1
,

e

γ
δ

Γ
(
1
−
δ
)

+
δ

∫
0

u
−
1

f
(
t
)

(
1
+
t
)

δ
−
1

d
t
,

a
m
p
;

u
>
1.

{u^\delta }f(u) = \left \{ {\begin {array}{*{20}{c}} {\frac {{{e^{{\gamma ^\delta }}}}}{{\Gamma (1 - \delta )}},} \hfill & {0 > u \leqslant 1,} \hfill \\ {\frac {{{e^{{\gamma ^\delta }}}}}{{\Gamma (1 - \delta )}} + \delta \int _0^{u - 1} {f(t){{(1 + t)}^{\delta - 1}}\;dt,} } \hfill & {u > 1.} \hfill \\ \end {array} } \right .

\]
This may also be viewed as a generalization of work by Buchstab and de Bruijn, who considered the case where

Q
Q

consisted of all primes.
Exact bounds for the stochastic upward matching problem
Authors: Rhee WanSoo T. , Talagrand Michel
Published: June 2010

We draw at random independently and according to the uniform distribution two sets ofnnpoints of the unit square. We consider a maximum matching of points of the first set with points of the second set with the restriction that a point can be matched only with a point located at its upper right. Then with probability close to one, the number of unmatched points is of ordern1/2(log⁡n)3/4{n^{1/2}}{(\log n)^{3/4}}.
Linear supergroup actions. I. On the defining properties
Authors: Sánchez-Valenzuela Oscar Adolfo
Published: June 2010

This paper studies the notions of linearity and bilinearity in the category of supermanifolds. Following the work begun by [OASV2], we deal with supermanifoldifications of supervector spaces. The

R

1

|

1

{{\mathbf {R}}^{1|1}}

-module operations are defined componentwise. The linearity and bilinearity properties are stated by requiring commutativity of some appropriate diagrams of supermanifold morphisms. It is proved that both linear and bilinear supermanifold morphisms are completely determined by their underlying continuous maps, which in turn have to be linear (resp., bilinear) in the usual sense. It is observed that whereas linear supermanifold morphisms are vector bundle maps, bilinear supermanifold morphisms are not. A natural generalization of the bilinear evaluation map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H o m left-parenthesis upper V comma upper W right-parenthesis times upper V right-arrow upper W left-parenthesis left-parenthesis upper F comma v right-parenthesis right-arrow from bar upper F left-parenthesis v right-parenthesis right-parenthesis">

Hom
⁡
(
V
,

W
)
×
V
→
W

(
(
F
,

v
)
↦
F
(
v
)
)

\operatorname {Hom} (V,\,W) \times V \to W\;((F,\,v) \mapsto F(v))

is given and some applications pointing toward the notions of linear supergroup actions and adjoint and coadjoint actions are briefly discussed.
The differential operator ring of an affine curve
Authors: Muhasky Jerry L.
Published: June 2010

The purpose of this paper is to investigate the structure of the ring

D
(
R
)

D(R)

of all linear differential operators on the coordinate ring of an affine algebraic variety

X
X

(possibly reducible) over a field

k
k

(not necessarily algebraically closed) of characteristic zero, concentrating on the case that dim

X
⩽
1

X \leqslant 1

. In this case, it is proved that

D
(
R
)

D(R)

is a (left and right) noetherian ring with (left and right) Krull dimension equal to dim

X
X

, that the endomorphism ring of any simple (left or right)

D
(
R
)

D(R)

-module is finite dimensional over

k
k

, that

D
(
R
)

D(R)

has a unique smallest ideal

L
L

essential as a left or right ideal, and that

D
(
R
)

/

L

D(R)/L

is finite dimensional over

k
k

. The following ring-theoretic tool is developed for use in deriving the above results. Let

D
D

be a subalgebra of a left noetherian

k
k

-algebra

E
E

such that

E
E

is finitely generated as a left

D
D

-module and all simple left

E
E

-modules have finite dimensional endomorphism rings (over

k
k

), and assume that

D
D

contains a left ideal

I
I

of

E
E

such that

E

/

I

E/I

has finite length. Then it is proved that

D
D

is left noetherian and that the endomorphism ring of any simple left

D
D

-module is finite dimensional over

k
k

.
A very singular solution of a quasilinear degenerate diffusion equation with absorption
Authors: Peletier L. A. , Wang Jun Yu
Published: June 2010

The object of this paper is to study the existence of a nonnegative solution of the Cauchy problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline equals d i v left-parenthesis StartAbsoluteValue nabla u EndAbsoluteValue Superscript p minus 2 Baseline nabla u right-parenthesis minus u Superscript q Baseline comma u left-parenthesis x comma 0 right-parenthesis equals 0 if x not-equals 0 comma">

u
t

=
div
⁡
(

|

∇
u

|

p
−
2

∇
u
)
−

u
q

,

u
(
x
,

0
)
=
0

if

x
≠
0
,

{u_t} = \operatorname {div} (|\nabla u{|^{p - 2}}\nabla u) - {u^q},\qquad u(x,\,0) = 0\quad {\text {if}}\;x \ne 0,

\]
which is more singular at

(
0
,

0
)

(0,\,0)

than the fundamental solution of the corresponding equation without the absorption term.
Generalized Chebyshev polynomials associated with affine Weyl groups
Authors: Hoffman Michael E. , Withers William Douglas
Published: June 2010

We begin with a compact figure that can be folded into smaller replicas of itself, such as the interval or equilateral triangle. Such figures are in one-to-one correspondence with affine Weyl groups. For each such figure in

n
n

-dimensional Euclidean space, we construct a sequence of polynomials

P
k

:

R

n

→

R

n

{P_k}:{{\mathbf {R}}^n} \to {{\mathbf {R}}^n}

so that the mapping

P
k

{P_k}

is conjugate to stretching the figure by a factor

k
k

and folding it back onto itself. If

n
=
1

n = 1

and the figure is the interval, this construction yields the Chebyshev polynomials (up to conjugation). The polynomials

P
k

{P_k}

are orthogonal with respect to a suitable measure and can be extended in a natural way to a complete set of orthogonal polynomials.
Hardy spaces of vector-valued functions: duality
Authors: Blasco Oscar
Published: June 2010

We prove here that the Hardy space of

B
B

-valued functions

H
1

(
B
)

{H^1}(B)

defined by using the conjugate function and the one defined in terms of

B
B

-valued atoms do not coincide for a general Banach space. The condition for them to coincide is the UMD property on

B
B

. We also characterize the dual space of both spaces, the first one by using

B

∗

{B^{\ast }}

-valued distributions and the second one in terms of a new space of vector-valued measures, denoted

B

M

O

(

B

∗

)

\mathcal {B}\mathcal {M}\mathcal {O}({B^{\ast }})

, which coincides with the classical

BMO
⁡
(

B

∗

)

\operatorname {BMO} ({B^{\ast }})

of functions when

B

∗

{B^{\ast }}

has the RNP.
𝑆𝑂(2)-equivariant vector fields on 3-manifolds: moduli of stability and genericity
Authors: dos Reis Genesio Lima , dos Santos Geovan Tavares
Published: June 2010

An open and dense class of vector fields on

3
3

-dimensional compact manifolds equivariant under the action of

SO
⁡
(
2
)

\operatorname {SO} (2)

is defined. Each such vector field has finite moduli of stability. We also exhibit an open and dense subset of the

SO
⁡
(
2
)

\operatorname {SO} (2)

-equivariant gradient vector fields which are structurally stable.
Geometry of the Severi variety
Authors: Diaz Steven , Harris Joe
Published: June 2010

This paper is concerned with the geometry of the Severi variety

W
W

parametrizing plane curves of given degree and genus, and specifically with the relations among various divisor classes on

W
W

. Two types of divisor classes on

W
W

are described: those that come from the intrinsic geometry of the curves parametrized, and those characterized by extrinsic properties such as the presence of cusps, tacnodes, hyperflexes, etc. The goal of the paper is to express the classes of the extrinsically defined divisors in terms of the intrinsic ones; this, along with other calculations such as the determination of the canonical class of

W
W

, is carried out by using various enumerative techniques. One corollary is that the variety of nodal curves of given degree and genus in the plane is affine.
Function spaces generated by blocks associated with spheres, Lie groups and spaces of homogeneous type
Authors: Založnik Aleš
Published: June 2010

Functions generated by blocks were introduced by M. Taibleson and G. Weiss in the setting of the one-dimensional torus

T
T

[TW1]. They showed that these functions formed a space "close" to the class of integrable functions for which we have almost everywhere convergence of Fourier series. Together with S. Lu [LTW] they extended the theory to the

n
n

-dimensional torus where this convergence result (for Bochner-Riesz means at the critical index) is valid provided we also restrict ourselves to

L
log
⁡
L

L\log L

. In this paper we show that this restriction is not needed if the underlying domain is a compact semisimple Lie group (or certain more general spaces of a homogeneous type). Other considerations (for example, these spaces form an interesting family of quasi-Banach spaces; they are connected with the notion of entropy) guide one in their study. We show how this point of view can be exploited in the setting of more general underlying domains.
On zeros of a system of polynomials and application to sojourn time distributions of birth-and-death processes
Authors: Sato Ken-iti
Published: June 2010

Zeros of the following system of polynomials are considered:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row upper P 0 left-parenthesis x right-parenthesis equals 1 comma 2nd Row upper P 1 left-parenthesis x right-parenthesis equals upper B 0 minus upper A 0 x comma 3rd Row upper P Subscript n plus 1 Baseline left-parenthesis x right-parenthesis equals left-parenthesis upper B Subscript n Baseline minus upper A Subscript n Baseline x right-parenthesis upper P Subscript n Baseline left-parenthesis x right-parenthesis minus upper C Subscript n Baseline upper P Subscript n minus 1 Baseline left-parenthesis x right-parenthesis for n greater-than-or-slanted-equals 1 period EndLayout">

{

P
0

(
x
)
=
1
,

P
1

(
x
)
=

B
0

−

A
0

x
,

P

n
+
1

(
x
)
=
(

B
n

−

A
n

x
)

P
n

(
x
)
−

C
n

P

n
−
1

(
x
)

for

n
⩾
1.

\left \{ \begin {gathered} {P_0}(x) = 1, \hfill \\ {P_1}(x) = {B_0} - {A_0}x, \hfill \\ {P_{n + 1}}(x) = ({B_n} - {A_n}x){P_n}(x) - {C_n}{P_{n - 1}}(x)\quad {\text {for}}\;n \geqslant 1. \hfill \\ \end {gathered} \right .

\]
Numbers of positive and negative zeros are determined and a separation property of the zeros of

P
m

(
x
)

{P_m}(x)

and

P
n

(
x
)

{P_n}(x)

is proved under the condition that

C
n

>
0

{C_n} > 0

and

P
n

(
0
)
>
0

{P_n}(0) > 0

for every

n
n

. No condition is imposed on

A
n

{A_n}

. These results are applied to determination of the distribution of a sojourn time with general (not necessarily positive) weight function for a birth-and-death process up to a first passage time. Unimodality and infinite divisibility of the distribution follow.
Positive quadratic differential forms and foliations with singularities on surfaces
Authors: Guíñez Víctor
Published: June 2010

To every positive

C
r

{C^r}

-quadratic differential form defined on an oriented two manifold is associated a pair of transversal one-dimensional

C
r

{C^r}

-foliations with common singularities. An open set of positive

C
r

{C^r}

-quadratic differential forms with structural stable associated foliations is characterized and it is proved that this set is dense in the space of positive

C
∞

{C^\infty }

-quadratic differential forms with

C
2

{C^2}

-topology. Also a realization theorem is established.
Stochastic perturbations to conservative dynamical systems on the plane. I. Convergence of invariant distributions
Authors: Wolansky G.
Published: June 2010

We consider a nonlinear system on the plane, given by an oscillator with homoclinic orbits. The above system is subjected to a perturbation, composed of a deterministic part and a random (white noise) part. Assuming the existence of a finite, invariant measure to the perturbed system, we deal with the convergence of the measures to a limit measure, as the perturbation parameter tends to zero. The limit measure is constructed in terms of the action function of the unperturbed oscillator, and the strong local

L
2

{L_2}

convergence of the associated densities is proved.
A Stone-type representation theorem for algebras of relations of higher rank
Authors: Andréka H. , Thompson R. J.
Published: June 2010

The Stone representation theorem for Boolean algebras gives us a finite set of equations axiomatizing the class of Boolean set algebras. Boolean set algebras can be considered to be algebras of unary relations. As a contrast here we investigate algebras of

n
n

-ary relations (originating with Tarski). The new algebras have more operations since there are more natural set theoretic operations on

n
n

-ary relations than on unary ones. E.g. the identity relation appears as a new constant. The Resek-Thompson theorem we prove here gives a finite set of equations axiomatizing the class of algebras of

n
n

-ary relations (for every ordinal

n
n

).
Field theories in the modern calculus of variations
Authors: Nowakowski Andrzej
Published: June 2010

Two methods of construction of fields of extremals ("geodesic coverings") in the generalized problem of Bolza are given and, as a consequence, sufficient conditions for optimality in a form similar to Weierstrass’ are formulated. The first field theory is an extension of Young’s field theory— "concourse of flights" for our problem; the other describes a nonclassical treatment of field theory which allows one to reject the "self-multiplier restriction".
On classical Clifford theory
Authors: Harris Morton E.
Published: June 2010

Let

k
k

be a field, let

N
N

be a normal subgroup of a finite group

H
H

and let

M
M

be a completely reducible

k
[
N
]

k[N]

-module. We give sufficient conditions for a finite dimensional (finite) group crossed product

k
k

-algebra to be a Frobenius or symmetric

k
k

-algebra. These results imply that

k
[
H
]

/

(
J
(
k
[
N
]
)
k
[
H
]
)

k[H]/(J(k[N])k[H])

and the endomorphism

k
k

-algebra,

End

k
[
H
]

(

M
H

)

{\operatorname {End} _{k[H]}}({M^H})

, of the induced module

M
H

{M^H}

are symmetric

k
k

-algebras. We also completely describe the

k
[
H
]

k[H]

-indecomposable decomposition of

M
H

{M^H}

. It follows that the head and socle of an indecomposable component of

M
H

{M^H}

are irreducible isomorphic

k
[
H
]

k[H]

-modules.
Amalgamation for inverse and generalized inverse semigroups
Authors: Hall T. E.
Published: June 2010

For any amalgam

(
S
,

T
;

U
)

(S,\,T;\,U)

of inverse semigroups, it is shown that the natural partial order on

S

∗

U

T

S{{\ast }_U}T

, the (inverse semigroup) free product of

S
S

and

T
T

amalgamating

U
U

, has a simple form on

S
∪
T

S \cup T

. In particular, it follows that the semilattice of

S

∗

U

T

S{{\ast }_U}T

is a bundled semilattice of the corresponding semilattice amalgam

(
E
(
S
)
,

E
(
T
)
;

E
(
U
)
)

(E(S),\,E(T);\,E(U))

; taken jointly with a result of Teruo Imaoka, this gives that the class of generalized inverse semigroups has the strong amalgamation property. Preserving finiteness is also considered.
Injectivity of operator spaces
Authors: Ruan Zhong-Jin
Published: June 2010

We study the structure of injective operator spaces and the existence and uniqueness of the injective envelopes of operator spaces. We give an easy example of an injective operator space which is not completely isometric to any

C
∗

{C^\ast }

-algebra. This answers a question of Wittstock [23]. Furthermore, we show that an operator space

E
E

is injective if and only if there exists an injective

C
∗

{C^\ast }

-algebra

A
A

and two projections

p
p

and

q
q

in

A
A

such that

E
E

is completely isometric to

p
A
q

pAq

.
The lifting problem for affine structures in nilpotent Lie groups
Authors: Boyom Nguiffo B.
Published: June 2010

Affine manifolds occur in several situations in pure and applied mathematics, (e.g. leaves of Lagrangian foliations, completely integrable Hamiltonian systems, linear representations of virtually polycyclic groups, geometric quantization and so on). This work is devoted to left invariant affinely flat structures in Lie groups. We are mainly concerned with the following situation. Let

G
G

and

G
0

{G_0}

be nilpotent Lie groups of dimension

n
+
1

n + 1

and

n
n

, respectively and let

h
:
G
→

G
0

h:G \to {G_0}

be a continuous homomorphism from

G
G

onto

G
0

{G_0}

. Given a left invariant affinely flat structure

(

G
0

,

∇
0

)

({G_0},{\nabla _0})

the lifting problem is to discover whether

G
G

has a left invariant affinely flat structure

(
G
,
∇
)

(G,\nabla )

such that

h
h

becomes an affine morphism. In the present work we answer positively when

(

G
0

,

∇
0

)

({G_0},{\nabla _0})

is "normal". Therefore the existence problem for a left invariant complete affinely flat structure in nilpotent Lie groups is solved by applying the following subsequent results. Let

A

f
(

G
0

)

\mathfrak {A}f({G_0})

be the set of left invariant affinely flat structures in the nilpotent Lie group

G
0

,
(

1
∘

)

A

f
(

G
0

)
≠
∅

{G_0},({1^ \circ })\;\mathfrak {A}f({G_0}) \ne \emptyset

implies the existence of normal structure
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper G 0 comma nabla Subscript 0 Baseline right-parenthesis element-of German upper A f left-parenthesis upper G 0 right-parenthesis semicolon left-parenthesis 2 Superscript ring Baseline right-parenthesis h colon upper G right-arrow upper G 0">

(

G
0

,

∇
0

)
∈

A

f
(

G
0

)
;
(

2
∘

)

h
:
G
→

G
0

({G_0},{\nabla _0}) \in \mathfrak {A}f({G_0});({2^ \circ })\;h:G \to {G_0}

being as above every normal structure

(

G
0

,

∇
0

)

({G_0},{\nabla _0})

has a normal lifted in

A

f
(
G
)

\mathfrak {A}f(G)

.
Optimal 𝐿^{𝑝} and Hölder estimates for the Kohn solution of the \overline∂-equation on strongly pseudoconvex domains
Authors: Chang Der-Chen E.
Published: June 2010

LetΩ\Omegabe an open, relatively compact subset inCn+1{{\mathbf {C}}^{n + 1}}, and assume the boundary ofΩ\Omega,∂Ω\partial \Omega, is smooth and strongly pseudoconvex. LetOp⁡(K)\operatorname {Op}(K)be an integral operator with mixed type homogeneities defined onΩ¯\overline \Omega: i.e.,KKhas the form as follows:\[∑k,l≥0EkHl,\sum \limits _{k,l \geq 0} {{E_k}{H_l},}\]whereEk{E_k}is a homogeneous kernel of degree−k- kin the Euclidean sense andHl{H_l}is homogeneous of degree−l- lin the Heisenberg sense. In this paper, we study the optimalLp{L^p}and Hölder estimates for the kernelKK. We also use Lieb-Range’s method to construct the integral kernel for the Kohn solution∂∗¯N\overline {{\partial ^\ast }} {\mathbf {N}}of the Cauchy-Riemann equation on the Siegel upper-half space and then apply our results to∂∗¯N\overline {{\partial ^\ast }} {\mathbf {N}}. On the other hand, we prove Lieb-Range’s kernel gains11in "good" directions (hence gains1/21/2in all directions) via Phong-Stein’s theory. We also discuss the transferred kernel from the Siegel upper-half space toΩ\Omega.
Singular relaxation moduli and smoothing in three-dimensional viscoelasticity
Authors: Desch Wolfgang , Grimmer Ronald
Published: June 2010

We develop a semigroup setting for linear viscoelasticity in three–dimensional space with tensor-valued relaxation modulus and give a criterion on the relaxation kernel for differentiability and analyticity of the solutions. The method is also extended to a simple problem in thermoviscoelasticity.
Uniform analyticity of orthogonal projections
Authors: Coifman R. R. , Murray Margaret A. M.
Published: June 2010

Let

X
X

denote the circle

T
T

or the interval

[
−
1
,
1
]

[ - 1,1]

, and let

d
μ

d\mu

denote a nonnegative, absolutely continuous measure on

X
X

. Under what conditions does the Gram-Schmidt procedure in the weighted space

L
2

(
X
,

ω
2

d
μ
)

{L^2}(X,{\omega ^2}\;d\mu )

depend analytically on the logarithm of the weight function

ω
\omega

? In this paper, we show that, in numerous examples of interest,

log
⁡
ω
∈
B
M
O

\log \omega \in BMO

is a sufficient (often necessary!) condition for analyticity of the Gram-Schmidt procedure. These results are then applied to establish the local analyticity of certain infinite-dimensional Toda flows.
Nearly representable operators
Authors: Kaufman R. , Petrakis Minos , Riddle Lawrence H. , Uhl J. J.
Published: June 2010

Among Bourgain’s many remarkable theorems is one from 1980 which states that if

T
T

is a non-Dunford-Pettis operator from

L
1

[
0
,
1
]

{L_1}[0,1]

into an arbitrary Banach space, then there is a Dunford-Pettis operator

D
D

from

L
1

[
0
,
1
]

{L_1}[0,1]

into

L
1

[
0
,
1
]

{L_1}[0,1]

such that the composition

T
∘
D

T \circ D

is not Bochner representable. This theorem sets up the following question: What are the operators

T
T

from

L
1

[
0
,
1
]

{L_1}[0,1]

into a Banach space

X
X

such that

T
∘
D

T \circ D

is Bochner representable for all Dunford-Pettis operators

D
:

L
1

[
0
,
1
]
→

L
1

[
0
,
1
]

D:{L_1}[0,1] \to {L_1}[0,1]

? We call such an operator nearly representable. In view of Bourgain’s theorem, all nearly representable operators are Dunford-Pettis. If

X
X

is a Banach space such that all nearly representable operators are, in addition, Bochner representable, then we say

X
X

has the near Radon-Nikodym property (NRNP) and ask which Banach spaces have the NRNP? This paper is an attempt to provide at least partial answers to these questions. The first section collects terminology, gives the introductory results and shows that the NRNP is a three space property. The second section studies a continuity property that implies near representability. Finally, the third section contains the main result of the paper, Theorem 15, which states that if

T
:

L
1

[
0
,
1
]
→

L
1

[
0
,
1
]

T:{L_1}[0,1] \to {L_1}[0,1]

is a nonrepresentable operator, there exists a Dunford-Pettis operator

D
:

L
1

[
0
,
1
]
→

L
1

[
0
,
1
]

D:{L_1}[0,1] \to {L_1}[0,1]

such that

T
∘
D

T \circ D

is also nonrepresentable. This implies that the

NRNP

{\text {NRNP}}

is shared by

L
1

[
0
,
1
]

{L_1}[0,1]

, lattices not containing

c
0

{c_0}

, and

L
1

(

T

)

/

H
0
1

{L_1}({\mathbf {T}})/H_0^1

.
Relations between 𝐻^{𝑝}ᵤ and 𝐿^{𝑝}ᵤ in a product space
Authors: Strömberg Jan-Olov , Wheeden Richard L.
Published: June 2010

Relations between

L
u
p

L_u^p

and

H
u
p

H_u^p

are studied for the product space

R

1

×

R

1

{{\mathbf {R}}^1} \times {{\mathbf {R}}^1}

in the case

1
>
p
>
∞

1 > p > \infty

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis x 1 comma x 2 right-parenthesis equals StartAbsoluteValue upper Q 1 left-parenthesis x 1 right-parenthesis EndAbsoluteValue Superscript p Baseline StartAbsoluteValue upper Q 2 left-parenthesis x 2 right-parenthesis EndAbsoluteValue Superscript p Baseline w left-parenthesis x 1 comma x 2 right-parenthesis">

u
(

x
1

,

x
2

)
=

|

Q
1

(

x
1

)

|

p

|

Q
2

(

x
2

)

|

p

w
(

x
1

,

x
2

)

u({x_1},{x_2}) = |{Q_1}({x_1}){|^p}|{Q_2}({x_2}){|^p}w({x_1},{x_2})

, where

Q
1

{Q_1}

and

Q
2

{Q_2}

are polynomials and

w
w

satisfies the

A
p

{A_p}

condition for rectangles. A description of the distributions in

H
u
p

H_u^p

is given. Questions about boundary values and about the existence of dense subsets of smooth functions satisfying appropriate moment conditions are also considered.
Infima of convex functions
Authors: Beer Gerald
Published: June 2010

Let

Γ
(
X
)

\Gamma (X)

be the lower semicontinuous, proper, convex functions on a real normed linear space

X
X

. We produce a simple description of what is, essentially, the weakest topology on

Γ
(
X
)

\Gamma (X)

such that the value functional

f
→
inf
f

f \to \inf f

is continuous on

Γ
(
X
)

\Gamma (X)

. When

X
X

is reflexive, convergence of a sequence in this topology is equivalent to Mosco plus pointwise convergence of the corresponding sequence of conjugate convex functions.
Hyperconvexity and nonexpansive multifunctions
Authors: Sine Robert
Published: June 2010

It is shown that a ball intersection valued nonexpansive multifunction on a hyperconvex space admits a nonexpansive point valued selection. This implies fixed point theorems for such multifunctions and to certain point valued nonexpansive maps. The result is used to study best approximation and to show the space of all nonexpansive maps of a bounded hyperconvex space is hyperconvex.
Rigidity for complete Weingarten hypersurfaces
Authors: Dajczer M. , Tenenblat K.
Published: June 2010

We classify, locally and globally, the ruled Weingarten hypersurfaces of the Euclidean space. As a consequence of the local classification and a rigidity theorem of Dajczer and Gromoll, it follows that a complete Weingarten hypersurface which does not contain an open subset of the form

L
3

×

R

n
−
3

{L^3} \times {{\mathbf {R}}^{n - 3}}

, where

L
3

{L^3}

is unbounded and

n
≥
3

n \geq 3

, is rigid.
A generalisation of the matroid lift construction
Authors: Whittle Geoff
Published: June 2010

This paper introduces a general matroid-theoretic construction which includes, as special cases, elementary lifts of matroids and bias matroids of biased graphs. To perform the construction on a matroid

M
M

, it is necessary (but not sufficient) to have a submodular function inducing

M
M

. Elementary lifts are obtained when the submodular function chosen is the rank function of

M
M

. We define what is meant by a

k
k

-induced matroid. These matroids simultaneously generalise matroids of graphs, transversal matroids and Dilworth truncations. They are induced by a particularly natural class of submodular functions. The effect of the above construction on

k
k

-induced matroids using these natural submodular functions is studied. Results on minors of

k
k

-induced matroids and the matroids obtained from them using the construction are given.
Higher-order Sugawara operators for affine Lie algebras
Authors: Goodman Roe , Wallach Nolan R.
Published: June 2010

Let

g

^

\hat {\mathfrak {g}}

be the affine Lie algebra associated to a simple Lie algebra

g

\mathfrak {g}

. Representations of

g

^

\hat {\mathfrak {g}}

are described by current fields

X
(
ζ
)

X(\zeta )

on the circle

T

(
X
∈

g

{\mathbf {T}}\;(X \in \mathfrak {g}

and

ζ
∈

T

)

\zeta \in {\mathbf {T}})

. In this paper a linear map

σ
\sigma

from the symmetric algebra

S
(

g

)

S(\mathfrak {g})

to (formal) operator fields on a suitable category of

g

^

\hat {\mathfrak {g}}

modules is constructed. The operator fields corresponding to

g

\mathfrak {g}

-invariant elements of

S
(

g

)

S(\mathfrak {g})

are called Sugawara fields. It is proved that they satisfy commutation relations of the form

(
∗
)

(\ast )

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket sigma left-parenthesis u right-parenthesis left-parenthesis zeta right-parenthesis comma upper X left-parenthesis eta right-parenthesis right-bracket equals c Subscript normal infinity Baseline upper D delta left-parenthesis zeta slash eta right-parenthesis sigma left-parenthesis nabla Subscript upper X Baseline u right-parenthesis left-parenthesis zeta right-parenthesis plus higher hyphen order terms">

[
σ
(
u
)
(
ζ
)
,
X
(
η
)
]
=

c
∞

D
δ
(
ζ

/

η
)
σ
(

∇
X

u
)
(
ζ
)
+

higher-order

terms

[\sigma (u)(\zeta ),X(\eta )] = {c_\infty }D\delta (\zeta /\eta )\sigma ({\nabla _X}u)(\zeta ) + {\text {higher-order}}\;{\text {terms}}

\]
with the current fields, where

c
∞

{c_\infty }

is a renormalization of the central element in

g

^

\hat {\mathfrak {g}}

and

D
δ

D\delta

is the derivative of the Dirac delta function. The higher-order terms in

(
∗
)

(\ast )

are studied using results from invariant theory and finite-dimensional representation theory of

g

\mathfrak {g}

. For suitably normalized invariants

u
u

of degree

4
4

or less, these terms are shown to be zero. This vanishing is also proved for

g

=

sl

(
n
,

C

)

\mathfrak {g} = {\text {sl}}(n,{\mathbf {C}})

and

u
u

running over a particular choice of generators for the symmetric invariants. The Sugawara fields defined by such invariants commute with the current fields whenever

c
∞

{c_\infty }

is represented by zero. This property is used to obtain the commuting ring, composition series, and character formulas for a class of highest-weight modules for

g

^

\hat {\mathfrak {g}}

.
On infinite root systems
Authors: Moody R. V. , Pianzola A.
Published: June 2010

We define in an axiomatic fashion the concept of a set of root data that generalizes the usual concept of root system of a Kac-Moody Lie algebra. We study these objects from a purely formal and geometrical point of view as well as in relation to their associated Lie algebras. This leads to a coherent theory of root systems, bases, subroot systems, Lie algebras defined by root data, and subalgebras.
Higher-dimensional analogues of the modular and Picard groups
Authors: Maclachlan C. , Waterman P. L. , Wielenberg N. J.
Published: June 2010

Clifford algebras are used to describe arithmetic groups which are generalizations of the modular and Picard groups.
Sarkovskiĭ’s theorem for hereditarily decomposable chainable continua
Authors: Minc Piotr , Transue W. R. R.
Published: June 2010

Sarkovskii’s theorem, which fails to hold for chainable continua, is shown to hold for hereditarily decomposable chainable continua.
Projections onto translation-invariant subspaces of 𝐻¹(𝑅)
Authors: Alspach Dale E. , Ullrich David C.
Published: June 2010

Recently I. Klemes has characterized the complemented translation-invariant subspaces of

H
1

(

T

)

{H^1}(\mathbb {T})

. In this paper we investigate the case of

H
1

(

R

)

{H^1}(\mathbb {R})

. The main results are that the hull of a complemented translation-invariant subspace is

ε
\varepsilon

-separated for some

ε
>
0

\varepsilon > 0

, and that an

ε
\varepsilon

-separated subset of

R

+

{\mathbb {R}^ + }

which is in the ring generated by cosets of closed subgroups of

R

\mathbb {R}

(intersected with

R

+

{\mathbb {R}^ + }

) and lacunary sequences is the hull of a complemented ideal.
Bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite-dimensional systems
Authors: Chow Shui-Nee , Deng Bo
Published: June 2010

Under some generic conditions, we show how a unique stable periodic orbit can bifurcate from a homoclinic orbit for semilinear parabolic equations and retarded functional differential equations. This is a generalization of a result of Šil’nikov for ordinary differential equations.
\overline∂_{𝑏}-equations on certain unbounded weakly pseudo-convex domains
Authors: Kang Hyeonbae
Published: June 2010

We found an explicit closed formula for the relative fundamental solution of

∂
¯

b

{\bar \partial _b}

on the surface

H
k

=
{
(

z
1

,

z
2

)
∈

C

2

:
Im
⁡

z
2

=

|

z
1

|

2
k

}

{H_k} = \{ ({z_1},{z_2}) \in {\mathbb {C}^2}:\operatorname {Im} {z_2} = |{z_1}{|^{2k}}\}

. We then make estimates of the relative fundamental solution in terms of the nonisotropic metric associated with the surface. The estimates lead us to the regularity results. We also study the problem of finding weights

ω
\omega

so that

∂
¯

b

{\bar \partial _b}

as an operator from

L
ω
2

L_\omega ^2

to

L
2

{L^2}

has a closed range. We find the best possible weight among radial weights.
Multiplicity of the adjoint representation in simple quotients of the enveloping algebra of a simple Lie algebra
Authors: Joseph Anthony
Published: June 2010

Let

g

\mathfrak {g}

be a complex simple Lie algebra,

h

\mathfrak {h}

a Cartan subalgebra and

U
(

g

)

U(\mathfrak {g})

the enveloping algebra of

g

\mathfrak {g}

. We calculate for each maximal two-sided ideal

J

max

(
λ
)
:
λ
∈

h

∗

{J_{\max }}(\lambda ):\lambda \in {\mathfrak {h}^{\ast }}

of

U
(

g

)

U(\mathfrak {g})

the number of times the adjoint representation occurs in

U
(

g

)

/

J

max

(
λ
)

U(\mathfrak {g})/{J_{\max }}(\lambda )

. This is achieved by reduction via the Kazhdan-Lusztig polynomials to the case when

λ
\lambda

lies on a corner, i.e. is a multiple of a fundamental weight. Remarkably in this case one can always present

U
(

g

)

/

J

max

(
λ
)

U(\mathfrak {g})/{J_{\max }}(\lambda )

as a (generalized) principal series module and here we also calculate its Goldie rank as a ring which is a question of independent interest. For some of the more intransigent cases it was necessary to use recent very precise results of Lusztig on left cells. The results are used to show how a recent theorem of Gupta established for "nonspecial"

λ
\lambda

can fail if

λ
\lambda

is singular. Finally we give a quite efficient procedure for testing if an induced ideal is maximal.
On the propagation of round-off errors in the numerical treatment of the wave equation
Authors: Lowan Arnold N.
Published: June 2010

An upper bound of the norm of the error vector after n time steps is

1
2

(
n
+
1
)
(
n
+
2
)
‖

δ

∗

‖

\tfrac {1}{2}(n + 1)(n + 2)\| {{\delta ^{\ast }}}\|

. For the explicit scheme

δ

∗

=
‖

δ

∗

‖
=
3
×

1
2

×

10

−
p

{\delta ^{\ast }} = \| {{\delta ^{\ast }}}\| = 3 \times \tfrac {1}{2} \times {10^{ - p}}

where p is the number of decimals carried in the computations. For the implicit scheme

δ

∗

=
‖

δ

∗

‖

{\delta ^{\ast }} = \| {{\delta ^{\ast }}}\|

is an upper bound of the errors which arise both from using approximations to

A

−
1

{A^{ - 1}}

and

A

−
1

B

{A^{ - 1}}B

in the determination of

u

k
+
1

{u_{k + 1}}

from equation (6

∗

^{*}

) and from rounding off the values of the products and quotients involved in the computation of the components of

u

k
+
1

{u_{k + 1}}

.
On the propagation of errors in the inversion of certain tridiagonal matrices
Authors: Lowan Arnold N.
Published: June 2010

When the differential equation of heat conduction is replaced by the implicit difference analog, one is led to the solution of Ay = b where A is a tridiagonal matrix whose elements on the principal diagonal are = 2 + 2r and whose elements off the principal diagonal are = -r. The system of equations may be solved by the following algorithm:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript k Baseline equals u equals r squared beta Subscript k minus 1 Superscript negative 1 Baseline comma beta 1 equals u 1 semicolon gamma Subscript k Baseline equals minus r beta Superscript negative 1 Baseline semicolon z Subscript k Baseline equals left-parenthesis b Subscript k Baseline plus r z Subscript k minus 1 Baseline right-parenthesis beta Subscript k Superscript negative 1 Baseline comma z 1 equals b 1 u Superscript negative 1 Baseline semicolon y Subscript k Baseline equals z Subscript k Baseline minus gamma Subscript k Baseline y Subscript k plus 1 Baseline comma y Subscript upper M Baseline equals z Subscript upper M Baseline period">

β
k

=

u

=

r
2

β

k
−
1

−
1

,

β
1

=

u
1

;

γ
k

=

−
r

β

−
1

;

z
k

=

(

b
k

+

r

z

k
−
1

)

β
k

−
1

,

z
1

=

b
1

u

−
1

;

y
k

=

z
k

−

γ
k

y

k
+
1

,

y
M

=

z
M

.

{\beta _k}\, = \,u\, = \,{r^2}\beta _{k - 1}^{ - 1},\,{\beta _1}\, = \,{u_1}\,;\qquad {\gamma _k}\, = \, - r{\beta ^{ - 1}};\qquad {z_k}\, = \,({b_k}\, + \,r{z_{k - 1}})\beta _k^{ - 1},\,{z_1}\, = \,{b_1}{u^{ - 1}};\qquad {y_k}\, = \,{z_k}\, - \,{\gamma _k}{y_{k + 1}},\qquad {y_M}\,=\,{z_M}.

\]
An upper bound of the round-off errors in the computed values of the

y
k

{y_k}

’s is obtained. An actual test case showed that the theoretical upper bound is about four times larger than the true round-off error. Moreover, the theoretical upper bound does not seem to vary appreciably with r.