bluuebunny/crossref_metadata_2025 huggingface dataset:
Bounds for the order of supersoluble automorphism groups of Riemann surfaces
Authors: Zomorrodian Reza
Published: June 2010

The maximal automorphism groups of compact Riemann surfaces for a class of groups positioned between nilpotent and soluble groups is investigated. It is proved that if

G
G

is any finite supersoluble group acting as the automorphism group of some compact Riemann surface

Ω
\Omega

of genus

g
≥
2

g \geq 2

, then: (i) If

g
=
2

g = 2

then

|

G

|

≤
24

|G| \leq 24

and equality occurs when

G
G

is the supersoluble group

D
4

⊗

Z

3

{D_4} \otimes {{\mathbf {Z}}_3}

that is the semidirect product of the dihedral group of order 8 and the cyclic group of order 3. This exceptional case occurs when the Fuchsian group

Γ
\Gamma

has the signature (0;2,4,6), and can cover only this finite supersoluble group of order 24. (ii) If

g
≥
3

g \geq 3

then

|

G

|

≤
18

(

g
−
1

)

|G| \leq 18\left ( {g - 1} \right )

, and if

|

G

|

=
18

(

g
−
1

)

|G| = 18\left ( {g - 1} \right )

then

(

g
−
1

)

\left ( {g - 1} \right )

must be a power of 3. Conversely if

(

g
−
1

)

=

3
n

,
n
≥
2

\left ( {g - 1} \right ) = {3^n},n \geq 2

, then there is at least one surface

Ω
\Omega

of genus

g
g

with an automorphism group of order

18

(

g
−
1

)

18\left ( {g - 1} \right )

which must be supersoluble since its order is of the form

2

3
m

2{3^m}

. This bound corresponds to a specific Fuchsian group given by the signature (0;2,3,18). The terms in the chief series of each of these Fuchsian groups to the point where a torsion-free subgroup is reached are computed.
Settling a number of questions about hyper-Archimedean lattice-ordered groups
Authors: Conrad Paul , Martinez Jorge
Published: June 2010

This note contains the answers to several long-standing questions about hyper-Archimedean lattice-ordered groups. In particular, we construct such a lattice-ordered group which admits no embedding into an Archimedean lattice-ordered group with a strong order unit.
On the surjectivity criterion for Buchsbaum modules
Authors: Goto Shiro
Published: June 2010

Let

R
R

be a Cohen-Macaulay local ring with maximal ideal

m
m

and suppose that

dim
⁡
R
≥
2

\dim R \geq 2

. Then

R
R

is regular if (and only if) for any Buchsbaum

R
R

-module

M
M

and for any integer

i
,
i
≠

dim
R

M

i,i \ne {\dim _R}M

, the canonical map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Ext Subscript upper R Superscript i Baseline left-parenthesis upper R slash m comma upper M right-parenthesis right-arrow upper H Subscript m Superscript i Baseline left-parenthesis upper M right-parenthesis colon equals limit Underscript StartLayout 1st Row right-arrow 2nd Row n EndLayout Endscripts normal upper E normal x normal t Subscript upper R Superscript i Baseline left-parenthesis upper R slash m Superscript n Baseline comma upper M right-parenthesis">

Ext

R
i

(

R

/

m
,
M

)

→

H
m
i

(
M
)

:=

lim

→

n

E
x
t

R
i

(
R

/

m
n

,
M
)

{\text {Ext}}_R^i\left ( {R/m,M} \right ) \to H_m^i\left ( M \right ): = \lim _{\substack {\to \\n}} \mathrm {Ext}_R^i \left (R/m^n, M \right )

is surjective. The hypothesis that

R
R

is Cohen-Macaulay is not superfluous. Two examples are given.
Defining Bernoulli polynomials in 𝑍/𝑝𝑍 (a generic regularity condition)
Authors: Granville Andrew , Shank H. S.
Published: June 2010

We consider the problem of whether Bernoulli polynomials are uniquely defined by certain interpolation equations. This leads to an interesting characterization of regular primes, a new insight into the

p
p

-divisibility of Fermat quotients, and a generalization of Voronoi’s congruences.
The semisimplicity problem for 𝑝-adic group algebras
Authors: Hare Kathryn E. , Shirvani Maziar
Published: June 2010

For a prime

p
p

let

Ω
=

Ω
p

\Omega = {\Omega _p}

denote the completion of the algebraic closure of the field of

p
p

-adic numbers with

p
p

-adic valuation

|
|

\left | \right |

. Given a group

G
G

consider the ring of formal sums
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l 1 left-parenthesis normal upper Omega comma upper G right-parenthesis equals left-brace sigma-summation Underscript x element-of upper G Endscripts alpha Subscript x Baseline x colon alpha Subscript x Baseline element-of normal upper Omega comma StartAbsoluteValue alpha Subscript x Baseline EndAbsoluteValue right-arrow 0 right-brace period">

l
1

(

Ω
,
G

)

=

{

∑

x
∈
G

α
x

x
:

α
x

∈
Ω
,

|

α
x

|

→
0

}

.

{l_1}\left ( {\Omega ,G} \right ) = \left \{ {\sum \limits _{x \in G} {{\alpha _x}x:{\alpha _x} \in \Omega ,\left | {{\alpha _x}} \right |} \to 0} \right \}.

\]
Motivated by the study of group rings and the complex Banach algebras

l
1

(

C

,
G

)

{l_1}\left ( {{\mathbf {C}},G} \right )

, we consider the problem of when this ring is semisimple (semiprimitive). Our main result is that for an Abelian group

G
,

l
1

(

Ω
,
G

)

G,{l_1}\left ( {\Omega ,G} \right )

is semisimple if and only if

G
G

does not contain a

C
p

∞

{C_p}\infty

subgroup. We also prove that

l
1

(

Ω
,
G

)

{l_1}\left ( {\Omega ,G} \right )

is semisimple if

G
G

is a nilpotent

p
′

p’

-group, an ordered group, or a torsion-free solvable group. We use a mixture of algebraic and analytic methods.
𝐿^{∞}-BMO boundedness for a singular integral operator
Authors: Namazi Javad
Published: June 2010

If

K

(
x
)

=
Ω

(
x
)

/

|

x

|

n

K\left ( x \right ) = \Omega \left ( x \right )/|x{|^n}

is a Calderón-Zygmund kernel and

b

(

|

x

|

)

b\left ( {|x|} \right )

is a bounded radial function, we find conditions on

b
b

such that the singular operator whose kernel is

b

(
x
)

K

(
x
)

b\left ( x \right )K\left ( x \right )

is bounded from

L
∞

(

R
n

)

{L^\infty }\left ( {{R^n}} \right )

to

BMO

(

R
n

)

{\text {BMO}}\left ( {{R^n}} \right )

, or equivalently from

H
1

(

R
n

)

{H^1}\left ( {{R^n}} \right )

into

L
1

(

R
n

)

{L^1}\left ( {{R^n}} \right )

.
Closure of invertible operators on a Hilbert space
Authors: Bouldin Richard
Published: June 2010

Although most of the characterizations of the closure of the invertible operators on a separable Hilbert space do not extend to a nonseparable Hilbert space, this note gives a characterization for an arbitrary Hilbert space that generalizes the separable case in a natural way. The new concept of essential nullity, which facilitates this characterization, should find other applications.
A generalization of a theorem of J. Holub
Authors: Abramovich Yuri
Published: June 2010

We present here a simple proof of the following result: Let

X
X

be an arbitrary

C
(
K
)

C(K)

or

L
1

(
μ
)

{L_1}(\mu )

space and let

T
:
X
→
X

T:X \to X

be an arbitrary linear continuous operator. Then for at least one choice of signs.
\[

‖

I
±
T

‖

=
1
+

‖
T
‖

.

\left \| {I \pm T} \right \| = 1 + \left \| T \right \|.

\]
This is a slightly generalized version of a recent result due to J. Holub [4].
Invariant subspace of strictly singular operators
Authors: Ruan Ji Shou
Published: June 2010

In this paper, we show that strictly singular operators are condensing maps. Moreover, we obtain a new result that every bounded linear operator

T
T

on a Banach space that commutes with a nonzero strictly singular operator

S
S

has a non-trivial invariant closed subspace.
A large deviation principle for the reduction of product representations
Authors: Duffield N. G.
Published: June 2010

A large deviation principle is proved for a family of measures

{

L

n

:
n
=
1
,
2
,
…

}

\left \{ {{\mathbb {L}_n}:n = 1,2, \ldots } \right \}

derived from the multiplicities occurring in the decomposition into irreducible components of

n
n

-fold tensor products of representations of arbitrary compact semisimple Lie groups.
The existence of absolutely continuous invariant measures for 𝐶^{1+𝜀} Jablonski transformation in 𝑅ⁿ
Authors: Lou You-Shi
Published: June 2010

Using a result of Rychlik, we present a sufficient condition for the existence of an absolutely continuous invariant measure for

C

1
+
ε

{C^{1 + \varepsilon }}

Jablonski transformation in

R
n

{R^n}

.
Mean ergodic theorems for nonlinear operators
Authors: Wittmann Rainer
Published: June 2010

The mean ergodic theorem is shown for nonlinear operators

T
:
K
→
K

T:K \to K

with

|

|

T
x
+
T
y

|

|

≤

|

|

x
+
y

|

|

||Tx + Ty|| \leq ||x + y||

for any

x
,
y
∈
K

x,y \in K

where

K
K

may be an arbitrary subset of a Hilbert space

H
H

.
Rationality of some genus 0 extensions of 𝐾(𝑋)
Authors: Deveney James K. , Yanik Joe
Published: June 2010

Let

L
L

be a quadratic extension of a field

K
K

with Galois group

{

e
,
Γ

}

\left \{ {e,\Gamma } \right \}

. Let

{

x
,
y

}

\left \{ {x,y} \right \}

be algebraically independent over

L
L

and let

Γ
\Gamma

be extended to an automorphism of

L
(
x
,
y
)

L(x,y)

so that

Γ
(
x
)
=
x

\Gamma (x) = x

and the extension has order 2. Then

L

(
x
,
y

)
Γ

L{(x,y)^\Gamma }

is a genus 0 extension of

K
(
x
)

K(x)

. This paper examines when

L

(
x
,
y

)
Γ

L{(x,y)^\Gamma }

will be pure transcendental over

K
K

. It is shown that some important examples from field theory can be realized by this construction. The main result shows that

L

(
x
,
y

)
Γ

L{(x,y)^\Gamma }

is pure transcendental over

K
K

if

Γ
(
y
)
=
(

x
2

+
b
x
+
c
)

/

y
(
char
⁡
K
≠
2
)

\Gamma (y) = ({x^2} + bx + c)/y(\operatorname {char} K \ne 2)

. An example illustrates that it is essential that the second degree polynomial be monic.
Relations between Banach function algebras and their uniform closures
Authors: Honary Taher G.
Published: June 2010

Let

A
A

be a Banach function algebra on a compact Hausdorff space

X
X

. In this paper we consider some relations between the maximal ideal space, the Shilov boundary and the Choquet boundary of

A
A

and its uniform closure

A
¯

\bar A

. As an application we determine the maximal ideal space, the Shilov boundary and the Choquet boundary of algebras of infinitely differentiable functions which were introduced by Dales and Davie in 1973.
Minimal surfaces with low index in the three-dimensional sphere
Authors: Urbano Francisco
Published: June 2010

In the present paper, the author gives a characterization of the Clifford torus among the minimal surfaces of the three-dimensional sphere in terms of its index.
Descriptions of nest algebras
Authors: Anoussis M. , Katsoulis E. G.
Published: June 2010

Every operator in the algebra of a continuous nest

N

\mathcal {N}

can be factored as a product of two operators which belong to certain diagonal disjoint ideals of

Alg
⁡

N

\operatorname {Alg} \mathcal {N}

. This factorization leads to a new description of nest algebras.
A property of purely infinite simple 𝐶*-algebras
Authors: Zhang Shuang
Published: June 2010

An alternative proof is given for the fact ([13]) that a purely infinite, simple

C
∗

{C^*}

-algebra has the FS property: the set of self-adjoint elements with finite spectrum is norm dense in the set of all self-adjoint elements. In particular, the Cuntz algebras

O
n

(
2
≤
n
≤
+
∞
)

{O_n}(2 \leq n \leq + \infty )

and the Cuntz-Krieger algebras

O
A

{O_A}

, if

A
A

is an irreducible matrix, have the FS property. This answers a question raised in [2, 2.10] concerning the structure of projections in the Cuntz algebras. Moreover, many corona algebras and multiplier algebras have the FS property.
Large compact separable spaces may all contain 𝛽𝑁
Authors: Dow Alan
Published: June 2010

In the Cohen model any compact separable space that does not contain

β
N

\beta N

has cardinality at most of the continuum.
An asymptotic bound for the iterates of certain real functions near a contractive fixed point
Authors: Wallen Lawrence J.
Published: June 2010

If

x
=
0

x = 0

is a contractive fixed point for the function

F
F

, then under certain conditions, the iterates

F
k

(
a
)

{F_k}(a)

are asymptotically equal to the numbers

ξ
k

{\xi _k}

defined by

k
=

∫

ξ
k

a

d
u

u
−
F
(
u
)

k = \int _{{\xi _k}}^a {\frac {{du}}{{u - F(u)}}}

. Using somewhat different hypotheses, we give a more precise bound on

F
k

(
a
)

/

ξ
k

{F_k}(a)/{\xi _k}

.
Betti numbers for modules of finite length
Authors: Charalambous Hara , Evans E. Graham , Miller Matthew
Published: June 2010

Let

R
R

be a Gorenstein local ring of dimension

d
>
5

d > 5

and let

M
M

be a module of finite length and finite projective dimension. If

M
M

is not isomorphic to

R
R

modulo a regular sequence, then the Betti numbers of

M
M

satisfy

β
i

(
M
)
>

(
i
d

)

{\beta _i}(M) > (_i^d)

for

0
>
i
>
d

0 > i > d

, and

∑

i
=
0

d

β
i

(
M
)
≥

2
d

+

2

d
−
1

\sum \nolimits _{i = 0}^d {{\beta _i}(M) \geq {2^d} + {2^{d - 1}}}

.
Lower bounds for the extrinsic total curvatures of a space-like codimension 2 surface in Minkowski space
Authors: Kossowski Marek
Published: June 2010

There are three invariant curvature functions defined on any smooth space-like

2
2

-surfaces in four-dimensional Minkowski space. (If the surface lies in a Euclidean hyperplane then the functions agree with

H
2

,

K
2

{H^2},{K^2}

, and

(

H
2

−
K

)

2

{\left ( {{H^2} - K} \right )^2}

. For each of these functions we show that there exists a space-like immersion of any oriented compact (or noncompact complete) surface with associated total curvature arbitrarily small.
PI semigroup algebras of linear semigroups
Authors: Okniński Jan , Putcha Mohan S.
Published: June 2010

It is well-known that if a semigroup algebra

K
[
S
]

K[S]

over a field

K
K

satisfies a polynomial identity then the semigroup

S
S

has the permutation property. The converse is not true in general even when

S
S

is a group. In this paper we consider linear semigroups

S
⊆

M

n

(
F
)

S \subseteq {\mathcal {M}_n}(F)

having the permutation property. We show then that

K
[
S
]

K[S]

has a polynomial identity of degree bounded by a fixed function of

n
n

and the number of irreducible components of the Zariski closure of

S
S

.
The relative form of Gersten’s conjecture for power series over a complete discrete valuation ring
Authors: Reid L. , Sherman C.
Published: June 2010

A relative form of Gersten’s Conjecture is established for a ring of formal power series over a complete discrete valuation ring. The main corollaries are that the absolute version of Gersten’s Conjecture is valid for such a ring if it is valid for arbitrary discrete valuation rings, and, consequently, that the conjecture is true for such a ring if we use

K
K

-theory with finite coefficients of order prime to the characteristic of the residue field.
Priestley spaces
Authors: Venugopalan P.
Published: June 2010

We give a purely order-theoretic characterization of complete lattices that are compact totally order-disconnected (Priestley) spaces with respect to the Lawson topology. We also characterize complete lattices that are Priestley spaces with respect to the interval topology.
Spectral pictures of hyponormal bilateral operator weighted shifts
Authors: Herrero Domingo A.
Published: June 2010

This note provides a complete description of the spectral picture of a hyponormal bilateral operator weighted shift of finite multiplicity. If the weights are

m
×
m

m \times m

matrices, then the operator cannot by cyclic, unless it is a normal operator of a very special kind. An example shows that, nevertheless, there exists a cyclic, nonnormal, hyponormal bilateral operator weighted shift whose weights are operators acting on an infinite dimensional Hilbert space. (This answers a question of J. B. Conway.)
On theta pairs for a maximal subgroup
Authors: Mukherjee N. P. , Bhattacharya Prabir
Published: June 2010

For a maximal subgroup

M
M

of a finite group

G
G

, a

Θ
\Theta

-pair is any pair of subgroups

(
C
,
D
)

(C,D)

of

G
G

such that (i)

D
◃
G
,
D
⊂
C

D \triangleleft G,D \subset C

, (ii)

⟨

M
,
C

⟩

=
G
,

⟨

M
,
D

⟩

=
M

\left \langle {M,C} \right \rangle = G,\left \langle {M,D} \right \rangle = M

and (iii)

C

/

D

C/D

has no proper normal subgroup of

G

/

D

G/D

. A natural partial ordering is defined on the family of

Θ
\Theta

-pairs. We obtain several results on the maximal

Θ
\Theta

-pairs which imply

G
G

to be solvable, supersolvable, and nilpotent.
𝑡-linked overrings as intersections of localizations
Authors: Dobbs David E. , Houston Evan G. , Lucas Thomas G. , Zafrullah Muhammad
Published: June 2010

This sequel to our work on

t
t

-linked overrings introduces, characterizes, and applies the

t
t

-theoretic analogues of integral domains satisfying the

Q
R

QR

- and

Q
Q
R

QQR

-properties. In particular, we show that, unlike the situation with

Q
Q
R

QQR

-domains, the

t
Q
Q
R

tQQR

-property is stable under the adjunction of an indeterminate.
Hall subgroup normalizers and character correspondences in 𝑀-groups
Authors: Isaacs I. M.
Published: June 2010

Let

H
H

be a Hall subgroup of an

M
M

-group

G
G

, and let

N
N

be its normalizer in

G
G

. It is shown that the group

N

/

H
′

N/H’

is an

M
M

-group. This involves the construction of an explicit bijection between the sets of irreducible characters of

G
G

and of

N
N

with degrees coprime to the order of

H
H

.
Topological rationalization of a class of meromorphic functions
Authors: Meziani A.
Published: June 2010

We prove that a class of germs at

0
∈

C

2

0 \in {{\mathbf {C}}^2}

of meromorphic functions can be transformed, via homeomorphisms, into rational functions.
Growth functions for residually torsion-free nilpotent groups
Authors: Alperin Roger C. , Peterson Brian L.
Published: June 2010

We show that G. Baumslag’s growth function for the free group is not algebraic.
Nonoscillatory solutions of second order differential equations with integrable coefficients
Authors: Naito Manabu
Published: June 2010

The asymptotic behavior of nonoscillatory solutions of the equation

x
+
a

(
t
)

|
x
|

γ

sgn
⁡
x
=
0
,
γ
>
0

x + a\left ( t \right ){\left | x \right |^\gamma }\operatorname {sgn} x = 0,\gamma > 0

, is discussed under the condition that

A

(
t
)

=

li

m

T
→
∞

∫
t
T

a

(
s
)

d
s

A\left ( t \right ) = {\text {li}}{{\text {m}}_{T \to \infty }}\int _t^T {a\left ( s \right )ds}

exists and

A

(
t
)

≥
0

A\left ( t \right ) \geq 0

for all

t
t

. For the sublinear case of

0
>
γ
>
1

0 > \gamma > 1

, the existence of at least one nonoscillatory solution is completely characterized.
A remark on strong maximum principle for parabolic and elliptic systems
Authors: Wang Xuefeng
Published: June 2010

We give a strong maximum principle for some nonlinear parabolic and elliptic systems with convex invariant regions. We also obtain a version of the Hopf boundary lemma for the systems.
Stable rank of subalgebras of the ball algebras
Authors: Rupp Rudolf
Published: June 2010

We show that every subalgebra (in the algebraic sense) of

A
1

(

B
n

)

{A^1}({B_n})

satisfying the weak Nullstellensatz has stable rank less than or equal to

n
n

. The results are sharp for

n
≤
2

n \leq 2

.
Hypercompletions of Riesz spaces
Authors: Filter Wolfgang
Published: June 2010

It is shown that each Riesz space with separating order continuous dual can be embedded in a unique "

e
e

-hypercompletion," where

e
e

is a fixed weak unit of the extended order continuous dual.
Nodal quintic threefolds and nodal octic surfaces
Authors: Borcea Ciprian
Published: June 2010

The relation between the nodes and the defect of nodal quintic threefolds or double solids ramified over nodal octic surfaces has implications on the existence of such varieties with fewer nodes.
Eigenvalue comparison for tubular domains
Authors: Lee Jeffrey M.
Published: June 2010

S. Y. Cheng’s eigenvalue comparison theorem for geodesic balls is generalized to certain tubular domains. This gives application to the infinitesimal volume comparison theory developed by E. Heintze and H. Karcher as well as A. Gray.
A Luecking-type subspace of ℒ¹ₐ and its dual
Authors: Ghatage Pratibha , Sun Shun Hua
Published: June 2010

The purpose of this investigation is to determine the extent to which Luecking’s decomposition of Bergman spaces

L

a
p

,
(
1
>
p
>
∞
)

\mathcal {L}_a^p,(1 > p > \infty )

[4] can be extended to

L

a
1

\mathcal {L}_a^1

. The set of functions for which an atomic decomposition (using the reproducing kernel of the Bergman space) is possible turns out to be only a small part of

L

a
1

\mathcal {L}_a^1

. In this note we equip each of such functions with a new norm and study the resulting Banach space. We describe its dual and predual.
The tightness of certain almost complex submanifolds
Authors: Sánchez Cristián U.
Published: June 2010

This paper contains a proof of the following fact. If a map

f
f

from a connected Riemannian manifold with an almost complex structure into a euclidean space has the following properties: (a)

f
f

is nondegenerate and (b) for almost all height functions, the linear Hessian of the composition with

f
f

commutes with the almost complex structure, then

f
f

is tight. This gives some information about the homology groups of the manifold. This result yields a new proof of a well-known theorem of R. Bott.
On some equivalent metrics for bounded operators on Hilbert space
Authors: Kittaneh Fuad
Published: June 2010

Several operator norm inequalities concerning the equivalence of some metrics for bounded linear operators on Hilbert space are given. In addition, some related inequalities for the Hilbert-Schmidt norm are presented.
The duals of harmonic Bergman spaces
Authors: Coffman Charles V. , Cohen Jonathan
Published: June 2010

In this paper we show that for

Ω
\Omega

, a starlike Lipschitz domain, the dual of the space of harmonic functions in

L
p

(
Ω
)

{L^p}(\Omega )

need not be the harmonic functions in

L
q

(
Ω
)

{L^q}(\Omega )

, where

1

/

p
+
1

/

q
=
1

1/p + 1/q = 1

. We show that, as a consequence, the harmonic Bergman projection for

Ω
\Omega

need not extend to a bounded operator on

L
p

(
Ω
)

{L^p}(\Omega )

for all

1
>
p
>
∞

1 > p > \infty

. The duality result is a partial answer to a question of Nakai and Sario [9] posed initially in the Proceedings of the London Mathematical Society in 1978. We treat the duality question as a biharmonic problem, and our result follows from the failure of uniqueness for the biharmonic Dirichlet problem in domains with sharp intruding corners.
Complex orbits of solvable groups
Authors: Snow Dennis M.
Published: June 2010

The following structure theorems are proved: An orbit of a real solvable Lie group in projective space that is a complex submanifold is isomorphic to

C

k

×

(

C

∗

)
m

×
Ω

{{\mathbf {C}}^k} \times {({{\mathbf {C}}^ * })^m} \times \Omega

, where

Ω
\Omega

is an open orbit of a real solvable Lie group in a projective rational variety. Also, any homogeneous space of a complex Lie group that is isomorphic to

C

n

{{\mathbf {C}}^n}

can be realized as an orbit in some projective space. As a consequence, left-invariant complex structures on real solvable Lie groups are always induced from complex orbits in projective space.
10₁₀₁ has no period 7: a criterion for periodic links
Authors: Traczyk Paweł
Published: June 2010

We prove that the prime knot

10

101

{10_{101}}

has no period 7 by analyzing its Jones polynomial.
States on 𝑊*-algebras and orthogonal vector measures
Authors: Hamhalter Jan
Published: June 2010

We show that every state on a

W
∗

{W^ * }

-algebra

A

\mathcal {A}

without type

I
2

{I_2}

direct summand is induced by an orthogonal vector measure on

A

\mathcal {A}

. This result may find an application in quantum stochastics

[
1
,
7
]

[1,7]

. Particularly, it allows us to find a simple formula for the transition probability between two states on

A

\mathcal {A}

[
3
,
8
]

[3,8]

.
On a Dirichlet series associated with a polynomial
Authors: Eie Min King
Published: June 2010

Let

P
(
x
)
=

∏

j
=
2

k

(
x
+

δ
j

)

P(x) = \prod \nolimits _{j = 2}^k {(x + {\delta _j})}

be a polynomial with real coefficients and

Re
⁡

δ
j

>
−
1
(
j
=
1
,
…
,
k
)

\operatorname {Re} {\delta _j} > - 1(j = 1, \ldots ,k)

. Define the zeta function

Z
p

(
s
)

{Z_p}(s)

associated with the polynomial

P
(
x
)

P(x)

as
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z Subscript upper P Baseline left-parenthesis s right-parenthesis equals sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts StartFraction 1 Over upper P left-parenthesis n right-parenthesis Superscript s Baseline EndFraction comma upper R e s greater-than 1 slash k period">

Z
P

(
s
)
=

∑

n
=
1

∞

1

P

(
n
)

s

,
Re
⁡
s
>
1

/

k
.

{Z_P}(s) = \sum \limits _{n = 1}^\infty {\frac {1}{{P{{(n)}^s}}}} ,\operatorname {Re} s > 1/k.

\]

Z
P

(
s
)

Z_P(s)

is holomorphic for

Re
⁡
s
>
1

/

k

\operatorname {Re} s > 1/k

and it has an analytic continuation in the whole complex

s
s

-plane with only possible simple poles at

s
=
j

/

k
(
j
=
1
,
0
,
−
1
,
−
2
,
−
3
,
…
)

s = j/k(j = 1,0, - 1, - 2, - 3, \ldots )

other than nonpositive integers. In this paper, we shall obtain the explicit value of

Z
P

(
−
m
)

{Z_P}( - m)

for any non-negative integer

m
m

, the asymptotic formula of

Z
P

(
s
)

{Z_P}(s)

at

s
=
1

/

k

s = 1/k

, the value

Z
P
′

(
0
)

{Z’_P}(0)

and its application to the determinants of elliptic operators.
A totally real three-sphere in 𝐶³ bounding a family of analytic disks
Authors: Forstnerič Franc
Published: June 2010

We construct a smoothly embedded totally real three-sphere

S
S

in

C

3

{{\mathbf {C}}^3}

and a one-parameter family of analytic disks in

C

3

{{\mathbf {C}}^3}

that have boundaries in

S
S

.
Cell-like maps that are shape equivalences
Authors: Choi Jung-In K.
Published: June 2010

Let

f
:

X
′

→
X

f:X’ \to X

be a cell-like map between metric spaces and set

N
f

=
{
x
∈
X
:

f

−
1

(
x
)
≠

point} 

{N_f} = \{ x \in X:{f^{ - 1}}(x) \ne {\text {point\} }}

. Even if

N
f

⊂

⋃

n
=
1

∞

B
n

{N_f} \subset \bigcup \nolimits _{n = 1}^\infty {{B_n}}

, where each

B
n

{B_n}

is closed and each
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f vertical-bar f Superscript negative 1 Baseline left-parenthesis upper B Subscript n Baseline right-parenthesis colon f Superscript negative 1 Baseline left-parenthesis upper B Subscript n Baseline right-parenthesis right-arrow upper B Subscript n Baseline">

f

|

f

−
1

(

B
n

)
:

f

−
1

(

B
n

)
→

B
n

f|{f^{ - 1}}({B_n}):{f^{ - 1}}({B_n}) \to {B_n}

is hereditary shape equivalence,

f
f

may not be a hereditary shape equivalence. Conditions are placed on the

B
n

{B_n}

’s to assure that

f
f

is a hereditary shape equivalence. For example, if

N
f

⊂

⋃

n
=
1

∞

B
n

{N_f} \subset \bigcup \nolimits _{n = 1}^\infty {{B_n}}

, where

B
n

{B_n}

is closed for each
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 1 comma 2 comma ellipsis comma f vertical-bar f Superscript negative 1 Baseline left-parenthesis upper B Subscript n Baseline right-parenthesis colon f Superscript negative 1 Baseline left-parenthesis upper B Subscript n Baseline right-parenthesis right-arrow upper B Subscript n Baseline">

n
=
1
,
2
,
…
,
f

|

f

−
1

(

B
n

)
:

f

−
1

(

B
n

)
→

B
n

n = 1,2, \ldots ,f|{f^{ - 1}}({B_n}):{f^{ - 1}}({B_n}) \to {B_n}

is a hereditary shape equivalence, and

B
n

{B_n}

has arbitrary small neighborhoods whose boundaries miss

⋃

i
=
1

∞

B
i

\bigcup \nolimits _{i = 1}^\infty {{B_i}}

then

f
f

is a hereditary shape equivalence. An immediate consequence is that if

{

B
n

}

n
=
1

∞

\{ {B_n}\} _{n = 1}^\infty

is a pairwise disjoint null-sequence and each

f

|

f

−
1

(

B
n

)

f|{f^{ - 1}}({B_n})

is a hereditary shape equivalence, then

f
f

is a hereditary shape equivalence. Previously G. Kozlowski showed that if

{

f

−
1

(

B
n

)

}

n
=
1

∞

\{ {f^{ - 1}}({B_n})\} _{n = 1}^\infty

is a pairwise disjoint null-sequence and each

f

|

f

−
1

(

B
n

)

f|{f^{ - 1}}({B_n})

is a hereditary shape equivalence, then

f
f

is a hereditary shape equivalence, which can be obtained as an immediate corollary of one of our results.
Quadrature formulae and functions of exponential type
Authors: Rahman Qazi I. , Schmeisser Gerhard
Published: June 2010

In this paper we obtain certain generalizations of the trapezoidal rule and the Euler-Maclaurin formula that involve derivatives. In the case of quadrature of functions of exponential type over infinite intervals we find conditions under which existence of the (improper) integral and convergence of the approximating series become equivalent. In the process, we also establish a best possible version of a theorem of R. P. Boas and A. C. Schaeffer.
Evaluation of multivariate polynomials and their derivatives
Authors: Carnicer J. , Gasca M.
Published: June 2010

An extension of Horner’s algorithm to the evaluation of m-variate polynomials and their derivatives is obtained. The schemes of computation are represented by trees because this type of graph describes exactly in which order the computations must be done. Some examples of algorithms for one and two variables are given.
Elliptic curves over the rationals with bad reduction at only one prime
Authors: Edixhoven Bas , de Groot Arnold , Top Jaap
Published: June 2010

A list is given of elliptic curves over Q having additive reduction at exactly one prime. It is also proved that for primes congruent to 5 modulo 12, no such curves having potentially good reduction exist. This enables one to find in a number of cases a complete list of all elliptic curves with bad reduction at only one prime.
A differential delay equation arising from the sieve of Eratosthenes
Authors: Cheer A. Y. , Goldston D. A.
Published: June 2010

The differential delay equation defined by

ω
(
u
)
=
1

/

u

\omega (u) = 1/u

for

1
≤
u
≤
2

1 \leq u \leq 2

and

(
u
ω
(
u
)

)
′

=
ω
(
u
−
1
)

(u\omega (u))’ = \omega (u - 1)

for

u
≥
2

u \geq 2

was introduced by Buchstab in connection with an asymptotic formula for the number of uncanceled terms in the sieve of Eratosthenes. Maier has recently used this result to show there is unexpected irregularity in the distribution of primes in short intervals. The function

ω
(
u
)

\omega (u)

is studied in this paper using numerical and analytical techniques. The results are applied to give some numerical constants in Maier’s theorem.
Multiconstrained variational problems of nonlinear eigenvalue type: new formulations and algorithms
Authors: Eydeland Alexander , Spruck Joel , Turkington Bruce
Published: June 2010

A new variational approach is proposed for a class of semilinear elliptic eigenvalue problems involving many eigenvalue parameters. These problems arise, for instance, in the modelling of magnetohydrodynamic equilibria with one spatial symmetry. In this case, the physical variational principle imposes a continuously infinite family of constraints, which prescribes the mass and helicity within every flux tube. The equilibrium equations therefore contain unspecified profile functions that are determined along with the solution as multipliers for those constraints. A prototype problem for this general class is formulated, and a natural discretization of its constraint family is introduced. The resulting multiconstrained minimization problem is solved by an iterative algorithm, which is based on relaxation of the given nonlinear equality constraints to linearized inequalities at each iteration. By appealing to convexity properties, the monotonicity and global convergence of this algorithm is proved. The explicit construction of the iterative sequence is obtained by a dual variational characterization.
On optimal high-order in time approximations for the Korteweg-de Vries equation
Authors: Karakashian Ohannes , McKinney William
Published: June 2010

We show that the well-known order reduction phenomenon affecting implicit Runge-Kutta methods does not occur when approximating periodic solutions of the Korteweg-de Vries equation.
Analysis of spectral projectors in one-dimensional domains
Authors: Maday Y.
Published: June 2010

In this paper we analyze a class of projection operators with values in a subspace of polynomials. These projection operators are related to the Hilbert spaces involved in the numerical analysis of spectral methods. They are, in the first part of the paper, the standard Sobolev spaces and, in the second part, some weighted Sobolev spaces, the weight of which is related to the orthogonality relation satisfied by the Chebyshev polynomials. These results are used to study the approximation of a model fourth-order problem.
On Stieltjes polynomials and Gauss-Kronrod quadrature
Authors: Peherstorfer Franz
Published: June 2010

Let D be a real function such that

D
(
z
)

D(z)

is analytic and

D
(
z
)
≠
0

D(z) \ne 0

for

|

z

|

≤
1

|z| \leq 1

. Furthermore, put

W
(
x
)
=

1
−

x
2

|

D
(

e

i
φ

)

|

2

W(x) = \sqrt {1 - {x^2}} |D({e^{i\varphi }}){|^2}

,

x
=
cos
⁡
φ

x = \cos \varphi

,

φ
∈
[
0
,
π
]

\varphi \in [0,\pi ]

, and denote by

p
n

(
∙
,
W
)

{p_n}( \bullet ,W)

the polynomial which is orthogonal on

[
−
1
,
+
1
]

[ - 1, + 1]

to

P

n
−
1

{\mathbb {P}_{n - 1}}

(

P

n
−
1

{\mathbb {P}_{n - 1}}

denotes the set of polynomials of degree at most

n
−
1

n - 1

) with respect to W. In this paper it is shown that for each sufficiently large n the polynomial

E

n
+
1

(
∙
,
W
)

{E_{n + 1}}( \bullet ,W)

(called the Stieltjes polynomial) of degree

n
+
1

n + 1

which is orthogonal on

[
−
1
,
+
1
]

[ - 1, + 1]

to

P

n

{\mathbb {P}_n}

with respect to the sign-changing function

p
n

(
∙
,
W
)
W

{p_n}( \bullet ,W)W

has

n
+
1

n + 1

simple zeros in

(
−
1
,
1
)

( - 1,1)

and that the interpolation quadrature formula (called the Gauss-Kronrod quadrature formula) based on nodes which are the

2
n
+
1

2n + 1

zeros of

E

n
+
1

(
∙
,
W
)

p
n

(
∙
,
W
)

{E_{n + 1}}( \bullet ,W){p_n}( \bullet ,W)

has all weights positive.
A note on Perrin pseudoprimes
Authors: Arno Steven
Published: June 2010

The cubic recurrence

A
(
n
+
3
)
=
A
(
n
)
+
A
(
n
+
1
)

A(n + 3) = A(n) + A(n + 1)

with initial conditions

A
(
0
)
=
3

A(0) = 3

,

A
(
1
)
=
0

A(1) = 0

,

A
(
2
)
=
2

A(2) = 2

, known as Perrin’s sequence, is associated with several types of pseudoprimes. In this paper we will explore a question of Adams and Shanks concerning the existence of the so-called Q and I Perrin pseudoprimes, and develop an algorithm to search for all such pseudoprimes below some specified limit. As an example, we show that none exist below

10

14

{10^{14}}

.
Large-time behavior of deterministic particle approximations to the Navier-Stokes equations
Authors: Cottet Georges-Henri
Published: June 2010

We prove that for a class of deterministic vortex methods for the Navier-Stokes equations in two dimensions, the numerical solution decays for large time with the same rate as the exact solution. We substantiate our result with numerical experiments and with a remark concerning the problem of reinitialization of a distribution of particles.
Error bounds for the method of good lattice points
Authors: Disney Shaun , Sloan Ian H.
Published: June 2010

New error bounds are obtained for the method of good lattice points for multidimensional quadrature, when m, the number of quadrature points, is prime. One of these bounds reduces the constant in Niederreiter’s asymptotic error bound, if the dimension exceeds 2. Together they give very much smaller numerical bounds for all values of m.
Convergence of a second-order scheme for semilinear hyperbolic equations in 2+1 dimensions
Authors: Glassey Robert , Schaeffer Jack
Published: June 2010

A second-order energy-preserving scheme is studied for the solution of the semilinear Cauchy Problem

u

t
t

−

u

x
x

−

u

y
y

+

u
3

=
0

(
t
>
0
;
x
,
y
∈

R

)

{u_{tt}} - {u_{xx}} - {u_{yy}} + {u^3} = 0\;(t > 0;x,y \in \mathbb {R})

. Smooth data functions of compact support are prescribed at

t
=
0

t = 0

. On any time interval [0, T], second-order convergence (up to logarithmic corrections) to the exact solution is established in both the energy and uniform norms.
Notes on integration and integer sublattices
Authors: Lyness J. N. , Sørevik T. , Keast P.
Published: June 2010

A lattice rule is a quadrature rule over an s-dimensional hypercube, using N abscissas located on an integration lattice. In this paper we study sublattices and superlattices of integration lattices and of integer lattices. We exploit the properties of generator matrices of a lattice to provide an easy and elegant description of the relation between a lattice and a sublattice of given order. We also obtain necessary and sufficient criteria for existence of sublattices and information about the number of these.
A polynomial approach to fast algorithms for discrete Fourier-cosine and Fourier-sine transforms
Authors: Steidl G. , Tasche M.
Published: June 2010

The discrete Fourier-cosine transform

(
cos
⁡

-DFT

)

(\cos {\text {-DFT}})

, the discrete Fourier-sine transform

(
sin
⁡

-DFT

)

(\sin {\text {-DFT}})

and the discrete cosine transform (DCT) are closely related to the discrete Fourier transform (DFT) of real-valued sequences. This paper describes a general method for constructing fast algorithms for the

(
cos
⁡

-DFT

)

(\cos {\text {-DFT}})

, the

(
sin
⁡

-DFT

)

(\sin {\text {-DFT}})

and the DCT, which is based on polynomial arithmetic with Chebyshev polynomials and on the Chinese Remainder Theorem.
On generating polynomials which are orthogonal over several intervals
Authors: Fischer Bernd , Golub Gene H.
Published: June 2010

We consider the problem of generating the recursion coefficients of orthogonal polynomials for a given weight function. The weight function is assumed to be the weighted sum of weight functions, each supported on its own interval. Some of these intervals may coincide, overlap or are contiguous. We discuss three algorithms. Two of them are based on modified moments, whereas the other is based on an explicit expression for the desired coefficients. Several examples, illustrating the numerical performance of the various methods, are presented.
Cyclotomic invariants for primes between 125000 and 150000
Authors: Ernvall R. , Metsänkylä T.
Published: June 2010

Computations by Iwasawa and Sims, by Johnson, and by Wagstaff have determined certain important cyclotomic invariants for all primes up to 125000. We extended their results to 150000, basing our work on a recently computed list of irregular primes and using a new method.
Amicable pairs of the form (𝑖,1)
Authors: Costello Patrick
Published: June 2010

A method is given for finding amicable pairs of a certain type. When implemented, thirteen new amicable pairs were discovered. Using methods for finding new pairs from known pairs, the thirteen new pairs generated 64 other new pairs.
A new Mersenne prime
Authors: Colquitt W. N. , Welsh L.
Published: June 2010

The number

2

110503

−
1

{2^{110503}} - 1

is a Mersenne prime. There are exactly two Mersenne exponents between 100000 and 139268, and there are no Mersenne exponents between 216092 and 353620. Thus, the number

2

132049

−
1

{2^{132049}} - 1

has been verified as the 30th Mersenne prime in order of size.
Improved techniques for lower bounds for odd perfect numbers
Authors: Brent R. P. , Cohen G. L. , te Riele H. J. J.
Published: June 2010

If N is an odd perfect number, and

q
k

|

|

N

{q^k}||N

, q prime, k even, then it is almost immediate that

N
>

q

2
k

N > {q^{2k}}

. We prove here that, subject to certain conditions verifiable in polynomial time, in fact

N
>

q

5
k

/

2

N > {q^{5k/2}}

. Using this and related results, we are able to extend the computations in an earlier paper to show that

N
>

10

300

N > {10^{300}}

.
Convergence results for pseudospectral approximations of hyperbolic systems by a penalty-type boundary treatment
Authors: Funaro Daniele , Gottlieb David
Published: June 2010

In a previous paper we have presented a new method of imposing boundary conditions in the pseudospectral Chebyshev approximation of a scalar hyperbolic equation. The novel idea of the new method is to collocate the equation at the boundary points as well as in the inner grid points, using the boundary conditions as penalty terms. In this paper we extend the above boundary treatment to the case of pseudospectral approximations to general constant-coefficient hyperbolic systems of equations, and we provide error estimates for the pseudospectral Legendre method. The same scheme can be implemented also in the general (even nonlinear) case.
Chebyshev-Vandermonde systems
Authors: Reichel Lothar , Opfer Gerhard
Published: June 2010

A Chebyshev-Vandermonde matrix
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V equals left-bracket p Subscript j Baseline left-parenthesis z Subscript k Baseline right-parenthesis right-bracket Subscript j comma k equals 0 Superscript n Baseline element-of double-struck upper C Superscript left-parenthesis n plus 1 right-parenthesis times left-parenthesis n plus 1 right-parenthesis">

V
=
[

p
j

(

z
k

)

]

j
,
k
=
0

n

∈

C

(
n
+
1
)
×
(
n
+
1
)

V = [{p_j}({z_k})]_{j,k = 0}^n \in {\mathbb {C}^{(n + 1) \times (n + 1)}}

\]
is obtained by replacing the monomial entries of a Vandermonde matrix by Chebyshev polynomials

p
j

{p_j}

for an ellipse. The ellipse is also allowed to be a disk or an interval. We present a progressive scheme for allocating distinct nodes

z
k

{z_k}

on the boundary of the ellipse such that the Chebyshev-Vandermonde matrices obtained are reasonably well-conditioned. Fast progressive algorithms for the solution of the Chebyshev-Vandermonde systems are described. These algorithms are closely related to methods recently presented by Higham. We show that the node allocation is such that the solution computed by the progressive algorithms is fairly insensitive to perturbations in the right-hand side vector. Computed examples illustrate the numerical behavior of the schemes. Our analysis can also be used to bound the condition number of the polynomial interpolation operator defined by Newton’s interpolation formula. This extends earlier results of Fischer and the first author.
Variational formulation of a model free-boundary problem
Authors: Saavedra Patricia , Scott L. Ridgway
Published: June 2010

The purpose of this work is to present an error analysis of the numerical approximation by a finite element method of a free-surface problem. The analysis has been done in an abstract model which has many of the features of a free-surface problem for a viscous liquid. We study in this paper how the numerical approximation of the free boundary affects the approximation of the other variables of the problem and vice versa.
Convergence of the two-point upstream weighting scheme
Authors: Viallon Marie-Claude
Published: June 2010

Convergence to the entropy solution of a second-order scheme for the resolution of scalar hyperbolic conservation laws is studied. We consider the case of the so-called two-point upstream weighting scheme, widely used in petroleum engineering, to solve fluid flows in porous media problems. We prove convergence of the two-point upstream weighting scheme to the entropy solution for both discrete and semidiscrete approximations.
A family of cyclic quartic fields arising from modular curves
Authors: Washington Lawrence C.
Published: June 2010

We study a family of cyclic quartic fields arising from the covering of modular curves

X
1

(
16
)
→

X
0

(
16
)

{X_1}(16) \to {X_0}(16)

. An integral basis and a fundamental system of units are found. It is shown that a root of the quartic polynomial we construct is a translate of a cyclotomic period by an integer of the quadratic subfield of the quartic field.
Counterexamples concerning a weighted 𝐿² projection
Authors: Xu Jinchao
Published: June 2010

Counterexamples are given to show that some results concerning a weighted

L
2

{L^2}

projection presented earlier by Bramble and the author are sharp, i.e., that certain error and stability estimates are impossible in some cases.
Radiation boundary conditions for the two-dimensional wave equation from a variational principle
Authors: Broeze Jan , van Daalen Edwin F. G.
Published: June 2010

A variational principle is used to derive a new radiation boundary condition for the two-dimensional wave equation. This boundary condition is obtained from an expression for the local energy flux velocity on the boundary in normal direction. The wellposedness of the wave equation with this boundary condition is analyzed by investigating the energy of the system. Results obtained with this (nonlinear) boundary condition are compared with those obtained with the (linear) first-order absorbing boundary condition suggested by Higdon. In an accompanying paper the underlying theory is presented.
Some divisibility properties of the subgroup counting function for free products
Authors: Grady Michael , Newman Morris
Published: June 2010

Let G be the free product of finitely many cyclic groups of prime order. Let

M
n

{M_n}

denote the number of subgroups of G of index n . Let

C
p

{C_p}

denote the cyclic group of order p , and

C
p
k

C_p^k

the free product of k cyclic groups of order p . We show that

M
n

{M_n}

is odd if

C
2
4

C_2^4

occurs as a factor in the free product decomposition of G . We also show that if

C
3
3

C_3^3

occurs as a factor in the free product decomposition of G and if

C
2

{C_2}

is either not present or occurs to an even power, then

M
n

≡
0

mod

3

{M_n} \equiv 0\;\bmod \, 3

if and only if

n
≡
2

mod

4

n \equiv 2\;\bmod \, 4

. If, on the other hand,

C
3
3

C_3^3

occurs as a factor, and

C
2

{C_2}

also occurs as a factor, but to an odd power, then all the

M
n

{M_n}

are

≡
1

mod

3

\equiv 1\;\bmod \, 3

. Several conjectures are stated.
Eigenvalue approximation by a mixed method for resonant inhomogeneous cavities with metallic boundaries
Authors: Levillain Vincent
Published: June 2010

For an inhomogeneous cavity bounded by a perfect conductor, we prove that the approximation of the eigenvalues for the Maxwell problem leads to a second-order rate of convergence when using mixed finite elements. If the cavity has a disconnected boundary, the problem has null eigenvalues. We verify the existence of null eigenvalues for the approximate problem. They do not mix with the others that still converge at the same rate.
The existence of efficient lattice rules for multidimensional numerical integration
Authors: Niederreiter Harald
Published: June 2010

In this contribution to the theory of lattice rules for multidimensional numerical integration, we first establish bounds for various efficiency measures which lead to the conclusion that in the search for efficient lattice rules one should concentrate on lattice rules with large first invariant. Then we prove an existence theorem for efficient lattice rules of rank 2 with prescribed invariants, which extends an earlier result of the author for lattice rules of rank 1.
Reconstruction techniques for classical inverse Sturm-Liouville problems
Authors: Rundell William , Sacks Paul E.
Published: June 2010

This paper gives constructive algorithms for the classical inverse Sturm-Liouville problem. It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. Two methods of solving this latter problem are then provided, and numerical examples are presented.
Some spectral approximations of two-dimensional fourth-order problems
Authors: Bernardi Christine , Coppoletta Giuseppe , Maday Yvon
Published: June 2010

This paper deals with the approximation of the biharmonic equation in a square domain with Dirichlet boundary conditions. Two types of discrete problems are presented, the numerical analysis is performed, and estimates for the error between the exact and approximate solutions are given.
A stochastic roundoff error analysis for the convolution
Authors: Calvetti Daniela
Published: June 2010

We study the accuracy of an algorithm which computes the convolution via Radix-2 fast Fourier transforms. Upper bounds are derived for the expected value and the variance of the accompanying linear forms in terms of the expected value and variance of the relative roundoff errors for the elementary operations of addition and multiplication. These results are compared with the corresponding ones for two algorithms computing the convolution directly, via Horner’s sums and using cascade summation, respectively.