bluuebunny/crossref_metadata_2025 huggingface dataset:
On two function spaces which are similar to 𝐿₀
Authors: Dilworth S. J. , Trautman D. A.
Published: June 2010

Let

Λ
0

{\Lambda _0}

consist of all functions

f
f

measurable on

(

0
,
∞

)

\left ( {0,\infty } \right )

with
\[

λ
{
s
:

|

f
(
s
)

|

>
t
}
>
∞

\lambda \{ s:|f(s)| > t\} > \infty

\]
for all

t
>
0

t > 0

, where

λ
\lambda

is Lebesgue measure, and let

L
0

(
0
,
∞
)

{L_0}(0,\infty )

consist of all measurable functions

f
f

with
\[

lim

t
→
∞

λ
{
s
:

|

f
(
s
)

|

>
t
}
=
0.

\lim \limits _{t \to \infty } \lambda \{ s:|f(s)| > t\} = 0.

\]
Let each space have the topology induced by convergence in measure. We show that every infinite-dimensional Banach subspace of

Λ
0

{\Lambda _0}

contains

c
0

{c_0}

or

l
p

{l_p}

for some

1
≤
p
>
∞

1 \leq p > \infty

. We also identify the duals of

Λ
0

{\Lambda _0}

and

L
0

(
0
,
∞
)

{L_0}(0,\infty )

.
Filtrations of rational representations of reductive groups of semisimple rank 1
Authors: Doty Stephen
Published: June 2010

A detailed study is made of the affine coordinate ring of the Chevalley group

S

L

2

{\text {S}}{{\text {L}}_2}

over the integers as base ring. Certain applications to the representation theory of groups of semisimple rank 1 are made, including the construction of a filtration on modules obtained by inducing a character of a maximal torus

T
T

from

T
T

up to the group. We show this filtration extends the Jantzen-Andersen filtration on the dual Weyl module with highest weight given by that character, in case the character in question is dominant.
Isoperimetric inequalities for convex cones
Authors: Lions Pierre-Louis , Pacella Filomena
Published: June 2010

We present here an isoperimetric inequality for sets contained in a convex cone. Some applications to symmetrization problems and Sobolev inequalities are also indicated.
A uniqueness condition for finite measures
Authors: Nymann J. E.
Published: June 2010

Let

μ
\mu

and

μ
′

\mu ’

be two finite measures on the same measurable space which have the property:

μ
(
A
)
=
μ
(
B
)

\mu (A) = \mu (B)

implies that

μ
′

(
A
)
=

μ
′

(
B
)

\mu ’(A) = \mu ’(B)

. If the range of

μ
\mu

is an interval, then there is a constant

α
\alpha

such that

μ
′

=
α
μ

\mu ’ = \alpha \mu

. This extends earlier results of Leth and Malitz on purely atomic measures.
A remark on 𝜆_{2𝑔-2}
Authors: Randol Burton
Published: June 2010

It is shown that if the first

2
g
−
3

2g - 3

Laplace eigenvalues on a compact Riemann surface of genus

≥
2

\geq 2

are small, then

λ

2
g
−
2

{\lambda _{2g - 2}}

is greater than

1
4

\frac {1}{4}

.
When is a flat algebra of finite type?
Authors: Schenzel Peter
Published: June 2010

Let

A
A

denote a commutative Noetherian domain. For an intermediate ring

A
⊆
B
⊆

A
x

A \subseteq B \subseteq {A_x}

flat over

A
A

, it is shown that

B
B

is an

A
A

-algebra of finite type. This is followed by an intrinsic description of the flatness of

B
B

over

A
A

and the asymptotic behavior of certain prime divisors. As an application, flat ideal-transforms are characterized.
Locally bounded noncontinuous linear forms on strong duals of nondistinguished Köthe echelon spaces
Authors: Bastin Françoise , Bonet José
Published: June 2010

In this note it is proved that if

λ
1

(
A
)

{\lambda _1}(A)

is any nondistinguished Köthe echelon space of order one and

K
∞

≃

(

λ
1

(
A
)

)
b
′

{K_\infty } \simeq {({\lambda _1}(A))’_b}

is its strong dual, then there is even a linear form :

K
∞

→

C

{K_\infty } \to {\mathbf {C}}

which is locally bounded (i.e. bounded on the bounded sets) but not continuous. It is also shown that every nondistinguished Köthe echelon space contains a sectional subspace with a particular structure.
The parameters of a chain sequence
Authors: Chihara T. S.
Published: June 2010

We give a method for constructing explicitly all parameter sequences for any chain sequence for which one parameter sequence is known. An application to orthogonal polynomials associated with birth and death processes is given.
On the structure of semiderivations in prime rings
Authors: Chuang Chen-Lian
Published: June 2010

Let

R
R

be a prime ring. By a semiderivation associated with a function

g
:
R
→
R

g:R \to R

, we mean an additive mapping

f
:
R
→
R

f:R \to R

such that, for all
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x comma y element-of upper R comma f left-parenthesis x y right-parenthesis equals f left-parenthesis x right-parenthesis g left-parenthesis y right-parenthesis plus x f left-parenthesis y right-parenthesis equals f left-parenthesis x right-parenthesis y plus g left-parenthesis x right-parenthesis f left-parenthesis y right-parenthesis">

x
,
y
∈
R
,
f
(
x
y
)
=
f
(
x
)
g
(
y
)
+
x
f
(
y
)
=
f
(
x
)
y
+
g
(
x
)
f
(
y
)

x,y \in R,f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y)

and

f
(
g
(
x
)
)
=
g
(
f
(
x
)
)

f(g(x)) = g(f(x))

. It is known that

g
g

must necessarily be a ring endomorphism. Here it is shown that

f
f

must be an ordinary derivation or of the form

f
(
x
)
=
λ
(
x
−
g
(
x
)
)

f(x) = \lambda (x - g(x))

for all

x
∈
R

x \in R

, where

λ
\lambda

is an element of the extended centroid of

R
R

.
Finite projective planes and a question about primes
Authors: Feit Walter
Published: June 2010

Let

n
n

be an even integer not divisible by 3. Suppose that

p
=

n
2

+
n
+
1

p = {n^2} + n + 1

is a prime and

2

n
+
1

≡
1

(

mod
p

)

{2^{n + 1}} \equiv 1\left ( {\bmod p} \right )

. The question is asked whether this can only occur if

n
n

is a power of 2. It is noted that an affirmative answer to this question implies that a finite projective plane with a flag transitive collineation group is Desarguesian.
Hodge theory of singular algebraic curves
Authors: Nagase Masayoshi
Published: June 2010

The spectrum of the Laplacian and the structure of the

L
2

{L^2}

-cohomology are studied on a singular algebraic curve with its singularity removed, which is hence an incomplete Kähler manifold.
Simply cyclic homogeneous non-tree-like curves decompose to solenoids
Authors: Rogers James T.
Published: June 2010

It is proved that if a one-dimensional, cyclic, homogeneous continuum

X
X

is the inverse limit of graphs each of which contains only one cycle, then

X
X

is a solenoid or

X
X

admits a decomposition into mutually homeomorphic, homogeneous, tree-like continua with quotient space a solenoid.
Sur la théorie spectrale locale et limite des nilpotents
Authors: Mbekhta Mostafa
Published: June 2010

Résumé. Dans ce travail, nous donnons une condition nécessaire et suffisante pour qu’un opérateur fermé dans un espace de Banach admette la propriété de l’extension unique. Pour cela nous introduisons la notions du coeur analytique d’un opérateur. Nous établissons plusieurs propriétés des opérateurs dont le coeur analytique est réduit à zéro (l’ensemble de ces opérateurs est noté

K

A

0

(
H
)

\mathcal {K}{\mathcal {A}_0}(H)

où

H
H

est un Hilbert), notamment nous montrons que si

T
∈

K

A

0

(
H
)

T \in \mathcal {K}{\mathcal {A}_0}(H)

alors

0
∈
σ
(

T

|

M

)

0 \in \sigma ({T_{|M}})

pour tout sous-espace invariant

M
≠
{
0
}

M \ne \{ 0\}

. En particulier

σ
p

(
T
)
⊆
{
0
}
,
σ
(
T
)

{\sigma _p}(T) \subseteq \{ 0\} ,\sigma (T)

est connexe et contient zéro et si

T
T

est décomposable alors

T
T

est quasinilpotent. Nous montrons aussi que l’ensemble

(

K

A

(
H
)
)

(\mathcal {K}\mathcal {A}(H))

des opérateurs

T
T

tels que

T
,

T
∗

∈

K

A

0

(
H
)

T,{T^ * } \in \mathcal {K}{\mathcal {A}_0}(H)

est inclus strictement dans l’adhérence des nilpotents

(

N

¯

)

(\overline {\mathcal {N}})

. Et pour tout

T
∈

N

¯

T \in \overline {\mathcal {N}}

et

ε
>
0

\varepsilon > 0

, il existe

K
ε

{K_\varepsilon }

opérateur compact tel que

‖

K
ε

‖

>
ε

\left \| {{K_\varepsilon }} \right \| > \varepsilon

et

T
+

K
ε

∈

K

A

(
H
)

T + {K_\varepsilon } \in \mathcal {K}\mathcal {A}(H)

.
Liapanov’s and related theorems
Authors: Wulbert Daniel
Published: June 2010

Four theorems related to vector-valued measures, are the consequence of a single approach.
On the theorem of Frullani
Authors: Arias-de-Reyna Juan
Published: June 2010

We prove that, for every function

f
:

R

+

→

C

f:{{\mathbf {R}}^ + } \to {\text {C}}

such that

(
f
(
a
x
)
−
f
(
b
x
)
)

/

x

(f(ax) - f(bx))/x

is Denjoy-Perron integrable on

[
0
,
+
∞
)

[0, + \infty )

for every pair of positive real numbers

a
,
b

a,b

, there exists a constant

A
A

(depending only on the values of

f
(
t
)

f(t)

in the neighborhood of 0 and

+
∞

+ \infty

) such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript normal infinity Baseline StartFraction f left-parenthesis a x right-parenthesis minus f left-parenthesis b x right-parenthesis Over x EndFraction d x equals upper A l o g StartFraction a Over b EndFraction period">

∫
0
∞

f
(
a
x
)
−
f
(
b
x
)

x

d
x
=
A
l
o
g

a
b

.

\int _0^\infty {\frac {{f(ax) - f(bx)}}{x}} dx = Alog \frac {a}{b}.

\]
To prove this assertion, we identify a Denjoy-Perron integrable function

f
:

R

→

C

f:{\mathbf {R}} \to {\text {C}}

with a distribution. In this way, we obtain the main result of this paper: The value at 0 (in Lojasiewicz sense) of the Fourier transform of the distribution

f
f

is the Denjoy-Perron integral of

f
f

. Assuming the Continuum Hypothesis, we construct an example of a non-Lebesgue measurable function that satisfies the hypotheses of the first theorem.
Principal eigenvalues for problems with indefinite weight function on 𝑅ⁿ
Authors: Brown K. J. , Cosner C. , Fleckinger J.
Published: June 2010

We investigate the existence of positive principal eigenvalues of the problem

−
Δ
u
(
x
)
=
λ
g
(
x
)
u

- \Delta u(x) = \lambda g(x)u

for

x
∈

R
n

;
u
(
x
)
→
0

x \in {R^n};u(x) \to 0

as

x
→
∞

x \to \infty

where the weight function

g
g

changes sign on

R
n

{R^n}

. It is proved that such eigenvalues exist if

g
g

is negative and bounded away from 0 at

∞
\infty

or if

n
≥
3

n \geq 3

and

|

g
(
x
)

|

|g(x)|

is sufficiently small at

∞
\infty

but do not exist if

n
=
1

or

2

n = 1\,{\text {or}}\,2

and

∫

R
n

g
(
x
)
d
x
>
0

\int _{{R^n}} {g(x)dx > 0}

.
Decreasing chains without lower bounds in the Rudin-Frolík order
Authors: Butkovičová Eva
Published: June 2010

We prove that for every uncountable cardinal less than continuum there exists a chain in the Rudin-Frolík order of

β
ω
−
ω

\beta \omega - \omega

isomorphic to the inverse order of this cardinal and without a lower bound.
Reflexivity of pairs of shifts
Authors: Ptak Marek
Published: June 2010

In the present paper the reflexivity of a WOT-closed algebra generated by certain pairs of commuting shifts, which are not necessarily doubly commuting, is proved.
Some rigidity phenomena for Einstein metrics
Authors: Shen Zhong Min
Published: June 2010

In this note we study the following problem: When must a complete Einstein metric

g
g

on an

n
n

-manifold with

Ric = 

(
n
−
1
)
λ
g

{\text {Ric = }}(n - 1)\lambda g

be a constant curvature metric of sectional curvature

λ
\lambda

?
Commuting and centralizing mappings in prime rings
Authors: Vukman J.
Published: June 2010

Let

R
R

be a ring. A mapping

F
:
R
→
R

F:R \to R

is said to be commuting on

R
R

if

[
F
(
x
)
,
x
]
=
0

[F(x),x] = 0

holds for all

x
∈
R

x \in R

. The main purpose of this paper is to prove the following result, which generalizes a classical result of E. Posner: Let

R
R

be a prime ring of characteristic not two. Suppose there exists a nonzero derivation

D
:
R
→
R

D:R \to R

, such that the mapping

x
↦
[
D
(
x
)
,
x
]

x \mapsto [D(x),x]

is commuting on

R
R

. In this case

R
R

is commutative.
Tychonoff reflection in products and the 𝜔-topology on function spaces
Authors: Watson Stephen
Published: June 2010

We show that if

X
X

is a topological space such that

C
R
(
X
)

CR(X)

, the topology on

X
X

generated by the cozero sets, is not locally compact, then there is a regular space

Y
Y

such that

C
R
(
X
×
Y
)
≠
C
R
(
X
)
×
C
R
(
Y
)

CR(X \times Y) \ne CR(X) \times CR(Y)

. We use the

ω
\omega

-topology on the space of continuous functions

C

(

X
,
Y

)

C\left ( {X,Y} \right )

(where

ω
\omega

is an open cover of

X
X

) which was defined by Arens and Dugundji in 1950.
Strong preinjective partitions and representation type of Artinian rings
Authors: Zimmermann-Huisgen Birge
Published: June 2010

It is shown that for every ring of left pure global dimension zero (i.e., for every ring all of whose left modules are direct sums of finitely generated modules), the finitely generated left modules can be grouped to a unique "strong preinjective partition" while the finitely presented right modules possess a "strong preprojective partition"; these strong partitions are upgraded versions of the partitions introduced by Auslander and Smaløfor Artin algebras. One direct consequence is that a ring of left pure global dimension zero has finite representation type if and only if there exist sufficiently many almost split maps among its finitely generated left modules. This provides a very elementary proof for Auslander’s theorem saying that for Artin algebras vanishing of the left pure global dimension is equivalent to finiteness of the representation type.
Commutators and Π-subgroups
Authors: Brandl Rolf
Published: June 2010

The following statement is shown to be true for a large class of groups

G
G

: if every commutator in

G
G

is a

∏
\prod

-element, then

G
′

G’

is a

∏
\prod

-group
Cancellation counterexamples in Krull dimension 1
Authors: Guralnick Robert M. , Levy Lawrence S. , Warfield Robert B.
Published: June 2010

We show that direct-sum cancellation can fail for modules in a single genus, over commutative noetherian rings of dimension 1. The example provides an application of noncommutative methods (including "wild representation type") to commutative ring theory.
The converse of the Minkowski’s inequality theorem and its generalization
Authors: Matkowski Janusz
Published: June 2010

Let (

Ω
\Omega

,

Σ
\Sigma

,

μ
\mu

) be a measure space with two sets

A
,
B
∈
Σ

A,B \in \Sigma

such that

0
>
μ
(
A
)
>
1
>
μ
(
B
)
>
∞

0 > \mu (A) > 1 > \mu (B) > \infty

, and let

φ
:

R

+

→

R

+

\varphi :{{\mathbf {R}}_ + } \to {{\mathbf {R}}_ + }

be bijective and

φ

−
1

{\varphi ^{ - 1}}

continuous at 0. We prove that if for all

μ
\mu

-integrable step functions

x
,
y
:
Ω
→

R

x,y:\Omega \to {\mathbf {R}}

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Superscript negative 1 Baseline left-parenthesis integral Underscript normal upper Omega Endscripts phi ring StartAbsoluteValue x plus y EndAbsoluteValue d mu right-parenthesis less-than-or-equal-to phi Superscript negative 1 Baseline left-parenthesis integral Underscript normal upper Omega Endscripts phi ring StartAbsoluteValue x EndAbsoluteValue d mu right-parenthesis plus phi Superscript negative 1 Baseline left-parenthesis integral Underscript normal upper Omega Endscripts phi ring StartAbsoluteValue y EndAbsoluteValue d mu right-parenthesis">

φ

−
1

(

∫
Ω

φ
∘

|

x
+
y

|

d
μ

)

≤

φ

−
1

(

∫
Ω

φ
∘

|

x

|

d
μ

)

+

φ

−
1

(

∫
Ω

φ
∘

|

y

|

d
μ

)

{\varphi ^{ - 1}}\left ( {\int _\Omega {\varphi \circ |x + y|d\mu } } \right ) \leq {\varphi ^{ - 1}}\left ( {\int _\Omega {\varphi \circ |x|d\mu } } \right ) + {\varphi ^{ - 1}}\left ( {\int _\Omega {\varphi \circ |y|d\mu } } \right )

\]
then

φ
(
t
)
=
φ
(
1
)

t
p

\varphi (t) = \varphi (1){t^p}

for some

p
≥
1

p \geq 1

. In the case of normalized measure we prove a generalization of Minkowski’s inequality theorem. The suitable results for the reversed inequality are also presented.
Pseudo-orbits and topological entropy
Authors: Barge Marcy , Swanson Richard
Published: June 2010

The topological entropy of a map of a compact metric space is equal to the exponential growth rate of the number of separated periodic pseudo-orbits of period

n
n

as

n
n

tends to infinity.
Mosco convergence and reflexivity
Authors: Beer Gerald , Borwein Jonathan M.
Published: June 2010

In this note we aim to show conclusively that Mosco convergence of convex sets and functions and the associated Mosco topology

τ
M

{\tau _M}

are useful notions only in the reflexive setting. Specifically, we prove that each of the following conditions is necessary and sufficient for a Banach space

X
X

to be reflexive: (1) whenever

A
,

A
1

,

A
2

,

A
3

,
…

A,{A_1},{A_2},{A_3}, \ldots

are nonempty closed convex subsets of

X
X

with

A
=

τ
M

−
lim

A
n

A = {\tau _M} - \lim {A_n}

, then

A
∘

=

τ
M

−
lim

A
n
∘

{A^ \circ } = {\tau _M} - \lim A_n^ \circ

; (2)

τ
M

{\tau _M}

is a Hausdorff topology on the nonempty closed convex subsets of

X
X

; (3) the arg min multifunction

f
⇉
{
x
∈
X
:
f
(
x
)
=
inf

X

f
}

f \rightrightarrows \{ x \in X:f(x) = \inf {}_Xf\}

on the proper lower semicontinuous convex functions on

X
X

, equipped with

τ
M

{\tau _M}

, has closed graph.
A stable/unstable “manifold” theorem for area preserving homeomorphisms of two manifolds
Authors: Baldwin Stewart , Slaminka Edward E.
Published: June 2010

The stable/unstable manifold theorem for hyperbolic diffeomorphisms has proven to be of extreme importance in differentiable dynamics. We prove a stable/unstable "manifold" theorem for area preserving homeomorphisms of orientable two manifolds having isolated fixed points of index less than 1. The proof relies upon the concept of free modification which was first developed by Morton Brown for homeomorphisms of two manifolds and later extended by Pelikan and Slaminka for area preserving homeomorphisms of two manifolds.
A characterization of certain measures using quasiconformal mappings
Authors: Nolder Craig A.
Published: June 2010

Suppose that

μ
\mu

is a finite positive measure on the unit disk. Carleson showed that the

L
2

(
μ
)

{L^2}(\mu )

-norm is bounded by the

H
2

{H^2}

-norm uniformly over the space of analytic functions on the unit disk if and only if

μ
\mu

is a Carleson measure. Analogues of this result exist for Bergmann spaces of analytic functions in the disk and in the unit ball of

C
n

{C^n}

. We prove here real variable analogues of certain Bergmann space results using quasiconformal and quasiregular mappings.
Geometric finiteness of certain Kleinian groups
Authors: Scott G. P. , Swarup G. A.
Published: June 2010

If

G
G

is a discrete subgroup of

PSL
⁡

(

2
;

C

)

\operatorname {PSL} \left ( {2;{\mathbf {C}}} \right )

representing a fibred

3
3

-manifold and

H
H

the subgroup of

G
G

corresponding to the fibre, we show that any finitely generated subgroup of infinite index in

H
H

is geometrically finite.
Leading terms in the heat invariants
Authors: Branson Thomas P. , Gilkey Peter B. , Ørsted Bent
Published: June 2010

Let

D
D

be a second-order differential operator with leading symbol given by the metric tensor on a compact Riemannian manifold. The asymptotics of the heat kernel based on

D
D

are given by homogeneous, invariant, local formulas. Within the set of allowable expressions of a given homogeneity there is a filtration by degree, in which elements of the smallest class have the highest degree. Modulo quadratic terms, the linear terms integrate to zero, and thus do not contribute to the asymptotics of the

L
2

{L^2}

trace of the heat operator; that is, to the asymptotics of the spectrum. We give relations between the linear and quadratic terms, and use these to compute the heat invariants modulo cubic terms. In the case of the scalar Laplacian, qualitative aspects of this formula have been crucial in the work of Osgood, Phillips, and Sarnak and of Brooks, Chang, Perry, and Yang on compactness problems for isospectral sets of metrics modulo gauge equivalence in dimensions 2 and 3.
The Haagerup type cross norm on 𝐶*-algebras
Authors: Itoh Takashi
Published: June 2010

Several cross norms between the projective

C
∗

{C^*}

-norm and the projective cross norm are introduced. Let

A
A

and

B
B

be

C
∗

{C^*}

-algebras. It is shown that

‖

‖

h

{\left \| {} \right \|_h}

is equivalent to

‖

‖

γ

{\left \| {} \right \|_\gamma }

on

A
⊗
B

A \otimes B

if and only if

A
A

or

B
B

is subhomogeneous.
Positive maps on self-dual cones
Authors: Iochum B.
Published: June 2010

Different properties of maps on a self-dual cone which preserve the order and the orthogonal decomposition are studied: polar decomposition, link with the order isomorphisms, normality, extremality within the order preserving maps. A characterization of derivation preserving maps is given. The results generalize a previous approach by S. Yamamuro et al.
Multiplier groups of planar difference sets and a theorem of Kantor
Authors: Ho Chat Yin , Pott Alexander
Published: June 2010

A recent result of W. Kantor followed by a work of W. Feit has rekindled interest in the longstanding conjecture of finite cyclic planes. In this paper we prove that the order of the multiplier group equals the odd part of the order of the automorphism group of a Singer group if and only if the order of the plane is 2, 3, or 8. This yields another proof for Feit’s result mentioned above.
An extreme absolutely continuous 𝑅𝑃-measure
Authors: McDonald John N.
Published: June 2010

We construct a measure on the torus

T
2

{T^2}

which is an extreme element of the set of RP-probability measures and is absolutely continuous with respect to Haar measure on

T
2

{T^2}

.
Unique subwords in nonperiodic words
Authors: Weinbaum C. M.
Published: June 2010

Let

A
,
D

A,D

be words over some alphabet.

D
D

has position

p
p

in the cyclic word

A
A

if the cyclic permutation of

A
A

which begins with the

p
p

th letter of

A
A

has an initial subword equal to

D
D

. It is proved that every nonperiodic word

A
A

of length

>
1

> 1

has a cyclic permutation which is a product BC for some nonempty subwords

B
,
C

B,C

having unique positions in the cyclic word

A
A

.
Evaluation of the infinite series ∑^{∞}_{𝑛=1,(\frac𝑛𝑝)=1}(\frac𝑛𝑝)₄𝑛⁻¹
Authors: Hardy Kenneth , Williams Kenneth S.
Published: June 2010

Let

p
p

be a prime

≡
1
(
mod
4
)

\equiv 1(\bmod 4)

. The value of the infinite series
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript Subscript left-parenthesis StartFraction n Over p EndFraction right-parenthesis equals 1 Superscript n equals 1 Baseline Overscript normal infinity Endscripts left-parenthesis StartFraction n Over p EndFraction right-parenthesis Subscript 4 Baseline n Superscript negative 1">

∑

(

n
p

)
=
1

n
=
1

∞

(

n
p

)

4

n

−
1

\sum \nolimits _{_{(\tfrac {n}{p}) = 1}^{n = 1}}^\infty {{{(\tfrac {n}{p})}_4}{n^{ - 1}}}

is given in finite terms.
On modular extensions
Authors: Sato Shizuka
Published: June 2010

M. E. Sweedler has proved that modular extensions of fields are characterized by a tensor product of primitive elements, and also by the equivalent condition that the ground field is the fixed field under higher derivations. In this paper we shall give an extension of his work about modular field extensions to modular integral domain extensions. Moreover, we shall prove that a modular extension is an extension that the derivation algebra is generated by components of higher derivations under some conditions. For example, in a finite extension of a field, a modular extension is characterized by the fact that the derivation algebra is generated by components of higher derivations.
A construction of a Dowker space
Authors: Watson Stephen
Published: June 2010

We devise a method of building Dowker spaces from copies of Bing’s example

G
G

. We show that, if there is a compact cardinal, then there is a

σ
\sigma

-discrete hereditarily normal Dowker space.
A note on Weinstein’s conjecture
Authors: Banyaga Augustin
Published: June 2010

We prove that the contact foliation of a compact contact manifold

(

M
,
α

)

\left ( {M,\alpha } \right )

has at least one compact leaf in the following two cases: (i)

α
\alpha

is a

K
K

-contact form and

M
M

is simply connected, (ii)

α
\alpha

is

C
2

{C^2}

-close to a regular contact form. This solves the Weinstein conjecture in those particular cases.
A closure theorem for 𝜎-compact subgroups of locally compact topological groups
Authors: Wu Ta Sun
Published: June 2010

We describe the closure of certain subgroups of a locally compact group.
The Poincaré conjecture is true in the product of any graph with a disk
Authors: Gillman David
Published: June 2010

We prove that the only compact

3
3

-manifold-with-boundary which has trivial rational homology, and which embeds in the product of a graph with a disk, is the

3
3

-ball. This implies that no punctured lens space embeds in the product of a graph with a disk. It also implies our title. The proof relies on a general position argument which enables us to perform surgery.
A brief proof of Jacobian hypothesis implies flatness
Authors: Miyanishi Masayoshi , Robbiano Lorenzo , Wang Stuart Sui Sheng
Published: June 2010

Let

S
S

be a polynomial ring in

n
n

variables over a field, and let

R
R

be the subring generated by

n
n

polynomials. We give a short proof of the fact that if the Jacobian determinant of these

n
n

polynomials is 1, then

S
S

is a flat

R
R

-module.
Oscillation in nonautonomous scalar differential equations with deviating arguments
Authors: Cheng Yuan Ji
Published: June 2010

In this paper, we study the oscillation in the nonautonomous scalar differential equations with deviating arguments and get some oscillation criteria. As an application of the results, we prove a conjecture posed by Brian R. Hunt and James A. Yorke in [1].
Notes on renewal systems
Authors: Williams Susan
Published: June 2010

A renewal system is a symbolic dynamical system generated by free concatenations of a finite set of words. Renewal systems are sofic, but not every sofic shift is topologically conjugate to a renewal system.
Subordination and 𝐻^{𝑝} functions
Authors: Younis Rahman
Published: June 2010

Let

ϕ
\phi

and

W
W

be inner functions with

ϕ
(
0
)
=
W
(
0
)
=
0

\phi (0) = W(0) = 0

. It is shown that if

F
F

is an exposed point of the unit ball of

H
1

{H^1}

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis upper W left-parenthesis e Superscript i t Baseline right-parenthesis right-parenthesis slash upper F left-parenthesis phi left-parenthesis e Superscript i t Baseline right-parenthesis right-parenthesis greater-than 0">

F
(
W
(

e

i
t

)
)

/

F
(
ϕ
(

e

i
t

)
)
>
0

F(W({e^{it}}))/F(\phi ({e^{it}})) > 0

\]
almost everywhere, then

F
∘
W
=
F
∘
ϕ

F \circ W = F \circ \phi

. If

f
=
z
F

f = zF

such that

F
F

and

1

/

F

1/F

are in

H
r

{H^r}

and

H
s

{H^s}

, respectively, where

1

/

r
+
1

/

s
≤
2

1/r + 1/s \leq 2

and

ϕ
\phi

is a finite Blaschke product, then a necessary and sufficient condition is provided in order for

f
(
W
(

e

i
t

)
)

/

f
(
ϕ
(

e

i
t

)
)

f(W({e^{it}}))/f(\phi ({e^{it}}))

to be positive almost everywhere.
Integral representations of univalent functions and singular measures
Authors: Bass Robert J.
Published: June 2010

In [5], T. MacGregor showed that not every univalent function has a proposed integral representation with respect to Borel measures on the unit circle. In this paper, we study the decomposition of measures which do give rise to univalent functions. The main results show that no measure with a continuous singular component can ever be associated with a univalent function.
A general multiplier theorem
Authors: Meda Stefano
Published: June 2010

We prove a "multiplier" result for functions of the infinitesimal generator

L

\mathcal {L}

of a symmetric semigroup, which generalizes some previous results by E. M. Stein and M. G. Cowling. As an application, we develop a functional calculus for

L

\mathcal {L}

in the case when the

L
p

{L^p}

-operator norm of

L

i
u

{\mathcal {L}^{iu}}

has polynomial growth at infinity. In particular, we prove a "multiplier" result of Marcinkiewicz type for functions of

L

\mathcal {L}

.
Syzygy pairs in a monomial algebra
Authors: Igusa Kiyoshi , Zacharia Dan
Published: June 2010

In this paper we construct the set of "syzygy pairs" of a finite-dimensional monomial algebra and use it to prove the following theorem.
Homological dimensions in a Morita context with applications to subidealizers and fixed rings
Authors: Loustaunau Philippe , Shapiro Jay
Published: June 2010

Given a Morita context

(
R
,
S
,
V
,
W
,
θ
,
ψ
)

(R,S,V,W,\theta ,\psi )

, we investigate the relationship between the various homological dimensions of the rings

R
R

and

S
S

. We then apply these results to two particular examples: subidealizers and fixed rings.
On the distance between normal matrices
Authors: Omladič Matjaž , Šemrl Peter
Published: June 2010

The least upper bound for the norm distance between two normal matrices is given in terms of their eigenvalues exclusively, thus solving a problem which appears to be long open.
Some questions of Erdős and Graham on numbers of the form ∑𝑔_{𝑛}/2^{𝑔_{𝑛}}
Authors: Borwein Peter , Loring Terry A.
Published: June 2010

Erdös in 1975 and Erdös and Graham in 1980 raised several questions concerning representing numbers as series of the form

Σ

n
=
1

∞

g
n

/

2

g
n

\Sigma _{n = 1}^\infty \;{g_n}/{2^{{g_n}}}

. For example, does the equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction n Over 2 Superscript n Baseline EndFraction equals sigma-summation Underscript k equals 1 Overscript upper T Endscripts StartFraction g Subscript k Baseline Over 2 Superscript g Super Subscript k Superscript Baseline EndFraction comma upper T greater-than 1 comma">

n

2
n

=

∑

k
=
1

T

g
k

2

g
k

,

T
>
1

,

\frac {n}{{{2^n}}} = \sum \limits _{k = 1}^T {\frac {{{g_k}}}{{{2^{{g_k}}}}},\quad T > 1} ,

\]
have a solution for infinitely many n ? The answer to this question is affirmative; in fact, we conjecture that the above equation is solvable for every n. This conjecture is based on a more general conjecture, namely that the algorithm
\[

a

n
+
1

=
2
(

a
n

mod
n
)

{a_{n + 1}} = 2({a_n}\bmod n)

\]
with initial condition

a
m

∈

Z

{a_m} \in {\mathbf {Z}}

always eventually terminates at zero. This, in turn, is based on an examination of how the "greedy algorithm" can be used to represent numbers in the form

∑

g
n

/

2

g
n

\sum {{g_n}/{2^{{g_n}}}}

. The analysis of this, reformulated as a "base change" algorithm, proves surprising. Some numbers have a unique representation, as above, others have uncountably many. Also, from this analysis we observe that

∑

g
n

/

2

g
n

\sum {{g_n}/{2^{{g_n}}}}

is irrational if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit sup Underscript n Endscripts left-parenthesis left-parenthesis g Subscript n plus 1 Baseline minus g Subscript n Baseline right-parenthesis slash log left-parenthesis g Subscript n plus 1 Baseline right-parenthesis right-parenthesis equals normal infinity">

lim

sup
n

(
(

g

n
+
1

−

g
n

)

/

log
⁡
(

g

n
+
1

)
)
=
∞

\lim {\sup _n}(({g_{n + 1}} - {g_n})/\log ({g_{n + 1}})) = \infty

and conjecture that this is best possible.
An algorithm based on the FFT for a generalized Chebyshev interpolation
Authors: Hasegawa Takemitsu , Torii Tatsuo , Sugiura Hiroshi
Published: June 2010

An algorithm for a generalized Chebyshev interpolation procedure, increasing the number of sample points more moderately than doubling, is presented. The FFT for a real sequence is incorporated into the algorithm to enhance its efficiency. Numerical comparison with other existing algorithms is given.
On the stability and convergence of higher-order mixed finite element methods for second-order elliptic problems
Authors: Suri Manil
Published: June 2010

We investigate the use of higher-order mixed methods for second-order elliptic problems by establishing refined stability and convergence estimates which take into account both the mesh size h and polynomial degree p. Our estimates yield asymptotic convergence rates for the p- and h — p-versions of the finite element method. They also describe more accurately than previously proved estimates the increased rate of convergence expected when the h-version is used with higher-order polynomials. For our analysis, we choose the Raviart-Thomas and the Brezzi-Douglas-Marini elements and establish optimal rates of convergence in both h and p (up to arbitrary

ε
>
0

\varepsilon > 0

).
The genus zeta function of hereditary orders in central simple algebras over global fields
Authors: Denert M.
Published: June 2010

Louis Solomon introduced the notion of a zeta function

ζ
Θ

(
s
)

{\zeta _\Theta }(s)

of an order

Θ
\Theta

in a finite-dimensional central simple K-algebra A, with K a number field or its completion

K
P

{K_P}

(P a non-Archimedean prime in K). In several papers, C. J. Bushnell and I. Reiner have developed the theory of zeta functions and they gave explicit formulae in some special cases. One important property of these zeta functions is the Euler product, which implies that in order to calculate

ζ
Θ

(
s
)

{\zeta _\Theta }(s)

, it is sufficient to consider the zeta function of local orders

Θ
P

{\Theta _P}

. However, since these local orders

Θ
P

{\Theta _P}

are in general not principal ideal domains, their zeta function is a finite sum of so-called ’partial zeta functions’. The most complicated term is the ’genus zeta function’,

Z

Θ
P

(
s
)

{Z_{{\Theta _P}}}(s)

, which is related to the free

Θ
P

{\Theta _P}

-ideals. I. Reiner and C. J. Bushnell calculated the genus zeta function for hereditary orders in quaternion algebras (i.e.,

[
A
:
K
]
=
4

[A:K] = 4

). The authors mention the general case but they remark that the calculations are cumbersome. In this paper we derive an explicit method to calculate the genus zeta function

Z

Θ
P

(
s
)

{Z_{{\Theta _P}}}(s)

of any local hereditary order

Θ
P

{\Theta _P}

in a central simple algebra over a local field. We obtain

Z

Θ
P

(
s
)

{Z_{{\Theta _P}}}(s)

as a finite sum of explicit terms which can be calculated with a computer. We make some remarks on the programming of the formula and give a short list of examples. The genus zeta function of the minimal hereditary orders (corresponding to the partition (1, 1, ... , 1) of n) seems to have a surprising property. In all examples, the nominator of this zeta function is a generating function for the q-Eulerian polynomials. We conclude with some remarks on a conjectured identity.
New algorithms for finding irreducible polynomials over finite fields
Authors: Shoup Victor
Published: June 2010

We present a new algorithm for finding an irreducible polynomial of specified degree over a finite field. Our algorithm is deterministic, and it runs in polynomial time for fields of small characteristic. We in fact prove the stronger result that the problem of finding irreducible polynomials of specified degree over a finite field is deterministic polynomial-time reducible to the problem of factoring polynomials over the prime field.
A note on discrete solutions of the Plateau problem
Authors: Tsuchiya Takuya
Published: June 2010

In this paper we prove theorems for convergence of discrete solutions of the Plateau problem under the assumption that the contour is rectifiable.
On the 𝑚𝑜𝑑2 reciprocation of infinite modular-part products and the parity of certain partition functions
Authors: Blecksmith Richard , Brillhart John , Gerst Irving
Published: June 2010

An infinite, modular-part (MP) product is defined to be a product of the form

Π

n
∈
S

(
1
−

x
n

)

{\Pi _{n \in S}}(1 - {x^n})

, where

S
=
{

n
∈

Z

+

:
n
≡

r
1

,
…
,

r
t

(
mod

m
)

}

S = \{ {n \in {{\mathbf {Z}}^ + }:n \equiv {r_1}, \ldots ,{r_t}\;\pmod m} \}

. Some products of this kind have a

mod

2

\bmod \, 2

reciprocal that is also an MP product, while others do not. A complete method is first developed which determines if a given MP product has an MP reciprocal modulo 2 and finds it if it does. Next, a graph-theoretic interpretation of this method is made from which a streamlined algorithm is derived for deciding whether the given MP product is such a reciprocal. This algorithm is then applied to the single-variable Jacobi triple product and the quintuple product to determine the cases when these products have an MP reciprocal

(
mod

2
)

\pmod 2

. When this occurs—and this occurs in infinitely many cases—the parity of the associated partition function can readily be found. A discussion is also made of the probability that a given MP product with modulus m has an MP reciprocal

(
mod

2
)

\pmod 2

.
Approximation by mediants
Authors: Bosma Wieb
Published: June 2010

The distribution is determined of some sequences that measure how well a number is approximated by its mediants (or intermediate continued fraction convergents). The connection with a theorem of Fatou, as well as a new proof of this, is given.
On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws
Authors: Johnson Claes , Szepessy Anders , Hansbo Peter
Published: June 2010

We extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock-capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the mesh size. With this term present, we prove a maximum norm bound for finite element solutions of Burgers’ equation and thus complete an earlier convergence proof for this equation. We further prove, using entropy variables, that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality associated with the entropy variables. Results of some numerical experiments for the time-dependent compressible Euler equations in two dimensions are also reported.
A computer search of rank-2 lattice rules for multidimensional quadrature
Authors: Sloan Ian H. , Walsh Linda
Published: June 2010

For certain lattice rules of ’rank 2’ it has been shown, in a recent paper, that a unique representation exists in a form suitable for computer evaluation. The present paper describes computer searches of such rules, reports results and identifies rules that appear promising for the numerical evaluation of practical multidimensional integrals.
Parallel multilevel preconditioners
Authors: Bramble James H. , Pasciak Joseph E. , Xu Jinchao
Published: June 2010

In this paper, we provide techniques for the development and analysis of parallel multilevel preconditioners for the discrete systems which arise in numerical approximation of symmetric elliptic boundary value problems. These preconditioners are defined as a sum of independent operators on a sequence of nested subspaces of the full approximation space. On a parallel computer, the evaluation of these operators and hence of the preconditioner on a given function can be computed concurrently. We shall study this new technique for developing preconditioners first in an abstract setting, next by considering applications to second-order elliptic problems, and finally by providing numerically computed condition numbers for the resulting preconditioned systems. The abstract theory gives estimates on the condition number in terms of three assumptions. These assumptions can be verified for quasi-uniform as well as refined meshes in any number of dimensions. Numerical results for the condition number of the preconditioned systems are provided for the new algorithms and compared with other well-known multilevel approaches.
A direct boundary element method for Signorini problems
Authors: Han Hou De
Published: June 2010

In this paper, a Signorini problem is reduced to a variational inequality on the boundary, and a direct boundary element method is presented for its solution. Furthermore, error estimates for the approximate solutions of Signorini problems are given. In addition, we show that the Signorini problem may be formulated as a saddle-point problem on the boundary.
Linear elliptic difference inequalities with random coefficients
Authors: Kuo Hung Ju , Trudinger Neil S.
Published: June 2010

We prove various pointwise estimates for solutions of linear elliptic difference inequalities with random coefficients. These estimates include discrete versions of the maximum principle of Aleksandrov and Harnack inequalities and Hölder estimates of Krylov and Safonov for elliptic differential operators with bounded coefficients.
Quadratic polynomials which have a high density of prime values
Authors: Fung G. W. , Williams H. C.
Published: June 2010

The University of Manitoba Sieve Unit is used to find several values of

A
(
>
0
)

A ( > 0)

such that the quadratic polynomial

x
2

+
x
+
A

{x^2} + x + A

will have a large asymptotic density of prime values. The Hardy-Littlewood constants which characterize this density are also evaluated.
Sieved partition functions and 𝑞-binomial coefficients
Authors: Garvan Frank , Stanton Dennis
Published: June 2010

The q-binomial coefficient is a polynomial in q. Given an integer t and a residue class r modulo t, a sieved q-binomial coefficient is the sum of those terms whose exponents are congruent to r modulo t. In this paper explicit polynomial identities in

q
t

{q^t}

are given for sieved q-binomial coefficients. As a limiting case, generating functions for the sieved partition function are found as multidimensional theta functions. A striking corollary of this representation is the proof of Ramanujan’s congruences

mod

5
,
7

\bmod \, 5, 7

, and 11 by exhibiting symmetry groups of orders 5, 7, and 11 of explicit quadratic forms. We also verify the Subbarao conjecture for

t
=
3

t = 3

,

t
=
5

t = 5

, and

t
=
10

t = 10

.
The Carmichael numbers to 10¹²
Authors: Jaeschke Gerhard
Published: June 2010

An algorithm is presented which determines all Carmichael numbers up to a given limit having a prescribed number of factors. An overview over all Carmichael numbers less than

10

12

{10^{12}}

is given.
Lower bounds for the discrepancy of inversive congruential pseudorandom numbers
Authors: Niederreiter Harald
Published: June 2010

The inversive congruential method is a uniform pseudorandom number generator which was introduced recently. For a prime modulus p the discrepancy

D
p

(
k
)

D_p^{(k)}

of k-tuples of successive pseudorandom numbers generated by this method determines the statistical independence properties of these pseudorandom numbers. It was shown earlier by the author that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript p Superscript left-parenthesis k right-parenthesis Baseline equals upper O left-parenthesis p Superscript negative 1 slash 2 Baseline left-parenthesis log p right-parenthesis Superscript k Baseline right-parenthesis for 2 less-than-or-equal-to k greater-than p period">

D
p

(
k
)

=
O
(

p

−
1

/

2

(
log
⁡
p

)
k

)

for

2
≤
k
>
p
.

D_p^{(k)} = O({p^{ - 1/2}}{(\log p)^k})\quad {\text {for}}\;2 \leq k > p.

\]
Here it is proved that this bound is essentially best possible. In fact, for a positive proportion of the admissible parameters in the inversive congruential method the discrepancy

D
p

(
k
)

D_p^{(k)}

is at least of the order of magnitude

p

−
1

/

2

{p^{ - 1/2}}

for all

k
≥
2

k \geq 2

.
Boundary integral equation methods for solving Laplace’s equation with nonlinear boundary conditions: the smooth boundary case
Authors: Atkinson Kendall E. , Chandler Graeme
Published: June 2010

A nonlinear boundary value problem for Laplace’s equation is solved numerically by using a reformulation as a nonlinear boundary integral equation. Two numerical methods are proposed and analyzed for discretizing the integral equation, both using product integration to approximate the singular integrals in the equation. The first method uses the product Simpson’s rule, and the second is based on trigonometric interpolation. Iterative methods (including two-grid methods) for solving the resulting nonlinear systems are also discussed extensively. Numerical examples are included.
Computing irreducible representations of finite groups
Authors: Babai László , Rónyai Lajos
Published: June 2010

We consider the bit-complexity of the problem stated in the title. Exact computations in algebraic number fields are performed symbolically. We present a polynomial-time algorithm to find a complete set of nonequivalent irreducible representations over the field of complex numbers of a finite group given by its multiplication table. In particular, it follows that some representative of each equivalence class of irreducible representations admits a polynomial-size description. We also consider the problem of decomposing a given representation

V

\mathcal {V}

of the finite group G over an algebraic number field F into absolutely irreducible constituents. We are able to do this in deterministic polynomial time if

V

\mathcal {V}

is given by the list of matrices

{

V

(
g
)
;
g
∈
G
}

\{ \mathcal {V}(g);g \in G\}

; and in randomized (Las Vegas) polynomial time under the more concise input

{

V

(
g
)
;
g
∈
S
}

\{ \mathcal {V}(g);g \in S\}

, where S is a set of generators of G.
Boolean methods for double integration
Authors: Delvos Franz-J.
Published: June 2010

This paper is concerned with numerical integration of continuous functions over the unit square

U
2

{U^2}

. The concept of the rth-order blending rectangle rule is introduced by carrying over the idea from Boolean interpolation. Error bounds are developed, and it is shown that rth-order blending rectangle rules are comparable with number-theoretic cubature rules. Moreover, rthorder blending midpoint rules are defined and compared with the rth-order blending rectangle rules.
Degree of adaptive approximation
Authors: DeVore Ronald A. , Yu Xiang Ming
Published: June 2010

We obtain various estimates for the error in adaptive approximation and also establish a relationship between adaptive approximation and free-knot spline approximation.
On the numerical condition of Bernstein-Bézier subdivision processes
Authors: Farouki R. T. , Neff C. A.
Published: June 2010

The linear map M that takes the Bernstein coefficients of a polynomial

P
(
t
)

P(t)

on a given interval [a, b] into those on any subinterval

[

a
¯

,

b
¯

]

[\bar a,\bar b]

is specified by a stochastic matrix which depends only on the degree n of

P
(
t
)

P(t)

and the size and location of

[

a
¯

,

b
¯

]

[\bar a,\bar b]

relative to [a, b]. We show that in the

‖
∙
‖

∞

{\left \| \bullet \right \|_\infty }

-norm, the condition number of M has the simple form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa Subscript normal infinity Baseline left-parenthesis bold upper M right-parenthesis equals left-bracket 2 f max left-parenthesis u Subscript m overbar Baseline comma v Subscript m overbar Baseline right-parenthesis right-bracket Superscript n">

κ
∞

(

M

)
=

[
2
f
max
(

u

m
¯

,

v

m
¯

)

]
n

{\kappa _\infty }({\mathbf {M}}) = {[2f\max ({u_{\bar m}},{v_{\bar m}})]^n}

, where

u

m
¯

=
(

m
¯

−
a
)

/

(
b
−
a
)

{u_{\bar m}} = (\bar m - a)/(b - a)

and

v

m
¯

=
(
b
−

m
¯

)

/

(
b
−
a
)

{v_{\bar m}} = (b - \bar m)/(b - a)

are the barycentric coordinates of the subinterval midpoint

m
¯

=

1
2

(

a
¯

+

b
¯

)

\bar m = \frac {1}{2}( {\bar a + \bar b} )

, and f denotes the "zoom" factor

(
b
−
a
)

/

(

b
¯

−

a
¯

)

(b - a)/(\bar b - \bar a)

of the subdivision map. This suggests a practical rule-of-thumb in assessing how far Bézier curves and surfaces may be subdivided without exceeding prescribed (worst-case) bounds on the typical errors in their control points. The exponential growth of

κ
∞

(

M

)

{\kappa _\infty }({\mathbf {M}})

with n also argues forcefully against the use of high-degree forms in computer-aided geometric design applications.
Nonlinear stability of multistep multiderivative methods
Authors: Li Shou Fu , Ruan Baogen
Published: June 2010

In this paper we examine nonlinear stability of multistep multiderivative methods for initial value problems of a class

K
φ

{K_\varphi }

in a Banach space. Stability criteria are established which extend results of R. Vanselow to this class of methods.