bluuebunny/crossref_metadata_2025 huggingface dataset:
Classification of homotopy torus knot spaces
Authors: Stevens Richard S.
Published: June 2010

The existence of nontrivial homotopy torus knot spaces is established as a corollary to the Theorem. Let

p
p

and

q
q

be two integers with

p
>
1
,
q
>
1

p > 1,q > 1

, and

(
p
,
q
)
=
1

(p,q) = 1

. Let

M

\mathfrak {M}

be a maximal set of topologically distinct compact orientable irreducible

3
3

-mainfolds with fundamental group presented by

⟨
a
,
b

|

a
p

b
q

⟩

\langle a,b|{a^p}{b^q}\rangle

. Then card

(

M

)
=
1

/

2
Φ
(
p
q
)

(\mathfrak {M}) = 1/2\Phi (pq)

, where

Φ
\Phi

denotes Euler’s function.
Branchings and partitions
Authors: Carlitz L. , Stanley Richard P.
Published: June 2010

A generating function is obtained for the number of partitions corresponding to a complete branching on a nonincreasing sequence of

n
n

integers. Complete branchings are shown to be related to certain types of plane partitions.
Rings whose cyclic modules are injective or projective
Authors: Geol S. C. , Jain S. K. , Singh Surjeet
Published: June 2010

The object of this paper is to prove Theorem. For a ring

R
R

the following are equivalent: (i) Every cyclic right

R
R

-module is injective or projective. (ii)

R
=
S
⊕
T

R = S \oplus T

where

S
S

is semisimple artinian and

T
T

is a simple right semihereditary right Öre-domain whose every proper cyclic right module is injective.
On the number of irreducible representations of degree ≤𝑛 of a Lie group
Authors: Rothaus O. S.
Published: June 2010

We give a proof of part of a result of Robert Cahn [1] on the asymptotic behavior of the number of irreducible representations of degree

≤
n

\leq n

of a semisimple Lie group. The argument is a general one and does not depend on classification.
Complete domains with respect to the Carathéodory distance. II
Authors: Kim Dong S.
Published: June 2010

In [1] we have obtained the following result: Let

D
D

be a bounded domain in

C

n

{{\text {C}}^n}

. Suppose there is a compact subset

K
K

of

D
D

such that for every

x
ϵ
D

x\epsilon D

there is an analytic automorphism

f
ϵ
Aut
⁡
(
D
)

f\epsilon \operatorname {Aut} (D)

and a point

a
ϵ
K

a\epsilon K

such that

f
(
x
)
=
a

f(x) = a

. Then

D
D

is a domain of bounded holomorphy, in the sense that

D
D

is the maximal domain on which every bounded holomorphic function on

D
D

can be continued holomorphically (cf. Narasimhan [2, Proposition 7, p. 127]). Here we shall give a stronger result: Under the same assumptions,

D
D

is

c
c

-complete. We note that a

c
c

-complete domain is a domain of bounded holomorphy, in particular, a domain of holomorphy. A domain of bounded holomorphy, however, need not be

c
c

-complete.
Approximation of functions by convexity preserving continuous linear operators
Authors: Tzimbalario Jean
Published: June 2010

In this note, extension of the ideas used by Korovkin, Bojanic and Roulier, Karlin and Studden, Lorentz and Schumaker are studied in order to treat various aspects of approximation of continuous functions by operators which preserve generalized convexity in the sense of Karlin and Studden, and Ziegler.
The Mackey topology as a mixed topology
Authors: Cooper J. B.
Published: June 2010

A theorem on the coincidence of a mixed topology and the Mackey topology is given. The techniques of partitions of unity are used. As a corollary, a result of LeCam and Conway on the strict topology is obtained. Similar methods are applied to obtain a theorem of Collins and Dorroh.
A class of 𝐿^{𝑝}-bounded pseudo-differential operators
Authors: Illner Reinhard
Published: June 2010

Pseudo-differential operators with symbol
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis x comma xi comma y right-parenthesis element-of upper S Subscript rho comma delta comma epsilon Superscript mu Baseline comma mu less-than-or-equal-to left-parenthesis rho minus 1 right-parenthesis left-parenthesis n plus 1 right-parenthesis">

p
(
x
,
ξ
,
y
)
∈

S

ρ
,
δ
,
ε

μ

,
μ
≤
(
ρ
−
1
)
(
n
+
1
)

p(x,\xi ,y) \in S_{\rho ,\delta ,\varepsilon }^\mu ,\mu \leq (\rho - 1)(n + 1)

, are proven to be generally

L
p

{L^p}

-bounded for

1
>
p
>
∞

1 > p > \infty

.
On the closure-preserving sum theorem
Authors: Singal M. K. , Arya Shashi Prabha
Published: June 2010

The closure-preserving sum theorem holds for a property

P

\mathcal {P}

if the following is satisfied: “if

{

F
α

:
α
ϵ
Ω
}

\{ {F_\alpha }:\alpha \epsilon \Omega \}

is a hereditarily closure-preserving closed covering of

X
X

such that each

F
α

{F_\alpha }

possesses the property

P

\mathcal {P}

, then

X
X

possesses

P

\mathcal {P}

". A general technique for proving this theorem is developed. The theorem is shown to hold for a large number of topological properties. As an application, three general sum theorems have also been obtained.
Extending continuous functions in zero-dimensional spaces
Authors: Warren Nancy M.
Published: June 2010

Suppose that

X
X

is a completely regular, zero-dimensional space and that a dense subset

S
S

of

X
X

is not

C

∗

{C^{\ast }}

-embedded in

X
X

; does there then exist a two-valued function in

C

∗

(
S
)

{C^{\ast }}(S)

with no continuous extension to

X
X

? The answer is negative even if

X
X

is a compact space. The question was raised by N. J. Fine and L. Gillman in Extension of continuous functions in

β
N

\beta N

, Bull. Amer. Math. Soc. 66 (1960), 376-381.
Łoś’ theorem and the Boolean prime ideal theorem imply the axiom of choice
Authors: Howard Paul E.
Published: June 2010

In this paper it is shown that Łoś’ theorem and the Boolean prime ideal theorem imply the axiom of choice. The possibility of eliminating the use of the Boolean prime ideal theorem from the proof is also discussed.
A large class of small varieties of lattice-ordered groups
Authors: Scrimger E. B.
Published: June 2010

We establish the existence of a countable collection of varieties, each of which covers the abelian variety in the lattice of varieties of lattice-ordered groups.
The map of the Witt ring of a domain into the Witt ring of its field of fractions
Authors: Craven Thomas C. , Rosenberg Alex , Ware Roger
Published: June 2010

Let

R
R

be an integral domain with field of fractions

K
K

. This paper studies the kernel of the map

W
(
R
)
→
W
(
K
)

W(R) \to W(K)

, where

W
W

is the Witt ring functor. In case

R
R

is regular and noetherian, it is shown that the kernel is a nilideal. The kernel is zero if

R
R

is a complete regular local noetherian ring with 2 a unit. Examples are given to show that the regularity assumptions are needed.
Cartan subalgebras and tori in prime characteristic
Authors: Wilson Robert Lee
Published: June 2010

Let

L
L

be a finite dimensional (not necessarily restricted) Lie algebra over a perfect field

F
F

of characteristic

p
>
0

p > 0

. Let

L
¯

\overline L

be any finite dimensional restricted Lie algebra containing

L
L

. The Cartan subalgebras of

L
L

are characterized as the centralizers of certain tori in

L
¯

\overline L

.
The intersection of Sylow subgroups
Authors: Mann Avionam
Published: June 2010

Let

G
G

be a finite soluble group. If the order of

G
G

is not divisible by any Fermat or Mersenne primes, then there exist Sylow

2
2

-subgroups,

P
P

and

Q
Q

, such that

P
∩
Q
=

O
p

(
G
)

P \cap Q = {O_p}(G)

. This improves on a result of Itô. A similar result is proved for nilpotent injectors.
𝑝-extremal length and 𝑝-measurable curve families
Authors: Hesse Joseph
Published: June 2010

It is well known that the reciprocal of

p
p

-extremal length, considered as a set function, is an outer measure. We show that if a curve family in euclidean

n
n

-space is measurable with respect to this outer measure, then the

p
p

-extremal length of the curve family is zero or infinite.
Two classical examples of Gross’ abstract Wiener measure
Authors: Finlayson Henry C.
Published: June 2010

Classical Wiener measure on a space of functions of two variables and Wiener measure on the space of sample paths of two dimensional Brownian motion are shown to be special cases of Gross’ abstract Wiener measure.
A general Hoffman-Wermer theorem for algebras of operator fields
Authors: Taylor D. C.
Published: June 2010

Let

A
A

be a closed separating subalgebra of

C
(
T
)

C(T)

that contains the identity. It is known that

Re
⁡
A

\operatorname {Re} A

is uniformly closed only if

A
=
C
(
T
)

A = C(T)

. In this note it is shown that this property characterizes all maximal full algebras of operator fields and not just

C
(
T
)

C(T)

.
Arcs defined by one-parameter semigroups of operators in Banach spaces with the Radon-Nikodym property
Authors: Byrne Charles L.
Published: June 2010

It is shown that a recent theorem of Junghenn and Taam concerning the domain of the infinitesimal generator of a strongly continuous one-parameter semigroup of operators on a reflexive, locally convex topological vector space remains valid if the domain of the operators is a Banach space with the Radon-Nikodym property. A partial result is obtained for general Banach spaces.
On the characterization of Souslin and Borel sets
Authors: Reiss R.-D.
Published: June 2010

Let

X

\mathfrak {X}

be an arbitrary family of sets in the basic set

X
X

. In this paper Souslin-

X

\mathfrak {X}

sets are represented as projections on

X
X

of certain

σ
δ

\sigma \delta

-sets in the cartesian product space of

X
X

and the Baire space (or

X
X

and the real line). For

L

 - 

X

L{\text { - }}\mathfrak {X}

sets (i.e. Souslin sets defined with disjoint unions; ensembles d’unicité) injective projections are considered. The results also apply to Borel-

X

\mathfrak {X}

sets since the system of

L

 - 

X

L{\text { - }}\mathfrak {X}

sets includes the Borel-

X

\mathfrak {X}

sets under suitable conditions.
On the existence of saturated models of stable theories
Authors: Harnik Victor
Published: June 2010

It is proven that a theory

T
T

stable in a power

λ
\lambda

,

λ
>

|

T

|

\lambda > |T|

, has a saturated model of cardinality

λ
\lambda

.
Partitions into specified parts which appear a specified number of times
Authors: Moore Elmo
Published: June 2010

Restrictions of this type, which are known to produce partition identities when any natural number may be used as a part, are shown to produce partition identities when only certain parts may be used.
An approximation theorem of Runge type for the heat equation
Authors: Jones B. Frank
Published: June 2010

If

Ω
\Omega

is an open subset of

R

n
+
1

{{\mathbf {R}}^{n + 1}}

, the approximation problem is to decide whether every solution of the heat equation on

Ω
\Omega

can be approximated by solutions defined on all of

R

n
+
1

{{\mathbf {R}}^{n + 1}}

. The necessary and sufficient condition on

Ω
\Omega

which insures this type of approximation is that every section of

Ω
\Omega

taken by hyperplanes orthogonal to the

t
t

-axis be an open set without “holes,” i.e., whose complement has no compact component. Part of the proof involves the Tychonoff counterexample for the initial value problem.
Logarithmic means and summability by the circle methods
Authors: Parameswaran M. R.
Published: June 2010

It is proved that if the logarithmic means of a sequence

{

s
n

}

\{ {s_n}\}

satisfy a certain order condition then the sequence

{

s
n

}

\{ {s_n}\}

will be summable by every circle method (Kreisverfahren) stronger than convergence; the condition is shown to be a best possible one, for even summability by the collective circle method. A second set of conditions leading to the conclusion of summability by every circle method is also proved.
Inverse cluster sets
Authors: Hamlett T. R. , Long Paul
Published: June 2010

For a function

f
:
X
→
Y

f:X \to Y

, the cluster set of

f
f

at

x
ϵ
X

x\epsilon X

is the set of all

y
ϵ
Y

y\epsilon Y

such that there exists a filter

F

\mathcal {F}

on

X
X

converging to

x
x

and the filter generated by

f
(

F

)

f(\mathcal {F})

converges to

y
y

. The inverse cluster set of

f
f

at

y
ϵ
Y

y\epsilon Y

is the set of all

x
ϵ
X

x\epsilon X

such that

y
y

belongs to the cluster set of

f
f

at

x
x

. General properties of inverse cluster sets are proved, including a necessary and sufficient condition for continuity. Necessary and sufficient conditions for functions to have a closed graph in terms of inverse cluster sets are also given. Finally, a known theorem giving a condition as to when a connected function is also a connectivity function is generalized and further investigated in terms of inverse cluster sets.
The regular continuous image of a minimal regular space is not necessarily minimal regular
Authors: Berriozabal Manuel P. , Blakemore Carroll F.
Published: June 2010

Herrlich has shown that the regular continuous image of a regular-closed space is regular-closed. In this paper, an example is given to show that Herrlich’s result cannot be extended to a corresponding result for minimal regular spaces. Also, a modification of this example shows that a continuous function from a minimal regular space onto a regular space is not necessarily a closed function.
Characteristic submanifolds of fixed point free involutions
Authors: Stone Ellen
Published: June 2010

According to López de Medrano, a manifold with a free involution is determined by its Browder-Livesay index and the normal cobordism class of its orbit space. Here we study the Browder-Livesay indices of some cobordant involutions, namely, the characteristic submanifolds of a fixed involution.
Essential extensions and intersection theorems
Authors: Schelter W.
Published: June 2010

If

R
R

is right and left noetherian, primitive factor rings are artinian, and

R
R

is right fully bounded, then a simple proof is given to show that finitely generated essential extensions of right artinian modules are artinian. An immediate corollary is that

∩

n
=
1

∞

J
n

=
0

\cap _{n = 1}^\infty {J^n} = 0

for such a ring.
On a generalized version of the Nakayama conjecture
Authors: Auslander Maurice , Reiten Idun
Published: June 2010

Nakayama proposed a conjecture which is equivalent to the following: If

Λ
\Lambda

is a finite dimensional algebra over a field and the dominant dimension of

Λ
\Lambda

is infinite, then

Λ
\Lambda

is self-injective. In this paper we study a generalized version of this conjecture.
A note on lifting Brauer characters
Authors: Gagola Stephen M.
Published: June 2010

A Brauer character of a finite group may be lifted to an ordinary character if it lies in a block whose defect groups are contained in a normal

p
p

-solvable subgroup.
The automorphism group of the Tits simple group ²𝐹₄(2)^{’}
Authors: Griess Robert L. , Lyons Richard
Published: June 2010

In this paper, we determine the automorphism groups of the finite groups

2

F
4

(
2
)

^2{F_4}(2)

and

2

F
4

(
2

)
′

^2{F_4}(2)’

. A consequence is that all

Aut
⁡
(
G
)

\operatorname {Aut} (G)

are known, where

G
G

is a nonabelian composition factor of a finite group of Lie type.
𝑆-transversality
Authors: Favaro L. A.
Published: June 2010

This paper will extend Thom’s transversality theorem to differentiable mappings between foliated manifolds, and deal with mappings with “nice” nontransversal points.
Random compact sets related to the Kakeya problem
Authors: Alexander Ralph
Published: June 2010

A

B
B

-set is defined to be a compact planar set of zero measure which contains a translate of any line segment lying in a disk of diameter one. A construction is given which associates a unique compact planar set with each sequence in a closed interval, and it is shown that for almost all such sequences a

B
B

-set is obtained. The construction depends on the measure properties of certain perfect linear sets. Several related problems of a subtler nature are also considered.
A one dimensional manifold is of cohomological dimension 2
Authors: Deo Satya
Published: June 2010

G. Bredon defines the cohomological Dimension of a topological space

X
X

to be the supremum of all cohomological

ϕ
\phi

-dimensions of

X
X

, where

ϕ
\phi

varies over the entire families of supports on

X
X

. He has proved that if

X
X

is a topological

n
n

-manifold then the cohomological Dimension of

X
X

is

n
n

or

n
+
1

n + 1

. He was not able to decide which one it is, even for a space as simple as the real line. The objective of this paper is to solve his problem for

n
=
1

n = 1

. In particular, we have shown that the cohomological Dimension of the real line is

2
2

.
On maximal torsion radicals. III
Authors: Beachy John A.
Published: June 2010

Maximal torsion radicals of a strongly semiprime ring correspond to minimal prime ideals and can be used to characterize both strongly semiprime and semiprime Goldie rings.
Holomorphic extensions of orthogonal projections into holomorphic functions
Authors: Kohn J. J.
Published: June 2010

A condition is given which insures that the orthogonal projection of a function into the holomorphic functions is holomorphically extendible across a given boundary point.
Intersections of commutants of analytic Toeplitz operators
Authors: Thomson James E.
Published: June 2010

In this paper we study the intersection of commutants of analytic Toeplitz operators. Our main result is that if

ϕ
\phi

is a finite Blaschke product and

Ψ
ϵ

H
∞

\Psi \epsilon {H^\infty }

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper T Subscript phi Baseline right-brace prime intersection left-brace upper T Subscript normal upper Psi Baseline right-brace Superscript prime Baseline equals left-brace upper T Subscript upper I Baseline right-brace prime">

{

T
ϕ

}
′

∩
{

T
Ψ

}
′

=
{

T
I

}
′

\{ {T_\phi }\} ’ \cap \{ {T_\Psi }\} ’ = \{ {T_I}\} ’

where

I
I

is a finite Blaschke product and

ϕ
\phi

and

Ψ
\Psi

are functions of

I
I

. The key step is a function-theoretic theorem describing the relationship between a finite Blaschke product and any

H
∞

{H^\infty }

function.
Lifting recursion properties through group homomorphisms
Authors: Al-Kutaibi S. H. , Rhodes F.
Published: June 2010

The problem of lifting recursion properties is studied for the case of homomorphisms between transformation groups with different acting groups. It is shown that if the spaces are the same and the group homomorphism is compact-covering then almost periodicity and recurrence lift. This result is applied to certain group actions on covering spaces and on fibre bundles with totally disconnected bundle groups induced by group actions on the base spaces. In particular, it is shown that under suitable conditions if a point in the base space of a fibre bundle with finite bundle group is almost periodic recurrent under the action of a group

G
G

then every point in the fibre over the given point is almost periodic recurrent under the induced action of a covering group

G
~

\tilde {G}

.
Evidence of a conspiracy among fixed point theorems
Authors: Rosenholtz Ira
Published: June 2010

Some generalizations of the Banach contraction theorem replace the global hypothesis that the function be a contraction with various local hypotheses. In this paper, we examine a few of these, and show that, in fact, the functions actually satisfied the global hypothesis after a suitable change of metric. Finally, the techniques developed are applied to prove a new fixed point theorem for locally expansive maps.
On pairs of nonintersecting faces of cell complexes
Authors: Wadler Philip L.
Published: June 2010

We show that, for all cell complexes whose underlying set is a manifold,

M
M

, an alternating sum of numbers of pairs of faces that do not intersect is a topological invariant. This is done by proving that it is a function of the Euler characteristic,

x
x

, of

M
M

.
Modular field extensions
Authors: Heerema Nickolas , Tucker David
Published: June 2010

Let

K
⊃
k

K \supset k

be fields having characteristic

p
≠
0

p \ne 0

. The following is proved. If

K
K

is algebraic over

k
k

then

K
K

is modular over

k
k

if and only if

K
=
S

⊗
k

M

K = S{ \otimes _k}M

where

S
S

is separably algebraic over

k
k

and

M
M

is purely inseparable, modular. If

K
K

is finitely generated over

k
k

(not necessarily algebraic), then

K
K

is modular over

k
k

if and only if

K
K

where

M
M

is finite, purely inseparable, modular over

k
k

, and

S
S

is a finitely generated, separable, extension of

k
k

. This leads immediately to the representation

K
=
(
S

⊗
k

M
)

⊗
S

R

K = (S{ \otimes _k}M){ \otimes _S}R

where

S
S

is finite separable over

k
,

M

k,\;M

is finite, purely inseparable, modular over

k
k

and

R
R

is a regular finitely generated extension of

S
S

, This last representation displays subfields of

K

/

k

K/k

related to recently obtained Galois theories. The above results are used to analyze transitivity properties of modularity.
A note on finite groups having a fixed-point-free automorphism
Authors: Pettet Martin R.
Published: June 2010

A fusion result of Glauberman has as a consequence the fact that a finite group admitting a fixed-point-free automorphism has a normal Sylow

2
2

-subgroup (and in particular, is solvable) if all nontrivial fixed-point subgroups have odd order.
Stability theory for Hill equations with generalized coefficient
Authors: Guggenheimer H.
Published: June 2010

A simple geometric proof is given for the existence of Ljapunov’s intervals of stability and instability for Hill equations with generalized coefficients.
A lemma of Élie Cartan
Authors: Maltz Robert
Published: June 2010

We provide in this paper an alternative proof of the lemma of E. Cartan which states that parallel translation of curvature and torsion locally determines an affine connection. Our proof uses covariant differentiation of tensor fields over mappings in place of Cartan’s exterior differential calculus.
Axiomatic shape theory
Authors: Bacon Philip
Published: June 2010

The notion of shape theory is so defined that, if

H

{\text {H}}

is a category and

W

{\text {W}}

is a subcategory of

H

{\text {H}}

, all shape theories on

(

H,

W

)

({\text {H,}}\;{\text {W}})

are isomorphic and, under a mild condition, a shape theory on

(

H,

W

)

({\text {H,}}\;{\text {W}})

always exists. Additional theorems facilitate the comparison of shape theories constructed by various means.
Remark on nilpotent orbits
Authors: Wolf Joseph A.
Published: June 2010

If

G
G

is a reductive Lie group and

O

f

=
Ad
⁡

(
G

)
∗

f

{\mathcal {O}_f} = \operatorname {Ad} {(G)^ \ast }f

is a nilpotent coadjoint orbit with invariant real polarization

p

\mathfrak {p}

, then

O

f

{\mathcal {O}_f}

is identified as an open

G
G

-orbit on the cotangent bundle of

G

/

P

G/P

.
Construction of the integral closure of a finite integral domain. II
Authors: Seidenberg A.
Published: June 2010

In a previous paper the problem of constructing the integral closure of a finite integral domain

k
[

x
1

,
…
,

x
n

]
=
k
[
x
]

k[{x_1}, \ldots ,{x_n}] = k[x]

was considered. A reduction to the case

d
t
k
(
x
)

/

k
=
1
,
k
(
x
)

/

k

dtk(x)/k = 1,k(x)/k

separable, and

n
=
2

n = 2

was made. A subsidiary problem was: if

k
[
x
]

k[x]

is not integrally closed, to find a

y
y

in

k
(
x
)

k(x)

integral over

k
[
x
]

k[x]

but not in it. This was done for

n
=
2

n = 2

, but should have been done for arbitrary

n
n

. The extra details are here given. For the convenience of the reader, the full argument is sketched.
A representable functor theorem for compact analytic spaces and applications
Authors: Sommese Andrew J.
Published: June 2010

It is shown that the functor given by maps from a given compact analytic space

X
X

into elements of a category

C
C

of analytic spaces is representable by a compact element of

C
C

if and only if

C
C

is closed under finite products and the taking of compact subspaces. Various specific choices are considered. Of special interest is the category of compact hyperbolic spaces which gives rise to a birational invariant when applied to projective manifolds.
Hilbert-Samuel functions of Cohen-Macaulay rings
Authors: Boratyński M. , Święcicka J.
Published: June 2010

Let

R
R

be a local ring with a maximal ideal

m

\mathfrak {m}

. It is proved that in case

R
R

is a Cohen-Macaulay (C.M.) ring and

dim
⁡

m

/

m

2

−
dim
⁡
R
=
1

\dim \mathfrak {m}/{\mathfrak {m}^2} - \dim R = 1

, then the multiplicity of

R
R

and its dimension determine uniquely the Hilbert-Samuel function of

R
R

. As a corollary we obtain that the C.M. property is determined by the Hilbert-Samuel function in case

dim
⁡

m

/

m

2

−
dim
⁡
R
=
1

\dim \mathfrak {m}/{\mathfrak {m}^2} - \dim R = 1

. An example is given which shows that it is not so in case

dim
⁡

m

/

m

2

−
dim
⁡
R
>
1

\dim \mathfrak {m}/{\mathfrak {m}^2} - \dim R > 1

.
Compactness of sets of integrable functions with values in a Banach space
Authors: Seierstad Atle
Published: June 2010

The present paper gives direct proofs of compactness (in a pairing topology) of bounded sets of strongly integrable functions with values in weakly compact subsets of a Banach space. The pairing in question is that one defined by the integral of the scalar product of two strongly integrable functions with values in the Banach space and its dual, respectively.
Linear interpolating bases in 𝐶[0,1] are not Besselian
Authors: Carpenter Ronn
Published: June 2010

It is shown that linear interpolating bases for

C
[
0
,
1
]

C[0,1]

cannot be

p
p

-Besselian.
Some characterizations of reflexivity
Authors: Singer Ivan
Published: June 2010

The results of R. C. James on characterizations of reflexivity of Banach spaces with an unconditional basis in terms of

c
0

{c_0}

and

l
1

{l^1}

are extended to arbitrary Banach spaces. Some consequences are obtained.
Epimorphisms in the category of abelian 𝑙-groups
Authors: Pedersen F. D.
Published: June 2010

By choosing the mappings for a category of

l
l

-groups carefully, some significant categorical results are obtainable. The intent of this paper is to investigate the reasons why epimorphisms in the category of

l
l

-homomorphisms between abelian

l
l

-groups are not “onto” functions.
Cylindricity of isometric immersions into Euclidean space
Authors: Maltz Robert
Published: June 2010

A simple geometric proof is given for the Hartman-Nirenberg cylindricity theorem and some generalizations. Then the following cylindricity theorem (unpublished) of S. Alexander is proved using the same idea. Theorem. Let

f
:
M
→

E
n

f:M \to {E^n}

be an isometric Euclidean immersion of the Riemannian product

M
=

M
1

×
…
×

M
k

×

E
m

M = {M_1} \times \ldots \times {M_k} \times {E^m}

where the

M
i

{M_i}

are not everywhere flat Riemannian manifolds, and

E
m

{E^m}

denotes Euclidean

m
m

-space. Then

C
⩾
k

C \geqslant k

, where

C
C

denotes the codimension of the immersion; and if

C
=
k

C = k

, then the immersion is cylindrical on the Euclidean factor.
The resultant of several homogeneous polynomials in two indeterminates
Authors: Kakié Kunio
Published: June 2010

We give one method of determining the degree of the highest common factor of several homogeneous polynomials in two indeterminates.
Maximal ideal transforms of Noetherian rings
Authors: Matijevic Jacob R.
Published: June 2010

Let

R
R

be a commutative Noetherian ring with unit. Let

T
T

be the set of all elements of the total quotient ring of

R
R

whose conductor to

R
R

contains a power of a finite product of maximal ideals of

R
R

. If

A
A

is any ring such that

R
⊂
A
⊂
T

R \subset A \subset T

, then

A

/

x
A

A/xA

is a finite

R
R

module for any non-zero-divisor

x
x

in

R
R

. It follows that if, in addition,

R
R

has no nonzero nilpotent elements, then any ring

A
A

such that

R
⊂
A
⊂
T

R \subset A \subset T

is Noetherian.
A property of groups of non-exponential growth
Authors: Rosset Shmuel
Published: June 2010

We prove that in a finitely generated nonexponential group a normal subgroup with a solvable quotient is finitely generated. This extends a theorem of Milnor which has the same conclusion if the group is also assumed to be solvable. The proof uses a lemma of Milnor, but in a different, simpler, way.
The commutants of certain analytic Toeplitz operators
Authors: Thomson James E.
Published: June 2010

In this paper we characterize the commutants of two classes of analytic Toeplitz operators. We show that if

F
F

in

H
∞

{H^\infty }

is univalent and nonvanishing, the

{

T

F
2

}
′

=
{

T
z

}
′

\{ {T_{{F^2}}}\} ’ = \{ {T_z}\} ’

. When

φ
\varphi

is the product of two Blaschke factors, we characterize

{

T
φ

}
′

\{ {T_\varphi }\} ’

in terms of algebraic combinations of Toeplitz and composition operators.
A simple proof of the Zolotareff-Frobenius theorem
Authors: Dressler Robert E. , Shult Ernest E.
Published: June 2010

We give a noncomputational, elementary group-theoretic proof of the Zolotareff-Frobenius Theorem. We use no results from the theory of quadratic residues.
The set where an approximate derivative is a derivative
Authors: O’Malley Richard J.
Published: June 2010

Let

f
:
[
0
,
1
]
→
R

f:[0,1] \to R

possess a finite approximate derivative

f

ap

′

f_{\operatorname {ap}}’

Let

E
E

be the set of points

x
x

where

f
f

is actually differentiable. It is shown that for every

λ
\lambda

if

{
x
:

f

ap

′

(
x
)
=
λ
}
≠
∅

\{ x:f_{\operatorname {ap}}’(x) = \lambda \} \ne \emptyset

, then

{
x
:

f

ap

′

(
x
)
=
λ
}
∩
E
≠
∅

\{ x:f_{\operatorname {ap}}’(x) = \lambda \} \cap E \ne \emptyset

. A strengthening of the mean value theorem associated with approximate derivatives is an immediate corollary.
Induced automorphisms and simple approximations
Authors: Goodson Geoffrey R.
Published: June 2010

A class of ergodic, measure preserving invertible point transformations, which are said to admit simple approximations is defined below. If

T
T

is an automorphism which admits a simple approximation, conditions are given on a set

A
A

so that the induced automorphisms

T
A

{T^A}

and

T
A

{T_A}

again admit simple approximations.
The starlike radius for classes of regular bounded functions
Authors: Sanders Robert W.
Published: June 2010

Let

B
0

(
a
)

{B_0}(a)

be the class of all functions

f
f

defined on

|

z

|

>
1

|z| > 1

such that (i)

f
(
z
)

f(z)

is regular, (ii)

|

f
(
z
)

|

>
1

|f(z)| > 1

(iii)

f
(
0
)
=
0

f(0) = 0

(iv)

0
>

|

f
′

(
0
)

|

=
a
⩽
1

0 > |f’(0)| = a \leqslant 1

. For fixed

R
,
a
⩽
R
>
1

R,a \leqslant R > 1

, let

B
0

(
a
;
R
)

{B_0}(a;R)

be that subclass having nonzero zeros at

z
=

z
k

,
k
=
1
,
2
,
…

z = {z_k},k = 1,2, \ldots

, such that

∏

|

z
k

|

=
R

\prod |{z_k}| = R

. The subclass having no nonzero zeros is designated as

B
0

(
a
;
1
)

{B_0}(a;1)

. A sharp lower bound for

Re
⁡
[
z

f
′

(
z
)

/

f
(
z
)
]

\operatorname {Re} [zf’(z)/f(z)]

for the class

B
0

(
a
;
R
)
,
a
⩽
R
⩽
1

{B_0}(a;R),a \leqslant R \leqslant 1

, is obtained, and the radius of starlikeness is found. A covering theorem for the class is also obtained.
On the behavior of meromorphic functions at the ideal boundary of a Riemann surface
Authors: Schiff J. L.
Published: June 2010

In a former work the author established an analog of a classical theorem of Painlevé in the context of an arbitrary resolutive compactification of a Riemann surface. In the same setting, a refinement of the argument used in the above yields an elementary proof of a theorem of Riesz-Luzin-Privaloff type: If a meromorphic function

f
f

tends to zero at each point of a subset

E
E

of the ideal boundary and

E
E

has positive harmonic measure, then

f
≡
0

f \equiv 0

on

R
R

. The well-known inclusion relations

U

H
B

⊂

O

M

B

∗

{U_{HB}} \subset {\mathcal {O}_{M{B^{\ast }}}}

and

U

H
D

∼

O

M

D

∗

{U_{HD}} \sim {\mathcal {O}_{M{D^{\ast }}}}

, are then established from the point of view of the resolutivity of the Wiener and Royden compactification respectively.
Harmonic analysis of harmonic functions in the plane
Authors: Rubel L. A.
Published: June 2010

A continuous function on the complex plane is harmonic if and only if the span of its compositions with entire functions is not dense in the space of continuous functions in the topology of uniform convergence on compact sets.
On direct sums of reflexive operators
Authors: Feintuch Avraham
Published: June 2010

Let

A
1

{A_1}

and

A
2

{A_2}

be reflexive operators on a Hilbert space

H
H

. If

A
2

{A_2}

is algebraic then

A
1

⊗

A
2

{A_1} \otimes {A_2}

is reflexive.
Approximation to fixed points of generalized nonexpansive mappings
Authors: Wong Chi Song
Published: June 2010

Let

K
K

be a convex subset of a uniformly convex Banach space. It is proved that if

K
K

is compact, then the fixed points of a continuous generalized nonexpansive self-mapping

T
T

on

K
K

can be approximated by the iterates of

T
t

{T_t}

with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t element-of left-parenthesis 0 comma 1 right-parenthesis comma upper T Subscript t Baseline left-parenthesis x right-parenthesis equals left-parenthesis 1 minus t right-parenthesis x plus t upper T left-parenthesis x right-parenthesis comma x element-of upper K semicolon upper T Subscript t Baseline">

t
∈
(
0
,
1
)
,

T
t

(
x
)
=
(
1
−
t
)
x
+
t
T
(
x
)
,
x
∈
K
;

T
t

t \in (0,1),{T_t}(x) = (1 - t)x + tT(x),x \in K;{T_t}

is asymptotically regular if

T
T

has a fixed point.
Positive sectional curvatures does not imply positive Gauss-Bonnet integrand
Authors: Geroch Robert
Published: June 2010

An example is given, in dimension six, of a curvature tensor having positive sectional curvatures and negative Gauss-Bonnet integrand.
The fundamental theorem of algebra on rational 𝐻-spaces
Authors: Weiss Laura
Published: June 2010

A form of the Fundamental Theorem of Algebra is proven for

Γ

 - 

H

\Gamma {\text { - }}H

structures.
Embedding contractible 2-complexes in 𝐸⁴
Authors: Freed Benjamin M.
Published: June 2010

If

L
L

is any figure eight complex or any complex of type (1,1,1), then there are infinitely many different embeddings of

L
L

in

E
4

{E^4}

.
A remark on derivations and skew-derivations on 𝒟(ℳ)
Authors: Suguri Tsuneo
Published: June 2010

We give here the corrections of the Proposition of S. Kobayashi and K. Nomizu concerning the derivation and skew-derivation of the algebra of differential forms on a differentiable manifold.
A note on generalizing alternative rings
Authors: Hentzel Irvin Roy , Piacentini Cattaneo Giulia Maria
Published: June 2010

Let

R
R

be a nonassociative ring of characteristic different from

2
2

and

3
3

which satisfies the following identities:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis i right-parenthesis left-parenthesis a b comma c comma d right-parenthesis plus left-parenthesis a comma b comma left-bracket c comma d right-bracket right-parenthesis equals a left-parenthesis b comma c comma d right-parenthesis plus left-parenthesis a comma c comma d right-parenthesis b comma">

(

i)

(
a
b
,
c
,
d
)
+
(
a
,
b
,
[
c
,
d
]
)
=
a
(
b
,
c
,
d
)
+
(
a
,
c
,
d
)
b
,

({\text {i)}}\;(ab,c,d) + (a,b,[c,d]) = a(b,c,d) + (a,c,d)b,

\]

\[

(

ii)

(
a
,
a
,
a
)
=
0
,

({\text {ii)}}\;(a,a,a) = 0,

\]

\[

(

iii)

(
a
,
b
∘
c
,
d
)
=
b
∘
(
a
,
c
,
d
)
+
c
∘
(
a
,
b
,
d
)

({\text {iii)}}\;(a,b \circ c,d) = b \circ (a,c,d) + c \circ (a,b,d)

\]
for all

a
,
b
,
c
,
d
∈
R

a,b,c,d \in R

and with

x
∘
y
=
(
x
y
+
y
x
)

/

2

x \circ y = (xy + yx)/2

. We prove that if

R
R

is semiprime, then

R
R

is alternative.
Conjugate powers in HNN groups
Authors: Anshel Michael
Published: June 2010

Our purpose is to show the conjugacy problem is solvable for certain

H
N
N

HNN

groups with many stable letters and in the process investigate conjugate powers in these groups.
Homological dimension of abelian groups over their endomorphism rings
Authors: Richman Fred , Walker Elbert A.
Published: June 2010

The projective dimension of an abelian group with torsion reduced part, as a module over its endomorphism ring, is determined. In particular, a group of projective dimension

2
2

is exhibited.
On torsion subgroups of Lie groups
Authors: Lee Dong Hoon
Published: June 2010

We are concerned with torsion subgroups of Lie groups. We extend the classical result of C. Jordan on the structure of finite linear groups to torsion subgroups of connected Lie groups.
Negative theorems on monotone approximation
Authors: Roulier John A.
Published: June 2010

In this paper we show that for

f
f

continuous on

[
−
1
,
+
1
]

[ - 1, + 1]

and satisfying

(
f
(

x
2

)
−
f
(

x
1

)
)

/

(

x
2

−

x
1

)
≧
δ
>
0

(f({x_2}) - f({x_1}))/({x_2} - {x_1}) \geqq \delta > 0

, it is possible to have infinitely many of the polynomials of best uniform approximation to

f
f

not increasing on

[
−
1
,
+
1
]

[ - 1, + 1]

.
On 𝐿¹ convergence of certain cosine sums
Authors: Garrett John W. , Stanojević Časlav V.
Published: June 2010

Rees and Stanojević introduced a new class of modified cosine sums
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace g Subscript n Baseline left-parenthesis x right-parenthesis equals one half sigma-summation Underscript k equals 0 Overscript n Endscripts normal upper Delta a left-parenthesis k right-parenthesis plus sigma-summation Underscript k equals 1 Overscript n Endscripts sigma-summation Underscript j equals k Overscript n Endscripts normal upper Delta a left-parenthesis j right-parenthesis cosine k x right-brace">

{

g
n

(
x
)
=

1
2

∑

k
=
0

n

Δ
a
(
k
)
+

∑

k
=
1

n

∑

j
=
k

n

Δ
a
(
j
)
cos
⁡
k
x
}

\{ {g_n}(x) = \tfrac {1} {2}\sum \nolimits _{k = 0}^n {\Delta a(k) + \sum \nolimits _{k = 1}^n {\sum \nolimits _{j = k}^n {\Delta a(j)\cos kx\} } } }

and found a necessary and sufficient condition for integrability of these modified cosine sums. Here we show that to every classical cosine series

f
f

with coefficients of bounded variation, a Rees-Stanojević cosine sum

g
n

{g_n}

can be associated such that

g
n

{g_n}

converges to

f
f

pointwise, and a necessary and sufficient condition for

L
1

{L^1}

convergence of

g
n

{g_n}

to

f
f

is given. As a corollary to that result we have a generalization of the classical result of this kind. Examples are given using the well-known integrability conditions.
The numerical range of an unbounded operator
Authors: Crabb M. J.
Published: June 2010

The numerical range of an unbounded linear operator on a complex Banach space is the whole complex plane.
An abelian ergodic theorem for semigroups in 𝐿_{𝑝} space
Authors: McGrath S. A.
Published: June 2010

The purpose of this paper is to prove individual and dominated ergodic theorems for Abel means of semigroups of positive

L
p

{L_p}

contractions,

1
>
p
>
∞

1 > p > \infty

.
Center of mass and 𝐺-local triviality of 𝐺-bundles
Authors: Grove Karsten
Published: June 2010

Riemannian geometry techniques are used to give a short and constructive proof that a differentiable

G
G

-fibre bundle with compact fibre is

G
G

-locally trivial when

G
G

is a compact Lie Group.
𝑃𝐿 involutions on the nonorientable 2-sphere bundle over 𝑆¹
Authors: Kim Paik Kee
Published: June 2010

We show that there are exactly nine distinct

PL

{\text {PL}}

involutions on the nonorientable

2
2

-sphere bundle over

S
1

{S^1}

, up to

PL

{\text {PL}}

equivalences. This, together with results of [1], [3] and [8], classifies all

PL

{\text {PL}}

involutions on the

2
2

-sphere bundles over

S
1

{S^1}

.
A counterexample concerning inseparable field extensions
Authors: Deveney James Kevin
Published: June 2010

Let

K
⊇
M
⊇
k

K \supseteq M \supseteq k

be a chain of fields of characteristic

p
≠
0

p \ne 0

where

K
K

is separable over

M
M

and

M
M

is purely inseparable over

k
k

. Recently it has been shown that if

K
K

has a separating transcendency basis over

M
M

or if

M
M

is of bounded exponent over

k
k

, then

K
=
M

⊗
k

S

K = M{ \otimes _k}S

where

S
S

is separable over

k
k

. This note presents an example to show that, in general, no such

S
S

need exist.
The general solution of a first order differential polynomial
Authors: Cohn Richard M.
Published: June 2010

A purely algebraic proof is given of a theorem, proved analytically by Ritt, which determines the number of derivations needed to find a basis for the perfect ideal of the general solution of an algebraically irreducible first order differential polynomial.
On nowhere monotone functions
Authors: Weil Clifford E.
Published: June 2010

The existence of everywhere differentiable but nowhere monotone functions is established using the Baire Category Theorem, and the relatively easy fact that there are nontrivial bounded derivatives with a dense set of zeros.
An analogue of some inequalities of P. Turán concerning algebraic polynomials having all zeros inside [-1,+1]
Authors: Varma A. K.
Published: June 2010

Let

P
n

(
x
)

{P_n}(x)

be an algebraic polynomial of degree

⩽
n

\leqslant n

having all its zeros inside

[
−
1
,
+
1
]

[ - 1, + 1]

; then we have
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative 1 Superscript 1 Baseline upper P Subscript n Superscript prime 2 Baseline left-parenthesis x right-parenthesis d x greater-than left-parenthesis n slash 2 right-parenthesis integral Subscript negative 1 Superscript 1 Baseline upper P Subscript n Superscript 2 Baseline left-parenthesis x right-parenthesis d x period">

∫

−
1

1

P
n

′

2

(
x
)
d
x
>
(
n

/

2
)

∫

−
1

1

P
n
2

(
x
)
d
x
.

\int _{ - 1}^1 {P_n^{’2}(x)dx > (n/2)\int _{ - 1}^1 {P_n^2(x)dx.} }

\]
The result is essentially best possible. Other related results are also proved.
A refinement for coefficient estimates of univalent functions
Authors: Horowitz David
Published: June 2010

By examining the coefficient inequalities of FitzGerald it is shown that if

f
(
z
)
=
z
+

a
2

z
2

+

a
3

z
3

+
…

f(z) = z + {a_2}{z^2} + {a_3}{z^3} + \ldots

is analytic and univalent in the unit disc, then

|

a
n

|

>
(
1.0691
)
n

|{a_n}| > (1.0691)n

.
Arithmetic means of Fourier coefficients
Authors: Sinha Rajendra
Published: June 2010

Given the Fourier coefficients of an even continuous function, we find a necessary and sufficient condition such that their arithmetic means are the Fourier coefficients of an odd continuous function. A similar result is shown for those Lipschitz classes whose elements are automatically equivalent to continuous functions.
A perturbation theorem for complete sets of complex exponentials
Authors: Young Robert M.
Published: June 2010

The purpose of this note is to show that the completeness of a set of complex exponentials

{

e

i

λ
n

t

}

\{ {e^{i{\lambda _n}t}}\}

in

L
2

(
−
π
,
π
)

{L^2}( - \pi ,\pi )

is preserved whenever the

λ
n

{\lambda _n}

are subjected to a suitable “lifting".
Idempotents of norm one and Banach algebra representations of compact groups
Authors: Kalton N. J. , Wood G. V.
Published: June 2010

Let

G
G

be a finite group of order

n
n

and let

A
A

be a (real or complex) Banach algebra. Rudin and Schneider [3] ask whether a mapping

f
:
G
→
A

f:G \to A

satisfying

|

|

f
(
x
)

|

|

=
1

||f(x)|| = 1

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals left-parenthesis 1 slash n right-parenthesis normal upper Sigma Subscript y element-of upper G Baseline f left-parenthesis x y Superscript negative 1 Baseline right-parenthesis f left-parenthesis y right-parenthesis">

f
(
x
)
=
(
1

/

n
)

Σ

y
∈
G

f
(
x

y

−
1

)
f
(
y
)

f(x) = (1/n){\Sigma _{y \in G}}f(x{y^{ - 1}})f(y)

is necessarily a homomorphism (Question 1, p. 602). They give an affirmative answer if

A
A

is either commutative and semisimple or strictly convex. Here, we prove this result for general Banach algebras, and at the same time prove the natural generalization to compact groups. This allows us to characterize norm one idempotents in generalized group algebras.
Homotopy trees with trivial classifying ring
Authors: Dyer Micheal N.
Published: June 2010

In this note we study a certain class of groups

π
\pi

for which the homotopy classification of

(
π
,
m
)

(\pi ,m)

-complexes is independent of the

k
k

-invariant for small

m
m

.
On properly embedding planes in 3-manifolds
Authors: Brown E. M. , Brown M. S. , Feustel C. D.
Published: June 2010

In this paper we prove an analog of the loop theorem for a certain class of noncompact

3
3

-manifolds. In particular, we show that the existence of a “nontrivial” proper map of a plane into a

3
3

-manifold implies the existence of a nontrivial proper embedding of a plane into a

3
3

-manifold.
Axial maps with further structure
Authors: Berrick A. J.
Published: June 2010

For

F
=

R

,

C

F = {\mathbf {R}},{\mathbf {C}}

or

H

{\mathbf {H}}

an

F
F

-axial map is defined to be an axial map

R

P
m

×

R

P
m

→

R

P

m
+
k

{\mathbf {R}}{P^m} \times {\mathbf {R}}{P^m} \to {\mathbf {R}}{P^{m + k}}

equivariant with respect to diagonal and trivial

F

∗

{F^{\ast }}

-actions. Analogously to the real case, it is shown that

C

{\mathbf {C}}

-axial maps correspond to immersions of

C

P
n

{\mathbf {C}}{P^n}

in

R

2
n
+
k

{{\mathbf {R}}^{2n + k}}

while (for

F
=

R

F = {\mathbf {R}}

and for

F
=

C

F = {\mathbf {C}}

,

k
k

odd) embeddings induce

F
F

-symmaxial maps. Examples are thereby given of symmaxial maps not induced by embeddings of

R

P
n

{\mathbf {R}}{P^n}

, and of

R

{\mathbf {R}}

-axial maps which are not

C

{\mathbf {C}}

-axial. Furthermore, the relationships which hold when

F
=

R

,

C

F = {\mathbf {R}},{\mathbf {C}}

are no longer valid for

F
=

H

F = {\mathbf {H}}

.
Note on a result of Kaplan
Authors: Williams Kenneth S.
Published: June 2010

A simple proof is given of a congruence (due to Kaplan) involving the number of solutions of a certain biquadratic congruence.
The existence of dual modules
Authors: Anderson D. D.
Published: June 2010

In this note we show that a Noetherian module has a dual module if and only if it satisfies

A
B

5

∗

AB{5^{\ast }}

. A connection between completeness and

A
B

5

∗

AB{5^{\ast }}

is also established.
Quasi-unmixedness and integral closure of Rees rings
Authors: Sawtelle Peter G.
Published: June 2010

For certain Rees rings

R

\mathcal {R}

of a local domain

R
R

, the quasi-unmixedness of

R
R

is characterized in terms of a certain transform of

R

\mathcal {R}

being contained in the integral closure of

R

\mathcal {R}

.
Strongly homogeneous torsion free abelian groups of finite rank
Authors: Arnold David M.
Published: June 2010

An abelian group is strongly homogeneous if for any two pure rank 1 subgroups there is an automorphism sending one onto the other. Finite rank torsion free strongly homogeneous groups are characterized as the tensor product of certain subrings of algebraic number fields with finite direct sums of isomorphic subgroups of

Q
Q

, the additive group of rationals. If

G
G

is a finite direct sum of finite rank torsion free strongly homogeneous groups, then any two decompositions of

G
G

into a direct sum of indecomposable subgroups are equivalent.
𝐿²(𝐺_{𝑄}\𝐺_{𝐴}) is not always multiplicity-free
Authors: Corwin Lawrence
Published: June 2010

We show that there are solvable adelic groups

G

A

{G_{\mathbf {A}}}

whose action on

L
2

(

G

Q

∖

G

A

)
,

G

Q

∖

G

A

{L^2}({G_{\mathbf {Q}}}\backslash {G_{\mathbf {A}}}),{G_{\mathbf {Q}}}\backslash {G_{\mathbf {A}}}

compact, it is not multiplicity-free.
On functions with summable approximate Peano derivative
Authors: Lee Cheng Ming
Published: June 2010

Let

n
n

be a positive integer and

F
F

a function defined on a closed interval

I
I

. For

x
x

in

I
I

, let the

n
n

th approximate Peano derivative of

F
F

at

x
x

, if it exists, be denoted as

F

(
n
)

(
x
)

{F_{(n)}}(x)

. For

n
=
1

n = 1

, the existence of

F

(
n
−
1
)

(
x
)

{F_{(n - 1)}}(x)

will simply mean that the function

F

(
0
)

(
≡
F
)

{F_{(0)}}( \equiv F)

is approximately continuous at

x
x

. Then the following theorem is proved, noting that the phrase “for nearly all

x
x

in

I
I

” means “for all

x
x

in

I
I

except perhaps for those points in a countable subset of

I
I

", Theorem

A

n

{{\mathbf {A}}_n}

. Let

F

(
n
−
1
)

(
x
)

{F_{(n - 1)}}(x)

exist finitely for all

x
x

in

I
I

. If

F

(
n
)

(
x
)

{F_{(n)}}(x)

exists finitely for nearly all

x
x

in

I
I

and is summable on

I
I

, then

F

(
n
−
1
)

{F_{(n - 1)}}

is absolutely continuous in

I
I

.