bluuebunny/crossref_metadata_2025 huggingface dataset:
Analytic properties of elliptic and conditionally elliptic operators.
Authors: Taylor Michael E.
Published: June 2010

In this note we give a short proof of a theorem of Kotake and Narasimhan to the effect that if

A
A

is a strongly elliptic operator of order

2
m

2m

with analytic coefficients and

|

|

A
j

u

|

|

≦

C

j
+
1

(
2
m
j
)
!

||{A^j}u|| \leqq {C^{j + 1}}(2mj)!

, where

|

|

|

|

||\;||

is some suitable norm, then

u
u

is analytic. (Actually Kotake and Narasimhan prove the theorem when

A
A

is elliptic, but the trick we use here requires some specialization.) This is applied to derive a short proof of a theorem proved by Gårding and Malgrange, in the constant coefficients case, concerning conditionally elliptic operators.
A commutativity criterion for closed subgroups of compact Lie groups.
Authors: Wolf Joseph A.
Published: June 2010

Let

Γ
\Gamma

be a closed subgroup of a compact Lie group

G
G

. If the identity component

Γ
0

{\Gamma _0}

is commutative, and if the order of

Γ

/

Γ
0

\Gamma /{\Gamma _0}

is prime to the order of the Weyl group of

G
G

, then it is shown that

Γ
\Gamma

is commutative. If

G
G

is a classical group this extends a theorem of Burnside on finite linear groups. If

G
G

is exceptional this gives some information on Cayley-Dickson algebras, Jordan algebras and the Cayley protective plane.
Rational approximation on the union of sets
Authors: Davie A. M. , Øksendal B. K.
Published: June 2010

A counterexample is given to a conjecture of Val’skiǔ that if K is a compact plane set with interior U and the continuous function f on K satisfies

f

|

U
¯

∈
R
(

U
¯

)

f|\bar U \in R(\bar U)

and

f

|

b
K
∈
R
(
b
K
)

f|bK \in R(bK)

then

f
∈
R
(
K
)

f \in R(K)

. The conjecture is shown to be true when U is a disc.
On continuous and measurable selections and the existence of solutions of generalized differential equations
Authors: Hermes Henry
Published: June 2010

Let

C

(

B
n

)

\mathcal {C}({B^n})

denote the space of nonempty compact subsets of some bounded set

B
n

{B^n}

in Euclidean n dimensional space

E
n

{E^n}

, topologized with the Hausdorff metric topology. The existence of a solution to the initial value problem for the generalized differential equation

d
x
(
t
)

/

d
t
∈
R
(
x
(
t
)
)

dx(t)/dt \in R(x(t))

is shown under the assumption that

R
:

E
n

→

C

(

B
n

)

R:{E^n} \to \mathcal {C}({B^n})

has bounded variation in some neighborhood of the initial value, and under a less restrictive condition on the variation of R. Included are continuous and Lipschitz continuous selection results for mappings

Q
:

E
1

→

C

(

B
n

)

Q:{E^1} \to \mathcal {C}({B^n})

which are, respectively, of bounded variation and Lipschitz continuous.
A priori bounds for boundary sets
Authors: Bebernes J. W. , Fraker Ross
Published: June 2010

Consider

y
=
f
(
t
,
y
,

y
′

)

y = f(t,y,y’)

with boundary conditions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma y left-parenthesis 0 right-parenthesis comma y prime left-parenthesis 0 right-parenthesis right-parenthesis element-of upper S 1 comma left-parenthesis 1 comma y left-parenthesis 1 right-parenthesis comma y prime left-parenthesis 1 right-parenthesis right-parenthesis element-of upper S 2">

(
0
,
y
(
0
)
,

y
′

(
0
)
)
∈

S
1

,
(
1
,
y
(
1
)
,

y
′

(
1
)
)
∈

S
2

(0,y(0),y’(0)) \in {S_1},(1,y(1),y’(1)) \in {S_2}

. It is shown that the boundary value problem has a solution for certain boundary sets

S
1

{S_1}

and

S
2

{S_2}

which depend on the assumed Nagumo condition for

f
(
t
,
y
,

y
′

)

f(t,y,y’)

.
Principal elements of lattices of ideals
Authors: McCarthy P. J.
Published: June 2010

The notion of principal element of a commutative multiplicative lattice was introduced by Dilworth. In this note the principal elements of the lattice of ideals of a commutative ring with unity R are characterized as those ideals of R which are finitely generated and locally principal ideals. It follows that a regular ideal of R is a principal element of the lattice of ideals of R if and only if it is invertible.
An algebraic proof of a theorem of A. Robinson
Authors: Gilmer Robert , Mott Joe L.
Published: June 2010

A. Robinson has used mathematical logic to obtain a theorem concerning systems of polynomial equations with only finitely many solutions; this paper contains an algebraic proof of Robinson’s theorem, based primarily on various equivalent forms of Hilbert’s Nullstellensatz.
Dense subgroups of compact groups
Authors: Wilcox Howard J.
Published: June 2010

A new short proof is given showing that infinite compact groups contain dense pseudocompact subgroups of small cardinality. Relationships between connectedness and divisibility in pseudocompact groups are given. It is shown that most compact connected Abelian groups contain proper dense connected subgroups.
Cohomology for the ergodic actions of countable groups
Authors: Westman Joel J.
Published: June 2010

Certain aspects of Mackey’s theory of virtual groups were fitted into a cohomology theory for ergodic groupoids in a previous paper by the author. Here we relate the groupoid cohomology for ergodic groupoids that arise (by Mackey’s construction) from ergodic actions of countable groups to the usual group cohomology. In case the countable group is a free group, we find that the groupoid cohomology in dimension

>
1

> 1

is

=
{
0
}

= \{ 0\}

.
On the conjugate points of fourth order, selfadjoint linear differential equations
Authors: Etgen G. J. , Scott J. B.
Published: June 2010

In a fundamental paper, W. Leighton and Z. Nehari studied the oscillation of solutions of fourth order, selfadjoint differential equations. One of their principal results is a characterization of the conjugate points. In this characterization, however, the proof of the essential uniqueness of the solutions determining the conjugate points is not complete. The purpose of this note is to complete their proof thereby establishing the essential uniqueness of the solutions involved in characterizing the conjugate points.
On the Arens product and annihilator algebras
Authors: Wong Pak-ken
Published: June 2010

The purpose of this paper is to generalize two results in a recent paper by B. J. Tomiuk and the author. Let A be a

B
∗

{B^ \ast }

-algebra and

M
(
A
)

M(A)

the algebra of double centralizers of A. We show that A is a dual algebra if and only if

M
(
A
)

M(A)

coincides with

A

∗
∗

{A^{ \ast \ast }}

. We also obtain that if A is an annihilator

A
∗

{A^\ast }

-algebra, then

π
A

(
A
)

{\pi _A}(A)

is a two-sided ideal of

A

∗
∗

{A^{ \ast \ast }}

.
Generalized pseudo-hermitian operators
Authors: Kritt Brian
Published: June 2010

An operator in a Banach space is called generalized pseudo-hermitian if it is a generalized scalar in the sense of Foiaş (i.e., admits a spectral distribution) and has a real spectrum. In this paper this class of operators is characterized by the condition that the real exponential group generated by such an operator has polynomial growth in the uniform operator norm.
Topological spaces with a 𝜎-point finite base
Authors: Aull C. E.
Published: June 2010

The principal results of the paper are as follows. A topological space with a

σ
\sigma

-point finite base has a

σ
\sigma

-disjoint base if it is either hereditarily collectionwise normal or hereditarily screenable. From a metrization theorem of Arhangel’skiĭ, it follows that a

T
1

{T_1}

-space with a

σ
\sigma

-point finite base is metrizable iff it is perfectly normal and collectionwise normal. A topological space with a

σ
\sigma

-point base is quasi-developable in the sense of Bennett. Consequently a theorem of Čoban follows that for a topological space

(
X
,
ℑ
)

(X,\Im )

the following are equivalent: (a)

(
X
,
ℑ
)

(X,\Im )

is a metacompact normal Moore space, (b)

(
X
,
ℑ
)

(X,\Im )

is a perfectly normal

T
1

{T_1}

-space with a

σ
\sigma

-point finite base.
An example of Hilton and Roitberg
Authors: Sieradski Allan J.
Published: June 2010

In [3], P. J. Hilton and J. Roitberg illustrated with several examples the failure of cancellation for products in the homotopy category of finite CW complexes. We reconstruct here these examples from a different point of view.
Covering isotopies of 𝑀ⁿ⁻¹ in 𝑁ⁿ
Authors: Wright Perrin
Published: June 2010

We show that a continuous family of locally flat separating embeddings of an

(
n
−
1
)

(n - 1)

-manifold

M

n
−
1

{M^{n - 1}}

into an n-manifold

N
n

{N^n}

, where the family is parametrized by a locally compact finite-dimensional metric space B, can be covered locally and sometimes globally by a continuous family of homeomorphisms of

N
n

{N^n}

onto itself, provided

n
≠
4

n \ne 4

. Furthermore, the covering family can be chosen to extend a preassigned covering family corresponding to a compact connected subset of B. We derive a stronger result for embeddings of

S

n
−
1

{S^{n - 1}}

in

S
n

{S^n}

, and show that the natural map from the space of orientation preserving homeomorphisms of

S
n

{S^n}

to the space of locally flat embeddings of

S

n
−
1

{S^{n - 1}}

into

S
n

,
n
≠
4

{S^n},n \ne 4

, is a Serre fibration and a weak homotopy equivalence.
Weak partition properties for infinite cardinals. I
Authors: Kleinberg E. M.
Published: June 2010

Partition properties are perhaps the most fruitful of the various methods for defining and discussing large cardinals in set theory. In this paper we weaken in a natural way the most well known of these partition properties and examine the extent to which the cardinals defined remain “large."
A covering condition for prime ideals
Authors: Smith W. W.
Published: June 2010

Commutative rings are studied which have the property that every ideal contained in the union of a collection of prime ideals is contained in one of the prime ideals of the collection. This property is shown to be equivalent to the condition that every prime ideal is the radical of a principal ideal.
On the centralizer of a subgroup of a Lie group
Authors: Lee Dong Hoon
Published: June 2010

In this paper, a theorem on the centralizer of a closed subgroup H of a Lie group G such that

G

/

H

G/H

admits a finite invariant measure is proved.
The joint spectrum of commuting nonnormal operators
Authors: Bunce John
Published: June 2010

We define the joint approximate spectrum of a finite family of bounded commuting operators on a Hilbert space and show that it is closed and nonempty. If the operators are all hyponormal, then the family need not be finite, and the joint approximate spectrum can be defined in terms of the multiplicative linear functionals on the generated

C
∗

{C^\ast }

-algebra. These definitions extend the usual definition of the joint spectrum of a family of commuting normal operators.
Subparacompactness and 𝐺_{𝛿}-diagonals in Arhangel′skiĭ’s class 𝑀𝑂𝐵𝐼
Authors: Bennett H. R. , Berney E. S.
Published: June 2010

In this note it is shown that a topological space in Arhangel’skiĭ’s class MOBI need not be subparacompact nor have a

G
δ

{G_\delta }

-diagonal.
Embedding the dual of 𝜋ₘ in the lattice of equational classes of commutative semigroups
Authors: Burris Stanley , Nelson Evelyn
Published: June 2010

The lattice of equational classes of commutative semigroups does not satisfy any special lattice laws.
On functions of bounded boundary rotation
Authors: Liu Ming-chit
Published: June 2010

Let

U
=
{
z
=
r

e

i
θ

|

r
>
1
}

U = \{ z = r{e^{i\theta }}\left | {r > 1\} } \right .

. For

k
≧
2

k \geqq 2

let

V
k

{V_k}

be the class of normalized analytic functions

f
(
z
)

f(z)

such that the boundary rotation of

f
(
U
)

f(U)

is at most

k
π

k\pi

. Let

A
(
r
)

A(r)

be the integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 2 pi Baseline integral Subscript 0 Superscript r Baseline StartAbsoluteValue f prime left-parenthesis rho e Superscript i theta Baseline right-parenthesis EndAbsoluteValue squared rho d rho d theta comma">

∫
0

2
π

∫
0
r

|

f
′

(
ρ

e

i
θ

)

|

2

ρ
d

ρ
d

θ
,

\int _0^{2\pi } {\int _0^r {\left | {f’(\rho {e^{i\theta }})} \right |} } {^2}\rho d\,\rho d\,\theta ,

\]

L
(
r
)

L(r)

the length of the image of the circle

|
z
|

=
r

\left | z \right | = r

under the mapping

f
(
z
)

f(z)

. In this paper the author proves that for

z
∈
U

z \in U

if

f
(
z
)
∈

V
k

f(z) \in {V_k}

then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit sup left-parenthesis sup Underscript f element-of upper V Subscript k Baseline Endscripts upper L left-parenthesis r right-parenthesis right-parenthesis left-parenthesis pi upper A left-parenthesis r right-parenthesis log left-parenthesis StartFraction 1 plus r Over 1 minus r EndFraction right-parenthesis right-parenthesis Superscript negative 1 slash 2 Baseline less-than-over-equals k period">

lim sup

r
→
1

(

sup

f
∈

V
k

L
(
r
)
)

(
π
A
(
r
)
log
⁡

(

1
+
r

1
−
r

)

)

−
1

/

2

≦
k
.

\limsup _{r \to 1} \left ( \sup _{f \in {V_k}} L(r) \right ) \left ( \pi A(r)\log \left ( \frac {1 + r}{1 - r} \right ) \right )^{-1/2} \leqq k.

\]
This generalizes to arbitrary

k
≧
2

k \geqq 2

the recent result of Nunokawa for the case

k
=
2

k = 2

.
Approximations of the identity operator by semigroups of linear operators
Authors: Pazy A.
Published: June 2010

Let

T
(
t
)
,
t
≧
0

T(t),t \geqq 0

, be a strongly continuous semigroup of linear operators on a Banach space X. It is proved that if for every

C
>
0

C > 0

there exists a

δ
c

>
0

{\delta _c} > 0

such that

‖

I
−
T
(
t
)

‖

≦
2
−
C
t
log
⁡
(
1

/

t
)

\left \| {I - T(t)} \right \| \leqq 2 - Ct\log (1/t)

for

0
>
t
>

δ
c

0 > t > {\delta _c}

then

A
T
(
t
)

AT(t)

is bounded for every

t
>
0

t > 0

. It is shown by means of an example that

‖

I
−
T
(
t
)

‖

≦
2
−
C
t

\left \| {I - T(t)} \right \| \leqq 2 - Ct

for a fixed C and all

0
>
t
>
δ

0 > t > \delta

is not sufficient to assure the boundedness of

A
T
(
t
)

AT(t)

for any

t
≧
0

t \geqq 0

.
Proper mappings and the minimum dimension of a compactification of a space
Authors: Keesling James
Published: June 2010

In this paper it is shown that for each positive integer n there is a locally compact Hausdorff space X having the property that

dim

X
=
n

\dim \,X = n

and in addition having the property that if

f
(
X
)
=
Y

f(X) = Y

is a proper mapping, then

dim

Y
≧
n

\dim \,Y \geqq n

. Using this result it is shown that there is a space Y having the property that

min

dim

Y
=
n

\min \, \dim \,Y = n

with a point

p
∈
Y

p \in Y

with

min

dim

Y
−
{
p
}
=
0

\min \, \dim \, Y - \{ p\} = 0

.
Splittings of Hochschild’s complex for commutative algebras
Authors: Fleury Patrick J.
Published: June 2010

Barr has shown that one may split Hochschild’s complex for commutative algebras into Harrison’s complex plus a shuffle subcomplex when working over a field of characteristic zero. We construct a splitting here for the above complex over a ring containing a field which does not have characteristic two and this splitting has Barr’s splitting as a special case.
Analyticity of determinants of operators on a Banach space
Authors: Howland James S.
Published: June 2010

If

F
(
z
)

F(z)

is an analytic family of operators on a Banach space which is of finite rank for each

z
z

, then rank

F
(
z
)

F(z)

is constant except for isolated points, and det

(
I
+
F
(
z
)
)

(I + F(z))

and tr

F
(
z
)

F(z)

are analytic. Similarly if

F
(
z
)

F(z)

is meromorphic.
𝐶-embedded subsets of products
Authors: Noble N.
Published: June 2010

It is shown that each dense subset of

R

n

{R^\mathfrak {n}}

is

z
z

-embedded, from which it follows that a dense subset is

C
C

-embedded if and only if it is

G
δ

{G_\delta }

-dense. These results extend to, for example, all products of separable metric spaces.
Algebraic algebras with involution
Authors: Montgomery Susan
Published: June 2010

The following theorem is proved: Let

R
R

be an algebra with involution over an uncountable field

F
F

. Then if the symmetric elements of

R
R

are algebraic,

R
R

is algebraic.
A note on 𝒵-realcompactifications
Authors: D’Aristotle Anthony J.
Published: June 2010

Orrin Frink showed that the real-valued functions over a Tychonoff space X which may be continuously extended to

ω
(

Z

)

\omega (\mathcal {Z})

, the Wallman-type compactification associated with a normal base

Z

\mathcal {Z}

for X, are those which are

Z

\mathcal {Z}

-uniformly continuous Let

Z

\mathcal {Z}

be a delta normal base on a Tychonoff space X, and let

η
(

Z

)

\eta (\mathcal {Z})

be the corresponding

Z

\mathcal {Z}

-realcompactification of X. In this note we show that countable

Z

\mathcal {Z}

-uniform continuity is a sufficient but not a necessary condition for continuously extending real-valued functions over X to

η
(

Z

)

\eta (\mathcal {Z})

.
Fixed point theorems for certain classes of multifunctions
Authors: Smithson R. E.
Published: June 2010

The following two fixed point theorems for multi-functions are proved: Theorem. If

X
X

is a tree and if

F
:
X
→
X

F:X \to X

is a lower semicontinuous multifunction such that

F
(
x
)

F(x)

is connected for each

x
∈
X

x \in X

, then

F
F

has a fixed point. Theorem. Let

X
X

be a topologically chained, acyclic space in which every nest of topological chains is contained in a topological chain. If

F
:
X
→
X

F:X \to X

is a point closed multi-function such that the image of a topological chain is chainable and such that

F

−
1

(
x
)

{F^{ - 1}}(x)

is either closed or chainable for each

x
∈
X

x \in X

, then

F
F

has a fixed point.
On Spanier’s higher order operations
Authors: Chen Yuh-ching
Published: June 2010

Given a stack

A
A

over a simplicial set

K
K

, we construct the generalized Eilenberg-Mac Lane complexes

K
(
A
,
n
)

K(A,n)

that represents a stack cohomology theory over

K
K

. We show that the higher order operations defined by Spanier are some sections of

K
(
A
,
n
)

K(A,n)

over

K
K

for some stack

A
A

.
Some applications of differential algebra to ring theory
Authors: Cozzens John , Johnson Joseph
Published: June 2010

It is shown that simple noetherian

V
V

-domains with a unique simple module can have any global homological dimension.
Unitary groups and commutators
Authors: Kauffman Robert M.
Published: June 2010

If H is a possibly unbounded selfadjoint operator and A is a closed operator in a Hilbert space, the relation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper U Subscript t Superscript negative 1 Baseline upper A upper U Subscript t Baseline f right-parenthesis prime equals i upper U Subscript t Superscript negative 1 Baseline left-parenthesis upper A upper H minus upper H upper A right-parenthesis upper U Subscript t Baseline f">

(

U
t

−
1

A

U
t

f

)
′

=
i

U
t

−
1

(
A
H
−
H
A
)

U
t

f

(U_t^{ - 1}A{U_t}f)’ = iU_t^{ - 1}(AH - HA){U_t}f

can be shown to hold under relatively reasonable hypotheses on A and f, where

U
t

=

e

i
H
t

{U_t} = {e^{iHt}}

. This relation can then be used to relate properties of the commutator

A
H
−
H
A

AH - HA

to properties of A and H.
On the comparability of 𝐴^{1/2} and 𝐴^{∗1/2}
Authors: McIntosh Alan
Published: June 2010

There exists a regularly accretive operator A in a Hilbert space H such that

A

1

/

2

{A^{1/2}}

and

A

∗
1

/

2

{A^{ \ast 1/2}}

have different domains. Consequently, the domain of the closed bilinear form corresponding to A is different from the domain of

A

1

/

2

{A^{1/2}}

.
Rigid pairs of long arcs
Authors: Miller Gary G.
Published: June 2010

Arcs (linearly ordered continua)

A
A

and

B
B

are constructed such that every map from

A
A

to

B
B

and every map from

B
B

to

A
A

is constant. The Generalized Continuum Hypothesis is sufficient for the existence of two such arcs each of cardinality

2
ℵ

{2^\aleph }

for each uncountable cardinal

ℵ
\aleph

.
A note on Lévy’s Brownian process on the Hilbert sphere
Authors: Strait Peggy Tang
Published: June 2010

Let

X
(
t
)

X(t)

, t in

l
2

{l_2}

, be Lévy’s separable Brownian process. Let S be the unit Hilbert sphere. It is shown that with probability 1, the image of

X
(
t
)
,
t
∈
S

X(t),t \in S

, is the entire real line.
Semigroups corresponding to algebroid branches in the plane
Authors: Bresinsky H.
Published: June 2010

The symmetric semigroups of nonnegative integers and their generators, corresponding to algebroid branches of the plane, are determined.
Ideals in halfrings
Authors: Stone H. E.
Published: June 2010

An analogue for halfrings of the Hilbert Basis Theorem is obtained by a study of the relationship between the ideals of a halfring and the ideals of its ring of differences. This technique also results in a characterization of those ideals for which the factor semiring has no nontrivial semi-isomorphic images.
Finite group schemes over fields
Authors: Hoobler Raymond T. , Magid Andy R.
Published: June 2010

A short proof that commutative group schemes over a field form an abelian category is given.
Measurability of functions in product spaces
Authors: Gowrisankaran Kohur
Published: June 2010

Let

f
f

be a function on a product space

X
×
Y

X \times Y

with values in a separable metrizable space such that it is measurable in one variable and continuous in the other. The joint measurability of such a function is proved under certain conditions on

X
X

and

Y
Y

.
On the coefficients of starlike functions
Authors: Holland Finbarr
Published: June 2010

Every probability measure

μ
\mu

on the circle group generates a function f that is starlike univalent on the open unit disc

Δ
\Delta

. In this note the relationship between

(

c
n

)

({c_n})

, the Fourier-Stieltjes coefficients of

μ
\mu

, and

(

a
n

)

({a_n})

, the Taylor coefficients of f, is examined. A number of theroems are presented which indicate (possibly in the presence of fairly mild restrictions) that the sequences

(

c
n

)

({c_n})

and

(
n

a
n

)

(n{a_n})

behave similarly. For example, it is shown that if

f
(
Δ
)

f(\Delta )

is finite, then

(

c
n

)

({c_n})

converges to zero if, and only if,

(
n

a
n

)

(n{a_n})

converges to zero, thereby completing a result of Pommerenke.
Double commutants of algebraic operators
Authors: Turner T. Rolf
Published: June 2010

The double commutant of an algebraic operator on a complex Hilbert space is equal to the algebra (with identity) generated by that operator.
A note on one-dimensional attracting sets in the three-sphere
Authors: Gibbons Joel C.
Published: June 2010

This paper is an application of Williams’ results for one-dimensional attracting sets to the three-sphere. Our objective is to classify up to

Ω
\Omega

-conjugacy all diffeomorphisms of

S
3

{S^3}

satisfying Smale’s axioms A and B and the condition that the nonwandering set consists of zero- and one-dimensional sinks and sources.
The bold strategy in presence of house limit
Authors: Wilkins J. Ernest
Published: June 2010

It is known that an optimal strategy for a gambler, who wishes to maximize the probability of winning an amount

a
−
x

a - x

in a subfair red-and-black casino if his initial capital is x, is the bold strategy in which the gambler wagers at each opportunity the minimum of his entire current capital

x
′

x’

and the amount

a
−

x
′

a - x’

required to reach the goal a if he wins the bet. If the casino imposes an upper limit L on wagers, we shall prove that the modified bold strategy of wagering

min
(

x
′

,
a
−

x
′

,
L
)

\min (x’,a - x’,L)

is optimal, at least in the important special case in which the goal a is an integral multiple of the house limit L.
On an identity of Eckford Cohen
Authors: Subbarao M. V. , Suryanarayana D.
Published: June 2010

We characterize all multiplicative arithmetical functions

f
k

(
r
)

{f_k}(r)

such that an identity of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript r equals 1 Overscript normal infinity Endscripts f Subscript k Baseline left-parenthesis r right-parenthesis c Subscript k Baseline left-parenthesis n comma r right-parenthesis equals q Subscript k Baseline left-parenthesis n right-parenthesis g left-parenthesis k right-parenthesis comma g left-parenthesis k right-parenthesis not-equals 0 comma">

∑

r
=
1

∞

f
k

(
r
)

c
k

(
n
,
r
)
=

q
k

(
n
)
g
(
k
)
,

g
(
k
)
≠
0
,

\sum \limits _{r = 1}^\infty {{f_k}(r){c_k}(n,r) = {q_k}(n)g(k),\quad g(k) \ne 0,}

\]
holds for all n, where

q
k

(
n
)

{q_k}(n)

is the characteristic function of the set of k-free integers and

c
k

(
n
,
r
)

{c_k}(n,r)

is the generalized Ramanujan sum. This characterization yields several arithmetical identities of the above form including an identity of Eckford Cohen, which occurs as a special case of our theorem on taking

f
k

(
r
)
=
μ
(
r
)

/

J
k

(
r
)

{f_k}(r) = \mu (r)/{J_k}(r)

and

g
(
k
)
=
ζ
(
k
)

g(k) = \zeta (k)

.
A note on 𝐾-commutativity of matrices
Authors: Dixon Edmond Dale , Wong Nancy
Published: June 2010

It is the purpose of this paper to find in terms of parameters the most general matrix X which is K-commutative with respect to a given matrix A. The proofs will yield a method of rational construction for such a matrix X.
A characterization of groups with isomorphic subgroups
Authors: Miller Donald W.
Published: June 2010

The concept of normalizer is generalized to derive a characterization of groups G which contain a proper subgroup isomorphic to G.
An extremal problem for functions with positive real part
Authors: Gupta R. S.
Published: June 2010

Let

P

\mathcal {P}

be the class of functions

P
(
z
)

P(z)

, normalized so that

P
(
0
)
=
1

P(0) = 1

which are regular in

|

z

|

>
1

|z| > 1

and have positive real part there. We obtain the minimum (maximum) real part of

e

i
α

(
z

P
′

/

P
)

{e^{i\alpha }}(zP’/P)

for fixed

α
\alpha

and

|

z

|

,
α
∈
[
0
,
2
π
]
,

|

z

|

>
1

|z|,\alpha \in [0,2\pi ],|z| > 1

and P running over the class

P

\mathcal {P}

.
Maximal independent collections of closed sets
Authors: Baker Harvy Lee
Published: June 2010

A theorem is proved which implies that if X is a separable metric space then there exists a countable maximal independent subset of the lattice of closed subsets of X. In the case where X has no isolated points this independent set is nontrivial in the sense that X does not belong to it and it contains no singletons. Furthermore, if X is a compact metric continuum such that

∪
{
o

|

o

\cup \{ o|o

is an open subset of X and O is homeomorphic to

E
n

{E^n}

for some

n
>
1
}

n > 1\}

is dense in X then there exists a countable maximal such collection whose elements are connected. This complements previous work by the author which characterized continua for which there are such collections of a specialized nature.
Darboux’s theorem fails for weak symplectic forms
Authors: Marsden J.
Published: June 2010

An example of a weak symplectic form on a Hilbert space for which Darboux’s theorem fails is given.
A note on model complete models and generic models
Authors: Shelah Saharon
Published: June 2010

We prove that there are many maximum model complete (= generic) models, and that there exists an (uncountable) theory with no generic models.
On perfect subfields over which a field is separably generated
Authors: Jacobson David
Published: June 2010

In this paper, we determine the perfect subfields over which a field is separably generated.
A characterization of general 𝑍.𝑃.𝐼.-rings
Authors: Levitz Kathleen B.
Published: June 2010

A commutative ring A is a general Z.P.I.-ring if each ideal of A can be represented as a finite product of prime ideals. We prove that a commutative ring A is a general Z.P.I.-ring if each finitely generated ideal of A can be represented as a finite product of prime ideals. We also give a characterization of Krull domains in terms of

∗

^ \ast

-operations, as defined by Gilmer.
A coefficient ring for finite non-commutative rings
Authors: Clark W. Edwin
Published: June 2010

We prove that every finite p-ring R contains a unique (up to isomorphism) subring S such that

S

/

p
S
≅
R

/

rad

R

S/pS \cong R/{\operatorname {rad}}\;R

. S is shown to be a direct sum of full matrix rings over rings of the form

Z

p
n

[
x
]

/

(
f
(
x
)
)

{Z_{{p^n}}}[x]/(f(x))

where

f
(
x
)

f(x)

is monic and irreducible modulo p.
A note on polynomial hulls
Authors: Alexander H.
Published: June 2010

Some properties of polynomially convex hulls of compact sets in

C
n

{C^n}

are deduced by simple topological arguments applied to known topological facts about polynomially convex sets.
An extension of a result by Talalyan on the representation of measurable functions by Schauder bases
Authors: Braun Ben-Ami
Published: June 2010

We define a class of Banach spaces which in particular include the

L
p

,
1
≦
p
>
∞

{L^p},1 \leqq p > \infty

, and the separable Orlicz spaces. We show that an arbitrary measurable function can be represented by a series of any Schauder basis chosen from that class of Banach spaces.
On the existence of invariant subspaces in spaces with indefinite metric
Authors: Masuda Kyûya
Published: June 2010

Let

P
1

,

P
2

{P_1},{P_2}

be complementary projections in Hilbert space H. Let U be a one-to-one and onto operator in H with

Q
(
U
x
)
=
Q
(
x
)

Q(Ux) = Q(x)

, where

Q
(
x
)
=

‖

P
1

x

‖

2

−

‖

P
2

x

‖

2

Q(x) = {\left \| {{P_1}x} \right \|^2} - {\left \| {{P_2}x} \right \|^2}

. The sufficient condition is given for the unique existence of maximal subspace L invariant under all operators commuting with U, and such that

Q
(
x
)
≧
0
,
x
∈
L

Q(x) \geqq 0,x \in L

. The result was obtained in the course of attacking the problem proposed by Phillips [5] (see also [1]).
A symmetrizable space that is not perfect
Authors: Bonnett Dennis A.
Published: June 2010

An example is given of a Hausdorff symmetrizable space which has a closed subset that is not a

G
δ

{G_\delta }

-subset (thus, it is not perfect) and which is not subparacompact. This example is then used in the construction of a symmetrizable

T
1

{T_1}

-space Y having a point

x
0

{x_0}

such that

{

x
0

}

\{ {x_0}\}

is not a

G
δ

{G_\delta }

-subset of Y.
The space of retractions of a 2-manifold
Authors: Wagner Neal R.
Published: June 2010

Let

M
2

{M^2}

be a 2-manifold and let

Λ
\Lambda

be the embedding of

M
2

{M^2}

into its space of retractions which maps each point to the constant retraction to that point. Denote by

L

(

M
2

)

\mathcal {L}({M^2})

the component containing the image of

Λ
\Lambda

. The embedding

Λ
\Lambda

, with range restricted to

L

(

M
2

)

\mathcal {L}({M^2})

, is shown to be a weak homotopy equivalence if

M
2

{M^2}

is compact, or if

M
2

{M^2}

is complete and the metric topology is used.
A necessary and sufficient condition for 𝛽𝑋\𝑋 to be an indecomposable continuum
Authors: Dickman R. F.
Published: June 2010

In his dissertation, David Bellamy has shown that if

I
=
[
0
,
1
)

I = [0,1)

, then

β
I
∖
I

\beta I\backslash I

is an indecomposable continuum, and R. G. Woods, in his dissertation, obtained the same result and in addition showed that for

m
>
1
,
β

R
m

∖

R
m

m > 1,\beta {R^m}\backslash {R^m}

is a decomposable continuum. In this note we give a necessary and sufficient condition for

β
X
∖
X

\beta X\backslash X

to be an indecomposable continuum when X is a locally connected generalized continuum.
Zeros of 𝜁^{’}(𝑠) in the critical strip
Authors: Spira Robert
Published: June 2010

It is shown that the abscissa of convergence for the Dirichlet series
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis negative 1 right-parenthesis Superscript k Baseline left-parenthesis 1 minus 2 Superscript 1 minus s Baseline right-parenthesis Superscript k plus 1 Baseline zeta Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis s right-parenthesis">

(
−
1

)
k

(
1
−

2

1
−
s

)

k
+
1

ζ

(
k
)

(
s
)

{( - 1)^k}{(1 - {2^{1 - s}})^{k + 1}}{\zeta ^{(k)}}(s)

is zero, where

ζ
(
s
)

\zeta (s)

is the Riemann zeta function. This implies the existence of infinitely many zeros of

ζ
′

(
s
)

\zeta ’(s)

in the critical strip.
An improved upper bound for global dimension of semigroup algebras
Authors: Nico William R.
Published: June 2010

An upper bound for the global dimension of the semigroup algebra of a finite regular monoid in terms of an ideal series for the monoid is determined by the partially ordered set of

I

\mathcal {I}

-classes of the monoid. In particular, if the monoid is combinatorial, the global dimension of the algebra is bounded by the sum of the global dimension of the coefficient ring and twice the length of the longest chain of

I

\mathcal {I}

-classes in the monoid.
On existence and a dominated convergence theorem for weighted 𝑔-summability
Authors: Wright Fred M. , Klasi Melvin L.
Published: June 2010

Let

(

w
1

,

w
2

,

w
3

)

({w_1},{w_2},{w_3})

be an ordered triple of real numbers such that

w
1

+

w
2

+

w
3

=
1

{w_1} + {w_2} + {w_3} = 1

. Let g be a real-valued function on the entire real axis which is of bounded variation on every closed interval. For f a real-valued function on the entire real axis which is bounded on a closed interval [a, b], we use the F. Riesz step function approach to define the concept of f being

(

w
1

,

w
2

,

w
3

)

({w_1},{w_2},{w_3})

g-summable over [a, b], and we define the integral
\[

[
F
,
(

w
1

,

w
2

,

w
3

)
]
s

∫
a
b

f
(
x
)
d
g
(
x
)

[F,({w_1},{w_2},{w_3})]s\int _a^b {f(x)dg(x)}

\]
when f has this property. We show that this integral extends the weighted refinement integral

[
F
,
(

w
1

,

w
2

,

w
3

)
]

∫
a
b

f
(
x
)
d
g
(
x
)

[F,({w_1},{w_2},{w_3})]\int _a^b {f(x)dg(x)}

for f’s as above. This paper generalizes the method of Pasquale Porcelli for the Stieltjes mean sigma integral. We present an existence theorem for the integral defined here involving saltus and continuous parts of g. We establish a convergence theorem for this integral which is analogous to the Lebesgue Dominated Convergence Theorem for the Lebesgue-Stieltjes integral.
Perturbations of dissipative operators with relative bound one
Authors: Chernoff Paul R.
Published: June 2010

Let A be the generator of a

(

C
0

)

({C_0})

contraction semigroup on a Banach space. Let B be a dissipative operator with densely defined adjoint. Assume that the inequality

‖

B
x

‖

≦

‖

A
x

‖

+
b

‖
x
‖

\left \| {Bx} \right \| \leqq \left \| {Ax} \right \| + b\left \| x \right \|

holds on the domain of A. Then the closure of

A
+
B

A + B

generates a

(

C
0

)

({C_0})

contraction semigroup.
Minimal surfaces with constant Gauss curvature
Authors: Chen Bang-yen
Published: June 2010

Minimal surfaces with constant Gauss curvature in real space forms are studied.
On an integral formula for closed hypersurfaces of the sphere
Authors: Houh Chorng-shi
Published: June 2010

In a compact oriented hypersurface

M
n

{M^n}

of the sphere

S

n
+
1

{S^{n + 1}}

the integral formula
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper M Superscript n Baseline Endscripts nabla upper K Subscript r Baseline d upper V equals n integral Underscript upper M Superscript n Baseline Endscripts left-parenthesis upper K Subscript r Baseline upper K 1 minus upper K Subscript r plus 1 Baseline right-parenthesis e d upper V">

∫

M
n

∇

K
r

d
V
=
n

∫

M
n

(

K
r

K
1

−

K

r
+
1

)
e
d
V

{\smallint _{{M^n}}}\nabla {K_r}dV = n{\smallint _{{M^n}}}({K_r}{K_1} - {K_{r + 1}})edV

is proved where

K
r

{K_r}

is the rth mean curvature, e is the unit normal of

M
n

{M^n}

in

S

n
+
1

{S^{n + 1}}

. Some applications are considered.
Simply-connected branched coverings of 𝑆³
Authors: Gordon C. McA. , Heil W.
Published: June 2010

It is shown that a simply connected covering of

S
3

{S^3}

branched over a torus knot or one of a certain class of links is

S
3

{S^3}

.
Two observations on the congruence extension property
Authors: Grätzer G. , Lakser H.
Published: June 2010

A pair of algebras

U

,

B

\mathfrak {U},\mathfrak {B}

with

B

\mathfrak {B}

a subalgebra of

U

\mathfrak {U}

is said to have the (Principal) Congruence Extension Property (abbreviated as PCEP and CEP, respectively) if every (principal) congruence relation of

B

\mathfrak {B}

can be extended to

U

\mathfrak {U}

. A pair of algebras

U

\mathfrak {U}

,

B

\mathfrak {B}

is constructed having PCEP but not CEP, solving a problem of A. Day. A result of A. Day states that if

B

\mathfrak {B}

is a subalgebra of

U

\mathfrak {U}

and if for any subalgebra

C

\mathfrak {C}

of

U

\mathfrak {U}

containing

B

\mathfrak {B}

, the pair

U

,

C

\mathfrak {U},\mathfrak {C}

has PCEP, then

U

,

B

\mathfrak {U},\mathfrak {B}

has CEP. A new proof of this theorem that avoids the use of the Axiom of Choice is also given.
The solvability of the equation 𝑎𝑥²+𝑏𝑦²=𝑐 in quadratic fields
Authors: Plotkin Neal
Published: June 2010

In a recent paper, L. J. Mordell gave necessary and sufficient conditions for the equation

a

x
2

+
b

y
2

=
c

a{x^2} + b{y^2} = c

to have algebraic integer solutions in the quadratic field

Q
(
√
(
−
n
)
)

Q(\surd ( - n))

. In this paper we drop the requirement that the solutions be algebraic integers. In particular, we prove that

a

x
2

+
b

y
2

=
c

a{x^2} + b{y^2} = c

has solutions in

Q
(
√
(
−
n
)
)

Q(\surd ( - n))

if and only if the quadratic form

a
b

t
2

−
b
c

u
2

−
a
c

v
2

−
n

w
2

ab{t^2} - bc{u^2} - ac{v^2} - n{w^2}

represents 0 over Q.
Quotient rings of endomorphism rings of modules with zero singular submodule
Authors: Hutchinson John , Zelmanowitz Julius
Published: June 2010

Throughout this paper (R, M, N, S) will denote a Morita context satisfying a certain nonsingularity condition. For such contexts we give necessary and sufficient conditions in terms of M and R for S to have a semisimple maximal left quotient ring; respectively a full linear maximal left quotient ring, a semisimple classical left quotient ring. In doing so we extend the corresponding well-known theorems for rings (employing them in the process) to endomorphism rings.
Geöcze area and a convergence property
Authors: Gariepy Ronald
Published: June 2010

Suppose f is a continuous mapping with finite Lebesgue area from a polyhedral region

X
⊂

R
k

X \subset {R^k}

into

R
n

,
2
≦
k
≦
n

{R^n},2 \leqq k \leqq n

. Let

f
=
l
∘
m

f = l \circ m

be the monotone-light factorization of f with middle space M. If f satisfies a “cylindrical condition” considered by T. Nishiura, then a current valued measure T over M can be associated with f by means of the Cesari-Weierstrass integral, and if

{

f
i

}

\{ {f_i}\}

is any sequence of quasi-linear maps

f
i

:
X
→

R
n

{f_i}:X \to {R^n}

converging uniformly to f with bounded areas, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis g right-parenthesis left-parenthesis phi right-parenthesis equals limit Underscript i right-arrow normal infinity Endscripts integral Underscript upper X Endscripts left-parenthesis g ring m right-parenthesis f Subscript i Superscript number-sign Baseline phi">

T
(
g
)
(
ϕ
)
=

lim

i
→
∞

∫
X

(
g
∘
m
)

f
i
#

ϕ

T(g)(\phi ) = \lim \limits _{i \to \infty } \int _X {(g \circ m)f_i^\# \phi }

\]
whenever

ϕ
\phi

is an infinitely differentiable k-form in

R
n

{R^n}

and g is a continuous real valued function on M which vanishes on

m
(
Bdry

X
)

m (\text {Bdry} \; X)

. The total variation measure of T, taken with respect to mass, coincides with the Geöcze area measure over M.
On the location of zeros of second-order differential equations
Authors: Komkov Vadim
Published: June 2010

The paper considers the location of zeros of the equation

(
α
(
t
)

x
′

)
′

+
γ
(
t
)
x
=
0
,
t
∈
[

t
0

,

t
1

]

(\alpha (t)x’)’ + \gamma (t)x = 0,t \in [{t_0},{t_1}]

. The following theorem is proved. Let

[
a
,
a
+
T
]
,
T
=
n
a

[a,a + T],T = na

(n a positive integer), be a subset of

[

t
0

,

t
1

]

[{t_0},{t_1}]

. Denote

ω
=
π

/

T

\omega = \pi /T

. Let the coefficient functions obey the inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript a Superscript a plus upper T Baseline left-brace gamma left-parenthesis t right-parenthesis minus omega squared alpha left-parenthesis t right-parenthesis sine squared left-parenthesis omega t right-parenthesis right-brace d t greater-than omega squared integral Subscript a Superscript a plus upper T Baseline left-brace alpha cosine 2 omega t right-brace d t">

∫
a

a
+
T

{
γ
(
t
)
−

ω
2

α
(
t
)

sin
2

(
ω
t
)
}
d
t
>

ω
2

∫
a

a
+
T

{
α
cos
⁡
2
ω
t
}
d
t

\smallint _a^{a + T}\{ \gamma (t) - {\omega ^2}\alpha (t){\sin ^2}(\omega t)\} dt > {\omega ^2}\smallint _a^{a + T}\{ \alpha \cos 2\omega t\} dt

. Then every solution of this equation will have a zero on

[
a
,
a
+
T
]

[a,a + T]

. A more general form of this theorem is also proved.
On the arithmetic mean of Fourier-Stieltjes coefficients
Authors: Georgakis Constantine
Published: June 2010

Let

{

a
n

}

n
=
0

∞

\{ {a_n}\} _{n = 0}^\infty

be the cosine Fourier-Stieltjes coefficients of the Borel measure

μ
\mu

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace a 0 comma left-parenthesis a 1 plus midline-horizontal-ellipsis plus a Subscript n Baseline right-parenthesis slash n right-brace Subscript n equals 1 Superscript normal infinity Baseline equals left-brace left-parenthesis upper T a right-parenthesis Subscript n Baseline right-brace Subscript n equals 0 Superscript normal infinity">

{

a
0

,
(

a
1

+
⋯
+

a
n

)

/

n

}

n
=
1

∞

=
{

(
T
a

)
n

}

n
=
0

∞

\{ {a_0},({a_1} + \cdots + {a_n})/n\} _{n = 1}^\infty = \{ {(Ta)_n}\} _{n = 0}^\infty

be the sequence of their arithmetic means. Then

∑

n
=
0

∞

(
T
a
)

n

cos
⁡
n
x

\sum \nolimits _{n = 0}^\infty {{{(Ta)}_n}\cos nx}

is a Fourier-Stieltjes series. Moreover, (a)

∑

n
=
0

∞

(
T
a
)

n

cos
⁡
n
x

\sum \nolimits _{n = 0}^\infty {{{(Ta)}_n}\cos nx}

is a Fourier series if and only if

(
T
a

)
n

→
0

{(Ta)_n} \to 0

at infinity or, equivalently, the measure

μ
\mu

is continuous at the origin, (b)

∑

n
=
1

∞

(
T
a
)

n

sin
⁡
n
x

\sum \nolimits _{n = 1}^\infty {{{(Ta)}_n}\sin nx}

is a Fourier series if and only if the function

x

−
1

μ
(
[
0
,
x
)
)

{x^{ - 1}}\mu ([0,x))

is in

L
1

[
0
,
π
]

{L^1}[0,\pi ]

. These results form the best possible analogue of a theorem of G. Goes, concerning arithmetic means of Fourier-Stieltjes sine coefficients, and improve considerably the theorems of L. Fejér and N. Wiener on the inversion and quadratic variation of Fourier-Stieltjes coefficients.
On the Laplace transform of a temperate distribution supported by a cone
Authors: Petersen Bent E.
Published: June 2010

The temperate distributions supported by a closed convex salient cone are characterized by explicit polynomial growth of the Laplace transform at infinity and at the boundary of the cylinder over the dual cone. This result is then used to characterize by their Laplace transforms the smooth kernels of degree

λ
−
n

\lambda - n

where

λ
\lambda

is positive.
On the Weyl spectrum of a Hilbert space operator
Authors: Baxley John V.
Published: June 2010

Using the perturbation definition of the Weyl spectrum, conditions are given on a closed (possibly unbounded) linear operator T in a Hilbert space which allow the Weyl spectrum to be characterized as a subset of the spectrum of T.
The convergence determining class of connected open sets in product spaces
Authors: Landers Dieter
Published: June 2010

It is proved in this paper that each sequence of measures with values in a topological group—defined on the Borel field of a finite or countable product of connected, locally connected, separable metric spaces—which is Cauchy convergent for all connected open sets is Cauchy convergent for all Borel sets, too.
On convolutions with the Möbius function
Authors: Segal S. L.
Published: June 2010

By using the results of [6], it is proved that for an extensive class of increasing functions h,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel sigma-summation Underscript 1 less-than-over-equals d less-than-over-equals x Endscripts StartFraction mu left-parenthesis d right-parenthesis Over d EndFraction h left-parenthesis StartFraction x Over d EndFraction right-parenthesis tilde x h prime left-parenthesis x right-parenthesis as x right-arrow normal infinity EndLayout">

(

∗

)

∑

1
≦
d
≦
x

μ
(
d
)

d

h

(

x
d

)

∼
x

h
′

(
x
)

as

x
→
∞

\begin{equation}\tag {$*$}\sum \limits _{1 \leqq d \leqq x} {\frac {{\mu (d)}}{d}h\left ( {\frac {x}{d}} \right )} \sim xh’(x)\quad {\text {as}}\;x \to \infty \end{equation}

where

μ
\mu

denotes the Möbius function. This result incidentally settles affirmatively Remark (iii) of [6], and refines the Tauberian Theorem 2 of that paper. It is also shown that one type of condition imposed in [6] is necessary to the truth of the cited Theorem 2, at least if some sort of quasi-Riemann hypothesis is true. Nevertheless, examples are given to show that on the one hand

(
∗

)

( ^\ast )

may be true for functions not covered by the first theorem of this paper, and on the other that some sort of nonnaïve condition on a function h is necessary to ensure the truth of

(
∗

)

(^\ast )

.
The Schwarzian derivative and univalent functions
Authors: Ozaki Shigeo , Nunokawa Mamoru
Published: June 2010

In this paper we prove under certain conditions the function

w
=
f
(
z
)

w = f(z)

is univalent in

|

z

|

>
1

|z| > 1

.
On modular complemented and annihilator algebras
Authors: Husain T. , Wong Pak Ken
Published: June 2010

Let A be a modular complemented algebra. We give some necessary and sufficient conditions for some special classes of algebras A to be annihilator or dual algebras.
A set-theoretic proposition implying the metrizability of normal Moore spaces
Authors: Tall Franklin D.
Published: June 2010

A set-theoretic proposition is shown to be equivalent to a topological statement implying that all first countable normal Hausdorff spaces are collectionwise normal, and hence that all normal Moore spaces are metrizable.
Representations of Euler classes
Authors: Osborn Howard
Published: June 2010

For any endomorphism

K
K

of an oriented module

F
F

with inner product there is an element

pf 

K

{\text {pf }}K

in the ground ring

R
R

, a constant multiple of the classical pfaffian in the case

F
=

R

2
n

F = {R^{2n}}

. If

R
R

is the algebra of even-dimensional differential forms on a smooth manifold, and if

F
F

is the tensor product of

R
R

and the module of sections of an oriented

2
n

2n

-plane bundle, then any connection in the bundle induces a curvature transformation

K
:
F
→
F

K:F \to F

for which

(
4
π

)

−
n

pf 

K

{(4\pi )^{ - n}}{\text {pf }}K

represents the Euler class. Properties of Euler classes are immediate consequences of corresponding properties of

pf

{\text {pf}}

.
Normal subgroups contained in the Frattini subgroup
Authors: Hill W. Mack , Wright Charles R. B.
Published: June 2010

Let H be a normal subgroup of the finite group G. If H has a subgroup K which is normal in G, satisfies

|

K

|

>

|

K
∩

Z
1

(
H
)

|

=
p

|K| > |K \cap {Z_1}(H)| = p

and is not of nilpotence class 2, then H is not contained in the Frattini subgroup of G.
A note on higher derivations and integral dependence
Authors: Brown William C.
Published: June 2010

In this note we prove the following: Theorem. Let R’ be an associative commutative ring with identity. Suppose R’ is an integral extension of R, and

δ
=
{

δ
i

}

\delta = \{ {\delta _i}\}

is a higher derivation on R’ which restricts to a higher derivation on R. Suppose p is a prime ideal in R which is differential under

δ
\delta

. Then there exists a prime ideal p’ in R’ such that p’ is

δ
\delta

-differential and

p
′

∩
R
=
p

p’ \cap R = p

.
On free products of finite abelian groups
Authors: Anshel Michael , Prener Robert
Published: June 2010

The purpose of this note is to show that if G is the free product of finitely many, finite abelian groups then the commutator subgroup is a finitely generated free group whose rank depends only on the number and orders of the factors. Moreover, we shall present a constructive procedure for obtaining a basis of this free group using the Kurosh rewriting process.