bluuebunny/crossref_metadata_2025 huggingface dataset:
𝐻₂ of the commutator subgroup of a knot group
Authors: Sumners D. W.
Published: June 2010

A short topological proof is given for the well-known theorem that if

G
G

is a knot group and

G
′

G’

its commutator subgroup, then

H
2

(

G
′

;
Z
)
=
0

{H_2}(G’;Z) = 0

.
Semilattices which are embeddable in a product of 𝑚𝑖𝑛 intervals
Authors: Stepp J. W.
Published: June 2010

In this paper we give necessary and sufficient conditions for a locally compact semilattice to be embeddable in a product of min intervals.
The categories of 𝑝-rings are equivalent
Authors: Stringall R. W.
Published: June 2010

Let p and q be prime numbers. Let

R
p

{R_p}

and

R
q

{R_q}

denote, respectively, the categories of p-rings and q-rings with ring homomorphisms as morphisms. Then

R
p

{R_p}

and

R
q

{R_q}

are equivalent categories. In particular, the category of all Boolean rings is equivalent to

R
p

{R_p}

.
On the arithmetic nature of definite integrals of rational functions.
Authors: Van der Poorten A. J.
Published: June 2010

A. Baker’s theorems on linear forms in the logarithms of algebraic numbers imply information on the arithmetic nature of definite integrals of rational functions. This paper pro vides a convenient formulation of these implied results.
A note on modules over a commutative regular ring
Authors: Teply Mark L.
Published: June 2010

An example is given of a commutative, von Neumann regular ring R, which has a module A satisfying the following conditions: (1)

T
(
A
)
=
{
a
∈
A

|

(
0
:
a
)

T(A) = \{ a \in A|(0:a)

is an essential ideal of R} is a cyclic R-module; (2)

A

/

T
(
A
)

A/T(A)

is a cyclic R-module; and (3)

T
(
A
)

T(A)

is not a direct summand of A. This answers in the negative a question raised by R. S. Pierce.
Cogenerator endomorphism rings
Authors: Wagoner Ronald L.
Published: June 2010

If

R
R

is a ring and

P
P

is a finitely generated projective right

R
R

-module, what properties of

R
R

does the

R
R

-endomorphism ring of

P
P

inherit? Rosenberg and Zelinsky have shown that if

R
R

is quasi-Frobenius, and

P
P

also has every simple epimorphic image isomorphic to a submodule, then the

R
R

-endomorphism ring of

P
P

is also quasi-Frobenius. In this paper we show that if

R
R

is a cogenerator ring, and

P
P

is a finitely generated projective right

R
R

-module with every simple epimorphic image isomorphic to a submodule of

P
P

, then the

R
R

-endomorphism ring of

P
P

is also a cogenerator ring.
Ideals in the modular group ring of a 𝑝-group
Authors: Hill E. T.
Published: June 2010

We show that if

G
G

has order

p
n

{p^n}

then the group ring has a chain of

p
n

+
1

{p^n} + 1

ideals and that the radical powers are canonical in the lattice of ideals. We then prove that if

G
G

is abelian,

G
G

is determined by the lattice of ideals.
On idempotent, commutative, and nonassociative groupoids
Authors: Grätzer G. , Padmanabhan R.
Published: June 2010

For an algebra

A

=

⟨

A
;
F

⟩

\mathfrak {A} = \left \langle {A;F} \right \rangle

and for

n
≧
2

n \geqq 2

, let

p
n

(

A

)

{p_n}(\mathfrak {A})

denote the number of essentially

n
n

-ary polynomials of

A

\mathfrak {A}

. J. Dudek has shown that if

A

\mathfrak {A}

is an idempotent and nonassociative groupoid then

p
n

(

A

)
≧
n

{p_n}(\mathfrak {A}) \geqq n

for all

n
>
2

n > 2

. In this paper this result is improved for the commutative case to show that for such groupoids
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper A comma p Subscript n Baseline left-parenthesis German upper A right-parenthesis greater-than-over-equals one third left-parenthesis 2 Superscript n Baseline minus left-parenthesis negative 1 right-parenthesis Superscript n Baseline right-parenthesis">

A

,

p
n

(

A

)
≧

1
3

(

2
n

−

(
−
1

)
n

)

\mathfrak {A},{p_n}(\mathfrak {A}) \geqq \frac {1}{3}({2^n} - {( - 1)^n})

for all

n
≧
2

n \geqq 2

(Theorem 1) and that this is the best possible result. Those groupoids for which this lower bound is attained are completely characterized. In fact, the relevant result proved below is much stronger (Theorem 3). From these and other known results it is deduced that the sequence

⟨

0
,
0
,
1
,
3

⟩

\left \langle {0,0,1,3} \right \rangle

has the minimal extension property.
Some remarks on injective envelopes
Authors: Garling D. J. H.
Published: June 2010

Another proof is given of the fact that every metric space (respectively every Banach space) has an injective envelope.
On special generators for properly infinite von Neumann algebras
Authors: Wogen W. R.
Published: June 2010

It is known that every properly infinite von Neumann algebra

A

\mathcal {A}

on a separable Hilbert space has a single generator. We show in this paper that a generator for

A

\mathcal {A}

may be chosen from some special classes of operators. In particular each of the following classes of operators contains a generator for

A

\mathcal {A}

: the hyponormals, the nilpotents, the transcendental quasinilpotents, and the unimodular contractions. We also show that a generator for

A

\mathcal {A}

may be chosen with arbitrarily prescribed spectrum.
Concordant mappings and the concordant-dissonant factorization of an arbitrary continuous function
Authors: Collins P. J.
Published: June 2010

The property of concordance (weaker than monotonicity) is introduced, and a characterization of concordant mappings using quotient spaces enables the derivation of a new factorization, the concordant-dissonant factorization of an arbitrary continuous function.
Irreducible Lie algebras of infinite type
Authors: Wilson Robert Lee
Published: June 2010

Let V be a finite dimensional vector space over an algebraically closed field of characteristic

≠
2
,
3
,
5

\ne 2,3,5

. It is shown that if

L
⊆

gl

(
V
)

L \subseteq {\text {gl}}(V)

is an irreducible Lie algebra of infinite type then either

V
=
2
r
≧
4

V = 2r \geqq 4

and

L
=

sp

(
V
)
,
dim

V
=
2
r
≧
4

L = {\text {sp}}(V),\dim \,V = 2r \geqq 4

and

L
=

csp

(
V
)

L = {\text {csp}}(V)

, or there exists

A
∈
L

A \in L

such that

A
≠
0
=

(

ad

A

)
2

A \ne 0 = {({\text {ad}}\;A)^2}

. As a corollary we obtain E. Cartan’s classification of the irreducible Lie algebras of infinite type over C.
A brief proof of Cauchy’s integral theorem
Authors: Dixon John D.
Published: June 2010

A short proof of Cauchy’s theorem for circuits homologous to 0 is presented. The proof uses elementary local properties of analytic functions but no additional geometric or topological arguments.
A class of commutative Banach algebras with unique complete norm topology and continuous derivations
Authors: Lindberg John A.
Published: June 2010

Let A be a semisimple commutative complex algebra with identity and

α
(
x
)

\alpha (x)

a monic polynomial over A. Two results are proved. If

B
=
A
[
x
]

/

(
α
(
x
)
)

B = A[x]/(\alpha (x))

is a Banach algebra under some norm, then B has a unique complete norm topology. Furthermore, B has nontrivial derivations if and only if B has a nontrivial radical.
The Weyl essential spectrum
Authors: Bouldin Richard
Published: June 2010

Using a modest geometric hypothesis the main theorem of these results classifies the Weyl essential spectrum and the Browder essential spectrum according to the standard terminology for the spectrum of a Hilbert space operator.
Every isometry is reflexive
Authors: Deddens James A.
Published: June 2010

A bounded linear operator

A
A

on a Hilbert space

H

\mathcal {H}

is called reflexive if any bounded linear operator which leaves invariant the invariant subspaces of

A
A

is a limit of polynomials in

A
A

in the weak operator topology. In this note we prove that every isometry

V
V

on a Hilbert space

H

\mathcal {H}

is reflexive.
Nonoscillation properties of a nonlinear differential equation
Authors: Hammett Michael E.
Published: June 2010

Sufficient conditions are given for the approach to zero of all nonoscillatory solutions of

(
p
(
t
)

x
′

)
′

+
q
(
t
)
g
(
x
)
=
f
(
t
)

(p(t)x’)’ + q(t)g(x) = f(t)

. The conditions are related to an oscillation theorem of N. P. Bhatia concerning the equation

(
p
(
t
)

x
′

)
′

+
q
(
t
)
g
(
x
)
=
0

(p(t)x’)’ + q(t)g(x) = 0

.
Property 𝑃 and direct integral decomposition of 𝑊*-algebras
Authors: Willig Paul
Published: June 2010

If

A

\mathcal {A}

is a

W

-

∗

W{{\text {-}}^\ast }

algebra on separable Hilbert space H, and if

A

(
λ
)

\mathcal {A}(\lambda )

are the factors in the direct integral decomposition of

A

\mathcal {A}

, then

P

=
{
λ

|

A

(
λ
)

\mathcal {P} = \{ \lambda |\mathcal {A}(\lambda )

has property P} is

μ
\mu

-measurable, and

A

\mathcal {A}

, has property P iff

A

(
λ
)

\mathcal {A}(\lambda )

has property P

μ
\mu

-a.e.
Conditions for continuity of certain open monotone functions
Authors: Hagan Melvin R.
Published: June 2010

In this paper continuity of certain open monotone functions is obtained by assuming for the domain and/or range various combinations of the properties of a metric continuum, regular metric continuum, semilocal connectedness, and hereditary local connectedness. An open monotone connected function from a hereditarily locally connected separable metric continuum onto a separable metric continuum is continuous. If the domain is a regular separable metric continuum, an upper semicontinuous decomposition and resulting monotone-light factorization yield continuity of an open monotone function with closed point inverses.
The canonical Schoenflies theorem
Authors: Gauld David B.
Published: June 2010

M. Brown has shown that if

f
:

S

n
−
1

×

I
′

→

S
n

f:{S^{n - 1}} \times I’ \to {S^n}

, where

I
′

=
[
−
1
,
1
]

I’ = [ - 1,1]

, is an embedding, then the closure of either complementary domain of

f
(

S

n
−
1

×
0
)

f({S^{n - 1}} \times 0)

is homeomorphic to

B
n

{B^n}

; in fact there is an embedding

g
:

B
n

→

S
n

g : {B^n} \to {S^n}

satisfying

g

|

S

n
−
1

=
f

|

S

n
−
1

×
0

g|{S^{n - 1}} = f|{S^{n - 1}} \times 0

. This paper shows that the choice of embedding

g
g

can be made to be “canonical,” i.e. if

f
′

f’

is an embedding near

f
f

in the compact-open sense, then the embedding

g
′

g’

corresponding to

f
′

f’

is near

g
g

.
The cardinality of ultrapowers—an example
Authors: Adler Andrew
Published: June 2010

Assume the axiom of measurable cardinals. If

D
D

is an

ω
\omega

-incomplete uniform ultrafilter on

I
I

, and

A
A

is infinite, it is still not necessarily the case that

A
I

/

D

{A^I}/D

has the same cardinality as

A
I

{A^I}

.
Rings generated by the inner-automorphisms of non-abelian groups
Authors: Chandy A. John
Published: June 2010

The endomorphisms of an abelian group form a ring. The characterization of groups in which endomorphisms generate a ring is still an open question. It is shown here that the inner-automorphisms of a group generate a ring if and only if the conjugate elements in the group commute.
Subharmonic versions of Fatou’s theorem
Authors: Meek James L.
Published: June 2010

A theorem of Fatou concerning the boundary behavior of bounded harmonic functions in the unit disk is extended to normal subharmonic functions. A question—which has been answered in the normal holomorphic and normal harmonic cases—concerning existence of Fatou points for normal subharmonic functions is posed.
Oscillation of nonlinear systems of matrix differential equations
Authors: Kartsatos A. G.
Published: June 2010

For systems of matrix equations of the form
\[

U
′

=
A
(
t
,
U
,
V
)
V
,

V
′

=
−
B
(
t
,
U
,
V
)

U’ = A(t,U,V)V,\quad V’ = - B(t,U,V)

\]
it is shown here that the oscillation problem can be reduced to the corresponding problem of “associated” (in some sense) scalar equations for which there exist numerous results. Furthermore, it is also shown that many of the existing results concerning the equation
\[

(
A
(
t
)

U
′

)
′

+
B
(
t
,
U
,

U
′

)
U
=
0

(A(t)U’)’ + B(t,U,U’)U = 0

\]
can be considerably improved by application of the above method.
Quasilinear systems with several periodic solutions
Authors: Cronin Jane
Published: June 2010

By using topological degree, it is proved that for a certain class of quasilinear systems of ordinary differential equations of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove x With dot equals upper A left-parenthesis t right-parenthesis x plus epsilon mu f left-parenthesis x comma t comma mu right-parenthesis plus mu g left-parenthesis x comma t comma mu right-parenthesis plus h left-parenthesis t right-parenthesis">

x
˙

=
A
(
t
)
x
+
ϵ
μ
f
(
x
,
t
,
μ
)
+
μ
g
(
x
,
t
,
μ
)
+
h
(
t
)

\dot x = A(t)x + \epsilon \mu f(x,t,\mu ) + \mu g(x,t,\mu ) + h(t)

\]
where

ϵ
,
μ

\epsilon ,\mu

are small parameters and A, f, g, h are periodic in t, there exist at least two periodic solutions.
Families of mutually complementary topologies
Authors: Anderson B. A.
Published: June 2010

Several lattices of topologies on an infinite set are considered and bounds are given for the sup of the set of cardinals d such that there is a family of d mutually complementary topologies. Large classes of

ℵ
0

{\aleph _0}

-topologies are shown to have

ℵ
0

{\aleph _0}

-complements, and an example is given to prove that complementation is not, in general, a very selective topological operation.
A class of arcwise connected continua
Authors: Hagopian Charles L.
Published: June 2010

It is known that every bounded semi-aposyndetic plane continuum which does not separate the plane is arcwise connected. To show that this theorem remains true if the phrase “does not separate the plane” is replaced by “does not have infinitely many complementary domains” is the primary purpose of this paper.
Sheltered modules and rings
Authors: Warner Seth
Published: June 2010

The problem of determining when a module (ring) admits an indiscrete, Hausdorff linear (ideal) topology is discussed in terms of sheltered modules (rings).
Subalgebras of 𝐵(𝑋)
Authors: Wilansky Albert
Published: June 2010

Two classes of subalgebras of the bounded operators on a Banach space X are introduced. This gives an abstract setting, generalizations, and some shorter proofs of results in summability which are the special case

X
=
c

X = c

.
On the absolute continuity of the limit random variable in the supercritical Galton-Watson branching process
Authors: Athreya Krishna B.
Published: June 2010

Let

{

Z
n

:
n
≧
0
}

\{ {Z_n}:n \geqq 0\}

be a simple Galton-Watson branching process with offspring distribution

{

p
j

}

\{ {p_j}\}

satisying

1
>
∑

j

p
j

>
∞

1 > \sum {j{p_j} > \infty }

. It is known that there exist constants

C
n

{C_n}

such that

W
n

≡

Z
n

C
n

{W_n} \equiv {Z_n}{C_n}

converges with probability one to a nondegenerate limit random variable W. Here we show that this W is always absolutely continuous on

(
0
,
∞
)

(0,\infty )

.
On the Fourier-algebra of a locally compact amenable group
Authors: Flory Volker
Published: June 2010

Let G be a locally compact amenable group. The elements of the Fourier-algebra A(G) are characterized with the aid of the theorem of Schoenberg-Eberlein-Eymard.
A convergence question in 𝐻^{𝑝}
Authors: Scheinberg Stephen
Published: June 2010

Let

ϕ
∈

H
p

\phi \in {H^p}

(unit disc),

0
>
p
>
∞

0 > p > \infty

. and let

ϕ
r

(
z
)
=
ϕ
(
r
z
)
,
r
>
1

{\phi _r}(z) = \phi (rz),r > 1

. If

ϕ
\phi

contains a nontrivial inner factor, it is known that

ϕ

/

ϕ
r

\phi /{\phi _r}

is unbounded in

H
p

{H^p}

-norm. We prove that if

ϕ
\phi

is analytic on the closed disc and has no zeros on the open disc, then

ϕ

/

ϕ
r

→
1

\phi /{\phi _r} \to 1

in

H
p

{H^p}

, as

r
→
1

r \to 1

. The same conclusion follows if

1

/

ϕ
∈

H
∞

1/\phi \in {H^\infty }

. We construct an outer function

ϕ
\phi

which is continuous on the closed disc, analytic for

z
≠
1

z \ne 1

, and such that

ϕ

/

ϕ
r

\phi /{\phi _r}

is unbounded in every

H
p

{H^p}

.
Uniqueness of solutions of certain boundary value problems for ultrahyperbolic equations
Authors: Diaz J. B. , Young Eutiquio C.
Published: June 2010

The paper gives necessary and sufficient conditions for the uniqueness of solutions of the Dirichlet and Neumann problems for ultrahyperbolic partial differential equations in cylindrical domains.
Spectrum of 𝑋 satisfying 0≤𝑋^{𝑝}≤𝑋
Authors: Gellar Ralph
Published: June 2010

A sharp estimate is found for the possible spectrum of an element satisfying the polynomial inequality

0
≦

X
p

≦
X

0 \leqq {X^p} \leqq X

in a Dedekind

σ
\sigma

-complete real linear algebra.
On a geometric property of the set of invariant means on a group
Authors: Chou Ching
Published: June 2010

If G is a discrete group and

x
∈
G

x \in G

then

x
∼

x^\sim

denotes the homeomorphism of

β
G

\beta G

onto

β
G

\beta G

induced by left multiplication by x. A subset K of

β
G

\beta G

is said to be invariant if it is closed, nonempty and

x
∼

∅
K
⊂
K

x^\sim \emptyset K \subset K

for each

x
∈
G

x \in G

. Let

M
L
(
G
)

ML(G)

denote the set of left invariant means on G. (They can be considered as measures on

β
G

\beta G

.) Let G be a countably infinite amenable group and let K be an invariant subset of

β
G

\beta G

. Then the nonempty

w
∗

{w^ \ast }

-compact convex set

M
(
G
,
K
)
=
{
ϕ
∈
M
L
(
G
)
:

suppt

ϕ
⊂
K
}

M(G,K) = \{ \phi \in ML(G):{\text {suppt}}\phi \subset K\}

has no exposed points (with respect to

w
∗

{w^ \ast }

-topology). Therefore, it is infinite dimensional.
Abstract martingales in Banach spaces
Authors: Uhl J. J.
Published: June 2010

The concept of martingale is generalized from probability theory to the setting of Banach spaces. Convergent martingales are characterized. An application to a Radon-Nikodym theorem for vector measures is given.
A characterization of order topologies by means of minimal 𝑇₀-topologies
Authors: Thron W. J. , Zimmerman Susan J.
Published: June 2010

In this article we give a purely topological characterization for a topology

ℑ
\Im

on a set

X
X

to be the order topology with respect to some linear order

R
R

on

X
X

, as follows. A topology

ℑ
\Im

on a set

X
X

is an order topology iff

(
X
,
ℑ
)

(X,\Im )

is a

T
1

{T_1}

-space and

ℑ
\Im

is the least upper bound of two minimal

T
0

{T_0}

-topologies [Theorem 1 ]. From this we deduce a purely topological description of the usual topology on the set of all real numbers. That is, a topological space

(
X
,
ℑ
)

(X,\Im )

is homeomorphic to the reals with the usual topology iff

(
X
,
ℑ
)

(X,\Im )

is a connected, separable,

T
1

{T_1}

-space, and

ℑ
\Im

is the least upper bound of two noncompact minimal

T
0

{T_0}

-topologies [Theorem 2].
A topological characterization of the dilation in 𝐸ⁿ
Authors: Husch L. S.
Published: June 2010

A topological characterization is given to determine whether a homeomorphism of Euclidean

n
n

-space,

n
≠
4
,
5

n \ne 4,5

, is topologically equivalent to the dilation

x
→

1
2

x

x \to \frac {1}{2}x

.
A characterization of hereditarily indecomposable continua
Authors: Crawford Albert L. , Jobe John
Published: June 2010

In this paper a characterization of a hereditarily indecomposable continuum is stated and proved. The motivation for this characterization is a theorem in a recent article by John Jobe.
Bases in 𝐿₂ spaces with applications to stochastic processes with orthogonal increments
Authors: Cambanis Stamatis
Published: June 2010

Orthonormal bases are explicitly given for

L
2

{L_2}

spaces on the real line. They are applied in obtaining orthogonal expansions for certain classes of stochastic processes and in constructing stochastic processes with orthogonal increments.
A differentiation theorem for functions defined on the dyadic rationals
Authors: Lindahl R. J.
Published: June 2010

In this paper we show that under certain conditions a real-valued function defined on an interval of dyadic rational numbers is a monotone function. One of these conditions involves a generalized differentiability property. From this result we offer a new proof of a conjecture of N. Fine concerning the uniqueness of solution of Walsh series.
Theorems of Accola type for higher dimensional manifolds
Authors: Chen Su-shing
Published: June 2010

Two theorems of Accola concerning automorphisms of Riemann surfaces can be extended to higher dimensional manifolds. Formulas are obtained concerning signatures of compact oriented 4k-dimensional differentiable manifolds and Euler-Poincaré characteristics of compact differentiable manifolds and compact complex manifolds.
A note on the Kleinecke-Shirokov theorem and the Wintner-Wielandt-Halmos theorem
Authors: Lin C.-S.
Published: June 2010

We extend the Kleinecke-Shirokov theorem to the almost commutative case. From this result we prove the Wintner-Wielandt-Halmos theorem.
Baire functions and their restrictions to special sets
Authors: Peek Darwin E.
Published: June 2010

A function f from a complete and separable metric space X into the real numbers is of Baire class 1 iff for every nonempty perfect subset H of X,

f

|

H

f|H

contains a point where

f

|

H

f|H

is continuous. This paper examines a similar idea obtained by changing “perfect subset H” to “union of a countable collection of perfect subsets” in the preceding characterization of Baire class 1 functions. This new idea is also characterized by using “condensation points” and “totally imperfect sets.” Functions of this new type are of Baire class 1. However, the converse is false.
Quickly oscillating solutions of autonomous ordinary differential equations
Authors: Bernfeld Stephen R. , Lasota A.
Published: June 2010

We are concerned here with the asymptotic behavior of quickly oscillating solutions of systems of differential equations. It is shown that the limit of the norm of any quickly oscillating solution exists and is either equal to infinity or zero. We then determine asymptotic bounds on the solutions by imposing certain growth conditions on the right-hand side of the equation. Our results, when applied to second order equations, yield asymptotic behavior of both the solutions and its derivatives.
Almost periodic solutions of Poisson’s equation
Authors: Sibuya Yasutaka
Published: June 2010

It is well known that if

f
(
t
)

f(t)

is almost periodic with respect to a real variable

t
t

, and if

F
(
t
)
=

∫
t

f
(
s
)
d
s

F(t) = {\smallint ^t}f(s)ds

is bounded for

−
∞
>
t
>
∞

- \infty > t > \infty

, then

F
(
t
)

F(t)

is also almost periodic in

t
t

. We shall prove a similar result for Poisson’s equation.
A class of hypo-Dirichlet algebras
Authors: Brandstein A. G.
Published: June 2010

A method is given of constructing a new class of hypo-Dirichlet algebras of given real codimension.
On the direct product of 𝑉-groups
Authors: Minassian Donald P.
Published: June 2010

Let G and H be ordered groups such that every full order on a subgroup extends to a full order on the group; then the direct product,

G
×
H

G \times H

, need not have this property. In fact a stronger result holds.
The Hardy class of a Bazilevič function and its derivative
Authors: Miller Sanford S.
Published: June 2010

The Bazilevič function

f
(
z
)

f(z)

defined in

Δ
:

|

z

|

>
1

\Delta :|z| > 1

by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis identical-to left-bracket alpha integral Subscript 0 Superscript z Baseline upper P left-parenthesis zeta right-parenthesis g left-parenthesis zeta right-parenthesis Superscript alpha Baseline zeta Superscript negative 1 Baseline d zeta right-bracket Superscript 1 slash alpha">

f
(
z
)
≡

[
α

∫
0
z

P
(
ζ
)
g

(
ζ

)
α

ζ

−
1

d
ζ

]

1

/

α

f(z) \equiv {[\alpha \smallint _0^zP(\zeta )g{(\zeta )^\alpha }{\zeta ^{ - 1}}d\zeta ]^{1/\alpha }}

where

g
(
ζ
)

g(\zeta )

is starlike in

Δ
\Delta

,

P
(
ζ
)

P(\zeta )

is regular with Re

P
(
ζ
)
>
0

P(\zeta ) > 0

in

Δ
\Delta

and

α
>
0

\alpha > 0

is univalent. The class of such functions contains many of the special classes of univalent functions. The author determines the Hardy classes to which

f
(
z
)

f(z)

and

f
′

(
z
)

f’(z)

belong. In addition if

f
(
z
)
=

∑
0
∞

a
n

z
n

f(z) = \sum \nolimits _0^\infty {{a_n}{z^n}}

the limiting value of

|

a
n

|

/

n

|{a_n}|/n

is obtained.
On rings satisfying [(𝑎,𝑏,𝑐),𝑑]=0
Authors: Thedy Armin
Published: June 2010

A simple nonassociative ring, in which the associators commute with all elements, is under mild additional assumptions either associative or commutative. This result cannot be extended to prime rings since a construction of semiprime rings gives counterexamples.
Sets accessible at each point only by wild arcs
Authors: Miller Gary Glenn
Published: June 2010

A positional pathology as described in the title is shown to occur in three-space. We construct an arcwise accessible point set

M
M

such that each arc to

M
M

is locally knotted at uncountably many points. In addition we give examples of connected, locally connected point sets

S
S

and

T
T

which are accessible at each point only by wild arcs and tame arcs respectively. The point set

M
M

has countably many components, and each of these is a tame finite

2
2

-complex. Moreover

E
3

−
M

{E^3} - M

is locally connected and arcwise connected.
On algebras satisfying the identity (𝑦𝑥)𝑥+𝑥(𝑥𝑦)=2(𝑥𝑦)𝑥
Authors: Chaffer Robert A.
Published: June 2010

Simple, strictly power-associative algebras satisfying the identity

(
y
x
)
x
+
x
(
x
y
)
=
2
(
x
y
)
x

(yx)x + x(xy) = 2(xy)x

over a field of characteristic not 2 or 3 have been classified by F. Kosier as commutative Jordan, quasi-associative, or of degree less than three. In the present paper those of degree three or greater are shown to be commutative, which eliminates the quasi-associative case mentioned above.
Pólya’s property 𝑊 and factorization—A short proof
Authors: Ristroph Robert
Published: June 2010

For an

n
n

th order linear differential expression, the equivalence of Pólya’s Property

W

{\text {W}}

and factorization into first order expressions is proven directly and briefly.
Stability of a scaler differential equation
Authors: Grimmer R.
Published: June 2010

Sufficient conditions are given for the stability and asymptotic stability of the zero solution of a scalar differential equation frequently encountered in the comparison method.
Skeleta of complexes with low 𝑀𝑈_{∗} projective dimension
Authors: Johnson David Copeland
Published: June 2010

Let

M

U
∗

(
X
)

M{U_ \ast }(X)

be the unitary bordism of a finite complex X. Let

X
p

{X^p}

be the p-skeleton of X. This note proves that certain properties of

M

U
∗

(
X
)

M{U_ \ast }(X)

are shared by

M

U
∗

(

X
p

)

M{U_\ast }({X^p})

when the projective dimension of

M

U
∗

(
X
)

M{U_\ast }(X)

as a

M

U
∗

M{U_\ast }

module is low (0, 1, or 2).
Separating 𝑝-bases and transcendental extension fields
Authors: Mordeson J. N. , Vinograde B.
Published: June 2010

Let

L

/

K

L/K

denote an extension field of characteristic

p
≠
0

p \ne 0

. It is known that if

L

/

K

L/K

has a finite separating transcendence base, then every relative

p
p

-base of

L

/

K

L/K

is a separating transcendence base of

L

/

K

L/K

. In this paper we show that when every relative

p
p

-base of

L

/

K

L/K

is a separating transcendence base of

L

/

K

L/K

, then the transcendence degree of

L

/

K

L/K

is finite. We also illustrate the connection between the finiteness of transcendence degree of

L

/

K

L/K

and the property that

L

/

K
(
X
)

L/K(X)

is separable algebraic for every relative

p
p

-base

X
X

of

L

/

K

L/K

.
Riemann surfaces in complex projective spaces
Authors: Chen Bang-yen , Ludden Gerald D.
Published: June 2010

The complex projective line and the complex quadric are the only compact Riemann surfaces in the complex projective plane with constant scalar normal curvature.
Band decompositions of semigroups
Authors: Putcha Mohan S. , Weissglass Julian
Published: June 2010

The existence of a band decomposition of a semigroup into semigroups having at most one idempotent and a group ideal whenever it has an idempotent is investigated. It is shown that S has such a band decomposition if and only if, for every c,

d
∈

S
1

,
a
∈
S

d \in {S^1},a \in S

, the idempotent multiples of cad and

c

a
2

d

c{a^2}d

coincide. The main result is used to characterize those semigroups which are bands of groups, extending a theorem of Clifford.
An eigenvalue problem for the Stieltjes mean sigma-integral related to Gronwall’s inequality
Authors: Kroll Jeffrey R. , Smith Keith P.
Published: June 2010

This paper is concerned with the solution of the eigenvalue problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda f left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript t Baseline f left-parenthesis s right-parenthesis d g left-parenthesis s right-parenthesis comma 0 less-than-over-equals t less-than-over-equals upper T">

λ
f
(
t
)
=

∫
0
t

f
(
s
)

d
g
(
s
)
,
0
≦
t
≦
T

\lambda f(t) = \smallint _0^t {f(s)\;dg(s),0 \leqq t \leqq T}

, where g is nondecreasing and right continuous on [0, T] and the integral is the Stieltjes mean sigma-integral. In addition to the solution of the general problem, the solution of the problem with positive eigenfunctions is given. The relationship of the eigenvalue problem to the failure of the Gronwall inequality for g is also established. Furthermore, the techniques used to solve this eigenvalue problem are readily adapted to solve the extended problem in which the function g is merely of bounded variation and right continuous.
Embedding partially ordered spaces in topological semilattices
Authors: Tucker Lloyd D.
Published: June 2010

A partial order

Γ
\Gamma

on a compact space S is called continuous if

Γ
\Gamma

is a closed subset of

S
×
S

S \times S

. In this paper, we define and study an embedding

Φ
\Phi

of the arbitrary compact continuously partially ordered space

(
S
,
Γ
)

(S,\Gamma )

into a corresponding compact topological semilattice

S
Γ

{S_\Gamma }

. We show that the structure of

S
Γ

{S_\Gamma }

entirely determines the structure of

(
S
,
Γ
)

(S,\Gamma )

. We prove that the inverse images under

Φ
\Phi

of components in

S
Γ

{S_\Gamma }

are the order components of

(
S
,
Γ
)

(S,\Gamma )

, where elements a and b of S are defined to be in the same order component of

(
S
,
Γ
)

(S,\Gamma )

if there exists no continuous monotonic map

f
:
(
S
,
Γ
)
→
{
0
,
1
}

f:(S,\Gamma ) \to \{ 0,1\}

which separates a and b. Finally, we show that

S
Γ

{S_\Gamma }

is connected if and only if

(
S
,
Γ
)

(S,\Gamma )

has only one order component.
An “extra” law for characterizing Moufang loops
Authors: Chein Orin , Robinson D. A.
Published: June 2010

Let

(
G
,
⋅
)

(G, \cdot )

be any loop and let

λ
,
δ
,
α

\lambda , \delta , \alpha

be mappings of G into G so that

x
λ
=
x
⋅
x
δ
=
x
(
x
α
⋅
x
)

x\lambda = x \cdot x\delta = x(x\alpha \cdot x)

for all

x
∈
G

x \in G

. It is shown that the following conditions are equivalent: (a)

(
x
y
⋅
z
)
x
α
=
x
(
y
(
z
⋅
x
α
)
)

(xy \cdot z)x\alpha = x(y(z \cdot x\alpha ))

for all

x
,
y
,
z
∈
G

x,y,z \in G

, (b)

(
G
,
⋅
)

(G, \cdot )

is Moufang and

x
δ

x\delta

is in the nucleus of

(
G
,
⋅
)

(G, \cdot )

for all

x
∈
G

x \in G

, (c)

(
x
y
)
(
z
⋅
x
λ
)
=
(
x
⋅
y
z
)
x
λ

(xy)(z \cdot x\lambda ) = (x \cdot yz)x\lambda

for all

x
,
y
,
z
∈
G

x,y,z \in G

. In particular, a loop

(
G
,
⋅
)

(G, \cdot )

is extra in that

(
x
y
⋅
z
)
x
=
x
(
y
⋅
z
x
)

(xy \cdot z)x = x(y \cdot zx)

for all

x
,
y
,
z
∈
G

x,y,z \in G

if and only if it satisfies the

M
3

{M_3}

-law in that

(
x
y
)
(
z
⋅

x
3

)
=
(
x
⋅
y
z
)

x
3

(xy)(z \cdot {x^3}) = (x \cdot yz){x^3}

for all

x
,
y
,
z
∈
G

x,y,z \in G

.
A theory of unbounded generalized scalar operators
Authors: Kritt Brian
Published: June 2010

In this paper is presented a theory of unbounded operators which admit spectral distributions. The compactness of the support in the bounded case has been replaced by certain growth conditions at infinity. Properties of these operators are studied, including the existence of a Spectral Mapping Theorem.
On compactness in function spaces
Authors: Vidossich Giovanni
Published: June 2010

There are shown some implications from pseudocompactness to compactness or sequential compactness. The latter, sequential compactness, is obtained via metrization.
Some Wallman compactifications determined by retracts
Authors: Bentley H. L.
Published: June 2010

Let Y be a Hausdorff compactification of a locally compact space X and let

K
=
Y
−
X

K = Y - X

be the remainder. Suppose that K is a regular Wallman space (i.e. K has a base for closed sets which is a ring and which consists of sets each of which is the closure of its interior) and suppose that K is a neighborhood retract of Y. Under these assumptions, it is proved below that Y is a Wallman compactification of X.
A note on homotopy equivalences
Authors: Vogt R. M.
Published: June 2010

Given a homotopy equivalence

f
:
X
→
Y

f:X \to Y

, a homotopy inverse g of f, and a homotopy

H
:
X
×
I
→
X

H:X \times I \to X

from

g
∘
f

g \circ f

to

1
X

{1_X}

. We show that there is a homotopy

K
:
Y
×
I
→
Y

K:Y \times I \to Y

from

f
∘
g

f \circ g

to

1
Y

{1_Y}

such that

f
∘
H
≃
K
∘
(
f
×

1
I

)

rel

X
×
∂
I

f \circ H \simeq K \circ (f \times {1_I})\,{\text {rel}}\,X \times \partial I

and

H
∘
(
g
×

1
I

)
≃
g
∘
K

rel

Y
×
∂
I

H \circ (g \times {1_I}) \simeq g \circ K\,{\text {rel}}\,Y \times \partial I

.
Balanced colourings and the four colour conjecture
Authors: Bondy J. A.
Published: June 2010

A conjectured property of bridgeless cubic planar graphs is shown to be equivalent to the four colour conjecture. In establishing this equivalence use is made of the König-Hall theorem on the existence of one-factors in bipartite graphs.
Union conditions in Noether lattices
Authors: Johnson Johnny A.
Published: June 2010

In this paper we introduce the strong union condition on elements and the strong union condition on primes in Noether lattices and prove some structural properties of Noether lattices which satisfy these conditions.
Regular rings and integral extension of a regular ring
Authors: Wong Edward T.
Published: June 2010

In this paper we show that a ring (not necessarily commutative) with identity element and without nonzero nilpotent elements is a von Neumann regular ring if every completely prime ideal is a maximal right ideal. Using this result, we show an integral extension (not necessarily commutative) without nonzero nilpotent elements of a regular ring is itself a regular ring.
Hereditary radicals in Jordan rings
Authors: Lewand Robert
Published: June 2010

The object of this paper is to examine some radical properties of quadratic Jordan algebras and to show that under certain conditions,

R
(

B

)
=

B

∩
R
(

J

)

R(\mathfrak {B}) = \mathfrak {B} \cap R(\mathfrak {J})

where

B

\mathfrak {B}

is an ideal of a quadratic Jordan algebra

J

,
R
(

B

)

\mathfrak {J},R(\mathfrak {B})

is the radical of

B

\mathfrak {B}

, and

R
(

J

)

R(\mathfrak {J})

is the radical of

J

\mathfrak {J}

.
On the Lebedev-Milin inequality
Authors: Zedek Mishael
Published: June 2010

A generalization is given of an inequality of Lebedev-Milin involving the coefficients of a power series f and those of

exp
⁡
(
f
)

\exp (f)

. It is proved that the exponential function can be replaced by certain, functions belonging to the family of functions whose Taylor expansions at the origin have nonnegative coefficients.
A note on polyhedral Banach spaces
Authors: Gleit Alan , McGuigan Robert
Published: June 2010

We give a sufficient condition for an infinitedimensional Banach space X to be polyhedral. If

X
∗

{X^\ast }

is an L-space this condition is also necessary.
Compact multilinear transformations
Authors: Krikorian Nishan
Published: June 2010

It is well known that if a

C
1

{C^1}

map between Banach spaces is compact, then its derivative is a compact operator. If the map is

C
r

{C^r}

, then what can be said about the compactness of its higher derivatives? This question leads us to a study of compact multilinear operators with the main result being that the higher derivatives of a compact map are such operators.
Closed ideals in subalgebras of Banach algebras. I
Authors: Burnham J. T.
Published: June 2010

In this note we obtain an elementary extension of the so-called Fundamental Theorem of Segal Algebras to a special class of noncommutative Banach algebras. In particular the case of noncommutative group algebras is settled.
Coarse uniformities on the rationals
Authors: Enos H. C.
Published: June 2010

There exist uniform spaces

μ
Q

\mu Q

homeomorphic with the space Q of rational numbers such that every homeomorphic uniform space admits a uniformly continuous homeomorphism upon

μ
Q

\mu Q

. If “homeomorphism” is replaced by “bijection", the resulting weaker property is equivalent to having as completion a Peano continuum. With “homeomorphism", a (countable) dense subspace of a Hilbert cube has the property, but not a dense subspace of an interval.
The probability of connectedness of an unlabelled graph can be less for more edges
Authors: Wright E. M.
Published: June 2010

We write

β
=
β
(
n
,
q
)

\beta = \beta (n,q)

for the probability that a graph on n unlabelled nodes with q edges is connected; that is

β
\beta

is the ratio of the number of connected graphs to the total number of graphs. We write

N
=
n
(
n
−
1
)

/

2

N = n(n - 1)/2

. For fixed n we might expect that

β
\beta

would increase with q, at least nonstrictly. On the contrary, we show that, for any given integer s, we have

β
(
n
,
q
+
1
)
>
β
(
n
,
q
)

\beta (n,q + 1) > \beta (n,q)

for

N
−
n
−
s
≦
q
≦
N
−
n

N - n - s \leqq q \leqq N - n

and

n
>

n
0

(
s
)

n > {n_0}(s)

. We can show that

β
(
n
,
q
+
1
)
>
β
(
n
,
q
)

\beta (n,q + 1) > \beta (n,q)

for a much longer range, but this requires much more elaborate arguments.
The converse to a theorem of Sharp on Gorenstein modules
Authors: Reiten Idun
Published: June 2010

Let A be a commutative local Noetherian ring with identity of Krull dimension n, m its maximal ideal. Sharp has proved that if A is Cohen-Macauley and a homomorphic image of a Gorenstein local ring, then A has a Gorenstein module M with

dim

A

/

m

Ext
n

⁡
(
A

/

m
,
M
)
=
1

{\dim _{A/m}}\operatorname {Ext}^n(A/m,M) = 1

. The aim of this note is to prove the converse to this theorem.
Operators associated with a pair of nonnegative matrices
Authors: Suchan Gerald E.
Published: June 2010

Let

A

m
×
n

,

B

m
×
n

,

X

n
×
1

{A_{m \times n}},{B_{m \times n}},{X_{n \times 1}}

, and

Y

m
×
1

{Y_{m \times 1}}

be matrices whose entries are nonnegative real numbers and suppose that no row of A and no column of B consists entirely of zeroes. Define the operators U, T and T’ by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper U upper X right-parenthesis Subscript i Baseline equals upper X Subscript i Superscript negative 1 Baseline left-bracket or left-parenthesis upper U upper Y right-parenthesis Subscript i Baseline equals upper Y Subscript i Superscript negative 1 Baseline right-bracket">

(
U
X

)
i

=

X
i

−
1

[

or

(
U
Y

)
i

=

Y
i

−
1

]

{(UX)_i} = X_i^{ - 1}[{\text {or}}{(UY)_i} = Y_i^{ - 1}]

,

T
=
U

B
t

U
A

T = U{B^t}UA

and

T
′

=
U
A
U

B
t

T’ = UAU{B^t}

. T is called irreducible if for no nonempty proper subset S of

{
1
,
⋯
,
n
}

\{ 1, \cdots ,n\}

it is true that

X
i

=
0
,
i
∈
S
;

X
i

≠
0
,
i
∉
S

{X_i} = 0,i \in S;{X_i} \ne 0,i \notin S

, implies
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper T upper X right-parenthesis Subscript i Baseline equals 0 comma i element-of upper S semicolon left-parenthesis upper T upper X right-parenthesis Subscript i Baseline not-equals 0 comma i not-an-element-of upper S">

(
T
X

)
i

=
0
,
i
∈
S
;

(
T
X

)
i

≠
0
,
i
∉
S

{(TX)_i} = 0,i \in S;{(TX)_i} \ne 0,i \notin S

. M. V. Menon proved the following Theorem. If T is irreducible, there exist row-stochastic matrices

A
1

{A_1}

and

A
2

{A_2}

, a positive number

θ
\theta

, and two diagonal matrices D and E with positive main diagonal entries such that

D
A
E
=

A
1

DAE = {A_1}

and

θ
D
B
E
=

A
2
t

\theta DBE = A_2^t

. Since an analogous theorem holds for T’, it is natural to ask if it is possible that T’ be irreducible if T is not. It is the intent of this paper to show that T’ is irreducible if and only if T is irreducible.
On oscillations for solutions of 𝑛th order differential equations
Authors: Onose H.
Published: June 2010

Necessary and sufficient conditions are given that all solutions of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript left-parenthesis n right-parenthesis Baseline plus f left-parenthesis t comma x comma x prime comma midline-horizontal-ellipsis comma x Superscript left-parenthesis n minus 2 right-parenthesis Baseline right-parenthesis equals 0">

x

(
n
)

+
f
(
t
,
x
,

x
′

,
⋯
,

x

(
n
−
2
)

)
=
0

{x^{(n)}} + f(t,x,x’, \cdots ,{x^{(n - 2)}}) = 0

are oscillatory for n even and are oscillatory or tend monotonically to zero as

t
→
∞

t \to \infty

for n odd. The results generalize recent results of J. S. W. Wong and G. H. Ryder and D. V. V. Wend.
On spaces with norms of negative and positive order
Authors: Peyser Gideon
Published: June 2010

The two Hilbert spaces

H
0

{H_0}

and

H
1

{H_1}

are defined to be a generating pair if

H
1

{H_1}

is a dense subspace of

H
0

{H_0}

and if the norm of an element in

H
1

{H_1}

is greater than or equal to the norm in

H
0

{H_0}

. It is shown that the pair generates a sequence of spaces

{

H
k

}
,
−
∞
>
k
>
∞

\{ {H_k}\}, - \infty > k > \infty

, such that any two spaces of the sequence form again a generating pair. Such a pair is shown to generate, in turn, a subsequence of

{

H
k

}

\{ {H_k}\}

. Also, representation theorems are derived for bounded linear functionals over the spaces of the sequence

{

H
k

}

\{ {H_k}\}

, generalizing the Lax representation theorem and the Lax-Milgram theorem.
The numerical range of a Toeplitz operator
Authors: Klein E. M.
Published: June 2010

In this paper we explicitly compute the numerical range of an arbitrary Toeplitz operator on the classical Hardy space

H
2

{H^2}

of the unit circle. In particular, we show that the numerical range depends only on the spectrum of the given Toeplitz operator. Several special cases are also considered.
The Fourier transform of an unbounded spectral distribution
Authors: Kritt Brian
Published: June 2010

The Fourier transform of an unbounded spectral distribution is studied: An explicit integral representation is obtained; connections are drawn to the associated generalized scalar operator. It is proved that every generalized pseudo-hermitian operator is the infinitesimal generator of a temperate

C
0

{C_0}

group.
On Weyl’s theorem and its converse
Authors: Lee Matthew C. Y.
Published: June 2010

In this paper we study Weyl’s theorem and von Neumann’s converse of Weyl’s theorem for the classes of all operators of the form

T
f

−
1

T
g

T_f^{ - 1}{T_g}

and of the form

T
g

T
f

−
1

{T_g}T_f^{ - 1}

, where

T
g

{T_g}

and

T
f

{T_f}

are Toeplilz operators such that

T
f

{T_f}

is invertible; and we can prove that Weyl’s theorem holds for

T
f

−
1

T
g

T_f^{ - 1}{T_g}

and for

T
g

T
f

−
1

{T_g}T_f^{ - 1}

.
An approximation theory for focal points and focal intervals
Authors: Gregory John
Published: June 2010

The theory of focal points and conjugate points is an important part of the study of problems in the calculus of variations and control theory. In previous works we gave a theory of focal points and of focal intervals for an elliptic form

J
(
x
)

J(x)

on a Hilbert space

A

\mathcal {A}

. These results were based upon inequalities dealing with the indices

s
(
σ
)

s(\sigma )

and

n
(
σ
)

n(\sigma )

of the elliptic form

J
(
x
;
σ
)

J(x;\sigma )

defined on the closed subspace

A

(
σ
)

\mathcal {A}(\sigma )

of

A

\mathcal {A}

, where

σ
\sigma

belongs to the metric space

(
Σ
,
ρ
)

(\Sigma ,\rho )

. In this paper we give an approximation theory for focal point and focal interval problems. Our results are based upon inequalities dealing with the indices

s
(
μ
)

s(\mu )

and

u
(
μ
)

u(\mu )

, where

μ
\mu

belongs to the metric space

(
M
,
d
)
,
M
=

E
1

×
Σ

(M,d),M = {E^1} \times \Sigma

. For the usual focal point problems we show that

λ
n

(
σ
)

{\lambda _n}(\sigma )

, the nth focal point, is a

ρ
\rho

continuous function of

σ
\sigma

. For the focal interval case we give sufficient hypotheses so that the number of focal intervals is a local minimum at

σ
0

{\sigma _0}

in

Σ
\Sigma

. Neither of these results seems to have been published before (under any setting) in the literature. For completeness an example is given for quadratic problems in a control theory setting.
The topological complementation theorem à la Zorn
Authors: Schnare Paul S.
Published: June 2010

Steiner’s topological complementation theorem is given a short simple proof using Zorn’s Lemma.
Discrete subsets of perfectly normal spaces
Authors: Stephenson R. M.
Published: June 2010

Following R. W. Heath and E. A. Michael [HM], we call a topological space perfect if each of its closed subsets is a

G
δ

{G_\delta }

. In this note we (i) give a characterization of compact perfect

T
1

{T_1}

spaces and (ii) obtain some results concerning the cardinality of discrete subsets of perfect spaces.
Homology of a fibration over an aspherical space
Authors: Nomura Yasutoshi
Published: June 2010

The aim of this paper is to establish, for a fibration over a space of type (Q, 1), an exact sequence involving the homology homomorphisms induced by the projection. Specializing to the case where the total space is aspherical, our sequence allows us to add some extra terms to the Eckmann-Stammbach exact sequence for a group extension with simple integer coefficient.
A note on the uniqueness of rings of coefficients in polynomial rings
Authors: Eakin Paul , Kubota K. K.
Published: June 2010

We say that the ring A is of transcendence degree n over its subfield k if for every prime

P
⊂
A

P \subset A

the transcendence degree of

A

/

P

A/P

over k is at most n and equality is attained for some P. In this paper we prove the following: Suppose A is a noetherian ring of transcendence degree one over its subfield k. Then if B is any ring such that the polynomial rings
\[

A
[

X
1

,
⋯
,

X
m

]

and

B
[

Y
1

,
⋯
,

Y
m

]

A[{X_1}, \cdots ,{X_m}]\quad {\text {and}}\quad B[{Y_1}, \cdots ,{Y_m}]

\]
are isomorphic, A is isomorphic to B. Moreover if A has no nontrivial idempotents then either A is isomorphic to the polynomials in one variable over a local artinian ring or, modulo the nil radical, the given isomorphism takes A onto B.
Retractions and quasi-monotone mappings of unicoherent spaces
Authors: McWaters M. M. , Reed J. H.
Published: June 2010

It is shown that retractions of connected, locally connected, unicoherent spaces are unieoherent, and that quasimonotone maps preserve the unicoherence of any connected unicoherent space.
Morse theory on Kähler homogeneous spaces
Authors: Parker George D.
Published: June 2010

By exploiting the Kähler structure, we find a Morse function f on the homogeneous space

G

/

C
(

T

′

)

\mathfrak {G}/C(\mathfrak {T}’)

. The homology is readily computed and is seen to be contained in the diagram of

G

\mathfrak {G}

. The Morse cells are shown to be complex analytic submanifolds and to coincide with those of a cell decomposition found by Borel in a different manner.
Extension of algebraic theories
Authors: Peake E. James , Peters Galen R.
Published: June 2010

The algebraic theories of Lawvere are extended in a natural way to small complete categories. These categories exhibit not only the operations and identities, but some of the homomorphisms, functions, objects and constructions which are encountered when working within the algebraic categories associated with the theories. The category of extended theories is isomorphic to the original category of theories. As an illustration, the extended theory of groups is used to construct commutator subgroups.