bluuebunny/crossref_metadata_2025 huggingface dataset:
Voiculescu’s double commutant theorem and the cohomology of 𝐶*-algebras
Authors: Phillips John , Raeburn Iain
Published: June 2010

In a previous paper, on the central cohomology of

C
∗

{C^ * }

-algebras [5], we outlined a proof of the following result: a separable, unital

C
∗

{C^ * }

-algebra has continuous trace if and only if all of its central cohomology groups for

n
≥
1

n \geq 1

vanish. Unfortunately, as was pointed out to us by Professors A. Ja. Helemskii and B. E. Johnson, the proof we outlined was incorrect. Our appeal to [3, Theorem 3.2] was invalid since the algebras we were interested in were not generally commutative. It is the purpose of this note to give a correct proof of this result as well as other interesting cohomological results. Our main tool will be D. Voiculescu’s celebrated double commutant theorem for separable

C
∗

{C^ * }

-subalgebras of the Calkin algebra [7].
Surjective mappings whose differential is nowhere surjective
Authors: Yomdin Y.
Published: June 2010

Examples of

C
k

{C^k}

-mappings

f
:

R

n

→

R

m

,
n
≥
m
>
2

f:{\mathbb {R}^n} \to {\mathbb {R}^m},n \geq m > 2

, are given, with rank

d
f
(
x
)
≤
s

df(x) \leq s

at any

x
∈

R

n

,
2
≤
s
>
m

x \in {\mathbb {R}^n},2 \leq s > m

, but

f
(

R

n

)
=

R

m

f({\mathbb {R}^n}) = {\mathbb {R}^m}

, for any

k
>
(
n
−
s
+
2
)

/

(
m
−
s
+
2
)

k > (n - s + 2)/(m - s + 2)

. Thus a weak form of Sard’s theorem (if all the points in the source are critical, the image has measure zero) does not hold for mappings of low smoothness.
Paradoxical decompositions and invariant measures
Authors: Zakrzewski Piotr
Published: June 2010

Suppose

G
G

is a certain group of bijections of a given set

X
X

. A subset

E
E

of

X
X

is countably

G
G

-paradoxical if it contains disjoint subsets

A
,
B

A,B

, each of which can be taken apart into countably many pieces that may be rearranged via

G
G

to form a partition of

E
E

. We prove that the existence of a countably additive measure on

P
(
X
)

P(X)

that normalizes

X
X

and vanishes on all countably

G
G

-paradoxical sets implies the existence of a countably additive,

G
G

-invariant measure on

P
(
X
)

P(X)

normalizing

X
X

.
Another characterization of trees
Authors: Ward L. E.
Published: June 2010

It is proved that a continuum is a tree if and only if each two connected subsets meet in a connected set.
Maximal Cohen-Macaulay modules and the quasihomogeneity of isolated Cohen-Macaulay singularities
Authors: Martsinkovsky Alex
Published: June 2010

We conjecture that a complete isolated Cohen-Macaulay singularity of dimension

≥
2

\geq 2

is graded if and only if sufficiently high syzygy modules of the residue field and of the transpose of the module of Kähler differentials are isomorphic. The "only if" part of the conjecture is proved for hypersurface singularities.
Linear independence of iterates of entire functions
Authors: Bernal González Luis
Published: June 2010

We prove the following result: The set

{

h
n

:
n
=
0
,
1
,
…

}

\left \{ {{h_n}:n = 0,1, \ldots } \right \}

is a linearly independent sequence of entire functions, where

h
0

=
1
,

h
1

=

g

1
,

h
2

=

g
1

∘

g
2

,

h
3

=

g
1

∘

g
2

∘

g
3

,
…
,

g
1

{h_0} = 1,{h_1} = {g_{1,}}{h_2} = {g_1} \circ {g_2},{h_3} = {g_1} \circ {g_2} \circ {g_3}, \ldots ,{g_1}

is a nonconstant entire function and

g
n

(
n
≥
2
)

{g_n}(n \geq 2)

are entire functions which are not polynomials of degree

≤
1

\leq 1

. Our theorem generalizes a previous one about linear independence of iterates.
Noninvertible transformations admitting no absolutely continuous 𝜎-finite invariant measure
Authors: Hawkins Jane M. , Silva Cesar E.
Published: June 2010

We study a family of

n
n

-to-1 conservative ergodic endomorphisms which we will show to admit no

σ
\sigma

-finite absolutely continuous invariant measure. We exhibit recurrent measures for these transformations and study their ratio sets; the examples can be realized as

C
∞

{C^\infty }

endomorphisms of the

2
2

-torus.
A central limit theorem on Heisenberg type groups
Authors: Ohring Peter
Published: June 2010

Heisenberg type groups are a class of

2
2

-step nilpotent Lie groups which have many similarities with Heisenberg groups. In this paper we consider a commutative Banach algebra of radial, bounded, Borel measures on such groups and prove a central limit theorem for measures in this algebra satisfying suitable growth conditions.
Separate convergence of general 𝑇-fractions
Authors: Thron W. J.
Published: June 2010

This article is concerned with the separate convergence of the sequences of numerators

{

A
n

(
z
)
}

\{ {A_n}(z)\}

and denominators

{

B
n

(
z
)
}

\{ {B_n}(z)\}

of the approximants

A
n

(
z
)

/

(

B
n

(
z
)

{A_n}(z)/({B_n}(z)

of the general

T

{\text {T}}

-fraction

\[
K\limits _{n = 1}^\infty \left ( {\frac {{{F_n}z}}{{1 + {G_n}z}}} \right ).\]
Convergence results for sequences

{

A
n

(
z
)

/

Γ
n

(
z
)
}

\{ {A_n}(z)/{\Gamma _n}(z)\}

and

{

B
n

(
z
)

/

Γ
n

(
z
)
}

\{ {B_n}(z)/{\Gamma _n}(z)\}

, where the sequence

{

Γ
n

(
z
)
}

\{ {\Gamma _n}(z)\}

is "sufficiently simple" are also derived.
A criterion on a subdomain of the disc for its harmonic measure to be comparable with Lebesgue measure
Authors: Vol′berg A. L.
Published: June 2010

A subdomain

O
O

of the disc

D

\mathbb {D}

is called a boundary layer if

ω
(
O
,
⋅
)
≥
α
⋅
m

\omega (O, \cdot ) \geq \alpha \cdot m

, where

ω
(
O
,
⋅
)

\omega (O, \cdot )

is the harmonic measure of

O
O

. The metric criterion in terms of

∂
O

\partial O

is given for the case when

α
\alpha

is near 1.
Some properties of noncommutative 𝐻¹ spaces
Authors: Arias Alvaro
Published: June 2010

In this paper it is proved that every

T
T

in the trace class operators,

c
1

{c_1}

, has a unique best approximant from

Δ
\Delta

, the set of upper triangular matrices in

c
1

{c_1}

; and that

c
1

/

Δ

{c_1}/\Delta

is weakly sequentially complete.
Multiple solutions for the 𝑝-Laplacian under global nonresonance
Authors: del Pino Manuel A. , Manásevich Raúl F.
Published: June 2010

Via the study of a simple Dirichlet boundary value problem associated with the one-dimensional

p
p

-Laplacian,

p
>
1

p > 1

, we show that in globally nonresonant problems for this differential operator the number of solutions may be arbitrarily large when

p
∈
(
1
,
∞
)
∖
{
2
}

p \in (1,\infty )\backslash \{ 2\}

. From this point of view

p
=
2

p = 2

turns out to be a very special case.
Some remarks on the homology of moduli space of instantons with instanton number 2
Authors: Kamiyama Yasuhiko
Published: June 2010

Let

M
2

{M_2}

be the framed moduli space of

SU(2)

{\text {SU(2)}}

instantons with instanton number 2. By combining the results of Boyer and Mann and the results of Hattori, we determine the structure of

H
∗

(

M
2

;

Z

2

)

{H^*}({M_2};{{\mathbf {Z}}_2})

.
Generating finite completely reducible linear groups
Authors: Kovács L. G. , Robinson Geoffrey R.
Published: June 2010

It is proved here that each finite completely reducible linear group of dimension

d
d

(over an arbitrary field) can be generated by

⌊

3
2

d

⌋

\left \lfloor {\frac {3}{2}d} \right \rfloor

elements. If a finite linear group

G
G

of dimension

d
d

is not completely reducible, then its characteristic is a prime,

p
p

say, and the factor group of

G
G

modulo the largest normal

p
p

-subgroup

O

p

(
G
)

{\mathbb {O}_p}\left ( G \right )

may be viewed as a completely reducible linear group acting on the direct sum of the composition factors of the natural module for

G
G

: consequently,

G

/

O

p

(
G
)

G/{\mathbb {O}_p}\left ( G \right )

can still be generated by

⌊

3
2

d

⌋

\left \lfloor {\frac {3}{2}d} \right \rfloor

elements.
𝜎-hereditarily closure-preserving 𝑘-networks and 𝑔-metrizability
Authors: Tanaka Yoshio
Published: June 2010

We show that a regular space is

g
g

-metrizable if and only if it is a weakly first countable space with a

σ
\sigma

-hereditarily closure-preserving

k
k

-network.
Integral domains with finitely generated groups of divisibility
Authors: Anderson D. D.
Published: June 2010

Let

D
D

be an integral domain with integral closure

D
¯

\overline D

. We show that the group of divisibility

G
(
D
)

G(D)

of

D
D

is finitely generated if and only if

G
(

D
¯

)

G(\overline D )

is finitely generated and

D
¯

/

[
D
:

D
¯

]

\overline D /[D:\overline D ]

is finite. We also show that

G
(
D
)

G(D)

is finitely generated if and only if the monoid of finitely generated fractional ideals of

D
D

(under multiplication) is finitely generated.
Around the relative center property in orthomodular lattices
Authors: Chevalier G.
Published: June 2010

The paper deals with the relative center property in orthomodular lattices (OMLs). The property holds in a large class of OMLs, including locally modular OMLs and projection lattices of

AW

∗

{{\text {AW}}^*}

- and

W

∗

{{\text {W}}^*}

-algebras, and it means that the center of any interval

[
0
,
a
]

[0,a]

is the set

{
a
∧
c

|

c

central

in

L
}

\{ a \wedge c|c\;{\text {central}}\;{\text {in}}\;L\}

. In §l we study the congruence lattice of an OML satisfying the Axiom of Comparability (A.C.) and, in §2, we prove that the central cover of an element can be expressed in many different ways in OMLs satisfying a certain condition (C). For complete OMLs, Axiom (A.C.) and condition (C) are equivalent to the relative center property. In §3, we give a coordinatization theorem for complete OMLs with the relative center property.
A proof of van Douwen’s right ideal theorem
Authors: Davenport Dennis E. , Hindman Neil
Published: June 2010

In 1979 Eric K. van Douwen announced a powerful theorem about the Stone-Čech compactification of a discrete semigroup which he called The Right Ideal Theorem. Its proof, however, was lost with his untimely death. In this paper we present a proof of the theorem and a derivation of some of its corollaries.
The cyclic homology of algebras with adjoined unit
Authors: Elliott George A. , Nest Ryszard , Rørdam Mikael
Published: June 2010

The effect on the cyclic homology of an algebra of adjoining a unit is calculated by means of an elementary analysis at the level of chains.
On indecomposable modules over directed algebras
Authors: Dräxler Peter
Published: June 2010

Generalizing a result of Bongartz we show that any nonsimple indecomposable module over a finite-dimensional

k
k

-algebra

A
A

is an extension of an indecomposable and a simple module provided

k
k

is a field with more than two elements and

A
A

is representation directed. Our proof is based on fibre sums over simple modules and some known classification results on socle projective modules over peak algebras. In case the global dimension of

A
A

is at most 2 our methods also yield a description of the dimension vectors of the indecomposable

A
A

-modules by the roots of the associated quadratic form.
Multiplicity theory and the outer boundary
Authors: Spraker John S.
Published: June 2010

Let

f
f

be a conformal map from the unit disk onto a simply-connected region

R
R

. Multiplication by

f
f

on

L
2

{L^2}

is a normal operator. In this paper it is shown that the outer boundary of

R
R

is a set of multiplicity one for this operator.
Hypersurfaces in 𝑅ⁿ whose unit normal has small BMO norm
Authors: Semmes Stephen
Published: June 2010

Let

M
M

be a hypersurface in

R

d
+
1

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

whose Gauss map has small BMO norm. This condition is closely related to (but much weaker than) the requirement that the principal curvatures of

M
M

have small

L
d

(
M
)

{L^d}\left ( M \right )

norm. (The relationship between these two conditions is a nonlinear geometrical analogue of a classical Sobolev embedding.) This paper deals with the problem of understanding the geometrical constraints imposed on

M
M

by the requirement that the Gauss map have small BMO norm.
Łojasiewicz type inequalities and Newton diagrams
Authors: Fukui Toshizumi
Published: June 2010

We give an upper estimation for the Lojasiewicz exponent

α
0

(
f
)

{\alpha _0}(f)

for a nondegenerate holomorphic function germ

f
f

using the Newton polygon of

f
f

.
𝑁-compactness and shape
Authors: Morón Manuel Alonso
Published: June 2010

In this paper we prove that two

N

\mathbb {N}

-compact spaces are homeomorphic if and only if they have the same shape. We also obtain a result concerning shape domination, and finally we give an answer to the problem of components in shape theory.
A note on the shift on the Cantor set
Authors: Fokkink Robbert
Published: June 2010

Let

O
(
x
)

O(x)

be the orbit of

x
x

under a homeomorphism

h
h

on a metric space

X
X

. It is shown that this motion can be found in the shift on the Cantor set; i.e. there is an embedding

j
=
O
(
x
)
→

{
0
,
1

}
Z

j = O(x) \to {\{ 0,1\} ^Z}

such that

σ
∘
j
=
j
∘
h

\sigma \circ j = j \circ h

.
Separation and von Neumann intersection theorems
Authors: Chang Shiow Yu
Published: June 2010

We give some separation theorems to extend the intersection theorem of von Neumann, Fan, and others, omitting hypotheses of convexity and local convexity for one of the coordinate spaces.
Meager-nowhere dense games. II. Coding strategies
Authors: Scheepers Marion
Published: June 2010

We consider three closely related infinite two person games in which the second player has a winning perfect information strategy and examine under what conditions it suffices for the second player to remember only the most recent two moves in the game in order to insure a win. Strategies depending on this information only are called coding strategies.
Topological equivalence of real binary forms
Authors: Weinberg David , Witte Dave
Published: June 2010

Necessary and sufficient conditions are given for two real binary homogeneous polynomials to be equivalent under a continuous change of variables, and for the forms to be equivalent under a continuous change of variables that is differentiable (or that is real-analytic) away from the origin.
Manifolds with finite first homology as codimension 2 fibrators
Authors: Daverman Robert J.
Published: June 2010

Given a map

f
:
M
→
B

f:M \to B

defined on an orientable

(
n
+
2
)

(n + 2)

-manifold with all point inverses having the homotopy type of a specified closed

n
n

-manifold

N
N

, we seek to catalog the manifolds

N
N

for which

f
f

is always an approximate fibration. Assuming

H
1

(
N
)

{H_1}(N)

finite, we deduce that the cohomology sheaf of

f
f

is locally constant provided

N
N

admits no self-map of degree

d
>
1

d > 1

when

H
1

(
N
)

{H_1}(N)

has a cyclic subgroup of order

d
d

. For manifolds

N
N

possessing additional features, we achieve the approximate fibration conclusion.
A Daniell-Stone approach to the general Denjoy integral
Authors: Leinenkugel Cornel
Published: June 2010

In this note we shall introduce a, as far as we know, new kind of derivative (diagonal derivative), characterizing a certain class of functions

E

d

{\mathcal {E}_d}

and a generalized Daniell integral

I
d

{I_d}

on this class. We follow Leinert and König to obtain a class of integrable functions

L

d
1

\mathcal {L}_d^1

belonging to

E

d

{\mathcal {E}_d}

, using the method of Daniell-Stone integration without the lattice condition as described in [1] or similarly in [3]. Our main purpose is to show that we obtain exactly the Denjoy integrable functions.
Remarks on quasiconvexity and stability of equilibria for variational integrals
Authors: Zhang Kewei
Published: June 2010

Let

F
:

R

n
N

→

R

F:{{\mathbf {R}}^{nN}} \to {\mathbf {R}}

be a uniformly strictly quasiconvex function (see [3, 4]) of class

C

2
+
α

,
(
0
>
α
>
1
)

{C^{2 + \alpha }},(0 > \alpha > 1)

, and be of polynomial growth. Then every smooth solution of the Euler-Lagrangian equation of the multiple integral

I

(

u
;
Ω

)

=

∫
Ω

F
(
D
u
(
x
)
)
d
x

I\left ( {u;\Omega } \right ) = {\smallint _\Omega }F(Du(x))dx

is a minimum of

I
I

for variations of sufficiently small supports contained in

Ω
\Omega

.
Real isometries between 𝐽𝐵*-triples
Authors: Dang T.
Published: June 2010

It is shown that except for a certain class of JB*-triples (for which the result is false), real linear surjective isometries preserve the triple product. In particular, unital real linear isometries of

C
∗

{C^*}

-algebras are real linear Jordan

∗

^*

-isomorphisms.
A study on a relation between two summability methods
Authors: Sulaiman W. T.
Published: June 2010

Bor (1985, 1986) gave a relation between the two summability methods

|

C
,
1

|

k

{\left | {C,1} \right |_k}

and

|

N
¯

,

p
n

|

k

|\overline N ,{p_n}{|_k}

of a series

Σ

a
n

\Sigma a_n

. These two methods are known to be independent. Generalizing the case, here we introduce relations between the two summability methods

|

C
,
α

|

k

{\left | {C,\alpha } \right |_k}

and

|

N
¯

,

p
n

|

k

|\overline N ,{p_n}{|_k}

using multipliers sequence

{

ε
n

}

\left \{ {{\varepsilon _n}} \right \}

.
Approximation of convex bodies by triangles
Authors: Lassak Marek
Published: June 2010

We show that for every plane convex body

C
C

there exist a triangle

T
1

{T_1}

and its image

T
2

{T_2}

under a homothety with ratio

5
2

\tfrac {5} {2}

such that

T
1

⊂
C
⊂

T
2

{T_1} \subset C \subset {T_2}

. We prove the conjecture of Grünbaum that if

C
C

is centrally symmetric, then

T
1

,

T
2

{T_1},{T_2}

can be chosen so that their centroids coincide with the center of

C
C

.
Haar measure for compact right topological groups
Authors: Milnes Paul , Pym John
Published: June 2010

Compact right topological groups arise in topological dynamics and in other settings. Following H. Furstenberg’s seminal work on distal flows, R. Ellis and I. Namioka have shown that the compact right topological groups of dynamical type always admit a probability measure invariant under the continuous left translations; however, this invariance property is insufficient to identify a unique probability measure (in contrast to the case of compact topological groups). In the present paper, we amplify on the proofs of Ellis and Namioka to show that a right invariant probability measure on the compact right topological group

G
G

exists provided

G
G

admits an appropriate system of normal subgroups, that it is uniquely determined and that it is also invariant under the continuous left translations. Using Namioka’s work, we show that

G
G

has such a system of subgroups if its topological centre contains a countable dense subset, or if it is a closed subgroup of such a group.
Homotopy and topological actions on spaces with few homotopy groups
Authors: Postol Michael S.
Published: June 2010

Cooke [7] studied the problem of replacing homotopy actions by topological actions. In this paper, we use Cooke’s results to show that this can always be done for a large class of spaces having few homotopy groups.
The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces
Authors: Tan Kok-Keong , Xu Hong Kun
Published: June 2010

Let

X
X

be a uniformly convex Banach space with a Frechet differentiable norm,

C
C

a bounded closed convex subset of

X
X

, and

T
:
C
→
C

T:C \to C

an asymptotically nonexpansive mapping. It is shown that for each

x
x

in

C
C

, the sequence

{

T
n

x
}

\{ {T^n}x\}

is weakly almost-convergent to a fixed point

y
y

of

T
T

, i.e.,

(
1

/

n
)

∑

i
=
0

n
−
1

T

k
+
i

x
→
y

(1/n)\sum \nolimits _{i = 0}^{n - 1} {{T^{k + i}}x \to y}

weakly as

n
n

tends to infinity uniformly in

k
=
0
,
1
,
2
,
…

k = 0,1,2, \ldots
Centric maps and realization of diagrams in the homotopy category
Authors: Dwyer W. G. , Kan D. M.
Published: June 2010

We study the problem of taking a diagram in the homotopy category and realizing it as a diagram in the category of spaces. Under some assumptions there is a relatively simple obstruction theory for doing this; these assumptions hold in two cases involving homotopy decomposition diagrams for the classifying spaces of compact Lie groups.
Blocks of small defect
Authors: Espuelas Alberto , Navarro Gabriel
Published: June 2010

A group of odd order

G
G

with

O
p

(
G
)
=
1

{O_p}(G) = 1

has a block of defect less than

[
n

/

2
]

[n/2]

, where

p
n

=

|

G

|

p

{p^n} = |G{|_p}

. In addition, if

G
G

is supersolvable by nilpotent,

G
G

has a block of defect zero.
Genus two Heegaard splittings
Authors: Hass Joel
Published: June 2010

It is shown that a

3
3

-manifold has a finite number of genus two Heegaard splittings. A corollary is that the mapping class group is finite if the manifold is non-Haken.
Normal spaces with prescribed Stone-Čech remainders
Authors: Porter Jack R. , Woods R. Grant
Published: June 2010

It is shown that if

X
X

is a locally compact Hausdorff space, then there is a normal space

Y
Y

such that

β
Y
∖
Y
≅
X

\beta Y\backslash Y \cong X

. Examples are given of a countable nonsequential space, and a sequential nonlocally compact space, each of which is the Stone-Čech remainder of a normal space. A method of constructing normal almost compact spaces is presented.
A generalization of the handle addition theorem
Authors: Wu Ying Qing
Published: June 2010

We will generalize Jaco’s Handle Addition Theorem to the

n
n

-compressibility of surfaces on the boundary of

3
3

-manifolds. Several corollaries are given that show how the theorem can be applied to different situations.
Some general local variational principles
Authors: Binding Paul , Ye Qiang
Published: June 2010

Local variational min-sup characterizations are presented for the real spectrum of a selfadjoint operator pencil. Instead of minimizing over all subspaces of fixed codimension as in the classical result, the new characterizations minimize over subspaces that are close to extremal subspaces. In this way, the entire real spectrum, including continuous spectrum, can be characterized.
The conjugacy problem for graph products with infinite cyclic edge groups
Authors: Lockhart Jody Meyer
Published: June 2010

Finite graph products of groups with solvable conjugacy problem and infinite cyclic edge groups are considered. It is shown that the graph product has solvable conjugacy problem if the images of the edge group generators in each vertex group

G
v

{G_v}

are powers of a common central element

c
c

, where the group generated by

c
c

has solvable generalized word problem in

G
v

{G_v}

.
A Riemann type definition of a variational integral
Authors: Pfeffer Washek F.
Published: June 2010

We present a Riemann type definition of a coordinate free integral for which a general divergence theorem holds. The definition is particularly simple in dimension one.
On a generalization of the notion of centralizing mappings
Authors: Brešar Matej
Published: June 2010

Let

R
R

be a ring with center

Z
Z

. A mapping

f
:
R
→
R

f:R \to R

is called centralizing (resp. commuting) if

[
f
(
x
)
,
x
]
∈
Z

[f(x),x] \in Z

(resp.

[
f
(
x
)
,
x
]
=
0

[f(x),x] = 0

) for all

x
∈
R

x \in R

. In this paper we consider a more general case where a mapping

f
:
R
→
R

f:R \to R

satisfies

[
[
f
(
x
)
,
x
]
,
x
]
=
0

[[f(x),x],x] = 0

for all

x
∈
R

x \in R

; it is shown that if

R
R

is a prime ring of characteristic not 2, then every additive mapping with this property is commuting.
Uniform and Sobolev extension domains
Authors: Herron David A. , Koskela Pekka
Published: June 2010

We prove that if a domain

D
⊂

R

n

D \subset {{\mathbf {R}}^n}

is quasiconformally equivalent to a uniform domain, then

D
D

is an extension domain for the Sobolev class

W
n
1

W_n^1

if and only if

D
D

is locally uniform. We provide examples which suggest that this result is best possible. We exhibit a list of equivalent conditions for domains quasiconformally equivalent to uniform domains, one of which characterizes the quasiconformal homeomorphisms between uniform and locally uniform domains.
The existence of left averaging functions that are not right averaging
Authors: Miao Tianxuan
Published: June 2010

Let

G
G

be a locally compact group. We show that

G
G

is amenable as a discrete group if and only if

∑

i
=
1

n

λ

i

x
i

f
∈

A

0

\sum \nolimits _{i = 1}^n {{\lambda _{i{x_i}}}} f \in {\mathcal {A}_0}

for any

f
0

∈

A

0

,

x
i

∈
G

{f_0} \in {\mathcal {A}_0},{x_i} \in G

, and

λ
i

>
0
(
i
=
1
,
2
,
…
,
n
)

{\lambda _i} > 0(i = 1,2, \ldots ,n)

with

∑

i
=
1

n

λ
i

=
1

\sum \nolimits _{i = 1}^n {{\lambda _i} = 1}

, where

A

0

{\mathcal {A}_0}

is the set of functions that left average to 0. We also confirm a conjecture of Rosenblatt and Yang that there is a left averaging function that is not right averaging if

G
G

is not amenable.
Relations between chain recurrent points and turning points on the interval
Authors: Li Shi Hai
Published: June 2010

If a point is in the

ω
\omega

-limit set and the

α
\alpha

-limit set of the same point, then we call it a

γ
\gamma

-limit point. Then a

γ
\gamma

-limit point is an

ω
\omega

-limit point and thus a nonwandering point. In this paper, we prove that, on the interval, a nonwandering point which is not a

γ
\gamma

-limit point is in the closure of the set of forward images of turning points, and such points are not always the forward images of turning points. But a nonwandering point which is not an

ω
\omega

-limit point forward image of some turning point. Two examples are given. One shows that a chain recurrent point which is not nonwandering, a

γ
\gamma

-limit point which is not recurrent and a recurrent point which is not periodic need not be in the closure of forward images of turning points. The other shows that an

ω
\omega

-limit point which is not a

γ
\gamma

-limit point can be a limit point of forward images of turning points but not a forward image nor an

ω
\omega

-limit point of any turning point.
A Galois type theorem in von Neumann algebras
Authors: Choda Hisashi
Published: June 2010

We shall give a simple proof for a Galois type theorem: Let

α
\alpha

be a dual free action of a discrete group

G

{\text {G}}

on a factor

M
M

. If an automorphism

θ
\theta

of

M
M

leaves the fixed point algebra

M
α

{M^\alpha }

pointwise invariant then there exists a

g
0

∈
G

{g_0} \in G

with

θ
=

α

g
0

\theta = {\alpha _{{g_0}}}

.
Compact-like totally dense subgroups of compact groups
Authors: Dikranjan Dikran N. , Shakhmatov Dmitrii B.
Published: June 2010

A subgroup

H
H

of a topological group

G
G

is (weakly) totally dense in

G
G

if for each closed (normal) subgroup

N
N

of

G
G

the set

H
∩
N

H \cap N

is dense in

N
N

. We show that no compact (or more generally,

ω
\omega

-bounded) group contains a proper, totally dense, countably compact subgroup. This yields that a countably compact Abelian group

G
G

is compact if and only if each continuous homomorphism

π
:
G
→
H

\pi :G \to H

of

G
G

onto a topological group

H
H

is open. Here "Abelian" cannot be dropped. A connected, compact group contains a proper, weakly totally dense, countably compact subgroup if and only if its center is not a

G
δ

{G_\delta }

-subgroup. If a topological group contains a proper, totally dense, pseudocompact subgroup, then none of its closed, normal

G
δ

{G_\delta }

-subgroups is torsion. Under Lusin’s hypothesis

2

ω
1

=

2
ω

{2^{{\omega _1}}} = {2^\omega }

the converse is true for a compact Abelian group

G
G

. If

G
G

is a compact Abelian group with nonmetrizable connected component of zero, then there are a dense, countably compact subgroup

K
K

of

G
G

and a proper, totally dense subgroup

H
H

of

G
G

with

K
⊆
H

K \subseteq H

(in particular,

H
H

is pseudocompact).
Dehomogenization of gradings to Zariskian filtrations and applications to invertible ideals
Authors: Li Hui Shi , Van Oystaeyen Freddy
Published: June 2010

The method of dehomogenizing graded rings has been used successfully in algebraic geometry, e.g., a determinental ring is a dehomogenization of a Schubert cycle. We extend this method to noncommutative graded rings, dehomogenizing suitably graded rings to Zariski filtered rings and deriving, in a very elementary way, homological properties related to Auslander regularity and the Gorenstein property for noncommutative rings. As an application we study the lifting of such properties from a quotient modulo an invertible ideal.
Sums of lattice homomorphisms
Authors: Bernau S. J. , Huijsmans C. B. , de Pagter B.
Published: June 2010

Let

E
E

and

F
F

be Riesz spaces and

T
1

,

T
2

,
…
,

T
n

{T_1},{T_2}, \ldots ,{T_n}

be linear lattice homomorphisms (henceforth called lattice homomorphisms) from

E
E

to

F
F

. If

T
=

∑

i
=
1

n

T
i

T = \sum \nolimits _{i = 1}^n {{T_i}}

, then it is easy to check that

T
T

is positive and that if

x
0

,

x
1

,
…

x
n

∈
E

{x_0},{x_1}, \ldots {x_n} \in E

and

x
i

∧

x
j

=
0

{x_i} \wedge {x_j} = 0

for all

i
≠
j

i \ne j

, then

∧

i
=
0

n

T

x
i

=
0

\wedge _{i = 0}^nT{x_i} = 0

. The purpose of this note is to show that if

F
F

is Dedekind complete, the above necessary condition for

T
T

to be be the sum of

n
n

lattice homomorphisms is also sufficient. The result extends to sums of disjointness preserving operators, thereby leading to a characterization of the ideal of order bounded operators generated by the lattice homomorphisms.
An example relevant to curvature pinching theorems for Riemannian foliations
Authors: Hebda James J.
Published: June 2010

We describe a one-parameter family of codimension-

2
2

Riemannian foliations of

S
3

{S^3}

which are not transversely homogeneous. The example shows that the comparison theorem for Riemannian foliations of Kamber-Ruh-Tondeur [4] fails without an additional hypothesis.
On the stable rank of 𝐻^{∞}
Authors: Holden Peter J.
Published: June 2010

We prove that if

f
1

,

f
2

{f_1},{f_2}

are corona data and

f
1

{f_1}

is the product of finitely many interpolating Blaschke products, then there exist corona solutions

g
1

,

g
2

{g_1},{g_2}

with

g
1

−
1

∈

H
∞

(
D
)

g_1^{ - 1} \in {H^\infty }\left ( D \right )

. This provides a partial result in the direction of proving the stable rank of the algebra of bounded analytic functions on the open unit disc is one.
On coefficient inequalities for meromorphic univalent functions
Authors: Liu Li Quan
Published: June 2010

We obtain some coefficient inequalities for the class

Σ
\Sigma

consisting of functions of the form

f
(
z
)
=
z
+

b
0

+

b
1

/

z
+
⋯

f(z) = z + {b_0} + {b_1}/z + \cdots

that are meromorphic and univalent in the exterior of the unit circle

|

z

|

=
1

|z| = 1

. These inequalities disprove two conjectures of Schober about linear functionals on

Σ
\Sigma

.
Banach spaces in which every compact lies inside the range of a vector measure
Authors: Piñeiro C. , Rodríguez-Piazza L.
Published: June 2010

We prove that the compact subsets of a Banach space

X
X

lie inside ranges of

X
X

-valued measures if and only if

X
∗

{X^*}

can be embedded in an

L
1

{L^1}

space. In these spaces we prove that every compact is, in fact, a subset of a compact range. We also prove that if every compact of

X
X

is a subset of the range of an

X
X

-valued measure of bounded variation, then

X
X

is finite dimensional. Thus we answer a question by R. Anantharaman and J. Diestel.
Generalized convex functions and best 𝐿_{𝑝} approximation
Authors: Mathsen Ronald M. , Ubhaya Vasant A.
Published: June 2010

Some properties of generalized convex functions significant to approximation theory are obtained. The existence of a bestLp{L_p}approximation(1≤p≤∞)(1 \leq p \leq \infty )from subsets of these functions is established under certain conditions. Special cases of these functions includenn-convex functions which are much investigated in the literature.
Constructive existence of Minkowski functionals
Authors: Ishihara Hajime
Published: June 2010

In Bishop’s constructive mathematics, the framework of this paper, there are many situations where we cannot easily prove the existence of functionals whose existence is a trivial consequence of classical logic. One such functional is the Minkowski functional of a convex absorbing set. We shall prove the existence of Minkowski functionals in various spaces, and apply the theorems to establish the locatedness of the kernel of linear mappings.
Stability and instability of positive solutions of semipositone problems
Authors: Tertikas Achilles
Published: June 2010

We consider the boundary value problem
\[

−
Δ
u
=
f
(
u
)

in

Ω

,

B
u
=
0

on

∂
Ω
,

- \Delta u = f(u)\,{\text {in}}\,\Omega {\text {,}}Bu = 0\,{\text {on}}\,\partial \Omega ,

\]
where

Ω
\Omega

is a bounded region in

R

n

{\mathbb {R}^n}

with smooth boundary. We prove stability and instability results of positive solutions under various choices of

f
f

.
A note on a fixed point theorem of Okhezin
Authors: Tominaga A.
Published: June 2010

In 1985 V. P. Okhezin proved that the cartesian product of a

B
B

-space

X
X

and a compact metric AR space has the fixed point property. In this paper it is shown that the cone over

X
X

and the suspension of

X
X

have the fixed point property.
Distance to invertible linear operators without separability
Authors: Bouldin Richard
Published: June 2010

The formula for the distance from a given operator to the invertible operators on a separable Hilbert space is not true if the underlying Hilbert space is not required to be separable. This paper obtains inequalities for that distance in the latter situation. This requires a new concept called the modulus of invertibility, and further study of the concepts of essential nullity and essential deficiency, which permitted us to characterize the closure of the invertible operators.
Complex sequences whose “moments” all vanish
Authors: Priestley W. M.
Published: June 2010

Must a sequence

{

z
k

}

\{ {z_k}\}

of complex numbers be identically zero if

∑

f
(

z
k

)
=
0

\sum {f({z_k}) = 0}

for every entire function

f
f

vanishing at the origin? Lenard’s example of a nonzero sequence of complex numbers whose power sums ("moments") all vanish is shown to give a negative answer to this question and to lead to a novel representation theorem for entire functions. On the positive side it is proved that if

{

z
k

}

\{ {z_k}\}

is in

l
p

{l^p}

where

p
>
∞

p > \infty

, then vanishing moments imply

{

z
k

}

\{ {z_k}\}

is identically zero. Virtually the same proof shows that, on a Hubert space, two compact normal operators

A
A

and

B
B

with trivial kernels are unitarily equivalent if some power of each belongs to the trace class and

tr
⁡
(

A
n

)
=
tr
⁡
(

B
n

)

\operatorname {tr}({A^n}) = \operatorname {tr}(B^n)

for all

n
n

in a set of positive integers with asymptotic density one.
On absolute weighted mean summability methods
Authors: Sarıgöl M. Ali
Published: June 2010

In this paper we have proved the converse of the Bor-Thorpe theorem [2] on absolute weighted mean summability.
d’Alembert functional equations in distributions
Authors: Deeba E. Y. , Koh E. L.
Published: June 2010

In this paper we shall develop a method to define and solve the D’Alembert functional equation in distributions. We shall also show that for regular distributions (i.e., locally integrable functions) the distributional solution reduces to the classical one.
The Weyl transform and 𝐿^{𝑝} functions on phase space
Authors: Simon Barry
Published: June 2010

This is primarily a negative paper showing that a bound of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper W left-parenthesis f right-parenthesis StartAbsoluteValue EndAbsoluteValue operator norm less-than-or-equal-to c StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue Subscript p Baseline">

|

|

W
(
f
)

|

|

operator norm

≤
c

|

|

f

|

|

p

||W(f)||{\text {operator norm}} \leq c||f|{|_p}

fails for the Weyl transform if

p
>
2

p > 2

.

L
p

{L^p}

properties of Wigner distribution functions are discussed as well as Cwikel’s theorem.
On the reflexivity of operators on function spaces
Authors: Seddighi K. , Yousefi B.
Published: June 2010

Let

Ω
\Omega

be a bounded plane domain. Sufficient conditions are given so that an operator

T
T

in the Cowen-Douglas class

B

n

(
Ω
)

{\mathcal {B}_n}(\Omega )

is reflexive. The operator

M
z

{M_z}

of multiplication by

z
z

on a Hilbert space of functions analytic on a finitely connected domain

Ω
\Omega

is shown to be reflexive whenever

σ
(

M
z

)
=

Ω
¯

\sigma ({M_z}) = \overline \Omega

is a spectral set.
Smooth perturbations of regular Dirichlet forms
Authors: Stollmann Peter
Published: June 2010

Given a regular Dirichlet form

h

\mathfrak {h}

, we prove that a measure

μ
\mu

is smooth iff the domain of

h

+
μ

\mathfrak {h} + \mu

is dense in the domain of

h

\mathfrak {h}

with respect to the form norm. The latter condition is in turn equivalent to the convergence of

h

+
a
μ

\mathfrak {h} + a\mu

to

h

\mathfrak {h}

.
A nonsolvable complex vector field with Hölder coefficients
Authors: Jacobowitz Howard
Published: June 2010

It is known that the equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential u Over partial-differential t EndFraction minus alpha left-parenthesis xi comma t right-parenthesis StartFraction partial-differential u Over partial-differential xi EndFraction equals f left-parenthesis xi comma tau right-parenthesis">

∂
u

∂
t

−
α
(
ξ
,
t
)

∂
u

∂
ξ

=
f
(
ξ
,
τ
)

\frac {{\partial u}}{{\partial t}} - \alpha (\xi ,t)\frac {{\partial u}}{{\partial \xi }} = f(\xi ,\tau )

\]
is solvable in a neighborhood of the origin provided Im

α
\alpha

does not change sign and

α
\alpha

is at least Lipschitz smooth. An example is given where solvability fails although

α
\alpha

is of Hölder class

λ
\lambda

for all

0
>
λ
>
1

0 > \lambda > 1

. Further, the only solutions to
\[

∂
u

∂
t

−
α
(
ξ
,
t
)

∂
u

∂
ξ

=
0

\frac {{\partial u}}{{\partial t}} - \alpha (\xi ,t)\frac {{\partial u}}{{\partial \xi }} = 0

\]
are the constant functions.
Simple restricted modules for the restricted contact Lie algebras
Authors: Holmes Randall R.
Published: June 2010

It is shown that, with a few exceptions, the simple restricted modules for a restricted contact Lie algebra are induced from those for the homogeneous component of degree zero.
On the Poincaré series for diagonal forms
Authors: Wang Jun
Published: June 2010

Let

p
p

be a fixed prime,

f
(

x
1

,
…
,

x
s

)

f({x_1}, \ldots ,{x_s})

a polynomial over

Z

p

{{\mathbf {Z}}_p}

, the

p
p

-adic integers,

c
n

{c_n}

the number of solutions of

f
=
0

f = 0

over

Z

/

p
n

Z

{\mathbf {Z}}/{p^n}{\mathbf {Z}}

, and

P
f

(
t
)
=

∑

n
=
0

∞

c
n

t
n

{P_f}(t) = \sum \nolimits _{n = 0}^\infty {{c_n}{t^n}}

the Poincaré series. Explicit formulas for

P
f

(
t
)

{P_f}(t)

are derived for diagonal forms.
Remarks on weak compactness of operators defined on certain injective tensor products
Authors: Emmanuele G.
Published: June 2010

We show that if

X
X

is a

L

∞

{\mathcal {L}_\infty }

-space with the Dieudonné property and

Y
Y

is a Banach space not containing

l
1

{l_1}

, then any operator

T
:
X

⊗
ε

Y
→
Z

T:X{ \otimes _\varepsilon }Y \to Z

, where

Z
Z

is a weakly sequentially complete Banach space, is weakly compact. Some other results of the same kind are obtained.
A generic property of the bounded syzygy solutions
Authors: Diacu Florin N.
Published: June 2010

For a set of masses having positive measure, excepting eventually a negligible set of initial conditions, every noncollinear bounded solution of the planar three-body problem that has a syzygy configuration encounters an infinity of such configurations. Along a noncollinear syzygy solution, the set of syzygy configuration instants is discrete.
On the Thom spectra over Ω(𝑆𝑈(𝑛)/𝑆𝑂(𝑛)) and Mahowald’s 𝑋_{𝑘} spectra
Authors: Yan Dung Yung
Published: June 2010

The Thom spectra

M
(
n
)

(

2
k

≤
n
≤

2

k
+
1

−
1
)

M(n)\quad ({2^k} \leq n \leq {2^{k + 1}} - 1)

induced from

Ω
(
SU
⁡
(
n
)

/

SO
⁡
(
n
)
)
→
B
O

\Omega (\operatorname {SU}(n)/\operatorname {SO}(n)) \to BO

is a wedge of suspensions of Mahowald’s

X
k

{X_k}

spectra that is induced from

Ω

J

2
k

−
1

S
2

→

Ω
2

S
3

→
B
O

\Omega {J_{{2^k} - 1}}{S^2} \to {\Omega ^2}{S^3} \to BO

, where

J
i

{J_i}

is the

i
i

th stage of the James construction.
𝐺-transversality revisited
Authors: Costenoble S. R. , Waner S.
Published: June 2010

We express the obstruction to

G
G

-transversality in two related ways: as an element in the quotient of homotopical equivariant normal bordism by geometric bordism and also as the obstruction in a homotopy lifting problem. In both cases, our formulation is given in terms of the initial data only, in contrast to earlier theories. We also obtain related obstructions to equivariant "near" transversality, i.e., transversality using arbitrarily small homotopies.
On an example of Ahern and Rudin
Authors: Anderson John T.
Published: June 2010

We show that the polynomial hull of a certain totally real three-sphere in

C

3

{\mathbb {C}^3}

constructed by Ahern and Rudin is the union of a two-parameter family of analytic disks.
Not all 𝜎-complete Boolean algebras are quotients of complete Boolean algebras
Authors: Dow A. , Vermeer J.
Published: June 2010

In Shelah’s model of no Borel lifting of the measure algebra we show that there is a

σ
\sigma

-complete Boolean algebra of cardinality

2
ω

{2^\omega }

that is not a quotient of a complete Boolean algebra. By Stone duality, there is a basically disconnected space of weight

2
ω

{2^\omega }

that cannot be embedded into an extremally disconnected space.
𝑅^{1|1}-supergroup actions and superdifferential equations
Authors: Ongay-Larios F. , Sánchez-Valenzuela O. A.
Published: June 2010

The problem of posing and solving ordinary differential equations on supermanifolds is addressed from the point of view of Lie’s theory. It is shown that no nonsingular, nondegenerate, odd supervector field can have a Lie supergroup action of

R

1

|

1

{{\mathbf {R}}^{1|1}}

as its flow. It is also shown that the class of integrable supervector fields goes far beyond homogeneity and rectifiability. The obstructions for the integral flow to be an

R

1

|

1

{{\mathbf {R}}^{1|1}}

-action are given by the Lie superbracket of the field with itself and the Lie bracket of its homogeneous components.
Norms on unitizations of Banach algebras
Authors: Gaur A. K. , Kovářík Z. V.
Published: June 2010

Equivalence of various norms on the unitization of a nonunital Banach algebra is established, with bounds (

1
1

and

6
exp
⁡
(
1
)

6\exp (1)

) uniform over the class of such algebras. A tighter bound,

3
3

, is obtained in

C

∗

{C^{\ast }}

-algebras for elements with Hermitian nonunital parts.
On the Jacobian conjecture
Authors: Kaliman Shulim
Published: June 2010

We show that the Jacobian conjecture can be reduced to a weaker conjecture in which all fibers of coordinate functions are irreducible.
Topological types of quasi-ordinary singularities
Authors: Oh Kyungho
Published: June 2010

A germ

(
X
,
x
)

(X,x)

of a complex analytic hypersurface in

C

d
+
1

{\mathbb {C}^{d + 1}}

is quasi-ordinary if it can be represented as the image of an open neighborhood of

0
0

in

C

d

{\mathbb {C}^d}

under the map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis s 1 comma ellipsis comma s Subscript d Baseline right-parenthesis right-arrow from bar left-parenthesis s 1 Superscript n Baseline comma ellipsis comma s Subscript d Superscript n Baseline comma zeta left-parenthesis s 1 comma ellipsis comma s Subscript d Baseline right-parenthesis right-parenthesis comma n greater-than 0">

(

s
1

,
…
,

s
d

)
↦
(

s
1
n

,
…
,

s
d
n

,
ζ
(

s
1

,
…
,

s
d

)
)
,

n
>
0

({s_1}, \ldots ,{s_d}) \mapsto (s_1^n, \ldots ,s_d^n,\zeta ({s_1}, \ldots ,{s_d})),\;n > 0

, where

ζ
\zeta

is a convergent power series. It is shown that the topological type of the singularity

(
X
,
x
)
⊂
(

C

d
+
1

,
0
)

(X,x) \subset ({\mathbb {C}^{d + 1}},0)

is determined by a certain set of fractional monomials, called the characteristic monomials, appearing in the fractional power series

ζ
(

t
1

1

/

n

,
…
,

t
d

1

/

n

)

\zeta (t_1^{1/n}, \ldots ,t_d^{1/n})

.
Boolean isomorphism between partial orderings of convergent and divergent series and infinite subsets of 𝑁
Authors: Vojtáš Peter
Published: June 2010

We prove that

RO
⁡
(

P

(

N

)

/

fin
,

⊆

∗

)

\operatorname {RO} (\mathcal {P}(\mathbb {N})/\operatorname {fin} ,\,{ \subseteq ^{\ast }})

is isomorphic under

p

=
cf
⁡
(

2

ℵ
0

)

\mathfrak {p} = \operatorname {cf} ({2^{{\aleph _0}}})

to the Boolean completion of the comparison ordering of the absolutely divergent series (downwards) and under

ℵ
1

=
cf
⁡
(

2

ℵ
0

)

{\aleph _1} = \operatorname {cf} ({2^{{\aleph _0}}})

to the Boolean completion of the ratio comparison ordering of absolutely convergent series (upwards).
A remark on weighted inequalities for general maximal operators
Authors: Pérez C.
Published: June 2010

Let

1
>
p
>
∞

1 > p > \infty

, and let

w
,

v

w,\;v

be two nonnegative functions. We give a sufficient condition on

w
,

v

w,\;v

for which the general maximal operator

M

B

{M_\mathcal {B}}

is bounded from

L
p

(
v
)

{L^p}(v)

into

L
p

(
w
)

{L^p}(w)

. Our condition is stronger but closely related to the

A

p
,

B

{A_{p,\mathcal {B}}}

condition for two weights.
Holomorphic mappings into Teichmüller spaces
Authors: Tanigawa Harumi
Published: June 2010

Let

T
(
F
)

T(F)

be the Teichmüller space of a finitely generated Fuchsian group

F
F

of the first kind. We shall show that there exists no nontrivial family of holomorphic proper mappings of the unit disc into

T
(
F
)

T(F)

with a complex analytic parameter. Using this result, we shall investigate a family of quasi-conformal mappings with a complex analytic parameter whose Beltrami differentials vanish on a fixed set of positive area.
Weak-polynomial convergence on a Banach space
Authors: Jaramillo J. A. , Prieto A.
Published: June 2010

We show that any super-reflexive Banach space is a

Λ
\Lambda

-space (i.e., the weak-polynomial convergence for sequences implies the norm convergence). We introduce the notion of

κ
\kappa

-space (i.e., a Banach space where the weak-polynomial convergence for sequences is different from the weak convergence) and we prove that if a dual Banach space

Z
Z

is a

κ
\kappa

-space with the approximation property, then the uniform algebra

A
(
B
)

A(B)

on the unit ball of

Z
Z

generated by the weak-star continuous polynomials is not tight.
A decomposition of elements of the free algebra
Authors: Ma Wen Xin
Published: June 2010

Let

f
f

be an element of

F
[
X
]

F[X]

, the free associative algebra over a field

F
F

and

n
n

the maximum of the degrees of the variables and the multiplicities of the degrees in

f
f

. A partial ordering on the homogeneous elements of

F
[
X
]

F[X]

is defined such that if

f
f

is homogeneous and

F
∤
n
!

F\nmid n!

, then

f
f

can be decomposed into a sum of two polynomials

f
0

{f_0}

and

f
1

{f_1}

such that for

0
>
m
⩽
n
,

f
0

0 > m \leqslant n,\;{f_0}

is symmetric or skew symmetric in all its arguments of degree

m
m

depending on whether

m
m

is even or odd and

f
1

{f_1}

is a consequence of polynomials of lower type than

f
f

. Osborn’s Theorem about the symmetry of the absolutely irreducible polynomial identities is obtained as a corollary. The same holds in the free nonassociative algebra. The proofs are combinatorial.