bluuebunny/crossref_metadata_2025 huggingface dataset:
Finitely generated right ideals of transformation near-rings
Authors: Scott S. D.
Published: June 2010

If V is an additive group,

M
0

(
V
)

{M_0}(V)

the near-ring of zero-fixing maps of V into V, then the finitely generated right ideals of

M
0

(
V
)

{M_0}(V)

are easily characterised. These are just the annihilators of subsets of V. Moreover, finitely generated right ideals of

M
0

(
V
)

{M_0}(V)

are generated by a single element.
Algebraic conditions leading to continuous lattices
Authors: Lawson Jimmie D.
Published: June 2010

This paper is concerned with sufficient conditions for a meet-continuous lattice L to be a continuous lattice. In §2 this is shown to be true if the prime elements order generate and

(
L
,
∨
)

(L, \vee )

is a compact topological semilattice. In §3 it is shown that a meet-continuous lattice with finite breadth is a continuous lattice.
Finite-dimensional Banach spaces with a.e. differentiable metric projection
Authors: Abatzoglou Theagenis
Published: June 2010

We prove that in a finite-dimensional Banach space, where the Hessian of the norm square is positive definite on the whole space, except possibly at 0, the metric projection onto any closed set is differentiable a.e.
A remark on the growth of solutions of first order algebraic differential equations
Authors: Strelitz Sh.
Published: June 2010

Two theorems on the growth of complex- and real-valued solutions of first order algebraic differential equations are proven.
The extension of 𝐻^{𝑝}-functions from certain hypersurfaces of a polydisc
Authors: Zarantonello Sergio E.
Published: June 2010

Let E be a subvariety of the open unit polydisc

U
n

,
n
⩾
2

{U^n},n \geqslant 2

, of pure dimension

n
−
1

n - 1

, satisfying the following conditions. There exists an annular domain
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Superscript n Baseline equals left-brace left-parenthesis z 1 comma ellipsis comma z Subscript n Baseline right-parenthesis element-of bold upper C Superscript n Baseline colon r greater-than StartAbsoluteValue z Subscript i Baseline EndAbsoluteValue greater-than 1 comma 1 less-than-or-slanted-equals i less-than-or-slanted-equals n right-brace">

Q
n

=
{
(

z
1

,
…
,

z
n

)
∈

C

n

:
r
>

|

z
i

|

>
1
,
1
⩽
i
⩽
n
}

{Q^n} = \{ ({z_1}, \ldots ,{z_n}) \in {{\mathbf {C}}^n}:r > |{z_i}| > 1,1 \leqslant i \leqslant n\}

, a continuous function

η
:
[
r
,
1
)
→
[
r
,
1
)

\eta :[r,1) \to [r,1)

, and a

δ
>
0

\delta > 0

, such that (i)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue z Subscript n Baseline EndAbsoluteValue less-than-or-slanted-equals eta left-parenthesis left-parenthesis StartAbsoluteValue z 1 EndAbsoluteValue plus midline-horizontal-ellipsis plus StartAbsoluteValue z Subscript n minus 1 Baseline EndAbsoluteValue right-parenthesis slash left-parenthesis n minus 1 right-parenthesis right-parenthesis">

|

z
n

|

⩽
η
(
(

|

z
1

|

+
⋯
+

|

z

n
−
1

|

)

/

(
n
−
1
)
)

|{z_n}| \leqslant \eta ((|{z_1}| + \cdots + |{z_{n - 1}}|)/(n - 1))

whenever

(

z
1

,
…
,

z
n

)
∈
E
∩

Q
n

({z_1}, \ldots ,{z_n}) \in E \cap {Q^n}

, (ii)

|

α
−
β

|

⩾
δ

|\alpha - \beta | \geqslant \delta

whenever

1
⩽
j
⩽
n

1 \leqslant j \leqslant n

and

(

ζ
′

,
α
,
ζ
)
≠
(

ζ
′

,
β
,
ζ
)

(\zeta ’,\alpha ,\zeta ) \ne (\zeta ’,\beta ,\zeta )

are both in

(

Q

j
−
1

×
U
×

Q

n
−
j

)
∩
E

({Q^{j - 1}} \times U \times {Q^{n - j}}) \cap E

. Theorem. Let

0
>
p
>
∞

0 > p > \infty

, let g be holomorphic on E and let u be the real part of a holomorphic function on E. If

|

g
(
z
)

|

p

⩽
u
(
z
)

|g(z){|^p} \leqslant u(z)

for all

z
∈
E

z \in E

, then g can be extended to a function in the Hardy space

H
p

(

U
n

)

{H^p}({U^n})

.
Characterization of the trace-class
Authors: Saworotnow Parfeny P.
Published: June 2010

We characterize the trace-class

τ
(
A
)

\tau (A)

associated with an

H
∗

{H^\ast }

-algebra A as well as the trace-class

(
τ
c
)

(\tau c)

of operators acting on a Hilbert space.
A note on integral geometry in the inversive plane
Authors: Fillmore Jay P.
Published: June 2010

The density for circles in the Euclidean plane, which is invariant under the group of similitudes, is in fact invariant under the inversive group. The fundamental invariants of the inversive plane, angle and inversive distance, can be obtained from the measures of certain sets of circles.
Developable spaces and nearness structures
Authors: Carlson John W.
Published: June 2010

Necessary and sufficient conditions on a nearness structure are provided for which the underlying topology is developable. It is also shown that a topology is developable if and only if it admits a compatible nearness structure with a countable base. Finally it is shown that a topological space is embeddable in a complete Moore space if and only if it admits a compatible nearness structure satisfying certain stated conditions. Moreover, the complete Moore space is homeomorphic to Herrlich’s completion of the space with the specified nearness structure.
Retracts in metric spaces
Authors: Pasicki Lech
Published: June 2010

In this paper we define S-contractibility and two classes of spaces connected with this notion. A space X is said to be S-contractible provided that S is a function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S colon upper X times mathematical left-angle 0 comma 1 mathematical right-angle times upper X left-parenthesis x comma alpha comma y right-parenthesis right-arrow from bar upper S Subscript x Baseline left-parenthesis alpha comma y right-parenthesis element-of upper X">

S
:
X
×
⟨
0
,
1
⟩
×
X
(
x
,
α
,
y
)
↦

S
x

(
α
,
y
)
∈
X

S:X \times \langle 0,1\rangle \times X (x,\alpha ,y) \mapsto {S_x}(\alpha ,y) \in X

that is continuous in

α
\alpha

and y, and for every

x
,
y
∈
X
,

S
x

(
0
,
y
)
=
y
,

S
x

(
1
,
y
)
=
x

x,y \in X,{S_x}(0,y) = y,{S_x}(1,y) = x

. This notion is close to equiconnectedness, which can be defined as follows. A space X is equiconnected if there exists a map S such that X is S-contractible and

S
x

(
α
,
x
)
=
x

{S_x}(\alpha ,x) = x

for all

x
∈
X

x \in X

and

α
∈
I

\alpha \in I

(cf. [4]). The results we obtain in the theory of retracts are close to those that are known for equiconnected spaces. Also the thickness of the neighborhood that can be retracted on a set in a metric space is estimated, which enables to prove a theorem belonging to fixed point theory.
Maximal symmetry and fully wound coverings
Authors: May Coy L.
Published: June 2010

A compact bordered Klein surface of genus

g
⩾
2

g \geqslant 2

is said to have maximal symmetry if its automorphism group is of order

12
(
g
−
1
)

12(g - 1)

, the largest possible. We show that for each value of k there are only finitely many topological types of bordered Klein surfaces with maximal symmetry that have exactly k boundary components. We also prove that there are no bordered Klein surfaces with maximal symmetry that have exactly p boundary components for any prime

p
⩾
5

p \geqslant 5

. These results are established using the concept of a fully wound covering, that is, a full covering

φ
:
X
→
Y

\varphi :X \to Y

of the bordered surface Y with the maximum possible boundary degree.
Bolzano’s theorem in several complex variables
Authors: Shih Mau Hsiang
Published: June 2010

Let

Ω
\Omega

be a bounded domain in

C
n

{C^n}

containing the origin. Let

f
:

Ω
¯

→

C
n

f:\bar \Omega \to {C^n}

be analytic in

Ω
\Omega

and continuous in

Ω
¯

\bar \Omega

, and

Re
⁡

z
¯

⋅
f
(
z
)
>
0

\operatorname {Re} \bar z \cdot f(z) > 0

for

z
∈
∂
Ω

z \in \partial \Omega

. It is shown that f has exactly one zero in

Ω
\Omega

.
A remark on Carleson’s characterization of BMO
Authors: Uchiyama Akihito
Published: June 2010

L. Carleson showed that if

φ
∈
BMO
(

R
n

)

\varphi \in \text {BMO}(R ^n)

and supp

φ
\varphi

is compact, then

φ
\varphi

can be written in the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis x right-parenthesis equals normal upper Sigma Subscript k equals 1 Superscript normal infinity Baseline integral upper P Subscript t Sub Subscript k Subscript left-parenthesis y right-parenthesis Baseline left-parenthesis x minus y right-parenthesis b Subscript k Baseline left-parenthesis y right-parenthesis d y plus b 0 left-parenthesis x right-parenthesis">

φ
(
x
)
=

Σ

k
=
1

∞

∫

P

t
k

(
y
)

(
x
−
y
)

b
k

(
y
)
d
y
+

b
0

(
x
)

\varphi (x) = \Sigma _{k = 1}^\infty \smallint {P_{{t_k}(y)}}(x - y){b_k}(y)dy + {b_0}(x)

where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Subscript k equals 0 Superscript normal infinity Baseline double-vertical-bar b Subscript k Baseline double-vertical-bar Subscript normal infinity Baseline less-than-or-slanted-equals upper C double-vertical-bar phi double-vertical-bar Subscript BMO Baseline comma t Subscript k Baseline left-parenthesis y right-parenthesis greater-than 0">

Σ

k
=
0

∞

‖

b
k

‖

∞

⩽
C

‖
φ
‖

BMO

,

t
k

(
y
)
>
0

\Sigma _{k = 0}^\infty {\left \| {{b_k}} \right \|_\infty } \leqslant C{\left \| \varphi \right \|_{{\text {BMO}}}},{t_k}(y) > 0

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript t Baseline left-parenthesis x right-parenthesis equals upper C Subscript n Baseline t left-parenthesis StartAbsoluteValue x EndAbsoluteValue squared plus t squared right-parenthesis Superscript minus left-parenthesis n plus 1 right-parenthesis slash 2">

P
t

(
x
)
=

C
n

t

(

|

x

|

2

+

t
2

)

−
(
n
+
1
)

/

2

{P_t}(x) = {C_n}t{(|x{|^2} + {t^2})^{ - (n + 1)/2}}

is the Poisson kernel. We show that we can take

b
2

=

b
3

=
⋯
=
0

{b_2} = {b_3} = \cdots = 0

. This can be generalized on the space of homogeneous type with certain assumptions.
On 𝐽-selfadjoint extensions of 𝐽-symmetric operators
Authors: Knowles Ian
Published: June 2010

A short proof is given (via the theory of conjugate-linear operators) of the fact that every J-symmetric operator in a Hilbert space

K

\mathcal {K}

has a J-selfadjoint extension in

K

\mathcal {K}

.
An example of a liminal 𝐶*-algebra
Authors: Lazar A. J. , Taylor D. C.
Published: June 2010

For each countable ordinal

γ
\gamma

there exists a unital separable liminal

C
∗

{C^ \ast }

-algebra

A
γ

{A_\gamma }

with the property that if

(

I
ρ

)

ρ
=
1

β

({I_\rho })_{\rho = 1}^\beta

is any composition sequence of

A
γ

{A_\gamma }

such that the spectra of the quotients

I

ρ
+
1

/

I
ρ

{I_{\rho + 1}}/{I_\rho }

are Hausdorff, then

β
⩾
γ
+
1

\beta \geqslant \gamma + 1

. Moreover, there is a composition sequence

(

I
ρ

)

ρ
=
1

γ
+
1

({I_\rho })_{\rho = 1}^{\gamma + 1}

of

A
γ

{A_\gamma }

such that the spectra of the quotients

I

ρ
+
1

/

I
ρ

{I_{\rho + 1}}/{I_\rho }

are Hausdorff.
Isomorphisms of locally compact groups and Banach algebras
Authors: Lau Anthony To Ming , McKennon Kelly
Published: June 2010

If G is a locally compact group, then

U
B
C

r

(
G

)
∗

\mathrm {UBC}_r (G)^*

, the dual of the space of bounded right uniformly continuous complex-valued functions on G, with the Arens product is a Banach algebra. We prove in this paper a result that will have as a consequence the following: Let

G
1

,

G
2

{G_1},{G_2}

be locally compact groups. Then the Banach algebras

U
B
C

r

(

G
1

)
∗

\mathrm {UBC}_r{({G_1})^ \ast }

and

U
B
C

r

(

G
2

)
∗

\mathrm {UBC}_r{({G_2})^\ast }

are isometric isomorphic if and only if

G
1

{G_1}

and

G
2

{G_2}

are topologically isomorphic.
Isometric immersions of complete Riemannian manifolds into Euclidean space
Authors: Baikousis Christos , Koufogiorgos Themis
Published: June 2010

Let M be a complete Riemannian manifold of dimension n, with scalar curvature bounded from below. If the isometric immersion of M into euclidean space of dimension

n
+
q
,
q
⩽
n
−
1

n + q,q \leqslant n - 1

, is included in a ball of radius

λ
\lambda

, then the sectional curvature K of M satisfies

lim

sup
M

K
⩾

λ

−
2

{\lim \, \sup _M}K \geqslant {\lambda ^{ - 2}}

. The special case where M is compact is due to Jacobowitz.
A counterexample related to a criterion for a function to be continuous
Authors: Steele J. Michael
Published: June 2010

A construction is given of a right-continuous function which satisfies an arithmetic continuity condition, but which is not continuous. The problem is motivated by a familiar criterion for the continuity of sample paths of a stochastic process.
Fundamental dimension and suspension of approximate fibrations
Authors: Duvall P. F. , Husch L. S.
Published: June 2010

Given an approximate fibration

f
:
X
→
Y

f:X \to Y

between ANR’s a new approximate fibration Sf is constructed with the property that the fibers of Sf are the suspensions of the fibers of f. This construction is used together with an approximation theorem of T. Chapman to give conditions under which

Fd

X
=

Fd

Y
+

Fd

f

−
1

(
x
)

{\text {Fd}}\;X = {\text {Fd}}\;Y + {\text {Fd}}\;{f^{ - 1}}(x)

.
Noncompact hereditarily strongly infinite-dimensional spaces
Authors: Rubin Leonard R.
Published: June 2010

It is shown that any strongly infinite dimensional space contains a strongly infinite dimensional subspace all of whose subspaces are either 0-dimensional or strongly infinite dimensional.
On semisimple semigroup rings
Authors: Teply Mark L. , Turman E. Geis , Quesada Antonio
Published: June 2010

Let

π
\pi

be a property of rings that satisfies the conditions that (i) homomorphic images of

π
\pi

-rings are

π
\pi

-rings and (ii) ideals of

π
\pi

-rings are

π
\pi

-rings. Let S be a semilattice P of semigroups

S
α

{S_\alpha }

. If each semigroup ring

R
[

S
α

]
(
α
∈
P
)

R[{S_\alpha }](\alpha \in P)

is

π
\pi

-semisimple, then the semigroup ring

R
[

S
α

]

R[{S_\alpha }]

is also

π
\pi

-semisimple. Conditions are found on P to insure that each

R
[

S
α

]
(
α
∈
P
)

R[{S_\alpha }](\alpha \in P)

is

π
\pi

-semisimple whenever S is a strong semilattice P of semigroups

S
α

{S_\alpha }

and

R
[
S
]

R[S]

is

π
\pi

-semisimple. Examples are given to show that the conditions on P cannot be removed. These results and examples answer several questions raised by J. Weissglass.
On finite division rings
Authors: Oehmke Robert H.
Published: June 2010

Herein it is shown that the set of right powers of a generic element of a finite division ring contains a basis of the ring as an algebra over a prime field. This result is then applied to finite flexible division rings of characteristic not 2 to obtain commutativity.
Positive definite matrices and Catalan numbers
Authors: Leighton Frank Thomson , Newman Morris
Published: June 2010

It is shown that the number of

n
×
n

n \times n

integral triple diagonal matrices which are unimodular, positive definite and whose sub and super diagonal elements are all one, is the Catalan number

(
n

2
n

)

/

(
n
+
1
)

(_n^{2n})/(n + 1)

. More generally, it is shown that if A is a fixed integral symmetric matrix and d is a fixed positive integer, then there are only finitely many integral diagonal matrices D such that

A
+
D

A + D

is positive definite and

det
(
A
+
D
)
=
d

\det (A + D) = d

.
No system of uncountable rank is purely simple
Authors: Okoh Frank
Published: June 2010

A pair of complex vector spaces (V, W) is a system if and only if there is a C-bilinear map

C

2

×
V

{{\mathbf {C}}^2} \times V

to W. The category of systems is equivalent to the category of modules over a certain subring of the ring of

3
×
3

3 \times 3

matrices over the complex numbers, and so module-theoretic concepts make sense for systems. A system is purely simple if it has no proper pure subsystem. Recently it has been shown that for every positive integer n, there exists a purely simple system of rank n but no system of rank greater than the cardinality of the continuum is purely simple. In this paper it is shown that no system of rank greater than

ℵ
0

{\aleph _0}

is purely simple. Necessary and sufficient conditions for a system of rank

ℵ
0

{\aleph _0}

to be purely simple are also given.
Extensions of difference specializations
Authors: Lando Barbara
Published: June 2010

Maximal difference specializations and difference places are defined. Let R be the domain of a difference specialization

ϕ
\phi

of a difference field K and

x
∈
K

x \in K

. Then

ϕ
\phi

can be extended to a specialization

x
→
0

x \to 0

if and only if

1
∉
[
x
]

1 \notin [x]

. This result applies to give a condition on a polynomial for the extension of a specialization to its generic zero. In a slightly different direction, a necessary and sufficient condition for the extension of a specialization to a larger difference field is given.
On the singularities of nonlinear Fredholm operators of positive index
Authors: Berger M. S. , Plastock R. A.
Published: June 2010

The singular set

B
=
{
x

|

F
′

(
x
)

B = \{ x|F’(x)

is not surjective} of a nonlinear Fredholm operator F of positive index (between Banach spaces

X
1

{X_1}

and

X
2

{X_2}

) is investigated. Under the assumption that the mapping is proper and has a locally Lipschitzian Fréchet derivative

F
′

(
x
)

F’(x)

, it is shown that the singular set B is nonempty. Furthermore, when the Banach spaces are infinite dimensional, B cannot be the countable union of compact sets nor can

F

−
1

(
F
(
B
)
)

{F^{ - 1}}(F(B))

contain isolated points.
Fredholm composition operators
Authors: Kumar Ashok
Published: June 2010

In this paper a necessary and sufficient condition for a composition operator

C
T

{C_T}

on

L
2

[
0
,
1
]

{L^2}[0,1]

to be a Fredholm operator is given. In addition, all Fredholm composition operators on

l
2

(
N
)

{l^2}(N)

are characterized.
A uniqueness theorem for fixed points
Authors: Smith H. L. , Stuart C. A.
Published: June 2010

In a recent paper, R. Kellogg [3] showed that if

F
:

D
¯

→

D
¯

F:\bar D \to \bar D

is a completely continuous map of the closure of a bounded, convex, open set D in a real Banach space X,

F
∈

C
1

(
D
)

F \in {C^1}(D)

, 1 is not an eigenvalue of

F
′

(
x
)

F’(x)

for

x
∈
D

x \in D

, and

F
(
x
)
≠
x

F(x) \ne x

for

x
∈
∂
D

x \in \partial D

, then F has a unique fixed point in D. More recently, L. Talman [7] extended this result to k-set contractions when

k
>
1

k > 1

. The main result of this note is to show that, if the dimension of X is larger than one, the result of Kellogg and its extension by Talman remain valid provided that the set

{
x
∈
D
:
1

\{ x \in D:1

is an eigenvalue of

F
′

(
x
)
}

F’(x)\}

has no accumulation points in D, the other assumptions remaining the same. This result is obtained as a corollary of a more general result which gives conditions under which the set of fixed points of F in D is connected.
On separable Banach spaces, universal for all separable reflexive spaces
Authors: Bourgain J.
Published: June 2010

It is shown that a separable Banach space which is universal for all separable reflexive spaces is also universal for all separable spaces.
Extension of vector-lattice homomorphisms
Authors: Lipecki Z.
Published: June 2010

Extensions of a positive linear operator on a vector lattice satisfying a certain approximation condition are considered. When the operator is a vector-lattice homomorphism, these extensions are also vector-lattice homomorphisms.
On fundamental sequences of translates
Authors: Zalik R. A.
Published: June 2010

We find Müntz-type theorems for sequences of the form

{
f
(
t
+

c
n

)
}

\{ f(t + {c_n})\}

or

{
exp
⁡
(
−

c
n

t
)
f
(
t
)
}

\{ \exp ( - {c_n}t)f(t)\}

on

[
0
,
∞
)

[0,\infty )

.
Branching and generalized-recursive inset entropies
Authors: Ebanks Bruce R.
Published: June 2010

In the framework of the mixed theory of information, the general form of branching entropies is determined. This result then leads to a characterization of regular, generalized-recursive entropies of randomized systems of events.
Properties of 𝛽𝑋-𝑋 for locally connected generalized continua
Authors: Cain George L.
Published: June 2010

Let X be a locally connected generalized continuum and let

β
X

\beta X

denote the Stone-Čech compactification of X. In this paper are given necessary and sufficient conditions for

β
X
−
X

\beta X - X

to be the union of a finite number of disjoint continua, and for each of these continua to be indecomposable.
𝑃̄=𝑅̄ for maps of the interval
Authors: Coven Ethan M. , Hedlund G. A.
Published: June 2010

We show that for continuous maps of a compact interval to itself the closure of the set of periodic points coincides with the closure of the set of recurrent points.
Sequential conditions and free topological groups
Authors: Ordman Edward T. , Smith-Thomas Barbara V.
Published: June 2010

Most of the results in this paper concern relationships between sequential properties of a pointed topological space (X, p) and sequential properties of the Graev free topological group on X. In particular, it is shown that the free group over a sequential

k
ω

{k_\omega }

-space is sequential, and that a nondiscrete sequential free group has sequential order equal to

ω
1

{\omega _1}

(the first uncountable ordinal). The free topological group on a space X which includes a convergent sequence contains a closed subspace homeomorphic to

S
ω

{S_\omega }

, a previously studied homogeneous, zero-dimensional sequential space. Finally, it is shown that there is no topological group homeomorphic to

S
ω

{S_\omega }

.
A note on extremally disconnected spaces
Authors: Noiri Takashi
Published: June 2010

A topological space X is said to be locally S-closed if each point of X has an open neighborhood which is an S-closed subspace of X. In this note it is shown that every locally S-closed weakly Hausdorff (or almost-regular) space is extremally disconnected.
A combinatorial proof of Schur’s 1926 partition theorem
Authors: Bressoud David M.
Published: June 2010

One of the partition theorems published by I. J. Schur in 1926 is an extension of the Rogers-Ramanujan identities to partitions with minimal difference three. This theorem of Schur is proved here by establishing a one-to-one correspondence between the two types of partitions counted.
An integro-differential equation
Authors: Burton T. A.
Published: June 2010

The vector equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime left-parenthesis t right-parenthesis equals upper A left-parenthesis t right-parenthesis x left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript t Baseline upper C left-parenthesis t comma s right-parenthesis upper D left-parenthesis x left-parenthesis s right-parenthesis right-parenthesis x left-parenthesis s right-parenthesis d s plus upper F left-parenthesis t right-parenthesis">

x
′

(
t
)
=
A
(
t
)
x
(
t
)
+

∫
0
t

C
(
t
,
s
)
D
(
x
(
s
)
)
x
(
s
)
d
s
+
F
(
t
)

x’(t) = A(t)x(t) + \int _0^t {C(t,s)D(x(s))x(s)ds + F(t)}

\]
is considered in which A is not necessarily a stable matrix, but

A
(
t
)
+
G
(
t
,
t
)
D
(
0
)

A(t) + G(t,t)D(0)

is stable where G is an antiderivative of C with respect to t. Stability and boundedness results are then obtained. We also point out that boundedness results of Levin for the scalar equation

u
′

(
t
)
=
−

∫
0
t

a
(
t
−
s
)
g
(
u
(
s
)
)
d
s

u’(t) = - \int _0^t {a(t - s)g(u(s))ds}

can be extended to a vector system

x
′

(
t
)
=
−

∫
0
t

H
(
t
,
s
)
x
(
s
)
d
s

x’(t) = - \int _0^t {H(t,s)x(s)ds}

.
Extremal solutions for a class of nonlinear differential equations
Authors: Kreith Kurt
Published: June 2010

Positive solutions of

y

(
n
)

±
f
(
t
,
y
)
=
0

{y^{(n)}} \pm f(t,y) = 0

are characterized as maximal or minimal depending on whether their asymptotic behavior is

O
(

t

n
−
1

)

O({t^{n - 1}})

or

O
(
1
)

O(1)

, respectively. Necessary and sufficient conditions for the existence of such extremal solutions are established.
Valence sequences
Authors: Goodman A. W.
Published: June 2010

Let

V
0

,

V
1

,
…
,

V
k

,
…

{V_0},{V_1}, \ldots ,{V_k}, \ldots

be an infinite sequence of positive integers (where we include

∞
\infty

as a possible value for

V
k

{V_k}

). The sequence is called a valence sequence if there is an

f
(
z
)

f(z)

regular in the unit disk such that for each

k
⩾
0

k \geqslant 0

the kth derivative

f

(
k
)

(
z
)

{f^{(k)}}(z)

has valence

V
k

{V_k}

in E. Most of the results obtained here about valence sequences are obvious, but we prove three theorems about valence sequences that are not trivial.
An integral analogue to parallelogram law
Authors: Penico A. J. , Stanojević Č. V.
Published: June 2010

It is shown that there is an integral analogue to the classical characterization of inner product spaces. The sharp bounds for the integral of the norm are found.
Extending a measure from a ring to a sigma-ring
Authors: Johnson Roy A.
Published: June 2010

For each measure on a ring, there is a smallest measure on the generated sigma-ring such that ring members of finite measure keep the same measure. Conditions are given under which this smallest semi-extension preserves sums and order.
Best 𝐿^{𝑝}-approximation of generalized biaxisymmetric potentials
Authors: McCoy Peter A.
Published: June 2010

Let F be a real-valued generalized biaxisymmetric potential (GBASP ) in

L
p

(
p
⩾
1
)

{L^p}(p \geqslant 1)

on

Σ
\Sigma

, the open unit sphere about the origin. Convergence of a sequence of best harmonic polynomial approximates to F in

L
p

{L^p}

identifies those F that harmonically continue as entire function GBASP and determines their order and type. The analysis utilizes the Bergman and Gilbert Integral Operator Method to extend results from classical function theory on the best polynomial approximation of analytic functions of one complex variable.
Idempotent multipliers on spaces of continuous functions with 𝑝-summable Fourier transforms
Authors: Bloom Lynette M. , Bloom Walter R.
Published: June 2010

Let G denote a compact abelian group, and

A
p

{A^p}

the space of functions continuous on G and having p-summable Fourier transforms. The idempotent multipliers from

A
p

{A^p}

to

A
q

{A^q}

are characterised for

p
,
q
∈
[
1
,
2
]

p,q \in [1,2]

.
Balayage defined by the nonnegative convex functions
Authors: Fischer P. , Holbrook J. A. R.
Published: June 2010

We study the Choquet order induced on measures on a linear space by the cone of nonnegative convex functions. We are concerned mainly with discrete measures, and the following result is typical. Let

x
1

,
…
,

x
r

,

y
1

,
…
,

y
n

{x_1}, \ldots ,{x_r},{y_1}, \ldots ,{y_n}

, where

r
⩽
n

r \leqslant n

, be points in

R

d

{{\mathbf {R}}^d}

. Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript 1 Overscript r Endscripts f left-parenthesis x Subscript k Baseline right-parenthesis less-than-or-slanted-equals sigma-summation Underscript 1 Overscript n Endscripts f left-parenthesis y Subscript k Baseline right-parenthesis">

∑
1
r

f
(

x
k

)
⩽

∑
1
n

f
(

y
k

)

\sum \limits _1^r {f({x_k}) \leqslant } \sum \limits _1^n {f({y_k})}

\]
for all nonnegative, continuous, convex functions f if, and only if, there exists a doubly stochastic matrix M such that
\[

x
j

=

∑

k
=
1

n

m

j
k

y
k

(
j
=
1
,
…
,
r
)
.

{x_j} = \sum \limits _{k = 1}^n {{m_{jk}}{y_k}\quad (j = 1, \ldots ,r).}

\]
In the case

d
=
1

d = 1

, this result may be found in the work of L. Mirsky; our methods allow us to place such results in a general setting.
Christensen zero sets and measurable convex functions
Authors: Fischer Pal , Słodkowski Zbigniew
Published: June 2010

A notion of measurability in abelian Polish groups related to Christensen’s Haar zero set is studied. It is shown that a measurable homomorphism or a measurable Jensen convex function defined on a real linear Polish space is continuous.
A holonomy proof of the positive curvature operator theorem
Authors: Poor W. A.
Published: June 2010

Extending work of Bochner-Yano and M. Berger, D. Meyer proved that if the curvature operator of a compact, oriented, Riemannian manifold M has positive eigenvalues, then M is a rational homology sphere. Here a proof is given using Chern’s holonomy formula for the Laplacian on M; for completeness, a quick proof of Chern’s formula is included.
A short proof for a.e. convergence of generalized conditional expectations
Authors: Landers D. , Rogge L.
Published: June 2010

Let

L
s

(
μ
)

{L_s}(\mu )

be the space of real valued random variables with

μ
(

|

f

|

s

)
>
∞
,
1
>
s
>
∞

\mu (|f{|^s}) > \infty ,1 > s > \infty

. Let

C
⊂

L
s

(
μ
)

C \subset {L_s}(\mu )

be a closed convex set. For each

f
∈

L
s

(
μ
)

f \in {L_s}(\mu )

there exists a unique element

μ
s

(
f

|

C
)

{\mu _s}(f|C)

with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f minus mu Subscript s Baseline left-parenthesis f vertical-bar upper C right-parenthesis double-vertical-bar Subscript s Baseline less-than-or-slanted-equals double-vertical-bar f minus c double-vertical-bar Subscript s">

‖

f
−

μ
s

(
f

|

C
)

‖

s

⩽

‖

f
−
c

‖

s

{\left \| {f - {\mu _s}(f|C)} \right \|_s} \leqslant {\left \| {f - c} \right \|_s}

for every

c
∈
C

c \in C

. Let

C
n

{C_n}

be a decreasing or increasing sequence of closed convex lattices converging to the closed convex lattice

C
∞

{C_\infty }

. We show that

μ
s

(
f

|

C
n

)
→

μ
s

(
f

|

C
∞

)
μ

{\mu _s}(f|{C_n}) \to {\mu _s}(f|{C_\infty })\mu

-a.e. for every

f
∈

L
s

(
μ
)

f \in {L_s}(\mu )

. This result contains the results of a.e. convergence of prediction sequences of Ando-Amemiya and the result of Brunk and Johansen of a.e. convergence of conditional expectations given

σ
\sigma

-lattices.
Orthocompactness and perfect mappings
Authors: Burke Dennis K.
Published: June 2010

An example is given which shows that orthocompactness is not preserved by perfect maps. Subparacompact pointwise star-orthocompact spaces are orthocompact; this shows that orthocompactness is preserved by closed maps in the presence of subparacompactness.
Darboux retracts
Authors: Roush Fred , Gibson Richard G. , Kellum Kenneth R.
Published: June 2010

Darboux retracts are defined, and it is proved that any continuum contained in a separable metric space is a Darboux retract. This leads immediately to a very general extension theorem.
On the height of the first Stiefel-Whitney class
Authors: Hiller Howard L.
Published: June 2010

Let

G
k

(

R

n
+
k

)

{G_k}({{\mathbf {R}}^{n + k}})

denote the grassmann manifold of k-planes in real

(
n
+
k
)

(n + k)

-space and

w
1

∈

H
1

(

G
k

(

R

n
+
k

)
;

Z

2

)

{w_1} \in {H^1}({G_k}({{\mathbf {R}}^{n + k}});{{\mathbf {Z}}_2})

the first Stiefel-Whitney class of the universal bundle. We determine, for many (k, n), the exact height of

w
1

{w_1}

in the cohomology ring. We also indicate the combinatorial significance of the complex analogue of these computations.
The cobordism classification of hypersurfaces in lens spaces
Authors: Creighton J. H. C.
Published: June 2010

In Theorem A we classify by cobordism type the codimension one submanifolds of lens spaces

L
d

2
n
+
1

L_d^{2n + 1}

(quotient of

S

2
n
+
1

{S^{2n + 1}}

by the action of the dth roots of unity). A related immersion result is also obtained.
A class of finite group-amalgams
Authors: Djoković Dragomir Ž.
Published: June 2010

Let

A

−
1

{A_{ - 1}}

and

A
1

{A_1}

be finite groups such that

A

−
1

∩

A
1

=

A
0

{A_{ - 1}} \cap {A_1} = {A_0}

is a common subgroup with

[

A

−
1

:

A
0

]
=
4

[{A_{ - 1}}:{A_0}] = 4

and

[

A
1

:

A
0

]
=
2

[{A_1}:{A_0}] = 2

. We further assume that only the trivial subgroup of

A
0

{A_0}

is normal in both

A

−
1

{A_{ - 1}}

and

A
1

{A_1}

. Let K be the intersection of all conjugates

x

A
0

x

−
1

x{A_0}{x^{ - 1}}

for

x
∈

A

−
1

x \in {A_{ - 1}}

. Then if

A
0

≠
{
1
}

{A_0} \ne \{ 1\}

we have

A

−
1

/

K
≅

D
4

,

A
4

{A_{ - 1}}/K \cong {D_4},{A_4}

, or

S
4

{S_4}

. We describe in detail all such amalgams

(

A

−
1

,

A
1

)

({A_{ - 1}},{A_1})

when

A

−
1

/

K
≅

D
4

{A_{ - 1}}/K \cong {D_4}

(dihedral group of order 8). There are infinitely many of them, while if

A

−
1

/

K
≅

A
4

{A_{ - 1}}/K \cong {A_4}

or

S
4

{S_4}

there are only finitely many amalgams.
On finite groups containing three 𝐶𝐶-subgroups
Authors: Arad Zvi , Ferguson Pamela
Published: June 2010

A finite group G has a self-centralization system of type

(
2

|

A
1

|

,
4

|

A
2

|

,
4

|

A
3

|

)

(2|{A_1}|,4|{A_2}|,4|{A_3}|)

if G contains three nonconjugate CC-subgroups

A
1

,

A
2

,

A
3

{A_1},{A_2},{A_3}

, such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper N Subscript upper G Baseline left-parenthesis upper A 1 right-parenthesis EndAbsoluteValue equals 2 StartAbsoluteValue upper A 1 EndAbsoluteValue comma StartAbsoluteValue upper N Subscript upper G Baseline left-parenthesis upper A 2 right-parenthesis EndAbsoluteValue equals 4 StartAbsoluteValue upper A 2 EndAbsoluteValue comma StartAbsoluteValue upper N Subscript upper G Baseline left-parenthesis upper A 3 right-parenthesis EndAbsoluteValue equals 4 StartAbsoluteValue upper A 3 EndAbsoluteValue">

|

N
G

(

A
1

)

|

=
2

|

A
1

|

,

|

N
G

(

A
2

)

|

=
4

|

A
2

|

,

|

N
G

(

A
3

)

|

=
4

|

A
3

|

|{N_G}({A_1})| = 2|{A_1}|,|{N_G}({A_2})| = 4|{A_2}|,|{N_G}({A_3})| = 4|{A_3}|

. The authors prove that if a finite group G has a self-centralization system of type

(
2

|

A
1

|

,
4

|

A
2

|

,
4

|

A
3

|

)

(2|{A_1}|,4|{A_2}|,4|{A_3}|)

and

|

G

|

⩽
3

|

A
1

|

2

|

A
2

|

2

|

A
3

|

2

|G| \leqslant 3|{A_1}{|^2}|{A_2}{|^2}|{A_3}{|^2}

, then G has a nilpotent normal subgroup N such that G/N is isomorphic to

S
z
(
q
)

Sz(q)

for suitable q.
A projective characterization for SKT-modules
Authors: Wick Brian D.
Published: June 2010

In this paper a class of abelian groups (SKT-modules), which includes the torsion totally projective groups, S-groups, and balanced projectives is shown to be a subclass of a projective class of groups with respect to a naturally defined class of short exact sequences called the ch-projective modules and ch-pure sequences, respectively. Every

Z
p

{Z_p}

-module has a ch-pure projective resolution and every reduced ch-projective module is a summand of a SKT-module. It is finally shown that a

Z
p

{Z_p}

-module M is ch-projective if and only if, for every ordinal

α
\alpha

, the two

Z
p

{Z_p}

-modules

p
α

M

{p^\alpha }M

and

M

/

p
α

M

M/{p^\alpha }M

are both ch-projective.
Divided differences and systems of nonharmonic Fourier series
Authors: Ullrich David
Published: June 2010

Suppose that

ω

n
,
0

,

ω

n
,
1

,
…
,

ω

n
,
k

{\omega _{n,0}},{\omega _{n,1}}, \ldots ,{\omega _{n,k}}

are distinct complex numbers with

|

n
−

ω

n
,
j

|

⩽
δ

|n - {\omega _{n,j}}| \leqslant \delta

for all

n
∈

Z

,
j
=
0
,
1
,
…
,
k

n \in {\mathbf {Z}},j = 0,1, \ldots ,k

. We show that if

δ
>
0

\delta > 0

is small enough then, given complex numbers

c

n
,
j

(
n
∈

Z

,
j
=
0
,
1
,
…
,
k
)

{c_{n,j}}(n \in {\mathbf {Z}},j = 0,1, \ldots ,k)

there exists

f
∈

L
2

(
−
(
k
+
1
)
π
,
(
k
+
1
)
π
)

f \in {L^2}( - (k + 1)\pi ,(k + 1)\pi )

with
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript minus left-parenthesis k plus 1 right-parenthesis pi Superscript left-parenthesis k plus 1 right-parenthesis pi Baseline f left-parenthesis t right-parenthesis e Superscript minus i t omega Super Subscript n comma j Superscript Baseline d t equals c Subscript n comma j Baseline for n element-of bold upper Z comma j equals 0 comma 1 comma ellipsis comma k">

∫

−
(
k
+
1
)
π

(
k
+
1
)
π

f
(
t
)

e

−
i
t

ω

n
,
j

d
t
=

c

n
,
j

for

n
∈

Z

,
j
=
0
,
1
,
…
,
k

\int _{ - (k + 1)\pi }^{(k + 1)\pi } {f(t){e^{ - it{\omega _{n,j}}}}} dt = {c_{n,j}}\quad {\text {for}}\;n \in {\mathbf {Z}},j = 0,1, \ldots ,k

\]
if and only if certain “divided differences” involving the

c

n
,
j

{c_{n,j}}

’s and the

ω

n
,
j

{\omega _{n,j}}

’s are square summable. This extends a classical theorem of Paley and Wiener, which is equivalent to the case

k
=
0

k = 0

above.
Tame measures on certain compact sets
Authors: Lee Hsuan Pei
Published: June 2010

A finite complex Borel measure

μ
\mu

on a compact subset

X
⊂

C

n

X \subset {{\mathbf {C}}^n}

is called tame if there exist finite measures

σ
1

,
…
,

σ
n

{\sigma _1}, \ldots ,{\sigma _n}

on X with
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper X Endscripts phi d mu equals integral Underscript upper X Endscripts sigma-summation Underscript 1 Overscript n Endscripts StartFraction partial-differential phi Over partial-differential z overbar Subscript j Baseline EndFraction d sigma Subscript j Baseline">

∫
X

ϕ
d
μ
=

∫
X

∑
1
n

∂
ϕ

∂

z
¯

j

d

σ
j

\int _X \phi d\mu = \int _X {\sum \limits _1^n {\frac {{\partial \phi }}{{\partial {{\bar z}_j}}}d{\sigma _j}} }

\]
for every

ϕ
∈

C
0
∞

(

C

n

)

\phi \in C_0^\infty ({{\mathbf {C}}^n})

. We define

X
T

=
{
(

z
1

,

z
2

)
:

|

z
1

|

2

+

|

z
2

|

2

=
1

{X_T} = \{ ({z_1},{z_2}):|{z_1}{|^2} + |{z_2}{|^2} = 1

and

z
1

∈
T
}

{z_1} \in T\}

, where T is a compact subset of

{

|

z
1

|

>
1
}

\{ |{z_1}| > 1\}

in

C

1

{{\mathbf {C}}^1}

. It is shown in this paper that tame measures form a weak-

∗

^ \ast

dense subset of

R

(

X
T

)
⊥

R{({X_T})^ \bot }

. It follows then, with the help of a theorem by Weinstock, that

R
(

X
T

)

R({X_T})

is a local algebra.
Essentially Hermitian operators in 𝐵(𝐿_{𝑝})
Authors: Allen G. D. , Legg D. A. , Ward J. D.
Published: June 2010

It is shown that on

L
p

[
0
,
1
]

{L_p}[0,1]

all bounded linear operators which are Hermitian in the Calkin algebra

B
(

L
p

)

/

C
(

L
p

)

B({L_p})/C({L_p})

, must be of the form “Hermitian plus compact". That is, essentially Hermitian operators have the form, real multiplier plus compact.
𝜅-finiteness and 𝜅-additivity of measures on sets and left invariant measures on discrete groups
Authors: Armstrong Thomas E. , Prikry Karel
Published: June 2010

For any cardinal

κ
\kappa

a possibly infinite measure

μ
⩾
0

\mu \geqslant 0

on a set X is strongly non-

κ
\kappa

-additive if X is partitioned into

κ
\kappa

or fewer

μ
\mu

-negligible sets. The measure

μ
\mu

is purely non-

κ
\kappa

-additive if it dominates no nontrivial

κ
\kappa

-additive measure. The properties and relationships of these types of measures are examined in relationship to measurable ideal cardinals and real-valued measurable cardinals. Any

κ
\kappa

-finite left invariant measure

μ
\mu

on a group G of cardinality larger than

κ
\kappa

is strongly non-

κ
\kappa

-additive. In particular,

σ
\sigma

-finite left invariant measures on infinite groups are strongly finitely additive.
The crossed product of a 𝐶*-algebra by an endomorphism
Authors: Paschke William L.
Published: June 2010

Let A be a unital, strongly amenable

C
∗

{C^ \ast }

-algebra,

σ
:
A
→
p
A
p

\sigma :A \to pAp

a

∗

^ \ast

-isomorphism (where p is a proper projection of A), and S an isometry such that

S
x

S
∗

=
σ
(
x
)

Sx{S^ \ast } = \sigma (x)

for all x in A. If A has no nontrivial

σ
\sigma

-invariant ideals, then

C
∗

(
A
,
S
)

{C^ \ast }(A,S)

is simple. Furthermore,

C
∗

(
A
,
S
)

{C^\ast }(A,S)

is isomorphic to a corner of the crossed product of

A
⊗

A \otimes

(compacts) by an automorphism.
Simple proof of a theorem of Erdős and Lax
Authors: Aziz Abdul , Mohammad Q. G.
Published: June 2010

An elementary new and simple proof of Erdös-Lax theorem is given which in essence involves no analysis.
Some characterizations of developable spaces
Authors: Brandenburg Harald
Published: June 2010

Two characterizations of developable spaces are proved which may be viewed as analogues, for developable spaces, of the Nagata-Smirnov metrization theorem or of the “double sequence metrization theorem” of Nagata respectively.
How many knots have the same group?
Authors: Simon Jonathan
Published: June 2010

Let K be a knot in

S
3

,
G
=

π
1

(

S
3

−
K
)
,
n
=

{S^3},G = {\pi _1}({S^3} - K),n =

number of prime factors of

K
,
ν
(
G
)
=

K,\nu (G) =

number of topologically different knot-complements with group G and

κ
(
G
)
=

\kappa (G) =

number of distinct knot types with group G. Theorem. If K is prime, then

ν
(
G
)
⩽
2

\nu (G) \leqslant 2

. If

n
⩾
2

n \geqslant 2

, then

ν
(
G
)
=
κ
(
G
)
⩽

2

n
−
1

\nu (G) = \kappa (G) \leqslant {2^{n - 1}}

. For each

n
⩾
2

n \geqslant 2

, the bound

2

n
−
1

{2^{n - 1}}

is the best possible. For K prime, we still have the conjecture

ν
(
G
)
=
κ
(
G
)
=
1

\nu (G) = \kappa (G) = 1

. If K is a cable-knot, then

κ
(
G
)
⩽
2

\kappa (G) \leqslant 2

.
The extension of norms on subgroups of free topological groups
Authors: Morris Sidney A. , Nickolas Peter
Published: June 2010

A norm on a group G is a nonnegative real-valued function N which is zero at the identity and satisfies

N
(
x

y

−
1

)
⩽
N
(
x
)
+
N
(
y
)

N(x{y^{ - 1}}) \leqslant N(x) + N(y)

, for

x
,
y
∈
G

x,y \in G

. Let

F
(
X
)

F(X)

be the free topological group on a space X. Bicknell and Morris have shown that any norm on a subgroup of

F
(
X
)

F(X)

generated by a finite subset of X may be extended to a continuous norm on the whole of

F
(
X
)

F(X)

. In this note a very direct and simple proof of this theorem is given.
Strong liftings which are not Borel liftings
Authors: Johnson Russell A.
Published: June 2010

The purpose of this note is to prove that any strong lifting of the circle commuting with translations cannot be a Borel lifting.
Eigenvalue distribution of Toeplitz matrices
Authors: Nevai Paul G.
Published: June 2010

A theorem of U. Grenander and G. Szegö on Toeplitz matrices is generalized. A new method is proposed for investigating eigenvalue distribution of Toeplitz matrices.
Distributional derivation of an asymptotic expansion
Authors: Wong R.
Published: June 2010

An alternative derivation of the asymptotic expansion of multiple Fourier transforms is presented. The present approach is based on the use of distributions. With some modifications, this method can also be applied to other integral transforms with oscillatory kernels such as the Hankel transform.
Boundary convergence and boundary limits of Blaschke products
Authors: Linden C. N.
Published: June 2010

For a given countable subset

γ
\gamma

of the unit circle, a method is given for the construction of Blaschke products

B
(
z
,
A
)

B(z,A)

which converge at all points of

γ
\gamma

and which, for each point

e

i
φ

{e^{i\varphi }}

of

γ
\gamma

, either (a) have no asymptotic value at

e

i
φ

{e^{i\varphi }}

or (b) have an asymptotic value at

e

i
φ

{e^{i\varphi }}

not equal to

B
(

e

i
φ

,
A
)

B({e^{i\varphi }},A)

.
A direct proof that the RC-integral is equivalent to the 𝐷*-integral
Authors: Kubota Yôto
Published: June 2010

We give a direct and short proof that the Riemann complete integral defined by R. Henstock is equivalent to the special Denjoy integral.
Brownian motion and a consequence of Harnack’s inequality: nodes of quantum wave functions
Authors: Hoffmann-Ostenhof M. , Hoffmann-Ostenhof T. , Simon B.
Published: June 2010

Using Brownian motion ideas, we give two proofs that for a large class of V’s,

(
−
Δ
+
V
)
u
=
E
u

( -\Delta + V)u = Eu

and

u
(

x
0

)
=
0

u({x_0}) = 0

implies that u changes sign near

x
0

{x_0}

.
Some consequences of (𝑉=𝐿) in the theory of analytic sets
Authors: Hansell R. W.
Published: June 2010

Following G. M. Reed’s definition of a Q-set, we define a

Q
A

{Q_A}

-set to be any non-

σ
\sigma

-discrete topological space with the property that each subset is (relatively) analytic (= Souslin-

F

\mathcal {F}

set). Clearly every Q-set is a

Q
A

{Q_A}

-set. The discrete irrational extension of the space of real numbers is an example of a first countable hereditarily paracompact (Hausdorff)

Q
A

{Q_A}

-set which is not a Q-set. Theorem.

(
V
=
L
)

(V = L)

Let X be a first countable normal space all of whose subsets are analytic. Then X is

σ
\sigma

-discrete if and only if the product of X with the space of irrational numbers is normal. A new structural property of analytic sets is developed in order to utilize a proof technique due to Reed. Several corollaries are obtained on properties of completely additive-analytic families of sets in general metric spaces.
Homotopy rigidity for Grassmannians
Authors: Back Allen
Published: June 2010

Two n-dimensional unitary representations which differ by complex conjugation or tensoring with a character induce topologically equivalent actions on the Grassmann manifold of complex m-planes in n-space. This paper shows under modest dimension hypotheses that only such projectively equivalent linear representations of compact connected Lie groups can give topologically conjugate actions.
Finite aspherical complexes with infinitely-generated groups of self-homotopy-equivalences
Authors: McCullough Darryl
Published: June 2010

A finite aspherical complex X is constructed whose group of homotopy classes of basepoint-preserving self-homotopy-equivalences is infinitely generated.
𝛽(𝑋) can be Fréchet
Authors: Berner Andrew J.
Published: June 2010

A class of spaces is defined which share many properties of Gillman and Jerison’s space

ψ
\psi

. These spaces are used to generalize a theorem of Malykhin, showing that certain one point compactifications are Stone-Čech compactifications. This is used to construct a space whose Stone-Čech compactification is a Fréchet space (under a set theoretic assumption which follows, for example, from the continuum hypothesis).
The Mazur-Orlicz bounded consistency theorem
Authors: Snyder A. K. , Wilansky A.
Published: June 2010

An elementary and computationally easy proof of the Mazur-Orlicz bounded consistency theorem.
Isomorphisms of sums of countable Boolean algebras
Authors: Trnková Věra
Published: June 2010

Denote by nA the sum of n copies of a Boolean algebra A. We prove that, for any countable Boolean algebra

A
,
n
A
≅
m
A

A,nA \cong mA

with

m
>
n

m > n

implies

n
A
≅
(
n
+
1
)
A

nA \cong (n + 1)A

.
The little Desarguesian theorem for algebras in modular varieties
Authors: Gumm H.-Peter
Published: June 2010

We show that the “Little Desarguesian Theorem” is true in the congruence-class geometry of algebras in modular varieties.
The stability of the cosine equation
Authors: Baker John A.
Published: June 2010

If

δ
>
0

\delta > 0

, G is an abelian group and f is a complex-valued function defined on G such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f left-parenthesis x plus y right-parenthesis plus f left-parenthesis x minus y right-parenthesis minus 2 f left-parenthesis x right-parenthesis f left-parenthesis y right-parenthesis EndAbsoluteValue less-than-or-slanted-equals delta">

|

f
(
x
+
y
)
+
f
(
x
−
y
)
−
2
f
(
x
)
f
(
y
)

|

⩽
δ

|f(x + y) + f(x - y) - 2f(x)f(y)| \leqslant \delta

for all

x
,
y
∈
G

x,y \in G

, then either

|

f
(
x
)

|

⩽
(
1
+

1
+
2
δ

)

/

2

|f(x)| \leqslant (1 + \sqrt {1 + 2\delta } )/2

for all

x
∈
G

x \in G

or

f
(
x
+
y
)
+
f
(
x
−
y
)
=
2
f
(
x
)
f
(
y
)

f(x + y) + f(x - y) = 2f(x)f(y)

for all

x
,
y
∈
G

x,y \in G

.
A lower bound for the spectral radius
Authors: Pták Vlastimil
Published: June 2010

We prove an inequality for a problem of Carathéodory type: given n inner functions

m
1

,

m
2

,
…
,

m
n

{m_1},{m_2}, \ldots ,{m_n}

, to find the smallest norm of an

H
∞

{H^\infty }

function such that the first n terms of its power series coincide with those of the product

m
1

⋯

m
n

{m_1} \cdots {m_n}

. As an application, we obtain a lower bound for the spectral radius of an n-dimensional operator on Hilbert space in terms of its norm and the norm of its nth power.
On the note of C. L. Belna
Authors: Waterman Daniel
Published: June 2010

The space of functions of ordered harmonic bounded variation (OHBV) has been shown by Belna to contain the space of functions of harmonic bounded variation (HBV) properly. OHBV is a Banach space and HBV is a first category subset. The ordered harmonic variation has continuity properties quite different from those of the harmonic variation. The relationship of these classes to the everywhere convergence of Fourier series is discussed.