bluuebunny/crossref_metadata_2025 huggingface dataset:
Mutual existence of product integrals
Authors: Helton Jon C.
Published: June 2010

Definitions and integrals are of the subdivision-refinement type, and functions are from

R
×
R

R \times R

to R, where R represents the real numbers. Let

O

M
∘

O{M^ \circ }

be the class of functions G such that

x

∏

y

(
1
+
G
)

_x\prod {^y} (1 + G)

exists for

a
≦
x
>
y
≦
b

a \leqq x > y \leqq b

and

∫
a
b

|

1
+
G
−
∏

(
1
+
G
)

|

=
0

\smallint _a^b|1 + G - \prod {(1 + G)| = 0}

. Let

O

P
∘

O{P^ \circ }

be the class of functions G such that

|

∏

q
=
i

j

(
1
+

G
q

)

|

|\prod \nolimits _{q = i}^j {(1 + {G_q})|}

is bounded for refinements

{

x
q

}

q
=
0

n

\{ {x_q}\} _{q = 0}^n

of a suitable subdivision of [a, b]. If F and G are functions from

R
×
R

R \times R

to R such that

F
∈
O

P
∘

F \in O{P^ \circ }

on [a, b],

lim

x
,
y
→

p
+

F
(
x
,
y
)

{\lim _{x,y \to {p^ + }}}F(x,y)

and

lim

x
,
y
→

p
−

F
(
x
,
y
)

{\lim _{x,y \to {p^ - }}}F(x,y)

exist and are zero for

p
∈
[
a
,
b
]

p \in [a,b]

, each of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript x right-arrow p Superscript plus Baseline Endscripts upper F left-parenthesis p comma x right-parenthesis comma limit Underscript x right-arrow p Superscript minus Baseline Endscripts upper F left-parenthesis x comma p right-parenthesis comma limit Underscript x right-arrow p Superscript plus Baseline Endscripts upper G left-parenthesis p comma x right-parenthesis">

lim

x
→

p
+

F
(
p
,
x
)
,

lim

x
→

p
−

F
(
x
,
p
)
,

lim

x
→

p
+

G
(
p
,
x
)

{\lim _{x \to {p^ + }}}F(p,x),{\lim _{x \to {p^ - }}}F(x,p),{\lim _{x \to {p^ + }}}G(p,x)

and

lim

x
→

p
−

G
(
x
,
p
)

{\lim _{x \to {p^ - }}}G(x,p)

exist for

p
∈
[
a
,
b
]

p \in [a,b]

, and G has bounded variation on [a, b], then any two of the following statements imply the other: (1)

F
+
G
∈
O

M
∘

F + G \in O{M^ \circ }

on [a, b], (2)

F
∈
O

M
∘

F \in O{M^ \circ }

on [a, b], and (3)

G
∈
O

M
∘

G \in O{M^\circ }

on [a, b].
The range of a vector-valued measure
Authors: Helmes Kurt
Published: June 2010

A short proof is given that the weak closure of the range of a totally nonatomic vector-valued measure is convex.
On analytic irreducibility at ∞ of a pencil of curves
Authors: Moh T. T.
Published: June 2010

In this article we establish that if a member of the pencil

f
(
x
,
y
)
+
λ

f(x,y) + \lambda

is analytic irreducible at

∞
\infty

then all members are.
Two characterizations of linear Baire spaces
Authors: Saxon Stephen A.
Published: June 2010

The Wilansky-Klee conjecture is equivalent to the (unproved) conjecture that every dense,

1
1

-codimensional subspace of an arbitrary Banach space is a Baire space (second category in itself). The following two characterizations may be useful in dealing with this conjecture: (i) A topological vector space is a Baire space if and only if every absorbing, balanced, closed set is a neighborhood of some point, (ii) A topological vector space is a Baire space if and only if it cannot be covered by countably many nowhere dense sets, each of which is a union of lines (

1
1

-dimensional subspaces). Characterization (i) has a more succinct form, using the definition of Wilansky’s text [8, p. 224]: a topological vector space is a Baire space if and only if it has the

t
t

property.
The Zygmund condition for polygonal approximation
Authors: Newman D. J.
Published: June 2010

We investigate the class of functions which can be uniformly approximated to within

O
(
1

/

n
)

O(1/n)

by a canonical choice of piecewise (

n
n

-piece) linear functions. The class turns out identical to the Zygmund class.
The valuation structure of homomorphic images of Prüfer domains
Authors: Boisen Monte B. , Sheldon Philip B.
Published: June 2010

Let

R
R

denote a Prüfer ring which is a homomorphic image of a Prüfer domain

D
D

. The purpose of this paper is to investigate the relationship between the valuation structure over

D
D

and the valuation structure over

R
R

. It is shown that there exists a one-to-one correspondence between the valuations over

R
R

and the valuations over

D
D

centered on primes containing the kernel of the homomorphism. This correspondence is shown to be natural in the sense that the value of an element of

R
R

is either infinity or the value of its pre-images under the corresponding valuation over

D
D

. Moreover, the value group of a valuation over

R
R

is an isolated subgroup of the value group of the corresponding valuation over

D
D

.
On the uniform ergodic theorem
Authors: Lin Michael
Published: June 2010

We give an elementary proof of the uniform ergodic theorem: “Let

T
T

be a linear operator on a Banach space with

|

|

T
n

/

n

|

|

→
0

||{T^n}/n|| \to 0

. The following are equivalent: (1)

N

−
1

∑

n
=
0

N
−
1

T
n

{N^{ - 1}}\sum \nolimits _{n = 0}^{N - 1} {{T^n}}

converges uniformly. (2)

(
I
−
T

)
2

X

{(I - T)^2}X

is closed. (3)

(
I
−
T
)
X

(I - T)X

is closed."
Invariant subspaces for products of Hermitian operators
Authors: Radjavi Heydar , Rosenthal Peter
Published: June 2010

It is shown that a nonscalar operator which is the product of a Hermitian operator and a positive operator has a nontrivial hyperinvariant subspace; this is a slight generalization of a result of Suzuki’s.
The Brauer group of an Amitsur field. II
Authors: Chang Chan Nan
Published: June 2010

The structure of the Brauer group over an Amitsur field with characteristic 0 has been determined in [5]. In this paper we extend these results to the Amitsur field with characteristic

p
>
0

p > 0

.
An ergodic super-property of Banach spaces defined by a class of matrices
Authors: Brunel A. , Fong H. , Sucheston L.
Published: June 2010

A matrix

(

a

n
i

)

({a_{ni}})

is called an

R
R

-matrix if (A)

Σ
i

a

n
i

↛
0

{\Sigma _i}{a_{ni}} \nrightarrow 0

, and (B)

lim
n

a

n
i

=
0

{\lim _n}{a_{ni}} = 0

for each

i
i

. A Banach space

X
X

is called

R
R

-ergodic if for each isometry

T
T

and each

x
∈
X

x \in X

, there is an

R
R

-matrix

(

a

n
i

)

({a_{ni}})

such that

Σ
i

a

n
i

T
i

x

→

W

{\Sigma _i}{a_{ni}}{T^i}x\xrightarrow {{\text {W}}}

(converges weakly). Given two Banach spaces

F
F

and

X
X

, write

F

 fr 

X

F{\text { fr }}X

if for each finite-dimensional subspace

F
′

F’

of

F
F

and

ϵ
>
0

\epsilon > 0

, there is an isomorphism

V
V

from

F
′

F’

onto a subspace of

X
X

such that

|

|

|

x

|

|

−

|

|

V
x

|

|

|

>
ϵ

\left | {||x|| - ||Vx||} \right | > \epsilon

for each

x
∈

F
′

x \in F’

with

|

|

x

|

|

≤
1

||x|| \leq 1

.

X
X

is called super-

R
R

-ergodic if

F
F

is

R
R

-ergodic for each

F

 fr 

X

F{\text { fr }}X

. Theorem.

X
X

is super-

R
R

-ergodic if and only if

X
X

is super-reflexive. The proof is based on the following: Theorem. Let

T
T

be a linear operator on

X
,
(

a

n
i

)

X,({a_{ni}})

a matrix satisfying (A),

x
∈
X

x \in X

such that

Σ
i

a

n
i

T
i

x

→

W

x
¯

{\Sigma _i}{a_{ni}}{T^i}x\xrightarrow {{\text {W}}}\bar x

. Then there is a constant

α
\alpha

such that

(
x
−
a

x
¯

)
∈

(
I
−
T
)
X

¯

(x - a\bar x) \in \overline {(I - T)X}

.
The supports of infinitely divisible distribution functions
Authors: Tucker Howard G.
Published: June 2010

An infinitely divisible probability measure over the real line whose associated Lévy spectral measure gives nonzero mass to every deleted neighborhood of the origin is shown to have as its support an interval of the form

(
−
∞
,
a
]
,
[
a
,
∞
)

( - \infty ,a],[a,\infty )

or

(
−
∞
,
∞
)

( - \infty ,\infty )

.
Lusin sets and well ordering the continuum
Authors: Pincus David , Prikry Karel
Published: June 2010

A Lusin set is constructed in a model of set theory which lacks a well ordering of the continuum.
Number fields with prescribed 𝑙-class groups
Authors: Gerth Frank
Published: June 2010

Let

G
G

be any finite elementary abelian

l
l

-group, where

l
l

is a rational prime. We show that there exist infinitely many number fields whose

l
l

-class groups are isomorphic to

G
G

.
Polynomial rings over a commutative von Neumann regular ring
Authors: Gilmer Robert
Published: June 2010

It is shown that the annihilator of each finitely generated ideal of

R
[

{

X
λ

}

λ
∈
Λ

]

R[{\{ {X_\lambda }\} _{\lambda \in \Lambda }}]

, where

R
R

is a commutative von Neumann regular ring with identity, is principal; this generalizes a recent result of P. J. McCarthy.
Pseudo-injective modules which are not quasi-injective
Authors: Teply Mark L.
Published: June 2010

For certain rings with infinitely many nonisomorphic simple left modules, a method is given for constructing pseudo-injective modules which are not quasi-injective. This method is used to produce examples of such modules over a commutative ring.
Groups with faithful blocks
Authors: Pahlings H.
Published: June 2010

A necessary and sufficient condition is given for a finite group to have a

p
p

-block with kernel 1. This extends a theorem of Gaschütz on the existence of a faithful irreducible representation of a finite group.
Conditions on a compact connected Lie group which insure a “Weyl character formula”
Authors: Shapiro Jack M.
Published: June 2010

A theorem showing the equivalence of three conditions on a compact connected Lie group is proved. Among the corollaries is an extended “Weyl character formula” as originally stated by Bott.
Smoothness of decaying wave motions
Authors: Rauch Jeffrey
Published: June 2010

We show that for compactly supported perturbations of the wave equation, solutions whose total energy decays to zero must be smooth, provided that there are no trapped rays.
Invariant means on almost periodic functions and equicontinuous actions
Authors: Lau Anthony To Ming
Published: June 2010

Let

S
S

be a topological semigroup such that the almost periodic functions on

S
S

have a left invariant mean (this is the case, for example, when

S
S

has finite intersection property for closed right ideals). Then whenever

S
S

acts equicontinuously on a compact Hausdorff space

X
X

, there exists a compact group

G
G

of homeomorphisms acting equicontinuously on a retract

Y
Y

of

X
X

such that

S
S

has a common fixed point in

X
X

if and only if

G
G

has a common fixed point in

Y
Y

. This result generalises some recent work of T. Mitchell. As an application, we show that whenever

S
S

acts equicontinuously on the closed unit interval

I
I

, then

I
I

contains a common fixed point for

S
S

.
On the nonexistence of unimodular functions in 𝑅²(𝑋,𝑑𝑥𝑑𝑦)
Authors: Brandstein Alfred G.
Published: June 2010

It is shown for certain planar sets, e.g., Brennan sets, with no interior, that if

f
∈

R
2

(
X
,
d
x
d
y
)

f \in {R^2}(X,dxdy)

and

|

f

|

=
1

a.e.

d
x
d
y

|f| = 1\; \text {a.e.}\; dxdy

then

f
≡
constant

f \equiv \text {constant}

.
Analytic domination by fractional powers with linear estimates
Authors: Sloan Alan D.
Published: June 2010

Conditions are given which imply the analytic domination of one operator by fractional powers of a positive selfadjoint operator. The conditions involve only linear estimates rather than the usual quadratic estimates.
A correction to the paper: “Semi-open sets and semi-continuity in topological spaces” (Amer. Math. Monthly 70 (1963), 36–41) by Norman Levine
Authors: Hamlett T. R.
Published: June 2010

A subset

A
A

of a topological space is said to be semi-open if there exists an open set

U
U

such that

U
⊆
A
⊆
Cl
⁡
(
U
)

U \subseteq A \subseteq \operatorname {Cl} (U)

where

Cl
⁡
(
U
)

\operatorname {Cl} (U)

denotes the closure of

U
U

. The class of semi-open sets of a given topological space

(
X
,

T

)

(X,\mathcal {T})

is denoted

S

.O

.

(
X
,

T

)

{\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T})

. In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if

T

\mathcal {T}

and

T

∗

{\mathcal {T}^ \ast }

are two topologies for a set

X
X

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S period upper O period left-parenthesis upper X comma script upper T right-parenthesis subset-of-or-equal-to upper S period upper O period left-parenthesis upper X comma script upper T Superscript asterisk Baseline right-parenthesis">

S

.O

.

(
X
,

T

)
⊆

S

.O

.

(
X
,

T

∗

)

{\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) \subseteq {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast })

, then

T

⊆

T

∗

\mathcal {T} \subseteq {\mathcal {T}^ \ast }

. In a corollary to this theorem, Levine states that if

S

.O

.

(
X
,

T

)
=

S

.O

.

(
X
,

T

∗

)

{\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) = {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast })

, then

T

=

T

∗

\mathcal {T} = {\mathcal {T}^ \ast }

. An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subsets can arise from functions, and points out some of the implications of two topologies being related in this manner.
A local characterization of Darboux ℬ functions
Authors: Gibson Richard G.
Published: June 2010

A. M. Bruckner and J. B. Bruckner gave the definition of Darboux

B

\mathcal {B}

functions and proved a theorem which is a local characterization of real-valued Darboux

B

\mathcal {B}

functions. The purpose of this paper is to generalize this theorem. To this end, the definition of a function being Darboux

B

\mathcal {B}

at a point is given which has a metric continuum as its range. Hence, the theorem that a function is Darboux

B

\mathcal {B}

if and only if it is Darboux

B

\mathcal {B}

at each point.
Practically perfect three-manifold groups
Authors: McMillan D. R.
Published: June 2010

Let

M
3

{M^3}

be a

3
3

-manifold containing no

2
2

-sided projective plane. Let

G
≠
{
1
}

G \ne \{ 1\}

be a finitely-generated subgroup of

π
1

(

M
3

)

{\pi _1}({M^3})

such that

G
G

is indecomposable relative to free product, and such that

G
G

abelianized is finite. (

G
G

is “practically perfect".) Then, it is shown that there is a compact

3
3

-submanifold

Z
3

⊂

M
3

{Z^3} \subset {M^3}

such that

π
1

(

Z
3

)

{\pi _1}({Z^3})

contains a subgroup of finite index conjugate to

G
G

, and

Z
3

{Z^3}

is bounded by a

2
2

-sphere. Some related extensions of this result are given, plus an application to compact absolute neighborhood retracts in

3
3

-manifolds.
On sequences with a prescribed number of pairwise sums
Authors: Choi S. L. G.
Published: June 2010

In this paper we obtain an estimate, which is substantially best possible, for the number of distinct pairs of subsets

A

,

B

\mathcal {A},\mathcal {B}

of

{
1
,
…
,
n
}

\{ 1, \ldots ,n\}

, each consisting of

k
k

integers, so that the number of elements in

A

+

B

\mathcal {A} + \mathcal {B}

satisfies a prescribed upper bound.
An upper bound for the permanent of a fully indecomposable matrix
Authors: Foregger Thomas H.
Published: June 2010

Let

A
A

be an

n
×
n

n \times n

fully indecompasable matrix with nonnegative integer entries and let

σ
(
A
)

\sigma (A)

denote the sum of the entries of

A
A

. We prove that

per

(
A
)
≤

2

σ
(
A
)
−
2
n

+
1

{\text {per}}(A) \leq {2^{\sigma (A) - 2n}} + 1

and give necessary and sufficient conditions for equality to hold.
Weighted norm inequalities for fractional integrals
Authors: Welland G. V.
Published: June 2010

A simpler proof of an inequality of Muckenhoupt and Wheeden is given. Let

T
α

f
(
x
)
=
∫
f
(
y
)

|

x
−
y

|

α
−
d

d
y

{T_\alpha }f(x) = \smallint f(y)|x - y{|^{\alpha - d}}dy

be given for functions defined in

R

d

{{\mathbf {R}}^d}

. Let

υ
\upsilon

be a weight function which satisfies
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis StartAbsoluteValue upper Q EndAbsoluteValue Superscript negative 1 Baseline integral Underscript upper Q Endscripts left-bracket upsilon left-parenthesis x right-parenthesis right-bracket Superscript q Baseline d x right-parenthesis Superscript 1 slash q Baseline left-parenthesis StartAbsoluteValue upper Q EndAbsoluteValue Superscript negative 1 Baseline integral Underscript upper Q Endscripts left-bracket upsilon left-parenthesis x right-parenthesis right-bracket Superscript minus p Super Superscript prime Superscript Baseline d x right-parenthesis Superscript 1 slash p Super Superscript prime Superscript Baseline less-than-or-equal-to upper K">

(

|

Q

|

−
1

∫
Q

[
υ
(
x
)
]

q

d
x

)

1

/

q

(

|

Q

|

−
1

∫
Q

[
υ
(
x
)
]

−

p
′

d
x

)

1

/

p
′

≤
K

(|Q{|^{ - 1}}\int _Q {{{[\upsilon (x)]}^q}dx{)^{1/q}}(|Q{|^{ - 1}}\int _Q {{{[\upsilon (x)]}^{ - p’}}dx{)^{1/p’}} \leq K} }

\]
for each cube,

Q
Q

, with sides parallel to a standard system of axes and

|

Q

|

|Q|

is the measure of such a cube. Suppose

1

/

q
=
1

/

p
−
α

/

d

1/q = 1/p - \alpha /d

and

0
>
α
>
d
,
1
>
p
>
d

/

α

0 > \alpha > d,1 > p > d/\alpha

. Then there exists a constant such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue left-parenthesis upper T Subscript alpha Baseline f right-parenthesis upsilon StartAbsoluteValue EndAbsoluteValue Subscript q Baseline less-than-or-equal-to upper C StartAbsoluteValue EndAbsoluteValue f upsilon StartAbsoluteValue EndAbsoluteValue Subscript p">

|

|

(

T
α

f
)
υ

|

|

q

≤
C

|

|

f
υ

|

|

p

||({T_\alpha }f)\upsilon |{|_q} \leq C||f\upsilon |{|_p}

. Certain results for

p
=
1

p = 1

and

q
=
∞

q = \infty

are also given.
Nonnormal products of 𝜔_{𝜇}-metrizable spaces
Authors: Vaughan J. E.
Published: June 2010

We describe a metric space

X
X

and an

ω
1

{\omega _1}

-metrizable space

Y
Y

whose product

X
×
Y

X \times Y

is not normal. This provides another example to show that the product of a metric space with a normal space need not be normal. The first example of this nature was given by E. Michael, and we compare our example to his. We use our example to answer several questions concerning the normality of finite products of linearly stratifiable spaces, and countable products of hereditarily paracompact spaces. We also use some related examples to answer a question of K. Morita concerning

P
(

m

)

P(\mathfrak {m})

-spaces.
A note on 𝐶_{𝑐}(𝑋)
Authors: Richardson G. D. , Kent D. C.
Published: June 2010

When

X
X

is a locally convex topological linear space, the function algebra

C
c

(
X
)

{C_c}(X)

(with continuous convergence) can have a closure operator which has infinitely many distinct iterations. The reverse situation is also possible:

X
X

can be a locally compact

c
c

-embedded convergence space whose closure operator has infinitely many distinct iterations, whereas

C
c

(
X
)

{C_c}(X)

is a topological space.
A fixed point criterion for compact 𝑇₂-spaces
Authors: Talman Louis A.
Published: June 2010

We prove a fixed point theorem which has as consequences some theorems of W. G. Dotson, Jr., [2], [3] and of K. Baron and J. Matkowski [1].
Some universally pierced arcs in 𝐸³
Authors: Loveland L. D.
Published: June 2010

A subset

X
X

of

E
3

{E^3}

is said to be universally pierced if each

2
2

-sphere containing

X
X

can be pierced by a tame arc at each point of

X
X

. We show that an arc

A
A

is universally pierced provided

A
A

has a shrinking point

p
p

such that either

p
p

lies in a tame arc in

A
A

or

E
3

−
A

{E^3} - A

has

1
1

-ALG at

p
p

. Applying this result we show the existence of infinitely many wild universally pierced arcs.
Monotone real analytic maps which are not homeomorphisms
Authors: Hsu A. C. , Timourian J. G.
Published: June 2010

Monotone, proper, real analytic, onto maps which have nonempty branch sets are exhibited. The smash product maps are shown to belong to this class of maps. Maps with branch set a submanifold and with discrete branch set images are classified (locally).
The cyclotomic numbers of order seven
Authors: Leonard Philip A. , Williams Kenneth S.
Published: June 2010

The cyclotomic numbers of order seven are given in terms of the solutions of a certain system of three quadratic diophantine equations. This is analogous to L. E. Dickson’s evaluation of the cyclotomic numbers of order five, and is a convenient approach for applications to the theory of power residues.
Hall subgroups and 𝑝-solvability
Authors: Gonçalves A. , Ho C. Y.
Published: June 2010

Let

G
G

be a finite group and let

π
(
G
)
=
{
p
,

q
1

,
…
,

q
r

}

\pi (G) = \{ p,{q_1}, \ldots ,{q_r}\}

be the set of all prime divisors of

G
G

. Suppose there is a

p
′

p’

-Hall subgroup

H
H

. If there are subgroups

P
,

Q
1

,
…
,

Q
r

P,{Q_1}, \ldots ,{Q_r}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P epsilon upper S y l Subscript p Baseline left-parenthesis upper G right-parenthesis comma upper Q Subscript i Baseline epsilon upper S y l Subscript q Sub Subscript i Subscript Baseline left-parenthesis upper H right-parenthesis">

P
ϵ

Syl
p

(
G
)
,

Q
i

ϵ

Syl

q
i

(
H
)

P\epsilon {\operatorname {Syl} _p}(G),{Q_i}\epsilon {\operatorname {Syl} _{{q_i}}}(H)

, and

L
i

=
P

Q
i

{L_i} = P{Q_i}

is a subgroup,

i
=
1
,
…
,
r

i = 1, \ldots ,r

, then

G
G

is

p
p

-solvable. Moreover, if the subgroup

H
H

is solvable, then

G
G

is solvable too.
The Baire order of the functions continuous almost everywhere
Authors: Mauldin R. Daniel
Published: June 2010

Let

S
S

be a complete and separable metric space and

μ
\mu

a

σ
\sigma

-finite, complete Borel measure on

S
S

with

μ
(
S
)
>
0

\mu (S) > 0

. Let

Φ
\Phi

be the family of all real-valued functions defined on

S
S

whose set of points of discontinuity is of

μ
\mu

-measure 0. Let

B
α

(
Φ
)

{B_\alpha }(\Phi )

be the functions of Baire’s class

α
\alpha

generated by

Φ
\Phi

. It is shown that

B
1

(
Φ
)
=

B
2

(
Φ
)

{B_1}(\Phi ) = {B_2}(\Phi )

if and only if

μ
\mu

is a purely atomic measure whose set of atoms forms a scattered subset of

S
S

and that if

B
1

(
Φ
)
≠

B
2

(
Φ
)

{B_1}(\Phi ) \ne {B_2}(\Phi )

, then the Baire order of

Φ
\Phi

is

ω
1

{\omega _1}

; in other words, if

0
≤
α
>

ω
1

0 \leq \alpha > {\omega _1}

, then

B
α

(
Φ
)
≠

B

α
+
1

(
Φ
)

{B_\alpha }(\Phi ) \ne {B_{\alpha + 1}}(\Phi )

. This answers a generalized version of a problem raised by Sierpinski and Felsztyn. An example is given of a normal space with Borel order 2 and Baire order

ω
1

{\omega _1}

.
Isolated points of the spectra of conservative matrices
Authors: Sharma N. K.
Published: June 2010

We show that the only possible isolated points in the spectrum of a conservative triangular matrix are its diagonal elements, and that any Hausdorff method corresponding to an absolutely continuous mass function is in the norm closure of the analytic methods.
Some problems of nonlinear integral equations in measure spaces
Authors: Sharma R. R.
Published: June 2010

An existence theorem is proved by the method of successive approximations for abstract nonlinear integral equations formulated in [1]. Some integral inequalities and extremal solutions are also considered.
A noncompact Choquet theorem
Authors: Edgar G. A.
Published: June 2010

The following noncompact analog of Choquet’s theorem is proved. Let

E
E

be a Banach space with the Radon-Nikodým property, let

C
C

be a separable, closed, bounded, convex subset of

E
E

, and let a be a point in

C
C

. Then there is a probability measure

μ
\mu

on the universally measurable sets in

C
C

such that

a
a

is the barycenter of

μ
\mu

and the set of extreme points of

C
C

has

μ
\mu

-measure 1.
Double normals and tangent normals for polygons
Authors: Halpern Benjamin
Published: June 2010

Given a polygonal closed plane curve

γ
\gamma

. Each segment of

γ
\gamma

has a tangent direction and a normal direction; each vertex of

γ
\gamma

has a cone of tangent directions and a cone of normal directions. Formulas are established connecting the numbers of various kinds of straight lines which either intersect

γ
\gamma

twice in a normal direction, or once in a normal direction and once in a tangent direction.
A reduction of the fundamental conjecture about locally compact 𝐴𝑁𝑅’s
Authors: Fathi A. , Visetti Y. M.
Published: June 2010

In this note, we show that the conjecture: the product of a locally compact ANR by the Hilbert cube is a Hilbert cube manifold, can be reduced to its compact AR case, i.e.: the product of a compact AR by the Hilbert cube is a Hilbert cube manifold.
Equivalence of two strong forms of determinacy
Authors: Blass Andreas
Published: June 2010

Determinacy of all games of length

ω
\omega

with moves from

R

{\mathbf {R}}

is equivalent to determinacy of all games of length

ω
2

{\omega ^2}

with moves from

ω
\omega

.
On 𝑘-free integers with small prime factors
Authors: Hazlewood D. G.
Published: June 2010

The object of this note is to give a nontrivial lower estimate for the function

ψ
k

(
x
,
y
;
h
)

{\psi _k}(x,y;h)

, defined to be the number of

k
k

-free integers

m
m

such that

1
⩽
m
>
x
,

(
m
,
h
)
=
1

1 \leqslant m > x,\,(m,h) = 1

, and

m
m

has no prime factor greater than or equal to

y
y

.
Zero divisors in tensor products of division algebras
Authors: Risman Lawence J.
Published: June 2010

Theorem. If a tensor product of a division algebra

D
D

with a quaternion algebra

Q
Q

is not a division algebra, then either

D
D

and

Q
Q

possess a common quadratic subfield or

D
D

contains a splitting field of

Q
Q

not quadratic over the base field. The above theorem generalizes a recently published result of Albert’s. Theorem. If the tensor product of two division algebras over a local or a global field

K
K

is not a division algebra, then they contain a common extension field of

K
K

.
Algebras satisfying congruence relations
Authors: Coughlin Raymond , Rich Michael , Thedy Armin
Published: June 2010

It is shown that the classical nonassociative algebras which have an identity element can be defined in terms of congruence relations modulo the base field.
Schur indices and sums of squares
Authors: Fein Burton
Published: June 2010

Let

G
G

be a finite group of exponent

n
,
F

n, F

a field of characteristic zero,

ε
\varepsilon

a primitive

n
n

th root of unity, and suppose that the Sylow

2
2

-subgroup of the Galois group of

F
(
ε
)

F(\varepsilon )

over

F
F

is cyclic. Let

χ
\chi

be an absolutely irreducible character of

G
G

. Strengthening a recent result of Goldschmidt and Isaacs, it is shown that if — 1 is a sum of two squares in

F
F

, then the Schur index of

χ
\chi

over

F
F

is odd.
Une caractérisation des ensembles des points de discontinuité des fonctions linéairement-continues
Authors: Grande Zbigniew
Published: June 2010

A function

f
:

R
2

→
R

f:{R^2} \to R

(where

R
R

is the set of real numbers) is called linearly-continuous if for each

x
x

and

y
y

the functions

f
x

{f_x}

and

f
y

{f^y}

given by

f
x

(
t
)
=
f
(
x
,
t
)

{f_x}(t) = f(x,t)

and

f
y

(
t
)
=
f
(
t
,
y
)

{f^y}(t) = f(t,y)

for

−
∞
>
t
>
∞

- \infty > t > \infty

are continuous. It is proven that: A set

A
⊂

R
2

A \subset {R^2}

is the set of points of discontinuity for a linearly-continuous function iff

A
A

is

F
σ

{F_\sigma }

contained in a cartesian product of two linear sets of first category. It is proven also that an analogous characterisation is not possible for an approximatively linearly-continuous function.
The radius of 𝛼-convexity for the class of starlike univalent functions, 𝛼-real
Authors: Mocanu Petru T. , Reade Maxwell O.
Published: June 2010

We use a result due to Gutljanskiĭ to obtain the radius of

α
\alpha

-convexity for the class

S
∗

{S^ \ast }

of starlike univalent functions for real

α
\alpha

.
On the lower order of an entire Dirichlet series
Authors: Singh J. P.
Published: June 2010

The lower order

λ
¯

s

{\overline \lambda _s}

of

f
(
s
)

f(s)

in each horizontal strip

S
(
π
a
)

S(\pi a)

, with

a
>

Δ

∗

a > {\Delta ^{\ast }}

, is equal to the lower order

λ
\lambda

of

f
(
s
)

f(s)

. The purpose of this note is to offer a proof of this result.
A category theorem for Tsuji functions
Authors: Colwell Peter
Published: June 2010

If

H
H

denotes the functions analytic in the open unit disk with the topology of uniform convergence on compact subsets, both the Tsuji functions in

H
H

and the functions in

H
H

with nonempty Tsuji sets comprise sets of first category in

H
H

. A question is posed about the category of a class of functions containing the Tsuji functions.
Characterizations of bounded mean oscillation
Authors: Berman Stephen Jay
Published: June 2010

Recall that an integrable function

f
f

on a cube

Q
0

{Q_0}

in

R

n

{{\mathbf {R}}^n}

is said to be of bounded mean oscillation if there is a constant

K
K

such that for every parallel subcube

Q
Q

of

Q
0

{Q_0}

there exists a constant

a
Q

{a_Q}

such that

∫
Q

|

f
−

a
Q

|

≤
K

|

Q

|

\int _Q {|f - {a_Q}| \leq K|Q|}

, where

|

Q

|

|Q|

denotes the volume of

Q
Q

. We prove here that if there is an integer

d
d

and a constant

K
K

such that for every parallel subcube

Q
Q

of

Q
0

{Q_0}

there exists a polynomial

p
Q

{p_Q}

of degree

≤
d

\leq d

such that

∫
Q

|

f
−

p
Q

|

≤
K

|

Q

|

\int _Q {|f - {p_Q}| \leq K|Q|}

, then

f
f

is of bounded mean oscillation.
Convex combinations of uniformly mean stable Markov operators
Authors: Sine Robert
Published: June 2010

A convex combination of commuting uniformly mean stable Markov operators acting on

C
(
X
)

C(X)

is shown to be uniformly mean stable. The proof is completely geometric.
Intersecting unions of convex sets in 𝑅ⁿ
Authors: Breen Marilyn
Published: June 2010

Let

C

=
{

C
α

:
α

\mathcal {C} = \{ {C_\alpha }:\alpha

in some index set

I
}

I\}

be a collection of convex sets, and let

M

=
{

C
α

∪

C
β

:
α
≠
β
,

C
α

,

C
β

\mathfrak {M} = \{ {C_\alpha } \cup {C_\beta }:\alpha \ne \beta ,{C_\alpha },{C_\beta }

in

C

}

\mathcal {C}\}

. In this paper, various decomposition theorems are obtained for the set

∩

M

\cap \mathfrak {M}

.
On Biesterfeldt’s completion axiom spaces
Authors: Gazik R. J.
Published: June 2010

It is proved that a Hausdorff, totally bounded Completion Axiom space is a uniform space. The method of proof shows that Hausdorff Completion Axiom spaces have completions (in the embedding sense) which are again Hausdorff Completion Axiom spaces; moreover these completions are uniformly regular, uniformly strict, and have the regular extension property.
𝑇-regular-closed convergence spaces
Authors: Kent D. C. , Richardson G. D. , Gazik R. J.
Published: June 2010

It is known that a convergence space which has a regular compactification is almost identical to a completely regular topological space. It is shown that a less restricitive class of convetgence spaces have

T
T

-regular-closed extensions with the universal property of the Stone-Čech compactification.
A note on the topolgy of 𝐶-convergence in hyperspaces
Authors: Morales Pedro
Published: June 2010

In this note we generalize and partially correct a recent Tychonoff theorem for hyperspaces of F. A. Chimenti [1].
Nonstratifiable regular quotients of separable stratifiable spaces
Authors: van Douwen Eric K.
Published: June 2010

A regular quotient of a countable

M
1

{M_1}

-space need not be stratifiable, and a regular first countable quotient of a first countable separable

M
1

{M_1}

-space need not be stratifiable either. This answers a question of Borges. The second example depends on a new, simple, example of a first countable separable nonmetrizable

M
1

{M_1}

-space.
A handlebody with one pillbox has no fake 3-cells
Authors: Berge J. O. , Brin Matthew G.
Published: June 2010

The main result establishes upper bounds on the rank of

H
1

{H_1}

of orientable

3
3

-manifolds with fake

3
3

-cells and on the rank of

π
1

{\pi _1}

of closed orientable

3
3

-manifolds with fake

3
3

-cells. As a consequence we get that a

3
3

-manifold obtained by sewing one pillbox on a handlebody of arbitrary genus cannot contain a fake

3
3

-cell.
Higher derivations on finitely generated integral domains. II
Authors: Brown William C.
Published: June 2010

We prove Theorem. Let

A
=
k
[

x
1

,
…
,

x
m

]

A = k[{x_1}, \ldots ,{x_m}]

be a finitely generated integral domain over a field

k
k

of characteristic zero. Then

A
A

regular, i.e. the local ring

A
q

{A_q}

is regular for every prime ideal

q
⊆
A

q \subseteq A

, is equivalent to the following two conditions: (1) no prime of

A
A

of height greater than one is differential, and (2) for all

ϕ
∈

Hom
k

(
A
,
A
)
,
ϕ
∈

Der
k
n

⁡
(
A
)

\phi \in {\operatorname {Hom} _k}(A,A),\phi \in \operatorname {Der} _k^n(A)

if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta phi element-of normal upper Sigma Subscript i equals 1 Superscript n minus 1 Baseline upper D e r Subscript k Superscript i Baseline left-parenthesis upper A right-parenthesis union upper D e r Subscript k Superscript n minus i Baseline left-parenthesis upper A right-parenthesis left-parenthesis n equals 1 comma 2 comma ellipsis right-parenthesis">

Δ
ϕ
∈

Σ

i
=
1

n
−
1

Der
k
i

⁡
(
A
)
∪

Der
k

n
−
i

⁡
(
A
)
(
n
=
1
,
2
,
…
)

\Delta \phi \in \Sigma _{i = 1}^{n - 1}\operatorname {Der} _k^i(A) \cup \operatorname {Der} _k^{n - i}(A)(n = 1,2, \ldots )

. Here

Δ
\Delta

denotes the Hochschild coboundary operator,

∪
\cup

denotes the cup product, and

Der
k
n

⁡
(
R
)

\operatorname {Der} _k^n(R)

is the module of higher derivations of rank

n
n

.
Applications of graph theory to matrix theory
Authors: Owens Frank W.
Published: June 2010

Let

A
1

,
…
,

A
k

{A_1}, \ldots ,{A_k}

be

n
×
n

n \times n

matrices over a commutative ring

R
R

with identity. Graph theoretic methods are established to compute the standard polynomial

[

A
1

,
…
,

A
k

]

[{A_1}, \ldots ,{A_k}]

. It is proved that if

k
>
2
n
−
2

k > 2n - 2

, and if the characteristic of

R
R

either is zero or does not divide

4
I
(
1

/

2
n
)
−
2

4I(1/2n) - 2

, where

I
I

denotes the greatest integer function, then there exist

n
×
n

n \times n

skew-symmetric matrices

A
1

,
…
,

A
k

{A_1}, \ldots ,{A_k}

such that

[

A
1

,
…
,

A
k

]
≠
0

[{A_1}, \ldots ,{A_k}] \ne 0

.
The freeness of a group based on a distributive lattice
Authors: Hill P. , Subramanian H.
Published: June 2010

Let

L
L

be a distributive lattice and

G
G

the abelian group with the following presentation. The generators of

G
G

are the elements of the lattice

L
L

, and the relations are

(
a
∨
b
)
+
(
a
∧
b
)
=
a
+
b

(a \vee b) + (a \wedge b) = a + b

where

a
a

and

b
b

are arbitrary elements of

L
L

. It is shown that

G
G

is free abelian. In particular,

G
G

is torsion free. The latter statement answers affirmatively a question posed several years ago by E. Weinberg.
A topological translation of a fundamental problem in cluster set theory
Authors: Eidswick J. A.
Published: June 2010

Two conjectures related to the problem of finding a “large” family of approach curves along which cluster sets can be preassigned are shown to be equivalent to special cases of the normal Moore space conjecture.
A proof of the homeomorphism of Lebesgue-Stieltjes measure with Lebesgue measure
Authors: Goffman Casper , Pedrick George
Published: June 2010

An elementary proof is given of the fact that nonatomic measures on

n
n

space, for which open sets have positive measure, are “homeomorphic". The proof is based on the fact that for such measures all hyperplanes in most directions have measure zero.
A sufficient condition for eventual disconjugacy
Authors: Trench William F.
Published: June 2010

It is known that the scalar equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript left-parenthesis n right-parenthesis Baseline plus p 1 left-parenthesis t right-parenthesis y Superscript left-parenthesis n minus 1 right-parenthesis Baseline plus midline-horizontal-ellipsis plus p Subscript n Baseline left-parenthesis t right-parenthesis y equals 0 comma t greater-than 0 comma n greater-than 1">

y

(
n
)

+

p
1

(
t
)

y

(
n
−
1
)

+
⋯
+

p
n

(
t
)
y
=
0
,
t
>
0
,
n
>
1

{y^{(n)}} + {p_1}(t){y^{(n - 1)}} + \cdots + {p_n}(t)y = 0,t > 0,n > 1

, is eventually disconjugate if

p
1

,
…
,

p
n

ϵ
C
[
0
,
∞
)

{p_1}, \ldots ,{p_n}\epsilon C[0,\infty )

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Superscript normal infinity Baseline StartAbsoluteValue p Subscript i Baseline left-parenthesis t right-parenthesis EndAbsoluteValue t Superscript i minus 1 Baseline d t greater-than normal infinity comma 1 less-than-or-slanted-equals i less-than-or-slanted-equals n">

∫

∞

|

p
i

(
t
)

|

t

i
−
1

d
t
>
∞
,
1
⩽
i
⩽
n

\int {^\infty |{p_i}(t)|{t^{i - 1}}dt > \infty ,1 \leqslant i \leqslant n}

. This paper presents a weaker integral condition which also implies that the given equation is eventually disconjugate.
Uniqueness for nonlinear Cauchy problems in Banach spaces
Authors: Goldstein Jerome A.
Published: June 2010

Recently Medeiros and Diaz and Weinacht have considered the question of uniqueness for the Cauchy problem for ordinary differential equations in a complex Hilbert space. The present paper extends their results to the case of equations in an arbitrary real or complex Banach space.
Remarks on dilations in 𝐿_{𝑝}-spaces
Authors: Akcoglu M. A. , Sucheston L.
Published: June 2010

Let

(
X
,

F

,

μ
)

(X,\;\mathcal {F},\;\mu )

be a nonatomic measure space. It is shown that there exists a unitary operator

U
U

on

L
2

=

L
2

(
X
,

F

,

μ
)

{L_2} = {L_2}(X,\;\mathcal {F},\;\mu )

, a function

f
ϵ

L
2

f\epsilon {L_2}

, and a nonnull set

A
A

in

F

\mathcal {F}

such that

n

−
1

Σ

i
=
1

n

U
i

f

{n^{ - 1}}\Sigma _{i = 1}^n{U^i}f

diverges on

A
A

.
Subspace of 𝐿𝐶(𝐻) and 𝐶_{𝑝}
Authors: Friedman Yaakov
Published: June 2010

This paper deals with certain properties of the subspaces of

L
C
(
H
)

LC(H)

and

C
p

{C_p}

and namely those connected with the reflexivity and with the property of containing classical spaces. It is proved that any subspace of

C
p

(
1
⩽
p
>
∞
)

{C_p}(1 \leqslant p > \infty )

is either isomorphic to Hilbert space or it contains a subspace isomorphic to

l
p

{l_p}

. For

C
1

{C_1}

and

L
C
(
H
)

LC(H)

the same results were obtained by J. R. Holub, cf. [4].
On an operator equation involving mappings of monotone type
Authors: Gupta Chaitan P.
Published: June 2010

Let

X
X

be a real reflexive Banach space and

A
:
X
→

2

X

∗

A:X \to {2^{{X^{\ast }}}}

a maximal monotone mapping such that the graph

G
(
A
)

G(A)

of

A
A

is weakly-closed in

X
×

X

∗

X \times {X^{\ast }}

and

0
ϵ
A
(
0
)

0\epsilon A(0)

. Also, let

T
:
X
→

2

X

∗

T:X \to {2^{{X^{\ast }}}}

be a quasi-bounded coercive mapping of type

(

M

)

({\text {M}})

such that the effective domain

D
(
T
)

D(T)

of

T
T

contains a dense linear subspace

X
0

{X_0}

of

X
X

. Then it is shown that for each

ω
ϵ

X

∗

\omega \epsilon {X^{\ast }}

there exists a

u
ϵ
X

u\epsilon X

such that

ω
ϵ
A
u
+
T
u

\omega \epsilon Au + Tu

and the subset

{
u
ϵ
X

|

ω
ϵ
A
u
+
T
u
}

\{ u\epsilon X|\omega \epsilon Au + Tu\}

is a weakly-compact subset of

X
X

. An application to an elliptic nonlinear boundary value problem of Neumann type is given.
Flow-invariant domains of Hölder continuity for nonlinear semigroups
Authors: Plant Andrew T.
Published: June 2010

Let

S
(
t
)

S(t)

be a nonlinear semigroup, on Banach space

X
X

, generated by an accretive set

A
A

. The set of

x
x

in

X
X

such that

t
→
S
(
t
)
x

t \to S(t)x

is Hölder continuous, with Hölder exponent

σ

ϵ

(
0
,

1
]

\sigma \,\epsilon \,(0,\;1]

, is flow-invariant and is characterised by the behaviour of the map

λ
→

(
I
+
λ
A

)

−
1

x

\lambda \to {(I + \lambda A)^{ - 1}}x

at

λ
=
0

\lambda = 0

.
A Sturm-Liouville theorem for some odd multivalued maps
Authors: Dias Jo ao-Paulo , Hernández Jesús
Published: June 2010

Let

T
:
H
→

2
H

T:H \to {2^H}

be the subdifferential of a real l. s. c. convex function on an infinite dimensional, separable, real Hilbert space

H
H

. Assuming that

T
T

is odd (i.e.

T
(
−
u
)
=
−
T
u
,

∀
u

ϵ
H
)

T( - u) = - Tu,\;\forall u\;\epsilon H)

),

0
ϵ
T
(
0
)
,

(
I
+
T

)

−
1

0\epsilon T(0),\;{(I + T)^{ - 1}}

is compact and

T
(
0
)

T(0)

satisfies a geometrical condition, we prove that

T
T

has an infinite sequence

{

λ
n

}

\{ {\lambda _n}\}

of eigenvalues such that

0
⩽

λ

n

n
→

+
∞

0 \leqslant {\lambda _{n\,\overrightarrow n }} + \infty

.
All 𝑛-uniform quasitranslation Hjelmslev planes are strongly 𝑛-uniform
Authors: Drake David A.
Published: June 2010

An affine

H
H

-plane

A

\mathcal {A}

is called a quasitranslation

H
H

-plane if it possesses a group of quasitranslations which is regular on the points of

A

\mathcal {A}

. Let

n
n

be any positive integer. Then every

n
n

-uniform quasitranslation

H
H

-plane is strongly

n
n

-uniform.
Closed mappings of 𝜎-locally compact metric spaces
Authors: Stricklen S. A.
Published: June 2010

We show that a metric space

M
M

is

σ
\sigma

-locally compact if and only if every image of

M
M

under a closed, continuous function is the countable union of closed, metrizable, locally compact subspaces. Several other theorems about closed, continuous images of metric spaces are given; one of these is that the closed, continuous image of a complete,

σ
\sigma

-locally compact metric space must contain a dense, metrizable open set.
Metric rigidity in 𝐸ⁿ
Authors: Scott Brian M. , Jones Ralph
Published: June 2010

In answer to a question rasied by L. Janos, it is shown (i) that

E
n

{E^n}

is not a finite union of metrically rigid subsets; (ii) that if

2
ω

>

κ
+

{2^\omega } > {\kappa ^ + }

, then

E
n

{E^n}

is not the union of

κ
\kappa

metrically rigid subsets, where

κ
\kappa

is an infinite cardinal; and (iii) that if

2
ω

=

κ
+

{2^\omega } = {\kappa ^ + }

, then

E
1

{E^1}

is the union of

κ
\kappa

metrically rigid subsets, and hence that

E
1

{E^1}

is a countable union of metrically rigid subsets iff the continuum hypothesis holds. (

A
⊆

E
n

A \subseteq {E^n}

is metrically rigid iff no two distinct two-point subsets of

A
A

are isometric.) Open question: assuming the continuum hypothesis, can

E
n

{E^n}

be written as a countable union of metrically rigid subsets if

n
>
1

n > 1

?
A fixed point theorem for hyperspaces of 𝜆 connected continua
Authors: Hagopian Charles L.
Published: June 2010

Suppose that the hyperspace of compact connected subsets

C

(
X
)

\mathcal {C}(X)

of a

λ
\lambda

connected continuum

X
X

can be

ϵ
\epsilon

-mapped (for each

ϵ
>
0

\epsilon > 0

) into the plane. We prove that

X
X

is either arc-like or circle-like. It follows from this theorem and results of J. T. Rogers, Jr. and J. Segal that

C

(
X
)

\mathcal {C}(X)

has the fixed point property.
On asymptotic behaviors of analytic mappings on the Martin boundary
Authors: Niimura Mikio
Published: June 2010

Some generalizations of the analogue of Collingwood and Cartwright in the large of Iversen’s theorem are given.
On the distance between zeroes
Authors: Patula William T.
Published: June 2010

For the equation

x
+
q
(
t
)
x
=
0

x + q(t)x = 0

, let

x
(
t
)

x(t)

be a solution with consecutive zeroes at

t
=
a

t = a

and

t
=
b

t = b

. A simple inequality is proven that relates not only

a
a

and

b
b

to the integral of

q
+

(
t
)

{q^ + }(t)

but also any point

c
ϵ
(
a
,
b
)

c\epsilon (a,b)

where

|

x
(
t
)

|

|x(t)|

is maximized. As a corollary, it is shown that if the above equation is oscillatory and if

q
+

(
t
)
ϵ

L
p

[
0
,
∞
)
,
1
⩽
p
>
∞

{q^ + }(t)\epsilon {L^p}[0,\infty ),1 \leqslant p > \infty

, then the distance between consecutive zeroes must become unbounded.
Oscillation properties of perturbed disconjugate equations
Authors: Trench William F.
Published: June 2010

Oscillation conditions are given for the equation

L
u

+
f
(
t
,
u
)
=
0

{L_u} + f(t,u) = 0

, where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L u equals StartFraction 1 Over beta Subscript n Baseline EndFraction StartFraction d Over d t EndFraction StartFraction 1 Over beta Subscript n minus 1 Baseline EndFraction midline-horizontal-ellipsis StartFraction d Over d t EndFraction StartFraction 1 Over beta 1 EndFraction StartFraction d Over d t EndFraction StartFraction u Over beta 0 EndFraction left-parenthesis n greater-than-or-slanted-equals 2 right-parenthesis comma">

L
u
=

1

β
n

d

d
t

1

β

n
−
1

⋯

d

d
t

1

β
1

d

d
t

u

β
0

(
n
⩾
2
)
,

Lu = \frac {1} {{{\beta _n}}}\frac {d} {{dt}}\frac {1} {{{\beta _{n - 1}}}} \cdots \frac {d} {{dt}}\frac {1} {{{\beta _1}}}\frac {d} {{dt}}\frac {u} {{{\beta _0}}}(n \geqslant 2),

\]
with

β
0

,
…
,

β
n

{\beta _0}, \ldots ,{\beta _n}

positive and continuous on
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma normal infinity right-parenthesis comma integral Superscript normal infinity Baseline beta Subscript i Baseline d t equals normal infinity left-parenthesis 1 less-than-or-slanted-equals i less-than-or-slanted-equals n minus 1 right-parenthesis">

(
0
,
∞
)
,
∫

∞

β
i

d
t
=
∞
(
1
⩽
i
⩽
n
−
1
)

(0,\infty ),\int {^\infty {\beta _i}dt = \infty (1 \leqslant i \leqslant n - 1)}

, and

f
f

subject to conditions which include

u
f
(
t
,
u
)
⩾
0

uf(t,u) \geqslant 0

. The results obtained include previously known oscillation conditions for the equation

u

(
n
)

+
f
(
t
,
u
)
=
0

{u^{(n)}} + f(t,u) = 0

for both linear and nonlinear cases.
On a theorem of Menchoff
Authors: Katznelson Y.
Published: June 2010

Construction of a continuous real-valued function which disagrees almost everywhere with any absolutely convergent Fourier series.
The metrizable linear extensions of metrizable sets in topological linear spaces
Authors: Drewnowski L.
Published: June 2010

Suppose a subset

X
X

of a Hausdorff [locally convex] topological linear space

(
E
,
τ
)

(E,\tau )

is metrizable in its relative topology

τ

|

X

\tau |X

. It is shown that if

τ

|

X

\tau |X

is separable, then there exists a metrizable [locally convex] linear topology

τ
0

{\tau _0}

on the subspace

V
V

generated by

X
X

such that

τ
0

⊂
τ

|

V

{\tau _0} \subset \tau |V

and

τ
0

|

X
=
τ

|

X

{\tau _0}|X = \tau |X

(Theorem 2). This is related to a recent result of Larman and Rogers which states that if, in addition,

X
X

is locally bounded, then

τ
0

{\tau _0}

can be chosen to be normable (but then not necessarily

τ
0

⊂
τ

|

V

{\tau _0} \subset \tau |V

) (Theorem 1). It is then observed that

τ
0

|

X
=
τ

|

X

{\tau _0}|X = \tau |X

does not mean the coincidence of the corresponding induced uniformities on

X
X

. However, this is the case if the invariant uniformity compatible with

τ
\tau

is metrizable on

X
X

and

X
X

is convex (Theorem 4).
On 𝐽-symmetric restricted shifts
Authors: Fuhrmann Paul A.
Published: June 2010

The restricted shift operators in proper left invariant subspaces of

H
2

{H^2}

that are

J
J

-symmetric are characterized and the signature of the corresponding operator

J
J

is determined.
The maps 𝐵𝑆𝑝(1)→𝐵𝑆𝑝(𝑛)
Authors: Mahmud Zafer
Published: June 2010

Let

S
p
(
n
)

Sp(n)

be the symplectic Lie group. Then it is known that given a map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper B upper S p left-parenthesis 1 right-parenthesis right-arrow upper B upper S p left-parenthesis 1 right-parenthesis comma f Superscript asterisk Baseline colon upper H Superscript 4 Baseline left-parenthesis upper B upper S p left-parenthesis 1 right-parenthesis comma bold upper Z right-parenthesis right-arrow upper H Superscript 4 Baseline left-parenthesis upper B upper S p left-parenthesis 1 right-parenthesis comma bold upper Z right-parenthesis">

f
:
B
S
p
(
1
)
→
B
S
p
(
1
)
,

f

∗

:

H
4

(
B
S
p
(
1
)
,

Z

)
→

H
4

(
B
S
p
(
1
)
,

Z

)

f:BSp(1) \to BSp(1),{f^{\ast }}:{H^4}(BSp(1),{\mathbf {Z}}) \to {H^4}(BSp(1),{\mathbf {Z}})

is zero or multiplication by the square of an odd integer. We generalise the latter part of this result using symplectic

K

∗

{K^{\ast }}

-theory.