bluuebunny/crossref_metadata_2025 huggingface dataset:
Exactness of the double dual
Authors: Colby R. R. , Fuller K. R.
Published: June 2010

Let

R
R

be an artinian ring. The double

R
R

-dual ( )** preserves monomorphisms, is left exact, or is right exact if and only if

R
R

is QF-

3
3

, has dominant dimension at least 2, or is QF, respectively. Furthermore, both double dual functors preserve epimorphisms of finitely generated (left and right) modules if and only if both dual functors ( )* are exact on short exact sequences of finitely generated torsionless modules, and these conditions are equivalent to finitely generated torsionless modules being reflexive.
Extensions of pure positive functionals on Banach *-algebras
Authors: Doran R. S. , Tiller Wayne
Published: June 2010

A known extension theorem for pure states on a Banach *-algebra with isometric involution is shown to hold for the wider class of Banach *-algebras with arbitrary, possibly discontinuous, involutions.
Some conditions for 𝑛-convex functions
Authors: Cross G. E.
Published: June 2010

It is shown that if

F
F

is a function defined and continuous on

[
a
,
b
]

[a,b]

such that (where

n
n

is an even integer): (a)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript left-parenthesis r right-parenthesis Baseline left-parenthesis x right-parenthesis equals limit Underscript h right-arrow 0 Endscripts left-parenthesis StartFraction r factorial Over h Superscript r Baseline EndFraction right-parenthesis left-parenthesis upper F left-parenthesis x plus h right-parenthesis minus upper F left-parenthesis x right-parenthesis minus sigma-summation Underscript k equals 1 Overscript r minus 1 Endscripts StartFraction h Superscript k Baseline upper F Subscript left-parenthesis k right-parenthesis Baseline left-parenthesis x right-parenthesis Over k factorial EndFraction right-parenthesis">

F

(
r
)

(
x
)
=

lim

h
→
0

(

r
!

h
r

)

(

F
(
x
+
h
)
−
F
(
x
)
−

∑

k
=
1

r
−
1

h
k

F

(
k
)

(
x
)

k
!

)

{F_{(r)}}(x) = \lim \limits _{h \to 0} \left ( {\frac {{r!}} {{{h^r}}}} \right )\left ( {F(x + h) - F(x) - \sum \limits _{k = 1}^{r - 1} {\frac {{{h^k}{F_{(k)}}(x)}} {{k!}}} } \right )

\]
exists and is finite in

(
a
,
b
)

(a,b)

for

1
⩽
r
⩽
n
−
2

1 \leqslant r \leqslant n - 2

; (b) for

x
∈
(
a
,
b
)
∖
E

x \in (a,b)\backslash E

, where

E
E

is countable,

F
F

is

n
n

-smooth, i.e.,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript h right-arrow 0 Endscripts left-parenthesis StartFraction n factorial Over h Superscript n minus 1 Baseline EndFraction right-parenthesis left-bracket StartFraction upper F left-parenthesis x plus h right-parenthesis plus upper F left-parenthesis x minus h right-parenthesis Over 2 EndFraction minus sigma-summation Underscript k equals 0 Overscript n slash 2 minus 1 Endscripts StartFraction h Superscript 2 k Baseline upper D Superscript 2 k Baseline upper F left-parenthesis x right-parenthesis Over left-parenthesis 2 k right-parenthesis factorial EndFraction right-bracket equals 0">

lim

h
→
0

(

n
!

h

n
−
1

)

[

F
(
x
+
h
)
+
F
(
x
−
h
)

2

−

∑

k
=
0

n

/

2
−
1

h

2
k

D

2
k

F
(
x
)

(
2
k
)
!

]

=
0

\lim \limits _{h \to 0} \left ( {\frac {{n!}} {{{h^{n - 1}}}}} \right )\left [ {\frac {{F(x + h) + F(x - h)}} {2} - \sum \limits _{k = 0}^{n/2 - 1} {\frac {{{h^{2k}}{D^{2k}}F(x)}} {{(2k)!}}} } \right ] = 0

\]
where

D

2
k

F
(
x
)

{D^{2k}}F(x)

denotes the (symmetric) de la Vallée-Poussin derivative; (c)

D
¯

n

F
(
x
)
⩾
0

{\bar D^n}F(x) \geqslant 0

a.e. in

(
a
,
b
)

(a,b)

; (d)

D
¯

n

F
(
x
)
>
−
∞

{\bar D^n}F(x) > - \infty

for

x
∈
(
a
,
b
)
∖
S

x \in (a,b)\backslash S

where

S
S

is countable and

F
(
x
)

F(x)

is

n
n

-smooth in

S
S

; then

F
(
x
)

F(x)

is

n
n

-convex in

[
a
,
b
]

[a,b]

. The same result holds for

n
n

odd. This is an improvement on the known result when

S
S

is a scattered set.
On a conjecture for oscillation of second-order ordinary differential systems
Authors: Mingarelli Angelo B.
Published: June 2010

We present here some results pertaining to the oscillatory behavior at infinity of the vector differential equation
\[

y
+
Q
(
t
)
y
=
0
,

t
∈
[
0
,
∞
)

y + Q(t)y = 0,\quad t \in [0,\infty )

\]
, where

Q
(
t
)

Q(t)

is a real continuous

n
×
n

n \times n

symmetric matrix function. It has been conjectured (cf., e.g. [6]) that the criterion
\[

lim

t
→
∞

λ
1

{

∫
0
t

Q
(
s
)

d
s

}

=
∞

\lim \limits _{t \to \infty } {\lambda _1}\left \{ {\int _0^t {Q(s)\;ds} } \right \} = \infty

\]
where

λ
1

(
⋅
)

{\lambda _1}( \cdot )

denotes the maximum eigenvalue of the matrix concerned, implies oscillation. We show that this is so under the tacit assumption
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit inf Underscript t right-arrow normal infinity Endscripts t Superscript negative 1 Baseline trace left-brace integral Subscript 0 Superscript t Baseline upper Q left-parenthesis s right-parenthesis d s right-brace greater-than negative normal infinity">

lim

inf

t
→
∞

t

−
1

tr

{

∫
0
t

Q
(
s
)

d
s

}

>
−
∞

\lim \inf \limits _{t \to \infty } {t^{ - 1}}{\text {tr}}\left \{ {\int _0^t {Q(s)\;ds} } \right \} > - \infty

\]
where

tr

(
⋅
)

{\text {tr}}( \cdot )

represents the trace of the matrix under consideration.
Elementary proof of the Rudin-Carleson and the F. and M. Riesz theorems
Authors: Doss Raouf
Published: June 2010

A very elementary proof is given of the theorem that on a set of measure zero on

T
T

, any continuous function is equal to a continuous function of analytic type. The same elementary method proves that a measure of analytic type is absolutely continuous.
Continuity of the spectrum and spectral radius
Authors: Murphy Gerard J.
Published: June 2010

Let

A
A

be a Banach algebra containing an element

x
x

. Topological conditions on the spectrum of

x
x

are given which are necessary and sufficient to ensure the continuity of the spectrum or spectral radius at

x
x

.
Completely regular and 𝜔-regular spaces
Authors: Kent D. C. , Richardson G. D.
Published: June 2010

The completely regular convergence spaces are characterized as those spaces having symmetric compactifications. The

ω
\omega

-regular convergence spaces are those which have regular compactifications.
A characterization of 𝐶-compact spaces
Authors: Clay Jesse P. , Joseph James E.
Published: June 2010

a characterization of

C
C

-compact spaces is given in terms of relations.
On the supercenter of a group over domains of characteristic 0
Authors: Merklen Héctor A.
Published: June 2010

Let

G
G

be a group and

R
R

a commutative ring with 1 and let

U
R

G

{U_R}G

be the group of units of the group ring

R
G

RG

. The

R
R

-supercenter of

G
G

,

S
R

G

{S_R}G

is the set of elements of

G
G

which have a finite number of conjugates under

U
R

G

{U_R}G

. The super-center is studied in the case where

R
R

is a domain of characteristic 0. It is shown that for the most important cases the supercenter coincides with the center of the group and with the intersection of all group bases of the group ring.
The Fuglede commutativity theorem modulo operator ideals
Authors: Weiss Gary
Published: June 2010

Let

H
H

denote a separable, infinite-dimensional complex Hilbert space. A two-sided ideal

I
I

of operators on

H
H

possesses the generalized Fuglede property (GFP) if, for every normal operator

N
N

and every

X
∈
L
(
H
)

X \in L(H)

,

N
X
−
X
N
∈
I

NX - XN \in I

implies

N
∗

X
−
X

N
∗

∈
I

{N^ * }X - X{N^ * } \in I

. Fuglede’s Theorem says that

I
=

{
0
}

I = \left \{ 0 \right \}

has the GFP. It is known that the class of compact operators and the class of Hilbert-Schmidt operators have the GFP. We prove that the class of finite rank operators and the Schatten

p
p

-classes for

0
>
p
>
1

0 > p > 1

fail to have the GFP. The operator we use as an example in the case of the Schatten

p
p

-classes is multiplication by

z
+
w

z + w

on

L
2

{L^2}

of the torus.
The equivariant extension theorem
Authors: Lashof R.
Published: June 2010

A simplified proof of Jaworowski’s equivariant extension theorem is given which enables one to generalize the domain to a class of

G
G

-spaces which include all (not necessarily compact)

G
G

-manifolds.
Stratifiable spaces as subspaces and continuous images of 𝑀₁-spaces
Authors: Heath Robert W. , Junnila Heikki J. K.
Published: June 2010

It is shown that every stratifiable space is the image of an

M
1

{M_1}

-space under a perfect retraction.
Interpolation spaces and unitary representations
Authors: Goodman Roe
Published: June 2010

Let

G
G

be a Lie group,

π
\pi

a unitary representation of

G
G

on a Hilbert space

H

(
π
)

\mathcal {H}(\pi )

, and

H

k

(
π
)

{\mathcal {H}^k}(\pi )

the subspace of

C
k

{C^k}

vectors for

π
\pi

. By quadratic interpolation there is a continuous scale

H

s

(
π
)

{\mathcal {H}^s}(\pi )

,

s
>
0

s > 0

, of

G
G

-invariant Hilbert spaces. When

G
=
H
⋅
K

G = H \cdot K

is a semidirect product of closed subgroups, then it is proved that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H Superscript s Baseline left-parenthesis pi right-parenthesis equals script upper H Superscript s Baseline left-parenthesis pi vertical-bar Subscript upper H Baseline right-parenthesis intersection script upper H Superscript s Baseline left-parenthesis pi vertical-bar Subscript upper K Baseline right-parenthesis">

H

s

(
π
)
=

H

s

(

π
|

H

)
∩

H

s

(

π
|

K

)

{\mathcal {H}^s}(\pi ) = {\mathcal {H}^s}({\left . \pi \right |_H}) \cap {\mathcal {H}^s}({\left . \pi \right |_K})

for

s
>
0

s > 0

. For solvable

G
G

this gives a characterisation of

H

s

(
π
)

{\mathcal {H}^s}(\pi )

in terms of smoothness along one-parameter subgroups, and an elliptic regularity result.
Applications of the 𝑢-closure operator
Authors: Espelie M. Solveig , Joseph James E. , Kwack Myung H.
Published: June 2010

Let

c

l

u

(
A
)

{\text {c}}{{\text {l}}_u}(A)

be the

u
u

-closure of a subset

A
A

of a space. We prove that a space is compact if and only if for each upper-semicontinuous multifunction

λ
\lambda

on the space, the multifunction

μ
\mu

defined on the space by

μ
(
x
)
=

c

l

u

(
λ
(
x
)
)

\mu (x) = {\text {c}}{{\text {l}}_u}(\lambda (x))

assumes a maximal value under set inclusion. We also prove that in a Urysohn-closed space any two subsets with disjoint

u
u

-closures are separated by disjoint open subsets. The quotient space induced by identifying those points with identical

u
u

-closures is investigated and shown to be

T
0

{T_0}

.
Flat bundles with solvable holonomy. II. Obstruction theory
Authors: Goldman William M.
Published: June 2010

Necessary and sufficient conditions for a connected solvable Lie group

G
G

are given so that every flat principal

G
G

-bundle over a

CW

{\text {CW}}

-complex is trivial after passing to a finite covering space.
On the equivariant homotopy type of 𝐺-ANRs
Authors: Kwasik Sławomir
Published: June 2010

I show that every metric

G
G

-

ANR

{\text {ANR}}

has the

G
G

-homotopy type of a

G
G

-

CW

{\text {CW}}

complex. Therefore I. James and G. Segal’s results concerning equivariant homotopy type are special cases of the Whitehead theorem for

G
G

-

CW

{\text {CW}}

complexes.
On atriodic tree-like continua
Authors: Oversteegen Lex G. , Tymchatyn E. D.
Published: June 2010

D. P. Bellamy has recently shown that atriodic tree-like continua do not have the fixed point property for homeomorphisms. J. B. Fugate and T. B. McLean showed that hereditarily indecomposable tree-like continua have the fixed point property for pointwise periodic homeomorphisms. In this paper the latter result is extended to the case of atriodic tree-like continua. In the course of the proof it is shown that the property of being an atriodic tree-like continuum is a Whitney property. In particular, it is shown that the hyperspace of an atriodic tree-like continuum is at most

2
2

-dimensional.
Involutions with fixed point set of constant codimension
Authors: Iwata Koichi
Published: June 2010

The cobordism classes of manifolds admitting involutions with fixed point set of codimension 5 are determined by means of Stiefel-Whitney classes.
𝐿₀ is 𝜔-transitive
Authors: Peck N. T. , Starbird T.
Published: June 2010

Let

L
0

{L_0}

be the space of measurable functions on the unit interval. Let

F
F

and

G
G

be two subspaces of

L
0

{L_0}

, each isomorphic to the space of all sequences. It is proved that there is a linear homeomorphism of

L
0

{L_0}

onto itself which takes

F
F

onto

G
G

. A corollary of this is a lifting theorem for operators into

L
0

/

F

{L_0}/F

, where

F
F

is a subspace of

L
0

{L_0}

isomorphic to the space of all sequences.
Spectral inclusion for doubly commuting subnormal 𝑛-tuples
Authors: Curto Raul E.
Published: June 2010

Let

S
=
(

S
1

,
…
,

S
n

)

S = ({S_1}, \ldots ,{S_n})

be a doubly commuting

n
n

-tuple of subnormal operators on a Hilbert space

H

\mathcal {H}

and

N
=
(

N
1

,
…
,

N
n

)

N = ({N_1}, \ldots ,{N_n})

be its minimal normal extension acting on a Hilbert space

K

⊃

H

\mathcal {K} \supset \mathcal {H}

. We show that

Sp
(
S
,

H

)
⊃
Sp
(
N
,

K

)

\text {Sp}(S,\mathcal {H}) \supset \text {Sp}(N,\mathcal {K})

and

Sp
(
S
,

H

)
⊂
p
.c

.h

.(Sp(

N
,

K

)
)

{\text {Sp}(S,\mathcal {H}) \subset \text {p}\text {.c}{\text {.h}}\text {.(Sp(}}N,\mathcal {K}))

, where

Sp

{\text {Sp}}

denotes Taylor spectrum and p.c.h. polynomially convex hull.
Selection and representation theorems for 𝜎-compact valued multifunctions
Authors: Srivastava S. M.
Published: June 2010

In this paper we give two applications of results of Shchegolkov and Saint-Raymond on Borel sets with

σ
\sigma

-compact sections. First we give a sufficient condition under which a partition of a Polish space into

σ
\sigma

-compact sets admits a Borel cross-section. Then a representation theorem for

σ
\sigma

-compact valued multifunctions, expressing them as unions of continuously indexed Borel graphs, is proved.
An inequality concerning three fundamental dimensions of paracompact 𝜎-spaces
Authors: Oka Shinpei
Published: June 2010

It is shown that Ind

Ind

X
⩽
dim
⁡
X
+

ind

X

{\text {Ind}}X \leqslant \dim X + {\text {ind}}X

for any nonempty paracompact

σ
\sigma

-space.
On a theorem from Skew field constructions
Authors: Cohn P. M.
Published: June 2010

Let

F
F

be a skew field and

C
C

a central subfield, then the free

F
F

-field on

X
X

centralizing

C
C

is denoted by

F
C

(
X
)

{F_C}(X)

. The object is to prove the following theorem. Let

F
F

be a skew field with a central subfield

C
C

, let

E
E

be a subfield of

F
F

and put

k
=
E
∩
C

k = E \cap C

; then there is a natural embedding of

E
k

(
X
)

{E_k}(X)

in

F
C

(
X
)

{F_C}(X)

if and only if

E
E

and

C
C

are linearly disjoint over

k
k

. This result replaces the erroneous Theorem 6.3.6 on p. 148 of the author’s Skew field constructions, a counterexample to the latter (due to G. M. Bergman) is also described. The paper also includes an improved form of the specialization lemma (1.c.).
On anisotropic solvable linear algebraic groups
Authors: Wang S. P.
Published: June 2010

A connected linear algebraic solvable group

G
G

defined over a field

k
k

is anisotropic over

k
k

if

G
G

has no

k
k

-subgroup splitting over

k
k

. A simple criterion for anisotropic solvable groups is presented when

k
k

is a local field.
View-obstruction problems. II
Authors: Cusick T. W.
Published: June 2010

Let

S
n

{S^n}

denote the region

0
>

x
i

>
∞
(
i
=
1
,
2
,
…
,
n
)

0 > {x_i} > \infty (i = 1,2, \ldots ,n)

of

n
n

-dimensional Euclidean space

E
n

{E^n}

. Suppose

C
C

is a closed convex body in

E
n

{E^n}

which contains the origin as an interior point. Define

α
C

\alpha C

for each real number

α
⩾
0

\alpha \geqslant 0

to be the magnification of

C
C

by the factor

α
\alpha

and define

C
+
(

m
1

,
…
,

m
n

)

C + ({m_1}, \ldots ,{m_n})

for each point

(

m
1

,
…
,

m
n

)

({m_1}, \ldots ,{m_n})

in

E
n

{E^n}

to be the translation of

C
C

by the vector

(

m
1

,
…
,

m
n

)

({m_1}, \ldots ,{m_n})

. Define the point set

Δ
(
C
,
α
)

\Delta (C,\alpha )

by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta left-parenthesis upper C comma alpha right-parenthesis equals left-brace alpha upper C plus left-parenthesis m 1 plus one half comma ellipsis comma m Subscript n Baseline plus one half right-parenthesis colon m 1 comma ellipsis comma m Subscript n Baseline">

Δ
(
C
,
α
)
=
{
α
C
+
(

m
1

+

1
2

,
…
,

m
n

+

1
2

)
:

m
1

,
…
,

m
n

\Delta (C,\alpha ) = \{ \alpha C + ({m_1} + \frac {1} {2}, \ldots ,{m_n} + \frac {1} {2}):{m_1}, \ldots ,{m_n}

nonnegative integers}. The view-obstruction problem for

C
C

is the problem of finding the constant

K
(
C
)

K(C)

defined to be the lower bound of those

α
\alpha

such that any half-line

L
L

given by

x
i

=

a
i

t
(
i
=
1
,
2
,
…
,
n
)

{x_i} = {a_i}t(i = 1,2, \ldots ,n)

, where the

a
i

(
1
⩽
i
⩽
n
)

{a_i}(1 \leqslant i \leqslant n)

are positive real numbers, and the parameter

t
t

runs through

[
0
,
∞
)

[0,\infty )

, intersects

Δ
(
C
,
α
)

\Delta (C,\alpha )

. The paper considers the case where

C
C

is the

n
n

-dimensional cube with side 1, and in this case the constant

K
(
C
)

K(C)

is known for

n
⩽
3

n \leqslant 3

. The paper gives a new proof for the case

n
=
3

n = 3

. Unlike earlier proofs, this one could be extended to study the cases with

n
⩾
4

n \geqslant 4

.
An algebraic proof of the Borsuk-Ulam theorem for polynomial mappings
Authors: Knebusch Manfred
Published: June 2010

An algebraic proof is given for the following theorem: Every system of

n
n

odd polynomials in

n
+
1

n + 1

variables over a real closed field

R
R

has a common zero on the unit sphere

S
n

(
R
)
⊂

R

n
+
1

{S^n}(R) \subset {R^{n + 1}}

.
Normal families of quasimeromorphic mappings
Authors: Miniowitz Ruth
Published: June 2010

We obtain several sufficient conditions for a family of quasiregular or quasimeromorphic mappings to be normal, which are generalizations to known results for analytic functions.
A stability property of a class of Banach spaces not containing a complemented copy of 𝑙₁
Authors: Saab Elias , Saab Paulette
Published: June 2010

Let

E
E

be a Banach space and

K
K

be a compact Hausdorff space. The space

C
(
K
,
E
)

C(K,E)

will stand for the Banach space of all continuous

E
E

-valued functions on

K
K

equipped with the sup norm. It is shown that the space

E
E

does not contain a complemented subspace isomorphic to

l
1

{l_1}

if and only if

C
(
K
,
E
)

C(K,E)

has the same property.
The first sign change of a cosine polynomial
Authors: Nulton James D. , Stolarsky Kenneth B.
Published: June 2010

It is reasonable to expect the first sign change of a real cosine polynomial to decrease when its smallest frequency is increased. Many cases in which this is true are exhibited, but it is shown that there exist (presumably unusual) cosine polynomials for which the first sign change may increase by an arbitrarily large amount.
Finite test sets and 𝑃-matrices
Authors: Kostreva Michael M.
Published: June 2010

The class of matrices with all principal minors positive, known as

P
P

-matrices, has been characterized by Murty and Tamir using a finite set of test vectors for the linear complementarity problem. This paper refines their characterizations by deriving a set of test vectors which has lower cardinality and vectors which are more easily tested.
Cardinal arithmetic and ℵ₁-Borel sets
Authors: Steprāns Juris
Published: June 2010

It is shown to be consistent with

2

ℵ
0

>

ℵ
1

{2^{{\aleph _0}}} > {\aleph _1}

that the smallest

ℵ
2

{\aleph _2}

-complete Boolean subalgebra of

P

(

R

)

\mathcal {P}({\mathbf {R}})

containing all closed sets is

P

(

R

)

\mathcal {P}({\mathbf {R}})

. Some related results are also proved.
Dihedral algebras are cyclic
Authors: Rowen Louis H. , Saltman David J.
Published: June 2010

Any central simple algebra of degree

n
n

split by a Galois extension with dihedral Galois group of degree

2
n

2n

is, in fact, a cyclic algebra. We assume that the centers of these algebras contain a primitive

n
n

th root of unity.
Characterization of approximately inner automorphisms
Authors: Choda Marie
Published: June 2010

Let

M
M

be a finite factor acting standardly on a Hilbert space

H
H

. An automorphism

θ
\theta

of

M
M

is approximately inner on

M
M

if and only if there exists a state

ϕ
\phi

on

B
(
H
)

B(H)

such that

ϕ
(
J
u
J
θ
(
u
)
)
=
1

\phi (JuJ\theta (u)) = 1

for every unitary

u
u

in

M
M

, where

J
J

is the canonical involution. Specially,

θ
\theta

is inner on

M
M

if and only if such a state is a vector state.
Finitely generated codings and the degrees r.e. in a degree 𝑑
Authors: Shore Richard A.
Published: June 2010

We introduce finitely generated (partial) lattices which can be used to code an arbitrary set

D
D

. Results of Lerman, Shore and Soare are used to embed these lattices in the degrees r.e. in

D
D

. Thus if the degrees r.e. in and above

d

{\mathbf {d}}

are isomorphic to those r.e. in and above

c

{\mathbf {c}}

,

d

{\mathbf {d}}

and

c

{\mathbf {c}}

are of the same arithmetic degree. Similar applications are given to generic degrees and general homogeneity questions.
Calculating the fundamental group of an orbit space
Authors: Armstrong M. A.
Published: June 2010

Suppose

G
G

acts effectively as a group of homeomorphisms of the connected, locally path connected, simply connected, locally compact metric space

X
X

. Let

G
¯

\overline G

denote the closure of

G
G

in

Homeo

(
X
)

{\text {Homeo}}(X)

, and

N
N

the smallest normal subgroup of

G
¯

\overline G

which contains the path component of the identity of

G
¯

\overline G

and all those elements of

G
¯

\overline G

which have fixed points. We show that

π
1

(
X

/

G
)

{\pi _1}(X/G)

is isomorphic to

G
¯

/

N

\overline G /N

subject to a weak path lifting assumption for the projection

X
→
X

/

G
¯

X \to X/\overline G

.
Sufficient conditions for a bundle-like foliation to admit a Riemannian submersion onto its leaf space
Authors: Escobales Richard H.
Published: June 2010

This note furnishes a necessary and sufficient condition for a bundlelike foliation to be defined globally by a Riemannian submersion.
Embedding the free group 𝐹(𝑋) into 𝐹(𝛽𝑋)
Authors: Fay Temple H. , Rajagopolan M. , Smith-Thomas Barbara V.
Published: June 2010

We show that for a Tychonoff space,

X
X

and the canonical embedding

β
X

:
X
→
β
X

{\beta _X}:X \to \beta X

, the induced homomorphism

F

β
X

:
F
(
X
)
→
F
(
β
X
)

F{\beta _X}:F(X) \to F(\beta X)

is an embedding between the free topological groups when

X
X

has the property that

X
n

{X^n}

is pseudo-compact for all

n
⩾
1

n \geqslant 1

. An application of this result is if

X
X

is such a space and

β
X

\beta X

is

0
0

-dimensional, then

F
(
X
)

F(X)

is

0
0

-dimensional.
A locally closed set with a smooth group structure is a Lie group
Authors: Machado Armando
Published: June 2010

We prove the following result. Let

V
V

be a smooth manifold and let

G
⊂
V

G \subset V

be a locally closed set with a group structure such that both multiplication and inversion are smooth maps; then

G
G

is an imbedded smooth submanifold of

V
V

. This result is a generalization of the well-known fact that a closed subgroup of a Lie group is itself a Lie group, because we are not assuming any group structure in the manifold

V
V

.
Completion of a Gaussian derivation
Authors: Ewell John A.
Published: June 2010

The author supplies evaluation of two coefficients of an identity partially derived by Gauss. Several special cases of this identity are then presented, the last of these identities yielding an alternate proof of Ramanujan’s theorem concerning divisibility of certain values of the partition function by the modulus 7.
Traces in 2-generator subgroups of 𝑆𝐿(2,𝐶)
Authors: Jørgensen Troels
Published: June 2010

An explicit formula for the trace polynomials of elements in

2
2

-generator subgroups of

SL

(
2
,

C

)

{\text {SL}}(2,{\mathbf {C}})

will be presented.
Triangularizable algebras of compact operators
Authors: Murphy G. J.
Published: June 2010

It is shown that a closed algebra

A
A

of compact operators is triangularizable if and only if the algebra

A

/

rad
⁡
(
A
)

A/\operatorname {rad} (A)

is commutative.
Unitary one-parameter groups with finite speed of propagation
Authors: Svendsen E. C.
Published: June 2010

Suppose that

ξ
\xi

is a Hermitian vector bundle over a Riemannian manifold and that

U
U

is a one-parameter group of linear operators on the set of smooth sections of

ξ
\xi

with compact support. We prove that if

U
U

satisfies a smoothness condition, is unitary, and propagates initial data with finite speed, then it can be constructed from the solutions of a first-order symmetric hyperbolic system of partial differential equations.
Oscillation theorems for nonlinear second order differential equations with damped term
Authors: Yeh Cheh Chih
Published: June 2010

Some new integral criteria for the oscillation of the nonlinear second order differential equation with damped term
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y left-parenthesis t right-parenthesis plus p left-parenthesis t right-parenthesis y prime left-parenthesis t right-parenthesis plus q left-parenthesis t right-parenthesis f left-parenthesis y left-parenthesis t right-parenthesis right-parenthesis equals 0">

y
(
t
)
+
p
(
t
)

y
′

(
t
)
+
q
(
t
)
f
(
y
(
t
)
)
=
0

y(t) + p(t)y’(t) + q(t)f(y(t)) = 0

are given.
Trees are contractible
Authors: Paulowich D. G.
Published: June 2010

Any hereditarily unicoherent, locally connected, compact connected Hausdorff space is contractible using an ordered continuum. An example is given of a hereditarily unicoherent, locally connected, first countable, compact connected Hausdorff space that does not admit the structure of a topological semigroup with zero and identity.
Mapping cylinder neighborhoods in the plane
Authors: Brechner Beverly , Brown Morton
Published: June 2010

We characterize those compact subsets of the plane which have mapping cylinder neighborhoods, describe the neighborhood closures, and show that such neighborhood closures are topologically unique. The proofs employ the notion of prime ends. We also show that if

U
U

is a mapping cylinder neighborhood of a pointlike continuum in

S
3

{S^3}

, then

U
¯

\overline U

is a

3
3

-cell.
Locally planar toroidal graphs are 5-colorable
Authors: Albertson Michael O. , Stromquist Walter R.
Published: June 2010

If a graph can be embedded in a torus in such a way that all noncontractible cycles have length at least 8, then its vertices may be

5
5

-colored. The conclusion remains true when some noncontractible cycles have length less than 8, if the exceptions are all homotopic. Essentially this hypothesis means that small neighborhoods of the graph are planar. No similar conclusion holds for

4
4

-colorability.
On the first factor of the class number of a cyclotomic field
Authors: Feng Ke Qin
Published: June 2010

Let

p
p

be an odd prime.

h
1

(
p
)

{h_1}(p)

is the first factor of the class number of field

Q
(

ζ
p

)

Q({\zeta _p})

. We proved that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h 1 left-parenthesis p right-parenthesis less-than-or-slanted-equals StartLayout Enlarged left-brace 1st Row 2 p left-parenthesis StartFraction p minus 1 Over 8 left-parenthesis 2 Superscript l slash 2 Baseline plus 1 right-parenthesis Superscript 4 slash l Baseline EndFraction right-parenthesis Superscript left-parenthesis p minus 1 right-parenthesis slash 4 Baseline comma if l is even comma 2nd Row 2 p left-parenthesis StartFraction p minus 1 Over 8 left-parenthesis 2 Superscript l Baseline minus 1 right-parenthesis Superscript 2 slash l Baseline EndFraction right-parenthesis Superscript left-parenthesis p minus 1 right-parenthesis slash 4 Baseline comma if l is odd period EndLayout">

h
1

(
p
)
⩽

{

2
p

(

p
−
1

8

(

2

l

/

2

+
1
)

4

/

l

)

(
p
−
1
)

/

4

,

if 

l

is even,

2
p

(

p
−
1

8

(

2
l

−
1
)

2

/

l

)

(
p
−
1
)

/

4

,

if 

l

is odd

.

{h_1}(p) \leqslant \left \{ \begin {gathered} 2p{\left ( {\frac {{p - 1}} {{8{{({2^{l/2}} + 1)}^{4/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is even,}} \hfill \\ 2p{\left ( {\frac {{p - 1}} {{8{{({2^l} - 1)}^{2/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is odd}}{\text {.}} \hfill \\ \end {gathered} \right .

\]
From that we obtain
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h 1 left-parenthesis p right-parenthesis less-than-or-slanted-equals 2 p left-parenthesis left-parenthesis p minus 1 right-parenthesis slash 31.997158 ellipsis right-parenthesis Superscript left-parenthesis p minus 1 right-parenthesis slash 4">

h
1

(
p
)
⩽
2
p

(
(
p
−
1
)

/

31.997158
…

)

(
p
−
1
)

/

4

{h_1}(p) \leqslant 2p{((p - 1)/31.997158 \ldots )^{(p - 1)/4}}

which is better than Carlitz’s and Metsänkyla’s results. For the fields

Q
(

ζ

2
n

)

Q({\zeta _{{2^n}}})

and

Q
(

ζ

p
n

)
(
n
⩾
2
)

Q({\zeta _{{p^n}}})(n \geqslant 2)

, we get the similar results.
A product theorem for ℱ_{𝓅} classes and an application
Authors: Pearce Kent
Published: June 2010

For Re

p
>
0

p > 0

let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F Subscript p Baseline equals left-brace f vertical-bar f left-parenthesis z right-parenthesis equals integral Underscript StartAbsoluteValue x EndAbsoluteValue equals 1 Endscripts left-parenthesis 1 minus x z right-parenthesis Superscript negative p Baseline d mu left-parenthesis x right-parenthesis">

F

p

=
{
f

|

f
(
z
)
=

∫

|

x

|

=
1

(
1
−
x
z
)

−
p

d
μ
(
x
)

{\mathcal {F}_p} = \{ f|f(z) = \int _{|x| = 1} {{{(1 - xz)}^{ - p}}d\mu (x)}

,

|

z

|

>
1

|z| > 1

,

μ
\mu

a probability measure on

|

x

|

=
1

|x| = 1

and let

F

p

⋅

F

q

=
{
f
g

|

f
∈

F

p

,
g
∈

F

q

}

{\mathcal {F}_p} \cdot {\mathcal {F}_q} = \{ fg|f \in {\mathcal {F}_p},g \in {\mathcal {F}_q}\}

. Brickman, Hallenbeck, MacGregor and Wilken proved a product theorem for the

F

p

{\mathcal {F}_p}

classes; they showed that if

p
>
0

p > 0

,

q
>
0

q > 0

, then

F

p

⋅

F

q

⊂

F

p
+
q

{\mathcal {F}_p} \cdot {\mathcal {F}_q} \subset {\mathcal {F}_{p + q}}

. We give an (essentially complete) converse for the result of Brickman et al., i.e., we show that if

F

p

⋅

F

q

⊂

F

p
+
q

{\mathcal {F}_p}\cdot {\mathcal {F}_q} \subset {\mathcal {F}_{p + q}}

, then

p
>
0

p > 0

,

q
>
0

q > 0

or else

p
=
q
=
1
+
i
t

p = q = 1 + it

for some

t
t

real. As an immediate consequence we disprove a conjecture about the extreme points of the closed convex hulls of the classes

Sp(

γ

)

{\text {Sp(}}\gamma {\text {)}}

,

0
>

|

γ

|

>
π

/

2

0 > |\gamma | > \pi /2

, of

γ
\gamma

-spirallike univalent functions, i.e., writing

m
=
1
+

e

−
2
i
γ

m = 1 + {e^{ - 2i\gamma }}

, we show
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet z slash left-parenthesis 1 minus x z right-parenthesis Superscript m Baseline StartAbsoluteValue EndAbsoluteValue x vertical-bar equals 1 EndSet subset-of-above-not-equals script upper E script upper K upper S p left-parenthesis gamma right-parenthesis">

{
z

/

(
1
−
x
z

)
m

|

|

x

|

=
1
}
⫋

E

K

Sp
⁡
(
γ
)

\{ z/(1 - xz)^m| |x| = 1\} \subsetneqq \mathcal {E}\mathcal {K} \operatorname {Sp}(\gamma )

,

0
>

|

γ

|

>
π

/

2

0 > |\gamma | > \pi /2

.
Subnormals in 𝐶*-algebras
Authors: Szafraniec F. H.
Published: June 2010

We prove, in a

C
∗

{C^* }

-algebra set-up, the Bram improvement of Halmos’ characterization of subnormals:

(
1
)
⇒
(
2
)

(1) \Rightarrow (2)

.
On certain comparison theorems for second order linear oscillation
Authors: Kwong Man Kam
Published: June 2010

It is shown that if

(
p

y
′

)
′

+
q
y
=
0

(py’)’ + qy = 0

on

[
0
,
∞
)

[0,\infty )

is oscillatory then

(
p

z
′

)
′

+
a
q
z
=
0

(pz’)’ + aqz = 0

is also oscillatory for functions satisfying

a
(
t
)
⩾
1

a(t) \geqslant 1

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 p left-parenthesis t right-parenthesis a prime left-parenthesis t right-parenthesis minus 3 integral Subscript 0 Superscript t Baseline p left-parenthesis s right-parenthesis a prime squared left-parenthesis s right-parenthesis a Superscript negative 1 Baseline left-parenthesis s right-parenthesis d s">

2
p
(
t
)

a
′

(
t
)
−
3

∫
0
t

p
(
s
)

a
′

2

(
s
)

a

−
1

(
s
)
d
s

2p(t)a’(t) - 3\int _0^t {p(s){{a’}^2}(s){a^{ - 1}}(s)ds}

\]
is nonincreasing.
Multipliers and asymptotic behaviour of the Fourier algebra of nonamenable groups
Authors: Nebbia Claudio
Published: June 2010

Let

G
G

be a locally compact group and

A
(
G
)

A(G)

the algebra of matrix coefficients of the regular representation. We prove that

G
G

is amenable if and only if there exist functions

u
∈
A
(
G
)

u \in A(G)

which vanish at infinity at any arbitrarily slow rate. The "only if" part of the result was essentially known. With the additional hypothesis that

G
G

be discrete, we deduce that

G
G

is amenable if and only if every multiplier of the algebra

A
(
G
)

A(G)

is a linear combination of positive definite functions. Again, the "only if" part of this result was known.
Index sets and Boolean operations
Authors: Miller Douglas E.
Published: June 2010

Hay’s thesis asserts that every naturally defined class of sets of natural numbers contains an index set which is

1
1

-complete for that class and that every index set is

1
1

-complete for some naturally defined class. We formalize "naturally defined" as "effective Boolean" and establish the thesis as a Metatheorem.
A symplectic fixed point theorem on open manifolds
Authors: Colvin Michael , Morrison Kent
Published: June 2010

In 1968 Bourgin proved that every measure-preserving, orientation-preserving homeomorphism of the open disk has a fixed point, and he asked whether such a result held in higher dimensions. Asimov, in 1976, constructed counterexamples in all higher dimensions. In this paper we answer a weakened form of Bourgin’s question dealing with symplectic diffeomorphisms: every symplectic diffeomorphism of an even-dimensional cell sufficiently close to the identity in the

C
1

{C^1}

-fine topology has a fixed point. This result follows from a more general result on open manifolds and symplectic diffeomorphisms.
A note on holomorphic imbeddings of the classical Cartan domains into the unit ball
Authors: Kubota Yoshihisa
Published: June 2010

Let

D
D

be a classical Cartan domain and let

B
B

be the unit ball.

We

{\text {We}}

find the exact value of the supremum of the set of positive numbers

ρ
\rho

satisfying the condition:

ρ
B
⊂
f
(
D
)

\rho B \subset f(D)

for a certain holomorphic imbedding

f
:
D
→
B

f:D \to B

.
On steady incompressible three-dimensional hydromagnetic flows
Authors: Prasad G. , Singh T.
Published: June 2010

In this paper certain theorems of theoretical interest have been established with the help of the geometrical properties of Faraday’s surface (which is spanned by the flow and field lines). These theorems shed light on the behaviour of steady incomprȩssible hydromagnetic flows. The complex-lamellar acceleration and simple geodesic motion admitted by such flows have also been studied.
Quadratic Morse-Smale vector fields which are not structurally stable
Authors: Chicone Carmen , Shafer Douglas S.
Published: June 2010

An example is given of a quadratic system in the plane which is Morse-Smale but not structurally stable. Also, it is proved that no such example exists for a quadratic system which is a gradient.
Modules whose endomorphism rings have isomorphic maximal left and right quotient rings
Authors: Khuri Soumaya Makdissi
Published: June 2010

Let

R

M

_RM

be a left

R
R

-module such that

Hom
R

(
M
,
U
)
≠
0

{\operatorname {Hom} _R}(M,U) \ne 0

for any nonzero submodule

U
U

of

M
M

, let

E
(
M
)

E(M)

denote the injective hull of

M
M

, and let

B
B

(resp.

A
A

) denote the ring of

R
R

-endomorphisms of

M
M

(resp.

E
(
M
)

E(M)

). It is known that if

M
M

is nonsingular then

B
B

is left nonsingular and

A
A

is the maximal left quotient ring of

B
B

. We give here necessary and sufficient conditions on

M
M

for

B
B

to be right nonsingular and for

A
A

to be the maximal right quotient ring of

B
B

.
Cauchy-type congruences for binomial coefficients
Authors: Hudson Richard H. , Williams Kenneth S.
Published: June 2010

In 1840 Cauchy [2] showed that for a prime

p
=
e
f
+
1

p = ef + 1

,

e
=
20

e = 20

,
\[

(

10
f

f

)

≡
±

(

10
f

3
f

)

(
mod
p
)
,

\left ( {\begin {array}{* {20}{c}} {10f} \\ f \\ \end {array} } \right ) \equiv \pm \left ( {\begin {array}{* {20}{c}} {10f} \\ {3f} \\ \end {array} } \right )\quad (\bmod p),

\]
and it was not until 1965 that Whiteman [6] succeeded in removing the sign ambiguity in this congruence. In this paper we show how the Davenport-Hasse relation [3] in the form given by Yamamoto [8] can be used to resolve the sign ambiguity in other Cauchy-type congruences. Details are given just for

e
=
8
,
12
,

and 

20

e = 8,12,{\text {and }}20

.
Two integrals of Ramanujan
Authors: Askey Richard
Published: June 2010

Two integrals of Ramanujan are evaluated.
The dependence of mass upon wind
Authors: Smith Brent
Published: June 2010

As the winding number of unimodular function on

R

{\mathbf {R}}

increases must the mass of the measure from which it arises (via the Fourier transform) increase?
Norm attaining operators and norming functionals
Authors: Bilyeu Russell G. , Lewis Paul W.
Published: June 2010

The question of whether a countably additive measure with values in a Banach space attains the diameter of its range was unresolved. In this paper an example is given of a countably additive vector measure, taking values in a

C
(
K
)

C(K)

space, for which the diameter of the range is not attained. A property stronger than the attainment of the diameter, but which is possessed by many measures taking values in

L
L

-spaces, is shown to fail for infinite-dimensional measures into a space having smooth dual. As an application of the concept of norming functional (the existence of which is equivalent to the attainment of diameter), a characterization is given of the countably additive measures into space having smooth dual.
Kakutani’s theorem for real-valued maps on a compact surface
Authors: Kronheimer E. H. , Kronheimer P. B.
Published: June 2010

Let

M
M

be a compact

2
2

-manifold (without boundary)

C
1

{C^1}

-embedded in

R

3

{{\mathbf {R}}^3}

. Then there exists positive

σ
\sigma

such that, given any positive

τ
⩽
σ

\tau \leqslant \sigma

and any continuous map

f
:
M
→

R

f:M \to {\mathbf {R}}

, there exist points

p
p

,

q
q

,

r
∈
M

r \in M

, satisfying

‖

q
−
r

‖

=

‖

r
−
p

‖

=

‖

p
−
q

‖

=
τ

\left \| {q - r} \right \| = \left \| {r - p} \right \| = \left \| {p - q} \right \| = \tau

in the euclidean norm, for which

f
(
p
)
=
f
(
q
)
=
f
(
r
)

f(p) = f(q) = f(r)

.
A characterization of the multiparameter Wiener process and an application
Authors: Dorea Chang C. Y.
Published: June 2010

In this note we give an application of a characterization of the multiparameter Wiener process and of the invariance principle for

R
k

{R^k}

-valued martingales.
On developability
Authors: Hung H. H.
Published: June 2010

The converse of an old result of Bing turns out to be a very interesting characterization of developability. It is interesting in the sense that it provides a unifying framework for certain classical results of Bing, F. B. Jones, Heath and Wicke and Worrell, strengthening some of them in the process, without much effort.
On the inertia groups of fibre bundles
Authors: Frame Michael
Published: June 2010

A subgroup

I
~

(
M
×

S
i

)

\tilde I(M \times {S^i})

of the inertia group

I
(
M
×

S
i

)

I(M \times {S^i})

is defined and shown to lie in

I
(
C
)

I(C)

for every fibre bundle

M
n

→
C
→

N
i

{M^n} \to C \to {N^i}

. For certain

M
M

, examples of nontrivial elements in

I
~

(
M
×

S
i

)

\tilde I(M \times {S^i})

are constructed using the

τ
\tau

-pairing of Milnor-Munkres-Novikov. For compact mapping tori

M
g

{M_g}

it is shown that

I
(

M
g

)
=
I
(
M
×

S
1

)

I({M_g}) = I(M \times {S^1})

if

π
1

M

{\pi _1}M

is finite and

Wh

(

π
1

M
)
=
0

{\text {Wh}}({\pi _1}M) = 0

.
An infinite class of periodic solutions of periodically perturbed Duffing equations at resonance
Authors: Ding Tung Ren
Published: June 2010

In this paper, by using a generalized form of the Poincaré-Birkhoff Theorem, we demonstrate that the Duffing equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d squared x Over d t squared EndFraction plus g left-parenthesis x right-parenthesis equals p left-parenthesis t right-parenthesis left-parenthesis identical-to p left-parenthesis t plus 2 pi right-parenthesis right-parenthesis">

d
2

x

d

t
2

+
g
(
x
)
=
p
(
t
)

(
≡
p
(
t
+
2
π
)
)

\frac {{{d^2}x}} {{d{t^2}}} + g(x) = p(t)\quad ( \equiv p(t + 2\pi ))

\]
may also admit an infinite number of

2
π

2\pi

-periodic solutions even in a resonance case.
Proper pseudocompact extensions of compact abelian group topologies
Authors: Comfort W. W. , Robertson Lewis C.
Published: June 2010

A compact Abelian group

G
G

admits a strictly finer pseudocompact topological group topology if and only if the weight of

G
G

is uncountable.
Differentiability of convex functions and Rybakov’s theorem
Authors: Bilyeu Russell G. , Lewis Paul W.
Published: June 2010

Rybakov proved that if

μ
:
Σ
→
X

\mu :\Sigma \to X

is a countably additive Banach space valued measure on a

σ
\sigma

-algebra

Σ
\Sigma

, then there is an element

x
∗

∈

X
∗

{x^ * } \in {X^ * }

so that

μ
≪

x
∗

μ

\mu \ll {x^ * }\mu

. In this note, we show that Rybakov’s theorem follows essentially from a classical result of Mazur on the Gateaux differentiability of convex functions.
On line integrals of rational functions of two complex variables
Authors: Azoff E. , Clancey K. , Gohberg I.
Published: June 2010

Let

γ
\gamma

be a simple rectifiable arc in the complex plane and

r
(
z
,
w
)

r(z,w)

a rational function of two complex variables. Set

r
γ

(
z
)
=

∫
γ

r
(
z
,
w
)

d
w

{r_\gamma }(z) = \int _\gamma {r(z,w)\;dw}

. The natural domain of

r
γ

{r_\gamma }

has countably many components, and

r
γ

{r_\gamma }

may vanish identically on infinitely many of these. It is shown however that unless

γ
\gamma

spirals in to one of its endpoints, only finitely many zeros of

r
γ

{r_\gamma }

are isolated.
𝐿¹-convergence of Fourier series with complex quasimonotone coefficients
Authors: Stanojevic Vera B.
Published: June 2010

A sequence of Fourier coefficients

{

f
^

(
n
)

}

\left \{ {\hat f(n)} \right \}

of a complex function in

L
1

(
T
)

{L^1}(T)

is said to be complex quasimonotone if there exists

θ
0

{\theta _0}

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta ModifyingAbove f With caret left-parenthesis n right-parenthesis plus StartFraction alpha Over n EndFraction ModifyingAbove f With caret left-parenthesis n right-parenthesis element-of StartSet z vertical-bar StartAbsoluteValue arg z EndAbsoluteValue less-than-or-slanted-equals theta 0 greater-than StartFraction pi Over 2 EndFraction EndSet">

Δ

f
^

(
n
)
+

α
n

f
^

(
n
)
∈

{

z

|

|

arg
⁡
z

|

⩽

θ
0

>

π
2

}

\Delta \hat f(n) + \frac {\alpha } {n}\hat f(n) \in \left \{ {z|\left | {\arg z} \right | \leqslant {\theta _0} > \frac {\pi } {2}} \right \}

\]
for some

α
⩾
0

\alpha \geqslant 0

and for all

n
n

. It is proved that Fourier series with asymptotically even and complex quasimonotone coefficients, satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts Overscript bar Endscripts n Superscript 1 slash q Baseline max Underscript n less-than-or-slanted-equals j less-than-or-slanted-equals left-bracket lamda n right-bracket Endscripts StartAbsoluteValue normal upper Delta ModifyingAbove f With caret left-parenthesis j right-parenthesis EndAbsoluteValue Superscript 1 slash q Baseline max Underscript n less-than-or-slanted-equals j less-than-or-slanted-equals left-bracket lamda n right-bracket Endscripts StartAbsoluteValue ModifyingAbove f With caret left-parenthesis j right-parenthesis EndAbsoluteValue Superscript 1 slash p Baseline equals o left-parenthesis 1 right-parenthesis comma lamda right-arrow 1 plus 0 comma StartFraction 1 Over p EndFraction plus StartFraction 1 Over q EndFraction equals 1 comma">

lim

n
→
∞

¯

n

1

/

q

max

n
⩽
j
⩽
[
λ
n
]

|

Δ

f
^

(
j
)

|

1

/

q

max

n
⩽
j
⩽
[
λ
n
]

|

f
^

(
j
)

|

1

/

p

=
o
(
1
)
,

λ
→
1
+
0
,

1
p

+

1
q

=
1
,

\overline {\lim \limits _{n \to \infty } } \;{n^{1/q}}\max \limits _{n \leqslant j \leqslant [\lambda n]} {\left | {\Delta \hat f(j)} \right |^{1/q}}\max \limits _{n \leqslant j \leqslant [\lambda n]} {\left | {\hat f(j)} \right |^{1/p}} = o(1),\; \lambda \to 1 + 0,\tfrac {1} {p} + \tfrac {1} {q} = 1,

\]
converges in

L
1

(
T
)

{L^1}(T)

-norm if and only if

f
^

(
n
)
lg
⁡

|
n
|

=
o
(
1
)

\hat f(n)\lg \left | n \right | = o(1)

,

n
→
∞

n \to \infty

. A recent result of Č V. Stanojević [3] is a special case of the corollary of the main theorem.
Nondiscriminating sets for 𝐻^{∞}
Authors: Marshall Donald E. , Stray Arne , Sundberg Carl
Published: June 2010

We characterize the subsets

S
S

of the unit disk

D
D

with the property that any function defined on

S
S

that has a bounded harmonic extension to

D
D

, must also have a bounded analytic extension to

D
D

.
A zero-one, Borel probability which admits of no countably additive extensions
Authors: Dubins Lester E.
Published: June 2010

There is a subsigma-field of the Borel subsets

B

\mathcal {B}

of the unit interval which supports a countably additive, two-valued, probability which cannot be extended to

B

\mathcal {B}

so as to remain countably additive.
Conformally flat spaces and a pinching problem on the Ricci tensor
Authors: Hasanis Th.
Published: June 2010

Recent results of S. I. Goldberg on conformally flat manifolds and hypersurfaces of Euclidean space are extended.
Appropriate locally convex domains for differential calculus
Authors: Graff Richard A. , Ruess Wolfgang M.
Published: June 2010

We make use of Grothendieck’s notion of quasinormability to produce a comprehensive class of locally convex spaces within which differential calculus may be developed along the same lines as those employed within the class of Banach spaces and which include the previously known examples of such classes. In addition, we show that there exist Fréchet spaces which do not belong to any possible such class.
Common fixed points for a class of commuting mappings on an interval
Authors: Cano J.
Published: June 2010

Let

C
C

be a family of continuous commuting functions of an interval

I
I

into itself. If each function, except for possibly one, has an interval

[
a
,
b
]
,
a
⩽
b

[a,b],a \leqslant b

, for its set of fixed points or does not have periodic points except fixed ones, then it is shown that

C
C

has a common fixed point. This result generalizes a previous theorem of T. Mitchell.
Conjugacy separability of certain 1-relator groups
Authors: Tang C. Y.
Published: June 2010

Let

G
=

⟨

a
,
b
;

(

a
l

b
m

)

t

⟩

G = \left \langle {a,b;{{({a^l}{b^m})}^t}} \right \rangle

where

l
l

,

m
m

,

t
t

are integers. We show that groups of this type are conjugacy separable.
A note on nonfinitely generated projective 𝑍𝜋-modules
Authors: Akasaki Takeo
Published: June 2010

Let

π
\pi

be a finite group and

Z

π

{\mathbf {Z}}\pi

be its integral group ring. It is shown that if

π
\pi

is not solvable, then there exists a nonfinitely generated projective

Z

π

{\mathbf {Z}}\pi

-module which is not free.