bluuebunny/crossref_metadata_2025 huggingface dataset:
Boundary zero sets of 𝐴^{∞} functions satisfying growth conditions
Authors: Taylor B. A. , Williams D. L.
Published: June 2010

Let A denote the algebra of functions analytic in the open unit disc D and continuous in D, and let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Superscript normal infinity Baseline equals left-brace f element-of upper A colon f Superscript left-parenthesis n right-parenthesis Baseline element-of upper A comma n equals 0 comma 1 comma 2 comma midline-horizontal-ellipsis right-brace period">

A
∞

=
{
f
∈
A
:

f

(
n
)

∈
A
,
n
=
0
,
1
,
2
,
⋯
}
.

{A^\infty } = \{ f \in A:{f^{(n)}} \in A,n = 0,1,2, \cdots \} .

\]
For

f
∈
A

f \in A

denote the set of zeros of f in D by

Z
0

(
f
)

{Z^0}(f)

, and for

f
∈

A
∞

f \in {A^\infty }

let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z Superscript normal infinity Baseline left-parenthesis f right-parenthesis equals intersection Subscript n equals 0 Superscript normal infinity upper Z Superscript 0 Baseline left-parenthesis f Superscript left-parenthesis n right-parenthesis Baseline right-parenthesis">

Z
∞

(
f
)
=
⋂

n
=
0

∞

Z

0

(

f

(
n
)

)

{Z^\infty }(f) = \bigcap {_{n = 0}^\infty Z{^0}({f^{(n)}})}

. We study the boundary zero sets of

A
∞

{A^\infty }

functions F satisfying, for some sequence

{

M
n

}

\{ {M_n}\}

and some

B
>
0

B > 0

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel StartAbsoluteValue upper F Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis z right-parenthesis EndAbsoluteValue less-than-over-equals n factorial upper B Superscript n Baseline upper M Subscript n Baseline comma z element-of upper D overbar comma n equals 0 comma 1 comma 2 comma midline-horizontal-ellipsis period EndLayout">

(

1

)

|

F

(
n
)

(
z
)

|

≦
n
!

B
n

M
n

,

z
∈

D
¯

,
n
=
0
,
1
,
2
,
⋯
.

\begin{equation}\tag {$1$}|{F^{(n)}}(z)| \leqq n!{B^n}{M_n},\quad z \in \bar D,n = 0,1,2, \cdots .\end{equation}

In particular, when

M
n

=
exp
⁡
(

n
p

)
,
p
>
1

{M_n} = \exp ({n^p}),p > 1

, it is shown that for E, a proper closed subset of

∂
D

\partial D

, there exists

F
∈

A
∞

F \in {A^\infty }

satisfying (1) and with

Z
0

(
F
)
=

Z
∞

(
F
)
=
E

{Z^0}(F) = {Z^\infty }(F) = E

if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative pi Superscript pi Baseline StartAbsoluteValue log rho left-parenthesis e Superscript i theta Baseline comma upper E right-parenthesis EndAbsoluteValue Superscript q Baseline upper D theta greater-than plus normal infinity">

∫

−
π

π

|

log
⁡
ρ
(

e

i
θ

,
E
)

|

q

D
θ
>
+
∞

\smallint _{ - \pi }^\pi |\log \rho ({e^{i\theta }},E){|^q}D\theta > + \infty

. Here

ρ
(
z
,
E
)

\rho (z,E)

is the distance from z to E and

(
1

/

p
)
+
(
1

/

q
)
=
1

(1/p) + (1/q) = 1

.
Generalizations of results of Agmon and Nirenberg to the Cauchy problem for evolution equations
Authors: Strauss Monty J.
Published: June 2010

Uniqueness and norm convexity for evolution equations are investigated by using tools developed by Agmon and Nirenberg.
Tangent bundles of homogeneous spaces are homogeneous spaces
Authors: Brockett R. W. , Sussmann H. J.
Published: June 2010

In this paper we describe how the tangent bundle of a homogeneous space can be viewed as a homogeneous space.
Permutable pronormal subgroups
Authors: Yen Ti
Published: June 2010

Let G be a finite solvable group. It is shown that a certain class of pronormal subgroups reducing a given Sylow system is a lattice of permutable subgroups.
A characterization of strongly measurable Pettis integrable functions
Authors: Uhl J. J.
Published: June 2010

Let

X

\mathfrak {X}

be a Banach space and

(
Ω
,
Σ
,
μ
)

(\Omega ,\Sigma ,\mu )

be a finite measure space. A strongly measurable

f
:
Ω
→

X

f:\Omega \to \mathfrak {X}

is Pettis integrable if and only if there exists a Young’s function

Φ
\Phi

with

lim

t
→
∞

Φ
(
t
)

/

t
=
∞

{\lim _{t \to \infty }}\Phi (t)/t = \infty

such that

x
∗

f
∈

L
Φ

(
μ
)

{x^ \ast }f \in {L^\Phi }(\mu )

for all

x
∗

∈

X

∗

{x^ \ast } \in {\mathfrak {X}^ \ast }

.
Henselian fields and solid 𝑘-varieties. II
Authors: Efroymson Gustave
Published: June 2010

Let k be a real closed or Henselian field. A k-variety X (affine) is said to be solid if X is determined by its k points. It is shown that a k-variety is solid if and only if it contains a nonsingular k point. Another condition for solidity is given and a dimension theorem indicated.
The free product of residually finite groups amalgamated along retracts is residually finite
Authors: Boler James , Evans Benny
Published: June 2010

It is shown that residual finiteness is preserved by the generalized free product provided that the amalgamated subgroups are retracts of their respective factors. This result is applied to knot groups. The outcome is that the question of residual finiteness for knot groups need only be answered for prime knots.
Spectrum of a composition operator
Authors: Ridge William C.
Published: June 2010

A composition operator is a linear operator induced on a subspace of

K
X

{K^X}

by a point transformation

ϕ
\phi

on a set X (where K denotes the scalar field) by the formula

T
f
(
x
)
=
f
∘
ϕ
(
x
)

Tf(x) = f \circ \phi (x)

. Familiar examples include translation operators on the real line and on topological groups, analytic functions which preserve the class of harmonic functions (and Green’s functions), ergodic transformations which induce unitary operators on

L
2

{L^2}

, shift and weighted shift operators. The spectrum, approximate point spectrum, and point spectrum of an

L
p

{L^p}

-composition operator have circular symmetry about 0, except on the unit circle, where they form unions of subgroups; certain consequences are derived from this.
On the Arens product and commutative Banach algebras
Authors: Wong Pak Ken
Published: June 2010

The purpose of this note is to generalize two recent results by the author for commutative Banach algebras. Let A be a commutative Banach algebra with carrier space

X
A

{X_A}

and

π
\pi

the canonical embedding of A into its second conjugate space

A

∗
∗

{A^{ \ast \ast }}

(with the Arens product). We show that if A is a semisimple annihilator algebra, then

π
(
A
)

\pi (A)

is a two-sided ideal of

A

∗
∗

{A^{ \ast \ast }}

. We also obtain that if A is a dense two-sided ideal of

C
0

(

X
A

)

{C_0}({X_A})

, then

π
(
A
)

\pi (A)

is a two-sided ideal of

A

∗
∗

{A^{ \ast \ast }}

if and only if A is a modular annihilator algebra.
An analytical criterion for the completeness of Riemannian manifolds
Authors: Gordon William B.
Published: June 2010

If M is a (not necessarily complete) riemannian manifold with metric tensor

g

i
j

{g_{ij}}

and f is any proper real valued function on M, then M is necessarily complete with respect to the metric
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g overTilde Subscript i j Baseline equals g Subscript i j Baseline plus left-parenthesis partial-differential f slash partial-differential x Superscript i Baseline right-parenthesis left-parenthesis partial-differential f slash partial-differential x Superscript j Baseline right-parenthesis">

g
~

i
j

=

g

i
j

+
(
∂
f

/

∂

x
i

)
(
∂
f

/

∂

x
j

)

{\tilde g_{ij}} = {g_{ij}} + (\partial f/\partial {x^i})(\partial f/\partial {x^j})

. Using this construction one can easily prove that a riemannian manifold is complete if and only if it supports a proper function whose gradient is bounded in modulus.
Compact-covering images of metric spaces
Authors: Michael E. , Nagami K.
Published: June 2010

Several kinds of compact-covering images of metric spaces are characterized.
The product of totally nonmeagre spaces
Authors: Aarts J. M. , Lutzer D. J.
Published: June 2010

In this note we give an example of a separable, pseudo-complete metric space

X
X

which is totally nonmeagre (= every closed subspace of

X
X

is a Baire space) and yet whose square

X
×
X

X \times X

not totally nonmeagre.
Injective compact distributive lattices
Authors: Choe Tae Ho
Published: June 2010

In the category of zero-dimensional compact distributive lattices and their continuous lattice-homomorphisms, it is firstly shown that the injective objects are precisely the compact Boolean algebras, and each object has an injective envelope. Secondly, it is shown that the category of compact distributive lattices and their continuous lattice-homomorphisms has no non-trivial injective object.
Finiteness conditions for matrix semigroups
Authors: Zalcstein Yechezkel
Published: June 2010

Classical theorems of Schur and Burnside on finiteness of matrix groups are generalized to regular and inverse matrix semigroups.
An 𝐻-space with finite dimensional homology whose loop space has torsion
Authors: Porter Richard D.
Published: June 2010

An example is given of a simply connected H-space whose integral homology is zero in dimensions greater than two and such that the integral homology of the loop space has torsion.
The Erdős-Rényi new law of large numbers for weighted sums
Authors: Book Stephen A.
Published: June 2010

From the partial sums

S
n

{S_n}

of the first

N
N

random variables of a sequence of independent, identically distributed random variables,

N
−
K
+
1

N - K + 1

averages of the form

K

−
1

(

S

n
+
K

−

S
n

)

{K^{ - 1}}({S_{n + K}} - {S_n})

can be constructed, one such average for each

n
n

between 0 and

N
−
K

N - K

, inclusive. If we denote by

Σ
(
N
,
K
)

\Sigma (N,K)

the maximum of those

N
−
K
+
1

N - K + 1

averages, then for a wide range of numbers

λ
\lambda

, Erdös and Rényi (1970) proved that, as

N
→
∞
,
Σ
(
N
,
K
)
→
λ

N \to \infty ,\Sigma (N,K) \to \lambda

a.e. for

K
=
[
C
(
λ
)
log
⁡
N
]

K = [C(\lambda )\log N]

, where

C
(
λ
)

C(\lambda )

is a constant depending only on

λ
\lambda

, not on

N
N

. The objective of the present article is to extend the Erdös-Rényi theorem to the case of weighted sums. The main theorem bears a relation to the law of large numbers for weighted sums of Jamison, Orey, and Pruitt (1965) similar to the one borne by the Erdös-Rényi theorem to the ordinary strong law of large numbers.
A note on solutions to the Wiener-Hopf equation with positive kernel
Authors: Mineka J.
Published: June 2010

For the Wiener-Hopf equation with positive kernel,

n
>
0

n > 0

an integer, either (1) there is a unique solution with bounded

n
n

th order differences, or (2) there is a unique solution with

n
n

th order differences approaching zero, or (3) there is no solution with bounded differences. Necessary and sufficient conditions for (1), (2) and (3) are formulated probabilistically.
On the inverse function theorem: A counterexample
Authors: Warren Hugh E.
Published: June 2010

The inverse function theorem for analytic functions has a generalization to

C
(
X
)

C(X)

which fails in the disk algebra.
A fixed point theorem for multifunctions in a locally convex space
Authors: Sehgal V. M. , Morrison Evelyn
Published: June 2010

Let

S
S

be a convex subset of a locally convex space

E
E

and

K
K

a compact subset of

S
S

. Let

f
:
S
→
F

f:S \to F

(a topological space) and

g
:
K
→
F

g:K \to F

be multifunctions. In this paper sufficient conditions are given for the existence of an

x
∈
K

x \in K

such that

f
(
x
)
∩
g
(
x
)
≠
∅

f(x) \cap g(x) \ne \emptyset

. The result generalizes a recent theorem of Himmelberg (J. Math. Anal. Appl. 38 (1972), 205-207) and in special cases extends some of the other well-known results.
Convergence properties of neighboring sequences
Authors: Sabella Ralph R.
Published: June 2010

First countable spaces and semimetrizable spaces are examples of topological spaces which can be characterized in terms of convergence properties of sequences “neighboring” a point. In this paper we consider conditions sufficient for metrizability of spaces defined in terms of convergence properties of “neighboring” sequences, in particular, those in which the set of cluster points of one sequence is a subset of that of any “neighboring” sequence. The special case in which the sets of cluster points are equal is shown to be a characterization of metrizability in

T
0

{T_0}

-spaces.
Absolutely structurally stable diffeomorphisms
Authors: Franks John
Published: June 2010

This paper gives a proof that if a diffeomorphism is structurally stable in a strong sense then it satisfies Axiom A of S. Smale. This provides a weakened converse of a theorem of J. Robbin on structural stability.
On the growth of the Taylor coefficients of automorphic forms
Authors: Metzger Thomas A.
Published: June 2010

The growth of the Taylor coefficients of an automorphic form of dimension -2 with respect to a Fuchsian group

Γ
\Gamma

is related to the area integral

∫

∫
U

|

F

|

s

(
1
−

|

z

|

2

)
t

d
x
d
y

\smallint {\smallint _U}|F{|^s}{(1 - |z{|^2})^t}dxdy

, and it is found that these coefficients must grow faster than a power of

n
n

. Moreover if

F
∈
H
(
p
,
Γ
)

F \in H(p,\Gamma )

then these coefficients must grow slower than a different power of

n
n

and, in fact,

a
n

/

n

{a_n}/n

is square summable if either

p
=
2

p = 2

or

1
>
p
>
∞

1 > p > \infty

and

Γ
\Gamma

is finitely generated of the second kind.
Zariski’s theorem on several linear systems
Authors: Ogus Arthur
Published: June 2010

We give a modern and fairly easy proof of (a slight improvement of) an important theorem of Zariski. The result gives conditions under which certain multigraded rings and modules associated with n linear systems are finitely generated, in a very strong sense.
Automorphism of von Neumann algebras as point transformations
Authors: Radin Charles
Published: June 2010

Given a concrete separable

C
∗

{C^ \ast }

-algebra

A

\mathfrak {A}

with unit and a faithful normal finite trace

τ
\tau

on

A

\mathfrak {A}

, we introduce a notion of

τ
\tau

-almost every state on

A

\mathfrak {A}

which has the proper relationship with Segal’s noncommutative integration theory. We then prove that any

∗
\ast

-automorphism of

A

\mathfrak {A}

is implemented by some point transformation in the state space of

A

\mathfrak {A}

, defined

τ
\tau

-almost everywhere. This generalizes the classical result of von Neumann-Maharam.
Generalized averaging operators and matrix summability
Authors: Atalla Robert E.
Published: June 2010

A bounded linear operator

T
T

on

C
(
X
)
,
X

C(X),X

compact, is a g.a.o. if it has associated with it a nonnegative projection

S
S

satisfying three conditions given below. An ordinary averaging operator is the case

T
=
S

T = S

. We show that if

T
T

is g.a.o., then the following problem has a fairly neat solution: What conditions on an operator

R
R

are necessary and sufficient for

kernel
⁡
(
T
)
⊂
kernel
⁡
(
R
)

\operatorname {kernel}(T) \subset \operatorname {kernel}(R)

? Application is made to the problem of the inclusion of one bounded convergence field in another, via the representation of regular matrices as linear operators on

C
(
β
N

/

N
)

C(\beta N/N)

.
Pro-nilpotent representation of homology types
Authors: Dror Emmanuel
Published: June 2010

The completion tower

R
n

X

{R_n}X

defined by Bousfield and Kan is shown to preserve the homology with

R
R

-coefficients. This property of preserving

R
R

-homology characterizes the tower completely [4].
Chebyshev estimates for Beurling generalized prime numbers
Authors: Diamond Harold G.
Published: June 2010

We consider a Beurling generalized number system for which the counting function of the integers satisfies
\[

N
(
x
)
=
c
x
+
O
(
x

log

−
γ

x
)

N(x) = cx + O(x{\log ^{ - \gamma }}x)

\]
for some positive

c
c

and

γ
\gamma

. Beurling showed that the prime number theorem must hold if

γ
>

3
2

\gamma > \tfrac {3}{2}

, but it can fail to hold if

γ
≦

3
2

\gamma \leqq \tfrac {3}{2}

. It was shown by Hall in the preceding article that the Chebyshev prime counting estimates
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than limit inf Underscript x right-arrow normal infinity Endscripts StartFraction psi left-parenthesis x right-parenthesis Over x EndFraction semicolon limit sup Underscript x right-arrow normal infinity Endscripts StartFraction psi left-parenthesis x right-parenthesis Over x EndFraction greater-than normal infinity">

0
>
lim

inf

x
→
∞

ψ
(
x
)

x

;

lim

sup

x
→
∞

ψ
(
x
)

x

>
∞

0 > \lim \inf \limits _{x \to \infty } \frac {{\psi (x)}}{x};\quad \lim \sup \limits _{x \to \infty } \frac {{\psi (x)}}{x} > \infty

\]
can fail if

γ
>
1

\gamma > 1

. Here we shall prove that these estimates hold for any system satisfying

γ
>
1

\gamma > 1

. The proof uses a convolution approximate inverse of the measure

d
N

dN

.
On uniform elementary estimates of arithmetic sums
Authors: Burr Stefan A.
Published: June 2010

A simple, elementary method is given for deriving estimates of sums of arithmetic functions, the estimates being in a certain sense uniform over a class of functions appearing in the summation. The method is particularly well suited to estimates needed in applications of Selberg’s sieve.
A group of exponent 4 with derived length at least 4
Authors: Bachmuth S. , Mochizuki H. Y. , Weston K.
Published: June 2010

There exists a group

G
G

of exponent 4 such that the third term of its derived series,

δ
3

(
G
)
=

G
m

{\delta _3}(G) = {G^m}

, is nontrivial. Furthermore, the third term of the derived series of

G
G

is not contained in the ninth term of its lower central series.
Odd dimensional manifolds with regular conjugate locus
Authors: Mercuri F.
Published: June 2010

We show that all odd dimensional manifolds, for which the first conjugate locus, with respect to some point, is regular, are homeomorphic to a sphere.
An alternate characterization of continuity
Authors: Smith Bruce D.
Published: June 2010

The notion of an almost continuous function has previously been defined in two different ways. This paper explores the relationship between these two notions, leading to an alternate characterization of continuity for real functions.
A theorem about the oscillation of sums of independent random variables
Authors: Hanson D. L. , Wright F. T.
Published: June 2010

Let

X
1

,

X
2

,
⋯

{X_1},{X_2}, \cdots

be i.i.d. random variables and let

S
n

=

X
1

+
⋯
+

X
n

{S_n} = {X_1} + \cdots + {X_n}

. The relationship between the tth moment of

X
1

{X_1}

and the convergence of the series

∑

n
=
1

∞

z
n

n

t
−
1

P
(

S
n

>
0
)

\sum \nolimits _{n = 1}^\infty {{z^n}{n^{t - 1}}P({S_n} > 0)}

is investigated in this paper. The convergence of the series above when

|

z

|

=
1

|z| = 1

but

z
≠
1

z \ne 1

is related to the oscillation of the sequence

{
P
(

S
n

>
0
)
}

\{ P({S_n} > 0)\}

and to the oscillation of the sequence

{

S
n

}

\{ {S_n}\}

about zero.
Compact semigroups with low dimensional orbit spaces
Authors: Hunter R. P.
Published: June 2010

If

S
S

is a compact connected semigroup,

G
G

a compact subgroup at the identity such that

S

/

G

S/G

is either two dimensional or embeddable in three space, then the normalizer conjecture holds.
On Borel measures and Baire’s class 3
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

. Let

Φ
\Phi

be the family of all real-valued functions, continuous

μ
\mu

-a.e. Let

B
α

(
Φ
)

{B_\alpha }(\Phi )

be the functions of Baire’s class

α
\alpha

generated by

Φ
\Phi

. It is shown that if

μ
\mu

is not a purely atomic measure whose set of atoms form a dispersed subset of

S
S

, then

B
2

(
Φ
)
≠

B

ω
1

(
Φ
)

{B_2}(\Phi ) \ne {B_{{\omega _1}}}(\Phi )

, where

ω
1

{\omega _1}

denotes the first uncountable ordinal.
Carathéodory distance and convexity with respect to bounded holomorphic functions
Authors: Chen Su Shing
Published: June 2010

A characterization of connected complex manifolds with complete Carathéodory distances is obtained. Some basic facts concerning convexity of complex manifolds with respect to their algebras of bounded holomorphic functions are also given.
Some totally real minimal surfaces in 𝐶𝑃²
Authors: Houh Chorng-shi
Published: June 2010

Totally real minimal surfaces with constant scalar normal curvature in

C

P
2

C{P^2}

are totally geodesic or nonpositive curved surfaces.
Schlais’ theorem extends to 𝜆 connected plane continua
Authors: Hagopian Charles L.
Published: June 2010

We call a nondegenerate metric space that is compact and connected a continuum. For each point

x
x

of a continuum

M
M

, F. B. Jones [2] defines

K
(
x
)

K(x)

to be the set consisting of all points

y
y

of

M
M

such that

M
M

is not aposyndetic at

x
x

with respect to

y
y

. H. E. Schlais [4] proved that if

M
M

is a hereditarily decomposable continuum, then for each point

x
x

of

M
M

, no nonempty open set in

M
M

is contained in

K
(
x
)

K(x)

. A continuum

M
M

is said to be

λ
\lambda

connected if any two of its points can be joined by a hereditarily decomposable continuum in

M
M

. Here we prove that if

M
M

is a

λ
\lambda

connected plane continuum, then for each point

x
x

of

M
M

, the set

K
(
x
)

K(x)

does not contain a nonempty open subset of

M
M

.
Vector bundles over finite 𝐶𝑊-complexes are algebraic
Authors: Lønsted Knud
Published: June 2010

It is proved that for any finite CW-complex

X
X

there exists a ring

A
A

of continuous functions on

X
X

, and natural 1-1 correspondences between the finitely generated projective

A
A

-modules (resp.

A

⊗
R

C

A{ \otimes _R}C

-modules), and the topological real vector bundles (resp. complex vector bundles) over

X
X

where

A
A

is Noetherian and has Krull-dimension equal to the topological dimension of

X
X

.
Symmetric homology over rings containing the rationals
Authors: Fleury Patrick J.
Published: June 2010

Let

R
R

be a commutative ring with unit which contains the rational numbers. Let

A
A

be a commutative

R
R

-algebra. In this paper we prove that the cotriple homology and cohomology modules of

R
R

for the symmetric algebra cotriple are the same as Harrison’s homology and cohomology modules.
On coefficients and zeros of sections of power series
Authors: Shaw J. K.
Published: June 2010

For a power series

f
(
z
)
=
∑

a
k

z
k

f(z) = \sum {a_k}{z^k}

let

s
n

{s_n}

denote the maximum modulus of the zeros of the

n
n

th partial sum

∑
0
n

a
k

z
k

\sum \nolimits _0^n {{a_k}} {z^k}

. Asymptotic bounds on the sequence

|

a
n

|

1

/

n

s
n

|{a_n}{|^{1/n}}{s_n}

are obtained for both entire functions and functions with finite radii of convergence. These extend the previous results of J. D. Buckholtz and J. K. Shaw. Finally, conjectures regarding best possible asymptotic bounds are stated.
On the zeros of certain confluent hypergeometric functions
Authors: Wynn P.
Published: June 2010

The theory of continued fractions is used to derive the following results which hold for

−

1
2

>
α
>
∞

- \tfrac {1}{2} > \alpha > \infty

: (1) If
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 1 Baseline upper F 1 left-parenthesis alpha semicolon 2 alpha plus 1 semicolon z right-parenthesis equals 0 left-bracket Subscript 1 Baseline upper F 1 left-parenthesis alpha plus 1 semicolon 2 alpha plus 1 semicolon z right-parenthesis equals 0 right-bracket">

1

F
1

(
α
;
2
α
+
1
;
z
)
=
0

[

1

F
1

(
α
+
1
;
2
α
+
1
;
z
)
=
0
]

_1{F_1}(\alpha ;2\alpha + 1;z) = 0\quad [{}_1{F_1}(\alpha + 1;2\alpha + 1;z) = 0]

\]
then

Re
⁡
(
z
)
>
0
[
>
0
]

\operatorname {Re} (z) > 0[ > 0]

. (It is deduced from this result that if

I

α

−
1

/

2

(
z
)
+

I

α

+
1

/

2

(
z
)
=
0

{I_{{\alpha ^{ - 1/2}}}}(z) + {I_{{\alpha ^{ + 1/2}}}}(z) = 0

then

Re
⁡
(
z
)
>
0

\operatorname {Re} (z) > 0

, and it is shown that if

α
\alpha

is an integer, an unbounded number of roots of this equation exists.) (2) The roots of the equ tions
\[

1

F
1

(
α
;
2
α
;
z
)

=
1

F
1

(
α
;
2
α
+
1
;
z
)

_1{F_1}(\alpha ;2\alpha ;z){ = _1}{F_1}(\alpha ;2\alpha + 1;z)

\]
and
\[

1

F
1

(
α
;
2
α
;
z
)

=
1

F
1

(
α
+
1
;
2
α
+
1
;
z
)

_1{F_1}(\alpha ;2\alpha ;z){ = _1}{F_1}(\alpha + 1;2\alpha + 1;z)

\]
are identical, pure imaginary, symmetrically distributed about the origin and unbounded in number. (3) Let

C
n

(
z
)
(
n
=
0
,
1
,
⋯
)

{C_n}(z)(n = 0,1, \cdots )

be the successive convergents of the continued fraction associated with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis z right-parenthesis equals Subscript 1 Baseline upper F 1 left-parenthesis alpha semicolon 2 alpha semicolon z right-parenthesis slash Subscript 1 Baseline upper F 1 left-parenthesis alpha semicolon 2 alpha plus 1 semicolon z right-parenthesis">

g
(
z
)

=
1

F
1

(
α
;
2
α
;
z
)

/

1

F
1

(
α
;
2
α
+
1
;
z
)

g(z){ = _1}{F_1}(\alpha ;2\alpha ;z){/_1}{F_1}(\alpha ;2\alpha + 1;z)

. Te oots

z
=
i

y
v

(
n
)

(
v
=
1
,
2
,
⋯
)

z = iy_v^{(n)}(v = 1,2, \cdots )

of the eqation

g
(
z
)
=

C
n

(
z
)

g(z) = {C_n}(z)

have the same properties as those described in (2). Furthermore, they interlace: subject to a suitable ordering,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Subscript v Superscript left-parenthesis n right-parenthesis Baseline greater-than y Subscript v Superscript left-parenthesis n plus 1 right-parenthesis Baseline greater-than y Subscript v plus 1 Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis v equals 1 comma 2 comma midline-horizontal-ellipsis right-parenthesis">

y
v

(
n
)

>

y
v

(
n
+
1
)

>

y

v
+
1

(
n
)

(
v
=
1
,
2
,
⋯
)

y_v^{(n)} > y_v^{(n + 1)} > y_{v + 1}^{(n)}(v = 1,2, \cdots )

. (A special case of this result concerns the function

e
z

{e^z}

and the convergents of its continued fraction expansion, and is an extension of the forula

e

2
i
v
π

=
1
(
v
=
1
,
2
,
⋯
)

{e^{2iv\pi }} = 1(v = 1,2, \cdots )

.)
Markuschevich bases and semireflexivity
Authors: Woods Paul Carlton
Published: June 2010

It is shown that a locally convex space E with a countable Markuschevich basis is semireflexive if each countable M-basis for E is shrinking or if each countable M-basis for E is boundedly complete.
Two ergodic theorems for convex combinations of commuting isometries
Authors: McGrath S. A.
Published: June 2010

Let

(
X
,

F

,
μ
)

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

be a measure space. In this paper we obtain

L
p

{L^p}

estimates for the supremum of the Cesàro averages of combinations of commuting isometries of

L
p

(
X
,

F

,
μ
)

{L^p}(X,\mathcal {F},\mu )

. In particular, we show that a convex combination of two invertible commuting isometries of

L
p

(
X
,

F

,
μ
)
,
p

{L^p}(X,\mathcal {F},\mu ),p

fixed,

1
>
p
>
∞
,
p
≠
2

1 > p > \infty ,p \ne 2

, admits of adominated estimate with constant

P

/

(
p
−
1
)

P/(p - 1)

. We also show that a convex combination of an arbitrary number of commuting positive invertible isometries of

L
2

(
X
,

F

,
μ
)

{L^2}(X,\mathcal {F},\mu )

admits of a dominated estimate with constant 2.
Inversion of normal operators by polynomial interpolation
Authors: Dunn J. C.
Published: June 2010

The inverse of a bounded normal linear operator on a Hilbert space is uniformly approximated by a sequence of Newton interpolation polynomials, provided the operator’s spectrum lies in either the right or left open half-planes.
Coverings of infinite-dimensional spheres
Authors: Cutler William H.
Published: June 2010

Let

F
F

be a normed linear space such that the countable infinite product of

F
F

is homeomorphic to a normed linear space. (This is true for all Hilbert spaces, for example.) Let

S
(
F
)

S(F)

denote the unit sphere in

F
F

. We prove the following Theorem 1. There is a countable cover of

S
(
F
)

S(F)

of open sets each of which contains no pair of antipodal points. Theorem 2. There is a countable collection of closed sets in

S
(
F
)

S(F)

the union of which contains exactly one member of each pair of antipodal points. Theorem 3. Let

F
F

be a Hilbert space. Then there is a countable collection of sets which cover

S
(
F
)

S(F)

and whose diameters are less than 2.
Some pathology involving pseudo 𝑙-groups as groups of divisibility
Authors: Martinez Jorge
Published: June 2010

In a partially ordered abelian group

G
G

, two elements

a
a

and

b
b

are pseudo-disjoint if

a
,
b
≧
0

a,b \geqq 0

and either one is zero, or both are strictly positive and each

o
o

-ideal which is maximal with respect to not containing

a
a

contains

b
b

, and vice versa.

G
G

is a pseudo lattice-group if every element of

G
G

can be written as a difference of pseudo-disjoint elements. We prove the following theorem: suppose

G
G

is an abelian pseudo lattice-group; if there is an

x
>
0

x > 0

and a finite set of pairwise pseudo-disjoint elements

x
1

,

x
2

,
⋯
,

x
k

{x_1},{x_2}, \cdots ,{x_k}

all of which exceed

x
x

, and in addition this set is maximal with respect to the above properties, then

G
G

is not a group of divisibility. The main consequence of this result is that every so-called “

v
v

-group”

V
(
Λ
,

R
λ

)

V(\Lambda ,{R_\lambda })

for a given partially ordered set

Λ
\Lambda

, and where

R
λ

{R_\lambda }

is a subgroup of the additive reals in their usual order, is a group of divisibility only if

Λ
\Lambda

is a root system, and hence

V
(
Λ
,

R
λ

)

V(\Lambda ,{R_\lambda })

is a lattice-ordered group. We do give examples of pseudo lattice-groups which are not lattice-groups, and yet are groups of divisibility. Finally, we compute for each integral domain

D
D

whose group of divisibility is a lattice-group, the group of divisibility of the polynomial ring

D
[
x
]

D[x]

in one variable.
Minimal primes of ideals and integral ring extensions
Authors: Heinzer William
Published: June 2010

It is shown that if

R
R

is a commutative ring with identity having the property that ideals in

R
R

have only a finite number of minimal primes, then a finite

R
R

-algebra again has this property. It is also shown that an almost finite integral extension of a noetherian integral domain has noetherian prime spectrum.
Planar Fourier transforms and Diophantine approximation
Authors: Kaufman R.
Published: June 2010

The radial behavior of its Fourier-Stieltjes transform in

R
2

{R^2}

is related to the modulus of continuity of a measure; in certain cases the Hausdorff dimension of an exceptional set of lines can be estimated. Converse results use the theory of Diophantine approximation established by Besicovitch and Jarník.
On a nonlinear stochastic integral equation of the Hammerstein type
Authors: Padgett W. J.
Published: June 2010

A nonlinear stochastic integral equation of the Hammerstein type in the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis t semicolon omega right-parenthesis equals h left-parenthesis t semicolon omega right-parenthesis plus integral Underscript s Endscripts k left-parenthesis t comma s semicolon omega right-parenthesis f left-parenthesis s comma x left-parenthesis s semicolon omega right-parenthesis right-parenthesis d mu left-parenthesis s right-parenthesis">

x
(
t
;
ω
)
=
h
(
t
;
ω
)
+

∫
s

k
(
t
,
s
;
ω
)
f
(
s
,
x
(
s
;
ω
)

)
d
μ
(
s
)

x(t;\omega ) = h(t;\omega ) + \int _s {k(t,s;\omega )f(s,x(s;\omega )} )d\mu (s)

\]
is studied where

t
∈
S
,
a

t \in S,a

, a

σ
\sigma

-finite measure space with certain properties,

ω
∈
Ω

\omega \in \Omega

, the supporting set of a probability measure space

(
Ω
,
A
,
P
)

(\Omega ,A,P)

, and the integral is a Bochner integral. A random solution of the equation is defined to be a second order vector-valued stochastic process

x
(
t
;
ω
)

x(t;\omega )

on

S
S

which satisfies the equation almost certainly. Using certain spaces of functions, which are spaces of second order vector-valued stochastic processes on

S
S

, and fixed point theory, several theorems are proved which give conditions such that a unique random solution exists.
The solution of an integral equation
Authors: Nasim C.
Published: June 2010

Various methods are developed to solve the integral equation

f
(
x
)
=

∫
0
∞

g
(
t
)

k
(
x
t
)
d
t

f(x) = \int _0^\infty {g(t)} k(xt)dt

, when the Mellin transform

K
(
s
)

K(s)

of the kernel function

k
(
x
)

k(x)

is decomposable. Each method corresponds to the way

K
(
s
)

K(s)

is decomposed: Namely (i)

K
(
s
)
=
1

/

L
(
1
−
s
)
M
(
1
−
s
)

K(s) = 1/L(1 - s)M(1 - s)

, (ii)

K
(
s
)
=
H
(
s
)

/

M
(
1
−
s
)

K(s) = H(s)/M(1 - s)

and (iii)

K
(
s
)
=
N
(
s
)
H
(
s
)

K(s) = N(s)H(s)

, where

L
,
M
,
N

L,M,N

and

H
H

are arbitrary functions of the complex variable

s
s

. Numerous special cases and examples are given to illustrate the technique and the advantage of these methods.
A generalization of the Banach-Stone theorem
Authors: Cengiz Bahattin
Published: June 2010

In this paper the following generalization of the Banach-Stone theorem is proved: If

ϕ
\phi

is a linear isomorphism of an extremely regular linear subspace of

C
0

(
X
)

{C_0}(X)

onto such a subspace of

C
0

(
Y
)

{C_0}(Y)

with

|

|

ϕ

|

|

|

|

ϕ

−
1

|

|

>
2

||\phi ||\;||{\phi ^{ - 1}}|| > 2

then

X
X

and

Y
Y

are homeomorphic.
A characterization of 2-dimensional spherical space
Authors: Loveland L. D. , Valentine J. E.
Published: June 2010

The midset of two distinct points

a
a

and

b
b

of a metric space is defined as the set of all points

x
x

in the space for which the distances

a
x

ax

and

b
x

bx

are equal. A metric space is said to have the

1
1

-WLMP if the midset of each two distinct points is a convex

1
1

-sphere having the property that each nonmaximal (with respect to inclusion) segment intersecting it twice lies in it. We show that a nondegenerate compact space

X
X

is isometric to a

2
2

-dimensional spherical space

S

2
,
α

{S_{2,\alpha }}

(a

2
2

-dimensional sphere of radius

α
\alpha

in euclidean

3
3

-space with the “shorter arc” metric) if and only if

X
X

has a metric with the

1
1

-WLMP.
A topological proof of a theorem of Kneser
Authors: Friberg Bjorn
Published: June 2010

We give an elementary topological proof that the orthogonal groups

O
(
2
)

O(2)

and

O
(
3
)

O(3)

are strong deformation retracts of the space of homeomorphisms (with the compact-open topology) of

R
2

{R^2}

and

S
2

{S^2}

, respectively. We also deform the space of bounded homeomorphisms of

R
2

{R^2}

to

S
1

{S^1}

.
A characteristic extremal property of simplices
Authors: Schneider Rolf
Published: June 2010

Among all

d
d

-dimensional convex bodies the simplices are characterized by the property that the volume of every circumscribed cylinder is not less than

d
d

times the volume of the body itself.
Certain subsets of products of 𝜃-refinable spaces are realcompact
Authors: Zenor Phillip
Published: June 2010

It is shown that the normal

T
1

{T_1}

-space

X
X

is realcompact if and only if (a) each discrete subset of

X
X

is realcompact and (b)

X
X

can be embedded as a closed subset in the product of a collection of regular

θ
\theta

-refinable spaces.
An exact sequence calculation for the second homotopy of a knot. II
Authors: Gutiérrez M. A.
Published: June 2010

This paper deals with the computation of the second homotopy of a knot as a module over its fundamental group.
An imbedding theorem for separable algebras
Authors: Nagahara Takasi
Published: June 2010

Let

S

/

R

S/R

be a ring extension, where

S
S

is a commutative ring. If

S

/

R

S/R

is strongly separable then it can be imbedded in a weakly Galois extension of

R
R

in the sense of [7, Definition 3.1].
The min cone over the circle group
Authors: L’Heureux J. E.
Published: June 2010

It is shown that the min cone over the circle group is the only inverse semigroup or Clifford semigroup on the

2
2

-cell with an identity whose set of idempotents has a cut point.
Zero sets of functions from non-quasi-analytic classes
Authors: Darst R. B.
Published: June 2010

Any closed subset

E
E

of the real numbers

R
R

is the zero set of some

C
∞

{C^\infty }

-function

f
f

. One can also specify the order

d
(
s
)

d(s)

of the zero of

f
f

at each element

s
s

of the set

S
S

of isolated points of

E
E

. The present note improves this result by showing that each non-quasi-analytic class

C
{

M
n

}

C\{ {M_n}\}

contains such functions.
Some regularity theorems for typically real functions
Authors: Leeman George B.
Published: June 2010

This paper studies the class of typically real functions and the subclass of typically real functions with fixed second coefficient. For each of these classes the author proves a few regularity theorems, which describe the coefficients’ asymptotic behavior.
Admissible exponential representations and topological indices for functions of bounded variation
Authors: Wright F. M. , Ling J. N.
Published: June 2010

In this paper we first prove a theorem concerning the composition

η
\eta

of an analytic complex-valued function

g
g

in a region of the complex plane with a continuous complex-valued function

ϕ
\phi

of bounded variation on the closed interval

[
a
,
b
]

[a,b]

of the real axis. We then relate this theorem to admissible exponential representations and topological indices introduced by Whyburn in his book Topological analysis. We also show how this theorem can be used to prove a result of interest in the study of the argument principle. Moreover, we look at the situation where

ϕ
\phi

is a complex-valued function of bounded variation but not necessarily continuous on a closed interval

[
a
,
b
]

[a,b]

of the real axis,

p
p

is a complex number not in the range of

ϕ
\phi

, and

u
u

is a complex-valued function on

[
a
,
b
]

[a,b]

such that

e

u
(
t
)

=
[
ϕ
(
t
)
−
p
]

{e^{u(t)}} = [\phi (t) - p]

for

t
t

in

[
a
,
b
]

[a,b]

. We present conditions for

u
u

to be of bounded variation on

[
a
,
b
]

[a,b]

.
The equation 𝐿(𝐸,𝑋**)=𝐿(𝐸,𝑋)** and the principle of local reflexivity
Authors: Dean David W.
Published: June 2010

A new derivation of the equation

L
(
E
,

X

∗
∗

)
=
L

(
E
,
X

)

∗
∗

L(E,{X^{ \ast \ast }}) = L{(E,X)^{ \ast \ast }}

is given, for

dim
⁡
(
E
)
>
∞

\dim (E) > \infty

and

X
X

a Banach space. From this equation the principle of local reflexivity is derived.
Bounded approximate units and bounded approximate identities
Authors: Wichmann Josef
Published: June 2010

In this paper we establish the equivalence of the general concept of bounded approximate units in a normed algebra with the traditionally used notion of a bounded approximate identity. Furthermore, we investigate pointwise-bounded approximate units in commutative normed algebras.
Compact 2-manifolds as maximal ideal spaces
Authors: Brandstein Alfred G.
Published: June 2010

It is shown that every compact

2
2

-manifold is (homeomorphic to) the maximal ideal space of an antisymmetric algebra which is Dirichlet on its Šilov boundary.
A global invariant of conformal mappings in space
Authors: White James H.
Published: June 2010

This paper shows that the total integral of the square of the mean curvature for a compact orientable surface in

E
3

{E^3}

is an invariant of a conformal space mapping. This result is then used to answer a problem raised by T. Willmore and B.-Y. Chen concerning embeddings of compact orientable surfaces, and in particular tori, for which this integral is a minimum.
Regular matrices and 𝑃-sets in 𝛽𝑁\𝑁
Authors: Atalla R. E.
Published: June 2010

A P-set is a closed set which is interior to any zero set (closed

G
δ

{G_\delta }

) which contains it. Henriksen and Isbell showed that the ’support set’ in

β
N
∖
N

\beta N\backslash N

of a nonnegative regular matrix is a P-set. We show that each such support set contains a family of

2
c

{2^c}

pairwise disjoint perfect nowhere dense P-sets, so that not every P-set comes from a matrix. Moreover, each of the P-sets produced is the support of a Borel probability measure on

β
N
∖
N

\beta N\backslash N

.
Terms in cylindric algebras
Authors: Pinter Charles
Published: June 2010

A new algebraic treatment of terms within the framework of cylindric algebras.
Boundedness properties for semigroups of operators
Authors: Terkelsen Frode
Published: June 2010

The note contains a characterization of an equicontinuity property for semigroups in metrizable spaces, and a result on essential boundedness for weakly measurable semigroups in metrizable locally convex spaces.
On the image of symplectic cobordism in unoriented cobordism
Authors: Roush Fred W.
Published: June 2010

It is shown in this paper that all 16th powers of unoriented cobordism classes can be represented by stably almost symplectic manifolds. The generators are coefficients in the expansion of a Conner-Floyd characteristic class of the symplectification of a tensor product of symplectic line bundles.
On hyperfinite 𝑊-* algebras
Authors: Willig Paul
Published: June 2010

If

A

\mathcal {A}

is a

W

−
∗

W{ - ^\ast }

algebra on separable Hilbert space

H
H

, and if

A

(
λ
)

\mathcal {A}(\lambda )

are the factors in the direct integral decomposition of

A

\mathcal {A}

, then

J

=
{
λ

|

A

(
λ
)

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

is hyperfinite} is

μ
\mu

-measurable, and

A

\mathcal {A}

is hyperfinite if and only if

A

(
λ
)

\mathcal {A}(\lambda )

is hyperfinite

μ
\mu

-a.e.
A quadratic eigenvalue problem
Authors: Greenlee W. M.
Published: June 2010

Let

P
,
Q

P,Q

be compact selfadjoint operators in a Hilbert space. It is proven that the characteristic and associated vectors of the quadratic eigenvalue problem,

x
=
λ
P
x
+
(
1

/

λ
)
Q
x

x = \lambda Px + (1/\lambda )Qx

, form a Riesz basis for the cartesian product of the closure of the range of

P
P

and the closure of the range of

Q
Q

.
Lie algebra representations of dimension 𝑝-1
Authors: Strade Helmut
Published: June 2010

A semisimple Lie algebra over an algebraically closed field of characteristic

p
>
2

p > 2

admitting a faithful representation of dimension

p
−
1

p - 1

is either a direct sum of classical algebras or the Witt algebra.
Compactification and completion as absolute closure
Authors: Hager Anthony W.
Published: June 2010

It is well known that a Tychonoff (respectively, Hausdorff uniform) space is compact (resp., complete) iff it is absolutely closed, i.e., dense in no other such space; we shall sketch a proof of this to our purpose, which is: Given

X
X

, we apply Zorn’s Lemma to obtain a space maximal with respect to the property of containing

X
X

densely, thus compact or complete. For uniform spaces, the “maximal extension” is automatically the completion; for Tychonoff spaces, we must, and do, explicitly arrange things so that the “maximal extension” has the universal mapping property describing the Stone-Čech compactification. A variant of the construction yields the Hewitt realcompactification. A crucial point in the proofs is (of course) the exhibition of an upper bound for a chain; this is, in essence, a direct limit construction.
Real homogeneous algebras
Authors: Djoković D. Ž.
Published: June 2010

Let

(
A
,
μ
)

(A,\mu )

be a finite dimensional real algebra (not necessarily associative) with multiplication

μ
≠
0

\mu \ne 0

. Assuming that

Aut
⁡
(
A
)

\operatorname {Aut}(A)

is transitive on one-dimensional subspaces we determine all such algebras. There are up to isomorphism only four such algebras, one in each of the dimensions 1, 3, 6, 7.
Sums of distances between points on a sphere. II
Authors: Stolarsky Kenneth B.
Published: June 2010

Given

N
N

points on a unit sphere in Euclidean

m
m

space,

m
≧
2

m \geqq 2

, we show that the sum of all distances which they determine plus their discrepancy is a constant. As applications we obtain (i) an upper bound for the sum of the distances which for

m
≧
5

m \geqq 5

is smaller than any previously known and (ii) the existence of

N
N

point distributions with small discrepancy. We make use of W. M. Schmidt’s work on the discrepancy of spherical caps.
Tensor products and joint numerical range
Authors: Dash A. T.
Published: June 2010

It is shown that the joint numerical range of the tensor product of several operators is the cartesian product of their numerical ranges.
Dimension theory
Authors: Zaks Abraham
Published: June 2010

The

T
T

-equivalence and

T
T

-iterations introduced for a bifunctor

T
T

enable one to deduce a “shifting property” from which one gets the “long exact sequence of homology.” Also, the appropriate lemma of Schanuel is proved, from which one can develop a

T
T

-dimension theory. These notions are useful in proving known duality homomorphisms and may serve to get some new ones. For one purpose, they unify the methods of studying the various common dimensions.
On retraceable sets with rapid growth
Authors: McLaughlin T. G.
Published: June 2010

We combine a refinement of a recent theorem of A. N. Degtev with a result of our own, in order to derive a general theorem about regressive sets which has the following Corollary. If

A
A

is any point-decomposable

π
1
0

\pi _1^0

set then

A
A

has an infinite

π
1
0

\pi _1^0

subset

B
B

such that

B
B

has “highly” dense-simple complement and, moreover, all infinite

π
1
0

\pi _1^0

subsets of

B
B

are effectively decomposable in a strong sense (namely, they are all retraceable).
On approximation in the Bers spaces
Authors: Chui Charles K.
Published: June 2010

Let

D
D

be a Jordan domain in the complex plane with rectifiable boundary

C
C

. Let

A
q

(
D
)

{A_q}(D)

denote the Bers space with norm

|

|

|

|

q

||\;|{|_q}

. We prove that if

f
∈

A
q

(
D
)
,
2
>
q
>
∞

f \in {A_q}(D),2 > q > \infty

, then there exist functions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s Subscript n Baseline left-parenthesis z right-parenthesis equals normal upper Sigma Subscript k equals 1 Superscript n Baseline 1 slash left-parenthesis z minus z Subscript n comma k Baseline right-parenthesis comma z Subscript n comma k Baseline element-of upper C for k equals 1 comma midline-horizontal-ellipsis comma n">

s
n

(
z
)
=

Σ

k
=
1

n

1

/

(
z
−

z

n
,
k

)
,

z

n
,
k

∈
C

 for 

k
=
1
,
⋯
,
n

{s_n}(z) = \Sigma _{k = 1}^n1/(z - {z_{n,k}}),\;{z_{n,k}} \in C{\text { for }}k = 1, \cdots ,n

, such that

|

|

s
n

−
f

|

|

q

→
0

||{s_n} - f|{|_q} \to 0

. This result does not hold for

1
>
q
≦
2

1 > q \leqq 2

even when

D
D

is a disc.
The strict topology for 𝑃-spaces
Authors: Wheeler Robert F.
Published: June 2010

A

P
P

-space is a completely regular Hausdorff space

X
X

in which every

G
δ

{G_\delta }

is open. It is shown that the generalized strict topologies

β
\beta

and

β
0

{\beta _0}

coincide on

C
∗

(
X
)

{C^\ast }(X)

, and that strong measuretheoretic properties hold; in particular,

(

C
∗

(
X
)
,
β
)

({C^\ast }(X),\beta )

is always a strong Mackey space. As an application, an example is constructed of a non-quasi-complete locally convex space in which closed totally bounded sets are compact.
A generalization of Tietze’s theorem on convex sets in 𝑅³
Authors: Stavrakas Nick M.
Published: June 2010

Let

S
⊂

R
3

S \subset {R^3}

and let

C
(
S
)

C(S)

denote the points of local convexity of

S
S

. One interesting result which is proven is Theorem. Let

S
⊂

R
3

S \subset {R^3}

be such that

S
⊂
cl
⁡
(
C
(
S
)
)
,
S

S \subset \operatorname {cl} (C(S)),S

not planar and

C
(
S
)

C(S)

is connected. Then

S
⊂
cl
⁡
(
int
⁡
S
)

S \subset \operatorname {cl} (\operatorname {int} S)

.
Projectivity of the Whitehead square
Authors: Randall Duane
Published: June 2010

We show that the Whitehead square of the generator of

π
n

(

S
n

)

{\pi _n}({S^n})

is not a projective homotopy class for any integer

n
n

with neither

n
n

or

n
+
1

n + 1

being a power of 2.