bluuebunny/crossref_metadata_2025 huggingface dataset:
Generating functions for some classes of univalent functions
Authors: Lewandowski Zdzisław , Miller Sanford , Złotkiewicz Eligiusz
Published: June 2010

Let

P
(
z
)
=

e

i
β

+

p
1

z
+

p
2

z
2

+
⋯

P(z) = {e^{i\beta }} + {p_1}z + {p_2}{z^2} + \cdots

be regular in the unit disc

Δ
\Delta

with

|

β

|

>
π

/

2

|\beta | > \pi /2

, and let

ψ
(
u
,
v
)

\psi (u,v)

be a continuous function defined in a domain of

C

×

C

{\mathbf {C}} \times {\mathbf {C}}

. With some very simple restrictions on

ψ
(
u
,
v
)

\psi (u,v)

the authors prove a lemma that

Re
⁡
ψ
(
p
(
z
)
,
z

p
′

(
z
)
)
>
0

\operatorname {Re} \psi (p(z),zp’(z)) > 0

implies

Re
⁡
p
(
z
)
>
0

\operatorname {Re} p(z) > 0

. This result is then used to generate subclasses of starlike, spirallike and close-to-convex functions.
A characterization of 𝐹⁺∩𝑁
Authors: Stoll M.
Published: June 2010

In this note we give a characterization of

F
+

∩
N

{F^ + } \cap N

, where

N
N

denotes the Nevanlinna class of functions of bounded characteristic and

F
+

{F^ + }

denotes the containing Fréchet space of

N
+

{N^ + }

. We show that a holomorphic function

f
∈

F
+

∩
N

f \in {F^ + } \cap N

if and only if

f
(
z
)
=
g
(
z
)

/

S
μ

(
z
)

f(z) = g(z)/{S_\mu }(z)

, where

g
∈

N
+

g \in {N^ + }

and

S
μ

{S_\mu }

is a singular inner function with respect to a nonnegative continuous singular measure

μ
\mu

.
Perturbations of limit-circle expressions
Authors: Read Thomas T.
Published: June 2010

It is shown that for any limit-circle expression

L
(
y
)
=

Σ

j
=
0

n

p
j

y

(
j
)

L(y) = \Sigma _{j = 0}^n {{p_j}{y^{(j)}}}

, any sequence of disjoint intervals

{
[

a
k

,

b
k

]

}

k
=
1

∞

\{ [{a_k},{b_k}]\} _{k = 1}^\infty

such that

a
k

→
∞

{a_k} \to \infty

as

k
→
∞

k \to \infty

, and any

i
⩽
n
−
1

i \leqslant n - 1

, there is an expression

M
(
y
)
=

Σ

j
=
0

n

q
j

y

(
j
)

M(y) = \Sigma _{j = 0}^n {{q_j}{y^{(j)}}}

such that

q
i

=

p
i

{q_i} = {p_i}

except on

∪
(

a
k

,

b
k

)
,

q
j

=

p
j

\cup ({a_k},{b_k}),{q_j} = {p_j}

for all

j
≠
i

j \ne i

, and such that

M
M

is not limit-circle.
On exposed points of the range of a vector measure. II
Authors: Anantharaman R.
Published: June 2010

If a weakly compact convex set

K
K

in a real Banach space

X
X

is strongly exposed by a dense set of functionals in

X
′

X’

, it is proved that the functionals which expose

K
K

form a residual set in

X
′

X’

. If

v
:

A

→
X

v:\mathcal {A} \to X

is a measure, it follows that the set of exposing functionals of its range is a residual

G
δ

{G_\delta }

in

X
′

X’

. This, in turn, is found to be equivalent to a theorem of B. Walsh on the residuality of functionals

x
′

∈

X
′

x’ \in X’

for which

x
′

∘
v
≡
v

x’ \circ v \equiv v

. If the set of exposed points of

v
(

A

)

v(\mathcal {A})

is weakly closed and

v
A

{v_A}

is the restriction of

v
v

to any set

A
∈

A

A \in \mathcal {A}

, it is further proved that every exposed point of the range of

v
A

{v_A}

is of the form

v
(
A
∩
E
)

v(A \cap E)

, where

E
∈

A

E \in \mathcal {A}

and

v
(
E
)

v(E)

is an exposed point of

v
(

A

)

v(\mathcal {A})

.
Perturbations of groups of automorphisms of von Neumann algebras
Authors: Reynolds M.
Published: June 2010

We show that if two uniformly continuous representations of a connected abelian group as

∗

{\ast }

-automorphisms of a von Neumann algebra are close in norm, then they are conjugate via a single automorphism close to the identity automorphism.
Positive operators on a Banach space and the Fejér-Riesz theorem
Authors: Miamee A. G. , Salehi H.
Published: June 2010

The Fejér and Riesz theorem on the factorization of nonnegative trigonometric polynomials is extended to the nonnegative operator valued trigonometric polynomials on a Banach space. The work is based on the analysis of quasi square roots of nonnegative operator valued functions on a Banach space.
A characterization of 𝐶^{∞}-sufficient 𝑘-jets
Authors: Kucharz Wojciech
Published: June 2010

We improve some results of Mather and Arnold and find several necessary and sufficient conditions of sufficiency of

k
k

-jets. As a corollary we prove that the set of

C
∞

{C^\infty }

-sufficient

k
k

-jets is a semialgebraic subset of the space of

k
k

-jets of

C
∞

{C^\infty }

mappings

F
:
(

R
n

,
0
)
→
(
R
,
0
)

F:({R^n},0) \to (R,0)

.
The structure of regulated functions
Authors: Goffman Casper , Moran Gadi , Waterman Daniel
Published: June 2010

It is shown that there is a nontrivial class of regulated functions each of which is a representable as the sum of a continuous function and a uniformly convergent series of jump functions whose jumps are those of the given function. The set of regulated functions is the union of the classes of functions of bounded

Φ
\Phi

-variation for convex

Φ
\Phi

.
Some inequalities for polynomials
Authors: Rahman Q. I.
Published: June 2010

Let

p
n

(
z
)

{p_n}(z)

be a polynomial of degree

n
n

. Given that

p
n

(
z
)

{p_n}(z)

has a zero on the circle

|

z

|

=
ρ
(
0
>
ρ
>
∞
)

|z| = \rho (0 > \rho > \infty )

we estimate

max

|

z

|

=
R
>
1

|

p
n

(
z
)

|

{\max _{|z| = R > 1}}|{p_n}(z)|

in terms of

max

|

z

|

=
1

|

p
n

(
z
)

|

{\max _{|z| = 1}}|{p_n}(z)|

. We also consider some other related problems.
On univalent polynomials
Authors: Quine J. R.
Published: June 2010

We define

V
n

⊆

C

n
−
1

{V_n} \subseteq {{\mathbf {C}}^{n - 1}}

to be the set of

(
n
−
1
)

(n - 1)

-tuples

(

a
2

,
…
,

a
n

)

({a_2}, \ldots ,{a_n})

such that the polynomial

p
(
z
)
=
z
+

a
2

z
2

+
⋯
+

a
n

z
n

p(z) = z + {a_2}{z^2} + \cdots + {a_n}{z^n}

is univalent, i.e., one-to-one in

|

z

|

>
1

|z| > 1

. In this paper we construct a real polynomial

h
h

of degree

4
(
2

(
n
−
1

)
2

−
1
)
(
n
−
1
)

4(2{(n - 1)^2} - 1)(n - 1)

such that if

(

a
2

,
…
,

a
n

)

({a_2}, \ldots ,{a_n})

is in the boundary of

V
n

{V_n}

then

h
(
Re
⁡

a
2

,
Im
⁡

a
2

,
…
,
Re
⁡

a
n

,
Im
⁡

a
n

)
=
0

h(\operatorname {Re} {a_2},\operatorname {Im} {a_2}, \ldots ,\operatorname {Re} {a_n},\operatorname {Im} {a_n}) = 0

. This shows that the boundary of

V
n

{V_n}

is a subset of an algebraic submanifold of

R

2
(
n
−
1
)

{R^{2(n - 1)}}

.
Holomorphic mappings between balls and polydiscs
Authors: Simha R. R.
Published: June 2010

A proof based only on Schwarz’s lemma is given for the fact that the polydisc and the ball are not holomorphically equivalent.
Nonatomic Banach lattices can have 𝑙₁ as a dual space
Authors: Lacey E. , Wojtaszczyk P.
Published: June 2010

Examples of nonatomic

M
M

spaces whose duals are

l
1

{l_1}

are constructed.
A geometric inequality for plane curves with restricted curvature
Authors: Chakerian G. D. , Johnson H. H. , Vogt A.
Published: June 2010

A geometric proof is given that a closed plane curve of length

L
L

and curvature bounded by

K
K

can be contained inside a circle of radius

L

/

4
−
(
π
−
2
)

/

2
K

L/4 - (\pi - 2)/2K

.
A note on the ergodic properties of homeomorphisms
Authors: Sine Robert
Published: June 2010

Powers of strongly ergodic transformations need not be strongly ergodic.
Duality and Eilenberg-Mac Lane spectra
Authors: Lin T. Y.
Published: June 2010

Stable cohomotopy groups of Eilenberg-Mac Lane spectra of finite groups are shown to be trivial. This implies that the stable homotopy category, which is large enough to represent ordinary cohomology theory, cannot be self-dual. It can also be interpreted as an evidence to support Freyd’s generating hypothesis and a proof of a stable version of a conjecture of D. Sullivan.
Some simple examples of symplectic manifolds
Authors: Thurston W. P.
Published: June 2010

This is a construction of closed symplectic manifolds with no Kaehler structure.
Join-principal element lattices
Authors: Johnson E. W.
Published: June 2010

Let

(

L

,
M
)

(\mathfrak {L},M)

be a local Noether lattice. If the maximal element

M
M

is meet principal, it is well known and easily seen that every element of

L

\mathfrak {L}

is meet principal. In this note, we obtain the corresponding result for

M
M

join-principal. We also consider join-principal elements generally under the assumption of the weak union condition and show, for example, that the square of a join-principal element is principal.
Monic and monic free ideals in a polynomial semiring
Authors: Dale Louis
Published: June 2010

Two classes of ideals are introduced in a polynomial semiring

S
[
x
]

S[x]

, where

S
S

is a commutative semiring with an identity. A structure theorem is given for each class.
Uniform structure in topological groups
Authors: Itzkowitz Gerald
Published: June 2010

A characterization is given of those locally compact groups having equivalent right and left uniform structures. It is shown that

α
\alpha

-compact locally compact groups have equivalent uniformities iff for each right uniformly discrete set

B
B

such that card

(
B
)
≤
α

(B) \leq \alpha

and each neighborhood

U
U

of

e
,

∩

x
∈
B

x
U

x

−
1

e,{ \cap _{x \in B}}xU{x^{ - 1}}

is a neighborhood of

e
e

. It is also shown that a group is not unimodular iff it contains an open

σ
\sigma

-compact group which is not unimodular.
Decomposition of 𝐶^{∞} intertwining operators for Lie groups
Authors: Penney R.
Published: June 2010

Let

U
U

be a unitary representation of a Lie group

G
G

in a Hilbert space

K

\mathcal {K}

and let

C
∞

(
U
)

{C^\infty }(U)

denote the space of differentiable vectors for

U
U

given its usual topology. A continuous operator on

C
∞

(
U
)

{C^\infty }(U)

is said to be a

C
∞

{C^\infty }

intertwining operator for

U
U

if it commutes with

U
U

. It is shown that if one decomposes

U
U

via a central decomposition into a direct integral of unitary representations, then every

C
∞

{C^\infty }

intertwining operator decomposes into a direct integral of unique

C
∞

{C^\infty }

intertwining operators. Furthermore, it is shown that if

U
U

is type I and primary, then every

C
∞

{C^\infty }

intertwining operator extends to unique bounded (in the sense of

K

\mathcal {K}

) intertwining operator defined on all of

K

\mathcal {K}

.
A sharp condition for univalence in Euclidean spaces
Authors: Gevirtz Julian
Published: June 2010

Let

B
⊂

K
k

B \subset {K^k}

be a ball. It is shown that if

f
:
B
→

E
k

f:B \to {E^k}

is a local homeomorphism for which the infinitesimal change in length is bounded above by

M
M

and for which the infinitesimal change in volume is bounded below by

m
k

{m^k}

, where

M

/

m
≤

2

1
∖
k

M/m \leq {2^{1\backslash k}}

, then

f
f

is univalent. This result is numerically sharp.
Central approximate units in a certain ideal of 𝐿¹(𝐺)
Authors: Kotzmann Ernst , Rindler Harald
Published: June 2010

In this paper we show that for a locally compact group

G
G

the ideal

L
0

(
G
)
=
{
f

|

f
∈

L
1

(
G
)
,
∫
f
=
0
}

{L^0}(G) = \{ f|f \in {L^1}(G),\smallint f = 0\}

of

L
1

(
G
)

{L^1}(G)

has multiple approximate units belonging to the center of

L
0

(
G
)

{L^0}(G)

iff

G
G

has a basis of invariant neighbourhoods of 1 and if all conjugacy classes of

G
G

are precompact, or, equivalently, the group of inner automorphisms is precompact in the group of all topological automorphisms. In a sense this is part of the problem to characterize certain classes of groups by properties of the group algebra.
Weak convergence of semigroups implies strong convergence of weighted averages
Authors: Fong Humphrey
Published: June 2010

For a fixed

p
,
1
⩽
p
>
∞

p,1 \leqslant p > \infty

, let

{

T
t

:
t
>
0
}

\{ {T_t}:t > 0\}

be a strongly continuous semigroup of positive contractions on

L
p

{L_p}

of a

σ
\sigma

-finite measure space. We show that weak convergence of

{

T
t

:
t
>
0
}

\{ {T_t}:t > 0\}

in

L
p

{L_p}

is equivalent with the strong convergence of the weighted averages

∫
0
∞

T
t

f

μ
n

(
d
t
)
(
n
→
∞
)

\int _0^\infty {{T_t}f{\mu _n}(dt)(n \to \infty )}

for every

f
∈

L
p

f \in {L_p}

and every sequence

(

μ
n

)

({\mu _n})

of signed measures on

(
0
,
∞
)

(0,\infty )

, satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript n Endscripts StartAbsoluteValue EndAbsoluteValue mu Subscript n Baseline StartAbsoluteValue EndAbsoluteValue greater-than normal infinity semicolon limit Underscript n Endscripts mu Subscript n Baseline left-parenthesis 0 comma normal infinity right-parenthesis equals 1">

sup
n

|

|

μ
n

|

|

>
∞
;

lim
n

μ
n

(
0
,
∞
)
=
1

{\sup _n}||{\mu _n}|| > \infty ;{\lim _n}{\mu _n}(0,\infty ) = 1

; and for each

d
>
0
,

lim
n

sup

c
⩾
0

|

μ
n

|

(
c
,
c
+
d
]
=
0

d > 0,{\lim _n}{\sup _{c \geqslant 0}}|{\mu _n}|(c,c + d] = 0

. The positivity assumption is not needed if

p
=
1

p = 1

or 2. We show that such a result can be deduced-not only in

L
p

{L_p}

, but in general Banach spaces-from the corresponding discrete parameter version of the theorem.
Ergodic theorems of weak mixing type
Authors: Jones Lee K. , Lin Michael
Published: June 2010

Given a linear contraction

T
T

on a Banach space

X
X

and

x
∈
X

x \in X

, the convergence
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="for-all x Superscript asterisk Baseline element-of upper X Superscript asterisk Baseline upper N Superscript negative 1 Baseline sigma-summation Underscript i equals 1 Overscript upper N Endscripts StartAbsoluteValue mathematical left-angle x Superscript asterisk Baseline comma upper T Superscript i Baseline x mathematical right-angle EndAbsoluteValue right-arrow Underscript upper N right-arrow normal infinity Overscript Endscripts 0">

∀

x
∗

∈

X
∗

N

−
1

∑

i
=
1

N

|

⟨

x
∗

,

T
i

x
⟩

|

→

N
→
∞

0

\forall {x^ \ast } \in {X^ \ast }{N^{ - 1}}\sum \limits _{i = 1}^N {|\langle {x^ \ast },{T^i}x\rangle |} \xrightarrow [{N \to \infty }]{}0

\]
is shown to be equivalent to the convergence
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript StartAbsoluteValue EndAbsoluteValue x Superscript asterisk Baseline StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals 1 Endscripts upper N Superscript negative 1 Baseline sigma-summation Underscript j equals 1 Overscript upper N Endscripts StartAbsoluteValue mathematical left-angle x Superscript asterisk Baseline comma upper T Superscript k Super Subscript j Superscript Baseline x mathematical right-angle EndAbsoluteValue right-arrow 0">

sup

|

|

x
∗

|

|

⩽
1

N

−
1

∑

j
=
1

N

|

⟨

x
∗

,

T

k
j

x
⟩

|

→
0

\sup \limits _{||{x^ \ast }|| \leqslant 1} {N^{ - 1}}\sum \limits _{j = 1}^N {|\langle {x^ \ast },{T^{{k_j}}}x\rangle | \to 0}

\]
for every subsequence with

k
j

/

j

{k_j}/j

bounded. A sufficient condition is that, for some

{

n
i

}
,

T

n
i

x
→
0

\{ {n_i}\} ,{T^{{n_i}}}x \to 0

weakly.
A characterization of the 2-sphere by eigenfunctions
Authors: Cheng Shiu Yuen
Published: June 2010

The spheres in

3

 - dim

3{\text { - dim}}

Euclidean space are characterized by the fact that they have three 1st eigenfunctions with square sum equal to a constant.
Characterizations of the sphere by the curvature of the second fundamental form
Authors: Simon Udo
Published: June 2010

On an ovaloid

S
S

with Gaussian curvature

K
(

I)

>
0

K({\text {I)}} > 0

in Euclidean three-space

E
3

{E^3}

the second fundamental form defines a nondegenerate Riemannian metric with curvature

K
(

II

)

K({\text {II}})

. R. Schneider [7] proved that the spheres in Euclidean space

E

n
+
1

{E^{n + 1}}

are the only closed hypersurfaces on which the second fundamental form defines a nondegenerate Riemannian metric of constant curvature. For surfaces in

E
3

{E^3}

we give a common generalization of Schneider’s theorem and the classical theorem of Liebmann [6] (which states that any ovaloid in

E
3

{E^3}

with constant Gaussian curvature is a sphere).
Clopen sets in hyperspaces
Authors: Bankston Paul
Published: June 2010

Let

X
X

be a space and let

H
(
X
)

H(X)

denote its hyperspace (= all nonempty closed subsets of

X
X

topologized via the Vietoris topology). Then

X
X

is Boolean (= totally disconnected compact Hausdorff) iff

H
(
X
)

H(X)

is Boolean; and if

B
B

denotes the characteristic algebra of clopen sets in

X
X

then the corresponding algebra for

H
(
X
)

H(X)

is the free algebra generated by

B
B

modulo the ideal which “remembers” the upper semilattice structure of

B
B

.
Hyperspaces of hereditarily indecomposable plane continua
Authors: Tymchatyn E. D.
Published: June 2010

In this note we prove that if

X
X

is a hereditarily indecomposable plane continuum then the hyperspace

C
(
X
)

C(X)

can be embedded in Euclidean

3
3

-space.
Countable paracompactness, normality, and Moore spaces
Authors: Wage Michael L.
Published: June 2010

In this paper we show that

MA

+
¬

CH

{\text {MA}} + \neg {\text {CH}}

implies that there exists a countably paracompact Moore space which is not normal. Further, if there is a model of set theory in which every countably paracompact Moore space is normal, then the normal Moore space conjecture is true in that model. Other examples are given, including a nonnormal space constructed with

⋄
\diamond

which is countably compact,

T
3

{T_3}

, first countable, locally compact, perfect, and hereditarily separable.
Some metrization theorems
Authors: Hung H. H.
Published: June 2010

We prove, using H. W. Martin’s result on metrizable symmetric spaces and a symmetric of P. W. Harley III’s construction, a theorem which is slightly stronger than a recent theorem of Nagata.
Symmetric overmaps
Authors: Noakes J. L.
Published: June 2010

We prove periodicity theorems for the degrees of fibre-preserving maps of sphere bundles, and of projective space bundles.
Fixed points of automorphisms
Authors: Currier A. W.
Published: June 2010

We prove that the fixed point set of a compact Lie group of automorphisms of a Lie group has finitely many components if the Lie group does.
Variational sums for additive processes
Authors: Hudson William N. , Mason J. David
Published: June 2010

Let

X
(
t
)
,
0
≦
t
≦
T

X(t),0 \leqq t \leqq T

, be an additive process, and let

X

n
k

{X_{nk}}

be the

k
k

th increment of

X
(
t
)

X(t)

associated with the partition

Π
n

{\Pi _n}

of

[
0
,
T
]

[0,T]

. Assume

|

|

Π
n

|

|

→
0

||{\Pi _n}|| \to 0

. Let

β
\beta

be the Blumenthal-Getoor index of

X
(
T
)

X(T)

and let

2
≧
γ
>
β

2 \geqq \gamma > \beta

. When the partitions are nested,

∑
k

|

X

n
k

|

γ

\sum \nolimits _k {|{X_{nk}}{|^\gamma }}

converges a.s. to

∑

{

|

J
(
s
)

|

γ

:
0
≦
s
≦
T
}

\sum {\{ |J(s){|^\gamma }:0 \leqq s \leqq T\} }

, where

J
(
s
)

J(s)

is the jump of

X
(
t
)

X(t)

at

s
s

. This convergence also holds when the partitions are not nested provided either

X
(
t
)

X(t)

has stationary increments or

1
≧
γ
>
β

1 \geqq \gamma > \beta

. This extends a result of P. W. Millar and completes a result of S. M. Berman.
Kernels in lattice-ordered groups
Authors: Byrd Richard D. , Lloyd Justin T.
Published: June 2010

It is shown that a lattice-ordered group has a largest representaba convex

l
l

-subgroup and a largest normal valued convex

l
l

-subgroup. Other kernels are discussed.
A note on some properties of 𝐴-functions
Authors: Sarbadhikari H.
Published: June 2010

This note deals with

(

M

,
∗
)

({\mathbf {M}},\ast )

functions for various families

M

{\mathbf {M}}

. It is shown that if

M

{\mathbf {M}}

is the family of Borel sets of additive class

α
\alpha

on a metric space

X
X

, then

(

M

,
∗
)

({\mathbf {M}},\ast )

functions are just the functions of the form

sup
y

g
(
x
,
y
)

{\sup _y}g(x,y)

where

g
:
X
×
R
→
R

g:X \times R \to R

is continuous in

y
y

and of class

α
\alpha

in

x
x

. If

M

{\mathbf {M}}

is the class of analytic sets in a Polish space

X
X

, then the

(

M

,
∗
)

({\mathbf {M}},\ast )

functions dominating a Borel function are just the functions

sup
y

g
(
x
,
y
)

{\sup _y}g(x,y)

where

g
g

is a real valued Borel function on

X
2

{X^2}

. It is also shown that there is an

A
A

-function

f
f

defined on an uncountable Polish space

X
X

and an analytic subset

C
C

of the real line such that

f

−
1

(
C
)
∉

{f^{ - 1}}(C) \notin

the

σ
\sigma

-algebra generated by the analytic sets on

X
X

.
Nets of extreme Banach limits
Authors: Nillsen Rodney
Published: June 2010

Let

N
N

be the set of natural numbers and let

σ
:
N
→
N

\sigma :N \to N

be an injection having no periodic points. Let

M
σ

{M_\sigma }

be the set of

σ
\sigma

-invariant means on

l
∞

{l_\infty }

. When

f
∈

l
∞

f \in {l_\infty }

let

d
¯

σ

(
f
)
=
sup
λ
(
f
)

{\overline d _\sigma }(f) = \sup \lambda (f)

, where the supremum is taken over all

λ
∈

M
σ

\lambda \in {M_\sigma }

. It is shown that when

f
∈

l
∞

f \in {l_\infty }

, there is a sequence

(

λ
s

)

s
=
2

∞

({\lambda _s})_{s = 2}^\infty

of extreme points of

M
σ

{M_\sigma }

which has no extreme weak

∗

^{\ast }

limit points and such that

λ
s

(
f
)
=

d
¯

σ

(
f
)

{\lambda _s}(f) = {\overline d _\sigma }(f)

for

s
=
2
,
3
,
…

s = 2,3, \ldots

. As a consequence, the extreme points of

M
σ

{M_\sigma }

are not weak

∗

^{\ast }

compact.
The signature of the fixed set of a map of odd period
Authors: Alexander J. P. , Hamrick G. C. , Vick J. W.
Published: June 2010

Let

T
T

be a diffeomorphism of odd period

n
n

on a closed smooth manifold

M

2
k

{M^{2k}}

. The Conner-Floyd analysis of fixed point data and the Atiyah-Singer Index Theorem are applied to prove there exist methods of orienting the components

F
F

of the fixed set of

T
T

, depending only on

n
n

, so that

Σ
F

sgn
⁡
F
≡
sgn
⁡
M
mod
4

{\Sigma _F}\operatorname {sgn} F \equiv \operatorname {sgn} M\bmod 4

whenever

T
∗

{T^ \ast }

is the identity on

H
k

(
M
;
Q
)

{H^k}(M;Q)

. Other special results of this type are obtained when assumptions are made restricting the possible eigenvalues in the normal bundle to the fixed set.
Actions of groups of odd order on compact, orientable manifolds
Authors: Sanchez Cristian U.
Published: June 2010

Differentiable actions of cyclic groups of odd order on compact, connected, orientable manifolds with only two fixed points are studied in order to obtain conditions under which the representations of the group on the tangent spaces at the fixed points are equivalent.
Embedding orthogonal partial Latin squares
Authors: Lindner Charles C.
Published: June 2010

Two partial latin squares are orthogonal provided that when they are superimposed any ordered pairs obtained are distinct. The purpose of this paper is to show that any collection of pairwise orthogonal finite partial latin squares can be embedded into pairwise orthogonal finite latin squares.
Dedekind sums and noncongruence subgroups of the Hecke groups 𝐺(√2) and 𝐺(√3)
Authors: Parson L. Alayne
Published: June 2010

An example is given of a character

χ
r

{\chi _r}

on a subgroup of

G
(

2

)

G(\sqrt 2 )

or

G
(

3

)

G(\sqrt 3 )

such that the kernel of

χ
r

{\chi _r}

is of finite index in

G
(

2

)

G(\sqrt 2 )

or

G
(

3

)

G(\sqrt 3 )

but is not a congruence subgroup.
A remark on the restriction map in field formation
Authors: Onishi Hironori
Published: June 2010

In this note we point out that in a field formation

(
G
,
{

G
F

}
,
A
)

(G,\{ {G_F}\} ,A)

, if

h
2

(
K

/

F
)
=

[
K
:
F

]
c

{h_2}(K/F) = {[K:F]^c}

for every normal layer

K

/

F

K/F

with a fixed integer

c
⩾
0

c \geqslant 0

, then for every tower

F
⊂
E
⊂
K

F \subset E \subset K

with

K

/

F

K/F

normal, the restriction map

H
2

(
K

/

F
)
→

H
2

(
K

/

E
)

{H^2}(K/F) \to {H^2}(K/E)

is surjective, and give an example with

c
=
2

c = 2

.
When is 𝐷+𝑀 coherent?
Authors: Dobbs David E. , Papick Ira J.
Published: June 2010

Let

V
V

be a valuation ring of the form

K
+
M

K + M

, where

K
K

is a field and

M
(
≠
0
)

M( \ne 0)

is the maximal ideal of

V
V

. Let

D
D

be a proper subring of

K
K

. Necessary and sufficient conditions are given that the ring

D
+
M

D + M

be coherent. The condition that a given ideal of

V
V

be

D
+
M

D + M

-flat is also characterized.
On linear planes
Authors: Sathaye Avinash
Published: June 2010

A linear plane over a ground field

k
k

is an algebraic surface in affine

3
3

-space over

k
k

which is biregular to the affine plane and whose equation is linear in one of the three variables of the

3
3

-space. In this note we give a concrete description of a linear plane over a field of characteristic zero, thereby proving it to be an embedded plane, i.e. we show that by an automorphism of the affine

3
3

-space, it can be transformed to a coordinate plane.
The pre-Frattini residual
Authors: Bechtell H.
Published: June 2010

The structure of a prefrattini subgroup in a finite solvable group is related to the structure of the prefrattini residual, that is, the residual for the formation of solvable

n
C

nC

-groups. In particular this structure can be identified with the system normalizers of a lower nilpotent series in the prefrattini residual. Moreover, the prefrattini subgroups form a complete conjugate class within this residual and also contain the system normalizers of the residual.
Gross’ abstract Wiener measure on 𝐶[0,∞).
Authors: Finlayson H. C.
Published: June 2010

Classical Wiener measure on

C
[
0
,
∞
)

C[0,\infty )

is obtained by the construction of Gross’ abstract Wiener measure on a suitable Banach subspace of

C
[
0
,
∞
)

C[0,\infty )

.
Interpolation for entire functions of exponential type and a related trigonometric moment problem
Authors: Young Robert M.
Published: June 2010

A classical theorem of Hausdorff-Young shows that when

1
>
p
>
2

1 > p > 2

, the system of equations

φ
^

(
n
)
=

c
n

(
−
∞
>
n
>
∞
)

\hat \varphi (n) = {c_n}( - \infty > n > \infty )

admits a solution

φ
\varphi

in

L
q

(
−
π
,
π
)

{L^q}( - \pi ,\pi )

whenever

{

c
n

}
∈

l
p

\{ {c_n}\} \in {l^p}

. Here, as usual,

φ
^

\hat \varphi

denotes the complex Fourier transform of

φ
\varphi

and

q
q

is the conjugate exponent given by

p

−
1

+

q

−
1

=
1

{p^{ - 1}} + {q^{ - 1}} = 1

. The purpose of this note is to show that if a set

{

λ
n

}

\{ {\lambda _n}\}

of real or complex numbers is “sufficiently close” to the inte gers, then the corresponding system

φ
^

(

λ
n

)
=

c
n

\hat \varphi ({\lambda _n}) = {c_n}

is also solvable for

φ
\varphi

whenever

{

c
n

}
∈

l
p

\{ {c_n}\} \in {l^p}

. The proof is accomplished by establishing a similar interpolation theorem for a related class of entire functions of exponential type.
The Levi form and local complex foliations
Authors: Freeman Michael
Published: June 2010

A short coordinate-free proof is given for some known results on the existence of local complex-analytic foliations of a real submanifold

M
M

of

C

n

{{\mathbf {C}}^n}

. The proof uses an explicit formulation of the equivalence between two definitions of the E. E. Levi form of

M
M

.
On the embedding of Schwartz spaces into product spaces
Authors: Randtke Daniel J.
Published: June 2010

Every Schwartz space is embeddable into some sufficiently high power

E
I

{E^I}

of a given Banach space

E
E

if and only if

E
E

contains

l
n
∞

l_n^\infty

uniformly.
Solvability of convolution equations in 𝒦’₁
Authors: Sznajder S. , Zieleźny Z.
Published: June 2010

Let

S
S

be a convolution operator in the space

K

1
′

{\mathcal {K}’_1}

of distributions of exponential growth. A condition on

S
S

introduced by O. von Grudzinski is proved to be equivalent to

S
∗

K

1
′

=

K

1
′

S \ast {\mathcal {K}’_1} = {\mathcal {K}’_1}

.
An embedding theorem for certain spaces with an equidistant property
Authors: Nadler Sam B.
Published: June 2010

It is shown that certain metric spaces with the unique equidistant property can be topologically embedded in the real line. Several examples are given which show that the spaces considered are nontrivial, and which indicate that the technique of proof is necessary.
Sullivan’s de Rham complex is definable in terms of its 0-forms
Authors: Kan Daniel M. , Miller Edward Y.
Published: June 2010

It is proved that for a simplicial complex Sullivan’s de Rham complex is definable in terms of its

0
0

-forms.
Relative collaring
Authors: Rubin Leonard R.
Published: June 2010

It was proved by Morton Brown that the boundary of a manifold has a collar neighborhood. Later Robert Connelly gave a different proof of that result, and his method of proof was used by T. Benny Rushing to prove a relative collaring theorem. This paper contains a relative collaring theorem that is more general than either of the two just mentioned.
Two characterizations of conditional probability
Authors: Rao M. M.
Published: June 2010

This note contains two types of functional equations characterizing the conditional probability measures as subclasses of vector measures on general Banach function spaces. One uses the theory of vector integration and the other employs a form of the Šidák identity.
Inert extensions of Krull domains
Authors: Costa Douglas L. , Johnson Jon L.
Published: June 2010

Let

A
⊆
B

A \subseteq B

be integral domains with

B
B

an inert extension of a Krull domain

A
A

. Let

P

(
A
)

\mathcal {P}(A)

be the set of height one primes of

A
A

, and let

T
=

∩

p
∈

P

(
A
)

B
⊗

A
p

T = { \cap _{p \in \mathcal {P}(A)}}B \otimes {A_p}

. When each

B
p

=
B
⊗

A
p

{B_p} = B \otimes {A_p}

is a UFD, a necessary and sufficient condition for

T
T

to be a Krull domain is obtained. If

T
T

is a Krull domain and each

B
p

{B_p}

is a UFD, then the divisor class groups of

A
A

and

T
T

are isomorphic under the natural mapping. These results are applied to

A
⊆
B

A \subseteq B

when

B
B

is a symmetric algebra over

A
A

and when

B
B

is locally a polynomial ring over

A
A

.
(𝐶𝐴) topological groups
Authors: Zerling David
Published: June 2010

A locally compact topological group

G
G

is called

(
C
A
)

(CA)

if the group of inner automorphisms of

G
G

is closed in the group of all bicontinuous automorphisms of

G
G

. We show that each non-

(
C
A
)

(CA)

locally compact connected group

G
G

can be written as a semidirect product of a

(
C

A
)

(C\,A)

locally compact connected group by a vector group. This decomposition yields a natural dense imbedding of

G
G

into a

(
C

A
)

(C\,A)

locally compact connected group

P
P

, such that each bicontinuous automorphism of

G
G

can be extended to a bicontinuous automorphism of

P
P

. This imbedding and extension property enables us to derive a sufficient condition for the normal part of a semidirect product decomposition of a

(
C

A
)

(C\,A)

locally compact connected group to be

(
C

A
)

(C\,A)

.
Some remarks on summability factors
Authors: Gavin Lloyd A.
Published: June 2010

Bosanquet [2] showed that a necessary and sufficient condition for

Σ

k
=
1

∞

x
k

y
k

\Sigma _{k = 1}^\infty {x_k}{y_k}

to be Cesàro summable of order

n
n

(

n
n

is a nonnegative integer) whenever

σ
k
n

(
y
)
=
o
(
k
)

\sigma _k^n(y) = o(k)

where

σ
k
n

(
y
)

\sigma _k^n(y)

is the

k
k

th Cesàro mean of

y
y

of order

n
n

is that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Subscript k equals 1 Superscript normal infinity Baseline k Superscript n plus 1 Baseline StartAbsoluteValue normal upper Delta Superscript n plus 1 Baseline x Subscript k Baseline EndAbsoluteValue greater-than normal infinity">

Σ

k
=
1

∞

k

n
+
1

|

Δ

n
+
1

x
k

|

>
∞

\Sigma _{k = 1}^\infty {k^{n + 1}}|{\Delta ^{n + 1}}{x_k}| > \infty

and

lim

k
→
0

k

x
k

=
0

{\lim _{k \to 0}}k{x_k} = 0

. The main result of this paper is to show that a necessary and sufficient condition for

Σ

k
=
1

∞

x
k

y
k

\Sigma _{k = 1}^\infty {x_k}{y_k}

to be Cesàro summable of order

n
n

(

n
n

is a nonnegative integer) whenever
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Subscript k equals 1 Superscript normal infinity Baseline k Superscript n plus 1 Baseline StartAbsoluteValue normal upper Delta Superscript n plus 1 Baseline x Subscript k Baseline EndAbsoluteValue greater-than normal infinity">

Σ

k
=
1

∞

k

n
+
1

|

Δ

n
+
1

x
k

|

>
∞

\Sigma _{k = 1}^\infty {k^{n + 1}}|{\Delta ^{n + 1}}{x_k}| > \infty

and

lim

k
→
∞

k

x
k

=
0

{\lim _{k \to \infty }}k{x_k} = 0

is that

σ
k
n

(
y
)
=
o
(
k
)

\sigma _k^n(y) = o(k)

.
Sequential convergence to invariance in 𝐵𝐶(𝐺)
Authors: Sine Robert
Published: June 2010

In this note it is shown that weak and strong convergence to invariance are equivalent for a sequence of probabilities acting on

B
C
(
G
)

BC(G)

of a noncompact locally compact group. This result was known for

G
=
Z

G = Z

. For other generalizations of the bounded sequences on

Z
Z

, say

B
U
C
(
G
)

BUC(G)

, the result does not hold.
𝐾₁ of the compact operators is zero
Authors: Brown L. G. , Schochet Claude
Published: June 2010

We prove that

K
1

{K_1}

of the compact operators is zero. This theorem has the following operator-theoretic formulation: any invertible operator of the form (identity)

+
+

(compact) is the product of (at most eight) multiplicative commutators

(

A
j

B
j

A
j

−
1

B
j

−
1

)

±
1

{({A_j}{B_j}A_j^{ - 1}B_j^{ - 1})^{ \pm 1}}

, where each

B
j

{B_j}

is of the form (identity)

+
+

(compact). The proof uses results of L. G. Brown, R. G. Douglas, and P. A. Fillmore on essentially normal operators and a theorem of A. Brown and C. Pearcy on multiplicative commutators.
Characterizations of completely Hausdorff-closed spaces
Authors: Herrington Larry L.
Published: June 2010

This paper gives several characterizations of completely Hausdorff-closed spaces.
Metric spaces in which minimal circuits cannot self-intersect
Authors: Sanders David
Published: June 2010

Definitions are given for self-intersecting polygons and cogeodesic points in terms of betweenness, and then it is proved that the metric spaces in which minimal polygons on a finite number of distinct noncogeodesic points are not self-intersecting are completely characterized as those metric spaces which have the following betweenness property for any four distinct points: if

b
b

is between

a
a

and

c
c

and between

a
a

and

d
d

then either

c
c

is between

a
a

and

d
d

or

d
d

is between

a
a

and

c
c

.
On the supplement to the law of biquadratic reciprocity
Authors: Williams Kenneth S.
Published: June 2010

A short proof is given of the supplement to the law of biquadratic reciprocity proved by Eisenstein in 1844.
A commutativity theorem for rings
Authors: Chacron M.
Published: June 2010

Let

R
R

be any associative ring. Suppose that for every pair

(

a
1

,

a
2

)
∈
R
×
R

({a_1},{a_2}) \in R \times R

there exists a pair

(

p
1

,

p
2

)

({p_1},{p_2})

such that the elements

a
i

−

a
i
2

p
i

(

a
i

)

{a_i} - a_i^2{p_i}({a_i})

commute, where the

p
i

{p_i}

’s are polynomials over the integers with one (central) indeterminate. It is shown here that the nilpotent elements of

R
R

form a commutative ideal

N
N

, and that the factor ring

R

/

N

R/N

is commutative. This result is obtained by the use of the concept of cohypercenter of a ring

R
R

, which concept parallels the hypercenter of a ring.
Fixed-point-free actions on a class of abelian groups
Authors: Curran P. M.
Published: June 2010

It is proved that all cohomology groups of a group

G
G

acting on an abelian group of a certain type vanish if the action of some subnormal abelian subgroup of

G
G

is fixed-point-free. This result is then applied to obtain results about group extensions and about the complete reducibility of linear groups.
Uniqueness of topology for the 𝑝-adic integers
Authors: Corwin Lawrence
Published: June 2010

It is shown that the only Hausdorff topologies on

Z

p

{{\mathbf {Z}}_p}

, the

p
p

-adic integers, which make it into a locally compact Abelian group are the

p
p

-adic and discrete topologies. The key ingredient in the proof is a structure theorem for certain LCA groups which may be of independent interest.
Inequalities for entire functions of exponential type
Authors: Genchev T.
Published: June 2010

This paper is concerned with a class of linear operators acting in the space of the trigonometric polynomials and preserving the inequalities of the form

|

S
(
θ
)

|

>

|

T
(
θ
)

|

|S(\theta )| > |T(\theta )|

in the half plane

Im

θ
>
0

{\text {Im}}\theta > 0

. Some inequalities for entire functions of exponential type and some theorems concerning the distribution of the zeros of the trigonometric polynomials, including an analogue to the Gauss-Lucas theorem, are derived.
On a uniqueness theorem for analytic mappings
Authors: Niimura Mikio
Published: June 2010

A generalization of the uniqueness theorem of Koebe-Nevanlinna for meromorphic functions of bounded type is given on Riemann surfaces.
Residual equisingularity
Authors: Becker Joseph , Stutz John
Published: June 2010

Let

V
V

be a complex analytic set and

Sg
⁡
V

\operatorname {Sg} V

the singular set of

V
V

be in codimension one; then the set of points of

Sg
⁡
V

\operatorname {Sg} V

for which

V
V

is not residually equisingular along

Sg
⁡
V

\operatorname {Sg} V

is a proper analytic subset of

Sg
⁡
V

\operatorname {Sg} V

.

V
V

is said to be residually equisingular along

Sg
⁡
V

\operatorname {Sg} V

if all one dimensional slices of

V
V

transverse to

Sg
⁡
V

\operatorname {Sg} V

have isomorphic resolutions.
Measurability and continuity conditions for nonlinear evolutionary processes
Authors: Ball John M.
Published: June 2010

This paper generalizes to nonlinear evolutionary processes on a metric space the well-known results connecting measurability and continuity properties with respect to time of linear semigroups of continuous operators on a Banach space.
Characterizations of Urysohn-closed spaces
Authors: Herrington Larry L.
Published: June 2010

This paper gives characterizations of Urysohn-closed and minimal Urysohn spaces, some of which make use of nets.
A characterization for the products of 𝑘- and ℵ₀-spaces and related results
Authors: Tanaka Yoshio
Published: June 2010

E. Michael introduced the notion of

ℵ
0

{\aleph _0}

-spaces and characterized spaces which are both

k
k

-spaces and

ℵ
0

{\aleph _0}

-spaces (or, briefly,

k
k

-and-

ℵ
0

{\aleph _0}

-spaces) as being precisely the quotient images of separable metric spaces. The purpose of this paper is to give a necessary and sufficient condition for the product of two

k
k

-and-

ℵ
0

{\aleph _0}

-spaces to be a

k
k

-and-

ℵ
0

{\aleph _0}

-space. Moreover, as related matters, we shall consider the products of

k
k

-spaces having other properties.
Homeomorphisms with many recurrent points
Authors: Halpern Benjamin
Published: June 2010

Let

X
X

be a topological space and

H
(
X
)

H(X)

the space of all homeomorphisms of

X
X

onto itself with the compact open topology. If

f
∈
H
(
X
)

f \in H(X)

and

p
∈
X

p \in X

, then

p
p

is a recurrent point of

f
f

provided

p
p

is in the closure of

{

f
n

(
p
)

|

n
⩾
1
}

\{ {f^n}(p)|n \geqslant 1\}

. It is shown that if

X
X

is Hausdorff and

V
V

is a nonempty open subset of

X
X

homeomorphic to Euclidean

n
n

-dimensional space with

n
⩾
1

n \geqslant 1

, then

{
f
∈
H
(
X
)

|

\{ f \in H(X)|

the recurrent points of

f
f

are dense in

V
V

is nowhere dense in

H
(
X
)

H(X)

.
Artin root numbers for real characters
Authors: Gechter Jerry
Published: June 2010

For

K
K

and

L
L

number fields,

χ
\chi

a real-valued character on

Gal
⁡
(
K

/

L
)

\operatorname {Gal} (K/L)

, the Artin root number

W
(
χ
)

W(\chi )

is

±
1

\pm 1

. We analyze the question of sign for

χ
\chi

a degree 2 character over

Q

{\mathbf {Q}}

induced from an abelian character on a quadratic extension.
Group rings with solvable 𝑛-Engel unit groups
Authors: Fisher J. L. , Parmenter M. M. , Sehgal S. K.
Published: June 2010

Let

K
G

KG

be the group ring of a group

G
G

over a field of characteristic

p
⩾
0
,

p
≠
2
,

3

p \geqslant 0,\;p \ne 2,\;3

. Suppose

G
G

contains no element of order

p
p

(if

p
>
0

p > 0

). Group algebras

K
G

KG

with unit group

U
(
K
G
)

U(KG)

solvable and

n
n

-Engel are characterized.
On the straightness of reduced Teichmüller space
Authors: Wason Judith C.
Published: June 2010

Under a natural injection which is shown to be an isometry, the image of the reduced Teichmüller space

T
#

(
G
)

{T^\# }(G)

in the open straight Teichmüller space of the Fuchsian model of

G
,
T
(
H
)

G,T(H)

, is an open straight subspace.
On the structure of Lindenbaum algebras: an approach using algebraic logic
Authors: Pinter Charles C.
Published: June 2010

The following problem of algebraic logic is investigated: to determine those Boolean algebras which admit the structure of a nondiscrete cylindric algebra. A partial solution is found, and is then used to give an algebraic characterization of the Lindenbaum algebras of formulas of several broad classes of countable theories.
Smoothness properties of generalized convex functions
Authors: Zalik R. A.
Published: June 2010

We present a concise and elementary proof of a theorem of Karlin and Studden concerning the smoothness properties of functions belonging to a generalized convexity cone.
Extremal and monogenic additive set functions
Authors: Plachky Detlef
Published: June 2010

The extreme points of the convex set of all additive set functions on a field, which coincide on a subfield are characterized by a simple approximation property. It is proved that a stronger approximation property is characteristic for a so-called monogenic additive set function on a field, which can be generated uniquely by an additive set function on a subfield. Finally it is shown that a simple decomposition property must hold if the convex set above has a finite number of extreme points.
A new flag transitive affine plane of order 27
Authors: Narayana Rao M. L. , Kuppuswamy Rao K.
Published: June 2010

A flag transitive affine plane of order

27
27

is constructed. The stabilizer of the origin of this plane contains a cyclic subgroup which is transitive on the lines through the origin. It is also shown that this plane is not isomorphic to the flag transitive plane constructed by Hering.
Hopf invariants for reduced products of spheres
Authors: Baues Hans Joachim
Published: June 2010

Let

S
m
n

S_m^n

be the

m
m

th reduced product complex of the even dimensional sphere

S
n

{S^n}

. Using ’cup’-products, James defined a Hopf invariant homomorphism
\[

H
m
n

:

π

m
n
−
1

(

S

m
−
1

n

)
→

Z

H_m^n:{\pi _{mn - 1}}(S_{m - 1}^n) \to {\mathbf {Z}}

\]
such that

H
2
n

H_2^n

is the classical Hopf invariant. Extending the result of Adams on

H
2
n

H_2^n

we determine the image of

H
m
n

H_m^n

. Partial calculations were made by Hardie and Shar.
A remark on the strong law of large numbers
Authors: Chen Robert
Published: June 2010

Let

X
1

,

X
2

,
…

{X_1},{X_2}, \ldots

be mutually independent random variables such that

E
(

X
n

)
=
0

E({X_n}) = 0

and

E
(

X
n
2

)
=

σ
n
2

=
1

E(X_n^2) = \sigma _n^2 = 1

for all

n
=
1
,
2
,
…

n = 1,2, \ldots

. For each

n
=
1
,
2
,
…

n = 1,2, \ldots

let

S
n

=

∑

j
=
1

n

X
j

{S_n} = \sum \nolimits _{j = 1}^n {{X_j}}

; then, by the Kolmogorov criterion for mutually independent random variables,

S
n

/

n

1

/

2
+
α

→
0

{S_n}/{n^{1/2 + \alpha }} \to 0

almost surely as

n
→
∞

n \to \infty

for any positive constant

α
\alpha

. A deeper understanding of this theorem will be facilitated if we know the order of magnitude of

E
{

N
∞

(
α
,
ε
)
}

E\{ {N_\infty }(\alpha ,\varepsilon )\}

as

ε
→

0
+

\varepsilon \to {0^ + }

, where

N
∞

(
α
,
ε
)

{N_\infty }(\alpha ,\varepsilon )

is the integer-valued random variable defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Subscript normal infinity Baseline left-parenthesis alpha comma epsilon right-parenthesis equals sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts chi Subscript left-parenthesis epsilon n Sub Superscript 1 slash 2 plus alpha Subscript comma normal infinity right-parenthesis Baseline left-parenthesis StartAbsoluteValue upper S Subscript n Baseline EndAbsoluteValue right-parenthesis">

N
∞

(
α
,
ε
)
=

∑

n
=
1

∞

χ

(
ε

n

1

/

2
+
α

,
∞
)

(

|

S
n

|

)

{N_\infty }(\alpha ,\varepsilon ) = \sum \nolimits _{n = 1}^\infty {{\chi _{(\varepsilon {n^{1/2 + \alpha }},\infty )}}(|{S_n}|)}

. The present note does the work for a wide class of random variables by using Esseen’s theorem and Katz-Petrov’s theorem.
A note on the continuity of local times
Authors: Geman Donald
Published: June 2010

Several conditions are given for a stochastic process

X
(
t
)

X(t)

on

[
0
,
1
]

[0,1]

to have a local time which is continuous in its time parameter (for example, in the Gaussian case, the integrability of

[
E

(
X
(
t
)
−
X
(
s
)

)
2

]

−
1

/

2

{[E{(X(t) - X(s))^2}]^{ - 1/2}}

over the unit square). Furthermore, for any Borel function

F
F

on

[
0
,
1
]

[0,1]

with a continuous local time, the approximate limit of

|

F
(
s
)
−
F
(
t
)

|

/

|

s
−
t

|

|F(s) - F(t)|/|s - t|

as

s
→
t

s \to t

is infinite for a.e.

t
∈
[
0
,
1
]

t \in [0,1]

and

s

|

F
(
s
)
=
F
(
t
)

s|F(s) = F(t)

is uncountable for a.e.

t
∈
[
0
,
1
]

t \in [0,1]

.
The Reuter-Ledermann representation for birth and death processes
Authors: Waymire Edward C.
Published: June 2010

The identification of the mass of the integrator at zero is made for the integral representation obtained by Reuter and Ledermann for the transition probabilities of birth and death processes. An ergodic theorem is given as an application of this result.
The \overline𝜇-invariants for groups
Authors: Gutiérrez M. A.
Published: June 2010

Given a presentation

(

P)

({\text {P)}}

for a group

G
G

, the cobar differentials

d
r

:

E

0
,
1

r

→

E

−
r
,
r

r

{d^r}:E_{0,1}^r \to E_{ - r,r}^r

are invariants of

(

P)

({\text {P)}}

. These invariants can be interpreted to be the Massey coproducts of

H

∗

(
G
)

{H_{\ast }}(G)

, and, if

(

P)

({\text {P)}}

is the Wirtinger presentation of a link group, they coincide with the

μ
¯

\overline \mu

-invariants of Milnor.
The conjugacy problem for HNN groups and the word problem for commutative semigroups
Authors: Anshel Michael
Published: June 2010

We show how a positive solution to the word problem for finitely generated commutative semigroups leads to a positive solution to the conjugacy problem for a class of

H
N
N

HNN

groups.
Left loops which satisfy the left Bol identity
Authors: Sharma B. L.
Published: June 2010

It is our purpose in this paper to initiate a study of the algebraic properties of a left loop

Q
(
⋅
)

Q( \cdot )

satisfying the identical relation

(1)

y
(
z
⋅
y
x
)
=
(
y
⋅
z
y
)
x

\begin{equation} \tag {1} y(z \cdot yx) = (y \cdot zy)x \end{equation}

for all

x
,

y
,

z
∈
Q

x,\;y,\;z \in Q

. It is shown that (1) implies right division in

Q
(
⋅
)

Q( \cdot )

. By introducing a new operation ’

∘
\circ

’ in

Q
Q

, the connection between the left loop

Q
(
⋅
)

Q( \cdot )

and Bol loop

Q
(
∘
)

Q( \circ )

is established. Further we show that the role of nuclei in the left loop theory is not the same as that in the loop theory. We conclude the paper by describing situations in which the left loop

Q
(
⋅
)

Q( \cdot )

is Moufang.
Lie groups isomorphic to direct products of unitary groups
Authors: Vidav Ivan , Legiša Peter
Published: June 2010

A criterion is given for a compact connected subgroup of

Gl
(
n
,

C

)

\text {Gl} (n,{\mathbf {C}})

to be isomorphic to a direct product of unitary groups. It implies that a compact connected subgroup of rank

n
n

in

Gl
(
n
,

C

)

\text {Gl} (n,{\mathbf {C}})

is isomorphic to a direct product of unitary groups.