bluuebunny/crossref_metadata_2025 huggingface dataset:
Algebras of iterated path integrals and fundamental groups
Authors: Chen Kuo-tsai
Published: July 2010

A method of iterated integration along paths is used to extend deRham cohomology theory to a homotopy theory on the fundamental group level. For every connected

C
∞

{C^\infty }

manifold

M

\mathfrak {M}

with a base point p, we construct an algebra

π
1

=

π
1

(

M

,
p
)

{\pi ^1} = {\pi ^1}(\mathfrak {M},p)

consisting of iterated integrals, whose value along each loop at p depends only on the homotopy class of the loop. Thus

π
1

{\pi ^1}

can be taken as a commutative algebra of functions on the fundamental group

π
1

(

M

)

{\pi _1}(\mathfrak {M})

, whose multiplication induces a comultiplication

π
1

→

π
1

⊗

π
1

{\pi ^1} \to {\pi ^1} \otimes {\pi ^1}

, which makes

π
1

{\pi ^1}

a Hopf algebra. The algebra

π
1

{\pi ^1}

relates the fundamental group to analysis of the manifold, and we obtain some analytical conditions which are sufficient to make the fundamental group nonabelian or nonsolvable. We also show that

π
1

{\pi ^1}

depends essentially only on the differentiable homotopy type of the manifold. The second half of the paper is devoted to the study of structures of algebras of iterated path integrals. We prove that such algebras can be constructed algebraically from the following data: (a) the commutative algebra A of

C
∞

{C^\infty }

functions on

M

\mathfrak {M}

; (b) the A-module M of

C
∞

{C^\infty }

1-forms on

M

\mathfrak {M}

; (c) the usual differentiation

d
:
A
→
M

d:A \to M

; and (d) the evaluation map at the base point p,

ε
:
A
→
K

\varepsilon :A \to K

, K being the real (or complex) number field.
On the signature of knots and links
Authors: Shinohara Yaichi
Published: July 2010

In 1965, K. Murasugi introduced an integral matrix M of a link and defined the signature of the link by the signature of

M
+

M
′

M + M’

. In this paper, we study some basic properties of the signature of links. We also describe the effect produced on the signature of a knot contained in a solid torus by a further knotting of the solid torus.
Stationary isotopies of infinite-dimensional spaces
Authors: Wong Raymond Y. T.
Published: July 2010

Let X denote the Hilbert cube or any separable infinite-dimensional Fréchet space. It has been shown that any two homeomorphisms f, g of X onto itself is isotopic to each other by means of an invertible-isotopy on X. In this paper we generalize the above results to the extent that if f, g are K-coincident on X (that is,

f
(
x
)
=
g
(
x
)

f(x) = g(x)

for

x
∈
K

x \in K

), then the isotopy can be chosen to be K-stationary provided K is compact and has property-Z in X. The main tool of this paper is the Stable Homeomorphism Extension Theorem which generalizes results of Klee and Anderson.
Theory of random evolutions with applications to partial differential equations
Authors: Griego Richard , Hersh Reuben
Published: July 2010

The selection from a finite number of strongly continuous semigroups by means of a finite-state Markov chain leads to the new notion of a random evolution. Random evolutions are used to obtain probabilistic solutions to abstract systems of differential equations. Applications include one-dimensional first order hyperbolic systems. An important special case leads to consideration of abstract telegraph equations and a generalization of a result of Kac on the classical n-dimensional telegraph equation is obtained and put in a more natural setting. In this connection a singular perturbation theorem for an abstract telegraph equation is proved by means of a novel application of the classical central limit theorem and a representation of the solution for the limiting equation is found in terms of a transformation formula involving the Gaussian distribution.
Grunsky-Nehari inequalities for a subclass of bounded univalent functions
Authors: DeTemple D. W.
Published: July 2010

Let

D
1

{D_1}

be the class of regular analytic functions

F
(
z
)

F(z)

in the disc

U
=
{
z
:

|

z

|

>
1
}

U = \{ z:|z| > 1\}

for which

F
(
0
)
>
0
,

|

F
(
z
)

|

>
1

F(0) > 0,|F(z)| > 1

, and

F
(
z
)
+
F
(
ζ
)
≠
0

F(z) + F(\zeta ) \ne 0

for all

z
,
ζ
∈
U

z,\zeta \in U

. Inequalities of the Grunsky-Nehari type are obtained for the univalent functions in

D
1

{D_1}

, the proof being based on the area method. By subordination it is shown univalency is unnecessary for certain special cases of the inequalities. Employing a correspondence between

D
1

{D_1}

and the class

S
1

{S_1}

of bounded univalent functions, the results can be reinterpreted to apply to this latter class.
Representation of vector valued nonlinear functions
Authors: Mizel Victor J. , Sundaresan K.
Published: July 2010

A representation theorem for “additive” nonlinear functional on spaces

L
p

(
μ
)

{L^p}(\mu )

is here extended to “additive” nonlinear functions from Lebesgue-Bochner function spaces

L
E
p

(
μ
)

L_E^p(\mu )

(

E
E

a separable Banach space) into Banach spaces

B
B

. A counterexample is provided to show that the restriction to separable

E
E

is essential.
On zonotopes
Authors: McMullen P.
Published: July 2010

In this paper is described a diagram technique for zonotopes, or vector sums of line segments, which is analogous to that of Gale diagrams for general polytopes, and central diagrams for centrally symmetric polytopes. The use of these new zonal diagrams leads to relationships between zonotopes with

n
n

zones of dimensions

d
d

and

n
−
d

n - d

, and enables one to enumerate all the combinatorial types of

d
d

-zonotopes with

n
≦
d
+
2

n \leqq d + 2

zones. The connexion between arrangements of hyperplanes in projective space and zonotopes leads to corresponding new results about arrangements.
Computability by probabilistic Turing machines
Authors: Santos Eugene S.
Published: July 2010

In the present paper, the definition of probabilistic Turing machines is extended to allow the introduction of relative computability. Relative computable functions, predicates and sets are discussed and their operations investigated. It is shown that, despite the fact that randomness is involved, most of the conventional results hold in the probabilistic case. Various classes of ordinary functions characterizable by computable random functions are introduced, and their relations are examined. Perhaps somewhat unexpectedly, it is shown that, in some sense, probabilistic Turing machines are capable of computing any given function. Finally, a necessary and sufficient condition for an ordinary function to be partially recursive is established via computable probabilistic Turing machines.
Linear representations of certain compact semigroups
Authors: Brown D. R. , Friedberg Michael
Published: July 2010

In this paper we initiate the study of representation theory of compact, not necessarily commutative, uniquely divisible semigroups. We show that a certain class of semigroups are all topologically isomorphic to real matrix semigroups. The proof utilizes a group embedding theorem and the standard results on homomorphisms of Lie groups into matrix groups.
Area measure and Radó’s lower area
Authors: Nishiura Togo
Published: July 2010

The theory of Geöcze area for two-dimensional surfaces in threedimensional space had been essentially completed by the mid 1950’s. The only hypothesis needed for all theorems in this case is the finiteness of the area. See [2] for an account of this theory. In the early 1960’s, H. Federer established, in his paper [6], fundamental facts concerning his integral geometric area for higher dimensional area theory by employing the theory of normal and integral currents. These facts employ not only the finiteness of area as a basic hypothesis but certain other hypotheses as well. The extensions of Geöcze type area to higher dimensions also employ not only the finiteness of area but certain added hypotheses. These hypotheses are of such a nature as to allow the use of the theory of quasi-additivity [3], [11]. The present paper concerns these added hypotheses which play such an important part of higher-dimensional area theory of today. It is shown that Radó’s lower area is the best Geöcze type area to describe these added hypotheses. That is, it is shown that the quasi-additivity hypotheses of Geöcze area in [11] imply the quasi-additivity hypotheses of lower area. Second, it is shown that the quasi-additivity hypotheses for lower area imply that the surface has the essential cylindrical property defined by J. Breckenridge in [5]. This essential cylindrical property is proved to be equivalent to the existence of area measures on the middle space of the mapping representing the surface. Finally, it is shown that the essential cylindrical property of a surface is equivalent to the quasi-additivity condition for lower area. Thus, an intrinsic property of the surface characterizes the quasi-additivity condition for the lower area of a surface.
Functions of finite 𝜆-type in several complex variables
Authors: Kujala Robert O.
Published: July 2010

If

λ
:

R

+

→

R

+

\lambda :{{\bf {R}}^ + } \to {{\bf {R}}^ + }

is continuous and increasing then a meromorphic function f on

C
k

{C^k}

is said to be of finite

λ
\lambda

-type if there are positive constants s, A, B and R such that

T
f

(
r
,
s
)
≦
A
λ
(
B
r
)

{T_f}(r,s) \leqq A\lambda (Br)

for all

r
>
R

r > R

where

T
f

(
r
,
s
)

{T_f}(r,s)

is the characteristic of f. It is shown that if

λ
(
B
r
)

/

λ
(
r
)

\lambda (Br)/\lambda (r)

is bounded for r sufficiently large and

B
>
1

B > 1

, then every meromorphic function of finite

λ
\lambda

-type is the quotient of two entire functions of finite

λ
\lambda

-type. This theorem is the result of a careful and detailed analysis of the relation between the growth of a function and the growth of its divisors. The central fact developed in this connection is: A nonnegative divisor

ν
\nu

on

C
k

{C^k}

with

ν
(

0

)
=
0

\nu ({\bf {0}}) = 0

is the divisor of an entire function of finite

λ
\lambda

-type if and only if there are positive constants A, B and R such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper N Subscript nu vertical-bar xi Baseline left-parenthesis r right-parenthesis less-than-over-equals upper A lamda left-parenthesis upper B r right-parenthesis comma 2nd Row StartAbsoluteValue StartFraction 1 Over p EndFraction sigma-summation Underscript s greater-than StartAbsoluteValue z EndAbsoluteValue less-than-over-equals r Endscripts nu vertical-bar xi left-parenthesis z right-parenthesis z Superscript negative p Baseline EndAbsoluteValue less-than-over-equals upper A lamda left-parenthesis upper B r right-parenthesis r Superscript negative p Baseline plus upper A lamda left-parenthesis upper B s right-parenthesis s Superscript negative p Baseline comma EndLayout">

N

ν

|

ξ

(
r
)
≦
A
λ
(
B
r
)
,

|

1
p

∑

s
>

|

z

|

≦
r

ν

|

ξ
(
z
)

z

−
p

|

≦
A
λ
(
B
r
)

r

−
p

+
A
λ
(
B
s
)

s

−
p

,

\begin {array}{*{20}{c}} {{N_{\nu |\xi }}(r) \leqq A\lambda (Br),} \\ {\left | {\frac {1}{p}\sum \limits _{s > |z| \leqq r} {\nu |\xi (z){z^{ - p}}} } \right | \leqq A\lambda (Br){r^{ - p}} + A\lambda (Bs){s^{ - p}},} \\ \end {array}

\]
for all

r
≧
s
>
R

r \geqq s > R

, all unit vectors

ξ
\xi

in

C
k

{C^k}

, and all natural numbers p. Here

ν

|

ξ

\nu |\xi

represents the lifting of the divisor

ν
\nu

to the plane via the map

z
↦
z
ξ

z \mapsto z\xi

and

N

ν

|

ξ

{N_{\nu |\xi }}

is the valence function of that divisor. Analogous facts for functions of zero

λ
\lambda

-type are also presented.
Homomorphisms on groups and induced maps on certain algebras of measures
Authors: Dunkl Charles F. , Ramirez Donald E.
Published: July 2010

Suppose that

φ
\varphi

is a continuous homomorphism of a locally compact group

G
G

into another such group,

H
H

, then

φ
\varphi

induces in a natural way a homomorphism

φ
∗

{\varphi ^ \ast }

of the measure algebra of

G
G

, called

M
(
G
)

M(G)

, into

M
(
H
)

M(H)

. The action of

φ
∗

{\varphi ^ \ast }

on the subspace

M
0

(
G
)

{M_0}(G)

is studied in this paper. The space

M
0

(
G
)

{M_0}(G)

is the nonabelian analogue to the space of measures on a locally compact abelian group whose Fourier-Stieltjes transforms vanish at infinity, and is defined herein. We prove that if

φ
\varphi

is an open homomorphism then

φ
∗

(

M
0

(
G
)
)
⊂

M
0

(
H
)

{\varphi ^ \ast }({M_0}(G)) \subset {M_0}(H)

. If

G
G

and

H
H

are abelian and

φ
\varphi

is not open, then

φ
∗

(
M
(
G
)
)
∩

M
0

(
H
)
=
{
0
}

{\varphi ^ \ast }(M(G)) \cap {M_0}(H) = \{ 0\}

. The main tool for this theorem is the fact, proved herein, that

φ
\varphi

is open if and only if its adjoint,

φ
^

:

H
^

→

G
^

\hat \varphi :\hat H \to \hat G

, is proper (where

G
^

,

H
^

\hat G,\hat H

are the character groups of

G
,
H

G,H

resp.). Further properties of

M
0

(
G
)

{M_0}(G)

for abelian or compact groups

G
G

are derived.
Representation of holomorphic functions by boundary integrals
Authors: Baernstein Albert
Published: July 2010

Let

K
K

be a compact locally connected set in the plane and let

f
f

be a function holomorphic in the extended complement of

K
K

with

f
(
∞
)
=
0

f(\infty ) = 0

. We prove that there exists a sequence of measures

{

μ
n

}

\{ {\mu _n}\}

on

K
K

satisfying

lim

n
→
∞

|

|

μ
n

|

|

1

/

n

=
0

{\lim _{n \to \infty }}||{\mu _n}|{|^{1/n}} = 0

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts integral Underscript upper K Endscripts left-parenthesis w minus z right-parenthesis Superscript negative n minus 1 Baseline d mu Subscript n Baseline left-parenthesis w right-parenthesis left-parenthesis z element-of upper K right-parenthesis">

f
(
z
)
=

∑

n
=
0

∞

∫
K

(
w
−
z
)

−
n
−
1

d

μ
n

(
w
)
(
z
∈
K
)

f(z) = \sum \nolimits _{n = 0}^\infty {\int _K {{{(w - z)}^{ - n - 1}}d{\mu _n}(w)(z \in K)} }

. It follows from the proof that two topologies for the space of functions holomorphic on

K
K

are the same. One of these is the inductive limit topology introduced by Köthe, and the other is defined by a family of seminorms which involve only the values of the functions and their derivatives on

K
K

. A key lemma is an open mapping theorem for certain locally convex spaces. The representation theorem and the identity of the two topologies is false when

K
K

is a compact subset of the unit circle which is not locally connected.
An asymptotic theorem for systems of linear differential equations.
Authors: Devinatz Allen
Published: July 2010

Asymptotic estimates are obtained for a complete linearly independent set of solutions of a linear system of differential equations of the form
\[

y
′

(
t
)
=
[
A
+
V
(
t
)
+
R
(
t
)
]
y
(
t
)
,

y’(t) = [A + V(t) + R(t)]y(t),

\]
where

A
A

is a constant

n
×
n

n \times n

matrix with

n
n

distinct eigenvalues,

R
(
t
)

R(t)

is an integrable matrix valued function on

(
0
,
∞
)

(0,\infty )

and

V
(
t
)

V(t)

is an

n
×
n

n \times n

matrix valued function having certain differentiability properties. The theorem that is obtained generalizes a theorem of N. Levinson, Duke Math. J. 15 (1948), 111-126.
An extension of the Weyl-von Neumann theorem to normal operators
Authors: Berg I. David
Published: July 2010

We prove that a normal operator on a separable Hilbert space can be written as a diagonal operator plus a compact operator. If, in addition, the spectrum lies in a rectifiable curve we show that the compact operator can be made Hilbert-Schmidt.
Products with closed projections. II
Authors: Noble N.
Published: July 2010

Conditions under which some or all of the projections on a product space will be closed or

z
z

-closed are studied, with emphasis on infinite products. These results are applied to characterize normal products up to countably many factors, to characterize closed product maps up to finitely many factors, and to give conditions under which products will be countably compact, Lindelöf, paracompact,

m

−

n

\mathfrak {m} - \mathfrak {n}

-compact, etc. Generalizations of these results to

n

\mathfrak {n}

-products and box products are also given. Our easily stated results include: All powers of a

T
1

{T_1}

space

X
X

are normal if and only if

X
X

is compact Hausdorff, all powers of a nontrivial closed map

p
p

are closed if and only if

p
p

is proper, the product of countably many Lindelöf

P
P

-spaces is Lindelöf; and the product of countably many countably compact sequential spaces is countably compact sequential.
An uncountable collection of chainable continua
Authors: Bellamy David P.
Published: July 2010

We demonstrate the existence of an uncountable collection of chainable continua, no member of which can be mapped onto any other member. Path connectedness considerations and techniques involving compactifications play an important role in the construction and argument.
Higher dimensional knots in tubes
Authors: Shinohara Yaichi
Published: July 2010

Let K be an n-knot in the

(
n
+
2
)

(n + 2)

-sphere and V a tubular neighborhood of K. Let

L
′

L’

be an n-knot contained in a tubular neighborhood

V
′

V’

of a trivial n-knot and L the image of

L
′

L’

under an orientation preserving diffeomorphism of

V
′

V’

onto V. The purpose of this paper is to show that the higher dimensional Alexander polynomial and the signature of the n-knot L are determined by those of K and

L
′

L’

.
Borel measurable mappings for nonseparable metric spaces
Authors: Hansell R. W.
Published: July 2010

The main object of this paper is the extension of part of the basic theory of Borel measurable mappings, from the “classical” separable metric case, to general metric spaces. Although certain results of the standard theory are known to fail in the absence of separability, we show that they continue to hold for the class of “

σ
\sigma

-discrete” mappings. This class is shown to be quite extensive, containing the continuous mappings, all mappings with a separable range, and any Borel measurable mappings whose domain is a Borel subset of a complete metric space. The last result is a consequence of our Basic Theorem which gives a topological characterization of those collections which are the inverse image of an open discrete collection under a Borel measurable mapping. Such collections are shown to possess a strong type of

σ
\sigma

-discrete refinement. The properties of

σ
\sigma

-discrete mappings together with the known properties of “locally Borel” sets allow us to extend, to general metric spaces, well-known techniques used for separable spaces. The basic properties of “complex” and “product” mappings, well known for separable spaces, are proved for general metric spaces for the class of

σ
\sigma

-discrete mappings. A consequence of these is a strengthening of the basic theorem of the structure theory of nonseparable Borel sets due to A. H. Stone. Finally, the classical continuity properties of Borel measurable mappings are extended, and, in particular, a generalization of the famous theorem of Baire on the points of discontinuity of a mapping of class 1 is obtained.
Monofunctors as reflectors
Authors: Ringel Claus Michael
Published: July 2010

In a well-powered and co-well-powered complete category

K

\mathcal {K}

with weak amalgamations, the class

M

\mathbfit {M}

of all reflective subcategories with a monofunctor as reflector forms a complete lattice; the limit-closure of the union of any class of elements of

M

\mathbfit {M}

belongs to

M

\mathbfit {M}

. If

K

\mathcal {K}

has injective envelopes, then the set-theoretical intersection of any class of elements of

M

\mathbfit {M}

belongs to

M

\mathbfit {M}

.
On the Wedderburn principal theorem for nearly (1,1) algebras
Authors: Miles T. J.
Published: July 2010

A nearly

(
1
,
1
)

(1,1)

algebra is a finite dimensional strictly power-associative algebra satisfying the identity

(
x
,
x
,
y
)
=
(
x
,
y
,
x
)

(x,x,y) = (x,y,x)

where the associator

(
x
,
y
,
z
)
=
(
x
y
)
z
−
x
(
y
z
)

(x,y,z) = (xy)z - x(yz)

. An algebra A has a Wedderburn decomposition in case A has a subalgebra

S
≅
A
−
N

S \cong A - N

with

A
=
S
+
N

A = S + N

(vector space direct sum) where N denotes the radical (maximal nil ideal) of A. D. J. Rodabaugh has shown that certain classes of nearly

(
1
,
1
)

(1,1)

algebras have Wedderburn decompositions. The object of this paper is to expand these classes. The main result is that a nearly

(
1
,
1
)

(1,1)

algebra A containing 1 over a splitting field of characteristic not 2 or 3 such that A has no nodal subalgebras has a Wedderburn decomposition.
Classification of generalized Witt algebras over algebraically closed fields
Authors: Wilson Robert Lee
Published: July 2010

Let

Φ
\Phi

be a field of characteristic

p
>
0

p > 0

and

m
,

n
1

,
…
,

n
m

m,{n_1}, \ldots ,{n_m}

be integers

≧
1

\geqq 1

. A Lie algebra

W
(
m
:

n
1

,
…
,

n
m

)

W(m:{n_1}, \ldots ,{n_m})

over

Φ
\Phi

is defined. It is shown that if

Φ
\Phi

is algebraically closed then

W
(
m
:

n
1

,
…
,

n
m

)

W(m:{n_1}, \ldots ,{n_m})

is isomorphic to a generalized Witt algebra, that every finite-dimensional generalized Witt algebra over

Φ
\Phi

is isomorphic to some

W
(
m
:

n
1

,
…
,

n
m

)

W(m:{n_1}, \ldots ,{n_m})

, and that

W
(
m
:

n
1

,
…
,

n
m

)

W(m:{n_1}, \ldots ,{n_m})

is isomorphic to

W
(
s
:

r
1

,
…
,

r
s

)

W(s:{r_1}, \ldots ,{r_s})

if and only if

m
=
s

m = s

and

r
i

=

n

σ
(
i
)

{r_i} = {n_{\sigma (i)}}

for

1
≦
i
≦
m

1 \leqq i \leqq m

where

σ
\sigma

is a permutation of

{
1
,
…
,
m
}

\{ 1, \ldots ,m\}

. This gives a complete classification of the finite-dimensional generalized Witt algebras over algebraically closed fields. The automorphism group of

W
(
m
:

n
1

,
…
,

n
m

)

W(m:{n_1}, \ldots ,{n_m})

is determined for

p
>
3

p > 3

.
Removable sets for pointwise subharmonic functions
Authors: Shapiro Victor L.
Published: July 2010

Pointwise subharmonic is defined in terms of the pointwise

L
1

{L^1}

total derivative of order 2. The class

A

(

x
∗

,

r
∗

)

\mathcal {A}({x^ \ast },{r_ \ast })

is introduced for the ball

B
(

x
∗

,

r
∗

)

B({x^ \ast },{r_ \ast })

, and the following theorem is established: Let

Q
Q

be a Borel set of Lebesgue measure zero contained in

B
(

x
∗

,

r
∗

)

B({x^ \ast },{r_ \ast })

. Then a necessary and sufficient condition that

Q
Q

be removable for pointwise subharmonic functions with respect to the class

A

(

x
∗

,

r
∗

)

\mathcal {A}({x^ \ast },{r_ \ast })

is that

Q
Q

be countable. It is also shown that the class

A

(

x
∗

,

r
∗

)

\mathcal {A}({x^ \ast },{r_ \ast })

is in a certain sense best possible for the sufficiency of the above theorem.
Segment-preserving maps of partial orders
Authors: Prins Geert
Published: July 2010

A bijective map from a partial order P to a partial order Q is defined to be segment-preserving if the image of every segment in P is a segment in Q. It is proved that a partial order P with 0-element admits nontrivial segment-preserving maps if and only if P is decomposable in a certain sense. By introducing the concept of “strong” segment-preserving maps further insight into the relations between segment-preserving maps and decompositions of partial orders is obtained.
Regular functions 𝑓(𝑧) for which 𝑧𝑓’(𝑧) is 𝛼-spiral
Authors: Libera Richard J. , Ziegler Michael R.
Published: July 2010

A function

f
(
z
)
=
z
+

Σ

n
=
2

∞

a
n

z
n

f(z) = z + \Sigma _{n = 2}^\infty {a_n}{z^n}

regular in the open unit disk

Δ
=
{
z
:

|

z

|

>
1
}

\Delta = \{ z:|z| > 1\}

is a (univalent)

α
\alpha

-spiral function for real

α
,

|

α

|

>
π

/

2

\alpha ,|\alpha | > \pi /2

, if

Re
⁡
{

e

i
α

z

f
′

(
z
)

/

f
(
z
)
}
>
0

\operatorname {Re} \{ {e^{i\alpha }}zf’(z)/f(z)\} > 0

for z in

Δ
\Delta

; in this case we write

f
(
z
)
∈

F

α

f(z) \in {\mathcal {F}_\alpha }

. A fundamental result of this paper shows that the transformation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript asterisk Baseline left-parenthesis z right-parenthesis equals StartFraction a z f left-parenthesis left-parenthesis z plus a right-parenthesis slash left-parenthesis 1 plus a overbar z right-parenthesis right-parenthesis Over f left-parenthesis a right-parenthesis left-parenthesis z plus a right-parenthesis left-parenthesis 1 plus a overbar z right-parenthesis Superscript e Super Superscript minus 2 i alpha Superscript Baseline EndFraction">

f
∗

(
z
)
=

a
z
f
(
(
z
+
a
)

/

(
1
+

a
¯

z
)
)

f
(
a
)
(
z
+
a
)

(
1
+

a
¯

z
)

e

−
2
i
α

{f_ \ast }(z) = \frac {{azf((z + a)/(1 + \bar az))}}{{f(a)(z + a){{(1 + \bar az)}^{{e^{ - 2i\alpha }}}}}}

\]
defines a function in

F

α

{\mathcal {F}_\alpha }

whenever

f
(
z
)

f(z)

is in

F

α

{\mathcal {F}_\alpha }

and a is in

Δ
\Delta

. If

g
(
z
)

g(z)

is regular in

Δ
,
g
(
0
)
=
0

\Delta ,g(0) = 0

and

g
′

(
0
)
=
1

g’(0) = 1

, then

g
(
z
)

g(z)

is in

G

α

{\mathcal {G}_\alpha }

if and only if

z

g
′

(
z
)

zg’(z)

is in

F

α

{\mathcal {F}_\alpha }

. The main result of the paper is the derivation of the sharp radius of close-to-convexity for each class

G

α

{\mathcal {G}_\alpha }

; it is given as the solution of an equation in r which is dependent only on

α
\alpha

. (Approximate solutions of this equation were made by computer and these suggest that the radius of close-to-convexity of the class

G

=

∪
α

G

α

\mathcal {G} = { \cup _\alpha }{\mathcal {G}_\alpha }

is approximately

.99097

+

.99097^{+}

.) Additional results are also obtained such as the radius of convexity of

G

α

{\mathcal {G}_\alpha }

, a range of

α
\alpha

for which

g
(
z
)

g(z)

in

G

α

{\mathcal {G}_\alpha }

is always univalent is given, etc. These conclusions all depend heavily on the transformation cited above and its analogue for

G

α

{\mathcal {G}_\alpha }

.
A reciprocity theorem for ergodic actions
Authors: Lange Kenneth
Published: July 2010

An analogue of the Frobenius Reciprocity Theorem is proved for virtual groups over a locally compact separable group G. Specifically, an ergodic analytic Borel G-space

M
(
V
π
)

M(V\pi )

is constructed from a virtual group V and a homomorphism

π
:
V
→
G

\pi :V \to G

of V into G. This construction proves to be functorial for the category of virtual groups over G; in fact, it is a left adjoint of the functor which takes an ergodic analytic Borel G-space T into the virtual group

T
×
G

T \times G

together with projection

ρ
:
T
×
G
→
G

\rho :T \times G \to G

onto G. Examples such as Kakutani’s induced transformation and flows under functions show the scope of this construction. A method for constructing the product of two virtual groups is also presented. Some of the structural properties of the product virtual group are deduced from those of the components. Finally, for virtual groups

π
1

:

V
1

→

G
1

{\pi _1}:{V_1} \to {G_1}

and

π
2

:

V
2

→

G
2

{\pi _2}:{V_2} \to {G_2}

over groups

G
1

{G_1}

and

G
2

{G_2}

respectively, the adjoint functor construction applied to

π
1

×

π
2

:

V
1

×

V
2

→

G
1

×

G
2

{\pi _1} \times {\pi _2}:{V_1} \times {V_2} \to {G_1} \times {G_2}

is shown to give the product of the

G
1

{G_1}

-space derived from

π
1

:

V
1

→

G
1

{\pi _1}:{V_1} \to {G_1}

and the

G
2

{G_2}

-space derived from

π
2

:

V
2

→

G
2

{\pi _2}:{V_2} \to {G_2}

, up to suitably defined isomorphism.
On the irreducibility of nonunitary induced representations of certain semidirect products
Authors: Thieleker Ernest
Published: July 2010

Let G be a connected Lie group which is a semidirect product of a compact subgroup K and a normal solvable subgroup S. Let

Λ
\Lambda

be a character of S, and let

M
Λ

{M_\Lambda }

be the stabilizer of

Λ
\Lambda

in K. Let

[
H
,

Λ
μ

]

[H,{\Lambda _\mu }]

be a finite-dimensional irreducible representation of the subgroup

S

M
Λ

S{M_\Lambda }

on the complex vector space H. In this paper we consider the induced representations of G on various Banach spaces, and study their topological irreducibility. The basic method used consists in studying the irreducibility of the Lie algebra representations which arise on the linear subspaces of K-finite vectors. The latter question then can be reduced to the problem of determining when certain modules over certain commutative algebras are irreducible. The method discussed in this paper leads to two theorems giving sufficient conditions on the character

Λ
\Lambda

that the induced representations be topologically irreducible. The question of infinitesimal equivalence of various induced representations is also discussed.
On time-free functions
Authors: Schwarz Gideon
Published: July 2010

By regarding as equivalent any two real-valued functions of a real variable that can be obtained from each other by a monotone continuous transformation of the independent variable, time-free functions are defined. A convenient maximal invariant is presented, and applied to some time-free functional equations.
Decompositions of 3-manifolds and pseudo-isotopies
Authors: Voxman William
Published: July 2010

In this paper we construct pseudo-isotopies which realize certain cellular decompositions of 3-manifolds. In general we show that the pseudo-isotopy may be defined so as to leave points fixed outside of a given open set containing the nondegenerate elements of the decomposition. For nondegenerately continuous decompositions it is shown that the pseudo-isotopy does not move the nondegenerate elements far from their original positions.
Martingales of strongly measurable Pettis integrable functions
Authors: Uhl J. J.
Published: July 2010

This paper deals with convergence theorems for martingales of strongly measurable Pettis integrable functions. First, a characterization of those martingales which converge in the Pettis norm is obtained. Then it is shown that a martingale which is convergent in the Pettis norm converges to its limit strongly in measure and, if the index set is the positive integers, it converges strongly almost everywhere to its limit. The second part of the paper deals with the strong measure and strong almost everywhere convergence of martingales which are not necessarily convergent in the Pettis norm. The resulting theorems here show that

L
1

{L^1}

-boundedness can be considerably relaxed to a weaker control condition on the martingale by the use of some facts on finitely additive vector measures.
Simple modules and centralizers
Authors: Dauns John
Published: July 2010

A class of modules generalizing the simple ones is constructed. Submodule structure and centralizers of quotient modules are completely determined. The above class of modules is used to study the primitive ideal structure of the tensor products of algebras.
Two theorems in the commutator calculus
Authors: Waldinger Hermann V.
Published: July 2010

Let

F
=
⟨
a
,
b
⟩

F = \langle a,b\rangle

. Let

F
n

{F_n}

be the nth subgroup of the lower central series. Let p be a prime. Let

c
3

>

c
4

>
⋯
>

c
z

{c_3} > {c_4} > \cdots > {c_z}

be the basic commutators of dimension

>
1

> 1

but

>
p
+
2

> p + 2

. Let

P
1

=
(
a
,
b
)
,

P
m

=
(

P

m
−
1

,
b
)

{P_1} = (a,b),{P_m} = ({P_{m - 1}},b)

for

m
>
1

m > 1

. Then

(
a
,

b
p

)
≡

∏

i
=
3

z

c
i

η
i

mod

F

p
+
2

(a,{b^p}) \equiv \prod \nolimits _{i = 3}^z {c_i^{{\eta _i}}\bmod {F_{p + 2}}}

. It is shown in Theorem 1 that the exponents

η
i

{\eta _i}

are divisible by p, except for the exponent of

P
p

{P_p}

which

=
1

= 1

. Let the group

G

\mathcal {G}

be a free product of finitely many groups each of which is a direct product of finitely many groups of order p, a prime. Let

G

′

\mathcal {G}’

be its commutator subgroup. It is proven in Theorem 2 that the “

G

\mathcal {G}

-simple basic commutators” of dimension

>
1

> 1

defined below are free generators of

G

′

\mathcal {G}’

.
Asymptotic behavior of functions with bounded boundary rotation
Authors: Noonan James W.
Published: July 2010

Fork≧2k \geqq 2denote byVk{V_k}the class of normalized functions, analytic in the unit disc, which have boundary rotation at mostkπk\pi. Letan{a_n}be thenth Taylor coefficient off(z)∈Vkf(z) \in {V_k}. LetIλ(r,f′){I_\lambda }(r,f’)andIλ(r,f){I_\lambda }(r,f)be theλ\lambda-integral mean off′(z)f’(z)andf(z)f(z)respectively. We determine asymptotic formulas forf′(z)f’(z), and these formulas are then applied to study the behavior of|an||{a_n}|asn→∞n \to \infty, and the behavior ofIλ(r,f′){I_\lambda }(r,f’)andIλ(r,f){I_\lambda }(r,f)asr→1r \to 1.
A unified approach to uniform real approximation by polynomials with linear restrictions
Authors: Chalmers Bruce L.
Published: July 2010

Problems concerning approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions have been studied by several authors. This paper is an attempt to provide a unified approach to these problems. In particular, the notion of restricted derivatives approximation is seen to fit into the theory and includes as special cases the notions of monotone approximation and restricted range approximation. Also bounded coefficients approximation,

ε
\varepsilon

-interpolator approximation, and polynomial approximation with interpolation fit into our scheme.
Diffusion semigroups on abstract Wiener space
Authors: Piech M. Ann
Published: July 2010

The existence of a semigroup of solution operators associated with a second order infinite dimensional parabolic equation of the form

∂
u

/

∂
t
=

L
x

u

\partial u/\partial t = {L_x}u

was previously established by the author. The present paper investigates the relationship between

L
x

{L_x}

and the infinitesimal generator

U

\mathcal {U}

of the semigroup. In particular, it is shown that

U

\mathcal {U}

is the closure of

L
x

{L_x}

in a natural sense. This in turn implies certain uniqueness results for both the semigroup and for solutions of the parabolic equation.
An equality for 2-sided surfaces with a finite number of wild points
Authors: Taylor Michael D. , Rosen Harvey
Published: July 2010

Let S be a 2-sided surface in a 3-manifold that is wild from one side U at just m points. It is shown that the minimal genus possible for all members of a sequence of surfaces in U converging to S (where these surfaces each separate the same point from S in

U
∪
S

U \cup S

) is equal to the sum of the genus of S and a certain multiple of the sum of m special topological invariants associated with the wild points. In this equality, the sum of these invariants is multiplied by just one of the numbers 0, 1, or 2, dependent upon the genus and orientability class of S and the value of m. As an application, an upper bound is given for the number of nonpiercing points that a 2-sided surface has with respect to one side.
Knots whose branched cyclic coverings have periodic homology
Authors: Gordon C. McA.
Published: July 2010

Let

M
k

{M_k}

be the

k
k

-fold branched cyclic covering of a (tame) knot of

S
1

{S^1}

in

S
3

{S^3}

. Our main result is that the following statements are equivalent: (1)

H
1

(

M
k

)

{H_1}({M_k})

is periodic with period

n
n

, i.e.

H
1

(

M
k

)
≅

H
1

(

M

k
+
n

)

{H_1}({M_k}) \cong {H_1}({M_{k + n}})

for all

k
k

, (2)

H
1

(

M
k

)
≅

H
1

(

M

(
k
,
n
)

)

{H_1}({M_k}) \cong {H_1}({M_{(k,n)}})

for all

k
k

, (3) the first Alexander invariant of the knot,

λ
1

(
t
)
=

Δ
1

(
t
)

/

Δ
2

(
t
)

{\lambda _1}(t) = {\Delta _1}(t)/{\Delta _2}(t)

, divides

t
n

−
1

{t^n} - 1

.
Equivariant bordism and Smith theory. III
Authors: Stong R. E.
Published: July 2010

The bordism classes of a manifold with involution is determined by means of Wu type classes in the Smith cohomology.
Hereditary properties and maximality conditions with respect to essential extensions of lattice group orders
Authors: Martinez Jorge
Published: July 2010

An l-group will be denoted by the pair (G, P), where G is the group and P is the positive cone. The cone Q is an essential extension of P if every convex l-subgroup of (G, Q) is a convex l-subgroup of (G, P). Q is very essential over P if it is essential over P and for each

0
≠
x
∈
G

0 \ne x \in G

and each Q-value D of x, there is a unique P-value C of x containing D. We seek conditions which are preserved by essential extensions; normal valuedness and the existence of a finite basis are so preserved. We then investigate l-groups which have the property that their positive cone has no proper very essential extensions. Q is a c-essential extension of P if Q is essential over P and every closed convex l-subgroup of (G, Q) is closed in (G, P). We show that a wreath product of totally ordered groups has no proper very c-essential extensions. We derive a sufficient condition for the nonexistence of such extensions in case the l-group has property (F): no positive element exceeds an infinite set of pairwise disjoint elements.
An extension of the theorem of Hartogs
Authors: Hunt L. R.
Published: July 2010

Hartogs proved that every function which is holomorphic on the boundary of the unit ball in

C
n

,
n
>
1

{C^n},n > 1

, can be extended to a function holomorphic on the ball itself. It is conjectured that a real k-dimensional

C

∞

{\mathcal {C}^\infty }

compact submanifold of

C
n

,
k
>
n

{C^n},k > n

, is extendible over a manifold of real dimension

(
k
+
1
)

(k + 1)

. This is known for hypersurfaces (i.e.,

k
=
2
n
−
1

k = 2n - 1

) and submanifolds of real codimension 2. It is the purpose of this paper to prove this conjecture and to show that we actually get C-R extendibility.
Some degeneracy theorems for entire functions with values in an algebraic variety
Authors: Carlson James A.
Published: July 2010

In the first part of this paper we prove the following extension theorem. Let

P
q
∗

P_q^ \ast

be a

q
q

-dimensional punctured polycylinder, i.e. a product of disks and punctured disks. Let

W
n

{W_n}

be a compact complex manifold such that the bundle of holomorphic

q
q

-forms is positive in the sense of Grauert. Let

f
:

P
q
∗

→

W
n

f:P_q^ \ast \to {W_n}

be a holomorphic map whose Jacobian determinant does not vanish identically. Then

f
f

extends as a rational map to the full polycylinder

P
q

{P_q}

. In the second half of the paper we prove the following generalization of the little Picard theorem to several complex variables: Let

V
⊂

P
n

V \subset {P_n}

be a hypersurface of degree

d
≧
n
+
3

d \geqq n + 3

whose singularities are locally normal crossings. Then any holomorphic map

f
:

C
n

→

P
n

−
V

f:{C^n} \to {P_n} - V

has identically vanishing Jacobian determinant.
A representation theorem and approximation operators arising from inequalities involving differential operators
Authors: Leviatan D.
Published: July 2010

A representation of functions as integrals of a kernel

ψ
(
t
;
x
)

\psi (t;x)

, which was introduced by Studden, with respect to functions of bounded variation in

[
0
,
∞
)

[0,\infty )

is obtained whenever the functions satisfy some conditions involving the differential operators

(
d

/

d
t
)
{
f
(
t
)

/

w
i

(
t
)
}
,
i
=
0
,
1
,
2
,
…

(d/dt)\{ f(t)/{w_i}(t)\} ,i = 0,1,2, \ldots

. The results are related to the concepts of generalized completely monotonic functions and generalized absolutely monotonic functions in

(
0
,
∞
)

(0,\infty )

. Some approximation operators for the approximation of continuous functions in

[
0
,
∞
)

[0,\infty )

arise naturally and are introduced; some sequence-to-function summability methods are also introduced.
Locally 𝐵*-equivalent algebras
Authors: Barnes Bruce A.
Published: July 2010

Let A be a Banach

∗

^ \ast

-algebra. A is locally

B
∗

{B^ \ast }

-equivalent if, for every selfadjoint element

t
∈
A

t \in A

, the closed

∗

^ \ast

-subalgebra of A generated by t is

∗

^\ast

-isomorphic to a

B
∗

{B^ \ast }

-algebra. In this paper it is shown that when A is locally

B
∗

{B^\ast }

-equivalent, and in addition every selfadjoint element in A has at most countable spectrum, then A is

∗

^ \ast

-isomorphic to a

B
∗

{B^ \ast }

-algebra.
Integral decomposition of functionals on 𝐶*-algebras
Authors: Halpern Herbert
Published: July 2010

The spectrum of the center of the weak closure of a

C
∗

{C^ \ast }

-algebra with identity on a Hilbert space is mapped into a set of quasi-equivalence classes of representations of the

C
∗

{C^ \ast }

-algebra so that every positive

σ
\sigma

-weakly continuous functional on the algebra can be written in a central decomposition as an integral over the spectrum of a field of states whose canonical representations are members of the respective quasi-equivalence classes except for a nowhere dense set. Various questions relating to disjointness of classes, factor classes, and uniformly continuous functionals are studied.
Locally uniform spaces
Authors: Williams James
Published: July 2010

The axioms for a locally uniform space

(
X
,

U

)

(X,\mathcal {U})

may be obtained by localizing the last axiom for a uniform space to obtain
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="for-all x element-of upper X comma for-all upper U element-of script upper U comma there-exists upper V element-of script upper V colon left-parenthesis upper V ring upper V right-parenthesis left-bracket x right-bracket subset-of-or-equal-to upper U left-bracket x right-bracket">

∀
x
∈
X
,
∀
U
∈

U

,
∃
V
∈

V

:
(
V
∘
V
)
[
x
]
⊆
U
[
x
]

\forall x \in X,\forall U \in \mathcal {U},\exists V \in \mathcal {V}:(V \circ V)[x] \subseteq U[x]

. With each locally uniform space one may associate a regular topology, just as one associates a completely regular topology with each uniform space. The topologies of locally uniform spaces with nested bases may be characterized using Boolean algebras of regular open sets. As a special case, one has that locally uniform spaces with countable bases have pseudo-metrizable topologies. Several types of Cauchy filters may be defined for locally uniform spaces, and a major portion of the paper is devoted to a study and comparison of their properties. For each given type of Cauchy filter, complete spaces are those in which every Cauchy filter converges; to complete a space is to embed it as a dense subspace in a complete space. In discussing these concepts, it is convenient to make the mild restriction of considering only those locally uniform spaces

(
X
,

V

)

(X,\mathcal {V})

in which each element of

V

\mathcal {V}

is a neighborhood of the diagonal in

X
×
X

X \times X

with respect to the relative topology; these spaces I have called NLU-spaces. With respect to the more general types of Cauchy filters, some NLU-spaces are not completable; this happens even though some completable NLU-spaces can still have topologies which are not completely regular. Examples illustrating these completeness situations and having various topological properties are obtained from a generalized construction. It is also shown that there is a largest class of Cauchy filters with respect to which each NLU-space has a completion that is also an NLU-space.
Interpolation to analytic data on unbounded curves
Authors: Thompson Maynard
Published: July 2010

This paper provides a method for constructing a family of sets of points on the boundary (assumed suitably smooth) of an unbounded Jordan region in the complex plane which is useful for certain interpolation problems. It is proved that if these sets are used as nodes for Lagrange interpolation to analytic data, then the resulting polynomials converge in the region, and the limit function is related in a natural way to the boundary data. Subsidiary results include an approximate quadrature formula for slowly decreasing functions on an infinite interval.
A generalized area theorem for harmonic functions on hermitian hyperbolic space
Authors: Putz Robert Byrne
Published: July 2010

Let

D
D

be the noncompact realization of hermitian hyperbolic space. We consider functions on

D
D

which are harmonic with respect to the Laplace-Beltrami operator. The principal result is a generalized area theorem which gives a necessary and sufficient condition for the admissible convergence of harmonic functions.
Complex structures on real product bundles with applications to differential geometry
Authors: Millman Richard S.
Published: July 2010

The purpose of this paper is to classify holomorphic principal fibre bundles which admit a smooth section (i.e. are real product bundles). This is accomplished if the structure group is solvable of type (E). In the general case, a sufficient condition is obtained for a real product bundle to be equivalent to the complex product bundle. A necessary and sufficient condition for the existence of a holomorphic connection on a real product bundle is also obtained. Using this criterion in the case where the structure group is abelian, a generalization of a theorem due to Atiyah (in the case the structure group is

C
∗

{C^ \ast }

) is obtained.
A notion of capacity which characterizes removable singularities
Authors: Harvey Reese , Polking John C.
Published: July 2010

In this paper the authors define a capacity for a given linear partial differential operator acting on a Banach space of distributions. This notion has as special cases Newtonian capacity, analytic capacity, and AC capacity. It is shown that the sets of capacity zero are precisely those sets which are removable sets for the corresponding homogeneous equation. Simple properties of the capacity are derived and special cases examined.
Multipliers for spherical harmonic expansions
Authors: Strichartz Robert S.
Published: July 2010

Sufficient conditions are given for an operator on the sphere that commutes with rotations to be bounded in

L
p

{L^p}

. The conditions are analogous to those of Hörmander’s well-known theorem on Fourier multipliers.
On the null-spaces of elliptic partial differential operators in 𝑅ⁿ
Authors: Walker Homer F.
Published: July 2010

The objective of this paper is to generalize the results of Lax and Phillips [4] and Walker [6] to include elliptic partial differential operators of all orders whose coefficients approach constant values at infinity with a certain swiftness. An example is given of an elliptic operator having an infinite-dimensional null-space whose coefficients slowly approach constant limiting values.
Diffusion and Brownian motion on infinite-dimensional manifolds
Authors: Kuo Hui Hsiung
Published: July 2010

The purpose of this paper is to construct certain diffusion processes, in particular a Brownian motion, on a suitable kind of infinite-dimensional manifold. This manifold is a Banach manifold modelled on an abstract Wiener space. Roughly speaking, each tangent space

T
x

{T_x}

is equipped with a norm and a densely defined inner product

g
(
x
)

g(x)

. Local diffusions are constructed first by solving stochastic differential equations. Then these local diffusions are pieced together in a certain way to get a global diffusion. The Brownian motion is completely determined by

g
g

and its transition probabilities are proved to be invariant under

d
g

{d_g}

-isometries. Here

d
g

{d_g}

is the almost-metric (in the sense that two points may have infinite distance) associated with

g
g

. The generalized Beltrami-Laplace operator is defined by means of the Brownian motion and will shed light on the study of potential theory over such a manifold.