bluuebunny/crossref_metadata_2025 huggingface dataset:
Fixed points of pointwise almost periodic homeomorphisms on the two-sphere
Authors: Mason W. K.
Published: July 2010

A homeomorphism

f
f

of the two-sphere

S
2

{S^2}

onto itself is defined to be pointwise almost periodic (p.a.p.) if the collection of orbit closures forms a decomposition of

S
2

{S^2}

. It is shown that if

f
:

S
2

→

S
2

f:{S^2} \to {S^2}

is p.a.p. and orientation-reversing then the set of fixed points of

f
f

is either empty or a simple closed curve; if

f
:

S
2

→

S
2

f:{S^2} \to {S^2}

is p.a.p. orientation-preserving and has a finite number of fixed points, then

f
f

is shown to have exactly two fixed points.
The Brauer group of graded Azumaya algebras. II. Graded Galois extensions
Authors: Childs Lindsay N.
Published: July 2010

This paper continues the study of the Brauer group

B
ϕ

(
R
,
G
)

{B_\phi }(R,G)

of

G
G

-graded Azumaya

R
R

-algebras begun in [5]. A group

Galz
ϕ

(
R
,
G
)

{\operatorname {Galz} _\phi }(R,G)

of graded Galois extensions is constructed which always contains, and often equals, the cokernel of

B
ϕ

(
R
,
G
)

{B_\phi }(R,G)

modulo the usual Brauer group of

R
R

. Sufficient conditions for equality are found. The structure of

Galz
ϕ

(
R
,
G
)

{\operatorname {Galz} _\phi }(R,G)

is studied, and

Galz
ϕ

(
R
,

(
Z

/

p
e

Z

)
r

)

{\operatorname {Galz} _\phi }(R,{(Z/{p^e}Z)^r})

is computed. These results are applied to give computations of a Brauer group of dimodule algebras constructed by F. W. Long.
Results on measures of irreducibility and full indecomposability
Authors: Hartfiel D. J.
Published: July 2010

This paper develops a notion of

k
k

th measure of irreducibility and

k
k

th measure of full indecomposability. The combinatorial properties of these notions, as well as relationships between these notions, are explored. The results are then used in converting results on positive matrices into results on nonnegative matrices.
Relationships between 𝜆-nuclearity and pseudo-𝜇-nuclearity
Authors: Robinson William B.
Published: July 2010

It is shown that for any Köthe space

λ
\lambda

,

λ
\lambda

-nuclearity coincides with pseudo-

λ

λ
×

\lambda {\lambda ^ \times }

-nuclearity. More particular results, including a Grothendieck-Pietsch criterion for

λ
\lambda

-nuclearity of sequence spaces, are given for Köthe spaces which are regular.
Conjugacy correspondences: a unified view
Authors: McLinden L.
Published: July 2010

As preparation for a duality theory for saddle programs, a partial conjugacy correspondence is developed among equivalence classes of saddle functions. Three known conjugacy correspondences, including Fenchel’s correspondence among convex functions and Rockafellar’s extension of it to equivalence classes of saddle functions, are shown to be degenerate special cases. Additionally, two new correspondences are brought to light as further special cases. Various questions are answered concerning the lower and upper closures and effective domain of the resulting equivalence class, as well as the effect the correspondence has on the related purely convex function and the subdifferential mapping.
The subclass algebra associated with a finite group and subgroup
Authors: Karlof John
Published: July 2010

Let

G
G

be a finite group and let

H
H

be a subgroup of

G
G

. If

g
∈
G

g \in G

, then the set

E
g

=
{
h
g

h

−
1

|

h
∈
H
}

{E_g} = \{ hg{h^{ - 1}}|h \in H\}

is the subclass of

G
G

containing

g
g

and

Σ

x
∈

E
g

x

{\Sigma _{x \in {E_g}}}x

is the subclass sum containing

g
g

. The algebra over the field of complex numbers generated by these subclass sums is called the subclass algebra (denoted by

S
S

) associated with

G
G

and

H
H

. The irreducible modules of

S
S

are demonstrated, and results about Schur algebras are used to develop formulas relating the irreducible characters of

S
S

to the irreducible characters of

G
G

and

H
H

.
Quantum logic and the locally convex spaces
Authors: Wilbur W. John
Published: July 2010

An important theorem of Kakutani and Mackey characterizes an infinite dimensional real (complex) Hilbert space as an infinite dimensional real (complex) Banach space whose lattice of closed subspaces admits an orthocomplementation. This result, also valid for quaternionic spaces, has proved useful as a justification for the unique role of Hilbert space in quantum theory. With a like application in mind, we present in the present paper a number of characterizations of real and complex Hilbert space in the class of locally convex spaces. One of these is an extension of the Kakutani-Mackey result from the infinite dimensional Banach spaces to the class of all infinite dimensional complete Mackey spaces. The implications for the foundations of quantum theory are discussed.
Generalization of right alternative rings
Authors: Hentzel Irvin Roy , Piacentini Cattaneo Giulia Maria
Published: July 2010

We study nonassociative rings

R
R

satisfying the conditions (1)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis a b comma c comma d right-parenthesis plus left-parenthesis a comma b comma left-bracket c comma d right-bracket right-parenthesis equals a left-parenthesis b comma c comma d right-parenthesis plus left-parenthesis a comma c comma d right-parenthesis b">

(
a
b
,
c
,
d
)
+
(
a
,
b
,
[
c
,
d
]
)
=
a
(
b
,
c
,
d
)
+
(
a
,
c
,
d
)
b

(ab,c,d) + (a,b,[c,d]) = a(b,c,d) + (a,c,d)b

for all

a
,
b
,
c
,
d
∈
R

a,b,c,d \in R

, and (2)

(
x
,
x
,
x
)
=
0

(x,x,x) = 0

for all

x
∈
R

x \in R

. We furthermore assume weakly characteristic not 2 and weakly characteristic not 3. As both (1) and (2) are consequences of the right alternative law, our rings are generalizations of right alternative rings. We show that rings satisfying (1) and (2) which are simple and have an idempotent

≠
0
,
≠
1

\ne 0, \ne 1

, are right alternative rings. We show by example that

(
x
,
e
,
e
)

(x,e,e)

may be nonzero. In general,

R
=

R
′

+
(
R
,
e
,
e
)

R = R’ + (R,e,e)

(additive direct sum) where

R
′

R’

is a subring and

(
R
,
e
,
e
)

(R,e,e)

is a nilpotent ideal which commutes elementwise with

R
R

. We examine

R
′

R’

under the added assumption of Lie admissibility: (3)

(
a
,
b
,
c
)
+
(
b
,
c
,
a
)
+
(
c
,
a
,
b
)
=
0

(a,b,c) + (b,c,a) + (c,a,b) = 0

for all

a
,
b
,
c
∈
R

a,b,c \in R

. We generate the Peirce decomposition. If

R
′

R’

has no trivial ideals contained in its center, the table for the multiplication of the summands is associative, and the nucleus of

R
′

R’

contains

R

10

′

+

R

01

′

{R’_{10}} + {R’_{01}}

. Without the assumption on ideals, the table for the multiplication need not be associative; however, if the multiplication is defined in the most obvious way to force an associative table, the new ring will still satisfy (1), (2), (3).
The radicals of a semigroup
Authors: Slover Rebecca
Published: July 2010

This paper investigates various radicals and radical congruences of a semigroup. A strongly prime ideal is defined. It is shown that the nil radical of a semigroup is the intersection of all strongly prime ideals of the semigroup. Furthermore, a semigroup with zero element is nil if and only if it has no strongly prime ideals. We investigate the question of when the left and right radical congruence relations of various radicals are equal. Some theorems analogous to theorems concerning the radicals of rings are also proved.
On symmetrically distributed random measures
Authors: Kallenberg Olav
Published: July 2010

A random measure

ξ
\xi

defined on some measurable space

(
S
,

S

)

(S,\mathcal {S})

is said to be symmetrically distributed with respect to some fixed measure

ω
\omega

on

S
S

, if the distribution of

(
ξ

A
1

,
⋯
,
ξ

A
k

)

(\xi {A_1}, \cdots ,\xi {A_k})

for

k
∈
N

k \in N

and disjoint

A
1

,
⋯
,

A
k

∈

S

{A_1}, \cdots ,{A_k} \in \mathcal {S}

only depends on

(
ω

A
1

,
⋯
,
ω

A
k

)

(\omega {A_1}, \cdots ,\omega {A_k})

. The first purpose of the present paper is to extend to such random measures (and then even improve) the results on convergence in distribution and almost surely, previously given for random processes on the line with interchangeable increments, and further to give a new proof of the basic canonical representation. The second purpose is to extend a well-known theorem of Slivnyak by proving that the symmetrically distributed random measures may be characterized by a simple invariance property of the corresponding Palm distributions.
Some 𝐻^{∞}-interpolating sequences and the behavior of certain of their Blaschke products
Authors: Weiss Max L.
Published: July 2010

Let

f
f

be a strictly increasing continuous real function defined near

0
+

{0^ + }

with

f
(
0
)
=

f
′

(
0
)
=
0

f(0) = f’(0) = 0

. Such a function is called a

K
K

-function if for every constant

k
,
f
(
θ
+
k
f
(
θ
)
)

/

f
(
θ
)
→
1

/

k,f(\theta + kf(\theta ))/f(\theta ) \to 1/

as

θ
→

0
+

\theta \to {0^ + }

. The curve in the open unit disc with corresponding representation

1
−
r
=
f
(
θ
)

1 - r = f(\theta )

is called a

K
K

-curve. Several analytic and geometric conditions are obtained for

K
K

-curves and

K
K

-functions. This provides a framework for some rather explicit results involving parts in the closure of

K
K

-curves,

H
∞

{H^\infty }

-interpolating sequences lying on

K
K

-curves and the behavior of their Blaschke products. In addition, a sequence of points in the disc tending upper tangentially to 1 with moduli increasing strictly to 1 and arguments decreasing strictly to 0 is proved to be interpolating if and only if the hyperbolic distance between successive points remains bounded away from zero.
The generalized Martin’s minimum problem and its applications in several complex variables
Authors: Matsuura Shozo
Published: July 2010

The objectives of this paper are to generalize the Martin’s

L

2

{\mathfrak {L}^2}

-minimum problem under more general additional conditions given by bounded linear functionals in a bounded domain

D
D

in

C
n

{C^n}

and to apply this problem to various directions. We firstly define the new

i
i

th biholomorphically invariant Kähler metric and the

i
i

th representative domain

(
i
=
0
,
1
,
2
,
…
)

(i = 0,1,2, \ldots )

, and secondly give estimates on curvatures with respect to the Bergman metric and investigate the asymptotic behaviors via an

A
A

-approach on the curvatures about a boundary point having a sort of pseudoconvexity. Further, we study (i) the extensions of some results recently obtained by K. Kikuchi on the Ricci scalar curvature, (ii) a minimum property on the reproducing subspace-kernel in

L

(
m
)

2

(
D
)

\mathfrak {L}_{(m)}^2(D)

, and (iii) an extension of the fundamental theorem of K. H. Look.
Submersive and unipotent group quotients among schemes of a countable type over a field 𝑘
Authors: Cherenack Paul
Published: July 2010

An algebraic group G is called submersive if every quotient in affine schemes

c
G

:

Spec

A
→

Spec

A
G

{c^G}:{\text {Spec}}\;A \to {\text {Spec}}\;{A^G}

which is surjective is also submersive. We prove that every unipotent group is submersive. Suppose G is submersive. We show that if

c
G

(

Spec

A
)

{c^G}({\text {Spec}}\;A)

is open in

Spec

A
G

{\text {Spec}}\;{A^G}

or if some restrictions on the action of G on A are made,

c
G

{c^G}

is a topological quotient. A criterion for semisimplicity of points is extended to the case where G is unipotent. Finally, applications of the theory are provided.
Fixed points in representations of categories
Authors: Adámek J. , Reiterman J.
Published: July 2010

Fixed points of endomorphisms of representations, i.e. functors into the category of sets, are investigated. A necessary and sufficient condition on a category K is given for each of its indecomposable representations to have the fixed point property. The condition appears to be the same as that found by Isbell and Mitchell for Colim:

Ab
K

→

Ab

{\text {Ab}^K} \to {\text {Ab}}

to be exact. A well-known theorem on mappings of Katětov and Kenyon is extended to transformations of functors.
Splitting an 𝛼-recursively enumerable set
Authors: Shore Richard A.
Published: July 2010

We extend the priority method in

α
\alpha

-recursion theory to certain arguments with no a priori bound on the required preservations by proving the splitting theorem for all admissible

α
\alpha

. THEOREM: Let

C
C

be a regular

α
\alpha

-r.e. set and

D
D

be a nonrecursive

α
\alpha

-r.e. set. Then there are regular

α
\alpha

-r.e. sets

A
A

and

B
B

such that

A
∪
B
=
C
,
A
∩
B
=
ϕ
,
A
,
B

≤
α

C

A \cup B = C,A \cap B = \phi ,A,B{ \leq _\alpha }C

and such that

D
D

is not

α
\alpha

-recursive in

A
A

or

B
B

. The result is also strengthened to apply to

≤

c
α

{ \leq _{c\alpha }}

, and various corollaries about the structure of the

α
\alpha

and

c
α

c\alpha

recursively enumerable degrees are proved.
The Gaussian law and the law of the iterated logarithm for lacunary sets of characters
Authors: Dudley E.
Published: July 2010

Salem and Zygmund showed that the Gaussian law holds for Hadamard sequences of real numbers while Mary Weiss proved a similar result for the law of the iterated logarithm. In the present paper, the author obtains corresponding results for lacunary sets of characters of an arbitrary infinite compact abelian group. It is shown that the laws are best satisfied for a certain class of lacunary sets but that modified results apply to more general classes.
Heegaard splittings of branched coverings of 𝑆³
Authors: Birman Joan S. , Hilden Hugh M.
Published: July 2010

This paper concerns itself with the relationship between two seemingly different methods for representing a closed, orientable 3-manifold: on the one hand as a Heegaard splitting, and on the other hand as a branched covering of the 3-sphere. The ability to pass back and forth between these two representations will be applied in several different ways: 1. It will be established that there is an effective algorithm to decide whether a 3-manifold of Heegaard genus 2 is a 3-sphere. 2. We will show that the natural map from 6-plat representations of knots and links to genus 2 closed oriented 3-manifolds is injective and surjective. This relates the question of whether or not Heegaard splittings of closed, oriented 3-manifolds are “unique” to the question of whether plat representations of knots and links are “unique". 3. We will give a counterexample to a conjecture (unpublished) of W. Haken, which would have implied that

S
3

{S^3}

could be identified (in the class of all simply-connected 3-manifolds) by the property that certain canonical presentations for

π
1

S
3

{\pi _1}{S^3}

are always “nice". The final section of the paper studies a special class of genus 2 Heegaard splittings: the 2-fold covers of

S
3

{S^3}

which are branched over closed 3-braids. It is established that no counterexamples to the “genus 2 Poincaré conjecture” occur in this class of 3-manifolds.
On Dedekind’s problem: the number of isotone Boolean functions. II
Authors: Kleitman D. , Markowsky G.
Published: July 2010

It is shown that

ψ
(
n
)

\psi (n)

, the size of the free distributive lattice on n generators (which is the number of isotone Boolean functions on subsets of an n element set), satisfies
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi left-parenthesis n right-parenthesis less-than-or-slanted-equals 2 Superscript left-parenthesis 1 plus upper O left-parenthesis log n slash n right-parenthesis right-parenthesis StartBinomialOrMatrix n Choose left-bracket n slash 2 right-bracket EndBinomialOrMatrix Baseline period">

ψ
(
n
)
⩽

2

(
1
+
O
(
log

n

/

n
)
)

(

n

[
n

/

2
]

)

.

\psi (n) \leqslant {2^{(1 + O(\log \;n/n))\left ( {\begin {array}{*{20}{c}} n \\ {[n/2]} \\ \end {array} } \right )}}.

\]
This result is an improvement by a factor

n

\sqrt n

in the 0 term of a previous result of Kleitman. In the course of deriving the main result, we analyze thoroughly the techniques used here and earlier by Kleitman, and show that the result in this paper is “best possible” (up to constant) using these techniques.
A codimension theorem for pseudo-Noetherian rings
Authors: McDowell Kenneth
Published: July 2010

M. Auslander and M. Bridger have shown that the depth of a Noetherian local ring is the sum of the Gorenstein dimension and the depth of any given nonzero finitely generated module of finite Gorenstein dimension. In this paper it is demonstrated that this result remains true when suitably interpreted for the class of coherent rings herein entitled pseudo-Noetherian rings. This class contains, among others, all Noetherian rings and valuation domains as well as non-Noetherian local rings of infinite depth.
Global dimension of differential operator rings. II
Authors: Goodearl K. R.
Published: July 2010

The aim of this paper is to find the global homological dimension of the ring of linear differential operators

R
[

θ
1

,
…
,

θ
u

]

R[{\theta _1}, \ldots ,{\theta _u}]

over a differential ring

R
R

with

u
u

commuting derivations. When

R
R

is a commutative noetherian ring with finite global dimension, the main theorem of this paper (Theorem 21) shows that the global dimension of

R
[

θ
1

,
…
,

θ
u

]

R[{\theta _1}, \ldots ,{\theta _u}]

is the maximum of

k
k

and

q
+
u

q + u

, where

q
q

is the supremum of the ranks of all maximal ideals

M
M

of

R
R

for which

R

/

M

R/M

has positive characteristic, and

k
k

is the supremum of the sums

r
a
n
k
(
P
)
+
d
i
f
f

d
i
m
(
P
)

rank(P) + diff\;dim(P)

for all prime ideals

P
P

of

R
R

such that

R

/

P

R/P

has characteristic zero. [The value

d
i
f
f

d
i
m
(
P
)

diff\;dim(P)

is an invariant measuring the differentiability of

P
P

in a manner defined in §3.] In case we are considering only a single derivation on

R
R

, this theorem leads to the result that the global dimension of

R
[
θ
]

R[\theta ]

is the supremum of gl

d
i
m
(
R
)

dim(R)

together with one plus the projective dimensions of the modules

R

/

J

R/J

, where

J
J

is any primary differential ideal of

R
R

. One application of these results derives the global dimension of the Weyl algebra in any degree over any commutative noetherian ring with finite global dimension.
Measures associated with Toeplitz matrices generated by the Laurent expansion of rational functions
Authors: Day K. Michael
Published: July 2010

Let

T
n

(
a
)
=
(

a

i
−
j

)

i
,
j
=
0

n

{T_n}(a) = ({a_{i - j}})_{i,j = 0}^n

be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function, and let

σ
n

=
{

λ

n
0

,
…
,

λ

n
n

}

{\sigma _n} = \{ {\lambda _{n0}}, \ldots ,{\lambda _{nn}}\}

be the corresponding sets of eigenvalues of

T
n

(
f
)

{T_n}(f)

. Define a sequence of measures
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Subscript n Baseline comma alpha Subscript n Baseline left-parenthesis upper E right-parenthesis equals left-parenthesis n plus 1 right-parenthesis Superscript negative 1 Baseline normal upper Sigma Subscript lamda Sub Subscript n i Subscript element-of upper E Baseline 1 comma lamda Subscript n i Baseline element-of sigma Subscript n Baseline">

α
n

,

α
n

(
E
)
=

(
n
+
1

)

−
1

Σ

λ

n
i

∈
E

1
,

λ

n
i

∈

σ
n

{\alpha _n},{\alpha _n}(E) = {(n + 1)^{ - 1}}{\Sigma _{{\lambda _{ni}} \in E}}1,{\lambda _{ni}} \in {\sigma _n}

, and

E
E

a set in the

λ
\lambda

-plane. It is shown that the weak limit

α
\alpha

of the measures

α
n

{\alpha _n}

is unique and possesses at most two atoms, and the function

f
f

which give rise to atoms are identified.
Nearness structures and proximity extensions
Authors: Gagrat M. S. , Thron W. J.
Published: July 2010

Proximity, contiguity and nearness structures are here studied from a unified point of view. In the discussion the role that grills can play in the theory is emphasized. Nearness structures were recently introduced by Herrlich and Naimpally. Thron pointed out the importance of grills in proximity theory. Nearness structures

v
v

are then used to generate proximity extensions

(
ϕ
,
(

X
v

,

Π
v

)
)

(\phi ,({X^v},{\Pi ^v}))

of a given LO-proximity space

(
X
,
Π
)

(X,\Pi )

, where

Π
v

=
Π

{\Pi _v} = \Pi

. Finally, the relation of the extensions

(
ϕ
,
(

X
v

,

Π
v

)
)

(\phi ,({X^v},{\Pi ^v}))

to arbitrary extensions

(
i
,
(
Y
,

Π
∗

)
)

(i,(Y,{\Pi ^ \ast }))

is investigated.
The bracket ring of a combinatorial geometry. I
Authors: White Neil L.
Published: July 2010

The bracket ring is a ring of generalized determinants, called brackets, constructed on an arbitrary combinatorial geometry

G
G

. The brackets satisfy several familiar properties of determinants, including the syzygies, which are equivalent to Laplace’s expansion by minors. We prove that the bracket ring is a universal coordinatization object for

G
G

in two senses. First, coordinatizations of

G
G

correspond to homomorphisms of the ring into fields, thus reducing the study of coordinatizations of

G
G

to the determination of the prime ideal structure of the bracket ring. Second,

G
G

has a coordinatization-like representation over its own bracket ring, which allows an interesting generalization of some familiar results of linear algebra, including Cramer’s rule. An alternative form of the syzygies is then derived and applied to the problem of finding a standard form for any element of the bracket ring. Finally, we prove that several important relations between geometries, namely orthogonality, subgeometry, and contraction, are directly reflected in the structure of the bracket ring.
Functions which are restrictions of 𝐿^{𝑝}-multipliers
Authors: Cowling Michael G.
Published: July 2010

Raouf Doss has given a sufficient condition for a measurable function

ϕ
\phi

on a measurable subset

Λ
\Lambda

of an LCA group

Γ
\Gamma

to be the restriction (l.a.e.) to

Λ
\Lambda

of the Fourier transform of a bounded measure, i.e., a Fourier multiplier of type (1, 1). We generalise Doss’ theorem, and prove that, if the measurable function

ϕ
\phi

on

Λ
\Lambda

is approximable on finite subsets of

Λ
\Lambda

by trigonometric polynomials which are Fourier multipliers of type (p, p) on

Γ
\Gamma

of norms no greater than C, then

ϕ
\phi

is equal locally almost everywhere to the restriction to

Λ
\Lambda

of a Fourier multiplier of type (p, p) and norm no greater than C.
Picard’s theorem and Brownian motion
Authors: Davis Burgess
Published: July 2010

Properties of the paths of two dimensional Brownian motion are used as the basis of a proof of the little Picard theorem and its analog for complex valued functions, defined on simply connected n dimensional manifolds, which map certain diffusions into Brownian motion.
Ramsey theorems for multiple copies of graphs
Authors: Burr S. A. , Erdős P. , Spencer J. H.
Published: July 2010

If

G
G

and

H
H

are graphs, define the Ramsey number

r
(
G
,
H
)

r(G,H)

to be the least number

p
p

such that if the edges of the complete graph

K
p

{K_p}

are colored red and blue (say), either the red graph contains

G
G

as a subgraph or the blue graph contains

H
H

. Let

m
G

mG

denote the union of

m
m

disjoint copies of

G
G

. The following result is proved: Let

G
G

and

H
H

have

k
k

and

l
l

points respectively and have point independence numbers of

i
i

and

j
j

respectively. Then

N
−
1
⩽
r
(
m
G
,
n
H
)
⩽
N
+
C

N - 1 \leqslant r(mG,nH) \leqslant N + C

, where

N
=
k
m
+
l
n
−
m
i
n
(
m
i
,
m
j
)

N = km + ln - min(mi,mj)

and where

C
C

is an effectively computable function of

G
G

and

H
H

. The method used permits exact evaluation of

r
(
m
G
,
n
H
)

r(mG,nH)

for various choices of

G
G

and

H
H

, especially when

m
=
n

m = n

or

G
=
H

G = H

. In particular,

r
(
m

K
3

,
n

K
3

)
=
3
m
+
2
n

r(m{K_3},n{K_3}) = 3m + 2n

when

m
⩾
n
,
m
⩾
2

m \geqslant n,m \geqslant 2

.
A formula for the tangent bundle of flag manifolds and related manifolds
Authors: Lam Kee Yuen
Published: July 2010

A formula is given for the tangent bundle of a flag manifold G in terms of canonically defined vector bundles over G. The formula leads to a unified proof of some parallelizability theorems of Stiefel manifolds. It can also be used to deduce some immersion theorems for flag manifolds.
The bracket ring of a combinatorial geometry. II. Unimodular geometries
Authors: White Neil L.
Published: July 2010

The bracket ring of a combinatorial geometry G is a ring of generalized determinants which acts as a universal coordinatization object for G. Our main result is the characterization of a unimodular geometry as a binary geometry such that the radical of the bracket ring is a prime ideal. This implies that a unimodular geometry has a universal coordinatization over an integral domain, which domain we construct explicitly using multisets. An ideal closely related to the radical, the coordinatizing radical, is also defined and proved to be a prime ideal for every binary geometry. To prove these results, we use two major preliminary theorems, which are of interest in their own right. The first is a bracket-theoretic version of Tutte’s Homotopy Theorem for Matroids. We then prove that any two coordinatizations of a binary geometry over a given field are projectively equivalent.
Hypercontractive semigroups and Sobolev’s inequality
Authors: Feissner George F.
Published: July 2010

If

H
⩾
0

H \geqslant 0

is the generator of a hypercontractive semigroup (HCSG), it is known that

(
H
+
1

)

−
1

/

2

{(H + 1)^{ - 1/2}}

is a bounded operator from

L
p

{L^p}

to

L
p

,
1
⩽
p
⩽
∞

{L^p},1 \leqslant p \leqslant \infty

. We prove that

(
H
+
1

)

−
1

/

2

{(H + 1)^{ - 1/2}}

is bounded from

L
2

{L^2}

to the Orlicz space

L
2

 I

n

+

L

{L^2}{\text { I}}{{\text {n}}^ + }L

, basing the proof on the uniform semiboundedness of the operator

H
+
V

H + V

, for suitable

V
V

. We also prove by an interpolation argument, that

(
H
+
1

)

−
1

/

2

{(H + 1)^{ - 1/2}}

is bounded from

L
p

{L^p}

to

L
p

 I

n

+

L
,
2
⩽
p
>
∞

{L^p}{\text { I}}{{\text {n}}^ + }L,2 \leqslant p > \infty

. Another interpolation argument shows that

(
H
+
1

)

−
1

/

2

{(H + 1)^{ - 1/2}}

is bounded from

L
p

(

I

n

+

L

)
m

{L^p}{({\text {I}}{{\text {n}}^ + }L)^m}

to

L
p

(

I

n

+

L

)

m
+
1

{L^p}{({\text {I}}{{\text {n}}^ + }L)^{m + 1}}

and

m
m

a positive integer. Finally, we identify the topological duals of the spaces mentioned above.
Some 𝐶*-alegebras with a single generator
Authors: Olsen Catherine L. , Zame William R.
Published: July 2010

This paper grew out of the following question: If X is a compact subset of

C
n

{C^n}

, is

C
(
X
)
⊗

M

n

C(X) \otimes {{\mathbf {M}}_n}

(the

C
∗

{C^\ast }

-algebra of

n
×
n

n \times n

matrices with entries from

C
(
X
)

C(X)

) singly generated? It is shown that the answer is affirmative; in fact,

A
⊗

M

n

A \otimes {{\mathbf {M}}_n}

is singly generated whenever A is a

C
∗

{C^\ast }

-algebra with identity, generated by a set of

n
(
n
+
1
)

/

2

n(n + 1)/2

elements of which

n
(
n
−
1
)

/

2

n(n - 1)/2

are selfadjoint. If A is a separable

C
∗

{C^\ast }

-algebra with identity, then

A
⊗
K

A \otimes K

and

A
⊗
U

A \otimes U

are shown to be singly generated, where K is the algebra of compact operators in a separable, infinite-dimensional Hilbert space, and U is any UHF algebra. In all these cases, the generator is explicitly constructed.
On the uniqueness of solutions to hyperbolic boundary value problems
Authors: Travis C. C.
Published: July 2010

The paper is concerned with the uniqueness of solutions to non-well-posed hyperbolic boundary value problems. Both regular and singular boundary value problems are considered. For the singular problem a class of boundary conditions is considered that has not appeared in the literature before in connection with this problem.
On the Stone-Čech compactification of the space of closed sets
Authors: Ginsburg John
Published: July 2010

For a topological space X, we denote by

2
X

{2^X}

the space of closed subsets of X with the finite topology. If X is normal and

T
1

{T_1}

, the map

F
→

cl

β
X

F

F \to {\text {cl}_{\beta X}}F

is an embedding of

2
X

{2^X}

onto a dense subspace of

2

β
X

{2^{\beta X}}

, and, in this way, we regard

2

β
X

{2^{\beta X}}

as a compactification of

2
X

{2^X}

. This paper is motivated by the following question. When can

2

β
X

{2^{\beta X}}

be identified as the Stone-Čech compactification of

2
X

{2^X}

? In [11], J. Keesling states that

β
(

2
X

)
=

2

β
X

\beta ({2^X}) = {2^{\beta X}}

implies

2
X

{2^X}

is pseudocompact. We give a proof of this result and establish the following partial converse. If

2
X

×

2
X

{2^X} \times {2^X}

is pseudocompact, then

β
(

2
X

)
=

2

β
X

\beta ({2^X}) = {2^{\beta X}}

. A corollary of this theorem is that

β
(

2
X

)
=

2

β
X

\beta ({2^X}) = {2^{\beta X}}

when X is

ℵ
0

{\aleph _0}

-bounded.
General position maps for topological manifolds in the 2\over3rds range
Authors: Dancis Jerome
Published: July 2010

For each proper map f of a topological m-manifold M into a topological q-manifold Q,

m
⩽
(
2

/

3
)
q
−
1

/

3

m \leqslant (2/3)q - 1/3

, we build an approximating map g such that the set of singularities S of g is a locally finite simplicial

(
2
m
−
q
)

(2m - q)

-complex locally tamely embedded in M,

g
(
S
)

g(S)

is another locally finite complex

g

|

:
S
↠
g
(
S
)

g|:S \twoheadrightarrow g(S)

is a piecewise linear map and g is a locally flat embedding on the complement of S. Furthermore if

f

|

∂
M

f|\partial M

is a locally flat embedding then we construct g so that it agrees with f on

∂
M

\partial M

even when

f
(
∂
M
)

f(\partial M)

meets

Int
⁡
Q
∩
f
(

Int

M
)

\operatorname {Int} Q \cap f({\operatorname {Int}}\;M)

. In addition we present two other general position lemmas. Also, we show that given two codimension

⩾
3

\geqslant 3

locally flat topological submanifolds M and V of a topological manifold Q,

dim

M
+
dim

V
−
dim

Q
⩽
3

\dim \;M + \dim \;V - \dim \;Q \leqslant 3

, then we can move M so that M and V are transverse in Q.
Estimates for the ∂̄-Neumann operator in weighted Hilbert spaces
Authors: Hantler Sidney L.
Published: July 2010

Estimates for the

∂
¯

\bar \partial

operator are used to derive estimates for the Neumann operator in weighted Hilbert spaces. The technique is similar to that used to prove regularity of solutions of elliptic partial differential equations. A priori estimates are first obtained for smooth compactly supported forms and these estimates are then extended by suitable approximation results. These estimates are applied to give new bounds for the reproducing kernels in the subspaces of entire functions.
Fixed point theorems for mappings satisfying inwardness conditions
Authors: Caristi James
Published: July 2010

Let X be a normed linear space and let K be a convex subset of X. The inward set,

I
K

(
x
)

{I_K}(x)

, of x relative to K is defined as follows:

I
K

(
x
)
=
{
x
+
c
(
u
−
x
)
:
c
⩾
1
,
u
∈
K
}

{I_K}(x) = \{ x + c(u - x):c \geqslant 1,u \in K\}

. A mapping

T
:
K
→
X

T:K \to X

is said to be inward if

T
x
∈

I
K

(
x
)

Tx \in {I_K}(x)

for each

x
∈
K

x \in K

, and weakly inward if Tx belongs to the closure of

I
K

(
x
)

{I_K}(x)

for each

x
∈
K

x \in K

. In this paper a characterization of weakly inward mappings is given in terms of a condition arising in the study of ordinary differential equations. A general fixed point theorem is proved and applied to derive a generalization of the Contraction Mapping Principle in a complete metric space, and then applied together with the characterization of weakly inward mappings to obtain some fixed point theorems in Banach spaces.
Existence of periodic solutions of nonlinear differential equations
Authors: Kannan R.
Published: July 2010

The nonlinear differential equation

x
=
f
(
t
,
x
(
t
)
)

x = f(t,x(t))

, f being

2
π

2\pi

-periodic in t, is considered for the existence of

2
π

2\pi

-periodic solutions. The equation is reduced to an equivalent system of two Hammerstein equations. The case of nonlinear perturbation at resonance is also discussed.
Representations of the 𝑙¹-algebra of an inverse semigroup
Authors: Barnes Bruce A.
Published: July 2010

In this paper the star representations on Hilbert space of the

l
1

{l^1}

-algebra of an inverse semigroup are studied. It is shown that the set of all irreducible star representations form a separating family for the

l
1

{l^1}

-algebra. Then specific examples of star representations are constructed, and some theory of star representations is developed for the

l
1

{l^1}

-algebra of a number of the most important examples of inverse semigroups.
Some inequalities for polynomials with a prescribed zero
Authors: Rahman Q. I. , Schmeisser G.
Published: July 2010

Let

f
(
z
)

f(z)

be a polynomial of degree n. Given that

f
(
z
)

f(z)

has a zero on the circle

|

z

|

=
ρ

(
0
>
ρ
>
∞
)

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

, we estimate

|

f
(
0
)

|

|f(0)|

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis left-parenthesis 2 pi right-parenthesis Superscript negative 1 Baseline integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue f left-parenthesis e Superscript i theta Baseline right-parenthesis EndAbsoluteValue squared d theta right-parenthesis Superscript 1 slash 2">

(

(
2
π

)

−
1

∫
0

2
π

|

f
(

e

i
θ

)

|

2

d
θ

)

1

/

2

{({(2\pi )^{ - 1}}\int _0^{2\pi }|f({e^{i\theta }}){|^2}d\theta )^{1/2}}

\]
in terms of

max

|

z

|

=
1

|

f
(
z
)

|

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

. We also consider some other related problems.
Slowly varying functions in the complex plane
Authors: Vuilleumier Monique
Published: July 2010

Let f be analytic and have no zeros in

|

arg
⁡
z

|

>
α
⩽
π

|\arg z| > \alpha \leqslant \pi

; f is called slowly varying if, for every

λ
>
0
,
f
(
λ
z
)

/

f
(
z
)
→
1

\lambda > 0,f(\lambda z)/f(z) \to 1

uniformly in

|

arg
⁡
z

|

⩽
β
>
α

|\arg z| \leqslant \beta > \alpha

, when

|

z

|

→
∞

|z| \to \infty

. One shows that f is slowly varying if and only if

z

f
′

(
z
)

/

f
(
z
)
→
0

zf’(z)/f(z) \to 0

uniformly in

|

arg
⁡
z

|

⩽
β
>
α

|\arg z| \leqslant \beta > \alpha

, when

|

z

|

→
∞

|z| \to \infty

.
Limit properties of Poisson kernels of Siegel domains of type II
Authors: Dickson Lawrence J.
Published: July 2010

The results of [1] concerning tight

C
0
∗

C_0^\ast

limit of the Poisson kernel of a tube domain, as its parameter converges to a point on the cone boundary, are extended under certain hypotheses to Siegel domains of type II. In the case where the domain is polytopic, almost everywhere convergence of the

L
p

{L^p}

Poisson integral to its boundary values is obtained. Examples and further conjectures conclude the paper.
Geodesics in piecewise linear manifolds
Authors: Stone David A.
Published: July 2010

A simplicial complex M is metrized by assigning to each simplex

a
∈

M

a \in {\mathbf {M}}

a linear simplex

a
∗

{a^\ast }

in some Euclidean space

R

k

{{\mathbf {R}}^k}

so that face relations correspond to isometries. An equivalence class of metrized complexes under the relation generated by subdivisions and isometries is called a metric complex; it consists primarily of a polyhedron M with an intrinsic metric

ρ

M

{\rho _{\mathbf {M}}}

. This paper studies geodesics in metric complexes. Let

P
∈

M

P \in {\mathbf {M}}

; then the tangent space

T
P

(

M

)

{T_P}({\mathbf {M}})

is canonically isometric to an orthogonal product of cones from

P
,

R

k

×

ν
P

(

M

)

P,{{\mathbf {R}}^k} \times {\nu _P}({\mathbf {M}})

; once k is as large as possible.

ν
P

(

M

)

{\nu _P}({\mathbf {M}})

is called the normal geometry at P in M. Let

P

X
¯

P\bar X

be a tangent direction at P in

ν
P

(

M

)

{\nu _P}({\mathbf {M}})

. I define numbers

κ
+

(
P

X
¯

)

{\kappa _ + }(P\bar X)

and

κ
−

(
P

X
¯

)

{\kappa _ - }(P\bar X)

, called the maximum and minimum curvatures at P in the direction

P

X
¯

P\bar X

. THEOREM. Let M be a complete, simply-connected metric complex which is a p.l. n-manifold without boundary. Assume

κ
+

(
P

X
¯

)
⩽
0

{\kappa _ + }(P\bar X) \leqslant 0

for all

P
∈

M

P \in {\mathbf {M}}

and all

P

X
¯

⊆

ν
P

(

M

)

P\bar X \subseteq {\nu _P}({\mathbf {M}})

. Then M is p.l. isomorphic to

R

n

{{\mathbf {R}}^n}

. This is analogous to a well-known theorem for smooth manifolds by E. Cartan and J. Hadamard. THEOREM (ROUGHLY). Let M be a complete metric complex which is a p.l. n-manifold without boundary. Assume (1) there is a number

κ
⩾
0

\kappa \geqslant 0

such that

κ
−

(
P

X
¯

)
⩾
κ

{\kappa _ - }(P\bar X) \geqslant \kappa

whenever P is in the

(
n
−
2
)

(n - 2)

-skeleton of M and whenever

P

X
¯

⊆

ν
P

(

M

)

P\bar X \subseteq {\nu _P}({\mathbf {M}})

; (2) the simplexes of M are bounded in size and shape. Then M is compact. This is analogous to a weak form of a well-known theorem of S. B. Myers for smooth manifolds.
Reversible diffeomorphisms and flows
Authors: Devaney Robert L.
Published: July 2010

We generalize the classical notion of reversibility of a mechanical system. The generic qualitative properties of symmetric orbits of such systems are studied using transversality theory. In particular, we prove analogues of the closed orbit, Liapounov, and homoclinic orbit theorems forR-reversible systems.
On the structure of certain subalgebras of a universal enveloping algebra
Authors: Kostant Bertram , Tirao Juan
Published: July 2010

The representation theory of a semisimple group G, from an algebraic point of view, reduces to determining the finite dimensional representation of the centralizer

U

k

{U^\mathfrak {k}}

of the maximal compact subgroup K of G in the universal enveloping algebra U of the Lie algebra

g

\mathfrak {g}

of G. The theory of spherical representations has been determined in this way since by a result of Harish-Chandra

U

k

{U^\mathfrak {k}}

modulo a suitable ideal I is isomorphic to the ring of Weyl group W invariants

U

(

a

)
W

U{(\mathfrak {a})^W}

in a suitable polynomial ring

U
(

a

)

U(\mathfrak {a})

. To deal with the general case one must determine the image of

U

k

{U^\mathfrak {k}}

in

U
(

k

)
⊗
U
(

a

)

U(\mathfrak {k}) \otimes U(\mathfrak {a})

, where

k

\mathfrak {k}

is the Lie algebra of K. We prove that if W is replaced by the Kunze-Stein intertwining operators

W
~

\tilde W

then

U

k

{U^\mathfrak {k}}

suitably localized and completed is indeed isomorphic to

U
(

k

)
⊗
U

(

a

)

W
~

U(\mathfrak {k}) \otimes U{(\mathfrak {a})^{\tilde W}}

suitably localized and completed.
Some special decompositions of 𝐸³
Authors: Bass Charles D.
Published: July 2010

A great deal of attention has been given to the question: which upper semicontinuous decompositions of

E
3

{E^3}

into pointlike continua give

E
3

{E^3}

. It has recently been determined that some decompositions of

E
3

{E^3}

into points and straight line segments give decomposition spaces which are topologically distinct from

E
3

{E^3}

. In this paper we apply a new condition to the set of nondegenerate elements of a decomposition which enables one to conclude that the resulting decomposition space is homeomorphic to

E
3

{E^3}

.
A special integral and a Gronwall inequality
Authors: Helton Burrell W.
Published: July 2010

This paper considers a special integral

(
I
)

∫
a
b

(
f
d
g
+
H
)

(I)\smallint _a^b(fdg + H)

which is a subdivision-refinement-type limit of the approximating sum
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript 1 Overscript n Endscripts left-brace f left-parenthesis t Subscript i Baseline right-parenthesis left-bracket g left-parenthesis x Subscript i Baseline right-parenthesis minus g left-parenthesis x Subscript i minus 1 Baseline right-parenthesis right-bracket plus upper H left-parenthesis x Subscript i minus 1 Baseline comma x Subscript i Baseline right-parenthesis right-brace comma">

∑
1
n

{
f
(

t
i

)
[
g
(

x
i

)
−
g
(

x

i
−
1

)
]
+
H
(

x

i
−
1

,

x
i

)
}
,

\sum \limits _1^n {\{ f({t_i})[g({x_i}) - g({x_{i - 1}})] + H({x_{i - 1}},{x_i})\} ,}

\]
where

x

i
−
1

>

t
i

>

x
i

{x_{i - 1}} > {t_i} > {x_i}

. The author shows, with appropriate restrictions, that

(
I
)

∫
a
b

(
f
d
g
+
H
)

(I)\smallint _a^b(fdg + H)

exists if and only if
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R right-parenthesis integral Subscript x Superscript y Baseline left-parenthesis f d g plus upper H minus upper A Superscript minus Baseline right-parenthesis equals left-parenthesis upper L right-parenthesis integral Subscript x Superscript y Baseline left-parenthesis f d g plus upper H plus upper A Superscript plus Baseline right-parenthesis">

(
R
)

∫
x
y

(
f
d
g
+
H
−

A
−

)
=
(
L
)

∫
x
y

(
f
d
g
+
H
+

A
+

)

(R)\smallint _x^y(fdg + H - {A^ - }) = (L)\smallint _x^y(fdg + H + {A^ + })

\]
for

a
⩽
x
>
y
⩽
b

a \leqslant x > y \leqslant b

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis p comma q right-parenthesis equals left-bracket f left-parenthesis q right-parenthesis minus f left-parenthesis p right-parenthesis right-bracket left-bracket g left-parenthesis q right-parenthesis minus g left-parenthesis p right-parenthesis right-bracket comma upper A Superscript minus Baseline left-parenthesis p comma q right-parenthesis equals upper A left-parenthesis q Superscript minus Baseline comma q right-parenthesis">

A
(
p
,
q
)
=
[
f
(
q
)
−
f
(
p
)
]
[
g
(
q
)
−
g
(
p
)
]
,

A
−

(
p
,
q
)
=
A
(

q
−

,
q
)

A(p,q) = [f(q) - f(p)][g(q) - g(p)],{A^ - }(p,q) = A({q^ - },q)

and

A
+

(
p
,
q
)
=
A
(
p
,

p
+

)

{A^ + }(p,q) = A(p,{p^ + })

. Furthermore, if either of the equivalent statements is true, then all the integrals are equal. These equivalent statements are used to prove an integration-by-parts theorem and to solve a Gronwall inequality involving this special integral. Product integrals are used in the solution of the Gronwall inequality.
Hypoellipticity of certain degenerate elliptic boundary value problems
Authors: Kannai Yakar
Published: July 2010

The concept of hypoellipticity for degenerate elliptic boundary value problems is defined, and its relation with the hypoellipticity of certain pseudo-differential operators on the boundary is discussed (for second order equations). A theorem covering smoothness of solutions of boundary value problems such as

a
(
x
)
∂
u

/

∂
n
+
b
(
x
)
u
=
f
(
x
)

a(x)\partial u/\partial n + b(x)u = f(x)

for the Laplace equation is proved. An almost complete characterization of hypoelliptic boundary value problems for elliptic second order equations in two dimensions is given via analysis of hypoelliptic pseudo-differential operators in one variable.
Asymptotic equipartition of energy for differential equations in Hilbert space
Authors: Goldstein Jerome A. , Sandefur James T.
Published: July 2010

Of concern are second order differential equations of the form

(
d

/

d
t
−
i

f
1

(
A
)
)
(
d

/

d
t
−
i

f
2

(
A
)
)
u
=
0

(d/dt - i{f_1}(A))(d/dt - i{f_2}(A))u = 0

. Here A is a selfadjoint operator and

f
1

,

f
2

{f_1},{f_2}

are real-valued Borel functions on the spectrum of A. The Cauchy problem for this equation is governed by a certain one parameter group of unitary operators. This group allows one to define the energy of a solution; this energy depends on the initial data but not on the time t. The energy is broken into two parts, kinetic energy

K
(
t
)

K(t)

and potential energy

P
(
t
)

P(t)

, and conditions on A,

f
1

,

f
2

{f_1},{f_2}

are given to insure asymptotic equipartition of energy:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript t right-arrow plus-or-minus normal infinity Endscripts upper K left-parenthesis t right-parenthesis equals limit Underscript t right-arrow plus-or-minus normal infinity Endscripts upper P left-parenthesis t right-parenthesis">

lim

t
→
±
∞

K
(
t
)
=

lim

t
→
±
∞

P
(
t
)

{\lim _{t \to \pm \infty }}K(t) = {\lim _{t \to \pm \infty }}P(t)

for all choices of initial data. These results generalize the corresponding results of Goldstein for the abstract wave equation

d
2

u

/

d

t
2

+

A
2

u
=
0

{d^2}u/d{t^2} + {A^2}u = 0

. (In this case,

f
1

(
λ
)
≡
λ
,

f
2

(
λ
)
≡
−
λ

{f_1}(\lambda ) \equiv \lambda ,{f_2}(\lambda ) \equiv - \lambda

.)
Maximal orders and reflexive modules
Authors: Cozzens J. H.
Published: July 2010

If R is a maximal two-sided order in a semisimple ring and

M
R

{M_R}

is a finite dimensional torsionless faithful R-module, we show that

m
=

End
R

M
∗

m = {\text {End}_R}\;{M^\ast }

is a maximal order. As a consequence, we obtain the equivalence of the following when

M
R

{M_R}

is a generator: 1. M is R-reflexive. 2.

k
=

End

M
R

k = {\text {End}}\;{M_R}

is a maximal order. 3.

k
=

End
R

M
∗

k = {\text {End}_R}\;{M^\ast }

where

M
∗

=

hom
R

(
M
,
R
)

{M^\ast } = {\hom _R}(M,R)

. When R is a prime maximal right order, we show that the endomorphism ring of any finite dimensional, reflexive module is a maximal order. We then show by example that R being a maximal order is not a property preserved by k. However, we show that

k
=

End

M
R

k = {\text {End}}\;{M_R}

is a maximal order whenever

M
R

{M_R}

is a maximal uniform right ideal of R, thereby sharpening Faith’s representation theorem for maximal two-sided orders. In the final section, we show by example that even if

R
=

End
k

V

R = {\text {End}_k}V

is a simple pli (pri)-domain, k can have any prescribed right global dimension

⩾
1

\geqslant 1

, can be right but not left Noetherian or neither right nor left Noetherian.
Tauberian theorems for a positive definite form, with applications to a Volterra equation
Authors: Staffans Olof J.
Published: July 2010

We study the relation between the condition
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript upper T greater-than 0 Endscripts integral Underscript left-bracket 0 comma upper T right-bracket Endscripts ModifyingAbove phi With bar left-parenthesis t right-parenthesis integral Underscript left-bracket t minus upper T comma t right-bracket Endscripts phi left-parenthesis t minus s right-parenthesis d v left-parenthesis s right-parenthesis d t greater-than normal infinity">

sup

T
>
0

∫

[
0
,
T
]

φ
¯

(
t
)

∫

[
t
−
T
,
t
]

φ
(
t
−
s
)

d
v
(
s
)

d
t
>
∞

\sup \limits _{T > 0} \int _{[0,T]} {\bar \varphi (t)} \int _{[t - T,t]} {\varphi (t - s)\;dv (s)\;dt > \infty }

\]
and the asymptotic behavior of the bounded function

φ
\varphi

when

ν
\nu

is a positive definite measure. Earlier we have proved that if

ν
\nu

is strictly positive definite and

φ
\varphi

satisfies a tauberian condition, then

φ
(
t
)
→
0

\varphi (t) \to 0

as

t
→
∞

t \to \infty

. Here we characterize the spectrum of the limit set of

φ
\varphi

in the case when

ν
\nu

is not strictly positive definite. Applying this theory to a nonlinear Volterra equation we get some new results on the asymptotic behavior of its bounded solutions.
Coincidence index and multiplicity
Authors: Laloux B. , Mawhin J.
Published: July 2010

This paper is devoted to the extension, in the frame of coincidence degree theory in normed spaces, of the concept of Leray-Schauder index of an isolated fixed point. The generalization includes basic properties of the coincidence index, Krasnosel’skiĭ type theorems for the case of noninvertible linear part and a Leray-Schauder’s type formula relating the index and spectral theory in the linear case. This last problem needs the introduction of the concept of characteristic value for some couples of linear mappings and of its multiplicity.
Piecewise linear bundles in the metastable range
Authors: Millett Kenneth C.
Published: July 2010

For numerable vector bundles a nonzero section determines a unique trivial line subbundle containing the section and this subbundle is a direct summand of the bundle. The main result, a consequence of concordance-isotopy theory, states that in the metastable range a nonzero section to a piecewise linear

R

n

{{\mathbf {R}}^n}

bundle determines a unique trivial line subbundle and that this is the best possible result. This fact is then compared with the known failure of the summand property below the stable range.