bluuebunny/crossref_metadata_2025 huggingface dataset:
A decomposition for combinatorial geometries
Authors: Brylawski Thomas H.
Published: July 2010

A construction based on work by Tutte and Grothendieck is applied to a decomposition on combinatorial pregeometries in order to study an important class of invariants. The properties of this Tutte decomposition of a pregeometry into a subgeometry

G
∖
e

G\backslash e

and contraction

G

/

e

G/e

is explored in a categorically integrated view using factored strong maps. After showing that direct sum decomposition distributes over the Tutte decomposition we construct a universal pair

(
R
,
t
)

(R,t)

where

R
R

is a free commutative ring with two generators corresponding to a loop and an isthmus; and

t
t

, the Tutte polynomial assigns a ring element to each pregeometry. Evaluations of

t
(
G
)

t(G)

give the Möbius function, characteristic polynomial, Crapo invariant, and numbers of subsets, bases, spanning and independent sets of

G
G

and its Whitney dual. For geometries a similar decomposition gives the same information as the chromatic polynomial throwing new light on the critical problem. A basis is found for all linear identities involving Tutte polynomial coefficients. In certain cases including Hartmanis partitions one can recover all the Whitney numbers of the associated geometric lattice

L
(
G
)

L(G)

from

t
(
G
)

t(G)

and conversely. Examples and counterexamples show that duals, minors, connected pregeometries, series-parallel networks, free geometries (on which many invariants achieve their upper bounds), and lower distributive pregeometries are all characterized by their polynomials. However, inequivalence, Whitney numbers, and representability are not always invariant. Applying the decomposition to chain groups we generalize the classical two-color theorem for graphs to show when a geometry can be imbedded in binary affine space. The decomposition proves useful also for graphical pregeometries and for unimodular (orientable) pregeometries in the counting of cycles and co-boundaries.
The structure of pseudocomplemented distributive lattices. III. Injective and absolute subretracts
Authors: Grätzer G. , Lakser H.
Published: July 2010

Absolute subretracts are characterized in the classes

B

n

,
n
⩽
ω

{\mathcal {B}_n},n \leqslant \omega

. This is applied to describe the injectives in

B

1

{\mathcal {B}_1}

(due to R. Balbes and G. Grätzer) and

B

2

{\mathcal {B}_2}

.
Modular representations of metabelian groups
Authors: Basmaji B. G.
Published: July 2010

The irreducible modular representations, the blocks, and the defect groups of finite metabelian groups are determined. Also the dimensions of the principal indecomposable modules are computed.
Automorphisms of 𝐺𝐿_{𝑛}(𝑅),𝑅 a local ring
Authors: Pomfret J. , McDonald B. R.
Published: July 2010

Let

R
R

denote a commutative local ring with maximal ideal

m
m

and residue field

k
=
R

/

m

k = R/m

. In this paper we determine the group automorphisms of the general linear group

G

L
n

(
R
)

G{L_n}(R)

when

n
⩾
3

n \geqslant 3

and the characteristic of

k
k

is not 2.
A Radon-Nikoým theorem for operator-valued measures
Authors: Maynard Hugh B.
Published: July 2010

The purpose of this paper is to obtain a characterization of indefinite integrals of vector-valued functions with respect to countably additive operator-valued measures with finite variation. This result is then specialized to several simpler situations.
Complex structures on Riemann surfaces
Authors: Kiremidjian Garo
Published: July 2010

Let

X
X

be a Riemann surface (compact or noncompact) with the property that the length of every closed geodesic is bounded away from zero. Then we show that sufficiently small complex structures on

X
X

can be described without making use of Schwarzian derivatives or the theory of quasiconformal mappings. Instead, we use methods developed in Kuranishi’s work on the existence of locally complete families of deformations of compact complex manifolds. We introduce norms

|

|

k

|\quad {|_k}

(

k
k

a positive integer) on the space of

C
∞

(
0
,
p
)

{C^\infty }(0,p)

-forms with values in the tangent bundle on

X
X

, which are similar to the usual Sobolev

|

|

|

|

k

||\quad |{|_k}

-norms. (In the compact case

|

|

k

|\quad {|_k}

is equivalent to

|

|

|

|

k

||\quad |{|_k}

.) Then we prove that certain properties of

|

|

|

|

k

||\quad |{|_k}

, crucial for Kuranishi’s approach, are also satisfied by

|

|

k

|\quad {|_k}

.
Two methods of integrating Monge-Ampère’s equations. II
Authors: Matsuda Michihiko
Published: July 2010

Generalizing the notion of an integrable system given in the previous note [2], we shall define an integrable system of higher order, and obtain the following results: 1. A linear hyperbolic equation is solved by integrable systems of order n if and only if its

(
n
+
1
)

(n + 1)

th Laplace invariant

H
n

{H_n}

vanishes. 2. An equation of Laplace type is solved by integrable systems of the second order if and only if the transformed equation by the associated Imschenetsky transformation is solved by integrable systems of the first order.
Ascent, descent, and commuting perturbations
Authors: Kaashoek M. A. , Lay D. C.
Published: July 2010

In the present paper we investigate the stability of the ascent and descent of a linear operator

T
T

when

T
T

is subjected to a perturbation by a linear operator

C
C

which commutes with

T
T

. The domains and ranges of

T
T

and

C
C

lie in some linear space

X
X

. The results are used to characterize the Browder essential spectrum of

T
T

. We conclude with a number of remarks concerning the notion of commutativity used in the present paper.
On antiflexible algebras
Authors: Rodabaugh David J.
Published: July 2010

In this paper we begin a classification of simple and semisimple totally antiflexible algebras (finite-dimensional) over splitting fields of char.

≠
2
,
3

\ne 2,3

. For such an algebra

A
A

, let

P
P

be the largest associative ideal in

A
+

{A^ + }

and let

N
+

{N^ + }

be the radical of

P
P

. We determine all simple and semisimple totally antiflexible algebras in which

N
⋅
N
=
0

N \cdot N = 0

. Defining

A
A

to be of type

(
m
,
n
)

(m,n)

if

N
+

{N^ + }

is nilpotent of class

m
m

with

dim
⁡
A
=
n

\dim A = n

, we then characterize all simple nodal totally anti-flexible algebras (over fields of char.

≠
2
,
3

\ne 2,3

) of types

(
n
,
n
)

(n,n)

and

(
n
−
1
,
n
)

(n - 1,n)

and give preliminary results for certain other types.
A new class of functions of bounded index
Authors: Shah S. M. , Shah S. N.
Published: July 2010

Entire functions of strongly bounded index have been defined and it is shown that functions of genus zero and having all negative zeros satisfying a one sided growth condition belong to this class.
Plurisubharmonic functions and convexity properties for general function algebras
Authors: Rickart C. E.
Published: July 2010

A “natural system” consists of a Hausdorff space

Σ
\Sigma

plus an algebra

A

\mathfrak {A}

of complex-valued continuous functions on

Σ
\Sigma

(which contains the constants and determines the topology in

Σ
\Sigma

) such that every continuous homomorphism of

A

\mathfrak {A}

onto

C

{\mathbf {C}}

is given by an evaluation at a point of

Σ
\Sigma

(compact-open topology in

A

\mathfrak {A}

). The prototype of a natural system is

[

C

n

,

P

]

[{{\mathbf {C}}^n},\mathfrak {P}]

, where

P

\mathfrak {P}

is the algebra of polynomials on

C

n

{{\mathbf {C}}^n}

. In earlier papers (Pacific J. Math. 18 and Canad. J. Math. 20), the author studied

A

\mathfrak {A}

-holomorphic functions, which are generalizations of ordinary holomorphic functions in

C

n

{{\mathbf {C}}^n}

, and associated concepts of

A

\mathfrak {A}

-analytic variety and

A

\mathfrak {A}

-holomorphic convexity in

Σ
\Sigma

. In the present paper, a class of extended real-valued functions, called

A

\mathfrak {A}

-subharmonic functions, is introduced which generalizes the ordinary plurisubharmonic functions in

C

n

{{\mathbf {C}}^n}

. These functions enjoy many of the properties associated with plurisubharmonic functions. Furthermore, in terms of the

A

\mathfrak {A}

-subharmonic functions, a number of convexity properties of

C

n

{{\mathbf {C}}^n}

associated with plurisubharmonic functions can be generalized. For example, if

G
G

is an open

A

\mathfrak {A}

-holomorphically convex subset of

Σ
\Sigma

and

K
K

is a compact subset of

G
G

, then the convex hull of

K
K

with respect to the continuous

A

\mathfrak {A}

-subharmonic functions on

G
G

is equal to its hull with respect to the

A

\mathfrak {A}

-holomorphic functions on

G
G

.
Analytic sets as branched coverings
Authors: Stutz John
Published: July 2010

In this paper we study the relation between the tangent structure of an analytic set V at a point p and the local representation of V as a branched covering. A prototype for our type of result is the fact that one obtains a covering of minimal degree by projecting transverse to the Zariski tangent cone

C
3

(
V
,
p
)

{C_3}(V,p)

. We show, for instance, that one obtains the smallest possible branch locus for a branched covering if one projects transverse to the cone

C
4

(
V
,
p
)

{C_4}(V,p)

. This and similar results show that points where the various tangent cones

C
i

(
V
,
p
)
,
i
=
4
,
5
,
6

{C_i}(V,p),i = 4,5,6

, have minimal dimension give rise to the simplest branched coverings. This observation leads to the idea of “Puiseux series normalization", generalizing the situation in one dimension. These Puiseux series allow us to strengthen some results of Hironaka and Whitney on the local structure of certain types of singularities.
Harmonic analysis on 𝐹-spaces with a basis
Authors: Kuelbs J. , Mandrekar V.
Published: July 2010

We establish Bochner’s theorem and the Levy continuity theorem in the case that the underlying space is a real

F
F

-space with a basis, and then examine the infinitely divisible probability measures on a class of such spaces.
Automorphisms of 𝜔₁-trees
Authors: Jech Thomas J.
Published: July 2010

The number of automorphisms of a normal

ω
1

{\omega _1}

-tree

T
T

, denoted by

σ
(
T
)

\sigma (T)

, is either finite or

2

ℵ
0

⩽
σ
(
T
)
⩽

2

ℵ
1

{2^{{\aleph _0}}} \leqslant \sigma (T) \leqslant {2^{{\aleph _1}}}

. Moreover, if

σ
(
T
)

\sigma (T)

is infinite then

σ

(
T

)

ℵ
0

=
σ
(
T
)

\sigma {(T)^{{\aleph _0}}} = \sigma (T)

. Moreover, if

T
T

has no Suslin subtree then

σ
(
T
)

\sigma (T)

is finite or

σ
(
T
)
=

2

ℵ
0

\sigma (T) = {2^{{\aleph _0}}}

or

σ
(
T
)
=

2

ℵ
1

\sigma (T) = {2^{{\aleph _1}}}

. It is consistent that there is a Suslin tree with arbitrary precribed

σ
(
T
)

\sigma (T)

between

2

ℵ
0

{2^{{\aleph _0}}}

and

2

ℵ
1

{2^{{\aleph _1}}}

, subject to the restriction above; e.g.

2

ℵ
0

=

ℵ
1

,

2

ℵ
1

=

ℵ

324

{2^{{\aleph _0}}} = {\aleph _1},{2^{{\aleph _1}}} = {\aleph _{324}}

and

σ
(
T
)
=

ℵ

17

\sigma (T) = {\aleph _{17}}

. We prove related results for Kurepa trees and isomorphism types of trees. We use Cohen’s method of forcing and Jensen’s techniques in

L
L

.
Orbits in a real reductive Lie algebra
Authors: Rothschild L. Preiss
Published: July 2010

The purpose of this paper is to give a classification of the orbits in a real reductive Lie algebra under the adjoint action of a corresponding connected Lie group. The classification is obtained by examining the intersection of the Lie algebra with the orbits in its complexification. An algebraic characterization of the minimal points in the closed orbits is also given.
On modification theorems
Authors: Rao Murali
Published: July 2010

Given a right continuous family

F
t

{F_t}

of complete

σ
\sigma

-fields and a bounded right continuous family

X
t

{X_t}

of random variables, we show in this paper that it is possible to modify the conditional expectations

E
(

X
t

|

F
t

)

E({X_t}|{F_t})

to be right continuous. When

X
t

=
X

{X_t} = X

, this reduces to a result of J. L. Doob.
A note on the geometric means of entire functions of several complex variables
Authors: Kamthan P. K.
Published: July 2010

Let

f
(

z
1

,
⋯
,

z
n

)

f({z_1}, \cdots ,{z_n})

be an entire function of

n
(
⩾
2
)

n( \geqslant 2)

complex variables. Recently Agarwal [Trans. Amer. Math. Soc. 151 (1970), 651-657] has obtained certain results involving geometric mean values of

f
f

. In this paper we have constructed examples to contradict some of the results of Agarwal and have thereafter given improvements and modifications of his results.
Oriented and weakly complex bordism algebra of free periodic maps
Authors: Shibata Katsuyuki
Published: July 2010

Free cyclic actions on a closed oriented (weakly almost complex, respectively) manifold which preserve the orientation (weakly complex structure) are considered from the viewpoint of equivariant bordism theory. The author gives an explicit presentation of the oriented bordism module structure and multiplicative structure of all orientation preserving (and reversing) free involutions. The odd period and weakly complex cases are also determined with the aid of the notion of formal group laws. These results are applied to a nonexistence problem for certain equivariant maps.
Krull dimension in power series rings
Authors: Arnold Jimmy T.
Published: July 2010

Let R denote a commutative ring with identity. If there exists a chain

P
0

⊂

P
1

⊂
⋯
⊂

P
n

{P_0} \subset {P_1} \subset \cdots \subset {P_n}

of

n
+
1

n + 1

prime ideals of R, where

P
n

≠
R

{P_n} \ne R

, but no such chain of

n
+
2

n + 2

prime ideals, then we say that R has dimension n. The power series ring

R
[
[
X
]
]

R[[X]]

may have infinite dimension even though R has finite dimension.
Envelopes of holomorphy and holomorphic convexity
Authors: Carmignani Robert
Published: July 2010

This paper is primarily a study of generalized notions of envelope of holomorphy and holomorphic convexity for special (algebraically restricted) subsets of

C

n

{{\mathbf {C}}^n}

and in part for arbitrary subsets of

C

n

{{\mathbf {C}}^n}

. For any special set S in

C

n

{{\mathbf {C}}^n}

, we show that every function holomorphic in a neighborhood of S not only can be holomorphically continued but also holomorphically extended to a neighborhood in

C

n

{{\mathbf {C}}^n}

of a maximal set

S
~

\tilde {S}

, the “envelope of holomorphy” of S, which is also a special set of the same type as S. Formulas are obtained for constructing

S
~

\tilde {S}

for any special set S. “Holomorphic convexity” is characterized for these special sets. With one exception, the only topological restriction on these special sets is connectivity. Examples are given which illustrate applications of the theorems and help to clarify the concepts of “envelope of holomorphy” and “holomorphic convexity."
Positive approximants
Authors: Bouldin Richard
Published: July 2010

Let

T
=
B
+
i
C

T = B + iC

with

B
=

B
∗

,
C
=

C
∗

B = {B^\ast },C = {C^\ast }

and let

δ
(
T
)

\delta (T)

denote the the distance of

T
T

to the set of nonnegative operators. We find upper and lower bounds for

δ
(
T
)

\delta (T)

. We prove that if

P
P

is any best approximation for

T
T

among nonnegative operators then

P
≤
B
+
(
(
δ
(
T
)

)
2

−

C
2

)

1

/

2

P \leq B + ((\delta (T))^2 - C^2)^{1/2}

. Provided

B
≥
0

B \geq 0

or

T
T

is normal we characterize those

T
T

which have a unique best approximation among the nonnegative operators. If

T
T

is normal we characterize its best approximating nonnegative operators which commute with it. We characterize those

T
T

for which the zero operator is the best approximating nonnegative operator.
𝜃-modular bands of groups
Authors: Spitznagel C.
Published: July 2010

The class of

θ
\theta

-modular bands of groups is defined by means of a type of modularity condition on the lattice of congruences on a band of groups. The main result characterizes

θ
\theta

-modularity as a condition on the multiplication in the band of groups. This result is then applied to the classes of normal bands of groups and orthodox bands of groups.
Barycenters, pinnacle points, and denting points
Authors: Khurana Surjit Singh
Published: July 2010

Some properties of probability measures, on closed convex bounded sets in locally convex spaces, having barycenters are obtained. Also some geometric and measure-theoretic characterizations of pinnacle points are given, and a result about denting points is proved.
𝑘-parameter semigroups of measure-preserving transformations
Authors: Fava Norberto Angel
Published: July 2010

An individual ergodic theorem is proved for semigroups of measure-preserving transformations depending on k real parameters, which generalizes N. Wiener’s ergodic theorem.
Differential equations on closed subsets of a Banach space
Authors: Martin R. H.
Published: July 2010

In this paper the problem of existence of solutions to the initial value problem

u
′

(
t
)
=
A
(
t
,
u
(
t
)
)
,
u
(
a
)
=
z

u’(t) = A(t,u(t)),u(a) = z

, is considered where

A
:
[
a
,
b
)
×
D
→
E

A:[a,b) \times D \to E

is continuous, D is a closed subset of a Banach space E, and

z
∈
D

z \in D

. With a dissipative type condition on A, we establish sufficient conditions for this initial value problem to have a solution. Using these results, we are able to characterize all continuous functions which are generators of nonlinear semigroups on D.
Asymptotic behavior of solutions of linear stochastic differential systems
Authors: Friedman Avner , Pinsky Mark A.
Published: July 2010

Following Kasminski, we investigate asymptotic behavior of solutions of linear time-independent Itô equations. We first give a sufficient condition for asymptotic stability of the zero solution. Then in dimension 2 we determine conditions for spiraling at a linear rate. Finally we give applications to the Cauchy problem for the associated parabolic equation by the use of a tauberian theorem.
Absolutely summing and dominated operators on spaces of vector-valued continuous functions
Authors: Swartz Charles
Published: July 2010

A. Pietsch has shown that the class of dominated linear operators on

C
(
S
)

C(S)

coincides with the class of absolutely summing operators. If the space

C
(
S
)

C(S)

is replaced by

C
X

(
S
)

{C_X}(S)

, where X is a Banach space, this is no longer the case. However, any absolutely summing operator is always dominated, and the classes of operators coincide exactly when X is finite dimensional. A characterization of absolutely summing operators on

C
X

(
S
)

{C_X}(S)

is given.
Almost spherical convex hypersurfaces
Authors: Moore John Douglas
Published: July 2010

Let

M
M

be a smooth compact hypersurface with positive sectional curvatures in

n
n

-dimensional euclidean space. This paper gives a sufficient condition for

M
M

to lie in the spherical shell bounded by concentric spheres of radius

1
−
ϵ

1 - \epsilon

and

1
+
ϵ

1 + \epsilon

. This condition is satisfied, in the case where

n
=
3

n = 3

, if the Gaussian curvature or the mean curvature of

M
M

is sufficiently pointwise close to one.
Homeomorphisms with polyhedral irregular sets. I
Authors: Duvall P. F. , Husch L. S.
Published: July 2010

Homeomorphisms on open manifolds with polyhedral irregular sets are studied. For high dimensions, necessary and sufficient conditions for certain codimension three irregular sets to be tame are given. Several examples of homeomorphisms with wild irregular sets are given.
Obstructions to embedding 𝑛-manifolds in (2𝑛-1)-manifolds
Authors: Maxwell J. W.
Published: July 2010

Suppose
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon left-parenthesis upper M Superscript n Baseline comma partial-differential upper M Superscript n Baseline right-parenthesis right-arrow left-parenthesis upper Q Superscript 2 n minus 1 Baseline comma partial-differential upper Q Superscript 2 n minus 1 Baseline right-parenthesis">

f
:
(

M
n

,
∂

M
n

)
→
(

Q

2
n
−
1

,
∂

Q

2
n
−
1

)

f:({M^n},\partial {M^n}) \to ({Q^{2n - 1}},\partial {Q^{2n - 1}})

is a proper PL map between PL manifolds

M
n

{M^n}

and

Q

2
n
−
1

{Q^{2n - 1}}

of dimension

n
n

and

2
n
−
1

2n - 1

respectively,

M
M

compact. J. F. P. Hudson has shown that associated with each such map

f
f

that is an embedding on

∂
M

\partial M

is an element

α
¯

(
f
)

\bar \alpha (f)

in

H
1

(
M
;

Z
2

)

{H_1}(M;{Z_2})

when

n
n

is odd and an element

β
¯

(
f
)

\bar \beta (f)

in

H
1

(
M
;
Z
)

{H_1}(M;Z)

when

n
n

is even. These elements are invariant under a homotopy relative to

∂
M

\partial M

. We show that, under slight additional assumptions on

M
,
Q

M,Q

and

f
,
f

f,f

is homotopic to an embedding if and only if

α
¯

(
f
)
=
0

\bar \alpha (f) = 0

for

n
n

odd and

β
¯

(
f
)
=
0

\bar \beta (f) = 0

for

n
n

even. This result is used to give a sufficient condition for extending an embedding

f
:
∂

M
n

→
∂

B

2
n
−
1

f:\partial {M^n} \to \partial {B^{2n - 1}}

(

B

2
n
−
1

{B^{2n - 1}}

denotes

(
2
n
−
1
)

(2n - 1)

-dimensional ball) to an embedding
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F colon left-parenthesis upper M Superscript n Baseline comma partial-differential upper M Superscript n Baseline right-parenthesis right-arrow left-parenthesis upper B Superscript 2 n minus 1 Baseline comma partial-differential upper B Superscript 2 n minus 1 Baseline right-parenthesis">

F
:
(

M
n

,
∂

M
n

)
→
(

B

2
n
−
1

,
∂

B

2
n
−
1

)

F:({M^n},\partial {M^n}) \to ({B^{2n - 1}},\partial {B^{2n - 1}})

.
Improbability of collisions in Newtonian gravitational systems. II
Authors: Saari Donald G.
Published: July 2010

It is shown that the set of initial conditions leading to collision in the inverse square force law has measure zero. For the inverse

q
q

force law the behavior of binary collisions for

1
>
q
>
3

1 > q > 3

and the behavior of any collision for

q
=
1

q = 1

is developed. This information is used to show that collisions are improbable in the inverse

q
q

force law where

q
>
17

/

7

q > 17/7

and that binary collisions are improbable for

q
>
3

q > 3

.
Extensions of functions and spaces
Authors: Viglino Giovanni
Published: July 2010

We investigate, for a given map

φ
\varphi

from a topological space X to a topological space Y (denoted by

[
X
,
φ
,
Y
]

[X,\varphi ,Y]

), those triples

[
E
,
Φ
,
Y
]

[E,\Phi ,Y]

where E is an extension of X and

Φ
\Phi

extends

φ
\varphi

to E. A maximal such extension, similar to the Katětov extension of a topological space, is examined.
Asymptotic properties of Gaussian random fields
Authors: Qualls Clifford , Watanabe Hisao
Published: July 2010

In this paper we study continuous mean zero Gaussian random fields

X
(
p
)

X(p)

with an N-dimensional parameter and having a correlation function

ρ
(
p
,
q
)

\rho (p,q)

for which

1
−
ρ
(
p
,
q
)

1 - \rho (p,q)

is asymptotic to a regularly varying (at zero) function of the distance

dis

(
p
,
q
)

{\text {dis}}\;(p,q)

with exponent

0
>
α
≤
2

0 > \alpha \leq 2

. For such random fields, we obtain the asymptotic tail distribution of the maximum of

X
(
p
)

X(p)

and an asymptotic almost sure property for

X
(
p
)

X(p)

as

|

p

|

→
∞

|p| \to \infty

. Both results generalize ones previously given by the authors for

N
=
1

N = 1

.
Bessel series expansions of the Epstein zeta function and the functional equation
Authors: Terras Audrey A.
Published: July 2010

For the Epstein zeta function of an n-ary positive definite quadratic form,

n
−
1

n - 1

generalizations of the Selberg-Chowla formula (for the binary case) are obtained. Further, it is shown that these

n
−
1

n - 1

formulas suffice to prove the functional equation of the Epstein zeta function by mathematical induction. Finally some generalizations of Kronecker’s first limit formula are obtained.
Differentials on quotients of algebraic varieties
Authors: Knighten Carol M.
Published: July 2010

The relations between differentials invariant with respect to a finite group acting on a variety and the differentials on the quotient variety are studied. If the quotient map is unramified in codimension 1 we have an isomorphism for Zariski differentials, but not in general for Käahler differentials. Necessary and sufficient conditions for isomorphism of the Zariski differentials are given when the finite group acts linearly. Examples illustrate the scope of the theorems and some open problems.
Fundamental theory of contingent differential equations in Banach space
Authors: Chow Shui Nee , Schuur J. D.
Published: July 2010

For a contingent differential equation that takes values in the closed, convex, nonempty subsets of a Banach space E, we prove an existence theorem and we investigate the extendability of solutions and the closedness and continuity properties of solution funnels. We consider first a space E that is separable and reflexive and then a space E with a separable second dual space. We also consider the special case of a point-valued or ordinary differential equation.
The commutant of analytic Toeplitz operators
Authors: Deddens James A. , Wong Tin Kin
Published: July 2010

In this paper we study the commutant of an analytic Toeplitz operator. For

ϕ

H
∞

\phi \;\;{H^\infty }

, let

ϕ
=
χ
F

\phi = \chi F

be its inner-outer factorization. Our main result is that if there exists

λ

ϵ

C

\lambda \;\epsilon \;{\text {C}}

such that X factors as

χ
=

χ
1

χ
2

⋯

χ
n

\chi = {\chi _1}{\chi _2} \cdots {\chi _n}

, each

χ
i

{\chi _i}

an inner function, and if

F
−
λ

F - \lambda

is divisible by each

χ
i

{\chi _i}

, then

{

T
ϕ

}
′

=
{

T
χ

}
′

∩
{

T
F

}
′

\{ {T_\phi }\} ’ = \{ {T_\chi }\} ’ \cap \{ {T_F}\} ’

. The key step in the proof is Lemma 2, which is a curious result about nilpotent operators. One corollary of our main result is that if

χ
(
z
)
=

z
n

,
n
≥
1

\chi (z) = {z^n},n \geq 1

, then

{

T
ϕ

}
′

=
{

T
χ

}
′

∩
{

T
F

}
′

\{ {T_\phi }\} ’ = \{ {T_\chi }\} ’ \cap \{ {T_F}\} ’

, another is that if

ϕ

ϵ

H
∞

\phi \;\epsilon {H^\infty }

is univalent then

{

T
ϕ

}
′

=
{

T
z

}
′

\{ {T_\phi }\} ’ = \{ {T_z}\} ’

. We are also able to prove that if the inner factor of

ϕ
\phi

is

χ
(
z
)
=

z
n

,
n
≥
1

\chi (z) = {z^n},n \geq 1

, then

{

T
ϕ

}
′

=
{

T

z
s

}
′

\{ {T_\phi }\} ’ = \{ {T_{{z^s}}}\} ’

where s is a positive integer maximal with respect to the property that

z
n

{z^n}

and

F
(
z
)

F(z)

are both functions of

z
s

{z^s}

. We conclude by raising six questions.
On the quasi-simple irreducible representations of the Lorentz groups
Authors: Thieleker Ernest
Published: July 2010

For

n
≥
2

n \geq 2

, let

G
(
n
)

G(n)

denote the generalized homogeneous Lorentz group of an

n
+
1

n + 1

-dimensional real vector space; that is,

G
(
n
)

G(n)

is the identity component of the orthogonal group of a real quadratic form of index

(
+
,
−
−
…
−
)

( + , - - \ldots - )

. Let

G
^

(
n
)

\hat G(n)

denote a two-fold covering group of

G
(
n
)

G(n)

, and let

S
^

(
n
)

M
^

(
n
)

\hat S(n)\hat M(n)

be a parabolic subgroup of

G
^

(
n
)

\hat G(n)

. We consider the induced representations of

G
^

(
n
)

\hat G(n)

, induced by the finite-dimensional irreducible representations of

S
^

(
n
)

M
^

(
n
)

\hat S(n)\hat M(n)

. By an extension of the methods used in a previous paper, we determine precise criteria for the topological irreducibility of these representations. Moreover, in the exceptional cases when these representations fail to be irreducible, we determine the irreducible subrepresentations of these induced representations. By means of some general results of Harish-Chandra together with the main results of this paper, we obtain a complete classification, up to infinitesimal equivalence, of the quasi-simple irreducible representations of the groups

G
^

(
n
)

\hat G(n)

.
Branched structures and affine and projective bundles on Riemann surfaces
Authors: Mandelbaum Richard
Published: July 2010

A classification for analytic branched G-structures on a Riemann surface M is provided by means of a map

ϕ
G

{\phi _G}

, into the moduli spaces of flat G-bundles on

M
.

(
G
=

GA

(
1
,

C

)

M.\;(G = {\text {GA}}(1,{\text {C}})

or

PL

(
1
,

C

)
.
)

{\text {PL}}(1,{\text {C}}).)

Conditions are determined under which

ϕ
G

{\phi _G}

is injective and these conditions are related to the total branching order of the G-structures. A decomposition of the space of analytic branched G-structures into a disjoint union of analytic varieties is exhibited and it is shown that

ϕ
G

{\phi _G}

is is fact holomorphic on each such variety.
On the points of Weierstrass in dimensions greater than one
Authors: Ogawa Roy H.
Published: July 2010

In this paper, the classical concept of Weierstrass points on a Riemann surface is generalized to the consideration of similar points associated with a holomorphic vector bundle E over a compact complex manifold M. These points are invariants of the pair (E, M). The study of these general Weierstrass points is then initiated in this paper by deriving some results about the relationship of the points to singular sets of holomorphic mappings of the manifold to Grassmann spaces associated with the vector space of sections of the vector bundle. The accessibility of the point sets are demonstrated with some examples.
Free products of von Neumann algebras
Authors: Ching Wai Mee
Published: July 2010

A new method of constructing factors of typeII1{\text {II}_1}, called free product, is introduced. It is a generalization of the group construction of factors of typeII1{\text {II}_1}when the given group is a free product of two groups. IfA1{A_1}andA2{A_2}are two von Neumann algebras with separating cyclic trace vectors and ortho-unitary bases, then the free productA1∗A2{A_1} \ast {A_2}ofA1{A_1}andA2{A_2}is a factor of typeII1{\text {II}_1}without propertyΓ\Gamma.
Commutative torsion theory
Authors: Cahen Paul-Jean
Published: July 2010

This paper links several notions of torsion theory with commutative concepts. The notion of dominant dimension [H. H. Storrer, Torsion theories and dominant dimensions, Appendix to Lecture Notes in Math., vol. 177, Springer-Verlag, Berlin and New York, 1971. MR 44 #1685.] is shown to be very close to the notion of depth. For a commutative ring A and a torsion theory such that the primes of A, whose residue field is torsion-free, form an open set U of the spectrum of A, Spec A, a concrete interpretation of the module of quotients is given: if M is an A-module, its module of quotients

Q
(
M
)

Q(M)

is isomorphic to the module of sections

M
~

(
U
)

\tilde M(U)

, of the quasi-coherent module

M
~

\tilde M

canonically associated to M. In the last part it is proved that the (T)-condition of Goldman is satisfied [O. Goldman, Rings and modules of quotients, J. Algebra 13 (1969), 10-47. MR 39 #6914.] if and only if the set of primes, whose residue field is torsion-free, is an affine subset of Spec A, together with an extra condirion. The extra, more technical, condition is always satisfied over a Noetherian ring, in this case also it is classical that the (T)-condition of Goldman means that the localization functor Q is exact. This gives a new proof to Serre’s theorem [J.-P. Serre, Sur la cohomologie des variétés algébriques, J. Math. Pures Appl. (9) 36 (1957), 1-16. MR 18,765.]. As an application, the affine open sets of a regular Noetherian ring are also characterized.
Inverse semigroups which are separated over a subsemigroup
Authors: McAlister D. B.
Published: July 2010

An inverse semigroup T is separated over a subsemigroup S if T is generated, as an inverse semigroup, by S and for each

a
,
b
,
ϵ
S

a,b,\epsilon S

there exists

x

ϵ

S
a
∩
S
b

x\;\epsilon \;Sa \cap Sb

such that

a

−
1

a

b

−
1

b
=

x

−
1

x

{a^{ - 1}}a{b^{ - 1}}b = {x^{ - 1}}x

and dually for right ideals. For example, if T is generated as an inverse semigroup by a semigroup S whose principal left and right ideals form chains under inclusion, then T is separated over S. In this paper we investigate the structure of inverse semigroups T which are separated over subsemigroups S.
Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces
Authors: Petryshyn W. V.
Published: July 2010

Let X be a real Banach space, D a bounded open subset of X, and

D
¯

\bar D

the closure of D. In §1 of this paper we establish a general fixed point theorem (see Theorem 1 below) for 1-set-contractions and 1-ball-contractions

T
:

D
¯

→
X

T:\bar D \to X

under very mild conditions on T. In addition to classical fixed point theorems of Schauder, Leray and Schauder, Rothe, Kransnoselsky, Altman, and others for T compact, Theorem 1 includes as special cases the earlier theorem of Darbo as well as the more recent theorems of Sadovsky, Nussbaum, Petryshyn, and others (see §1 for further contributions and details) for T k-set-contractive with

k
>
1

k > 1

, condensing, and 1-set-contractive. In §§2, 3, 4, and 5 of this paper Theorem 1 is used to deduce a number of known, as well as some new, fixed point theorems for various special classes of mappings (e.g. mappings of contractive type with compact or completely continuous perturbations, mappings of semicontractive type introduced by Browder, mappings of pseudo-contractive type, etc.) which have been recently extensively studied by a number of authors and, in particular, by Browder, Krasnoselsky, Kirk, and others (see §1 for details),
Inequalities associated with regular and singular problems in the calculus of variations
Authors: Bradley J. S. , Everitt W. N.
Published: July 2010

An inequality of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript a Superscript b Baseline left-bracket p StartAbsoluteValue f prime EndAbsoluteValue squared plus q StartAbsoluteValue f EndAbsoluteValue squared right-bracket greater-than-or-equal-to mu 0 integral Subscript a Superscript b Baseline StartAbsoluteValue f EndAbsoluteValue squared left-parenthesis f element-of upper D right-parenthesis">

∫
a
b

[
p

|

f
′

|

2

+
q

|

f

|

2

]
≥

μ
0

∫
a
b

|

f

|

2

(
f
∈
D
)

\smallint _a^b[p|f’{|^2} + q|f{|^2}] \geq {\mu _0}\smallint _a^b|f{|^2}\;(f \in D)

is established, where p and q are real-valued coefficient functions and f is a complex-valued function in a set D so chosen that both sides of the inequality are finite. The interval of integration is of the form

−
∞
>
a
>
b
≤
∞

- \infty > a > b \leq \infty

. The inequality is first established for functions in the domain of an operator in the Hilbert function space

L
2

(
a
,
b
)

{L^2}(a,b)

that is associated with the differential equation

−
(
p

y
′

)
′

+
q
y
=
λ
y

- (py’)’ + qy = \lambda y

, and the number

μ
0

{\mu _0}

in the inequality is the smallest number in the spectrum of this operator. An approximation theorem is given that allows the inequality to be established for the larger set of functions D. An extension of some classical results from the calculus of variations and some spectral theory is then used to give necessary and sufficient conditions for equality and to show that the constant

μ
0

{\mu _0}

is best possible. Certain consequences of these conclusions are also discussed.
Constructing isotopes on noncompact 3-manifolds
Authors: Brown Marianne S.
Published: July 2010

We consider the question “When are two homeomorphisms of a noncompact

3
3

-manifold onto itself isotopic?” Roughly, the answer is when they are homotopic to each othet. More precisely, this paper deals with the question for irreducible

3
3

-manifolds which either have an infinite hierarchy or have such a hierarchy after the removal of a compact set. Manifolds having the first property are called end-irreducible; the others are called eventually endirreducible. There are two results fot each type of manifold depending on whether the homotopy between the two homeomorphisms sends the boundary of the manifold into itself or not.
Additive set functions on lattices of sets
Authors: DeBoth Gene A.
Published: July 2010

This paper is concerned with properties of additive set functions defined on lattices of sets. Extensions of results of Brunk and Johansen, Darst, Johansen, and Uhl are obtained. Two fundamental approximation properties for lattices of sets (established in another paper) permit us to translate the setting and consider countably additive set functions defined on sigma lattices of sets. Thereby results for countably additive set functions defined on sigma lattices of sets are used to obtain alternate derivations and extensions of Darst’s results for additive set functions defined on lattices of sets, i.e., we consider the Radon-Nikodym derivative, conditional expectation, and martingale convergence for lattices of sets.
An invariance principle for a class of 𝑑-dimensional polygonal random functions
Authors: Gorostiza Luis G.
Published: July 2010

A class of random functions is formulated, which represent the motion of a point in d-dimensional Euclidean space

(
d
≥
1
)

(d \geq 1)

undergoing random changes of direction at random times while maintaining constant speed. The changes of direction are determined by random orthogonal matrices that are irreducible in the sense of not having an almost surely invariant nontrivial subspace if

d
≥
2

d \geq 2

, and not being almost surely nonnegative if

d
=
1

d = 1

. An invariance principle stating that under certain conditions a sequence of such random functions converges weakly to a Gaussian process with stationary and independent increments is proved. The limit process has mean zero and its covariance matrix function is given explicitly. It is shown that when the random changes of direction satisfy an appropriate condition the limit process is Brownian motion. This invariance principle includes central limit theorems for the plane, with special distributions of the random times and direction changes, that have been proved by M. Kac, V. N. Tutubalin and T. Watanabe by methods different from ours. The proof makes use of standard methods of the theory of weak convergence of probability measures, and special results due to P. Billingsley and B. Rosén, the main problem being how to apply them. For this, renewal theoretic techniques are developed, and limit theorems for sums of products of independent identically distributed irreducible random orthogonal matrices are obtained.
On factorized groups
Authors: Buchthal David C.
Published: July 2010

The effect on a finite group G by the imposition of the condition that G is factorized by each of its maximal subgroups has been studied by Huppert, Deskins, Kegel, and others. In this paper, the effect on G brought about by the condition that G is factorized by a normalizer of a Sylow p-subgroup for each

p
∈
π
(
G
)

p \in \pi (G)

is studied. Through an extension of a classical theorem of Burnside, it is shown that certain results in the case where the factors are maximal subgroups continue to hold under the new conditions. Definite results are obtained in the case where the supplements of the Sylow normalizers are cyclic groups of prime power order or are abelian Hall subgroups of G.
Measurable transformations on compact groups
Authors: Choksi J. R.
Published: July 2010

For an arbitrary finite Baire measure

μ
\mu

on an arbitrary compact group G, it is shown that every automorphism of the measure algebra of

μ
\mu

can be induced by an invertible completion Baire measurable point transformation of G. If

μ
\mu

is Haar measure, the point transformation is completion Borel measurable.
Multipliers and linear functionals for the class 𝑁⁺
Authors: Yanagihara Niro
Published: July 2010

Multipliers for the classes

H
p

{H^p}

are studied recently by several authors, see Duren’s book, Theory of

H
p

{H^p}

spaces, Academic Press, New York, 1970. Here we consider corresponding problems for the class

N
+

{N^ + }

of holomorphic functions in the unit disk such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript r right-arrow 1 Endscripts integral Subscript 0 Superscript 2 pi Baseline log Superscript plus Baseline StartAbsoluteValue f left-parenthesis r e Superscript i theta Baseline right-parenthesis EndAbsoluteValue d theta equals integral Subscript 0 Superscript 2 pi Baseline log Superscript plus Baseline StartAbsoluteValue f left-parenthesis e Superscript i theta Baseline right-parenthesis EndAbsoluteValue d theta greater-than normal infinity period">

lim

r
→
1

∫
0

2
π

log

+

|

f
(
r

e

i
θ

)

|

d
θ
=

∫
0

2
π

log

+

|

f
(

e

i
θ

)

|

d
θ
>
∞
.

\lim \limits _{r \to 1} \int _0^{2\pi } {{{\log }^ + }} |f(r{e^{i\theta }})|d\theta = \int _0^{2\pi } {{{\log }^ + }|f({e^{i\theta }})|} d\theta > \infty .

\]
Our results are: 1.

N
+

{N^ + }

is an

F
F

-space in the sense of Banach with the distance function
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho left-parenthesis f comma g right-parenthesis equals StartFraction 1 Over 2 pi EndFraction integral Subscript 0 Superscript 2 pi Baseline log left-parenthesis 1 plus StartAbsoluteValue f left-parenthesis e Superscript i theta Baseline right-parenthesis minus g left-parenthesis e Superscript i theta Baseline right-parenthesis EndAbsoluteValue right-parenthesis d theta period">

ρ
(
f
,
g
)
=

1

2
π

∫
0

2
π

log
⁡
(
1
+

|

f
(

e

i
θ

)
−
g
(

e

i
θ

)

|

)

d
θ
.

\rho (f,g) = \frac {1}{{2\pi }}\int _0^{2\pi } {\log (1 + |f({e^{i\theta }}) - g({e^{i\theta }})|)} d\theta .

\]
2. A complex sequence

Λ
=
{

λ
n

}

\Lambda = \{ {\lambda _n}\}

is a multiplier for

N
+

{N^ + }

into

H
q

{H^q}

for a fixed

q
,
0
>
q
>
∞

q,0 > q > \infty

, if and only if

λ
n

=
O
(
exp
⁡
[
−
c

n

]
)

{\lambda _n} = O(\exp [ - c\sqrt n ])

for a positive constant

c
c

. 3. A continuous linear functional

ϕ
\phi

on the space

N
+

{N^ + }

is represented by a holomorphic function

g
(
z
)
=
Σ

b
n

z
n

g(z) = \Sigma {b_n}{z^n}

which satisfies

b
n

=
O
(
exp
⁡
[
−
c

n

]
)

{b_n} = O(\exp [ - c\sqrt n ])

for a positive constant

c
c

. Conversely, such a function

g
(
z
)
=
Σ

b
n

z
n

g(z) = \Sigma {b_n}{z^n}

defines a continuous linear functional on the space

N
+

{N^ + }

.
Wandering out to infinity of diffusion processes
Authors: Friedman Avner
Published: July 2010

Let

ξ
(
t
)

\xi (t)

be a diffusion process in

R
n

{R^n}

, given by

d
ξ
=
b
(
ξ
)
d
t
+
σ
(
ξ
)
d
w

d\xi = b(\xi )dt + \sigma (\xi )dw

. Conditions are given under which either

|

ξ
(
t
)

|

→
∞

|\xi (t)| \to \infty

as

t
→
∞

t \to \infty

with probability 1, or

ξ
(
t
)

\xi (t)

visits any neighborhood at a sequence of times increasing to infinity, with probability 1. The results are obtained both in case (i)

σ
(
x
)

\sigma (x)

is nondegenerate, and (ii)

σ
(
x
)

\sigma (x)

is degenerate at a finite number of points and hypersurfaces.
The 𝐿^{𝑝} norm of sums of translates of a function
Authors: Marek Kanter
Published: July 2010

For p not an even integer,

p
>
0

p > 0

, we prove that knowledge of the

L
p

{L^p}

norm of all linear combinations of translates of a real valued function in

L
p

(
R
)

{L^p}(R)

determines the function up to translation and multiplication by

±
1

\pm 1

.
The genera of amalgamations of graphs
Authors: Alpert Seth R.
Published: July 2010

If

p
≤
m

p \leq m

, n then

K
m

∨

K
p

K
n

{K_m}{ \vee _{{K_p}}}{K_n}

is the graph obtained by identify ing a copy of

K
p

{K_p}

contained in

K
m

{K_m}

with a copy of

K
p

{K_p}

contained in

K
n

{K_n}

. It is shown that for all integers

p
≤
m

p \leq m

, n the genus

g
(

K
m

∨

K
p

K
n

)

g({K_m}{ \vee _{{K_p}}}{K_n})

of

K
m

∨

K
p

K
n

{K_m}{ \vee _{{K_p}}}{K_n}

is less than or equal to

g
(

K
m

)
+
g
(

K
n

)

g({K_m}) + g({K_n})

. Combining this fact with the lower bound obtained from the Euler formula, one sees that for

2
≤
p
≤
5
,
g
(

K
m

∨

K
p

K
n

)

2 \leq p \leq 5,g({K_m}{ \vee _{{K_p}}}{K_n})

is either

g
(

K
m

)
+
g
(

K
n

)

g({K_m}) + g({K_n})

or else

g
(

K
m

)
+
g
(

K
n

)
−
1

g({K_m}) + g({K_n}) - 1

. Except in a few special cases, it is determined which of these values is actually attained.
Monotonically normal spaces
Authors: Heath R. W. , Lutzer D. J. , Zenor P. L.
Published: July 2010

This paper begins the study of monotone normality, a common property of linearly ordered spaces and of Borges’ stratifiable spaces. The concept of monotone normality is used to give necessary and sufficient conditions for stratifiability of a

T
1

{T_1}

-space, to give a new metrization theorem for p-spaces with

G
δ

{G_\delta }

-diagonals, and to provide an easy proof of a metrization theorem due to Treybig. The paper concludes with a list of examples which relate monotone normality to certain familiar topological properties.
Existence theorems for matroid designs
Authors: Young H. Peyton
Published: July 2010

A study is made of matroids in which the hyperplanes have equal cardinality. Fundamental constructions of such matroids are exhibited, and existence theorems are proved for large parametric classes of them.
Scattering theory for hyperbolic systems with coefficients of Gevrey type
Authors: Goodhue William L.
Published: July 2010

Using the techniques developed by P. D. Lax and R. S. Phillips, qualitative results on the location of the poles of the scattering matrix for symmetric, hyperbolic systems are obtained. The restrictions placed on the system are that the coefficient matrices be of Gevrey type and that the bicharacteristic rays tend to infinity.
On a compactness property of topological groups
Authors: Wang S. P.
Published: July 2010

A density theorem of semisimple analytic groups acting on locally compact groups is presented.
An equiconvergence theorem for a class of eigenfunction expansions
Authors: Pitts C. G. C.
Published: July 2010

A recent result of Muckenhoupt concerning the convergence of the expansion of an arbitrary function in terms of the Hermite series of orthogonal polynomials is generalised to a class of orthogonal expansions which arise from an eigenfunction problem associated with a second-order linear differential equation.
Wave equations with finite velocity of propagation
Authors: Berman Stephen J.
Published: July 2010

If B is a selfadjoint translation-invariant operator on the space

L
2

{L^2}

of complex-valued functions on n-dimensional Euclidean space which are square-summable with respect to Lebesgue measure, then the wave equation

d
2

F

/

d

t
2

+

B
2

F
=
0

{d^2}F/d{t^2} + {B^2}F = 0

has the solution

F
(
t
)
=
(
cos
⁡
t
B
)
f
+
(
(
sin
⁡
t
B
)

/

B
)
g

F(t) = (\cos tB)f + ((\sin tB)/B)g

, for f and g in

L
2

{L^2}

. In the classical case in which

−

B
2

- {B^2}

is the Laplacian, this solution has finite velocity of propagation in the sense that (letting supp denote support of a function)

supp

F
(
t
)
⊂
(

supp

f
∪

supp

g
)
+

K
t

{\text {supp}}\;F(t) \subset ({\text {supp}}\;f \cup {\text {supp}}\;g) + {K_t}

for all f and g and some compact set

K
t

{K_t}

independent of f and g. We show that a converse holds, namely, if

cos

t
B

\cos \;tB

has finite velocity of propagation (that is, if

supp

(
cos
⁡
t
B
)
f
⊂

supp

f
+

K
t

{\text {supp}}\,(\cos tB)f \subset {\text {supp}}\;f + {K_t}

for all f and some compact

K
t

{K_t}

) for three values of t whose reciprocals are independent over the rationals, then

B
2

{B^2}

must be a second order differential operator. If Euclidean space is replaced by a locally compact abelian group which does not contain the real line as a subgroup, then

cos

t
B

\cos \;tB

has finite velocity of propagation for all t if and only if it is convolution with a distribution

T
t

{T_t}

such that all

T
t

{T_t}

are supported on a compact open subgroup. Problems of a similar nature are discussed for compact connected abelian groups and for the nonabelian group

SL

(
2
,

R

)

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

.