bluuebunny/crossref_metadata_2025 huggingface dataset:
Stability theorems for some functional equations
Authors: MacCamy R. C. , Wong J. S. W.
Published: July 2010

Functional-differential equations of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove u With dot left-parenthesis t right-parenthesis equals minus integral Subscript 0 Superscript t Baseline upper A left-parenthesis t minus tau right-parenthesis g left-parenthesis u left-parenthesis tau right-parenthesis right-parenthesis d tau plus f left-parenthesis t comma u left-parenthesis t right-parenthesis right-parenthesis">

u
˙

(
t
)
=
−

∫
0
t

A
(
t
−
τ
)
g
(
u
(
τ
)
)
d
τ
+
f
(
t
,
u
(
t
)
)

\dot u(t) = - \int _0^t {A(t - \tau )g(u(\tau ))d\tau + f(t,u(t))}

\]
are considered. Here

u
(
t
)

u(t)

is to be an element of a Hilbert space

H

,
A
(
t
)

\mathcal {H},A(t)

a family of bounded symmetric operators on

H

\mathcal {H}

and g an operator with domain in

H

\mathcal {H}

. g may be unbounded. A is called strongly positive if there exists a semigroup exp St, where S is symmetric and

(
S
ξ
,
ξ
)
≦
−
m

‖
ξ
‖

2

,
m
>
0

(S\xi ,\xi ) \leqq - m{\left \| \xi \right \|^2},m > 0

, such that

A
∗

=
A
−
exp

{A^ \ast } = A - \exp

St is positive, that is,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript upper T Baseline left-parenthesis v left-parenthesis t right-parenthesis comma integral Subscript 0 Superscript t Baseline upper A Superscript asterisk Baseline left-parenthesis t minus tau right-parenthesis v left-parenthesis tau right-parenthesis right-parenthesis d tau greater-than-over-equals 0 comma">

∫
0
T

(

v
(
t
)
,

∫
0
t

A
∗

(
t
−
τ
)
v
(
τ
)

)

d
τ
≧
0
,

\int \nolimits _0^T \left ( {v(t),\int _0^t {{A^\ast }(t - \tau )v(\tau )} } \right )d\tau \geqq 0,

\]
for all smooth

v
(
t
)

v(t)

. It is shown that if A is strongly positive, and g and f are suitably restricted, then any solution which is weakly bounded and uniformly continuous must tend weakly to zero. Examples are given of both ordinary and partial differential-functional equations.
On replacing proper Dehn maps with proper embeddings
Authors: Feustel C. D.
Published: July 2010

In this paper we develop algebraic and geometric conditions which imply that a given proper Dehn map can be replaced by an embedding. The embedding, whose existence is implied by our theorem, retains most of the algebraic and geometric properties required in the original proper Dehn map.
Polynomials defined by generating relations
Authors: Mittal Hari Ballabh
Published: July 2010

Various authors have made efforts for finding new generating functions for known polynomial sets. In the present paper, by making use of the operatorTk=x(k+xD){T_k} = x(k + xD), a number of generating functions and characterizations have been obtained for various polynomials in a systematic manner.
Transversals to the flow induced by a differential equation on compact orientable 2-dimensional manifolds
Authors: Hartzman Carl S.
Published: July 2010

Every treatment of the theory of differential equations on a torus uses the fact that given a differential equation on a torus of class

C
k

{C^k}

, there is a non-null-homotopic closed Jordan curve

Γ
\Gamma

of class

C
k

{C^k}

which is transverse to the trajectories of the differential equation that pass through points of

Γ
\Gamma

. Such a curve necessarily cannot separate the torus. Here, we prove that given a differential equation on an n-fold torus

T
n

{T_n}

of class

C
k

{C^k}

, possessing only “simple” singularities of negative index there is a non-null-homotopic closed Jordan curve

Γ
\Gamma

of class

C
k

{C^k}

which is a transversal. The nonseparating property, however, does not follow immediately. For the particular case

T
2

{T_2}

, we prove the existence of such a transversal that does not separate

T
2

{T_2}

.
Bounded linear operators on Banach function spaces of vector-valued functions
Authors: Gretsky N. E. , Uhl J. J.
Published: July 2010

Representations of bounded linear operators on Banach function spaces of vector-valued functions to Banach spaces are given in terms of operator-valued measures. Then spaces whose duals are Banach function spaces are characterized. With this last information, reflexivity of this type of space is discussed. Finally, the structure of compact operators on these spaces is studied, and an observation is made on the approximation problem in this context.
A local spectral theory for operators. III. Resolvents, spectral sets and similarity
Authors: Stampfli J. G.
Published: July 2010

Let

T
T

be a bounded linear operator on a Hilbert space and assume

T
T

has thin spectrum. When is

T
T

similar to a normal operator? This problem is studied in a variety of situations and sufficient conditions are given in terms of characteristic functions, resolvents, spectral sets, and spectral resolutions. By contrast, the question “When is

T
T

normal?” has a relatively simple answer since in that case a necessary and sufficient condition can be given in terms of the resolvent alone.
Geodesic flow in certain manifolds without conjugate points
Authors: Eberlein Patrick
Published: July 2010

A complete simply connected Riemannian manifold H without conjugate points satisfies the uniform Visibility axiom if the angle subtended at a point p by any geodesic

γ
\gamma

of H tends uniformly to zero as the distance from p to

γ
\gamma

tends uniformly to infinity. A complete manifold M is a uniform Visibility manifold if it has no conjugate points and if the simply connected covering H satisfies the uniform Visibility axiom. We derive criteria for the existence of uniform Visibility manifolds. Let M be a uniform Visibility manifold, SM the unit tangent bundle of M and

T
t

{T_t}

the geodesic flow on SM. We prove that if every point of SM is nonwandering with respect to

T
t

{T_t}

then

T
t

{T_t}

is topologically transitive on SM. We also prove that if

M
′

M’

is a normal covering of M then

T
t

{T_t}

is topologically transitive on

S

M
′

SM’

if

T
t

{T_t}

is topologically transitive on SM.
On character sums and power residues
Authors: Norton Karl K.
Published: July 2010

Sharp estimates are given for a double sum involving Dirichlet characters. These are applied to the problem of estimating certain sums whose values give a measure of the average distance between successive power residues to an arbitrary modulus. A particularly good result of the latter type is obtained when the modulus is prime.
Oscillation and nonoscillation of solutions of generalized Emden-Fowler equations
Authors: Coffman C. V. , Wong J. S. W.
Published: July 2010

This paper treats the ordinary differential equation

y
+
y
F
(

y
2

,
x
)
=
0
,
x
>
0

y + yF({y^2},x) = 0,x > 0

, where

y
F
(

y
2

,
x
)

yF({y^2},x)

is continuous in (y, x) for

x
>
0
,

|

y

|

>
∞

x > 0,|y| > \infty

, and

F
(
t
,
x
)

F(t,x)

is non-negative; the equation is assumed to be either of sublinear or superlinear type. Criteria are given for the equation to be oscillatory, to be nonoscillatory, to possess oscillatory solutions or to possess nonoscillatory solutions. An attempt has been made to unify the methods of treatment of the sublinear and superlinear cases. These methods consist primarily of comparison with linear equations and the use of “energy” functions. An Appendix treats the questions of continuability and uniqueness of solutions of the equation considered in the main text.
Classification of all hereditarily indecomposable circularly chainable continua
Authors: Fearnley Lawrence
Published: July 2010

In a recent paper the author has established an affirmative solution to a well-known and previously unsolved problem raised by R. H. Bing in 1951 concerning whether or not the pseudo-circle is topologically unique. Now in this present paper, as the natural culmination of the study initiated in the earlier paper, complete topological and mapping classification theorems are established for all hereditarily indecomposable circularly chainable continua. The principal topological classification results of this paper are the theorems that hereditarily indecomposable circularly chainable continua are characterized set-theoretically by their equivalence classes of fundamental sequences and are characterized algebraic-topologically by their Čech cohomology groups. These topological classification theorems are then used in establishing mapping classification theorems for all hereditarily indecomposable circularly chainable continua and in proving that the mapping hierarchy of hereditarily indecomposable circularly chainable continua constitutes a lattice. Among the consequences of the foregoing primary results of the paper are the additional theorems that two hereditarily indecomposable circularly chainable continua are topologically equivalent if and only if each of them is a continuous image of the other, and that every

k
k

-adic pseudo-solenoid is topologically unique.
Open mappings of the universal curve onto continuous curves
Authors: Wilson David C.
Published: July 2010

A criterion for the existence of an open mapping from one compact metric space onto another is established in this paper. This criterion is then used to establish the existence of a monotone open mapping of the universal curve onto any continuous curve and the existence of a light open mapping of the universal curve onto any nondegenerate continuous curve. These examples show that if

f
f

is a monotone open or a light open mapping of one compact space

X
X

onto another

Y
Y

, then it will not necessarily be the case that

dim
⁡
Y
≦
dim
⁡
X
+
k

\dim Y \leqq \dim X + k

, where

k
k

is some positive integer.
Elementary statements over large algebraic fields
Authors: Jarden Moshe
Published: July 2010

We prove here the following theorems: A. If k is a denumerable Hilbertian field then for almost all

(

σ
1

,
…
,

σ
e

)
∈

G

(

k
s

/

k

)
e

({\sigma _1}, \ldots ,{\sigma _e}) \in \mathcal {G}{({k_s}/k)^e}

the fixed field of

{

σ
1

,
…
,

σ
e

}
,

k
s

(

σ
1

,
…
,

σ
e

)

\{ {\sigma _1}, \ldots ,{\sigma _e}\} ,{k_s}({\sigma _1}, \ldots ,{\sigma _e})

, has the following property: For any non-void absolutely irreducible variety V defined over

k
s

(

σ
1

,
…
,

σ
e

)

{k_s}({\sigma _1}, \ldots ,{\sigma _e})

the set of points of V rational over K is not empty. B. If E is an elementary statement about fields then the measure of the set of

σ
∈

G

(

Q
~

/

Q
)

\sigma \in \mathcal {G}(\tilde Q/Q)

(Q is the field of rational numbers) for which E holds in

Q
~

(
σ
)

\tilde Q(\sigma )

is equal to the Dirichlet density of the set of primes p for which E holds in the field

F
p

{F_p}

of p elements.
A note on quadratic Jordan algebras of degree 3
Authors: Racine M. L.
Published: July 2010

McCrimmon has defined a class of quadratic Jordan algebras of degree 3 obtained from a cubic form, a quadratic mapping and a base point. The structure of such an algebra containing no absolute zero-divisor is determined directly. A simple proof of Springer’s Theorem on isomorphism of reduced simple exceptional quadratic Jordan algebras is given.
Analytic continuation of Eisenstein series
Authors: Lewittes Joseph
Published: July 2010

The classical Eisenstein series are essentially of the form

Σ

m
,
n

′

(
(
m
+

r
1

)
z
+
n
+

r
2

)

−
s

\Sigma ’_{m, n} \left ( (m + r_1) z + n + r_2\right )^{-s}

,

m
m

,

n
n

ranging over integer values,

Im
⁡
z
>
0

\operatorname {Im} z > 0

,

r
1

r_1

,

r
2

r_2

rational and

s
s

an integer

>
2

> 2

. In this paper we show that if

s
s

is taken to be complex the series, with

r
1

,

r
2

{r_1},{r_2}

any real numbers, defines an analytic function of

(
z
,
s
)

(z,s)

for

Im
⁡
z
>
0

\operatorname {Im} z > 0

,

Re
⁡
s
>
2

\operatorname {Re} s > 2

. Furthermore this function has an analytic continuation over the entire

s
s

plane, exhibted explicitly by a convergent Fourier expansion. A formula for the transformation of the function when

z
z

is subjected to a modular transformation is obtained and the special case of

s
s

an integer is studied in detail.
The Hochschild homology of complete intersections
Authors: Wolffhardt Klaus
Published: July 2010

Let

R
~

\tilde {R}

be the algebra of all convergent (or of all strictly convergent) power series in

N
N

variables over a commutative field

K
K

of characteristic 0 with a valuation, e.g.

R
~

=
K
[

X
1

,
⋯
,

X
N

]

\tilde {R} = K[{X_1}, \cdots ,{X_N}]

. With each

K
K

-algebra

R
≅

R
~

/

a

R \cong \tilde {R}/\mathfrak {a}

we associate a bigraded

R
R

-algebra

E
E

. By the powers of

a

\mathfrak {a}

a filtration of the Poincaré complex of

R
~

\tilde {R}

is induced, and

E
E

is the first term of the corresponding spectral sequence. If

a

\mathfrak {a}

is generated by a prime sequence in

R
~

,
R

\tilde {R},R

is called a complete intersection, and

E
E

—with an appropriate simple grading—is isomorphic to the Hochschild homology of

R
R

. The result is applied to hypersurfaces.
A comparison principle for terminal value problems in ordinary differential equations
Authors: Hallam Thomas G.
Published: July 2010

A comparison principle for a terminal value problem of an ordinary differential equation is formulated. Basic related topics such as the existence of maximal and minimal solutions of terminal value problems are investigated. The close relationship between the existence of the extremal solutions of a terminal value problem and the concept of asymptotic equilibrium of the differential equation is explored. Several applications of the terminal comparison principle are given.
Topological types of polynomial differential equations
Authors: Markus L.
Published: July 2010

Consider a first order system of real ordinary differential equations, with polynomial coefficients, having no critical points in the number space

R
n

{R^n}

. Two such differential systems are called topologically equivalent in case there exists a homeomorphism of

R
n

{R^n}

onto itself carrying the sensed (not parametrized) solutions of the first system onto the solution family of the second system. Let

B
n

(
m
)

{B^n}(m)

be the cardinal number of topological equivalence classes for systems in

R
n

{R^n}

with polynomial coefficients of degree at most

m
m

. The author proves that

B
2

(
m
)

{B^2}(m)

is finite and obtains explicit upper and lower bounds in terms of

m
m

. Also examples are given to show that

B
n

(
m
)

{B^n}(m)

is noncountable for

n
⩾
3

n \geqslant 3

and

m
⩾
6

m \geqslant 6

.
Irreducible representations of the 𝐶*-algebra generated by an 𝑛-normal operator
Authors: Bunce John W. , Deddens James A.
Published: July 2010

For

A
A

an

n
n

-normal operator on Hilbert space, we determine the irreducible representations of

C
∗

(
A
)

{C^ \ast }(A)

, the

C
∗

{C^ \ast }

-algebra generated by

A
A

and the identity. For

A
A

a binormal operator, we determine an explicit description of the topology on the space of unitary equivalence classes of irreducible representations of

C
∗

(
A
)

{C^ \ast }(A)

.
A non-Noetherian factorial ring
Authors: David John
Published: July 2010

This paper supplies a counterexample to the conjecture that factorial implies Noetherian in finite Krull dimension. The example is the integral closure of a three-dimensional Noetherian ring, and is the union of Noetherian domains, which are proven to be factorial by means of derivation techniques.
Associo-symmetric algebras
Authors: Coughlin Raymond , Rich Michael
Published: July 2010

Let A be an algebra over a field F satisfying

(
x
,
x
,
x
)
=
0

(x,x,x) = 0

with a function

g
:
A
×
A
×
A
→
F

g:A \times A \times A \to F

such that

(
x
y
)
z
=
g
(
x
,
y
,
z
)
x
(
y
z
)

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

for all x, y, z in A. If

g
(

x
1

,

x
2

,

x
3

)
=
g
(

x

1
π

,

x

2
π

,

x

3
π

)

g({x_1},{x_2},{x_3}) = g({x_{1\pi }},{x_{2\pi }},{x_{3\pi }})

for all

π
\pi

in

S
3

{S_3}

and all

x
1

,

x
2

,

x
3

{x_1},{x_2},{x_3}

in A then A is called an associo-symmetric algebra. It is shown that a simple associo-symmetric algebra of degree

>
2

> 2

or degree

=
1

= 1

over a field of characteristic

≠
2

\ne 2

is associative. In addition a finite-dimensional semisimple algebra in this class has an identity and is a direct sum of simple algebras.
Iterated fine limits and iterated nontangential limits
Authors: Gowrisankaran Kohur
Published: July 2010

Let

Ω
k

,
k
=
1

 to 

n

{\Omega _k},k = 1{\text { to }}n

, be harmonic spaces of Brelot and

u
k

>
0

{u_k} > 0

harmonic functions on

Ω
k

{\Omega _k}

. For each

w
w

in a class of multiply superharmonic functions it is shown that the iterated fine limits of

[
w

/

u
1

⋯

u
n

]

[w/{u_1} \cdots {u_n}]

exist up to a set of measure zero for the product of the canonical measures corresponding to

u
k

{u_k}

and are independent of the order of iteration. This class contains all positive multiply harmonic functions on the product of

Ω
k

{\Omega _k}

’s. For a holomorphic function

f
f

in the Nevanlinna class of the polydisc

U
n

{U^n}

, it is shown that the

n
n

th iterated fine limits exist and equal almost everywhere on

T
n

{T^n}

the

n
n

th iterated nontangential limits of

f
f

, for any fixed order of iteration. It is then deduced that, with the exception of a set of measure zero on

T
n

{T^n}

, the absolute values of the different iterated limits of

f
f

are equal. It is also shown that the

n
n

th iterated nontangential limits are equal almost everywhere on

T
n

{T^n}

for any

f
f

in

N
1

(

U
n

)

{N_1}({U^n})

.
Lattice-ordered injective hulls
Authors: Steinberg Stuart A.
Published: July 2010

It is well known that the injective hull of a lattice-ordered group (

l
l

-group)

M
M

can be given a lattice order in a unique way so that it becomes an

l
l

-group extension of

M
M

. This is not the case for an arbitrary

f
f

-module over a partially ordered ring (po-ring). The fact that it is the case for any

l
l

-group is used extensively to get deep theorems in the theory of

l
l

-groups. For instance, it is used in the proof of the Hahn-embedding theorem and in the characterization of

ℵ
a

{\aleph _a}

-injective

l
l

-groups. In this paper we give a necessary and sufficient condition on the injective hull of a torsion-free

f
f

-module

M
M

(over a directed essentially positive po-ring) for it to be made into an

f
f

-module extension of

M
M

(in a unique way). An

f
f

-module is called an

i
−
f

i - f

-module if its injective hull can be made into an

f
f

-module extension. The class of torsion-free

i
−
f

i - f

-modules is closed under the formation of products, sums, and Hahn products of strict

f
f

-modules. Also, an

l
l

-submodule and a torsion-free homomorphic image of a torsion-free

i
−
f

i - f

-module are

i
−
f

i - f

-modules. Let

R
R

be an

f
f

-ring with zero right singular ideal whose Boolean algebra of polars is atomic. We show that

R
R

is a

q
f

qf

-ring (i.e.,

R
R

{R_R}

is an

i
−
f

i - f

-module) if and only if each torsion-free

R
−
f

R - f

-module is an

i
−
f

i - f

-module. There are no injectives in the category of torsion-free

R
−
f

R - f

-modules, but there are

ℵ
a

{\aleph _a}

-injectives. These may be characterized as the

f
f

-modules that are injective

R
R

-modules and

ℵ
a

{\aleph _a}

-injective

l
l

-groups. In addition, each torsion-free

f
f

-module over

R
R

can be embedded in a Hahn product of

l
l

-simple

Q
(
R
)
−
f

Q(R) - f

-modules. We note, too, that a totally ordered domain has an

i
−
f

i - f

-module if and only if it is a right Ore domain.
Some results on parafree groups
Authors: Roitberg Yael
Published: July 2010

We obtain some theorems concerning parafree groups in certain varieties, which are analogs of corresponding theorems about free groups in these varieties. Our principal results are: (1) A normal subgroup

N
N

of a parafree metabelian group

P
P

of rank

⩾
2

\geqslant 2

such that

N
⋅

γ
2

P

N \cdot {\gamma _2}P

has infinite index in

P
P

is not finitely generated unless it is trivial. (2) If

x
x

and

y
y

are elements of a parafree group

P
P

in any variety containing the variety of all metabelian groups which are independent modulo

γ
2

P

{\gamma _2}P

, then the commutator

[
x
,
y
]

[x,y]

is not a proper power.
A method of symmetrization and applications. II
Authors: Aharonov Dov , Kirwan W. E.
Published: July 2010

In this paper we make use of a new method of symmetrization introduced in [1] to study various covering properties of univalent functions. More specifically, we introduce a generalization of a classical problem raised by Fekete and give a partial solution.
The study of commutative semigroups with greatest group-homomorphism
Authors: Tamura Takayuki , Hamilton Howard B.
Published: July 2010

This paper characterizes commutative semigroups which admit a greatest group-homomorphism in various ways. One of the important theorems is that a commutative semigroup

S
S

has a greatest group-homomorphic image if and only if for every

a
∈
S

a \in S

there are

b
,
c
∈
S

b,c \in S

such that

a
b
c
=
c

abc = c

. Further the authors study a relationship between

S
S

and a certain cofinal subsemigroup and discuss the structure of commutative separative semigroups which have a greatest group-homomorphic image.
Π⁰₁ classes and degrees of theories
Authors: Jockusch Carl G. , Soare Robert I.
Published: July 2010

Using the methods of recursive function theory we derive several results about the degrees of unsolvability of members of certain

Π
1
0

\Pi _1^0

classes of functions (i.e. degrees of branches of certain recursive trees). As a special case we obtain information on the degrees of consistent extensions of axiomatizable theories, in particular effectively inseparable theories such as Peano arithmetic,

P

{\mathbf {P}}

. For example: THEOREM 1. If a degree

a

{\mathbf {a}}

contains a complete extension of

P

{\mathbf {P}}

, then every countable partially ordered set can be embedded in the ordering of degrees

⩽

a

\leqslant {\mathbf {a}}

. (This strengthens a result of Scott and Tennenbaum that no such degree

a

{\mathbf {a}}

is a minimal degree.) THEOREM 2. If

T

{\mathbf {T}}

is an axiomatizable, essentially undecidable theory, and if

{

a

n

}

\{ {{\mathbf {a}}_n}\}

is a countable sequence of nonzero degrees, then

T

{\mathbf {T}}

has continuum many complete extensions whose degrees are pairwise incomparable and incomparable with each

a

n

{{\mathbf {a}}_n}

. THEOREM 3. There is a complete extension

T

{\mathbf {T}}

of

P

{\mathbf {P}}

such that no nonrecursive arithmetical set is definable in

T

{\mathbf {T}}

. THEOREM 4. There is an axiomatizable, essentially undecidable theory

T

{\mathbf {T}}

such that any two distinct complete extensions of

T

{\mathbf {T}}

are Turing incomparable. THEOREM 5. The set of degrees of consistent extensions of

P

{\mathbf {P}}

is meager and has measure zero.
Characteristic classes of stable bundles of rank 2 over an algebraic curve
Authors: Newstead P. E.
Published: July 2010

Let

X
X

be a complete nonsingular algebraic curve over

C

{\mathbf {C}}

and

L
L

a line bundle of degree 1 over

X
X

. It is well known that the isomorphism classes of stable bundles of rank 2 and determinant

L
L

over

X
X

form a nonsingular projective variety

S
(
X
)

S(X)

. The Betti numbers of

S
(
X
)

S(X)

are also known. In this paper we define certain distinguished cohomology classes of

S
(
X
)

S(X)

and prove that these classes generate the rational cohomology ring. We also obtain expressions for the Chern character and Pontrjagin classes of

S
(
X
)

S(X)

in terms of these generators.
On a variation of the Ramsey number
Authors: Chartrand Gary , Schuster Seymour
Published: July 2010

Let

c
(
m
,
n
)

c(m,n)

be the least integer

p
p

such that, for any graph

G
G

of order

p
p

, either

G
G

has an

m
m

-cycle or its complement

G
¯

\bar G

has an

n
n

-cycle. Values of

c
(
m
,
n
)

c(m,n)

are established for

m
,
n
⩽
6

m,n \leqslant 6

and general formulas are proved for

c
(
3
,
n
)
,
c
(
4
,
n
)

c(3,n),c(4,n)

, and

c
(
5
,
n
)

c(5,n)

.
0≤𝑋²≤𝑋
Authors: Gellar Ralph
Published: July 2010

This paper studies the structure of elements

X
X

satisfying

0
⩽

X
2

⩽
X

0 \leqslant {X^2} \leqslant X

in a Dedekind

σ
\sigma

-complete partially ordered real linear algebra. The lollipop-shaped possible spectrum of

X
X

had been described previously. Three basic example types are described, each with possible spectrum a characteristic part of the lollipop and the possibility of splitting

X
X

into a sum of these types is considered. The matrix case is scrutinized. There are applications to operator theory. Contributions to the theory of convergence in partially ordered algebras are developed for technical purposes.
Canonical neighborhoods for topologically embedded polyhedra
Authors: Craggs Robert
Published: July 2010

D. R. McMillan has shown that in any neighborhood of a compact two sided surface in a

3
3

-manifold there is a closed neighborhood of the surface which is the sum of a solid homeomorphic to the cartesian product of the surface with the unit interval and some small disjoint cubes-with-handles each of which intersects the cartesian product in a disk on its boundary. In the present paper the author generalizes this notion of canonical neighborhood so that it applies to topological embeddings of arbitrary polyhedra in

3
3

-manifolds. This is done by replacing the cartesian products by small regular neighborhoods of polyhedral approximations to the topological embeddings.
On a convolution theorem for 𝐿(𝑝,𝑞) spaces
Authors: Blozinski A. P.
Published: July 2010

The principal result of this paper is a proof of the Convolution Theorem based on the definition of a convolution operator as presented by E. M. Stein and R. O’Neil. Closely related are earlier versions and special cases of the Convolution Theorem, which are

L
(
p
,
q
)

L(p,q)

analogues of an inequality of W. H. Young, given in papers by R. O’Neil, L. Y. H. Yap, R. Hunt, and B. Muckenhoupt and E. M. Stein.
Iterated limits in 𝑁*(𝑈ⁿ)
Authors: Davis Carl Stephen
Published: July 2010

It is shown that if f is in

N
∗

(

U
n

)

{N^\ast }({U^n})

, then the iterated limits of f are almost everywhere independent of the order of iteration. In fact, the iterated limit and the radial limit are equal almost everywhere.
Summability of Jacobi series
Authors: Askey Richard
Published: July 2010

The positivity of some Cesàro mean is proven for Jacobi series
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma a Subscript n Baseline upper P Subscript n Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis x right-parenthesis comma alpha comma beta greater-than-over-equals negative one half">

Σ

a
n

P
n

(
α
,
β
)

(
x
)
,
α
,
β
≧
−

1
2

\Sigma {a_n}P_n^{(\alpha ,\beta )}(x),\alpha ,\beta \geqq - \tfrac {1}{2}

. This has applications to the mean convergence of Lagrange interpolation at the zeros of Jacobi polynomials. The positivity of the

(
C
,
α
+
β
+
2
)

(C,\alpha + \beta + 2)

means is conjectured and proven for some

(
α
,
β
)

(\alpha ,\beta )

. One consequence of this conjecture would be the complete monotonicity of

x

−
c

(

x
2

+
1

)

−
c

,
c
≧
1

{x^{ - c}}{({x^2} + 1)^{ - c}},c \geqq 1

.
Asymptotic abelianness of infinite factors
Authors: Glaser M. S.
Published: July 2010

Studying Pukánszky’s type III factor,M2{M_2}, we show that it does not have the property of asymptotic abelianness and discuss how this property is related to property L. We also prove that there are no asymptotic abelianII∞{\text {II}_\infty }factors. The extension (by ampliation) of central sequences in a finite factor,N, toM⊗NM \otimes Nis shown to be central. Also, we give two examples of the reduction (by equivalence) of a central sequence inM⊗NM \otimes Nto a sequence inN. Finally, applying the definition of asymptotic abelianness ofC∗{C^\ast }-algebras toW∗{W^\ast }-algebras leads to the conclusion that all factors satisfying this property are abelian.
Submanifolds and a pinching problem on the second fundamental tensors
Authors: Okumura Masafumi
Published: July 2010

This paper gives a sufficient condition for a submanifold of a Riemannian manifold of nonnegative constant curvature to be totally umbilical. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.
Decreasing rearrangements and doubly stochastic operators
Authors: Day Peter W.
Published: July 2010

In this paper generalizations to measurable functions on a finite measure space

(
X
,
Λ
,
μ
)

(X,\Lambda ,\mu )

of some characterizations of the Hardy-Littlewood-Pólya preorder relation

≺
\prec

are considered. Let

ρ
\rho

be a saturated, Fatou function norm such that

L
∞

⊂

L
ρ

⊂

L
1

{L^\infty } \subset {L^\rho } \subset {L^1}

, and let

L
ρ

{L^\rho }

be universally rearrangement invariant. The following equivalence is shown to hold for all

f
∈

L
ρ

f \in {L^\rho }

iff

(
X
,
Λ
,
μ
)

(X,\Lambda ,\mu )

is nonatomic or discrete:

g
≺
f

g \prec f

iff g is in the

ρ
\rho

-closed convex hull of the set of all rearrangements of f. Finally, it is shown that

g
≺
f
∈

L
1

g \prec f \in {L^1}

iff g is the image of f by a doubly stochastic operator.
𝑘-congruence orders for 𝐸_{𝑘}
Authors: Murphy Grattan P.
Published: July 2010

This paper generalizes the notion of congruence order for metric spaces to k-metric (k-dimensional metric) spaces. The k-congruence order of

E
k

{E_k}

with respect to the class of oriented semi k-metric spaces is determined. An example shows that this result is sharp.
Fatou theorems for eigenfunctions of the invariant differential operators on symmetric spaces
Authors: Michelson H. Lee
Published: July 2010

On a Riemannian symmetric space of noncompact type we introduce a generalization of the Poisson kernel which may be used to generate simultaneous eigenfunctions of the invariant differential operators with eigenvalues not necessarily zero. We investigate the boundary behavior of our generalized Poisson integrals, extending to them many of the Fatou-type theorems known for harmonic functions.
𝛼_{𝑇} is finite for ℵ₁-categorical 𝑇
Authors: Baldwin John T.
Published: July 2010

Let

T
T

be a complete countable

ℵ
1

{\aleph _1}

-categorical theory. Definition. If

A

\mathcal {A}

is a model of

T
T

and

A
A

is a

1
1

-ary formula in

L
(

A

)

L(\mathcal {A})

then

A
A

has rank 0 if

A
(

A

)

A(\mathcal {A})

is finite.

A
(

A

)

A(\mathcal {A})

has rank

n
n

degree

m
m

iff for every set of

m
+
1

m + 1

formulas

B
1

,
⋯
,

B

m
+
1

∈

S
1

(
L
(

A

)
)

{B_1}, \cdots ,{B_{m + 1}} \in {S_1}(L(\mathcal {A}))

which partition

A
(

A

)

A(\mathcal {A})

some

B
i

(

A

)

{B_i}(\mathcal {A})

has rank

⩽
n
−
1

\leqslant n - 1

. Theorem. If

T
T

is

ℵ
1

{\aleph _1}

-categorical then for every

A

\mathcal {A}

a model of

T
T

and every

A
∈

S
1

(
L
(

A

)
)
,
A
(

A

)

A \in {S_1}(L(\mathcal {A})),A(\mathcal {A})

has finite rank. Corollary.

α
T

{\alpha _T}

is finite. The methods derive from Lemmas 9 and 11 in “On strongly minimal sets” by Baldwin and Lachlan.

α
T

{\alpha _T}

is defined in “Categoricity in power” by Michael Morley.
Boundary values in the four color problem
Authors: Albertson Michael O. , Wilf Herbert S.
Published: July 2010

Let

G
G

be a planar graph drawn in the plane so that its outer boundary is a

k
k

-cycle. A four coloring of the outer boundary

γ
\gamma

is admissible if there is a four coloring of

G
G

which coincides with

γ
\gamma

on the boundary. If

ψ
\psi

is the number of admissible boundary colorings, we show that the 4CC implies

ψ
⩾
3
⋅

2
k

\psi \geqslant 3 \cdot {2^k}

for

k
=
3
,
⋯
,
6

k = 3, \cdots ,6

. We conjecture this to be true for all

k
k

and show

ψ
\psi

is

⩾
c

(
(
1
+

5

1

/

2

)

/

2

)
k

\geqslant c{((1 + {5^{1/2}})/2)^k}

. A graph is totally reducible (t.r.) if every boundary coloring is admissible. There are triangulations of the interior of a

k
k

-cycle which are t.r. for anv

k
k

. We investigate a class of graphs called annuli, characterize t.r. annuli and show that annuli satisfy the above conjecture.
Regularity properties of the element of closest approximation
Authors: Shapiro Harold S.
Published: July 2010

Given an element

f
ϵ

L
p

(
T
)
,
1
>
p
>
∞

f \epsilon {L^p}(T),1 > p > \infty

, and a closed translation invariant subspace

S
S

of

L
p

(
T
)

{L^p}(T)

, we investigate the regularity (smoothness) properties of the element of

S
S

which is closest to

f
f

. The regularity of this element is in general less than that of

f
f

. The problem reveals a surprising connection with a hitherto unstudied class of extremal Fourier multipliers.
Generalized Laplacians and multiple trigonometric series
Authors: Kohn M. J.
Published: July 2010

V. L. Shapiro gave a

k
k

-variable analogue for Riemann’s theorem on formal integration of trigonometric series. This paper derives Shapiro’s results with weaker conditions on the coefficients of the series and extends the results to series which are Bochner-Riesz summable of larger order.
Self-dual axioms for many-dimensional projective geometry
Authors: Esser Martinus
Published: July 2010

Proposed and compared are four equivalent sets R, S, T, D of self-dual axioms for projective geometries, using points, hyperplanes and incidence as primitive elements and relation. The set R is inductive on the number of dimensions. The sets S, T, D all include the axiom “on every n points there is a plane", the dual of this axiom, one axiom on the existence of a certain configuration, and one or several axioms on the impossibility of certain configurations. These configurations consist of

(
n
+
1
)

(n + 1)

points and

(
n
+
1
)

(n + 1)

planes for sets S, T, but of

(
n
+
2
)

(n + 2)

points and

(
n
+
2
)

(n + 2)

planes for set D. Partial results are obtained by a preliminary study of self-dual axioms for simplicial spaces (spaces which may have fewer than 3 points per line).
Kernels in dimension theory
Authors: Aarts J. M. , Nishiura T.
Published: July 2010

All spaces are metrizable. A conjecture of de Groot states that a weak inductive dimension theory beginning with the class of compact spaces will characterize those spaces which can be extended to a compact space by the adjunction of a set of dimension not exceeding n. Nagata has proposed a variant of this conjecture as a means of finding insights into the original conjecture. (See Internat. Sympos. on Extension Theory, Berlin, 1967, pp. 157-161.) The proposed variant replaces compact with

σ
\sigma

-compact. The present paper concerns a study of strong inductive dimension theory beginning with an arbitrary class of spaces. The study is motivated by the above two conjectures. It indicates that a theory of kernels is a more natural by-product of inductive theory than a theory of extensions. An example has resulted which, with the aid of the developed theory and the Baire category theorem, resolves the second conjecture in the negative. The original conjecture is still unresolved. It is also shown that the notion of kernels results in a further generalization of Lelek’s form of the dimension lowering map theorem (Colloq. Math. 12 (1964), 221-227. MR 31 #716).
Free 𝑍₈ actions on 𝑆³
Authors: Ritter Gerhard X.
Published: July 2010

This paper is devoted to the problem of classifying periodic homeomorphisms which act freely on the

3
3

-sphere. The main result is the classification of free period eight actions and a generalization to free actions whose squares are topologically equivalent to orthogonal transformations. The result characterizes those

3
3

-manifolds which have the

3
3

-sphere as universal covering space and the cyclic group of order eight as fundamental group.
Equivariant cobordism and duality
Authors: Hook Edward C.
Published: July 2010

We consider equivariant cobordism theory, defined by means of an equivariant Thorn spectrum; in particular, we investigate the relationship between this theory and the more geometric equivariant bordism theory, showing that there is a Poincaré-Lefschetz duality theorem which is valid in this setting.
Orbits of families of vector fields and integrability of distributions
Authors: Sussmann Héctor J.
Published: July 2010

Let

D
D

be an arbitrary set of

C
∞

{C^\infty }

vector fields on the

C
∞

{C^\infty }

manifold

M
M

. It is shown that the orbits of

D
D

are

C
∞

{C^\infty }

submanifolds of

M
M

, and that, moreover, they are the maximal integral submanifolds of a certain

C
∞

{C^\infty }

distribution

P
D

{P_D}

. (In general, the dimension of

P
D

(
m
)

{P_D}(m)

will not be the same for all

m
∈
M

m \in M

.) The second main result gives necessary and sufficient conditions for a distribution to be integrable. These two results imply as easy corollaries the theorem of Chow about the points attainable by broken integral curves of a family of vector fields, and all the known results about integrability of distributions (i.e. the classical theorem of Frobenius for the case of constant dimension and the more recent work of Hermann, Nagano, Lobry and Matsuda). Hermann and Lobry studied orbits in connection with their work on the accessibility problem in control theory. Their method was to apply Chow’s theorem to the maximal integral submanifolds of the smallest distribution

Δ
\Delta

such that every vector field

X
X

in the Lie algebra generated by

D
D

belongs to

Δ
\Delta

(i.e.

X
(
m
)
∈
Δ
(
m
)

X(m) \in \Delta (m)

for every

m
∈
M

m \in M

). Their work therefore requires the additional assumption that

Δ
\Delta

be integrable. Here the opposite approach is taken. The orbits are studied directly, and the integrability of

Δ
\Delta

is not assumed in proving the first main result. It turns out that

Δ
\Delta

is integrable if and only if

Δ
=

P
D

\Delta = {P_D}

, and this fact makes it possible to derive a characterization of integrability and Chow’s theorem. Therefore, the approach presented here generalizes and unifies the work of the authors quoted above.
Upper bounds for vertex degrees of planar 5-chromatic graphs
Authors: Johnson Lee W.
Published: July 2010

Upper bounds are given for the degrees of vertices in planar

5
5

-chromatic graphs. Some inequalities are derived for irreducible graphs which restrict the type of planar graphs that can be irreducible.
Curvature tensors in Kaehler manifolds
Authors: Sitaramayya Malladi
Published: July 2010

Curvature tensors of Kaehler type (or type K) are defined on a hermitian vector space and it has been proved that the real vector space

L

K

(
V
)

{\mathcal {L}_K}(V)

of curvature tensors of type K on V is isomorphic with the vector space of sym metric endomorphisms of the symmetric product of

V
+

{V^ + }

, where

V

C

=

V
+

⊕

V
−

{V^{\text {C}}} = {V^ + } \oplus {V^ - }

(Theorem 3.6). Then it is shown that

L

K

(
V
)

{\mathcal {L}_K}(V)

admits a natural orthogonal decomposition (Theorem 5.1) and hence every

L
∈

L

K

(
V
)

L \in {\mathcal {L}_K}(V)

is expressed as

L
=

L
1

+

L
W

+

L
2

L = {L_1} + {L_W} + {L_2}

. These components are explicitly determined and then it is observed that

L
W

{L_W}

is a certain formal tensor introduced by Bochner. We call

L
W

{L_W}

the Bochner-Weyl part of L and the space of all these

L
W

{L_W}

is called the Weyl subspace of

L

K

(
V
)

{\mathcal {L}_K}(V)

.
Equivariant bordism of maps
Authors: Stong R. E.
Published: July 2010

This note computes the bordism classification of equivariant maps between closed manifolds with action of a cyclic group of prime order.
On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. I
Authors: Ho Chung Wu
Published: July 2010

Let

K
K

be a proper rectilinear triangulation of a

2
2

-simplex

S
S

in the plane and

L
(
K
)

L(K)

be the space of all homeomorphisms of

S
S

which are linear on each simplex of

K
K

and are fixed on

Bd
(
S
)

\text {Bd}(S)

. The author shows in this paper that

L
(
K
)

L(K)

with the compact open topology is simply-connected. This is a generalization of a result of S. S. Cairns in 1944 that the space

L
(
K
)

L(K)

is pathwise connected. Both results will be used in Part II of this paper to show that

π
0

(

L
2

)
=

π
1

(

L
2

)
=
0

{\pi _0}({L_2}) = {\pi _1}({L_2}) = 0

where

L
n

{L_n}

is a space of p.l. homeomorphisms of an

n
n

-simplex, a space introduced by

R

{\mathbf {R}}

. Thom in his study of the smoothings of combinatorial manifolds.
On derived functors of limit
Authors: Latch Dana May
Published: July 2010

If

A

\mathcal {A}

is a cocomplete category with enough projectives and

C

{\mathbf {C}}

is a

↓
\downarrow

-finite small category, then there is a spectral sequence which shows that the cardinality of

C

{\mathbf {C}}

and colimits over finite initial subcategories

C
′

{\mathbf {C’}}

of

C

{\mathbf {C}}

are determining factors for computation of derived functors of colimit. Applying a recent result of Mitchell to this spectral sequence we show that if the cardinality of

C

{\mathbf {C}}

is at most

ℵ

n

\aleph _{n}

, and the flat dimension of

Δ
∗

Z

{\Delta ^ \ast }Z

(constant diagram of type

C

op

{{\mathbf {C}}^{{\text {op}}}}

with value

Z
Z

) is

k
k

, then the derived functors of

lim

C

:

A

b

C

→

A

b

{\lim _{\mathbf {C}}}:\mathcal {A}{b^{\mathbf {C}}} \to \mathcal {A}b

vanish above dimension

n
+
1
+
k

n + 1 + k

.
Bicohomology theory
Authors: Van Osdol Donovan H.
Published: July 2010

Given a triple T and a cotriple G on a category

D

\mathcal {D}

such that T preserves group objects in

D

\mathcal {D}

, let P and M be in

D

\mathcal {D}

with M an abelian group object. Applying the “hom functor”

D

(
−
,
−
)

\mathcal {D}( - , - )

to the (co)simplicial resolutions

G
∗

P

{G^ \ast }P

and

T
∗

M

{T^ \ast }M

yields a double complex

D

(

G
∗

P
,

T
∗

M
)

\mathcal {D}({G^ \ast }P,{T^ \ast }M)

. The nth homology group of this double complex is denoted

H
n

(
P
,
M
)

{H^n}(P,M)

, and this paper studies

H
0

{H^0}

and

H
1

{H^1}

. When

D

\mathcal {D}

is the category of bialgebras arising from a triple, cotriple, and mixed distributive law, a complete description of

H
0

{H^0}

and

H
1

{H^1}

is given. The applications include a solution of the singular extension problem for sheaves of algebras.
Oscillation, continuation, and uniqueness of solutions of retarded differential equations
Authors: Burton T. , Grimmer R.
Published: July 2010

In this paper we present a number of results on continuation and uniqueńess of solutions of the n-dimensional system
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime left-parenthesis t right-parenthesis equals f left-parenthesis t comma x left-parenthesis t right-parenthesis right-parenthesis plus g left-parenthesis t comma x left-parenthesis t minus tau left-parenthesis t right-parenthesis right-parenthesis right-parenthesis">

x
′

(
t
)
=
f
(
t
,
x
(
t
)
)
+
g
(
t
,
x
(
t
−
τ
(
t
)
)
)

x’(t) = f(t,x(t)) + g(t,x(t - \tau (t)))

for

τ
(
t
)
≥
0

\tau (t) \geq 0

. We then give some necessary, some sufficient, and some necessary and sufficient conditions for oscillation of solutions of the second order equation

x
+
a
(
t
)
f
(
x
(
t
−
τ
(
t
)
)
)
=
0

x + a(t)f(x(t - \tau (t))) = 0

.
Product integral solutions for hereditary systems
Authors: Reneke James A.
Published: July 2010

Hereditary systems which satisfy a Lipschitz condition are solved in terms of product integrals. Realizations of this type of hereditary system are provided from functional differential and integral equations.
𝐶²-preserving strongly continuous Markovian semigroups
Authors: Priestley W. M.
Published: July 2010

Let

X
X

be a compact

C
2

{C^2}

-manifold. Let

|

|

|

|

,

|

|

|

|

′

||\;||,||\;||’

denote the supremum norm and the

C
2

{C^2}

-norm, respectively, and let

{

P
t

}

\{ {P^t}\}

be a Markovian semigroup on

C
(
X
)

C(X)

. The semigroup’s infinitesimal generator

A
A

, with domain

D

\mathcal {D}

, is defined by

A
f
=

lim

t
→
0

t

−
1

(

P
t

f
−
f
)

Af = {\lim _{t \to 0}}{t^{ - 1}}({P^t}f - f)

, whenever the limit exists in

|

|

|

|

||\;||

. Theorem. Assume that

{

P
t

}

\{ {P^t}\}

preserves

C
2

{C^2}

-functions and that the restriction of

{

P
t

}

\{ {P^t}\}

to

C
2

(
X
)
,

|

|

|

|

′

{C^2}(X),||\;||’

is strongly continuous. Then

C
2

(
X
)
⊂

D

{C^2}(X) \subset \mathcal {D}

and

A
A

is a bounded operator from

C
2

(
X
)
,

|

|

|

|

′

{C^2}(X),||\;||’

to

C
(
X
)
,

|

|

|

|

C(X),||\;||

. From the conclusion is obtained a representation of

A
f
⋅
(
x
)

Af \cdot (x)

as an integrodifferential operator on

C
2

(
X
)

{C^2}(X)

. The representation reduces to that obtained by Hunt [Semi-groups of measures on Lie groups, Trans. Amer. Math. Soc. 81 (1956), 264-293] in case

X
X

is a Lie group and

P
t

{P^t}

commutes with translations. Actually, a stronger result is proved having the above theorem among its corollaries.
Overrings of commutative rings. III. Normal pairs
Authors: Davis Edward D.
Published: July 2010

A pair of integral domains (A, B) is a normal (resp., QR-) pair provided that A is a subring of B and all intermediate rings are normal in B (resp., rings of quotients of A). The special case of B the field of fractions of A (e.g., Prüfer domains and Dedekind domains with torsion class group) has been studied in detail. It is shown that any domain A possesses a unique overring B maximal with respect to forming a normal (resp., QR-) pair with A. An explicit description of this overring and all the intermediate rings in terms of localizations A is obtained, and further details are provided in the presence of a noetherian-like condition on A. In addition, the “overring” characterizations of Prüfer domains are extended to “intermediate ring” characterizations of normal pairs.
A characteristic zero non-Noetherian factorial ring of dimension three
Authors: David John
Published: July 2010

This paper shows the previously unknown existence of a finite dimensional non-Noetherian factorial ring in characteristic zero. The example, called “

J
J

", contains a field of characteristic zero and is contained in a pure transcendental extension of degree three of that field.

J
J

is seen to be an ascending union of polynomial rings and degree functions are introduced on each of the polynomial rings. These are the basic facts that enable it to be seen that two extensions of

J
J

are Krull. One of these extensions is a simple one and the other is a localization of

J
J

at a prime ideal

P
P

. In the case of the latter extension, it is necessary to show that the intersection of the powers of

P
P

is zero. As

J
J

is the intersection of these two extensions, a theorem of Nagata is all that is needed to show then that

J
J

is factorial. It is easily proved that

J
J

is non-Noetherian once it is known to be factorial.
Almost maximal integral domains and finitely generated modules
Authors: Brandal Willy
Published: July 2010

We present a class of integral domains with all finitely generated modules isomorphic to direct sums of cyclic modules. This class contains all previously known examples (i.e., the principal ideal domains and the almost maximal valuation rings) and, by an example, at least one more domain. The class consists of the integral domains satisfying (1) every finitely generated ideal is principal (obviously a necessary condition) and (2) every proper homomorphic image of the domain is linearly compact. We call an integral domain almost maximal if it satisfies (2). This is one of eleven conditions which, for valuation rings, is equivalent of E. Matlis’ “almost maximal.” An arbitrary integral domain R is almost maximal if and only if it is h-local and

R
M

{R_M}

is almost maximal for every maximal ideal M of R. Finally, equivalent conditions for a Prüfer domain to be almost maximal are studied, and in the process some conjectures of E. Matlis are answered.