bluuebunny/crossref_metadata_2025 huggingface dataset:
On bilateral derivates and the derivative
Authors: Garg K. M.
Published: July 2010

In this paper we prove a new result on the monotonicity of a function in terms of its bilateral derivates, and obtain from it extensions of several existing results on such derivates and the derivative of a function. Let

f
:
R
→
R

f:R \to R

, where

R
R

denotes the set of real numbers. If its lower derivate

D
_

f
>
0

\underline {D} f > 0

at a nonmeager set of points, we prove

f
f

to be “adequately” increasing in some interval, viz. even the function

f
(
x
)
−
α
x

f(x) - \alpha x

is increasing for some

α
>
0

\alpha > 0

. When

f
f

is nowhere adequately monotone, it follows that there exists a residual set of points where

f
f

has a zero “median” derivate, i.e. either

D
_
f
⩽
0
⩽

D
−

f

D\_f \leqslant 0 \leqslant {D^ - }f

or

D
+

f
⩽
0
⩽

D
+

f

{D_ + }f \leqslant 0 \leqslant {D^ + }f

. These results remain valid for functions defined on an arbitrary set

X
⊂
R

X \subset R

under a mild continuity hypothesis, e.g. the absence of ordinary discontinuity at the unilateral limit points of

X
X

. The last result leads to a new version of A. P. Morse’s theorem for median derivates, and this in turn yields an improved version of the Goldowski-Tonelli theorem. We also obtain some necessary and sufficient conditions for a function to be nondecreasing, and extensions of the mean-value theorem and the Denjoy and other properties of a derivative. If

f
:
X
→
R

f:X \to R

, where

X
⊂
R

X \subset R

, and both the derivates of

f
f

are finite at a set of points that is not meager in

X
X

, then

f
f

is further proved to satisfy the Lipschitz condition on some portion of

X
X

. When

f
f

has a finite derivate almost everywhere and

X
X

has a finite measure, it is shown that

f
f

can be made Lipschitz by altering its values on a set with arbitrarily small measure. Some results on singular functions are also strengthened. The results and the methods of this paper further provide extensions of some results of Young, Tolstoff, Kronrod, Zahorski, Brudno, Fort, Hájek, Filipczak, Neugebauer and Lipiński on derivates and the derivability of a function.
A generalized topological measure theory
Authors: Kirk R. B. , Crenshaw J. A.
Published: July 2010

The theory of measures in a topological space, as developed by V. S. Varadarajan for the algebra

C
b

{C^b}

of bounded continuous functions on a completely regular topological space, is extended to the context of an arbitrary uniformly closed algebra

A
A

of bounded real-valued functions. Necessary and sufficient conditions are given for

A
∗

{A^ \ast }

to be represented in the natural way by a space of regular finitely-additive set functions. The concepts of additivity and tightness for these set functions are considered and some remarks about weak convergence are made.
On some classes of multivalent starlike functions
Authors: Leach Ronald J.
Published: July 2010

Classes of multivalent functions analogous to certain classes of univalent starlike functions are defined and studied. Estimates on coefficients and distortion are made, using a variety of techniques.
Stable positive definite functions
Authors: Parthasarathy K. R. , Schmidt K.
Published: July 2010

This paper investigates the stability of positive definite functions on locally compact groups under one parameter groups of automorphisms. As an application of this it is shown that the only probability distributions on

R
n

{R^n}

which are stable under the full automorphism group

GL
(
n
,
R
)

\text {GL}(n,R)

of

R
n

{R^n}

are the nondegenerate Gaussian distributions. It is furthermore shown that there are no nondegenerate probability distributions of

R
n

{R^n}

which are stable under

SL

(
n
,
R
)

{\text {SL}}(n,R)

.
On sums over Gaussian integers
Authors: Hazlewood D. G.
Published: July 2010

The object of this paper is to give asymptotic estimates for some number theoretic sums over Gaussian integers. As a consequence of general estimates, asymptotic estimates with explicit error terms for the number of Gaussian integers with only “large” prime factors and for the number of Gaussian integers with only “small” prime factors are given.
Sobolev-Galpern equations of order 𝑛+2 in 𝑅^{𝑚}×𝑅, 𝑚≥2
Authors: Rao V. R. Gopala
Published: July 2010

Equations with mixed time and space derivatives play an important role in several branches of physics. Here we establish existence and uniqueness results for such equations. In addition, we also prove a regularity result which employs a regularity result for nonhomogeneous elliptic equations whose proof is also included.
Concerning first countable spaces. III
Authors: Reed G. M.
Published: July 2010

The primary purpose of this paper is (1) to provide a “real” example of a regular first countable

T
1

{T_1}

-space which has no dense developable subspace and (2) to provide a new technique for producing Moore spaces which fail to have dense metrizable subspaces. Related results are established which produce new examples of noncompletable Moore spaces and which show that each regular hereditary

M
M

-space with a

G
δ

{G_\delta }

-diagonal has a dense metrizable subspace.
Compactness properties of topological groups. III
Authors: Wang S. P.
Published: July 2010

Compactness properties of topological groups and finiteness of Haar measure on homogeneous spaces are studied. Some concrete structure theorems are presented.
On minimal immersions of 𝑆² into 𝑆^{2𝑚}
Authors: Barbosa Jo ao Lucas Marquês
Published: July 2010

The study of minimal immersions of the

2
2

-sphere into the standard

n
n

-sphere of the euclidean space has been better accomplished by associating to each such immersion a certain holomorphic curve. This has been done in several ways in the literature. In the present paper we explore this technique applying some knowledge about the topological and analytical invariants of the particular set of holomorphic curves used to obtain further results. Some new examples are provided, a beginning of a general description of such immersions is given and a rigidity theorem is proved.
Isotropic immersions and Veronese manifolds
Authors: Itoh T. , Ogiue K.
Published: July 2010

An

n
n

-dimensional Veronese manifold is defined as a minimal immersion of an

n
n

-sphere of curvature

n

/

2
(
n
+
1
)

n/2(n + 1)

into an

{
n
(
n
+
3
)

/

2
−
1
}

\{ n(n + 3)/2 - 1\}

-dimensional unit sphere. The purpose of this paper is to give some characterizations of a Veronese manifold in terms of isotropic immersions.
Monotone and open mappings on manifolds. I
Authors: Walsh John J.
Published: July 2010

Sufficient conditions are given for the existence of open mappings from a p. 1. manifold

M
m

,
m
⩾
3

{M^m},m \geqslant 3

, onto a polyhedron

Q
Q

. In addition, it is shown that a mapping

f
f

from

M
m

,
m
⩾
3

{M^m},m \geqslant 3

, to

Q
Q

is homotopic to a monotone mapping of

M
M

onto

Q
Q

iff

f
∗

:

π
1

(
M
)
→

π
1

(
Q
)

{f_ \ast }:{\pi _1}(M) \to {\pi _1}(Q)

is onto. Finally, it is shown that a monotone mapping of

M
m

,
m
⩾
3

{M^m},m \geqslant 3

, onto

Q
Q

can be approximated by a monotone open mapping of

M
M

onto

Q
Q

.
Results on sums of continued fractions
Authors: Hlavka James L.
Published: July 2010

Let

F
(
m
)

F(m)

be the (Cantor) set of infinite continued fractions with partial quotients no greater than m and let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis m right-parenthesis plus upper F left-parenthesis n right-parenthesis equals left-brace alpha plus beta colon alpha element-of upper F left-parenthesis m right-parenthesis comma beta element-of upper F left-parenthesis n right-parenthesis right-brace">

F
(
m
)
+
F
(
n
)
=
{
α
+
β
:
α
∈
F
(
m
)
,
β
∈
F
(
n
)
}

F(m) + F(n) = \{ \alpha + \beta :\alpha \in F(m),\beta \in F(n)\}

. We show that

F
(
3
)
+
F
(
4
)

F(3) + F(4)

is an interval of length 1.14 ... so every real number is the sum of an integer, an element of

F
(
3
)

F(3)

and an element of

F
(
4
)

F(4)

. Similar results are given for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis 2 right-parenthesis plus upper F left-parenthesis 7 right-parenthesis comma upper F left-parenthesis 2 right-parenthesis plus upper F left-parenthesis 2 right-parenthesis plus upper F left-parenthesis 4 right-parenthesis comma upper F left-parenthesis 2 right-parenthesis plus upper F left-parenthesis 3 right-parenthesis plus upper F left-parenthesis 3 right-parenthesis">

F
(
2
)
+
F
(
7
)
,
F
(
2
)
+
F
(
2
)
+
F
(
4
)
,
F
(
2
)
+
F
(
3
)
+
F
(
3
)

F(2) + F(7),F(2) + F(2) + F(4),F(2) + F(3) + F(3)

and

F
(
2
)
+
F
(
2
)
+
F
(
2
)
+
F
(
2
)

F(2) + F(2) + F(2) + F(2)

. The techniques used are applicable to any Cantor sets in R for which certain parameters can be evaluated.
On subnormal operators
Authors: Radjabalipour Mehdi
Published: July 2010

Let T be the adjoint of a subnormal operator defined on a Hilbert space H. For any closed set

δ
\delta

, let

X
T

(
δ
)
=
{
x
∈
H

{X_T}(\delta ) = \{ x \in H

: there exists an analytic function

f
x

:

C

∖
δ
→
H

{f_x}:{\text {C}}\backslash \delta \to H

such that

(
z
−
T
)

f
x

(
z
)
≡
x
}

(z - T){f_x}(z) \equiv x\}

. It is shown that T is decomposable (resp. normal) if

X
T

(
∂

G
α

)

{X_T}(\partial {G_\alpha })

is closed (resp. if

X
T

(
∂

G
α

)
=
{
0
}
)

{X_T}(\partial {G_\alpha }) = \{ 0\} )

for a certain family

{

G
α

}

\{ {G_\alpha }\}

of open sets. Some of the results are extended to the case that T is the adjoint of the restriction of a spectral or decomposable operator to an invariant subspace.
Two applications of twisted wreath products to finite soluble groups
Authors: Hawkes Trevor O.
Published: July 2010

The group construction sometimes known as the twisted wreath product is used here to answer two questions in the theory of finite, soluble groups: first to show that an arbitrary finite, soluble group may be embedded as a subgroup of a group whose upper nilpotent series is a chief series; second to construct an A-group whose Carter subgroup is “small” relative to its nilpotent length.
Homology with multiple-valued functions applied to fixed points
Authors: Jerrard Richard
Published: July 2010

Certain multiple-valued functions (m-functions) are defined and a homology theory based upon them is developed. In this theory a singular simplex is an m-function from a standard simplex to a space and an m-function from one space to another induces a homomorphism of homology modules. In a family of functions

f
x

:
Y
→
Y

{f_x}:Y \to Y

indexed by

x
∈
X

x \in X

the fixed points of

f
x

{f_x}

are taken to be the images at x of a multiple-valued function

ϕ
:
X
→
Y

\phi :X \to Y

. In certain circumstances

ϕ
\phi

is an m-function, giving information about the behavior of the fixed points of

f
x

{f_x}

as x varies over X. These facts are applied to self-maps of products of compact polyhedra and the question of whether such a product has the fixed point property for continuous functions is essentially reduced to the question of whether one of its factors has the fixed point property for m-functions. Some light is thrown on the latter problem by using the homology theory to prove a Lefschetz fixed point theorem for m-functions.
Group presentations and formal deformations
Authors: Wright Perrin
Published: July 2010

Formal deformations (expansions and collapses) of dimension

⩽
3

\leqslant 3

among

2
2

-dimensional polyhedra are explained in terms of a certain collection of operations on finite group presentations. The results are valid for any simple homotopy type of

2
2

-dimensional polyhedra, and simplifications are possible within the simply connected simple homotopy types.
Circle actions on homotopy spheres not bounding spin manifolds
Authors: Schultz Reinhard
Published: July 2010

Smooth circle actions are constructed on odd-dimensional homotopy spheres that do not bound spin manifolds. Examples are given in every dimension for which exotic spheres of the described type exist.
Some one-sided theorems on the tail distribution of sample sums with applications to the last time and largest excess of boundary crossings
Authors: Chow Y. S. , Lai T. L.
Published: July 2010

In this paper, we prove certain one-sided Paley-type inequalities and use them to study the convergence rates for the tail probabilities of sample sums. We then apply our results to find the limiting moments and the limiting distribution of the last time and the largest excess of boundary crossings for the sample sums, generalizing the results previously obtained by Robbins, Siegmund and Wendel. Certain one-sided limit theorems for delayed sums are also obtained and are applied to study the convergence rates of tail probabilities.
Gelfond’s method for algebraic independence
Authors: Brownawell W. Dale
Published: July 2010

This paper extends Gelfond’s method for algebraic independence to fields

K
K

with transcendence type

⩽
τ

\leqslant \tau

. The main results show that the elements of a transcendence basis for

K
K

and at least two more numbers from a prescribed set are algebraically independent over

Q
Q

. The theorems have a common hypothesis:

{

α
1

,
…
,

α
M

}
,
{

β
1

,
…
,

β
N

}

\{ {\alpha _1}, \ldots ,{\alpha _M}\} ,\{ {\beta _1}, \ldots ,{\beta _N}\}

are sets of complex numbers, each of which is

Q
Q

-linearly independent. THEOREM A. If

(
2
τ
−
1
)
>
M
N

(2\tau - 1) > MN

, then at least two of the numbers
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Subscript i Baseline comma beta Subscript j Baseline comma exp left-parenthesis alpha Subscript i Baseline beta Subscript j Baseline right-parenthesis comma 1 less-than-or-slanted-equals i less-than-or-slanted-equals upper M comma 1 less-than-or-slanted-equals j less-than-or-slanted-equals upper N">

α
i

,

β
j

,
exp
⁡
(

α
i

β
j

)
,
1
⩽
i
⩽
M
,
1
⩽
j
⩽
N

{\alpha _i},{\beta _j},\exp ({\alpha _i}{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N

, are algebraically dependent over

K
K

. THEOREM B. If

2
τ
(
M
+
N
)
⩽
M
N
+
M

2\tau (M + N) \leqslant MN + M

, then at least two of the numbers
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Subscript i Baseline comma exp left-parenthesis alpha Subscript i Baseline comma beta Subscript j Baseline right-parenthesis comma 1 less-than-or-slanted-equals i less-than-or-slanted-equals upper M comma 1 less-than-or-slanted-equals j less-than-or-slanted-equals upper N">

α
i

,
exp
⁡
(

α
i

,

β
j

)
,
1
⩽
i
⩽
M
,
1
⩽
j
⩽
N

{\alpha _i},\exp ({\alpha _i},{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N

, are algebraically dependent over

K
K

. THEOREM C. If

2
τ
(
M
+
N
)
⩽
M
N

2\tau (M + N) \leqslant MN

, then at least two of the numbers

1
⩽
i
⩽
M
,
1
⩽
j
⩽
N

1 \leqslant i \leqslant M,1 \leqslant j \leqslant N

, are algebraically dependent over

K
K

.
Adjoint groups, regular unipotent elements and discrete series characters
Authors: Lehrer G. I.
Published: July 2010

It is shown that if G is a finite Chevalley group or twisted type over a field of characteristic p and U is a maximal p-subgroup of G then any nonlinear irreducible character of U vanishes on regular elements. For groups of adjoint type the linear content of the restriction to U of a discrete series character J of G is calculated and it is deduced that J takes the value 0 or

(
−
1

)
s

{( - 1)^s}

on regular elements of U

(
s
=

rank

G
)

(s = {\text {rank}}\;G)

.
𝑜-weakly compact mappings of Riesz spaces
Authors: Dodds P. G.
Published: July 2010

A characterization is given of linear mappings from a Riesz space to a Banach space which map order intervals to relatively weakly compact sets. The characterization is based on recent results of Burkinshaw and Fremlin. A number of applications are made to known results concerning weakly compact mappings and to results in the theory of Banach space-valued measures.
Weighted shifts and covariance algebras
Authors: O’Donovan Donal P.
Published: July 2010

The

C
∗

{C^ \ast }

-algebras generated by bilateral and unilateral shifts are studied in terms of certain covariance algebras. This enables one to obtain an answer to the question of when such shifts are G.C.R., or not, or even when they are N.G.C.R.. In addition these shifts are classified to within algebraic equivalence.
Some open mapping theorems for marginals
Authors: Eifler Larry Q.
Published: July 2010

Let S and T be compact Hausdorff spaces and let

P
(
S
)
,
P
(
T
)

P(S),P(T)

and

P
(
S
×
T
)

P(S \times T)

denote the collection of probability measures on S, T and

S
×
T

S \times T

, respectively. Given a probability measure

μ
\mu

on

S
×
T

S \times T

, set

π
μ
=
(
α
,
β
)

\pi \mu = (\alpha ,\beta )

where

α
\alpha

and

β
\beta

are the marginals of

μ
\mu

on S and T. We prove that the mapping

π
:
P
(
S
×
T
)
→
P
(
S
)
×
P
(
T
)

\pi :P(S \times T) \to P(S) \times P(T)

is norm open and

weak
∗

{\text {weak}^\ast }

open. An analogous result for

L
1

(
X
×
Y
,
μ
×
ν
)

{L_1}(X \times Y,\mu \times \nu )

where

(
X
,
μ
)

(X,\mu )

and

(
Y
,
ν
)

(Y,\nu )

are probability spaces is established.
Diffeomorphisms obtained from endomorphisms
Authors: Block Louis
Published: July 2010

It is shown that if f is a differentiable map of a compact manifold, and the singularities of f satisfy a certain condition, then there is a diffeomorphism (of a different manifold) whose orbit structure is closely related to that of f. This theorem is then used to extend several results on the orbit structure of diffeomorphisms to the noninvertible case.
Fourier analysis on the sphere
Authors: Sherman Thomas O.
Published: July 2010

A new approach to harmonic analysis on the unit sphere in

R

d
+
1

{{\mathbf {R}}^{d + 1}}

is given, closer in form to Fourier analysis on

R

d

{{\mathbf {R}}^d}

than the usual development in orthonormal polynomials. Singular integrals occur in the transform formulae. The results generalize to symmetric space.
An improved version of the noncompact weak canonical Schoenflies theorem
Authors: Brakes W. R.
Published: July 2010

The main result of this paper is that any proper collared embedding of

R

n
−
1

{R^{n - 1}}

in

R
n

{R^n}

can be extended to a homeomorphism of

R
n

{R^n}

such that the extension depends continuously on the original embedding in a stronger sense than previously known. Analogous results are proved for proper embeddings of

R
k

{R^k}

in

R
n

{R^n}

(with the usual homotopy conditions when

k
=
n
−
2

k = n - 2

). An alternative proof of the usual compact form of the weak canonical Schoenflies theorem is also obtained.
Differential geometry on simplicial spaces
Authors: Penna Michael A.
Published: July 2010

A simplicial space M is a separable Hausdorff topological space equipped with an atlas of linearly related charts of varying dimension; for example every polyhedron is a simplicial space in a natural way. Every simplicial space possesses a natural structure complex of sheaves of piecewise smooth differential forms, and the homology of the corresponding de Rham complex of global sections is isomorphic to the real cohomology of M. A cosimplicial bundle is a continuous surjection

ξ
:
E
→
M

\xi :E \to M

from a topological space E to a simplicial space M which satisfies certain criteria. There is a category of cosimplicial bundles which contains a subcategory of vector bundles. To every simplicial space M a cosimplicial bundle

τ
(
M
)

\tau (M)

over M is associated;

τ
(
M
)

\tau (M)

is the cotangent object of M since there is an isomorphism between the module of global piecewise smooth one-forms on M and sections of

τ
(
M
)

\tau (M)

.
Minimal covers and hyperdegrees
Authors: Simpson Stephen G.
Published: July 2010

Every hyperdegree at or above that of Kleene’s

O
O

is the hyperjump and the supremum of two minimal hyperdegrees (Theorem 3.1). There is a nonempty

Σ
1
1

\Sigma _1^1

class of number-theoretic predicates each of which has minimal hyperdegree (Theorem 4.7). If

V
=
L

V = L

or a generic extension of

L
L

, then there are arbitrarily large hyperdegrees which are not minimal over any hyperdegree (Theorems 5.1, 5.2). If

O
#

{O^\# }

exists, then there is a hyperdegree such that every larger hyperdegree is minimal over some hyperdegree (Theorem 5.4). Several other theorems on hyperdegrees and degrees of nonconstructibility are presented.
On 𝜋₃ of a finite 𝐻-space
Authors: Hubbuck J. R. , Kane R.
Published: July 2010

The third homotopy group of a finite H-space is shown to have no torsion.
Endomorphism rings and direct sums of torsion free abelian groups
Authors: Arnold D. M. , Lady E. L.
Published: July 2010

Properties of abelian groups related to a given finite rank torsion free abelian group A are analyzed in terms of End (A), the endomorphism ring of A. This point of view gives rise to generalizations of some classical theorems by R. Baer and examples of pathological direct sum decompositions of finite rank torsion free abelian groups.
On the double suspension homomorphism
Authors: Mahowald Mark
Published: July 2010

This paper studies the family of unstable Adams spectral sequences,

E
2

s
,
t

(

S

2
n
+
1

)

E_2^{s,t}({S^{2n + 1}})

. The main results deal with the range of filtrations for which these groups stabilize and for which the groups

E
2

s
,
t

(

Ω
2

S

2
n
+
1

,

S

2
n
−
1

)

E_2^{s,t}({\Omega ^2}{S^{2n + 1}},{S^{2n - 1}})

stabilize.
Function algebras and flows. IV
Authors: Muhly Paul S.
Published: July 2010

The automorphisms of the algebra

A

\mathfrak {A}

of analytic functions associated with a flow (without periodic orbits) are completely determined. This result extends earlier work of Arens who determined the automorphisms of

A

\mathfrak {A}

when the flow is almost periodic. The Choquet boundary of the maximal ideal space of

A

\mathfrak {A}

is also determined under the hypothesis that the flow has no fixed points.
Automorphisms of 𝐺𝐿_{𝑛}(𝑅)
Authors: McDonald Bernard R.
Published: July 2010

Let R be a commutative ring and S a multiplicatively closed subset of R having no zero divisors. The pair

⟨
R
,
S
⟩

\langle R,S\rangle

is said to be stable if the ring of fractions of R,

S

−
1

R

{S^{ - 1}}R

, defined by S is a ring for which all finitely generated projective modules are free. For a stable pair

⟨
R
,
S
⟩

\langle R,S\rangle

assume 2 is a unit in R and V is a free R-module of dimension

⩾
3

\geqslant 3

. This paper examines the action of a group automorphism of

G
L
(
V
)

GL(V)

(the general linear group) on the elementary matrices relative to a basis B of V. In the case that R is a local ring, a Euclidean domain, a connected semilocal ring or a Dedekind domain whose quotient field is a finite extension of the rationals, we obtain a description of the action of the automorphism on all elements of

G
L
(
V
)

GL(V)

.
Almost isolated spectral parts and invariant subspaces
Authors: Putnam C. R.
Published: July 2010

Let T be an operator with spectrum

σ
(
T
)

\sigma (T)

on a Hilbert space. A compact subset E of

σ
(
T
)

\sigma (T)

is called a disconnected part of

σ
(
T
)

\sigma (T)

if, for some bounded open set A, E is the closure of

σ
(
T
)
∩
A

\sigma (T) \cap A

and

σ
(
T
)
−
E

\sigma (T) - E

is the union of the isolated parts of

σ
(
T
)

\sigma (T)

lying completely outside the closure of A. The set E is called an almost isolated part of

σ
(
T
)

\sigma (T)

if, in addition, the set A can be chosen so as to have a rectifiable boundary

∂
A

\partial A

on which the subset

σ
(
T
)
∩
∂
A

\sigma (T) \cap \partial A

has arc length measure 0. The following results are obtained. If T is subnormal and if E is a disconnected part of

σ
(
T
)

\sigma (T)

then there exists a reducing subspace

M

\mathfrak {M}

of T for which

σ
(
T

|

M

)
=
E

\sigma (T|\mathfrak {M}) = E

. If

T
∗

{T^\ast }

is hyponormal and if E is an almost isolated part of

σ
(
T
)

\sigma (T)

then there exists an invariant subspace

M

\mathfrak {M}

of T for which

σ
(
T

|

M

)
=
E

\sigma (T|\mathfrak {M}) = E

. An example is given showing that if T is arbitrary then an almost isolated part of

σ
(
T
)

\sigma (T)

need not be the spectrum of the restriction of T to any invariant subspace.
Convolution, differential equations, and entire functions of exponential type
Authors: Mugler Dale H.
Published: July 2010

The Whittaker-Shannon interpolation formula, or “cardinal series", is a special case of the more general linear integro-differential equation with constant complex coefficients
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Subscript n equals 0 Superscript normal infinity Baseline a Subscript n Baseline f Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis z right-parenthesis equals integral f left-parenthesis z minus t right-parenthesis d mu left-parenthesis t right-parenthesis">

Σ

n
=
0

∞

a
n

f

(
n
)

(
z
)
=
∫
f
(
z
−
t
)
d
μ
(
t
)

\Sigma _{n = 0}^\infty {a_n}{f^{(n)}}(z) = \smallint f(z - t)d\mu (t)

where the integral is taken over the whole real line with respect to the measure

μ
\mu

. In this study, I show that many of these equations provide representations for particular classes of entire functions of exponential type. That is, every function in the class satisfies the equation and conversely every solution of the equation is a member of the class of functions. When the measure in the convolution integral above is chosen to be discrete, a particular form of the above type of equation is an equation of periodicity

f
(
z
)
=
f
(
z
+
τ
)

f(z) = f(z + \tau )

. Following an extensive treatment of the general equation written above, the study concludes by offering a generalization in terms of these convolution equations of a classical theorem in complex analysis concerning periodic entire functions.
An order topology in ordered topological vector spaces
Authors: Carter Lyne H.
Published: July 2010

An order topology

Ω
\Omega

that can be defined on any partially-ordered space has as its closed sets those that contain the (o)-limits of all their (o)-convergent nets. In this paper we study the situation in which a topological vector space with a Schauder basis is ordered by the basis cone. In a Fréchet space

(
E
,
τ
)

(E,\tau )

, we obtain necessary and sufficient conditions both for

τ
⊂
Ω

\tau \subset \Omega

and for

τ
=
Ω

\tau = \Omega

. Characterizations of (o)- and

Ω
\Omega

-convergence and of

Ω
\Omega

-closed sets are obtained. The equality of the order topology with the strong topology in certain dual Banach spaces is related to weak sequential completeness through the concept of a shrinking basis.
Finite groups as isometry groups
Authors: Asimov D.
Published: July 2010

We show that given any finite group G of cardinality

k
+
1

k + 1

, there is a Riemannian sphere

S

k
−
1

{S^{k - 1}}

(imbeddable isometrically as a hypersurface in

R

k

{{\mathbf {R}}^k}

) such that its full isometry group is isomorphic to G. We also show the existence of a finite metric space of cardinality

k
(
k
+
1
)

k(k + 1)

whose full isometry group is isomorphic to G.
Essential embeddings of annuli and Möbius bands in 3-manifolds
Authors: Cannon James W. , Feustel C. D.
Published: July 2010

In this paper we give conditions when the existence of an “essential” map of an annulus or Möbius band into a 3-manifold implies the existence of an “essential” embedding of an annulus or Möbius band into that 3-manifold. Let

λ
1

{\lambda _1}

and

λ
2

{\lambda _2}

be disjoint simple “orientation reversing” loops in the boundary of a 3-manifold M and A an annulus. Let

f
:
(
A
,
∂
A
)
→
(
M
,
∂
M
)

f:(A,\partial A) \to (M,\partial M)

be a map such that

f
∗

:

π
1

(
A
)
→

π
1

(
M
)

{f_\ast }:{\pi _1}(A) \to {\pi _1}(M)

is monic and

f
(
∂
A
)
=

λ
1

∪

λ
2

f(\partial A) = {\lambda _1} \cup {\lambda _2}

. Then we show that there is an embedding

g
:
(
A
,
∂
A
)
→
(
M
,
∂
M
)

g:(A,\partial A) \to (M,\partial M)

such that

g
(
∂
A
)
=

λ
1

∪

λ
2

g(\partial A) = {\lambda _1} \cup {\lambda _2}

.
On the torus theorem for closed 3-manifolds
Authors: Feustel C. D.
Published: July 2010

In this paper we give the appropriate generalization of the torus theorem to closed, sufficiently large, irreducible, orientable 3-manifolds.
Breadth two topological lattices with connected sets of irreducibles
Authors: Lea J. W.
Published: July 2010

Breadth two topological lattices with connected sets of irreducible elements are characterized by these sets.
Topologically defined classes of going-down domains
Authors: Papick Ira J.
Published: July 2010

Let R be an integral domain. Our purpose is to study GD (going-down) domains which arise topologically; that is, we investigate how certain going-down assumptions on R and its overrings relate to the topological space

Spec

(
R
)

{\text {Spec}}(R)

. Many classes of GD domains are introduced topologically, and a systematic study of their behavior under homomorphic images, localization and globalization, integral change of rings, and the “

D
+
M

D + M

construction” is undertaken. Also studied, is the algebraic and topological relationships between these newly defined classes of GD domains.
Compactifications of spaces of functions and integration of functionals
Authors: Grinblat L. Š.
Published: July 2010

For a locally compact space there exists a compactification such that all its points are effectively describable, namely, Alexandroff’s onepoint compactification. The effective construction of compactifications for numerous standard separable metric spaces is already a very nontrivial problem. We propose a method of compactification which enables us to effectively construct compactifications of some spaces of functions (for example, of a ball in

L
p

(
−
∞
,
∞
)

{L_p}( - \infty ,\infty )

). It will be shown that the study of compactifications of spaces of functions is of principal importance in the theory of integration of functionals and in limit theorems for random processes.
Generalized Kloosterman sums and the Fourier coefficients of cusp forms
Authors: Parson L. Alayne
Published: July 2010

Certain generalized Kloosterman sums connected with congruence subgroups of the modular group and suitably restricted multiplier systems of half-integral degree are studied. Then a Fourier coefficient estimate is obtained for cusp forms of half-integral degree on congruence subgroups of the modular group and the Hecke groups

G
(

2

)

G(\sqrt 2 )

and

G
(

3

)

G(\sqrt 3 )

.
Continuous cohomology for compactly supported vectorfields on 𝑅ⁿ
Authors: Shnider Steven
Published: July 2010

In this paper we study the Gelfand-Fuks cohomology of the Lie algebra of compactly supported vectorfields on

R

n

{{\mathbf {R}}^n}

and establish the degeneracy of a certain spectral sequence at the

E
1

{E_1}

level. We apply this result to the study of another spectral sequence introduced by Resetnikov for the cohomology of the algebra of vectorfields on

S
n

{S^n}

.
Mean convergence of Fourier series on compact Lie groups
Authors: Stanton Robert J.
Published: July 2010

The main result is an

L
p

{L^p}

mean convergence theorem for the partial sums of the Fourier series of a class function on a compact semi-simple Lie group. A central element in the proof is a Lie group-Lie algebra analog of the theorems in classical Fourier analysis that allow one to pass back and forth between multiplier operators for Fourier series in several variables and multiplier operators for the Fourier transform in Euclidean space. To obtain the

L
p

{L^p}

mean convergence theorem, the theory of the Hilbert transform with weight function is needed.
Evolution system approximations of solutions to closed linear operator equations
Authors: Purdom Seaton D.
Published: July 2010

With S a linearly ordered set with the least upper bound property, with g a nonincreasing real-valued function on S, and with A a densely defined dissipative linear operator, an evolution system M is developed to solve the modified Stieljes integral equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis s comma t right-parenthesis x equals x plus upper A left-parenthesis left-parenthesis upper L right-parenthesis integral Subscript upper S Superscript t Baseline d g upper M left-parenthesis dot comma t right-parenthesis x right-parenthesis">

M
(
s
,
t
)
x
=
x
+
A
(
(
L
)

∫
S
t

d
g
M
(
⋅
,
t
)
x
)

M(s,t)x = x + A((L)\smallint _S^t dgM( \cdot ,t)x)

. An affine version of this equation is also considered. Under the hypothesis that the evolution system associated with the linear equation is strongly (resp. weakly) asymptotically convergent, an evolution system is used to approximate strongly (resp. weakly) solutions to the closed operator equation

A
y
=
−
z

Ay = - z

.
𝑆𝐾₁ of 𝑛 lines in the plane
Authors: Roberts Leslie G.
Published: July 2010

We calculateSK1(A)S{K_1}(A)whereAis the coordinate ring of the reduced affine variety consisting ofnstraight lines in the plane.
Sets definable over finite fields: their zeta-functions
Authors: Kiefe Catarina
Published: July 2010

Sets definable over finite fields are introduced. The rationality of the logarithmic derivative of their zeta-function is established, an application of purely algebraic content is given. The ingredients used are a result of Dwork on algebraic varieties over finite fields and model-theoretic tools.
Weak bases and metrization
Authors: Martin Harold W.
Published: July 2010

Several weak base (in the sense of A. V. Arhangel’skiĭ) metrization theorems are established, including a weak base generalization of the Nagata-Smirnov Metrization Theorem.
Comparison of eigenvalues associated with linear differential equations of arbitrary order
Authors: Gentry R. D. , Travis C. C.
Published: July 2010

Existence and comparison theorems for eigenvalues of

(
k
,
n
−
k
)

(k,n - k)

-focal point and

(
k
,
n
−
k
)

(k,n - k)

-conjugate point problems are proved for a class of nth order linear differential equations for arbitrary n.
Finite groups with prime 𝑝 to the first power
Authors: Chang Zon I
Published: July 2010

Earlier D. G. Higman classified the finite groups of order n, such that n is divisible by 3 to the first power, with the assumption that the centralizer

C
G

(
X
)

{C_G}(X)

of X, where X is a subgroup of order 3, is a cyclic trivial intersection set of even order 3s. In this paper the theorem is generalized to include all prime numbers greater than 3. With an additional assumption:

|

N
G

(
X
)
:

C
G

(
X
)

|

=
2

|{N_G}(X):{C_G}(X)| = 2

, we have proved that one of the following holds for these groups, hereafter designated as G: (A) G is isomorphic to

L
2

(
q
)

{L_2}(q)

, where

q
=
2
p
s
±
1

q = 2ps \pm 1

; (B) there exists a normal subgroup

G
0

{G_0}

of odd index in G, and a normal subgroup N of

G
0

{G_0}

of index 2 such that

G
=
N
⟨
σ
⟩

G = N\langle \sigma \rangle

where

C
G

(
X
)
=
X
×
⟨
σ
⟩

{C_G}(X) = X \times \langle \sigma \rangle

.
Singularities in the nilpotent scheme of a classical group
Authors: Hesselink Wim
Published: July 2010

If

(
X
,
x
)

(X,x)

is a pointed scheme over a ring k, we introduce a (generalized) partition

ord

(
x
,
X

/

k
)

{\text {ord}}(x,X/k)

. If G is a reductive group scheme over k, the existence of a nilpotent subscheme

N
(
G
)

N(G)

of

Lie

(
G
)

{\text {Lie}}(G)

is discussed. We prove that

ord

(
x
,
N
(
G
)

/

k
)

{\text {ord}}(x,N(G)/k)

characterizes the orbits in

N
(
G
)

N(G)

, their codimension and their adjacency structure, provided that G is

G

l
n

G{l_n}

, or

S

p
n

S{p_n}

and

1

/

2
∈
k

1/2 \in k

. For

S

O
n

S{O_n}

only partial results are obtained. We give presentations of some singularities of

N
(
G
)

N(G)

. Tables for its orbit structure are added.
A Paley-Wiener theorem for locally compact abelian groups
Authors: Liepins Gunar E.
Published: July 2010

Extending the Paley-Wiener theorem to locally compact Abelian groups involves both finding a suitable Laplace transform and a suitable analogue for analytic functions. The Laplace transform is defined in terms of complex characters, and differentiability is defined with use of one-parameter subgroups. The resulting theorem is much as conjectured by Mackey [7],(

1

^{1}

) the major differences being that the theorem is very much an

L
2

{L^2}

theorem and that the problem exhibits a surprising finite dimensional nature.
Periodic homeomorphisms of 3-manifolds fibered over 𝑆¹
Authors: Tollefson Jeffrey L.
Published: July 2010

Two problems concerning periodic homeomorphisms of 3-manifolds are considered. The first is that of obtaining systems of incompressible surfaces invariant under a given involution. The second problem is the realization by a periodic homeomorphism of an element of finite order in the mapping class group of a 3-manifold. Solutions to both problems are obtained in certain instances.
Involutions on homotopy spheres and their gluing diffeomorphisms
Authors: Liang Chao Chu
Published: July 2010

Let

h
S
(

P

2
n
+
1

)

hS({P^{2n + 1}})

denote the set of equivalence classes of smooth fixed-point free involutions on

(
2
n
+
1
)

(2n + 1)

-dimensional homotopy spheres. Browder and Livesay defined an invariant

σ
(

Σ

2
n
+
1

,
T
)

\sigma ({\Sigma ^{2n + 1}},T)

for each

(

Σ

2
n
+
1

,
T
)
∈
h
S
(

P

2
n
+
1

)

({\Sigma ^{2n + 1}},T) \in hS({P^{2n + 1}})

, where

σ
∈
Z

\sigma \in Z

if n is odd,

σ
∈

Z
2

\sigma \in {Z_2}

if n is even. They showed that for

n
⩾
3
,
σ
(

Σ

2
n
+
1

,
T
)
=
0

n \geqslant 3,\sigma ({\Sigma ^{2n + 1}},T) = 0

if and only if

(

Σ

2
n
+
1

,
T
)

({\Sigma ^{2n + 1}},T)

admits a codim 1 invariant sphere. For any

(

Σ

2
n
+
1

,
T
)

({\Sigma ^{2n + 1}},T)

, there exists an A-equivariant diffeomorphism f of

S
n

×

S
n

{S^n} \times {S^n}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Sigma Superscript 2 n plus 1 Baseline comma upper T right-parenthesis equals left-parenthesis upper S Superscript n Baseline times upper D Superscript n plus 1 Baseline comma upper A right-parenthesis union Subscript f Baseline left-parenthesis upper D Superscript n plus 1 Baseline times upper S Superscript n Baseline comma upper A right-parenthesis">

(

Σ

2
n
+
1

,
T
)
=
(

S
n

×

D

n
+
1

,
A
)

∪
f

(

D

n
+
1

×

S
n

,
A
)

({\Sigma ^{2n + 1}},T) = ({S^n} \times {D^{n + 1}},A){ \cup _f}({D^{n + 1}} \times {S^n},A)

, where A denotes the antipodal map. Let

β
(
f
)
=
σ
(

Σ

2
n
+
1

,
T
)

\beta (f) = \sigma ({\Sigma ^{2n + 1}},T)

. In the case n is odd, we can show that the Browder-Livesay invariant is additive:

β
(
f
g
)
=
β
(
f
)
+
β
(
g
)

\beta (fg) = \beta (f) + \beta (g)

. But if n is even, then there exists f and g such that

β
(
g
f
)
=
β
(
g
)
+
β
(
f
)
≠
β
(
f
g
)

\beta (gf) = \beta (g) + \beta (f) \ne \beta (fg)

. Let

D
0

(

S
n

×

S
n

,
A
)

{D_0}({S^n} \times {S^n},A)

be the group of concordance classes of A-equivariant diffeomorphisms which are homotopic to the identity map of

S
n

×

S
n

{S^n} \times {S^n}

. We can prove that “For

n
≡
0
,
1
,
2
mod
4
,
h
S
(

P

2
n
+
1

)

n \equiv 0,1,2 \bmod 4, hS({P^{2n + 1}})

is in 1-1 correspondence with a subgroup of

D
0

(

S
n

×

S
n

,
A
)

{D_0}({S^n} \times {S^n},A)

. As an application of these theorems, we demonstrated that “Let

Σ
0

8
k
+
3

\Sigma _0^{8k + 3}

denote the generator of

b

P

8
k
+
4

b{P_{8k + 4}}

. Then the number of

(

Σ
0

8
k
+
3

,
T
)

(\Sigma _0^{8k + 3},T)

’s with

σ
(

Σ
0

8
k
+
3

,
T
)
=
0

\sigma (\Sigma _0^{8k + 3},T) = 0

is either 0 or equal to the number of

(

S

8
k
+
3

,
T
)

({S^{8k + 3}},T)

’s with

σ
(

S

8
k
+
3

,
T
)
=
0

\sigma ({S^{8k + 3}},T) = 0

, where

S

8
k
+
3

{S^{8k + 3}}

denotes the standard sphere".
Convolution equations for vector-valued entire functions of nuclear bounded type
Authors: Dwyer Thomas A. W.
Published: July 2010

Given two complex Banach spaces E and F, convolution operators “with scalar coefficients” are characterized among all convolution operators on the space

H

N
b

(

E
′

;
F
)

{H_{Nb}}(E’;F)

of entire mappings of bounded nuclear type of E’ into F. The transposes of such operators are characterized as multiplication operators in the space

E
x
p
(
E
;

F
′

)

Exp(E;F’)

of entire mappings of exponential type of E into F’. The division theorem for entire functions of exponential type of Malgrange and Gupta is then extended to the case when one factor is vector-valued. With this tool the following “vector-valued” existence and approximation theorems for convolution equations are proved: THEOREM 1. Nonzero convolution operators “of scalar type” are surjective on

H

N
b

(

E
′

;
F
)

{H_{Nb}}(E’;F)

. THEOREM 2. Solutions of homogeneous convolution equations of scalar type can be approximated in

H

N
b

(

E
′

;
F
)

{H_{Nb}}(E’;F)

by exponential-polynomial solutions.
Critical groups having central monoliths of a nilpotent by abelian product variety of groups
Authors: Woeppel James J.
Published: July 2010

Let

N

\mathfrak {N}

be a variety of groups which has nilpotency class two and finite odd exponent. Let

A

\mathfrak {A}

be an abelian variety of groups with finite exponent relatively prime to the exponent of

N

\mathfrak {N}

. The existence in the product variety

N

A

\mathfrak {N}\mathfrak {A}

of nonnilpotent critical groups having central monoliths is established. The structure of these critical groups is studied. This structure is shown to depend on an invariant, k. The join-irreducible subvariety of

N

A

\mathfrak {N}\mathfrak {A}

generated by the nonnilpotent critical groups of

N

A

\mathfrak {N}\mathfrak {A}

having central monoliths is determined, in particular, for k odd.
The fundamental form of an inseparable extension
Authors: Gerstenhaber Murray
Published: July 2010

If K is a finite purely inseparable extension of a field k, then the symmetric multiderivations of K (symmetric maps

f
:
K
×
⋯
×
K

(
n

times

)
→
K

f:K \times \cdots \times K\;(n\;{\text {times}}) \to K

which are derivations as functions of each single variable) form a ring under the symmetrized cup product. This ring contains an element

Γ
(
K

/

k
)

\Gamma (K/k)

called the fundamental form of K over k, which is defined up to multiplication by a nonzero element of K and has the property that if B is any intermediate field between K and k, then

Γ
(
K

/

B
)

\Gamma (K/B)

divides

Γ
(
K

/

k
)

\Gamma (K/k)

.
A relation between two biharmonic Green’s functions on Riemannian manifolds
Authors: Hada Dennis
Published: July 2010

The biharmonic Green’s function

γ
\gamma

whose values and Laplacian are identically zero on the boundary of a region and the biharmonic Green’s function

Γ
\Gamma

whose values and normal derivative vanish on the boundary originated in the investigation of thin plates whose edges are simply supported or clamped, respectively. A relation between these two biharmonic Green’s functions known for planar regions is extended to Riemannian manifolds thereby establishing that any Riemannian manifold for which

γ
\gamma

exists must also carry

Γ
\Gamma

.
Addendum to: “Some polynomials defined by generating relations” (Trans. Amer. Math. Soc. 205 (1975), 360–370)
Authors: Srivastava H. M. , Buschman R. G.
Published: July 2010

Certain constraints are explicitly specified for the validity of a recent result involving a multivariate generating function, due to the present authors [1, p. 369, Theorem 6]. It is also indicated how this result can be further generalized. See Theorem

6
∗

{6^\ast }

below.
𝐾-theory and Steenrod homology: applications to the Brown-Douglas-Fillmore theory of operator algebras
Authors: Kaminker Jerome , Schochet Claude
Published: July 2010

The remarkable work of L. G. Brown, R. Douglas and P. Fillmore on operators with compact self-commutators once again ties together algebraic topology and operator theory. This paper gives a comprehensive treatment of certain aspects of that connection and some adjacent topics. In anticipation that both operator theorists and topologists may be interested in this work, additional background material is included to facilitate access.
Zero-one laws and the minimum of a Markov process
Authors: Millar P. W.
Published: July 2010

If

{

X
t

,
t
>
0
}

\{ {X_t},t > 0\}

is a real strong Markov process whose paths assume a (last) minimum at some time M strictly before the lifetime, then conditional on I, the value of this minimum, the process

{
X
(
M
+
t
)
,
t
>
0
}

\{ X(M + t),t > 0\}

is shown to be Markov with stationary transitions which depend on I. For a wide class of Markov processes, including those obtained from Lévy processes via time change and multiplicative functional, a zero-one law is shown to hold at M in the sense that

∩

t
>
0

σ
{
X
(
M
+
s
)
,
s
⩽
t
}
=
σ
{
X
(
M
)
}

{ \cap _{t > 0}}\sigma \{ X(M + s),s \leqslant t\} = \sigma \{ X(M)\}

, modulo null sets. When such a law holds, the evolution of

{
X
(
M
+
t
)
,
t
⩾
0
}

\{ X(M + t),t \geqslant 0\}

depends on events before M only through

X
(
M
)

X(M)

and I.
A new characterization of Cesàro-Perron integrals using Peano derivates
Authors: Bergin J. A.
Published: July 2010

The

Z
n

{Z_n}

-integrals are defined according to the method of Perron using Peano derivates. The properties of the integrals are given including the essential integration by parts theorem. The integrals are then shown to be equivalent to the Cesàro-Perron integrals of Burkill.
Examples for the nonuniqueness of the equilibrium state
Authors: Hofbauer Franz
Published: July 2010

In this paper equilibrium states on shift spaces are considered. A uniqueness theorem for equilibrium states is proved. Then we study a particular class of continuous functions. We characterize the functions of this class which satisfy Ruelle’s Perron-Frobenius condition, those which admit a measure determined by a homogeneity condition, and those which have unique equilibrium state. In particular, we get examples for the nonuniqueness of the equilibrium state.
On the zeros of Stieltjes and Van Vleck polynomials
Authors: Zaheer Neyamat , Alam Mahfooz
Published: July 2010

Stieltjes and Van Vleck polynomials arise in the study of the polynomial solutions of the generalized Lamé differential equation. Our object is to generalize a theorem due to Marden on the location of the zeros of Stieltjes and Van Vleck polynomials. In fact, our generalization is two-fold: Firstly, we employ sets which are more general than the ones used by Marden for prescribing the location of the complex constants occurring in the Lamé differential equation; secondly, Marden deals only with the standard form of the said differential equation, whereas our result is equally valid for yet another form of the same differential equation. The part of our main theorem concerning Stieltjes polynomials may also be regarded as a generalization of Lucas’ theorem to systems of partial fraction sums.
The classifying space of a permutation representation
Authors: Blowers James V.
Published: July 2010

In this article the concept of classifying space of a group is generalized to a classifying space of an arbitrary permutation representation. An example of this classifying space is given by a generalization of the infinite join construction that defines the standard example of a classifying space of a group. In a previous paper of the author, the join of two permutation representations was defined, and it was shown that the cohomology ring of the join was trivial. In this paper the classifying space of the join of two permutation representations is shown to be the topological join of the two classifying spaces and from this the triviality of the cup-product is derived topologically.