bluuebunny/crossref_metadata_2025 huggingface dataset:
𝐿-functions of a quadratic form
Authors: Callahan T. , Smith R. A.
Published: July 2010

Let Q be a positive definite integral quadratic form in n variables, with the additional property that the adjoint form

Q
†

{Q^\dagger }

is also integral. Using the functional equation of the Epstein zeta function, we obtain a symmetric functional equation of the L-function of Q with a primitive character

ω
mod
q

\omega \bmod q

(additive or multiplicative) defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma omega left-parenthesis upper Q left-parenthesis x right-parenthesis right-parenthesis upper Q left-parenthesis x right-parenthesis Superscript negative s Baseline comma upper R e left-parenthesis s right-parenthesis greater-than n slash 2">

Σ
ω
(
Q
(

x

)
)
Q

(

x

)

−
s

,
Re
⁡
(
s
)
>
n

/

2

\Sigma \omega (Q({\text {x}}))Q{({\text {x}})^{ - s}},\operatorname {Re} (s) > n/2

, where the summation extends over all

x

∈

Z
n

,

x

≠
0

{\text {x}} \in {Z^n},{\text {x}} \ne 0

; our result does not depend upon the usual restriction that q be relatively prime to the discriminant of Q, but rather on a much milder restriction.
Cartan subspaces of symmetric Lie algebras
Authors: Lepowsky J. , McCollum G. W.
Published: July 2010

A symmetric Lie algebra is defined, following J. Dixmier, to be a Lie algebra

g

\mathfrak {g}

with a decomposition

g

=

k

⊕

p

\mathfrak {g} = \mathfrak {k} \oplus \mathfrak {p}

such that

k

\mathfrak {k}

is a subalgebra of

g

,
[

k

,

p

]
⊂

p

\mathfrak {g},[\mathfrak {k},\mathfrak {p}] \subset \mathfrak {p}

and

[

p

,

p

]
⊂

k

[\mathfrak {p},\mathfrak {p}] \subset \mathfrak {k}

. A definition of Cartan subspace of a symmetric Lie algebra is given, and a theory is presented which parallels the standard theory of Cartan subalgebras of Lie algebras, and which generalizes the classical results for real and complex semisimple symmetric Lie algebras.
Principal quadratic functionals
Authors: Tomastik E. C.
Published: July 2010

The general theory of the existence of a minimum limit in the fixed and variable end point problems of singular quadratic functionals of n dependent variables is developed, generalizing the one dimensional results of Marston Morse, Walter Leighton and A. D. Martin and completing a phase of the n-dimensional theory initiated recently by the author.
Complex space forms immersed in complex space forms
Authors: Nakagawa H. , Ogiue K.
Published: July 2010

We determine all the isometric immersions of complex space forms into complex space forms. Our result can be considered as the local version of a well-known result of Calabi.
Yoneda products in the Cartan-Eilenberg change of rings spectral sequence with applications to 𝐵𝑃_{∗}(𝐵𝑂(𝑛))
Authors: Ming Ronald
Published: July 2010

Yoneda product structure is defined on a Cartan-Eilenberg change of rings spectral sequence. Application is made to a factorization theorem for the

E
2

{E_2}

-term of the Adams spectral sequence for Brown-Peterson homology of the classifying spaces

B
O
(
n
)

BO(n)

.
On the existence of compact metric subspaces with applications to the complementation of 𝑐₀
Authors: Chapman William H. , Randtke Daniel J.
Published: July 2010

A topological space X has property

σ
−

CM

\sigma - {\text {CM}}

if for every countable family F of continuous scalar valued functions on X there is a compact metrizable subspace M of X such that

f
(
X
)
=
f
(
M
)

f(X) = f(M)

for every f in F. Every compact metric space, every weakly compact subset of a Banach space and every closed ordinal space has property

σ
−

CM

\sigma - {\text {CM}}

. Every continuous image of an arbitrary product of spaces having property

σ
−

CM

\sigma - {\text {CM}}

also has property

σ
−

CM

\sigma - {\text {CM}}

. If X has property

σ
−

CM

\sigma - {\text {CM}}

, then every copy of

c
0

{c_0}

in

C
(
X
)

C(X)

is complemented in

C
(
X
)

C(X)

. If a locally convex space E belongs to the variety of locally convex spaces generated by the weakly compactly generated Banach spaces, then every copy of

c
0

{c_0}

in E is complemented in E.
Some results on orientation preserving involutions
Authors: Gibbs David E.
Published: July 2010

The bordism of orientation preserving differentiable involutions is studied by use of the signature-like invariant

ab

:

O

∗

(

Z
2

)
→

W
0

(

Z
2

;
Z
)

{\text {ab}}: {\mathcal {O}_\ast }({Z_2}) \to {W_0}({Z_2};Z)

. The equivariant Witt ring

W
0

(

Z
2

;
Z
)

{W_0}({Z_2};Z)

is calculated and is shown to be isomorphic under ab to the effective part of

O

4

(

Z
2

)

{\mathcal {O}_4}({Z_2})

. Modulo 2 relations are established between the representation of the involution on

H

2
k

(

M

4
k

;
Z
)

/

torsion

{H^{2k}}({M^{4k}};Z)/{\operatorname {torsion}}

and

χ
0

(
F
)

{\chi _0}(F)

and

χ
2

(
F
)

{\chi _2}(F)

, where

χ
i

(
F
)

{\chi _i}(F)

is the Euler characteristic of those components of the fixed point set with dimensions congruent to i modulo 4. For manifolds of dimension

4
k
+
2

4k + 2

, it is shown that

χ
0

(
F
)
≡

χ
2

(
F
)
≡
0

(
mod
2
)

{\chi _0}(F) \equiv {\chi _2}(F) \equiv 0\;(\bmod 2)

. Finally the ideal

E
0

(

Z
2

;
Z
)

{E_0}({Z_2};Z)

consisting of those elements of

W
0

(

Z
2

;
Z
)

{W_0}({Z_2};Z)

admitting a representative of type II is determined.
Simplicial geometry and transportation polytopes
Authors: Bolker Ethan D.
Published: July 2010

The classical transportation problem is the study of the set of nonnegative matrices with prescribed nonnegative row and column sums. It is aesthetically satisfying and perhaps potentially useful to study more general higher dimensional rectangular arrays whose sums on some subarrays are specified. We show how such problems can be rewritten as problems in homology theory. That translation explains the appearance of bipartite graphs in the study of the classical transportation problem. In our generalization, higher dimensional cell complexes occur. That is why the general problem requires a substantial independent investigation of simplicial geometry, the name given to the class of theorems on the geometry of a cell complex which depend on a particular cellular decomposition. The topological invariants of the complex are means, not ends. Thus simplicial geometry attempts to do for complexes what graph theory does for graphs. The dual title of this paper indicates that we shall spend as much time studying simplicial geometry for its own sake as applying the results to transportation problems. Our results include formulas for inverting the boundary operator of an acyclic cell complex, and some information on the number of such subcomplexes of a given complex.
Asymptotic relations for partitions
Authors: Richmond L. B.
Published: July 2010

Asymptotic relations are obtained for the number

p
A

(
n
)

{p_A}(n)

of partitions of the integer n into summands from a set A of integers. The set A is subject to certain conditions; however the only arithmetic condition is that A have property

P
k

{P_k}

of Bateman and Erdös. A conjecture of Bateman and Erdös concerning the kth differences of

p
A

(
n
)

{p_A}(n)

may be verified using these asymptotic relations.
On the topology of a compact inverse Clifford semigroup
Authors: Yeager D. P.
Published: July 2010

A description of the topology of a compact inverse Clifford semigroup S is given in terms of the topologies of its subgroups and that of the semilattice X of idempotents. It is further shown that the category of compact inverse Clifford semigroups is equivalent to a full subcategory of the category whose objects are inverse limit preserving functors

F
:
X
→
G

F:X \to G

, where X is a compact semilattice and G is the category of compact groups and continuous homomorphisms, and where a morphism from

F
:
X
→
G

F:X \to G

to

G
:
Y
→
G

G:Y \to G

is a pair

(
ε
,
w
)

(\varepsilon ,w)

such that

ε
\varepsilon

is a continuous homomorphism of X into Y and w is a natural transformation from F to

G
ε

G\varepsilon

. Simpler descriptions of the topology of S are given in case the topology of X is first countable and in case the bonding maps between the maximal subgroups of S are open mappings.
On analytic independence
Authors: Abhyankar S. S. , Moh T. T.
Published: July 2010

This article examines the concept of “analytic independence". Several illustrative examples have been included. The main results are Theorems 1-4 which state the relations between analytic independence and the degree of field extensions, transcendence degree, order of poles and “gap” respectively.
Asymmetric maximal ideals in 𝑀(𝐺)
Authors: Saeki Sadahiro
Published: July 2010

Let G be a nondiscrete LCA group,

M
(
G
)

M(G)

the measure algebra of G, and

M
0

(
G
)

{M_0}(G)

the closed ideal of those measures in

M
(
G
)

M(G)

whose Fourier transforms vanish at infinity. Let

Δ
G

,

Σ
G

{\Delta _G},{\Sigma _G}

and

Δ
0

{\Delta _0}

be the spectrum of

M
(
G
)

M(G)

, the set of all symmetric elements of

Δ
G

{\Delta _G}

, and the spectrum of

M
0

(
G
)

{M_0}(G)

, respectively. In this paper this is shown: Let

Φ
\Phi

be a separable subset of

M
(
G
)

M(G)

. Then there exist a probability measure

τ
\tau

in

M
0

(
G
)

{M_0}(G)

and a compact subset X of

Δ
0

∖

Σ
G

{\Delta _0}\backslash {\Sigma _G}

such that for each

|

c

|

⩽
1

|c| \leqslant 1

and each
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu element-of normal upper Phi Card StartSet f element-of upper X colon ModifyingAbove tau With caret left-parenthesis f right-parenthesis equals c and StartAbsoluteValue ModifyingAbove nu With caret left-parenthesis f right-parenthesis EndAbsoluteValue equals r left-parenthesis nu right-parenthesis EndSet greater-than-or-slanted-equals 2 Superscript c Baseline period">

ν
∈
Φ

Card

{
f
∈
X
:

τ
^

(
f
)
=
c

and

|

ν
^

(
f
)

|

=
r
(
ν
)
}
⩾

2

c

.

\nu \in \Phi \;{\text {Card}}\;\{ f \in X:\hat \tau (f) = c\;{\text {and}}\;|\hat \nu (f)| = r(\nu )\} \geqslant {2^{\text {c}}}.

\]
Here

r
(
ν
)
=
sup
{

|

ν
^

(
f
)

|

:
f
∈

Δ
G

∖

G
^

}

r(\nu ) = \sup \{ |\hat \nu (f)|:f \in {\Delta _G}\backslash \hat G\}

. As immediate consequences of this result, we have (a) every boundary for

M
0

(
G
)

{M_0}(G)

is a boundary for

M
(
G
)

M(G)

(a result due to Brown and Moran), (b)

Δ
G

∖

Σ
G

{\Delta _G}\backslash {\Sigma _G}

is dense in

Δ
G

∖

G
^

{\Delta _G}\backslash \hat G

, (c) the set of all peak points for

M
(
G
)

M(G)

is

G
^

\hat G

if G is

σ
\sigma

-compact and is empty otherwise, and (d) for each

μ
∈
M
(
G
)

\mu \in M(G)

the set

μ
^

(

Δ
0

∖

Σ
G

)

\hat \mu ({\Delta _0}\backslash {\Sigma _G})

contains the topological boundary of

μ
^

(

Δ
G

∖

G
^

)

\hat \mu ({\Delta _G}\backslash \hat G)

in the complex plane.
Simplicial triangulation of noncombinatorial manifolds of dimension less than 9
Authors: Scharlemann Martin
Published: July 2010

Necessary and sufficient conditions are given for the simplicial triangulation of all noncombinatorial manifolds in the dimension range

5
⩽
n
⩽
7

5 \leqslant n \leqslant 7

, for which the integral Bockstein of the combinatorial triangulation obstruction is trivial. A weaker theorem is proven in case

n
=
8

n = 8

. The appendix contains a proof that a map between PL manifolds which is a TOP fiber bundle can be made a PL fiber bundle.
Symmetrizable and related spaces
Authors: Harley Peter W. , Stephenson R. M.
Published: July 2010

A study is made of a family of spaces which contains the symmetrizable spaces as well as many of the well-known examples of perfectly normal spaces.
A Hausdorff measure inequality
Authors: Ernst Lawrence R. , Freilich Gerald
Published: July 2010

We prove that the Hausdorff

(
m
+
k
)

(m + k)

-measure of a product set is no less than the product of the Hausdorff m-measure of the (measurable) first component set in

R

m

{{\mathbf {R}}^m}

and the (finite) Hausdorff k-measure of the second component in

R

n

{{\mathbf {R}}^n}

.
Pointwise bounded approximation and analytic capacity of open sets
Authors: Jacobson Steven
Published: July 2010

We examine the semi-additivity question for analytic capacity by studying the relation between the capacities of bounded open sets and their closures.
Universal properties of Prym varieties with an application to algebraic curves of genus five
Authors: Masiewicki Leon
Published: July 2010

It is proved that every morphism of a curve with an involution into an Abelian variety, anticommuting with the involution, factors through the associated Prym variety. This result is used to show that Jacobians of curves of genus five arise as Prym varieties associated to a certain class of curves.
The 2-transitive permutation representations of the finite Chevalley groups
Authors: Curtis Charles W. , Kantor William M. , Seitz Gary M.
Published: July 2010

The permutation representations in the title are all determined, and no surprises are found to occur.
On the second Hankel determinant of areally mean 𝑝-valent functions
Authors: Noonan J. W. , Thomas D. K.
Published: July 2010

In this paper we determine the growth rate of the second Hankel determinant of an areally mean p-valent function. This result both extends and unifies previously known results concerning this problem.
Algebras of functions on semitopological left-groups
Authors: Berglund John F. , Milnes Paul
Published: July 2010

We consider various algebras of functions on a semitopological left-group

S
=
X
×
G

S = X \times G

, the direct product of a left-zero semigroup X and a group G. In §1 we examine various analogues to the theorem of Eberlein that a weakly almost periodic function on a locally compact abelian group is uniformly continuous. Several appealing conjectures are shown by example to be false. In the second section we look at compactifications of products

S
×
T

S \times T

of semitopological semigroups with right identity and left identity, respectively. We show that the almost periodic compactification of the product is the product of the almost periodic compactifications, thus generalizing a result of deLeeuw and Glicksberg. The weakly almost periodic compactification of the product is not the product of the weakly almost periodic compactifications except in restrictive circumstances; for instance, when T is a compact group. Finally, as an application, we define and study analytic weakly almost periodic functions and derive the theorem, analogous to a classical theorem about almost periodic functions, that an analytic function which is weakly almost periodic on a single line is analytic weakly almost periodic on a whole strip.
An approximation theory for generalized Fredholm quadratic forms and integral-differential equations
Authors: Gregory J. , Lopez G. C.
Published: July 2010

An approximation theory is given for a very general class of elliptic quadratic forms which includes the study of 2nth order (usually in integrated form), selfadjoint, integral-differential equations. These ideas follows in a broad sense from the quadratic form theory of Hestenes, applied to integral-differential equations by Lopez, and extended with applications for approximation problems by Gregory. The application of this theory to a variety of approximation problem areas in this setting is given. These include focal point and focal interval problems in the calculus of variations/optimal control theory, oscillation problems for differential equations, eigenvalue problems for compact operators, numerical approximation problems, and finally the intersection of these problem areas. In the final part of our paper our ideas are specifically applied to the construction and counting of negative vectors in two important areas of current applied mathematics: In the first case we derive comparison theorems for generalized oscillation problems of differential equations. The reader may also observe the essential ideas for oscillation of many nonsymmetric (indeed odd order) ordinary differential equation problems which will not be pursued here. In the second case our methods are applied to obtain the “Euler-Lagrange equations” for symmetric tridiagonal matrices. In this significant new result (which will allow us to reexamine both the theory and applications of symmetric banded matrices) we can construct in a meaningful way, negative vectors, oscillation vectors, eigenvectors, and extremal solutions of classical problems as well as faster more efficient algorithms for the numerical solution of differential equations. In conclusion it appears that many physical problems which involve symmetric differential equations are more meaningful presented as integral differential equations (effects of friction on physical processes, etc.). It is hoped that this paper will provide the general theory and present examples and methods to study integral differential equations.
Stability theorems in shape and pro-homotopy
Authors: Edwards David A. , Geoghegan Ross
Published: July 2010

Conditions are given under which a topological space has the pointed shape of a CW complex. These are derived from analogous conditions in pro-homotopy.
On the jump of an 𝛼-recursively enumerable set
Authors: Shore Richard A.
Published: July 2010

We discuss the proper definition of the jump operator in

α
\alpha

-recursion theory and prove a sample theorem: There is an incomplete

α
\alpha

-r.e. set with jump

0
0

unless there is precisely one nonhyperregular

α
\alpha

-r.e. degree. Thus we have a theorem in the first order language of Turing degrees with the jump which fails to generalize to all admissible

α
\alpha

.
Holomorphic convexity of compact sets in analytic spaces and the structure of algebras of holomorphic germs
Authors: Zame William R.
Published: July 2010

Let

(
X
,

O

X

)

(X,{\mathcal {O}_X})

be a reduced analytic space and let K be a compact, holomorphically convex subset of X. It is shown that analogs of Cartan’s Theorems A and B are valid for coherent analytic sheaves on K. This result is applied to the study of the algebra of germs on K of functions holomorphic near K. In particular, characterizations of finitely generated ideals, prime ideals and homomorphisms are obtained.
The Mackey problem for the compact-open topology
Authors: Wheeler Robert F.
Published: July 2010

Let

C
c

(
T
)

{C_c}(T)

denote the space of continuous real-valued functions on a completely regular Hausdorff space T, endowed with the compact-open topology. Well-known results of Nachbin, Shirota, and Warner characterize those T for which

C
c

(
T
)

{C_c}(T)

is bornological, barrelled, and infrabarrelled. In this paper the question of when

C
c

(
T
)

{C_c}(T)

is a Mackey space is examined. A convex strong Mackey property (CSMP), intermediate between infrabarrelled and Mackey, is introduced, and for several classes of spaces (including first countable and scattered spaces), a necessary and sufficient condition on T for

C
c

(
T
)

{C_c}(T)

to have CSMP is obtained.
Hypoelliptic convolution equations in 𝐾’_{𝑝}, 𝑝>1
Authors: Sampson G. , Zieleźny Z.
Published: July 2010

We consider convolution equations in the space

K
p
′

,
p
>
1

{K’_p},p > 1

, of distributions which “grow” no faster than

exp
⁡
(
k

|

x

|

p

)

\exp (k|x{|^p})

for some constant k. Our main result is a complete characterization of hypoelliptic convolution operators in

K
p
′

{K’_p}

in terms of their Fourier transforms.
A Banach algebra of functions with bounded 𝑛th differences
Authors: Daly John T. , Downum Philip B.
Published: July 2010

Several characterizations are given for the Banach algebra of

(
n
−
1
)

(n - 1)

-times continuously differentiable functions whose

(
n
−
1
)

(n - 1)

st derivative satisfies a bounded Lipschitz condition. The structure of the closed primary ideals is investigated and spectral synthesis is shown to be satisfied.
Inseparable finite solvable groups
Authors: Bechtell Homer
Published: July 2010

A finite group is called inseparable if the only proper normal subgroup over which it splits is the identity element. The E-residual, for the formation E of groups in which all Sylow subgroups are elementary abelian, appears to control the action of splitting. In this article, inseparable solvable groups are identified that have a metacyclic Fitting subgroup and the E-residual a p-group.
On the blocks of 𝐺𝐿(𝑛,𝑞). I
Authors: Olsson Jørn B.
Published: July 2010

A study is made of the distribution of the ordinary irreducible characters of

GL

(
n
,
q
)

{\text {GL}}(n,q)

into p-blocks for primes different from the characteristic. The paper gives a description of all possible defect groups for

p
≠
2

p \ne 2

and their normalizers. Various other results are obtained, including a classification of the blocks of defect 0.
Spherical distributions on Lie groups and 𝐶^{∞} vectors
Authors: Penney R.
Published: July 2010

Given a Lie group G (not necessarily unimodular) and a subgroup K of G (not necessarily compact), it is shown how to associate with every finite-dimensional unitary irreducible representation

δ
\delta

of K a class of distributions analogous to the class of spherical functions of height

δ
\delta

familiar from the unimodular-maximal compact case. The two concepts agree as nearly as possible. A number of familiar theorems are generalized to our situation. As an application we obtain a generalization of the Frobenius reciprocity theorem and of Plancherel’s theorem to arbitrary induced representations of Lie groups.
Positive definite measures with applications to a Volterra equation
Authors: Staffans Olof J.
Published: July 2010

We study the asymptotic behavior of the solutions of the nonlinear Volterra integrodifferential equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel x prime left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript t Baseline g left-parenthesis x left-parenthesis t minus tau right-parenthesis right-parenthesis d mu left-parenthesis tau right-parenthesis equals f left-parenthesis t right-parenthesis comma EndLayout">

(

∗

)

x
′

(
t
)
+

∫
0
t

g
(
x
(
t
−
τ
)
)

d
μ
(
τ
)
=
f
(
t
)
,

\begin{equation}\tag {$\ast $} x’(t) + \int _0^t {g(x(t - \tau ))\;d\mu (\tau ) = f(t),}\end{equation}

with a positive definite kernel

μ
\mu

. In particular, we give new sufficient conditions on the kernel

μ
\mu

, which together with standard assumptions on f and g yield results on boundedness and asymptotic behavior of the solutions of

(
∗
)

(\ast )

. Our proofs are based on the theory of distribution Fourier transforms.
Consistency results concerning supercompactness
Authors: Menas Telis K.
Published: July 2010

A general framework for proving relative consistency results with regard to supercompactness is developed. Within this framework we prove the relative consistency of the assertion that every set is ordinal definable with the statement asserting the existence of a supercompact cardinal. We also generalize Easton’s theorem; the new element in our result is that our forcing conditions preserve supercompactness.
Free 𝑆³-actions on 2-connected nine-manifolds
Authors: Resch Richard I.
Published: July 2010

In this paper a classification of free

S
3

{S^3}

-actions on 2-connected 9-manifolds is obtained by examining the corresponding principal

S
3

{S^3}

-bundles. The orbit spaces that may occur are determined and it is proved that there are exactly two homotopy classes of maps from each of these spaces into the classifying space for principal

S
3

{S^3}

-bundles. It is shown that the total spaces of the corresponding bundles are distinct, yielding the main result that for each nonnegative integer k, there exist exactly two 2-connected 9-manifolds which admit free

S
3

{S^3}

-actions and, furthermore, the actions on each of these manifolds are unique.
A development of contraction mapping principles on Hausdorff uniform spaces
Authors: Lee Cheng Ming
Published: July 2010

Certain generalized Banach’s contraction mapping principles on metric spaces are unified and/or extended to Hausdorff uniform spaces. Also given are some relationships between the set of all cluster points of the Picard iterates and the set of all fixed points for the mapping. These are obtained by assuming that the latter set is nonempty and by considering certain “quasi"-contractive conditions. The ("quasi") contractive conditions are defined by using a suitable family of pseudometrics on the uniform space.
On composite abstract homogeneous polynomials
Authors: Zaheer Neyamat
Published: July 2010

We study the null-sets of composite abstract homogeneous polynomials obtained from a pair of abstract homogeneous polynomials defined on a vector space over an algebraically closed field of characteristic zero. First such study for ordinary polynomials in the complex plane was made by Szegö, Cohn, and Egerváry and Szegö’s theorem was later generalized to fields and vector spaces, respectively, by Zervos and Marden. Our main theorem in this paper further generalizes their results and, in the complex plane, improves upon Szegö’s theorem and some other classical results. The method of proof is purely algebraic and utilizes the author’s vector space analogue [Trans. Amer. Math. Soc. 218 (1976), 115-131] of Grace’s theorem on apolar polynomials. We also show that our results cannot be further generalized in certain directions.
Oscillation and a class of linear delay differential equations
Authors: Lovelady David Lowell
Published: July 2010

The differential equation

u

(
m
)

(
t
)
+
p
(
t
)
u
(
g
(
t
)
)
=
0

{u^{(m)}}(t) + p(t)u(g(t)) = 0

. where P is one-signed, is broken into four cases, according to the parity of m and the sign of p. In each case, an analysis is given of the effect g can have on oscillation properties, and oscillation and nonoscillation criteria are given.
Discrete analytic functions of exponential growth
Authors: Zeilberger Doron
Published: July 2010

Analogues of classical representation formulas for entire functions of exponential type are proved in the class of discrete analytic functions.
Bochner identities for Fourier transforms
Authors: Strichartz Robert S.
Published: July 2010

Let G be a compact Lie group and R an orthogonal representation of G acting on

R

n

{{\mathbf {R}}^n}

. For any irreducible unitary representation

π
\pi

of G and vector v in the representation space of

π
\pi

define

S

(
π
,
v
)

\mathcal {S}(\pi ,v)

to be those functions in

S

(

R

n

)

\mathcal {S}({{\mathbf {R}}^n})

which transform (under the action R) according to the vector v. The Fourier transform

F

\mathcal {F}

preserves the class

S

(
π
,
v
)

\mathcal {S}(\pi ,v)

. A Bochner identity asserts that for different choices of G, R,

π
,
v

\pi ,v

the Fourier transform is the same (up to a constant multiple). It is proved here that for G, R,

π
,
v

\pi ,v

and

G
′

,

R
′

,

π
′

,

v
′

G’,R’,\pi ’,v’

and a map

T
:

S

(
π
,
v
)
→

S

(

π
′

,

v
′

)

T:\mathcal {S}(\pi ,v) \to \mathcal {S}(\pi ’,v’)

which has the form: restriction to a subspace followed by multiplication by a fixed function, a Bochner identity

F

′

T
f
=
c
T

F

f

\mathcal {F}’Tf = cT\mathcal {F}f

for all

f
∈

S

(
π
,
v
)

f \in \mathcal {S}(\pi ,v)

holds if and only if

Δ
′

T
f
=

c
1

T
Δ
f

\Delta ’Tf = {c_1}T\Delta f

for all

f
∈

S

(
π
,
v
)

f \in \mathcal {S}(\pi ,v)

. From this result all known Bochner identities follow (due to Harish-Chandra, Herz and Gelbart), as well as some new ones.
Resolvents and bounds for linear and nonlinear Volterra equations
Authors: Levin J. J.
Published: July 2010

The asymptotic behavior of the resolvent of a linear Volterra equation is investigated without the assumption that the kernel of the equation is in

L
1

(
0
,
∞
)

{L^1}(0,\infty )

. A lower bound is obtained on the solutions of a related nonlinear Volterra equation. A special case of the latter result is employed in the proof of the former result.
Structure of subalgebras between 𝐿^{∞} and 𝐻^{∞}
Authors: Chang Sun Yung A.
Published: July 2010

Let B be a closed subalgebra of

L
∞

{L^\infty }

of the unit circle which contains

H
∞

{H^\infty }

properly. Let

C
B

{C_B}

be the

C
∗

{C^\ast }

-algebra generated by the inner functions that are invertible in B. It is shown that the linear span

H
∞

+

C
B

{H^\infty } + {C_B}

is equal to B. Also, a closed subspace (called

V
M

O
B

VM{O_B}

) of BMO (space of functions of bounded mean oscillation) is identified to which B bears the same relation as

L
∞

{L^\infty }

does to BMO.
Module structure of certain induced representations of compact Lie groups
Authors: Funderburk E. James
Published: July 2010

Let G be a compact connected Lie group and assume a choice of maximal torus and positive roots has been made. Given a dominant weight

λ
\lambda

, the Borel-Weil Theorem shows how to construct a holomorphic line bundle on whose sections G acts so that the holomorphic sections provide a realization of the irreducible representation of G with highest weight

λ
\lambda

. This paper studies the G-module structure of the space

Γ
\Gamma

of square integrable sections of the Borel-Weil line bundle. It is found that

Γ
=

lim

n
→
∞

Γ
(
n
)

\Gamma = {\lim _{n \to \infty }}\Gamma (n)

, where

Γ
(
n
)
⊂
Γ
(
n
+
1
)
⊂
Γ

\Gamma (n) \subset \Gamma (n + 1) \subset \Gamma

and

Γ
(
n
)

\Gamma (n)

is isomorphic, as G-module, to
\[

V
(
λ
+
n
λ
)
⊗
V
(
n

λ
∗

)
,

V(\lambda + n\lambda ) \otimes V(n{\lambda ^\ast }),

\]
where

V
(
μ
)

V(\mu )

denotes the irreducible representation of highest weight

μ
\mu

, ’+’ is the Cartan semigroup operation, and ’

∗

^\ast

’ is the contragredient operation. Similar formulas hold for powers of the Borel-Weil line bundle.
Ulm’s theorem for partially ordered structures related to simply presented abelian 𝑝-groups
Authors: Rogers Laurel A.
Published: July 2010

If we have an abelian p-group G, a multiplication by p for each element of G is defined by setting

p
x
=
x
+
x
+
⋯
+
x

px = x + x + \cdots + x

, where p is the number of terms in the sum. If we forget about the addition on G, and just keep the multiplication by p, we have the algebraic structure called a p-basic tree. A natural partial order can be defined, the graph of which is a tree with 0 as root. A p-basic tree generates a simply presented abelian p-group, and provides a natural direct sum decomposition for it. Ulm invariants may be defined directly for a p-basic tree so that they are equal to the Ulm invariants of the corresponding group. A central notion is that of a stripping function between two p-basic trees. Given a stripping function from X onto Y we can construct an isomorphism between the groups corresponding to X and Y; in particular, X and Y have the same Ulm invariants. Conversely, if X and Y have the same Ulm invariants, then there is a map from X onto Y that is the composition of two stripping functions and two inverses of stripping functions. These results constitute Ulm’s theorem for p-basic trees, and provide a new proof of Ulm’s theorem for simply presented groups.
The equivalence of complete reductions
Authors: Hindley R.
Published: July 2010

This paper is about two properties of the

λ
β

\lambda \beta

-calculus and combinatory reduction, namely (E): all complete reductions

ρ
\rho

and

σ
\sigma

of the residuals of a set of redexes in a term X have the same end; and

(

E

+

)
:
ρ

({{\text {E}}^ + }):\rho

and

σ
\sigma

leave the same residuals of any other redex in X. Property (E) is deduced from abstract assumptions which do not imply

(

E

+

)

({{\text {E}}^ + })

. Also

(

E

+

)

({{\text {E}}^ + })

is proved for the usual extensions of combinatory and

λ
β

\lambda \beta

-reduction, and a weak but natural form of

(

E

+

)

({{\text {E}}^ + })

is proved for

λ
β
η

\lambda \beta \eta

-reduction.
Disintegration of measures on compact transformation groups
Authors: Johnson Russell A.
Published: July 2010

Let G be a compact metrizable group which acts freely on a locally compact Hausdorff space X. Let X,

μ
\mu

be a measure on

X
,
π
:
X
→
X

/

G
≡
Y

X,\pi :X \to X/G \equiv Y

the projection,

ν
=
π
(
μ
)

\nu = \pi (\mu )

. We show that there is a

ν
\nu

-Lusin-measurable disintegration of

μ
\mu

with respect to it. We use this result to prove a structure theorem concerning T-ergodic measures on bitransformation groups (G, X, T) with G metric and X compact. We finish with some remarks concerning the case when G is not metric.
Zeroes of holomorphic vector fields and Grothendieck duality theory
Authors: O’Brian N. R.
Published: July 2010

The holomorphic fixed point formula of Atiyah and Bott is discussed in terms of Grothendieck’s theory of duality. The treatment is valid for an endomorphism of a compact complex-analytic manifold with arbitrary isolated fixed points. An expression for the fixed point indices is then derived for the case where the endomorphism belongs to the additive group generated by a holomorphic vector field with isolated zeroes. An application and some examples are given. Two generalisations of these results are also proved. The first deals with holomorphic vector bundles having sufficient homogeneity properties with respect to the action of the additive group on the base manifold, and the second with additive group actions on algebraic varieties.
Ideals of coadjoint orbits of nilpotent Lie algebras
Authors: Godfrey Colin
Published: July 2010

For f a linear functional on a nilpotent Lie algebra g over a field of characteristic 0, let

J
(
f
)

J(f)

be the ideal of all polynomials in

S
(
g
)

S(g)

vanishing on the coadjoint orbit through f in

g
∗

{g^\ast }

, and let

I
(
f
)

I(f)

be the primitive ideal of Dixmier in the universal enveloping algebra

U
(
g
)

U(g)

, corresponding to the orbit. An inductive method is given for computing generators

P
1

,
…
,

P
r

{P_1}, \ldots ,{P_r}

of

J
(
f
)

J(f)

such that

φ

P
1

,
…
,
φ

P
r

\varphi {P_1}, \ldots ,\varphi {P_r}

generate

I
(
f
)
,
φ

I(f),\varphi

being the symmetrization map from

S
(
g
)

S(g)

to

U
(
g
)

U(g)

. Upper bounds are given for the number of variables in the polynomials

P
i

{P_i}

and a counterexample is produced for upper bounds proposed by Kirillov.
Parametrizations of Titchmarsh’s 𝑚(𝜆)-functions in the limit circle case
Authors: Fulton Charles T.
Published: July 2010

For limit-circle eigenvalue problems the so-called′m(λ)′’m(\lambda )’-functions of Titchmarsh [15] are introduced in such a fashion that their parametrization is built into the definition.
One-parameter groups of isometries on Hardy spaces of the torus: spectral theory
Authors: Berkson Earl , Porta Horacio
Published: July 2010

The spectral theory of the infinitesimal generator of an arbitrary one-parameter group of isometries on

H
p

{H^p}

of the torus,

1
⩽
p
>
∞
,
p
≠
2

1 \leqslant p > \infty ,p \ne 2

, is considered. In particular, the spectrum of the generator is determined.
Vector fields on polyhedra
Authors: Penna Michael A.
Published: July 2010

This paper presents a bundle theory for studying vector fields and their integral flows on polyhedra

∗

_ \ast

and applications. Every polyhedron has a tangent object in the category of simplicial bundles in much the same way as every smooth manifold has a tangent object in the category of smooth vector bundles. One can show that there is a correspondence between piecewise smooth flows on a polyhedron P and sections of the tangent object of P (i.e., vector fields on P); using this result one can prove existence results for piecewise smooth flows on polyhedra. Finally an integral formula for the Euler characteristic of a closed, oriented, even-dimensional combinatorial manifold is given; as a consequence of this result one obtains a representation of Euler classes of such combinatorial manifolds in terms of piecewise smooth forms.
A law of the iterated logarithm for stable summands
Authors: Pakshirajan R. P. , Vasudeva R.
Published: July 2010

Let

X
1

,

X
2

,
…

{X_1},{X_2}, \ldots

be a sequence of independent indentically distributed stable random variables with parameters

α

(
0
>
α
>
2
)

\alpha \;(0 > \alpha > 2)

and

β
(

|

β

|

⩽
1
)

\beta (|\beta | \leqslant 1)

. Let

S
n

=

∑

i
=
1

n

X
i

{S_n} = \sum \nolimits _{i = 1}^n {{X_i}}

. Suppose that

(

S

1
,
n

)

({S_{1,n}})

and

(

S

2
,
n

)

({S_{2,n}})

are independent copies of the sequence

(

S
n

)

({S_n})

. In this paper we obtain the set of all limit points in the plane of the sequence
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet StartAbsoluteValue n Superscript negative 1 slash alpha Baseline left-parenthesis upper S Subscript 1 comma n Baseline minus a Subscript n Baseline right-parenthesis EndAbsoluteValue Superscript 1 slash left-parenthesis log log n right-parenthesis Baseline comma StartAbsoluteValue n Superscript negative 1 slash alpha Baseline left-parenthesis upper S Subscript 2 comma n Baseline minus a Subscript n Baseline right-parenthesis EndAbsoluteValue Superscript 1 slash left-parenthesis log log n right-parenthesis Baseline EndSet">

{

|

n

−
1

/

α

(

S

1
,
n

−

a
n

)

|

1

/

(
log
⁡
log
⁡
n
)

,

|

n

−
1

/

α

(

S

2
,
n

−

a
n

)

|

1

/

(
log
⁡
log
⁡
n
)

}

\left \{ {|{n^{ - 1/\alpha }}({S_{1,n}} - {a_n}){|^{1/(\log \log n)}},|{n^{ - 1/\alpha }}({S_{2,n}} - {a_n}){|^{1/(\log \log n)}}} \right \}

\]
where

(

a
n

)

({a_n})

is zero if

α
≠
1

\alpha \ne 1

and is

(
2
β
n
log
⁡
n
)

/

π

(2\beta n\log n)/\pi

if

α
=
1

\alpha = 1

.
Topological irreducibility of nonunitary representations of group extensions
Authors: Williams Floyd L.
Published: July 2010

A functional-analytic approach to the study of the topological irreducibility of certain nonunitary induced representations is set forth. The methods contrast, and in some sense, encompass those first initiated by E. Thieleker in [4], and are amenable to complete irreducibility questions as well. Several sufficient conditions for topological irreducibility are established. A sufficient condition for reducibility is also presented—the latter serving to explain an interesting counterexample due to J. M. G. Fell.
On inner ideals and ad-nilpotent elements of Lie algebras
Authors: Benkart Georgia
Published: July 2010

An inner ideal of a Lie algebra L over a commutative ring k is a k-submodule B of L such that

[
B
[
B
L
]
]
⊆
B

[B[BL]] \subseteq B

. This paper investigates properties of inner ideals and obtains results relating ad-nilpotent elements and inner ideals. For example, let L be a simple Lie algebra in which

D
y
2

=
0

D_y^2 = 0

implies

y
=
0

y = 0

, where

D
y

{D_y}

denotes the adjoint mapping determined by y. If L satisfies the descending chain condition on inner ideals and has proper inner ideals, then L contains a subalgebra

S
=
⟨
e
,
f
,
h
⟩

S = \langle e,f,h\rangle

, isomorphic to the split 3-dimensional simple Lie algebra, such that

D
e
3

=

D
f
3

=
0

D_e^3 = D_f^3 = 0

. Lie algebras having such 3-dimensional subalgebras decompose into the direct sum of two copies of a Jordan algebra, two copies of a special Jordan module, and a Lie subalgebra of transformations of the Jordan algebra and module. The main feature of this decomposition is the correspondence between the Lie and the Jordan structures. In the special case when L is a finite dimensional, simple Lie algebra over an algebraically closed field of characteristic

p
>
5

p > 5

this decomposition yields: Theorem. L is classical if and only if there is an

x
≠
0

x \ne 0

in L such that

D
x

p
−
1

=
0

D_x^{p - 1} = 0

and if

D
y
2

=
0

D_y^2 = 0

implies

y
=
0

y = 0

. The proof involves actually constructing a Cartan subalgebra which has 1-dimensional root spaces for nonzero roots and then using the Block axioms.
The closed leaf index of foliated manifolds
Authors: Conlon Lawrence , Goodman Sue
Published: July 2010

For M a closed, connected, oriented 3-manifold, a topological invariant is computed from the cohomology ring

H
∗

(
M
;

Z

)

{H^\ast }(M;{\mathbf {Z}})

that provides an upper bound to the number of topologically distinct types of closed leaves any smooth transversely oriented foliation of M can contain. In general, this upper bound is best possible.
On characterizing the standard quantum logics
Authors: Wilbur W. John
Published: July 2010

Let

L

\mathcal {L}

be a complete projective logic. Then

L

\mathcal {L}

has a natural representation as the lattice of

⟨

⋅
,
⋅

⟩

\langle { \cdot , \cdot } \rangle

-closed subspaces of a left vector space V over a division ring D, where

⟨

⋅
,
⋅

⟩

\langle {\cdot ,\cdot } \rangle

is a definite

θ
\theta

-bilinear symmetric form on V,

θ
\theta

being some involutive antiautomorphism of D. Now a well-known theorem of Piron states that if D is isomorphic to the real field, the complex field or the sfield of quaternions, if

θ
\theta

is continuous, and if the dimension of

L

\mathcal {L}

is properly restricted, then

L

\mathcal {L}

is just one of the standard Hilbert space logics. Here we also assume

L

\mathcal {L}

is a complete projective logic. Then if every

θ
\theta

-fixed element of D is in the center of D and can be written as

±

d
θ
(
d
)

\pm \,d\theta (d)

, some

d
∈
D

d \in D

, and if the dimension of

L

\mathcal {L}

is properly restricted, we show that

L

\mathcal {L}

is just one of the standard Hilbert space logics over the reals, the complexes, or the quaternions. One consequence is the extension of Piron’s theorem to discontinuous

θ
\theta

. Another is a purely lattice theoretic characterization of the lattice of closed subspaces of separable complex Hilbert space.
The level structure of a residual set of continuous functions
Authors: Bruckner A. M. , Garg K. M.
Published: July 2010

Let C denote the Banach space of continuous real-valued functions on

[
0
,
1
]

[0,1]

with the uniform norm. The present article is devoted to the structure of the sets in which the graphs of a residual set of functions in C intersect with different straight lines. It is proved that there exists a residual set A in C such that, for every function

f
∈
A

f \in A

, the top and the bottom (horizontal) levels of f are singletons, in between these two levels there are countably many levels of f that consist of a nonempty perfect set together with a single isolated point, and the remaining levels of f are all perfect. Moreover, the levels containing an isolated point correspond to a dense set of heights between the minimum and the maximum values assumed by the function. As for the levels in different directions, there exists a residual set B in C such that, for every function

f
∈
B

f \in B

, the structure of the levels of f is the same as above in all but a countable dense set of directions, and in each of the exceptional nonvertical directions the level structure of f is the same but for the fact that one (and only one) of the levels has two isolated points in place of one. For a general function

f
∈
C

f \in C

a theorem is proved establishing the existence of singleton levels of f, and of the levels of f that contain isolated points.
Normally flat deformations
Authors: Bennett Bruce
Published: July 2010

We study flat families

Z

/

T

Z/T

, together with a section

σ
:
T
→
Z

\sigma :T \to Z

such that the normal cone to the image of

σ
\sigma

in Z is flat over T. Such a family is called a “normally flat deformation (along

σ
\sigma

)"; it corresponds intuitively to a deformation of a singularity which preserves the Hilbert-Samuel function. We construct the versal normally flat deformation of an isolated singularity (X,x) in terms of the flat strata of the relative jets of the “usual” versal deformation of X. We give explicit criteria, in terms of equations, for a flat family to be normally flat along a given section. These criteria are applied to demonstrate the smoothness of normally flat deformation theoryand of the canonical map from it to the cone deformation theory of the tangent cone-in the case of strict complete intersections. Finally we study the tangent space to the normally flat deformation theory, expressing it as the sum of two spaces: The first is a piece of a certain filtration of the tangent space to the usual deformation theory of X; the second is the tangent space to the special fibre of the canonical map

N
→
S

N \to S

, where N (resp. S) is the parameter space for the versal normally flat deformation of (X, x) (resp. for the versal deformation of X). We discuss the relation of this second space to infinitesimal properties of sections.
Potential processes
Authors: Chacon R. V.
Published: July 2010

The prototype of a potential process is a stochastic process which visits the same points in the same order as a Markov process, but at a rate obtained from a nonanticipating time change. The definition of a potential process may be given intrinsically and most generally without mention of a Markov process, in terms of potential theory. The definition may be given more directly and less generally in terms of potentials which arise from Markov processes, or more directly than this, as suitably time-changed Markov processes. The principal purpose of studying the class of potential processes, which may be shown to include martingales as well as Markov processes themselves, is to give a unified treatment to a wide class of processes which has potential theory at its core. That it is possible to do so suggests that potential rather than martingale results are central to the study of Markov processes. Furthermore, this also suggests that it is not the Markov property itself which makes Markov processes tractable, but rather the potential structure which can be constructed with the assistance of the Markov property. The general theory of potential processes is developed in a forthcoming paper. It will be shown there that a Markov process subject to an ordinary continuous nonanticipating time change is a local potential process. It may be seen, by examining examples, that it is necessary to consider randomized stopping times and randomized nonanticipating time changes in the general case. In the forthcoming paper a more general notion than randomized nonanticipating time changes is used to obtain a characterization of potential processes. It is an open problem whether randomization itself is sufficient in the general case, and whether ordinary nonanticipating time changes are sufficient for continuous parameter martingales and Brownian motion on the line. The emphasis in the present paper will be on developing the theory of discrete parameter martingales as a special case of the general theory.
Functions satisfying elementary relations
Authors: Singer Michael F.
Published: July 2010

In this paper we deal with the following problems: When do the solutions of a collection of differential equations satisfy an elementary relation, that is, when is there an equation of the form

R
=
0

R = 0

where R is some algebraic combination of logarithmic, exponential and algebraic functions involving solutions of our differential equations? If such relations exist, what can they look like? These problems are given an algebraic setting and general forms for such relations are exhibited. With these, we are able to show that certain classes of functions satisfy no elementary relations.
A superposition theorem for unbounded continuous functions
Authors: Doss Raouf
Published: July 2010

Let

R
n

{R^n}

be the n-dimensional Euclidean space. We prove that there are 4n real functions

φ
q

{\varphi _q}

continuous on

R
n

{R^n}

with the following property: Every real function f, not necessarily bounded, continuous on

R
n

{R^n}

, can be written
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals normal upper Sigma Subscript q equals 1 Superscript 2 n plus 1 Baseline g left-parenthesis phi Subscript q Baseline left-parenthesis x right-parenthesis right-parenthesis plus normal upper Sigma Subscript q equals 2 n plus 2 Superscript 4 n Baseline h left-parenthesis phi Subscript q Baseline left-parenthesis x right-parenthesis right-parenthesis comma x element-of upper R Superscript n">

f
(
x
)
=

Σ

q
=
1

2
n
+
1

g
(

φ
q

(
x
)
)
+

Σ

q
=
2
n
+
2

4
n

h
(

φ
q

(
x
)
)
,
x
∈

R
n

f(x) = \Sigma _{q = 1}^{2n + 1}g({\varphi _q}(x)) + \Sigma _{q = 2n + 2}^{4n}h({\varphi _q}(x)),x \in {R^n}

, where g, h are 2 real continuous functions of one variable, depending on f.