bluuebunny/crossref_metadata_2025 huggingface dataset:
On the ideal structure of 𝐶(𝑋)
Authors: Dietrich William E.
Published: June 2010

The ideal structure of

C
(
X
)

C(X)

, the algebra of continuous functions from a completely regular Hausdorff space

X
X

to the scalars is analyzed by examining for fixed

A
⊂
β
X

A \subset \beta X

(the Stone-Čech compactification of

X
X

) the structure of the quotient

I
A

/

F
A

{I^A}/{F^A}

, where

I
A

[

F
A

]

{I^A}[{F^A}]

is the ideal of maps

f
∈
C
(
X
)

f \in C(X)

for which
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A subset-of cl Subscript beta upper X Baseline upper Z left-parenthesis f right-parenthesis left-bracket upper A subset-of i n t Subscript beta upper X Baseline cl Subscript beta upper X Baseline upper Z left-parenthesis f right-parenthesis right-bracket period">

A
⊂

cl

β
X

Z
(
f
)

[
A
⊂

int

β
X

cl

β
X

Z
(
f
)
]
.

A \subset {\text {cl} _{\beta X}}Z(f)\quad [A \subset {\operatorname {int} _{\beta X}}{\text {cl} _{\beta X}}Z(f)].

\]
Unless it vanishes,

I
A

/

F
A

{I^A}/{F^A}

has no minimal or maximal ideals, and its Krull dimension is infinite. If

J
J

is an ideal of

C
(
X
)

C(X)

strictly between

F
A

{F^A}

and

I
A

{I^A}

, there are ideals

J
_

\underline {J}

and

J
¯

\bar J

of

C
(
X
)

C(X)

for which

F
A

⊂

J
_

⊂
J
⊂

J
¯

⊂

I
A

{F^A} \subset \underline {J} \subset J \subset \bar J \subset {I^A}

with all inclusions proper. For

K
⊂
C
(
X
)

K \subset C(X)

, let

Z
(
K
)
=
∩
{

cl

β
X

Z
(
f
)
:
f
∈
K
}

Z(K) = \cap \{ {\text {cl} _{\beta X}}Z(f):f \in K\}

. If

J
⫋
I

J \subsetneqq I

are ideals of

C
(
X
)

C(X)

with

Z
(
J
)
=
Z
(
I
)

Z(J) = Z(I)

and if

I
I

is semiprime, there is an ideal

K
K

strictly between

J
J

and

I
I

. If

I
I

and

J
J

are

Z
Z

-ideals,

K
K

can be chosen to be of the form

P
∩
I
,
P

P \cap I, P

a prime ideal of

C
(
X
)

C(X)

. The maximal ideals of a semiprime ideal

I
I

of

C
(
X
)

C(X)

are of the form

I
q

∩
I
,
q
∈
β
X
−
Z
(
I
)

{I^q} \cap I,q \in \beta X - Z(I)

. If

A
⊂
β
X

A \subset \beta X

is closed,

I
A

{I^A}

is a finitely generated ideal iff

A
A

is open.
Global dimension of orders
Authors: Tarsy Richard B.
Published: June 2010

We prove that the finitistic global dimension (fGD) of an order in a quaternion algebra over the quotient field of a Dedekind domain is one. Examples are given of orders of global dimension

n
−
1

n - 1

in

n
×
n

n \times n

matrices over the quotient field of a discrete valuation ring.
The strict topology for double centralizer algebras
Authors: Taylor Donald Curtis
Published: June 2010

Sufficient conditions are given for a double centralizer algebra under the strict topology to be a Mackey space.
On the classification of symmetric graphs with a prime number of vertices
Authors: Chao Chong-yun
Published: June 2010

We determine all the symmetric graphs with a prime number of vertices. We also determine the structure of their groups.
Matching theory for combinatorial geometries
Authors: Aigner Martin , Dowling Thomas A.
Published: June 2010

Given two combinatorial (pre-) geometries and an arbitrary binary relation between their point sets, a matching is a subrelation which defines a bijection between independent sets of the geometries. The theory of matchings of maximum cardinality is developed in two directions, one of an algorithmic, the other of a structural nature. In the first part, the concept of an augmenting chain is introduced to establish as principal results a min-max type theorem and a generalized Marriage Theorem. In the second part, Ore’s notion of a deficiency function for bipartite graphs is extended to determine the structure of the set of critical sets, i.e. those with maximum deficiency. The two parts of the investigation are then connected using the theory of Galois connections.
On the 𝐶*-algebra of Toeplitz operators on the quarterplane
Authors: Douglas R. G. , Howe Roger
Published: June 2010

Using the device of the tensor product ofC∗{C^ \ast }-algebras, we study theC∗{C^ \ast }-algebra generated by the Toeplitz operators on the quarter-plane. We obtain necessary and sufficient conditions for such an operator to be Fredholm, but show in this case that not all such operators are invertible.
Applications of extreme point theory to classes of multivalent functions
Authors: Hallenbeck David J. , Livingston Albert E.
Published: June 2010

Extreme points of the closed convex hulls of several classes of multivalent functions are determined. These are then used to determine the precise bounds on the coefficients of a function majorized by or subordinate to a function in any of the classes.

L
q

{L^q}

means are also discussed and subordination theorems are considered. The classes we consider are generalizations of the univalent starlike, convex and close-to-convex functions in addition to others.
On chain varieties of linear algebras
Authors: Artamonov V. A.
Published: June 2010

In the present paper we study varieties of linear k-algebras over a commutative associative Noetherian ring k with 1, whose subvarieties form a chain. We describe these varieties in terms of identities in the following cases: residually nilpotent varieties, varieties of alternative, Jordan and

(
−
1
,
1
)

( - 1,1)

-algebras.
Two extremal problems
Authors: Harrington Andrew , Ortel Marvin
Published: June 2010

The class H consists of all functions f which are analytic in the disk D and satisfy
\[

∫
D

∫

|

f
(
z
)

|

d
x

d
y

>
∞
.

\int \limits _D {\int {|f(z)|dx\;dy} > \infty .}

\]
Linear extremal problems for functionals of the type
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda left-parenthesis f right-parenthesis equals integral integral Underscript upper D Endscripts f left-parenthesis z right-parenthesis phi left-parenthesis z right-parenthesis d x d y comma phi element-of upper L Superscript normal infinity Baseline left-parenthesis upper D right-parenthesis comma f element-of upper H">

Λ
(
f
)
=
∫

∫
D

f
(
z
)
ϕ
(
z
)

d
x

d
y
,
ϕ
∈

L
∞

(
D
)
,
f
∈
H

\Lambda (f) = \smallint {\smallint _D}f(z)\phi (z)\;dx\;dy,\phi \in {L^\infty }(D),f \in H

, are studied.
The infinitesimal stability of semigroups of expanding maps
Authors: Sacksteder Richard
Published: June 2010

The concept of

C
∞

{C^\infty }

infinitesimal stability for representations of a semigroup by

C
∞

{C^\infty }

maps is defined. In the case of expanding linear maps of the torus

T
d

{T^d}

it is shown that certain algebraic conditions assure such stability.
Partially ordered linear algebras with multiplicative diagonal map
Authors: Dai Taen Yu , DeMarr Ralph
Published: June 2010

The diagonal of the product of two triangular matrices is the product of the diagonals of each matrix. This idea is used to characterize partially ordered linear algebras which have order properties similar to an algebra of real triangular matrices.
Coefficient multipliers of Bloch functions
Authors: Anderson J. M. , Shields A. L.
Published: June 2010

The class

B

\mathcal {B}

of Bloch functions is the class of all those analytic functions in the open unit disc for which the maximum modulus is bounded by

c

/

(
1
−
r
)

c/(1 - r)

on

|

z

|

⩽
r

|z| \leqslant r

. We study the absolute values of the Taylor coefficients of such functions. In particular, we find all coefficient multipliers from

l
p

{l^p}

into

B

\mathcal {B}

and from

B

\mathcal {B}

into

l
p

{l^p}

. We find the second Köthe dual of

B

\mathcal {B}

and show its relevance to the multiplier problem. We identify all power series

∑

a
n

z
n

\sum {a_n}{z^n}

such that

∑

w
n

a
n

z
n

\sum {w_n}{a_n}{z^n}

is a Bloch function for every choice of the bounded sequence

{

w
n

}

\{ {w_n}\}

. Analogous problems for

H
p

{H^p}

spaces are discussed briefly.
Natural limits for harmonic and superharmonic functions
Authors: Diederich J. R.
Published: June 2010

In this paper it is shown that Fatou’s theorem holds for superharmonic functions in certain Liapunov domains if mean continuous limits are used in place of nontangential limits for which Fatou’s theorem fails. Also, existence of mean continuous limits is established for certain semi-linear elliptic equations in Liapunov domains.
Deformation of open embeddings of 𝑄-manifolds
Authors: Fathi A. , Visetti Y. M.
Published: June 2010

We prove here for Hilbert cube manifolds the full analogue of the Černavski-Edwards-Kirby Theorem concerning the deformation Principle for open embeddings of topological manifolds.
Unique factorization in modules and symmetric algebras
Authors: Costa Douglas L.
Published: June 2010

Necessary and sufficient conditions are given for a torsion-free module M over a UFD D to admit a smallest factorial module containing it. This factorial hull is

∩

M
P

\cap {M_P}

, the intersection taken over all height one primes of D. In case M is finitely generated, the hull is

M

∗
∗

{M^{ \ast \ast }}

, the bidual of M. It is shown that if the symmetric algebra

S
D

(
M
)

{S_D}(M)

admits a hull, then the hull is the smallest graded UFD containing

S
D

(
M
)

{S_D}(M)

.

S
D

(
M
)

{S_D}(M)

is a UFD if and only if it is a factorial D-module. If M is finitely generated over D, but not necessarily torsion-free, then

⊕

i
⩾
0

(

S
i

(
M
)

)

∗
∗

{ \oplus _{i \geqslant 0}}{({S^i}(M))^{ \ast \ast }}

is a graded UFD. Examples are given to show that any finite number of symmetric powers of M may be factorial without

S
D

(
M
)

{S_D}(M)

being factorial.
Fringe families in stable homotopy
Authors: Zahler Raphael S.
Published: June 2010

It is shown how to detect and construct elements in the stable homotopy groups of spheres corresponding to the

i

{_i}

family of Toda. The only tools used are Brown-Peterson cohomology and the Adams spectral sequence.
Inclusions and noninclusion of spaces of convolution operators
Authors: Cowling Michael G. , Fournier John J. F.
Published: June 2010

Let G be an infinite, locally compact group. Denote the space of convolution operators, on G, of strong type

(
p
,
q
)

(p,q)

by

L
p
q

(
G
)

L_p^q(G)

. It is shown that, if

|

1

/

q
−
1

/

2

|

>

|

1

/

p
−
1

/

2

|

|1/q - 1/2| > |1/p - 1/2|

, then

L
q
q

(
G
)

L_q^q(G)

is not included in

L
p
p

(
G
)

L_p^p(G)

. This result follows from estimates on the norms, in these spaces, of Rudin-Shapiro measures. The same method leads to a simple example of a convolution operator that is of strong type (q, q) for all q in the interval

(
p
,

p
′

)

(p,p’)

but is not of restricted weak type (p, p) or of restricted weak type

(

p
′

,

p
′

)

(p’,p’)

. Other statements about noninclusion among the spaces

L
p
q

(
G
)

L_p^q(G)

also follow from various assumptions about G. For instance, if G is unimodular, but not compact,

1
⩽
p
,
q
,
r
,
s
⩽
∞

1 \leqslant p,q,r,s \leqslant \infty

, with

p
⩽
q

p \leqslant q

, and

min
(
s
,

r
′

)
>
min
(
q
,

p
′

)

\min (s,r’) > \min (q,p’)

, then

L
p
q

(
G
)

L_p^q(G)

is not included in

L
r
s

(
G
)

L_r^s(G)

. Using Zafran’s multilinear interpolation theorem for the real method, it is shown that, if

1
>
p
>
2

1 > p > 2

, then there exists a convolution operator on G that is of weak type (p, p) but not of strong type (p, p); it is not known whether such operators exist when

p
>
2

p > 2

, but it is shown that if

p
≠
1
,
2
,
∞

p \ne 1,2,\infty

, then there exists a convolution operator that is of restricted weak type (p, p) but is not of weak type (p, p). Many of these results also hold for the spaces of operators that commute with left translation rather than right translation. Further refinements are presented in three appendices.
Asymptotic behavior of solutions of nonlinear functional differential equations in Banach space
Authors: Haddock John R.
Published: June 2010

Let X be a Banach space and let

C
=
C
(
[
−
r
,
0
]
,
X
)

C = C([ - r,0],X)

denote the space of continuous functions from

[
−
r
,
0
]

[ - r,0]

to X. In this paper the problem of convergence in norm of solutions of the nonlinear functional differential equation

x
˙

=
F
(
t
,

x
t

)

\dot x = F(t,{x_t})

is considered where

F
:
[
0
,
∞
)
×
C
→
X

F:[0,\infty ) \times C \to X

. As a special case of the main theorem, stability results are given for the equation

x
˙

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

x
t

)

\dot x(t) = f(t,x(t)) + g(t,{x_t})

, where

−
f
(
t
,
⋅
)
−
α
(
t
)
I

- f(t, \cdot ) - \alpha (t)I

satisfies certain accretive type conditions and

g
(
t
,
⋅
)

g(t, \cdot )

is Lipschitzian with Lipschitz constant

β
(
t
)

\beta (t)

closely related to

α
(
t
)

\alpha (t)

.
Cyclic purity versus purity in excellent Noetherian rings
Authors: Hochster Melvin
Published: June 2010

A characterization is given of those Noetherian rings

R
R

such that whenever

R
R

is ideally closed (

≡
\equiv

cyclically pure) in an extension algebra

S
S

, then

R
R

is pure in

S
S

. In fact,

R
R

has this property if and only if the completion

(
A
,
m
)

(A,m)

of each local ring of

R
R

at a maximal ideal has the following two equivalent properties: (i) For each integer

N
>
0

N > 0

there is an

m
m

-primary irreducible ideal

I
N

⊂

m
N

{I_N} \subset {m^N}

. (ii) Either

dim

A
=
0

\dim \;A = 0

and

A
A

is Gorenstein or else depth

A
⩾
1

A \geqslant 1

and there is no

P
∈

Ass

(
A
)

P \in {\operatorname {Ass}}(A)

such that

dim
⁡
(
A

/

P
)
=
1

\dim (A/P) = 1

and

(
A

/

P
)
⊕
(
A

/

P
)

(A/P) \oplus (A/P)

is embeddable in

A
A

. It is then shown that if

R
R

is a locally excellent Noetherian ring such that either

R
R

is reduced (or, more generally, such that

R
R

is generically Gorenstein), or such that Ass(

R
R

) contains no primes of coheight

⩽
1

\leqslant 1

in a maximal ideal, and

R
R

is ideally closed in

S
S

, then

R
R

is pure in

S
S

. Matlis duality and the theory of canonical modules are utilized. Module-theoretic analogues of condition (i) above are, of necessity, also analyzed. Numerous related questions are studied. In the non-Noetherian case, an example is given of a ring extension

R
→
S

R \to S

such that

R
R

is pure in

S
S

but

R
[
[
T
]
]

R[[T]]

is not even cyclically pure in

S
[
[
T
]
]

S[[T]]

.
The broken-circuit complex
Authors: Brylawski Tom
Published: June 2010

The broken-circuit complex introduced by H. Wilf (Which polynomials are chromatic?, Proc. Colloq. Combinational Theory (Rome, 1973)) of a matroid G is shown to be a cone over a related complex, the reduced broken-circuit complex

C

′

(
G
)

\mathcal {C}’(G)

. The topological structure of

C

′

(
G
)

\mathcal {C}’(G)

is studied, its Euler characteristic is computed, and joins and skeletons are shown to exist in the class of all such complexes. These computations and constructions are compared with analogous results in the theory of the independent set complex of a matroid. Reduced broken-circuit complexes are partially characterized by conditions concerning which subcomplexes are pure (i.e., have equicardinal maximal simplices). In particular, every matroid complex is a reduced broken-circuit complex. Three proofs are given that the simplex numbers of the cone of

C

′

(
G
)

\mathcal {C}’(G)

are the coefficients which appear in the characteristic polynomial

χ
(
G
)

\chi (G)

of G. This relates the work of W. Tutte on externally inactive bases to that of H. Whitney on sets which do not contain broken circuits. One of these proofs gives a combinatorial correspondence between these sets. Properties of the characteristic polynomial are then given topological proofs.
Efficient generation of maximal ideals in polynomial rings
Authors: Davis E. D. , Geramita A. V.
Published: June 2010

The cardinality of a minimal basis of an ideal I is denoted

ν
(
I
)

\nu (I)

. Let A be a polynomial ring in

n
>
0

n > 0

variables with coefficients in a noetherian (commutative with

1
≠
0

1 \ne 0

) ring R, and let M be a maximal ideal of A. In general
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu left-parenthesis upper M upper A Subscript upper M Baseline right-parenthesis plus 1 greater-than-or-slanted-equals nu left-parenthesis upper M right-parenthesis greater-than-or-slanted-equals nu left-parenthesis upper M upper A Subscript upper M Baseline right-parenthesis">

ν
(
M

A
M

)
+
1
⩾
ν
(
M
)
⩾
ν
(
M

A
M

)

\nu (M{A_M}) + 1 \geqslant \nu (M) \geqslant \nu (M{A_M})

. This paper is concerned with the attaining of equality with the lower bound. It is shown that equality is attained in each of the following cases: (1)

A
M

{A_M}

is not regular (valid even if A is not a polynomial ring), (2)

M
∩
R

M \cap R

is maximal in R and (3)

n
>
1

n > 1

. Equality may fail for

n
=
1

n = 1

, even for R of dimension 1 (but not regular), and it is an open question whether equality holds for R regular of dimension

>
1

> 1

. In case

n
=
1

n = 1

and

dim
⁡
(
R
)
=
2

\dim (R) = 2

the attaining of equality is related to questions in the K-theory of projective modules. Corollary to (1) and (2) is the confirmation, for the case of maximal ideals, of one of the Eisenbud-Evans conjectures; namely,

ν
(
M
)
⩽
max
{
ν
(
M

A
M

)
,
dim
⁡
(
A
)
}

\nu (M) \leqslant \max \{ \nu (M{A_M}),\dim (A)\}

. Corollary to (3) is that for R regular and

n
>
1

n > 1

, every maximal ideal of A is generated by a regular sequence—a result well known (for all

n
⩾
1

n \geqslant 1

) if R is a field (and somewhat less well known for R a Dedekind domain).
On 𝑛-widths in 𝐿^{∞}
Authors: Micchelli Charles A. , Pinkus Allan
Published: June 2010

The n-width in

L
∞

{L^\infty }

of certain sets determined by matrices and integral operators is determined. The notion of total positivity is essential in the analysis.
Of regulated and steplike functions
Authors: Moran Gadi
Published: June 2010

Let C denote the class of regulated real-valued functions on the unit interval vanishing at the origin, whose positive and negative jumps sum to infinity in every nontrivial subinterval of I. Goffman [2] showed that every f in C is (essentially) a sum

g
+
s

g + s

where g is continuous and s is steplike. In this sense, a function in C is like a function of bounded variation, that has a unique such g and s. The import of this paper is that for f in C the representation

f
=
g
+
s

f = g + s

is not only not unique, but by far the opposite holds: g can be chosen to be any continuous function on I vanishing at 0, at the expense of a rearrangement of s.
Ergodic equivalence relations, cohomology, and von Neumann algebras. I
Authors: Feldman Jacob , Moore Calvin C.
Published: June 2010

Let

(
X
,

B

)

(X,\mathcal {B})

be a standard Borel space,

R
⊂
X
×
X

R \subset X \times X

an equivalence relation

∈

B

×

B

\in \mathcal {B} \times \mathcal {B}

. Assume each equivalence class is countable. Theorem 1:

∃
\exists

a countable group G of Borel isomorphisms of

(
X
,

B

)

(X,\mathcal {B})

so that

R
=
{
(
x
,
g
x
)
:
g
∈
G
}

R = \{ (x,gx):g \in G\}

. G is far from unique. However, notions like invariance and quasi-invariance and R-N derivatives of measures depend only on R, not the choice of G. We develop some of the ideas of Dye [1], [2] and Krieger [1]-[5] in a fashion explicitly avoiding any choice of G; we also show the connection with virtual groups. A notion of “module over R” is defined, and we axiomatize and develop a cohomology theory for R with coefficients in such a module. Surprising application (contained in Theorem 7): let

α
,
β

\alpha ,\beta

be rationally independent irrationals on the circle

T

\mathbb {T}

, and f Borel:

T

→

T

\mathbb {T} \to \mathbb {T}

. Then

∃
\exists

Borel

g
,
h
:

T

→

T

g,h:\mathbb {T} \to \mathbb {T}

with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals left-parenthesis g left-parenthesis a x right-parenthesis slash g left-parenthesis x right-parenthesis right-parenthesis left-parenthesis h left-parenthesis beta x right-parenthesis slash h left-parenthesis x right-parenthesis right-parenthesis">

f
(
x
)
=
(
g
(
a
x
)

/

g
(
x
)
)
(
h
(
β
x
)

/

h
(
x
)
)

f(x) = (g(ax)/g(x))(h(\beta x)/h(x))

a.e. The notion of “skew product action” is generalized to our context, and provides a setting for a generalization of the Krieger invariant for the R-N derivative of an ergodic transformation: we define, for a cocycle c on R with values in the group A, a subgroup of A depending only on the cohomology class of c, and in Theorem 8 identify this with another subgroup, the “normalized proper range” of c, defined in terms of the skew action. See also Schmidt [1].
Commutativity in series of ordinals: a study of invariants
Authors: Hickman J. L.
Published: June 2010

It is well known that two ordinals are additively commutative if and only if they are finite multiples of some given ordinal, and it is very easy to extend this result to any finite sequence of ordinals. However, no necessary and sufficient conditions for the commutativity of a series of ordinals seem to be known when the length of that series is infinite, although sufficient conditions for certain cases have been given by Sierpiński and Ginsburg. In this paper we present such necessary and sufficient conditions. The general problem is split into five distinct cases: those in which the length of the series is a regular initial ordinal, a singular initial ordinal, an infinite, noninitial prime component, an infinite successor ordinal, and an infinite limit ordinal that is not a prime component. These are dealt with respectively in the second through to the sixth sections of the paper, and it turns out that in every case our criteria can be expressed in terms of an ordinal parameter, which is in fact an invariant of the series in question. This concept of invariance is introduced in the first section, which also contains several lemmas and a slight strengthening of the original Sierpiński-Ginsburg result. The final section of this paper differs from the preceding four sections in two aspects. Firstly, the proofs of its two main results are merely sketched, since they contain no arguments that have not previously appeared in some form or other. Secondly, we have not given any explicit determination of the ordinal parameter introduced in this section, since we felt that such a determination would prolong the paper intolerably and encroach upon work done by J. A. H. Anderson: we have therefore simply referred to Anderson’s interesting paper.
A class of Schur algebras
Authors: Brender M.
Published: June 2010

This paper delineates a class of Schur algebras over a finite group G, parametrized by two subgroups

K

◃

H

⊂

G

K\, \triangleleft \,H\, \subset \,G

. The constructed Schur algebra

C

[
G
]

K
H

{\text {C}}\left [ G \right ]_K^H

is maximal for the two properties (a) centralizing the elements of H, and (b) containing the elements of K in the identity. Most commonly considered examples of Schur algebras fall into this class. A complete set of characters of

C

[
G
]

K
H

{\text {C}}\left [ G \right ]_K^H

is given in terms of the spherical functions on the group G with respect to the subgroup H. Necessary and sufficient conditions are given for this Schur algebra to be commutative, in terms of a condition on restriction multiplicities of characters. This leads to a second-orthogonality-type relation among a subset of the spherical functions. Finally, as an application, a particular Schur algebra of this class is analyzed, and shown to be a direct sum of centralizer rings.
Fourier inversion of invariant integrals on semisimple real Lie groups
Authors: Herb Rebecca A.
Published: June 2010

Let G be a connected, semisimple real Lie group with finite center. Associated with every regular semisimple element g of G is a tempered invariant distribution

Λ
g

{ \Lambda _g}

given by an orbital integral. This paper gives an inductive formula for computing the Fourier transform of

Λ
g

{ \Lambda _g}

in terms of the space of tempered invariant eigendistributions of G.
The module of indecomposables for mod 2 finite 𝐻-spaces
Authors: Kane Richard
Published: June 2010

The module of indecomposables obtained from the mod 2 cohomology of a finite H-space is studied. It is shown that this module is trivial in dimensions

≡

0

\equiv \,0

(mod 4).
𝐾-theory of hyperplanes
Authors: Dayton Barry H. , Weibel Charles A.
Published: June 2010

Let R be the coordinate ring of a union of N hyperplanes in general position in

A

k

n
+
1

{\textbf {A}}_k^{n + 1}

. Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript i Baseline left-parenthesis upper R right-parenthesis equals upper K Subscript i Baseline left-parenthesis k right-parenthesis circled-plus StartBinomialOrMatrix upper N minus 1 Choose n plus 1 EndBinomialOrMatrix upper K Subscript n plus i Baseline left-parenthesis k right-parenthesis period">

K
i

(
R
)

=

K
i

(
k
)

⊕

(

N

−

1

n

+

1

)

K

n
+
i

(
k
)
.

{K_i}(R)\, = \,{K_i}(k)\, \oplus \,\left ( {\begin {array}{*{20}{c}} {N\, - \,1} \\ {n\, + \,1} \\ \end {array} } \right )\,{K_{n + i}}(k).

\]
This formula holds for

K
0

,

K
1

,

K
i

(
i
>
0
)

{K_0},\,{K_1},\,{K_i}\,(i > 0)

, and for the Karoubi-Villamayor groups

K

V
i

(
i
∈

Z

)

K{V_i}\,(i \in \,{\mathbf {Z}})

. For

K
2

{K_2}

there is an extra summand

R
¯

/

R

\bar R/R

, where

R
¯

\bar R

is the normalization of R. For

K
3

{K_3}

the above is a quotient of

K
3

(
R
)

{K_3}(R)

. In §4 we show that

K
1

{K_1}

-regularity implies

K
0

{K_0}

-regularity, answering a question of Bass. We also show that

K
i

{K_i}

-regularity is equivalent to Laurent

K
i

{K_i}

-regularity for

i
⩽
1

i \leqslant 1

. The results of this section are independent of the rest of the paper.
𝑝-adic gamma functions and Dwork cohomology
Authors: Boyarsky Maurizio
Published: June 2010

The relations of Gross and Koblitz between gauss sums and the p-adic gamma function is reexamined from the point of view of Dwork’s formulation of p-adic cohomology. Some higher dimensional generalizations are proposed.
Decomposition of nonnegative group-monotone matrices
Authors: Jain S. K. , Kwak Edward K. , Goel V. K.
Published: June 2010

A decomposition of nonnegative matrices with nonnegative group inverses has been obtained. This decomposition provides a new approach to the solution of problems relating to nonnegative matrices with nonnegative group inverses. As a consequence, a number of results are derived. Our results, among other things, answer a question of Berman, extend the theorems of Berman and Plemmons, DeMarr and Flor.
A spectral theorem for 𝐽-nonnegative operators
Authors: Harvey Bernard N.
Published: June 2010

A J-space is a Hilbert space with the usual inner product denoted

[
x
,
y
]

[x,y]

and an indefinite inner product defined by

(
x
,
y
)

=

[
J
x
,
y
]

(x,y)\, =\, [Jx,y]

where J is a bounded selfadjoint operator whose square is the identity. We define a J-adjoint

T
+

{T^ + }

of an operator T with respect to the indefinite inner product in the same way as the regular adjoint

T

∗

T^{\ast }

is defined with respect to

[
x
,
y
]

[x,y]

. We say T is J-selfadjoint if

T
=

T
+

T = {T^ + }

. An operator-valued function is called a J-spectral function with critical point zero if it is defined for all

t
≠
0

t \ne 0

, is bounded, J-selfadjoint and has the properties of a resolution of the identity on its domain. It has been proved by M. G. Krein and Ju. P. Smul’jan that bounded Jselfadjoint operators A with

(
A
x
,
x
)
⩾
0

(Ax,x) \geqslant 0

for all x can be represented as a strongly convergent improper integral of t with respect to a J-spectral function with critical point zero plus a nilpotent of index 2. Further, the product of the nilpotent with the J-spectral function on intervals not containing zero is zero. The present paper extends this theory to the unbounded case. We show that unbounded J-selfadjoint operators with

(
A
x
,
x
)
⩾
0

(Ax,x) \geqslant 0

are a direct sum of an operator of the above mentioned type and the inverse of a bounded operator of the same type whose nilpotent part is zero.
Approximate torus fibrations of high dimensional manifolds can be approximated by torus bundle projections
Authors: Goad R. E.
Published: June 2010

In this paper, we prove that approximate torus fibrations of high dimensional manifolds can be approximated by torus bundle projections. The principal tools are the torus trick developed by Kirby and Siebenmann, a surgery theorem concerning homotopy structures on torii due to Hsiang and Wall, a theorem on the space of homeomorphisms of the torus due to Hamstrom and a generalization of hereditary homotopy equivalence developed by the author.
On the existence of eigenvalues of differential operators dependent on a parameter
Authors: Strelitz Sh. , Abramovich S.
Published: June 2010

In this paper we obtain results about the existence of eigenvalues for a system which depends polynomially on

λ
\lambda

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column u prime Subscript k Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript j equals 1 Overscript n Endscripts b Subscript k j Baseline left-parenthesis x comma lamda right-parenthesis u Subscript j Baseline left-parenthesis x right-parenthesis comma 2nd Column a m p semicolon sigma-summation Underscript i equals 0 Overscript p Endscripts sigma-summation Underscript j equals 1 Overscript upper N Endscripts a Subscript k j Superscript i Baseline u Subscript j Baseline left-parenthesis x Subscript i Baseline right-parenthesis equals 0 comma EndLayout">

u
′

k

(
x
)

=

∑

j

=

1

n

b

k
j

(
x
,

λ
)

u
j

(
x
)
,

a
m
p
;

∑

i

=

0

p

∑

j

=

1

N

a

k
j

i

u
j

(

x
i

)

=

0
,

\begin {array}{*{20}{c}} {{{u’}_k}(x)\, = \,\sum \limits _{j\, = \,1}^n {{b_{kj}}(x,\,\lambda ){u_j}(x),} } & {\sum \limits _{i\, = \,0}^p {\sum \limits _{j\, = \,1}^N {a_{kj}^i{u_j}({x_i})\, = \,0,} } } \\ \end {array}

\]
,

k

=

1
,
.
.
.
,

N

k\, = \,1,...,\,N

. In order to get these results we prove that this system can be reduced to a standard system of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column y prime Subscript k Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript j equals 1 Overscript n Endscripts a Subscript k j Baseline left-parenthesis x comma lamda right-parenthesis y Subscript j Baseline left-parenthesis x right-parenthesis comma 2nd Column a m p semicolon y Subscript k Baseline left-parenthesis 0 right-parenthesis equals a Subscript k Baseline left-parenthesis lamda right-parenthesis comma 3rd Column a m p semicolon y Subscript n Baseline left-parenthesis 1 right-parenthesis equals 0 comma EndLayout">

y
′

k

(
x
)

=

∑

j
=
1

n

a

k
j

(
x
,

λ
)

y
j

(
x
)

,

a
m
p
;

y
k

(
0
)

=

a
k

(
λ
)
,

a
m
p
;

y
n

(
1
)

=

0
,

\begin {array}{*{20}{c}} {{{y’}_k}(x)\, = \,\sum \limits _{j = 1}^n {{a_{kj}}(x,\,\lambda )\,{y_j}(x)} ,} & {{y_k}(0)\, = \,{a_k}(\lambda ),} & {{y_n}(1)\, = \,0,} \\ \end {array}

\]

k

=

1
,
.
.
.
,

n

k\, = \,1,...,\,n

.
Parametrizations of 𝐺_{𝛿}-valued multifunctions
Authors: Sarbadhikari H. , Srivastava S. M.
Published: June 2010

Let T, X be Polish spaces,

J

\mathcal {J}

a countably generated sub-

σ
\sigma

-field of

B

T

{\mathcal {B}_T}

, the Borel

σ
\sigma

-field of T, and

F
:

T

→

X

F:\,T\, \to \,X

a multifunction such that

F
(
t
)

F(t)

is a

G
δ

{G_\delta }

in X for each

t

∈

T

t\, \in \,T

. F is

J

\mathcal {J}

-measurable and

Gr

(
F
)

∈

J

⊗

B

X

{\text {Gr}}(F)\, \in \,J\, \otimes \,{\mathcal {B}_X}

, where

Gr

(
F
)

{\text {Gr}}(F)

denotes the graph of F. We prove the following three results on F. (I) There is a map

f
:

T

×

Σ

→

X

f:\,T\, \times \,\Sigma \, \to \,X

such that for each

t

∈

T
,

f
(
t
,

⋅
)

t\, \in \,T,\,f(t,\, \cdot )

is a continuous, open map from

Σ
\Sigma

onto

F
(
t
)

F(t)

and for each

σ

∈

Σ
,

f
(
⋅
,

σ
)

\sigma \, \in \,\Sigma ,\,f( \cdot ,\,\sigma )

is

J

\mathcal {J}

-measurable, where

Σ
\Sigma

is the space of irrationals. (II) The multifunction F is of Souslin type. (III) If X is uncountable and

F
(
t
)
,

t

∈

T

F(t),\,t\, \in \,T

, are all dense-in-itself then there is a

J

⊗

B

X

\mathcal {J}\, \otimes \,{\mathcal {B} _X}

-measurable map

f
:

T

×

X

→

X

f:\,T\, \times \,X\, \to \,X

such that for each

t

∈

T
,

f
(
t
,

⋅
)

t\, \in \,T,\,f(t,\, \cdot )

is a Borel isomorphism of X onto

F
(
t
)

F(t)

.
A nonlinear Volterra equation with rapidly decaying solutions
Authors: Staffans Olof J.
Published: June 2010

We study the asymptotic behavior of the solutions of a nonlinear integrodifferential Volterra equation with a convolution kernel. More specifically, we give conditions which imply that a solution x satisfies

x
(
t
)

=

O
(

t

−
p

)

(
t
→
∞
)

x(t)\, = \,O({t^{ - p}})\,(t \to \infty )

, where p is an arbitrary, positive real number.
Companionship of knots and the Smith conjecture
Authors: Myers Robert
Published: June 2010

This paper studies the Smith Conjecture in terms of H. Schubert’s theory of companionship of knots. Suppose J is a counterexample to the Smith Conjecture, i.e. is the fixed point set of an action of

Z

p

{{\textbf {Z}}_p}

on

S
3

{S^3}

. Theorem. Every essential torus in an invariant knot space

C
(
J
)

C(J)

of J is either invariant or disjoint from its translates. Since the companions of J correspond to the essential tori in

C
(
J
)

C(J)

, this often allows one to split the action among the companions and satellites of J. In particular: Theorem. If J is composite, then each prime factor of J is a counterexample, and conversely. Theorem. The Smith Conjecture is true for all cabled knots. Theorem. The Smith Conjecture is true for all doubled knots. Theorem. The Smith Conjecture is true for all cable braids. Theorem. The Smith Conjecture is true for all nonsimple knots with bridge number less than five. In addition we show: Theorem. If the Smith Conjecture is true for all simple fibered knots, then it is true for all fibered knots. Theorem. The Smith Conjecture is true for all nonfibered knots having a unique isotopy type of incompressible spanning surface.
On the Hardy-Littlewood maximal function and some applications
Authors: Neugebauer C. J.
Published: June 2010

With a monotone family

F

=

{

S
α

}
,

S
α

⊂

R

n

F\, = \,\{ {S_\alpha }\} ,\,{S_\alpha }\, \subset \,{{\textbf {R}}^n}

, we associate the Hardy-Littlewood maximal function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript upper F Baseline f left-parenthesis x right-parenthesis equals sup Underscript alpha Endscripts left-parenthesis 1 slash StartAbsoluteValue upper S Subscript alpha Baseline EndAbsoluteValue right-parenthesis integral Underscript upper S Subscript alpha Baseline plus x Endscripts StartAbsoluteValue f EndAbsoluteValue">

M
F

f
(
x
)

=

sup
α

(
1

/

|

S
α

|

)

∫

S
α

+

x

|
f
|

{M_F}f(x)\, = \,{\sup _\alpha }(1/\left | {{S_\alpha }} \right |)\int _{{S_\alpha }\, + \,x} {\left | f \right |}

. In general,

M
F

{M_F}

is not weak type (1.1). However, if we replace in the denominator

S
α

{S_\alpha }

by

S
F

∗

=

{
x

−

y
:

x
,

y

∈

S
α

}

S_F^ {\ast } \, = \,\{ x\, - \,y:\,x,\,y\, \in \,{S_\alpha }\}

, and denote the resulting maximal function by

M
F

∗

M_F^ {\ast }

, then

M
F

∗

M_F^ {\ast }

is weak type (1, 1) with weak type constant 1.
Generic deformations of varieties
Authors: Wan Yieh Hei
Published: June 2010

Typical families of varieties, which are defined by families of smooth maps, are studied through the method from the singularity theory of differentiable maps. It is proved that generic families of varieties (of certain types) are stable in an appropriate sense.
Subspaces of basically disconnected spaces or quotients of countably complete Boolean algebras
Authors: van Douwen Eric K. , van Mill Jan
Published: June 2010

Under

MA

 + 

2

ω

=

ω
2

{\text {MA}}\,{\text { + }}\,{{\text {2}}^\omega }\, = \,{\omega _2}

there is a (compact) strongly zero-dimensional F-space of weight

2
ω

{2^\omega }

which cannot be embedded in any basically disconnected space. Dually, under

MA

+

2
ω

=

ω
2

{\text {MA}}\, + \,{2^\omega }\, = \,{\omega _2}

there is a weakly countably complete (or almost

σ
\sigma

-complete, or countable separation property) Boolean algebra of cardinality

2
ω

{2^\omega }

which is not a homomorphic image of any countably complete Boolean algebra. The key to our construction is the observation that if X is a subspace of a basically disconnected space and

β
ω

⊆

X

\beta \omega \, \subseteq \,X

then

β
ω

\beta \omega

is a retract of X. Dually, if B is a homomorphic image of a countably complete Boolean algebra, and if h is a homomorphism from B onto

P

(
ω
)

\mathcal {P}(\omega )

, the field of subsets of w, then there is an embedding

e
:

P

(
ω
)
→

B

e:\,\mathcal {P}(\omega ) \to \,B

such that

h

∘

e

=

i

d

P

(
ω
)

h\, \circ \,e\, = \,{\text {i}}{{\text {d}}_{\mathcal {P}(\omega )}}

.
Lifting surgeries to branched covering spaces
Authors: Hilden Hugh M. , Montesinos José María
Published: June 2010

It is proved that if

M
n

{M^n}

is a branched covering of a sphere, branched over a manifold, so is

M
n

×

S
m

{M^n}\, \times \,{S^m}

, but the number of sheets is one more. In particular, the n-dimensional torus is an n-fold simple covering of

S
n

{S^n}

branched over an orientable manifold. The proof involves the development of a new technique to perform equivariant handle addition. Other consequences of this technique are given.
Equivariant dynamical systems
Authors: Field M. J.
Published: June 2010

The basic properties of vector fields and diffeomorphisms invariant under the action of a compact Lie group are presented. A Kupka-Smale density theorem for equivariant dynamical systems and an existence theorem for equivariant Morse-Smale systems on an arbitrary compact G-manifold are proved.
Linear operators on 𝐿_{𝑝} for 0<𝑝<1
Authors: Kalton N. J.
Published: June 2010

If

0

>

p

>

1

0\, > \,p\, > \,1

we classify completely the linear operators

T
:

L
p

→

X

T:\,{L_p}\, \to \,X

where X is a p-convex symmetric quasi-Banach function space. We also show that if

T
:

L
p

→

L
0

T:\,{L_p}\, \to \,{L_0}

is a nonzero linear operator, then for

p

>

q

⩽

2

p\, > \,q\, \leqslant \,2

there is a subspace Z of

L
p

{L_p}

, isomorphic to

L
q

{L_q}

, such that the restriction of T to Z is an isomorphism. On the other hand, we show that if

p

>

q

>

∞

p\, > \,q\, > \,\infty

, the Lorentz space

L
(
p
,

q
)

L(p,\,q)

is a quotient of

L
p

{L_p}

which contains no copy of

l
p

{l_p}

.
An application of homological algebra to the homotopy classification of two-dimensional CW-complexes
Authors: Dyer Micheal N.
Published: June 2010

Let

π
\pi

be

Z
m

×

Z
n

{Z_m}\, \times \,{Z_n}

. In this paper the homotopy types of finite connected two dimensional CW-complexes with fundamental group

π
\pi

are shown to depend only on the Euler characteristic. The basic method is to study the structure of the group

Ext

Z
π

1

(
I

π
2

,

Z
)

{\text {Ext}}_{Z\pi }^1(I{\pi ^2},\,Z)

as a principal

End(

I

π
2

)

{\text {End(}}I{\pi ^2}{\text {)}}

-module.
Homological algebra on a complete intersection, with an application to group representations
Authors: Eisenbud David
Published: June 2010

Let R be a regular local ring, and let

A

=

R

/

(
x
)

A\, = \,R/(x)

, where x is any nonunit of R. We prove that every minimal free resolution of a finitely generated A-module becomes periodic of period 1 or 2 after at most

dim

A

\operatorname {dim} \, A

steps, and we examine generalizations and extensions of this for complete intersections. Our theorems follow from the properties of certain universally defined endomorphisms of complexes over such rings.
Nonstandard extensions of transformations between Banach spaces
Authors: Tacon D. G.
Published: June 2010

Let

X
X

and

Y
Y

be (infinite-dimensional) Banach spaces and denote their nonstandard hulls with respect to an

ℵ
1

\aleph _1

-saturated enlargement by

X
^

\hat X

and

Y
^

\hat Y

respectively. If

B

(
X
,
Y
)

\mathcal {B}(X,Y)

denotes the space of bounded linear transformations then a subset

S
S

of elements of

B

(
X
,
Y
)

\mathcal {B}(X,Y)

extends naturally to a subset

S
^

\hat S

of

B

(

X
^

,

Y
^

)

\mathcal {B}(\hat {X}, \hat {Y})

. This paper studies the behaviour of various kinds of transformations under this extension and introduces, in this context, the concepts of super weakly compact, super strictly singular and socially compact operators. It shows that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove script upper B left-parenthesis upper X comma upper Y right-parenthesis With caret subset-of-above-not-equals script upper B left-parenthesis ModifyingAbove upper X With caret comma ModifyingAbove upper Y With caret right-parenthesis">

B

(
X
,
Y
)

^

⫋

B

(

X
^

,

Y
^

)

\widehat {\mathcal {B}(X,Y)} \subsetneqq \mathcal {B}(\hat {X}, \hat {Y})

provided

X
X

and

Y
Y

are infinite dimensional and contrasts this with the inclusion

K

(

H
^

)
⫋

K

(
H
)

^

\mathcal {K}(\hat H) \subsetneqq \widehat {\mathcal {K}(H)}

where

K

(
H
)

\mathcal {K}(H)

denotes the space of compact operators on a Hilbert space.
Distinguished subfields
Authors: Deveney James K. , Mordeson John N.
Published: June 2010

Let L be a finitely generated nonalgebraic extension of a field K of characteristic

p

≠

0

p\, \ne \,0

. A maximal separable extension D of K in L is distinguished if

L

⊆

K

p

−

n

(
D
)

L\, \subseteq \,{K^{{p^{ - \,n}}}}(D)

for some n. Let d be the transcendence degree of L over K. If every maximal separable extension of K in L is distinguished, then every set of d relatively p-independent elements is a separating transcendence basis for a distinguished subfield. Conversely, if

K
(

L
p

)

K({L^p})

is separable over K, this condition is also sufficient. A number of properties of such fields are determined and examples are presented illustrating the results.
A representation theorem and applications to topological groups
Authors: Belley J.-M.
Published: June 2010

We show that, given a set S dense in a compact Hausdorff space X and a complex-valued bounded linear functional

Λ
\Lambda

on the space

C
(
X
)

C(X)

of continuous complex-valued functions on X with uniform norm, there exist an algebra

A

{\mathcal {A}}

of sets in S and a unique bounded finitely additive set function

μ
:

A

→

C

\mu :\,{\mathcal {A}}\, \to \,{\textbf {C}}

which is inner and outer regular with respect to the zero and cozero sets respectively and such that

∫
s

f

|
S

d
μ

\int _s {f\left | S \right .\,d\mu }

exists and is equal to

Λ
(
f
)

\Lambda (f)

for all

f

∈

C
(
X
)

f\, \in \,C(X)

. In the context of topological groups, this theorem permits us to obtain (1) a concrete representation theorem for bounded complex-valued linear functionals on the space of almost periodic functions with uniform norm, (2) a representation theorem for (not necessarily continuous) positive definite functions, (3) a characterization of the space B of finite linear combinations of positive definite functions, and (4) a necessary and sufficient condition to have a linear transformation from B to B.
The embeddings of the discrete series in the principal series for semisimple Lie groups of real rank one
Authors: Baldoni Silva M. Welleda
Published: June 2010

We consider the problem of finding all the “embeddings” of a discrete series representation in the principal series in the case of a simple real Lie group G of real rank one. More precisely, we solve the problem when G is
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p i n left-parenthesis 2 n comma 1 right-parenthesis comma SU left-parenthesis n comma 1 right-parenthesis comma SP left-parenthesis n comma 1 right-parenthesis or upper F 4 left-parenthesis n greater-than-or-slanted-equals 2 right-parenthesis">

Spin
⁡
(
2
n
,

1
)
,

SU

(
n
,

1
)
,

SP

(
n
,

1
)

or

F
4

(
n

⩾

2
)

\operatorname {Spin} (2n,\,1),{\text {SU}}(n,\,1),\,{\text {SP}}(n,\,1)\,{\text {or}}\,{F_4}\,(n\, \geqslant \,2)

. The problem is reduced to considering only discrete series representations with trivial infinitesimal character, by means of tensoring with finite dimensional representations. Various other techniques are employed.
Octonion planes over local rings
Authors: Bix Robert
Published: June 2010

Let

D

\mathcal {D}

be an octonion algebra which is a free module over a local ring R and let

J
=
H
(

D

3

,
γ
)

J = H({\mathcal {D}_3},\gamma )

be the quadratic Jordan algebra of Hermitian 3-by-3 matrices over R. We define the octonion plane determined by J and prove that every collineation is induced by a norm semisimilarity of J. We classify the subgroups of the collineation group normalized by the little projective group.
Composition factors of the principal series representations of the group 𝑆𝑝(𝑛,1)
Authors: Baldoni Silva M. W. , Kraljević H.
Published: June 2010

Using Vogan’s algorithm the composition factors of any principal series representation of the group

S
p
(
n
,

1
)

Sp(n,\,1)

are determined.
Regularity of certain small subharmonic functions
Authors: Fenton P. C.
Published: June 2010

Suppose that u is subharmonic in the plane and that

lim
_

r
→
∞

B
(
r
)

/

(
log

r

)

2

=

σ

>

∞

{\underline {\lim } _{r \to \infty }}B(r)/{(\log \,r)^{2\,}}\, = \,\sigma \, > \,\infty

. It is known that, given

ε
>
0

\varepsilon > 0

, there are arbitrarily large values of r such that

A
(
r
)

>

B
(
r
)

−
(
σ

+

ε
)

π
2

A(r)\, > \,B(r)\, - (\sigma \, + \,\varepsilon ){\pi ^2}

. The following result is proved. Let u be subharmonic and let

σ
\sigma

be any positive number. Then either

A
(
r
)

>

B
(
r
)

−

π
2

σ

A(r)\, > \,B(r)\, - {\pi ^2}\sigma

for certain arbitrarily large values of r or, if this is false, then
\[

lim

r
→
∞

(

B

(
r
)

−

σ

(

log

r

)

2

)

/

log

r

\lim \limits _{r \to \infty } \left ( {B\left ( r \right )\, - \,\sigma {{\left ( {\log \,r} \right )}^2}} \right )/\log \,r

\]
exists and is either

+

∞

+ \,\infty

or finite.
On the singularities of Gegenbauer (ultraspherical) expansions
Authors: Zayed Ahmed I.
Published: June 2010

The results of Gilbert on the location of the singular points of an analytic function

f
(
z
)

f(z)

given by Gegenbauer (ultraspherical) series expansion

f
(
z
)

=

Σ

n

=

0

∞

a
n

C
n
μ

(
z
)

f(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}\,C_n^\mu (z)

are extended to the case where the series converges to a distribution. On the other hand, this generalizes Walter’s results on distributions given by Legendre series:

f
(
z
)

=

Σ

n

=

0

∞

a
n

C
n

1

/

2

(
z
)

f(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}\,C_n^{1/2}(z)

. The singularities of the analytic representation of

f
(
z
)

f(z)

are compared to those of the associated power series

g
(
z
)

=

Σ

n

=

0

∞

a
n

z
n

g(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}{z^n}

. The notion of value of a distribution at a point is used to study the boundary behavior of the associated power series. A sufficient condition for Abel summability of Gegenbauer series is also obtained in terms of the distribution to which the series converges.
On locally and globally conformal Kähler manifolds
Authors: Vaisman Izu
Published: June 2010

Some relations between the locally conformal Kähler (l.c.K.) and the globally conformal Kähler (g.c.K.) properties are established. Compact l.c.K. manifolds which are not g.c.K. do not have Kähler metrics. l.c.K. manifolds which are not g.c.K. are analytically irreducible. Various curvature restrictions on l.c.K. manifolds imply the g.c.K. property. Total spaces of induced Hopf fibrations are l.c.K. and not g.c.K. manifolds. Conjecture. A compact l.c.K. manifold which is not g.c.K. has at least one odd odd-dimensional Betti number.