bluuebunny/crossref_metadata_2025 huggingface dataset:
On the Picard group of a continuous trace 𝐶*-algebra
Authors: Raeburn Iain
Published: June 2010

Let

A
A

be a continuous trace

C
∗

{C^\ast }

-algebra with paracompact spectrum

T
T

, and let

C
(
T
)

C(T)

be the algebra of bounded continuous functions on

T
T

, so that

C
(
T
)

C(T)

acts on

A
A

in a natural way. An

A
−
A

A - A

bimodule

X
X

is an

A

−

C
(
T
)

A

A{ - _{C(T)}}A

imprimitivity bimodule if it is an

A
−
A

A - A

imprimitivity bimodule in the sense of Rieffel and the induced actions of

C
(
T
)

C(T)

on the left and right of

X
X

agree. We denote by

Pi

c

C
(
T
)

A

{\text {Pi}}{{\text {c}}_{C(T)}}A

the group of isomorphism classes of

A

−

C
(
T
)

A

A{ - _{C(T)}}A

imprimitivity bimodules under

⊗
A

{ \otimes _A}

. Our main theorem asserts that

Pi

c

C
(
T
)

A
≅

Pi

c

C
(
T
)

C
0

(
T
)

{\text {Pi}}{{\text {c}}_{C(T)}}A \cong {\text {Pi}}{{\text {c}}_{C(T)}}{C_0}(T)

. This result is well known to algebraists if

A
A

is an

n
n

-homogeneous

C
∗

{C^\ast }

-algebra with identity, and if

A
A

is separable it can be deduced from two recent descriptions of the automorphism group

Au

t

C
(
T
)

A

{\text {Au}}{{\text {t}}_{C(T)}}A

due to Brown, Green and Rieffel on the one hand and Phillips and Raeburn on the other. Our main motivation was to provide a direct link between these two characterisations of

Au

t

C
(
T
)

A

{\text {Au}}{{\text {t}}_{C(T)}}A

.
Monotone decompositions of 𝜃_{𝑛}-continua
Authors: Grace E. E. , Vought Eldon J.
Published: June 2010

We prove the following theorem for a compact, metric

θ
n

{\theta _n}

-continuum (i.e., a compact, connected, metric space that is not separated into more than

n
n

components by any subcontinuum). The continuum

X
X

admits a monotone, upper semicontinuous decomposition

D

\mathfrak {D}

such that the elements of

D

\mathfrak {D}

have void interiors and the quotient space

X

/

D

X/\mathfrak {D}

is a finite graph, if and only if, for each nowhere dense subcontinuum

H
H

of

X
X

, the continuum

T
(
H
)
=
{
x

|

T(H) = \{ x|

if

K
K

is a subcontinuum of

X
X

and

x
∈

K
∘

x \in {K^ \circ }

, then

K
∩
H
≠
∅
}

K \cap H \ne \emptyset \}

is nowhere dense. The elements of the decomposition are characterized in terms of the set function

T
T

. An example is given showing that the condition that requires

T
(
x
)

T(x)

to have void interior for all

x
∈
X

x \in X

is not strong enough to guarantee the decomposition.
Basic sequences and subspaces in Lorentz sequence spaces without local convexity
Authors: Popa Nicolae
Published: June 2010

After some preliminary results

(
§
1
)

(\S 1)

, we give in

§
2

\S 2

another proof of the result of N. J. Kalton [5] concerning the unicity of the unconditional bases of

l
p

{l_p}

,

0
>
p
>
1

0 > p > 1

. Using this result we prove in §3 the unicity of certain bounded symmetric block bases of the subspaces of the Lorentz sequence spaces

d
(
w
,
p
)

d(w,p)

,

0
>
p
>
1

0 > p > 1

. In

§
4

\S 4

we show that every infinite dimensional subspace of

d
(
w
,
p
)

d(w,p)

contains a subspace linearly homeomorphic to

l
p

{l_p}

,

0
>
p
>
1

0 > p > 1

. Unlike the case

p
⩾
1

p \geqslant 1

there are subspaces of

d
(
w
,
p
)

d(w,p)

,

0
>
p
>
1

0 > p > 1

, which contain no complemented subspaces of

d
(
w
,
p
)

d(w,p)

linearly homeomorphic to

l
p

{l_p}

. In fact there are spaces

d
(
w
,
p
)

d(w,p)

,

0
>
p
>
1

0 > p > 1

, which contain no complemented subspaces linearly homeomorphic to

l
p

{l_p}

. We conjecture that this is true for every

d
(
w
,
p
)

d(w,p)

,

0
>
p
>
1

0 > p > 1

. The answer to the previous question seems to be important: for example we can prove that a positive complemented sublattice

E
E

of

d
(
w
,
p
)

d(w,p)

,

0
>
p
>
1

0 > p > 1

, with a symmetric basis is linearly homeomorphic either to

l
p

{l_p}

or to

d
(
w
,
p
)

d(w,p)

; consequently, a positive answer to this question implies that

E
E

is linearly homeomorphic to

d
(
w
,
p
)

d(w,p)

. In

§
5

\S 5

we are able to characterise the sublattices of

d
(
w
,
p
)

d(w,p)

,

p
=

k

−
1

p = {k^{ - 1}}

(however under a supplementary restriction concerning the sequence

(

w
n

)

n
=
1

∞

)

({w_n})_{n = 1}^\infty )

, which are positive and contractive complemented, as being the order ideals of

d
(
w
,
p
)

d(w,p)

. Finally, in

§
6

\S 6

, we characterise the Mackey completion of

d
(
w
,
p
)

d(w,p)

also in the case

p
=

k

−
1

p = {k^{ - 1}}

,

k
∈

N

k \in {\mathbf {N}}

.
Boundary crossing probabilities for stationary Gaussian processes and Brownian motion
Authors: Cuzick Jack
Published: June 2010

Let

X
(
t
)

X(t)

be a stationary Gaussian process,

f
(
t
)

f(t)

a continuous function, and

T
T

a finite or infinite interval. This paper develops asymptotic estimates for

P
(
X
(
t
)
⩾
f
(
t
)

P(X(t) \geqslant f(t)

, some

t
∈
T

t \in T

when this probability is small. After transformation to an Ornstein Uhlenbeck process the results are also applicable to Brownian motion. In that special case, if

W
(
t
)

W(t)

is Brownian motion,

f
f

is continuously differentiable, and

T
=
[
0
,
T
]

T = [0,T]

our estimate for

P
(
W
(
t
)
⩾
f
(
t
)

P(W(t) \geqslant f(t)

, some

t
∈
T
)

t \in T)

is
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda equals integral Subscript 0 Superscript upper T Baseline left-parenthesis 2 t right-parenthesis Superscript negative 1 Baseline left-parenthesis f left-parenthesis t right-parenthesis slash t Superscript 1 slash 2 Baseline right-parenthesis phi left-parenthesis f left-parenthesis t right-parenthesis slash t Superscript 1 slash 2 Baseline right-parenthesis d t plus upper I Subscript StartSet left-parenthesis f left-parenthesis t right-parenthesis slash t Sub Superscript 1 slash 2 Subscript right-parenthesis prime vertical-bar Sub Subscript t equals upper T Subscript greater-than 0 EndSet Baseline normal upper Phi Superscript asterisk Baseline left-parenthesis f left-parenthesis upper T right-parenthesis slash upper T Superscript 1 slash 2 Baseline right-parenthesis">

Λ
=

∫
0
T

(
2
t
)

−
1

(
f
(
t
)

/

t

1

/

2

)
ϕ
(
f
(
t
)

/

t

1

/

2

)

d
t
+

I

{
(
f
(
t
)

/

t

1

/

2

)
′

|

t
=
T

>
0
}

Φ
∗

(
f
(
T
)

/

T

1

/

2

)

\Lambda = \int _0^T {{{(2t)}^{ - 1}}(f(t)/{t^{1/2}})\phi (f(t)/{t^{1/2}})} dt + {I_{\{ (f(t)/{t^{1/2}})’{|_{t = T}} > 0\} }}{\Phi ^ \ast }(f(T)/{T^{1/2}})

\]
provided

Λ
\Lambda

is small. Here

ϕ
\phi

is the standard normal density and

Φ
∗

{\Phi ^ \ast }

is its upper tail distribution. Our approach is to find an approximate first passage density and then compute crossing probabilities as a one-dimensional integral. In the case of boundaries without cusps, our results unify and extend separate results for crossings of constant levels developed by Pickands, and Qualls-Watanabe, and crossings of rapidly increasing barriers studied by Berman. Applications are also briefly explored.
The genus of a map
Authors: Hurvitz Sara
Published: June 2010

The elements

[

f
′

]
(

f
′

:

X
′

→

Y
′

)

[f’](f’:X’ \to Y’)

of the genus

−
G
(
f
)

- G(f)

of a map

f
:
X
→
Y

f:X \to Y

are equivalence classes of homotopy classes of maps

f
′

f’

which satisfy: For every prime

p
p

there exist homotopy equivalences

h
p

:

X
p
′

→

X
p

{h_p}:{X’_p} \to {X_p}

and

k
p

:

Y
p
′

→

Y
p

{k_p}:{Y’_p} \to {Y_p}

so that

f
p

h
p

∼

k
p

f
p
′

{f_p}{h_p} \sim {k_p}{f’_p}

. The genus of

f
f

under

X
−

G
X

(
f
)

X - {G^X}(f)

and the genus of

f
f

over

Y
−

G
Y

(
f
)

Y - {G_Y}(f)

are defined similarly. In this paper we prove that under certain conditions on

f
f

, the sets

G
(
f
)

G(f)

,

G
X

(
f
)

{G^X}(f)

and

G
Y

(
f
)

{G_Y}(f)

are finite and admit an abelian group structure. We also compare the genus of

f
f

to those of

X
X

and

Y
Y

, calculate it for some principal fibrations of the form

K
(
G
,
n
−
1
)
→
X
→
Y

K(G,n - 1) \to X \to Y

, and deal with the noncancellation phenomenon.
Schur products of operators and the essential numerical range
Authors: Stout Quentin F.
Published: June 2010

Let

E

=
{

e
n

}

n
=
1

∞

\mathcal {E} = \{ {e_n}\} _{n = 1}^\infty

be an orthonormal basis for a Hilbert space

H

\mathcal {H}

. For operators

A
A

and

B
B

having matrices

(

a

i
j

)

i
,

j
=
1

∞

({a_{ij}})_{i,\;j = 1}^\infty

and

(

b

i
j

)

i
,

j

∞

=
1

({b_{ij}})_{i,\;j}^\infty = 1

, their Schur product is defined to be

(

a

i
j

b

i
j

)

i
,

j

∞

=
1

({a_{ij}}{b_{ij}})_{i,\:j}^\infty = 1

. This gives

B

(

H

)

\mathcal {B}(\mathcal {H})

a new Banach algebra structure, denoted

P

E

{\mathcal {P}_\mathcal {E}}

. For any operator

T
T

it is shown that

T
T

is in the kernel (hull(compact operators)) in some

B

E

{\mathcal {B}_\mathcal {E}}

iff

0
0

is in the essential numerical range of

T
T

. These conditions are also equivalent to the property that there is a basis such that Schur multiplication by

T
T

is a compact operator mapping Schatten classes into smaller Schatten classes. Thus we provide new results linking

B

(

H

)

\mathcal {B}(\mathcal {H})

,

B

E

{\mathcal {B}_\mathcal {E}}

and

B

(

B

(

H

)
)

\mathcal {B}(\mathcal {B}(\mathcal {H}))

.
Characterizations of the Fischer groups. I, II, III
Authors: Parrott David
Published: June 2010

B. Fischer, in his work on finite groups which contain a conjugacy class of

3
3

-transpositions, discovered three new sporadic finite simple groups, usually denoted

M
(
22
)

M(22)

,

M
(
23
)

M(23)

and

M
(
24

)
′

M(24)’

. In Part I two of these groups,

M
(
22
)

M(22)

and

M
(
23
)

M(23)

, are characterized by the structure of the centralizer of a central involution. In addition, the simple groups

U
6

(
2
)

{U_6}(2)

(often denoted by

M
(
21
)
)

M(21))

and

P
Ω
(
7
,
3
)

P\Omega (7,3)

, both of which are closely connected with Fischer’s groups, are characterized by the same method. The largest of the three Fischer groups

M
(
24
)

M(24)

is not simple but contains a simple subgroup

M
(
24

)
′

M(24)’

of index two. In Part II we give a similar characterization by the centralizer of a central involution of

M
(
24
)

M(24)

and also a partial characterization of the simple group

M
(
24

)
′

M(24)’

. The purpose of Part III is to complete the characterization of

M
(
24

)
′

M(24)’

by showing that our abstract group

G
G

is isomorphic to

M
(
24

)
′

M(24)’

. We first prove that

G
G

contains a subgroup

X
≅
M
(
23
)

X \cong M(23)

and then we construct a graph (on the cosets of

X
X

) which is shown to be isomorphic to the graph for

M
(
24
)

M(24)

.
Tensegrity frameworks
Authors: Roth B. , Whiteley W.
Published: June 2010

A tensegrity framework consists of bars which preserve the distance between certain pairs of vertices, cables which provide an upper bound for the distance between some other pairs of vertices and struts which give a lower bound for the distance between still other pairs of vertices. The present paper establishes some basic results concerning the rigidity, flexibility, infinitesimal rigidity and infinitesimal flexibility of tensegrity frameworks. These results are then applied to a number of questions, problems and conjectures regarding tensegrity frameworks in the plane and in space.
An elementary proof of the local Kronecker-Weber theorem
Authors: Rosen Michael
Published: June 2010

Let

K
K

be a local field. Lubin and Tate have shown how to explicitly construct an abelian extension of

K
K

which they prove to be the maximal abelian extension. Their proof of this result uses local class field theory. When

K
K

is a

p
p

-adic field we give an elementary proof which even avoids the use of higher ramification groups. Instead we rely on facts about the principal units in a finite abelian extension of

K
K

as a module for the Galois group.
Absolute continuity and uniqueness of measures on metric spaces
Authors: Mattila Pertti
Published: June 2010

We show that if two Borel regular measures on a separable metric space are in a suitable sense homogeneous, then they are mutually absolutely continuous. We use such absolute continuity theorems together with some density theorems to prove the uniqueness of measures more general than Haar measures.
A counterexample to the bounded orbit conjecture
Authors: Boyles Stephanie M.
Published: June 2010

A long outstanding problem in the topology of Euclidean spaces is the Bounded Orbit Conjecture, which states that every homeomorphism of

E
2

{E^2}

onto itself, with the property that the orbit of every point is bounded, must have a fixed point. It is well known that the conjecture is true for orientation preserving homeomorphisms. We provide a counterexample to the conjecture by constructing a fixed point free orientation reversing homeomorphism which satisfies the hypothesis of the conjecture.
Symmetry properties of the zero sets of nil-theta functions
Authors: Goodman Sharon
Published: June 2010

Let

N
N

denote the three dimensional Heisenberg group, and let

Γ
\Gamma

be a discrete two-generator subgroup of

N
N

such that

N

/

Γ

N/\Gamma

is compact. Then we may decompose

L
2

(
N

/

Γ
)

{L^2}(N/\Gamma )

into primary summands with respect to the right regular representation

R
R

of

N
N

on

L
2

(
N

/

Γ
)

{L^2}(N/\Gamma )

as follows:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis upper N slash normal upper Gamma right-parenthesis equals circled-plus sigma-summation Underscript m element-of bold upper Z Endscripts upper H Subscript m Baseline left-parenthesis normal upper Gamma right-parenthesis">

L
2

(
N

/

Γ
)
=
⊕

∑

m
∈

Z

H
m

(
Γ
)

{L^2}(N/\Gamma ) = \oplus \sum \nolimits _{m \in {\mathbf {Z}}} {{H_m}(\Gamma )}

. It can be shown that for

m
≠
0
,

H
m

(
Γ
)

m \ne 0,{H_m}(\Gamma )

is a multiplicity space for the representation

R
R

of multiplicity

|
m
|

\left | m \right |

. The distinguished subspace theory of

L

{\text {L}}

. Auslander and

J

{\text {J}}

. Brezin singles out a finite number of the decompositions of

H
m

(
Γ
)
,
m
≠
0

{H_m}(\Gamma ),m \ne 0

, which are in some ways nicer than the others. They define algebraically an integer valued function, called the index, on the set

Ω
m

{\Omega _m}

of irreducible closed

R
R

-invariant subspaces of

H
m

(
Γ
)

{H_m}(\Gamma )

such that the distinguished subspaces have index one. In this paper, we give an analytic-geometric interpretation of the index. Every space in

Ω
m

{\Omega _m}

contains a unique (up to constant multiple) special function, called a nil-theta function, that arises as a solution of a certain differential operator on

N

/

Γ

N/\Gamma

. These nil-theta functions have been shown to be closely related to the classical theta functions. Since the classical theta functions are determined (up to constant multiple) by their zero sets, it is natural to attempt to classify the spaces in

Ω
m

{\Omega _m}

using various properties of the zero sets of the nil-theta functions lying in these spaces. We define the index of a nil-theta function in

H
m

(
Γ
)

{H_m}(\Gamma )

using the symmetry properties of its zero set. Our main theorem asserts that the algebraic index of a space in

Ω
m

{\Omega _m}

equals the index of the unique nil-theta function lying in that space. We have thus an analytic-geometric characterization of the index. We then use these results to give a complete description of the zero sets of those nil-theta functions of a fixed index. We also investigate the behavior of the index under the multiplication of nil-theta functions; i.e. we discuss how the index of the nil-theta function

F
G

FG

relates to the indices of the nil-theta functions

F
F

and

G
G

.
Weak and pointwise compactness in the space of bounded continuous functions
Authors: Wheeler Robert F.
Published: June 2010

Let

T
T

be a completely regular Hausdorff space,

C
b

(
T
)

{C_b}(T)

the space of bounded continuous real-valued functions on

T
T

,

M
(
T
)

M(T)

the Banach space dual of

C
b

(
T
)

{C_b}(T)

. Let

H

\mathcal {H}

denote the family of subsets of

C
b

(
T
)

{C_b}(T)

which are uniformly bounded and relatively compact for the topology

J

p

{\mathfrak {J}_p}

of pointwise convergence. The basic question considered here is: what is the largest subspace

Z
Z

of

M
(
T
)

M(T)

such that every member of

H

\mathcal {H}

is relatively

σ
(

C
b

,
Z
)

\sigma ({C_b},Z)

-compact? Classical results of Grothendieck and Ptak show that

Z
=
M
(
T
)

Z = M(T)

if

T
T

is pseudocompact. In general,

M
t

⊂
Z
⊂

M
s

;

{M_t} \subset Z \subset {M_s};

assuming Martin’s Axiom, a deep result of Talagrand improves the lower bound to

M
τ

{M_\tau }

. It is frequently, but not always, true that

Z
=

M
s

;

Z = {M_s};

counterexamples are given which use Banach spaces in their weak topologies to construct the underlying

T
T

’s.
On the genus of symmetric groups
Authors: Proulx Viera Krňanová
Published: June 2010

A new method for determining genus of a group is described. It involves first getting a bound on the sizes of the generating set for which the corresponding Cayley graph could have smaller genus. The allowable generating sets are then examined by methods of computing average face sizes and by voltage graph techniques to find the best embeddings. This method is used to show that genus of the symmetric group

S
5

{S_5}

is equal to four. The voltage graph method is used to exhibit two new embeddings for symmetric groups on even number of elements. These embeddings give us a better upper bound than that previously given by A. T. White.
Algebraic extensions of power series rings
Authors: Arnold Jimmy T.
Published: June 2010

Let

D
D

and

J
J

be integral domains such that

D
⊂
J

D \subset J

and

J
[
[
X
]
]

J[[X]]

is not algebraic over

D
[
[
X
]
]

D[[X]]

. Is it necessarily the case that there exists an integral domain

R
R

such that

D
[
[
X
]
]
⊂
R
⊆
J
[
[
X
]
]

D[[X]] \subset R \subseteq J[[X]]

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R approximately-equals upper D left-bracket left-bracket upper X right-bracket right-bracket left-bracket left-bracket left-brace upper Y Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity Baseline right-bracket right-bracket">

R
≅
D
[
[
X
]
]
[
[
{

Y
i

}

i
=
1

∞

]
]

R \cong D[[X]][[\{ {Y_i}\} _{i = 1}^\infty ]]

? While the general question remains open, the question is answered affirmatively in a number of cases. For example, if

D
D

satisfies any one of the conditions (1)

D
D

is Noetherian, (2)

D
D

is integrally closed, (3) the quotient field

K
K

of

D
D

is countably generated as a ring over

D
D

, or (4)

D
D

has Krull dimension one, then an affirmative answer is given. Further, in the Noetherian case it is shown that

J
[
[
X
]
]

J[[X]]

is algebraic over

D
[
[
X
]
]

D[[X]]

if and only if it is integral over

D
[
[
X
]
]

D[[X]]

and necessary and sufficient conditions are given on

D
D

and

J
J

in order that this occur. Finally if, for every positive integer

n
n

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D left-bracket left-bracket upper X 1 comma ellipsis comma upper X Subscript n Baseline right-bracket right-bracket subset-of upper R subset-of-or-equal-to upper J left-bracket left-bracket upper X 1 comma ellipsis comma upper X Subscript n Baseline right-bracket right-bracket">

D
[
[

X
1

,
…
,

X
n

]
]
⊂
R
⊆
J
[
[

X
1

,
…
,

X
n

]
]

D[[{X_1}, \ldots ,{X_n}]] \subset R \subseteq J[[{X_1}, \ldots ,{X_n}]]

implies that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R approximately-but-not-actually-equals upper D left-bracket left-bracket upper X 1 comma ellipsis comma upper X Subscript n Baseline right-bracket right-bracket left-bracket left-bracket left-brace upper Y Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity Baseline right-bracket right-bracket">

R
≆
D
[
[

X
1

,
…
,

X
n

]
]
[
[
{

Y
i

}

i
=
1

∞

]
]

R \ncong D[[{X_1}, \ldots ,{X_n}]][[\{ {Y_i}\} _{i = 1}^\infty ]]

, then it is shown that

J
[
[

X
1

,
…
,

X
n

]
]

J[[{X_1}, \ldots ,{X_n}]]

is algebraic over

D
[
[

X
1

,
…
,

X
n

]
]

D[[{X_1}, \ldots ,{X_n}]]

for every

n
n

.
Derivatives and Lebesgue points via homeomorphic changes of scale
Authors: Hancock Don L.
Published: June 2010

Let

I
I

be a closed interval, and suppose

K

\mathcal {K}

,

H

\mathcal {H}

, and

Λ
\Lambda

denote, respectively, the class of homeomorphisms of

I
I

onto itself, the class of homeomorphisms of the line onto itself, and the class of real functions on

I
I

for which each point is a Lebesgue point. Maximoff proved that

Λ
∘

K

\Lambda \circ \mathcal {K}

is exactly the class of Darboux Baire

1
1

functions, where

Λ
∘

K

=
{
f
∘
k
:
f
∈
Λ
,
k
∈

K

}

\Lambda \circ \mathcal {K} = \{ f \circ k:f \in \Lambda ,k \in \mathcal {K}\}

. The present paper is devoted primarily to a study of

H

∘
Λ
=
{
h
∘
f
:
f
∈
Λ
,
h
∈

H

}

\mathcal {H} \circ \Lambda = \{ h \circ f:f \in \Lambda ,h \in \mathcal {H}\}

. The characterizations of this class which are obtained show that a function is a member of

H

∘
Λ

\mathcal {H} \circ \Lambda

if and only if, in addition to the obvious requirement of approximate continuity, it satisfies certain growth and density-like conditions. In particular, any approximately continuous function with countably many non-Lebesgue points belongs to

H

∘
Λ

\mathcal {H} \circ \Lambda

. It is also established that

H

∘
Λ

\mathcal {H} \circ \Lambda

is a uniformly closed algebra properly containing the smallest algebra generated from

Λ
\Lambda

, and a characterization of the latter algebra is provided.
On analytic diameters and analytic centers of compact sets
Authors: Kobayashi Shōji , Suita Nobuyuki
Published: June 2010

In this paper several results on analytic diameters and analytic centers are obtained. We show that the extremal function for analytic diameter is unique and that there exist compact sets with many analytic centers. We answer negatively several problems posed by F. Miinsker.
Invariant connections and Yang-Mills solutions
Authors: Itoh Mitsuhiro
Published: June 2010

A condition on the self-duality and the stability of Yang-Mills solutions are discussed. The canonical invariant

G
G

-connections on

S
4

{S^4}

and

P
2

(

C

)

{P_2}({\mathbf {C}})

are considered as Yang-Mills solutions. The non-self-duality of the connections requires the injectivity of the isotropy homomorphisms. We construct examples of non-self-dual connections on

G
G

-vector bundles (

G
G

is a compact simple group). Under a certain property of the isotropy homomorphism, these canonical connections are not weakly stable.
The Hilbert transform and maximal function for approximately homogeneous curves
Authors: Weinberg David A.
Published: June 2010

Let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H Subscript gamma Baseline f left-parenthesis x right-parenthesis equals p period v period integral Subscript negative 1 Superscript 1 Baseline f left-parenthesis x minus gamma left-parenthesis t right-parenthesis right-parenthesis d t slash t">Hγf(x)=p.v.∫−11f(x−γ(t))dt/t{\mathcal {H}_\gamma }f(x) = {\text {p}}{\text {.v}}{\text {.}}\int _{ - 1}^1 {f(x - \gamma (t))dt/t}and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper M Subscript gamma Baseline f left-parenthesis x right-parenthesis equals sup Underscript 1 greater-than-or-slanted-equals h greater-than 0 Endscripts h Superscript negative 1 Baseline integral Subscript 0 Superscript h Baseline StartAbsoluteValue f left-parenthesis x minus gamma left-parenthesis t right-parenthesis right-parenthesis EndAbsoluteValue d t">Mγf(x)=sup1⩾h>0h−1∫0h|f(x−γ(t))|dt{\mathfrak {M}_\gamma }f(x) = {\sup _{1 \geqslant h > 0}}{h^{ - 1}}\int _0^h {|f(x - \gamma (t))|dt}. It is proved that forf∈S(Rn)f \in \mathcal {S}({{\mathbf {R}}^n}), the Schwartz class, and for an approximately homogeneous curveγ(t)∈Rn\gamma (t) \in {{\mathbf {R}}^n},‖Hγf‖2⩽C‖f‖2{\left \| {{\mathcal {H}_\gamma }f} \right \|_2} \leqslant C{\left \| f \right \|_2},‖Mγf‖2⩽C‖f‖2{\left \| {{\mathfrak {M}_\gamma }f} \right \|_2} \leqslant C{\left \| f \right \|_2}. A homogeneous curve is one which satisfies a differential equationγ1′(t)=(A/t)γ1(t){\gamma ’_1}(t) = (A/t){\gamma _1}(t),0>t>∞0 > t > \infty, whereAAis a nonsingular matrix all of whose eigenvalues have positive real part. An approximately homogeneous curveγ(t)\gamma (t)has the formγ1(t)+γ2(t){\gamma _1}(t) + {\gamma _2}(t), whereγ2(t){\gamma _2}(t)is a carefully specified "error", such thatγ2(j)\gamma _2^{(j)}is also restricted forj=2,…,n+1j = 2, \ldots ,n + 1. The approximately homogeneous curves generalize the curves of standard type treated by Stein and Wainger.
Prime knots and tangles
Authors: Lickorish W. B. Raymond
Published: June 2010

A study is made of a method of proving that a classical knot or link is prime. The method consists of identifying together the boundaries of two prime tangles. Examples and ways of constructing prime tangles are explored.
Morse theory by perturbation methods with applications to harmonic maps
Authors: Uhlenbeck K.
Published: June 2010

There are many interesting variational problems for which the Palais-Smale condition cannot be verified. In cases where the Palais-Smale condition can be verified for an approximating integral, and the critical points converge, a Morse theory is valid. This theory applies to a class of variational problems consisting of the energy integral for harmonic maps with a lower order potential.
Fundamental solutions for differential equations associated with the number operator
Authors: Lee Yuh Jia
Published: June 2010

Let

(
H
,

B
)

(H,\,B)

be an abstract Wiener space. If

u
u

is a twice

H
H

-differentiable function on

B
B

such that

D
u
(
x
)
∈

B

∗

Du(x) \in {B^{\ast }}

and

D
2

u
(
x
)

{D^2}u(x)

is of trace class, then we define

N

u
(
x
)
=
−
Δ
u
(
x
)
+
(
x
,

D
u
(
x
)
)

\mathfrak {N}u(x) = - \Delta u(x) + (x,\,Du(x))

, where

Δ
u
(
x
)
=

trace
H

D
2

u
(
x
)

\Delta u(x) = {\operatorname {trace} _H}\,{D^2}u(x)

is the Laplacian and

(
⋅
,

⋅
)

( \cdot ,\, \cdot )

denotes the

B
B

-

B

∗

{B^{\ast }}

pairing. The closure

N

¯

\overline {\mathfrak {N}}

of

N

\mathfrak {N}

is known as the number operator. In this paper, we investigate the existence, uniqueness and regularity of solutions for the following two types of equations: (1)

u
t

=
−

N

u

{u_t} = - \mathfrak {N}u

(initial value problem) and (2)

N

k

u
=
f
(
k
⩾
1
)

{\mathfrak {N}^k}u = f(k \geqslant 1)

. We show that the fundamental solutions of (1) and (2) exist in the sense of measures and we represent their solutions by integrals with respect to these measures.
The Dror-Whitehead theorem in prohomotopy and shape theories
Authors: Singh S.
Published: June 2010

Many analogues of the classical Whitehead theorem from homotopy theory are now available in pro-homotopy and shape theories. E. Dror has significantly extended the homology version of the Whitehead theorem from the well-known simply connected case to the more general, for instance, nilpotent case. We prove a full analogue of Dror’s theorems in pro-homotopy and shape theories. More specifically, suppose

f
_

:

X
_

→

Y
_

\underline f :\underline X \to \underline Y

is a morphism in the pro-homotopy category of pointed and connected topological spaces which induces isomorphisms of the integral homology pro-groups. Then

f
_

\underline f

induces isomorphisms of the homotopy pro-groups, for instance, when

X
_

\underline X

and

Y
_

\underline Y

are simple, nilpotent, complete, or

H
_

\underline H

-objects; these notions are well known in homotopy theory and we have naturally extended them to pro-homotopy and shape theories.
A hypersurface defect relation for a class of meromorphic maps
Authors: Biancofiore Aldo
Published: June 2010

LetD1,…,Dq{D_1}, \ldots ,{D_q}be hypersurfaces of degreeppinPn{{\mathbf {P}}_n}with normal crossings. We prove for a certain class of meromorphic mapsf:Cm→Pnf:{{\mathbf {C}}^m} \to {{\mathbf {P}}_n}a defect relation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta Subscript f Baseline left-parenthesis upper D 1 right-parenthesis plus midline-horizontal-ellipsis plus delta Subscript f Baseline left-parenthesis upper D Subscript q Baseline right-parenthesis less-than-or-slanted-equals left-parenthesis n plus 1 right-parenthesis slash p">δf(D1)+⋯+δf(Dq)⩽(n+1)/p{\delta _f}\left ( {{D_1}} \right ) + \cdots + {\delta _f}({D_q}) \leqslant (n + 1)/pconjectured by Ph. Griffiths and B. Shiffman.
Duality between logics and equivalence relations
Authors: Mundici Daniele
Published: June 2010

Assuming

ω
\omega

is the only measurable cardinal, we prove: (i) Let

∼
\sim

be an equivalence relation such that

∼

=

≡
L

\sim \, = \,{ \equiv _L}

for some logic

L
⩽

L

∗

L \leqslant {L^{\ast }}

satisfying Robinson’s consistency theorem (with

L

∗

{L^{\ast }}

arbitrary); then there exists a strongest logic

L
+

⩽

L

∗

{L^ + } \leqslant {L^{\ast }}

such that

∼

=

≡

L
+

\sim \, = \,{ \equiv _{{L^ + }}}

; in addition,

L
+

{L^ + }

is countably compact if

∼

≠

≅

\sim \, \ne \, \cong

. (ii) Let

∼
˙

\dot \sim

be an equivalence relation such that

∼

=

≡

L
0

\sim \, = \,{ \equiv _{{L^0}}}

for some logic

L
0

{L^0}

satisfying Robinson’s consistency theorem and whose sentences of any type

τ
\tau

are (up to equivalence) equinumerous with some cardinal

κ
τ

{\kappa _\tau }

; then

L
0

{L^0}

is the unique logic

L
L

such that

∼

=

≡
L

\sim \, = \,{ \equiv _L}

; furthermore,

L
0

{L^0}

is compact and obeys Craig’s interpolation theorem. We finally give an algebraic characterization of those equivalence relations

∼
\sim

which are equal to

≡
L

{ \equiv _L}

for some compact logic

L
L

obeying Craig’s interpolation theorem and whose sentences are equinumerous with some cardinal.
A note on complete intersections
Authors: Bhatwadekar S. M.
Published: June 2010

Let

R
R

be a regular local ring and let

R
[
T
]

R[T]

be a polynomial algebra in one variable over

R
R

. In this paper the author proves that every maximal ideal of

R
[
T
]

R[T]

is complete intersection in each of the following cases: (1)

R
R

is a local ring of an affine algebra over an infinite perfect field, (2)

R
R

is a power series ring over a field.
The reciprocal of an entire function of infinite order and the distribution of the zeros of its second derivative
Authors: Rossi John
Published: June 2010

Let

f
f

be a real entire function of infinite order whose zeros together with those of

f
′

f’

are all real. It is proved that

(
1

/

f
)

(1/f)

has an infinity of nonreal zeros. The location of the zeros of

f
f

and

(
1

/

f
)

(1/f)

is also investigated. The result complements a finite order result of Hellerstein and Williamson.
The rank 3 permutation representations of the finite classical groups
Authors: Kantor William M. , Liebler Robert A.
Published: June 2010

The permutation representations in the title are all determined, and no surprises are found to occur.
The Hodge theory of flat vector bundles on a complex torus
Authors: Hoffman Jerome William
Published: June 2010

We study the Hodge spectral sequence of a local system on a compact, complex torus by means of the theory of harmonic integrals. It is shown that, in some cases, Baker’s theorems concerning linear forms in the logarithms of algebraic numbers may be applied to obtain vanishing theorems in cohomology. This is applied to the study of Betti and Hodge numbers of compact analytic threefolds which are analogues of hyperelliptic surfaces. Among other things, it is shown that, in contrast to the two-dimensional case, some of these varieties are nonalgebraic.
Analogues of the Denjoy-Young-Saks theorem
Authors: Belna C. L. , Cargo G. T. , Evans M. J. , Humke P. D.
Published: June 2010

In this paper, an analogue of the Denjoy-Young-Saks theorem concerning the almost everywhere classification of the Dini derivates of an arbitrary real function is established in both the case where the exceptional set is of first category and the case where it is

σ
\sigma

-porous. Examples are given to indicate the sharpness of these results.
Derivations into the integral closure
Authors: Draper Richard , Fischer Klaus
Published: June 2010

Let

A
A

be a reduced analytical

k
k

-algebra of dimension

r
r

and

A
′

A’

its integral closure in the full ring of quotients of

A
A

. We investigate the condition on

A
A

that there exist

r
r

elements

x
1

,
…
,

x
r

{x_1}, \ldots ,{x_r}

in

A
A

and

k
k

-derivations

d
1

,
…
,

d
r

{d_1}, \ldots ,{d_r}

from

A
A

into

A
′

A’

so that

d
i

(

x
j

)

{d_i}({x_j})

is the

r
×
r

r \times r

identity matrix and so that

d
1

,
…
,

d
r

{d_1}, \ldots ,{d_r}

freely generate

Der
k

(
A
,

A
′

)

{\operatorname {Der} _k}(A,\,A’ )

. We show this is equivalent to a number of other conditions. If

A
A

is a complete intersection, then the above is equivalent to the Jacobian ideal

J
J

becoming principal in

A
′

A’

. If

A

/

J

A / \sqrt J

is regular of dimension

r
−
1

r - 1

and satisfies the above condition, then

A
′

A’

is regular.
Self-maps of projective spaces
Authors: McGibbon C. A.
Published: June 2010

The classical projective

n
n

-spaces (real, complex, and quaternionic) are studied in terms of their self maps, from a homotopy point of view. Self maps of iterated suspensions of these spaces are also considered. The goal in both cases is to classify, up to homology, all such maps. This goal is achieved in the stable case. Some partial results are obtained in the unstable case. The results from both cases are used to compute the genus groups and the stable genus groups of the classical projective spaces. Applications to other spaces are also given.
Nilpotent inverse semigroups with central idempotents
Authors: Kowol G. , Mitsch H.
Published: June 2010

An inverse semigroup

S
S

with central idempotents, i.e. a strong semilattice of groups, will be called nilpotent, if it is finite and if for each prime divisor

p
i

{p_i}

of the orders of the structure groups of

S
S

the sets
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript i Baseline equals left-brace s element-of upper S vertical-bar o left-parenthesis s right-parenthesis equals p Subscript i Superscript k Super Subscript s Superscript Baseline comma k Subscript s Baseline greater-than-or-slanted-equals 0 right-brace">

P
i

=
{
s
∈
S

|

o
(
s
)
=

p
i

k
s

,

k
s

⩾
0
}

{P_i} = \{ s \in S|o(s) = p_i^{{k_s}},\,{k_s} \geqslant 0\}

are subsemigroups of

S
S

. If

S
S

is a group, then

P
i

{P_i}

are exactly the Sylow

p
i

{p_i}

-subgroups of the group. A theory similar to that given by W. Burnside for finite nilpotent groups is developed introducing the concepts of ascending resp. descending central series in an inverse semigroup, and it is shown that almost all of the well-known properties of finite nilpotent groups do hold also for the class of finite inverse semigroups with central idempotents.
Transitivity of families of invariant vector fields on the semidirect products of Lie groups
Authors: Bonnard B. , Jurdjevic V. , Kupka I. , Sallet G.
Published: June 2010

In this paper we give necessary and sufficient conditions for a family of right (or left) invariant vector fields on a Lie group

G
G

to be transitive. The concept of transitivity is essentially that of controllability in the literature on control systems. We consider families of right (resp. left) invariant vector fields on a Lie group

G
G

which is a semidirect product of a compact group

K
K

and a vector space

V
V

on which

K
K

acts linearly. If

F

\mathcal {F}

is a family of right-invariant vector fields, then the values of the elements of

F

\mathcal {F}

at the identity define a subset

Γ
\Gamma

of

L
(
G
)

L(G)

the Lie algebra of

G
G

. We say that

F

\mathcal {F}

is transitive on

G
G

if the semigroup generated by

∪

X
∈
Γ

{
exp
⁡
(
t
X
)
:
t
⩾
0
}

{ \cup _{X \in \Gamma }}\{ \exp (tX):t \geqslant 0\}

is equal to

G
G

. Our main result is that

F

\mathcal {F}

is transitive if and only if

Lie
⁡
(
Γ
)

\operatorname {Lie} (\Gamma )

, the Lie algebra generated by

Γ
\Gamma

, is equal to

L
(
G
)

L(G)

.
Minimal immersions of closed Riemann surfaces
Authors: Sacks J. , Uhlenbeck K.
Published: June 2010

Let

M
M

be a closed orientable surface of genus larger than zero and

N
N

a compact Riemannian manifold. If

u
:
M
→
N

u:M \to N

is a continuous map, such that the map induced by it between the fundamental groups of

M
M

and

N
N

contains no nontrivial element represented by a simple closed curve in its kernel, then there exists a conformal branched minimal immersion

s
:
M
→
N

s:M \to N

having least area among all branched immersions with the same action on

π
1

(
M
)

{\pi _1}(M)

as

u
u

. Uniqueness within the homotopy class of

u
u

fails in general: It is shown that for certain

3
3

-manifolds which fiber over the circle there are at least two geometrically distinct conformal branched minimal immersions within the homotopy class of any inclusion map of the fiber. There is also a topological discussion of those

3
3

-manifolds for which uniqueness fails.
Minimum simplicial complexes with given abelian automorphism group
Authors: Miller Zevi
Published: June 2010

Let

K
K

be a pure

n
n

-dimensional simplicial complex. Let

Γ
0

(
K
)

{\Gamma _0}(K)

be the automorphism group of

K
K

, and let

Γ
n

(
K
)

{\Gamma _n}(K)

be the group of permutations on

n
n

-cells of

K
K

induced by the elements of

Γ
0

(
K
)

{\Gamma _0}(K)

. Given an abelian group

A
A

we consider the problem of finding the minimum number of points

M
0

(
n
)

(
A
)

M_0^{(n)}(A)

in

K
K

such that

Γ
0

(
K
)
≅
A

{\Gamma _0}(K) \cong A

, and the minimum number of

n
n

-cells

M
1

(
n
)

(
A
)

M_1^{(n)}(A)

in

K
K

such that

Γ
n

(
K
)
≅
A

{\Gamma _n}(K) \cong A

. Write

A
=

∏

p
α

Z

p
α

e
(

p
α

)

A = {\prod _{{p^\alpha }}}{\mathbf {Z}}_{{p^\alpha }}^{e({p^\alpha })}

, where each factor

Z

p
α

{{\mathbf {Z}}_{{p^\alpha }}}

appears

e
(

p
α

)

e({p^\alpha })

times in the canonical factorization of

A
A

. For

A
A

containing no factors

Z

p
α

{{\mathbf {Z}}_{{p^\alpha }}}

satisfying

p
α

>
17

{p^\alpha } > 17

we find that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M 1 Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis upper A right-parenthesis equals upper M 0 Superscript left-parenthesis 2 right-parenthesis Baseline left-parenthesis upper A right-parenthesis equals sigma-summation Underscript p Superscript alpha Baseline Endscripts p Superscript alpha Baseline e left-parenthesis p Superscript alpha Baseline right-parenthesis">

M
1

(
n
)

(
A
)
=

M
0

(
2
)

(
A
)
=

∑

p
α

p
α

e
(

p
α

)

M_1^{(n)}(A) = M_0^{(2)}(A) = {\sum _{{p^\alpha }}}{p^\alpha }e({p^\alpha })

when

n
⩾
4

n \geqslant 4

, and we derive upper bounds for

M
1

(
n
)

(
A
)

M_1^{(n)}(A)

and

M
0

(
n
)

(
A
)

M_0^{(n)}(A)

in the remaining possibilities for

A
A

and

n
n

.
Brownian motion with partial information
Authors: McConnell Terry R.
Published: June 2010

We study the following problem concerning stopped

N
N

-dimensional Brownian motion: Compute the maximal function of the process, ignoring those times when it is in some fixed region

R
R

. Suppose this modified maximal function belongs to

L
q

{L^q}

. For what regions

R
R

can we conclude that the unrestricted maximal function belongs to

L
q

{L^q}

? A sufficient condition on

R
R

is that there exist

p
>
q

p > q

and a function

u
u

, harmonic in

R
R

, such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue x EndAbsoluteValue Superscript p Baseline less-than-or-slanted-equals u left-parenthesis x right-parenthesis less-than-or-slanted-equals upper C StartAbsoluteValue x EndAbsoluteValue Superscript p Baseline plus upper C comma x element-of upper R comma">

|

x

|

p

⩽
u
(
x
)
⩽
C

|

x

|

p

+
C
,

x
∈
R
,

|x{|^p} \leqslant u(x) \leqslant C|x{|^p} + C,\qquad x \in R,

\]
for some constant

C
C

. We give applications to analytic and harmonic functions, and to weak inequalities for exit times.
Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators
Authors: Lagarias J. C.
Published: June 2010

This paper defines the notion of a best simultaneous Diophantine approximation to a vector

α
\alpha

in

R
n

R^n

with respect to a norm

‖

⋅

‖

\left \| \,\cdot \, \right \|

on

R
n

R^n

. Suppose

α
\alpha

is not rational and order the best approximations to

α
\alpha

with respect to

‖

⋅

‖

\left \|\, \cdot \, \right \|

by increasing denominators

1
=

q
1

>

q
2

>
⋯

1=q_1 > q_2 > \cdots

. It is shown that these denominators grow at least at the rate of a geometric series, in the sense that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis alpha comma double-vertical-bar dot double-vertical-bar right-parenthesis equals limit inf left-parenthesis q Subscript k Baseline right-parenthesis Superscript 1 slash k Baseline greater-than-or-equal-to 1 plus StartFraction 1 Over 2 Superscript n plus 1 Baseline EndFraction">

g

(

α
,

‖

⋅

‖

)

=

lim inf

k
→
∞

(

q
k

)

1

/

k

≥
1
+

1

2

n
+
1

g\left ( {\alpha ,\,\left \| {\,\cdot \,} \right \|} \right ) = \liminf \limits _{k \to \infty } {({q_k})^{1/k}} \geq 1 + \frac {1}{{{2^{n + 1}}}}

\]
. Let

g

(

‖

⋅

‖

)

g\left ( {\left \|\, \cdot \, \right \|} \right )

denote the infimum of

g

(

α
,

‖

⋅

‖

)

g\left ( {\alpha ,\,\left \| {\,\cdot \,} \right \|} \right )

over all

α
\alpha

in

R
n

R^n

with an irrational coordinate. For the sup norm

‖

⋅

‖

s

\left \|\, \cdot \,\right \|_s

on

R
2

R^2

it is shown that

g

(

‖

⋅

‖

s

)

≥
θ
=

1.270

+

g\left ( {\left \| \, \cdot \, \right \|}_s \right )\ge \theta =1.270^{+}

where

θ
4

=

θ

2

+
1

\theta ^4=\theta ^{2}+1

.
On actions of regular type on complex Stiefel manifolds
Authors: Wang McKenzie Y.
Published: June 2010

The usual unitary representations of the special unitary, symplectic, or special orthogonal groups define a sequence of smooth actions on the complex Stiefel manifolds called the regular linear models. If one of the above groups acts smoothly on the complex Stiefel manifold of orthonormal

2
2

-frames in

C

n

\mathbf C^n

for odd

n
n

, and if the identity component of the principal isotropy type is of regular type, then it is shown under mild dimension restrictions that the orbit structure and the cohomology structure of the fixed point varieties (over the

mod

2

\mod 2

Steenrod algebra) resemble those of the regular linear models. The resemblance is complete in the cases of the special unitary and symplectic groups. There is an obstruction to complete resemblance in the case of the special orthogonal groups. An application of the above regularity theorems is given.
Multiple critical points of perturbed symmetric functionals
Authors: Rabinowitz Paul H.
Published: June 2010

Variational problems which are invariant under a group of symmetries often possess multiple solutions. This paper studies the effect of perturbations which are not small and which destroy the symmetry for two classes of such problems and shows how multiple solutions persist despite the perturbation.
Asymptotic expansions of some integral transforms by using generalized functions
Authors: Zayed Ahmed I.
Published: June 2010

The technique devised by Wong to derive the asymptotic expansions of multiple Fourier transforms by using the theory of Schwartz distributions is extended to a large class of integral transforms. The extension requires establishing a general procedure to extend these integral transforms to generalized functions. Wong’s technique is then applied to some of these integral transforms to obtain their asymptotic expansions. This class of integral transforms encompasses, among others, the Laplace, the Airy, the

K
K

and the Hankel transforms.
Cohomology of nilmanifolds and torsion-free, nilpotent groups
Authors: Lambe Larry A. , Priddy Stewart B.
Published: June 2010

Let

M
M

be a nilmanifold, i.e.

M
=
G

/

D

M = G/D

where

G
G

is a simply connected, nilpotent Lie group and

D
D

is a discrete uniform, nilpotent subgroup. Then

M
≃
K
(
D
,
1
)

M \simeq K(D,1)

. Now

D
D

has the structure of an algebraic group and so has an associated algebraic group Lie algebra

L
(
D
)

L(D)

. The integral cohomology of

M
M

is shown to be isomorphic to the Lie algebra cohomology of

L
(
D
)

L(D)

except for some small primes depending on

D
D

. This gives an effective procedure for computing the cohomology of

M
M

and therefore the group cohomology of

D
D

. The proof uses a version of form cohomology defined for subrings of

Q

{\mathbf {Q}}

and a type of Hirsch Lemma. Examples, including the important unipotent case, are also discussed.
Contractive projections on 𝐶₀(𝐾)
Authors: Friedman Yaakov , Russo Bernard
Published: June 2010

We show that the range of a norm one projection on a commutative

C
∗

{C^\ast }

-algebra has a ternary product structure (Theorem 2). We describe and characterize all such projections in terms of extreme points in the unit ball of the image of the dual (Theorem 1). We give necessary and sufficient conditions for the range to be isometric to a

C
∗

{C^\ast }

-algebra (Theorem 4) and we show that the range is a

C
σ

{C_\sigma }

-space (Theorem 5).
𝑞-extension of the 𝑝-adic gamma function. II
Authors: Koblitz Neal
Published: June 2010

Taylor series and asymptotic expansions are developed for

q
q

-extensions of the

p
p

-adic psi (derivative of log-gamma) function "twisted" by roots of unity. Connections with

p
p

-adic

L
L

-functions and

q
q

-expansions of Eisenstein series are discussed. The

p
p

-adic series are compared with the analogous classical expansions.
When is the natural map 𝑋→ΩΣ𝑋 a cofibration?
Authors: Lewis L. Gaunce
Published: June 2010

It is shown that a map

f
:
X
→
F
(
A
,
W
)

f:X \to F(A,W)

is a cofibration if its adjoint

f
:
X
∧
A
→
W

f:X \wedge A \to W

is a cofibration and

X
X

and

A
A

are locally equiconnected (LEC) based spaces with

A
A

compact and nontrivial. Thus, the suspension map

η
:
X
→
Ω
∑
X

\eta :X \to \Omega \sum X

is a cofibration if

X
X

is LEC. Also included is a new, simpler proof that C.W. complexes are LEC. Equivariant generalizations of these results are described.
Local blow-up of stratified sets up to bordism
Authors: Buoncristiano S. , Dedò M.
Published: June 2010

Homological obstructions are given, whose vanishing is a necessary and sufficient condition for the existence of a blow-up of an abstract prestratification

V
V

along a ’locally top-dimensional’ substratification

V
′

V’

.
Resolvent operators for integral equations in a Banach space
Authors: Grimmer R. C.
Published: June 2010

Conditions are given which ensure the existence of a resolvent operator for an integrodifferential equation in a Banach space. The resolvent operator is similar to an evolution operator for nonautonomous differential equations in a Banach space. As in the finite dimensional case, this operator is used to obtain a variation of parameters formula which can be used to obtain results concerning the asymptotic behaviour of solutions and weak solutions.
Partition algebras for almost-disjoint families
Authors: Baumgartner James E. , Weese Martin
Published: June 2010

A set

a
⊆
ω

a \subseteq \omega

is a partitioner of a maximal almost-disjoint faculty

F
F

of subsets of

ω
\omega

if every element of

F
F

is almost contained in or almost-disjoint from

a
a

. The partition algebra of

F
F

is the quotient of the Boolean algebra of partitioners modulo the ideal generated by

F
F

and the finite sets. We show that every countable algebra is a partition algebra, and that CH implies every algebra of cardinality

≤

2

ℵ
0

\leq {2^{{\aleph _0}}}

is a partition algebra. We also obtain consistency and independence results about the representability of Boolean algebras as partition algebras.