bluuebunny/crossref_metadata_2025 huggingface dataset:
On secondary bifurcations for some nonlinear convolution equations
Authors: Comets F. , Eisele Th. , Schatzman M.
Published: June 2010

On the

d
d

-dimensional torus

T

d

=

(

R

/

Z

)
d

{{\mathbf {T}}^d} = {({\mathbf {R}}/{\mathbf {Z}})^d}

, we study the nonlinear convolution equation
\[

u
(
t
)
=
g
{
λ
⋅
w
∗
u
(
t
)
}
,

t
∈

T

d

,
λ
>
0.

u(t) = g\{ \lambda \cdot w \ast u(t)\} , \quad t \in {{\mathbf {T}}^d}, \lambda > 0.

\]
where

∗
\ast

is the convolution on

T

d

{{\mathbf {T}}^d}

,

w
w

is an integrable function which is not assumed to be even, and

g
g

is bounded, odd, increasing, and concave on

R

+

{{\mathbf {R}}^ + }

. A typical example is

g
=

th

g = {\text {th}}

. For a general function

w
w

, we show by the standard theory of local bifurcation that, if the eigenspace of the linearized problem is of dimension

2
2

, a branch of solutions bifurcates at

λ
=

(
g
′
(
0
)

w
^

(
p
)

)

−
1

\lambda = {(g\prime (0)\hat w(p))^{ - 1}}

from the zero solution, and we show that it can be extended to infinity. For special simple forms of

w
w

, we show that the first bifurcating branch has no secondary bifurcation, but the other branches can. These results are related to the theory of spin models on

T

d

{{\mathbf {T}}^d}

in statistical mechanics, where they allow one to show the existence of a secondary phase transition of first order, and to some models of periodic structures in the brain in neurophysiology.
Above and below subgroups of a lattice-ordered group
Authors: Ball Richard N. , Conrad Paul , Darnel Michael
Published: June 2010

In an

l
l

-group

G
G

, this paper defines an

l
l

-subgroup

A
A

to be above an

l
l

-subgroup

B
B

(or

B
B

to be below

A
A

) if for every integer

n
n

,

a
∈
A

a \in A

, and

b
∈
B

b \in B

,

n
(

|

a

|

∧

|

b

|

)
⩽

|

a

|

n(|a| \wedge |b|) \leqslant |a|

. It is shown that for every

l
l

-subgroup

A
A

, there exists an

l
l

-subgroup

B
B

maximal below

A
A

which is closed, convex, and, if the

l
l

-group

G
G

is normal-valued, unique, and that for every

l
l

-subgroup

B
B

there exists an

l
l

-subgroup

A
A

maximal above

B
B

which is saturated: if

0
=
x
∧
y

0 = x \wedge y

and

x
+
y
∈
A

x + y \in A

, then

x
∈
A

x \in A

. Given

l
l

-groups

A
A

and

B
B

, it is shown that every lattice ordering of the splitting extension

G
=
A
×
B

G = A \times B

, which extends the lattice orders of

A
A

and

B
B

and makes

A
A

lie above

B
B

, is uniquely determined by a lattice homomorphism

π
\pi

from the lattice of principal convex

l
l

-subgroups of

A
A

into the cardinal summands of

B
B

. This extension is sufficiently general to encompass the cardinal sum of two

l
l

-groups, the lex extension of an

l
l

-group by an

o
o

-group, and the permutation wreath product of two

l
l

-groups. Finally, a characterization is given of those abelian

l
l

-groups

G
G

that split off below: whenever

G
G

is a convex

l
l

-subgroup of an

l
l

-group

H
H

,

H
H

is then a splitting extension of

G
G

by

A
A

for any

l
l

-subgroup

A
A

maximal above

G
G

in

H
H

.
Eigenvalues below the essential spectra of singular elliptic operators
Authors: Evans W. D. , Lewis Roger T.
Published: June 2010

A new technique is developed for determining if the number of eigenvalues below the essential spectrum of a singular elliptic differential operator is finite. A method is given for establishing lower bounds for the least point of the essential spectrum in terms of the behavior of the coefficients and weight near the singularities. Higher-order operators are included in these results as well as second-order Schrödinger operators.
Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees
Authors: Ash C. J.
Published: June 2010

We show that, under certain assumptions of recursiveness in

A

\mathfrak {A}

, the recursive structure

A

\mathfrak {A}

is

Δ
α
0

\Delta _\alpha ^0

-stable for

α
>

ω
1

C
K

\alpha > \omega _1^{CK}

if and only if there is an enumeration of

A

\mathfrak {A}

using a

Σ
α
0

\Sigma _\alpha ^0

set of recursive

Σ
α

{\Sigma _\alpha }

infinitary formulae and finitely many parameters from

A

\mathfrak {A}

. This extends the results of [1]. To do this, we first obtain results concerning

Δ
α
0

\Delta _\alpha ^0

paths in recursive labelling systems, also extending results of [1]. We show, more generally, that a path and a labelling can simultaneously be defined, when each node of the path is to be obtained by a

Δ
α
0

\Delta _\alpha ^0

function from the previous node and its label.
Around Effros’ theorem
Authors: Charatonik J. J. , Maćkowiak T.
Published: June 2010

Some stronger versions of the Effros theorem are discussed in the paper, not only for homeomorphisms but also for some other mappings, e.g. for open ones. Equivalent formulations of the theorem are presented as the

ε
\varepsilon

-push property and the existence of a so-called Effros’ metric.
Invariants of the Lusternik-Schnirelmann type and the topology of critical sets
Authors: Clapp Mónica , Puppe Dieter
Published: June 2010

We introduce and study in detail generalizations of the notion of Lusternik-Schnirelmann category which give information about the topology of the critical set of a differentiable function. We also improve a result of T. Ganea about the equality of the strong category and the category (even in the classical case).
Construction of high-dimensional knot groups from classical knot groups
Authors: Lien Magnhild
Published: June 2010

In this paper we study constructions of high dimensional knot groups from classical knot groups. We study certain homomorphic images of classical knot groups. Specifically, let

K
K

be a classical knot group and

w
w

any element in

K
K

. We are interested in the quotient groups

G
G

obtained by centralizing

w
w

, i.e.

G
=
K

/

[
K
,
w
]

G = K/[K,w]

, and ask whether

G
G

is itself a knot group. For certain

K
K

and

w
w

we show that

G
G

can be realized as the group of a knotted

3
3

-sphere in

5
5

-space, but

G
G

is not realizable by a

2
2

-sphere in

4
4

-space. By varying

w
w

, we also obtain quotients that are groups of knotted

2
2

-spheres in

4
4

-space, but they cannot be realized as the groups of classical knots. We have examples of quotients

K

/

[
K
,
w
]

K/[K,w]

that have nontrivial second homology. Hence these groups cannot be realized as knot groups of spheres in any dimension. However, we show that these groups are groups of knotted tori in

S
4

{S^4}

.
Diffuse sequences and perfect 𝐶*-algebras
Authors: Akemann Charles A. , Anderson Joel , Pedersen Gert K.
Published: June 2010

The concept of a diffuse sequence in a

C

∗

{C^{\ast }}

-algebra is introduced and exploited to complete the classification of separable, perfect

C

∗

{C^{\ast }}

-algebras. A

C

∗

{C^{\ast }}

-algebra is separable and perfect exactly when the closure of the pure state space consists entirely of atomic states.
Norms of Hankel operators and uniform algebras
Authors: Nakazi Takahiko
Published: June 2010

Two generalizations of the classical Hankel operators are defined on an abstract Hardy space that is associated with a uniform algebra. In this paper the norms of Hankel operators are studied. This has applications to weighted norm inequalities for conjugation operators, and invertible Topelitz operators. The results in this paper have applications to concrete uniform algebras, for example, a polydisc algebra and a uniform algebra which consists of rational functions.
Endomorphisms of right ideals of the Weyl algebra
Authors: Stafford J. T.
Published: June 2010

Let

A
=
A
(
k
)

A = A(k)

be the first Weyl algebra over an infinite field

k
k

, let

P
P

be any noncyclic, projective right ideal of

A
A

and set

S
=
End
⁡
(
P
)

S = \operatorname {End} (P)

. We prove that, as

k
k

-algebras,

S
≇
A

S\not \cong A

. In contrast, there exists a noncyclic, projective right ideal

Q
Q

of

S
S

such that

S
≅
End
⁡
(
Q
)

S \cong \operatorname {End} (Q)

. Thus, despite the fact that they are Morita equivalent,

S
S

and

A
A

have surprisingly different properties. For example, under the canonical maps,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u t Subscript k Baseline left-parenthesis upper A right-parenthesis approximately-equals upper P i c Subscript k Baseline left-parenthesis upper A right-parenthesis approximately-equals upper P i c Subscript k Baseline left-parenthesis upper S right-parenthesis">

Aut
k

(
A
)
≅

Pic
k

(
A
)
≅

Pic
k

(
S
)

{\operatorname {Aut} _k}(A) \cong {\operatorname {Pic} _k}(A) \cong {\operatorname {Pic} _k}(S)

. In contrast,

Aut
k

(
S
)

{\operatorname {Aut} _k}(S)

has infinite index in

Pic
k

(
S
)

{\operatorname {Pic} _k}(S)

.
A closed separable subspace of 𝛽𝑁 which is not a retract
Authors: Simon Petr
Published: June 2010

We shall exhibit a countable subset,

X
X

, of

N

∗

{{\mathbf {N}}^{\ast }}

whose closure is not a retract of

β

N

\beta {\mathbf {N}}

. The points of

X
X

are constructed in

c
c

steps with the aid of an independent matrix of subsets of

ω
\omega

.
A classification of simple Lie modules having a 1-dimensional weight space
Authors: Britten D. J. , Lemire F. W.
Published: June 2010

Let

L
L

denote a simple Lie algebra over the complex numbers. In this paper, we classify and construct all simple

L
L

modules which may be infinite dimensional but have at least one

1
1

-dimensional weight space. This completes the study begun earlier by the authors for the case of

L
=

A
n

L = {A_n}

. The approach presented here relies heavily on the results of Suren Fernando whose dissertation dealt with simple weight modules and their weight systems.
Some moduli spaces for rank 2 stable reflexive sheaves on 𝑃³
Authors: Miró-Roig Rosa M.
Published: June 2010

In [Ma], Maruyama proved that the set

M
(

c
1

,

c
2

,

c
3

)

M({c_1},{c_2},{c_3})

of isomorphism classes of rank

2
2

stable reflexive sheaves on

P

3

{{\mathbf {P}}^3}

with Chern classes

(

c
1

,

c
2

,

c
3

)

({c_1},{c_2},{c_3})

has a natural structure as an algebraic scheme. Until now, there are no general results about these schemes concerning dimension, irreducibility, rationality, etc. and only in a few cases a precise description of them is known. This paper is devoted to the following cases: (i)

M
(
−
1
,

c
2

,

c
2
2

−
2
r

c
2

+
2
r
(
r
+
1
)
)

M( - 1,{c_2},c_2^2 - 2r{c_2} + 2r(r + 1))

with

c
2

⩾
4

{c_2} \geqslant 4

,

1
⩽
r
⩽
(
−
1
+

4

c
2

−
7

)

/

2

1 \leqslant r \leqslant ( - 1 + \sqrt {4{c_2} - 7} )/2

; and (ii)

M
(
−
1
,

c
2

,

c
2
2

−
2
(
r
−
1
)

c
2

)

M( - 1,{c_2},c_2^2 - 2(r - 1){c_2})

with

c
2

⩾
8

{c_2} \geqslant 8

,

2
⩽
r
⩽
(
−
1
+

4

c
2

−
7

)

/

2

2 \leqslant r \leqslant ( - 1 + \sqrt {4{c_2} - 7} )/2

.
Dimension de Hausdorff des ensembles de zéros et d’interpolation pour 𝐴^{∞}(𝐷)
Authors: Chaumat Jacques , Chollet Anne-Marie
Published: June 2010

Soit

D
D

un domaine borné strictement pseudoconvexe dans

C

n

{{\mathbf {C}}^n}

à frontière régulière

∂
D

\partial D

et soit

A
∞

(
D
)

{A^\infty }(D)

la classe des fonctions holomorphes dans

D
D

, indéfiniment dérivables dans

D
¯

\overline D

. Un sous-ensemble compact

E
E

de

∂
D

\partial D

est un ensemble de zéros pour

A
∞

(
D
)

{A^\infty }(D)

s’il existe une fonction de

A
∞

(
D
)

{A^\infty }(D)

s’annulant seulement sur

E
E

. C’est un ensemble d’interpolation d’ordre infini pour

A
∞

(
D
)

{A^\infty }(D)

si, pour toute fonction

f
f

de classe

C
∞

{C^\infty }

dans

C

n

{{\mathbf {C}}^n}

telle que

∂
¯

f

\overline \partial f

soit plate sur

E
E

, il existe une fonction

F
F

de

A
∞

(
D
)

{A^\infty }(D)

telle que

F
−
f

F - f

soit plate sur

E
E

. On construit ici des ensembles de dimension de Hausdorff

n
n

. Ce résultat est le meilleur possible dans le cas d’ensembles totalement réels. Le point de vue utilisé pour montrer qu’un sous-ensemble

E
E

de

∂
D

\partial D

est d’interpolation d’ordre infini pour

A
∞

(
D
)

{A^\infty }(D)

est de vérifier qu’il a la propriété de division par

A
∞

(
D
)

{A^\infty }(D)

, c’est-á-dire, que, pour toute famille de fonctions

(

f
i

)

i
∈

N

{({f_i})_{i \in {\text {N}}}}

de

C
∞

(

D
¯

)

{C^\infty }(\overline D )

, plates sur

E
E

, il existe une fonction

F
F

de

A
∞

(
D
)

{A^\infty }(D)

, plate sur

E
E

et nulle seulement sur

E
E

et une famille de fonctions

(

k
i

)

i
∈

N

{({k_i})_{i \in {\text {N}}}}

de

C
∞

(

D
¯

)

{C^\infty }(\overline D )

, plates sur

E
E

, telles que l’on ait, pour tout

i
i

dans

N

{\mathbf {N}}

,

f
i

=
F

k
i

{f_i} = F{k_i}

.
The structure of the critical set in the mountain pass theorem
Authors: Pucci Patrizia , Serrin James
Published: June 2010

We show that the critical set generated by the Mountain Pass Theorem of Ambrosetti and Rabinowitz must have a well-defined structure. In particular, if the underlying Banach space is infinite dimensional then either the critical set contains a saddle point of mountain-pass type, or the set of local minima intersects at least two components of the set of saddle points. Related conclusions are also established for the finite dimensional case, and when other special conditions are assumed. Throughout the paper, no hypotheses of nondegeneracy are required on the critical set.
On inductive limits of matrix algebras of holomorphic functions
Authors: Peters Justin
Published: June 2010

Let

A

\mathfrak {A}

be a UHF algebra and

A

(

D

)

\mathcal {A}({\mathbf {D}})

the disk algebra. If

A

=

[

∪

n
≥
1

A

n

]

−

\mathfrak {A} = {\left [ {{ \cup _{n \geq 1}}{\mathfrak {A}_n}} \right ]^ - }

and

α
\alpha

is a product-type automorphism of

A

\mathfrak {A}

which leaves each

A

n

{\mathfrak {A}_n}

invariant, then

α
\alpha

defines an embedding
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper A Subscript n Baseline circled-times script upper A left-parenthesis bold upper D right-parenthesis right-arrow with hook Overscript modifying above d with tless i Subscript n Baseline Endscripts German upper A Subscript n plus 1 Baseline circled-times script upper A left-parenthesis bold upper D right-parenthesis">

A

n

⊗

A

(

D

)

↪

ı
n

A

n
+
1

⊗

A

(

D

)

\mathfrak {A}_n \otimes \mathcal {A}({\mathbf {D}}) \stackrel {\imath _n}{\hookrightarrow } {\mathfrak {A}_{n + 1}} \otimes \mathcal {A}({\mathbf {D}})

\]
. The inductive limit of this system is a Banach algebra whose maximal ideal space is closely related to that of the disk algebra if the Connes spectrum

Γ
(
α
)

\Gamma (\alpha )

is finite.
First layer formulas for characters of 𝑆𝐿(𝑛,𝐶)
Authors: Stembridge John R.
Published: June 2010

Some problems concerning the decomposition of certain characters of

S
L
(
n
,

C

)

SL(n,\,{\mathbf {C}})

are studied from a combinatorial point of view. The specific characters considered include those of the exterior and symmetric algebras of the adjoint representation and the Euler characteristic of Hanlon’s so-called “Macdonald complex.” A general recursion is given for computing the irreducible decomposition of these characters. The recursion is explicitly solved for the first layer representations, which are the irreducible representations corresponding to partitions of

n
n

. In the case of the exterior algebra, this settles a conjecture of Gupta and Hanlon. A further application of the recursion is used to give a family of formal Laurent series identities that generalize the (equal parameter)

q
q

-Dyson Theorem.
Best constants in norm inequalities for the difference operator
Authors: Kaper Hans G. , Spellman Beth E.
Published: June 2010

Let

ξ
=

(

ξ
m

)

m
∈

Z

\xi = {({\xi _m})_{m \in {\mathbf {Z}}}}

be an arbitrary element of the sequence space

l
∞

(

Z

)

{l^\infty }({\mathbf {Z}})

, and let

Δ
\Delta

be the difference operator on
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Superscript normal infinity Baseline left-parenthesis bold upper Z right-parenthesis colon normal upper Delta xi equals left-parenthesis xi Subscript m plus 1 Baseline minus xi Subscript m Baseline right-parenthesis Subscript m element-of bold upper Z Baseline">

l
∞

(

Z

)
:
Δ
ξ
=

(

ξ

m
+
1

−

ξ
m

)

m
∈

Z

{l^\infty }({\mathbf {Z}}):\Delta \xi = {({\xi _{m + 1}} - {\xi _m})_{m \in {\mathbf {Z}}}}

. The object of this investigation is the best possible value
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis n comma k right-parenthesis equals sup left-brace upper Q Subscript n comma k Baseline left-parenthesis xi right-parenthesis colon xi element-of l Superscript normal infinity Baseline left-parenthesis bold upper Z right-parenthesis comma normal upper Delta Superscript n Baseline xi not-equals 0 right-brace">

C
(
n
,

k
)
=

sup

{

Q

n
,
k

(
ξ
)
:
ξ
∈

l
∞

(

Z

)
,

Δ
n

ξ
≠
0
}

C(n,\,k) = {\operatorname {sup}}\{ {Q_{n,k}}(\xi ):\xi \in {l^\infty }({\mathbf {Z}}),\,{\Delta ^n}\xi \ne 0\}

\]
of the quotient
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Subscript n comma k Baseline left-parenthesis xi right-parenthesis equals StartFraction double-vertical-bar normal upper Delta Superscript k Baseline xi double-vertical-bar Over double-vertical-bar xi double-vertical-bar Superscript left-parenthesis n minus k right-parenthesis slash n Baseline double-vertical-bar normal upper Delta Superscript n Baseline xi double-vertical-bar Superscript k slash n Baseline EndFraction">

Q

n
,
k

(
ξ
)
=

‖

Δ
k

ξ

‖

‖
ξ
‖

(
n
−
k
)

/

n

‖

Δ
n

ξ

‖

k

/

n

{Q_{n,k}}(\xi ) = \frac {{\left \| {{\Delta ^k}\xi } \right \|}} {{{{\left \| \xi \right \|}^{(n - k)/n}}{{\left \| {{\Delta ^n}\xi } \right \|}^{k/n}}}}

\]
, where

n
=
2
,

3
,
…

n = 2,\:3, \ldots

;

k
=
1
,
…
,

n
−
1

k = 1, \ldots ,\,n - 1

. It is shown that

C
(
n
,

k
)

C(n,\,k)

is at least equal to the corresponding constant

K
(
n
,

k
)

K(n,\,k)

, determined by Kolmogorov [Moscov. Gos. Univ. Uchen. Zap. Mat. 30 (1939), 3-13; Amer. Math. Soc. Transl. (1) 2 (1962), 233-243] for the differential operator

D
D

on

L
∞

(

R

)

{L^\infty }({\mathbf {R}})

, and exactly equal to

K
(
n
,

k
)

K(n,\,k)

if

k
=
n
−
1

k = n - 1

. Lower bounds for

C
(
n
,

k
)

C(n,\,k)

are derived that show that

C
(
n
,

k
)

C(n,\,k)

is generally greater than

K
(
n
,

k
)

K(n,\,k)

. The values of

C
(
n
,

k
)

C(n,\,k)

,

k
=
1
,
…
,

n
−
1

k = 1, \ldots ,\,n - 1

, are computed for

n
=
2
,
…
,
5

n = 2, \ldots ,5

.
𝐾-theory and multipliers of stable 𝐶*-algebras
Authors: Mingo J. A.
Published: June 2010

The main theorem is that if

A
A

is a

C

∗

C^{\ast }

-algebra with a countable approximate identity consisting of projections, then the unitary group of

M
(
A
⊗
K
)

M(A \otimes K)

is contractible. This gives a realization, via the index map, of

K
0

(
A
)

{K_0}(A)

as components in the set of Fredholm operators on

H
A

{H_A}

.
Eisenstein series and the Selberg trace formula. II
Authors: Jacquet H. , Zagier D.
Published: June 2010

The integral of the kernel of the trace formula against an Eisenstein series is investigated. The analytic properties of this integral imply the divisibility of the convolution

L
L

-function attached to a form by the zeta function of the field.
Orthogonal polynomials, measures and recurrences on the unit circle
Authors: Nevai Paul
Published: June 2010

New characterizations are given for orthogonal polynomials on the unit circle and the associated measures in terms of the reflection coefficients in the recurrence equation satisfied by the polynomials.
Oscillatory phenomena associated to semilinear wave equations in one spatial dimension
Authors: Cazenave T. , Haraux A.
Published: June 2010

Let

g
g

be a nonincreasing, odd

C
1

{C^1}

function and

l
>
0

l > 0

. We establish that for any solution

u
∈
C
(

R

;

H
0
1

(
0
,
l
)
)

u \in C({\mathbf {R}};H_0^1(0,l))

of the equation

u

t
t

−

u

x
x

+
g
(
u
)
=
0

{u_{tt}} - {u_{xx}} + g(u) = 0

and any

x
0

∈
]
0
,
l
[

{x_0} \in ]0,l[

, the function

t
↦
u
(
t
,

x
0

)

t \mapsto u(t,{x_0})

satisfies the following alternative: either

u
(
t
,

x
0

)
=
0
,
∀
t
∈

R

u(t,{x_0}) = 0,\forall t \in {\mathbf {R}}

, or

∀
a
∈

R

\forall a \in {\mathbf {R}}

, there exist

t
1

{t_1}

and

t
2

{t_2}

in

[
a
,
a
+
2
l
]

[a,a + 2l]

such that

u
(

t
1

,

x
0

)
>
0

u({t_1},{x_0}) > 0

and

u
(

t
2

,

x
0

)
>
0

u({t_2},{x_0}) > 0

. We study the structure of the set of points satisfying the first possibility. We give analogous results for

u
x

{u_x}

and for some other homogeneous boundary conditions.
Sets of uniqueness in compact, 0-dimensional metric groups
Authors: Grubb D. J.
Published: June 2010

An investigation is made of sets of uniqueness in a compact

0
0

-dimensional space. Such sets are defined by pointwise convergence of sequences of functions that generalize partial sums of trigonometric series on Vilenkin groups. Several analogs of classical uniqueness theorems are proved, including a version of N. Bary’s theorem on countable unions of closed sets of uniqueness.
On scrambled sets for chaotic functions
Authors: Bruckner A. M. , Hu Thakyin
Published: June 2010

Some recent research has raised questions concerning the possible sizes of scrambled sets for chaotic functions. We answer these questions by showing that a scrambled set can have full measure, but cannot be residual although a scrambled set can be second category in every interval. We also indicate relationships that exist between chaotic functions and transitive functions.
Convergence of series of scalar- and vector-valued random variables and a subsequence principle in 𝐿₂
Authors: Dilworth S. J.
Published: June 2010

Let

(

d
n

)

n
=
1

∞

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

be a martingale difference sequence in

L
0

(
X
)

{L_0}(X)

, where

X
X

is a uniformly convex Banach space. We investigate a necessary condition for convergence of the series

∑

n
=
1

∞

a
n

d
n

\sum {_{n = 1}^\infty {a_n}{d_n}}

. We also prove a related subsequence principle for the convergence of a series of square-integrable scalar random variables.
The isometry groups of manifolds and the automorphism groups of domains
Authors: Saerens Rita , Zame William R.
Published: June 2010

We prove that every compact Lie group can be realized as the (full) automorphism group of a strictly pseudoconvex domain and as the (full) isometry group of a compact, connected, smooth Riemannian manifold.
Bordism of semifree circle actions on Spin manifolds
Authors: Borsari Lucília Daruiz
Published: June 2010

Using traditional methods in bordism theory, an almost complete description of the rational bordism groups of semifree circle actions on Spin manifolds is given. The single remaining problem, to describe the ideal of

Ω
∗

Spin

⊗

Q

\Omega _ \ast ^{{\operatorname {Spin}}}\, \otimes \,\mathbf {Q}

, generated by bordism classes of Spin manifolds admitting a semifree action of odd type, has been recently solved by S. Ochanine

[

O

]

[\mathbf {O}]

.
Lattice embeddings in the recursively enumerable truth table degrees
Authors: Haught Christine Ann
Published: June 2010

It is shown that every finite lattice, and in fact every recursively presentable lattice, can be embedded in the r.e.

tt
\text {tt}

-degrees by a map preserving least and greatest elements. The decidability of the

1
1

-quantifier theory of the r.e.

tt
\text {tt}

-degrees in the language with

⩽
,

∨
,

∧
,

0

\leqslant ,\, \vee ,\, \wedge ,\,0

, and 1 is obtained as a corollary.
Stability of harmonic maps and eigenvalues of the Laplacian
Authors: Urakawa Hajime
Published: June 2010

The index and nullity of the Hessian of the energy for every harmonic map are estimated above by a geometric quantity. The stability theory of harmonic maps is developed and as an application, the Kähler version of the Lichnerowicz-Obata theorem about the first eigenvalue of the Laplacian is proved.
Supersymmetry, twistors, and the Yang-Mills equations
Authors: Eastwood Michael
Published: June 2010

This article investigates a supersymmetric proof due to Witten of the twistor description of general Yang-Mills fields due to Green, Isenberg, and Yasskin. In particular, some rigor is added and the rather complicated calculations are given in detail.
Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions
Authors: Gross Kenneth I. , Richards Donald St. P.
Published: June 2010

Hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis, multivariate statistics, quantum physics, molecular chemistry, and number theory. This paper presents a general theory of such functions for real division algebras. These functions, which generalize the classical hypergeometric functions, are defined by infinite series on the spaceS=S(n,F)S = S(n,\,\mathbf {F})of alln×nn \times nHermitian matrices over the division algebraF\mathbf {F}. The theory depends intrinsically upon the representation theory of the general linear groupG=GL(n,F)G = GL(n,\,\mathbf {F})of invertiblen×nn \times nmatrices overF\mathbf {F}, and the theme of this work is the full exploitation of the inherent group theory. The main technique is the use of the method of “algebraic induction” to realize explicitly the appropriate representations ofGG, to decompose the space of polynomial functions onSS, and to describe the “zonal polynomials” from which the hypergeometric functions are constructed. Detailed descriptions of the convergence properties of the series expansions are given, and integral representations are provided. Future papers in this series will develop the fine structure of these functions.
Prime ideals in enveloping rings
Authors: Passman D. S.
Published: June 2010

Let

L
L

be a Lie algebra over the field

K
K

of characteristic

0
0

and let

U
(
L
)

U(L)

denote its universal enveloping algebra. If

R
R

is a

K
K

-algebra and

L
L

acts on

R
R

as derivations, then there is a natural ring generated by

R
R

and

U
(
L
)

U(L)

which is denoted by

R
#
U
(
L
)

R\# U(L)

and called the smash product of

R
R

by

U
(
L
)

U(L)

. The aim of this paper is to describe the prime ideals of this algebra when it is Noetherian. Specifically we show that there exists a twisted enveloping algebra

U
(
X
)

U(X)

on which

L
L

acts and a precisely defined one-to-one correspondence between the primes

P
P

of

R
#
U
(
L
)

R\#U(L)

with

P
∩
R
=
0

P \cap R = 0

and the

L
L

-stable primes of

U
(
X
)

U(X)

. Here

X
X

is a Lie algebra over some field

C
⊇
K

C \supseteq K

.
On the central limit theorem for dynamical systems
Authors: Burton Robert , Denker Manfred
Published: June 2010

Given an aperiodic dynamical system

(
X
,
T
,
μ
)

(X,T,\mu )

then there is an

f
∈

L
2

(
μ
)

f \in {L^2}(\mu )

with

∫
f
d
μ
=
0

\smallint fd\mu = 0

satisfying the Central Limit Theorem, i.e. if

S
m

f
=
f
+
f
∘
T
+
⋯
+
f
∘

T

m
−
1

{S_m}f = f + f \circ T + \cdots + f \circ {T^{m - 1}}

and

σ
m

=

‖

S
m

f

‖

2

{\sigma _m} = {\left \| {{S_m}f} \right \|_2}

then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu left-brace x vertical-bar StartFraction upper S Subscript m Baseline f left-parenthesis x right-parenthesis Over sigma Subscript m Baseline EndFraction greater-than u right-brace right-arrow left-parenthesis 2 pi right-parenthesis Superscript negative 1 slash 2 Baseline integral Subscript negative normal infinity Superscript u Baseline exp left-bracket StartFraction minus upsilon squared Over 2 EndFraction right-bracket d upsilon period">

μ

{

x

|

S
m

f
(
x
)

σ
m

>
u

}

→

(
2
π

)

−
1

/

2

∫

−
∞

u

exp

[

−

υ
2

2

]

d
υ
.

\mu \left \{ {x|\frac {{{S_m}f(x)}}{{{\sigma _m}}} > u} \right \} \to {(2\pi )^{ - 1/2}}\int _{ - \infty }^u {{\text {exp}}} \left [ {\frac {{ - {\upsilon ^2}}}{2}} \right ]d\upsilon .

\]
The analogous result also holds for flows.
On the Stickelberger ideal and the relative class number
Authors: Kimura Tatsuo , Horie Kuniaki
Published: June 2010

Letkkbe any imaginary abelian field,RRthe integral group ring ofG=Gal(k/Q)G = {\text {Gal}}(k/\mathbb {Q}), andSSthe Stickelberger ideal ofkk. Roughly speaking, the relative class numberh−{h^ - }ofkkis expressed as the index ofSSin a certain idealAAofRRdescribed by means ofGGand the complex conjugation ofk;c−h−=[A:S]k;{c^ - }{h^ - } = [A:S], with a rational numberc−{c^ - }in12N={n/2;n∈N}\frac {1} {2}\mathbb {N} = \{ n/2;n \in \mathbb {N}\}, which can be described withouth−{h^ - }and is of lower thanh−{h^ - }if the conductor ofkkis sufficiently large (cf. [6, 9, 10]; see also [5]). We shall prove that2c−2{c^ - }, a natural number, divides2([k:Q]/2)[k:Q]/22{([k:\mathbb {Q}]/2)^{[k:\mathbb {Q}]/2}}. In particular, ifkkvaries through a sequence of imaginary abelian fields of degrees bounded, thenc−{c^ - }takes only a finite number of values. On the other hand, it will be shown thatc−{c^ - }can take any value in12N\frac {1} {2}\mathbb {N}whenkkranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.
An infinite-dimensional Hamiltonian system on projective Hilbert space
Authors: Bloch Anthony M.
Published: June 2010

We consider here the explicit integration of a Hamiltonian system on infinite-dimensional complex projective space. The Hamiltonian, which is the restriction of a linear functional to this projective space, arises in the problem of line fitting in complex Hilbert space (or, equivalently, the problem of functional approximation) or as the expectation value of a model quantum mechanical system. We formulate the system here as a Lax system with parameter, showing how this leads to an infinite set of conserved integrals associated with the problem and to an explicit formulation of the flow in action-angle form via an extension of some work of J. Moser. In addition, we find the algebraic curve naturally associated with the system.
An étale cohomology duality theorem for number fields with a real embedding
Authors: Bienenfeld Mel
Published: June 2010

The restriction on

2
2

-primary components in the Artin-Verdier duality theorem [2] has been eliminated by Zink [9], who has shown that the sheaf of units for the étale topology over the ring of integers of any number field acts as a dualizing sheaf for a modified cohomology of sheaves. The present paper provides an alternate means of removing the

2
2

-primary restriction. Like Zink’s, it involves a topology which includes infinite primes, but it avoids modified cohomology and will be more directly applicable in the proof of a theorem of Lichtenbaum regarding zeta- and

L
L

-functions [4, 5]. Related results—including the cohomology of units sheaves, the norm theorem, and punctual duality theorem of Mazur [6]—are also affected by the use of a topology including the infinite primes. The corresponding results in the new setting are included here.
Strong Ramsey theorems for Steiner systems
Authors: Nešetřil Jaroslav , Rödl Vojtěch
Published: June 2010

It is shown that the class of partial Steiner

(
k
,

l
)

(k,\,l)

-systems has the edge Ramsey property, i.e., we prove that for every partial Steiner

(
k
,

l
)

(k,\,l)

-system

G

\mathcal {G}

there exists a partial Steiner

(
k
,

l
)

(k,\,l)

-system

H

\mathcal {H}

such that for every partition of the edges of

H

\mathcal {H}

into two classes one can find an induced monochromatic copy of

G

\mathcal {G}

. As an application we get that the class of all graphs without cycles of lengths

3
3

and

4
4

has the edge Ramsey property. This solves a longstanding problem in the area.
On the generalized spectrum for second-order elliptic systems
Authors: Cantrell Robert Stephen , Cosner Chris
Published: June 2010

We consider the system of homogeneous Dirichlet boundary value problems

(

∗

)

({\ast })

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 1 u equals lamda left-bracket a 11 left-parenthesis x right-parenthesis u plus a 12 left-parenthesis x right-parenthesis v right-bracket comma upper L 2 v equals mu left-bracket a 12 left-parenthesis x right-parenthesis u plus a 22 left-parenthesis x right-parenthesis v right-bracket">

L
1

u
=
λ
[

a

11

(
x
)
u
+

a

12

(
x
)
v
]
,

L
2

v
=
μ
[

a

12

(
x
)
u
+

a

22

(
x
)
v
]

{L_1}u = \lambda [{a_{11}}(x)u + {a_{12}}(x)v],\quad {L_2}v = \mu [{a_{12}}(x)u + {a_{22}}(x)v]

\]
in a smooth bounded domain

Ω
⊆

R

N

\Omega \subseteq {{\mathbf {R}}^N}

, where

L
1

{L_1}

and

L
2

{L_2}

are formally self-adjoint second-order strongly uniformly elliptic operators. Using linear perturbation theory, continuation methods, and the Courant-Hilbert variational eigenvalue characterization, we give a detailed qualitative and quantitative description of the real generalized spectrum of

(

∗

)

({\ast })

, i.e., the set

(
λ
,

μ
)
∈

R

2

:

(

∗

)

(\lambda ,\,\mu ) \in {{\mathbf {R}}^2}:\,({\ast })

has a nontrivial solution. The generalized spectrum, a term introduced by Protter in 1979, is of considerable interest in the theory of linear partial differential equations and also in bifurcation theory, as it is the set of potential bifurcation points for associated semilinear systems.
Rings of differential operators on invariant rings of tori
Authors: Musson Ian M.
Published: June 2010

Let

k
k

be an algebraically closed field of characteristic zero and

G
G

a torus acting diagonally on

k
s

{k^s}

. For a subset

β
\beta

of

s

=
{
1
,

2
,
…
,

s
}

{\mathbf {s}} = \{ 1,\,2, \ldots ,\,s\}

, set

U
β

=
{
u
∈

k
s

|

u
j

≠
0

if

j
∈
β
}

{U_\beta } = \{ u \in {k^s}|{u_j} \ne 0\;{\text {if}}\;j \in \beta \}

. Then

G
G

acts on

O

(

U
β

)

\mathcal {O}({U_\beta })

, the ring of regular functions on

U
β

{U_\beta }

, and we study the ring

D
(

O

(

U
β

)
G

)

D(\mathcal {O}{({U_\beta })^G})

of all differential operators on the invariant ring. More generally suppose that

Δ
\Delta

is a set of subsets of s, such that each invariant ring

O

(

U
β

)
G

\mathcal {O}{({U_\beta })^G}

,

β
∈
Δ

\beta \in \Delta

, has the same quotient field. We prove that

∩

β
∈
Δ

D
(

O

(

U
β

)
G

)

{ \cap _{\beta \in \Delta }}D(\mathcal {O}{({U_\beta })^G})

is Noetherian and finitely generated as a

k
k

-algebra. Now

G
G

acts on each

D
(

O

(

U
β

)
)

D(\mathcal {O}({U_\beta }))

and there is a natural map
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta colon intersection Underscript beta element-of normal upper Delta Endscripts upper D left-parenthesis script upper O left-parenthesis upper U Subscript beta Baseline right-parenthesis right-parenthesis Superscript upper G Baseline right-arrow intersection Underscript beta element-of normal upper Delta Endscripts upper D left-parenthesis script upper O left-parenthesis upper U Subscript beta Baseline right-parenthesis Superscript upper G Baseline right-parenthesis equals upper D left-parenthesis upper Y Subscript normal upper Delta Baseline slash upper G right-parenthesis">

θ
:

⋂

β
∈
Δ

D

(

O

(

U
β

)
)

G

→

⋂

β
∈
Δ

D
(

O

(

U
β

)

G

)
=
D
(

Y
Δ

/

G
)

\theta :\bigcap \limits _{\beta \in \Delta } {D{{(\mathcal {O}({U_\beta }))}^G} \to \bigcap \limits _{\beta \in \Delta } {D(\mathcal {O}{{({U_\beta })}^G}) = D({Y_\Delta } / G)} }

\]
obtained by restriction of the differential operators. We find necessary and sufficient conditions for

θ
\theta

to be surjective and describe the kernel of

θ
\theta

. The algebras

∩

β
∈
Δ

D

(

O

(

U
β

)

)
G

{ \cap _{\beta \in \Delta }}D{(\mathcal {O}({U_\beta }))^G}

and

∩

β
∈
Δ

D
(

O

(

U
β

)
G

)

{ \cap _{\beta \in \Delta }}D(\mathcal {O}{({U_\beta })^G})

carry a natural filtration given by the order of the differential operators. We show that the associated graded rings are finitely generated commutative algebras and are Gorensetin rings. We also determine the centers of

∩

β
∈
Δ

D

(

O

(

U
β

)

)
G

{ \cap _{\beta \in \Delta }}D{(\mathcal {O}({U_\beta }))^G}

and

∩

β
∈
Δ

D
(

O

(

U
β

)
G

)

{ \cap _{\beta \in \Delta }}D(\mathcal {O}{({U_\beta })^G})

.
The Morava 𝐾-theories of some classifying spaces
Authors: Kuhn Nicholas J.
Published: June 2010

Let

P
P

be a finite abelian

p
p

-group with classifying space

B
P

BP

. We compute, in representation theoretic terms, the Morava

K
K

-theories of the stable wedge summands of

B
P

BP

. In particular, we obtain a simple, and purely group theoretic, description of the rank of

K

(
s

)

∗

(
B
G
)

K{(s)^{\ast }}(BG)

for any finite group

G
G

with an abelian

p
p

-Sylow subgroup. A minimal amount of topology quickly reduces the problem to an algebraic one of analyzing truncated polynomial algebras as modular representations of the semigroup

M
n

(

Z

/

p
)

{M_n}({\mathbf {Z}} / p)

.
Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations
Authors: Lieberman Gary M.
Published: June 2010

We consider solutions (and subsolutions and supersolutions) of the boundary value problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row a Superscript i j Baseline left-parenthesis x comma u comma upper D u right-parenthesis upper D Subscript i j Baseline u plus a left-parenthesis x comma u comma upper D u right-parenthesis equals 0 in normal upper Omega comma 2nd Row beta Superscript i Baseline left-parenthesis x right-parenthesis upper D Subscript i Baseline u plus gamma left-parenthesis x right-parenthesis u equals g left-parenthesis x right-parenthesis on partial-differential normal upper Omega EndLayout">

a

i
j

(
x
,

u
,

D
u
)

D

i
j

u
+
a
(
x
,

u
,

D
u
)
=
0

in

Ω
,

β
i

(
x
)

D
i

u
+
γ
(
x
)
u
=
g
(
x
)

on

∂
Ω

\begin {array}{*{20}{c}} {{a^{ij}}(x,\,u,\,Du){D_{ij}}u + a(x,\,u,\,Du) = 0\quad {\text {in}}\;\Omega ,} \\ {{\beta ^i}(x){D_i}u + \gamma (x)u = g(x)\quad {\text {on}}\;\partial \Omega } \\ \end {array}

\]
for a Lipschitz domain

Ω
\Omega

, a positive-definite matrix-valued function

[

a

i
j

]

[{a^{ij}}]

, and a vector field

β
\beta

which points uniformly into

Ω
\Omega

. Without making any continuity assumptions on the known functions, we prove Harnack and Hölder estimates for

u
u

near

∂
Ω

\partial \Omega

. In addition we bound the

L
∞

{L^\infty }

norm of

u
u

near

∂
Ω

\partial \Omega

in terms of an appropriate

L
p

{L^p}

norm and the known functions. Our approach is based on that for the corresponding interior estimates of Trudinger.
Classifying spaces for foliations with isolated singularities
Authors: Greenberg Peter
Published: June 2010

Let

Γ
a

⊂
Γ

{\Gamma ^a} \subset \Gamma

be transitive pseudogroups on

R

n

{{\mathbf {R}}^n}

, such that, for any element

g
:

U
→
V

g:\,U \to V

of

Γ
\Gamma

, there is a locally finite subset

S
⊂
U

S \subset U

, such that

g

|

U
−
S

g{|_{U - S}}

is an element of

Γ
a

{\Gamma ^a}

. We construct

B
Γ

B\Gamma

, up to weak homotopy type, from

B

Γ
a

B{\Gamma ^a}

and the classifying spaces of certain groups of germs. As an application, the classifying space of the pseudogroup of orientation-preserving, piecewise linear homeomorphisms between open subsets of

R

{\mathbf {R}}

is weakly homotopy equivalent to

B

R

∗

B

R

B{\mathbf {R}}{\ast }B{\mathbf {R}}

.
Borel classes and closed games: Wadge-type and Hurewicz-type results
Authors: Louveau A. , Saint-Raymond J.
Published: June 2010

For each countable ordinal

ξ
\xi

and pair

(

A
0

,

A
1

)

({A_0},\,{A_1})

of disjoint analytic subsets of

2
ω

{2^\omega }

, we define a closed game

J
ξ

(

A
0

,

A
1

)

{J_\xi }({A_0},\,{A_1})

and a complete

Π
ξ
0

\Pi _\xi ^0

subset

H
ξ

{H_\xi }

of

2
ω

{2^\omega }

such that (i) a winning strategy for player I constructs a

∑
ξ
0

\sum _\xi ^0

set separating

A
0

{A_0}

from

A
1

{A_1}

; and (ii) a winning strategy for player II constructs a continuous map

φ
:

2
ω

→

A
0

∪

A
1

\varphi :{2^\omega } \to {A_0} \cup {A_1}

with

φ

−
1

(

A
0

)
=

H
ξ

{\varphi ^{ - 1}}({A_0}) = {H_\xi }

. Applications of this construction include: A proof in second order arithmetics of the statement "every

Π
ξ
0

\Pi _\xi ^0

non

∑
ξ
0

\sum _\xi ^0

set is

Π
ξ
0

\Pi _\xi ^0

-complete"; an extension to all levels of a theorem of Hurewicz about

∑
2
0

\sum _2^0

sets; a new proof of results of Kunugui, Novikov, Bourgain and the authors on Borel sets with sections of given class; extensions of results of Stern and Kechris. Our results are valid in arbitrary Polish spaces, and for the classes in Lavrentieff’s and Wadge’s hierarchies.
Weak limits of projections and compactness of subspace lattices
Authors: Wagner Bruce H.
Published: June 2010

A strongly closed lattice of projections on a Hilbert space is compact if the associated algebra of operators has a weakly dense subset of compact operators. If the lattice is commutative, there are necessary and sufficient conditions for compactness, one in terms of the structure of the lattice, and the other in terms of a measure on the lattice. There are many examples of compact lattices, and two main types of examples of noncompact lattices. Compactness is also related to the study of weak limits of certain projections.
Linear series with cusps and 𝑛-fold points
Authors: Schubert David
Published: June 2010

A linear series

(
V
,

L

)

(V,\,\mathcal {L})

on a curve

X
X

has an

n
n

-fold point along a divisor

D
D

of degree

n
n

if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension left-parenthesis upper V intersection upper H Superscript 0 Baseline left-parenthesis upper X comma script upper L left-parenthesis negative upper D right-parenthesis right-parenthesis right-parenthesis greater-than-or-slanted-equals dimension left-parenthesis upper V right-parenthesis minus 1">

dim
⁡
(
V
∩

H
0

(
X
,

L

(
−
D
)
)
)
⩾
dim
⁡
(
V
)
−
1

\dim (V \cap {H^0}(X,\,\mathcal {L}( - D))) \geqslant \dim (V) - 1

. The linear series has a cusp of order

e
e

at a point

P
P

if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension left-parenthesis upper V intersection upper H Superscript 0 Baseline left-parenthesis upper X comma script upper L left-parenthesis minus left-parenthesis e plus 1 right-parenthesis upper P right-parenthesis right-parenthesis right-parenthesis greater-than-or-slanted-equals dimension left-parenthesis upper V right-parenthesis minus 1">

dim
⁡
(
V
∩

H
0

(
X
,

L

(
−
(
e
+
1
)
P
)
)
)
⩾
dim
⁡
(
V
)
−
1

\dim (V \cap {H^0}(X,\,\mathcal {L}( - (e + 1)P))) \geqslant \dim (V) - 1

. Linear series with cusps and

n
n

-fold points are shown to exist if certain inequalities are satisfied. The dimensions of the families of linear series with cusps are determined for general curves.
On maximal functions associated to hypersurfaces and the Cauchy problem for strictly hyperbolic operators
Authors: Sogge Christopher D.
Published: June 2010

In this paper we prove a maximal Fourier integral theorem for the types of operators which arise in the study of maximal functions associated to averaging over hypersurfaces and also the Cauchy problem for hyperbolic operators. We apply the Fourier integral theorem to generalize Stein’s spherical maximal theorem (see [8]) and also to prove a sharp theorem for the almost everywhere convergence to

L
p

{L^p}

initial data of solutions to the Cauchy problem for second order strictly hyperbolic operators. Our results improve those of Greenleaf [3] and Ruiz [6]. We also can prove almost everywhere convergence to

L
2

{L^2}

initial data for operators of order

m
⩾
3

m \geqslant 3

.
Spectral measures, boundedly 𝜎-complete Boolean algebras and applications to operator theory
Authors: Ricker Werner J.
Published: June 2010

A systematic study is made of spectral measures in locally convex spaces which are countably additive for the topology of uniform convergence on bounded sets, briefly, the bounded convergence topology. Even though this topology is not compatible for the duality with respect to the pointwise convergence topology it turns out, somewhat surprisingly, that the corresponding

L
1

{L^1}

-spaces for the spectral measure are isomorphic as vector spaces. This fact, together with I. Kluvanek’s notion of closed vector measure (suitably developed in our particular setting) makes it possible to extend to the setting of locally convex spaces a classical result of W. Bade. Namely, it is shown that if

B
B

is a Boolean algebra which is complete (with respect to the bounded convergence topology) in Bade’s sense, then the closed operator algebras generated by

B
B

with respect to the bounded convergence topology and the pointwise convergence topology coincide.
Interpolation of Besov spaces
Authors: DeVore Ronald A. , Popov Vasil A.
Published: June 2010

We investigate Besov spaces and their connection with dyadic spline approximation in

L
p

(
Ω
)

{L_p}(\Omega )

,

0
>
p
⩽
∞

0 > p \leqslant \infty

. Our main results are: the determination of the interpolation spaces between a pair of Besov spaces; an atomic decomposition for functions in a Besov space; the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines.
Fonctions sphériques des espaces symétriques compacts
Authors: Clerc Jean-Louis
Published: June 2010

An integral formula, similar to Harish-Chandra’s formula for spherical functions on a noncompact Riemannian symmetric space

G

/

K

G/K

is given for the spherical functions of the compact dual

U

/

K

U/K

. As a consequence, an asymptotic expansion, as the parameter tends to infinity, is obtained, by using the (complex) stationary phase method. RÉSUMÉ. On démontre une formule intégrale pour les fonctions sphériques d’un espace symétrique de type compact

U

/

K

U/K

, analogue de la formule d’Harish-Chandra pour le dual non-compact

G

/

K

G/K

. En conséquence on obtient un équivalent asymptotique lorsque le paramètre tend vers l’infini, en utilisant la méthode de la phase stationnaire complexe.
On inductive limits of certain 𝐶*-algebras of the form 𝐶(𝑋)⊗𝐹
Authors: Pasnicu Cornel
Published: June 2010

A certain class of

∗

{\ast }

-homomorphisms

C
(
X
)
⊗
A
→
C
(
Y
)
⊗
B

C(X) \otimes A \to C(Y) \otimes B

, called compatible with a map defined on

Y
Y

with values in the set of all closed nonempty subsets of

X
X

, is studied. A local description of

∗

{\ast }

-homomorphisms

C
(
X
)
⊗
A
→
C
(
Y
)
⊗
B

C(X) \otimes A \to C(Y) \otimes B

is given considering separately the cases

X
=

point

X = {\text {point}}

and

A
=

C

A = {\mathbf {C}}

; this is done in terms of continuous "quasifields" of

C

∗

{C^{\ast }}

-algebras. Conditions under which an inductive limit

lim
→

(
C
(

X
k

)
⊗

A
k

,

Φ
k

)

\underrightarrow {\lim }(C({X_k}) \otimes {A_k},\,{\Phi _k})

, where each

Φ
k

{\Phi _k}

is of the above type, is

∗

{\ast }

-isomorphic with the tensor product of a commutative

C

∗

{C^{\ast }}

-algebra with an AF algebra are given. For such inductive limits the isomorphism problem is considered.
𝑣₁-periodic 𝐸𝑥𝑡 over the Steenrod algebra
Authors: Davis Donald M. , Mahowald Mark
Published: June 2010

For a large family of modules

M
M

over the

mod
2

\bmod 2

Steenrod algebra

A
A

,

Ext
A

s
,
t

⁡
(
M
,

Z

2

)

\operatorname {Ext} _A^{s,t}(M,\,{{\mathbf {Z}}_2})

is periodic for

t
>
4
s

t > 4s

with respect to operators

v
1

2
n

v_1^{2n}

of period

(

2
n

,

3
⋅

2
n

)

({2^n},\,3 \cdot {2^n})

for varying

n
n

.

v
1

−
1

Ext
A

s
,
t

⁡
(
M
,

Z

2

)

v_1^{ - 1}\operatorname {Ext} _A^{s,t}(M,\,{{\mathbf {Z}}_2})

can be defined by extending this periodic behavior outside this range. We calculate this completely when

M
=

H

∗

(
Y
)

M = {H^{\ast }}(Y)

, where

Y
Y

is the suspension spectrum of

R

P
2

∧

C

P
2

{\mathbf {R}}{P^2} \wedge {\mathbf {C}}{P^2}

.
A class of nonlinear Sturm-Liouville problems with infinitely many solutions
Authors: Schaaf Renate , Schmitt Klaus
Published: June 2010

This paper is concerned with the existence of solutions of nonlinear Sturm-Liouville problems whose linear part is at resonance. It is shown that such problems may have infinitely many solutions if the nonlinear perturbations are periodic.