bluuebunny/crossref_metadata_2025 huggingface dataset:
Banach space bifurcation theory
Authors: Westreich David
Published: July 2010

We consider the bifurcation problem for the nonlinear operator equation

x
=
λ
L
x
+
T
(
λ
,
x
,
y
)

x = \lambda Lx + T(\lambda ,x,y)

in a real Banach space

X
X

. Here

λ
0

{\lambda _0}

is an eigenvalue of the bounded linear operator

L
,
X
=
N
(
I
−

λ
0

L
)
⊕
R
(
I
−

λ
0

L
)
,
T
∈

C
1

L,X = N(I - {\lambda _0}L) \oplus R(I - {\lambda _0}L),T \in {C^1}

and

T
T

is of higher order in

x
x

. New techniques are developed to simplify the solution of the bifurcation problem. When

λ
0

{\lambda _0}

is a simple eigenvalue,

λ
0

{\lambda _0}

is shown to be a bifurcation point of the homogeneous equation (i.e.

y
≡
0

y \equiv 0

) with respect to 0. All solutions near

(

λ
0

,
0
)

({\lambda _0},0)

are shown to be of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis lamda left-parenthesis epsilon right-parenthesis comma x left-parenthesis epsilon right-parenthesis right-parenthesis comma 0 less-than-or-slanted-equals StartAbsoluteValue epsilon EndAbsoluteValue greater-than epsilon 0 comma lamda left-parenthesis epsilon right-parenthesis">

(
λ
(
ϵ
)
,
x
(
ϵ
)
)
,
0
⩽

|

ϵ

|

>

ϵ
0

,
λ
(
ϵ
)

(\lambda (\epsilon ),x(\epsilon )),0 \leqslant |\epsilon | > {\epsilon _0},\lambda (\epsilon )

and

x
(
ϵ
)

x(\epsilon )

are continuous and

λ
(
ϵ
)

\lambda (\epsilon )

and

x
(
ϵ
)

x(\epsilon )

are in

C
n

{C^n}

or real analytic as

T
T

is in

C

n
+
1

{C^{n + 1}}

or is real analytic. When

T
T

is real analytic and

λ
(
ϵ
)

λ
0

\lambda (\epsilon ){\lambda _0}

then there are at most two solution branches, and each branch is an analytic function of

λ
\lambda

for

λ
≠

λ
0

\lambda \ne {\lambda _0}

. If

T
T

is odd and analytic, for each

λ
∈
(

λ
0

−
δ
,

λ
0

)

\lambda \in ({\lambda _0} - \delta ,{\lambda _0})

(or

λ
∈
(

λ
0

,

λ
0

+
δ
)

\lambda \in ({\lambda _0},{\lambda _0} + \delta )

) there exist two nontrivial solutions near 0 and there are no solutions near 0 for

λ
∈
(

λ
0

,

λ
0

+
δ
)

\lambda \in ({\lambda _0},{\lambda _0} + \delta )

(or

λ
∈
(

λ
0

−
δ
,

λ
0

)

\lambda \in ({\lambda _0} - \delta ,{\lambda _0})

). We then demonstrate that in each sufficiently small neighborhood of a solution of the homogeneous bifurcation problem there are solutions of the nonhomogeneous equation (i.e.

y
≢
0

y \not \equiv 0

) depending continuously on a real parameter and on

y
y

. If

λ
0

{\lambda _0}

is an eigenvalue of odd multiplicity we prove it is a point of bifurcation of the homogeneous equation. With a strong restriction on the projection of

T
T

onto the null space of

I
−

λ
0

L

I - {\lambda _0}L

we show

λ
0

{\lambda _0}

is a bifurcation point of the homogeneous equation when

λ
0

{\lambda _0}

is a double eigenvalue. Counterexamples to some of our results are given when the hypotheses are weakened.
Radial limit sets on the torus
Authors: Hoffmann Laurence D.
Published: July 2010

Let

U
N

{U^N}

denote the unit polydisc and

T
N

{T^N}

the unit torus in the space of

N
N

complex variables. A subset

A
A

of

T
N

{T^N}

is called an (RL)-set (radial limit set) if to each positive continuous function

ρ
\rho

on

T
N

{T^N}

, there corresponds a function

f
f

in

H
∞

(

U
N

)

{H^\infty }({U^N})

such that the radial limit

|

f

|

∗

|f{|^ \ast }

of the absolute value of

f
f

equals

ρ
\rho

, a.e. on

T
N

{T^N}

and everywhere on

A
A

. If

N
>
1

N > 1

, the question of characterizing (RL)-sets is open, but two positive results are obtained. In particular, it is shown that

T
N

{T^N}

contains an (RL)-set which is homeomorphic to a cartesian product

K
×

T

N
−
1

K \times {T^{N - 1}}

, where

K
K

is a Cantor set. Also, certain countable unions of “parallel” copies of

T

N
−
1

{T^{N - 1}}

are shown to be (RL)-sets in

T
N

{T^N}

. In one variable, every subset of

T
T

is an (RL)-set; in fact, there is always a zero-free function

f
f

in

H
∞

(
U
)

{H^\infty }(U)

with the required properties. It is shown, however, that there exist a circle

A
⊂

T
2

A \subset {T^2}

and a positive continuous function

ρ
\rho

on

T
2

{T^2}

to which correspond no zero-free

f
f

in

H
∞

(

U
2

)

{H^\infty }({U^2})

with

|

f

|

∗

=
ρ

|f{|^ \ast } = \rho

a.e. on

T
2

{T^2}

and everywhere on

A
A

.
Mappings from 3-manifolds onto 3-manifolds
Authors: Wright Alden
Published: July 2010

Let f be a compact, boundary preserving mapping from the 3-manifold

M
3

{M^3}

onto the 3-manifold

N
3

{N^3}

. Let

Z
p

{Z_p}

denote the integers mod a prime p, or, if

p
=
0

p = 0

, the integers. (1) If each point inverse of f is connected and strongly 1-acyclic over

Z
p

{Z_p}

, and if

M
3

{M^3}

is orientable for

p
>
2

p > 2

, then all but a locally finite collection of point inverses of f are cellular. (2) If the image of the singular set of f is contained in a compact set each component of which is strongly acyclic over

Z
p

{Z_p}

, and if

M
3

{M^3}

is orientable for

p
≠
2

p \ne 2

, then

N
3

{N^3}

can be obtained from

M
3

{M^3}

by cutting out of

Int

M
3

\operatorname {Int} \;{M^3}

a compact 3-manifold with 2-sphere boundary, and replacing it by a

Z
p

{Z_p}

-homology 3-cell. (3) If the singular set of f is contained in a 0-dimensional set, then all but a locally finite collection of point inverses of f are cellular.
Torsion differentials and deformation
Authors: Rim D. S.
Published: July 2010

Let

S
S

-scheme

X
X

be a Schlessinger deformation of a curve

X
0

{X_0}

defined over a field

k
k

. In §§1 and 2, the dimension of the parameter space

S
S

, the relative differentials of

X
X

over

S
S

, and the fibres with singularity were studied, in case when

X
0

{X_0}

is locally complete-intersection. In §3 we show that if

k
k

-scheme

X
0

{X_0}

is a specialization of a smooth

k
k

-scheme, then the punctured spectrum

Spex
⁡
(

O

X

0
,
x

)

\operatorname {Spex} ({O_{{X_{0,x}}}})

has to be connected for every point

x
∈

X
0

x \in {X_0}

such that

dim
⁡

O

X

0
,
x

⩾
2

\dim {O_{{X_{0,x}}}} \geqslant 2

. In turn we construct a rigid singularity on a surface. In the last section a few conjectures amplifying those of P. Deligne are made.
Analytic capacity and approximation problems
Authors: Davie A. M.
Published: July 2010

We consider some problems concerning analytic capacity as a set function, which are relevant to approximation problems for analytic functions on plane sets. In particular we consider the question of semiadditivity of capacity. We obtain positive results in some special cases and give applications to approximation theory. In general we establish some equivalences among various versions of the semiadditivity question and certain questions in approximation theory.
Some open mapping theorems for measures
Authors: Ditor Seymour , Eifler Larry Q.
Published: July 2010

Given a compact Hausdorff space X, let

C
(
X
)

C(X)

be the Banach space of continuous real valued functions on X with sup norm and let

M
(
X
)

M(X)

be its dual considered as finite regular Borel measures on X. Let

U
(
X
)

U(X)

denote the closed unit ball of

M
(
X
)

M(X)

and let

P
(
X
)

P(X)

denote the nonnegative measures in

M
(
X
)

M(X)

of norm 1. A continuous map

φ
\varphi

of X onto another compact Hausdorff space Y induces a natural linear transformation

π
\pi

of

M
(
X
)

M(X)

onto

M
(
Y
)

M(Y)

defined by setting

π
(
μ
)
(
g
)
=
μ
(
g
∘
φ
)

\pi (\mu )(g) = \mu (g \circ \varphi )

for

μ
∈
M
(
X
)

\mu \in M(X)

and

g
∈
C
(
Y
)

g \in C(Y)

. It is shown that

π
\pi

is norm open on

U
(
X
)

U(X)

and on

A
⋅
P
(
X
)

A \cdot P(X)

for any subset A of the real numbers. If

φ
\varphi

is open, then

π
\pi

is

w
e
a
k

∗

\mathrm {weak}^*

open on

A
⋅
P
(
X
)

A \cdot P(X)

. Several examples are given which show that generalization in certain directions is not possible. The paper concludes with some remarks about continuous selections.
Geodesic flows on negatively curved manifolds. II
Authors: Eberlein Patrick
Published: July 2010

Let M be a complete Riemannian manifold with sectional curvature

K
≤
0

K \leq 0

, SM the unit tangent bundle of M,

T
t

{T_t}

the geodesic flow on SM and

Ω
⊆
S
M

\Omega \subseteq SM

the set of nonwandering points relative to

T
t

{T_t}

.

T
t

{T_t}

is topologically mixing (respectively topologically transitive) on SM if for any open sets 0, U of SM there exists

A
>
0

A > 0

such that

|
t
|

≥
A

\left | t \right | \geq A

implies

T
t

(
O
)
∩
U
≠
∅

{T_t}(O) \cap U \ne \emptyset

(respectively there exists

t

ε

R

t\;\varepsilon \;R

such that

T
t

(
O
)
∩
U
≠
∅

{T_t}(O) \cap U \ne \emptyset

). For each vector

v

ε

S
M

v\;\varepsilon \;SM

we define stable and unstable sets

W
s

(
v
)
,

W

s
s

(
v
)
,

W
u

(
v
)

{W^s}(v),{W^{ss}}(v),{W^u}(v)

and

W

u
u

(
v
)

{W^{uu}}(v)

, and we relate topological mixing (respectively topological transitivity) of

T
t

{T_t}

to the existence of a vector

v

∈

S
M

v\; \in \;SM

such that

W

s
s

(
v
)

{W^{ss}}(v)

(respectively

W
s

(
v
)

{W^s}(v)

) is dense in SM. If M is a Visibility manifold (implied by

K
≤
c
>
0

K \leq c > 0

) and if

Ω
=
S
M

\Omega = SM

then

T
t

{T_t}

is topologically mixing on SM. Let

S
n

=

{S_n} =

{Visibility manifolds M of dimension n such that

T
t

{T_t}

is topologically mixing on SM}. For each

n
≥
2

n \geq 2

,

S
n

{S_n}

is closed under normal (Galois) Riemannian coverings. If

M

∈

S
n

M\; \in \;{S_n}

we classify {

v

∈

S
M
:

W

s
s

(
v
)

v\; \in \;SM:\;{W^{ss}}(v)

is dense in SM}, and M is compact if and only if this set = SM. We also consider the case where

Ω
\Omega

is a proper subset of SM.
Isotopic unknotting in 𝐹×𝐼
Authors: Feustel C. D.
Published: July 2010

This paper is essentially a generalization of Unknotting in

M
2

×
I

{M^2} \times I

, by E. M. Brown. The major results in this paper concern the existence of ambient isotopies of unknotted arcs (families of arcs) properly embedded in

F
×
I

F \times I

.
Dual operations on saddle functions
Authors: McLinden L.
Published: July 2010

Dual operations on convex functions play a central role in the analysis of constrained convex optimization problems. Our aim here is to provide tools for a similar analysis of constrained concave-convex minimax problems. Two pairs of dual operations on convex functions, including addition and infimal convolution, are extended to saddle functions. For the resulting saddle functions much detailed information is given, including subdifferential formulas. Also, separable saddle functions are defined and some basic facts about them established.
On a theorem of Chern
Authors: Sampson J. H.
Published: July 2010

A new proof is given for Chern’s theorem showing that the Laplace operator for differential forms commutes with decomposition of forms associated with G-structures admitting a suitable connection. An analogous result is proved for symmetric tensor fields, and an application is made to determine all harmonic symmetric fields on a compact space of constant negative curvature. Vector-valued forms are also discussed.
On symmetric orders and separable algebras
Authors: Fossum T. V.
Published: July 2010

Let

K
K

be an algebraic number field, and let

Λ
\Lambda

be an

R
R

-order in a separable

K
K

-algebra

A
A

, where

R
R

is a Dedekind domain with quotient field

K
K

; let

Δ
\Delta

denote the center of

Λ
\Lambda

. A left

Λ
\Lambda

-lattice is a finitely generated left

Λ
\Lambda

-module which is torsion free as an

R
R

-module. For left

Λ
\Lambda

-modules

M
M

and

N
,

Ext
Λ
1

⁡
(
M
,
N
)

N, \operatorname {Ext} _\Lambda ^1(M,N)

is a module over

Δ
\Delta

. In this paper we examine ideals of

Δ
\Delta

which are the annihilators of

Ext
Λ
1

⁡
(
M
,
_
)

\operatorname {Ext} _\Lambda ^1(M,\_)

for certain classes of left

Λ
\Lambda

-lattices

M
M

related to the central idempotents of

A
A

, and we compute these ideals explicitly if

Λ
\Lambda

is a symmetric

R
R

-algebra. For a group algebra, these ideals determine the defect of a block. We then compare these annihilator ideals with another set of ideals of

Δ
\Delta

which are closely related to the homological different of

Λ
\Lambda

, and which in a sense measure deviation from separability. Finally we show that, for

Λ
\Lambda

to be separable over

R
R

, it is necessary and sufficient that

Λ
\Lambda

is a symmetric

R
R

-algebra,

Δ
\Delta

is separable over

R
R

, and the center of each localization of

Λ
\Lambda

at the maximal ideals of

R
R

maps onto the center of its residue class algebra.
Dual spaces of groups with precompact conjugacy classes
Authors: Liukkonen John R.
Published: July 2010

We show that a second countable locally compact type I group with a compact invariant neighborhood of the identity is CCR, and has a Hausdorff dual if and only if its conjugacy classes are precompact. We obtain sharper results if the group is almost connected or has a fundamental system of invariant neighborhoods of the identity. Along the way we show that for a locally compact abelian group

A
A

and a group

B
B

of topological group automorphisms of

A
,
A

A, A

has small

B
B

invariant neighborhoods at 1 if and only if

A
^

\hat A

has precompact orbits under the dual actions of

B
B

.
Addendum to: “Differential-boundary operators”
Authors: Krall Allan M.
Published: July 2010

The proof of a lemma and the statement of another were omitted from an earlier paper. This corrects that omission.
The Šilov boundary of 𝑀₀(𝐺)
Authors: Moran William
Published: July 2010

Let G be a locally compact abelian group and let

M
0

(
G
)

{M_0}(G)

be the convolution algebra consisting of those Radon measures on G whose Fourier-Stieltjes transforms vanish at infinity. It is shown that the Šilov boundary of

M
0

(
G
)

{M_0}(G)

is a proper subset of the maximal ideal space of

M
0

(
G
)

{M_0}(G)

. The measures constructed to prove this theorem are also used to obtain a stronger result for the full measure algebra

M
(
G
)

M(G)

.
The trace-class of a full Hilbert algebra
Authors: Kervin Michael R. W.
Published: July 2010

The trace-class of a full Hilbert algebra A is the set

τ
(
A
)
=
{
x
y

|

x
∈
A
,
y
∈
A
}

\tau (A) = \{ xy|x \in A,y \in A\}

. This set is shown to be a

∗
\ast

-ideal of A, and possesses a norm

τ
\tau

defined in terms of a positive hermitian linear functional on

τ
(
A
)

\tau (A)

. The norm

τ
\tau

is in general both incomplete and not an algebra norm, and is also not comparable with the Hilbert space norm

‖
‖

\left \|\right \|

on

τ
(
A
)

\tau (A)

. However, a one-sided ideal of

τ
(
A
)

\tau (A)

is closed with respect to one norm if and only if it is closed with respect to the other. The topological dual of

τ
(
A
)

\tau (A)

with respect to the norm

τ
\tau

is isometrically isomorphic to the set of left centralizers on A.
Differential geometric structures on principal toroidal bundles
Authors: Blair David E. , Ludden Gerald D. , Yano Kentaro
Published: July 2010

Under an assumption of regularity a manifold with an

f
f

-structure satisfying certain conditions analogous to those of a Kähler structure admits a fibration as a principal toroidal bundle ovet a Kähler manifold. In some natural special cases, additional information about the bundle space is obtained. Finally, curvature relations between the bundle space and the base space are studied.
Sets of formulas valid in finite structures
Authors: Selman Alan L.
Published: July 2010

A function

V

\mathcal {V}

is defined on the set of all subsets of

ω
\omega

so that for each set K, the value,

V

K

{\mathcal {V}_K}

, is the set of formulas valid in all structures of cardinality in K. An analysis is made of the dependence of

V

K

{\mathcal {V}_K}

on K, For any set K, let

d

(
K
)

{\text {d}}(K)

be the Kleene-Post degree to which K belongs. It is easily seen that for all infinite sets K,

d

(
K
)
∨
1
≤
d
(

V

K

)
≤

d

(
K

)
′

{\text {d}}(K) \vee 1 \leq d({\mathcal {V}_K}) \leq {\text {d}}(K)’

. On the other hand, we prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d left-parenthesis script upper V Subscript upper K logical-or upper J Baseline right-parenthesis equals d left-parenthesis script upper V Subscript upper K Baseline right-parenthesis logical-or d left-parenthesis script upper V Subscript upper J Baseline right-parenthesis">

d

(

V

K
∨
J

)
=

d

(

V

K

)
∨

d

(

V

J

)

{\text {d}}({\mathcal {V}_{K \vee J}}) = {\text {d}}({\mathcal {V}_K}) \vee {\text {d}}({\mathcal {V}_J})

, and use this to prove that, for any two degrees a and b,

a

≥
1
,

a

≤

b

≤

a

′

{\text {a}} \geq 1,{\text {a}} \leq {\text {b}} \leq {\text {a}}’

, and b r.e. a, there exists a set K so that

d

(
K
)
=

a

{\text {d}}(K) = {\text {a}}

and

d

(

V

K

)
=
b

{\text {d}}({\mathcal {V}_K}) = b

. Various similar results are also included.
Intersections of quasi-local domains
Authors: Prekowitz Bruce
Published: July 2010

Let

R
=
⋂

V
i

R = \bigcap {{V_i}}

be an intersection of quasi-local domains with a common quotient field

K
K

. Our goal is to find conditions on the

V
i

{V_i}

’s in order to get some or all of

V
i

{V_i}

’s to be localizations of

R
R

. We show for example that if

V
1

{V_1}

is a

1
1

-dimensional valuation domain and if

V
1

⊉

V
2

{V_1} \nsupseteq {V_2}

, then both

V
1

{V_1}

and

V
2

{V_2}

are localizations of

R
=

V
1

∩

V
2

R = {V_1} \cap {V_2}

.
A Laurent expansion for solutions to elliptic equations
Authors: Harvey Reese , Polking John C.
Published: July 2010

Let

P
(
ξ
)

P(\xi )

be a homogeneous elliptic polynomial of degree

m
m

. Let

E
E

be a fundamental solution for the partial differential operator

P
(
D
)

P(D)

. Suppose

Ω
\Omega

is a neighborhood of 0 in

R

n

{{\mathbf {R}}^n}

. Suppose

f
∈

C
∞

(
Ω
∼
{
0
}
)

f \in {C^\infty }(\Omega \sim \{ 0\} )

satisfies

P
(
D
)
f
=
0

P(D)f = 0

in

Ω
∼
{
0
}

\Omega \sim \{ 0\}

. It is shown that there is a differential operator

H
(
D
)

H(D)

(perhaps of infinite order) and a function

g
∈

C
∞

(
Ω
)

g \in {C^\infty }(\Omega )

satisfying

P
(
D
)
g
=
0

P(D)g = 0

in

Ω
\Omega

, such that

f
=
H
(
D
)
E
+
g

f = H(D)E + g

in

Ω
∼
{
0
}

\Omega \sim \{ 0\}

. This analog of the Laurent expansion for

f
f

is made unique by requiring that the Cauchy principal value of

H
(
D
)
E

H(D)E

be equal to

H
(
D
)
E

H(D)E

.
The measure algebra of a locally compact hypergroup
Authors: Dunkl Charles F.
Published: July 2010

A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. This paper deals only with commutative hypergroups. §1 contains definitions, a discussion of invariant measures, and a characterization of idempotent probability measures. §2 deals with the characters of a hypergroup. §3 is about hypergroups, which have generalized translation operators (in the sense of Levitan), and subhypergroups of such. In this case the set of characters provides much information. Finally §4 discusses examples, such as the space of conjugacy classes of a compact group, certain compact homogeneous spaces, ultraspherical series, and finite hypergroups.
Optimal arcs and the minimum value function in problems of Lagrange
Authors: Rockafellar R. Tyrrell
Published: July 2010

Existence theorems are proved for basic problems of Lagrange in the calculus of variations and optimal control theory, in particular problems for arcs with both endpoints fixed. Emphasis is placed on deriving continuity and growth properties of the minimum value of the integral as a function of the endpoints of the arc and the interval of integration. Control regions are not required to be bounded. Some results are also obtained for problems of Bolza. Conjugate convex functions and duality are used extensively in the development, but the problems themselves are not assumed to be especially “convex". Constraints are incorporated by the device of allowing the Lagrangian function to be extended-real-valued. This necessitates a new approach to the question of what technical conditions of regularity should be imposed that will not only work, but will also be flexible and general enough to meet the diverse applications. One of the underlying purposes of the paper is to present an answer to this question.
Variational problems within the class of solutions of a partial differential equation
Authors: Delver Robert
Published: July 2010

The subject of this paper is the optimization of a multiple integral over a domain

G
G

of a function, containing as arguments the independent variables, the unknown function and its partial derivatives up to order

l
l

, within the class of all sufficiently smooth solutions in

G
G

of a given partial differential equation of order greater than or equal to

2
l

2l

. Necessary conditions in the form of a boundary value problem are derived. A physical application occurs in the control with boundary and initial conditions of a process in

G
G

that is described by a specific partial differential equation.
Topological spaces and lattices of lower semicontinuous functions
Authors: Thornton M. C.
Published: July 2010

Lower semicontinuous real-valued functions on a space

X
X

form a conditionally complete distributive lattice

L
(
X
)

L(X)

. Those lattices which can be represented as

L
(
X
)

L(X)

for some

X
X

are characterized algebraically. All spaces producing isomorphic lattices ate determined. The class of spaces which are determined by their function lattices is introduced.
Induced flows
Authors: Petersen Karl , Shapiro Leonard
Published: July 2010

The construction of induced transformations is considered in the setting of topological dynamics. Sufficient conditions are given for induced flows to be topologically weakly mixing, and it is proved that Toeplitz flows and certain Sturmian flows satisfy these conditions and give rise to new and easily constructed classes of flows which have entropy zero and are uniquely ergodic, minimal, and topologically weakly mixing. An example is given of a weakly mixing minimal flow which is not topologically strongly mixing.
Adjoining inverses to commutative Banach algebras
Authors: Bollobás Béla
Published: July 2010

Let

A
A

be a commutative unital Banach algebra. Suppose

G
⊂
A

G \subset A

is such that

|

|

a

|

|

⩽

|

|

g
a

|

|

||a|| \leqslant ||ga||

for all

g
∈
G
,
a
∈
A

g \in G,a \in A

. Two questions are considered in the paper. Does there exist a superalgebra

B
B

of

A
A

in which every

g
∈
G

g \in G

is invertible? Can one always have also

|

|

g

−
1

|

|

⩽
1

||{g^{ - 1}}|| \leqslant 1

if

g
∈
G

g \in G

? Arens proved that if

G
=
{
g
}

G = \{ g\}

then there is an algebra containing

g

−
1

{g^{ - 1}}

, with

|

|

g

−
1

|

|

⩽
1

||{g^{ - 1}}|| \leqslant 1

. In the paper it is shown that if

G
G

is countable

B
B

exists, but if

G
G

is uncountable, this is not necessarily so. The answer to the second question is negative even if

G
G

consists of only two elements.
Cobordism invariants, the Kervaire invariant and fixed point free involutions
Authors: Browder William
Published: July 2010

Conditions are found which allow one to define an absolute version of the Kervaire invariant in

Z
2

{Z_2}

of a

Wu - 

(
q
+
1
)

{\text {Wu - }}(q + 1)

oriented 2q-manifold. The condition is given in terms of a new invariant called the spectral cobordism invariant. Calculations are then made for the Kervaire invariant of the n-fold disjoint union of a manifold M with itself, which are then applied with

M
=

P

2
q

M = {P^{2q}}

, the real protective space. These give examples where the Kervaire invariant is not defined, and other examples where it has value

1
∈

Z

2

1 \in {{\mathbf {Z}}_2}

. These results are then applied to construct examples of smooth fixed point free involutions of homotopy spheres of dimension

4
k
+
1

4k + 1

with nonzero desuspension obstruction, of which some Brieskorn spheres are examples (results obtained also by Berstein and Giffen). The spectral cobordism invariant is also applied directly to these examples to give another proof of a result of Atiyah-Bott. The question of which values can be realized as the sequence of Kervaire invariants of characteristic submanifolds of a smooth homotopy real projective space is discussed with some examples. Finally a condition is given which yields smooth embeddings of homotopy

P
m

{P^m}

’s in

R

m
+
k

{R^{m + k}}

(which has been applied by E. Rees).
The polynomial identities of the Grassmann algebra
Authors: Krakowski D. , Regev A.
Published: July 2010

By using the theory of codimensions the

T
T

-ideal of polynomial identities of the Grassmann (exterior) algebra is computed.
The automorphism group of an abelian 𝑝-group and its normal 𝑝-subgroups
Authors: Hausen Jutta
Published: July 2010

Let

Γ

{\mathbf {\Gamma }}

be the automorphism group of a nonelementary reduced abelian p-group,

p
≥
5

p \geq 5

. It is shown that every noncentral normal subgroup of

Γ

{\mathbf {\Gamma }}

contains a noncentral normal subgroup

Δ

{\mathbf {\Delta }}

of

Γ

{\mathbf {\Gamma }}

such that

Δ

p

=
1

{{\mathbf {\Delta }}^p} = 1

. Furthermore, every cyclic normal subgroup of

Γ

{\mathbf {\Gamma }}

is contained in the center of

Γ

{\mathbf {\Gamma }}

.
Existence of solutions of ordinary differential equations with generalized boundary conditions
Authors: Bernfeld Stephen R. , Lakshmikantham V.
Published: July 2010

An investigation of the existence of solutions of the nonlinear boundary value problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime equals f left-parenthesis t comma x comma y right-parenthesis comma y prime equals g left-parenthesis t comma x comma y right-parenthesis comma upper A upper V left-parenthesis a comma x left-parenthesis a right-parenthesis comma y left-parenthesis a right-parenthesis right-parenthesis plus upper B upper W left-parenthesis a comma x left-parenthesis a right-parenthesis comma y left-parenthesis a right-parenthesis right-parenthesis equals upper C 1 comma upper C upper V left-parenthesis b comma x left-parenthesis b right-parenthesis comma y left-parenthesis b right-parenthesis right-parenthesis plus upper D upper W left-parenthesis b comma x left-parenthesis b right-parenthesis comma y left-parenthesis b right-parenthesis right-parenthesis equals upper C 2">

x
′

=
f
(
t
,
x
,
y
)
,

y
′

=
g
(
t
,
x
,
y
)
,
A
V
(
a
,
x
(
a
)
,
y
(
a
)
)
+
B
W
(
a
,
x
(
a
)
,
y
(
a
)
)
=

C
1

,
C
V
(
b
,
x
(
b
)
,
y
(
b
)
)
+
D
W
(
b
,
x
(
b
)
,
y
(
b
)
)
=

C
2

x’ = f(t,x,y),y’ = g(t,x,y),AV(a,x(a),y(a)) + BW(a,x(a),y(a)) = {C_1},CV(b,x(b),y(b)) + DW(b,x(b),y(b)) = {C_2}

, is made. Here we assume

g
,
f
:
[
a
,
b
]
×

R
p

×

R
q

→

R
p

g,f:[a,b] \times {R^p} \times {R^q} \to {R^p}

are continuous, and

V
,
W
:
[
a
,
b
]
×

R
p

×

R
q

→
R

V,W:[a,b] \times {R^p} \times {R^q} \to R

are continuous and locally Lipschitz. The main techniques used are the theory of differential inequalities and Lyapunov functions.
The existence and uniqueness of nonstationary ideal incompressible flow in bounded domains in 𝑅₃
Authors: Swann H. S. G.
Published: July 2010

It is shown here that the mixed initial-boundary value problem for the Euler equations for ideal flow in bounded domains of

R
3

{R_3}

has a unique solution for a small time interval. The existence of a solution is shown by converting the equations to an equivalent system involving the vorticity and applying Schauder’s fixed point theorem to an appropriate mapping.
A metric characterization of cells
Authors: Nunnally Ellard
Published: July 2010

We examine finite dimensional compact convex metric spaces each having the property that the union of two line segments in the space, having more than one point in common, is a line segment. The question has been asked (Borsuk; Bing) whether each such space is a cell. The answer is yes if the dimension of the space is

≤
2

\leq 2

(Lelek and Nitka) or 3 (Rolfsen). Here we provide an affirmative answer for arbitrary finite dimension provided the space has the additional property that the join of any point to any line segment in the space is a convex set.
Cross-sections of symplectic Stiefel manifolds
Authors: Sigrist François , Suter Ulrich
Published: July 2010

The cross-section problem for the symplectic Stiefel manifolds is solved, using the now-proved Adams conjecture.
Torsion in 𝐾-theory and the Bott maps
Authors: Lundell Albert T.
Published: July 2010

The nonstable Bott maps

b

n
′

:
U
(
n
)
→

Ω
2

U
(
n
+
1
)

b{’_n}:U(n) \to {\Omega ^2}U(n + 1)

for the unitary group are studied as to their behavior under iteration. They are then used to define and compute the coefficients of a spectrum. The corresponding cohomology theory is developed and compared with reduced complex K-theory. In this context the Chern character is induced by a map of spectra. The complex e-invariant appears as a coboundary in the long exact sequence of a cofibration.
Infinite particle systems
Authors: Port Sidney C. , Stone Charles J.
Published: July 2010

We consider a system of denumerably many particles that are distributed at random according to a stationary distribution P on some closed subgroup X of Euclidean space. We assume that the expected number of particles in any compact set is finite. We investigate the relationship between P and the distribution Q of particles as viewed from a particle selected “at random” from some set. The distribution Q is called the tagged particle distribution. We give formulas for computing P in terms of Q and Q in terms of P and show that, with the appropriate notion of convergences,

P
n

→
P

{P_n} \to P

implies

Q
n

→
Q

{Q_n} \to Q

and vice versa. The particles are allowed to move in an appropriate translation invariant manner and we show that the tagged particle distribution Q’ at a later time 1 is the same as the distribution of particles at time 1 as viewed from a particle selected “at random” from those initially in some set. We also show that Q’ is the same as the distribution of particles at time 1 as viewed from a particle selected at random from those at the origin, when initially the particles are distributed according to Q. The one-dimensional case is treated in more detail. With appropriate topologies, we show that in this case there is a homeomorphism between the collection of stationary distributions P and tagged particle distributions Q. A stationary spacings distribution

Q
0

{Q_0}

related to Q is introduced, and we show that with the appropriate topology the map taking Q to

Q
0

{Q_0}

is a homeomorphism. Explicit expressions are found for all these maps and their inverses. The paper concludes by using the one-dimensional results to find stationary distributions for a class of motions of denumerably many unit intervals and to establish criteria for convergence to one of these distributions.
On the principal series of 𝐺𝑙_{𝑛} over𝑝-adic fields
Authors: Howe Roger E.
Published: July 2010

The entire principal series of

G
=
G

l
n

(
F
)

G = G{l_n}(F)

, for a p-adic field F, is analyzed after the manner of the analysis of Bruhat and Satake for the spherical principal series. If K is the group of integral matrices in

G

l
n

(
F
)

G{l_n}(F)

, then a “principal series” of representations of K is defined. It is shown that precisely one of these occurs, and only once, in a given principal series representation of G. Further, the spherical function algebras attached to these representations of K are all shown to be abelian, and their explicit spectral decomposition is accomplished using the principal series of G. Computation of the Plancherel measure is reduced to MacDonald’s computation for the spherical principal series, as is computation of the spherical functions themselves.
Universal generators for varieties of nuclear spaces
Authors: Rosenberger B.
Published: July 2010

It is shown that a product of several copies of

Λ
(

β
ϕ

)

\Lambda ({\beta ^\phi })

is a universal

ϕ
\phi

-nuclear space if the power series space

Λ
(

β
ϕ

)

\Lambda ({\beta ^\phi })

with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript k Superscript phi Baseline equals minus log left-parenthesis phi Superscript negative 1 Baseline left-parenthesis 1 slash StartRoot k plus 1 EndRoot right-parenthesis right-parenthesis comma k epsilon left-brace 0 comma 1 comma 2 comma midline-horizontal-ellipsis right-brace">

β
k
ϕ

=
−
log
⁡
(

ϕ

−
1

(
1

/

k
+
1

)
)
,
k

ϵ

{
0
,
1
,
2
,
⋯
}

\beta _k^\phi = - \log ({\phi ^{ - 1}}(1/\sqrt {k + 1} )),k\;\epsilon \;\{ 0,1,2, \cdots \}

, is

ϕ
\phi

-nuclear; here

ϕ
=
[
0
,
∞
)
→
[
0
,
∞
)

\phi = [0,\infty ) \to [0,\infty )

is a continuous, strictly increasing subadditive function with

ϕ
(
0
)
=
0

\phi (0) = 0

. In case

Λ
(

β
ϕ

)

\Lambda ({\beta ^\phi })

is not

ϕ
\phi

-nuclear the sequence space

Λ
(

l
ϕ
+

)

\Lambda (l_\phi ^ + )

is a

ϕ
\phi

-nuclear space with the property that every

ϕ
\phi

-nuclear space is isomorphic to a subspace of a product of

Λ
(

l
ϕ
+

)

\Lambda (l_\phi ^ + )

if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit sup Underscript t right-arrow 0 Endscripts left-parenthesis phi left-parenthesis t right-parenthesis right-parenthesis Superscript negative 1 Baseline phi left-parenthesis StartRoot t EndRoot right-parenthesis greater-than normal infinity">

lim

sup

t
→
0

(
ϕ
(
t
)

)

−
1

ϕ
(

t

)
>
∞

{\lim \;\sup _{t \to 0}}{(\phi (t))^{ - 1}}\phi (\sqrt t ) > \infty

.
Decomposition theorems of Riemannian manifolds
Authors: Wang Pyng
Published: July 2010

Given two complementary orthogonal parallel foliations on a complete connected Riemannian manifold M, a necessary and sufficient condition for the direct product of the two leaves through a point m being a covering manifold of M is obtained. It is shown that the direct product of the two leaves through m of the two foliations is a Riemannian covering of M if the two leaves are regular at m. Moreover, if one of the foliations is regular and the intersection of the two leaves through m contains only the point m, then M is isometric to the direct product of the two leaves.
On Green’s function of an 𝑛-point boundary value problem
Authors: Das K. M. , Vatsala A. S.
Published: July 2010

The Green’s function

g
n

(
x
,
s
)

{g_n}(x,s)

for an n-point boundary value problem,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis x right-parenthesis equals 0 comma y left-parenthesis a 1 right-parenthesis equals y left-parenthesis a 2 right-parenthesis equals midline-horizontal-ellipsis equals y left-parenthesis a Subscript n Baseline right-parenthesis equals 0">

y

(
n
)

(
x
)
=
0
,
y
(

a
1

)
=
y
(

a
2

)
=
⋯
=
y
(

a
n

)
=
0

{y^{(n)}}(x) = 0,y({a_1}) = y({a_2}) = \cdots = y({a_n}) = 0

is explicitly given. As a tool for discussing

sgn
⁡

g
n

(
x
,
s
)

\operatorname {sgn} g_n(x,s)

on the square

[

a
1

,

a
n

]
×
[

a
1

,

a
n

]

[{a_1},{a_n}] \times [{a_1},{a_n}]

, some results about polynomials with coefficients as symmetric functions of a’s are obtained. It is shown that
\[

∫

a
1

a
n

|

g
n

(
x
,
s
)

|

d
s

\int _{{a_1}}^{{a_n}} {|{g_n}(x,s)|ds}

\]
is a suitable polynomial in x. Applications to n-point boundary value problems and lower bounds for

a
m

(
m
≥
n
)

{a_m}\;(m \geq n)

are included.
Perfect open and distinguishable multivalued maps
Authors: Braude Eric John
Published: July 2010

It is shown that perfect open multivalued maps preserve

Z

\mathcal {Z}

-analytic sets (which include compact zero sets) as well as other objects of descriptive set theory. The concept of “distinguishability", introduced by Frolík, is applied to multivalued maps, yielding a new class of such maps with similar preservation properties. That the projection of a compact zero set is a zero set is one corollary, and another is a generalized

G

δ

{\mathcal {G}_\delta }

diagonal metrization theorem.
Generalized semigroups of quotients
Authors: Hinkle C. V.
Published: July 2010

For S a semigroup with 0 and

M
S

{M_S}

a right S-set, certain classes of sub S-sets called right quotient filters are defined. A study of these right quotient filters is made and examples are given including the classes of intersection large and dense sub S-sets respectively. The general semigroup of right quotients Q corresponding to a right quotient filter on a semigroup S is developed and basic properties of this semigroup are noted. A nonzero regular semigroup S is called primitive dependent if each nonzero right ideal of S contains a 0-minimal right ideal of S. The theory developed in the paper enables us to characterize all primitive dependent semigroups having singular congruence the identity in terms of subdirect products of column monomial matrix semigroups over groups.
The degree of approximation by Chebyshevian splines
Authors: DeVore R. , Richards F.
Published: July 2010

This paper studies the connections between the smoothness of a function and its degree of approximation by Chebyshevian splines. This is accomplished by proving companion direct and inverse theorems which give a characterization of smoothness in terms of degree of approximation. A determination of the saturation properties is included.
Maximal regular right ideal space of a primitive ring. II
Authors: Koh Kwangil , Luh Hang
Published: July 2010

If

R
R

is a ring, let

X
(
R
)

X(R)

be the set of maximal regular right ideals of

R
R

. For each nonempty subset

E
E

of

R
R

, define the hull of

E
E

to be the set

{
I
ϵ

X
(
R
)

|

 
E
⊆
I
}

\{ I \epsilon \, X(R)|\ E \subseteq I\}

and the support of

E
E

to be the complement of the hull of

E
E

. Topologize

X
(
R
)

X(R)

by taking the supports of right ideals of

R
R

as a subbase. If

R
R

is a right primitive ring, then

X
(
R
)

X(R)

is homeomorphic to an open subset of a compact space

X
(

R
#

)

X({R^\# })

of a right primitive ring

R
#

{R^\# }

, and

X
(
R
)

X(R)

is a discrete space if and only if

X
(
R
)

X(R)

is a compact Hausdorff space if and only if either

R
R

is a finite ring or a division ring. Call a closed subset

F
F

of

X
(
R
)

X(R)

a line if

F
F

is the hull of

I
∩
J

I \cap J

for some two distinct elements

I
I

and

J
J

in

X
(
R
)

X(R)

. If

R
R

is a semisimple ring, then every line contains an infinite number of points if and only if either

R
R

is a division ring or

R
R

is a dense ring of linear transformations of a vector space of dimension two or more over an infinite division ring such that every pair of simple (right)

R
R

-modules are isomorphic.
Algebraic cohomology of topological groups
Authors: Wigner David
Published: July 2010

A general cohomology theory for topological groups is described, and shown to coincide with the theories of C. C. Moore [12] and other authors. We also recover some invariants from algebraic topology.
Global regularity for ∂̄ on weakly pseudo-convex manifolds
Authors: Kohn J. J.
Published: July 2010

Let

M
′

M’

be a complex manifold and let

M
⊂⊂

M
′

M \subset \subset M’

be an open pseudo-convex submanifold with a smooth boundary which can be exhausted by strongly pseudo-convex submanifolds. The main result of this paper is the following: Given a

∂
¯

\overline \partial

-closed

(
p
,
q
)

(p,q)

-form

α
\alpha

, which is

C
∞

{C^\infty }

on

M
¯

\overline M

and which is cohomologous to zero on

M
M

then for every

m
m

there exists a

(
p
,
q
−
1
)

(p,q - 1)

-form

u

(
m
)

{u_{(m)}}

which is

C
m

{C^m}

on

M
¯

\overline M

such that

∂
¯

u

(
m
)

=
α

\overline \partial {u_{(m)}} = \alpha

.
The representation of norm-continuous multipliers on 𝐿^{∞}-spaces
Authors: Hively Gregory A.
Published: July 2010

If G is a group and

L

∞

(
G
,

S

)

{\mathcal {L}^\infty }(G,\mathcal {S})

is an appropriate space of bounded measurable functions on G, a representation is obtained for the algebra of norm-continuous multipliers on

L

∞

(
G
,

S

)

{\mathcal {L}^\infty }(G,\mathcal {S})

as an algebra of bounded additive set functions on G. If G is a locally compact group, a representation of the norm-continuous multipliers on the quotient space

L

∞

(
G
)

{\mathcal {L}^\infty }(G)

is obtained in terms of a quotient algebra of bounded additive set functions on G.
Existence of sum and product integrals
Authors: Helton Jon C.
Published: July 2010

Functions are from

R
×
R

R \times R

to R, where R represents the set of real numbers. If c is a number and either (1)

∫
a
b

G
2

\smallint _a^b{G^2}

exists and

∫
a
b

G

\smallint _a^bG

exists, (2)

∫
a
b

G

\smallint _a^bG

exists and

a

Π

b

(
1
+
G
)

_a{{\mathbf {\Pi }}^b}(1 + G)

exists and is not zero or (3) each of

a

Π

b

(
1
+
G
)

_a{{\mathbf {\Pi }}^b}(1 + G)

and

a

Π
b

(
1
−
G
)

_a{\Pi ^b}(1 - G)

exists and is not zero, then

∫
a
b

c
G

\smallint _a^bcG

exists,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript a Superscript b Baseline StartAbsoluteValue c upper G minus integral c upper G EndAbsoluteValue equals 0 comma Subscript x Baseline bold upper Pi Superscript y Baseline left-parenthesis 1 plus c upper G right-parenthesis">

∫
a
b

|

c
G
−
∫
c
G

|

=
0

,
x

Π

y

(
1
+
c
G
)

\smallint _a^b|cG - \smallint cG| = 0{,_x}{{\mathbf {\Pi }}^y}(1 + cG)

exists for

a
≤
x
>
y
≤
b

a \leq x > y \leq b

and

∫
a
b

|

1
+
c
G
−

Π

(
1
+
c
G
)

|

=
0

\smallint _a^b|1 + cG - {\mathbf {\Pi }}(1 + cG)| = 0

. Furthermore, if H is a function such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript x right-arrow p Superscript minus Baseline Endscripts upper H left-parenthesis x comma p right-parenthesis comma limit Underscript x right-arrow p Superscript plus Baseline Endscripts upper H left-parenthesis p comma x right-parenthesis comma limit Underscript x comma y right-arrow p Superscript minus Baseline Endscripts upper H left-parenthesis x comma y right-parenthesis">

lim

x
→

p
−

H
(
x
,
p
)
,

lim

x
→

p
+

H
(
p
,
x
)
,

lim

x
,
y
→

p
−

H
(
x
,
y
)

{\lim _{x \to {p^ - }}}H(x,p),{\lim _{x \to {p^ + }}}H(p,x),{\lim _{x,y \to {p^ - }}}H(x,y)

and

lim

x
,
y
→

p
+

H
(
x
,
y
)

{\lim _{x,y \to {p^ + }}}H(x,y)

exist for each

p
∈
[
a
,
b
]
,
n
≥
2

p \in [a,b],n \geq 2

is an integer, and G satisfies either (1), (2) or (3) of the above, then

∫
a
b

H

G
n

\smallint _a^bH{G^n}

exists,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript a Superscript b Baseline StartAbsoluteValue upper H upper G Superscript n Baseline minus integral upper H upper G Superscript n Baseline EndAbsoluteValue equals 0 comma Subscript x Baseline bold upper Pi Superscript y Baseline left-parenthesis 1 plus upper H upper G Superscript n Baseline right-parenthesis">

∫
a
b

|

H

G
n

−
∫
H

G
n

|

=
0

,
x

Π

y

(
1
+
H

G
n

)

\smallint _a^b|H{G^n} - \smallint H{G^n}| = 0{,_x}{{\mathbf {\Pi }}^y}(1 + H{G^n})

exists for

a
≤
x
>
y
≤
b

a \leq x > y \leq b

and

∫
a
b

|

1
+
H

G
n

−

Π

(
1
+
H

G
n

)

|

=
0

\smallint _a^b|1 + H{G^n} - {\mathbf {\Pi }}(1 + H{G^n})| = 0

.
The Rudin-Keisler ordering of 𝑃-points
Authors: Blass Andreas
Published: July 2010

The Stone-Čech compactification

β
ω

\beta \omega

of the discrete space

ω
\omega

of natural numbers is weakly ordered by the relation “D is the image of E under the canonical extension

β
f
:
β
ω
→
β
ω

\beta f:\beta \omega \to \beta \omega

of some map

f
:
ω
→
ω

f:\omega \to \omega

.” We shall investigate the structure, with respect to this ordering, of the set of P-points of

β
ω
−
ω

\beta \omega - \omega

.
A second quadrant homotopy spectral sequence
Authors: Bousfield A. K. , Kan D. M.
Published: July 2010

For each cosimplicial simplicial set with basepoint, the authors construct a homotopy Spectral sequence generalizing the usual spectral sequence for a second quadrant double chain complex. For such homotopy spectral sequences, a uniqueness theorem and a general multiplicative pairing are established. This machinery is used elsewhere to show the equivalence of various unstable Adams spectral sequences and to construct for them certain composition pairings and Whitehead products.
Pairings and products in the homotopy spectral sequence
Authors: Bousfield A. K. , Kan D. M.
Published: July 2010

Smash and composition pairings, as well as Whitehead products are constructed in the unstable Adams spectral sequence; and these pairings and products are described homologically on the

E
2

{E_2}

. level. In the special case of the Massey-Peterson spectral sequence, the composition action is given homologically by the Yoneda product, while the Whitehead product vanishes. It is also shown that the unstable Adams spectral sequence over the rationals, with its Whitehead products, is given by the primitive elements in the rational cobar spectral sequence.
Homeomorphisms of Hilbert cube manifolds
Authors: Chapman T. A.
Published: July 2010

It is shown in this paper that the homeomorphism group of any compact Hubert cube manifold is locally contractible. The proof uses some standard infinite-dimensional techniques along with an infinite-dimensional version of the torus-homeomorphism idea which was used by Edwards and Kirby to establish a corresponding finite-dimensional result.
The lattice triple packing of spheres in Euclidean space
Authors: Purdy G. B.
Published: July 2010

We say that a lattice

Λ
\Lambda

in

n
n

-dimensional Euclidean space

E
n

{E_n}

provides a

k
k

-fold packing for spheres of radius 1 if, when open spheres of radius 1 are centered at the points of

Λ
\Lambda

, no point of space lies in more than

k
k

spheres. The multiple packing constant

Δ
k

(
n
)

\Delta _k^{(n)}

is the smallest determinant of any lattice with this property. In the plane, the first three multiple packing constants

Δ
2

(
2
)

,

Δ
3

(
2
)

\Delta _2^{(2)},\Delta _3^{(2)}

, and

Δ
4

(
2
)

\Delta _4^{(2)}

are known, due to the work of Blundon, Few, and Heppes. In

E
3

,

Δ
2

(
3
)

{E_3},\Delta _2^{(3)}

is known, because of work by Few and Kanagasabapathy, but no other multiple packing constants are known. We show that

Δ
3

(
3
)

⩽
8

38

/

27

\Delta _3^{(3)} \leqslant 8\sqrt {38} /27

and give evidence that

Δ
3

(
3
)

=
8

38

/

27

\Delta _3^{(3)} = 8\sqrt {38} /27

. We show, in fact, that a lattice with determinant

8

38

/

27

8\sqrt {38} /27

gives a local minimum of the determinant among lattices providing a

3
3

-fold packing for the unit sphere in

E
3

{E_3}

.
Remarks on global hypoellipticity
Authors: Greenfield Stephen J. , Wallach Nolan R.
Published: July 2010

We study differential operators D which commute with a fixed normal elliptic operator E on a compact manifold M. We use eigenfunction expansions relative to E to obtain simple conditions giving global hypoellipticity. These conditions are equivalent to D having parametrices in certain spaces of functions or distributions. An example is given by M = compact Lie group and and E = Casimir operator, with D any invariant differential operator. The connections with global subelliptic estimates are investigated.
Quadratic expressions in a free boson field
Authors: Klein Abel
Published: July 2010

Quadratic expressions in a massive spinless free Boson field are treated by an appropriate extension of the method of second quantization. A certain class of these second quantized operators is shown to generate semigroups that act on a suitable scale of

L
p

{L_p}

-spaces, obtained through the diagonalization of the field at a fixed time, in a particularly regular fashion. The techniques are developed first in an abstract setting, and then applied to the neutral scalar free field. The locally correct generator of Lorentz transformations for

P

(
φ

)
2

P{(\varphi )_2}

is studied in detail, and essential selfadjointness is shown. These techniques are also used to solve explicitly the

(

φ
2

)
n

{({\varphi ^2})_n}

model.
Some results on the length of proofs
Authors: Parikh R. J.
Published: July 2010

Given a theory T, let

⊢
T
k

A

\vdash _T^kA

mean “A has a proof in T of at most k lines". We consider a formulation

P

A
∗

P{A^\ast }

of Peano arithmetic with full induction but addition and multiplication being ternary relations. We show that

⊢
k

A

{ \vdash ^k}A

is decidable for

P

A
∗

P{A^\ast }

and hence

P

A
∗

P{A^\ast }

is closed under a weak

ω
\omega

-rule. An analogue of Gödel’s theorem on the length of proofs is an easy corollary.