bluuebunny/crossref_metadata_2025 huggingface dataset:
Pseudojump operators. I. The r.e. case
Authors: Jockusch Carl G. , Shore Richard A.
Published: June 2010

Call an operator

J
J

on the power set of

ω
\omega

a pseudo jump operator if

J
(
A
)

J(A)

is uniformly recursively enumerable in

A
A

and

A
A

is recursive in

J
(
A
)

J(A)

for all subsets

A
A

of

ω
\omega

. Thus the (Turing) jump operator is a pseudo jump operator, and any existence proof in the theory of r.e. degrees yields, when relativized, one or more pseudo jump operators. Extending well-known results about the jump, we show that for any pseudo jump operator

J
J

, every degree

⩾

0

′

\geqslant {\mathbf {0}}’

has a representative in the range of

J
J

, and that there is a nonrecursive r.e. set

A
A

with

J
(
A
)

J(A)

of degree

0

′

{\mathbf {0}}’

. The latter result yields a finite injury proof in two steps that there is an incomplete high r.e. degree, and by iteration analogous results for other levels of the

H
n

{H_n}

,

L
n

{L_n}

hierarchy of r.e. degrees. We also establish a result on pairs of pseudo jump operators. This is combined with Lachlan’s result on the impossibility of combining splitting and density for r.e. degrees to yield a new proof of Harrington’s result that

0

′

{\mathbf {0}}’

does not split over all lower r.e. degrees.
Stochastic waves
Authors: Dynkin E. B. , Vanderbei R. J.
Published: June 2010

Let

ϕ
\phi

be a real valued function defined on the state space of a Markov process

x
t

{x_t}

. Let

τ
t

{\tau _t}

be the first time

x
t

{x_t}

gets to a level set of

ϕ
\phi

which is

t
t

units higher than the one on which it started. We call the time changed process

x
~

t

=

x

τ
t

\tilde {x}_{t} = x_{{\tau _t}}

a stochastic wave. We give conditions under which this process is Markovian and we evaluate its infinitesimal operator.
Structural stability and group cohomology
Authors: Fleming Philip J.
Published: June 2010

We prove a version of the theorem of Stowe concerning the stability of stationary points of a differentiable group action which is valid on Hilbert manifolds. This result is then used to show that the vanishing of certain cohomology groups is sufficient to guarantee structural semistability for a differentiable action of a group of finite type on a closed smooth manifold. We then apply this to groups of diffeomorphisms of the circle.
Krull dimension of differential operator rings. III. Noncommutative coefficients
Authors: Goodearl K. R. , Lenagan T. H.
Published: June 2010

This paper is concerned with the Krull dimension (in the sense of Gabriel and Rentschler) of a differential operator ring

S
[
θ
;
δ
]

S[\theta ;\delta ]

, where

S
S

is a right noetherian ring with finite Krull dimension

n
n

and

δ
\delta

is a derivation on

S
S

. The main theorem states that

S
[
θ
;
δ
]

S[\theta ;\delta ]

has Krull dimension

n
n

unless there exists a simple right

S
S

-module

A
A

such that

A

⊗
S

S
[
θ
;
δ
]

A{ \otimes _S}S[\theta ;\delta ]

is not simple (as an

S
[
θ
;
δ
]

S[\theta ;\delta ]

-module) and

A
A

has height

n
n

in the sense that there exist critical right

S
S

-modules

A
=

A
0

,

A
1

,
…
,

A
n

A = {A_0},{A_1},\ldots ,{A_n}

such that each

A
i

⊗
s

S
[
θ
;
δ
]

{A_i} \otimes _s S[\theta ;\delta ]

is a critical

S
[
θ
;
δ
]

S[\theta ;\delta ]

-module, each

A
i

{A_i}

is a minor subfactor of

A

i
+
1

{A_{i + 1}}

and

A
n

{A_n}

is a subfactor of

S
S

. If such an

A
A

does exist, then

S
[
θ
;
δ
]

S[\theta ;\delta ]

has Krull dimension

n
+
1

n + 1

. This criterion is simplified when

S
S

is fully bounded, in which case it is shown that

S
[
θ
;
δ
]

S[\theta ;\delta ]

has Krull dimension

n
n

unless

S
S

has a maximal ideal

M
M

of height

n
n

such that either

char(

S

/

M
)
>
0

{\text {char(}}S/M) > 0

or

δ
(
M
)
⊆
M

\delta (M) \subseteq M

, and in these cases

S
[
θ
;
δ
]

S[\theta ;\delta ]

has Krull dimension

n
+
1

n + 1

.
On the grössencharacter of an abelian variety in a parametrized family
Authors: Rumely Robert S.
Published: June 2010

We consider families of abelian varieties parametrized by classical theta-functions, and show that specifying the family and a CM point in Siegel space determines the grössencharacter of the corresponding CM abelian variety. We associate an adelic group to the family, and describe the kernel of the grössencharacter as the pull-back of the group under the map in Shimura’s Reciprocity Law.
The approximation property for some 5-dimensional Henselian rings
Authors: Becker Joseph , Denef J. , Lipshitz L.
Published: June 2010

Let

k
k

be a field of characteristic

0
0

,

k
[
[

X
1

,

X
2

]
]

k[[{X_1},{X_2}]]

the ring of formal power series and

R
=
k
[
[

X
1

,

X
2

]
]

[

X
3

,

X
4

,

X
5

]
∼

R = k[[{X_1},{X_2}]]{[{X_3},{X_4},{X_5}]^ \sim }

the algebraic closure of

k
[
[

X
1

,

X
2

]
]
[

X
3

,

X
4

,

X
5

]

k[[{X_1},{X_2}]][{X_3},{X_4},{X_5}]

in

k
[
[

X
1

,
…
,

X
5

]
]

k[[{X_1},\ldots ,{X_5}]]

. It is shown that

R
R

has the Approximation Property.
On the generalized Seidel class 𝑈
Authors: Hwang Jun Shung
Published: June 2010

As usual, we say that a function

f
∈
U

f \in U

if

f
f

is meromorphic in

|

z

|

>
1

| z | > 1

and has radial limits of modulus

1
1

a.e. (almost everywhere) on an arc

A
A

of

|
z
|

=
1

\left | z \right | = 1

. This paper contains three main results: First, we extend our solution of A. J. Lohwater’s problem (1953) by showing that if

f
∈
U

f \in U

and

f
f

has a singular point

P
P

on

A
A

, and if

υ
\upsilon

and

1

/

υ
¯

1/\bar {\upsilon }

are a pair of values which are not in the range of

f
f

at

P
P

, then one of them is an asymptotic value of

f
f

at some point of

A
A

near

P
P

. Second, we extend our solution of J. L. Doob’s problem (1935) from analytic functions to meromorphic functions, namely, if

f
∈
U

f \in U

and

f
(
0
)
=
0

f(0) = 0

, then the range of

f
f

over

|
z
|

>
1

\left | z \right | > 1

covers the interior of some circle of a precise radius depending only on the length of

A
A

. Finally, we introduce another class of functions. Each function in this class has radial limits lying on a finite number of rays a.e. on

|
z
|

=
1

\left | z \right | = 1

, and preserves a sector between domain and range. We study the boundary behaviour and the representation of functions in this class.
A generalization of minimal cones
Authors: Ejiri Norio
Published: June 2010

Let

R
+

{R_ +}

be a positive real line,

S
n

{S^n}

an

n
n

-dimensional unit sphere. We denote by

R
+

×

S
n

{R_+} \times {S^n}

the polar coordinate of an

(
n
+
1
)

(n + 1)

-dimensional Euclidean space

R

n
+
1

{R^{n + 1}}

. It is well known that if

M
M

is a minimal submanifold in

S
n

{S^n}

, then

R
+

×
M

{R_ +} \times M

is minimal in

R

n
+
1

{R^{n + 1}}

.

R
+

×
M

{R_+} \times M

is called a minimal cone. We generalize this fact and give many minimal submanifolds in real and complex space forms.
Generalized intersection multiplicities of modules
Authors: Dutta Sankar P.
Published: June 2010

In this paper we study intersection multiplicities of modules as defined by Serre and prove that over regular local rings of

dim
⩽
5

\dim \leqslant 5

, given two modules

M
,
N

M,N

with

l
(
M

⊗

R

N
)
>
∞

l(M\otimes _{R}N) > \infty

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension upper M plus dimension upper N greater-than dimension upper R comma chi left-parenthesis upper M comma upper N right-parenthesis equals sigma-summation Underscript i equals 0 Overscript dimension upper R Endscripts left-parenthesis negative 1 right-parenthesis Superscript i Baseline l left-parenthesis upper T o r Subscript i Superscript upper R Baseline left-parenthesis upper M comma upper N right-parenthesis right-parenthesis equals 0">

dim

M
+
dim

N
>
dim

R
,
χ
(
M
,
N
)
=

∑

i
=
0

dim

R

(
−
1

)
i

l
(

Tor
i
R

⁡
(
M
,
N
)
)
=
0

\dim \;M + \dim \;N > \dim \;R,\chi (M,N) = \sum \nolimits _{i = 0}^{\dim \; R}( - 1)^i l(\operatorname {Tor}_i^R(M,N)) = 0

. We also study multiplicity in a more general set up. Finally we extend Serre’s result from pairs of modules to pairs of finite free complexes whose homologies are killed by

I
n

,

J
n

{I^n},{J^n}

, respectively, for some

n
>
0

n > 0

, with

dim

R

/

I
+
dim

R

/

J
>
dim

R

\dim \,R/I + \dim \,R/J > \dim \,R

.
Shrinking countable decompositions of 𝑆³
Authors: Denman Richard , Starbird Michael
Published: June 2010

Conditions are given which imply that a countable, cellular use decomposition

G
G

is shrinkable. If the embedding of each element in

G
G

has the bounded nesting property, defined in this paper, then

S
3

/

G

{S^3}/G

is homeomorphic to

S
3

{S^3}

. The bounded nesting property is a condition on the defining sequence of cells for an element of

G
G

which implies that

G
G

satisfies the Disjoint Disk criterion for shrinkability [

S
1

{\mathbf {S1}}

, Theorem 3.1]. From this result, one deduces that countable, star-like equivalent use decompositions of

S
3

{S^3}

are shrinkable—a result proved independently by E. Woodruff [

W

{\mathbf {W}}

]. Also, one deduces the shrinkability of countable bird-like equivalent use decompositions (a generalization of the star-like result), and the recently proved theorem that if each element of a countable use decomposition

G
G

of

S
3

{S^3}

has a mapping cylinder neighborhood, then

G
G

is shrinkable [

E

{\mathbf {E}}

;

S
1

{\mathbf {S1}}

, Theorem 4.1;

S

-

W

{\mathbf {S}}\text {-}{\mathbf {W}}

, Theorem 1].
Finitely generic abelian lattice-ordered groups
Authors: Saracino Dan , Wood Carol
Published: June 2010

The authors characterize the finitely generic abelian lattice-ordered groups and make application of this characterization to specific examples.
On neighbourly triangulations
Authors: Sarkaria K. S.
Published: June 2010

A simplicial complex is called

d
d

-neighbourly if any

d
+
1

d + 1

vertices determine a

d
d

-simplex. We give methods for constructing

1
1

-neighbourly triangulations of

3
3

- and

4
4

-manifolds; further we discuss some relationships between

d
d

-neighbourly triangulations, chromatic numbers and the problem of finding upper and lower bounds on the number of simplices and locating the zeros of the characteristic polynomial of a triangulation. A triangulation of an orientable manifold is called order-orientable if there exists some ordering of the vertices which orients the manifold. We give necessary conditions for their existence; also we construct such triangulations on

3
3

-dimensional handlebodies and discuss the problem of recognising finite monotone subsets of an affine space by using these ideas.
On lexicographically shellable posets
Authors: Björner Anders , Wachs Michelle
Published: June 2010

Lexicographically shellable partially ordered sets are studied. A new recursive formulation of

CL

{\text {CL}}

-shellability is introduced and exploited. It is shown that face lattices of convex polytopes, totally semimodular posets, posets of injective and normal words and lattices of bilinear forms are

CL

{\text {CL}}

-shellable. Finally, it is shown that several common operations on graded posets preserve shellability and

CL

{\text {CL}}

-shellability.
Real-analytic submanifolds which are local uniqueness sets for holomorphic functions of 𝐶³
Authors: Harris Gary A.
Published: June 2010

The following problem is considered. Given a real-analytic two-dimensional submanifold,

M
M

, of complex Euclidean three-space, are ambient holomorphic functions determined by their values on

M
?

M?

For a large class of submanifolds a necessary and sufficient condition is found for

M
M

to be a local uniqueness set for holomorphic functions on complex three-space. Finally, the general problem is shown to be related to two-dimensional Nevanlinna theory.
Quadratic spaces over Laurent extensions of Dedekind domains
Authors: Parimala Raman
Published: June 2010

Let

R
R

be a Dedekind domain in which

2
2

is invertible. We show in this paper that any isotropic quadratic space over

R
[
T
,

T

−
1

]

R[T,{T^{ - 1}}]

is isometric to

q
1

⊥
T

q
2

{q_1} \perp T{q_2}

where

q
1

,

q
2

{q_1},{q_2}

are quadratic spaces over

R
R

. We give an example to show that this result does not hold for anisotropic spaces.
Some examples of square integrable representations of semisimple 𝑝-adic groups
Authors: Lusztig George
Published: June 2010

We construct irreducible representations of the Hecke algebra of an affine Weyl group analogous to Kilmoyer’s reflection representation corresponding to finite Weyl groups, and we show that in many cases they correspond to a square integrable representation of a simple

p
p

-adic group.
Projections onto translation-invariant subspaces of 𝐿₁(𝑅)
Authors: Alspach Dale E. , Matheson Alec
Published: June 2010

The complemented translation-invariant subspaces of

L
1

(

R

)

{L_1}({\mathbf {R}})

are characterized. This completes an investigation begun by H. P. Rosenthal.
Spectral decomposition with monotonic spectral resolvents
Authors: Erdélyi I. , Wang Sheng Wang
Published: June 2010

The spectral decomposition problem of a Banach space over the complex field entails two kinds of constructive elements: (1) the open sets of the field and (2) the invariant subspaces (under a given linear operator) of the Banach space. The correlation between these two structures, in the framework of a spectral decomposition, is the spectral resolvent concept. Special properties of the spectral resolvent determine special types of spectral decompositions. In this paper, we obtain conditions for a spectral resolvent to have various monotonic properties.
On quasiprojective covers
Authors: Faticoni Theodore G.
Published: June 2010

The main results determine the Goldie dimension of superfluous submodules of semiprime left Goldie rings and apply this to the study of quasiprojective covers of torsion free modules. Conditions are given to guarantee that a quasiprojective cover of a torsion free module is an isomorphism. A class of nonperfect rings is given such that finitely generated singular modules possess quasiprojective covers.
Toral actions on 5-manifolds
Authors: Oh Hae Soo
Published: June 2010

We are mainly concerned with closed orientable manifolds of dimension

5
5

supporting effective three-dimensional torus actions. We obtain a complete classification of simply-connected manifolds of this type and a partial classification for the nonsimply-connected case.
The nest subgroups of Kleinian groups
Authors: Sasaki Takehiko
Published: June 2010

The residual limit points of finitely generated Kleinian groups are classified into two types: the first kind and the second kind. To each point of the second kind, Abikoff associated a web subgroup. We shall classify the points of the first kind into two types and associate to each point of one type a finitely generated subgroup, the nest subgroup. To the points of the other type we shall give a significance showing that they are important for the sets of generators.
Poles of a two-variable 𝑃-adic complex power
Authors: Strauss Leon
Published: June 2010

For almost all

P
P

-adic completions of an algebraic number field, if

s
∈

C

s \in {\mathbf {C}}

is a pole of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript s Baseline equals integral Subscript Superscript Baseline integral Subscript Superscript Baseline StartAbsoluteValue f left-parenthesis x comma y right-parenthesis EndAbsoluteValue Superscript s Baseline StartAbsoluteValue d x EndAbsoluteValue Subscript upper K Sub Subscript p Baseline StartAbsoluteValue d y EndAbsoluteValue Subscript upper K Sub Subscript p Baseline">

f
s

=

∫

∫

|

f
(
x
,
y
)

|

s

|

d
x

|

K
p

|

d
y

|

K
p

{f^s} = \int _{}^{} {\int _{}^{} {|f(x,y){|^s}|dx{|_{{K_p}}}|dy{|_{{K_p}}}} }

, where

f
f

is a polynomial whose only singular point is the origin,

f
(
0
,
0
)
=
0

f(0,0) = 0

, and

f
f

is irreducible in

K
¯

[
[
x
,
y
]
]

\overline K [[x,y]]

, then

Re
⁡
(
s
)

\operatorname {Re} (s)

is

−
1

- 1

or one of an explicitly given set of rational numbers, whose cardinality is the number of characteristic exponents of

f
=
0

f = 0

.
Support points of families of analytic functions described by subordination
Authors: Hallenbeck D. J. , MacGregor T. H.
Published: June 2010

We determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination. The families we study include the functions with a positive real part, the typically-real functions, and the functions which are subordinate to a given majorant. If the majorant

F
F

is univalent then each support point has the form

F
∘

ϕ

F \circ \;\phi

, where

ϕ
\phi

is a finite Blaschke product and

ϕ
(
0
)
=
0

\phi (0) = 0

. This completely characterizes the set of support points when

F
F

is convex. The set of support points is found for some specific majorants, including

F
(
z
)
=

(
(
1
+
z
)

/

(
1
−
z
)

)
p

F(z) = {((1 + z)/(1 - z))^p}

where

p
>
1

p > 1

. Let

K
K

and

St

{\text {St}}

denote the set of normalized convex and starlike mappings, respectively. We find the support points of the families

K

∗

{K^{\ast } }

and

St

∗

{\text {St}}^{\ast }

defined by the property of being subordinate to some member of

K
K

or

St

{\text {St}}

, respectively.
Embedding 𝐿¹ in 𝐿¹/𝐻¹
Authors: Bourgain J.
Published: June 2010

It is proved that

L
1

{L^1}

is isomorphic to a subspace of

L
1

/

H
1

{L^1}/{H^1}

. More precisely, there exists a diffuse

σ
\sigma

-algebra

S

\mathfrak {S}

on the circle such that the corresponding expectation

E

:

H
∞

→

L
∞

(

C

)

{\mathbf {E}}:{H^\infty } \to {L^\infty }({\mathbf {C}})

is onto. The method consists in studying certain martingales on the product

∏

N

{\prod ^{\mathbf {N}}}

.
Mean values of subsolutions of elliptic and parabolic equations
Authors: Ziemer William P.
Published: June 2010

Integral averages of weak subsolutions (and supersolutions) in

R
n

{R^n}

of quasilinear elliptic and parabolic equations are investigated. The important feature is that these integral averages are defined in terms of measures that reflect interesting geometric phenomena. Harnack type inequalities are established in terms of these integral averages.
Gradings of 𝐵_{𝑛} and 𝐶_{𝑛} of finite representation type
Authors: Assem Ibrahim , Roldán Oscar
Published: June 2010

It was shown by Bongartz and Gabriel that the classification of simplyconnected algebras (i.e. finite-dimensional, basic, of finite representation type and with a simply-connected Auslander-Reiten graph) can be reduced to the study of certain numerical functions, called gradings, operating on a tree. Here, we classify in terms of their bounden species the simply-connected algebras arising from gradings of the Dynkin trees

B

n

{{\mathbf {B}}_n}

and

C

n

{{\mathbf {C}}_n}

, and show that these are exactly the tilted algebras of types

B

n

{{\mathbf {B}}_n}

and

C

n

{{\mathbf {C}}_n}

, respectively.
On a question of Quillen
Authors: Bhatwadekar S. M. , Rao R. A.
Published: June 2010

Let

R
R

be a regular local ring, and

f
f

a regular parameter of

R
R

. Quillen asked whether every projective

R
f

{R_f}

-module is free. We settle this question when

R
R

is a regular local ring of an affine algebra over a field

k
k

. Further, if

R
R

has infinite residue field, we show that projective modules over Laurent polynomial extensions of

R
f

{R_f}

are also free.
A strong type (2,2) estimate for a maximal operator associated to the Schrödinger equation
Authors: Kenig Carlos E. , Ruiz Alberto
Published: June 2010

Let

T

∗

f
(
x
)
=

sup

t
>
0

|

T
t

f
(
x
)

|

{T^{\ast } }f(x) = \sup _{t > 0}|{T_t}f(x)|

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper T Subscript t Baseline f right-parenthesis Superscript ModifyingAbove With caret Baseline left-parenthesis xi right-parenthesis equals e Superscript i t StartAbsoluteValue xi EndAbsoluteValue squared Baseline ModifyingAbove f With caret left-parenthesis xi right-parenthesis slash StartAbsoluteValue xi EndAbsoluteValue Superscript 1 slash 4">

(

T
t

f

)

^

(
ξ
)
=

e

i
t

|

ξ

|

2

f
^

(
ξ
)

/

|

ξ

|

1

/

4

({T_t}f)^{\hat {}}(\xi ) = {e^{it|\xi |^2}}\hat f(\xi )/|\xi {|^{1/4}}

. We show that, given any finite interval

I
I

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper I Endscripts StartAbsoluteValue upper T Superscript asterisk Baseline f EndAbsoluteValue squared d x less-than-or-slanted-equals upper C Subscript upper I Baseline integral Underscript bold upper R Endscripts StartAbsoluteValue f left-parenthesis x right-parenthesis EndAbsoluteValue squared d x">

∫
I

|

T

∗

f

|

2

d
x
⩽

C
I

∫

R

|

f
(
x
)

|

2

d
x

\int _I {|{T^{\ast } }f{|^2}\;dx \leqslant {C_I}\int _{\mathbf {R}} {|f(x){|^2}\;dx} }

, and that the above inequality is false with

2
2

replaced by any

p
>
2

p > 2

. This maximal operator is related to solutions of the Schrödinger equation.
The 𝑆¹-transfer in surgery theory
Authors: Munkholm H. J. , Pedersen E. K.
Published: June 2010

Let

S
1

→
X
→
Y

{S^1} \to X \to Y

be an

S
1

{S^1}

-bundle of Poincaré spaces. If

f
:
N
→
Y

f:N \to Y

is a surgery problem then so is the pullback

f
^

:
M
→
X

\hat f:M \to X

. We define algebraically a homomorphism
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Superscript factorial Baseline colon upper L Subscript n Baseline left-parenthesis bold upper Z pi 1 left-parenthesis upper Y right-parenthesis right-parenthesis right-arrow upper L Subscript n plus 1 Baseline left-parenthesis bold upper Z pi 1 left-parenthesis upper X right-parenthesis right-parenthesis">

φ
!

:

L
n

(

Z

π
1

(
Y
)
)
→

L

n
+
1

(

Z

π
1

(
X
)
)

{\varphi ^!}:{L_n}({\mathbf {Z}}{\pi _1}(Y)) \to {L_{n + 1}}({\mathbf {Z}}{\pi _1}(X))

and prove that it maps the surgery obstruction for

f
f

to the one for

f
^

\hat f

.
Recursivity in quantum mechanics
Authors: Baez John C.
Published: June 2010

The techniques of effective descriptive set theory are applied to the mathematical formalism of quantum mechanics in order to see whether it actually provides effective algorithms for the computation of various physically significant quantities, e.g. matrix elements. Various Hamiltonians are proven to be recursive (effectively computable) and shown to generate unitary groups which act recursively on the Hilbert space of physical states. In particular, it is shown that the

n
n

-particle Coulombic Hamiltonian is recursive, and that the time evolution of

n
n

-particle quantum Coulombic systems is recursive.
Zero distribution for pairs of holomorphic functions with applications to eigenvalue distribution
Authors: Shkalikov A. A.
Published: June 2010

Let

f
f

and

g
g

be holomorphic in an angle

Λ
\Lambda

. Theorem 1 shows that the zero-distributions of

f
f

and

g
g

are comparable if, near

∂
Λ

\partial \Lambda

,

f
f

and

g
g

grow similarly. This result is applied to analyse the asymptotic behavior of eigenvalues of certain perturbed normal operators.
An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees
Authors: Ambos-Spies Klaus , Jockusch Carl G. , Shore Richard A. , Soare Robert I.
Published: June 2010

We specify a definable decomposition of the upper semilattice of recursively enumerable (r.e.) degrees

R

\mathbf {R}

as the disjoint union of an ideal

M

\mathbf {M}

and a strong filter

N
C

\mathbf {NC}

. The ideal

M

\mathbf {M}

consists of

0

\mathbf {0}

together with all degrees which are parts of r.e. minimal pairs, and thus the degrees in

N
C

\mathbf {NC}

are called noncappable degrees. Furthermore,

N
C

\mathbf {NC}

coincides with five other apparently unrelated subclasses of

R
:
E
N
C

\mathbf {R: ENC}

, the effectively noncappable degrees;

P
S

\mathbf {PS}

, the degrees of promptly simple sets;

L
C

\mathbf {LC}

, the r.e. degrees cuppable to

0

′

{\mathbf {0}}’

by a low r.e. degree;

S
P

H
¯

{\mathbf {SP\bar H}}

, the degrees of non-

h
h

hh

-simple r.e. sets with the splitting property; and

G

\mathbf {G}

, the degrees in the orbit of an r.e. generic set under automorphisms of the lattice of r.e. sets.
James maps and 𝐸_{𝑛} ring spaces
Authors: Cohen F. R. , May J. P. , Taylor L. R.
Published: June 2010

We parametrize by operad actions the multiplicative analysis of the total James map given by Caruso and ourselves. The target of the total James map
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j equals sigma-summation j Subscript q Baseline colon upper C left-parenthesis upper R Superscript n Baseline comma upper X right-parenthesis right-arrow product Underscript q greater-than-or-slanted-equals 0 Endscripts upper Q upper D Subscript q Baseline left-parenthesis upper R Superscript n Baseline comma upper X right-parenthesis">

j
=
∑

j
q

:
C
(

R
n

,
X
)
→

∏

q
⩾
0

Q

D
q

(

R
n

,
X
)

j = \sum {{j_q}} :C({R^n},X) \to \prod \limits _{q \geqslant 0} {Q{D_q}({R^n},X)}

\]
is an

E
n

{E_n}

ring space and

j
j

is a

C

n

{\mathcal {C}_n}

-map, where

C

n

{\mathcal {C}_n}

is the little

n
n

-cubes operad. This implies that

j
j

has an

n
n

-fold delooping with domain

Σ
n

X

{\Sigma ^n}X

. It also implies an algorithm for the calculation of

j

∗

{j_{\ast }}

and thus of each

(

j
q

)

∗

{({j_q})_{\ast }}

on

mod

p

\bmod \, p

homology. When

n
=
∞

n = \infty

and

p
=
2

p = 2

, this algorithm is the essential starting point for Kuhn’s proof of the Whitehead conjecture.
Nonstable reflexive sheaves on 𝑃³
Authors: Sauer Timothy
Published: June 2010

The spectrum is defined for nonstable rank two reflexive sheaves on

P

3

{{\mathbf {P}}^3}

and is used to establish vanishing theorems for intermediate cohomology in terms of the Chern classes and the order of nonstability. These results are shown to be best possible and the extremal cases are classified. Some applications to Cohen-Macaulay generically local complete intersection curves in

P

3

{{\mathbf {P}}^3}

are given.
Approximation in the mean by solutions of elliptic equations
Authors: Bagby Thomas
Published: June 2010

A result analogous to the Vituškin approximation theorem is proved for mean approximation by solutions of certain elliptic equations.
Equivalence problems in projective differential geometry
Authors: Yang Kichoon
Published: June 2010

Equivalence problems for abstract, and induced, projective structures are investigated. (i) The notion of induced projective structures on submanifolds of a projective space is rigorously defined. (ii) Equivalence problems for such structures are discussed; in particular, it is shown that nonplanar surfaces in

R

P
3

\mathbf {R}{P^3}

are all projectively equivalent to each other. (iii) The imbedding problem of abstract projective structures is studied; in particular, we show that every abstract projective structure on a

2
2

-manifold arises as an induced structure on an arbitrary nonplanar surface in

R

P
3

\mathbf {R}{P^3}

; this result should be contrasted to that of Chern (see [6]).
Codimension 1 orbits and semi-invariants for the representations of an oriented graph of type 𝒜_{𝓃}
Authors: Abeasis S.
Published: June 2010

We consider the Dynkin diagram

A

n

\mathcal {A}_n

with an arbitrary orientation

Ω
\Omega

. For a given dimension

d
=
(

d
1

,
…
,

d
n

)

d = ({d_1}, \ldots ,{d_n})

we consider the corresponding variety

L
d

{L_d}

of all the representations of

(

A

n

,
Ω
)

(\mathcal {A}_n,\Omega )

on which a group

G
d

{G_d}

acts naturally. In this paper we determine the maximal orbit and the codim.

1
1

orbits of this action, giving explicitly their decomposition in terms of the irreducible representations of

A

n

\mathcal {A}_n

. We also deduce a set of algebraically independent semi-invariant polynomials which generate the ring of semi-invariants.
Orthogonal polynomials on the sphere with octahedral symmetry
Authors: Dunkl Charles F.
Published: June 2010

For any finite reflection group

G
G

acting on

R

N

{{\mathbf {R}}^N}

there is a family of

G
G

-invariant measures (

(

h
2

d
ω

({h^2}d\omega

, where

h
h

is a certain product of linear functions whose zero-sets are the reflecting hyperplanes for

G
G

) on the unit sphere and an associated partial differential operator (

L
h

f
:=
Δ
(
f
h
)
−
f
Δ
h

{L_h}f: = \Delta (fh) - f\Delta h

;

Δ
\Delta

is the Laplacian). Analogously to spherical harmonics, there is an orthogonal (with respect to

h
2

d
ω

{h^2}d\omega

) decomposition of homogeneous polynomials, that is, if

p
p

is of degree

n
n

then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis x right-parenthesis equals sigma-summation Underscript j equals 0 Overscript left-bracket n slash 2 right-bracket Endscripts StartAbsoluteValue x EndAbsoluteValue Superscript 2 j Baseline p Subscript n minus 2 j Baseline left-parenthesis x right-parenthesis comma">

p
(
x
)
=

∑

j
=
0

[
n

/

2
]

|

x

|

2
j

p

n
−
2
j

(
x
)
,

p(x) = \sum \limits _{j = 0}^{[n/2]} {|x{|^{2j}}{p_{n - 2j}}(x),}

\]
where

L
h

p
i

=
0

{L_h}{p_i} = 0

and

deg

p
i

=
i

{\operatorname {deg}}{p_i} = i

for each

i
i

, but with the restriction that

p
p

and

p
i

{p_i}

must all be

G
G

-invariant. The main topic is the hyperoctahedral group with
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h left-parenthesis x right-parenthesis equals left-parenthesis x 1 x 2 midline-horizontal-ellipsis x Subscript upper N Baseline right-parenthesis Superscript alpha Baseline left-parenthesis product Underscript i greater-than j Endscripts left-parenthesis x Subscript i Superscript 2 Baseline minus x Subscript j Superscript 2 Baseline right-parenthesis right-parenthesis Superscript beta Baseline period">

h
(
x
)
=

(

x
1

x
2

⋯

x
N

)
α

(

∏

i
>
j

(

x
i
2

−

x
j
2

)

)

β

.

h(x) = {({x_1}{x_2} \cdots {x_N})^\alpha }{\left ( {\prod \limits _{i > j} {(x_i^2 - x_j^2)} } \right )^\beta }.

\]
The special case

N
=
2

N = 2

leads to Jacobi polynomials. A detailed study of the case

N
=
3

N = 3

is made; an important result is the construction of a third-order differential operator that maps polynomials associated to

h
h

with indices

(
α
,
β
)

(\alpha ,\beta )

to those associated with

(
α
+
2
,
β
+
1
)

(\alpha + 2,\beta + 1)

.
Asymptotic behavior of solutions of second order differential equations with integrable coefficients
Authors: Naito Manabu
Published: June 2010

The differential equation

x
+
a
(
t
)
f
(
x
)
=
0

x + a(t)f(x) = 0

,

t
>
0

t > 0

, is considered under the condition that

lim

t
→
∞

∫
t

a
(
s
)
d
s

{\lim _{t \to \infty }}{\int ^t}a(s)ds

exists and is finite, and necessary and/or sufficient conditions are given for this equation to have solutions which behave asymptotically like nontrivial linear functions

c
1

+

c
2

t

{c_1} + {c_2}t

.
Pure states on some group-invariant 𝐶*-algebras
Authors: Price Geoffrey L.
Published: June 2010

Let

A

\mathfrak {A}

be a UHF algebra of Glimm type

n
∞

{n^\infty }

, i.e.,

A

=

⊗

k
⩾
1

∗

N
k

\mathfrak {A} = \otimes _{k \geqslant 1}^{\ast }{N_k}

, where

N
=

N
1

=

N
2

=
⋯

N = {N_1} = {N_2} = \cdots

are

n
×
n

n \times n

matrix algebras. We define an AF-subalgebra

A

G

{\mathfrak {A}^G}

of

A

\mathfrak {A}

, consisting of those elements of

A

\mathfrak {A}

invariant under a group of automorphisms

{

α
g

:
g
∈
G
=
SU
⁡
(
n
)
}

\{ {\alpha _g}:g \in G = \operatorname {SU} (n)\}

of product type.

A

G

{\mathfrak {A}^G}

is shown to be generated by an embedding of

S
(
∞
)

S(\infty )

, the discrete group of finite permutations on countably many symbols. Let

ω
\omega

be a pure product state on

A

\mathfrak {A}

,

ω
G

{\omega ^G}

its restriction to

A

G

{\mathfrak {A}^G}

. Let

e
∈
N

e \in N

be a one-dimensional projection with corresponding projections

e
k

∈

N
k

{e^k} \in {N_k}

. Then if both (i)

Σ

k
⩾
1

ω
(

e
k

)
=
∞

{\Sigma _{k \geqslant 1}}\omega ({e^k}) = \infty

, and (ii)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than normal upper Sigma Subscript k greater-than-or-slanted-equals 1 Baseline omega left-parenthesis e Superscript k Baseline right-parenthesis left-bracket 1 minus omega left-parenthesis e Superscript k Baseline right-parenthesis right-bracket greater-than normal infinity">

0
>

Σ

k
⩾
1

ω
(

e
k

)
[
1
−
ω
(

e
k

)
]
>
∞

0 > {\Sigma _{k \geqslant 1}}\omega ({e^k})[1 - \omega ({e^k})] > \infty

hold,

ω
G

{\omega ^G}

is not pure.

ω
G

{\omega ^G}

is shown to be pure if there exist orthogonal one-dimensional projections

{

p
i

:
1
⩽
i
⩽
n
}

\{ {p_i}:1 \leqslant i \leqslant n\}

of

N
N

with corresponding projections

p
i
k

∈

N
k

p_i^k \in {N_k}

such that

ω
(

p
i
k

)
=
0

\omega (p_i^k) = 0

or

1
1

,

1
⩽
i
⩽
n
,

k
⩾
1

1 \leqslant i \leqslant n,\,k \geqslant 1

, and

0
>

Σ

k
⩾
1

ω
(

p
i
k

)
>
∞

0 > {\Sigma _{k \geqslant 1}}\omega (p_i^k) > \infty

for at most one

i
i

.
Tangential equivalence of group actions
Authors: Kwasik Sławomir
Published: June 2010

We consider the problem of tangential equivalence of group actions on manifolds. In particular we discuss a conjecture of B. Mazur and its modifications. The negative answer to this conjecture is presented. On the other hand we prove that the "isovariant" version of this conjecture, as well as the modified one, remains true. As an application some results on the tangential equivalence of

Z
p

{Z_p}

-actions on homotopy spheres are obtained.
Contraction semigroups for diffusion with drift
Authors: Seeley R.
Published: June 2010

Recently Dodziuk, Karp and Li, and Strichartz have given results on existence and uniqueness of contraction semigroups generated by the Laplacian

Δ
\Delta

on a manifold

M
M

; earlier, Yau gave related results for

L
=
Δ
+
V

L = \Delta + V

for a vector field

V
V

. The present paper considers

L
=
Δ
−
V
−
c

L = \Delta - V - c

, with

c
c

a real function, and gives conditions for (a) uniqueness of semigroups on the bounded continuous functions, (b) preservation of

C
0

{C_0}

(functions vanishing at

∞
\infty

) by the minimal semigroup, and (c) existence and uniqueness of contraction semigroups on

L
p

(
μ
)
,

1
⩽
p
>
∞

{L^p}(\mu ),\;1 \leqslant p > \infty

, for an arbitrary smooth density

μ
\mu

on

M
M

. The conditions concern

L
ρ

/

ρ

L\rho /\rho

, where

ρ
\rho

is a smooth function,

ρ
→
∞

\rho \to \infty

as

x
→
∞

x \to \infty

. They variously extend, strengthen, and complement the previous results mentioned above.
Some sequence spaces and absolute almost convergence
Authors: Das G. , Kuttner B. , Nanda S.
Published: June 2010

The object of this paper is to introduce a new concept of absolute almost convergence which emerges naturally as an absolute analogue of almost convergence, in the same way as convergence leads to absolute convergence.
Weak solutions of the Gellerstedt and the Gellerstedt-Neumann problems
Authors: Aziz A. K. , Schneider M.
Published: June 2010

We consider the question of existence of weak and semistrong solutions of the Gellerstedt problem
\[

u

|

Γ
0

∪

Γ
1

∪

Γ
2

=
0

u{|_{{\Gamma _0} \cup {\Gamma _1} \cup {\Gamma _2}}} = 0

\]
and the Gellerstedt-Neumann problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d Subscript n Baseline u equals k left-parenthesis y right-parenthesis u Subscript x Baseline d y minus u Subscript y Baseline d x StartAbsoluteValue equals 0 comma u EndAbsoluteValue Subscript normal upper Gamma 0 Baseline Subscript normal upper Gamma 1 union normal upper Gamma 2 Baseline equals 0 right-parenthesis">

(

d
n

u
=
k
(
y
)

u
x

d
y
−

u
y

d
x

|

Γ
0

=
0
,

u

|

Γ
1

∪

Γ
2

=
0
)

({d_n}u = k(y){u_x}dy - {u_y}dx{|_{{\Gamma _0}}} = 0,\qquad u{|_{{\Gamma _1} \cup {\Gamma _2}}} = 0)

\]
for the equation of mixed type
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-bracket u right-bracket identical-to k left-parenthesis y right-parenthesis u Subscript x x Baseline plus u Subscript y y Baseline plus lamda u equals f left-parenthesis x comma y right-parenthesis comma lamda equals c o n s t greater-than 0">

L
[
u
]
≡
k
(
y
)

u

x
x

+

u

y
y

+
λ
u
=
f
(
x
,
y
)
,

λ
=
const
>
0

L[u] \equiv k(y){u_{xx}} + {u_{yy}} + \lambda u = f(x,y),\qquad \lambda = \operatorname {const} > 0

\]
in a region

G
G

bounded by a piecewise smooth curve

Γ
0

{\Gamma _0}

lying in the half-plane

y
>
0

y > 0

and intersecting the line

y
=
0

y = 0

at the points

A
(
−
1
,
0
)

A( - 1,0)

and

B
(
1
,
0
)

B(1,0)

. For

y
>
0

y > 0

,

G
G

is bounded by the characteristic curves

γ
1

(
x
>
0
)

{\gamma _1}(x > 0)

and

γ
2

(
x
>
0
)

{\gamma _2}(x > 0)

of (1) through the origin and the characteristics

Γ
1

{\Gamma _1}

and

Γ
2

{\Gamma _2}

through

A
A

and

B
B

which intersect

γ
1

{\gamma _1}

and

γ
2

{\gamma _2}

at the points

P
P

and

Q
Q

, respectively. Using a variation of the energy integral method, we give sufficient conditions for the existence of weak and semistrong solutions of the boundary value problems (Theorems 4.1, 4.2, 5.1).
On the Weil-Petersson metric on Teichmüller space
Authors: Fischer A. E. , Tromba A. J.
Published: June 2010

Teichmüller space for a compact oriented surface

M
M

without boundary is described as the quotient

A

/

D

0

\mathcal {A}/{\mathcal {D}_0}

, where

A

\mathcal {A}

is the space of almost complex structures on

M
M

(compatible with a given orientation) and

D

0

{\mathcal {D}_0}

are those

C
∞

{C^\infty }

diffeomorphisms homotopic to the identity. There is a natural

D

0

{\mathcal {D}_0}

invariant

L
2

{L_2}

Riemannian structure on

A

\mathcal {A}

which induces a Riemannian structure on

A

/

D

0

\mathcal {A}/{\mathcal {D}_0}

. Infinitesimally this is the bilinear pairing suggested by Andre Weil—the Weil-Petersson Riemannian structure. The structure is shown to be Kähler with respect to a naturally induced complex structure on

A

/

D

0

\mathcal {A}/{\mathcal {D}_0}

.
Rate of approach to minima and sinks—the Morse-Smale case
Authors: Wisniewski Helena S.
Published: June 2010

The dynamical systems herein are Morse-Smale diffeomorphisms and flows on

C
∞

{C^\infty }

compact manifolds. We show the asymptotic rate of approach of orbits to the sinks of the systems to be bounded by an expression of the form

K

exp
⁡
(
−
D
N
)

K\;\exp ( - DN)

, where

D
D

may be any number smaller than

C
=

min
p

{
1

/

m

log

Jac

D
P

f
m

|

W
u

(
P
)
}

C = {\min _p}\{ 1/m\;\log \;\operatorname {Jac}\;{D_P}\,{f^m}|{W^u}(P)\}

. Here the minimum is taken over all nonsink

P
P

in the nonwandering set of

f
f

, and

m
m

is the period of

P
P

. We extend our theorems to the entire manifold, so that there is no restriction on the location of the initial points of trajectories.
Involutions with isolated fixed points on orientable 3-dimensional flat space forms
Authors: Luft E. , Sjerve D.
Published: June 2010

In this paper we completely classify (up to conjugacy) all involutions

ι
:
M
→
M

\iota : M \to M

, where

M
M

is an orientable connected flat

3
3

-dimensional space form, such that

ι
\iota

has fixed points but only finitely many. If

M
1

,
…
,

M
6

M_1,\ldots ,M_6

are the

6
6

space forms then only

M
1

,

M
2

,

M
6

M_1, M_2, M_6

admit such involutions. Moreover, they are unique up to conjugacy. The main idea behind the proof is to find incompressible tori

T
⊆
M

T \subseteq M

so that either

ι
(
T
)
=
T

\iota (T) = T

or

ι
(
T
)
∩
T
=
∅

\iota (T) \cap T = \varnothing

and then cut

M
M

into simpler pieces. These results lead to a complete classification of

3
3

-manifolds containing

Z

⊕

Z

⊕

Z

\mathbf {Z} \oplus \mathbf {Z} \oplus \mathbf {Z}

in their fundamental groups.
Infinitesimally rigid polyhedra. I. Statics of frameworks
Authors: Whiteley Walter
Published: June 2010

From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation of the natural faces on these vertices, such polyhedra are infinitesimally rigid. In this paper the dual (and equivalent) concept of static rigidity for frameworks is used to describe the behavior of bar and joint frameworks built around convex (and other) polyhedra. The static techniques introduced provide a new simplified proof of Alexandrov’s theorem, as well as an essential extension which characterizes the static properties of frameworks built with more general patterns on the faces, including frameworks with vertices interior to the faces. The static techniques are presented and employed in a pattern appropriate to the extension of an arbitrary statically rigid framework built around any polyhedron (nonconvex, toroidal, etc.). The techniques are also applied to derive the static rigidity of tensegrity frameworks (with cables and struts in place of bars), and the static rigidity of frameworks projectively equivalent to known polyhedral frameworks. Finally, as an exercise to give an additional perspective to the results in

3
3

-space, detailed analogues of Alexandrov’s theorem are presented for convex

4
4

-polytopes built as bar and joint frameworks in

4
4

-space.
Normal structure and weakly normal structure of Orlicz sequence spaces
Authors: Landes Thomas
Published: June 2010

For a convex Orlicz functionφ:R+→R+∪{∞}\varphi :{{\bf {R}}_ + } \to {{\bf {R}}_ + } \cup \{ \infty \}and the associated Orlicz sequence spacelφ{l_\varphi }, we consider the following five properties: (1)lφ{l_\varphi }has a subspace isometric tol1{l_1}. (2)lφ{l_\varphi }is Schur. (3)lφ{l_\varphi }has normal structure. (4) Every weakly compact subset oflφ{l_\varphi }has normal structure. (5) Every bounded sequence inlφ{l_\varphi }has a subsequence(xn)({x_n})which is pointwise and almost convergent tox∈lφx \in {l_\varphi }, i.e.,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit sup Underscript n right-arrow normal infinity Endscripts parallel-to x Subscript n Baseline minus x parallel-to greater-than limit inf Underscript n right-arrow normal infinity Endscripts parallel-to x Subscript n Baseline minus y parallel-to">limsupn→∞∥xn−x∥φ>liminfn→∞∥xn−y∥φ\lim \,{\sup _{n \to \infty }}\parallel {x_n} - x{\parallel _{\varphi }} > \lim \inf _{n \to \infty }\parallel {x_n} - y{\parallel _\varphi }for ally≠xy \ne x. Our results are: (1)⇔φ\Leftrightarrow \;\varphiis either linear at0(φ(s)/s=c>0,0>s⩽t)0\;(\varphi (s)/s = c > 0,0 > s \leqslant t)or does not satisfy theΔ2{\Delta _2}-condition at00. (2)⇔lφ\Leftrightarrow \;{l_\varphi }is isomorphic tol1⇔φ′(0)=limt→0φ(t)/t>0{l_1}\; \Leftrightarrow \;\varphi ’(0) = {\lim _{t \to 0}}\,\varphi \,(t)/t > 0. (3)⇔φ\Leftrightarrow \varphisatisfies theΔ2{\Delta _2}-condition at0,φ0, \varphiis not linear at00andC(φ)=sup{φ(t)>1}>12C(\varphi ) = \sup \,\{ \varphi \,(t) > 1\} > \frac {1}{2}. (4)⇔φ\Leftrightarrow \,\varphisatisfies theΔ2{\Delta _2}-condition at00andC(φ)>12orφ′(0)>0C\,(\varphi ) > \frac {1}{2}\;{\rm {or}}\;\varphi ’(0) > 0. (5)⇔φ\Leftrightarrow \;\varphisatisfies theΔ2{\Delta _2}-condition at00andC(φ)=1C(\varphi ) = 1. The last equivalence contains a result of Lami-Dozo [10].
Interpolation and Gleason parts in 𝐿-domains
Authors: Behrens Michael Frederick
Published: June 2010

We describe the closure of

[
−
1

/

2
,
0
)

[ - 1/2,0)

in the maximal ideal space

M

(

D

)

\mathcal {M}(\mathcal {D})

of

H
∞

(

D

)

{H^\infty }(\mathcal {D})

) for an arbitrary

L
L

-domain

D

\mathcal {D}

. For

L
L

-domains satisfying

sup
(

c

n
+
1

/

c
n

)
>
1

\sup ({c_{n + 1}}/{c_n}) > 1

and

Σ

(

r
n

/

c
n

)
p

>
∞

\Sigma {({r_n}/{c_n})^p} > \infty

, some

p
⩾
1

p \geqslant 1

, we describe all interpolation sequences for

H
∞

(

D

)

{H^\infty }(\mathcal {D})

, we show that a homomorphism (except the distinguished homomorphism, when it exists) lies in a nontrivial Gleason part if and only if it is contained in the closure of an interpolating sequence, and we describe all the analytic structure occurring in

M

(

D

)

\mathcal {M}(\mathcal {D})

.
The amalgamation property for varieties of lattices
Authors: Day Alan , Ježek Jaroslav
Published: June 2010

There are precisely three varieties of lattices that satisfy the amalgamation property: trivial lattices, distributive lattices, and all lattices.
On subalgebras of simple Lie algebras of characteristic 𝑝>0
Authors: Weisfeiler B.
Published: June 2010

The main results of the paper are Theorems I.5.1, II.1.3 and III.2.1. Theorem I.5.1 states that if a maximal subalgebra

M
M

of a simple finite-dimensional Lie algebra

G
G

has solvable quotients of dimension

⩾
2

\geqslant 2

, then every nilpotent element of

H
H

acts nilpotently on

G
G

. Theorem II.1.3 states that if such a simple Lie algebra

G
G

contains a maximal subalgebra which is solvable, then

G
G

is Zassenbaus-Witt algebra. Theorem III.2.1 states that certain

Z

{\mathbf {Z}}

-graded finite-dimensional simple Lie algebras are either classical or the difference between the number of nonzero positive and negative homogeneous components is large.
Multi-invariant sets on compact abelian groups
Authors: Berend Daniel
Published: June 2010

Let

G
G

be a finite-dimensional connected compact abelian group. Generalizing previous results, dealing with the case of finite-dimensional tori, a full characterization is given herewith of those commutative semigroups

Σ
\Sigma

of continuous endomorphisms of

G
G

which satisfy the following property: The only infinite closed

Σ
\Sigma

-invariant subset of

G
G

is

G
G

itself.
Tempered ultradistributions as boundary values of analytic functions
Authors: Pathak R. S.
Published: June 2010

The spaces

S

a
k

{S_{{a_k}}}

,

S

b
q

{S^{{b_q}}}

and

S

a

k

b
q

S_{{a_{k}}}^{{b_q}}

were introduced by I. M. Gel’fand as a generalization of the test function spaces of type

S
S

; the elements of the corresponding dual spaces are called tempered ultradistributions. It is shown that a function which is analytic in a tubular radial domain and satisfies a certain nonpolynomial growth condition has a distributional boundary value in the weak topology of the tempered ultradistribution space

(

S

b

k

a

q

)
′

(S_{{b_{k}}}^{{a_{q}}})\prime

, which is the space of Fourier transforms of elements in

(

S

a

k

b

q

)
′

(S_{{a_{k}}}^{{b_{q}}})\prime

. This gives rise to a representation of the Fourier transform of an element

U
∈
(

S

a

k

b

q

)
′

U \in (S_{{a_{k}}}^{{b_{q}}})\prime

having support in a certain convex set as a weak limit of the analytic function. Converse results are also obtained. These generalized Paley-Wiener-Schwartz theorems are established by means of a number of new lemmas concerning

S

a

k

b

q

S_{{a_{k}}}^{{b_{q}}}

and its dual. Finally, in the appendix the equality

S

a
k

b
q

=

S

a
k

∩

S

b
q

S_{{a_k}}^{{b_q}} = {S_{{a_k}}} \cap {S^{{b_q}}}

is proved.
Hermitian forms in function theory
Authors: Leverenz Christine R.
Published: June 2010

Let

f
f

and

g
g

be analytic in the unit disk

|

z

|

>
1

|z|\; > 1

. We give a new derivation of the positive semidefinite Hermitian form equivalent to

|

g
(
z
)

|

≤

|

f
(
z
)

|

|g(z)| \leq |f(z)|

, for

|

z

|

>
1

| z | > 1

, and use it to derive Hermitian forms for various classes of univalent functions. Sharp coefficient bounds for these classes are obtained from the Hermitian forms. We find the specific functions required to make the Hermitian forms equal to zero.