bluuebunny/crossref_metadata_2025 huggingface dataset:
Quotients of 𝐿^{∞} by Douglas algebras and best approximation
Authors: Luecking Daniel H. , Younis Rahman M.
Published: June 2010

We show that

L
∞

/

A

{L^\infty }/A

is not the dual space of any Banach space when

A
A

is a Douglas algebra of a certain type. We do this by showing its unit ball has no extreme points. The method used requires that any function in

L
∞

{L^\infty }

has a nonunique best approximation in

A
A

. We therefore also show that the Douglas algebra

H
∞

+

L
F
∞

{H^\infty } + L_F^\infty

, when

F
F

is an open subset of the unit circle, permits best approximation. We use a method originating in Hayashi [6] and independently obtained by Marshall and Zame.
Approximation by smooth multivariate splines
Authors: de Boor C. , DeVore R.
Published: June 2010

The degree of approximation achievable by piecewise polynomial functions of given total order on certain regular grids in the plane is shown to be adversely affected by smoothness requirements—in stark contrast to the univariate situation. For a rectangular grid, and for the triangular grid derived from it by adding all northeast diagonals, the maximum degree of approximation (as the grid size

1

/

n

1/n

goes to zero) to a suitably smooth function is shown to be

O
(

n

−
ρ
−
2

)

O({n^{- \rho - 2}})

in case we insist that the approximating functions are in

C
ρ

{C^\rho }

. This only holds as long as

ρ
⩽
(
r
−
3
)

/

2

\rho \leqslant (r - 3)/2

and

ρ
⩽
(
2
r
−
4
)

/

3

\rho \leqslant (2r - 4)/3

, respectively, with

r
r

the total order of the polynomial pieces. In the contrary case, some smooth functions are not approximable at all. In the discussion of the second mesh, a new and promising kind of multivariate

B

{\text {B}}

-spline is introduced.
Scattering theory and the geometry of multitwistor spaces
Authors: Ginsberg Matthew L.
Published: June 2010

Existing results which show the zero rest mass field equations to be encoded in the geometry of projective twistor space are extended, and it is shown that the geometries of spaces of more than one twistor contain information concerning the scattering of such fields. Some general constructions which describe spacetime interactions in terms of cohomology groups on subvarieties in twistor space are obtained and are used to construct a purely twistorial description of spacetime propagators and of first order

ϕ
4

{\phi ^4}

scattering. Spacetime expressions concerning these processes are derived from their twistor counterparts, and a physical interpretation is given for the twistor constructions.
Conformally invariant variational integrals
Authors: Granlund S. , Lindqvist P. , Martio O.
Published: June 2010

Let

f
:
G
→

R
n

f:G \to {R^n}

be quasiregular and

I
=
∫

F
(
x
,
∇

u
)

d
m

I = \int {F(x,\nabla \,u)\,dm}

a conformally invariant variational integral. Hölder-continuity, Harnack’s inequality and principle are proved for the extremals of

I
I

. Obstacle problems and their connection to subextremals are studied. If

u
u

is an extremal or a subextremal of

I
I

, then

u
∘
f

u \circ f

is again an extremal or a subextremal if an appropriate change in

F
F

is made.
Trace class self-commutators
Authors: Berger C. A. , Ben-Jacob Marion Glazerman
Published: June 2010

This paper extends earlier results of Berger and Shaw to all

W
∗

{W^\ast }

algebras. The multiplicity of an operator in a

W
∗

{W^\ast }

algebra is defined in terms of the trace on the

W
∗

{W^\ast }

-algebra, and it is shown that if

T
T

is a hyponormal operator in such an algebra, the trace of its self-commutator is bounded by this multiplicity times the area of the spectrum of

T
T

, divided by

π
\pi

.
Almost convergent and weakly almost periodic functions on a semigroup
Authors: Dzinotyiweyi Heneri A. M.
Published: June 2010

Let

S
S

be a topological semigroup,

US

(
S
)

{\text {US}}(S)

the set of all bounded uniformly continuous functions on

S
,

WAP(

S
)

S,{\text {WAP(}}S)

the set of all (bounded) weakly almost periodic functions on
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S comma upper E 0 left-parenthesis upper S right-parenthesis colon equals left-brace f element-of UC left-parenthesis upper S right-parenthesis colon m left-parenthesis StartAbsoluteValue f EndAbsoluteValue right-parenthesis equals 0">

S
,

E
0

(
S
)
:=
{
f
∈

UC(

S
)
:
m
(

|

f

|

)
=
0

S,{E_0}(S): = \{ f \in {\text {UC(}}S):m(|f|) = 0

for each left and right invariant mean

m
m

on

UC(

S
)
}

{\text {UC(}}S)\}

and

W
0

(
S
)
:=
{
f
∈

WAP

(
S
)
:

m
(

|

f

|

)
=
0

{W_0}(S): = \{ f \in {\text {WAP}}(S):\:m(|f|) = 0

for each left and right invariant mean

m
m

on

WAP(

S
)
}

{\text {WAP(}}S)\}

. Among other results, for a large class of noncompact locally compact topological semigroups

S
S

, we show that the quotient space

E
0

(
S
)

/

W
0

(
S
)

{E_0}(S)/{W_0}(S)

contains a linear isometric copy of

l
∞

{l^\infty }

and so is nonseparable.
Twisting cochains and duality between minimal algebras and minimal Lie algebras
Authors: Hain Richard M.
Published: June 2010

An algebraic duality theory is developed between

1
1

-connected minimal cochain algebras of finite type and connected minimal chain Lie algebras of finite type by means of twisting cochains. The duality theory gives a concrete relationship between Sullivan’s minimal models, Chen’s power series connections and the various Lie algebra models of a

1
1

-connected topological space defined by Quillen, Allday, Baues-Lemaire and Neisendorfer. It can be used to compute the Lie algebra model of a space from the algebra model of the space and vice versa.
On the detonation of a combustible gas
Authors: Gardner Robert A.
Published: June 2010

This paper is concerned with the existence of detonation waves for a combustible gas. The equations are those of a viscous, heat conducting, polytropic gas coupled with an additional equation which governs the evolution of the mass fraction of the unburned gas (see (1)). The reaction is assumed to be of the simplest form:

A
→
B

A \to B

, i.e., there is a single product and a single reactant. The main result (see Theorem 2.1) is a rigorous existence theorem for strong, and under certain conditions, weak detonation waves for explicit ranges of the viscosity, heat conduction, and species diffusion coefficients. In other words, a class of admissible "viscosity matrices" is determined. The problem reduces to finding an orbit of an associated system of four ordinary differential equations which connects two distinct critical points. The proof employs topological methods, including Conley’s index of isolated invariant sets.
Tangent cones and quasi-interiorly tangent cones to multifunctions
Authors: Thibault Lionel
Published: June 2010

R. T. Rockafellar has proved a number of rules of subdifferential calculus for nonlocally lipschitzian real-valued functions by investigating the Clarke tangent cones to the epigraphs of such functions. Following these lines we study in this paper the tangent cones to the sum and the composition of two multifunctions. This will be made possible thanks to the notion of quasi-interiorly tangent cone which has been introduced by the author for vector-valued functions in [29] and whose properties in the context of multifunctions are studied. The results are strong enough to cover the cases of real-valued or vector-valued functions.
Nonanalytic solutions of certain linear PDEs
Authors: Zachmanoglou E. C.
Published: June 2010

It is shown that if

P
P

is a linear partial differential operator with analytic coefficients, and if

M
M

is an analytic submanifold of codimensions

3
3

in

R

n

{{\mathbf {R}}^n}

, which is partially characteristic with respect to

P
P

and satisfies certain additional conditions, then one can find, in a neighborhood of any point of

M
M

, solutions of the equation

P
u
=
0

Pu = 0

which are flat or singular precisely on

M
M

. The additional condition requires that a nonhomogeneous Laplace equation in two variables possesses a solution with a strong extremum at the origin. The right side of this nonhomogeneous equation is a homogeneous polynomial in two variables with coefficients being repeated Poisson brackets of the real and imaginary parts of the principal symbol of

P
P

.
Small into-isomorphisms between spaces of continuous functions. II
Authors: Benyamini Yoav
Published: June 2010

We construct two compact Hausdorff spaces,

X
X

and

Y
Y

, so that

C
(
X
)

C(X)

does not embed isometrically into

C
(
Y
)

C(Y)

, but for each

ε
>
0

\varepsilon > 0

, there is an isomorphism

T
ε

{T_\varepsilon }

from

C
(
X
)

C(X)

into

C
(
Y
)

C(Y)

satisfying

∥
f
∥⩽∥

T
ε

f

∥⩽
(
1
+
ε
)
∥
f
∥

\parallel f\parallel \leqslant \parallel {T_\varepsilon }f\;\parallel \leqslant (1 + \varepsilon )\parallel f\parallel

for all

f
∈
C
(
X
)

f \in C(X)

.
An extension of the Fuglede commutativity theorem modulo the Hilbert-Schmidt class to operators of the form ∑𝑀_{𝑛}𝑋𝑁_{𝑛}
Authors: Weiss Gary
Published: June 2010

We study the operators

Δ
(
X
)
=

∑
1
n

M
n

X

N
n

\Delta (X) = \sum \nolimits _1^n {{M_n}X{N_n}}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Superscript asterisk Baseline left-parenthesis upper X right-parenthesis equals sigma-summation Underscript 1 Overscript n Endscripts upper M Subscript n Superscript asterisk Baseline upper X upper N Subscript n Superscript asterisk">

Δ

∗

(
X
)
=

∑
1
n

M
n

∗

X

N
n

∗

{\Delta ^{\ast }}(X) = \sum \nolimits _1^n {M_n^{\ast }XN_n^{\ast }}

which map the algebra of all bounded linear operators on a separable Hubert space to itself, where

⟨

M
n

⟩
1
m

\langle {M_n}\rangle _1^m

and

⟨

N
n

⟩
1
m

\langle {N_n}\rangle _1^m

are separately commuting sequences of normal operators. We prove that (1) when

m
⩽
2

m \leqslant 2

, the Hilbert-Schmidt norms of

Δ
(
X
)

\Delta (X)

and

Δ

∗

(
X
)

{\Delta ^{\ast }}(X)

are equal (finite or infinite); (2) for

m
⩾
3

m \geqslant 3

, if

Δ
(
X
)

\Delta (X)

and

Δ

∗

(
X
)

{\Delta ^{\ast }}(X)

are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (3) if

Δ
,

Δ

∗

\Delta ,{\Delta ^{\ast }}

have the property that for each

X
,
Δ
(
X
)
=
0

X,\Delta (X) = 0

implies

Δ

∗

(
X
)
=
0

{\Delta ^{\ast }}(X) = 0

, then for each

X
X

, if

Δ
(
X
)

\Delta (X)

is a Hilbert-Schmidt operator then

Δ

∗

2

(
X
)

{\Delta ^{\ast }}^2(X)

is also and the latter has the same Hilbert-Schmidt norm as

Δ
2

(
X
)

{\Delta ^2}(X)

. Note that Fuglede’s Theorem is immediate from

(
1
)

(1)

in the case

m
=
2
,

M
1

=

N
2

m = 2,{M_1} = {N_2}

and

N
1

=
I
=
−

M
2

{N_1} = I = - {M_2}

. The proofs employ the duality between the trace class and the class of all bounded linear operators and, unlike the early proofs of Fuglede’s Theorem, they are free of complex function theory.
Uniqueness of Γ_{𝑝} in the Gross-Koblitz formula for Gauss sums
Authors: Adolphson Alan
Published: June 2010

It is determined what continuous functions besides the

p
p

-adic gamma function make the Gross-Koblitz formula valid.
A Boolean algebra with few subalgebras, interval Boolean algebras and retractiveness
Authors: Rubin Matatyahu
Published: June 2010

Using

♢

ℵ
1

{\diamondsuit _{{\aleph _1}}}

we construct a Boolean algebra

B
B

of power

ℵ
1

{\aleph _1}

, with the following properties: (a)

B
B

has just

ℵ
1

{\aleph _1}

subalgebras. (b) Every uncountable subset of

B
B

contains a countable independent set, a chain of order type

η
\eta

, and three distinct elements

a
,
b

a,b

and

c
c

, such that

a
∩
b
=
c

a \cap b = c

. (a) refutes a conjecture of J. D. Monk, (b) answers a question of R. McKenzie.

B
B

is embeddable in

P
(
ω
)

P(\omega )

. A variant of the construction yields an almost Jónson Boolean algebra. We prove that every subalgebra of an interval algebra is retractive. This answers affirmatively a conjecture of

B
\text {B}

. Rotman. Assuming

MA
\text {MA}

or the existence of a Suslin tree we find a retractive

BA
\text {BA}

not embeddable in an interval algebra. This refutes a conjecture of B. Rotman. We prove that an uncountable subalgebra of an interval algebra contains an uncountable chain or an uncountable antichain. Assuming

CH
\text {CH}

we prove that the theory of Boolean algebras in Magidor’s and Malitz’s language is undecidable. This answers a question of M. Weese.
On the generators of the first homology with compact supports of the Weierstrass family in characteristic zero
Authors: Kato Goro C.
Published: June 2010

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper W Subscript bold upper Q Baseline equals upper P r o j left-parenthesis bold upper Q left-bracket g Subscript 2 Baseline comma g 3 comma upper X comma upper Y comma upper Z right-bracket slash left-parenthesis homogeneous ideal generated by minus upper Y squared upper Z plus 4 upper X cubed minus g 2 upper X upper Z squared minus g 3 upper Z cubed right-parenthesis right-parenthesis">

W

Q

=
Proj
⁡
(

Q

[

g

2

,

g
3

,
X
,
Y
,
Z
]

/

(

homogeneous ideal generated by 

−

Y
2

Z
+
4

X
3

−

g
2

X

Z
2

−

g
3

Z
3

)
)

{{\mathbf {W}}_{\mathbf {Q}}} = \operatorname {Proj}({\mathbf {Q}}[{{\text {g}}_2},{g_3},X,Y,Z]/({\text {homogeneous ideal generated by }} - {Y^2}Z + 4\,{X^3} - {g_2}\,X{Z^2} - {g_3}\,{Z^3}))

. This is said to be the Weierstrass Family over the field

Q

{\mathbf {Q}}

. Then the first homology with compact supports of the Weierstrass Family is computed explicitly, i.e., it is generated by

{

C

−
k

d
X

∧

d
Y

}

k
⩾
1

{\{ {C^{ - k}}\,dX\, \wedge \;dY\}_{k \geqslant 1}}

and

{
X

C

−
k

d
X
∧

d
Y

}

k

⩾
1

{\{ X{C^{ - k}}dX \wedge \,dY\}_{k\, \geqslant 1}}

over the ring

Q

[

g

2

,

g
3

]

{\mathbf {Q}}[{g_{2}},{g_3}]

, where

C
C

is a polynomial

Y
2

−
4

X
3

+

g
2

X
+

g
3

{Y^2} - 4{X^3} + {g_2}X + {g_3}

. When one tensors the homology of the Weierstrass Family with

Δ

−
1

Q

[

g
2

,

g
3

]

{\Delta ^{ - 1}}\,{\mathbf {Q}}[{g_2},{g_3}]

, being localized at the discriminant

Δ
=

g
2
3

−
27

g
3
2

\Delta = g_2^3 - 27g_3^2

, over

Q

[

g

2

,

g
3

]

{\mathbf {Q}}[{{\text {g}}_2},{g_3}]

, the first homology is generated by

C

−
1

d
X

∧

d
Y

{C^{ - 1}}dX\; \wedge \;dY

and

X

C

−
1

d
X

∧
d
Y

X{C^{ - 1}}dX\, \wedge dY

. One also obtains the first homologies with compact supports of singular fibres over

℘
=
(

g
2

=

g

3

=
0
)

\wp = ({g_2} = {g_{3}} = 0)

and

℘
=

(

g
2

=
3
,

g

3

=
1
)

\wp = \,({g_2} = 3,{g_{3}} = 1)

as corollaries.
Uniformly exhaustive submeasures and nearly additive set functions
Authors: Kalton N. J. , Roberts James W.
Published: June 2010

Every uniformly exhaustive submeasure is equivalent to a measure. From this, we deduce that every vector measure with compact range in an

F
F

-space has a control measure. We also show that

c
0

{c_0}

(or any

L

∞

{\mathcal {L}_\infty }

-space) is a

K

\mathcal {K}

-space, i.e. cannot be realized as the quotient of a nonlocally convex

F
F

-space by a one-dimensional subspace.
Markov neighborhoods for zero-dimensional basic sets
Authors: Pixton Dennis
Published: June 2010

We extend the local stable and unstable laminations for a zero-dimensional basic set to semi-invariant laminations of a neighborhood, and use these extensions to construct the appropriate analog of a Markov partition, which we call a Markov neighborhood. The main applications we give are in the perturbation theory for stable and unstable manifolds; in particular, we prove a transversality theorem. For these applications we require not only that the basic sets be zero dimensional but that they satisfy certain tameness assumptions. This leads to global results on improving stability properties via small isotopies.
Espaces 𝑙^{𝑝} dans les sous-espaces de 𝐿¹
Authors: Guerre S. , Levy M.
Published: June 2010

It is shown that every subspace

E
E

of

L
1

{L^1}

contains a subspace isomorphic to

l

p
(
E
)

{l^{p(E)}}

, where

p
(
E
)

p(E)

is the upper bound of the set of real

p
p

’s such that

E
E

is of type

p
p

-Rademacher. As

p
(
E
)

p(E)

is also the upper bound of the set of real

p
p

’s such that

E
E

embeds into

L
p

{L^p}

, this result answers a question of H. P. Rosenthal. The proof uses the theory of stable Banach spaces developed by J. L. Krivine and B. Maurey.
Some applications of direct integral decompositions of 𝑊*-algebras
Authors: Sarian Edward
Published: June 2010

Let

A

\mathcal {A}

be a

W

∗

{W^{\ast }}

-algebra and let

A
∈

A

A \in \mathcal {A}

.

K

(

A

)

\mathcal {K}(\mathcal {A})

and

C
(
A
)

C(A)

represent certain convex subsets of

A

\mathcal {A}

. We prove the following via direct integral theory: (1) If

A

\mathcal {A}

is of type

I

∞

{{\text {I}}_\infty }

,

II

∞

{\text {II}}_\infty

, or III, then

C
(
A
)
=
{
0
}

C(A) = \{ 0\}

iff

A

∈

K

(

A

)

{\text {A}} \in \mathcal {K}(\mathcal {A})

. (2) If

A

\mathcal {A}

is of type I or II, then

K

(

A

)

\mathcal {K}(\mathcal {A})

is strongly dense in

A

\mathcal {A}

. (3) If

A

\mathcal {A}

is of type

I

∞

{{\text {I}}_\infty }

,

II

∞

{\text {II}}_\infty

, or III and

B

\mathcal {B}

is a

W

∗

{W^{\ast }}

-subalgebra of

A

\mathcal {A}

, we give sufficient conditions for a Schwartz map

P
P

of

A

\mathcal {A}

into

B

\mathcal {B}

to annihilate

K

(

A

)

\mathcal {K}(\mathcal {A})

. Several preliminary lemmas that are useful for direct integral theory are also proved.
Linearization of second-order nonlinear oscillation theorems
Authors: Kwong Man Kam , Wong James S. W.
Published: June 2010

The problem of oscillation of super- and sublinear Emden-Fowler equations is studied. Established are a number of oscillation theorems involving comparison with related linear equations. Recent results on linear oscillation can thus be used to obtain interesting oscillation criteria for nonlinear equations.
Spectral properties of a certain class of complex potentials
Authors: Guillemin V. , Uribe A.
Published: June 2010

In this paper we discuss spectral properties of the Schroedinger operator

−
Δ
+
q

- \Delta + q

on compact homogeneous spaces for certain complex valued potentials

q
q

. We show, for instance, that for these potentials the spectrum of

−
Δ
+
q

- \Delta + q

is identical with the spectrum of

−
Δ

- \Delta

.
The properties *-regularity and uniqueness of 𝐶*-norm in a general *-algebra
Authors: Barnes Bruce A.
Published: June 2010

In this paper two properties of a

∗

^{\ast }

-algebra

A
A

are considered which are concerned with the relationship between the algebra and its

C

∗

{C^{\ast }}

-enveloping algebra. These properties are that

A
A

have a unique

C

∗

{C^{\ast }}

-norm, and that

A
A

be

∗

^{\ast }

-regular. Both of these concepts are closely involved with the representation theory of the algebra.
Controlled boundary and ℎ-cobordism theorems
Authors: Chapman T. A.
Published: June 2010

In this paper two theorems are established which are consequences of some earlier approximation results of the author. The first theorem is a controlled boundary theorem for finite-dimensional manifolds. By this we mean an ordinary boundary theorem plus small

ε
\varepsilon

-control in a given parameter space. The second theorem is a controlled

h
h

-cobordism theorem for finite-dimensional manifolds with small

ε
\varepsilon

-control in a given parameter space. These results generalize the End Theorem and the Thin

h
h

-Corbordism of Quinn.
Local spectra of seminormal operators
Authors: Clancey Kevin F. , Wadhwa Bhushan L.
Published: June 2010

The local spectral theory of seminormal operators is studied by examining the connection between two naturally occurring contractive operator functions. These results are used to control the local spectra of cohyponormal operators. An invariant subspace result for seminormal operators whose real part has thin spectra is provided.
Complex and integral laminated lattices
Authors: Conway J. H. , Sloane N. J. A.
Published: June 2010

In an earlier paper we studied real laminated lattices (or

Z

{\mathbf {Z}}

-modules)

Λ
n

{\Lambda _n}

, where

Λ
1

{\Lambda _1}

is the lattice of even integers, and

Λ
n

{\Lambda _n}

is obtained by stacking layers of a suitable

(
n
−
1
)

(n - 1)

-dimensional lattice

Λ

n
−
1

{\Lambda _{n - 1}}

as densely as possible, keeping the same minimal norm. The present paper considers two generalizations: (a) complex and quaternionic lattices, obtained by replacing

Z

{\mathbf {Z}}

-module by

J
J

-module, where

J
J

may be the Eisenstein, Gaussian or Hurwitzian integers, etc., and (b) integral laminated lattices, in which

Λ
n

{\Lambda _n}

is required to be an integral lattice with the prescribed minimal norm. This enables us to give a partial answer to a question of J. G. Thompson on integral lattices, and to explain some of the computer-generated results of Plesken and Pohst on this problem. Also a number of familiar lattices now arise in a canonical way. For example the Coxeter-Todd lattice is the

6
6

-dimensional integral laminated lattice over

Z

[
ω
]

{\mathbf {Z}}[ \omega ]

of minimal norm

2
2

. The paper includes tables of the best real integral lattices in up to

24
24

dimensions.
Secant functions, the Reiss relation and its converse
Authors: Green Mark L.
Published: June 2010

Generalizing a classical Euclidean theorem for the circle, certain meromorphic functions on

P

1

{{\mathbf {P}}_1}

relating to the geometry of algebraic plane curves are shown to be constant. Differentiated twice, this gives a new proof of the Reiss relation and its converse. The relation of these functions to Abel’s Theorem is discussed, and a generalization of secant functions to space curves is given, for which the Chow form arises in a natural way.
Representations of generic algebras and finite groups of Lie type
Authors: Howlett R. B. , Lehrer G. I.
Published: June 2010

The complex representation theory of a finite Lie group

G
G

is related to that of certain "generic algebras". As a consequence, formulae are derived ("the Comparison Theorem"), relating multiplicities in

G
G

to multiplicities in the Weyl group

W
W

of

G
G

. Applications include an explicit description of the dual (see below) of an arbitrary irreducible complex representation of

G
G

.
Sets of essentially unitary operators
Authors: Lange Ridgley
Published: June 2010

Let

U
e

{U_e}

be the set of essentially unitary operators on a separable Hilbert space

H
H

; for

1
⩽
p
⩽
∞

1 \leqslant p \leqslant \infty

, let

U
p

{U_p}

be the set of operators

T
T

such that

I
−

T

∗

T

I - T^{\ast }T

lies in the Schatten

p
p

-ideal and the spectrum of

T
T

does not fill the unit disc; and let

U
e
n

U_e^n

be the set of operators in

U
e

{U_e}

of Fredholm index

n
n

. The author proves that each

U
e
n

U_e^n

is closed and path connected, that

U
p

{U_p}

is dense in

U
e

0

{U_e}^0

and

U
p

{U_p}

is path connected for each

p
p

, and that all these sets are invariant under Cayley transform. It is proved that the spectrum is continuous on

U
∞

{U_\infty }

but not on

U
e

{U_e}

, while the spectral radius is continuous on

U
e

{U_e}

. Sufficient conditions that an operator in

U
e

{U_e}

have a nontrivial hyperinvariant subspace are given, and it is proved that the general hyperinvariant subspace problem can be reduced to that problem for perturbations of the bilateral shift. The product of commuting operators in

U
p

{U_p}

is

U
p

{U_p}

, but this result is false in general. Quasisimilarity in

U
e

{U_e}

is also studied; quasisimilar operators in

U
e

∖

U
∞

{U_e}\backslash {U_\infty }

are unitarily equivalent modulo the ideal of compacts, and this result also holds in

U
∞

{U_\infty }

if the spectrum is also preserved.
Further results on convergence acceleration for continued fractions 𝐾(𝑎_{𝑛}/1)
Authors: Jacobsen Lisa
Published: June 2010

If

K
(

a
n
′

/

1
)

K(a_n’/1)

is a convergent continued fraction with known tails, it can be used to construct modified approximants

f
n

∗

f_n^{\ast }

for other continued fractions

K
(

a
n

/

1
)

K({a_n}/1)

with unknown values. These modified approximants may converge faster to the value

f
f

of

K
(

a
n

/

1
)

K({a_n}/1)

than the ordinary approximants

f
n

{f_n}

do. In particular, if

a
n

−

a
n
′

→
0

{a_n} - a_n’ \to 0

fast enough, we obtain

|

f
−

f
n

∗

|

/

|

f
−

f
n

|

→
0

|f - f_n^{\ast }|/|f - {f_n}| \to 0

; i.e. convergence acceleration. the present paper also gives bounds for this ratio of the two truncation errors, in many cases.
A linear homogenization problem with time dependent coefficient
Authors: Mascarenhas Maria Luisa
Published: June 2010

We consider: the homogenization problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column left-parenthesis partial-differential u epsilon slash partial-differential t right-parenthesis left-parenthesis x comma t right-parenthesis plus beta Subscript epsilon Baseline left-parenthesis x right-parenthesis u Subscript epsilon Baseline left-parenthesis x comma t right-parenthesis equals 0 comma 2nd Column a m p semicolon t less-than-or-slanted-equals 0 comma 2nd Row 1st Column u Subscript epsilon Baseline left-parenthesis x comma 0 right-parenthesis equals phi left-parenthesis x right-parenthesis comma EndLayout">

{

(
∂
u
ε

/

∂
t
)
(
x
,
t
)
+

β
ε

(
x
)

u
ε

(
x
,
t
)
=
0
,

a
m
p
;
t
⩽
0
,

u
ε

(
x
,
0
)
=
ϕ
(
x
)
,

\begin {cases} (\partial u\varepsilon /\partial t)(x,t) + \beta _\varepsilon (x) u_\varepsilon (x,t) = 0, & t\leqslant 0, \\ u_\varepsilon (x,0) = \phi (x), \end {cases}

\]
where

β
\beta

is a strictly positive bounded real function, periodic of period

1
1

, and

β
ε

(
x
)
=
β
(
x

/

ε
)

{\beta _\varepsilon }(x) = \beta (x/\varepsilon )

; the equivalent integral equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript epsilon Baseline left-parenthesis x comma t right-parenthesis plus integral Subscript 0 Superscript t Baseline beta Subscript epsilon Baseline left-parenthesis x right-parenthesis u Subscript epsilon Baseline left-parenthesis x comma s right-parenthesis d s equals phi left-parenthesis x right-parenthesis semicolon">

u
ε

(
x
,
t
)
+

∫
0
t

β
ε

(
x
)

u
ε

(
x
,
s
)

d
s
=
ϕ
(
x
)

;

{u_\varepsilon }(x,t) + \int _0^t {{\beta _\varepsilon }(x)\,{u_\varepsilon }(x,s)\;ds = \phi (x)};

\]
and the homogenized equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u 0 left-parenthesis x comma t right-parenthesis plus integral Subscript 0 Superscript t Baseline upper K left-parenthesis t minus s right-parenthesis u 0 left-parenthesis s right-parenthesis d s equals phi left-parenthesis x right-parenthesis comma">

u
0

(
x
,
t
)
+

∫
0
t

K
(
t
−
s
)

u
0

(
s
)

d
s
=
ϕ
(
x
)

,

{u_0}(x,t) + \int _0^t {K(t - s)\,{u_0}(s)\,ds = \phi (x)},

\]
where

K
K

is a unique, well-defined function depending on

β
\beta

. We study this problem for a time dependent

β
\beta

, and characterize a two-variable function

K
(
s
,
t
)

K(s,t)

satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u 0 left-parenthesis x comma t right-parenthesis plus integral Subscript 0 Superscript t Baseline upper K left-parenthesis s comma t minus s right-parenthesis u 0 left-parenthesis x comma s right-parenthesis d s equals phi left-parenthesis x right-parenthesis">

u
0

(
x
,
t
)
+

∫
0
t

K
(
s
,
t
−
s
)

u
0

(
x
,
s
)

d
s
=
ϕ
(
x
)

{u_0}(x,t) + \int _0^t {K(s,t - s)\,{u_0}(x,s)\;ds = \phi (x)}

\]
and study its uniqueness.
James maps, Segal maps, and the Kahn-Priddy theorem
Authors: Caruso J. , Cohen F. R. , May J. P. , Taylor L. R.
Published: June 2010

The standard combinatorial approximation

C
(

R
n

,
X
)

C({R^n},X)

to

Ω
n

Σ
n

X

{\Omega ^n}{\Sigma ^n}X

is a filtered space with easily understood filtration quotients

D
q

(

R
n

,
X
)

{D_q}({R^n},X)

. Stably,

C
(

R
n

,
X
)

C({R^n},X)

splits as the wedge of the

D
q

(

R
n

,
X
)

{D_q}({R^n},X)

. We here analyze the multiplicative properties of the James maps which give rise to the splitting and of various related combinatorially derived maps between iterated loop spaces. The target of the total James map

\[
j = (j_q): \Omega ^n \Sigma ^n X \bigtimes _{q \geqslant 0} \Omega ^{2nq} \Sigma ^{2nq} D_q(R^n, X) \]
is a ring space, and

j
j

is an exponential

H
H

-map. There is a total Segal map

\[
s = \bigtimes _{q \geqslant 0} \;{s_{q}}:\bigtimes _{q \geqslant 0} \;{\Omega ^{2nq}}\,{\Sigma ^{2nq}}{D_q}({R^{n}},X)\; \bigtimes _{q \geqslant 0} \;\Omega ^{3nq}\,\Sigma ^{3nq}{X^{[q]}}\]
which is a ring map between ring spaces. There is a total partial power map

\[
k = ({k_q}): {\Omega ^{n}}\,{\Sigma ^{n}}X \to \bigtimes _{q \geqslant 0} \;{\Omega ^{n\,q}}\,{\Sigma ^{n\,q}}{X^{[q]}}\]
which is an exponential

H
H

-map. There is a noncommutative binomial theorem for the computation of the smash power
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Superscript n Baseline normal upper Sigma Superscript n Baseline upper X right-arrow normal upper Omega Superscript n q Baseline normal upper Sigma Superscript n q Baseline upper X Superscript left-bracket q right-bracket">

Ω
n

Σ
n

X
→

Ω

n
q

Σ

n
q

X

[
q
]

{\Omega ^n}{\Sigma ^n}X \to {\Omega ^{nq}}{\Sigma ^{nq}}{X^{[q]}}

in terms of the

k
m

{k_m}

for

m
⩽
q

m \leqslant q

. The composite of

s
s

and

j
j

agrees with the composite of

k
k

and the natural inclusion

\[
\bigtimes _{q \geqslant 0} \;{\Omega ^{n\,q}}\,{\Sigma ^{n\,q}}{X^{[q]}} \to \bigtimes _{q \geqslant 0} \,{\Omega ^{3\,n\,q}}\,{\Sigma ^{3\,n\,q}}{X^{[q]}}.\]
This analysis applies to essentially arbitrary spaces

X
X

. When specialized to

X
=

S
0

X = {S^0}

, it implies an unstable version of the Kahn-Priddy theorem. The exponential property of the James maps leads to an analysis of the behavior of loop addition with respect to the stable splitting of

Ω
n

Σ
n

X

{\Omega ^n}{\Sigma ^n}X

when

X
X

is connected, and there is an analogous analysis relating loop addition to the stable splitting of

Q
(

X
+

)

Q({X^ + })

.
An algebraic classification of some even-dimensional spherical knots. II
Authors: Farber M. Š.
Published: June 2010

The paper reduces the problem of classification of simple even-dimensional spherical knots of codimension two to an algebraic problem.
Finitely generated extensions of partial difference fields
Authors: Evanovich Peter
Published: June 2010

A proof of the following theorem is given: If

M

\mathcal {M}

is a finitely generated extension of a partial difference field

K

\mathcal {K}

then every subextension of

M

/

K

\mathcal {M}/\mathcal {K}

is finitely generated. An integral measure of partial difference field extensions having properties similar to the dimension of field extensions and the limit degree of ordinary difference field extensions and a new method of computing transformal transcendence degree are developed.
Free products of inverse semigroups
Authors: Jones Peter R.
Published: June 2010

A structure theorem is provided for the free productSinvTS\,{\operatorname {inv}}\,Tof inverse semigroupsSSandTT. Each element ofSinvTS\,{\operatorname {inv}}\,Tis uniquely expressible in the formε(A)a\varepsilon (A)a, whereAAis a certain finite set of “left reduced” words and eithera=1a = 1ora=a1⋯ama = {a_1} \cdots {a_m}is a “reduced” word withaam−1∈Aaa_m^{ - 1} \in A. (The worda1⋯am{a_1} \cdots {a_m}inSsgpTS\,{\operatorname {sgp}}\,Tis called reduced if no letter is idempotent, and left reduced if exactlyam{a_m}is idempotent; the notationε(A)\varepsilon (A)stands forΠ{aa−1:a∈A}\Pi \{ a{a^{ - 1}}:\,a \in A\}.) Under a product remarkably similar to Scheiblich’s product for free inverse semigroups, the corresponding pairs(A,a)(A,\,a)form an inverse semigroup isomorphic withSinvTS\,{\operatorname {inv}}\,T. This description enables various properties ofSinvTS\,{\operatorname {inv}}\,Tto be determined. For example(SinvT)∖(S∪T)(S\:{\operatorname {inv}}\:T)\backslash (S \cup T)is always completely semisimple and each of its subgroups is isomorphic with a finite subgroup ofSSorTT. If neitherSSnorTThas a zero then(SinvT)(S\:{\operatorname {inv}}\:T)is fundamental, but in general fundamentality itself is not preserved.
Ford and Dirichlet regions for discrete groups of hyperbolic motions
Authors: Nicholls P. J.
Published: June 2010

It is shown that for a discrete group of hyperbolic motions of the unit ball of

R

n

{\mathbf {R}^n}

, there is a single construction of fundamental regions which gives the Ford and Dirichlet regions as special cases and which also yields fundamental regions based at limit points. It is shown how the region varies continuously with the construction. The construction is connected with a class of limit points called Garnett points. The size of the set of such points is investigated.
Approximation of infinite-dimensional Teichmüller spaces
Authors: Gardiner Frederick P.
Published: June 2010

By means of an exhaustion process it is shown that Teichmüller’s metric and Kobayashi’s metric are equal for infinite dimensional Teichmüller spaces. By the same approximation method important estimates coming from the Reich-Strebel inequality are extended to the infinite dimensional cases. These estimates are used to show that Teichmüller’s metric is the integral of its infinitesimal form. They are also used to give a sufficient condition for a sequence to be an absolute maximal sequence for the Hamilton functional. Finally, they are used to give a new sufficient condition for a sequence of Beltrami coefficients to converge in the Teichmüller metric.
Discontinuous translation invariant functionals
Authors: Saeki Sadahiro
Published: June 2010

Let

G
G

be an infinite

σ
\sigma

-compact locally compact group. We shall study the existence of many discontinuous translation invariant linear functionals on a variety of translation invariant Fréchet spaces of Radon measures on

G
G

. These spaces include the convolution measure algebra

M
(
G
)

M(G)

, the Lebesgue spaces

L
p

(
G
)

{L^p}(G)

, where

1
≤
p
≤
∞

1 \leq p \leq \infty

, and certain combinations thereof. Among other things, it will be shown that

C
(
G
)

C(G)

has many discontinuous translation invariant functionals, provided that

G
G

is amenable. This solves a problem of G. H. Meisters.
Matrix localizations of 𝑛-firs. II
Authors: Malcolmson Peter
Published: June 2010

In a previous paper by this author and with a similar title, it was shown that adjoining universal inverses for all

k
×
k

k \times k

full matrices over an

n
n

-fir results in the localized ring being an

(
n
−
2
k
)

(n - 2k)

-fir. In this note a counterexample is used to show that the result is best possible in general. Techniques of the previous paper are strengthened and a result on a kind of finite inertia of certain rings within their localizations is obtained.
Haefliger structures and linear homotopy
Authors: Bracho Javier
Published: June 2010

The notion of linear-homotopy into a classifying space is introduced and used to give a precise classification of Haefliger structures. Appendix on the product theorem for simplicial spaces and realizations of bisimplicial spaces.
Degrees of recursively saturated models
Authors: Macintyre Angus , Marker David
Published: June 2010

Using relativizations of results of Goncharov and Peretyat’kin on decidable homogeneous models, we prove that if

M
M

is

S
S

-saturated for some Scott set

S
S

, and

F
F

is an enumeration of

S
S

, then

M
M

has a presentation recursive in

F
F

. Applying this result we are able to classify degrees coding (i) the reducts of models of PA to addition or multiplication, (ii) internally finite initial segments and (iii) nonstandard residue fields. We also use our results to simplify Solovay’s characterization of degrees coding nonstandard models of Th(N).
Entropy via random perturbations
Authors: Kifer Yuri
Published: June 2010

The entropy of a dynamical system

S
t

{S^t}

on a hyperbolic attractor with respect to the Bowen-Ruelle-Sinaǐ measure is obtained as a limit of entropy characteristics of small random perturbations

x
t
ε

x_t^\varepsilon

of

S
t

{S^t}

. Both the case of perturbations only in some neighborhood of an attractor and global perturbations of a flow with hyperbolic attracting sets are considered.
Intermediate normalizing extensions
Authors: Heinicke A. G. , Robson J. C.
Published: June 2010

Relationships between the prime ideals of a ring

R
R

and of a normalizing extension

S
S

have been studied by several authors recently. In this work, most of these known results are extended to give relationships between the prime ideals of

R
R

and of

T
T

where

T
T

is a ring with

R
⊂
T
⊂
S

R \subset T \subset S

, and

S
S

is a normalizing extension of

R
R

: such rings

T
T

are called intermediate normalizing extensions of

R
R

. One result ("Cutting Down") asserts that for any prime ideal

J
J

of

T
T

,

J
∩
R

J \cap R

is the intersection of a finite set of prime ideals

P
i

{P_i}

of

R
R

, uniquely defined by

J
J

, whose corresponding factor rings

R

/

P
i

R/{P_i}

are mutually isomorphic. The minimal members of the family of

P
i

{P_i}

’s are the primes of

R
R

minimal over

J
∩
R

J \cap R

, and an "incomparability" theorem is proved which shows that no two comparable primes of

T
T

can have their intersections with

R
R

share a common minimal prime. Other results include versions of the "lying over" and "going up" theorems, proofs that chain conditions such as right Goldie or right Noetherian pass between

T

/

J

T/J

and each of the rings

R

/

P
i

R/{P_i}

, and a demonstration that the "additivity principle" holds.
The equivariant Dolbeault lemma
Authors: Schmid Wilfried , Wolf Joseph A.
Published: June 2010

A form of the Dolbeault Lemma is obtained for circular domains

D
⊂

C

n

D \subset {{\mathbf {C}}^n}

, which is equivariant for the subgroup of

GL
⁡
(
n
;

C

)

\operatorname {GL} (n;{\mathbf {C}})

that stabilizes

D
D

.
An infinite graph of girth 12
Authors: Weiss Asia Ivić
Published: June 2010

From the regular hyperbolic honeycomb

{
3
,
6
,
3
}

\{ 3,6,3\}

we derive regular honeycombs with finite numbers of toroidal cells. Joining centers of faces of these honeycombs to the midpoints of its edges we obtain trivalent symmetrical graphs. We investigate the relation between these honeycombs, their groups and the graphs embedded in them.
Tameness of pairs of nuclear power series spaces and related topics
Authors: Nyberg Kaisa
Published: June 2010

The equivalence of the following six assertions is proved: (i) The set of the finite limit points of the ratios

α
m

/

β
n

,
n
,
m
∈

N

{\alpha _m}/{\beta _n},n,m \in {\mathbf {N}}

, is bounded, (ii) Every operator from

Λ
∞

(
β
)

{\Lambda _\infty }(\beta )

to

Λ
1

(
α
)

{\Lambda _1}(\alpha )

is compact, (iii) The pair

(

Λ
∞

(
β
)
,

Λ
1

(
α
)
)

({\Lambda _\infty }(\beta ),\,{\Lambda _1}(\alpha ))

is tame, i.e., for every operator

T
T

from

Λ
∞

(
β
)

{\Lambda _\infty }(\beta )

to

Λ
1

(
α
)

{\Lambda _1}(\alpha )

there is a positive integer

a
a

such that for every

k
∈

N

k \in {\mathbf {N}}

there is a constant

C
k

{C_k}

such that

|

|

T
x

|

|

k

⩽

C
k

|

x

|

a
k

||Tx|{|_k} \leqslant {C_k}|x{|_{ak}}

for every

x
∈

Λ
∞

(
β
)

x \in {\Lambda _\infty }(\beta )

. (iv) Every short exact sequence

0
→

Λ
τ

(
β
)
→
X
→

Λ
1

(
α
)
→
0

0 \to {\Lambda _\tau }(\beta ) \to X \to {\Lambda _1}(\alpha ) \to 0

, where

τ
=
1

\tau = 1

or

∞
\infty

, splits. (v)

Λ
1

(
α
)
×

Λ
∞

(
β
)

{\Lambda _1}(\alpha ) \times {\Lambda _\infty }(\beta )

has a regular basis, (vi)

Λ
1

(
α
)
⊗

Λ
∞

(
β
)

{\Lambda _1}(\alpha ) \otimes {\Lambda _\infty }(\beta )

has a regular basis. Also the finite type power series spaces that contain subspaces isomorphic to an infinite type power series space are characterized as well as the infinite type power series spaces that have finite type quotient spaces.
On certain elementary extensions of models of set theory
Authors: Enayat Ali
Published: June 2010

In

§
1

\S 1

we study two canonical methods of producing models of

ZFC
\operatorname {ZFC}

with no elementary end extensions.

§
2

\S 2

is devoted to certain "completeness" theorems dealing with elementary extensions, e.g., using

♢

ω
1

{\diamondsuit _{{\omega _1}}}

we show that for a consistent

T
⊇
ZFC

T \supseteq \operatorname {ZFC}

the property "Every model

A

\mathfrak {A}

of

T
T

has an elementary extension fixing

ω

A

{\omega ^\mathfrak {A}}

" is equivalent to

T
⊢

T\vdash

"There exists an uncountable measurable cardinal". We also give characterizations of

T
⊢

T\vdash

"

κ
\kappa

is weakly compact" and

T
⊢

T\vdash

"

κ
\kappa

is measurable" in terms of elementary extensions.
Sieved ultraspherical polynomials
Authors: Al-Salam Waleed , Allaway W. R. , Askey Richard
Published: June 2010

The continuous

q
q

-ultraspherical polynomials contain a number of important examples as limiting or special cases. One of these arose in Allaway’s Ph.D. thesis. In a previous paper we solved a characterization problem essentially equivalent to Allaway’s and showed that these polynomials arose from the

q
q

-ultraspherical polynomials when

q
q

approached a root of unity. A second class of such polynomials is found, and the recurrence relation and orthogonality relation are found for each of these polynomials. The orthogonality is interesting because the weight function has a finite number of zeros in

(
−
1
,
1
)

(-1, 1)

. Generating functions and other formulas are also found.
Localization of equivariant cohomology rings
Authors: Duflot J.
Published: June 2010

The main result of this paper is the "calculation" of the Borel equivariant cohomology ring

H

∗

(
E
G

×
G

X
,

Z

/

p

Z

)

{H^{\ast } }(EG \times _G\,X,{\mathbf {Z}}/p{\mathbf {Z}})

localized at one of its minimal prime ideals. In case

X
X

is a point, the work of Quillen shows that the minimal primes

P

A

{\mathfrak {P}_A}

are parameterized by the maximal elementary abelian

p
p

-subgroups

A
A

of

G
G

and the result is
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript asterisk Baseline left-parenthesis upper B upper G comma bold upper Z slash p bold upper Z right-parenthesis Subscript German upper P Sub Subscript upper A Baseline approximately-equals upper H Superscript asterisk Baseline left-parenthesis upper B upper C Subscript upper G Baseline left-parenthesis upper A right-parenthesis comma bold upper Z slash p bold upper Z right-parenthesis Subscript German upper P Sub Subscript upper A Superscript upper W Super Subscript upper G Superscript left-parenthesis upper A right-parenthesis">

H

∗

(
B
G
,

Z

/

p

Z

)

P

A

≅

H

∗

(
B

C
G

(
A
)
,

Z

/

p

Z

)

P

A

W
G

(
A
)

{H^{\ast } }{(BG,{\mathbf {Z}}/p{\mathbf {Z}})_{{\mathfrak {P}_A}}} \cong {H^{\ast } }(B{C_G}(A),{\mathbf {Z}}/p{\mathbf {Z}})_{{\mathfrak {P}_A}}^{{W_G}(A)}

\]
. Here,

C
G

(
A
)

{C_G}(A)

is the centralizer of

A
A

in

G
G

, and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript upper G Baseline left-parenthesis upper A right-parenthesis equals upper N Subscript upper G Baseline left-parenthesis upper A right-parenthesis slash upper C Subscript upper G Baseline left-parenthesis upper A right-parenthesis">

W
G

(
A
)
=

N
G

(
A
)

/

C
G

(
A
)

{W_G}(A) = {N_G}(A)/{C_G}(A)

, where

N
G

(
A
)

{N_G}(A)

is the normalizer of

A
A

in

G
G

. An example is included.
On the universal theory of classes of finite models
Authors: Tulipani S.
Published: June 2010

First order theories for which the truth of a universal sentence on their finite models implies the truth on all models are investigated. It is proved that an equational theory has such a property if and only if every finitely presented model is residually finite. The most common classes of algebraic structures are discussed.
Positive-definiteness and its applications to interpolation problems for holomorphic functions
Authors: Beatrous Frank , Burbea Jacob
Published: June 2010

Holomorphic interpolation problems of the Pick-Nevanlinna and Loewner types as well as abstract interpolation theorems on functional Hilbert spaces are considered. Various characterizations are presented for restrictions of bounded holomorphic functions. In addition, certain norm estimates for restrictions and extensions of holomorphic functions are obtained.
Singular Vietoris-Begle theorems for relations
Authors: Bourgin D. G. , Nehs Robert M.
Published: June 2010

The Vietoris-Begle theorem with singularities, for three spaces

X
X

,

Y
Y

,

T
T

, is extended to the case that a closed relation replaces a continuous map and more generally to set valued maps. The developments are carried out based on modification of the topology of

T
T

so that in general it is no longer even Hausdorff. This entails interpretation of dimension of singulars sets in terms of considertions in

Y
Y

rather than

T
T

. The techniques are those of sheaf and spectral sequence theory.
Deformation and linkage of Gorenstein algebras
Authors: Kustin Andrew R. , Miller Matthew
Published: June 2010

General double linkage of Gorenstein algebras is defined. Rigidity, genericity, and regularity up to codimension six all pass across general double linkage. Rigid strongly unobstructed codimension four Gorenstein algebras which lie in different Herzog classes are produced.
Nondegenerate symmetric bilinear forms on finite abelian 2-groups
Authors: Miranda Rick
Published: June 2010

Let

B

2

{\mathcal {B}_2}

be the semigroup of isomorphism classes of finite abelian

2
2

-groups with a nondegenerate symmetric bilinear form having values in

Q

/

Z

Q/{\mathbf {Z}}

. Generators for

B

2

{\mathcal {B}_2}

were given by C. T. C. Wall and the known relations among these generators were proved to be complete by A. Kawauchi and S. Kojima. In this article we describe a normal form for such bilinear forms, expressed in terms of Wall’s generators, and as a by-product we obtain a simpler proof of the completeness of the known relations.
Infinite crossed products and group-graded rings
Authors: Passman D. S.
Published: June 2010

In this paper, we precisely determine when a crossed product

R

∗

G

R\;\ast \;G

is semiprime or prime. Indeed we show that these conditions ultimately depend upon the analogous conditions for the crossed products

R

∗

N

R\;\ast \;N

of the finite subgroups

N
N

of

G
G

and upon the interrelationship between the normalizers of these subgroups and the ideal structure of

R
R

. The proof offered here is combinatorial in nature, using the

Δ
\Delta

-methods, and is entirely self-contained. Furthermore, since the argument applies equally well to strongly

G
G

-graded rings, we have opted to work in this more general context.
Representations of compact groups on Banach algebras
Authors: Gurarie David
Published: June 2010

Let a compact group

U
U

act by automorphisms of a commutative regular and Wiener Banach algebra

A

\mathcal {A}

. We study representations

R
ω

{R^\omega }

of

U
U

on quotient spaces

A

/

I
(
ω
)

\mathcal {A}/I(\omega )

, where

ω
\omega

is an orbit of

U
U

in the Gelfand space

X
X

of

A

\mathcal {A}

and

I
(
ω
)

I(\omega )

is the minimal closed ideal with hull

ω
⊂
X

\omega \subset X

. The main result of the paper is: if

A

=

A

ρ

(
X
)

\mathcal {A} = \,{\mathcal {A}_\rho }(X)

is a weighted Fourier algebra on a LCA group

X
=

A
^

X = \hat A

with a subpolynomial weight

ρ
\rho

on

A
A

, and

U
U

acts by affine transformations on

X
X

, then for any orbit

ω
⊂
X

\omega \subset X

the representation

R
ω

{R^\omega }

has finite multiplicity. Precisely, the multiplicity of

π
∈

U
^

\pi \in \hat U

in

R
ω

{R^\omega }

is estimated as

k
(
π
;

R
ω

)
≤
c
⋅
deg
⁡
(
π
)

∀
π
∈

U
^

k(\pi ;{R^\omega }) \leq c \cdot \deg (\pi )\;\forall \pi \in \hat U

with a constant

c
c

depending on

A
A

and

ρ
\rho

. Applications of this result are given to topologically irreducible representations of motion groups and primary ideals of invariant subalgebras.