bluuebunny/crossref_metadata_2025 huggingface dataset:
Torsion in the bordism of oriented involutions
Authors: Rowlett Russell J.
Published: June 2010

In the bordism theory

Ω
∗

(

Z
2
k

)

{\Omega _ \ast }(Z_2^k)

of smooth, orientation-preserving

Z
2
k

Z_2^k

-actions all torsion has order two. Furthermore, the torsion classes inject in the unoriented theory

N
∗

(

Z
2
k

)

{N_ \ast }(Z_2^k)

, and any class represented by a stationary-point free action has infinite order. In addition, a procedure is given for producing Smith constructions in some generality.
On ordinary linear 𝑝-adic differential equations
Authors: Dwork B. , Robba P.
Published: June 2010

We study the solutions of ordinary linear differential equations whose coefficients are analytic elements. As one application we show nonexistence of index for certain linear differential operators with rational function coefficients.
Transversality in 𝐺-manifolds
Authors: Field M. J.
Published: June 2010

A definition of transversality is given for the category of G-manifolds (G, a compact Lie group). Transversality density and isotopy theorems are shown to hold for this definition. An example is given to show that we cannot require differential stability of intersections.
Compact perturbations of certain von Neumann algebras
Authors: Plastiras Joan K.
Published: June 2010

Let

E

\mathcal {E}

be a sequence of mutually orthogonal, finite dimensional projections whose sum is the identity on a Hilbert space

H

\mathcal {H}

. If we denote the commutant of

E

\mathcal {E}

by

D

(

E

)

\mathcal {D}(\mathcal {E})

and the ideal of compact operators on

H

\mathcal {H}

by

C

(

H

)

\mathcal {C}(\mathcal {H})

, then it is easily verified that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D left-parenthesis script upper E right-parenthesis plus script upper C left-parenthesis script upper H right-parenthesis equals left-brace upper T plus upper K colon upper T element-of script upper D left-parenthesis script upper E right-parenthesis comma upper K element-of script upper C left-parenthesis script upper H right-parenthesis right-brace">

D

(

E

)
+

C

(

H

)
=
{
T
+
K
:
T
∈

D

(

E

)
,
K
∈

C

(

H

)
}

\mathcal {D}(\mathcal {E}) + \mathcal {C}(\mathcal {H}) = \{ T + K:T \in \mathcal {D}(\mathcal {E}),K \in \mathcal {C}(\mathcal {H})\}

is a

C
∗

{C^\ast }

-algebra. In this paper we classify all such algebras up to

∗

^\ast

-isomorphism and characterize them by examining their relationship to certain quasidiagonal and quasitriangular algebras.
Inclusion relations between power methods and matrix methods of limitation
Authors: Ziv Abraham
Published: June 2010

A matrix method of limitation is a generalization of both ordinary Toeplitz methods and semicontinuous methods. A power method is a generalization of both Abel’s method and Borel’s exponential method. The main concern of this paper is to find necessary and sufficient conditions for the field of a given power method to be included in the field of a given matrix method. The problem is solved for a wide family of power methods which includes all Abel type methods, the logarithmic method, all Borel type methods and others (also nonregular power methods). Preliminary results, which serve as tools in the solution of the main problem, clarify some aspects of the nature of the field of a power method as an FK space.
Group presentations corresponding to spines of 3-manifolds. II
Authors: Osborne R. P. , Stevens R. S.
Published: June 2010

Let

ϕ
=
⟨

a
1

,
…
,

a
n

|

R
1

,
…
,

R
m

⟩

\phi = \langle {a_1}, \ldots ,{a_n}|{R_1}, \ldots ,{R_m}\rangle

denote a group presentation. Let

K
ϕ

{K_\phi }

denote the corresponding 2-complex. It is well known that every compact 3-manifold has a spine of the form

K
ϕ

{K_\phi }

for some

ϕ
\phi

, but that not every

K
ϕ

{K_\phi }

is a spine of a compact 3-manifold. Neuwirth’s algorithm (Proc. Cambridge Philos. Soc. 64 (1968), 603-613) decides whether

K
ϕ

{K_\phi }

can be a spine of a compact 3-manifold. However, it is impractical for presentations of moderate length. In this paper a simple planar graph-like object, called a RR-system (railroad system), is defined. To each RR-system corresponds a whole family of compact orientable 3-manifolds with spines of the form

K
ϕ

{K_\phi }

, where

ϕ
\phi

has a particular form (e.g.,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle a comma b a Superscript m Baseline b Superscript n Baseline a Superscript p Baseline b Superscript n Baseline comma a Superscript m Baseline b Superscript n Baseline a Superscript m Baseline b Superscript q Baseline mathematical right-angle">

⟨
a
,
b

a
m

b
n

a
p

b
n

,

a
m

b
n

a
m

b
q

⟩

\langle a,b{a^m}{b^n}{a^p}{b^n},{a^m}{b^n}{a^m}{b^q}\rangle

), subject only to certain requirements of relative primeness of certain pairs of exponents. Conversely, every

K
ϕ

{K_\phi }

which is a spine of some compact orientable 3-manifold can be obtained in this way. An equivalence relation on RR-systems is defined so that equivalent RR-systems determine the same family of manifolds. Results of Zieschang are applied to show that the simplest spine of 3-manifolds arises from a particularly simple kind of RR-system called a reduced RR-system. Following Neuwirth, it is shown how to determine when a RR-system gives rise to a collection of closed 3-manifolds.
The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension
Authors: Knerr Barry F.
Published: June 2010

We consider the Cauchy problem in one space dimension for a nonlinear degenerate parabolic partial differential equation. The connectedness of the support of the solution and estimates of the growth of the support as

t

→

∞

t \,\to \,\infty

are established.
Fourier inversion for unipotent invariant integrals
Authors: Barbasch Dan
Published: June 2010

Consider G a semisimple Lie group and

Γ

⊆

G

\Gamma \, \subseteq \,G

a discrete subgroup such that

vol(

G

/

Γ
)

>

∞

{\text {vol(}}G/\Gamma )\, > \,\infty

. An important problem for number theory and representation theory is to find the decomposition of

L
2

(
G

/

Γ
)

{L^2}(G/\Gamma )

into irreducible representations. Some progress in this direction has been made by J. Arthur and G. Warner by using the Selberg trace formula, which expresses the trace of a subrepresentation of

L
2

(
G

/

Γ
)

{L^2}(G/\Gamma )

in terms of certain invariant distributions. In particular, measures supported on orbits of unipotent elements of G occur. In order to obtain information about representations it is necessary to expand these distributions into Fourier components using characters of irreducible unitary representations of G. This problem is solved in this paper for real rank

G

=

1

G\, =\, 1

. In particular, a relationship between the semisimple orbits and the nilpotent ones is made explicit generalizing an earlier result of R. Rao.
The sharpness of Lorentz’s theorem on incomplete polynomials
Authors: Saff E. B. , Varga R. S.
Published: June 2010

For any fixed

θ
\theta

with

0
>
θ
>
1

0 > \theta > 1

, G. G. Lorentz recently showed that bounded sequences
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace normal upper Sigma Subscript theta n Sub Subscript i Subscript less-than-or-slanted-equals k less-than-or-slanted-equals n Sub Subscript i Subscript Baseline a Subscript k Baseline left-parenthesis i right-parenthesis left-parenthesis 1 plus t right-parenthesis Superscript k Baseline right-brace Subscript i equals 1 Baseline Superscript normal infinity Baseline">

{

Σ

θ

n
i

⩽
k
⩽

n
i

a
k

(
i
)

(
1
+
t
)

k

}

i
=
1

∞

\{\Sigma _{\theta {n_i} \leqslant k \leqslant {n_i}} {{a_k}(i){{(1 + t)}^k}\} _{i = 1}^\infty }

of incomplete polynomials on

[
−
1
,
+
1
]

[ - 1, + 1]

tend uniformly to zero on closed intervals of

[
−
1
,
Δ
(
θ
)
)

[ - 1,\Delta (\theta ))

, where

2

θ
2

−
1
⩽
Δ
(
θ
)
>
2
θ
−
1

2{\theta ^2} - 1 \leqslant \Delta (\theta ) > 2\theta - 1

. In this paper, we show that

Δ
(
θ
)
=
2

θ
2

−
1

\Delta (\theta ) = 2{\theta ^2} - 1

is best possible, and that the geometric convergence to zero of such sequences on closed intervals

[

t
0

,

t
1

]

[{t_0},{t_1}]

can be precisely bounded above as a function of

t
j

{t_j}

and

θ
\theta

. Extensions of these results to the complex plane are also included.
𝜎-idéaux engendrés par des ensembles fermés et théorèmes d’approximation
Authors: Louveau Alain
Published: June 2010

This paper is motivated by the study of *-games on

ω
\omega

, and by a question of D. A. Martin on the strength of the hypothesis

AD

ω

∗

{\text {AD}}_\omega ^{\ast }

that every *-game on

ω
ω

{\omega ^\omega }

is determined. A general study of the

σ
\sigma

-ideals of subsets of

ω
ω

{\omega ^\omega }

generated by closed sets encompasses the *-games and the “perfect set property". Using associated games, we extend for these ideals many properties known for countable sets, under various hypotheses of determinacy. Our methods thus apply also to other examples of regularity properties, such as those introduced by A. S. Kechris. Finally, a general theorem of approximation by analytic sets in Solovay’s model is proved which, together with the preceding results, gives the solution of Martin’s problem:

AD

ω

∗

{\text {AD}}_\omega ^{\ast }

is true in Solovay’s model.
Affine extensions of a Bernoulli shift
Authors: Feldman J. , Rudolph D. J. , Moore C. C.
Published: June 2010

(a) For any automorphism

ϕ
\phi

of a compact metric group G, and any

a
>
0

a > 0

, we show the existence of a free finite measure-preserving (m.p.) action of the twisted product

Z

×
ϕ

G

Z{ \times ^\phi }\,G

whose restriction to Z is Bernoulli with entropy

a

+

h
(
ϕ
)

a\, + \,h(\phi )

,

h
(
ϕ
)

h(\phi )

being the entropy of

ϕ
\phi

on G with Haar measure. (b) A classification is given of all free finite m.p. actions of

Z

×

ϕ

G

Z\, \times {\,^\phi }\,G

such that the action of Z on the

σ
\sigma

-algebra of invariant sets of G is a Bernoulli action. (c) The classification of (b) is extended to “quasifree” actions: those for which the isotropy subgroups are in a single conjugacy class within G. An existence result like that of (a) holds in this case, provided certain necessary and sufficient algebraic conditions are satisfied; similarly, an isomorphism theorem for such actions holds, under certain necessary and sufficient conditions. (d) If G is a Lie group, then all actions of

Z

×

ϕ

G

Z\, \times {\,^\phi }\,G

are quasifree; if G is also connected, then the second set of additional algebraic conditions alluded to in (c) is always satisfied, while the first will be satisfied only in an obvious case. (e) Examples are given where the isomorphism theorem fails: by violation of the algebraic conditions in the quasifree case, for other reasons in the non-quasifree case.
Destructible and indestructible Blaschke products
Authors: Morse H. Stephen
Published: June 2010

A special case of destructibility for Blaschke products is introduced and studied. An example is given of a destructible Blaschke product which becomes indestructible when a single point is deleted from its zero-set.
The dependence of the generalized Radon transform on defining measures
Authors: Quinto Eric Todd
Published: June 2010

Guillemin proved that the generalized Radon transform R and its dual

R
t

{R^t}

are Fourier integral operators and that

R
t

R

{R^t}R

is an elliptic pseudodifferential operator. In this paper we investigate the dependence of the Radon transform on the defining measures. In the general case we calculate the symbol of

R
t

R

{R^t}R

as a pseudodifferential operator in terms of the measures and give a necessary condition on the defining measures for

R
t

R

{R^t}R

to be invertible by a differential operator. Then we examine the Radon transform on points and hyperplanes in

R

n

{\textbf {R}^n}

with general measures and we calculate the symbol of

R
t

R

{R^t}R

in terms of the defining measures. Finally, if

R
t

R

{R^t}R

is a translation invariant operator on

R

n

{\textbf {R}^n}

then we prove that

R
t

R

{R^t}R

is invertible and that our condition is equivalent to

(

R
t

R

)

−
1

{({R^t}R)^{ - 1}}

being a differential operator.
The free boundary for elastic-plastic torsion problems
Authors: Friedman Avner , Pozzi Gianni A.
Published: June 2010

Consider the variational inequality: Find

u

∈

K

u\, \in \, K

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper Q Endscripts nabla u dot nabla left-parenthesis upsilon minus u right-parenthesis greater-than-or-slanted-equals mu integral Underscript upper Q Endscripts left-parenthesis upsilon minus u right-parenthesis left-parenthesis mu greater-than 0 right-parenthesis">

∫
Q

∇
u
⋅

∇

(
υ

−

u
)

⩾

μ

∫
Q

(
υ

−

u
)

(
μ

>

0
)

\int _Q {\nabla u \cdot \, \nabla \, (\upsilon \, -\, u)\, \geqslant \, \mu \, \int _Q\, {(\upsilon \, -\, u)\,(\mu \, >\, 0)} }

for any

υ

∈

K

\upsilon \, \in \, K

, where

K

=

{
w

∈

H
0
1

(
Q
)
;

|

∇

w

|

⩽

1
}

K\, =\, \{ w\, \in \, H_0^1(Q);\,\left | {\nabla \, w} \right |\, \leqslant \, 1\}

and Q is a 2-dimensional simply connected domain in

R
2

{R^2}

with piecewise

C
3

{C^3}

boundary. The solution u represents the stress function in a torsion problem of an elastic-plastic bar with cross section Q. The sets

E

=

{
x

∈

Q
;

|

∇

u
(
x
)

|

>

1
}

E\, =\, \{ x\, \in \, Q;\,\left | {\nabla \, u(x)} \right |\, >\, 1\}

,

P

=

{
x

∈

Q
;

|

∇

u
(
x
)

|

=

1
}

P\, =\, \{ x\, \in \, Q;\,\left | {\nabla \, u(x)} \right |\, =\, 1\}

are the elastic and plastic sets respectively. The purpose of this paper is to study the free boundary

∂
E

∩

Q

\partial E\, \cap \, Q

; more specifically, an estimate is derived on the number of points of local maximum of the free boundary.
A lower central series for split Hopf algebras with involution
Authors: Jordan Bruce W.
Published: June 2010

A lower central series is defined for split Hopf algebras with involution over a field k. Various structure theorems for coalgebras and Hopf algebras are established.
Continuously translating vector-valued measures
Authors: Tewari U. B. , Dutta M.
Published: June 2010

Let G be a locally compact group and A an arbitrary Banach space.

L
p

(
G
,
A
)

{L^p}(G,A)

will denote the space of p-integrable A-valued functions on G.

M
(
G
,
A
)

M(G,A)

will denote the space of regular A-valued Borel measures of bounded variation on G. In this paper, we characterise the relatively compact subsets of

L
p

(
G
,
A
)

{L^p}(G,A)

. Using this result, we prove that if

μ

∈

M
(
G
,
A
)

\mu \, \in \, M(G,A)

, such that either

x

→

μ
x

x\, \to \, {\mu _x}

or

x

→
x

μ

x{ \to _x}\mu

is continuous, then

μ

∈

L
1

(
G
,
A
)

\mu \, \in \, {L^1}(G,A)

.
Asymptotic completeness for classes of two, three, and four particle Schrödinger operators
Authors: Hagedorn George A.
Published: June 2010

Formulas for the resolvent

(
z

−

H

)

−
1

{(z\, - \,H)^{ - 1}}

are derived, where

H

=

H
0

+

Σ

i
>
j

λ

i
j

V

i
j

H\, = \,{H_0}\, + \,{\Sigma _{i > j}}{\lambda _{ij}}{V_{ij}}

is an N particle Schrödinger operator with the center of mass motion removed. For a large class of two-body potentials and generic couplings

{

λ

i
j

}

\{ {\lambda _{ij}}\}

, these formulas are used to prove asymptotic completeness in the

N

⩽

4

N\, \leqslant \,4

body problem when the space dimension is

m

⩾

3

m\, \geqslant \,3

. The allowed potentials belong to a space of dilation analytic multiplication operators which fall off more rapidly than

r

−
2
−
ε

{r^{ - 2 - \varepsilon }}

at

∞
\infty

. In particular, Yukawa potentials, generalized Yukawa potentials, and potentials of the form

(
1

+

r

)

−
2
−
ε

{(1\, + \,r)^{ - 2 - \varepsilon }}

are allowed.
Fibrewise localization and completion
Authors: May J. P.
Published: June 2010

The behavior of fibrewise localization and completion on the classifying space level is analyzed. The relationship of these constructions to fibrewise joins and smash products and to orientations of spherical fibrations is also analyzed. This theory is essential to validate Sullivan’s proof of the Adams conjecture.
Matrix-valued special functions and representation theory of the conformal group. I. The generalized gamma function
Authors: Gross Kenneth I. , Holman Wayne J.
Published: June 2010

This article examines in detail the matrix-valued gamma function
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Superscript lamda Super Superscript 0 Baseline left-parenthesis alpha right-parenthesis equals integral Underscript upper P Endscripts e Superscript minus trace left-parenthesis r right-parenthesis Baseline lamda Superscript 0 Baseline left-parenthesis r comma r overbar right-parenthesis left-parenthesis det r right-parenthesis Superscript alpha minus 2 Baseline d r">

Γ

λ
0

(
α
)

=

∫
P

e

−

tr

(
r
)

λ
0

(
r
,

r
¯

)

(
det

r
)

α
−
2

d
r

{\Gamma ^{{\lambda ^0}}}\,(\alpha )\, = \,\int _P {{e^{ - {\text {tr}}(r)}}{\lambda ^0}(r,\,\bar r){{(\det \,r)}^{\alpha - 2}}\,} dr

\]
associated to the conformal group

G

=

U
(
2
,

2
)

G\, = \,U(2,\,2)

. Here,

α
\alpha

is a continuous complex parameter,

λ
0

{\lambda ^0}

runs through a family of “weights” of

K

=

U
(
2
)

×

U
(
2
)

K\, = \,U(2)\, \times \,U(2)

, P is the cone of

2

×

2

2\, \times \,2

positive-definite Hermitian matrices, and the integral is well known to converge absolutely for

Re

(
α
)

>

1

{\text {Re}}(\alpha )\, > \,1

. However, until now very little has been known about the analytic continuation for the general weight

λ
0

{\lambda ^0}

. The results of this paper include the following: The complete analytic continuation of

Γ

λ
0

{\Gamma ^{{\lambda ^0}}}

is determined for all weights

λ
0

{\lambda ^0}

. In analogy to the case of the classical gamma function it is proved that for any weight

λ
0

{\lambda ^0}

the mapping

α

→

Γ

λ
0

(
α

)

−
1

\alpha \, \to \,{\Gamma ^{{\lambda ^0}}}\,{(\alpha )^{ - 1}}

is entire. A new integral formula is given for the inverse of the gamma function. An explicit calculation is given for the normalized variant of the gamma matrix that arises in the reproducing kernel for the spaces in which the holomorphic discrete series of G is realized, and one observes that the behavior of the analytic continuation for weights “in general position” is markedly different from the special cases in which the gamma function “is scalar". The full analytic continuation of the holomorphic discrete series for G is determined. The gamma function for the forward light cone (the boundary orbit) is found, and the associated Hardy space of vector-valued holomorphic functions is described. Analogs are given for some of the well-known formulas for the classical gamma function. As an epilogue, applications of the matrix-valued gamma function, such as generalizations to

2

×

2

2\, \times \,2

matrix space of the classical binomial theorem, are announced. These applications require a detailed understanding of the (generalized) Bessel functions associated to the conformal group that will be treated in the sequel to this paper.
Some undecidability results concerning Radon measures
Authors: Gardner R. J. , Pfeffer W. F.
Published: June 2010

We show that in metalindelöf spaces certain questions about Radon measures cannot be decided within the Zermelo-Fraenkel set theory, including the axiom of choice.
On generalized harmonic analysis
Authors: Lau Ka Sing , Lee Jonathan K.
Published: June 2010

Motivated by Wiener’s work on generalized harmonic analysis, we consider the Marcinkiewicz space

M

p

(

R

)

{{\mathcal {M}}^p}({\textbf {R}})

of functions of bounded upper average p power and the space

V

p

(

R

)

{{\mathcal {V}}^p}({\textbf {R}})

of functions of bounded upper p variation. By identifying functions whose difference has norm zero, we show that

V

p

(

R

)

{{\mathcal {V}}^p}({\textbf {R}})

,

1

>

p

>
∞

1\, > \,p\, > \infty

, is a Banach space. The proof depends on the result that each equivalence class in

V

p

(

R

)

{{\mathcal {V}}^p}({\textbf {R}})

contains a representative in

L
p

(

R

)

{L^p}({\textbf {R}})

. This result, in turn, is based on Masani’s work on helixes in Banach spaces. Wiener defined an integrated Fourier transformation and proved that this transformation is an isometry from the nonlinear subspace

W

2

(

R

)

{{\mathcal {W}}^2}({\textbf {R}})

of

M

2

(

R

)

{{\mathcal {M}}^2}({\textbf {R}})

consisting of functions of bounded average quadratic power, into the nonlinear subspace

U

2

(

R

)

{\mathcal {U}^2}({\textbf {R}})

of

V

2

(

R

)

{{\mathcal {V}}^2}({\textbf {R}})

consisting of functions of bounded quadratic variation. By using two generalized Tauberian theorems, we prove that Wiener’s transformation W is actually an isomorphism from

M

2

(

R

)

{{\mathcal {M}}^2}({\textbf {R}})

onto

V

2

(

R

)

{{\mathcal {V}}^2}({\textbf {R}})

. We also show by counterexamples that W is not an isometry on the closed subspace generated by

W

2

(

R

)

{{\mathcal {W}}^2}({\textbf {R}})

.
Littlewood-Paley and multiplier theorems on weighted 𝐿^{𝑝} spaces
Authors: Kurtz Douglas S.
Published: June 2010

The Littlewood-Paley operator

γ
(
f
)

\gamma (f)

, for functions f defined on

R

n

{{\textbf {R}}^n}

, is shown to be a bounded operator on certain weighted

L
p

{L^p}

spaces. The weights satisfy an

A
p

{A_p}

condition over the class of all n-dimensional rectangles with sides parallel to the coordinate axes. The necessity of this class of weights demonstrates the 1-dimensional nature of the operator. Results for multipliers are derived, including weighted versions of the Marcinkiewicz Multiplier Theorem and Hörmander’s Multiplier Theorem.
Statically tame periodic homeomorphisms of compact connected 3-manifolds. II. Statically tame implies tame
Authors: Moise Edwin E.
Published: June 2010

Let f be a periodic homeomorphism

M

↔

M

M\, \leftrightarrow \,M

, where M is a compact connected 3-manifold (without boundary). Suppose that for each i, the fixed-point set of

f
i

{f^i}

is a tame set. Then f is simplicial, relative to some triangulation of M.
Quasilinear evolution equations in Banach spaces
Authors: Murphy Michael G.
Published: June 2010

This paper is concerned with the quasi-linear evolution equation

u
′

(
t
)

+

A
(
t
,

u
(
t
)
)
u
(
t
)

=

0

u’(t)\, + \,A(t,\,u(t))u(t)\, = \,0

in

[
0
,

T
]
,

u
(
0
)

=

x
0

[0,\,T],\,u(0)\, = \,{x_0}

in a Banach space setting. The spirit of this inquiry follows that of T. Kato and his fundamental results concerning linear evolution equations. We assume that we have a family of semigroup generators that satisfies continuity and stability conditions. A family of approximate solutions to the quasi-linear problem is constructed that converges to a “limit solution.” The limit solution must be the strong solution if one exists. It is enough that a related linear problem has a solution in order that the limit solution be the unique solution of the quasi-linear problem. We show that the limit solution depends on the initial value in a strong way. An application and the existence aspect are also addressed.
Complementary series for 𝑝-adic groups. I
Authors: Silberger Allan J.
Published: June 2010

Let

Ω
\Omega

be a nonarchimedean local field, G the group of

Ω
\Omega

-points of a connected reductive algebraic group defined over

Ω
\Omega

. This paper establishes that to each zero of the Plancherel measure of G one can associate complementary series. Our result is the analogue for p-adic groups of a similar statement, announced separately by Knapp-Stein and Harish-Chandra, for real groups.
Flows on fibre bundles
Authors: Noakes J. L.
Published: June 2010

Conditions are given under which a fibrewise flow on a fibre bundle must have a nonempty catastrophe space.
The Minakshisundaram-Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature
Authors: Miatello Roberto J.
Published: June 2010

We use harmonic analysis on semisimple Lie groups to determine the Minakshisundaram-Pleijel asymptotic expansion for the trace of the heat kernel on natural vector bundles over compact, locally symmetric spaces of strictly negative curvature.
Shellable and Cohen-Macaulay partially ordered sets
Authors: Björner Anders
Published: June 2010

In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to R. Stanley, is generalized to posets and shown to imply shellability, while Stanley’s main theorem on the Jordan-HÖlder sequences of such labelings remains valid. Further, we show a number of ways in which shellable posets can be constructed from other shellable posets and complexes. These results give rise to several new examples of Cohen-Macaulay posets. For instance, the lattice of subgroups of a finite group G is Cohen-Macaulay (in fact shellable) if and only if G is supersolvable. Finally, it is shown that all the higher order complexes of a finite planar distributive lattice are shellable.
Homotopy operations for simplicial commutative algebras
Authors: Dwyer W. G.
Published: June 2010

The indicated operation algebra is studied by methods dual to the usual ones for studying the Steenrod algebra. In particular, the operations are constructed using higher symmetries of the shuffle map and their “Adem relations” are computed using the transfer map in the cohomology of symmetric groups.
𝑞-extension of the 𝑝-adic gamma function
Authors: Koblitz Neal
Published: June 2010

p-adic functions depending on a parameter q,

0

>

|

q

−

1

|

p

>

1

0\, > \,|q\, - \,1{|_p}\, > \,1

, are defined which extend Y. Morita’s p-adic gamma function and the derivative of J. Diamond’s p-adic log-gamma function in the same way as the classical q-gamma function

Γ
q

(
x
)

{\Gamma _q}(x)

extends

Γ
(
x
)

\Gamma (x)

. Properties of these functions which are analogous to the basic identities satisfied by

Γ
q

(
x
)

{\Gamma _q}(x)

are developed.
Minimal skew products
Authors: Glasner S.
Published: June 2010

Let

(
σ
,

Z
)

(\sigma ,\,Z)

be a metric minimal flow. Let Y be a compact metric space and let

g

\mathcal {g}

be a pathwise connected group of homeomorphisms of Y. We consider a family of skew product flows on

Z

×

Y

=

X

Z\, \times \,Y\, = \,X

and show that when

(

g

,

Y
)

(\mathcal {g},\,Y)

is minimal most members of this family have the property of being disjoint from every minimal flow which is disjoint from

(
σ
,

Z
)

(\sigma ,\,Z)

. From this and some further results about skew product flows we deduce the existence of a minimal metric flow which is disjoint from every weakly mixing minimal flow but is not PI.
A 𝑍×𝑍 structurally stable action
Authors: Grossi Sad P. R.
Published: June 2010

We consider in the product of spheres

S
m

×

S
n

{S^m}\, \times \,{S^n}

the

Z

×

Z

Z\, \times \,Z

-action generated by two simple Morse-Smale diffeomorphisms; if they have some kind of general position, the action is shown to be stable. An application is made to foliations.
Factorization of curvature operators
Authors: Vilms Jaak
Published: June 2010

Let V be a real finite-dimensional vector space with inner product and let R be a curvature operator, i.e., a symmetric linear map of the bivector space

Λ

2

V

\Lambda {\,^2}V

into itself. Necessary and sufficient conditions are given for R to admit factorization as

R

=

Λ

2

L

R\, = \,\Lambda {\,^2}L

, with L a symmetric linear map of V into itself. This yields a new characterization of Riemannian manifolds that admit local isometric embedding as hypersurfaces of Euclidean space.
Fourier inversion on Borel subgroups of Chevalley groups: the symplectic group case
Authors: Lipsman Ronald L.
Published: June 2010

In recent papers, the author and J. A. Wolf have developed the Plancherel theory of parabolic subgroups of real reductive Lie groups. This includes describing the irreducible unitary representations, computing the Plancherel measure, and-since parabolic groups are nonunimodular-explicating the (unbounded) Dixmier-Pukanszky operator that appears in the Plancherel formula. The latter has been discovered to be a special kind of pseudodifferential operator. In this paper, the author considers the problem of extending this analysis to parabolic subgroups of semisimple algebraic groups over an arbitrary local field. Thus far he has restricted his attention to Borel subgroups (i.e. minimal parabolics) in Chevalley groups (i.e. split semisimple groups). The results he has obtained are described in this paper for the case of the symplectic group. The final result is (perhaps surprisingly), to a large extent, independent of the local field over which the group is defined. Another interesting feature of the work is the description of the “pseudodifferential” Dixmier-Pukanszky operator in the nonarchimedean situation.
Invariant solutions to the oriented Plateau problem of maximal codimension
Authors: Bindschadler David
Published: June 2010

The principal result gives conditions which imply that a solution to the Plateau problem inherits the symmetries of its boundary. Specifically, let G be a compact connected Lie subgroup of

SO

(
n
)

{\text {SO}}(n)

. Assume the principal orbits have dimension m, there are no exceptional orbits and the distribution of

(
n

−

m
)

(n\, - \,m)

-planes orthogonal to the principal orbits is involutive. We show that if B is a finite sum of oriented principal orbits, then every absolutely area minimizing current with boundary B is invariant. As a consequence of the methods used, the above Plateau problems are shown to be equivalent to 1-dimensional variational problems in the orbit space. Some results concerning invariant area minimizing currents in Riemannian manifolds are also obtained.
Approximation results in Hilbert cube manifolds
Authors: Chapman T. A.
Published: June 2010

The purpose of this paper is to answer questions of the following type for maps

f
:

M
→

N

f:\,M \to \,N

between Q-manifolds: When is f close to an approximate fibration? When is f close to a fiber bundle map? When is f close to a block bundle map? For the second and third questions a series of obstructions are encountered in the lower algebraic K-theoretic functors

K

−

i

{K_{ - \,i}}

of Bass.
The category of 𝐷-completely regular spaces is simple
Authors: Heldermann N. C.
Published: June 2010

In a recent paper H. Brandenburg characterized the objects of the epireflective hull of all developable spaces-that are those spaces which are homeomorphic to a subspace of a product of developable spaces-by intrinsic properties. It is shown here that these spaces, called D-completely regular, can be generated from a single second countable developable space D which has the same cardinality as the reals. As an application of this result we obtain a new characterization of D-normal spaces analogous to Urysohn’s lemma and a new (external) characterization of perfect spaces (meaning every closed set is a

G
δ

{G_\delta }

).
Linear spaces with an 𝐻*-algebra-valued inner product
Authors: Saworotnow Parfeny P.
Published: June 2010

The paper deals with a particular class of VH-spaces of Loynes [5] whose inner product assumes its values in a trace-algebra associated with an

H

∗

H^{\ast }

-algebra. It is shown that these spaces admit a structure of a “nonassociative module", and this structure could be used to characterize such spaces. Also we characterize other related spaces.
Degenerations of 𝐾3 surfaces of degree 4
Authors: Shah Jayant
Published: June 2010

A generic

K
3

K3

surface of degree

4
4

may be embedded as a nonsingular quartic surface in

P

3

{{\mathbf {P}}_3}

. Let

f
:
X
→
Spec

C

[
[
t
]
]

f:X \to \operatorname {Spec} \;{\mathbf {C}}[[t]]

be a family of quartic surfaces such that the generic fiber is regular. Let

Σ
0

{\Sigma _0}

,

Σ
2
0

{\Sigma _2^0}

,

Σ
4

{\Sigma _4}

be respectively a nonsingular quadric in

P

3

{{\mathbf {P}}_3}

, a cone in

P

3

{{\mathbf {P}}_3}

over a nonsingular conic and a rational, ruled surface in

P

9

{{\mathbf {P}}_9}

which has a section with self intersection

−
4

- 4

. We show that there exists a flat, projective morphism

f
′

:

X
′

→

Spec

C

[
[
t
]
]

f’:X’ \to {\text {Spec}}\;{\mathbf {C}}[[t]]

and a map

ρ
:

Spec

C

[
[
t
]
]
→

Spec

C

[
[
t
]
]

\rho :{\text {Spec}}\:{\mathbf {C}}[[t]] \to {\text {Spec}}\:{\mathbf {C}}[[t]]

such that (i) the generic fiber of

f
′

f’

and the generic fiber of the pull-back of

f
f

via

ρ
\rho

are isomorphic, (ii) the fiber

X
0
′

{X’_0}

of

f
′

f’

over the closed point of

Spec

C

[
[
t
]
]

{\text {Spec}}\;{\mathbf {C}}[[t]]

has only insignificant limit singularities and (iii)

X
0
′

{X’_0}

is either a quadric surface or a double cover of

Σ
0

{\Sigma _0}

,

Σ
2
0

{\Sigma _2^0}

or

Σ
4

{\Sigma _4}

. The theorem is proved using the geometric invariant theory.
Product-convolution operators and mixed-norm spaces
Authors: Busby Robert C. , Smith Harvey A.
Published: June 2010

Conditions for boundedness and compactness of product-convolution operators

g
→

P
h

C
f

g
=
h
⋅
(
f
∗
g
)

g \to {P_h}{C_f}g = h \cdot (f\ast g)

on spaces

L
p

(
G
)

{L_p}(G)

are studied. It is necessary for boundedness to define a class of "mixed-norm" spaces

L

(
p
,
q
)

(
G
)

{L_{(p,q)}}(G)

interpolating the

L
p

(
G
)

{L_p}(G)

spaces in a natural way

(

L

(
p
,
p
)

=

L
p

)

({L_{(p,p)}} = {L_p})

. It is then natural to study the operators acting between

L

(
p
,
q
)

(
G
)

{L_{(p,q)}}(G)

spaces, where

G
G

has a compact invariant neighborhood. The theory of

L

(
p
,
q
)

(
G
)

{L_{(p,q)}}(G)

is developed and boundedness and compactness conditions of a nonclassical type are obtained. It is demonstrated that the results extend easily to a somewhat broader class of integral operators. Several known results are strengthened or extended as incidental consequences of the investigation.
Spherical bundles adapted to a 𝐺-fibration
Authors: Hodgson J. P. E.
Published: June 2010

A spherical fibration

p
:
E
→
B

p:E \to B

is said to be adapted to a

G
G

-fibration

π
:
E
→
E

/

G

\pi :E \to E/G

if there is a fibration

q
:
E

/

G
→
B

q:E/G \to B

with fibre the quotient of a sphere by a free

G
G

-action and such that the composition

q
∘
π
=
p

q \circ \pi = p

. In this paper it is shown that for spherical bundles in the PL, TOP or Homotopy categories that are adapted to

Z
2

{Z_2}

- or

S
1

{S^1}

-fibrations there is a procedure analogous to the splitting principle for vector bundles that enables one to define characteristic classes for these fibrations and to relate them to the usual characteristic classes. The methods are applied to show that a spherical fibration over a

4
4

-connected base which is adapted to an

S
1

{S^1}

-fibration admits a PL structure.
Derivations and automorphisms of nonassociative matrix algebras
Authors: Benkart G. M. , Osborn J. M.
Published: June 2010

This paper studies the derivation algebra and the automorphism group ofMn(A){M_n}(A),n×nn \times nmatrices over an arbitrary nonassociative algebraAAwith multiplicative identity11. The investigation also includes results on derivations and automorphisms of the algebras obtained fromMn(A){M_n}(A)using the Lie product[xy]=xy−yx[xy] = xy - yx, and the Jordan productx∘y=12(xy+yx)x \circ y = \tfrac {1} {2}(xy + yx).
Some countability conditions on commutative ring extensions
Authors: Gilmer Robert , Heinzer William
Published: June 2010

IfSSis a finitely generated unitary extension ring of the commutative ringRR, thenSScannot be expressed as the union of a strictly ascending sequence{Rn}n=1∞\{ {R_n}\} _{n = 1}^\inftyof intermediate subrings. A primary concern of this paper is that of determining the class of commutative ringsTTfor which the converse holds—that is, each unitary extension ofTTnot expressible as∪1∞Ti\cup _1^\infty {T_i}is finitely generated overTT.
The order convergence of martingales indexed by directed sets
Authors: Astbury Kenneth A.
Published: June 2010

We obtain a condition on the underlying family of

σ
\sigma

-algebras which is properly weaker than the Vitali property but which is also a sufficient condition for the order convergence of martingales of semibounded variation. We also obtain a sufficient condition for the order convergence of martingales of semibounded variation in terms of the finiteness of the extreme order limits of martingales of bounded variation.
Lifting cycles to deformations of two-dimensional pseudoconvex manifolds
Authors: Laufer Henry B.
Published: June 2010

Let

M
M

be a strictly pseudoconvex manifold with exceptional set

A
A

. Let

D
⩾
0

D \geqslant 0

be a cycle on

A
A

. Let

ω
:

M

→
Q

\omega :\mathfrak {M} \to Q

be a deformation of

M
M

. Kodaira’s theory for deforming submanifolds of

M

\mathfrak {M}

is extended to the subspace

D
D

. Let

J

\mathfrak {J}

be the sheaf of germs of infinitesimal deformations of

D
D

. Suppose that

H
1

(
D
,

J

)
=
0

{H^1}(D,\mathfrak {J}) = 0

. If

ω
\omega

is the versal deformation, then

D
D

lifts to above a submanifold of

Q
Q

. This lifting is a complete deformation of

D
D

with a smooth generic fiber. If all of the fibers of

M

\mathfrak {M}

are isomorphic, then

ω
\omega

is the trivial deformation. If

M
M

has no exceptional curves of the first kind, then there exists

ω
\omega

such that only any given irreducible component of

A
A

disappears as part of the exceptional set.
Hulls of deformations in 𝐶ⁿ
Authors: Alexander H.
Published: June 2010

A problem of

E

{\text {E}}

. Bishop on the polynomially convex hulls of deformations of the torus is considered. Let the torus

T
2

{T^2}

be the distinguished boundary of the unit polydisc in

C

2

{{\mathbf {C}}^2}

. If

t
↦

T
t
2

t \mapsto T_t^2

is a smooth deformation of

T
2

{T^2}

in

C

2

{{\mathbf {C}}^2}

and

g
0

{g_0}

is an analytic disc in

C

2

{{\mathbf {C}}^2}

with boundary in

T
2

{T^2}

, a smooth family of analytic discs

t
↦

g
t

t \mapsto {g_t}

, is constructed with the property that the boundary of

g
t

{g_t}

lies in

T
t
2

T_t^2

. This construction has implications for the polynomially convex hulls of the tori

T
t
2

T_t^2

. An analogous problem for a

2
2

-sphere in

C

2

{{\mathbf {C}}^2}

is also considered.
Algebraic fiber bundles
Authors: Landsburg Steven E.
Published: June 2010

When

X
X

is a finite simplicial complex and

G
G

is any of a certain class of groups, a classification of

G
G

-principal bundles over

X
X

in terms of projective modules over a ring

R
(
G
,
X
)

R(G,X)

is given. This generalizes Swan’s classification of vector bundles and uses the results of Mulvey. Often,

R
R

can be taken to be noetherian; in this case

Spec

(
R
)

{\text {Spec}}(R)

is usually reducible with "cohomologically trivial" irreducible components. Information is derived concerning the nature of projective modules over such rings, and some results are obtained indicating how such information reflects information about

X
X

.
Baire category principle and uniqueness theorem
Authors: Hwang J. S.
Published: June 2010

Applying a theorem of Bagemihl and Seidel (1953), we prove that if

S
2

{S_2}

is a set of second category in

(
α
,
β
)

(\alpha ,\beta )

, where

0
⩽
α
>
β
⩽
2
π

0 \leqslant \alpha > \beta \leqslant 2\pi

, and if

f
(
z
)

f(z)

is a function meromorphic in the sector
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta left-parenthesis alpha comma beta right-parenthesis equals left-brace z colon 0 greater-than StartAbsoluteValue z EndAbsoluteValue greater-than normal infinity comma alpha greater-than arg z greater-than beta right-brace">

Δ
(
α
,
β
)
=
{
z
:
0
>

|
z
|

>
∞
,
α
>
arg
⁡
z
>
β
}

\Delta (\alpha ,\beta ) = \{ z:0 > \left | z \right | > \infty ,\alpha > \arg z > \beta \}

for which

lim

_

r
→
∞

|

f
(
r

e

i
θ

)

|

>
0

{\underline {{\operatorname {lim}}} _{r \to \infty }}\left | {f(r{e^{i\theta }})} \right | > 0

, for all

θ
∈

S
2

\theta \in {S_2}

, then there exists a sector

Δ
(

α
′

,

β
′

)
⊆
Δ
(
α
,
β
)

\Delta (\alpha ’,\beta ’) \subseteq \Delta (\alpha ,\beta )

such that

(

α
′

,

β
′

)
⊆

S
¯

2

,

S
2

(\alpha ’,\beta ’) \subseteq {\bar S_2},{S_2}

is second category in

(

α
′

,

β
′

)

(\alpha ’,\beta ’)

, and

f
(
z
)

f(z)

has no zero in

Δ
(

α
′

,

β
′

)

\Delta (\alpha ’,\beta ’)

. Based on this property, we prove several uniqueness theorems.
On the zeros of Dirichlet 𝐿-functions. II
Authors: Fujii Akio
Published: June 2010

Some consequences of the main theorem of On the zeros of Dirichlet

L
L

-functions. I, Trans. Amer. Math. Soc. 196 (1974), 225-235 are proved.
Subspaces of 𝐿¹, via random measures
Authors: Aldous David J.
Published: June 2010

It is shown that every subspace ofL1{L^1}contains a subspace isomorphic to somelq{l_q}. The proof depends on a fixed point theorem for random measures.
Some universal sets of terms
Authors: Taylor Walter
Published: June 2010

For every

Π
2
1

\Pi _2^1

class of cardinals containing

0
0

and

1
1

, there exists a finite set

T
T

of terms, such that

X
X

is precisely the class of cardinals in which

T
T

is universal.
Conditional expectations in 𝐶*-crossed products
Authors: Itoh Shigeru
Published: June 2010

Let

(
A
,

G
,

α
)

(A,\,G,\,\alpha )

be a

C

∗

{C^{\ast }}

-dynamical system. Let

B
B

be a

C

∗

{C^{\ast }}

-subalgebra of

A
A

and

P
P

be a conditional expectation of

A
A

onto

B
B

such that

α
t

P
=
P

α
t

{\alpha _t}P = P{\alpha _t}

for each

t
∈
G

t \in G

. Then it is proved that there exists a conditional expectation of

C

∗

(
G
,

A
,

α
)

{C^{\ast }}(G,\,A,\,\alpha )

onto

C

∗

(
G
,

B
,

α
)

{C^{\ast }}(G,\,B,\,\alpha )

. In particular, if

G
G

is amenable and

A
A

is unital, then there always exists a conditional expectation of

C

∗

(
G
,

A
,

α
)

{C^{\ast }}(G,\,A,\,\alpha )

onto

C

∗

(
G
)

{C^{\ast }}(G)

. Some related results are also obtained.
Les modèles dénombrables d’une théorie ayant des fonctions de Skolem
Authors: Lascar Daniel
Published: June 2010

Let

T
T

be a countable complete theory having Skolem functions. We prove that if all the types over finitely generated models are definable (this is the case for example if

T
T

is stable), then either

T
T

has

2

ℵ
0

{2^{{\aleph _0}}}

countable models or all its models are homogeneous. The proof makes heavy use of stability techniques.
Uncountable admissibles. I. Forcing
Authors: Friedman Sy D.
Published: June 2010

Assume

V
=
L

V = L

. Let

κ
\kappa

be a regular cardinal and for

X
⊆
κ

X \subseteq \kappa

let

α
(
X
)

\alpha (X)

denote the least ordinal

α
\alpha

such that

L
α

[
X
]

{L_\alpha }[X]

is admissible. In this paper we characterize those ordinals of the form

α
(
X
)

\alpha (X)

using forcing and fine structure of

L
L

techniques. This generalizes a theorem of Sacks which deals with the case

κ
=
ω

\kappa = \omega

.
Calibers of compact spaces
Authors: Argyros S. , Tsarpalias A.
Published: June 2010

LetXXbe a compact Hausdorff space andκ\kappaits Souslin number.2^{2}We prove that ifα\alphais a cardinal such that eitherα\alphaandcf⁡(α)\operatorname {cf} (\alpha )are greater thanκ\kappaand stronglyκ\kappa-inaccessible or elseα\alphais regular and greater thanκ\kappa, thenXXhas(α,ακ⏝)(\alpha , \sqrt [\underparen {\kappa }]{\alpha })caliber. Restricting our interest to the category of compact spacesXXwithS(X)=ω+S(X) = {\omega ^ + }(i.e.XXsatisfy the countable chain condition), the above statement takes, under G.C.H., the following form. For any compact spaceXXwithS(X)=ω+S(X) = {\omega ^ + }, we have that (a) ifα\alphais a cardinal andcf⁡(α)\operatorname {cf} (\alpha )does not have the formβ+{\beta ^ + }withcf⁡(β)=ω\operatorname {cf} (\beta ) = \omega, thenα\alphais caliber for the spaceXX. (b) Ifε=β+\varepsilon = {\beta ^ + }andcf⁡(β)=ω\operatorname {cf} (\beta ) = \omegathen(α,β)(\alpha ,\,\beta )is caliber forXX. A related example shows that the result of (b) is in a sense the best possible.
Closures of conjugacy classes in classical real linear Lie groups. II
Authors: Djoković Dragomir Ž.
Published: June 2010

By a classical group we mean one of the groups

G

L
n

(
R
)

G{L_n}(R)

,

G

L
n

(
C
)

G{L_n}(C)

,

G

L
n

(
H
)

G{L_n}(H)

,

U
(
p
,

q
)

U(p,\,q)

,

O
n

(
C
)

{O_n}(C)

,

O
(
p
,

q
)

O(p,\,q)

,

S

O

∗

(
2
n
)

S{O^{\ast }}(2n)

,

S

p

2
n

(
C
)

S{p_{2n}}(C)

,

S

p

2
n

(
R
)

S{p_{2n}}(R)

, or

S
p
(
p
,

q
)

Sp(p,\,q)

. Let

G
G

be a classical group and

L
L

its Lie algebra. For each

x
∈
L

x \in L

we determine the closure of the orbit

G
⋅
x

G \cdot x

(for the adjoint action of

G
G

on

L
L

). The problem is first reduced to the case when

x
x

is nilpotent. By using the exponential map we also determine the closures of conjugacy classes of

G
G

.
Banach bundles of continuous functions and an integral representation theorem
Authors: Seda Anthony Karel
Published: June 2010

A construction is given of a Banach bundle

p
:
A
→
X

p:A \to X

whose fibres are spaces of continuous functions which vanish at infinity. A Riesz type integral representation theorem is established which describes all functional on

A
A

.
Szegő limit theorems for Toeplitz operators on compact homogeneous spaces
Authors: Hirschman I. I. , Liang D. S. , Wilson E. N.
Published: June 2010

Letffbe a real valued integrable function on a compact homogeneous spaceM=K∖GM = K\backslash GandMf{M_f}the operator of pointwise multiplication byff. The authors consider families of Toeplitz operatorsTf,P=PMfP{T_{f,P}} = P{M_f}PasPPranges over a net of orthogonal projections fromL2(M){L^2}(M)to finite dimensionalGG-invariant subspaces. Necessary and sufficient conditions are given on the net in order that the distribution of eigenvalues of these Toeplitz operators is asymptotic to the distribution of values offfin the sense of Szegö’s classical theorem for the circle. Explicit sequences satisfying these conditions are constructed for all compact Lie groups and for all Riemannian symmetric compact spaces.
Formal spaces with finite-dimensional rational homotopy
Authors: Félix Yves , Halperin Stephen
Published: June 2010

Let

S
S

be a simply connected space. There is a certain principal fibration

K
1

→
E

→

π

K
0

{K_1} \to E \stackrel {\pi }{\to } K_0

in which

K
1

{K_1}

and

K
0

{K_0}

are products of rational Eilenberg-Mac Lane spaces and a continuous map

ϕ
:
S
→
E

\phi :S \to E

such that in particular

ϕ
0

=
π
∘
ϕ

{\phi _0} = \pi \circ \phi

maps the primitive rational homology of

S
S

isomorphically to that of

K
0

{K_0}

. A main result of this paper is the Theorem. If

dim
⁡

π

∗

(
S
)
⊗

Q

>
∞

\dim \pi _{\ast }(S) \otimes {\mathbf {Q}} > \infty

then

ϕ
\phi

is a rational homotopy equivalence if and only if all the primitive homology in

H

∗

(
S
;

Q

)

H_{\ast }(S;\,{\mathbf {Q}})

and

H

∗

(

K
0

,

S
;

Q

)

H_{\ast }({K_0},\,S;\,{\mathbf {Q}})

can (up to integral multiples) be represented by spheres and disk-sphere pairs. Corollary. If

S
S

is formal,

ϕ
\phi

is a rational homotopy equivalence.
The nonfinite generation of 𝐴𝑢𝑡(𝐺), 𝐺 free metabelian of rank 3
Authors: Bachmuth S. , Mochizuki H. Y.
Published: June 2010

The group of automorphisms of the free metabelian group of rank

3
3

is not finitely generated.