bluuebunny/crossref_metadata_2025 huggingface dataset:
On linear Volterra equations of parabolic type in Banach spaces
Authors: Prüss Jan
Published: June 2010

Linear integrodifferential equations of Volterra type in a Banach space are studied in case the main part of the equation generates an analytic

C
0

{C_0}

-semigroup. Under very general assumptions it is shown that a resolvent operator exists and that many of the solution properties of parabolic evolution equations are inherited. The results are then applied to integro-partial differential equations of parabolic type.
Hardy spaces of heat functions
Authors: Bear H. S.
Published: June 2010

We consider spaces of solutions of the one-dimensional heat equation on appropriate bounded domains in the

(
x
,

t
)

(x,\,t)

-plane. The domains we consider have the property that they are parabolically star-shaped at some point; i.e., each downward half-parabola from some center point intersects the boundary exactly once. We introduce parabolic coordinates

(
r
,

θ
)

(r,\,\theta )

in such a way that the curves

θ
=
constant

\theta =\text {constant}

are the half-parabolas, and dilation by multiplying by

r
r

preserves heat functions. An integral kernel is introduced by specializing to this situation the very general kernel developed by Gleason and the author for abstract harmonic functions. The combination of parabolic coordinates and kernel function provides a close analogy with the Poisson kernel and polar coordinates for harmonic functions on the disc, and many of the Hardy space theorems for harmonic functions generalize to this setting. Moreover, because of the generality of the Bear-Gleason kernel, much of this theory extends nearly verbatim to other situations where there are polar-type coordinates (such that the given space of functions is preserved by the “radial” expansion) and the maximum principle holds. For example, most of these theorems hold unchanged for harmonic functions on a radial star in

R

n

{\mathbf {R}^n}

. As ancillary results we give a simple condition that a boundary point of a plane domain be regular, and give a new local Phragmén-Lindelöf theorem for heat functions.
The multiplicity of isolated two-dimensional hypersurface singularities
Authors: Laufer Henry B.
Published: June 2010

Consider an isolated two-dimensional complex analytic hypersurface singularity

(
V
,
p
)

(V,p)

. A relation is given between the abstract topology of

(
V
,
p
)

(V,p)

and the multiplicity of

(
V
,
p
)

(V,p)

, yielding an upper bound for the multiplicity. This relation is a necessary condition for a Gorenstein singularity to be a hypersurface.
Proof of a conjecture of Kostant
Authors: Đoković Dragomir Ž.
Published: June 2010

Let

g

0

=

k

0

+

p

0

{\mathfrak {g}_0} = {\mathfrak {k}_0} + {\mathfrak {p}_0}

be a Cartan decomposition of a semisimple real Lie algebra and

g

=

k

+

p

\mathfrak {g} = \mathfrak {k} + \mathfrak {p}

its complexification. Denote by

G
G

the adjoint group of

g

\mathfrak {g}

and by

G
0

,
K
,

K
0

{G_0},K,{K_0}

the connected subgroups of

G
G

with respective Lie algebras

g

0

,

k

,

k

0

{\mathfrak {g}_0},\mathfrak {k},{\mathfrak {k}_0}

. A conjecture of Kostant asserts that there is a bijection between the

G
0

{G_0}

-conjugacy classes of nilpotent elements in

g

0

{\mathfrak {g}_0}

and the

K
K

-orbits of nilpotent elements in

p

\mathfrak {p}

which is given explicitly by the so-called Cayley transformation. This conjecture is proved in the paper.
Une minoration de la norme de l’opérateur de Cauchy sur les graphes lipschitziens
Authors: David Guy
Published: June 2010

It was shown by T. Murai that the norm of the operator defined by the Cauchy kernel on the graph of a Lipschitz function

A
A

is less than

C

(
1
+

‖

A
′

‖

∞

)

1

/

2

C{(1 + {\left \| {A’} \right \|_\infty })^{1/2}}

. We use Garnett’s example to show that this estimate is optimal.
Stopping times and Γ-convergence
Authors: Baxter J. , Dal Maso G. , Mosco U.
Published: June 2010

The equation

∂
u

/

∂
t
=
Δ
u
−
μ
u

\partial u/\partial t = \Delta u - \mu u

represents diffusion with killing. The strength of the killing is described by the measure

μ
\mu

, which is not assumed to be finite or even

σ
\sigma

-finite (to illustrate the effect of infinite values for

μ
\mu

, it may be noted that the diffusion is completely absorbed on any set

A
A

such that

μ
(
B
)
=
∞

\mu (B) = \infty

for every nonpolar subset

B
B

of

A
A

). In order to give rigorous mathematical meaning to this general diffusion equation with killing, one may interpret the solution

u
u

as arising from a variational problem, via the resolvent, or one may construct a semigroup probabilistically, using a multiplicative functional. Both constructions are carried out here, shown to be consistent, and applied to the study of the diffusion equation, as well as to the study of the related Dirichlet problem for the equation

Δ
u
−
μ
u
=
0

\Delta u - \mu u = 0

. The class of diffusions studied here is closed with respect to limits when the domain is allowed to vary. Two appropriate forms of convergence are considered, the first being

γ
\gamma

-convergence of the measures

μ
\mu

, which is defined in terms of the variational problem, and the second being stable convergence in distribution of the multiplicative functionals associated with the measures

μ
\mu

. These two forms of convergence are shown to be equivalent.
Scalar curvature and warped products of Riemann manifolds
Authors: Dobarro F. , Lami Dozo E.
Published: June 2010

We establish the relationship between the scalar curvature of a warped product

M
×

f

N

M \times {}_fN

of Riemann manifolds and those ones of

M
M

and

N
N

. Then we search for weights

f
f

to obtain constant scalar curvature on

M
×

f

N

M \times {}_fN

when

M
M

is compact.
Smale flows on the three-sphere
Authors: de Rezende Ketty
Published: June 2010

In this paper, a complete classification of Smale flows on

S
3

{S^3}

is obtained. This classification is presented by means of establishing a concise set of properties that must be satisfied by an (abstract) Lyapunov graph associated to a Smale flow and a Lyapunov function. We show that these properties are necessary, that is, given a Smale flow and a Lyapunov function, its Lyapunov graph satisfies this set of properties. We also show that these properties are sufficient, that is, given an abstract Lyapunov graph

L
′

L’

satisfying this set of properties, it is possible to realize a Smale flow on

S
3

{S^3}

that has a graph

L
L

as its Lyapunov graph where

L
L

is equal to

L
′

L’

up to topological equivalence. The techniques employed in proving that the conditions imposed on the graph are necessary involve some use of homology theory. Geometrical methods are used to construct the flow on

S
3

{S^3}

associated to the given graph and therefore establish the sufficiency of the above conditions. The main theorem in this paper generalizes a result of Franks [8] who classified nonsingular Smale flows on

S
3

{S^3}

.
Homogeneous random measures and a weak order for the excessive measures of a Markov process
Authors: Fitzsimmons P. J.
Published: June 2010

Let

X
=
(

X
t

,

P
x

)

X = ({X_t},\,{P^x})

be a right Markov process and let

m
m

be an excessive measure for

X
X

. Associated with the pair

(
X
,

m
)

(X,\,m)

is a stationary strong Markov process

(

Y
t

,

Q
m

)

({Y_t},\,{Q_m})

with random times of birth and death, with the same transition function as

X
X

, and with

m
m

as one dimensional distribution. We use

(

Y
t

,

Q
m

)

({Y_t},\,{Q_m})

to study the cone of excessive measures for

X
X

. A "weak order" is defined on this cone: an excessive measure

ξ
\xi

is weakly dominated by

m
m

if and only if there is a suitable homogeneous random measure

κ
\kappa

such that

(

Y
t

,

Q
ξ

)

({Y_t},\,{Q_\xi })

is obtained by "birthing"

(

Y
t

,

Q
m

)

({Y_t},\,{Q_m})

, birth in

[
t
,

t
+
d
t
]

[t,\,t + dt]

occurring at rate

κ
(
d
t
)

\kappa (dt)

. Random measures such as

κ
\kappa

are studied through the use of Palm measures. We also develop aspects of the "general theory of processes" over

(

Y
t

,

Q
m

)

({Y_t},\,{Q_m})

, including the moderate Markov property of

(

Y
t

,

Q
m

)

({Y_t},\,{Q_m})

when the arrow of time is reversed. Applications to balayage and capacity are suggested.
Squares of conjugacy classes in the infinite symmetric groups
Authors: Droste Manfred
Published: June 2010

Using combinatorial methods, we will examine squares of conjugacy classes in the symmetric groups

S
ν

{S_\nu }

of all permutations of an infinite set of cardinality

ℵ
ν

{\aleph _\nu }

. For arbitrary permutations

p
∈

S
ν

p \in {S_\nu }

, we will characterize when each element

s
∈

S
ν

s \in {S_\nu }

with finite support can be written as a product of two conjugates of

p
p

, and if

p
p

has infinitely many fixed points, we determine when all elements of

S
ν

{S_\nu }

are products of two conjugates of

p
p

. Classical group-theoretical theorems are obtained from similar results.
The Radon-Nikodým property and the Kreĭn-Milman property are equivalent for strongly regular sets
Authors: Schachermayer Walter
Published: June 2010

The result announced in the title is proved. As corollaries we obtain that RNP and KMP are equivalent for subsets of spaces with an unconditional basis and for

K
K

-convex Banach spaces. We also obtain a sharpening of a result of R. Huff and P. Morris: A dual space has the RNP iff all separable subspaces have the KMP.
A finiteness theorem in the Galois cohomology of algebraic number fields
Authors: Raskind Wayne
Published: June 2010

In this note we show that if

k
k

is an algebraic number field with algebraic closure

k
¯

\overline k

and

M
M

is a finitely generated, free

Z

l

{{\mathbf {Z}}_l}

-module with continuous

Gal
⁡
(

k
¯

/

k
)

\operatorname {Gal} (\overline k /k)

-action, then the continuous Galois cohomology group

H
1

(
k
,

M
)

{H^1}(k,\,M)

is a finitely generated

Z

l

{{\mathbf {Z}}_l}

-module under certain conditions on

M
M

(see Theorem 1 below). Also, we present a simpler construction of a mapping due to S. Bloch which relates torsion algebraic cycles and étale cohomology.
Contributions to the theory of set valued functions and set valued measures
Authors: Papageorgiou Nikolaos S.
Published: June 2010

Measurable multifunctions and multimeasures with values in a Banach space are studied. We start by proving a variation of the known Dunford theorem for weak compactness in

L
1

(
X
)

{L^1}(X)

. With a similar technique we prove that the range of certain vector valued integrals that appear in applications is

w
w

-compact and convex. Also we obtain Dunford-Pettis type theorems for sequences of integrably bounded multifunctions. Some pointwise

w
w

-compactness theorems are also obtained for certain families of measurable multifunctions. Then we prove a representation theorem for additive, set valued operators defined on

L
1

(
X
)

{L^1}(X)

. Finally, in the last section, a detailed study of transition multimeasures is conducted and several representation theorems are proved.
Visibility and rank one in homogeneous spaces of 𝐾≤0
Authors: Druetta María J.
Published: June 2010

In this paper we study relationships between the visibility axiom and rank one in homogeneous spaces of nonpositive curvature. We obtain a complete classification (in terms of rank) of simply connected homogeneous spaces of nonpositive curvature and dimension

⩽
4

\leqslant 4

. We provide examples, in every

dim
⩾
4

\dim \geqslant 4

, of simply connected, irreducible homogeneous spaces

(
K
⩽
0
)

(K \leqslant 0)

which are neither visibility manifolds nor symmetric spaces.
Traveling waves in combustion processes with complex chemical networks
Authors: Heinze Steffen
Published: June 2010

The existence of traveling waves for laminar flames with complex chemistry is proved. The crucial assumptions are that all reactions have to be exothermic and that no cycles occur in the graph of the reaction network. The method is to solve the equations first in a bounded interval by a degree argument and then taking the infinite domain limit.
Spaces of geodesic triangulations of the sphere
Authors: Awartani Marwan , Henderson David W.
Published: June 2010

We study questions concerning the homotopy-type of the space

GT
⁡
(
K
)

\operatorname {GT} (K)

of geodesic triangulations of the standard

n
n

-sphere which are (orientation-preserving) isomorphic to

K
K

. We find conditions which reduce this question to analogous questions concerning spaces of simplexwise linear embeddings of triangulated

n
n

-cells into

n
n

-space. These conditions are then applied to the

2
2

-sphere. We show that, for each triangulation

K
K

of the

2
2

-sphere, certain large subspaces of

GT
⁡
(
K
)

\operatorname {GT} (K)

are deformable (in

GT
⁡
(
K
)

\operatorname {GT} (K)

) into a subsapce homeomorphic to

SO
⁡
(
3
)

\operatorname {SO} (3)

. It is conjectured that (for

n
=
2

n = 2

)

GT
⁡
(
K
)

\operatorname {GT} (K)

has the homotopy of

SO
⁡
(
3
)

\operatorname {SO} (3)

. In a later paper the authors hope to use these same conditions to study the homotopy type of spaces of geodesic triangulations of the

n
n

-sphere,

n
>
2

n > 2

.
The equivariant Conner-Floyd isomorphism
Authors: Costenoble Steven R.
Published: June 2010

This paper proves two equivariant generalizations of the Conner-Floyd isomorphism relating unitary cobordism and

K
K

-theory. It extends a previous result of Okonek for abelian groups to all compact Lie groups. We also show that the result for finite groups is true using either the geometric or homotopical versions of cobordism.
Representations of hyperharmonic cones
Authors: Eriksson Sirkka-Liisa
Published: June 2010

Hyperharmonic cones are ordered convex cones possessing order properties similar to those of hyperharmonic functions on harmonic spaces. The dual of a hyperharmonic cone is defined to be the set of extended real-valued additive and left order-continuous mappings

(
≢
∞
)

(\not \equiv \infty )

. The second dual gives a representation of certain hyperharmonic cones in which suprema of upward directed families are pointwise suprema, although infima of pairs of functions are not generally pointwise infima. We obtain necessary and sufficient conditions for the existence of a representation of a hyperharmonic cone in which suprema of upward directed families are pointwise suprema and infima of pairs of functions are pointwise infima.
Classification of continuous 𝐽𝐵𝑊*-triples
Authors: Horn G. , Neher E.
Published: June 2010

We show that every

J
B

W

∗

JB{W^{\ast }}

-triple without a direct summand of type I is isometrically isomorphic to an

l
∞

{l^\infty }

-sum

R

⊗
∞

H
(
A
,

α
)

\mathcal {R}{ \otimes ^\infty }H(A,\,\alpha )

where

R

\mathcal {R}

is a

w

∗

{w^{\ast }}

-closed right ideal in a

W

∗

{W^{\ast }}

-algebra

B
B

and

H
(
A
,

α
)

H(A,\,\alpha )

are the elements of a

W

∗

{W^{\ast }}

-algebra

A
A

which are symmetric under a C-linear involution

α
\alpha

of

A
A

. Both

A
A

and

B
B

do not have a direct (

W

∗

{W^{\ast }}

-algebra) summand of type I. In order to refine the decomposition

R

⊗
∞

H
(
A
,

α
)

\mathcal {R}{ \otimes ^\infty }H(A,\,\alpha )

we define and characterize types of

J
B

W

∗

JB{W^{\ast }}

-triples.
Exceptional boundary sets for solutions of parabolic partial differential inequalities
Authors: Hile G. N. , Yeh R. Z.
Published: June 2010

Let

M

\mathcal {M}

be a second order, linear, parabolic partial differential operator with coefficients defined in a domain

D

=
Ω
×
(
0
,

T
)

\mathcal {D} = \Omega \times (0,\,T)

in

R

n

×

R

{{\mathbf {R}}^n} \times {\mathbf {R}}

, with

Ω
\Omega

a domain in

R

n

{{\mathbf {R}}^n}

. Let

u
u

be a suitably regular real function in

D

\mathcal {D}

such that

u
u

is bounded below and

M

u

\mathcal {M}u

is bounded above in

D

\mathcal {D}

. If

u
⩾
0

u \geqslant 0

on

Ω
×
{
0
}

\Omega \times \{ 0\}

except on a set

Γ
×
{
0
}

\Gamma \times \{ 0\}

, with

Γ
\Gamma

a subset of

Ω
\Omega

of suitably restricted Hausdorff dimension, then necessarily

u
⩾
0

u \geqslant 0

also on

Γ
×
{
0
}

\Gamma \times \{ 0\}

. The allowable Hausdorff dimension of

Γ
\Gamma

depends on the coefficients of

M

\mathcal {M}

. For example, if

M

\mathcal {M}

is the heat operator

Δ
−
∂

/

∂
t

\Delta - \partial /\partial t

, the Hausdorff dimension of

Γ
\Gamma

needs to be smaller than the number of space dimensions

n
n

. Analogous results are valid for exceptional boundary sets on the lateral boundary,

∂
Ω
×
(
0
,

T
)

\partial \Omega \times (0,\,T)

, of

D

\mathcal {D}

.
Construction of cohomology of discrete groups
Authors: Tong Y. L. , Wang S. P.
Published: June 2010

A correspondence between Hermitian modular forms and vector valued harmonic forms in locally symmetric spaces associated to

U
(
p
,

q
)

U(p,\,q)

is constructed and also shown in general to be nonzero. The construction utilizes Rallis-Schiffmann type theta functions and simplified arguments to circumvent differential geometric calculations used previously in related problems.
Derivatives of meromorphic functions of finite order
Authors: Kohs Werner P. , Williamson Jack
Published: June 2010

Let

F
F

be a nonentire, meromorphic function of finite order with only real zeros and real poles such that

F
′

F’

has no zeros. We classify all such real

F
F

and all such strictly nonreal

F
F

whose poles are of bounded multiplicities. We also give examples of such

F
F

which are strictly nonreal and whose poles are of unbounded multiplicities.
Primeness and sums of tangles
Authors: Eudave Muñoz Mario
Published: June 2010

We consider knots and links obtained by summing a rational tangle and a prime tangle. For a given prime tangle, we show that there are at most three rational tangles that will induce a composite or splittable link. In fact, we show that there is at most one rational tangle that will give a splittable link. These results extend Scharlemann’s work.
Characteristic multipliers and stability of symmetric periodic solutions of 𝑥̇(𝑡)=𝑔(𝑥(𝑡-1))
Authors: Chow Shui-Nee , Walther Hans-Otto
Published: June 2010

We study the scalar delay differential equation

x
˙

(
t
)
=
g
(
x
(
t
−
1
)
)

\dot x(t) = g(x(t - 1))

with negative feedback. We assume that the nonlinear function

g
g

is odd and monotone. We prove that periodic solutions

x
(
t
)

x(t)

of slowly oscillating type satisfying the symmetry condition

x
(
t
)
=
−
x
(
t
−
2
)

x(t) = - x(t - 2)

,

t
∈

R

t \in {\mathbf {R}}

, are nondegenerate and have all nontrivial Floquet multipliers strictly inside the unit circle. This says that the periodic orbit

{

x
t

:
t
∈

R

}

\{ {x_t}:t \in {\mathbf {R}}\}

in the phase space

C
[
−
1
,

0
]

C[ - 1,\,0]

is orbitally exponentially asymptotically stable.
Conditional gauge and potential theory for the Schrödinger operator
Authors: Cranston M. , Fabes E. , Zhao Z.
Published: June 2010

This paper extends the Conditional Gauge Theorem to more general operators and less regular domains than in previous works. We use this to obtain potential-theoretic results for the Schrödinger equation.
Constant isotropic submanifolds with 4-planar geodesics
Authors: Pak Jin Suk , Sakamoto Kunio
Published: June 2010

Let

f
f

be an isometric immersion of a Riemannian manifold

M
M

into

M
¯

\overline M

. We prove that if

f
f

is constant isotropic,

4
4

-planar geodesic and

M
¯

\overline M

is a Euclidean sphere, then

M
M

is isometric to one of compact symmetric spaces of rank equal to one and

f
f

is congruent to a direct sum of standard minimal immersions. We also determine constant isotropic,

4
4

-planar geodesic, totally real immersions into a complex projective space of constant holomorphic sectional curvature.
A classification of a class of 3-branchfolds
Authors: Takeuchi Yoshihiro
Published: June 2010

An

n
n

-orbifold is a topological space provided with a local modelling on (an open set in

R

n

{{\mathbf {R}}^n}

)/(a finite group action). Mainly, we deal with

3
3

-branchfolds (i.e.

3
3

-orbifolds with

1
1

-dimensional singular locus). We define a map between two

3
3

-branchfolds. With respect to this map, we prove some facts parallel to

3
3

-manifold theorems. Using the facts, we classify a class of

3
3

-branchfolds, analogous to Waldhausen’s classification theorem of Haken manifolds.
Products of involution classes in infinite symmetric groups
Authors: Moran Gadi
Published: June 2010

Let

A
A

be an infinite set. Denote by

S
A

{S_A}

the group of all permutations of

A
A

, and let

R
i

{R_i}

, denote the class of involutions of

A
A

moving

|

A

|

|A|

elements and fixing

i
i

elements

(
0
⩽
i
⩽

|

A

|

)

(0 \leqslant i \leqslant |A|)

. The products

R
i

R
j

{R_i}{R_j}

were determined in [M1]. In this article we treat the products

R

i
1

⋯

R

i
n

{R_{{i_1}}} \cdots {R_{{i_n}}}

for

n
⩾
3

n \geqslant 3

. Let INF denote the set of permutations in

S
A

{S_A}

moving infinitely many elements. We show: (1)

R

i
1

⋯

R

i
n

=

S
A

{R_{{i_1}}} \cdots {R_{{i_n}}} = {S_A}

for

n
⩾
4

n \geqslant 4

. (2)(a)

R
i

R
j

R
k

=
INF

{R_i}{R_j}{R_k} = \operatorname {INF}

if

{
i
,

j
,

k
}

\{ i,\,j,\,k\}

contains two integers of different parity; (b)

R
i

R
j

R
k

=

S
A

{R_i}{R_j}{R_k} = {S_A}

if

i
+
j
+
k
>
0

i + j + k > 0

and all integers in

{
i
,

j
,

k
}

\{ i,\,j,\,k\}

have the same parity. (3)

R
0
3

=

S
A

∖
E

R_0^3 = {S_A}\backslash E

, where

θ
∈
E

\theta \in E

iff

θ
\theta

satisfies one of the following three conditions: (i)

θ
\theta

moves precisely three elements. (ii)

θ
\theta

moves precisely five elements. (iii)

θ
\theta

moves precisely seven elements and has order

12
12

. These results were announced in 1973 in [MO]. (1) and part of (2)(a) were generalized recently by Droste [D1, D2].
Monotone reducibility over the Cantor space
Authors: Dougherty Randall
Published: June 2010

Define the partial ordering

⩽
\leqslant

on the Cantor space

ω

2

{}^\omega 2

by

x
⩽
y

x \leqslant y

iff

∀
n

x
(
n
)
⩽
y
(
n
)

\forall n\,x(n) \leqslant y(n)

(this corresponds to the subset relation on the power set of

ω
\omega

). A set

A
⊆

ω

2

A \subseteq {}^\omega 2

is monotone reducible to a set

B
⊆

ω

2

B \subseteq {}^\omega 2

iff there is a monotone (i.e.,

x
⩽
y
⇒
f
(
x
)
⩽
f
(
y
)

x \leqslant y \Rightarrow f(x) \leqslant f(y)

) continuous function

f
:

ω

2
→

ω

2

f:{}^\omega 2 \to {}^\omega 2

such that

x
∈
A

x \in A

iff

f
(
x
)
∈
B

f(x) \in B

. In this paper, we study the relation of monotone reducibility, with emphasis on two topics: (1) the similarities and differences between monotone reducibility on monotone sets (i.e., sets closed upward under

⩽
\leqslant

) and Wadge reducibility on arbitrary sets; and (2) the distinction (or lack thereof) between ‘monotone’ and ‘positive,’ where ‘positive’ means roughly ‘a priori monotone’ but is only defined in certain specific cases. (For example, a

Σ
2
0

\Sigma _2^0

-positive set is a countable union of countable intersections of monotone clopen sets.) Among the main results are the following: Each of the six lowest Wadge degrees contains one or two monotone degrees (of monotone sets), while each of the remaining Wadge degrees contains uncountably many monotone degrees (including uncountable antichains and descending chains); and, although ‘monotone’ and ‘positive’ coincide in a number of cases, there are classes of monotone sets which do not match any notion of ‘positive.’
Multilinear convolutions defined by measures on spheres
Authors: Oberlin Daniel M.
Published: June 2010

Let

σ
\sigma

be Lebesgue measure on

Σ

n
−
1

{\Sigma _{n - 1}}

and write

σ
=
(

σ
1

,
…
,

σ
n

)

\sigma = ({\sigma _1}, \ldots ,{\sigma _n})

for an element of

Σ

n
−
1

{\Sigma _{n - 1}}

. For functions

f
1

,
…
,

f
n

{f_1}, \ldots ,{f_n}

on

R

{\mathbf {R}}

, define
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis f 1 comma ellipsis comma f Subscript n Baseline right-parenthesis left-parenthesis x right-parenthesis equals integral Underscript normal upper Sigma Subscript n minus 1 Baseline Endscripts f 1 left-parenthesis x minus sigma 1 right-parenthesis midline-horizontal-ellipsis f Subscript n Baseline left-parenthesis x minus sigma Subscript n Baseline right-parenthesis d sigma comma x element-of bold upper R period">

T
(

f
1

,
…
,

f
n

)
(
x
)
=

∫

Σ

n
−
1

f
1

(
x
−

σ
1

)
⋯

f
n

(
x
−

σ
n

)

d
σ
,

x
∈

R

.

T({f_1}, \ldots ,{f_n})(x) = \int _{{\Sigma _{n - 1}}} {{f_1}(x - {\sigma _1}) \cdots {f_n}(x - {\sigma _n})\,d\sigma ,\qquad x \in {\mathbf {R}}.}

\]
This paper partially answers the question: for which values of

p
p

and

q
q

is there an inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper T left-parenthesis f 1 comma ellipsis comma f Subscript n Baseline right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript q Baseline less-than-or-slanted-equals upper C StartAbsoluteValue EndAbsoluteValue f 1 StartAbsoluteValue EndAbsoluteValue Subscript p Baseline midline-horizontal-ellipsis StartAbsoluteValue EndAbsoluteValue f Subscript n Baseline StartAbsoluteValue EndAbsoluteValue Subscript p Baseline question-mark">

|

|

T
(

f
1

,
…
,

f
n

)

|

|

q

⩽
C

|

|

f
1

|

|

p

⋯

|

|

f
n

|

|

p

?

||T({f_1}, \ldots ,{f_n})|{|_q} \leqslant C||{f_1}|{|_p} \cdots ||{f_n}|{|_p}?

\]
Generalized quotients in Coxeter groups
Authors: Björner Anders , Wachs Michelle L.
Published: June 2010

For (

W
W

,

S
S

) a Coxeter group, we study sets of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W slash upper V equals StartSet w element-of upper W vertical-bar l left-parenthesis w v right-parenthesis equals l left-parenthesis w right-parenthesis plus l left-parenthesis v right-parenthesis for all v element-of upper V EndSet comma">

W

/

V
=
{
w
∈
W

|

l
(
w
v
)
=
l
(
w
)
+
l
(
v
)

for all

v
∈
V
}
,

W/V = \{ w \in W|l(wv) = l(w) + l(v)\;{\text {for all}}\;v \in V\} ,

\]
, where

V
⊆
W

V \subseteq W

. Such sets

W

/

V

W/V

, here called generalized quotients, are shown to have much of the rich combinatorial structure under Bruhat order that has previously been known only for the case when

V
⊆
S

V \subseteq S

(i.e., for minimal coset representatives modulo a parabolic subgroup). We show that Bruhat intervals in

W

/

V

W/V

, for general

V
⊆
W

V \subseteq W

, are lexicographically shellable. The Möbius function on

W

/

V

W/V

under Bruhat order takes values in

{
−
1
,

0
,

+
1
}

\{ - 1,\,0,\, + 1\}

. For finite groups

W
W

, generalized quotients are the same thing as lower intervals in the weak order. This is, however, in general not true. Connections with the weak order are explored and it is shown that

W

/

V

W/V

is always a complete meet-semilattice and a convex order ideal as a subset of

W
W

under weak order. Descent classes
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript upper I Baseline equals left-brace w element-of upper W vertical-bar l left-parenthesis w s right-parenthesis greater-than l left-parenthesis w right-parenthesis left right double arrow s element-of upper I comma for all s element-of upper S right-brace">

D
I

=
{
w
∈
W

|

l
(
w
s
)
>
l
(
w
)
⇔
s
∈
I
,

for all

s
∈
S
}

{D_I} = \{ w \in W|l(ws) > l(w) \Leftrightarrow s \in I,\;{\text {for all}}\;s \in S\}

,

I
⊆
S

I \subseteq S

, are also analyzed using generalized quotients. It is shown that each descent class, as a poset under Bruhat order or weak order, is isomorphic to a generalized quotient under the corresponding ordering. The latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics. For instance, the set of standard Young tableaux of a fixed shape or the set of linear extensions of a rooted forest, suitably interpreted, form generalized quotients. We prove a factorization result for the quotients that come from rooted forests, which shows that algebraically these quotients behave as a system of minimal "coset" representatives of a subset which is in general not a subgroup. We also study the rank generating function for certain quotients in the symmetric group.
On the complete 𝐺𝐿(𝑛,𝐶)-decomposition of the stable cohomology of 𝑔𝑙_{𝑛}(𝐴)
Authors: Hanlon Phil
Published: June 2010

Let

A
A

be a graded, associative

C

{\mathbf {C}}

-algebra. For each

n
n

let

g

l
n

(
A
)

g{l_n}(A)

denote the Lie algebra of

n
×
n

n \times n

matrices with entries from

A
A

. In this paper we extend the Loday-Quillen theorem to nontrivial isotypic components of

G
L
(
n
,

C

)

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

acting on the Lie algebra cohomology of

g

l
n

(
A
)

g{l_n}(A)

. For

α
\alpha

and

β
\beta

partitions of some nonnegative integer

m
m

let

[
α
,

β

]
n

∈

Z

n

{[\alpha ,\,\beta ]_n} \in {{\mathbf {Z}}^n}

denote the maximal

G
L
(
n
,

C

)

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

-weight given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket alpha comma beta right-bracket Subscript n Baseline equals sigma-summation Underscript i Endscripts alpha Subscript i Baseline e Subscript i Baseline minus sigma-summation Underscript j Endscripts beta Subscript j Baseline e Subscript n plus 1 minus j Baseline period">

[
α
,

β

]
n

=

∑
i

α
i

e
i

−

∑
j

β
j

e

n
+
1
−
j

.

{[\alpha ,\,\beta ]_n} = \sum \limits _i {{\alpha _i}{e_i}} - \sum \limits _j {{\beta _j}{e_{n + 1 - j}}.}

\]
We show that the

[
α
,

β

]
n

{[\alpha ,\,\beta ]_n}

-isotypic component of the Lie algebra cohomology of

g

l
n

(
A
)

g{l_n}(A)

stabilizes when

n
→
∞

n \to \infty

and is equal to
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H upper R upper C Superscript asterisk Baseline left-parenthesis upper A right-parenthesis circled-times left-parenthesis upper H overTilde Superscript asterisk Baseline left-parenthesis upper A semicolon bold upper C right-parenthesis Superscript circled-times m Baseline circled-times upper S Superscript alpha Baseline circled-times upper S Superscript beta Baseline right-parenthesis s Subscript m">

H
R

C

∗

(
A
)
⊗
(

H
~

∗

(
A
;

C

)

⊗
m

⊗

S
α

⊗

S
β

)

s
m

HR{C^{\ast }}(A) \otimes ({\tilde H^{\ast }}{(A;\,{\mathbf {C}})^{ \otimes m}} \otimes {S^\alpha } \otimes {S^\beta }){s_m}

\]
where

H
~

∗

(
A
;

C

)

{\tilde H^{\ast }}(A;\,{\mathbf {C}})

is the reduced Hochschild cohomology of

A
A

with trivial coefficients, where

H
R

C

∗

(
A
)

HR{C^{\ast }}(A)

is the graded exterior algebra generated by the cyclic cohomology of

A
A

, where

S
α

{S^\alpha }

and

S
β

{S^\beta }

are the irreducible

S
m

{S_m}

-modules indexed by

α
\alpha

and

β
\beta

and where the action of

S
m

{S_m}

on

H
~

(
A
;

C

)

⊗
m

\tilde H{(A;\,{\mathbf {C}})^{ \otimes m}}

is the exterior action.
Iwasawa’s 𝜆⁻-invariant and a supplementary factor in an algebraic class number formula
Authors: Horie Kuniaki
Published: June 2010

Letllbe a prime number andkkan imaginary abelian field. Sinnott [12] has shown that the relative class number ofkkis expressed by the so-called index of the Stickelberger ideal ofkk, with a "supplementary factor"c−{c^ - }inN/2={n/2|n∈N}\mathbb {N}/2 = \{ n/2|n \in \mathbb {N}\}, and that ifkkvaries through the layers of the basicZl{\mathbb {Z}_l}-extension over an imaginary abelian field, thenc−{c^ - }becomes eventually constant. On the other hand,c−{c^ - }can take any value inN/2\mathbb {N}/2askkranges over the imaginary abelian fields (cf. [10]). In this paper, we shall study relations between the supplementary factorc−{c^ - }and Iwasawa’sλ−{\lambda ^ - }-invariant for the basicZl{\mathbb {Z}_l}-extension overkk, our discussion being based upon some formulas of Kida [8, 9], those of Sinnott [12], and fundamental results concerning a finite abelianll-group acted on by a cyclic group. As a consequence, we shall see that theλ−{\lambda ^ - }-invariant goes to infinity wheneverkkranges over a sequence of imaginary abelian fields such that thell-part ofc−{c^ - }goes to infinity.
Orthogonal polynomials on several intervals via a polynomial mapping
Authors: Geronimo J. S. , Van Assche W.
Published: June 2010

Starting from a sequence

{

p
n

(
x
;

μ
0

)
}

\{ {p_n}(x;\,{\mu _0})\}

of orthogonal polynomials with an orthogonality measure

μ
0

{\mu _0}

supported on

E
0

⊂
[
−
1
,

1
]

{E_0} \subset [ - 1,\,1]

, we construct a new sequence

{

p
n

(
x
;

μ
)
}

\{ {p_n}(x;\,\mu )\}

of orthogonal polynomials on

E
=

T

−
1

(

E
0

)

E = {T^{ - 1}}({E_0})

(

T
T

is a polynomial of degree

N
N

) with an orthogonality measure

μ
\mu

that is related to

μ
0

{\mu _0}

. If

E
0

=
[
−
1
,

1
]

{E_0} = [ - 1,\,1]

, then

E
=

T

−
1

(
[
−
1
,

1
]
)

E = {T^{ - 1}}([ - 1,\,1])

will in general consist of

N
N

intervals. We give explicit formulas relating

{

p
n

(
x
;

μ
)
}

\{ {p_n}(x;\,\mu )\}

and

{

p
n

(
x
;

μ
0

)
}

\{ {p_n}(x;\,{\mu _0})\}

and show how the recurrence coefficients in the three-term recurrence formulas for these orthogonal polynomials are related. If one chooses

T
T

to be a Chebyshev polynomial of the first kind, then one gets sieved orthogonal polynomials.
Ordinal rankings on measures annihilating thin sets
Authors: Kechris Alexander S. , Lyons Russell
Published: June 2010

We assign a countable ordinal number to each probability measure which annihilates all

H
H

-sets. The descriptive-set theoretic structure of this assignment allows us to show that this class of measures is coanalytic non-Borel. In addition, it allows us to quantify the failure of Rajchman’s conjecture. Similar results are obtained for measures annihilating Dirichlet sets.
Some inequalities for singular convolution operators in 𝐿^{𝑝}-spaces
Authors: Seeger Andreas
Published: June 2010

Suppose that a bounded function

m
m

satisfies a localized multiplier condition
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript t greater-than 0 Endscripts StartAbsoluteValue EndAbsoluteValue phi m left-parenthesis t Superscript upper P Baseline dot right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript upper M Sub Subscript p Baseline greater-than normal infinity">

sup

t
>
0

|

|

ϕ
m
(

t
P

⋅
)

|

|

M
p

>
∞

{\sup _{t > 0}}||\phi m({t^P} \cdot )|{|_{{M_p}}} > \infty

, for some bump function

ϕ
\phi

. We show that under mild smoothness assumptions

m
m

is a Fourier multiplier in

L
p

{L^p}

. The approach uses the sharp maximal operator and Littlewood-Paley-theory. The method gives new results for lacunary maximal functions and for multipliers in Triebel-Lizorkin-spaces.
Umbral calculus, binomial enumeration and chromatic polynomials
Authors: Ray Nigel
Published: June 2010

We develop the concept of partition categories, in order to extend the Mullin-Rota theory of binomial enumeration, and simultaneously to provide a natural setting for recent applications of the Roman-Rota umbral calculus to computations in algebraic topology. As a further application, we describe a generalisation of the chromatic polynomial of a graph.
On the dual of an exponential solvable Lie group
Authors: Currey Bradley N.
Published: June 2010

Let

G
G

be a connected, simply connected exponential solvable Lie group with Lie algebra

g

\mathfrak {g}

. The Kirillov mapping

η
:

g

∗

/

Ad

∗

(
G
)
→

G
^

\eta :\,\,\mathfrak {g}{\ast }/\operatorname {Ad} {\ast }(G) \to \hat G

gives a natural parametrization of

G
^

\hat G

by co-adjoint orbits and is known to be continuous. In this paper a finite partition of

g

∗

/

Ad

∗

(
G
)

\mathfrak {g}{\ast }/\operatorname {Ad} {\ast }(G)

is defined by means of an explicit construction which gives the partition a natural total ordering, such that the minimal element is open and dense. Given

π
∈

G
^

\pi \in \hat G

, elements in the enveloping algebra of

g

c

{\mathfrak {g}_c}

are constructed whose images under

π
\pi

are scalar and give crucial information about the associated orbit. This information is then used to show that the restriction of

η
\eta

to each element of the above-mentioned partition is a homeomorphism.
Seifert matrices and 6-knots
Authors: Hillman J. A. , Kearton C.
Published: June 2010

A new classification of simple

Z

{\mathbf {Z}}

-torsion-free

2
q

2q

-knots,

q
⩾
3

q \geqslant 3

, is given in terms of Seifert matrices modulo an equivalence relation. As a result the classification of such

2
q

2q

-knots,

q
⩾
4

q \geqslant 4

, in terms of

F
F

-forms is extended to the case

q
=
3

q = 3

.
Spherical functions on Cartan motion groups
Authors: Rader Cary
Published: June 2010

This paper gives a reasonably complete treatment of harmonic analysis on Cartan motion groups. Included is an explicit parameterization of irreducible spherical functions of general

K
K

-type, and of the nonunitary dual (and its topology). Also included is the explicit Plancherel measure, the Paley Wiener theorem, and an asymptotic expansion of general matrix entries. (These are generalized Bessel functions.) However the main result is Theorem 19, a technical result which measures the size of the centralizer of

K
K

in the universal enveloping algebra of the corresponding reductive group.
The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set
Authors: Gerstenhaber Murray , Schack Samuel D.
Published: June 2010

To each presheaf (over a poset) of associative algebras

A

\mathbb {A}

we associate an algebra

A

!

\mathbb {A}!

. We define a full exact embedding of the category of (presheaf)

A

\mathbb {A}

-bimodules in that of

A

!

\mathbb {A}!

-bimodules. We show that this embedding preserves neither enough (relative) injectives nor enough (relative) projectives, but nonetheless preserves (relative) Yoneda cohomology. The cohomology isomorphism links the deformations of manifolds, algebraic presheaves, and algebras. It also implies that the cohomology of any triangulable space is isomorphic to the Hochschild cohomology of an associative algebra. (The latter isomorphism preserves all known cohomology operations.) We conclude the paper by exhibiting for each associative algebra and triangulable space a "product" which is again an associative algebra.
An Erdős-Wintner theorem for differences of additive functions
Authors: Hildebrand Adolf
Published: June 2010

An Erdös-Wintner type criterion is given for the convergence of the distributions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript x Baseline left-parenthesis z right-parenthesis equals left-bracket x right-bracket Superscript negative 1 Baseline number-sign StartSet 1 less-than-or-slanted-equals n less-than-or-slanted-equals x colon f left-parenthesis n plus 1 right-parenthesis minus f left-parenthesis n right-parenthesis less-than-or-slanted-equals z EndSet">

D
x

(
z
)
=

[
x

]

−
1

#
{
1
⩽
n
⩽
x
:

f
(
n
+
1
)
−
f
(
n
)
⩽
z
}

{D_x}(z) = {[x]^{ - 1}}\# \{ 1 \leqslant n \leqslant x:\,f(n + 1) - f(n) \leqslant z\}

, where

f
f

is a real-valued additive function. A corollary of this result is that an additive function

f
f

, for which

f
(
n
+
1
)
−
f
(
n
)

f(n + 1) - f(n)

tends to zero on a set of density one, must be of the form

f
=
λ
log

f = \lambda \log

for some constant

λ
\lambda

. This had been conjectured by Erdős.
Approximation properties for orderings on *-fields
Authors: Craven Thomas C.
Published: June 2010

The goal of this paper is to extend the main theorems on approximation properties of the topological space of orderings from formally real fields to skew fields with an involution

∗

^{\ast }

. To accomplish this, the concept of

∗

^{\ast }

-semiordering is developed and new theorems are obtained for lifting

∗

^{\ast }

orderings from the residue class field of a real valuation.
Singularities of the scattering kernel and scattering invariants for several strictly convex obstacles
Authors: Petkov Vesselin M. , Stojanov Luchezar N.
Published: June 2010

Let

Ω
⊂

R

n

\Omega \subset {{\mathbf {R}}^n}

be a domain such that

R

n

∖
Ω

{{\mathbf {R}}^n}\backslash \Omega

is a disjoint union of a finite number of compact strictly convex obstacles with

C
∞

{C^\infty }

smooth boundaries. In this paper the singularities of the scattering kernel

s
(
t
,
θ
,
ω
)

s(t,\theta ,\omega )

, related to the wave equation in

R

×
Ω

{\mathbf {R}} \times \Omega

with Dirichlet boundary condition, are studied. It is proved that for every

ω
∈

S

n
−
1

\omega \in {S^{n - 1}}

there exists a residual subset

R

(
ω
)

\mathcal {R}(\omega )

of

S

n
−
1

{S^{n - 1}}

such that for each

θ
∈

R

(
ω
)
,
θ
≠
ω

\theta \in \mathcal {R}(\omega ),\theta \ne \omega

\[

singsupp

s
(
t
,
θ
,
ω
)
=

{
−

T
γ

}
γ

,

{\text {singsupp}}\,s(t,\theta ,\omega ) = {\{ - {T_\gamma }\} _\gamma },

\]
where

γ
\gamma

runs over the scattering rays in

Ω
\Omega

with incoming direction

ω
\omega

and with outgoing direction

θ
\theta

having no segments tangent to

∂
Ω

\partial \Omega

, and

T
γ

{T_\gamma }

is the sojourn time of

γ
\gamma

. Under some condition on

Ω
\Omega

, introduced by M. Ikawa, the asymptotic behavior of the sojourn times of the scattering rays related to a given configuration, as well as the precise rate of the decay of the coefficients of the main singularity of

s
(
t
,
θ
,
ω
)

s(t,\theta ,\omega )

, is examined.
Local properties of secant varieties in symmetric products. I
Authors: Huibregtse Mark E. , Johnsen Trygve
Published: June 2010

Let

L
L

be a line bundle on an abstract nonsingular curve

C
C

, let

V
⊂

H
0

(
C
,
L
)

V \subset {H^0}(C,L)

be a linear system, and denote by

C

(
d
)

{C^{(d)}}

the symmetric product of

d
d

copies of

C
C

. There exists a canonically defined

C

(
d
)

{C^{(d)}}

-bundle map:
\[

σ
:
V
⊗

O

C

(
d
)

→

E
L

,

\sigma :V \otimes {\mathcal {O}_{{C^{(d)}}}} \to {E_L},

\]
where

E
L

{E_L}

is a bundle of rank

d
d

obtained from

L
L

by a so-called symmetrization process. The various degenerary loci of

σ
\sigma

can be considered as subsecant schemes of

C

(
d
)

{C^{(d)}}

. Our main result, Theorem 4.2, is given in

§
4

\S 4

, where we obtain a local matrix description of

σ
\sigma

valid (also) at points on the diagonal in

C

(
d
)

{C^{(d)}}

, and thereby we can determine the completions of the local rings of the secant schemes at arbitrary points. In

§
5

\S 5

we handle the special case of giving a local scheme structure to the zero set of

σ
\sigma

.
Well-quasi-ordering infinite graphs with forbidden finite planar minor
Authors: Thomas Robin
Published: June 2010

We prove that given any sequence

G
1

,

G
2

,
…

{G_1},{G_2}, \ldots

of graphs, where

G
1

{G_1}

is finite planar and all other

G
i

{G_i}

are possibly infinite, there are indices

i
,
j

i,j

such that

i
>
j

i > j

and

G
i

{G_i}

is isomorphic to a minor of

G
j

{G_j}

. This generalizes results of Robertson and Seymour to infinite graphs. The restriction on

G
1

{G_1}

cannot be omitted by our earlier result. The proof is complex and makes use of an excluded minor theorem of Robertson and Seymour, its extension to infinite graphs, Nash-Williams’ theory of better-quasi-ordering, especially his infinite tree theorem, and its extension to something we call tree-structures over

QO

{\text {QO}}

-categories, which includes infinitary version of a well-quasi-ordering theorem of Friedman.
Cauchy integral equalities and applications
Authors: Choe Boo Rim
Published: June 2010

We study bounded holomorphic functions

π
\pi

on the unit ball

B
n

{B_n}

of

C

n

{\mathbb {C}^n}

satisfying the following so-called Cauchy integral equalities:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column upper C left-bracket pi Superscript m plus 1 Baseline pi overbar right-bracket equals gamma Subscript m Baseline pi Superscript m Baseline 2nd Column a m p semicolon left-parenthesis m equals 0 comma 1 comma 2 comma ellipsis right-parenthesis EndLayout">

C
[

π

m
+
1

π
¯

]
=

γ
m

π
m

a
m
p
;

(
m
=
0
,
1
,
2
,
…
)

\begin {array}{*{20}{c}} {C[{\pi ^{m + 1}}\bar \pi ] = {\gamma _m}{\pi ^m}} & {(m = 0,1,2, \ldots )} \\ \end {array}

\]
for some sequence

γ
m

{\gamma _m}

depending on

π
\pi

. Among the applications are the Ahern-Rudin problem concerning the composition property of holomorphic functions on

B
n

{B_n}

, a projection theorem about the orthogonal projection of

H
2

(

B
n

)

{H^2}({B_n})

onto the closed subspace generated by holomorphic polynomials in

π
\pi

, and some new information about the inner functions. In particular, it is shown that if we interpret

BMOA

(

B
n

)

{\text {BMOA}}({B_n})

as the dual of

H
1

(

B
n

)

{H^1}({B_n})

, then the map

g
→
g
∘
π

g \to g \circ \pi

is a linear isometry of

BMOA

(

B
1

)

{\text {BMOA}}({B_1})

into

BMOA

(

B
n

)

{\text {BMOA}}({B_n})

for every inner function

π
\pi

on

B
n

{B_n}

such that

π
(
0
)
=
0

\pi (0) = 0

.
Lie groups that are closed at infinity
Authors: Hoke Harry F.
Published: June 2010

A noncompact Riemannian manifold

M
M

is said to be closed at infinity if no bounded volume form which is also bounded away from zero can be written as the exterior derivative of a bounded form on

M
M

. The isoperimetric constant of

M
M

is defined by

h
(
M
)
=
inf
{

vol

(
∂
S
)

/

vol

(
S
)
}

h(M) = \inf \{ {\text {vol}}(\partial S)/{\text {vol}}(S)\}

where

S
S

ranges over compact domains with boundary in

M
M

. It is shown that a Lie group

G
G

with left invariant metric is closed at infinity if and only if

h
(
G
)
=
0

h(G) = 0

if and only if

G
G

is amenable and unimodular. This result relates these geometric invariants of

G
G

to the algebraic structure of

G
G

since the conditions amenable and unimodular have algebraic characterizations for Lie groups.

G
G

is amenable if and only if

G
G

is a compact extension of a solvable group and

G
G

is unimodular if and only if

Tr
⁡
(

ad

X
)
=
0

\operatorname {Tr}({\text {ad}}\,X) = 0

for all

X
X

in the Lie algebra of

G
G

. An application is the clarification of relationships between several conditions for the existence of transversal invariant measures for a foliation of a compact manifold by the orbits of a Lie group action.