bluuebunny/crossref_metadata_2025 huggingface dataset:
Universal spaces for 𝑅-trees
Authors: Mayer John C. , Nikiel Jacek , Oversteegen Lex G.
Published: June 2012

R

{\mathbf {R}}

-trees arise naturally in the study of groups of isometries of hyperbolic space. An

R

{\mathbf {R}}

-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. It follows that an

R

{\mathbf {R}}

-tree is locally arcwise connected, contractible, and one-dimensional. Unique and local arcwise connectivity characterize

R

{\mathbf {R}}

-trees among metric spaces. A universal

R

{\mathbf {R}}

-tree would be of interest in attempting to classify the actions of groups of isometries on

R

{\mathbf {R}}

-trees. It is easy to see that there is no universal

R

{\mathbf {R}}

-tree. However, we show that there is a universal separable

R

{\mathbf {R}}

-tree

T

ℵ
0

{T_{{\aleph _0}}}

. Moreover, for each cardinal

α
,
3
≤
α
≤

ℵ
0

\alpha ,3 \leq \alpha \leq {\aleph _0}

, there is a space

T
α

⊂

T

ℵ
0

{T_\alpha } \subset {T_{{\aleph _0}}}

, universal for separable

R

{\mathbf {R}}

-trees, whose order of ramification is at most

α
\alpha

. We construct a universal smooth dendroid

D
D

such that each separable

R

{\mathbf {R}}

-tree embeds in

D
D

; thus, has a smooth dendroid compactification. For nonseparable

R

{\mathbf {R}}

-trees, we show that there is an

R

{\mathbf {R}}

-tree

X
α

{X_\alpha }

, such that each

R

{\mathbf {R}}

-tree of order of ramification at most

α
\alpha

embeds isometrically into

X
α

{X_\alpha }

. We also show that each

R

{\mathbf {R}}

-tree has a compactification into a smooth arboroid (a nonmetric dendroid). We conclude with several examples that show that the characterization of

R

{\mathbf {R}}

-trees among metric spaces, rather than, say, among first countable spaces, is the best that can be expected.
Hilbert’s tenth problem for rings of algebraic functions in one variable over fields of constants of positive characteristic
Authors: Shlapentokh Alexandra
Published: June 2012

The author builds an undecidable model of integers with certain relations and operations in the rings of

S
S

-integers of algebraic function fields in one variable over fields of constants of positive characteristic, in order to show that Hilbert’s Tenth Problem has no solution there.
Groups of piecewise linear homeomorphisms
Authors: Stein Melanie
Published: June 2012

In this paper we study a class of groups which may be described as groups of piecewise linear bijections of a circle or of compact intervals of the real line. We use the action of these groups on simplicial complexes to obtain homological and combinatorial information about them. We also identify large simple subgroups in all of them, providing examples of finitely presented infinite simple groups.
Free actions on ℝ-trees
Authors: Rimlinger Frank
Published: June 2012

We characterize the free minimal actions of finitely generated groups on

R

\mathbb {R}

-trees in terms of certain equivalence relations on compact metric graphs.
Spectral multiplicity for 𝐺𝑙_{𝑛}(𝑅)
Authors: Huntley Jonathan
Published: June 2012

We study the behavior of the cuspidal spectrum of

Γ
∖

H

\Gamma \backslash \mathcal {H}

, where

H

\mathcal {H}

is associated to

Gl
n

⁡
(
R
)

\operatorname {Gl}_n(R)

and

Γ
\Gamma

is cofinite but not compact. By a technique that modifies the Lax-Phillips technique and uses ideas from wave equation techniques, if

r
r

is the dimension of

H

\mathcal {H}

,

N
α

(
λ
)

{N_\alpha }(\lambda )

is the counting function for the Laplacian attached to a Hilbert space

H
α

{H_\alpha }

,

M
α

(
λ
)

{M_\alpha }(\lambda )

is the multiplicity function, and

H
0

{H_0}

is the space of cusp forms, we obtain the following results: Theorem 1. There exists a space of functions

H
1

{H^1}

, containing all cusp forms, such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N prime left-parenthesis lamda right-parenthesis equals upper C Subscript r Baseline left-parenthesis Vol upper X right-parenthesis lamda Superscript StartFraction r Over 2 EndFraction Baseline plus upper O left-parenthesis lamda Superscript StartFraction r minus 1 Over 2 EndFraction Baseline lamda Superscript StartFraction 1 Over n plus 1 EndFraction Baseline left-parenthesis log lamda right-parenthesis Superscript n minus 1 Baseline right-parenthesis period">

N
′
(
λ
)
=

C
r

(

Vol

X
)

λ

r
2

+
O
(

λ

r
−
1

2

λ

1

n
+
1

(
log
⁡
λ

)

n
−
1

)
.

N\prime (\lambda ) = {C_r}({\text {Vol}}\;X){\lambda ^{\frac {r} {2}}} + O({\lambda ^{\frac {{r - 1}} {2}}}{\lambda ^{\frac {1} {{n + 1}}}}{(\log \lambda )^{n - 1}}).

\]
Theorem 2.
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M 0 left-parenthesis lamda right-parenthesis equals upper O left-parenthesis lamda Superscript StartFraction r minus 1 Over 2 EndFraction Baseline lamda Superscript StartFraction 1 Over n plus 1 EndFraction Baseline left-parenthesis log lamda right-parenthesis Superscript n minus 1 Baseline right-parenthesis period">

M
0

(
λ
)
=
O
(

λ

r
−
1

2

λ

1

n
+
1

(
log
⁡
λ

)

n
−
1

)
.

{M_0}(\lambda ) = O({\lambda ^{\frac {{r - 1}} {2}}}{\lambda ^{\frac {1} {{n + 1}}}}{(\log \lambda )^{n - 1}}).

\]
The uniqueness and stability of the solution of the Riemann problem of a system of conservation laws of mixed type
Authors: Fan Hai Tao
Published: June 2012

We establish the uniqueness and stability of the similarity solution of the Riemann problem for a

2
×
2

2 \times 2

system of conservation laws of mixed type, with initial data separated by the elliptic region, which satisfies the viscosity-capillarity travelling wave admissibility criterion.
Complemented ideals in the Fourier algebra and the Radon Nikodým property
Authors: Forrest Brian
Published: June 2012

Necessary and sufficient conditions are given for an ideal

I
(
H
)

I(H)

of the Fourier algebra to be complemented when

H
H

is a closed subgroup of

G
G

. Using the Radon Nikodym property, an example of a group

G
G

with a normal abelian subgroup

H
H

for which

I
(
H
)

I(H)

is not complemented is presented.
Adjoint transform, overconvexity and sets of constant width
Authors: Bavaud François
Published: June 2012

The properties of the adjoint transform (associating to a set the intersection of all disks of given radius centered in the set) are systematically investigated, in particular its relationship with the overconvex, the parallelisation and completion of sets. Sets conjugate by the transform can be characterised in a new way as the union or the intersection of all completions of the reference body. New relationships satisfied by their areas and perimeters are derived. Two applications in problems of random intersection of disks are finally treated.
Capacitability theorem in measurable gambling theory
Authors: Maitra A. , Purves R. , Sudderth W.
Published: June 2012

A player in a measurable gambling house

Γ
\Gamma

defined on a Polish state space

X
X

has available, for each

x
∈
X

x \in X

, the collection

Σ
(
x
)

\Sigma (x)

of possible distributions

σ
\sigma

for the stochastic process

x
1

,

x
2

,
…

{x_1},{x_2}, \ldots

of future states. If the object is to control the process so that it will lie in an analytic subset

A
A

of

H
=
X
×
X
×
⋯

H = X \times X \times \cdots

, then the player’s optimal reward is
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis upper A right-parenthesis left-parenthesis x right-parenthesis equals sup left-brace right-brace colon of sigma sigma left-parenthesis right-parenthesis upper A colon element-of element-of sigma of upper Sigma upper Sigma left-parenthesis right-parenthesis x period">

M
(
A
)
(
x
)
=
sup
{
σ
(
A
)
:
σ
∈
Σ
(
x
)
}
.

M(A)(x) = \sup \{ \sigma (A):\sigma \in \Sigma (x)\}.

\]
The operator

M
(
∙
)
(
x
)

M( \bullet )(x)

is shown to be regular in the sense that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis upper A right-parenthesis left-parenthesis x right-parenthesis equals inf upper M left-parenthesis left-brace tau greater-than normal infinity right-brace right-parenthesis left-parenthesis x right-parenthesis comma">

M
(
A
)
(
x
)
=
inf
M
(
{
τ
>
∞
}
)
(
x
)
,

M(A)(x) = \inf M(\{ \tau > \infty \} )(x),

\]
where the infimum is over Borel stopping times

τ
\tau

such that

A
⊆
{
τ
>
∞
}

A \subseteq \{ \tau > \infty \}

. A consequence of this regularity property is that the value of

M
(
A
)
(
x
)

M(A)(x)

is unchanged if, as in the gambling theory of Dubins and Savage, the player is allowed to use nonmeasurable strategies. This last result is seen to hold for bounded, Borel measurable payoff functions including that of Dubins and Savage.
Single loop space decompositions
Authors: Anick David J.
Published: June 2012

The method of single loop space decompositions, in which

Ω
X

\Omega X

is factored into minimal factors, is an important one for understanding the unstable homotopy of many simply-connected spaces

X
X

. This paper begins with a survey of the major known theorems along these lines. We then give a necessary and sufficient condition for

Ω
X

\Omega X

to be decomposable as a product of spaces belonging to a certain list. We conclude with a nontrivial instance of an application of this condition.
Characteristic matrices and spectral properties of evolutionary systems
Authors: Kaashoek M. A. , Verduyn Lunel S. M.
Published: June 2012

In this paper we introduce the notion of a characteristic matrix for a large class of unbounded operators and study the precise connection between characteristic matrices and spectral properties of evolutionary systems. In particular, we study so-called multiplicity theorems. Several examples will illustrate our results.
Sieved orthogonal polynomials. VII. Generalized polynomial mappings
Authors: Charris Jairo A. , Ismail Mourad E. H.
Published: June 2012

Systems of symmetric orthogonal polynomials whose recurrence relations are given by compatible blocks of second-order difference equations are studied in detail. Applications are given to the theory of the recently discovered sieved orthogonal polynomials. The connection with polynomial mappings is examined. An example of a family of orthogonal polynomials having discrete masses in the interior of the spectrum is included.
Inverse monoids, trees and context-free languages
Authors: Margolis Stuart W. , Meakin John C.
Published: June 2012

This paper is concerned with a study of inverse monoids presented by a set

X
X

subject to relations of the form

e
i

=

f
i

{e_i} = {f_i}

,

i
∈
I

i \in I

, where

e
i

{e_i}

and

f
i

{f_i}

are Dyck words, i.e. idempotents of the free inverse monoid on

X
X

. Some general results of Stephen are used to reduce the word problem for such a presentation to the membership problem for a certain subtree of the Cayley graph of the free group on

X
X

. In the finitely presented case the word problem is solved by using Rabin’s theorem on the second order monadic logic of the infinite binary tree. Some connections with the theory of rational subsets of the free group and the theory of context-free languages are explored.
Higher order commutators for vector-valued Calderón-Zygmund operators
Authors: Segovia Carlos , Torrea José L.
Published: June 2012

Weighted norm estimates for higher order commutators are obtained. The proof, that remain valid in the vector-valued case, are obtained as an application of some extrapolation results. The vector-valued version of the commutator theorem is applied to the Carleson operator, U.M.D. Banach spaces, approximate identities and maximal operators.
A classification of the finite extensions of a multidimensional Bernoulli shift
Authors: Kammeyer Janet Whalen
Published: June 2012

The finite extensions of a multidimensional Bernoulli shift are classified completely, up to factor isomorphism, and up to isomorphism. If such an extension is weakly mixing then it must be Bernoulli; otherwise, it has a finite rotation factor, which has a Bernoulli complementary algebra. This result is extended to multidimensional Bernoulli flows and Bernoulli shifts of infinite entropy.
Reye constructions for nodal Enriques surfaces
Authors: Conte A. , Verra A.
Published: June 2012

A classical Reye congruence

X
X

is an Enriques surface of rational equivalence class

(
3
,
7
)

(3,7)

in the grassmannian

G
(
1
,
3
)

G(1,3)

of lines of

P

3

{{\mathbf {P}}^3}

.

X
X

is the locus of lines of

P

3

{{\mathbf {P}}^3}

which are included in two quadrics of

W
=

W=

web of quadrics. A generalization to

G
(
1
,
t
)

G(1,t)

is given (1) for each

t
>
2

t > 2

there exist Enriques surfaces

X
X

of class

(
t
,
3
t
−
2
)

(t,3t - 2)

in

G
(
1
,
t
)

G(1,t)

, (2) the determinant of the dual of the universal bundle on

X
X

is

O

X

(
2
E
+
R
+

K
X

)

{\mathcal {O}_X}(2E + R + {K_X})

, with

E
=

E=

isolated elliptic curve,

R
2

=
−
2

{R^2} = - 2

,

E
⋅
R
=
t

E \cdot R = t

, (3)

X
X

parameterizes lines of

P

t

{{\mathbf {P}}^t}

which are included in a codimension

2
2

subsystem of

W
W

,

W
=

W=

linear system of quadrics of dimension

(

t

2

)

\left ( \begin {array}{*{20}{c}} t \\ 2 \\ \end {array} \right )

. The paper includes a description of the variety of trisecant lines to a smooth Enriques surface of degree

10
10

in

P

5

{{\mathbf {P}}^5}

.
Self-similar measures and their Fourier transforms. II
Authors: Strichartz Robert S.
Published: June 2012

A self-similar measure on

R

n

{{\mathbf {R}}^n}

was defined by Hutchinson to be a probability measure satisfying

(

∗

)

({\ast })

\[

μ
=

∑

j
=
1

m

a
j

μ
∘

S
j

−
1

\mu = \sum \limits _{j = 1}^m {{a_j}\mu \circ S_j^{ - 1}}

\]
, where

S
j

x
=

ρ
j

R
j

x
+

b
j

{S_j}x = {\rho _j}{R_j}x + {b_j}

is a contractive similarity

(
0
>

ρ
j

>
1
,

R
j

(0 > {\rho _j} > 1,{R_j}

orthogonal) and the weights

a
j

{a_j}

satisfy

0
>

a
j

>
1
,

∑

j
=
1

m

a
j

=
1

0 > {a_j} > 1,\sum \nolimits _{j = 1}^m {{a_j} = 1}

. By analogy, we define a self-similar distribution by the same identity

(

∗

)

( {\ast } )

but allowing the weights

a
j

{a_j}

to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to

(

∗

)

( {\ast } )

among distributions of compact support, and show that the space of such solutions is always finite dimensional. If

F
F

denotes the Fourier transformation of a self-similar distribution of compact support, let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis upper R right-parenthesis equals StartFraction 1 Over upper R Superscript n minus beta Baseline EndFraction integral Underscript StartAbsoluteValue x EndAbsoluteValue less-than-or-equal-to upper R Endscripts StartAbsoluteValue upper F left-parenthesis x right-parenthesis EndAbsoluteValue squared d x comma">

H
(
R
)
=

1

R

n
−
β

∫

|

x

|

≤
R

|

F
(
x
)

|

2

d
x
,

H(R) = \frac {1}{{{R^{n - \beta }}}}\int _{|x| \leq R} {|F(x){|^2}dx,}

\]
where

β
\beta

is defined by the equation

∑

j
=
1

m

ρ
j

−
β

|

a
j

|

2

=
1

\sum \nolimits _{j = 1}^m {\rho _j^{ - \beta }|{a_j}{|^2} = 1}

. If

ρ
j

ν
j

=
ρ

\rho _j^{{\nu _j}} = \rho

for some fixed

ρ
\rho

and

ν
j

{\nu _j}

positive integers we say the

{

ρ
j

}

\{ {\rho _j}\}

are exponentially commensurable. In this case we prove (under some additional hypotheses) that

H
(
R
)

H(R)

is asymptotic (in a suitable sense) to a bounded function

H
~

(
R
)

\tilde H(R)

that is bounded away from zero and periodic in the sense that

H
~

(
ρ
R
)
=

H
~

(
R
)

\tilde H(\rho R) = \tilde H(R)

for all

R
>
0

R > 0

. If the

{

ρ
j

}

\{ {\rho _j}\}

are exponentially incommensurable then

lim

R
→
∞

H
(
R
)

{\lim _{R \to \infty }}H(R)

exists and is nonzero.
On injectivity in locally presentable categories
Authors: Adámek Jiří , Rosický Jiří
Published: June 2012

Classes of objects injective w.r.t. specified morphisms are known to be closed under products and retracts. We prove the converse: a class of objects in a locally presentable category is an injectivity class iff it is closed under products and retracts. This result requires a certain large-cardinal principle. We characterize classes of objects injective w.r.t. a small collection of morphisms: they are precisely the accessible subcategories closed under products and

κ
\kappa

-filtered colimits. Assuming the (large-cardinal) Vopênka’s principle, the accessibility can be left out. As a corollary, we solve a problem of

L

{\text {L}}

. Fuchs concerning injectivity classes of abelian groups. Finally, we introduce a weak concept of reflectivity, called cone reflectivity, and we prove that under Vopênka’s principle all subcategories of locally presentable categories are cone reflective. Several open questions are formulated, e.g., does each topological space have a largest (non-

T
2

{T_2}

) compactification?
Witt’s extension theorem for mod four valued quadratic forms
Authors: Wood Jay A.
Published: June 2012

The

mod

4

\bmod \,4

valued quadratic forms defined by E. H. Brown, Jr. are studied. A classification theorem is proven which states that these forms are determined by two things: whether or not their associated bilinear form is alternating, and the

σ
\sigma

-invariant of Brown (which generalizes the Arf invariant of an ordinary quadratic form). Particular attention is paid to a generalization of Witt’s extension theorem for quadratic forms. Some applications to selforthogonal codes are sketched, and an exposition of some unpublished work of E. Prange on Witt’s theorem is provided in an appendix.
Hyperbolic structures for surfaces of infinite type
Authors: Basmajian Ara
Published: June 2012

Our main objective is to understand the geometry of hyperbolic structures on surfaces of infinite type. In particular, we investigate the properties of surfaces called flute spaces which are constructed from infinite sequences of "pairs of pants," each glued to the next along a common boundary geodesic. Necessary and sufficient conditions are supplied for a flute space to be constructed using only "tight pants," along with sufficient conditions on when the hyperbolic structure is complete. An infinite version of the Klein-Maskit combination theorem is derived. Finally, using the above constructions a number of applications to the deformation theory of infinite type hyperbolic surfaces are examined.
𝑊^{2,𝑝}-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients
Authors: Chiarenza Filippo , Frasca Michele , Longo Placido
Published: June 2012

We prove a well-posedness result in the class

W

2
,
p

∩

W
0

1
,
p

{W^{2,p}} \cap W_0^{1,p}

for the Dirichlet problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column upper L u equals f 2nd Column a m p semicolon a period e period in normal upper Omega comma 2nd Row 1st Column u equals 0 2nd Column a m p semicolon on partial-differential normal upper Omega period EndLayout">

{

L
u
=
f

a
m
p
;

a.e.

in

Ω

,

u
=
0

a
m
p
;

on

∂
Ω

.

\left \{ {\begin {array}{*{20}{c}} {Lu = f} & {{\text {a.e.}}\;{\text {in}}\;\Omega }, \\ {u = 0} & {{\text {on}}\;\partial \Omega }. \\ \end {array} } \right .

\]
We assume the coefficients of the elliptic nondivergence form equation that we study are in

VMO

∩

L
∞

{\text {VMO}} \cap {L^\infty }

.
Elementary duality of modules
Authors: Herzog Ivo
Published: June 2012

Let

R
R

be a ring. A formula

φ
(

x

)

\varphi ({\mathbf {x}})

in the language of left

R
R

-modules is called a positive primitive formula (ppf) if it is of the form

∃

y

(

A
B

)

(

x

y

)

=
0

\exists {\mathbf {y}}\left ({AB} \right )\left (\begin {array}{*{20}{c}}x\\y\\\end {array} \right ) = 0

where

A
A

and

B
B

are matrices of appropriate size with entries in

R
R

. We apply Prest’s notion of

D
φ
(

x

)

D\varphi ({\mathbf {x}})

, the ppf in the language of right

R
R

-modules dual to

φ
\varphi

, to show that the model theory of left

R
R

-modules as developed by Ziegler [Z] is in some sense dual to the model theory of right

R
R

-modules. We prove that the topologies on the left and right Ziegler spectra are "isomorphic" (Proposition 4.4). When the lattice of ppfs is well behaved, there is a homeomorphism

D
D

between the left and right Ziegler spectra which assigns to a given pure-injective indecomposable left

R
R

-module

U
U

the dual pure-injective indecomposable right

R
R

-module

D
U

DU

. Theorem 6.6 asserts that given a complete theory

T
T

of left

R
R

-modules, there is a dual complete theory

D
T

DT

of right

R
R

-modules with corresponding Baur-Garavaglia-Monk invariants. In the end, we give some conditions on a pure-injective indecomposable

R

U

_RU

which ensure that its dual

D
U

DU

may be represented as a hom set of the form

Hom
S

(
R

U
S

,

E
S

)

{\operatorname {Hom}_S}{(_R}{U_S},{E_S})

where

S
S

is some ring making

R

U
S

_R{U_S}

into a bimodule and

E
S

{E_S}

is injective.
Failure of cancellation for direct sums of line bundles
Authors: Swan Richard G.
Published: June 2012

In answer to a question of Murthy and Wiegand, examples are given of finitely generated projective modules

L
L

of rank

1
1

over a commutative ring

R
R

such that

L
⊕

L

−
1

L \oplus {L^{ - 1}}

is stably free but not free. Examples are also given of projective modules for which the determinant map det:

Aut
⁡
(
P
)
→

R

∗

\operatorname {Aut}(P) \to {R^{\ast } }

is not onto. Some related topological questions are also discussed.
Toeplitz operators and weighted Wiener-Hopf operators, pseudoconvex Reinhardt and tube domains
Authors: Salinas Norberto
Published: June 2012

The notion of weighted Wiener-Hopf operators is introduced. Their relationship with Toeplitz operators acting on the space of holomorphic functions which are square integrable with respect to a given "symmetric" measure is discussed. The groupoid approach is used in order to present a general program for studying the

C

∗

{C^{\ast } }

-algebra generated by weighted Wiener-Hopf operators associated with a solid cone of a second countable locally compact Hausdorff group. This is applied to the case when the group is the dual of a connected locally compact abelian Lie group and the measure is "well behaved" in order to produce a geometric groupoid which is independent of the representation. The notion of a Reinhardt-tube domain

Ω
\Omega

appears thus naturally, and a decomposition series of the corresponding

C

∗

{C^{\ast } }

-algebra is presented in terms of groupoid

C

∗

{C^{\ast } }

-algebras associated with various parts of the boundary of the domain

Ω
\Omega

.
Small solutions to inhomogeneous linear equations over number fields
Authors: O’Leary Robbin , Vaaler Jeffrey D.
Published: June 2012

We consider a system of

M
M

independent, inhomogeneous linear equations in

N
>
M

N > M

variables having coefficients in an algebraic number field

k
k

. We give a best possible lower bound on the inhomogeneous height of a solution vector in

k
N

{k^N}

and determine when a solution exists in

(

O

S

)
N

{({\mathcal {O}_S})^N}

, where

O

S

{\mathcal {O}_S}

is the ring of

S
S

-integers in

k
k

. If such a system has a solution vector in

(

O

S

)
N

{({\mathcal {O}_S})^N}

, we show that it has a solution

ζ
→

\vec \zeta

in

(

O

S

)
N

{({\mathcal {O}_S})^N}

such that the inhomogeneous height of

ζ
→

\vec \zeta

is relatively small. We give an explicit upper bound for this height in terms of the heights of the matrices defining the linear system. Our method uses geometry of numbers over adele spaces and local to global arguments.
New invariant Einstein metrics on generalized flag manifolds
Authors: Arvanitoyeorgos Andreas
Published: June 2012

A generalized flag manifold (or a Kählerian

C
C

-space) is a homogeneous space

G

/

K

G/K

whose isotropy subgroup

K
K

is the centralizer of a torus in

G
G

. These spaces admit a finite number of Kähler-Einstein metrics. We present new non-Kahler Einstein metrics for certain quotients of

U
(
n
)

U(n)

,

S
O
(
2
n
)

SO(2n)

and

G
2

{G_2}

. We also examine the isometry question for these metrics.
Nonstandard topology on function spaces with applications to hyperspaces
Authors: Render Hermann
Published: June 2012

In this paper the techniques of Nonstandard Analysis are used to study topologies on the set of all continuous functions. We obtain nonstandard characterizations for conjoining and splitting topologies and we give a complete description of the monads of the compact-open topology which leads to very elegant and simple proofs of some important results. For example we prove a generalized Ascoli Theorem where the image space is only Hausdorff or regular. Then we apply our results to the hyperspace and solve questions of Arens and Dugundji, Wattenberg and Topsøe. Finally we discuss real compact spaces and the continuity of the diagonal function.
Groups of dualities
Authors: Dimov Georgi D. , Tholen Walter
Published: June 2012

For arbitrary categories

A

\mathcal {A}

and

B

\mathcal {B}

, the "set" of isomorphism-classes of dualities between

A

\mathcal {A}

and

B

\mathcal {B}

carries a natural group structure. In case

A

\mathcal {A}

and

B

\mathcal {B}

admit faithful representable functors to Set, this structure can often be described quite concretely in terms of "schizophrenic objects" (in the sense of Johnstone’s book on "Stone Spaces"). The general theory provided here allows for a concrete computation of that group in case

A

=

B

=

C

\mathcal {A} = \mathcal {B} = \mathcal {C}

is the category of all compact and all discrete abelian groups: it is the uncountable group of algebraic automorphisms of the circle

R

/

Z

\mathbb {R}/\mathbb {Z}

, modulo its subgroup

Z

2

{\mathbb {Z}_2}

of continuous automorphisms.
Algebraic shift equivalence and primitive matrices
Authors: Boyle Mike , Handelman David
Published: June 2012

Motivated by symbolic dynamics, we study the problem, given a unital subring

S
S

of the reals, when is a matrix

A
A

algebraically shift equivalent over

S
S

to a primitive matrix? We conjecture that simple necessary conditions on the nonzero spectrum of

A
A

are sufficient, and establish the conjecture in many cases. If

S
S

is the integers, we give some lower bounds on sizes of realizing primitive matrices. For Dedekind domains, we prove that algebraic shift equivalence implies algebraic strong shift equivalence.
A qualitative uncertainty principle for unimodular groups of type 𝐼
Authors: Hogan Jeffrey A.
Published: June 2012

It has long been known that if

f
∈

L
2

(

R

n

)

f \in {L^2}({{\mathbf {R}}^n})

and the supports of and its Fourier transform

f
^

\hat f

are bounded then

f
=
0

f = 0

almost everywhere. More recently it has been shown that the same conclusion can be reached under the weaker condition that the supports of

f
f

and

f
^

\hat f

have finite measure. These results may be thought of as qualitative uncertainty principles since they limit the "concentration" of the Fourier transform pair

(
f
,

f
^

)

(f,\hat f)

. Little is known, however, of analogous results for functions on locally compact groups. A qualitative uncertainty principle is proved here for unimodular groups of type I.
On Dehn functions and products of groups
Authors: Brick Stephen G.
Published: June 2012

If

G
G

is a finitely presented group then its Dehn function—or its isoperimetric inequality—is of interest. For example,

G
G

satisfies a linear isoperimetric inequality iff

G
G

is negatively curved (or hyperbolic in the sense of Gromov). Also, if

G
G

possesses an automatic structure then

G
G

satisfies a quadratic isoperimetric inequality. We investigate the effect of certain natural operations on the Dehn function. We consider direct products, taking subgroups of finite index, free products, amalgamations, and

HNN
\text {HNN}

extensions.
Cayley-Bacharach schemes and their canonical modules
Authors: Geramita Anthony V. , Kreuzer Martin , Robbiano Lorenzo
Published: June 2012

A set of

s
s

points in

P

d

{\mathbb {P}^d}

is called a Cayley-Bacharach scheme (

CB

{\text {CB}}

-scheme), if every subset of

s
−
1

s - 1

points has the same Hilbert function. We investigate the consequences of this "weak uniformity." The main result characterizes

CB

{\text {CB}}

-schemes in terms of the structure of the canonical module of their projective coordinate ring. From this we get that the Hilbert function of a

CB

{\text {CB}}

-scheme

X
X

has to satisfy growth conditions which are only slightly weaker than the ones given by Harris and Eisenbud for points with the uniform position property. We also characterize

CB

{\text {CB}}

-schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points. Applications include efficient algorithms for checking whether a given set of points is a

CB

{\text {CB}}

-scheme, results about generic hyperplane sections of arithmetically Cohen-Macaulay curves and inequalities for the Hilbert functions of Cohen-Macaulay domains.
On the characterization of a Riemann surface by its semigroup of endomorphisms
Authors: Erëmenko A.
Published: June 2012

Suppose

D
1

{D_1}

and

D
2

{D_2}

be Riemann surfaces which have bounded nonconstant holomorphic functions. Denote by

E
(

D
i

)

E({D_i})

,

i
=
1
,
2

i = 1,2

, the semigroups of all holomorphic endomorphisms. If

ϕ
:
E
(

D
1

)
→
E
(

D
2

)

\phi :E({D_1}) \to E({D_2})

is an isomorphism of semigroups then there exists a conformal or anticonformal isomorphism

ψ
:

D
1

→

D
2

\psi :{D_1} \to {D_2}

such that

ϕ
\phi

is the conjugation by

ψ
\psi

. Also the semigroup of injective endomorphisms as well as some parabolic surfaces are considered.
Ricci flow, Einstein metrics and space forms
Authors: Ye Rugang
Published: June 2012

The main results in this paper are: (1) Ricci pinched stable Riemannian metrics can be deformed to Einstein metrics through the Ricci flow of R. Hamilton; (2) (suitably) negatively pinched Riemannian manifolds can be deformed to hyperbolic space forms through Ricci flow; and (3)

L
2

{L^2}

-pinched Riemannian manifolds can be deformed to space forms through Ricci flow.
Probing L-S category with maps
Authors: Jessup Barry
Published: June 2012

For any map

X

→

f

Y

X\xrightarrow {f}Y

, we introduce two new homotopy invariants,

dcat

f

{\text {dcat}}\;f

and

rcat

f

{\text {rcat}}\;f

. The classical category

cat

f

{\text {cat}}\;f

is a lower bound for both, while

dcat

f
≤

cat

X

{\text {dcat}}\;f \leq {\text {cat}}\;X

and

rcat

f
≤

cat

Y

{\text {rcat}}\;f \leq {\text {cat}}\;Y

. When

Y
Y

is an Eilenberg-Mac Lane space,

f
f

represents a cohomology class and

dcat

f

{\text {dcat}}\;f

often gives a good estimate for

cat

X

{\text {cat}}\;X

. We prove that if

Ω
∈

H
n

(
M
;

Z

)

\Omega \in {H^n}(M;\mathbb {Z})

is the fundamental class of a compact, simply connected

n
n

-manifold, then

dcat

Ω
=

cat

M

{\text {dcat}}\;\Omega = {\text {cat}}\;M

. Similarly, when

X
X

is sphere, then

f
f

is a homotopy class and while

cat

f
=
1

{\text {cat}}\;f = 1

,

rcat

f

{\text {rcat}}\;f

can be a good approximation to

cat

Y

{\text {cat}}\;Y

. We show that if

α
∈

π
2

(

C

P
n

)

\alpha \in {\pi _2}(\mathbb {C}{P^n})

is nonzero, then

rcat

α
=
n

{\text {rcat}}\;\alpha = n

. Rational analogues are introduced and we prove that for

u
∈

H
∗

(
X
;

Q

)

u \in {H^\ast }(X;\mathbb {Q})

,

dcat
0

u
=
1
⇔

u
2

=
0

{\text {dcat}_0}\;u = 1 \Leftrightarrow {u^2} = 0

and

u
u

is spherical.
Wavelets in wandering subspaces
Authors: Goodman T. N. T. , Lee S. L. , Tang W. S.
Published: June 2012

Mallat’s construction, via a multiresolution approximation, of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions. The connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in Hilbert space is exploited in the general setting. An example of multiresolution approximation generated by cardinal Hermite

B
B

-splines is constructed.
An indirect method in the calculus of variations
Authors: Clarke F. H.
Published: June 2012

This article presents a new approach to the issue of the existence of solutions to the basic problem in the calculus of variations. The method is indirect, and applies to certain classes of of problems with slow or no growth, in addition to those satisfying the traditional coercivity condition. The proof hinges upon showing with the help of nonsmooth analysis that a certain value function is constant. Examples are given to illustrate the applicability of the results and the necessity of the hypotheses.
Totally monotone functions with applications to the Bergman space
Authors: Korenblum B. , O’Neil R. , Richards K. , Zhu K.
Published: June 2012

Using a theorem of S. Bernstein [1] we prove a special case of the following maximum principle for the Bergman space conjectured by B. Korenblum [3]: There exists a number

δ
∈
(
0
,
1
)

\delta \in (0,1)

such that if

f
f

and

g
g

are analytic functions on the open unit disk

D

{\mathbf {D}}

with

|

f
(
z
)

|

≤

|

g
(
z
)

|

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

on

δ
≤

|

z

|

>
1

\delta \leq |z| > 1

then

‖
f
‖

2

≤

‖
g
‖

2

{\left \| f \right \|_2} \leq {\left \| g \right \|_2}

, where

‖

‖

2

{\left \| {} \right \|_2}

is the

L
2

{L^2}

norm with respect to area measure on

D

{\mathbf {D}}

. We prove the above conjecture when either

f
f

or

g
g

is a monomial; in this case we show that the optimal constant

δ
\delta

is greater than or equal to

1

/

3

1/\sqrt 3

.
Localizing with respect to self-maps of the circle
Authors: Casacuberta Carles , Peschke Georg
Published: June 2012

We describe a general procedure to construct idempotent functors on the pointed homotopy category of connected

CW

{\text {CW}}

-complexes, some of which extend

P
P

-localization of nilpotent spaces, at a set of primes

P
P

. We focus our attention on one such functor, whose local objects are

CW

{\text {CW}}

-complexes

X
X

for which the

p
p

th power map on the loop space

Ω
X

\Omega X

is a self-homotopy equivalence if

p
∉
P

p \notin P

. We study its algebraic properties, its behaviour on certain spaces, and its relation with other functors such as Bousfield’s homology localization, Bousfield-Kan completion, and Quillen’s plus-construction.
On manifolds with nonnegative curvature on totally isotropic 2-planes
Authors: Seaman Walter
Published: June 2012

We prove that a compact orientable

2
n

2n

-dimensional Riemannian manifold, with second Betti number nonzero, nonnegative curvature on totally isotropic

2
2

-planes, and satisfying a positivity-type condition at one point, is necessarily Kähler, with second Betti number

1
1

. Using the methods of Siu and Yau, we prove that if the positivity condition is satisfied at every point, then the manifold is biholomorphic to complex projective space.
Harmonic measures on covers of compact surfaces of nonpositive curvature
Authors: Brin M. , Kifer Y.
Published: June 2012

Let

M
M

be the universal cover of a compact nonflat surface

N
N

of nonpositive curvature. We show that on the average the Brownian motion on

M
M

behaves similarly to the Brownian motion on negatively curved manifolds. We use this to prove that harmonic measures on the sphere at infinity have positive Hausdorff dimension and if the geodesic flow on

N
N

is ergodic then the harmonic and geodesic measure classes at infinity are singular unless the curvature is constant.
On automorphisms of matrix invariants
Authors: Reichstein Zinovy
Published: June 2012

Let

Q

m
,
n

{Q_{m,n}}

be the space of

m
m

-tuples of

n
×
n

n \times n

-matrices modulo the simultaneous conjugation action of

P
G

L
n

PG{L_n}

. Let

Q

m
,
n

(
τ
)

{Q_{m,n}}(\tau )

be the set of points of

Q

m
,
n

{Q_{m,n}}

of representation type

τ
\tau

. We show that for

m
≥
n
+
1

m \geq n + 1

the group

Aut
⁡
(

Q

m
,
n

)

\operatorname {Aut}({Q_{m,n}})

of representation type preserving algebraic automorphisms of

Q

m
,
n

{Q_{m,n}}

acts transitively on each

Q

m
,
n

(
τ
)

{Q_{m,n}}(\tau )

. Moreover, the action of

Aut
⁡
(

Q

m
,
n

)

\operatorname {Aut}({Q_{m,n}})

on the Zariski open subset

Q

m
,
n

(
1
,
n
)

{Q_{m,n}}(1,n)

of

Q

m
,
n

{Q_{m,n}}

is

s
s

-transitive for every positive integer

s
s

. We also prove slightly weaker analogues of these results for all

m
≥
3

m \geq 3

.
Weighted norm inequalities for Vilenkin-Fourier series
Authors: Young Wo-Sang
Published: June 2012

Let

S
n

f

{S_n}f

be the

n
n

th partial sum of the Vilenkin-Fourier series of

f
∈

L
1

f \in {L^1}

. For

1
>
p
>
∞

1 > p > \infty

, we characterize all weight functions

w
w

such that if

f
∈

L
p

(
w
)

f \in {L^p}(w)

,

S
n

f

{S_n}f

converges to

f
f

in

L
p

(
w
)

{L^p}(w)

. We also determine all weight functions

w
w

such that

{

S
n

}

\{ {S_n}\}

is uniformly of weak type

(
1
,
1
)

(1,1)

with respect to

w
w

.
Continuous dependence of nonmonotonic discontinuous differential equations
Authors: Biles Daniel C.
Published: June 2012

Continuous dependence of solutions for a class of nonmonotonic, discontinuous differential equations is studied. First, a local existence theorem due to

Z
Z

. Wu is extended to a larger class. Then, a result concerning continuous dependence for this larger class is proven. This employs a type of convergence similar to Gihman’s Convergence Criterion, which is defined to be
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="For all a comma b and y limit Underscript n right-arrow normal infinity Endscripts integral Subscript a Superscript b Baseline f Subscript n Baseline left-parenthesis s comma y right-parenthesis d s equals integral Subscript a Superscript b Baseline f Subscript normal infinity Baseline left-parenthesis s comma y right-parenthesis d s period">

For all

a
,
b

and

y

lim

n
→
∞

∫
a
b

f
n

(
s
,
y
)
d
s
=

∫
a
b

f
∞

(
s
,
y
)

d
s

.

{\text {For all}}\;a,b\;{\text {and}}\;y\quad \lim \limits _{n \to \infty } \int _a^b {{f_n}(s,y)ds = } \int _a^b {{f_\infty }(s,y)\,ds}.

\]
The significance of Gihman’s Convergence Criterion is that for certain classes of differential equations it has been found to be necessary and sufficient for continuous dependence. Finally, examples are presented to motivate and clarify this continuous dependence result.
Inverse scattering for singular potentials in two dimensions
Authors: Sun Zi Qi , Uhlmann Gunther
Published: June 2012

We consider the Schrödinger equation for a compactly supported potential having jump type singularities at a subdomain of

R

2

{\mathbb {R}^2}

. We prove that knowledge of the scattering amplitude at a fixed energy, determines the location of the singularity as well as the jump across the curve of discontinuity. This result follows from a similar result for the Dirichlet to Neumann map associated to the Schrödinger equation for a compactly supported potential with the same type of singularities.
On the discrete series of generalized Stiefel manifolds
Authors: Li Jian-Shu
Published: June 2012

A study of the discrete series of generalized Stiefel manifolds is made using the oscillator representation. New infinite families of such discrete series are constructed.
Sandwich matrices, Solomon algebras, and Kazhdan-Lusztig polynomials
Authors: Putcha Mohan S.
Published: June 2012

Sandwich matrices have proved to be of importance in semigroup theory for the last 50 years. The work of the author on algebraic monoids leads to sandwich matrices in group theory. In this paper, we find some connections between sandwich matrices and the Hecke algebras (for monoids) introduced recently by Louis Solomon. At the local level we then obtain an explicit isomorphism between Solomon’s Hecke algebra and the complex monoid algebra of the Renner monoid. In the simplest case of monoids associated with a Borel subgroup, we find that the entries of the inverse of the sandwich matrix, as well as those of the related structure matrix of Solomon’s Hecke algebra are ’almost’ the polynomials

R

x
,
y

{R_{x,y}}

associated with the Kazhdan-Lusztig polynomials.
Weighted norm inequalities for general operators on monotone functions
Authors: Lai Shanzhong
Published: June 2012

In this paper we characterize the weights

w
,
v

w,v

for which

‖

S
ϕ

f

‖

p
,
w

≤
C

‖
f
‖

q
,
v

{\left \| {{S_\phi }f} \right \|_{p,w}} \leq C{\left \| f \right \|_{q,v}}

, for

f
f

nonincreasing, where

S
ϕ

f
=

∫
0
∞

ϕ
(
x
,
y
)
f
(
y
)
d
y

{S_\phi }f = \smallint _0^\infty {\phi (x,y)f(y)dy}

.
The trace of the heat kernel in Lipschitz domains
Authors: Brown Russell M.
Published: June 2012

We establish the existence of an asymptotic expansion as

t
→

0
+

t \to {0^ + }

for the trace of the heat kernel for the Neumann Laplacian in a bounded Lipschitz domain. The proof of an asymptotic expansion for the heat kernel for the Dirichlet Laplacian is also sketched. The treatment of the Dirichlet Laplacian extends work of Brossard and Carmona who obtained the same result in

C
1

{C^1}

-domains.
Baire class 1 selectors for upper semicontinuous set-valued maps
Authors: Srivatsa V. V.
Published: June 2012

Let

T
T

be a metric space and

X
X

a Banach space. Let

F
:
T
→
X

F:T \to X

be a set-valued map assuming arbitrary values and satisfying the upper semicontinuity condition:

{
t
∈
T
:
F
(
t
)
∩
C
≠
∅
}

\{ t \in T:F(t) \cap C \ne \emptyset \}

is closed for each weakly closed set

C
C

in

X
X

. Then there is a sequence of norm-continuous functions converging pointwise (in the norm) to a selection for

F
F

. We prove a statement of similar precision and generality when

X
X

is a metric space.
Uniqueness in bounded moment problems
Authors: Kellerer Hans G.
Published: June 2012

Let

(
X
,

A

,
μ
)

(X,\mathfrak {A},\mu )

be a

σ
\sigma

-finite measure space and

K

\mathcal {K}

be a linear subspace of

L

1

(
μ
)

{\mathcal {L}_1}(\mu )

with

K

=
X

\mathcal {K} = X

. The following inverse problem is treated: Which sets

A
∈

A

A \in \mathfrak {A}

are "

K

\mathcal {K}

-determined" within the class of all functions

g
∈

L

∞

(
μ
)

g \in {\mathcal {L}_\infty }(\mu )

satisfying

0
≤
g
≤
1

0 \leq g \leq 1

, i.e. when is

g
=

1
A

g = {1_A}

the unique solution of

∫
f
g

d
μ
=
∫
f

1
A

d
μ

\smallint fg\;d\mu = \smallint f{1_A}\;d\mu

,

f
∈

K

?

f \in \mathcal {K}?

Recent results of Fishburn et al. and Kemperman show that the condition

A
=
{
f
≥
0
}

A = \{ f \geq 0\}

for some

f
∈

K

f \in \mathcal {K}

is sufficient but not necessary for uniqueness. To obtain a complete characterization of all

K

\mathcal {K}

-determined sets,

K

\mathcal {K}

has to be enlarged to some hull

K

∗

{\mathcal {K}^{\ast } }

by extending the usual weak convergence to limits not in

L

1

(
μ
)

{\mathcal {L}_1}(\mu )

. Then one of the main results states that

A
A

is

K

\mathcal {K}

-determined if and only if there is a representation

A
=
{

f

∗

>
0
}

A = \{ {f^{\ast } } > 0\}

and

X
∖
A
=
{

f

∗

>
0
}

X\backslash A = \{ {f^{\ast } } > 0\}

for some

f

∗

∈

K

∗

{f^{\ast }} \in {\mathcal {K}^{\ast } }

.
Calcul du spectre d’une nilvariété de rang deux et applications
Authors: Pesce Hubert
Published: June 2012

Résumé. On calcule, en utilisant la théorie des orbites de Kirillov, le spectre d’une nilvariété compacte de rang deux. Puis on utilise ce calcul pour étudier et caractériser les déformations isospectrales de ces variétés.
The Koebe semigroup and a class of averaging operators on 𝐻^{𝑝}(𝐷)
Authors: Siskakis Aristomenis G.
Published: June 2012

We study on the Hardy space

H
p

{H^p}

the operators

T
F

{T_F}

given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript upper F Baseline left-parenthesis f right-parenthesis left-parenthesis z right-parenthesis equals StartFraction 1 Over z EndFraction integral Subscript 0 Superscript z Baseline f left-parenthesis zeta right-parenthesis StartFraction 1 Over upper F left-parenthesis zeta right-parenthesis EndFraction d zeta">

T
F

(
f
)
(
z
)
=

1
z

∫
0
z

f
(
ζ
)

1

F
(
ζ
)

d
ζ

{T_F}(f)(z) = \frac {1} {z}\int _0^z {f(\zeta )\frac {1} {{F(\zeta )}}\;d\zeta }

\]
where

F
(
z
)

F(z)

is analytic on the unit disc

D

\mathbb {D}

and has

Re
⁡
F
(
z
)
≥
0

\operatorname {Re} F(z) \geq 0

. Each such operator is closely related to a strongly continuous semigroup of weighted composition operators. By studying first an extremal such semigroup (the Koebe semigroup) we are able to obtain the upper bound
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper T Subscript upper F Baseline double-vertical-bar Subscript p Baseline less-than-or-equal-to 2 p upper R e left-parenthesis 1 slash upper F left-parenthesis 0 right-parenthesis right-parenthesis plus StartAbsoluteValue upper I m left-parenthesis 1 slash upper F left-parenthesis 0 right-parenthesis right-parenthesis EndAbsoluteValue">

‖

T
F

‖

p

≤
2
p
Re
⁡
(
1

/

F
(
0
)
)
+

|

Im
⁡
(
1

/

F
(
0
)
)

|

{\left \| {{T_F}} \right \|_p} \leq 2p\operatorname {Re} (1/F(0)) + |\operatorname {Im} (1/F(0))|

for the norm. We also show that

T
F

{T_F}

is compact on

H
p

{H^p}

if and only if the measure

μ
\mu

in the Herglotz representation of

1

/

F

1/F

is continuous.
Branched surfaces and attractors. I. Dynamic branched surfaces
Authors: Christy Joe
Published: June 2012

We show how, using ideas of R. F. Williams about branched surfaces, hyperbolic attractors of flows on three manifolds may be classified up to topological equivalence on an isolating neighborhood by a finite combinatorial object, a swaddled graph.
A generalization of the Airy integral for 𝑓”-𝑧ⁿ𝑓=0
Authors: Gundersen Gary G. , Steinbart Enid M.
Published: June 2012

It is well known that the Airy integral is a solution of the Airy differential equation

f
−
z
f
=
0

f - zf = 0

and that the Airy integral is a contour integral function with special properties. We show that there exist analogous special contour integral solutions of the more general equation

f
−

z
n

f
=
0

f - {z^n}f = 0

where

n
n

is any positive integer. Related results are given.
Extensions of étale by connected group spaces
Authors: Jaffe David B.
Published: June 2012

The main theorem, in rough terms, asserts the following. Let

K
K

and

D
D

be group spaces over a scheme

S
S

. Assume that

K
K

has connected fibers and that

D
D

is finite and étale over

S
S

. Assume that there exists a single finite, surjective, étale, Galois morphism

S
¯

→
S

\overline S \to S

which decomposes (scheme-theoretically) every extension of

D
D

by

K
K

. Let

π
=
Aut
⁡
(

S
¯

/

S
)

\pi = \operatorname {Aut}(\overline S /S)

. Then group space extensions of

D
D

with kernel

K
K

are in bijective correspondence with pairs

(
ξ
,
χ
)

(\xi ,\chi )

consisting of a

π
\pi

-group extension
\[

ξ
:
1
→
K
(

S
¯

)
→
X
→
D
(

S
¯

)
→
1

\xi :1 \to K(\overline S) \to X \to D(\overline S ) \to 1

\]
and a

π
\pi

-group homomorphism

χ
:
X
→
Aut
⁡
(

K
¯

)

\chi :X \to \operatorname {Aut}(\overline K )

which lifts the conjugation map

X
→
Aut
⁡
(
K
(

S
¯

)
)

X \to \operatorname {Aut}(K(\overline S ))

and which agrees with the conjugation map

K
(

S
¯

)
→
Aut
⁡
(

K
¯

)

K(\bar S) \to \operatorname {Aut}(\overline K )

. In this way, the calculation of group space extensions is reduced to a purely group-theoretic calculation.
The number of irreducible factors of a polynomial. I
Authors: Pinner Christopher G. , Vaaler Jeffrey D.
Published: June 2012

Let

F
(
x
)

F(x)

be a polynomial with coefficients in an algebraic number field

k
k

. We estimate the number of irreducible cyclotomic factors of

F
F

in

k
[
x
]

k[x]

, the number of irreducible noncyclotomic factors of

F
F

, the number of

n
n

th roots of unity among the roots of

F
F

, and the number of primitive

n
n

th roots of unity among the roots of

F
F

. All of these quantities are counted with multiplicity and estimated by expressions which depend explicitly on

k
k

, on the degree of

F
F

and height of

F
F

, and (when appropriate) on

n
n

. We show by constructing examples that some of our results are essentially sharp.