bluuebunny/crossref_metadata_2025 huggingface dataset:
Small rational model of subspace complement
Authors: Yuzvinsky Sergey
Published: July 2002

This paper concerns the rational cohomology ring of the complement

M
M

of a complex subspace arrangement. We start with the De Concini-Procesi differential graded algebra that is a rational model for

M
M

. Inside it we find a much smaller subalgebra

D
D

quasi-isomorphic to the whole algebra.

D
D

is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice

L
L

whence connecting the De Concini-Procesi model with the Goresky-MacPherson formula for the additive structure of

H
∗

(
M
)

H^*(M)

. The algebra

D
D

has a natural integral version that is a good candidate for an integral model of

M
M

. If the rational local homology of

L
L

can be computed explicitly we obtain an explicit presentation of the ring

H
∗

(
M
,

Q

)

H^*(M,{\mathbf Q})

. For example, this is done for the cases where

L
L

is a geometric lattice and where

M
M

is a

k
k

-equal manifold.
A problem of Foldes and Puri on the Wiener process
Authors: Shi Z.
Published: July 2002

Let

W
W

be a real-valued Wiener process starting from 0, and

τ
(
t
)

\tau (t)

be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript tau left-parenthesis t right-parenthesis Baseline double-struck 1 Subscript left-brace StartAbsoluteValue upper W left-parenthesis u right-parenthesis EndAbsoluteValue less-than-or-equal-to alpha t right-brace Baseline d u">

X
(
t
)
=

∫
0

τ
(
t
)

1

{

|

W
(
u
)

|

≤
α
t
}

d
u

X(t)=\int _0^{\tau (t)} \mathbb {1}_{\{|W(u)| \le \alpha t\}}\,du

as

t
t

tends to infinity, i.e. they ask: how long does

W
W

stay in a tube before “crossing very much” a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for

X
X

.
On the variances of occupation times of conditioned Brownian motion
Authors: Zhang Biao
Published: July 2002

We extend some bounds on the variance of the lifetime of two–dimensional Brownian motion, conditioned to exit a planar domain at a given point, to certain domains in higher dimensions. We also give a short “analytic” proof of some existing results.
A cascade decomposition theory with applications to Markov and exchangeable cascades
Authors: Waymire Edward , Williams Stanley
Published: July 2002

A multiplicative random cascade refers to a positive

T
T

-martingale in the sense of Kahane on the ultrametric space

T
=

{
0
,
1
,
…
,
b
−
1
}

N

.

T = { \{ 0,1,\dots ,b-1 \} }^{\mathbf {N}}.

A new approach to the study of multiplicative cascades is introduced. The methods apply broadly to the problems of: (i) non-degeneracy criterion, (ii) dimension spectra of carrying sets, and (iii) divergence of moments criterion. Specific applications are given to cascades generated by Markov and exchangeable processes, as well as to homogeneous independent cascades.
Polynomial structures for nilpotent groups
Authors: Dekimpe Karel , Igodt Paul , Lee Kyung
Published: July 2002

If a polycyclic-by-finite rank-

K
K

group

Γ
\Gamma

admits a faithful affine representation making it acting on

R

K

\mathbb {R}{K}

properly discontinuously and with compact quotient, we say that

Γ
\Gamma

admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups

Γ
\Gamma

. Very recently examples have been obtained showing that, even for torsion-free, finitely generated nilpotent groups

N
N

, affine structures do not always exist. It looks natural to consider affine structures as examples of polynomial structures of degree one. We introduce the concept of a canonical type polynomial structure for polycyclic-by-finite groups. Using the algebraic framework of the Seifert Fiber Space construction and a nice cohomology vanishing theorem, we prove the existence and uniqueness (up to conjugation) of canonical type polynomial structures for virtually finitely generated nilpotent groups. Applying this uniqueness to a result going back to Mal′cev, it follows that, for torsion-free, finitely generated nilpotent groups, each canonical polynomial structure is expressed in polynomials of limited degree. The minimal degree needed for obtaining a polynomial structure will determine the “affine defect number”. We prove that the known counterexamples to Milnor’s question have the smallest possible affine defect, i.e. affine defect number equal to one.
Gelfand-Kirillov dimension in Jordan Algebras
Authors: Martinez C.
Published: July 2002

In this paper we study Gelfand-Kirillov dimension in Jordan algebras. In particular we will relate Gelfand-Kirillov (GK for short) dimensions of a special Jordan algebra and its associative enveloping algebra and also the GK dimension of a Jordan algebra and the GK dimension of its universal multiplicative enveloping algebra.
Nonnegative Radix Representations for the Orthant 𝑅ⁿ₊
Authors: Lagarias Jeffrey , Wang Yang
Published: July 2002

Let

A
A

be a nonnegative real matrix which is expanding, i.e. with all eigenvalues

|

λ

|

>
1

|\lambda | > 1

, and suppose that

|

det
(
A
)

|

|\det (A)|

is an integer. Let

D

{\mathcal D}

consist of exactly

|

det
(
A
)

|

|\det (A)|

nonnegative vectors in

R

n

\mathbb {R}^n

. We classify all pairs

(
A
,

D

)

(A, {\mathcal D})

such that every

x
x

in the orthant

R

+
n

\mathbb {R}^n_+

has at least one radix expansion in base

A
A

using digits in

D

{\mathcal D}

. The matrix

A
A

must be a diagonal matrix times a permutation matrix. In addition

A
A

must be similar to an integer matrix, but need not be an integer matrix. In all cases the digit set

D

\mathcal D

can be diagonally scaled to lie in

Z

n

\mathbb {Z}^n

. The proofs generalize a method of Odlyzko, previously used to classify the one–dimensional case.
Distinguished representations and quadratic base change for 𝐺𝐿(3)
Authors: Jacquet Herve , Ye Yangbo
Published: July 2002

Let

E

/

F

E/F

be a quadratic extension of number fields. Suppose that every real place of

F
F

splits in

E
E

and let

H
H

be the unitary group in 3 variables. Suppose that

Π
\Pi

is an automorphic cuspidal representation of

G
L
(
3
,

E

A

)

GL(3,E_{\mathbb {A}})

. We prove that there is a form

ϕ
\phi

in the space of

Π
\Pi

such that the integral of

ϕ
\phi

over

H
(
F
)
∖
H
(

F

A

)

H(F)\setminus H(F_{\mathbb {A}})

is non zero. Our proof is based on earlier results and the notion, discussed in this paper, of Shalika germs for certain Kloosterman integrals.
Spectral averaging, perturbation of singular spectra, and localization
Authors: Combes J. , Hislop P. , Mourre E.
Published: July 2002

A spectral averaging theorem is proved for one-parameter families of self-adjoint operators using the method of differential inequalities. This theorem is used to establish the absolute continuity of the averaged spectral measure with respect to Lebesgue measure. This is an important step in controlling the singular continuous spectrum of the family for almost all values of the parameter. The main application is to the problem of localization for certain families of random Schrödinger operators. Localization for a family of random Schrödinger operators is established employing these results and a multi-scale analysis.
Compact self-dual Hermitian surfaces
Authors: Apostolov Vestislav , Davidov Johann , Muskarov Oleg
Published: July 2002

In this paper, we obtain a classification (up to conformal equivalence) of the compact self-dual Hermitian surfaces. As an application, we prove that every compact Hermitian surface of pointwise constant holomorphic sectional curvature with respect to either the Riemannian or the Hermitian connection is Kähler.
Geometrizing Infinite Dimensional Locally Compact Groups
Authors: Plaut Conrad
Published: July 2002

We study groups having invariant metrics of curvature bounded below in the sense of Alexandrov. Such groups are a generalization of Lie groups with invariant Riemannian metrics, but form a much larger class. We prove that every locally compact, arcwise connected, first countable group has such a metric. These groups may not be (even infinite dimensional) manifolds. We show a number of relationships between the algebraic and geometric structures of groups equipped with such metrics. Many results do not require local compactness.
Continuous-trace groupoid 𝐶*-algebras. III
Authors: Muhly Paul , Renault Jean , Williams Dana
Published: July 2002

Suppose that

G

\mathcal {G}

is a second countable locally compact groupoid with a Haar system and with abelian isotropy. We show that the groupoid

C

∗

C^{*}

-algebra

C
∗

(

G

,
λ
)

C^*(\mathcal {G},\lambda )

has continuous trace if and only if there is a Haar system for the isotropy groupoid

A

\mathcal {A}

and the action of the quotient groupoid

G

/

A

\mathcal {G}/\mathcal {A}

is proper on the unit space of

G

\mathcal {G}

.
Projectively bounded Fréchet measures
Authors: Blei Ron
Published: July 2002

A scalar valued set function on a Cartesian product of

σ
\sigma

-algebras is a Fréchet measure if it is a scalar measure independently in each coordinate. A basic question is considered: is it possible to construct products of Fréchet measures that are analogous to product measures in the classical theory? A Fréchet measure is said to be projectively bounded if it satisfies a Grothendieck type inequality. It is shown that feasibility of products of Fréchet measures is linked to the projective boundedness property. All Fréchet measures in a two dimensional framework are projectively bounded, while there exist Fréchet measures in dimensions greater than two that are projectively unbounded. A basic problem is considered: when is a Fréchet measure projectively bounded? Some characterizations are stated. Applications to harmonic and stochastic analysis are given.
Bott’s vanishing theorem for regular Lie algebroids
Authors: Kubarski Jan
Published: July 2002

Differential geometry has discovered many objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups. For example: — differential groupoids, — principal bundles, — vector bundles, — actions of Lie groups on manifolds, — transversally complete foliations, — nonclosed Lie subgroups, — Poisson manifolds, — some complete closed pseudogroups. We carry over the idea of Bott’s Vanishing Theorem to regular Lie algebroids (using the Chern-Weil homomorphism of transitive Lie algebroids investigated by the author) and, next, apply it to new situations which are not described by the classical version, for example, to the theory of transversally complete foliations and nonclosed Lie subgroups in order to obtain some topological obstructions for the existence of involutive distributions and Lie subalgebras of some types (respectively).
Tensor products over abelian 𝑊*-algebras
Authors: Magajna Bojan
Published: July 2002

Tensor products of C

∗

−

^*-

algebras over an abelian W

∗

−

^*-

algebra

Z
Z

are studied. The minimal C

∗

−

^*-

norm on

A

⊙
Z

B

A\odot _ZB

is shown to be just the quotient of the minimal C

∗

−

^*-

norm on

A
⊙
B

A\odot B

if

A
A

or

B
B

is exact.
On representations of affine Kac-Moody groups and related loop groups
Authors: Chen Yu
Published: July 2002

We demonstrate a one to one correspondence between the irreducible projective representations of an affine Kac-Moody group and those of the related loop group, which leads to the results that every non-trivial representation of an affine Kac-Moody group must have its degree greater than or equal to the rank of the group and that the equivalence appears if and only if the group is of type

A

n

(
1
)

A_{n}^{(1)}

for some

n
≥
1

n\ge 1

. Moreover the characteristics of the base fields for the non-trivial representations are found being always zero.
Strong laws for 𝐿- and 𝑢-statistics
Authors: Aaronson J. , Burton R. , Dehling H. , Gilat D. , Hill T. , Weiss B.
Published: July 2002

Strong laws of large numbers are given for

L
L

-statistics (linear combinations of order statistics) and for

U
U

-statistics (averages of kernels of random samples) for ergodic stationary processes, extending classical theorems of Hoeffding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.
A variational problem for surfaces in Laguerre geometry
Authors: Musso Emilio , Nicolodi Lorenzo
Published: July 2002

We consider the variational problem defined by the functional

∫

H

2

−
K

K

d
A

\int {\frac {{H^{2}-K}}{{K}}}dA

on immersed surfaces in Euclidean space. Using the invariance of the functional under the group of Laguerre transformations, we study the extremal surfaces by the method of moving frames.
Abstract functions with continuous differences and Namioka spaces
Authors: Basit Bolis , Günzler Hans
Published: July 2002

Let

G
G

be a semigroup and a topological space. Let

X
X

be an Abelian topological group. The right differences

△

h

φ

\triangle _{h} \varphi

of a function

φ
:
G
→
X

\varphi : G \to X

are defined by

△

h

φ
(
t
)
=
φ
(
t
h
)
−
φ
(
t
)

\triangle _{h}\varphi (t) = \varphi (th) - \varphi (t)

for

h
,
t
∈
G

h,t \in G

. Let

△

h

φ

\triangle _{h} \varphi

be continuous at the identity

e
e

of

G
G

for all

h
h

in a neighbourhood

U
U

of

e
e

. We give conditions on

X
X

or range

φ
\varphi

under which

φ
\varphi

is continuous for any topological space

G
G

. We also seek conditions on

G
G

under which we conclude that

φ
\varphi

is continuous at

e
e

for arbitrary

X
X

. This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.
The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations
Authors: Fels M.
Published: July 2002

A simple invariant characterization of the scalar fourth-order ordinary differential equations which admit a variational multiplier is given. The necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth-order equation through the application of Cartan’s equivalence method. The solution to the inverse problem for fourth-order scalar equations provides the solution to an equivalence problem for second-order Lagrangians, as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated Lagrangian.
Separable Banach space theory needs strong set existence axioms
Authors: Humphreys A. , Simpson Stephen
Published: July 2002

We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom,

Π
1
1

\Pi ^1_1

comprehension, is needed to prove such basic facts as the existence of the weak-

∗
*

closure of any norm-closed subspace of

ℓ
1

=

c
0
∗

\ell _1=c_0^*

. This is in contrast to earlier work in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for

Π
2
0

\Pi ^0_2

sentences. En route to our main results, we prove the Krein-Šmulian theorem in

A
C
A

0

\mathsf {ACA}_0

, and we give a new, elementary proof of a result of McGehee on weak-

∗
*

sequential closure ordinals.
Characterizations of Kadec-Klee properties in symmetric spaces of measurable functions
Authors: Chilin V. , Dodds P. , Sedaev A. , Sukochev F.
Published: July 2002

We present several characterizations of Kadec-Klee properties in symmetric function spaces on the half-line, based on the

K
K

-functional of J. Peetre. In addition to the usual Kadec-Klee property, we study those symmetric spaces for which sequential convergence in measure (respectively, local convergence in measure) on the unit sphere coincides with norm convergence.
Confluence of cycles for hypergeometric functions on 𝑍_{2,𝑛+1}
Authors: Haraoka Yoshishige
Published: July 2002

The hypergeometric function of general type, which is a generalization of the classical confluent hypergeometric functions, admits an integral representation derived from a character of a linear abelian group. For the hypergeometric function on the space of

2
×
(
n
+
1
)

2\times (n+1)

matrices, a basis of cycles for the integral is constructed by a limit process, which is called a process of confluence. The determinant of the period matrix is explicitly evaluated to show the independence of the cycles.
The local dimensions of the Bernoulli convolution associated with the golden number
Authors: Hu Tian-You
Published: July 2002

Let

X
1

,

X
2

,
…

X_1,X_2,\dotsc

be a sequence of i.i.d. random variables each taking values of 1 and

−
1

-1

with equal probability. For

1

/

2
>
ρ
>
1

1/2>\rho >1

satisfying the equation

1
−
ρ
−
⋯
−

ρ
s

=
0

1-\rho -\dotsb -\rho ^s=0

, let

μ
\mu

be the probability measure induced by

S
=

∑

i
=
1

∞

ρ
i

X
i

S=\sum _{i=1}^\infty \rho ^iX_i

. For any

x
x

in the range of

S
S

, let
\[

d
(
μ
,
x
)
=

lim

r
→

0
+

log
⁡
μ
(
[
x
−
r
,
x
+
r
]
)

/

log
⁡
r

d(\mu ,x)=\lim _{r\to 0^+}\log \mu ([x-r,x+r])/\log r

\]
be the local dimension of

μ
\mu

at

x
x

whenever the limit exists. We prove that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Superscript asterisk Baseline equals minus StartFraction log 2 Over log rho EndFraction and alpha Subscript asterisk Baseline equals minus StartFraction log delta Over s log rho EndFraction minus StartFraction log 2 Over log rho EndFraction comma">

α
∗

=
−

log
⁡
2

log
⁡
ρ

and

α
∗

=
−

log
⁡
δ

s
log
⁡
ρ

−

log
⁡
2

log
⁡
ρ

,

\alpha ^*=-\frac {\log 2}{\log \rho }\quad \text {and}\quad \alpha _*=-\frac {\log \delta }{s\log \rho }-\frac {\log 2}{\log \rho },

\]
where

δ
=
(

5

−
1
)

/

2

\delta =(\sqrt {5}-1)/2

, are respectively the maximum and minimum values of the local dimensions. If

s
=
2

s=2

, then

ρ
\rho

is the golden number, and the approximate numerical values are

α
∗

≈
1.4404

\alpha ^*\approx 1.4404

and

α
∗

≈
0.9404

\alpha _*\approx 0.9404

.
Analytic subgroups of 𝑡-modules
Authors: Tubbs Robert
Published: July 2002

In this paper we study the structure of analytic subgroups and of

t
t

-submodules of

t
t

-modules.
Locally conformal Kähler structures in quaternionic geometry
Authors: Ornea Liviu , Piccinni Paolo
Published: July 2002

We consider compact locally conformal quaternion Kähler manifolds

M
M

. This structure defines on

M
M

a canonical foliation, which we assume to have compact leaves. We prove that the local quaternion Kähler metrics are Ricci-flat and allow us to project

M
M

over a quaternion Kähler orbifold

N
N

with fibers conformally flat 4-dimensional real Hopf manifolds. This fibration was known for the subclass of locally conformal hyperkähler manifolds; in this case we make some observations on the fibers’ structure and obtain restrictions on the Betti numbers. In the homogeneous case

N
N

is shown to be a manifold and this allows a classification. Examples of locally conformal quaternion Kähler manifolds (some with a global complex structure, some locally conformal hyperkähler) are the Hopf manifolds quotients of

H

n

−
{
0
}

\mathbb H^n-\{0\}

by the diagonal action of appropriately chosen discrete subgroups of

C

O
+

(
4
)

CO^+(4)

.
On the cohomology of split extensions of finite groups
Authors: Siegel Stephen
Published: July 2002

LetG=H⋊QG=H\rtimes Qbe a split extension of finite groups. A theorem of Charlap and Vasquez gives an explicit description of the differentialsd2d_2in the Lyndon-Hochschild-Serre spectral sequence of the extension with coefficients in a fieldkk. We generalize this to give an explicit description of all thedrd_r(r≥2r\geq 2) in this case. The generalization is obtained by associating to the group extension a new twisting cochain, which takes values in thekGkG-endomorphism algebra of the minimalkHkH-projective resolution induced fromHHtoGG. This twisting cochain not only determines the differentials, but also allows one to construct an explicitkGkG-projective resolution ofkk.
Some combinatorial aspects of reduced words in finite Coxeter groups
Authors: Stembridge John
Published: July 2002

We analyze the structure of reduced expressions in the Coxeter groups

A

n

A_{n}

,

B

n

B_{n}

and

D

n

D_{n}

. Several special classes of elements are singled out for their connections with symmetric functions or the theory of

P
P

-partitions. Membership in these special classes is characterized in a variety of ways, including forbidden patterns, forbidden subwords, and by the form of canonically chosen reduced words.
The homology representations of the 𝑘-equal partition lattice
Authors: Sundaram Sheila , Wachs Michelle
Published: July 2002

We determine the character of the action of the symmetric group on the homology of the induced subposet of the lattice of partitions of the set

{
1
,
2
,
…
,
n
}

\{1,2,\ldots ,n\}

obtained by restricting block sizes to the set

{
1
,
k
,
k
+
1
,
…
}

\{1,k,k+1,\ldots \}

. A plethystic formula for the generating function of the Frobenius characteristic of the representation is given. We combine techniques from the theory of nonpure shellability, recently developed by Björner and Wachs, with symmetric function techniques, developed by Sundaram, for determining representations on the homology of subposets of the partition lattice.
Tetragonal curves, scrolls and 𝐾3 surfaces
Authors: Brawner James
Published: July 2002

In this paper we establish a theorem which determines the invariants of a general hyperplane section of a rational normal scroll of arbitrary dimension. We then construct a complete intersection surface on a four-dimensional scroll and prove it is regular with a trivial dualizing sheaf. We determine the invariants for which the surface is nonsingular, and hence a

K
3

K3

surface. A general hyperplane section of this surface is a tetragonal curve; we use the first theorem to determine for which tetragonal invariants such a construction is possible. In particular we show that for every genus

g
≥
7

g\geq 7

there is a tetragonal curve of genus

g
g

that is a hyperplane section of a

K
3

K3

surface. Conversely, if the tetragonal invariants are not sufficiently balanced, then the complete intersection must be singular. Finally we determine for which additional sets of invariants this construction gives a tetragonal curve as a hyperplane section of a singular canonically trivial surface, and discuss the connection with other recent results on canonically trivial surfaces.
Boundary value maps, Szegö maps and intertwining operators
Authors: Barchini L.
Published: July 2002

We consider one series of unitarizable representations, the cohomological induced modules

A

q

(
λ
)

A_{\mathfrak {q}}(\lambda )

with dominant regular infinitesimal character. The minimal

K
K

-type

(
τ
,
V
)

(\tau , V)

of

A

q

(
λ
)

A_{\mathfrak {q}}(\lambda )

determines a homogeneous vector bundle

V

τ

⟶
G

/

K

V_{\tau } \longrightarrow G/K

. The derived functor modules can be realized on the solution space of a first order differential operator

D

l

λ

\mathcal {D}_{\mathfrak {l}}^{\lambda }

on

V

τ

V_{\tau }

. Barchini, Knapp and Zierau gave an explicit integral map

S

\mathcal {S}

from the derived functor module, realized in the Langlands classification, into the space of smooth sections of the vector bundle

V

τ

⟶
G

/

K

V_{\tau } \longrightarrow G/K

. In this paper we study the asymptotic behavior of elements in the image of

S

\mathcal {S}

. We obtain a factorization of the standard intertwining opeartors into the composition of the Szegö map

S

\mathcal {S}

and a passage to boundary values.
Nonexistence of global solutions of a nonlinear hyperbolic system
Authors: Deng Keng
Published: July 2002

Consider the initial value problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column u Subscript t t Baseline equals normal upper Delta u plus StartAbsoluteValue v EndAbsoluteValue Superscript p Baseline comma 2nd Column a m p semicolon v Subscript t t Baseline equals normal upper Delta v plus StartAbsoluteValue u EndAbsoluteValue Superscript q Baseline comma 3rd Column a m p semicolon x element-of double-struck upper R Superscript n Baseline comma 4th Column a m p semicolon t greater-than 0 comma 2nd Row 1st Column u left-parenthesis x comma 0 right-parenthesis equals f left-parenthesis x right-parenthesis comma 2nd Column a m p semicolon v left-parenthesis x comma 0 right-parenthesis equals h left-parenthesis x right-parenthesis comma 3rd Column a m p semicolon 4th Column a m p semicolon 3rd Row 1st Column u Subscript t Baseline left-parenthesis x comma 0 right-parenthesis equals g left-parenthesis x right-parenthesis comma 2nd Column a m p semicolon v Subscript t Baseline left-parenthesis x comma 0 right-parenthesis equals k left-parenthesis x right-parenthesis comma EndLayout">

u

t
t

=
Δ
u
+
|
v

|

p

,

a
m
p
;

v

t
t

=
Δ
v
+
|
u

|

q

,

a
m
p
;
x
∈

R

n

,

a
m
p
;
t
>
0
,

u
(
x
,
0
)
=
f
(
x
)
,

a
m
p
;
v
(
x
,
0
)
=
h
(
x
)
,

a
m
p
;

a
m
p
;

u

t

(
x
,
0
)
=
g
(
x
)
,

a
m
p
;

v

t

(
x
,
0
)
=
k
(
x
)
,

\begin{equation*} \begin {array}{llll} u_{tt} = \Delta u+\vert v\vert ^{p}, & v_{tt} = \Delta v +\vert u\vert ^{q}, &x\in \mathbb {R}^{n},&t>0,\\ u(x,0)=f(x),&v(x,0)=h(x),&{}&{}\\ u_{t}(x,0) = g(x), &v_{t}(x,0) = k(x), \end{array} \end{equation*}

with

1
≤
n
≤
3

1\le n\le 3

and

p
,
q
>
0

p,q>0

. We show that there exists a bound

B
(
n
)
(
≤
∞
)

B(n) (\le \infty )

such that if

1
>
p
q
>
B
(
n
)

1>pq>B(n)

all nontrivial solutions with compact support blow up in finite time.
On the number of geodesic segments connecting two points on manifolds of non-positive curvature
Authors: Horja Paul
Published: July 2002

We prove that on a complete Riemannian manifold

M
M

of dimension

n
n

with sectional curvature

K
M

>
0

K_M > 0

, two points which realize a local maximum for the distance function (considered as a function of two arguments) are connected by at least

2
n
+
1

2n+1

geodesic segments. A simpler version of the argument shows that if one of the points is fixed and

K
M

≤
0

K_M \leq 0

then the two points are connected by at least

n
+
1

n+1

geodesic segments. The proof uses mainly the convexity properties of the distance function for metrics of negative curvature.
The Floer homotopy type of height functions on complex Grassmann manifolds
Authors: Hurtubise David
Published: July 2002

A family of Floer functions on the infinite dimensional complex Grassmann manifold is defined by taking direct limits of height functions on adjoint orbits of unitary groups. The Floer cohomology of a generic function in the family is computed using the Schubert calculus. The Floer homotopy type of this function is computed and the Floer cohomology which was computed algebraically is recovered from the Floer homotopy type. Certain non-generic elements of this family of Floer functions were shown to be related to the symplectic action functional on the universal cover of the loop space of a finite dimensional complex Grassmann manifold in the author’s preprint The Floer homotopy type of complex Grassmann manifolds.
Non-Archimedean Nevanlinna theory in several variables and the non-Archimedean Nevanlinna inverse problem
Authors: Cherry William , Ye Zhuan
Published: July 2002

Cartan’s method is used to prove a several variable, non-Archimedean, Nevanlinna Second Main Theorem for hyperplanes in projective space. The corresponding defect relation is derived, but unlike in the complex case, we show that there can only be finitely many non-zero non-Archimedean defects. We then address the non-Archimedean Nevanlinna inverse problem, by showing that given a set of defects satisfying our conditions and a corresponding set of hyperplanes in projective space, there exists a non-Archimedean analytic function with the given defects at the specified hyperplanes, and with no other defects.
Cohen-Macaulay Section Rings
Authors: Caijun Zhou
Published: July 2002

In this paper, we study the section rings of sheaves of Cohen-Macaulay algebras (over a field

F
F

) on a ranked poset. A necessary and sufficient condition for these rings to be Cohen-Macaulay will be given. This is a further generalization of a result of Yuzvinsky, which generalizes Reisner’s theorem concerning Stanley-Reisner rings.
An infinite dimensional Morse theory with applications
Authors: Kryszewski Wojciech , Szulkin Andrzej
Published: July 2002

In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the one-dimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations.
Some ramifications of a theorem of Boas and Pollard concerning the completion of a set of functions in 𝐿²
Authors: Kazarian K. , Zink Robert
Published: July 2002

About fifty years ago, R. P. Boas and Harry Pollard proved that an orthonormal system that is completable by the adjunction of a finite number of functions also can be completed by multiplying the elements of the given system by a fixed, bounded, nonnegative measurable function. In subsequent years, several variations and extensions of this theorem have been given by a number of other investigators, and this program is continued here. A mildly surprising corollary of one of the results is that the trigonometric and Walsh systems can be multiplicatively transformed into quasibases for

L

1

[
0
,
1
]

L^{1}[0,1]

.
Realizing homology boundary links with arbitrary patterns
Authors: Bellis Paul
Published: July 2002

Homology boundary links have become an increasingly important class of links, largely due to their significance in the ongoing concordance classification of links. Tim Cochran and Jerome Levine defined an algebraic object called a pattern associated to a homology boundary link which can be used to study the deviance of a homology boundary link from being a boundary link. Since a pattern is a set of

m
m

elements which normally generates the free group of rank

m
m

, any invariants which detect non-trivial patterns can be applied to the purely algebraic question of when such a set is a set of conjugates of a generating set for the free group. We will give a constructive geometric proof that all patterns are realized by some homology boundary link

L
n

L^n

in

S

n
+
2

S^{n+2}

. We shall also prove an analogous existence theorem for calibrations of

E

\mathbb {E}

-links, a more general and less understood class of links tha homology boundary links.
Relativity of the spectrum and discrete groups on hyperbolic spaces
Authors: Mandouvalos N.
Published: July 2002

We give a simple proof of the analytic continuation of the resolvent kernel for a convex cocompact Kleinian group.
Tight contact structures on solid tori
Authors: Makar-Limanov Sergei
Published: July 2002

In this paper we study properties of tight contact structures on solid tori. In particular we discuss ways of distinguishing two solid tori with tight contact structures. We also give examples of unusual tight contact structures on solid tori. We prove the existence of a

Z

\mathbb {Z}

-valued and a

R

/

2
π

Z

\mathbb {R}/2\pi \mathbb {Z}

-valued invariant of a closed solid torus. We call them the self-linking number and the rotation number respectively. We then extend these definitions to the case of an open solid torus. We show that these invariants exhibit certain monotonicity properties with respect to inclusion. We also prove a number of results which give sufficient conditions for two solid tori to be contactomorphic. At the same time we discuss various ways of constructing a tight contact structure on a solid torus. We then produce examples of solid tori with tight contact structures and calculate self-linking and rotation numbers for these tori. These examples show that the invariants we defined do not give a complete classification of tight contact structure on open solid tori. At the end, we construct a family of tight contact structure on a solid torus such that the induced contact structure on a finite-sheeted cover of that solid torus is no longer tight. This answers negatively a question asked by Eliashberg in 1990. We also give an example of tight contact structure on an open solid torus which cannot be contactly embedded into a sphere with the standard contact structure, another example of unexpected behavior.
Contiguous relations, continued fractions and orthogonality
Authors: Gupta Dharma , Masson David
Published: July 2002

We examine a special linear combination of balanced very-well-poised

10

ϕ

9

{_{10} \phi _{9}}

basic hypergeometric series that is known to satisfy a transformation. We call this

Φ
\Phi

and show that it satisfies certain three-term contiguous relations. From two of these contiguous relations for

Φ
\Phi

we obtain fifty-six pairwise linearly independent solutions to a three-term recurrence that generalizes the recurrence for Askey-Wilson polynomials. The associated continued fraction is evaluated using Pincherle’s theorem. From this continued fraction we are able to derive a discrete system of biorthogonal rational functions. This ties together Wilson’s results for rational biorthogonality, Watson’s

q
q

-analogue of Ramanujan’s Entry 40 continued fraction, and a conjecture of Askey concerning the latter. Some new

q
q

-series identities are also obtained. One is an important three-term transformation for

Φ
\Phi

’s which generalizes all the known two- and three-term

8

ϕ

7

{_{8} \phi _{7}}

transformations. Others are new and unexpected quadratic identities for these very-well-poised

8

ϕ

7

{_{8} \phi _{7}}

’s.
Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials
Authors: Przytycki Feliks
Published: July 2002

We prove that for every rational map on the Riemann sphere

f
:

C

¯

→

C

¯

f:\overline {\mathbb {C}} \to \overline {\mathbb {C}}

, if for every

f
f

-critical point

c
∈
J

c\in J

whose forward trajectory does not contain any other critical point, the growth of

|

(

f

n

)
′

(
f
(
c
)
)

|

|(f^{n})’(f(c))|

is at least of order

exp
⁡
Q

n

\exp Q \sqrt n

for an appropriate constant

Q
Q

as

n
→
∞

n\to \infty

, then

HD

ess

⁡
(
J
)
=

α

0

=
HD
⁡
(
J
)

\operatorname {HD}_{\operatorname {ess}} (J)=\alpha _{0}=\operatorname {HD} (J)

. Here

HD

ess

⁡
(
J
)

\operatorname {HD}_{\operatorname {ess}} (J)

is the so-called essential, dynamical or hyperbolic dimension,

HD
⁡
(
J
)

\operatorname {HD} (J)

is Hausdorff dimension of

J
J

and

α

0

\alpha _{0}

is the minimal exponent for conformal measures on

J
J

. If it is assumed additionally that there are no periodic parabolic points then the Minkowski dimension (other names: box dimension, limit capacity) of

J
J

also coincides with

HD
⁡
(
J
)

\operatorname {HD}(J)

. We prove ergodicity of every

α
\alpha

-conformal measure on

J
J

assuming

f
f

has one critical point

c
∈
J

c\in J

, no parabolic, and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts StartAbsoluteValue left-parenthesis f Superscript n Baseline right-parenthesis prime left-parenthesis f left-parenthesis c right-parenthesis right-parenthesis EndAbsoluteValue Superscript negative 1 Baseline greater-than normal infinity">

∑

n
=
0

∞

|

(

f

n

)
′

(
f
(
c
)
)

|

−
1

>
∞

\sum _{n=0}^{\infty }|(f^{n})’(f(c))|^{-1} >\infty

. Finally for every

α
\alpha

-conformal measure

μ
\mu

on

J
J

(satisfying an additional assumption), assuming an exponential growth of

|

(

f

n

)
′

(
f
(
c
)
)

|

|(f^{n})’(f(c))|

, we prove the existence of a probability absolutely continuous with respect to

μ
\mu

,

f
f

-invariant measure. In the Appendix we prove

HD

ess

⁡
(
J
)
=
HD
⁡
(
J
)

\operatorname {HD}_{\operatorname {ess}} (J)=\operatorname {HD} (J)

also for every non-renormalizable quadratic polynomial

z
↦

z

2

+
c

z\mapsto z^{2}+c

with

c
c

not in the main cardioid in the Mandelbrot set.
𝑛-unisolvent sets and flat incidence structures
Authors: Polster Burkard
Published: July 2002

For the past forty years or so topological incidence geometers and mathematicians interested in interpolation have been studying very similar objects. Nevertheless no communication between these two groups of mathematicians seems to have taken place during that time. The main goal of this paper is to draw attention to this fact and to demonstrate that by combining results from both areas it is possible to gain many new insights about the fundamentals of both areas. In particular, we establish the existence of nested orthogonal arrays of strength

n
n

, for short nested

n
n

-OAs, that are natural generalizations of flat affine planes and flat Laguerre planes. These incidence structures have point sets that are “flat” topological spaces like the Möbius strip, the cylinder, and strips of the form

I
×

R

I \times \mathbb {R}

, where

I
I

is an interval of

R

\mathbb {R}

. Their circles (or lines) are subsets of the point sets homeomorphic to the circle in the first two cases and homeomorphic to

I
I

in the last case. Our orthogonal arrays of strength

n
n

arise from

n
n

-unisolvent sets of half-periodic functions,

n
n

-unisolvent sets of periodic functions, and

n
n

-unisolvent sets of functions

I
→

R

I\to \mathbb {R}

, respectively. Associated with every point

p
p

of a nested

n
n

-OA,

n
>
1

n>1

, is a nested

(
n
−
1
)

(n-1)

-OA—the derived

(
n
−
1
)

(n-1)

-OA at the point

p
p

. We discover that, in our examples that arise from

n
n

-unisolvent sets of

n
−
1

n-1

times differentiable functions that solve the Hermite interpolation problem, deriving in our geometrical sense coincides with deriving in the analytical sense.
Geometry of families of nodal curves on the blown-up projective plane
Authors: Greuel Gert-Martin , Lossen Christoph , Shustin Eugenii
Published: July 2002

Let

P

r
2

\mathbb P^2_r\,

be the projective plane blown up at

r
r

generic points. Denote by

E
0

,

E
1

,
…
,

E
r

E_0,E_1,\ldots ,E_r

the strict transform of a generic straight line on

P

2

\mathbb P^2

and the exceptional divisors of the blown–up points on

P

r
2

\mathbb P^2_r

respectively. We consider the variety

V

i
r
r

(
d
;

d
1

,
…
,

d
r

;

k
)

V_{irr}(d;\,d_1,\ldots ,d_r;\,k)

of all irreducible curves

C
C

in

|

d

E
0

−

∑

i
=
1

r

d
i

E
i

|

|dE_0-\sum _{i=1}^{r} d_iE_i|

with

k
k

nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non–emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility to families of reducible curves. For

r
≤
9

r\leq 9

we give the complete answer concerning the existence of nodal curves in

V

i
r
r

(
d
;

d
1

,
…
,

d
r

;

k
)

V_{irr}(d;\,d_1,\ldots ,d_r;\,k)

.
Factorisation in nest algebras. II
Authors: Anoussis M. , Katsoulis E.
Published: July 2002

The main result of this paper is Theorem 5, which provides a necessary and sufficient condition on a positive operator

A
A

for the existence of an operator

B
B

in the nest algebra

A
l
g
N

AlgN

of a nest

N
N

satisfying

A
=
B

B

∗

A=BB^{*}

(resp.

A
=

B

∗

B
)

A=B^{*}B)

. In Section 3 we give a new proof of a result of Power concerning outer factorisation of operators. We also show that a positive operator

A
A

has the property that there exists for every nest

N
N

an operator

B
N

B_N

in

A
l
g
N

AlgN

satisfying

A
=

B
N

B
N

∗

A=B_NB_N^{*}

(resp.

A
=

B
N

∗

B
N

A=B_N^{*}B_N

) if and only if

A
A

is a Fredholm operator. In Section 4 we show that for a given operator

A
A

in

B
(
H
)

B(H)

there exists an operator

B
B

in

A
l
g
N

AlgN

satisfying

A

A

∗

=
B

B

∗

AA^{*}=BB^{*}

if and only if the range

r
(
A
)

r(A)

of

A
A

is equal to the range of some operator in

A
l
g
N

AlgN

. We also determine the algebraic structure of the set of ranges of operators in

A
l
g
N

AlgN

. Let

F
r

(
N
)

F_r(N)

be the set of positive operators

A
A

for which there exists an operator

B
B

in

A
l
g
N

AlgN

satisfying

A
=
B

B

∗

A=BB^{*}

. In Section 5 we obtain information about this set. In particular we discuss the following question: Assume

A
A

and

B
B

are positive operators such that

A
≤
B

A\leq B

and

A
A

belongs to

F
r

(
N
)

F_r(N)

. Which further conditions permit us to conclude that

B
B

belongs to

F
r

(
N
)

F_r(N)

?
Linearization, Dold-Puppe stabilization, and Mac Lane’s 𝑄-construction
Authors: Johnson Brenda , McCarthy Randy
Published: July 2002

In this paper we study linear functors, i.e., functors of chain complexes of modules which preserve direct sums up to quasi-isomorphism, in order to lay the foundation for a further study of the Goodwillie calculus in this setting. We compare the methods of Dold and Puppe, Mac Lane, and Goodwillie for producing linear approximations to functors, and establish conditions under which these methods are equivalent. In addition, we classify linear functors in terms of modules over an explicit differential graded algebra. Several classical results involving Dold-Puppe stabilization and Mac Lane’s

Q
Q

-construction are extended or given new proofs.
Geometric families of constant reductions and the Skolem property
Authors: Green Barry
Published: July 2002

Let

F

|

K

F|K

be a function field in one variable and

V

\mathcal V

be a family of independent valuations of the constant field

K
.

K.

Given

v
∈

V

,

v\in \mathcal V ,

a valuation prolongation

v

\mathrm v

to

F
F

is called a constant reduction if the residue fields

F

v

|

K
v

F\mathrm v |Kv

again form a function field of one variable. Suppose

t
∈
F

t\in F

is a non-constant function, and for each

v
∈

V

v\in \mathcal V

let

V

t

V_{t}

be the set of all prolongations of the Gauß valuation

v

t

v_{t}

on

K
(
t
)

K(t)

to

F
.

F.

The union of the sets

V

t

V_{t}

over all

v
∈

V

v\in \mathcal V

is denoted by

V

t

.

\mathbfit {V}_{t}.

The aim of this paper is to study families of constant reductions

V

\mathbfit {V}

of

F
F

prolonging the valuations of

V

\mathcal V

and the criterion for them to be principal, that is to be sets of the type

V

t

.

\mathbfit {V}_{t}.

The main result we prove is that if either

V

\mathcal V

is finite and each

v
∈

V

v\in \mathcal V

has rational rank one and residue field algebraic over a finite field, or if

V

\mathcal V

is any set of non-archimedean valuations of a global field

K
K

satisfying the strong approximation property, then each geometric family of constant reductions

V

\mathbfit {V}

prolonging

V

\mathcal V

is principal. We also relate this result to the Skolem property for the existence of

V

\mathcal V

-integral points on varieties over

K
,

K,

and Rumely’s existence theorem. As an application we give a birational characterization of arithmetic surfaces

X

/

S

\mathcal X /S

in terms of the generic points of the closed fibre. The characterization we give implies the existence of finite morphisms to

P

S

1

.

\mathbb P ^{1}_{S}.
Copies of 𝑐₀ and ℓ_{∞} in topological Riesz spaces
Authors: Drewnowski Lech , Labuda Iwo
Published: July 2002

The paper is concerned with order-topological characterizations of topological Riesz spaces, in particular spaces of measurable functions, not containing Riesz isomorphic or linearly homeomorphic copies of

c

0

c_{0}

or

ℓ

∞

\ell _{\infty }

.
Homoclinic Solutions and Chaos in Ordinary Differential Equations with Singular Perturbations
Authors: Gruendler Joseph
Published: July 2002

Ordinary differential equations are considered which contain a singular perturbation. It is assumed that when the perturbation parameter is zero, the equation has a hyperbolic equilibrium and homoclinic solution. No restriction is placed on the dimension of the phase space or on the dimension of intersection of the stable and unstable manifolds. A bifurcation function is established which determines nonzero values of the perturbation parameter for which the homoclinic solution persists. It is further shown that when the vector field is periodic and a transversality condition is satisfied, the homoclinic solution to the perturbed equation produces a transverse homoclinic orbit in the period map. The techniques used are those of exponential dichotomies, Lyapunov-Schmidt reduction and scales of Banach spaces. A much simplified version of this latter theory is developed suitable for the present case. This work generalizes some recent results of Battelli and Palmer.
Poincaré embedding of the diagonal
Authors: Byun Yanghyun
Published: July 2002

There is a Poincaré embedding structure on the diagonal

X
→
X
×
X

X\rightarrow X\times X

under the conditions: i)

X
X

is formed by gluing two compact smooth manifolds along their boundaries using a homotopy equivalence and ii) a square-root closed condition is satisfied by the fundamental groupoid of the boundary.
A growth dichotomy for o-minimal expansions of ordered groups
Authors: Miller Chris , Starchenko Sergei
Published: July 2002

Let

R

\mathfrak {R}

be an o-minimal expansion of a divisible ordered abelian group

(
R
,
>
,
+
,
0
,
1
)

(R,>,+,0,1)

with a distinguished positive element

1
1

. Then the following dichotomy holds: Either there is a

0
0

-definable binary operation

⋅
\cdot

such that

(
R
,
>
,
+
,
⋅
,
0
,
1
)

(R,>,+,\cdot ,0,1)

is an ordered real closed field; or, for every definable function

f
:
R
→
R

f:R\to R

there exists a

0
0

-definable

λ
∈
{
0
}
∪
Aut
⁡
(
R
,
+
)

\lambda \in \{0\}\cup \operatorname {Aut}(R,+)

with

lim

x
→
+
∞

[
f
(
x
)
−
λ
(
x
)
]
∈
R

\lim _{x\to +\infty }[f(x)-\lambda (x)]\in R

. This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure

M

:=
(
M
,
>
,
…
)

\mathfrak {M}:=(M,>,\dots )

there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order)

M

\mathfrak {M}

-definable groups with underlying set

M
M

.
An algorithm for calculating the Nielsen number on surfaces with boundary
Authors: Wagner Joyce
Published: July 2002

Let

f
:
M
→
M

f:M\to M

be a self-map of a hyperbolic surface with boundary. The Nielsen number,

N
(
f
)

N(f)

, depends only on the induced map

f

#

f_{\#}

of the fundamental group, which can be viewed as a free group on

n
n

generators,

a
1

,
…
,

a
n

a_1,\dotsc ,a_n

. We determine conditions for fixed points to be in the same fixed point class and if these conditions are enough to determine the fixed point classes, we say that

f

#

f_{\#}

is

W
W

-characteristic. We define an algebraic condition on the

f

#

(

a
i

)

f_{\#}(a_i)

and show that “most” maps satisfy this condition and that all maps which satisfy this condition are

W
W

-characteristic. If

f

#

f_{\#}

is

W
W

-characteristic, we present an algorithm for calculating

N
(
f
)

N(f)

and prove that the inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper L left-parenthesis f right-parenthesis minus chi left-parenthesis upper M right-parenthesis EndAbsoluteValue less-than-or-equal-to upper N left-parenthesis f right-parenthesis minus chi left-parenthesis upper M right-parenthesis">

|

L
(
f
)
−
χ
(
M
)

|

≤
N
(
f
)
−
χ
(
M
)

|L(f)-\chi (M)|\le N(f)-\chi (M)

holds, where

L
(
f
)

L(f)

denotes the Lefschetz number of

f
f

and

χ
(
M
)

\chi (M)

the Euler characteristic of

M
M

, thus answering in part a question of Jiang and Guo.