bluuebunny/crossref_metadata_2025 huggingface dataset:
Iterated integrals and higher order automorphic forms
Authors: Diamantis Nikolaos , Sreekantan Ramesh
Published: December 2009

Higher order automorphic forms have recently been introduced to study important questions in number theory and mathematical physics. We investigate the connection between these functions and Chen's iterated integrals. Then using Chen's theory, we prove a structure theorem for automorphic forms of all orders. This allows us to define an analogue of a mixed Hodge structure on a space of higher order automorphic forms.
Growth exponent of generic groups
Authors: Ollivier Yann
Published: December 2009

In [GrH97], Grigorchuk and de la Harpe ask for conditions under which some group presentations have growth rate close to that of the free group with the same number of generators. We prove that this property holds for a generic group (in the density model of random groups). Namely, for every positive

\epsilon

, the property of having growth exponent at least

1-\epsilon

(in base

2m-1

where

m

is the number of generators) is generic in this model. In particular this extends a theorem of Shukhov [Shu99]. More generally, we prove that the growth exponent does not change much through a random quotient of a torsion-free hyperbolic group.
Semi-continuity of the first $\ell^2$-Betti number on the space of finitely generated groups
Authors: Pichot Michael
Published: December 2009

To each finitely generated group is associated a sequence

(\beta_0, \beta_1,\beta_2,\dots)

of non negative real numbers, its

\ell^2

-Betti numbers. On the other hand, the set of finitely generated groups
desingularizes
into a usual topological space, the space

\operatorname{MG}

of finitely generated
marked
groups. It is proved in this note that if a sequence

\Gamma_n\in \operatorname{MG}

of marked groups converges to a group

\Gamma

, then

\overline\lim \beta_1(\Gamma_n)\leq \beta_1(\Gamma)

.
Higher-order linking forms for knots
Authors: Leidy Constance
Published: December 2009

We construct examples of knots that have isomorphic

n

th-order Alexander modules, but non-isomorphic

n

th-order linking forms, showing that the linking forms provide more information than the modules alone. This generalizes work of Trotter [T], who found examples of knots that have isomorphic classical Alexander modules, but non-isomorphic classical Blanchfield linking forms.
Some Novikov rings that are von Neumann finite and knot-like groups
Authors: Kochloukova Dessislava H.
Published: December 2009

We show that for a finitely generated group

G

and for every discrete character

\chi\colon G \rightarrow \mathbb{Z}

any matrix ring over the Novikov ring

\widehat{\mathbb{Z}G}_{\chi}

is von Neumann finite. As a corollary we obtain that if

G

is a non-trivial discrete group with a finite

K(G,1)

CW-complex

Y

of dimension

n

and Euler characteristics zero and

N

is a normal subgroup of

G

of type

FP_{n-1}

containing the commutator subgroup

G'

and such that

G/N

is cyclic-by-finite, then

N

is of homological type

FP_n

and

G/N

has finite virtual cohomological dimension

vcd(G/N) = cd(G) - cd(N).

This completes the proof of the Rapaport Strasser conjecture that for a knot-like group

G

with a finitely generated commutator subgroup

G'

the commutator subgroup

G'

is always free and generalises an earlier work by the author where the case when

G'

is residually finite was proved. Another corollary is that a finitely presentable group

G

with

def(G) > 0

and such that

G'

is finitely generated and perfect can be only

\mathbb{Z}

or

\mathbb{Z}^2

, a result conjectured by A. J. Berrick and J. Hillman in [1].
Isometric immersions into 3-dimensional homogeneous manifolds
Authors: Daniel Benoît
Published: December 2009

We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous Riemannian manifold with a 4-dimensional isometry group. The condition is expressed in terms of the metric, the second fundamental form, and data arising from an ambient Killing field. This class of 3-manifolds includes in particular the Berger spheres, the Heisenberg group Nil3, the universal cover of the Lie group PSL2(R) and the product spaces S2×R and H2×R. We give some applications to constant mean curvature (CMC) surfaces in these manifolds; in particular we prove the existence of a generalized Lawson correspondence, i.e., a local isometric correspondence between CMC surfaces in homogeneous 3-manifolds.
A fixed point theorem for deformation spaces of $G$-trees
Authors: Clay Matt T.
Published: December 2009

For a finitely generated free group

F_n

, of rank at least 2, any finite subgroup of

\operatorname{Out}(F_n)

can be realized as a group of automorphisms of a graph with fundamental group

F_n

. This result, known as

\operatorname{Out}(F_n)

realization, was proved by Zimmermann, Culler and Khramtsov. This theorem is comparable to Nielsen realization as proved by Kerckhoff: for a closed surface with negative Euler characteristic, any finite subgroup of the mapping class group can be realized as a group of isometries of a hyperbolic surface. Both of these theorems have restatements in terms of fixed points of actions on spaces naturally associated to

\operatorname{Out}(F_n)

and the mapping class group respectively. For a nonnegative integer

n

we define a class of groups

(\textit{GVP}(n))

and prove a similar statement for their outer automorphism groups.
Laminar free hyperbolic 3-manifolds
Authors: Fenley Sérgio R.
Published: December 2009

The purpose of the article is to prove that there are infinitely many closed hyperbolic 3-manifolds which do not admit essential laminations. The manifolds are obtained by Dehn surgery on torus bundles over the circle. This gives a definitive negative answer to a fundamental question posed by Gabai and Oertel when they introduced essential laminations. The proof is obtained by analysing group actions on on trees and showing that certain 3-manifold groups only have trivial actions on trees. There are corollaries concerning the existence of Reebless foliations and pseudo-Anosov flows.
Topologie locale des espaces de feuilletages en surfaces des variétés fermées
de dimension 3
Authors: Larcanché Audrey
Published: December 2009

Dans cet article, on montre que, sur toute variété fermée et orientable de dimension 3, deux feuilletages de codimension 1 qui sont orientables, tendus et suffisamment proches sont homotopes parmi des feuilletages présentant (en général) des composantes de Reeb.
The isomorphism problem for residually torsion-free nilpotent groups
Authors: Baumslag Gilbert , Miller III Charles F.
Published: December 2009

Both the conjugacy and isomorphism problems for finitely generated nilpotent groups are recursively solvable. In some recent work, the first author, with a tiny modification of work in the second author's thesis, proved that the conjugacy problem for finitely presented, residually torsion-free nilpotent groups is recursively unsolvable. Here we complete the algorithmic picture by proving that the isomorphism problem for such groups is also recursively unsolvable.
Reflections on the residual finiteness of one-relator groups
Authors: Baumslag Gilbert , Miller III Charles F. , Troeger Douglas
Published: December 2009

Let G = < a, b, … | r = 1 > be a one-relator group equipped with at least two generators. For all w which do not commute with r in the ambient free group on the generators a, b, …, the groups G(r,w) = < a,b, … | r r w = r 2 > are not residually finite and have the same finite images as G. The existence of this family of one-relator groups which are not residually finite reinforces what is becoming more obvious with time, that one-relator groups can be extremely complicated. This not only serves to underline the complexity of one-relator groups but provides us with the opportunity to raise a number of problems about these groups in the hope that they will stimulate further work on the conjugacy and isomorphism problems for one-relator groups as a whole.
Elementary abelian 2-subgroups of Sidki-type in finite groups
Authors: Aschbacher Michael , Guralnick Robert M. , Segev Yoav
Published: December 2009

Let

G

be a finite group. We say that a nontrivial elementary abelian 2-subgroup

V

of

G

is of
Sidki-type
in

G

, if for each involution

i

in

G

,

C_V(i) ≠ 1

. A conjecture due to S. Sidki (J. Algebra 39, 1976) asserts that if

V

is of Sidki-type in

G

, then

V ∩ O_2(G) ≠ 1

. In this paper we prove a stronger version of Sidki's conjecture. As part of the proof, we also establish weak versions of the saturation results of G. Seitz (Invent. Math. 141, 2000) for involutions in finite groups of Lie type in characteristic

2

. Seitz's results apply to elements of order

p

in groups of Lie type in characteristic

p

, but only when

p

is a good prime, and

2

is usually not a good prime.
Virtual endomorphisms of nilpotent groups
Authors: Berlatto Adilson A. , Sidki Said N.
Published: December 2009

A virtual endomorphism of a group

G

is a homomorphism

f : H→ G

where

H

is a subgroup of

G

of finite index

m

. The triple

(G,H,f)

produces a state-closed (or, self-similar) representation

φ

of

G

on the

1

-rooted

m

-ary tree. This paper is a study of properties of the image

G^φ

when

G

is nilpotent. In particular, it is shown that if

G

is finitely generated, torsion-free and nilpotent then

G^φ

has solvability degree bounded above by the number of prime divisors of

m

.
On kernels of cellular covers
Authors: Farjoun Emmanuel D. , Göbel Rüdiger , Segev Yoav , Shelah Saharon
Published: December 2009

In the present paper we continue to examine cellular covers of groups, focusing on the cardinality and the structure of the kernel

K

of the cellular map

G → M

. We show that in general a torsion free reduced abelian group

M

may have a proper class of non-isomorphic cellular covers. In other words, the cardinality of the kernels is unbounded. In the opposite direction we show that if the kernel of a cellular cover of any group

M

has certain “freeness” properties, then its cardinality is bounded by

|M|

.
Rips construction and Kazhdan property (T)
Authors: Belegradek Igor , Osin Denis
Published: December 2009

We show that for any non-elementary hyperbolic group

H

and any finitely presented group

Q

, there exists a short exact sequence

1 → N → G → Q → 1

, where

G

is a hyperbolic group and

N

is a quotient group of

H

. As an application we construct a hyperbolic group that has the same

n

-dimensional complex representations as a given finitely generated group, show that adding relations of the form

x^n = 1

to a presentation of a hyperbolic group may drastically change the group even in case

n ≫ 1

, and prove that some properties (e.g. properties (T) and FA) are not recursively recognizable in the class of hyperbolic groups. A relatively hyperbolic version of this theorem is also used to generalize results of Ollivier–Wise on outer automorphism groups of Kazhdan groups.
Finiteness of arithmetic hyperbolic reflection groups
Authors: Agol Ian , Belolipetsky Mikhail , Storm Peter , Whyte Kevin
Published: December 2009

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.
Infinite conjugacy classes in groups acting on trees
Authors: de Cornulier Yves
Published: December 2009

We characterize amalgams and HNN extensions with infinite conjugacy classes.
Combination of quasiconvex subgroups of relatively hyperbolic groups
Authors: Martínez-Pedroza Eduardo
Published: December 2009

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two quasiconvex subgroups

Q_1

and

Q_2

is quasiconvex and isomorphic to

Q_1 ∗_{Q_1 ∩ Q_2} Q_2

. Our results generalize known combination theorems for quasiconvex subgroups of word hyperbolic groups. Some applications are presented.
Copies of one-ended groups in mapping class groups
Authors: Dahmani François , Fujiwara Koji
Published: December 2009

We establish that, given Σ a compact orientable surface and G a finitely presented one-ended group, the set of copies of G in the mapping class group MCG(Σ) consisting of only pseudo-Anosov elements except identity is finite up to conjugacy. This relies on a result of Bowditch on the same problem for images of surfaces groups. He asked us whether we could reduce the case of one-ended groups to his result; this is a positive answer. Our work involves analogues of Rips and Sela’s canonical cylinders in curve complexes and an argument of Delzant to bound the number of images of a group in a hyperbolic group.
Square-free words as products of commutators
Authors: Duncan Andrew J. , Vdovina Alina
Published: December 2009

Elements of the commutator subgroup of a free group

\mathbb F

can be presented as values of canonical forms, called Wicks forms. We show that, starting from sufficiently high genus

g

, there is a sequence of words

w_g

which can be presented by

f(g)

distinct Wicks forms, where

f(g) > g!

. Moreover we may choose these words

w_g

to be square-free.
Subequivalence relations and positive-definite functions
Authors: Ioana Adrian , Kechris Alexander S. , Tsankov Todor
Published: December 2009

We study a positive-definite function associated with a countable, measure-preserving equivalence relation, which can be used to measure quantitatively the proximity of subequivalence relations. Combined with a co-inducing construction introduced by Epstein and earlier work of Ioana, this can be used to construct many mixing actions of countable groups and establish the non-classifiability, in a strong sense, of orbit equivalence of actions of non-amenable groups. We also discuss connections with percolation on Cayley graphs and the theory of costs.
A hyperbolic Out($F_n$)-complex
Authors: Bestvina Mladen , Feighn Mark
Published: December 2009

For any finite collection

f_i

of fully irreducible automorphisms of the free group

F_n

we construct a connected

δ

-hyperbolic Out(

F_n

)-complex in which each

f_i

has positive translation length.
Stable commutator length in word-hyperbolic groups
Authors: Calegari Danny , Fujiwara Koji
Published: December 2009

In this paper we obtain uniform positive lower bounds on the stable commutator length of elements in word-hyperbolic groups and certain groups acting on hyperbolic spaces (namely the mapping class group acting on the complex of curves, and an amalgamated free product acting on an associated Bass–Serre tree). If G is a word-hyperbolic group that is δ-hyperbolic with respect to a symmetric generating set S, then there is a positive constant C depending only on δ and on |S| such that every element of G either has a power which is conjugate to its inverse, or else the stable commutator length of the element is at least equal to C. By Bavard’s theorem, these lower bounds on stable commutator length imply the existence of quasimorphisms with uniform control on the defects; however, we show how to construct such quasimorphisms directly.  We also prove various separation theorems on families of elements in such groups, constructing homogeneous quasimorphisms (again with uniform estimates) which are positive on some prescribed element while vanishing on some family of independent elements whose translation lengths are uniformly bounded.  Finally, we prove that the first accumulation point for stable commutator length in a torsion-free word-hyperbolic group is contained between 1/12 and 1/2. This gives a universal sense of what it means for a conjugacy class in a hyperbolic group to have a small stable commutator length, and can be thought of as a kind of “homological Margulis lemma”.
Connectedness at infinity of systolic complexes and groups
Authors: Osajda Damian
Published: December 2009

By studying connectedness at infinity of systolic groups we distinguish them from some other classes of groups, in particular from the fundamental groups of manifolds covered by Euclidean space of dimension at least three. We also study semistability at infinity for some systolic groups.
A quasilinear parabolic singular perturbation problem
Authors: Lederman Claudia , Oelz Dietmar
Published: December 2009

We study a singular perturbation problem for a quasilinear uniformly parabolic operator of interest in combustion theory. We obtain uniform estimates, we pass to the limit and we show that, under suitable assumptions, the limit function

u

is a solution to the free boundary problem

{\rm div } F(\nabla u)-\partial_{t}u=0

in

\{ u>0 \}

,

u_\nu=\alpha(\nu, M)

on

\partial\{ u>0 \}

, in a pointwise sense and in a viscosity sense. Here

\nu

is the inward unit spatial normal to the free boundary

\partial\{ u>0 \}

and

M

is a positive constant. Some of the results obtained are new even when the operator under consideration is linear.
On a phase-ﬁeld model for electrowetting
Authors: Eck Christof , Fontelos Marco Antonio , Grün Günther , Klingbeil F. , Vantzos O.
Published: December 2009

The term electrowetting is commonly used for phenomena where shape and wetting behavior of liquid droplets are changed by the application of electric ﬁelds. We develop and analyze a model for electrowetting that combines the Navier–Stokes system for ﬂuid ﬂow, a phase-ﬁeld model of Cahn–Hilliard type for the movement of the interface, a charge transport equation, and the potential equation of electrostatics. The model is derived with the help of a variational principle due to Onsager and conservation laws. A modiﬁcation of the model with the Stokes system instead of the Navier– Stokes system is also presented. The existence of weak solutions is proved for several cases in two and three space dimensions, either with non-degenerate or with degenerate electric conductivity vanishing in the droplet exterior. Some numerical examples in two space dimensions illustrate the applicability of the model.
A multi-phase transition model for dislocations with interfacial microstructure
Authors: Cacace Simone , Garroni Adriana
Published: December 2009

We study, by means of
Γ
-convergence, the asymptotic behavior of a variational model for dislocations moving on a slip plane. The variational functional is a two-dimensional multi-phase transition-type energy given by a nonlocal term and a nonlinear potential which penalizes noninteger values for the components of the phase. In the limit we obtain an anisotropic sharp interfaces model. The relevant feature of this problem is that optimal sequences in general are not given by a onedimensional proﬁle, but they can create microstructure.
An anti-classification theorem for ergodic measure preserving transformations
Authors: Foreman Matthew , Weiss Benjamin
Published: December 2009

Despite many notable advances the general problem of classifying ergodic measure preserving transformations (MPT) has remained wide open. We show that the action of the whole group of MPT's on ergodic actions by conjugation is turbulent in the sense of G. Hjorth. The type of classifications ruled out by this property include countable algebraic objects such as those that occur in the Halmos-von Nemann theorem classifying ergodic MPT's with pure point spectrum. We treat both the classical case of

\bf Z

as well as the case of general countable amenable groups.
Giant component and vacant set for random walk on a discrete torus
Authors: Benjamini Itai , Sznitman Alain-Sol
Published: December 2009

We consider random walk on a discrete torus

E

of side-length

N

, in sufficiently high dimension

d

. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time

uN^d

. We show that when

u

is chosen small, as

N

tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const

\log N

. Moreover, this connected component occupies a non-degenerate fraction of the total number of sites

N^d

of

E

, and any point of

E

lies within distance

N^\beta

of this component, with

\beta

an arbitrary positive number.
Finite projective planes, Fermat curves, and Gaussian periods
Authors: Thas Koen , Zagier Don B.
Published: December 2009

One of the oldest and most fundamental problems in the theory of finite projective planes is to classify those having a group which acts transitively on the incident point-line pairs (flags). The conjecture is that the only ones are the Desarguesian projective planes (over a finite field). In this paper, we show that non-Desarguesian finite flag-transitive projective planes exist if and only if certain Fermat surfaces have no nontrivial rational points, and formulate several other equivalences involving Fermat curves and Gaussian periods. In particular, we show that a non-Desarguesian flag-transitive projective plane of order

n

exists if and only if

n>8

, the number

p=n^2+n+1

is prime, and the square of the absolute value of the Gaussian period

\sum_{a\in\mathcal D_n}\zeta^a\,

(

\zeta=

primitive

p

th root of unity,

\mathcal D_n=

group of

n

th powers in

\mathbb F_p^\times

) belongs to

\mathbb Z

. We also formulate a conjectural classification of all pairs

(p,n)

with

p

prime and

n\mid p-1

having this latter property, and give an application to the construction of symmetric designs with flag-transitive automorphism groups. Numerical computations are described verifying the first conjecture for

p<4\times10^{22}

and the second for

p<10^7

.
Homological category weights and estimates for $\mathrm{cat}^1(X,\xi)$
Authors: Farber Michael , Schütz Dirk
Published: December 2009

In this paper we study a new notion of category weight of homology classes developing further the ideas of E. Fadell and S. Husseini. In the case of closed smooth manifolds the homological category weight is equivalent to the cohomological category weight of E. Fadell and S. Husseini but these two notions are distinct already for Poincaré complexes. An important advantage of the homological category weight is its homotopy invariance. We use the notion of homological category weight to study various generalizations of the Lusternik - Schnirelmann category which appeared in the theory of closed one-forms and have applications in dynamics. Our primary goal is to compare two such invariants

\mathrm{cat}(X,\xi)

and

\mathrm{cat}^1(X,\xi)

which are defined similarly with reversion of the order of quantifiers. We compute these invariants explicitly for products of surfaces and show that they may differ by an arbitrarily large quantity. The proof of one of our main results uses an algebraic characterization of homology classes

z\in H_i(\tilde X;\mathbb{Z})

(where

\tilde X\to X

is a free abelian covering) which are movable to infinity of

\tilde X

with respect to a prescribed cohomology class

\xi\in H^1(X;\mathbb{R})

. This result is established in Part II which can be read independently of the rest of the paper.
Infinitesimal automorphisms and deformations of parabolic geometries
Authors: Cap Andreas
Published: December 2009

We show that infinitesimal automorphisms and infinitesimal deformations of parabolic geometries can be nicely described in terms of the twisted de Rham sequence associated to a certain linear connection on the adjoint tractor bundle. For regular normal geometries, this description can be related to the underlying geometric structure using the machinery of BGG sequences. In the locally flat case, this leads to a deformation complex, which generalizes the well known complex for locally conformally flat manifolds.

Recently, a theory of subcomplexes in BGG sequences has been developed. This applies to certain types of torsion free parabolic geometries including quaternionic structures, quaternionic contact structures and CR structures. We show that for these structures one of the subcomplexes in the adjoint BGG sequence leads (even in the curved case) to a complex governing deformations in the subcategory of torsion free geometries. For quaternionic structures, this deformation complex is elliptic.
Rational symplectic field theory over $\mathbb{Z}_2$ for exact Lagrangian cobordisms
Authors: Ekholm Tobias
Published: December 2009

We construct a version of rational Symplectic Field Theory for pairs

(X,L)

, where

X

is an exact symplectic manifold, where

L\subset X

is an exact Lagrangian submanifold with components subdivided into

k

subsets, and where both

X

and

L

have cylindrical ends. The theory associates to

(X,L)

a

\Z

-graded chain complex of vector spaces over

\Z_2

, filtered with

k

filtration levels. The corresponding

k

-level spectral sequence is invariant under deformations of

(X,L)

and has the following property: if

(X,L)

is obtained by joining a negative end of a pair

(X',L')

to a positive end of a pair

(X'',L'')

, then there are natural morphisms from the spectral sequences of

(X',L')

and of

(X'',L'')

to the spectral sequence of

(X,L)

. As an application, we show that if

\Lambda\subset Y

is a Legendrian submanifold of a contact manifold then the spectral sequences associated to

(Y\times\R,\Lambda_k^s\times\R)

, where

Y\times\R

is the symplectization of

Y

and where

\Lambda_k^s\subset Y

is the Legendrian submanifold consisting of

s

parallel copies of

\Lambda

subdivided into

k

subsets, give Legendrian isotopy invariants of

\Lambda

.
Lieb–Thirring inequalities on the half-line with critical exponent
Authors: Ekholm Tomas , Frank Rupert L.
Published: December 2009

We consider a Schrödinger operator on the half-line with a Dirichlet boundary condition at the origin and show that moments of its negative eigenvalues can be estimated by the part of the potential that is larger than the critical Hardy weight. The estimate is valid for the critical value of the moment parameter.
Horocyclic products of trees
Authors: Bartholdi Laurent , Neuhauser Markus , Woess Wolfgang
Published: December 2009

Let

T_1,\dots, T_d

be homogeneous trees with degrees

q_1+1, \dots, q_d+1 \ge 3,

respectively. For each tree, let

\mathfrak h:T_j \to \mathbb Z

be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of

T_1,\dots, T_d

is the graph

\mathsf{DL}(q_1,\dots,q_d)

consisting of all

d

-tuples

x_1 \cdots x_d \in T_1 \times \dots \times T_d

with

\mathfrak h(x_1)+\dots+\mathfrak h(x_d)=0

, equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If

d=2

and

q_1=q_2=q

then we obtain a Cayley graph of the lamplighter group (wreath product)

\mathfrak Z_q \wr \mathbb Z

. If

d = 3

and

q_1 = q_2 = q_3 = q

then

\mathsf{DL}

is the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also when

d\ge 4

and

q_1 = \dots = q_d = q

is such that each prime power in the decomposition of

q

is larger than

d-1

, we show that

\mathsf{DL}

is a Cayley graph of a finitely presented group. This group is of type

F_{d-1}

, but not

F_d

. It is not automatic, but it is an automata group in most cases. On the other hand, when the

q_j

do not all coincide,

\mathsf{DL}(q_1,\dots,q_d)

is a vertex-transitive graph, but is not the Cayley graph of a finitely generated group. Indeed, it does not even admit a group action with finitely many orbits and finite point stabilizers. The

\ell^2

-spectrum of the “simple random walk” operator on

\mathsf{DL}

is always pure point. When

d=2

, it is known explicitly from previous work, while for

d=3

we compute it explicitly. Finally, we determine the Poisson boundary of a large class of group-invariant random walks on

\mathsf{DL}

. It coincides with a part of the geometric boundary of

\mathsf{DL}

.
Hardy's uncertainty principle, convexity and Schrödinger evolutions
Authors: Escauriaza Luis , Kenig Carlos E. , Ponce Gustavo , Vega Luis
Published: December 2009

We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrödinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy's version of the uncertainty principle. We also obtain corresponding results for heat evolutions.
Some aspects of the variational nature of mean curvature flow
Authors: Bellettini Giovanni , Mugnai Luca
Published: December 2009

We show that the classical solution of the heat equation can be seen as the minimizer of a suitable functional defined in space-time. Using similar ideas, we introduce a functional

\mathcal F

on the class of space-time tracks of moving hypersurfaces, and we study suitable minimization problems related with

\mathcal F

. We show some connections between minimizers of

\mathcal F

and mean curvature flow.
Multiplication operators on $L(L_p)$ and $\ell_p$-strictly singular operators
Authors: Johnson William B. , Schechtman Gideon
Published: December 2009

A classification of weakly compact multiplication operators on

L(L_p)

,

1<p<\infty

, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of

\ell_p

-strictly singular operators, and we also investigate the structure of general

\ell_p

-strictly singular operators on

L_p

. The main result is that if an operator

T

on

L_p

,

1<p<2

, is

\ell_p

-strictly singular and

T_{|X}

is an isomorphism for some subspace

X

of

L_p

, then

X

embeds into

L_r

for all

r<2

, but

X

need not be isomorphic to a Hilbert space. It is also shown that if

T

is convolution by a biased coin on

L_p

of the Cantor group,

1\le p <2

, and

T_{|X}

is an isomorphism for some reflexive subspace

X

of

L_p

, then

X

is isomorphic to a Hilbert space. The case

p=1

answers a question asked by Rosenthal in 1976.
Convergence of singular integrals with general measures
Authors: Mattila Pertti , Verdera Joan
Published: December 2009

We show that

L^2

-bounded singular integrals in metric spaces with respect to general measures and kernels converge weakly. This implies a kind of average convergence almost everywhere. For measures with zero density we prove the almost everywhere existence of principal values.
The regular inverse Galois problem over non-large fields
Authors: Koenigsmann Jochen
Published: December 2009

By a celebrated theorem of Harbater and Pop, the regular inverse Galois problem is solvable over any field containing a large field. Using this and the Mordell conjecture for function fields, we construct the first example of a field

K

over which the regular inverse Galois problem can be shown to be solvable, but such that

K

does not contain a large field. The paper is complemented by model-theoretic observations on the diophantine nature of the regular inverse Galois problem.
Convergence of a two-grid algorithm for the control of the wave equation
Authors: Ignat Liviu I. , Zuazua Enrique
Published: December 2009

We analyze the problem of boundary observability of the finite-difference space semi-discretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the mesh-size) boundary observability for the solutions obtained by the two-grid preconditioner introduced by Glowinski [9]. Our method uses previously known uniform observability inequalities for low frequency solutions and a dyadic spectral time decomposition. As a consequence we prove the convergence of the two-grid algorithm for computing the boundary controls for the wave equation. The method can be applied in any space dimension, for more general domains and other discretization schemes.
Cycles on algebraic models of smooth manifolds
Authors: Kucharz Wojciech
Published: December 2009

Every compact smooth manifold

M

is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of

M

. We study modulo

2

homology classes represented by algebraic subsets of

X

, as

X

runs through the class of all algebraic models of

M

. Our main result concerns the case where

M

is a spin manifold.
Which 3-manifold groups are Kähler groups?
Authors: Dimca Alexandru , Suciu Alexander I.
Published: December 2009

The question in the title, first raised by Goldman and Donaldson, was partially answered by Reznikov. We give a complete answer, as follows: if
G
can be realized as both the fundamental group of a closed 3-manifold and of a compact Kähler manifold, then
G
must be finite—and thus belongs to the well-known list of finite subgroups of O(4), acting freely on S3.
Existence of rational points on smooth projective varieties
Authors: Poonen Bjorn
Published: December 2009

Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and also an algorithm for computing X(k) for any k-variety X for which X(k) is finite. The proof involves the construction of a one-parameter algebraic family of Châtelet surfaces such that exactly one of the surfaces fails to have a k-point.
Positive solutions for nonlinear Schrödinger equations with deepening potential well
Authors: Wang Zhengping , Zhou Huan-Song
Published: December 2009

Consider the following nonlinear Schrödinger equation:

\tag{$\ast$}
-Δu + (1 + λg(x))u = f(u) \text{ and } u> 0 \text{ in } ℝ^N,\ u \in H^1(ℝ^N),\ N ≥ 3,

where

λ ≥ 0

is a parameter,

g \in L^∞(ℝ^N)

vanishes on a bounded domain in

ℝ^N

, and the function

f

is such that

\lim_{s→0} f(s)/s = 0 \text{ and } 1 ≤ α + 1 = \lim_{s→∞} f(s)/s < ∞.

We are interested in whether problem (

\ast

) has a solution for any given

α, λ ≥ 0

. It is shown in [14] and [31] that problem (

\ast

) has solutions for some

α

and

λ

. In this paper, we establish the existence of solution of (

\ast

) for all

α

and

λ

by using a variant of the Mountain Pass Theorem. Based on these results, we give a diagram in the

(λ,α)

-plane showing how the solvability of problem (

\ast

) depends on the parameters

α

and

λ

.
Limits of Calabi–Yau metrics when the Kähler class degenerates
Authors: Tosatti Valentino
Published: December 2009

We study the behavior of families of Ricci-ﬂat Kähler metrics on a projective Calabi– Yau manifold when the Kähler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-ﬂat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ricci-ﬂat metric. The limit can also be understood from algebraic geometry.
Saddle-shaped solutions of bistable diffusion equations in all of $\mathbb{R}^{2m}$
Authors: Cabré Xavier , Terra Joana
Published: December 2009

We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation −∆
u
=
f
(
u
) in the whole ℝ2_m_, where
f
is of bistable type. It is known that in dimension 2_m_ = 2 there exists a saddle-shaped solution. This is a solution which changes sign in ℝ2 and vanishes only on {|

x_1 | = |
x_2 |}. It is also known that this solution is unstable. In this article we prove the existence of saddle-shaped solutions in every even dimension, as well as their instability in the case of dimension 2_m
= 4. More precisely, our main result establishes that if 2_m

= 4, every solution vanishing on the Simons cone {(_x_1, _x_2) ∈ ℝ2 × ℝ2 : |_x_1| = |_x_2|} is unstable outside every compact set and, as a consequence, has inﬁnite Morse index. These results are relevant in connection with a conjecture of De Giorgi extensively studied in recent years and for which the existence of a counter-example in high dimensions is still an open problem.
Holomorphic functions and subelliptic heat kernels over Lie groups
Authors: Driver Bruce K. , Gross Leonard , Saloff-Coste Laurent
Published: December 2009

A Hermitian form

q

on the dual space,

\mathfrak g^*

, of the Lie algebra,

\mathfrak g

, of a Lie group,

G

, determines a sub-Laplacian,

\Delta

, on

G

. It will be shown that Hörmander’s condition for hypoellipticity of the sub-Laplacian holds if and only if the associated Hermitian form, induced by

q

on the dual of the universal enveloping algebra,

\mathcal U'

, is non-degenerate. The subelliptic heat semigroup,

e^{t\Delta/4}

, is given by convolution by a

C^\infty

probability density

\rho_t

. When

G

is complex and

u: G \to \mathbb C

is a holomorphic function, the collection of derivatives of

u

at the identity in

G

gives rise to an element,

\hat u(e)\in \mathcal U'

. We will show that if

G

is complex, connected, and simply connected then the “Taylor” map,

u\mapsto \hat u(e)

, deﬁnes a unitary map from the space of holomorphic functions in

L^2(G, \rho_t)

onto a natural Hilbert space lying in

\mathcal U'

.
Resonant normal form for even periodic FPU chains
Authors: Henrici Andreas , Kappeler Thomas
Published: December 2009

We investigate periodic FPU chains with an even number of particles. We show that near the equilibrium point, any such chain admits a resonant Birkhoff normal form of order four which is
completely integrable
—an important fact which helps explain the numerical experiments of Fermi, Pasta, and Ulam. We analyze the moment map of the integrable approximation of an even FPU chain. Unlike the case of odd FPU chains these integrable systems (generically) exhibit hyperbolic dynamics. As an application we prove that any FPU chain with Dirichlet boundary conditions admits a Birkhoff normal form up to order four and show that a KAM theorem applies.
Local rigidity in quaternionic hyperbolic space
Authors: Kim Inkang , Pansu Pierre
Published: December 2009

We study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more ﬂexible in quaternionic hyperbolic plane than in complex hyperbolic plane.
Almost localization and almost reducibility
Authors: Avila Artur , Jitomirskaya Svetlana
Published: December 2009

We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schrödinger cocycles associated to a non-perturbatively small analytic potential and Diophantine frequency. In particular, we establish the full version of Eliasson’s reducibility theory in this regime (our approach actually leads to improvements even in the perturbative regime: we are able to show, for all energies, “almost reducibility” in some band of analyticity). We also prove 1/2-Hölder continuity of the integrated density of states. For the almost Mathieu operator, our results hold through the entire regime of subcritical coupling and imply also the dry version of the Ten Martini Problem for the relevant parameters.
Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity
Authors: Ambrosetti Antonio , Felli Veronica , Malchiodi Andrea
Published: December 2009

We deal with a class on nonlinear Schrödinger equations (NLS) with potentials

V(x)\sim |x|^{-\alpha}

,

0<\alpha<2

, and

K(x)\sim |x|^{-\beta}

,

\beta>0

. Working in weighted Sobolev spaces, the existence of ground states

v_{\varepsilon}

belonging to

W^{1,2}(\mathbb R^n)

is proved under the assumption that

\sigma<p<(N+2)/(N-2)

for some

\sigma=\sigma_{N,\alpha,\beta}

. Furthermore, it is shown that

v_{\varepsilon}

are
spikes
concentrating at a minimum of

{\mathcal A}=V^{\theta}K^{-2/(p-1)}

, where

\theta= (p+1)/(p-1)-1/2

.
Recovering an algebraic curve using its projections from different points. Applications to static and dynamic computational vision
Authors: Kaminski Jeremy Yirmeyahu , Fryers Michael , Teicher Mina
Published: December 2009

We study some geometric configurations related to the projection of an irreducible algebraic curve embedded in

\mathbb C \mathbb P^3

onto embedded projective planes. These configurations are motivated by applications to static and dynamic computational vision. More precisely, we study how an irreducible closed algebraic curve

X

embedded in

\mathbb C \mathbb P^3

, which degree is

d

and genus

g

, can be recovered using its projections from points onto embedded projective planes. The different embeddings are unknown. The only input is the defining equation of each projected curve. We show how both the embeddings and the curve in

\mathbb C \mathbb P^3

can be recovered modulo some action of the group of projective transformations of

\mathbb C \mathbb P^3

. In particular in the case of two projections, we show how in a generic situation, a characteristic matrix of the pair of embeddings can be recovered. In the process we address dimensional issues and as a result establish the minimal number of irreducible algebraic curves required to compute this characteristic matrix up to a finite-fold ambiguity, as a function of their degrees and genus. Then we use this matrix to recover the class of the couple of maps and as a consequence to recover the curve. For a generic situation, two projections define a curve with two irreducible components. One component has degree

d(d-1)

and the other has degree

d

, being the original curve. Then we consider another problem.

N

projections, with known projections operators and

N >> 1

, are considered as an input and we want to recover the curve. The recovery can be done by linear computations in the dual space and in the Grassmannian of lines in

\mathbb C \mathbb P^3

. Those computations are respectively based on the dual variety and on the variety of intersecting lines. In both cases a simple lower bound for the number of necessary projections is given as a function of the degree and the genus. A closely related question is also considered. Each point of a finite closed subset of an irreducible algebraic curve, is projected onto a plane from a point. For each point the center of projection is different. The projections operators are known. We show when and how the recovery of the algebraic curve is possible, in function of the degree of the curve, and of the degree of the curve of minimal degree generated by the centers of projection. Eventually we show how these questions were motivated by applications to static and dynamic computational vision. A second part of this work is devoted to several applications to this field. The results in this paper solve a long standing problem in computer vision that could not have been solved without algebraic-geometric methods.
Mean curvature properties for p-Laplace phase transitions
Authors: Sciunzi Berardino , Valdinoci Enrico
Published: December 2009

This paper deals with phase transitions corresponding to an energy which is the sum of a kinetic part of

p

-Laplacian type and a double well potential

h_0

with suitable growth conditions. We prove that level sets of solutions of

\Delta_p u=h_0'(u)

possessing a certain decay property satisfy a mean curvature equation in a suitable weak viscosity sense. From this, we show that, if the above level sets approach uniformly a hypersurface, the latter has zero mean curvature.
Arbitrary Number of Positive Solutions For an Elliptic Problem with Critical Nonlinearity
Authors: Rey Olivier , Wei Juncheng
Published: December 2009

We show that the critical nonlinear elliptic Neumann problem

\Delta u -\mu u + u^{7/3} = 0 \text{ in } \Omega,\ u >0 \ \text{ in } \Omega \text{ and } \frac{\partial u}{\partial \nu} = 0 \text{ on } \partial \Omega

where

\Omega

is a bounded and smooth domain in

\R^5

, has arbitrarily many solutions, provided that

\mu>0

is small enough. More precisely, for any positive integer

K

, there exists

\mu_K >0

such that for

0 <\mu < \mu_K

, the above problem has a nontrivial solution which blows up at

K

interior points in

\Omega

, as

\mu \to 0

. The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, geometry nor topology of the domain is needed.
A remark on the bifurcation diagrams of superlinear elliptic equations
Authors: Bahri Abbas
Published: December 2009

We prove a formula relating the index of a solution and the rotation number of a certain complex vector along bifurcation diagrams.
On the principal eigenvalue of elliptic operators in $\R^N$ and applications
Authors: Berestycki Henri , Rossi Luca
Published: December 2009

Two generalizations of the notion of principal eigenvalue for elliptic operators in

\R^N

are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; self-adjoint operators; and “limit periodic” operators. These results apply to questions of existence and uniqueness for some semi-linear problems in all of space. We also indicate several outstanding open problems and formulate some conjectures.
Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials
Authors: Byeon Jaeyoung , Wang Zhi-Qiang
Published: December 2009

For singularly perturbed Schrödinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeroes of the potentials. The results generalize some recent work of Ambrosetti–Malchiod–Ni [3] which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in [3].
A maximum principle for mean-curvature type elliptic inequalities
Authors: Serrin James
Published: December 2009

Consider the divergence structure elliptic inequality

\tag{1}
\mathrm{div}\{\boldsymbol A(x,u,Du)\} + B(x,u,Du) \ge 0

in a bounded domain

\Omega\subset \mathbb R^n

. Here

\boldsymbol A(x,z,\boldsymbol \xi): \,K \to \mathbb R^n, \qquad B(x,z,\boldsymbol \xi): K \to \mathbb R, \qquad K = \Omega \times \mathbb R^+ \times \mathbb R^n,

and

\boldsymbol A

,

B

satisfy the following conditions

\langle\boldsymbol A(\boldsymbol \xi) , \boldsymbol \xi \rangle \ge |\xi| - c(x)z - a(x), \qquad |\boldsymbol A(x,z,\boldsymbol \xi)| \le \text{Const.},\qquad
B(x,z,\boldsymbol \xi) \le b(x),

for all

(x,z,\boldsymbol \xi)) \in K

, where

a(x)

,

b(x),\ c(x)

are given non-negative functions. Our interest is in the validity of the maximum principle for solutions of (1), that is, the statement that

any solution which satisfies

u\le 0

on

\partial\Omega

must be a priori bounded above in

\Omega

.

This question arises, in particular, when one is interested in the mean curvature equation

\mathrm{div}\frac{Du}{\sqrt{1+|Du|^2}} = nH(x).
On sum-product representations in $\Bbb Z_q$
Authors: Chang Mei-Chu
Published: December 2009

The purpose of this paper is to investigate efficient representations of the residue classes modulo

q

, by performing sum and product set operations starting from a given subset

A

of

\mathbb Z_q

. We consider the case of very small sets

A

and composite

q

for which not much seemed known (nontrivial results were recently obtained when

q

is prime or when

\log |A| \sim \log q

). Roughly speaking we show that all residue classes are obtained from a

k

-fold sum of an

r

-fold product set of

A

, where

r\ll \log q

and

\log k \ll \log q

, provided the residue sets

\pi_{q'} (A)

are large for all large divisors

q'

of

q

. Even in the special case of prime modulus

q

, some results are new, when considering large but bounded sets

A

. It follows for instance from our estimates that one can obtain

r

as small as

r\sim\log q/\log|A|

with similar restriction on

k

, something not covered by earlier work of Konyagin and Shparlinski. On the technical side, essential use is made of Freiman's structural theorem on sets with small doubling constant. Taking for

A=H

a possibly very small multiplicative subgroup, bounds on exponential sums and lower bounds on

\min_{a \in \mathbb Z_q^*} \max_{x \in H} \Vert ax/q\Vert

are obtained. This is an extension to the results obtained by Konyagin, Shparlinski and Robinson on the distribution of solutions of

x^m = a

(mod

q)

to composite modulus

q

.