DSpace Collection:
http://hdl.handle.net/2117/3229
Thu, 23 Oct 2014 11:56:05 GMT20141023T11:56:05Zwebmaster.bupc@upc.eduUniversitat Politècnica de Catalunya. Servei de Biblioteques i DocumentaciónoLyubeznik numbers of local rings and linear strands of graded ideals
http://hdl.handle.net/2117/24185
Title: Lyubeznik numbers of local rings and linear strands of graded ideals
Authors: Álvarez Montaner, Josep; Yanagawa, Kohji
Abstract: n this work we intro duce a new set of invariants asso ciated to the linear
strands of a minimal free resolution of a
Z
graded ideal
I
R
=

[
x
1
;:::;x
n
]
. We
also prove that these invariants satisfy some prop erties analogous to those of Lyub eznik
numb ers of lo cal rings. In particular, they satisfy a consecutiveness prop erty that we
prove rst for Lyub eznik numb ers. For the case of squarefree monomial ideals we get
more insight on the relation b etween Lyub eznik numb ers and the linear strands of their
asso ciated Alexander dual ideals. Finally, we prove that Lyub eznik numb ers of Stanley
Reisner rings are not only an algebraic invariant but also a top ological invariant, meaning
that they dep end on the homeomorphic class of the geometric realization of the asso ciated
simplicial complex and the characteristic of the base fieldTue, 30 Sep 2014 09:46:24 GMThttp://hdl.handle.net/2117/2418520140930T09:46:24ZÁlvarez Montaner, Josep; Yanagawa, Kohjinon this work we intro duce a new set of invariants asso ciated to the linear
strands of a minimal free resolution of a
Z
graded ideal
I
R
=

[
x
1
;:::;x
n
]
. We
also prove that these invariants satisfy some prop erties analogous to those of Lyub eznik
numb ers of lo cal rings. In particular, they satisfy a consecutiveness prop erty that we
prove rst for Lyub eznik numb ers. For the case of squarefree monomial ideals we get
more insight on the relation b etween Lyub eznik numb ers and the linear strands of their
asso ciated Alexander dual ideals. Finally, we prove that Lyub eznik numb ers of Stanley
Reisner rings are not only an algebraic invariant but also a top ological invariant, meaning
that they dep end on the homeomorphic class of the geometric realization of the asso ciated
simplicial complex and the characteristic of the base fieldA methodology for obtaining asymptotic estimates for the exponentially small splitting of separatrices to whiskered tori with quadratic frequencies
http://hdl.handle.net/2117/24155
Title: A methodology for obtaining asymptotic estimates for the exponentially small splitting of separatrices to whiskered tori with quadratic frequencies
Authors: Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, PereThu, 25 Sep 2014 08:12:02 GMThttp://hdl.handle.net/2117/2415520140925T08:12:02ZDelshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, PerenoContinuation of the exponentially small lower bounds for the splitting of separatrices to a whiskered torus with silver ratio
http://hdl.handle.net/2117/24138
Title: Continuation of the exponentially small lower bounds for the splitting of separatrices to a whiskered torus with silver ratio
Authors: Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
Abstract: We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast
frequencies in nearlyintegrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega=\sqrt21$. We show that the oincareMelnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide
asymptotic estimates for the tranversality of the splitting whose dependence on the perturbation parameter $\varepsilon$ satisffies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilonTue, 23 Sep 2014 09:33:01 GMThttp://hdl.handle.net/2117/2413820140923T09:33:01ZDelshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Perenotransverse homoclinic orbits, splitting of separatrices, Melnikov integrals, silver ratioWe study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast
frequencies in nearlyintegrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega=\sqrt21$. We show that the oincareMelnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide
asymptotic estimates for the tranversality of the splitting whose dependence on the perturbation parameter $\varepsilon$ satisffies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilonCàlcul numèric. Manual de pràctiques
http://hdl.handle.net/2117/24072
Title: Càlcul numèric. Manual de pràctiques
Authors: Lázaro Ochoa, José Tomás; Ollé Torner, Mercè; Pacha Andújar, Juan RamónWed, 17 Sep 2014 07:45:15 GMThttp://hdl.handle.net/2117/2407220140917T07:45:15ZLázaro Ochoa, José Tomás; Ollé Torner, Mercè; Pacha Andújar, Juan RamónnoDecomposition spaces, incidence algebras and Möbius inversion
http://hdl.handle.net/2117/23130
Title: Decomposition spaces, incidence algebras and Möbius inversion
Authors: Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, AndrewTue, 03 Jun 2014 08:45:51 GMThttp://hdl.handle.net/2117/2313020140603T08:45:51ZGálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, AndrewnoAlgebraic topology, CombinatoricsExponentially small lower bounds for the splitting of separatrices to whiskered tori with frequencies of constant type
http://hdl.handle.net/2117/22687
Title: Exponentially small lower bounds for the splitting of separatrices to whiskered tori with frequencies of constant type
Authors: Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
Abstract: We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearlyintegrable
Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We
consider a 2dimensional torus with a fast frequency vector $\omega/v\epsilon$, with $\epsilon=(1,\Omega)$ where $\Omega$ is an irrational number of constant type, i.e. a number whose continued fraction has bounded
entries. Applying the Poincar´e–Melnikov method, we find exponentially small lower bounds for
the maximal splitting distance between the stable and unstable invariant manifolds associated to
the invariant torus, and we show that these bounds depend strongly on the arithmetic properties
of the frequencies.Thu, 24 Apr 2014 09:32:24 GMThttp://hdl.handle.net/2117/2268720140424T09:32:24ZDelshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Perenosplitting of separatrices, Melnikov integrals, numbers of constant typeWe study the splitting of invariant manifolds of whiskered tori with two frequencies in nearlyintegrable
Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We
consider a 2dimensional torus with a fast frequency vector $\omega/v\epsilon$, with $\epsilon=(1,\Omega)$ where $\Omega$ is an irrational number of constant type, i.e. a number whose continued fraction has bounded
entries. Applying the Poincar´e–Melnikov method, we find exponentially small lower bounds for
the maximal splitting distance between the stable and unstable invariant manifolds associated to
the invariant torus, and we show that these bounds depend strongly on the arithmetic properties
of the frequencies.Generalized CliffordSeveri inequality and the volume of irregular varieties
http://hdl.handle.net/2117/22384
Title: Generalized CliffordSeveri inequality and the volume of irregular varieties
Authors: Barja Yáñez, Miguel Ángel
Abstract: We give a sharp lower bound for the selfintersection of a nef li
ne bundle L on an irregular variety X in terms of its continuous global sections and the Albanese dim
ension of X, which we call the Generalized CliffordSeveri inequality. We also extend the result to nef
vector bundles and give a slope inequality for fibred irregular varieties. As a byproduct we obtain a lower b
ound for the volume of irregular varieties; when X is of maximal Albanese dimension the bound is vol(X)=2n!¿¿X
and it is sharp.
Description: Preprint. Acceptat per publicar a Duke Math. J.Tue, 25 Mar 2014 20:15:26 GMThttp://hdl.handle.net/2117/2238420140325T20:15:26ZBarja Yáñez, Miguel ÁngelnoSeveri inequality
Slope
maximal Albanese Varieties
VolumeWe give a sharp lower bound for the selfintersection of a nef li
ne bundle L on an irregular variety X in terms of its continuous global sections and the Albanese dim
ension of X, which we call the Generalized CliffordSeveri inequality. We also extend the result to nef
vector bundles and give a slope inequality for fibred irregular varieties. As a byproduct we obtain a lower b
ound for the volume of irregular varieties; when X is of maximal Albanese dimension the bound is vol(X)=2n!¿¿X
and it is sharp.Local description of phylogenetic groupbased models
http://hdl.handle.net/2117/22383
Title: Local description of phylogenetic groupbased models
Authors: Casanellas Rius, Marta; Fernández Sánchez, Jesús; Michalek, Mateusz
Abstract: Motivated by phylogenetics, our aim is to obtain a system of equations that
de ne a phylogenetic variety on an open set containing the biologically meaningful points. In
this paper we consider phylogenetic varieties de ned via groupbased models. For any nite
abelian group G, we provide an explicit construction of codimX phylogenetic invariants
(polynomial equations) of degree at most jGj that de ne the variety X on a Zariski open set
U. The set U contains all biologically meaningful points when G is the group of the Kimura
3parameter model. In particular, our main result con rms [Mic12, Conjecture 7.9] and, on
the set U, Conjectures 29 and 30 of [SS05].Tue, 25 Mar 2014 19:26:25 GMThttp://hdl.handle.net/2117/2238320140325T19:26:25ZCasanellas Rius, Marta; Fernández Sánchez, Jesús; Michalek, Mateusznogroupbased model, phylogenetic invariant, toric varietyMotivated by phylogenetics, our aim is to obtain a system of equations that
de ne a phylogenetic variety on an open set containing the biologically meaningful points. In
this paper we consider phylogenetic varieties de ned via groupbased models. For any nite
abelian group G, we provide an explicit construction of codimX phylogenetic invariants
(polynomial equations) of degree at most jGj that de ne the variety X on a Zariski open set
U. The set U contains all biologically meaningful points when G is the group of the Kimura
3parameter model. In particular, our main result con rms [Mic12, Conjecture 7.9] and, on
the set U, Conjectures 29 and 30 of [SS05].Testing of the three multiplicatively closed (Lie Markov) model heirarchies which respect purine/pyrimidine, WatsonCrick, and amino/keto nucleotide groupings
http://hdl.handle.net/2117/22382
Title: Testing of the three multiplicatively closed (Lie Markov) model heirarchies which respect purine/pyrimidine, WatsonCrick, and amino/keto nucleotide groupings
Authors: Woodhams, Michael D.; Fernández Sánchez, Jesús; Sumner, Jeremy
Abstract: We present three hierarchies of Lie Markov models of DNA sequence evolution. These models are
(locally) “multiplicatively closed,” meaning that the composition of two Markov matrices in the
model results, with some (rare) exceptions, in a third Markov matrix that is still in the model.
Additionally, the models in each hierarchy respectively distinguish between (i) purines and pyrimadines
(RY), (ii) WatsonCrick pairs (WS), and (iii) amino/keto pairs (MK), but otherwise treat
the four nucleotides without distinction. The multiplicative closure property allows mathematically
consistent modeling of timeinhomogeneous scenarios, unlike models that are not closed, such
as the general timereversible model (GTR) and many of its submodels. We derive the nesting
relationships of the three model hierarchies and present software implementing the models. For a
diverse range of biological data sets, we perform Bayesian information criterion model comparision
analogous to that of the ModelTest framework. We find that our models outperform the GTR
model in some (but not all) cases.Tue, 25 Mar 2014 17:59:41 GMThttp://hdl.handle.net/2117/2238220140325T17:59:41ZWoodhams, Michael D.; Fernández Sánchez, Jesús; Sumner, JeremynoLie Markov models, multiplicative closure, DNA evolutionWe present three hierarchies of Lie Markov models of DNA sequence evolution. These models are
(locally) “multiplicatively closed,” meaning that the composition of two Markov matrices in the
model results, with some (rare) exceptions, in a third Markov matrix that is still in the model.
Additionally, the models in each hierarchy respectively distinguish between (i) purines and pyrimadines
(RY), (ii) WatsonCrick pairs (WS), and (iii) amino/keto pairs (MK), but otherwise treat
the four nucleotides without distinction. The multiplicative closure property allows mathematically
consistent modeling of timeinhomogeneous scenarios, unlike models that are not closed, such
as the general timereversible model (GTR) and many of its submodels. We derive the nesting
relationships of the three model hierarchies and present software implementing the models. For a
diverse range of biological data sets, we perform Bayesian information criterion model comparision
analogous to that of the ModelTest framework. We find that our models outperform the GTR
model in some (but not all) cases.Lie Markov models with purine/pyrimidine symmetry
http://hdl.handle.net/2117/22381
Title: Lie Markov models with purine/pyrimidine symmetry
Authors: Fernández Sánchez, Jesús; Sumner, Jeremy; Jarvis, Peter; Woodhams, Michael D.
Abstract: Continuoustime Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying timeindependent rates of substitutions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situation. If a model is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with timedependent rates satisfying the same constraints. It is then useful to require that there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of "Lie Markov models", which are precisely the class of models where such an average exists. These models form Lie algebras and hence concepts from Lie group theory are central to their derivation. In this paper, we concentrate on applications to phylogenetics and nucleotide evolution, and derive the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines  that is, models with purine/pyrimidine symmetry. We also discuss how to handle the subtleties of applying Lie group methods, most naturally defined over the complex field, to the stochastic case of a Markov process, where parameter values are restricted to be real and positive. In particular, we explore the geometric embedding of the cone of stochastic rate matrices within the ambient space of the associated complex Lie algebra.Tue, 25 Mar 2014 17:13:58 GMThttp://hdl.handle.net/2117/2238120140325T17:13:58ZFernández Sánchez, Jesús; Sumner, Jeremy; Jarvis, Peter; Woodhams, Michael D.noevolutionary model, group representation theory, Lie algebraContinuoustime Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying timeindependent rates of substitutions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situation. If a model is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with timedependent rates satisfying the same constraints. It is then useful to require that there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of "Lie Markov models", which are precisely the class of models where such an average exists. These models form Lie algebras and hence concepts from Lie group theory are central to their derivation. In this paper, we concentrate on applications to phylogenetics and nucleotide evolution, and derive the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines  that is, models with purine/pyrimidine symmetry. We also discuss how to handle the subtleties of applying Lie group methods, most naturally defined over the complex field, to the stochastic case of a Markov process, where parameter values are restricted to be real and positive. In particular, we explore the geometric embedding of the cone of stochastic rate matrices within the ambient space of the associated complex Lie algebra.Periodic orbits of planar integrable birational maps
http://hdl.handle.net/2117/21748
Title: Periodic orbits of planar integrable birational maps
Authors: Gálvez Carrillo, Maria Immaculada; Mañosa Fernández, Víctor
Abstract: A birational planar map F possessing a rational ﬁrst integral preserves a
foliation of the plane given by algebraic curves which, if F is not globally periodic,
is given by a foliation of curves that have generically genus 0 or 1. In the genus 1
case, the group structure of the foliation characterizes the dynamics of any birational
map preserving it. We will see how to take advantage of this structure to ﬁnd periodic
orbits of such maps.Tue, 25 Feb 2014 12:09:41 GMThttp://hdl.handle.net/2117/2174820140225T12:09:41ZGálvez Carrillo, Maria Immaculada; Mañosa Fernández, VíctornoDiscrete dynamical systems, Algebraic geometry, Birational maps, Integrable maps, Elliptic curves, Periodic orbits.A birational planar map F possessing a rational ﬁrst integral preserves a
foliation of the plane given by algebraic curves which, if F is not globally periodic,
is given by a foliation of curves that have generically genus 0 or 1. In the genus 1
case, the group structure of the foliation characterizes the dynamics of any birational
map preserving it. We will see how to take advantage of this structure to ﬁnd periodic
orbits of such maps.Symplectic topology of bsymplectic manifolds
http://hdl.handle.net/2117/21516
Title: Symplectic topology of bsymplectic manifolds
Authors: Miranda Galcerán, Eva; Martinez Torres, David; Frejlich, Pedro
Abstract: A Poisson manifold (M2n; ) is bsymplectic if
Vn is transverse
to the zero section. In this paper we apply techniques of Symplectic Topology
to address global questions pertaining to bsymplectic manifolds. The main
results provide constructions of: bsymplectic submanifolds a la Donaldson,
bsymplectic structures on open manifolds by Gromov's hprinciple, and of
bsymplectic manifolds with a prescribed singular locus, by means of surgeries.Tue, 11 Feb 2014 13:34:06 GMThttp://hdl.handle.net/2117/2151620140211T13:34:06ZMiranda Galcerán, Eva; Martinez Torres, David; Frejlich, PedronoA Poisson manifold (M2n; ) is bsymplectic if
Vn is transverse
to the zero section. In this paper we apply techniques of Symplectic Topology
to address global questions pertaining to bsymplectic manifolds. The main
results provide constructions of: bsymplectic submanifolds a la Donaldson,
bsymplectic structures on open manifolds by Gromov's hprinciple, and of
bsymplectic manifolds with a prescribed singular locus, by means of surgeries.An extension problem for the CR fractional Laplacian
http://hdl.handle.net/2117/21457
Title: An extension problem for the CR fractional Laplacian
Authors: Frank, Rupert L.; González Nogueras, María del Mar; Monticelli, Dario D.; Tan, Jinggang
Abstract: We show that the conformally invariant fractional powers of the subLaplacian
on the Heisenberg group are given in terms of the scattering operator for an extension
problem to the Siegel upper halfspace. Remarkably, this extension problem is di erent
from the one studied, among others, by Ca arelli and SilvestreWed, 05 Feb 2014 10:52:39 GMThttp://hdl.handle.net/2117/2145720140205T10:52:39ZFrank, Rupert L.; González Nogueras, María del Mar; Monticelli, Dario D.; Tan, JinggangnoWe show that the conformally invariant fractional powers of the subLaplacian
on the Heisenberg group are given in terms of the scattering operator for an extension
problem to the Siegel upper halfspace. Remarkably, this extension problem is di erent
from the one studied, among others, by Ca arelli and SilvestreSingularities for a fully nonlinear elliptic equation in conformal geometry
http://hdl.handle.net/2117/20686
Title: Singularities for a fully nonlinear elliptic equation in conformal geometry
Authors: González Nogueras, María del Mar; Mazzieri, Lorenzo
Abstract: We construct some radially symmetric solutions of the constant
k
equation on
R
n
n
R
p
, which blow
up exactly at the submanifold
R
p
R
n
. These are the basic models to the problem of nding
complete metrics of constant
k
{curvature on a general subdomain of the sphere
S
n
n
p
that blow up
exactly at the singular set
p
and that are conformal to the canonical metric. More precisely, we look
at the case
k
= 2 and 0
<p<
p
2
:=
n
p
n
2
2
. The main result is the understanding of the precise
asymptotics of our solutions near the singularity and their decay away from the singularity. The rst
aspect will insure the completeness of the metric about the singular locus, whereas the second aspect
will guarantee that the model solutions can be locally transplanted to the original metric on
S
n
, and
hence they can be used to deal with the general problem on
S
n
n
pThu, 21 Nov 2013 10:52:41 GMThttp://hdl.handle.net/2117/2068620131121T10:52:41ZGonzález Nogueras, María del Mar; Mazzieri, LorenzonoWe construct some radially symmetric solutions of the constant
k
equation on
R
n
n
R
p
, which blow
up exactly at the submanifold
R
p
R
n
. These are the basic models to the problem of nding
complete metrics of constant
k
{curvature on a general subdomain of the sphere
S
n
n
p
that blow up
exactly at the singular set
p
and that are conformal to the canonical metric. More precisely, we look
at the case
k
= 2 and 0
<p<
p
2
:=
n
p
n
2
2
. The main result is the understanding of the precise
asymptotics of our solutions near the singularity and their decay away from the singularity. The rst
aspect will insure the completeness of the metric about the singular locus, whereas the second aspect
will guarantee that the model solutions can be locally transplanted to the original metric on
S
n
, and
hence they can be used to deal with the general problem on
S
n
n
pInstability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
http://hdl.handle.net/2117/20671
Title: Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
Authors: Delshams Valdés, Amadeu; de la Llave, Rafael; MartínezSeara Alonso, M. Teresa
Abstract: Abstract.
We consider models given by Hamiltonians of the form
H
(
I;';p;q;t
;
"
) =
h
(
I
)+
n
X
j
=1
1
2
p
2
j
+
V
j
(
q
j
)
+
"Q
(
I;';p;q;t
;
"
)
where
I
2I
R
d
;'
2
T
d
,
p;q
2
R
n
,
t
2
T
1
. These are higher di
mensional analogues, both in the center and hyperbolic directions,
of the models studied in [DLS03, DLS06a, GL06a, GL06b]. All
these models present the
large gap problem
.
We show that, for 0
< "
1, under regularity and explicit non
degeneracy conditions on the model, there are orbits whose action
variables
I
perform rather arbitrary excursions in a domain of size
O
(1). This domain includes resonance lines and, hence, large gaps
among
d
dimensional KAM tori.
The method of proof follows closely the strategy of [DLS03,
DLS06a]. The main new phenomenon that appears when the di
mension
d
of the center directions is larger than one, is the exis
tence of multiple resonances. We show that, since these multiple
resonances happen in sets of codimension greater than one in the
space of actions
I
, they can be contoured. This corresponds to
the mechanism called
di usion across resonances
in the Physics
literature.
The present paper, however, di ers substantially from [DLS03,
DLS06a]. On the technical details of the proofs, we have taken
advantage of the theory of the scattering map [DLS08], not avail
able when the above papers were written. We have analyzed the
conditions imposed on the resonances in more detail.
More precisely, we have found that there is a simple condition
on the Melnikov potential which allows us to conclude that the res
onances are crossed. In particular, this condition does not depend
on the resonances. So that the results are new even when applied
to the models in [DLS03, DLS06a]Wed, 20 Nov 2013 12:48:04 GMThttp://hdl.handle.net/2117/2067120131120T12:48:04ZDelshams Valdés, Amadeu; de la Llave, Rafael; MartínezSeara Alonso, M. TeresanoAbstract.
We consider models given by Hamiltonians of the form
H
(
I;';p;q;t
;
"
) =
h
(
I
)+
n
X
j
=1
1
2
p
2
j
+
V
j
(
q
j
)
+
"Q
(
I;';p;q;t
;
"
)
where
I
2I
R
d
;'
2
T
d
,
p;q
2
R
n
,
t
2
T
1
. These are higher di
mensional analogues, both in the center and hyperbolic directions,
of the models studied in [DLS03, DLS06a, GL06a, GL06b]. All
these models present the
large gap problem
.
We show that, for 0
< "
1, under regularity and explicit non
degeneracy conditions on the model, there are orbits whose action
variables
I
perform rather arbitrary excursions in a domain of size
O
(1). This domain includes resonance lines and, hence, large gaps
among
d
dimensional KAM tori.
The method of proof follows closely the strategy of [DLS03,
DLS06a]. The main new phenomenon that appears when the di
mension
d
of the center directions is larger than one, is the exis
tence of multiple resonances. We show that, since these multiple
resonances happen in sets of codimension greater than one in the
space of actions
I
, they can be contoured. This corresponds to
the mechanism called
di usion across resonances
in the Physics
literature.
The present paper, however, di ers substantially from [DLS03,
DLS06a]. On the technical details of the proofs, we have taken
advantage of the theory of the scattering map [DLS08], not avail
able when the above papers were written. We have analyzed the
conditions imposed on the resonances in more detail.
More precisely, we have found that there is a simple condition
on the Melnikov potential which allows us to conclude that the res
onances are crossed. In particular, this condition does not depend
on the resonances. So that the results are new even when applied
to the models in [DLS03, DLS06a]