DSpace Collection:
http://hdl.handle.net/2117/3229
2014-09-16T19:26:01ZDecomposition 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, Andrew2014-06-03T08:45:51ZExponentially 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 2-dimensional 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.2014-04-24T09:32:24ZGeneralized Clifford-Severi inequality and the volume of irregular varieties
http://hdl.handle.net/2117/22384
Title: Generalized Clifford-Severi 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 Clifford-Severi 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.2014-03-25T20:15:26ZLocal description of phylogenetic group-based models
http://hdl.handle.net/2117/22383
Title: Local description of phylogenetic group-based 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 group-based 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
3-parameter model. In particular, our main result con rms [Mic12, Conjecture 7.9] and, on
the set U, Conjectures 29 and 30 of [SS05].2014-03-25T19:26:25ZTesting of the three multiplicatively closed (Lie Markov) model heirarchies which respect purine/pyrimidine, Watson-Crick, 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, Watson-Crick, 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) Watson-Crick 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 time-inhomogeneous scenarios, unlike models that are not closed, such
as the general time-reversible 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.2014-03-25T17:59:41ZLie 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: Continuous-time Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying time-independent 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 time-dependent 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.2014-03-25T17:13:58ZPeriodic 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.2014-02-25T12:09:41ZSymplectic topology of b-symplectic manifolds
http://hdl.handle.net/2117/21516
Title: Symplectic topology of b-symplectic manifolds
Authors: Miranda Galcerán, Eva; Martinez Torres, David; Frejlich, Pedro
Abstract: A Poisson manifold (M2n; ) is b-symplectic if
Vn is transverse
to the zero section. In this paper we apply techniques of Symplectic Topology
to address global questions pertaining to b-symplectic manifolds. The main
results provide constructions of: b-symplectic submanifolds a la Donaldson,
b-symplectic structures on open manifolds by Gromov's h-principle, and of
b-symplectic manifolds with a prescribed singular locus, by means of surgeries.2014-02-11T13:34:06ZAn 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 sub-Laplacian
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 Silvestre2014-02-05T10:52:39ZSingularities for a fully non-linear elliptic equation in conformal geometry
http://hdl.handle.net/2117/20686
Title: Singularities for a fully non-linear 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
p2013-11-21T10:52:41ZInstability 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; Martínez-Seara 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]2013-11-20T12:48:04ZToric actions on b-symplectic manifolds
http://hdl.handle.net/2117/20405
Title: Toric actions on b-symplectic manifolds
Authors: Miranda Galcerán, Eva; Pires, Ana Rita; Guillemin, Victor; Scott, Geoffrey
Abstract: We study Hamiltonian actions on b-symplectic manifolds with a focus on the e ective case of half the dimension of the manifold.
In particular, we prove a Delzant-type theorem that classi es these manifolds using polytopes that reside in a certain enlarged and decorated version of the dual of the Lie algebra of the torus. At the end of the
paper we suggest further avenues of study, including an example of a toric action on a
b 2-manifold and applications of our ideas to integrable systems on b-manifolds2013-10-17T16:13:15ZExponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies
http://hdl.handle.net/2117/20156
Title: Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies
Authors: Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
Abstract: We study the splitting of invariant manifolds of whiskered t
ori with two or three frequencies in nearly-integrable
Hamiltonian systems. We consider 2-dimensional tori with a
frequency vector
ω
= (1
,
Ω) where Ω is a quadratic
irrational number, or 3-dimensional tori with a frequency v
ector
ω
= (1
,
Ω
,
Ω
2
) where Ω is a cubic irrational number.
Applying the Poincar ́e–Melnikov method, we find exponentia
lly small asymptotic estimates for the maximal splitting
distance between the stable and unstable manifolds associa
ted to the invariant torus, showing that such estimates
depend strongly on the arithmetic properties of the frequen
cies. In the quadratic case, we use the continued fractions
theory to establish a certain arithmetic property, fulfille
d in 24 cases, which allows us to provide asymptotic estimate
s
in a simple way. In the cubic case, we focus our attention to th
e case in which Ω is the so-called cubic golden number
(the real root of
x
3
+
x
−
1 = 0), obtaining also asymptotic estimates. We point out the
similitudes and differences
between the results obtained for both the quadratic and cubi
c cases.2013-09-18T12:06:15ZLyubeznik table of sequentially Cohen-Macaulay rings
http://hdl.handle.net/2117/19625
Title: Lyubeznik table of sequentially Cohen-Macaulay rings
Authors: Álvarez Montaner, Josep
Abstract: We prove that sequentially Cohen-Macaulay rings in positive characteristic,
as well as sequentially Cohen-Macaulay Stanley-Reisner rings in any characteristic, have
trivial Lyubeznik table. Some other con gurations of Lyubeznik tables are also provided
depending on the de ciency modules of the ring.2013-06-25T08:18:53ZAddendum to "Frobenius and Cartier algebras of Stanley-Reisner rings" [J.Algebra 358 (2012), 162-177]
http://hdl.handle.net/2117/19624
Title: Addendum to "Frobenius and Cartier algebras of Stanley-Reisner rings" [J.Algebra 358 (2012), 162-177]
Authors: Álvarez Montaner, Josep; Yanagawa, Kohji
Abstract: We give a purely combinatorial characterization of complete Stanley-Reisner
rings having a principally generated (equivalently, nitely generated) Cartier algebra2013-06-25T08:12:20Z