Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems

Many problems in Physics are described by dynamical systems that are conformally symplectic (e.g., mechanical systems with a friction proportional to the velocity, variational problems with a small discount or thermostated systems). Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. The limit of small dissipation, which is the object of the present study, is particularly interesting. We provide all details for maps, but we present also the modifications needed to obtain a direct proof for the case of differential equations. We consider a family of conformally symplectic maps $f_{\mu, \epsilon}$ defined on a $2d$-dimensional symplectic manifold $\mathcal M$ with exact symplectic form $\Omega$; we assume that $f_{\mu,\epsilon}$ satisfies $f_{\mu,\epsilon}^*\Omega=\lambda(\epsilon) \Omega$. We assume that the family depends on a $d$-dimensional parameter $\mu$ (called drift) and also on a small scalar parameter $\epsilon$. Furthermore, we assume that the conformal factor $\lambda$ depends on $\epsilon$, in such a way that for $\epsilon=0$ we have $\lambda(0)=1$ (the symplectic case). We study the domains of analyticity in $\epsilon$ near $\epsilon=0$ of perturbative expansions (Lindstedt series) of the parameterization of the quasi--periodic orbits of frequency $\omega$ (assumed to be Diophantine) and of the parameter $\mu$. Notice that this is a singular perturbation, since any friction (no matter how small) reduces the set of quasi-periodic solutions in the system. We prove that the Lindstedt series are analytic in a domain in the complex $\epsilon$ plane, which is obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The radii of the excluded balls decrease faster than any power of the distance of the center to the origin

Genome-wide profiling of uncapped mRNA

Gene transcripts are under extensive posttranscriptional regulation, including the regulation of their stability. A major route for mRNA degradation produces uncapped mRNAs, which can be generated by decapping enzymes, endonucleases, and small RNAs. Profiling uncapped mRNA molecules is important for the understanding of the transcriptome, whose composition is determined by a balance between mRNA synthesis and degradation. In this chapter, we describe a method to profile these uncapped mRNAs at the genome scale

Health of Philippine Emigrants Study (HoPES): study design and rationale.

BackgroundImmigrants to the United States are usually healthier than their U.S.-born counterparts, yet the health of immigrants declines with duration of stay in the U.S. This pattern is often seen for numerous health problems such as obesity, and is usually attributed to acculturation (the adoption of "American" behaviors and norms). However, an alternative explanation is secular trends, given that rates of obesity have been rising globally. Few studies of immigrants are designed to distinguish the effects of acculturation versus secular trends, in part because most studies of immigrants are cross-sectional, lack baseline data prior to migration, and do not have a comparison group of non-migrants in the country of origin. This paper describes the Health of Philippine Emigrants Study (HoPES), a study designed to address many of these limitations.MethodsHoPES is a dual-cohort, longitudinal, transnational study. The first cohort consisted of Filipinos migrating to the United States (n = 832). The second cohort consisted of non-migrant Filipinos who planned to remain in the Philippines (n = 805). Baseline data were collected from both cohorts in 2017 in the Philippines, with follow-up data collection planned over 3 years in either the U.S. for the migrant cohort or the Philippines for the non-migrant cohort. At baseline, interviewers administered semi-structured questionnaires that assessed demographic characteristics, diet, physical activity, stress, and immigration experiences. Interviewers also measured weight, height, waist and hip circumferences, blood pressure, and collected dried blood spot samples.DiscussionMigrants enrolled in the study appear to be representative of recent Filipino migrants to the U.S. Additionally, migrant and non-migrant study participants are comparable on several characteristics that we attempted to balance at baseline, including age, gender, and education. HoPES is a unique study that approximates a natural experiment from which to study the effects of immigration on obesity and other health problems. A number of innovative methodological strategies were pursued to expand the boundaries of current immigrant health research. Key to accomplishing this research was investment in building collaborative relationships with stakeholders across the U.S. and the Philippines with shared interest in the health of migrants

Whiskered KAM tori of conformally symplectic systems

We investigate the existence of whiskered tori in some dissipative systems, called \sl conformally symplectic \rm systems, having the property that they transform the symplectic form into a multiple of itself. We consider a family $f_\mu$ of conformally symplectic maps which depend on a drift parameter $\mu$. We fix a Diophantine frequency of the torus and we assume to have a drift $\mu_0$ and an embedding of the torus $K_0$, which satisfy approximately the invariance equation $f_{\mu_0} \circ K_0 - K_0 \circ T_\omega$ (where $T_\omega$ denotes the shift by $\omega$). We also assume to have a splitting of the tangent space at the range of $K_0$ into three bundles. We assume that the bundles are approximately invariant under $D f_{\mu_0}$ and that the derivative satisfies some "rate conditions". Under suitable non-degeneracy conditions, we prove that there exists $\mu_\infty$, $K_\infty$ and splittings, close to the original ones, invariant under $f_{\mu_\infty}$. The proof provides an efficient algorithm to construct whiskered tori. Full details of the statements and proofs are given in [CCdlL18].Comment: 15 pages, 1 figur

Boundaries of Siegel Disks: Numerical Studies of their Dynamics and Regularity

Siegel disks are domains around fixed points of holomorphic maps in which the maps are locally linearizable (i.e., become a rotation under an appropriate change of coordinates which is analytic in a neighborhood of the origin). The dynamical behavior of the iterates of the map on the boundary of the Siegel disk exhibits strong scaling properties which have been intensively studied in the physical and mathematical literature. In the cases we study, the boundary of the Siegel disk is a Jordan curve containing a critical point of the map (we consider critical maps of different orders), and there exists a natural parametrization which transforms the dynamics on the boundary into a rotation. We compute numerically this parameterization and use methods of harmonic analysis to compute the global Holder regularity of the parametrization for different maps and rotation numbers. We obtain that the regularity of the boundaries and the scaling exponents are universal numbers in the sense of renormalization theory (i.e., they do not depend on the map when the map ranges in an open set), and only depend on the order of the critical point of the map in the boundary of the Siegel disk and the tail of the continued function expansion of the rotation number. We also discuss some possible relations between the regularity of the parametrization of the boundaries and the corresponding scaling exponents. (C) 2008 American Institute of Physics.NSFMathematic

Origin of morphological depressions on the Guadalquivir Bank uplifted area (Gulf of Cadiz middle slope)

We have investigated the origin of morphological depressions (circular-elliptical depressions, amphitheatre-shaped escarpments and valleys) on the Guadalquivir Bank uplifted area (Gulf of Cadiz middle slope). This work is based on swath bathymetry and high- and mid-resolution reflection seismic datasets. Depressions occur on the distal (depositional) sector of the Gulf of Cadiz Contourite Depositional System, which has been developed under the influence of the Mediterranean Outflow Water (MOW). The Guadalquivir Bank is a NE-oriented relief that was uplifted along the Neogene and Quaternary. It forms the southern limit of the Bartolomeu Dias and Faro Sheeted Drift (SD) plateaus that are separated by the NW-trending Diego Cao Contourite Channel. Circular-elliptical depressions occur on the Bartolomeu Dias SD plateau, aligned parallel to the rim of the Diego Cao Channel. Irregular, crescent-shaped depressions occur to the SE of the study area and a valley surrounds the Guadalquivir Bank. The origin of these features is interpreted as the result of the interplay between oceanographic, mass-wasting, tectonic and fluid-escape processes. Four stages define the development of these features: 1) Onset of a contourite mounded drift associated with a proto-Diego Cao moat originated by a weak MOW circulation as it interacted with the structural features of the Guadalquivir Bank during the Lower Pliocene; 2) Evolution to a more complex multi-crest drift and moat system, probably as a result of an enhanced MOW and increased deformation of the underlying structures during the Upper Pliocene-Early Quaternary; 3) Event of enhanced tectonic activity that provoked widespread mass-wasting events along middle slope sheeted drift plateaus during the Mid Pleistocene. It was recorded in a prominent erosive surface under the present-day Diego Cao channel western rim and numerous slide scars displaying amphitheatre shapes on the limits of the plateaus; 4) Final stage (Late Quaternary) when the Mediterranean Intermediate Branch started flowing towards the N-NW along the deep gateway that was opened as a result of the mass-wasting event and/or structural adjustments. The contourite system evolved, due to tectonic events, to the present-day channel and a complex separated drift that includes circular depressions. They result from the interaction between the bottom current and the irregular basal surface created by the slide scars. During this phase, crescent-shaped depressions were created, probably by the interplay between bottom currents and fluid escape processes, and the marginal valley around the Guadalquivir Bank resulted from current reworking of the irregular topography of contouritic deposits affected by slide scars

Linearization of Cohomology-free Vector Fields

We study the cohomological equation for a smooth vector field on a compact manifold. We show that if the vector field is cohomology free, then it can be embedded continuously in a linear flow on an Abelian group

The Destruction of Tori in Volume-Preserving Maps

Invariant tori are prominent features of symplectic and volume preserving maps. From the point of view of chaotic transport the most relevant tori are those that are barriers, and thus have codimension one. For an $n$-dimensional volume-preserving map, such tori are prevalent when the map is nearly "integrable," in the sense of having one action and $n-1$ angle variables. As the map is perturbed, numerical studies show that the originally connected image of the frequency map acquires gaps due to resonances and domains of nonconvergence due to chaos. We present examples of a three-dimensional, generalized standard map for which there is a critical perturbation size, $\epsilon_c$, above which there are no tori. Numerical investigations to find the "last invariant torus" reveal some similarities to the behavior found by Greene near a critical invariant circle for area preserving maps: the crossing time through the newly destroyed torus appears to have a power law singularity at $\epsilon_c$, and the local phase space near the critical torus contains many high-order resonances.Comment: laTeX, 16 figure