Understanding singularities in Cartan's and NSF geometric structures

In this work we establish a relationship between Cartan's geometric approach to third order ODEs and the 3-dim Null Surface Formulation (NSF). We then generalize both constructions to allow for caustics and singularities that necessarily arise in these formalisms.Comment: 22 pages, 2 figure

Perturbation theory and singular perturbations for input-to-state multistable systems on manifolds

We consider the notion of Input-to-State Multistability, which generalizes ISS to nonlinear systems evolving on Riemannian manifolds and possessing a finite number of compact, globally attractive, invariant sets, which in addition satisfy a specific condition of acyclicity. We prove that a parameterized family of dynamical systems whose solutions converge to those of a limiting system inherits such Input-to-State Multistability property from the limiting system in a semi-global practical fashion. A similar result is also established for singular perturbation models whose boundary-layer subsystem is uniformly asymptotically stable and whose reduced subsystem is Input-to-State Multistable. Known results in the theory of perturbations, singular perturbations, averaging, and highly oscillatory control systems, are here generalized to the multistable setting by replacing the classical asymptotic stability requirement of a single invariant set with attractivity and acyclicity of a decomposable invariant one

Input-to-state stability for cascade systems with multiple invariant sets

In a recent paper Angeli and Efimov (2015), the notion of Input-to-State Stability (ISS) has been generalized for systems with decomposable invariant sets and evolving on Riemannian manifolds. In this work, we analyze the cascade interconnection of such ISS systems and we characterize the finest possible decomposition of its invariant set for three different scenarios: 1. the driving system exhibits multistability (convergence to fixed points only); 2. the driving system exhibits multi-almost periodicity (convergence to fixed points as well as periodic and almost-periodic orbits) and the driven system is assumed to be incremental ISS; 3. the driving system exhibits multiperiodicity (convergence to fixed points and periodic orbits) whereas the driven system is ISS in the sense of Angeli and Efimov (2015). Furthermore, we provide marginal results on the backward/forward asymptotic behavior of incremental ISS systems and on the response of a contractive system under asymptotically almost-periodic forcing. Three examples illustrate the potentiality of the proposed framework

Asymmetric Effects of Monetary Policy Easing and Tightening

Monetary policy easing and tightening have asymmetric effects: a policy easing has large effects on prices but small effects on real activity variables. The opposite is found for a policy tightening: large real effects but small effects on prices. Non-linearities are estimated using a new and simple procedure based on linear Structural Vector Autoregressions with exogenous variables (SVARX). We rationalize the results through the lenses of a simple model with downward nominal wage rigidities

HOW NATURAL SELECTION SHAPED DIVERSITY AT IMMUNE RESPONSE GENES AND AUTOIMMUNE RISK ALLELES DURING MAMMALIAN EVOLUTION

ABSTRACT INTRODUCTION Genetic diversity is generated by a combination of different evolutionary processes, including mutation, genetic drift, migration, and natural selection.It is well known that natural selection acts on a specific locus\variant, whereas demographic effects act on all loci in the same way; also the selection is expected to be focused on genomic positions that have a functional role. Importantly, the selected variants targeted by selection may not only have a functional role but can correlate with predisposition or protection to some specific diseases. They can therefore be prioritized in screenings for association with diseases and infections; indeed, genetic variants that are advantageous tend to increase in frequency in the population, while deleterious mutations tend to be eliminated. To identify selection, intra and inter species approaches are usually applied; comparing orthologous genes among different species is a successful approach to detect positive selection acting over long evolutionary timescales; on the other hand, comparing genetic variation within human populations may underline more recent adaptive events. Genes related to immune system are among the most studied genes from an evolutionary point of view: it is now established that infections have been acting as a major selective pressure on humans and, most likely, on all living organisms. Thus, the interactions between hosts and pathogens have shaped the genetic diversity over time on both sides: moreover the continuous arm race between hosts and pathogens creates a competition of co-evolving genes that develop adaptations and counter-adaptations against each other. Therefore, it is important to evaluate the level of genetic variation that determines advantageous phenotypic traits to identify genomic regions/positions underlying diversity and adaptation. My first study was focused on molecules involved in the regulation of T-cell activation. The activation of T lymphocytes is a complex phenomenon that is mediated by the interaction of a number of proteins expressed on the surface of T lymphocytes and antigen presenting cells (APC).Several pathogens have evolved strategies that specifically target these genes to either invade the host or to reduce the response of the immune system. Thus, on one hand,these genes have been engaged in a constant conflict with a large number of pathogens and play a fundamental role during infections; on the other hand, genetic variation at these loci has a potential impact on the development of autoimmune and inflammatory conditions. In the second study I focused on molecules involved in the antigen processing and presentation pathway (APP).Whatever the nature of the presenting molecule, the limited dimension of its cleft makes it impossible for macro molecules to be presented: only fragments (antigens) derived from the lysis of such molecules can be nested in the cleft. This antigenic repertoire is generated by the antigen processing and presentation pathway. Another study focused on the contact system and the molecules involved in this pathway. In particular this pathway represents a link between the coagulation and inflammatory responses, two systems central to host survival in the face of tissue damage and infection. Finally, the last molecules analyzed were the proteins responsible for nucleic acids recognition and the activation of the immune response against virus and bacteria, as the RIG-I like proteins and the AIM-2 like proteins AIM OF THE WORK The aim of all the studies was to investigate the evolutionary history of the genes involved in different pathways, both at the inter- and intra- specific level, and to exploit this information to provide novel insight into the functional role of these molecules in human health and disease. Also i wanted to evaluate the history of autoimmune risk alleles and how they spread in human populations. MATERIALS AND METHODS Mammalian coding sequences were retrieved from the Ensembl website and aligned using the The RevTrans 2.0 utility. For detecting the action of positive selection I used the PAML software with different evolutionary models. Sites under selection were identified using Bayes empirical Bayes and Mixed Effects Model of Evolution analyses.Genotype Data from the Pilot 1 phase of the 1000 Genomes Project were retrieved from the dedicated website for the genes analyzed and for 1,200 randomly selected RefSeq genes (control set) for three populations with different ancestry: European, African and East Asian. These data were used to calculate nucleotide diversity parameters as well as some site frequency spectrum-based statistics. Data from the control gene set were used to calculate empirical distributions of these parameters. FST, a measure of population genetic differentiation, and the DIND test, a test based on haplotype homozygosity, were calculated for all SNPs analyzed. I calculated statistical significance of these tests by obtaining an empirical distribution for variants located within the control genes. RESULTS AND DISCUSSION In my analyses I found that genes involved in the regulation of T cell activation have represented selection targets both along mammalian evolution and during the history of human populations; I also found that variants in these genes related to human diseases to be preferential targets of pathogen-driven selection. These results showed that an allele can spread in a population because it confers higher protection against some infectious agent, indicating adaptation to infection as the underlying explanation for the maintenance of a set of autoimmune risk alleles. These result has a relevance for the hygiene hypothesis, and support the idea that human adaptation to an environment with reduced presence of pathogens has determined the spread of some risk alleles for autoimmune and inflammatory diseases. We then presented a comprehensive analysis of the selective events acting on the antigen processing and presentation pathway across different evolutionary timescales, revealing a high proportion of genes under positive selection in mammalian species. Data also indicate a continuum in selective pressure acting on different timescales for some of these genes analyzed, and we also demonstrated that the selected variants in human populations were always located within regions with regulatory function and can have a role in modulating human phenotypes. The evolutionary analysis we performed about contact system genes indicated that mammalian kininogen has been a target of long-lasting and strong selective pressures. In particular our results reinforced the possibility that kininogen plays a central role in the modulation of immune response and is a target of different pathogen species. Finally our study of two different families of nucleic acid receptors showed that a proportion of these genes have been engaged in host-virus genetic conflict leading to a continuous host\u2013pathogen arms race scenario, and again our results provide functional information about variants that might affect immunologic phenotypes. CONCLUSIONS Results in all these studies showed how natural selection shaped diversity in different pathway involved in the immune response, and selected sites involve positions of fundamental importance to the protein function. These novel data give rise to a number of experimentally testable hypothesis concerning the role of specific sites or regions as modulators of immunological phenotypes; they also suggest caution when extrapolating results from specific experiments in model organisms, as a considerable portion of genetic diversity in these molecules has accumulated not as a result of neutral processes but in response to adaptive events

Square-tiled cyclic covers

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichm\"uller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichm\"uller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in \cite{Forni:Matheus}) of a Teichm\"uller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example found previously by Forni in genus three also corresponds to a square-tiled cyclic cover \cite{ForniSurvey}). We present several new examples of Teichm\"uller curves in strata of holomorphic and meromorphic quadratic differentials with maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichm\"uller curves with maximally degenerate spectrum. We prove that this is indeed the case within the class of square-tiled cyclic covers.Comment: 34 pages, 6 figures. Final version incorporating referees comments. In particular, a gap in the previous version was corrected. This file uses the journal's class file (jmd.cls), so that it is very similar to published versio