On local and global aspects of the 1:4 resonance in the conservative cubic H\'enon maps

We study the 1:4 resonance for the conservative cubic H\'enon maps $\mathbf{C}_\pm$ with positive and negative cubic term. These maps show up different bifurcation structures both for fixed points with eigenvalues $\pm i$ and for 4-periodic orbits. While for $\mathbf{C}_-$ the 1:4 resonance unfolding has the so-called Arnold degeneracy (the first Birkhoff twist coefficient equals (in absolute value) to the first resonant term coefficient), the map $\mathbf{C}_+$ has a different type of degeneracy because the resonant term can vanish. In the last case, non-symmetric points are created and destroyed at pitchfork bifurcations and, as a result of global bifurcations, the 1:4 resonant chain of islands rotates by $\pi/4$. For both maps several bifurcations are detected and illustrated.Comment: 21 pages, 13 figure

Caracterização geológica e petrológica preliminar do vulcanismo ácido da região de Campos Novos, Santa Catarina

Geological mapping on a scale of 1:50,000 was carried out in an area of 1800 km2, located in the central part of the Videira and Barracão Sheets in the Campos Novos, Celso Ramos and Anita Garibaldi municipalities, Santa Catarina. Five basic flows followed by six acidic units, similar to ignimbrites, were recognized and comprise a 500- to 600- m thick pile. Petrographic and major element geochemical data reveal the tholeitic character of the association and its relationship with low-titanium basalts and Palmas-type acidic rocks. Preliminary petrogenetic study suggests that the acid rocks originated by partial melting of a gabbroic parent which may be the intrusive equivalent of the basic flows of the study area.Trabalhos de mapeamento geológico em escala 1:50.000 foram realizados numa área de 1800 km2, localizada na região centro-sul do estado de Santa Catarina, cobrindo parte das folhas de Videira e Barracão, nos municípios de Campos Novos, Celso Ramos e Anita Garibaldi. Foram reconhecidos 5 derrames básicos sobrepostos por 6 unidades ácidas do tipo ignimbrito, perfazendo um pacote com espessura de 500 a 600 metros. Estudos petrográficos e litoquímicos de elementos maiores revelam o caráter toleítico da associação e sua correlação com os basaltos baixo-titânio e ácidas tipo Palmas. Um ensaio petrogenético preliminar sugere a origem das rochas ácidas por fusão parcial de uma fonte gabróica que pode ser o correspondente intrusivo dos derrames básicos registrados na área estudada

OPTIMIZATION OF TISSUE SAMPLES FOR THE DEVELOPMENT AND VALIDATION OF DISEASE BIOMARKERS: THE OPTIMARK PROJECT

Europe Biobank Week, Sept 13-16, 2016. Vienna, AustriaS

Dynamics of 4 D symplectic maps near a double resonance

We study the dynamics of a family of 4 D symplectic mappings near a doubly resonant elliptic fixed point. We derive and discuss algebraic properties of the resonances required for the analysis of a Takens type normal form. In particular, we propose a classification of the double resonances adapted to this problem, including cases of both strong and weak resonances. Around a weak double resonance (a junction of two resonances of two different orders, both being larger than 4) the dynamics can be described in terms of a simple (in general non-integrable) Hamiltonian model. The non-integrability of the normal form is a consequence of the splitting of the invariant manifolds associated with a normally hyperbolic invariant cylinder. We use a 4 D generalisation of the standard map in order to illustrate the difference between a truncated normal form and a full 4 D symplectic map. We evaluate numerically the volume of a 4 D parallelotope defined by 4 vectors tangent to the stable and unstable manifolds respectively. In good agreement with the general theory this volume is exponentially small with respect to a small parameter and we derive an empirical asymptotic formula which suggests amazing similarity to its 2 D analog. Different numerical studies point out that double resonances play a key role to understand Arnold diffusion. This paper has to be seen, also, as a first step in this direction. © 2012 Elsevier B.V. All rights reserved

Stochastic modelling of the eradication of the HIV-1 infection by stimulation of latently infected cells in patients under highly active anti-retroviral therapy

HIV-1 infected patients are effectively treated with highly active anti-retroviral therapy (HAART). Whilst HAART is successful in keeping the disease at bay with average levels of viral load well below the detection threshold of standard clinical assays, it fails to completely eradicate the infection, which persists due to the emergence of a latent reservoir with a half-life time of years and is immune to HAART. This implies that life-long administration of HAART is, at the moment, necessary for HIV-1-infected patients, which is prone to drug resistance and cumulative side effects as well as imposing a considerable financial burden on developing countries, those more afflicted by HIV, and public health systems. The development of therapies which specifically aim at the removal of this latent reservoir has become a focus of much research. A proposal for such therapy consists of elevating the rate of activation of the latently infected cells: by transferring cells from the latently infected reservoir to the active infected compartment, more cells are exposed to the anti-retroviral drugs thus increasing their effectiveness. In this paper, we present a stochastic model of the dynamics of the HIV-1 infection and study the effect of the rate of latently infected cell activation on the average extinction time of the infection. By analysing the model by means of an asymptotic approximation using the semi-classical quasi steady state approximation (QSS), we ascertain that this therapy reduces the average life-time of the infection by many orders of magnitudes. We test the accuracy of our asymptotic results by means of direct simulation of the stochastic process using a hybrid multi-scale Monte Carlo scheme