Weierstrass integrability of differential equations

The integrability problem consists in finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define {\it Weierstrass integrability} and we determine which Weierstrass integrable systems are Liouvillian integrable. Inside this new class of integrable systems there are non--Liouvillian integrable systems.

Interaccion among systems of finite size in predictive relativistic mechanics -II. Electromagnetic and gravitational interactions

We explicitly construct a closed system of differential equations describing the electromagnetic and gravitational interactions among bodies to first order in the coupling constants, retaining terms up to order c-2. The Breit and Barker and O'Connell Hamiltonians are recovered by means of a coordinate transformation. The method used throws light on the meaning of these coordinates.

Dynamical properties of nonmarkovian stochastic differential equations

We study nonstationary non-Markovian processes defined by Langevin-type stochastic differential equations with an OrnsteinUhlenbeck driving force. We concentrate on the long time limit of the dynamical evolution. We derive an approximate equation for the correlation function of a nonlinear nonstationary non-Markovian process, and we discuss its consequences. Non-Markovicity can introduce a dependence on noise parameters in the dynamics of the correlation function in cases in which it becomes independent of these parameters in the Markovian limit. Several examples are discussed in which the relaxation time increases with respect to the Markovian limit. For a Brownian harmonic oscillator with fluctuating frequency, the non-Markovicity of the process decreases the domain of stability of the system, and it can change an infradamped evolution into an overdamped one.

On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities

"Vegeu el resum a l´inici del document del fitxer adjunt."

Symmetries and fixed point stability of stochastic differential equations modeling self-organized criticality

A stochastic nonlinear partial differential equation is constructed for two different models exhibiting self-organized criticality: the Bak-Tang-Wiesenfeld (BTW) sandpile model [Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988)] and the Zhang model [Phys. Rev. Lett. 63, 470 (1989)]. The dynamic renormalization group (DRG) enables one to compute the critical exponents. However, the nontrivial stable fixed point of the DRG transformation is unreachable for the original parameters of the models. We introduce an alternative regularization of the step function involved in the threshold condition, which breaks the symmetry of the BTW model. Although the symmetry properties of the two models are different, it is shown that they both belong to the same universality class. In this case the DRG procedure leads to a symmetric behavior for both models, restoring the broken symmetry, and makes accessible the nontrivial fixed point. This technique could also be applied to other problems with threshold dynamics.

Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions

In this paper, a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for viscosity solutions of semi-linear stochastic partial differential equations with a Neumann boundary condition is given.

Emotional attention as a modulatory system of perception

Report for the scientific sojourn carried out at the University Medical Center, Swiss, from 2010 to 2012. Abundant evidence suggests that negative emotional stimuli are prioritized in the perceptual systems, eliciting enhanced neural responses in early sensory regions as compared with neutral information. This facilitated detection is generally paralleled by larger neural responses in early sensory areas, relative to the processing of neutral information. In this sense, the amygdala and other limbic regions, such as the orbitofrontal cortex, may play a critical role by sending modulatory projections onto the sensory cortices via direct or indirect feedback.The present project aimed at investigating two important issues regarding these mechanisms of emotional attention, by means of functional magnetic resonance imaging. In Study I, we examined the modulatory effects of visual emotion signals on the processing of task-irrelevant visual, auditory, and somatosensory input, that is, the intramodal and crossmodal effects of emotional attention. We observed that brain responses to auditory and tactile stimulation were enhanced during the processing of visual emotional stimuli, as compared to neutral, in bilateral primary auditory and somatosensory cortices, respectively. However, brain responses to visual task-irrelevant stimulation were diminished in left primary and secondary visual cortices in the same conditions. The results also suggested the existence of a multimodal network associated with emotional attention, presumably involving mediofrontal, temporal and orbitofrontal regions Finally, Study II examined the different brain responses along the low-level visual pathways and limbic regions, as a function of the number of retinal spikes during visual emotional processing. The experiment used stimuli resulting from an algorithm that simulates how the visual system perceives a visual input after a given number of retinal spikes. The results validated the visual model in human subjects and suggested differential emotional responses in the amygdala and visual regions as a function of spike-levels. A list of publications resulting from work in the host laboratory is included in the report.

Emotional attention as a modulatory system of perception

Report for the scientific sojourn carried out at the University Medical Center, Swiss, from 2010 to 2012. Abundant evidence suggests that negative emotional stimuli are prioritized in the perceptual systems, eliciting enhanced neural responses in early sensory regions as compared with neutral information. This facilitated detection is generally paralleled by larger neural responses in early sensory areas, relative to the processing of neutral information. In this sense, the amygdala and other limbic regions, such as the orbitofrontal cortex, may play a critical role by sending modulatory projections onto the sensory cortices via direct or indirect feedback.The present project aimed at investigating two important issues regarding these mechanisms of emotional attention, by means of functional magnetic resonance imaging. In Study I, we examined the modulatory effects of visual emotion signals on the processing of task-irrelevant visual, auditory, and somatosensory input, that is, the intramodal and crossmodal effects of emotional attention. We observed that brain responses to auditory and tactile stimulation were enhanced during the processing of visual emotional stimuli, as compared to neutral, in bilateral primary auditory and somatosensory cortices, respectively. However, brain responses to visual task-irrelevant stimulation were diminished in left primary and secondary visual cortices in the same conditions. The results also suggested the existence of a multimodal network associated with emotional attention, presumably involving mediofrontal, temporal and orbitofrontal regions Finally, Study II examined the different brain responses along the low-level visual pathways and limbic regions, as a function of the number of retinal spikes during visual emotional processing. The experiment used stimuli resulting from an algorithm that simulates how the visual system perceives a visual input after a given number of retinal spikes. The results validated the visual model in human subjects and suggested differential emotional responses in the amygdala and visual regions as a function of spike-levels. A list of publications resulting from work in the host laboratory is included in the report.

Bistability of mitochondrial respiration underlies paradoxical reactive oxygen species generation induced by anoxia

Increased production of reactive oxygen species (ROS) in mitochondria underlies major systemic diseases, and this clinical problem stimulates a great scientific interest in the mechanism of ROS generation. However, the mechanism of hypoxia-induced change in ROS production is not fully understood. To mathematically analyze this mechanism in details, taking into consideration all the possible redox states formed in the process of electron transport, even for respiratory complex III, a system of hundreds of differential equations must be constructed. Aimed to facilitate such tasks, we developed a new methodology of modeling, which resides in the automated construction of large sets of differential equations. The detailed modeling of electron transport in mitochondria allowed for the identification of two steady state modes of operation (bistability) of respiratory complex III at the same microenvironmental conditions. Various perturbations could induce the transition of respiratory chain from one steady state to another. While normally complex III is in a low ROS producing mode, temporal anoxia could switch it to a high ROS producing state, which persists after the return to normal oxygen supply. This prediction, which we qualitatively validated experimentally, explains the mechanism of anoxia-induced cell damage. Recognition of bistability of complex III operation may enable novel therapeutic strategies for oxidative stress and our method of modeling could be widely used in systems biology studies.

Effects of differential taxation on factor accumulation and growth

In this paper we try to analyze the role of fiscal policy in fostering a higher participation of the different production factors in the human capital production sector in the long-run. Introducing a tax on physical capital and differentiating both a tax on raw labor wage and a tax on skills or human capital we also attempt to present a way to influence inequality as measured by the skill premium, thus trying to relate the increase in human capital with the decrease in income inequality. We will do that in the context of a non-scale growth model.The model here is capable to alter the shares of private factors devoted to each of the two production sectors, final output and human capital, and affect inequality in a different way according to the different tax changes. The simulation results derived in the paper show how a human capital (skills) tax cut, which could be interpreted as a reduction in progressivity, ends up increasing both the shares of labor and physical capital devoted to the production of knowledge and decreasing inequality. Moreover, a raw labor wage tax decrease, which could also be interpreted as an increase in the progressivity of the system, increases the share of labor devoted to the production of final output and increases inequality. Finally, a physical capital tax decrease reduces the share of physical capital devoted to the production of knowledge and allows for a lower inequality value. Nevertheless, none of the various types of taxes ends up changing the share of human capital in the knowledge production, which will deserve our future attention

Effects of differential taxation on factor accumulation and growth

In this paper we try to analyze the role of fiscal policy in fostering a higher participation of the different production factors in the human capital production sector in the long-run. Introducing a tax on physical capital and differentiating both a tax on raw labor wage and a tax on skills or human capital we also attempt to present a way to influence inequality as measured by the skill premium, thus trying to relate the increase in human capital with the decrease in income inequality. We will do that in the context of a non-scale growth model.The model here is capable to alter the shares of private factors devoted to each of the two production sectors, final output and human capital, and affect inequality in a different way according to the different tax changes. The simulation results derived in the paper show how a human capital (skills) tax cut, which could be interpreted as a reduction in progressivity, ends up increasing both the shares of labor and physical capital devoted to the production of knowledge and decreasing inequality. Moreover, a raw labor wage tax decrease, which could also be interpreted as an increase in the progressivity of the system, increases the share of labor devoted to the production of final output and increases inequality. Finally, a physical capital tax decrease reduces the share of physical capital devoted to the production of knowledge and allows for a lower inequality value. Nevertheless, none of the various types of taxes ends up changing the share of human capital in the knowledge production, which will deserve our future attention

Differential equations driven by fractional Brownian motion

A global existence and uniqueness result of the solution for multidimensional, time dependent, stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H> is proved. It is shown, also, that the solution has finite moments. The result is based on a deterministic existence and uniqueness theorem whose proof uses a contraction principle and a priori estimates.

Stochastic differential equations with random coefficients

In this paper we establish the existence and uniqueness of a solution for different types of stochastic differential equation with random initial conditions and random coefficients. The stochastic integral is interpreted as a generalized Stratonovich integral, and the techniques used to derive these results are mainly based on the path properties of the Brownian motion, and the definition of the Stratonovich integral.

Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H 1/2

We consider the Cauchy problem for a stochastic delay differential equation driven by a fractional Brownian motion with Hurst parameter H>¿. We prove an existence and uniqueness result for this problem, when the coefficients are sufficiently regular. Furthermore, if the diffusion coefficient is bounded away from zero and the coefficients are smooth functions with bounded derivatives of all orders, we prove that the law of the solution admits a smooth density with respect to Lebesgue measure on R.

Anticipating linear stochastic differential equations driven by a Lévy process

In this paper we study the existence of a unique solution for linear stochastic differential equations driven by a Lévy process, where the initial condition and the coefficients are random and not necessarily adapted to the underlying filtration. Towards this end, we extend the method based on Girsanov transformations on Wiener space and developped by Buckdahn [7] to the canonical Lévy space, which is introduced in [25].