An Overview on Some Results Concerning the Transport Equation and its Applications to Conservation Laws

We provide an informal overview on the theory of transport equations with non smooth velocity fields, and on some applications of this theory to the well-posedness of hyperbolic systems of conservation laws.Comment: 12 page

On the singular local limit for conservation laws with nonlocal fluxes

We give an answer to a question posed in [P. Amorim, R. Colombo, and A. Teixeira, ESAIM Math. Model. Numerics. Anal. 2015], which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one in general does not have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.Comment: 26 page

Initial-boundary value problems for continuity equations with BV coefficients

We establish well-posedness of initial-boundary value problems for continuity equations with BV (bounded total variation) coefficients. We do not prescribe any condition on the orientation of the coefficients at the boundary of the domain. We also discuss some examples showing that, regardless the orientation of the coefficients at the boundary, uniqueness may be violated as soon as the BV regularity deteriorates at the boundary.Comment: 25 pages, 5 figure

Brain interaction during cooperation: Evaluating local properties of multiple-brain network

Subjects’ interaction is the core of most human activities. This is the reason why a lack of coordination is often the cause of missing goals, more than individual failure. While there are different subjective and objective measures to assess the level of mental effort required by subjects while facing a situation that is getting harder, that is, mental workload, to define an objective measure based on how and if team members are interacting is not so straightforward. In this study, behavioral, subjective and synchronized electroencephalographic data were collected from couples involved in a cooperative task to describe the relationship between task difficulty and team coordination, in the sense of interaction aimed at cooperatively performing the assignment. Multiple-brain connectivity analysis provided information about the whole interacting system. The results showed that averaged local properties of a brain network were affected by task difficulty. In particular, strength changed significantly with task difficulty and clustering coefficients strongly correlated with the workload itself. In particular, a higher workload corresponded to lower clustering values over the central and parietal brain areas. Such results has been interpreted as less efficient organization of the network when the subjects’ activities, due to high workload tendencies, were less coordinated

A new perspective for the training assessment: Machine learning-based neurometric for augmented user's evaluation

Inappropriate training assessment might have either high social costs and economic impacts, especially in high risks categories, such as Pilots, Air Traffic Controllers, or Surgeons. One of the current limitations of the standard training assessment procedures is the lack of information about the amount of cognitive resources requested by the user for the correct execution of the proposed task. In fact, even if the task is accomplished achieving the maximum performance, by the standard training assessment methods, it would not be possible to gather and evaluate information about cognitive resources available for dealing with unexpected events or emergency conditions. Therefore, a metric based on the brain activity (neurometric) able to provide the Instructor such a kind of information should be very important. As a first step in this direction, the Electroencephalogram (EEG) and the performance of 10 participants were collected along a training period of 3 weeks, while learning the execution of a new task. Specific indexes have been estimated from the behavioral and EEG signal to objectively assess the users' training progress. Furthermore, we proposed a neurometric based on a machine learning algorithm to quantify the user's training level within each session by considering the level of task execution, and both the behavioral and cognitive stabilities between consecutive sessions. The results demonstrated that the proposed methodology and neurometric could quantify and track the users' progresses, and provide the Instructor information for a more objective evaluation and better tailoring of training programs. © 2017 Borghini, Aricò, Di Flumeri, Sciaraffa, Colosimo, Herrero, Bezerianos, Thakor and Babiloni

Nonlocal gravity and the diffusion equation

We propose a nonlocal scalar-tensor model of gravity with pseudodifferential operators inspired by the effective action of p-adic string and string field theory on flat spacetime. An infinite number of derivatives act both on the metric and scalar field sector. The system is localized via the diffusion equation approach and its cosmology is studied. We find several exact dynamical solutions, also in the presence of a barotropic fluid, which are stationary in the diffusion flow. In particular, and contrary to standard general relativity, there exist solutions with exponential and power-law scale factor also in an open universe, as well as solutions with sudden future singularities or a bounce. Also, from the point of view of quantum field theory, spontaneous symmetry breaking can be naturally realized in the class of actions we consider.Comment: 18 pages, 5 figures. v2: typos corrected, references added. Major changes are an expansion of the discussion of homogeneous perturbations and the inclusion of cosmological fluids in the dynamic

Thinking beyond organism energy use: A trait-based bioenergetic mechanistic approach for predictions of life history traits in marine organisms

The functional trait-based bioenergetic approach is emergent in many ecological spectra, from the conservation of natural resources to mitigation and adaptation strategies in a global climate change context. Such an approach relies on being able to exploit mechanistic rules to connect environmental human-induced variability to functional traits (i.e. all those specific traits defining species in terms of their ecological roles) and use these to provide estimates of species life history traits (LH; e.g. body size, fecundity per life span, number of reproductive events). LHs are species-specific and proximate determinants of population characteristics in a certain habitat. They represent the most valuable quantitative information to investigate how broad potential distributional boundaries of a species are, and to feed predictive population models. There is much to be found in the current literature that describes mechanistic functional trait-based bioenergetics models, using them to test ecological hypotheses, but a mathematical framework often renders interpretation and use complicated. Here, we wanted to present a simpler interpretation and description of one of the most important recent mechanistic bioenergetic theories: the dynamic energy budget theory by Kooijman (Dynamic Energy Budget Theory for Metabolic Organisation, 2010, Cambridge University Press, Cambridge). Our main aim was to disentangle those aspects that at first reading may seem too mathematically challenging to many marine biologists, ecologists and environmental scientists, and present them for use in mechanistic applications

Test particle motion in a gravitational plane wave collision background

Test particle geodesic motion is analysed in detail for the background spacetimes of the degenerate Ferrari-Ibanez colliding gravitational wave solutions. Killing vectors have been used to reduce the equations of motion to a first order system of differential equations which have been integrated numerically. The associated constants of the motion have also been used to match the geodesics as they cross over the boundary between the single plane wave and interaction zones.Comment: 11 pages, 6 Postscript figure