Proyectos

The main goal of this proposal is to study different particle systems interacting through a empirical mean term and the corresponding mean field descriptions when the number of individuals is going to infinity. In particular, we are interested in well posedness questions for those nonlinear SDEs, with coefficients depending on the law of the unknown process itself. Classically, the existence of those solutions can be addressed via propagation of chaos arguments. By using tightness arguments on the particle system solutions one can find weak solutions to the mean field nonlinear processes using the consistency of equations. In that sense, we are concerned in revisit some classical techniques and adapt them to some modern problems involving L ́evy, Hawkes and Asymmetric noisy components. Thus, we are interested in stating a new framework to englobe some applications that are not rigorously treated in literature (specially in theoretical biology, neuroscience and ecology). Mathematically speaking, the proposed methodology is quite standard. We start by well known problems and modify the proofs to attack perturbations of the seminal equations. In doing so, we will find out what are the key constraints and get some insights of the elements we need to develop in order to study the new questions stated. To fill the gaps between classic and modern applications, there are some very interesting prints stating general BDG inequalities for L ́evy processes and central limit theorems for Hawkes processes (see Proposed Research for more details). The proposed research was motivated by the final goal of having process with memory, but when Markov Property no longer apply there is no much we can solve directly. Thus, we will try to restraint first to the Markovian framework by using compactness assumptions in past/delay dependance and/or regular approximations of the functions involved in the SDEs. The goals and expected results are in particular: 1. Toy model 1: The justification of the mean-field limit process for a general model of interacting particles connected through Poisson measures. The main novelty of the equation is presence of the unknown in the size and intensity of the mean field jump part. By imposing a non mass creation hypothesis, solutions still upper bounded and the tightness method of Sznitman for proving the chaos propagation holds. 2. Toy model 2: The analysis of a model composed by a small world network of interacting particles and a large scale interaction in an open system approach. To use the coupling technique and prove the consistence between the particle system and the nonlinear process of a system driven by one (or in general a sequence) of compound Poisson process(es). This result can be done by using the arguments of Jourdain et al. 2008, in particular, by using the BDG type of inequality stated there. 3. Study numerically the shape of the invariant measures in some ecologically inspired models of type 1 and 2. This equations are naturally found by using logistic growth dynamics and competitive/cooperative interaction functions (Lotka-Volterra kind of interactions). To find resolvable toy models and prove stability of the steady states. ˆ 4. L ́evy particle system: to use the L ́evy-Ito decomposition theorem to write the randomness as the sum of a continuous part, a compound Poisson process and a square integrable pure jump part. To control the deterministic part of the dynamics by using a Lipschitz hypothesis. The stochastic components, can be treated by using classical arguments on diffusive particles, the toy-model 2 and again the BDG type of inequality of Jourdain et al. 2008. 5. Hawkes particle system: to prove an equivalence between Hawkes processes and a system of equations involving Poisson measures by using the method of Delattre et al. 2016. To solve the well posedness of the new particle system by using the arguments of the toy-model 1. By using the results of Chevallier et al. 2015 for the pure point process case, to solve our limit equation. With the well posedness of both systems, to use the consistence part of the chaoticity to conclude the convergence in law towards the mean field equation. 6. Asymmetric particle system: to use the regularity approximation of LeGall to justify the passage from an equation involving local times, to a classical diffusive equation. Solve the well-posedness and the chaos propagation questions for this new system which is an application of the classical version of the chaos property. To take the limit on the approximation functions for the new mean-field equation and study the convergence towards an asymmetric nonlinear process.

Problems that cannot be solved by classical computers in reasonable time due to their high computational cost arise in many research areas. In general, the evaluation of conjunctive queries over relational databases belongs to those problems. Conjunctive queries form the core of the Structured Query Language (SQL) which became a de facto standard for querying and maintaining relational databases. This work is about developing new approximation techniques for conjunctive queries which cannot be evaluated in reasonable time. Our new approximation techniques should lead to significant improvements for data aided decision making, e.g., for early warning system which are based on the analysis of big data or to make business-critical decisions by analyzing big data. In the last decades, a very good understanding of the classes of conjunctive queries which can be evaluated in reasonable time has been gained and it has been proven that an under-approximation of a query always exists within each of those classes. However this approach is rather strict and some of the under-approximations can be rather uninformative, i.e., the under-approximation might return the empty result set while the original query would not. over-approximations might be helpful when this happens, as they return all answers to a query. One of our goals is to study the foundational aspects of over-approximations, including the existence problem and the problem of computing an approximation. Unfortunately, over-approximations do not always exist (within a class of queries which can be evaluated in reasonable time), and it is not even known to be decidable whether a conjunctive query admits an over-approximation. Therefore, another goal of the proposed work is the development of more liberal approximation techniques that yield some kind of quantitative guarantees. This means that they should guarantee that the result of the approximation is not too “far” from the result of the original query over a set of databases of interest. Therefore we need to define a measure of disagreement between queries and/or results. For conjunctive query evaluation, such measures do not exist up until now. Based on that measure, we study approximations whose disagreement with the result of the query they approximate is below a certain threshold. Furthermore, we investigate how the underlying data of a database can help us to find better approximations. It has been shown that there are close relations between the approximation of conjunctive queries over relational databases and some classes of Semantic Web queries over semi-structured data. We also study possible connections between our approximation techniques and approximating Semantic Web queries.

Waves are ubiquitous in nature. They are all around us in our daily lives, we find them in several contexts, in particular in fluids. They usually involve a complex variety of interaction processes, and di↵erent mechanisms. Of our particular inter- est is the case of waves at the interface between two fluids when they are perturbed. When strongly forced, the nonlinear interactions can produce a turbulent-like regime called wave turbulence. Theoretical, numerical and experimental studies have made a great deal of progress on this subject, and yet, there are several aspects that have not been properly addressed, namely the role of viscosity on the energy flux as it cascades through di↵erent scales or the physical origin of the intermittency phenomenon. In this proposal, we will consider the problem of capillary wave turbulence from an experimental and numerical point of view. One of the main complications to study surface wave turbulence is that, in the same system, there is involvement of di↵erent types of waves, such as the case of gravito-capillary wave turbulence. Thus, it becomes of foremost importance to study wave turbulence on the presence of only one type of waves. Thereby, in order to study pure capillary wave turbulence, gravity waves must be negligible. We propose to study a system of capillary surface waves at the interface of two immiscible and incompressible fluids, water and silicon oil, of almost equal densities and layer depths, thus preventing the action of gravity. By changing the kinematic viscosity ⌫, and density ⇢ of both fluids, it is possible to control the relation between injected and dissipated power, thus exploring several regimes in a simple and controlled way. We pose to implement the technique called Free-surface synthetic Schlieren that allows a reconstruction of the instantaneous surface topography. Velocity fields will be explored by using the standard Particle Image Velocimetry. We will make use of an already existing experimental setup which will be modified in order to accomplish these techniques adequately. With these measurements we will be able to compute the spectrum in frequency f and wavevector k, hence accessing to statistical and dynamical properties of capillary wave turbulence, such as intermittency, or the function of the injected power on the system. We also propose to use the open source solver GERRIS and make a systematic study on the role of viscosity on the cascading of the energy flux.

Institución: Universidad de Concepción, Concepción, Chile Descripción: Fondo para comprar un amplificador de potencia para realizar experiencias hardware- in-the-loop. Tareas principales: . Escritura del proyecto. . Dise˜ no del setup experimental y especificación de equipamiento para realizar experiencias hardware- in-the-loop.

The Chilean Coastal Orographic Precipitation Experiment (CCOPE) was conducted during the austral winter of 2015 (May–August) in the Nahuelbuta Mountains (peak elevation 1.3 km MSL) of southern Chile (38ºS). CCOPE used soundings, two profiling Micro Rain Radars, a Parsivel disdrometer, and a rain gauge network to characterize warm and ice-initiated rain regimes and explore their consequences for orographic precipitation