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Research Centre for Mathematics and Applications

Centro de Investigação em Matemática e Aplicações

Reference
UIDB/04674/2020
Project Start Date
2020-01-01
Project End Date
2024-12-31
Principal Investigator
Funding Program
Concurso de avaliação no âmbito do Programa Plurianual de Financiamento de Unidades de I&D (2017/2018) - Financiamento Base

Abstract

10.2 Summary in English for general dissemination purposes Research in the CIMA in the period 2018-2022 will develop mainly, but not exclusively, in the following areas: A. On Boundary Value Problems of Fractional order (FBVP) the following tasks will be considered: * Find sufficient conditions on nonlinearities, depending on several derivatives and some parameters, in order to guarantee, the existence, non existence, unicity and/or multiplicity of solutions for higher order fractional boundary value problems. * Obtain the functional framework for fractional problems on unbounded domains. * Conditions for existence of homoclinic and/or heteroclinic solutions for (FBVP). * Study functional FBVP, that is, where there is a global variation of the unknown function and some of the derivatives. * Study conditions for the solvability of coupled systems of FBVP with coupled nonlinearities and different boundary conditions. B. The area of dynamical systems possesses a group of researchers specialized in topological and symbolic dynamics. The following topics will be developed: kneading theory, Markov chains, thermodynamic formalism, statistical physics methods, ergodic theory, fractal geometry, renormalization, operator theory, algebraic invariants, hyperbolic geometry, combinatorics (enumerative, braids, knots, group and graph theory), qualitative theory of ODE and equations from mathematical-physics. C. In the field of Logic, Algebra and Geometry we have the following lines: C.1 Modal Logic and Belief Change: Definition and axiomatic characterizations of operators for belief change with respect to revision and contraction. C.2 External sets and uncertainty: Asymptotic equations, determination of eigenvalues with floating point computation. C.3 Numerical semigroups: Frobenius problem for some classes of semigroups. C.4 Exterior differential system of Riemannian geometry: Applications to differential forms and the geometry of tangent bundle manifolds, like Cartan structural equations. C.5 Algebraic geometry: Cohomological characterization of monads under generalized conditions, minimal decomposition of skew-symmetric tensors. D. On Stochastic Processes and Statistics the following themes are studied : D.1 Self-avoiding random paths of (non-)Gaussian processes applied to polymer physics. Form factors and energy functions, analytics, asymptotic decay. Periodic and branching trajectories. Fractional kinetics in a general fractional calculus. D.2 Applications of Stochastic Differential Equations (SDE) to: Individual cattle growth for parameters dependent on the individual animal genetic values. Study stochastic epidemic models, vaccination strategies for disease eradication and optimization of public health costs. D.3 Construct a new combined adaptive and predetermined sampling intervals method in statistical quality control. D.4 Population and community dynamics, development and applications of models to estimate parameters of natural populations. 10.3 Summary in English for evaluation Research in the CIMA in the period 2018-2022 will develop mainly, but not exclusively, in the following areas: A. The research on the field of Boundary Value Problems of Fractional order (FBVP) will follow the goals: Fractional-order differential equations with boundary value problems sprung up intensively, with different types of boundary conditions: Dirichlet, Neumann, two point and multipoint, nonlocal, among others. However many problems remain open. Our aim is to contribute for finding answers for some of them, namely pursuing the following goals: * Find sufficient conditions on nonlinearities, depending on several derivatives and some parameters, in order to guarantee, the existence, non existence, unicity and/or multiplicity of solutions for higher order fractional boundary value problems. * Obtain the functional framework for the regularity conditions to have the solvability of fractional problems on unbounded domains, for example with infinite time. * Set necessary and/or sufficient conditions for the existence of homoclinic and/or heteroclinic solutions for fractional boundary value problems. * Study functional FBVP, that is, where there is a global variation of the unknown function and some of the derivatives. In this way, it would be possible to consider max and/or min arguments, nonlocal and nonlinear boundary conditions. This is a completely new approach for FBVP. * Study conditions for the solvability of coupled systems of fractional differential equations with coupled nonlinearities and different types of boundary conditions. Some tasks to be carried out on the Dynamical Systems area: B. In the area of dynamical systems CIMA possesses a group of researchers specialized in topological and symbolic dynamics. Using these techniques the following topics will be developed during 2018-2022: kneading theory, Markov chains, thermodynamic formalism, statistical physics methods, ergodic theory, fractal geometry, renormalization, operator theory (C*algebras, Von Neumann), algebraic invariants, dimension groups, hyperbolic geometry, combinatorics (enumerative, braids, knots, group and graph theory), nonlinear boundary value problems, qualitative theory of ODE (Poincaré section) and equations from mathematical-physics. Additional strength of the group is the ability to articulate the development of pure mathematics with the applications to natural and social sciences and the ability to use experimental procedures. Therefore, methods will be developed to apply symbolic dynamics to other scientific areas such computation, biology, sociology, economy, urban planning, materials and geology, according to several ongoing projects. C. In the field of Logic, Algebra and Geometry we have the following lines: C.1. Modal Logic and Belief Change (Carmo, Reis, Garapa): Definition and axiomatic characterizations of operators for belief change with respect to revision and contraction, both for theories and belief bases. C.2. External sets and uncertainty (Van den Berg, Justino): Asymptotic equations, determination of eigenvalues, floating point computation, singular perturbations with canards in unpredictable directions, topological dimension theorem. C.3. Numerical semigroups (Branco, Torrão): Frobenius problem for some classes of semigroups, including almost symmetric semigroups, relation to the Wilfs Conjecture. Determination of the set of all numerical semigroups with given multiplicity and gender. C.4. Exterior differential system of Riemannian geometry (Albuquerque): Applications to differential forms and the geometry of tangent bundle and tangent sphere bundle manifolds, like Cartan structural equations and structures in different settings. C.5. Algebraic geometry (Marques, Soares): Cohomological characterization of monads under generalized conditions, and with new types of monads, Artinian Gorenstein algebras and Gorenstein sequences, minimal decomposition of skew-symmetric tensors. D. On Stochastic Processes and Statistics the following themes are studied : D.1. Study of (weakly) self-avoiding random paths and non-Gaussian processes applied to polymer physics, in particular form factors and energy functions, their analytics, asymptotic decay, near-to-near attraction-repulsion. Renormalization group techniques for scaling indices. Dirichlet forms for Edwards measures. Periodic and branching trajectories. Fractional kinetics in a general fractional calculus. P-Adic infinite dimensional analysis applied to polymers chains. D.2. Extension of the monophasic and biphasic SDE models of individual cattle growth for parameters dependent on the individual animal genetic values. Allee effects on SDE applied to population extinction, special for endangered species. Use of SDE on profit optimization issues under different harvesting fishing policies for a sustainable activity. Study stochastic epidemic models vaccination strategies for disease eradication time and optimization of public health costs. Human mortality rates in view of pension plans and life insurance. Other applications of SDE to human mortality rates in view of pension plans and life insurance. Statistical Modeling of Pulmonary Tuberculosis data in Portugal. Study of road accidents in Angola through Dynamic Temporal Regression Models. D.3. Construct a new combined adaptive and predetermined sampling intervals method in statistical quality control. Study of the effect of using an incorrect model on the performance of risk-adjusted control charts for different models. Develop other measures to evaluate the impact of parameter estimation error. D.4. Population and community dynamics, development and applications of models to estimate parameters of populations with evasive behaviour, on natural populations. Capture-recapture methodology. Distance sampling methodology. Modelling and parameter estimation of vector-borne diseases in tropical and subtropical areas with dengue, chikungunya and zika.

Institutions

Main Institutions

  • Universidade de Évora Centro de Investigação em Matemática e Aplicações (CIMA)
  • Universidade de Évora (Uévora)

Other Institutions

  • Instituto Politécnico de Lisboa Instituto Superior de Engenharia de Lisboa (IPL ISEL)
  • Universidade da Madeira (UMA)

Funding 916.500,00 €

Fundação para a Ciência e a Tecnologia (FCT) - Portugal

916.500,00 €