Recent Advances in Pure and Applied Mathematics

Description

This volume comprises high-quality works in pure and applied mathematics from the mathematical communities in Spain and Brazil. A wide range of subjects are covered, ranging from abstract algebra, including Lie algebras, commutative semigroups, and differential geometry, to optimization and control in real world problems such as fluid mechanics, the numerical simulation of cancer PDE models, and the stability of certain dynamical systems. The book is based on contributions presented at the Second Joint Meeting Spain-Brazil in Mathematics, held in Cádiz in December 2018, which brought together more than 330 delegates from around the world. All works were subjected to a blind peer review process. The book offers an excellent summary of the recent activity of Spanish and Brazilian research groups and will be of interest to researchers, PhD students, and graduate scholars seeking up-to-date knowledge on these pure and applied mathematics subjects. Preface Contents On the Control of the Navier-Stokes Equations and Related Systems 1 Introduction 1.1 The Navier-Stokes Equations 1.2 A Little Bit of History 1.3 Fundamentals of Control Theory 1.4 Some Applications of Control Theory 2 Controlling the Navier-Stokes System 2.1 Three Motivations 2.2 A Bi-objective Control Problem 2.3 A Null Controllability Problem References Asymptotic Stability in Some Generic Classes of Three-Dimensional Discontinuous Dynamical Systems 1 Introduction 2 Preliminaries 2.1 Filippov's Convention 2.2 Contact Between a Smooth Vector Field X and a Codimension One Submanifold M of R3 2.3 Classification of Singularities of ZΩr 2.4 Properties of Some Families of 3D PSVF's 2.5 Subclasses of Ω0 and Ω1 3 Main Results 4 Asymptotic Stability in Ω0Ωr 5 Asymptotic Stability in Ω1Ωr References Regularisation in Ejection-Collision Orbits of the RTBP 1 Introduction 2 Ejection-Collision in the RTBP 2.1 The RTBP 2.2 Two Regularisations 2.2.1 McGehee Regularisation 2.2.2 Levi-Civita Regularisation 2.3 The Collision Manifold 2.4 Results for 1-EC Orbits 2.4.1 Existence of Four 1-EC Orbits 2.4.2 Results for n-EC Orbits References On Local Algebras of Maximal Algebras of Jordan Quotients 1 Introduction 2 Preliminaries 2.1 Essential Ideals 2.2 Special Jordan Algebras 2.3 Algebras of Quotients and Local Algebras 2.4 The Socle of Nondegenerate Jordan Algebras 3 Algebras of Quotients of Local Algebras at Non-PI-elements 4 Algebras of Quotients of Local Algebras at PI-Elements 5 Maximal Algebras of Quotients of Jordan Algebras with Nonzero PI-Elements References On the Numerical Behavior of a Chemotaxis Model with Linear Production Term 1 The Model 1.1 Chemorepulsion Production Systems 1.2 Some Properties 2 Numerical Results 2.1 The Numerical Difficulties 2.2 Numerical Approach Using (P) 2.3 Numerical Approach Using the Scheme US 2.4 Numerical Approach Using the Scheme UZSW 2.5 Numerical Simulations 2.5.1 Approximated Positivity 2.5.2 Energy Stability 3 Conclusions References The Thin-Sandwich Problem in General Relativity 1 Introduction 2 The Initial Value Formulation of GR 3 The Reduced Thin Sandwich Equations 3.1 The Compact Case 3.1.1 Existence of Reference Solutions 3.1.2 Solutions Around Symmetric Data 3.1.3 Neighbourhoods of Umbilical Reference Solutions with Conformal Killing Fields 3.2 The Asymptotically Euclidean Case 3.3 Existence of Reference Solutions 4 Ricci-Flat Embedding of Lorentzian Manifolds References Parametric Solutions to a Static Fourth-Order Euler–Bernoulli Beam Equation in Terms of Lamé Functions 1 Introduction 2 Exact General Solution to Eq.(2) 3 A Three-Parametric Family of Solutions Corresponding to K0=0 4 Three-Parameter Families of Solutions Corresponding to K0 = 1627 5 Graphical Representations of Some Solutions 6 Concluding Remarks References On Large Orbits of Actions of Finite Soluble Groups: Applications 1 Introduction 2 Regular Orbits 3 Prefrattini Subgroups and Normalisers References Poisson Algebras and Graphs 1 Introduction 2 Poisson Algebras Admitting a Multiplicative Basis, and Graphs 3 Relating the Graphs Given by Different Choices of Bases 4 Minimality and Weak Symmetry References Graphs with Weight of Fold Gauss Maps 1 Introduction 2 Stable Gauss Maps and its Graphs 3 Fold Gauss Maps and its Graph 4 Lips and Beaks Transition and Surgeries 5 Graphs with Weight of Fold Gauss Maps References A Note on Spacelike Hypersurfaces and Timelike Conformal Vectors 1 Introduction 2 The Geometric Setting 2.1 Doubly Warped Product Spacetimes 3 Rigidity of H-Hypersurfaces with H≥H References Naturally Graded Quasi-Filiform Associative Algebras 1 Introduction 2 Preliminaries 3 Naturally Graded Quasi-Filiform Associative Algebras 3.1 Associative Algebra with Characteristic Sequence (n-2,2) 3.2 Associative Algebra with Characteristic Sequence (n-2,1.1) References Spacelike Hypersurfaces in Conformally Stationary Spacetimes 1 Introduction 2 Preliminaries 3 The Case of Compact Spacelike Hypersurfaces 4 The Case of Complete Noncompact Spacelike Hypersurfaces 5 The Case of Higher Order Mean Curvatures References Geodesic Completeness and the Quasi-Einstein Equation for Locally Homogeneous Affine Surfaces 1 Affine Geometry 2 Strongly Projectively Flat Geometries 3 The Quasi-Einstein Equation References