Recent advances in pure and applied mathematics

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