Quaternions in collective dynamics

We introduce a model of multi-agent dynamics for self-organised motion; individuals travel at a constant speed while trying to adopt the averaged body attitude of their neighbours. The body attitudes are represented through unitary quaternions. We prove the correspondance with the model presented in Ref. [16] where the body attitudes are represented by rotation matrices. Differently from this previous work, the individual based model (IBM) introduced here is based on nematic (rather than polar) alignment. From the IBM, the kinetic and macroscopic equations are derived. The benefit of this approach, in contrast to Ref. [16], is twofold: firstly, it allows for a better understanding of the macroscopic equations obtained and, secondly, these equations are prone to numerical studies, which is key for applications

Quaternionic approach to dual Magneto-hydrodynamics of dyonic cold plasma

The dual magneto-hydrodynamics of dyonic plasma describes the study of electrodynamics equations along with the transport equations in the presence of electrons and magnetic monopoles. In this paper, we formulate the quaternionic dual fields equations, namely, the hydro-electric and hydro-magnetic fields equations which are an analogous to the generalized Lamb vector field and vorticity field equations of dyonic cold plasma fluid. Further, we derive the quaternionic Dirac-Maxwell equations for dual magneto-hydrodynamics of dyonic cold plasma. We also obtain the quaternionic dual continuity equations that describe the transport of dyonic fluid. Finally, we establish an analogy of Alfven wave equation which may generate from the flow of magnetic monopoles in the dyonic field of cold plasma. The present quaternionic formulation for dyonic cold plasma is well invariant under the duality, Lorentz and CPT transformations.Comment: 20 pages, Revised versio

Methods for suspensions of passive and active filaments

Flexible filaments and fibres are essential components of important complex fluids that appear in many biological and industrial settings. Direct simulations of these systems that capture the motion and deformation of many immersed filaments in suspension remain a formidable computational challenge due to the complex, coupled fluid--structure interactions of all filaments, the numerical stiffness associated with filament bending, and the various constraints that must be maintained as the filaments deform. In this paper, we address these challenges by describing filament kinematics using quaternions to resolve both bending and twisting, applying implicit time-integration to alleviate numerical stiffness, and using quasi-Newton methods to obtain solutions to the resulting system of nonlinear equations. In particular, we employ geometric time integration to ensure that the quaternions remain unit as the filaments move. We also show that our framework can be used with a variety of models and methods, including matrix-free fast methods, that resolve low Reynolds number hydrodynamic interactions. We provide a series of tests and example simulations to demonstrate the performance and possible applications of our method. Finally, we provide a link to a MATLAB/Octave implementation of our framework that can be used to learn more about our approach and as a tool for filament simulation

A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks

In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called $\mathcal{B}$-projection functions. Broadly speaking, a $\mathcal{B}$-projection function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley-Dickson algebras

Geometry of discrete-time spin systems

Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space $(S^2)^n$. In this paper we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features, that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on $T^*\mathbf{R}^{2n}$ for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with K\"ahler geometry, this provides another geometric proof of symplecticity.Comment: 17 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1402.333

Supercharges, Quantum States and Angular Momentum for N=4 Supersymmetric Monopoles

We revisit the moduli space approximation to the quantum mechanics of monopoles in N=4 supersymmetric Yang-Mills-Higgs theory with maximal symmetry breaking. Starting with the observation that the set of fermionic zero-modes in N=4 supersymmetric Yang-Mills-Higgs theory can be viewed as two copies of the set of fermionic zero-modes in the N=2 version, we build a model to describe the quantum mechanics of N=4 supersymmetric monopoles, based on our previous paper [1] on the N=2 case, in which this doubling of fermionic zero-modes is manifest throughout. Our final picture extends the familiar result that quantum states are described by differential forms on the moduli space and that the Hamiltonian operator is the Laplacian acting on forms. In particular, we derive a general expression for the total angular momentum operator on the moduli space which differs from the naive candidate by the adjoint action of the complex structures. We also express all the supercharges in terms of (twisted) Dolbeault operators and illustrate our results by discussing, in some detail, the N=4 supersymmetric quantum dynamics of monopoles in a theory with gauge group SU(3) broken to U(1) x U(1).Comment: Updated references, included a derivation of the angular momentum operator, 32 page