Perturbed Three Vortex Dynamics

It is well known that the dynamics of three point vortices moving in an ideal fluid in the plane can be expressed in Hamiltonian form, where the resulting equations of motion are completely integrable in the sense of Liouville and Arnold. The focus of this investigation is on the persistence of regular behavior (especially periodic motion) associated to completely integrable systems for certain (admissible) kinds of Hamiltonian perturbations of the three vortex system in a plane. After a brief survey of the dynamics of the integrable planar three vortex system, it is shown that the admissible class of perturbed systems is broad enough to include three vortices in a half-plane, three coaxial slender vortex rings in three-space, and `restricted' four vortex dynamics in a plane. Included are two basic categories of results for admissible perturbations: (i) general theorems for the persistence of invariant tori and periodic orbits using Kolmogorov-Arnold-Moser and Poincare-Birkhoff type arguments; and (ii) more specific and quantitative conclusions of a classical perturbation theory nature guaranteeing the existence of periodic orbits of the perturbed system close to cycles of the unperturbed system, which occur in abundance near centers. In addition, several numerical simulations are provided to illustrate the validity of the theorems as well as indicating their limitations as manifested by transitions to chaotic dynamics.Comment: 26 pages, 9 figures, submitted to the Journal of Mathematical Physic

On the minimum number of neighbours for good routing performance in MANETs

In a mobile ad hoc network, where nodes are deployed without any wired infrastructure and communicate via multihop wireless links, the network topology is based on the nodes’ locations and transmission ranges. The nodes communicate through wireless links, with each node acting as a relay when necessary to allow multihop communications. The network topology can have a major impact on network performance. We consider the impact of number and placement of neighbours on mobile network performance. Specifically, we consider how neighbour node placement affects the network overhead and routing delay. We develop an analytical model, verified by simulations, which shows widely varying performance depending on source node speed and, to a lesser extent, number of neighbour nodes

The integrability of Lie-invariant geometric objects generated by ideals in the Grassmann algebra

We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E. Cartan. Especially, the E. Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame work of the Wahlquist Estabrook prolongation structures on jet-manifolds and Cartan-Ehresmann connection theory on fibered spaces. General structure of integrable one-forms augmenting the two-forms associated with a closed ideal in the Grassmann algebra is studied in great detail. An effective Maurer-Cartan one-forms construction is suggested that is very useful for applications. As an example of application the developed Lie-invariant geometric object theory for the Burgers nonlinear dynamical system is considered having given rise to finding an explicit form of the associated Lax type representation

Reduced pre-Lie algebraic structures, the weak and weakly deformed Balinsky-Novikov type symmetry algebras and related Hamiltonian operators

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky–Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler–Kostant–Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky–Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky–Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky–Novikov algebras, including their fermionic version and related multiplicative and Lie structures

The differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann type hierarchy revisited

A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic hierarchy is revisited, its new Lax type representation and Poisson structures constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of the generalized Riemann type hierarchy are also discussed.Comment: 18 page

The nonabelian Liouville-Arnold integrability by quadratures problem: a symplectic approach

A symplectic theory approach is devised for solving the problem of algebraic-analytical construction of integral submanifold imbeddings for integrable (via the nonabelian Liouville-Arnold theorem) Hamiltonian systems on canonically symplectic phase spaces

A Memetic Analysis of a Phrase by Beethoven: Calvinian Perspectives on Similarity and Lexicon-Abstraction

This article discusses some general issues arising from the study of similarity in music, both human-conducted and computer-aided, and then progresses to a consideration of similarity relationships between patterns in a phrase by Beethoven, from the first movement of the Piano Sonata in A flat major op. 110 (1821), and various potential memetic precursors. This analysis is followed by a consideration of how the kinds of similarity identified in the Beethoven phrase might be understood in psychological/conceptual and then neurobiological terms, the latter by means of William Calvin’s Hexagonal Cloning Theory. This theory offers a mechanism for the operation of David Cope’s concept of the lexicon, conceived here as a museme allele-class. I conclude by attempting to correlate and map the various spaces within which memetic replication occurs

Influences on academics' approaches to development: voices from below

The purpose of this qualitative case study research was to explore faculty-based academics’ views on what influences their behaviours and attitudes towards their development. Informed by critical realist ontology, the data collection was carried out through narrative interviews with academics in two contrasting English Universities. Findings, or areas for reflection, have emerged about the constraints and enablements academics perceive in respect of their professional development. In particular, themes such as the significance of professional status; misaligned initiatives and priorities; the influence of supportive networks; and emergent personal, individual concerns have surfaced. The conclusion is drawn that the significance of agency raises the importance of responding to the ‘voices from below’

Policy, Performativity and Partnership: an Ethical Leadership Perspective

This article identifies the need to think differently about educational partnerships in a changing and turbulent post compulsory policy environment in England. The policy and institutional contexts in which universities and colleges currently operate seem to be fuelling performativity at the expense of educational values. There appears to be a sharp interruption in the steady increase in educational partnerships as a vehicle for increasing and widening participation in higher education. We are witnessing a marked change in university / college relationships that appears to be a consequence of government calling a halt to increased participation in higher education, creating an increasingly competitive market for a more limited pool of student places. The implication that educational policy at the national level determines a particular pattern or mode of leadership decision making throughout an institution should however be resisted. Policy developments that challenge the moral precepts of education should not be allowed to determine how a leader acts, rather they should prompt actions that are truly educational, rooted in morality, and atached to identifiable educational values. Educational leaders have agency to resist restricted discourses in favour of ethical and principled change strategies that are a precondition for sustainable transformative partnerships in post compulsory education. University leaders in particular are called upon to use their considerable influence to resist narrow policy or managerial instrumentalism or performativity and embrace alternatives that are both educationally worthwhile and can enhance institutional resilience