Some New Addition Formulae for Weierstrass Elliptic Functions

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These new results were inspired by new addition formulae found in the case of an equianharmonic curve, which we can now observe as a specialisation of the results here. The new formulae, and the techniques used to find them, also follow the recent work for the generalisation of Weierstrass' functions to curves of higher genus.Comment: 20 page

Interplay of linear and nonlinear impurities in the formation of stationary localized states

Formation of stationary localized states in one-dimensional chain as well as in a Cayley tree due to a linear impurity and a nonlinear impurity is studied. Furthermore, a one-dimensional chain with linear and nonlinear site energies at the alternate sites is studied and rich phase diagrams of SL states are obtained for all systems we considered. The results are compared with those of the linear and nonlinear systems.Comment: 7 pages, Latex, 7 figure

Optimal escape from circular orbits around black holes

Using the theory of optimal rocket trajectories in general relativity, recently developed in arXiv:1105.5235, we show that the "obvious" manoeuvre of using a tangential instantaneous acceleration to escape a stable circular orbit in the Schwarzschild spacetime satisfies the optimality conditions if and only if the magnitude of the acceleration is smaller than a certain bound.Comment: 7 page

Linearized Impulsive Fixed-Time Fuel-Optimal Space rendezvous: A New Numerical Approach

International audienceThis paper focuses on the fixed-time minimum-fuel rendezvous between close elliptic orbits of an active spacecraft with a passive target spacecraft, assuming a linear impulsive setting and a Keplerian relative motion. Following earlier works developed in the 1960s, the original optimal control problem is transformed into a semi-infinite convex optimization problem using a relaxation scheme and duality theory in normed linear spaces. A new numerical convergent algorithm based on discretization methods is designed to solve this problem. Its solution is then used in a general simple procedure dedicated to the computation of the optimal velocity increments and optimal impulses locations. It is also shown that the semi-infinite convex programming has an analytical solution for the out-of-plane rendezvous problem. Different realistic numerical examples illustrate these results

Acceleration and localization of matter in a ring trap

A toroidal trap combined with external time-dependent electric field can be used for implementing different dynamical regimes of matter waves. In particular, we show that dynamical and stochastic acceleration, localization and implementation of the Kapitza pendulum can be originated by means of proper choice of the external force

Incremental support structures for housing and urbanisation

South Africa is experiencing unprecedented population growth due to rapid urbanisation. This growth often overwhelms the current planning and developmental capacities of city-regions acutely impacting informal settlement areas. As a result the city's most vulnerable citizens experience poor service delivery and poor living conditions. This project proposal challenges the current approach to housing delivery and the upgrading of informal settlements in urban areas of South Africa. It is positioned within a complex informal housing environment with poor basic infrastructure and high exposure to the risk of fire and flooding in winter. Based on the research of this project, the Barney Molokana Section in Khayelitsha was selected as the conditions above were evident in this informal settlement. The project comprises three parts; the first is a proposal for an infrastructural intervention aimed to act as a settlement organisational device, the second is a public amenities building that promotes an active public interface and a didactic architecture and the third a series of support structures that further promote the concept of incremental housing development. The process learnt from existing spatial configurations and transformations within informal settlements allowed the working backwards to discover the minimal elements or support structures from which a settlement can grow incrementally

An analytical study of resonant transport of Bose-Einstein condensates

We study the stationary nonlinear Schr\"odinger equation, or Gross-Pitaevskii equation, for a one--dimensional finite square well potential. By neglecting the mean--field interaction outside the potential well it is possible to discuss the transport properties of the system analytically in terms of ingoing and outgoing waves. Resonances and bound states are obtained analytically. The transmitted flux shows a bistable behaviour. Novel crossing scenarios of eigenstates similar to beak--to--beak structures are observed for a repulsive mean-field interaction. It is proven that resonances transform to bound states due to an attractive nonlinearity and vice versa for a repulsive nonlinearity, and the critical nonlinearity for the transformation is calculated analytically. The bound state wavefunctions of the system satisfy an oscillation theorem as in the case of linear quantum mechanics. Furthermore, the implications of the eigenstates on the dymamics of the system are discussed.Comment: RevTeX4, 16 pages, 19 figure

Optimal time travel in the Godel universe

Using the theory of optimal rocket trajectories in general relativity, recently developed in arXiv:1105.5235, we present a candidate for the minimum total integrated acceleration closed timelike curve in the Godel universe, and give evidence for its minimality. The total integrated acceleration of this curve is lower than Malament's conjectured value (Malament, 1984), as was already implicit in the work of Manchak (Manchak, 2011); however, Malament's conjecture does seem to hold for periodic closed timelike curves.Comment: 16 pages, 2 figures; v2: lower bound in the velocity and reference adde

On the generalized continuity equation

A generalized continuity equation extending the ordinary continuity equation has been found using quanternions. It is shown to be compatible with Dirac, Schrodinger, Klein-Gordon and diffusion equations. This generalized equation is Lorentz invariant. The transport properties of electrons are found to be governed by Schrodinger-like equation and not by the diffusion equation.Comment: 9 Latex pages, no figure

Hamiltonian flows on null curves

The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the trajectories of a certain particle model on null curves described by a Lagrangian linear in the curvature. In addition, it is shown that the curvature of a null curve which evolves by similarities can be computed in terms of the solutions of the second Painlev\'e equation.Comment: 14 pages, v2: final version; minor changes in the expositio