Covering a line segment with variable radius discs

The paper addresses the problem of locating sensors with a circular field of view so that a given line segment is under full surveillance, which is termed as the Disc Covering Problem on a Line. The cost of each sensor includes a fixed component f, and a variable component that is a convex function of the diameter of the field-of- view area. When only one type of sensor or, in general, one type of disc, is available, then a simple polynomial algorithm solves the problem. When there are different types of sensors, the problem becomes hard. A branch-and-bound algorithm as well as an efficient heuristic are developed for the special case in which the variable cost component of each sensor is proportional to the square of the measure of the field-of-view area. The heuristic very often obtains the optimal solution as shown in extensive computational testing

Broadcasting Automata and Patterns on Z^2

The Broadcasting Automata model draws inspiration from a variety of sources such as Ad-Hoc radio networks, cellular automata, neighbourhood se- quences and nature, employing many of the same pattern forming methods that can be seen in the superposition of waves and resonance. Algorithms for broad- casting automata model are in the same vain as those encountered in distributed algorithms using a simple notion of waves, messages passed from automata to au- tomata throughout the topology, to construct computations. The waves generated by activating processes in a digital environment can be used for designing a vari- ety of wave algorithms. In this chapter we aim to study the geometrical shapes of informational waves on integer grid generated in broadcasting automata model as well as their potential use for metric approximation in a discrete space. An explo- ration of the ability to vary the broadcasting radius of each node leads to results of categorisations of digital discs, their form, composition, encodings and gener- ation. Results pertaining to the nodal patterns generated by arbitrary transmission radii on the plane are explored with a connection to broadcasting sequences and ap- proximation of discrete metrics of which results are given for the approximation of astroids, a previously unachievable concave metric, through a novel application of the aggregation of waves via a number of explored functions

Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs

We prove that generic homologically nontrivial $(2n-1)$-parameter family of analytic discs attached by their boundaries to a CR manifold $\Omega$ in $\mathbb C^n, n \le 2$ tests CR functions: if a smooth function on $\Omega$ extends analytically inside each analytic disc then it satisfies the tangential CR equations. In particular, we answer, in real analytic category, two open questions: on characterization of analytic functions in planar domains (the strip-problem), and on characterization of boundary values of holomorphic functions in domains in $\mathbb C^n$ (a conjecture of Globevnik and Stout). We also characterize complex curves in $\mathbb C^2$ as real 2-manifolds admitiing homologically nontrivial 1-parameter families of attached analytic discs. The proofs are based on reduction to a problem of propagation of degeneracy of CR foliations of torus-like manifolds.Comment: The version accepted in Advances in Mathematic

Investigation of the Nicole model

We study soliton solutions of the Nicole model - a non-linear four-dimensional field theory consisting of the CP^1 Lagrangian density to the non-integer power 3/2 - using an ansatz within toroidal coordinates, which is indicated by the conformal symmetry of the static equations of motion. We calculate the soliton energies numerically and find that they grow linearly with the topological charge (Hopf index). Further we prove this behaviour to hold exactly for the ansatz. On the other hand, for the full three-dimensional system without symmetry reduction we prove a sub-linear upper bound, analogously to the case of the Faddeev-Niemi model. It follows that symmetric solitons cannot be true minimizers of the energy for sufficiently large Hopf index, again in analogy to the Faddeev-Niemi model.Comment: Latex, 35 pages, 1 figur

Non-LTE models for the gaseous metal component of circumstellar discs around white dwarfs

Gaseous metal discs around single white dwarfs have been discovered recently. They are thought to develop from disrupted planetary bodies. Spectroscopic analyses will allow us to study the composition of extrasolar planetary material. We investigate in detail the first object for which a gas disc was discovered (SDSS J122859.93+104032.9). Therefor we perform non-LTE modelling of viscous gas discs by computing the detailed vertical structure and line spectra. The models are composed of carbon, oxygen, magnesium, silicon, calcium, and hydrogen with chemical abundances typical for Solar System asteroids. Line asymmetries are modelled by assuming spiral-arm and eccentric disc structures as suggested by hydrodynamical simulations. The observed infrared Ca II emission triplet can be modelled with a hydrogen-deficient metal gas disc located inside of the tidal disruption radius, with an effective temperature of about 6000 K and a surface mass density of 0.3 g/cm^2. The inner radius is well constrained at about 0.64 Solar radii. The line profile asymmetry can be reproduced by either a spiral-arm structure or an eccentric disc, the latter being favoured by its time variability behaviour. Such structures, reaching from 0.64 to 1.5 Solar radii, contain a mass of about 3 to 6*10^21 g, the latter equivalent to the mass of a 135-km diameter Solar System asteroid.Comment: 7 pages, 10 figures, accepted for publication in A&

Hyperbolic entire functions and the Eremenko–Lyubich class: Class B or not class B?

Hyperbolicity plays an important role in the study of dynamical systems, and is a key concept in the iteration of rational functions of one complex variable. Hyperbolic systems have also been considered in the study of transcendental entire functions. There does not appear to be an agreed definition of the concept in this context, due to complications arising from the non-compactness of the phase space. In this article, we consider a natural definition of hyperbolicity that requires expanding properties on the preimage of a punctured neighbourhood of the isolated singularity. We show that this definition is equivalent to another commonly used one: a transcendental entire function is hyperbolic if and only if its postsingular set is a compact subset of the Fatou set. This leads us to propose that this notion should be used as the general definition of hyperbolicity in the context of entire functions, and, in particular, that speaking about hyperbolicity makes sense only within the Eremenko–Lyubich classB of transcendental entire functions with a bounded set of singular values. We also considerably strengthen a recent characterisation of the class B, by showing that functions outside of this class cannot be expanding with respect to a metric whose density decays at most polynomially. In particular, this implies that no transcendental entire function can be expanding with respect to the spherical metric. Finally we give a characterisation of an analogous class of functions analytic in a hyperbolic domain