1,031 research outputs found
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
On the Number of Clumps Resulting from the Overlap of Randomly Placed Figures in a Plane
When two-dimensional figures, called laminae, are randomly placed on a plane domains result that can either be aggregates or individual laminae. The intersection of the union, U, of these domains with a specified field of view, F, in the plane is considered. The separate elements of the intersection are called clumps; they may be laminae, aggregates or partial laminae and aggregates. A formula is derived for the expected number of clumps minus enclosed voids. For bounded laminae homeomorphic to a closed disc with isotropic random direction the formula contains only their mean area and mean perimeter, the area and perimeter of F, and the intensity of the Poisson process
Dowker-type theorems for hyperconvex discs
A hyperconvex disc of radius r is a planar set with nonempty interior that is the intersection of closed circular discs of radius r . A convex disc-polygon of radius r is a set with nonempty interior that is the intersection of a finite number of closed circular discs of radius r . We prove that the maximum area and perimeter of convex disc- n -gons of radius r contained in a hyperconvex disc of radius r are concave functions of n , and the minimum area and perimeter of disc- n -gons of radius r containing a hyperconvex disc of radius r are convex functions of n . We also consider hyperbolic and spherical versions of these statements
A central limit theorem for random disc-polygons in smooth convex discs
In this paper we prove a quantitative central limit theorem for the area of
uniform random disc-polygons in smooth convex discs whose boundary is .
We use Stein's method and the asymptotic lower bound for the variance of the
area proved by Fodor, Gr\"unfelder and V\'igh (2022)
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