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1

On the Fermat-Weber Point of a Polygonal Chain and its Generalizations

Bhattacharya Bhargab.B.

Fundamenta Informaticae

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2011

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Vol. 107, nr 4

331-343

EN

In this paper, we study the properties of the Fermat-Weber point for a set of fixed points, whose arrangement coincides with the vertices of a regular polygonal chain. A k-chain of a regular n-gon is the segment of the boundary of the regular n-gon formed by a set of k (≤n) consecutive vertices of the regular n-gon. We show that for every odd positive integer k, there exists an integer N(k), such that the Fermat-Weber point of a set of k fixed points lying on the vertices a k-chain of a n-gon coincides with a vertex of the chain whenever n≥ N(k). We also show that πm(m + 1) - .π^2/4. . N(k)≤πm(m + 1) + 1., where k (= 2m + 1) is any odd positive integer. We then extend this result to a more general family of point set, and give an O(hk log k) time algorithm for determining whether a given set of k points, having h points on the convex hull, belongs to such a family.

2

Tresses of polygons

Łupiński A., Petelczyc K., Prażmowski K.

Demonstratio Mathematica

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2007

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Vol. 40, nr 2

419-439

EN

A class of configurations which can be considered as series of suitably inscribed closed polygons is introduced and some fundamental properties of them are established.

3

Register Cellular Automata in the Hyperbolic Plane

Grigorieff S., Margenstern M.

Fundamenta Informaticae

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2004

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Vol. 61, nr 1

19--27

EN

In this paper, we consider cellular automata on special grids of the hyperbolic plane: the grids are constructed on infinigons, i.e. polygons with infinitely many sides. We show that the truth of arithmetical formulas can be decided in finite time with infinite initial recursive configurations. Next, we define a new kind of cellular automata, endowed with data and more powerful operations which we call register cellular automata. This time, starting from finite configurations, it is possible to decide the truth of arithmetic formulas in linear time with respect to the size of the formula.