On Buffon Needle Problem for an Irregular Lattice

AUTHOR(S):

TITLE

On Buffon Needle Problem for an Irregular Lattice

ABSTRACT

In the previous papers [1] and [6] the authors introduced in the Buffon-Laplace type problems so-called obstacles. They considered two lattices and considering a classic Buffon type problem introducing in the first moment the maximum value of probability, i.e. reducing the probability interval and in the second considering an irregular lattice. In [5] Caristi and Ferrara considered also a Buffon type problem considering the possibles deformations of the lattice and in [2] Caristi, Puglisi and Stoka considered another particular regular lattices with eight sides. Fengfan and Deyi [4] study similar problem using two concepts, the generalized support function and restricted chord function, both referring to the convex set, which were introduced by Delin in [3]. In this paper, we consider another particular irregular lattice (see fig. 1) and considering the formula of the kinematic measure of Poincar´e [7] and the result of Stoka [9] we study a Buffon problem for this irregular lattice. We determine the probability of intersection of a body test needle of length l, l < a/3.

KEYWORDS

Geometric probability, integral geometry, Buffon problem, lattice of regions, kinematic measure_x000D_
2000 MRS Classification: 53C65; 52A22

REFERENCES

[1] D. Barilla, G. Caristi, A. Puglisi and M. Stoka, A Buffon-Laplace type problems for an irregular lattice with maximum probability, Applied Mathematical Sciences, vol. 8 (2014),no. 165, pp. 8287-8293.

[2] G. Caristi, A. Puglisi and M. Stoka, A Laplace type problem for regular lattices with octagonal cell, Far East Journal of Mathematical Sciences, vol. 48, issue 1, January 2011, pp. 103-118.

[3] R. Delin, Topics in Integral Geometry, Singapore, New Jersey, London, and Hongkong: World Scientic, 1994.

[4] X. Fengfan and L. Deyi, On generalized Buffon Needle problem for lattices, Acta Mathematica Scientia 2011, 31B(1), pp. 303-308.

[5] G. Caristi and M. Ferrara, On Buffon’s problem for a lattice and its deformations. Beitrage zur Algebra und Geometrie, 2004, 45(1), pp. 13-20.

[6] G. Caristi and M. Stoka, A Buffon-Laplace type problem for an irregular lattice with ”body test” rectangle, Applied Mathematical Sciences, vol. 8 (2014), pp. 8395–8401.

[7] H. Poincar'e, Calcul des probabilit'es, 2nd ed., Gauthier-Villard, Paris, 1912.

[8] L. A. Santal'o, Integral Geometry and Geometric Probability, London: Addison-Wesley Publishing Company, 1976.

Cite this paper

D. Barilla, G. Caristi, A. Puglisi. (2018) On Buffon Needle Problem for an Irregular Lattice. International Journal of Economics and Management Systems, 3, 36-38