A Laplace Type Problem for a Regular Lattice with Convex-Concave Cell with Obstacles Rhombus

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1 International Journal of Conteporary Matheatical Sciences Vol. 9, 14, no. 1, HIKARI Ltd, A Laplace Type Proble for a Regular Lattice with Convex-Concave Cell with Obstacles Rhobus A. Puglisi, E. Saitta Univerisity of Messina, Departent S.E.A.M. Via dei Verdi, 75, 91 - Messina, Italy M. Stoka Accadeia delle Scienze di Torino, Italy Copyright c 14 A. Puglisi, E. Saitta and M. Stoka. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. Abstract In the previous papers, 1,, 3, 4, 5, 6, 7,, 9, the authors studies soe Laplace proble for different lattices and different obstacles. In this paper we consider two regular lattices with the cell represented as in figure 1, and we copute the probability that a rando segent of costant lenght intersects a side of lattice. In particular we obtain the probability deterinated in the previous work, then the Laplace probability. Keywords: Geoetric Probability, stochastic geoetry, rando sets, rando convex sets and integral geoetry 1 Cell with obstacles rhobus. Let R 1 a, b, ; α be the regular lattice with the fundaental cell C 1 is represented in the figure

2 4 A. Puglisi, E. Saitta and M. Stoka A B / B 1 B / F 1 / B 5 F 5 F / F a a C C 5 5 / C / C 1 D E E E 1 E fig.1 where < a, b and α π different types. We have: an angle. The obstacles are rhobus of two area C = ab sin α sin α. 1 Considering a rando segent s of constant leght l with l < in a, b and we copute the probabiltiy P 1 int that this segent intersects a side of lattice, then the probability that the segent s intersects the side of the fundaental cell C 1. We consider a segents of rando position, deterinated by his iddle point and by the angle ϕ that it foras with the side CD of C 1. To copute the probability P 1 int we consider the liiting positions of the segent s for a deterinated value of ϕ and let Ĉ1 ϕ be the deterinated as in fig. :

3 Laplace type proble 41 A B / B 1 b b 1 B B' B 3 b11 A 1 1 b F 4 F 1 b 9 F F F 3 B 4 b 3 b a D 1 C / C / C 3 C 4 b 4 C 5 C 1 b b 6 5 D E 4 E' b 7 E 5 E E 3 E 1 E fig. We can write: We have that areaĉ ϕ = areac + areab 1 ϕ + areab ϕ areab 11 ϕ. areab 1 ϕ + areab ϕ = l sin ϕ sin α ϕ sin α areab 3 ϕ = a l sin ϕ sin α sin α. 3 4 l sin α ϕ. 4 areab 4 ϕ = l sin α α cos ϕ sin α. 5 areab 9 = l cos α α sin + ϕ sin α. 6 areab 7 ϕ = l sin α ϕ sin α ϕ sin α areab 6 ϕ = areab 5 ϕ = b sin α. 7 b l sin ϕ. l sin α ϕ l sin α ϕ. 9 sin α

4 4 A. Puglisi, E. Saitta and M. Stoka and areab ϕ = We obtain that + l a areab 1 = areab 11 ϕ = l sin α ϕ l sin α ϕ. 1 sin α b l sin α ϕ 11 b l sin α ϕ l sin ϕ. 1 sin α areaĉ1 ϕ = areac 1 { l a sin α + b sin α cos ϕ + b + a cos α b cos α } sin ϕ l sin α ϕ 5 sin α. 13 Denoting with M 1 the set of all segents s which have the center point in C 1 and with N 1 the set of all segents s copletely contained in C 1, we have that : P 1 int = 1 µ N 1 µ M We have that: Therefore µ M 1 = dϕ { x,y C 1 and, considering the 45, µ N 1 = dϕ } dxdy = { } x,y Ĉ1 ϕ dxdy = areac 1 dϕ = αareac 1 15 area Ĉ1 ϕ dϕ = { αareac 1 l a sin α + b sin α sin ϕ

5 Laplace type proble 43 l b + a cos α b cos α } cos ϕ l 4 cos α ϕ 5 sin α α ϕ = { αareac 1 a sin α + b sin α sin α+ + b + b a cos α cos α 1 cos α} l The relation 14, 15 and 16 give us: 1 cos α l 5 α sin α P 1 int = 1 { a sin α + b sin α sin α+ 17 α ab sin α sin α b + b a cos α } l cos α 1 cos α 1 cos α l 5 α sin α. 4 For α = π, the fundaental cell becoe a rectangle with sides a and b and with four square obstacles with side and the probability 17 becoe: P 1 = a + b l l 5π π ab For we have the Laplace probability: P = a + b l l. πab

6 44 A. Puglisi, E. Saitta and M. Stoka References 1 D. Barilla, G. Caristi, A. Puglisi, A Laplace type proble for a regular lattice with six obstacles, Recent Advances in Applied & Bioedical Inforatics and Coputational Engineering in Syste Applications Proceedings of the 11th WSEAS International Conference on Applied Inforatics and Counications AIC 11 Florence Italy, ISBN , pp. 3-35, 11. D. Barilla, G. Caristi, A. Puglisi, M. Stoka, A Laplace type proble for two hexagonal lattices of Delone with obstacles, Applied Matheatical Sciences, Vol. 7, no. 9, 13, pp D. Barilla, A. Fein, A. Puglisi, E. Saitta, B. Toader, A Laplace type proble for regular lattice with circular sections obstacles, Quality - Access to success 14 suppl pp G. Caristi, A. Puglisi, E. Saitta, A Laplace Type Proble for Regular Lattices with Convex-Concave Cell and Obstacles Rhobus, Applied Matheatical Sciences, Vol. 7, 13, no., pp G. Caristi, M. Stoka, A Laplace type proble for Delone Sessadecagonal lattice with obstacles, Matheatical Models & Methods in Applied Sciences Proceedings of the 13th WSEAS International Conference on Matheatics and Coputers in Business and Econoics MCBE 1, ISBN , pp G. Caristi, M. Stoka, A Laplace type proble for a Dirichlet-Voronoi lattice different obstacles, Applied Matheatical Sciences vol. 5 n. 3, 11, pp G. Caristi, E. L. Sorte, M. Stoka, Laplace probles for regular lattices with three different types of obstacles, Applied Matheatical Sciences vol. 5 n , pp G. Caristi, M. Stoka, A Laplace type proble for a lattice with axial syetric and different obstacles I, Far East Journal of Matheatical Sciences vol. 5 n. 1, pp G. Caristi, M. Stoka, A Laplace type proble for lattice with axial syetric and different obstacles typeii, Far East Journal of Matheatical Sciences vol. 64 n. pp Poincaré H., Calcul des probabilitiés, ed., Carré, Paris, 191.

7 Laplace type proble Stoka M., Probabilités géoétriques de type Buffon dans le plan euclidien, Atti Acc. Sci. Torino, T. 11, pp , Received: June 3, 14