Applied Mathematical Sciences, Vol. 8, 1, no. 165, 879-886 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/ams.1.11915 A Laplace Type Problems for a Lattice with Cell Composed by Three Quadrilaterals and with Maximum Probability D. Barilla, G. Caristi and E. Saitta Department SEAM, University of Messina Via dei Verid, 75, 981 Messina, Italy Dedicated to Professor Marius Stoka on the occasion of his 8 th birthday Copyright c 1 D. Barilla, G. Caristi and E. Saitta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In some previous papers [1], [], [3], [], [5], [6], [7], [8], [9] and [1] the authors studies same Laplace problems with different fundamental cells. In this paper we consider a lattice with cell represented as in figure 1 and we compute the probability that a segment of random position and constant length intersects a side of lattice. Then we prove that there are values for parameters that determine the lattice and the length of segment for which the probability determined is maximum. Mathematics Subject Classification: primary: 3C5, 3C8, secondary: 3D Keywords: Geometric Probability, stochastic geometry, random sets, random convex sets and integral geometry Let R a; α) the lattice with the fundamental cell C = C 1 C C 3 represented in fig.1
88 D. Barilla, G. Caristi and E. Saitta A a/ a/ C 1 F E G a/ a/ C C 3 B α D α C fig.1 where α is an angle with αɛ [π/, π/3]. By this figure we have the following relations AD = a sin α, BD = CD = a cos α, BC = a cos α, AG = a sin α, DG = a sin α 1 ), F G = EG = a sin α ctgα; 1) areac = a sin α. ) We want to compute the probability that a segment s with random position and of constant length l < a intersects a side of lattice, i.e. the probability P int that s intersects a side of cell C. The position of the segment s is determinated by middle point and by the angle ϕ that s forms with line BC. To compute the probability P int we consider the limit positions of segment s, for a fixed value of ϕ, situated in the cells C 1, C, C 3. So, we have the fig.
A Laplace type problems with maximum probability. 881 A a 1 A A 3 A 1 a a 6 Ĉ 1 ϕ) F b 1 F 1 F 3 b 6 a 3 G a 5 G3 E a c E 6 E E 6 3 E 1 E F c 1 E 5 G 1 G c 5 b ) Ĉ ϕ b 5 Ĉ 3 ϕ) C B B1 D ϕ b 3 3 D 1 D D D c 3 C 1 C ϕ 3 c C fig. and Considering the fig., areaĉ1 ϕ) = areac 1 areaĉ ϕ) = areac areaĉ3 ϕ) = areac 3 areaa i ϕ), 3) areab i ϕ), ) areac i ϕ). 5) areaa 1 ϕ) = l sin α ϕ) sin ϕ + α), 6) sin α areaa ϕ) = al sin α ϕ) l sin α ϕ) sin ϕ + α), 7) sin α areaa 5 ϕ) = l sin ϕ + α), 8)
88 D. Barilla, G. Caristi and E. Saitta areaa 6 ϕ) = al sin ϕ + α) l sin ϕ + α) l sin α ϕ) sin ϕ + α), 9) sin α areaa ϕ) = l areaa 3 ϕ) = al ctgα cos α ϕ), 1) al sin ϕ + α) ctgα cos ϕ + α). 11) From relations 6), 7), 8), 9) 1) and 11) follows that A 1 ϕ) = In the same way areaa i ϕ) = al l sin ϕ l sin α cos ϕ + cos α sin ϕ) +.cos ϕ + cos α sin α 1) areab 1 ϕ) = l sin α ϕ), 13) areab ϕ) = al sin α ϕ) l sin α ϕ), 1) areab ϕ) = l sin ϕ, 15) areab 3 ϕ) = al cos α areab 5 ϕ) = al sin α 1 ) areab 6 ϕ) = alctgα sin α sin ϕ l sin ϕ, 16) cos ϕ l sin ϕ, 17) cos α ϕ) l sin α ϕ). 18) By relations 13), 1), 15) 16) 17) and 18) we get A ϕ) = areab i ϕ) = al cos α sin ϕ+ al sin α cos ϕ l sin ϕ l sin α ϕ) 19)
A Laplace type problems with maximum probability. 883 At the end areac ϕ) = l sin ϕ sin ϕ + α), ) sin α areac 3 ϕ) = al cos α sin ϕ l sin ϕ sin ϕ + α), 1) sin α ) areac 1 ϕ) = l cos ϕ sin ϕ + α π sin α areac ϕ) = al sin α 1 ) sin α cos ϕ l cos ϕ sin ϕ + α π sin α, ) ), 3) areac 5 ϕ) = al sin ϕ + α) l sin ϕ sin ϕ + α), ) sin α areac 6 ϕ) = l cos ϕ cos ϕ + α) sin α al ctgα cos ϕ + α). 5) By formulas ), 1), ) 3) ) and 5) we have that: A 3 ϕ) = areac i ϕ) = al sin α 1 ) sin α cos ϕ + al cos α sin ϕ l sin ϕ ctgα cos ϕ). 6) We denote with M i i = 1,, 3) the set of segments s that they have the middle point in the cell C i and with N i the set of segments s full content in C i, we have [1]: P int = 1 3 µ N i) 3 µ M i), 7) where µ is the Lebesgue measure in the Euclidean plane. To compute the measure µ M i ) and µ N i ) we use the kinematic measure of Poincaré [11]: dk = dx dy dϕ, where x, y are the coordinate of middle point of s and ϕ the fixed angle ϕɛ [, α] we have µ M i ) = dϕ dxdy = {x,y)ɛc i }
88 D. Barilla, G. Caristi and E. Saitta then areac i ) dϕ = αareac i, i = 1,, 3). Moreover 3 µ M i ) = α µ N i ) = 3 areac i = αareac = a α sin α. 8) dϕ {x,y)ɛĉiϕ)} dxdy = [ ] areaĉi ϕ) dϕ = then, [areac i A i ϕ)] dϕ = αareac i 3 µ N i ) = a α sin α [A i ϕ) dϕ], i = 1,, 3), [ 3 ] A i ϕ) dϕ. 9) The formulas 7), 8) and 9) give us [ α 3 ] P int = A a i ϕ) dϕ. 3) α sin α We have that [ 3 ] A i ϕ) dϕ = al 3 + 5 cos α 9 cos α ) l 8 5 1 cos α + αctgα ). P int = Replacing this relation in the 6) follows [ 1 al 3 + 5 cos α 9 cos α ) l 5 1 cos α + αctgα )]. a α sin α Considering f α) = al 3 + 5 cos α 9 cos α) l 5 1 cos α + αctgα), α sin α we have
A Laplace type problems with maximum probability. 885 f α) = { [ 1 α sin α sin α al sin α 18 cos α 5) l 5 sin α + ctgα α )] α sin α [ sin α + α cos α) al 3 + 5 cos α 9 cos α ) l 5 1 cos α + αctgα )]}. If [ α sin α al sin α 18 cos α 5) l 5 sin α + ctgα α )] sin α [ sin α + α cos α) al 3 + 5 cos α 9 cos α ) l 5 1 cos α + αctgα )] =, for α = π we have f α) =. We prove that for these values we have then the probability is maximum. f α) <, P int = 1 a f α) References [1] D. Barilla, M. Bisaia, E. Saitta, M. Stoka, A Laplace type problems for a lattice with cell composed by two trapezium and a triangle, Applied Mathematical Sciences, Vol. 8, 1, no. 9, 3-39. [] D. Barilla, M. Bisaia, G. Caristi and A. Puglisi, On Laplace type problems I), Journal of Pure and Applied Mathematics: Advances and Applications, vol. 6, n.1, 11, pp. 51-7. [3] D. Barilla, M. Bisaia, G. Caristi and A. Puglisi, On Laplace type problems II), Far East Journal of Mathematical Sciences, vol. 58, n., 11, pp. 15-155.
886 D. Barilla, G. Caristi and E. Saitta [] D. Barilla, G. Caristi, A. Puglisi, M. Stoka, A Laplace type problem for two hexagonal lattices of Delone with obstacles, Applied Mathematical Sciences, Vol. 7, no. 9, pp. 571-581. [5] D. Barilla, G. Caristi, A. Puglisi, Laplace type for an irregular trapetium lattice and body test rectangle, Applied Mathematical Sciences vol. 7 n. 1 pp. 87-51. [6] D. Barilla, G. Caristi, E. Saitta, M. Stoka, Laplace type problem for lattice with cell composed by two quadrilaterals and one triangle, Applied Mathematical Sciences, Vol. 8, 1, no. 16, 789-8. [7] D. Barilla, A. Puglisi, E. Saitta, M. Stoka, A Laplace type problems for a lattice with cell composed by two triangles and a trapezium, Applied Mathematical Sciences, Vol. 8, 1, no. 9, 1-61. [8] D. Barilla, A. Puglisi, E. Saitta, Laplace type problems for irregular lattice with cell Pentagon + Triangle, International Mathematical Forum, Vol. 8, no. 3, pp. 1131-11. [9] G. Caristi, M. Stoka, A Laplace type problem for Delone Sessadecagonal lattice with obstacles, Mathematical Models and Methods in Applied Sciences Proceedings of the 13th WSEAS International Conference on Mathematics and Computers in Business and Economics MCBE 1) pp. 1-13. [1] G. Caristi, M. Stoka, Buffon-Laplace type problems for three regular lattices and body test a parallelogram, International Mathematical Forum, vol. 8 n. 6, pp. 5-71. [11] H. Poincarè, Calcul des probabilitès, nd ed., Gauthier-Villard, Paris, 191. [1] M. Stoka, Probabilités géométriques de type Buffon dans le plan euclidien, Atti Accd. Sci. Torino, T, 11. Received: September 1, 1; Published: November 5, 1