A Buffon - Laplace Type Problems for an Irregular Lattice and with Maximum Probability

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1 Applied Mathematical Sciences, Vol. 8,, no. 65, HIKARI Ltd, A Buffon - Laplace Type Problems for an Irregular Lattice and with Maximum Probability D. Barilla, G. Caristi and A. Puglisi Department SEAM, University of Messina Via dei Verid, 75, 98 Messina, Italy Dedicated to Professor Marius Stoka on the occasion of his 8 th birthday Copyright c D. Barilla, G. Caristi and A. Puglisi. 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 [], [], [3], [], [5], [6], [7], [8], [9] and [] the authors studies same Laplace problems with different fundamental cells. In this paper we consider a lattice with fundamental cell represented in fig. 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 fundamental cell 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; b; α) where α is an angle with αɛ], Arctg b ] and with the fundamental cell C represented in a fig.

2 888 D. Barilla, G. Caristi and A. Puglisi A b D α α a B E fig. F C By this figure we have AE = DF = a, cos α BE = CF = atgα, EF = b atgα, () areac = ab We want to compute the probability that a segment s with random position and of constant length l < min (atgα, b atgα) intersects a side of lattice R, i.e. the probability P int that the segment s intersects a side of the fundamental cell C. The position of the segment s is determinated by middle point and by the angle ϕ that s forms with line BC (o AD). To compute the probability P int we consider the limit positions of segment s, for a fixed value of ϕ, in the cells C i, (i =,, 3). A c A5 A A 6 c 6 D D b a D 3 A A3 A c Ĉ 3 ( ϕ) c 5 D D B a Ĉ ( ϕ) E a 5 ϕba a E 3 3 E E c 3 fig. F F 3 c F F b5 b Ĉ ( ϕ) C C 3 F b 3 b C C

3 A Buffon-Laplace type problems for an irregular lattice 889 and the relations areaĉ (ϕ) = areac areaĉ (ϕ) = areac areaĉ3 (ϕ) = areac 3 To compute areaĉ (ϕ) considering fig. areaa i (ϕ), () areab i (ϕ), (3) 6 areac i (ϕ). () areaa (ϕ) = l cos ϕ cos (ϕ α). (5) sin α areaa (ϕ) = al cos ϕ l cos ϕ cos (ϕ α). (6) sin α The relations (7) and (8) give us areaa (ϕ) + area (ϕ) = al cos ϕ. (7) areaa (ϕ) = l sin ϕ cos (ϕ α). (8) areaa 3 (ϕ) = al tgα sin ϕ l sin ϕ cos (ϕ α). (9) l areaa 5 (ϕ) = al cos (ϕ α) l cos ϕ cos (ϕ α) sin ϕ cos (ϕ α). () sin α By the formulas (5), (6), (7), (8) and () we have A (ϕ) = areaa i (ϕ) = al (cos ϕ + tgα sin ϕ) l In the same way [ sin ϕ + (ctgα tgα) cos ϕ + tgα + ctgα]. () areab (ϕ) = l cos ϕ cos (ϕ + α). () sin α

4 89 D. Barilla, G. Caristi and A. Puglisi areab (ϕ) = al cos (ϕ + α) l cos ϕ cos (ϕ + α). (3) sin α areab (ϕ) = l sin ϕ. () areab 3 (ϕ) = al tgα sin ϕ l sin ϕ. (5) areab 5 (ϕ) = al cos ϕ l ( + cos ϕ) ctgα. (6) By the relations (), (3), (), (5) and (6) follows A (ϕ) = areab i (ϕ) = al cos ϕ l ctgα ( + cos ϕ). (7) i= Now we compute areaĉ3 (ϕ). areac (ϕ) = al areac (ϕ) = l sin ϕ cos (ϕ α). (8) al cos ϕ tgα sin ϕ l sin ϕ cos (ϕ α). (9) areac (ϕ) = l sin ϕ cos (ϕ + α). () areac 3 (ϕ) = (b atgα) l sin ϕ l sin ϕ cos (ϕ + α). () areac 6 (ϕ) = bl sin ϕ l sin ϕ cos (ϕ α). () areac 5 (ϕ) = al al cos ϕ tgα sin ϕ l sin ϕ cos (ϕ + α). (3) Considering the relations (8), (9), (), (), () and (3) we can write A 3 (ϕ) = 6 areac i (ϕ) = al cos ϕ + (b atgα) l sin ϕ l sin ϕ. () We denote with M i (i =,, 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 []:

5 A Buffon-Laplace type problems for an irregular lattice 89 P int = 3 µ (N i) 3 µ (M i), (5) 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 []: dk = dx dy dϕ, where x, y are the coordinate of middle point of s and ϕ the fixed angle ϕ[, π we have µ (M i ) = α dϕ dxdy = {(x,y)ɛc i } then α (areac i ) dϕ = α ) areac i, (i =,, 3), then Moreover µ (N i ) = α 3 µ (M i ) = dϕ {(x,y)ɛĉi(ϕ)} dκdy = α ) ab. (6) α ) α α areac i [A i (ϕ)] dy, 3 µ (N i ) = ) α ab The formulas (5), (6), and (7) give us Denote P int = α) ab α [areac i A i (ϕ)] dy = [ 3 ] A i (ϕ) dϕ. (7) {[3a cos α + b ( sin α) ] l l [ )]} 3 cos α + + (ctgα + tgα) α. (8)

6 89 D. Barilla, G. Caristi and A. Puglisi f (α) = we have [ 3a cos α + b ( sin α)] [ ( l l 3 cos α + + (ctgα + tgα) π α)] π α, f (α) = π {( 3a α sin α b ) cos α l l [ 6 sin α ctgα tgα+ )]} α ) ( cos α sin α + α) {[3a cos α+ For ] b [ ( ( sin α) l l π )] } 3 cos α + + (ctgα + tgα) α. α = π, b = 3a and we have 3 ( + π ) a + ( 7 π ) l = f (α) =. Then, with a simple calculation we show that for these values of parameters a, b, l, α we have f (α) <. Consequently, for these values the probability P int is maximum. References [] 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,, no. 9, [] 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.,, pp. 5-7.

7 A Buffon-Laplace type problems for an irregular lattice 893 [3] D. Barilla, M. Bisaia, G. Caristi and A. Puglisi, On Laplace type problems (II), Far East Journal of Mathematical Sciences, vol. 58, n.,, pp [] 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 [5] D. Barilla, G. Caristi, A. Puglisi, Laplace type for an irregular trapetium lattice and body test rectangle, Applied Mathematical Sciences vol. 7 n. pp [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,, no. 6, [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,, no. 9, - 6. [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 [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 3th WSEAS International Conference on Mathematics and Computers in Business and Economics (MCBE ) pp. -3. [] 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 [] H. Poincarè, Calcul des probabilitès, nd ed., Gauthier-Villard, Paris, 9. [] M. Stoka, Probabilités géométriques de type Buffon dans le plan euclidien, Atti Accd. Sci. Torino, T,. Received: September, ; Published: November 5,