Extensions of Laplace Type Problems in the Euclidean Space

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1 Internationa Mathematica Forum, Vo. 9, 214, no. 26, HIKARI Ltd, Extensions of Lapace Type Probems in the Eucidean Space Giuseppe Caristi Univerisity of Messina, Department S.E.A.M. Via dei Verdi, 75, Messina, Itay Marius Stoka Accademia dee Scienze di Torino, Itay Copyright c 214 Giuseppe Caristi and Marius Stoka. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. Abstract The cassica Buffon neede probem is to find the probabiity that a neede of ength n when dropped on a foor made of boards of width b wi cross a crack between the boards. This probem can be soved by evauating a simpe singe integra. In his extension of the probem, Lapace considered a foor tied by congruent rectanges and considered the probabiity of the neede crossing one or two of the cracks bettween the rectanges. In 1974 M. Stoka studies an extension of the Buffon- Lapace neede probem in the space R n. In this paper we consider two extension of the Lapace probem in E 3 Keywords: Geometric Probabiity, stochastic geometry, random sets, random convex sets and integra geometry 1 The Buffon-Lapace neede probem In Buffon s origina work the partice was considered to have ength and the mesh separation distance d [2, 3]. Henceforth a engths are in units of the ine spacing d. The soution is found by considering ony two ines, one at

2 1254 Giuseppe Caristi and Marius Stoka y = and the other at y = 1, and the neede s center of mass may ie at any height between and 1. The neede orientation takes any ange θ ± with uniform probabiity. An intersection is avoided when the center of mass position is within y [ sinθ /2, 1 sinθ /2]. Averaging this range over a orientations resuts in the intersection probabiity P int > 1) = sin θ ) dθ = 2. 1) When the neede ength equas the ine spacing = 1) the intersection probabiity is 2/ =.637, indicating non-intersecting orientations about θ =. The soution for > 1 is found by restricting the range in θ to θ ±arcsin1/), with intersection probabiity P int > 1) = 1 1 arcsin1/) 1 sin θ ) dθ = 2) arcsin1/) 2 ) sec 1. 3) This expression has the correct arge asymptote of P ) = 1. This anaysis has been used to find experimenta vaues for by dropping needes and counting the intersections, with at east one suspected case of fraud [1]. Whie Lapace s generaization of Buffon s work [5, 6] was rectanguar meshes of arbitrary ength and width, we ony consider the square mesh of sideength. The center of mass sampes an awabe A θ) = 1 cos θ) 1 sin θ ), with the resut that P int > 1) = 1 1 A θ) dθ = 4 ). 4) It does not appear that Lapace cacuated the probabiities for onger engths, athough it is cear that not unti = 2 is an intersection guaranteed. Changing the imits in the integra in equation 4) to ±arcsin1/) as in equation 2)) resuts in an intersection probabiity of P int > 1) = 1 1 arcsin1/) A θ) dθ = arcsin1/)

3 Extensions of Lapace type probems [arcsin 1/) arccos 1/) ]. 5) ) This expression has the correct imit that P int = 2 = 1. To make the Buffon Lapace resuts appicabe to actua partices, we first generaize the anaysisto spherocyinders [7] - cyinders dimensioness diameter σ) with hemispherica endcaps. A points on the surface of the spherocyinder are a distance σ/2 from the centra ine segment. The tota ength of the spherocyinder is + σ and its aspect ratio 1 + /σ, with = corresponding to a sphere. The finite partice width reduces the tota area avaiabe to the center of mass to A θ, σ) = 1 σ cos θ) 1 σ sin θ ), 6) and the intersection probabiity is found to be P int < 1) = σ [2 4) σ 3) 2]. 7) Now we considering the Stoka paper [8] we prove two anothers extensions. 2 First extension We consider a attice Ra 1, a 2, a 3 ; λ) with foundamenta ce P a paraeepiped with sides a 1, a 2, a 3 e angoo λ ], 2 ]. We have that: vop = a 1 a 2 a 3 sin λ. 8) Let s a random segment of constant enght with < infa 1, a 2, a 3 ). Our aim is to compute the probabiity that this segment intersects a side of attice R.To be more specific,we denote by O the center of s, the position of the segment is determineted by the center and the direction cosines of the ine support: α 1 = cos θ, α 2 = sin θ cos ϕ, α 3 = sin θ sin ϕ. To compute the probabiity P int we consider the paraeepiped P with a vertix in A cos θ, sin θ cos ϕ, sin θ sin ϕ), with the sides parae to the sides of P and with sides then a 1 cos θ, a 2 sin θ cos ϕ, a 3 sin θ sin ϕ,

4 1256 Giuseppe Caristi and Marius Stoka vop = a 1 cos θ)a 2 sin θ cos ϕ)a 3 sin θ sin ϕ) sin λ. 9) Denoting by M 1 the set of a segments s which have their center in P. Denote ikewise by N 1 the set of a segments s in competey contained in P. In view of [8]: P int = 1 µn 1) µm 1 ), 1) where µ is the Lebesgue measure in the space E 3. We compute the measures µm 1 ) and µn 1 ) using the cinematic measure of Baschke in the space E 3 [2]. dk = dψ sin θ dθ dx dy dz, 11) where x, y, z are the coordinates of the center of s, ϕ and θ are the anges and ψ the rotation ange. We have By 8), 11) and 12) we have that ϕ, θ, ψ 2. 12) 2 µm 1 ) = 2 dψ 2 2 sin θdθ dxdydz = 2 a 1 a 2 a 3 sin λ. 13) {x,y,z) P } Considering 9), we have µn 1 ) = 2 dψ 2 2 sin θdθ {x,y,z) P } dxdydz = 2 sin λ 2 2 sin θa 1 cos θ)a 2 sin θ cos ϕ)a 3 sin θ sin ϕ)dθ = [ 2 sin λ 2 a 1a 2 a 3 ] 4 a 1a 2 + a 1 a 3 + a 2 a 3 ) a 1 + a 2 + a 3 ) 3. 14) 8

5 Extensions of Lapace type probems 1257 By cobining 1), with 13) and 14) we obtain that P int = a 1 a 2 a 3 [ 2 a 1a 2 + a 1 a 3 + a 2 a 3 ) ] 3 a 1 + a 2 + a 3 ) 2. 15) 8 This probabiity is indipendent by λ, i.e. is equa with the probabiity for a right paraeepiped, that was find in a previous paper [8]. Now we want to show that if the sides a 1, a 2, a 3 and mutipy with the same factor µ, the probabiity P int is unchanged. In fact, if repacing with a 1, a 2, a 3, µa 1, µa 2, µa 3, µ, µ > ), the probabiity 15) remains unchanged. 3 Second extension Denoting by S the atera surface area L of the paraeepiped P we have that S = 2 a 1 a 2 + a 2 a 3 + a 1 a 3 sin λ) 16) and if S is the atera surface area L of the paraeepiped P, we have that S = [2 a 1 cos θ) a 2 sin θ cos ϕ) + a 2 sin θ cos ϕ) a 3 sin θ sin ϕ) + a 1 cos θ) a 3 sin θ sin ϕ) sin λ]. 17) Denoting by M 2 the set of a segments s which are in one of the two panes which dtermine the paraeepiped P and which have the center in one of these piani. Denote ikewise by N 2 the set of a segments s in competey contained in atera surface of P. In view of [8]: By 11), 12) and 16) we have that P int = 1 µ N 2) µ M 2 ). 18) 2 µ M 2 ) = dψ sin θdθ dxdydz = {x,y,z) L }

6 1258 Giuseppe Caristi and Marius Stoka 2 S sin θ) dθ = 2 2 a 1 a 2 + a 2 a 3 + a 1 a 3 sin λ). 19) In the same way and by 17) we have that: 2 µ N 2 ) = dψ sin θdθ dxdydz = 2 {x,y,z) L } S sin θ) dθ = 2 2 a 1 a 2 + a 2 a 3 + a 1 a 3 sin λ) [a sin λ) + a ) + ] 2 2 a sin λ) 2 2 sin λ). 2) 2 By the 18), 19) and 2) we obtain that: P int = 2 2 a 1 a 2 + a 2 a 3 + a 1 a 3 sin λ) [a sin λ) + a ) a sin λ) + ] 2 2 sin λ). 21) Repacing a 1, a 2, a 3, with µa 1, µa 2, µa 3, µ, µ > ), the probabiity 21) remains unchanged. Finay, for λ =, i.e. for a right paraeepiped, the probabiity 21) became P int = [2a a ) + a 3 + ]. a 1 a 2 + a 2 a 3 + a 1 a 3 ) 2 2 References [1] Badger L, Lazzarini s ucky approximation of, 1994 MAth. Mag [2] Baschke, W.: Voresungen über Integraegeometrie, 3 ed., V.E.B. Deutscher Verag der Wiss., Berin, [3] Buffon G., Editor s note concerning a ecture given 1733 by Mr. e Cerc de Buffon to the Roya Academy of Sciences in Paris, 1733 Hist. Acad. R. Sci [4] Buffon G., Essai d arithmétique morae, 1777 His. Nat. Gen. Part. 4,

7 Extensions of Lapace type probems 1259 [5] Lapace P-S, 1812 Théorie Anaytique des Probabiité Paris: Veuve Courcier) [6] Lapace P-S, 182 Théorie anaytique des probabiités 3rd edn Paris: Veuve Courcier) [7] Pournin L, Weber M, Tsukahara M, Ferrez J. A., Ramaioi M. and Libing Th M, Three-dimensiona distinct eement simuation of spherocyinder crystaization, 25 Granu. Matter [8] Stoka M.: Une extension du probème de Buffon dans espace Eucidien R n, Boettino U.M.I. 4), ). pp Received: June 3, 214

Buffon-Laplace Type Problem for an Irregular Lattice

Applied Mathematical Sciences Vol. 11 17 no. 15 731-737 HIKARI Ltd www.m-hikari.com https://doi.org/1.1988/ams.17.783 Buffon-Laplace Type Problem for an Irregular Lattice Ersilia Saitta Department of Economics