A Laplace Type Problem for a Lattice with Non-convex Cells and with a Rectangle Body Test

Transcription

1 Interntionl Journl of Mthemticl Anlysis Vol. 9, 15, no. 38, HIKARI Ltd, A Lplce Type Problem for Lttice with Non-convex Cells nd with Rectngle Body Test A. Puglisi Deprtment S.E.A.M. University of Messin Vi dei Verdi, 75, 981 Messin, Itly D. Qurtrone University of Messin Pizz Puglitti, 1, 981 Messin, Itly Copyright c 15 A. Puglisi nd D. Qurtrone. This rticle is distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct In this pper we consider lttice with cell represented in fig. 1 nd we compute the probbility tht rndom rectngle nd constnt dimensions intersect side of lttice. Let R () the lttice with the fundmentl cell C rppresented in fig.1 A H C B G F C D 3 E

2 1896 A. Puglisi nd D. Qurtrone fig.1 We wnt to compute the probbility tht rndom rectngle r nd with sides of length m < l < intersects side of lttice. This probbility is equl to the probbility Pint tht r intersects side of cell C. The position of rectngle r is determined by its center nd the ngle tht the side of length l formed with the line DE (or AH). To compute the probbility Pint we consider the limiting positions, for fixed ngle ϕ, of rectngle r. We obtin the fig. A ϕ 1 A O1 A1 A7 H7 H A5 A H 5 1 ϕ H H O1 H1 A6 H3 A3 H B6 C C 6 5 O C1 7 C C7 8 D5 C6 B5 G5 18 G G O8 G7 G3 B3 O5 9 D3 1 F F7 O9 B O3 B7 B D O G6 Cˆ (ϕ ) G1 17 F3 16 ϕ 15 F O7 F6 G F1 F F5 1 E 13 E3 D6 D1 D C5 C3 ϕ B B1 E1 O6 E6 11 ϕ 1 D E E fig. nd the reltion b (ϕ) = rec 1lm rec X rei (ϕ). (1) i=1 By fig. we obtin 1 = 6 = 1, then l re1 (ϕ) = re6 (ϕ) = re1 (ϕ) = sin ϕ, moreover = 7 = 13 = 18, nd then re (ϕ) = re7 (ϕ) = re13 (ϕ) = re18 (ϕ) = m tgϕ.

3 Lplce type problem for lttice 1897 re 3 (ϕ) = 1 ( (l cos ϕ + m sin ϕ) l sin ϕ m ), cos ϕ re 9 (ϕ) = re 15 (ϕ) = re (ϕ) = m sin ϕ, re (ϕ) = re 1 (ϕ) = re 16 (ϕ) = re 1 (ϕ) = l tgϕ. re 5 (ϕ) = (l sin ϕ + m cos ϕ) l sin ϕ lm cos ϕ 3lm l tgϕ. re 8 (ϕ) = re 1 (ϕ) = (l cos ϕ + m sin ϕ) l + m sin ϕ + lm. re 11 (ϕ) = 3 (l sin ϕ + m cos ϕ) l + m sin ϕ l tgϕ lm. re 17 (ϕ) = (l sin ϕ + m cos ϕ) m sin ϕ + lm cos ϕ l tgϕ 3lm. re 19 (ϕ) = (l cos ϕ + m sin ϕ) m sin ϕ ml cos ϕ m tgϕ 3lm. re (ϕ) = (l sin ϕ + m cos ϕ) l + m sin ϕ l tgϕ lm. Then, we cn write tht A (ϕ) = re i (ϕ) = (l cos ϕ + m sin ϕ) + 3 (l sin ϕ + m cos ϕ) i=1 3 ( l + m ) sin ϕ lm. () Replcing this formul in reltion (1) we obtin

4 1898 A. Puglisi nd D. Qurtrone reĉ (ϕ) = rec 1lm A (ϕ). (3) We denote with M, the set of ll rectngle r tht they hve center in the cell C, nd with N the set of ll rectngle r entirely conteined in thge cell C o,we hve [16]: P int = 1 µ (N) µ (M), () where µ is the Lebesgue mesure in the eucliden plne. To compute the mesure µ (M) nd µ (N) we use the kinemtic mesure of Poincrè [15]: dk = dx dy dϕ, where κ, y re the coordinte of center of rectngle r nd ϕ the fixed ngle. For ϕ [, π ] we cn write µ (M) = dϕ dxdy = (rec ) dϕ = π {(x,y)ɛc } rec. (5) In the sme wy, considering formul () we hve µ (N) = dϕ {(x,y)ɛĉ(ϕ)} = [rec 1lm A (ϕ)] dϕ = π rec 5πlm [ ] reĉ (ϕ) dϕ = [A (ϕ)] dϕ. (6) Replcing the formuls (6), (7) in (5) we hve [ ] π P int = 5πlm + [A (ϕ)] dϕ. (7) πrec The formul (3) give us [A (ϕ)] dy = 5l (l + m) 3 (l + m ) πlm. (8) By reltions (8) nd (9) nd considering tht rec =, we hve P int = 1 [ 5 (l + m) 3 ] (l + m ) 3πlm. πα

5 Lplce type problem for lttice 1899 For m = the rctngle r become segment of lenght l nd probbility P int we cn write References P int = 5 l π 3 ( ) l. 8π [1] D. Brill, M. Bisi, G. Cristi, A. Puglisi, On Lplce type problems (II), Fr Est Journl of Mthemticl Sciences 58 (11), [] D. Brill, G. Cristi, A. Puglisi, A Lplce Type Problem for n Irregulr Trpetium Lttice nd Body Test Rectngle, Applied Mthemticl Sciences, 7 (13), [3] D. Brill, G. Cristi, A. Puglisi, A Buffon - Lplce type problems for n irregulr lttice nd with mximum probbility, Applied Mthemticl Sciences, 8 (1), [] D. Brill, G. Cristi, Mrius Stok, A. Puglisi, A Lplce type problem for two hexgonl lttices of Delone with obstcles, Applied Mthemticl Sciences, 9 (13), [5] D. Brill, G. Cristi, A. Puglisi, M. Stok, Lplce type problems for tringulr lttice nd different body test, Applied Mthemticl Sciences, 8 (1), [6] D. Brill, G. Cristi, E. Sitt, M. Stok, Lplce type problem for lttice with cell composed by two qudrilterls nd one tringle, Applied Mthemticl Sciences, 8 (1), [7] D. Brill, A. Puglisi, E. Sitt, M. Stok, A Lplce type problems for lttice with cell composed by two tringles nd trpezium, Applied Mthemticl Sciences, 8 (1), [8] D. Brill, A. Puglisi, E. Sitt, A Buffon type problem for lttice with fundmentl cell composed by two tringles nd two trpeziums, Applied Mthemticl Sciences, 8 (1),

6 19 A. Puglisi nd D. Qurtrone [9] D. Brill, A. Puglisi, E. Sitt, Lplce type problems for irregulr lttice with cell Pentgon + Tringle, Interntionl Mthemticl Forum, 8 (13), [1] G. Cristi, Lplce Problem for Regulr Lttices, Recent Advnces in Applied Mthemtics, editors Corneli A. Bulce, Vleri Mldenov, Monic Leb, Nikos Mstorkis, proceedings of the 1th WSEAS Interntionl Conference on Applied Mthemtics, Puerto de l Cruz, Spin, 13, ISBN , [11] G. Cristi, Buffon-Lplce Type Problems for Three Regulr Lttices nd Body Test Prllelogrm, Interntionl Mthemticl Forum, 8 (13), [1] G. Cristi, A. Puglisi, M. Stok, A Lplce type problem for regulr lttices with octgonl cell, Fr Est Journl Mthemticl Sciences, 8 (11), [13] G. Cristi, M. Stok, A Lplce type problem for regulr lttices with irregulr hexgonl cell, Fr Est Journl Mthemticl Sciences, 5 (11), [1] G. Cristi, M. Stok, A Buffon - Lplce type problem for n irregulr lttice nd body test rectngle, Applied Mthemticl Sciences, 8 (13), [15] H. Poincré, Clcul des probbilités, ed., Guthier Villrs, Pris, 191. [16] M. Stok, Probbilités géométriques de type Buffon dns le pln euclidien, Atti Acc. Sci. Torino, 11 ( ), Received: April, 15; Published: July 17, 15