M. Mollard and C. Payan. Some progress in the packing of equal circles in a square. Discrete Mathematics, 84:303-307, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....square determine a distance 1 2 sec 15 o . This problem for up to nine circles (n = 2; 9) was solved in 1965 [43, 45] although the rst proof for n = 6 was reported in 1970 [47] and for n = 7 in 1996 [33] Between 1970 and 1990, at least ten papers have reported solutions for n = 10 [10, 15, 16, 17, 29, 30, 40, 44, 46, 48], but the optimal solution was given in 1990 [16] R. Peikert et al. 39] proposed an elimination procedure and found optimal results for n = 10; 20. Until recently the problem has been solved for n 27 [34, 39] using computer aided methods for This work was supported by the ....

....These numerical results were obtained using several di erent strategies; for instance, using billiard simulation [13, 20, 21] minimization of the energy function [32] standard BFGS quasi Newton algorithm [39] nonlinear programming solver (MINOS 5. 3) 23] or the Cabri G eom etre software in [30]. Although it is not known whether these solutions are optimal, in some cases these numerical results can help to nd better solutions (if any) when they are used as lower bounds of the optimal (maximal) solutions. For instance, several strategies described in the packing literature, as that ....

M. Mollard and C. Payan. Some progress in the packing of equal circles in a square. Discrete Mathematics, 84:303-307, 1990.

....K.J. Nurmela and P.R.J. Osterg ard [11] built a minimization of energy function strategy, standard BFGS quasi Newton algorithm were used by R. Peikert et al. 13] nonlinear programming solver (MINOS 5.3) by C.D. Maranas et al. in [6] or Cabri G eom etre software by M. Mollard and C. Payan in [9]. Most of the algorithmic suggestions used for solving (1) are based on Global Optimization stochastic approaches, nevertheless, there have been several papers ( 5] 13] published with the description of deterministic approaches. Deterministic methods can ensure that their solutions are optimal ....

M. Mollard and C. Payan. Some progress in the packing of equal circles in a square. Discrete Mathematics, 84:303-307, 1990.

....K.J. Nurmela and P.R.J. Osterg ard [11] built a minimization of energy function strategy, standard BFGS quasi Newton algorithm were used by R. Peikert et al. 13] nonlinear programming solver (MINOS 5.3) by C.D. Maranas et al. in [6] or Cabri G eom etre software by M. Mollard and C. Payan in [9]. Most of the algorithmic proposals used to solve (1) are based on Global Optimization stochastic approaches, nevertheless several papers ( 5] 13] describing deterministic approaches have been published. Deterministic methods can ensure that their solutions are optimal solutions to (1) and ....

M. Mollard and C. Payan. Some progress in the packing of equal circles in a square. Discrete Mathematics, 84:303-307, 1990.

....square determine a distance 1 2 sec 15 o . This problem for up to nine circles (n = 2; 9) was solved in 1965 [44, 46] although the rst proof for n = 6 was reported in 1970 [48] and for n = 7 in 1996 [34] Between 1970 and 1990, at least ten papers have reported solutions for n = 10 [11, 16, 17, 18, 30, 31, 41, 45, 47, 49], but the optimal solution was given in 1990 [17] R. Peikert et al. 40] proposed an elimination procedure and found optimal results for n = 10; 20. Until recently the problem has been solved for n 27 [35, 40] using computer aided methods for This work was supported by the ....

....These numerical results were obtained by using several di erent strategies; for instance, using billiard simulation [14, 21, 22] minimization of the energy function [33] standard BFGS quasi Newton algorithm [40] nonlinear programming solver (MINOS 5. 3) 24] or the Cabri G eom etre software in [31]. Although it is not known whether these solutions are optimal, in some cases these numerical results can help to nd better solutions (if any) when they are used as lower bounds of the optimal (maximal) solutions. For instance, several strategies described in the packing literature, as that of R. ....

M. Mollard and C. Payan. Some progress in the packing of equal circles in a square. Discrete Mathematics, 84:303-307, 1990.

....K.J. Nurmela and P.R.J. Osterg ard [11] built a minimization of energy function strategy, standard BFGS quasi Newton algorithm were used by R. Peikert et al. 13] nonlinear programming solver (MINOS 5.3) by C.D. Maranas et al. in [6] or Cabri G eom etre software by M. Mollard and C. Payan in [9]. Most of the algorithmic suggestions used to solve (1) are based on Global Optimization stochastic approaches, nevertheless there has been published several papers ( 5] 13] with the description of deterministic approaches. Deterministic methods can ensure that their solutions are optimal ....

M. Mollard and C. Payan. Some progress in the packing of equal circles in a square. Discrete Mathematics, 84:303-307, 1990.

....determine a distance 1 2 sec 15 o . This problem for up to nine circles (n = 2; 9) was solved in 1965 [51, 53] although the rst proof for n = 6 was reported in 1970 [55] and for n = 7 in 1996 [40] Between 1970 and 1990, at least ten papers have reported solutions for n = 10 [12, 17, 18, 19, 35, 36, 47, 52, 54, 58], but the optimal solution was given in 1990 [18] R. Peikert et al. 46] proposed an elimination procedure and found optimal results for n = 10; 20. Until recently the problem has been solved for n 27 [41, 46] using computer aided methods for optimality proofs [40, 46] and the n = 36 ....

....These numerical results were obtained using several di erent strategies; for instance, using billiard simulation [15, 24, 25] minimization of the energy function [39] standard BFGS quasi Newton algorithm [46] nonlinear programming solver (MINOS 5. 3) 27] or the CabriG eom etre software in [36]. Although it is not known whether these solutions are optimal, in some cases these numerical results can help to nd better solutions (if any) when they are used as lower bounds of the optimal (maximal) solutions. For instance, several strategies described in the packing literature, as that ....

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M. Mollard and C. Payan. Some progress in the packing of equal circles in a square. Discrete Mathematics, 84:303-307, 1990.

.... Goldberg [4] and Schaer [5] However, the currently best known solution has first been reported by Schluter [6] Subsequently, Milano [7] and Valette [8] came up with less dense solutions and lately the best configuration has been published again independently by Grunbaum [9] and Mollard and Payan [10]. Recently, de Groot et al. 11] by using an elimination algorithm proved that the solution first given by Schluter [6] is indeed exact. The most thorough work on this problem has been published by Goldberg [4] in which conjectural optimal arrangements were provided for n 27 as well as for some n ....

....algorithm proved that the solution first given by Schluter [6] is indeed exact. The most thorough work on this problem has been published by Goldberg [4] in which conjectural optimal arrangements were provided for n 27 as well as for some n 27. For n = 11 and n = 13 Mollard and Payan [10] lately reported better solutions, for n = 14 first Wengerodt [12] and then Mollard and Payan [10] provided the same improved solution. 1 Author to whom all correspondence should be addressed. Finally, for n = 16; 25 and 36, Wengerodt [13, 14, 15] matched the solutions given by Goldberg [4] ....

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M. Mollard and C. Payan, Some progress in the packing of equal circles in a square, Discrete Math. 84 (1990) 303--307.

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