Montreal Math ClubMathematics students in Montreal - Étudiants de mathématiques à Montréal

 Montreal Math Club :: Mathematics :: Set theory and Topology Share

# Perfect Shells

AuthorMessage

Euclid

Posts : 95
Join date : 2009-09-15
Age : 56
Location : Alexandria

 Subject: Perfect Shells   Fri Oct 16, 2009 12:00 pm ===== 3 (+1) gangs on 3 members, two pieces ======={1, 2, 3} optional{ 1, 2 } { 1, 3 }{ 2, 3 }===== 7 (+1) gangs on 4 members, four pieces ======={ 1, 2, 3, 4 } optional{ 1, 2, 3}{ 1, 3, 4 }{ 1, 2, 4 }{ 2, 3, 4 }{ 1, 2 }{ 1, 3 }{ 1, 4 } xor { 2, 3 } ===== 15 (+1) gangs on 5 members, four pieces ======={ 1, 2, 3, 4, 5 } optional{ 1, 2, 3, 4 }{ 1, 2, 3, 5 }{ 1, 2, 4, 5 }{ 1, 3, 4, 5 }{ 2, 3, 4, 5 }{ 1, 2, 3 }{ 1, 2, 4 }{ 1, 2, 5 }{ 1, 3, 4 }{ 1, 3, 5 }{ 1, 4, 5 }{ 2, 3, 4 }{ 2, 3, 5 }{ 2, 4, 5 }{ 1, 2 } xor { 3, 4, 5 }===== 21 gangs on 6 members =======ab bc acab1 bc1 ac1ab2 bc2 ac2ab3 bc3 ac3ab12 bc12 ac12ab23 bc13 ac23ab13 bc23 ac13Last edited by nick on Sun Oct 18, 2009 12:41 pm; edited 8 times in total

Euclid

Posts : 95
Join date : 2009-09-15
Age : 56
Location : Alexandria

 Subject: Infinite sequences of perfect shells   Fri Oct 16, 2009 12:18 pm Two infinite sequences of 15,16, 63,64, 255,2564^N - 1 (+1, optional) gangs on 2N+1 members,example for N=6 generalization of of the above 15-shell, N>= 2 ---------------------------the whole base of 13, optional : 1all 12-sets ...............: C(13,1) = C(12,0) + C(12,1)all 11-sets containing a or b ..: C(12,2) + C(12,2) - C(11,2)all 10-sets containing a or b ..: C(12,3) + C(12,3) - C(11,3)all 9-sets containing a or b ....: C(12,4) + C(12,4) - C(11,4)all 8-sets containing a or b ....: C(12,5) + C(12,5) - C(11,5)all 7-sets containing a or b ....: C(12,6) + C(12,6) - C(11,6)all 6-sets containing a and b ..: C(11,4)all 5-sets containing a and b ..: C(11,3)all 4-sets containing a and b ..: C(11,2)all 3-sets containing a and b ..: C(11,1)the set { a , b } xor its complement : 1Two infinite sequences of 31,32, 127,128, 511,5124^N/2 - 1 (+1, optional) gangs on 2N members, N>= 3, example for N=7---------------------------the whole base of 14, optional : 1all 13-sets ...............: C(14,1) = C(13,0) + C(13,1)all 12-sets containing a or b ..: C(13,2) + C(13,2) - C(12,2)all 11-sets containing a or b ..: C(13,3) + C(13,3) - C(12,3)all 10-sets containing a or b ..: C(13,4) + C(13,4) - C(12,4)all 9-sets containing a or b ....: C(13,5) + C(13,5) - C(12,5)all 8-sets containing a or b ....: C(13,6) + C(13,6) - C(12,6)all 7-sets containing a .........: C(13,6)all 6-sets containing a and b ..: C(12,4)all 5-sets containing a and b ..: C(12,3)all 4-sets containing a and b ..: C(12,2)all 3-sets containing a and b ..: C(12,1)the set { a , b } xor its complement : 1Last edited by nick on Sun Oct 18, 2009 1:32 pm; edited 7 times in total

Euclid

Posts : 95
Join date : 2009-09-15
Age : 56
Location : Alexandria

 Subject: Infinite colections of perfect shells   Sun Oct 18, 2009 12:46 pm Let A be a set of individuals (members). A family is a collections of sets of powerset(A) (without the emptyset); When every two set in this family have a non-empty intersection, the family is an intersecting family, or a shell (my wording).The sets inside a shell are gangs; A pivot is an individual common to all sets of a family. Its family is a stared family.The trail of a family F is another family that contains all the subsets of the sets of F; ex:{1, 2} , {1, 3, 4 } produces the trail {1}, {2}, {3} {1,2} , {1,3}, {1,4}, {1,3,4}.A star is a family that has a pivot and contains some of its trail, i.e. those sets in the trail that contain the pivot ex:{1}, {1,2}, {1,3},{1,2.3}, {1,4} has the pivot 1. A perfect shell is a shell that has the same cardinality with the stars generated by each member ex| {1,2},{2,3},{1,3} | = | {1},{1,2},{1,3} | = Star(1) = | {2},{2,1},{2,3} | = Star(2) = | {3},{3,1},{3,2} | = Star(3) I think it is funny to search for perfect shells.[ to be continued...]

Euclid

Posts : 95
Join date : 2009-09-15
Age : 56
Location : Alexandria

 Subject: union product   Thu Oct 22, 2009 10:38 pm Take A = powerset (1, 2, 3) Take B = {a, b}, {a, c}, {b, c}, {a, b, c}Take A xu B, the union product of A and B := the family of all reunions of two sets, first in A and the second in B.{ 1, 2, 3, a, b, c } is optional, and we get a 31-32 perfect shell on 6 members, which contains the 21 previous shell. In fact we have 3x7 = 21 and (3+1)x(7+1)-1 = 31ab bc ac abcab1 bc1 ac1 abc1ab2 bc2 ac2 abc2ab3 bc3 ac3 abc3ab12 bc12 ac12 abc12ab23 bc13 ac23 abc23ab13 bc23 ac13 abc13ab123 bc123 ac123 (abc123 is optional)==== Until now ====Let HP[n] denote the perfect shell built by previous method (vivat Maple)(HP means Half Power)HP[2]* = {a}HP[2] = {a} {a,b} HP[3]* = {a, b}, {b, c}, {a, c}HP[3] = {a, b}, {b, c}, {a, c}, {a, b, c} HP[4]* = {a, b}, {a, c}, {a, d}, {a, b, c} , {a, b, d}, {a, c, d}, {b, c, d}HP[4] = HP[4]* and {a, b, c, d}and so on.i) A method to build new perfect shells is to take the union product of a perfect shell with a power set. Example:HP[3]* xu Powerset [3]* ------> the previous 21 perfect shell.ii) Another method is by factorization, taking a = b in a shell;iii) Also, by changing small sets in the HP series with their complementaries we get a bundle of new perfect shells. The extrema is a new series of half powers:- for n odd, take all subsets larger than (n+1)/2- for n even, take all subsets larger than n/2 + 1, then add half of the n/2-subsets : for each non-intersecting couple, choose one of the couple.

 Subject: Re: Perfect Shells

 Perfect Shells
 Page 1 of 1
 Similar topics
» While waiting to get that perfect job!!
» THE ECONOMIC COLLAPSE - MICHAEL MOORE EXPLAINS WHY DONALD TRUMP WILL WIN IN NOVEMBER | AND IT ACTUALLY MAKES PERFECT SENSE
» A CALL FOR AN UPRISING - ED SHEERAN ILLUMINATI EXPOSED! MUSIC VIDEO EXPOSED (SHAPE OF YOU, PERFECT, GALWAY GIRL, SAVE MYSELF)

Permissions in this forum:You cannot reply to topics in this forum
 Jump to: Select a forum||--Math Club|   |--Mathematical Sundays|   |--Suggestions|   |--Introduce yourself|   |--Mathematics|   |--Problem Solving|   |   |--Solved problems|   |   |   |--Calculus and Analysis|   |--Combinatorics|   |--Geometry|   |--Number Theory|   |--Probabilities and statistics|   |--Set theory and Topology|   |--Online Resources|   |--Chit chat    |--The Water Cooler    |--Mathematical poems