Talk:Combinatorial species/Archive 1

message: It would help to know in what areas the term is used. It's linked to from Species - is "combinatorial species" a biological term also? Mathematical? Or both? Tannin
Actually, it didn't occur to me that I had not made that clear by using the word "combinatorial".  But now I've made it more explicit.  (For now, it's still only a stub article.)  -- Mike Hardy
:The description of the cycle index series seems slightly ambiguous. Namely, where is the dependence of \left( \sum_{\sigma \in S_A} |\mathrm{Fix}(F[\sigma])| x_1^{\sigma_1} x_2^{\sigma_2} \cdots \right) on n? - Gauge 02:15, 10 January 2006 (UTC)
replies: text: For whatever reason, it was commented out with a completely unrelated edit summary: . Now all images are shown again. --Daniel5Ko (talk) 09:16, 12 July 2012 (UTC)

message: The combinatorial species Lin does not necessarily stands for a Linear Order, in the very same way that Ens does not stands for a Boolean Algebra.
A word xyzt has the very same combinatorial structure as { βx, 3y, Az, Πt } where β, 3, A, Π are specific "slots". Nicolae-boicu (talk) 20:21, 4 August 2012 (UTC)
thread_title:  Note on the Lin species 

message: I have an example of actually using a cyc.
McKay's proof of Cauchy's theorem (1959, AMM) uses a p-tuples of elements of G 
( x1, x2,....,xp ) and x1.x2......xp = 1
Well, this is a cyc ! (cyclic order does not matter) as usual, when species are present it turns spectaculary.Nicolae-boicu (talk) 05:54, 5 August 2012 (UTC)
thread_title:  Example of concrete use of a cyc 

message: In terms of permutation groups, primitives of molecular species are called transitive extensions. This does not close the issue, since (UV)'= U'V+UV'Nicolae-boicu (talk) 21:44, 5 August 2012 (UTC)
thread_title:  Note on integration 

message: Since three basic operations are defined by splitting the set A with n elements like n = (n-1) + 1 (differentiation), n = m1 + m2 (product) or n = m1 + m2 +...+ mk (partitional composition) one should understand the transport of structure as a transport of partitional stuctures. Nicolae-boicu (talk) 17:37, 15 August 2012 (UTC)
thread_title:  Note on the transport of structures 

message: Pointing is about introducing coordinates. One cannot do too much math with non-coordinated structures. In a combinatorial set, all elements are 'total' conjugates without any other determination. For example, if |A|=3 and |B|=5, all we can write about A and B inside Species Theory may be expressed as 5 ' ' = 3 (Sym5 is a two point extension of Sym3). 
To define a (mathematical) function on the (mathematical) set A, first one must point three times A toward •••A, thus obtaining an X.X.X; and then define the function on X.X.X, only after the elements of A got their own identity and they become distinct within respect to any conjugation.
X.X.X...X may be seen as a 'solid' structure, a 'zero-entropy' structure, with no possibility to make confusions between conjugate elements. 
In Geometry, the coordination is implicitly introduced when choosing points or quadrangles or whatever. After choosing two points, all other points of an affine line are well determined (modulo the Galois conjugation)(unlike sets, the geometrical structures are efficiently coordinated with less pointing operations; after several pointing operations they became X.X.X...X's). In the corresponding algebraic structures, 0 and 1 are technologically introduced by the very first axioms, and all other numbers become well determined : one element-one sign, without confusions. 
The pointing verb, the key to coordination, is 'let it be' (French 'soit'). For example, to totally coordinate the complex field and to eliminate the Galois confusion, one must point the i : let i be such that i.i = 1. Nicolae-boicu (talk) 12:00, 15 January 2013 (UTC)
thread_title:  Note on pointing 

message: based on a Wikipedia tabel. I apologize, I still search a scholastic example.
1) US Drone Strike Statistics estimate according to the New America Foundation.
(As of 17 April 2013)
Year 	Number ofAttacks	Number Killed	Number Killed
Year 	Number ofAttacks	Min.         	Max.
2004 	1               	5            	8
2005 	3               	12           	13
2006 	2               	90           	102
2007 	4               	48           	77
2008 	36              	219          	344
2009 	54              	350          	721
2010 	122             	608          	1,028
2011 	72              	366          	599
2012 	48              	222          	349
2013 	12              	62           	73
Total	354             	1,982        	3,314
Let X be the sort of attacks, Y be the sort of casualties.
Each line will be encoded in one species e.g. Line2007(X,Y). On has
Line2004 ( X, Y ) = X × [ Ens5 ( Y ) + Ens6 ( Y ) + Ens7 ( Y ) + Ens8 ( Y ) ]
Line2005 ( X, Y ) = Ens3 ( X ) × [ Ens12 ( Y ) +  Ens13 ( Y ) ]
...
Line2013 ( X, Y ) = Ens12 ( X ) × [ Ens62 ( Y ) +...+  Ens73 ( Y ) ]
The unknown number between min and max entered the machinery as a sum of possibilities.
The whole table may be encoded in species terms as a product of its lines:
Table ( X, Y ) = Line2004 ( X, Y ) × Line2005 ( X, Y ) ×...× Line2013 ( X, Y ) /Nicolae-boicu (talk) 10:02, 14 July 2013 (UTC)
thread_title:  note on sum and product rules 

message: By interrupting one of L1 and L2 signals the output COM gets L2 or L1. 
Nicolae-boicu (talk) 12:20, 16 February 2014 (UTC)
thread_title:  note on electrical interpretation 

message: the And/or conjunction fits to the combinatorial multisorting. "He will eat cake, pie, and/or brownies" easily becomes cake + pie + brownies, Plate(X+Y+Z) Nicolae-boicu (talk) 09:30, 8 February 2015 (UTC)
thread_title:  the and/or conjunction 

message: In one word, the generalized differentiation means sampling. Let's say that John has 3 chickens and 2 ducks and he wants to remove 2 birds.
 { C.C.C.D.D\over  2(C+D)   }
 = { C.C.C.D.D\over { 2(C) + C.D + 2(D) }  } 
 = { C.C.C.D.D\over { 2(C)} }  + {C.C.C.D.D\over {C.D}} + { C.C.C.D.D\over {2(D)}} 
 =  {3\choose 2} .C.D.D + 3.2.C.C.D + {2\choose 2} .C.C.C  
  = 3.C.D.D + 6.C.C.D + 1.C.C.C 
One may conclude, for example, that the probability for John to remove two birds of different kind is 6/10.
in the above I have noted the differentiation by a ratio line and a set of 3 chickens by 3(C). 
Eventually, we have reached in our research the formula of binomial coefficient.
 { X.X.....X.X \over  \{ X,X,...,X \}   }={   X^n \over k(X)  } = {n\choose k}.X^{n-k} 
Undersum (talk) 00:59, 8 July 2016 (UTC)
thread_title:  the generalized differentiation 

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thread_title:  External links modified 

message: Suppose we have to investigate claims like the ones in Fano plane article :
*There are 7 lines, and 24 symmetries fixing any line.
*There are 28 triangles, which correspond one-for-one with the 28 bitangents of a quartic (Manivel 2006). For each triangle there are six symmetries fixing it, one for each permutation of the points within the triangle.
There are two ways of doing this; one is to use the generalized differentiation of a Fano species within respect to Ens3, by applying to ZFano the differentiator 
 {d_1.d_1.d_1 \over 6}  + d_1.d_2 + d_3  that comes from   Z_Ens3 (d_1, 2.d_2,..., i.d_1,...)  
The result is
 {Fano \over  { \{ X,X,X \} } } =
{1 \over 24}( a_1^4 + 3a_2^2  + 6 a_1^2a_2    + 8 a_1 a_3 + 6 a_4) + a_1.{1 \over 6}(a_1^3 + 3 a_1a_2  + 2 a_3)  = Ens_4 + X.Ens_3 
A faster way is to use directly the cycle index definition. We consider the action of Fano group on 3-sets and we slightly modify the definition by taking the Fano group insted of Sn
 {1 \over |Fano|} \sum_{\sigma in Fano} fix(\sigma) . a^\sigma   
then we evaluate fix(σ) for each type; the result is the very same.
type of σ      	fixes 3-sets	total	difference
a_1^7          	35          	35   	35.a_1^4
a_1a_1a_1a_2a_2	1 + 3.2 = 7 	21×7 	21.a_2^2 + 126.a_1^2a_2
a_1 a_2 a_4    	1           	42   	42.a_4
a_1 a_3 a_3    	2           	56×2 	112.a_1 a_3
a_7            	-           	-    	-
               	            	336  	
Nboyku (talk) 20:49, 31 May 2018 (UTC)
thread_title:  Note on generalized differentiation