A Combinatorial Construction of a Nonmeasurable Set

1985 ◽  
Vol 92 (6) ◽  
pp. 421 ◽  
Author(s):  
Robin Thomas
Keyword(s):  
Nonmeasurable Set ◽  
Combinatorial Construction

scholarly journals On Negligible and Absolutely Nonmeasurable Subsets of the Euclidean Plane

2004 ◽  
Vol 11 (3) ◽  
pp. 479-487
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Plane ◽  
Identity Transformation ◽  
Extension Problem ◽  
Measure Extension ◽  
Nonmeasurable Set

Abstract The notions of a negligible set and of an absolutely nonmeasurable set are introduced and discussed in connection with the measure extension problem. In particular, it is demonstrated that there exist subsets of the plane 𝐑2 which are 𝑇2-negligible and, simultaneously, 𝐺-absolutely nonmeasurable. Here 𝑇2 denotes the group of all translations of 𝐑2 and 𝐺 denotes the group generated by {𝑔} ∪ 𝑇2, where 𝑔 is an arbitrary rotation of 𝐑2 distinct from the identity transformation and all central symmetries of 𝐑2.

scholarly journals An Algorithm for Recognising the Exterior Square of a Multiset

2000 ◽  
Vol 3 ◽  
pp. 96-116 ◽  
Author(s):  
Catherine Greenhill
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Combinatorial Construction ◽  
Exterior Square

AbstractThe exterior square of a multiset is a natural combinatorial construction which is related to the exterior square of a vector space. We consider multisets of elements of an abelian group. Two properties are defined which a multiset may satisfy: recognisability and involution-recognisability. A polynomial-time algorithm is described which takes an input multiset and returns either (a) a multiset which is either recognisable or involution-recognisable and whose exterior square equals the input multiset, or (b) the message that no such multiset exists. The proportion of multisets which are neither recognisable nor involution-recognisable is shown to be small when the abelian group is finite but large. Some further comments are made about the motivating case of multisets of eigenvalues of matrices.

scholarly journals Overlapping Pfaffians

10.37236/1263 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Donald E. Knuth
Keyword(s):  
Symmetric Matrix ◽  
History Of ◽  
Combinatorial Construction ◽  
Skew Symmetric Matrix

A combinatorial construction proves an identity for the product of the Pfaffian of a skew-symmetric matrix by the Pfaffian of one of its submatrices. Several applications of this identity are followed by a brief history of Pfaffians.

Visualizing a Nonmeasurable Set

2003 ◽  
Vol 76 (2) ◽  
pp. 137-140
Author(s):  
Saleem Watson ◽  
Arthur Wayman
Keyword(s):  
Nonmeasurable Set
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