Systems of distinct representatives, II
نویسندگان
چکیده
منابع مشابه
Systems of Distinct Representatives and Linear Algebra *
So me purposes of thi s paper are: (1) To take se riously the term , " term rank. " (2) To ma ke an issue of not " rea rra nging rows a nd colu mns" by not "a rranging" the m in the firs t place. (3) To promote the nu merica l use of Cra mer 's rul e. (4) To ill us tra te that the re levance of " numbe r of s teps" to "a mount of wo rk" depends on the amount of work in a step. (5) To ca ll a tt...
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A transversal generated by a system of distinct representatives (SDR) for a collection of sets consists of an element from each set (its representative) such that the representative uniquely identifies the set it belongs to. Theorem 1 gives a necessary and sufficient condition that an arbitrary collection, finite or infinite, of sets, finite or infinite, have an SDR. The proof is direct, short....
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Clearly (1) has an SDR provided that |Ai| > i for all i = 1, · · · , n, in particular an SDR of (1) exists if |A1| = · · · = |An| > n or 0 < |A1| < · · · < |An|. Let G be an additive abelian group and A1, · · · , An its subsets. We associate any SDR (2) of (1) with the sum ∑n i=1 ai and set (4) S({Ai}i=1) = S(A1, · · · , An) = {a1 + · · ·+ an : {ai}i=1 forms an SDR of {Ai}i=1} . Of course, S(A1...
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Consider a finite collection of subsets of a metric space and ask for a system of representatives which are pairwise at a distance at least q, where q is a parameter of the problem. In discrete spaces this generalizes the well known problem of distinct representatives, while in Euclidean metrics the problem reduces to finding a system of disjoint balls. This problem is closely related to practi...
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The set A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer is the sum of h elements of A . For example, the squares form an asymptotic basis of order 4 and the square-free numbers form an asymptotic basis of order 2 . If A is an asymptotic basis of order h, but no proper subset of A is an asymptotic basis of order h, then A is a minimal asymptotic ba...
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 1970
ISSN: 0022-247X
DOI: 10.1016/0022-247x(70)90314-8