Let n be a large number. A subset A of Zn is complete if SA = Zn, where SA is the collection of the subset sums of A. Olson proved that if n is prime and |A| > 2n1/2, then SA is complete. We show that a similar result for the case when n is a composite number, using a different approach.

Consider the following problem which we call Maximum k-Subset Intersection (MSI): Given a collection C = {S1, . . . , Sm} of m subsets over a finite set of elements E = {e1, . . . , en}, and a positive integer k, the objective is to select exactly k subsets Sj1 , . . . , Sjk whose intersection size |Sj1 ∩ . . . ∩ Sjk | is maximum. In [2], Clifford and Popa studied a related problem and left as ...

A set S of positive integers has distinct subset sums if there are 2|S| distinct elements of the set {∑ x∈X x : X ⊂ S } . Let f(n) = min{maxS : |S| = n and S has distinct subset sums}. Erdős conjectured f(n) ≥ c2n for some constant c. We give a construction that yields f(n) < 0.22002 · 2n for n sufficiently large. This now stands as the best known upper bound on f(n).

In this paper we study the conjugacy problem in polycyclic groups. Our main result is that we create polycyclic groups Gn whose conjugacy problem is at least as hard as the subset sum problem with n indeterminates. As such, the conjugacy problem over the groups Gn is NP-complete where the parameters of the problem are taken in terms of n and the length of the elements given on input.

in this paper a first step in classifying the fuzzy normalsubgroups of a finite group is made. explicit formulas for thenumber of distinct fuzzy normal subgroups are obtained in theparticular cases of symmetric groups and dihedral groups.

We obtain a characterization of the proximal normal cone to prox-regular subset Riemannian manifold and some properties Bouligand tangent cones these sets are presented. Moreover, we show that on an open neighborhood set, metric projection is locally Lipschitz it directionally differentiable at boundary points set. Finally, necessary condition for curve be minimizing in set derived.

IAl > ((l/k) + E)n then there is a subset B L A such that 0 < 1 BI 0, let snd(nt) denote the smallest integer that does not divide PI. We prove that for every I-: > 0 there is a constant c = ~(8:) z I, such that for every n > 0 and every rn, n ' +' 6 WI < n'llog'n...

background: the wilm’s tumor gene 1 (wt1) encodes a zinc finger transcription factor that is inactivated in a subset of wilm’s tumors. it plays a crucial role in growth, proliferation and development of some embryonic and adult organs. wt1 is expressed as a tumor associated antigen (taa) in various types of solid and hematopoietic malignancies and can be employed as a useful marker for targeted...