let $r$ be an infinite ring. here we prove that if $0_r$ belongs to ${x_1x_2cdots x_n ;|; x_1,x_2,dots,x_nin x}$ for every infinite subset $x$ of $r$, then $r$ satisfies the polynomial identity $x^n=0$. also we prove that if $0_r$ belongs to ${x_1x_2cdots x_n-x_{n+1} ;|; x_1,x_2,dots,x_n,x_{n+1}in x}$ for every infinite subset $x$ of $r$, then $x^n=x$ for all $xin r$.

this paper uses the l1−norm and the concept of the non-dominated vector, topropose a method to find a well-dispersed subset of non-dominated (wdsnd) vectorsof a multi-objective mixed integer linear programming (momilp) problem.the proposed method generalizes the proposed approach by tohidi and razavyan[tohidi g., s. razavyan (2014), determining a well-dispersed subset of non-dominatedvectors of...

We study the following question: for given d ⩾ 2 , n and k ⩽ what is largest value c ( ) such that from any set of unit vectors in R we may select with corresponding signs ±1 so their signed sum has norm at least ? The problem dual to classical vector minimization balancing questions, which have been studied over a century. give asymptotically sharp estimates general case. In several special ca...

BACKGROUND Lymphocyte subsets enumeration is considered prominent in the management of primary and acquired immunodeficiency disorders. Because of local variations due to race, age, gender, and environmental conditions on lymphocyte subsets, and to improve the accuracy of interpretation of laboratory findings, reference intervals must be determined in every population. OBJECTIVE To establish ...

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 ...