In this paper, we aresupposed to introduce the definitions of n-fold commutative, andimplicative hyper K-ideals. These definitions are thegeneralizations of the definitions of commutative, andimplicative hyper K-ideals, respectively, which have been definedin [12]. Then we obtain some related results. In particular wedetermine the relationships between n-fold implicative hyperK-ideal and n-fol...

in this paper, we aresupposed to introduce the definitions of n-fold commutative, andimplicative hyper k-ideals. these definitions are thegeneralizations of the definitions of commutative, andimplicative hyper k-ideals, respectively, which have been definedin [12]. then we obtain some related results. in particular wedetermine the relationships between n-fold implicative hyperk-ideal and n-fold...

Let k > 1 be an integer. A set A ⊂ Z is a k-fold Sidon set if A has only trivial solutions to each equation of the form c1x1 + c2x2 + c3x3 + c4x4 = 0 where 0 6 |ci| 6 k, and c1 + c2 + c3 + c4 = 0. We prove that for any integer k > 1, a k-fold Sidon set A ⊂ [N ] has at most (N/k)1/2 + O((Nk)1/4) elements. Indeed we prove that given any k positive integers c1 < · · · < ck, any set A ⊂ [N ] that c...

The K-fold Cross Validation (KCV) technique is one of the most used approaches by practitioners for model selection and error estimation of classifiers. The KCV consists in splitting a dataset into k subsets; then, iteratively, some of them are used to learn the model, while the others are exploited to assess its performance. However, in spite of the KCV success, only practical rule-of-thumb me...