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...

‎Let $p(z)$ be a polynomial of degree $n$ and for a complex number $alpha$‎, ‎let $D_{alpha}p(z)=np(z)+(alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $alpha$‎. ‎Dewan et al proved‎ ‎that if $p(z)$ has all its zeros in $|z| leq k, (kleq‎ ‎1),$ with $s$-fold zeros at the origin then for every‎ ‎$alphainmathbb{C}$ with $|alpha|geq k$‎, ‎begin{align*}‎ ‎max_{|z|=...

In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank \(1\) boolean intervals whose minimal faces together form subcomplex. further conjectured complexes with higher notion of acyclicity could in similar way using rank. We provide an explicit counterexample to this conjecture. also prove version the conjecture fo...