Let F be a graph.A hypergraph H is Berge-F if there is a bijection f:E(F)→E(H)such that e■f(e)for every e∈E(F).A hypergraph is Berge-F-free if it does not contain a subhypergraph isomorphic to a Berge-F hypergraph.The authors denote the maximum number of hyperedges in an n-vertex r-uniform Berge-F-free hypergraph by ex_(r)(n,Berge-F).A(k,p)-fan,denoted by F_(k,p),is a graph on k(p-1)+1 vertices consisting of k cliques with p vertices that intersect in exactly one common vertex.In this paper they determine the bounds of ex_(r)(n,Berge-F)when F is a(k,p)-fan for k≥2,p≥3 and r≥3.

Let F be a graph. A hypergraph H is Berge-F if there is a bijection f : E（F） →E（H） such that e ? f（e） for every e ∈ E（F）. A hypergraph is Berge-F-free if it does not contain a subhypergraph isomorphic to a Berge-F hypergraph. The authors denote the maximum number of hyperedges in an n-vertex r-uniform Berge-F-free hypergraph by exr（n, Berge-F）.A（k, p）-fan, denoted by Fk,p, is a graph on k（p-1） + 1 vertices consisting of k cliques with p vertices that intersect in exactly one common vertex. In this paper they determine the bounds of exr（n, Berge-F） when F is a（k, p）-fan for k ≥ 2, p ≥ 3 and r ≥ 3.

Let F be a graph.A hypergraph H is Berge-F if there is a bijection f:E(F)→E(H)such that e ? f(e)for every e ∈ E(F).A hypergraph is Berge-F-free if it does not contain a subhypergraph isomorphic to a Berge-F hypergraph.The authors denote the maximum number of hyperedges in an n-vertex r-uniform Berge-F-free hypergraph by exr(n,Berge-F).A(k,p)-fan,denoted by Fk,p,is a graph on k(p-1)+1 vertices consisting of k cliques with p vertices that intersect in exactly one common vertex.In this paper they determine the bounds of exr(n,Berge-F)when F is a(k,p)-fan for k≥2,p≥3 and r≥3.

We disprove some power sum conjectures of Turan that would have implied the density hypothesis of the Riemann zeta-function if true.

We disprove some power sum conjectures of Tur谩n that would have implied the density hypothesis of the Riemann zeta-function if true.

Our aim is to prove the inequalities 1 x2 Pn(x) Pn (x) 1 x2 hn +1 , x [1,1], n = 1,2,..., n(n + 1) Pn (x) 1 Pn(x) 2 n where hn := 1 and (Pn) are the Legendre polynomials . At the same time, it is shown k=1 k n=0 that the sequence having as general term Pn(x) Pn (x) +1 n(n + 1) Pn (x) 1 Pn(x) is non-decreasing for x [1, 1].

The L-m extremal polynomials in an explicit form with respect to the weights (1 - x)(-1/2) (1 + X)((m-1)/2) and (1 - x)((m-1)/2) (1 + x)(-1/2) for even m are given. Also, an explicit representation for the Cotes numbers of the corresponding Turan quadrature formulas and their asymptotic behavior is provided. (C) 1999 Academic Press.