Let I be a non-empty family of a-subsets of an n-element set and 1 a non-empty family of b-subsets satisfying A A B# 0 for all A ES/, BEG. Suppose that n>a+b, baa. It is proved that in this case ~~~+~.4Y~<(~)-(" ;L1)+l holds. Various extensions of this result are proved. Two new proofs of the Hilton-Milner theorem on non-trivial intersection families are given as well.

We construct a (Lipschitz) differentiability space which has at generic points a disconnected tangent and thus does not contain positive measure subsets isometric to positive measure subsets of spaces admitting a Poincaré inequality. We also prove that l-valued Lipschitz maps are differentiable a.e., but there are also Lipschitz maps taking values in some other Banach spaces having the Radon-Ni...

We consider the problem of variational interpolation of subsets of Euclidean spaces by curves such that the L norm of the second derivative is minimized. It is well known that the resulting curves are cubic spline curves. We study geometric boundary conditions arising for various types of subsets such as subspaces, polyhedra, and submanifolds, and we indicate how solutions can be computed in th...

1 Preliminaries In the following, W will be a symmetric n × n matrix with non-negative elements. The assumption is that W ij are weights corresponding to the edges ij of a (complete) graph (V, E) with vertices indexed by the numbers 1,. .. n. The indices k, l will be used to index subsets of V in a partition; we will call these subsets clusters. The indices i, j will index elements of V .

Multilevel scattered data interpolation requires decomposing the given data into a hierarchy of nested subsets. This paper concerns the eecient construction of such hierarchies. To this end, a recursive lter scheme for scattered data is proposed which generates hierarchies of locally optimal nested subsets. The scheme is a composition of greedy thinning, a recursive point removal strategy, and ...

number of k-subsets of an n-set so that the size of the intersection of any two among them is in L. It is proven that for every rational number r 2 1 there is a choice of k and L so that cn'<m(n, k, L) idn', where c, d depend on k and L but not on n. 0 1986 Academic Press. Inc

Let L = {1 b1 , . . . , as bs }, where for every i ∈ [s], ai bi ∈ [0, 1) is an irreducible fraction. Let F = {A1, . . . , Am} be a family of subsets of [n]. We say F is a fractional L-intersecting family if for every distinct i, j ∈ [m], there exists an ab ∈ L such that |Ai ∩Aj | ∈ {ab |Ai|, ab |Aj |}. In this paper, we introduce and study the notion of fractional L-intersecting families.

Alon, Babai and Suzuki proved the following theorem: Let p be a prime and let K , L be two disjoint subsets of {0, 1, . . . , p − 1}. Let |K | = r, |L| = s, and assume r(s − r + 1) ≤ p − 1 and n ≥ s + kr where kr is the maximal element of K . Let F be a family of subsets of an n-element set. Suppose that (i) |F | ∈ K (mod p) for each F ∈ F; (ii) |E ∩ F | ∈ L (mod p) for each pair of distinct se...

We consider approximation of L p functions by Hardy functions on subsets of the circle. We rst derive some properties of traces of Hardy classes on such subsets, and then turn to a generalization of classical extremal problems involving norm constraints on the complementary subset. Approximation dans l'espace de Hardy de fonctions L p sur un sous{ensemble du cercle R esum e : Nous etudions l'ap...