We construct a version of rational Symplectic Field Theory for pairs (X, L), where X is an exact symplectic manifold, where L ⊂ X is an exact Lagrangian submanifold with components subdivided into k subsets, and where both X and L have cylindrical ends. The theory associates to (X, L) a Z-graded chain complex of vector spaces over Z2, filtered with k filtration levels. The corresponding k-level...

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 .

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...

Partial remission in patients newly diagnosed with type 1 diabetes is a period of good glucose control that can last from several weeks to over a year. The clinical significance of the remission period is that patients might be more responsive to immunotherapy if treated within this period. This article provides clinical data that indicates the level of glucose control and insulin-secreting β-c...