In this paper we prove a basic theorem which says that if f : Fpn → [0, 1] has few ‘large’ Fourier coefficients, and if the sum of the norms squared of the ‘small’ Fourier coefficients is ‘small’ (i.e. the L2 mass of f̂ is not concetrated on small Fourier coefficients), then Σn,df(n)f(n+ d)f(n + 2d) is ‘large’. If f is the indicator function for some set S, then this would be saying that the set...

In this paper, we prove the following strong convergence theorem: Let C be a closed convex subset of a Hilbert space H. Let {T (t) : t ≥ 0} be a strongly continuous semigroup of nonexpansive mappings on C such that ⋂ t≥0 F ( T (t) ) 6= ∅. Let {αn} and {tn} be sequences of real numbers satisfying 0 < αn < 1, tn > 0 and limn tn = limn αn/tn = 0. Fix u ∈ C and define a sequence {un} in C by un = (...

In this article we describe the notion of affinely indepedent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine indepedence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set of vectors which is an intersection of al...

For a standard path of connections going to a generic point at infinity in the moduli spaceMDR of connections on a compact Riemann surface, we show that the Laplace transform of the family of monodromy matrices has an analytic continuation with locally finite branching. In particular the convex subset representing the exponential growth rate of the monodromy is a polygon, whose vertices are in ...

The max-k-sum of a set of real scalars is the maximum sum of a subset of size k, or alternatively the sum of the k largest elements. We study two extensions: First, we show how to obtain smooth approximations to functions that are pointwise max-k-sums of smooth functions. Second, we discuss how the max-k-sum can be defined on vectors in a finite-dimensional real vector space ordered by a closed...

Given a finite set F of estimators, the problem of aggregation is to construct a new estimator whose risk is as close as possible to the risk of the best estimator in F . It was conjectured that empirical minimization performed in the convex hull of F is an optimal aggregation method, but we show that this conjecture is false. Despite that, we prove that empirical minimization in the convex hul...

Abstract Given a finite set F of estimators, the problem of aggregation is to construct a new estimator that has a risk as close as possible to the risk of the best estimator in F . It was conjectured that empirical minimization performed in the convex hull of F is an optimal aggregation method, but we show that this conjecture is false. Despite that, we prove that empirical minimization in the...

*Correspondence: [email protected] 1Department of Mathematics, Faculty of Science For Girls, King Abdulaziz University, Jeddah, 21593, Saudi Arabia Full list of author information is available at the end of the article Abstract Let (X ,‖ · ‖) be a Banach space. Let C be a nonempty, bounded, closed, and convex subset of X and T : C→ C be a monotone nonexpansive mapping. In this paper, it is ...

(0.1) F∞(u, U) := ‖|Du|‖L∞(U) among all such functions. Here |Du| is the Euclidean length of the gradient Du of u. We will also be interested in the “Lipschitz constant” functional as well. If K is any subset of IR and u : K → IR, its least Lipschitz constant is denoted by (0.2) Lip(u,K) := inf {L ∈ IR : |u(x)− u(y)| ≤ L|x− y| ∀ x, y ∈ K} . Of course, inf ∅ = +∞. Likewise, if any definition suc...

We prove that a preference relation which is continuous on every straight line has a utility representation if its domain is a convex subset of a finite dimensional vector space. Our condition on the domain of a preference relation is stronger than Eilenberg (1941) and Debreu (1959, 1964), but our condition on the continuity of a preference relation is strictly weaker than theirs. JEL classific...