Given nonnegative real sequences a = (αk)k∈Z and b = (βk)k∈Z we study the generated summation operator Sa,b(x) := ( αk [∑ l<k βlxl ]) k∈Z , x = (xk)k∈Z , regarded as a mapping from `p(Z) to `q(Z). We give necessary and sufficient conditions for the boundedness of Sa,b and prove optimal estimates for its entropy numbers relative to the summation properties of a and b. Our results are applied to ...

Technical Lemma 1. Assume that (H1) is satisfied and that L ≡ 0 for all cuts in some given cell A. Then the regression function m is constant on A. Proof of Technical Lemma 1. We start by proving the result in dimension p = 1. Letting A = [a, b] (0 ≤ a < b ≤ 1), and recalling that = − 1 (b − a) 2 b a m(t)dt 2 + 1 (b − a)(z − a) z a m(t)dt 2 + 1 (b − a)(b − z) b z m(t)dt 2. Let C = b a m(t)dt an...

The illustration shown here appears in the master’s thesis of Mary Emily Sinclair (1878–1955). Her thesis, supervised by Oscar Bolza at the University of Chicago in 1903, deals with quintic polynomials p(t) = t + xt + yt + z. Every (real) triple (x, y, z) yields such a polynomial, and Sinclair describes the sets of all (x, y, z) such that the corresponding polynomial has a given number of real ...

Sections 3, 4 and 5 give results under Lp assumptions without any further assumptions on the decay rates. Section 3 deals with the case r ∈ Lp, p > 2. Section 4 generalises to p > 1. Section 5 covers the case p = 1 for mean zero observables. There is still the general p = 1 case to consider! The optimal result in the stretched exponential case is given in Section 6. Section 7 gives an alternati...

We study a class of Markovian optimal stochastic control problems in which the controlled process Z is constrained to satisfy an a.s. constraint Z(T ) ∈ G ⊂ R P − a.s. at some final time T > 0. When the set is of the form G := {(x, y) ∈ R × R : g(x, y) ≥ 0}, with g non-decreasing in y, we provide a Hamilton-Jacobi-Bellman characterization of the associated value function. It gives rise to a sta...

We use semianalytic models of structure formation to interpret gravitational lensing measurements of substructure in galaxy cluster cores ( h 1 kpc) at . The dynamic range of the lensing-based substructure R ≤ 250 z p 0.2 fraction measurements is well matched to the theoretical predictions, both spanning . The structure f ∼ 0.05–0.65 sub formation model predicts that is correlated with cluster ...

In the convergence theory of Padé approximation, one needs to estimate the size of a set on which a suitably normalized polynomial q is small. For example, one needs to estimate the size of the set of r 2 [0; 1] for which max jtj=1 jq (t)j =min jtj=r jq (t)j is not “too large”. We discuss some old and new problems of this type, and the methods used to solve them. 1. Introduction Let f be a func...

Let Ei be a collection of i.i.d. exponential random variables. Bouchaud’s model on Z is a Markov chain X(t) whose transition rates are given by wij = ν exp(−β((1 − a)Ei − aEj)) if i, j are neighbours in Z. We study the behaviour of two correlation functions: P[X(tw + t) = X(tw)] and P [ X(t) = X(tw)∀t′ ∈ [tw, tw + t] ] . We prove the (sub)aging behaviour of these functions when β > 1 and a ∈ [0...

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let U = {z ∈ C : |z| < 1} be the unit disc in the complex plane.Let Σ k be the class of meromorphic functions f in U having the form: f (z) = 1 z + α k z k + · · · , 0 < |z| < 1, k ≥ 0 A function f ∈ Σ = Σ 0 is called starlike...

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