1 Extension of a Ring by a Module Let C be a ring and N an ideal of C with N = 0. If C ′ = C/N then the C-module N has a canonical C ′-module structure. In a sense analogous with the notion of extension for modules, the data C,N is an “extension” of the ring C ′. Definition 1. Let C ′ be a ring and N a C ′-module. An extension of C ′ by N is a triple (C, ε, i) consisting of a ring C, a surjecti...

We give a nonstandard variant of Jordan’s proof of the Jordan curve theorem which is free of the defects his contemporaries criticized and avoids the epsilontic burden of the classical proof. The proof is selfcontained, except for the Jordan theorem for polygons taken for granted.

Part 1. Chains as pseudo–orbits 8 2. Pesin charts 8 2.1. Non-uniform hyperbolicity 8 2.2. Lyapunov change of coordinates 8 2.3. Pesin Charts 10 2.4. Distortion compensating bounds 11 2.5. NUHχ (f) 11 3. Overlapping charts 11 3.1. The overlap condition 12 3.2. The form of f in overlapping charts 14 3.3. Coarse graining 16 4. ε–chains and an infinite-to-one Markov extension of f 18 4.1. Double ch...

In this paper we consider the problem of maximizing any submodular function subject to d knapsack constraints, where d is a fixed constant. We establish a strong relation between the discrete problem and its continuous relaxation, obtained through extension by expectation of the submodular function. Formally, we show that, for any non-negative submodular function, an α-approximation algorithm f...

In this paper we present an asymptotic analysis of the three-dimensional problem for a thin linearly elastic cantilever Ωε = ωε × (0, l) with rectangular cross-section ωε of sides ε and ε 2, as ε goes to zero. Under suitable assumptions on the given loads, we show that the threedimensional problem converges in a variational sense to the classical one-dimensional model for extension, flexure and...

Let D be the open unit disk in the complex plane. For ε > 0 we consider the sector Σε = {z ∈ C : | arg z| < ε}. We prove that for every α ≥ 0 and for each ε > 0 there is a constant K > 0 depending only on α and ε such that for any function f in the weighted Bergman space Aα univalent on D, and f(0) = 0, then ∫ f−1(Σε) |f(z)|dAα(z) > K‖f‖1,α. This result extends a theorem of Marshall and Smith i...

We prove that for every ε ∈ (0, 1) there exists Cε ∈ (0,∞) with the following property. If (X, d) is a compact metric space and μ is a Borel probability measure on X then there exists a compact subset S ⊆ X that embeds into an ultrametric space with distortion O(1/ε), and a probability measure ν supported on S satisfying ν (Bd(x, r)) 6 (μ(Bd(x,Cεr)) 1−ε for all x ∈ X and r ∈ (0,∞). The dependen...