Maximum Likelihood for Matrices with Rank Constraints

Maximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their likelihood functions. This led to the discovery of maximum likelihood duality between matrices of complementary ranks, a result proved subsequently by Draisma and Rodriguez.Comment: 22 pages, 1 figur

The topology of Stein fillable manifolds in high dimensions II

We continue our study of contact structures on manifolds of dimension at least five using complex surgery theory. We show that in each dimension 2q+1 > 3 there are 'maximal' almost contact manifolds to which there is a Stein cobordism from any other (2q+1)-dimensional contact manifold. We show that the product M x S^2 admits a weakly fillable contact structure provided M admits a weak symplectic filling. We also study the connection between Stein fillability and connected sums: we give examples of almost contact manifolds for which the connected sum is Stein fillable, while the components are not. Concerning obstructions to Stein fillings, we show that the (8k-1)-dimensional sphere has an almost contact structure which is not Stein fillable once k > 1. As a consequence we deduce that any highly connected almost contact (8k-1)-manifold (with k > 1) admits an almost contact structure which is not Stein fillable. The proofs rely on a new number-theoretic result about Bernoulli numbers.Comment: We corrected mistakes in the proofs of Lemma 2.9 and Corollary 2.10. This lead to an assumption being removed from the statement of Theorem 1.3. The paper is now published in Geometry and Topology. The appendix was written by Bernd C. Kellne

Production, Bonding and Application of Metal Matrix Composite Hot Forging Tool Components

Metal matrix composite materials are of high interest for their increased stiffness, strength or wear resistance. Wear resistant composites contain hard ceramic particles to reduce microcutting and grooving of the metal matrix surface. In this paper, a gas atomised hot work tool steel X40CrMoV5-1 (1.2344/AISI H13) was combined with fused tungsten carbide (FTC) particles in order to create forging tools with increased abrasive wear resistance. For that purpose, tool components were manufactured by sinter-forging of stacked powder layers to build up a graded hard phase concentration of up to 10 vol.-%. Subsequently, sinter-forged specimens were combined with basic hot work tool steel components and joined by diffusion bonding to assemble the complete tool. In order to evaluate their performance, the tools were examined in a hot backward can extrusion process of low-alloyed steel. Optical geometry measurements, light microscopy and scanning electron microscopy of the worn tool radii indicated a significant decrease in abrasive wear when using FTC-reinforced tools rather than conventional hardened tool steel

Social Structure of Sperm Whales in the Northern Gulf of Mexico

Sperm whales exhibit highly structured social behavior that depends on sex, age, and possibly local ecological characteristics. We analyzed sighting data collected between 1994 and 2005 to determine the social structure of sperm whale groups in the northern Gulf of Mexico (714 good-quality photographs of 285 individual whales). Average typical group size was approximately eight when estimated with mark-recapture techniques and using data from 2003 to 2005. Lagged association rate analyses including data from 1994 to 2004 indicated average group sizes of 11.41. Therefore, groups in the Gulf are considerably smaller than groups in the Pacific Ocean, but similar to those from the Caribbean Sea. Similarly, groups in the Gulf of Mexico remained stable for longer periods (62.5 d, SE = 47.62) than Pacific groups, but were comparable to groups from the Gulf of California. Such differences and similarities between populations could be due to adaptations to local conditions, indicating that Gulf of Mexico sperm whales may live in ecological conditions more similar to those of the Caribbean and the Sea of Cortez than to the Pacific

Parts, Wholes, and Quantity in Euclid’s Elements

This paper develops a novel methodology, combining history of mathematics, philology, philosophy of mathematics, and logic. We develop a formal logical treatment of Euclid’s Elements , in which set theory plays no role, but the logic of part and whole does. We first consider a controversy about the nature of Euclid’s Elements Book II. For Euclid, the part-whole relation plays roles that are now played by arithmetic operations. This shows one crucial limitation of the controversial interpretation of this text as geometrical algebra. Returning to the beginning, we present a formal language for stating Propositions 1 through 10 (omitting 7) and proofs of them. Surprisingly, this has never been done (except for one recent approach, which differs from ours in an essential way). We conclude by sketching several significant ways in which this project can be further develope

A Cacciopoli-Type Inequality to Prove Coercivity of a Bilinear Form Associated with Spatial Hysteresis Internal Damping for an Euler-Bernoulli Beam

We prove an inequality that resembles Cacciopoli inequalities in that it bounds the norm of the derivative of a function by using the norm of the function. Unlike in Cacciopoli inequalities, there is no restriction on the function, a fact made up for by adding an extra term to the norm of the function. The inequality arose in the proof that a bilinear form associated with spatial hysteresis internal damping for an Euler-Bernoulli beam is coercive

Report No. 29: The Mobility and Integration of People with Disabilities into the Labour Market

Report based on a study conducted for the European Parliament, Bonn 2010 (95 pages)