Developing a Mathematical Model for Bobbin Lace

Bobbin lace is a fibre art form in which intricate and delicate patterns are created by braiding together many threads. An overview of how bobbin lace is made is presented and illustrated with a simple, traditional bookmark design. Research on the topology of textiles and braid theory form a base for the current work and is briefly summarized. We define a new mathematical model that supports the enumeration and generation of bobbin lace patterns using an intelligent combinatorial search. Results of this new approach are presented and, by comparison to existing bobbin lace patterns, it is demonstrated that this model reveals new patterns that have never been seen before. Finally, we apply our new patterns to an original bookmark design and propose future areas for exploration.Comment: 20 pages, 18 figures, intended audience includes Artists as well as Computer Scientists and Mathematician

Instrumental Variable Estimators for Binary Outcomes

Instrumental variables (IVs) can be used to construct estimators of exposure effects on the outcomes of studies affected by non-ignorable selection of the exposure. Estimators which fail to adjust for the effects of non-ignorable selection will be biased and inconsistent. Such situations commonly arise in observational studies, but even randomised controlled trials can be affected by non-ignorable participant non-compliance. In this paper, we review IV estimators for studies in which the outcome is binary. Recent work on identification is interpreted using an integrated structural modelling and potential outcomes framework, within which we consider the links between different approaches developed in statistics and econometrics. The implicit assumptions required for bounding causal effects and point-identification by each estimator are highlighted and compared within our framework. Finally, the implications for practice are discussed.bounds, causal inference, generalized method of moments, local average treatment effects, marginal structural models, non-compliance, parameter identification, potential outcomes, structural mean models, structural models

Are adverse selection models of debt robust to changes in market structure?

Many adverse selection models of standard one-period debt contracts are based on the following seemingly innocuous assumptions. First, entrepreneurs have private information about the quality of their return distributions. Second, return distributions are ordered by the monotone likelihood-ratio property. Third, financiers’ payoff functions are restricted to be monotonically non-decreasing in firm profits. Fourth, financial markets are competitive. We argue that debt is not an optimal contract in these models if there is only one (monopoly) financier rather than an infinite number of competitive financiers.security design; adverse selection; monotonic contracts; monotone likelihood ratio; first-order stochastic dominance

Moving to Extremal Graph Parameters

Which graphs, in the class of all graphs with given numbers n and m of edges and vertices respectively, minimizes or maximizes the value of some graph parameter? In this paper we develop a technique which provides answers for several different parameters: the numbers of edges in the line graph, acyclic orientations, cliques, and forests. (We minimize the first two and maximize the third and fourth.) Our technique involves two moves on the class of graphs. A compression move converts any graph to a form we call fully compressed: the fully compressed graphs are split graphs in which the neighbourhoods of points in the independent set are nested. A second consolidation move takes each fully compressed graph to one particular graph which we call H(n,m). We show monotonicity of the parameters listed for these moves in many cases, which enables us to obtain our results fairly simply. The paper concludes with some open problems and future directions