Measuring edge importance: a quantitative analysis of the stochastic shielding approximation for random processes on graphs

Mathematical models of cellular physiological mechanisms often involve random walks on graphs representing transitions within networks of functional states. Schmandt and Gal\'{a}n recently introduced a novel stochastic shielding approximation as a fast, accurate method for generating approximate sample paths from a finite state Markov process in which only a subset of states are observable. For example, in ion channel models, such as the Hodgkin-Huxley or other conductance based neural models, a nerve cell has a population of ion channels whose states comprise the nodes of a graph, only some of which allow a transmembrane current to pass. The stochastic shielding approximation consists of neglecting fluctuations in the dynamics associated with edges in the graph not directly affecting the observable states. We consider the problem of finding the optimal complexity reducing mapping from a stochastic process on a graph to an approximate process on a smaller sample space, as determined by the choice of a particular linear measurement functional on the graph. The partitioning of ion channel states into conducting versus nonconducting states provides a case in point. In addition to establishing that Schmandt and Gal\'{a}n's approximation is in fact optimal in a specific sense, we use recent results from random matrix theory to provide heuristic error estimates for the accuracy of the stochastic shielding approximation for an ensemble of random graphs. Moreover, we provide a novel quantitative measure of the contribution of individual transitions within the reaction graph to the accuracy of the approximate process.Comment: Added one reference, typos corrected in Equation 6 and Appendix C, added the assumption that the graph is irreducible to the main theorem (results unchanged

Community Heritage Work in Africa: Village-Based Preservation and Development

This paper examines alternatives to top-down approaches to heritage management and development. One of the key issues facing communities around the globe today is the Authorized Heritage Discourse (AHD)--the determination of heritage values by “experts” and government officials on behalf of the people. It is all too common to find local people alienated by such practices and searching for ways in which they can take ownership of their own heritage. Community-based research that shares power and is participatory is one avenue that is quickly developing in many regions around the globe.In Africa, a number of villages and other small communities have taken the initiative to preserve and develop their heritage, free of outside control. Important lessons may be drawn from these experiences, particularly the use of discourse-based research that captures how the people define and live out their heritages through everyday practice

Reefing of Quarter Spherical Ribbon Parachutes Used in the Ares I First Stage Deceleration System

This paper introduces the parachutes that have been drop tested in support of the Ares I first stage deceleration system development. The results of the tests show that the reefing ratios for these quarter spherical ribbon parachutes provide the same reefed drag area as historical conical ribbon parachutes. Two sources are investigated for properly normalizing the parachutes relative to their suspension line length, and one is found to be superior

The analysis of implied volatilities

The analysis of volatility in financial markets has become a first rank issue in modern financial theory and practice: Whether in risk management, portfolio hedging, or option pricing, we need to have a precise notion of the market's expectation of volatility. Much research has been done on the analysis of realized historic volatilities, Roll (1977) and references therein. However, since it seems unsettling to draw conclusions from past to expected market behavior, the focus shifted to implied volatilities, Dumas, Fleming and Whaley (1998). To derive implied volatilities the Black and Scholes (BS) formula is solved for the constant volatility parameter a using observed option prices. This is a more natural approach as the option value is decisively determined by the market's assessment of current and future volatility. Hence implied volatility may be used as an indicator for market expectations over the remaining lifetime of the option. It is well known that the volatilities implied by observed market prices exhibit a pattern that is far different from the flat constant one used in the BS formula. Instead of finding a constant volatility across strikes, implied volatility appears to be non flat, a stylized fact which has been called smile effect. In this chapter we illustrate how implied volatilities can be analyzed. We focus first on a static and visual investigation of implied volatilities, then we concentrate on a dynamic analysis with two variants of principal components and interpret the results in the context of risk management