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

Lunar drilling

Lunar drilling studie

Performance with and without inlet radial distortion of a transonic fan stage designed for reduced loading in the tip region

A transonic compressor stage designed for a reduced loading in the tip region of the rotor blades was tested with and without inlet radial distortion. The rotor was 50 cm in diameter and designed for an operating tip speed of 420 m/sec. Although the rotor blade loading in the tip region was reduced to provide additional operating range, analysis of the data indicates that the flow around the damper appears to be critical and limited the stable operating range of this stage. For all levels of tip and hub radial distortion, there was a large reduction in the rotor stall margin

A model for orientation effects in electron‐transfer reactions

A method for solving the single‐particle Schrödinger equation with an oblate spheroidal potential of finite depth is presented. The wave functions are then used to calculate the matrix element T_BA which appears in theories of nonadiabatic electron transfer. The results illustrate the effects of mutual orientation and separation of the two centers on TBA. Trends in these results are discussed in terms of geometrical and nodal structure effects. Analytical expressions related to T_BA for states of spherical wells are presented and used to analyze the nodal structure effects for T_BA for the spheroidal wells