The Synthesis of Arbitrary Stable Dynamics in Non-linear Neural Networks II: Feedback and Universality

We wish to construct a realization theory of stable neural networks and use this theory to model the variety of stable dynamics apparent in natural data. Such a theory should have numerous applications to constructing specific artificial neural networks with desired dynamical behavior. The networks used in this theory should have well understood dynamics yet be as diverse as possible to capture natural diversity. In this article, I describe a parameterized family of higher order, gradient-like neural networks which have known arbitrary equilibria with unstable manifolds of known specified dimension. Moreover, any system with hyperbolic dynamics is conjugate to one of these systems in a neighborhood of the equilibrium points. Prior work on how to synthesize attractors using dynamical systems theory, optimization, or direct parametric. fits to known stable systems, is either non-constructive, lacks generality, or has unspecified attracting equilibria. More specifically, We construct a parameterized family of gradient-like neural networks with a simple feedback rule which will generate equilibrium points with a set of unstable manifolds of specified dimension. Strict Lyapunov functions and nested periodic orbits are obtained for these systems and used as a method of synthesis to generate a large family of systems with the same local dynamics. This work is applied to show how one can interpolate finite sets of data, on nested periodic orbits.

Homosexuality and the United Methodist Church: An Ecclesiological Dilemma

Since 1968 The United Methodist Church has publicly debated the status and roles of homosexual persons in the life of the Church, creating considerable conflict within the Denomination. Academic research on the question of homosexuality and the Church has often focused on theological understandings of homosexuality and on the ways the conflict reflects broader "culture wars" in society. Yet little attention has been given to how the Church's concrete practices and polity toward homosexual persons reflect underlying tensions within the ecclesiological identity of the Denomination. This dissertation proposes that the issue of homosexuality is a critically important case study for exploring the practical ecclesiology of The United Methodist Church. In an effort to identify tensions within contemporary United Methodism's practical ecclesiology, it traces in detail the history of the denominational debate over homosexuality since 1968 and articulates the diverse and often conflicting ecclesiological commitments embedded within that debate. Focusing on the debate itself as a practice of the Church, this dissertation illustrates the ways in which the controversy over sexuality reflects the Denomination's conflicted practical ecclesiology. By examining the rhetoric of the sexuality debates in The United Methodist Church from 1968 to 2008, and by articulating the ecclesiological commitments embedded in those debates, the dissertation reveals a fundamental conflict over interpretations of ecclesial unity. Moreover, the dissertation explores the extent to which the conflict over unity reflects ecclesiological tensions present in John Wesley's own practical ecclesiology; and it asks whether or not contemporary interpretations of United Methodist ecclesiology might provide a normative framework for assessing and resolving the underlying ecclesial conflict at work in sexuality debates. The dissertation concludes by exploring the practice of public narrative as a concrete strategy that might be employed by the Denomination to reconcile the diverging ecclesiological visions within the contemporary church so that a clear and consensual ecclesiology might emerge.