From "Rational Utopia" to "Will-to-Utopia". On the "Post-modern" Turn in the Recent Work of Agnes Heller

Agnes Heller recently described her position as 'postmodernist', suggesting a move from a political radical to a politically liberal or 'neoconservative' position. The aim of this paper is to assess the degree to which Heller can still be regarded as a radical political thinker through an evaluation of her work on autonomy, democracy and contingency all of which remain key concepts in her thinking about the political. We find in each case that whilst many of the motifs of her critical Marxist period recur in her recent work, they are losing their oppositional or 'negative' character in the sense that making these motifs operational would require changes to the structure or functioning of liberal-capitalism. Whils remaining in some sense a radical thinker Heller has moved from the advocacy of a 'rational utopia' to a form of theorising which I describe as 'will-to-utopia': radical at the surface yet conservative at the core.

Moral Requirements are still not Rational Requirements

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Efficient Solution of Rational Conics

We present efficient algorithms for solving Legendre equations over Q (equivalently, for finding rational points on rational conics) and parametrizing all solutions. Unlike existing algorithms, no integer factorization is required, provided that the prime factors of the discriminant are known.

Modeling electrocortical activity through improved local approximations of integral neural field equations

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular we are able to treat "patchy'" connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a "lattice-directed" traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs. Article published and (c) American Physical Society 2007

A Pre-processing Moving Mesh Method for Discontinuous Galerkin Approximations of Advection-Diffusion-Reaction Problems

We propose a pre-processing mesh re-distribution algorithm based upon harmonic maps employed in conjunction with discontinuous Galerkin approximations of advection-diffusion-reaction problems. Extensive two-dimensional numerical experiments with different choices of monitor functions, including monitor functions derived from goal-oriented a posteriori error indicators are presented. The examples presented clearly demonstrate the capabilities and the benefits of combining our pre-processing mesh movement algorithm with both uniform, as well as, adaptive isotropic and anisotropic mesh refinement.

An A Posteriori Error Indicator for Discontinuous Galerkin Approximations of Fourth Order Elliptic Problems

We introduce a residual-based a posteriori error indicator for discontinuous Galerkin discretizations of the biharmonic equation with essential boundary conditions. We show that the indicator is both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm, under minimal regularity assumptions. We validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems.