Formal Affinities Research Articles

Loss and retention of accuracy in affine scaling methods

This paper considers affine scaling methods for solving linear l1 problems of the form The paper explains the motivation behind the method and describes its basic steps. Analysis of the method reveals several interesting features that characterize its asymptotic behaviour. Special attention is given to propagation of rounding errors. It is shown that the formal affine scaling algorithm has an inherent drawback: At the kth iterationk= 1,2,..., the algorithm solves a linear least squares problem and uses its residual vector, rk, to compute a search direction uk. However, as kincreases tends to zero, while the rounding errors in ukgrow faster than where e denotes the machine precision in our computations. That is, the smaller the larger the error. Thus, although in theory in practice ukcontains an error vector that belongs to Range(A). This error component is the major reason for accumulation of rounding errors. Cancelling this component avoids most of the damage. The paper explains how to overcome this difficulty. Numerical experiments illustrate the deteriorating effects of small residuals and the usefulness of the proposed safeguards.

Hamiltonians and zero-curvature equations for integrable partial differential equations

We discuss the relationship between two approaches to integrable partial differential equations, one using formal affine Lie algebras and the other Banach–Lie groups. In the first approach integrability of the equations follows from commutativity of Hamiltonian flows on the Lie algebra, while in the second it follows from commutativity of certain flows induced by an action on the Banach–Lie group. We show that these two methods are essentially equivalent since one can calculate one type of the flows from the other. The classes of solutions encompassed by the two methods, however, vary significantly. We demonstrate this relationship specifically with the nonlinear Schrödinger equation.

Rosenzweig's Aesthetic Theory and Jewish Unheimlichkeit

The aesthetic theory in Franz Rosenzweig's Star of Redemption remains a conundrum to Rosenzweig's interpreters, both in terms of its relation to the Jewish philosophical tradition and in terms of its German cultural and philosophical context.1 Can the Star's emphasis on aesthetics be reconciled with the Jewish philosophical tradition's, and most particularly with the German-Jewish, aversion to image-making? Does the Star even have a unified aesthetic theory, or are there several aesthetic theories reflected in the three parts of the Star? If the Star can be said to have a unified aesthetic theory, is it significantly different from any of a number of Rosenzweig's Romantic influences, and Schelling's most specifically?2 This essay attempts to address these issues by way of two interrelated claims. First, I argue that the Star's aesthetic theory bears its closest resemblance to Hermann Cohen's, in form though not in content. Second, recognizing this formal affinity between what I label Rosenzweig's and Cohen's ethical monotheisms points to the connection between Rosenzweig's

Fernand Léger and the spectacle of objects

Abstract Fernand Leger customarily presented his own esthetic project in terms of a central modernist goal: the transformation of quotidian appearances into highly organized, intensified and formalized esthetic matter. The painter pursued this aim by isolating and sometimes fragmenting represented forms — industrial, organic or human - and deploying them in systems of comparison or opposition designed to suggest unexpected thematic and formal affinities. Leger called this visual scenario ‘object-spectacle.’ The idea of object-spectacle provided a logic for Leger's estheticism. The painter also offered this same concept — the artistic objectification of the mundane — as an intervention in modernist cultural politics. I propose, first, to investigate the concept of object-spectacle as the ideological crux of Leger's work and thought during the years between the two World Wars. Second, I will trace selected aspects of the critical debate surrounding that same concept as that debate took shape toward the mid-...

Figures of Repetition in Sidney's Astrophil and Stella and in the Scenic Form of Measure for Measure

ew critics have noted any connection between Measure for Measure and the poetry of Sir Philip Sidney.' Yet some formal affinities between Shakespeare's play and Astrophil and Stella correspond with unexpected congruences of theme and method. Indeed, a reflection in Measure for Measure of certain rhetorical forms found also in Sidney's love sonnet sequence is pertinent to a famous difficulty of the play, its problematic transition from extreme dubiety concerning erotic life to a married solution. To pursue this intertextuality^ let us first consider aspects of Sidneys sonnets, and then of Shakespeare's play.

Negation and Subordination in Arizona Tewa: Discourse Pragmatics Influencing Syntax

0. Introduction. This article explores the formal affinity of negation and subordination in Arizona Tewa. Comparative evidence from other Tanoan languages reveals that while this association of grammatical processes is a subfamily trait, Arizona Tewa has realized it in noncognate form. In an attempt to explain this innovation, the synchronic form of Arizona Tewa negatives is viewed as the grammaticalization of a discourse-pragmatic association with subordination. Lacking detailed historical documentation, the present account proceeds by inference from the contemporary structure and use of negatives in Arizona Tewa and other Tanoan languages.2 Hypotheses generated from these comparative findings receive significant support from analogous Australian language materials. In the concluding section, I suggest the implications of my findings beyond the data at hand, considering their more general typological and methodological relevance.