Tangent lines, inflections, and vertices of closed curves

We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. We also show that 2(P'+I)+V >3, where P' is the number of pairs of concordant parallel tangent lines. The proofs, which employ curve shortening flow with surgery, are based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's "tennis ball theorem".Comment: Minor revisions; To appear in Duke Math.

Phasage: a phase based account of English existential constructions

English existentials have received much attention in the generative literature as they exhibit certain properties that are difficult to capture under standard theoretical assumptions. The aim of this paper is to provide, in one fell swoop, an account of two pressing issues regarding English existentials: the non-canonical subject position, and their aspectual properties

A space for inflections: following up on JMM's special issue on mathematical theories of voice leading

Journal of Mathematics and Music's recent special issue 7(2) reveals substantial common ground between mathematical theories of harmony advanced by Tymoczko, Hook, Plotkin, and Douthett. This paper develops a theory of scalar inflection as a kind of voice-leading distance using quantization in voice-leading geometries, which combines the best features of different approaches represented in the special issue: it is grounded in the concrete sense of voice-leading distance promoted by Tymoczko, invokes scalar contexts in a similar way as filtered point-symmetry, and abstracts the circle of fifths like Hook's signature transformations. The paper expands upon Tymoczko's ‘generalized signature transform’ showing the deep significance of generalized circles of fifths to voice-leading properties of all collections. Analysis of Schubert's Notturno for Piano Trio and ‘Nacht und Träume’ demonstrate the musical significance of inflection as a kind of voice leading, and the value of a robust geometrical understanding of it.Accepted manuscrip

Typologies of agreement: some problems from Kayardild

In this paper I describe a number of agreement-type phenomena in the Australian language Kayardild, and assess them against existing definitions, stating both the boundaries of what is to be considered agreement, and characteristics of prototypical agreement phenomena. Though conforming, prima facie, to definitions of agreement that stress semantically based covariance in inflections on different words, the Kayardild phenomena considered here pose a number of challenges to accepted views of agreement: the rich possibilities for stacking case-like agreement inflections emanating from different syntactic levels, the fact that inflections resulting from agreement may change the word class of their host, and the semantic categories involved, in particular tense/aspect/mood, which have been claimed not to be agreement categories on nominals. Two types of inflection, in particular - 'modal case' and 'associating case' - lie somewhere between prototypical agreement and prototypical government. Like agreement, but unlike government, they are triggered by inflectional rather than lexical features of the head, and appear on more than one constituent; like government, but unlike agreement, the semantic categories on head and dependent are not isomorphic. Other types of inflection, though unusual in the categories involved, the possibility of recursion, and their effects on the host's word class, are close to prototypical in terms of how they fare in Corbett's proposed tests for canonical agreement

On geodesic envelopes and caustics

We give a global description of envelopes of geodesic tangents of regular curves in (not necessarily convex) Riemannian surfaces. We prove that such an envelope is the union of the curve itself, its inflectional geodesics and its tangential caustics (formed by the conjugate points to those of the initial curve along the tangent geodesics). Stable singularities of tangential caustics and geodesic envelopes are discussed. We also prove the (global) stability of tangential caustics of close curves in convex closed surfaces under small deformations of the initial curve and of the ambient metric.Comment: 7 pp. 1 figure. 2nd versio