Independent systems of representatives in weighted graphs

The following conjecture may have never been explicitly stated, but seems to have been floating around: if the vertex set of a graph with maximal degree Δ is partitioned into sets V _i of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each V _i at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR’s). En route, we give a survey of some recent developments in the theory of ISR’s. The research of the first author was supported by grant no 780/04 from the Israel Science Foundation, and grants from the M. & M. L. Bank Mathematics Research Fund and the fund for the promotion of research at the Technion. The research of the third author was supported by the Sacta-Rashi Foundation.     

Singular vortices in regular flows

A review of the theory of quasigeostrophic singular vortices embedded in regular flows is presented with emphasis on recent results. The equations governing the joint evolution of singular vortices and regular flow, and the conservation laws (integrals) yielded by these equations are presented. Using these integrals, we prove the nonlinear stability of a vortex pair on the f-plane with respect to any small regular perturbation with finite energy and enstrophy. On the β-plane, a new exact steady-state solution is presented, a hybrid regular-singular modon comprised of a singular vortex and a localized regular component. The unsteady drift of an individual singular β-plane vortex confined to one layer of a two-layer fluid is considered. Analysis of the β-gyres shows that the vortex trajectory is similar to that of a barotropic monopole on the β-plane. Non-stationary behavior of a dipole interacting with a radial flow produced by a point source in a 2D fluid is examined. The dipole always survives after collision with the source and accelerates (decelerates) in a convergent (divergent) radial flow.     

The (dimension + 2)-secant Lemma

A. P. Sloan fellow; partially supported by NSF     

Traffic jams and ordering far from thermal equilibrium

The recently suggested correspondence between domain dynamics of traffic models and the asymmetric chipping model is reviewed. It is observed that in many cases traffic domains perform the two characteristic dynamical processes of the chipping model, namely chipping and diffusion. This correspondence indicates that jamming in traffic models in which all dynamical rates are non-deterministic takes place as a broad crossover phenomenon, rather than a sharp transition. Two traffic models are studied in detail and analyzed within this picture.     

Gene Expression on DNA Biochips Patterned with Strand‐Displacement Lithography

Lithographic patterning of DNA molecules enables spatial organization of cell‐free genetic circuits under well‐controlled experimental conditions. Here, we present a biocompatible, DNA‐based resist termed “Bephore”, which is based on commercially available components and can be patterned by both photo‐ and electron‐beam lithography. The patterning mechanism is based on cleavage of a chemically modified DNA hairpin by ultraviolet light or electrons, and a subsequent strand‐displacement reaction. All steps are performed in aqueous solution and do not require chemical development of the resist, which makes the lithographic process robust and biocompatible. Bephore is well suited for multistep lithographic processes, enabling the immobilization of different types of DNA molecules with micrometer precision. As an application, we demonstrate compartmentalized, on‐chip gene expression from three sequentially immobilized DNA templates, leading to three spatially resolved protein‐expression gradients.     

Jacobi Cohomology, Local Geometry of Moduli Spaces, and Hitchin Connections

We develop some cohomological tools for the study of the localgeometry of moduli and parameter spaces in complex AlgebraicGeometry. Notably, we develop canonical formulae for the differentialoperators of arbitrary order and their natural action on suitable‘natural’ modules (for example, functions); in particular,we obtain a formula, in terms of the moduli problem, for theLie bracket of vector fields on a moduli space. As an application,we obtain another construction and proof of flatness for thefamiliar KZW or Hitchin connection on moduli spaces of curves.2000 Mathematics Subject Classification 14D15, 32G05.     

Bend, break and count

This paper gives a formula for the number of members of a given ‘nice’ family of rational curves on a surface passing through the appropriate number of general points, expressing this number in terms of reducible members of the family. Similar formulae have been obtained previously using methods of quantum cohomology, but the present method is by contrast completely elementary, relying merely on some simple geometry on ruled surfaces.