The intersection of a matroid and a simplicial complex

A classical theorem of Edmonds provides a min-max formula relating the maximal size of a set in the intersection of two matroids to a ``covering" parameter. We generalize this theorem, replacing one of the matroids by a general simplicial complex. One application is a solution of the case of a matroidal version of Ryser's conjecture. Another is an upper bound on the minimal number of sets belonging to the intersection of two matroids, needed to cover their common ground set. This, in turn, is used to derive a weakened version of a conjecture of Rota. Bounds are also found on the dual parameter--the maximal number of disjoint sets, all spanning in each of two given matroids. We study in detail the case in which the complex is the complex of independent sets of a graph, and prove generalizations of known results on ``independent systems of representatives" (which are the special case in which the matroid is a partition matroid). In particular, we define a notion of -matroidal colorability of a graph, and prove a fractional version of a conjecture, that every graph is -matroidally colorable.

The methods used are mostly topological.

     

A globally linearly convergent method for pointwise quadratically supportable convex–concave saddle point problems

We study the Proximal Alternating Predictor–Corrector (PAPC) algorithm introduced recently by Drori, Sabach and Teboulle [8] to solve nonsmooth structured convex–concave saddle point problems consisting of the sum of a smooth convex function, a finite collection of nonsmooth convex functions and bilinear terms. We introduce the notion of pointwise quadratic supportability, which is a relaxation of a standard strong convexity assumption and allows us to show that the primal sequence is R-linearly convergent to an optimal solution and the primal-dual sequence is globally Q-linearly convergent. We illustrate the proposed method on total variation denoising problems and on locally adaptive estimation in signal/image deconvolution and denoising with multiresolution statistical constraints.     

Irreducible Apollonian Configurations and Packings

An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. One well-studied method of forming an Apollonian configuration is to start with three mutually tangent circles and fill a curvilinear triangle with a new circle, then repeat with each newly created curvilinear triangle. More generally, we can start with three mutually tangent circles and a rule (or rules) for how to fill a curvilinear triangle with circles.     

Tiling Polygons with Lattice Triangles

Given a simple polygon with rational coordinates having one vertex at the origin and an adjacent vertex on the x-axis, we look at the problem of the location of the vertices for a tiling of the polygon using lattice triangles (i.e., triangles which are congruent to a triangle with the coordinates of the vertices being integer). We show that the coordinates of the vertices in any tiling are rationals with the possible denominators odd numbers dependent on the cotangents of the angles in the triangles.     

Nuclear Magnetic Resonance (NMR) Spectroscopy: A Review and a Look at Its Use as a Probative Tool in Deamination Chemistry

Abstract

This year (2006) represents the 60th anniversary of nuclear magnetic resonance (NMR) spectroscopy (discovered independently by Nobel laureates Edward Purcell and Felix Bloch). It is therefore appropriate and indeed valuable to reflect on how this versatile methodology has developed, expanded, and evolved into a cornerstone of chemical research since 1946. No doubt multiple reviews discussing various aspects of NMR technology will emerge over the course of this year, but the field has grown so exponentially since its inception that it would be impossible for a single review to meaningfully encompass all features of the NMR methodology. This work, therefore, is not meant to provide a comprehensive review of NMR spectroscopy (such an undertaking would prove unwieldy and is inapt in the current context). Instead, it will provide an overview of NMR spectroscopy including the basic principles of NMR (the NMR phenomenon, instrumentation, and spectral interpretation) the historical development of the field, and a few unique applications of the methodology. Finally, illustrations of the utility and application of NMR spectroscopy as a probative tool in the intriguing field of deamination chemistry will be examined. Among the examples highlighted are the elucidation of the mechanism of N‐nitrosoamide conversion to the trans‐diazotate ester, denitrosation under near‐neutral conditions, elucidation of the bond‐forming step of Friedel‐Crafts benzylation, and the identification of novel electronic (π?‐acceptor agostic‐type interaction) and steric (persisteric) effects.     

GLSMs for partial flag manifolds

In this paper we outline some aspects of nonabelian gauged linear sigma models. First, we review how partial flag manifolds (generalizing Grassmannians) are described physically by nonabelian gauged linear sigma models, paying attention to realizations of tangent bundles and other aspects pertinent to (0, 2) models. Second, we review constructions of Calabi–Yau complete intersections within such flag manifolds, and properties of the gauged linear sigma models. We discuss a number of examples of nonabelian GLSMs in which the Kähler phases are not birational, and in which at least one phase is realized in some fashion other than as a complete intersection, extending previous work of Hori–Tong. We also review an example of an abelian GLSM exhibiting the same phenomenon. We tentatively identify the mathematical relationship between such non-birational phases, as examples of Kuznetsov’s homological projective duality. Finally, we discuss linear sigma model moduli spaces in these gauged linear sigma models. We argue that the moduli spaces being realized physically by these GLSMs are precisely Quot and hyperquot schemes, as one would expect mathematically.     

Interaction of self-assembled monolayers of DNA with electrons: HREELS and XPS studies

We present results from high-resolution electron energy loss spectroscopy (HREELS) and XPS studies of self-assembled monolayers of DNA. The monolayers are well-organized and display sharp vibrational peaks in the HREEL spectra. The electrons interact mainly with the backbone of the DNA. The XPS results indicate that, in most of the samples studied, the phosphates on the DNA are not charged.