Inelastic electron tunneling spectroscopy

Inelastic electron tunneling spectroscopy (IETS) is a relatively new form of vibrational spectroscopy which is able to address problems previously unsolved by either IR or Raman. It is particularly useful for surface analysis.     

Parameters Tied to Treewidth

Treewidth is a graph parameter of fundamental importance to algorithmic and structural graph theory. This article surveys several graph parameters tied to treewidth, including separation number, tangle number, well‐linked number, and Cartesian tree product number. We review many results in the literature showing these parameters are tied to treewidth. In a number of cases we also improve known bounds, provide simpler proofs, and show that the inequalities presented are tight.     

Woodhouse, babbage, peacock, and modern Algebra

In a recent article, J. M. Dubbey [Historia Mathematica 4 (1977), 295–302] showed that George Peacock's A Treatise on Algebra (1830) was similar to an unpublished work written by Charles Babbage in 1821. Evidently perplexed about the absence of a dispute over priority, Dubbey concluded that Peacock had unconsciously assimilated Babbage's ideas, and that Babbage was too busy with other activities to be concerned. The thesis of this article is that the innovative aspects of the work of both Babbage and Peacock are extensions of ideas put forth in 1803 by Robert Woodhouse, and that probably neither Babbage nor Peacock was overly concerned with acknowledgments because their approach to algebra was not unique at Cambridge.     

Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces

We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic‐symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation‐equivalent to the Hilbert scheme of length‐3 subschemes of a K3 surface. The class of the projective space in the cohomology ring has prescribed intersection properties, which translate into Diophantine equations. Possible homology classes correspond to integral points on an explicit elliptic curve; our proof entails showing that the only such point is two‐torsion. © 2011 Wiley Periodicals, Inc.     

Robust optimization of contaminant sensor placement for community water systems

We present a series of related robust optimization models for placing sensors in municipal water networks to detect contaminants that are maliciously or accidentally injected. We formulate sensor placement problems as mixed-integer programs, for which the objective coefficients are not known with certainty. We consider a restricted absolute robustness criteria that is motivated by natural restrictions on the uncertain data, and we define three robust optimization models that differ in how the coefficients in the objective vary. Under one set of assumptions there exists a sensor placement that is optimal for all admissible realizations of the coefficients. Under other assumptions, we can apply sorting to solve each worst-case realization efficiently, or we can apply duality to integrate the worst-case outcome and have one integer program. The most difficult case is where the objective parameters are bilinear, and we prove its complexity is NP-hard even under simplifying assumptions. We consider a relaxation that provides an approximation, giving an overall guarantee of near-optimality when used with branch-and-bound search. We present preliminary computational experiments that illustrate the computational complexity of solving these robust formulations on sensor placement applications.     

A Survey of Calculator Usage in High Schools

This survey investigated the current status of calculator use in classrooms and schools. The results indicated that the prevailing policy in the sample of high schools is to allow the use of calculators during classroom learning activities and tests. Scientific calculators are more frequently used than graphing calculators in algebra I and geometry; whereas, graphing calculators are more frequently used in algebra II and precalculus/trigonometry. At the time of this survey, school policies regarding the use of graphing calculators with symbolic algebra capabilities were still not determined.     

Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm

We define an infinite sequence of new invariants, δ_n, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. They give lower bounds for the Thurston norm which provide better estimates than the bound established by McMullen using the Alexander norm. We also show that the δ_n give obstructions to a 3-manifold fibering over S¹ and to a 3-manifold being Seifert fibered. Moreover, we show that the δ_n give computable algebraic obstructions to a 4-manifold of the form X×S¹ admitting a symplectic structure even when the obstructions given by the Seiberg-Witten invariants fail. There are also applications to the minimal ropelength and genera of knots and links in S³.     

Boundaries of varieties in projective manifolds

We give a characterization of the boundaries of holomorphic chains in complex projective space. This extends previous work of the authors and complements results of Dolbeault and Henkin.     

Gårding's Theory of Hyperbolic Polynomials

This paper presents a simple, self‐contained account of Gårding's theory of hyperbolic polynomials, together with a recent convexity result of Bauschke‐Güler‐Lewis‐Sendov and an inequality of Gurvits. This account begins by establishing some new results. The first concerns the existence of a pointwise arrangement of the eigenvalues so that they become global real analytic functions. The second asserts that the associated “branches” are independent of the choice of hyperbolic direction. © 2013 Wiley Periodicals, Inc.     

Decomposition of the optical transfer function: wavefront coding imaging systems

We describe the mapping of the optical transfer function (OTF) of an incoherent imaging system into a geometrical representation. We show that for defocused traditional and wavefront-coded systems the OTF can be represented as a generalized Cornu spiral. This representation provides a physical insight into the way in which wavefront coding can increase the depth of field of an imaging system and permits analytical quantification of salient OTF parameters, such as the depth of focus, the location of nulls, and amplitude and phase modulation of the wavefront-coding OTF.